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# Quasinormal Modes of Scalar Field in Five-dimensional Lovelock Black Hole
Spacetime
Juhua Chen1,2 [email protected] Yongjiu Wang1 College of Physics and
Information Science, Hunan Normal University, Changsha, Hunan 410081, P. R.
China.
Department of Physics & Astronomy, University of Missouri, Columbia, MO 65211,
USA.
###### Abstract
In this paper using the third-order WKB approximation, a numerical method
devised by Schutz, Will and Iyer, we investigate the quasinormal frequencies
of the scalar field in the background of five-dimensional Lovelock black hole.
We find that the ultraviolet correction to Einstein theory in the Lovelock
theory makes the scalar field decay more slowly and makes the scalar field
oscillate more quickly, and the cosmological constant makes the scalar field
decay more slowly and makes the scalar field oscillate more slowly in Lovelock
black hole backgroud. On the other hand we also find that quasinormal
frequencies depend very weakly on the angular quantum number $l$.
###### pacs:
04.30.-w, 04.62.+v, 97.60.Lf.
## I Introduction
The quasinormal modesKonoplya , depending only on a black hole parameters, are
of great importance in gravitational-wave astrophysics, and might be observed
in existing or advanced gravitational-wave detectors. Furthermore, black holes
are often used as a testing ground for ideas in quantum gravity, and their
quasinormal modes are obvious candidates for an interpretation in terms of
quantum levelsMaggiore . Because it is so important for black hole physics and
gravitational-wave astrophysics, there are a lot of authors who are focus on
the quasinormal modes of matter fields in different black hole background in
the past decade. Such as: Quasinormal modes of black holes in anti-de Sitter
spaceMorgan ; the Dirac field quasinormal modesSayan and the scalar field
quasinormal modesWang ; Chakrabarti in different backgrouds. In recently some
scholars investigated effects of dark energy and dark matter on quasinormal
modesHe and some extended the investigation of the.quasinormal modes to
higher dimensional spacetimesOrtega .
LovelockLovelock extended the Einstein tensor, which is the only symmetric
and conserved tensor depending on the metric and its derivatives up to the
second order, to the most general tensor. They obtained tensor is non linear
in the Riemann tensor and differs from the Einstein tensor only if the space-
time has more than 4 dimensions. Therefore, the Lovelock theory is the most
natural extension of general relativity in higher dimensional space-times. On
the other hand, Lovelock theory resembles also string inspired models of
gravity as its action contains, among others, the quadratic Gauss-Bonnet term,
which is the dimensionally extended version of the four-dimensional Euler
density. This quadratic term is present in the low energy effective action of
heterotic string theoryCallan . Since the Lovelock theory represents a very
interesting scenario to study how the physics of gravity results corrected at
short distance due to the presence of higher order curvature terms in the
action. C. Garraffo et al Garraffo gave a black hole solutions of this
theory, and discussed how short distance corrections to black hole physics
substantially change the qualitative features. And M. Aiello et al Aiello
presented the exact five-dimensional charged black hole solution in Lovelock
gravity coupled to Born- Infeld electrodynamics. In their paper they also
investigated thermodynamical properties of lovelock black hole spacetime.
Further-more, M. H. Dehghani and R. Pourhasan Dehghani focused on the
temperature of the uncharged black holes of third order lovelock gravity and
the entropy through the use of first law of thermodynamics. They analyzed
thermodynamical stability and found that there exists an intermediate
thermodynamically unstable phase for black holes with hyperbolic horizon. R.
A. Konoplya et alAbdalla presented analysis of the scalar perturbations in
the background of Bauss-Bonnet black hole spacetimes and its (in)stability in
high dimensionsRoman .
The aim of this paper is to study the quasinormal mode of a scalar field in
the Lovelock black hole spacetime in five-dimensional for different angular
quantum number $l$ by using the third-order WKB approximation, a numerical
method devised by Schutz, Will and Iyer Schutz . The paper is organized as
follows: In section II we will give a brief review on the Lovelock black hole
spacetime in five dimensions. In Section III a detail analysis on the
quasinormal mode of a scalar field in the Lovelock black hole spacetime in
five-dimensional is performed. In the last section a brief conclusion is
given.
## II Lovelock Black hole spacetime in five dimensions
The Lovelock Lagrangian density in $D$ dimensions is Lovelock
$\displaystyle L=\sum_{k=0}^{N}\alpha_{k}\lambda^{2(k-1)}L_{k},$ (1)
where $N=\frac{D}{2}-1$ (for even $D$) and $N=\frac{D-1}{2}$ (for odd $D$). In
(1), $\alpha_{k}$ and $\lambda$ are constants which represent the coupling of
the terms in the whole Lagrangian and give the proper dimensions.
In Eq. (1) $L_{k}$ is
$\displaystyle
L_{k}=\frac{1}{2^{k}}\sqrt{-g}\delta^{i_{1}...i_{2k}}_{j_{1}...j_{2k}}R^{j_{1}j_{2}}_{i_{1}i_{2}}...\>R^{j_{2k-1}j_{2k}}_{i_{2k-1}i_{2k}},$
(2)
where ${R^{\mu}\>_{\nu\rho\gamma}}$ is the Riemann tensor in $D$ dimensions,
$R^{\mu\nu}\>_{\rho\sigma}=g^{\nu\delta}\>R^{\mu}\>_{\delta\rho\sigma}$, $g$
is the determinant of the metric $g_{\mu\nu}$ and
$\delta^{i_{1}...i_{2k}}_{j_{1}...j_{2k}}$ is the generalized Kronecker delta
of order $2k$ Misner .
The Lagrangian up to order 2 are given by Lanczos
$\displaystyle L_{0}$ $\displaystyle=$ $\displaystyle\sqrt{-g},$ (3)
$\displaystyle L_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{-g}\delta^{i_{1}i_{2}}_{j_{1}j_{2}}R^{j_{1}j_{2}}_{i_{1}i_{2}}=\sqrt{-g}R,$
(4) $\displaystyle L_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sqrt{-g}\delta^{i_{1}i_{2}i_{3}i_{4}}_{j_{1}j_{2}j_{3}j_{4}}R^{j_{1}j_{2}}_{i_{1}i_{2}}R^{j_{3}j_{4}}_{i_{3}i_{4}}=\sqrt{-g}(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}),$
(5)
where we recognize the usual Lagrangian for the cosmological term, the
Einstein-Hilbert Lagrangian and the Lanczos Lagrangian Lanczos , respectively.
For dimensions $D=5$ and $D=6$ the Lovelock Lagrangian is a linear combination
of the Einstein-Hilbert and Lanczos Lagrangian.
Hence, the geometric action is written as
$\displaystyle S=\int Ld^{D}x.$ (6)
In this paper we only consider the spacetime in five dimensions, so the
Lagrangian is a linear combination of the Einstein-Hilbert and the Lanczos
ones, and the Lovelock tensor results
$\displaystyle{\cal
G}_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}\>R\>g_{\mu\nu}+\Lambda\>g_{\mu\nu}$ (7)
$\displaystyle-\alpha\>\\{\frac{1}{2}\>g_{\mu\nu}\>(R_{\rho\delta\gamma\lambda}\>R^{\rho\delta\gamma\lambda}-4\>R_{\rho\delta}\>R^{\rho\delta}+R^{2})-$
(8) $\displaystyle
2\>R\>R_{\mu\nu}+4\>R_{\mu\rho}\>R^{\rho}_{\nu}+4\>R_{\rho\delta}\>R^{\rho\delta}_{\mu\nu}-2\>R_{\mu\rho\delta\gamma}\>R_{\nu}^{\rho\delta\gamma}\\}.$
(9)
The five-dimension Lovelock theory mainly corresponds to Einstein gravity
coupled to the dimensional extension of four dimensional Euler density, that’s
to say, the theory referred as Einstein-Gauss-Bonnet theory. The spherically
symmetric solution in five dimensions take as the follow form:
$\displaystyle ds^{2}=-N(r)dt^{2}+N^{-1}(r)dr^{2}+r^{2}d\Omega^{2}_{3},$ (10)
where $d\Omega^{2}_{3}$ is the metric of a unitary 3-sphere, and
$\displaystyle N(r)=\frac{4\alpha-4M+2r^{2}-\Lambda
r^{4}/3}{4\alpha+r^{2}+\sqrt{r^{4}+\frac{4}{3}\alpha\Lambda
r^{4}+16M\alpha}},$ (11)
where $M,\Lambda$ are ADM mass, cosmological constant, respectively, and
$\alpha$ is the coupling constant of additional term that presents the
ultraviolet correction to Einstein theory.
## III quasinormal mode of a scalar field in the Lovelock Black hole
spacetime
Figure 1: The behavior of the effective potential $V(r)$ vs $r$ for the
Lovelock Black hole by fixed parameters $l=1,M=1,\Lambda=0.1$ and coupling
constants $\alpha=0.1(red),0.4(yellow),0.7(blue)$.
Figure 2: The behavior of the effective potential $V(r)$ vs $r$ for the
Lovelock black hole by fixed parameters $l=1,M=1,\alpha=0.1$ and cosmological
constants $\Lambda=0(red),0.3(yellow),0.6(blue)$.
Figure 3: The behavior of the effective potential $V(r)$ vs $r$ for the
Lovelock black hole by fixed parameters $M=1,\Lambda=\alpha=0.1$ and angular
quantum numbers $l=1(red),2(yellow),3(blue)$. Figure 4: The peak point
($r=r_{p}$)of the effective potential vs the parameters of the Lovelock black
hole for different angular quantum numbers. The left corresponds to Fig.1 and
the right corresponds to Fig.2. Figure 5: Variation of the real parts (the
above row) and imaginary parts (the bottom row) of quasinormal frequencies of
the scalar field in the Lovelock black hole spacetime with parameters
$M=1,\Lambda=0.1$. Figure 6: Variation of the real parts (the above row) and
imaginary parts (the bottom row) of quasinormal frequencies of the scalar
field in the Lovelock black hole spacetime with parameters $M=1,\alpha=0.1$.
The general perturbation equation for the massless scalar field in the curve
spacetime is given by
$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})\psi=0,$
(12)
where $\psi$ is the scalar field.
Introducing the variables $\psi=\frac{e^{-i\omega
t}\Phi(r)}{r}Y(\theta,\varphi)$ and
$r_{*}=\int{\frac{4\alpha-4M+2r^{2}-\Lambda
r^{4}/3}{4\alpha+r^{2}+\sqrt{r^{4}+\frac{4}{3}\alpha\Lambda
r^{4}+16M\alpha}}dr}$, and substituting Eq.(11) into Eq.(12), we obtain a
radial perturbation equation
$\displaystyle\frac{d^{2}\Phi(r)}{dr_{*}^{2}}+(\omega^{2}-V(r))\Phi(r)=0,$
(13)
where
$\displaystyle V(r)$ $\displaystyle=$
$\displaystyle\frac{4\alpha-4M+2r^{2}-\Lambda
r^{4}/3}{4\alpha+r^{2}+\sqrt{r^{4}+\frac{4}{3}\alpha\Lambda
r^{4}+16M\alpha}}[\frac{l(l+2)}{r^{2}}$ (14) $\displaystyle+$
$\displaystyle\frac{3}{4r^{2}}\frac{4\alpha-4M+2r^{2}-\Lambda
r^{4}/3}{4\alpha+r^{2}+\sqrt{r^{4}+\frac{4}{3}\alpha\Lambda r^{4}+16M\alpha}}$
$\displaystyle+$ $\displaystyle\frac{1}{4\alpha}(3-\frac{3+4\alpha
M}{\sqrt{9r^{4}+12\alpha\Lambda r^{4}+144M\alpha}}r^{2})].$
It is obvious that the effective potential $V$ depends only on the value of
$r$, angular quantum number $l$, ADM mass $M$, cosmological constant $\Lambda$
and coupling constant $\alpha$, respectively. Fig.1 and the left one of Fig.4
show the variation of the effective potential and its peak point $r_{p}$ with
respect to the coupling constant $\alpha$. From these two figures we can find
that the peak value of potential barrier gets lower and the location of the
peak ($r=r_{p}$) moves along the right when the coupling constant $\alpha$
decreases. In Fig.2 and the right one of Fig.4 we give the variation of the
effective potential and the its peak point $r_{p}$ with respect to the
cosmological constant $\Lambda$. On the other side, from these two figures we
can find that the peak value of potential barrier gets lower and the location
of the peak ($r=r_{p}$) moves along the right when the coupling constant
$\Lambda$ increases, which is different from the coupling constant $\alpha$.
But from Fig.3 we can see that the peak value of potential barrier gets upper
and the location of the peak point ($r=r_{p}$) moves along the right when the
angular quantum number $l$ increases.
From effective potential $V(r)$, i.e., Eq.(14) and Fig.1,2, we find that the
quasinormal frequencies depend on the coupling constant $\alpha$ and the
cosmological constant $\Lambda$. In this paper, we plan to investigate the
relationship between the quasinormal mode and the coupling constant $\alpha$
and the cosmological constant $\Lambda$, respectively. For convenience we take
$M=1$ in our calculation. In order to evaluate the quasinormal frequencies for
the massless scalar field in the Lovelock black hole spacetime (10), we use
the third-order WKB approximation, a numerical method devised by Schutz, Will
and Iyer Schutz . This method has been used extensively in evaluating
quasinormal frequencies of various black holes because of its considerable
accuracy for lower-lying modes. In this approximate method, the formula for
the complex quasinormal frequencies $\omega$ is
$\displaystyle\omega^{2}=[V_{0}+(-2V^{{}^{\prime\prime}}_{0})^{1/2}\Lambda]-i(n+\frac{1}{2})(-2V^{{}^{\prime\prime}}_{0})^{1/2}(1+\Omega),$
(15)
where
$\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle\frac{1}{(-2V^{{}^{\prime\prime}}_{0})^{1/2}}\left\\{\frac{1}{8}\left(\frac{V^{(4)}_{0}}{V^{{}^{\prime\prime}}_{0}}\right)\left(\frac{1}{4}+N^{2}\right)-\frac{1}{288}\left(\frac{V^{{}^{\prime\prime\prime}}_{0}}{V^{{}^{\prime\prime}}_{0}}\right)^{2}(7+60N^{2})\right\\},$
(16) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\frac{1}{(-2V^{{}^{\prime\prime}}_{0})^{1/2}}\bigg{\\{}\frac{5}{6912}\left(\frac{V^{{}^{\prime\prime\prime}}_{0}}{V^{{}^{\prime\prime}}_{0}}\right)^{4}(77+188N^{2})$
(17) $\displaystyle-$
$\displaystyle\frac{1}{384}\left(\frac{V^{{}^{\prime\prime\prime
2}}_{0}V^{(4)}_{0}}{V^{{}^{\prime\prime
3}}_{0}}\right)(51+100N^{2})+\frac{1}{2304}\left(\frac{V^{(4)}_{0}}{V^{{}^{\prime\prime}}_{0}}\right)^{2}(67+68N^{2})$
$\displaystyle+$
$\displaystyle\frac{1}{288}\left(\frac{V^{{}^{\prime\prime\prime}}_{0}V^{(5)}_{0}}{V^{{}^{\prime\prime
2}}_{0}}\right)(19+28N^{2})-\frac{1}{288}\left(\frac{V^{(6)}_{0}}{V^{{}^{\prime\prime}}_{0}}\right)(5+4N^{2})\bigg{\\}},$
and
$\displaystyle
N=n+\frac{1}{2},\;\;\;\;\;V^{(n)}_{0}=\frac{d^{n}V}{dr^{n}_{*}}\bigg{|}_{\;r_{*}=r_{*}(r_{p})}.$
(18)
Substituting the effective potential (14) into the formula above, we can
obtain the quasinormal frequencies of the scalar field in the background of
five-dimensional Lovelock black hole. Fig.5 and Table.I show the real and
imagine parts of quasinormal frequencies for the scalar field with the
variation of coupling constant $\alpha$ and angle quantum number $l$. By
analyzing these data and curves, we can find that, when the coupling constant
$\alpha$ (i.e. the additional term presents the ultraviolet correction to
Einstein theory) increases, the real part quasinormal frequencies of the
scalar field increases, while the imaginary part decreases, which means that
the ultraviolet correction makes the scalar field decay more slowly and makes
the scalar oscillate more quickly. Fig.6 and Table.II show the real and
imagine parts of quasinormal frequencies for the scalar field with the
variation of the cosmological constant $\Lambda$ and angle quantum number $l$.
Base on the data, we can make a conclusion that, when the cosmological
constant $\Lambda$ increases, the real part and the imaginary part of
quasinormal frequencies of the scalar field decreases, that’s to say which
means that the cosmological constant makes the scalar field decay more slowly
and makes the scalar oscillate more slowly.
Moreover, The Re($\omega$) increases (decreases the oscillatory time scale)
and the Im($\omega$) decreases (increases the damping time scale) as the
angular quantum number $l$ increases for fixed n, quasinormal frequencies
depend very weakly on the angular quantum number $l$, which is the same as
Jing’sJing .
Table 1: Quasinormal frequencies of the scalar field in the Lovelock black hole spacetime with parameters $M=1,\Lambda=0.1$ and $n=0$. $\alpha$ | $\omega\;(l=0)$ | $\omega\;(l=1)$ | $\omega\;(l=2)$ | $\omega\;(l=3)$
---|---|---|---|---
0.1 | 0.373800-0.303847i | 0.715847-0.249250i | 1.07354-0.238997i | 1.43010-0.236352i
0.2 | 0.374204-0.288632i | 0.725245-0.237874i | 1.08845-0.228468i | 1.45024-0.226095i
0.3 | 0.374958-0.275545i | 0.736564-0.226962i | 1.10557-0.217960i | 1.47295-0.215606i
0.4 | 0.376224-0.263901i | 0.749615-0.215928i | 1.12509-0.207052i | 1.49867-0.204530i
0.5 | 0.377815-0.253200i | 0.764359-0.204153i | 1.14737-0.195167i | 1.52804-0.192338i
0.6 | 0.379676-0.243030i | 0.780916-0.190888i | 1.17301-0.181496i | 1.56212-0.178243i
0.7 | 0.381678-0.233081i | 0.799620-0.175064i | 1.20317-0.164778i | 1.60272-0.161014i
Table 2: Quasinormal frequencies of the scalar field in the Lovelock black hole spacetime with parameters $M=1,\alpha=0.1$ and $n=0$. $\Lambda$ | $\omega\;(l=0)$ | $\omega\;(l=1)$ | $\omega\;(l=2)$ | $\omega\;(l=3)$
---|---|---|---|---
0 | 0.375051-0.304596i | 0.718738-0.249821i | 1.07808-0.239750i | 1.43618-0.237140i
0.1 | 0.373800-0.303847i | 0.715847-0.249250i | 1.07354-0.238997i | 1.43010-0.236352i
0.2 | 0.372561-0.303102i | 0.712923-0.248564i | 1.06905-0.238251i | 1.42409-0.235571i
0.3 | 0.371335-0.302359i | 0.710035-0.247882i | 1.06462-0.237511i | 1.41815-0.234797i
0.4 | 0.370118-0.301619i | 0.707182-0.247203i | 1.06024-0.236776i | 1.41229-0.234030i
0.5 | 0.368915-0.300883i | 0.704363-0.246530i | 1.05591-0.236049i | 1.40650-0.233271i
0.6 | 0.367723-0.300150i | 0.701578-0.245860i | 1.05164-0.235327i | 1.40078-0.232518i
## IV conclusions
Using the third-order WKB approximation, a numerical method devised by Schutz,
Will and Iyer, we obtained the quasinormal frequencies of the scalar field in
the background of five-dimensional Lovelock black hole in further detail. we
can find that the ultraviolet correction to Einstein theory in the Lovelock
theory makes the scalar field decay more slowly and makes the scalar field
oscillate more quickly, and the cosmological constant makes the scalar field
decay more slowly and makes the scalar field oscillate more slowly in Lovelock
black hole backgroud. At the same time we also find that quasinormal
frequencies depend very weakly on the angular quantum number $l$.
## V Acknowledgments
J.H. Chen is supported by National Natural Science Foundation of
China(Grant:10873004), Scientific Research Fund of Hunan Provincial Education
Department(Grant:08B051), program for excellent talents in Hunan Normal
University and State Key Development Program for Basic Research Program of
China (Grant: 2003CB716300).
## References
* (1) R. A. Konoplya, Phy. Rev. D 68, 024018 (2003)
* (2) M. Maggiore, Phy. Rev. Lett. 100, 141301 (2008)
* (3) J. Morgan, V. Cardoso, A. S. Miranda, C. Molina, V. T. Zanchin, arXiv: 0906.0064[gr-qc]; E. Berti, V. Cardoso, P. Pani, arXiv: 0903.5311[gr-qc]; G. Festuccia, Hong Liu, arXiv: 0811.1033[gr-qc]; R. G. Daghigh, M. D. Green, arXiv: 0808.1596[gr-qc]; J. Alsup, G. Siopsis, Phys. Rev. D78: 086001(2008); J. J. Friess, S. S. Gubser, G. Michalogiorgakis, S. S. Pufu, JHEP 0704, 080(2007); S.F.J. Chan, R.B. Mann Phys.Rev. D55 7546(1997)/arXiv:9612026[gr-qc]; S. F. J. Chan., R.B. Mann Phys.Rev. D59 064025(1999); R. G. Daghigh, JHEP, 0904:045(2009) /arXiv:0901.2353[gr-qc]; Vitor Cardoso, Roman Konoplya, Jose’ P. S. Lemos, Phys.Rev.D68:044024,2003/arXiv:gr-qc/0305037;
* (4) S. K. Chakrabarti, Eur. Phys. J. C. 61:477 (2009); Jiliang Jing, Qiyuan Pan, Nucl.Phys. B728 109 (2005); Jiliang Jing, Phys. Rev. D71 124006(2005); Jiliang Jing, Qiyuan Pan, Phys. Rev. D71 124011 (2005); M. Giammatteo, Jiliang Jing, Phys.Rev. D71 024007(2005).
* (5) Bin Wang, C.Y. Lin, C. Molina, Phys. Rev. D70 064025 (2004)/arXiv:hep-th/0407024; Da-Ping Du, Bin Wang, Ru-Keng Su, Phys. Rev. D70 064024(2004); E. Berti, K. D. Kokkotas, Phys. Rev. D67(2003)064020; C. R. Ma, Y. X. Gui, W. Wang, F. J. Wang, arXiv: 0611146[gr-qc]; B.R. Iyer, S.Iyer, and C.V.Vishveshwara, Class.Quant.Grav.6:1627,1989; R.G. Daghigh, G.Kunstatter, and J.Ziprick, Class.Quant.Grav.24:1981-1992,2007/e-Print: gr-qc/0611139
* (6) S. K. Chakrabarti, Gen. Rel. Grav.39:567(2007)/arXiv:hep-th/0603123; S. K. Chakrabarti, K. S. Gupta, Int. J. Mod. Phys. A21 3565(2006)/arXiv:hep-th/0506133;.
* (7) Xi He, Bin Wang, Shao-Feng Wu, Chi-Yong Lin, Phys. Lett. B673, 156(2009)/arXiv:0901.0034[gr-qc]; Yun Soo Myung, Yong-Wan Kim, Young-Jai Park, Eur. Phys. J. C58, 617(2008)/arXiv:0809.1933[gr-qc]; Songbai Chen, Jiliang Jing, Class. Quant. Grav. 22 4651(2005); Yu Zhang, Yuanxing Gui, Fei Yu, Fenglin Li, Gen. Rel. Grav.39:1003(2007)/arXiv:gr-qc/0612010[gr-qc]; Yu Zhang, Y X Gui, Class. Quant. Grav. 23:6141(2006)/arXiv:gr-qc/0612009[gr-qc]; Ping Xi, Astrophys. Space Sci. 321 47(2009)/arXiv:0810.4022[gr-qc]; Nijo Varghese, V. C. Kuriakose, arXiv:0802.1397[gr-qc]; Yu Zhang, Yuan-Xing Gui, Fei Yu, Chin. Phys. Lett.26:030401(2009).
* (8) A. L pez-Ortega, arXiv:0905.0073[gr-qc]; S. K. Chakrabarti, Eur. Phys. J C 61:477(2009)/arXiv:0809.1140[gr-qc]; Hsien-chung Kao, Dan Tomino, Phys. Rev. D77:127503(2008)/arXiv:0801.4195[gr-qc]; Vitor Cardoso, Jose’ P. S. Lemos, Shijun Yoshida, Phys.Rev. D69 (2004) 044004/arXiv:gr-qc/0309112; Vitor Cardoso, Jose’ P.S. Lemos, Shijun Yoshida, JHEP 0312 (2003) 041/arXiv:hep-th/0311260.
* (9) D. Lovelock, J. Math. Phys. 12, 498 (1971); N. Deruelle, Phys. Rev. D 41, 3696 (1990).
* (10) C. Callan, I. Klebanov and M. Perry, Nucl. Phys. B278 78(1986); P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 46(1985); D. Gross and J. Sloan, Nucl. Phys. B291 41 (1987).
* (11) C. Garraffo and G. Giribet, Mod. Phys. Lett. A23 1801(2008)/arXiv:0805.3537[gr-qc]; S. Habib Mazharimousavi, M. Halilsoy, Phys. Lett. B665, 125(2008)/ arXiv:0801.1726; Rong-Gen Cai, Nobuyoshi Ohta, Phys.Rev. D74 064001 (2006).
* (12) M. Aiello, R. Ferraro, G. Giribet, Class. Quant. Grav. 22 2579(2005)/arXiv:0408078[gr-qc]
* (13) M. H. Dehghani, R. Pourhasan, Phys. Rev. D79 064015(2009)
* (14) C.W. Misner, J.A. Wheeler and K. S. Thorne, Gravitation (Freeman, San Francisco) (1973).
* (15) C. Lanczos, Phys. 73, 147 (1932); Lanczos C., Ann. Math. 39, 842 (1938).
* (16) E. Abdalla, R. A. Konoplya, C. Molina, Phys.Rev. D72 084006 (2005)/arXiv:hep-th/0507100; Roman Konoplya, Phys.Rev. D71 024038(2005)/arXiv:hep-th/0410057.
* (17) R. A. Konoplya, A. Zhidenko, Phys.Rev.D77:104004,2008 /arXiv:hep-th/0802.0267.
* (18) B. F. Schutz and C. M. Will, Astrophys. J. Lett. Ed. 291, L33 (1985); S. Iyer and C. M. Will, Phys. Rev. D 35, 3621 (1987); S. Iyer, Phys. Rev. D 35, 3632 (1987).
* (19) Jiliang Jing, Qiyuan Pan, Phys. Rev. D71 124011(2005).
|
arxiv-papers
| 2009-06-07T01:32:36 |
2024-09-04T02:49:03.178122
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juhua Chen, Yongjiu Wang",
"submitter": "Juhua Chen",
"url": "https://arxiv.org/abs/0906.1318"
}
|
0906.1336
|
# Controlling entanglement sudden death and birth in cavity QED
Jian-Song Zhang Jing-Bo Xu [email protected] Zhejiang Institute of Modern
Physics and Physics Department,
Zhejiang University, Hangzhou 310027, China
###### Abstract
We present a scheme to control the entanglement sudden birth and death in
cavity quantum electrodynamics system, which consists of two noninteracting
atoms each locally interacting with its own vacuum field, by applying and
adjusting classical driving fields.
###### pacs:
03.67.Mn; 03.65.Ud
## I INTRODUCTION
In recent years, entanglement has been considered as a key resource of quantum
information processing 1 ; 2 ; 3 ; 4 . A cavity quantum electrodynamics(QED)
system is a useful tool to create the entanglement between atoms in cavities
and establish quantum communications between different optical cavities.
Recently, the manipulation of quantum entanglement for the system of cavity
QED has been extensively investigated5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 .
Many efforts have been devoted to the study of the evolution of the
entanglement under the influence of the environment 14 ; 15 ; 16 ; Yu2009 ; 17
; 18 ; 19 . It is pointed out by Yu and Eberly 14 that the entanglement of an
entangled two-qubit interacting with uncorrelated reservoirs may disappear
within a finite time during the dynamics evolution. This phenomenon, called
entanglement sudden death (ESD) has been observed in experiment 20 ; 21 .
Recently, the entanglement sudden birth(ESB) in cavity QED has been discussed
by Yönac, Yu, and Eberly Yonac2006 ; Yonac2007 . More recently, Lopez _et al_.
22 have studied the entanglement dynamics of a quantum system consisting of
two cavities interacting with two independent reservoirs and shown that ESD in
a bipartite system independently coupled to reservoirs is related to the ESB.
It has been pointed out that the cavity coherent state can be used to control
the ESB and ESD in cavity QEDYonac2008 .
In the present paper, we propose a scheme to control ESB and ESD of a quantum
system consisting of two noninteracting atoms each locally interacting with
its own vacuum field. The two atoms, which are initially prepared in entangled
states, are driven by two classical fields additionally. It is shown that ESB
and ESD phenomenon may appear in this system and the time of ESB and ESD can
be controlled by classical driving fields. In addition, the amount of the
entanglement of the two atoms or cavities can be significantly increased by
applying classical fields.
## II Effective Hamiltonian
Now, we consider a system consisting of a two-level atom inside a single mode
cavity. The atom is driven by a classical field additionally. The Hamiltonian
of the system can be described by 12
$\displaystyle H$ $\displaystyle=$ $\displaystyle\omega
a^{{\dagger}}a+\frac{\omega_{0}}{2}\sigma_{z}+g(\sigma_{+}a+\sigma_{-}a^{{\dagger}})$
(1)
$\displaystyle+\lambda(e^{-i\omega_{c}t}\sigma_{+}+e^{i\omega_{c}t}\sigma_{-}),$
where $\omega$, $\omega_{0}$ and $\omega_{c}$ are the frequency of the cavity,
atom and classical field, respectively. The operators $\sigma_{z}$ and
$\sigma_{\pm}$ are defined by $\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle
g|$, $\sigma_{+}=|e\rangle\langle g|$, and $\sigma_{-}=\sigma_{+}^{{\dagger}}$
where $|e\rangle$ and $|g\rangle$ are the excited and ground states of the
atom. Here, $a$ and $a^{{\dagger}}$ are the annihilation and creation
operators of the cavity; g and $\lambda$ are the coupling constants of the
interactions of the atom with the cavity and with the classical driving field,
respectively. Note that we have set $\hbar=1$ throughout this paper.
In the rotating reference frame the Hamiltonian of the system is transformed
to the Hamiltonian $H_{1}$ under a unitary transformation
$U_{1}=\exp{(-i\omega_{c}t\sigma_{z}/2)}$
$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle
U_{1}^{{\dagger}}HU_{1}-iU_{1}^{{\dagger}}\frac{\partial U_{1}}{\partial t}$
(2) $\displaystyle=$ $\displaystyle H_{1}^{(1)}+H_{1}^{(2)},$
with
$\displaystyle H_{1}^{(1)}$ $\displaystyle=$ $\displaystyle\omega
a^{{\dagger}}a+g(e^{i\omega_{c}t}\sigma_{+}a+e^{-i\omega_{c}t}\sigma_{-}a^{{\dagger}}),$
$\displaystyle H_{1}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\Delta_{1}}{2}\sigma_{z}+\lambda(\sigma_{+}+\sigma_{-}),$
(3)
and $\Delta_{1}=\omega_{0}-\omega_{c}$. Using the method similar to that used
in Ref.23 , diagonalizing the Hamiltonian $H_{1}^{(2)}$, and neglecting the
terms which do not conserve energies (rotating wave approximation), we can
recast the Hamiltonian $H_{1}$ as follows:
$\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle\omega
a^{{\dagger}}a+\frac{\Omega_{1}\sin{\theta}}{2}(\sigma_{+}+\sigma_{-})+g\cos^{2}{\frac{\theta}{2}}[e^{i\omega_{c}t}$
(4)
$\displaystyle\times(-\frac{\sin{\theta}}{2}\sigma_{z}+\cos^{2}{\frac{\theta}{2}}\sigma_{+}-\sin^{2}{\frac{\theta}{2}}\sigma_{-})a+h.c],$
with $\theta=\arctan{(\frac{2\lambda}{\Delta_{1}})}$. Here $h.c$ stands for
Hermitian conjugation.
The Hamiltonian $H_{1}$ can be diagonalized by a final unitary transformation
$U_{2}$ with $U_{2}=\exp{[\frac{i\omega_{c}t}{2}(\sigma_{+}+\sigma_{-})]}$.
Then, we can rewrite the Hamiltonian of the system
$\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle\omega
a^{{\dagger}}a+\frac{\omega^{\prime}\sin{\theta}}{2}(\sigma_{+}+\sigma_{-})+g^{\prime}[(-\frac{\sin{\theta}}{2}\sigma_{z}$
(5)
$\displaystyle+\cos^{2}{\frac{\theta}{2}}\sigma_{+}-\sin^{2}{\frac{\theta}{2}}\sigma_{-})a+h.c],$
where $\omega^{\prime}=\sqrt{\Delta_{1}^{2}+4\lambda^{2}}+\omega_{c}$ and
$g^{\prime}=g\cos^{2}{\frac{\theta}{2}}$. It is worth noting that the unitary
transformations $U_{1}$ and $U_{2}$ are both local unitary transformations. As
we known the entanglement of a quantum system does not change under local
unitary transformations 24 . Thus, the entanglement of the system considered
here will not be changed by applying the local unitary transformations $U_{1}$
and $U_{2}$.
## III Controlling entanglement sudden death and birth
In this section, we investigate ESD and ESB of a quantum system consisting of
two noninteracting atoms each locally interacting with its own vacuum field.
Each atom interacts with its own vacuum field where the interaction of the
system is described by $H_{2}$. We show how to control entanglement sudden
death and birth of a quantum system formed by two two-level atoms and two
cavities via classical driving fields. Assume the two-level atoms are prepared
in entangled states and the cavities are prepared in vacuum states, i.e., the
whole system is initially prepared in the state
$\displaystyle|\psi(0)\rangle=(\alpha|-_{a_{1}}\rangle|-_{a_{2}}\rangle+\beta|+_{a_{1}}\rangle|+_{a_{2}}\rangle)|0_{c_{1}}\rangle|0_{c_{2}}\rangle,$
(6)
where the subscripts $a_{1}$, $a_{2}$, $c_{1}$, and $c_{2}$ refer to atom 1,
atom 2, cavity 1, and cavity 2, respectively. Here, $|\pm\rangle$ can be
interpreted as the dressed states of the two-level atom. They are defined as
follows:
$\displaystyle|+\rangle$ $\displaystyle=$
$\displaystyle\cos{\frac{\theta}{2}}|e\rangle+\sin{\frac{\theta}{2}}|g\rangle,$
$\displaystyle|-\rangle$ $\displaystyle=$
$\displaystyle-\sin{\frac{\theta}{2}}|e\rangle+\cos{\frac{\theta}{2}}|g\rangle.$
(7)
After some algebra, we find the state of the whole system at time t is
$\displaystyle|\psi(t)\rangle$ $\displaystyle=$
$\displaystyle\alpha|-_{a_{1}}\rangle|-_{a_{2}}\rangle|0_{c_{1}}\rangle|0_{c_{2}}\rangle$
(8) $\displaystyle+\beta
f^{2}_{1}(t)|+_{a_{1}}\rangle|+_{a_{2}}\rangle|0_{c_{1}}\rangle|0_{c_{2}}\rangle$
$\displaystyle+\beta
f^{2}_{2}(t)|-_{a_{1}}\rangle|-_{a_{2}}\rangle|1_{c_{1}}\rangle|1_{c_{2}}\rangle$
$\displaystyle+\beta
f_{1}(t)f_{2}(t)(|+_{a_{1}}\rangle|-_{a_{2}}\rangle|0_{c_{1}}\rangle|1_{c_{2}}\rangle$
$\displaystyle+|-_{a_{1}}\rangle|+_{a_{2}}\rangle|1_{c_{1}}\rangle|0_{c_{2}}\rangle),$
with
$\displaystyle f_{1}(t)$ $\displaystyle=$ $\displaystyle
e^{i\Delta_{2}t/2}[\cos{(\Omega t)}-\frac{i\Delta_{2}}{2\Omega}\sin{(\Omega
t)}],$ $\displaystyle f_{2}(t)$ $\displaystyle=$ $\displaystyle-
ig\cos^{2}{\frac{\theta}{2}}e^{-i\Delta_{2}t/2}\sin{(\Omega t)}/\Omega,$
$\displaystyle\Delta_{2}$ $\displaystyle=$
$\displaystyle\sqrt{(\omega_{0}-\omega_{c})^{2}+4\lambda^{2}}+\omega_{c}-\omega,$
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\Delta_{2}^{2}}{4}+(g\cos^{2}{\frac{\theta}{2}})^{2}}.$
(9)
Tracing over the degrees of the freedom of cavities, we obtain the reduced
density matrix of two atoms
$\displaystyle\rho_{a_{1}a_{2}}(t)$ $\displaystyle=$
$\displaystyle[|\alpha|^{2}+|\beta
f^{2}_{2}(t)|^{2}]|-_{a_{1}}\rangle|-_{a_{2}}\rangle\langle-_{a_{1}}|\langle-_{a_{2}}|$
(10) $\displaystyle+|\beta
f^{2}_{1}(t)|^{2}|+_{a_{1}}\rangle|+_{a_{2}}\rangle\langle+_{a_{1}}|\langle+_{a_{2}}|$
$\displaystyle+|\beta
f_{1}(t)f_{2}(t)|^{2}(|+_{a_{1}}\rangle|-_{a_{2}}\rangle\langle+_{a_{1}}|\langle-_{a_{2}}|$
$\displaystyle+|-_{a_{1}}\rangle|+_{a_{2}}\rangle\langle-_{a_{1}}|\langle+_{a_{2}})$
$\displaystyle+[\alpha\beta^{*}f_{1}^{*2}(t)|-_{a_{1}}\rangle|-_{a_{2}}\rangle\langle+_{a_{1}}|\langle+_{a_{2}}|+h.c].$
Similarly, the reduced density matrix of two cavities is
$\displaystyle\rho_{c_{1}c_{2}}(t)$ $\displaystyle=$
$\displaystyle[|\alpha|^{2}+|\beta
f^{2}_{1}(t)|^{2}]|-_{a_{1}}\rangle|-_{a_{2}}\rangle\langle-_{a_{1}}|\langle-_{a_{2}}|$
(11) $\displaystyle+|\beta
f^{2}_{2}(t)|^{2}|+_{a_{1}}\rangle|+_{a_{2}}\rangle\langle+_{a_{1}}|\langle+_{a_{2}}|$
$\displaystyle+|\beta
f_{1}(t)f_{2}(t)|^{2}(|+_{a_{1}}\rangle|-_{a_{2}}\rangle\langle+_{a_{1}}|\langle-_{a_{2}}|$
$\displaystyle+|-_{a_{1}}\rangle|+_{a_{2}}\rangle\langle-_{a_{1}}|\langle+_{a_{2}})$
$\displaystyle+[\alpha\beta^{*}f_{2}^{*2}(t)|-_{a_{1}}\rangle|-_{a_{2}}\rangle\langle+_{a_{1}}|\langle+_{a_{2}}|+h.c].$
In order to study the entanglement of above system described by density matrix
$\rho$, we adopt the measure concurrence which is defined as 25
$C=\max{\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\}},$ (12)
where the $\lambda_{i}$(i=1,2,3,4) are the square roots of the eigenvalues in
decreasing order of the magnitude of the “spin-flipped” density matrix
operator
$R=\rho(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$ and
$\sigma_{y}$ is the Pauli Y matrix, i.e.,
$\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right)$.
Particularly, for a density matrix of the form
$\displaystyle\rho=\left(\begin{array}[]{cccc}a&0&0&0\\\ 0&b&z&0\\\
0&z^{*}&c&0\\\ 0&0&0&d\end{array}\right),$ (17)
the concurrence is
$\displaystyle C=2\max\\{0,|z|-\sqrt{ad}\\}.$ (18)
Combing the above equation with the reduced density matrix, we find that the
concurrence of two atoms is
$\displaystyle C_{a_{1}a_{2}}(t)=2|f_{1}(t)|^{2}\max\\{0,|\alpha\beta|-|\beta
f_{2}(t)|^{2}\\},$ (19)
and the concurrence of two cavities is
$\displaystyle C_{c_{1}c_{2}}(t)=2|f_{2}(t)|^{2}\max\\{0,|\alpha\beta|-|\beta
f_{1}(t)|^{2}\\}.$ (20)
In Fig.1, the evolution of two-qubit concurrence for different partitions
$C_{a_{1}a_{2}}$ (solid line) and $C_{c_{1}c_{2}}$ (dotted line) are plotted
with $\alpha=1/\sqrt{10},\beta=3/\sqrt{10},\omega=3,\omega_{0}=2,g=1$. For
simplicity, we sometimes choose the special case of
$\omega:\omega_{0}:\omega_{c}=3:2:1$. On the one hand, the concurrence of two
atoms $C_{a_{1}a_{2}}$ will disappear within a finite time during the dynamics
evolution(ESD). On the other hand, the concurrence of two cavities
$C_{c_{1}c_{2}}$ can appear during the dynamics evolution(see the dotted line
in Fig.1). It is not difficult to see that the time for which ESD($t_{ESD}$)
and ESB($t_{ESB}$) occur could be adjusted by controlling the frequency
$\omega_{c}$ and strength $\lambda$ of classical driving fields. In addition,
the amount of entanglement between two cavities can also be controlled by
classical driving fields.
In order to show this more clearly, we plot the two-qubit concurrence for
different partitions $C_{a_{1}a_{2}}$ (solid line) and $C_{c_{1}c_{2}}$
(dotted line) with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$
in Fig.2. Comparing Fig.1 and Fig.2, one can see time of ESD($t_{ESD}$) and
ESB($t_{ESB}$) depend on the parameters $\alpha$ and $\beta$. In the case of
$\alpha=1/\sqrt{10}$ and $\beta=3/\sqrt{10}$, $t_{ESD}<t_{ESB}$, that is, ESB
appears after ESD. However, when $\alpha=\sqrt{3}/\sqrt{10}$ and
$\beta=\sqrt{7}/\sqrt{10}$, $t_{ESD}>t_{ESB}$, that is, ESB appears before
ESD. Again, the time of ESD and ESB and the amount of entanglement between two
cavities can be controlled by adjusting classical driving fields.
We now turn to show the influence of classical driving fields on the
distribution of entanglement in the present system. The bipartite entanglement
of $a_{1}\otimes a_{2}$, $c_{1}\otimes c_{2}$, $a_{1}\otimes c_{2}$, and
$c_{1}\otimes a_{2}$ are displayed in Fig.3. It is not difficult to see that
the concurrence $C_{a_{1}a_{2}}$, $C_{c_{1}c_{2}}$, $C_{a_{1}c_{2}}$, and
$C_{c_{1}a_{2}}$ are periodic functions of time t. The periods of them depend
on the strength and the frequencies of classical driving fields. Comparing the
right panel and the left panel of Fig.3, we find that the time of ESB and ESD
and the amount of the entanglement of two qubits can be controlled by
classical driving fields. For example, $t_{ESD}$ and the amount of
$C_{c_{1}c_{2}}$(dashed line) of the right panel are larger than that of the
left panel.
## IV CONCLUSIONS
In summary, we have considered a quantum system consisting of two
noninteracting atoms each locally interacting with its own vacuum field. The
two atoms, which are driven by two classical fields, are initially prepared in
entangled states. We find that classical driving fields can increase the
amount of entanglement of the two-atom system. It is worth noting that the
time of ESB and ESD can be controlled by the classical driving fields. The
approach presented in the present Letter may have potential applications in
quantum information processing.
## ACKNOWLEDGEMENTS
This project was supported by the National Natural Science Foundation of China
(Grant No.10774131) and the National Key Project for Fundamental Research of
China (Grant No. 2006CB921403).
## References
* (1) J. I. Cirac, P. Zoller, Nature 404 (2000) 579.
* (2) M.A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
* (3) C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, Phys. Rev. Lett. 70 (1993) 1895\.
* (4) V. Vedral, M. B. Plenio, Phys. Rev. A 57 (2000) 1619.
* (5) J. M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys 73 (2001) 565.
* (6) S. Bose, I. Fuentes-Guridi, P. L. Knight, V. Vedral, Phys. Rev. Lett. 87 (2001) 050401.
* (7) M. S. Kim, Jinhyoung Lee, D. Ahn, P. L. Knight, Phys. Rev. A 65 (2002) 040101(R).
* (8) M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, M. Aspelmeyer, Phys. Rev. Lett. 99 (2007) 250401.
* (9) D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, M. Aspelmeyer, Phys. Rev. Lett. 98 (2007) 030405.
* (10) S. B. Li and J. B. Xu, Phys. Rev. A 72 (2005) 022332.
* (11) J. B. Xu and S. B. Li, New. J. Phys. 7 (2005) 72.
* (12) E. Solano, G. S. Agarwal, and H. Walther, Phys. Rev. Lett. 90 (2003) 027903\.
* (13) S. B. Li and J. B. Xu, Phys. Lett. A 337 (2005) 321\.
* (14) Y. Yu and J. H. Eberly, Phys. Rev. Lett. 93 (2004) 140404; Y. Yu and J. H. Eberly, Opt. Commun. 264 (2006) 393.
* (15) Z. Ficek and R. Tanaś, Rev. Rev. A 74 (2006) 024304.
* (16) T. Yu and J. H. Eberly, Quantum Inf. Comput. 7 (2007) 459.
* (17) T. Yu and J. H. Eberly, Science(London) 323 (2009) 598\.
* (18) M. Ikram, Fu-Li Li, and M. S. Zubairy, Phys. Rev. A 75 (2007) 062336.
* (19) B. Bellomo, R. Lo France, and G. Compagno, Phys. Rev. A 77 (2008) 032342.
* (20) A. R. P. Rau, Mazhar Ali and G. Alber, Europhys. Lett. 82 (2008) 40002; J.-S. Zhang, J.-B. Xu, Opt. Commun. (2009), doi:10.1016/j.optcom.2009.03.015.
* (21) M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, Science(London) 316 (2007) 579\.
* (22) J. Laurat, K. S. Choi, H. Deng, C. W. Chou, and H. J. Kimble, Phys. Rev. Lett. 99 (2007) 180504.
* (23) M. Yönac, T. Yu, J. H. Eberly, J. Phys. B 39 (2006) S621 .
* (24) M. Yönac, T. Yu, J. H. Eberly, J. Phys. B 40 (2007) S45.
* (25) C. E. Lopez, G. Romero, F. Lastra, E. Solano, and J. C. Retamal, Phys. Rev. Lett. 101 (2008) 080503.
* (26) M. Yönac and J.H. Eberly, Opt. Lett. 33 (2008) 270.
* (27) Yu-Xi Liu, C. P. Sun, and Franco Nori, Phys. Rev. A 74 (2006) 052321.
* (28) V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78 (1997) 2275\.
* (29) W. K. Wootters, Phys. Rev. Lett. 80 (1998) 2245.
List of captions
FIG.1 The concurrence of two atoms (solid line) and two caviteis (dotted line)
are plotted as a function of t with
$\alpha=1/\sqrt{10},\beta=3/\sqrt{10},\omega=3,\omega_{0}=2,g=1$. Right panel:
$\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.
FIG.2 The concurrence of two atoms (solid line) and two caviteis (dotted line)
are plotted as a function of t with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$.
Right panel: $\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.
FIG.3 The concurrence of two qubits for different partitions are plotted as a
function of t with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$.
Right panel: $\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.
Figure 1: The concurrence of two atoms (solid line) and two caviteis (dotted
line) are plotted as a function of t with
$\alpha=1/\sqrt{10},\beta=3/\sqrt{10},\omega=3,\omega_{0}=2,g=1$. Right panel:
$\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.
Figure 2: The concurrence of two atoms (solid line) and two caviteis (dotted
line) are plotted as a function of t with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$.
Right panel: $\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.
Figure 3: The concurrence of two qubits for different partitions are plotted
as a function of t with
$\alpha=\sqrt{3}/\sqrt{10},\beta=\sqrt{7}/\sqrt{10},\omega=3,\omega_{0}=2,g=1$.
Right panel: $\omega_{c}=\lambda=0$. Left panel: $\omega_{c}=\lambda=1$.
|
arxiv-papers
| 2009-06-07T07:32:19 |
2024-09-04T02:49:03.183301
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian-Song Zhang",
"submitter": "Zhang Jian-Song",
"url": "https://arxiv.org/abs/0906.1336"
}
|
0906.1365
|
# HIGGS PHYSICS AND BEYOND THE STANDARD MODEL AT CMS/ATLAS
N. DE FILIPPIS
Prospective searches about Higgs physics and beyond the Standard Model are
presented for the CMS and ATLAS experiments. Possible excesses of events in
real data could be an indication of the existence of new particles, even with
few hundred $\mathrm{pb^{-1}}$ of integrated luminosity. In this paper the
focus is on the current analyses strategies and on the potential both for a
discovery and/or for an exclusion of the Standard Model Higgs boson in the
main decay channels. The searches for some supersymmetric and exotic particles
predicted by several theoretical models are also discussed.
## 1 Introduction
The field of high energy physics is approaching an important period of its
history with the start of the operations of the Large Hadron Collider (LHC) at
CERN, the world’s largest and highest-energy particle accelerator. The LHC
will collide opposing beams of protons or lead ions, each carrying energies
per nucleon up to 2.76 TeV. LHC started to operate with the injection of first
beams in the beam pipe in fall 2008\. The LHC has been built with the purpose
of exploring new frontiers of particle physics, giving evidence of the
existence of the Higgs boson and/or a wide spectrum of new particles predicted
by supersymmetry and exotic models.
In general, the experiments at the LHC could provide answers or ingredients to
answer some of the most fundamental open questions in particle physics, such
as: the reality of the Higgs mechanism for generating gauge bosons and
fermions masses, the problem of the hierarchy between the electroweak gauge
boson scale and the Grand Unification or Planck scale, the existence of a
supersymmetry which implies that the known Standard Model (SM) particles have
supersymmetric partners, the existence of extra dimensions as predicted by
various models inspired e.g. by string theory.
CMS and ATLAS are the two general purpose experiments built at the LHC aimed
to provide answers to those fundamental questions. Prospective studies have
been performed over the last years in the physics groups of the CMS and ATLAS
collaborations to optimize strategies for the search of the Higgs boson(s), of
the supersymmetric particles and of some exotic particles predicted by several
models, at the center-of-mass energies of the LHC collider, both with low and
high integrated luminosity.
## 2 Prospective searches at CMS/ATLAS
The search for Higgs and supersymmetric particles has been the major guide to
define the detector requirements and performance that are detailed in Ref.
$\\!{}^{{\bf?}}$ and Ref. $\\!{}^{{\bf?}}$ for CMS and ATLAS.
Detailed simulations of the detector closest to the real experimental set-up
with miscalibration/misalignment conditions at start-up luminosity have been
used in the CMS and ATLAS studies.
Advanced Monte Carlo physics generators has been used for signal and
background simulation with the estimation of NLO QCD and electroweak
corrections.
### 2.1 Searches for Standard Model Higgs
Direct searches for the SM Higgs particle at the LEP $\mathrm{e^{+}e^{-}}$
collider have led to a lower mass bound of ${m_{\rm
H}>114.4\,\mathrm{GeV/c^{2}}}$ at 95% C.L. $\\!{}^{{\bf?}}$. On-going direct
searches at the Tevatron $\mathrm{p\bar{p}}$ collider by the D0 and CDF
experiments set constraints on the production cross-section for a SM-like
Higgs boson in a mass range extending up to about $200\,\mathrm{GeV/c^{2}}$
and allow to exclude his existence $\\!{}^{{\bf?}}$ with mass between 160 and
170 $\mathrm{GeV/c^{2}}$.
The main production mechanisms for SM Higgs particle at LHC are the gluon
-gluon fusion mechanism, the associated production with W/Z bosons, the weak
vector boson fusion processes and the associated Higgs production with heavy
top or bottom quarks, as detailed in Ref. $\\!{}^{{\bf?}}$. The gluon fusion
mechanism dominates especially at low Higgs mass and the cross section at NLO
is in between 0.1 and 50 pb depending on the Higgs mass; the cross section of
Higgs production via the vector boson fusion is generally one order of
magnitude lower with respect to gluon fusion while the other contributions are
much less important.
SM Higgs couples to fermions, gauge bosons and to itself. In the low mass
region (namely $\mathrm{m_{H}<130\,GeV/c^{2}}$) the dominant decay is in
$\mathrm{b\bar{b}}$ with a branching ratio between 60 and 90 $\%$;
$\mathrm{H\rightarrow\tau^{+}\tau^{-},\,c\bar{c},\,\gamma\gamma}$ contribution
to the total width is less that few %. In the high mass range the decay
channels $\mathrm{H\rightarrow WW^{(*)}}$ and $\mathrm{H\rightarrow ZZ^{(*)}}$
play the main role given a clear signature of multi leptons in the final
state.
A prospective analysis about the $\mathrm{H\rightarrow WW\rightarrow
ll\nu\nu}$ decay chain was performed both in CMS $\\!{}^{{\bf?}}$ and in ATLAS
$\\!{}^{{\bf?}}$. The signature consists of two isolated high momentum leptons
and missing energy related to the neutrinos escaping the detection. No hard
jet in the central region of the acceptance is expected and it is not possible
to reconstruct the Higgs mass peak due to the neutrinos. The main background
comes from $\mathrm{t\bar{t}}$ and di-boson events, di-leptons a la Drell-Yan,
$\mathrm{tW}$ and W+jets events in the topologies including two leptons in the
final state.
The analysis mainly consists of selecting events with high transverse momentum
leptons and sufficient missing energy; a central jet veto strategy is used to
select events with no hard jet in central rapidity region and the angular
correlation between the leptons coming from Higgs decays is used as a
discriminating observable.
Both a cut-based and neural net-based approaches were used to gain
discrimination between signal and background. The distribution of the output
result of the neural net for signal and background is reported in Fig. 1
(left), for $\mathrm{1\,fb^{-1}}$ of integrated luminosity. Strategies to
control the efficiency of leptons and jet reconstruction, the rate of jets
faking leptons, the measurement of the missing energy and the estimation of
$\mathrm{t\bar{t}}$ and WW background rates from data were also developed.
The significance for the signal observation in CMS with $\mathrm{1\,fb^{-1}}$
of integrated luminosity as a function of the Higgs mass hypothesis is
reported in Fig. 1 (right); that is converted in an equivalent number of one-
sided tail $\mathrm{\sigma}$ of the Gaussian distribution and it is larger
than 3 for Higgs masses between 155 and 185 $\mathrm{GeV/c^{2}}$.
Figure 1: The distribution of the output result of the neural net (left) for
signal and background in the $\mathrm{H\rightarrow WW\rightarrow 2l2\nu}$
search, with $\mathrm{1\,fb^{-1}}$ of integrated luminosity; significance of
the signal observation (right) in the $\mathrm{H\rightarrow WW\rightarrow
2l2\nu}$ with an integrated luminosity of $1\,\mathrm{fb^{-1}}$.
In the case of $\mathrm{H\rightarrow ZZ}$ decay channel the topology of four
leptons in the final state (electron and/or muons) was studied with an
integrated luminosity of 1 and 30 $\mathrm{fb^{-1}}$ for CMS $\\!{}^{{\bf?}}$
and ATLAS $\\!{}^{{\bf?}}$ respectively; the irreducible background comes from
the ZZ events with four leptons in the final state while $Z\mathrm{b\bar{b}}$
and $\mathrm{t\bar{t}}$ events could be reduced.
A preselection strategy aimed to get rid of QCD related background with jets
faking leptons was developed in the CMS collaboration; that is based on
electron identification techniques, loose isolation on leptons and a minimal
cuts on di-lepton and four-lepton invariant mass. $Z\mathrm{b\bar{b}}$ and
$\mathrm{t\bar{t}}$ events were substantially reduced with a tight isolation
on leptons and cuts on their impact parameters at the closest approach point.
Another powerful observable is the mass of the reconstructed off-mass shell Z.
With the purpose of providing a robust baseline strategy for the observation
of the Higgs, the complete selection is cut-based and $m_{H}$-independent.
Strategies to control efficiencies of lepton reconstruction and estimate the
rate of ZZ and $\mathrm{Zb\bar{b}}$ events from data were also developed.
The four-lepton invariant mass spectrum obtained in the case $2e2\mu$ final
state at the end of the selection is reported in Fig. 2 (left). The
significance for the signal observation with an integrated luminosity of 1
$\mathrm{fb^{-1}}$ is reported in Fig. 2 (right), as obtained by the CMS
collaboration. The significance of such an observation needs to be further de-
rated by about 1s unit to take into account the probability of a random
fluctuation anywhere in the mass spectrum (the so-called look-elsewhere
effect); when taking into account that effect, it is unlikely that an
integrated luminosity of 1 $\mathrm{fb^{-1}}$ will yield an observation of a
mass peak with an overall significance above $\mathrm{2\sigma}$.
Figure 2: $\mathrm{2e2\mu}$ invariant mass (left) after the full selection,
corresponding to an integrated luminosity of $1\,\mathrm{fb^{-1}}$;
significance for the signal observation (right) in the $\mathrm{H\rightarrow
ZZ\rightarrow 4l}$ channel with an integrated luminosity of
$1\,\mathrm{fb^{-1}}$.
Even if the branching ratio of the decay in two photons
$\mathrm{H\rightarrow\gamma\gamma}$ is less than % at low Higgs mass the clear
signature of the final state makes that topology very promising. Background
events come from the production of two isolated photons, which are usually
referred to as irreducible, while reducible background sources are events with
at least one fake photon. Fake photons are mostly due to the presence of a
leading $\mathrm{\pi^{0}}$ resulting from the fragmentation of a quark or a
gluon.
The performance of the electromagnetic calorimeter and of the photon
reconstruction, identification (to reject background from jets faking photons)
and calibration are fundamental to disentangle the signal from the background.
Considering Higgs boson decays with photons within the acceptance, about 57%
of the selected events have at least one true conversion with a radius smaller
than 80 cm in the ATLAS detector. Conversions are reconstructed by a vertexing
algorithm using the reconstructed particle tracks. Among the reconstructed
photons passing the identification cuts, the two with highest transverse
momentum are assumed to come from the Higgs boson decay so the vertex position
of that is reconstructed. The invariant mass distributions for photons pairs
from 120 $\mathrm{GeV/c^{2}}$ mass Higgs boson decays after trigger and
identification cuts is reported in Fig. 3 (left).
In the ATLAS collaboration, in addition to the inclusive
$\mathrm{H\rightarrow\gamma\gamma}$ search, many topologies with one or two
jets, with missing transverse energy and isolated leptons or with only missing
transverse energy, were also studied $\\!{}^{{\bf?}}$. The significance in the
$\mathrm{H\rightarrow\gamma\gamma}$ as a function of the Higgs mass is
reported in Fig. 3 (right); a significance based on event counting of 2.6 with
10 $\mathrm{fb^{-1}}$ for $\mathrm{m_{H}=120\,GeV}$ is obtained in the case of
inclusive analysis.
Figure 3: Invariant mass distributions (left) for photons pairs from Higgs
boson decays with Higgs mass of 120 $\mathrm{GeV/c^{2}}$ after trigger and
identification cuts; signal significance (right) in
$\mathrm{H\rightarrow\gamma\gamma}$ channel as a function of the Higgs mass
for 10 $\mathrm{fb^{-1}}$ of integrated luminosity . The solid circles
correspond to the sensitivity of the inclusive analysis by using event
counting. The open circles display the event counting significance when the
Higgs boson plus jet analyses are included. The squares markers correspond to
the sensitivity obtained using a combined analysis.
Statistical procedures for combination of results were used in the ATLAS
collaboration to derive the potential of discovery and exclusion from
independent searches: $\mathrm{H\rightarrow\tau^{+}\tau^{-}}$,
$\mathrm{H\rightarrow WW\rightarrow e\nu\mu\nu}$,
$\mathrm{H\rightarrow\gamma\gamma}$ and $\mathrm{H\rightarrow ZZ\rightarrow
4l}$, as detailed in Ref. $\\!{}^{{\bf?}}$. The level of compatibility between
data that give an observed value of a given estimator (typically a likelihood
ratio) and a given hypothesis (background only or signal+bagkround) is
quantified by giving the p-value that is the probability, under the assumption
of a given hypothesis, of seeing data with equal or greater incompatibility,
relative to the data actually obtained. Any p-value below 0.05 indicates an
exclusion; the median p-value obtained for excluding a SM Higgs Boson for the
various channels as well as the combination with integrated luminosity of 2
$\mathrm{fb^{-1}}$ is reported in Fig. 4; ATLAS has the median sensitivity to
exclude a SM Higgs boson with a mass in a 115-460 GeV range at 95 % C.L..
Figure 4: The median p-value obtained for excluding a SM Higgs Boson for the
various channels as well as the combination for (left) the lower mass range
(right) for masses up to 600 GeV with integrated luminosity of 2
$\mathrm{fb^{-1}}$.
### 2.2 Searches for supersymmetric particles
Hints of supersymmetry $\\!{}^{{\bf?}}$ are looked for at LHC via the
production of squarks and gluinos, the supersymmetric partners of quark and
gluons of the SM. The final state topologies of the supersymmetric events at
LHC consist of multiple jets, often very energetic, with possibly some leptons
and missing energy in the final state or simply with many leptons and missing
energy.
Most of the studies performed in CMS and ATLAS were done in the context of the
Minimal Supersymmetric Standard Model (MSSM) with R-parity conservation and in
the scenario of heavy squarks and gluinos. in order to reduce the number of
free parameters of MSSM the hypotheses of minimal Supergravity (mSUGRA
$\\!{}^{{\bf?}}$) are used, in particular by assuming a common sfermion mass
at GUT scale ($\mathrm{m_{0}}$) and a common gaugino mass
($\mathrm{m_{1/2}}$).
Prospective analyses were developed to search for final states including jets,
leptons and missing energy both in CMS $\\!{}^{{\bf?}}$ and in the ATLAS
collaboration $\\!{}^{{\bf?}}$. Typically some benchmark points of the
parameter space of MSSM with mSUGRA hypotheses are used as starting points and
scans of parameters around them is performed to derive conservative limits.
Concerning ATLAS analyses, one possible inclusive signature is consist of four
jets and missing energy. The main backgrounds are $\mathrm{t\bar{t}}$ and
W/Z+jets events. Simple selection cuts are applied on the total transverse
momentum of the jets, on the missing transverse energy, on the angle between
the jet and the missing energy directions and on the effective mass of
transverse momentum of the jets and leptons and missing transverse energy.
Final state topologies with less than four jets and with one or more leptons
were also studied.
In the Fig. 5 (left) is reported the $\mathrm{5\sigma}$ discovery reach in the
plane ($\mathrm{m_{0}},m_{1/2}$) in the case of four jets with one or more
leptons in the final state and missing energy; zero-lepton mode can probe
close to 1.5 TeV for the minimum between the squark and the gluino mass, with
1 $\mathrm{fb^{-1}}$ of integrated luminosity; the four-jets topology seems to
give the best results in zero-lepton mode, as derived by Fig. 5 (right).
Therefore ATLAS could discover signals with gluino and squark masses less than
O(1 TeV) after having accumulated an integrated luminosity of about
$\mathrm{1\,fb^{-1}}$.
Figure 5: The $\mathrm{5\sigma}$ discovery reach in the plane
($\mathrm{m_{0},m_{1/2}}$) in the case of four jets with one or more leptons
in the final state and missing energy (left) and in the case of two, three and
four jets with zero lepton (right).
### 2.3 Searches for exotic particles
Exotic massive gauge bosons are expected in several theoretical models beyond
the SM. In the sequential Standard Model $\\!{}^{{\bf?}}$ (SSM) a Z-like
boson, called Z’, with the same couplings of the Z to fermions and gauge
bosons and with O(TeV) mass is predicted. Other exotic scenarios based on
extra dimension $\\!{}^{{\bf?}}$ predict the existence of a graviton with
O(TeV) mass decaying in $\mathrm{e^{+}e^{-}}$.
Searches for high mass gauge bosons decaying in $\mathrm{e^{+}e^{-}}$ pair
were performed both in CMS $\\!{}^{{\bf?}}$ and ATLAS $\\!{}^{{\bf?}}$. The
cross section times the branching ratio is between few fb to few hundred fb
depending on the mass of the resonance and the theoretical model. Main
backgrounds for those searches were di-electron events produced via Drell Yan
mechanism, $\mathrm{t\bar{t}}$ events with two electrons in the final state,
QCD with jets faking electrons, W+jets, $\mathrm{\gamma}$+jets,
$\mathrm{\gamma\gamma}$.
Concerning the CMS analysis, an important aspect of the analysis was the usage
of high threshold trigger patterns to tag those events
($\mathrm{E_{T}>80\,GeV}$ and loose isolation on leptons with
$\mathrm{E_{T}>200\,GeV}$ in electromagnetic calorimeter). Saturation occurs
in the electromagnetic calorimeter electronics for very high energy deposits
in a single ECAL crystal ($\mathrm{>\,1.7\,TeV}$ for the barrel and
$\mathrm{>\,3.0\,TeV}$ for the endcaps); the energy in the saturated crystal
can be reconstructed, with a resolution of about 7%, using the energy deposit
distribution in the surrounding crystals, as detailed in Ref.
$\\!{}^{{\bf?}}$.
The di-electron invariant mass spectrum for signal and background at 100
$\mathrm{pb^{-1}}$ is reported in Fig. 6 (left); at high mass only few
background events survive the selection giving an optimal signal to background
rejection.
At the end of the analysis, after computing the integrated luminosity for
5$\mathrm{\sigma}$ discovery at $\mathrm{\sqrt{s}=14\,TeV}$ as a function of
the Z’ mass, it could be shown that few hundred $\mathrm{pb^{-1}}$ of
integrated luminosity are needed to discover the Z’ with O(1 TeV) mass with
5$\mathrm{\sigma}$.
Search for di-muon resonances at O(1TeV) mass were addressed too by CMS and
ATLAS $\\!{}^{{\bf?}}$. Sources of background are di-muons from Drell Yan
events and W+jets, Z+jets.
At large transverse momentum ($\mathrm{>\,100\,GeV}$), an important
contribution to the muon momentum resolution is related to the misalignment of
the muon spectrometer. A detailed study was carried out in order to determine
the effect of possible larger uncertainties in the position of the muon
chambers to the Z’ search; in addition to the ideal case of no misalignment,
several different hypotheses of misalignment were simulated. Muon chamber
misalignment has an important effect causing a loss of Z’ mass resolution that
degrade the determination of the charge of muon.
In the Fig. 6 (right) is reported the luminosity needed for a
5$\mathrm{\sigma}$ discovery of Z’ as predicted by the SSM. That luminosity
ranges from 20 to 40 $\mathrm{pb^{-1}}$, which makes the di-muon channel
competitive with the di-electron channel. The inclusion of the effect of
misalignment and all the systematics makes the prediction less powerful and
the result worst.
Figure 6: Di-electron invariant mass spectrum (left) for a 100
$\mathrm{pb^{-1}}$ integrated luminosity with 1 $\mathrm{TeV/c^{2}}$ Z’
signal, compared to SM background estimates for the Drell-Yan process,
$\mathrm{t\bar{t}}$, QCD di-jet, W+jet, $\mathrm{\gamma}$+jet and
$\mathrm{\gamma\gamma}$; $\mathrm{1-CL_{b}}$ distribution (right) obtained as
a function of the integrated luminosity for the Z’ expected in the SSM at mass
of 1 $\mathrm{TeV/c^{2}}$, if the muon spectrometer is aligned with a
precision of 300 $\mathrm{\mu m}$. The effect of the systematic uncertainty on
the trigger selection and on the knowledge of the SM Drell-Yan cross-section
is also displayed.
## References
## References
* [1] The CMS experiment at the CERN LHC, CMS Collaboration, JINST 3 S08004, 2008.
* [2] The ATLAS experiment at the CERN LHC, ATLAS Collaboration, JINST 3 S08003, 2008.
* [3] R. Barate et al., LEP Working Group for Higgs boson searches, Search for the standard model Higgs boson at LEP, Phys. Lett. B 565, 61 (2003), arXiv:hep-ex/0306033.
* [4] The TEVNPH Working Group for the CDF and D0 Collaborations, Combined CDF and D0 Upper Limits on Standard Model Higgs-Boson Production with up to 4.2 $fb^{-1}$ of Data, FERMILAB-PUB-09-060-E, CDF Note 9713, D0 Note 5889, arXiv:0903.4001v1.
* [5] The Anatomy of Electro Weak Symmetry Breaking, LPT Orsay 05 17 March 2005, arXiv:hep-ph/0503172v2.
* [6] Search Strategy for a Standard Model e Higgs Boson Decaying to Two W Bosons in the Fully Leptonic Final State, CMS Collaboration, CMS PAS HIG-08-006.
* [7] Expected Performance of the ATLAS Experiment : Detector, Trigger and Physics, ATLAS Collaboration, CERN-OPEN-2008-020.
* [8] Search strategy for the Higgs boson in the $ZZ^{*}$ decay 5 channel with the CMS experiment, CMS Collaboration, CMS PAS HIG-08-003.
* [9] P. Fayet, Phys. Lett. B 64, 159 (196); P. Fayet, Phys. Lett. B 69, 489 (1977), Phys. Lett. B 84, 416 (1979); G.R. Farrar and P. Fayet, Phys. Lett. B 76, 575 (1978).
* [10] H.P. Nilles, Phys. Rev. C 110, 1 (1984).
* [11] SUSY searches with dijet events, CMS Collaboration, CMS PAS SUS-08-005.
* [12] Prospects for SUSY discovery based on inclusive searches with the ATLAS detector, ATLAS Collaboration, ATL-PHYS-PROC-2009-038; ATL-COM-PHYS-2009-035.
* [13] S. Dimopoulos and H. Georgi, NPB 193, 150 (1981)
* [14] A Large Mass Hierarchy from a Small Extra Dimension, Phys. Rev. Lett. 83, 3370 (1999).
* [15] Search for high mass resonance production decaying into an electron pair in the CMS experiment, CMS Collaboration, CMS PAS EXO-08-001.
* [16] B. Clerbaux, T. Mahmoud, C.Collard, M.-C. Lemaire and V. Litvin, TeV electron and photon saturation studies, CMS Collaboration, CMS NOTE, 2006-004 (2006).
|
arxiv-papers
| 2009-06-07T16:25:52 |
2024-09-04T02:49:03.188680
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. De Filippis, for CMS and ATLAS Collaboration (Laboratoire Leprince\n Ringuet - Ecole Polytechnique - IN2P3/CNRS, Palaiseau, France)",
"submitter": "Nicola De Filippis",
"url": "https://arxiv.org/abs/0906.1365"
}
|
0906.1367
|
11institutetext: Institut d’Astrophysique Spatiale, CNRS & Université Paris
Sud, Orsay, France.
# Electron density in the quiet solar coronal transition region from
SoHO/SUMER measurements of S vi line radiance and opacity
E. Buchlin J.-C. Vial
(Received : / Revised date : )
###### ?abstractname?
Context. The sharp temperature and density gradients in the coronal transition
region are a challenge for models and observations.
Aims. We set out to get the average electron density $\langle
n_{\textrm{e}}\rangle$ in the region emitting the S vi lines. We use two
different techniques which allow to derive linearly-weighted (opacity method)
and quadratically-weighted (Emission Measure method) electron density along
the line-of-sight, in order to estimate a filling factor or to derive a
thickness of the layer at the formation temperature of the lines.
Methods. We analyze SoHO/SUMER spectroscopic observations of the S vi lines,
using the center-to-limb variations of radiance, the center-to-limb ratios of
radiance and line width, and the radiance ratio of the
$93.3$–$94.4\>\mathrm{nm}$ doublet to derive the opacity. We also use the
Emission Measure derived from radiance at disk center.
Results. We get an opacity $\tau_{0}$ at S vi 93.3 nm line center of the order
of $0.05$. The resulting average electron density $\langle
n_{\textrm{e}}\rangle$, under simple assumptions concerning the emitting
layer, is $2.4\cdot 10^{16}\>\mathrm{m^{-3}}$ at $T=2\cdot
10^{5}\>\mathrm{K}$. This value is higher than (and incompatible with) the
values obtained from radiance measurements ($2\cdot
10^{15}\>\mathrm{m^{-3}}$). The last value leads to an electron pressure of
$10^{-2}\>\mathrm{Pa}$. Conversely, taking a classical value for the density
leads to a too high value of the thickness of the emitting layer.
Conclusions. The pressure derived from the Emission Measure method compares
well with previous determinations. It implies a low opacity of $5\,10^{-3}$ to
$10^{-2}$. The fact that a direct derivation leads to a much higher opacity
remains unexplained, despite tentative modeling of observational biases.
Further measurements, in S vi and other lines emitted at a similar
temperature, need to be done, and more realistic models of the transition
region need to be used.
###### Key Words.:
Sun : atmosphere – Sun: transition region – Sun: UV radiation
††offprints: E. Buchlin, [email protected]
## 1 Introduction
In the simplest description of the solar atmosphere, where it is considered as
a series of concentric spherical layers of plasma at different densities and
temperatures, the transition region (hereafter TR) between the chromosphere
and the corona is the thin interface between the high-density and low-
temperature chromosphere (a few $10^{16}\>\mathrm{m^{-3}}$ hydrogen density at
about $10^{4}\>\mathrm{K}$) and the low-density and high-temperature corona
(about $10^{14}\>\mathrm{m^{-3}}$ at $10^{6}\>\mathrm{K}$). The variation of
temperature $T$ and electron number density $n_{\textrm{e}}$ has been mostly
derived from the modelling of this transition region, where radiative losses
are balanced by thermal conduction (e.g. Mariska 1993; Avrett & Loeser 2008).
Measurements of the electron density usually rely either on estimation of the
Emission Measure or on line ratios. On one hand, using absolute line
radiances, the Emission Measure (EM) and Differential Emission Measure (DEM)
techniques provide $\langle n_{\textrm{e}}^{2}\rangle$ at the formation
temperature of a line (or as a function of temperature if several lines
covering some range of temperatures are measured). On the other hand, the
technique of line radiance ratios provides a wealth of values of
$n_{\textrm{e}}$ (Mason 1998) with the assumption of uniform density along the
line-of-sight, and with an accuracy limited by the accuracy of the two
respective radiance measurements: typically, a 15 % uncertainty on line
radiance measurement leads to 30 % uncertainty on the line ratio and then to
about a factor 3 uncertainty on the density. However, for a given pair of
lines, this technique only works in a limited range of densities. Let us add
that the accuracy is also limited by the precision of atomic physics data.
Here we propose to use also the concept of opacity (or optical thickness) in
order to derive the population of the low (actually the ground) level $i$ of a
given transition $i\rightarrow j$, and then the electron density. At a given
wavelength, the opacity of a column of plasma corresponds indeed to the sum of
the absorption coefficients of photons by the individual ions in the column.
The opacity can be derived by different complementary techniques (Dumont et
al. 1983) if many measurements are available with spatial (preferably center-
to-limb) information. This is the case in a full-Sun observations program by
the SoHO/SUMER UV spectro-imager (Wilhelm et al. 1995; Peter 1999; Peter &
Judge 1999) run in 1996. In particular, thanks to a specific “compressed”
mode, a unique dataset of 36 full-Sun observations in S vi lines has been
obtained; this makes possible to derive at the same time $\langle
n_{\textrm{e}}\rangle$ from opacity measurements and $\langle
n_{\textrm{e}}^{2}\rangle$ from line radiance measurements (via the EM).
We have already used this data set in order to get properties of turbulence in
the TR (Buchlin et al. 2006). Note that here, contrary to Peter (1999); Peter
& Judge (1999); Buchlin et al. (2006), we are not interested in the resolved
directed velocities or in the non-thermal velocities but in the line
radiances, peak spectral radiances and widths. Also note that, along with the
modelling work of Avrett & Loeser (2008), we do not distinguish network and
internetwork (anyway a difficult task at the limb) and aim at a precise
determination of the properties of an average TR.
This paper is organized as follows: we first present the data set we use, then
we determine opacities and radiances of S vi 93.3 nm, we get two
determinations of density in the region emitting the S vi 93.3 nm line, we
discuss the disagreement between the two determinations (especially possible
biases), and we conclude.
## 2 Data
### 2.1 Data sets
We use the data from a SoHO/SUMER full-Sun observation program in S vi 93.3
nm, S vi 94.4 nm and Ly $\varepsilon$ designed by Philippe Lemaire. The
spectra, obtained with detector A of SUMER and an exposure time of
$3\>\mathrm{s}$, were not sent to the ground (except for context spectra) but
5 parameters (“moments”) of 3 lines were computed on-board for each position
on the Sun:
* •
(1) peak spectral radiance, (2) Doppler shift, and (3) width of the line S vi
93.3 nm,
* •
(4) line radiance (integrated spectral radiance) of the line Ly $\varepsilon$
93.8 nm,
* •
(5) line radiance of the line S vi 94.4 nm. It must be noted that this line is
likely to be blended with Si viii.
The detailed characteristics of these lines can be found in Table 1. A list of
the 36 observations of this program run throughout year 1996, close to solar
minimum, can be found in Table 1 of Buchlin et al. (2006). These original data
constitute the main data set we use in this paper, hereafter DS1. They are
complemented by a set of 22 context observations from the same observation
program, that we use when we need the full profiles of the spectral lines
close to disk center: the full SUMER detector ($1024\times 360$ pixels) has
been recorded at a given position on the Sun at less than
$40\>\mathrm{arcsec}$ from disk center and with an exposure time of
$300\>\mathrm{s}$. This data is calibrated using the Solar Software procedure
`sum_read_corr_fits` (including correction of the flat field, as measured on
23 September 1996, and of distortion), and it will hereafter be referred to as
DS2.
?tablename? 1: Spectral lines present in the data sets, with parameters computed by CHIANTI and given by previous observations. | | CHIANTI111Using the “Arnaud & Raymond” ionization fractions file, the “Sun coronal” abundance file and the “Quiet Sun” DEM file. CHIANTI does not include data for the Hydrogen lines (Ly $\varepsilon$ in particular). | Curdt et al. (2001)
---|---|---|---
Ion | Transition $j\rightarrow i$ | $\log T_{\rm max}$ (K) | Wavelength (Å) | Radiance222Radiances are given in $\>\mathrm{W\,m^{-2}sr^{-1}}$, and peak spectral radiances are given in $\>\mathrm{W\,m^{-2}sr^{-1}nm^{-1}}$. | Wavelength (Å) | Peak radianceb
S vi | $2\mathrm{p}^{6}\;3\mathrm{p}\;{}^{2}\mathrm{P}_{3/2}\rightarrow 2\mathrm{p}^{6}\;3\mathrm{s}\;{}^{2}\mathrm{S}_{1/2}$ | $5.3$ | $933.3800$ | $3.81\cdot 10^{-3}$ | $933.40$ | $0.57$
Ly $\varepsilon$ | $6\mathrm{p}\;{}^{2}\mathrm{P}_{3/2}\rightarrow 1\mathrm{s}\;{}^{2}\mathrm{S}_{1/2}$ | — | — | — | $937.80$ | $1.07$
Si viii | $2\mathrm{s}^{2}\;2\mathrm{p}^{3}\;{}^{2}\mathrm{P}_{3/2}\rightarrow 2\mathrm{s}^{2}\;2\mathrm{p}^{3}\;{}^{4}\mathrm{S}_{3/2}$ | $5.9$ | $944.4670$ | $5.24\cdot 10^{-3}$ | $944.34$ | $0.14$
S vi | $2\mathrm{p}^{6}\;3\mathrm{p}\;{}^{2}\mathrm{P}_{1/2}\rightarrow 2\mathrm{p}^{6}\;3\mathrm{s}\;{}^{2}\mathrm{S}_{1/2}$ | $5.3$ | $944.5240$ | $1.91\cdot 10^{-3}$ | $944.55$ | $0.29$
?figurename? 1: Raw line profiles from the context spectrum taken on 4 May
1996 at 07:32 UT at disk center with an exposure time of $300\>\mathrm{s}$.
The profiles are averaged over pixels 50 to 299 along the slit ($1\times
300\>\mathrm{arcsec}$, detector A), with no prior destretching of the data.
### 2.2 Averages of the data as a function of distance to disk center
In order to obtain averages of the radiances in data set DS1 as a function of
the radial distance $r$ to the disk center, and as a function of $\mu$, the
cosine of the angle between the normal to the solar “surface” and the line-of-
sight, we apply the following method, assuming that the Sun is spherical:
* •
We detect the limb automatically by finding the maximum of the S vi 93.3 nm
radiance at each solar-$y$ position in two detection windows in the solar-$x$
direction, corresponding to the approximate expected position of the limb.
This means that the limb is found in a TR line and is actually approximately
$3\>\mathrm{arcsec}$ above the photosphere. However, this is the relevant limb
position for the geometry of the S vi 93.3 nm emission region.
* •
We fit these limb positions to arcs of a circle described by $x(y)$ functions,
and we get the real position $(a,b)$ of the solar disk center in solar
coordinates $(x,y)$ given by SUMER, and the solar radius $R_{\astrosun}$ (this
changes as a function of the time of year due to the eccentricity of SoHO’s
orbit around the Sun). The solar radius is evaluated for the observed
wavelength of $93.3\>\mathrm{nm}$.
* •
We choose to exclude zones corresponding to active regions, as the aim of this
paper is to obtain properties of the TR in the Quiet Sun.
* •
For each of the remaining pixels, we get values of the radial distance
$r=\sqrt{(x-a)^{2}+(y-b)^{2}}$ to disk center and of
$\mu=\sqrt{1-(r/R_{\astrosun})^{2}}$.
* •
We compute the averages of each moment (radiances and widths) in bins of
$r/R_{\astrosun}$ and in bins of $1/\mu$.
The resulting averages as a function of $r/R_{\astrosun}$ and of $1/\mu$ are
plotted in Fig. 2 (except for the S vi 93.3 nm Doppler shift, which will not
be used in this paper). The radiances are approximately linear functions of
$1/\mu$ for small $1/\mu$, as expected from optically thin lines in a plane-
parallel geometry. Such a behavior actually validates the consideration of a
“mean” plane-parallel transition region, at least for $1/\mu<10$ or
$\theta<84$°.
?figurename? 2: Average of the data as a function of $r/R_{\astrosun}$ (top
panels) and as a function of $1/\mu$ (bottom panels).
## 3 Determination of opacities
### 3.1 Using center-to-limb variations
We follow here the method A proposed by Dumont et al. (1983). Assuming that
the TR is spherically symmetric and that it can be considered as plane-
parallel when not seen too close to the limb, that the lines are optically
thin, and that the source function $S$ is constant in the region where the
line is formed333We release this assumption in Sec. 5., the spectral radiance
is:
$I_{0}(\mu)=S(1-\exp(-\tau_{0}/\mu))$ (1)
where the subscript $0$ is for the line center and $\tau$ is the opacity of
the emitting layer at disk center. Then:
$I_{0}(\mu)=I_{0}(1)\frac{1-\exp(-\tau_{0}/\mu)}{1-\exp(-\tau_{0})}$ (2)
and a fit of the observed $I_{0}(\mu)$ by this function, with $I_{0}(1)$ and
$\tau_{0}$ as parameters444Note that, contrary to Dumont et al. (1983), we
take $I_{0}(1)$ as an additional parameter. This is because by doing so, we
avoid the sensitivity of $I_{0}(1)$ to structures close to disk center, and
because the first data bin _starts_ at $1/\mu=1$ instead of being centered on
$1/\mu=1$, gives an estimate of $\tau_{0}$.
For the lines for which only the line radiance $E$ is known (S vi 94.4 nm and
Ly $\varepsilon$), we need to fit this function, with $\tau_{0}$ and $E(1)$ as
parameters555We take here $E(1)$ as a parameter for the same reason as we did
before for $I_{0}(1)$.:
$E(\mu)=E(1)\frac{\int_{\mathbb{R}}\left(1-\exp\left(-\frac{\tau_{0}}{\mu}\,e^{-u^{2}}\right)\right)\,\text{d}u}{\int_{\mathbb{R}}\left(1-\exp\left(-\tau_{0}\,e^{-u^{2}}\right)\right)\,\text{d}u}$
(3)
This expression comes from Dumont et al. (1983) and assumes a Doppler
absorption profile $\exp(-u^{2})$ with $u=\Delta\lambda/\Delta\lambda_{D}$.
Here, contrary to the case of the peak spectral radiance ratio, the function
and its derivative with respect to $\tau_{0}$ and $E(1)$ cannot be computed
analytically anymore, and we need to estimate them numerically; this is done
by a fast method, using a Taylor expansion of the outermost exponential of
both the numerator and denominator of Eq. (3).
These theoretical functions of $\mu$ are then plotted for different values of
the parameter $\tau_{0}$ over the observations in Fig. 3, for all three lines
(either for the peak spectral radiance or the line radiance, depending on the
data). We have performed a non-linear least-squares fit using the Levenberg-
Marquardt algorithm as implemented in the Interactive Data Language (IDL); it
gives the parameter $\tau_{0}$. The uncertainties on each point of the
$E(\mu)$ or $I(\mu)$ functions (an average on $N_{d}$ pixels) that we take as
input to the fitting procedure come mainly from the possible presence of
coherent structures such as bright points: the number of such possible
structures is of order $N_{d}/N_{s}$, where $N_{s}$ is the size of such a
structures (we take $N_{s}=100$ pixels), and then the uncertainty on $I$ or
$E$ is $\sigma/\sqrt{N_{d}/N_{s}}$ where $\sigma$ is the standard deviation of
the data points (in each pixel of a $1/\mu$ bin). Compared to this
uncertainty, the photon noise is negligible.
The results of the fits on the interval $1/\mu\in[1,5]$ are shown in Fig. 3:
as far as $\tau_{0}$ is concerned, they are $0.113$ for moment (1) (S vi 93.3
nm peak spectral radiance) and $0.244$ for moment (5) (S vi 94.4 nm radiance,
blended with Si viii). The approximations we used in writing Eq. (1) are not
valid for the optically thick Ly $\varepsilon$ line, hence the bad fit. On the
other hand, these approximations are valid for both the S vi lines, as long as
$1/\mu$ is small enough. For large $1/\mu$ there is an additional uncertainty
resulting from the determination of the limb.
These results are somewhat sensitive to the limb fitting: a $10^{-3}$ relative
error in the determination of the solar radius leads to a $7\;10^{-2}$
relative error on $\tau_{0}$. As $10^{-3}$ is a conservative upper limit of
the error on the radius from the limb fitting, we can consider that
$7\;10^{-2}$ is a conservative estimate of the relative error on $\tau_{0}$
resulting from the limb fitting.
?figurename? 3: Diamonds: average profiles of the radiance data (moments 1, 4
and 5) as a function of $1/\mu$, normalized to their values at disk center.
Dotted lines: theoretical profiles for different values of $\tau_{0}$. Plain
lines: fits of the theoretical profiles to the data, giving the values for
$\tau_{0}$: $0.113$ for (1) and $0.244$ for (5). The fit for Ly $\varepsilon$
is bad because this line is optically thick.
### 3.2 Using center-to-limb ratios of S vi 93.3 nm width and radiance
The variation with position of the S vi 93.3 nm line width (see Fig. 2) can be
interpreted as an opacity saturation of the S vi 93.3 nm line at the limb, and
then method B of Dumont et al. (1983) can be applied. This method relies on
the measurement of the ratio $d=\Delta\lambda^{*}_{l}/\Delta\lambda^{*}_{c}$
of the FWHM at the limb and at the disk center: the optical thickness at line
center $t_{0}$ at the limb is given by solving
$2\left(1-\exp\left(-t_{0}\,e^{-d^{2}\ln 2}\right)\right)=1-\exp(-t_{0})$ (4)
(this is Eq. 4 of Dumont et al. 1983 where a sign error has been corrected)
and then the opacity at line center $\tau_{0}$ is given by solving
$\frac{I_{0}(\mu=1)}{I_{0}(\mu=0)}=\frac{1-\exp(-\tau_{0})}{1-\exp(-t_{0})}$
(5)
Using the full-Sun S vi 93.3 nm compressed data set DS1666Although not obvious
from the data headers, moment (3) corresponds to the deconvoluted FWHM of S vi
93.3 nm, as is confirmed by a comparison with the width obtained from the full
profiles in data set DS2 and deconvoluted using the Solar Software procedure
con_width_4., we find that the ratio $d$ is $1.274$ and then $t_{0}$ is
$1.53$. Finally, we use the S vi 93.3 nm peak spectral radiance ratio
$I_{0}(\mu=1)/I_{0}(\mu=0)=0.062$ to get $\tau_{0}=0.05$.
### 3.3 Using the S vi 94.4 – 93.3 line ratio
The theoretical dependence of the S vi 94.4 – 93.3 peak radiance line ratio as
a function of the line opacities and source functions is:
$K=\frac{I_{0,933}}{2\,I_{0,944}}=\frac{S_{933}}{2\,S_{944}}\frac{1-\exp(-\tau_{0,933})}{1-\exp(-\tau_{0,944})}$
(6)
For this doublet, we assume $S_{933}=S_{944}$ and $\tau_{0,933}=2\tau_{0,944}$
(because the oscillator strengths are in the proportion $f_{933}=2f_{944}$).
Then $K$ reduces to
$K=\frac{1}{2}\frac{1-\exp(-\tau_{0,933})}{1-\exp(-\tau_{0,933}/2)}=\frac{1+\exp(-\tau_{0,933}/2)}{2}$
(7)
and we get $\tau_{0,933}$ from the observed value of $K$:
$\tau_{0,933}=-2\ln(2K-1)$ (8)
The difficulty comes from the S vi 94.4 nm blend with the Si viii line. In
order to remove this blend, we have analyzed the line profiles available in
data set DS2. After averaging the line profiles over the 60 central pixels
along the slit, we have fitted the S vi 93.3 nm line by a Gaussian with
uniform background and the S vi 94.4 nm line blend by two Gaussians with
uniform background. We have then computed the Gaussian amplitude from these
fits for both S vi lines, and this gives $I_{0,933}$ and $I_{0,944}$, and then
$K$, that we average over all observations. From this method we get
$\tau_{0,933}=0.089$.
The same kind of method could in theory be used for the S vi 94.4 – 93.3 line
radiance ratio
$K=\frac{E_{933}}{2\,E_{944}}=\frac{S_{933}}{2\,S_{944}}\frac{\int_{\mathbb{R}}\left(1-\exp\left(-\tau_{0,933}\,e^{-u^{2}}\right)\right)\,\mathrm{d}u}{\int_{\mathbb{R}}\left(1-\exp\left(-\tau_{0,944}\,e^{-u^{2}}\right)\right)\,\mathrm{d}u}$
(9)
with, again, $S_{933}=S_{944}$ and $\tau_{0,933}=2\tau_{0,944}$. As for method
A, the integral makes it necessary to invert this function of $\tau_{0,933}$
numerically, in order to recover $\tau_{0,933}$ for a given observed value of
$K$. As $K$ is decreasing as a function of $\tau_{0,933}$, this is possible by
a simple dichotomy. However, the average $K$ from the observations is greater
than $1$, which makes it impossible to invert the function and get a value for
$\tau_{0}$.
### 3.4 Discussion on opacity determination
It is clear that the three methods provide different values of the opacity at
disk center. We confirm the result of Dumont et al. (1983), obtained in
different lines, by which the method of center-to-limb ratios of width and
radiance (Sec. 3.2, or method B in Dumont et al. 1983) provides the smallest
value of the opacity. As mentioned by these authors, the center-to-limb
variations method (Sec. 3.1, or method A) overestimates the opacity for
different reasons described in Dumont et al. (1983), among which the curvature
of the layers close to the limb and their roughness. The method of line ratios
(Sec.3.3, or method C) also provides larger values of the opacity, although
free from geometrical assumptions; Dumont et al. (1983) interpret them as
resulting from a difference between the source functions of the lines of the
doublet.
This does not mean that there are no additional biases. For instance, we have
adopted a constant Doppler width from center to limb; actually this is not
correct since at the limb the observed layer is at higher altitude, where the
temperature and turbulence are higher than in the emitting layer as viewed at
disk center. Consequently, the excessive line width is wrongly interpreted as
only an opacity effect. However, it seems improbable that a $27.4\%$ increase
of Doppler width from center to limb can be entirely interpreted in terms of
temperature (because of the square-root temperature variation of Doppler
width) and turbulence (as the emitting layer is — a posteriori — optically not
very thick).
## 4 First estimates of densities
### 4.1 Densities using the opacities
The line-of-sight opacity at line center of the S vi 93.3 nm line is given by
$\tau_{0}=\int k_{\nu_{0}}\,n_{\textrm{{S {vi}}},i}(s)\,\textrm{d}s$ (10)
where the integration is along the line-of-sight. The variable $n_{\textrm{{S
{vi}}},i}$ is the numerical density of S vi in its level $i$, which can be
written as
$n_{\textrm{{S {vi}}},i}=\frac{n_{\textrm{{S {vi}}},i}}{n_{\textrm{{S
{vi}}}}}\frac{n_{\textrm{{S
{vi}}}}}{n_{\textrm{S}}}\mathop{\mathrm{Abund}}(\textrm{S})\frac{n_{\textrm{H}}}{n_{\textrm{e}}}n_{\textrm{e}}$
(11)
where $\mathop{\mathrm{Abund}}(\textrm{S})=n_{\textrm{S}}/n_{\textrm{H}}$ is
the Sulfur abundance in the corona ($10^{-4.73}$ according to the CHIANTI
database, Dere et al. 1997; Landi et al. 2006), $n_{\textrm{{S
{vi}}},i}/n_{\textrm{{S {vi}}}}$ is the proportion of S vi at level $i$,
$n_{\textrm{{S {vi}}}}/n_{\textrm{S}}$ is the ionization fraction (known as a
function of temperature) and $n_{\textrm{H}}/n_{\textrm{e}}=0.83$ is constant
in a fully ionized medium as the upper transition region. In this work $i$ is
the ground state $i=1$, and as in this region $n_{\textrm{{S
{vi}}},1}/n_{\textrm{{S {vi}}}}$ is very close to $1$, we will drop this term
from now. The variable $k_{\nu_{0}}$ is the absorption coefficient at line
center frequency $\nu_{0}$ for each S vi ion, given by:
$k_{\nu_{0}}=\frac{h\nu_{0}}{4\pi}B_{ij}\frac{1}{\sqrt{\pi}\,\Delta\nu_{D}}$
(12)
where $B_{ij}$ is the Einstein absorption coefficient for the transition
$i\rightarrow j$ (i.e.,
$2\mathrm{p}^{6}\;3\mathrm{s}\;{}^{2}\mathrm{S}_{1/2}\rightarrow
2\mathrm{p}^{6}\;3\mathrm{p}\;{}^{2}\mathrm{P}_{3/2}$) at
$\lambda_{0}=93.3\>\mathrm{nm}$ and integration over a Gaussian Doppler shift
distribution has been done ($\Delta\nu_{D}$ is the Doppler width in
frequency). Using:
$B_{ij}=\frac{g_{j}}{g_{i}}B_{ji}=\frac{g_{j}}{g_{i}}\frac{A_{ji}}{2h\nu_{0}^{3}/c^{2}}$
(13)
with $g_{j}/g_{i}=2$ and $\lambda_{0}=c/\nu_{0}$, this gives:
$k_{\nu_{0}}=\frac{\lambda_{0}^{4}A_{ji}}{4\pi^{3/2}c\,\Delta\lambda_{D}}$
(14)
Finally, for an emitting layer of thickness $\Delta s$ and average electron
density $\langle n_{\textrm{e}}\rangle$, we have:
$\tau_{0}=\frac{\lambda_{0}^{4}A_{ji}}{4\pi^{3/2}c\,\Delta\lambda_{D}}\frac{n_{\textrm{{S
{vi}}}}}{n_{\textrm{S}}}\mathop{\mathrm{Abund}}(\textrm{S})\frac{n_{\textrm{H}}}{n_{\textrm{e}}}\langle
n_{\textrm{e}}\rangle\,\Delta s$ (15)
Taking $\tau_{0}=0.05$, we get $\langle n_{\textrm{e}}\rangle\,\Delta
s=4.9\cdot 10^{21}\>\mathrm{m^{-2}}$. Then, with $\Delta s=206\>\mathrm{km}$
(the altitude interval corresponding to the FWHM of the S vi 93.3 nm
contribution function $G(T)$ as computed by CHIANTI), this gives $\langle
n_{\textrm{e}}\rangle=2.4\cdot 10^{16}\>\mathrm{m^{-3}}$.
### 4.2 Squared densities using the contribution function
The average S vi 93.3 nm line radiance at disk center obtained from data set
DS2 (excluding the 5% higher values which are considered not to be part of the
quiet Sun) is $E=1.4\cdot 10^{-2}\>\mathrm{W\,m^{-2}sr^{-1}}$ (to be compared
to the value $3.81\cdot 10^{-3}$ given by CHIANTI with a Quiet Sun DEM — see
Table 1). This can be used to estimate $\langle
n_{\textrm{e}}^{2}\rangle\,\Delta s$ in the emitting region of thickness
$\Delta s$, as
$E=\int G(T(s))\,n_{\textrm{e}}^{2}(s)\;\,\mathrm{d}s\approx G(\langle
T\rangle)\,\langle n_{\textrm{e}}^{2}\rangle\,\Delta s$ (16)
where $G(T)$ is the contribution function and the integral is on the line-of-
sight and where we have made the assumption that $\tau_{0}\ll 1$. We take the
average temperature in the emitting region to be $\langle T\rangle=T_{\rm
max}=10^{5.3}\>\mathrm{K}$, and, for densities of the order of
$10^{16}\>\mathrm{m^{-3}}$, the `gofnt` function of CHIANTI gives $G(\langle
T\rangle)=1.8\cdot 10^{-37}\>\mathrm{W\,m^{3}sr^{-1}}$. We finally get
$\langle n_{\textrm{e}}^{2}\rangle\,\Delta s=8.4\cdot
10^{35}\>\mathrm{m^{-5}}$ (17)
With again $\Delta s=206\>\mathrm{km}$, we get $\langle
n_{\textrm{e}}\rangle_{\text{RMS}}=2.0\cdot 10^{15}\>\mathrm{m^{-3}}$.
Assuming an uncertainty of $20\%$ on $E$, the uncertainty on $\langle
n_{\textrm{e}}\rangle_{\text{RMS}}$ would be $10\%$ for a given $\Delta s$.
## 5 Discussion of biases in the method
One of our aims when starting this work was to determine a filling factor777We
explain this definition of the filling factor in Appendix A.
$f=\frac{\langle n_{\textrm{e}}\rangle^{2}}{\langle
n_{\textrm{e}}^{2}\rangle}$ (18)
in the S vi-emitting region. This initial objective needs to be revised, since
we get $f=144$, an impossible value as it is more than $1$. Our values of
densities can be compared to the density at $\log T=5.3$ in the Avrett &
Loeser (2008) model ($1.7\cdot 10^{15}\>\mathrm{m^{-3}}$): our value of
$\langle n_{\textrm{e}}\rangle$ is an order of magnitude higher, while
$\langle n_{\textrm{e}}\rangle_{\text{RMS}}=\sqrt{\langle
n_{\textrm{e}}^{2}\rangle}$ is about the same (while it should be higher than
$\langle n_{\textrm{e}}\rangle$). Our value of intensity is compatible with
average values from other sources, such as Del Zanna et al. (2001) (see their
Fig. 1).
Given the same measurements of $\tau_{0}$ and $E$, one can instead start from
the assumption of a filling factor $f\in[0,1]$ and deduce $\Delta s$:
$\Delta s=\frac{1}{f}\frac{(\langle n_{\textrm{e}}\rangle\,\Delta
s)^{2}}{\langle n_{\textrm{e}}^{2}\rangle\,\Delta s}$ (19)
where the numerator and denominator of the second fraction are deduced from
Eq. (15) and (16) respectively. With the values from Sec. 4, this gives
$\Delta s>29\>\mathrm{Mm}=0.04R_{\astrosun}$, a value much larger than
expected.
In any case, there seems to be some inconsistencies around $\log T=5.3$
between our new observations of opacities on one hand, and transition region
models and observations of intensities on the other hand. We propose now to
discuss the possible sources of these discrepancies, while releasing, when
needed, some of the simplistic assumptions we have made until now.
### 5.1 Assumption of a uniform emitting layer
#### 5.1.1 Bias due to this assumption
When computing the average densities from the S vi 93.3 nm opacity and
radiance, we have assumed a uniform emitting layer at the temperature of
maximum emission and of thickness $\Delta s$ given by the width of
contribution function $G(T)$. However, the different dependences in the
electron density of Eqs. (10) and (16) — the first is linear while the second
is quadratic — means that the slope of the $n_{\textrm{e}}(s)$ function
affects differently the weights on the integrals of Eqs. (10) and (16): a
bias, different for $\tau_{0}$ and $E$, can be expected, and here we explore
this effect starting from the Avrett & Loeser (2008) model, which has the
merit of giving average profiles of temperature and density (among other
variables) as a function of altitude $s$.
##### Opacity.
Using the Avrett & Loeser (2008) profiles and atomic physics data, we get
$\tau_{0}=0.008$. Then, using the same simplistic method as for observations
(still with a uniform layer of thickness $\Delta s=206\>\mathrm{km}$), we
obtain $\langle n_{\textrm{e}}\rangle=2.4\cdot 10^{15}\>\mathrm{m^{-3}}$, a
value only 40% higher than the density at $\log T=5.3$ in this model
($1.7\cdot 10^{15}\>\mathrm{m^{-3}}$).
##### Radiance.
Using the same Avrett & Loeser (2008) profiles and the CHIANTI contribution
function $G(T)$, we get $E=1.3\cdot 10^{-2}\>\mathrm{W\,m^{-2}sr^{-1}}$.
Then, using the same simplistic method as for observations, we obtain $\langle
n_{\textrm{e}}\rangle_{\text{RMS}}=1.9\cdot 10^{15}\>\mathrm{m^{-3}}$, a value
12% higher than the density at $\log T=5.3$ in this model.
We see then that the assumption of a uniform emitting layer has a bias towards
high densities, which is stronger for the opacity method than for the radiance
method. A filling factor computed from these values would be $f=1.5$, while it
has been assumed to be $1$ when computing $\tau_{0}$ and $E$ from the Avrett &
Loeser (2008) model: this can be one of the reasons contributing to our too
high observed filling factor.
This differential bias acts in a surprising way as, due to the
$n_{\textrm{e}}^{2}$ term in Eq. (16) one would rather expect the bias to be
stronger for $E$ than for $\tau_{0}$; however, it can be understood by
comparing the effective temperatures for $\tau_{0}$ and $E$, which are
respectively:
$\displaystyle T_{\text{eff},\tau_{0}}=\frac{\int
T(s)\,n_{\textrm{e}}(s)\,K(T(s))\;\,\mathrm{d}s}{\int
n_{\textrm{e}}(s)\,K(T(s))\;\,\mathrm{d}s}=10^{5.38}\>\mathrm{K}$ (20)
$\displaystyle T_{\text{eff},E}=\frac{\int
T(s)\,n_{\textrm{e}}^{2}(s)\,G(T(s))\;\,\mathrm{d}s}{\int
n_{\textrm{e}}^{2}(s)\,G(T(s))\;\,\mathrm{d}s}=10^{5.40}\>\mathrm{K}$ (21)
where $K(T)=k_{\nu_{0}}(T)\,n_{\textrm{{S {vi}}}}/n_{\textrm{e}}$, while
$T(s)$ and $n_{\textrm{e}}(s)$ are from Avrett & Loeser (2008). The higher
effective temperature for $E$ than for $\tau_{0}$ means that the bias is more
affected by the respective shapes of the high-temperature wings of $G(T)$ and
$K(T)$ than by the exponent of $n_{\textrm{e}}$ in the integrals of Eqns. (12)
and (16).
It can be pointed out here that the difference between the $K(T)$ and $G(T)$
kernels lies in the fact that $G(T)$ (unlike $K(T)$) not only takes into
account the ionization equilibrium of S vi, but also the collisions from $i$
to $j$ levels of S vi ions.
#### 5.1.2 Releasing this assumption: a tentative estimate of the density
gradient around $\log T=5.3$
In Sec. 5.1 we have incidentally shown that the radiance computed with the
Avrett & Loeser (2008) profiles and the CHIANTI contribution function $G(T)$
is a factor $3$ higher than the radiance computed directly by CHIANTI using
the standard Quiet Sun DEM (see Table 1). This is simply because the DEM
computed from the temperature and density profiles of the Avrett & Loeser
(2008) model is different888The reason for this is that the Avrett & Loeser
(2008) model is determined from theoretical energy balance and needs further
improvements in order to reproduce the observed DEM (E. Avrett, private
communication). than the CHIANTI DEM, as can be seen in Fig. 4. In particular,
the Avrett & Loeser (2008) DEM is missing the dip around $\log T=5.5$ that is
obtained from most observations; at $\log T=5.3$ it is a factor $3$ higher
than the CHIANTI Quiet Sun DEM.
We model the upper transition region locally around $\log T_{0}=5.3$ and
$s_{0}=2.346\>\mathrm{Mm}$ (chosen because $T(s_{0})=T_{0}$ in the Avrett &
Loeser 2008 model) by a vertically stratified plasma at pressure
$P_{0}=1.91n_{0}k_{B}T_{0}$ (we consider a fully ionized coronal plasma) and:
$\frac{T(s)}{T_{0}}=\frac{n_{0}}{n_{\textrm{e}}(s)}=\sqrt{\frac{s-s_{T}}{s_{0}-s_{T}}}\quad\text{for}\quad
s>s_{T}$ (22)
These equations were chosen to provide a good approximation of a transition
region, with some symmetry between the opposite curvatures of the variations
of $T$ and $n_{\textrm{e}}$ with altitude. The parameters of this model
atmosphere are the pressure $P_{0}$ and $s_{T}$ (with $s_{T}<s_{0}$), which
can be interpreted as the altitude of the base of the transition region. Given
the constraint $T(s_{0})=T_{0}$ that we imposed when building the model, with
$T_{0}$ and $s_{0}$ fixed, $s_{T}$ actually controls the derivative of $T(s)$
at $s=s_{0}$:
$T^{\prime}(s_{0})=\frac{T_{0}}{2(s_{0}-s_{T})}\quad\text{or}\quad
s_{T}=s_{0}-\frac{T_{0}}{2T^{\prime}(s_{0})}$ (23)
We plot in Fig. 5 some temperature profiles from this simple transition region
model, for different model parameters $T^{\prime}(s_{0})$ ($P_{0}$ only
affects the scale of $n_{\textrm{e}}(s)$). For the Avrett & Loeser (2008)
model, $P_{0}=8.7\cdot 10^{-3}\>\mathrm{Pa}$ and
$T^{\prime}(s_{0})=0.45\>\mathrm{K\,m^{-1}}$, and the corresponding model
profile is also shown.
We propose to use such models along with atomic physics data and the equations
of Sec. 4 to compute $\tau_{0}$ and $E$ as a function of model parameters
$P_{0}$ and $T^{\prime}(s_{0})$, as shown in Fig. 6. As the slopes of the
level lines are different in the $\tau_{0}(P_{0},T^{\prime}(s_{0}))$ and
$E(P_{0},T^{\prime}(s_{0}))$ plots, one would in theory be able to estimate
the parameters $(P_{0},T^{\prime}(s_{0}))$ of the best model for the
observation of $(\tau_{0,\text{obs}},E_{\text{obs}})$ by simply finding the
crossing between the level lines
$\tau_{0}(P_{0},T^{\prime}(s_{0}))=\tau_{0,\text{obs}}$ and
$E(P_{0},T^{\prime}(s_{0}))=E_{\text{obs}}$.
In practice however, the level lines for our observations of $\tau_{0}$ and
$E$ do not intersect in the range of parameters plotted in Fig. 6,
corresponding to realistic values of the parameters. As a consequence, it is
not possible to tell from these measurements (from a single spectral line,
here S vi 93.3 nm), what is the temperature slope and the density of the TR
around the formation of this line.
If we now extend the range of $T^{\prime}(s_{0})$ to unrealistically low
values, a crossing of the level lines can be found below $\log P_{0}=-3.5$ and
$T^{\prime}(s_{0})=5\>\mathrm{mK/m}$. Given the width of $G(T)$ for S vi 93.3
nm, this corresponds to $\Delta s>20\>\mathrm{Mm}$, a value consistent with
the one obtained from Eq. (19) and which is also much larger than expected.
Let us note that Keenan (1988) derived a much lower S vi 93.3 nm opacity value
($\tau_{0}=1.1\,10^{-4}$ at disk center) from a computation implying the cells
of the network model of Gabriel (1976). However, while our value of $\tau_{0}$
seems to be too high, the level lines in Fig. 6 show that an opacity value
$\tau_{0}=1.1\,10^{-4}$ would be too low: from this figure we expect that a
value compatible with radiance measurements and with realistic values of the
temperature gradient would be in the range $5\,10^{-3}$ to $10^{-2}$.
?figurename? 4: Quiet Sun standard DEM from CHIANTI (plain line) and DEM
computed from the Avrett & Loeser (2008) temperature and density profiles. The
dotted lines give the DEMs for $\log T=5.3$, the maximum emission temperature
of the ${S\textsc{vi}}$ lines. ?figurename? 5: Temperature as a function of
altitude in our local transition region simple models around $T_{0}=10^{5.3}$
and $s_{0}=2.346\>\mathrm{Mm}$. The temperature profile from Avrett & Loeser
(2008) is shown with the diamonds signs, and the simple model with the same
temperature slope is shown with a dashed line.
?figurename? 6: S vi 93.3 nm opacity $\tau_{0}$ (top panel) and line radiance
$E$ (middle panel) as a function of model parameters $P_{0}$ and
$T^{\prime}(s_{0})$. The level lines close to our actual observations are
shown as plain lines for $\tau_{0}$ and as dashed lines for $E$. The bottom
panel reproduces these level lines together in the same plot. The parameters
$(P_{0},T^{\prime}(s_{0}))$ estimated from the Avrett & Loeser (2008) model at
$T=T_{0}$ are shown with the diamond sign on each plot.
### 5.2 Anomalous behavior of Na-like ions
Following works such as Dupree (1972) for Li-like ions, Judge et al. (1995)
report that standard DEM analysis fails for ions of the Li and Na
isoelectronic sequences; in particular, for S vi (which is Na-like), Del Zanna
et al. (2001) find that the atomic physics models underestimate the S vi 93.3
nm line radiance $E$ by a factor $3$. This fully explains the difference
between our observation of $E$ and the value computed by CHIANTI (Table 1).
However, this means also that where $G(T)$ from CHIANTI is used, as in Eq.
(16), it presumably needs to be multiplied by $3$. As a result, one can expect
$\langle n_{\textrm{e}}\rangle_{\text{RMS}}$ to be lower by a factor $1.7$,
resulting into a filling factor of 415 (actually worse than our initial
result).
The reasons for the anomalous behavior of these ions for $G(T)$, which could
be linked to the ionization equilibrium or to collisions, are still unknown.
As a result, it is impossible to tell whether these reasons also produce an
anomalous behavior of these ions for $K(T)$, hence on our measurements of
opacities and on our estimations of densities: this could again reduce the
filling factor.
### 5.3 Cell-and-network pattern
When analyzing our observations, we have not made the distinction between the
network lanes and the cells of the chromospheric supergranulation. Here we try
to evaluate the effect of the supergranular pattern on our measurements, by
using a 2D model emitting layer with a simple “paddle wheel” cell-and-network
pattern: in polar coordinates $(r,\theta)$, the emitting layer is defined by
$R_{1}<r<R_{2}$; in the emitting layer, the network lanes are defined by
$\theta\in[0,\delta\theta]\mod\Delta\theta$ and the cells are the other parts
of the emitting layer, with $\Delta\theta$ the pattern angular cell size (an
integer fraction of $2\pi$) and $\delta\theta$ the network lane angular size.
The network lanes and cells are characterized by different (but uniform)
source functions $S$, densities $n_{\textrm{{S {vi}}}}$ and absorption
coefficients $k_{\nu_{0}}$. We then solve the radiative transfer equations for
$\lambda_{0}$ along rays coming from infinity through the emitting layer to
the observer.
As the opacity is obtained by a simple integration of
$k_{\nu_{0}}n_{\textrm{{S {vi}}},i}$, the average line-of-sight opacity
$t_{0}$ as a function of $\mu$ for the “paddle-wheel” pattern is the same as
for a uniform layer with the same average $k_{\nu_{0}}n_{\textrm{{S
{vi}}},i}$. However, as seen in Fig. 7, still for the same average $S$ and
$k_{\nu_{0}}n_{\textrm{{S {vi}}},i}$, the effect of opacity (a decrease in
intensity) is higher in the “paddle-wheel” case, in particular for
intermediate values of $1/\mu$. As a result, neglecting the cell-and-network
pattern of the real TR leads to overestimating the opacity when using method
A.
?figurename? 7: Average spectral radiance at line center $I_{0}$ as a function
of $1/\mu$ for a uniform layer (dashed line) and for a model layer with a
simple cell-and-network pattern (plain line). Both models have the same
average opacity and source function. The factor-$2$ jump at $1/\mu=11.3$
corresponds to the limb of the opaque solar disk; the reference radius used to
compute $\mu$ corresponds to the middle of the emitting layer. The
oscillations are artefacts of the averaging process.
### 5.4 Roughness and fine structure
In order to explain the high values of opacity (as derived from their method
A), Dumont et al. (1983) introduce the concept of roughness of the TR: as the
TR plasma is not perfectly vertically stratified (there is some horizontal
variation), method A leads to an overestimated value of $\tau_{0}$. This could
reconcile the values obtained following our application of methods A and B.
We model the roughness of the transition region by incompressible vertical
displacements of any given layer (at given optical depth) from its average
vertical position, in the geometry shown in Fig. 8. The layer then forms an
angle $\alpha$ with the horizontal and has still the same vertical thickness
$\,\mathrm{d}s$; the thickness along the LOS is
$\,\mathrm{d}s\cos\alpha/\cos(\theta+\alpha)$, as can be deduced from Fig. 8.
If we assume that $\theta+\alpha$ remains sufficiently small for the plane-
parallel approximation to hold (and so that the LOS crosses one given layer
only once), the opacity is
$\displaystyle t_{0}=\int
n_{\textrm{e}}(s)K(T(s))\frac{\cos\alpha\,\mathrm{d}s}{\cos(\theta+\alpha)}$
(24)
The angle $\alpha$ is a random variable, with some given distribution
${\Pr}(\alpha)$. We compute the average of $t_{0}$ as a function of $\theta$
and of $\Pr(\alpha)$:
$\displaystyle\left\langle t_{0}(\theta,\Pr(\alpha))\right\rangle_{\alpha}$
$\displaystyle={\displaystyle\iint
n_{\textrm{e}}(s)K(T(s))\frac{\cos\alpha\,\mathrm{d}s}{\cos(\theta+\alpha)}\Pr(\alpha)\,\mathrm{d}\alpha}$
(25)
$\displaystyle={\displaystyle\frac{\tau_{0}}{\mu}\left\langle\frac{\cos\theta\cos\alpha}{\cos(\theta+\alpha)}\right\rangle}_{\alpha}\equiv{\displaystyle\frac{\tau_{0}}{\mu}\beta(\theta,\Pr(\alpha))}$
(26)
The opacity $t_{0}=\tau_{0}/\mu$ is corrected by the factor
$\beta(\theta,\Pr(\alpha))$ defined in the previous equation. We recover
$\beta=1$ for $\Pr(\alpha)=\delta(\alpha)$, i.e., when there is no roughness.
We immediately see that $\beta=1$ for $\theta=0$, for any $\Pr(\alpha)$:
roughness (as modelled here by incompressible vertical displacements) does not
change the optical thickness at disk center. Nevertheless, the estimate of
optical thickness at disk center from observations in Sec. 3.1 (method A of
Dumont et al. 1983) is affected by this roughness effect.
Coming back to $\langle t_{0}\rangle$, we take
$\Pr(\alpha)=\cos^{2}(\pi\alpha/2A)/A$, and we compute $\beta$ numerically
($A$ represents the width of $\Pr(\alpha)$ and can be thought as a
quantitative measurement of the roughness). The results, shown in Fig. 9,
indicate for example that the modelled roughness with $A=\pi/5$ increases the
opacity by $9\%$ at $1/\mu=1.5$ (corresponding to $\theta=45$°). This is a
significant effect, and we can evaluate its influence on the estimate of
$\tau_{0}$ in Sec. 3.1: in the theoretical profiles of $I_{0}(\mu)$ and
$E(\mu)$ (Eq. 2–3), $\tau_{0}/\mu$ needs to be replaced by
$\tau_{0}/\mu\times\beta$. As $\beta>1$ for a rough corona, this means that
the value of $\tau_{0}$ determined from the fit of observed radiances to Eq.
(2)–(3) is overestimated by a factor corresponding approximately to the mean
value of $\beta$ on the fitting range.
In this way, we have given a quantitative value for the overestimation factor
of $\tau_{0}$ by the method of Sec. 3.1, thus extending the qualitative
discussion on roughness found in Dumont et al. (1983). This factor, of the
order of $1.1$ may seem modest, but one needs to remember that the fit for
obtaining $\tau_{0}$ in Sec. 3.1 was done on a wide range ($1/\mu$ from $1$ to
$5$, or $\theta$ from $0$ to $78$ degrees) that our roughness model cannot
reproduce entirely999For high values of the $\Pr(\alpha)$ width $A$, the
correction factor $\beta$ cannot be computed for high values of $1/\mu$ (high
angles $\theta$) because the values of $\alpha$ in the wings of $\Pr(\alpha)$
fall in the range where $|\theta+\alpha|\nll\pi/2$: the plane-parallel
approximation is not valid anymore. This explains the limited range of the
$\beta(1/\mu)$ curves in Fig. 9..
One can think of different roughness models representing the strong
inhomogeneity of the TR, for instance with a different and very peculiar
roughness model Pecker et al. (1988) obtain an overestimation factor of more
than 10 under some conditions. This means that our values of $\tau_{0}$ may
need to be decreased by a large factor due to a roughness effect.
Roughness models can be seen as simplified models of the fine structure of the
TR, which is known to be heterogeneous at small scales. Indeed, in addition to
the chromospheric network pattern that we have already modelled in Sec. 5.3,
the TR contains parts of different structures, with different plasma
properties, like the base of large loops and coronal funnels, smaller loops
(Dowdy et al. 1986; Peter 2001), and spicules. Furthermore, the loops
themselves are likely to be composed of strands, which can be heated
independently (Cargill & Klimchuk 2004; Parenti et al. 2006). The magnetic
field in these structures inhibits perpendicular transport, and as a
consequence the horizontal inhomogeneities are not smoothed out efficiently.
?figurename? 8: Geometry of a TR layer (plain contour), displaced from its
average position (dashed contour) while retaining its original vertical
thickness $\,\mathrm{d}s$, and locally forming an angle $\alpha$ with the
average layer. The line-of-sight (LOS) forms an angle $\theta$ to the vertical
(normal to the average layer). ?figurename? 9: Multiplicative coefficient to
$t_{0}$ due to roughness, for different roughness parameters $A$.
## 6 Conclusion
We have first derived the average electron density in the TR from the opacity
$\tau_{0}$ of the S vi 93.3 nm line, obtained by three different methods from
observations of the full Sun: center-to-limb variation of radiance, center-to-
limb ratios of radiance and line width, and radiance ratio of the
$93.3$–$94.4\>\mathrm{nm}$ doublet. Assuming a spherically symmetric plane-
parallel layer of constant source function, we find a S vi 93.3 nm opacity of
the order of $0.05$. The derived average electron density is of the order of
$2.4\cdot 10^{16}\>\mathrm{m^{-3}}$.
We have then used the line radiance (by an EM method) in order to get the RMS
average electron density in the S vi 93.3 nm-emitting region: we obtain
$2.0\cdot 10^{15}\>\mathrm{m^{-3}}$. This corresponds to a total pressure of
$10^{-2}\>\mathrm{Pa}$, slightly higher than the range of pressures found by
Dumont et al. (1983) ($1.3$ to $6.3\cdot 10^{-3}\>\mathrm{Pa}$, as deduced
from their Sec. 4.2), but lower than the value given in Mariska (1993)
($2\cdot 10^{-2}\>\mathrm{Pa}$).
The average electron densities obtained from these methods (opacity on one
hand, radiance on the other hand) are incompatible, as can be seen either from
a direct comparison of the values of $\langle n_{\textrm{e}}\rangle$ and
$\langle n_{\textrm{e}}^{2}\rangle$ for a given thickness $\Delta s$ of a
uniform emitting layer, or by computing the $\Delta s$ that would reconcile
the measurements of $\langle n_{\textrm{e}}\rangle\,\Delta s$ and $\langle
n_{\textrm{e}}^{2}\rangle\,\Delta s$. Furthermore, we have seen that the
density obtained from the opacity method is also incompatible with standard
DEMs of the Quiet Sun (see Sec. 4.2) and with semi-empirical models of the
temperature and density profiles in the TR (see Sec. 5.1.2).
We investigated several possible sources of biases in the determination of
$\tau_{0}$: the approximation of a constant temperature in the S vi emitting
layer, the anomalous behavior of the S vi ion, the chromospheric network
pattern, and the roughness of the TR. Some of these could help explain partly
the discrepancy between the average densities deduced from opacities and from
radiances, but there is still a long way to go to fully understand this
discrepancy and to reconcile the measurements. At this stage, we can only
encourage colleagues to look for similar discrepancies in lines formed around
$\log T=5.3$ (like C iv and O vi), Na-like and not Na-like, and to repeat
similar S vi center-to-limb measurements.
In Sec. 5.1.2 we have tried to combine opacity and radiance information to
compute the gradient of temperature. This appeared to be impossible (if
restricting ourselves to a realistic range of parameters) because of the
above-mentioned incompatibility. We have estimated that a value $\tau_{0}$ of
the S vi 93.3 nm opacity compatible with radiance measurements and with
realistic values of the temperature gradient would be in the range
$5\,10^{-3}$ to $10^{-2}$.
In spite of the difficulties we met, we still think that the combination of
opacity and radiance information should be a powerful tool for investigating
the thermodynamic properties and the fine structure of the TR. For instance
the excess opacity derived from observations and a plane-parallel model could
be used to evaluate models of roughness and fine structure of the TR. Clearly,
progress in modelling the radiative output of the complex structure of the TR
needs to be done in order to achieve this.
###### Acknowledgements.
The authors thank G. del Zanna, E. H. Avrett and Ph. Lemaire for interesting
discussions and the anonymous referee for suggestions concerning this paper.
The SUMER project is supported by DLR, CNES, NASA and the ESA PRODEX Programme
(Swiss contribution). SoHO is a project of international cooperation between
ESA and NASA. Data was provided by the MEDOC data center at IAS, Orsay. EB
thanks CNES for financial support, and the ISSI group on Coronal Heating (S.
Parenti). CHIANTI is a collaborative project involving the NRL (USA), RAL
(UK), MSSL (UK), the Universities of Florence (Italy) and Cambridge (UK), and
George Mason University (USA).
## ?appendixname? A About the filling factor
In this paper we have defined the filling factor as
$f=\frac{\langle n_{\textrm{e}}\rangle^{2}}{\langle
n_{\textrm{e}}^{2}\rangle}$ (27)
while it is usually obtained, from solar observations (e.g. Judge 2000;
Klimchuk & Cargill 2001), from
$f=\frac{EM}{\Delta s\,n_{0}^{2}}$ (28)
where $EM$ is the emission measure, $\Delta s$ is the thickness of the plasma
layer and $n_{0}$ is the electron density (usually determined from line
ratios) _in the non-void parts of the plasma layer_.
It may seem surprising that the $EM$ is at the numerator of this second
expression, while it provides an estimate for $\langle
n_{\textrm{e}}^{2}\rangle$ which appears at the denominator of the first
expression. However, we can show that these both expressions, despite looking
very different, give actually the same result for a given plasma.
We take a plasma with a differential distribution $\xi(n_{\textrm{e}},T)$ for
the density and temperature:
$\xi(n_{\textrm{e}},T)\;\,\mathrm{d}n_{\textrm{e}}\;\,\mathrm{d}T$ is the
proportion of any given volume occupied by plasma at a density between
$n_{\textrm{e}}$ and $n_{\textrm{e}}+\,\mathrm{d}n_{\textrm{e}}$ and a
temperature between $T$ and $T+\,\mathrm{d}T$.
The contributions to the line radiance $E$ and to the opacity at line center
$\tau_{0}$ from a volume $V$ with this plasma distribution are
$\displaystyle\frac{E}{V}=\iint
n_{\textrm{e}}^{2}G(n_{\textrm{e}},T)\,\xi(n_{\textrm{e}},T)\;\,\mathrm{d}n_{\textrm{e}}\;\,\mathrm{d}T$
(29) $\displaystyle\frac{\tau_{0}}{V}=\iint
n_{\textrm{e}}K(n_{\textrm{e}},T)\,\xi(n_{\textrm{e}},T)\;\,\mathrm{d}n_{\textrm{e}}\;\,\mathrm{d}T$
(30)
with the notations of our article.
The usual assumption (e.g. Judge 2000) is that $G(T,n_{\textrm{e}})$ “selects”
a narrow range of temperatures around $T=T_{\textrm{max}}$ and does not depend
on $n_{\textrm{e}}$, i.e.,
$G(n_{\textrm{e}},T)\approx\tilde{G}(T_{\textrm{max}})\,\delta(T-T_{\textrm{max}})$.
Similarly, we can consider that
$K(n_{\textrm{e}},T)\approx\tilde{K}(T_{\textrm{max}})\,\delta(T-T_{\textrm{max}})$.
Then
$\displaystyle\frac{E}{V}$
$\displaystyle\approx\tilde{G}(T_{\textrm{max}})\int
n_{\textrm{e}}^{2}\,\xi(n_{\textrm{e}},T_{\textrm{max}})\;\,\mathrm{d}n_{\textrm{e}}=\tilde{G}(T_{\textrm{max}})\,\langle
n_{\textrm{e}}^{2}\rangle_{T=T_{\textrm{max}}}$ (31)
$\displaystyle\frac{\tau_{0}}{V}$
$\displaystyle\approx\tilde{K}(T_{\textrm{max}})\int
n_{\textrm{e}}\,\xi(n_{\textrm{e}},T_{\textrm{max}})\;\,\mathrm{d}n_{\textrm{e}}=\tilde{K}(T_{\textrm{max}})\,\langle
n_{\textrm{e}}\rangle_{T=T_{\textrm{max}}}$ (32)
The line ratio $R_{ij}=E_{i}/E_{j}$ is, following Judge (2000) and with the
assumption
$G(n_{\textrm{e}},T)=\hat{G}(n_{\textrm{e}},T_{\textrm{max}})\,\delta(T-T_{\textrm{max}})$:
$\displaystyle R_{ij}=\frac{E_{i}}{E_{j}}$ $\displaystyle=\frac{\iint
n_{\textrm{e}}^{2}G_{i}(n_{\textrm{e}},T)\,\xi(n_{\textrm{e}},T)\;\,\mathrm{d}n_{\textrm{e}}\;\,\mathrm{d}T}{\iint
n_{\textrm{e}}^{2}G_{j}(n_{\textrm{e}},T)\,\xi(n_{\textrm{e}},T)\;\,\mathrm{d}n_{\textrm{e}}\;\,\mathrm{d}T}$
(33) $\displaystyle\approx\frac{\int
n_{\textrm{e}}^{2}\hat{G}_{i}(n_{\textrm{e}},T_{\textrm{max}})\,\xi(n_{\textrm{e}},T_{\textrm{max}})\;\,\mathrm{d}n_{\textrm{e}}}{\int
n_{\textrm{e}}^{2}\hat{G}_{j}(n_{\textrm{e}},T_{\textrm{max}})\,\xi(n_{\textrm{e}},T_{\textrm{max}})\;\,\mathrm{d}n_{\textrm{e}}}$
(34)
When homogeneity is assumed, i.e.,
$\xi(n_{\textrm{e}},T)=\delta(n_{\textrm{e}}-n_{0})\,\tilde{\xi}(T)$, this
becomes
$R_{ij}\approx\frac{n_{0}^{2}G_{i}(n_{0},T_{\textrm{max}})\,\tilde{\xi}(T_{\textrm{max}})}{n_{0}^{2}G_{j}(n_{0},T_{\textrm{max}})\,\tilde{\xi}(T_{\textrm{max}})}=\frac{G_{i}(n_{0},T_{\textrm{max}})}{G_{j}(n_{0},T_{\textrm{max}})}\equiv
g_{ij}(n_{0})$ (35)
and inverting this function allows to recover $n_{0}$ from the observed value
of $R_{ij}$.
The fundamental point is that $R_{ij}$ does not depend on the proportion $f$
(the filling factor) of the volume actually occupied by the plasma: $n_{0}$ is
the density in the non-void region only. For example, for
$\xi_{f}(n_{\textrm{e}},T)$ defined by
$f\delta(n_{\textrm{e}}-n_{0})+(1-f)\delta(n_{\textrm{e}})$, the line ratio
$R_{ij}$ is $g_{ij}(n_{0})$ which is independent on $f$, while $\langle
n_{\textrm{e}}^{2}\rangle_{T=T_{\textrm{max}}}$ determined from $E/V$ would be
$fn_{0}^{2}$ and $\langle n_{\textrm{e}}\rangle_{T=T_{\textrm{max}}}$
determined from $\tau_{0}/V$ would be $fn_{0}$. One can see in this case that
$f$ can (equivalently) either be recovered from
$\frac{\langle
n_{\textrm{e}}^{2}\rangle_{T=T_{\textrm{max}}}}{(n_{0})^{2}}=\frac{(fn_{0}^{2})}{(n_{0})^{2}}=f$
(36)
(corresponding to Judge 2000) or from
$\frac{\langle n_{\textrm{e}}\rangle_{T=T_{\textrm{max}}}^{2}}{\langle
n_{\textrm{e}}^{2}\rangle_{T=T_{\textrm{max}}}}=\frac{(fn_{0})^{2}}{(fn_{0}^{2})}=f$
(37)
(corresponding to our method).
## ?refname?
* Avrett & Loeser (2008) Avrett, E. H. & Loeser, R. 2008, Astrophys. J. Suppl. Ser., 175, 229
* Buchlin et al. (2006) Buchlin, E., Vial, J.-C., & Lemaire, P. 2006, Astron. Astrophys., 451, 1091
* Cargill & Klimchuk (2004) Cargill, P. J. & Klimchuk, J. A. 2004, Astrophys. J., 605, 911
* Curdt et al. (2001) Curdt, W., Brekke, P., Feldman, U., et al. 2001, Astron. Astrophys., 375, 591
* Del Zanna et al. (2001) Del Zanna, G., Bromage, B. J. I., & Mason, H. E. 2001, in American Institute of Physics Conference Series, Vol. 598, Joint SOHO/ACE workshop ”Solar and Galactic Composition”, ed. R. F. Wimmer-Schweingruber, 59–64
* Dere et al. (1997) Dere, K. P., Landi, E., Mason, H. E., Monsignori Fossi, B. C., & Young, P. 1997, Astron. Astrophys. Suppl. Ser., 125, 149
* Dowdy et al. (1986) Dowdy, J. F., Rabin, D., & Moore, R. L. 1986, Sol. Phys., 105, 35
* Dumont et al. (1983) Dumont, S., Pecker, J.-C., Mouradian, Z., Vial, J.-C., & Chipman, E. 1983, Sol. Phys., 83, 27
* Dupree (1972) Dupree, A. K. 1972, Astrophys. J., 178, 527
* Gabriel (1976) Gabriel, A. 1976, Royal Society of London Philosophical Transactions Series A, 281, 339
* Judge (2000) Judge, P. G. 2000, Astrophys. J., 531, 585
* Judge et al. (1995) Judge, P. G., Woods, T. N., Brekke, P., & Rottman, G. J. 1995, Astrophys. J., 455, L85+
* Keenan (1988) Keenan, F. P. 1988, Sol. Phys., 116, 279
* Klimchuk & Cargill (2001) Klimchuk, J. A. & Cargill, P. J. 2001, Astrophys. J., 553, 440
* Landi et al. (2006) Landi, E., Del Zanna, G., Young, P. R., et al. 2006, Astrophys. J. Suppl. Ser., 162, 261
* Mariska (1993) Mariska, J. T. 1993, The Solar Transition Region (Cambridge University Press)
* Mason (1998) Mason, H. E. 1998, in Lecture Notes in Physics, Berlin Springer Verlag, Vol. 507, Space Solar Physics: Theoretical and Observational Issues in the Context of the SOHO Mission, ed. J. C. Vial, K. Bocchialini, & P. Boumier, 143–+
* Parenti et al. (2006) Parenti, S., Buchlin, E., Cargill, P. J., Galtier, S., & Vial, J.-C. 2006, Astrophys. J., 651, 1219
* Pecker et al. (1988) Pecker, J.-C., Dumont, S., & Mouradian, Z. 1988, Astron. Astrophys., 196, 269
* Peter (1999) Peter, H. 1999, Astrophys. J., 516, 490
* Peter (2001) Peter, H. 2001, Astron. Astrophys., 374, 1108
* Peter & Judge (1999) Peter, H. & Judge, P. G. 1999, Astrophys. J., 522, 1148
* Wilhelm et al. (1995) Wilhelm, K., Curdt, W., Marsch, E., et al. 1995, Sol. Phys., 162, 189
|
arxiv-papers
| 2009-06-07T16:28:31 |
2024-09-04T02:49:03.194911
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Buchlin, J.-C. Vial",
"submitter": "Eric Buchlin",
"url": "https://arxiv.org/abs/0906.1367"
}
|
0906.1406
|
# The Drazin inverse of the linear combinations of two idempotents in the
Banach algebra ††thanks: This project is supported by Natural Science Found of
China (10771191, 10471124 and 10771034).
Shifang Zhang, Junde Wu Corresponding author: [email protected]
Abstract In this paper, some Drazin inverse representations of the linear
combinations of two idempotents in Banach algebra are obtained.
Key Words. Drazin inverse, idempotent, linear combinations.
AMS classification. 46C05, 46C07
## 1 Introduction
Let $\mathscr{A}$ be a Banach$-*$ algebra with the unit $e$. An element
$P\in\mathscr{A}$ is said to be an idempotent if $P^{2}=P$ and a projection if
$P^{2}=P=P^{*}$. The set $\mathscr{P}(\mathscr{A})$ of all idempotents in
$\mathscr{A}$ is invariant under similarity, that is, if
$P\in\mathscr{P}(\mathscr{A})$ and $S\in\mathscr{A}$ is an invertible element,
then $S^{-1}PS$ is still an idempotent.
Let us recall that the Drazin inverse of $A\in\mathscr{A}$ is the element
$B\in\mathscr{A}$ (denoted by $A^{D}$) which satisfies
$BAB=B,\,\,\,AB=BA,\,\,\,A^{k+1}B=A^{k}$ (1)
for some nonnegative integer $k$ ([1]). The least such $k$ is the index of
$A$, denoted by ind$(A)$. It is well-known that if $A$ is Drazin invertible,
then the Drazin inverse is unique and $(aA)^{D}=\frac{1}{a}A^{D}$ for each
nonzero scalar $a$. In particular, for invertible operator $A$, the Drazin
inverse $A^{D}$ coincide with the usual inverse $A^{-1}$ and ind$(a)=0$. The
conditions (1) are also equivalent to
$BAB=B,\,\,\,AB=BA,\,\,\,A-A^{2}B\makebox{\,\,is nilpotent.}$ (2)
The Drazin inverse of an operator in $\mathscr{A}$ is similarly invariant,
that is, if $T$ is Drazin invertible and $S\in\mathscr{A}$ is an invertible
element, then $S^{-1}TS$ is still Drazin invertible and
$(S^{-1}TS)^{D}=S^{-1}T^{D}S.$ If $P\in\mathscr{P}(\mathscr{A})$, it is easy
to verify that $P^{D}=P$.
This paper is concerned with the Drazin inverses $(aP+bQ)^{D}$ of the linear
combinations of two idempotents in $\mathscr{A}$ for nonzero scalars $a$ and
$b$. In recent years, many authors paid much attention to properties of linear
combinations of idempotents or projections (see [2-7,9-14]). In [7], Deng has
discussed the drazin inverses of the products and differences of two
projections. Motivated by this paper, A. Böttcher and I. M. Spitkovsky wrote
[1] and in that paper they proved that the Drazin invertibility of the sum
$P+Q$ of two projections $P$ and $Q$ is equivalent to the Drazin invertibility
of any linear combination $aP+bQ$ where $ab\not=0,a+b\not=0.$ However, without
some additional conditions, it is difficult to discuss the Drazin
invertibility of linear combinations of two idempotents, even if the sum of
them. More recently, under some conditions, Deng in [8] gave the Drazin
inverses of sums and differences of idempotents on the Hilbert space. The
methods used in [8] are the space decompositions and operator matrix
representations which are not avail for general Banach$-*$ algebra, or general
Banach algebra.
In this paper, by using the direct calculation methods, we obtained some
formulae for the Drazin inverse $(aP+bQ)^{D}$ of the linear combinations of
idempotents $P$ and $Q$ in Banach algebra $\mathscr{A}$ under some conditions,
we also study the index ind$(aP+bQ)$.
## 2 Main results
In this section, we always suppose that $\mathscr{A}$ is a Banach algebra with
the unit $I$, $aP+bQ$ is the linear combinations of two idempotents $P$ and
$Q$ in $\mathscr{A}$ with nonzero scalars $a$ and $b$. In order to prove
$(aP+bQ)^{D}$ is Drazin invertible, it follows from the definition of Drazin
inverse that we only need to find out some $M\in\mathscr{A}$ satisfies that
$(aP+bQ)M=M(aP+bQ),M^{2}(aP+bQ)=M,(aP+bQ)^{k+1}M=(aP+bQ)^{k}$ (3)
for some nonnegative integer $k$.
Theorem 2.1. Let $P$ and $Q$ be the idempotents in Banach algebra
$\mathscr{A}$ and $PQP=0$. Then $aP+bQ$ is Drazin invertible for any nonzero
scalars $a$ and $b$, ind$(aP+bQ)\leq 1$ and
$(aP+bQ)^{D}=\frac{1}{a}P+\frac{1}{b}Q-(\frac{1}{a}+\frac{1}{b})PQ-(\frac{1}{a}+\frac{1}{b})QP+(\frac{1}{a}+\frac{2}{b})QPQ.$
Moreover, ind$(aP+bQ)=0$ if and only if $P+Q+QPQ=I+PQ+QP.$
Proof. We first prove that
$(aP+Q)^{D}=\frac{1}{a}P+Q-(\frac{1}{a}+1)PQ-(\frac{1}{a}+1)QP+(\frac{1}{a}+2)QPQ.$
For this, let
$M=\frac{1}{a}P+Q-(\frac{1}{a}+1)PQ-(\frac{1}{a}+1)QP+(\frac{1}{a}+2)QPQ.$ By
the assumption that $PQP=0$, we have
$\begin{array}[]{ll}&M(aP+Q)\\\
=&(\frac{1}{a}P+Q-(\frac{1}{a}+1)PQ-(\frac{1}{a}+1)QP+(\frac{1}{a}+2)QPQ)(aP+Q)\\\
=&[P+aQP-(a+1)P-(a+1)QP]+[\frac{1}{a}PQ+Q-(\frac{1}{a}+1)PQ-(\frac{1}{a}+1)QPQ+(\frac{1}{a}+2)QPQ]\\\
=&P+Q-PQ-QP+QPQ\end{array}$
and
$\begin{array}[]{ll}&(aP+Q)M\\\
=&(aP+Q)(\frac{1}{a}P+Q-(\frac{1}{a}+1)PQ-(\frac{1}{a}+1)QP+(\frac{1}{a}+2)QPQ)\\\
=&[P+aPQ-(a+1)PQ]+[\frac{1}{a}QP+Q-(\frac{1}{a}+1)QPQ-(\frac{1}{a}+1)QP+(\frac{1}{a}+2)QPQ]\\\
=&P+Q-PQ-QP+QPQ.\end{array}$
Therefore, $M(aP+Q)=(aP+Q)M.$ Moreover, a direct calculation shows that
$\begin{array}[]{ll}&M(aP+Q)M\\\ =&(P+Q-PQ-
QP+QPQ)[\frac{1}{a}P+Q-(\frac{1}{a}+1)PQ-(\frac{1}{a}+1)QP+(\frac{1}{a}+2)QPQ]\\\
=&[\frac{1}{a}P+PQ-(\frac{1}{a}+1)PQ]+[\frac{1}{a}QP+Q-(\frac{1}{a}+1)QPQ-(\frac{1}{a}+1)QP+(\frac{1}{a}+2)QPQ]\\\
&-PQ-\frac{1}{a}QP-QPQ+(\frac{1}{a}+1)QPQ+QPQ\\\
=&\frac{1}{a}P+Q-(\frac{1}{a}+1)PQ-(\frac{1}{a}+1)QP+(\frac{1}{a}+2)QPQ=M\end{array}$
and
$\begin{array}[]{ll}&M(aP+Q)^{2}\\\
=&\\{\frac{1}{a}P+Q-(\frac{1}{a}+1)PQ-(\frac{1}{a}+1)QP+(\frac{1}{a}+2)QPQ\\}(aP+Q)^{2}\\\
=&\\{P+Q-PQ-QP+QPQ\\}(aP+Q)\\\ =&aP+aQP-aQP+PQ+Q-PQ-QPQ+QPQ=aP+Q.\end{array}$
Thus, from (3) we get that $(aP+Q)^{D}=M.$ So we have
$\begin{array}[]{ll}(aP+bQ)^{D}&=(b(\frac{a}{b}P+Q))^{D}\\\
&=\frac{1}{b}(\frac{a}{b}P+Q)^{D}\\\
&=\frac{1}{b}\\{\frac{b}{a}P+Q-(\frac{b}{a}+1)PQ-(\frac{b}{a}+1)QP+(\frac{b}{a}+2)QPQ\\}\\\
&=\frac{1}{a}P+\frac{1}{b}Q-(\frac{1}{a}+\frac{1}{b})PQ-(\frac{1}{a}+\frac{1}{b})QP+(\frac{1}{a}+\frac{2}{b})QPQ.\end{array}$
Moreover, since ind$(aP+Q)\leq 1$ proved above and the fact that
ind$(aT)=$ind$(T)$ when $T$ is Drazin invertible, it follows that
ind$(aP+bQ)=$ind$(\frac{b}{a}P+Q)\leq 1.$ In addition, a direct calculation
shows that
$(aP+bQ)^{D}(aP+bQ)=P+Q-PQ-QP+QPQ.$
Note that ind$(aP+bQ)=0$ if and only if $(aP+bQ)^{D}(aP+bQ)=I$, so
ind$(aP+bQ)=0$ if and only if $I=P+Q-PQ-QP+QPQ$. This completed the proof.
Theorem 2.2. Let $P$ and $Q$ be the idempotents in Banach algebra
$\mathscr{A}$ and $PQP=P$. Then $aP+bQ$ is Drazin invertible for any nonzero
scalars $a$ and $b$, and
$(aP+bQ)^{D}=\left\\{\begin{array}[]{ll}\frac{a^{2}}{(a+b)^{3}}P+\frac{1}{b}Q+\frac{ab}{(a+b)^{3}}(PQ+QP)+(\frac{b^{2}}{(a+b)^{3}}-\frac{1}{b})QPQ,&\makebox{\,if\,\,}a+b\not=0;\\\
\frac{1}{a}Q(P-I)Q,&\makebox{\,if\,\,}a+b=0.\end{array}\right.$
Moreover, ind$(aP-aQ)\leq 3$ and ind$(aP+bQ)\leq 2$ when $a+b\not=0$.
Proof. Case (1). Suppose that $a+b\not=0.$ Firstly, we shall show that when
$a\not=-1$, we have
$(aP+Q)^{D}=\frac{a^{2}}{(a+1)^{3}}P+Q+\frac{a}{(a+1)^{3}}(PQ+QP)+(\frac{1}{(a+1)^{3}}-1)QPQ.$
To do this, let
$M=\frac{a^{2}}{(a+1)^{3}}P+Q+\frac{a}{(a+1)^{3}}(PQ+QP)+(\frac{1}{(a+1)^{3}}-1)QPQ.$
By the assumption that $PQP=P$, we have
$\begin{array}[]{ll}&M(aP+Q)\\\
=&\\{\frac{a^{2}}{(a+1)^{3}}P+Q+\frac{a}{(a+1)^{3}}(PQ+QP)+(\frac{1}{(a+1)^{3}}-1)QPQ\\}(aP+Q)\\\
=&\frac{a^{3}}{(a+1)^{3}}P+aQP+\frac{a^{2}}{(a+1)^{3}}P+\frac{a^{2}}{(a+1)^{3}}QP+a(\frac{1}{(a+1)^{3}}-1)QP+\frac{a^{2}}{(a+1)^{3}}PQ\\\
&+Q+\frac{a}{(a+1)^{3}}PQ+\frac{a}{(a+1)^{3}}QPQ+(\frac{1}{(a+1)^{3}}-1)QPQ\\\
=&\frac{a^{2}}{(a+1)^{2}}P+Q+\frac{a}{(a+1)^{2}}(PQ+QP)+(\frac{1}{(a+1)^{2}}-1)QPQ\end{array}$
and
$\begin{array}[]{ll}&(aP+Q)M\\\
=&(aP+Q)\\{\frac{a^{2}}{(a+1)^{3}}P+Q+\frac{a}{(a+1)^{3}}(PQ+QP)+(\frac{1}{(a+1)^{3}}-1)QPQ\\}\\\
=&\frac{a^{3}}{(a+1)^{3}}P+aPQ+\frac{a^{2}}{(a+1)^{3}}PQ+\frac{a^{2}}{(a+1)^{3}}P+a(\frac{1}{(a+1)^{3}}-1)PQ+\frac{a^{2}}{(a+1)^{3}}QP\\\
&+Q+\frac{a}{(a+1)^{3}}QP+\frac{a}{(a+1)^{3}}QPQ+(\frac{1}{(a+1)^{3}}-1)QPQ\\\
=&\frac{a^{2}}{(a+1)^{2}}P+Q+\frac{a}{(a+1)^{2}}(PQ+QP)+(\frac{1}{(a+1)^{2}}-1)QPQ.\end{array}$
Thus,
$(aP+Q)M=M(aP+Q).$ (4)
Since
$\begin{array}[]{ll}&M(aP+Q)^{3}\\\
=&\\{\frac{a^{2}}{(a+1)^{3}}P+Q+\frac{a}{(a+1)^{3}}(PQ+QP)+(\frac{1}{(a+1)^{3}}-1)QPQ\\}(aP+Q)^{3}\\\
=&\\{\frac{a^{2}}{(a+1)^{2}}P+Q+\frac{a}{(a+1)^{2}}(PQ+QP)+(\frac{1}{(a+1)^{2}}-1)QPQ\\}(aP+Q)^{2}\\\
=&\\{\frac{a^{2}}{(a+1)}P+Q+\frac{a}{(a+1)}(PQ+QP)+(\frac{1}{(a+1)}-1)QPQ\\}(aP+Q)\\\
=&a^{2}P+Q+a(PQ+QP)\\\ =&(aP+Q)^{2},\end{array}$
so,
$(aP+Q)^{3}M=(aP+Q)^{2}.$ (5)
Moreover, by calculating, we get that
$\begin{array}[]{ll}&M(aP+Q)M\\\
=&(\frac{a^{2}}{(a+1)^{2}}P+Q+\frac{a}{(a+1)^{2}}(PQ+QP)+(\frac{1}{(a+1)^{2}}-1)QPQ)\times\\\
&(\frac{a^{2}}{(a+1)^{3}}P+Q+\frac{a}{(a+1)^{3}}(PQ+QP)+(\frac{1}{(a+1)^{3}}-1)QPQ)\\\
=&\frac{a^{4}}{(a+1)^{5}}P+\frac{a^{2}}{(a+1)^{3}}QP+\frac{a^{3}}{(a+1)^{5}}QP+\frac{a^{3}}{(a+1)^{5}}P+(\frac{1}{(a+1)^{2}}-1)\frac{a^{2}}{(a+1)^{3}}QP+\\\
&\frac{a^{2}}{(a+1)^{2}}PQ+Q+\frac{a}{(a+1)^{2}}QPQ+\frac{a}{(a+1)^{2}}PQ+(\frac{1}{(a+1)^{2}}-1)QPQ+\\\
&\frac{a^{3}}{(a+1)^{5}}PQ+\frac{a}{(a+1)^{3}}QPQ+\frac{a^{2}}{(a+1)^{5}}QPQ+\frac{a^{2}}{(a+1)^{5}}PQ+(\frac{1}{(a+1)^{2}}-1)\frac{a}{(a+1)^{3}}QPQ+\\\
&\frac{a^{3}}{(a+1)^{5}}P+\frac{a}{(a+1)^{3}}QP+\frac{a^{2}}{(a+1)^{5}}QP+\frac{a^{2}}{(a+1)^{5}}P+(\frac{1}{(a+1)^{2}}-1)\frac{a}{(a+1)^{3}}QP+\\\
&\frac{a^{2}}{(a+1)^{2}}(\frac{1}{(a+1)^{3}}-1)PQ+\frac{a}{(a+1)^{2}}(\frac{1}{(a+1)^{3}}-1)QPQ+(\frac{1}{(a+1)^{3}}-1)QPQ+\\\
&\frac{a}{(a+1)^{2}}(\frac{1}{(a+1)^{3}}-1)PQ+(\frac{1}{(a+1)^{2}}-1)(\frac{1}{(a+1)^{3}}-1)QPQ\par\par\\\
=&(\frac{a^{4}}{(a+1)^{5}}+\frac{a^{3}}{(a+1)^{5}}+\frac{a^{3}}{(a+1)^{5}}+\frac{a^{2}}{(a+1)^{5}})P+Q+\\\
&(\frac{a^{2}}{(a+1)^{3}}+\frac{a^{3}}{(a+1)^{5}}+(\frac{1}{(a+1)^{2}}-1)\frac{a^{2}}{(a+1)^{3}}+\frac{a}{(a+1)^{3}}+\frac{a^{2}}{(a+1)^{5}}+(\frac{1}{(a+1)^{2}}-1)\frac{a}{(a+1)^{3}})QP+\\\
&(\frac{a^{2}}{(a+1)^{2}}+\frac{a}{(a+1)^{2}}+\frac{a^{3}}{(a+1)^{5}}+\frac{a^{2}}{(a+1)^{5}}+\frac{a^{2}}{(a+1)^{2}}(\frac{1}{(a+1)^{3}}-1)+\frac{a}{(a+1)^{2}}(\frac{1}{(a+1)^{3}}-1))PQ+\\\
&\\{\frac{a}{(a+1)^{2}}+(\frac{1}{(a+1)^{2}}-1)+\frac{a}{(a+1)^{3}}+\frac{a^{2}}{(a+1)^{5}}+(\frac{1}{(a+1)^{2}}-1)\frac{a}{(a+1)^{3}}+\frac{a}{(a+1)^{2}}(\frac{1}{(a+1)^{3}}-1)+\\\
&(\frac{1}{(a+1)^{3}}-1)+(\frac{1}{(a+1)^{2}}-1)(\frac{1}{(a+1)^{3}}-1)\\}QPQ\\\
=&(\frac{a^{2}}{(a+1)^{3}}P+Q+\frac{a^{3}+2a^{2}+a}{(a+1)^{5}}PQ+\frac{a^{3}+2a^{2}+a}{(a+1)^{5}}QP+\\{\frac{a^{2}}{(a+1)^{5}}+\frac{1}{(a+1)^{2}}\frac{a}{(a+1)^{3}}+\\\
&\frac{a}{(a+1)^{2}}\frac{1}{(a+1)^{3}}+(\frac{1}{(a+1)^{3}}-1)+(\frac{1}{(a+1)^{2}}-1)\frac{1}{(a+1)^{3}}\\}QPQ\\\
=&\frac{a^{2}}{(a+1)^{3}}P+Q+\frac{a}{(a+1)^{3}}(PQ+QP)+(\frac{1}{(a+1)^{3}}-1)QPQ=M.\end{array}$
That is,
$M(aP+Q)M=M.$ (6)
It follows from equations (4), (5) and (6) that $aP+Q$ is Drazin invertible,
$(aP+Q)^{D}=M$ and ind$(aP+Q)\leq 2$ when $a\neq 1$. Similar to the
disscussion in Theorem 2.1, when $a+b\not=0$, we have
$\begin{array}[]{ll}(aP+bQ)^{D}&=\frac{1}{b}(\frac{a}{b}P+Q)^{D}\\\
&=\frac{1}{b}\\{\frac{(\frac{a}{b})^{2}}{(\frac{a}{b}+1)^{3}}P+Q+\frac{\frac{a}{b}}{(\frac{a}{b}+1)^{3}}(PQ+QP)+(\frac{1}{(\frac{a}{b}+1)^{3}}-1)QPQ\\}\\\
&=\frac{a^{2}}{(a+b)^{3}}P+\frac{1}{b}Q+\frac{ab}{(a+b)^{3}}(PQ+QP)+(\frac{b^{2}}{(a+b)^{3}}-\frac{1}{b})QPQ\end{array}$
and ind$(aP+bQ)=$ind$(\frac{a}{b}P+Q)\leq 2$.
Case (2). Suppose that $a+b=0.$ By calculating, we have
$(aP-aQ)\frac{1}{a}Q(P-I)Q=\frac{1}{a}Q(P-I)Q(aP-aQ)=Q-QPQ,$
$(aP-aQ)(\frac{1}{a}Q(P-I)Q)^{2}=(Q-QPQ)\frac{1}{a}Q(P-I)Q=\frac{1}{a}(QPQ-Q-
QPQ+QPQ)=\frac{1}{a}Q(P-I)Q,$
and
$\begin{array}[]{ll}(aP-aQ)^{4}(\frac{1}{a}Q(P-I)Q)&=(Q-QPQ)(aP-aQ)^{3}\\\
&=a(QPQ-Q)(aP-aQ)^{2}\\\ &=a^{2}(Q-QPQ)(aP-aQ)\\\ &=a^{3}(QPQ-Q)\\\
&=a^{2}(P-PQ-QP+Q)(aP-aQ)\\\ &=(aP-aQ)^{3}.\\\ \end{array}$
Therefore, $(aP-aQ)^{D}=\frac{1}{a}Q(P-I)Q$, $(aP-aQ)^{4}(aP-aQ)^{D}=(aP-
aQ)^{3}$ and ind$(aP-aQ)\leq 3.$ This completed the proof.
Remark 2.3. Under the assumption of Theorem 2.2, we have ind$(aP-aQ)=3$ if and
only if $P+QPQ\not=PQ+QP$. For this, we only need to note that $(aP-
aQ)^{3}(aP-aQ)^{D}=a^{2}(Q-QPQ)$ and $(aP-aQ)^{2}=a^{2}(P-PQ-QP+Q).$
Theorem 2.4. Let $P$ and $Q$ be the idempotents in Banach algebra
$\mathscr{A}$ and $PQ=QP$. Then $aP+bQ$ is Drazin invertible for any nonzero
scalars $a$ and $b$, ind$(aP+bQ)\leq 1$ and
$(aP+bQ)^{D}=\left\\{\begin{array}[]{ll}\frac{1}{a}P+\frac{1}{b}Q+(\frac{1}{a+b}-\frac{1}{a}-\frac{1}{b})PQ,&\makebox{\,if\,\,}a+b\not=0;\\\
\frac{1}{a}(P-Q),&\makebox{\,if\,\,}a+b=0.\end{array}\right.$ (7)
Moreover, when $a+b\not=0$, $ind(aP+bQ)=0$ if and only if $P+Q=I+PQ$; while
ind$(aP-aQ)=0$ if and only if $P+Q=I+2PQ$.
Proof. We first prove that when $a\not=-1,$
$(aP+Q)^{D}=\frac{1}{a}P+Q+(\frac{1}{a+1}-\frac{1}{a}-1)PQ.$
For this, let $M=\frac{1}{a}P+Q+(\frac{1}{a+1}-\frac{1}{a}-1)PQ$. By the
assumption that $PQ=QP$, a direct calculation shows that
$\begin{array}[]{ll}(aP+Q)M&=M(aP+Q)\\\
&=(\frac{1}{a}P+Q+(\frac{1}{a+1}-\frac{1}{a}-1)PQ)(aP+Q)\\\
&=P+aPQ+(\frac{a}{a+1}-1-a)PQ+\frac{1}{a}PQ+Q+(\frac{1}{a+1}-\frac{1}{a}-1)PQ\\\
&=P+Q-PQ.\\\ \end{array}$
Moreover, it is easy to check that
$\begin{array}[]{ll}(aP+Q)^{2}M&=(aP+Q)(P+Q-PQ)\\\ &=aP+aPQ-aPQ+PQ+Q-PQ\\\
&=aP+Q\end{array}$
and
$\begin{array}[]{ll}&M(aP+Q)M\\\
&=(P+Q-PQ)(\frac{1}{a}P+Q+(\frac{1}{a+1}-\frac{1}{a}-1)PQ)\\\
&=\frac{1}{a}P+\frac{1}{a}PQ-\frac{1}{a}PG+PQ+Q-PQ+(1+1-1)(\frac{1}{a+1}-\frac{1}{a}-1)PQ\\\
&=\frac{1}{a}P+Q+(\frac{1}{a+1}-\frac{1}{a}-1)PQ=M.\end{array}$
So,
$(aP+Q)^{D}=\frac{1}{a}P+Q+(\frac{1}{a+1}-\frac{1}{a}-1)PQ.$
If $a+b\not=0$, then
$\begin{array}[]{ll}(aP+bQ)^{D}&=(b(\frac{a}{b}P+Q))^{D}\\\
&=\frac{1}{b}(\frac{a}{b}P+Q)^{D}\\\
&=\frac{1}{b}\\{\frac{b}{a}P+Q+(\frac{b}{a+b}-\frac{b}{a}-1)PQ\\}\\\
&=\frac{1}{a}P+\frac{1}{b}Q+(\frac{1}{a+b}-\frac{1}{a}-\frac{1}{b})PQ.\end{array}$
Moreover, we can show that ind$(aP+bQ)\leq 1$ and when $a+b\not=0$,
$(aP+bQ)^{D}(aP+bQ)=P+Q-PQ.$
So, ind$(aP+bQ)=0$ if and only if $I=P+Q-PQ$.
On the other hand, note that $PQ=QP$, so we have
$(P-Q)^{2}=P+Q-2PQ\makebox{\,\, and \,\,}(P-Q)^{3}=P-Q,$
this implied that $(P-Q)^{D}=P-Q$. Thus, when $a+b=0$, we have
$(aP+bQ)^{D}=\frac{1}{a}(P-Q)$ and ind$((aP+bQ)^{D})\leq 1$. It is clear that
ind$(aP-aQ)=0$ if and only if $P+Q=I+2PQ$. This completed the proof.
Noting that $PQP=Q$ implies that $Q=QP=PQ$, so, it follows from Theorem 2.4
immediately:
Corollary 2.5. Let $P$ and $Q$ be the idempotents in Banach algebra
$\mathscr{A}$ and $PQP=Q$. Then $aP+bQ$ is Drazin invertible for any nonzero
scalars $a$ and $b$, ind$(aP+bQ)\leq 1$ and
$(aP+bQ)^{D}=\left\\{\begin{array}[]{ll}\frac{1}{a}P+(\frac{1}{a+b}-\frac{1}{a})Q,&\makebox{\,if\,\,}a+b\not=0;\\\
\frac{1}{a}(P-Q),&\makebox{\,if\,\,}a+b=0.\end{array}\right.$
Remark 2.6. (1). It follows from Corollary 2.5 that if $PQP=Q$, then
$(P-Q)^{D}=P-Q$. Moreover, we can prove that $(P-Q)^{D}=P-Q$ if and only if
$PQP=QPQ$.
(2). Our results recovered most of the main conclusions in [8], but our
methods are very different from the methods used in [8], in particular, the
methods used in [8] cannot obtain any information about the Drazin index.
The group inverse of $A\in\mathscr{A}$ ([16-19]) is the element
$B\in\mathscr{A}$ (denoted by $A^{g}$) which satisfies
$BAB=B,\,\,\,AB=BA,\,\,\,ABA=A.$ (8)
Obviously, $A$ has group inverse if and only if $A$ has Drazin inverse with
ind$(A)\leq 1$.
Before giving the revised versions of theorems 3.2 and 3.3 in [15], let us see
the following two interesting counter-examples.
Example 2.7 Let $A=\left(\begin{array}[]{cc}S&0\\\ 0&0\end{array}\right)\in
B(l_{2}\oplus l_{2})$ and $B=\left(\begin{array}[]{cc}0&0\\\
T&0\end{array}\right)\in B(l_{2}\oplus l_{2})$ with $S$ and $T$ in $B(l_{2})$
such that $TS\neq 0$. Considering operator
$P=\left(\begin{array}[]{cc}I&0\\\ 0&0\end{array}\right)\in B(H_{2}\oplus
H_{2}),\,\,\,\,\,\,Q=\left(\begin{array}[]{cc}I&A\\\ B&0\end{array}\right)\in
B(H_{2}\oplus H_{2}),$
where $H_{2}=l_{2}\oplus l_{2}$.
Direct calculations shows that
$BA\neq 0,\,\,\,(BA)^{2}=AB=0.$
Hence we have $P^{2}=P,Q^{2}=Q,PQP=P$. From Theorem 2.2 , we know that $P+Q$
has Drazin inverse and
$(P+Q)^{D}=\frac{1}{8}P+Q+\frac{1}{8}(PQ+QP)-(\frac{7}{8})QPQ$. Hence
$(P+Q)-(P+Q)^{2}(P+Q)^{D}=(P+Q)-(\frac{1}{2}P+Q+\frac{1}{2}(PQ+QP)-\frac{1}{2}QPQ)=\frac{1}{2}(P+QPQ)-\frac{1}{2}(PQ+QP)=\frac{1}{2}\left(\begin{array}[]{cc}0&0\\\
0&BA\end{array}\right)\neq 0$, which implies that ind$(P+Q)>1$. Together this
with Theorem 2.2, it is clear that ind$(P+Q)=2.$ So the group inverse
$(P+Q)^{g}$ of $P+Q$ does not exist.
Example 2.8 Define operators $p$ and $q$ in $B(\mathbb{C}^{5})$ by
$p=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\end{array}\right)$ and
$q=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&0&0&0\\\
0&1&0&0&0\\\ 0&0&0&0&1\end{array}\right)$, respectively. Obviously,
$p^{2}=p,q^{2}=q,pqp=p=pq.$
This means that $p$ and $q$ are idempotents in $B(\mathbb{C}^{5})$. Then it
results from Theorem 2.2 that $(p-q)^{D}=q(p-1)q$. But a direct calculation
shows that $(p-q)^{2}(p-q)^{D}=qpq-q=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&-1\end{array}\right)\neq p-q$,
this mean that ind$(p-q)>1$. So the group inverse $(p-q)^{g}$ of $p-q$ does
not exist.
The above two examples illustrate not only Theorem 3.2, but also part (ii) of
Theorem 3.3 in [15] are not always true. Now we present the modified versions
as follows
Theorem 3.2′ Let $P$ and $Q$ be the idempotents in Banach algebra
$\mathscr{A}$ and $PQP=P$ Then
(i)$(P+Q)^{D}=\frac{1}{8}P+Q+\frac{1}{8}(PQ+QP)-(\frac{7}{8})QPQ$,
(ii) $(P-Q)^{D}=Q(P-1)Q$,
(iii) $P+Q$ has group inverse if and only if $P+QPQ=PQ+QP$ ,
(iv) $P-Q$ has group inverse if and only if $P=QPQ$.
Proof. Since the results of part (i) and part (ii) is a special case of
Theorem 2.2 , it suffice to show part (iii) and part (iv). For this, we only
need to note that $(P+Q)-(P+Q)^{2}(P+Q)^{D}=\frac{1}{2}(P+QPQ-PQ-QP)$ and that
$(P-Q)-(P-Q)^{2}(P-Q)^{D}=P-QPQ$, which can be obtained by direct
calculations. This completed the proof.
Theorem 3.3′ Let $P$ and $Q$ be the idempotents in Banach algebra
$\mathscr{A}$ and $PQP=PQ$. Then
$(P+Q)^{g}=P+Q-2QP-\frac{3}{4}PQ+\frac{5}{4}QPQ,$ $(P-Q)^{D}=P-Q-PQ+QPQ.$
Moreover, ind$(P-Q)\leq 2$ and $P-Q$ has group inverse if and only if
$PQ=QPQ$.
Proof. Since the group inverse of $P+Q$ can by checked directly, its proof is
omitted. Now let $M=P-Q-PQ+QPQ$. By direct calculations we have that
$M(P-Q)M=M,(P-Q)^{2}M=M,$ (9)
and that
$(P-Q)^{3}M=(P-Q)^{2}=(P-Q)M=M(P-Q)=P-PQ-QP+Q.$
This implies that $(P-Q)^{D}=P-Q-PQ+QPQ$ and that ind$(P-Q)\leq 2$. In this
case, from equation (9) and the definition of group inverse, we know that
$P-Q$ has group inverse if and only if
$(P-Q)^{2}(P-Q)^{D}=(P-Q)=(P-Q)^{D}=P-Q-PQ+QPQ.$ This completed the proof.
## References
* [1] A. Böttcher, I. M. Spitkovsky, Drazin inversion in the von Neumann algebra generated by two orthogonal projections. To appear.
* [2] J. K. Baksalary, O. M. Baksalary, H. Ozdemir, A note on linear combinations of commuting tripotent matrices, Linear Algebra Appl. 388 (2004) 45–51.
* [3] J. K. Baksalary, O. M. Baksalary, Idempotency of linear combinations of two idempotent matrices, Linear Algebra Appl. 321 (2000) 3–7.
* [4] J. K. Baksalary, O.M. Baksalary, G.P.H. Styan, Idempotency of linear combinations of an idempotent matrix and a tripotent matrix, Linear Algebra Appl. 54 (2002) 21–34.
* [5] O. M. Baksalary, J. Benítez, Idempotency of linear combinations of three idempotent matrices, two of which are commuting, Linear Algebra Appl. 424(2007) 320–337.
* [6] J. Benítez, N. Thome, Idempotency of linear combinations of an idempotent matrix and a t-potent matrix that commute, Linear Algebra Appl. 403 (2005) 414–418.
* [7] C. Y. Deng, The Drazin inverses of products and differences of orthogonal projections, J. Math. Anal. Appl. 335 (2007), 64–71.
* [8] C. Y. Deng, The Drazin inverses of sum and difference of idempotents, LinearAlgebra Appl.,430 (2009) 1282–1291.
* [9] H. Du, X. Yao, C. Deng, Invertibility of linear combinations of two idempotents, Proc. Amer. Math. Soc. 134 (2006), 1451–1457.
* [10] J.J.Koliha, V. Rakč cevič , The nullity and rank of linear combinations of idempotent matrices, Linear Algebra Appl. 418 (2006), 11–14.
* [11] J.J. Koliha, V. Rakč cevič , Stability theorems for linear combinations of idempotents, Integral Equations Operator Theory, to appear.
* [12] J.J. Koliha, V. Rakč cevič , I. Stra skraba, The difference and sum of projectors, Linear Algebra Appl. 388 (2004), 279–288.
* [13] H. Özdemir, A.Y. Özban, On idempotency of linear combinations of idempotent matrices, Appl. Math. Comput. 159 (2004) 439–448.
* [14] M. Sarduvan, H. Özdemir, On linear combinations of two tripotent, idempotent, and involutive matrices, Appl. Math. Comput. 200 (2008) 401–406.
* [15] D.S. Cvetkovi ć-IIić, C. Y. Deng, Some results on the Drazin invertibility and idempotents, J. Math. Anal. Appl. 359 (2009) 731-738.
* [16] K. P. S. Bhaskara Rao, The theory of generalized inverses over commutative rings, Taylor and Francis,London and NewYork, 2002.
* [17] C. D. Meyer, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Review, 17 (1975) 443-464.
* [18] C. J. Bu, J. M. Zhao, J. S. Zheng, Group inverse for a class $2\times 2$ block matrices over skew fields, Applied Mathematics and Computation 204(2008)45-49.
|
arxiv-papers
| 2009-06-08T02:58:56 |
2024-09-04T02:49:03.204661
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhang Shifang, Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0906.1406"
}
|
0906.1412
|
# On Fixed Points of Lüders Operation††thanks: This project is supported by
Natural Science Found of China (10771191 and 10471124).
Liu Weihua, Wu Junde
Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
Corresponding author E-mail: [email protected]
Abstract. In this paper, we give a concrete example of a Lüders operation
$L_{{\cal A}}$ with $n=3$ such that $L_{{\cal A}}(B)=B$ does not imply that
$B$ commutes with all $E_{1},E_{2}$ and $E_{3}$ in $\cal A$, this example
answers an open problem of Professor Gudder.
Key words. Hilbert space, Lüders operation, fixed point.
Let $H$ be a complex Hilbert space, ${\cal B}(H)$ be the bounded linear
operator set on $H$, ${\cal E}(H)=\\{A:0\leq A\leq I\\}$, ${\cal
A}=\\{E_{i}\\}_{i=1}^{n}\subseteq{\cal E}(H)$ and $\sum_{i=1}^{n}E_{i}=I$,
where $1\leq n\leq\infty$. The famous Lüders operation $L_{{\cal A}}$ is a map
which is defined on ${\cal B}(H)$ by:
$L_{{\cal
A}}:A\rightarrow\sum\limits_{i=1}^{n}E_{i}^{\frac{1}{2}}AE_{i}^{\frac{1}{2}}.$
A question related to a celebrated theorem of Lüders operation is whether
$L_{{\cal A}}=A$ for some $A\in{\cal E}(H)$ implies that $A$ commutes with all
$E_{i}$ for $i=1,2,\cdots,n$ ([1]). The answer to this question is positive
for $n=2$ ([2]), and negative for $n=5$ ([1]). In this paper it is shown, by
using a simple derivation of the example of Arias-Gheondea-Gudder in [1], that
the answer is negative as well for $n=3$, a question raised by Gudder in 2005
([3]).
First, we denote ${\cal B}(H)^{L_{{\cal A}}}=\\{B\in{\cal B}(H):L_{{\cal
A}}(B)=B\\}$ is the fixed point set of $L_{{\cal A}}$, ${\cal A}^{\prime}$ is
the commutant of ${\cal A}$.
Lemma 1 ([1]). If ${\cal B}(H)^{L_{{\cal A}}}={\cal A}^{\prime}$, then ${\cal
A}^{\prime}$ is injective.
Lemma 2 ([1]). Let $F_{2}$ be the free group generated by two generators
$g_{1}$ and $g_{2}$ with identity $e$, $\mathbb{C}$ be the complex numbers set
and $H=l_{2}(F_{2})$ be the separable complex Hilbert space
$H=l_{2}(F_{2})=\\{f|f:F_{2}\rightarrow\mathbb{C},\sum|f(x)|^{2}<\infty\\}.$
For $x\in F_{2}$ define $\delta_{x}:F_{2}\rightarrow C$ by $\delta_{x}(y)$
equals $0$ for all $y\neq x$ and $1$ when $y=x$. Then $\\{\delta_{x}|x\in
F_{2}\\}$ is an orthonormal basis for $H$. Define the unitary operators
$U_{1}$ and $U_{2}$ on $H$ by $U_{1}(\delta_{x})=\delta_{g_{1}x}$ and
$U_{2}(\delta_{x})=\delta_{g_{2}x}$. Then the von Neumann algebra
$\mathscr{N}$ which is generated by $U_{1}$ and $U_{2}$ and its commutant
$\mathscr{N}^{\prime}$ are not injective.
Now, we follow the Lemma 1 and Lemma 2 to prove our main result:
Let $\mathbb{C}_{1}$ be the unite circle in $\mathbb{C}$ and $h$ be a Borel
function be defined on the $\mathbb{C}_{1}$ as following:
$h(e^{i\theta})=\theta$ for $\theta\in[0,2\pi)$. Then $A_{1}=h(U_{1})$ and
$A_{2}=h(U_{2})$ are two positive operators in $\mathcal{N}$. If take the real
and imagine parts of $U_{1}=V_{1}+iV_{2}$ and $U_{2}=V_{3}+iV_{4}$, then
$\mathcal{N}$ is generated by the self-adjoint operators
$\\{V_{1},V_{2},V_{3},V_{4}\\}$ ([1]). Since functions $\cos$ and $\sin$ are
two Borel functions, so we have
$V_{1}=\frac{1}{2}(U_{1}+U_{1}^{*})=\cos(A_{1})$, $V_{2}=\sin(A_{1})$,
$V_{3}=\cos(A_{2})$ and $V_{4}=\sin(A_{2})$. Thus $\mathscr{N}$ is contained
in the von Neumann algebra which is generated by $A_{1}$ and $A_{2}$.
On the other hand, it is clear that the von Neumann algebra which is generated
by $A_{1}$ and $A_{2}$ is contained in $\mathscr{N}$. So $\mathscr{N}$ is the
von Neumann algebra which is generated by $A_{1}$ and $A_{2}$. Let
$E_{1}=\frac{A_{1}}{2\|A_{1}\|}$, $E_{2}=\frac{A_{2}}{2\|A_{2}\|}$ and
$E_{3}=I-E_{1}-E_{2}$. Then ${\cal A}=\\{E_{1},E_{2},E_{3}\\}\subseteq{\cal
E}(H)$ and $E_{1}+E_{2}+E_{3}=I$.
Now, we define the Lüders operation on ${\cal B}(H)$ by
$L_{{\cal A}}(B)=\sum\limits_{i=1}^{3}E_{i}^{1/2}BE_{i}^{1/2}.$
It is clear that the Von Neumann algebra which is generated by
$\\{E_{1},E_{2},E_{3}\\}$ is $\mathscr{N}$, so it follows from Lemma 1 and
Lemma 2 that $B(H)^{L_{{\cal A}}}\supsetneq{\cal A}^{\prime}$, thus there
exists a $D\in B(H)^{L_{{\cal A}}}\setminus{\cal A}^{\prime}$. Now, the real
part or the imaginary part $D_{1}$ of $D$ also satisfies $D_{1}\in
B(H)^{L_{{\cal A}}}\setminus{\cal A}^{\prime}$. Let $D_{2}=||D_{1}||I-D_{1}$.
Then $D_{2}\geq 0$. Let $D_{3}=\frac{D_{2}}{||D_{2}||}$. Then $D_{3}\in{\cal
E}(H)$ and $D_{3}\in B(H)^{L_{{\cal A}}}\setminus{\cal A}^{\prime}$. Thus, we
proved the following theorem which answered the question in [3].
Theorem 1. Let $H=l_{2}(F_{2})$, ${\cal A}=\\{E_{i}\\}_{i=1}^{3}$ be defined
as above. Then there is a $B\in{\cal E}(H)$ such that $L_{{\cal A}}(B)=B$, but
$B$ does not commute with all $E_{1},E_{2}$ and $E_{3}$.
Acknowledgement. The authors wish to express their thanks to the referee for
his (her) important comments and suggestions.
References
[1] A. Arias, A. Gheondea, S. Gudder. Fixed points of quantum operations. J.
Math. Phys., 43, 2002, 5872-5881
[2] P. Busch, J. Singh. Lüders theorem for unsharp quantum measurements. Phys.
Letter A, 249, 1998, 10-12
[3] S. Gudder. Open problems for sequential effect algebras. Inter. J. Theory.
Physi. 44, 2005, 2199-2205
|
arxiv-papers
| 2009-06-08T06:05:49 |
2024-09-04T02:49:03.209955
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liu Weihua, Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0906.1412"
}
|
0906.1482
|
# A new current algebra and the reflection equation
P. Baseilhac Laboratoire de Mathématiques et Physique Théorique CNRS/UMR
6083, Fédération Denis Poisson, Université de Tours, Parc de Grammont, 37200
Tours, FRANCE [email protected] and K. Shigechi Institute for
Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, THE NETHERLANDS
[email protected]
###### Abstract.
We establish an explicit algebra isomorphism between the quantum reflection
algebra for the $U_{q}(\widehat{sl_{2}})$ $R$-matrix and a new type of current
algebra. These two algebras are shown to be two realizations of a special case
of tridiagonal algebras ($q-$Onsager).
MSC: 81R50; 81R10; 81U15.
Keywords: Current algebra; Reflection equation; $q-$Onsager algebra; Quantum
integrable models
## 1\. Introduction
Discovered in the context of the quantum inverse scattering method for solving
quantum integrable systems, quantum groups appeared in the literature through
different ways (see [Cha] for references). On one hand, starting from the
fundamental independent discovery of Drinfeld [Dr1] and Jimbo [Jim] the
quantum affine algebras $U_{q}({\widehat{g}})$ were initially formulated using
a $q-$deformed version of the commutation relations between the elements of
the Chevalley presentation of ${\widehat{g}}$. Later on [Dr2], Drinfeld
proposed a new realization of $U_{q}({\widehat{g}})$ in terms of elements
$\\{x_{i,k}^{\pm},\varphi_{i,m},\psi_{i,n}|i=1,...,l;k\in{\mathbb{Z}},m\in-{\mathbb{Z}}_{+},n\in{\mathbb{Z}}_{+}\\}$
with $l=rank(g)$ generated through operator-valued functions
$x^{\pm}_{i}(u),\varphi_{i}(u),\psi_{i}(u)$ of the formal variable $u$, the
so-called currents. In some sense, the Drinfeld’s realization is a quantum
analogue of the loop realization of affine Lie algebras. Although Drinfeld
stated the isomorphism between the two realizations, the proof only appeared
later on [Be, Jin]. In particular, in [Be] (see also [Dam]) Lusztig’s theory
of braid group action [L] on the quantum enveloping algebras was used from
which an explicit homomorphism from Drinfeld’s new realization [Dr2] to the
initial one [Dr1, Jim] was obtained. On the other hand, an alternative
realization of quantum affine algebras $U_{q}({\widehat{g}})$ by means of
solutions of the quantum Yang-Baxter equation [KRS, KS, F1] \- called the
$R-$matrix - and the “RLL” algebraic relations of the quantum inverse
scattering method was proposed by Reshetikhin and Semenov-Tian-Shansky in
[RS], extending the previous results of Faddeev-Reshetikhin-Takhtajan [FRT1]
for finite dimensional simple Lie algebra $g$.
In view of these realizations, in [DiF] Ding and Frenkel exhibited an explicit
isomorphism between the “RLL” formulation and Drinfeld’s second realization.
Namely, $L-$operators were shown to admit a unique (Gauss) decomposition in
terms of Drinfeld’s currents $x^{\pm}_{i}(u),\varphi_{i}(u),\psi_{i}(u)$. So,
all these different realizations may be summarized by the following picture
which provides an unifying algebraic scheme for quantum affine algebras:
“RLL” algebra [FRT1] Yang-Baxter equation [DiF] Current algebra [Dr2]
Drinfeld’s presentation $\\{x_{i,k}^{\pm},\varphi_{i,m},\psi_{i,n}\\}$
$U_{q}({\widehat{g}})$ [RS],[DiF] [Be] [Jin] Drinfeld-Jimbo [Dr1], [Jim]
Beyond the interest of the algebraic structures involved, the explicit
relation between the two different realizations (“RLL” and Drinfeld’s one) of
$U_{q}({\widehat{g}})$ has found many interesting applications in the study of
quantum integrable systems and representation theory.
For quantum integrable systems with boundaries, Cherednik [Cher] and later on
[Sk] introduced another example of quadratic algebra associated with the so-
called reflection equations. In this case, given an $R-$matrix associated with
$U_{q}(\widehat{g})$ one is looking for a $K-$operator (sometimes called a
Sklyanin’s operator) satisfying “RKRK” algebraic relations. Motivated by the
study of related integrable systems, several examples of $K-$operators acting
on finite dimensional representations have been constructed. However, a
formulation of $K-$operators in terms of current algebras i.e. a “boundary” -
in reference to boundary integrable models - version of Ding-Frenkel [DiF]
analysis has never been explicitly presented, nor a “boundary” analogue of
Drinfeld’s presentation even in the simplest case $U_{q}(\widehat{sl_{2}})$.
In this paper, we argue that the $q-$Onsager algebra ${\mathbb{T}}$ (a type of
tridiagonal algebra) which independently appeared in the context of orthogonal
polynomial association schemes [Ter2] and hidden symmetries of boundary
integrable models [Bas] admits analogously two alternative realizations. One
realization is given in terms of a $K-$operator satisfying “RKRK” defining
relations for the $U_{q}(\widehat{sl_{2}})$ $R$-matrix, and the other
realization in terms of a new type of current algebra associated with the
generating set $\\{{\cal W}_{-k},{\cal W}_{k+1},{\cal G}_{k+1},{\tilde{\cal
G}}_{k+1}|k\in{\mathbb{Z}}_{+}\\}$ introduced in [BasK]. A new algebraic
scheme follows, which extends to the family of reflection equation algebras
the standard scheme relating the Faddeev-Reshetikhin-Takhtajan, Jimbo and
Drinfeld (first and second) realizations of quantum affine algebras (see above
picture). Although it is not considered here, the extension of our work to
other classical Lie algebra - technically more complicated - is an interesting
and open problem.
The paper is organized as follows. In Section 2, a new current algebra -
denoted $O_{q}(\widehat{sl_{2}})$ below - with generators $\cal
W_{\pm}(u),\cal G_{\pm}(u)$ and formal variable $u$ is introduced. It is shown
to be isomorphic to the “RKRK” algebra. A coaction map, the analogue of the
coproduct for Hopf’s algebras, is also explicitly derived. In Section 3, the
new currents are found to be generating functions in the symmetric variable
$U=(qu^{2}+q^{-1}u^{-2})/(q+q^{-1})$ which coefficients coincide with the
elements of the infinite dimensional algebra - denoted ${\cal A}_{q}$ below -
introduced in [BasK]. In the last section, based on the commuting properties
of the $K-$operator with the two generators of the $q-$Onsager algebra we
establish the isomorphism between ${\mathbb{T}}$ and the “RKRK” algebra. A new
algebraic scheme unifying these realizations is then proposed.
###### Notation .
In this paper, ${\mathbb{R}}$, ${\mathbb{C}}$, ${\mathbb{Z}}$ denote the field
of real, complex numbers and integers, respectively. We denote
${\mathbb{R}}^{*}={\mathbb{R}}\backslash\\{0\\}$,
${\mathbb{C}}^{*}={\mathbb{C}}\backslash\\{0\\}$,
${\mathbb{Z}}^{*}={\mathbb{Z}}\backslash\\{0\\}$ and ${\mathbb{Z}}_{+}$ for
nonnegative integers. We introduce the $q-$commutator
$\big{[}X,Y\big{]}_{q}=qXY-q^{-1}YX$ where $q$ is the deformation parameter,
assumed not to be a root of unity.
## 2\. A new current algebra
Let ${\cal V}$ be a finite dimensional space. Let the operator-valued function
$R:{\mathbb{C}}^{*}\mapsto\mathrm{End}({\cal V}\otimes{\cal V})$ be the
intertwining operator (quantum $R-$matrix) between the tensor product of two
fundamental representations ${\cal V}={\mathbb{C}}^{2}$ associated with the
algebra $U_{q}(\widehat{sl_{2}})$. The element $R(u)$ depends on the
deformation parameter $q$ and is defined by [Baxter]
(2.5) $\displaystyle R(u)=\left(\begin{array}[]{cccc}uq-u^{-1}q^{-1}&0&0&0\\\
0&u-u^{-1}&q-q^{-1}&0\\\ 0&q-q^{-1}&u-u^{-1}&0\\\
0&0&0&uq-u^{-1}q^{-1}\end{array}\right)\ ,$
where $u$ is called the spectral parameter. Then $R(u)$ satisfies the quantum
Yang-Baxter equation in the space ${\cal V}_{1}\otimes{\cal V}_{2}\otimes{\cal
V}_{3}$. Using the standard notation $R_{ij}(u)\in\mathrm{End}({\cal
V}_{i}\otimes{\cal V}_{j})$, it reads
(2.6) $\displaystyle
R_{12}(u/v)R_{13}(u)R_{23}(v)=R_{23}(v)R_{13}(u)R_{12}(u/v)\ \qquad\forall
u,v.$
Let us now consider an extension related with the reflection equation or
boundary quantum Yang-Baxter equation - which was first introduced in the
context of boundary quantum inverse scattering theory (see [Cher],[Sk] for
details) -. For simplicity and without loosing generality we consider the
simplest case, i.e. the $U_{q}(\widehat{sl_{2}})$ $R-$matrix defined above.
###### Definition 2.1 (“RKRK” Reflection equation algebra).
Define $R(u)$ to be (2.5). $B_{q}(\widehat{sl_{2}})$ is an associative algebra
with unit $1$ and generators $K_{11}(u)\equiv A(u)$, $K_{12}(u)\equiv B(u)$,
$K_{21}(u)\equiv C(u)$, $K_{22}(u)\equiv D(u)$ considered as the elements of
the $2\times 2$ square matrix $K(u)$ which obeys the defining relations
$\forall u,v$
(2.7) $\displaystyle R_{12}(u/v)\ (K(u)\otimes I\\!\\!I)\ R_{12}(uv)\
(I\\!\\!I\otimes K(v))\ =\ (I\\!\\!I\otimes K(v))\ R_{12}(uv)\ (K(u)\otimes
I\\!\\!I)\ R_{12}(u/v)\ .$
It is known that given a solution $K(u)$ of the reflection equation (2.7), one
can construct by induction other solutions using suitable combinations of Lax
operators $L(u)$. This is sometimes named as the “dressing” procedure. In
particular, for the simplest case one has:
###### Proposition 2.1 (see [Sk]).
Given $R(u)$ defined by (2.5), let $L(u)$ be a solution of the quantum Yang-
Baxter algebra with defining relations $\forall u,v$
(2.8) $\displaystyle R(u/v)(L(u)\otimes I\\!\\!I)(I\\!\\!I\otimes L(v))\ =\
(I\\!\\!I\otimes L(v))(L(u)\otimes I\\!\\!I)R(u/v)\ .$
Let $K(u)$ be a solution of (2.7). Then, the matrix $L(u)K(u)L^{-1}(u^{-1})$
is a solution of the reflection equation (2.7).
For instance, using the generating set $\\{S_{\pm},s_{3}\\}$ of the quantum
algebra $U_{q}(sl_{2})$ with defining relations $[s_{3},S_{\pm}]=\pm S_{\pm}$
and $[S_{+},S_{-}]=(q^{2s_{3}}-q^{-2s_{3}})/(q-q^{-1})$ , it is known that the
Lax operator
(2.11) $\displaystyle{L}(u)=\left(\begin{array}[]{cc}uq^{{1\over
2}}q^{s_{3}}-u^{-1}q^{-{1\over 2}}q^{-s_{3}}&(q-q^{-1})S_{-}\\\
(q-q^{-1})S_{+}&uq^{{1\over 2}}q^{-s_{3}}-u^{-1}q^{-{1\over 2}}q^{s_{3}}\\\
\end{array}\right)\ $
satisfies the quantum Yang-Baxter algebra (2.8). In quantum integrable lattice
models with boundaries, the “dressing” procedure is often used. Starting from
an elementary solution with $c-$number entries (associated with one boundary
of the system) and dressing the $K-$operator with a product of $N$
$L-$operators acting on different quantum spaces, one reconstructs a whole
spin chain with $N$ sites including inhomogeneities, if necessary [Sk].
In order to exhibit the new current algebra starting from the “RKRK”
reflection equation algebra, based on previous works on boundary quantum
integrable systems on the lattice [Bas, BasK] it seems rather natural to write
the elements $A(u)$, $B(u)$, $C(u)$, $D(u)$ in terms of new currents as
follows. It may be important to stress that Proposition 2.1 plays an essential
role (see [Bas, BasK]) in suggesting such a decomposition.
###### Lemma 2.1.
Suppose $q\neq 1$, $u\neq q^{-1}$ and $k_{\pm}\in{\mathbb{C}}^{*}$. Any
solution of the reflection equation algebra $B_{q}(\widehat{sl_{2}})$ admits
the following decomposition in terms of new elements $\cal W_{\pm}(u)$, $\cal
G_{\pm}(u)$:
(2.12) $\displaystyle A(u)=uq\cal W_{+}(u)-u^{-1}q^{-1}\cal W_{-}(u)\ ,$
(2.13) $\displaystyle D(u)=uq\cal W_{-}(u)-u^{-1}q^{-1}\cal W_{+}(u)\ ,$
(2.14) $\displaystyle B(u)=\frac{1}{k_{-}(q+q^{-1})}\cal
G_{+}(u)+\frac{k_{+}(q+q^{-1})}{(q-q^{-1})}\ ,$ (2.15) $\displaystyle
C(u)=\frac{1}{k_{+}(q+q^{-1})}\cal
G_{-}(u)+\frac{k_{-}(q+q^{-1})}{(q-q^{-1})}\ .$
Given the elements $A(u),B(u),C(u)$ of this form, this decomposition is
unique.
###### Proof.
The reflection equation being satisfied for arbitrary $u,v\in{\mathbb{C}}^{*}$
and generic $q$, in view of (2.5) the elements $A(u)$, $B(u)$, $C(u)$, $D(u)$
are a priori formal power series in $u$. With no restrictions, let us choose
$A(u)$, $B(u)$, $C(u)$ to be (2.12), (2.14), (2.15), respectively. We have to
show that $D(u)$ is uniquely defined by (2.13). To prove it, assume the set
$\\{A,B,C,D\\}$ given by (2.12)-(2.15) satisfies the reflection equation
algebra with (2.5). In terms of these elements, explicitly (2.7) reads
$\displaystyle(i)$ $\displaystyle
a_{-}c_{+}\left(BC^{\prime}-B^{\prime}C\right)+a_{-}a_{+}[A,A^{\prime}]=0\ ,$
$\displaystyle(i^{\prime})$ $\displaystyle
a_{-}c_{+}\left(CB^{\prime}-C^{\prime}B\right)+a_{-}a_{+}[D,D^{\prime}]=0\ ,$
$\displaystyle(ii)$ $\displaystyle
b_{-}b_{+}[A,D^{\prime}]+c_{-}c_{+}[D,D^{\prime}]+\
c_{-}a_{+}\big{(}CB^{\prime}-C^{\prime}B\big{)}=0\ ,$
$\displaystyle(ii^{\prime})$ $\displaystyle
b_{-}b_{+}[D,A^{\prime}]+c_{-}c_{+}[A,A^{\prime}]+\
c_{-}a_{+}\big{(}BC^{\prime}-B^{\prime}C\big{)}=0\ ,$ $\displaystyle(iii)$
$\displaystyle
c_{-}b_{+}\big{(}DA^{\prime}-D^{\prime}A\big{)}+b_{-}c_{+}\big{(}AA^{\prime}-D^{\prime}D\big{)}+\
b_{-}a_{+}[B,C^{\prime}]=0\ ,$ $\displaystyle(iii^{\prime})$ $\displaystyle
c_{-}b_{+}\big{(}AD^{\prime}-A^{\prime}D\big{)}+b_{-}c_{+}\big{(}DD^{\prime}-A^{\prime}A\big{)}+\
b_{-}a_{+}[C,B^{\prime}]=0\ ,$ $\displaystyle(iv)$ $\displaystyle
b_{-}b_{+}AC^{\prime}+c_{-}c_{+}DC^{\prime}+\ c_{-}a_{+}CA^{\prime}-\
a_{-}a_{+}C^{\prime}A-\ a_{-}c_{+}D^{\prime}C=0\ ,$ $\displaystyle(v)$
$\displaystyle b_{-}b_{+}B^{\prime}A+\ c_{-}c_{+}B^{\prime}D+\
c_{-}a_{+}A^{\prime}B-\ a_{-}a_{+}AB^{\prime}-\ a_{-}c_{+}BD^{\prime}=0\ ,$
$\displaystyle(vi)$ $\displaystyle b_{-}b_{+}C^{\prime}D+\
c_{-}c_{+}C^{\prime}A+\ c_{-}a_{+}D^{\prime}C-\ a_{-}a_{+}DC^{\prime}-\
a_{-}c_{+}CA^{\prime}=0\ ,$ $\displaystyle(vii)$ $\displaystyle
b_{-}b_{+}DB^{\prime}+\ c_{-}c_{+}AB^{\prime}+\ c_{-}a_{+}BD^{\prime}-\
a_{-}a_{+}B^{\prime}D-\ a_{-}c_{+}A^{\prime}B=0\ ,$ $\displaystyle(viii)$
$\displaystyle b_{-}a_{+}BD^{\prime}+\ c_{-}b_{+}DB^{\prime}+\
b_{-}c_{+}AB^{\prime}-\ a_{-}b_{+}D^{\prime}B=0\ ,$ $\displaystyle(ix)$
$\displaystyle b_{-}a_{+}A^{\prime}B+\ c_{-}b_{+}B^{\prime}A+\
b_{-}c_{+}B^{\prime}D-\ a_{-}b_{+}BA^{\prime}=0\ ,$ $\displaystyle(x)$
$\displaystyle b_{-}a_{+}D^{\prime}C+\ c_{-}b_{+}C^{\prime}D+\
b_{-}c_{+}C^{\prime}A-\ a_{-}b_{+}CD^{\prime}=0\ ,$ $\displaystyle(xi)$
$\displaystyle b_{-}a_{+}CA^{\prime}+\ c_{-}b_{+}AC^{\prime}+\
b_{-}c_{+}DC^{\prime}-\ a_{-}b_{+}A^{\prime}C=0\ ,$ $\displaystyle(xii)$
$\displaystyle a_{-}b_{+}[B,B^{\prime}]=0\ ,$ $\displaystyle(xiii)$
$\displaystyle a_{-}b_{+}[C,C^{\prime}]=0\ ,$
where $a(u)=uq-u^{-1}q^{-1}$, $b(u)=u-u^{-1}$, $c_{\pm}=q-q^{-1}$ and we used
the shorthand notations $a_{-}=a(u/v)$, $a_{+}=a(uv)$ and similarly for $b$.
Also $A=A(u)$, $A^{\prime}=A(v)$ and similarly for $B,C$ and $D$. Now,
consider another set, say $\\{A,B,C,{\overline{D}}\\}$,
${\overline{D}(u)}=D(u)+f(u)$ where $f(u)$ is an unknown function of $u$ and
the elements of the reflection equation algebra. If
$\\{A,B,C,{\overline{D}}\\}$ is also a solution of the reflection equation
algebra, then $f(u)\equiv f(A,B,C,D;u)$ \- the equations $(i)-(xiii)$ being
the complete set of defining relations. Replacing ${\overline{D}(u)}$ in
$(iv)-(xi)$, we obtain $B(u)f(v)=f(u)B(v)=C(u)f(v)=f(u)C(v)=0$ $\forall u,v$.
On the other hand, from $(i)-(iii^{\prime})$ one gets
$\big{[}A(u),f(v)\big{]}=0$. Acting with the l.h.s of $(ix)$ on $f(w)$ and
using previous equations it follows $\big{[}D(u),f(w)\big{]}=0$ $\forall u,w$.
All these equations imply that $f(u)\equiv 0$ $\forall u$. ∎
The next step is to prove the equivalence between the (sixteen in total)
independent equations coming from the reflection equation algebra (2.7) with
(2.5) and a closed system of commutation relations among the currents. The
relations below are among the main results of the paper.
###### Definition 2.2 (Current algebra).
$O_{q}(\widehat{sl_{2}})$ is an associative algebra with unit $1$, current
generators $\cal W_{\pm}(u)$, $\cal G_{\pm}(u)$ and parameter
$\rho\in{\mathbb{C}}^{*}$. Define the formal variables
$U=(qu^{2}+q^{-1}u^{-2})/(q+q^{-1})$ and $V=(qv^{2}+q^{-1}v^{-2})/(q+q^{-1})$
$\forall u,v$. The defining relations are:
(2.16) $\displaystyle\big{[}{\cal W}_{\pm}(u),{\cal W}_{\pm}(v)\big{]}=0\
,\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (2.17) $\displaystyle\big{[}{\cal
W}_{+}(u),{\cal W}_{-}(v)\big{]}+\big{[}{\cal W}_{-}(u),{\cal
W}_{+}(v)\big{]}=0\ ,\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (2.18)
$\displaystyle(U-V)\big{[}{\cal W}_{\pm}(u),{\cal
W}_{\mp}(v)\big{]}=\frac{(q-q^{-1})}{\rho(q+q^{-1})}\left({\cal
G}_{\pm}(u){\cal G}_{\mp}(v)-{\cal G}_{\pm}(v){\cal
G}_{\mp}(u)\right)\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1}{(q+q^{-1})}\big{(}{\cal
G}_{\pm}(u)-{\cal G}_{\mp}(u)+{\cal G}_{\mp}(v)-{\cal G}_{\pm}(v)\big{)}\ ,$
(2.19) $\displaystyle{\cal W}_{\pm}(u){\cal W}_{\pm}(v)-{\cal W}_{\mp}(u){\cal
W}_{\mp}(v)+\frac{1}{\rho(q^{2}-q^{-2})}\big{[}{\cal G}_{\pm}(u),{\cal
G}_{\mp}(v)\big{]}\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\
\frac{1-UV}{U-V}\big{(}{\cal W}_{\pm}(u){\cal W}_{\mp}(v)-{\cal
W}_{\pm}(v){\cal W}_{\mp}(u)\big{)}=0\ ,$ (2.20) $\displaystyle U\big{[}{\cal
G}_{\mp}(v),{\cal W}_{\pm}(u)\big{]}_{q}-V\big{[}{\cal G}_{\mp}(u),{\cal
W}_{\pm}(v)\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{\mp}(u){\cal
G}_{\mp}(v)-{\cal W}_{\mp}(v){\cal G}_{\mp}(u)\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\
\rho\big{(}U{\cal W}_{\pm}(u)-V{\cal W}_{\pm}(v)-{\cal W}_{\mp}(u)+{\cal
W}_{\mp}(v)\big{)}=0\ ,$ (2.21) $\displaystyle U\big{[}{\cal W}_{\mp}(u),{\cal
G}_{\mp}(v)\big{]}_{q}-V\big{[}{\cal W}_{\mp}(v),{\cal
G}_{\mp}(u)\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{\pm}(u){\cal
G}_{\mp}(v)-{\cal W}_{\pm}(v){\cal G}_{\mp}(u)\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\
\rho\big{(}U{\cal W}_{\mp}(u)-V{\cal W}_{\mp}(v)-{\cal W}_{\pm}(u)+{\cal
W}_{\pm}(v)\big{)}=0\ ,$ (2.22) $\displaystyle\big{[}{\cal
G}_{\epsilon}(u),{\cal W}_{\pm}(v)\big{]}+\big{[}{\cal W}_{\pm}(u),{\cal
G}_{\epsilon}(v)\big{]}=0\ ,\quad\forall\epsilon=\pm\
,\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (2.23) $\displaystyle\big{[}{\cal
G}_{\pm}(u),{\cal G}_{\pm}(v)\big{]}=0\
,\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (2.24) $\displaystyle\big{[}{\cal
G}_{+}(u),{\cal G}_{-}(v)\big{]}+\big{[}{\cal G}_{-}(u),{\cal
G}_{+}(v)\big{]}=0\ .\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ $
###### Remark 1.
There exists an automorphism $\Omega$ defined by:
(2.25) $\displaystyle\Omega(\cal W_{\pm}(u))=\cal W_{\mp}(u)\
,\qquad\Omega(\cal G_{\pm}(u))=\cal G_{\mp}(u)\ .$
Contrary to all known examples of Drinfeld currents associated with quantum
affine Lie algebras or superalgebras, it is important to notice that the
variables $u,v$ only arise through the symmetric $(qx^{2}\leftrightarrow
q^{-1}x^{-2},\ \forall x\in u,v$) combinations $U,V$, respectively. In view of
the connections with algebraic structures that appear in boundary quantum
integrable models [Bas, Bas2], such a fact is not surprising although not
obvious from (2.7). We now turn to the derivation of all equations above.
###### Theorem 1.
The map $\Phi:B_{q}(\widehat{sl_{2}})\mapsto O_{q}(\widehat{sl_{2}})$ defined
by (2.12-2.15) is an algebra isomorphism.
###### Proof.
First, according to Lemma 2.1 we have to show that the map $\Phi$ defined by
(2.12-2.15) is an algebra homomorphism from $B_{q}(\widehat{sl_{2}})$ to
$O_{q}(\widehat{sl_{2}})$. Set $\rho\equiv k_{+}k_{-}(q+q^{-1})^{2}$ and
define
$\displaystyle\qquad\qquad X_{1}\equiv\big{[}{\cal W}_{+}(u),{\cal
W}_{+}(v)\big{]}\ ,\qquad X_{2}\equiv\big{[}{\cal W}_{-}(u),{\cal
W}_{-}(v)\big{]}\ ,$ $\displaystyle\qquad\qquad X_{3}\equiv\big{[}{\cal
W}_{+}(u),{\cal W}_{-}(v)\big{]}+\big{[}{\cal W}_{-}(u),{\cal
W}_{+}(v)\big{]}\ ,$ $\displaystyle\qquad\qquad X_{4}\equiv\big{[}{\cal
G}_{+}(u),{\cal G}_{-}(v)\big{]}+\big{[}{\cal G}_{-}(u),{\cal
G}_{+}(v)\big{]}\ ,$ $\displaystyle\qquad\qquad
X_{5}\equiv(q+q^{-1})(U-V)\big{[}{\cal W}_{+}(u),{\cal
W}_{-}(v)\big{]}-\frac{(q-q^{-1})}{\rho}\left({\cal G}_{+}(u){\cal
G}_{-}(v)-{\cal G}_{+}(v){\cal G}_{-}(u)\right)\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\big{(}{\cal
G}_{+}(u)-{\cal G}_{-}(u)+{\cal G}_{-}(v)-{\cal G}_{+}(v)\big{)}\ ,$
where the variables $U\equiv(qu^{2}+q^{-1}u^{-2})/(q+q^{-1})$ and similarly
for $V$ are introduced. In terms of the combinations $X_{i}$, it is
straightforward to show that the equations
$(i),(i^{\prime}),(ii),(ii^{\prime})$ above can be simply written,
respectively, as
$\displaystyle(i)$ $\displaystyle\Leftrightarrow\quad
a(uv)uvq^{2}X_{1}+a(uv)u^{-1}v^{-1}q^{-2}X_{2}-a(uv)u^{-1}vX_{3}-X_{5}=0\ ,$
$\displaystyle(i^{\prime})$ $\displaystyle\Leftrightarrow\quad
a(uv)uvq^{2}X_{2}+a(uv)u^{-1}v^{-1}q^{-2}X_{1}-a(uv)uv^{-1}X_{3}+\frac{q-q^{-1}}{\rho}X_{4}+X_{5}=0\
,$ $\displaystyle(ii)$
$\displaystyle\Leftrightarrow\quad\big{(}b(u/v)b(uv)uv^{-1}-(q-q^{-1})^{2}u^{-1}v^{-1}q^{-2}\big{)}X_{1}$
$\displaystyle\qquad+\big{(}b(u/v)b(uv)u^{-1}v-(q-q^{-1})^{2}uvq^{2}\big{)}X_{2}$
$\displaystyle\qquad-\big{(}b(u/v)b(uv)u^{-1}v^{-1}q^{-2}-(q-q^{-1})^{2}uv^{-1}\big{)}X_{3}$
$\displaystyle\qquad-a(uv)\frac{(q-q^{-1})}{\rho}X_{4}-a(uv)X_{5}=0\ ,$
$\displaystyle(ii^{\prime})$
$\displaystyle\Leftrightarrow\quad\big{(}b(u/v)b(uv)u^{-1}v-(q-q^{-1})^{2}uvq^{2}\big{)}X_{1}$
$\displaystyle\qquad+\big{(}b(u/v)b(uv)uv^{-1}-(q-q^{-1})^{2}u^{-1}v^{-1}q^{-2}\big{)}X_{2}$
$\displaystyle\qquad-\big{(}b(u/v)b(uv)uvq^{2}-(q-q^{-1})^{2}u^{-1}v\big{)}X_{3}$
$\displaystyle\qquad-a(uv)X_{5}=0\ .$
Simplifying these expressions, in particular it follows
$\displaystyle a(uv)(i)-(ii^{\prime})$ $\displaystyle\Leftrightarrow\quad
v^{2}q^{2}X_{1}+v^{-2}q^{-2}X_{2}-X_{3}=0\ ,$ $\displaystyle
a(uv)(i^{\prime})-(ii)$ $\displaystyle\Leftrightarrow\quad
v^{2}q^{2}X_{2}+v^{-2}q^{-2}X_{1}-X_{3}=0\ .$
Considering both equations for $v$ arbitrary, it implies $X_{1}=X_{2}$. Then
it is important to notice that the combinations $X_{i}|_{u\leftrightarrow
v}=-X_{i}$ for $i=1,2,3$. As now $X_{3}=(v^{2}q^{2}+v^{-2}q^{-2})X_{1}$ and
$u$ is arbitrary, it immediately follows $X_{3}\equiv X_{1}\equiv X_{2}\equiv
0$. Plugged into $(ii)$, $(ii^{\prime})$ we obtain $X_{4}\equiv X_{5}\equiv
0$. In terms of the currents, these equalities lead to the commutation
relations (2.16), (2.17), (2.18), (2.24).
As a consequence of these relations, after some straightforward calculations
one finds that the equations $(iii),(iii^{\prime})$ drastically simplify into
the relations (2.19).
Let us now consider the equations $(iv),(vi),(x),(xi)$ above. Proceeding
similarly, let us introduce
$\displaystyle Y_{1}\equiv(q+q^{-1})\big{(}U\big{[}C(v),{\cal
W}_{+}(u)\big{]}_{q}-V\big{[}C(u),{\cal
W}_{+}(v)\big{]}_{q}+(q-q^{-1})\big{(}{\cal W}_{-}(v)C(u)-{\cal
W}_{-}(u)C(v)\big{)}\big{)}\ ,$ $\displaystyle
Y_{2}\equiv(q+q^{-1})\big{(}U\big{[}{\cal
W}_{-}(u),C(v)\big{]}_{q}-V\big{[}{\cal
W}_{-}(v),C(u)\big{]}_{q}+(q-q^{-1})\big{(}{\cal W}_{+}(v)C(u)-{\cal
W}_{+}(u)C(v)\big{)}\big{)}\ ,$ $\displaystyle Y_{3}\equiv\big{[}C(u),{\cal
W}_{+}(v)\big{]}+\big{[}{\cal W}_{+}(u),C(v)\big{]}\ ,$ $\displaystyle
Y_{4}\equiv\big{[}C(u),{\cal W}_{-}(v)\big{]}+\big{[}{\cal
W}_{-}(u),C(v)\big{]}\ .$
In terms of these combinations, the equations $(iv),(vi),(x),(xi)$ become,
respectively,
$\displaystyle(iv)$ $\displaystyle\Leftrightarrow\quad
u\big{(}qY_{1}+q(v^{2}+v^{-2})Y_{3}+(q-q^{-1})Y_{4})\big{)}$
$\displaystyle\qquad\quad+\
u^{-1}\big{(}q^{-1}Y_{2}-q^{-1}(v^{2}+v^{-2})Y_{4}+(q-q^{-1})Y_{3})\big{)}=0\
,$ $\displaystyle(vi)$ $\displaystyle\Leftrightarrow\quad
u\big{(}qY_{2}-q(v^{2}+v^{-2})Y_{4}+q^{2}(q-q^{-1})Y_{3})\big{)}$
$\displaystyle\qquad\quad+\
u^{-1}\big{(}q^{-1}Y_{1}+q^{-1}(v^{2}+v^{-2})Y_{3}+q^{-2}(q-q^{-1})Y_{4})\big{)}=0\
,$ $\displaystyle(x)$ $\displaystyle\Leftrightarrow\quad
v\big{(}Y_{2}-q(q+q^{-1})UY_{4}+(q^{2}-q^{-2})Y_{3})\big{)}$
$\displaystyle\qquad\quad+\
v^{-1}\big{(}Y_{1}+q^{-1}(q+q^{-1})UY_{3}+(q^{2}-q^{-2})Y_{4})\big{)}=0\ ,$
$\displaystyle(xi)$ $\displaystyle\Leftrightarrow\quad
v\big{(}Y_{1}+q(q+q^{-1})UY_{3})\big{)}$ $\displaystyle\qquad\quad+\
v^{-1}\big{(}Y_{2}-q^{-1}(q+q^{-1})UY_{4})\big{)}=0\ .$
The variables $u,v$ and deformation parameter $q$ being arbitrary,
compatibility of these equations implies $Y_{1}\equiv Y_{2}\equiv Y_{3}\equiv
Y_{4}\equiv 0$. Replacing the explicit expression of $C(u)$ into $Y_{i}$, one
ends up with the commutation relations (2.20), (2.21), (2.22) for the current
${\cal G}_{-}(u)$. Similar analysis for the remaining equations
$(v),(vii),(viii),(ix)$ imply (2.20), (2.21), (2.22) for ${\cal G}_{+}(u)$.
Finally, from $(xii),(xiii)$ we immediately obtain (2.23).
Surjectivity of the map being shown, the injectivity of the homomorphism
follows from the fact that $\Phi$ is invertible for $u$ generic. This
completes the proof. ∎
Quantum affine algebras are known to be Hopf algebras, thanks to the existence
of a coproduct, counit and antipode actions. Although the explicit Hopf
algebra isomorphism between Drinfeld’s new realization (currents) and
Drinfeld-Jimbo construction is still an open problem, several results are
already known (see for instance [DiF]). For the new current algebra
(2.16)-(2.24), it is also important to exhibit analogous properties. Actually,
solely using the results of [Sk] a coaction map [Cha] can be easily
identified.
###### Proposition 2.2.
For any $k_{\pm},w\in{\mathbb{C}}^{*}$, there exists an algebra homomorphism
$\delta_{w}:O_{q}(\widehat{sl_{2}})\mapsto U_{q}(sl_{2})\times
O_{q}(\widehat{sl_{2}})$ such that
$\displaystyle\delta_{w}(\cal W_{\pm}(u))\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left((q-q^{-1})^{2}S_{\pm}S_{\mp}-q(q^{\pm
2s_{3}}-q^{\mp 2s_{3}})\right)\otimes{\cal
W}_{\mp}(u)-(w^{2}+w^{-2})I\\!\\!I\otimes{\cal W}_{\pm}(u)$
$\displaystyle\\!\\!\\!\\!+\frac{(q-q^{-1})}{k_{+}k_{-}(q+q^{-1})}\left(k_{+}w^{\pm
1}q^{\pm 1/2}S_{+}q^{\pm s_{3}}\otimes{\cal G}_{+}(u)+k_{-}w^{\mp 1}q^{\mp
1/2}S_{-}q^{\pm s_{3}}\otimes{\cal G}_{-}(u)\right)$
$\displaystyle\\!\\!\\!\\!+(q+q^{-1})\left((k_{+}w^{\pm 1}q^{\pm
1/2}S_{+}q^{\pm s_{3}}+k_{-}w^{\mp 1}q^{\mp 1/2}S_{-}q^{\pm s_{3}})\otimes
I\\!\\!I+q^{\pm 2s_{3}}\otimes U{\cal W}_{\pm}(u)\right),$
$\displaystyle\delta_{w}(\cal G_{\pm}(u))$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{k_{\mp}}{k_{\pm}}(q-q^{-1})^{2}S_{\mp}^{2}\otimes{\cal
G}_{\mp}(u)-(w^{2}q^{\pm 2s_{3}}+w^{-2}q^{\mp 2s_{3}})\otimes{\cal
G}_{\pm}(u)+I\\!\\!I\otimes(q+q^{-1})U{\cal G}_{\pm}(u)$ $\displaystyle\ \ +\
(q+q^{-1})^{2}(q-q^{-1})\left(k_{\mp}w^{\pm 1}q^{\mp
1/2}S_{\mp}q^{s_{3}}\otimes(U{\cal W}_{+}(u)-{\cal W}_{-}(u))\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+\left.k_{\mp}w^{\mp 1}q^{\pm
1/2}S_{\mp}q^{-s_{3}}\otimes(U{\cal W}_{-}(u)-{\cal W}_{+}(u))\right)$
$\displaystyle\ \ +\
\frac{k_{+}k_{-}(q+q^{-1})^{2}}{(q-q^{-1})}\left((q+q^{-1})U+\frac{k_{\mp}}{k_{\pm}}(q-q^{-1})^{2}S_{\mp}^{2}-(w^{2}q^{\pm
2s_{3}}+w^{-2}q^{\mp 2s_{3}}+1)\right)\otimes I\\!\\!I\ .$
###### Proof.
According to [Sk] (see Proposition 2.1) and the Lax operator (2.11),
$L(uw)K(u)L(uw^{-1})$ is a solution $\forall w$ of (2.7). Expanding this
expression using (2.12)-(2.15), the new entries of $L(uw)K(u)L(uw^{-1})$ are
found to take the form (2.12)-(2.15) replacing $\cal
W_{\pm}(u)\rightarrow\delta_{w}(\cal W_{\pm}(u))$, $\cal
G_{\pm}(u)\rightarrow\delta_{w}(\cal G_{\pm}(u))$. For more details, we refer
the reader to [BasK] where similar calculations have been performed. ∎
## 3\. Another presentation
In [BasK], an infinite dimensional algebra denoted below ${\cal A}_{q}$ was
proposed in order to solve boundary integrable systems with hidden symmetries
related with a coideal subalgebra of $U_{q}(\widehat{sl_{2}})$. However, its
defining relations were essentially conjectured based on the commutation
relations and properties of certain operators acting on irreducible finite
dimensional tensor product of evaluation representations. The aim of this
Section is to construct an analogue of Drinfeld’s presentation for the current
algebra (2.16)-(2.24). As a consequence, it provides a rigorous derivation of
the relations conjectured in [BasK].
###### Definition 3.1 ([BasK]).
${\cal A}_{q}$ is an associative algebra with parameter
$\rho\in{\mathbb{C}}^{*}$, unit $1$ and generators $\\{{\cal W}_{-k},{\cal
W}_{k+1},{\cal G}_{k+1},{\tilde{\cal G}}_{k+1}|k\in{\mathbb{Z}}_{+}\\}$
satisfying the following relations:
(3.1) $\displaystyle\big{[}{\cal W}_{0},{\cal W}_{k+1}\big{]}=\big{[}{\cal
W}_{-k},{\cal W}_{1}\big{]}=\frac{1}{(q+q^{-1})}\big{(}{\tilde{\cal
G}_{k+1}}-{{\cal G}_{k+1}}\big{)}\ ,$ (3.2) $\displaystyle\big{[}{\cal
W}_{0},{\cal G}_{k+1}\big{]}_{q}=\big{[}{\tilde{\cal G}}_{k+1},{\cal
W}_{0}\big{]}_{q}=\rho{\cal W}_{-k-1}-\rho{\cal W}_{k+1}\ ,$ (3.3)
$\displaystyle\big{[}{\cal G}_{k+1},{\cal W}_{1}\big{]}_{q}=\big{[}{\cal
W}_{1},{\tilde{\cal G}}_{k+1}\big{]}_{q}=\rho{\cal W}_{k+2}-\rho{\cal W}_{-k}\
,$ (3.4) $\displaystyle\big{[}{\cal W}_{-k},{\cal W}_{-l}\big{]}=0\
,\quad\big{[}{\cal W}_{k+1},{\cal W}_{l+1}\big{]}=0\ ,\quad$ (3.5)
$\displaystyle\big{[}{\cal W}_{-k},{\cal W}_{l+1}\big{]}+\big{[}{{\cal
W}}_{k+1},{\cal W}_{-l}\big{]}=0\ ,$ (3.6) $\displaystyle\big{[}{\cal
W}_{-k},{\cal G}_{l+1}\big{]}+\big{[}{{\cal G}}_{k+1},{\cal W}_{-l}\big{]}=0\
,$ (3.7) $\displaystyle\big{[}{\cal W}_{-k},{\tilde{\cal
G}}_{l+1}\big{]}+\big{[}{\tilde{\cal G}}_{k+1},{\cal W}_{-l}\big{]}=0\ ,$
(3.8) $\displaystyle\big{[}{\cal W}_{k+1},{\cal G}_{l+1}\big{]}+\big{[}{{\cal
G}}_{k+1},{\cal W}_{l+1}\big{]}=0\ ,$ (3.9) $\displaystyle\big{[}{\cal
W}_{k+1},{\tilde{\cal G}}_{l+1}\big{]}+\big{[}{\tilde{\cal G}}_{k+1},{\cal
W}_{l+1}\big{]}=0\ ,$ (3.10) $\displaystyle\big{[}{\cal G}_{k+1},{\cal
G}_{l+1}\big{]}=0\ ,\quad\big{[}{\tilde{\cal G}}_{k+1},\tilde{{\cal
G}}_{l+1}\big{]}=0\ ,$ (3.11) $\displaystyle\big{[}{\tilde{\cal
G}}_{k+1},{\cal G}_{l+1}\big{]}+\big{[}{{\cal G}}_{k+1},\tilde{{\cal
G}}_{l+1}\big{]}=0\ .$
A natural ordering of ${\cal A}_{q}$ arises from the study of the commutation
relations above. Indeed, starting from monomials of lowest $k=0,1,...$ and
using (3.1) possible definitions of ${\cal G}_{1},{\tilde{\cal G}}_{1}$ are
such that $\mathrm{d}[{\cal G}_{1}]=\mathrm{d}[{\tilde{\cal G}}_{1}]\leq 2$,
where $\mathrm{d}$ denotes the degree of the monomials in the elements ${\cal
W}_{0},{\cal W}_{1}$. By induction, from (3.2), (3.3) with (3.1) one
immediately gets:
###### Corollary 3.1.
The elements of ${\cal A}_{q}$ are monomials in ${\cal W}_{0},{\cal W}_{1}$ of
degree:
(3.12) $\displaystyle\qquad\qquad\mathrm{d}[{\cal W}_{-k}]=\mathrm{d}[{{\cal
W}}_{k+1}]\leq 2k+1\qquad\mbox{and}\qquad\mathrm{d}[{\cal
G}_{k+1}]=\mathrm{d}[{\tilde{\cal G}}_{k+1}]\leq 2k+2\ ,\qquad
k\in{\mathbb{Z}}_{+}.$
Note that writing explicitly all higher elements of ${\cal A}_{q}$ in terms of
${\cal W}_{0},{\cal W}_{1}$ is essentially related with the construction of a
Poincare-Birkoff-Witt basis for the algebra considered in the next Section, a
problem that will be considered elsewhere.
###### Remark 2.
According to the ordering (3.12), the elements ${\cal G}_{1},{\tilde{\cal
G}}_{1}\in{\cal A}_{q}$ are uniquely determined:
(3.13) $\displaystyle{\cal G}_{1}=\big{[}{\cal W}_{1},{\cal
W}_{0}\big{]}_{q}+\alpha\qquad\mbox{and}\qquad{\tilde{\cal
G}}_{1}=\big{[}{\cal W}_{0},{\cal W}_{1}\big{]}_{q}+\alpha\qquad\forall\alpha\
\in{\mathbb{C}}\ .$
For the derivation of the second theorem, several other equalities will be
required which can all be deduced from the relations above and (3.13). Indeed,
let us show the following.
###### Proposition 3.1.
If (3.1)-(3.11) are satisfied, then the following relations hold:
(3.14) $\displaystyle\qquad\qquad\quad\big{[}{\cal W}_{-k-1},{\cal
W}_{l+1}\big{]}-\big{[}{\cal W}_{-k},{\cal
W}_{l+2}\big{]}=\frac{q-q^{-1}}{\rho(q+q^{-1})}\big{(}{\cal
G}_{k+1}\tilde{{\cal G}}_{l+1}-{\cal G}_{l+1}\tilde{{\cal G}}_{k+1}\big{)}\ ,$
(3.15) $\displaystyle\qquad\qquad\quad-{\cal W}_{-k}{\cal W}_{0}+{\cal
W}_{k+1}{\cal W}_{1}-{\cal W}_{-k-1}{\cal W}_{1}+{\cal W}_{0}{\cal
W}_{k+2}-\frac{1}{\rho(q^{2}-q^{-2})}\big{[}{\cal G}_{k+1},{\tilde{\cal
G}}_{1}\big{]}=0\ ,$ (3.16) $\displaystyle\qquad\qquad\quad{\cal
W}_{-k-1}{\cal W}_{-l}-{\cal W}_{k+2}{\cal W}_{l+1}-{\cal W}_{-k}{\cal
W}_{-l-1}+{\cal W}_{k+1}{\cal W}_{l+2}$ $\displaystyle\qquad\qquad\quad+{\cal
W}_{-k}{\cal W}_{l+1}-{\cal W}_{-l}{\cal W}_{k+1}-{\cal W}_{-k-1}{\cal
W}_{l+2}+{\cal W}_{-l-1}{\cal W}_{k+2}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\frac{1}{\rho(q^{2}-q^{-2})}\big{(}\big{[}{\cal
G}_{k+2},{\tilde{\cal G}}_{l+1}\big{]}-\big{[}{\cal G}_{k+1},{\tilde{\cal
G}}_{l+2}\big{]}\big{)}=0\ ,$ (3.17)
$\displaystyle\qquad\qquad\quad\big{[}{\cal G}_{l+1},{\cal
W}_{k+2}\big{]}_{q}-\big{[}{\cal G}_{k+1},{\cal
W}_{l+2}\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{-k}{\cal G}_{l+1}-{\cal
W}_{-l}{\cal G}_{k+1}\big{)}=0\ ,$ (3.18)
$\displaystyle\qquad\qquad\quad\big{[}{\cal W}_{-k-1},{\cal
G}_{l+1}\big{]}_{q}-\big{[}{\cal W}_{-l-1},{\cal
G}_{k+1}\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{k+1}{\cal G}_{l+1}-{\cal
W}_{l+1}{\cal G}_{k+1}\big{)}=0\ ,$ (3.19)
$\displaystyle\qquad\qquad\quad\big{[}{\tilde{\cal G}}_{l+1},{\cal
W}_{-k-1}\big{]}_{q}-\big{[}{\tilde{\cal G}_{k+1}},{\cal
W}_{-l-1}\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{k+1}{\tilde{\cal
G}}_{l+1}-{\cal W}_{l+1}{\tilde{\cal G}_{k+1}}\big{)}=0\ ,$ (3.20)
$\displaystyle\qquad\qquad\quad\big{[}{\cal W}_{k+2},{\tilde{\cal
G}}_{l+1}\big{]}_{q}-\big{[}{\cal W}_{l+2},{\tilde{\cal
G}}_{k+1}\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{-k}{\tilde{\cal
G}}_{l+1}-{\cal W}_{-l}{\tilde{\cal G}}_{k+1}\big{)}=0\ .$
###### Proof.
To show (3.14), let us consider the first commutator. Expand it using (3.2).
Combining ${\cal W}_{0}$ and ${\cal W}_{l+1}$ using (3.1), one finds:
$\displaystyle\big{[}{\cal W}_{-k-1},{\cal W}_{l+1}\big{]}$ $\displaystyle=$
$\displaystyle\frac{q}{\rho(q+q^{-1})}\big{(}\tilde{{\cal G}}_{l+1}{\cal
G}_{k+1}-\tilde{{\cal G}}_{k+1}{\cal G}_{l+1}\big{)}$
$\displaystyle+\frac{q^{-1}}{\rho(q+q^{-1})}\big{(}{\cal G}_{l+1}\tilde{{\cal
G}}_{k+1}-{\cal G}_{k+1}\tilde{{\cal G}}_{l+1}\big{)}+\big{[}{\cal
W}_{-l-1},{\cal W}_{k+1}\big{]}\ .$
Then, using (3.5) and (3.11) one obtains (3.14).
Consider now (3.15). Introduce (3.13) in the last commutator, and expand using
(3.2) and (3.3). Collecting terms and simplifying, one obtains (3.15).
Equation (3.16), although technically slightly more complicated, is derived
along the same line.
To show (3.17)-(3.20), the same procedure will be used so we only explain
(3.17). Consider the two commutators and expand using (3.3). Then, using (3.8)
and (3.11), one verifies that (3.17) is indeed satisfied. ∎
By analogy with Drinfeld’s construction, we are now looking for an infinite
dimensional set of elements of an algebra in terms of which the currents $\cal
W_{\pm}(u)$, $\cal G_{\pm}(u)$ can be expanded. According to the structure of
the equations (2.16)-(2.24) defining the current algebra - in particular the
dependence in the formal variable $U,V$ \- we obtain the second main result of
the paper.
###### Theorem 2.
Define the formal variable $U=(qu^{2}+q^{-1}u^{-2})/(q+q^{-1})$. Let
$\Psi:O_{q}(\widehat{sl_{2}})\mapsto{\cal A}_{q}$ be the map defined by
(3.21) $\displaystyle{\cal W}_{+}(u)=\sum_{k\in{\mathbb{Z}}_{+}}{\cal
W}_{-k}U^{-k-1}\ ,\quad{\cal W}_{-}(u)=\sum_{k\in{\mathbb{Z}}_{+}}{\cal
W}_{k+1}U^{-k-1}\ ,$ (3.22) $\displaystyle\quad{\cal
G}_{+}(u)=\sum_{k\in{\mathbb{Z}}_{+}}{\cal G}_{k+1}U^{-k-1}\ ,\quad{\cal
G}_{-}(u)=\sum_{k\in{\mathbb{Z}}_{+}}\tilde{{\cal G}}_{k+1}U^{-k-1}\ .$
Then, $\Psi$ is an algebra isomorphism between $O_{q}(\widehat{sl_{2}})$ and
${\cal A}_{q}$.
###### Proof.
Plugging (3.21), (3.22) into (2.16)-(2.24), expanding and identifying terms of
same order in $U^{-k}V^{-l}$ one finds all defining relations (3.1)-(3.11),
together with the set of higher relations (3.14)-(3.20). From Proposition 3.1,
it follows that the sixteen independent algebraic relations (3.1)-(3.11) are
sufficient i.e. the map is surjective. The currents being analytic in the
variable $U\in{\mathbb{C}}$, according to Cauchy’s theorem any element of
${\cal A}_{q}$ is uniquely determined from the currents using contour
integrals. The injectivity of the map follows, which completes the proof. ∎
It is important to stress that in [BasK], commutation relations among the so-
called transfer matrix were used to derive some of the relations (3.1)-(3.11).
However, the derivation described above uses solely the reflection equation
algebra. Consequently, this theorem establishes a rigorous proof of the
relations conjectured in [BasK]. In addition, for the case of the reflection
equation algebra with the $U_{q}(\widehat{sl_{2}})$ $R$-matrix it shows that
the presentation $\\{{\cal W}_{-k},{\cal W}_{k+1},{\cal G}_{k+1},{\tilde{\cal
G}}_{k+1}|k\in{\mathbb{Z}}_{+}\\}$ is the “boundary” analogue of Drinfeld’s
one.
## 4\. intertwiner of the $q-$Onsager (tridiagonal) algebra
and the reflection equation
The purpose of this Section is to exhibit an intertwiner $K(u)$ of the
$q-$Onsager algebra, to show its uniqueness and that it coincides exactly with
the solution $K(u)$ of the reflection equation (2.7). The final aim is
actually to establish the isomorphism between the new current algebra and the
$q-$Onsager algebra. Although the reader may be familiar with the ideas of
[Jim], it will be useful to first recall some well-known results. Indeed, the
procedure we follow to construct the intertwiner is analogous to the one
described in [Jim]. In the context of quantum integrable systems, note that
for finite dimensional representations intertwiners have already been obtained
along the same line in [MN, DeMS, Nep, DeG, DeM].
a. The $R-$matrix as an intertwiner of $U_{q}(\widehat{sl_{2}})$ [Jim].
In [Jim], Jimbo pointed out that intertwiners $R$ of quantum loop algebras
lead to trigonometric solutions of the quantum Yang-Baxter equation (2.8). Any
tensor product of two evaluation representations with generic evaluation
parameters $u$ and $v$ being indecomposable, by Schur’s lemma the solution $R$
is unique up to an overall scalar factor. In particular, considering the
quantum affine algebra $U_{q}(\widehat{sl_{2}})$ the construction of the
solution $R(u)$ given by (2.5) goes as follow.
First, we need to recall the realization of the quantum affine algebra
$U_{q}(\widehat{sl_{2}})$ in the Chevalley presenation
$\\{H_{j},E_{j},F_{j}\\}$, $j\in\\{0,1\\}$ (see e.g [Cha]):
###### Definition 4.1.
Define the extended Cartan matrix $\\{a_{ij}\\}$ ($a_{ii}=2$, $a_{ij}=-2$ for
$i\neq j$). The quantum affine algebra $U_{q}(\widehat{sl_{2}})$ is generated
by the elements $\\{H_{j},E_{j},F_{j}\\}$, $j\in\\{0,1\\}$ which satisfy the
defining relations
$\displaystyle[H_{i},H_{j}]=0\ ,\quad[H_{i},E_{j}]=a_{ij}E_{j}\
,\quad[H_{i},F_{j}]=-a_{ij}F_{j}\
,\quad[E_{i},F_{j}]=\delta_{ij}\frac{q^{H_{i}}-q^{-H_{i}}}{q-q^{-1}}\ $
together with the $q-$Serre relations
(4.1) $\displaystyle[E_{i},[E_{i},[E_{i},E_{j}]_{q}]_{q^{-1}}]=0\
,\quad\mbox{and}\quad[F_{i},[F_{i},[F_{i},F_{j}]_{q}]_{q^{-1}}]=0\ .$
The sum ${\it K}=H_{0}+H_{1}$ is the central element of the algebra. The Hopf
algebra structure is ensured by the existence of a comultiplication
$\Delta:U_{q}(\widehat{sl_{2}})\mapsto U_{q}(\widehat{sl_{2}})\otimes
U_{q}(\widehat{sl_{2}})$, antipode ${\cal S}:U_{q}(\widehat{sl_{2}})\mapsto
U_{q}(\widehat{sl_{2}})$ and a counit ${\cal
E}:U_{q}(\widehat{sl_{2}})\mapsto{\mathbb{C}}$ with
$\displaystyle\Delta(E_{i})$ $\displaystyle=$ $\displaystyle E_{i}\otimes
q^{-H_{i}/2}+q^{H_{i}/2}\otimes E_{i}\ ,$ $\displaystyle\Delta(F_{i})$
$\displaystyle=$ $\displaystyle F_{i}\otimes q^{-H_{i}/2}+q^{H_{i}/2}\otimes
F_{i}\ ,$ (4.2) $\displaystyle\Delta(H_{i})$ $\displaystyle=$ $\displaystyle
H_{i}\otimes I\\!\\!I+I\\!\\!I\otimes H_{i}\ ,$ $\displaystyle{\cal
S}(E_{i})=-E_{i}q^{-H_{i}}\ ,\quad{\cal S}(F_{i})=-q^{H_{i}}F_{i}\ ,\quad{\cal
S}(H_{i})=-H_{i}\qquad{\cal S}({I\\!\\!I})=1\ $
and
$\displaystyle{\cal E}(E_{i})={\cal E}(F_{i})={\cal E}(H_{i})=0\ ,\qquad{\cal
E}({I\\!\\!I})=1\ .$
Note that the opposite coproduct $\Delta^{\prime}$ can be similarly defined
with $\Delta^{\prime}\equiv\sigma\circ\Delta$ where the permutation map
$\sigma(x\otimes y)=y\otimes x$ for all $x,y\in U_{q}(\widehat{sl_{2}})$ is
used.
Then, by definition the intertwiner $R(u/v):{\cal V}_{u}\otimes{\cal
V}_{v}\mapsto{\cal V}_{v}\otimes{\cal V}_{u}$ between two fundamental
$U_{q}(\widehat{sl_{2}})-$evaluation representations obeys
(4.3) $\displaystyle
R(u/v)(\pi_{u}\times\pi_{v})\big{[}\Delta(x)\big{]}=(\pi_{u}\times\pi_{v})\big{[}\Delta^{\prime}(x)\big{]}R(u/v)\qquad\forall
x\in U_{q}(\widehat{sl_{2}})\ ,$
where the (evaluation) endomorphism
$\pi_{u}:U_{q}(\widehat{sl_{2}})\mapsto\mathrm{End}({\cal V}_{u})$ is chosen
such that $({\cal V}\equiv{\mathbb{C}}^{2})$
$\displaystyle\pi_{u}[E_{1}]=uq^{1/2}\sigma_{+}\ ,\qquad\ \ \ \ \
\pi_{u}[E_{0}]=uq^{1/2}\sigma_{-}\ ,$
$\displaystyle\pi_{u}[F_{1}]=u^{-1}q^{-1/2}\sigma_{-}\
,\qquad\pi_{u}[F_{0}]=u^{-1}q^{-1/2}\sigma_{+}\ ,$ (4.4)
$\displaystyle\pi_{u}[q^{H_{1}}]=q^{\sigma_{3}}\ ,\qquad\qquad\quad\
\pi_{u}[q^{H_{0}}]=q^{-\sigma_{3}}\ $
in terms of the Pauli matrices $\sigma_{\pm},\sigma_{3}$:
(4.11) $\displaystyle\sigma_{+}=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)\ ,\qquad\sigma_{-}=\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right)\ ,\qquad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\ .$
As one can easily check, the matrix $R(u)$ given by (2.5) indeed satisfies the
required conditions (4.3). The tensor product ${\cal V}_{u}\otimes{\cal
V}_{v}$ being indecomposable with respect to $U_{q}(\widehat{sl_{2}})$
evaluation representations for generic evaluation parameters $u,v$, the
intertwiner $R(u)$ is unique (up to an overall scalar factor). As a
consequence, it automatically satisfies the Yang-Baxter equation (2.8) which
may be depicted by the following commutative diagram setting $w=1$:
(4.12) $\begin{CD}{\cal V}_{u}\otimes{\cal V}_{v}\otimes{\cal
V}_{w}@>{R(u/v)\,\otimes\,\text{id}}>{}>{\cal V}_{v}\otimes{\cal
V}_{u}\otimes{\cal V}_{w}@>{id\,\otimes\,R(u/w)}>{}>{\cal V}_{v}\otimes{\cal
V}_{w}\otimes{\cal V}_{u}\\\ @V{}V{\text{id}\otimes
R(v/w)}V@V{R(v/w)\otimes\text{id}}V{}V\\\ {\cal V}_{u}\otimes{\cal
V}_{w}\otimes{\cal V}_{v}@>{R(u/w)\,\otimes\,\text{id}}>{}>{\cal
V}_{w}\otimes{\cal V}_{u}\otimes{\cal V}_{v}@>{id\,\otimes\,R(u/v)}>{}>{\cal
V}_{w}\otimes{\cal V}_{v}\otimes{\cal V}_{u}\end{CD}$
b. The $K-$matrix as an intertwiner of ${\mathbb{T}}$.
Tridiagonal algebras have been introduced and studied in [Ter1, ITTer, Ter2],
where they first appeared in the context of $P-$ and $Q-$polynomial
association schemes. A tridiagonal algebra is an associative algebra with unit
which consists of two generators A and ${\textsf{A}}^{*}$ called the standard
generators. In general, the defining relations depend on five scalars
$\rho,\rho^{*},\gamma,\gamma^{*}$ and $\beta$. In the following, we will focus
on the reduced parameter sequence $\gamma=0,\gamma^{*}=0$,
$\beta=q^{2}+q^{-2}$ and $\rho=\rho^{*}$ which exhibits all interesting
properties that can be extended to more general parameter sequences. We call
the corresponding algebra the $q-$Onsager algebra denoted ${\mathbb{T}}$, in
view of its closed relationship with the Onsager algebra [Ons] and the Dolan-
Grady relations [DoG]. In particular, the isomorphism between the Onsager and
Dolan-Grady algebraic structures has been studied in [Pe, AMPT, Dav] and shown
explicitly in [DaRo].
###### Definition 4.2 (see also [Ter2]).
The $q-$Onsager algebra $\mathbb{T}$ is the associative algebra with unit and
standard generators $\textsf{A},\textsf{A}^{*}$ subject to the following
relations
(4.13)
$\displaystyle[\textsf{A},[\textsf{A},[\textsf{A},\textsf{A}^{*}]_{q}]_{q^{-1}}]=\rho[\textsf{A},\textsf{A}^{*}]\
,\qquad[\textsf{A}^{*},[\textsf{A}^{*},[\textsf{A}^{*},\textsf{A}]_{q}]_{q^{-1}}]=\rho[\textsf{A}^{*},\textsf{A}]\
.$
###### Remark 3.
For $\rho=0$ the relations (4.13) reduce to the $q-$Serre relations of
$U_{q}(\widehat{sl_{2}})$. For $q=1$, $\rho=16$ they coincide with the Dolan-
Grady relations [DoG].
By analogy with the construction described above for the $R-$matrix and along
the lines described in [DeM, DeG], an intertwiner for ${\mathbb{T}}$ can be
easily constructed. Before, we need to introduce the concept of comodule
algebra using the analogue of the Hopf’s algebra coproduct action called the
coaction map.
###### Definition 4.3 ([Cha]).
Given a Hopf algebra ${\cal H}$ with comultiplication $\Delta$ and counit
${\cal E}$, ${\cal I}$ is called a left ${\cal H}-$comodule if there exists a
left coaction map $\delta:\ \ {\cal I}\rightarrow{\cal H}\otimes{\cal I}$ such
that
(4.14) $\displaystyle(\Delta\times id)\circ\delta=(id\times\delta)\circ\delta\
,\qquad({\cal E}\times id)\circ\delta\cong id\ .$
Right ${\cal H}-$comodules are defined similarly.
###### Proposition 4.1 (see also [Bas]).
Let $k_{\pm}\in{\mathbb{C}}^{*}$ and set $\rho\equiv
k_{+}k_{-}(q+q^{-1})^{2}$. The q-Onsager algebra ${\mathbb{T}}$ is a left
$U_{q}(\widehat{sl_{2}})-$comodule algebra with coaction map
$\delta:{\mathbb{T}}\rightarrow U_{q}(\widehat{sl_{2}})\otimes{\mathbb{T}}$
such that
$\displaystyle\delta({\textsf{A}})$ $\displaystyle=$
$\displaystyle(k_{+}E_{1}q^{H_{1}/2}+k_{-}F_{1}q^{H_{1}/2})\otimes
1+q^{H_{1}}\otimes{\textsf{A}}\ ,$ (4.15)
$\displaystyle\delta({\textsf{A}}^{*})$ $\displaystyle=$
$\displaystyle(k_{-}E_{0}q^{H_{0}/2}+k_{+}F_{0}q^{H_{0}/2})\otimes
1+q^{H_{0}}\otimes{\textsf{A}}^{*}\ .$
###### Proof.
The verification of the comodule algebra axioms (4.14) is immediate using
(4.2). One also has to check that $\delta$ is an algebra homomorphism i.e
$\delta({\textsf{A}}),\delta({\textsf{A}}^{*})$ satisfy (4.13). This
calculation is rather long but straightforward, so we omit the details (see
also [Bas2, Bas3]). ∎
Having identified such a coaction map, we are now in position to consider an
intertwiner relating representations of ${\mathbb{T}}$, a key ingredient in
relating the $q-$Onsager algebra and the reflection equation algebra.
###### Proposition 4.2.
Let $\pi_{u}:U_{q}(\widehat{sl_{2}})\mapsto\mathrm{End}({\cal V}_{u})$ be the
evaluation endomorphism for ${\cal V}\equiv{\mathbb{C}}^{2}$. Let $W$ denote a
vector space over ${\mathbb{C}}$ on which the elements of ${\mathbb{T}}$ act.
There exists an intertwinner
$\displaystyle K(u):{\cal V}_{u}\otimes W\mapsto{\cal V}_{u^{-1}}\otimes W$
such that
(4.16) $\displaystyle K(u)(\pi_{u}\times
id)\big{[}\delta(a)\big{]}=(\pi_{u^{-1}}\times id)\big{[}\delta(a)\big{]}K(u)\
,\qquad\forall a\in{\mathbb{T}}\ .$
It is unique (up to an overall scalar factor), and it satisfies the reflection
equation (2.7).
###### Proof.
First, let us identify one solution of (4.16). By definition, ${\cal V}_{u}$
is a two-dimensional vector space. Then $K(u)$ is a $2\times 2$ matrix, which
entries are formal power series in the variable $u$ in view of (4.4) and
(4.15). Define
(4.19) $\displaystyle K(u)=\left(\begin{array}[]{cc}A(u)&B(u)\\\
C(u)&D(u)\end{array}\right)\ .$
Replacing $K(u)$ in (4.16), we find that the entries must satisfy the
following system of equations
$\displaystyle\big{[}{\textsf{A}},A(u)\big{]}=q^{-1}u^{-1}\big{(}k_{-}B(u)-k_{+}C(u)\big{)}\
,$ (4.20)
$\displaystyle\big{[}{\textsf{A}},D(u)\big{]}=-qu\big{(}k_{-}B(u)-k_{+}C(u)\big{)}\
,$
$\displaystyle\big{[}{\textsf{A}},B(u)\big{]}_{q}=k_{+}\big{(}uA(u)-u^{-1}D(u)\big{)}\
,$
$\displaystyle\big{[}{\textsf{A}},C(u)\big{]}_{q^{-1}}=-k_{-}\big{(}uA(u)-u^{-1}D(u)\big{)}\
$
and similar relations for ${\textsf{A}}^{*}$, provided one substitutes
$q\rightarrow q^{-1},u\rightarrow u^{-1}$ in (4.20). Then, using (3.13) in
(3.2) for $k=0$ it is easy to notice that the defining relations (4.13) are
nothing but (3.4) for $k=0,l=1$, provided we consider the following
homomorphism
(4.21) $\displaystyle{\textsf{A}}\mapsto{\cal W}_{0}\
,\qquad{\textsf{A}}^{*}\mapsto{\cal W}_{1}\ .$
Now, identify the entries of $K(u)$ with (2.12)-(2.15). Expanding and using
the defining relations (3.1)-(3.3) of the algebra ${\cal A}_{q}$, it is easy
to check (4.20) as well as all other relations for ${\textsf{A}}^{*}$. So, at
least one solution $K(u)$ exists and it is written in terms of elements of
${\cal A}_{q}$. For generic $u$, the tensor product $\mathrm{End}({\cal
V}_{u})\otimes W$ is not decomposable with respect to ${\mathbb{T}}$
representations. By Schur’s lemma, this means that given $W$, the solution to
the intertwining relation (4.16) is unique (up to an overall scalar factor).
It remains to show that $K(u)$ satisfying (4.16) is automatically a solution
of the reflection equation algebra (2.7). To this end, let us recall that
$K(u):{\cal V}_{u}\otimes W\mapsto{\cal V}_{u^{-1}}\otimes W$ and
$R(u/v):{\cal V}_{u}\otimes{\cal V}_{v}\mapsto{\cal V}_{v}\otimes{\cal
V}_{u}$. Then, the proof that this solution $K(u)$ satisfies the reflection
equation (2.7) follows from the commutativity of the following diagram (up to
an overall scalar factor):
$\begin{CD}{\cal V}_{u}\otimes{\cal V}_{v}\otimes
W@>{id\,\otimes\,K(v)}>{}>{\cal V}_{u}\otimes{\cal V}_{v^{-1}}\otimes
W@>{R(uv)\,\otimes\,id}>{}>{\cal V}_{v^{-1}}\otimes{\cal V}_{u}\otimes W\\\
@V{}V{R(u/v)\otimes id}V@V{id\otimes K(u)}V{}V\\\ {\cal V}_{v}\otimes{\cal
V}_{u}\otimes W\qquad{\cal V}_{v^{-1}}\otimes{\cal V}_{u^{-1}}\otimes W\\\
@V{}V{id\otimes K(u)}V@V{R(u/v)\otimes id}V{}V\\\ {\cal V}_{v}\otimes{\cal
V}_{u^{-1}}\otimes W@>{R(uv)\,\otimes\,\text{id}}>{}>{\cal
V}_{u^{-1}}\otimes{\cal V}_{v}\otimes W@>{id\,\otimes\,K(v)}>{}>{\cal
V}_{u^{-1}}\otimes{\cal V}_{v^{-1}}\otimes W\end{CD}$
∎
Combining previous results, we obtain the third main result of the paper:
###### Theorem 3.
The $q-$Onsager algebra ${\mathbb{T}}$ and the current algebra
$O_{q}(\widehat{sl_{2}})$ are isomorphic.
###### Proof.
According to Proposition 4.2, $K(u)$ with (2.12)-(2.15) is the unique
intertwiner of ${\mathbb{T}}$ satisfying (4.16). Also, it satisfies the
reflection equation algebra (2.7). So, $K(u)$ establishes the isomorphism
between ${\mathbb{T}}$ and the reflection equation algebra (2.7) for the
$U_{q}(\widehat{sl_{2}})$ $R-$matrix. Theorem 1 then establishes the
isomorphism between the reflection equation algebra (2.7) and
$O_{q}(\widehat{sl_{2}})$, which supports the claim. ∎
Although the isomorphism between ${\mathbb{T}}$ and
$O_{q}({\widehat{sl_{2}}})\cong{\cal A}_{q}$ is now established, an
interesting problem remains to construct an explicit homomorphism from ${\cal
A}_{q}$ to ${\mathbb{T}}$, i.e. to write all higher elements of ${\cal A}_{q}$
solely in terms of ${\cal W}_{0},{\cal W}_{1}$. This problem will be
considered elsewhere.
To conclude, the $q-$Onsager algebra ${\mathbb{T}}$ admits two different
realizations: one [see Proposition 4.2] in terms of the reflection equation
algebra for the $U_{q}({\widehat{sl_{2}}})$ $R-$matrix and another one in
terms [see Theorems 1, 2, 3] of the current algebra
$O_{q}({\widehat{sl_{2}}})\cong{\cal A}_{q}$. Previous results are resumed by
the picture below.
“RKRK” algebra [Cher, Sk] Reflection equation for $U_{q}(\widehat{sl_{2}})$
Theorems 1,2 Current algebra (Def. 2.2) Presentation $\\{{\cal W}_{-k},{\cal
W}_{k+1},{\cal G}_{k+1},{\tilde{\cal G}}_{k+1}\\}$ [BasK]
$O_{q}({\widehat{sl_{2}}})$ Proposition 4.2 Theorem 3 $q-$Onsager algebra
${\mathbb{T}}$ [Ter2] Figure 1. An algebraic scheme for
$O_{q}({\widehat{sl_{2}}})$
Acknowledgements: Part of this work has been supported by the ANR Research
project “Boundary integrable models: algebraic structures and correlation
functions”, contract number JC05-52749. P.B thanks S. Pakuliak for detailed
explanations and notes about Drinfeld’s construction at the early stage of
this work, as well as P. Terwilliger for reading the manuscript and helpful
comments.
## References
* [AMPT] H. Au-Yang, B.M. McCoy, J.H.H. Perk and S. Tang, Solvable models in statistical mechanics and Riemann surfaces of genus greater than one in Algebraic Analysis, Vol. 1, M. Kashiwara and T. Kawai, eds., Academic Press, San Diego, 1988, 29-40.
* [Bas] P. Baseilhac, Deformed Dolan-Grady relations in quantum integrable models, Nucl.Phys. B 709 (2005) 491-521, arXiv:hep-th/0404149.
* [Bas2] P. Baseilhac, An integrable structure related with tridiagonal algebras, Nucl.Phys. B 705 (2005) 605-619, arXiv:math-ph/0408025.
* [Bas3] P. Baseilhac, A family of tridiagonal pairs and related symmetric functions, J. Phys. A 39 (2006) 11773-11791, arXiv:math-ph/0604035v3.
* [BasK] P. Baseilhac and K. Koizumi, A new (in)finite dimensional algebra for quantum integrable models, Nucl. Phys. B 720 (2005) 325-347, arXiv:math-ph/0503036.
* [Baxter] R. Baxter, Exactly solvable models in statistical mechanics, New York, Academic Press.
* [Be] J. Beck, Braid group action and quantum affine algebras, Commun. Math. Phys. 165 (1994) 555-568.
* [Cha] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge (1994).
* [Cher] I.V. Cherednik, Factorizing particles on the half-line and root systems, Teor. Mat. Fiz. 61 (1984) 35-44.
* [Dam] I. Damiani, A basis of type Poincaré-Birkhoff-Witt for the quantum algebra of $\widehat{sl_{2}}$, J. Algebra 161 (1993) 291-310.
* [DaRo] E. Date and S. S. Roan, The structure of quotients of the Onsager algebra by closed ideals, J. Phys. A: Math. Gen. 33 (2000) 3275-3296, math.QA/9911018;
E. Date and S. S. Roan, The algebraic structure of the Onsager algebra, Czech.
J. Phys. 50 (2000) 37-44, cond-mat/0002418.
* [Dav] B. Davies, Onsager’s algebra and superintegrability, J. Phys. A 23 (1990) 2245-2261;
B. Davies, Onsager’s algebra and the Dolan-Grady condition in the non-self-
dual case, J. Math. Phys. 32 (1991) 2945-2950.
* [DeG] Gustav W. Delius and Alan George, Quantum affine reflection algebras of type $d_{n}^{(1)}$ and reflection matrices, Lett. Math. Phys. 62 (2002) 211-217, arXiv:math/0208043.
* [DeM] G.W Delius and N.J. MacKay, Quantum group symmetry in sine-Gordon and affine Toda field theories on the half-line, Commun. Math. Phys. 233 (2003) 173-190, arXiv:hep-th/0112023.
* [DeMS] G. W. Delius, N. J. MacKay and B. J. Short, Boundary remnant of Yangian symmetry and the structure of rational reflection matrices, Phys. Lett. B 522 (2001) 335-344; Erratum-ibid. B 524 (2002) 401, arXiv:hep-th/0109115v2.
* [DiF] J. Ding and I.B. Frenkel, Isomorphism of two realizations of quantum affine algebra $U_{q}(\widehat{gl(n)})$, Commun. Math. Phys. 156 (1993) 277-300.
* [DoG] L. Dolan and M. Grady, Conserved charges from self-duality, Phys. Rev. D 25 (1982) 1587-1604.
* [Dr1] V. G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Doklady 32 (1985) 254-258.
* [Dr2] V. G. Drinfeld, A new realization of Yangians and quantum affine algebras, Sov. Math. Doklady 36 (1988) 212-216.
* [F1] L.D. Faddeev, Integrable models in (1+1)-dimensional quantum field theory in Recent advances in Field Theory and Statistical Mechanics, Les Houches 1982, J-B. Zuber and R. Stora Eds., Amsterdam North-Holland, (1984) pp. 561-608.
* [FRT1] L.D. Faddeev, N. Yu. Reshetikhin and L.A. Takhtajan, Quantization of Lie groups and Lie algebras, Yang-Baxter equation and quantum integrable systems, Advanced Series in Mathematical Physics, Vol. 10, Singapore, World Scientific, (1989) 299-309.
* [ITTer] T. Ito, K. Tanabe and P. Terwilliger, Some algebra related to $P$\- and $Q$-polynomial association schemes, Codes and association schemes (Piscataway, NJ, 1999), 167-192, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 56, Amer. Math. Soc., Providence, RI, (2001); arXiv:math/0406556v1.
* [Jim] M. Jimbo, A $q-$difference analogue of $U(g)$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985) 63-69;
M. Jimbo, A $q-$analog of $U(gl(N+1))$, Hecke algebra and the Yang-Baxter
equation, Lett. Math. Phys. 11 (1986) 247-252.
* [Jin] N. Jing, On Drinfeld realization of quantum affine algebras, Proceedings of Conf. on Lie Alg. at Ohio State Univ., May 1996; in Monster and Lie Algebras, eds. J. Ferrar and K. Harada, OSU Math. Res. Inst. Publ. 7, de Gruyter, Berlin (1998) pp.195-206 .
* [KS] P.P. Kulish and E.K. Sklyanin, Solutions of the Yang-Baxter equation, J. Soviet. Math. 19 (1982) 1596-1620;
P.P. Kulish and E.K. Sklyanin, Quantum spectral transform method. Recent
developments in Integrable quantum field theories, Tvarminne (1981), J.
Hietarinta and C. Montonen eds; Lecture Notes in Physics 151 61-119, Springer,
Berlin.
* [KRS] P.P. Kulish, N.Yu. Reshetikhin and E. K. Sklyanin, Yang-Baxter equations and representation theory: I, Lett. Math. Phys. 5 (1981) 393-403.
* [L] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988) 237-249.
* [MN] L. Mezincescu and R.I. Nepomechie, Fractional-Spin Integrals of Motion for the Boundary Sine-Gordon Model at the Free Fermion Point, Int. J. Mod. Phys. A 13 (1998) 2747-2764, arXiv:hep-th/9709078v1.
* [Nep] R.I. Nepomechie, Boundary quantum group generators of type A, Lett. Math. Phys. 62 (2002) 83-89, arXiv:hep-th/0204181.
* [Ons] L. Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65 (1944) 117-149.
* [Pe] J.H.H. Perk, Star-triangle equations, quantum Lax operators, and higher genus curves, Proceedings 1987 Summer Research Institute on Theta functions, Proc. Symp. Pure. Math. Vol. 49, part 1. (Am. Math. Soc., Providence, R.I., 1989), 341-354.
* [RS] N.Yu. Reshetikhin and M. A. Semenov-Tian-Shansky, Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990) 133-142.
* [Sk] E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988) 2375-2389.
* [Ter1] P. Terwilliger, The subconstituent algebra of an association scheme. III., J. Algebraic Combin., 2 (1993) 177-210.
* [Ter2] P. Terwilliger,Two relations that generalize the $q-$Serre relations and the Dolan-Grady relations, Proceedings of the Nagoya 1999 International workshop on physics and combinatorics. Editors A. N. Kirillov, A. Tsuchiya, H. Umemura. pp 377–398, math.QA/0307016.
|
arxiv-papers
| 2009-06-08T12:45:46 |
2024-09-04T02:49:03.215289
|
{
"license": "Public Domain",
"authors": "P. Baseilhac and K. Shigechi",
"submitter": "Pascal Baseilhac",
"url": "https://arxiv.org/abs/0906.1482"
}
|
0906.1518
|
# Two classes of algebras with infinite Hochschild homology
Andrea Solotar Departamento de Matemática, Facultad de Ciencias Exactas y
Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 1 1428,
Buenos Aires, Argentina [email protected] and Micheline Vigué-Poirrier
Laboratoire Analyse, Géométrie & Applications, UMR CNRS 7539, Institut
Galilée, Université Paris 13 F-93430 Villetaneuse, France [email protected]
paris13.fr
(Date: June 8, 2009.)
###### Abstract.
We prove without any assumption on the ground field that higher Hochschild
homology groups do not vanish for two large classes of algebras whose global
dimension is not finite.
###### Key words and phrases:
global dimension, Hochschild homology theory
###### 2000 Mathematics Subject Classification:
Primary 16E40, 16W50
This work has been supported by the projects UBACYTX212 and PIP-CONICET 5099.
The first author is a research member of CONICET (Argentina) and a Regular
Associate of ICTP Associate Scheme. The second author is a research member of
University of Paris 13, CNRS, UMR 7539 (LAGA)
## 1\. Introduction
Let $k$ be a fixed field. All the algebras we consider are associative unital
$k$-algebras. We will denote $\otimes=\otimes_{k}$.
It is well known that the homological properties of an algebra are related to
the properties of its Hochschild (co)homology groups. For example, if a finite
dimensional algebra over an algebraically closed field has finite global
dimension, then all its higher Hochschild cohomology groups vanish. In [12],
D. Happel conjectured that the converse would be true. However, it has been
shown in [5] that the conjecture does not hold for algebras of type
$A_{q}=k\langle x,y\rangle/(x^{2},y^{2},xy-qyx)$, where $q\in k$.
In [11], Han proved that the total Hochschild homology of the algebras $A_{q}$
is infinite dimensional. This fact led him to suggest the following
conjecture:
Conjecture(Han): Let $A$ be a finite dimensional $k$-algebra. If the total
Hochschild homology of $A$ is finite dimensional, then $A$ has finite global
dimension.
In the same paper, Han provided a proof of this statement for monomial finite
dimensional algebras.
Avramov and Vigué’s computations in [1] show that Han’s conjecture holds in
the commutative case not only for finite dimensional algebras but for
essentially finitely generated ones, see also [18].
In [4], Han’s conjecture is proved for graded local algebras, Koszul algebras
and graded cellular algebras, provided the characteristic of the ground field
is zero. The proof relies on the properties of the graded Cartan matrix and
the logarithm and strongly uses the hypothesis on the characteristic of the
field.
In [3], the authors compute the Hochschild homology groups of quantum complete
intersections, that is algebras of type $A=k\langle
x,y\rangle/(x^{a},y^{b},xy-qyx)$, where $q\in k^{*}$ is not a root of unity
and $a,b\geq 2$ are fixed integers. In particular they prove Han’s conjecture
for this class of finite dimensional algebras.
The main purpose of this paper is to prove that higher Hochschild homology
groups do not vanish for two large classes of algebras whose global dimension
is not finite, without any assumption on the ground field.
In Theorem I, the algebras we consider are generalizations of quantum complete
intersections and they are not assumed to be finite dimensional.
On the other hand, the algebras satisfying the hypotheses of Theorem II are,
in some sense, opposite to quantum complete intersections, since we assume
that they have two generators $x$ and $y$ such that $xy=yx=0$.
Now we state both main theorems.
Theorem I: Let $A=k\langle x_{1},\dots,x_{n}\rangle/(f_{1},\dots,f_{p})$ be a
finitely generated $k$-algebra, such that $f_{1}$ belongs to $k[x_{1}]$ and,
for $i\geq 2$, $f_{i}$ belongs to the two-sided ideal $(x_{2},\dots,x_{n})$.
If $B=k[x_{1}]/(f_{1})$ is not smooth, then the Hochschild homology groups
$HH_{n}(A)$ are not zero for an infinite increasing sequence of integers.
For example Theorem I is valid if $f_{1}=x_{1}^{2}g_{1}$, with $g_{1}\in
k[x_{1}]$ and $f_{2},\dots,f_{p}$ satisfying the hypothesis of the theorem.
Theorem II: Let $A=\bigoplus_{n\geq 0}A^{n}$ be a finite dimensional graded
$k$-algebra with $A^{0}=k$ and such that $\overline{A}=\bigoplus_{n\geq
1}A^{n}$ is not zero. Assume that there exist two generators $x$ and $y$ of
the algebra $A$ verifying $xy=yx=0$. Then the total Hochschild homology of $A$
is not finite dimensional.
###### Remark 1.1.
This theorem is valid for very large classes of graded local algebras since
relations between the other generators play no role.
The proof of Theorem I follows without any computation from the well known
result for commutative algebras.
The methods used in the proof of Theorem II rely on differential homological
algebra. In fact, we will work with the cobar construction on the graded
coalgebra $\bigoplus_{n\geq 0}{\mathrm{Hom}}_{k}(A^{n},k)$. We denote it
$(\Omega^{*}A,d)$. The Hochschild homology groups of the differential graded
algebra $(\Omega^{*}A,d)$ are dual, as vector spaces, to the Hochschild
homology groups of the graded $k$-algebra $A$. Since $(\Omega^{*}A,d)$ is a
tensor algebra, a short complex is available to compute its Hochschild
homology.
The paper is organized as follows:
1. (1)
Introduction.
2. (2)
Proof of Theorem I.
3. (3)
Interpretation in terms of quivers.
4. (4)
Hochschild homology in the differential graded case.
5. (5)
Proof of Theorem II.
## 2\. Proof of Theorem I
Let $A$ be an associative unital $k$-algebra. The definition of the Hochschild
homology groups, $HH_{n}(A)$, $n\geq 0$ is well known (see for example [13]).
We have
$HH_{n}(A):={\mathrm{Tor}}_{n}^{A^{e}}(A,A)=H_{n}(C_{*}(A),b)$
where $(C_{*}(A),b)$ is the Hochschild complex of $A$. Clearly, $HH_{n}(A)$ is
a $k$-vector space for all $n\geq 0$.
In this section we assume that $A=k\langle
x_{1},\dots,x_{n}\rangle/(f_{1},\dots,f_{p})$ where $n,p\geq 1$, $f_{1}$,
which we may suppose monic, belongs to $k[x_{1}]$ and, for $i\geq 2$, $f_{i}$
belongs to the two-sided ideal $(x_{2},\dots,x_{n})$. Let us consider the
$k$-algebra $B=k[x_{1}]/(f_{1})$ and the maps
$\iota:B\to A\hbox{ with }\iota(x_{1})=x_{1},$ $\pi:A\to B\hbox{ with
}\pi(x_{1})=x_{1},\pi(x_{i})=0,\hbox{ for }i\geq 2.$
The following lemma is easy to prove.
###### Lemma 2.1.
The maps $\iota$ and $\pi$ are morphisms of $k$-algebras and satisfy
$\pi\circ\iota=id_{B}$.
Now, Theorem I is an immediate consequence of the following facts:
* •
the morphisms $\iota$ and $\pi$ induce by functoriality $k$-linear maps
$HH_{*}(\iota):HH_{*}(B)\to HH_{*}(A)\hbox{ and }HH_{*}(\pi):HH_{*}(A)\to
HH_{*}(B)$
satisfying $HH_{*}(\pi)\circ HH_{*}(\iota)=id_{HH_{*}(B)}$,
* •
using a result of [1], $HH_{n}(B)$ is non zero for an infinite sequence of
integers $n$.
Another proof can be given using the computations for $HH_{n}(B)$ in [6]: if
$f_{1}$ and $f_{1}^{\prime}$ are not coprime, then $HH_{n}(B)\neq 0$ for all
$n\in{\mathbb{N}}$.
###### Example 2.2.
If $f_{1}=x_{1}^{a}$, with $a\geq 2$, and $f_{i}\in(x_{2},\dots,x_{n})$, then
Theorem I holds. This covers the case of quantum complete intersections.
An interesting question is to know if the algebras $A$ considered in Theorem I
have infinite global dimension. In the commutative case, it is well known that
this is true. Also, if $A=k\langle
x_{1},\dots,x_{n}\rangle/(f_{1},\dots,f_{p})$ is a finite dimensional
$k$-vector space, Happel’s result [12] implies that
${\mathrm{gldim}}(A)=\infty$, where ${\mathrm{gldim}}$ denotes the global
dimension of the algebra.
It follows from Serre’s theorem in page 37 of [15] that if $B$ is not smooth,
then its global dimension is not finite. In the general case, we cannot ensure
that if we have $k$-algebras $A$ and $B$ as above with
${\mathrm{gldim}}(B)=\infty$, then ${\mathrm{gldim}}(A)=\infty$.
However, we can use the algebra map $\iota:B\to A$ to obtain that the global
dimension of $A$ is not finite in some cases: Suppose that $\iota$ endows $A$
with a structure of flat $B$-module. In this situation, Corollary 4.4 of [2]
says that ${\mathrm{gldim}}(A)=\infty$. This is the case, for example, of
quantum complete intersections.
## 3\. Interpretation in terms of quivers
Let $A$ be a finite dimensional basic $k$-algebra, then there exist a quiver
$Q^{A}$ and an admissible ideal $I^{A}$ such that $A$ is isomorphic to
$kQ^{A}/I^{A}$. In other words, if we denote by
$Q_{0}^{A}=\\{e_{1},\dots,e_{r}\\}$ the set of vertices of $Q^{A}$ and by
$Q_{1}^{A}$ its set of arrows, then $kQ_{0}^{A}$ is an algebra, $kQ_{1}^{A}$
is a $kQ_{0}^{A}$ two-sided ideal and $A=T_{kQ_{0}^{A}}kQ_{1}^{A}/I^{A}$,
where $I^{A}\subseteq(kQ_{1}^{A})^{2}$.
Suppose that there exist $e_{i}\in kQ_{0}^{A}$ and $x\in
e_{i}(kQ_{1}^{A})e_{i}$. In fact, since $A$ is finite dimensional and $I^{A}$
is admissible, if such a loop $x$ exists then $x^{n}=0$ for some integer
$n{\geq 2}$.
Let $B$ be the $k$-algebra $k[x]/\langle x^{n}\rangle$, then
$B=T_{kQ_{0}^{B}}kQ_{1}^{B}/I^{B}$, where $Q_{0}^{B}=\\{e_{i}\\}$,
$Q_{1}^{B}=\\{x\\}$ and $I^{B}=\langle x^{n}\rangle$.
We may consider the morphisms of algebras of the previous section. In this
case the map $\iota$ is completely determined by its values on $e_{i}$ and
$x$. It sends $e_{i}$ to $e_{1}+\dots+e_{r}$ and $x$ to $x$. Clearly, it is
well defined.
On the other hand, the morphism $\pi:A\to B$ is given as follows,
$\pi(e_{j})=\delta_{ij}e_{i}$, for $1\leq j\leq r$, and the restriction of
$\pi$ to the arrows of $A$ is given by $\pi(y)=\delta_{yx}x$, where $\delta$
is the Kronecker delta. If we assume that $I^{A}=\langle
x^{n},f_{2}\dots,f_{s}\rangle$ is admissible and that $f_{i}$ belongs to the
two-sided ideal generated by $Q_{1}^{A}-\\{x\\}$, then it is straightforward
to check that $\pi$ is also well defined and $\pi\circ\iota=id_{B}$.
As a consequence of the results of Section 2, we see that the Hochschild
homology dimension, denoted ${\mathrm{hhdim}}(B)$, is infinite and so the same
holds for $A$. Being both $k$-finite dimensional, their global dimensions
cannot be finite.
It is interesting to note that analogous situations hold in several cases, for
example, using results of [11], each time we have $char(k)=0$, $B$ monomial
and ${\mathrm{hhdim}}(B)\neq 0$.
## 4\. Hochschild homology and cobar construction
In this section we deal with finite dimensional algebras.
### 4.1. Notation
We use the methods of differential graded algebra of [7]. In particular an
element of lower degree $i\in{\mathbb{Z}}$ is, by the classical convention, of
upper degree $-i$. All the algebras considered from now on are unital,
associative, with a differential of degree $-1$. We recall that if
$V=\bigoplus_{i\in{\mathbb{Z}}}V_{i}$ is a graded $k$-vector space, then the
suspended graded $k$-vector space $sV$ has homogeneous components
$(sV)_{i}=V_{i-1}$, for $i\in{\mathbb{Z}}$. The $k$-algebra $TV$ will denote
the tensor algebra on $V$. The degree of an element $v\in V$ is denoted $|v|$.
For any differential graded algebra $A$, let $A^{op}$ be the opposite graded
algebra, and $A^{e}=A\otimes A^{op}$ be the enveloping algebra. The categories
of graded $A$-bimodules and of left (or right) differential graded
$A^{e}$-modules are equivalent.
### 4.2. Bar resolution and Hochschild homology
Let $(A,d)$ be an augmented algebra and
$\overline{A}={\mathrm{Ker}}(\epsilon:A\to k)$. The normalized bar resolution
of $A$, denoted $B(A,A,A)$, is the differential graded $A^{e}$-module
$(A\otimes T(s\overline{A})\otimes A,D_{0}+D_{1})$, where $D_{0}$ is the
differential induced by $d$ on the tensor product of complexes and $D_{1}$ is
defined as follows (see for example [9], 2.2.)
$\displaystyle D_{1}(a\otimes sa_{1}\otimes\dots\otimes sa_{n}\otimes b)=$
$\displaystyle(-1)^{|a|}aa_{1}\otimes sa_{2}\otimes\dots\otimes sa_{n}\otimes
b$ $\displaystyle\pm\sum_{i=1}^{n-1}a\otimes sa_{1}\otimes\dots\otimes
s(a_{i}a_{i+1})\otimes\dots\otimes sa_{n}\otimes b$ $\displaystyle\pm a\otimes
sa_{1}\otimes\dots\otimes sa_{n-1}\otimes a_{n}b.$
The Hochschild homology of the differential graded algebra $(A,d)$ is, by
definition, the graded vector space
$\mathcal{HH}_{*}(A)={\mathrm{Tor}}^{A^{e}}_{*}(A,A)$ in the differential
sense of [14].
###### Lemma 4.1.
[7] The canonical map $m:B(A,A,A)\to A$ defined by $0$ on $A\otimes T^{\geq
1}(s\overline{A})\otimes A$, and by multiplication on $A\otimes A$ is a
semifree resolution of $A$ as an $A^{e}$-module.
Consequently we have,
$\mathcal{HH}_{*}(A,d)=H_{*}(\mathcal{C}_{*}(A),\delta)$
with
$\mathcal{C}_{*}(A)=A\otimes_{A^{e}}B(A,A,A)=A\otimes T(s\overline{A}),$
and $\delta=\delta_{0}+\delta_{1}$, where $\delta_{0}$ and $\delta_{1}$ are
obtained by tensorization.
Explicitly,
$\displaystyle\delta_{1}(a\otimes sa_{1}\otimes\dots\otimes sa_{n})=$
$\displaystyle(-1)^{|a|}aa_{1}\otimes sa_{2}\otimes\dots\otimes sa_{n}$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{\epsilon_{i}}a\otimes
sa_{1}\otimes\dots\otimes s(a_{i}a_{i+1})\otimes\dots\otimes sa_{n}$
$\displaystyle+(-1)^{\epsilon_{n}}a_{n}a\otimes sa_{1}\otimes\dots\otimes
sa_{n-1},$
where the $\epsilon_{i}$’s are integers depending on the degrees of the
elements $a_{i}$; if all these degrees are even, then $\epsilon_{i}=i$.
In the rest of this paper we consider only differential graded algebras
$(A,d)$ satisfying either condition (a) or condition (b) below.
* (a)
$A_{n}=0$ for $n<0$ and $A_{0}=k$, so that $\mathcal{C}_{n}(A)=0$ for $n<0$;
* (b)
$A_{n}=0$ for $n>0$, $A_{0}=k$, $A_{-1}=0$, so that $\mathcal{C}_{n}(A)=0$ for
$n>0$.
In both cases, we have $\mathcal{C}_{0}(A)=k$.
### 4.3. Cobar construction and duality construction in Hochschild homology
We next recall the definition of the cobar construction described in Section
19 of [8]. Let $(C,d_{C})$ be a coaugmented differential graded coalgebra with
comultiplication $\Delta$, and $\overline{C}={\mathrm{Ker}}(\epsilon:C\to k)$.
We denote $(\Omega C,d)$ the augmented differential graded algebra defined as
follows:
* •
$\Omega C=T(s^{-1}\overline{C})$, as augmented graded algebra,
* •
$d=d_{0}+d_{1}$, with $d_{0}(s^{-1}c)=-s^{-1}(d_{C}(c))$, if
$c\in\overline{C}$, and $d_{1}$ is defined from $\Delta$.
Suppose now that $(A,d_{A})$ is a finite dimensional differential graded
algebra, then the graded dual $A^{\vee}=Hom_{k}(A,k)$ is a differential graded
coalgebra with differential $d_{A}^{\vee}$, the transpose of $d_{A}$.
###### Definition 4.2.
$(\Omega^{*}A,d):=(\Omega(A^{\vee}),d)$, where $d$ is defined from
$d_{A}^{\vee}$ and the comultiplication of $A^{\vee}$ as above.
We have $\Omega^{*}A=T(V)$ with $V=Hom_{k}(s\overline{A},k)$. If $(A,d_{A})$
satisfies condition (b) above, then
$V=\bigoplus_{n\geq 1}V_{n},\hbox{ with }V_{n}=Hom_{k}(A_{-n-1},k)$
and then $(\Omega^{*}A,d)$ satisfies condition (a). Similarly, if $(A,d_{A})$
satisfies condition (a), then $(\Omega^{*}A,d)$ satisfies condition (b).
The first ingredient used to prove Theorem II is the following duality
property.
###### Theorem 4.3.
[10], [16]: Let $(A,d_{A})$ be a finite dimensional algebra satisfying either
condition (a) or (b) above, then for all $n\in\mathbb{Z}$ we have:
${\mathrm{Hom}}_{k}(\mathcal{HH}_{-n}(A),k)=\mathcal{HH}_{n}(\Omega^{*}A).$
Consequently, the computation of the graded vector space $\mathcal{HH}_{n}(A)$
can be replaced by the computation of the Hochschild homology of a quasifree
differential graded algebra $(T(V),d)$.
### 4.4. A short complex for the computation of the Hochschild homology
Now, we want to compute the Hochschild homology of $(T(V),d)$, with
$V=\bigoplus_{n\geq 1}V_{n}$.
We recall here the main results of [17]. Put $(T(V),d)=(B,d)$ and let
$P=(B\otimes B)\oplus(B\otimes(sV)\otimes B)$, we define a differential $D$ on
$P$, which is the tensor product of the differentials on $B\otimes B$, and
$D(a\otimes sv\otimes b)=da\otimes sv\otimes b\pm(av\otimes b-a\otimes
vb)+S(a\otimes sv\otimes b),$
where $S(a\otimes sv\otimes b)\in B\otimes sV\otimes B,$ for $a\in B,b\in B$
and $v\in V$.
###### Proposition 4.4.
(Thm. 1.4 in [17]) The canonical map $m:(P,D)\to B$ defined as $0$ on
$B\otimes sV\otimes B$ and as multiplication on $B\otimes B$ is a semifree
resolution of $B$ as $B^{e}$-module.
As a consequence,
$\mathcal{HH}_{*}(T(V),d)=H_{*}(B\otimes_{B^{e}}P,\delta),$
with differential $\delta=d\otimes_{B^{e}}D$ that will be precised in the next
section. We have:
* •
$\delta_{|T(V)}=d$,
* •
$\delta(a\otimes sv)=da\otimes
sv+(-1)^{|a|}(av-(-1)^{|v|\times|a|}va)-\sigma(a\otimes dv)$, where
$\sigma(a\otimes dv)$ belongs to $T(V)\otimes sV$, for $a\in T(V),v\in V$.
Put $Q_{*}:=B\otimes_{B^{e}}P=T(V)\oplus(T(V)\otimes sV)$.
###### Theorem 4.5.
(Thm. 1.5 of [17]) With the above notations,
$\mathcal{HH}_{*}(T(V),d)=H_{*}(Q_{*},\delta).$
In the following section we will use the complex $(Q_{*},\delta)$ to compute
the Hochschild homology of a finite dimensional graded algebra
$A=\bigoplus_{n\geq 0}A^{n}$, with $A^{0}=k$. In this case, the graded vector
space $V$ is also finite dimensional, and the differential $\delta$ has good
properties.
## 5\. Proof of Theorem II
We work with a finite dimensional graded algebra with $A^{0}=k$. We may assume
without loss of generality that $A$ is graded in even degrees,
$A=k\oplus\left(\bigoplus_{n\geq 2}A^{n}\right)$, and
$\overline{A}=\bigoplus_{n\geq 2}A^{n}$ is non zero.
### 5.1. Relations between $HH_{*}(A)$ and $\mathcal{HH}_{*}(A,0)$
Using the conventions recalled at the beginning of the previous section, we
consider $A$ as a differential graded algebra with differential $0$ and
$A_{-n}=A^{n}$.
Since $A$ is graded, the ordinary Hochschild homology $HH_{*}(A)$ defined in
Section 2 is graded, and there is a decomposition
$HH_{*}(A)=\bigoplus_{p,q\geq 0}HH_{p}(A)^{q}.$
Since $A$ is finite dimensional, $HH_{p}(A)$ is finite dimensional for all
$p$.
###### Lemma 5.1.
Let $A$ be an algebra as above. Then,
1. (1)
$\mathcal{HH}_{*}(A,0)=\bigoplus_{n\geq 0}\mathcal{HH}_{-n}(A)$ and
$\mathcal{HH}_{-n}(A)=\bigoplus_{p}HH_{p}(A)^{p+n}$.
2. (2)
$HH_{p}(A)^{p+n}=0$ if $p>n$ or $p<\frac{n-N}{N-1}$, where
$N=sup\\{n|A^{n}\neq 0\\}$.
###### Corollary 5.2.
If there exists an increasing sequence of integers $n_{i}$ such that
$\mathcal{HH}_{-n_{i}}(A)\neq 0$, then $HH_{*}(A)$ is not finite dimensional.
The strategy now is to focus our attention on $\mathcal{HH}_{*}(\Omega^{*}A)$,
using Theorem 4.3. But Theorem 4.5 allows us to use the short complex
$(Q_{*},\delta)$ to compute $\mathcal{HH}_{*}(\Omega^{*}A)$, so we will work
with this last one.
### 5.2. Description of $(Q_{*},\delta)$
Let $A=k\oplus\left(\bigoplus_{n\geq 2}A^{n}\right)$ be a finite dimensional
graded algebra. We fix a homogeneous linear basis $(a_{i})_{i\in I}$ for
$\overline{A}=\bigoplus_{n\geq 2}A^{n}$. This choice determines the structure
constants $\alpha^{i}_{jk}$ by the equalities
$a_{j}a_{k}=\sum\alpha^{i}_{jk}a_{i}$.
In this situation, $(\overline{A})^{\vee}={\mathrm{Hom}}_{k}(\overline{A},k)$
is endowed with the dual basis $(b_{i})_{i\in I}$ satisfying $\langle
b_{i},a_{j}\rangle=\delta_{ij}$. Notice that $A^{\vee}$ is a graded coalgebra
with comultiplication $\Delta$, and $\Delta
b_{i}=\sum_{j.k}\beta^{jk}_{i}b_{j}\otimes b_{k}$, where
$\alpha^{i}_{jk}=(-1)^{|a_{j}|\times|a_{k}|}\beta^{jk}_{i}$.
We have already defined $(\Omega^{*}A,d)=(\Omega(A^{\vee}),d)=(T(V),d)$. Now,
put $v_{i}=s^{-1}b_{i}$, then $|v_{i}|=n-1$ if $a_{i}\in A^{n}$. We check that
$dv_{i}=\sum_{j,k}(-1)^{|a_{j}|+|a_{j}|\times|a_{k}|}\alpha^{i}_{jk}v_{j}\otimes
v_{k}.$
So $(\Omega^{*}A,d)=(T(V),d)$ is a tensor algebra with a quadratic
differential.
Furthermore, we have assumed without loss of generality that $A$ is graded in
even degrees, so that $V$ is graded only in odd degrees. In this case, we give
an explicit formula for the differential $\delta$ on $Q_{*}$ (cf. Subsection
4.4 ).
Put $\overline{V}=sV$, then $Q_{*}=T(V)\oplus T(V)\otimes\overline{V}$. Let
$v$ be an element in $V$, and $dv=\sum_{j,k}\lambda_{jk}v_{j}\otimes v_{k}$,
with $\lambda_{jk}\in k$. Let $a$ be an element in $T(V)$.
We have:
$\delta(a\otimes\overline{v})=da\otimes\overline{v}+(-1)^{|a|}(av-(-1)^{|a|}va)-\sigma(a\otimes
dv),$
where
$\sigma(a\otimes
dv)=-(-1)^{|a|}\sum_{j,k}\lambda_{jk}av_{j}\otimes\overline{v}_{k}+\sum_{j,k}\lambda_{jk}v_{k}a\otimes\overline{v}_{j}.$
### 5.3. A nice homogeneous basis $(a_{i})$ for $\overline{A}$
Since $A=k\oplus\overline{A}$, the projection
$\overline{A}\to\overline{A}/\overline{A}^{2}=U$ has a section $\rho$ that
extends to a morphism of algebras $T(U)\to A$ whose kernel is contained in
$T^{\geq 2}(U)$. This implies that $(x_{i})_{1\leq i\leq p}$ are generators of
the algebra $A$ if and only if their images in $\overline{A}/\overline{A}^{2}$
form a basis of this vector space.
As vector spaces,
$\overline{A}=\overline{A}/\overline{A}^{2}\oplus\overline{A}^{2}$, and we
will consider a homogeneous basis of $\overline{A}/\overline{A}^{2}$ and a
basis of $\overline{A}^{2}$. If $a_{i}\in\overline{A}/\overline{A}^{2}$, then
the corresponding $v_{i}$ in $(\Omega^{*}A,d)$ satisfies $dv_{i}=0$.
We will now prove the following result.
###### Theorem 5.3.
Let $A=\bigoplus_{n\geq 0}A^{n}$ be a finite dimensional graded $k$-algebra
with $A^{0}=k$, such that $\overline{A}=\bigoplus_{n\geq 1}A^{n}$ is not zero.
Assume that there exist two generators $x$ and $y$ of the algebra $A$
satisfying $xy=yx=0$, then $H_{n_{i}}(Q_{*},\delta)\neq 0$ for a strictly
increasing sequence of integers $(n_{i})$.
###### Proof.
We can associate to $x$ and $y$ two elements $a_{1}$ and $a_{2}$, linearly
independent in $\overline{A}$. We denote by $v_{1}$ and $v_{2}$ the
corresponding elements in a dual basis of $V$. If $(a_{1},\dots,a_{n})$ is a
linear basis of $\overline{A}$ and $(v_{1},\dots,v_{n})$ is the corresponding
basis of $V$, then we have $dv_{1}=0$, $dv_{2}=0$ and for $i\geq 3$,
$dv_{i}=\sum_{j,k}\alpha^{i}_{jk}v_{j}\otimes v_{k}.$
The fact that $xy=yx=0$ implies that, for $i\geq 3$,
$\alpha^{i}_{12}=\alpha^{i}_{21}=0$.
For $n\geq 1$, consider:
$X_{n}=v_{1}\otimes v_{2}\otimes v_{1}\otimes v_{2}\otimes\dots\otimes
v_{1}\otimes\overline{v}_{2}-v_{2}\otimes v_{1}\otimes v_{2}\otimes
v_{1}\otimes\dots\otimes v_{2}\otimes\overline{v}_{1}\in
V^{\otimes(2n-1)}\otimes\overline{V}.$
It is easy to see that $|X_{n}|=n(|v_{1}|+|v_{2}|)+1$ and that $\delta
X_{n}=0$.
If $X_{n}$ was a boundary, it should exist $Y,b_{i}\in T(V)$ such that
$X_{n}=\delta(Y+\sum_{i}b_{i}\otimes\overline{v}_{i})$ and
$X_{n}=dY+\sum_{i}db_{i}\otimes\overline{v}_{i}+\sum_{i}(b_{i}v_{i}-v_{i}b_{i})+\sum_{i}\alpha^{i}_{jk}b_{i}v_{j}\otimes\overline{v}_{k}-\sum_{i}\alpha^{i}_{jk}v_{k}b_{i}\otimes\overline{v}_{j}.$
Such elements cannot exist since, for all $i$,
$dv_{i}=\sum_{j,k}\alpha^{i}_{jk}v_{j}\otimes v_{k}\hbox{ with
}\alpha^{i}_{12}=\alpha^{i}_{21}=0.$
∎
###### Example 5.4.
Let $A=k\langle x,y,z\rangle/(xy,yx,x^{2}-y^{2},x^{2}-z^{2},xz-qzx,yz-qzy)$
where $q\in k$, $q^{2}\neq 1$ and $-1$ is not a square in $k$. This example is
not covered by Theorem I.
## References
* [1] Avramov, L.; Vigué-Poirrier, M. Hochschild homology criteria for smoothness. Internat. Math. Res. Notices 1 (1992), 17–25.
* [2] Bavula, V. V. Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras, Bull. Sci. Math., 120 (1996), no. 3, 293–335.
* [3] Bergh, P. A.; Erdmann, K. Homology and cohomology of quantum complete intersections. Algebra Number Theory 2 (2008), no. 5, 501–522
* [4] Bergh, P. A.; Madsen, D. Hochschild homology and global dimension. Bull. London Math. Soc., to appear. arXiv:0803.3550
* [5] Buchweitz, R.; Green, E.; Madsen, D.; Solberg, O. Hochschild cohomology without finite global dimension. Math. Res. Let. 12 (2005), 805–816.
* [6] Buenos Aires Cyclic Homology Group. Cyclic homology of algebras with one generator. J. A. Guccione, J. J. Guccione, M. J. Redondo, A. Solotar and O. Villamayor participated in this research. $K$-Theory 5 (1991), 51–69.
* [7] Félix, Y.; Halperin, S.; Thomas, J.-C. Differential graded algebras in topology. Handbook of algebraic topology, 829–865, North-Holland, Amsterdam, 1995.
* [8] Félix, Y.; Halperin, S.; Thomas, J.-C. Rational homotopy theory. Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001.
* [9] Félix, Y.; Thomas, J.-C.; Vigué-Poirrier, M. The Hochschild cohomology of a closed manifold. Publ. Math. Inst. Hautes Études Sci. 99 (2004), 235–252.
* [10] Halperin, S.; Vigué-Poirrier, M. The homology of a free loop space. Pacific J. Math. 147 (1991), no. 2, 311–324.
* [11] Han, Y. Hochschild (co)homology dimension. J. London Math. Soc. (2) 73 (2006), no. 3, 657–668.
* [12] Happel, D. Hochschild cohomology of finite-dimensional algebras. Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988), 108–126, Lecture Notes in Math., 1404, Springer, Berlin, 1989.
* [13] Loday, J.-L. Cyclic homology. Appendix E by M. Ronco. Second edition. Chapter 13 by the author in collaboration with Teimuraz Pirashvili. Grundlehren der Mathematischen Wissenschaften, vol. 301, Springer-Verlag, Berlin, 1998.
* [14] MacLane, S. Homology. Reprint of the first edition. Die Grundlehren der mathematischen Wissenschaften, vol. 114. Springer-Verlag, Berlin-New York, 1967.
* [15] Serre, J.-P. Algèbre locale. Multiplicités. Lecture Notes in Math. 11 (1965). Springer-Verlag, Berlin.
* [16] Solotar, A. Cyclic homology of a free loop space. Comm. Algebra 21 (1993), no. 2, 575–582.
* [17] Vigué-Poirrier, M. Homologie de Hochschild et homologie cyclique des algèbres différentielles graduées. International Conference on Homotopy Theory (Marseille-Luminy, 1988). Astérisque, vol. 191, Soc. Math. France, 1990, pp. 255–267.
* [18] Vigué-Poirrier, M. Critères de nullité pour l’homologie des algèbres graduées. C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 7, 647–649.
|
arxiv-papers
| 2009-06-08T15:21:27 |
2024-09-04T02:49:03.223580
|
{
"license": "Public Domain",
"authors": "Andrea Solotar and Micheline Vigu\\'e-Poirrier",
"submitter": "Andrea Solotar",
"url": "https://arxiv.org/abs/0906.1518"
}
|
0906.1563
|
# Investigating Dark Energy with Black Hole Binaries
Laura Mersini-Houghton
Adam Kelleher [UNCCH][DAMTP] Department of Physics and Astronomy, The
University of North Carolina at Chapel Hill, Phillips Hall, CB # 3255, Chapel
Hill, NC 27599-3255, USA DAMTP, Center for Mathematical Sciences, University
of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
###### Abstract
The accelerated expansion of the universe is ascribed to the existence of dark
energy. Black holes accretion of dark energy induces a mass change
proportional to the energy density and pressure of the background dark energy
fluid. The time scale during which the mass of black holes changes
considerably is too long relative to the age of the universe, thus beyond
detection possibilities. We propose to take advantage of the modified black
hole masses for exploring the equation of state $w[z]$ of dark energy, by
investigating the evolution of supermassive black hole binaries on a dark
energy background. Deriving the signatures of dark energy accretion on the
evolution of binaries, we find that dark energy imprints on the emitted
gravitational radiation and on the changes in the orbital radius of the binary
can be within detection limits for certain supermassive black hole binaries.
In this talk I describe how binaries can provide a useful tool in obtaining
complementary information on the nature of dark energy, based on the work done
with A.Kelleher.
## 1 INTRODUCTION
Dark energy drives the universe into an accelerated expansion. A decade after
its discovery, the nature and origin of dark energy remain as elusive as ever.
The detection of time variations in the equation of state $w[z]$ of dark
energy, given by the ratio $w[z]=p/\rho$ of its pressure $p$ to the energy
density $\rho$, is notoriously hard to constrain. A precise measurement of
$w[z]$ will guide our theoretical exploration and shed light into the type of
dark energy, thereby helping us understand our present universe and its future
evolution. For these reasons, a precision measurement of $w[z]$ is of the
highest priority in cosmology. Our current bounds are derived by a combination
of data from cosmic microwave background radiation (CMBR), Large Scale
Structure (LSS) surveys, baryon acoustic oscillation (BAO) and Supernova $1A$
(Sn1A) data. Current data analysis constrains the dark energy equation of
state to $-0.14<1+w<0.12$ [2]. While this represents progress, we would like
to know if $1+w$ is positive, negative or $0$, and whether or not w changes
with time. Each case corresponds to quite different types of this mysterious
energy and, it leads to dramatically different predictions for the future
evolution of the universe.
Dark energy evolves slowly with time for the cases when $1+w$ is positive or
negative. However, since the bounds on its equation of state are close to a
pure cosmological constant, $w=-1$, then unlike matter dark energy does not
cluster. For these reasons a universe filled with dark energy can be
reasonably assumed to be a background of a perfect cosmic fluid. T.Jacobson
and other authors [14, 6] took dark energy to be a cosmic ’fluid’ and showed
that the mass of a black hole changes due to accretion of this background
’fluid’. The mass of the black hole increases or decreases, depending on the
sign of $1+w$. This effect would be hard to observe in a single black hole, as
the time scale for this phenomena is quite slow relative to the age of the
universe. For example the evolution time scale $\tau$ for a solar mass black
hole is about $10^{32}yrs$. We proposed in [1] to use the evolution of
supermassive black hole binaries instead of single black hole accretion to
probe dark energy. The reason for our proposed method relies on the fact that,
instead of measurements of the evolution time scale of a single black hole,
the ’footprints’ of $w[z]$ for supermassive black hole binaries can be
observed and tracked down through the modifications introduced by dark energy
accretion in the the orbital separation of the binary and its emitted
gravitational waves. For certain binaries dark energy modifications can be
detectable by gravitational waves or X-ray and radio measurements. The
information obtained by these modifications should help increase our bounds on
$w[z]$. The use of binaries for probing dark energy is a different approach
from our current methods of large scale experiments, since the observation
utilizes localized systems, thereby avoiding noise inherited by propagation of
signals through the vast structures of the universe.
## 2 Dark Energy Accretion by Black Holes
### 2.1 Evolution of a Single Black Hole on the Dark Energy Background:
Review
Let us assume that dark energy corresponds to a perfect fluid with an energy
density $\rho$, pressure $p$ and equation of state $w[z]$ as a function of the
redshift $z$, related by $p=w[z]\rho$. The case of a pure vacuum energy would
have $\rho=-p$. As long as $w[z]$ is close to -1 and varies slowly with time,
the metric solution from Einstein’s equations is taken to be approximately
that of a De Sitter geometry, i.e. for $w=-1$.
A particularly interesting case is that of a Schwarzschild black hole in the
background of the dark energy cosmic fluid. This case was studied in [14, 3,
6, 7] and the fluid accretion flowing through the black hole horizon was
solved analytically. The perfect fluid energy-momentum tensor with equation of
state $p=w\rho$ is assumed to be
$T^{\mu\nu}=\rho(1+w)u^{\mu}u^{\nu}+w\rho g^{\mu\nu}$ (1)
where $g^{\mu\nu}$ is the inverse of the Schwarzschild metric,
$g_{\mu\nu}=diag(-(1-2M/r),(1-2M/r)^{-1},r^{2},r^{2}sin^{2}\theta)$, and for
the fluid 4-velocity, $u^{\mu}u_{\mu}=-1$. Integrating the energy-momentum
conservation equation and the projection of the fluid 4-momentum into it leads
to the expression
$u\left(\frac{M}{r}\right)^{2}exp\left[\int_{\rho_{\infty}}^{\rho}\frac{d\rho^{\prime}}{\rho^{\prime}+p(\rho^{\prime})}\right]=-A$
(2)
and
$\displaystyle\left(\rho+p\right)\\!\left(1-\frac{2M}{r}+u^{2}\right)^{1/2}\\!\\!$
(3) $\displaystyle\times
exp\left[-\int_{\rho_{\infty}}^{\rho}\frac{d\rho^{\prime}}{\rho^{\prime}+p(\rho^{\prime})}\right]=C$
Following Babichev et al [7], these expressions are manipulated into one for
$r^{2}T_{0}^{r}$. Then, integrating the conservation law over the volume
within the event horizon yields
$\dot{M}=-4\pi r^{2}T_{0}^{r}=4\pi
AM^{2}\left[\rho_{\infty}+p(\rho_{\infty})\right]$ (4)
This expression can be integrated to give M as a function of time, neglecting
the cosmological time evolution of $\rho_{\infty}$:
$M(t)=\frac{M(0)}{1-\frac{t}{\tau}}.$ (5)
The timescale for the accretion of dark energy Eq. 5 is given by the parameter
$\tau$, and is $\tau=\frac{1}{(4\pi AM(0)[\rho_{\infty}+p(\rho_{\infty})])}$.
For a black hole of a mass $m=am_{s}$ which is $a$ times larger than a solar
mass $m_{s}$, the evolution timescale is roughly $\tau=10^{32}/a$ years. This
is usually much longer than the age of the universe, thus beyond observational
feasibility. But the detection possibilities of the dark energy accretion
improve dramatically for the case of black hole binaries. The reason is that
the evolution of the black hole binaries in the background of dark energy is
different from that of a single hole. Therefore, as described below, detection
of the modifications to the orbital radius and the emission of gravitational
radiation from these binaries in the background of the dark energy fluid is
within reach, [1] .
### 2.2 Evolution of Black Hole Binaries in the Background of Dark Energy
The Hulse-Taylor effect predicts a decrease of the orbital radius of the
binary due the energy lost by the emission of gravitational radiation. This
prediction has been succesfully tested. The evolution of black hole binaries
in the background of dark energy is modified relative to the Hulse-Taylor
effect, due to the accretion of dark energy by the stars in the binary. Since
the change in the mass of the stars, Eq.5, has a direct dependence on the
parameters of the dark energy being accreted, specifically on $w[z]$, then
this information is carried out on the amplitude and the power of
gravitational radiation produced [1] by the binary. The dark energy accretion
also imprints modifications in the orbital separation and merging time.
The modifications induced from the background dark energy fluid onto the
evolution of the binary, namely on the frequency $\omega$ of gravitational
waves emitted by the binary and, on the orbital separation $R$, can be derived
as follows: the change of the gravitational energy of the binary is equal to
the power lost due to gravitational radiation[5]:
$\frac{d}{dt}\left(m_{1}+m_{2}-\frac{1}{2}\frac{m_{1}m_{2}}{R}\right)=P_{GW}$
(6)
where
$P_{GW}=\frac{-32G^{4}}{5c^{5}}\left[\frac{m_{1}^{2}m_{2}^{2}(m_{1}+m_{2})}{R^{5}}\right]$
(7)
This is a temporal equation. Power losses via emission of gravitational
radiation drives the binary’s configuration to a new gravitational equlibrium
separation. As a result the orbital radius $R$ decreases and eventually the
stars inspiral and merge. For binaries immersed in the dark energy fluid, it
should be noticed that the point of gravitationally stable configurations of
the binary at each moment is now driven by two effects: the usual loss of
energy via gravitational waves emission; and, the stars changing mass (leading
to a change of the gravitational energy of the binary) due to dark energy
accretion. The masses of the two black holes in the binary, $m_{1}$ and
$m_{2}$, are increasing or decreasing with time, Eq.5, depending on $(1+w)$
being positive or negative. Therefore the two terms that induce changes in the
orbital radius and period, namely: modifications due to dark energy accretion,
and modifications due to energy losses from gravitational radiation compete
with each other and determine the evolution of the orbit.
Without loss of generality, this expression can be algebraically simplified by
taking the two stars to be of equal mass, $m_{1}=m_{2}=m$ with $m_{0}=m(0)$.
Replacing the constant mass of a black hole with the new expression, the mass
rate of change of the black holes induced by the dark energy accretion from
Eq.5, leads to a differential equation for the evolution of the binary R(t).
$\displaystyle{}R^{3}\frac{dR}{dt}=-\frac{64}{5}\frac{2m_{0}^{3}}{\left[1-2Ltm_{0}\right]^{3}}$
(8)
$\displaystyle{}-\\!\\!\left[\frac{-4LR^{4}m_{0}}{\left[1-2Ltm_{0}\right]}+8LR^{6}\right]$
where the parameter $L$ denotes $L=\frac{c^{3}}{2G^{2}m_{0}\tau}$. This
parameter contains all the modifications induced by the dark energy background
and the new modification terms due to dark energy to the orbital radius can be
tracked down from all the terms in Eqs.9 that contain $L$. It should be noted
that $L\simeq(1+w)$ is positive for quintessence type fluids; $L$ is negative
for phantom type fluids ($1+w<0$); and, it becomes identically zero for
$w=-1$. In the latter case, all the modifications due to dark energy on the
orbital radius vanish. The sign of $L\neq 0$ determines the behaviour of the
orbit, i.e. whether it grows or decreases with time for the cases when the
’$L-terms$’ dominate over the conventional gravitational waves term.
An approximate solution to Eq.2.2 for the case when dark energy changes
adiabatically is [1]
$\displaystyle R(t,w)=\\!R_{0}[1+16Lm_{0}\left(\frac{G^{2}}{c^{3}}\right)t$
(9)
$\displaystyle-32LR_{0}\left(\frac{G}{c}\right)t-\frac{64}{5}\left(4\frac{G^{3}}{c^{5}}\right)\left[\frac{8tm_{0}^{3}}{R_{0}^{4}}\right]]^{1/4},$
The fourth term in the expression Eq.9 corresponds to the conventional Hulse-
Taylor term. As a consistency check, the Hulse-Taylor equation is recovered in
the limit when our universe approaches a DeSitter geometry $w\rightarrow-1$,
(i.e. $L\rightarrow 0$). The Hulse-Taylor term describes the changes in the
orbital radius that result for the energy losses of a binary from the emitted
gravitational radiation. The terms proportional to $L$ are the new
modification terms to the evolution of black hole binaries. They account for
the effects of dark energy accretion by the system. It should be noticed that
one of the dark energy modification terms is of opposite sign to the Hulse-
Taylor term; and, the type of dark energy with $(1+w)$ positive or negative
leads to different types of evolution for the binary. An analysis of the
solution for $R[z]$ shows that the dark energy terms can dominate the
evolution of the orbital radius, Eq.9, for certain cases of supermassive black
hole binaries with large separation, quantified below. The different time
evolution of the two terms, dark energy and the conventional Hulse-Taylor one,
in Eq.9 on $R$ and $m$, especially the linear dependence of $\dot{R}$ on the
equation of state of dark energy $1+w$, allow us to discriminate among the
modifications to the orbit induced by dark energy and for probing the dark
energy equation of state $w[z]$ by observing the rate at which the orbit
change $\dot{R}$.
In order to quantify the analysis of the above expression Eq.9 and discuss the
interplay between the dark energy and Hulse-Taylor types of modifications in
the orbital radius we can parameterize the binary as follows: let the initial
mass of the star, (before modifications due to accretion), be $m_{0}=am_{s}$
where $m_{s}$ is a solar mass and $a$ a parameter; and the orbital radius be
$\beta$ times larger than the Schwarzchild radius of each star
$R_{0}=2m_{0}\beta\frac{G}{c^{2}}$. Then, the ratio of the two correction
terms to the binary’s orbit $R$ in Eq.9, the Hulse-Taylor correction due to
the emission of gravitational waves (GW), and corrections due to dark energy
(DE) accretion (terms containing $L$), is
$\frac{GWcorrection}{DEcorrection}=\frac{10^{45}}{(2\beta)^{5}(1+w)a^{2}}\geq
1$ (10)
We can use Eq.10 to quantify the classes of binaries for which the dark energy
correction terms dominate over the Hulse-Taylor term. Clearly for large enough
separations ($\beta$) or masses of the black holes ($a\simeq 10^{18}$) the
corrections due to dark energy can dominate over gravitational radiation.
Observing this effect via gravitational waves experiments, we need both
parameters of the binary to be such that they favor the dark energy
corrections over the Hulse-Taylor corrections to $R$, while at the same time
being within observable ranges of frequency windows. The frequency and
amplitude of gravitational radiation from these systems are given by
$f=\frac{10^{5}}{(2\beta)^{3/2}a}$ (11)
and
$h=\frac{1}{r}\frac{2}{\beta}a10^{3},$ (12)
where r is the distance of the binary from the observer. Eq.12 constrains the
second parameter $\beta$ not to be too large. For example LIGO is designed to
detect radiation around 150 Hz optimally, and with an amplitude greater than
$h=10^{-23}Hz^{1/2}$.[8] LISA can detect much lower frequency radiation, down
to $\sim 10^{-5}Hz$.[9]. Since for both GW experiments, the binary radius
$\beta$ can not be too large, then our requirement of Eq.10 for using binaries
to probe dark energy via their modifications on the orbital radius $R$, can be
fulfilled by considering supermassive black holes, $a\gg 1$, (see [1] for
specific examples).
### 2.3 Observation Candidates?
There have now been observations of black hole binary systems, and we can
assess the effects of dark energy accretion on these systems. One example is
galaxy 0402+379, observed in 2007 with VLBA. It has parameters $a=210^{8}$,
$2\beta=10^{6}$, and $r=10^{26}m$. This case, while not observable through
gravitational radiation, is one that is sensitive to the sign of $1+w$, and
it’s merging time is either greatly accelerated (1,000 years instead of 60,000
years) by the effect of dark energy accretion when $1+w>0$, or it is driven
apart when $1+w<0$.
Another example is Radio Galaxy OJ287, observed in 2008 with VLBA [10]. This
system is more complicated, since one of the black holes is more massive than
the other. The parameters for this system are $R_{0}\simeq 10^{20}m$, or
$2\beta\simeq 10^{7}$ and $r\simeq 3.5Mly\simeq 10^{22}m$. While it is
straightforward to treat this system with the above equations for $m_{1}\neq
m_{2}$, the point is well illustrated by a simpler system of comparable
parameters. We will use $a\simeq 10^{9}$. This gives $f\simeq 10^{-9}Hz$ and
$h\simeq 10^{-}20$. Again, this system falls outside of observable ranges for
gravitational radiation, but the merging time is shortened by three orders of
magnitude when $1+w>0$, and the black holes are pulled apart for $1+w<0$.
## 3 Conclusions
It is remarkable that localized systems like black hole binaries can be used
to provide information about dark energy. The method described here provides a
complementary way of probing the dark energy equation of state by using
supermassive black hole binaries. Its strength lies on the fact that it avoids
the noise inherited by the signal propagating through the vastness of
structure in the universe and it takes advantage of existing experiments,
initially designed to investigate gravitational waves or structure.
Supermassive black hole binaries could soon help to shed light on the nature
of dark energy.
By observing gravitational radiation from black hole binaries, we might
distinguish the waveform from a system accreting dark energy from one that
does not. Observing changes in the orbital radius over a fraction of the
binary’s period with X-ray and Radio measurement is entirely possible with our
current experiments and provides a wealth of information on $w[z]$ through the
dark energy correction terms in Eq.8,9. This method should helps us pinpoint
at least whether dark energy is a quintessence or a phantom type, or simply a
cosmological constant. Observing how the waveform differs from the
cosmological constant case gives further information about the sign of $1+w$
[11]. If $1+w>0$, the masses of the black holes will increase, they will
spiral in faster, and this will result also in a faster increase in frequency
of their gravitational radiation. If $1+w<0$ [12], the masses decrease and the
system can be pulled apart by the effects of dark energy accretion.
It is possible we are close to collecting evidence from existing observed
supermassive black hole binaries that the phantom type $(1+w)<0$ which rips
the binary apart may be already disfavored. One such binary of supermassive
black holes was recently observed [15]. An interesting question is whether
this method can be used to test theories of modified gravity and to
discriminate those from the dark energy models. The evolution of binaries on
the background of modified gravity is currently under investigation.
## References
* [1] L. Mersini-Houghton and Adam Kelleher, arXiv: gr-qc/0808.3419v1
* [2] G. Hinshaw, et al., 2009, ApJS, 180, 225-245; A. Melchiorri, L. Mersini-Houghton, C. J. Odman, M. Trodden, Phys.Rev.D68:043509.
* [3] F.C. Michel, Ap. Sp. Sc. 15, 153, (1972).
* [4]
* [5] Miser, Thorne and Wheeler, Gravitation. W.H. Freeman and co., 1973. p. 986-9.
* [6] E. Babichev, V. Dokuchaev, Y. Eroshenko, Phys. Rev. Lett. 93. arXiv:gr-qc/0402089v3
* [7] E. Babichev, V. Dokychaev, Y. Eroshenko (Moscow, INR), J. Exp. Theor. Phys. 100:528-538 (2005). arXiv:astro-ph/0505618v1
* [8] LIGO Scientific Collaboration, ”LIGO: The Laser Interferometer Gravitational-Wave Observatory.” [gr-qc/0711.3041].
* [9] Michele Vallisneri, ”LISA: Laser Interferometer Space Antenna.” http://lisa.nasa.gov, retrieved 20 May, 2009.
* [10] G.B. Taylor et al. (2006). ”Imaging Compact Supermassive Binary Black Holes with Very Long Baseline Interferometry.” Proceedings of the International Astronomical Union, 2, pp269-272.
* [11] Y. Wang, M. Tegmark, Phys. Rev. Lett.bf 92:241302,2004; D. Huterer and M. S. Turner, Phys. Rev. D 64, 123527 ( 2001); E. Linder, Phys. Rev. Lett.bf 90, 091301 (2003).
* [12] R. R. Caldwell, Phys. Lett. B545, 23 (2002); R. R. Caldwell,M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003
* [13] V. Faraoni, W. Israel, Phys. Rev. D71:064017,2005, [gr-qc/0503005]; M. Bouhmadi-Lopez, J. Jimenez Madrid, JCAP 0505:005, (2005),[astro-ph/0404540]; L. Chimento, R. Lazkoz, Mod. Phys. Lett. A19:2479-2484,(2004), [gr-qc/0405020].
* [14] T. Jacobson, Phys. Rev. Lett. 83, 2699 (1999).
* [15] T. A. Boroson, T. Lauer, [arXiv:0901.3779].
|
arxiv-papers
| 2009-06-08T18:57:04 |
2024-09-04T02:49:03.230478
|
{
"license": "Public Domain",
"authors": "Laura Mersini-Houghton, Adam Kelleher",
"submitter": "Adam Kelleher",
"url": "https://arxiv.org/abs/0906.1563"
}
|
0906.1645
|
# Stochastic Quantization of the Hořava Gravity
Fu-Wen Shu [email protected] College of Mathematics and Physics, Chongqing
University of Posts and Telecommunications, Chongqing, 400065, China Yong-Shi
Wu [email protected] Department of Physics and Astronomy, University of
Utah, Salt Lake City, UT 84112, USA
###### Abstract
The stochastic quantization method is applied to the recent proposal by Hořava
for gravity. We show that in contrast to General Relativity, the Hořava’s
action, satisfying the detailed balance condition, has a stable, non-
perturbative quantum vacuum when the DeWitt parameter $\lambda$ is not greater
than $1/3$, providing a possible candidate for consistent quantum gravity.
###### pacs:
04.60.NC
Introduction. The goal of formulating a consistent and renormalizable quantum
theory of gravity has been pursued for more than half century. Attempts of
quantizing General Relativity (Einstein’s theory of gravitation) have met
tremendous difficulties. On one hand, the canonical quantization is shown to
be perturbatively non-renormalizable in four dimensionsthooft ; weinberg and,
therefore, loses predictability, because the Einstein-Hilbert action is non-
polynomial. On the other hand, the Euclidean path integral approach suffers
the indefiniteness problemhawking : Namely the Einstein-Hilbert action is not
positive-definite, because conformal transformations can make the action
arbitrarily negative.
A recent effort attempting to overcome these difficulties is the proposal made
by Hořavahorava09 . (For the ideas that led to this proposal, see also refs.
horava1 ; horava2 .) This proposal is a non-Lorentz invariant theory of
gravity in 3+1 dimensions, inspired by the Lifshitz modellifshitz studied in
condensed matter physics. At the microscopic (ultraviolet) level this model
exhibits anisotropic scaling between space and time, with the dynamical
critical exponent $z$ set equal to 3. (Namely, the action is invariant under
the scaling $x^{i}\rightarrow bx^{i}(i=1,2,3),t\rightarrow b^{z}t$, where
$z\neq 1$ violates the Lorentz symmetry.) The action is assumed to satisfy the
so-called detailed balance condition, and is renormalizable by power counting.
It is argued that the renormalization group flow in the model approaches an
infrared (IR) fixed point theory with $z=1$, thus Einstein’s General
Relativity (with local Lorentz symmetry) is naturally emergent or recovered at
the macroscopic level. It is this perspective that has enabled the proposal to
attract a lot of interests in recent literature. Many papers have appeared to
study the classical solutions or consequences of the Hořava’s proposal (e.g.
see refs. sm ; calcagni ; kiritsis ). A number of fundamental questions
remained unanswered. In this letter we report a study of the most fundamental
questions on Hořava gravity: whether the action can really be quantized in a
consistent and non-perturbative manner? If yes, whether this will put any
constraint(s) on the parameters appearing in the action or not? (A recent
paperorlando on the renormalizability of the model did not address these
issues, assuming no problem with quantization.)
Among the three existing – canonical, path integral and stochastic –
quantization approaches, only the last (stochastic quantization) is
constructive through stochastic differential equation, so that the question of
whether a stable vacuum (ground state) really exists or not can be easily
investigated and answered. Also it has the great advantagewu of no need for
gauge-fixing when applied to theories with gauge symmetry. In this letter we
apply stochastic quantization to the Hořava gravity, where the gauge symmetry
is spatial diffeomorphisms. We will show that the quantized theory with a
stable vacuum indeed exists only when the parameter $\lambda$ in the DeWitt
metric in the space of three-metrics is not greater than a critical value 1/3:
(i.e. $\lambda\leq\lambda_{c}=1/3$). This is the range of the values of
$\lambda$ for which Hořava’s action may make sense for a consistent quantum
theory of gravity. (In contrast, stochastic quantization of General Relativity
does not lead to a stable vacuum (ground) state. See below.)
Preliminaries. Assume the spacetime allows a $(3+1)$-decomposition:
$ds_{4}^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}-N^{i}dt)(dx^{j}-N^{j}dt)\,,$ (1)
where $g_{ij}(i,j=1,2,3)$ is the 3-metric, $N$ and $N_{i}$ are the lapse
function and shift vector, respectively. The Hořava action with $z=3$ is given
byhorava09
$S=\int
dtd^{3}x\sqrt{g}N\left[\frac{2}{\kappa^{2}}K_{ij}\mathcal{G}^{ijkl}K_{kl}+\frac{\kappa^{2}}{8}E^{ij}\mathcal{G}_{ijkl}E^{kl}\right],$
(2)
where $g$ denotes the determinant of the 3-metric $g_{ij}$ and $\kappa^{2}$ is
the coupling constant, to be identified with $32\pi Gc$ in the IR regime with
$z=1$ ($G$ and $c$ the Newton’s gravitational constant and the speed of light,
respectively). The extrinsic curvature $K_{ij}$ and the DeWitt metric
$\mathcal{G}^{ijkl}$ in (2) are defined by
$\displaystyle K_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2N}(\dot{g}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i}),$ (3)
$\displaystyle\mathcal{G}^{ijkl}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(g^{ik}g^{jl}+g^{il}g^{jk}\right)-\lambda
g^{ij}g^{kl}$ (4)
with $\lambda$ a free parameter. The potential term in (2), when $E^{ij}$ is
given by $\sqrt{g}E^{ij}=\frac{\delta W}{\delta g_{ij}},$ is said to satisfy
the so-called detailed balance condition. Hořava took $W$ to be
$W=\frac{1}{w^{2}}\int\omega_{3}(\Gamma)+\mu\int
d^{3}x\sqrt{g}(R-2\Lambda_{W}).$ (5)
Here $\mu\,,w$ and $\Lambda_{W}$ are coupling constants, and $\omega_{3}$ is
the gravitational Chern-Simons term:
$\omega_{3}\equiv\mbox{Tr}\left(\Gamma\wedge
d\Gamma+\frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma\right),$ (6)
with $\Gamma$ the Christoffel symbols. Simple dimensional analysis for the
coupling constants shows that the theory is ultraviolet (UV)
renormalizablecalcagni . The renormalizability beyond the power counting of
this theory has recently been confirmed in orlando , assuming no problem with
quantization. Here we will examine the more fundamental question of the non-
perturbative existence of quantum vacuum.
Stochastic Quantization. Stochastic quantizationwu has been proved to be an
effective tool for quantizing a field theory, in particular a gauge
theoryhuffel ; namiki . Stochastic quantization is based on the principle that
quantum dynamics of a $d$-dimensional system is equivalent to classical
equilibrium statistical mechanics of a $(d+1)$-dimensional system. The essence
of stochastic quantization is to use a stochastic evolution – the Langevin
equation – in fictitious time, driven by white noises, to construct the
equilibrium state corresponding to the quantum ground state. The existence of
an equilibrium state can be proved or disproved by studying the corresponding
Fokker-Planck equation associated with the Langevin equation.In this spirit,
we start with the Langevin equation of the Hořava gravity:
$\displaystyle\begin{cases}\dot{N}=-\frac{1}{\sqrt{g}}\frac{\delta
S_{E}}{\delta N}+\eta,\\\ \dot{N_{i}}=-\frac{1}{\sqrt{g}}\frac{\delta
S_{E}}{\delta N^{i}}+\zeta_{i},\\\
\dot{g}^{I}=-\mathcal{G}^{IJ}\partial_{J}S_{E}+\xi^{I},\end{cases}$ (7)
where the dot represents derivative with respect to the fictitious time $\tau$
and following notations have been introduced:
$g_{ij}\equiv g^{I},\ \ \ \ \mathcal{G}^{IJ}\equiv\mathcal{G}_{ijkl},\ \ \ \
\partial_{I}S_{E}\equiv\frac{1}{\sqrt{g}}\frac{\delta S_{E}}{\delta g_{ij}}.$
In eq. (7), $\eta^{I}$, $\zeta_{i}$ and $\xi^{I}$ are noises, and $S_{E}$ is
the Euclidean version of the action (2).
Note that the indices $I$, $J$ (=1,2,…6) are raised and lowered by
$\mathcal{G}^{IJ}$ and its inverse $\mathcal{G}_{IJ}$. The stochastic
correlation of a gauge invariant functional $\mathcal{F}(N,N_{i},g_{I})$ is
defined as the expectation value of the functional with respect to the noises
$<\mathcal{F}(N,N_{i},g_{I})>\sim\int\mathcal{D}[\eta]\mathcal{D}[\zeta]\mathcal{D}[\xi]\mathcal{F}(N,N_{i},g_{I})\exp\left[-\frac{1}{4}\int
d\mbox{\psyra
t}d^{3}xd\tau\sqrt{g}N(\eta^{2}+g^{ij}\zeta_{i}\zeta_{j}+\mathcal{G}^{IJ}\xi_{I}\xi_{J})\right]$
(8)
where $g^{ij}$ and $\mathcal{G}^{IJ}$ are solutions of the Langevin equation
(7) and hence are functions of $\zeta^{i}$ and $\xi^{I}$, respectively. The
Wick rotation to imaginary time t has been applied and $\tau$ is the
fictitious time. Eq. (8) indicates that the noises $\zeta_{i}$ and $\xi_{I}$
are not Gaussian. As suggested in orlando , one can overcome this difficulty
by introducing a set of new noises via vielbein. That is $\zeta^{a}\equiv
e^{a}{}_{i}\zeta^{i}$, $\xi^{A}\equiv E^{A}{}_{I}\xi^{I},$ and its inverse
$\zeta^{i}=e^{i}{}_{a}\zeta^{a}$, $\xi^{I}=E^{I}{}_{A}\xi^{A},$ where
$e^{a}{}_{i}$ and $E_{A}{}^{I}$ are the vielbeins. The following relations
hold
$\displaystyle e_{a}{}^{i}e_{b}{}^{j}g_{ij}=\delta_{ab},\ \ \
E_{A}{}^{I}E_{B}{}^{J}\mathcal{G}_{IJ}=\delta_{AB},$ (9) $\displaystyle
e_{a}{}^{i}e_{b}{}^{j}\delta^{ab}=g^{ij},\ \ \
E_{A}{}^{I}E_{B}{}^{J}\delta^{AB}=\mathcal{G}^{IJ}.$ (10)
The new noises turn out to be Gaussian and we have
$\displaystyle<\eta(x,\tau)>=0,\ \ <\zeta^{a}(x,\tau)>=0,\ \
<\xi^{A}(x,\tau)>=0,$ (11)
$\displaystyle<\eta(x,\tau)\eta(y,\tau^{\prime})>=2\delta(x-y)\delta(\tau-\tau^{\prime}),$
(12)
$\displaystyle<\zeta^{a}(x,\tau)\zeta^{b}(y,\tau^{\prime})>=2\delta^{ab}\delta(x-y)\delta(\tau-\tau^{\prime}),$
(13)
$\displaystyle<\xi^{A}(x,\tau)\xi^{B}(y,\tau^{\prime})>=2\delta^{AB}\delta(x-y)\delta(\tau-\tau^{\prime}).$
(14)
(Here $x$ stands for Euclidean coordinates $(x^{i},\mbox{\psyra t})$.) The
Langevin equation (7) then becomes
$\displaystyle\begin{cases}\dot{N}=-\frac{1}{\sqrt{g}}\frac{\delta
S_{E}}{\delta N}+\eta,\\\ \dot{N_{i}}=-\frac{1}{\sqrt{g}}\frac{\delta
S_{E}}{\delta N^{i}}+\zeta_{a}e^{a}{}_{i},\\\
\dot{g}^{I}=-\mathcal{G}^{IJ}\partial_{J}S_{E}+\xi^{A}E_{A}{}^{I},\end{cases}$
(15)
and the correlation functional is redefined with respect to $\eta$,
$\zeta^{a}$ and $\xi^{A}$ by
$<\mathcal{F}(N,N_{i},g_{I})>\sim\int\mathcal{D}[\eta]\mathcal{D}[\zeta]\mathcal{D}[\xi]\mathcal{F}(N,N_{i},g_{I})\exp\left[-\frac{1}{4}\int
d\mbox{\psyra
t}d^{3}xd\tau\sqrt{g}N(\eta^{2}+\zeta^{a}\zeta_{a}+\xi^{A}\xi_{A})\right],$
(16)
which is obviously Gaussian as desired.
To study whether the Langevin process (15) really converges to a stationary
equilibrium distribution, we examine the probability density functional
associated with it:
$P(N,N^{i},g_{I},\tau)=\frac{\exp\left[-\frac{1}{4}\int d\mbox{\psyra
t}d^{3}xd\tau\sqrt{g}N(\eta^{2}+\zeta^{a}\zeta_{a}+\xi^{A}\xi_{A})\right]}{\int\mathcal{D}[\eta]\mathcal{D}[\zeta]\mathcal{D}[\xi]\exp\left[-\frac{1}{4}\int
d\mbox{\psyra
t}d^{3}xd\tau\sqrt{g}N(\eta^{2}+\zeta^{a}\zeta_{a}+\xi^{A}\xi_{A})\right]}.$
(17)
We introduce
$Q(N,N^{i},g_{I},\tau)\equiv P(N,N^{i},g_{I},\tau)e^{S_{E}/2}.$ (18)
and the Fokker-Planck equation for the probability distribution is
$\displaystyle\frac{\partial
Q(N,N^{i},g_{I},\tau)}{\partial\tau}=-\mathcal{H}_{FP}Q(N,N^{i},g_{I},\tau),$
(19)
where the Fokker-Planck Hamiltonian $\mathcal{H}_{FP}$ is of the form
$\displaystyle\mathcal{H}_{FP}=a^{\dagger}a+g^{ij}a_{i}{}^{\dagger}a_{j}+\mathcal{G}^{IJ}\mathcal{A}_{I}{}^{\dagger}\mathcal{A}_{J}.$
(20)
Here
$a=i\pi+\frac{1}{2}\frac{1}{\sqrt{g}}\frac{\delta S_{E}}{\delta N},\ \
a^{i}=i\pi^{i}+\frac{1}{2}\frac{1}{\sqrt{g}}\frac{\delta S_{E}}{\delta
N_{i}},\ \ \mathcal{A}^{I}=i\pi^{I}+\frac{1}{2}\partial^{I}S_{E},$
with $\pi$, $\pi^{i}$ and $\pi^{I}$, respectively, the conjugate momenta of
$N$, $N^{i}$ and $g^{I}$: $\pi=-i\frac{1}{\sqrt{g}}\frac{\delta}{\delta N}$,
$\pi^{i}=-i\frac{1}{\sqrt{g}}\frac{\delta}{\delta N_{i}}$,
$\pi_{I}=-i\partial_{I}$. The time independent eigenvalue equation associated
with Eq. (19) is
$\displaystyle\mathcal{H}_{FP}Q_{n}(N,N^{i},g_{I},\tau)=E_{n}Q_{n}(N,N^{i},g_{I},\tau).$
(21)
The solutions of Eq. (19) lead to the general solution
$\displaystyle
P(N,N^{i},g_{I},\tau)=\sum_{n=0}^{\infty}a_{n}Q_{n}(N,N^{i},g_{I},\tau)e^{-S_{E}/2-E_{n}\tau}.$
(22)
The stationary candidate for the equilibrium state is given by
$Q_{0}(N,N^{i},g_{I})=e^{-S_{E}/2}$ with $E_{0}=0$. From the above formula we
see that the theory will approach an equilibrium state for large $\tau$ if and
only if all other $E_{n}>0$. To find the condition(s) under which the Fokker-
Planck Hamiltonian (20) is non-negative definite, we note that the sum of the
first two terms $(a^{\dagger}a+g^{ij}a_{i}{}^{\dagger}a_{j})$ always respects
this property. So we only need to find condition(s) under which the
eigenvalues of the DeWitt metric $\mathcal{G}^{IJ}$ are all non-negative. By a
straightforward computation, the desired condition is found to be $\lambda\leq
1/3$: When $\lambda<1/3$, $\mathcal{G}^{IJ}$ is positive definite; if
$\lambda>1/3$, one and only one eigenvalue of $\mathcal{G}^{IJ}$ becomes
negative. At the critical value $\lambda=1/3$, one eigenvalue of
$\mathcal{G}^{IJ}$ becomes zero, while all others remain positive.
Thus the Fokker-Planck Hamiltonian is non-negative definite if $\lambda\leq
1/3$, and the theory approaches an equilibrium in this case: It follows from
eq. (22) that
$\displaystyle\lim_{\tau\rightarrow\infty}P(N,N^{i},g_{I},\tau)=a_{0}e^{-S_{E}},$
(23)
where $a_{0}$ is determined by the normalization condition. Note that this
result is independent of the initial conditions. Any equal-time correlation
function (16), if invariant under spatial diffeomorphisms, tends to its
equilibrium value for large time $\tau$. Therefore, though the solution given
by (23) is always a stationary state for the Fokker-Planck equation, it
represents an equilibrium state (or a stable ground state) reached at large
time $\tau$ only when $\lambda\leq 1/3$. In contrast, a similar result would
not be obtained with stochastic quantization of Einstein’s gravity, which
corresponds to $\lambda=1>1/3$, since the associated Fokker-Planck Hamiltonian
is not positive definite and hence does not lead to an equilibrium state at
large fictitious times.
In the above derivation, the detailed balance condition is crucial for the
Hořava gravity to have a stable vacuum when $\lambda<1/3$. In fact, with the
detailed balance condition satisfied at short distances, $S_{E}$ is of the
form
$S_{E}=\int\mathcal{G}^{IJ}(K_{I}K_{J}+\alpha E_{I}E_{J}),$
where $\alpha>0$ and $E_{I}=\partial_{I}W$ with $W$ given by (5). $S_{E}$ has
a similar structure to eq. (20), so it is positive definite for $\lambda<1/3$
and indefinite for $\lambda>1/3$. As a consequence, the state (23) is a
physical ground state for $\lambda<1/3$ and is unstable for $\lambda>1/3$.
We have seen that $\lambda_{c}=1/3$ is a critical value for the theory: Above
it the quantized theory does not make sense, while the opposite is true below
it. Exactly at $\lambda=\lambda_{c}$, extra zero modes develop for the DeWitt
metric $\mathcal{G}^{IJ}$ and, hence, for eq. (19) as well. This implies that
the gauge symmetry of the theory is enhanced, which now includes local Weyl
transformations as already observed in ref. horava09 . It would be extremely
interesting to understand the fate of the enhanced gauge symmetry in the
quantized theory. Anyway, in principle stochastic quantization method should
be applicable at $\lambda=1/3$, and the appearance of extra zero modes does
not destroy the stability of the new vacuum, though there are subtle issues to
be resolved.
Conclusions and Discussions. In summary, we have applied stochastic
quantization to the Hořava gravity. By analyzing the associated Fokker-Planck
equation, we have found that with $\lambda<1/3$ the system will approach to
equilibrium as the fictitious time goes to infinity, giving rise to a stable
vacuum state for the quantized theory. The key to this property is the
detailed balance condition obeyed by the Hořava action. When $\lambda>1/3$,
stochastic quantization does not make sense because of development of a
negative mode. The $\lambda=1/3$ case would be probably alright, but needs
more careful examination.
In ref. horava09 , to make sense of the speed of light in the IR regime with
$z=1$, one needs $\Lambda_{W}/(1-3\lambda)$ to be positive. Our constraint
$\lambda<1/3$ for the stability of gravity vacuum further constrains the
cosmological constant to be positive: $\Lambda_{W}>0$. This agrees with
cosmological observationswmap5 .
Our suggestion opens the door for using stochastic quantization to numerically
study the quantized Hořava gravity, in particular to check whether the
renormalization group would indeed change the value of $z$ from $z=3$ in the
UV regime to $z=1$ in the IR regime.
Finally, it should be noted that the stochastic quantization applied in this
letter is the standard one that introduces a fictitious time. This is
different from the one used in ref. orlando , where the time for stochastic
evolution is identified with the real time. In this reference, for the purpose
of studying the renormalizability of Hořava gravity, they have explored the
fact that like any Lifshitz-type models, the Hořava gravity can be viewed as
stochastic quantization of a lower dimensional theorydijkgraaf , which in the
present case is three-dimensional topological massive gravity.
Acknowledgments. This work is partially supported by a grant from FQXi. One of
us (F.W.) thanks Department of Physics and Astronomy, University of Utah for
warm hospitality, where this work was done. F.W. is supported by a grant from
CQUPT. YSW is supported by US NSF grant PHY-0756958.
## References
* (1) G. ’t Hooft and M. Veltman, Ann. Inst. Henri Poincaré 20, 69 (1974).
* (2) S. Weinberg, Ultraviolet Divergences in Quantum Theories of Gravitation, in: General Relativity. An Einstein Centenary Survey (Cambridge University Press, 1980) eds: S. W. Hawking and W. Israel.
* (3) G.W. Gibbons, S.W. Hawking and M.J. Perry, Nucl. Phys. B138 141 (1978).
* (4) P. Hořava, Phys. Rev. D 79 084008 (2009).
* (5) P. Hořava, JHEP 0903, 020 (2009).
* (6) P. Hořava, Phys. Rev. Lett. 102 161301 (2009)
* (7) E.M. Lifshitz, On the Theory of Second-order Phase transition I & II, Zh. Eksp. Teor. Fis. 11, 255 & 269 (1941).
* (8) S. Mukohyama, Scale-invariant cosmological perturbations from Hořava-Lifshitz gravity without inflation arXiv: 0904. 2190 [hep-th]
* (9) G. Calcagni, Cosmology of the Lifshitz universe, arXiv:0904.0829 [hep-th].
* (10) E. Kiritsis and G. Kofinas, Hořava-Lifshitz cosmology, arXiv:0904.1334 [hep-th]; T. Takahashi and J. Soda, Chiral primordial gravitational waves from a Lifshitz point, arXiv:0904.0554 [hep-th]; J. Kluson, Branes at quantum criticality, arXiv:0904.1343 [hep-th]; R. Brandenberger, Matter Bounce in Horava-Lifshitz Cosmology , arXiv: 0904.2835 [hep-th]; H. Nikolic, Horava-Lifshitz gravity, absolute time, and objective particles in curved space, arXiv: 0904.3412 [hep-th]; R.-G. Cai, Y. Liu, Y.-W. Sun, On the $z=4$ Horava-Lifshitz Gravity, arXiv: 0904.4104 [hep-th]. B. Chen, S. Pi, J.Z. Tang, Scale Invariant Power Spectrum in Hořava-Lifshitz Gravity without Matter, arXiv: 0905.2300 [hep-th].
* (11) D. Orlando and S. Reffert, On the renormalizability of Hořava-Lifshitz-type Gravtities, arXiv:0905. 0301 [hep-th].
* (12) G. Parisi and Y.-S. Wu, Sci. Sin. 24 483 (1981).
* (13) P. H. Damgaard and H. Hüffel, Phys. Rep. 152 227 (1987).
* (14) M. Namiki, Stochatic Quantization, (Springer Verlag, Heidelberg, 1992)
* (15) For example, for a recent complition of data, see E. Komatsu et al., Astrophys. J. Suppl. 180, 330 (2009).
* (16) R. Dijkgraaf, D. Orlando and S. Reffert, ”Relating Field Theories via Stochastic Quantization”, arXiv: 0903.0732.
|
arxiv-papers
| 2009-06-09T07:03:51 |
2024-09-04T02:49:03.246002
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fu-Wen Shu and Yong-Shi Wu",
"submitter": "Fu-Wen Shu",
"url": "https://arxiv.org/abs/0906.1645"
}
|
0906.1755
|
10–17
# Astronomy and the Media: a love story?
Henri M.J. Boffin ESO
Karl-Schwarzschild-str. 2, 85748 Garching, Germany
email: [email protected]
(2009)
###### Abstract
With the availability of nice images and amazing, dramatic stories, the
fundamental questions it addresses, and the attraction is exerces on many, it
is often assumed that astronomy is an obvious topic for the media. Looking
more carefully, however, one realises that the truth is perhaps not as
glamorous as one would hope, and that, although well present in the media,
astronomy’s coverage is rather tiny, and often, limited to the specialised
pages or magazines.
###### keywords:
science communication, media, astronomical topics, press releases
††volume: 260††journal: The Rôle of Astronomy in Society and Culture††editors:
D. Valls-Gabaud & A. Boksenberg, eds.
## 1 Introduction
In astronomy as in other scientific or societal fields, communication is an
important aspect that no single organisation can overlook. Especially public
research organisations should be accountable to the public for the tax money
they use. This is only possible if the public is well informed. But this is
even more crucial in order to secure additional funding for new projects. As
one scientist said, perhaps a little bit too provocatively, “the one percent
spent on outreach allows one to get the 99 percent to have the project done”.
This is most likely too strong a statement but the general idea is there.
Communication is also important to entertain the necessary excellent relations
with the local communities – some of the large astronomical observatories know
a lot about this. Communication is also essential for astronomy to fulfil a
fundamental role in modern society: attracting bright youngsters to scientific
careers. Although girls and boys are more and more moving away from science,
there is a great need for future scientists. And even if the young people
won’t become scientists, it is important that they are sensitive to science as
a whole: as grown-ups, they won’t be able to avoid relying on science in their
daily life, and they will have to take decisions with a scientific dimension.
For all these reasons, the communication of research organisation will address
various target groups: general public, scientists, policy-makers, educators,
and the industry. But with limited resources, one needs amplifying outlets to
reach a significant fraction of the targeted audience. It is indeed impossible
to prepare all types of communication material, with different emphasis, at
all levels of complexity, and in all languages. One needs to rely on specific
amplifiers. Media outlets are one of these. Indeed, not only are journalists
trained to adapt the material to their public, which they know very well, but
the public get informed about science through these channels. The 2007
Eurobarometer on “Scientific research in the media” (Eurobarometer, 2007)
shows for example that 61% of respondents in the European Union get informed
about science watching TV programmes, 49% reading science articles in general
newspapers and magazines, 28% through the internet, 26% listening to radio,
and 22% buying specialised press. Similar numbers are observed in the US.
Obviously, the media are an important channel to communicate science. However,
there are caveats. The first one is that science on TV represents at most 2%
of all news shown. The other is that studies have revealed that only a quarter
of all adults can read and understand the stories in the science sections of
quality newspapers.
The crucial question is nevertheless whether the media are indeed a efficient
channel for communicating astronomy. This is clearly a difficult question
which can be answered in several ways. Before briefly attempting to do so, let
me make a general remark. As shown above and in various studies, there is no
doubt that the media play a a very important role by raising public awareness
about science and its results, but it is doubtful how much the media are
really able to teach science to the wide public ([West, 2004]). And one should
realise that this is not an easy task. In their study of the public
understanding of scientific terms and concepts, the US National Science
Foundation ([NSF, 2004]) found that less than 15% of people understand the
term molecule while less than 50% know that the Earth goes around the Sun once
a year! Starting to talk about gamma-ray bursts, redshifts, galaxies or
interferometry represent thus formidable challenges. Scientists and science
communicators must thus set realistic goals when interacting with the media
and the public, and recognise that other activities are required to transform
curiosity into knowledge such as the internet, public events, science centres,
and so on. A nice example of such programme, trying to exploit several avenues
was the Venus Transit Programme (Boffin & West, 2004, 2005). Other examples
have been successfully organised in the framework of the International Year of
Astronomy 2009.
Coming back to our main question, at first sight there are many reasons to be
optimistic and to think that astronomy and the media have a love affair. For
example, the American reference newspaper The New York Times online science
section has two specific subsections, one on environment and the other on
space & cosmos! Similarly, the British magazine New Scientist has a rather
successful specific Space section, and one should not forget that the BBC Sky
at Night programme is the longest running television series, existing since
1957, although to be fair, one should admit that it is no more shown during
prime time but very late in the evening. Here again there is an important
caveat, which is that often space and astronomical news are put together, but
their share is far from equal. The NSF 2008 study of S&T attitudes and
understanding reveal that the NASA Space Shuttle Programme has taken a very
large share of all science related news in 2005 and 2006, but this is of
course not astronomy as such. Another important unfortunate aspect is the
general tendency for media to cut down on science coverage. As a journalist of
the french newspaper “Le Monde” told me, from the 10 journalists working for
the science section in 1998, only 4 are still in place ten years later. All
others where moved to other sections.
## 2 Does astronomy sell?
In order to try to be a little bit more quantitative, I looked at the US
magazine Time. Since this main street magazine exists, astronomy has been
featured no less than 12 times on its cover. About once every five years or
so. This would be a nice result per se, especially when by comparison, biology
had only 4 covers in the same period, and chemistry only 9 (in the latter
case, most of them having appeared before 1965). However, when looking at
other scientific fields, things start to be less exceptional. History was
featured 24 times, and environment took the seat 90 times. The overall winner
is definitively medicine which was featured on 248 covers. This is 20 times as
much as astronomy! The same trend can be seen in the number of articles
dealing with the various topics that appear in the magazine. With 598 articles
published from 1923 till nowadays, astronomy comes well behind most other
scientific topics. Archeology, biology, chemistry, physics, environment, all
do better with, respectively, 1031, 1503, 2240, 2290, and 7764 articles. And
again medicine is the great winner with no less than 11814 articles, 20 times
as many as the one devoted to astronomy.
This first superficial quantitative study clearly illustrates that while the
media do not hesitate to talk about the greatest discoveries in astronomy, it
is far from being the most loved of journalists and editors. Is there any
logic behind this? Given what I stated above, that journalists know their
readers, I would assume so. Looking back at an Eurobarometer – from 2005 this
time ([Eurobarometer, 2005]), it is interesting to see that when asked “which
science and technology developments are you most interested in?”, astronomy
takes only the 6th place, with 23% of respondents choosing it. People are more
interested in economics and social sciences (24%), the internet (29%),
humanities (30%), the environment (47%), and medicine (61%). There is clearly
a logic, although one could invoke the ubiquitous chicken and egg problem as a
reason for this situation. Are journalists providing stories on subjects that
are most interesting to people or are people interested in the stories given
by the journalists? As always, the truth must lie in the middle, but it is
perhaps no such surprise that what interests the majority of people is their
health. A cause for optimism can be found however in the fact that the
comparison between the 2001 and 2005 Eurobarometer surveys reveals an increase
of 6% over 4 years in the percentage of people interested in astronomy. Let us
hope that the International Year of Astronomy, with its florilege of
activities, will lead to a continuation of this trend.
## 3 Astronomy topics
The New York Times science writer John Noble Wilford (as cited by Maran et
al., 2000) stated that the topics most likely to cause public impact are
mysterious and catastrophic subjects. Astronomy is not devoid of these and
likely candidates would be subjects such as dark matter, black holes,
exoplanets, or Near-Earth Objects on collision course with our planets.
It seems that press offices are not unaware of this and already make a pre-
selection along these lines, although some subjects seem more difficult to
deal with than others. Here is the distribution of subjects in the 144 press
releases distributed on the American Astronomical Society (AAS) mailing list,
to which about 1300 science journalists are subscribed worldwide, in September
and in November 2008:
Solar System 52 New Facilities 15 Exoplanets 12 Awards, fellowships, contests
10 Stars, supernovae 10 Black holes 9 Press photos 6 Galaxies 4 Dark matter 2
Cosmology 2
Among the press releases distribued by the AAS, one can note the large place
taken by the major players. Out of the 144 press releases mentioned above, 61
were issued by NASA (or related to NASA), 17 by ESA and 15 by ESO. The large
presence of NASA and ESA could also explain the predominance of solar system
stories, as these organisations tend to also devote a large part of their
communication to their solar system space missions. But again, things appear
more tricky. Looking at the distribution of topics in all ESO press releases
issued between 2004 and 2008 (for a total of 228), one can see that the solar
system is also taking its lion’s share:
Solar System 52 Press photos 43 Awards, organisation, contests 42 Stars,
supernovae 38 New Facilities 30 Exoplanets 20 Distant Universe 11 Gamma-ray
bursts 9 Galaxies 9 Black holes 7 Milky Way 6 Dark matter 2
ESO being the European intergovernmental organisation for ground-based
astronomy, with observatories located in Chile, the space mission argument
does not hold here.
Another way to get into the news (Maran et al. 2000) is to to use a
superlative: biggest, most distant, closest, brightest, and so on. Looking at
the titles of a few press releases mentioned above show that press officers
and scientists are not shy of using these, as shown in the following list:
* •
Closest Look Ever at the Edge of A Black Hole
* •
Analysis Begins on Phoenix Lander’s Deepest Soil Sample
* •
First Picture of Likely Planet Around Sun-Like Star
* •
Most Dark Matter-Dominated Galaxy in Universe
* •
The Deepest Ultraviolet Image of The Universe Yet
* •
Gemini Releases Historic Discovery image of Planetary “First Family”
* •
Gamma-Ray Burst was Aimed Squarely at Earth.
## 4 The place of astronomy
The AAS mailing list is an important source of information for science
journalists on astronomy and can clearly serve as a good representation of
what science journalists are exposed to. This is particularly relevant as
Madsen (2001, 2003) has shown that the astronomy covered in the media finds
most of the time its origin in press releases. The first thing to remark is
that journalists have a large choice of stories. The AAS distributes typically
80 press releases per month, or about 4 per working day. One should realise,
however, that this is still only a very tiny fraction of all scientific press
releases received by journalists. Looking at the European science agency
Alphagalileo, astronomy covers only about 10% of all scientific press releases
they distribute. Journalism is really about making a choice and in such
conditions, one can be happy if some astronomical news get covered. This is of
course not the only place where choices are made. Taking the example of ESO,
the European Southern Observatory, one can note that in 2008 there were more
than 700 refereed scientific papers published, while there were only 50 press
releases, of which only half were based on a scientific paper (the others
being organisation news, instrumentation news or press photos). At the sourcce
there is thus already a selection by a factor 30! All in all, the chances that
an astronomical scientific paper will be reported upon in the media is less
than one in a few hundreds.
Madsen (2001) in his study “Stars in the Media”, in which he looked at the
coverage of astronomy and space science in broadsheet papers in the United
Kingdom, France, Spain, Sweden and Denmark, provided some useful conclusions:
* •
The choice of topics is influenced by national (cultural, political) aspects,
but the narrative (story, rhetoric) is rather uniform;
* •
Fundamental research is reported within a narrow scientific frame;
* •
Articles on astrophysics/space currently occupy approximatively 0.1% of
leading European newspapers;
* •
There is much more emphasis on health/environment than on astronomy;
* •
Science is mostly presented in special sections.
Madsen also emphasised that more effort should be invested to show the role of
fundamental research for societal development and general culture, and that
this may also attract more interest. I can’t agree more. Despite the fact that
astronomy may be considered humankind’s boldest attempt to understand the
world in which we live, addressing fundamental questions such as “are we
alone?”, “what is the Universe made of?”, and “how did it all begin?”, which
have deep philosophical, religious, and societal impacts, astronomy is too
often limited to the science sections that are accessed by a small audience.
We need to bring the message home to the editors that astronomy is not for
‘geeks’ only, but deserves a more prominent place in the media.
## 5 Conclusion
When looking at the presence of astronomy and the media, it is also
interesting to have the opinion of the journalists themselves. I have
therefore conducted a small survey via e-mail to all journalists subscribed to
the ESO media mailing list. This is by no means supposed to be a
scientifically accurate survey, but is useful to get a first glance at the
‘other side’. I submitted to the journalists a series of 5 questions, which
are indicated below as well as their answers. Some interesting facts come out.
Most journalists said that they run between 2 and 3 astronomical stories per
month, with some running a few more. Representatives of the online media were
generally running more than 5 stories per month. This is due to the fact, as
one journalist put it, that “online, space is infinite” and there is not so
much struggle with other subjects. This illustrates that given time and space,
journalists do find astronomy stories interesting. The majority of journalists
said that they have no a priori about the possible topics to be run, and that
the most important when selecting the story is the subject and the
availability of a nice image or a video. Some journalists highlighted
nevertheless exoplanets and the solar system as the topics they will most
likely write about. It seems also that once journalists have made their mind
into writing a story, it is not difficult to convince their editor to run it.
They also acknowledge, however, that “they don’t make it to the front page”
and are often confined in special sections. And, finally, it is perhaps
revealing that almost two-third of the journalists thought that astronomy has
the place it deserves in the media. It is also important to note that the
journalists said that they won’t necessarily increase their coverage of
astronomy just because it is the International Year of Astronomy. Good stories
is what they want and need.
Small survey of science journalists
1\. How often would you run a story related to astronomy per month?
1 11% 2-3 37% 4-5 19% More than 5 33%
2\. What are the topics most likely to be run? (several answers possible)
None in particular 55% Exoplanets 33% Solar system 29% Cosmology 15% Stars and
nebulae 11% Galaxies 7%
3\. What is most important when selecting the story? (several answers
possible)
Subject 92% Availability of a nice image or video 74% Nationalities of the
scientists involved 22% Names and host institutions of the scientists involved
9% Institution issuing the press release 0%
4\. Do you find it difficult to run an astronomical story past the chief
editor?
NO 80% YES 20%
5\. Do you think astronomy has the place it deserves in the media?
YES 62% NO 38%
It is therefore clear that journalists appear to be keen to cover astronomy in
the media and that most major breakthroughs are covered. ESO estimates a
yearly readership in newspapers and magazines of tens of million people
worldwide, while it appeared in hundreds of TV news reports or documentaries,
potentially reaching hundreds of millions of viewers. The impact is
undeniable. This shouldn’t hide the fact that more efforts should be done for
astronomy to be dealt outside of the special science sections, taking into
account its important societal and cultural aspects.
## References
* [Boffin & West, 2004] Boffin, H., West, R. 2004, The Messenger, 116, 39
* [Boffin & West, 2005] Boffin, H., West, R. 2005, in IAU Commission 55: Communicating Astronomy with the Public 2005, I. Robson & L. L. Christensen (eds), p. 266
* [Eurobarometer, 2005] Eurobarometer 2005, Special Eurobarometer on Europeans, Science and Technology
* [Eurobarometer, 2007] Eurobarometer 2007, Scientific research in the media
* [Maran et al., 2000] Maran, S.P., Cominsky, L.R., Marschall. L.A. 2000, in Information Handling in Astronomy, A. Heck, ed., Kluwer, p. 13
* [Madsen, 2001] Madsen, C. 2001, Stars in the Media, Open University
* [Madsen, 2003] Madsen, C. 2003 Astronomy and Space Science in the European Print Media, in Astronomy Communication, ed. André Heck & Claus Madsen, Kluwer
* [NSF, 2004] NSF Science and Engineering Indicators 2004, National Science Board, Chapter 7, Science and Technology: Public Attitudes and Understanding
* [West, 2004] West, R. 2004, priv. comm.
|
arxiv-papers
| 2009-06-09T15:03:43 |
2024-09-04T02:49:03.265908
|
{
"license": "Public Domain",
"authors": "Henri M.J. Boffin",
"submitter": "Henri M. J. Boffin",
"url": "https://arxiv.org/abs/0906.1755"
}
|
0906.1822
|
# The Dark Matter Annihilation Signal from Dwarf Galaxies and Subhalos
Michael Kuhlen Institute for Advanced Study, School of Natural Science
Einstein Lane, Princeton, NJ 08540
Email address: [email protected]
###### Abstract
Dark Matter annihilation holds great potential for directly probing the
clumpiness of the Galactic halo that is one of the key predictions of the Cold
Dark Matter paradigm of hierarchical structure formation. Here we review the
$\gamma$-ray signal arising from dark matter annihilation in the centers of
Galactic subhalos. We consider both known Galactic dwarf satellite galaxies
and dark clumps without a stellar component as potential sources. Utilizing
the Via Lactea II numerical simulation, we estimate fluxes for 18 Galactic
dwarf spheroidals with published central densities. The most promising source
is Segue 1, followed by Ursa Major II, Ursa Minor, Draco, and Carina. We show
that if any of the known Galactic satellites can be detected, then at least
ten times more subhalos should be visible, with a significant fraction of them
being dark clumps.
## I Introduction
A decade has gone by since the emergence of the “Missing Satellite Problem”
Klypin et al. (1999); Moore et al. (1999), which refers to the apparent
discrepancy between the observed number of Milky Way satellite galaxies, 23 by
latest count Mateo (1998); Willman et al. (2005a, b); Belokurov et al. (2006);
Zucker et al. (2006a, b); Sakamoto and Hasegawa (2006); Belokurov et al.
(2007); Irwin et al. (2007); Walsh et al. (2007), and the predicted number of
dark matter (DM) subhalos that should be orbiting in the Milky Way’s halo. The
latest cosmological numerical simulations Diemand et al. (2008); Springel et
al. (2008a); Stadel et al. (2008) resolve close to 100,000 individual self-
bound clumps of DM within the Galactic virial volume – remnants of the
hierarchical build-up of the Milky Way’s DM halo. A consensus seems to be
emerging that this discrepancy is not a short-coming of the otherwise
tremendously successful Cold Dark Matter (CDM) hypothesis White and Rees
(1978); Blumenthal et al. (1984), but instead reflects the complicated
baryonic physics that determines which subhalos are able to host a luminous
stellar component and which aren’t Dekel and Silk (1986); Bullock et al.
(2000); Kravtsov et al. (2004); Mayer et al. (2006); Madau et al. (2008);
Koposov et al. (2009); Maccio’ et al. (2009). If this explanation is correct,
then an immediate consequence is that the Milky Way dark halo should be filled
with clumps on all scales down to the CDM free-streaming scale of $10^{-12}$
to $10^{-4}\,\rm M_{\odot}$ Profumo et al. (2006); Bringmann (2009). At the
moment there is little empirical evidence for or against this prediction, and
this has motivated searches for new signals that could provide tests of this
hypothesis, and ultimately help to constrain the nature of the DM particle.
One of the most promising such signals is DM annihilation Bergström et al.
(1999). In regions of sufficiently high density, for example in the centers of
Galactic subhalos, the DM pair annihilation rate might become large enough to
allow for a detection of neutrinos, energetic electrons and positrons, or
$\gamma$-ray photons, which are the by-products of the annihilation process.
This is one of the few ways in which the dark sector can be coupled to
ordinary matter and radiation amenable to astronomical observation. Belying
its commonly used name of “indirect detection”, DM annihilation is really the
only way we can hope to directly probe the clumpiness of the Galactic DM
distribution. One could argue that it is a more “direct” method than trying to
constrain DM clumpiness from its effects on strong gravitational lensing (see
Zackrisson & Riehm’s contribution in this special edition), or from the
kinematics of stars orbiting in DM-dominated potentials Strigari et al.
(2008a), or from perturbations of cold stellar structures like globular
cluster tidal streams Ibata et al. (2002); Johnston et al. (2002); Peñarrubia
et al. (2006); Siegal-Gaskins and Valluri (2008) or the heating of the Milky
Way’s stellar disk Toth and Ostriker (1992); Read et al. (2008); Kazantzidis
et al. (2009), although all of these are also worthwhile approaches to take.
The only trouble with the DM annihilation signal is that so far there have
been no undisputed claims of its detection. Recently there have been several
reports of “anomalous” features in the local cosmic ray flux: the PAMELA
satellite reported an increasing positron fraction at energies between 10 and
100 GeV Adriani et al. (2009), where standard models of cosmic ray propagation
predict a decreasing fraction; the ATIC Chang et al. (2008) and PPB-BETS Torii
et al. (2008) balloon-borne experiments reported a surprisingly large total
electron and positron flux at $\sim 500$ GeV, although recent Fermi Abdo et
al. (2009) and H.E.S.S. data H. E. S. S. Collaboration: F. Aharonian (2009)
appear to be inconsistent with it. Either of these cosmic ray anomalies might
be the long sought after signature of local DM annihilation. However, since
the currently available data can equally well be explained by conventional
astrophysical sources (e.g. nearby pulsars or supernova remnants), they hardly
provide incontrovertible evidence for DM annihilation. The next few years hold
great potential for progress, since the recently launched Fermi Gamma-ray
Space Telescope will conduct a blind survey of the $\gamma$-ray sky at
unprecedented sensitivity, energy extent, and angular resolution. At the same
time, Atmospheric Cerenkov Telescopes, such as H.E.S.S., VERITAS, MAGIC, and
STACEE, are greatly increasing their sensitivity, and have only recently begun
to search for a DM annihilation signal from the centers of nearby dwarf
satellite galaxies Aharonian et al. (2008); Hui (2008); Albert et al. (2008);
Aliu et al. (2009); Driscoll et al. (2008).
The purpose of this paper is to provide an overview of the potential DM
annihilation signal from individual Galactic DM subhalos, either as dwarf
satellite galaxies or as dark clumps. It does not cover a number of very
interesting and closely related topics, which are actively being researched
and deserve to be examined in equal detail. These include the diffuse flux
from Galactic substructure and its anisotropies (e.g. Siegal-Gaskins, 2008;
Ando, 2009; Fornasa et al., 2009), the relative strength of the signal from
individual subhalos compared with that from the Galactic Center or an annulus
around it Stoehr et al. (2003); Springel et al. (2008b), the effect of a
nearby DM subhalo on the amplitude and spectrum of the local high energy
electron and positron flux Brun et al. (2009); Kuhlen and Malyshev (2009), and
the role of the Sommerfeld enhancement Arkani-Hamed et al. (2009) on the DM
annihilation rate and its implications for substructure signals Lattanzi and
Silk (2009); Pieri et al. (2009a); Kuhlen et al. (2009).
This paper is organized as follows: we first review the basic physics of DM
annihilation, briefly touching on the relic density calculation, the “WIMP
miracle”, DM particle candidates, and, in more detail, the sources of
$\gamma$-rays from DM annihilation. In the following section we review what
numerical simulations have revealed about the basic properties of DM subhalos
that are relevant for the annihilation signal. We go on to consider known
Milky Way dwarf spheroidal galaxies as sources, using the Via Lactea II
simulation to infer the most likely annihilation fluxes from published values
of the dwarfs’ central masses. Next we discuss the possibility of a DM
annihilation signal from dark clumps, halos that have too low a mass to host a
luminous stellar component. Lastly, we briefly discuss the role of the
substructure boost factor for the detectability of individual DM subhalos.
## II Dark Matter Annihilation
If DM is made up of a so-called “thermal relic” particle111An alternative DM
candidate is the axion, a non-thermal relic particle motivated as a solution
to the strong CP problem Turner (1990). Since it doesn’t produce an
annihilation signal today, we don’t further consider it here., its abundance
today is set by its annihilation cross section in the early universe. The
thermal relic abundance calculation relating today’s abundance of DM to the
properties of the DM particle (its mass and annihilation cross section) is
straightforward and elegant, and has been described in pedagogical detail
previously Kolb and Turner (1990); Jungman et al. (1996); Bertone et al.
(2005). We briefly summarize the story here.
At sufficiently early times, the DM particles are in thermal equilibrium with
the rest of the universe. As long as they remain relativistic ($T\gg
m_{\chi}$), their creation and destruction rates are balanced, and hence their
co-moving abundance remains constant. Once the universe cools below the DM
particle’s rest-mass ($T<m_{\chi}$), its equilibrium abundance is suppressed
by a Boltzmann factor $\exp(-m_{\chi}/T)$. If equilibrium had been maintained
until today, the DM particles would have completely annihilated away. Instead
the expansion of the universe comes to the rescue and causes the DM particles
to fall out of equilibrium once the expansion rate (given by $H(a)$, the
Hubble constant at cosmological scale factor $a$) exceeds the annihilation
rate $\Gamma(a)=n\langle\sigma v\rangle$, i.e. when DM particles can no longer
find each other to annihilate. The co-moving number density of DM particles is
then fixed at a “freeze-out” temperature that turns out to be approximately
$T_{f}\simeq m_{\chi}/20$, with only a weak additional logarithmic dependence
on the mass and cross section of the DM particle. A back of the envelope
calculation results in the following relation between $\Omega_{\chi}$, the
relic mass density in units of the critical density of the universe $\rho_{\rm
crit}=3H_{0}^{2}/8\pi G$, and $\langle\sigma v\rangle$, the thermally averaged
velocity-weighted annihilation cross section:
$\omega_{\chi}=\Omega_{\chi}h^{2}=\frac{3\times 10^{-27}\;{\rm cm}^{3}\;{\rm
s}^{-1}}{\langle\sigma v\rangle}.$ (1)
Note that this relation is independent of $m_{\chi}$. The WMAP satellite’s
measurement of the DM density is $\omega_{\chi}=0.1131\pm 0.0034$ Hinshaw et
al. (2009), implying a value of
$\langle\sigma v\rangle\approx 3\times 10^{-26}\;{\rm cm}^{3}\;{\rm s}^{-1}.$
(2)
A more accurate determination of $\langle\sigma v\rangle$ must rely on a
numerical solution of the Boltzmann equation in an expanding universe, taking
into account the full temperature dependence of the annihilation rate,
including the possibilities of resonances and co-annihilations into other,
nearly degenerate dark sector particles (e.g. Griest and Seckel, 1991; Gondolo
and Gelmini, 1991). It is a remarkable coincidence that this value of
$\langle\sigma v\rangle$ is close to what one expects for a cross section set
by the weak interaction. This is the so-called “WIMP miracle”, and it is the
main motivation for considering weakly interacting massive particles (WIMPs)
as prime DM candidates.
The Standard Model of particle physics actually provides one class of WIMPs,
massive neutrinos. Although neutrinos thus constitute a form of DM, they
cannot make up the bulk of it, since their small mass, $\sum m_{\nu}<0.63$ eV
Hinshaw et al. (2009), implies a cosmological mass density of only
$\omega_{\nu}=7.1\times 10^{-3}$. The attention thus turns to extensions of
the Standard Model, which themselves are theoretically motivated by the
hierarchy problem (the enormous disparity between the weak and Planck scales)
and the quest for a unification of gravity and quantum mechanics. The most
widely studied class of such models consists of supersymmetric extensions of
the Standard Model, although models with extra dimensions have received a lot
of attention in recent years as well. Both of these approaches offer good DM
particle candidates: the lightest supersymmetric particle (LSP), typically a
neutralino in R-parity conserving supersymmetry, and the lightest Kaluza-Klein
particle (LKP), typically the $B^{(1)}$ particle, the first Kaluza-Klein
excitation of the hypercharge gauge boson, in Universal Extra Dimension
models. For much more information, we recommend the comprehensive recent
review of particle DM candidates by Bertone, Hooper & Silk Bertone et al.
(2005).
Figure 1: A schematic of the different sources and energy distributions of
$\gamma$-rays from WIMP annihilation. Top: Secondary photons arising from the
decay of neutral pions produced in the hadronization of primary annihilation
products. Middle: Internal bremsstrahlung photons associated with charged
annihilation products, either in the form of final state radiation (FSR) from
external legs or as virtual internal bremsstrahlung (VIB) from the exchange of
virtual charged particles. Bottom: Mono-chromatic line signals from the prompt
annihilation into two photons or a photon and $Z$ boson. This process occurs
only at loop level, and hence is typically strongly suppressed.
The direct products of the annihilation of two DM particles are strongly model
dependent. Typical channels include annihilations into charged leptons
($e^{+}e^{-},\mu^{+}\mu^{-},\tau^{+}\tau^{-}$), quark-antiquark pairs, and
gauge and Higgs bosons ($W^{+}W^{-},Z,h$). In the end, however, the decay and
hadronization of these annihilation products results in only three types of
emissions: (i) high energy neutrinos, (ii) relativistic electrons and protons
and their anti-particles, and (iii) $\gamma$-ray photons. Additional lower
energy photons can result from the interaction of the relativistic electrons
with magnetic fields (synchrotron radiation), with interstellar material
(bremsstrahlung), and with the CMB and stellar radiation fields (inverse
Compton scattering). In the following we will focus on the $\gamma$-rays,
since they are likely the strongest signal from Galactic DM substructure.
$\gamma$-rays are produced in DM annihilations in three ways (see accompanying
Fig. 1)
* (i)
Since the DM particle is neutral, there is no direct coupling to photons.
Nevertheless, copious amounts of secondary $\gamma$-ray photons can be
produced through the decay of neutral pions, $\pi^{0}\rightarrow\gamma\gamma$,
arising in the hadronization of the primary annihilation products. Since the
DM particles are non-relativistic, their annihilation results in a pair of
mono-energetic particles with energy equal to $m_{\chi}$, which fragment and
decay into $\pi$-meson dominated “jets”. In this way a single DM annihilation
event can produce several tens of $\gamma$-ray photons. The result is a broad
spectrum with a cutoff around $m_{\chi}$.
* (ii)
An important additional contribution at high energies ($E\lesssim m_{\chi}$)
arises from the internal bremsstrahlung process Bringmann et al. (2008), which
may occur with any charged annihilation product. One can distinguish between
final state radiation, in which the photon is radiated from an external leg,
and virtual internal bremsstrahlung, arising from the exchange of a charged
virtual particle. Note that neither of these processes requires an external
electromagnetic field (hence the name internal bremsstrahlung). The resulting
$\gamma$-ray spectrum is peaked towards $E\sim m_{\chi}$ and exhibits a sharp
cutoff. Although it is suppressed by one factor of the coupling $\alpha$
compared to pion decays, it can produce a distinctive spectral feature at high
energies. This could aide the confirmation of a DM annihilation nature of any
source and might allow a direct determination of $m_{\chi}$.
* (iii)
Lastly, it is possible for DM particles to directly produce $\gamma$-ray
photons, but one has to go to loop-level to find contributing Feynman
diagrams, and hence this flux is typically strongly suppressed by two factors
of $\alpha$ (although exceptions exist Gustafsson et al. (2007)). On the other
hand, the resulting photons would be mono-chromatic, and a detection of such a
line signal would provide strong evidence of a DM annihilation origin of any
signal. While annihilations directly into two photons,
$\chi\chi\rightarrow\gamma\gamma$, would produce a narrow line at
$E=m_{\chi}$, in some models it is also possible to annihilate into a photon
and a $Z$ boson, $\chi\chi\rightarrow\gamma Z$, and this process would result
in a somewhat broadened (due to the mass of the $Z$) line at $E\sim
m_{\chi}(1-m_{Z}^{2}/4m_{\chi}^{2}$).
The relative importance of these three $\gamma$-ray production channels and
the resulting spectrum $dN_{\gamma}/dE$ depend on the details of the DM
particle model under consideration. For any given model, realistic
$\gamma$-ray spectra can be calculated using sophisticated and publicly
available computer programs, such as the PYTHIA Monte-Carlo event generator
Sjöstrand (1994), which is also contained in the popular DarkSUSY package
Gondolo et al. (2004).
## III Dark Matter Substructure as Discrete $\gamma$-ray Sources
DM subhalos as individual discrete $\gamma$-ray sources hold great potential
for providing a “smoking gun” signature of DM annihilation Bergström et al.
(1999); Calcáneo-Roldán and Moore (2000); Baltz et al. (2000); Tasitsiomi and
Olinto (2002); Stoehr et al. (2003); Aloisio et al. (2004); Evans et al.
(2004); Koushiappas et al. (2004); Koushiappas (2006); Diemand et al. (2007);
Pieri et al. (2008); Kuhlen et al. (2008); Strigari et al. (2008b). Compared
to diffuse $\gamma$-ray annihilation signals, these discrete sources should be
easier to distinguish from astrophysical backgrounds and foregrounds Baltz et
al. (2007), since a) typical astrophysical sources of high energy
$\gamma$-rays, such as pulsars and supernova remnants, are very rare in dwarf
galaxies, owing to their predominantly old stellar populations, b) the DM
annihilation flux should be time-independent, c) angularly extended, and d)
not exhibit any (or only very weak) low energy emission due to the absence of
strong magnetic fields or stellar radiation fields.
We can distinguish between DM subhalos hosting a Milky Way dwarf satellite
galaxy and dark clumps that, for whatever reason, don’t host a luminous
stellar population, or one that is too faint to have been detected up to now.
Before we go on to discuss the prospects of detecting a DM annihilation signal
from these two classes of sources, we review the basic properties of DM
subhalos common to both.
Figure 2: A comparison of NFW and Einasto ($\alpha=0.17$) radial profiles of
density (top, dark lines, left axis), circular velocity (top, light lines,
right axis), enclosed annihilation luminosity (bottom, dark lines, left axis),
enclosed mass (bottom, light lines, right axis). The density profiles have
been normalized to have the same $V_{\rm max}$ and $r_{\rm Vmax}$.
Numerical simulations have shown that pure DM (sub)halos have density profiles
that are well described by a Navarro, Frenk & White (NFW) Navarro et al.
(1997) profile over a wide range of masses Macciò et al. (2007); Diemand et
al. (2004),
$\rho_{\rm NFW}(r)=\frac{4\rho_{s}}{(r/r_{s})(1+r/r_{s})^{2}}.$ (3)
The parameter $r_{s}$ indicates the radius at which the logarithmic slope
$\gamma(r)\equiv\frac{d\ln\rho}{d\ln r}=-2$, and $\rho(r_{s})=\rho_{s}$. The
very highest resolution simulations have recently provided some indications of
a flattening of the density profile in the innermost regions Navarro et al.
(2008); Stadel et al. (2008). In this case a so-called Einasto profile may
provide a better overall fit,
$\rho_{\rm
Einasto}(r)=\rho_{s}\exp{\left[-\frac{2}{\alpha}\left(\left(\frac{r}{r_{s}}\right)^{\alpha}-1\right)\right]}.$
(4)
Here the additional parameter $\alpha$ governs how fast the profile rolls
over, and has been found to have a value of $\alpha\approx 0.17\pm 0.03$ in
numerical simulations Navarro et al. (2008). Note that the two density
profiles actually do not differ very much until $r\ll r_{s}$ (cf. top panel of
Figure 2). Simulated DM halos are of course not perfectly spherically
symmetric, and instead typically exhibit prolate or triaxial iso-density
contours that become more elongated towards the center Allgood et al. (2006).
The degree of prolateness decreases with mass, and galactic subhalos have axis
ratios of $\gtrsim 0.7$ Kuhlen et al. (2007).
The “virial” radius $R_{\rm vir}$ of a halo is defined as the radius enclosing
a mean density equal to $\Delta_{\rm vir}\rho_{0}$, where $\Delta_{\rm
vir}\approx 389$ Bryan and Norman (1998) and $\rho_{0}$ is the mean density of
the universe. The corresponding virial mass $M_{\rm vir}$ is the mass within
$R_{\rm vir}$, and a halo’s concentration can then be defined as $c=R_{\rm
vir}/r_{s}$. While these quantities are well defined for isolated halos and
commonly used in analytic models, they are somewhat less applicable to
galactic subhalos, since the outer radius of a subhalo is set by tidal
truncation, which depends on the subhalo’s location within its host halo.
Furthermore, in numerical simulations it is difficult to resolve $r_{s}$ in
low mass subhalos. For this reason we prefer to work with $V_{\rm max}$, the
maximum of the circular velocity curve $V_{c}(r)^{2}=GM(<r)/r$ and a proxy for
a subhalo’s mass, and $r_{\rm Vmax}$, the radius at which $V_{\rm max}$
occurs. These quantities are much more robustly determined for subhalos in
numerical simulations than $(M,c)$. Note that even $(V_{\rm max},r_{\rm
Vmax})$ can be affected by tidal interactions with the host halo, especially
for subhalos close to the host halo center. For this reason we also sometimes
consider $V_{\rm peak}$, the largest value of $V_{\rm max}$ that a subhalo
ever acquired during its lifetime (i.e. before tidal stripping began to lower
its $V_{\rm max}$) and $r_{\rm Vpeak}$, the corresponding radius.
Since DM annihilation is a two body process, its rate is proportional to the
square of the local density, and the annihilation “luminosity” is given by the
volume integral of $\rho(r)^{2}$,
$\mathcal{L}(<r)\equiv\int_{0}^{r}\rho^{2}\;dV.$ (5)
$\mathcal{L}$ has dimensions of (mass)2 (length)-3, and we express it in units
of $\,\rm M_{\odot}^{2}$ pc-3. In order to convert to a conventional
luminosity, one must multiply by a particle physics term,
$L=c^{2}\frac{\langle\sigma v\rangle}{m_{\chi}}\mathcal{L},$ (6)
where $m_{\chi}$ is the mass of the DM particle and $\langle\sigma v\rangle$
the thermally averaged velocity-weighted annihilation cross section discussed
in the previous section. This is the total luminosity, but we are interested
here only in the fraction emitted as $\gamma$-rays. Furthermore, a given
detector is only sensitive to $\gamma$-rays above a threshold energy of
$E_{\rm th}$ and below a maximum energy of $E_{\rm max}$. In that case the
effective $\gamma$-ray luminosity is
$L^{\rm eff}_{\gamma}=\left[\frac{\langle\sigma
v\rangle}{2m_{\chi}^{2}}\int_{E_{\rm th}}^{E_{\rm
max}}\\!\\!\\!E\frac{dN_{\gamma}}{dE}dE\right]\mathcal{L},$ (7)
where $dN_{\gamma}/dE$ is the spectrum of $\gamma$-ray photons produced in a
single annihilation event.
A comparison of the enclosed luminosity and mass profiles is shown in the
bottom panel of Figure 2. Clearly, $\mathcal{L}$ is much more centrally
concentrated than $M$: $\sim 90\%$ of the total luminosity is produced within
$r_{s}$, compared with only 10% of the total mass. In terms of $(V_{\rm
max},r_{\rm Vmax})$, the total luminosity of a halo is given by
$\mathcal{L}=f\frac{V_{\rm max}^{4}}{G^{2}r_{\rm Vmax}},$ (8)
where $f$ is an $\mathcal{O}(1)$ numerical factor that depends on the shape of
the density profile; for an NFW profile $f=1.227$, and for an $\alpha=0.17$
Einasto profile $f=1.735$. In physical units, the total annihilation
luminosity is
$\mathcal{L}=\begin{array}[]{c}1.1\\\ 1.5\end{array}\times 10^{7}\;\,\rm
M_{\odot}^{2}\;{\rm pc}^{-3}\left(\frac{V_{\rm max}}{20\,{\rm km}\;{\rm
s}^{-1}}\right)^{4}\left(\frac{r_{\rm Vmax}}{1{\rm kpc}}\right)^{-1},$ (9)
for NFW (top) and $\alpha=0.17$ Einasto (bottom). Note that even though the
slope of the Einasto profile is shallower than NFW in the very center, the
total luminosity exceeds that of an NFW halo with the same $(V_{\rm
max},r_{\rm Vmax})$. This is due to the fact that the Einasto profile rolls
over less rapidly than the NFW profile, and actually has slightly higher
density than NFW between $r_{s}$ and a cross-over point at $\sim
10^{-3}r_{s}$.
## IV Milky Way Dwarf Spheroidal Galaxies
Name | $D$ | $M_{0.3}$ | $V_{\rm max}$ | $r_{\rm Vmax}$ | $V_{\rm peak}$ | $r_{\rm Vpeak}$
---|---|---|---|---|---|---
| [kpc] | [$10^{7}M_{\odot}$] | [km s-1] | [kpc] | [km s-1] | [kpc]
Segue 1 | 23 | $1.58^{+3.30}_{-1.11}$ | $10\;(^{17}_{8.4})$ | $0.43\;(^{0.89}_{0.29})$ | $26\;(^{55}_{13})$ | $2.4\;(^{33}_{1.4})$
Ursa Major II | 32 | $1.09^{+0.89}_{-0.44}$ | $13\;(^{17}_{11})$ | $0.59\;(^{0.89}_{0.31})$ | $27\;(^{33}_{17})$ | $3.3\;(^{14}_{2.4})$
Wilman 1 | 38 | $0.77^{+0.89}_{-0.42}$ | $8.3\;(^{11}_{7.5})$ | $0.38\;(^{0.62}_{0.29})$ | $15\;(^{27}_{10})$ | $2.0\;(^{3.9}_{0.90})$
Coma Berenices | 44 | $0.72^{+0.36}_{-0.28}$ | $9.1\;(^{12}_{8.2})$ | $0.42\;(^{0.62}_{0.31})$ | $15\;(^{25}_{11})$ | $1.9\;(^{3.4}_{0.97})$
Ursa Minor | 66 | $1.79^{+0.37}_{-0.59}$ | $18\;(^{21}_{15})$ | $0.81\;(^{1.8}_{0.61})$ | $30\;(^{56}_{21})$ | $3.8\;(^{9.7}_{2.8})$
Draco | 80 | $1.87^{+0.20}_{-0.29}$ | $19\;(^{22}_{17})$ | $0.86\;(^{2.4}_{0.81})$ | $28\;(^{37}_{26})$ | $3.8\;(^{32}_{2.4})$
Sculptor | 80 | $1.20^{+0.11}_{-0.37}$ | $13\;(^{15}_{12})$ | $0.64\;(^{1.0}_{0.54})$ | $20\;(^{25}_{16})$ | $2.9\;(^{5.6}_{1.6})$
Sextans | 86 | $0.57^{+0.45}_{-0.14}$ | $9.7\;(^{12}_{8.5})$ | $0.52\;(^{0.89}_{0.37})$ | $14\;(^{19}_{11})$ | $1.6\;(^{3.0}_{0.97})$
Carina | 101 | $1.57^{+0.19}_{-0.10}$ | $17\;(^{22}_{16})$ | $1.00\;(^{2.3}_{0.69})$ | $30\;(^{42}_{24})$ | $3.8\;(^{32}_{3.3})$
Ursa Major I | 106 | $1.10^{+0.70}_{-0.29}$ | $14\;(^{17}_{13})$ | $0.84\;(^{1.3}_{0.61})$ | $20\;(^{30}_{16})$ | $3.2\;(^{6.8}_{1.6})$
Fornax | 138 | $1.14^{+0.09}_{-0.12}$ | $15\;(^{16}_{14})$ | $1.1\;(^{1.3}_{0.64})$ | $20\;(^{24}_{18})$ | $3.0\;(^{6.1}_{1.9})$
Hercules | 138 | $0.72^{+0.51}_{-0.21}$ | $11\;(^{14}_{9.4})$ | $0.69\;(^{1.1}_{0.45})$ | $14\;(^{20}_{12})$ | $1.9\;(^{3.8}_{1.2})$
Canes Venatici II | 151 | $0.70^{+0.53}_{-0.25}$ | $11\;(^{13}_{8.9})$ | $0.67\;(^{1.1}_{0.44})$ | $14\;(^{19}_{11})$ | $1.8\;(^{3.7}_{1.1})$
Leo IV | 158 | $0.39^{+0.50}_{-0.29}$ | $5.0\;(^{7.2}_{4.2})$ | $0.35\;(^{0.57}_{0.22})$ | $6.7\;(^{10}_{5.0})$ | $0.84\;(^{1.7}_{0.48})$
Leo II | 205 | $1.43^{+0.23}_{-0.15}$ | $18\;(^{21}_{16})$ | $1.5\;(^{2.1}_{0.93})$ | $24\;(^{28}_{19})$ | $4.1\;(^{8.2}_{2.4})$
Canes Venatici I | 224 | $1.40^{+0.18}_{-0.19}$ | $18\;(^{20}_{16})$ | $1.5\;(^{2.1}_{1.0})$ | $22\;(^{29}_{18})$ | $2.9\;(^{6.1}_{2.1})$
Leo I | 250 | $1.45^{+0.27}_{-0.20}$ | $19\;(^{21}_{17})$ | $1.7\;(^{3.1}_{1.1})$ | $25\;(^{27}_{19})$ | $2.9\;(^{6.3}_{2.1})$
Leo T | 417 | $1.30^{+0.88}_{-0.42}$ | $16\;(^{21}_{13})$ | $1.2\;(^{2.4}_{0.85})$ | $19\;(^{26}_{17})$ | $2.4\;(^{6.1}_{1.6})$
Table 1: The properties of likely DM subhalos of the 18 Milky Way dSph galaxies for which $M_{0.3}$ values (column 3) have been published Strigari et al. (2008a). $V_{\rm max}$ and $r_{\rm Vmax}$ are the maximum circular velocity and its radius, $V_{\rm peak}$ and $r_{\rm Vpeak}$ the largest $V_{\rm max}$ a subhalo ever acquired and its corresponding radius. The first number is the median over all Via Lactea II subhalos matching the dSph’s distance and $M_{0.3}$, the numbers in parentheses the 16th and 84th percentiles. (See text for details.) Name | $D$ | $\mathcal{L}_{\rm tot}$ | $\mathcal{L}_{0.3}$ | $\mathcal{F}_{\rm tot}$ | $\mathcal{F}_{c}$
---|---|---|---|---|---
| [kpc] | [$10^{6}M_{\odot}^{2}\;{\rm pc}^{-3}$] | [$10^{6}M_{\odot}^{2}\;{\rm pc}^{-3}$] | [$10^{-5}M_{\odot}^{2}\;{\rm pc}^{-5}$] | [$10^{-5}M_{\odot}^{2}\;{\rm pc}^{-5}$]
Segue 1 | 23 | $2.8\;(^{7.2}_{0.93})$ | $2.5\;(^{6.1}_{0.89})$ | $41\;(^{110}_{14})$ | $12\;(^{34}_{5.6})$
Ursa Major II | 32 | $3.5\;(^{7.2}_{2.8})$ | $3.1\;(^{6.1}_{2.5})$ | $28\;(^{56}_{21})$ | $9.5\;(^{18}_{7.7})$
Ursa Minor | 66 | $6.2\;(^{9.4}_{5.1})$ | $4.7\;(^{7.3}_{3.1})$ | $11\;(^{17}_{9.3})$ | $5.2\;(^{8.4}_{2.5})$
Draco | 80 | $7.0\;(^{9.9}_{6.0})$ | $5.6\;(^{8.2}_{3.1})$ | $8.8\;(^{12}_{7.4})$ | $4.3\;(^{6.4}_{1.7})$
Carina | 101 | $7.0\;(^{9.4}_{4.8})$ | $5.6\;(^{7.3}_{3.5})$ | $5.5\;(^{7.3}_{3.7})$ | $3.1\;(^{3.8}_{1.6})$
Wilman 1 | 38 | $0.88\;(^{2.9}_{0.55})$ | $0.85\;(^{2.7}_{0.53})$ | $4.9\;(^{16}_{3.0})$ | $2.6\;(^{6.4}_{1.5})$
Coma Berenices | 44 | $1.2\;(^{2.8}_{0.78})$ | $1.1\;(^{2.5}_{0.70})$ | $4.8\;(^{11}_{3.2})$ | $2.5\;(^{5.1}_{1.6})$
Sculptor | 80 | $2.9\;(^{3.7}_{2.3})$ | $2.5\;(^{3.3}_{2.0})$ | $3.7\;(^{4.6}_{2.8})$ | $2.0\;(^{2.8}_{1.6})$
Ursa Major I | 106 | $3.3\;(^{5.4}_{2.3})$ | $2.5\;(^{4.5}_{1.9})$ | $2.3\;(^{3.8}_{1.6})$ | $1.3\;(^{2.4}_{0.91})$
Fornax | 138 | $3.5\;(^{4.4}_{3.0})$ | $2.9\;(^{3.3}_{2.3})$ | $1.4\;(^{1.8}_{1.3})$ | $1.00\;(^{1.2}_{0.74})$
Sextans | 86 | $1.2\;(^{2.0}_{0.77})$ | $1.1\;(^{1.8}_{0.69})$ | $1.3\;(^{2.1}_{0.83})$ | $0.86\;(^{1.4}_{0.55})$
Leo II | 205 | $4.6\;(^{6.5}_{3.8})$ | $3.1\;(^{4.7}_{2.1})$ | $0.88\;(^{1.2}_{0.73})$ | $0.55\;(^{0.85}_{0.37})$
Canes Venatici I | 224 | $4.6\;(^{7.9}_{3.8})$ | $3.1\;(^{5.0}_{2.3})$ | $0.73\;(^{1.3}_{0.60})$ | $0.48\;(^{0.79}_{0.35})$
Leo I | 250 | $5.2\;(^{7.9}_{3.9})$ | $3.2\;(^{5.4}_{2.3})$ | $0.66\;(^{1.0}_{0.50})$ | $0.41\;(^{0.73}_{0.31})$
Hercules | 138 | $1.4\;(^{2.6}_{0.94})$ | $1.2\;(^{2.2}_{0.80})$ | $0.57\;(^{1.1}_{0.39})$ | $0.42\;(^{0.74}_{0.28})$
Canes Venatici II | 151 | $1.2\;(^{2.5}_{0.79})$ | $1.1\;(^{2.0}_{0.68})$ | $0.44\;(^{0.88}_{0.27})$ | $0.33\;(^{0.59}_{0.21})$
Leo T | 417 | $3.5\;(^{8.2}_{2.4})$ | $2.2\;(^{4.1}_{1.7})$ | $0.16\;(^{0.38}_{0.11})$ | $0.12\;(^{0.24}_{0.093})$
Leo IV | 158 | $0.14\;(^{0.43}_{0.063})$ | $0.13\;(^{0.39}_{0.060})$ | $0.043\;(^{0.14}_{0.020})$ | $0.039\;(^{0.12}_{0.018})$
Table 2: Estimated luminosities and fluxes for the 18 dSph from Table 1.
$\mathcal{L}_{\rm tot}$ is the total luminosity and $\mathcal{L}_{\rm 0.3}$
the luminosity from within the central 0.3 kpc. $\mathcal{F}_{\rm
tot}=\mathcal{L}_{\rm tot}/4\pi D^{2}$ is the total flux and $\mathcal{F}_{c}$
the flux from a central region subtending $0.15^{\circ}$ (about the angular
resolution of Fermi above 3 GeV). The first number is the median over all
subhalos matching the dSph distance and $M_{0.3}$, the numbers in parentheses
are the 16th and 84th percentiles. The table is ordered by decreasing
$\mathcal{F}_{\rm tot}$.
There are several advantages of known dwarf satellite galaxies as DM
annihilation sources: firstly, the kinematics of individual stars imply mass-
to-light ratios of up to several hundred Kleyna et al. (2005); Muñoz et al.
(2006); Martin et al. (2007); Simon and Geha (2007), and hence there is an a
priori expectation of high DM densities; secondly, since we know their
location in the sky, it is possible to directly target them with sensitive
atmospheric Cerenkov telescopes (ACT) such as H.E.S.S., VERITAS, MAGIC, and
STACEE, whose small field of view makes blind searches impractical; lastly,
our approximate knowledge of the distances to many dwarf satellites would
allow a determination of the absolute annihilation rate, which may lead to a
direct constraint on the annihilation cross section, if the DM particle mass
can be independently measured (from the shape of the spectrum, for example).
Recent observational progress utilizing the Sloan Digital Sky Survey (SDSS)
has more than doubled the number of known dwarf spheroidal (dSph) satellite
galaxies orbiting the Milky Way Willman et al. (2005a, b); Belokurov et al.
(2006); Zucker et al. (2006a, b); Sakamoto and Hasegawa (2006); Belokurov et
al. (2007); Irwin et al. (2007); Walsh et al. (2007), raising the total from
the 9 “classical” ones to 23. Many of the newly discovered satellites are so-
called “ultra-faint” dSph’s, with luminosities as low as $1,000L_{\odot}$ and
only tens to hundreds of spectroscopically confirmed member stars. Simply
accounting for the SDSS sky coverage (about 20%), the total number of luminous
Milky Way satellites can be estimated to be at least 70. Taking into account
the SDSS detection limits Koposov et al. (2008) and a radial distribution of
DM subhalos motivated by numerical simulations, this estimate can grow to
several hundreds of satellite galaxies in total Tollerud et al. (2008); Walsh
et al. (2009).
In order to assess the strength of the DM annihilation signal from these
dSph’s, it is necessary to have an estimate of the total dynamical mass, or at
least $V_{\rm max}$, of the DM halo hosting the galaxies. Owing to the extreme
faintness of these objects and their lack of a detectable gaseous component
Grcevich and Putman (2009), it has been very difficult to obtain kinematic
information that allows for such measurements. Progress has been made through
spectroscopic observations of individual member stars, whose line-of-sight
velocity dispersions have confirmed that these objects are in fact strongly DM
dominated Kleyna et al. (2005); Muñoz et al. (2006); Martin et al. (2007);
Simon and Geha (2007). Such data best constrain the enclosed dynamical mass
within the stellar extent, which on average is about 0.3 kpc for current data
sets. A recent analysis has determined $M_{0.3}\equiv M(<0.3\;{\rm kpc})$ for
18 of the Milky Way dSph’s, and found that, surprisingly, they all have
$M_{0.3}\approx 10^{7}\,\rm M_{\odot}$ to within a factor of two Strigari et
al. (2008a).
State-of-the-art cosmological numerical simulations of the formation of the DM
halo of a Milky Way scale galaxy, such as those of the Via Lactea Project
Diemand et al. (2007, 2008) and the Aquarius Project Springel et al. (2008a),
have now reached an adequate mass and force resolution to directly determine
$M_{0.3}$ in their simulated subhalos. This makes it possible to infer the
most likely values of $(V_{\rm max},r_{\rm Vmax})$ for a Milky Way dSph of a
given $M_{0.3}$ and Galacto-centric distance $D$, by identifying all simulated
subhalos with comparable $M_{0.3}$ and $D$ and averaging over their $(V_{\rm
max},r_{\rm Vmax})$. This analysis was performed for the 9 “classical” dwarfs
using the Via Lactea I simulation Madau et al. (2008), and we extend it here
to all 18 dwarfs published in Strigari et al. (2008a) and with the more recent
and higher resolution Via Lactea II (VL2) simulation.
We randomly generated 100 observer locations at 8 kpc from the VL2 host halo
center, and selected, for each Milky Way dSph in Strigari et al. (2008a)
separately, all simulated subhalos with distances within 40% and numerically
determined $M_{0.3}$ within the published $1-\sigma$ error bars. We then
determined the median value and the 16th and 84th percentiles of $(V_{\rm
max},r_{\rm Vmax})$ and $(V_{\rm peak},r_{\rm Vpeak})$ for each dSph. These
values are given in Table 1. The median values of $V_{\rm max}$ range from 5.0
km s-1 (Leo IV) to 19 km s-1 (Draco, Leo I), and of $V_{\rm peak}$ from 6.7 km
s-1 (Leo IV) to 30 km s-1 (Ursa Minor, Carina). Note that, as expected, dSph’s
closer to the Galactic Center typically show a larger reduction from $V_{\rm
peak}$ to $V_{\rm max}$, sometimes by more than a factor of 2.
In the same fashion, we then determine the most likely annihilation
luminosities for the 18 dSph’s by using Eq. (9) for an NFW profile to
calculate the total luminosity $\mathcal{L}_{\rm tot}$ for every simulated
subhalo. Additionally we also determine $\mathcal{L}_{0.3}$, the luminosity
within 0.3 kpc from the center, motivated by the fact that we only have
dynamical evidence for a DM dominated potential out to this radius. Lastly we
also consider two measures of the brightness of each halo: $\mathcal{F}_{\rm
tot}=\mathcal{L}_{\rm tot}/4\pi D^{2}$, the total expected flux from the dSph,
and $\mathcal{F}_{c}$, the flux from a central region subtending
$0.15^{\circ}$, which is comparable to the angular resolution of Fermi above 3
GeV. $\mathcal{F}_{c}$ thus corresponds to the brightest “pixel” in a Fermi
$\gamma$-ray image of a subhalo. These numbers are given in Table 2.
### IV.1 Current observational constraints
Several ACT have performed observations of a handful of dSph’s.
* •
The H.E.S.S. array (consisting of four 107 m2 telescopes with a 5∘ field of
view and an energy threshold of $160$ GeV Bernlöhr et al. (2003)) has obtained
an 11 hour exposure of the Sagittarius dwarf galaxy. No $\gamma$-ray signal
was detected, resulting in a flux limit of $3.6\times 10^{-12}$ cm-2 s-1 (95%
confidence) at $E>250$ GeV, and a corresponding limit on the cross section of
$\langle\sigma v\rangle\lesssim 10^{-23}\;{\rm cm}^{3}\;{\rm s}^{-1}$ for an
NFW profile and $\langle\sigma v\rangle\lesssim 2\times 10^{-25}\;{\rm
cm}^{3}\;{\rm s}^{-1}$ for a cored profile (for a $m_{\chi}=100\;{\rm
GeV}-1\;{\rm TeV}$ neutralino) Aharonian et al. (2008). Note that the
Sagittarius dwarf is undergoing a strong tidal interaction with the Milky Way
galaxy Martínez-Delgado et al. (2004), and no confident determination of
$M_{0.3}$ has been possible.
* •
The VERITAS array (consisting of four 144 m2 telescopes with a 3.5∘ field of
view and an energy threshold of $100$ GeV Holder et al. (2008)) has conducted
a 15 hours observation of Willman 1 and 20 hour observations each of Draco and
Ursa Minor Hui (2008). No $\gamma$-ray signal was detected at a flux limit of
$\sim 1\%$ of the flux from the Crab Nebula, corresponding to a limit of
$2.4\times 10^{-12}$ cm-2 s-1 (95% confidence) at $E>200$ GeV Essig et al.
(2009).
* •
Additionally the MAGIC Albert et al. (2008); Aliu et al. (2009) and STACEE
Driscoll et al. (2008) telescopes have reported observations of Willman 1 and
Draco, resulting in comparable or slightly higher flux limits.
To convert the values of $\mathcal{F}_{c}$ in Table 2 into physical fluxes
that can directly be compared to these observational limits, it would be
necessary to obtain values of the particle physics term of Eq. (7) by
performing a scan of the DM model parameter space. This is beyond the scope of
this work, but a similar analysis has been performed by others Strigari et al.
(2008b); Bovy (2009); Martinez et al. (2009); Pieri et al. (2009b); Essig et
al. (2009). Current ACT observations of dSph’s are beginning to directly
constrain DM models, and future longer exposure time observations of
additional dSph’s (in particular Segue 1 and Ursa Major II) with a lower
threshold energy hold great potential. We also eagerly await the first Fermi
data on fluxes from the known dSph galaxies.
Figure 3: The annihilation flux $\mathcal{F}_{\rm tot}$ from subhalos in the
Via Lactea II simulation versus their $M_{0.3}$, $V_{\rm max}$, and $V_{\rm
peak}$. The gray shaded areas indicate regions containing subhalos with
$\mathcal{F}_{\rm tot}$ as least as high as the fifth-brightest Milky Way dSph
galaxy (Carina), but with $M_{0.3},V_{\rm max},V_{\rm peak}$ below that of the
known dSph’s, i.e. probable dark clumps. Only one of the 100 random observer
locations used in the analysis is shown here.
## V Dark Clumps
An annihilation signal from dark clumps not associated with any known luminous
stellar counterpart would provide evidence for one of the fundamental
implications of the CDM paradigm of structure formation: abundant Galactic
substructure. Barring a serendipitous discovery with an ACT, the discovery of
such a source will have to rely on all-sky surveys, such as provided by Fermi.
Of course even a weak and tentative identification of a dark clump with Fermi
could be followed up with an ACT.
Unlike for known dSph galaxies, for which we at least have some astronomical
observations to guide us, we must rely entirely upon numerical simulations to
quantify the prospects of detecting the annihilation signal from dark clumps.
Recent significant progress Governato et al. (2007) notwithstanding, it is at
present not yet possible to perform realistic cosmological hydrodynamic
galaxy-formation simulations, which include, in addition to the DM dynamics,
all the relevant baryonic physics of gas cooling, star formation, supernova
and AGN feedback, etc. that may have a significant impact on the DM
distribution at the centers of massive halos. Instead we make use of the
extremely high resolution, purely collisionless DM-only Via Lactea II (VL2)
simulation Diemand et al. (2008), which provides an exquisite view of the
clumpiness of the Galactic DM distribution, but at the expense of not
capturing all the relevant physics at the baryon-dominated Galactic center.
For the abundance, distribution, and internal properties of the DM subhalos
that are the focus of this work, the neglect of baryonic physics is less of a
problem, since they are too small to allow for much gas cooling and
significant baryonic effects (this is supported by the high mass-to-light
ratios observed in the Milky Way dSph’s), although tidal interactions with the
Galactic stellar and gaseous disk might significantly affect the population of
nearby subhalos.
With a particle mass of $4,100\,\rm M_{\odot}$ and a force softening of 40 pc,
the VL2 simulation resolves over 50,000 subhalos today within the host’s
$r_{\rm 200}=402$ kpc (the radius enclosing an average density 200 times the
mean matter value). Above $\sim 200$ particles per halo, the differential
subhalo mass function is well-fit by a single power law, $dN/dM\sim M^{-1.9}$,
and the cumulative $V_{\rm max}$ function is $N(>V_{\rm max})\sim V_{\rm
max}^{-3}$ Diemand et al. (2008). The radial distribution of subhalos is
“anti-biased” with respect to the host halo’s density profile, meaning that
the mass distribution becomes less clumpy as one approaches the host’s center
Kuhlen et al. (2007); Diemand et al. (2008). Similar results have been
obtained by the Aquarius group Springel et al. (2008a); Navarro et al. (2008).
Typical subhalo concentrations, defined as $\Delta_{V}=\langle\rho(<r_{\rm
Vmax})\rangle/\rho_{\rm crit}$, grow towards the center, owing to a
combination of earlier formation times Diemand et al. (2005); Moore et al.
(2006) and stronger tidal stripping of central subhalos: VL2 subhalos on
average have a 60 times higher $\Delta_{V}$ at 8 kpc than at 400 kpc Diemand
et al. (2008). Note that this also implies $\sim 7$ times higher annihilation
luminosities for central subhalos, since $\mathcal{L}\sim V_{\rm
max}^{4}/r_{\rm Vmax}\sim V_{\rm max}^{3}\sqrt{\Delta_{V}}$. The counter-
acting trends of decreasing relative abundance of subhalos and increasing
annihilation luminosity towards the center makes it more difficult for
(semi-)analytical methods to accurately assess the role of subhalos in the
Galactic annihilation signal, and motivate future, even higher resolution,
numerical simulations of the formation and evolution of Galactic DM
(sub-)structure. A direct analysis of the VL2 simulations in terms of the
detectability with Fermi of individual subhalos was performed by Kuhlen et al.
(2008). They found that for reasonable particle physics parameters a handful
of subhalos should be able to outshine the astrophysical backgrounds and would
be detected at more than $5\sigma$ significance over the lifetime of the Fermi
mission.
As discussed in the previous section, we have directly calculated the
annihilation luminosities for all VL2 subhalos using Eq. (9) and assuming an
NFW density profile. The luminosities would be $\sim 40\%$ higher if an
Einasto ($\alpha=0.17$) profile had been adopted instead. We then converted
these luminosities to fluxes by dividing by $4\pi D^{2}$, where the distances
$D$ were determined for 100 randomly chosen observer locations 8 kpc from the
host halo center. The resulting values of $\mathcal{F}_{\rm tot}$ are plotted
in Figure 3, for just one of the 100 observer positions, as a function of the
subhalos’ $M_{0.3}$, $V_{\rm max}$, and $V_{\rm peak}$. Although the
distributions show quite a bit of scatter, in all three cases a clear trend is
apparent of more massive subhalos having higher fluxes. This trend could
simply be the result of the higher luminosities of more massive halos, but one
might have expected smaller mass subhalos to be brighter, since their greater
abundance should result in lower typical distances and hence higher fluxes.
This latter effect could be artificially suppressed in the numerical
simulations, if smaller mass subhalos, whose dense centers are not as well
resolved, were more easily tidally disrupted closer to the Galactic Center, or
if the subhalo finding algorithm had trouble identifying low mass halos in the
high background density central region. In Figure 4 we plot the subhalos’
$V_{\rm max}$ against their distance to the host halo center $\hat{D}$. There
appears to be a dearth of the lowest $V_{\rm max}$ subhalos ($V_{\rm
max}\lesssim 2$ km s-1) at small distances, but at the moment it is not clear
whether this suppression is a real effect or a numerical artifact. It’s also
worth noting that such small subhalos might be more susceptible to disruption
by interactions with the Milky Way’s stellar and gaseous disk. At any rate, we
can obtain an analytic estimate of the scaling of the typical subhalo flux
with $V_{\rm max}$ by noting that the luminosity scales as $\mathcal{L}\sim
V_{\rm max}^{3}\sqrt{\Delta_{V}}$ and the typical distance as $D\sim
n^{-1/3}\sim V_{\rm max}^{4/3}$ (since $dn/dV_{\rm max}\sim V_{\rm
max}^{-4}$). The typical flux should thus scale as
$\mathcal{F}\sim\mathcal{L}/D^{2}\sim V_{\rm max}^{1/3}\sqrt{\Delta_{V}}$, and
would be higher for more massive subhalos at a fixed $\Delta_{V}$. Actually
lower $V_{\rm max}$ subhalos might be expected to have higher $\Delta_{V}$ due
to their earlier formation times, but it remains to be seen to what degree
this expectation is borne out in numerical simulations.
Figure 4: VL2 subhalo $V_{\rm max}$ vs. distance to host halo $\hat{D}$.
Figure 5: Top: The cumulative number of subhalos with flux exceeding
$\mathcal{F}_{\rm tot}$, $\mathcal{F}_{c}$. Bottom: The fraction of dark
clumps, i.e. subhalos likely not hosting any stars and defined by
$M_{0.3}<5\times 10^{6}\,\rm M_{\odot}$, $V_{\rm max}<8$ km s-1, or $V_{\rm
peak}<14$ km s-1, as a function of limiting flux $\mathcal{F}_{\rm tot}$.
These distributions are averages over 100 randomly chosen observer locations 8
kpc from the host halo center.
The points in Figure 3 can be directly compared with the values for the known
Milky Way dSph’s in Tables 1 and 2: it appears that there are many DM subhalos
at least as bright as the known Milky Way dSph’s. This impression is confirmed
by the top panel of Figure 5, in which we show the cumulative number of
subhalos with fluxes greater than $\mathcal{F}_{\rm tot}$ and
$\mathcal{F}_{c}$. These distributions were obtained by averaging over 100
randomly chosen observer locations 8 kpc from the host halo center. The mean
number of DM subhalos with $\mathcal{F}_{\rm tot}$ greater than that of
(Carina, Draco, Ursa Minor, Ursa Major, Segue 1) is (90, 54, 43, 17, 13), and
the corresponding numbers for $\mathcal{F}_{c}$ are (96, 62, 49, 24, 19). This
demonstrates that if a DM annihilation signal from any of the known Milky Way
dSph’s is detected, then many more DM subhalos should be visible. The plot
also implies that Segue 1, the dSph with the highest $\mathcal{F}_{\rm tot}$
and $\mathcal{F}_{c}$ of the currently known sample, is unlikely to be the
brightest DM subhalo in the sky. Of course some of these additional bright
sources could very well have stellar counterparts that have simply been missed
so far, due to the limited sky coverage of current surveys or insufficiently
deep exposures. To assess what fraction of high flux sources are likely to be
genuinely dark clumps without any stars, we split the sample by a limiting
value of $M_{0.3}=5\times 10^{6}\,\rm M_{\odot}$, $V_{\rm max}=8$ km s-1, and
$V_{\rm max}=14$ km s-1. We assume that DM subhalos below these limits are too
small to have been able to form any stars, and hence are truly dark clumps. Of
the known dSph’s listed in Table 1 only Leo IV falls below these limits. In
the bottom panel of Figure 5 we plot $f_{\rm dark}(>\mathcal{F}_{\rm tot})$,
the fraction of subhalos without stars, as a function of the limiting
annihilation flux $\mathcal{F}_{\rm tot}$. $f_{\rm dark}$ falls monotonically
with $\mathcal{F}_{\rm tot}$, which makes sense given that higher flux sources
are typically more massive and hence more likely to host stars. Between 30 and
40% of all DM subhalos brighter than Carina are expected to be dark clumps.
This fraction drops to 10% for subhalos brighter than Segue 1.
### V.1 Boost Factor?
The analysis presented here so far has been limited to known dSph galaxies and
clumps resolved in the VL2 simulation, whose resolution limit is set by the
available computational resources, and has nothing to do with fundamental
physics. Indeed, the CDM expectation is that the clumpiness should continue
all the way down to the cut-off in the matter power spectrum, set by
collisional damping and free streaming in the early universe (Green et al.,
2005; Loeb and Zaldarriaga, 2005). For typical WIMP DM, this cut-off occurs at
masses of $m_{0}=10^{-12}$ to $10^{-4}\,\rm M_{\odot}$ (Profumo et al., 2006;
Bringmann, 2009), some 10 to 20 orders of magnitude below VL2’s mass
resolution. Since the annihilation rate goes as $\rho^{2}$ and
$\langle\rho^{2}\rangle>\langle\rho\rangle^{2}$, this sub-resolution
clumpiness will lead to an enhancement of the total luminosity compared to the
smooth mass distribution in the simulation.
The magnitude of this so-called substructure boost factor depends sensitively
on the properties of subhalos below the simulation’s resolution limit, in
particular on the behavior of the concentration-mass relation. A simple power
law extrapolation of the contribution of simulated subhalos to the total
luminosity of the host halo leads to boosts on order of a a few hundred
Springel et al. (2008b). More sophisticated (semi-)analytical models,
accounting for different possible continuations of the concentration-mass
relation to lower masses, typically find smaller boosts of around a few tens
Strigari et al. (2007); Pieri et al. (2008); Kuhlen et al. (2008); Martinez et
al. (2009).
More importantly, this boost refers to the enhancement of the total
annihilation luminosity of a subhalo, but this is not likely the quantity most
relevant for detection. At the distances where subhalos might be detectable as
individual sources, their projected size exceeds the angular resolution of
today’s detectors. The surface brightness profile from annihilations in the
smooth DM component would be strongly peaked towards the center (yet probably
still resolved by Fermi Kuhlen et al. (2008)), owing to the $\rho(r)^{2}$
dependence of the annihilation rate. The luminosity contribution from a
subhalo’s sub-substructure population (i.e. its boost), however, is much less
centrally concentrated: at best it follows the subhalo’s mass density profile
$\rho(r)$, although it might very well even be anti-biased. This implies that
substructure would preferentially boost the outer regions of a subhalo, where
the surface brightness typically remains below the level of astrophysical
backgrounds and hence doesn’t contribute much to the detection significance.
In other words, the boost factor might apply to $\mathcal{F}_{\rm tot}$, but
much less (or not at all) to $\mathcal{F}_{c}$; yet it is $\mathcal{F}_{c}$
that is likely to determine whether a given subhalo can be detected with Fermi
or an ACT. It thus seems unlikely that the detectability of Galactic subhalos
would be significantly enhanced by their own substructure222This is in
contrast to many previous claims in the literature, including some by the
present author (e.g. Kuhlen et al., 2008). A re-analysis of that work (in
progress) with an improved treatment of the angular dependence of the
substructure boost, indeed finds that the boost only weakly increases the
number of detectable subhalos.. On the other hand, a substructure boost could
be very important for diffuse DM annihilation signals, either from
extragalactic sources, where the boost would simply increase the overall
amplitude Ullio et al. (2002), or from Galactic DM, where the boost could
affect the amplitude and angular profile of the signal, as well as the power
spectrum of its anisotropies Pieri et al. (2008); Kuhlen et al. (2008);
Siegal-Gaskins (2008); Springel et al. (2008b); Fornasa et al. (2009); Ando
(2009).
## VI Summary and Conclusions
In this work we have reviewed the DM annihilation signal from Galactic
subhalos. After going over the basics of the annihilation process with a focus
on the resulting $\gamma$-ray output, we summarized the properties of DM
subhalos relevant for estimating their annihilation luminosity. In the
remainder of the paper we used the Via Lactea II simulation to assess the
strength of the annihilation flux from both known Galactic dSph galaxies as
well as from dark clumps not hosting any stars. By matching the distances $D$
and central masses $M_{0.3}$ of simulated subhalos to the corresponding
published values of 18 known dSph’s, we were able to infer most probable
values, and the 1-$\sigma$ scatter around them, for $V_{\rm max}$ and $r_{\rm
Vmax}$, and hence for the annihilation luminosity $\mathcal{L}$ and flux
$\mathcal{F}$ of all dwarfs. According to this analysis, the recently
discovered dSph Segue 1 should be the brightest of the known dSph’s, followed
by Ursa Major II, Ursa Minor, Draco, and Carina. Further, we showed that if
any of the known Galactic dSph’s are bright enough to be detected, then at
least 10 times more subhalos should appear as visible sources. Some of these
would be as-of-yet undiscovered luminous dwarf galaxies, but a significant
fraction should correspond to dark clumps not hosting any stars. The fraction
of dark clump sources is 10% for subhalos at least as bright as Segue 1 and
grows to 40% for subhalos brighter than Carina. Lastly, we briefly considered
the role that a substructure boost factor should play in the detectability of
individual Galactic dSph’s and other DM subhalos. We argued that any boost is
unlikely to strongly increase their prospects for detection, since its
shallower angular dependence would preferentially boost the outer regions of
subhalos, which typically don’t contribute much to the detection significance.
Several caveats to these findings are in order. Probably the most important of
these is that our simulation completely neglects the effects of baryons. Gas
cooling, star formation, and the associated feedback processes are unlikely to
strongly affect most subhalos, owing to their low masses. However, tidal
interactions with the baryonic components of the Milky Way galaxy might do so.
The Sagittarius dSph, for example, is thought to be in the process of complete
disruption from tidal interactions with the Milky Way. A second caveat is that
our analysis is based on only one, albeit very high resolution, numerical
simulation, and so we cannot assess the importance of cosmic variance, or the
dependence on cosmological parameters such as $\sigma_{8}$ and $n_{s}$. Other
work has found considerable halo-to-halo scatter Reed et al. (2005); Springel
et al. (2008a); Ishiyama et al. (2009), with a factor of $\sim 2$ variance in
the total subhalo abundance, for example.
These caveats motivate further study and future, higher resolution numerical
simulations, including the effects of baryonic physics. The characterization
of the Galactic DM annihilation signal is of crucial importance in guiding
observational efforts to shed light on the nature of DM. We are hopeful that
in the next few years the promise of a DM annihilation signal will come to
fruition, and will help us to unravel this puzzle.
## VII Acknowledgments
Support for this work was provided by the William L. Loughlin Fellowship at
the Institute for Advanced Study. I would like to thank my collaborators from
the Via Lactea Project for their expertise and invaluable contributions.
## References
* Klypin et al. (1999) A. Klypin, A. V. Kravtsov, O. Valenzuela, and F. Prada, ApJ 522, 82 (1999), eprint arXiv:astro-ph/9901240.
* Moore et al. (1999) B. Moore, S. Ghigna, F. Governato, G. Lake, T. Quinn, J. Stadel, and P. Tozzi, ApJ 524, L19 (1999), eprint arXiv:astro-ph/9907411.
* Mateo (1998) M. L. Mateo, ARA&A 36, 435 (1998), eprint arXiv:astro-ph/9810070.
* Willman et al. (2005a) B. Willman, M. R. Blanton, A. A. West, J. J. Dalcanton, D. W. Hogg, D. P. Schneider, N. Wherry, B. Yanny, and J. Brinkmann, AJ 129, 2692 (2005a), eprint arXiv:astro-ph/0410416.
* Willman et al. (2005b) B. Willman, J. J. Dalcanton, D. Martinez-Delgado, A. A. West, M. R. Blanton, D. W. Hogg, J. C. Barentine, H. J. Brewington, M. Harvanek, S. J. Kleinman, et al., ApJ 626, L85 (2005b), eprint arXiv:astro-ph/0503552.
* Belokurov et al. (2006) V. Belokurov, D. B. Zucker, N. W. Evans, G. Gilmore, S. Vidrih, D. M. Bramich, H. J. Newberg, R. F. G. Wyse, M. J. Irwin, M. Fellhauer, et al., ApJ 642, L137 (2006), eprint arXiv:astro-ph/0605025.
* Zucker et al. (2006a) D. B. Zucker, V. Belokurov, N. W. Evans, M. I. Wilkinson, M. J. Irwin, T. Sivarani, S. Hodgkin, D. M. Bramich, J. M. Irwin, G. Gilmore, et al., ApJ 643, L103 (2006a), eprint arXiv:astro-ph/0604354.
* Zucker et al. (2006b) D. B. Zucker, V. Belokurov, N. W. Evans, J. T. Kleyna, M. J. Irwin, M. I. Wilkinson, M. Fellhauer, D. M. Bramich, G. Gilmore, H. J. Newberg, et al., ApJ 650, L41 (2006b), eprint arXiv:astro-ph/0606633.
* Sakamoto and Hasegawa (2006) T. Sakamoto and T. Hasegawa, ApJ 653, L29 (2006), eprint arXiv:astro-ph/0610858.
* Belokurov et al. (2007) V. Belokurov, D. B. Zucker, N. W. Evans, J. T. Kleyna, S. Koposov, S. T. Hodgkin, M. J. Irwin, G. Gilmore, M. I. Wilkinson, M. Fellhauer, et al., ApJ 654, 897 (2007), eprint arXiv:astro-ph/0608448.
* Irwin et al. (2007) M. J. Irwin, V. Belokurov, N. W. Evans, E. V. Ryan-Weber, J. T. A. de Jong, S. Koposov, D. B. Zucker, S. T. Hodgkin, G. Gilmore, P. Prema, et al., ApJ 656, L13 (2007), eprint arXiv:astro-ph/0701154.
* Walsh et al. (2007) S. M. Walsh, H. Jerjen, and B. Willman, ApJ 662, L83 (2007), eprint 0705.1378.
* Diemand et al. (2008) J. Diemand, M. Kuhlen, P. Madau, M. Zemp, B. Moore, D. Potter, and J. Stadel, Nature 454, 735 (2008), eprint 0805.1244.
* Springel et al. (2008a) V. Springel, J. Wang, M. Vogelsberger, A. Ludlow, A. Jenkins, A. Helmi, J. F. Navarro, C. S. Frenk, and S. D. M. White, MNRAS 391, 1685 (2008a), eprint 0809.0898.
* Stadel et al. (2008) J. Stadel, D. Potter, B. Moore, J. Diemand, P. Madau, M. Zemp, M. Kuhlen, and V. Quilis, ArXiv e-prints (2008), eprint 0808.2981.
* White and Rees (1978) S. D. M. White and M. J. Rees, MNRAS 183, 341 (1978).
* Blumenthal et al. (1984) G. R. Blumenthal, S. M. Faber, J. R. Primack, and M. J. Rees, Nature 311, 517 (1984).
* Dekel and Silk (1986) A. Dekel and J. Silk, ApJ 303, 39 (1986).
* Bullock et al. (2000) J. S. Bullock, A. V. Kravtsov, and D. H. Weinberg, ApJ 539, 517 (2000), eprint arXiv:astro-ph/0002214.
* Kravtsov et al. (2004) A. V. Kravtsov, O. Y. Gnedin, and A. A. Klypin, ApJ 609, 482 (2004), eprint arXiv:astro-ph/0401088.
* Mayer et al. (2006) L. Mayer, C. Mastropietro, J. Wadsley, J. Stadel, and B. Moore, MNRAS 369, 1021 (2006), eprint arXiv:astro-ph/0504277.
* Madau et al. (2008) P. Madau, J. Diemand, and M. Kuhlen, ApJ 679, 1260 (2008), eprint 0802.2265.
* Koposov et al. (2009) S. E. Koposov, J. Yoo, H.-W. Rix, D. H. Weinberg, A. V. Macciò, and J. M. Escudé, ApJ 696, 2179 (2009), eprint 0901.2116.
* Maccio’ et al. (2009) A. V. Maccio’, X. Kang, F. Fontanot, R. S. Somerville, S. E. Koposov, and P. Monaco, ArXiv e-prints (2009), eprint 0903.4681.
* Profumo et al. (2006) S. Profumo, K. Sigurdson, and M. Kamionkowski, Physical Review Letters 97, 031301 (2006), eprint arXiv:astro-ph/0603373.
* Bringmann (2009) T. Bringmann, ArXiv e-prints (2009), eprint 0903.0189.
* Bergström et al. (1999) L. Bergström, J. Edsjö, P. Gondolo, and P. Ullio, Phys. Rev. D 59, 043506 (1999), eprint arXiv:astro-ph/9806072.
* Stoehr et al. (2003) F. Stoehr, S. D. M. White, V. Springel, G. Tormen, and N. Yoshida, MNRAS 345, 1313 (2003), eprint arXiv:astro-ph/0307026.
* Calcáneo-Roldán and Moore (2000) C. Calcáneo-Roldán and B. Moore, Phys. Rev. D 62, 123005 (2000), eprint arXiv:astro-ph/0010056.
* Koushiappas et al. (2004) S. M. Koushiappas, A. R. Zentner, and T. P. Walker, Phys. Rev. D 69, 043501 (2004), eprint arXiv:astro-ph/0309464.
* Strigari et al. (2008a) L. E. Strigari, J. S. Bullock, M. Kaplinghat, J. D. Simon, M. Geha, B. Willman, and M. G. Walker, Nature 454, 1096 (2008a), eprint 0808.3772.
* Ibata et al. (2002) R. A. Ibata, G. F. Lewis, M. J. Irwin, and T. Quinn, MNRAS 332, 915 (2002), eprint arXiv:astro-ph/0110690.
* Johnston et al. (2002) K. V. Johnston, D. N. Spergel, and C. Haydn, ApJ 570, 656 (2002), eprint arXiv:astro-ph/0111196.
* Peñarrubia et al. (2006) J. Peñarrubia, A. J. Benson, D. Martínez-Delgado, and H. W. Rix, ApJ 645, 240 (2006), eprint arXiv:astro-ph/0512507.
* Siegal-Gaskins and Valluri (2008) J. M. Siegal-Gaskins and M. Valluri, ApJ 681, 40 (2008), eprint 0710.0385.
* Toth and Ostriker (1992) G. Toth and J. P. Ostriker, ApJ 389, 5 (1992).
* Read et al. (2008) J. I. Read, G. Lake, O. Agertz, and V. P. Debattista, MNRAS 389, 1041 (2008), eprint 0803.2714.
* Kazantzidis et al. (2009) S. Kazantzidis, A. R. Zentner, A. V. Kravtsov, J. S. Bullock, and V. P. Debattista, ArXiv e-prints (2009), eprint 0902.1983.
* Adriani et al. (2009) O. Adriani, G. C. Barbarino, G. A. Bazilevskaya, R. Bellotti, M. Boezio, E. A. Bogomolov, L. Bonechi, M. Bongi, V. Bonvicini, S. Bottai, et al., Nature 458, 607 (2009), eprint 0810.4995.
* Chang et al. (2008) J. Chang, J. H. Adams, H. S. Ahn, G. L. Bashindzhagyan, M. Christl, O. Ganel, T. G. Guzik, J. Isbert, K. C. Kim, E. N. Kuznetsov, et al., Nature 456, 362 (2008).
* Torii et al. (2008) S. Torii, T. Yamagami, T. Tamura, K. Yoshida, H. Kitamura, K. Anraku, J. Chang, M. Ejiri, I. Iijima, A. Kadokura, et al., ArXiv e-prints (2008), eprint 0809.0760.
* Abdo et al. (2009) A. A. Abdo, M. Ackermann, M. Ajello, W. B. Atwood, M. Axelsson, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, M. Battelino, et al., Physical Review Letters 102, 181101 (2009), eprint 0905.0025.
* H. E. S. S. Collaboration: F. Aharonian (2009) H. E. S. S. Collaboration: F. Aharonian, ArXiv e-prints (2009), eprint 0905.0105.
* Aharonian et al. (2008) F. Aharonian, A. G. Akhperjanian, A. R. Bazer-Bachi, M. Beilicke, W. Benbow, D. Berge, K. Bernlöhr, C. Boisson, O. Bolz, V. Borrel, et al., Astroparticle Physics 29, 55 (2008), eprint 0711.2369.
* Hui (2008) C. M. Hui, in _American Institute of Physics Conference Series_ , edited by F. A. Aharonian, W. Hofmann, and F. Rieger (2008), vol. 1085 of _American Institute of Physics Conference Series_ , pp. 407–410.
* Albert et al. (2008) J. Albert, E. Aliu, H. Anderhub, P. Antoranz, M. Backes, C. Baixeras, J. A. Barrio, H. Bartko, D. Bastieri, J. K. Becker, et al., ApJ 679, 428 (2008), eprint 0711.2574.
* Aliu et al. (2009) E. Aliu, H. Anderhub, L. A. Antonelli, P. Antoranz, M. Backes, C. Baixeras, S. Balestra, J. A. Barrio, H. Bartko, D. Bastieri, et al., ApJ 697, 1299 (2009), eprint 0810.3561.
* Driscoll et al. (2008) D. D. Driscoll, C. E. Covault, J. Ball, J. E. Carson, A. Jarvis, R. A. Ong, J. Zweerink, D. S. Hanna, J. Kildea, T. Lindner, et al., Phys. Rev. D 78, 087101 (2008).
* Siegal-Gaskins (2008) J. M. Siegal-Gaskins, Journal of Cosmology and Astro-Particle Physics 10, 40 (2008), eprint 0807.1328.
* Ando (2009) S. Ando, ArXiv e-prints (2009), eprint 0903.4685.
* Fornasa et al. (2009) M. Fornasa, L. Pieri, G. Bertone, and E. Branchini, ArXiv e-prints (2009), eprint 0901.2921.
* Springel et al. (2008b) V. Springel, S. D. M. White, C. S. Frenk, J. F. Navarro, A. Jenkins, M. Vogelsberger, J. Wang, A. Ludlow, and A. Helmi, Nature 456, 73 (2008b), eprint 0809.0894.
* Brun et al. (2009) P. Brun, T. Delahaye, J. Diemand, S. Profumo, and P. Salati, ArXiv e-prints (2009), eprint 0904.0812.
* Kuhlen and Malyshev (2009) M. Kuhlen and D. Malyshev, ArXiv e-prints (2009), eprint 0904.3378.
* Arkani-Hamed et al. (2009) N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. Weiner, Phys. Rev. D 79, 015014 (2009), eprint 0810.0713.
* Lattanzi and Silk (2009) M. Lattanzi and J. Silk, Phys. Rev. D 79, 083523 (2009), eprint 0812.0360.
* Pieri et al. (2009a) L. Pieri, M. Lattanzi, and J. Silk, ArXiv e-prints (2009a), eprint 0902.4330.
* Kuhlen et al. (2009) M. Kuhlen, P. Madau, and J. Silk (2009), submitted to Science.
* Turner (1990) M. S. Turner, Phys. Rep. 197, 67 (1990).
* Kolb and Turner (1990) E. W. Kolb and M. S. Turner, _The early universe_ (1990).
* Jungman et al. (1996) G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rep. 267, 195 (1996), eprint arXiv:hep-ph/9506380.
* Bertone et al. (2005) G. Bertone, D. Hooper, and J. Silk, Phys. Rep. 405, 279 (2005), eprint arXiv:hep-ph/0404175.
* Hinshaw et al. (2009) G. Hinshaw, J. L. Weiland, R. S. Hill, N. Odegard, D. Larson, C. L. Bennett, J. Dunkley, B. Gold, M. R. Greason, N. Jarosik, et al., ApJS 180, 225 (2009), eprint 0803.0732.
* Griest and Seckel (1991) K. Griest and D. Seckel, Phys. Rev. D 43, 3191 (1991).
* Gondolo and Gelmini (1991) P. Gondolo and G. Gelmini, Nuclear Physics B 360, 145 (1991).
* Bringmann et al. (2008) T. Bringmann, L. Bergström, and J. Edsjö, Journal of High Energy Physics 1, 49 (2008), eprint 0710.3169.
* Gustafsson et al. (2007) M. Gustafsson, E. Lundström, L. Bergström, and J. Edsjö, Physical Review Letters 99, 041301 (2007), eprint arXiv:astro-ph/0703512.
* Sjöstrand (1994) T. Sjöstrand, Computer Physics Communications 82, 74 (1994).
* Gondolo et al. (2004) P. Gondolo, J. Edsjö, P. Ullio, L. Bergström, M. Schelke, and E. A. Baltz, Journal of Cosmology and Astro-Particle Physics 7, 8 (2004), eprint arXiv:astro-ph/0406204.
* Diemand et al. (2007) J. Diemand, M. Kuhlen, and P. Madau, ApJ 657, 262 (2007), eprint arXiv:astro-ph/0611370.
* Strigari et al. (2008b) L. E. Strigari, S. M. Koushiappas, J. S. Bullock, M. Kaplinghat, J. D. Simon, M. Geha, and B. Willman, ApJ 678, 614 (2008b), eprint 0709.1510.
* Pieri et al. (2008) L. Pieri, G. Bertone, and E. Branchini, MNRAS 384, 1627 (2008), eprint 0706.2101.
* Baltz et al. (2000) E. A. Baltz, C. Briot, P. Salati, R. Taillet, and J. Silk, Phys. Rev. D 61, 023514 (2000), eprint arXiv:astro-ph/9909112.
* Tasitsiomi and Olinto (2002) A. Tasitsiomi and A. V. Olinto, Phys. Rev. D 66, 083006 (2002), eprint arXiv:astro-ph/0206040.
* Aloisio et al. (2004) R. Aloisio, P. Blasi, and A. V. Olinto, ApJ 601, 47 (2004), eprint arXiv:astro-ph/0206036.
* Evans et al. (2004) N. W. Evans, F. Ferrer, and S. Sarkar, Phys. Rev. D 69, 123501 (2004), eprint arXiv:astro-ph/0311145.
* Koushiappas (2006) S. M. Koushiappas, Physical Review Letters 97, 191301 (2006), eprint arXiv:astro-ph/0606208.
* Kuhlen et al. (2008) M. Kuhlen, J. Diemand, and P. Madau, ApJ 686, 262 (2008), eprint 0805.4416.
* Baltz et al. (2007) E. A. Baltz, J. E. Taylor, and L. L. Wai, ApJ 659, L125 (2007), eprint arXiv:astro-ph/0610731.
* Navarro et al. (1997) J. F. Navarro, C. S. Frenk, and S. D. M. White, ApJ 490, 493 (1997), eprint arXiv:astro-ph/9611107.
* Macciò et al. (2007) A. V. Macciò, A. A. Dutton, F. C. van den Bosch, B. Moore, D. Potter, and J. Stadel, MNRAS 378, 55 (2007), eprint arXiv:astro-ph/0608157.
* Diemand et al. (2004) J. Diemand, B. Moore, and J. Stadel, MNRAS 353, 624 (2004), eprint arXiv:astro-ph/0402267.
* Navarro et al. (2008) J. F. Navarro, A. Ludlow, V. Springel, J. Wang, M. Vogelsberger, S. D. M. White, A. Jenkins, C. S. Frenk, and A. Helmi, ArXiv e-prints (2008), eprint 0810.1522.
* Allgood et al. (2006) B. Allgood, R. A. Flores, J. R. Primack, A. V. Kravtsov, R. H. Wechsler, A. Faltenbacher, and J. S. Bullock, MNRAS 367, 1781 (2006), eprint arXiv:astro-ph/0508497.
* Kuhlen et al. (2007) M. Kuhlen, J. Diemand, and P. Madau, ApJ 671, 1135 (2007), eprint 0705.2037.
* Bryan and Norman (1998) G. L. Bryan and M. L. Norman, ApJ 495, 80 (1998), eprint arXiv:astro-ph/9710107.
* Kleyna et al. (2005) J. T. Kleyna, M. I. Wilkinson, N. W. Evans, and G. Gilmore, ApJ 630, L141 (2005), eprint arXiv:astro-ph/0507154.
* Muñoz et al. (2006) R. R. Muñoz, J. L. Carlin, P. M. Frinchaboy, D. L. Nidever, S. R. Majewski, and R. J. Patterson, ApJ 650, L51 (2006), eprint arXiv:astro-ph/0606271.
* Martin et al. (2007) N. F. Martin, R. A. Ibata, S. C. Chapman, M. Irwin, and G. F. Lewis, MNRAS 380, 281 (2007), eprint 0705.4622.
* Simon and Geha (2007) J. D. Simon and M. Geha, ApJ 670, 313 (2007), eprint 0706.0516.
* Koposov et al. (2008) S. Koposov, V. Belokurov, N. W. Evans, P. C. Hewett, M. J. Irwin, G. Gilmore, D. B. Zucker, H.-W. Rix, M. Fellhauer, E. F. Bell, et al., ApJ 686, 279 (2008), eprint 0706.2687.
* Tollerud et al. (2008) E. J. Tollerud, J. S. Bullock, L. E. Strigari, and B. Willman, ApJ 688, 277 (2008), eprint 0806.4381.
* Walsh et al. (2009) S. M. Walsh, B. Willman, and H. Jerjen, AJ 137, 450 (2009), eprint 0807.3345.
* Grcevich and Putman (2009) J. Grcevich and M. E. Putman, ApJ 696, 385 (2009), eprint 0901.4975.
* Bernlöhr et al. (2003) K. Bernlöhr, O. Carrol, R. Cornils, S. Elfahem, P. Espigat, S. Gillessen, G. Heinzelmann, G. Hermann, W. Hofmann, D. Horns, et al., Astroparticle Physics 20, 111 (2003), eprint arXiv:astro-ph/0308246.
* Martínez-Delgado et al. (2004) D. Martínez-Delgado, M. Á. Gómez-Flechoso, A. Aparicio, and R. Carrera, ApJ 601, 242 (2004), eprint arXiv:astro-ph/0308009.
* Holder et al. (2008) J. Holder, V. A. Acciari, E. Aliu, T. Arlen, M. Beilicke, W. Benbow, S. M. Bradbury, J. H. Buckley, V. Bugaev, Y. Butt, et al., in _American Institute of Physics Conference Series_ , edited by F. A. Aharonian, W. Hofmann, and F. Rieger (2008), vol. 1085 of _American Institute of Physics Conference Series_ , pp. 657–660.
* Essig et al. (2009) R. Essig, N. Sehgal, and L. E. Strigari, ArXiv e-prints (2009), eprint 0902.4750.
* Bovy (2009) J. Bovy, Phys. Rev. D 79, 083539 (2009), eprint 0903.0413.
* Martinez et al. (2009) G. D. Martinez, J. S. Bullock, M. Kaplinghat, L. E. Strigari, and R. Trotta, ArXiv e-prints (2009), eprint 0902.4715.
* Pieri et al. (2009b) L. Pieri, A. Pizzella, E. M. Corsini, E. Dalla Bontã, and F. Bertola, A&A 496, 351 (2009b), eprint 0812.1494.
* Governato et al. (2007) F. Governato, B. Willman, L. Mayer, A. Brooks, G. Stinson, O. Valenzuela, J. Wadsley, and T. Quinn, MNRAS 374, 1479 (2007), eprint arXiv:astro-ph/0602351.
* Diemand et al. (2005) J. Diemand, P. Madau, and B. Moore, MNRAS 364, 367 (2005), eprint arXiv:astro-ph/0506615.
* Moore et al. (2006) B. Moore, J. Diemand, P. Madau, M. Zemp, and J. Stadel, MNRAS 368, 563 (2006), eprint arXiv:astro-ph/0510370.
* Green et al. (2005) A. M. Green, S. Hofmann, and D. J. Schwarz, Journal of Cosmology and Astro-Particle Physics 8, 3 (2005), eprint arXiv:astro-ph/0503387.
* Loeb and Zaldarriaga (2005) A. Loeb and M. Zaldarriaga, Phys. Rev. D 71, 103520 (2005), eprint arXiv:astro-ph/0504112.
* Strigari et al. (2007) L. E. Strigari, S. M. Koushiappas, J. S. Bullock, and M. Kaplinghat, Phys. Rev. D 75, 083526 (2007), eprint arXiv:astro-ph/0611925.
* Ullio et al. (2002) P. Ullio, L. Bergström, J. Edsjö, and C. Lacey, Phys. Rev. D 66, 123502 (2002), eprint arXiv:astro-ph/0207125.
* Reed et al. (2005) D. Reed, F. Governato, T. Quinn, J. Gardner, J. Stadel, and G. Lake, MNRAS 359, 1537 (2005), eprint arXiv:astro-ph/0406034.
* Ishiyama et al. (2009) T. Ishiyama, T. Fukushige, and J. Makino, ApJ 696, 2115 (2009), eprint 0812.0683.
|
arxiv-papers
| 2009-06-09T21:04:35 |
2024-09-04T02:49:03.280122
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Kuhlen (IAS, Princeton)",
"submitter": "Michael Kuhlen",
"url": "https://arxiv.org/abs/0906.1822"
}
|
0906.1948
|
# Invariants of open books of links of surface singularities
András Némethi Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u.
13–15, Hungary [email protected] http://www.renyi.hu/~nemethi and Meral Tosun
Galatasaray University, Departement of Mathematics, 34257 Ortakoy-Istanbul,
Turkiye [email protected] http://math.gsu.edu.tr/tosun/index.html
###### Abstract.
If $M$ is the link of a complex normal surface singularity, then it carries a
canonical contact structure $\xi_{can}$, which can be identified from the
topology of the 3–manifold $M$. We assume that $M$ is a rational homology
sphere. We compute the support genus, the binding number and the norm
associated with the open books which support $\xi_{can}$, provided that we
restrict ourself to the case of (analytic) Milnor open books. In order to do
this, we determine monotoneity properties of the genus and the Milnor number
of all Milnor fibrations in terms of the Lipman cone.
We generalize results of [3] valid for links of rational surface
singularities, and we answer some questions of Etnyre and Ozbagci [7, section
8] regarding the above invariants.
###### Key words and phrases:
Surface singularity, Milnor open book, semigroup of divisors.
###### 2000 Mathematics Subject Classification:
Primary: 32S25 Secondary. 32S50, 57R17
The first author is partially supported by OTKA grants, the second author by
Galatasaray University Research fund, and both authors by the BudAlgGeo
project, in the framework of the European Community’s ‘Structuring the
European Research Area’ programme.
## 1\. Introduction
Let $M$ be an oriented 3-dimensional manifold. By a result of Giroux [8] there
is a one-to-one correspondence between open book decompositions of $M$ (up to
stabilization) and contact structures on $M$ (up to isotopy). In [7] Etnyre
and Ozbagci consider three invariants associated with a fixed contact
structure $\xi$ defined in terms of all open book decompositions supporting
it:
$\bullet$ the support genus sg$(\xi)$ is the minimal possible genus for a page
of an open book that supports $\xi$;
$\bullet$ the binding number bn$(\xi)$ is the minimal number of of binding
components for an open book supporting $\xi$ and that has pages of genus
sg$(\xi)$;
$\bullet$ the norm $\mathfrak{n}(\xi)$ of $\xi$ is the negative of the maximal
(topological) Euler characteristic of a page of an open book that supports
$\xi$.
In the present article we determine and characterize completely the above
invariants under the following restrictions: $M$ will be a rational homology
sphere which can be realized as the link of a complex surface singularity
$(S,0)$. Moreover, we will restrict ourselves to the collection of those open
book decompositions which can be realized as Milnor fibrations determined by
some analytic germ (the so-called Milnor open books). Notice that by [5], all
the Milnor open book decompositions define the same contact structure on $M$,
the canonical contact structure $\xi_{can}$. This structure is also induced by
any complex structure $(S,0)$ realized on the topological type, and it can be
characterized completely from the topology of $M$.
Hence our results will be applied exactly for the canonical contact structure
$\xi_{can}$, and for (analytic) Milnor open books, cf. section 5. The
corresponding invariants are denoted by sg${}_{an}(\xi_{can})$,
bn${}_{an}(\xi_{can})$ and $\mathfrak{n}_{an}(\xi_{can})$.
The present article generalize results of [3] valid for links of rational
surface singularities, and we answer some questions of [7, section 8]
regarding the above invariants.
## 2\. Preliminaries
### 2.1. Invariants associated with a resolution.
In what follows we assume that $(S,0)$ is a complex normal surface singularity
whose link is a rational homology sphere. Let $\pi:X\longrightarrow S$ be a
good resolution. We will denote by $E_{1},\ldots,E_{n}$ the smooth irreducible
components of the exceptional curve $E:=\pi^{-1}(0)$ and by $\Gamma$ its dual
graph. By our assumption, each $E_{i}$ has genus 0 and $\Gamma$ is a tree.
Consider the free group ${\mathcal{G}}:=H_{2}(X,{\mathbb{Z}})$ generated by
the irreducible components of $E$, i.e.
${\mathcal{G}}=\\{D=\sum_{i=1}^{n}m_{i}E_{i}\mid m_{i}\in{\mathbb{Z}}\\}$. On
${\mathcal{G}}$ there is a natural intersection pairing $(\cdot,\cdot)$ and a
natural partial ordering:
$\sum_{i}m^{\prime}_{i}E_{i}\leq\sum_{i}m^{\prime\prime}_{i}E_{i}$ if and only
if $m^{\prime}_{i}\leq m^{\prime\prime}_{i}$ for all $i$.
We denote the Lipman cone (semi-group) by
${\mathcal{E}}^{+}=\\{D\in{\mathcal{G}}\mid(D,E_{i})\leq 0\ \ \mbox{for any
$i$}\\}.$
It is known (see e.g. [2, 10]) that if $D=\sum m_{i}E_{i}\in{\mathcal{E}}^{+}$
then $m_{i}\geq 0$ for all $i$, and $m_{i}>0$ for all $i$ whenever
$D\in{\mathcal{E}}^{+}\setminus\\{0\\}$. Moreover,
${\mathcal{E}}^{+}\setminus\\{0\\}$ admits a unique minimal element (the so-
called Artin, or fundamental cycle), denoted by $Z_{min}$.
The definition of ${\mathcal{E}^{+}}$ is motivated by the following fact. Let
$f:(S,0)\to({\mathbb{C}},0)$ be a germ of an analytic function. Then the
divisor $(\pi^{*}(f))$ in $X$ of $f\circ\pi$ can be written as
$D_{\pi}(f)+S_{\pi}(f)$, where $D_{\pi}(f)$, called the compact part of
$(\pi^{*}(f))$, is supported on $E$, and $S_{\pi}(f)$ is the strict transform
by $\pi$ of $\\{f=0\\}$. The collection of compact parts (when $f$ runs over
${\mathcal{O}}_{S,0}$) forms a semi-group too, it will be denoted by
${\mathcal{A}^{+}}$. It is a sub-semi-group of ${\mathcal{E}^{+}}$ (since
$(\pi^{*}(f))\cdot E_{i})=0$ and $(S_{\pi}(f)\cdot E_{i})\geq 0$ for all $i$).
The subset ${\mathcal{A}}^{+}\setminus\\{0\\}$ also has a unique minimal
element $Z_{max}$, the maximal ideal divisor. It is the divisor of the generic
hyperplane section. By definitions $Z_{min}\leq Z_{max}$.
For rational singularities one has ${\mathcal{A}}^{+}={\mathcal{E}}^{+}$
(hence $Z_{max}=Z_{min}$ too). But, in general, these equalities do not hold.
The fundamental cycle $Z_{min}$ can be obtained by Laufer’s (combinatorial)
algorithm (cf. [9]), but the structure of ${\mathcal{A}}^{+}$ (and even of
$Z_{max}$ too) can be very difficult, it depends essentially on the analytic
structure of $(S,0)$.
### 2.2. (Milnor) open books.
Assume that $f:(S,0)\to({\mathbb{C}},0)$ defines an isolated singularity. Let
$M$ be the link of $(S,0)$ and $L_{f}:=f^{-1}(0)\cap M\subset M$ the
(transversal) intersection of $f^{-1}(0)$ with $M$. Then the Milnor fibration
of $f$ defines an open book decomposition of $M$ with binding $L_{f}$. One has
the following facts:
1. (1)
For any $f$, consider an embedded good resolution $\pi$ of the pair
$(S,f^{-1}(0))$. Then the strict transform $S_{\pi}(f)$ intersects $E$
transversally, and the number of intersection points $(S_{\pi}(f),E_{i})$
(i.e. the number of binding components associated with $E_{i}$) is exactly
$-(D_{\pi}(f),E_{i})$. Since the intersection form is negative definite, the
collection of binding components $\\{(S_{\pi}(f),E_{i})\\}_{i=1}^{n}$ and
$D_{\pi}(f)\in{\mathcal{A}^{+}}$ determine each other perfectly.
Moreover, by classical results of Stallings and Waldhausen, the (topological
type of the) binding $L_{f}\subset M$ determines completely the open book up
to an isotopy, provided that $M$ is a rational homology sphere. ([6, page 34]
provides two different arguments for this fact, one of them based on [4], the
other one on [14]. For counterexamples for the statement in the general
situation, see e.g. [11].)
Notice that the classification of all the (Milnor) open books associated with
a fixed analytic type of $(S,0)$ and analytic germs $f\in{\mathcal{O}}_{S,0}$
can be a very difficult problem (in fact, as difficult as the determination of
${\mathcal{A}}^{+}$).
2. (2)
Therefore, from a topological points of view, it is more natural to consider
the open books of all the analytic germs associated with all the analytic
structures supported by the topological type of $(S,0)$.
Notice that for a fixed topological type of $(S,0)$, in any (negative
definite) plumbing graph of $M$ one can also define the cone
${\mathcal{E}^{+}}$. The point is that for any non-zero element $D$ of
${\mathcal{E}^{+}}$ there is a convenient analytic structure on $(S,0)$ and an
analytic germ $f$, such that the plumbing graph can be identified with a dual
resolution graph (which serves as an embedded resolution graph for the pair
$(S,f^{-1}(0))$ too), and $D$ is the compact part $D_{\pi}(f)$, see [13, 12].
Hence, changing the analytic structure of $(S,0)$, we fill by the collections
${\mathcal{A}^{+}}$ all the semi-group ${\mathcal{E}^{+}}$.
In particular, for any $Z\in{\mathcal{E}}^{+}\setminus\\{0\\}$, there is an
open book decomposition (well-defined up to an isotopy) realized as Milnor
open book (by a convenient choice of the analytic objects).
3. (3)
For any fixed analytic type $(S,0)$, the open book associated with $Z_{max}$
is the Milnor fibration of the generic hyperplane section, in particular this
open book is (resolution) graph-independent. Similarly, for a fixed
topological type of $(S,0)$, the open book associated with $Z_{min}$ is also
graph-independent. It depends only on the topology of the link.
### 2.3. Invariants of Milnor open books.
Let us fix $M$, a plumbing (or, a dual resolution) graph $\Gamma$. Let us
consider a Milnor open book associated with an element
$Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$, cf. (2.2). In the sequel we will
consider the following numerical invariants of it:
1. (1)
The number of binding components $\beta(Z)$ is given by $-(Z,E)$ (which is
$\geq 1$).
2. (2)
Let $F$ be the fiber of the open book. It is an oriented connected surface
with $-(Z,E)$ boundary components. Let $g(Z)$ be its genus (the so-called
page-genus of the open book) and $\mu(Z)$ be the first Betti-number of $F$
(the so-called Milnor number). Clearly:
(2.3.1) $\mu(Z)=2\cdot g(Z)-1+\beta(Z)=2\cdot g(Z)-1-(Z,E)\geq 2g(Z).$
We will also write $\nu_{i}=(E_{i},E-E_{i})$, the number of components of
$E-E_{i}$ meeting $E_{i}$.
### 2.4. The ‘monotoneity’ property.
The main results of the next sections targets the ‘monotoneity’ property of
invariants listed in (2.3).
###### Definition 2.4.1.
Assume that for any resolution $\pi$ of $(S,0)$ one has a map
$I_{\pi}:{\mathcal{E}^{+}}\setminus\\{0\\}\to{\mathbb{Z}}_{\geq 0}$. We say
that $I=\\{I_{\pi}\\}_{\pi}$ is monotone if for any two cycles
$Z_{i}\in{\mathcal{E}^{+}}\setminus\\{0\\}$ ($i=1,2$) with $Z_{1}\leq Z_{2}$
one has $I_{\pi}(Z_{1})\leq I_{\pi}(Z_{2})$ for any $\pi$.
###### Remark 2.4.2.
Assume that the collection of invariants $\\{I_{\pi}\\}_{\pi}$ can be
transformed into (or comes from) an invariant $I$ which associates with any
(Milnor) open book $\mathfrak{m}$ of the link a non-negative integer. For any
fixed analytic type, let $\mathfrak{m}_{\max}$ be the Milnor open book
associated with $Z_{max}$ (considered in any resolution). Similarly, for any
topological type, let $\mathfrak{m}_{min}$ be the Milnor open book associated
with $Z_{min}$ (in any resolution of an analytic structure conveniently
chosen); cf. (2.2)(3).
Then, whenever $\\{I_{\pi}\\}_{\pi}$ is monotone, one has automatically the
next consequences:
1. (1)
Fix an analytic singularity $(S,0)$ and consider all the Milnor open books
associated with all isolated holomorphic germs $f\in{\mathcal{O}}_{S,0}$. Then
the minimum of integers $I(\mathfrak{m})$ of all these Milnor open books
$\mathfrak{m}$ is realized by the generic hyperplane section, i.e. by
$I(\mathfrak{m}_{max})$.
2. (2)
Fix a topological type of a normal surface singularity, and consider the open
books associated with all the isolated holomorphic germs of all the possible
analytic structures supported by the fixed topological type. Then the minimum
of all integers $I(\mathfrak{m})$ of all these Milnor open books
$\mathfrak{m}$ is realized by the open book associated with the Artin cycle,
i.e. by $I(\mathfrak{m}_{min})$.
## 3\. The monotoneity of the genus
### 3.1. The relation between the genus and the Euler-characteristic.
For any fixed graph $\Gamma$, we consider the ‘canonical cycle’
$K\in{\mathcal{G}}\otimes{\mathbb{Q}}$ defined by the (adjunction formulas)
$(K+E_{i},E_{i})+2=0$ for all $i$. Then the (holomorphic) Euler-characteristic
of any element $D\in{\mathcal{G}}$ is given by
(3.1.1) $\chi(D):=-\frac{1}{2}(D,D+K)\in{\mathbb{Z}}.$
###### Proposition 3.1.2.
Fix $Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$. Then
(3.1.3) $g(Z)=1+(Z,E)+\chi(-Z).$
###### Proof.
For any $1\leq i\leq n$ consider $k_{i}:=-(Z,E_{i})$ (the number of binding
components associated with $E_{i}$). Write also $Z=\sum_{i}m_{i}E_{i}$. Then
by the A’Campo’s formula (cf. [1]) $1-\mu=\sum_{i}(2-\nu_{i}-k_{i})m_{i}$.
Then use (2.3.1) and (3.1.1).∎
###### Remark 3.1.4.
Since $\chi(-Z)+\chi(Z)+Z^{2}=0$, one also has $g(Z)=1+(Z,E-Z)-\chi(Z).$ Since
for any $Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$ one gets $Z\geq E$, one has
$(Z,E-Z)\geq 0$ too. In particular:
(3.1.5) $g(Z)\geq 1-\chi(Z).$
Recall that rational singularities are characterized by $\chi(Z_{min})=1$ [2].
If additionally, $(S,0)$ is a minimal (i.e. if $Z_{min}=E$), then
$g(Z_{min})=0$. For arbitrary rational germs one has
$g(Z_{min})=(Z_{min},E-Z_{min})\geq 0$. This number, in general, might be non-
zero: e.g. in the case of the $E_{8}$-singularity it is 1. Considering
arbitrary singularities, $\chi(Z_{min})$ tends to $-\infty$ as the complexity
of the topological type of the germ increases, hence by (3.1.5) $g(Z_{min})$
tends to infinity too.
### 3.2. The “virtual genus” and its positivity.
The formula (3.1.3) motivates the following definition. For
$D=\sum_{i}m_{i}E_{i}\in{\mathcal{G}}$, let $|D|$ be the support
$\sum_{i\,:\,m_{i}\not=0}E_{i}$ of $D$ and $\\#(D)$ the number of connected
components of $|D|$.
###### Definition 3.2.1.
For $D\in{\mathcal{G}}$, $D\geq 0$, we define the “virtual genus” of $D$ by
(3.2.2) $g(D)=\\#(D)+(D,|D|)+\chi(-D).$
Since for any $Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$ one has $|Z|=E$, and $E$
is connected, (3.2.2) extends (3.1.3). Moreover, for any such
$Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$, by its definition, $g(Z)\geq 0$.
###### Theorem 3.2.3.
The virtual genus of any $D\in{\mathcal{G}}$, $D\geq 0$, is positive:
$g(D)\geq 0$.
###### Proof.
Assume that the statement is not true at least for one such a cycle. Since
$g(E_{i})=1+E_{i}^{2}+\chi(-E_{i})=0$, there exist a minimal cycle $D>0$ with
$g(D)<0$. Clearly, we can assume that $|D|$ is connected (and replacing
$\Gamma$ by its subgraph supported on $|D|$) that $|D|=E$. Write
$D=\sum_{i}m_{i}E_{i}$. Hence we have:
(3.2.4) $1+(D,E)+\chi(-D)<0.$
and, using the notation $\\#_{i}$ for the number of components of $|D-E_{i}|$:
(3.2.5) $\\#_{i}+(D-E_{i},|D-E_{i}|)+\chi(-D+E_{i})\geq 0$
for all $E_{i}$. Since $\chi(A+B)=\chi(A)+\chi(B)-(A,B)$, the two inequalities
can easily be compared. Indeed, first assume that $m_{i}=1$ for some $i$. Then
$|D-E_{i}|=E-E_{i}$ and $\\#_{i}=\nu_{i}$, hence (3.2.4) and (3.2.5)
contradict each other. Therefore, $m_{i}\geq 2$ for all $i$. In that case,
$|D-E_{i}|=E$ and $\\#_{i}=1$, hence (3.2.4) and (3.2.5) lead to
$(D-E,E_{i})\geq 0$ for all $i$. Hence $(D-E,D-E)$ is also non-negative by
summation. Since the intersection form is negative definite, this implies
$D=E$. This contradicts the fact that $D$ is non-reduced (and also with the
fact that $g(E)=0$). ∎
### 3.3. The monotoneity of the genus
The main result of this section is the following inequality:
###### Theorem 3.3.1.
Consider two cycles $Z$ and $Z+D$, where
$Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$ and $D\in{\mathcal{G}}$, $D\geq 0$.
Then the (virtual) genera satisfy $g(Z)\leq g(Z+D)$.
###### Proof.
By (3.1.3), one has
$\displaystyle g(Z+D)-g(Z)$ $\displaystyle=$
$\displaystyle(D,E)+\chi(-D)-(D,Z)$ $\displaystyle=$ $\displaystyle
g(D)+(D,E-|D|)-\\#(D)-(D,Z).$
If $|D|=E$ then $\\#(D)=1$ and $-(D,Z)\geq 1$ (otherwise we would have
$(Z,E_{i})=0$ for all $i$, or $Z=0$). If $|D|<E$, then $-(D,Z)\geq 0$ and
$(D,E-|D|)\geq(|D|,E-|D|)\geq\\#(D)$ by the connectivity of $\Gamma$. Hence,
in both cases, the right–hand side is $\geq g(D)$. Since $g(D)\geq 0$ by
(3.2.3), the inequality follows. ∎
###### Corollary 3.3.2.
The genus is monotone: for any $Z_{1}$ and $Z_{2}$ from ${\cal
E}^{+}\setminus\\{0\\}$ with $Z_{1}\leq Z_{2}$ one has $g(Z_{1})\leq
g(Z_{2})$. In particular, the statements of (2.4.2) also hold.
## 4\. The Milnor number and the number of boundary components
### 4.1. The monotoneity of the Milnor number
If one combines (2.3.1) and (3.1.3), one gets for any
$Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$:
(4.1.1) $\displaystyle\mu(Z)$ $\displaystyle=$ $\displaystyle
1+(Z,E)+2\cdot\chi(-Z)$ (4.1.2) $\displaystyle=$ $\displaystyle
g(Z)+\chi(-Z).$
Again, we extend the above formula (in a compatible way with (4.1.2)) for any
$D\geq 0$ by considering the ‘virtual Milnor number’ $\mu(D)$ as
$g(D)+\chi(-D)$, defined via the virtual genus $g(D)$.
Clearly, $\mu(Z)\geq 0$ for any $Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$, since
$\mu(Z)$ stays for a Betti number. Moreover, for any rational graph $\Gamma$,
one has $\min\chi=0$, hence for them the virtual invariants satisfy
$\mu(D)\geq g(D)\geq 0$ too. The next theorem generalizes this for a general
$\Gamma$.
###### Theorem 4.1.3.
Set $D\in{\mathcal{G}}$ with $D\geq 0$. Then the following inequalities hold:
1. (1)
$\chi(-D)\geq 0$;
2. (2)
$\mu(D)\geq g(D)\geq 0$;
3. (3)
$\mu(Z+D)\geq\mu(Z)$ for any $Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$.
###### Proof.
The proof of (1) is well-known for specialist, for the convenience of the
reader we provide it. We claim that for any $D>0$ there exists at least one
$E_{i}$ with $E_{i}\leq D$ such that $\chi(-D+E_{i})\leq\chi(-D)$. This by
induction shows that $\chi(-D)\geq 0$. The proof of the claim runs as follows.
Assume that it is not true for some $D>0$. Then for any $E_{i}$ from its
support one has $\chi(-D+E_{i})\geq\chi(-D)+1$. This is equivalent with
$(D,E_{i})\geq 0$, hence by summation one gets $D^{2}\geq 0$. This implies
$D=0$, a contradiction.
(2) follows from (4.1.2), part (1) and (3.2.3). For (3) notice that by (4.1.2)
$\mu(Z+D)-\mu(Z)=g(Z+D)-g(Z)+\chi(-D)-(Z,D).$
Notice that $g(Z+D)\geq g(Z)$ by (3.3.1), $\chi(-D)\geq 0$ by (1), and
$-(Z,D)\geq 0$ since $Z\in{\mathcal{E}^{+}}$. ∎
###### Corollary 4.1.4.
The Milnor number is monotone: for any $Z_{1}$ and $Z_{2}$ from ${\cal
E}^{+}\setminus\\{0\\}$ with $Z_{1}\leq Z_{2}$ one has
$\mu(Z_{1})\leq\mu(Z_{2})$. In particular, the statements of (2.4.2) also hold
for $\mu$.
### 4.2. The number of binding components
Recall that the number of binding components of the open book associated with
some $Z\in{\mathcal{E}^{+}}\setminus\\{0\\}$ is $\beta(Z)=-(Z,E)$. We wish to
understand the variation of this number in the realm of (Milnor) open books
with page-genus fixed. In order to do this, let us consider the following
subsets of ${\mathcal{E}^{+}}$:
${\mathcal{E}}^{+}_{min}:=\\{Z\,|\,g(Z)=g(Z_{min})\\},\ \mbox{and}\
{\mathcal{E}}^{+}_{g=a}:=\\{Z\,|\,g(Z)=a\\},$
where $a\in{\mathbb{Z}}$. Since $\mu(Z)-\beta(Z)=2g(Z)-1$, we get:
###### Lemma 4.2.1.
For any $a$, the restrictions of $\mu$ and $\beta$ to
${\mathcal{E}}^{+}_{g=a}$ take their minima on the same elements of
${\mathcal{E}}^{+}_{g=a}$. In particular, the restriction of $\mu$ (resp. of
$\beta$) on ${\mathcal{E}}^{+}_{min}$ is $\mu(Z_{min})$ (resp.
$\beta(Z_{min})$).
## 5\. Application to the canonical contact structure of the link
Our application targets the invariants sg${}_{an}(\xi_{can})$,
bn${}_{an}(\xi_{can})$ and $\mathfrak{n}_{an}(\xi_{can})$; for notations, see
Introduction. Indeed, the previous results read as follows:
$\mbox{sg}_{an}(\xi_{can})=g(Z_{min});$
$\mbox{bn}_{an}(\xi_{can})=\beta(Z_{min});$
$\mathfrak{n}_{an}(\xi_{can})=\mu(Z_{min})-1.$
In particular,
$\mathfrak{n}_{an}(\xi_{can})-\mbox{bn}_{an}(\xi_{can})=2\cdot\mbox{sg}_{an}(\xi_{can})-2.$
These facts answer some of the questions of [7], section 8, at least in the
realm of Milnor open books.
## References
* [1] A’Campo, N.: Sur la monodromie des singularités isolées d’hypersurfaces complexes, Invent. Math. 20 (1973), 147-169.
* [2] Artin, A.: On isolated rational singularities of surfaces, Amer. J. Math. 88 (1) (1966), 129-136.
* [3] Altinok, S. and Bhupal, M.: Minimal page-genus of Milnor open books on links of rational surface singularities, Singularities II, Contemp. Math. 475 Amer. Math. Soc., Providence, RI, 2008, 1-10.
* [4] Blank, S. and Laudenbach, F.: Isotopie des formes fermées en dimension 3, Inv. Math. 54 (1979), 103-177.
* [5] Caubel, C., Némethi, A., Popescu-Pampu, P.: Milnor open books and Milnor fillable contact 3-manifolds, Topology 45 (2006), 673-689.
* [6] Eisenbud, D. and Neumann, W.: Three–dimensional link theory and invariants of plane curve singularities, Annals of Math. Studies 110, Princeton University Press, 1985.
* [7] Etnyre, J. and Ozbagci, B.: Invariants of contact structures from open books, Trans. AMS 360 (6) (2008), 3133-3151.
* [8] Giroux, E.: Géometrie de contact: de la dimension trois les dimensions supérieures, Proc. of the International Congress of Math. (Beijing 2002), Vol. II, 405-414.
* [9] Laufer, H.: On rational singularities, Amer. J. Math. 94 (1972), 597-608.
* [10] Lipman, J.: Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Etudes Sci. Publ. Math. 36 (1969), 195-279.
* [11] Némethi, A.: The resolution of some surface singularities, I., (cyclic coverings), Contemporary Mathematics 266, Singularities in Algebraic and Analytic Geometry, (C. G. Melles and R. I. Michler Editors), American Math. Soc. 2000, 89-128.
* [12] Neumann W.D. and Pichon, A.: Complex analytic realization of links, Proceedings of the international conference, “Intelligence of Low Dimensional Topology 2006”, Series on Knots and Everything n 40, World Scientific Publishing Co.
* [13] Pichon, A.: Fibrations sur le cercle et surfaces complexes, Annales de l’Institut Fourier 51 (2001), 337-374.
* [14] Waldhausen, F.: On irreducible 3–manifolds that are sufficiently large, Ann. of Math. 87 (1968), 56-88.
|
arxiv-papers
| 2009-06-10T14:31:35 |
2024-09-04T02:49:03.292113
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Nemethi and M. Tosun",
"submitter": "Meral Tosun",
"url": "https://arxiv.org/abs/0906.1948"
}
|
0906.2014
|
# A categorical approach to Weyl modules
Vyjayanthi Chari Department of Mathematics, University of California,
Riverside, CA 92521, USA [email protected] , Ghislain Fourier
Mathematisches Institut, Universität zu Köln, Germany [email protected]
koeln.de and Tanusree Khandai Harish-Chandra Research Institute, Allahabad,
India [email protected]
###### Abstract.
Global and local Weyl Modules were introduced via generators and relations in
the context of affine Lie algebras in [CP2] and were motivated by
representations of quantum affine algebras. In [FL] a more general case was
considered by replacing the polynomial ring with the coordinate ring of an
algebraic variety and partial results analogous to those in [CP2] were
obtained. In this paper, we show that there is a natural definition of the
local and global Weyl modules via homological properties. This
characterization allows us to define the Weyl functor from the category of
left modules of a commutative algebra to the category of modules for a simple
Lie algebra. As an application we are able to understand the relationships of
these functors to tensor products, generalizing results in [CP2] and [FL]. We
also analyze the fundamental Weyl modules and show that unlike the case of the
affine Lie algebras, the Weyl functors need not be left exact.
VC was partially supported by the NSF grant DMS-0500751
G.F. was supported by the DFG-project “Kombinatorische Beschreibung von
Macdonald und Kostka-Foulkes Polynomen ”
## 1\. Introduction
The category of finite–dimensional representations of affine and quantum
affine Lie algebras has been intensively studied in recent years. One of the
reasons that this category has proved to be interesting is the fact that it is
not semi-simple. Moreover, it was proved in [CP2] that irreducible
representations of the quantum affine algebra specialized to reducible
indecomposable representations of the affine Lie algebra. This phenomenon is
analogous to the one observed in modular representation theory where an
irreducible finite–dimensional representation in characteristic zero becomes
reducible on passing to characteristic $p$ and is called a Weyl module.
The definition of Weyl modules (global and local) in [CP2] for affine algebras
was motivated by this analogy. Thus given any dominant integral weight of the
semisimple Lie algebra $\mathfrak{g}$, one can define an infinite–dimensional
left module $W(\lambda)$ for the corresponding affine (in fact for the loop)
algebra via generators and relations. The module $W(\lambda)$ is a direct sum
of finite–dimensional $\mathfrak{g}$–modules and it was shown in [CP2] that it
is also a right module for a polynomial algebra $\mathbb{A}_{\lambda}$ which
is canonically associated with $\lambda$. The local Weyl modules are obtained
by tensoring the global Weyl modules with irreducible modules for
$\mathbb{A}_{\lambda}$ or equivalently can be given via generators and
relations. A necessary and sufficient condition for the tensor product of
local Weyl modules to be a local Weyl module was given. Using this fact, the
character of the local Weyl module was conjectured in [CP2] and the conjecture
was heavily influenced by the connection with quantum affine algebras. In
particular, the conjecture implied that the dimension of the local Weyl module
was independent of the choice of the irreducible
$\mathbb{A}_{\lambda}$–module, i.e that the global Weyl module is a free
module for $\mathbb{A}_{\lambda}$. The character formula was proved in [CP2]
for $\mathfrak{sl_{2}}$, in [CL] for $\mathfrak{sl}_{r+1}$, in [FoL] for
simply–laced algebras and the general case can be deduced by passing to the
quantum case by using the work of [K] and [BN].
In [FL], Feigin and Loktev extended the notion of Weyl modules to the
higher–dimensional case, i.e. instead of the loop algebra they worked with the
Lie algebra $\mathfrak{g}\otimes A$ where $A$ is the coordinate ring of an
algebraic variety and obtained analogs of some of the results of [CP2]. For
instance when $\mathfrak{g}$ is of type $\mathfrak{sl}_{2}$ and $A$ is the
polynomial ring in two variables they compute the dimension of the Weyl
module. They also give a necessary and sufficient condition for the tensor
product of local Weyl modules to be a local Weyl module analogous to the one
in [CP2]. However, they do not define the algebra $\mathbb{A}_{\lambda}$ and
the bi–module structure on $W(\lambda)$ and hence do not say much about the
structure of the global Weyl module.
In this paper, we take a more general functorial approach to Weyl modules
associated to the algebra $\mathfrak{g}\otimes A$, where $A$ is a commutative
associative algebra (with unit) over the complex numbers. This approach (as
also the approach in [CG1], [CG2]) is motivated by the methods used to study
another well–known category in representation theory: the BGG-category $\cal
O$ for semi–simple Lie algebras. As a result we are able to extend the
definition of Weyl modules to a more general situation and allows us to do a
deeper analysis of the global Weyl modules. We also give the classification
and description of irreducible modules for $\mathfrak{g}\otimes A$ for an
arbitrary finitely generated algebra which is analogous to the one given in
[C1],[CP1],[L],[R] in the case when $A$ is a polynomial algebra.
We now explain our results in some detail. Let $\cal I_{A}$ be the category of
$\mathfrak{g}\otimes A$–modules which are integrable as
$\mathfrak{g}$–modules. For $\lambda\in P^{+}$ we let $\cal I^{\lambda}_{A}$
be the full subcategory of $\cal I_{A}$ consisting of objects whose weights
are bounded above by $\lambda$. Given $\lambda\in P^{+}$, one can define in a
canonical way a projective module $P_{A}(\lambda)\in\cal I_{A}$ and we prove
that the global Weyl module $W_{A}(\lambda)$ is the largest quotient of
$P_{A}(\lambda)$ that lies in $\cal I^{\lambda}_{A}$. We then define a right
action of the algebra $\mathbf{U}(\mathfrak{h}\otimes A)$ on $W_{A}(\lambda)$
where $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ which is
compatible with the left action of $\mathfrak{g}\otimes A$. Let
$\mathbf{A}_{\lambda}$ be the quotient of $\mathbf{U}(\mathfrak{h}\otimes A)$
by the torsion ideal for this action so that $W_{A}(\lambda)$ can be regarded
as a bi-module for $(\mathfrak{g}\otimes A,\mathbf{A}_{\lambda})$. We prove
that the bimodule structure is functorial in $A$.
Let $\mathbf{W}^{\lambda}_{A}$ be the right exact functor
$W_{A}(\lambda)\otimes_{\mathbf{A}_{\lambda}}$ from the category
$\operatorname{mod}\mathbf{A}_{\lambda}$ of left modules for
$\mathbf{A}_{\lambda}$ to $\cal I^{\lambda}_{A}$. The local Weyl modules are
then just $\mathbf{W}^{\lambda}_{A}M$ where $M$ is an irreducible object of
$\operatorname{mod}\mathbf{A}_{\lambda}$. In section 3, we prove that one can
define a functor $\mathbf{R}^{\lambda}_{A}$ which is exact and right adjoint
to $\mathbf{W}^{\lambda}_{A}$. That allows us to give a categorical
characterization of the local Weyl modules and more generally of the modules
$\mathbf{W}_{A}^{\lambda}M$, $M\in\operatorname{mod}\mathbf{A}_{\lambda}$.
Namely we prove that these modules are given by the vanishing of
$\operatorname{Hom}_{\cal I^{\lambda}_{A}}$ and $\operatorname{Ext}^{1}_{\cal
I^{\lambda}_{A}}$ and we show also that the functors
$\mathbf{W}^{\lambda}_{A}$ are left exact iff we have vanishing of
$\operatorname{Ext}^{2}_{\cal I^{\lambda}_{A}}$.
In section 4 we prove that the algebra $\mathbf{A}_{\lambda}$ is finitely
generated iff $A$ is finitely generated. We use the results of section 3 to
study the relationship between the functors $\mathbf{W}^{\lambda+\mu}_{A\oplus
B}$ and $\mathbf{W}^{\lambda}_{A}\otimes\mathbf{W}^{\mu}_{B}$ when $A,B$ are
finite–dimensional algebras. In section 5, we give a necessary and sufficient
condition for the tensor product
$\mathbf{W}_{A}^{\lambda}M\otimes\mathbf{W}_{A}^{\mu}N$ to be isomorphic to
$\mathbf{W}^{\lambda+\mu}_{A}(M\otimes N)$ when $A$ is finitely generated and
$M,N\in\operatorname{mod}\mathbf{A}_{\lambda}$.
In section 6 we assume that $A$ is finitely generated and that the Jacobson
radical of $A$ is $0$. We prove that the algebra $\mathbf{A}_{\lambda}$ is
isomorphic to the ring of invariants of a subgroup $S_{\lambda}$ of the
symmetric group on $d_{\lambda}$ letters acting on $A^{\otimes d_{\lambda}}$.
Here $d_{\lambda}$ is a positive integer naturally associated with $\lambda$.
This implies that the irreducible modules in
$\operatorname{mod}\mathbf{A}_{\lambda}$ are determined (up to isomorphism) by
the orbits of this action.
The tensor product results of Sections 4 and 5 imply that to understand the
local Weyl modules it is enough to understand local Weyl modules corresponding
to certain special orbits. In section 7, we consider the case when $\xi$ is
the orbit of a point in $A^{\otimes d_{\lambda}}$ which has trivial stabilizer
under the entire symmetric group $S_{d_{\lambda}}$. In this case
$\mathbf{W}^{\lambda}_{A}M_{\xi}$ is a tensor product of the local fundamental
Weyl modules and we describe the character of these modules completely for any
finitely generated algebra $A$ and for the classical simple Lie algebras.
The results of section 7 show that there are many important differences
between the study of Weyl modules for the polynomial algebra in one variable
and the more general case considered here. The dimension of the local
fundamental Weyl modules associated to $A$ depends on $\xi$ if the variety
associated to $A$ is not smooth. It also proves that the dimension of
$\mathbf{W}_{A}^{\lambda}M_{\xi}$ is not independent of $\xi$ even if $A$ is
an irreducible smooth variety and $\xi$ is the orbit of a point in $A^{\otimes
d_{\lambda}}$ with trivial stabilizer for the $S_{\lambda}$-action. In
particular, this proves that the global Weyl module is not projective as a
right $\mathbf{A}_{\lambda}$–module (and hence the Weyl functors not exact)
even when $A$ is the polynomial ring in two variables. There are thus, many
natural and interesting algebraic and geometric questions that arise as a
result of this paper which will be studied elsewhere.
Acknowledgements: We would like to thank Wee Liang Gan, Michael Ehrig,
Friederich Knop, Peter Littelmann for many discussions on the algebra
$\mathbb{A}_{\lambda}$. We are grateful to Peter Russell for his patience with
our long discussions and our not always well-formulated questions on group
actions, homological algebra and commutative algebra. Finally, particular
thanks are due to Shrawan Kumar for sharing with us, his result (Proposition
Proposition ) on extensions between tensor products of modules for direct sums
of Lie algebras.
## 2\. Preliminaries
### 2.1.
Throughout the paper $\mathbf{C}$ denotes the set of complex numbers and
$\mathbf{Z}_{+}$ the set of non–negative integers. Let $\mathfrak{g}$ be a
finite–dimensional simple Lie algebra of rank $n$ with Cartan matrix
$(a_{ij})_{i,j\in I}$ where $I=\\{1,\cdots,n\\}$. Fix a Cartan subalgebra
$\mathfrak{h}$ of $\mathfrak{g}$ and let $R$ denote the corresponding set of
roots. Let $\\{\alpha_{i}\\}_{i\in I}$ (resp. $\\{\omega_{i}\\}_{i\in I}$) be
a set of simple roots (resp. fundamental weights) and $Q$ (resp. $Q^{+}$), $P$
(resp. $P^{+}$) be the integer span (resp. $\mathbf{Z}_{+}$–span) of the
simple roots and fundamental weights respectively. Denote by $\leq$ the usual
partial order on $P$,
$\lambda,\mu\in P,\ \ \lambda\leq\mu\ \iff\ \mu-\lambda\in Q^{+}.$
Set $R^{+}=R\cap Q^{+}$ and let $\theta$ be the unique maximal element in
$R^{+}$ with respect to the partial order.
Let $x^{\pm}_{\alpha}$, $h_{i}$, $\alpha\in R^{+}$, $i\in I$ be a Chevalley
basis of $\mathfrak{g}$ and set $x_{i}^{\pm}=x^{\pm}_{\alpha_{i}}$,
$h_{\alpha}=[x^{+}_{\alpha},x^{-}_{\alpha}]$ and note that
$h_{i}=h_{\alpha_{i}}$. For each $\alpha\in R^{+}$, the subalgebra of
$\mathfrak{g}$ spanned by $\\{x^{\pm}_{\alpha},h_{\alpha}\\}$ is isomorphic to
$\mathfrak{sl}_{2}$. Define subalgebras $\mathfrak{n}^{\pm}$ of
$\mathfrak{g}$, by
$\mathfrak{n}^{\pm}=\bigoplus_{\alpha\in R^{+}}\mathbf{C}x^{\pm}_{\alpha},$
and note that
$\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}.$
Given any Lie algebra $\mathfrak{a}$, let $\mathbf{U}(\mathfrak{a})$ be the
universal enveloping algebra of $\mathfrak{a}$. The map $x\to x\otimes
1+1\otimes x$, $x\in\mathfrak{a}$ extends to an algebra homomorphism
$\Delta:\mathbf{U}(\mathfrak{a})\to\mathbf{U}(\mathfrak{a})\otimes\mathbf{U}(\mathfrak{a})$.By
the Poincare Birkhoff Witt theorem, we know that if $\mathfrak{b}$ and
$\mathfrak{c}$ are Lie subalgebras of $\mathfrak{a}$ such that
$\mathfrak{a}=\mathfrak{b}\oplus\mathfrak{c}$ as vector spaces then
$\mathbf{U}(\mathfrak{a})\cong\mathbf{U}(\mathfrak{b})\otimes\mathbf{U}(\mathfrak{c})$
as vector spaces.
### 2.2.
Let $A$ be a commutative associative algebra with unity over $\mathbf{C}$ and
let $A_{+}$ be a fixed vector space complement to the subspace $\mathbf{C}$ of
$A$. Given a Lie algebra $\mathfrak{a}$ define a Lie algebra structure on
$\mathfrak{a}\otimes A$, by
$[x\otimes a,y\otimes b]=[x,y]\otimes ab,\ \ x,y\in\mathfrak{g},\ \ a,b\in A.$
If $\phi:B\to A$ is a homomorphism of associative algebras, there exists a
corresponding homomorphism $\phi_{\mathfrak{a}}:\mathfrak{a}\otimes
B\to\mathfrak{a}\otimes A$ of Lie algebras, which is injective (resp.
surjective) if $\phi$ is injective (resp. surjective). In particular, if $B$
is a subalgebra of $A$, the Lie algebra $\mathfrak{a}\otimes B$ can be
regarded naturally as a Lie subalgebra of $\mathfrak{a}\otimes A$ and we
identify $\mathfrak{a}$ with the Lie subalgebra
$\mathfrak{a}\otimes\mathbf{C}$ of $\mathfrak{a}\otimes A$. Similarly, if
$\mathfrak{b}$ is a Lie subalgebra of $\mathfrak{a}$, then
$\mathfrak{b}\otimes A$ is naturally isomorphic to a subalgebra of
$\mathfrak{a}\otimes A$. Finally we denote by $\mathbf{U}(\mathfrak{g}\otimes
A_{+})$ the subspace of $\mathbf{U}(\mathfrak{g}\otimes A)$ spanned by
monomials in the elements $x\otimes a$ where $x\in\mathfrak{g}$, $a\in A_{+}$.
The following is elementary but we include a proof for the reader convenience
and because it is used repeatedly throughout the paper.
###### Lemma.
Let $\mathfrak{g}$ be a finite–dimensional simple Lie algebra and $A$ a
commutative associative algebra with unity over $\mathbf{C}$. Then any ideal
of $\mathfrak{g}\otimes A$ is of the form $\mathfrak{g}\otimes S$ for some
ideal $S$ of $A$ and $[\mathfrak{g}\otimes A/S,\mathfrak{g}\otimes
A/S]=\mathfrak{g}\otimes A/S.$
###### Proof.
Let $\mathfrak{i}$ be an ideal in $\mathfrak{g}\otimes A$ and set
$S=\\{a\in A:\mathfrak{g}\otimes a\subset\mathfrak{i}\\}.$
Since $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$ we see that $S$ is an ideal
on $A$. The Lemma follows if we prove that $\mathfrak{g}\otimes
S=\mathfrak{i}.$ Let $x\in\mathfrak{i}$ and write
$x=\sum_{\alpha\in R}x_{\alpha}\otimes a_{\alpha}+\sum_{i\in I}h_{i}\otimes
a_{i},$
for some $a_{\alpha},a_{i}\in A$. We proceed by induction on
$r=\\#\\{\alpha\in R:a_{\alpha}\neq 0\\},$
to show that $\mathfrak{g}\otimes a_{\alpha}\subset\mathfrak{i}$ and
$\mathfrak{g}\otimes a_{i}\subset\mathfrak{i}$ for all $\alpha\in R$, $i\in
I$. If $r=0$, we have
$[\sum_{i\in I}h_{i}\otimes a_{i},x_{j}^{+}]=x_{j}^{+}\otimes\sum_{i\in
I}\alpha_{j}(h_{i})a_{i}\in\mathfrak{i},\ \ j\in I.$
Since the Cartan matrix of $A$ is invertible, it follows now that
$x_{j}^{+}\otimes a_{i}\in\mathfrak{i}$ for all $i,j\in I$ and since
$\mathfrak{g}$ is simple we see that $\mathfrak{g}\otimes
a_{i}\in\mathfrak{i}$ for all $i\in I$.
Suppose now that we have proved the result when $0\leq r<k$ and suppose that
$a_{\beta_{1}},\cdots,a_{\beta_{k}}$ are the non–zero elements. Choose
$h\in\mathfrak{h}$ such that $\beta_{k}(h)\neq 0$ and $\beta_{k-1}(h)=0$. Then
$0\neq[h,x]=\sum_{s=1}^{k-2}\beta_{s}(h)x_{\alpha}\otimes
a_{\beta_{s}}+\beta_{k}(h)x_{\beta_{k}}\otimes a_{\beta_{k}}\in\mathfrak{i}.$
The induction hypothesis applies to $[h,x]$ and we find that
$a_{\beta_{k}}\in S,\ \ \ x-x_{\beta_{k}}\otimes
a_{\beta_{k}}\in\mathfrak{i}.$
The induction hypothesis again applies to $x-(x_{\beta_{k}}\otimes
a_{\beta_{k}})$ and we get the result. ∎
### 2.3.
Let $V$ be any $\mathfrak{g}$–module. We say that $V$ is locally
finite–dimensional if any element of $V$ lies in a finite–dimensional
$\mathfrak{g}$–submodule of $V$. This means that $V$ is isomorphic to a direct
sum of irreducible finite–dimensional $\mathfrak{g}$–modules and hence we can
write
$V=\bigoplus_{\lambda\in\mathfrak{h}^{*}}V_{\lambda},$
where $V_{\lambda}=\\{v\in V:hv=\lambda(h)v,\ \ \forall\ h\in\mathfrak{h}\\}$.
We set
$\operatorname{wt}(V)=\\{\lambda\in\mathfrak{h}^{*}:V_{\lambda}\neq 0\\}.$
For $\lambda\in P^{+}$, let $V(\lambda)$ be the simple $\mathfrak{g}$–module
which is generated by an element $v_{\lambda}\in V(\lambda)$ satisfying the
defining relations:
$\mathfrak{n}^{+}v_{\lambda}=0,\quad
hv_{\lambda}=\lambda(h)v_{\lambda},\quad(x^{-}_{i})^{\lambda(h_{i})+1}v_{\lambda}=0,$
for all $h\in\mathfrak{h}$, $i\in I$. Then,
$\operatorname{wt}(V(\lambda))\subset\lambda-Q^{+},\ \ \dim
V(\lambda)<\infty.$
Moreover any irreducible locally finite–dimensional $\mathfrak{g}$–module is
isomorphic to $V(\lambda)$ for some $\lambda\in P^{+}$. The following can be
found in [B].
###### Lemma.
Let $\mathfrak{a}$ be a Lie algebra such that
$[\mathfrak{a},\mathfrak{a}]=\mathfrak{a}$ and assume that $\mathfrak{a}$ has
a faithful finite–dimensional irreducible representation. Then $\mathfrak{a}$
is a semi–simple Lie algebra.
### 2.4.
Suppose that $\mathfrak{g}$ is a finite–dimensional semisimple Lie algebra and
that $\mathfrak{g}_{1}$, $\mathfrak{g}_{2}$ are ideals of $\mathfrak{g}$ such
that
$\mathfrak{g}\cong\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$
as Lie algebras. Then $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are also
semisimple Lie algebras and it is standard that any irreducible
finite–dimensional representation of ${\mathfrak{g}}$ is isomorphic to a
tensor product of irreducible representations of $\mathfrak{g}_{1}$ and
$\mathfrak{g}_{2}$.
###### Proposition.
Let $A$ and $B$ be commutative associative algebras. Any finite–dimensional
irreducible representation $V$ of $\mathfrak{g}\otimes(A\oplus B)$ is
isomorphic to a tensor product $V_{1}\otimes V_{2}$ where $V_{1}$ and $V_{2}$
are irreducible representations of $\mathfrak{g}\otimes A$ and
$\mathfrak{g}\otimes B$ respectively.
###### Proof.
Let $\rho:\mathfrak{g}\otimes(A\oplus B)\to\operatorname{End}(V)$ be an
irreducible finite–dimensional representation. Then $\ker\rho$ is an ideal of
finite codimension in $\mathfrak{g}\otimes(A\oplus B)$ and hence
$\ker\rho=\mathfrak{g}\otimes M,$
for some ideal $M$ of $A\oplus B$. Since any ideal of $A\oplus B$ is of the
form $M_{1}\oplus M_{2}$ where $M_{1},M_{2}$ are ideals in $A$ and $B$
respectively, we see that $V$ is a faithful irreducible representation of
$\tilde{\mathfrak{g}}=\mathfrak{g}\otimes(A/M_{1}\oplus B/M_{2})$. Lemma Lemma
implies that $\tilde{\mathfrak{g}}$ is a finite–dimensional semi-simple Lie
algebra. The result now follows by the comments preceding the statement of
this proposition. ∎
### 2.5.
We shall need the following result due to Shrawan Kumar [Ku].
###### Proposition.
For $r=1,2$, let $\mathfrak{g}_{r}$ be a finite–dimensional Lie algebra and
assume that $U_{r},V_{r}$ are finite dimensional $\mathfrak{g}_{r}$–modules.
For all $m\geq 0$ we have
$\operatorname{Ext}^{m}_{\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}}(U_{1}\otimes
U_{2},V_{1}\otimes
V_{2})\cong\bigoplus_{p+q=m}\operatorname{Ext}^{p}_{\mathfrak{g}_{1}}(U_{1},V_{1})\otimes\operatorname{Ext}^{q}_{\mathfrak{g}_{2}}(U_{2},V_{2}).$
## 3\. The category $\mathcal{I}_{A}$
### 3.1.
Let $\cal I_{A}$ be the category whose objects are modules for
$\mathfrak{g}\otimes A$ which are locally finite–dimensional
$\mathfrak{g}$–modules and morphisms
$\operatorname{Hom}_{\cal
I_{A}}(V,V^{\prime})=\operatorname{Hom}_{\mathfrak{g}\otimes
A}(V,V^{\prime}),\ \ V,V^{\prime}\in\cal I_{A}.$
Clearly $\cal I_{A}$ is an abelian category and is closed under tensor
products. We shall use the following elementary result often without mention
in the rest of the paper.
###### Lemma.
Let $V\in\operatorname{Ob}\cal I_{A}$.
* (i)
If $V_{\lambda}\neq 0$ and $\operatorname{wt}V\subset\lambda-Q^{+}$, then
$\lambda\in P^{+}$ and
$(\mathfrak{n}^{+}\otimes A)V_{\lambda}=0,\ \
(x_{i}^{-})^{\lambda(h_{i})+1}V_{\lambda}=0,\ \ i\in I.$
If in addition, $V=\mathbf{U}(\mathfrak{g}\otimes A)V_{\lambda}$ and $\dim
V_{\lambda}=1$, then $V$ has a unique irreducible quotient.
* (ii)
If $V=\mathbf{U}(\mathfrak{g}\otimes A)V_{\lambda}$ and
$(\mathfrak{n}^{+}\otimes A)V_{\lambda}=0$, then
$\operatorname{wt}(V)\subset\lambda-Q^{+}$.
* (iii)
If $V\in\cal I_{A}$ is irreducible and finite–dimensional, then there exists
$\lambda\in\operatorname{wt}V$ such that
$\dim V_{\lambda}=1,\ \ \operatorname{wt}(V)\subset\lambda-Q^{+}.$
∎
### 3.2.
Regard $\mathbf{U}(\mathfrak{g}\otimes A)$ as a right $\mathfrak{g}$–module
via right multiplication and given a left $\mathfrak{g}$–module $V$, set
$P_{A}(V)=\mathbf{U}(\mathfrak{g}\otimes
A)\otimes_{\mathbf{U}(\mathfrak{g})}V.$
Then $P_{A}(V)$ is a left $\mathfrak{g}\otimes A$–module by left
multiplication and we have an isomorphism of vector spaces
$P_{A}(V)\cong\mathbf{U}(\mathfrak{g}\otimes A_{+})\otimes_{\mathbf{C}}V.$
(3.1)
###### Proposition.
Let $V$ be a locally finite–dimensional $\mathfrak{g}$–module. Then $P_{A}(V)$
is a projective object of $\cal I_{A}$. If in addition $V\in\cal I_{A}$, then
the map $P_{A}(V)\to V$ given by $u\otimes v\to uv$ is a surjective morphism
of objects in $\cal I_{A}$. Finally, if $\lambda\in P^{+}$, then
$P_{A}(V(\lambda))$ is generated as a $\mathbf{U}(\mathfrak{g}\otimes
A)$–module by the element $p_{\lambda}=1\otimes v_{\lambda}$ with defining
relations
$\mathfrak{n}^{+}p_{\lambda}=0,\quad
hp_{\lambda}=\lambda(h)p_{\lambda},\quad(x^{-}_{i})^{\lambda(h_{i})+1}p_{\lambda}=0,\
\ i\in I,\ h\in\mathfrak{h}.$ (3.2)
###### Proof.
For $x\in\mathfrak{g}$, we have
$x(u\otimes v)=[x,u]\otimes v+u\otimes xv,\ \
u\in\mathbf{U}(\mathfrak{g}\otimes A),\ \ v\in V.$
Since the adjoint action of $\mathfrak{g}$ on $\mathfrak{g}\otimes A$ (and
hence on $\mathbf{U}(\mathfrak{g}\otimes A)$) is locally finite, it is
immediate that $P_{A}(V)\in\cal I_{A}$. The proof that it is projective is
standard. It is clear that the element $p_{\lambda}\in P_{A}(V(\lambda))$
satisfies the relations in (3.2) and the fact that they are the defining
relations follows by using the isomorphism in (3.1).∎
For $\nu\in P^{+}$ and $V\in\operatorname{Ob}\cal I_{A}$, let
$V^{\nu}\in\operatorname{Ob}\cal I_{A}$ be the unique maximal
$\mathfrak{g}\otimes A$ quotient of $V$ satisfying
$\operatorname{wt}(V^{\nu})\subset\nu-Q^{+},$ (3.3)
or equivalently,
$V^{\nu}=V/\sum_{\mu\nleq\nu}\mathbf{U}(\mathfrak{g}\otimes A)V_{\mu}.$
A morphism $\pi:V\to V^{\prime}$ of objects in $\cal I_{A}$ clearly induces a
morphism $\pi^{\nu}:V^{\nu}\to(V^{\prime})^{\nu}$. Let $\cal I_{A}^{\nu}$ be
the full subcategory of objects $V\in\cal I_{A}$ such that $V=V^{\nu}$. It
follows from the theory of finite–dimensional representations of simple Lie
algebras that
$V\in\cal I^{\nu}_{A}\implies\\#\operatorname{wt}V<\infty.$ (3.4)
The following is immediate.
###### Corollary.
Let $\nu\in P^{+}$ and $V\in\cal I_{A}^{\nu}$. Then $P_{A}(V)^{\nu}$ is a
projective object of $\cal I_{A}^{\nu}$.
### 3.3.
For $\lambda\in P^{+}$, set
$W_{A}(\lambda)=P_{A}(V(\lambda))^{\lambda},$
and let $w_{\lambda}$ be the image of $p_{\lambda}$ in $W_{A}(\lambda)$. The
following proposition is essentially an immediate consequence of Proposition
Proposition and gives an alternative definition of $W_{A}(\lambda)$ via
generators and relations. In fact this was the original definition given in
[CP2] when $A$ is the ring of Laurent polynomials and later generalized in
[FL].
###### Proposition.
For $\lambda\in P^{+}$, the module $W_{A}(\lambda)$ is generated by
$w_{\lambda}$ with defining relations:
$(\mathfrak{n}^{+}\otimes A)w_{\lambda}=0,\quad
hw_{\lambda}=\lambda(h)w_{\lambda},\quad(x^{-}_{i})^{\lambda(h_{i})+1}w_{\lambda}=0,\
\ i\in I,\ h\in\mathfrak{h}.$ (3.5)
###### Proof.
Since $\operatorname{wt}W_{A}(\lambda)\subset\lambda-Q^{+}$ it follows that
$(\mathfrak{n}^{+}\otimes A)w_{\lambda}=0$. The other relations are clear
since they are already satisfied by $p_{\lambda}$. To see that these are all
the relations, let $W^{\prime}_{A}(\lambda)$ be the module generated by an
element $w_{\lambda}$ with the relations in (3.5). By Proposition Proposition
we see that $W^{\prime}_{A}(\lambda)$ is a quotient of $P_{A}(V(\lambda))$. On
the other hand
$\operatorname{wt}(W^{\prime}_{A}(\lambda))\subset\lambda-Q^{+}$ which implies
that $W^{\prime}_{A}(\lambda)$ satisfies (3.3). It follows by the maximality
of $W_{A}(\lambda)$ that $W^{\prime}_{A}(\lambda)$ is a quotient of
$W_{A}(\lambda)$ and the proposition is proved. ∎
Set
$\displaystyle\operatorname{Ann}_{\mathfrak{g}\otimes
A}(w_{\lambda})=\\{u\in\mathbf{U}(\mathfrak{g}\otimes A):uw_{\lambda}=0\\},\ \
\operatorname{Ann}_{\mathfrak{h}\otimes
A}(w_{\lambda})=\operatorname{Ann}_{\mathfrak{g}\otimes
A}(w_{\lambda})\cap\mathbf{U}(\mathfrak{h}\otimes A).$
Clearly $\operatorname{Ann}_{\mathfrak{h}\otimes A}(w_{\lambda})$ is an ideal
in $\mathbf{U}(\mathfrak{h}\otimes A)$ and we denote by $\mathbf{A}_{\lambda}$
the quotient of $\mathbf{U}(\mathfrak{h}\otimes A)$ by the ideal
$\operatorname{Ann}_{\mathfrak{h}\otimes A}(w_{\lambda})$.
### 3.4.
Regard $W_{A}(\lambda)$ as a right module for $\mathfrak{h}\otimes A$ as
follows:
$(uw_{\lambda})(h\otimes a)=u(h\otimes a)w_{\lambda},\ \
u\in\mathbf{U}(\mathfrak{g}\otimes A),\ h\in\mathfrak{h},a\in A.$
To see that this map is well defined, one must prove that:
$\displaystyle(\mathfrak{n}^{+}\otimes A)(h\otimes a)w_{\lambda}=0,\ \
(h^{\prime}-\lambda(h^{\prime}))(h\otimes a)w_{\lambda}=0,$
$\displaystyle(x_{i}^{-})^{\lambda(h_{i})+1}(h\otimes a)w_{\lambda}=0,$
for all $i\in I$, $a\in A$ and $h,h^{\prime}\in\mathfrak{h}$. The first two
are obvious. The third follows from the fact that $x_{i}^{+}((h\otimes
a)\otimes v_{\lambda})=0$ and that $W_{A}(\lambda)\in\cal I_{A}$. Thus, we
have proved that $W_{A}(\lambda)$ is a bi–module for the pair
$(\mathfrak{g}\otimes A,\mathfrak{h}\otimes A)$.
For all $\mu\in P$, the subspaces $W_{A}(\lambda)_{\mu}$ are
$\mathfrak{h}\otimes A$–submodules for both the left and right actions and
$\operatorname{Ann}_{\mathfrak{h}\otimes
A}(w_{\lambda})=\\{u\in\mathbf{U}(\mathfrak{h}\otimes
A):w_{\lambda}u=0=uw_{\lambda}\\}=\\{u\in\mathbf{U}(\mathfrak{h}\otimes
A):W_{A}(\lambda)u=0\\}.$
Then $W_{A}(\lambda)$ is a $(\mathfrak{g}\otimes
A,\mathbf{A}_{\lambda})$–bimodule and each subspace $W_{A}(\lambda)_{\mu}$ is
a right $\mathbf{A}_{\lambda}$–module. Moreover $W_{A}(\lambda)_{\lambda}$ is
a $\mathbf{A}_{\lambda}$–bimodule and we have an isomorphism of bimodules,
$W_{A}(\lambda)_{\lambda}\cong\mathbf{A}_{\lambda}.$
Let $\operatorname{mod}\mathbf{A}_{\lambda}$ be the category of left
$\mathbf{A}_{\lambda}$–modules. Let
$\mathbf{W}^{\lambda}_{A}:\operatorname{mod}\mathbf{A}_{\lambda}\to
I_{A}^{\lambda}$ be the right exact functor given by
$\mathbf{W}_{A}^{\lambda}M=W_{A}(\lambda)\otimes_{\mathbf{A}_{\lambda}}M,\
\qquad\ \mathbf{W}_{A}^{\lambda}f=1\otimes f,$
where $M\in\operatorname{mod}\mathbf{A}_{\lambda}$ and
$f\in\operatorname{Hom}_{\mathbf{A}_{\lambda}}(M,M^{\prime})$ for some
$M^{\prime}\in\operatorname{mod}\mathbf{A}_{\lambda}$. Note that since
$W_{A}(\lambda)\in\cal I_{A}$, it is clear that the $\mathfrak{g}$–action on
$\mathbf{W}_{A}^{\lambda}M$ is also locally finite and so
$\mathbf{W}_{A}^{\lambda}M\in\operatorname{Ob}\cal I_{A}^{\lambda}$. The
preceding discussion also shows that
$\mathbf{W}^{\lambda}_{A}\mathbf{A}_{\lambda}\cong_{\mathfrak{g}\otimes
A}W_{A}(\lambda),\qquad\ (\mathbf{W}_{A}^{\lambda}M)_{\mu}\cong
W_{A}(\lambda)_{\mu}\otimes_{\mathbf{A}_{\lambda}}M,\ \ \mu\in P,\ \
M\in\operatorname{mod}\mathbf{A}_{\lambda}.$
### 3.5.
###### Lemma.
For all $\lambda\in P^{+}$ and $V\in\cal I_{A}^{\lambda}$ we have
$\operatorname{Ann}_{\mathfrak{h}\otimes A}(w_{\lambda})V_{\lambda}=0$.
###### Proof.
By Lemma Lemma and Proposition Proposition we see that given $v\in
V_{\lambda}$ there exists a morphism of $\mathfrak{g}\otimes A$–modules
$W_{A}(\lambda)\to\mathbf{U}(\mathfrak{g}\otimes A)v$ which maps
$w_{\lambda}\to v$. Hence $uv=0$ for all
$u\in\operatorname{Ann}_{\mathbf{U}(\mathfrak{h}\otimes A)}(w_{\lambda})$ ∎
As a consequence of the Lemma we see that the left action of
$\mathbf{U}(\mathfrak{h}\otimes A)$ on $V_{\lambda}$ induces a left action of
$\mathbf{A}_{\lambda}$ on $V_{\lambda}$ and we denote the resulting
$\mathbf{A}_{\lambda}$–module by $\mathbf{R}^{\lambda}_{A}V$. Given
$\pi\in\operatorname{Hom}_{\cal I_{A}^{\lambda}}(V,V^{\prime})$ the
restriction of $\pi_{\lambda}:V_{\lambda}\to V^{\prime}_{\lambda}$ is a
morphism of $\mathbf{A}_{\lambda}$–modules and
$V\to\mathbf{R}^{\lambda}_{A}V,\ \
\pi\to\mathbf{R}^{\lambda}_{A}\pi=\pi_{\lambda}$
defines a functor $\mathbf{R}^{\lambda}_{A}:\cal
I_{A}^{\lambda}\to\operatorname{mod}\mathbf{A}_{\lambda}$ which is exact since
restriction $\pi$ to a weight space is exact. If
$M\in\operatorname{Ob}\operatorname{mod}\mathbf{A}_{\lambda}$, we have an
isomorphism of left $\mathbf{A}_{\lambda}$–modules,
$\mathbf{R}^{\lambda}_{A}\mathbf{W}_{A}^{\lambda}M=(\mathbf{W}_{A}^{\lambda}M)_{\lambda}=W_{A}(\lambda)_{\lambda}\otimes_{\mathbf{A}_{\lambda}}M\cong
w_{\lambda}\mathbf{A}_{\lambda}\otimes_{\mathbf{A}_{\lambda}}M\cong M,$
and hence an isomorphism of functors
$\operatorname{id}_{\mathbf{A}_{\lambda}}\cong\mathbf{R}^{\lambda}_{A}\mathbf{W}_{A}^{\lambda}$.
### 3.6.
###### Proposition.
Let $\lambda\in P^{+}$ and $V\in\cal I_{A}^{\lambda}$. There exists a
canonical map of $\mathfrak{g}\otimes A$–modules
$\eta_{V}:\mathbf{W}_{A}^{\lambda}\mathbf{R}^{\lambda}_{A}V\to V$ such that
$\eta:\mathbf{W}_{A}^{\lambda}\mathbf{R}^{\lambda}_{A}\Rightarrow\operatorname{id}_{\cal
I_{A}^{\lambda}}$ is a natural transformation of functors and
$\mathbf{R}^{\lambda}_{A}$ is a right adjoint to $\mathbf{W}_{A}^{\lambda}$.
###### Proof.
Regard $W_{A}(\lambda)\otimes_{\mathbf{C}}V_{\lambda}$ as a left
$\mathfrak{g}\otimes A$–module via the action of $\mathfrak{g}\otimes A$ on
$W_{A}(\lambda)$. Lemma Lemma implies that the assignment
$W_{A}(\lambda)\otimes_{\mathbf{C}}V_{\lambda}\to V$ given by
$gw_{\lambda}\otimes v\to gv$ is a well–defined map of left
$\mathfrak{g}\otimes A$–modules. To see that this map factors through to a map
$\eta_{V}:\mathbf{W}_{A}^{\lambda}V_{\lambda}\to V$ it suffices to observe
that
$gw_{\lambda}(h\otimes a)\otimes v-gw_{\lambda}\otimes(h\otimes a)v=g(h\otimes
a)w_{\lambda}\otimes v-gw_{\lambda}\otimes(h\otimes a)v\mapsto 0$
for all $g\in\mathbf{U}(\mathfrak{g}\otimes A)$, $h\in\mathfrak{h}$ and $a\in
A$. It is now clear that the collection $\\{\eta_{V};V\in\operatorname{Ob}\cal
I_{A}^{\lambda}\\}$ defines a natural transformation
$\eta:\mathbf{W}_{A}^{\lambda}\mathbf{R}^{\lambda}_{A}\Rightarrow\operatorname{id}_{\cal
I_{A}^{\lambda}}$.
To check that $\operatorname{\mathbf{R}^{\lambda}_{A}}$ is right adjoint to
$\mathbf{W}_{A}^{\lambda}$ we must prove that there exists a natural
isomorphism of abelian groups
$\tau=\tau_{M,V}:\operatorname{Hom}_{\cal
I_{A}^{\lambda}}(\mathbf{W}_{A}^{\lambda}M,V)\cong\operatorname{Hom}_{\mathbf{A}_{\lambda}}(M,\mathbf{R}^{\lambda}_{A}V),$
for all $M\in\operatorname{mod}\mathbf{A}_{\lambda}$ and $V\in\cal
I_{A}^{\lambda}$, such that the the following diagram commutes for all
$f\in\operatorname{Hom}_{\mathbf{A}_{\lambda}}(M,M^{\prime})$,
$\pi\in\operatorname{Hom}_{\cal I_{A}^{\lambda}}(V,V^{\prime})$:
$\begin{CD}\operatorname{Hom}_{\cal
I_{A}^{\lambda}}(\mathbf{W}_{A}^{\lambda}M^{\prime},V)@>{\mathbf{W}_{A}^{\lambda}f^{*}}>{}>\operatorname{Hom}_{\cal
I_{A}^{\lambda}}(\mathbf{W}_{A}^{\lambda}M,V)@>{\pi_{*}}>{}>\operatorname{Hom}_{\cal
I_{A}^{\lambda}}(\mathbf{W}_{A}^{\lambda}M,V^{\prime})\\\
@V{}V{\tau}V@V{}V{\tau}V@V{}V{\tau}V\\\
\operatorname{Hom}_{\mathbf{A}_{\lambda}}(M^{\prime},\mathbf{R}^{\lambda}_{A}V)@>{f^{*}}>{}>\operatorname{Hom}_{\mathbf{A}_{\lambda}}(M,\mathbf{R}^{\lambda}_{A}V)@>{\operatorname{\mathbf{R}^{\lambda}_{A}}\pi_{*}}>{}>\operatorname{Hom}_{\mathbf{A}_{\lambda}}(M,\mathbf{R}^{\lambda}_{A}V^{\prime}).\end{CD}$
Define $\tau_{M,V}$ by
$\tau_{M,V}(\pi)=\pi_{\lambda}.$
Since $\mathbf{W}_{A}^{\lambda}M$ is generated by $M$ as a
$\mathfrak{g}\otimes A$–module, it follows that $\tau(\pi)=\tau(\pi^{\prime})$
implies $\pi=\pi^{\prime}$. For
$f\in\operatorname{Hom}_{\mathbf{A}_{\lambda}}(M,\mathbf{R}^{\lambda}_{A}V)$
it is easily seen that
$\tau_{M,V}(\eta_{V}\circ\mathbf{W}_{A}^{\lambda}f)=f,$
and hence $\tau$ is an isomorphism. The fact that the diagram commutes is
straightforward. ∎
The following is a standard consequence of properties of adjoint functors.
###### Corollary.
The functor $\mathbf{W}_{A}^{\lambda}$ maps projective objects to projective
objects.
### 3.7.
The next result gives a categorical definition of $\mathbf{W}_{A}^{\lambda}M$.
###### Theorem.
Let $V\in\cal I_{A}^{\lambda}$. Then
$V\cong\mathbf{W}_{A}^{\lambda}\mathbf{R}^{\lambda}_{A}V$ iff for all
$U\in\cal I_{A}^{\lambda}$ with $U_{\lambda}=0$, we have
$\operatorname{Hom}_{\cal I_{A}^{\lambda}}(V,U)=0,\ \
\operatorname{Ext}^{1}_{\cal I_{A}^{\lambda}}(V,U)=0.$ (3.6)
###### Proof.
Suppose first that $M\in\operatorname{mod}\mathbf{A}_{\lambda}$. Then
$(\mathbf{W}^{\lambda}_{A}M)_{\lambda}=w_{\lambda}\otimes M$ generates
$\mathbf{W}_{A}^{\lambda}M$ and hence
$\operatorname{Hom}_{\cal I_{A}^{\lambda}}(\mathbf{W}_{A}^{\lambda}M,U)=0,\ \
{\rm{if}}\ \ U_{\lambda}=0.$
Let
$P_{1}\to P_{0}\to M\to 0$
be a right exact sequence of modules in
$\operatorname{mod}\mathbf{A}_{\lambda}$, with $P_{0},P_{1}$ projective and
consider the corresponding right exact sequence
$\mathbf{W}_{A}^{\lambda}P_{1}\to\mathbf{W}_{A}^{\lambda}P_{0}\to\mathbf{W}_{A}^{\lambda}M\to
0$
in $\cal I_{A}^{\lambda}$. Let $K$ be the image of
$\mathbf{W}_{A}^{\lambda}P_{1}$ in $\mathbf{W}_{A}^{\lambda}P_{0}$ (or
equivalently the kernel of
$\mathbf{W}_{A}^{\lambda}P_{0}\to\mathbf{W}^{\lambda}_{A}M$). Then $K$ is
generated as $\mathbf{U}(\mathfrak{g}\otimes A)$–module by $K_{\lambda}$ and
hence $\operatorname{Hom}_{\cal I_{A}^{\lambda}}(K,U)=0$ if $U\in\cal
I_{A}^{\lambda}$ and $U_{\lambda}=0$. By Corollary Proposition we see that
$\mathbf{W}_{A}^{\lambda}P_{0}$ is projective and it now follows by applying
$\operatorname{Hom}_{\cal I_{A}^{\lambda}}(-,U)$ to the short exact sequence
$0\to K\to\mathbf{W}_{A}^{\lambda}P_{0}\to\mathbf{W}_{A}^{\lambda}M\to 0.$
that $\operatorname{Ext}^{1}_{\cal
I_{A}^{\lambda}}(\mathbf{W}_{A}^{\lambda}M,U)=0$.
Conversely suppose that we are given $V\in\cal I_{A}^{\lambda}$ satisfying
(3.6). Let $V^{\prime}=\mathbf{U}(\mathfrak{g}\otimes A)V_{\lambda}$ and note
that
$V/V^{\prime}\in\cal I_{A}^{\lambda},\ \ (V/V^{\prime})_{\lambda}=0.$
It follows from (3.6) that
$\operatorname{Hom}_{\cal I_{A}^{\lambda}}(V,V/V^{\prime})=0.$
This proves that $V=V^{\prime}=\mathbf{U}(\mathfrak{g}\otimes A)V_{\lambda}$
and hence that the map
$\eta_{V}:\mathbf{W}_{A}^{\lambda}\mathbf{R}_{A}^{\lambda}V\to V$ defined in
Proposition Proposition is surjective. Moreover if we set $U=\ker\eta_{V}$,
then we have $\mathbf{R}^{\lambda}_{A}U=0$. Consider the short exact sequence
$0\to U\to\mathbf{W}_{A}^{\lambda}V_{\lambda}\to V\to 0.$
Applying $\operatorname{Hom}_{\cal I_{A}^{\lambda}}(-,U)$ now gives
$0\to\operatorname{Hom}_{\cal I_{A}^{\lambda}}(U,U)\to 0,$
and hence $U=0$ and the proof is complete. ∎
###### Corollary.
The functor $\mathbf{W}_{A}^{\lambda}$ is exact iff for all $U\in\cal
I_{A}^{\lambda}$ with $U_{\lambda}=0$, we have
$\operatorname{Ext}^{2}_{\cal I_{A}^{\lambda}}(\mathbf{W}_{A}^{\lambda}M,U)=0\
\ \forall\ M\in\operatorname{mod}\mathbf{A}_{\lambda}.$ (3.7)
###### Proof.
Assume that (3.7) is satisfied. Let $0\to M^{\prime\prime}\to M\to
M^{\prime}\to 0$ be a short exact sequence of modules in
$\operatorname{mod}\mathbf{A}_{\lambda}$ and consider the induced short exact
sequence
$0\to K\to\mathbf{W}_{A}^{\lambda}M\to\mathbf{W}_{A}^{\lambda}M^{\prime}\to
0.$
Apply $\operatorname{Hom}(-,U)$ to the preceding short exact sequence and
using Theorem Theorem and (3.7) we find that
$\operatorname{Hom}_{\cal I_{A}^{\lambda}}(K,U)=0,\ \
\operatorname{Ext}^{1}_{\cal I_{A}^{\lambda}}(K,U)=0,\ \ \forall\ \
U\in\operatorname{Ob}\cal I_{A}^{\lambda}\text{ with }U_{\lambda}=0$
Hence $K\cong\mathbf{W}_{A}^{\lambda}K_{\lambda}$. Applying the functor
$\mathbf{R}^{\lambda}_{A}$ and using the fact that
$\mathbf{R}^{\lambda}_{A}\mathbf{W}_{A}^{\lambda}$ is naturally isomorphic to
the identity functor, we see that if $V$ is the kernel of
$\mathbf{W}_{A}^{\lambda}M^{\prime\prime}\to K$ then $V_{\lambda}=0$. Applying
$\operatorname{Hom}_{\cal I_{A}^{\lambda}}(-,V)$ to the short exact sequence
$0\to V\to\mathbf{W}_{\lambda}M^{\prime\prime}\to K\to 0,$
proves that $V=0$.
For the converse, suppose that $\mathbf{W}_{A}^{\lambda}$ is exact. Let
$M\in\operatorname{Ob}\operatorname{mod}\mathbf{A}_{\lambda}$ and let
$P\in\operatorname{Ob}\operatorname{mod}\mathbf{A}_{\lambda}$ be projective
such that we have an exact sequence $0\to M^{\prime}\to P\to M\to 0.$ This
gives us
$0\to\mathbf{W}_{A}^{\lambda}M^{\prime}\to\mathbf{W}_{A}^{\lambda}P\to\mathbf{W}_{A}^{\lambda}M\to
0.$
Applying $\operatorname{Hom}_{\cal I_{A}^{\lambda}}(-,U)$ with $U\in\cal
I^{\lambda}_{A}$, $U_{\lambda}=0$ and recalling that
$\mathbf{W}_{A}^{\lambda}P$ is projective in $\cal I_{A}^{\lambda}$ we get a
piece of the long exact sequence
$0\to\operatorname{Ext}^{2}(\mathbf{W}_{A}^{\lambda}M,U)\to 0,$
and the converse is established. ∎
## 4\. The structure of $W_{A}(\lambda)$
### 4.1.
We begin by proving that the construction of $W_{A}(\lambda)$ is functorial in
$A$. Assume that $B$ is a commutative associative algebra and let $f:A\to B$
be a homomorphism of algebras. Then $(1\otimes f):\mathfrak{g}\otimes
A\to\mathfrak{g}\otimes B$ is a homomorphism of Lie algebras and given any
$\mathfrak{g}\otimes B$–module $V$ we can regard it as a $\mathfrak{g}\otimes
A$–module via $f$ and we denote this module by $f^{*}V$.
###### Proposition.
Let $\lambda\in P^{+}$ and let $f:A\to B$ be a homomorphism of associative
algebras. Then $f$ induces a canonical homomorphism
$f_{\lambda}:\mathbf{A}_{\lambda}\to\mathbf{B}_{\lambda}$ of associative
algebras and a canonical map of ($\mathfrak{g}\otimes
A,\mathbf{A}_{\lambda})$-bimodules $f_{\lambda}^{*}:W_{A}(\lambda)\to
f^{*}(W_{B}(\lambda))$. Moreover, $f_{\lambda}$ and $f_{\lambda}^{*}$ are
surjective if $f$ is surjective.
###### Proof.
The action of $\mathfrak{g}\otimes A$ on $f^{*}(W_{B}(\lambda))$ is given by
$(x\otimes a)\circ w_{\lambda,B}=(x\otimes f(a))w_{\lambda,B}$
and it follows immediately from Proposition Proposition that there is a
well–defined map of left $\mathfrak{g}\otimes A$–modules
$W_{A}(\lambda)\to f^{*}(W_{B}(\lambda)),\ \ \ \ w_{\lambda,A}\to
w_{\lambda,B}.$
Since $(1\otimes f)$ maps $\mathfrak{h}\otimes A$ to $\mathfrak{h}\otimes B$
this is also a map of right $\mathbf{U}(\mathfrak{h}\otimes A)$–modules. The
proof of the proposition is complete if we prove that
$u\in\operatorname{Ann}_{\mathfrak{h}\otimes
A}(w_{\lambda,A})\implies(1\otimes
f)(u)\in\operatorname{Ann}_{\mathfrak{h}\otimes B}(w_{\lambda,B}).$
But this is clear since
$w_{\lambda,A}u=uw_{\lambda,A}\to(1\otimes
f)(u)w_{\lambda,B}=w_{\lambda,B}(1\otimes f)(u).$
∎
Let $A,B$ and $f:A\to B$ be as in the proposition and given
$M\in\operatorname{mod}\mathbf{B}_{\lambda}$, let
$f_{\lambda}^{*}M\in\operatorname{mod}\mathbf{A}_{\lambda}$ be the
corresponding $\mathbf{A}_{\lambda}$–module.
###### Corollary.
There exists a natural morphism of $\mathfrak{g}\otimes A$–modules
$\mathbf{W}^{\lambda}_{A}f_{\lambda}^{*}M\to f^{*}\mathbf{W}^{\lambda}_{B}M$
which is surjective if $f$ is surjective. In particular we have a morphism of
$\mathfrak{g}\otimes A$–modules
$\mathbf{W}^{\lambda}_{A}f_{\lambda}^{*}\mathbf{B}_{\lambda}\to
f^{*}\mathbf{W}^{\lambda}_{B}\mathbf{B}_{\lambda}\cong f^{*}(W_{B}(\lambda)),$
(4.1)
which is surjective if $f$ is surjective.
###### Proof.
It is clear that there exists a map $f^{*}\otimes f_{\lambda}^{*}$ of
$\mathfrak{g}\otimes A$–modules
$W_{A}(\lambda)\otimes_{\mathbf{A}_{\lambda}}f_{\lambda}^{*}M=\mathbf{W}^{\lambda}_{A}f_{\lambda}^{*}M\longrightarrow
f^{*}W_{B}(\lambda)\otimes_{\mathbf{A}_{\lambda}}f_{\lambda}^{*}M.$
Composing with the map of $\mathfrak{g}\otimes A$–modules,
$f^{*}W_{B}(\lambda)\otimes_{\mathbf{A}_{\lambda}}f_{\lambda}^{*}M\to
f^{*}\mathbf{W}^{\lambda}_{B}M=f^{*}(W_{B}(\lambda)\otimes_{\mathbf{B}_{\lambda}}M),\
\qquad u\otimes m\to u\otimes m$
proves the corollary. ∎
### 4.2.
The next proposition begins an analysis of the behaviour of the modules
$W_{A}(\lambda)$ and the functors $\mathbf{W}^{\lambda}_{A}$ under tensor
products. We shall assume from now on that an unadorned $\otimes$ denotes the
tensor product of vector spaces over $\mathbf{C}$.
###### Proposition.
Let $\lambda,\mu\in P^{+}$.
* (i)
There exists a homomorphism of $\mathfrak{g}\otimes A$–modules
$\tau_{\lambda,\mu}:W_{A}(\lambda+\mu)\to W_{A}(\lambda)\otimes W_{A}(\mu),$
such that $\tau_{\lambda,\mu}(w_{\lambda+\mu})=\ w_{\lambda}\otimes w_{\mu}.$
* (ii)
The homomorphism $\Delta:\mathbf{U}(\mathfrak{h}\otimes
A)\to\mathbf{U}(\mathfrak{h}\otimes A)\otimes\mathbf{U}(\mathfrak{h}\otimes
A)$ induces a canonical homomorphism
$\Delta_{\lambda,\mu}:\mathbf{A}_{\lambda+\mu}\to\mathbf{A}_{\lambda}\otimes\mathbf{A}_{\mu}$
and
$\Delta_{\lambda,\mu}=\sigma_{\mu,\lambda}\circ\Delta_{\mu,\lambda},\ \
(1\otimes\Delta_{\mu,\nu})\circ\Delta_{\lambda,\mu+\nu}=(\Delta_{\lambda,\mu}\otimes
1)\circ\Delta_{\lambda+\mu,\nu},\ \ \nu\in P^{+}.$
where
$\sigma_{\lambda,\mu}:\mathbf{A}_{\lambda}\otimes\mathbf{A}_{\mu}\longrightarrow\mathbf{A}_{\mu}\otimes\mathbf{A}_{\lambda}$
denotes the flip map.
* (iii)
The tensor product $W_{A}(\lambda)\otimes W_{A}(\mu)$ is canonically a
$(\mathfrak{g}\otimes A,\mathbf{A}_{\lambda}\otimes\mathbf{A}_{\mu})$–bimodule
and hence also a $(\mathfrak{g}\otimes A,\mathbf{A}_{\lambda+\mu})$–bimodule.
* (iv)
The map $\tau_{\lambda,\mu}$ is a map of $(\mathfrak{g}\otimes
A,\mathbf{A}_{\lambda+\mu})$–bimodules and for
$M\in\operatorname{mod}\mathbf{A}_{\lambda}$,
$N\in\operatorname{mod}\mathbf{A}_{\mu}$ we have an induced map of
$\mathfrak{g}\otimes A$-modules
$\tau_{M,N}:\mathbf{W}_{A}^{\lambda+\mu}\Delta_{\lambda,\mu}^{*}(M\otimes
N)\to\mathbf{W}_{A}^{\lambda}M\otimes\mathbf{W}^{\mu}_{A}N.$
###### Proof.
Part (i) is immediate from Proposition Proposition. It follows that
$u\in\operatorname{Ann}_{\mathfrak{h}\otimes
A}(w_{\lambda+\mu})\implies\Delta(u)(w_{\lambda}\otimes w_{\mu})=0,$
i.e., that
$\Delta(u)\in\operatorname{Ann}_{\mathfrak{h}\otimes
A}(w_{\lambda})\otimes\mathbf{U}(\mathfrak{h}\otimes
A)+\mathbf{U}(\mathfrak{h}\otimes
A)\otimes\operatorname{Ann}_{\mathfrak{h}\otimes A}(w_{\mu}),$
and hence we have an induced map
$\Delta_{\lambda,\mu}:\mathbf{A}_{\lambda+\mu}\to\mathbf{A}_{\lambda}\otimes\mathbf{A}_{\mu}$.
The remaining statements in (ii) follow from the co-commutativity and co-
associativity of $\Delta$. The right action of $\mathbf{A}_{\lambda}$ on
$W_{A}(\lambda)$ and of $\mathbf{A}_{\mu}$ on $W_{A}(\mu)$ defines a right
action of $\mathbf{A}_{\lambda}\otimes\mathbf{A}_{\mu}$ on
$W_{A}(\lambda)\otimes W_{A}(\mu)$ in the obvious pointwise way and part (iii)
now follows easily. To prove (iv), note that we clearly have a map
$\mathbf{W}^{\lambda+\mu}_{A}\Delta_{\lambda,\mu}^{*}(M\otimes
N)\to\left(W_{A}(\lambda)\otimes
W_{A}(\mu)\right)\otimes_{\mathbf{A}_{\lambda+\mu}}\Delta_{\lambda,\mu}^{*}(M\otimes
N).$
Since there exist canonical maps of $\mathfrak{g}\otimes A$–modules
$\displaystyle\left(W_{A}(\lambda)\otimes
W_{A}(\mu)\right)\otimes_{\mathbf{A}_{\lambda+\mu}}\Delta_{\lambda,\mu}^{*}(M\otimes
N)\to\left(W_{A}(\lambda)\otimes
W_{A}(\mu)\right)\otimes_{\mathbf{A}_{\lambda}\otimes\mathbf{A}_{\mu}}(M\otimes
N)$
and a map
$\displaystyle\left(W_{A}(\lambda)\otimes
W_{A}(\mu)\right)\otimes_{\mathbf{A}_{\lambda}\otimes\mathbf{A}_{\mu}}(M\otimes
N)\to\mathbf{W}^{\lambda}_{A}M\otimes\mathbf{W}^{\mu}_{A}N,$
$\displaystyle(w\otimes w^{\prime})\otimes(m\otimes n)\to(w\otimes
m)\otimes(w^{\prime}\otimes n),$
the result follows. ∎
### 4.3.
Given two commutative associative algebras $A$ and $B$ the direct sum
$C=A\oplus B$ is canonically an associative algebra and let $p_{A}$ (resp.
$p_{B}$) be the projection onto $A$ (resp. $B$). By Proposition Proposition
any $M\in\operatorname{mod}\mathbf{A}_{\lambda}$ (resp.
$N\in\operatorname{mod}\mathbf{B}_{\mu}$) can be regarded as a module for
$\mathbf{C}_{\lambda}$ (resp. $\mathbf{C}_{\mu}$) and hence the tensor product
$M\otimes N$ can be viewed as a module for
$\mathbf{C}_{\lambda}\otimes\mathbf{C}_{\mu}$. Pulling this module back by
$\Delta_{\lambda,\mu}$ we get a $\mathbf{C}_{\lambda+\mu}$–module which by
abuse of notation, we shall just denote by $M\otimes N$ and we shall see that
the context is such that no confusion arises from this abuse of notation. The
following is immediate from Corollary Proposition and Proposition
Proposition(iv).
###### Corollary.
For
$M\in\operatorname{mod}\mathbf{A}_{\lambda},N\in\operatorname{mod}\mathbf{B}_{\mu}$,
there exists a surjective homomorphism of $\mathfrak{g}\otimes C$–modules
$\mathbf{W}_{C}^{\lambda+\mu}(M\otimes
N)\twoheadrightarrow\mathbf{W}_{A}^{\lambda}M\otimes\mathbf{W}_{B}^{\mu}N.$
### 4.4.
###### Theorem.
Assume that $A$ is a finitely generated algebra.
* (i)
For all $\lambda\in P^{+}$, the algebra $\mathbf{A}_{\lambda}$ is finitely
generated and $W_{A}(\lambda)$ is a finitely generated right
$\mathbf{A}_{\lambda}$–module.
* (ii)
If $M\in\operatorname{mod}\mathbf{A}_{\lambda}$ is a finitely generated (resp.
finite–dimensional) then $\mathbf{W}_{A}^{\lambda}M$ is a finitely generated
(resp. finite-dimensional) $\mathfrak{g}\otimes A$–module.
* (iii)
Suppose that $A$ and $B$ are finite–dimensional commutative, associative
algebras and let $\lambda,\mu\in P^{+}$. For
$M\in\operatorname{mod}\mathbf{A}_{\lambda}$,
$N\in\operatorname{mod}\mathbf{B}_{\mu}$ with $\dim M<\infty$ and $\dim
N<\infty$ we have,
$\mathbf{W}_{A\oplus B}^{\lambda+\mu}(M\otimes
N)\cong\mathbf{W}^{\lambda}_{A}M\otimes\mathbf{W}^{\mu}_{B}N,$
as $\mathfrak{g}\otimes(A\oplus B)$–modules.
We prove the theorem in the rest of the section.
### 4.5.
Let $u$ be an indeterminate and for $a\in A$, $\alpha\in R^{+}$, define a
power series $\mathbf{p}_{a,\alpha}(u)$ in $u$ with coefficients in
$\mathbf{U}(\mathfrak{h}\otimes A)$ by
$\mathbf{p}_{a,\alpha}(u)=\exp\left(-\sum_{r=1}^{\infty}\frac{h_{\alpha}\otimes
a^{r}}{r}u^{r}\right).$
For $s\in\mathbf{Z}_{+}$, let $p_{a,\alpha}^{s}$ be the coefficient of $u^{s}$
in $\mathbf{p}_{a,\alpha}(u)$. The following formula is proved in [G] in the
case when $A$ is the polynomial ring $\mathbf{C}[t]$ and $a=t$. Applying the
Lie algebra homomorphism
$\mathfrak{g}\otimes\mathbf{C}[t]\to\mathfrak{g}\otimes A,\ \ \ x\otimes
t^{r}\to x\otimes a^{r},\ \ r\in\mathbf{Z}_{+},\ \ x\in\mathfrak{g},$
gives the result for $\mathfrak{g}\otimes A$.
###### Lemma.
Let $r\in\mathbf{Z}_{+}$. Then,
$\displaystyle(x^{+}_{\alpha}\otimes a)^{r}(x^{-}_{\alpha}\otimes
1)^{r+1}-\sum_{s=0}^{r}(x_{\alpha}^{-}\otimes
a^{r-s})p^{s}_{a,\alpha}\in\mathbf{U}(\mathfrak{g}\otimes
A)(\mathfrak{n}^{+}\otimes A),$ $\displaystyle(x^{+}_{\alpha}\otimes
a)^{r+1}(x^{-}_{\alpha}\otimes
1)^{r+1}-p^{r+1}_{a,\alpha}\in\mathbf{U}(\mathfrak{g}\otimes
A)(\mathfrak{n}^{+}\otimes A)$
∎
### 4.6.
Part (i) of the theorem was proved in the case when $A$ is the polynomial ring
in one variable in [CP2]. The proof in the general case is very similar, and
we only give a brief sketch here. Let $a_{1},\cdots,a_{m}$ be a set of
generators for $A$. Using the defining relations of $W_{A}(\lambda)$ and Lemma
Lemma, we see that
$(x^{+}_{i}\otimes a_{k})^{n_{i}}(x^{-}_{i}\otimes
1)^{n_{i}+1}w_{\lambda}=\sum_{s=0}^{n_{i}}(x_{i}^{-}\otimes
a_{k}^{n_{i}-s})p^{s}_{a_{k},\alpha_{i}}w_{\lambda}=0$
for all $i\in I$, $1\leq k\leq m$ and $n_{i}=\lambda(h_{i}).$ Applying
$x_{i}^{+}\otimes a$, $a\in A$, to both sides of the equation, we get
$\left(h_{i}\otimes aa_{k}^{n_{i}}+\sum_{s=1}^{n_{i}}(h_{i}\otimes
aa_{k}^{n_{i}-s})p^{s}_{a_{k},\alpha_{i}}\right)w_{\lambda}=0.$
It is now straightforward to see by using an iteration of this argument that
for all $i\in I$, $(r_{1},\cdots,r_{m})\in\mathbf{Z}_{+}^{m}$, we have
$h_{i}\otimes(a_{1}^{r_{1}}\cdots
a_{m}^{r_{m}})w_{\lambda}=H(i,r_{1},\cdots,r_{m})w_{\lambda}$
for some $H(i,r_{1},\cdots,r_{m})$ in the subalgebra of
$\mathbf{U}(\mathfrak{h}\otimes A)$ generated by the elements of the set
$\\{h_{i}\otimes a_{1}^{s_{1}}\cdots a_{m}^{s_{m}}:0\leq s_{\ell}\leq n_{i},\
1\leq\ell\leq m,\ i\in I\\}.$
In other words, we have proved that $\mathbf{A}_{\lambda}$ is the quotient of
a finitely generated algebra.
Let $\\{\beta_{1},\cdots,\beta_{N}\\}$ be an enumeration of $R^{+}$ and set
$S=\\{a_{1}^{s_{1}}\cdots
a_{m}^{s_{m}}:(s_{1},\cdots,s_{m})\in\mathbf{Z}_{+}^{M}\\}.$
Using the PBW theorem, we see that elements of the set,
$\left\\{(x^{-}_{\beta_{i_{1}}}\otimes
b_{1})\cdots(x^{-}_{\beta_{i_{\ell}}}\otimes b_{\ell})w_{\lambda}:1\leq
i_{1}\leq\cdots\leq i_{\ell}\leq N,\ \ \ell\in\mathbf{Z}_{+},\
b_{1},\cdots,b_{\ell}\in S\right\\}$ (4.2)
generate $W_{A}(\lambda)$ as a right module for $\mathbf{A}_{\lambda}$. Using
Lemma Lemma and the defining relations for $W_{A}(\lambda)$ we see that
$(x^{+}_{\alpha}\otimes a_{r})^{n_{\alpha}}(x^{-}_{\alpha}\otimes
1)^{n_{\alpha}+1}w_{\lambda}=\sum_{s=0}^{n_{\alpha}}x_{\alpha}^{-}\otimes
a_{r}^{n_{\alpha}-s}p^{s}_{a_{r},\alpha}w_{\lambda}=0,\ \ 1\leq r\leq m,$
for all $\alpha\in R^{+}$ and $n_{\alpha}=\lambda(h_{\alpha})$. That implies
$(x^{-}_{\alpha}\otimes
a_{r}^{s})w_{\lambda}\in{\rm{sp}}\\{(x^{-}_{\alpha}\otimes
a_{r}^{\ell})w_{\lambda}\mathbf{A}_{\lambda}:\ \
0\leq\ell<\lambda(h_{\alpha})\\}.$
Applying $h_{\alpha}\otimes a^{k}_{p}$ with $r\neq p$ to the preceding
equation gives,
$\displaystyle(x^{-}_{\alpha}\otimes
a_{r}^{s}a_{p}^{k})w_{\lambda}\in{\rm{sp}}\\{(x^{-}_{\alpha}\otimes
a_{r}^{\ell}a_{p}^{k})w_{\lambda}\mathbf{A}_{\lambda}:\ \
0\leq\ell<\lambda(h_{\alpha})\\}$
$\displaystyle\subset{\rm{sp}}\\{x^{-}_{\alpha}\otimes
a_{r}^{\ell}a_{p}^{\ell^{\prime}}W_{A}(\lambda)_{\lambda},\ \
0\leq\ell,\ell^{\prime}<n_{\alpha}\\}.$
It is now clear that more generally we have
$(x^{-}_{\alpha}\otimes
A)w_{\lambda}\subset{\rm{sp}}\\{(x^{-}_{\alpha}\otimes(a_{1}^{r_{1}}\cdots
a_{m}^{r_{m}})w_{\lambda}\mathbf{A}_{\lambda}:0\leq r_{\ell}<n_{\alpha})\\}.$
An induction on the length of the monomials in (4.2) identical to the one used
in [CP2] now proves that $W_{A}(\lambda)$ is a finitely generated
$\mathbf{A}_{\lambda}$–module. Part (ii) of the theorem is now immediate by
using (3.4).
### 4.7.
To prove (iii), we begin with the following refinement of Theorem Theorem.
###### Proposition.
* (i)
Let $\lambda,\nu\in P^{+}$ be such that $\lambda\nleq\nu$ and
$\nu\nleq\lambda$. Let $U\in\cal I^{\nu}_{A}$ be irreducible and assume that
$U_{\nu}\neq 0$. Then
$\operatorname{Ext}^{m}_{\cal I_{A}}(\mathbf{W}^{\lambda}_{A}M,U)=0,\ \
m=0,1,$
for all $M\in\operatorname{Ob}\operatorname{mod}\mathbf{A}_{\lambda}$.
* (ii)
Let $V\in\cal I^{\lambda}_{A}$ be such that $\dim V_{\lambda}<\infty$. Then
$\mathbf{W}^{\lambda}_{A}\mathbf{R}^{\lambda}_{A}V_{\lambda}\cong V$ iff
$\operatorname{Ext}^{m}_{\mathfrak{g}\otimes A}(V,U)=0,\ \ m=0,1$ (4.3)
for all $U\in\operatorname{Ob}\cal I^{\lambda}_{A}$ with $\dim U<\infty$ and
$U_{\lambda}=0$.
###### Proof.
For (i), observe that since $U$ is irreducible any non–zero morphism
$\eta:W_{A}(\lambda)\to U$ must be surjective. But this is impossible since
$(\mathbf{W}^{\lambda}_{A}M)_{\nu}=0$. Suppose next that
$0\to U\to V\to\mathbf{W}^{\lambda}_{A}M\to 0$
is a short exact sequence of objects in $\cal I_{A}$. Then
$V_{\lambda}\neq 0,\ \ \ \
\operatorname{wt}V\subset(\nu-Q^{+})\cup(\lambda-Q^{+}),$
and since $\lambda\nleq\nu$ we see that $(\mathfrak{n}^{+}\otimes
A)V_{\lambda}=0.$ Set $V^{\prime}=\mathbf{U}(\mathfrak{g}\otimes
A)V_{\lambda}$ so that $\operatorname{wt}V\subset\lambda-Q^{+}$. To prove that
the sequence splits, it suffices to prove that
$V^{\prime}\cap U=\\{0\\}.$
Otherwise since $U$ is irreducible we would have $U\cap V^{\prime}=U$ which
would imply that $\nu\in\operatorname{wt}V^{\prime}$ contradicting
$\nu\nleq\lambda$.
A simple induction on the length of $U$ shows that it suffices to to prove
that $\mathbf{W}^{\lambda}_{A}V_{\lambda}\cong V$ if (4.3) holds for all
irreducible modules $U\in\operatorname{Ob}\cal I^{\lambda}_{A}$ with
$U_{\lambda}=0$. As in the proof of Theorem Theorem we have
$V=\mathbf{U}(\mathfrak{g}\otimes A)V_{\lambda}$ and hence a short exact
sequence
$0\to K\to\mathbf{W}^{\lambda}_{A}V_{\lambda}\to V\to 0.$
By part (ii) of Theorem Theorem we have
$\dim\mathbf{W}^{\lambda}_{A}V_{\lambda}<\infty$ and hence we have
$\dim K<\infty,\ \ K_{\lambda}=0.$
If $K\neq 0$, then $\operatorname{Hom}_{\mathfrak{g}\otimes A}(K,U)\neq 0$ for
some irreducible module $U\in\cal I^{\lambda}_{A}$ with $U_{\lambda}=0$.
Applying $\operatorname{Hom}_{\cal I^{\lambda}_{A}}(-,U)$ and using the fact
that $\operatorname{Hom}_{\mathfrak{g}\otimes
A}(\mathbf{W}^{\lambda}_{A},U)=0$, we get
$0\to\operatorname{Hom}_{\mathfrak{g}\otimes
A}(K,U)\to\operatorname{Ext}^{1}_{\mathfrak{g}\otimes A}(V,U)$
which is impossible since $V$ satisfies (4.3). Hence $K=0$ and the proof of
(ii) is complete. ∎
### 4.8.
The proof of part(iii) of the Theorem is completed as follows. By Corollary
Corollary we have a surjective map of $\mathfrak{g}\otimes(A\oplus
B)$–modules,
$\mathbf{W}_{A\oplus B}^{\lambda+\mu}(M\otimes
N)\longrightarrow\mathbf{W}^{\lambda}_{A}M\otimes\mathbf{W}^{\mu}_{B}N\to 0.$
To prove that it is an isomorphism it suffices by Proposition Proposition(ii)
to prove that
$\operatorname{Ext}^{m}_{\cal I^{\lambda+\mu}_{A\oplus
B}}(\mathbf{W}^{\lambda}_{A}M\otimes\mathbf{W}^{\mu}_{B}N,U)=0,\ \ m=0,1,$
for all irreducible $U\in\operatorname{Ob}\cal I^{\lambda+\mu}_{A\oplus B}$
with $U_{\lambda+\mu}=0$. By Proposition Proposition we may write such a
module as a tensor product,
$U\cong U_{A}\otimes U_{B},\ \ U_{A}\in\operatorname{Ob}\cal I_{A},\ \
U_{B}\in\operatorname{Ob}\cal I_{B},$
where $U_{A}$ and $U_{B}$ are irreducible. Let $\nu_{A}$ (resp. $\nu_{B}$) be
the highest weight of $U_{A}$ (resp. $U_{B}$) and note that
$\nu_{A}+\nu_{B}\in\operatorname{wt}U\subset\lambda+\mu-Q^{+}$. Since
$\mathbf{W}^{\lambda}_{A}M$, $\mathbf{W}^{\mu}_{B}N$ and $U$ are all
finite–dimensional modules for finite–dimensional Lie algebras, we have for
$m=0,1$,
$\displaystyle\operatorname{Ext}^{m}_{\mathfrak{g}\otimes(A\oplus
B)}(\mathbf{W}^{\lambda}_{A}M\otimes\mathbf{W}^{\mu}_{B}N,U)\cong\operatorname{Ext}^{m}_{\cal
I^{\lambda+\mu}_{A\oplus
B}}(\mathbf{W}^{\lambda}_{A}M\otimes\mathbf{W}^{\mu}_{B}N,U),$
$\displaystyle\operatorname{Ext}^{m}_{\mathfrak{g}\otimes
A}(\mathbf{W}^{\lambda}_{A}M,U_{A})\cong\operatorname{Ext}^{m}_{\cal
I^{\lambda}_{A}}(\mathbf{W}^{\lambda}_{A}M,U_{A}),\qquad\operatorname{Ext}^{m}_{\mathfrak{g}\otimes
B}(\mathbf{W}^{\mu}_{B}N,U_{B})\cong\operatorname{Ext}^{m}_{\cal
I^{\lambda}_{b}}(\mathbf{W}^{\mu}_{B}N,U_{B}).$
By Proposition Proposition it suffices to prove that either
$\operatorname{Ext}^{m}_{\cal
I^{\lambda}_{A}}(\mathbf{W}^{\lambda}_{A}M,U_{A})=0,\ \\\ \ {\rm{or}}\ \
\operatorname{Ext}^{m}_{\cal I^{\mu}_{B}}(\mathbf{W}^{\mu}_{B}N,U_{B})=0,\ \
m=0,1.$ (4.4)
If $U_{A}\in\operatorname{Ob}\cal I^{\lambda}_{A}$ or
$U_{B}\in\operatorname{Ob}\cal I^{\nu}_{B}$ then (4.4) follows from
Proposition Proposition(ii). Otherwise we have
$\nu_{A}\nleq\lambda,\ \qquad\ \nu_{B}\nleq\mu.$
Since $\nu_{A}+\nu_{B}<\lambda+\mu$, it follows now that $\lambda\nleq\nu_{A}$
and now (4.4) follows from Proposition Proposition(i).
## 5\. Further results on tensor products
Throughout this section, we assume that $A$ is finitely generated.
### 5.1.
Let $\rm{irr}\operatorname{mod}\mathbf{A}_{\lambda}$ be the set of irreducible
representations of $\mathbf{A}_{\lambda}$. Since $\mathbf{A}_{\lambda}$ is a
commutative finitely generated algebra it follows that if
$M\in\rm{irr}\operatorname{mod}\mathbf{A}_{\lambda}$ then $\dim M=1$. By
Theorem Theorem we see that
$\dim\mathbf{W}_{A}^{\lambda}M<\infty,\ \
\mathbf{R}_{A}^{\lambda}\mathbf{W}^{\lambda}_{A}M=M,\ \text{ for }\
M\in\rm{irr}\operatorname{mod}\mathbf{A}_{\lambda},$
and we denote by $\mathbf{V}^{\lambda}_{A}M$ the unique irreducible quotient
of $\mathbf{W}_{A}^{\lambda}M$ (see Lemma Lemma). It now follows from Lemma
Lemma and Lemma Lemma that there exists an ideal of finite–codimension
$\tilde{K}^{\lambda}_{M}$ of $A$ such that $\mathfrak{g}\otimes
A/\tilde{K}^{\lambda}_{M}$ is a semisimple Lie algebra and
$(x\otimes a)\mathbf{V}_{A}^{\lambda}M=0\ \ \forall\ \ x\in\mathfrak{g},\ \
a\in\tilde{K}^{\lambda}_{M}.$
Suppose that $M\in\operatorname{mod}\mathbf{A}_{\lambda}$ is finite
dimensional of length $r$, $M_{1},\cdots,M_{r}$ be the irreducible
constituents of $M$ and set
$\tilde{K}^{\lambda}_{M}=\prod_{s=1}^{r}\tilde{K}^{\lambda}_{M_{s}}.$
### 5.2.
The next result shows that any irreducible module in $\cal I_{A}^{\lambda}$ is
isomorphic to $\mathbf{V}^{\mu}_{A}M$ for some $\mu\in P^{+}$.
###### Lemma.
Let $\lambda\in P^{+}$ and assume that $V\in\cal I_{A}^{\lambda}$ is
irreducible. There exists $\mu\in P^{+}\cap\cal(\lambda-Q^{+})$ such that
$\operatorname{wt}V\subset\mu-Q^{+},\ \ \dim V_{\mu}=1.$
In particular, $V$ is the unique irreducible quotient of
$\mathbf{W}_{A}^{\mu}\mathbf{R}_{A}^{\mu}V$ and hence $\dim V<\infty$. If
$V^{\prime}\in\operatorname{Ob}\cal I_{A}$ we have $V\cong V^{\prime}$ as
$\mathfrak{g}\otimes A$–modules iff
$\mathbf{R}^{\mu}_{A}V\cong\mathbf{R}^{\mu^{\prime}}_{A}V^{\prime}$ as
$\mathbf{A}_{\mu}$–modules.
###### Proof.
Since $V\in\cal I^{\lambda}_{A}$, it follows that there exists
$\mu\in\lambda-Q^{+}$ with
$V_{\mu}\neq 0,\ \ \ \ (\mathfrak{n}^{+}\otimes A)V_{\mu}=0.$
It is immediate from Proposition Proposition that $V$ is a quotient of
$\mathbf{W}_{A}^{\mu}\mathbf{R}_{A}^{\mu}$. If
$V_{\mu}^{\prime}=\mathbf{U}(\mathfrak{h}\otimes A)V_{\mu}$ is a proper
$\mathfrak{h}\otimes A$–submodule of $V_{\mu}$, then
$V^{\prime}=\mathbf{U}(\mathfrak{g}\otimes A)V_{\mu}^{\prime}$ is a proper
submodule of $V$ which is a contradicton. Hence $\mathbf{R}_{A}^{\mu}V$ is an
irreducible $\mathbf{A}_{\mu}$–module which implies that $\dim V_{\mu}=1$.
Theorem Theorem now implies that
$\dim\mathbf{W}_{A}^{\mu}\mathbf{R}_{A}^{\mu}V<\infty$ and hence $\dim
V<\infty$. The proof that $V$ is the unique irreducible quotient of
$\mathbf{W}^{A}_{\mu}\mathbf{R}_{A}^{\mu}V$ is standard since
$\mathbf{R}_{A}^{\mu}\mathbf{W}^{\mu}_{A}\mathbf{R}_{A}^{\mu}V\cong V_{\mu}$.
The final statement of the lemma is now trivial. ∎
### 5.3.
The main result of this section is the following.
###### Theorem.
Let $\lambda,\mu\in P^{+}$ and let $M,N$ be irreducible modules for
$\mathbf{A}_{\lambda}$ and $\mathbf{A}_{\mu}$ respectively and assume that
$A/\tilde{K}^{\lambda}_{M}\tilde{K}^{\lambda}_{N}\cong
A/\tilde{K}^{\lambda}_{M}\oplus A/\tilde{K}^{\lambda}_{N}.$ (5.1)
Then
$\displaystyle\mathbf{V}_{A}^{\lambda+\mu}(M\otimes
N)\cong_{\mathfrak{g}\otimes
A}\mathbf{V}_{A}^{\lambda}M\otimes\mathbf{V}^{\mu}_{A}N,\qquad\
\tilde{K}^{\lambda+\mu}_{M\otimes
N}=\tilde{K}^{\lambda}_{M}\tilde{K}^{\mu}_{N},$ (5.2)
$\displaystyle\mathbf{W}_{A}^{\lambda+\mu}(M\otimes
N)\cong_{\mathfrak{g}\otimes
A}\mathbf{W}_{A}^{\lambda}M\otimes\mathbf{W}^{\mu}_{A}N.$ (5.3)
### 5.4.
To prove (5.2) recall that $M\otimes N$ is an irreducible
$\mathbf{A}_{\lambda}\otimes\mathbf{A}_{\mu}$–module with the action being
pointwise and hence also an irreducible $\mathbf{A}_{\lambda+\mu}$–module (via
$\Delta_{\lambda,\mu}$). By Lemma Lemma we see that it suffices to prove that
$\mathbf{V}_{A}^{\lambda}M\otimes\mathbf{V}^{\mu}_{A}N$ is the irreducible
$\mathfrak{g}\otimes A$ quotient of $\mathbf{W}^{\lambda+\mu}_{A}(M\otimes
N)$. Clearly, $\mathbf{V}_{A}^{\lambda}M\otimes\mathbf{V}^{\mu}_{A}N$ is an
irreducible module for the semisimple Lie algebra
$\mathfrak{g}\otimes(A/\tilde{K}^{\lambda}_{M}\oplus
A/\tilde{K}^{\lambda}_{N})$ and hence using (5.1) it is an irreducible module
for $\mathfrak{g}\otimes A/\tilde{K}^{\lambda}_{M}\tilde{K}^{\lambda}_{N}$ and
so for $\mathfrak{g}\otimes A$ as well. Since
$\mathbf{R}_{A}^{\lambda+\mu}(\mathbf{V}_{A}^{\lambda}M\otimes\mathbf{V}^{\mu}_{A}N)\cong
M\otimes N,$
we see from Lemma Lemma that
$\mathbf{V}_{A}^{\lambda}M\otimes\mathbf{V}^{\mu}_{A}N$ is a quotient of
$\mathbf{W}^{\lambda+\mu}_{A}(M\otimes N)$ and the first isomorphism in (5.2)
is proved. For the second, observe that by definition if $S$ is any ideal in
$A$ such that
$(\mathfrak{g}\otimes S)\mathbf{V}_{A}^{\lambda}M=0,$
then $S\subset\tilde{K}^{\lambda}_{M}$ and similarly for
$\tilde{K}^{\mu}_{N}$. One deduces easily from (5.1) that
$\tilde{K}^{\lambda}_{M}\tilde{K}^{\lambda}_{N}$ is the largest ideal in $A$
such that
$(\mathfrak{g}\otimes\tilde{K}^{\lambda}_{M}\tilde{K}^{\lambda}_{N})\mathbf{V}_{A}^{\lambda}M\otimes\mathbf{V}^{\mu}_{A}N=0.$
Since $\tilde{K}^{\lambda+\mu}_{M\otimes N}$ is maximal with the property that
$(\mathfrak{g}\otimes\tilde{K}^{\lambda+\mu}_{M\otimes
N})\mathbf{V}_{A}^{\lambda+\mu}(M\otimes N)=0$
we now get that $\tilde{K}^{\lambda+\mu}_{M\otimes
N}=\tilde{K}^{\lambda}_{M}\tilde{K}^{\lambda}_{N}$.
### 5.5.
We need several results to prove (5.3). Theorem Theorem and Lemma Lemma imply
that given $\lambda\in P^{+}$ and $M\in\operatorname{mod}\mathbf{A}_{\lambda}$
with $\dim M<\infty$, there exists an ideal of finite codimension
$K^{\lambda}_{M}$ in $A$ which is maximal with the property that
$(\mathfrak{g}\otimes K^{\lambda}_{M})\mathbf{W}_{A}^{\lambda}M=0.$
If $0\to M^{\prime}\to M\to M^{\prime\prime}\to 0,$ is a short exact sequence
of modules in $\mathbf{A}_{\lambda}$ then since the functor
$\mathbf{W}^{\lambda}_{M}$ is right exact, we see that
$K^{\lambda}_{M^{\prime}}K^{\lambda}_{M^{\prime\prime}}\subset
K^{\lambda}_{M}\subset K^{\lambda}_{M^{\prime\prime}}.$ (5.4)
Let $K\subset K^{\lambda}_{M}$ be an ideal in $A$ and set $A/K=B$. It is clear
that $\mathbf{W}^{\lambda}_{A}M$ is a module for $\mathfrak{g}\otimes B$ and
since
$\mathbf{R}^{\lambda}_{B}\mathbf{W}^{\lambda}_{A}M=M,$
we get by Lemma Lemma that $M$ is also a $\mathbf{B}_{\lambda}$–module.
###### Lemma.
Let $\lambda\in P^{+}$ and $M\operatorname{mod}\mathbf{A}_{\lambda}$ be
finite–dimensional. For all ideals $K\subset K^{\lambda}_{M}$, we have an
isomorphism of $\mathfrak{g}\otimes A$ (or equivalently $\mathfrak{g}\otimes
A/K$) modules,
$\mathbf{W}^{\lambda}_{A}M\cong\mathbf{W}^{\lambda}_{A/K}M.$ (5.5)
###### Proof.
By Corollary Proposition and the discussion preceding the statement of the
Lemma we see that we have a surjective map of $\mathfrak{g}\otimes A$–modules
$\mathbf{W}_{A}^{\lambda}M\to\mathbf{W}^{\lambda}_{B}M\to 0,\ \
w_{\lambda}\otimes m\to w_{\lambda}\otimes m.$
On the other hand by Proposition Proposition we have a map of
$\mathfrak{g}\otimes B$–modules
$\mathbf{W}^{\lambda}_{B}M\cong\mathbf{W}^{\lambda}_{B}\mathbf{R}^{\lambda}_{B}\mathbf{W}^{\lambda}_{A}M\longrightarrow\mathbf{W}^{\lambda}_{A}M,\
w_{\lambda}\otimes m\to w_{\lambda}\otimes m$
and hence (5.5) is proved. ∎
### 5.6.
###### Proposition.
Let $\lambda\in P^{+}$ and $M\in\operatorname{mod}\mathbf{A}_{\lambda}$ be
finite–dimensional. We have
$(\tilde{K}^{\lambda}_{M})^{\lambda(h_{\theta})}\subset K^{\lambda}_{M}.$
###### Proof.
It suffices by (5.4) to consider the case when $M$ is irreducible. Using Lemma
Lemma we see as in the proof of Theorem Theorem that
$0=(x^{+}_{\theta}\otimes
a)(x^{-}_{\theta})^{\lambda(h_{\theta})+1}(w_{\lambda}\otimes
m)=\sum_{s=0}^{\lambda(h_{\theta})}(x^{-}_{\theta}\otimes
a^{r-s})p^{s}_{a,\theta}(w_{\lambda}\otimes m).$
If $a\in\tilde{K}^{\lambda}_{M}$ then $(h\otimes a)(w_{\lambda}\otimes m)=0$
and since $p^{s}_{a,\theta}$ is in the subalgebra generated by the elements
$\\{h_{\theta}\otimes a^{p}:p\in\mathbf{Z}_{+},p>0\\}$ with constant term
zero, we see that $p^{s}_{a,\theta}(w_{\lambda}\otimes m)=0$ for all $s>0$.
This implies that
$(x^{-}_{\theta}\otimes a^{\lambda(h_{\theta})})(w_{\lambda}\otimes m)=0.$
Since $[x^{-}_{\theta},\mathfrak{n}^{-}]=0$ we get
$(x^{-}_{\theta}\otimes a^{\lambda(h_{\theta})})\mathbf{W}_{A}^{\lambda}M=0.$
Since $\mathfrak{g}$ is generated by $x^{-}_{\theta}$ as a
$\mathfrak{g}$–module the result follows.
∎
### 5.7.
By part (i) of the theorem and Proposition Proposition, we may choose $r\geq
1$ so that
$(\tilde{K}^{\lambda}_{M})^{r}(\tilde{K}_{N}^{\mu})^{r}=(\tilde{K}_{M\otimes
N}^{\lambda+\mu})^{r}\subset(K_{M}^{\lambda}K^{\mu}_{M})\cap
K^{\lambda+\mu}_{M\otimes N}.$
Set $C=A/(\tilde{K}^{\lambda}_{M}\tilde{K}^{\mu}_{N})^{r}$ and note that
$C=A/(\tilde{K}^{\lambda}_{M})^{r}\oplus A/(\tilde{K}^{\mu}_{N})^{r}.$
By Theorem Theorem(ii), we have an isomorphism of $\mathfrak{g}\otimes
A/C$–modules
$\mathbf{W}_{C}^{\lambda+\mu}(M\otimes
N)\cong\mathbf{W}_{A/(\tilde{K}^{\lambda}_{M})^{r}}^{\lambda}M\otimes\mathbf{W}^{\mu}_{A/(\tilde{K}^{\mu}_{N})^{r}}N.$
Lemma 5.5 now proves that we have isomorphisms of $\mathfrak{g}\otimes
A$–modules,
$\mathbf{W}_{C}^{\lambda+\mu}(M\otimes
N)\cong\mathbf{W}^{\lambda+\mu}_{A}(M\otimes N),\ \ \
\mathbf{W}_{A/(\tilde{K}^{\lambda}_{M})^{r}}M\cong\mathbf{W}^{\lambda}_{A}M,\
\ \
\mathbf{W}^{\mu}_{A/(\tilde{K}^{\mu}_{N})^{r}}N\cong\mathbf{W}^{\mu}_{A}N,$
and (5.3) is proved.
### 5.8.
The statement of (5.3) can be strengthened as follows by using Proposition
Proposition.
###### Corollary.
Let $M\in\operatorname{mod}\mathbf{A}_{\lambda}$ and
$N\in\operatorname{mod}\mathbf{A}_{\mu}$ be finite–dimensional and assume that
$A/\tilde{K}^{\lambda}_{M}\tilde{K}^{\lambda}_{N}\cong
A/\tilde{K}^{\lambda}_{M}\oplus A/\tilde{K}^{\lambda}_{N}.$ (5.6)
Then
$\mathbf{W}^{\lambda+\mu}_{A}(M\otimes
N)\cong\mathbf{W}^{\lambda}_{A}M\otimes\mathbf{W}^{\mu}_{A}N.$ (5.7)
## 6\. The algebra $\mathbf{A}_{\lambda}$
We continue to assume that $A$ is a finitely generated commutative associative
algebra over $\mathbf{C}$. Denote by $\max A$ the set of maximal ideals of $A$
and let ${\rm}J(A)$ be the Jacobson radical of $A$. In this section we shall
identify the max spectrum of $\mathbf{A}_{\lambda}$ and if $J(A)=0$ we shall
also identify the algebra $\mathbf{A}_{\lambda}$. As a consequence we also
obtain a classification of the irreducible finite dimensional modules in $\cal
I_{\lambda}^{A}$. Special cases of this classification were proved earlier in
[C1], [CP1] for $A=\mathbf{C}[t,t^{-1}]$, in [L] and [R] in the case when $A$
is the polynomial ring in $k$ variables.
### 6.1.
For $r\in\mathbf{Z}_{+}$ the symmetric group $S_{r}$ acts naturally on
$A^{\otimes r}$ and $\max(A)^{\times r}$ and we let $(A^{\otimes r})^{S_{r}}$
be the corresponding ring of invariants and $\max(A)^{\times r}/S_{r}$ the set
of orbits. If $r=r_{1}+\cdots+r_{n}$, then we regard
$S_{r_{1}}\times\cdots\times S_{r_{n}}$ as a subgroup of $S_{r}$ in the
canonical way, i.e $S_{r_{1}}$ permutes the first $r_{1}$ letters, $S_{r_{2}}$
the next $r_{2}$ letters and so on. Given $\lambda=\sum_{i\in
I}r_{i}\omega_{i}\in P^{+}$, set
$\displaystyle r_{\lambda}=\sum_{i\in I}r_{i},\ \
S_{\lambda}=S_{r_{1}}\times\cdots\times S_{r_{n}},\ \ \ \
\mathbb{A}_{\lambda}=(A^{\otimes r_{\lambda}})^{S_{\lambda}},$ (6.1)
$\displaystyle\max(\mathbb{A}_{\lambda})=(\max(A)^{r_{1}}/S_{r_{1}})\times\cdots\times(\max(A^{r_{n}})/S_{r_{n}}).$
(6.2)
The algebra $\mathbb{A}_{\lambda}$ is clearly finitely generated. For
$\mathbb{M}\in\max\mathbb{(}A_{\lambda})$, let
$\operatorname{ev}_{\mathbb{M}}:\mathbb{A}_{\lambda}\to\mathbf{C}$ be the
corresponding algebra homomorphism.
We shall prove the following in the rest of the section.
###### Theorem.
* (i)
There exists a canonical bijection
$\max\mathbb{A}_{\lambda}\to\max\mathbf{A}_{\lambda}$
* (ii)
Assume that $\rm{J}(A)=0$ and let $\lambda\in P^{+}$. There exists an
isomorphism of algebras
$\tau_{\lambda}:\mathbf{A}_{\lambda}\to\mathbb{A}_{\lambda}.$
### 6.2.
Let $\Xi$ be the monoid of finitely supported functions $\xi:\max(A)\to
P^{+}$, where for $\xi,\xi^{\prime}\in\Xi$ and $S\in\max A$, we define
$\displaystyle(\xi+\xi^{\prime})(S)=\xi(S)+\xi^{\prime}(S),\qquad\operatorname{supp}\xi=\\{S\in\max(A):\xi(S)\neq
0\\},\qquad\operatorname{wt}(\xi)=\sum_{S\in\max(A)}\xi(S).$
Clearly $\operatorname{wt}:\Xi\to P^{+}$ is a morphism of monoids and we set
$\Xi_{\lambda}=\\{\xi\in\Xi:\operatorname{wt}\xi=\lambda\\}.$
Given $\xi\in\Xi_{\lambda}$, let
$K_{\xi}=\prod_{S\in\operatorname{supp}\xi}\\!\\!S,\
\quad\mathfrak{g}_{\xi}=\mathfrak{g}\otimes A/K_{\xi},\
\quad\mathbf{V}_{\xi}=\bigotimes_{S\in\operatorname{supp}\xi}V(\xi(S)).$
Then $\mathfrak{g}_{\xi}$ is a finite–dimensional semi–simple Lie algebra and
$\mathbf{V}_{\xi}$ is an irreducible finite–dimensional representation of
$\mathfrak{g}_{\xi}$ and hence of $\mathfrak{g}\otimes A$ with action given by
$(x\otimes a)(v_{1}\otimes\cdots\otimes
v_{r})=\sum\limits_{k=1}^{r}\operatorname{ev}_{S_{k}}(a)(v_{1}\otimes\cdots\otimes
xv_{k}\otimes\cdots\otimes v_{r}),$ (6.3)
where $S_{1},\cdots,S_{r}$ is an enumeration of $\operatorname{supp}\xi$. Set
$M_{\xi}=\mathbf{R}_{A}^{\lambda}\mathbf{V}_{\xi}.$ By Lemma 5.2 we see that
$\mathbf{V}_{\xi}$ is the unique irreducible quotient of
$\mathbf{W}^{\lambda}_{A}M_{\xi}$ and hence
$\mathbf{V}_{\xi}\cong\mathbf{V}^{\lambda}_{A}M_{\xi}.$
Let $\lambda\in P^{+}$ and
$M\in\rm{irr}\operatorname{mod}\mathbf{A}_{\lambda}$. Since
$A/\tilde{K}^{\lambda}_{M}$ is a finite–dimensional semi–simple algebra we
know that
$\tilde{K}^{\lambda}_{M}=S_{1}\cdots S_{r},\ \ \ \ r\in\mathbf{Z}_{+},$
where $S_{1},\cdots,S_{r}$ are (uniquely defined up to permutation) maximal
ideals in $A$. Moreover $\mathbf{V}^{\lambda}_{A}M$ is a representation for
the semi-simple Lie algebra
$\mathfrak{g}_{M}=\oplus_{k=1}^{r}\mathfrak{g}\otimes A/S_{i}$. So there exist
unique elements $\mu_{1},\cdots,\mu_{r}\in P^{+}$ such that
$\mathbf{V}^{\lambda}_{A}M\cong_{\mathfrak{g}_{M}}V(\mu_{1})\otimes\cdots\otimes
V(\mu_{r}).$
Define $\xi_{M}\in\Xi_{\lambda}$ by
$\xi_{M}(S_{k})=\mu_{k},\ \ 1\leq k\leq r,\ \ \xi(S)=0,\ \ {\rm{otherwise}}.$
Then $\mathbf{V}^{\lambda}_{A}M\cong\mathbf{V}_{\xi}$ as $\mathfrak{g}\otimes
A$–modules. Summarizing, we have proved that:
###### Proposition.
The assignment $\xi\to M_{\xi}$, (resp. $\xi\to\mathbf{V}_{\xi}$) defines a
natural bijection between $\Xi_{\lambda}$ and the set of isomorphism classes
of irreducible representations of $\mathbf{A}_{\lambda}$ (resp. isomorphism
classes of irreducible objects in $\cal I_{A}^{\lambda})$. Moreover this
bijection is compatible with the functor $\mathbf{V}_{A}^{\lambda}$, in the
sense that
$\mathbf{V}_{\xi}\cong\mathbf{V}^{\lambda}_{A}M_{\xi}.$
∎
Given $\xi\in\Xi_{\lambda}$, define
$\operatorname{ev}_{\xi}:\mathbf{U}(\mathfrak{h}\otimes A)\to\mathbf{C}$ by
extending
$\operatorname{ev}_{\xi}(h\otimes a)=\sum_{S\in\max
A}\operatorname{ev}_{S}(a)\xi(S)(h).$
###### Corollary.
Let $\lambda\in P^{+}$. Then
$\operatorname{Ann}_{\mathfrak{h}\otimes
A}w_{\lambda}\subset\bigcap_{\xi\in\Xi_{\lambda}}\ker\operatorname{ev}_{\xi}.$
###### Proof.
Let $u\in\mathbf{U}(\mathfrak{h}\otimes A)$ and assume that $uw_{\lambda}=0$.
Since $\mathbf{V}_{\xi}$ is a quotient of $W_{A}(\lambda)$ it follows that
$u(\mathbf{V}_{\xi})_{\lambda}=0$. On the other hand it is clear from the
definition of $\mathbf{V}_{\xi}$ that
$(h\otimes a)(\mathbf{V}_{\xi})_{\lambda}=\operatorname{ev}_{\xi}(h\otimes
a)(\mathbf{V}_{\xi})_{\lambda},$
and the corollary follows. ∎
### 6.3.
The set $\Xi_{\lambda}$ also parametrizes the set $\max\mathbb{A}_{\lambda}$
as follows. Let $\mathbb{M}\in\max(\mathbb{A}_{\lambda})$ be the orbit of an
element $(S_{1},\cdots,S_{r_{\lambda}})\in\max(A)^{\times r_{\lambda}}$.
Define $\xi(\mathbb{M})\in\Xi_{\lambda}$ by
$\displaystyle\xi(\mathbb{M})(S)=\sum_{i\in I}p_{i}(S)\omega_{i},\ \
S\in\max(A),$ $\displaystyle
p_{i}(S)=\\#\\{p:\sum_{k=1}^{i-1}r_{k}<p\leq\sum_{k=1}^{i}r_{k},\ \ \
S_{p}=S\\}.$
It is easily seen that the assignment $\mathbb{M}\to\xi(\mathbb{M})$ is
well–defined bijection of sets $\max(\mathbb{A}_{\lambda})\to\Xi_{\lambda}$
and part (i) of the Theorem is established.
### 6.4.
The algebra $\mathbb{A}_{\lambda}$ is generated by elements of the form
$\operatorname{sym}^{i}_{\lambda}(a)=1^{\otimes(r_{1}+\cdots
r_{i-1})}\otimes\left(\sum_{k=0}^{r_{i}-1}1^{\otimes k}\otimes a\otimes
1^{\otimes(r_{i}-k-1)}\right)\otimes 1^{\otimes(r_{i+1}+\cdots r_{n})},\ \
a\in A,\ i\in I.$ (6.4)
It is clear that the assignment
$\tilde{\tau}_{\lambda}(h_{i}\otimes a)=\operatorname{sym}^{i}_{\lambda}(a),\
\ i\in I,a\in A$
extends to a surjective algebra homomorphism
$\tilde{\tau}_{\lambda}:\mathbf{U}(\mathfrak{h}\otimes
A)\mapsto\mathbb{A}_{\lambda}$. Moreover it is easily checked that
$\operatorname{ev}_{\xi(\mathbb{M})}(h\otimes
a)=\operatorname{ev}_{\mathbb{M}}\tilde{\tau}_{\lambda}(h\otimes a),\ \
h\in\mathfrak{h},\ a\in A.$ (6.5)
###### Lemma.
We have
$\ker\tilde{\tau}_{\lambda}=\bigcap_{\mathbb{M}\in\max{\mathbb{A}_{\lambda}}}\ker\operatorname{ev}_{\mathbb{M}}\tilde{\tau_{\lambda}}=\bigcap_{\xi\in\Xi_{\lambda}}\ker\operatorname{ev}_{\xi},$
(6.6)
and hence $\tilde{\tau}_{\lambda}$ induces a surjective homomorphism of
algebras $\tau_{\lambda}:\mathbf{A}_{\lambda}\to\mathbb{A}_{\lambda}$.
###### Proof.
The first equality in (6.6) is trivial since $\rm{J}(\mathbb{A}_{\lambda})=0$
if $\rm{J}(A)=0$. The second equality is immediate from (6.5) and the fact
that $\mathbb{M}\to\xi(\mathbb{M})$ is bijective. The final statement of the
Lemma is immediate from Corollary Corollary. ∎
### 6.5.
It remains to prove that $\tau_{\lambda}$ is injective. To do this we adapt an
argument in [FL]. Thus, we identify a natural spanning set of
$\mathbf{A}_{\lambda}$ and prove that its image in $\mathbb{A}_{\lambda}$ is a
basis. Fix an ordered countable basis $\\{a_{r}:r\in\mathbf{Z}_{+}\\}$ of $A$
with $a_{0}=1$ and $a_{r}\in A_{+}$ for $r\geq 1$.
###### Lemma.
The elements
$\\{\prod_{i=1}^{n}\prod_{s=1}^{q_{i}}(h_{i}\otimes
a_{i,s})w_{\lambda}:a_{0}<a_{i,1}\leq\cdots\leq a_{i,q_{i}},\ \ i\in I,\ \
q_{i}\leq\lambda(h_{i})\\}$
span $W_{A}(\lambda)_{\lambda}$.
###### Proof.
It is clearly enough to prove that for each $i\in I$ and elements $1\leq
p_{1}\leq\cdots\leq p_{\ell}$,
$\prod_{s=1}^{\ell}(h_{i}\otimes
a_{p_{s}})w_{\lambda}\in{\rm{span}}\left\\{\prod_{s=1}^{m}(h_{i}\otimes
a_{r_{s}})w_{\lambda}:1\leq r_{1}\leq r_{2}\leq\cdots\leq r_{m},\ \
m\leq\lambda(h_{i})\right\\}.$
Since
$0=\prod_{s=1}^{\ell}(x_{i}^{+}\otimes a_{p_{s}})(x_{i}^{-}\otimes
1)^{\ell}=\prod_{s=1}^{\ell}(h_{i}\otimes a_{p_{s}})w_{\lambda}+Hw_{\lambda},\
\ \ell\geq\lambda(h_{i})+1,$
where $H$ is in the span of elements of the form $\prod_{s=1}^{r}(h_{i}\otimes
a_{p_{j_{s}}})$ with $r<\ell$, the Lemma follows by a simple induction on
$\ell$. ∎
### 6.6.
As a result of the Lemma we see that $\mathbf{A}_{\lambda}$ is spanned by the
image of the set
$\\{\prod_{i=1}^{n}\prod_{s=1}^{m_{i}}(h_{i}\otimes a_{i,s}):a_{i,s}\in
A_{+},a_{i,1}\leq\cdots\leq a_{i,m_{i}},i\in I,\ \
m_{i}\leq\lambda(h_{i})\\}.$
The proof that $\tau_{\lambda}$ is injective follows if we prove that the set
$\left\\{\bigotimes_{s=1}^{m_{1}}\operatorname{sym}^{1}_{\lambda}(a_{1,s})\bigotimes\cdots\bigotimes_{s=1}^{m_{n}}\operatorname{sym}^{n}_{\lambda}(a_{n,s}):a_{i,s}\in
A_{+},a_{i,1}\leq\cdots\leq a_{i,m_{i}},i\in I,\ \
m_{i}\leq\lambda(h_{i})\right\\}$
is linearly independent in $\mathbb{A}_{\lambda}$. Since the tensor product of
linearly independent sets is linearly independent it is enough to prove the
following. Let $N\in\mathbf{Z}_{+}$ and for $b_{1},\cdots,b_{N}\in A$ let
$\operatorname{sym}_{N}(b_{1}\otimes\cdots\otimes b_{N})=\sum_{\sigma\in
S_{N}}(b_{\sigma(1)}\otimes\cdots\otimes b_{\sigma(N)}).$
###### Lemma.
The elements
$\operatorname{sym}_{N}(a_{r_{1}}\otimes 1^{\otimes
N-1})\operatorname{sym}_{N}(a_{r_{2}}\otimes 1^{\otimes
N-1})\cdots\operatorname{sym}_{N}(a_{r_{m}}\otimes 1^{\otimes N-1}),1\leq
r_{1}\leq\cdots\leq r_{m},\ \ m\leq N$ (6.7)
are linearly independent in $A^{\otimes N}$.
###### Proof.
Set
$\mathbb{U}=\bigoplus_{0\leq m\leq N}A_{+}^{\otimes m}\otimes
1^{\otimes(N-m)},$
and let $\mathbf{p}:A^{\otimes N}\to\mathbb{U}$ be the canonical projection.
The projection onto $\mathbb{U}$ of the elements in (6.7) are
$\operatorname{sym}_{r_{m}}(a_{r_{1}}\otimes a_{r_{2}}\otimes\cdots
a_{r_{m}})\otimes 1^{N-m},\ \ 1\leq r_{1}\leq\cdots\leq r_{m},\ \ m\leq N$
and these are clearly linearly independent in $\mathbb{U}$ and the Lemma is
proved. ∎
## 7\. The fundamental Weyl modules
We use the notation of the previous sections freely. Throughout this section
we shall assume that $A$ is finitely generated. Theorem Theorem(i) applies and
we have bijections
$\max(\mathbf{A}_{\lambda})\to\Xi_{\lambda}\to\max(\mathbb{A}_{\lambda})$.
Recall that $\max\mathbb{A}_{\lambda}$ is the set of orbits of the group
$S_{\lambda}$ acting on $(\max A)^{\otimes r_{\lambda}}$. The orbits of
maximal size (i.e those coming from an element of $(\max A)^{\otimes
r_{\lambda}}$ with trivial stabilizer under the $S_{\lambda}$ action)
correspond under this bijection to the subset
$\Xi_{\lambda}^{\rm{ns}}=\\{\xi\in\Xi_{\lambda}:\xi(S)=\sum_{i\in
I}m_{i}\omega_{i},\ \ m_{i}\leq 1\ \forall\ \ S\in\max A\\}$
of $\Xi$. The group $S_{r_{\lambda}}$ also acts on $(\max A)^{\otimes
r_{\lambda}}$ by permutations and the orbits of this action can be naturally
identified with a subset of $\max\mathbb{A}_{\lambda}$. The orbit of points
with trivial stabilizer under the $S_{r_{\lambda}}$ action corresponds further
to the subset
${}_{1}\Xi_{\lambda}^{\rm{ns}}=\\{\xi\in\Xi_{\lambda}:\xi(S)\in\\{0,\omega_{1},\cdots,\omega_{n}\\},\
\ \forall\ \ S\in\max A\\},$
of $\Xi_{\lambda}^{ns}$. Clearly
$\Xi_{\omega_{i}}={}_{1}\Xi^{ns}_{\omega_{i}}.$
In this section we shall analyze the modules
$\mathbf{W}^{\lambda}_{A}M_{\xi}$, $\xi\in{}_{1}\Xi_{\lambda}^{\rm{ns}}$ when
$\mathfrak{g}$ is an algebra of classical type. By Theorem Theorem we see that
$\mathbf{W}^{\lambda}_{A}M_{\xi}\cong\bigotimes_{S\in\operatorname{supp}\xi}\mathbf{W}_{A}^{\xi(S)}M_{\xi_{S}},\
\ \operatorname{supp}\xi_{S}=\\{S\\},\ \ \xi_{S}(S)=\xi(S).$ (7.1)
This means that if $\xi\in{}_{1}\Xi^{\rm{ns}}_{\lambda}$, it is enough to
analyze the modules $\mathbf{W}^{\omega_{i}}_{A}M_{\xi}$, $i\in I$,
$\xi\in\Xi_{\omega_{i}}$.
### 7.1.
Assume from now on that $\mathfrak{g}$ is of type $A_{n}$, $B_{n}$, $C_{n}$ or
$D_{n}$. Assume also that the nodes of the Dynkin diagram of $\mathfrak{g}$
are numbered as in [B]. Define a subset $J_{0}$ of $I$ as follows:
$J_{0}=\begin{cases}I,\ \ \mathfrak{g}\ \ {\rm{of\ type}}\ A_{n},\ C_{n},\\\
\\{n\\},\ \ \mathfrak{g}\ \ {\rm{of\ type}}\ B_{n},\\\ \\{n-1,n\\},\ \
\mathfrak{g}\ \ {\rm{of\ type}}\ \ D_{n}.\end{cases}$
Given $m,k\in\mathbf{Z}_{+}$, let $\mathbf{c}(m)$ be the dimension of the
space of polynomials of degree $m$ in $k$–variables, i.e
$\mathbf{c}(m)=\\#\\{\mathbf{s}=(s_{1},\cdots,s_{k})\in\mathbf{Z}_{+}^{k}:s_{1}+\cdots+s_{k}=m\\}.$
For $S\in\max A$ and $i\in I$ let $\xi^{i}_{S}\in\Xi_{\omega_{i}}$ be given by
requiring $\operatorname{supp}\xi=S$.
###### Theorem.
Assume that $S\in\max A$ and that $\dim S/S^{2}=k$. We have an isomorphism of
$\mathfrak{g}$–modules,
$\displaystyle\mathbf{W}_{A}^{\omega_{i}}M_{\xi^{i}_{S}}\cong_{\mathfrak{g}}V(\omega_{i}),\
\ i\in J_{0},$ (7.2)
$\displaystyle\mathbf{W}_{A}^{\omega_{i}}M_{\xi^{i}_{S}}\cong_{\mathfrak{g}}\bigoplus_{\\{j:i-2j\geq
0\\}}V(\omega_{i-2j})^{\oplus\mathbf{c}(j)},\ \ i\notin J_{0}.$ (7.3)
###### Remark.
The theorem was proved when $A$ is the polynomial ring in one variable in
[C2],[CM].
### 7.2.
Before proving the theorem, we note the following. Let
$\dim_{\lambda}:\Xi_{\lambda}\to\mathbf{Z}_{+}$ be the function
$\xi\to\dim\mathbf{W}_{A}^{\lambda}M_{\xi}$.
###### Corollary.
Let $A$ be a smooth irreducible algebraic variety. The restriction of
$\dim_{\lambda}$ to ${}_{1}\Xi^{{\rm{ns}}}_{\lambda}$ is constant.
###### Proof.
Since $A$ is smooth and irreducible, it follows that $\dim S/S^{2}$ is
independent of $S$ and hence by Theorem Theorem we see that the corollary is
true for $\omega_{i}$. The general case now follows from (7.1). ∎
###### Remark.
In the special case when $A=\mathbf{C}[t]$ the function $\dim_{\lambda}$ is
constant on $\Xi_{\lambda}$. This was conjectured in [CP2] and proved there
for $\mathfrak{sl}_{2}$. It was later proved in [CL] for
$\mathfrak{sl}_{r+1}$, in [FoL] for algebras of type $A,D,E$. The general case
can be deduced by passing to the quantum group situation and using results in
[K], [BN]. No self–contained algebraic proof of this fact has been given for
the non–simply laced algebras.
However, it is not true that if $A$ is an arbitrary smooth irreducible
variety, then $\dim_{\lambda}$ is constant on $\Xi_{\lambda}^{\rm{ns}}$. As an
example take $\mathfrak{g}=\mathfrak{sl}_{3}$, $A=\mathbf{C}[t_{1},t_{2}]$ and
consider $\lambda=\omega_{1}+\omega_{2}$. Let $S,S^{\prime}$ be the maximal
ideals in $A$ corresponding to distinct points $(z_{1},z_{2},)$ and
$(z_{1}^{\prime},z_{2}^{\prime})$. Let $\xi,\xi^{\prime}\in\Xi_{\lambda}$ be
given by
$\xi(S)=\omega_{1},\ \ \xi(S^{\prime})=\omega_{2},\ \
\xi^{\prime}(S)=\lambda.$
Then by Theorem Theorem
$\mathbf{W}^{\lambda}_{A}M_{\xi}\cong_{\mathfrak{g}}V(\omega_{1})\otimes
V(\omega_{2}),$
and hence is nine–dimensional.
On the other hand the following argument proves that
$\mathbf{W}^{\lambda}_{A}(M_{\xi}^{\prime})$ is at least 10–dimensional.
Recall that
$V(\omega_{1}+\omega_{2})\cong_{\mathfrak{g}}\mathfrak{g}_{\operatorname{ad}}$
where $\mathfrak{g}_{\operatorname{ad}}$ is the adjoint representation of
$\mathfrak{g}$ and hence has dimension eight. Let $<\ ,\ >$ be the Killing
form of $\mathfrak{g}$. A relatively straightforward check shows that if we
set $W=\mathfrak{g}_{\operatorname{ad}}\oplus\mathbf{C}\oplus\mathbf{C}$ and
define an action of $\mathfrak{g}\otimes A$ on $W$ by
$(x\otimes f)(y,z,z^{\prime})=(f(z^{\prime}_{1},z^{\prime}_{2})[x,y],\
\frac{df}{dt_{1}}(z^{\prime}_{1},z^{\prime}_{2})<x,y>,\ \
\frac{df}{dt_{2}}(z^{\prime}_{1},z^{\prime}_{2})<x,y>),$
then $W$ is a quotient of $\mathbf{W}^{\lambda}_{A}(M_{\xi}^{\prime})$.
### 7.3.
The rest of the section is devoted to proving the theorem. We shall repeatedly
use the following
$(\mathfrak{h}\otimes S)(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})=0.$ (7.4)
Given $\alpha\in R^{+}$, let $\varepsilon_{i}(\alpha)\in\\{0,1,2\\}$ be the
coefficient of $\alpha_{i}$ in $\alpha$ and set
$\operatorname{ht}\alpha=\sum_{j\in I}\varepsilon_{j}(\alpha),\
\qquad\mathfrak{n}^{-}_{r}=\bigoplus_{\\{\alpha\in
R^{+}:\varepsilon_{i}(\alpha)=r\\}}\mathfrak{g}_{-\alpha}.$
It is a simple matter to check that
$[\mathfrak{n}^{-}_{0},\mathfrak{n}^{-}_{0}]=\mathfrak{n}^{-}_{0},\qquad[\mathfrak{n}^{-}_{0},\mathfrak{n}^{-}_{1}]=\mathfrak{n}^{-}_{1},\qquad[\mathfrak{n}_{1}^{-},\mathfrak{n}^{-}_{1}]=\mathfrak{n}_{2}^{-}.$
(7.5)
###### Lemma.
We have
$\displaystyle\left((\mathfrak{n}^{-}_{0}\otimes
A)\oplus(\mathfrak{n}^{-}_{1}\otimes S)\oplus(\mathfrak{n}^{-}_{2}\otimes
S^{2})\right)(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})=0.$
In particular $(\mathfrak{g}\otimes
S^{2})\mathbf{W}^{\omega_{i}}_{A}M_{\xi^{i}_{S}}=0$, i.e. $S^{2}\subset
K^{\omega_{i}}_{M_{\xi^{i}_{S}}}$.
###### Proof.
It is trivial that
$\mathfrak{n}^{+}(x^{-}_{j}\otimes A)(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})=0,\ \ j\neq i,\qquad\ \mathfrak{n}^{+}(x^{-}_{i}\otimes
S)(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})=0$
Since $\omega_{i}-\alpha_{j}\notin P^{+}$ for all $i\in I$, it follows by
elementary representation theory that
$(x^{-}_{j}\otimes A)(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})=0,\ \ j\neq
i,\qquad\ (x^{-}_{i}\otimes S)(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})=0.$
Using (7.5) we see that a straightforward induction on
$\operatorname{ht}\alpha$ proves the lemma. ∎
### 7.4.
We now prove by using Lemma 5.5 and Lemma Lemma that it suffices to prove
Theorem Theorem in the case when $A$ is the polynomial ring in finitely many
variables. For this, suppose that $B$ is a finitely generated algebra and let
$S_{B}$ a maximal ideal in $B$. Let $t_{1},\ldots,t_{k}\in S$ be such that the
images of these elements form a basis of $S_{B}/S_{B}^{2}$. Let
$A=\mathbf{C}[x_{1},\ldots,x_{k}]$, and define an algebra homomorphism
$A\longrightarrow B$ by extending the assignment $x_{i}\mapsto t_{i}.$ Let
$S_{A}$ be the ideal in $A$ generated by $x_{1},\cdots,x_{k}$. Clearly $S_{A}$
maps to $S_{B}$ and we have a homomorphism of algebras $\phi:A/S_{A}^{2}\to
B/S_{B}^{2}$. Moreover, since $t_{1},\cdots,t_{k}$ are linearly independent in
$S_{B}/S_{B}^{2}$ it follows that $\phi$ is injective. Further, since
$\dim A/S_{A}^{2}=\dim B/S_{B}^{2}=k+1,$
it follows that $\phi$ is an isomorphism of algebras. We now have
$\mathbf{W}_{B}^{\omega_{i}}M_{\xi^{i}_{S_{B}}}\cong\mathbf{W}_{B/S_{B}^{2}}^{\omega_{i}}M_{\xi^{i}_{S_{B}}}\cong\mathbf{W}_{A/(S_{A})^{2}}^{\omega_{i}}M_{\xi^{i}_{S_{A}}}\cong\mathbf{W}_{A}^{\omega_{i}}M_{\xi^{i}_{S_{A}}},$
where the first and last isomorphisms follow from Lemma 5.5 and the
isomorphism in the middle is induced by $\phi$.
### 7.5.
From now on we shall assume that $A=\mathbf{C}[t_{1},\dots,t_{k}]$ is the
polynomial ring in $k$ variables. Moreover since the theorem is proved for
$k=1$ in [C2],[CM], we shall assume that $k>1$. In addition we may assume that
$S$ is the maximal ideal generated by $t_{1},\cdots,t_{k}$. There is no loss
of generality in doing this for the following reason. Suppose that
$S^{\prime}$ is another maximal ideal corresponding to the point
$\mathbf{z}=(z_{1},\cdots,,z_{k})\in\mathbf{C}^{k}$. Consider the automorphism
of $\phi_{\mathbf{z}}:\mathfrak{g}\otimes A\to\mathfrak{g}\otimes A$ given by
$x\otimes t_{r}\to x\otimes(t_{r}-z_{r})$, $x\in\mathfrak{g}$, $1\leq r\leq
k$. It is not hard to check that
$\mathbf{W}^{\omega_{i}}_{A}M_{\xi^{i}_{S}}\cong\phi_{\mathbf{z}}^{*}\mathbf{W}^{\omega_{i}}_{A}M_{\xi^{i}_{S^{\prime}}}.$
### 7.6.
Let $A_{+}$ be the subspace of polynomials with constant term zero. Since
$\mathfrak{g}\otimes A_{+}$ is an ideal in $\mathfrak{g}\otimes A$, to prove
(7.2) it suffices to show that for all $\alpha\in R^{+}$ and $a\in A_{+}$,
$(x^{-}_{\alpha}\otimes a)(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})\in\mathbf{U}(\mathfrak{g})(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}}).$ (7.6)
Let $C=\mathbf{C}[t]$, where $t$ is an indeterminante. Consider the map
$\mathfrak{g}\otimes C\to\mathfrak{g}\otimes A$ given by $x\otimes t\to
x\otimes a$. By Proposition Proposition there exists a map of
$\mathfrak{g}\otimes C$–modules
$\mathbf{W}^{\omega_{i}}_{C}M_{\xi^{i}_{S}}\to\mathbf{W}^{\omega_{i}}_{A}M_{\xi^{i}_{S}}$.
Since the theorem is known for $C$, it follows that
$(x_{\alpha}^{-}\otimes t)(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})\in\mathbf{U}(\mathfrak{g})(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})\subset\mathbf{W}^{\omega_{i}}_{C}M_{\xi^{i}_{S}},$
which proves (7.6).
### 7.7.
The rest of the section is devoted to proving (7.3) and hence we may and will
assume that $\mathfrak{g}$ is of type $B_{n}$ or $D_{n}$. For $j\in I$, $j\geq
2$, set $\omega_{j}-\omega_{j-2}=\theta_{j}$. Then one checks easily [H]
$\theta_{j}\in R^{+},\
\quad\theta_{j-2}-\theta_{j}=\alpha_{j-3}+2\alpha_{j-2}+\alpha_{j-1},\quad\theta_{j}-\alpha_{r}\in
R^{+}\ \ \iff r=j.$
where we understand that $\alpha_{-1}=0$.
###### Proposition.
Let $i\in I$, $1\leq\ell,m\leq k$, and set
$v_{\ell}=(x^{-}_{\theta_{i}}\otimes t_{\ell})(w_{\omega_{i}}\otimes
M_{\xi_{S}^{i}})$. Then
$\displaystyle(\mathfrak{n}^{+}\otimes A)v_{\ell}=0,\ \
\qquad(\mathfrak{h}\otimes S)v_{\ell}=0.$ (7.7)
In particular, the $\mathfrak{g}\otimes A$–submodule of
$\mathbf{W}^{\omega_{i}}_{A}M_{\xi^{i}_{S}}$ generated by $v_{\ell}$ is a
quotient of $\mathbf{W}^{\omega_{i-2}}_{A}M_{\xi^{i-2}_{S}}$. Further, we have
$\displaystyle(x^{-}_{\theta_{i-2}}\otimes
t_{m})v_{\ell}=(x^{-}_{\theta_{i-2}}\otimes t_{\ell})v_{m}.$ (7.8)
###### Proof.
Note that $(\mathfrak{n}^{+}\otimes S)v_{\ell}$ and $(\mathfrak{h}\otimes
S)v_{\ell}$ are both contained in $(\mathfrak{g}\otimes
S^{2})(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})$ and hence by Lemma Lemma
$(\mathfrak{n}^{+}\otimes S)v_{\ell}\ \ =0\ \ =\ (\mathfrak{h}\otimes
S)v_{\ell}.$
Since $S$ is maximal, (7.7) follows if we prove that
$(\mathfrak{n}^{+}\otimes 1)v_{\ell}=0.$
Since
$[x^{+}_{j},x^{-}_{\theta_{i}}]=0,\ j\neq i,\ \ {\rm{and}}\ \
\varepsilon_{i}(\theta_{i}-\alpha_{i})=1,$
we see that Lemma Lemma gives $(x_{j}^{+}\otimes 1)v_{\ell}=0$ for all $j\in
I$. The second statement of the proposition is now clear. Hence we have by
Lemma Lemma that
$(x^{-}_{\alpha}\otimes S)v_{\ell}=0\ \qquad{\rm{if}}\ \
\varepsilon_{i-2}(\alpha)\neq 2.$
Writing
$x^{-}_{\theta_{i-2}}=[x^{-}_{i-2},[x^{-}_{\alpha_{i-3}+\alpha_{i-2}+\alpha_{i-1}},x^{-}_{\theta_{i}}]],$
and using Lemma Lemma we get
$\displaystyle(x^{-}_{\theta_{i-2}}\otimes
t_{m})v_{\ell}=(x^{-}_{\theta_{i-2}}\otimes t_{m})(x^{-}_{\theta_{i}}\otimes
t_{\ell})(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})=(x^{-}_{\theta_{i}}\otimes
t_{\ell})(x^{-}_{\theta_{i-2}}\otimes t_{m})(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})$ $\displaystyle=(x^{-}_{\theta_{i}}\otimes
t_{\ell})x^{-}_{i-2}x^{-}_{\alpha_{i-3}+\alpha_{i-2}+\alpha_{i-1}}(x^{-}_{\theta_{i}}\otimes
t_{m})(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})$
$\displaystyle=(x^{-}_{\theta_{i-2}}\otimes
t_{\ell})v_{m}+X(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}}),$
where $X$ is a linear combination of the elements
$\displaystyle
x^{-}_{i-2}x^{-}_{\alpha_{i-3}+\alpha_{i-2}+\alpha_{i-1}}(x^{-}_{\theta_{i}}\otimes
t_{\ell})(x^{-}_{\theta_{i}}\otimes t_{m}),\ \
x_{i-2}^{-}(x^{-}_{\theta_{i}+\alpha_{i-3}+\alpha_{i-2}+\alpha_{i-1}}\otimes
t_{\ell})(x^{-}_{\theta_{i}}\otimes t_{m}),$ $\displaystyle
x^{-}_{\alpha_{i-3}+\alpha_{i-2}+\alpha_{i-1}}(x^{-}_{\theta_{i}+\alpha_{i-2}}\otimes
t_{\ell})(x^{-}_{\theta_{i}}\otimes t_{m}).$
But by Lemma Lemma all these terms act as zero on $(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})$, since $(x^{-}_{\theta_{i}}\otimes
t_{m})(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})$ generates a quotient of
$\mathbf{W}_{A}^{\omega_{i-2}}M_{\xi_{S}^{i-2}}$ and
$\varepsilon_{i-2}(\theta_{i}+\alpha_{i-2}+\alpha_{i-1}+\alpha_{i-3})=1=\varepsilon_{i-2}(\theta_{i}+\alpha_{i-2}).$
∎
The following is now immediate.
###### Corollary.
Given $i,\ell\in I$ with $2\ell\leq i$, and $r_{s}\in\\{1,\cdots,k\\}$, $1\leq
s\leq\ell$, the elements,
$v(r_{1},\cdots,r_{\ell})=(x^{-}_{\theta_{i-2\ell}}\otimes
t_{r_{\ell}})\cdots(x^{-}_{\theta_{i-2}}\otimes
t_{r_{2}})(x^{-}_{\theta_{i}}\otimes t_{r_{1}}).(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})$
generate a submodule of $\mathbf{W}_{A}^{\omega_{i}}M_{\xi^{i}_{S}}$ which is
a quotient of $\mathbf{W}_{A}^{\omega_{i-2\ell}}M_{\xi_{S}^{i-2\ell}}$.
Moreover if $\sigma\in S_{\ell}$, we have,
$v(r_{1},\cdots,r_{\ell})=v(r_{\sigma(1)},\cdots,r_{\sigma(\ell}).$
### 7.8.
Suppose that $\alpha\in R^{+}$ is such that $\varepsilon_{i}(\alpha)=2$. Then
we can write $\alpha=\gamma+\beta+\theta_{i}$ for some $\beta,\gamma\in R^{+}$
with $\varepsilon_{i}(\beta)=\varepsilon_{i}(\gamma)=0$. This implies that
$x^{-}_{\alpha}=c[x^{-}_{\beta},[x^{-}_{\gamma},x^{-}_{\theta_{i}}]],$ for
some non–zero $c\in\mathbf{C}$ and hence
$(x^{-}_{\alpha}\otimes t_{\ell})(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})=c[x^{-}_{\beta},[x^{-}_{\gamma},x^{-}_{\theta_{i}}\otimes
t_{\ell}]](w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})\in\mathbf{U}(\mathfrak{g})(x^{-}_{\theta_{i}}\otimes
t_{\ell})(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}}).$
Proposition Proposition now gives,
$\mathbf{W}_{A}^{\omega_{i}}M_{\xi^{i}_{S}}=\mathbf{U}(\mathfrak{g})(w_{\omega_{i}}\otimes
M_{\xi^{i}_{S}})\oplus\sum_{\ell=1}^{k}\mathbf{U}(\mathfrak{g}\otimes
A)(x^{-}_{\theta_{i}}\otimes t_{\ell})(w_{\omega_{i}}\otimes M_{\xi^{i}_{S}})$
as $\mathfrak{g}$–modules. Using Corollary Proposition we find
$\mathbf{W}_{A}^{\omega_{i}}M_{\xi^{i}_{S}}=\mathbf{U}(\mathfrak{g})(w_{\omega_{i}}\otimes
M_{\xi_{S}^{i}})\bigoplus_{0\leq 2l\leq i}\;\;\left(\sum_{0\leq
r_{1}\leq\cdots\leq
r_{\ell}}\mathbf{U}(\mathfrak{g})v(r_{1},\cdots,r_{\ell})\right),$
which proves that
$\operatorname{Hom}_{\mathfrak{g}}(V(\mu),\mathbf{W}^{\omega_{i}}_{A}M_{\xi_{S}^{i}})=0,\
\ \mu\neq i-2j,\ \
\dim\operatorname{Hom}_{\mathfrak{g}}(V(\omega_{i-2j}),\mathbf{W}^{\omega_{i}}_{A}M_{\xi_{S}^{i}})\leq\mathbf{c}(j).$
### 7.9.
To complete the proof it suffices to prove that the elements
$v(r_{1},\cdots,r_{l})$ are linearly independent for all $i,\ell\in I$ with
$2\ell\leq i$ and $r_{s}\in\\{1,\cdots,k\\}$, $1\leq s\leq\ell$. We do this as
in [CM] by explicitly constructing a module which is a quotient of
$\mathbf{W}^{\omega_{i}}_{A}M_{\xi_{S}^{i}}$ and where these elements are
linearly independent. Suppose that $V_{s}$ for $0\leq s\leq\ell$ are
$\mathfrak{g}$–modules such that
$\operatorname{Hom}_{\mathfrak{g}}(\mathfrak{g}\otimes V_{s},V_{s+1})\neq 0,\
\ \operatorname{Hom}_{\mathfrak{g}}(\wedge^{2}(\mathfrak{g})\otimes
V_{s},V_{s+1})=0.$ (7.9)
Set $V=\oplus_{s=0}^{\ell}V_{s}$ and fix non–zero elements
$p_{s}\in\operatorname{Hom}_{\mathfrak{g}}(\mathfrak{g}\otimes V_{s},V_{s+1})$
for $0\leq s\leq k$. Define a $\mathfrak{g}\otimes A$–module structure on
$V\otimes A$ by:
$\displaystyle(x\otimes 1)(v\otimes a)=xv\otimes a,\ \ (x\otimes
t_{r})(v\otimes a)=p_{s}(x\otimes v)\otimes at_{r},\ \ x\in\mathfrak{g},\ \
a\in A\ \ 1\leq r\leq k,$ $\displaystyle(x\otimes S^{2})(v\otimes a)=0.$
To see that this is an action, the only non–trivial part is to notice that,
$\displaystyle[x\otimes t_{r},y\otimes t_{m}](v\otimes c)=p_{s+1}(x\otimes
p_{s}(y\otimes v))\otimes t_{r}t_{m}c-p_{s+1}(y\otimes p_{s}(x\otimes
v))\otimes t_{r}t_{m}c,$ $\displaystyle=p_{s+1}(p_{s}\otimes 1)((x\otimes
y-y\otimes x)\otimes v)\otimes t_{r}t_{\ell}c=0,$
where the last equality follows by noticing that $p_{s+1}(p_{s}\otimes
1)\in\operatorname{Hom}_{\mathfrak{g}}(\mathfrak{g}\otimes\mathfrak{g}\otimes
V_{s},V_{s+1})$ and using (7.9).
It was shown in [CM] that the modules $V(\omega_{i-2s})$, $0\leq 2s\leq i$
satisfy (7.9) and also that
$p_{s}(x^{-}_{\theta_{i-2s-2}}\otimes v_{\omega_{i-2s}})=v_{\omega_{i-2s-2}}.$
and hence we can apply the preceding construction to this family of modules.
Consider the $\mathbf{U}(\mathfrak{g}\otimes A)$–module $\bar{W}$ generated by
$v_{\omega_{i}}\otimes 1$. It is clear that
$(\mathfrak{n}^{+}\otimes A)(v_{\omega_{i}}\otimes 1)\ =\ 0\
=(\mathfrak{h}\otimes S)(v_{\omega_{i}}\otimes 1),$
since $\omega_{i-2}<\omega_{i}.$ Hence $\bar{W}$ is a quotient of
$\mathbf{W}^{\omega_{i}}_{A}M_{\xi_{S}^{i}}$. Moreover, it is simple to check
now that
$(x^{-}_{\theta_{i-2\ell}}\otimes
t_{r_{\ell}})\cdots(x^{-}_{\theta_{i-2}}\otimes
t_{r_{2}})(x^{-}_{\theta_{i}}\otimes
t_{r_{1}}).v_{\omega_{i}}=v_{\omega_{i-2\ell}}\otimes t_{r_{1}}\cdots
t_{r_{\ell}}.$
Since these elements are manifestly linearly independent the result follows.
## References
* [BN] J. Beck and H. Nakajima, _Crystal bases and two-sided cells of quantum affine algebras_ , Duke Math. J. 123 (2004), no. 2, 335–402.
* [B] N. Bourbaki, _El ements de math ematique: Groupes et algèbres de Lie_ , Hermann, Paris, 1958.
* [C1] V. Chari, _Integrable representations of affine Lie-algebras_ , Invent. Math. 85 (1986), 317 -335.
* [C2] V. Chari, _On the fermionic formula and the Kirillov-Reshetikhin conjecture_ , Internat. Math. Res. Notices (2001), 12, 629–654.
* [CG1] V. Chari and J. Greenstein, Current algebras, highest weight categories and quivers, Adv. in Math. 216 (2007), no. 2, 811–840.
* [CG2] V. Chari and J. Greenstein, _A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras_ , preprint, RT/08081463.
* [CL] V. Chari and S. Loktev, _Weyl, Demazure and fusion modules for the current algebra of $sl_{r+1}$ _, Adv. in Math., 207, (2006), Issue 2, 928–960.
* [CM] V. Chari and A. Moura, _The restricted Kirillov-Reshetikhin modules for the current and twisted current algebras_ , Comm. Math. Phys. 266 (2006), 431 -454.
* [CP1] V. Chari and A. Pressley, _New unitary representations of loop groups_ , Math. Ann. 275 (1986), 87 -104.
* [CP2] V. Chari and A. Pressley, _Weyl modules for classical and quantum affine algebras_ , Represent. Theory 5 (2001), 191–223 (electronic).
* [FL] B. Feigin and S. Loktev, _Multi-dimensional Weyl modules and symmetric functions_ , Comm. Math. Phys. 251 (2004) pp. 427–445
* [FoL] G. Fourier and P. Littelmann, _Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions_ Advances in Mathematics, 211(2007) 566 -593. 567.
* [G] H. Garland, _The arithmetic theory of loop algebras_ , J. Algebra, 53, (1978), 480 -551.
* [H] J.E.Humphreys, _Introduction to Lie Algebras and Representation Theory_ , Springer, 1968
* [K] M. Kashiwara, _On level-zero representations of quantized affine algebras_ , Duke Math. J. 112 (2002), no. 1, 117- 175
* [Ku] S. Kumar, private correspondence
* [L] M. Lau, _Representations of multi–loop algebras,_ math. RT/ arXiv:0811.2011v2.
* [N] H. Nakajima, _Quiver varieties and finite-dimensional representations of quantum affine algebras_ , J. Amer. Math. Soc. 14 (2001), 145 -238,
* [R] S.E. Rao, _On representations of loop algebras,_ Comm. Algebra 21 (1993), 2131 -2153.
|
arxiv-papers
| 2009-06-11T19:13:21 |
2024-09-04T02:49:03.298750
|
{
"license": "Public Domain",
"authors": "Vyjayanthi Chari, Ghislain Fourier, Tanusree Khandai",
"submitter": "Ghislain Fourier",
"url": "https://arxiv.org/abs/0906.2014"
}
|
0906.2336
|
# Spin amplitude modulation driven magnetoelectic coupling in the new
multiferroic FeTe2O5Br
M. Pregelj Institute ”Jozef Stefan”, Jamova 39, 1000 Ljubljana, Slovenia O.
Zaharko [email protected] Laboratory for Neutron Scattering, ETHZ & PSI,
CH-5232 Villigen, Switzerland A. Zorko Institute ”Jozef Stefan”, Jamova 39,
1000 Ljubljana, Slovenia Z. Kutnjak Institute ”Jozef Stefan”, Jamova 39,
1000 Ljubljana, Slovenia P. Jeglič Institute ”Jozef Stefan”, Jamova 39, 1000
Ljubljana, Slovenia P. J. Brown Institut Laue-Langevin, 156X, 38042 Grenoble
Cedex, France M. Jagodič Institute of Mathematics, Physics and Mechanics,
Jadranska 19, 1000 Ljubljana, Slovenia Z. Jagličić Institute of Mathematics,
Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia Faculty of
Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, 1000
Ljubljana, Slovenia H. Berger Institute of Physics of Complex Matter, EPFL,
1015 Lausanne, Switzerland D. Arčon [email protected] Institute ”Jozef
Stefan”, Jamova 39, 1000 Ljubljana, Slovenia Faculty of Mathematics and
Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
###### Abstract
Magnetic and ferroelectric properties of layered geometrically frustrated
cluster compound FeTe2O5Br were investigated with single-crystal neutron
diffraction and dielectric measurements. Incommensurate transverse amplitude
modulated magnetic order with the wave vector $\bf{q}$=(${\frac{1}{2}}$,
0.463, 0) develops below $T_{N}=10.6(2)\,{\rm K}$. Simultaneously, a
ferroelectric order due to exchange striction involving polarizable Te4+ lone-
pair electrons develops perpendicular to ${\bf q}$ and to Fe3+ magnetic
moments. The observed magnetoelectric coupling is proposed to originate from
the temperature dependent phase difference between neighboring amplitude
modulation waves.
###### pacs:
75.25.+z, 75.80.+q
††preprint: APS
Switching ferroelectric polarization by magnetic field Kimura or, conversely,
controling magnetic order with the electric field Lottermoser in
magnetoelectric materials has been for a long time hampered by a very small
magnitude of the magnetoelectric coupling. Recently, strong magnetoelectric
coupling has been discovered in several multiferroic oxides ($R$MnO3,
$R$Mn2O5, Ni3V2O8, $\ldots$ , $R$ = rare earth) where ferroelectricity exists
only in a magnetically ordered state Nmat07 ; Fiebig05 ; ScottNature ; Kimura
; Lottermoser . In these systems, spiral magnetic order, such as cycloidal or
transverse conical structures Kasuga , breaks the inversion symmetry and
removes strict symmetry restrictions for the existence of the magnetoelectric
coupling. Since spiral order often results from magnetic frustration, the
current focus is on materials with geometrically frustrated lattices. However,
it remains to be seen whether the strong magnetoelectric effect is restricted
to spiral magnetic structures or it can be found also in other spatially
modulated magnetic arrangements.
In recent years several geometrically frustrated spin-cluster oxyhalide
compounds $M$-Te-O-$X$ ($M$ = Cu, Ni, Fe; $X$ = Cl, Br, I) have been
synthesized. Because of their reduced magnetic dimensionality and frustrated
lattices they frequently exhibit a complex magnetic order, having low magnetic
symmetry. Moreover, these systems contain Te4+ ions with lone-pair electrons
($5s^{2}5p^{0}$), which are highly polarizable LonePair . Thus the
$M$-Te-O-$X$ family represents a new class of materials, where magnetic and
polar order may coexist. We have focused our investigations on FeTe2O5Br with
a crystal structure that implies magnetic frustration Becker . This system
crystallizes in a monoclinic unit cell (space group $P21/c$) and adopts a
layered structure. The layers, which are stacked along the crystal
$a^{*}$-axis, consist of triangularly arranged [Fe4O16]20- clusters linked by
[Te4O10Br2]6- units. Within each iron tetramer cluster there are two
crystallographically non-equivalent Fe3+ ($S$ = 5/2) ions (Fe1 and Fe2 on
4($e$) sites) coupled through competing antiferromagnetic exchange
interactions. In this letter we show that below the Neel transition
temperature $T_{N}=10.6\,{\rm K}$ the Fe3+ magnetic moments order almost
collinearly with an incommensurate amplitude modulation. A spontaneous
electric polarization associated with the polarizable Te4+ lone-pair electrons
appears simultaneously with the long-range magnetic order. We propose that the
phase difference between coupled modulation waves is responsible for the
magnetoelectric effect in FeTe2O5Br and possibly also in other incommensurate
amplitude modulated magnetic structures.
Single crystals of FeTe2O5Br were grown by standard chemical vapor phase
method. Single-crystal X-ray diffraction measurements ($\lambda=$0.64 Å) were
performed at the BM01A Swiss-Norwegian Beamline of ESRF, France using closed-
cycle He cryostat mounted on a six-circle kappa diffractometer KUMA. Data sets
collected in the temperature range 4.5 to 35 K were refined using the SHELXL
program SHELXL . Neutron integrated intensities were collected on a $5\times
4\times 1$ mm3 single crystal at 5 K on the single crystal diffractometer
TriCS ($\lambda$ =2.32 Å) at the Swiss Neutron Spallation Source, Switzerland.
Spherical neutron polarimetry measurements on a $7\times 5\times 1.6$ mm3
single crystal were carried out at 1.8 K with CRYOPAD II installed on the IN20
spectrometer ($\lambda$=2.34 Å) at the Institute Laue-Langevin, France. The
crystal was mounted with the $c$-axis perpendicular to the scattering plane.
The complex dielectric constant
$\epsilon^{*}(T,\omega)=\epsilon^{\prime}(T,\omega)-i\epsilon^{\prime\prime}(T,\omega)$
was measured as a function of temperature and frequency with the HP4282A
precision LCR meter. The quasistatic polarization $P$ was determined by
electrometer charge accumulation measurements eps1 ; eps2 in a field cooling
run (a bias field of 10 kV/cm).
Table 1: Neutron polarization matrices $P_{ij}$ ($i$ \- incoming, $j$ \- outcoming component of polarization) for two representative reflections measured at $T=1.8\,{\rm K}$. ${h~{}~{}~{}~{}~{}~{}~{}~{}k~{}~{}~{}~{}~{}~{}~{}~{}l}$ | ${P_{i}}$ | ${P_{ix}}$ | ${P_{iy}}$ | ${P_{iz}}$
---|---|---|---|---
${{\frac{1}{2}}}$ | -0.463 | 0 | $x$ | -0.85(2) | 0.05(1) | 0.04(1)
| | | $y$ | 0.03(1) | 0.83(1) | -0.09(1)
| | | $z$ | -0.00(1) | -0.10(1) | -0.77(1)
${{\frac{3}{2}}}$ | 1.537 | 0 | $x$ | -0.927(4) | 0.05(1) | 0.01(1)
| | | $y$ | 0.01(1) | 0.823(6) | 0.34(1)
| | | $z$ | -0.04(1) | 0.38(1) | -0.843(6)
Figure 1: The agreement between experimental and calculated quantities E:
(left) components of neutron polarization matrices E=$P$ and (right) magnetic
structure factors E=$F$. The reliability factors are defined as:
R1=$\Sigma\Delta E/\Sigma E$ and $\chi^{2}=(\Delta
E)^{2}/(N_{observables}-N_{parameters})$.
In FeTe2O5Br three-dimensional long-range magnetic ordering sets in at
$T_{N}=10.6(2)\,{\rm K}$, where a pronounced change in the temperature
dependence of $\chi$ is evident Becker . Our neutron diffraction measurements
reveal that the magnetic reflections emerge at the incommensurate positions
described by the wave vector $\bf{q}$=(${\frac{1}{2}}$, 0.463, 0). Close
inspection of polarization matrices obtained from neutron polarimetry
measurements (Table 1) indicates that the magnetic arrangement is neither a
spiral, nor a cycloid or strongly canted. The absence of the $P_{yx}$ and
$P_{zx}$ components and almost full polarization of the scattered beam implies
that chiral magnetic scattering is negligible. The off-diagonal components
$P_{yz}$ and $P_{zy}$ increase with increasing $h$ or $k$ suggesting a small
$c$-component of magnetic moment.
Table 2: Parameters of the magnetic structure deduced from neutron diffraction experiments. The sites Fe12-Fe14 are obtained from Fe11 ($0.1184(6)$, $-0.001(1)$, $-0.0243(7)$) and Fe22-Fe24 from Fe21 ($0.9386(6)$, $0.296(1)$, $0.8568(6)$) by symmetry elements $2_{1y}$, $i$, and $2_{1y}i$. Angles $\theta$ and $\phi$, which describe the orientation of the iron magnetic moments, are defined with respect to the $a^{*}bc$ coordinate system. Additionally, each spin has individual phase $\psi_{kl}$ [deg], where index $k$ =1, 2 counts the sites and the second index $l$ = 1-4 counts the atoms within the site. | $\theta$ | $\phi$ | $\psi_{11}$ | $\psi_{12}$ | $\psi_{13}$ | $\psi_{14}$
---|---|---|---|---|---|---
Fe11-14 | 100(1) | -52(3) | 0 | 55(5) | 17(4) | 260(10)
| | | $\psi_{21}$ | $\psi_{22}$ | $\psi_{23}$ | $\psi_{24}$
Fe21-24 | 100(1) | -45(3) | 10(5) | 113(5) | -10(11) | 274(10)
Figure 2: Low temperature magnetic structure. Two different colors of arrows
are used for the two sites, the Fe3+ ions are labeled as in table 2, the
tetramers are shown schematically.
The combined refinement of polarization components and integrated magnetic
intensities (25 and 41 independent reflections, respectively) using the CCSL
code CCSL yields excellent agreement between the experimental and calculated
quantities (Fig. 1). The best solution is the amplitude modulated model
$S(i,k,l)=S_{0}\cos({\bf{q}}\cdot{\bf{r}}_{i}+\psi_{kl})$ with ${\bf{r}}_{i}$
being the vector defining the origin of the $i$-th cell. The modulation
amplitude $S_{0}=4.02(9)\,\mu_{B}$ is the same for all iron sites in the unit
cell, though each atom has its individual phase $\psi_{kl}$ (Table 2).
Magnetic moments on the same site in adjacent cells are collinear (Fig. 2) and
almost orthogonal to the wave vector $\bf{q}$, but their directions on Fe1 and
Fe2 sites are inclined at a small angle 7(3) deg (Table 2). There are two
equally populated domains related by the $2_{1y}$ axis. We note that the
incommensurate long-range magnetic order in FeTe2O5Br is most probably due to
competing interactions within the geometrically frustrated iron tetramers.
Evidently, the magnetic structure has no inversion center. This removes the
symmetry restriction for the coexistence of ferroelectric and magnetic order.
We therefore decided to measure the temperature dependence of $\epsilon$ and
spontaneous polarization $P$. An extremely sharp peak in $\epsilon^{\prime}$
at $T_{N}=10.5(1)\,{\rm K}$ (Fig. 3a) announces a transition to a long-range
ferroelectric state. At the same time, $\epsilon^{\prime\prime}$ is very small
and frequency independent, proving intrinsic nature of the observed
transition. The ferroelectric state is unambiguously confirmed by the
emergence of $P$ at $T_{N}$ (Fig. 3b) and its reversal with the electric field
(inset to Fig. 3a). $P$ is the largest along the crystal $c-$axis,
$P(c)=8.5(2)\mu{\rm C/m}^{2}$. It is almost an order of magnitude smaller
along $a^{*}$, $P(a^{*})=1.0(1)\,\mu{\rm C/m}^{2}$, while for the $b$
direction it is below the sensitivity of our experimental equipment. Comparing
the temperature dependence of $P$ to the intensity of the magnetic
(${\frac{1}{2}}$, 1.537, 0) peak, $I$, it is obvious that the two transitions
coincide precisely (Fig. 3b). When applying the magnetic field along the
$a^{*}$ direction both the Neel-transition and the ferroelectric-transition
temperatures simultaneously decrease to $T_{N}=9.4(3)$ K in the 9 T magnetic
field. This strong magnetic filed dependence provide additional evidence for
the magnetoelectric coupling in FeTe2O5Br .
The phenomenological explanation for the occurrence of magnetoelectric effect
in incommensurate helical or spiral magnetic phases has been given with
thermodynamic potential terms type
${\bf{P}}\cdot\left[{\bf{M}}(\nabla\cdot\bf{M})-({\bf{M}}\cdot\nabla)\bf{M}\right]$
Most . For our magnetic structure (Table II) we calculate that $P$ should lay
in the $ab$ plane in striking contrast to the experimentally observed $P(c)$
component. We next extended calculations by additional
${\bf{P}}\cdot\nabla\left({\bf{M}}^{2}\right)$ term, which is important when
$P$ is a sum of homogeneous and spatially modulated contributions Bet .
However, this additional term also cannot reproduce the correct ${\bf{P}}$
direction. Hence, it appears that coupling terms, which work very well for
helical or spiral magnetic orderings, cannot explain the appearance and the
correct direction of the ferroelectric polarization in FeTe2O5Br.
Figure 3: (a) Temperature dependence of the change in the dielectric constant
$\Delta\epsilon^{\prime}=\epsilon^{\prime}(T)-\epsilon^{\prime}(14K)$ measured
for $E||c$. Inset: Ferroelectric hysteresis loop measured at $T=5$ K. (b)
Temperature dependence of the spontaneous electric polarization, $P$, for
$E||c$ (open circles, right scale) and the intensity of the (${\frac{1}{2}}$,
1.537, 0) neutron diffraction magnetic peak, $I$ (solid circles, left scale).
$I$ and $P$ calculated from Eq. (1) are presented with solid and dashed line
respectively for $\beta$ = 0.15. Inset: A linear correlation between
$\sqrt{I}$ and $P$. Table 3: Results of representation analysis for
${\bf{q}}$=(${\frac{1}{2}}$, 0.463, 0) in $P2_{1}/c$. Top: Irreducible
representations (IRR), bottom: complex basis vectors of magnetic moments for
atoms 1 (x, y, z) and 2 (-x, y+1/2, -z+1/2) from the same orbit. $\eta=cos(\pi
q_{y}),\epsilon=sin(\pi q_{y})$.
IRR | (1$\mid$0) | ($2_{1y}\mid 00{\frac{1}{2}}$)
---|---|---
$\Gamma_{1}$ | 1 | $\eta$
$\Gamma_{2}$ | 1 | -$\eta$
Irrep | Atom | Re | Im
---|---|---|---
$\Gamma_{1}$ | 1 | 1 0 0 | 0 1 0 | 0 0 1 | 0 0 0 | 0 0 0 | 0 0 0
| 2 | -$\eta$ 0 0 | 0 $\eta$ 0 | 0 0 -$\eta$ | $\epsilon$ 0 0 | 0 -$\epsilon$ 0 | 0 0 $\epsilon$
$\Gamma_{2}$ | 1 | 1 0 0 | 0 1 0 | 0 0 1 | 0 0 0 | 0 0 0 | 0 0 0
| 2 | $\eta$ 0 0 | 0 -$\eta$ 0 | 0 0 $\eta$ | -$\epsilon$ 0 0 | 0 $\epsilon$ 0 | 0 0 -$\epsilon$
In order to better understand the magnetoelectric coupling in FeTe2O5Br we
have performed representation analysis. The star of the wave vector is formed
by the two vectors ${\bf{q}}$ and $-{\bf{q}}$, defining the little group,
which is composed of two elements: identity $1$ and two-fold screw axis
$2_{1y}$. It has two one-dimensional irreducible representations, $\Gamma_{1}$
and $\Gamma_{2}$ and the 4($e$) sites split into two orbits (Table 3). Since
the refined phase shift between the two magnetic moments from the same orbit
(Table 2) differs from the $\pi q_{y}$ = 83 deg value expected from the
symmetry relations, we conclude that our magnetic model is a combination of
both $\Gamma_{1}$ and $\Gamma_{2}$. The important coupling term, which already
takes into account observed orientations of ${\bf{P}}$ and Fe3+ magnetic
moments as well as the symmetry operations of the little group, is written as
$V=i\sum_{\alpha,\beta}\varepsilon_{\alpha\beta}\left(S_{\alpha}({\bf{q}},1)S^{*}_{\beta}({\bf{q}},2)-S^{*}_{\alpha}({\bf{q}},1)S_{\beta}({\bf{q}},2)\right)P_{c}\,.$
(1)
Here $\varepsilon_{\alpha\beta}$ is the magnetoelectric coupling tensor,
$\alpha,\beta=x,y$ and $S_{\alpha}({\bf{q}},i)$ is the Fourier component of
the magnetic moments for Fe atoms $i=1,2$ (Table III). For each irreducible
representation we define a complex magnetic order parameter, whose magnitude
in the vicinity of the phase transition can be described with the simple power
law ansatz $(T_{N}-T)^{\beta}$. Phase difference between the two order
parameters define the phases of individual amplitude modulation waves
$\psi_{kl}$ (Table II). The temperature dependence of $I$ and $P$ is simulated
(Fig. 3b) by assuming temperature dependent $\psi_{kl}$ approaching low-
temperature values obtained from the neutron diffraction experiments. The
agreement with the experiment is much worse, if $\psi_{kl}$ are kept constant.
The above analysis suggests that sliding of the individual amplitude
modulation waves, which also removes the center of inversion at the magnetic
phase transition, is responsible for the magnetoelectric effect in FeTe2O5Br.
Opposed to the $P\propto I$ dependence reported for representative
magnetically incommensurate systems Fox ; Yasui ; Kenzelmann07 we find here
the unusual proportionality between $\sqrt{I}$ and $P$ (inset to Fig. 3b).
Similar dependence in the low-temperature incommensurate spiral phase of
Ni3V2O8 NiVO was explained with the saturation of the high-temperature
magnetic order parameter already in the paraelectric phase. In contrast, the
observed $P\propto\sqrt{I}$ scaling in FeTe2O5Br is reproduced within our
model as a direct consequence of the temperature dependence of the amplitude
modulation wave phases.
Figure 4: Variation of the selected interatomic distances in the temperature
range 4.5 K - 35 K from single crystal x-ray diffraction. The labeling of the
atoms corresponds to Ref.Becker, .
To shed some additional light on microscopic picture of ferroelectricity and
the magnetoelectric coupling we performed low-temperature single-crystal
synchrotron X-ray diffraction experiments. On cooling through the magnetic
transition the deviations from the high-temperature crystallographic symmetry
are very small and bellow the resolution of our XRD experiment. However,
clearly distinguishable changes of the Fe-Te interatomic distances (Fig. 4)
can be seen. This finding is important, because (i) Te4+ ions bridge the
intercluster exchange interactions and (ii) Te4+ ions have lone-pair
electrons. The observed structural anomalies therefore suggest the
polarization of the Te4+ lone-pair electrons and may thus explain the
ferroelectricity in the magnetic phase. We note that tetramer Fe-O interatomic
distances also change slightly at the magnetic transition implying that the
coupling between polar and magnetic order parameters is likely mediated
through Fe-O-Te-O-Fe intercluster exchange. The standard spin-current SC and
”inverse Dzyaloshinskii-Moriya” IDM models developed for spiral magnetic
structures are unlikely to be active in FeTe2O5Br, since magnetic moments vary
in amplitude and not in direction along ${\bf q}$. Alternatively, exchange-
striction model was frequently applied to magnetoelectrics with collinear
magnetic order ES ; ES1 ; ES2 ; ES3 . If exchange-striction model applies to
FeTe2O5Br then the above coupling term (Eq. (1)) suggests that the spin phonon
coupling is provoked by the difference in the individual phases of spin
modulation waves. Additional experimental and theoretical investigations are
necessary to validate this suggestion.
In summary, we have discovered simultaneous emergence of ferroelectric and
magnetic order in FeTe2O5Br in the state with nearly transverse amplitude
modulated incommensurate magnetic structure described by the wave vector
$\bf{q}$=(${\frac{1}{2}}$, 0.463, 0). The ferroelectricity is ascribed to the
polarization of Te4+ lone-pair electrons. The magnetoelectric effect and the
unusual temperature dependence of the magnetic and ferroelectric properties
are explained with the sliding of neighbouring amplitude modulation waves
opening the possibility for the exchange-striction in the Fe-O-Te-O-Fe
intercluster exchange bridges. Our results suggest to look for new
magnetoelectrics in the vast family of $M$-$T$-O-$X$ compounds ($M$ = Cu, Ni,
Fe; $X$ = Cl, Br, I, $T$ = Te, Se, Sb, Bi, Pb), as they frequently posses
strong magnetic frustration complemented by the presence of $T$ ions with
lone-pair electrons.
We acknowledge fruitful discussions with J.F. Scott and M. Kenzelmann. We
thank Ya. Filinchuk and D. Chernyshov for settling up the x-ray diffraction
experiment. The sample preparation was supported by the NCCR research pool
MaNEP of the Swiss NSF.
## References
* (1) T. Kimura et al., Nature 426, 558 (2003).
* (2) T. Lottermoser et al., Nature 430, 541 (2004).
* (3) S. W. Cheong and M. Mostovoy, Nature Mater. 6, 13 (2007).
* (4) M. Fiebig, J. Phys. D: Appl. Phys. 38, R123 (2005).
* (5) W. Eerenstein et al., Nature 442, 759 (2006).
* (6) H. Katsura et al., Phys. Rev. Lett. 101, 187207 (2008).
* (7) R. Seshardi and N. A. Hill, Chem. Mater. 13, 2892 (2001).
* (8) R. Becker et al., J. Am. Chem. Soc. 128, 15469 (2006).
* (9) G. M. Sheldrick, SHELXL97, University of Göttingen: Göttingen, Germany, 1997.
* (10) Z. Kutnjak et al., Nature 441, 956 (2006).
* (11) Z. Kutnjak and R. Blinc, Phys. Rev. B 76, 104102 (2007).
* (12) P. J. Brown, J. C. Matthewman, CCSL, 1897, (2008).
* (13) D. L. Fox et al. Phys. Rev. B 21, 2926 (1980).
* (14) Y. Yasui et al., J. Phys. Soc. Jpn. 77, 023712 (2008).
* (15) M. Kenzelmann et al., Phys. Rev. Lett. 98, 267205 (2007).
* (16) M. Mostovoy Phys. Rev. Lett. 96, 067601, (2006).
* (17) J. J. Betouras et al. Phys. Rev. Lett. 98, 257602, (2007).
* (18) G. Lawes et al., Phys. Rev. Lett. 95, 087205 (2005).
* (19) H. Katsura et al., Phys. Rev. Lett. 95, 057205 (2005).
* (20) I.A. Sergienko and E. Dagotto, Phys. Rev. B 73, 094434 (2006).
* (21) A. B. Harriset al., Phys. Rev. B 73, 184433 (2006).
* (22) L. C. Chapon et al., Phys. Rev. Lett., 93, 177402 (2004).
* (23) N. Aliouane et al., Phys. Rev. B 73, 020102(R) (2006).
* (24) I. A. Sergienko et al., Phys. Rev. Lett., 97, 227204 (2006).
|
arxiv-papers
| 2009-06-12T13:36:32 |
2024-09-04T02:49:03.315273
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Pregelj, O. Zaharko, A. Zorko, Z. Kutnjak, P. Jeglic, P.J. Brown,\n M. Jagodic, Z. Jaglicic, H. Berger, D. Arcon",
"submitter": "Denis Arcon",
"url": "https://arxiv.org/abs/0906.2336"
}
|
0906.2406
|
NYU-TH-09/06/15
Strongly Coupled Condensate of High Density Matter
Gregory Gabadadze
Center for Cosmology and Particle Physics, Department of Physics,
New York University, New York, NY 10003, USA
###### Abstract
Arguments are summarized, that neutral matter made of helium, carbon, etc.,
should form a quantum liquid at the above-atomic but below-nuclear densities
for which the charged spin-0 nuclei can condense. The resulting substance has
distinctive features, such as a mass gap in the bosonic sector and a gap-less
spectrum of quasifermions, which determine its thermodynamic properties. I
discuss an effective field theory description of this substance, and as an
example, consider its application to calculation of a static potential between
heavy charged impurities. The potential exhibits a long-range oscillatory
behavior in which both the fermionic and bosonic low-energy degree of freedom
contribute. Observational consequences of the condensate for cooling of
helium-core white dwarf stars are briefly discussed.
Based on a talk given at the international workshop
“Crossing the boundaries: Gauge dynamics at strong coupling”
honoring the 60th birthday of M.A. Shifman
Minneapolis, May 14-17, 2009
## Foreword
Like many in the audience, I first met Misha on the pages of journal
publications, before meeting him in person. While working on an undergraduate
thesis at Moscow University, I came across Misha’s review paper “Anomalies and
Low-Energy Theorems of Quantum Chromodynamics” [1]. Impressions of that work
were very distinct – a clear exposition of subtle field theory aspects of the
quantum anomalies, culminating in creative applications to low-energy hadron
phenomenology. The work stood out by its originality, depth, inspiration and
balance of the formalism and applications – the remarkable signatures of
Misha’s enormous contribution to theoretical physics at the forefront of both
field theory and particle phenomenology.
I met Misha in person in Minneapolis in 1998. The discussion with him was very
inspiring. Soon, in Aspen, we started to work on a project. A bit later I
ceased the opportunity to get exposed to two years of a unique FTPI
experience. We continued to work on and off on various projects since then. I
value those works very highly, and feel privileged, as I’m sure many of you do
too, for having such a collaborator.
Happy 60th Birthday Misha!
## Description of charged condensate
Consider a neutral system of a large number of nuclei each having charge $Z$,
and neutralizing electrons. If average inter-particle separations in this
system are much smaller than the atomic scale, $\sim 10^{-8}~{}cm$, while
being much larger than the nuclear scale, $\sim 10^{-13}~{}cm$, neither the
atomic nor nuclear effects will play any significant role. Moreover, the
nuclei can also be treated as point-like particles.
In what follows we focus on spin-0 nuclei with $Z\leq 8$ (helium, carbon,
oxygen), and consider the electron number-density in the interval
$J_{0}\simeq(0.1-5~{}MeV)^{3}$. Then the electron Fermi energy will exceed the
electron-electron and electron-nucleus Coulomb interaction energy. Moreover,
at temperatures below $\sim 10^{7}~{}K$, which are of interest here, the
system of electron represents a degenerate Fermi gas.
Since the nuclei (we also call them ions below) are heavier, temperature at
which they’ll start to exhibit quantum properties will be lower. Let us define
the “critical” temperature $T_{c}$, at which the de Broglie wavelengths of the
ions begin to overlap
$T_{c}\simeq\frac{4\pi^{2}}{3m_{H}d^{2}}\,,~{}~{}~{}~{}~{}d\equiv\left(3Z\over
4\pi J_{0}\right)^{1/3}\,,$ (1)
where, $m_{H}$ denotes the mass of the ion (the subscript
${}^{\prime\prime}H^{\prime\prime}$ stands for heavy), and $d$ denotes the
average separation between the ions111The de Broglie wavelength above is
defined as $\lambda_{dB}=2\pi/|{\bf k}|$, where ${\bf
k}^{2}/2m_{H}=3k_{B}T/2$. We define $T_{c}$ as the temperature at which
$\lambda_{dB}\simeq d$. Note that this differs by a numerical factor of
$\sqrt{2\pi/3}$ from the standard definition of the thermal de Broglie
wavelength, $\Lambda\equiv\sqrt{2\pi/mk_{B}T}$, that appears in the partition
function of an ideal gas of number-density $n$ in the dimensionless
combination $\Lambda^{3}n$..
Somewhat below $T_{c}$ quantum-mechanical uncertainties in the ion positions
become greater than an average inter-ion separation. Hence the latter concept
looses its meaning as a microscopic characteristic of the system; the ions
enter a quantum-mechanical regime of indistinguishability. Then, the many-body
wavefunction of the spin-0 ions should be symmetrized, and this would
unavoidably lead to probabilistic “attraction” of the bosons to condense,
i.e., to occupy one and the same quantum state. We refer the system of
condensed nuclei and electrons as charged condensate.
In the condensate the scalars occupy a quantum state with zero momentum.
Moreover, small fluctuations of the bosonic sector happen to have a mass gap,
$m_{\gamma}=(Ze^{2}J_{0}/m_{H})^{1/2}$, which exceeds $T_{c}$ by more than an
order of magnitude. Therefore, once bosons are in the charged condensate,
their phonons cannot be thermally excited. However, the gap-less fermionic
degrees of freedom are thermally excited, and carry the most of the entropy of
the entire system [2]-[5].
For further discussions it is useful to rewrite the expression for $T_{c}$ in
terms of the mass density $\rho\equiv m_{H}J_{0}$ measured in $g/cm^{3}$:
$\displaystyle T_{c}=\rho^{2/3}\,\left({3.5\cdot 10^{2}\over
Z^{5/3}}\right)~{}K\,,$ (2)
where the baryon number of an ion was assumed to equal twice the number of
protons, $A=2Z$ (true for helium, carbon, oxygen…). Thus, for
$\rho=10^{6}~{}g/cm^{3}$ and helium-4 nuclei we get $T_{c}\simeq 10^{6}~{}K$,
while for the carbon nuclei with the same mass density $T_{c}\simeq 2\cdot
10^{5}~{}K$.
Temperature at which the condensation phase transition takes place,
$T_{condens}$, need not coincide with $T_{c}$. Moreover, we would expect
$T_{condens}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}T_{c}$. Calculation of $T_{condens}$
from the fundamental principles of this theory is hard. However, we can obtain
an interval in which $T_{condens}$ should fit. For this we introduce the
following parametrization:
$\displaystyle T_{condens}=\zeta\,T_{c}\,,$ (3)
where $\zeta$ is an unknown dimensionless parameter that should depend on
density more mildly than $T_{c}$ does. Numerically, however, this parameter
should vary in the interval $0.1\ll\zeta\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}1$: The point
$\zeta=0.1$ would corresponds to the temperature of the Bose-Einstein (BE)
condensation of a free gas for which, $T^{BE}_{condens}\simeq 1.3/m_{H}d^{2}$,
is known from the fundamental principles. The condensation temperature in our
system should be higher than $T^{BE}_{condens}$ since the repulsion makes
easier for the condensation to take place [6]. In our case, repulsive
interactions between the bosons are strong – the Coulomb energy is at least an
order of magnitude greater that any other energy scale in the system. Hence,
we should expect $\zeta\gg 0.1$. On the other hand, given the definition of
$T_{c}$, the parameter $\zeta$ cannot be greater than unity. In what follows
we will retain $\zeta$ in our expressions, but use $\zeta\simeq 1$ when it
comes to numerical estimates.
The condensation will take place after gradual cooling, only if $T_{condens}$
is greater than the temperature at which the substance could crystallize. A
classical plasma crystallizes when the Coulomb energy becomes about $\sim 180$
times greater than the average thermal energy per particle [7, 8, 9]. This
gives the following crystallization temperature222The presented formula for
the crystallization temperature is entirely classical. The temperature scale
that determines the classical versus quantum nature of the crystallization
transition is the Debye temperature $\theta_{D}\simeq 4\cdot
10^{3}\rho^{1/2}~{}K$. Often, $\theta_{D}$ may significantly exceed
$T_{\text{cryst}}$ [10]. In such cases, quantum zero-point oscillations should
be taken into account. This seems to delay the formation of quantum crystal,
lowering $T_{\text{cryst}}$ from its classical value at most by about $\sim
10\%$ [11]. Since this is a small change, we will ignore it in our estimates.
$T_{\text{cryst}}\simeq\rho^{1/3}\left(0.8\cdot 10^{3}Z^{5/3}\right)~{}K\,.$
(4)
Note that the density dependence of $T_{c}$ is different from that of
$T_{\text{cryst}}$ – for higher densities $T_{c}$ grows faster, making
condensation more and more favorable! One can define the “equality” density
for which $T_{condens}=T_{\text{cryst}}$:
$\displaystyle\rho_{\rm
eq}=\left({2.3\over\zeta}\right)^{3}Z^{10}\,g/cm^{3}\,.$ (5)
For helium, $Z=2$, and $\rho_{\rm eq}\simeq 10^{4}~{}g/cm^{3}$, while for
carbon, $Z=6$, and $\rho_{\rm eq}\simeq 10^{9}~{}g/cm^{3}$ (as mentioned
above, we use $\zeta\simeq 1$). These results are very sensitive to the value
of $\zeta$; for instance, $\rho_{\rm eq}$ could be an order of magnitude
higher if $\zeta\simeq 0.5$. Irrespective of this uncertainty, however, the
obtained densities are in the right ballpark of average densities present in
helium-core white dwarfs $\sim 10^{6}~{}g/cm$, (for carbon dwarfs, they’re
closer to those expected in high density regions only [5].)
Is the charged condensate a ground state of the system at hand? For the higher
values of the density interval considered, the crystal would not exist due to
strong zero-point oscillations. At lower densities, the crystalline state has
lower free energy (at least near zero temperature) due to more favorable
Coulomb binding. Hence, the condensate can only be a metastable state. The
question arises whether after condensation at $\sim T_{condens}$ the system
could transition at lower temperatures $\sim T_{\rm cryst}$ to the crystal
state, as soon as the latter becomes available.
In the condensate, the boson positions are entirely uncertain while their
momenta equal to zero. In order for such a system to crystallize later on,
each of the bosons should acquire energy of the zero-point oscillations of
crystal ions. As long as this energy, $\sim(Ze^{2}J_{0}/m_{H})^{1/2}$, is much
greater than $T_{\rm cryst}$, no thermal fluctuations can excite the condensed
bosons to transition to the crystalline state. The latter condition is well-
satisfied for all the densities considered in this work. There could, however,
exist a spontaneous transition of a region of size $R_{c}$ to the crystallized
state via tunneling. The value of $R_{c}$, and the rate of this transition,
will be determined, among other things, by tension of the interface between
the condensate and crystal state, which is hard to evaluate. However, for
estimates the following qualitative arguments should suffice: the height of
the barrier for each particle is $(Ze^{2}J_{0}/m_{H})^{1/2}=m_{\gamma}$, while
the number of bosons in the $R_{c}$ region $\sim R_{c}^{3}J_{0}/Z$. Hence, the
transition rate should scale as ${\rm exp}(-m_{\gamma}J_{0}R^{4}_{c}/Z)$.
Since we expect that $R_{c}>1/m_{\gamma}$, the rate is strongly suppressed for
the parameters at hand.
## Effective field theory description
We use a low-energy effective field theory description to study the charged
condensate. Even though realistic temperatures in the system may be well above
zero, we focus on the zero-temperature limit. The relevance of this limit is
justified a posteriori and goes as follows: the spin-0 nuclei undergo the
condensation to the zero-momentum state; their phonons cannot be excited since
their gap, $m_{\gamma}$, is greater than $T_{c}$. On the other hand, gap-less
near-the-Fermi-surface quasielectrons will be excited. Therefore, all the
thermal fluctuations will end up being stored in the fermionic quasiparticles.
For the latter, however, the finite temperature effects aren’t significant
since their Fermi energy is so much higher, $T/J_{0}^{1/3}\ll 10^{-2}$. We
note that the finite temperature effects, in a general setup with condensed
bosons, were calculated in Refs. [12, 13].
We begin at scales that are well below the heavy mass scale $m_{H}$, but
somewhat above the scale set by ${\rm max}[\mu_{f},m_{e}]$, where $\mu_{f}$
and $m_{e}$ are the electron chemical potential and mass respectively. Hence
the electrons are described by their Dirac Lagrangian, while for the
description of the nuclei we will use a charged scalar order parameter
$\Phi(x)$. As it was shown in [4], in a non-relativistic approximation for the
nuclei, the effective Lagrangian proposed by Greiter, Wilczek and Witten (GWW)
[14] in a context of superconductivity, is also applicable here, given that an
appropriate reinterpretation of its variables and parameters is made.
The construction of the GWW Lagrangian is based on the following fundamental
principles: it is consistent with the translational, rotational, Galilean and
the global $U(1)$ symmetries, preserves the algebraic relation between the
charged current density and momentum density, gives the Schrödinger equation
for the order parameter in the lowest order, and is gauge invariant [14].
Combined with the electron dynamics the GWW Lagrangian reads (we omit for
simplicity the Maxwell term):
$\displaystyle{\cal L}_{eff}={\cal P}\left({i\over
2}(\Phi^{*}D_{0}\Phi-(D_{0}\Phi)^{*}\Phi)-{|D_{j}\Phi|^{2}\over
2m_{H}}\right)\,+{\bar{\psi}}(i\gamma^{\mu}D^{f}_{\mu}-m_{f})\psi,$ (6)
where we use the standard notations for covariant derivatives with the
appropriate charge assignments: $D_{0}\equiv(\partial_{0}-iZeA_{0})$,
$D_{j}\equiv(\partial_{j}-iZeA_{j})$, $D^{f}_{\mu}=\partial_{\mu}+ieA_{\mu}$,
while ${\cal P}(x)$ stands for a general polynomial function of its argument.
The coefficients of this polynomial, ${\cal
P}(x)=\sum^{\infty}_{n=0}C_{n}x^{n}$, are dimensionful parameters that are
inversely proportional to powers of a short-distance cutoff of the effective
field theory333In general one should also add to the Lagrangian terms
$\mu_{NR}\Phi^{*}\Phi$, $\lambda(\Phi^{*}\Phi)^{2}/m_{H}^{2}$,
$\lambda_{1}(\Phi^{*}\Phi){\bar{\psi}}\psi/(m_{H}J_{0}^{1/3})$, and other
higher dimensional operators that are consistent with all the symmetries and
conditions that lead to (6) (the Yukawa term is not). Here $\mu_{NR}$ denotes
a non-relativistic chemical potential for the scalars. These terms are not
important for the low-temperature spectrum of small perturbations we’re
interested in, as long as $\lambda,\lambda_{1}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}1$ and $J_{0}\ll
m^{3}_{H}$. However, near the phase transition point it is the sign of
$\mu_{NR}$ that would distinguish between the broken and symmetric phases, so
these terms should be included for the discussion of the symmetry restoration.
We also note that the scalar part of (6) is somewhat similar to the Ginzburg-
Landau (GL) Lagrangian for superconductivity. However, there are significant
differences between them, one such difference being that the coherence length
in the GL theory is many orders of magnitude greater than the average
interelectron separation, while in the present case, the “size of the scalar”
$\Phi$ is smaller that the average interparticle distance..
Once the basic Lagrangian is fixed, we introduce the electron chemical
potential term $\mu_{f}\psi^{+}\psi\,$, where
$\mu_{f}=\epsilon_{F}=[(3\pi^{2}J_{0})^{2/3}+m_{f}^{2}]^{1/2}$. This is the
only term that at the tree level sets a frame in which the electron total
momentum is zero.
There exists a homogeneous solution of the equations of motion that follow
from the effective Lagrangian (6) [3]:
$\displaystyle Z|\Phi|^{2}=J_{0}\,,~{}~{}~{}A_{\mu}=0,~{}~{}~{}~{}{\cal
P}^{\prime}(0)=1\,.$ (7)
(We use the unitary gauge $\Phi=|\Phi|$). The condition ${\cal
P}^{\prime}(0)=1$ is satisfied by any polynomial functions ${\cal P}(x)$ for
which the first coefficient is normalized to unity
$\displaystyle{\cal P}(x)=x+C_{2}x^{2}+...\,.$ (8)
The above solution describes a neutral system of negatively charged electrons
of charge density $-eJ_{0}$, and positively charged scalar condensate of
charge density $Ze\Phi^{+}\Phi=eJ_{0}$ [4, 5].
Calculation of the spectrum of small perturbations is straightforward. The
Lagrangian density for the fluctuations in the quadratic approximation reads
[2]
$\displaystyle{\cal L}_{2}=-{1\over 4}F_{\mu\nu}^{2}+{1\over
2}m_{0}^{2}A_{0}^{2}-{1\over 2}m_{\gamma}^{2}A_{j}^{2}+{1\over
2}\,A_{0}{(2m_{H}m_{\gamma})^{2}\over-\Delta}A_{0}\,,$ (9)
where $\Delta$ denotes the Laplacian, and the last term emerged due to mixing
of $A_{0}$ with the fluctuation of the $|\Phi|$, which we integrated out. As
before,
$\displaystyle m_{\gamma}^{2}\equiv{Ze^{2}J_{0}\over m_{H}}\,,$ (10)
and $m_{0}^{2}=m_{\gamma}^{2}+C_{2}e^{2}J^{2}_{0}$. At this stage we retained
the fermionic fluctuations only in the Thomas-Fermi approximation [3]; an
important refinement of this approximation, discussed in [4], will be included
below.
That there are no pathologies in (9), such as ghost and/or tachyons, can be
seen by calculating the Hamiltonian density:
$\displaystyle{\cal H}={\pi_{j}^{2}\over 2}+{F^{2}_{ij}\over 4}+{1\over
2}(\partial_{j}\pi_{j})\left(m_{0}^{2}+{4M^{4}\over-\Delta}\right)^{-1}(\partial_{j}\pi_{j})+{1\over
2}m_{\gamma}^{2}A_{j}^{2}\,.$ (11)
Here, $M^{2}\equiv m_{H}m_{\gamma}$ and $\pi_{j}\equiv-F_{0j}$. The
Hamiltonian is positive semi-definite. Moreover, the spectrum has a mass gap
determined by $m_{\gamma}$ (10). There are two transverse polarizations of a
massive photon, as well as the longitudinal mode, the phonon, with the same
mass $m_{\gamma}$ [2].
The massive bosonic collective excitations give rise to exponentially
suppressed contributions to the value of specific heat of the charged
condensate since typically $m_{\gamma}\gg T_{c}$. The suppression scales as
${\rm exp}(-m_{\gamma}/T)$, where $T\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}T_{c}$. This is in
contrast with the crystal, where the dominant contribution to the specific
heat comes from a gap-less phonon, and scales with temperature as $T^{3}$.
As to the electrons, their behavior is similar in both crystal and condensate
cases. At temperatures of interest they form a degenerate Fermi gas with gap-
less excitations near the Fermi surface. Their contribution to the specific
heat scales linearly with temperature. In the case of crystallized substance
this is sub-dominant to the specific heat due to the crystal phonon. For the
charged condensate, however, the (quasi)electron fluctuations are the dominant
contributors to the specific heat.
To study the effects of collective bosonic and fermionic modes, as an
interesting example, we look at a potential between two impurity nuclei (say
hydrogen, or helium-3) of charge $Q_{1}$ and $Q_{2}$. The calculation of the
propagator that involves the light collective modes (for relativistic
fermions) gives the following result [4]:
$\displaystyle V_{stat}=\alpha_{\rm em}{Q_{1}Q_{2}}\left({e^{-Mr}\over\,r}{\rm
cos}(Mr)\,+{4\alpha_{\rm em}\over\pi}{k_{F}^{5}{\rm sin}(2k_{F}r)\over
M^{8}r^{4}}\right)\,.$ (12)
The first, exponentially suppressed term modulated by a periodic function, is
due to cancellation between the screened Coulomb potential and that of a
phonon [4]. The fact of such a cancellation, and that it could give rise to
the oscillatory behavior of the exponentially screened potential was pointed
out before in Ref. [15] in the context of superconductivity444I’d like to
thank Ki-Myeong Lee who recently brought the paper [15] to my attention..
Most important, however, is the second term in (12) that has a long-range [4].
It dominates over the exponentially suppressed term in (12) for scales of
physical interest, and exhibits the power-like behavior modulated by a
periodic function.
The potential (12) is a generalization of the Friedel potential to the case
when in addition to the fermionic excitations there are also collective modes
due to the charged condensate. The long-range oscillating term in (12) is also
a result of a subtraction between the conventional Friedel term and the long-
range oscillating term due to a phonon. As a result, its magnitude is
suppressed compared to what it would have been in a theory without the
condensed charged bosons [4] (see, [16] for the discussion of the conventional
Friedel potential, and Ref. [13] for its recent detailed study in the presence
of the charged condensate at finite temperature.)555Note that for spin-
dependent interactions the same effects of the charged condensate would give a
generalization of the Ruderman-Kittel-Kasuya-Yosida (RKKY) potential [17].
The potential (12) is not sign-definite. In particular, it can give rise to
attraction between like charges; this attraction is due to collective
excitations of both fermionic and bosonic degrees of freedom. This represents
a generalization of the Kohn-Luttinger [18] effect to the case where on top of
the fermionic excitations the collective modes of the charged condensate are
also contributing666In the charged condensate Cooper pairs of electrons can
also be formed, however, the corresponding transition temperature, and the
magnitude of the gap, are suppressed by a factor ${\rm exp}(-1/e_{eff}^{2})$,
where $e_{eff}^{2}$ is proportional to the value of the inter-electron
potential that contains both screened Coulomb and phonon exchange. The fact
that this potential has attractive domain, but is very small, is suggested by
the static potential found in [4] (see also eq. (12) above); the latter is
suppressed by a power of a large scale $M$. Furthermore, taking into account
the frequency dependence of the propagator in the Eliashberg equation does not
seem to change qualitatively the conclusion on a strong suppression of the
Green’s function and pairing temperature. Hence, even though the bosonic
sector (condensed nuclei) is superconducting at reasonably high temperatures
$\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{6}~{}K$, interactions with gap-less fermions could
dissipate the superconducting currents. Only at extremely low temperatures,
exponentially close to the absolute zero, the electrons could also form a gap
leading to superconductivity of the whole system. In the present work we
consider temperatures at which electrons are not condensed into Cooper pairs,
and ignore the finite temperature effects..
## Applications to White Dwarfs
The above described system of electrons and nuclei constitutes cores of white
dwarf stars. Up to a factor of a few, these are roughly Earth’s size solar-
mass objects; their mass density may range over
$\sim(10^{6}-10^{10})~{}g/cm^{3}$, most of them being near the lower edge of
this interval. Since the dwarf stars exhausted thermonuclear fuel in their
cores already, they evolve by cooling [19]; the ones that we consider in this
work cool from $\sim 10^{7}~{}K$ down to lower temperatures.
As a typical dwarf star cools down, the Coulomb interaction energy in a
classical plasma of charged nuclei will significantly exceed their classical
thermal energy, and the nuclei, in order to minimize energy, would organize
themselves into a crystal lattice [20]. In most of these cases quantum effects
of the nuclei should be negligible; for instance, the Debye temperature should
be less than the temperature at which crystallization takes place, and the de
Broglie wavelengths of the nuclei should be much smaller than the average
internuclear separations. This indeed is the case in majority of white dwarf
stars, the cores of which are composed of carbon and/or oxygen nuclei and span
the interval of mass densities around $\sim(10^{6}-10^{8})~{}g/cm^{3}$.
However, there exists a class of dwarf stars in which the nuclei enter the
quantum regime before the classical crystallization process sets in [10, 11].
Among these, furthermore, there is a relatively small subclass of the dwarf
stars for which the temperature $T_{c}$, is higher than the would-be
crystallization temperature $T_{cryst}$ [5]. In such dwarf cores the charged
condensation should be expected to take place.
White dwarfs composed of helium constitute a smaller sub-class of dwarf stars
(see, [21, 22] are references therein); they exhibit best conditions for the
charged condensation. Most of helium dwarfs are believed to be formed in
binary systems, where the removal of the envelope off the dwarf progenitor red
giant by its binary companion happened before helium ignition, producing a
remnant that evolves to a white dwarf with a helium core. Helium dwarf masses
range from $\sim 0.5~{}M_{\odot}$ down to as low as $(0.18-0.19)~{}M_{\odot}$,
while their envelopes are mainly composed of hydrogen.
Using the approach of [23], and following [5] we will consider an over-
simplified model of a reference helium star of mass $M=0.5~{}M_{\odot}$ with
the atmospheric mass fractions of the hydrogen, and heavy elements
(metallicity) respectively equal to
$\displaystyle X\simeq 0.99,\quad\quad Z_{m}\simeq(0.0002-0.002)~{}.$ (13)
The lower value of the metallicity $Z_{m}\simeq 0.0002$ is appropriate for the
recently discovered 24 He WDs in NGC 6397 [22], but for completeness, we
consider a wider range for this parameter.
It is straightforward to find the following expression for the cooling time of
a star in the classical regime [23]
$\displaystyle
t_{He}=\frac{k_{B}}{CAm_{u}}\left[\frac{3}{5}(T_{f}^{-\frac{5}{2}}-T_{0}^{-\frac{5}{2}})+Z\frac{\pi^{2}}{3}\frac{k_{B}}{E_{F}}(T_{f}^{-\frac{3}{2}}-T_{0}^{-\frac{3}{2}})\right],$
(14)
where $T_{f}$ and $T_{0}$ denote the final and initial core temperatures. The
first term in the bracket on the right hand side corresponds to cooling due to
classical gas of the ions and the second term corresponds to the contribution
coming from the Fermi sea. The latter is sub-dominant in the range of final
temperatures we are interested in (the factor Z in front of this term is due
to $Z$ electrons per ion). Since $T_{f}\ll T_{0}$, the age of a dwarf star
typically doesn’t depend on the initial temperature. Neglecting the fermion
contribution, we find time that is needed to cool down to critical temperature
$T_{f}=T_{c}$
$\displaystyle
t_{He}=\frac{3}{5}\frac{k_{B}T_{c}M}{Am_{u}L(T_{c})}\simeq(0.76-7.6)~{}\text{Gyr}\,.$
(15)
Where an order of magnitude interval in (15) is due to the interval in the
envelope metallicity composition given in (13). We also find the corresponding
luminosities
$\displaystyle
L(T_{c})\simeq(10^{8}~{}erg/s)\frac{M}{M_{\odot}}\left(\frac{T_{c}}{\text{K}}\right)^{{7/2}}\simeq
1.5\cdot(10^{-4}-10^{-5})L_{\odot}\,,$ (16)
which are in the range of observable luminosities ($L_{\odot}\simeq 3.84\cdot
10^{33}~{}erg/s$).
After the condensation, specific heat of the system dramatically drops as the
collective excitations of the condensed nuclei become massive and “get
extinct”. A contribution from the Fermi sea, which is strongly suppressed by
the value of Fermi energy, becomes the dominant one. The phase transition
itself would take some time to complete, and the drop-off in specific heat
will not be instantaneous.
In the zeroth approximation, we can regard the transition to be very fast, and
retain only the fermion contribution to specific heat below $T_{c}$. Then, the
expression for the age of the star for $T_{f}<T_{c}$, reads as follows
$\displaystyle
t_{He}^{\prime}=\frac{k_{B}}{CAm_{u}}\left[\frac{3}{5}(T_{c}^{-\frac{5}{2}}-T_{0}^{-\frac{5}{2}})+Z\frac{\pi^{2}}{3}\frac{k_{B}}{E_{F}}(T_{f}^{-\frac{3}{2}}-T_{0}^{-\frac{3}{2}})\right].$
(17)
Notice the difference of (17) from (14) – in the former $T_{f}<T_{c}$ and it
is $T_{f}$ that enters as final temperature in the fermionic part, while
$T_{c}$ should be taken as the final temperature in the bosonic part.
From the ratio of ages, $\eta={t_{He}/t_{He}^{\prime}}$, for two identical
helium dwarf stars, with and without the interior condensation, we deduce that
the charged condensation substantially increases the rate of cooling– the age
could be twenty times less than it would have been without the condensation
phase [5].
The condensation of the core would induce significant deviations from the
classical curve for helium white dwarfs. What is independent of the
uncertainties involved in these discussions, is the fact that the luminosity
function (LF) will experience a significant drop-off after the charged
condensation phase transition is complete. This is due to the “extinction” of
the bosonic quasiparticles below the phase transition point. In fact, the LF
will drop by a factor of $\sim 200$. This may be relevant for an explanation
of the observed termination of a sequence of the 24 He WD’s found in [22]. See
Ref. [5] for more details.
Finally, the magnetic properties of the charged condensate, which are similar
to those of type II superconductor, and in particular admit the presence of
Abrikosov’s vortices, were studied in Ref. [24]. As was shown there, only very
strong magnetic fields, $\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}10^{7}~{}Gauss$, will be able to
penetrate the dwarf cores in the vortices, while weaker fields will be
entirely expelled from it.
Acknowledgments
The above-reported results constitute a part of the work done in collaboration
with Rachel A. Rosen and David Pirtskhalava [2]-[5], [24]. I’d like to thank
Paul Chaikin, Daniel Eisenstein, Leonid Glazman, Andrei Gruzinov, Stefan
Hofmann, Andrew MacFadyen, Juan Maldacena, Aditi Mitra, Slava Mukhanov, Hector
Rubinstein, Malvin Ruderman and Arkady Vainshtein for useful discussions and
correspondence on these topics. The work was supported by the NSF grant
PHY-0758032.
## References
* [1] M.A. Shifman, “Anomalies and Low-Energy Theorems of Quantum Chromodynamics,” Phys. Rept. 209, 341 (1991) [Sov. Phys. Usp. 32, 289 (1989)]
* [2] G. Gabadadze and R. A. Rosen, Phys Lett. B 658 (2008), 266; ibid. B 666, 277 (2008)
* [3] G. Gabadadze and R. A. Rosen, JCAP 0810, 030 (2008)
* [4] G. Gabadadze and R. A. Rosen, JCAP 0902, 016 (2009)
* [5] G. Gabadadze and D. Pirtskhalava, JCAP 0905, 017 (2009)
* [6] K. Huang, Phys. Rev. Lett. 83, 3770 (1999)
* [7] D.Q. Lamb and H.M. Van Horn, Ap.J. 200:306 (1975)
* [8] S. Ichimaru, Rev. Mod. Phys. 54:1017 (1982)
* [9] G.S. Stringfellow, H.E. DeWitt, W.I. Slattery, Phys. Lett. A 41, 1105 (1990)
* [10] G. Chabrier, N.W. Ashcroft, and H.E. DeWitt, Nature, 360,48 (1992)
* [11] G. Chabrier, Ap.J. 414:695 (1993)
* [12] A. D. Dolgov, A. Lepidi and G. Piccinelli, JCAP 0902, 027 (2009)
* [13] A. D. Dolgov, A. Lepidi and G. Piccinelli, arXiv:0905.4422 [hep-ph]
* [14] M. Greiter, F. Wilczek and E. Witten, Mod. Phys. Lett. B 3, 903 (1989)
* [15] K. Lee and O. Tchernyshyov, Mod. Phys.Lett. A13 (1998) 987 [cond-mat/9707202]
* [16] A.L. Fetter, J.D. Walecka, “Quantum Theory of Many-Particle Systems”, McGraw-Hill, 1971
* [17] M.A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954); T. Kasuya, Prog. Theor. Phys. 16, 45 (1956); K. Yosida, Phys. Rev. 106, 893 (1957)
* [18] W. Kohn, J.H. Luttinger, Phys. Rev. Lett. 15, 524 (1965)
* [19] L. Mestel, 1952, MNRAS, 112, 583
* [20] L. Mestel and M.A. Ruderman, MNRAS, 136:27 (1967)
* [21] J. Liebert, P. Bergeron, D. Eisenstein, H.C. Harris, S.J. Kleinman, A. Nitta, J. Krzesinski, ApJ, 606, L147, (2004), [astro-ph/040429]
* [22] R. R. Strickler, A.M. Cool, J. Anderson, H. N. Cohn, P. M. Lugger, A.M. Serenelli, arXiv:0904.3496, astro-phGA
* [23] S.L. Shapiro and S. A. Teukolsky, “Black Holes, White Dwarfs, and Neutron Stars”, John Wiley & Sons, (1983)
* [24] G. Gabadadze and R. A. Rosen, arXiv:0905.2444 [hep-th]
|
arxiv-papers
| 2009-06-12T19:26:39 |
2024-09-04T02:49:03.321411
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gregory Gabadadze",
"submitter": "Gregory Gabadadze",
"url": "https://arxiv.org/abs/0906.2406"
}
|
0906.2441
|
Nonmass Eigenstates of Fermion and Boson Fields
Xin-Bing Huang***[email protected]
Shanghai United Center for Astrophysics (SUCA),
Shanghai Normal University, No.100 Guilin Road, Shanghai 200234, China
Abstract
It appears natural to consider the four dimensional relativistic massive field
as a five dimensional massless field. If the fifth coordinate is interpreted
as the proper time, then the fifth moment can be understood as the rest mass.
After introducing the rest mass operator, we define the mass eigenstate and
the nonmass eigenstate. The general equations of spin-0, spin-$\frac{1}{2}$
and spin-1 fields are obtained respectively. It is shown that the Klein-Gordon
equation, the Dirac equation and the Proca equation describe the mass
eigenstates only. The rest mass of spin-$\frac{1}{2}$ field and the rest mass
squared of Boson fields are calculated. The $U(1)$ gauge field that couples to
the nonmass eigenstates is studied carefully, whose gauge boson can be
massive.
PACS numbers: 12.15.Ff, 11.10.Kk, 12.60.Cn
What is the origin of the rest mass? How to distinguish the massive field and
the massless field explicitly from the mathematical point of view? What is the
essential difference between the flavor eigenstates and the mass eigenstates?
All those problems are fundamental, in which the problem of the rest mass has
been studied widely from different viewpoints. In the standard model of
electroweak interactions [1, 2], the rest masses of leptons and quarks
originate from their Yukawa couplings with the Higgs field [3], and the
mismatch between the flavor eigenstates and the mass eigenstates of leptons
and quarks is caused by the Higgs interactions [4]. In the 5-dimensional
Kaluza-Klein theory [5, 6, 7] and the 11-dimensional string theory (or called
M-theory) [8], the quantum field can obtain the rest mass via the
compactification of extra dimension. It is studied in Ref.[9] to consider the
4-dimensional relativistic particle as a 5-dimensional massless particle and
interpret the fifth coordinate as the particle’s proper time while the fifth
moment can be understood as the mass. In this letter, we use the proper time
to define the rest mass operator and discuss the nonmass eigenstates of Boson
and Fermion fields.
Consider a Minkowskian spacetime with the following metric tensor (covariant
components)
$\eta_{\mu\nu}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{array}\right)~{},~{}~{}~{}~{}(\mu,\nu=0,1,2,3)~{}.$ (1)
In this letter we will use the contravariant three-vector
$x^{i}=\\{x^{1},x^{2},x^{3}\\}\equiv\\{x,y,z\\}~{},~{}~{}~{}~{}(i=1,2,3)~{},$
(2)
and four-vector
$x^{\mu}=\\{x^{0},x^{1},x^{2},x^{3}\\}\equiv\\{ct,x,y,z\\}~{},$ (3)
for the description of the spacetime coordinates, where the timelike component
is denoted as zero component. The proper time $s$ can be given by
$s^{2}=x^{\mu}x_{\mu}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}~{},$ (4)
where $c$ is the speed of light in vacuum, which is invariant under the
Lorentz transformations. From (4) one can acquire that
$sds=ctd(ct)-xdx-ydy-zdz~{},$ (5)
therefore
$\frac{1}{c}\frac{\partial s}{\partial t}=\frac{ct}{s}~{},~{}~{}\frac{\partial
s}{\partial x}=-\frac{x}{s}~{},~{}~{}\frac{\partial s}{\partial
y}=-\frac{y}{s}~{},~{}~{}\frac{\partial s}{\partial z}=-\frac{z}{s}~{},$ (6)
furthermore
$\frac{\partial}{\partial s}=\frac{s}{c^{2}t}\frac{\partial}{\partial
t}-\frac{s}{x}\frac{\partial}{\partial x}-\frac{s}{y}\frac{\partial}{\partial
y}-\frac{s}{z}\frac{\partial}{\partial z}={\bf
n}^{\mu}\partial_{\mu}~{},~{}~{}\partial_{\mu}\equiv\frac{\partial}{\partial
x^{\mu}}~{},$ (7)
where the contravariant vector ${\bf n}^{\mu}$ has been defined by
${\bf
n}^{\mu}=\left\\{\frac{s}{ct},-\frac{s}{x},-\frac{s}{y},-\frac{s}{z}\right\\}~{}.$
(8)
In order to discuss the rest mass we start by considering a free particle with
the relativistic relation
$\frac{E^{2}}{c^{2}}=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+m_{0}^{2}c^{2}~{},$ (9)
here $m_{0}$ is the rest mass of the particle. Obviously the rest mass plays
the equal role with the component of momentum in above equation, they
therefore should have the similar quantization. In elementary quantum
mechanics [10], the energy $E$ and the component of momentum $p_{i}$ are
quantized by
$\hat{E}=i\hbar\frac{\partial}{\partial
t}~{},~{}~{}~{}~{}\hat{p}_{i}=-i\hbar\frac{\partial}{\partial x^{i}}~{},$ (10)
which are Hermitian operators, namely, $\hat{E}^{{\dagger}}=\hat{E}$ and
$\hat{p}^{{\dagger}}_{i}=\hat{p}_{i}$. If the rest mass is quantized by a
Hermitian operator also, then the real number $m_{0}$ in (9) must be one of
the eigenvalues of this Hermitian operator. According to the relativistic
relation (9) and the operators of energy and momentum (10), we define the rest
mass operator as follows
$\hat{m}=-i\frac{\hbar}{c}\frac{\partial}{\partial s}~{},$ (11)
which is invariant under the Lorentz transfromations. It is easy to prove that
the rest mass operator is Hermitian if and only if $s$ is timelike.
Assume that a set of eigenfunctions
$\sigma_{j}(x^{\mu})~{}(j=1,2,\cdot\cdot\cdot,n)$ constitutes an
$n$-dimensional complete Hilbert space, which are the eigenfunctions of
$\hat{m}$, and $m_{j}$ are the corresponding eigenvalues, which are real and
nonnegative–as the rest mass, of course, must be. We reexpress this assumption
in the mathematical language
$\hat{m}\sigma_{j}(x^{\mu})=-i\frac{\hbar}{c}\frac{\partial\sigma_{j}(x^{\mu})}{\partial
s}=m_{j}\sigma_{j}(x^{\mu})~{},~{}~{}~{}~{}(j=1,2,\cdot\cdot\cdot,n)~{}.$ (12)
Above assumption shows that the mass eigenstate has been defined by the
eigenfunction of the rest mass operator. According to quantum mechanics [10],
we can then define the nonmass eigenstate $\sigma(x^{\mu})$ by
$\sigma(x^{\mu})=\sum_{j=1}^{n}a_{j}\sigma_{j}(x^{\mu})~{},~{}~{}~{}~{}\sum_{j=1}^{n}a_{j}a_{j}^{*}=1~{},$
(13)
where $a_{j}$ is complex and $a_{j}^{*}$ the complex conjugate of $a_{j}$.
In quantum field theories, the mass operator had been introduced and the
nonmass eigenstates had also been used to denote the off-shell states [11,
12]. But we give a quite different definition for the rest mass operator. The
nonmass eigenstate discussed by us will devote it to a better understanding of
particle mixing.
From elementary quantum mechanics it is known that the schrödinger equation
corresponds to the non-relativistic energy relation in operator form. we
therefore replace the energy, the momentum and the rest mass in the
relativistic relation (9) by their corresponding operators (10) and (11) to
get a relativistic equation
$\left(\frac{\partial^{2}}{c^{2}\partial t^{2}}-\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-\frac{\partial^{2}}{\partial
z^{2}}-\frac{\partial^{2}}{\partial{s}^{2}}\right)\phi(x^{\mu})=0~{},$ (14)
here $\phi(x^{\mu})$ is a nonmass eigenstate of the free scalar fields. This
equation is invariant under the Lorentz transformations. For a mass-squared
eigenstate $\phi(x^{\mu})$, we obtain
$\hat{m}^{2}\phi(x^{\mu})=-\frac{\hbar^{2}}{c^{2}}\frac{\partial^{2}\phi(x^{\mu})}{\partial{s}^{2}}=m_{\phi}^{2}\phi(x^{\mu})~{}.$
(15)
Then (14) reduces to
$\left(\frac{\partial^{2}}{c^{2}\partial t^{2}}-\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-\frac{\partial^{2}}{\partial
z^{2}}+\frac{m_{\phi}^{2}c^{2}}{\hbar^{2}}\right)\phi(x^{\mu})=0~{}.$ (16)
The definition of the rest mass operator $\hat{m}$ shows that (15) is the
eigen equation of the operator $\hat{m}^{2}$. Therefore, the function
$\phi(x^{\mu})$ should be called the mass-squared eigenstate of a scalar
field. (16) is the Klein-Gordon equation.
Assume that a set of mass-squared eigenstates $\phi_{j}(x^{\mu})$ given by
(15) constitutes an $n$-dimensional complete Hilbert space, where $m_{j}$ is
the corresponding rest mass. Then the nonmass eigenstate of spin-$0$ fields is
generally given by
$\phi(x^{\mu})=\sum_{j=1}^{n}a_{j}\phi_{j}(x^{\mu})~{},~{}~{}~{}~{}\sum_{j=1}^{n}a_{j}a_{j}^{*}=1~{}.$
(17)
Since $\phi_{j}(x^{\mu})$ are mass-squared eigenstates, one can only obtain
the square rest mass of a nonmass eigenstate defined by (17) as follows
$m^{2}=\sum_{j=1}^{n}a_{j}a_{j}^{*}m_{j}^{2}~{}.$ (18)
Following the historical approach of Dirac who, in 1928, obtained a
relativistic covariant wave equation for spin-$\frac{1}{2}$ field, we give the
relativistic covariant equation for the nonmass eigenstate of
spin-$\frac{1}{2}$ fields with a general potential $V(x^{\mu})$
$i\hbar\frac{\partial\psi(x^{\mu})}{\partial t}=\left[-i\hbar
c\hat{\alpha}^{i}\frac{\partial}{\partial x^{i}}-i\hbar
c\hat{\beta}\frac{\partial}{\partial s}+V\right]\psi(x^{\mu})~{},$ (19)
where $\psi(x^{\mu})$ is a $4\times 1$ matrix, and
$\hat{\alpha}^{i},\hat{\beta}$ are $4\times 4$ Hermitian matrices defined by
Dirac. One can prove the covariance of this equation by noticing that the
operator $\frac{\partial}{\partial s}$ is invariant under the Lorentz
transformations.
For a mass eigenstate $\psi(x^{\mu})$, we get
$\hat{m}\psi(x^{\mu})=-i\frac{\hbar}{c}\frac{\partial\psi(x^{\mu})}{\partial
s}=m_{\psi}\psi(x^{\mu})~{},$ (20)
and
$\left(i\hbar\gamma^{\mu}\frac{\partial}{\partial
x^{\mu}}-m_{\psi}c\right)\psi(x^{\mu})=\frac{V}{c}\gamma^{0}\psi(x^{\mu})~{},$
(21)
here we have adopted the definitions of
$\gamma^{\mu}=\\{\gamma^{0},\gamma^{1},\gamma^{2},\gamma^{3}\\}$ and
$\gamma^{0}=\hat{\beta},~{}\gamma^{i}=\hat{\beta}\hat{\alpha}^{i}$. From above
analysis, we can draw a conclusion that the mass eigenstate of the
spin-$\frac{1}{2}$ field is described by the well-known Dirac equation (21).
From (19) we find that the Lagrange density of a nonmass eigenstate of free
spin-$\frac{1}{2}$ particles has the form†††In this letter ${\cal L}_{1n}$
denotes the Lagrange density of one nonmass eigenstate and ${\cal L}_{1m}$
denotes the Lagrange density of one mass eigenstate.
$\displaystyle{\cal L}_{1n}={\bar{\psi}}(x^{\mu})\left(i\hbar
c\gamma^{\mu}\frac{\partial}{\partial x^{\mu}}+i\hbar
c\frac{\partial}{\partial s}\right){\psi}(x^{\mu})~{}.$ (22)
$\bar{\psi}\equiv{\psi}^{{\dagger}}\gamma^{0}$ is called the spinor adjoint to
$\psi$. The variation of above ${\cal L}_{1n}$ with respect to
${\bar{\psi}}(x,z)$ yields the general equation for a nonmass eigenstate of
free spin-$\frac{1}{2}$ Fermions
$\displaystyle\left(\gamma^{\mu}\frac{\partial}{\partial
x^{\mu}}+\frac{\partial}{\partial s}\right)\psi(x^{\mu})=0~{}.$ (23)
The above equation is (19) with $V=0$. The Dirac equation in the models of
flat $1+4$ dimensional spacetime is generally given by [6, 7, 13, 14]
$\displaystyle\left(i\hbar\gamma^{\mu}\frac{\partial}{\partial
x^{\mu}}+i\hbar\gamma^{5}\frac{\partial}{\partial
x^{5}}-mc\right)\Psi(x,x^{5})=0~{},$ (24)
where the metric of 5-dimensional spacetime is of the signature $(+,-,-,-,-)$
and $\gamma^{5}=-\gamma_{5}\equiv-i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$.
Obviously (23) can be treated as the massless 5-dimensional Dirac equation.
We can obtain the Lagrange density of a mass eigenstate of the free
spin-$\frac{1}{2}$ particle from (22), that is
$\displaystyle{\cal
L}_{1m}={\bar{\psi}}(x^{\mu})\left(ic\hbar\gamma^{\mu}\frac{\partial}{\partial
x^{\mu}}-m_{\psi}c^{2}\right)\psi(x^{\mu})~{},~{}~{}~{}~{}\bar{\psi}\equiv{\psi}^{{\dagger}}\gamma^{0}~{}.$
(25)
Assume that a set of mass eigenstates $\psi_{j}(x^{\mu})$ given by (20)
constitutes an $n$-dimensional complete Hilbert space, where $m_{j}$ is the
corresponding rest mass. Then the nonmass eigenstate of spin-$\frac{1}{2}$
fields must be
$\psi(x^{\mu})=\sum_{j=1}^{n}a_{j}\psi_{j}(x^{\mu})~{},~{}~{}~{}~{}\sum_{j=1}^{n}a_{j}a_{j}^{*}=1~{}.$
(26)
According to quantum mechanics, the rest mass of nonmass eigenstate
$\psi(x^{\mu})$ given by (26) is therefore of the form
$m=\sum_{j=1}^{n}a_{j}a_{j}^{*}m_{j}~{}.$ (27)
In a word, we have given the equation for a nonmass eigenstate of
spin-$\frac{1}{2}$ fields and the rest mass of a nonmass eigenstate.
The gauge field theories [2, 15, 16, 17, 18] tell us a good method to
introduce the spin-1 fields in our framework. In quantum field theories, we
are familiar with the electromagnetic field, which is a massless $U(1)$ gauge
field. From the Lagrange densities (22) and (25), we find that both of them
admit the introduction of a $U(1)$ gauge field. Since ${\cal L}_{1m}$ can be
treated as the Lagrangian of a free charged spin-$\frac{1}{2}$ particle in
quantum field theory, introducing a $U(1)$ gauge field into the Lagrange
density (25) will directly give an electromagnetic field. The total Lagrangian
${\cal L}_{1mt}={\bar{\psi}}\left(ic\hbar\gamma^{\mu}\frac{\partial}{\partial
x^{\mu}}-e\gamma^{\mu}A_{\mu}-m_{\psi}c^{2}\right)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$
(28)
is invariant under the following local $U(1)$ transformation
${\psi}^{\prime}(x^{\mu})=e^{i\theta(x^{\mu})}\psi(x^{\mu})~{},~{}~{}~{}~{}A^{\prime}_{\mu}=A_{\mu}-\frac{\hbar
c}{e}\frac{\partial\theta(x^{\mu})}{\partial x^{\mu}}~{},$ (29)
where $\theta(x^{\mu})$ is a function of $x^{\mu}$, and $A_{\mu}$ is the
electromagnetic field. The electromagnetic field strength tensor
$F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial
A_{\mu}}{\partial x^{\nu}}$ is invariant under above transformation as well.
From now on we will pay more attention to introducing a $U(1)$ gauge field
into the Lagrange density ${\cal L}_{1n}$. To make the gauge invariance
explicit, we formally define
$x^{4}=-x_{4}=s~{},~{}~{}~{}~{}\gamma^{4}=-\gamma_{4}={\bf 1}~{}.$ (30)
Thus the Lagrange density of a nonmass eigenstate of free spin-$\frac{1}{2}$
fields (22) becomes
$\displaystyle{\cal
L}_{1n}={\bar{\psi}}(x^{\mu})\left(ic\hbar\gamma^{\alpha}\frac{\partial}{\partial
x^{\alpha}}\right)\psi(x^{\mu})~{},~{}~{}~{}~{}\alpha=0,1,2,3,4~{}.$ (31)
Let us multiply the nonmass eigenstate $\psi(x^{\mu})$ by a local phase
$e^{i\Theta(x^{\mu})}$, that is
$\psi^{\prime}(x^{\mu})=~{}e^{i\Theta(x^{\mu})}\psi(x^{\mu})~{}.$ (32)
We can reexpress the above equation as doing a $U(1)$ transformation on
$\psi(x^{\mu})$ because the phase factor $e^{i\Theta(x^{\mu})}$ is an element
of $U(1)$ group. So naturally
${\bar{\psi}}^{\prime}(x^{\mu})=~{}e^{-i\Theta(x^{\mu})}{\bar{\psi}}(x^{\mu})~{}.$
(33)
The crucial result is that the total Lagrange density
$\displaystyle{\cal
L}_{1nt}={\bar{\psi}}(x^{\mu})\left(ic\hbar\gamma^{\alpha}\frac{\partial}{\partial
x^{\alpha}}-g\gamma^{\alpha}{\bf
A}_{\alpha}(x^{\mu})\right)\psi(x^{\mu})-\frac{1}{4}{\bf
F}_{\alpha\beta}(x^{\mu}){\bf F}^{\alpha\beta}(x^{\mu})$ (34)
is invariant under a group of local gauge transformations, given by (32), (33)
and
${\bf A}^{\prime}_{\alpha}(x^{\mu})={\bf A}_{\alpha}(x^{\mu})-\frac{\hbar
c}{g}\frac{\partial\Theta(x^{\mu})}{\partial x^{\alpha}}~{},$ (35)
$g$ in above equations is the coupling constant. The strength tensor of $U(1)$
gauge field is of the form
${\bf F}_{\alpha\beta}(x^{\mu})=\frac{\partial{\bf
A}_{\beta}(x^{\mu})}{\partial x^{\alpha}}-\frac{\partial{\bf
A}_{\alpha}(x^{\mu})}{\partial x^{\beta}}~{},$ (36)
which is invariant under the transformations of (32), (33) and (35) as well.
Now we will prove that ${\bf A}_{\alpha}(x^{\mu})$ is a four dimensional
covariant vector. To do this, we decompose ${\bf A}_{\alpha}(x^{\mu})$ into
two parts
${\bf A}_{\alpha}\equiv\\{{\bf A}_{\mu},{\bf A}_{s}\\}~{}.$ (37)
Considering (7) and (34) together, we get
${\bf A}_{s}=\frac{s}{ct}{\bf A}_{0}-\frac{s}{x}{\bf A}_{1}-\frac{s}{y}{\bf
A}_{2}-\frac{s}{z}{\bf A}_{3}={\bf n}^{\mu}{\bf A}_{\mu}~{}.$ (38)
Combining (35) with (38) gives
${\bf A}_{s}^{\prime}={\bf n}^{\mu}{\bf A}_{\mu}^{\prime}={\bf
n}^{\mu}\left({\bf A}_{\mu}-\frac{\hbar c}{g}\frac{\partial\Theta}{\partial
x^{\mu}}\right)={\bf A}_{s}-\frac{\hbar c}{g}\frac{\partial\Theta}{\partial
s}~{}.$ (39)
Therefore ${\bf A}_{\alpha}$ is a four-vector and (38) is in accordance with
(39).
Compare the Lagrange density (34) with (28), we find that the $U(1)$ gauge
field ${\bf A}_{\alpha}(x^{\mu})$ can be treated as a 5-dimensional Maxwell’s
electromagnetic field. Therefore
$\frac{\partial{\bf F}_{\alpha\beta}(x^{\mu})}{\partial
x^{\gamma}}+\frac{\partial{\bf F}_{\gamma\alpha}(x^{\mu})}{\partial
x^{\beta}}+\frac{\partial{\bf F}_{\beta\gamma}(x^{\mu})}{\partial
x^{\alpha}}=0~{},$ (40)
and
$\frac{\partial{\bf F}_{\alpha\beta}(x^{\mu})}{\partial x_{\alpha}}={\bf
J}_{\beta}(x^{\mu})~{},$ (41)
where ${\bf J}_{\beta}(x^{\mu})$ is the 5-dimensional current.
Substituting (36) into (41) we find that ${\bf A}_{\alpha}(x^{\mu})$ satisfies
$\frac{\partial}{\partial x_{\alpha}}\frac{\partial{\bf
A}_{\beta}(x^{\mu})}{\partial x^{\alpha}}-\frac{\partial}{\partial
x^{\beta}}\frac{\partial{\bf A}_{\alpha}(x^{\mu})}{\partial x_{\alpha}}={\bf
J}_{\beta}(x^{\mu})~{}.$ (42)
We may now make use of the freedom (35) and choose a particular
$\Theta(x^{\mu})$ so that the transformed ${\bf A}_{\alpha}(x^{\mu})$
satisfies the following gauge condition:
$\frac{\partial{\bf A}_{\alpha}}{\partial x_{\alpha}}=0~{}.$ (43)
In this special “choice of gauge”, (42) becomes
$\frac{\partial}{\partial x_{\alpha}}\frac{\partial{\bf
A}_{\beta}(x^{\mu})}{\partial x^{\alpha}}={\bf J}_{\beta}(x^{\mu})~{}.$ (44)
In vacuo, there are no current, namely ${\bf J}_{\beta}(x^{\mu})=0$, (44)
reduces to
$\left(\frac{\partial^{2}}{c^{2}\partial t^{2}}-\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-\frac{\partial^{2}}{\partial
z^{2}}-\frac{\partial^{2}}{\partial{s}^{2}}\right){\bf A}_{\alpha}=0~{}.$ (45)
This is the general equation of a nonmass eigenstate of free spin-1 fields.
Under this special choice of gauge, let us consider a mass-squared eigenstate
of ${\bf A}_{\mu}$, which is given by
$\hat{m}^{2}{\bf A}_{\mu}=-\frac{\hbar^{2}}{c^{2}}\frac{\partial^{2}{\bf
A}_{\mu}}{\partial{s}^{2}}=m_{{\bf A}}^{2}{\bf A}_{\mu}~{},~{}~{}~{}~{}m_{{\bf
A}}\geq 0~{}.$ (46)
Then (45) reduces to
$\left(\frac{\partial^{2}}{c^{2}\partial t^{2}}-\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-\frac{\partial^{2}}{\partial
z^{2}}+\frac{m_{{\bf A}}^{2}c^{2}}{\hbar^{2}}\right){\bf A}_{\mu}=0~{}.$ (47)
Hence we have shown that the mass-squared eigenstates (or simply called “the
mass eigenstates”) of free vector field satisfies Proca equation (47).
Therefore the gauge boson of $U(1)$ gauge field that couples to the nonmass
eigenstates can be massive.
Assume that a set of mass-squared eigenstates $\left[{\bf
A}_{\mu}(x^{\mu})\right]_{j}$ given by (46) constitutes an $n$-dimensional
complete Hilbert space, where $m_{j}$ is the corresponding rest mass. Then the
nonmass eigenstate of vector fields is expressed by
${\bf A}_{\mu}(x^{\mu})=\sum_{j=1}^{n}a_{j}\left[{\bf
A}_{\mu}(x^{\mu})\right]_{j}~{},~{}~{}~{}~{}\sum_{j=1}^{n}a_{j}a_{j}^{*}=1$
(48)
It is proved that $\left[{\bf A}_{\alpha}(x,z)\right]_{j}$ are mass-squared
eigenstates, one can calculate the square rest mass of the nonmass eigenstate
defined by (48), namely
$m^{2}=\sum_{j=1}^{n}a_{j}a_{j}^{*}m_{j}^{2}~{}.$ (49)
Obviously the vector field has the same rest mass formula as that of the
scalar field.
It is well-known that the mass term in Lagrangian of a charged particle must
be invariant under the Lorentz transformations and the local gauge
transformations, therefore, for a spin-$\frac{1}{2}$ nonmass eigenstate who
couples with a $U$(1) gauge field, the rest mass operator must be invariant
not only under the Lorentz transformations but also under the $U$(1) gauge
transformations. Hence, the rest mass operator of $\psi(x^{\mu})$ in
Lagrangian (34) should be
$\hat{M}=-i\frac{\hbar}{c}\left(\frac{\partial}{\partial s}+i\frac{g}{\hbar
c}{\bf A}_{s}\right)~{}.$ (50)
One can easily prove that $\bar{\psi}(x^{\mu})\hat{M}\psi(x^{\mu})$ is
invariant under the Lorentz transformations and the local $U$(1) gauge
transformations.
In the Standard Model, the Yukawa interactions of the quarks with the Higgs
condensate cause the mismatch between the flavor eigenstates $d_{k}^{\prime}$
and the mass eigenstates $d_{l}$, namely
$d_{k}^{\prime}\equiv\sum_{l}V_{kl}d_{l},~{}k,l=1,2,3,$ and $V$ is the
Cabibbo-Kobayashi-Maskawa mixing matrix [19, 20], which is unitary
$V^{{\dagger}}V=1$. In our framework, the mass eigenstates of the quarks must
be written as $d_{l}$, where $m_{l}$ is the corresponding mass of the quark.
Then the nonmass eigenstates of the quarks are of the form
$d_{k}^{\prime\prime}=\sum_{l=1}^{3}V_{kl}d_{l}~{}.$ (51)
In the forthcoming papers [21, 22] we will prove that the nonmass eigenstates
play an important role in constructing an electroweak model without Higgs
mechanism.
By interpreting the proper time as the fifth coordinate, we define the
operator of the rest mass and give the concepts of mass eigenstate and nonmass
eigenstate. The general equations for nonmass eigenstates of free spin-0,
spin-$\frac{1}{2}$ and spin-1 fields are obtained. It is found that there are
two kinds of $U(1)$ gauge fields: The $U(1)$ gauge field of first kind merely
couples to mass eigenstates, in which the gauge boson is massless. The second
kind of $U(1)$ gauge field couples to nonmass eigenstates, whose gauge boson
may be massive.
Acknowledgement: I am grateful to Prof. Chao-Jun Feng and Prof. Dao-Jun Liu
for their enlightening discussions. It is very important that Prof. Chao-Jun
Feng indicated that the rest mass operator could be well defined by proper
time without introducing an extra dimension.
## References
* [1] S. L. Glashow, Nucl. Phys. 22, 579 (1960); J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2, 1285 (1970).
* [2] W. Greiner and B. Müller, Gauge Theory of Weak Interactions, (3rd. edition), (Springer-Verlag, 2000).
* [3] P. W. Higgs, Phys. Lett. 12, 132 (1964); P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964); P. W. Higgs, Phys. Rev. 145, 1156 (1966); T. W. Kibble, Phys. Rev. 155, 1554 (1967); F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).
* [4] Z. Z. Xing, Int. J. Mod. Phys. A 19, 1 (2004).
* [5] O. M. Lecian and G. Montani, Int. J. Mod. Phys. D 15, 717 (2006).
* [6] K. S. Soh and P. Y. Pac, Phys. Rev. D 35, 544 (1987).
* [7] A. Macias and H. Dehnen, Class. Quantum Grav. 8, 203 (1991).
* [8] J. Polchinski, String Theory , Vols. I, II (Cambridge University Press, 1998).
* [9] L. Freidel, F. Girelli and E. R. Livine, Phys. Rev. D 75, 105016 (2007)[arXiv:hep-th/0701113].
* [10] D. J. Griffiths, Introduction to Quantum Mechanics, (2nd. edition), (Pearson Education, 2005).
* [11] S. Weinberg, The Quantum Theory of Fields, Vol. I, (Cambridge University Press, 1995).
* [12] N. Straumann, “Unitary Representations of the Inhomogeneous Lorentz Group and Their Significance in Quantum Physics”, [arXiv:math-ph/0809.4942].
* [13] S. Ichinose, Phys. Rev. D 66, 104015 (2002)[arXiv:hep-th/0206187].
* [14] S.-Q. Wu, Phys. Rev. D 78, 064052 (2008)[arXiv:hep-th/0807.2114].
* [15] F. Scheck, Electroweak and Strong Interactions, (2nd. edition), (Springer-Verlag, 1996).
* [16] P. H. Frampton, Gauge Field Theories, (3rd. edition), (WILEY-VCH Verlag GmbH & Co. KGaA, 2008).
* [17] T.-P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics, (Clarendon Press, 1984).
* [18] L. H. Ryder, Quantum Field Theory, (2nd. edition), (Cambridge University Press, 1996).
* [19] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).
* [20] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
* [21] X.-B. Huang, “Massive Gauge Bosons in Yang-Mills Theory without Higgs Mechanism”, [arXiv:hep-ph/0906.2584].
* [22] X.-B. Huang, “An Electroweak Model without Higgs Mechanism”, in preparation.
|
arxiv-papers
| 2009-06-13T02:34:47 |
2024-09-04T02:49:03.328214
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin-Bing Huang",
"submitter": "Xin-Bing Huang",
"url": "https://arxiv.org/abs/0906.2441"
}
|
0906.2446
|
# Ftklipse – Design and Implementation of an Extendable Computer Forensics
Environment
Software Requirements Specification Document
Marc-André Laverdière Serguei A. Mokhov Suhasini Tsapa Djamel Benredjem
(April 2006)
## Chapter 1 Introduction
### 1.1 Purpose
The purpose behind this document is to describe the features of ftklipse, an
extendable platform for computer forensics. This document will explain the
product for the customer, as well as provide a detailed specification for the
developer.
### 1.2 Scope
Ftklipse is a thick-client solution for forensics investigation. It allows to
collect and preserve evidence, to analyze it and to report on it.
It supports chain of custody management, access control policies and batch
operation of its included tools in order to facilitate and accelerate the
investigation. The environment itself and its tools are configurable as well.
### 1.3 Definitions and Acronyms
Cryptographic Hash Function
Function mapping input data of an arbitrary size to a fixed-sized output that
is highly collision resistant.
JVM
The Java Virtual Machine. Program and framework allowing the execution of
program developed using the Java programming language.
GUI
Graphical User Interface.
### 1.4 Compliance
This document was written based on [So98].
## Chapter 2 Overall Description
### 2.1 Product Perspective
* •
Ftklipse is meant to be a stand-alone product, depending on a variety of
standard tools organized as plug-ins.
* •
Ftklipse is meant to be extendable using plug-ins that will add evidence
gathering and analysis properties
* •
The product has only one interface, a graphical user interface residing on the
client computer
#### 2.1.1 System interfaces
The only interface to the system will be its GUI.
#### 2.1.2 User Interfaces
Ftlipse implements a user interfaces that is evidence-centric. It offers
wizards for each of its features for ease of use. It allows investigators to
record notes for each piece of evidence as well as to record additional
reporting information. Please refer to Figure 2.1 and Figure 2.2 for an
example of the look and feel of the application.
Figure 2.1: User Interface Showing the Case Introduction Figure 2.2: User
Interface Showing the Evidence Information and Notes
#### 2.1.3 Software Interfaces
The product must expose a software interface for plug-in developers to use.
The interfaces provided must allow to:
* •
Register the plug-in
* •
Extend the Graphical User Interface’s tool menus (window, pop-up, etc.)
* •
Offer an interface for the plug-in to implement to allow callbacks enabling
execution
### 2.2 Product Functions
The system will implement the following functionalities:
* •
Creation of cases
* •
Evidence Gathering using integrated and plug-in tools
* •
Evidence Integrity validation using a hash function
* •
Evidence Import from any media to an existing case
* •
Logging of all operations performed on the evidence
* •
Validation of integrity of evidence after each operation over it
* •
Display of evidence in read-only mode either in ASCII, Unicode or Hex formats
* •
Recording of investigative notes for each piece of evidence
* •
Capability to extract a part of the evidence into another file
* •
Capability to copy and rename the copy of the evidence
* •
Generation of reports in PDF and LaTeX2e formats that includes listing of the
evidence in the case, a printout of selected parts of the evidence, the
investigative notes related to selected parts of the evidence and a customized
executive summary, introduction, and conclusion. It also integrates the chain
of custody information for each part of the evidence displaying the principal,
time stamp and operation performed on the evidence.
* •
An extendable set of tools through a plug-in architecture
* •
Tool-specific defaults and configuration screens
### 2.3 User Characteristics
Users are cyber forensics investigators. They are experienced using existing
sets of tools, and will be trained in the use of ftklipse before its
deployment.
Indirect users are investigators, prosecutors, judges and laypersons, which
will consult the reports generated. They expect reports of high quality which
demonstrate objectivity and methodology.
### 2.4 Constraints
#### 2.4.1 Hardware Constraints
Any computer able to operate the Eclipse platform can be used to operate
Ftklipse.
#### 2.4.2 Software Constraints
It is assumed that the investigator’s computer supports and includes the
following programs:
* •
JVM, version 5 or higher
* •
LaTeX2e, preferably pdflatex
Other tools are not assumed to be present, as they are integrated in each
plug-in.
In the case of using Ftklipse for evidence collection only, only the JVM is
required.
### 2.5 Assumptions and Dependencies
The software assumes a non-hostile environment (i.e. not aiming at disturbing
its operation).
### 2.6 Apportioning of requirements
Some features are to be implemented in later versions of Ftklipse, notably:
* •
Integration of the Access Control framework with administrator screens
* •
LaTeXoutput of reports
* •
Object-specific logging
* •
Hexadecimal and image display
* •
Evidence Extraction
## Chapter 3 Specific Requirements
### 3.1 External Interfaces
The product must expose a software interface for plug-in developers to use.
The interfaces provided must allow to:
* •
Register the plug-in
* •
Extend the Graphical User Interface’s tool menus (window, pop-up, etc.)
* •
Offer an interface for the plug-in to implement to allow callbacks enabling
execution
### 3.2 Functional Requirements
#### 3.2.1 Domain Model
Our domain model is a traditional police investigation one, augmented with
some information specific to cyber forensics and our requirements[Deb]. It is
summarized in Figure 3.1.
Figure 3.1: Domain Model for Ftklipse
#### 3.2.2 Use Case Model
The use case model for Ftklipse is illustrated in Figure 3.2.
Figure 3.2: Use Case Diagram for Ftklipse
### 3.3 Requirements Description
#### 3.3.1 Creation of cases
##### Description
Ftklipse allows the creation of cases with their associated metadata, as
specified in section 3.5.
##### Criticality
This feature is critical to the software
##### Technical Issues
None
##### Dependencies with Other Requirements
None
#### 3.3.2 Evidence Gathering
##### Description
Ftklipse allows to run different tools in order to perform evidence collection
on a live system.
##### Criticality
This feature is critical to the software.
##### Technical Issues
The collection of the output of the gathering tool can be problematic,
considering the variety of tools and their working. The redirection of the
tool’s standard input and output in a manner useful to the investigator should
be considered.
##### Dependencies with Other Requirements
None
#### 3.3.3 Evidence Analysis
##### Description
Ftklipse allows to run different tools on one or more selected evidences, as
well as to operate a batch analysis. In the latter case, the system must offer
a GUI to the user that allows the selection of the evidence and operations to
perform on it.
##### Criticality
The ability to analyze the evidence is critical. However, the automated
analysis of multiple pieces of evidence is not critical.
##### Technical Issues
The development of a generic programming interface for the variety of analysis
tools is likely to be complex.
##### Dependencies with Other Requirements
None
#### 3.3.4 Evidence Integrity Validation
##### Description
Ftklipse records the SHA-1 signature of every piece of evidence and ensures
that the evidence is kept correct during the investigation. In the case of a
corruption of the evidence, Ftklipse detects it and records which operation
caused this corruption.
##### Criticality
This feature is important to the operation of the software, although not
critical.
##### Technical Issues
##### Dependencies with Other Requirements
#### 3.3.5 Evidence Import
##### Description
Ftklipse allows to import evidence that was collected outside of itself. The
evidence must be accompanied by a SHA-1 digest that is correct in order to
import the evidence in the system.
##### Criticality
This feature is important, although not critical.
##### Technical Issues
The encoding and format of the SHA-1 signature can vary from one tool to
another.
##### Dependencies with Other Requirements
#### 3.3.6 Logging
##### Description
All operations are logged globally by Ftklipse. Furthermore, all operations
related to a given piece of evidence are logged for that evidence
specifically.
##### Criticality
The global logging is critical to Ftklipse. The specific logging is important,
but not essential.
##### Technical Issues
##### Dependencies with Other Requirements
#### 3.3.7 Evidence Display
##### Description
The evidence can be visualized, if authorized, in read-only mode either in
ASCII, Unicode or Hex formats. Furthermore, images can be viewed within
Ftklipse and can be opened in an external viewer program.
##### Criticality
This function is critical to the operation of the software in ASCII.
##### Technical Issues
##### Dependencies with Other Requirements
#### 3.3.8 Recording of Investigative Notes
##### Description
The investigator must be able to record information regarding each piece of
evidence, as well as report-specific information.
##### Criticality
This function is critical to the operation of Ftklipse.
##### Technical Issues
##### Dependencies with Other Requirements
#### 3.3.9 Evidence Extraction
##### Description
The investigator must be able to select a subset of the viewed evidence and
extract it into another file, which will then be treated as evidence itself.
Ftklipse must record this operation and keep relationship information in the
database of evidence.
##### Criticality
This feature is of moderate importance.
##### Technical Issues
##### Dependencies with Other Requirements
#### 3.3.10 Evidence Cloning
##### Description
The investigator must be able to copy a piece of evidence in full and
optionally to rename the copy.
##### Criticality
This feature is nice to have.
##### Technical Issues
##### Dependencies with Other Requirements
#### 3.3.11 Report Generation
##### Description
The investigator must be able to generate a report for a selected case that
includes all evidence, their notes, as well as other report-specific data. The
output formats can be PDF or LaTeX2e.
##### Criticality
This feature is critical.
##### Technical Issues
##### Dependencies with Other Requirements
#### 3.3.12 Plug-in Architecture
##### Description
Ftklipse allows third-party developers to create plug-ins that can be added at
configuration time by system administrators.
##### Criticality
This feature is critical.
##### Technical Issues
##### Dependencies with Other Requirements
#### 3.3.13 Access Control Management
##### Description
Ftklipse operates with an access control list for each case, piece of
evidence, and report information. Each user must be authenticated and each
operation must be authorized in the view of the user’s access rights.
Notably, the rights that must be implemented are:
* •
View rights over a case or piece of evidence. This defines if the user is
authorized to be aware of the existence of a given case or piece of evidence.
* •
Read rights over a case or piece of evidence. This defines if the user, being
previously granted view rights over the object, is able to read the case’s
information or visualize or operate on a piece of evidence.
* •
Write rights over a case or piece of evidence. This defines if the user is
authorized to add to the general case notes or the evidence notes. This also
defines if the user is allowed to add evidence to a given case.
By default, Ftklipse must offer default access rights based on the user’s
role, as well as default access rights for different categories of objects.
Ftklipse must provide GUI tools to manage the both user and object rights.
##### Criticality
This feature is important, not critical.
##### Technical Issues
The implementation of the access control algorithm can be complex.
Furthermore, some administration functions (such as the impact of a
redefinition of default rights) require some thought to ensure that no
previously confidential information becomes publicly available.
##### Dependencies with Other Requirements
#### 3.3.14 Tool-specific defaults and configuration screens
##### Description
Each tool is responsible to maintain its state, notably regarding its default
settings which must be modifiable by the user and preserved from one run of
ftklipse to another.
Each tool must supply a screen that allows to set the proper parameters before
the operation of the tool.
Default options are to be used on direct invocation of the tool.
##### Criticality
This feature is important
##### Technical Issues
##### Dependencies with Other Requirements
### 3.4 Performance Requirements
Ftklipse does not have any particular performance requirements
### 3.5 Logical Database Requirements
A database is required in order to store the case management and chain of
custody information.
The database must be able to store:
* •
The relationship between parts of the evidence
* •
The operations done on the evidence, including its time stamp, its description
and the investigator that performed it.
The information that must be tracked by the database is the following:
* •
The case’s meta-information (ID, details, description, timestamps,
investigators)
* •
The case’s evidence.
* •
The user credentials.
* •
The object access control lists.
* •
The chain of custody over every piece of evidence. This includes the
cryptographic hash sums, the operations performed on the evidence and the
principal who performed it.
### 3.6 Design Constraints
The design must take in consideration that the base implementation language is
Java. It also must take in consideration the different options of the tools
that can be plugged into it.
### 3.7 Software System Attributes
In this section, we describe the non-functional attributes of Ftklipse.
#### 3.7.1 Security
#### 3.7.2 Reliability
The software must behave correctly during 20 continuous hours of operation.
#### 3.7.3 Availability
There are no availability constraints.
#### 3.7.4 Maintainability
The software must allow for tool plug-ins to be integrated automatically. The
software must also be self-updatable.
#### 3.7.5 Portability
The software must operate on POSIX and Windows systems. Tools integrated in
the software must be adjusted accordingly.
## Bibliography
* [Deb] M. Debbabi. Course notes from inse 6150.
* [So98] S. Standards and C. of. the ieee. ieee recommended practice for software requirements specifications, 1998.
## Chapter 4 Supporting Information
###### Contents
1. 1 Introduction
1. 1.1 Purpose
2. 1.2 Scope
3. 1.3 Definitions and Acronyms
4. 1.4 Compliance
2. 2 Overall Description
1. 2.1 Product Perspective
1. 2.1.1 System interfaces
2. 2.1.2 User Interfaces
3. 2.1.3 Software Interfaces
2. 2.2 Product Functions
3. 2.3 User Characteristics
4. 2.4 Constraints
1. 2.4.1 Hardware Constraints
2. 2.4.2 Software Constraints
5. 2.5 Assumptions and Dependencies
6. 2.6 Apportioning of requirements
3. 3 Specific Requirements
1. 3.1 External Interfaces
2. 3.2 Functional Requirements
1. 3.2.1 Domain Model
2. 3.2.2 Use Case Model
3. 3.3 Requirements Description
1. 3.3.1 Creation of cases
2. 3.3.2 Evidence Gathering
3. 3.3.3 Evidence Analysis
4. 3.3.4 Evidence Integrity Validation
5. 3.3.5 Evidence Import
6. 3.3.6 Logging
7. 3.3.7 Evidence Display
8. 3.3.8 Recording of Investigative Notes
9. 3.3.9 Evidence Extraction
10. 3.3.10 Evidence Cloning
11. 3.3.11 Report Generation
12. 3.3.12 Plug-in Architecture
13. 3.3.13 Access Control Management
14. 3.3.14 Tool-specific defaults and configuration screens
4. 3.4 Performance Requirements
5. 3.5 Logical Database Requirements
6. 3.6 Design Constraints
7. 3.7 Software System Attributes
1. 3.7.1 Security
2. 3.7.2 Reliability
3. 3.7.3 Availability
4. 3.7.4 Maintainability
5. 3.7.5 Portability
4. 4 Supporting Information
###### List of Figures
1. 2.1 User Interface Showing the Case Introduction
2. 2.2 User Interface Showing the Evidence Information and Notes
3. 3.1 Domain Model for Ftklipse
4. 3.2 Use Case Diagram for Ftklipse
|
arxiv-papers
| 2009-06-13T04:46:52 |
2024-09-04T02:49:03.333418
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marc-Andr\\'e Laverdi\\`ere, Serguei A. Mokhov, Suhasini Tsapa, and\n Djamel Benredjem",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/0906.2446"
}
|
0906.2447
|
# Ftklipse – Design and Implementation of an Extendable Computer Forensics
Environment
Specification Design Document
Marc-André Laverdière Serguei A. Mokhov Suhasini Tsapa Djamel Benredjem
(April 24, 2006)
###### Contents
1. 1 Introduction
1. 1.1 Purpose
2. 1.2 Scope
3. 1.3 Definitions and Acronyms
2. 2 System Overview
1. 2.1 Architectural Strategies
2. 2.2 System Architecture
1. 2.2.1 Module View
3. 2.3 Execution View
1. 2.3.1 Runtime Entities
2. 2.3.2 Communication Paths
3. 2.3.3 Execution Configuration
4. 2.4 Coding Standards and Project Management
3. 3 Detailed System Design
1. 3.1 Class Diagrams
2. 3.2 Data Storage Format
1. 3.2.1 Entity Relationship Diagram
2. 3.2.2 External Systems and Databases
3. 3.2.3 Log File Format
3. 3.3 Directory and Package Organization
4. 3.4 Plug-Ins
5. 3.5 User Interface Design
1. 3.5.1 Appearance
4. 4 Conclusion
1. 4.1 Summary of Technologies Used
2. 4.2 Summary of Tools Added
3. 4.3 Summary of Difficulties
4. 4.4 Limitations and Technological Restrictions
5. 4.5 Future Work and Work-In-Progress
6. 4.6 Acknowledgments
## Chapter 1 Introduction
This chapter briefly presents the purpose and the scope of the work on the
Ftklipse project with a subset of relevant definitions and acronyms. All these
aspects are detailed to some extent later through the document.
### 1.1 Purpose
To design and implement a plugin-based environment that allows to integrate
forensic tools working together to support programming tasks and addition of
new tools. Integration is done through GUI components.
### 1.2 Scope
The end-product enviroment must have user friendly GUI, configuration
capabilities, plug-in capabilities to insert/inject new tools, case
management, and chain of custody capabilities, along with evidence gathering
capabilities, evidence preservation capabilities, and, finally report
generation capabilities. A subset of these requirements has been implemented
in Ftklipse, which is detailed throughout the rest of this document.
### 1.3 Definitions and Acronyms
Cryptographic Hash Function
Function mapping input data of an arbritary size to a fixed-sized output that
is highly collision resistant.
Digital evidence
Information stored or transmitted in binary form that may be relied upon in
court.
dcfldd
Enhanced DD imager with built-in hashing, works like dd from command line.
Hashing on-the-fly
dcfldd can hash the input data as it is being transferred, helping to ensure
data integrity.
Status output
dcfldd can update the user of its progress in terms of the amount of data
transferred and how much longer operation will take.
Flexible disk wipes
dcfldd can be used to wipe disks quickly and with a known pattern if desired.
Image/wipe verify
dcfldd can verify that a target drive is a bit-for-bit match of the specified
input file or pattern.
Multiple outputs
dcfldd can output to multiple files or disks at the same time.
Split output
dcfldd can split output to multiple files with more configurability than the
split command.
Piped output and logs
dcfldd can send all its log data and output to commands as well as files
natively.
Documentation
Written notes, audio/videotapes, printed forms, sketches, and/or photographs
that form a detailed record of the scene, evidence recovered, and actions
taken during the search of the scene.
JVM
The Java Virtual Machine. Program and framework allowing the execution of
program developped using the Java programming language.
Magnetic media
A disk, tape, cartridge, diskette, or cassette that is used to store data
magnetically.
Steganography
It simply takes one piece of information and hides it within another. Computer
files (images, sounds recordings, even disks) contain unused or insignificant
areas of data. Steganography takes advantage of these areas, replacing them
with information (encrypted mail, for instance). The files can then be
exchanged without anyone knowing what really lies inside of them. For example,
an image of the space shuttle landing might contain a private letter to a
friend. A recording of a short sentence might contain your company’s plans for
a secret new product. Steganography can also be used to place a hidden
“trademark” in images, music, and software, a technique referred to as
watermarking.
SWT
The Standard Widget Toolkit [Con06c], a set of graphical user interface
components provided by the Eclipse framework.
Temporary and swap files
Many computers use operating systems and applications that store data
temporarily on the hard drive. These files, which are generally hidden and
inaccessible,may contain information that the investigator finds useful.
## Chapter 2 System Overview
In this chapter, we examine the architecture of our implementation of
Ftklipse. We first introduce our architectural philosophy before informing the
reader about the Siemens Four View Model, an architectural methodology for the
conception of large-scale software systems. Afterwards, we examine each of the
view, as architected for our system. Finally, we conclude with other software
engineering matters that were of high importance in the development of our
implementation.
### 2.1 Architectural Strategies
Our principles are:
Platform independence
We target systems that are capabale of running a JVM.
The Eclipse plug-in based environment
slightly imitating the MVC (Model-view -Controller) pattern, to map the
traditional input, processing, output roles into the GUI realm. In Eclipse
model, a plug-in may be related to another plug-in by one of two
relationships:
Dependency
The roles in this relationship are dependent plug-in and prerequsite plug-in.
A prerequisite plug-in supports the function of a dependent plug-in.
Extension
The roles in this relationship are host plug-in and extender plug-in. An
extender plug-in extends the functions of a host plug -in.
Database independent API
will allows us to swap database engines on-the-fly.
Reasonable Efficiency
We will architect and implement an efficient system, but will avoid advanced
programming tricks that improve the efficiency at the cost of maintainability
and readability.
Simplicity And Maintainability
We will target a simplistic and easy to maintain organization of the source.
Architectural Consistency
We will consistently implement our architectural approach.
Separation of Concerns
We will isolate separate concerns between modules and within modules to
encourage reuse and code simplicity.
### 2.2 System Architecture
#### 2.2.1 Module View
##### Layering
We divided our application between layers. The top level has a front-end and a
back-end. The frontend comprised a collection of GUI modules provided by and
customized from eclipse as well as custom-designd by the team. The backend
consists of supporting functionality and specifically database management,
report generation, and external tool invocation.
##### Interface Design
Several interfaces had to be designed for the architecture to work All the
backend modules have an interface they expose to the frontend to use. Thus,
there are interfaces between, GUI-to-External-Tools, GUI-to-Database, and GUI-
to-Report-Generation. All these are designed to be swappable and highly
modular so any component series can be replaced at any time with little or no
change to the code. The interfaces (FtklipseCommonDatabaseBridge and
IDatabaseAdapter, ITool and IToolExecutor, and IReportGenerator and
ReportGeneratorFactory) are presented in the detiled design chapter.
### 2.3 Execution View
#### 2.3.1 Runtime Entities
In the case of our application, there is hosting run-time environment that of
Eclipse. The application can run within Eclipse IDE or be a stand-alone with a
minimal subset of the Eclipse run-time. By nature, a JVM machine is executing
all the environment and all GUI-based applications are multi-threaded to avoid
blockage on user’s input. Additionally, depending on the database engine used
behind the scenes, it may as well be multi-threaded to provide concurrent
access and connection pooling.
#### 2.3.2 Communication Paths
It was resolved that the modules would all communicate through message passing
between methods. Communication to the database depends on the database
adapter, and in our sample implementation is done through and in-process JDBC
driver. Additionally, Java’s reflection is used to discover instantiation
communication paths at run-time for pluggable modules.
#### 2.3.3 Execution Configuration
Execution configuration in Ftklipse has to do with where its data directory
is. The data directory is always local to where the application was ran from.
The directory contains the main case database in the ftklipsedb.* files as
well as numerical directorys with case ID with imported evidence files.
Additionall configuration for application is located in plugin.properties and
plugin.xml files.
### 2.4 Coding Standards and Project Management
In order to produce high-quality code, we decided to normalize on the OpenBSD
style. We also decided to use javadoc source code documentation style for its
completeness and the automated tool support. We used Subversion (svn) [Col07]
in order to manage the source code, makefile, and documentation revisions
provided by SourceForge.net.
## Chapter 3 Detailed System Design
* •
Case management: Investigations are organized by cases, which can contain one
or more evidences. Each evidence can contain one or more file system images to
analyze;
* •
Evidence Gathering using integrated and plug-in tools;
* •
Evidence Integrity validation using a hash function;
* •
Evidence Import from any media to an existing case;
* •
Logging of all operations performed on the evidence;
* •
Validation of integrity of evidence after each operation over it;
* •
Display of evidence in read-only mode either in ASCII, Unicode or Hex formats;
* •
Recording of investigative notes for each piece of evidence;
* •
Capability to extract a part of the evidence into another file;
* •
Capability to copy and rename the copy of the evidence;
* •
Generation of reports in PDF and LaTeX2e formats that includes listing of the
evidence in the case, a printout of selected parts of the evidence, the
investigative notes related to selected parts of the evidence and a customized
executive summary, introduction, and conclusion. It also integrates the chain
of custodity information for each part of the evidence displaying the
principal, timestamp and operation performed on the evidence.
* •
An extendable set of tools through a plug-in architecture;
* •
General as well as tool-specific defaults and configuration screens;
### 3.1 Class Diagrams
We have a number of class diagrams representing the majore modules and their
relationships. Please located the detailed descriprion of the modules in the
generated HTML of javadoc or the javadoc comments themeselves in the
doc/javadoc directory.
The basic UI classes are in Figure 3.1. The prototype internal access control
classes are in Figure 3.2. The main database abstraction is in Figure 3.3.
Next, concrete database adatpters are in Figure LABEL:fig:uml:dbadapters..
Further, the database- and UI-indepedent database objects data structures are
in Figure 3.5. The report generation-related API is in Figure 3.6. Finally,
the external tools invocation framework is in Figure 3.7.
Figure 3.1: Class Diagram for the basic User Interface Figure 3.2: Class
Diagram for Access Control Framework Figure 3.3: Class Diagram for the
Database Root Package Figure 3.4: Class Diagram for the Database Adapters
Figure 3.5: Class Diagram for the Database Objects Figure 3.6: Class Diagram
for the Report Generation Figure 3.7: Class Diagram for the Backend Tools
Framework
### 3.2 Data Storage Format
This section is about data storage issues and the details on the chosen
undelying implementation and ways of addressing those issues.
#### 3.2.1 Entity Relationship Diagram
The ER diagram of the underlying SQL engine we chose is in Figure 3.8. The
database is pretty simple as the case_data field is a BLOB to which the Case
data structure is serialized. The id_count table is simply there to contain
the maximum ID used accros the database objects in the application. It is
updated on application close, so when the application is loaded back again, it
sets its internal ID from the database properly for newly created cases and
other objects.
Figure 3.8: Simple ER Diagram of the Internal Database
The database is slatted for extension with some code map data for the UI as
well as log facilities later on for better reporting, like who, what, when,
etc.
#### 3.2.2 External Systems and Databases
The database engine the Ftklispe application talks to is abstracted away so
that the actual engine particularities (e.g. SQL queries or XML atoms) are not
visible to the application thus making it database-engine independent. The
provision was made to have SQL, XML, JavaSpaces [Mam05], or raw object
serialization databases. The actual external database engine used in the demo
version of the toolkit, is the HSQLDB [The08] database, which is implemented
in Java itself and has an in-process execution capability. This database
engine is started automatically within the same process as an application when
a first connection is made. It is shudown when application exits. This choice
is justified by simplicity and does not require an external database server to
be set up. This external implementation of the engine is in lib/hsqldb.jar.
The database-produced files are stored in the data directory relative to the
current execution environment. The files are ftklipsedb.properties and
ftklipsedb.script. The former describes the global database settings and the
latter is the serilized database itself, including DDL DML statements to
reproduce the database. Both are managed by the HSQLDB engine itself.
Originally when deploying the application, neither may present. They will be
created if not present when Ftklipse starts.
Another external system we rely on in the form of library is the PDF
generation library iText [LS06] [LS06], which is in lib/itext.jar. This
library is used in PDFReportGenerator to produce a PDF copy of the case data
stored in the database.
#### 3.2.3 Log File Format
The log is saved in the ftklipse.application.log. As of this version, the file
is produced with the help of the Logger class that has been imported from MARF
[The09]. (Another logging facility that was considered but not yet implemented
is the Log4J tool [AGS+06], which has a full-fledged logging engine.) The log
file produced by Logger has a classical format of [ time stamp ]: message. The
logger intercepts all attempts to write to STDOUT or STDERR and makes a copy
of them to the file.
### 3.3 Directory and Package Organization
In this section, we introduce the reader to the structure of the folders for
ftklipse. Please note that Java, by default, converts sub-packages into
subfolders, which is what we see in Figure 3.9.
Please also refer to Table 3.1 and Table 3.2 for description of the data
contained in the folders and the package organization, respectively.
Figure 3.9: Folder Structure of the Project Folder | Description
---|---
bin | Directory containing the compiled files. All package names described here are also present under this directory.
data | Directory containing the case database as well as subdirectories for each of the cases.
doc | Project’s documentation
example_evidence | Demo evidence that can be used in the projects
icons | icons useable for branding and decorating the application
lib | External libraries used by ftklipse
META-INF | Project’s meta-information that would be included in a JAR bundle
references | Some useful references on the web on Eclipse development
schema | Project’s extension point definitions
src | Directory containing the source code files. All package names described here are underneath this directory
tools | Precompiled tools to use. Also organized hierarchically.
Table 3.1: Details on folder structure Package | Description
---|---
ca.concordia.ciise.ftklipse | Ftklipse’s root package name
ca.concordia.ciise.ftklipse.accesscontrol | Ftklipse’s access control model
ca.concordia.ciise.ftklipse.database | Ftklipse’s database module
ca.concordia.ciise.ftklipse.database.adapters | Database adapters
ca.concordia.ciise.ftklipse.database.connection | Database connection objects
ca.concordia.ciise.ftklipse.database.objects | Object model that is saved and restored from the database
ca.concordia.ciise.ftklipse.database.reporting | Reporting sub-module
ca.concordia.ciise.ftklipse.database.util | Database utility classes
ca.concordia.ciise.ftklipse.junit | Some JUnit tests
ca.concordia.ciise.ftklipse.tools | Tool execution module, not including GUI screens
ca.concordia.ciise.ftklipse.tools.executors | Tool execution adapters for the underlying platform
ca.concordia.ciise.ftklipse.tools.linux | Tool adapters for Linux tools
ca.concordia.ciise.ftklipse.tools.windows | Tool adapters for Windows tools
ca.concordia.ciise.ftklipse.ui | Ftklipse’s user interface classes
ca.concordia.ciise.ftklipse.ui.actions | Eclise actions for the menu and right-click menu
ca.concordia.ciise.ftklipse.ui.tools | User interfaces for the tools provided by default
ca.concordia.ciise.ftklipse.util | Utility classes
Table 3.2: Package organization
### 3.4 Plug-Ins
In order to allow tools to be plugged in, we use Eclipse’s default mechanism,
which requires to define and export and extension point. The extension point
Table 3.3 defines a set of properties that are mostly used to populate the
user interface as well as providing the interfaces that must be implemented in
order to contribute a plug-in to ftklipse.
Attribute | Type | Summary
---|---|---
id | string | unique identifier for the tool
name | string | name of the tool. Not currently used
class | ITool | class implementing our standard interface for the tool execution
type | enumeration | one of collection, analysis or other. Used for structuring tools in menus
parameter | string | for future use, allowing a tool to register more than once but with different paramters that would let it act differently.
outputfile | string | for future use, allowing a tool to register and specify a default output file for its operation
category | string | for future use, in order to group tools for batch collection or batch analysis of data
platform | enumeration | either win or unix. To specify on which platform the tool operates
inBatchMenu | boolean | whether the plug-in requires to be registered in batch processing menus
inRightClickMenu | boolean | whether the plug-in requires to be registered in the right-click menu
friendlyName | string | short name of the tool, for displaying the user
uiclass | ITooUI | class implementing our standard interface for the tool execution
Table 3.3: Extension Point for Third-Party Plug-Ins
Any third party can contribute a plug-in tool in ftklipse by creating an
Eclipse plug-in project that chooses to extend ca.concordia.ciise.
ftklipse.ftklipse_tools. Those plug-ins can afterwards be installed manually
in the Eclipse folder’s sub-root, or using Eclipse’s built-in installer and
updater. When installed properly, ftklipse will detect them without the need
to update any configuration file or perform other similar adminsitrative
works.
Each plug-in is responsible for implementing its own dialog(s) and may
optionally define its own parameters persistence mechanism, although our API
strongly sugests the use of Eclipse’s technology to do so.
In order that all tools can have access to information from the user
interface, and that the user interface can have access to information about
all tools, we used a set of registry singletons which are responsible to
conserve single instances of the information.
Plug-in developpers would thus find the WidgetRegistrySingleton to be very
helpful, as it notably returns a reference to the case and evidence tree,
which can be queried to find the active evidence and active projects.
As such, we do not implement a strict Model-View-Controller (MVC)
architecture, but merely a model that is similar to it, as the plug-ins are
trusted not to modify and user interface elements.
### 3.5 User Interface Design
#### 3.5.1 Appearance
Ftklipse is implemented using JFace and SWT, technologies provided within the
Eclipse framework. It consists of a single window composed of a menu bar on
the top, a tree structure on the left-hand side, and a multiply-tabbed area at
the centre.
This central area displays information about the currently opened evidence
file or case information from the case database. Please refer to Figure 3.10
and Figure 3.11 for screenshots of the implementation.
Figure 3.10: User Interface Showing the Case Introduction Figure 3.11: User
Interface Showing the Evidence Information and Notes
## Chapter 4 Conclusion
Despite the technological difficulties and limitations the chosen approach
seems very promising. Highly modular design allows also swapping module
implementaions from one technology to another if need be making it very
extensible. Case management, very strong backend architecture for Tools,
Database, and Report Generation. Eclipse UI integration are strong points of
this project.
### 4.1 Summary of Technologies Used
The following were the most prominent technologies used throughout the
project:
* •
Eclipse IDE[E+08]
* •
iText PDF generation library [LS06]
* •
HSQLDB lighweight embedded Java SQL engine [The08]
* •
Visual Editor for Eclipse [Con06e]
### 4.2 Summary of Tools Added
The number of testing tools is not large and many more could be added from
various resources [htt06], however, there were enough for many test cases
given time limitations. The following Linux tools were used for testing and
worked:
* •
stegdetect [Pro04], stegbreak, stegdeimage, magic2mime,
* •
file,
* •
strings,
* •
dcfldd.
### 4.3 Summary of Difficulties
Learning curve for Eclipse plug-in and UI frameworks [Bol03, Con06b, Gal02,
KFL02, Pro05, Bur06, Con06f, Con06d, Con06a] with large volumes of APIs and
documentation was overwhelming at the beginning and making things like right-
and double-click to work as well as SWT-based [Con06c]. UIs was sometimes non-
trivial.
### 4.4 Limitations and Technological Restrictions
The Eclipse framework imposes some technological restrictions in user
interface programming on two major areas that impacted our design.
The first restriction is that the menu items are populated by ‘Actions’, and
that it is impractical to have a different Action instance for each menu item
for each possible item the menu can interact with. For example, the right-
click menu, although capable of being dynamically generated every time,
requires to perform an action based on the currently selected item. Re-
creating the menu on each right-click from new objects is expensive both in
memory and computationally, risking to create an interface with a high
response time to the user, which impacts negatively on usability. Another
option is to create a cache of such items and change internal data members
related to the selected widget before displaying the menu. This approach
increases complexity and was not considered to be a good solution in our
context, due to the complexity of propagating this strategy to existing and
future options. Finally, we considered having a central access point to the
information on the selected items that would be opaque to the underlying data
types creating the tree hierarchy. This last approach, altough less ‘pure’
object-oriented design, was retained for its ease of use in prototyping new
features, as well as the assumed atomicity of GUI operation (i.e. it should
not be possible to change the selection while the handling of the right-click
on the selection is running).
The second restriction is Eclipse’s all-or-nothing approach to plug-in
development. As far as we understood the framework, it is possible to use
Eclipse’s internal data types and existing advanced widgets only when
extending the framework in our plug-in. A plug-in that would choose not to
follow Eclipse’s organization (which is our case) could thus not have access
to pre-existing file browsers and variety of editors. As such, the tree
hierarchy, mouse handling, and data visualization needed to be reimplemented
from lower-level SWT components.
### 4.5 Future Work and Work-In-Progress
Allow addition of tools dynamically though GUI Improve case management with
full chain of custody (backend is done for this) Integration of the
hexadecimal editor plugin [Pal06]
### 4.6 Acknowledgments
* •
Dr. Mourad Debbabi for the excellent course.
* •
Open Source community for Eclipse, HSQLDB, iText
* •
Dr. Peter Grogono for LaTeX introductory tutorial [Gro01]
## Bibliography
* [AGS+06] N. Asokan, Ceki Gulcu, Michael Steiner, IBM Zurich Research Laboratory, and OSS Contributors. log4j, Hierachical Logging Service for Java. apache.org, 2006. http://logging.apache.org/log4j/.
* [Bol03] Azad Bolour. Notes on the Eclipse Plug-in Architecture. eclipse.org, July 2003. http://www.eclipse.org/articles/Article-Plug-in-architecture/plugin_arc%hitecture.html.
* [Bur06] Ed Burnette. Rich Client Tutorial. eclipse.org, February 2006. http://www.eclipse.org/articles/Article-RCP-1/tutorial1.html.
* [Col07] Inc. CollabNet. Subversion (SVN). tigris.org, 2007. http://subversion.tigris.org/.
* [Con06a] Contributors. Creating and using Extension Points. refractions.net, 2006. http://udig.refractions.net/confluence/display/DEV/1+Creating+and+Using%+Extension+Points.
* [Con06b] Contributors. Eclipse Plugin Central - Forums. eclipseplugincentral.com, 2006. http://www.eclipseplugincentral.com/PNphpBB2+file-viewforum-f-74.html.
* [Con06c] Contributors. SWT: The Standard Widget Toolkit. eclipse.org, 2006. http://www.eclipse.org/swt/.
* [Con06d] Contributors. User Guide: Building a Rich Client Platform application. eclipse.org, 2006. http://help.eclipse.org/help31/index.jsp?topic=/org.eclipse.platform.do%c.isv/guide/rcp.htm.
* [Con06e] Contributors. Visual Editor Project. eclipse.org, 2006. http://wiki.eclipse.org/index.php/Visual_Editor_Project.
* [Con06f] Contributors. Workbench User Guide: Plugging into the workbench. eclipse.org, 2006. http://help.eclipse.org/help31/index.jsp?topic=/org.eclipse.platform.do%c.isv/guide/workbench.htm.
* [E+08] Eclipse contributors et al. Eclipse Platform. eclipse.org, 2000-2008. http://www.eclipse.org, last viewed April 2008.
* [Gal02] David Gallardo. Developing Eclipse plug-ins. ibm.com, December 2002. http://www-128.ibm.com/developerworks/opensource/library/os-ecplug/?Ope%n&ca=daw-ec-dr.
* [Gro01] Peter Grogono. A LaTeX2e Gallimaufry. Techniques, Tips, and Traps. Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada, March 2001. http://www.cse.concordia.ca/~grogono/Writings/gallimaufry.pdf, last viewed May 2008.
* [htt06] http://www.dmares.com. Software Links for Forensics Investigative Tasks. 2006\. http://www.dmares.com/maresware/SITES/tasks.htm.
* [KFL02] Dan Kehn, Scott Fairbrother, and Cam-Thu Le. Internationalizing your Eclipse plug-in. ibm.com, June 2002. http://www-128.ibm.com/developerworks/opensource/library/os-i18n/.
* [LS06] Bruno Lowagie and Paulo Soares. iText, a Free Java-PDF library. lowagie.com, 2006. http://www.lowagie.com/iText/.
* [Mam05] Qusay H. Mamoud. Getting Started With JavaSpaces Technology: Beyond Conventional Distributed Programming Paradigms. Sun Microsystems, Inc., July 2005. http://java.sun.com/developer/technicalArticles/tools/JavaSpaces/.
* [Pal06] Marcel Palko. Eclipse Hex Editor Plugin. sourceforge.net, 2006. http://ehep.sourceforge.net/.
* [Pro04] Niels Provos. Steganography detection with stegdetect, 2004. http://www.outguess.org/detection.php.
* [Pro05] Emmanuel Proulx. Eclipse Plugins Exposed. onjava.com, February 2005. http://www.onjava.com/pub/a/onjava/2005/02/09/eclipse.html.
* [The08] The hsqldb Development Group. HSQLDB – lightweight 100% Java SQL database engine v.1.8.0.10. hsqldb.org, 2001–2008. http://hsqldb.org/.
* [The09] The MARF Research and Development Group. The Modular Audio Recognition Framework and its Applications. SourceForge.net, 2002–2009. http://marf.sf.net, last viewed December 2008.
## Index
* API
* Case §3.2.1
* FtklipseCommonDatabaseBridge §2.2.1
* IDatabaseAdapter §2.2.1
* IReportGenerator §2.2.1
* ITool §2.2.1
* IToolExecutor §2.2.1
* Logger §3.2.3, §3.2.3
* PDFReportGenerator §3.2.2
* ReportGeneratorFactory §2.2.1
* Data Storage Format §3.2
* ER Diagram §3.2.1
* External Systems and Databases §3.2.2
* Log File Format §3.2.3
* Design Chapter 3
* Class Diagrams §3.1
* Data Storage Format §3.2
* Detailed System Design Chapter 3
* Directory Structure §3.3
* Files
* data §2.3.3, §2.3.3, §3.2.2
* doc/javadoc §3.1
* ftklipse.application.log §3.2.3
* ftklipsedb.* §2.3.3
* ftklipsedb.properties §3.2.2
* ftklipsedb.script §3.2.2
* lib/hsqldb.jar §3.2.2
* lib/itext.jar §3.2.2
* plugin.properties §2.3.3
* plugin.xml §2.3.3
* Introduction Chapter 1
* Definitions and Acronyms §1.3
* Purpose §1.1
* Scope §1.2
* Java §3.2.2
* Libraries
* HSQLDB §3.2.2
* iText §3.2.2
* JavaSpaces §3.2.2
* MARF §3.2.3
* Package Organization §3.3
* System Overview Chapter 2
* Architectural Strategies §2.1
* Tools
* dcfldd item dcfldd, item dcfldd, item Hashing on-the-fly, item Status output, item Flexible disk wipes, item Image/wipe verify, item Multiple outputs, item Split output, item Piped output and logs, 4th item
* dd item dcfldd
* file 2nd item
* javadoc §2.4
* magic2mime 1st item
* MARF §3.2.3
* stegbreak 1st item
* stegdeimage 1st item
* stegdetect 1st item
* strings 3rd item
* svn §2.4
|
arxiv-papers
| 2009-06-13T05:06:22 |
2024-09-04T02:49:03.337947
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marc-Andr\\'e Laverdi\\`ere, Serguei A. Mokhov, Suhasini Tsapa, and\n Djamel Benredjem",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/0906.2447"
}
|
0906.2483
|
# Many-body instability of Coulomb interacting bilayer graphene: RG approach
Oskar Vafek Kun Yang National High Magnetic Field Laboratory and Department
of Physics, Florida State University, Tallahassee, Florida 32306, USA
###### Abstract
Low-energy electronic structure of (unbiased) bilayer graphene is made of two
Fermi points with quadratic dispersions, if trigonal-warping and other high
order contributions are ignored. We show that as a result of this qualitative
difference from single-layer graphene, short-range (or screened Coulomb)
interactions are marginally relevant. We use renormalization group to study
their effects on low-energy properties of the system, and show that the two
quadratic Fermi points spontaneously split into four Dirac points, at zero
temperature. This results in a nematic state that spontaneously breaks the
six-fold lattice rotation symmetry (combined with layer permutation) down to a
two-fold one, with a finite transition temperature. Critical properties of the
transition and effects of trigonal warping are also discussed.
The ability to predict the nature of the low temperature state of an
interacting quantum system is one of the main goals of condensed matter
theory. Nevertheless, despite ongoing effort, no single method has proved
universally sufficient and experimental input is essentially inevitable.
Under special circumstances, however, progress can be made. In particular, in
non-interacting systems with susceptibilities diverging as the temperature
approaches zero, the inclusion of arbitrarily small interaction can be shown
to lead to a finite, but also arbitrarily small transition temperature. The
method of choice in this case is the renormalization group (RG), which has the
virtue of unbiased determination of the leading instabilityShankar (1994).
In this paper we apply the RG method to the bilayer graphene with Bernal
stackingNovoselov et al. (2006); McCann and Fal’ko (2006); Neto et al. (2009);
Geim and MacDonald (2007). While in general, the motion of the non-interacting
electrons in such potential does not lead to diverging susceptibilities since
the energy spectrum has two sets of four Dirac points in the corners of the
Brillouin zone (due to trigonal warping)McCann and Fal’ko (2006); Neto et al.
(2009), if only nearest neighbor hopping is considered each set of four Dirac
points merges into a single degenerate point with parabolic dispersion (See
Fig. 1). As the nearest neighbor hopping amplitudes are the largest, the
latter is the natural starting point of theoretical analysisNilsson et al.
(2006); Min et al. (2008).
|
---|---
---
Figure 1: (Upper left) Honeycomb bilayer unit cell. Atoms in the lower layer
(2) are marked as empty (black) circles, atoms in the upper layer (1) are
filled (red) circles. As a starting point, only the intralayer nearest
neighbor hopping amplitudes $t$ and the interlayer hopping amplitudes
$t_{\perp}$ are considered. (Upper right) Constant energy contours of the
resulting dispersion, with minima at $K=\frac{4\pi}{\sqrt{3}3a}\hat{y}$ and
$K^{\prime}$ points and maximum at $\Gamma$ point. (Lower left) The energy
dispersion of the four bands along the vertical cut in the Brillouin zone. The
band splitting at the $K$ (and $K^{\prime}$) points is $t_{\perp}$. (Lower
right) Magnification of the dispersion (in units of $t$) near the degeneracy
point (solid black) as well as the dispersion in the nematic state (dashed
red) with $\Delta_{x}\neq 0$ (See Eq.35).
We start with the tight-binding Hamiltonian for electrons hopping on the
bilayer honeycomb lattice with Bernal stacking
$\mathcal{H}=\sum_{\langle{\bf r}{\bf r}^{\prime}\rangle}\left[t_{{\bf r}{\bf
r}^{\prime}}c_{\sigma}^{\dagger}({\bf r})c_{\sigma}({\bf
r}^{\prime})+h.c.\right]+\frac{1}{2}\sum_{{\bf r}{\bf
r}^{\prime}}\delta\hat{n}({\bf r})V({\bf r}-{\bf
r}^{\prime})\delta\hat{n}({\bf r}^{\prime}),$ (1)
where, in the nearest neighbor approximation, the (real) hopping amplitudes
$t$ connect the in-plane nearest neighbor sites belonging to different
sublattices and, for one of the sublattices, also the sites vertically above
it with amplitude $t_{\perp}$. Since there are four sites in the unit cell,
there are four bands whose dispersion for the above model comes from the
solution of the eigenvalue problem:
$\displaystyle\left[\begin{array}[]{cccc}0&d^{*}_{{\bf k}}&t_{\perp}&0\\\
d_{{\bf k}}&0&0&0\\\ t_{\perp}&0&0&d_{{\bf k}}\\\ 0&0&d^{*}_{{\bf
k}}&0\end{array}\right]\left[\begin{array}[]{c}b_{1}({\bf k})\\\ a_{1}({\bf
k})\\\ a_{2}({\bf k})\\\ b_{2}({\bf k})\end{array}\right]=E({\bf
k})\left[\begin{array}[]{c}b_{1}({\bf k})\\\ a_{1}({\bf k})\\\ a_{2}({\bf
k})\\\ b_{2}({\bf k})\end{array}\right].$ (14)
We find $E({\bf k})=\pm\left(\frac{1}{2}t_{\perp}\pm\sqrt{|d_{{\bf
k}}|^{2}+\frac{1}{4}t^{2}_{\perp}}\right)$, with $d_{{\bf
k}}=t\left[2\cos\left(\frac{\sqrt{3}}{2}k_{y}a\right)e^{-\frac{i}{2}k_{x}a}+e^{ik_{x}a}\right]$.
Two of the bands are gapped (at ${\bf K},{\bf K}^{\prime}$ by $t_{\perp}$) and
become separated from the low energy pair which touches at ${\bf k}=0$ (See
Fig.1). The resulting density of states at zero energy is therefore finite.
The repulsive interaction $V({\bf r}-{\bf r}^{\prime})$ in Eq.(1) is taken to
have a finite range $\xi$ which is however much larger than the lattice
spacing $a$. This is assumed to be the correct starting point, since the full
Coulomb interactions is screenedHwang and Sarma (2008) at low energy due to
the finite density of states. The analysis starting from the $1/|{\bf r}-{\bf
r}^{\prime}|$ interaction will be postponed to a future publication.
| | | |
---|---|---|---|---
Figure 2: Diagrams appearing at 1-loop RG. The vertices are either
$\delta_{\alpha\beta}$ or $\Sigma^{\mu}_{\alpha\beta}$.
Following Nilsson et. al.Nilsson et al. (2008) we project out the gapped
bands. The resulting low energy effective (imaginary time) action (which
includes both $K$ and $K^{\prime}$ valleys) is
$\displaystyle\mathcal{S}$ $\displaystyle=$ $\displaystyle\int d\tau d^{2}{\bf
r}\left[\psi^{\dagger}\left(\frac{\partial}{\partial\tau}+\sum_{a=x,y}\Sigma^{a}d^{a}_{{\bf
p}}\psi\right)\right]$ (15) $\displaystyle+$
$\displaystyle\frac{1}{2}g_{1}\int d\tau d^{2}{\bf r}\psi^{\dagger}\psi({\bf
r},\tau)\psi^{\dagger}\psi({\bf r},\tau)$ $\displaystyle+$
$\displaystyle\frac{1}{2}g_{2}\int d\tau d^{2}{\bf
r}\psi^{\dagger}\Sigma^{z}\psi({\bf r},\tau)\psi^{\dagger}\Sigma^{z}\psi({\bf
r},\tau)$ $\displaystyle+$ $\displaystyle\frac{1}{2}g_{3}\int d\tau d^{2}{\bf
r}\sum_{a=x,y}\psi^{\dagger}\Sigma^{a}\psi({\bf
r},\tau)\psi^{\dagger}\Sigma^{a}\psi({\bf r},\tau)$
where the four component Fermi (Grassman) fields
$\displaystyle\psi({\bf r},\tau)=\int^{\Lambda}_{0}\frac{d^{2}{\bf
k}}{(2\pi)^{2}}e^{i{\bf k}\cdot{\bf r}}\left[\begin{array}[]{c}a_{1}({\bf
K}+{\bf k},\tau)\\\ b_{2}({\bf K}+{\bf k},\tau)\\\ a_{1}({\bf K}^{\prime}+{\bf
k},\tau)\\\ b_{2}({\bf K}^{\prime}+{\bf k},\tau)\end{array}\right]$ (20)
and
$\displaystyle d^{x}_{{\bf k}}$ $\displaystyle=$
$\displaystyle\frac{k^{2}_{x}-k^{2}_{y}}{2m},\;\;\;d^{y}_{{\bf
k}}=\frac{2k_{x}k_{y}}{2m},$ (21) $\displaystyle\Sigma^{x}$ $\displaystyle=$
$\displaystyle
1\sigma^{x},\;\;\Sigma^{y}=\tau^{z}\sigma^{y},\;\;\Sigma^{z}=\tau^{z}\sigma^{z}.$
(22)
The Pauli matrices $\sigma_{j}$ act on the layer indices $1$-$2$ and the
$\tau$ matrices act on the valley indices ${\bf K}$-${\bf K}^{\prime}$. The
effective mass is $m=2t_{\perp}/(9t^{2})$, and $\psi$ represents
$\frac{N}{2}-$copies of the four component pseudo-spinor. $N=4$ for spin
$1/2$, and e.g. for $s=1,\ldots N$, $\psi^{\dagger}\Sigma^{z}\psi({\bf
r},\tau)=\psi_{\alpha s}^{\dagger}\Sigma_{\alpha\beta}^{z}\psi_{\beta s}.$
Note that $\Sigma^{\prime}s$ have the same multiplication table as the Pauli
$\sigma^{\prime}s$:
$\Sigma^{\mu}\Sigma^{\nu}=1_{4}\delta_{\mu\nu}+i\epsilon_{\mu\nu\lambda}\Sigma^{\lambda}$
and are traceless, too. $\Lambda$ is a momentum cutoff which restricts the
modes to the vicinity of the ${\bf K}$-${\bf K}^{\prime}$ points and whose
order of magnitude is $\lesssim\sqrt{2mt_{\perp}}$.
The coupling constant $g_{1}=\int d^{2}{\bf r}V({\bf r})$, i.e. it is the
${\bf q}=0$ Fourier component of $V({\bf r})$. The coupling constants $g_{2}$
and $g_{3}$ are zero in the starting action, but as will be shown next, they
get generated in the momentum-shell RGShankar (1994), and therefore they are
made explicit in the original action.
From simple power-counting, the (engineering) scaling dimension of the field
$\psi$ is $L^{-1}$ and $L^{2}$ for $\tau$. This makes $g_{1}$, $g_{2}$ and
$g_{3}$ marginal (at the tree-level) and the question is how they flow upon
inclusion of the loop corrections. To answer this we note that all possible
Wick contractionsShankar (1994) of four-fermion operators correspond to the
diagrams in the Figure (2). The RG equations obtained by integrating fermion
modes within a thin shell $\Lambda$ and $\Lambda/s$ (centered at the $K$
point), and $\int^{\infty}_{-\infty}\frac{d\omega}{2\pi}$, are:
$\displaystyle\frac{dg_{1}}{d\ln s}$ $\displaystyle=$
$\displaystyle\left[-4g_{1}g_{3}\right]\frac{m}{4\pi}$ (23)
$\displaystyle\frac{dg_{2}}{d\ln s}$ $\displaystyle=$
$\displaystyle\left[-4(N-1)g^{2}_{2}+4g^{2}_{3}+4g_{1}g_{2}-12g_{2}g_{3}\right]\frac{m}{4\pi}$
(24) $\displaystyle\frac{dg_{3}}{d\ln s}$ $\displaystyle=$
$\displaystyle\left[-(g_{1}-g_{3})^{2}-(g_{2}-g_{3})^{2}-2(N+1)g^{2}_{3}\right]\frac{m}{4\pi}$
(25)
Figure 3: RG flow diagram of the ratios $g_{1}/g_{3}$ and $g_{2}/g_{3}$ for
$g_{3}<0$. While the ratio $g_{1}/g_{3}$ flows to zero (even if the starting
point is $g_{2}=g_{3}=0$ and $g_{1}\neq 0$), the ratio $g_{2}/g_{3}$ flows to
a fixed value, indicating two stable and one unstable rays with slopes
$m_{1}\approx-0.525$, $m_{3}\approx 13.98$ and $m_{2}\approx 0.545$,
respectively.
While the above equations cannot be solved in a closed form, it is possible to
fully analyze the qualitative nature of the RG flows. Such analysis is
facilitated by the observation that
$\frac{dg_{3}}{d\ln s}\leq 0$
which means that, unless $g_{1}=g_{2}=g_{3}=0$ when the equality holds,
$g_{3}$ strictly decreases under RG rescaling. We can therefore trade the
parametric dependence on $s$ of $g_{1}$ and $g_{2}$ for their dependence on
$g_{3}$ and retain the direction of the RG flow. For $g_{3}<0$ ($>0$), an
increase in $d\log s$ therefore corresponds to an increase (decrease) in
$\frac{dg_{3}}{g_{3}}$. Since the system is autonomous, we can eliminate $\log
s$ and arrive at a system
$\displaystyle\frac{dg_{1}}{dg_{3}}=f\left(\frac{g_{1}}{g_{3}},\frac{g_{2}}{g_{3}}\right)$
(26)
$\displaystyle\frac{dg_{2}}{dg_{3}}=g\left(\frac{g_{1}}{g_{3}},\frac{g_{2}}{g_{3}}\right)$
(27)
where
$\displaystyle f\left(x,y\right)$ $\displaystyle=$
$\displaystyle\frac{-4x}{-x^{2}-y^{2}-2(N+2)+2x+2y}$ (28) $\displaystyle
g\left(x,y\right)$ $\displaystyle=$
$\displaystyle\frac{-4(N-1)y^{2}+4+4xy-12y}{-x^{2}-y^{2}-2(N+2)+2x+2y}$ (29)
The system of Eqs.(26)-(27) is in turn homogeneous and can therefore be
written as
$\displaystyle
g_{3}\frac{d\frac{g_{1}}{g_{3}}}{dg_{3}}=-\frac{g_{1}}{g_{3}}+f\left(\frac{g_{1}}{g_{3}},\frac{g_{2}}{g_{3}}\right)$
(30) $\displaystyle
g_{3}\frac{d\frac{g_{2}}{g_{3}}}{dg_{3}}=-\frac{g_{2}}{g_{3}}+g\left(\frac{g_{1}}{g_{3}},\frac{g_{2}}{g_{3}}\right).$
(31)
The above system has three fixed points, all of which have $g_{1}/g_{3}=0$,
while $g_{2}/g_{3}=m_{1},m_{2},m_{3}$. As shown in the Fig.(3),
$m_{1}\approx-0.525$ and $m_{3}\approx 13.98$ are sinks, while $m_{2}\approx
0.545$ has one attractive direction and one repulsive. This means that once
$g_{3}$ gets to be negative, only $g_{2}$ and $g_{3}$ become important (their
ratio being fixed) while $g_{1}$ is too small compared to $g_{3}$. To see that
this is indeed what happens if the starting point is $g_{1}(s=1)>0$ and
$g_{2}(s=1)=g_{3}(s=1)=0$, note that the Eqs.(23-25) imply that finite $g_{1}$
generates finite and negative $g_{3}$ upon first iteration while $g_{2}$
remains zero until the second iteration. This means that we start with
$g_{1}/g_{3}\rightarrow-\infty$ and $g_{2}/g_{3}=0$ which is below the (red)
separatrix, thus the flow is into the region of attraction of $m_{1}$
(Fig.(3)).
$\psi^{\dagger}\tau^{\mu}\sigma^{\nu}\psi$ | $\nu=0$ | $\nu=x$ | $\nu=y$ | $\nu=z$
---|---|---|---|---
$\mu=0$ | $0,0,0$ | $1,-1,-2N$ | $1,-1,0$ | $2,2,-4$
$\mu=x$ | $1,-1,0$ | $0,0,0$ | $2,2,-4$ | $1,-1,0$
$\mu=y$ | $1,-1,0$ | $0,0,0$ | $2,2,-4$ | $1,-1,0$
$\mu=z$ | $0,0,0$ | $1,-1,0$ | $1,-1,-2N$ | $2,2-4N,-4$
Table 1: The susceptibility coefficients $A,B,C$ in Eq.(Many-body instability of Coulomb interacting bilayer graphene: RG approach) for different particle-hole order parameters $\psi^{\dagger}\mathcal{O}_{i}\psi$. In the physical case $N=4$. $\psi_{\alpha s}(\tau^{\mu}\sigma^{\nu})_{\alpha\beta}\psi_{\beta s^{\prime}}$ | $\nu=0$ | $\nu=x$ | $\nu=y$ | $\nu=z$
---|---|---|---|---
$\mu=0$ | $-1,-1,0$ | $-2,2,-4$ | $0,0,0$ | $-1,-1,0$
$\mu=x$ | $-2,2,-4$ | $-1,-1,0$ | $-1,-1,0$ | $0,0,0$
$\mu=y$ | $-2,2,-4$ | $-1,-1,0$ | $-1,-1,0$ | $0,0,0$
$\mu=z$ | $-1,-1,0$ | $-2,2,-4$ | $0,0,0$ | $-1,-1,0$
Table 2: The susceptibility coefficients $A^{\prime},B^{\prime},C^{\prime}$ in
Eq.(Many-body instability of Coulomb interacting bilayer graphene: RG
approach) for different particle-particle order parameters
$\psi_{\alpha\sigma}\mathcal{O}^{(i)}_{\alpha\beta}\psi_{\beta\sigma^{\prime}}$.
Figure 4: Numerical integration of the susceptibilities in Eq.(Many-body
instability of Coulomb interacting bilayer graphene: RG approach) for
$g_{1}(s=1)=0.01$ and $g_{2}(s=1)=g_{3}(s=1)=0$. The strongest divergence is
towards the nematic order. (Inset) Numerically determined nematic transition
temperature in units of cutoff $T_{\Lambda}\lesssim t_{\perp}$ as a function
of the dimensionless coupling $g_{1}\frac{m}{4\pi}$.
From Eqs.(23-25) we see for the fixed ratios $g_{1}/g_{3}=0$ and
$g_{2}/g_{3}=m_{j}$, $g_{3}$ becomes large and negative, indicating a runaway
flow. Given the flow of the coupling constants we can determine the
susceptibilities towards the formation of ordered states. In particular, we
consider coupling the fermions to external sources, which correspond to the
possible broken symmetry states. We therefore have additional terms in the
action:
$\displaystyle\Delta{\mathcal{S}}$ $\displaystyle=$
$\displaystyle-\Delta_{ph}^{\mathcal{O}_{i}}\int d\tau d^{2}{\bf
r}\psi^{\dagger}\mathcal{O}_{i}\psi({\bf r},\tau)$ (32) $\displaystyle-$
$\displaystyle\Delta^{\mathcal{O}_{i}}_{pp}\int d\tau d^{2}{\bf
r}\psi_{\alpha\sigma}\mathcal{O}^{i}_{\alpha\beta}\psi_{\beta\sigma^{\prime}}({\bf
r},\tau)$
Such terms, with infinitesimal $\Delta$’s explicitly break the symmetry and so
are relevant operators. The question of instability is answered by finding the
renormalization of the verticesChubukov (2009). The one which diverges first
determines the broken symmetry states. After a straigthforward calculation we
find that for a general particle-hole order parameter
$\mathcal{O}_{i}=\tau^{\mu}\sigma^{\nu}$ where $\mu,\nu=0,1,2,3$ and
$\tau_{0}=\sigma_{0}=1$,
$\displaystyle\Delta_{ph,ren}^{\tau^{\mu}\sigma^{\nu}}$ $\displaystyle=$
$\displaystyle\Delta_{ph}^{\tau^{\mu}\sigma^{\nu}}\left(1+\left[Ag_{1}+Bg_{2}+Cg_{3}\right]\frac{m}{4\pi}\ln
s\right)$
where the coefficients $A$, $B$, and $C$ are given in the Table 1. Similarly,
for a general particle-particle order parameter
$\psi_{\alpha\sigma}\mathcal{O}^{(i)}_{\alpha\beta}\psi_{\beta\sigma^{\prime}}$
$\displaystyle\Delta_{pp,ren}^{\tau^{\mu}\sigma^{\nu}}$ $\displaystyle=$
$\displaystyle\Delta_{pp}^{\tau^{\mu}\sigma^{\nu}}\left(1+\left[A^{\prime}g_{1}+B^{\prime}g_{2}+C^{\prime}g_{3}\right]\frac{m}{4\pi}\ln
s\right)$
where the coefficients $A^{\prime}$, $B^{\prime}$, and $C^{\prime}$ are given
in the Table 2.
The instability towards a particular order occurs at an energy scale (i.e.
temperature) at which the corresponding coefficient of the $\ln s$ in
Eqs.(Many-body instability of Coulomb interacting bilayer graphene: RG
approach-Many-body instability of Coulomb interacting bilayer graphene: RG
approach) diverges. Since $N=4$ and the fixed point value of
$g_{2}/g_{3}\approx-0.525$, with $g_{3}$ large and negative, it can be seen
from Table 1 that the instability appears in the $\Sigma^{x,y}$ channel, which
as we discuss next corresponds to a nematic order. The numerical integration
of the RG equations (23-25) starting with $g_{1}(s=1)>0$ and
$g_{2}(s=1)=g_{3}(s=1)=0$ shown in Fig.(4) indeed confirms that the
susceptibility diverges fastest in this channel. Within the continuum model
and in weak coupling, the instability is therefore towards the order
parameter, which we can parametrize by a complex field
$\Delta_{nem}({\bf r})\equiv\Delta_{x}({\bf r})+i\Delta_{y}({\bf
r})=\langle\psi^{\dagger}({\bf r})\left(\Sigma^{x}+i\Sigma^{y}\right)\psi({\bf
r})\rangle.$
To see that this is indeed a nematic order, note that at ${\bf q}=0$ (1) it is
translationally invariant and (2) even under rotations by $\pi$. In fact, as
the low energy Hamitonian is invariant under arbitrary rotations by an angle
$\alpha$, i.e. $U^{\dagger}(\alpha)\mathcal{H}U(\alpha)=\mathcal{H}$, where
$U_{\alpha}=e^{-i\alpha\hat{L}_{z}}e^{-i\alpha\Sigma^{z}},\;\;L_{z}=x\frac{\partial}{\partial
y}-y\frac{\partial}{\partial x}$, we find that under a rotation by $\alpha$
$\Delta_{nem}({\bf r})\rightarrow\Delta_{nem}({\bf r})e^{2i\alpha}.$
This shows that the order parameter is even under rotations by $\pi$ and odd
under rotations by $\pi/2$, which makes it nematic. For uniform
$\Delta_{nem}({\bf r})$ the quadratic degeneracy point is split into two
(massless) Dirac points by an amount proportional to the magnitude of the
order parameter and the direction given by the nematic director.
The presence of the underlaying lattice further breaks the full rotational
symmetry of the long distance effective Hamiltonian down to hexagonal symmetry
centered on $a_{2}-b_{1}$ site, where the standard operations of $C_{6v}$ must
be accompanied by the appropriate layer permutations. The two components of
the order parameter, which give finite expectation values of, for instance,
$\Delta_{x}({\bf r})=$
$\displaystyle\left\langle a^{\dagger}_{1\sigma}({\bf
r})\left(b_{2\sigma}({\bf r}-a\hat{x})-\frac{1}{2}\sum_{s=\pm}b_{2\sigma}({\bf
r}+\frac{a}{2}\hat{x}s\frac{\sqrt{3}}{2}\hat{y})\right)+h.c.\right\rangle$
(35) $\displaystyle\mbox{and}\;\;\Delta_{y}({\bf r})=$
$\displaystyle\left\langle a^{\dagger}_{1\sigma}({\bf
r})\left(\frac{\sqrt{3}}{2}\sum_{s=\pm}sb_{2\sigma}({\bf
r}+\frac{a}{2}\hat{x}+s\frac{\sqrt{3}}{2}\hat{y})\right)+h.c.\right\rangle$
(36)
form a two dimensional representation of the hexagonal group. Note that the
nematic order parameter remains even under $\pi$-rotation followed by the
layer permutation.
From the arguments above we expect that the lattice has an important effect on
the critical nature of the phase transition, which would otherwise be of
Kosterlitz-Thousless kind. The reason is the existence of the third order
invariant $\Delta^{3}_{x}-3\Delta_{x}\Delta^{2}_{y}$. As a result the finite
temperature phase transition should be described by the effective Hamiltonian
$\displaystyle\mathcal{H}_{nem}=\sum_{\langle{\bf x}{\bf
x}^{\prime}\rangle}-J\cos[2(\theta({\bf x})-\theta({\bf
x}^{\prime}))]+h\sum_{{\bf x}}\cos[6\theta({\bf x})].$ (37)
where $\Delta_{x}({\bf x})+i\Delta_{y}({\bf x})=e^{2i\theta({\bf x})}$,
$\theta\in(0,2\pi]$ and the sum runs over the vertices of the triangular sub-
lattice spanned by $a_{1}$ sites. This corresponds to the $p=3$ case of the
two dimensional planar model studied by Jose et.al.José et al. (1977) and the
concomitant absence of the Gaussian spin-wave phase. Instead there is a
continuous transition between the low temperature phase where the director
locks into one of three values and a high temperature phase where vortices
unbind. Such transition is believed to belong to the 2D three-state Potts
model universality classNelson (2002) with exponentsWu (1982) $\nu=5/6$ and
$\eta=4/15$.
Finally, we discuss the effects of the trigonal warping which splits each of
the quadratic degeneracies into four massless Dirac points, which were ignored
up to now. If we denote the energy scale associated with such terms as
$T_{trig}$, below which the dispersion must be modified, then the transition
will still occur provided that the mean-field transition temperature $T_{c}$
estimated from the above model and plotted in the inset of Fig.(4) satisfies
$T_{c}\gg T_{trig}$. For screened Coulomb interactionsHwang and Sarma (2008)
$g_{1}\frac{m}{4\pi}\sim\mathcal{O}(1)$, leading to $T_{c}\lesssim t_{\perp}$.
Since the current estimates of $T_{trig}$ are of the same order of
magnitudeZhang et al. (2008), the ultimate test is experimental.
Acknowledgements: While this paper was in preparation, we became aware of
Ref.Sun et al. (2009) where lattices with fourfold and sixfold rotational
symmetry are constructed in either case the parabolic degeneracy points are
protected by the point group symmetry. In there, the degeneracy point maps
unto itself under time reversal, unlike our $K$ and $K^{\prime}$, and nematic
was found to be stabilized (within mean-field) only at finite coupling. This
work is supported in part by NSF grant No. DMR-0704133 (KY). Part of this work
was carried out while the authors were visiting Kavli Institute for
Theoretical Physics (KITP). The work at KITP is supported in part by NSF grant
No. PHY-0551164.
## References
* Shankar (1994) R. Shankar, Rev. Mod. Phys. 66, 129 (1994).
* Novoselov et al. (2006) K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim, Nature Physics 2, 177 (2006).
* McCann and Fal’ko (2006) E. McCann and V. I. Fal’ko, Physical Review Letters 96, 086805 (2006).
* Neto et al. (2009) A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Reviews of Modern Physics 81, 109 (2009).
* Geim and MacDonald (2007) A. K. Geim and A. H. MacDonald, Physics Today 60, 35 (2007).
* Nilsson et al. (2006) J. Nilsson, A. H. C. Neto, N. M. R. Peres, and F. Guinea, Physical Review B 73, 214418 (2006).
* Min et al. (2008) H. Min, G. Borghi, M. Polini, and A. H. MacDonald, Physical Review B 77, 041407 (2008).
* Hwang and Sarma (2008) E. H. Hwang and S. D. Sarma, Physical Review Letters 101, 156802 (2008).
* Nilsson et al. (2008) J. Nilsson, A. H. C. Neto, F. Guinea, and N. M. R. Peres, Physical Review B 78, 045405 (2008).
* Chubukov (2009) A. V. Chubukov (2009), arXiv:0902.4188.
* José et al. (1977) J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).
* Nelson (2002) D. R. Nelson, _Defects and Geometry in Condensed Matter Physics_ (Cambridge University Press, Cambridge, UK, 2002), p.56.
* Wu (1982) F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982).
* Zhang et al. (2008) L. M. Zhang, Z. Q. Li, D. N. Basov, M. M. Fogler, Z. Hao, and M. C. Martin, Physical Review B 78, 235408 (2008).
* Sun et al. (2009) K. Sun, H. Yao, E. Fradkin, and S. A. Kivelson (2009), arXiv:0905.0907.
|
arxiv-papers
| 2009-06-13T16:03:59 |
2024-09-04T02:49:03.344884
|
{
"license": "Public Domain",
"authors": "Oskar Vafek and Kun Yang",
"submitter": "Oskar Vafek",
"url": "https://arxiv.org/abs/0906.2483"
}
|
0906.2512
|
# On Implementation of a Safer C Library, ISO/IEC TR 24731.
Technical Report
CIISE Security Investigation Initiative
Represented by:
Marc-André Laverdière-Papineau
Serguei A. Mokhov
Djamel Benredjem
{ma_laver,mokhov,d_benred}@ciise.concordia.ca
Montréal, Québec, Canada
(April 2006)
###### Contents
1. 1 Introduction
1. 1.1 Security Problems in C Standard Functions
2. 1.2 Introducing ISO/IEC
3. 1.3 ISO/IEC TR 24731
2. 2 Architecture
1. 2.1 Principles and Philosophy
2. 2.2 Summary of Siemens Four View Model
1. 2.2.1 Conceptual View
2. 2.2.2 Module View
3. 2.2.3 Execution View
4. 2.2.4 Code View
5. 2.2.5 Conceptual View
1. 2.2.5.1 Conceptual Overview
2. 2.2.5.2 Configurations
3. 2.2.5.3 Protocols
4. 2.2.5.4 Resource Budgeting
6. 2.2.6 Module View
1. 2.2.6.1 Layering
2. 2.2.6.2 Interface Design
3. 2.3 Execution View
1. 2.3.1 Runtime Entities
2. 2.3.2 Communication Paths
3. 2.3.3 Execution Configuration
4. 2.4 Code View
1. 2.4.1 Source Components
2. 2.4.2 Intermediate Components
3. 2.4.3 Deployment Components
4. 2.4.4 Make Process
5. 2.4.5 Configuration Management
5. 2.5 Example for One Module
6. 2.6 Iterations
7. 2.7 Coding Standards
3. 3 Implementation
1. 3.1 Run-time Constraint Handling API
2. 3.2 Constraint Violation Information Encapsulation API
3. 3.3 Constraint Enumeration and Validator
4. 3.4 Constraint Handling Example
4. 4 Results
1. 4.1 Implemented API
1. 4.1.1 Library
1. 4.1.1.1 Data Types
2. 4.1.1.2 Functions
2. 4.1.2 Private Constraint Handling API
1. 4.1.2.1 Data Types
2. 4.1.2.2 Functions
2. 4.2 Constraint Handling In Action – stdio
3. 4.3 Constraint Handling In Action – string
5. 5 Conclusions
1. 5.1 Summary of the Difficulties
2. 5.2 Limitations So Far
3. 5.3 Acknowledgments
4. 5.4 Future Work
## Chapter 1 Introduction
### 1.1 Security Problems in C Standard Functions
The functions standardized as part of ISO C 1999 and their addendums improved
very little the security options from the previously available library.
The largest flaw remained that no function asked for the buffer size of
destination buffers for any function copying data into a user-supplied buffer.
According to earlier research we performed [PMB], we know that error condition
handling was the first solution to security vulnerabilities, followed by
precondition validation. The standard C functions typically perform little
precondition validation and error handling, allowing for a wide range of
security issues to be introduced in their use.
For example:
char *strncat(char *dest, const char *src, size_t n); does not null-terminate,
can still overflow char *strtok(char * restrict s1, const char * restrict s2);
not reentrant size_t strlen(const char *s); can iterate in the memory up to an
invalid page and cause a program crash
This effort remained not enough, and many projects developed additional
functions, namely:
* •
OpenBSD strlcpy family [MdR99]
* •
GNU C extensions [pro]
* •
Microsoft strsafe.h and others [Cor05]
### 1.2 Introducing ISO/IEC
The International Standardization Organization (ISO) and the International
Electrotechnical Commission (IEC) are standard-making bodies headquartered in
Geneva (Switzerland) [Sec05].
Both organizations are constituted from an international membership, with
local member organizations involved in standard-making activities as well. For
example, we have the Standards Council of Canada, American National Standards
Institute, Deutsches Institut für Normung, and Association française de
normalisation [ISO06b].
ISO and IEC collaborate closely on standards related to computer equipment and
information technologies.
These organizations established a hierarchical structure under JTC 1 - Joint
Technical Committee on Information Technology. JTC 1 is subdivided in 17
subcommittees, one of which (SC 22) deals with programming languages, with a
working group for each programming languages [ISO06a].
The C language is normalized by ISO/IEC JTC 1/ SC 22/ WG 14. Its members
include representatives from Microsoft, SEI/CMU, Cisco, Intel, etc [SC2]. The
Computer Security Laboratory of CIISE, through Pr. Debbabi, is a member of
this Working Group as Canadian representative with voting rights.
### 1.3 ISO/IEC TR 24731
In the ISO jargon, TR 24731 [WG106] is a Technical Report Type 2 [ISO05, ISO],
which means that the document is not a standard, but a direction for future
normalization. This specification is currently in the draft state.
Titled “TR 24731: Safer C library functions”, it defines 41 new library
functions for memory copying, string handling (both for normal and wide
character strings), time printing, sorting, searching etc. Another inovation
it brings is a constraint handling architecture, forcing error handling when
certain security-related preconditions are violated when the functions are
called. It also specifies the null-termination of all strings manipulated
through its function and introduces a new unsigned integer type that helps
preventing integer overflows and underflows. It is currently implemented by
Microsoft as part of their Visual Studio 2005 [Sea05].
## Chapter 2 Architecture
In this chapter, we examine the architecture of our implementation of ISO/IEC
TR 24731. We first introduce our architectural philosophy before informing the
reader about the Siemens Four View Model, an architectural methodology for the
conception of large-scale software systems.
Afterwards, we examine each of the view, as architected for our library.
Finally, we conclude with other software engineering matters that were of high
importance in the development of our implementation.
### 2.1 Principles and Philosophy
The library specification imposes that the functions be in addition of other
standard functions, in the same header files. However, we do not want our
implementation to re-implement the standard C library, nor do we want to
augment an existing implementation and be bound to a specific platform. The
compromise solution is to organize the code to be using the low-level
implementation of any existing C library, such as the one from GNU [gp],
FreeBSD or OpenBSD.
In short, our principles are:
Platform independence
We target systems complying with the POSIX standard.
Standards Compliance
We will implement the library using features available only in ISO C99 and
POSIX.
C Library Independence
We will architect in a way that prevents us being tied to the underlying C
library.
Realistic Compiler Indepedence
A corollary of standards compliance, we will avoid compiler-specific macros
and optimizations as possible. This means that the source code should be free
of such dependencies, but that the build process may be bound to the compiler.
Reasonable Efficiency
We will architect and implement an efficient library, but will avoid advanced
programming tricks that improve the efficiency at the cost of maintainability
and readability.
Simplicity And Maintainability
We will target a simplistic and easy to maintain organization of the source.
Architectural Consistency
We will consistently implement our architectural approach. We could say that
we will have a “template” approach.
Separation of Concerns
We will isolate separate concerns between modules and within modules to
encourage reuse and code simplicity.
Functional Grouping
We will have functional coupling between within a module, meaning that
functions will have a commonality of type, such as I/O, string manipulations,
etc.
### 2.2 Summary of Siemens Four View Model
We decided to use the Siemens Four View Model for our architectural
description, mostly due to previous experience using the technology. A
detailed description with case studies can be found in [HNS00]. This
methodology is introduced by scientists of Siemens Corporate Research and has
been successfully applied to a variety of systems, many of which with real-
time and embedded requirements. Please refer to Figure 2.1 for an abstract
presentation of the view. In the context of this project report, we inform the
reader of the basic principles of each view.
Figure 2.1: High-Level Description of the Four Views [Cha]
#### 2.2.1 Conceptual View
The conceptual view defines the conceptual components and their conceptual
connectors, as well as their conceptual configuration. The architect, in this
phase, must also specify resource budgets.
#### 2.2.2 Module View
In the module view, the architect maps the conceptually-defined elements into
modules and layers. The architect must then define the interface to the
modules.
#### 2.2.3 Execution View
In the execution view, the architect must define the runtime entities, the
communication paths and the execution configuration. It essentially means the
mapping of modules to threads and processes, and to define the inter-process
communication mechanisms to be used. In the case of our library, there are no
threads of execution per se, as such, this architectural step was skipped.
#### 2.2.4 Code View
In the code view, the architect must define the source components,
intermediate components and deployment components, followed by the build
procedure and configuration management.
#### 2.2.5 Conceptual View
##### 2.2.5.1 Conceptual Overview
We divided our architecture in a few modules, isolating the core
functionalities from each library and grouping functions per library. We also
decided to use wrappers to a C library implementation. This overview can be
seen in Figure Figure 2.2.
Figure 2.2: Conceptual Modules
##### 2.2.5.2 Configurations
The only options in configuration relate to the wrappers to use for a specific
C library. Since this is decided at compilation time and that only one is
possible in any case, we decided not to further specify configurations.
##### 2.2.5.3 Protocols
Because of the simplistic nature of the components (direction function calls)
and that all components need to be re-entrant, we conclude that no protocols
are necessary.
##### 2.2.5.4 Resource Budgeting
We did not specify any resource budget for any components due to the lack of
specific constraints.
#### 2.2.6 Module View
##### 2.2.6.1 Layering
We divided our library between layers when we found significant redundancy for
some operations. We decided to keep the memory copying for wide characters
apart from the one without wide characters due to the risk of integer
overflows that could result in faulty logic, and thus deserving a
centralization of the functionality.
The complete view of our layering is included in Figure Figure 2.3.
Figure 2.3: Layering
##### 2.2.6.2 Interface Design
The only interface that needed to be designed was related to the constraint
validation. Please refer to the specification given in section Section 3.1
### 2.3 Execution View
#### 2.3.1 Runtime Entities
In the case of our library, there are no threads of execution per se, as the
thread is provided by the calling program(s).
#### 2.3.2 Communication Paths
It was resolved that the functions would all communicate through message
passing.
#### 2.3.3 Execution Configuration
Coherently with previous decisions, there are no specific execution
configurations.
### 2.4 Code View
#### 2.4.1 Source Components
Figure 2.4: Directory Tree
Each of stdio.h, stdlib.h, string.h, time.h, and wchar.h are mapped into a
corresponding directory, as a module. The folder named test copies the
previous structure, and contains test programs exclusively. The include
directory contains the .h files to be included by external programs linking to
the library. The folder named adapters holds a sub-directory per C library
implementation to adapt to, and each of those adapters mirror the base
directory structure. Each interface function was implemented in a file with
its name, to facilitate maintainability. Related functions are grouped in the
file to which it is the most logically related.
We established a naming convention for functions as follows:
function
The function’s interface for the user.
__function_impl
The function’s implementation, used within the library.
__function_validate_preconditions
The function’s precondition validation component.
__function_component
refactored sub-component of a function or common code within functions.
#### 2.4.2 Intermediate Components
In order to facilitate the building process, we decided that each directory of
functionality and adapters will assemble all intermediary (.o) files into an
archive (.a).
#### 2.4.3 Deployment Components
The only deployment component will be a .so file that includes our
implementation and the underlying library.
#### 2.4.4 Make Process
The build process is organized as a hierarchical organization of makefiles.
Each makefile cross-reference the makefiles for its dependencies. The
intermediary objects defined are the standard .o that are also grouped in an
archive (.a). The testers are built separately from the library itself for
efficiency and faster compilation.
The make files are designed to adapt to both Linux and Windows/Cygwin
platforms by detecting the presence of Cygwin and using different compiler
options consequently.
The documentation generation fits outside of the normal make process and is
generated by the doxygen tool itself.
#### 2.4.5 Configuration Management
We used Subversion (svn) [Tig] in order to manage the source code, makefile,
and documentation revisions.
### 2.5 Example for One Module
In Figure 2.5 and Figure 2.6, we show the example of function call and source
file dependencies for a function implementing vfprintf_s.
Figure 2.5: Call dependencies for function vfprintf_s Figure 2.6: File
dependencies for function vfprintf_s
### 2.6 Iterations
In order to efficiently reach our architectural goal, we divided the final
objectives of the project in the following steps:
1. 1.
Implementation of the body, without precondition validations
2. 2.
Implementation of precondition validation and integration
3. 3.
Validation of compliance for all testable cases
4. 4.
Implementation of adapters to a C library
5. 5.
Redefinition of insecure function calls to our function calls
### 2.7 Coding Standards
In order to produce high-quality code, we decided to normalize on the OpenBSD
style. We also decided to use Doxygen [vH] source code documentation style for
its completeness and the automated tool support.
## Chapter 3 Implementation
This chapter gives concrete implementation details for the constraint handling
API and examples of its usage. An excessive amount of work was done to have a
decent precondition validation and this chapter mostly focuses on the API
aspect of this implementatition. A particular achievement was to fully enable
parsing and restraining of %s and %n modifiers with the flex-based scanner.
### 3.1 Run-time Constraint Handling API
For precondition validation we deviced an API to encasulate run-time
constraint check and violation information and an appropriate currently
registered constraint handler. Later on this was abstracted with inline
function calls reducing the clutter for the libc_s programmers such as
ourselves and for those who might maintain it after us.
The standard defines this type to allow custom constraint handlers in
stdlib.h:
⬇
typedef void (*constraint_handler_t)(const char* restrict msg, void* restrict
ptr, errno_t error);
constraint_handler_t set_constraint_handler_s(constraint_handler_t handler);
void abort_handler_s(const char * restrict msg, void * restrict ptr, errno_t
error);
void ignore_handler_s(const char * restrict msg, void * restrict ptr, errno_t
error);
Listing 1: Standard API for Constraint Handling
In our implementation we mark abort_handler_s as the default handler for all
contraint violations. That means, after erroring out, the handler calls
exit(0) and the application build around libc_s terminates.
### 3.2 Constraint Violation Information Encapsulation API
This is the capsule enclosing the error information we defined. The structure
depicts some meta information about a paramater and its value. An instance of
this structure is created for each input parameter to be validated.
Listing 2: Synopsys: param_validation_status_t
⬇
#include "stdlib_implementation.h"
typedef struct _param_validation_status_t
{
e_errcheck errtype;
const void* value;
const void* pairvalue;
const void* result;
const char * restrict function_name;
const char * restrict param_name;
const char * restrict pair_param_name;
bool error_present;
} param_validation_status_t;
Some helper data structures allow us to describe more complex types. The
param_range_t struct allows specifying the range for a domain value. A
reference to the instance of this struct is passed in the
param_validation_status_t.pairvalue field. The object_range_t struct is
supposed to contain the data describing a memory objec with its starting
address and length. The purpose of this is to help with validation of ranged
objects that they do not overlap in memory. The implementor of the library
should provide the two intances of this struct as references in value and
pairvalue of the two ranged objects. Which object reference goes to where is
unimportant as the implementation takes care of figuring out the object
precedence.
Listing 3: Synopsys: param_range_t
⬇
typedef struct _param_range_t
{
size_t min;
size_t max;
} param_range_t;
Listing 4: Synopsys: object_range_t
⬇
typedef struct _object_range_t
{
const void* restrict object_ptr;
size_t object_length;
} object_range_t;
### 3.3 Constraint Enumeration and Validator
The enumeration in Listing 5 defines most common error types to check for and
report. These correspond to the index for the human readable error messages.
⬇
typedef enum
{
E_NOERROR = 0,
E_NULL_PARAMETER_NOT_ALLOWED = 1,
E_PARAMETER_OUT_OF_RANGE = 2,
E_ENVIRONMENTAL_LIMIT_NOT_MET = 3,
E_INVALID_FORMAT_PARAMETER_S = 4, /* %s */
E_INVALID_FORMAT_PARAMETER_N = 5, /* %n */
E_RSIZE_MAX_EXCEEDED = 6,
E_NOT_ZERO = 7,
E_OBJECTS_OVERLAP = 8,
E_NOT_IMPLEMENTED = 9,
E_TOKEN_END_NOT_FOUND = 10
} e_errcheck;
Listing 5: Enumeration of most common error types to check for.
⬇
errno_t __error_out(param_validation_status_t * restrict status);
errno_t __contraint_validator_s(param_validation_status_t * restrict status);
errno_t
__constraint_validator_object_overlap(const char * restrict function_name,
const char * restrict parameterNames, const void * restrict object1Start,
const size_t object1Size, const void * restrict object2Start, const size_t
object2Size);
errno_t
__constraint_validator_value_inrange(const char * restrict function_name,
const char * restrict parameterName, const size_t value, const size_t
lowerBound, const size_t upperBound);
errno_t
__constraint_validator_not_null(const char * restrict function_name, const
char * restrict parameter_name, const void * restrict value_ptr);
errno_t
__constraint_validator_not_null_args(const char * restrict function_name,
const char * restrict parameter_name, const char * restrict format, const
va_list args);
errno_t
__constraint_validator_s_format(const char * restrict function_name, const
char * restrict parameter_name, const char * restrict format, const va_list
args);
errno_t
__constraint_validator_n_format(const char * restrict function_name, const
char * restrict parameter_name, const char * restrict format, const va_list
args);
errno_t
__constraint_validator_not_zero(const char * restrict function_name, const
char * restrict parameter_name, const size_t value);
errno_t
__constraint_validator_rsize_limit(const char * restrict function_name, const
char * restrict parameter_name, const rsize_t value);
void
__report_constraint_violation_end_of_token_not_present(const char * restrict
function_name, const char * restrict parameter_name);
Listing 6: Constraint Validator API
### 3.4 Constraint Handling Example
Our implementation of the API (in Listing 6) does something similar to the
code snippet presented in Listing 7.
⬇
...
param_validation_status_t format_string_validation;
__memory_zero_fill_range(&format_string_validation,
sizeof(param_validation_status_t));
format_string_validation.errtype = E_INVALID_FORMAT_PARAMETER_S;
format_string_validation.value = format;
format_string_validation.pairvalue = &arg;
format_string_validation.param_name = "format/arg null %s";
format_string_validation.function_name = "vfprintf_s";
error = __contraint_validator_s(&format_string_validation);
if(error != OK)
{
errno = error;
return error;
}
...
Listing 7: Validation Code
## Chapter 4 Results
This chapter summarizes the result achieved as of this writing. This includes
implemented API to this point as well as some concrete results demonstraiting
correctness of implementation.
### 4.1 Implemented API
This is the summary of the implemented API from the library and our internal
constraint handling. We summarized the functions and data types added in
ISO/IEC TR 24731 in this section for the sake of reference.
#### 4.1.1 Library
##### 4.1.1.1 Data Types
Added the following data types: rsize_t, errno_t, constraint_handler_t that
were necessary to add.
##### 4.1.1.2 Functions
The functions in Listing 8 were to the large extend implemented by our team as
of this writing. Likewise, Listing 9 lists API not yet addressed. Finally,
Listing 10 lists API implemented half-way through.
⬇
int fprintf_s(FILE * restrict stream, const char * restrict format, ...);
int fscanf_s(FILE * restrict stream, const char * restrict format, ...);
int printf_s(const char * restrict format, ...);
int scanf_s(const char * restrict format, ...);
int snprintf_s(char * restrict s, rsize_t n, const char * restrict format,
...);
int sprintf_s(char * restrict s, rsize_t n, const char * restrict format,
...);
int sscanf_s(const char * restrict s, const char * restrict format, ...);
int vfprintf_s(FILE * restrict stream, const char * restrict format, va_list
arg);
int vfscanf_s(FILE * restrict stream, const char * restrict format, va_list
arg);
int vprintf_s(const char * restrict format, va_list arg);
int vscanf_s(const char * restrict format, va_list arg);
int vsnprintf_s(char * restrict s, rsize_t n, const char * restrict format,
va_list arg);
int vsprintf_s(char * restrict s, rsize_t n, const char * restrict format,
va_list arg);
int vsscanf_s(const char * restrict s, const char * restrict format, va_list
arg);
constraint_handler_t set_constraint_handler_s(constraint_handler_t handler);
void abort_handler_s(const char * restrict msg, void * restrict ptr, errno_t
error);
void ignore_handler_s(const char * restrict msg, void * restrict ptr, errno_t
error);
errno_t wctomb_s(int * restrict status, char * restrict s, rsize_t smax,
wchar_t wc);
errno_t mbstowcs_s(size_t * restrict retval, wchar_t * restrict dst, rsize_t
dstmax, const char * restrict src, rsize_t len);
errno_t wcstombs_s(size_t * restrict retval, char * restrict dst, rsize_t
dstmax, const wchar_t * restrict src, rsize_t len);
errno_t memcpy_s(void * restrict s1, rsize_t s1max, const void * restrict s2,
rsize_t n);
errno_t memmove_s(void *s1, rsize_t s1max, const void *s2, rsize_t n);
errno_t strcpy_s(char * restrict s1, rsize_t s1max, const char * restrict s2);
errno_t strncpy_s(char * restrict s1, rsize_t s1max, const char * restrict s2,
rsize_t n);
errno_t strcat_s(char * restrict s1, rsize_t s1max, const char * restrict s2);
errno_t strncat_s(char * restrict s1, rsize_t s1max, const char * restrict s2,
rsize_t n);
char *strtok_s(char * restrict s1, rsize_t * restrict s1max, const char *
restrict s2, char ** restrict ptr);
int fwprintf_s(FILE * restrict stream, const wchar_t * restrict format, ...);
int fwscanf_s(FILE * restrict stream, const wchar_t * restrict format, ...);
int snwprintf_s(wchar_t * restrict s, rsize_t n, const wchar_t * restrict
format, ...);
int swprintf_s(wchar_t * restrict s, rsize_t n, const wchar_t * restrict
format, ...);
int swscanf_s(const wchar_t * restrict s, const wchar_t * restrict format,
...);
int vfwprintf_s(FILE * restrict stream, const wchar_t * restrict format,
va_list arg);
int vfwscanf_s(FILE * restrict stream, const wchar_t * restrict format,
va_list arg);
int vsnwprintf_s(wchar_t * restrict s, rsize_t n, const wchar_t * restrict
format, va_list arg);
int vswprintf_s(wchar_t * restrict s, rsize_t n, const wchar_t * restrict
format, va_list arg);
int vswscanf_s(const wchar_t * restrict s, const wchar_t * restrict format,
va_list arg);
int vwprintf_s(const wchar_t * restrict format, va_list arg);
int vwscanf_s(const wchar_t * restrict format, va_list arg);
int wprintf_s(const wchar_t * restrict format, ...);
int wscanf_s(const wchar_t * restrict format, ...);
Listing 8: Implemented Safer C Library API
⬇
char *gets_s(char *s, rsize_t n);
errno_t getenv_s(size_t * restrict len, char * restrict value, rsize_t
maxsize, const char * restrict name);
void *bsearch_s(const void *key, const void *base, rsize_t nmemb, rsize_t
size, int (*compar)(const void *k, const void *y, void *context), void
*context);
errno_t qsort_s(void *base, rsize_t nmemb, rsize_t size, int (*compar)(const
void *x, const void *y, void *context), void *context);
errno_t strerror_s(char *s, rsize_t maxsize, errno_t errnum);
size_t strerrorlen_s(errno_t errnum);
size_t strnlen_s(const char *s, size_t maxsize);
errno_t asctime_s(char *s, rsize_t maxsize, const struct tm *timeptr);
errno_t ctime_s(char *s, rsize_t maxsize, const time_t *timer);
struct tm *gmtime_s(const time_t * restrict timer, struct tm * restrict
result);
struct tm *localtime_s(const time_t * restrict timer, struct tm * restrict
result);
errno_t wcscpy_s(wchar_t * restrict s1, rsize_t s1max, const wchar_t *
restrict s2);
errno_t wcsncpy_s(wchar_t * restrict s1, rsize_t s1max, const wchar_t *
restrict s2, rsize_t n);
errno_t wmemcpy_s(wchar_t * restrict s1, rsize_t s1max, const wchar_t *
restrict s2, rsize_t n);
errno_t wmemmove_s(wchar_t *s1, rsize_t s1max, const wchar_t *s2, rsize_t n);
errno_t wcscat_s(wchar_t * restrict s1, rsize_t s1max, const wchar_t *
restrict s2);
errno_t wcsncat_s(wchar_t * restrict s1, rsize_t s1max, const wchar_t *
restrict s2, rsize_t n);
wchar_t *wcstok_s(wchar_t * restrict s1, rsize_t * restrict s1max, const
wchar_t * restrict s2, wchar_t ** restrict ptr);
size_t wcsnlen_s(const wchar_t *s, size_t maxsize);
errno_t wcrtomb_s(size_t * restrict retval, char * restrict s, rsize_t smax,
wchar_t wc, mbstate_t * restrict ps);
errno_t mbsrtowcs_s(size_t * restrict retval, wchar_t * restrict dst, rsize_t
dstmax, const char ** restrict src, rsize_t len, mbstate_t * restrict ps);
errno_t wcsrtombs_s(size_t * restrict retval, char * restrict dst, rsize_t
dstmax, const wchar_t **restrict src, rsize_t len, mbstate_t * restrict ps);
Listing 9: Not Implemented Safer C Library API
⬇
errno_t tmpfile_s(FILE * restrict * restrict streamptr);
errno_t tmpnam_s(char *s, rsize_t maxsize);
errno_t fopen_s(FILE * restrict * restrict streamptr, const char * restrict
filename, const char * restrict mode);
errno_t freopen_s(FILE * restrict * restrict newstreamptr, const char *
restrict filename, const char * restrict mode, FILE * restrict stream);
Listing 10: Partially Implemented Safer C Library API
#### 4.1.2 Private Constraint Handling API
##### 4.1.2.1 Data Types
Added the following data types: param_validation_status_t, param_range_t,
object_range_t that were necessary to add.
##### 4.1.2.2 Functions
The functions in Listing 11 were to the large extend implemented by our team
as of this writing. Likewise, Listing 12 lists API not yet addressed.
⬇
errno_t __validate_n_format(const char * restrict format, va_list args);
errno_t __validate_s_format(const char * restrict format, va_list args);
errno_t __validate_sn_format(const char * restrict format, va_list arg);
/* constraint validator; constraint_validator_s.c */
errno_t __error_out(bool errflag, param_validation_status_t * restrict
status);
errno_t __constraint_validator_s(param_validation_status_t * restrict status);
errno_t
__constraint_validator_object_overlap(const char * restrict function_name,
const char * restrict parameterNames, const void * restrict object1Start,
const size_t object1Size, const void * restrict object2Start, const size_t
object2Size);
errno_t
__constraint_validator_value_inrange(const char * restrict function_name,
const char * restrict parameterName, const size_t value, const size_t
lowerBound, const size_t upperBound);
errno_t
__constraint_validator_not_null(const char * restrict function_name, const
char * restrict parameter_name, const void * restrict value_ptr);
errno_t
__constraint_validator_s_format(const char * restrict function_name, const
char * restrict parameter_name, const char * restrict format, const va_list
args);
errno_t
__constraint_validator_n_format(const char * restrict function_name, const
char * restrict parameter_name, const char * restrict format, const va_list
args);
errno_t
__constraint_validator_not_zero(const char * restrict function_name, const
char * restrict parameter_name, const size_t value);
errno_t
__constraint_validator_rsize_limit(const char * restrict function_name, const
char * restrict parameter_name, const rsize_t value);
void
__report_constraint_violation_end_of_token_not_present(const char * restrict
function_name, const char * restrict parameter_name);
Listing 11: Implemented Constraint Handling API
⬇
errno_t __validate_args_not_null(const char * restrict format, va_list args);
errno_t
__constraint_validator_not_null_args(const char * restrict function_name,
const char * restrict parameter_name, const char * restrict format, const
va_list args);
Listing 12: Not Implemented Constraint Handling API
### 4.2 Constraint Handling In Action – stdio
Test code is in Listing 13.
⬇
/*
* Sloppy Programming Test Cases
*/
#define __STDC_WANT_LIB_EXT1__ 1
#include "stdio.h"
#include "stdlib.h"
#include <unistd.h>
#include <string.h>
int
main(int argc, char** argv)
{
int valid = 1;
set_constraint_handler_s(ignore_handler_s);
if(argc > 1)
{
printf("Sloppy programming zone: [[%s]]\n", argv[1]);
printf_s(argv[1]);
printf("\n\n");
}
printf_s("valid s = [%s]\n", "valid");
printf_s(" valid n1 = [%%%%n]\n\n", &valid);
printf_s("invalid n2 = [%%%n]\n\n", &valid);
printf_s(" valid n3 = [%%n]\n\n", &valid);
printf_s("invalid n4 = [%n]\n\n", &valid);
printf_s("invalid s = [%s]\n", NULL);
printf_s("invalid n = [%n]\n\n", &argc);
printf("return value for %%n: [%d]\n", printf_s("%n\n", &argc));
printf("return value for %%s: [%d]\n", printf_s("%s\n", NULL));
return 0;
}
/* EOF */
Listing 13: Example of a Test Program for stdio to test and reject invalid %s
and %n cases.
Output is in Listing 14.
⬇
bash-2.05b$ test/stdio/test %n
Sloppy programming zone: [[%n]]
printf_s(): invalid format parameter (%n is disallowed) : format/args %n
valid s = [%s]
valid s = [valid]
valid n1 = [n]
valid n1 = [%%n]
invalid n2 = [printf_s(): invalid format parameter (%n is disallowed) :
format/args %n
valid n3 = [n]
valid n3 = [%n]
invalid n4 = [printf_s(): invalid format parameter (%n is disallowed) :
format/args %n
printf_s(): invalid format parameter (NULL argument for %s) : format/args null
%s
invalid n = [printf_s(): invalid format parameter (%n is disallowed) :
format/args %n
printf_s(): invalid format parameter (%n is disallowed) : format/args %n
return value for %n: [22]
printf_s(): invalid format parameter (NULL argument for %s) : format/args null
%s
return value for %s: [22]
bash-2.05b$
Listing 14: Output
### 4.3 Constraint Handling In Action – string
Test code is in Listing 15.
⬇
#define __STDC_WANT_LIB_EXT1__ 1
#include "string.h"
#include <errno.h>
#include <stdio.h>
#include <string.h>
#include "stdlib.h"
int main(int argc, char** argv){
char buffer1[1024];
char buffer2[1024];
set_constraint_handler_s(ignore_handler_s);
/*Failure tests*/
printf("strcpy_s failure test:\t");
strcpy_s(buffer1, 1024, NULL);
printf("strncpy_s failure test:\t");
strncpy_s(buffer1, 10, buffer2, 50);
printf("strncat_s failure test:\t");
strcat_s(buffer1, -1, buffer2);
printf("strncat_s failure test:\t");
strncat_s(NULL, 1024, NULL, 50);
printf("strncat_s failure test:\t");
strncat_s(buffer1, 1024, buffer1-10, 50);
printf("memcpy_s failure test:\t");
memcpy_s(buffer1, 1024, buffer2, -1);
printf("memmove_s failure test:\t");
memmove_s(buffer1, 1023, NULL, 1024);
/*Normal operation tests*/
strcpy_s(buffer1, 1024, "test string");
printf("strcpy_s: %s \n", buffer1);
strncpy_s(buffer2, 1024, buffer1, 1024);
printf("strncpy_s: %s \n", buffer2);
strcat_s(buffer1, 1024, buffer2);
printf("strcat_s: %s \n", buffer1);
...
printf("strnlen_s(buffer1): %u \n", strlength);
size_t errlen = strerrorlen_s(EINVAL);
printf("strerrorlen_s(EINVAL): %u\n", errlen);
strerror_s(buffer2, 1014, EINVAL);
printf("%s\n", buffer2);
rsize_t l= strnlen_s (buffer1, 100);
char * strktokresult = NULL;
char * token = strtok_s(buffer1, &l, " ", &strktokresult);
printf("strtok_s token: %s, remaining length: %u, remaining substring: %s\n",
token, l, strktokresult);
token = strtok_s(NULL, &l, "gt", &strktokresult); /*Should be found*/
...
/*Move printfs to a real test file*/
size_t s = strnlen_s("12345", 10);
printf("size: %u\n", s);
}
Listing 15: Example of a Test Program for string to test and reject invalid
cases.
Output is in Listing 16.
⬇
bash-2.05b$ test/string/test
strcpy_s failure test: strcpy_s(): has invalid NULL pointer argument : s2
strncpy_s failure test: strncat_s failure test: strcat_s(): rsize_t value
exceeds RSIZE_MAX : s1max
strncat_s failure test: strncat_s(): has invalid NULL pointer argument : s1
strncat_s failure test: strncat_s(): two data structures overlap in memory :
s1 and s2
memcpy_s failure test: memcpy_s(): rsize_t value exceeds RSIZE_MAX : n
memmove_s failure test: memmove_s(): has invalid NULL pointer argument : s2
strcpy_s: test string
strncpy_s: test string
strcat_s: test stringtest string
strncat_s: test stringtest stringtest string
memmove_s: test stringtest stringtest string
memcpy_s: test stringtest stringtest string
strnlen_s(buffer1): 33
strerrorlen_s(EINVAL): 0
test string
strtok_s token: test, remaining length: 28, remaining substring: stringtest
stringtest string
strtok_s token: strin, remaining length: 21, remaining substring: est
stringtest string
strtok_s(): token end not found within defined bounds : *ptr
strtok_s token: est stringtest string, remaining length: 0, remaining
substring:
size: 5
bash-2.05b$
Listing 16: Output
## Chapter 5 Conclusions
Here were briefly address the following topics:
* •
Difficulties
* •
Limitations
* •
Acknowledgments
* •
Future Work
### 5.1 Summary of the Difficulties
1. 1.
Parsing/processing of varargs and %n in particular
2. 2.
Deciding on default values
3. 3.
Implementing strtok_s and its wide character equivalent
### 5.2 Limitations So Far
* •
Incomplete implementation (of approx. 45%) of the entire API
* •
Lack of thorough testing for all the implemented API
### 5.3 Acknowledgments
* •
Dr. Prabir Bhattacharya
* •
Dr. Mourad Debbabi
* •
ISO
* •
Open Source Community and the GLIBC Team [gp]
### 5.4 Future Work
This project, being a derivative of the standard C library, will see a major
effort put into the testing of the library. Furthermore, a large part of the
project (transforming the obsolete calls to wrappers to the newer ones) makes
the assumption that the buffer free size is easily obtainable, an assumption
which may not necessarily hold true in all circumstances. As such, it is
possible that we will have to develop novel algorithms to ensure that this
portability is possible, or it may also be that this portability is not 100%
attainable. Furthermore, we will need to investigate good potential software
for performance and security testing of our improved solution. Thus, we will
focus on:
* •
Completion of implementation,
* •
Addition more comprehensive test cases by the developers and the OSS
community,
* •
Application for EAL5,
* •
Inclusion into the Linux kernel as a standard.
## Bibliography
* [Cha] Dr. P. Chalin. Soen 344 slides. http://www.cs.concordia.ca/~chalin/courses/06W/SOEN344/.
* [Cor05] Microsoft Corp. Using the strsafe.h functions, 2005. http://msdn.microsoft.com/library/default.asp?url=/library/en-us/winui/%winui/windowsuserinterface/resources/strings/usingstrsafefunctions.asp.
* [gp] glibc project. Gnu c library. http://www.gnu.org/software/libc/.
* [HNS00] Christine Hofmeister, Robert Nord, and Dilip Soni. Applied Software Architecture. Addison-Wesley, 2000.
* [ISO] ISO. ISO/TR Technical Report. http://www.iso.org/iso/en/stdsdevelopment/whowhenhow/proc/deliverables/%iso_tr.html.
* [ISO05] ISO/IEC. ISO/IEC Directives, Part 2: Rules for the structure and drafting of International Standards, 5th edition, 2005. http://isotc.iso.org/livelink/livelink.exe/4230517/ISO_IEC_Directives__%Part_2__Rules_for_the_structure_and_drafting_of_International_Standards__2004_%_5th_edition___pdf_format_.pdf?func=doc.Fetch&nodeid=4230517.
* [ISO06a] ISO. JTC 1, 2006. http://www.iso.org/iso/en/stdsdevelopment/tc/tclist/TechnicalCommitteeD%etailPage.TechnicalCommitteeDetail?COMMID=1.
* [ISO06b] ISO. Member bodies, 2006. http://www.iso.org/iso/en/aboutiso/isomembers/MemberList.MemberSummary?%MEMBERCODE=10.
* [MdR99] T. C. Miller and T. de Raadt. strlcpy and strlcat—consistent, safe string copy and concatenation. In Proceedings of the FREENIX Track, 1999 USENIX Annual Technical Conference, pages 175–178. USENIX Association, 1999. http://www.usenix.org/publications/library/proceedings/usenix99/full_papers/millert/millert.pdf.
* [PMB] M-A Laverdière Papineau, S. Mokhov, and D. Benredjem. Statistical classification of vulnerability solutions in the linux kernels 2.4/2.6. Submitted to IEEE COMPSAC, pending acceptance.
* [pro] GCC project. Extensions to the c language family. http://gcc.gnu.org/onlinedocs/gcc-3.3.1/gcc/C-Extensions.html.
* [SC2] ISO/IEC JTC1 SC22/WG14. Draft minutes for 25-28 september 2005 meeting of iso/iec jtc1 sc22/wg14 and incits j11. http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1145.pdf.
* [Sea05] R. Seacord. Secure Coding in C and C++. SEI Series. Addison-Wesley, 2005.
* [Sec05] ISO Central Secretariat. ISO in brief, 2005. http://www.iso.org/iso/en/prods-services/otherpubs/pdf/isoinbrief_2005-%en.pdf.
* [Tig] Tigris. Subversion. http://subversion.tigris.org/.
* [vH] Dimitry van Heesch. Doxygen manual for version 1.4.6. ftp://ftp.stack.nl/pub/users/dimitri/doxygen_manual-1.4.6.pdf.zip.
* [WG106] ISO/IEC JTC1 SC22 WG14. Programming language c - specification for safer more secure c library functions. Technical Report ISO/IEC TR 24731, ISO, 2006. Draft status at time of writing.
## Index
* API
* abort_handler_s §3.1
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|
arxiv-papers
| 2009-06-14T17:00:24 |
2024-09-04T02:49:03.350159
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marc-Andr\\'e Laverdi\\`ere, Serguei A. Mokhov, and Djamel Benredjem",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/0906.2512"
}
|
0906.2584
|
Massive Gauge Bosons in Yang-Mills Theory without Higgs Mechanism
Xin-Bing Huang***[email protected]
Shanghai United Center for Astrophysics (SUCA),
Shanghai Normal University, No.100 Guilin Road, Shanghai 200234, China
Abstract
Two kinds of Yang-Mills fields are found upon the concepts of mass eigenstate
and nonmass eigenstate. The Yang-Mills fields of the first kind were proposed
by Yang and Mills, which couple to the mass eigenstates with the same rest
mass, whose gauge bosons are massless. I find that there are second kind of
Yang-Mills fields, which are constructed on a five-dimensional manifold. Only
the nonmass eigenstates couple to the Yang-Mills fields of the second kind,
which are the nonmass eigenstates as well and composed of mass eigenstates of
gauge bosons. The mass eigenstates of the Yang-Mills fields of the second kind
live in the four-dimensional spacetime, the corresponding gauge bosons of
which may be massive. The $SU(2)\times U(1)$ gauge fields of the second kind
are studied carefully, whose gauge bosons, which are the mass eigenstates, are
the $W^{\pm}$, $Z^{0}$ and photon fields. The rest masses of $W^{\pm}$ and
$Z^{0}$ obtained are the same as that given by the Glashow-Salam-Weinberg
model of electroweak interactions. It is discussed that this model should be
renormalizable.
PACS numbers: 11.15.-q, 11.10.Kk, 12.60.-i
55 years ago, Yang and Mills constructed the gauge field theory of non-Abelian
group, which has become the most fundamental content in quantum field theory.
Upon the principle that physical laws should be covariant under the local
isospin rotation they proposed the $SU(2)$ Yang-Mills theory [1]. But they
could not obtain the massive gauge bosons then. About 10 years later, an
ingenious trick called the Higgs mechanism was independently invented by Higgs
and Englert and Brout [2], who introduced a scalar field and the spontaneous
symmetry broken mechanism of vacuum by fixing a vacuum expectation value of
the scalar field and make the intermediate vector bosons obtain masses.
Based on the Yang-Mills fields and the Higgs mechanism, Glashow, Salam and
Weinberg etc. proposed a renormalizable theory unifying the weak and
electromagnetic interactions, namely $SU_{L}(2)\times U_{Y}(1)$ gauge theory
[3]. Although this electroweak theory had predicted the masses of intermediate
vector bosons, which were confirmed by experiments, there are still several
unconfirmed predictions or conflicting phenomena in it. e.g. Firstly,
experimenters have not found any hints of the Higgs boson till now; Secondly,
a lot of recent experiments imply that the neutrinos should be massive and be
mixed [4]. Here I discuss a model to give the massive gauge bosons in Yang-
Mills theory without Higgs mechanism.
In this letter, the signature of spacetime metric
$\eta_{\mu\nu}(\mu,\nu=0,1,2,3)$ is $(+,-,-,-)$, and the spacetime coordinates
are described by the contravariant four-vector $x^{\mu}$ ($\hbar=c=1$ is
adopted). In Ref.[5], the rest mass operator†††I use
$\partial_{z}\equiv\frac{\partial}{\partial
z}~{},~{}\partial_{\mu}\equiv\frac{\partial}{\partial x^{\mu}}$ and
$\partial_{\alpha}\equiv\frac{\partial}{\partial x^{\alpha}}$.
$\hat{m}=-i\partial_{z}$ (1)
is defined by introducing an extra parameter $z$ besides of the spacetime
coordinates $x^{\mu}$. From the mathematical point of view, $z$ and $x^{\mu}$
establish a five-dimensional manifold. The definition of the rest mass
operator leads to a theorem that a field ${\cal F}(x,z)$ is massless if and
only if ${\cal F}(x,z)$ is $z$-independent [5]. Hence the massless
gravitational field, the electromagnetic field and $SU(3)$ gauge fields in
$QCD$ are all $z$-independent, who live in the $z=0$ brane of five-dimensional
manifold.
The Lagrangian of a nonmass eigenstate ${\Phi}(x,z)$ of free
spin-$\frac{1}{2}$ fields is of the form‡‡‡${\cal L}_{1n}$, ${\cal L}_{1m}$
denote the Lagrangian of one nonmass eigenstate or one mass eigenstate
respectively. ${\cal L}_{2n}$, ${\cal L}_{2m}$ have the similar meanings.
$\displaystyle{\cal
L}_{1n}={\bar{\Phi}}(x,z)\left(i\gamma^{\mu}{\partial}_{\mu}+i{\partial}_{z}\right)\Phi(x,z)~{},$
(2)
here $\bar{\Phi}\equiv{\Phi}^{{\dagger}}\gamma^{0}$ is called the spinor
adjoint to $\Phi$. I indicated that the mass eigenstate of a
spin-$\frac{1}{2}$ field satisfies $\Phi(x,z)=e^{imz}\phi(x)$ in Ref.[5],
where $m$ is the rest mass. Therefore one can obtain the Lagrangian of the
mass eigenstate of a free spin-$\frac{1}{2}$ field from (2), that is
$\displaystyle{\cal
L}_{1m}={\bar{\phi}}(x)\left(i\gamma^{\mu}{\partial}_{\mu}-m\right)\phi(x)~{},$
(3)
where $\bar{\phi}\equiv{\phi}^{{\dagger}}\gamma^{0}$ is the spinor adjoint to
$\phi$.
Let’s consider a quantum field system in which two different nonmass
eigenstates $\Psi_{1}(x,z)$ and $\Psi_{2}(x,z)$ of free spin-$\frac{1}{2}$
fields form an isospin doublet as follows
$\displaystyle\Psi(x,z)=\left(\begin{array}[]{c}\Psi_{1}(x,z)\\\
\Psi_{2}(x,z)\end{array}\right)~{}.$ (6)
So the Lagrangian of two nonmass eigenstates of free spin-$\frac{1}{2}$ fields
is
$\displaystyle{\cal
L}_{2n}={\bar{\Psi}}(x,z)\left(i\gamma^{\mu}{\partial}_{\mu}+i{\partial}_{z}\right)\Psi(x,z)~{}.$
(7)
Here Let’s first consider a special case: if $\Psi_{1}(x,z)$ and
$\Psi_{2}(x,z)$ are mass eigenstates with the same rest mass, then
$\Psi_{1}(x,z)=e^{imz}\psi_{1}(x)$, and $\Psi_{2}(x,z)=e^{imz}\psi_{2}(x)$,
where $m$ is the rest mass. I can therefore obtain $\Psi(x,z)=e^{imz}\psi(x)$
by defining
$\displaystyle\psi(x)=\left(\begin{array}[]{c}\psi_{1}(x)\\\
\psi_{2}(x)\end{array}\right)~{}.$ (10)
Hence the Lagrangian of a quantum field system where two mass eigenstates
$e^{imz}\psi_{1}(x)$ and $e^{imz}\psi_{2}(x)$ form an isospin doublet is
acquired from (7), namely
$\displaystyle{\cal
L}_{2m}={\bar{\psi}}(x)\left(i\gamma^{\mu}{\partial}_{\mu}-m\right)\psi(x)~{}.$
(11)
The largest inner gauge symmetry group in this system is obviously
$SU(2)\times U(1)$. The total Lagrangian of this system reads
$\displaystyle{\cal
L}_{2mt}=i{\bar{\psi}}\gamma^{\mu}(\partial_{\mu}-ig^{\prime}{\bf T}\cdot{\bf
B}_{\mu})\psi-e{\bar{\psi}}\gamma^{\mu}A_{\mu}\psi$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-m{\bar{\psi}}\psi-\frac{1}{4}~{}{\tilde{\bf
F}}_{\mu\nu}\cdot{\tilde{\bf
F}}^{\mu\nu}-~{}\frac{1}{4}~{}{\tilde{E}}_{\mu\nu}{\tilde{E}}^{\mu\nu}~{},$
(12)
where $e$, $g^{\prime}$ are the coupling constants of $U(1)$ and $SU(2)$ gauge
fields respectively, and the dot “$\cdot$” denotes a scalar product in the
isospace. In this case, ${\bf T}\cdot{\bf B}_{\mu}$ means
$\displaystyle{\bf T}\cdot{\bf
B}_{\mu}=T^{1}B^{1}_{\mu}+T^{2}B^{2}_{\mu}+T^{3}B^{3}_{\mu}~{},$ (13)
where $T^{a},~{}a=1,2,3$ are the generators of $SU(2)$ group, which are
written as
$T^{a}=\frac{1}{2}\tau^{a}~{},$ (14)
where $\tau^{a}$ are the traceless matrices
$\displaystyle\tau^{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)~{},~{}~{}\tau^{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)~{},~{}~{}\tau^{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)~{},$ (21)
known as the Pauli matrices. They obey the commutation relations
$[\tau^{a},\tau^{b}]=2i\sum^{3}_{c=1}\varepsilon_{abc}\tau^{c}~{}.$ (22)
Here $\varepsilon_{abc}$ is the totally antisymmetry tensor in 3-dimensions.
In Yang-Mills theory [1], ${\bf T}\cdot{\bf B}_{\mu}$ is called the $SU(2)$
gauge field, and its field strength tensor is of the form
$\displaystyle{\tilde{\bf F}}_{\mu\nu}\cdot{\bf T}=\partial_{\mu}({\bf
B}_{\nu}\cdot{\bf T})-\partial_{\nu}({\bf B}_{\mu}\cdot{\bf
T})-{i}g^{\prime}[{\bf B}_{\mu}\cdot{\bf T},{\bf B}_{\nu}\cdot{\bf T}]~{}.$
(23)
Hence the field strength ${\tilde{\bf F}}_{\mu\nu}$ satisfies
${\tilde{\bf F}}_{\mu\nu}=\partial_{\mu}{\bf B}_{\nu}-\partial_{\nu}{\bf
B}_{\mu}+g^{\prime}{\bf B}_{\mu}\times{\bf B}_{\nu}~{}.$ (24)
I use $A_{\mu}$ to denote the $U(1)$ gauge field. The field strength of the
$U(1)$ gauge field is defined by
${\tilde{E}}_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}~{}.$ (25)
We are very familiar with above $SU(2)$ gauge fields and $U(1)$ gauge field
which have been the fundamental content of quantum field theories. The gauge
bosons are massless. From a viewpoint of the rest mass operator, above $SU(2)$
gauge fields couple to the mass eigenstates with the same rest mass. I call
this Yang-Mills fields the first kind. The $SU(3)$ gauge fields in $QCD$
obviously belong to this kind.
From now on I will study another kind of Yang-Mills fields carefully. To make
the gauge invariance explicit, let’s formally introduce the extra dimension
$x^{4}$ as follows
$x^{4}=-x_{4}=z~{},~{}~{}~{}~{}\gamma^{4}=-\gamma_{4}={\bf 1}~{}.$ (26)
Hence the Lagrangian ${\cal L}_{2n}$ is rewritten as
$\displaystyle{\cal
L}_{2n}=i{\bar{\Psi}}(x,z)\gamma^{\alpha}{\partial}_{\alpha}\Psi(x,z)~{},~{}~{}~{}~{}\alpha=0,1,2,3,4~{}.$
(27)
Since two different nonmass eigenstates $\Psi_{1}(x,z)$ and $\Psi_{2}(x,z)$
form an isospin doublet, I can consider a local isospin rotation logically
similar to what Yang and Mills did in their original paper. That is
$\displaystyle{\Psi}^{\prime}(x,z)=S(x,z)\Psi(x,z)~{},$ (28)
where $S(x,z)$ is a $2\times 2$ matrix. To make sure that the probability
density ${\bar{\Psi}}(x,z)\Psi(x,z)$ is invariant under above rotation (28),
the matrix $S(x,z)$ must be unitary with unit determinant
$\displaystyle S^{{\dagger}}(x,z)S(x,z)=1~{}.$ (29)
All the matrices satisfy this condition generate the group $SU(2)$, which is a
non-Abelian Lie group. The transformation (28) directly means that
$\displaystyle{\bar{\Psi}}^{\prime}(x,z)={\bar{\Psi}}(x,z)S^{{\dagger}}(x,z)~{}.$
(30)
The matrix $S(x,z)$ can be written in the form
$\displaystyle
S(x,z)=\exp\left(i\sum_{a=1}^{3}\frac{{\tau}^{a}}{2}{\Theta}^{a}(x,z)\right)~{}.$
(31)
To discuss the gauge invariance, here I introduce the gauge-invariant
derivative
$\displaystyle\hat{D}_{\alpha}=\partial_{\alpha}-ig_{1}{\bf T}\cdot{\bf
W}_{\alpha}(x,z)~{},$ (32)
where $g_{1}$ is the coupling constant of $SU(2)$ gauge fields, and
$\displaystyle{\bf T}\cdot{\bf W}_{\alpha}(x,z)=\sum_{a=1}^{3}T^{a}{\bf
W}_{\alpha}^{a}(x,z)~{}.$ (33)
Invariance requires that
$\displaystyle(\partial_{\alpha}-ig_{1}{\bf T}\cdot{\bf
W}_{\alpha}^{\prime}){\Psi}^{\prime}=S(\partial_{\alpha}-ig_{1}{\bf
T}\cdot{\bf W}_{\alpha})\Psi~{}.$ (34)
Combining (28) and (34), I obtain the gauge transformation on ${\bf
W}_{\alpha}$:
$\displaystyle{\bf T}\cdot{\bf W}_{\alpha}^{\prime}=S{\bf T}\cdot{\bf
W}_{\alpha}S^{-1}+\frac{i}{g_{1}}S\left(\partial_{\alpha}S^{-1}\right)~{}.$
(35)
In analogy to the procedure of obtaining gauge invariant field strengths in
electromagnetic case, I define now
$\displaystyle{{\bf F}}_{\alpha\beta}\cdot{\bf T}=\sum_{a=1}^{3}{\bf
F}^{a}_{\alpha\beta}{T}^{a}=\hat{D}_{\alpha}({\bf T}\cdot{\bf
W}_{\beta})-\hat{D}_{\beta}({\bf T}\cdot{\bf W}_{\alpha})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}=\partial_{\alpha}({\bf
W}_{\beta}\cdot{\bf T})-\partial_{\beta}({\bf W}_{\alpha}\cdot{\bf
T})-{i}g_{1}\left[{\bf W}_{\alpha}\cdot{\bf T},{\bf W}_{\beta}\cdot{\bf
T}\right]$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}=(\partial_{\alpha}{\bf
W}_{\beta})\cdot{\bf T}-(\partial_{\beta}{\bf W}_{\alpha})\cdot{\bf
T}+g_{1}\sum_{abc}{\bf W}^{a}_{\alpha}{\bf
W}^{b}_{\beta}\varepsilon_{abc}T^{c}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}=(\partial_{\alpha}{\bf
W}_{\beta}-\partial_{\beta}{\bf W}_{\alpha}+g_{1}{\bf W}_{\alpha}\times{\bf
W}_{\beta})\cdot{\bf T}~{}.$ (36)
Therefore the isovector of field strengths is
$\displaystyle{{\bf F}}_{\alpha\beta}=\partial_{\alpha}{\bf
W}_{\beta}-\partial_{\beta}{\bf W}_{\alpha}+g_{1}{\bf W}_{\alpha}\times{\bf
W}_{\beta}~{}.$ (37)
One easily shows from the equation (35) that
$\displaystyle{\bf F}^{\prime}_{\alpha\beta}\cdot{\bf T}=S{{\bf
F}}_{\alpha\beta}\cdot{\bf T}S^{-1}~{}.$ (38)
I obtain a gauge invariant Lagrangian by performing the trace over the isospin
indices:
$\displaystyle{\cal L}_{SU(2)}=-\frac{1}{2}{\rm Tr}\\{({{\bf
F}}_{\alpha\beta}\cdot{\bf T})({{\bf F}}^{\alpha\beta}\cdot{\bf T})\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=-\frac{1}{4}{{\bf
F}}_{\alpha\beta}\cdot{{\bf F}}^{\alpha\beta}=-\frac{1}{4}\sum_{a=1}^{3}{{\bf
F}}^{a}_{\alpha\beta}{{\bf F}}^{a\alpha\beta}~{}.$ (39)
Considering the couplings between fermions and gauge bosons and the self-
couplings of gauge bosons, one can get the complete Lagrangian as follows
$\displaystyle{\cal L}_{2nt}={\cal L}_{2n}+{\cal L}_{int}+{\cal L}_{SU(2)}$
$\displaystyle~{}~{}~{}~{}~{}~{}=i{\bar{\Psi}}\gamma^{\alpha}(\partial_{\alpha}-ig_{1}{\bf
T}\cdot{\bf W}_{\alpha})\Psi-\frac{1}{4}{{\bf F}}_{\alpha\beta}\cdot{{\bf
F}}^{\alpha\beta}~{}.$ (40)
In order to build a foundation for setting up an electroweak model without
Higgs mechanism, I discuss the $SU(2)\times U(1)$ gauge fields in this letter.
The $U(1)$ gauge field that couples to a nonmass eigenstate has been studied
in my preceding paper [5]. The Lagrangian (27) shows me that the maximal gauge
groups for this quantum field system are $SU(2)\times U(1)$. I have introduced
the $SU(2)$ gauge fields in this system, now I put in the $U(1)$ gauge field.
Let us multiply the nonmass eigenstates $\Psi(x,z)$ by a local phase
$e^{i\Theta(x,z)}$, namely
$\Psi^{\prime\prime}=~{}e^{i\Theta(x,z)}\Psi~{},~{}{\bar{\Psi}}^{\prime\prime}=~{}e^{-i\Theta(x,z)}{\bar{\Psi}}~{}.$
(41)
According to the discussion in Ref.[5], I introduce the $U(1)$ gauge field of
the second kind, that is ${\bf X}_{\alpha}(x,z)$. Under the transformation of
(41), ${\bf X}_{\alpha}(x,z)$ transforms as
${\bf X}^{\prime\prime}_{\alpha}(x,z)={\bf
X}_{\alpha}(x,z)+\frac{1}{g_{2}}{\partial_{\alpha}}\Theta(x,z)~{},$ (42)
here $g_{2}$ is the coupling constant of $U(1)$. The strength tensor of $U(1)$
gauge field is of the form
${\bf E}_{\alpha\beta}(x,z)={\partial}_{{\alpha}}{\bf
X}_{\beta}(x,z)-{\partial}_{\beta}{\bf X}_{\alpha}(x,z)~{},$ (43)
which is invariant under the transformations of (41) and (42). Therefore the
total Lagrangian including the $U(1)$ gauge field of the second kind is
written as
$\displaystyle{\cal L}_{total}={\cal L}_{2n}+{\cal L}_{int}+{\cal
L}_{U(1)}+{\cal L}_{SU(2)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}=i{\bar{\Psi}}\gamma^{\alpha}(\partial_{\alpha}-ig_{2}{\bf
X}_{\alpha}-ig_{1}{\bf T}\cdot{\bf W}_{\alpha})\Psi$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{1}{4}{{\bf
E}}_{\alpha\beta}{\bf E}^{\alpha\beta}-\frac{1}{4}{{\bf
F}}_{\alpha\beta}\cdot{{\bf F}}^{\alpha\beta}~{}.$ (44)
Obviously the total Lagrangian is also invariant under the transformations of
(41) and (42). The gauge covariance requires that ${\bf X}_{\alpha}$ and ${\bf
T}\cdot{\bf W}_{\alpha}$ are all nonmass eigenstates.
Till now I merely constructed the $SU(2)\times U(1)$ gauge fields on a five-
dimensional manifold, which is quite the same as the gauge fields proposed by
Yang and Mills. Yes, from the five-dimensional point of view, the gauge bosons
are massless in above discussed $SU(2)\times U(1)$ gauge fields since there
are no mass term in the total Lagrangian (44). But, things will be quite
different when I discuss them from the viewpoint of $z=0$ brane.
I have pointed out that the gravitational field, the electromagnetic field and
$SU(3)$ gauge fields in $QCD$ are living in the $z=0$ brane of five-
dimensional manifold. Also I have proved that the $z$-independent
electromagnetic field, gravitational field and $SU(3)$ gauge fields only
couple to the mass eigenstates. Therefore I can find that the mass eigenstates
coupled by the gravitation, the electromagnetic field and the gluon fields are
also living in the $z=0$ 4-dimensional brane.
It is indicated that the nonmass eigenstate is composed of mass eigenstates
[5]. To discuss the physical properties of the mass eigenstates who compose
the gauge fields ${\bf X}_{\alpha}$ and ${\bf W}_{\alpha}$, I write out the
spacetime component and $z$-related component of gauge fields separately.
Therefore
$\displaystyle{\bf X}_{\alpha}(x,z)\equiv({\bf X}_{\mu}(x,z),{\bf
X}_{z}(x,z))~{},$ (45) $\displaystyle{\bf W}_{\alpha}(x,z)\equiv({\bf
W}_{\mu}(x,z),{\bf W}_{z}(x,z))~{}.$ (46)
To list the components of ${\bf W}_{\alpha}(x,z)$ manifestly, I rewrite (46)
as
${\bf W}^{a}_{\alpha}(x,z)\equiv({\bf W}^{a}_{\mu}(x,z),{\bf
W}^{a}_{z}(x,z))~{},~{}a=1,2,3~{}.$ (47)
After that, the strength tensor ${\bf E}_{\alpha\beta}(x,z)$ is
correspondingly divided into three parts
$\displaystyle{\bf E}_{\mu\nu}(x,z)=\partial_{\mu}{\bf
X}_{\nu}(x,z)-\partial_{\nu}{\bf X}_{\mu}(x,z)~{},$ (48) $\displaystyle{\bf
E}_{\mu z}(x,z)=-{\bf E}_{z\mu}(x,z)=\partial_{\mu}{\bf
X}_{z}(x,z)-\partial_{z}{\bf X}_{\mu}(x,z)~{},$ (49) $\displaystyle{\bf
E}_{zz}(x,z)=\partial_{z}{\bf X}_{z}(x,z)-\partial_{z}{\bf X}_{z}(x,z)\equiv
0~{}.$ (50)
Surely one can also get the decomposition of the strength tensor ${\bf
F}_{\alpha\beta}(x,z)$ as follows
$\displaystyle{{\bf F}}_{\mu\nu}(x,z)=\partial_{\mu}{\bf
W}_{\nu}-\partial_{\nu}{\bf W}_{\mu}+g_{1}{\bf W}_{\mu}\times{\bf
W}_{\nu}~{},$ (51) $\displaystyle{\bf F}_{\mu z}(x,z)=-{\bf
F}_{z\mu}(x,z)=\partial_{\mu}{\bf W}_{z}-\partial_{z}{\bf W}_{\mu}+g_{1}{\bf
W}_{\mu}\times{\bf W}_{z}~{},$ (52) $\displaystyle{\bf F}_{zz}(x,z)\equiv
0~{}.$ (53)
Now let’s consider the movement of gauge bosons. Firstly, the interaction term
${\cal L}_{int}$ in total Lagrangian (44) shows that the movement of gauge
bosons is decided by the momentum of $\Psi$ and $\bar{\Psi}$. The $\Psi$ and
$\bar{\Psi}$ are nonmass eigenstates, who are composed of mass eigenstates
that are living in the $z=0$ brane and moving along the $z=0$ brane, hence
gauge bosons must move along the $z=0$ brane. Secondly, once the gauge bosons
are produced, they are constrained by gravitation, which is living in the
$z=0$ brane. Consequently
${\bf W}_{z}(x,z)=0~{},~{}{\bf X}_{z}(x,z)=0~{}.$ (54)
Hence the equations (49) and (52) reduce to
$\displaystyle{\bf E}_{\mu z}(x,z)=-{\bf E}_{z\mu}(x,z)=-\partial_{z}{\bf
X}_{\mu}(x,z)~{},$ (55) $\displaystyle{\bf F}_{\mu z}(x,z)=-{\bf
F}_{z\mu}(x,z)=-\partial_{z}{\bf W}_{\mu}(x,z)~{}.$ (56)
The spacetime components ${\bf X}_{\mu}(x,z)$ and ${\bf W}_{\mu}(x,z)$ are
nonmass eigenstates, which are linear combinations of mass eigenstates. I
define the mass eigenstates of bosons $W^{\pm}$ by
$\displaystyle{\bf W}^{+}_{\mu}(x,z)=\frac{1}{\sqrt{2}}\left({\bf
W}^{1}_{\mu}(x,z)-i{\bf W}^{2}_{\mu}(x,z)\right)~{},$ (57) $\displaystyle{\bf
W}^{-}_{\mu}(x,z)=\frac{1}{\sqrt{2}}\left({\bf W}^{1}_{\mu}(x,z)+i{\bf
W}^{2}_{\mu}(x,z)\right)~{}.$ (58)
When the mass eigenstates of $W^{\pm}$ are expressed by
$\displaystyle{\bf W}^{+}_{\mu}(x,z)=e^{im_{W}z}{W}^{+}_{\mu}(x)~{},~{}{\bf
W}^{-}_{\mu}(x,z)=e^{im_{W}z}{W}^{-}_{\mu}(x)~{},$ (59)
$m_{W}$ being the rest mass of $W^{\pm}$, the nonmass eigenstates ${\bf
W}^{1}_{\mu}(x,z)$ and ${\bf W}^{2}_{\mu}(x,z)$ are manifestly given by
$\displaystyle{\bf W}^{1}_{\mu}(x,z)=\frac{1}{\sqrt{2}}\left({\bf
W}^{+}_{\mu}(x,z)+{\bf W}^{-}_{\mu}(x,z)\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{1}{\sqrt{2}}e^{im_{W}z}\left({W}^{+}_{\mu}(x)+{W}^{-}_{\mu}(x)\right)~{},$
(60)
and
$\displaystyle{\bf W}^{2}_{\mu}(x,z)=\frac{i}{\sqrt{2}}\left({\bf
W}^{+}_{\mu}(x,z)-{\bf W}^{-}_{\mu}(x,z)\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{i}{\sqrt{2}}e^{im_{W}z}\left({W}^{+}_{\mu}(x)-{W}^{-}_{\mu}(x)\right)~{}.$
(61)
The boson fields ${\bf W}^{+}_{\mu}(x,z)$, ${\bf W}^{-}_{\mu}(x,z)$, ${\bf
Z}_{\mu}(x,z)$ and photon field $A_{\mu}(x)$, which are mass eigenstates,
constitute a complete Hilbert space. From Ref.[5], I know that
$\displaystyle{\bf Z}_{\mu}(x,z)=e^{im_{Z}z}Z_{\mu}(x)~{},$ (62)
here $m_{Z}$ is the rest mass of boson $Z^{0}$. The nonmass eigenstates ${\bf
W}^{3}_{\mu}(x,z)$ and ${\bf X}_{\mu}(x,z)$ are the linear combinations of
${\bf Z}_{\mu}(x,z)$ and $A_{\mu}(x)$, namely
$\displaystyle{\bf W}^{3}_{\mu}(x,z)=\sin\theta_{W}A_{\mu}+\cos\theta_{W}{\bf
Z}_{\mu}(x,z)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\sin\theta_{W}A_{\mu}+\cos\theta_{W}e^{im_{Z}z}Z_{\mu}~{},$
(63) $\displaystyle{\bf X}_{\mu}(x,z)=\cos\theta_{W}A_{\mu}-\sin\theta_{W}{\bf
Z}_{\mu}(x,z)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\cos\theta_{W}A_{\mu}-\sin\theta_{W}e^{im_{Z}z}Z_{\mu}~{},$
(64)
where $\theta_{W}$ is the Weinberg angle.
The nonmass eigenstates ${\bf W}_{\alpha}^{a}(a=1,2,3)$ in ${\bf T}\cdot{\bf
W}_{\alpha}$ must have the same rest mass because of two reasons: Each ${\bf
W}_{\alpha}^{a}$ plays the quite equal role in gauge field ${\bf T}\cdot{\bf
W}_{\alpha}$; The model must be $SU(2)$ gauge invariant. Consequently
$\displaystyle m_{{\bf W}^{1}_{\mu}}=m_{{\bf W}^{2}_{\mu}}=m_{{\bf
W}^{3}_{\mu}}~{}.$ (65)
In Ref.[5], it is indicated that the rest mass squared of nonmass eigenstate
of vector fields can be calculated, that is
$\displaystyle m_{{\bf V}_{\mu}}^{2}=\sum_{j=1}^{n}a_{j}a^{*}_{j}m_{j}^{2}$
(66)
is right if and only if
$\displaystyle{\bf V}_{\mu}=\sum_{j=1}^{n}a_{j}[{\bf
V}_{\mu}]_{j}=\sum_{j=1}^{n}a_{j}e^{im_{j}z}[V_{\mu}]_{j}~{}.$ (67)
Then one can easily obtain the rest masses of ${\bf W}^{1}_{\mu}$ and ${\bf
W}^{2}_{\mu}$ from (60) and (61) respectively
$\displaystyle m_{{\bf W}^{1}_{\mu}}^{2}=m_{{\bf
W}^{2}_{\mu}}^{2}=m_{W}^{2}~{},$ (68)
also get the rest mass of ${\bf W}^{3}_{\mu}$ from (63)
$\displaystyle m_{{\bf W}^{3}_{\mu}}^{2}=m_{Z}^{2}(\cos\theta_{W})^{2}~{}.$
(69)
Therefore combining (65), (68) and (69), I obtain the following relation
$\displaystyle m_{W}=m_{Z}\cos\theta_{W}~{}.$ (70)
In the total Lagrangian (44), the kinetic terms of Fermions and the
interaction terms will be discussed carefully in my forthcoming paper [6], in
this letter I only discuss the self-couplings of $SU(2)\times U(1)$ gauge
fields, namely the terms ${\cal L}_{U(1)}+{\cal L}_{SU(2)}$ in (44). It has
been pointed out that the gauge bosons in my model merely propagate along the
$z=0$ brane, therefore ${\bf W}_{z}(x,z)=0,~{}{\bf X}_{z}(x,z)=0$. In this
case, substituting (64) into (55) yields
$\displaystyle{{\bf E}}_{z\mu}=-im_{Z}\sin\theta_{W}e^{im_{Z}z}Z_{\mu}~{}.$
(71)
Substituting (60), (61) and (63) into (56) respectively, I obtain
$\displaystyle{{\bf
F}}_{z\mu}^{1}=\frac{i}{\sqrt{2}}m_{W}e^{im_{W}z}\left({W}^{+}_{\mu}+{W}^{-}_{\mu}\right)~{},$
(72) $\displaystyle{{\bf
F}}_{z\mu}^{2}=-\frac{1}{\sqrt{2}}m_{W}e^{im_{W}z}\left({W}^{+}_{\mu}-{W}^{-}_{\mu}\right)~{},$
(73) $\displaystyle{{\bf
F}}_{z\mu}^{3}=im_{Z}\cos\theta_{W}e^{im_{Z}z}Z_{\mu}~{}.$ (74)
Substituting (71), (72), (73) and (74) into ${\cal L}_{U(1)}+{\cal
L}_{SU(2)}$, I find that the self-coupling terms of $SU(2)\times U(1)$ gauge
fields become
$\displaystyle~{}~{}~{}~{}{\cal L}_{U(1)}+{\cal L}_{SU(2)}$
$\displaystyle=-\frac{1}{4}{{\bf E}}_{\alpha\beta}{\bf
E}^{\alpha\beta}-\frac{1}{4}{{\bf F}}_{\alpha\beta}\cdot{{\bf
F}}^{\alpha\beta}$ $\displaystyle=-\frac{1}{4}{{\bf E}}_{\mu\nu}{\bf
E}^{\mu\nu}-\frac{1}{4}{{\bf F}}_{\mu\nu}\cdot{{\bf
F}}^{\mu\nu}-\frac{1}{2}\left({{\bf E}}_{z\mu}{\bf E}^{z\mu}+{{\bf
F}}_{z\mu}\cdot{{\bf F}}^{z\mu}\right)$ $\displaystyle=-\frac{1}{4}{{\bf
E}}_{\mu\nu}{\bf E}^{\mu\nu}-\frac{1}{4}{{\bf F}}_{\mu\nu}\cdot{{\bf
F}}^{\mu\nu}-\frac{1}{2}\left({{\bf E}}_{z\mu}{\bf E}^{z\mu}+{\bf
F}^{1}_{z\mu}{{\bf F}}^{1z\mu}+{\bf F}^{2}_{z\mu}{{\bf F}}^{2z\mu}+{\bf
F}^{3}_{z\mu}{{\bf F}}^{3z\mu}\right)$ $\displaystyle=-\frac{1}{4}{{\bf
E}}_{\mu\nu}{\bf E}^{\mu\nu}-\frac{1}{4}{{\bf F}}_{\mu\nu}\cdot{{\bf
F}}^{\mu\nu}+\frac{1}{2}m_{Z}^{2}e^{2im_{Z}z}Z_{\mu}Z^{\mu}+m_{W}^{2}e^{2im_{W}z}W_{\mu}^{+}W^{\mu-}~{}.$
(75)
It is indicated that all the mass eigenstates coupled by the gravitation, the
electromagnetic field and the gluon fields are living in the $z=0$ brane.
Hence, expressed by the mass eigenstates in the $z=0$ brane, the self-coupling
terms of $SU(2)\times U(1)$ gauge fields reduce to
$\displaystyle~{}~{}~{}~{}{\cal L}_{U(1),z=0}+{\cal L}_{SU(2),z=0}$
$\displaystyle=-\frac{1}{4}{E}_{\mu\nu}{E}^{\mu\nu}-\frac{1}{4}{F}_{\mu\nu}\cdot{F}^{\mu\nu}+\frac{1}{2}m_{Z}^{2}Z_{\mu}Z^{\mu}+m_{W}^{2}W_{\mu}^{+}W^{\mu-}~{},$
(76)
where
${F}_{\mu\nu}\equiv\\{{F}^{1}_{\mu\nu},{F}^{2}_{\mu\nu},{F}^{3}_{\mu\nu}\\}$,
and ${E}_{\mu\nu}$ is formulated by
$\displaystyle{E}_{\mu\nu}(x)=\partial_{\mu}{X}_{\nu}(x)-\partial_{\nu}{X}_{\mu}(x)~{},$
(77)
and ${F}_{\mu\nu}$ is given by
$\displaystyle{F}_{\mu\nu}(x)=\partial_{\mu}{W}_{\nu}(x)-\partial_{\nu}{W}_{\mu}(x)+g_{1}{W}_{\mu}(x)\times{W}_{\nu}(x)~{},$
(78)
in which
${W}_{\mu}(x)\equiv\\{{W}^{1}_{\mu}(x),{W}^{2}_{\mu}(x),{W}^{3}_{\mu}(x)\\}$.
The four-dimensional fields ${X}_{\mu}(x)$ and ${W}_{\mu}(x)$ are composed of
mass eigenstates which are constrained in the $z=0$ brane. From (60), (61),
(63) and (64), one can easily obtain the expressions of them in the following
$\displaystyle{W}^{1}_{\mu}(x)=\frac{1}{\sqrt{2}}\left({W}^{+}_{\mu}(x)+{W}^{-}_{\mu}(x)\right)~{},$
(79)
$\displaystyle{W}^{2}_{\mu}(x)=\frac{i}{\sqrt{2}}\left({W}^{+}_{\mu}(x)-{W}^{-}_{\mu}(x)\right)~{},$
(80)
$\displaystyle{W}^{3}_{\mu}(x)=\sin\theta_{W}A_{\mu}(x)+\cos\theta_{W}Z_{\mu}(x)~{},$
(81)
$\displaystyle{X}_{\mu}(x)=\cos\theta_{W}A_{\mu}(x)-\sin\theta_{W}Z_{\mu}(x)~{}.$
(82)
Obviously they have the same forms as the definitions of gauge bosons of
$SU(2)\times U(1)$ gauge fields in the Glashow-Salam-Weinberg model [7]. The
fields ${W}^{+}_{\mu}(x)$, ${W}^{-}_{\mu}(x)$, $Z_{\mu}(x)$ and $A_{\mu}(x)$
in above expressions are mass eigenstates that are constrained in the $z=0$
brane.
The $SU(2)\times U(1)$ gauge fields of the second kind merely couple to the
nonmass eigenstates, which are the nonmass eigenstates as well, hence cannot
be observed directly. The nonmass eigenstates of gauge fields are composed of
the mass eigenstates that are constrained in the $z=0$ brane. When I reexpress
the Lagrangian ${\cal L}_{U(1)}+{\cal L}_{SU(2)}$ by the mass eigenstates of
gauge bosons who live in four-dimensional spacetime, I find that the fields
${W}^{+}_{\mu}(x)$, ${W}^{-}_{\mu}(x)$ and $Z_{\mu}(x)$ can be treated as
massive gauge bosons from the four-dimensional point of view since their mass
terms automatically appear in the four-dimensional Lagrangian (76).
The mass terms of gauge bosons who are living in four-dimensional spacetime
aren’t inserted by hands, which is produced automatically. From five-
dimensional point of view, the $SU(2)\times U(1)$ gauge fields of the second
kind are massless, which are the usual gauge fields that we are very familiar
with, since there are no mass term in the total Lagrangian (44). Therefore,
the gauge fields in this model should be renormalizable.
let me explicitly explain this model again: The general equation of a nonmass
eigenstate of spin-$\frac{1}{2}$ fields is built on a five-dimensional
manifold, therefore the gauge fields of the second kinds, who couple to the
nonmass eigenstates of spin-$\frac{1}{2}$ fields, are constructed on a five-
dimensional manifold as well. But the nonmass eigenstates are composed of mass
eigenstates, which are physically observable. The mass eigenstates are merely
coupled by the electromagnetic field, the gravitation and the gluon fields,
who are living in the $z=0$ brane. Hence the initial momentum of the nonmass
eigenstates of spin-$\frac{1}{2}$ fields are along the $z=0$ brane, which
decides that the gauge fields of the second kind must propagate along the
$z=0$ spacetime as well. When the gauge fields of the second kind are
reexpressed by their mass eigenstates that living in the four-dimensional
spacetime, it is found that the gauge bosons who are mass eigenstates can be
treated as massive vector fields.
## References
* [1] C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).
* [2] P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964); P. W. Higgs, Phys. Rev. 145, 1156 (1966); F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).
* [3] S. L. Glashow, Nucl. Phys. 22, 579 (1960); J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2, 1285 (1970).
* [4] For recent reviews on neutrino masses and mixing angles, see Z. Z. Xing, Int. J. Mod. Phys. A 19, 1 (2004); R. D. McKeown and P. Vogel, Phys. Rept. 394, 315 (2004); M. C. Gonzalez-Garcia and Y. Nir, Rev. Mod. Phys. 75, 345 (2003); M.-C. Chen and K. T. Mahanthappa, Int. J. Mod. Phys. A 18, 5819 (2003).
* [5] X.-B. Huang, “Nonmass Eigenstates of Boson and Fermion Fields”, [arXiv:hep-th/0906.2441].
* [6] X.-B. Huang, “An Electroweak Model without Higgs Mechanism”, in preparation.
* [7] W. Greiner and B. Müller, Gauge Theory of Weak Interactions, (3rd. edition), (Springer-Verlag, 2000).
|
arxiv-papers
| 2009-06-15T00:51:25 |
2024-09-04T02:49:03.357281
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin-Bing Huang",
"submitter": "Xin-Bing Huang",
"url": "https://arxiv.org/abs/0906.2584"
}
|
0906.2602
|
# Probing the Intermediate-Age Globular Clusters in NGC 5128 from Ultraviolet
Observations
Soo-Chang Rey11affiliation: Department of Astronomy and Space Science,
Chungnam National University, Daejeon 305-764, Korea , Sangmo T.
Sohn22affiliation: Center for Space Astrophysics, Yonsei University, Seoul
120-749, Korea 33affiliation: California Institute of Technology, MC 405-47,
1200 East California Boulevard, Pasadena, CA 91125 , Michael A.
Beasley44affiliation: Instituto de Astrofisica de Canarias, Via Lactea,
E-38200 La Laguna, Tenerife, Spain , Young-Wook Lee22affiliation: Center for
Space Astrophysics, Yonsei University, Seoul 120-749, Korea , R. Michael
Rich55affiliation: Department of Physics and Astronomy, University of
California, Los Angeles, CA 90095 , Suk-Jin Yoon22affiliation: Center for
Space Astrophysics, Yonsei University, Seoul 120-749, Korea , Sukyoung K.
Yi22affiliation: Center for Space Astrophysics, Yonsei University, Seoul
120-749, Korea , Luciana Bianch66affiliation: Department of Physics and
Astronomy, The Johns Hopkins University, Homewood Campus, Baltimore, MD 21218
, Yongbeom Kang11affiliation: Department of Astronomy and Space Science,
Chungnam National University, Daejeon 305-764, Korea , Kyeongsook
Lee11affiliation: Department of Astronomy and Space Science, Chungnam National
University, Daejeon 305-764, Korea , Chul Chung22affiliation: Center for Space
Astrophysics, Yonsei University, Seoul 120-749, Korea , Tom A.
Barlow33affiliation: California Institute of Technology, MC 405-47, 1200 East
California Boulevard, Pasadena, CA 91125 , Karl Foster33affiliation:
California Institute of Technology, MC 405-47, 1200 East California Boulevard,
Pasadena, CA 91125 , Peter G. Friedman33affiliation: California Institute of
Technology, MC 405-47, 1200 East California Boulevard, Pasadena, CA 91125 , D.
Christopher Martin33affiliation: California Institute of Technology, MC
405-47, 1200 East California Boulevard, Pasadena, CA 91125 , Patrick
Morrissey33affiliation: California Institute of Technology, MC 405-47, 1200
East California Boulevard, Pasadena, CA 91125 , Susan G. Neff77affiliation:
Laboratory for Astronomy and Solar Physics, NASA Goddard Space Flight Center,
Greenbelt, MD 20771 , David Schiminovich88affiliation: Department of
Astronomy, Columbia University, New York, NY 10027 , Mark
Seibert99affiliation: Observatories of the Carnegie Institution of Washington,
813 Santa Barbara St., Pasadena, CA 91101 , Ted K. Wyder33affiliation:
California Institute of Technology, MC 405-47, 1200 East California Boulevard,
Pasadena, CA 91125 , Jose Donas1010affiliation: Laboratoire d’Astrophysique de
Marseille, BP 8, Traverse du Siphon, 13376 Marseille Cedex 12, France ,
Timothy M. Heckman66affiliation: Department of Physics and Astronomy, The
Johns Hopkins University, Homewood Campus, Baltimore, MD 21218 , Barry F.
Madore99affiliation: Observatories of the Carnegie Institution of Washington,
813 Santa Barbara St., Pasadena, CA 91101 , Bruno Milliard1010affiliation:
Laboratoire d’Astrophysique de Marseille, BP 8, Traverse du Siphon, 13376
Marseille Cedex 12, France , Alex S. Szalay66affiliation: Department of
Physics and Astronomy, The Johns Hopkins University, Homewood Campus,
Baltimore, MD 21218 , Barry Y. Welsh1111affiliation: Space Sciences
Laboratory, University of California at Berkeley, 601 Campbell Hall, Berkeley,
CA 94720
###### Abstract
We explore the age distribution of the globular cluster (GC) system of the
nearby elliptical galaxy NGC 5128 using ultraviolet (UV) photometry from
Galaxy Evolution Explorer (GALEX) observations, with UV$-$optical colors used
as the age indicator. Most GCs in NGC 5128 follow the general trends of GCs in
M31 and Milky Way in UV$-$optical color-color diagram, which indicates that
the majority of GCs in NGC 5128 are old similar to the age range of old GCs in
M31 and Milky Way. A large fraction of spectroscopically identified
intermediate-age GC (IAGC) candidates with $\sim$ 3$-$8 Gyr are not detected
in the FUV passband. Considering the nature of intermediate-age populations
being faint in the far-UV (FUV) passband, we suggest that many of the
spectroscopically identified IAGCs may be truly intermediate in age. This is
in contrast to the case of M31 where a large fraction of spectroscopically
suggested IAGCs are detected in FUV and therefore may not be genuine IAGCs but
rather older GCs with developed blue horizontal branch stars. Our UV
photometry strengthens the results previously suggesting the presence of GC
and stellar subpopulation with intermediate age in NGC 5128. The existence of
IAGCs strongly indicates the occurrence of at least one more major star
formation episode after a starburst at high redshift.
###### Subject headings:
galaxies: individual (NGC 5128) — galaxies: star clusters — globular clusters:
general — ultraviolet: galaxies
## 1\. Introduction
Globular cluster (GC) systems provide the signatures of formation and assembly
histories of their host galaxies assuming that major star formations in
galaxies are accompanied with global GC formation. Several scenarios have been
proposed to account for the observational properties obtained for the GC
systems (see a comprehensive review of Brodie & Strader 2006). Many aspects of
those scenarios are in favor of the currently accepted hierarchical galaxy
formation theory (Press & Schechter 1974) rather than the monolithic formation
at high redshift (Eggen et al. 1962; Larson 1974). In this galaxy formation
paradigm, constituent of galaxy mass including GCs is predicted to form
through quiescent as well as merger/interaction-driven star formation (Kaviraj
et al. 2007b).
One of the best templates in the local universe for testing this scenario is
the elliptical galaxy NGC 5128 due to its proximity. There have been several
pieces of evidence supporting the picture that the NGC 5128 is the prototype
for a postmerger elliptical galaxy (see Israel 1998 and references therein).
Previous photometric and spectroscopic observations of GCs also suggest that
merging and/or interaction events have played an important role in shaping its
star cluster system (Peng, Ford, & Freeman 2004a, b; Woodley et al. 2007;
Beasley et al. 2008).
Constraining the formation scenario of the NGC 5128 GC system requires the
understanding of its global age distribution. Clusters younger than the bulk
of ancient Galactic counterparts are of particular interest because these
objects represent the later stages of star formation histories in galaxies.
Recent spectroscopic observations suggest that NGC 5128 hosts a cluster
population significantly younger than the old GCs in the Milky Way and M31
(Peng et al. 2004b). Based on the spectroscopic observations for an increased
sample of GCs, Beasley et al. (2008) reported the discovery of metal-rich,
intermediate-age GCs (IAGCs) with ages of $\sim 3-8$ Gyr in NGC 5128. They
propose that this population may be the byproduct formed during merging events
and/or interactions involving star formation and GC formation several
gigayears ago.
However, it is important to note that age-dating of GCs via integrated spectra
is hampered by the degeneracy between age and the existence of hot old stellar
population (e.g., blue horizontal branch [HB] stars) affecting the strength of
age-sensitive line indices (Lee, Yoon, & Lee 2000; Maraston et al. 2003;
Thomas, Maraston, & Bender 2003; Schiavon et al. 2004; Lee & Worthey 2005;
Trager et al. 2005; Cenarro et al. 2007). The effect of old blue HB stars in
the integrated spectra can mimic young ages for old GCs, raising a cause of
concern that may cast doubt on the intermediate age nature of the GC in some
galaxies.
The UV colors (e.g. FUV$-V$ and FUV$-$NUV), on the other hand, are known to
provide robust age estimation of simple stellar populations (e.g., Yi 2003;
Rey et al. 2005, 2007; Kaviraj et al. 2007a; Bianchi et al. 2007). Kaviraj et
al. (2007a) found that the age constraint is far superior when UV photometry
is added to the optical colors and its quality is comparable or marginally
better than the case of utilizing spectroscopic indices.
With the new approach using UV observations, in this letter, we take advantage
of the combination of available optical photometry and the GALEX (Galaxy
Evolution Explorer) UV photometry to confirm the existence of IAGCs and to
explore the age distribution of the NGC 5128 GC system. In the following
sections, we emphasize the importance of the UV photometry as a probe of IAGCs
in general. Comparing with GCs in M31 and the Milky Way with the aid of our
population models, we describe the overall age distribution of GCs and
identification of IAGCs in NGC 5128. In this paper, we denote IAGCs as those
having ages $\sim$ 3 $-$ 8 Gyrs.
## 2\. Observations and Data Analysis
GALEX (Martin et al. 2005) imaged one 1.25 deg circular field centered on 26
arcmin East and 7 arcmin North of the NGC 5128 core in two UV bands: FUV (1350
– 1750Å) and NUV (1750 – 2750Å). The images were obtained on April 2004, and
are included in the GALEX fourth and fifth data release
(GR4/GR5)111http://galex.stsci.edu/gr4. Total integration times were 30,428
sec and 20,072 sec for NUV and FUV, respectively. Preproccessing and
calibrations were performed via the GALEX pipeline (Morrissey et al. 2005,
2007). GALEX image has a sampling of 1.5 arcsec pixel-1 which corresponds to
19 pc at the distance of NGC 5128 (3.9 Mpc, Woodley et al. 2007)
Using the DAOPHOTII/ALLSTAR package (Stetson 1987), we performed aperture
photometry for all detected point sources in the GALEX NGC 5128 field.
Aperture corrections were derived using moderately bright, isolated objects.
Flux calibrations were applied to bring all measurements into the AB magnitude
system (Oke 1990; Morrissey et al. 2005, 2007).
Point sources in our GALEX photometry were cross-matched using a matching
radius of 3 arcsec with the catalog of Woodley et al. (2007). This catalog
provides positions as well as optical magnitudes and mean radial velocities
for 415 GCs in NGC 5128. All spurious and ambiguous sources were rejected
based on visual inspection. The final sample of visually confirmed GCs are 157
and 35 in NUV and FUV, respectively. We adopted a foreground reddening value
of $E(B-V)$ = 0.11 for NGC 5128 (Woodley et al. 2007) and use the reddening
law of Cardelli, Clayton, & Mathis (1989). The full UV catalog and discussion
of the UV properties of GCs in NGC 5128 will be presented in a forthcoming
paper. Figure 1 shows the optical color-magnitude diagram (CMD) of GCs in NGC
5128 detected in the NUV and FUV bandpasses. For comparison, we overplot GCs
in M31 detected from GALEX observations (Rey et al. 2007). The CMD shows that
most of the UV-detected objects in NGC 5128 and M31 have similar distribution
and are confined to $V-I<1.05$.
## 3\. Ultraviolet as a Probe of Intermediate-Age Globular Clusters
FUV flux plays an important role in identifying IAGCs. Young ($<1$ Gyr)
stellar populations emit a substantial portion of their flux in the UV. Metal-
poor old ($>10$ Gyr) stellar populations also show large FUV to optical flux
ratio due to the contribution of hot HB stars. On the contrary, intermediate-
age ($\sim$ 3$-$8 Gyr) populations emit negligible amount of FUV flux since
the constituent stars are not hot enough to produce a significantly large FUV
flux (see Fig. 1 of Kaviraj et al. 2007a). Consequently, if the IAGC
candidates identified by spectroscopic observations are truly intermediate in
age, they should be very faint or not detected in our GALEX FUV photometry
given our integration time and the detection limit (Lee & Worthey 2005; Rey et
al. 2007; Kaviraj et al. 2007a).
The first use of UV color as a tool for identifying IAGCs was demonstrated in
our M31 study (see Rey et al. 2007). Spectroscopic observations of M31
clusters have suggested the existence of IAGCs with mean age $\sim$ 5 Gyr
(Burstein et al. 2004; Beasley et al. 2005; Puzia et al. 2005). However, based
on GALEX FUV detections of more than half of M31 IAGC candidates, Rey et al.
(2007) suggested that a large fraction of the spectroscopically identified
IAGCs may not be truly intermediate in age but are rather old GCs with a
developed blue HB sequence. Among the 42 GCs in M31 whose ages are estimated
by Kaviraj et al. (2007a), we find that four IAGC candidates turn out to be
old GCs with $>12$ Gyr. By comparing of mass-to-light ratios of three IAGC
candidates in M31 with those of old GCs, Strader et al. (2009) also found no
evidence that M31 IAGC candidates are of intermediate in age.
The most direct way to identify genuine IAGCs is to inspect CMDs of the
clusters of interest. In the case of M31, $HST$ CMDs of two IAGC candidates
B311 and B058 exhibit clearly developed blue HB sequences (Rich et al. 2005).
In a separate study, Chandar et al. (2006) showed that a star cluster in M33,
C38, is a genuine IAGC with age $\sim 2$–5 Gyr based on the HST CMD and Balmer
line measurements. It is important to note that this cluster is also confirmed
to be a genuine IAGC using the GALEX FUV observations of M33 (S. T. Sohn et
al. 2009, in prep). In any case, UV$-$optical color can be used to
discriminate genuine IAGCs from the old GCs masquerading as IAGCs.
## 4\. Age Distribution of Globular Clusters in NGC 5128
### 4.1. Old Globular Clusters
Figure 2 shows the $V-I$ versus UV$-V$ diagrams. We compare our NGC 5128
sample with those of the Milky Way (crosses, Sohn et al. 2006) and M31 (open
circles, Rey et al. 2007) GCs whose age distributions are reasonably well
constrained. We also show our simple stellar population (SSP) models
constructed using the Yonsei Evolutionary Population Synthesis (YEPS) code
(Lee, Yoon, & Lee 2000; Lee et al. 2005; Rey et al. 2005, 2007; Yoon et al.
2006, 2008).
In Fig. 2, NGC 5128 GCs appear to show tight distribution around 12 Gyr model
line similar to that of Milky Way, while GCs in M31 are rather scattered in
$V-I$. This is partly due to the detection limit of optically red GCs in NGC
5128 (see Fig. 1) and insufficient sample of Milky Way GCs obtained from
previous UV observations of various satellites (see Sohn et al. 2006).
Furthermore, Rey et al. (2007) reported the existence of UV-bright metal-rich
GCs with extreme hot blue HB stars in M31 (e.g., NGC 6388 and NGC 6441 in the
Milky Way, Rich et al. 1997). In this regard, some of the red ($V-I>1.0$) M31
GCs that show UV excess with respect to the 14 Gyr model line may be such
peculiar objects. Considering these points, at a fixed $V-I$, the majority of
GCs in three galaxies show similar spread in the UV$-V$ colors and are well
accounted for by the 10–14 Gyr model lines. This suggests that the mean age
and age spread, at least, for old ($\geq 10$ Gyr) GCs are similar among GC
systems of different galaxies, Milky Way, M31, and NGC 5128.
### 4.2. Intermediate-Age Globular Clusters
Beasley et al. (2008) found a population of intermediate-age and predominantly
metal-rich ([Z/H] $>-1.0$) GCs (15 % of the sample) from their spectroscopic
observations. Among the 21 IAGC candidates (age $\sim 3-8$ Gyr) identified by
Beasley et al. (2008), we detect only two in the GALEX FUV passband.
In Figure 3, we show the $V-I$ vs. $FUV-V$ diagram for the spectroscopically
identified IAGC candidates in NGC 5128 (filled squares) and M31 (filled
circles) detected in GALEX FUV passband. Population model lines covering range
of intermediate (3 and 8 Gyr) and old (10, 12, and 14 Gyr) ages are
overplotted for guidance. It is immediately apparent that all of the IAGC
candidates of NGC 5128 and M31 detected in the FUV show similar distribution
to those of old GCs with $>10$ Gyr, i.e., all FUV-detected IAGC candidates
have significantly bluer $FUV-V$ colors than the 3 and 8 Gyr model lines. This
indicates that IAGC candidates detected in the FUV are in fact old GCs ($\geq
10$ Gyr) containing developed blue HB populations that contribute to the
strong Balmer absorption lines.
It is important to note that, as shown in Fig. 3, most M31 IAGC candidates
with $E(B-V)<0.16$ are detected in the GALEX FUV (6 out of 7, see Rey et al.
2007 for the details). If we restrict the sample of M31 IAGC candidates to
match the observed optical brightness and color range ($M_{V}<-8$ and
$V-I<1.05$, see Fig. 1) of the FUV-detected sample of NGC 5128 GCs, 4 out of 5
M31 IAGC candidates are detected in FUV. In the case of NGC 5128 GCs, only two
out of 9 IAGC candidates are detected in the FUV. Since all of the NGC 5128
GCs detected in the FUV cover similar range of $(FUV-V)_{o}$ colors of FUV-
detected IAGC candidates in M31, most, if not all, spectroscopically
identified IAGC candidates in NGC 5128 are not likely to be as bright as those
in M31.
Among the 21 IAGC candidates identified by Beasley et al. (2008), 12 GCs are
detected in the GALEX NUV but not in the FUV. Whereas the FUV flux of old
($>8$ Gyr) GC is almost entirely dominated by stars in the hot HB sequence,
the NUV flux is influenced by both the HB stars and those on the main-sequence
turnoff. In this regard, we cannot rule out that some of the NUV-detected IAGC
candidates are truly intermediate in age, despite the fact that NUV$-V$ is
relatively insensitive to age variations compared to the FUV$-V$ (see Fig. 2).
To test this hypothesis, in Fig. 3, we show the bluer limits of the NUV-
detected IAGC candidates having similar $V$ magnitudes of FUV-detected IAGCs.
Most of the color limits are consistent with the NUV-detected IAGC candidates
being $\sim 3-8$ Gyr in age. In summary, our UV photometry suggests that NGC
5128 does possess a non-negligible fraction of IAGCs that are intrinsically
faint in the FUV as proposed by previous spectroscopic studies.
## 5\. Discussion and Conclusions
In this work, we explored the age distribution of GCs in the giant elliptical
galaxy NGC 5128 using the UV colors. The majority of NGC 5128 GCs show age
ranges similar to old GCs in M31 and the Galactic halo. Our most important
result is that a large fraction of IAGCs identified by the spectroscopic
observations are not detected in the GALEX FUV passband and therefore may be
truly intermediate in age. This is in contrast to the case of M31 GCs where
the majority of IAGC candidates turned out to be old GCs with developed HB
sequence based on their FUV$-V$ colors (see Rey et al. 2007).
The existence of IAGCs in NGC 5128 supports the galaxy formation scenario
accompanied with at least two major star formation episodes; e.g.,
hierarchical assembly of the protogalactic fragments or disks (Bekki et al.
2003; Beasley et al. 2002, 2003; Yi et al. 2004; Kaviraj et al. 2005). In
these models, some of the metal-rich GCs are formed from pre-enriched gas
clouds and are on average younger than the metal-poor GCs. Based on the
kinematic analysis in combination with the age distribution of GCs, an
alternative mechanism may have taken place where the NGC 5128 formed its main
body at early times and has gradually built up by minor mergers and gas-rich
satellite accretions accompanied by star formation episodes (Woodley 2006;
Woodley et al. 2007).
The presence of IAGCs in NGC 5128 has an interesting implication for the
recent star formation (RSF) recently discovered using the large GALEX UV
sample of early-type galaxies at different redshifts ($0<z<1$; e.g., Yi et al.
2005; Kaviraj et al. 2007b, 2008; Schawinski et al. 2007). Kaviraj et al.
(2008) found that high-redshift early-type galaxies in the range of $0.5<z<1$
exhibit typical RSFs in addition to the case of low-redshift ($0<z<0.1$)
early-type galaxies. This provides a compelling evidence that RSFs in early-
type galaxies are non-negligible over the last 8 billion years. Furthermore,
Kaviraj et al. (2008) suggest that up to 10$-$15% of the mass of luminous
($-23<M_{V}<-20.5$) early-type galaxies such as NGC 5128 ($M_{V}=-21.08$, Gil
de Paz et al. 2007) may have formed after $z=1$. These results imply that
early-type galaxies in the local Universe are likely to possess intermediate-
age stellar populations. In this respect, IAGCs in NGC 5128 may be considered
as relics of residual star formations that occurred during the last few
billion years.
UV observations of the GC systems have been shown to provide important
insights into the identification of IAGCs which is at present difficult to be
identified solely by spectroscopic observations. In particular, the Balmer
line strengths themselves cannot reliably pin down the age of GCs because of
the degeneracy between age and HB morphology. FUV colors, on the other hand,
can verify the contribution from hot stellar populations in GCs and help
identify the true IAGCs. Deep UV observations are highly anticipated for other
galaxies with IAGC candidates identified by various spectroscopic and near-
infrared photometric observations.
We thank Sugata Kaviraj for useful suggestions on the manuscript. This work
was supported by the Korea Research Foundation Grant funded by the Korean
Government (MOEHRD) (KRF-2005-202-C00158) and the Korea Science and
Engineering Foundation (KOSEF) through the Astrophysical Research Center for
the Structure and Evolution of the Cosmos (ARCSEC). GALEX (Galaxy Evolution
Explorer) is a NASA Small Explorer, launched in April 2003. We gratefully
acknowledge NASA’s support for construction, operation, and science analysis
for the GALEX mission, developed in cooperation with the Centre National
d’Etudes Spatiales of France and the Korean Ministry of Science and
Technology.
## References
* (1) Barmby, P., Huchra, J. P. Brodie, J. P., Forbes, D. A., Schroder, L. L. & Grillmair, C. J. 2000 AJ, 119, 727
* (2) Beasley, M. A., Baugh, C. M., Forbes, D. A., Sharples, R. M., & Frenk, C. S. 2002, MNRAS, 333, 383
* (3) Beasley, M. A., Harris, W. E., Harris, G. L. H., & Forbes, D. A. 2003, MNRAS, 340, 341
* (4) Beasley, M. A., Brodie, J. P., Strader, J., et al. 2005, AJ, 129, 1412
* (5) Beasley, M. A., Bridges, T., Peng, E., Harris, W. E., Harris, G. L. H., Forbes, D. A., & Mackie, G. 2008, MNRAS, 386, 1443
* (6) Bekki, K., Harris, W. E., & Harris, G. L. H. 2003, MNRAS, 338, 587
* (7) Bianchi, L., et al. 2007, ApJS, 173, 659
* (8) Brodie, J. P., & Strader, J. 2006, ARA&A, 44, 193
* (9) Burstein, D., Li, Y., Freeman, K. C., et al. 2004, ApJ, 614, 158
* (10) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
* (11) Cenarro, A. J., Beasley, M. A., Strader, J., Brodie, J. P., & Forbes, D. A. 2007, AJ, 134, 391
* (12) Chandar, R., Puzia, T. H., Sarajedini, A., & Goudfrooij, P. 2006, ApJ, 646, L107
* (13) Eggen, O. J., Lynden-Bell, D., & Sandage, A. R. 1962, ApJ, 136, 748
* (14) Gil de Paz, A., et al. 2007, ApJS, 173, 185
* (15) Israel, F. P. 1998, A&A Rev., 8, 237
* (16) Kaviraj, S., Ferreras, I. Yoon, S.-J., & Yi, S. K. 2005, A&A, 439, 913
* (17) Kaviraj, S., Rey, S.,-C., Rich, R. M., Yoon, S.-J., & Yi, S. K. 2007a, MNRAS, 381, 74
* (18) Kaviraj, S., et al. 2007b, ApJS, 173, 619
* (19) Kaviraj, S., et al. 2008, MNRAS, 388, 67
* (20) Larson, R. B. 1974, MNRAS, 166, 585
* (21) Lee, H.-c., Yoon, S.-J., & Lee, Y.-W. 2000, AJ, 120, 998
* (22) Lee, H.-c., & Worthey, G. 2005, ApJS, 160, 176
* (23) Lee, Y.-W. et al. 2005, ApJ, 621, 57L
* (24) Maraston, C., Greggio, L., Renzini, A., Ortolani, S., Saglia, R. P., Puzia, T. H., & Kissler-Patig, M. 2003, A&A, 400, 823
* (25) Martin, D. C., et al. 2005, ApJ, 619, L1
* (26) Morrissey, P., et al. 2005, ApJ, 619, L7
* (27) Morrissey, P., et al. 2007, ApJS, 173, 682
* (28) Oke, J. B. 1990, AJ, 99, 1621
* (29) Peng, E. W., Ford, H. C., & Freeman, K. C. 2004a, ApJS, 150, 367
* (30) Peng, E. W., Ford, H. C., & Freeman, K. C. 2004b, ApJ, 602, 705
* (31) Press, W. H., & Schechter, P. 1974, ApJ, 187, 425
* (32) Puzia, T. H., Perrett, K. M., & Bridges, T. J. 2005, A&A, 434, 909
* (33) Rey, S.-C. et al. 2005, ApJ, 619, L119
* (34) Rey, S.-C. et al. 2007, ApJS, 173, 643
* (35) Rich, R. M., et al. 1997, ApJ, 484, L25
* (36) Rich, R. M., et al. 2005, AJ, 129, 2670
* (37) Schawinski, K., et al. 2007, ApJS, 173, 512
* (38) Schiavon, R. P., Rose, J. A., Courteau, S., & MacArthur, L. A. 2004, ApJ, 608, 33
* (39) Sohn, S. T., et al. 2006, AJ, 131, 866
* (40) Stetson, P. B. 1987, PASP, 99, 191
* (41) Strader, J., Smith, G. H., Larsen, S., Brodie, J. P., & Huchra, J. P. 2009, AJ, in press (arXiV:0906.0397)
* (42) Thomas, D., Maraston, C., & Bender, R. 2003, MNRAS, 339, 897
* (43) Trager, S. C., Worthey, G., Faber, S. M., & Dressler, A. 2005, MNRAS, 362, 2
* (44) Woodley, K. A. 2006, AJ, 132, 2424
* (45) Woodley, K. A., Harris, W. E., Beasley, M. A., Peng, E. W., Bridges, T. I., Forbes, D. A., & Harris, G. L. H. 2007, AJ, 134, 494
* (46) Yi, S. K. 2003, ApJ, 582, 202
* (47) Yi, S. K., Peng, E., Ford, H., Kaviraj, S., & Yoon, S.-J. 2004, MNRAS, 349, 1493
* (48) Yi, S. K., et al. 2005, ApJ, 619, L111
* (49) Yoon, S.-J., Yi, S. K., & Lee, Y.-W. 2006, Science, 311, 1129
* (50) Yoon, S.-J., Joo, S.-J., Ree, C. H., Han, S.-I., Kim, D.-G, & Lee, Y.-W. 2008, ApJ, 677, 1080
Figure 1.— $M_{V}$ vs. $(V-I)_{o}$ color-magnitude diagram of GALEX UV-
detected GCs in NGC 5128 (squares) and M31 (circles, Rey et al. 2007). Open
and filled symbols are objects detected in NUV and FUV, respectively. We note
that all FUV-detected GCs in NGC 5128 are detected in NUV. The small dots
indicate GCs in NGC 5128 that are not detected in GALEX UV observations.
Figure 2.— $(V-I)_{o}$ vs. $(UV-V)_{o}$ diagrams of NGC 5128 (filled squares),
Milky Way (crosses), and M31 (open circles) GCs. Large and small squares
indicate NGC 5128 GCs with small and large magnitude errors in the UV passband
(0.2 mag for NUV and 0.3 mag for FUV as the border line), respectively. Large
circles are M31 GCs with E$(B-V)<0.16$ from Barmby et al. (2000). Small
circles are M31 GCs with no available reddening information in Barmby et al.,
assuming that they are only affected by the Galactic foreground reddening of
E$(B-V)$=0.10. We superpose our simple stellar population model lines with old
(10, 12, and 14 Gyr; solid lines from bottom to top) and young (long dashed
line for 1 Gyr) ages. The dotted lines represent iso-metallicity lines varying
from [Fe/H] = -2.0 to +0.5 dex (from bottom to top). There is no significant
difference of distribution between red [$(V-I)_{o}$ $>$ 0.8] and old GCs in
the three galaxies. Figure 3.— $(V-I)_{o}$ vs. $(FUV-V)_{o}$ color-color
diagram for the spectroscopically classified IAGC candidates in NGC 5128
(filled squares) and M31 (filled circles) detected in the GALEX FUV passband.
The model lines for intermediate ages (solid line for 3 Gyr and long dashed
line for 8 Gyr) are overplotted in addition to the old (10, 12, and 14 Gyr;
dotted lines from bottom to top) ones. All IAGC candidates of NGC 5128 and M31
detected in the FUV show similar distribution to that of old GCs (open circles
and squares) with $>$ 10 Gyr. The color limit for each IAGC candidate of NGC
5128 not detected in the FUV is plotted with a vertical bar and a horizontal
arrow pointing to the redder color. Color limits were determined by adopting
the flux of the faintest FUV-detected GC in NGC 5128.
|
arxiv-papers
| 2009-06-15T04:26:41 |
2024-09-04T02:49:03.363142
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Soo-Chang Rey, Sangmo T. Sohn, Michael A. Beasley, Young-Wook Lee, R.\n Michael Rich, Suk-Jin Yoon, Sukyoung K. Yi, Luciana Bianch, Yongbeom Kang,\n Kyeongsook Lee, Chul Chung, Tom A. Barlow, Karl Foster, Peter G. Friedman, D.\n Christopher Martin, Patrick Morrissey, Susan G. Neff, David Schiminovich,\n Mark Seibert, Ted K. Wyder, Jose Donas, Timothy M. Heckman, Barry F. Madore,\n Bruno Milliard, Alex S. Szalay, Barry Y. Welsh",
"submitter": "Yongbeom Kang",
"url": "https://arxiv.org/abs/0906.2602"
}
|
0906.2722
|
This paper has been withdrawn by the author.
|
arxiv-papers
| 2009-06-15T15:31:25 |
2024-09-04T02:49:03.369068
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuqing Zhang",
"submitter": "Yuqing Zhang",
"url": "https://arxiv.org/abs/0906.2722"
}
|
0906.2853
|
File: mtqm100323.tex, printed: 2024-8-27, 18.27
# On Mori’s theorem for
quasiconformal maps in the $n$-space
B.A. Bhayo Department of Mathematics, University of Turku, FI-20014 Turku,
Finland [email protected] and M. Vuorinen Department of Mathematics, University
of Turku, FI-20014 Turku, Finland [email protected]
###### Abstract.
R. Fehlmann and M. Vuorinen proved in 1988 that Mori’s constant $M(n,K)$ for
$K$-quasiconformal maps of the unit ball in $\mathbf{R}^{n}$ onto itself
keeping the origin fixed satisfies $M(n,K)\to 1$ when $K\to 1\,.$ We give here
an alternative proof of this fact, with a quantitative upper bound for the
constant in terms of elementary functions. Our proof is based on a refinement
of a method due to G.D. Anderson and M. K. Vamanamurthy. We also give an
explicit version of the Schwarz lemma for quasiconformal self-maps of the unit
disk. Some experimental results are provided to compare the various bounds for
the Mori constant when $n=2\,.$
###### Key words and phrases:
Quasiconformal mappings, Hölder continuity
###### 2000 Mathematics Subject Classification:
Primary 30C65
In memoriam: M.K. Vamanamurthy, 5 September 1934– 6 April 2009
## 1\. Introduction
Distortion theory of quasiconformal and quasiregular mappings in the Euclidean
$n$-space $\mathbf{R}^{n}$ deals with estimates for the modulus of continuity
and change of distances under these mappings. Some of the examples are the
Hölder continuity, the quasiconformal counterpart of the Schwarz lemma, and
Mori’s theorem. The investigation of these topics started in the early 1950’s
for the case $n=2$ and ten years later for the case $n\geq 3\,.$ Many authors
have contributed to the distortion theory, for some historical remarks see
[Vu1, 11.50].
As in [FV] we define Mori’s constant $M(n,K)$ in the following way. Let
$QC_{K},\,K\geq 1,$ stand for the family of all $K$-quasiconformal maps of the
unit ball $\mathbf{B}^{n}$ onto itself keeping the origin pointwise fixed.
Note that it is a well-known basic fact that an element in the set $QC_{K}$
can be extended by reflection to a $K$-quasiconformal map of the whole space
$\overline{\mathbf{R}}^{n}={\mathbf{R}}^{n}\cup\\{\infty\\}$ onto itself
keeping the point $\infty$ fixed. Then for all $K\geq 1,\,n\geq 2\,,$ there
exists a least constant $M(n,K)\geq 1$ such that
(1.1) $|f(x)-f(y)|\leq M(n,K)|x-y|^{\alpha},\quad\alpha=K^{1/(1-n)}\,,$
for all $f\in QC_{K},x,y\in\mathbf{B}^{n}\,.$
L. V. Ahlfors [A1] proved in 1954 that $M(2,K)\leq 12^{K^{2}}$ and this
property was refined by A. Mori [Mo] in 1956 to the effect that $M(2,K)\leq
16$ and $16$ cannot be replaced by a smaller constant independent of $K\,.$
This result can also be found in [A2], [FM], and [LV]. On the other hand the
trivial observation that $16$ fails to be a sharp constant for $K=1$ led to
the following conjecture, which is still open in 2009.
###### 1.2 The Mori Conjecture.
$M(2,K)=16^{1-1/K}.$
O. Lehto and K.I. Virtanen demonstrated in 1973 [LV, pp. 68] that $M(2,K)\geq
16^{1-1/K}$ (this lower bound was not given in the 1965 German edition of the
book). It is natural to expect that for a fixed $n\geq 2,$ $M(n,K)\to 1$ when
$K\to 1$ and this convergence result with an explicit upper bound for $M(n,K)$
was proved by R. Fehlmann and M. Vuorinen [FV]. A counterpart of this result
for the chordal metric was proved recently by P. Hästö in [H].
###### 1.3 Theorem.
[FV, Theorem 1.3] Let $f$ be a $K$-quasiconformal mapping of $\mathbf{B}^{n}$
onto $\mathbf{B}^{n}$, $n\geq 2$, $f(0)=0$. Then
(1.4) $|f(x)-f(y)|\leq M(n,K)|x-y|^{\alpha}$
for all $x,y\in\mathbf{B}^{n}$ where $\alpha=K^{1/(1-n)}$ and the constant
$M(n,K)$ has the following three properties:
1. (1)
$M(n,K)\to 1$ as $K\to 1$, uniformly in $n$ ,
2. (2)
$M(n,K)$ remains bounded for fixed $K$ and varying $n$ ,
3. (3)
$M(n,K)$ remains bounded for fixed $n$ and varying $K$ .
For $n=2\,,$ the first majorants with the convergence property in 1.3(1) were
proved only in the mid 1980s and for $n\geq 3$ in [FV]. In [FV] a survey of
the various known bounds for $M(n,K)$ when $n\geq 2$ can be found – that
survey reflects what was known at the time of publication of [FV]. Some
earlier results on Hölder continuity had been proved in [G], [MRV], [R], [S].
Step by step the bound for Mori’s constant was reduced during the past twenty
years. As far as we know, the best upper bound known today for $n=2$ is
$M(2,K)\leq 46^{1-1/K}$ due to S.-L. Qiu [Q] (1997). Refining the parallel
work [FV], G.D. Anderson and M. K. Vamanamurthy proved the following theorem
in [AV].
###### 1.5 Theorem.
For $n\geq 2,K\geq 1$,
$M(n,K)\leq 4\lambda_{n}^{2(1-\alpha)}\,,$
where $\alpha=K^{1/(1-n)}\,$ and $\lambda_{n}\in[4,2e^{n-1})\,,\lambda_{2}=4,$
is the Grötzsch ring constant [AN], [Vu1, p.89].
The first main result of this paper is Theorem 1.6 which improves on Theorem
1.5.
###### 1.6 Theorem.
(1) For $n\geq 2,K\geq 1$, $M(n,K)\leq T(n,K)$
(1.7) $T(n,K)\leq\inf\\{h(t):t\geq 1\\}\,,\quad
h(t)=(3+\lambda_{n}^{\beta-1}t^{\beta})t^{-\alpha}\lambda_{n}^{2(1-\alpha)},\;t\geq
1\,,$
where $\alpha=K^{1/(1-n)}=1/\beta,$ and $\lambda_{n}$ is as in Theorem 1.5.
(2) There exists a number $K_{1}>1$ such that for all $K\in(1,K_{1})$ the
function $h$ has a minimum at a point $t_{1}$ with $t_{1}>1$ and
(1.8) $T(n,K)\leq
h(t_{1})=\left[\frac{3^{1-\alpha^{2}}(\beta-\alpha)^{\alpha^{2}}}{\alpha^{\alpha^{2}}}\lambda_{n}^{\alpha-\alpha^{2}}+\lambda_{n}^{\beta-1}\left(\frac{(3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}}{(\beta-\alpha)^{\alpha}}\right)^{\beta-\alpha}\right]\lambda_{n}^{2(1-\alpha)}\,.$
Moreover, for $\beta\in(1,2)$ we have
(1.9) $h(t_{1})\leq
3^{1-\alpha^{2}}2^{5(1-\alpha)}K^{5}\left(\frac{3}{2}\sqrt[4]{\beta-\alpha}+\exp(\sqrt{\beta^{2}-1})\right).$
In particular, $h(t_{1})\to 1$ when $K\to 1\,.$
The last statement shows that Theorem 1.6 is better than the result of
Anderson and Vamanamurthy, Theorem 1.5, at least for values of $K$ close to
the critical value $1$, because the constant of Theorem 1.5 satisfies
$4\lambda_{n}^{2(1-\alpha)}\geq 4.$
The main method of our proof is to replace the argument of Anderson and
Vamanamurthy by a more refined inequality from [Vu2] and to introduce an
additional parameter ($t$ in the above theorem) which will be chosen in an
optimal way. The fact that this refined inequality is essentially sharp for
values of $t$ large enough, was recently proved by V. Heikkala and M. Vuorinen
in [HV]. This gave us a hint that the inequality from [Vu2] might lead to an
improvement of the results in [AV]. For the case $n=2$ a numerical comparison
of our bound (1.8) to Mori’s conjectured bound, to the bound in Theorem 1.5
and to the bound in [FV] is presented in tabular and graphical form at the end
of the paper.
We conclude this paper by discussing the Schwarz lemma for plane
quasiconformal self-mappings of the unit disk, formulated in terms of the
hyperbolic metric. The long history of this result is summarized in [Vu1,
p.152, 11.50]. An up-to-date form of the Schwarz lemma was given in [Vu1,
Theorem 11.2] and it will be stated for convenient reference also below as
Theorem 4.4. A particular case, formula (4.6), was rediscovered by D.B.A.
Epstein, A. Marden and V. Markovic [EMM, Thm 5.1].
We use the notations ch, th, arch and arth as in [Vu1], to denote the
hyperbolic cosine, tangent and their inverse functions, resp. The second main
result of this paper is an explicit form of the Schwarz lemma for quasiregular
mappings, Theorem 1.10. We believe that in this simple form the result is new
and perhaps of independent interest. The constant $c(K)$ below involves the
transcendental function $\varphi_{K}$ defined in Section 4.
###### 1.10 Theorem.
If $f:\mathbf{B}^{2}\to\mathbf{R}^{2}$ is a non-constant $K$-quasiregular
mapping with $f\mathbf{B}^{2}\subset\mathbf{B}^{2}$, and $\rho$ is the
hyperbolic metric of $\mathbf{B}^{2}\,,$ then
$\rho(f(x),f(y))\leq c(K)\max\\{\rho(x,y),\rho(x,y)^{1/K}\\}$
for all $x,y\in\mathbf{B}^{2}$ where $c(K)=2{\rm arth}(\varphi_{K}({\rm
th}\frac{1}{2}))\,$ and
$K\leq u(K-1)+1\leq\log({\rm ch}(K{\rm arch}(e)))\leq c(K)\leq v(K-1)+K$
with $u={\rm arch}(e){\rm th}({\rm arch}(e))>1.5412$ and
$v=\log(2(1+\sqrt{1-1/e^{2}}))<1.3507$. In particular, $c(1)=1\,.$
Acknowledgments. The first author is indebted to the Graduate School of
Mathematical Analysis and its Applications for support. Both authors wish to
acknowledge the kind help of Prof. G.D. Anderson in the proof of Lemma 4.8,
the valuable help of the referee for the improvement of the manuscript, as
well as the expert help of Dr. H. Ruskeepää in the use of Mathematica [Ru].
## 2\. The main results
We shall follow here the standard notation and terminology for
$K$-quasiconformal and $K$-quasiregular mappings in the Euclidean $n$-space
$\mathbf{R}^{n}\,,$ see e.g. [V], [Vu1], and we also recall some basic
notation. For the modulus $M(\Gamma)$ of a curve family $\Gamma$ and its basic
properties see [V] and [Vu1].
Let $D$ and $D^{{}^{\prime}}$ be domains in $\overline{\mathbf{R}}^{n},K\geq
1$, and let $f:D\to D^{{}^{\prime}}$ be a homeomorphism. Then $f$ is
$K$-quasiconformal if
$M(\Gamma)/K\leq M(f\Gamma)\leq KM(\Gamma)$
for every curve family $\Gamma$ in $D$ [V].
For subsets $E,F,D\subset\overline{\mathbf{R}}^{n}$ we denote by
$\Delta(E,F;D)$ the family of all curves joining $E$ and $F$ in $D$. For
brevity we write $\Delta(E,F)=\Delta(E,F;{\mathbf{R}}^{n})\,.$ A ring is a
domain in ${\mathbf{R}}^{n}$, whose complement consists of two compact and
connected sets. If these sets are $E$ and $F$, then the ring is denoted by
$R(E,F)\,.$ The capacity of a ring $R(E,F)$ is
${\rm cap}R(E,F)=M(\Delta(E,F)).$
The complementary components of the Grötzsch ring $R_{G,n}(s)$ are
$\overline{\mathbf{B}}^{n}$ and $[se_{1},\infty],s>1$, while those of the
Teichmüller ring $R_{T,n}(t)$ are $[-e_{1},0]$ and $[te_{1},\infty],t>0$. The
conformal capacities of $R_{G,n}(s)$ and $R_{T,n}(t)$ are denoted by
$\left\\{\begin{array}[]{lll}\gamma_{n}(s)=M(\Delta(\overline{\mathbf{B}}^{n},[se_{1},\infty]))\\\
\tau_{n}(t)=M(\Delta([-e_{1},0],[te_{1},\infty]))\end{array}\right.$
respectively. Here $\gamma_{n}:(1,\infty)\to(0,\infty)$ and
$\tau_{n}:(0,\infty)\to(0,\infty)$ are decreasing homeomorphisms and they
satisfy the fundamental identity
(2.1) $\gamma_{n}(s)=2^{n-1}\tau_{n}(s^{2}-1),\quad t>1\,,$
see e.g. [Vu1, 5.53].
For $n\geq 2$ and $K>0$, the distortion function $\varphi_{K,n}:[0,1]\to[0,1]$
is a homeomorphism. It is defined by
(2.2)
$\varphi_{K,n}(t)=\displaystyle\frac{1}{\gamma_{n}^{-1}(K\gamma_{n}(1/t))},\quad
t\in(0,1),$
and $\varphi_{K,n}(0)=0\,,$ $\varphi_{K,n}(1)=1\,.$ For $n\geq 2,K\geq 1$ and
$0\leq r\leq 1$
(2.3)
$\varphi_{K,n}(r)\leq\lambda_{n}^{1-\alpha}r^{\alpha},\quad\alpha=K^{1/(1-n)}\,,$
(2.4)
$\varphi_{1/K,n}(r)\geq\lambda_{n}^{1-\beta}r^{\beta},\quad\beta=K^{1/(n-1)}\,,$
by [Vu1, Theorem 7.47] and where $\lambda_{n}\geq 4$ is as in Theorem 1.5.
###### 2.5 Lemma.
Suppose that $f:\mathbf{B}^{n}\to\mathbf{B}^{n}$ is a $K$-quasiconformal
mapping with $f\mathbf{B}^{n}=\mathbf{B}^{n}$, $f(0)=0,$ and let
$h:\overline{\mathbf{R}}^{n}\to\overline{\mathbf{R}}^{n}$ be the inversion
$h(x)=x/|x|^{2}\,,h(\infty)=0,h(0)=\infty,$ and define
$g:\overline{\mathbf{R}}^{n}\to\overline{\mathbf{R}}^{n}$ by $g(x)=f(x)$ for
$x\in\mathbf{B}^{n},g(x)=h(f(h(x)))$ for
$x\in\mathbf{R}^{n}\setminus\overline{\mathbf{B}}^{n}$ and $g(x)=\lim_{z\to
x}f(z)$ for $x\in\partial\mathbf{B}^{n},g(\infty)=\infty$. Then $g$ is a
$K$-quasiconformal mapping, and we have for $x\in\mathbf{B}^{n}$
(2.6) $\varphi_{1/K,n}(|x|)\leq|f(x)|\leq\varphi_{K,n}(|x|).$
For $x\in\mathbf{R}^{n}\setminus\overline{\mathbf{B}}^{n}$
(2.7) $1/\varphi_{K,n}(1/|x|)\leq|g(x)|\leq 1/\varphi_{1/K,n}(1/|x|).$
###### Proof.
It is well-known that the above definition defines $g$ as a $K$-quasiconformal
homeomorphism. The formula (2.6) is well-known (see [AVV2, Theorem 4.2]) and
(2.7) follows easily. ∎
###### 2.8 Lemma.
[Vu1, Lemma 7.35] Let $R=R(E,F)$ be a ring in $\overline{\mathbf{R}}^{n}$ and
let $a,b\in E,c,d\in F$ be distinct points. Then
$\text{cap}R=M(\Delta(E,F))\geq\tau_{n}\left(\frac{|a-c||b-d|}{|a-b||c-d|}\right).$
Equality holds if $b=t_{1}e_{1},a=t_{2}e_{1},c=t_{3}e_{1},d=t_{4}e_{1}$ and
$t_{1}<t_{2}<t_{3}<t_{4}$.
We consider Teichmüller’s extremal problem, which will be used to provide a
key estimate in what follows. For
$x\in\mathbf{R}^{n}\setminus\\{0,e_{1}\\},n\geq 2$, define
$p_{n}(x)=\inf_{E,F}M(\Delta(E,F))$
where the infimum is taken over all the pairs of continua $E$ and $F$ in
$\overline{\mathbf{R}}^{n}$ with $0,e_{1}\in E$ and $x,\infty\in F$. Note that
Lemma 2.8 gives the lower bound for $p_{n}(x)$ in Lemma 2.9.
###### 2.9 Lemma.
[Vu2, Theorem 1.5] For $z\in\mathbf{R}^{n},|z|>1$, the following inequalities
hold:
$\tau_{n}(|z|)=p_{n}(-|z|e_{1})\leq p_{n}(z)\leq
p_{n}(|z|e_{1})=\tau_{n}(|z|-1)$
where $p_{n}(z)$ is the Teichmüller function. Furthermore, for
$z\in\mathbf{R}^{n}\setminus\\{0,e_{1}\\}$, there exists a circular arc $E$
with $0,e_{1}\in E$ and a ray $F$ with $z,\infty\in F$ such that
(2.10)
$p_{n}(z)\leq\tau_{n}\left(\frac{|z|+|z-e_{1}|-1}{2}\right)=M(\Delta(E,F))\leq\tau_{n}(|z|-1)$
with equality in the first inequality both for $z=-se_{1},s>0$, and for
$z=se_{1},s>1\,.$
###### 2.11.
Notation. For $t>0,x,y\in\mathbf{B}^{n}\,,$ we write
$D(t,x,y)=|x+t\frac{y}{|y|}|\,\,\,\mathrm{if\,\,}y\neq 0,\quad
D(t,x,0)=|x+e_{1}|\,.$
By the triangle inequality we have
(2.12) $t-|x|\leq D(t,x,y)\leq t+|x|\,.$
###### 2.13 Theorem.
For $n\geq 2,K\geq 1$, let
$f:\overline{\mathbf{R}}^{n}\to\overline{\mathbf{R}}^{n}$ be a
$K$-quasiconformal mapping, with $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$,
$f(0)=0$ and $f(\infty)=\infty$. Then for $t\geq 1\,,$
$x,y\in\mathbf{B}^{n}\setminus\\{0\\}\,,$ we have
$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq$
$\displaystyle(3+\varphi_{1/K,n}(1/t)^{-1})\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{1/2}\right)$
$\displaystyle\leq$
$\displaystyle(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{\alpha}\,,\;\alpha=K^{1/(1-n)}=1/\beta,$
where $s_{1}=\displaystyle\max\\{a,b\\},a=t+|x|+D(t,y,x),b=t+|y|+D(t,x,y)$.
###### Proof.
Let $\Gamma$ be the family $\Delta(E,F)$ and let $E$ and $F$ be connected sets
as in Lemma 2.9 with $x,y\in E,z,\infty\in F$, where $z=-tx/|x|$ and
$\Gamma^{{}^{\prime}}=f(\Gamma)$. By Lemma 2.8 and (2.10), we have
$\displaystyle\tau_{n}\left(\frac{|f(z)-f(x)|}{|f(x)-f(y)|}\right)\leq
M(\Gamma^{{}^{\prime}})\leq KM(\Gamma)$ $\displaystyle\leq$ $\displaystyle
K\tau_{n}(u-1)\,,\quad
u=\displaystyle\frac{|x-z|+|z-y|-|x-y|+2|x-y|}{2|x-y|}\,.$
The basic identity ($\ref{1})$ yields
(2.14)
$\gamma_{n}\left(\left(\displaystyle\frac{|f(z)-f(y)|+|f(x)-f(y)|}{|f(x)-f(y)|}\right)^{1/2}\right)\leq
K\gamma_{n}\left(\left(u\right)^{1/2}\right)$
$=K\gamma_{n}\left(\left(\displaystyle\frac{t+|x|+D(t,y,x)+|x-y|}{2|x-y|}\right)^{1/2}\right).$
Applying $\gamma_{n}^{-1}$ to (2.14) we have
Figure 1. Geometrical meaning of the proof of Theorem 2.13.
$\displaystyle\frac{|f(z)-f(y)|+|f(x)-f(y)|}{|f(x)-f(y)|}\geq\displaystyle\left(\gamma_{n}^{-1}\left(K\gamma_{n}\left(\left(\frac{a+|x-y|}{2|x-y|}\right)^{1/2}\right)\right)\right)^{2}=v.$
Because $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$, by (2.6) and (2.4) we know
that
$|f(z)-f(y)|+|f(x)-f(y)|\leq 3+\varphi_{1/K,n}(1/t)^{-1}\leq
3+\lambda_{n}^{(\beta-1)}t^{\beta},$ (2.15)
$\frac{|f(x)-f(y)|}{3+\varphi_{1/K,n}(1/t)^{-1}}\leq\frac{|f(x)-f(y)|}{|f(z)-f(y)|+|f(x)-f(y)|}\leq
1/v,$
also
$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq$
$\displaystyle(3+\varphi_{1/K,n}(1/t)^{-1})\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{a+|x-y|}\right)^{1/2}\right)$
$\displaystyle\leq$
$\displaystyle(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{a+|x-y|}\right)^{\alpha}$
by inequalities (2.2) and (2.3). Exchanging the roles of $x$ and $y$ we see
that
$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq$
$\displaystyle(3+\varphi_{1/K,n}(1/t)^{-1})\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{1/2}\right)$
$\displaystyle\leq$
$\displaystyle(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{\alpha}.$
∎
Setting $t=1$, we get the following corollary.
###### 2.16 Corollary.
For $n\geq 2,K\geq 1$, let
$f:\overline{\mathbf{R}}^{n}\to\overline{\mathbf{R}}^{n}$ be a
$K$-quasiconformal mapping, with $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$,
$f(0)=0$ and $f(\infty)=\infty$. Then for all
$x,y\in\mathbf{B}^{n}\setminus\\{0\\}\,,$
$|f(x)-f(y)|\leq
4\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{s+|x-y|}\right)^{\alpha}\,,$
where $\alpha=K^{1/(1-n)}$ and
$s=\displaystyle\max\\{a,b\\},a=1+|x|+D(1,y,x),b=1+|y|+D(1,x,y)\,.$
###### Proof.
The proof is similar to the above proof except that here we consider the
particular case $t=1$. Because $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$, we know
that $|f(z)-f(y)|+|f(x)-f(y)|\leq 4$,
$\displaystyle\frac{|f(x)-f(y)|}{4}$ $\displaystyle\leq$
$\displaystyle\frac{|f(x)-f(y)|}{|f(z)-f(y)|+|f(x)-f(y)|}$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{1}{\left(\gamma_{n}^{-1}\left(K\gamma_{n}\left(\left(\displaystyle\frac{a+|x-y|}{2|x-y|}\right)^{1/2}\right)\right)\right)^{2}},$
or
$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq$ $\displaystyle
4\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{a+|x-y|}\right)^{1/2}\right)$
$\displaystyle\leq$ $\displaystyle
4\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{a+|x-y|}\right)^{\alpha}$
by inequalities (2.2) and (2.3). Exchanging the roles of $x$ and $y$ we get
$|f(x)-f(y)|\leq
4\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{\max\\{a,b\\}+|x-y|}\right)^{\alpha}\,.$
∎
###### 2.17 Corollary.
For $n\geq 2,K\geq 1,t\geq 1$, let $f$ be as in Theorem 2.13. Then
(2.18)
$|f(x)-f(y)|\leq(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{2t+||x|-|y||+|x-y|}\right)^{\alpha},$
for all $x,y\in\mathbf{B}^{n}\,,$
(2.19)
$|f(x)-f(y)|\leq(3+\lambda_{n}^{\beta-1}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{|x-y|}{\max\\{t+|x|,t+|y|\\}}\right)^{\alpha},$
for all $x,y\in\mathbf{B}^{n}\,,$ and
(2.20)
$|f(x)-f(y)|\leq(3+\lambda_{n}^{(\beta-1)}t^{\beta})\lambda_{n}^{2(1-\alpha)}\left(\frac{|x-y|}{t+|x|+(|x-y|)/2}\right)^{\alpha},$
if $D(t,y,x)>t+|x|,x,y\in\mathbf{B}^{n}$.
###### Proof.
Inequality $(\ref{11a})$ follows because by (2.11) $D(t,y,x)>t-|y|$ and
$D(t,x,y)>t-|x|$ for $x,y\in\mathbf{B}^{n}$, and hence, in the notation of
Theorem 2.13,
$s_{1}\geq\max\\{2t+|x|-|y|,2t+|y|-|x|\\}=2t+||x|-|y||\,.$
It is also clear that $D(t,y,x)\geq t+|x|-|x-y|$, and this implies that
$s_{1}\geq\max\\{2(t+|x|)-|x-y|,2(t+|y|)-|x-y|\\}=2\max\\{t+|x|,t+|y|\\}-|x-y|$
and hence the inequality $(\ref{11aa})$ follows. In the case of $(\ref{12a})$
we have $D(t,y,x)>t+|x|$ and see that, in the notation of Corollary 2.16,
$s>2(t+|x|)$ and $(\ref{12a})$ holds. ∎
###### 2.21 Corollary.
For $n\geq 2,K\geq 1$, let $f$ be as in Theorem 2.13. Then
(2.22) $|f(x)-f(y)|\leq
4\lambda_{n}^{2(1-\alpha)}\left(\frac{2|x-y|}{2+||x|-|y||+|x-y|}\right)^{\alpha},$
for all $x,y\in\mathbf{B}^{n}\setminus\\{0\\}\,.$
###### 2.23 Remark.
(1) In several of the above results we have supposed that
$x,y\in\mathbf{B}^{n}\setminus\\{0\\}\,.$ If one of the points $x,y$ were
equal to $0\,,$ then we would have a better result from the Schwarz lemma
estimate (4.7).
(2) Corollary 2.21 is an improvement of the Anderson-Vamanamurthy theorem 1.5
.
## 3\. Comparison with earlier bounds
###### 3.1.
Proof of Theorem 1.6. (1) The inequality (1.7) follows easily from the
inequality (2.19).
(2) We see that the function $h$ has a local minimum at
$t_{1}=(3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}(\beta-\alpha)^{-\alpha}\,.$ If
$t_{1}\geq 1\,,$ then the inequality (2.19) yields the desired conclusion. The
upper bound for $T(n,K)$ follows by substituting the argument $t_{1}$ in the
expression of $h\,.$
We next show that the value $K_{1}=4/3$ will do. Fix $K\in(1,K_{1})\,.$ Then
$\alpha=K^{1/(1-n)}\geq 3/4$ and $\alpha/(1-\alpha^{2})>1$.
Because $\lambda_{n}^{\alpha-1}\geq 2^{1/K-1}K^{-1}$ by [Vu1, Lemma 7.50(1)],
with $d=(6/K)^{1/K}/2K$ we have
$t_{1}=(3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}(\beta-\alpha)^{-\alpha}\geq(3/K)^{1/K}2^{1/K-1}K^{-1}\left(\frac{\alpha}{1-\alpha^{2}}\right)^{\alpha}$
$=d\left(\frac{\alpha}{1-\alpha^{2}}\right)^{\alpha}\geq
d\left(\frac{\alpha}{1-\alpha^{2}}\right)^{3/4}$
$=\left(2r(K)\frac{\alpha}{1-\alpha^{2}}\right)^{3/4}\,;r(K)=d^{4/3}/2\,.$
It suffices to observe that $t_{1}>1$ certainly holds if
$2r(K)(\frac{\alpha}{1-\alpha^{2}})>1$ which holds for
$\alpha>1/(r(4/3)+\sqrt{1+r(4/3)^{2}})=0.53...\,,$ in particular, $t_{1}>1$
holds in the present case $\alpha>3/4\,.$
For the proof of (1.9) we give the following inequalites
(3.2) $\lambda_{n}^{\alpha-\alpha^{2}}\leq 2^{\alpha(1-\alpha)}K^{\alpha}\leq
2^{1-\alpha}K^{\alpha},\quad K\geq 1\,,$ (3.3)
$\lambda_{n}^{\beta-\alpha}=\lambda_{n}^{\beta+1-1-\alpha}=\lambda_{n}^{\beta(1-\alpha)+1-\alpha}=\lambda_{n}^{(\beta+1)(1-\alpha)}\leq(2^{1-\alpha}K)^{3},\quad\beta\in(1,2)\,,$
see [Vu1, Lemma 7.50(1)]. The formula (1.8) for $h(t_{1})$ has two terms. We
estimate separately each term as follows
$\displaystyle\frac{3^{1-\alpha^{2}}(\beta-\alpha)^{\alpha^{2}}}{\alpha^{\alpha^{2}}}\lambda_{n}^{\alpha-\alpha^{2}}\lambda_{n}^{2(1-\alpha)}$
$\displaystyle\leq$
$\displaystyle\frac{3^{(1-\alpha)(1+\alpha)}2^{\alpha(1-\alpha)}2^{2(1-\alpha)}K^{2}(\beta-\alpha)^{\alpha^{2}}}{\alpha^{\alpha^{2}}}K^{\alpha}$
$\displaystyle\leq$ $\displaystyle\frac{(9\cdot 2\cdot
4)^{1-\alpha}K^{2}(\beta-\alpha)^{\alpha^{2}}}{\alpha^{\alpha^{2}}}K^{\alpha}$
$\displaystyle=$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}K^{\alpha}\exp(-\alpha^{2}\log\alpha)$
$\displaystyle\leq$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}K^{\alpha}\exp(-\alpha\log\alpha)$
$\displaystyle=$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}\exp((\log K-\log\alpha)\alpha)$
$\displaystyle=$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}\exp\left(\left(1+\frac{1}{n-1}\log
K\right)\alpha\right)$ $\displaystyle=$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}\exp\left(\frac{n}{n-1}\alpha\log
K\right)$ $\displaystyle\leq$ $\displaystyle
72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{2}\exp(2\log K)$ $\displaystyle=$
$\displaystyle 72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{4}$
by inequality (3.2),
$\displaystyle\lambda_{n}^{2(1-\alpha)}\lambda_{n}^{\beta-1}\left(\frac{(3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}}{(\beta-\alpha)^{\alpha}}\right)^{\beta-\alpha}$
$\displaystyle=$
$\displaystyle\lambda_{n}^{2(1-\alpha)}\lambda_{n}^{\beta-1}\left((3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}\right)^{\beta-\alpha}(\beta-\alpha)^{-\alpha(\beta-\alpha)}$
$\displaystyle\leq$
$\displaystyle(2^{1-\alpha}K)^{2}\lambda_{n}^{\beta-\alpha}\left((3\alpha)^{\alpha}\lambda_{n}^{\alpha-1}\right)^{\beta-\alpha}\left(\frac{\beta^{2}-1}{\beta}\right)^{-\alpha((\beta^{2}-1)/\beta)}$
$\displaystyle\leq$
$\displaystyle(2^{1-\alpha}K)^{2}\left(3^{\alpha}\lambda_{n}\right)^{\beta-\alpha}\beta^{\alpha^{2}}(\beta^{2}-1)^{-\alpha^{2}(\beta^{2}-1)}$
$\displaystyle\leq$
$\displaystyle(2^{1-\alpha}K)^{2}3^{\alpha(\beta-\alpha)}\lambda_{n}^{(\beta+1)(1-\alpha)}\exp\left(\frac{2\alpha^{2}}{e}\sqrt{\beta^{2}-1}\right)$
$\displaystyle\leq$ $\displaystyle
3^{1-\alpha^{2}}(2^{1-\alpha}K)^{2}(2^{1-\alpha}K)^{(\beta+1)}\exp\left(\frac{2\alpha^{2}}{e}\sqrt{\beta^{2}-1}\right)$
$\displaystyle\leq$ $\displaystyle
3^{1-\alpha^{2}}(2^{1-\alpha}K)^{5}\exp(\sqrt{\beta^{2}-1}),$
here we assume that $\beta\in(1,2)$ which implies that $\alpha\in(1/2,1)$.
Also the inequalities $(K-1)^{-(K-1)}\leq\exp((2/e)\sqrt{K-1})$ and (3.3) were
used, and we get
(3.4)
$h(t_{1})\leq\left[72^{1-\alpha}(\beta-\alpha)^{\alpha^{2}}K^{4}+3^{\beta-\alpha}(2^{1-\alpha}K)^{5}\exp(\sqrt{\beta^{2}-1})\right].$
Because $(\beta-\alpha)\in(0,\frac{3}{2})$ this implies that
$\frac{2}{3}(\beta-\alpha)\in(0,1)$ and $\alpha^{2}\in(\frac{1}{4},1)$ and
further
$(\frac{2}{3}(\beta-\alpha))^{\alpha^{2}}\leq(\frac{2}{3}(\beta-\alpha))^{1/4}$,
and finally
$(\beta-\alpha)^{\alpha^{2}}\leq(2/3)^{-\alpha^{2}}(\frac{2}{3}(\beta-\alpha))^{1/4}\leq(3/2)^{3/4}\sqrt[4]{\beta-\alpha}$
$=(3/2)^{3/4}\sqrt[4]{\beta-\alpha}<(3/2)\sqrt[4]{\beta-\alpha}\,.$
Next we prove that
(3.5) $72^{1-\alpha}\leq 3^{1-\alpha^{2}}2^{5(1-\alpha)}K\,.$
This inequality is equivalent to
$2^{2(\alpha-1)}3^{(1-\alpha)^{2}}\leq K\Longleftrightarrow-(1-\alpha)\log
4+(1-\alpha)^{2}\log 3\leq\log K\,.$
This last inequality holds because the left hand side is negative. Now from
(3.4) and (3.5) we get the desired inequality (1.9). $\square$
###### 3.6.
Graphical and numerical comparision of various bounds. The above bounds
involve the Grötzsch ring constant $\lambda_{n},$ which is known only for
$n=2,\lambda_{2}=4.$ Therefore only for $n=2$ we can compute the values of the
bounds. Solving numerically the equation ${4\cdot 16}^{1-1/K}=h(t_{1})$ for
$K$ we obtain $K=1.3089\,.$ We give numerical and graphical comparison of the
various bounds for the Mori constant.
Tabulation of the various upper bounds for Mori’s constant when $n=2$ and
$\lambda_{2}=4$ as a function of $K$: (a) Mori’s conjectured bound
$16^{1-1/K}$, (b) the Anderson-Vamanamurthy bound $4\cdot 16^{1-1/K}$, (c) the
bound from (1.8). For $K\in(1,1.3089)$ the upper bound in (1.8) is better than
the Anderson-Vamanamurthy bound and for $K>1.5946$ the upper bound in (1.8) is
better than the bound of Fehlmann and Vuorinen. Numerical values of the [FV]
bound given in the table were computed with the help of the algorithm for
$\varphi_{K,2}(r)$ attached with [AVV1, p. 92, 439].
Figure 2. Graphical illustration of the various upper bounds for Mori’s
constant when $n=2$ and $\lambda_{2}=4$ as a function of $K$: (a) Mori’s
conjectured bound $16^{1-1/K}$, (b) the Anderson-Vamanamurthy bound $4\cdot
16^{1-1/K}$, (c) the bound from (1.8). For $K\in(1,1.3089)$ the upper bound in
(1.8) is better than the Anderson-Vamanamurthy bound.
$\begin{array}[]{|c|c|c|c|c|}\hline\cr K&\log({16}^{1-1/K})&\log({4\cdot
16}^{1-1/K})&\log(FV)&\displaystyle\log(h(t_{1}))\\\ \hline\cr
1.1&0.2521&1.6384&0.7051&1.0188\\\ 1.2&0.4621&1.8484&1.2485&1.6058\\\
1.3&0.6398&2.0261&1.7046&2.0107\\\ 1.4&0.7922&2.1785&2.0913&2.3061\\\
1.5&0.9242&2.3105&2.4221&2.5296\\\ 1.6&1.0397&2.4260&2.7094&2.7031\\\
1.7&1.1417&2.5280&2.9633&2.8409\\\ 1.8&1.2323&2.6186&3.1921&2.9521\\\
1.9&1.3133&2.6996&3.4020&3.0433\\\ 2.0&1.3863&2.7726&3.5979&3.1192\\\
\hline\cr\end{array}$
For graphing and tabulation purposes we use the logarithmic scale. Note that
the upper bound for $M(2,K)$ given in [FV, Theorem 2.29] also has the
desirable property that it converges to $1$ when $K\to 1\,,$ see Figure 2.
Figure 3. Graphical comparison of various bounds when $n=2$ and
$\lambda_{2}=4\,,$ as a function of $K$: (a) the bound from (1.8), (b) the
Fehlmann and Vuorinen bound [FV]
$M(2,K)\leq\left(1+\varphi_{K,2}\left(\frac{K^{2}-1}{K^{2}+1}\right)\right)2^{2K-3/K}\frac{(K^{2}+1)^{(K+1/K)/2}}{(K^{2}-1)^{(K-1/K)/2}}.$
For $K>1.5946$ the upper bound in (1.8) is better than the Fehlmann-Vuorinen
bound.
###### 3.7.
Comparison of estimates for the Hölder quotient. For a $K$-quasiconformal
mapping $f:\mathbf{B}^{n}\to f\mathbf{B}^{n}=\mathbf{B}^{n}\,,$ we call the
expression
$HQ(f)=\sup\\{|f(x)-f(y)|/|x-y|^{\alpha}:\;x,y\in\mathbf{B}^{n},f(0)=0\,\;x\neq
y\\},$
the Hölder coefficient of $f$. Clearly $HQ(f)\leq M(n,K)$. Theorem 2.13
yields, after dividing the both sides of the inequality in 2.13 by
$|x-y|^{\alpha}\,,$ the upper bound $HQ(f)\leq HQ(K)$ for the Hölder quotient
with
(3.8) $HQ(K)=\sup\\{\inf\\{U(t,x,y):\;t\geq 1\\}:\;x,y\in\mathbf{B}^{n}\\}\,,$
$U(t,x,y)=(3+\varphi_{1/K,n}(1/t)^{-1})\varphi_{K,n}^{2}\left(\left(\frac{2|x-y|}{s_{1}+|x-y|}\right)^{1/2}\right)\frac{1}{|x-y|^{\alpha}}\,.$
For $n=2$ we compare $HQ(K)$ to several other bounds (a) Mori’s conjectured
bound, (b) the FV bound, (c) the AV bound and give the results as a table and
Figure 3. Because the supremum and infimum in (3.8) cannot be explicitly found
we use numerical methods that come with Mathematica software. For the
numerical tests we used for the supremum a sample of $100,000$ random points
of the unit disk.
Figure 4. Graphical comparison of various bounds when $n=2$ and
$\lambda_{2}=4\,,$ as a function of $K$: (a) the bound from (3.8), (b) the
Fehlmann and Vuorinen bound [FV]
$M(2,K)\leq\left(1+\varphi_{K,2}\left(\frac{K^{2}-1}{K^{2}+1}\right)\right)2^{2K-3/K}\frac{(K^{2}+1)^{(K+1/K)/2}}{(K^{2}-1)^{(K-1/K)/2}},$
(c) the bound of the Mori conjecture. Note that the bound (3.8), based on a
simulation with $100,000$ random points, gives the best estimate in the cases
considered in the picture.
$\begin{array}[]{|c|c|c|c|c|}\hline\cr K&\log({16}^{1-1/K})&\log({4\cdot
16}^{1-1/K})&\log(FV)&\log(HQ(K))\\\ \hline\cr
1.1&0.2521&1.6384&0.7051&1.0171\\\ 1.2&0.4621&1.8484&1.2485&1.5940\\\
1.3&0.6398&2.0261&1.7046&1.9712\\\ 1.4&0.7922&2.1785&2.0913&2.1668\\\
1.5&0.9242&2.3105&2.4221&2.2928\\\ 1.6&1.0397&2.4260&2.7094&2.4003\\\
1.7&1.1417&2.5280&2.9633&2.4922\\\ 1.8&1.2323&2.6186&3.1921&2.5706\\\
1.9&1.3133&2.6996&3.4020&2.6371\\\ 2.0&1.3863&2.7726&3.5979&2.6934\\\
\hline\cr\end{array}$
## 4\. An explicit form of the Schwarz lemma
Recall that the hyperbolic metric $\rho(x,y),x,y\in\mathbf{B}^{n}\,,$ of the
unit ball is given by (cf. [KL], [Vu1])
(4.1) ${\rm
th}^{2}\frac{\rho(x,y)}{2}=\frac{|x-y|^{2}}{|x-y|^{2}+t^{2}}\,,\quad
t^{2}=(1-|x|^{2})(1-|y|^{2})\,.$
Next, we consider a decreasing homeomorphism
$\mu:(0,1)\longrightarrow(0,\infty)$ defined by
(4.2) $\mu(r)=\frac{\pi}{2}\,\frac{{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}}(r^{\prime})}{{\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}}(r)},\quad{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp
K}}}(r)=\int_{0}^{1}\frac{dx}{\sqrt{(1-x^{2})(1-r^{2}x^{2})}}\,,$
where ${\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp K}}}(r)$ is Legendre’s complete elliptic integral of the
first kind and $r^{\prime}=\sqrt{1-r^{2}},$ for all $r\in(0,1)$.
The Hersch-Pfluger distortion function is an increasing homeomorphism
$\varphi_{K}:(0,1)\longrightarrow(0,1)$ defined by setting
(4.3) $\varphi_{K}(r)=\mu^{-1}(\mu(r)/K)\,,\,r\in(0,1),\,\,K>0.$
Note that with the notation of Section 2, $\gamma_{2}(1/r)=2\pi/\mu(r)$ and
$\varphi_{K}(r)=\varphi_{K,2}(r)\,$ for $r\in(0,1)\,.$
###### 4.4 Theorem.
[Vu1, 11.2] Let $f:\mathbf{B}^{n}\to\mathbf{R}^{n}$ be a nonconstant
$K$-quasiregular mapping with $f\mathbf{B}^{n}\subset\mathbf{B}^{n}$ and let
$\alpha=K^{1/(1-n)}\,.$ Then
(4.5) ${\rm th}\frac{\rho(f(x),f(y))}{2}\leq\varphi_{K,n}({\rm
th}\frac{\rho(x,y)}{2})\leq\lambda_{n}^{1-\alpha}\left({\rm
th}\frac{\rho(x,y)}{2}\right)^{\alpha}\,,$ (4.6) $\rho(f(x),f(y))\leq
K(\rho(x,y)+\log 4)\,,$
for all $x,y\in\mathbf{B}^{n}\,,$ where $\lambda_{n}$ is the same constant as
in (1.5). If $f(0)=0\,,$ then
(4.7) $|f(x)|\leq\lambda_{n}^{1-\alpha}|x|^{\alpha}\,,$
for all $x\in\mathbf{B}^{n}\,.$
In the case of quasiconformal mappings with $n=2$ formulas (4.5) and (4.7)
also occur in [LV, p. 65] and formula (4.6) was rediscovered in [EMM, Theorem
5.1]. Comparing Theorem 4.4 to Theorem 1.10 we see that for $n=2$ the
expression $K(\rho(x,y)+\log 4)$ may be replaced with
$c(K)\max\\{\rho(x,y),\rho(x,y)^{1/K}\\}\,,$ which tends to $0$ when $x\to
y\,$ and to $\rho(x,y)$ when $K\to 1\,,$ as expected.
###### 4.8 Lemma.
For $K>1$ the function
$t\mapsto\frac{2{\rm arth}(\varphi_{K}({\rm
th}\frac{t}{2}))}{\max\\{t,t^{1/K}\\}}\,,$
is monotone increasing on $(0,1)$ and decreasing on $(1,\infty)\,.$
###### Proof.
(1) Fix $K>1$ and consider
$f(t)=\frac{2{\rm arth}(\varphi_{K}({\rm th}\frac{t}{2}))}{t},\quad t>0.$
Let $r={\rm th}\frac{t}{2}$. Then $t/2={\rm arth}r$, and $t$ is an increasing
function of $r$ for $0<r<1$. Then
$f(t)=\frac{2{\rm arth}(\varphi_{K}({\rm th}\frac{t}{2}))}{t/2}=\frac{{\rm
arth}(\varphi_{K}(r))}{{\rm arth}r}=F(r).$
Then by [AVV1, Theorem 10.9(3)], $F(r)$ is strictly decreasing from $(0,1)$
onto $(K,\infty)$. Hence $f(t)$ is strictly decreasing from $(0,\infty)$ onto
$(K,\infty)$.
(2) Next consider
$g(t)=\frac{2{\rm arth}(\varphi_{K}({\rm th}\frac{t}{2}))}{t^{1/K}},$
and let $r={\rm th}\frac{t}{2}$. Then $t=2{\rm arth}r$ and
$g(t)=\frac{2{\rm arth}s}{2^{1/K}({\rm arth}r)^{1/K}}=\frac{2^{1-1/K}{\rm
arth}s}{({\rm arth}r)^{1/K}}\,,$
where $s=\varphi_{K}(r)$. We next apply [AVV1, Theorem 1.25]. We know
$\frac{d}{dr}({\rm arth}r)=1/(1-r^{2})$.
Writing $r^{\prime}=\sqrt{1-r^{2}},s^{\prime}=\sqrt{1-s^{2}},$ we obtain the
quotient of the derivatives
$\displaystyle\frac{2^{1-1/K}(1/(1-s^{2}))\frac{ds}{dr}}{\frac{1}{K}({\rm
arth}r)^{1/K-1}(1/(1-r^{2})}$ $\displaystyle=$ $\displaystyle
2^{1-1/K}\,K\,({\rm
arth}r)^{1-1/K}\frac{r^{{}^{\prime}2}}{s^{{}^{\prime}2}}\frac{1}{K}\frac{ss^{{}^{\prime}2}\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(s)^{2}}{rr^{{}^{\prime}2}\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r)^{2}}$ $\displaystyle=$ $\displaystyle
2^{1-1/K}({\rm arth}r)^{1-1/K}\frac{s\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(s)^{2}}{r\mathchoice{\hbox{\,\fFt
K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r)^{2}}$
by [AVV1, appendix E(23)]. By [AVV1, Lemma 10.7(3)],
$\frac{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt K}}{\hbox{\,\fFa
K}}{\hbox{\,\fFp K}}(s)^{2}}{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp K}}(r)^{2}}$ is increasing, since $K>1$,
$({\rm arth}r)^{1/K-1}$ is increasing. Finally, $s/r$ is increasing by [AVV1,
Theorem 1.25] and E(23). So $g(t)$ is increasing in $t$ on $(0,\infty)$.
(3) Fix $K>1$. Clearly
$\max\\{t,t^{1/K}\\}=\left\\{\begin{array}[]{lll}t^{1/K}\quad{\rm for}\quad
0\leq t\leq 1\\\ t\quad{\rm for}\quad 1\leq t<\infty.\end{array}\right.$
Thus
$h(t)=\frac{2{\rm arth}(\varphi_{K}({\rm
th}\frac{t}{2}))}{\max\\{t,t^{1/K}\\}},\;$
increases on $(0,1)$ and decreases on $(1,\infty)$. ∎
Figure 5. Graphical comparison of lower and upper bounds for $c(K)$ with
$b(K)=\log({\rm ch}(K{\rm arch}(e)))$.
###### 4.9.
Proof of Theorem 1.10. The maximum value of the function considered in Lemma
4.8 is $c(K)=2{\rm arth}(\varphi_{K}({\rm th}\frac{1}{2}))$. The inequality
now follows from Lemma 4.8.$\qquad\square$
###### 4.10.
Bounds for the constant $c(K)$. In order to give upper and lower bounds for
$c(K)\,,$ we observe that the identity [AVV1, Theorem 10.5(2)] yields the
following formula
$c(K)=2{\rm
arth}\left(\varphi_{K}\left(\frac{1-1/e}{1+1/e}\right)\right)=2{\rm
arth}\left(\frac{1-\varphi_{1/K}(1/e)}{1+\varphi_{1/K}(1/e)}\right)\,.$
A simplification leads to
$c(K)=-\log\varphi_{1/K}(1/e)\,.$
Next, from the inequality $\varphi_{1/K}(r)\geq
2^{1-K}(1+r^{{}^{\prime}})^{1-K}r^{K}$ for $K\geq 1,r\in(0,1)$ (cf. [AVV1,
Corollary 8.74(2)]) we get with $v=\log(2(1+\sqrt{1-1/e^{2}}))<1.3507$
$\displaystyle c(K)$ $\displaystyle=$
$\displaystyle-\log\varphi_{1/K}(1/e)\leq-\log\left(2^{1-K}(1+\sqrt{1-1/e^{2}})^{1-K}e^{-K}\right)$
$\displaystyle=$ $\displaystyle v(K-1)+K<1.3507(K-1)+K.$
In order to estimate the constant $c(K)$ from below we need an upper bound for
$\varphi_{1/K,2}(r),\;K>1$, from above. For this purpose we prove the
following lemma.
###### 4.11 Lemma.
For every integer $n\geq 2$ and each $K>1,\;r\in(0,1)$, there exists
$K$-quasiconformal maps $g:\mathbf{B}^{n}\to\mathbf{B}^{n}$ and
$h:\mathbf{B}^{n}\to\mathbf{B}^{n}$ with
$(a)\qquad
g(0)=0,\;g\mathbf{B}^{n}=\mathbf{B}^{n},\;h(0)=0,\;h\mathbf{B}^{n}=\mathbf{B}^{n}$
$(b)\qquad
g(re_{1})=\displaystyle\frac{2r^{\alpha}}{(1+r^{{}^{\prime}})^{\alpha}+(1-r^{{}^{\prime}})^{\alpha}},\;h(re_{1})=\displaystyle\frac{2r^{\beta}}{(1+r^{{}^{\prime}})^{\beta}+(1-r^{{}^{\prime}})^{\beta}}$
where $r^{{}^{\prime}}=\sqrt{1-r^{2}}$ and $\alpha=K^{1/(1-n)}=1/\beta$.
In particular, for $n=2$ and $K>1,\;r\in(0,1)$
$(c)\qquad\varphi_{1/K}(r)\leq\displaystyle\frac{2r^{K}}{(1+r^{{}^{\prime}})^{K}+(1-r^{{}^{\prime}})^{K}}\;;\;\;\varphi_{K}(r)\geq\displaystyle\frac{2r^{1/K}}{(1+r^{{}^{\prime}})^{1/K}+(1-r^{{}^{\prime}})^{1/K}}$.
###### Proof.
Fix $r\in(0,1)$. Let $T_{a}:\mathbf{B}^{n}\to\mathbf{B}^{n}$ be a Möbius
automorphism with $T_{a}(a)=0$ and $T_{a}(\mathbf{B}^{n})=\mathbf{B}^{n}$.
Choose $s\in(0,r)$ such that $T_{se_{1}}(0)=-T_{se_{1}}(re_{1})$. Then
$\rho(0,re_{1})=2\rho(0,se_{1})$ [Vu1, (2.17)], or equivalently,
$(1+r)/(1-r)=((1+s)/(1-s))^{2}$ and hence $s=r/(1+r^{{}^{\prime}})$. Consider
the $K$-quasiconformal mapping $f:\mathbf{B}^{n}\to\mathbf{B}^{n}$,
$f(x)=|x|^{\alpha-1}x,\;\alpha=K^{1/(1-n)}$. Then $f(\pm se_{1})=\pm
s^{\alpha}e_{1}$. The mapping $g=T_{-s^{\alpha}e_{1}}\circ f\circ
T_{se_{1}}:\mathbf{B}^{n}\to\mathbf{B}^{n}$ satisfies $g(0)=0$,
$g(re_{1})=te_{1}$ where
$\rho(-s^{\alpha}e_{1},s^{\alpha}e_{1})=\rho(0,te_{1})$ and hence
$t=2r^{\alpha}/((1+r^{{}^{\prime}})^{\alpha}+(1-r^{{}^{\prime}})^{\alpha})$ by
[Vu1, (2.17)]. The proof for $g$ is complete. For the map $h$ the proof is
similar except that we use the $K$-quasiconformal mapping
$m:x\mapsto|x|^{\beta-1}x,\;\beta=1/\alpha$. Note that $m=f^{-1}$ and
$t=1/{\rm ch}(\alpha\;{\rm arch}(1/r))$. For the proof of $(c)$ we apply
$(a),\;(b)$ together with [LV, (3.4), p.64]. ∎
###### 4.12 Lemma.
For $K>1,$ $c(K)\geq\log({\rm ch}(K{\rm arch}(e)))\geq u(K-1)+1,$ where
$u={\rm arch}(e){\rm th}({\rm arch}(e))>1.5412$.
###### Proof.
From Lemma 4.11(c), we know that
$\displaystyle\varphi_{1/K}(1/e)$ $\displaystyle\leq$
$\displaystyle\frac{2/e^{K}}{(1+\sqrt{1-1/e^{2}})^{K}+(1-\sqrt{1-1/e^{2}})^{K}}$
$\displaystyle=$
$\displaystyle\frac{2}{(e+\sqrt{e^{2}-1})^{K}+(e-\sqrt{e^{2}-1})^{K}},$
hence
$\displaystyle c(K)$ $\displaystyle=$
$\displaystyle-\log\varphi_{1/K}(1/e)\geq-\log\left(\frac{2}{(e+\sqrt{e^{2}-1})^{K}+(e-\sqrt{e^{2}-1})^{K}}\right)$
$\displaystyle=$
$\displaystyle\log\left(\frac{(e+\sqrt{e^{2}-1})^{K}+(e-\sqrt{e^{2}-1})^{K}}{2}\right)$
$\displaystyle=$ $\displaystyle\log({\rm ch}(K{\rm arch}(e)))\geq u(K-1)+1,$
where the last inequality follows easily from the mean value theorem, applied
to the function $p(K)=\log({\rm ch}(K{\rm arch}(e)))\,.$ ∎
## References
* [A1] L. V. Ahlfors: On quasiconformal mappings, J. Analyse Math. 3, (1954). 1–58; correction, 207–208, also: pp. 2-61 in Collected papers. Vol. 2. 1954–1979. Edited with the assistance of Rae Michael Shortt. Contemporary Mathematicians. Birkhäuser, Boston, Mass., 1982\. xix+515 pp. ISBN: 3-7643-3076-7.
* [A2] L. V. Ahlfors: Lectures on quasiconformal mappings. Second edition. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. University Lecture Series, 38. American Mathematical Society, Providence, RI, 2006. viii+162 pp. ISBN: 0-8218-3644-7.
* [AN] G. Anderson: Dependence on dimension of a constant related to the Grötzsch ring, Proc. Amer. Math. Soc. 61 (1976), no. 1, 77–80 (1977).
* [AV] G. Anderson and M. Vamanamurthy: Hölder continuity of quasiconformal mappings of the unit ball, Proc. Amer. Math. Soc. 104 (1988), no. 1, 227–230.
* [AVV1] G. D. Anderson, M. K. Vamanamurthy, and M. K. Vuorinen: Conformal invariants, inequalities and quasiconformal maps, J. Wiley, 1997, 505 pp.
* [AVV2] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen: Dimension-free quasiconformal distortion in $n$-space, Trans. Amer. Math. Soc. 297 (1986), 687–706.
* [EMM] D. B. A. Epstein, A. Marden, and V. Markovic: Quasiconformal homeomorphisms and the convex hull boundary. Ann. of Math. (2) 159 (2004), no. 1, 305–336.
* [FV] R. Fehlmann and M. Vuorinen: Mori’s theorem for $n$-dimensional quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), no. 1, 111–124.
* [FM] A. Fletcher and V. Markovic: Quasiconformal maps and Teichmüller theory. Oxford Graduate Texts in Mathematics, 11. Oxford University Press, Oxford, 2007. viii+189 pp. ISBN: 978-0-19-856926-8; 0-19-856926-2.
* [G] F. W. Gehring: Rings and quasiconformal mappings in space. Trans. Amer. Math. Soc. 103 (1962) 353–393.
* [H] P. Hästö: Distortion in the spherical metric under quasiconformal mappings. (English summary) Conform. Geom. Dyn. 7 (2003), 1–10.
* [HV] V. Heikkala and M. Vuorinen: Teichmüller’s extremal ring problem, Math. Z. 254(2006), no. 3, 509–529.
* [KL] L. Keen and N. Lakic: Hyperbolic geometry from a local viewpoint. London Mathematical Society Student Texts, 68. Cambridge University Press, Cambridge, 2007.
* [LV] O. Lehto and K.I. Virtanen: Quasiconformal mappings in the plane. Second edition. Translated from the German by K. W. Lucas. Die Grundlehren der mathematischen Wissenschaften, Band 126. Springer-Verlag, New York-Heidelberg, 1973. viii+258 pp.
* [MRV] O. Martio, S. Rickman, and J. Väisälä: Distortion and singularities of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I No. 465 (1970) 13 pp.
* [Mo] A. Mori: On an absolute constant in the theory of quasi-conformal mappings, J. Math. Soc. Japan 8 (1956), 156–166.
* [Q] S.-L. Qiu: On Mori’s theorem in quasiconformal theory. A Chinese summary appears in Acta Math. Sinica 40 (1997), no. 2, 319. Acta Math. Sinica (N.S.) 13 (1997), no. 1, 35–44.
* [R] Yu. G. Reshetnyak: Estimates of the modulus of continuity for certain mappings. (Russian) Sibirsk. Mat. Ž. 7 (1966) 1106–1114.
* [Ru] H. Ruskeepää: Mathematica Navigator. 3rd ed. Academic Press, 2009.
* [S] B. V. Shabat: On the theory of quasiconformal mappings in space. Dokl. Akad. Nauk SSSR 132 1045–1048 (Russian); translated as Soviet Math. Dokl. 1 (1960) 730–733.
* [V] J. Väisälä: Lectures on $n$-dimensional quasiconformal mappings. Lecture Notes in Mathematics, Vol. 229. Springer-Verlag, Berlin-New York, 1971. xiv+144 pp.
* [Vu1] M. Vuorinen: Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics 1319, Springer, Berlin, 1988.
* [Vu2] M. Vuorinen: Conformally invariant extremal problems and quasiconformal maps, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 172, 501–514.
|
arxiv-papers
| 2009-06-16T07:33:10 |
2024-09-04T02:49:03.374431
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Barkat Ali Bhayo and Matti Vuorinen",
"submitter": "Matti Vuorinen",
"url": "https://arxiv.org/abs/0906.2853"
}
|
0906.2993
|
# Impurity induced spin gap asymmetry in nanoscale graphene
Julia Berashevich and Tapash Chakraborty [email protected] Department
of Physics and Astronomy, University of Manitoba, Winnipeg, Canada, R3T 2N2
###### Abstract
We propose a unique way to control both bandgap and the magnetic properties of
nanoscale graphene, which might prove highly beneficial for application in
nanoelectronic and spintronic devices. We have shown that chemical doping by
nitrogen along a single zigzag edge breaks the sublattice symmetry of
graphene. This leads to the opening of a gap and a shift of the molecular
orbitals localized on the doped edge in such a way that the spin gap
asymmetry, which can lead to half-metallicity under certain conditions, is
obtained. The spin-selective behavior of graphene and tunable spin gaps help
us to obtain semiconductor diode-like current-voltage characteristics, where
the current flowing in one direction is preferred over the other. The doping
in the middle of the graphene layer results in an impurity level between the
HOMO and LUMO orbitals of pure graphene (again, much like in semiconductor
systems) localized on the zigzag edges thus decreasing the bandgap and adding
unpaired electrons, and this can also be used to control graphene
conductivity.
## I Introduction
Applications of graphene with its unique physical properties nov1 ; expt2 ;
tapash ; tapash1 ; david in nanoelectronics chen ; kern , magnetism and
spintronics cohen ; rudb ; esq ; cho ; karpan , hang crucially on its bandgap
and spin ordering at the zigzag edges. A bandgap can be opened in graphene by
breaking the certain symmetries. For example, interaction of graphene with its
substrate, such as SiC, leads to the charge exchange between them which breaks
the sublattice symmetry zhou . Moreover, the quantum confinement effect also
has been found to introduce a small bandgap in graphene nanoribbons kim , just
as was predicted earlier theoretically cohen1 ; nak ; lee ; pisani . The
effect of bandgap opening and spin ordering between the zigzag edges are found
to be directly linked to each other harrison . When the spins align along the
zigzag edges and spin states localized at opposite edges have the same spin
orientation, then symmetry of graphene is preserved and the system is gapless.
Otherwise, if the spin-up states are localized along one zigzag edge and the
spin-down along the other, the sublattice symmetry is broken which leads to a
gap. In the light of a recent breakthrough in fabrication of nanoribbons of
required size through unzipping of carbon nanotubes, the nanoribbons and
nanoscale graphene are the most promising systems for application in
nanoelectronics nature .
Manipulation of the spin ordering is important for both graphene magnetism and
its electronic properties. There are several approaches which have been
proposed recently to control the spin ordering along the edges fer ; my ; gun
; bouk ; li ; hod ; kan . One of them is the termination of the zigzag edges
by functional groups fer . This has the advantage that one can achieve half
metallicity in this process. However, there are some serious issues involved
here. Firstly, many of these functional groups are placed out of the graphene
plane thus making the whole structure non-planar and, most importantly,
termination was applied to every second edge cell, which makes its
technological application very difficult. In fact, we found that the strong
interactions of the graphene lattice with some of the functional groups, such
as NH2 and NO2, lead to buckling of the graphene layer and twisting of the
functional groups, which subsequently may result in the disappearance of the
half-metallicity of graphene. The curling of the graphene layer has been seen
in earlier studies as well kan , where the boundary conditions were found to
control the graphene planarity, namely the curling occurs for stand-alone
systems. For nanoscale graphene the ferromagnetic ordering of the spin states
along the zigzag edges can be also achieved as subsequence of adsorption of
gas and water molecules on the graphene surface, as we have shown in our
previous study my . The adsorption leads to pushing of the $\alpha$\- and
$\beta$-spin states to the opposite zigzag edges thereby breaking sublattice
symmetry and opening a gap. In some cases the spin asymmetry can occur. For
example, the adsorption of HF gas molecule provide the HOMO-LUMO gap of 2.1 eV
for $\alpha$-spin state and of 1.2 eV for $\beta$-spin state. However, due to
the weak interaction between adsorbant and graphene surface the phenomena of
the spin alignment along the edges takes place locally, thus limiting its
application.
The connection of the phenomena of bandgap opening and of the spin ordering
with the sublattice symmetry lead us to conclude that breaking of this
symmetry is the main direction to achieve the required semiconductor-type
bandgap in graphene and a tunable spin ordering. Here we make a proposal that
the symmetry breaking can be done by chemical doping along a single zigzag
edge. This method is far superior to the earlier approaches involving edge
termination by functional groups because doping can be done for every unit
cell along the zigzag edge and thus preserve the planarity of graphene. Doping
not only breaks the graphene symmetry, but also can induce the spin gap
asymmetry due to the energetic shift of the molecular orbitals localized on
the doped edge. In a structure with broken symmetry, the HOMOα and LUMOβ
orbital states are localized at one edge, while HOMOβ and LUMOα are at the
other. Suppose the doping shifts the HOMOα and LUMOβ orbital states localized
at one edge down, then the HOMOα-LUMOα bandgap ($\Delta_{\alpha}$) is
increased, while $\Delta_{\beta}$, in contrast, will be reduced. If a certain
type of impurities can cause a significant shift of the bands, then the half-
metallicity of graphene may occur. This is what we set out to investigate
here. An important advantage of this approach is that we expect insensitivity
of spin selective behavior to the quality of the edges, when the band shift
induced by the impurities is stronger than the contribution from the edge
defects. We also investigate the possibility of obtaining an impurity level in
the middle of the graphene bandgap by doping (in analogy to semiconductors),
which has a lot of technological implications as well. Our study of nanoscale
graphene was based on the quantum chemistry methods using the spin-polarized
density functional theory with the semilocal gradient corrected functional
(UB3LYP/6-31G) performed in the Jaguar 7.5 program jaguar .
## II Symmetry of nanoscale graphene
Bulk graphene has hexagonal symmetry, while the highest possible symmetry of
nanoscale graphene would be the D2h planar symmetry with an inversion center.
The D2h symmetry results in structurally identical corners exhibiting
ferromagnetic ordering of the spin-polarized states localized at the corners,
as presented in Fig. 1 (a). According to the NBO (natural bond orbital)
analysis, the localized electrons at the corners are unpaired $sp$ electrons
belonging to non-bonded orbitals, which are located at the bottom of
conduction band or top of the valence band. For this symmetry, both $\alpha-$
and $\beta$-spin states of the HOMO and LUMO orbitals are localized on the
zigzag edges but their spin density is equally distributed between two edges.
For nanoscale graphene of D2h symmetry the HOMO-LUMO gap appears due to
confinement and edge effects cohen1 . The degeneracy of the $\alpha$\- and
$\beta$-spin states belonging to the HOMO and LUMO orbitals depends on the
edge configuration, i.e., $\alpha$\- and $\beta$ states can be non-degenerate
or degenerate depending on number of the carbon rings along the zigzag and
armchair edges my . The degeneracy reappears for large structures, such as
$n\geq 8$, $m\geq 7$ (see notation in Fig. 1 (a)). The increase of the
graphene size leads to disapperance of the confinement effect and as a result
closing of the gap. Thus, for $n=4$ and $m=5$ the gap is $\sim$0.5 eV and
already for $n=6$ and $m=7$ the gap is suppressed to $\sim$0.19 eV. The
influence of the confinement effect on the graphene gap has been already
confirmed experimentally kim .
Figure 1: (color online). The spin density distribution: (a) for nanoscale
graphene optimized with the D2h point-group symmetry, and (b) for the case
when one edge is doped by nitrogen, where the highest symmetry is the planar
symmetry. Different colors indicate the $\alpha$– (light) and $\beta$–spin
(dark) states. The spin density is plotted for isovalues of $\pm 0.01$ e/Å3.
The $n$ and $m$ are introduced to identify the the number of the carbon rings
along the zigzag edge ($n$) and along the armchair edge ($m$).
However, the state of D2h symmetry is not the ground state for nanoscale
graphene. Graphene, optimized with C2v symmetry, where the mirror plane of
symmetry is parallel to the armchair edges, has a total energy lower than that
for the D2h symmetry. For the C2v symmetry, the HOMO and LUMO orbitals are
characterized by the $\alpha-$ and $\beta$-spin states localized on the
opposite zigzag edges. Because the carbon atoms at the opposite zigzag edges
belong to different sublattices, such spin distribution breaks sublattice
symmetry and opens a gap ($\sim 1.63$ eV for $n=4$ and $m=5$). The size of the
gap is comparable to that found for nanoribbons harrison . The large gap of
nanoscale graphene obtained here is a result of significant contribution of
the confinement effect, as the nanoscale graphene is confined in all
directions.
The localization of the $\alpha-$ and $\beta$-spin states belonging to HOMO
and LUMO at the opposite zigzag edges is important for the application of
graphene in spintronics cohen ; rudb ; hod1 ; dutta . However, the C2v
symmetry state is a highly metastable state. Its total energy is comparable to
that of D2h symmetry with a difference of $\sim-0.5$ eV for small structures
such as $n=4$ and $m=5$, but the difference decreases exponentially down to
$\sim-0.02$ eV with increasing the structure size up to $n\geq 6$ and $m\geq
7$, that has good agreement with earlier work lee , and disappears when $n>8$
and $m>8$ . The competition of the C2v state with C1 symmetry, which is not
constrained to have spin ordering along the zigzag edge, is even more crucial
because of almost identical magnitude of their total energy. However, we found
that the distortion or dissimilarity induced along a single zigzag edge not
only breaks the sublattice symmetry of the graphene, but can control the spin
ordering, thereby stabilizing the ground state of the C2v symmetry. The
highest possible symmetry of the doped graphene is lowered from D2h to C2v
symmetry as a result of the edge dissimilarity. The spin density distribution
for the nanoscale graphene with one zigzag edge doped by nitrogen is presented
in Fig. 1 (b). The localized states along the zigzag edges are formed by
unpaired electrons belonging to the natural non-bonded orbitals, which
participate in formation of HOMO and LUMO orbitals. The $\alpha$– and
$\beta$–spin states of HOMO and LUMO orbitals are spatially separated, i.e.
localized at opposite zigzag edges. The (HOMO-1) and (LUMO+1) orbitals usually
correspond to the surface states, redistributed over the entire graphene
structure. The surface states are important for conductivity of graphene in a
transverse electric field, because the charge transfer between the spatially
separated HOMO and LUMO orbitals may occurs through participation of the
surface states. The electron density distribution for the edge states and the
surface states is presented in Fig. 2. The slight difference between the
$\alpha$\- and $\beta$-spin surface states (HOMO-1) is due to doping of the
left zigzag edge. The $\alpha$\- and $\beta$-spin states remain spatially
separated with increasing structure size.
Figure 2: (color online). Spin polarizations in nanoscale graphene where the
left edge is doped by nitrogen. Different colors correspond to different signs
of the molecular orbital lobes. The electron densities are plotted for
isovalues of $\pm 0.02$ e/Å3: (a) $\alpha$-state of HOMO ($E_{\rm HOMO}=-6.04$
eV) (b) $\beta$-state of HOMO ($E_{\rm HOMO}=-5.43$ eV) (c) $\alpha$-state of
(HOMO-1)($E_{\rm HOMO-1}=-6.38$ eV) (d) $\beta$-state of (HOMO-1) ($E_{\rm
HOMO-1}=-6.43$ eV). The HOMO and LUMO are found to be localized at the single
zigzag edges (edge states), while (HOMO-1) and (LUMO+1) – delocalized over the
entire graphene structure (surface states). Bottom pictures show the
representation of the localized and surface states.
## III Half-metallicity of graphene
The edge dissimilarity allows us to explore the required properties, such as
the semiconductor-type bandgap and localization of $\alpha$– and $\beta$–spin
states at opposite zigzag edges. Moreover, the spatial separation of the
$\alpha$– and the $\beta$–spin states resulted from doping of the single
zigzag edge is stable in comparison to the water adsorption my . Doping of a
single edge shifts the band energies of the orbitals which are strongly
localized at this edge. Such a shift provides an opportunity to obtain another
useful property which is important for spintronics – the half-metallicity of
graphene. For the HOMO or LUMO orbitals, which are shown to be localized at
the edges, doping can create a strong non-degeneracy of the $\alpha$– and
$\beta$–spin states, because these states are spatially separated and
localized at the opposite edges. Moreover, the HOMOα and LUMOβ orbitals are
localized at one edge, while HOMOβ and LUMOα at the other. If doping increases
the bandgap $\Delta_{\alpha}$ for the $\alpha$-spin state, then the bandgap
$\Delta_{\beta}$ for the $\beta$-spin state, in contrast will be reduced, and
vice versa. Therefore, doping induces the spin gap asymmetry in graphene.
Materials exhibiting asymmetric gaps for the $\alpha$\- and $\beta$-spin
states where one gap is of semiconductor type while the other is an insulator,
are known as half-semiconductor materials, but if one of them is metallic, the
system is half-metallic. Therefore, by choosing the right doping we can
achieve a stable half-metallicity in graphene which will be an important step
forward for applications in spintronics.
Figure 3: (color online). Schematic diagrams showing the distribution of the
edge states and surface states in the energy scale and over the graphene
structure (see the bottom pictures in Fig. 2 for pictorial description of the
states). The structures at the bottom demonstrate the spin distribution with
isovalues of $\pm 0.01$ e/Å3. (a) nanoscale graphene optimized with the D2h
symmetry, (b) with left edge terminated by hydrogen and (c) with the left
zigzag edge doped by nitrogen. For the localized states the energy levels
(HOMO and LUMO) show density distribution (schematic), particularly
delocalization of the orbitals between the two edges if the D2h symmetry is
preserved, and their localization on the zigzag edges when sublattice symmetry
is broken and C2v symmetry becomes to be highest possible symmetry. The
surface states ((HOMO-1) and (LUMO+1)) are delocalized over the entire
graphene structure (see for example Fig. 2 (c,d)).
We have investigated the transformation of the electronic structure of
nanoscale graphene due to the induced edge dissimilarities. The results are
schematically presented in Fig. 3. For nanoscale graphene with the D2h
symmetry, a small bandgap occurs due to the confinement effect. The HOMO and
LUMO orbitals in this case are localized at the zigzag edges, but their
electron density is equally redistributed over both edges (see Fig. 3 (a)).
Termination of the left zigzag edge by hydrogen (see Fig. 3 (b)) opens a gap
as a result of breaking of the sublattice symmetry, thereby lowering D2h
symmetry to a stable ground state of C2v symmetry. The hydrogenation leads to
saturation of the dangling $\sigma$ bonds at the terminated edge but does not
significantly change the energy of the HOMOα and LUMOβ states localized at
this edge. The resulting non-degeneracy of the $\alpha$\- and $\beta$-spin
states is not large, and the HOMO-LUMO gap of the $\alpha$-spin state
($\Delta_{\alpha}$ =1.8 eV) is almost identical to that of the $\beta$-spin
state ($\Delta_{\beta}$=2.1 eV). The doping of the left zigzag edge by
nitrogen (see Fig. 3 (c)) shifts down the orbital energies of the HOMOα and
LUMOβ states localized at the doped edge and results in a strong non-
degeneracy of the orbitals. This leads to a slight enhancement of the HOMO-
LUMO gap for the $\alpha$-spin state up to $\Delta_{\alpha}$=2.2 eV, but
significantly decreases the HOMO-LUMO gap for the $\beta$-spin state down to
$\Delta_{\beta}$ =0.8 eV. The length of the nitrogen-carbon bond at the edges
is found to be $d_{N-C}$=1.35 Å, which is similar to the carbon-carbon bonds
$d_{C-C}$=1.39 Å. Similar results are obtained for phosphorus impurities,
where the gaps are $\Delta_{\alpha}$=2.0 eV and $\Delta_{\beta}$=0.9 eV.
Phosphorus is, however, less useful because of the large phosphorus-carbon
bond ($d_{P-C}$=1.78 Å) which can lead to destruction of the lattice. We have
also investigated the possibility to dope the single zigzag edge of the
nanoscale graphene by other impurities, such as oxygen and boron, but they are
not as effective as nitrogen. The oxygen doping leads to strong delocalization
of the electron density of the orbitals localized at the edges. The doping by
boron leads to even more troubles due to the long boron-carbon bonds at the
edges ($d_{B-C}$=1.42 Å) and shifting of the states localized at the edges
from the HOMO-LUMO gap deeper into the conduction and valence bands.
For the nanoscale graphene structure investigated in this work, the spin
asymmetry is achieved but bandgap magnitude for $\alpha$\- and $\beta$-spin
states corresponds to the half-semiconductor behavior ($\Delta_{\alpha}$=2.2
eV,$\Delta_{\beta}$ =0.8 eV). Increasing the size of the graphene results in a
decrease of both the $\Delta_{\alpha}$ and $\Delta_{\beta}$ gaps due to the
diminishing of the confinement effect. Therefore, for graphene structures
doped by nitrogen or phosphorus of size $n\geq 6$ and $m\geq 7$, the
$\Delta_{\beta}$ gap is closer to metallic type. Thus, for $n=6$ and $m=7$ the
gap for $\alpha$-spin state is suppressed down to 1.13 eV while for
$\beta$-spin state down to 0.19 eV, which corresponds to the half-metallic
behavior of graphene. An external electric field applied between the zigzag
edges has been shown cohen ; rudb ; hod1 ; dutta to shift the band of
graphene with spatially separated and degenerated $\alpha$\- and $\beta$-spin
states. The electric field shifts the bands in such a way that for the
$\alpha$-spin the HOMO and LUMO levels move closer to each other in the energy
scale, while for $\beta$-spin they move apart. At a certain electric field
$\Delta_{\alpha}$ vanishes, thereby creating a metallic behavior of graphene.
If $+E_{c}$ electric field can close the bandgap for the $\alpha$-spin state,
the $-E_{c}$ leads to bandgap disappearance for the $\beta$-spin state.
Therefore, the current voltage characteristic of such a structure will be
symmetrical because the $\Delta_{\alpha}$ equals $\Delta_{\beta}$ and for both
spin states the switch from the semiconductor behavior to metallic occurs at
the same critical electric field $\pm E_{c}$. The advantage of graphene with
spin gap asymmetry, i.e. different $\Delta_{\alpha}$ and $\Delta_{\beta}$
gaps, found in this work is the different values of the critical electric
field required to close these gaps, such that $\mid E_{c(\beta)}\mid<\mid
E_{c(\alpha)}\mid$ when $\Delta_{\beta}<\Delta_{\alpha}$. Therefore, this
structure will be characterized by the spin-polarized current and by a non-
symmetric current-voltage characteristics as for a semiconductor diode, when
the current flow in one direction is preferable to the other.
## IV Doping of graphene
We have also investigated the influence of impurities on the electronic
structure of graphene in the case when they are not embedded at the zigzag
edges. Replacing carbon atoms by nitrogen atoms in a graphene lattice results
in the appearance of impurity levels inside of both the $\Delta_{\alpha}$ and
$\Delta_{\beta}$ gaps. The energy diagram of localization of the molecular
orbitals for the doped graphene is presented in Fig. 4. As we mentioned
earlier, in pure graphene the HOMO and LUMO orbitals are localized at the
zigzag edges. The applied doping creates one extra occupied orbital (HOMO)
which is localized at the embedded nitrogen atoms and located above the
occupied orbital belonging to edges, which becomes HOMO-1. The NBO analysis
has shown that this extra orbital is formed by the unpaired $sp$ electron
localized on each nitrogen atoms. This reduces the HOMO-LUMO gap
($\Delta_{\alpha}$=1.1 eV and $\Delta_{\beta}$=0.7 eV) while preserving the
spin asymmetry.
In an applied in-plane electric field the charge transfer occurs between the
orbitals localized on the opposite zigzag edges, i.e., in our case such
transfer occurs between the HOMO-1 and LUMO orbitals, which is a multi-step
process with participation of HOMO. Because the gap is decreased and each
nitrogen atom adds an unpaired electron into the system due to the doping, the
conductivity of graphene would be significantly enhanced, and can be
controlled as it is done in semiconductor devices.
Figure 4: (color online). Schematic diagram showing the distribution of the
edge states (LUMO, HOMO-1) and states localized by the dopant in the middle of
the graphene structure (HOMO) (see the bottom pictures in Fig. 2 for pictorial
description of the states). The HOMOα and HOMOβ are extra impurity levels that
appear due to the doping and replaces the occupied orbital localized on the
left carbon edge by shifting it dipper into the valence band. The inset
picture (in brackets) demonstrates the electron density distribution for the
HOMOα with isovalues of $\pm 0.01$ e/Å3.
## V Conclusion
We have investigated the possibility to control the electronic and magnetic
properties of nanoscale graphene. We found that if pure graphene can be
characterized by a small bandgap and no spin ordering at the zigzag edges the
dissimilarity of the edges induced by doping impurities lowers the highest
possible symmetry to C2v, which is characterized by the spin ordering along
the zigzag edges and their antiparallel alignment between opposite zigzag
edges. Moreover, impurities embedded at a single zigzag edge shifts in the
energy scale the molecular orbitals localized at this edge, thereby decreasing
the bandgap for one spin channel and increasing the other. Under these
conditions, the half-metallic behavior can be achieved. Nitrogen doping in the
middle of the graphene surface is found to have the prospect for application
in nanoelectronics due to the appearance of the occupied impurity levels in
the bandgap. The impurity level results in a decrease of the bandgap of $\sim$
2.0 eV by one half and contains unpaired electrons, which should lead to an
enhancement of the conductivity. Therefore, both the conductivity of the
nanoscale graphene and its magnetic properties can be controlled by the
impurities.
## VI Acknowledgments
The work was supported by the Canada Research Chairs Program and the NSERC
Discovery Grant.
## References
* (1) A.K. Geim and K.S. Novoselov, Nature Materials 6, 183 (2007).
* (2) Y. Zhang, Y.-W. Tan, H.L. Stormer, and P. Kim, Nature438, 201 (2005).
* (3) V.M. Apalkov and T. Chakraborty, Phys. Rev. Lett. 97, 126801 (2006).
* (4) X.-F. Wang and T. Chakraborty, Phys. Rev. B 75, 041404 (2007).
* (5) D.S.L. Abergel and T. Chakraborty, Phys. Rev. Lett. 102, 056807 (2009).
* (6) Z. Chen, Y.-M. Lin, M.J. Rooks, and P. Avouris, Physica E 40, 228 (2007).
* (7) E.J.H. Lee, K. Balasubramanian, R. T. Weitz and M. Burghard, Nature Nanotechnology 3, 486 (2008).
* (8) Y.-W. Son, M. L. Cohen and S.G. Louie, Nature 444, 347 (2006).
* (9) E. Rudberg, P. Salek, and Y. Luo, Nano Letters 7, 2211 (2007).
* (10) P. Esquinazi, D. Spemann, R. Høhne, A. Setzer, K.-H. Han and T. Butz, Phys. Rev. Lett. 91, 227201 (2003).
* (11) S. Cho, Y. Chen, and M.S. Fuhrer, Appl. Phys. Lett. 91, 123105 (2007).
* (12) V.M. Kapran, G. Giovannetti, P.A. Khomyakov, M. Talanana, A.A. Starikov, M.Zwierzycki, J. van der Brink, G. Brocks and P.J. Kelly, Phys. Rev. Lett. 99, 176602 (2007).
* (13) S.Y. Zhou, D.A. Siegel, A.V. Fedorov and A. Lanzara, Phys. Rev. Lett. 101, 086402 (2008).
* (14) M.Y. Han, B. Özyilmaz, Y. Zhang and P. Kim, Phys. Rev. Lett 98, 206805 (2007).
* (15) Y.-W. Son, M. L. Cohen and S.G. Louie, Phys. Rev. Lett. 97, 216803 (2006).
* (16) K. Nakada, M. Fujita, G. Dresselhaus and M.S. Dresselhaus, Phys. Rev. B 54, 17954 (1996).
* (17) H. Lee, Y.-W. Son, N. Park, S. Han and J. Yu, Phys. Rev. B 72, 174431(2005).
* (18) L. Pisani, B. Montanari and N.M. Harrison,New Journal of Physics 10, 033002 (2008).
* (19) D. V. Kosynkin, A. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev, B. K. Price and J. Tour, Nature 458, 872 (2009); L. Jiao, L. Zhang, X. Wang, G. Diankov and H. Dai, Nature 458, 877 (2009).
* (20) L. Pisani, J.A. Chan, B. Montanari and N.M. Harrison, Phys. Rev. B 75, 064418 (2007).
* (21) F. Cervantes-Sodi, G. Csányi, S. Piscanec, and A.C. Ferrari, Phys. Rev. B 77, 165427 (2008).
* (22) J. Berashevich and T. Chakraborty, arXiv:0901.4956.
* (23) D. Gunlycke, J.Li, J.W. Mintmire and C.T. White, Appl. Phys. Lett.91, 112108 (2007).
* (24) D.W. Boukhvalov and M.I. Katsnelson, Nano Letters 8, 4373 (2008).
* (25) Z. Li, J. Yang and J.G. Hou, J. Am. Chem. Soc.130, 4224 (2008).
* (26) O. Hod, V. Barone, J.E. Peralta and G.E. Scuseria, Nano Letters 7, 2295 (2007).
* (27) E.-J. Kan, X. Wu, Z. Li, X.C. Zeng, J. Yang and J.G. Hou, J. Chem. Phys. 129, 084712 (2008).
* (28) Jaguar, version 7.5. Schrödinger 2007. LLC: New York, NY.
* (29) O. Hod, V. Barone and G. E. Scuseria, Phys. Rev. B 77, 035411 (2008).
* (30) S. Dutta and S. K. Pati,J. Phys. Chem. B 112, 1333 (2008).
|
arxiv-papers
| 2009-06-16T18:39:37 |
2024-09-04T02:49:03.383666
|
{
"license": "Public Domain",
"authors": "Julia Berashevich and Tapash Chakraborty",
"submitter": "Julia Berashevich",
"url": "https://arxiv.org/abs/0906.2993"
}
|
0906.3062
|
# Infinite-Dimensional Hamiltonian Description of Dissipative Mechanical
Systems
Tianshu Luo [email protected] Institute of Applied Mechanics, Department
of Applied Mechanics, Zhejiang University,
Hangzhou, Zhejiang, 310027, P.R.China
Yimu Guo [email protected] Institute of Applied Mechanics, Department of
Applied Mechanics, Zhejiang University,
Hangzhou, Zhejiang, 310027, P.R.China
###### Abstract
In this paper an approach is proposed to define an infinite-dimensional
Hamiltonian formalism to represent dissipative mechanical systems. This
approach is based upon below viewpoints: for any non-conservative classical
mechanical system and any initial condition, there exists a conservative one;
the two systems share one and only one common phase curve; the value of the
Hamiltonian of the conservative system is equal to the sum of the total energy
of the non-conservative system on the aforementioned phase curve and a
constant depending on the initial condition. We called the conservative system
as the substituting conservative system. The infinite-dimensional Hamilton’s
description of the ideal fluid in Lagrangian and Poisson-Vlasov equation
motivate us to consider a dissipative mechanical system as a special fluid in
a domain $D$ of the phase space, viz. a collection of particles in the domain.
By comparing the description of the ideal fluid in Lagrangian coordinates, the
Hamiltonian and the Lagrangian can be thought of as the integrals of the
Hamiltonian and the Lagrangian of the substituting conservative system over
the initial value space and a new Poisson bracket is defined to represent the
Hamilton’s equation. The advantage of the approach is: the value of the
canonical momentum density $\pi$ is identical with that of the mechanical
momentum and the value of canonical coordinate $q$ is identical with that of
the coordinate of the dissipative mechanical system.
Hamiltonian formalism, dissipation, non-conservative system, damping
###### pacs:
45.20.Jj
††preprint: AIP/123-QED
## I Introduction
Since Hamilton originated Hamilton equations of motion and Hamiltonian
formalism, it has been stated in most classical textbooks that the Hamiltonian
formalism focuses on solving conservative problems.
In 1960s, Hori and Brouwer (1961) utilized the classical Hamiltonian formalism
and a perturbation theory to solve a non-conservative problem. They did not
attempt to derive the Hamiltonian formalism for non-conservative problems.
Several authors have attempted to enlarge the scope of Hamiltonian formalism
to dissipative problems. Some significant works in this area were reported by
Vujanovic (1970, 1978) and Djukic (1973); Djukic and Vujanovic (1975); Djukic
(1975). They have proposed a technique for systems with gauge variant
Lagrangian. A.Mukherjee and A.Dasgupta (2006) considered that the technique is
rather algebraic in nature. To overcome the limitations, A.Mukherjee (1994)
proposed a modified equation with an introduction of an additional time like
variable called ’umbra time’ and extending this notion to the co-kinetic
kinetic, potential, complimentary energies as well as Lagrangian itself.
Amalendu Mukherjee (1997) introduced a procedure for getting umbra-Lagrangian
through system bond graphs and extended the basic idea of Karnopp (1977).
Mukherjee (2001) consolidated this idea and presented an important idea of
invariants of motion. A gauge variant Lagrangian implies a new definition of
canonical momentum, which might not be identical with mechanical momentum.
Some other literature of Jerrold E. Marsden (1994) and Morrison (2006, 1998),
Salmon (1988) in the geometrical mechanics field focused on the conservative
system or some special dissipative systems, e.g. an oscillator with gyroscopic
damping. Morrison (1998) had written so: ’the ideal fluid description is one
in which viscosity or other phenomenological terms are neglected. Thus, as is
the case for systems governed by Newton’s second law without dissipation, such
fluid descriptions posses Lagrangian and Hamiltonian descriptions.’
Krechetnikov and Marsden (2007) and other researchers applied the equations as
below to the problem of stability of dissipative system,
$\displaystyle\dot{p}_{i}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial q_{i}}+\bm{F}\left(\frac{\partial{r}}{\partial q_{i}}\right)$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial p_{i}},$ (1)
where $\\{q,p\\}$ denote the coordinate and momentum, and the position vector
$r$ depends on the canonical variable $\\{q,p\\}$, i.e. $r(q,p)$, $H$ denotes
Hamiltonian, $\bm{F}(\partial{r}/\partial{q_{i}})$ denotes a generalized force
in direction $i$. Marsden considered that Eqs. (1) was composed of a
conservative part and a non-conservative part. Eq. (1) apparently is not a
Hamilton’s equation but only a representation of dissipative mechanical
systems in the phase space.
In this paper an $n$-dimensional dissipative mechanical system as the
following is considered:
$\ddot{\bm{q}}+\mathsfsl{c}\dot{\bm{q}}+\mathsfsl{k}\bm{q}=0,$ (2)
where $\mathsfsl{c}$ denotes the damping coefficient matrix, $\mathsfsl{k}$
denotes the stiffness coefficient matrix. In light of the proposition proposed
by Luo and Guo (2010) an attempt is made to represent the dissipative
mechanical system (2) as an infinite-dimensional Hamilton’s equation. This
proposition asserts that for any non-conservative classical mechanical system
and any initial condition, there exists a conservative one; the two systems
share one and only one common phase curve; the Hamiltonian of the conservative
system is the sum of the total energy of the non-conservative system on the
aforementioned phase curve and a constant depending on the initial condition.
In sec. II the demonstration of the proposition is first reported. Analogous
to Hamiltonian description of ideal fluid in Lagrangian variables and that of
Poisson-Vlasov equations, we attempt to define Lagrangian and Hamiltonian as
an integral over the entire initial value space. The generalized coordinates
and the canonical momentum will be thought of as the function of the initial
value and time. A new Poisson bracket will be defined to represent Eq. (2) as
an infinite-dimensional Hamilton’s Equation. This process will be in detail
presented in Sec. III.
## II Corresponding Conservative Mechanical Systems
### II.1 Common Phase Flow Curve
First we represented Eq. (2) as Eq. (1).Under general circumstances, the force
$\bm{F}$ is a damping force that depends on the variable set
$q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n}$. We denote by $F_{i}$ the
components of the generalized force $\bm{F}$.
$F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})=\bm{F}\left(\frac{\partial{r}}{\partial
q_{i}}\right).$ (3)
Thus we can reformulate the Eq. (2) as follows:
$\displaystyle\dot{p}_{i}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial q_{i}}+F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial p_{i}}.$ (4)
Suppose the Hamiltonian quantity of a conservative system without damping is
$\hat{H}$. Thus we may write a Hamilton’s equation of the conservative system:
$\displaystyle\dot{p}_{i}$ $\displaystyle=$
$\displaystyle-\frac{\partial{\hat{H}}}{\partial q_{i}}$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$
$\displaystyle\frac{\partial\hat{H}}{\partial p_{i}}.$ (5)
We do not intend to change the definition of momentum in classical mechanics,
but we do require that a special solution of Eq. (5) is the same as that of
Eq. (4). We may therefore assume a phase curve $\gamma$ of Eq. (4) coincides
with that of Eq. (5). The phase curve $\gamma$ corresponds to an initial
condition $q_{i0},p_{i0}$. Consequently by comparing Eq. (4) and Eq. (5), we
have
$\displaystyle\left.\frac{\partial{\hat{H}}}{\partial{q_{i}}}\right|_{\gamma}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial H}{\partial
q_{i}}\right|_{\gamma}-\left.F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\right|_{\gamma}$
$\displaystyle\left.\frac{\partial{\hat{H}}}{\partial{p_{i}}}\right|_{\gamma}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial H}{\partial
p_{i}}\right|_{\gamma},$ (6)
where
$\left.\frac{\partial{\hat{H}}}{\partial{q_{i}}}\right|_{\gamma},\left.\frac{\partial
H}{\partial
q_{i}}\right|_{\gamma},\left.\frac{\partial{\hat{H}}}{\partial{p_{i}}}\right|_{\gamma}$
and $\left.\frac{\partial H}{\partial p_{i}}\right|_{\gamma}$ denote the
values of these partial derivatives on the phase curve $\gamma$ and
$\left.F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\right|_{\gamma}$
denotes the value of the force $F_{i}$ on the phase curve $\gamma$. In
classical mechanics the Hamiltonian $H$ of a conservative mechanical system is
mechanical energy and can be written as:
$H=\int_{\gamma}\left(\frac{\partial{H}}{\partial{q_{i}}}\right)\mathrm{d}q_{i}+\int_{\gamma}\left(\frac{\partial
H}{\partial p_{i}}\right)\mathrm{d}p_{i}+const_{1},$ (7)
where $const_{1}$ is a constant that depends on the initial condition
described above. The mechanical energy $H$ of the system (4) can be evaluated
via Eq. (7) too. If $q_{i}=0,p_{i}=0$, then $const_{1}=0$. The Einstein
summation convention has been used this section. Thus an attempt has been made
to find $\left.\hat{H}\right|_{\gamma}$ through line integral along the phase
curve $\gamma$ of the dissipative system
$\displaystyle\int_{\gamma}\left(\frac{\partial{\hat{H}}}{\partial{q_{i}}}\right)\mathrm{d}q_{i}$
$\displaystyle=$ $\displaystyle\int_{\gamma}\left[\left(\frac{\partial
H}{\partial
q_{i}}\right)-F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\right]\mathrm{d}q_{i}$
$\displaystyle\int_{\gamma}\left(\frac{\partial\hat{H}}{\partial
p_{i}}\right)\mathrm{d}p_{i}$ $\displaystyle=$
$\displaystyle\int_{\gamma}\left(\frac{\partial H}{\partial
p_{i}}\right)\mathrm{d}p_{i}.$ (8)
Analogous to Eqs.(7), we have
$\hat{H}=\int_{\gamma}\left(\frac{\partial{\hat{H}}}{\partial{\hat{q}_{i}}}\right)_{\hat{q}\hat{p}}\mathrm{d}\hat{q}_{i}+\int_{\gamma}\left(\frac{\partial{\hat{H}}}{\partial\hat{p}_{i}}\right)_{\hat{q}\hat{p}}\mathrm{d}\hat{p}_{i}+const_{2},$
(9)
where $const_{2}$ is a constant which depends on the initial condition.
Substituting Eqs.(7)and Eqs.(8) into Eq. (9), we have
$\left.\hat{H}\right|_{\gamma}=H-\int_{\gamma}F_{i}(q_{1},\cdots,q_{n},\dot{q}_{1},\cdots,\dot{q}_{n})\mathrm{d}q_{i}+const.$
(10)
where $const=const_{2}-const_{1}$, and $H=\left.H\right|_{\gamma}$ because $H$
is mechanical energy of the non-conservative system (4). According to the
physical meaning of Hamiltonian, $const_{1}$, $const_{2}$ and $const$ are
added into Eq. (7)(9)(10) respectively such that the integral constant
vanishes in the Hamiltonian quantity. Arnold. (1997) had presented the Newton-
Laplace principle of determinacy as, ’This principle asserts that the state of
a mechanical system at any fixed moment of time uniquely determines all of its
(future and past) motion.’ In other words, in the phase space the position
variable and the velocity variable are determined only by the time $t$.
Therefore, we can assume that we have already a solution of Eq. (4)
$\displaystyle q_{i}$ $\displaystyle=$ $\displaystyle q_{i}(t)$
$\displaystyle\dot{q_{i}}$ $\displaystyle=$ $\displaystyle\dot{q_{i}}(t),$
(11)
where the solution satisfies the initial condition. We can divide the whole
time domain into a group of sufficiently small domains and in these domains
$q_{i}$ is monotone, and hence we can assume an inverse function $t=t(q_{i})$.
If $t=t(q_{i})$ is substituted into the non-conservative force
$\left.F_{i}\right|_{\gamma}$, we can assume that:
$\left.F_{i}(q_{1}(t(q_{i})),\cdots,q_{n}(t(q_{i})),\dot{q}_{1}(t(q_{i})),\cdots,\dot{q}_{n}(t(q_{i})))\right|_{\gamma}=\mathcal{F}_{i}(q_{i}),$
(12)
where $\mathcal{F}_{i}$ is a function of $q_{i}$ alone. In Eq. (12) the
function $F_{i}$ is restricted on the curve $\gamma$, such that a new function
$\mathcal{F}_{i}(q_{i})$ yields. Thus we have
$\displaystyle\int_{\gamma}F_{i}\mathrm{d}q_{i}$ $\displaystyle=$
$\displaystyle\int_{q_{i0}}^{q_{i}}\mathcal{F}_{i}(q_{i})\mathrm{d}q_{i}=W_{i}(q_{i})-W_{i}(q_{i0}).$
(13)
According to Eq. (13) the function $\mathcal{F}_{i}$ is path independent, and
therefore $\mathcal{F}_{i}$ can be regarded as a conservative force. For that
Eq. (12) represents an identity map from the non-conservative force $F$ on the
curve $\gamma$ to the conservative force $\mathcal{F}_{i}$ which is distinct
from $F_{i}$. Eq. (12) is tenable only on the phase curve $\gamma$.
Consequently the function form of $\mathcal{F}_{i}$ depends on the
aforementioned initial condition; from other initial conditions
$\mathcal{F}_{i}$ with different function forms will yield.
According to the physical meaning of Hamiltonian, $const$ is added to Eq. (10)
such that the integral constant vanishes in Hamiltonian quantity. Hence
$const=-W_{i}(q_{i0})$. Substituting Eq. (13) and $const=-W_{i}(q_{i0})$ into
Eq. (10), we have
$\left.\hat{H}\right|_{\gamma}=H-W_{i}(q_{i})$ (14)
where $-W_{i}(q_{i})$ denotes the potential of the conservative force
$\mathcal{F}_{i}$ and $W_{i}(q_{i})$ is equal to the sum of the work done by
the non-conservative force $F$ and $const$. In Eq. (14) $\hat{H}$ and $H$ are
both functions of $q_{i}$ and $W_{i}(q_{i})$ a function of $q_{i}$. Eq. (14)
and Eq. (10) can be thought of as a map from the total energy of the
dissipative system (4) to the Hamiltonian of the conservative system (5).
Indeed, $\left.\hat{H}\right|_{\gamma}$ and the total energy differ in the
constant $const=-W_{i}(q_{i0})$. When the conservative system takes a
different initial condition, if one does not change the function form of
$\left.\hat{H}\right|_{\gamma}$, one can consider
$\left.\hat{H}\right|_{\gamma}$ as a Hamiltonian quantity $\hat{H}$,
$\hat{H}=\left.\hat{H}\right|_{\gamma}=H-W_{i}(q_{i})$ (15)
and the conservative system (5) can be thought of as an entirely new
conservative system.
Based on the above, the following proposition is made:
###### Proposition II.1.
For any non-conservative classical mechanical system and any initial
condition, there exists a conservative one; the two systems share one and only
one common phase curve; the value of the Hamiltonian of the conservative
system is equal to the sum of the total energy of the non-conservative system
on the aforementioned phase curve and a constant depending on the initial
condition.
###### Proof.
First we must prove the first part of the Proposition II.1, i.e. that a
conservative system with Hamiltonian presented by Eq. (15) shares a common
phase curve with the non-conservative system represented by Eq. (4). In other
words the Hamiltonian quantity presented by Eq. (15) satisfies Eq. (6) under
the same initial condition. Substituting Eq. (15) into the left side of Eq.
(6), we have
$\displaystyle\frac{\partial{\hat{H}(q_{i},p_{i})}}{\partial{q_{i}}}$
$\displaystyle=$ $\displaystyle\frac{\partial
H(q_{i},p_{i})}{\partial{q_{i}}}-\frac{\partial{W_{j}(q_{j})}}{\partial{q_{i}}}$
$\displaystyle\frac{\partial{\hat{H}(q_{i},p_{i})}}{\partial{p_{i}}}$
$\displaystyle=$ $\displaystyle\frac{\partial
H(q_{i},p_{i})}{\partial{p_{i}}}-\frac{\partial{W_{j}(q_{j})}}{\partial{p_{i}}}.$
(16)
It must be noted that although $q_{i}$ and $p_{i}$ are considered as distinct
variables in Hamilton’s mechanics, we can consider $q_{i}$ and $\dot{q_{i}}$
as dependent variables in the process of constructing of $\hat{H}$. At the
trajectory $\gamma$ we have
$\displaystyle\frac{\partial{{W_{j}(q_{j})}}}{\partial{q_{i}}}$
$\displaystyle=$
$\displaystyle\frac{\partial{(\int_{q_{j0}}^{q_{j}}\mathcal{F}_{j}(q_{j})\mathrm{d}q_{j}+W_{i}(q_{i0}))}}{\partial{q_{i}}}=\mathcal{F}_{i}(q_{i})$
$\displaystyle\frac{\partial{{W_{j}(q_{j})}}}{\partial{p_{i}}}$
$\displaystyle=0,$ (17)
where $\mathcal{F}_{i}(q_{i})$ is equal to the damping force $F_{i}$ on the
phase curve $\gamma$. Hence under the initial condition $q_{0},p_{0}$, Eq. (6)
is satisfied. As a result, we can state that the phase curve of Eq. (5)
coincides with that of Eq. (4) under the initial condition; and $\hat{H}$
represented by Eq. (15) is the Hamiltonian of the conservative system
represented by Eq. (5).
Then we must prove the second part of Proposition II.1: the uniqueness of the
common phase curve.
We assume that Eq. (5) shares two common phase curves, $\gamma_{1}$ and
$\gamma_{2}$, with Eq. (4). Let a point of $\gamma_{1}$ at the time $t$ be
$z_{1}$, a point of $\gamma_{2}$ at the time $t$ $z_{2}$, and $g^{t}$ the
Hamiltonian phase flow of Eq. (5). Suppose a domain $\Omega$ at $t$ which
contains only points $z_{1}$ and $z_{2}$, and $\Omega$ is not only a subset of
the phase space of the non-conservative system (4) but also that of the phase
space of the conservative system (5). Hence there exists a phase flow
$\hat{g}^{t}$ composed of $\gamma_{1}$ and $\gamma_{2}$, and $\hat{g}^{t}$ is
the phase flow of Eq. (4) restricted by $\Omega$. According to the following
Louisville’s theorem in the book of Arnold. (1978):
###### Theorem II.1.
The phase flow of Hamilton’s equations preserves volume: for any region $D$ in
the phase space we have
$volume\ of\ g^{t}D=volume\ of\ D$
where $g^{t}$ is the one-parameter group of transformations of phase space
$g^{t}:(p(0),q(0))\longmapsto:(p(t),q(t))$
$g^{t}$ preserves the volume of $\Omega$. This implies that the phase flow of
Eq. (4) $\hat{g}^{t}$ preserves the volume of $\Omega$ too. But the system (4)
is not conservative, which conflicts with Louisville’s theorem; hence only a
phase curve of Eq. (5) coincides with that of Eq. (4).
∎∎
### II.2 Obtaining the Equivalent Stiffness Matrix $\tilde{K}$
According to Proprostion II.1, an attempt is made to find a new conservative
mechanical system which is corresponding to the dissipative system (2) and an
initial condition. Under the initial condition, the dissipative system (2)
posses a phase curve $\gamma$. As in Eq. (12) we can consider that the damping
forces are equal to some conservative force on the phase curve $\gamma$
$\begin{array}[]{ccc}c_{11}\dot{q}_{1}=\varrho_{11}(q_{1})&\dots&c_{1n}\dot{q}_{n}=\varrho_{1n}(q_{1})\\\
\vdots&\ddots&\vdots\\\
c_{n1}\dot{q}_{1}=\varrho_{21}(q_{n})&\dots&c_{nn}\dot{q}_{n}=\varrho_{nn}(q_{n}).\end{array}$
(18)
For convenience, these conservative forces can be thought of as elastic
restoring forces:
$\begin{array}[]{ccc}\varrho_{11}(q_{1})=\kappa_{11}(q_{1})q_{1}&\dots&\varrho_{1n}(q_{1})=\kappa_{1n}(q_{1})q_{1}\\\
\vdots&\ddots&\vdots\\\
\varrho_{n1}(q_{1})=\kappa_{n1}(q_{n})q_{n}&\dots&\varrho_{nn}(q_{n})=\kappa_{nn}(q_{n})q_{n}.\end{array}$
(19)
An equivalent stiffness matrix $\mathsfsl{\tilde{K}}$ is obtained, which is a
diagonal matrix
$\mathsfsl{\tilde{K}}_{ii}=\sum_{l=1}^{n}\kappa_{il}(q_{l}).$ (20)
Consequently an $n$-dimensional conservative system is obtained
$\bm{\ddot{q}}+(\mathsfsl{K}+\mathsfsl{\tilde{K}})\bm{q}=0$ (21)
which shares the common phase curve $\gamma$ with the $n$-dimensional damping
system (2). In this paper, the conservative system is named as substituting
conservative system. The Lagrangian of Eqs.(21) is
$\hat{L}=\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}-\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}-\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(22)
and the Hamiltonian of Eqs.(21) is
$\hat{H}=\frac{1}{2}\bm{p}^{T}\bm{p}+\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}+\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(23)
where $\bm{0}$ is a zero vector, $\bm{p}=\dot{\bm{q}}$. $\hat{H}$ in Eq. (23)
is the mechanical energy of the conservative system (21), because
$\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}$ is a
potential function such that $\hat{H}$ doest not depend on any path of Eq.
(22).
## III Definition of a Generalized Hamilton’s Equation
In this section Proposition II.1 would be represented as a uniform infinite-
dimensional Hamilton’s equation. IN infinite-dimensional Hamiltonian
formalism, techniques of functional derivative must be devoted. Morrison
(1998) introduced the definition the functional derivative simply. We would
report the introduction.
### III.1 Introduction of Functional Derivative and Canonical Hamiltonian
Description of the Ideal Fluid in Lagrangian variables
Consider a functional $K[u]$. The first change in $K$ induced by $\delta u$ is
called the first variation, $\delta K$, and is given by
$\displaystyle\delta K[u;\delta u]$ $\displaystyle:=$
$\displaystyle\lim_{\varepsilon\rightarrow 0}\frac{K[u+\varepsilon
u]-K[u]}{\varepsilon}$ (24) $\displaystyle=$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\left.K[u+\varepsilon
u]\right|_{\varepsilon=0}$ $\displaystyle=:$
$\displaystyle\int_{x_{0}}^{x_{1}}\delta u\frac{\delta K}{\delta
u(x)}=:\langle\frac{\delta K}{\delta u},\delta u\rangle$
The quantity $\delta K/\delta u(x)$ of Eq. (24) is the functional derivative
of the functional $K$. Consider a now a more general functional, one of the
form
$\hat{F}[u]=\int_{x_{0}}^{x_{1}}\mathcal{\hat{F}}(x,u,u_{x},u_{xx},\dots)\mathrm{d}x$
(25)
where $\mathcal{\hat{F}}$ is an ordinary, sufficiently differentiable,
function of its arguments. Note $u_{x}=\mathrm{d}u/\mathrm{d}x$, etc. This
first variation of Eq. (25) yields
$\delta\hat{F}[u,\delta
u]=\int_{x_{0}}^{x_{1}}\left(\frac{\partial\mathcal{\hat{F}}}{\partial
u}\delta u+\frac{\partial\mathcal{\hat{F}}}{\partial u_{x}}\delta
u_{x}+\frac{\partial\mathcal{\hat{F}}}{\partial u_{xx}}\delta
u_{xx}+\cdots\right)\mathrm{d}x,$ (26)
which upon integration by parts becomes
$\displaystyle\delta\hat{F}[u,\delta u]$ $\displaystyle=$
$\displaystyle\int_{x_{0}}^{x_{1}}\delta
u\left(\frac{\partial\mathcal{\hat{F}}}{\partial
u}-\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial\mathcal{\hat{F}}}{\partial
u_{x}}+\frac{\mathrm{d}^{2}}{\mathrm{d}^{2}x}\frac{\partial\mathcal{\hat{F}}}{\partial
u_{xx}}\delta u_{xx}-\cdots\right)\mathrm{d}x$ (27)
$\displaystyle+\left.\left(\frac{\partial\mathcal{\hat{F}}}{\partial
u_{x}}\delta u+\cdots\right)\right|_{x_{0}}^{x_{1}}.$
Usually the variations $\delta u$ are chosen so that the last term, the
boundary term, vanishes; e.g., $\delta u(x_{0})=\delta u(x_{1})=0,\ \ \delta
u_{x}(x_{0})=\delta u_{x}(x_{1})=0$, etc. Sometimes the boundary term vanishes
without a condition on $\delta u$ because of the form of $\mathcal{\hat{F}}$.
When this happens the boundary conditions are called natural. Assuming, for
one reason or the other, that the boundary term vanishes, Eq. (27) becomes
$\delta\hat{F}[u;\delta u]=\langle\frac{\delta\hat{F}}{\delta u},\delta
u\rangle,$ (28)
where the functional derivative
$\frac{\delta F}{\delta u}=\frac{\partial\mathcal{\hat{F}}}{\partial
u}-\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial\mathcal{\hat{F}}}{\partial
u_{x}}+\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\frac{\partial\mathcal{\hat{F}}}{\partial
u_{xx}}-\dots.$ (29)
The main objective of the calculus of variations is the extremization of
functionals. A common terminology is to call a function $\hat{u}$, which is a
point in the domain, an extremal point if $\delta\hat{F}[u]/\delta
u|_{u=\hat{u}}=0$. It could be a maxi- mum, a minimum, or an inflection point.
If the extremal point $\hat{u}$ is a minimum or maximum, then such a point is
called an extremum.
An example is the functional defined by evaluating the function $u$ at the
point $x$ . This can be written as
$u(x^{\prime})=\int_{x_{0}}^{x_{1}}\delta(x-x^{\prime})u(x)\mathrm{d}x,$ (30)
where $\delta(x-x^{\prime})$ is the Dirac delta function and where we have
departed from the $[]$ notion. Applying the definition of Eq. (24) yields
$\frac{\delta u(x^{\prime})}{\delta u(x)}=\delta(x-x^{\prime}).$ (31)
This is the infinite-dimensional or continuum analog of $\partial
x_{i}/\partial x_{j}=\delta_{ij}$, where $\delta_{ij}$ is is the Kronecker
delta function. Eq. (30) shows why it is sometimes useful to display the
argument of the function in the functional derivative.
The generalizations of the above ideas to functionals of more than one
function and to more than a single spatial variable are straightforward. An
example is given by the kinetic energy of a three-dimensional compress- ideal
fluid,
$T(\rho,\bm{v})=\int_{D}\frac{1}{2}\rho\bm{v}^{2}\mathrm{d}^{3}\bm{x}$ (32)
where the velocity has three rectangular components
$\bm{v}=\\{v_{1},v_{2},v_{3}\\}$ that depend upon
$\bm{x}=\\{x_{1},x_{2},x_{3}\\}\in D$ and
$\bm{v}^{2}=\bm{x}\cdot\bm{x}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$. The functional
derivatives are
$\frac{\delta T}{\delta v_{i}}=\rho v_{i},\ \ \frac{\delta
T}{\delta\rho}=\frac{\bm{v}^{2}}{2}.$ (33)
For a more general functional $\hat{F}[\bm{\psi}]$, where
$\bm{\psi}(\bm{x})=(\psi_{1},\psi_{2},\cdots,\psi_{n})$ and
$\bm{x}=(x_{1},x_{2},\cdots,x_{n})$, the analog of Eq. (24) is
$\delta\hat{F}[\bm{\psi};\delta\bm{\psi}]=\int_{D}\delta\psi_{i}\frac{\delta\hat{F}}{\delta\psi_{i}(\bm{x})}\mathrm{d}^{n}\bm{x}=:\langle\frac{\delta\hat{F}}{\delta\bm{\psi}},\delta\bm{\psi}\rangle.$
(34)
Salmon (1988) and Morrison (1998) described the Hamiltonian formalism of ideal
fluid in Lagrangian variables in detail. In order to state the infinite-
dimensional formalism for dissipative mechanical system, we repeat the
representation of Salmon (1988) and Morrison (1998).
In the the Hamiltonian description, a fluid is described as a collection of
fluid particles or elements. Suppose the coordinate of a fluid particle at
time $t$
$\bm{q}=\bm{q}(\bm{a},t),$ (35)
where $\bm{q}=\\{q_{1},q_{2},q_{3}\\}$,$\bm{a}=\\{a_{1},a_{2},a_{3}\\}$ is the
coordinate of the particle at the initial time $t=t_{0}$. We assume that
$\bm{a}$ varies over a fixed domain $D$, which is completely filled with
fluid, and that the functions $q$ map $D$ onto itself.
In Lagrangian variables $\bm{a}$ the Lagrangian quantity of the fluid particle
is considered as Lagrangian density
$\mathcal{L}_{f}(\bm{q},\dot{\bm{q}},\partial\bm{q}/\partial\bm{a},t)=\frac{1}{2}\rho_{0}\dot{\bm{q}}^{2}-\rho_{0}E(s_{0},\rho_{0}/\mathsfsl{\mathcal{J}})-\phi,$
(36)
where $\rho_{0}=\rho_{0}(\bm{a})$ is a given initial density distribution,
$\dot{\bm{q}}$ is the velocity of the fluid particle, a shorthand
$\dot{\bm{q}}^{2}=\delta_{ij}q_{i}q_{j}$ is used, $E$ is the energy per unit
mass, $s_{0}$ is the entropy per unit mass at the time $t_{0}$,
$\mathsfsl{\mathcal{J}}=\det(\partial{q}^{i}/\partial{a}^{j})$, $\phi$ is a
potential function for external conservative forces. The intensive quantities,
pressure and temperature, are obtained as follows:
$T=\frac{\partial U}{\partial s}(s,\rho),\ \ p=\rho^{2}\frac{\partial
U}{\partial\rho}(s,\rho)$ (37)
Therefore, we have the Lagrangian functional of the fluid particles of the
domain $D$:
$L_{f}[\bm{q},\dot{\bm{q}}]=\int_{D}\mathcal{L}_{f}\mathrm{d}^{3}\bm{a}=\int_{D}\left[\frac{1}{2}\rho_{0}\dot{\bm{q}}^{2}-\rho_{0}E(s_{0},\rho_{0}/\mathsfsl{\mathcal{J}})-\phi\right]\mathrm{d}^{3}\bm{a},$
(38)
where $\mathrm{d}^{3}\bm{a}=\mathrm{d}a_{1}\mathrm{d}a_{2}\mathrm{d}a_{3}$.
Thus the action functional is given by
$S_{f}[\bm{q}]=\int_{t_{0}}^{t^{1}}\mathrm{d}t\int_{D}L_{f}[\bm{q},\dot{\bm{q}}]\mathrm{d}^{3}\bm{a}=\int_{t_{0}}^{t^{1}}\mathrm{d}t\int_{D}\left[\frac{1}{2}\rho_{0}\dot{\bm{q}}^{2}-\rho_{0}E-\phi\right]\mathrm{d}^{3}\bm{a}$
(39)
Observe that this action functional is like that for finite-degree-of-freedom
systems, as treated above, except that the sum over particles is replaced by
integration over $D$, i.e.,
$\int_{D}\mathrm{d}^{3}\bm{a}\leftrightarrow\sum_{i}$ (40)
By a Legendre transform, we have a canonical momentum density
$\bm{\varpi}_{i}(\bm{a},t)=\frac{\delta
L_{f}}{\delta\dot{\bm{q}}_{i}(\bm{a},t)}=\rho_{0}\dot{\bm{q}}_{i},$ (41)
and a generalized Hamiltonian quantity
$H_{f}[\bm{q},\bm{\varpi}]=\int_{D}\left[\bm{\varpi}\dot{\bm{q}}-\mathcal{L}_{f}\right]\mathrm{d}^{3}{\bm{a}}=\int_{D}\left[\frac{\dot{\bm{\varpi}^{2}}}{2\rho_{0}}+E+\phi\right]\mathrm{d}^{3}{\bm{a}},$
(42)
where $\rho_{0}\dot{\bm{q}}^{2}/2+E+\phi=\mathcal{H}_{f}$ can be consider as a
Hamiltonian density. A generalized Hamilton’s equation is
$\dot{\varpi}_{i}=-\frac{\delta H_{f}}{\delta q_{i}},\ \ \ \
\dot{q}_{i}=\frac{\delta H_{f}}{\delta\bm{\varpi}_{i}}.$ (43)
These equations can also be written in terms of the Poisson bracket (see
Morrison (1998)),
$\\{F,G\\}=\int_{D}\left[\frac{\delta F}{\delta\bm{q}}\cdot\frac{\delta
G}{\delta\bm{\varpi}}-\frac{\delta G}{\delta q}\cdot\frac{\delta
F}{\delta\bm{\varpi}}\right]\mathrm{d}^{3}\bm{a}$ (44)
viz.,
$\dot{\bm{\varpi}}_{i}=\\{\bm{\varpi}_{i},H_{f}\\},\ \ \ \
\dot{\bm{q}}_{i}=\\{\bm{q}_{i},H_{f}\\}$ (45)
Here $\delta q_{i}(\bm{a}/)\delta
q_{j}(\bm{a}^{\prime})=\delta_{ij}\delta(\bm{a}-\bm{a}^{\prime})$ has been
used, where $\delta(\bm{a}-\bm{a}^{\prime})$ is a three-dimensional Dirac
delta function(recall Eq. (31).
### III.2 Derivation of Hamiltonian Description of Dissipative Mechanical
Systems
The Hamiltonian description of the ideal fluid is infinite-dimensional, and
the Hamiltonian quantity and Lagranian is the integrals over the domain $D$ in
the initial configuration space. In addition, Morrison (1980) proposed the
Hamiltonian description of Poisson-Vlasov equations with Hamiltonian quantity,
which is an integral over the phase space. These ideas of Salmon (1988),
Morrison (1998) and Morrison (1980) motivate us to consider the mechanical
system (2) as a special fluid which is a collection of fluid particles in the
phase space. In general case Hamilton’s quantity is an energy function.
Although the total energy of the oscillator with damping is conservative, the
total energy depends on the initial condition. Consequently there is a path-
dependency problem. It is well known that the energy per unit mass $E$ is the
origin of the pressure in the fluid. The mechanical system (2) describes that
a particle moves in the configuration space. One can also consider that
individual particles of the special fluid moves without interaction.
Therefore, one can assume that no internal energy function $E$. exists in the
Lagrangian density of the system (2); the Lagrangian variable of the special
fluid in a fixed domain $D$ is
$\bm{a}=(\bm{q}_{0},\dot{\bm{q}}_{0})=(q_{0}^{1},\dots,q_{0}^{n},\dot{q}_{0}^{1},\dots,\dot{q}_{0}^{n})$
(46)
; the coordinate of a particle in the configuration space is
$\bm{q}=\bm{q}(\bm{a},t)=(q_{1}(\bm{a},t),\dots,q_{n}(\bm{a},t);$ (47)
$\rho_{o}=1$. By comparing the generalized Hamilton’s equation(43) and
Hamilton’s equation in odd dimensional phase space, one can find that the
Hamiltonian density does not need to satisfy the path independency requirement
fully, according to to Eq. (29) we have
$\displaystyle\dot{\varpi}_{i}(\bm{a})=-\frac{\delta H_{f}}{\delta
q_{i}(\bm{a})}=-\frac{\partial\mathcal{H}_{f}}{\partial q_{i}(\bm{a})},$
where $q_{i}(\bm{a})$ is the value of $q_{i}$ on the path of the particle
$\bm{a}$ in the configuration space. Therefore, analogous to Eq. (36), one can
consider $\hat{L}$ in Eq. (22) as a Lagrangian density of the system (2)
$\mathcal{L}=\hat{L}=\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}-\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}-\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(48)
and consider $\hat{H}$ in Eq. (23) as a Hamiltonian density of the system (2)
$\mathcal{H}=\hat{H}=\frac{1}{2}\bm{p}^{T}\bm{p}+\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}+\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q},$
(49)
where $q_{i}(\bm{a})$ is the value of $q_{i}$ on the path of the particle
$\bm{a}$ in the phase space, suth that one can avoid the afore-mentioned path-
dependency problem. Thus the Lagrangian functional of Eq. (2) can be presented
as following:
$L[q,\dot{q}]=\int_{D}\mathcal{L}\mathrm{d}^{2n}\bm{a}=\int_{D}\left[\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}-\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}-\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}\right]\mathrm{d}^{2n}\bm{a},$
(50)
where
$\mathrm{d}^{2n}=\mathrm{d}^{n}\bm{q}_{0}\mathrm{d}^{n}\dot{\bm{q}}_{0}=\mathrm{d}q^{1}_{0}\dots\mathrm{d}q^{n}_{0}\mathrm{d}\dot{q}^{1}_{0}\dots\mathrm{d}\dot{q}^{n}_{0}$.
The Lagrangian functional Thus the action functional can be presented as
following:
$S[q]=\int^{t1}_{t0}L[q,\dot{q}]\mathrm{d}t=\int^{t1}_{t0}\mathrm{d}t\int_{D}\left[\frac{1}{2}\dot{\bm{q}}^{T}\dot{\bm{q}}-\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}-\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}\right]\mathrm{d}^{2n}\bm{a}$
(51)
According to Hamiltonian theorem, we have the functional derivative $\delta
S/\delta\bm{q}(a,t)=0$, according to the generalization Eq. (29):
$\displaystyle\frac{\delta S}{\delta\bm{q}(\bm{a},t)}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\bm{q}(\bm{a},t)}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\mathcal{L}}{\partial\dot{\bm{q}}(\bm{a},t)}$
(52) $\displaystyle=$
$\displaystyle-\ddot{\bm{q}}(\bm{a},t)-\mathsfsl{k}\bm{q}-\tilde{\mathsfsl{K}}\bm{q}=0$
The equation above implies that under the initial condition $\bm{a}$ a
conservative system exists, the control equation of which is Eq. (21), the
phase curve of which coincides with that of the oscillator with damping.
Define a canonical momentum density for the dissipative system (2) is
$\pi_{i}(\bm{a},t)=\frac{\delta
L}{\delta\dot{q}_{i}(\bm{a})}=\dot{\bm{q}_{i}},$ (53)
which is a functional derivative, while classical canonical momentum is
defined as a partial derivative. By a Legendre transform, we have the
generalized Hamiltonian $\hat{K}$ is
$\hat{K}[\bm{\pi},\bm{q}]=\int_{D}\mathrm{d}^{2n}\bm{a}\left[\bm{\pi}\cdot\dot{\bm{q}}-\mathcal{L}\right]=\int_{D}\mathrm{d}^{2n}\bm{a}\left[\frac{1}{2}\bm{p}^{T}\bm{p}+\frac{1}{2}\bm{q}^{T}\mathsfsl{K}\bm{q}+\int_{\bm{0}}^{\bm{q}}(\tilde{\mathsfsl{K}}\bm{q})^{T}\mathrm{d}\bm{q}\right],$
(54)
where $\bm{q}=\bm{q}(\bm{a},t)$. Thus the generalized Hamilton’s equations of
the dissipative system (2) are
$\dot{\pi}_{i}=-\frac{\delta\hat{K}}{\delta q_{i}},\ \
\dot{q}_{i}=\frac{\delta\hat{K}}{\delta\pi_{i}}.$ (55)
###### Definition III.1.
For two functionals $F[\bm{\pi}(\bm{a}),\bm{q}(a)]$ and
$G[\bm{\pi}(a),\bm{q}(a)]$ in a domain $D$ of the phase space exists a
functional
$\\{F,G\\}[\bm{\pi}(\bm{a}),\bm{q}(\bm{a})]=\int_{D}\left[\frac{\delta
F}{\delta\bm{q}(\bm{a}^{\prime})}\cdot\frac{\delta
G}{\delta\bm{\pi}(\bm{a}^{\prime})}-\frac{\delta
G}{\delta\bm{q}(\bm{a}^{\prime})}\cdot\frac{\delta
F}{\delta\bm{\pi}(\bm{a}^{\prime})}\right]\mathrm{d}^{2n}\bm{a},$ (56)
where the functional derivative $\delta F/\delta\bm{q}(\bm{a}^{\prime})$ is
defined analogues to Eq. (24) and Eq. (34) as:
$\left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\right|_{\varepsilon=0}[\bm{q}(\bm{a}^{\prime})+\varepsilon\delta\bm{q}(\bm{a}^{\prime}),\bm{\pi}(\bm{a}^{\prime})]=\int_{D}\frac{\delta
F}{\delta\bm{q}(\bm{a}^{\prime})}\mathrm{d}^{2}n\bm{a}$
The Hamilton’s equations can also be represented in terms of the Poisson
bracket (56) viz.,
$\dot{\pi}_{i}=\\{\pi_{i},\hat{K}\\},\dot{q}_{i}=\\{q_{i},\hat{K}\\}.$ (57)
Expand $\\{\pi_{i},\hat{K}\\}$, we have
$\displaystyle\\{\pi_{i}(\bm{a}),\hat{K}\\}$ $\displaystyle=$
$\displaystyle\frac{\delta\pi_{i}(\bm{a})}{\delta
q_{j}(\bm{a}^{\prime})}\frac{\delta\hat{K}}{\delta\pi_{j}(\bm{a}^{\prime})}-\frac{\delta\pi_{i}(\bm{a})}{\delta\pi_{j}(\bm{a}^{\prime})}\frac{\delta\hat{K}}{\delta
q_{j}(\bm{a}^{\prime})}$ (58) $\displaystyle=$
$\displaystyle-\delta_{ij}\delta(\bm{a}-\bm{a}^{\prime})\frac{\delta\hat{K}}{\delta
q_{j}(\bm{a}^{\prime})}$ $\displaystyle=$
$\displaystyle-\frac{\delta\hat{K}}{\delta q_{i}(\bm{a})},$
Here $\delta q_{i}(\bm{a}/)\delta
q_{j}(\bm{a}^{\prime})=\delta_{ij}\delta(\bm{a}-\bm{a}^{\prime})$ has been
used, where $\delta(\bm{a}-\bm{a}^{\prime})$ is a three-dimensional Dirac
delta function(recall Eq. (31). Analogous to Eq. (40), we have
$\int_{D}\mathrm{d}^{2n}\bm{a}\leftrightarrow\sum_{i},\ \
\hat{K}=\sum_{i}\mathcal{H}(\bm{a})$ (59)
According to Eq. (59), from Eq. (58) we can derive
$\dot{\pi}_{i}(\bm{a})=\\{\pi_{i}(\bm{a}),\hat{K}\\}=-\frac{\delta\hat{K}}{\delta
q_{i}(\bm{a})}=-\frac{\partial\mathcal{H}(\bm{a})}{\partial q_{i}(\bm{a})}$
(60)
In the similiar way
$\dot{q}_{i}(\bm{a})=\\{q_{i}(\bm{a}),\hat{K}\\}=\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}=\frac{\partial\mathcal{H}(\bm{a})}{\partial\pi_{i}(\bm{a})}$
(61)
Therefore, we can assert that Eq. (60) and Eq. (61) describes a phase curve
which is a common phase curve of the dissipative system and a conservative
system under the initial condition $\bm{a}$.
From the Hamilton’s equation(55), we can derive the total energy conservative
principle
$\displaystyle\delta\hat{K}$ $\displaystyle=$
$\displaystyle\int_{D}\left[\frac{\delta\hat{K}}{\delta q_{i}(\bm{a})}\delta
q_{i}(\bm{a})+\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}\delta\pi_{i}(\bm{a})\right]\mathrm{d}^{2n}\bm{a}$
$\displaystyle=$ $\displaystyle\int_{D}\left[\frac{\delta\hat{K}}{\delta
q_{i}(\bm{a})}\frac{\mathrm{d}q_{i}(\bm{a})}{\mathrm{d}t}\mathrm{d}t+\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}\frac{\mathrm{d}\pi_{i}(\bm{a})}{\mathrm{d}t}\mathrm{d}t\right]\mathrm{d}^{2n}\bm{a}$
$\displaystyle=$ $\displaystyle\int_{D}\left[\frac{\delta\hat{K}}{\delta
q_{i}(\bm{a})}\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}\mathrm{d}t-\frac{\delta\hat{K}}{\delta\pi_{i}(\bm{a})}\frac{\delta\hat{K}}{\delta
q_{i}(\bm{a})}\mathrm{d}t\right]\mathrm{d}^{2n}\bm{a}$ $\displaystyle=$
$\displaystyle 0$
## IV Conclusion
The following conclusions can be drawn. The infinite-dimensional
description(53),(54), (55),(56,(57) can describe a dissipative mechanical
system based on the propositon II.1: For any non-conservative classical
mechanical system and any initial condition, there exists a conservative one;
the two systems share one and only one common phase curve; the value of the
Hamiltonian of the conservative system is equal to the sum of the total energy
of the non-conservative system on the aforementioned phase curve and a
constant depending on the initial condition. In fact, if the generalized
Hamilton’s equation (55) and (57) is constrained at a initial condition
$\bm{a}$, the generalized Hamilton’s equation is a phase curve of the afore-
mentioned conservative system (21). As the classical Hamilton’s equation
represents the conservation of mechanical energy principle, the generalized
Hamilton’s equation(55,57) describes the conservation of total energy
principle. One can assert that the generalized Hamilton’s equation(55,57) are
the generalization of the classic Hamilton’s equations.
## References
* Amalendu Mukherjee (1997) Amalendu Mukherjee, Arun Kumar Samantaray, “Umbra lagrange’s equations through bondgraphs,” in _Simulation Series_ , Vol. 29 (International Conference on Bond Graph Modeling and Simulation, Phoenix, 1997) pp. 168–174.
* A.Mukherjee (1994) A.Mukherjee, “Junction structures of bond graph theory from analytical viewpoint,” in _Proc of CISS-1st_ (Conference of International Simulation Societies, Zuerich, Switzerland, 1994) pp. 661–666.
* A.Mukherjee and A.Dasgupta (2006) A.Mukherjee, V.Rastogi and A.Dasgupta, “A procedure for finding invariants of motions for general class of unsymmetric systems with gauge-variant Umbra-Lagrangian generated by bond graphs,” SIMULATION, Transactions of the Society for Modeling and Simulation International 82, 207–226 (2006).
* Arnold. (1978) Arnold., V. I., _Mathematical Methods of classical Mechanics, second edition_ (Springer-Verlag, Berlin, 1978).
* Arnold. (1997) Arnold., V. I., _Mathematical aspects of classical and celestial mechanics_ (Springer-Verlag, Berlin, 1997).
* Djukic (1973) Djukic, D., “A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians,” International Journal of Non Linear Mechanics 8, 479–488 (1973).
* Djukic (1975) Djukic, Dj., “Integral invariants in classical nonconservative mechanics,” Acta Mechanica 23, 291–296 (1975), http://dx.doi.org/10.1007/BF01174025.
* Djukic and Vujanovic (1975) Djukic, Dj. and Vujanovic, B. D., “Noether’s theory in classical nonconservative mechanics,” Acta Mechanica 23, 17–27 (1975).
* Hori and Brouwer (1961) Hori, G. and Brouwer, D., “Theoretical evaluation of atmospheric drag effects in the motion of an artificial satellite,” Astron. J 66, 193–225 (1961).
* Jerrold E. Marsden (1994) Jerrold E. Marsden, Tudor S. Ratiu, _Introduction to Mechanics and Symmetry_ (Springer-Verlag, New York, 1994).
* Karnopp (1977) Karnopp, D.C., “Lagrange’s equations for complex bond graph systems,” in _Trans. ASME_ (Journal of the Dynamic Systems, Measurement, and Control, 1977) pp. 300–306.
* Krechetnikov and Marsden (2007) Krechetnikov, R. and Marsden, J. E., “Dissipation-induced instabilities in finite dimensions,” Reviews of Modern Physics 79, 519–553 (2007).
* Luo and Guo (2010) Luo, T. and Guo, Y., “An Examination of the Time-Centered Difference Scheme for Dissipative Mechanical Systems from a Hamiltonian Perspective,” ArXiv e-prints(2010), arXiv:1007.2709 [math-ph].
* Morrison (1980) Morrison, P. J., “The maxwell-vlasov equations as a continuous hamiltonian system..” Phys. Lett. A 80, 383–386 (1980).
* Morrison (1998) Morrison, P. J., “Hamiltonian description of the ideal fluid,” Rev. Mod. Phys. 70, 467–521 (1998).
* Morrison (2006) Morrison, P.J., “Hamiltonian fluid dynamics,” in _Encyclopedia of Mathematical Physics_, edited by Jean-Pierre Fran oise, Gregory L. Naber, , and Tsou Sheung Tsun (Academic Press, Oxford, 2006) pp. 593–600, ISBN 978-0-12-512666-3, http://www.sciencedirect.com/science/article/B7T7D-4KF807K-7Y/2/a5086cc%c96edae361e0a4c005562290a.
* Mukherjee (2001) Mukherjee, A., “The issue of invariants of motion for general class of symmetric systems through bond graphs and umbra-langrangian,” in _Simulation Series_ , Vol. 33 (International Conference on Bond Graph Modeling ICBGM’01, Phoenix, Arizona, 2001) pp. 295–304.
* Salmon (1988) Salmon, Rick, “Hamiltonian fluid mechanics,” Ann. Rev. Fluid Mechanics 20, 225–256 (1988).
* Vujanovic (1970) Vujanovic, B., “A group-variational procedure for finding first integrals of dynamical systems,” International Journal of Non Linear Mechanics 5, 269–278 (1970).
* Vujanovic (1978) Vujanovic, B., “Conservation laws of dynamical systems via d’alembert’s principle,” International Journal of Non Linear Mechanics 13, 185–197 (1978).
|
arxiv-papers
| 2009-06-17T02:36:27 |
2024-09-04T02:49:03.389941
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tianshu Luo, Yimu Guo",
"submitter": "Tianshu Luo",
"url": "https://arxiv.org/abs/0906.3062"
}
|
0906.3303
|
# On Zero Controllability of Evolution Equations
B. Shklyar
###### Abstract
The exact controllability to the origin for linear evolution control equation
is considered.The problem is investigated by its transformation to infinite
linear moment problem.
Conditions for the existence of solution for infinite linear moment problem
has been obtained. The obtained results are applied to the zero
controllability for control evolution equations.
## Introduction
Let $X$ be a separable complex Hilbert space.
Given sequences $\left\\{c_{n},n=1,2\ldots,\ \right\\}$ and $\left\\{x_{n}\in
X,n=1,2\ldots,\ \right\\}$ find necessary and sufficient conditions for the
existence of an element $g\in X$ such that
$c_{n}=\left(x_{n},g\right),n=1,2\ldots,\ .$
The problem formulated above is called the linear moment problem. It has a
long history and many applications in geometry, physics, mechanics.
The goal of this paper is to establish necessary and sufficient conditions of
exact null-controllability for linear evolution control equations with
unbounded input operator by transformation of exact null-controllability
problem (controllability to the origin) to linear infinite moment problem.
It is well-known, that if the sequence
$\left\\{x_{n},n=1,2,...,\right\\}$forms a Riesz basic in the closure of its
linear span, the linear moment problem has a solution if and only if
$\sum_{n=1}^{\infty}\left|c_{n}\right|^{2}<\infty$ and vice-versa [3], [7],
[16], [17]. This well-known fact is one of main tools for the controllability
analysis of various partial hyperbolic control equations and functional
differential control systems of neutral type.
However the sequence $\left\\{x_{n},n=1,2,...,\right\\}$ doesn’t need to be a
Riesz basic for the solvability of linear moment problem. This case appears
under the investigation of the controllability of parabolic control equations
or hereditary functional differential control systems. In this paper we
consider the zero controllability of control evolution equations for the case
when the sequence $\left\\{x_{n},n=1,2,...,\right\\}$ of the moment problem
obtained by the transformation of the source control problem doesn’t form a
Riesz basic in its closed linear span.
## 1 Problem statement
Let $X,U$ be complex Hilbert spaces, and let $\,A$ be infinitesimal generator
of strongly continuous $C_{0}$-semigroups $S\left(t\right)$ in $X$ [8],[10].
Consider the abstract evolution control equation [8], [10]
$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right),x\left(0\right)=x_{0},\;\,0\leq
t<+\infty,$ (1.1)
where $x\left(t\right),\;x_{0}\in X,u\left(t\right),u_{0}\in
U,\;B:U\rightarrow X$ is a linear possibly unbounded operator, $W\subset
X\subset V$ are Hilbert spaces with continuous dense injections, where
$W=D\left(A\right)~{}$equipped with graphic norm,$~{}V=W^{\ast}$, the operator
$B$ is a bounded operator from $U$ to $V$ (see more details in [14], [4],
[11], [15]).
It is well-known that [4], [11],[14], [15]), etc. :
$\bullet$ for each $t\geq 0$ the operator $S\left(t\right)$ has an unique
continuous extension $\mathcal{S}\left(t\right)$ on the space $V$ and the
family of operators $\mathcal{S}\left(t\right):V\rightarrow V$ is the
semigroup in the class $C_{0}$ with respect to the norm of $V$ and the
corresponding infinitesimal generator $\mathcal{A}$ of the semigroup
$\mathcal{S}\left(t\right)$ is the closed dense extension of the operator $A$
on the space $V$ with domain $D\left(\mathcal{A}\right)=X$;
$\bullet$ the sets of eigenvalues and of generalized eigenvectors of operators
$\mathcal{A},\mathcal{A}^{\ast}$ and $A,\,A^{\ast}$ are the same;
$\bullet$ for each $\mu\notin\sigma\left(A\right)\,$ the resolvent operator
$R_{A}\left(\mu\right)\,$ has a unique continuous extension to the resolvent
operator $\mathcal{R}_{A}\left(\mu\right):V\rightarrow X$;
$\bullet$ a mild solution $x\left(t,x_{0},u\left(\cdot\right)\right)$ of
equation (1.1) with initial condition $x\left(0\right)=x_{0}$ is obtained by
the following representation formula
$x(t,x_{0},u(\cdot))=S(t)x_{0}+\int\limits_{0}^{t}\mathcal{S}(t-\tau)Bu(\tau)d\tau,$
(1.2)
where the integral in (2.3) is understood in the Bochner’s sense [8]. To
assure $x(t,x_{0},u(\cdot))\in X,~{}\forall x_{0}\in X,u(\cdot)\in
L_{2}^{\mathrm{loc}}\left[0,+\infty\right),t\geq 0,$ we assume that
$\int\limits_{0}^{t}\mathcal{S}(t-\tau)Bu(\tau)d\tau\in X$ for any
$u(\cdot)\in L_{2}^{\mathrm{loc}}\left[0,+\infty\right),t\geq 0$ [14], [15].
###### Definition 1.1
Equation (1.1) is said to be exact null-controllable on $\left[0,t_{1}\right]$
by controls vanishing after time moment $t_{2}$ if for each $x_{0}\in X$ there
exists a control $u\left(\cdot\right)\in
L_{2}\left(\left[0,t_{2}\right],U\right),u\left(t\right)=0$ a.e. on
$[t_{2},+\infty)$ such that
$x\left(t_{1},x_{0},u\left(\cdot\right)\right)=0.$ (1.3)
### 1.1 The assumptions
The assumptions on $A$ are listed below.
1. 1.
The operators $A$ has purely point spectrum $\sigma$ with no finite limit
points. Eigenvalues of $A$ have finite multiplicities.
2. 2.
There exists $T\geq 0$ such that all mild solutions of the equation
$\dot{x}\left(t\right)=Ax\left(t\right)$ are expanded in a series of
generalized eigenvectors of the operator $A$ converging uniformly for any
$t\in\left[T_{1},T_{2}\right],T<T_{1}<T_{2}.$
## 2 Main results
### 2.1 One input case
For the sake of simplicity we consider the following:
1. 1.
The operator $A$ has all the eigenvalues with multiplicity $1$.
2. 2.
$U=\mathbb{R}$ (one input case). It means that the possibly unbounded operator
$B:U\rightarrow\mathbb{R}$ is defined by an element $b\in V$, i.e. equation
(1.1) can be written in the form
$\dot{x}\left(t\right)=Ax\left(t\right)+bu\left(t\right),x\left(0\right)=x_{0},b\in
V,\;\,0\leq t<+\infty.$ (2.1)
The operator defined by $b\in V$ is bounded if and only if $b\in X.$
Let the eigenvalues $\lambda_{j}\in\sigma,j=1,2,\ldots$ of the operator $A$ be
enumerated in the order of non-decreasing of their absolute values, and let
$\varphi_{j},\psi_{j},j=1,2,\ldots,$be eigenvectors of the operator $A$ and
the adjoint operator $A^{\ast}$ respectively. It is well-known, that
$(\varphi_{{}_{k},}\psi_{j})=\delta_{kj},\ j,k=1,2\ldots,\ $ (2.2)
where $\delta_{kj},\ j,k=1,2\ldots$is the Kroneker delta.
Denote:
$x_{j}\left(t\right)=\left(x\left(t,x_{0},u\left(\cdot\right)\right),\psi_{j}\right),~{}x_{0j}=\left(x_{0},\psi_{j}\right),~{}b_{j}=\left(b,\psi_{j}\right),~{}j=1,2,....$
(2.3)
All scalar products in (2.3) are correctly defined, because $\psi_{j}\in W,$
$b\in V=W^{\ast}.$
###### Theorem 2.1
For equation (1.1) to be exact null-controllable on $\left[0,t_{1}\right],$
$t_{1}>T,$ by controls vanishing after time moment $t_{1}-T$, it is necessary
and sufficient that the following infinite moment problem
$x_{0j}=-\int_{0}^{t_{1}-T}e^{-\lambda_{j}\tau}b_{j}u\left(\tau\right)d\tau,~{}j=1,2,...$
(2.4)
with respect to $u\left(\cdot\right)\in L_{2}\left[0,t_{1}-T\right]$ is
solvable for any $x_{0}\in X$ .
Proof. Necessity. Multiplying (1.1) by $\psi_{j},$ $j=1,2,...,$and using (2.3)
we obtain
$\displaystyle\dot{x}_{j}\left(t\right)$ $\displaystyle=$
$\displaystyle\left(Ax\left(t\right),\psi_{j}\right)+b_{j}u\left(t\right)=\left(x\left(t\right),A^{\ast}\psi_{j}\right)+b_{j}u\left(t\right)=$
(2.5) $\displaystyle=$
$\displaystyle\lambda_{j}x_{j}\left(t\right)+b_{j}u\left(t\right),j=1,2,...,.$
Here $x_{j}\left(t\right),\dot{x}_{j}\left(t\right)\ $and $b_{j},j=1,2,...,$
are well-defined because $\psi_{j}\in W,\
\dot{x}\left(t\right),Ax\left(t\right),b\in V=W^{\ast}.$
From (2.5) it follows that
$x_{j}\left(t\right)=e^{\lambda_{j}t}\left(x_{j0}+\int_{0}^{t}e^{-\lambda_{j}t}b_{j}u\left(\tau\right)d\tau\right),j=1,2,...,.$
(2.6)
In accordance with the definition of exact null-controllability there exists
$u\left(\cdot\right)\in
L_{2}\left(\left[0,t_{1}-T\right],U\right),u\left(t\right)=0$ a.e. on
$[t_{1}-T,+\infty)$ such that (1.3) holds. Using $u\left(t\right)$ and $t_{1}$
in (2.6), we obtain by (1.3) and (2.5), that
$x_{j}\left(t_{1}\right)=e^{\lambda_{j}t_{1}}\left(x_{j0}+\int_{0}^{t_{1}-T}e^{-\lambda_{j}t}b_{j}u\left(\tau\right)d\tau\right)=0,j=1,2,...,.$
(2.7)
Hence we have (2.4) to be true. This proves the necessity.
Sufficiency. Let the control $u\left(\cdot\right)\in
L_{2}\left(\left[0,t_{1}-T\right],U\right),u\left(t\right)=0$ a.e. on
$[t_{1}-T,+\infty)$ satisfies (2.4). It follows from (2.4) and (2.7) that
$x_{j}\left(t_{1}-T\right)=\left(x\left(t_{1}-T\right),\psi_{j}\right)=0,j=1,2,....$
(2.8)
Denote $z\left(t\right)=x\left(t+t_{1}-T\right),~{}t\geq T.$Obviously,
$z\left(t\right)$ is a mild solution of the equation
$\dot{z}\left(t\right)=Az\left(t\right)$ with initial condition
$z\left(0\right)=x\left(t_{1}-T\right)~{}.$By assumption 3 (see the list of
assumptions) $z\left(t\right)$ is expanded in a series
$z\left(t\right)=\sum_{j=1}^{\infty}e^{\lambda_{j}t}\left(x\left(t_{1}-T\right),\psi_{j}\right),t\geq
T,$ (2.9)
so by (2.8) and (2.9) we obtain
$z\left(t\right)=x\left(t+t_{1}-T,x_{0},u\left(\cdot\right)\right)\equiv
0,t\geq T\Leftrightarrow x\left(t,x_{0},u\left(\cdot\right)\right)\equiv
0,t\geq t_{1}.$
This proves the sufficiency.
### 2.2 Solution of moment problem (2.4)
The solvability of moment problem (2.4) for each $x_{0}\in X\ $essentially
depends on the properties of eigenvalues $\lambda_{j},$ $~{}j=1,2,...,.$
If the sequence of exponents
$\left\\{e^{-\lambda_{n}t}b_{n},n=1,2,...,\right\\}$forms a Riesz basic in
$L_{2}\left[0,t_{1}-T\right],$then the moment problem
$c_{j}=-\int_{0}^{t_{1}-T}e^{-\lambda_{j}\tau}b_{j}u\left(\tau\right)d\tau,~{}j=1,2,...$
(2.10)
is solvable if and only if
$\sum_{j=1}^{\infty}\left|c_{j}\right|^{2}<\infty$ (2.11)
There are very large number of papers and books devoted to conditions for
sequence of exponents to be a Riesz basic. All these conditions can be used
for sufficient conditions of zero controllability of equation (1.1). They are
very useful for the investigation of the zero controllability of hyperbolic
partial control equations and functional differential control systems of
neutral type [13].
However moment problem (2.10) may also be solvable when the sequence
$\left\\{e^{-\lambda_{n}t}b_{n},n=1,2,...,\right\\}$ doesn’t form a Riesz
basic in $L_{2}\left[0,t_{1}-T\right].$ Below we will try to find more
extended controllability conditions which are applicable for the case when the
sequence $\left\\{e^{-\lambda_{n}t}b_{n},n=1,2,...,\right\\}$ doesn’t form a
Riesz basic in $L_{2}\left[0,t_{1}-T\right].$
###### Definition 2.1
The sequence $\left\\{x_{j}\in X,j=1,2,...,\right\\}$ is said to be minimal,
if there no element of the sequence belonging to the closure of the linear
span of others. By other words,
$x_{j}\notin\overline{\mathrm{span}}\left\\{x_{k}\in X,k=1,2,...,k\neq
j\right\\}.$
The investigation of the controllability problem defined above is based on the
following result of Boas [2] (see also [3] and [18]).
Theorem Let $x_{j}\in X,j=1,2,...,.$ The linear moment problem
$c_{j}=\left(x_{j},g\right),j=1,2,...$
has a solution $g\in X$ for each square summable sequence
$\left\\{c_{j},j=1,2,...\right\\}$ if and only if there exists a positive
constant $\gamma$ such that all the inequalities
$\gamma\sum_{k=1}^{n}\left|c_{k}\right|^{2}\leq\left\|\sum_{j=1}^{n}c_{j}x_{j}\right\|^{2},n=1,2,...,.$
(2.12)
are valid.
Let $\left\\{x_{j}\in X,j=1,2,...,\right\\}$ a sequence of elements of $X$ ,
and let
$G_{n}=\left\\{\left(x\,_{i},x_{j}\right),i,j=1,2,...,n\right\\}$
be the Gram matrix of $n$ first elements $\left\\{x_{1},...,x_{n}\right\\}$ of
above sequence. Denote by $\gamma_{n}^{\min}$ the minimal eigenvalue of the
$n\times n$-matrix $G_{n}.$Each minimal sequence $\left\\{x_{j}\in
X,j=1,2,...,\right\\}$ is linear independent, hence any first $n~{}$elements
$\left\\{x_{1},...,x_{n}\right\\},$ $n=1,2,...,$ of this sequence are linear
independent, so $\gamma_{n}^{\min}>0,$ $\forall n=1,2,...,.~{}~{}$It is easily
to show that the sequence $\left\\{\gamma_{n}^{\min},n=1,2,...,\right\\}$
decreases , so there exists
$\lim\limits_{n\rightarrow\infty}\gamma_{n}^{\min}\geq 0.$
###### Definition 2.2
The sequence $\left\\{x_{j}\in X,j=1,2,...,\right\\}$ is said to be strongly
minimal, if
$\gamma^{\mathrm{\min}}=\lim\limits_{n\rightarrow\infty}\gamma_{n}^{\min}>0.$
It is well-known that for Hermitian $n\times n$-matrix
$G_{n}=\left\\{\left(x_{j},x_{k}\right),~{}j,k=1,2,...,n\right\\}$
$\gamma_{n}^{\mathrm{\min}}\sum_{k=1}^{n}\left|c_{k}\right|^{2}\leq\sum_{j=1}^{n}\sum_{k=1}^{n}c_{j}\left(x_{j},x_{k}\right)\overline{c_{k}},n=1,2,...,.$
(2.13)
From the well-known formula
$\sum_{j=1}^{m}\sum_{k=1}^{m}c_{j}\left(x_{j},x_{k}\right)\overline{c_{k}}=\left\|\sum_{j=1}^{m}c_{j}x_{j}\right\|^{2},$(2.12)
and the inequality $\gamma_{n}^{\mathrm{\min}}\geq$ $\gamma^{\mathrm{\min}}>0$
it follows that
$\gamma^{\mathrm{\min}}\sum_{k=1}^{n}\left|c_{k}\right|^{2}\leq\left\|\sum_{j=1}^{n}c_{j}x_{j}\right\|^{2}$
(2.14)
Hence the above theorem can be reformulated as follows
###### Theorem 2.2
The linear moment problem
$c_{j}=\left(x_{j},g\right),j=1,2,...$ (2.15)
has a solution $g\in X$ for any sequence $\left\\{c_{n},n=1,2,...\right\\},$
$\sum\limits_{j=1}^{\infty}c_{j}^{2}<\infty$ if and only if the sequence
$\left\\{x_{n},n=1,2,..,\right\\}$is strongly minimal.
## 3 Solution of the exact null-controllability problem.
###### Theorem 3.1
For equation (1.1) to be exact null-controllable on $\left[0,t_{1}\right],$
$t_{1}>T,$ by controls vanishing after time moment $t_{1}-T$, it is necessary,
that the sequence
$\left\\{e^{-\lambda_{j}\tau}b_{j},t\in\left[0,t_{1}-T\right],~{}j=1,2,...,\right\\}$
(3.1)
is minimal, and sufficient , that:
* •
the sequence
$\left\\{e^{-\lambda_{j}\tau}b_{j},t\in\left[0,t_{1}-T\right],~{}j=1,2,...\right\\}$is
strongly minimal;
* •
$\sum_{j=1}^{\infty}\left|\left(x_{0},\psi_{j}\right)\right|^{2}<+\infty,\forall
x_{0}\in X.$ (3.2)
Proof. Necessity. If the problem (2.4) has a solution for any $x_{0}\in
X,$then it has a solution for any eigenvector $\varphi_{k},k=1,2,...,$ of the
operator $A,$ so for each $k=1,2,...,$ there exists a function
$u_{k}\left(\cdot\right)\in L_{2}\left[0,t_{1}-T\right]$ such that
$\left(\varphi_{k},\psi_{j}\right)=-\int_{0}^{t_{1}-T}e^{-\lambda_{j}\tau}b_{j}u_{k}\left(\tau\right)d\tau,~{}~{}~{}j=1,2,...,.$
(3.3)
The sequence $\left\\{\varphi_{k},k=1,2,...,\right\\}$ of eigenvectors of the
operator $A$ is biorthogonal to the sequence
$\left\\{\psi_{k},k=1,2,...,\right\\}$ of eigenvectors of the operator
$A^{\ast}.$ Hence it follows from (3.3) and (2.2) that
$\delta_{jk}=\left(\varphi_{k},\psi_{j}\right)=-\int_{0}^{t_{1}-T}e^{-\lambda_{j}\tau}b_{j}u_{k}\left(\tau\right)d\tau,j=1,2,...,.$
i.e. the sequence
$\left\\{-u_{k}\left(t\right),t\in\left[0,t_{1}-T\right],~{}k=1,2,...,\right\\}$
is biorthogonal to the
sequence$~{}\left\\{e^{-\lambda_{j}t}b_{j},t\in\left[0,t_{1}-T\right],~{}j=1,2,...,\right\\}.$
It proves the necessity.
Sufficiency. The sufficiency follows immediately from (3.2) and Theorem 2.2.
It proves the theorem.
### 3.1 The case of the strongly minimal sequence of eigenvectors of the
operator $A$.
Obviously the sequence of eigenvectors of the operator $A$ being considered is
a minimal sequence.
Below we consider the operator $A$ having the strongly minimal sequence of
eigenvectors.
###### Theorem 3.2
Let the sequence $\left\\{\varphi_{j},j=1,2,...\right\\}$ of eigenvectors of
the operator $A$ be strongly minimal.
For equation (1.1) to be exact null-controllable on $\left[0,t_{1}\right],$
$t_{1}>T,$ by controls vanishing after time moment $t_{1}-T$, it is necessary,
that the sequence
$\left\\{e^{-\lambda_{j}\tau}b_{j},t\in\left[0,t_{1}-T\right],~{}j=1,2,...\right\\}$
is minimal, and sufficient, that $\mathop{\mathrm{R}e}\lambda_{j}\geq\beta$
for some $\beta\in\mathbb{R}$ and the
sequence$\left\\{e^{-\lambda_{j}t}b_{j},t\in\left[0,t_{1}-T\right],~{}j=1,2,...\right\\}$
is strongly minimal.
Proof. The necessity follows from Theorem 3.1.
Sufficiency. By Assumption 3 of the list of assumptions the series
$\sum\limits_{j=1}^{\infty}\left(x_{0},\psi_{j}\right)e^{\lambda
jt}\varphi_{j},\forall t>T$ (3.4)
converges. Since the sequence $\left\\{\varphi_{j},j=1,2,...\right\\}$ of
eigenvectors of the operator $A$ is strongly minimal, then on account of
property (2.10 there exists a number $\alpha$ such that
$\displaystyle\alpha^{2}\sum\limits_{j=1}^{n}\left|\left(x_{0},\psi_{j}\right)\right|^{2}e^{2\mathop{\mathrm{R}e}\lambda_{j}t}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{n}\sum_{k=1}^{n}\left(x_{0},\psi_{j}\right)e^{\lambda
jt}\left(\varphi_{j},\varphi_{k}\right)\overline{\left(x_{0},\psi_{k}\right)}e^{\overline{\lambda_{k}}t},$
(3.5) $\displaystyle\forall x_{0}$ $\displaystyle\in$ $\displaystyle
X,~{}\forall n\in\mathbb{N},~{}\forall t>T.$
It follows from (3.4) and (3.5) that
$\sum\limits_{j=1}^{\infty}\left|\left(x_{0},\psi_{j}\right)\right|^{2}e^{2\mathop{\mathrm{R}e}\lambda_{j}t}<+\infty,\forall
x_{0}\in X,\forall t>T.$ (3.6)
As $\mathop{\mathrm{R}e}\lambda_{j}\geq\beta$ for some $\beta\in\mathbb{R},$we
have by (3.6) that (3.2) holds.
In accordance with Theorem 3.1 condition (3.2) and the strong minimality of
the sequence (3.1) imply the exact null-controllability of equation (1.1). It
proves the theorem.
### 3.2 The case when the eigenvectors of the operator $A$ form a Riesz basic
One of the important problems of the operator theory is the case when the
generalized eigenvectors of the operator $A$ being considered form a Riesz
basic in $X.$ The problem of expansion into a Riesz basic of eigenvectors of
the operator $A$ is widely investigated in the literature (see, for example,
[1], [6], [7], [12] and references therein). Obviously the sequence of these
vectors is strongly minimal. In this case one can set $T=0,$ so the Theorems
3.1, 3.2 and Lemma 3.1 can be proven with $T=0.$
###### Theorem 3.3
Let the sequence of operator $A$ forms a Riesz basic in $X.$
For equation (1.1) to be exact null-controllable on
$\left[0,t_{1}\right],t_{1}>T,$ by controls vanishing after time moment
$t_{1}-T$, it is necessary and sufficient, that the sequence
sequence$\left\\{e^{-\lambda_{j}t}b_{j},t\in\left[0,t_{1}-T\right],~{}j=1,2,...\right\\}$
is strongly minimal .
Proof. Let $\left\\{c_{j},j=1,2,...,\right\\}$ be any complex sequence
satisfying the condition $\sum_{j=1}^{\infty}\left|c_{j}\right|^{2}<\infty.$
Since the sequence $\left\\{\varphi_{j},j=1,2,...,\right\\}$ of eigenvectors
of the operator $A$ forms the Riesz basic, there exists a vector $x_{0}\in X$
such that
$c_{j}=\left(x_{0},\psi_{j}\right),j=1,2,...,$
so in virtue of Theorem 2.1 the exact null controllability being considered in
the paper is equivalent to the solvability of the linear moment problem
$c_{j}=\int_{0}^{t_{1}-T}e^{-\lambda_{j}\tau}b_{j}u\left(\tau\right)d\tau,~{}j=1,2,...,$
(3.7)
for any complex sequence $\left\\{c_{j},j=1,2,...,\right\\}~{}$satisfying the
condition $\sum_{j=1}^{\infty}\left|c_{j}\right|^{2}<\infty.$
By above mentioned results of [2] and [3] the linear moment problem (3.7) is
solvable for any complex sequence $\left\\{c_{j},j=1,2,...,\right\\}$
satisfying the condition $\sum_{j=1}^{\infty}\left|c_{j}\right|^{2}<\infty\
$if and only if the sequence
$\left\\{e^{-\lambda_{j}t}b_{j},t\in\left[0,t_{1}-T\right],~{}j=1,2,...\right\\}$
is strongly minimal . It proves the theorem.
Obviously, the condition $b_{j}\neq 0,j=1,2,...,$ is the necessary condition
for the solvability of the moment problem (2.1).
###### Lemma 3.1
If the sequence
$\left\\{e^{-\lambda_{j}t},t\in\left[0,t_{1}-T\right],~{}j=1,2,...\right\\}$
(3.8)
is strongly minimal and
$\inf\limits_{n\in\mathbb{N}}\left|b_{n}\right|=\beta>0$ (3.9)
holds, then the sequence
$\left\\{e^{-\lambda_{j}t}b_{j},t\in\left[0,t_{1}-T\right],~{}j=1,2,...\right\\}$is
also strongly minimal.
Proof. Let the sequence
$\left\\{e^{-\lambda_{j}t},t\in\left[0,t_{1}-T\right],~{}j=1,2,...\right\\}$
be strongly minimal. From (2.12) it follows that
$\alpha\sum_{k=1}^{n}\left|c_{k}\right|^{2}\left|b_{j}\right|^{2}\leq\int_{0}^{t_{1}-T}\left|\sum_{j=1}^{n}c_{j}e^{-\lambda_{j}t}b_{j}\right|^{2}dt$
(3.10)
for some positive $\alpha\ $and for every finite sequence
$\left\\{c_{1},c_{2},...,c_{n}\right\\}.$ By (3.9) and (3.10) we have
$\gamma\sum_{k=1}^{n}\left|c_{k}\right|^{2}\leq\int_{0}^{t_{1}-T}\left|\sum_{j=1}^{n}c_{j}e^{-\lambda_{j}t}b_{j}\right|^{2}dt,n=1,2,...,\gamma=\alpha\beta>0.$
(3.11)
where $\gamma=\alpha\beta>0.$ It proves the lemma.
Example of strongly minimal sequence. Below we will prove that the sequence
$\left\\{e^{n^{2}\pi^{2}t},n=1,2,...,t\in\left[0,t_{1}\right]\right\\}$ is
strongly minimal for any $t_{1}>0.$
Let $t_{1}=2t_{2}.$ The series $\sum_{n=1}^{\infty}\frac{1}{n^{2}\pi^{2}}$
converges and $\left(n+1\right)^{2}-n^{2}\geq 1$, so the sequence
$\left\\{e^{n^{2}\pi^{2}t},n=1,2,...,t\in\left[0,t_{2}\right]\right\\}$is
minimal [5]. In virtue of Theorem 1.5 of [5] for each $\varepsilon>0$ there
exists a positive constant $K_{\varepsilon}$ such that the biorthogonal
sequence
$\left\\{w_{n}\left(t\right),n=1,2,...,t\in\left[0,t_{2}\right]\right\\}$
satisfies the condition
$\left\|w_{n}\left(\cdot\right)\right\|<K_{\varepsilon}e^{\varepsilon
n^{2}\pi^{2}},n=1,2,...,.$ (3.12)
The positive constant $\varepsilon$ can be chosen such that
$t_{2}-\varepsilon>0.$
By the Minkowsky inequality and (3.12) one can show that
$\sum_{n=1}^{p}\sum_{m=1}^{p}c_{n}e^{-n^{2}\pi^{2}t_{2}}\left(\int_{0}^{t_{2}}w_{n}\left(t\right)w_{m}\left(t\right)dt\right)e^{-m^{2}\pi^{2}t_{2}}c_{m}=\int_{0}^{t_{2}}\left(\sum_{n=1}^{p}c_{n}e^{-n^{2}\pi^{2}t_{2}}w_{n}\left(t\right)dt\right)^{2}dt\leq$
$\leq\int_{0}^{t_{2}}\sum_{n=1}^{p}\left|c_{n}\right|^{2}\sum_{n=1}^{p}\left|e^{-n^{2}\pi^{2}t_{2}}w_{n}\left(t\right)\right|^{2}dt=\sum_{n=1}^{p}\left|c_{n}\right|^{2}\sum_{n=1}^{p}e^{-2n^{2}\pi^{2}t_{2}}\int_{0}^{t_{2}}\left|w_{n}\left(t\right)\right|^{2}dt\leq$
$\leq\sum_{n=1}^{p}\left|c_{n}\right|^{2}\sum_{n=1}^{p}e^{-2n^{2}\pi^{2}t_{2}}\left\|w_{n}\left(\cdot\right)\right\|^{2}\leq
K_{\varepsilon}^{2}\sum_{n=1}^{p}\left|c_{n}\right|^{2}\sum_{n=1}^{p}e^{-2n^{2}\pi^{2}\left(t_{2}-\varepsilon\right)}.$
The series
$\sum_{n=1}^{\infty}e^{-2n^{2}\pi^{2}\left(t_{2}-\varepsilon\right)}$
converges for any $t_{2},\varepsilon,t_{2}>\varepsilon,~{}$so
$\sum_{n=1}^{p}e^{-2n^{2}\pi^{2}\left(t_{2}-\varepsilon\right)}\leq M$, where
$M$ is a positive constant.
Hence
$\sum_{n=1}^{p}\sum_{m=1}^{p}c_{n}e^{-n^{2}\pi^{2}t_{2}}\left(\int_{0}^{t_{2}}w_{n}\left(t\right)w_{m}\left(t\right)dt\right)e^{-m^{2}\pi^{2}t_{2}}c_{m}\leq
K_{\varepsilon}^{2}M\sum_{n=1}^{p}\left|c_{n}\right|^{2}$ (3.13)
for every finite sequence $\left\\{c_{1},c_{2},...,c_{p}\right\\}.$ Obviously
the sequence
$\left\\{h_{n}\left(t\right)=\left\\{\begin{array}[]{cc}e^{-n^{2}\pi^{2}t_{2}}w_{n}\left(t-t_{2}\right),&t\in\left[t_{2},2t_{2}\right],\\\
0,&t\in\left[0,t_{2}\right)\end{array}\right.,n=1,2,...,\right\\}$is the
biorthogonal sequence to the sequence
$\left\\{e^{n^{2}\pi^{2}t},n=1,2,...,t\in\left[0,t_{1}\right]\right\\},$ and
$\left(\int_{0}^{t_{1}}h_{n}\left(t\right)h_{m}\left(t\right)dt\right)=e^{-n^{2}\pi^{2}t_{2}}\left(\int_{t_{2}}^{2t_{2}}w_{n}\left(t-t_{2}\right)w_{m}\left(t-t_{2}\right)dt\right)e^{-m^{2}\pi^{2}t_{2}}=e^{-n^{2}\pi^{2}t_{2}}\left(\int_{0}^{t_{2}}w_{n}\left(t\right)w_{m}\left(t\right)dt\right)e^{-m^{2}\pi^{2}t_{2}},$
so it follows from (3.13) that
$\sum_{n=1}^{p}\sum_{m=1}^{p}c_{n}\left(\int_{0}^{t_{1}}h_{n}\left(t\right)h_{m}\left(t\right)dt\right)c_{m}\leq
K_{\varepsilon}^{2}M\sum_{n=1}^{p}\left|c_{n}\right|^{2}.$
Hence [9]
$\sum_{n=1}^{p}\sum_{m=1}^{p}c_{n}\left(\int_{0}^{2t_{1}}e^{n^{2}\pi^{2}\tau}e^{m^{2}\pi^{2}\tau}\right)c_{m}d\tau\geq\gamma\sum_{n=1}^{p}\left|c_{n}\right|^{2},p=1,2,...,$
(3.14)
for every finite sequence $\left\\{c_{1},c_{2},...,c_{p}\right\\},$where
$\gamma=\frac{1}{K_{\varepsilon}^{2}M}>0.$ It proves that the sequence
$\left\\{e^{n^{2}\pi^{2}t},t\in\left[0,t_{1}\right],~{}n=1,2,...\right\\}$ is
strongly minimal for any $~{}t_{1}>0$.
## 4 Approximation Theorems
As was said at the end of the previous section the condition
$\lim\limits_{n\rightarrow\infty}\lambda_{n}^{\min}$ $>0$ in general can be
checked by numerical methods. The problem appears to be rather difficult in
general.
However there are sequences for which the validity of above inequality can be
easily established. For example, every orthonormal sequence is strongly
minimal.
Below we will show that if the sequence
$\left\\{y_{j}\in X,j=1,2,...\right\\}$
can be approximated in the some sense by strongly minimal sequence
$\left\\{x_{j}\in X,j=1,2,...\right\\},$
then it is also strongly minimal.
###### Theorem 4.1
If the sequence $\left\\{x_{j}\in X,j=1,2,...\right\\}$ is strongly minimal,
let the sequence $\left\\{y_{j}\in X,j=1,2,...\right\\}$ be such that the
sequence $\left\\{P_{n}y_{j}-x_{j},j=1,2,...\right\\}$ is linear independent
and
$\left\|\sum_{j=1}^{n}c_{j}\left(y_{j}-x_{j}\right)\right\|\leq
q\left\|\sum_{j=1}^{n}c_{j}x_{j},\right\|,n=1,2,...~{},$ (4.1)
where $\left\\{c_{j},j=1,2,...\right\\}$ is any sequence of complex numbers,
$q$ is a constant, $0<q<1,$ then the sequence $\left\\{y_{j}\in
X,j=1,2,...\right\\}$ also is strongly minimal.
Proof. Let $\left\\{c_{k},k=1,2,...\right\\}$ be an arbitrary sequence of
complex number. Denote:
$x^{0}=\sum_{k=1}^{n}c_{k}x_{k},~{}x^{1}=\sum_{k=1}^{n}c_{k}\left(x_{k}-y_{k}\right).$
(4.2)
From (4.2) it follows, that
$x^{0}=x^{1}+\sum_{k=1}^{n}c_{k}y_{k},~{}n=\ 1,2,....$ (4.3)
By (4.1) we obtain that
$\left\|x^{1}\right\|\leq q\left\|x^{0}\right\|.$ (4.4)
Hence using (4.4) in (4.3) we obtain
$\left\|x^{0}\right\|\leq\frac{1}{1-q}\left\|\sum_{k=1}^{n}c_{k}y_{k}\right\|,~{}n=\
1,2,....$ (4.5)
Since the sequence $\left\\{x_{j}\in X,j=1,2,...\right\\}$ is strongly minimal
and $x^{0}$ $=\sum_{k=1}^{n}c_{k}x_{k}$, we have
$\sum_{k=1}^{n}\left|c_{k}\right|^{2}\leq\frac{1}{\alpha^{2}}\left\|x^{0}\right\|^{2},~{}n=1,2,...,$
(4.6)
for some $\alpha>0.$
By (4.6) and (4.5) we obtain
$\alpha^{2}\sum_{k=1}^{n}\left|c_{k}\right|^{2}\leq\frac{1}{1-q}\left\|\sum_{k=1}^{n}c_{k}y_{k}\right\|,~{}n=\
1,2,...,$ so
$\alpha^{2}\left(1-q\right)^{2}\left(\sum_{k=1}^{n}\left|c_{k}\right|^{2}\right)\leq\left\|\sum_{k=1}^{n}c_{k}y_{k}\right\|,~{}n=\
1,2,...,.$ (4.7)
Using in (4.7) the formula (2.14) we obtain
$\gamma\left(\sum_{k=1}^{n}\left|c_{k}\right|^{2}\right)\leq\sum_{k=1}^{n}\sum_{l=1}^{n}c_{k}\left(y_{k},y_{l}\right)\overline{c_{l}},\gamma=\alpha^{2}\left(1-q\right)^{2}>0$
(4.8)
Let $\mu_{\min}^{\left[n\right]}$ be a minimal eigenvalue of the Gram matrix
$G_{n}=\left\\{\left(y_{k},y_{l}\right),k,l=1,2....\right\\}$ for the sequence
$\left\\{y_{j},j=1,2,...,n\right\\}.$ From (4.8), it follows that
$\lim_{n\rightarrow\infty}\mu_{\min}^{\left[n\right]}\geq\gamma>0.$
This proves the theorem.
### 4.1 Example
Let $X=$ $l_{2}$ be the Hilbert space of square summable sequences. Consider
the evolution system
$\left\\{\begin{array}[]{ccc}\dot{x}_{k}\left(t\right)=\lambda_{k}x_{k}\left(t\right)+u\left(t\right),&k=1,2,...,&0<t<t_{1},\\\
x_{k}\left(0\right)=x_{k0},n=1,2,...,&k=1,2,...,&\end{array}\right.$ (4.9)
where $u\left(t\right),0<t<t_{1}$ is a scalar control function,
$\left\\{x_{k}\left(t\right),k=1,2,...,\right\\},\left\\{x_{k0},k=1,2,...,\right\\}\in
l^{2},$ the complex numbers $\lambda_{k},$ $k=1,2,...,$belong to the strip
$\left\\{z\in\mathbb{C}:\left|\mathop{\mathrm{R}e}z\right|\leq\gamma\right\\},$
i.e. $\left|\mathop{\mathrm{R}e}\lambda_{k}\right|\leq\gamma,k=1,2,...,$ .
###### Definition 4.1
Equation (4.9) is said to be exact null-controllable on $\left[0,t_{1}\right]$
by controls vanishing after time moment $t_{2},$ if for each
$x_{0}\left(\cdot\right)=\left\\{x_{k0},k=1,2,...,\right\\}\in l_{2}$ there
exists a control $u\left(\cdot\right)\in
L_{2}\left[0,t_{2}\right],u\left(t\right)=0$ a.e. on $[t_{2},+\infty)$ such
that
$x_{k}\left(t\right)\equiv 0,~{}k=1,2,...,\forall t\geq t_{1}.$
Control problem (4.9) can be written in the form of (1.1), where
$x\left(t\right)=\left\\{x_{k}\left(t\right),k=1,2,...,\right\\}\in
l^{2},u\left(\cdot\right)\in L_{2}\left[0,t_{1}\right]$; the self-adjoint
operator $A:l_{2}\rightarrow l_{2}$ is defined for
$x=\left\\{x_{k},k=1,2,...,\right\\}\in l_{2}~{}$by
$Ax=\left\\{\lambda_{k}x_{k},k=1,2,...,\right\\}$ (4.10)
with domain $D\left(A\right)=\left\\{x\in l_{2}:Ax\in l_{2}\right\\}$, and the
unbounded operator $B$ is defined by
$Bu=bu,u\in\mathbb{R},$ (4.11)
where $b=\\{1,1,...,1,...\\}\notin l_{2}$.
One can show that all the assumptions imposed on equation (1.1) are fulfilled
for equation (4.9) with $T=0$.
Obviously, the numbers $\lambda_{k},$ $k=1,2,...,$ are eigenvalues of the
operator $A$ defined above; the sequences
$e_{k}=\left\\{\underset{1~{}\mathrm{on~{}}k\text{-{th
place}}}{\underbrace{0,...,0,1,0,...,0}}\right\\}$ are corresponding
eigenvectors, forming the Riesz basic of $l_{2},$ so $b_{j}=1,j=1,2,...,.$
Together with system (4.9) consider the other evolution system
$\left\\{\begin{array}[]{ccc}\dot{x}_{k}\left(t\right)=\mu_{k}x_{k}\left(t\right)+u\left(t\right),&n=1,2,...,&0<t<t_{1},\\\
x_{k}\left(0\right)=x_{k0},k=1,2,...,&n=1,2,...,&\end{array}\right.$ (4.12)
where
$\mu_{k}=\lambda_{k}+O\left(\frac{1}{k}\right),k=1,2,...,.$ (4.13)
###### Proposition 1
If system (4.9) is exact null-controllable on $\left[0,t_{1}\right]$ by
controls vanishing after time moment $t_{2},$then the same is valid for system
(4.12).
Proof. From the Caushy-Schvartz inequality it follows that
$\int_{0}^{t_{2}}\left|\sum_{k=1}^{n}c_{k}\left(e^{-\mu_{k}t}-e^{-\lambda_{k}t}\right)\right|^{2}dt\leq\sum_{k=1}^{n}\left|c_{k}\right|^{2}\int_{0}^{t_{2}}\sum_{k=1}^{n}\left|e^{-\mu_{k}t}-e^{-\lambda_{k}t}\right|^{2}dt=$
$=\sum_{k=1}^{n}\left|c_{k}\right|^{2}\int_{0}^{t_{2}}\sum_{k=1}^{n}e^{-2\lambda_{k}t}\left|e^{O\left(\frac{1}{k}\right)t}-1\right|^{2}dt\leq\sum_{k=1}^{n}\left|c_{k}\right|^{2}\int_{0}^{t_{2}}e^{2\gamma
t}\sum_{k=1}^{n}\left|e^{O\left(\frac{1}{k}\right)t}-1\right|^{2}dt.$
The series
$\sum_{k=1}^{\infty}\left|e^{O\left(\frac{1}{k}\right)t}-1\right|^{2}$
converges for any $t\geq 0$. Denote
$M\left(t_{2}\right)=\int_{0}^{t_{2}}e^{2\gamma
t}\sum_{k=1}^{\infty}\left|e^{O\left(\frac{1}{k}\right)t}-1\right|^{2}dt.$
(4.14)
Hence
$\int_{0}^{t_{2}}\left|\sum_{k=1}^{n}c_{k}\left(e^{-\mu_{k}t}-e^{-\lambda_{k}t}\right)\right|^{2}dt\leq
M\left(t_{2}\right)\sum_{k=1}^{n}\left|c_{k}\right|^{2}.$ (4.15)
By Theorem 3.2 we have the
sequence$\left\\{e^{-\lambda_{j}t},t\in\left[0,t_{2}\right],~{}j=1,2,...\right\\}$
to be strongly minimal, so
$\sum_{k=1}^{n}\left|c_{k}\right|^{2}\leq\frac{1}{\alpha^{2}}\int_{0}^{t_{2}}\left|\sum_{k=1}^{n}c_{k}e^{-\lambda_{k}t}\right|^{2}dt~{}\mathrm{for~{}some~{}}\alpha>0.$
(4.16)
Joining (4.15) and (4.16) we obtain
$\int_{0}^{t_{2}}\left|\sum_{k=1}^{n}c_{k}\left(e^{-\mu_{k}t}-e^{-\lambda_{k}t}\right)\right|^{2}dt\leq
q\int_{0}^{t_{2}}\left|\sum_{k=1}^{n}c_{k}e^{-\lambda_{k}t}\right|^{2}dt,~{}$
(4.17)
where $q=\frac{M\left(t_{2}\right)}{\alpha}.$
Since from (4.14) it follows that
$\lim\limits_{t_{1}\rightarrow\infty}M\left(t_{2}\right)=0,$ one can choose
the number $t_{2}$ such that $0<q<1.$ Hence conditions (4.17) are the same as
(4.1) for
$x_{k}=e^{-\lambda_{k}t},y_{k}=e^{-\mu_{k}t},k=1,2,...,t\in\left[0,t_{2}\right];q=\frac{M\left(t_{2}\right)}{\alpha^{2}}.$
As it was said abov by Theorem 3.2 we have the
sequence$\left\\{e^{-\lambda_{j}t},t\in\left[0,t_{2}\right],~{}j=1,2,...\right\\}$
to be strongly minimal .
In accordance with Theorem 4.1 the sequence
$\left\\{y_{k}=e^{-\mu_{k}t},k=1,2,...,t\in\left[0,t_{2}\right]\right\\}$ is
also strongly minimal, provided that $t_{2}$ is chosen such that
$\frac{M\left(t_{2}\right)}{\alpha^{2}}<1.$ In accordance with Theorem 3.1 the
strong minimality of the sequence
$\left\\{y_{k}=e^{-\mu_{k}t},k=1,2,...,t\in\left[0,t_{2}\right]\right\\}$
provides the zero controllability of equation (4.11) on $\left[0,t_{1}\right]$
by controls vanishing after time moment
$t_{2},~{}\frac{M\left(t_{2}\right)}{\alpha^{2}}<1,$ for any $t_{1}\geq
t_{2}$.
## References
* [1] N. Ahiezer, I. Glazman, _Linear Operator Theory in Hilbert Spaces_ , Moscow, Nauka Publisher, 1966 (Russian).
* [2] R. Boas, A general moment problem, Amer. J. Math., 63(1941), 361—370.
* [3] N. Bari, Biorthogonal sequences and bases in Hilbert spaces. Uchen. Zap. Mosk. Univ., 148, Nat, 4(1951), 69—107.
* [4] Da Pratto, Abstract differential equations and extrapolation spaces, Lecture Notes in Mathematics, 1184, Springer-Berlag, Berlin, New York, 1984.
* [5] H. Fattorini, D. Russel, Uniform bounds on biorthogonal functions for real exponents with an application to the control theory of parabolic equations, Quart. Appl.Math., 1074, 45 — 69.
* [6] Gen Qi Xu, Siu Pang Yung, The expansion of semogroup and a Riesz basic criterion, J. Diff. Eqn., 210(2005), 1 — 24.
* [7] I. Gohberg, M. Krein, Introduction to the Theory of Linear Nonselfadjoint operators, Transl. math. Monogr., 18, AMS, Providence, RI, 1969.
* [8] E. Hille, R. Philips, Functional Analysis and Semi-Groups, AMS, 1957.
* [9] S. Kaczmarz, H. Steinhaus, Theory of orthogonal series Monographs Mat., Bd. 6, (PWN, Warsaw), 1958
* [10] M. Krein, Linear Differential Equations in Banach Spaces, Moscow, Nauka Publisher, 1967 (in Russian).
* [11] R. Nagel, One-parameter semigroups of positive operators, Lecture Notes in Notes in Mathematics, 1184, Springer-Berlag, Berlin, New York, 1984.
* [12] M. Naimark, Linear differential Operators, Moscow, Nauka Publisher, 1969 (in Russian).
* [13] R. Rabah, G. Sklyar, Thw analysis od exact controllability of neutral-type systems by the moment problem approach, SIAM J. Contr. Optimiz., 36 (2007), 2148 — 2181.
* [14] D. Salamon, Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach, Trans. Amer. Math. Soc., 300(1987), 383 — 431.
* [15] G. Weiss, Admissibility of unbounded control operators, SIAM J. Contr. and Optimiz., 27(1989), 527 — 545.
* [16] D. Ullrich, Divided differences and systems of nonharmonic Fourier series, Proc. Amer. Math. Soc., 80(1980), 47 — 57\.
* [17] R. Young, An Introduction to Nonharmonic Analysis, Academic Press, New York, 1980.
* [18] R. Young, On a class of Riesz-Fisher sequences, Proceedings of AMS, 126(1998), 1139—1142.
|
arxiv-papers
| 2009-06-17T20:31:20 |
2024-09-04T02:49:03.399628
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "B. Shklyar",
"submitter": "Benzion Shklyar",
"url": "https://arxiv.org/abs/0906.3303"
}
|
0906.3336
|
This paper has been withdrawn since the results are not satisfied.
|
arxiv-papers
| 2009-06-18T02:19:29 |
2024-09-04T02:49:03.405319
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guangyue Huang, Bingqing Ma",
"submitter": "Huang Guangyue",
"url": "https://arxiv.org/abs/0906.3336"
}
|
0906.3382
|
# Scattering for the focusing ${\dot{H}}^{1/2}$-critical Hartree equation with
radial data
Yanfang Gao1, Changxing Miao2 and Guixiang Xu2
1 Institute of Mathematics, Jilin University, Changchun, China, 130012
2 Institute of Applied Physics and Computational Mathematics
P. O. Box 8009, Beijing, China, 100088
( [email protected], [email protected],
[email protected])
###### Abstract
We investigate the focusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger
equation (NLS) of Hartree type $i\partial_{t}u+\Delta
u=-(|\cdot|^{-3}\ast|u|^{2})u$ with $\dot{H}^{1/2}$ radial data in dimension
$d=5$. It is proved that if the maximal life-span solution obeys
$\sup_{t}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{2}<\frac{\sqrt{6}}{3}\big{\|}|\nabla|^{\frac{1}{2}}Q\big{\|}_{2}$,
where $Q$ is the positive radial solution to the elliptic equation with
nonlocal operator (1.4) which corresponds to a new variational structure. Then
the solution is global and scatters.
Key Words: Hartree equation, scattering, profiles decomposition, almost
periodic solution, concentration compactness
AMS Classification: 35Q40, 35Q55, 47J35.
## 1 Introduction
Consider the Cauchy problem for the $\dot{H}^{1/2}$-critical Hartree equation
$i\partial_{t}u+\Delta u=F(u)$ (1.1)
in $\mathbb{R}^{5}$, where $F(u)=-(|\cdot|^{-3}\ast|u|^{2})u$, $u$ is a
complex-valued function defined on some spacetime slab
$I\times\mathbb{R}^{5}$. The Hartree equation arises in the study of boson
stars and other physical phenomena, see, for instance, [25].
The term $\dot{H}^{1/2}$-critical means that the scaling
$u_{\lambda}(t,x)=\lambda^{-2}u(\lambda^{-2}t,\lambda^{-1}x)$ (1.2)
leaves both the equation and the initial data of $\dot{H}^{1/2}_{x}$\- norm
invariant. By a function $u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ is a
solution to (1.1), it means that $u\in
C_{t}^{0}\dot{H}^{1/2}_{x}(K\times\mathbb{R}^{5})\cap
L_{t}^{3}L_{x}^{15/4}(K\times\mathbb{R}^{5})$ for any compact $K\subset I$,
and $u$ obeys the Duhamel formula
$u(t)=e^{i(t-t_{0})\Delta}u(t_{0})-i\int_{t_{0}}^{t}e^{i(t-t^{\prime})\Delta}F(u(t^{\prime}))\,\mathrm{d}t^{\prime}$
for all $t,\,t_{0}\in I$. We call $I$ the life-span of $u$. If $I$ can not be
extended strictly larger, we say $I$ is the maximal life-span of $u$, and $u$
is a maximal life-span solution. If $I=\mathbb{R}$, then $u$ is global.
###### Definition 1.1 (Blow up).
Let $u:I\times\mathbb{R}^{d}\mapsto\mathbb{C}$ be a solution to (1.1). Say $u$
blows up forward in time if there exists $t_{1}\in I$ such that
$\|u\|_{L_{t}^{3}L_{x}^{15/4}([t_{1},\;\sup
I)\times\mathbb{R}^{5})}=\infty\,;$
and $u$ blows up backward in time if there exists $t_{1}$ such that
$\|u\|_{L_{t}^{3}L_{x}^{15/4}((\inf I,\;t_{1}]\times\mathbb{R}^{5})}=\infty.$
Throughout the paper, we write
$\|u\|_{S(I)}:=\|u\|_{L_{t}^{3}L^{15/4}_{x}(I\times\mathbb{R}^{5})},\quad\|u\|_{X(I)}:=\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{3}L_{x}^{30/11}(I\times\mathbb{R}^{5})}.$
The local theory for (1.1) was established by Cazenave and Weissler [3], [4].
Using a fixed point argument together with Strichartz’s estimates in the
framework of Besov spaces, they constructed local in time solution for
arbitrary initial data. However, due to the critical nature of the equation,
the existence time depends on the profile of the initial data and not merely
on its $\dot{H}^{1/2}_{x}$-norm. They also proved the global existence for
small data.
###### Theorem 1.1 (Local theory, [3], [4]).
Let $u_{0}\in\dot{H}_{x}^{1/2}(\mathbb{R}^{5}),\,t_{0}\in\mathbb{R}$, there
exists a unique maximal life-span solution
$u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ to $(1.1)$ with initial data
$u(t_{0})=u_{0}$. This solution also has the following properties:
* •
(Local existence) $I$ is an open neighborhood of $t_{0}$.
* •
(Blow up criterion) If $\sup I$ is finite, then $u$ blows up forward in time;
if $\inf I$ is finite, then $u$ blows up backward in time.
* •
(Scattering) If $\sup I=+\infty$, and $u$ does not blow up forward in time,
then $u$ scatters forward in time, that is, there exists a unique
$u_{+}\in\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ such that
$\lim_{t\to+\infty}\|u(t)-e^{it\Delta}u_{+}\|_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}=0.$
(1.3)
Conversely, given $u_{+}\in\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$, there exists a
unique solution to $(1.1)$ in a neighborhood of infinity such that $(1.3)$
holds.
* •
(Small data scattering) If $\big{\|}|\nabla|^{\frac{1}{2}}u_{0}\big{\|}_{2}$
is sufficiently small, then $u$ scatters in both time directions. Indeed,
$\|u\|_{S(\mathbb{R})}\lesssim\big{\|}|\nabla|^{\frac{1}{2}}u_{0}\big{\|}_{2}$.
* •
(Radial symmetry) If $u_{0}$ is radially symmetric, then $u$ remains radially
symmetric for all time.
From Theorem 1.1, a solution to (1.1) with small data must be scattering.
However, the result is unknown for arbitrary data, even in the defocusing
case. In [10], Kenig and Merle proved for the defocusing cubic NLS that the
solution is global and scatters if it remains uniformly bounded in
$\dot{H}^{1/2}_{x}$ on its maximal life-span. The assumption that the solution
is uniformly bounded in $\dot{H}^{1/2}_{x}$ plays a role of the missing
conservation law. The argument presented there applies to the corresponding
defocusing Hartree equation without difficulty. As to the focusing case, there
has been no result on the line of scattering, neither NLS nor of Hartree type.
Our primary goal in this paper is to establish scattering result for the
focusing Hartree equation, and we believe that the argument can be adapted to
the focusing NLS.
For the Cauchy problem of $(1.1)$, there is a stationary solution
$e^{it}\bar{Q}$ that is global but blows up both forward and backward. Here
$\bar{Q}$ is the unique positive radial Schwartz solution to
$\Delta\bar{Q}+(|\cdot|^{-3}\ast|\bar{Q}|^{2})\bar{Q}=\bar{Q}.$
In the focusing energy/mass critical case, the corresponding stationary
solution/ground state play the role of an obstruction to the global well-
posedness and scattering. Indeed, the global existence follows so long as the
kinetic energy/mass of the initial data is strictly less than that of the
stationary solution/ground state. In [17], Li-Zhang classify the minimal
blowup solutions of the focusing mass-critical Hartree equation. However,
wether the solution $u$ to $(1.1)$ on its maximal life-span with
$\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}}<\|\bar{Q}\|_{\dot{H}^{1/2}_{x}}$
implies global existence is still open. In this paper we will introduce a new
elliptic equation:
$\Delta Q+(|\cdot|^{-3}\ast|Q|^{2})Q=(-\Delta)^{1/2}Q,$ (1.4)
which corresponds to a new variational structure, and prove that if the
solution $u$ to $(1.1)$ satisfies
$\|u\|_{L_{t}^{\infty}\dot{H}_{x}^{1/2}}<\frac{\sqrt{6}}{3}\|Q\|_{\dot{H}^{1/2}_{x}}$,
then the solution is global and scatters.
Solutions to critical NLS and of Hartree type have been intensively studied,
especially those of energy critical equations. Scattering results for the
defocusing energy-critical equations have been completely established. These
were accomplished by Bourgain [2], Grillakis [7], Tao [23], Colliander-Keel-
Staffilani-Takaoka-Tao [5], Ryckman-Visan [24], and Visan [29], Miao-Xu-Zhao
[21]. As will be discussed later, the focusing energy-critical NLS theory has
also been well established by Kenig-Merle and Killip-Visan, except for
dimensions 3 and 4. For the focusing Hartree, it was proved by Li-Miao-Zhang
[16], and Miao-Xu-Zhao[23].
Another kind of critical NLS and of Hartree type which receives lots of
attention is the mass-critical one. Results in earlier work which is devoted
to global well-posedness were usually obtained under the assumption of the
$H^{1}_{x}$ initial data. See, e.g., [3], [30]. In [30], Weinstein first
observed the role of the ground state for the focusing mass-critical NLS
despite finite energy. As far as $L_{x}^{2}$ initial data is concerned, Tao-
Visan-Zhang [27] proved the scattering results for the defocusing case for
large spherically symmetric data in dimensions three and higher. More recent
and nice work on scattering results for $L_{x}^{2}$ data were done by Killip-
Tao-Visan [13], Killip-Visan-Zhang [15], and Miao-Xu-Zhao [22] with spherical
symmetry assumption.
The recent progress in studying those equations is due to a new and highly
efficient approach based on a concentration compactness idea to provide a
linear profile decomposition. This approach arises from investigating the
defect of compactness for the Strichatz estimates. Based on a refined Sobolev
inequality, Kerrani [12] obtained a linear profile decomposition for solutions
of free NLS with $H^{1}_{x}$ data. It was Kenig and Merle who first introduced
Kerrani’s linear profile decomposition to obtain scattering results. They
treated the focusing energy-critical NLS in dimensions 3, 4, 5 in [9]. Using
the same decomposition, Killip and Visan [14] dealt with the focusing energy-
critical NLS in dimensions five and higher without radial assumption. Using
the decomposition of [19], Tao-Visan-Zhang [28] made a reduction for failure
of scattering. And by combining the reduction with an in/out decomposition
technique, [13], [15] settled the scattering problem for the mass-critical NLS
with spherically symmetric data.
A linear profile decomposition for general $\dot{H}^{s}$ data was proved by
Shao [26]. Unlike Kerrani’s approach which is based on a refined Sobolev
inequality, Shao took advantage of the existing $L_{x}^{2}$ linear profile
decomposition and the Galilean transform, and managed to eliminate the
frequency parameter from the decomposition. In this paper, we will use Shao’s
linear profile decomposition, and our main result is:
###### Theorem 1.2.
Let $u_{0}\in\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$, radially symmetric,
$t_{0}\in\mathbb{R}$, $I$ is a time interval containing $t_{0}$. Let
$u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ be a maximal life-span solution to
$(1.1)$. Assume $\sup_{t\in
I}\big{\|}|\nabla|^{\frac{1}{2}}u(t)\big{\|}_{2}<\frac{\sqrt{6}}{3}\big{\|}|\nabla|^{\frac{1}{2}}Q\big{\|}_{2}$.
Then $u$ is global and scatters with
$\|u\|_{L_{t}^{3}L_{x}^{15/4}(\mathbb{R}\times\mathbb{R}^{5})}^{3}=\int_{\mathbb{R}}\left(\int_{\mathbb{R}^{5}}|u(t,x)|^{15/4}\,\mathrm{d}x\right)^{4/5}\,\mathrm{d}t<\infty.$
###### Remark 1.1.
It is an interesting problem to describe the correspondence between $Q$ and
$\bar{Q}$, and thus leading to some investigation with the gap. It is also an
interesting problem that wether the solution blows up so long as
$\sup\limits_{t\in
I}\big{\|}|\nabla|^{\frac{1}{2}}u(t)\big{\|}_{2}\geq\frac{\sqrt{6}}{3}\big{\|}|\nabla|^{\frac{1}{2}}Q\big{\|}_{2}$.
The concentration compactness argument reduces matters to the study of almost
periodic solutions modulo symmetries.
###### Definition 1.2 (Almost periodic modulo scaling).
Let $u$ be a solution to $(1.1)$ with maximal life-span $I$. Say $u$ is almost
periodic modulo scaling if there exist functions $N:I\mapsto\mathbb{R}^{+}$,
$C:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$ such that for all $\eta>0$, $t\in I$
$\int_{|x|\geq
C(\eta)/{N(t)}}\big{|}|\nabla|^{\frac{1}{2}}u(t,x)\big{|}^{2}\,\mathrm{d}x\leq\eta$
and
$\int_{|\xi|\geq
C(\eta)N(t)}|\xi||\hat{u}(t,\xi)|^{2}\,\mathrm{d}\xi\leq\eta.$
We refer to $N(t)$ as the frequency scale function for the solution, and $C$
the compactness modulus function.
###### Remark 1.2.
By the Arzela-Ascoli theorem, a family of functions is precompact in
$\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ if and only if it is norm-bounded and
there exists a compactness modulus function $C$ so that
$\int_{|x|\geq
C(\eta)}\big{|}|\nabla|^{\frac{1}{2}}f(x)\big{|}^{2}\,\mathrm{d}x+\int_{|\xi|\geq
C(\eta)}|\xi||\hat{f}(\xi)|^{2}\,\mathrm{d}\xi\leq\eta$
for all functions in the family and all $\eta>0$. Thus, $u$ is almost periodic
modulo scaling if and only if there exists a compact subset $K$ of
$\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ such that
$\big{\\{}\,u(t):t\in
I\,\big{\\}}\subseteq\big{\\{}\,\lambda^{-2}f(\lambda^{-1}x):\lambda\in(0,+\infty),f\in
K\,\big{\\}}.$
By Sobolev’s embedding theorem, any solution
$u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ to $(1.1)$ that is almost periodic
modulo scaling also satisfies
$\int_{|x|\geq C(\eta)/{N(t)}}|u(t,x)|^{\frac{5}{2}}\,\mathrm{d}x\leq\eta$
(1.5)
for all $t\in I$ and all $\eta>0$.
By the compactness modulo scaling, there also exists a function
$c:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$ such that
$\int_{|x|\leq
c(\eta)/{N(t)}}\big{|}|\nabla|^{\frac{1}{2}}u(t,x)\big{|}^{2}\,\mathrm{d}x+\int_{|\xi|\leq
c(\eta)N(t)}|\xi||\hat{u}(t,\xi)|^{2}\,\mathrm{d}\xi\leq\eta$ (1.6)
for all $t\in I$ and all $\eta>0$.
We now present the process of reduction. If Theorem 1.2 failed, then there
must be an almost periodic solution. More precisely, we have:
###### Theorem 1.3.
Suppose Theorem $1.2$ failed for radially symmetric data. Then there exists a
maximal life-span solution $u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ to
$(1.1)$ with
$\sup_{t}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{2}<\frac{\sqrt{6}}{3}\big{\|}|\nabla|^{\frac{1}{2}}Q\big{\|}_{2}$.
$u$ is almost periodic modulo scaling, blows up both forward and backward.
Moreover, the frequency scale function $N(t)$ and the maximal life-span $I$
match one of the following scenarios :
I. (Finite-time blowup) Either $|\inf I|<\infty$ or $\sup I<\infty$.
II. (Low-to-high cascade) $I=\mathbb{R}$,
$\inf N(t)\geq 1\quad\textrm{for
all}\,\,t\in\mathbb{R},\quad\textrm{and}\quad\limsup_{t\to+\infty}N(t)=+\infty.$
III. (Soliton-like solution) $I=\mathbb{R}$, $N(t)\equiv 1$ for all
$t\in\mathbb{R}$.
The delicate relationship between the frequency scale function and the maximal
life-span for almost periodic solution was first discovered by Killip, Tao,
and Visan in [13] for mass-critical NLS. The argument was adapted to the
energy-critical case in [14]. This latter argument is directly applicable to
the setting of this paper.
To prove Theorem 1.2, it suffices to preclude the three scenarios in Theorem
1.3. We adapt ideas in [13], [14]. However, when precluding the finite-time
blowup, Plancherel’s theorem and Hardy’s inequality are not enough to obtain a
decay for the localized mass, especially for large scales, as we are working
in the fractional Sobolev space. To surmount this, we take advantage of the
intrinsic description of fractional derivatives, estimate the integral formula
in cases according to the spatial scales. Some negative regularity is needed
for disproving the rest two scenarios, and our discussions are somewhat
involved due to the nonlocal nonlinearity and low regularity. We shall make
full use of the frequency localization. For instance, in the proof of Lemma
6.1, we should firstly use Bernstein’s inequality to obtain a positive gain in
estimating the high frequency components and the medium frequency components,
such that the Gronwall’s inequality is applicable. What we would also like to
emphasize in particular is that as the $\dot{H}^{1/2}$-critical equation
enjoys no conservation law, beside proving the negative regularity, we have to
gain additional regularity of at least 1 order differentiability, which means
that the soliton-like solution has conserved energy; and thus allows us to
apply virial-type argument to disprove it. We also obtain the local spacetime
bounds in terms of the frequency scale function for all
$\dot{H}^{1/2}$-admissible pairs and of those $L^{2}$-admissible pairs
$(q,\,r)$ with $q\geq 3$, $r\leq 30/11$.
The following lemma plays an important role in proving the negative and
additional regularity. See [28] for a proof.
###### Lemma 1.1.
Let $u$ be an almost periodic solution to $(1.1)$ on its maximal life-span
$I$. Then, for all $t\in I$
$\displaystyle u(t)$ $\displaystyle=$ $\displaystyle\lim\limits_{T\nearrow\sup
I}i\int_{t}^{T}e^{i(t-t^{\prime})\Delta}F(u(t^{\prime}))\,\mathrm{d}t^{\prime}$
(1.7) $\displaystyle=$ $\displaystyle-\lim\limits_{T\searrow\inf
I}\int_{T}^{t}e^{i(t-t^{\prime})\Delta}F(u(t^{\prime}))\,\mathrm{d}t^{\prime}$
as weak limits in $\dot{H}^{1/2}_{x}$.
The rest of paper is organized as follows. In Section 2, we list out some
notations and known results that we use repeatedly in the paper. In Section 3,
the sharp constant for a Hardy-Littlewood-Sobolev type inequality is obtained,
and a sufficient condition for global existence of $(1.1)$ with finite energy
initial data is given. In Section 4, we first prove a Palais-Smale condition
modulo scaling, and then Theorem 1.3. In Section 5, we preclude the finite-
time blowup scenario. In Section 6, we prove the negative regularity for
global case. In Section 7, we disprove the low-to-high cascade. In Section 8,
we prove an additional regularity for the soliton-like solution. In Section 9,
we preclude the soliton-like solution. In Section 10, we prove Proposition
1.1.
## 2 Preliminaries
### 2.1 Notations
For any spacetime slab $I\times\mathbb{R}^{5}$, we use
$L_{t}^{q}L_{x}^{r}(I\times\mathbb{R}^{d})$ to denote the Banach space with
norm
$\|u\|_{L_{t}^{q}L_{x}^{r}}:=\left(\int_{I}\left(\int_{\mathbb{R}^{d}}|u(t,x)|^{r}\,\mathrm{d}x\right)^{q/r}\,\mathrm{d}t\right)^{1/q},$
with the usual modifications when $q$ or $r$ are infinity. When $q=r$ we
abbreviate $L_{t}^{q}L_{x}^{r}$ as $L_{t,x}^{q}$.
We use the ‘Japanese bracket’ convention $\langle
x\rangle:=(1+|x|^{2})^{1/2}$.
We use $X\lesssim Y$ or $Y\gtrsim X$ whenever $X\leq CY$ for some constant
$C>0$. If $C$ depends on some parameters, we will indicate this with
subscripts; for example, $X\lesssim_{u}Y$ denote the assertion that $X\leq
C_{u}Y$ for some $C_{u}$ depending on $u$. We denote by $X{\pm}$ any quantity
of the form $X\pm\varepsilon$ for any $\varepsilon>0$. we define the Fourier
transform on $\mathbb{R}^{d}$ by
$\hat{f}(\xi):=(2\pi)^{-\frac{d}{2}}\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}f(x)\,\mathrm{d}x.$
For $s\in\mathbb{R}$, we define the fractional differential/integral operators
$\widehat{|\nabla|^{s}f}(\xi):=|\xi|^{s}\hat{f}(\xi)$
and the homogeneous Sobolev norm
$\|f\|_{\dot{H}^{s}_{x}(\mathbb{R}^{d})}:=\big{\|}|\nabla|^{s}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d})}.$
The next following lemma is a form of Gronwall’s inequality that we will use
to handle some bootstrap argument below.
###### Lemma 2.1 (Gronwall’s inequality).
Given $\gamma>0$, $0<\eta<\frac{1}{2}(1-2^{-\gamma})$ and $\\{b_{k}\\}\in
l^{\infty}(\mathbb{Z}^{+})$. Let $\\{x_{k}\\}\in l^{\infty}(\mathbb{Z}^{+})$
be a non-negative sequence obeying
$x_{k}\leq b_{k}+\eta\sum_{l=0}^{\infty}2^{-\gamma|k-l|}x_{l}\quad\textrm{for
all}\,\,k\geq 0.$
Then
$x_{k}\lesssim\sum_{l=0}^{\infty}r^{|k-l|}b_{l}\quad\textrm{for all}\,\,k\geq
0$ (2.1)
for some $r=r(\eta)\in(2^{-\gamma},1)$. Moreover, $r\downarrow 2^{-\gamma}$ as
$\eta\downarrow 0$.
### 2.2 Basic harmonic analysis
Let $\varphi(\xi)$ be a radial bump function supported in the ball
$\\{\,\xi\in\mathbb{R}^{d}:|\xi|\leq\frac{11}{10}\,\\}$ and equal to 1 on the
ball $\\{\,\xi\in\mathbb{R}^{d}:|\xi|\leq 1\,\\}$. For each number $N>0$, we
define the Fourier multipliers
$\displaystyle\widehat{P_{\leq N}f}(\xi):=\varphi(\xi/N)\hat{f}(\xi),$
$\displaystyle\widehat{P_{>N}f}(\xi):=(1-\varphi(\xi/N))\hat{f}(\xi),$
$\displaystyle\widehat{P_{N}f}(\xi):=\psi(\xi/N)\hat{f}(\xi)=(\varphi(\xi/N)-\varphi(2\xi/N))\hat{f}(\xi)$
and similarly $P_{<N}$ and $P_{\geq N}$. We also define
$P_{M<\cdot\leq N}:=P_{\leq N}-P_{\leq M}=\sum_{M<N^{\prime}\leq
N}P_{N^{\prime}}$
for $M<N$. We will use these multipliers when $M$ and $N$ are dyadic numbers;
in particular, all summations over $N$ or $M$ are understood to be over dyadic
numbers. Nevertheless, it will occasionally be convenient to allow $M$ and $N$
to not be the power of 2. Note that, $P_{N}$ is not truly a projection; to get
around this, define
$\tilde{P}_{N}:=P_{N/2}+P_{N}+P_{2N}.$
These obey $\tilde{P}_{N}P_{N}=P_{N}\tilde{P}_{N}=P_{N}$.
The Littlewood-Paley operators commute with the propagator $e^{it\Delta}$, as
well as with differential operators such as $i\partial_{t}+\Delta$. We will
use basic properties of these operators many many times. First, we introduce
###### Lemma 2.2 (Bernstein).
For $1\leq p\leq q\leq\infty$,
$\displaystyle\big{\|}|\nabla|^{\pm
s}P_{N}f\big{\|}_{L_{x}^{q}(\mathbb{R}^{d})}\thicksim N^{\pm
s}\|P_{N}f\|_{L_{x}^{p}(\mathbb{R}^{d})},$ $\displaystyle\|P_{\leq
N}f\|_{L_{x}^{q}(\mathbb{R}^{d})}\lesssim N^{\frac{d}{p}-\frac{d}{q}}\|P_{\leq
N}f\|_{L_{x}^{p}(\mathbb{R}^{d})},$
$\displaystyle\|P_{N}f\|_{L_{x}^{q}(\mathbb{R}^{d})}\lesssim
N^{\frac{d}{p}-\frac{d}{q}}\|P_{N}f\|_{L_{x}^{p}(\mathbb{R}^{d})}.$
We also need the following fractional Leibniz rule, [11].
###### Lemma 2.3 (Fractional Leibniz rule).
Let $\alpha\in(0,\,1),\,\alpha_{1},\,\alpha_{2}\in[0,\alpha]$ with
$\alpha=\alpha_{1}+\alpha_{2}$. Let
$1<p,\,p_{1},\,p_{2},\,q,\,q_{1},\,q_{2}<\infty$ be such that
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$,
$\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$. Then
$\big{\|}D^{\alpha}(fg)-gD^{\alpha}f-fD^{\alpha}g\big{\|}_{L_{t}^{q}L_{x}^{p}}\lesssim\big{\|}D^{\alpha_{1}}f\big{\|}_{L_{t}^{q_{1}}L_{x}^{p_{1}}}\|D^{\alpha_{2}}g\|_{L_{t}^{q_{2}}L_{x}^{p_{2}}}.$
If $\alpha_{1}=0$, $q_{1}=\infty$ is allowed.
### 2.3 Strichartz’s estimates
Let $e^{it\Delta}$ be the free Schrödinger evolution. From the explicit
formula
$e^{it\Delta}f(x)=\frac{1}{(4\pi
it)^{d/2}}\int_{\mathbb{R}^{d}}e^{i|x-y|^{2}/4t}f(y)\,\mathrm{d}y,$
we deduce the standard dispersive inequality
$\|e^{it\Delta}f\|_{L_{x}^{\infty}(\mathbb{R}^{d})}\lesssim\frac{1}{|t|^{d/2}}\|f\|_{L_{x}^{1}(\mathbb{R}^{d})}$
for all $t\neq 0$.
Finer bounds on (frequency localized) linear propagator can be derived using
stationary phase:
###### Lemma 2.4 (Kernel estimates, [13]).
For any $m\geq 0$, the kernel of the linear propagator obeys the following
estimates:
$|(P_{N}e^{it\Delta})(x,y)|\lesssim_{m}\begin{cases}|t|^{-d/2},&|x-y|\thicksim
N|t|\\\ \dfrac{N^{d}}{|Nt|^{m}\langle
N|x-y|\rangle^{m}},&\textrm{otherwise}\end{cases}$
for $|t|\geq N^{-2}$ and
$|(P_{N}e^{it\Delta})(x,y)|\lesssim_{m}N^{d}\langle N|x-y|\rangle^{-m}$
for $|t|\leq N^{-2}$.
The standard Strichartz’s estimate reads:
###### Lemma 2.5 (Strichartz).
Let $k\geq 0$, $d\geq 3$. Let $I$ be a compact time interval, $t_{0}\in I$.
Then the function $u$ defined by
$u(t):=e^{i(t-t_{0})\Delta}u(t_{0})-i\int_{t_{0}}^{t}e^{i(t-t^{\prime})\Delta}f(t^{\prime})\,\mathrm{d}t^{\prime}$
(2.2)
obeys
$\|u\|_{\dot{S}^{k}(I)}\lesssim\|u(t_{0})\|_{\dot{H}^{k}_{x}}+\|f\|_{\dot{N}^{k}(I)}$
for any $t_{0}\in I$, where $\dot{S}^{k}(I)$ is the Strichartz norm, and
$\dot{N}^{k}(I)$ is its dual norm.
Proof. See, for example, [6], [8]. For a textbook treatment, see [20].
We also need the following weighted Strichartz’s inequality. It is very useful
in regions of space far from the origin.
###### Lemma 2.6 (Weighted Strichartz, [15]).
Let $I$ be an interval, $t_{0}\in I$, $u_{0}\in L_{x}^{2}(\mathbb{R}^{d})$,
$f\in L_{t,x}^{2(d+2)/(d+4)}(I\times\mathbb{R}^{d})$ be radially symmetric.
Then the function $u$ defined by $(\ref{e022})$ obeys the estimate
$\big{\|}|x|^{\frac{2(d-1)}{q}}u\big{\|}_{L_{t}^{q}L_{x}^{\frac{2q}{q-4}}(I\times\mathbb{R}^{d})}\lesssim\|u_{0}\|_{L_{x}^{2}(\mathbb{R}^{d})}+\|f\|_{L_{t}^{2}L_{x}^{2d/(d+2)}(I\times\mathbb{R}^{d})}$
for all $4\leq q\leq\infty$.
### 2.4 In/out decomposition
For a radially symmetric function $f$, we define the projection onto outgoing
spherical waves by
$[P^{+}f](r)=\frac{1}{2}\int_{0}^{\infty}r^{\frac{2-d}{2}}H_{\frac{d-2}{2}}^{(1)}(kr)\hat{f}(k)k^{\frac{d}{2}}\,\mathrm{d}k$
and the projection onto incoming spherical waves by
$[P^{-}f](r)=\frac{1}{2}\int_{0}^{\infty}r^{\frac{2-d}{2}}H_{\frac{d-2}{2}}^{(2)}(kr)\hat{f}(k)k^{\frac{d}{2}}\,\mathrm{d}k$
where $H_{\frac{d-2}{2}}^{(1)}$ denotes the Hankle function of the first kind
with order $\frac{d-2}{2}$ and $H_{\frac{d-2}{2}}^{(2)}$ denotes the Hankle
function of the second kind with the same order. We write $P_{N}^{\pm}$ for
the product $P^{\pm}P_{N}$, then we have
###### Lemma 2.7 (Kernel estimates, [15]).
For $|x|\gtrsim N^{-1}$ and $|t|\gtrsim N^{-2}$, the integral kernel obeys
$\big{|}[P_{N}^{\pm}e^{\mp
it\Delta}](x,y)\big{|}\lesssim\begin{cases}(|x||y|)^{-\frac{d-1}{2}}|t|^{-\frac{1}{2}},&|y|-|x|\thicksim
N|t|\\\ \dfrac{N^{d}}{(N|x|)^{\frac{d-1}{2}}\langle
N|y|\rangle^{\frac{d-1}{2}}}\langle
N^{2}t+N|x|-N|y|\rangle^{-m},&\textrm{otherwise}\end{cases}$
for any $m\geq 0$. For $|x|\gtrsim N^{-1}$ and $|t|\lesssim N^{-2}$, the
integral kernel obeys
$\big{|}[P_{N}^{\pm}e^{\mp
it\Delta}](x,y)\big{|}\lesssim\frac{N^{d}}{(N|x|)^{\frac{d-1}{2}}\langle
N|y|\rangle^{\frac{d-1}{2}}}\langle N|x|-N|y|\rangle^{-m}$
for any $m\geq 0$.
###### Lemma 2.8 (Properties of $P^{\pm}$, [15]).
We have:
* •
$P^{+}+P^{-}$ acts as the identity on $L_{rad}^{2}(\mathbb{R}^{d})$.
* •
Fix $N>0$, for any radially symmetric function $f\in
L_{x}^{2}(\mathbb{R}^{d})$,
$\|P^{\pm}P_{\geq
N}f\|_{L_{x}^{2}(|x|\geq\frac{1}{100N})}\lesssim\|f\|_{L_{x}^{2}(\mathbb{R}^{d})},$
with an $N$-independent constant.
### 2.5 Concentration compactness
In this subsection we record the linear profile decomposition statement due to
Shao [26]. We first recall the symmetries of the solutions to equation $(1.1)$
which fix the initial surface $t=0$.
###### Definition 2.1 (Symmetry group).
For any phase $\theta\in\mathbb{R}/2\pi\mathbb{Z}$, position
$x_{0}\in\mathbb{R}^{5}$, and scaling parameter $\lambda>0$, we define the
unitary transformation
$g_{\theta,x_{0},\lambda}:\dot{H}^{1/2}_{x}(\mathbb{R}^{5})\mapsto\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$
by
$[g_{\theta,x_{0},\lambda}f](x):=\lambda^{-2}e^{i\theta}f(\lambda^{-1}(x-x_{0})).$
Let $G$ denotes the collection of such transformations. For a function
$u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$, define
$T_{g_{\theta,x_{0},\lambda}}u:\lambda^{2}I\times\mathbb{R}^{5}\mapsto\mathbb{C}$
by
$[T_{g_{\theta,x_{0},\lambda}}u](t,x):=\lambda^{-2}e^{i\theta}u(\lambda^{-2}t,\lambda^{-1}(x-x_{0}))$
where $\lambda^{2}I:=\\{\,\lambda^{2}t:\,t\in I\,\\}$.
Let $G_{rad}\subset G$ denotes the collection of transformations in $G$ which
preserves radial symmetry, or more precisely
$G_{rad}:=\\{\,g_{\theta,0,\lambda}:\theta\in\mathbb{R}/2\pi\mathbb{Z},\,\lambda>0\,\\}.$
###### Remark 2.1.
$u$ is a maximal life-span solution to $(1.1)$ if and only if $T_{g}u$ is a
maximal life-span solution to $(1.1)$. Moreover,
$\|T_{g}u\|_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}=\|u\|_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})},\quad\|T_{g}u\|_{S(\lambda^{2}I)}=\|u\|_{S(I)},\quad\textrm{for
all}\;\;g\in G.$
We are now ready to state the linear profile decomposition.
###### Lemma 2.9 (Linear profiles, [26]).
Let $\\{u_{n}\\}_{n\geq 1}$ be a bounded sequence of functions in
$\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$. Then after passing to a subsequence if
necessary, there exist a sequence of functions $\\{\phi^{j}\\}_{j\geq
1}\subset\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$, group elements $g_{n}^{j}\in G$,
and times $t_{n}^{j}\in\mathbb{R}$ such that we have the decomposition
$u_{n}=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+\omega_{n}^{J}$
(2.3)
for all $J\geq 1$; $\omega_{n}^{J}\in\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$
obeying
$\lim_{J\to\infty}\limsup_{n\to\infty}\|e^{it\Delta}\omega_{n}^{J}\|_{L_{t}^{3}L_{x}^{15/4}(\mathbb{R}\times\mathbb{R}^{5})}=0.$
(2.4)
Moreover, for any $j^{\prime}\neq j$, we have the following orthogonal
property
$\lim_{n\to\infty}\left(\frac{\lambda_{n}^{j}}{\lambda_{n}^{j^{\prime}}}+\frac{\lambda_{n}^{j^{\prime}}}{\lambda_{n}^{j}}+\frac{|x_{n}^{j}-x_{n}^{j^{\prime}}|}{\lambda_{n}^{j}}+\frac{|t_{n}^{j}-t_{n}^{j^{\prime}}|}{(\lambda_{n}^{j})^{2}}\right)=0.$
(2.5)
For any $J\geq 1$
$\lim_{n\to\infty}\Big{[}\big{\|}|\nabla|^{\frac{1}{2}}u_{n}\big{\|}_{2}^{2}-\sum_{j=1}^{J}\big{\|}|\nabla|^{\frac{1}{2}}\phi^{j}\big{\|}_{2}^{2}-\big{\|}|\nabla|^{\frac{1}{2}}\omega_{n}^{J}\big{\|}_{2}^{2}\Big{]}=0.$
(2.6)
When $\\{u_{n}\\}$ is assumed to be radially symmetric, one can choose
$\phi^{j},\omega_{n}^{J}$ to be radially symmetric and $g_{n}^{j}\in G_{rad}$.
The error term also satisfies the following lemma
###### Lemma 2.10.
For all $J\geq 1,\,1\leq j\leq J$, the sequence
$e^{-it_{n}^{j}\Delta}[(g_{n}^{j})^{-1}\omega_{n}^{J}]$ converges weakly to
zero in $\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ as $n\to\infty$.
Proof. The proof is an analogue to that in [14], [9].
We end this section with a perturbation theorem
###### Theorem 2.1 (Long time perturbation theory).
Let $I\subset\mathbb{R}$ be a compact time interval and let $t_{0}\in I$. Let
$\tilde{u}:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ be a near-solution to
$(1.1)$ in the sense that
$i\partial_{t}\tilde{u}+\Delta\tilde{u}=F(\tilde{u})+e$
for some function $e$. Suppose $\tilde{u}$ satisfies
$\sup_{t\in I}\|\tilde{u}\|_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}\leq
A,\quad\|\tilde{u}\|_{S(I)}\leq M,\quad\|\tilde{u}\|_{X(I)}<+\infty,$
for some constant $M,\,A>0$. Assume also that
$\displaystyle\|u_{0}-\tilde{u}(t_{0})\|_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}\leq
A^{\prime},$
$\displaystyle\big{\|}|\nabla|^{1/2}e\big{\|}_{L_{t}^{1}L_{x}^{2}(I\times\mathbb{R}^{5})}\leq\varepsilon,$
$\displaystyle\big{\|}e^{i(t-t_{0})\Delta}(u_{0}-\tilde{u}(t_{0}))\big{\|}_{S(I)}\leq\varepsilon.$
Then, there exists a solution $u:I\times\mathbb{R}^{5}$ to $(1.1)$ with
$u(t_{0})=u_{0}$ such that
$\sup_{t\in
I}\|u-\tilde{u}(t)\|_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}+\|u-\tilde{u}\|_{S(I)}+\|u-\tilde{u}\|_{X(I)}\leq\varepsilon.$
## 3 Sharp constant for a Hardy-Littlewood-Sobolev type inequality
In this section we find the best constant to the following Hardy-Littlewood-
Sobolev type inequality
$\iint_{\mathbb{R}^{5}\times\mathbb{R}^{5}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|^{3}}\,\mathrm{d}x\,\mathrm{d}y\leq
C_{5}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{2}^{2}\big{\|}\nabla
u\big{\|}_{2}^{2},$ (3.1)
and obtain a sufficient condition for global existence of equation $(1.1)$
with initial data in
$\dot{H}^{1}_{x}(\mathbb{R}^{5})\cap\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$. We
find that the best constant
$C_{5}=2\big{\|}|\nabla|^{\frac{1}{2}}Q\big{\|}_{2}^{-2}$, where $Q$ is the
solution to $(\ref{e14})$. The approach is essentially from [30].
Consider the Weinstein functional
$J(u)=\dfrac{\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{2}^{2}\|\nabla
u\|_{2}^{2}}{\int_{\mathbb{R}^{5}}(|\cdot|^{-3}\ast|u|^{2})|u|^{2}\,\mathrm{d}x}\,,\qquad\forall
u\in\dot{H}^{1}_{x}(\mathbb{R}^{5})\cap\dot{H}^{1/2}_{x}(\mathbb{R}^{5}).$
First observe that if we set $u_{a,b}=au(bx)$, then
$J(u_{a,b})=J(u),\qquad\big{\|}|\nabla|^{\frac{1}{2}}u_{a,b}\big{\|}_{2}^{2}=a^{2}b^{-4}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{2}^{2},\qquad\|\nabla
u_{a,b}\|_{2}^{2}=a^{2}b^{-3}\|\nabla u\|_{2}^{2}.$
###### Theorem 3.1.
$C_{5}^{-1}=\inf_{u\in\dot{H}^{1}_{x}(\mathbb{R}^{5})\cap\dot{H}^{1/2}_{x}(\mathbb{R}^{5})\setminus\\{0\\}}J(u)$
can be obtained at some
$Q\in\dot{H}^{1}_{x}(\mathbb{R}^{5})\cap\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$. In
addition, $C_{5}=2\big{\|}|\nabla|^{\frac{1}{2}}Q\|_{2}^{-2}$.
Before proving the theorem, we present some compactness tools.
###### Lemma 3.1 (Radial Lemma).
Let $d\geq 3$, $u\in\dot{H}^{1}_{\rm
rad}(\mathbb{R}^{d})\cap\dot{H}^{1/2}_{\rm rad}(\mathbb{R}^{d})$ be a radially
symmetric function. Then
$\sup_{x\in\mathbb{R}^{d}}|x|^{\frac{2d-3}{4}}|u(x)|\lesssim\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{2}^{\frac{1}{2}}\|\nabla
u\|_{2}^{\frac{1}{2}}.$ (3.2)
Proof. Suppose first $u\in C^{\infty}_{c}(\mathbb{R}^{d})$. We have
$\displaystyle r^{\frac{2d-3}{2}}u(r)^{2}$
$\displaystyle=-\int_{r}^{\infty}\frac{\mathrm{d}}{\mathrm{d}s}\big{(}s^{\frac{2d-3}{2}}u(s)^{2}\big{)}\mathrm{d}s$
$\displaystyle\leq-2\int_{r}^{\infty}s^{\frac{2d-3}{2}}u(s)u^{\prime}(s)\mathrm{d}s$
$\displaystyle\lesssim\big{\|}|x|^{-\frac{1}{2}}u\|_{2}\big{\|}\nabla
u\|_{2},$
$(\ref{a2})$ follows from Hardy’s inequality. The general case then follows by
the density argument.
###### Lemma 3.2 (Compactness Lemma).
$\dot{H}^{1}_{\rm rad}(\mathbb{R}^{d})\cap\dot{H}^{1/2}_{\rm
rad}(\mathbb{R}^{d})\hookrightarrow L^{p}(\mathbb{R}^{d})\quad\textrm{for
all}\quad\frac{2d}{d-1}<p<\frac{2d}{d-2}.$
Proof. Let $\\{u_{k}\\}$ be a bounded sequence in $\dot{H}^{1}_{\rm
rad}\cap\dot{H}^{1/2}_{\rm rad}$, then by the weak compactness principle,
there exists $u\in\dot{H}^{1}_{\rm rad}\cap\dot{H}^{1/2}_{\rm rad}$ such that
$u_{k}\rightharpoonup u$ weakly in $\dot{H}^{1}_{\rm
rad}\cap\dot{H}^{1/2}_{\rm rad}$.
For $\varepsilon>0$, let $R>0$ to be chosen later. Given $p$ as in the
statement, we have
$\displaystyle\|u_{k}-u\|_{L^{p}(\mathbb{R}^{d})}$
$\displaystyle\leq\|u_{k}-u\|_{L^{p}(B_{R})}+\|u_{k}-u\|_{L^{p}(\\{\,x\,:\,|x|>R\,\\})}$
$\displaystyle\leq\|u_{k}-u\|_{L^{p}(B_{R})}+\|u_{k}-u\|_{L^{\infty}(\\{\,x\,:\,|x|>R\,\\})}^{\frac{p(d-1)-2d}{(d-1)p}}\|u_{k}-u\|_{L^{\frac{2d}{d-1}}(\mathbb{R}^{d})}^{\frac{2d}{(d-1)p}}.$
By Lemma 3.1, we first choose $R$ large enough so that
$\|u_{k}-u\|_{L^{\infty}(\\{\,x\,:\,|x|>R\,\\})}^{\frac{p(d-1)-2d}{(d-1)p}}\|u_{k}-u\|_{L^{\frac{2d}{d-1}}(\mathbb{R}^{d})}^{\frac{2d}{(d-1)p}}\leq\frac{\varepsilon}{2}.$
On the other hand, it follows from Rellich’s compactness lemma that
$\|u_{k}-u\|_{L^{p}(B_{R})}\leq\frac{\varepsilon}{2}$
for large $k$ and so $\|u_{k}-u\|_{L^{p}(\mathbb{R}^{d})}\leq\varepsilon$.
This proves the lemma.
Proof of Theorem 3.1 . Since $J(u)\geq 0$, we may find a minimizing sequence
$\\{u_{k}\\}\subset\dot{H}^{1}\cap\dot{H}^{1/2}$ such that
$C_{5}^{-1}=\inf J(u)=\lim_{k\to\infty}J(u_{k}).$
By symmetric rearrangement technique, we may assume $u_{k}>0$ and is radially
symmetric for all $k$.
Set $a_{k}=\big{\|}|\nabla|^{\frac{1}{2}}u_{k}\big{\|}_{2}^{3}/{\|\nabla
u_{k}\|_{2}^{4}}$,
$b_{k}=\big{\|}|\nabla|^{\frac{1}{2}}u_{k}\big{\|}_{2}^{2}/{\|\nabla
u_{k}\|_{2}^{2}}$, and $Q_{k}=a_{k}u(b_{k}x)$. Then $Q_{k}\geq 0$, is radially
symmetric. Moreover, we have
$\big{\|}|\nabla|^{\frac{1}{2}}Q_{k}\big{\|}_{2}=\|\nabla
Q_{k}\|_{2}=1,\quad\lim_{k\to\infty}J(Q_{k})=C_{5}^{-1}.$
Since $\\{Q_{k}\\}\subset\dot{H}^{1}_{\rm rad}\cap\dot{H}^{1/2}_{\rm rad}$ is
uniformly bounded, up to a subsequence, $Q_{k}\rightharpoonup Q^{*}$ in
$\dot{H}^{1}_{\rm rad}\cap\dot{H}^{1/2}_{\rm rad}$, and
$\big{\|}|\nabla|^{\frac{1}{2}}Q^{*}\big{\|}_{2}\leq 1$, $\|\nabla
Q^{*}\|_{2}\leq 1$. From Lemma $3.2$, $Q_{k}\to Q^{*}$ in
$L^{p}(\mathbb{R}^{5})$ for $\frac{5}{2}<p<\frac{10}{3}$. Furthermore, we have
$\iint_{\mathbb{R}^{5}\times\mathbb{R}^{5}}\frac{|Q_{k}(x)|^{2}|Q_{k}(y)|^{2}}{|x-y|^{3}}\,\mathrm{d}x\mathrm{d}y\longrightarrow\iint_{\mathbb{R}^{5}\times\mathbb{R}^{5}}\frac{|Q^{*}(x)|^{2}|Q^{*}(y)|^{2}}{|x-y|^{3}}\,\mathrm{d}x\mathrm{d}y\quad\textrm{as}\,k\to\infty.$
This is easily checked by a direct computation using the Hardy-Littlewood-
Sobolev inequality.
Thus
$C_{5}^{-1}\leq
J(Q^{*})\leq\frac{1}{\int_{\mathbb{R}^{5}}(|\cdot|^{-3}\ast|Q^{*}|^{2})|Q^{*}|^{2}\,\mathrm{d}x}=\lim_{k\to\infty}J(Q_{k})=C_{5}^{-1}.$
This implies that $\big{\|}|\nabla|^{\frac{1}{2}}Q^{*}\big{\|}_{2}^{2}\|\nabla
Q^{*}\|_{2}^{2}=1$, which further gives
$\big{\|}|\nabla|^{\frac{1}{2}}Q^{*}\big{\|}_{2}=\|\nabla Q^{*}\|_{2}=1$.
Since $Q^{*}$ is a minimizer, it satisfies the Euler-Lagrangian equation
$\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\Big{|}_{\varepsilon=0}J(Q^{*}+\varepsilon\phi)=0\quad\textrm{for
all}\,\phi\in C_{0}^{\infty}(\mathbb{R}^{5}).$
Taking into account the fact that
$\big{\|}|\nabla|^{\frac{1}{2}}Q^{*}\big{\|}_{2}=\|\nabla Q^{*}\|_{2}=1$, we
have
$-\Delta
Q^{*}+(-\Delta)^{1/2}Q^{*}-2C_{5}^{-1}(|\cdot|^{-3}\ast|Q^{*}|^{2})Q^{*}=0.$
Let $Q^{*}=\sqrt{C_{5}/2}Q$, then $Q$ solves $(\ref{e14})$.
By the fact that $\big{\|}|\nabla|^{\frac{1}{2}}Q^{*}\big{\|}_{2}=1$, it
yields $C_{5}=2\big{\|}|\nabla|^{\frac{1}{2}}Q\big{\|}_{2}^{-2}$. $\square$
###### Proposition 3.1.
Let
$u_{0}\in\dot{H}^{1}_{x}(\mathbb{R}^{5})\cap\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$.
Suppose
$\sup_{t}\||\nabla|^{\frac{1}{2}}u\|_{2}<\||\nabla|^{\frac{1}{2}}Q\|_{2}$,
then the solution to $(1.1)$ is global.
Proof. It is a consequence of the energy conservation
$E(u(t))=\frac{1}{2}\int_{\mathbb{R}^{5}}|\nabla
u|^{2}\,\mathrm{d}x-\frac{1}{4}\iint_{\mathbb{R}^{5}\times\mathbb{R}^{5}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|^{3}}\,\mathrm{d}x\mathrm{d}y,$
and $(\ref{a1})$.
## 4 Reduction to almost periodic solution
In this section we will prove Theorem 1.3. The main step toward this end is to
prove a Palais-Smale condition modulo scaling.
For any $A>0$, define
$L(A)=\sup\left\\{\|u\|_{S(I)}:\,\,u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}\,\textrm{such
that }\,\sup_{t\in I}\|u\|_{\dot{H}^{1/2}_{x}}\leq A\right\\}.$
Here, the supremum is taken over all solutions
$u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ to (1.1) satisfying $\sup_{t\in
I}\|u\|_{\dot{H}^{1/2}_{x}}\leq A$. Note that $L(A)$ is non-decreasing and
left-continuous. On the other hand, from Theorem 1.1,
$L(A)\lesssim A\quad\textrm{for }\quad A\leq\delta_{0},$
where $\delta_{0}$ is the threshold from the small data global well-posedness
theory. Theorem 1.2 states that for each
$A<\frac{\sqrt{6}}{3}\|Q\|_{\dot{H}^{1/2}}$, $L(A)<\infty$. Therefore, if
Theorem 1.2 failed, there exists
$\delta_{0}<A_{c}<\frac{\sqrt{6}}{3}\|Q\|_{\dot{H}^{1/2}}$ such that
$L(A)<+\infty$ for $A<A_{c}$, $L(A)=+\infty$ for $A\geq A_{c}$.
Convention: In this section and the rest sections, we write $|x|^{-3}\ast$ as
$|\nabla|^{-2}$ since they are equivalent up to a constant. Moreover, we
ignore the distinction between a function and its conjugation as they make no
difference in our discussion.
### 4.1 Palais-Smale condition modulo scaling
###### Proposition 4.1.
Let $u_{n}:I_{n}\times\mathbb{R}^{5}\mapsto\mathbb{C}$ be a sequence of
solutions to $(1.1)$ such that
$\limsup_{n\to\infty}\sup_{t\in I_{n}}\|u_{n}(t)\|_{\dot{H}^{1/2}_{x}}=A_{c}.$
(4.1)
Let $t_{n}\in I_{n}$ be a time sequence such that
$\lim_{n\to\infty}\|u_{n}\|_{S(-\infty,\;t_{n})}=\lim_{n\to\infty}\|u_{n}\|_{S(t_{n},\;\infty)}=\infty.$
Then there exists a subsequence of $u_{n}(t_{n})$, which converges in
$\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ modulo scaling.
The proof of this Proposition is achieved through several steps.
Proof. By time-translation invariant of (1.1), we may set $t_{n}=0$ for all
$n\geq 1$. Then
$\lim_{n\to\infty}\|u_{n}\|_{S(-\infty,\;0)}=\lim_{n\to\infty}\|u_{n}\|_{S(0,\;\infty)}=\infty.$
(4.2)
Now applying Lemma 2.9 to the sequence $u_{n}(0)$, and up to a subsequence, we
obtain a decomposition
$u_{n}(0)=\sum_{j=1}^{J}g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}+\omega_{n}^{J}$
for any $J\geq 1$, $n\geq 1$.
By passing to a further subsequence, we may assume $t_{n}^{j}$ converges to
some $t^{j}\in[-\infty,+\infty]$ for each $j$. If $t^{j}$ is finite, then
replacing $\phi^{j}$ by $e^{it^{j}\Delta}\phi^{j}$, we may set $t^{j}=0$.
Adding $e^{it_{n}^{j}\Delta}\phi^{j}-\phi^{j}$ to the error term
$\omega_{n}^{J}$, we may assume $t_{n}^{j}\equiv 0$ . Thus, we only need to
deal with $t_{n}^{j}\equiv 0$ and $t_{n}^{j}\to\pm\infty$.
For each $\phi^{j}$ and $t_{n}^{j}$, define nonlinear profile
$v^{j}:I^{j}\times\mathbb{R}^{5}\mapsto\mathbb{C}$ as follows:
* •
If $t_{n}^{j}\equiv 0$, then $v^{j}$ is the maximal life-span solution to
(1.1) with initial data $v^{j}(0)=\phi^{j}$.
* •
If $t_{n}^{j}\to\infty$, then $v^{j}$ is the maximal life-span solution to
(1.1) that scatters forward to $e^{it\Delta}\phi^{j}$.
* •
If $t_{n}^{j}\to-\infty$, then $v^{j}$ is the maximal life-span solution to
(1.1) that scatters backward to $e^{it\Delta}\phi^{j}$.
For each $j$, $n\geq 1$, define
$v_{n}^{j}:I_{n}^{j}\times\mathbb{R}^{5}\mapsto\mathbb{C}$ by
$v_{n}^{j}(t):=T_{g_{n}^{j}}[v^{j}(\cdot+t_{n}^{j})](t),$
where $I_{n}^{j}:=\\{\,t\in\mathbb{R}:\,(\lambda_{n}^{j})^{-2}t+t_{n}^{j}\in
I^{j}\,\\}$. Then for each $j$, $v_{n}^{j}$ is also a maximal life-span
solution to (1.1) with initial data $v_{n}^{j}(0)=g_{n}^{j}v^{j}(t_{n}^{j})$,
and with maximal life-span $I_{n}^{j}=(-T^{-}_{n,j},\;T^{+}_{n,j})$,
$-\infty\leq-T_{n,j}^{-}<0<T_{n,j}^{+}\leq+\infty$.
With these preliminaries out of the way, we first have
Step 1: There exists $J_{0}\geq 1$ such that, for all $j\geq J_{0}$, $n$
sufficiently large
$\sup_{t\in\mathbb{R}}\|v_{n}^{j}(t)\|_{\dot{H}^{1/2}_{x}}+\|v_{n}^{j}\|_{S(\mathbb{R})}+\|v_{n}^{j}\|_{X(\mathbb{R})}\lesssim\|\phi^{j}\|_{\dot{H}^{1/2}_{x}}.$
(4.3)
Proof. From $(\ref{e24})$, there exists $J_{0}\geq 1$ such that for
sufficiently large $n$
$\|\phi^{j}\|_{\dot{H}^{1/2}_{x}}\leq\delta_{0}\quad\textrm{for all}\quad
j\geq J_{0}$
where $\delta_{0}$ is the threshold from the small data theory. Hence, by
Theorem 1.1, $v_{n}^{j}$ is global and
$\sup_{t\in\mathbb{R}}\|v_{n}^{j}\|_{\dot{H}^{1/2}_{x}}+\|v_{n}^{j}\|_{X(\mathbb{R})}+\|v_{n}^{j}\|_{S(\mathbb{R})}\lesssim\|\phi^{j}\|_{\dot{H}^{1/2}_{x}}.$
for all $j\geq J_{0}$ and all $n$ sufficiently large.
Step 2: There exists $1\leq j_{0}<J_{0}$ such that
$\limsup_{n\to\infty}\|v_{n}^{j_{0}}\|_{S(0,\;T_{n,j_{0}}^{+})}=\infty.$
Proof. Suppose to the contrary that for all $1\leq j<J_{0}$
$\limsup_{n\to\infty}\|v_{n}^{j}\|_{S(0,\;T_{n,j}^{+})}\leq M<\infty$ (4.4)
for some $M>0$. This implies that $T_{n,j}^{+}=\infty$ for all $1\leq j<J_{0}$
and all sufficiently large $n$. Given $\eta>0$, divide $(0,\infty)$ into
subintervals $I_{k}$ such that on each $I_{k}$,
$\|v_{n}^{j}\|_{S(I_{k})}\leq\eta$. By Strichartz’s estimate, we have for all
$1\leq j<J_{0}$ and all large $n$ that
$\|v_{n}^{j}\|_{X(0,\infty)}<\infty.$ (4.5)
Indeed, let $\eta>0$, divide $(0,\infty)$ into subintervals
$I_{k}=[t_{k},t_{k+1}]$ such that on each $I_{k}$ we have
$\|v_{n}^{j}\|_{S(I_{k})}\leq\eta$. Note that, there are at most
$\eta^{-1}\times M$ such intervals. Applying the Strichartz estimate
$\displaystyle\|v_{n}^{j}\|_{X(I_{k})}$ $\displaystyle\lesssim$
$\displaystyle\|v^{j}_{n}(t_{k})\|_{\dot{H}^{1/2}_{x}}+\big{\|}|\nabla|^{\frac{1}{2}}F(v_{n}^{j})\big{\|}_{L_{t}^{1}L_{x}^{2}}$
$\displaystyle\lesssim$ $\displaystyle
A_{c}+\|v_{n}^{j}\|^{2}_{S(I_{k})}\|v_{n}^{j}\|_{X(I_{k})}.$
If we choose $\eta>0$ sufficiently small, then
$\|v_{n}^{j}\|_{X(I_{k})}\lesssim A_{c}.$
Summing over all $I_{k}$, we achieve $(\ref{e35})$.
Combining $(\ref{e34})$ with Step 1, and then using $(\ref{e24})$ and
$(\ref{e31})$, we have that for all sufficiently large $n$,
$\sum_{j\geq
1}\sup_{t\in(0,\infty)}\|v_{n}^{j}\|_{\dot{H}^{1/2}_{x}}+\|v_{n}^{j}\|_{S(0,\infty)}+\|v_{n}^{j}\|_{X(0,\infty)}\lesssim
1+A_{c}.$ (4.6)
Next, we will use perturbation theorem to obtain a bound on
$\|u_{n}\|_{S(0,\,\infty)}$ for $n$ sufficiently large.
Define an approximation to $u_{n}$ by
$u_{n}^{J}(t):=\sum_{j=1}^{J}v_{n}^{j}(t)+e^{it\Delta}\omega_{n}^{J}.$ (4.7)
Then, by the definition of nonlinear profile
$\displaystyle\limsup_{n\to\infty}\|u_{n}^{J}(0)-u_{n}(0)\|_{\dot{H}^{1/2}_{x}}$
$\displaystyle=\limsup_{n\to\infty}\Big{\|}\sum_{j=1}^{J}g_{n}^{j}v^{j}(t_{n}^{j})-g_{n}^{j}e^{it_{n}^{j}\Delta}\phi^{j}\Big{\|}_{\dot{H}^{1/2}_{x}}$
$\displaystyle\lesssim\limsup_{n\to\infty}\sum_{j=1}^{J}\|v^{j}(t_{n}^{j})-e^{it_{n}^{j}\Delta}\phi^{j}\|_{\dot{H}^{1/2}_{x}}=0.$
Note that $(\ref{e23})$ with a few computations yields that for all $j\geq 1$
$\limsup_{n\to\infty}\|v_{n}^{j^{\prime}}v_{n}^{j}\|_{S(0,\;\infty)}=0\quad$
(4.8)
for any $j^{\prime}\neq j$.(Such an asymptotic orthogonal property was well
developed in [12], [26], we refer to them for details.)
Thus, by $(\ref{e22})$, $(\ref{e36})$ and $(\ref{e37})$
$\displaystyle\lim_{J\to\infty}\limsup_{n\to\infty}\|u_{n}^{J}\|_{S(0,\;\infty)}$
$\displaystyle\lesssim$
$\displaystyle\lim\limits_{J\to\infty}\limsup\limits_{n\to\infty}\Big{(}\Big{\|}\sum_{j=1}^{J}v_{n}^{j}\Big{\|}_{S(0,\;\infty)}+\big{\|}e^{it\Delta}\omega_{n}^{J}\big{\|}_{S(0,\;\infty)}\Big{)}$
(4.9) $\displaystyle\lesssim$
$\displaystyle\lim\limits_{J\to\infty}\limsup\limits_{n\to\infty}\sum_{j=1}^{J}\|v_{n}^{j}\|_{S(0,\;\infty)}\lesssim
1+A_{c}.$
By the same argument as that to derive $(\ref{e35})$ from $(\ref{e34})$, we
obtain
$\lim_{J\to\infty}\limsup_{n\to\infty}\|u_{n}^{J}\|_{X(0,\;\infty)}<\infty.$
Now, we have to verify that
$\lim_{J\to\infty}\limsup_{n\to\infty}\Big{\|}|\nabla|^{\frac{1}{2}}\big{[}(i\partial_{t}+\Delta)u_{n}^{J}+F(u_{n}^{J})\big{]}\Big{\|}_{L_{t}^{1}L_{x}^{2}((0,\infty)\times\mathbb{R}^{5})}=0.$
Using the triangle inequality, we need to show on
$(0,\infty)\times\mathbb{R}^{5}$ that
$\lim_{J\to\infty}\limsup_{n\to\infty}\Big{\|}|\nabla|^{\frac{1}{2}}\Big{[}\sum_{j=1}^{J}F(v_{n}^{j})-F(\sum_{j=1}^{J}v_{n}^{j})\Big{]}\Big{\|}_{L_{t}^{1}L_{x}^{2}}=0$
(4.10)
and
$\lim_{J\to\infty}\limsup_{n\to\infty}\Big{\|}|\nabla|^{\frac{1}{2}}\big{(}F(u_{n}^{J}-e^{it\Delta}\omega_{n}^{J})-F(u_{n}^{J})\big{)}\Big{\|}_{L_{t}^{1}L_{x}^{2}}=0.$
(4.11)
We first consider $(\ref{e39})$. By expanding out the nonlinearity
$\displaystyle\Big{|}|\nabla|^{\frac{1}{2}}\Big{[}\sum_{j=1}^{J}F(v_{n}^{j})-F(\sum_{j=1}^{J}v_{n}^{j})\Big{]}\Big{|}$
$\displaystyle\leq$
$\displaystyle\sum_{j_{1},j_{2},j_{3}=1}^{J}\Big{|}|\nabla|^{\frac{1}{2}}\big{[}\big{(}|\nabla|^{-2}(v_{n}^{j_{1}}{v_{n}^{j_{2}}})\big{)}v_{n}^{j_{3}}\big{]}\Big{|},$
where at least two of $j_{1},j_{2},j_{3}$ are different.
Note that the nonlocal action (i.e. convolution) break up the spatial
orthogonality, whereas time orthogonality will be preserved. Recalling the
radial assumption, we may assume $j_{2}\neq j_{1}$. Thus, using the fractional
Leibniz rule, Hölder’s inequality,the Hardy-Littlewood-Sobolev inequality, and
$(\ref{e37})$, we obtain on $(0,\infty)\times\mathbb{R}^{5}$ that
$\displaystyle\lim_{J\to\infty}\limsup_{n\to\infty}\Big{\|}|\nabla|^{\frac{1}{2}}\Big{[}\sum_{j=1}^{J}F(v_{n}^{j})-F(\sum_{j=1}^{J}v_{n}^{j})\Big{]}\Big{\|}_{L_{t}^{1}L_{x}^{2}}$
$\displaystyle\lesssim_{J}$
$\displaystyle\lim_{J\to\infty}\limsup_{n\to\infty}\sum_{j_{1},j_{2},j_{3}=1}^{J}\Big{(}\big{\|}|\nabla|^{\frac{1}{2}}\big{(}|\nabla|^{-2}(v_{n}^{j_{1}}{v_{n}^{j_{2}}})\big{)}v_{n}^{j_{3}}\big{\|}_{L_{t}^{1}L_{x}^{2}}+\big{\|}\big{(}|\nabla|^{-2}(v_{n}^{j_{1}}{v_{n}^{j_{2}}})\big{)}|\nabla|^{\frac{1}{2}}v_{n}^{j_{3}}\big{\|}_{L_{t}^{1}L_{x}^{2}}\Big{)}$
$\displaystyle\lesssim_{J}$
$\displaystyle\lim_{J\to\infty}\limsup_{n\to\infty}\sum_{j_{1},j_{2},j_{3}=1}^{J}\Big{(}\big{\|}|\nabla|^{\frac{1}{2}}\big{(}|\nabla|^{-2}(v_{n}^{j_{1}}{v_{n}^{j_{2}}})\big{)}\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{30}{7}}}\|v_{n}^{j_{3}}\|_{S(0,\infty)}$
$\displaystyle\hskip
113.81102pt+\big{\|}|\nabla|^{-2}(v_{n}^{j_{1}}{v_{n}^{j_{2}}})\big{\|}_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{15}{2}}}\|v_{n}^{j_{3}}\|_{X(0,\infty)}\Big{)}$
$\displaystyle\lesssim_{J}$
$\displaystyle\lim_{J\to\infty}\limsup_{n\to\infty}\sum_{j_{1},j_{2},j_{3}=1}^{J}\|v_{n}^{j_{1}}{v_{n}^{j_{2}}}\|_{L_{t}^{\frac{3}{2}}L_{x}^{\frac{15}{8}}}=0,$
where the last limit is also a consequence of the orthogonality.
For $(\ref{e310})$, note that on $(0,\infty)\times\mathbb{R}^{5}$
$\displaystyle\big{\|}|\nabla|^{\frac{1}{2}}(F(u_{n}^{J}-e^{it\Delta}\omega_{n}^{J})-F(u_{n}^{J}))\big{\|}_{L_{t}^{1}L_{x}^{2}}$
$\displaystyle\lesssim$
$\displaystyle\big{\|}|\nabla|^{\frac{1}{2}}[\big{(}|\nabla|^{-2}(u_{n}^{J}{e^{it\Delta}\omega_{n}^{J}})\big{)}u_{n}^{J}]\big{\|}_{L_{t}^{1}L_{x}^{2}}+\big{\|}|\nabla|^{\frac{1}{2}}[(|\nabla|^{-2}(u_{n}^{J}{e^{it\Delta}\omega_{n}^{J}}))e^{it\Delta}\omega_{n}^{J}]\big{\|}_{L_{t}^{1}L_{x}^{2}}$
$\displaystyle+\big{\|}|\nabla|^{\frac{1}{2}}[(|\nabla|^{-2}|u_{n}^{J}|^{2})e^{it\Delta}\omega_{n}^{J}]\big{\|}_{L_{t}^{1}L_{x}^{2}}+\big{\|}|\nabla|^{\frac{1}{2}}[(|\nabla|^{-2}|e^{it\Delta}\omega_{n}^{J}|^{2})e^{it\Delta}\omega_{n}^{J}]\big{\|}_{L_{t}^{1}L_{x}^{2}}$
$\displaystyle+\big{\|}|\nabla|^{\frac{1}{2}}[(|\nabla|^{-2}|e^{it\Delta}\omega_{n}^{J}|^{2})u_{n}^{J}]\big{\|}_{L_{t}^{1}L_{x}^{2}}.$
Using $(\ref{e22})$, Hölder’s inequality, the Hardy-Littlewood-Sobolev
inequality, the above terms on the right hand side will go to zero as $J$, $n$
tend to $\infty$, except
$\big{\|}|\nabla|^{\frac{1}{2}}[(|\nabla|^{-2}|u_{n}^{J}|^{2})e^{it\Delta}\omega_{n}^{J}]\big{\|}_{L_{t}^{1}L_{x}^{2}((0,\;\infty)\times\mathbb{R}^{5})}.$
By the fractional Leibniz rule and the triangle inequality, it suffices to
estimate
$\big{\|}|\nabla|^{\frac{1}{2}}(|\nabla|^{-2}|u_{n}^{J}|^{2})e^{it\Delta}\omega_{n}^{J}\big{\|}_{L_{t}^{1}L_{x}^{2}((0,\;\infty)\times\mathbb{R}^{5})}$
and
$\big{\|}(|\nabla|^{-2}|u_{n}^{J}|^{2})|\nabla|^{\frac{1}{2}}e^{it\Delta}\omega_{n}^{J}\big{\|}_{L_{t}^{1}L_{x}^{2}((0,\;\infty)\times\mathbb{R}^{5})}.$
Using Hölder’s, the Hardy-Littlewood-Sobolev inequality, and $(\ref{e22})$,
the first integral goes to zero when $J$, $n$ go to infinity. Then, we are
reduced to showing that the second integral has limit zero with $J$, $n$.
Replace $u_{n}^{J}$ with its definition formula $(\ref{e00})$ to get on
$(0,\infty)\times\mathbb{R}^{5}$
$\displaystyle\big{\|}(|\nabla|^{-2}|u_{n}^{J}|^{2})|\nabla|^{\frac{1}{2}}e^{it\Delta}\omega_{n}^{J}\big{\|}_{L_{t}^{1}L_{x}^{2}}$
$\displaystyle\lesssim$
$\displaystyle\sum_{j=1}^{J}\big{\|}(|\nabla|^{-2}|v_{n}^{j}|^{2})|\nabla|^{\frac{1}{2}}e^{it\Delta}\omega_{n}^{J}\big{\|}_{L_{t}^{1}L_{x}^{2}}+\sum_{j^{\prime}\neq
j}\big{\|}(|\nabla|^{-2}(v_{n}^{j}{v_{n}^{j^{\prime}}}))|\nabla|^{\frac{1}{2}}e^{it\Delta}\omega_{n}^{J}\big{\|}_{L_{t}^{1}L_{x}^{2}}$
$\displaystyle+\sum_{j=1}^{J}\big{\|}(|\nabla|^{-2}(v_{n}^{j}{e^{it\Delta}\omega_{n}^{J}}))|\nabla|^{\frac{1}{2}}e^{it\Delta}\omega_{n}^{J}\big{\|}_{L_{t}^{1}L_{x}^{2}}:={\rm
I_{1}+I_{2}+I_{3}}.$
By $(\ref{e23})$, ${\rm I}_{2}$ will go to zero as $J$, $n$ go to infinity.
Using $(\ref{e22})$, ${\rm I}_{3}$ vanishes as $J$, $n$ tend to infinity. So,
We only need to show that ${\rm I}_{1}$ also vanishes.
For arbitrary $\eta>0$, from $(\ref{e38})$, there exists $J^{\prime}(\eta)\geq
1$ such that
$\sum_{j\geq J^{\prime}}\|v_{n}^{j}\|_{S(0,\;\infty)}\leq\eta.$
Thus, we are reduced to proving that
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}(|\nabla|^{-2}|v_{n}^{j}|^{2})|\nabla|^{\frac{1}{2}}e^{it\Delta}\omega_{n}^{J}\big{\|}_{L_{t}^{1}L_{x}^{2}((0,\;\infty)\times\mathbb{R}^{5})}=0\quad\textrm{for
all}\quad 1\leq j\leq J^{\prime}.$
Fix $1\leq j\leq J^{\prime}$. A change of variables yields
$\big{\|}(|\nabla|^{-2}|v_{n}^{j}|^{2})|\nabla|^{\frac{1}{2}}e^{it\Delta}\omega_{n}^{J}\big{\|}_{L_{t}^{1}L_{x}^{2}}=\big{\|}\big{(}|\nabla|^{-2}|v^{j}|^{2}\big{)}|\nabla|^{\frac{1}{2}}\big{[}T_{(g_{n}^{j})^{-1}}(e^{it\Delta}\omega_{n}^{J})\big{]}(\cdot-
t_{n}^{j})\big{\|}_{L_{t}^{1}L_{x}^{2}}.$
Let
$\tilde{\omega}_{n}^{J}:=[T_{(g_{n}^{j})^{-1}}(e^{it\Delta}\omega_{n}^{J})](\cdot-
t_{n}^{j})$, $\mathcal{I}:v^{j}\mapsto(|\nabla|^{-2}|v^{j}|^{2})$. Note that
$\|\tilde{\omega}_{n}^{J}\|_{S(0,\;\infty)}=\|e^{it\Delta}\omega_{n}^{J}\|_{S(0,\infty)},\quad\|\tilde{\omega}_{n}^{J}\|_{X(0,\infty)}=\|e^{it\Delta}\omega_{n}^{J}\|_{X(0,\;\infty)}.$
(4.12)
Using Hölder’s inequality, the interpolation theorem, we see
$\displaystyle\big{\|}\mathcal{I}(v^{j})|\nabla|^{\frac{1}{2}}\tilde{\omega}_{n}^{J}\big{\|}_{L_{t}^{1}L_{x}^{2}}$
$\displaystyle\lesssim$
$\displaystyle\|\mathcal{I}(v^{j})\|_{L_{t}^{12/7}L_{x}^{15}}\big{\|}|\nabla|^{\frac{1}{2}}\tilde{\omega}_{n}^{J}\big{\|}_{L_{t}^{12/5}L_{x}^{30/13}}$
$\displaystyle\lesssim$
$\displaystyle\|v^{j}\|_{L_{t}^{24/7}L_{x}^{30/7}}\big{\|}\tilde{\omega}_{n}^{J}\big{\|}_{X(0,\infty)}^{1/2}\big{\|}|\nabla|^{\frac{1}{2}}\tilde{\omega}_{n}^{J}\big{\|}_{L_{t,x}^{2}}^{1/2}.$
By density, we may assume $\mathcal{I}(v_{n}^{j})\in
C_{c}^{\infty}(\mathbb{R}\times\mathbb{R}^{5})$. It thus suffices to verify
$\lim_{J\to\infty}\limsup_{n\to\infty}\big{\|}|\nabla|^{\frac{1}{2}}\tilde{\omega}_{n}^{J}\big{\|}_{L_{t,x}^{2}(K)}=0$
for any compact $K\subset\mathbb{R}\times\mathbb{R}^{5}$. This is a
consequence of $(\ref{e22})$ and the following lemma:
###### Lemma 4.1.
Let $\phi\in\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$. Then
$\big{\|}|\nabla|^{\frac{1}{2}}e^{it\Delta}\phi\big{\|}_{L_{t,x}^{2}(\,[-T,\;T]\times\\{\,x:\,|x|\leq
R\,\\}\,)}^{2}\lesssim
T^{\frac{1}{6}}R^{\frac{5}{3}}\|e^{it\Delta}\phi\|_{L_{t}^{3}L_{x}^{15/4}}\big{\|}|\nabla|^{\frac{1}{2}}\phi\big{\|}_{L_{x}^{2}}.$
Proof. The proof is analogous to the one of Lemma 2.5 in [14].
Now, applying perturbation theorem with $\tilde{u}=u_{n}^{J}$,
$e=(i\partial_{t}+\Delta)u_{n}^{J}-F(u_{n}^{J})$, and using $(\ref{e38})$, we
obtain
$\|u_{n}^{J}\|_{S(0,\;\infty)}\lesssim 1+A_{c}$
for all sufficiently large $n$. This contradicts $(\ref{e32})$, which
concludes Step 2.
Combining Step 1 with Step 2, and rearranging the indices, we may find $1\leq
J_{1}\leq J_{0}$ such that
$\displaystyle\limsup_{n\to\infty}\|v_{n}^{j}\|_{S(0,\;T_{n,j}^{+})}=\infty\quad\textrm{for}\,\,1\leq
j\leq J_{1},$
$\displaystyle\limsup_{n\to\infty}\|v_{n}^{j}\|_{S(0,\;T_{n,j}^{+})}<\infty\quad\textrm{for}\,\,j>J_{1}.$
For $m\in\mathbb{N}$, $n\geq 1$, define an interval $K_{n}^{m}$ of the form
$[0,\tau]$ by
$\sup_{1\leq j\leq J_{1}}\|v_{n}^{j}\|_{S(K_{n}^{m})}=m.$
Then, $v_{n}^{j}$ is defined on $K_{n}^{m}$ for all $j\geq 1$ and
$\|v_{n}^{j}\|_{S(K_{n}^{m})}$ is finite for all $j\geq 1$.
Since $u_{n}^{J}$ is a good approximation to $u_{n}$, using the same argument
as in Step 2, we may obtain
$\lim_{J\to\infty}\limsup_{n\to\infty}\sup_{t\in
K_{n}^{m}}\|u_{n}^{J}-u_{n}\|_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}=0$ (4.13)
for each $m\geq 1$.
By the definition of $K_{n}^{m}$, we may choose $1\leq j_{0}=j_{0}(m,n)\leq
J_{1}$ such that
$\|v_{n}^{j_{0}(m,n)}\|_{S(K_{n}^{m})}=m.$ (4.14)
Moreover, there are infinitely many $m$ satisfying $j_{0}(m,n)=j_{0}$ for
infinitely many $n$.
By the definition of $A_{c}$, we have
$\limsup_{m\to\infty}\limsup_{n\to\infty}\sup_{t\in
K_{n}^{m}}\|v_{n}^{j_{0}}\|_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}\geq A_{c}.$
(4.15)
Step 3: For all $J\geq 1$ and $m\geq 1$
$\lim_{n\to\infty}\sup_{t\in
K_{n}^{m}}\Big{|}\|u_{n}^{J}(t)\|^{2}_{\dot{H}^{1/2}_{x}}-\sum_{j=1}^{J}\|v_{n}^{j}(t)\|^{2}_{\dot{H}^{1/2}_{x}}-\|\omega_{n}^{J}\|^{2}_{\dot{H}^{1/2}_{x}}\Big{|}=0.$
(4.16)
Proof. Note that for all $J\geq 1$, $m\geq 1$
$\displaystyle\|u_{n}^{J}(t)\|_{\dot{H}^{1/2}_{x}}^{2}$
$\displaystyle=\big{\langle}\,|\nabla|^{\frac{1}{2}}u_{n}^{J}(t),|\nabla|^{\frac{1}{2}}u_{n}^{J}(t)\,\big{\rangle}$
$\displaystyle=\sum_{j=1}^{J}\big{\|}|\nabla|^{\frac{1}{2}}v_{n}^{j}\big{\|}_{\dot{H}^{1/2}_{x}}^{2}+\|\omega_{n}^{J}\|^{2}_{\dot{H}^{1/2}_{x}}+\sum_{j^{\prime}\neq
j}\big{\langle}\,|\nabla|^{\frac{1}{2}}v_{n}^{j}(t),|\nabla|^{\frac{1}{2}}v_{n}^{j^{\prime}}(t)\,\big{\rangle}$
$\displaystyle\hskip
24.0pt+\sum_{j=1}^{J}\Big{(}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it\Delta}\omega_{n}^{J},|\nabla|^{\frac{1}{2}}v_{n}^{j}(t)\,\big{\rangle}+\big{\langle}\,|\nabla|^{\frac{1}{2}}v_{n}^{j}(t),|\nabla|^{\frac{1}{2}}e^{it\Delta}\omega_{n}^{J}\,\big{\rangle}\Big{)}.$
Thus, to establish $(\ref{e314})$, it suffices to show that for all $t_{n}\in
K_{n}^{m}$,
$\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}v_{n}^{j}(t_{n})\,,\,|\nabla|^{\frac{1}{2}}v_{n}^{j^{\prime}}(t_{n})\,\big{\rangle}=0$
(4.17)
and
$\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}\Delta}\omega_{n}^{J}\,,\,|\nabla|^{\frac{1}{2}}v_{n}^{j}(t_{n})\,\big{\rangle}=0$
(4.18)
for all $1\leq j,\,j^{\prime}\leq J$, $j\neq j^{\prime}$.
We only deal with $(\ref{e316})$, as $(\ref{e315})$ can be done in the same
manner, using $(\ref{e23})$.
Do a change of variables, the formula in $(\ref{e316})$ becomes
$\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}(\lambda_{n}^{j})^{-2}\Delta}[(g_{n}^{j})^{-1}\omega_{n}^{J}]\,,\,|\nabla|^{\frac{1}{2}}v^{j}(t_{n}^{j}+t_{n}(\lambda_{n}^{j})^{-2})\,\big{\rangle}.$
(4.19)
Since $t_{n}\in K_{n}^{m}\subset[0,T_{n,j}^{+})$ for all $1\leq j\leq J_{1}$
and $v_{j}$ has maximal-life span $I^{j}=\mathbb{R}$ for $j>J_{1}$, we have
$t_{n}(\lambda_{n}^{j})^{-2}+t_{n}^{j}\in I^{j}$ for all $j\geq 1$. By passing
to a subsequence in $n$, we may assume
$t_{n}(\lambda_{n}^{j})^{-2}+t_{n}^{j}\to\tau^{j}$.
If $\tau^{j}$ is finite, then by the continuity of the flow,
$v^{j}(t_{n}(\lambda_{n}^{j})^{-2}+t_{n}^{j})\to v^{j}(\tau^{j})$ in
$\dot{H}^{1/2}_{x}$.
From $(\ref{e24})$, we have
$\displaystyle\lim_{n\to\infty}\big{\|}e^{it_{n}(\lambda_{n}^{j})^{-2}\Delta}[(g_{n}^{j})^{-1}\omega_{n}^{J}]\big{\|}_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}=\lim_{n\to\infty}\|\omega_{n}^{J}\|_{\dot{H}^{1/2}_{x}}\lesssim
A_{c}.$
Combining this with $(\ref{e317})$, and using Lemma 2.10, we obtain
$\displaystyle\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}\Delta}\omega_{n}^{J}\,,\,|\nabla|^{\frac{1}{2}}v_{n}^{j}(t_{n}^{j})\,\big{\rangle}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}(\lambda_{n}^{j})^{-2}\Delta}[(g_{n}^{j})^{-1}\omega_{n}^{J}]\,,\,|\nabla|^{\frac{1}{2}}v^{j}(\tau^{j})\,\big{\rangle}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{-it_{n}^{j}\Delta}[(g_{n}^{j})^{-1}\omega_{n}^{J}]\,,\,|\nabla|^{\frac{1}{2}}e^{-i\tau^{j}\Delta}v^{j}(\tau^{j})\,\big{\rangle}$
$\displaystyle=$ $\displaystyle\,\,0,$
which concludes $(\ref{e314})$.
If $\tau^{j}=+\infty$, then since $t_{n}(\lambda_{n}^{j})^{-2}\geq 0$, we must
have $\sup I^{j}=\infty$ and $v^{j}$ scatters forward in time. Therefore,
there exists $\psi^{j}\in\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ such that
$\lim_{n\to\infty}\big{\|}v^{j}(t_{n}^{j}+t_{n}(\lambda_{n}^{j})^{2})-e^{i(t_{n}(\lambda_{n}^{j})^{-2}+t_{n}^{j})\Delta}\psi^{j}\big{\|}_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}=0.$
Thus, together with $(\ref{e317})$ and Lemma 2.10 yields
$\displaystyle\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}\Delta}\omega_{n}^{J}\,,\,|\nabla|^{\frac{1}{2}}v^{j}_{n}(t_{n}^{j})\,\big{\rangle}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}(\lambda_{n}^{j})^{-2}\Delta}[(g_{n}^{j})^{-1}\omega_{n}^{J}]\,,\,e^{i(t_{n}(\lambda_{n}^{j})^{-2}+t_{n}^{j})\Delta}|\nabla|^{\frac{1}{2}}\psi^{j}\,\big{\rangle}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}^{j}\Delta}[(g_{n}^{j})^{-1}\omega_{n}^{J}],\,|\nabla|^{\frac{1}{2}}\psi^{j}\,\big{\rangle}$
$\displaystyle=$ $\displaystyle\,\,0.$
If $\tau^{j}=-\infty$, then we must have $t_{n}^{j}\to-\infty$ as
$n\to\infty$. Indeed, since $t_{n}(\lambda_{n}^{j})^{-2}\geq 0$ and $\inf
I^{j}<\infty$, $t_{n}^{j}$ can not converges to $+\infty$; if $t_{n}^{j}\equiv
0$, then since $\inf I^{j}<0$, we have $t_{n}(\lambda_{n}^{j})^{-2}\leq 0$,
which contradicts $t_{n}\in K_{n}^{m}\subset[0,T_{n,j}^{+})$. Hence, $\inf
I^{j}=-\infty$. By the definition of nonlinear profile, $v^{j}$ scatters
backward in time to $e^{it\Delta}\phi^{j}$.
$\lim_{n\to\infty}\big{\|}v^{j}(t_{n}^{j}+t_{n}(\lambda_{n}^{j})^{2})-e^{i(t_{n}(\lambda_{n}^{j})^{-2}+t_{n}^{j})\Delta}\phi^{j}\big{\|}_{\dot{H}^{1/2}_{x}(\mathbb{R}^{5})}=0.$
Combining this with $(\ref{e317})$ gives
$\displaystyle\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}\Delta}\omega_{n}^{J}\,,\,|\nabla|^{\frac{1}{2}}v^{j}_{n}(t_{n}^{j})\,\big{\rangle}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}(\lambda_{n}^{j})^{-2}\Delta}[(g_{n}^{j})^{-1}\omega_{n}^{J}]\,,\,e^{i(t_{n}(\lambda_{n}^{j})^{-2}+t_{n}^{j})\Delta}|\nabla|^{\frac{1}{2}}\phi^{j}\,\big{\rangle}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\big{\langle}\,|\nabla|^{\frac{1}{2}}e^{it_{n}^{j}\Delta}[(g_{n}^{j})^{-1}\omega_{n}^{J}]\,,\,|\nabla|^{\frac{1}{2}}\phi^{j}\,\big{\rangle}$
$\displaystyle=$ $\displaystyle\,\,0.$
This completes the proof of Step 3.
From $(\ref{e31})$, $(\ref{e313})$, $(\ref{e314})$
$A_{c}^{2}\geq\limsup_{n\to\infty}\sup_{t\in
K_{n}^{m}}\|u_{n}(t)\|^{2}_{\dot{H}^{1/2}_{x}}\geq\lim_{n\to\infty}\sup_{t\in
K_{n}^{m}}\Big{(}\sum_{j=1}^{J}\|v_{n}^{j}\|^{2}_{\dot{H}^{1/2}}+\|\omega_{n}^{J}\|^{2}_{\dot{H}^{1/2}}\Big{)}.$
Invoking $(\ref{e312})$ that
$\limsup_{m\to\infty}\limsup_{n\to\infty}\sup_{t\in
K_{n}^{m}}\|v_{n}^{j_{0}}(t)\|_{\dot{H}^{1/2}_{x}}\geq A_{c},$
we conclude that $v_{n}^{j}\equiv 0$ for all $j\neq j_{0}$, and
$\omega_{n}^{j_{0}}\to 0$ in $\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$. Thus,
$u_{n}(0)=g_{n}e^{i\tau_{n}\Delta}\phi+\omega_{n}$ (4.20)
for some $g_{n}\in G_{rad}$, $\tau_{n}\in\mathbb{R}$, $\phi$,
$\omega_{n}\in\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ with $\omega_{n}\to 0$ in
$\dot{H}^{1/2}$. We also have $\tau_{n}\equiv 0$ or $\tau_{n}\to\pm\infty$.
If $\tau_{n}\equiv 0$, then $u_{n}(0)\to\phi$ in $\dot{H}^{1/2}_{x}$ modulo
scaling. This proves Proposition 4.1.
If $\tau_{n}\to\pm\infty$, by time-reversal symmetry, we only consider
$\tau_{n}\to+\infty$. In this case, by the Strichartz estimate, we have
$\|e^{it\Delta}\phi\|_{S(\mathbb{R}^{+})}<\infty$. By a change of variables,
$\lim_{n\to\infty}\|e^{it\Delta}e^{-i\tau_{n}\Delta}\phi\|_{S(\mathbb{R}^{+})}=0.$
Taking the group action yields
$\lim_{n\to\infty}\|e^{it\Delta}g_{n}e^{-i\tau_{n}\Delta}\phi\|_{S(\mathbb{R}^{+})}=0.$
From $(\ref{e318})$, $(\ref{e22})$, we deduce
$\lim_{n\to\infty}\|e^{it\Delta}u_{n}(0)\|_{S(\mathbb{R}^{+})}=0.$
Invoking perturbation theorem, we obtain
$\lim_{n\to\infty}\|u_{n}\|_{S(\mathbb{R}^{+})}=0,$
which contradicts $(\ref{e32})$. This completes the proof of Proposition 4.1.
$\square$
### 4.2 Proof of Theorem 1.3
Proof. Suppose Theorem 1.2 failed. Then
$A_{c}<\frac{\sqrt{6}}{3}\|Q\|_{\dot{H}^{1/2}_{x}}$, and by the definition of
$A_{c}$, we can find a sequence of solutions
$u_{n}:I_{n}\times\mathbb{R}^{5}\mapsto\mathbb{C}$ to $(1.1)$ with $I_{n}$
compact,
$\sup_{n\geq 1}\sup_{t\in
I_{n}}\big{\|}|\nabla|^{\frac{1}{2}}u_{n}(t)\big{\|}_{2}=A_{c},\quad\lim_{n\to\infty}\|u_{n}\|_{S(I_{n})}=\infty.$
(4.21)
Then exists $t_{n}\in I_{n}$ such that
$\lim_{n\to\infty}\|u_{n}\|_{S(-\infty,\;t_{n})}=\lim_{n\to\infty}\|u_{n}\|_{S(t_{n},\;\infty)}=\infty.$
(4.22)
By time-translation symmetry, we set all $t_{n}=0$. Applying Proposition 4.1,
there exists (up to a subsequence) $g_{n}\in G_{rad}$ and a function
$u_{0}\in\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ such that $g_{n}u_{n}(0)\to u_{0}$
in $\dot{H}^{1/2}_{x}$. By taking group action $T_{g_{n}}$ to the solution
$u_{n}$, we may make $g_{n}$ be the identity. Thus $u_{n}(0)\to u_{0}$ in
$\dot{H}^{1/2}_{x}$.
Let $u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ be the maximal-life span
solution to $(1.1)$ with initial data $u(0)=u_{0}$. Then, Theorem 1.1 implies
$I\subseteq\liminf I_{n}$ and
$\lim_{n\to\infty}\sup_{t\in K}\|u_{n}(t)-u(t)\|_{\dot{H}^{1/2}_{x}}=0$
for all compact $K\subset I$.
Thus, from $(\ref{e319})$
$\sup_{t\in I}\|u(t)\|_{\dot{H}^{1/2}_{x}}\leq A_{c}.$ (4.23)
On the other hand, we claim that $u$ blows up both froward and backward in
time. If not, $\|u\|_{S(0,\;\infty)}<\infty$, $\|u\|_{S(-\infty,\;0)}<\infty$.
From perturbation theorem, $\|u_{n}\|_{S(0,\;\infty)}<\infty$,
$\|u_{n}\|_{S(-\infty,\;0)}<\infty$ for $n$ large enough, which contradicts
$(\ref{e320})$.
So, by the definition of $A_{c}$
$\sup_{t\in I}\|u(t)\|_{\dot{H}^{1/2}_{x}}\geq A_{c}$
which together with $(\ref{e321})$ yields
$\sup_{t\in I}\|u(t)\|_{\dot{H}^{1/2}_{x}}=A_{c}.$
Next, we prove that $u$ is almost periodic modulo scaling. For arbitrary
sequence $\tau_{n}\in I$, we have
$\|u\|_{S(-\infty,\;\tau_{n})}=\|u\|_{S(\tau_{n},\;\infty)}=\infty,$
since $u$ blows up both forward and backward. From Proposition 4.1,
$u(\tau_{n})$ has a subsequence which converges in
$\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ modulo scaling. Thus $\\{u(t):t\in I\\}$
is precompact in $\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$ modulo $G_{rad}$(Remark
1.2). This completes the proof of the first part of Theorem 1.3.
An almost periodic blowup solution which obeys the three scenarios in Theorem
1.3 can be extracted from the above solution by renormalization and
subsequential limits. As we’ve pointed out, the process is similar to that in
[13], [14], and we refer the readers to these papers for a detailed
discussion.
## 5 No finite-time blow up
In this section, we prove
###### Theorem 5.1.
There exists no such maximal life-span solution
$u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ to $(1.1)$ that is almost periodic
modulo scaling and
$\sup_{t\in
I}\|u(t)\|_{\dot{H}^{1/2}_{x}}<\frac{\sqrt{6}}{3}\|Q\|_{\dot{H}^{1/2}_{x}},\quad\|u\|_{S(I)}=\infty$
(5.1)
and either $|\inf I|<\infty$ or $\sup I<\infty$.
Proof. Assume for a contradiction that there existed such a solution. Without
loss of generality, we may assume $\sup I<\infty$. We claim that
$\liminf_{t\nearrow\sup I}N(t)=\infty.$ (5.2)
If not, we may find a time sequence $t_{n}\in I$ such that $t_{n}\nearrow\sup
I$, $\liminf\limits_{n}N(t_{n})<\infty$. For each $n\geq 1$, define
$v_{n}:I_{n}\times\mathbb{R}^{5}\mapsto\mathbb{C}$ by
$v_{n}(t,x):=u(t_{n}+tN(t_{n})^{-2},\;xN(t_{n})^{-1})$
with $I_{n}:=\\{\,t\in\mathbb{R}:t_{n}+tN(t_{n})^{-2}\in I\,\\}$. Then $v_{n}$
is also a solution to $(1.1)$, $\\{v_{n}(0)\\}$ is precompact in
$\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$. After passing to a subsequence, we may
assume $v_{n}(0)\to v_{0}$ in $\dot{H}^{1/2}_{x}(\mathbb{R}^{5})$. Since
$\|v_{n}(0)\|_{\dot{H}^{1/2}_{x}}=\|u(t_{n})\|_{\dot{H}^{1/2}_{x}}$, $v_{0}$
is not identically zero.
Let $v$ be the maximal life-span solution to $(1.1)$ with initial data
$v_{0}$, and maximal life-span $(-T_{-},\;T_{+})$, $-\infty\leq
T_{-}<0<T_{+}\leq\infty$. For any compact $J\subset(-T_{-},\;T_{+})$, from
local wellposedness theory, for sufficiently large $n$, $v_{n}$ is wellposed
on $J$ and $\|v_{n}\|_{S(J)}<\infty$. Thus, $u$ is wellposed on the interval
$J_{n}=\\{\,t_{n}+tN(t_{n})^{-2}:t\in J\,\\}$ and $\|u\|_{S(J_{n})}<\infty$.
But $\liminf_{t\nearrow\sup I}N(t)<\infty$ implies that $\|u\|_{S}$ is finite
beyond $\sup I$, which contradicts the assumption that $u$ blows up on $I$.
Next, we will prove that for all $R>0$
$\limsup_{t\nearrow\sup I}\int_{|x|\leq R}|u(t,x)|^{2}\,\mathrm{d}x=0.$ (5.3)
Let $\eta>0$, $t\in I$. Using Hölder’s inequality, Sobolev’s embedding
theorem, $(\ref{e42})$
$\displaystyle\int_{|x|\leq R}|u(t,x)|^{2}\,\mathrm{d}x$
$\displaystyle\leq\int_{|x|\leq\eta R}|u(t,x)|^{2}\,\mathrm{d}x+\int_{\eta
R\leq|x|\leq R}|u(t,x)|^{2}\,\mathrm{d}x$ $\displaystyle\leq\eta
R\left(\int|u(t,x)|^{5/2}\,\mathrm{d}x\right)^{4/5}+R\left(\int_{|x|\geq\eta
R}|u(t,x)|^{5/2}\,\mathrm{d}x\right)^{4/5}$ $\displaystyle\lesssim\eta
R\|u(t)\|_{\dot{H}^{1/2}_{x}}^{2}+R\left(\int_{|x|\geq\eta
R}|u(t,x)|^{5/2}\,\mathrm{d}x\right)^{4/5}.$
The first term will go to zero as $\eta$ tends to zero. On the other hand,
from $(\ref{e42})$, almost periodic modulo scaling, and (1.5), we have
$\limsup_{t\nearrow\sup I}\int_{|x|\geq
R}|u(t,x)|^{5/2}\,\mathrm{d}x\leq\limsup_{t\nearrow\sup I}\int_{|x|\geq
C(\eta)/N(t)}|u(t,x)|^{5/2}\,\mathrm{d}x=0.$
Thus $(\ref{e43})$ holds.
We will prove from $(\ref{e43})$ that $u$ is identically zero.
For $t\in I$, define
$M_{R}(t):=\int_{\mathbb{R}^{5}}\phi\big{(}\frac{|x|}{R}\big{)}|u(t,x)|^{2}\,\mathrm{d}x$
where $\phi$ is a smooth, radial function with
$\phi(r)=\begin{cases}1,&r\leq 1\\\ 0,&r\geq 2.\end{cases}$
By $(\ref{e43})$,
$\limsup_{t\nearrow\sup I}M_{R}(t)=0\quad\textrm{for all }\,\,R>0.$ (5.4)
A direct computation involving Plancherel, Hardy’s inequality and
$(\ref{e41})$ yields
$\displaystyle\big{|}\partial_{t}M_{R}(t)\big{|}$ $\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{R}^{5}}\bigg{(}\Big{|}{\frac{x}{R^{2}}\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}\bar{u}}\Big{|}\bigg{)}^{\widehat{}}(\xi)|\xi||\hat{u}|\,\mathrm{d}\xi$
$\displaystyle\lesssim$
$\displaystyle\Big{\|}|\nabla|^{\frac{1}{2}}\big{(}\frac{x}{R^{2}}\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}\bar{u}\big{)}\Big{\|}_{2}\big{\|}|\xi|^{\frac{1}{2}}\hat{u}\big{\|}_{2}$
$\displaystyle\lesssim_{u}$
$\displaystyle\Big{\|}|\nabla|^{\frac{1}{2}}\big{(}\frac{x}{R^{2}}\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}\big{)}\bar{u}\Big{\|}_{2}+\Big{\|}\frac{x}{R^{2}}\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}|\nabla|^{\frac{1}{2}}\bar{u}\Big{\|}_{2}$
$\displaystyle\lesssim_{u}$
$\displaystyle\Big{\|}|x|^{\frac{1}{2}}|\nabla|^{\frac{1}{2}}\big{(}\frac{x}{R^{2}}\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}\big{)}\Big{\|}_{L^{\infty}}\Big{\|}\frac{\bar{u}}{|x|^{1/2}}\Big{\|}_{2}+\Big{\|}\frac{x}{R^{2}}\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}\Big{\|}_{L^{\infty}}\||\nabla|^{\frac{1}{2}}u\|_{2}.$
$\displaystyle\lesssim_{u}$
$\displaystyle\Big{\|}|x|^{\frac{1}{2}}|\nabla|^{\frac{1}{2}}\big{(}\frac{x}{R^{2}}\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}\big{)}\Big{\|}_{L^{\infty}}+\frac{1}{R}.$
Furthermore, if $|x|\leq 4R$, then by our chosen of $\phi$
$\Big{\|}|x|^{\frac{1}{2}}|\nabla|^{\frac{1}{2}}\Big{(}\frac{x}{R^{2}}\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}\Big{)}\Big{\|}_{L^{\infty}}\lesssim\frac{1}{R}.$
If $|x|>4R$, then using the intrinsic description of derivatives, we have the
following
$\displaystyle\frac{|x|^{\frac{1}{2}}}{R^{2}}|\nabla|^{\frac{1}{2}}\Big{(}x\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}\Big{)}=$
$\displaystyle\,\frac{1}{R^{2}}\int_{\mathbb{R}^{5}}\frac{|x|^{\frac{1}{2}}\big{[}x\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}-y\phi^{\prime}\big{(}\frac{|y|}{R}\big{)}\big{]}}{|x-y|^{5+\frac{1}{2}}}\,\mathrm{d}y$
$\displaystyle=$
$\displaystyle\frac{1}{R^{2}}\int_{|x-y|\geq\frac{1}{2}|x|}\frac{|x|^{\frac{1}{2}}\big{[}x\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}-y\phi^{\prime}\big{(}\frac{|y|}{R}\big{)}\big{]}}{|x-y|^{5+\frac{1}{2}}}\,\mathrm{d}y$
$\displaystyle+\frac{1}{R^{2}}\int_{|x-y|<\frac{1}{2}|x|}\frac{|x|^{\frac{1}{2}}\big{[}x\phi^{\prime}\big{(}\frac{|x|}{R}\big{)}-y\phi^{\prime}\big{(}\frac{|y|}{R}\big{)}\big{]}}{|x-y|^{5+\frac{1}{2}}}\,\mathrm{d}y.$
It is easily to check that the first integration has a bound $R^{-1}$, since
$|x-y|\geq\frac{1}{2}|x|\geq 2R$. For the second one, we have
$|y|>|x|-|x-y|>\frac{1}{2}|x|>2R$, and by the property of $\phi$, it follows
that the integration is equal to zero.
From the above, we obtain
$\big{|}\partial_{t}M_{R}(t)\big{|}\lesssim_{u}\frac{1}{R}.$
Thus, by the Fundamental Theorem of Calculus
$M_{R}(t_{1})\lesssim
M_{R}(t_{2})+\int_{t_{2}}^{t_{1}}\partial_{t}M_{R}(t)\,\mathrm{d}t\lesssim
M_{R}(t_{2})+\frac{1}{R}|t_{1}-t_{2}|$
for all $t_{1},t_{2}\in I$ and $R>0$.
Let $t_{2}\nearrow\sup I$ and from $(\ref{e44})$, we obtain
$M_{R}(t_{1})\lesssim_{u}\frac{1}{R}|\sup I-t_{1}|.$
Let $R\to\infty$, then we deduce that $M(u(t))=0$ for all $t\in I$. This
implies that $u\equiv 0$, which contradicts $\|u\|_{S(I)}=\infty$. This
completes the proof of Theorem 5.1.
## 6 Negative regularity
In this section, we prove the following
###### Theorem 6.1 (Negative regularity in the global case).
Let $u$ be a global radially symmetric solution to $(1.1)$ which is almost
periodic modulo scaling. Suppose also that
$\sup_{t\in\mathbb{R}}\|u(t)\|_{\dot{H}^{1/2}_{x}}<\frac{\sqrt{6}}{3}\|Q\|_{\dot{H}^{1/2}_{x}}$
(6.1)
and
$\inf_{t\in\mathbb{R}}N(t)\gtrsim 1.$ (6.2)
Then, $u\in
L_{t}^{\infty}\dot{H}^{-\varepsilon}(\mathbb{R}\times\mathbb{R}^{5})$ for some
$\varepsilon>0$. In particular, $u\in L_{t}^{\infty}L_{x}^{2}$.
In order to prove Theorem 6.1, we first establish a recurrence formula.
Given $\eta>0$, from Remark 1.1, there exists $N_{0}=N_{0}(\eta)$ such that
$\|u_{\leq N_{0}}(t)\|_{\dot{H}^{1/2}_{x}}\leq\eta.$ (6.3)
Now, define
$A(N):=N^{-\frac{3}{4}}\sup_{t\in\mathbb{R}}\|u_{N}(t)\|_{L_{x}^{4}}$
for all $N\leq 8N_{0}$.
Note that by Bernstein’s inequality, Sobolev’s embedding theorem
$A(N)\lesssim
N^{-\frac{3}{4}}N^{\frac{3}{4}}\|u_{N}\|_{L_{t}^{\infty}L_{x}^{5/2}}\leq\|u\|_{L_{t}^{\infty}\dot{H}^{1/2}_{x}}<\infty.$
Moreover, $A(N)$ satisfies the following recurrence formula
###### Lemma 6.1.
For $N\leq 8N_{0}$
$A(N)\lesssim_{u}\left(\frac{N}{N_{0}}\right)^{\frac{1}{2}}+\eta^{2}\sum_{8N\leq
N_{1}\leq
N_{0}}\left(\frac{N}{N_{1}}\right)^{\frac{1}{8}}A(N_{1})+\eta^{2}\sum_{N_{1}\leq
8N}\left(\frac{N_{1}}{N}\right)^{\frac{3}{4}}A(N_{1}).$ (6.4)
Proof. We only need to prove that for all $t\in\mathbb{R}$
$N^{-\frac{3}{4}}\|u_{N}(t)\|_{L_{x}^{4}}\lesssim\,\,\textrm{RHS
of}\,(\ref{e54}).$
By the time-translation symmetry, it reduces to prove
$N^{-\frac{3}{4}}\|u_{N}(0)\|_{L_{x}^{4}}\lesssim\,\,\textrm{RHS
of}\,(\ref{e54}).$
By the Duhamel formula $(\ref{e16})$, the triangle, Bernstein’s and the
dispersive inequality, we have
$\displaystyle N^{-\frac{3}{4}}\|u_{N}(0)\|_{L_{x}^{4}}$ $\displaystyle\leq
N^{-\frac{3}{4}}\Big{\|}\int_{0}^{N^{-2}}e^{-it\Delta}P_{N}F(u(t))\,\mathrm{d}t\Big{\|}_{L_{x}^{4}}$
$\displaystyle\quad+N^{-\frac{3}{4}}\Big{\|}\int_{N^{-2}}^{\infty}e^{-it\Delta}P_{N}F(u(t))\,\mathrm{d}t\Big{\|}_{L_{x}^{4}}$
$\displaystyle\lesssim
N^{\frac{1}{2}}\Big{\|}\int_{0}^{N^{-2}}e^{-it\Delta}P_{N}F(u(t))\,\mathrm{d}t\Big{\|}_{L_{x}^{2}}$
$\displaystyle\quad+N^{-\frac{3}{4}}\|P_{N}F(u)\|_{L_{t}^{\infty}L_{x}^{4/3}}\int_{N^{-2}}^{\infty}t^{-\frac{5}{4}}\,\mathrm{d}t$
$\displaystyle\lesssim
N^{-\frac{3}{2}}\|P_{N}F(u)\|_{L_{t}^{\infty}L_{x}^{2}}+N^{-\frac{1}{4}}\|P_{N}F(u)\|_{L_{t}^{\infty}L_{x}^{4/3}}$
$\displaystyle\lesssim
N^{-\frac{1}{4}}\|P_{N}F(u)\|_{L_{t}^{\infty}L_{x}^{4/3}}.$
Decompose $u$ as
$u:=u_{\geq N_{0}}+u_{\frac{N}{8}\leq\cdot<N_{0}}+u_{<\frac{N}{8}},$
and then make a corresponding expansion of $F(u)$, we obtain terms constitute
$F(u)$ of the following types
1\. At least one high frequency, i.e. $|\nabla|^{-2}(uu_{\geq N_{0}})u$, or
$|\nabla|^{-2}(u^{2})u_{\geq N_{0}}$;
2\. Non-high frequency component and at least one lower frequency:
$|\nabla|^{-2}(u_{<\frac{N}{8}}u_{\leq N_{0}})u_{\leq
N_{0}},\quad|\nabla|^{-2}(u_{\leq N_{0}}^{2})u_{<\frac{N}{8}};$
3\. All medium components:
$|\nabla|^{-2}(u_{\frac{N}{8}\leq\cdot<N_{0}}^{2})u_{\frac{N}{8}\leq\cdot<N_{0}}$.
Case 1(At least one high frequency). Using Bernstein’s inequality, discarding
the projector $P_{N}$, and then using the Hardy-Littlewood-Sobolev, Hölder’s
and Bernstein’s inequality, Sobolev embedding, we have
$\displaystyle N^{-\frac{1}{4}}\big{\|}P_{N}(|\nabla|^{-2}(uu_{\geq
N_{0}})u)\big{\|}_{L_{t}^{\infty}L_{x}^{4/3}}$ $\displaystyle\lesssim
N^{\frac{1}{2}}\big{\|}|\nabla|^{-2}(uu_{\geq
N_{0}})u\big{\|}_{L_{t}^{\infty}L_{x}^{10/9}}$
$\displaystyle\lesssim_{u}N^{\frac{1}{2}}\big{\|}|\nabla|^{-2}(uu_{\geq
N_{0}})\|_{L_{t}^{\infty}L_{x}^{2}}\|u\|_{L_{t}^{\infty}L_{x}^{5/2}}$
$\displaystyle\lesssim_{u}N^{\frac{1}{2}}\|uu_{\geq
N_{0}}\|_{L_{t}^{\infty}L_{x}^{10/9}}$
$\displaystyle\lesssim_{u}N^{\frac{1}{2}}\|u\|_{L_{t}^{\infty}L_{x}^{5/2}}\|u_{\geq
N_{0}}\|_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}N^{\frac{1}{2}}N_{0}^{-\frac{1}{2}},$
$\displaystyle N^{-\frac{1}{4}}\big{\|}P_{N}(|\nabla|^{-2}(u^{2})u_{\geq
N_{0}})\big{\|}_{L_{t}^{\infty}L_{x}^{4/3}}$ $\displaystyle\lesssim
N^{\frac{1}{2}}\big{\|}|\nabla|^{-2}(u^{2})u_{\geq
N_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{10/9}}$ $\displaystyle\lesssim
N^{\frac{1}{2}}\big{\|}|\nabla|^{-2}(u^{2})\big{\|}_{L_{t}^{\infty}L_{x}^{5/2}}\|u_{\geq
N_{0}}\|_{L_{t}^{\infty}L_{x}^{2}}$ $\displaystyle\lesssim
N^{\frac{1}{2}}\|u\|^{2}_{L_{t}^{\infty}L_{x}^{5/2}}\|u_{\geq
N_{0}}\|_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}N^{\frac{1}{2}}N_{0}^{-\frac{1}{2}};$
Case 2(Lower frequency components). By the triangle, Bernstein’s inequality,
Sobolev’s embedding theorem, Hölder’s and the Hardy-Littlewood-Sobolev
inequality
$\displaystyle\qquad
N^{-\frac{1}{4}}\big{\|}P_{N}(|\nabla|^{-2}(u_{<\frac{N}{8}}u_{\leq
N_{0}})u_{\leq N_{0}})\big{\|}_{L_{t}^{\infty}L_{x}^{4/3}}$
$\displaystyle\lesssim
N^{-\frac{1}{4}}\big{\|}P_{>\frac{N}{8}}\big{(}|\nabla|^{-2}(u_{<\frac{N}{8}}u_{\leq
N_{0}})\big{)}u_{\leq N_{0}}\big{\|}_{L_{t}^{\infty}L^{4/3}}$
$\displaystyle\quad+N^{-\frac{1}{4}}\big{\|}|\nabla|^{-2}(u_{<\frac{N}{8}}u_{\leq
N_{0}})P_{>\frac{N}{8}}u_{\leq N_{0}}\big{\|}_{L_{t}^{\infty}L^{4/3}}$
$\displaystyle\lesssim
N^{-\frac{1}{4}}\big{\|}P_{>\frac{N}{8}}|\nabla|^{-2}(u_{<\frac{N}{8}}u_{\leq
N_{0}})\big{\|}_{L_{t}^{\infty}L_{x}^{20/7}}\|u_{\leq
N_{0}}\|_{L_{t}^{\infty}L_{x}^{5/2}}$
$\displaystyle\quad+N^{-\frac{1}{4}}\big{\|}|\nabla|^{-2}(u_{<\frac{N}{8}}u_{\leq
N_{0}})\big{\|}_{L_{t}^{\infty}L_{x}^{4}}\|P_{>\frac{N}{8}}u_{\leq
N_{0}}\|_{L_{t}^{\infty}L_{x}^{2}}$ $\displaystyle\lesssim\eta
N^{-\frac{3}{4}}\big{\|}u_{<\frac{N}{8}}u_{\leq
N_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{20/13}}+N^{-\frac{3}{4}}\big{\|}u_{<\frac{N}{8}}u_{\leq
N_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{20/13}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\leq
N_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\eta^{2}\sum_{N_{1}\leq\frac{N}{8}}\left(\frac{N_{1}}{N}\right)^{\frac{3}{4}}A(N_{1}),$
$\displaystyle N^{-\frac{1}{4}}\big{\|}P_{N}\big{(}|\nabla|^{-2}(u_{\leq
N_{0}}^{2})u_{<\frac{N}{8}}\big{)}\big{\|}_{L_{t}^{\infty}L_{x}^{4/3}}\leq
N^{-\frac{1}{4}}\big{\|}P_{>\frac{N}{4}}|\nabla|^{-2}(u_{\leq
N_{0}}^{2})u_{<\frac{N}{8}}\big{\|}_{L_{t}^{\infty}L_{x}^{4/3}}$
$\displaystyle\lesssim N^{-\frac{3}{4}}\big{\|}|\nabla|^{-\frac{3}{2}}(u_{\leq
N_{0}}^{2})\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\|u_{<\frac{N}{8}}\|_{L_{t}^{\infty}L_{x}^{4}}\lesssim
N^{-\frac{3}{4}}\|u_{\leq
N_{0}}^{2}\|_{L_{t}^{\infty}L_{x}^{5/4}}\|u_{<\frac{N}{8}}\|_{L_{t}^{\infty}L_{x}^{4}}$
$\displaystyle\lesssim\eta^{2}\sum_{N_{1}\leq\frac{N}{8}}\left(\frac{N_{1}}{N}\right)^{\frac{3}{4}}A(N_{1});$
Case 3(Medium components). By Bernstein’s, the Hardy-Littlewood-Sobolev, the
triangle and Hölder’s inequality
$\displaystyle\quad
N^{-\frac{1}{4}}\big{\|}P_{N}(|\nabla|^{-2}(u_{\frac{N}{8}\leq\cdot<N_{0}}^{2})u_{\frac{N}{8}\leq\cdot<N_{0}})\big{\|}_{L_{t}^{\infty}L_{x}^{4/3}}$
$\displaystyle\lesssim
N^{\frac{1}{8}}\big{\|}|\nabla|^{-2}(u_{\frac{N}{8}\leq\cdot<N_{0}}^{2})u_{\frac{N}{8}\leq\cdot<N_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{40/33}}$
$\displaystyle\lesssim\sum_{\frac{N}{8}\leq N_{1}\leq N_{2},\,N_{3}\leq
N_{0}}N^{\frac{1}{8}}\big{\|}|\nabla|^{-2}(u_{N_{1}}u_{N_{2}})u_{N_{3}}\big{\|}_{L_{t}^{\infty}L_{x}^{40/33}}$
$\displaystyle\quad+\sum_{\frac{N}{8}\leq N_{3}\leq N_{1}\leq N_{2}\leq
N_{0}}N^{\frac{1}{8}}\big{\|}|\nabla|^{-2}(u_{N_{1}}u_{N_{2}})u_{N_{3}}\big{\|}_{L_{t}^{\infty}L_{x}^{40/33}}$
$\displaystyle\lesssim\sum_{\frac{N}{8}\leq N_{1}\leq N_{2},N_{3}\leq
N_{0}}N^{\frac{1}{8}}\big{\|}|\nabla|^{-2}(u_{N_{1}}u_{N_{2}})\big{\|}_{L_{t}^{\infty}L_{x}^{40/13}}\big{\|}u_{N_{3}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\quad+\sum_{\frac{N}{8}\leq N_{3}\leq N_{1}\leq N_{2}\leq
N_{0}}N^{\frac{1}{8}}\big{\|}|\nabla|^{-2}(u_{N_{1}}u_{N_{2}})\big{\|}_{L_{t}^{\infty}L_{x}^{40/23}}\big{\|}u_{N_{3}}\big{\|}_{L_{t}^{\infty}L_{x}^{4}}$
$\displaystyle\lesssim_{u}\sum_{\frac{N}{8}\leq N_{1}\leq N_{2},N_{3}\leq
N_{0}}N^{\frac{1}{8}}\|u_{N_{1}}u_{N_{2}}\|_{L_{t}^{\infty}L_{x}^{40/29}}N_{3}^{-\frac{1}{2}}$
$\displaystyle\quad+\sum_{\frac{N}{8}\leq N_{3}\leq N_{1}\leq N_{2}\leq
N_{0}}N^{\frac{1}{8}}\|u_{N_{1}}u_{N_{2}}\|_{L_{t}^{\infty}L_{x}^{40/39}}\|u_{N_{3}}\|_{L_{t}^{\infty}L_{x}^{4}}$
$\displaystyle\lesssim_{u}\sum_{\frac{N}{8}\leq N_{1}\leq N_{2},N_{3}\leq
N_{0}}N^{\frac{1}{8}}\|u_{N_{1}}\|_{L_{t}^{\infty}L_{x}^{4}}\|u_{N_{2}}\|_{L_{t}^{\infty}L_{x}^{40/19}}N_{3}^{-\frac{1}{2}}$
$\displaystyle\quad+\sum_{\frac{N}{8}\leq N_{3}\leq N_{1}\leq N_{2}\leq
N_{0}}N^{\frac{1}{8}}\|u_{N_{1}}\|_{L_{t}^{\infty}L_{x}^{2}}\|u_{N_{2}}\|_{L_{t}^{\infty}L_{x}^{40/19}}\|u_{N_{3}}\|_{L_{t}^{\infty}L_{x}^{4}}$
$\displaystyle\lesssim_{u}\eta^{2}\sum_{\frac{N}{8}\leq N_{1}\leq
N_{2},N_{3}\leq
N_{0}}N^{\frac{1}{8}}N_{2}^{-\frac{3}{8}}N_{3}^{-\frac{1}{2}}\|u_{N_{1}}\|_{L_{t}^{\infty}L_{x}^{4}}$
$\displaystyle\quad+\eta^{2}\sum_{\frac{N}{8}\leq N_{3}\leq N_{1}\leq
N_{2}\leq
N_{0}}N^{\frac{1}{8}}N_{1}^{-\frac{1}{2}}N_{2}^{-\frac{3}{8}}\|u_{N_{3}}\|_{L_{t}^{\infty}L_{x}^{4}}$
$\displaystyle\lesssim_{u}\eta^{2}\sum_{\frac{N}{8}\leq N_{1}\leq
N_{0}}\left(\frac{N}{N_{1}}\right)^{\frac{1}{8}}A(N_{1}).$
This concludes the proof of Lemma 6.1.
###### Proposition 6.1.
Let $u$ be as in Theorem $6.1$. Then
$u\in L_{t}^{\infty}L_{x}^{p}\quad\textrm{for}\,\,\frac{22}{9}\leq
p<\frac{5}{2},$
Furthermore, by the Hardy-Littlewood-Sobolev inequality
$|\nabla|^{\frac{1}{2}}F(u)\in
L_{t}^{\infty}L_{x}^{r}\quad\textrm{for}\,\,\frac{110}{101}\leq
r<\frac{10}{9}.$
Proof. Let $N=8\cdot 2^{-k}N_{0}$, applying Lemma 2.1 with $b_{k}=(8\cdot
2^{-k})^{\frac{1}{8}}$, $x_{k}=A(8\cdot 2^{-k}N_{0})$, we obtain
$\|u_{N}\|_{L_{t}^{\infty}L_{x}^{4}}\lesssim_{u}N^{7/8+}\quad\textrm{for all
}\quad N\leq 8N_{0}.$
By the interpolation theorem, Bernstein’s inequality, and $(\ref{e51})$
$\displaystyle\|u_{N}\|_{L_{t}^{\infty}L^{p}_{x}}$
$\displaystyle\lesssim\|u_{N}\|_{L_{t}^{\infty}L_{x}^{4}}^{2-\frac{4}{p}}\|u_{N}\|^{\frac{4}{p}-1}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}N^{\frac{7}{8}(2-\frac{4}{p})+}N^{-\frac{1}{2}(\frac{4}{p}-1)}$
$\displaystyle\lesssim_{u}N^{\frac{9}{4}-\frac{11}{2p}+}$
for all $N\leq 8N_{0}$.
Thus, using Bernstein’s inequality together with $(\ref{e51})$, we have
$\|u\|_{L_{t}^{\infty}L_{x}^{p}}\leq\|u_{\leq
N_{0}}\|_{L_{t}^{\infty}L_{x}^{p}}+\|u_{>N_{0}}\|_{L_{t}^{\infty}L_{x}^{p}}\lesssim_{u}\sum_{N\leq
N_{0}}N^{\frac{9}{4}-\frac{11}{2p}+}+\sum_{N>N_{0}}N^{2-\frac{5}{p}}\lesssim_{u}1.$
###### Proposition 6.2 ( Some negative regularity).
Let $u$ be as in Theorem 6.1. Assume also that $|\nabla|^{s}F(u)\in
L_{t}^{\infty}L_{x}^{r}$ for some $\frac{110}{101}\leq r<\frac{10}{9}$ and
some $0\leq s\leq\frac{1}{2}$. Then there exists $s_{0}=s_{0}(r)>0$ such that
$u\in L_{t}^{\infty}\dot{H}_{x}^{s-s_{0}+}$.
Proof. It only needs to prove that
$\big{\|}|\nabla|^{s}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\lesssim
N^{s_{0}}\quad\textrm{for all}\quad N>0,\,s_{0}:=\frac{5}{r}-\frac{9}{2}.$
(6.5)
In fact, by Bernstein’s inequality and $(\ref{e51})$
$\displaystyle\big{\|}|\nabla|^{s-s_{0}+}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\leq\big{\|}|\nabla|^{s-s_{0}+}u_{\leq
1}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\big{\|}|\nabla|^{s-s_{0}+}u_{>1}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim\sum_{N\leq
1}\big{\|}|\nabla|^{s-s_{0}+}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}+\sum_{N>1}\big{\|}|\nabla|^{s-s_{0}+}u_{N}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\sum_{N\leq
1}N^{s_{0}}N^{-s_{0}+}+\sum_{N>1}N^{(s-s_{0}+)-\frac{1}{2}}\lesssim_{u}1.$
To prove $(\ref{e55})$, by time-translation invariant, we only need to show
that
$\big{\|}|\nabla|^{s}u_{N}(0)\big{\|}_{L_{x}^{2}}\lesssim_{u}N^{s_{0}}\quad\textrm{for
all}\quad N>0,\,s_{0}:=\frac{5}{r}-\frac{9}{2}>0.$
Using Duhamel formula $(\ref{e16})$ both forward and backward, we have
$\displaystyle\big{\|}|\nabla|^{s}u_{N}(0)\big{\|}_{L_{x}^{2}}$
$\displaystyle=\Big{\langle}i\int_{0}^{\infty}e^{it\Delta}|\nabla|^{s}P_{N}F(u(t))\,\mathrm{d}t,-i\int_{-\infty}^{0}e^{i\tau\Delta}|\nabla|^{s}P_{N}F(u(\tau))\,\mathrm{d}\tau\Big{\rangle}$
$\displaystyle\leq\int_{0}^{\infty}\int_{-\infty}^{0}\Big{|}\big{\langle}e^{it\Delta}|\nabla|^{s}P_{N}F(u(t)),\;e^{i\tau\Delta}|\nabla|^{s}P_{N}F(u(\tau))\big{\rangle}\Big{|}\,\mathrm{d}t\,\mathrm{d}\tau.$
By Hölder’s and the dispersive inequality
$\displaystyle\quad\Big{|}\big{\langle}e^{it\Delta}|\nabla|^{s}P_{N}F(u(t)),e^{i\tau\Delta}|\nabla|^{s}P_{N}F(u(\tau))\big{\rangle}\Big{|}$
$\displaystyle=$
$\displaystyle\quad\Big{|}\big{\langle}|\nabla|^{s}P_{N}F(u(t)),e^{i(\tau-t)\Delta}|\nabla|^{s}P_{N}F(u(\tau))\big{\rangle}\Big{|}$
$\displaystyle\leq$
$\displaystyle\quad\big{\|}|\nabla|^{s}P_{N}F(u(t))\big{\|}_{L_{x}^{r}}\big{\|}e^{i(\tau-t)\Delta}|\nabla|^{s}P_{N}F(u(\tau))\big{\|}_{L_{x}^{r^{\prime}}}$
$\displaystyle\lesssim$
$\displaystyle\quad|\tau-t|^{5(\frac{1}{2}-\frac{1}{r})}\big{\|}|\nabla|^{s}P_{N}F(u)\big{\|}_{L_{x}^{r}}^{2}.$
On the other hand, from Bernstein’s inequality
$\displaystyle\quad\Big{|}\big{\langle}e^{it\Delta}|\nabla|^{s}P_{N}F(u(t)),e^{i\tau\Delta}|\nabla|^{s}P_{N}F(u(\tau))\big{\rangle}\Big{|}$
$\displaystyle\leq$
$\displaystyle\quad\big{\|}|\nabla|^{s}P_{N}F(u)\big{\|}_{L_{x}^{2}}^{2}$
$\displaystyle\lesssim$ $\displaystyle\quad
N^{10(\frac{1}{r}-\frac{1}{2})}\big{\|}|\nabla|^{s}P_{N}F(u)\big{\|}_{L_{x}^{r}}^{2}.$
Thus
$\displaystyle\,\,\int_{0}^{\infty}\int_{-\infty}^{0}\Big{|}\big{\langle}e^{it\Delta}|\nabla|^{s}P_{N}F(u(t)),e^{i\tau\Delta}|\nabla|^{s}P_{N}F(u(\tau))\big{\rangle}\Big{|}\,\mathrm{d}t\,\mathrm{d}\tau$
$\displaystyle\lesssim$
$\displaystyle\,\,\big{\|}|\nabla|^{s}F(u)\big{\|}_{L_{t}^{\infty}L_{x}^{r}}^{2}\int_{0}^{\infty}\int_{-\infty}^{0}\min\\{|\tau-t|^{5(\frac{1}{2}-\frac{1}{r})},N^{10(\frac{1}{r}-\frac{1}{2})}\\}\,\mathrm{d}t\,\mathrm{d}\tau$
$\displaystyle\lesssim$
$\displaystyle\,\,\big{\|}|\nabla|^{s}F(u)\big{\|}_{L_{t}^{\infty}L_{x}^{r}}^{2}N^{2(\frac{5}{r}-\frac{9}{2})}=\,\,\big{\|}|\nabla|^{s}F(u)\big{\|}_{L_{t}^{\infty}L_{x}^{r}}^{2}N^{2s_{0}},$
where we use the fact that $\frac{5}{2}-\frac{5}{r}<-2$.
With these propositions, we are now ready to complete the proof of Theorem
6.1. First, applying Proposition 6.2 with $s=\frac{1}{2}$, we obtain $u\in
L_{t}^{\infty}\dot{H}^{\frac{1}{2}-s_{0}+}_{x}$ for some $s_{0}+>0$. By
fractional chain rule and $(\ref{e51})$, we have
$|\nabla|^{\frac{1}{2}-s_{0}+}F(u)\in L_{t}^{\infty}L_{x}^{r}$ for some
$\frac{110}{101}\leq r<\frac{10}{9}$. Again using Proposition 6.2 with
$s=\frac{1}{2}-s_{0}+$, we have $u\in
L_{t}^{\infty}\dot{H}^{\frac{1}{2}-2s_{0}+}_{x}$. By doing this with finite
times, we will obtain $u\in L_{t}^{\infty}\dot{H}^{-\varepsilon}_{x}$ for some
$0<\varepsilon<2s_{0}+$. This proves Theorem 6.1.
## 7 Low-to-high cascade
In this section we prove
###### Theorem 7.1 (Absence of cascade).
There can not exist a global solution to $(1.1)$ which is almost periodic
modulo scaling, blows up both forward and backward and is low-to-high cascade
in the sense of Theorem $1.3$.
Proof. We argue by contradiction. Assume there exists such an $u$. Then, by
Theorem 6.1, $u\in L_{t}^{\infty}L_{x}^{2}$ and
$0\leq
M(u)=M(u(t))=\int_{\mathbb{R}^{5}}|u(t,x)|^{2}\,\mathrm{d}x<\infty\quad\textrm{for
all }\quad t\in\mathbb{R}.$
Fix $t\in\mathbb{R}$. Let $\eta>0$ be sufficiently small. From
$(\ref{e15})$(Remark 1.1)
$\int_{|\xi|\leq
c(\eta)N(t)}|\xi||\hat{u}(t,\xi)|^{2}\,\mathrm{d}\xi\leq\eta.$
Since $u\in L_{t}^{\infty}\dot{H}^{-\varepsilon}_{x}(\varepsilon>0)$, we see
that
$\int_{|\xi|\leq
c(\eta)N(t)}|\xi|^{-2\varepsilon}|\hat{u}(t,\xi)|^{2}\,\mathrm{d}\xi\lesssim
1.$
Thus, by the interpolation theorem, we obtain
$\int_{|\xi|\leq
c(\eta)N(t)}|\hat{u}(t,\xi)|^{2}\,\mathrm{d}\xi\lesssim_{u}\eta^{\frac{2\varepsilon}{1+2\varepsilon}}.$
(7.1)
Meanwhile, it follows from the assumption $(\ref{e51})$ that
$\displaystyle\int_{|\xi|\geq c(\eta)N(t)}|\hat{u}(t,\xi)|^{2}\,\mathrm{d}\xi$
$\displaystyle\leq[c(\eta)N(t)]^{-1}\int|\xi||\hat{u}(t,\xi)|^{2}\,\mathrm{d}\xi$
$\displaystyle\lesssim_{u}[c(\eta)N(t)]^{-1}.$
This together with $(\ref{e61})$ and Plancherel’s theorem yields
$M(u)\lesssim[c(\eta)N(t)]^{-1}+\eta^{\frac{2\varepsilon}{1+2\varepsilon}}\quad\textrm{for
all}\quad t\in\mathbb{R}.$
As $u$ is a low-to-high cascade solution, there exists $t_{n}\to\infty$ such
that $N(t_{n})\to\infty$. Since $\eta$ is arbitrarily small, we conclude that
$M(u)\equiv 0$. Thus, $u\equiv 0$, contradicting $\|u\|_{S(\mathbb{R})}=0$.
## 8 Additional regularity for soliton
In order to preclude the final enemy, namely the soliton-like solution, we
need to gain additional regularity to make the virial-type argument available.
###### Theorem 8.1.
Let $u$ be a global radially symmetric solution to $(1.1)$ that is almost
periodic modulo scaling. Suppose also that $N(t)\equiv 1$ for all
$t\in\mathbb{R}$. Then $u\in L_{t}^{\infty}\dot{H}^{s}_{x}$ for all
$s\geq\frac{1}{2}$.
To prove Theorem 8.1, we first develop some properties of the soliton-like
solution.
###### Lemma 8.1 (Compactness in $L_{x}^{2}$).
Let $u$ be a soliton solution to $(1.1)$ in the sense of Theorem $1.3$. Then
for any $\eta>0$, there exists $C(\eta)>0$ such that
$\sup_{t\in\mathbb{R}}\int_{|x|\geq
C(\eta)}|u(t,x)|^{2}\,\mathrm{d}x\leq\eta.$ (8.1)
Proof. By negative regularity(Theorem 6.1),
$\|u_{<N}(t)\|_{L_{x}^{2}(|x|\geq R)}\leq\|u_{<N}(t)\|_{L_{x}^{2}}\leq
N^{\varepsilon}\big{\|}|\nabla|^{-\varepsilon}u\big{\|}_{L_{t}^{\infty}L_{x}^{2}}\lesssim_{u}N^{\varepsilon}.$
This can be made smaller than $\eta$ by choosing $N=N(\eta)$ sufficiently
small.
To estimate the contribution of high frequency, using Schur’s test lemma
$\big{\|}\chi_{|x|\geq
2R}(-\Delta)^{-\frac{1}{2}}|\nabla|^{\frac{1}{2}}P_{\geq N}\chi_{|x|\leq
R}\big{\|}_{L^{2}\to L^{2}}\lesssim N^{-\frac{1}{2}}\langle RN\rangle^{-m}.$
On the other hand, by Bernstein’s inequality
$\big{\|}\chi_{|x|\geq
2R}(-\Delta)^{-\frac{1}{2}}|\nabla|^{\frac{1}{2}}P_{\geq N}\chi_{|x|\geq
R}\big{\|}_{L^{2}\to L^{2}}\lesssim N^{-\frac{1}{2}}.$
Thus,
$\displaystyle\,\,\int_{|x|\geq 2R}|u_{\geq N}(t,x)|^{2}\,\mathrm{d}x$
$\displaystyle\lesssim$ $\displaystyle\,\,\int_{|x|\geq
2R}\big{|}(-\Delta)^{-\frac{1}{2}}|\nabla|^{\frac{1}{2}}P_{\geq N}\chi_{\leq
R}|\nabla|^{\frac{1}{2}}u_{\geq N}\big{|}^{2}\,\mathrm{d}x$
$\displaystyle\quad+\int_{|x|\geq
2R}\big{|}(-\Delta)^{-\frac{1}{2}}|\nabla|^{\frac{1}{2}}P_{\geq N}\chi_{\geq
R}|\nabla|^{\frac{1}{2}}u_{\geq N}\big{|}^{2}\,\mathrm{d}x$
$\displaystyle\lesssim_{u}$ $\displaystyle\,\,N^{-1}\langle
RN\rangle^{-2m}+N^{-1}\int_{|x|\geq
2R}\big{|}|\nabla|^{\frac{1}{2}}u\big{|}^{2}\,\mathrm{d}x.$
Choosing $R$ sufficiently large, the first term on the right hand side can be
made smaller than $\eta$. By Definition 1.2, the second term can also be
smaller that $\eta$. Thus, it concludes $(\ref{e71})$.
###### Lemma 8.2 (Spacetime bounds).
Let $u:I\times\mathbb{R}^{5}\mapsto\mathbb{C}$ be a maximal life-span solution
to $(1.1)$ which is almost periodic modulo scaling. Let $J$ be any subinterval
of $I$. Then for any $L^{2}$-admissible pair $(q,r)$
$\int_{J}N(t)^{2}\,\mathrm{d}t\lesssim\int_{J}\Big{(}\int_{\mathbb{R}^{5}}\big{|}|\nabla|^{\frac{1}{2}}u(t,x)\big{|}^{r}\,\mathrm{d}x\Big{)}^{q/r}\,\mathrm{d}t\lesssim
1+\int_{J}N(t)^{2}\,\mathrm{d}t$ (8.2)
Proof. As noted, the proof can be found in [13], [14]. For the sake of
convenience, we give a self-contained argument using the ideas in them.
We first prove the second inequality. Let $\eta>0$ be chosen later, divide $J$
into subintervals $I_{j}$ such that on each $I_{j}$
$\int_{I_{j}}N(t)^{2}\,\mathrm{d}t\leq\eta.$
By pigeonhole principle, there are at most
$m\leq\eta^{-1}\times\big{(}1+\int_{J}N(t)^{2}\,\mathrm{d}t\big{)}$
subintervals. For each $j$, choose $t_{j}$ such that
$N(t_{j})^{2}|I_{j}|\leq 2\eta.$ (8.3)
By Strichartz’s estimate, the Hardy-Littlewood-Sobolev, and Hölder’s,
Sobolev’s inequality, we have on $I_{j}\times\mathbb{R}^{5}$ that
$\displaystyle\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{q}L_{x}^{r}}\leq$
$\displaystyle\quad\big{\|}e^{i(t-t_{j})\Delta}|\nabla|^{\frac{1}{2}}u(t_{j})\big{\|}_{L_{t}^{q}L_{x}^{r}}$
$\displaystyle\qquad+\Big{\|}\int_{t_{j}}^{t}e^{i(t-\tau)\Delta}|\nabla|^{\frac{1}{2}}F(u(\tau))\,\mathrm{d}\tau\Big{\|}_{L_{t}^{q}L_{x}^{r}}$
$\displaystyle\lesssim$
$\displaystyle\quad\big{\|}|\nabla|^{\frac{1}{2}}u_{\geq
N_{0}}(t_{j})\big{\|}_{2}+\big{\|}e^{i(t-t_{j})\Delta}|\nabla|^{\frac{1}{2}}u_{\leq
N_{0}}(t_{j})\big{\|}_{L_{t}^{q}L_{x}^{r}}$
$\displaystyle\qquad+\big{\|}|\nabla|^{\frac{1}{2}}F(u)\big{\|}_{L_{t}^{\tilde{q}^{\prime}}L_{x}^{\tilde{r}^{\prime}}}$
$\displaystyle\lesssim$
$\displaystyle\quad\big{\|}|\nabla|^{\frac{1}{2}}u_{\geq
N_{0}}(t_{j})\big{\|}_{2}+|I_{j}|^{1/q}N_{0}^{2/q}\big{\|}|\nabla|^{\frac{1}{2}}u_{<N_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\qquad+\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}^{3}_{L_{t}^{q}L_{x}^{r}},$
where $\tilde{q}^{\prime}=q/3$, $\tilde{r}^{\prime}=(15-3r)/5r$. From the
definition of almost periodic modulo scaling, choosing $N_{0}$ as a large
multiple of $N(t_{j})$, then the first term on the right hand side can be made
as small as we wish. Invoking $(\ref{s2})$ and choosing $\eta$ sufficiently
small, the second term can also be made sufficiently small. Thus, by bootstrap
argument, we obtain
$\int_{I_{j}}\Big{(}\int_{\mathbb{R}^{5}}\big{|}|\nabla|^{\frac{1}{2}}u(t,x)\big{|}^{r}\,\mathrm{d}x\Big{)}^{q/r}\,\mathrm{d}t\leq\eta.$
Recalling the bound on subinterval number, we have
$\int_{J}\Big{(}\int_{\mathbb{R}^{5}}\big{|}|\nabla|^{\frac{1}{2}}u(t,x)\big{|}^{r}\,\mathrm{d}x\Big{)}^{q/r}\,\mathrm{d}t\leq
1+\int_{J}N(t)^{2}\,\mathrm{d}t.$
For the first inequality, note that by Definition 1.2, we must have
$\int_{|x|\leq
C(\eta)N(t)^{-1}}\big{|}|\nabla|^{\frac{1}{2}}u(t,x)\big{|}^{2}\,\mathrm{d}x\gtrsim_{u}1.$
Using Hölder’s inequality
$\Big{(}\int_{\mathbb{R}^{5}}\big{|}|\nabla|^{\frac{1}{2}}u(t,x)\big{|}^{r}\,\mathrm{d}x\Big{)}^{1/r}\gtrsim\Big{(}\int_{|x|\leq
C(\eta)N(t)^{-1}}\big{|}|\nabla|^{\frac{1}{2}}u(t,x)\big{|}^{2}\,\mathrm{d}x\Big{)}^{1/2}N(t)^{2/q}\gtrsim_{u}N(t)^{2/q}.$
Integrating the above inequality on $J$, we have
$\int_{J}\Big{(}\int_{\mathbb{R}^{5}}\big{|}|\nabla|^{\frac{1}{2}}u(t,x)\big{|}^{r}\,\mathrm{d}x\Big{)}^{q/r}\,\mathrm{d}t\gtrsim_{u}\int_{J}N(t)^{2}\,\mathrm{d}t.$
###### Remark 8.1.
We have for all $\dot{H}^{1/2}$-admissible pairs $(q,\,r)$ that
$\int_{J}N(t)^{2}\,\mathrm{d}t\lesssim\|u\|_{L_{t}^{q}L_{x}^{r}(J\times\mathbb{R}^{5})}^{q}\lesssim
1+\int_{J}N(t)^{2}\,\mathrm{d}t.$
Indeed,
$\displaystyle\|u\|_{L_{t}^{q}L_{x}^{r}}\leq$
$\displaystyle\quad\big{\|}e^{i(t-t_{j})\Delta}|\nabla|^{\frac{1}{2}}u(t_{j})\big{\|}_{L_{t}^{q}L_{x}^{r}}$
$\displaystyle\qquad+\Big{\|}\int_{t_{j}}^{t}e^{i(t-\tau)\Delta}|\nabla|^{\frac{1}{2}}F(u(\tau))\,\mathrm{d}\tau\Big{\|}_{L_{t}^{q}L_{x}^{r}}$
$\displaystyle\lesssim$
$\displaystyle\quad\big{\|}|\nabla|^{\frac{1}{2}}u_{\geq
N_{0}}(t_{j})\big{\|}_{2}+|I_{j}|^{1/q}N_{0}^{2/q}\big{\|}|\nabla|^{\frac{1}{2}}u_{<N_{0}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\qquad+\|u\|_{L_{t}^{q}L_{x}^{r}}^{2}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{q_{1}}L_{x}^{r_{1}}},$
where $(q_{1},\,r_{1})$ is an $L^{2}$-admissible pair. Using the same argument
as that in proving $(\ref{s1})$, we easily get the bounds.
Due to this proposition, we could obtain some local estimates for the soliton-
like solution. Specifically, we have for $L^{2}$-admissible pair $(q,r)$ and
$\dot{H}^{1/2}$-admissible pair $(\tilde{q},\tilde{r})$ that
$\|u\|_{L_{t}^{\tilde{q}}L_{x}^{\tilde{r}}(J\times\mathbb{R}^{5})}\lesssim_{u}\langle|J|\rangle^{\frac{1}{\tilde{q}}},\quad\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{q}L_{x}^{r}(J\times\mathbb{R}^{5})}\lesssim_{u}\langle|J|\rangle^{\frac{1}{q}}.$
(8.4)
By the Hardy-Littlewood-Sobolev inequality and the interpolation
$\displaystyle\|F(u)\|_{L_{t}^{2}L_{x}^{10/7}}$
$\displaystyle\leq\|(|\cdot|^{-3}\ast|u|^{2})\|_{L_{t}^{2}L_{x}^{10/3}}\|u\|_{L_{t}^{\infty}L_{x}^{5/2}}\lesssim_{u}\|u\|_{L_{t}^{4}L_{x}^{20/7}}^{2}$
(8.5)
$\displaystyle\lesssim_{u}\|u\|_{L_{t}^{4}L_{x}^{10/3}}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{4}L_{x}^{5/2}}\lesssim_{u}\langle|J|\rangle^{\frac{1}{2}}.$
By the weighted Strichartz estimate
$\big{\|}|x|^{2}u\big{\|}_{L^{4}_{t}L_{x}^{\infty}}\lesssim_{u}\langle|J|\rangle^{\frac{1}{2}}.$
(8.6)
From Definition 1.2
$\lim_{N\to\infty}\|u_{\geq
N}\|_{L_{t}^{\infty}\dot{H}^{1/2}_{x}(\mathbb{R}\times\mathbb{R}^{5})}=0.$
(8.7)
Now, define
$G(N):=\|u_{\geq
N}\|_{L_{t}^{\infty}\dot{H}^{1/2}_{x}(\mathbb{R}\times\mathbb{R}^{5})}$ (8.8)
Note that
$\lim_{N\to\infty}G(N)=0.$ (8.9)
To prove Theorem 8.1, it suffices to prove that $G(N)\lesssim_{u}N^{-s}$ holds
for any $s>0$ and any sufficiently large $N$, since we consequently have
$\|u_{N}\|_{L_{t}^{\infty}\dot{H}_{x}^{s+1/2}}\lesssim
N^{s}\|u_{N}\|_{L_{t}^{\infty}\dot{H}^{1/2}_{x}}\lesssim_{u}1$. This will be
achieved by iterating the following proposition with sufficiently small
$\eta$.
###### Proposition 8.1.
Let $u$ be as in Theorem $8.1$. Let $\eta>0$ be sufficiently small. Then, for
sufficiently large $N=N(\eta,u)$, we have
$G(N)\lesssim_{u}\eta G\big{(}\frac{N}{16}\big{)}.$ (8.10)
To prove the proposition, it suffices to prove
$\|u_{\geq N}(t)\|_{\dot{H}^{1/2}_{x}}\lesssim_{u}\eta
G\left(\frac{N}{16}\right)$ (8.11)
for all $t\in\mathbb{R}$ and all $N$ sufficiently large. By time-translation
invariant, we may set $t=0$. Using Duhamel formula $(\ref{e16})$ and the
in/out decomposition
$\displaystyle|\nabla|^{\frac{1}{2}}u_{\geq N}(0)$ $\displaystyle=$
$\displaystyle(P^{+}+P^{-})|\nabla|^{\frac{1}{2}}u_{\geq N}(0)$ (8.12)
$\displaystyle=$
$\displaystyle\lim_{T\to\infty}i\int_{0}^{T}P^{+}e^{-it\Delta}P_{\geq
N}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}t$
$\displaystyle-\lim_{T\to\infty}\int_{-T}^{0}P^{-}e^{-it\Delta}P_{\geq
N}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}t$
as weak limits in $L_{x}^{2}$. Using the property of weak closedness for unit
ball, namely
$f_{T}\rightharpoonup
f\quad\Longrightarrow\quad\|f\|\leq\liminf_{T}\|f_{T}\|,$
we are reduced to proving that RHS of $(\ref{e711})$ $\lesssim_{u}$ RHS of
$(\ref{e710})$.
Note that $P^{\pm}$ are singular at $x=0$; to get around this, we introduce
the cutoff $\psi_{N}(x):=\psi(N|x|)$, where $\psi$ is the characteristic
function of $[1,\infty)$. As the short times and large times will be treated
differently, we rewrite $(\ref{e711})$ as
$\displaystyle|\nabla|^{\frac{1}{2}}u_{\geq N}(0)$ $\displaystyle=$
$\displaystyle[\psi_{N}(x)+(1-\psi_{N}(x))]|\nabla|^{\frac{1}{2}}u_{\geq
N}(0)$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\int_{0}^{T}\psi_{N}(x)P^{+}e^{-it\Delta}P_{\geq
N}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}t$
$\displaystyle-\lim_{T\to\infty}i\int_{-T}^{0}\psi_{N}(x)P^{-}e^{-it\Delta}P_{\geq
N}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}t$
$\displaystyle+\lim_{T\to\infty}i\int_{0}^{T}(1-\psi_{N}(x))e^{-it\Delta}P_{\geq
N}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}t,$
$\displaystyle|\nabla|^{\frac{1}{2}}u_{\geq N}(0)$ $\displaystyle=$
$\displaystyle i\int_{0}^{\delta}\psi_{N}(x)P^{+}e^{-it\Delta}P_{\geq
N}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}t$ (8.13)
$\displaystyle-i\int_{-\delta}^{0}\psi_{N}(x)P^{-}e^{-it\Delta}P_{\geq
N}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}t$
$\displaystyle+i\int_{0}^{\delta}(1-\psi_{N}(x))e^{-it\Delta}P_{\geq
N}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}t$
$\displaystyle+\lim_{T\to\infty}\sum_{M\geq
N}i\int_{\delta}^{T}\int_{\mathbb{R}^{5}}\psi_{N}[P^{+}_{M}e^{-it\Delta}](x,y)\tilde{P}_{M}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}y\,\mathrm{d}t$
$\displaystyle-\lim_{T\to\infty}\sum_{M\geq
N}i\int_{-T}^{-\delta}\int_{\mathbb{R}^{5}}\psi_{N}[P^{-}_{M}e^{-it\Delta}](x,y)\tilde{P}_{M}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}y\,\mathrm{d}t$
$\displaystyle+\lim_{T\to\infty}\sum_{M\geq
N}i\int_{\delta}^{T}\int_{\mathbb{R}^{5}}(1-\psi_{N})[\tilde{P}_{M}e^{-it\Delta}](x,y)P_{M}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}y\,\mathrm{d}t$
$\displaystyle:=$ $\displaystyle I_{1}-I_{2}+I_{3}+I_{4}-I_{5}+I_{6}.$
Note that we used the identity
$P_{\geq N}=\sum_{M\geq N}P_{M}\tilde{P}_{M},$
where $\tilde{P}_{M}:=P_{M/2}+P_{M}+P_{2M}$.
For integrals over short times, namely $I_{1}$, $I_{2}$, $I_{3}$, we have the
following estimate, that is
###### Lemma 8.3 (Local estimate).
For any sufficiently small $\eta>0$, there exists $\delta=\delta(u,\eta)>0$
such that
$\Big{\|}\int_{0}^{\delta}e^{-it\Delta}P_{\geq
N}|\nabla|^{\frac{1}{2}}F(u(t))\,\mathrm{d}t\Big{\|}_{L_{x}^{2}}\lesssim_{u}\eta
G\left(\frac{N}{8}\right)$
for sufficiently large $N$ depending on $u$ and $\eta$. An analogous estimate
holds for integration over $[-\delta,0]$ and after pre-multiplication by
$P^{\pm}$.
Proof. By Strichartz’s estimate, it only needs to prove
$\big{\|}|\nabla|^{\frac{1}{2}}P_{\geq
N}F(u)\big{\|}_{L_{t}^{2}L_{x}^{10/7}(J\times\mathbb{R}^{5})}\lesssim_{u}\eta
G\left(\frac{N}{8}\right)$
for any time interval $J$ with $|J|\leq\delta$.
From $(\ref{e78})$, for any $\eta>0$, there exists $N_{0}=N_{0}(u,\eta)$ such
that
$\|u_{\geq N_{0}}\|_{L_{t}^{\infty}\dot{H}^{1/2}_{x}}\leq\eta.$ (8.14)
Let $N\geq 8N_{0}$. Decompose $u$ as
$u:=u_{\geq\frac{N}{8}}+u_{N_{0}\leq\cdot<\frac{N}{8}}+u_{<N_{0}},$
and make a corresponding expansion of $P_{\geq N}F(u)$. Note that any term in
the resulting expansion does not contain $u_{\geq\frac{N}{8}}$ vanishes.
We first consider a term with two factors of the form $u_{<N_{0}}$. Using
Hölder’s inequality, the fractional Leibniz rule, the Hardy-Littlewood-
Sobolev, and Bernstein’s inequality
$\displaystyle\quad\big{\|}|\nabla|^{\frac{1}{2}}(|\nabla|^{-2}(u_{<N_{0}}^{2})u_{\geq\frac{N}{8}})\big{\|}_{L_{t}^{2}L_{x}^{10/7}{(J\times\mathbb{R}^{5})}}$
$\displaystyle\leq\quad\big{\|}|\nabla|^{-2}(u_{<N_{0}}^{2})\big{\|}_{L_{t}^{2}L_{x}^{5}{(J\times\mathbb{R}^{5})}}\big{\|}|\nabla|^{\frac{1}{2}}u_{\geq\frac{N}{8}}\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\qquad+\big{\|}|\nabla|^{-\frac{3}{2}}(u_{<N_{0}}^{2})\big{\|}_{L_{t}^{2}L_{x}^{10/3}{(J\times\mathbb{R}^{5})}}\|u_{\geq\frac{N}{8}}\|_{L_{t}^{\infty}L_{x}^{5/2}}$
$\displaystyle\lesssim\quad\|u_{<N_{0}}^{2}\|_{L_{t}^{2}L_{x}^{5/3}(J\times\mathbb{R}^{5})}G\big{(}\frac{N}{8}\big{)}+\|u_{<N_{0}}^{2}\|_{L_{t}^{2}L_{x}^{5/3}(J\times\mathbb{R}^{5})}G\big{(}\frac{N}{8}\big{)}$
$\displaystyle\lesssim_{u}\quad|J|^{\frac{1}{2}}N_{0}G\big{(}\frac{N}{8}\big{)},$
and
$\displaystyle\quad\big{\|}|\nabla|^{\frac{1}{2}}(|\nabla|^{-2}(u_{<N_{0}}u_{\geq\frac{N}{8}})u_{<N_{0}})\big{\|}_{L_{t}^{2}L_{x}^{10/7}{(J\times\mathbb{R}^{5})}}$
$\displaystyle\leq\quad\big{\|}|\nabla|^{-2}(u_{<N_{0}}u_{\geq\frac{N}{8}})\big{\|}_{L_{t}^{4}L_{x}^{10/3}}\big{\|}|\nabla|^{\frac{1}{2}}u_{<N_{0}}\big{\|}_{L_{t}^{4}L_{x}^{5/2}}$
$\displaystyle\qquad+\big{\|}|\nabla|^{-\frac{3}{2}}(u_{<N_{0}}u_{\geq\frac{N}{8}})\big{\|}_{L_{t}^{4}L_{x}^{5/2}}\|u_{<N_{0}}\|_{L_{t}^{4}L_{x}^{10/3}}$
$\displaystyle\lesssim_{u}\quad\|u_{<N_{0}}u_{\geq\frac{N}{8}}\|_{L_{t}^{4}L_{x}^{10/7}}|J|^{\frac{1}{4}}N_{0}^{\frac{1}{2}}+\|u_{<N_{0}}u_{\geq\frac{N}{8}}\|_{L_{t}^{4}L_{x}^{10/7}}|J|^{\frac{1}{4}}N_{0}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}\quad\|u_{<N_{0}}\|_{L_{t}^{4}L_{x}^{10/3}}\|u_{\geq\frac{N}{8}}\|_{L_{t}^{\infty}L_{x}^{5/2}}|J|^{\frac{1}{4}}N_{0}^{\frac{1}{2}}$
$\displaystyle\lesssim_{u}\quad|J|^{\frac{1}{2}}N_{0}G\big{(}\frac{N}{8}\big{)}.$
Choosing $\delta$ sufficiently small depending on $\eta$ and $N_{0}$, we see
they are acceptable.
Now, we have to estimate those components of $P_{\geq N}F(u)$ which involve
$u_{\geq\frac{N}{8}}$ and at least one of the other terms is not $u_{<N_{0}}$.
Using Hölder’s inequality, the fractional Leibniz rule, the Hardy-Littlewood-
Sobolev, Bernstein’s inequality, $(\ref{e73})$, $(\ref{e713})$,
$\displaystyle\quad\big{\|}|\nabla|^{\frac{1}{2}}(|\nabla|^{-2}(u_{\geq
N_{0}}u_{\geq\frac{N}{8}})u)\big{\|}_{L_{t}^{2}L_{x}^{10/7}(J\times\mathbb{R}^{5})}$
$\displaystyle\lesssim\quad\big{\|}|\nabla|^{-\frac{3}{2}}(u_{\geq
N_{0}}u_{\geq\frac{N}{8}})\big{\|}_{L_{t}^{4}L_{x}^{5/2}(J\times\mathbb{R}^{5})}\|u\|_{L_{t}^{4}L_{x}^{10/3}(J\times\mathbb{R}^{5})}$
$\displaystyle\qquad+\big{\|}|\nabla|^{-2}(u_{\geq
N_{0}}u_{\geq\frac{N}{8}})\big{\|}_{L_{t}^{\infty}L_{x}^{5}(J\times\mathbb{R}^{5})}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{L_{t}^{2}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\quad\|u_{\geq
N_{0}}u_{\geq\frac{N}{8}}\|_{L_{t}^{4}L_{x}^{10/7}(J\times\mathbb{R}^{5})}\langle|J|\rangle^{\frac{1}{4}}+|J|^{\frac{1}{2}}\|u_{\geq
N_{0}}u_{\geq\frac{N}{8}}\|_{L_{t}^{\infty}L_{x}^{5/3}(J\times\mathbb{R}^{5})}$
$\displaystyle\lesssim_{u}\quad\|u_{\geq\frac{N}{8}}\|_{L_{t}^{\infty}L_{x}^{5/2}}\|u_{\geq
N_{0}}\|_{L_{t}^{4}L_{x}^{10/3}(J\times\mathbb{R}^{5})}\langle|J|\rangle^{\frac{1}{4}}+|J|^{\frac{1}{2}}\|u_{\geq\frac{N}{8}}\|_{L_{t}^{\infty}L_{x}^{5/2}}\|u_{\geq
N_{0}}\|_{L_{t}^{\infty}L_{x}^{5}(J\times\mathbb{R}^{5})}$
$\displaystyle\lesssim_{u}\quad\langle|J|\rangle^{\frac{1}{4}}\|u_{\geq
N_{0}}\|_{L_{t}^{4}L_{x}^{10/3}(J\times\mathbb{R}^{5})}G\big{(}\frac{N}{8}\big{)}+\eta|J|^{\frac{1}{2}}N_{0}G\big{(}\frac{N}{8}\big{)}$
By $(\ref{e73})$
$\|u_{\geq
N_{0}}\|_{L_{t}^{2}L_{x}^{5}(J\times\mathbb{R}^{5})}\lesssim\langle|J|\rangle^{\frac{1}{2}}.$
Hence, interpolating with $(\ref{e713})$, we have
$\|u_{\geq
N_{0}}\|_{L_{t}^{4}L_{x}^{10/3}(J\times\mathbb{R}^{5})}\lesssim\eta^{\frac{1}{2}}\langle|J|\rangle^{\frac{1}{4}}.$
Thus, we obtain
$\big{\|}|\nabla|^{\frac{1}{2}}(|\nabla|^{-2}(u_{\geq
N_{0}}u_{\geq\frac{N}{8}})u)\big{\|}_{L_{t}^{2}L_{x}^{10/7}(J\times\mathbb{R}^{5})}\lesssim_{u}\eta^{\frac{1}{2}}\langle|J|\rangle^{\frac{1}{2}}G\big{(}\frac{N}{8}\big{)}+\eta|J|^{\frac{1}{2}}N_{0}G\big{(}\frac{N}{8}\big{)},$
which is acceptable.
In the same manner, we estimate
$\displaystyle\quad\big{\|}|\nabla|^{\frac{1}{2}}(|\nabla|^{-2}(u_{\geq\frac{N}{8}}u)u_{\geq
N_{0}})\big{\|}_{L_{t}^{2}L_{x}^{10/7}(J\times\mathbb{R}^{5})}$
$\displaystyle\lesssim\big{\|}|\nabla|^{-\frac{3}{2}}(u_{\geq\frac{N}{8}}u)\big{\|}_{L_{t}^{2}L_{x}^{10/3}(J\times\mathbb{R}^{5})}\|u_{\geq
N_{0}}\|_{L_{t}^{\infty}L_{x}^{5/2}}$
$\displaystyle\qquad+\big{\|}|\nabla|^{-2}(u_{\geq\frac{N}{8}}u)\big{\|}_{L_{t}^{\infty}L_{x}^{5}(J\times\mathbb{R}^{5})}\big{\|}|\nabla|^{\frac{1}{2}}u_{\geq
N_{0}}\big{\|}_{L_{t}^{2}L_{x}^{2}}$
$\displaystyle\lesssim_{u}\eta\|u\|_{L_{t}^{2}L_{x}^{5}(J\times\mathbb{R}^{5})}\|u_{\geq\frac{N}{8}}\|_{L_{t}^{\infty}L_{x}^{5/2}}\lesssim_{u}\eta\langle|J|\rangle^{\frac{1}{2}}G\big{(}\frac{N}{8}\big{)}.$
Another term $\big{\|}|\nabla|^{\frac{1}{2}}\big{(}|\nabla|^{-2}(u_{\geq
N_{0}}u)u_{\geq\frac{N}{8}}\big{)}\big{\|}_{L_{t}^{2}L_{x}^{10/7}(J\times\mathbb{R}^{5})}$
can be estimated similarly. This concludes the proof of Lemma 8.3.
We now turn our attention to $I_{4},\,I_{5},\,I_{6}$, namely the integrations
over large times: $|t|\geq\delta$. Making use of the properties of the kernels
$P_{M}e^{-it\Delta}$, $P_{M}^{\pm}e^{-it\Delta}$(see Lemma 2.4, Lemma 2.7), we
break the regions of $(t,y)$ integration into two pieces: $|y|\gtrsim M|t|$
and $|y|\ll M|t|$. when $|x|\geq N^{-1}$, we use the kernel
$P_{M}^{\pm}e^{-it\Delta}$; in this case $|y|-|x|\thicksim M|t|$ implies
$|y|\gtrsim M|t|$ for $|t|\geq\delta\geq N^{-2}$. When $|x|\leq N^{-1}$, we
use $P_{M}e^{-it\Delta}$; in this case $|y-x|\thicksim M|t|$ implies
$|y|\gtrsim M|t|$ for $|t|\geq\delta\geq N^{-2}$. The condition $\delta\geq
N^{-2}$ can be satisfied under our statement $N$ sufficiently large depending
on $u$ and $\eta$.
Define $\chi_{k}$ as the characteristic function of the set
$\\{\,(t,y):2^{k}\delta\leq|t|\leq 2^{k+1}\delta,|y|\gtrsim M|t|\,\\}.$
Then we have the following estimate
###### Lemma 8.4 (Main contribution).
Let $\eta>0$ be a small number and $\delta$ be as in Lemma $8.3$. Then
$\sum_{M\geq
N}\sum_{k=0}^{\infty}\Big{\|}\int_{2^{k}\delta}^{2^{k+1}\delta}\int_{\mathbb{R}^{5}}[P_{M}e^{-it\Delta}](x,y)\chi_{k}(t,y)[\tilde{P}_{M}|\nabla|^{\frac{1}{2}}F(u(t))](y)\,\mathrm{d}y\,\mathrm{d}t\Big{\|}_{L_{x}^{2}}\lesssim_{u}\eta
G\big{(}\frac{N}{16}\big{)}$ (8.15)
for all $N$ sufficiently large depending on $u$ and $\eta$. An analogous
estimate holds for integration over $[-2^{k+1}\delta,-2^{k}\delta]$ and with
$P_{M}$ replaced by $P_{M}^{\pm}$.
Proof. By Strichartz’s estimates
$\displaystyle\Big{\|}\int_{2^{k}\delta}^{2^{k+1}\delta}\int_{\mathbb{R}^{5}}[P_{M}e^{-it\Delta}](x,y)\chi_{k}(t,y)[\tilde{P}_{M}|\nabla|^{\frac{1}{2}}F(u(t))](y)\,\mathrm{d}y\,\mathrm{d}t\Big{\|}_{L_{x}^{2}}$
$\displaystyle\lesssim\big{\|}\chi_{k}\tilde{P}_{M}(|\nabla|^{\frac{1}{2}}F(u))\big{\|}_{L_{t}^{2}L_{y}^{10/7}([2^{k}\delta,\,2^{k+1}\delta]\times\mathbb{R}^{5})}$
Using the fractional Leibniz rule, we turn to estimate
${\rm
II_{1}}=\big{\|}\chi_{k}\tilde{P}_{M}(|\nabla|^{-\frac{3}{2}}(|u|^{2})u)\big{\|}_{L_{t}^{2}L_{x}^{10/7}([2^{k}\delta,\,2^{k+1}\delta]\times\mathbb{R}^{5})},$
${\rm
II_{2}}=\big{\|}\chi_{k}\tilde{P}_{M}(|\nabla|^{-2}(|u|^{2})|\nabla|^{\frac{1}{2}}u)\big{\|}_{L_{t}^{2}L_{x}^{10/7}([2^{k}\delta,\,2^{k+1}\delta]\times\mathbb{R}^{5})}.$
We only estimate ${\rm II_{1}}$, since ${\rm II_{2}}$ can be treated
similarly, using the fact that $u\in L_{t}^{\infty}H^{1/2}$.
Write $u$ as $u:=u_{\leq{\frac{M}{16}}}+u_{>\frac{M}{16}}$. In what follows,
all spacetime norms are taken on the slab
$[2^{k}\delta,\;2^{k+1}\delta]\times\mathbb{R}^{5}$, unless noted otherwise.
Using the support property of $\tilde{P}_{M}$, ${\rm II_{11}}$ can be
controlled by
$\displaystyle{\rm II_{1}}$ $\displaystyle\lesssim$
$\displaystyle\big{\|}\chi_{k}|\nabla|^{-\frac{3}{2}}(u^{2})u_{>\frac{M}{16}}\big{\|}_{L_{t}^{2}L_{x}^{10/7}}+\big{\|}\chi_{k}|\nabla|^{-\frac{3}{2}}(uu_{>\frac{M}{16}})u_{\leq\frac{M}{16}}\big{\|}_{L_{t}^{2}L_{x}^{10/7}}$
$\displaystyle:=$ $\displaystyle{\rm II_{11}+II_{12}}.$
Using Hölder’s inequality, and $(\ref{e75})$, we have
$\displaystyle\big{\|}\chi_{k}|\nabla|^{-\frac{3}{2}}(u^{2})u_{>\frac{M}{16}}\big{\|}_{L_{t}^{2}L_{x}^{10/7}}\leq\big{\|}u_{>\frac{M}{16}}\big{\|}_{L_{t}^{\infty}L_{x}^{5/2}}\big{\|}\chi_{k}|\nabla|^{-\frac{3}{2}}(u^{2})\big{\|}_{L_{t}^{2}L_{x}^{10/3}}$
$\displaystyle\lesssim$ $\displaystyle
G\big{(}\frac{M}{16}\big{)}\left(\Big{\|}\chi_{k}\int_{|x-y|\geq\frac{|y|}{2}}\frac{|u(x)|^{2}}{|x-y|^{7/2}}\,\mathrm{d}x\Big{\|}_{L_{t}^{2}L_{y}^{10/3}}+\Big{\|}\chi_{k}\int_{|x-y|<\frac{|y|}{2}}\frac{|u(x)|^{2}}{|x-y|^{7/2}}\,\mathrm{d}x\Big{\|}_{L_{t}^{2}L_{y}^{10/3}}\right)$
$\displaystyle\lesssim$ $\displaystyle
G\big{(}\frac{M}{16}\big{)}\left(\|\chi_{k}|y|^{-\frac{7}{2}}\|_{L_{t}^{2}L_{y}^{10/3}}\|u\|_{L_{t}^{\infty}L_{x}^{2}}+\Big{\|}\chi_{k}|y|^{-\frac{16}{5}}\int_{|x-y|<\frac{|y|}{2}}\frac{|y|^{16/5}|u|^{2}}{|x-y|^{7/2}}\,\mathrm{d}x\Big{\|}_{L_{t}^{2}L_{y}^{10/3}}\right)$
$\displaystyle\lesssim_{u}$ $\displaystyle
G\big{(}\frac{M}{16}\big{)}\left(M^{-2}(2^{k}\delta)^{-\frac{3}{2}}+\Big{\|}\chi_{k}|y|^{-\frac{16}{5}}\big{\|}1_{\leq\frac{|y|}{2}}|\cdot|^{-\frac{7}{2}}\big{\|}_{L_{x}^{5/4}}\big{\|}|y|^{2}u\|^{\frac{8}{5}}_{L_{x}^{\infty}}\|u\|_{L_{x}^{2}}^{\frac{2}{5}}\Big{\|}_{L_{t}^{2}L_{y}^{10/3}}\right)$
$\displaystyle\lesssim_{u}$ $\displaystyle
G\big{(}\frac{M}{16}\big{)}\left(M^{-2}(2^{k}\delta)^{-\frac{3}{2}}+\big{\|}\chi_{k}|y|^{-\frac{27}{10}}\big{\|}_{L_{t}^{10}L_{y}^{10/3}}\big{\|}|y|^{2}u\big{\|}_{L_{t}^{4}L_{x}^{\infty}}^{\frac{8}{5}}\right)$
$\displaystyle\lesssim_{u}$ $\displaystyle
G\big{(}\frac{M}{16}\big{)}\left(M^{-2}(2^{k}\delta)^{-\frac{3}{2}}+M^{-\frac{6}{5}}(2^{k}\delta)^{-\frac{11}{10}}\langle
2^{k}\delta\rangle^{\frac{4}{5}}\right).$
Using the Hardy-Littlewood-Sobolev, Hölder’s inequality, $(\ref{e75})$, we
estimate ${\rm II_{12}}$ as the following :
$\displaystyle{\rm II_{12}}$ $\displaystyle\leq$
$\displaystyle\|\chi_{k}u\|_{L_{t}^{2}L_{x}^{5}}\big{\|}|\nabla|^{-\frac{3}{2}}(uu_{>\frac{M}{16}})\big{\|}_{L_{t}^{\infty}L_{x}^{2}}$
$\displaystyle\lesssim$
$\displaystyle\|\chi_{k}|y|^{-2}\|_{L_{t}^{4}L_{x}^{5}}\big{\|}|y|^{2}u\big{\|}_{L_{t}^{4}L_{x}^{\infty}}\|uu_{>\frac{M}{16}}\|_{L_{t}^{\infty}L_{x}^{5/4}}$
$\displaystyle\lesssim_{u}$ $\displaystyle
M^{-1}(2^{k}\delta)^{-\frac{3}{4}}\langle
2^{k}\delta\rangle^{\frac{1}{2}}G\big{(}\frac{M}{16}\big{)}.$
Thus, the left hand side of $(\ref{e714})$ can be bounded by:
$\big{(}N^{-\frac{6}{5}}\delta^{-\frac{11}{10}}+N^{-\frac{6}{5}}\delta^{-\frac{3}{10}}+N^{-2}\delta^{-\frac{3}{2}}+N^{-1}\delta^{-\frac{3}{4}}+N^{-1}\delta^{-\frac{1}{2}}\big{)}G\big{(}\frac{N}{16}\big{)}.$
This is acceptable by choosing $N$ sufficiently large depending on $\delta$
and $\eta$.
The last claim follows from the time reversal symmetry and the
$L_{x}^{2}$-boundedness of $P^{\pm}$.
We now turn to the region of $(t,y)$ integration where $|y|\ll M|t|$. To begin
with, we recall the bounds in [15] for the kernels of the propagators in the
region $|x|\leq N^{-1}$, $|y|\ll M|t|$, $|t|\geq\delta\gg N^{-2}$; and the
region $|x|\geq N^{-1}$, $y$ and $t$ as above:
$\displaystyle\big{|}P_{M}e^{-it\Delta}(x,y)\big{|}+\big{|}P_{M}^{\pm}e^{-it\Delta}(x,y)\big{|}\lesssim\frac{1}{(M^{2}|t|)^{50}}K_{M}(x,y),$
where
$K_{M}(x,y):=\dfrac{M^{5}}{\langle M(x-y)\rangle^{50}}+\dfrac{M^{5}}{\langle
Mx\rangle^{2}\langle My\rangle^{2}\langle M|x|-M|y|\rangle^{50}}$
be bounded on $L_{x}^{2}$.
Let $\tilde{\chi}_{k}$ be the characteristic function of the set
$\\{\,(t,y):2^{k}\delta\leq|t|\leq 2^{k+1}\delta,\,|y|\ll M|t|\,\\}.$
###### Lemma 8.5 (The tail).
Let $\eta>0$ be a small number and $\delta$ be as in Lemma $8.3$. Then
$\sum_{M\geq
N}\sum_{k=0}^{\infty}\Big{\|}\int_{2^{k}\delta}^{2^{k+1}\delta}\int_{\mathbb{R}^{5}}\frac{K_{M}(x,y)}{(M^{2}|t|)^{50}}\tilde{\chi}_{k}(t,y)[\tilde{P}_{M}|\nabla|^{\frac{1}{2}}F(u(t))](y)\,\mathrm{d}y\,\mathrm{d}t\Big{\|}_{L_{x}^{2}}\lesssim_{u}\eta
G\big{(}\frac{N}{16}\big{)}$
for sufficiently large $N$ depending on $u$ and $\eta$.
Proof. By Minkowski’s inequality, the boundedness of $K_{M}$, the support
property of $\tilde{P}_{M}$, Hölder’s and the Hardy-Littlewood-Sobolev
inequality
$\displaystyle\Big{\|}\int_{2^{k}\delta}^{2^{k+1}\delta}\int_{\mathbb{R}^{5}}\frac{K_{M}(x,y)}{(M^{2}|t|)^{50}}\tilde{\chi}_{k}(t,y)[\tilde{P}_{M}|\nabla|^{\frac{1}{2}}F(u(t))](y)\,\mathrm{d}y\,\mathrm{d}t\Big{\|}_{L_{x}^{2}}$
$\displaystyle\lesssim$
$\displaystyle(M^{2}2^{k}\delta)^{-50}\big{\|}\tilde{\chi}_{k}(t,y)[\tilde{P}_{M}|\nabla|^{\frac{1}{2}}F(u)]\big{\|}_{L_{t}^{1}L_{y}^{2}}$
$\displaystyle\lesssim$ $\displaystyle(M^{2}2^{k}\delta)^{-50}2^{k}\delta
M^{\frac{1}{2}}\Big{\|}\tilde{P}_{M}\big{(}|\nabla|^{-2}\big{(}\,\big{|}u_{\leq\frac{M}{16}}+u_{>\frac{M}{16}}\big{|}^{2}\,\big{)}(u_{\leq\frac{M}{16}}+u_{>\frac{M}{16}})\big{)}\Big{\|}_{L_{t}^{\infty}L_{y}^{2}}$
$\displaystyle\lesssim$ $\displaystyle(M^{2}2^{k}\delta)^{-50}2^{k}\delta
M^{\frac{1}{2}}\Big{(}\big{\|}|\nabla|^{-2}(u^{2})u_{>\frac{M}{16}}\big{\|}_{L_{t}^{\infty}L_{y}^{2}}+\big{\|}|\nabla|^{-2}(uu_{>\frac{M}{16}})u_{\leq\frac{M}{16}}\big{\|}_{L_{t}^{\infty}L_{y}^{2}}\Big{)}$
$\displaystyle\lesssim$ $\displaystyle(M^{2}2^{k}\delta)^{-50}2^{k}\delta
M^{\frac{1}{2}}\Big{(}\big{\|}|\nabla|^{-2}(u^{2})\big{\|}_{L_{t}^{\infty}L_{x}^{5/2}}\|u_{>\frac{M}{16}}\|_{L_{t}^{\infty}L_{x}^{10}}$
$\displaystyle\hskip
113.81102pt+\big{\|}|\nabla|^{-2}(uu_{>\frac{M}{16}})\big{\|}_{L_{t}^{\infty}L_{x}^{10}}\|u\|_{L_{t}^{\infty}L_{x}^{5/2}}\Big{)}$
$\displaystyle\lesssim_{u}$ $\displaystyle(M^{2}2^{k}\delta)^{-50}2^{k}\delta
M^{\frac{1}{2}}\Big{(}\|u\|^{2}_{L_{t}^{\infty}L_{x}^{5/2}}M^{\frac{3}{2}}G\big{(}\frac{M}{16}\big{)}+\|u_{>\frac{M}{16}}\|_{L_{t}^{\infty}L_{x}^{10}}\|u\|_{L_{t}^{\infty}L_{x}^{5/2}}\Big{)}$
$\displaystyle\lesssim_{u}$ $\displaystyle(M^{2}2^{k}\delta)^{-50}2^{k}\delta
M^{2}G\big{(}\frac{M}{16}\big{)}$
Summing first over $k\geq 0$ and then $M\geq N$, we obtain
$\displaystyle\sum_{M\geq
N}\sum_{k=0}^{\infty}\Big{\|}\int_{2^{k}\delta}^{2^{k+1}\delta}\int_{\mathbb{R}^{5}}\frac{K_{M}(x,y)}{(M^{2}|t|)^{50}}\tilde{\chi}_{k}(t,y)[\tilde{P}_{M}|\nabla|^{\frac{1}{2}}F(u(t))](y)\,\mathrm{d}y\,\mathrm{d}t\Big{\|}_{L_{x}^{2}}$
$\displaystyle\lesssim_{u}(N^{2}\delta)^{-49}G\big{(}\frac{N}{16}\big{)}.$
Choosing $N$ sufficiently large depending on $\delta,\eta$, we get the desired
result.
From $(\ref{e710})$, $(\ref{e712})$, Lemma 8.3, Lemma 8.4, Lemma 8.5, it
concludes Proposition 8.1, which in turn proves Theorem 8.1.
## 9 No soliton
In this section we prove
###### Theorem 9.1.
There exists no non-zero soliton-like solution in the sense of Theorem $1.3$.
Proof. We argue by contradiction. Assume that there exists such a soliton
solution, then by Theorem 6.1, Theorem 8.1, $u\in
L_{t}^{\infty}H^{s}_{x}(s\geq 1)$, and $u$ has the energy of the form
$E(u(t))=\frac{1}{2}\int_{\mathbb{R}^{5}}|\nabla
u|^{2}\,\mathrm{d}x-\frac{1}{4}\iint_{\mathbb{R}^{5}\times\mathbb{R}^{5}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|^{3}}\,\mathrm{d}x\mathrm{d}y.$
Now, define
$M_{a}(t):=2{\rm Im}\int_{\mathbb{R}^{5}}\bar{u}(t,x)\vec{a}(x)\cdot\nabla
u(t,x)\,\mathrm{d}x,$
where $a(x)=x\psi\big{(}\frac{|x|}{R}\big{)}$, $\psi$ is a smooth, radial
function such that
$\psi(r)=\begin{cases}1,&r\leq 1\\\ 0,&r\geq 2.\end{cases}$
Then, by the Cauchy-Schwarz inequality, we have
$|M_{a}(t)|\leq R\|u\|_{2}\|\nabla u\|_{2}\lesssim_{u}R.$ (9.1)
We should prove by our assumption
$\sup_{t\in\mathbb{R}}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{2}<\frac{\sqrt{6}}{3}\big{\|}|\nabla|^{\frac{1}{2}}Q\big{\|}_{2}$
that $M_{a}(t)$ is an increasing function of time, i.e.,
$\partial_{t}M_{a}(t)>0$. Thus, a contradiction with $(\ref{e81})$
A few computations with equation $(1.1)$ yields
$\displaystyle\partial_{t}M_{a}(t)$ $\displaystyle=$ $\displaystyle
12E(u(t))-2\int_{\mathbb{R}^{5}}|\nabla u|^{2}\,\mathrm{d}x$ (9.5)
$\displaystyle-\int_{\mathbb{R}^{5}}\Big{[}\frac{24}{R|x|}\psi^{\prime}\big{(}\frac{|x|}{R}\big{)}+\frac{11}{R^{2}}\psi^{{}^{\prime\prime}}\big{(}\frac{|x|}{R}\big{)}+\frac{|x|}{R^{3}}\psi^{{}^{\prime\prime\prime}}\big{(}\frac{|x|}{R}\big{)}\Big{]}|u(t,x)|^{2}\,\mathrm{d}x$
$\displaystyle+4\int_{\mathbb{R}^{5}}\Big{[}\psi\big{(}\frac{|x|}{R}\big{)}-1+\frac{|x|}{R}\psi^{\prime}\big{(}\frac{|x|}{R}\big{)}\Big{]}|\nabla
u(t,x)|^{2}\,\mathrm{d}x$
$\displaystyle-3\iint_{\mathbb{R}^{5}\times\mathbb{R}^{5}}\Big{[}x\psi\big{(}\frac{|x|}{R}\big{)}-y\psi\big{(}\frac{|y|}{R}\big{)}-(x-y)\Big{]}\cdot\frac{x-y}{|x-y|^{5}}|u(t,x)|^{2}|u(t,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y.$
We will prove that $(\ref{e82})$, $(\ref{e83})$, $(\ref{e84})$ are
sufficiently small compared to $(\ref{e80})$.
Note that $(\ref{e82})$ has a trivial bound $R^{-2}$.
Now, let $\eta>0$ be a small number to be chosen later. From Lemma 8.1, there
exists $R=R(\eta)$ such that for all $t\in\mathbb{R}$
$\int_{|x|\geq\frac{R}{4}}|u(t,x)|^{2}\,\mathrm{d}x\leq\eta.$ (9.6)
Define $\chi$ as a smooth cutoff to the region $|x|\geq\frac{R}{2}$ with
$\nabla\chi$ be bounded by $R^{-1}$ and supported on $\\{|x|\thicksim R\\}$.
Since $u\in C_{t}^{0}H^{s}(s>1)$, using the interpolation theorem and
$(\ref{e85})$, we deduce
$\displaystyle|(\ref{e83})|\lesssim\|\chi\nabla u(t)\|^{2}_{2}\lesssim$
$\displaystyle\|\nabla(\chi u)\|_{2}^{2}+\|u\nabla\chi\|_{2}^{2}\lesssim\|\chi
u(t)\|_{2}^{\frac{2(s-1)}{s}}\|u(t)\|_{H^{s}}^{\frac{2}{s}}+\eta$
$\displaystyle\lesssim_{u}\eta^{\frac{s-1}{s}}+\eta.$
It remains to estimate $(\ref{e84})$. We divide the integration into three
parts.
$\displaystyle(\ref{e84})$ $\displaystyle=$ $\displaystyle
2\mu\int\\!\\!\\!\int_{\begin{subarray}{c}|x|\geq R\\\ |y|\geq
R\end{subarray}}\bigg{(}x\Big{(}\psi\big{(}\frac{|x|}{R}\big{)}-1\Big{)}-y\Big{(}\psi\big{(}\frac{|y|}{R}\big{)}-1\Big{)}\bigg{)}\cdot\frac{x-y}{|x-y|^{5}}|u(t,x)|^{2}|u(t,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle+2\mu\iint_{\begin{subarray}{c}|x|\geq R\\\
|y|<R\end{subarray}}x\Big{(}\psi\big{(}\frac{|x|}{R}\big{)}-1\Big{)}\cdot\frac{x-y}{|x-y|^{5}}|u(t,x)|^{2}|u(t,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle-2\mu\iint_{\begin{subarray}{c}|x|<R\\\ |y|\geq
R\end{subarray}}y\Big{(}\psi\big{(}\frac{|y|}{R}\big{)}-1\Big{)}\cdot\frac{x-y}{|x-y|^{5}}|u(t,x)|^{2}|u(t,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle:=I_{1}+I_{2}+I_{3}.$
We first estimate $I_{1}$. By the Gagliardo-Nirenberg inequality of
convolution type and $(\ref{e85})$
$|I_{1}|\lesssim\iint_{\begin{subarray}{c}|x|\geq R\\\ |y|\geq
R\end{subarray}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|^{3}}\,\mathrm{d}x\,\mathrm{d}y\lesssim\|\chi
u\|_{2}\|\nabla u\|_{2}^{3}\lesssim_{u}\eta^{1/2}.$
To estimate $I_{2}$, using the Hardy-Littlewood-Sobolev inequality, Lemma 3.1,
Sobolev’s embedding theorem,
$\displaystyle|I_{2}|$ $\displaystyle\lesssim$
$\displaystyle\iint_{\begin{subarray}{c}|x|>2R\\\
|y|<R\end{subarray}}|x|\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|^{4}}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle+\iint_{\begin{subarray}{c}R<|x|\leq 2R\\\
|y|<R\end{subarray}}\bigg{|}x\Big{(}\psi\big{(}\frac{|x|}{R}\big{)}-1\Big{)}\bigg{|}\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|^{4}}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\lesssim$
$\displaystyle\iint_{\mathbb{R}^{5}\times\mathbb{R}^{5}}\frac{|\chi
u(x)|^{2}|u(y)|^{2}}{|x-y|^{3}}\,\mathrm{d}x\,\mathrm{d}y+R^{-\frac{3}{4}}\iint_{\mathbb{R}^{5}\times\mathbb{R}^{5}}\frac{|x|^{7/4}|u|\cdot|\chi
u(x)||u(y)|^{2}}{|x-y|^{4}}\,\mathrm{d}x\,\mathrm{d}y$ $\displaystyle\lesssim$
$\displaystyle\|\chi u\|_{2}\|\nabla(\chi u)\|_{2}\|\nabla
u\|_{2}^{2}+R^{-\frac{3}{4}}\big{\|}|x|^{7/4}u\big{\|}_{L^{\infty}_{x}}\|\chi
u\|_{2}\|u\|_{H^{1}_{x}}^{2}$ $\displaystyle\hskip 6.0pt\lesssim_{u}$
$\displaystyle\eta^{\frac{2s-1}{2s}}+R^{-\frac{3}{4}}\eta^{\frac{1}{2}}.$
Note that in the last inequality, we used the interpolation as that to
estimate $(\ref{e83})$.
$I_{3}$ can be estimated in the same argument.
Thus, choosing $\eta$ sufficiently small depending on $u$, $R$ sufficiently
large depending on $u$ and $\eta$, we have
$|(\ref{e82})|+|(\ref{e83})|+|(\ref{e84})|\lesssim\frac{1}{100}\times\bigg{[}12E(u(t))-2\int_{\mathbb{R}^{5}}|\nabla
u|^{2}\,\mathrm{d}x\bigg{]}.$
On the other hand, as
$\sup_{t\in\mathbb{R}}\big{\|}|\nabla|^{\frac{1}{2}}u\big{\|}_{2}<\frac{\sqrt{6}}{3}\big{\|}|\nabla|^{\frac{1}{2}}Q\big{\|}_{2}$,
using the Hardy-Littlewood-Sobolev type inequality $(\ref{a1})$, we see
$(\ref{e80})>0$. Hence $\partial_{t}M_{a}(t)>0$. This concludes the proof of
Theorem 9.1.
Acknowledgements: The authors would like to thank Prof. B. Pausader for his
invaluable comments and suggestions. The authors are partly supported by the
NSF of China (No. 10725102 and No. 10726053).
## References
* [1] P. Begout, A. Vargas, Mass concentration phenomena for the $L^{2}$-critical nonlinear Schrödinger equation, Tans. Amer. Math. Soc. 359 (2007), 5257-5282.
* [2] J. Bourgain, Global well-posedness of defocuing 3D critical NLS in the radial case, J. Amer. Math. Soc. 12 (1999), 145-171.
* [3] T. Cazenave, F. B. Weissler, The Cauchy problem for the critical NLS in $H^{s}$, Nonlinear Anal., Theory Methods Appl. 14 (1990), 807-836.
* [4] T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, 2003.
* [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy critical NLS in $\mathbb{R}^{3}$, Ann. of Math. 167 (2008), 767-865.
* [6] J. Ginibre, G. Velo, Smoothing prpperties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144 (1992), 163-188.
* [7] G. Grillakis, On Nonlinear Schrödinger equations, Comm.PDE 25 (2000),1827-1844.
* [8] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980.
* [9] C. E. Kenig, F. Merle, Global well-posedness, scattering and blow up for the energy-critical, focusing NLS in the radial case, Invent. Math. 166 (2006), 645-675.
* [10] C. E. Kenig, F. Merle, Scattering for $\dot{H}^{1/2}$ bounded solutions to cubic, defocusing NLS in dimension 3, arXiv: 0712.1834.
* [11] C. E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure. Appl. Math 46 (1993), 527-620.
* [12] S. Kerrani, On the defect of compactness for the Strichartz estimates for the Schrödinger equations, J.Diff. Eq. 175 (2001), 353-392.
* [13] R. Killip, T. Tao, M. Visan, The cubic nonlinear Schödinger equation in two dimensions with radial data, arXiv: 0707.3188.
* [14] R. Killip, M. Visan, The focusing energy-critical NLS in dimensions five and higher, arXiv: 0804.1018.
* [15] R. Killip, M. Visan, X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, arXiv: 0708.0849.
* [16] D. Li, C. Miao, and X. Zhang, The focusing energy-critical Hartree equation, J. Diff. Eq. 246 (2009),1139-1163.
* [17] D. Li, X. Zhang, On the classification of minimal mass blowup solutions of the focusing mass-critical Hartree equation, Adv. in Math. 220 (2009), 1171-1192.
* [18] E. H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, Vol.14, American Mathematical Society, Providence, RI, 2001.
* [19] F. Merle, L. Vega, Compactness at blow-up time for $L^{2}$ solutions of the critical nonlinear Schrödinger equation in 2D, Int. Math. Res. Note 8 (1998), 399-425.
* [20] C. Miao, Harmonic analysis and its application to PDE (in Chinese), second edition, Science Press, 2004.
* [21] C. Miao, G. Xu, and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Func. Anal. 253(2) (2007), 605-627.
* [22] C. Miao, G. Xu, and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation for radial data , J. Math. Pures Appl. 91 (2009), 49-79
* [23] C. Miao, G. Xu, and L. Zhao, Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloqu. Math. 114(2) (2009), 213-236.
* [24] E. Ryckman, M. Visan, Global well-posedness and scattering for the defosuing energy-critical NLS in $\mathbb{R}^{1+4}$, J. Amer. Math. 129 (2007), 1-60.
* [25] L. P. Pitaevskii, Vortex lines in an imperfect Bose gases, Sov. Phys. JETP 13 (1961), 451-454.
* [26] S. Shao, Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation, arXiv: 0809.0153.
* [27] T. Tao, M. Visan, and X. Zhang, Global well-posedness and scattering for the mass-critical nonlinear Schödinger equation for radial data in high dimensions, Duke Math. J. 140 (2007), 165-202.
* [28] T. Tao, M. Visan, and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, to appear in Forum Math.
* [29] M. Visan, The defocusing energy-critical NLS in higher dimensions, J. Duke Math. 138 (2007), 218-374.
* [30] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567-576.
|
arxiv-papers
| 2009-06-18T09:00:23 |
2024-09-04T02:49:03.410165
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yanfang Gao, Changxing Miao and Guixiang Xu",
"submitter": "Changxing Miao",
"url": "https://arxiv.org/abs/0906.3382"
}
|
0906.3583
|
2009 March 31 2009 June 16 $\langle$publication date$\rangle$ T. Yuasa et
al.The origin of an extended X-ray emission of 47 Tuc
Galaxy: globular clusters: individual (47 Tuc) — X-rays: ISM
# The origin of an extended X-ray emission apparently associated with the
globular cluster 47 Tucanae
Takayuki Yuasa 11affiliation: Department of Physics, School of Science,
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 Kazuhiro Nakazawa
11affiliationmark: and Kazuo Makishima11affiliationmark: 22affiliation:
Cosmic Radiation Laboratory, The Institute of Physical and Chemical Research
(RIKEN), 2-1 Hirosawa, Wako, Saitama 351-0198 Last update:
[email protected]
###### Abstract
Using the Suzaku X-ray Imaging Spectrometer, we performed a 130 ks observation
of an extended X-ray emission, which was shown by ROSAT and Chandra
observations to apparently associate with the globular cluster 47 Tucanae. The
obtained $0.5-6$ keV spectrum was successfully fitted with a redshifted thin
thermal plasma emission model whose temperature and redshift are
$2.2^{+0.2}_{-0.3}~{}$keV (at the rest frame) and $0.34\pm 0.02$,
respectively. Derived parameters, including the temperature, redshift, and
luminosity, indicate that the extended X-ray source is a background cluster of
galaxies, and its projected location falls, by chance, on the direction of the
proper motion of 47 Tucanae.
## 1 Introduction
Many globular clusters in the Galaxy move through the Galactic halo with a
typical velocity of $\sim 200$ km s-1, which exceeds the sound velocity (a few
tens of km s-1 to 150 km s-1) specified by roughly estimated halo plasma
temperatures ($T\sim 0.7-1.4\times 10^{5}$ K by [Savage & de Boer (1981)];
$T\sim 1.5-1.6\times 10^{6}$ K by [Pietz et al. (1998)]). Then, a bow shock is
expected to form between the halo plasma and any gas (intra-cluster gas) in a
moving globular cluster (Ruderman & Spiegel, 1971). Since the temperature of
the post-shock plasma should then become $\sim 10^{6}$ K (e.g., Okada et al.
(2007)), we expect to detect diffuse X-ray emission with a shape that traces
the bow shock. Several X-ray satellites have been observing globular clusters
in search for such bow-shock-heated X-ray emitting plasmas. Indeed, using the
Einstein satellite, Hartwick, Cowley, & Grindlay (1982) first detected such
diffuse emissions around 47 Tucanae (hereafter 47 Tuc), $\rm{\omega}$
Centauri, and M22, although a part of the emissions was resolved into point
sources by later observations (Koch-Miramond & Auriére (1987); Krockenberger &
Grindlay (1995); Gendre, Barret, & Webb (2003)). Subsequently, Krockenberger &
Grindlay (1995) newly reported diffuse emissions in 47 Tuc, followed by,
possible detections of such a diffuse emission in several globular clusters;
e.g., Hopwood et al. (2000) in NGC 6779, and Okada et al. (2007) in 47 Tuc,
NGC 6752, and M5.
Among the extended emissions so far detected, those in the globular clusters
47 Tuc and NGC 6752 are of particular interest. They spatially coincide with
the directions of projected proper motions of these globular clusters
(Krockenberger & Grindlay, 1995; Okada et al., 2007), and were hence
considered to have physical relationships with the globular clusters.
According to Okada (2005) and Okada et al. (2007), the 0.5-4.5 keV Chandra
spectra from these extended sources are so hard that they require power-law
models with photon indices of $\Gamma=2.1\pm 0.3$ (47 Tuc) and $\Gamma=2.0\pm
0.2$ (NGC 6752), or a thermal plasma emission model with temperatures of
$kT=3.7^{+2.7}_{-1.3}$ keV (47 Tuc) and $kT=2.9^{+1.0}_{-0.7}$ keV (NGC 6752)
which largely exceed values expected from the bow shock heating ($\sim 10^{6}$
K). While the two X-ray sources have no optical identifications (Krockenberger
& Grindlay, 1995; Okada et al., 2007), Okada et al. (2007) reported that both
have possible radio counterparts in the 843 MHz Sydney University Molongo Sky
Survey (SUMSS; Bock et al. (1999)).
From these properties, the extended emissions apparently associated with 47
Tuc and NGC 6752 were considered to arise via inverse Compton scattering
(Krockenberger & Grindlay, 1995) or non-thermal bremsstrahlung (Okada et al.,
2007) of high energy ($E\sim 20-100$ keV) electrons that are stochastically
accelerated in the bow shock. The interpretation is attractive because the
shock is expected to be a moderate one with a Mach value of $\sim 10$, and the
condition is much different from those in the more typical acceleration sites
such as supernova remnants and jets of active galactic nuclei.
As an alternative explanation to those extended X-ray sources, Krockenberger &
Grindlay (1995) and Okada et al. (2007) also considered a chance coincidence
with a background cluster of galaxies that is not related to the globular
clusters. This alternative must be kept in mind, even though its possibility
was estimated low ($<1\%$ by Krockenberger & Grindlay (1995) and Okada et al.
(2007)) in 47 Tuc.
In the previous spectral analysis of the Chandra data from the extended
emission in 47 Tuc and NGC 6752, Okada et al. (2007) were unable to
distinguish a power-law model from a thermal emission model because of rather
large statistical errors. In the present paper, we utilize the larger
effective area and lower background level of Suzaku, to perform detailed
spectral analysis of the extended emission of 47 Tuc. Based on the model
fitting result, we conclude that the emission is from a background galaxy
cluster with a redshift of 0.3, and the spatial coincidence between the
extended emission and the globular cluster is accidental.
Throughout the paper, cosmological parameters of $\Omega_{\rm{M}}=0.28$ and
$H_{0}=70$ km s-1 Mpc-1 are used in calculations.
## 2 Observation and Data Reduction
The globular cluster 47 Tucanae was observed with Suzaku (Mitsuda et al.,
2007) on 2007 June 10–12 (observation ID 502048010). Since the target of this
observation is the extended emission (EE) at the north east region of 47 Tuc,
we set the nominal pointing of the satellite at
$(\timeform{00h24m50s},\timeform{-71D59^{\prime}46^{\prime\prime}})$, which is
$\sim\timeform{6^{\prime}}$ off the cluster center.
In the present study, we focus on the data taken with the X-ray Imaging
Spectrometer (XIS; Koyama et al. (2007)), which comprises four X-ray charge
coupled device (CCD) sensors each placed on the focal plane of the X-ray
Telescope (XRT; Serlemitsos et al. (2007)). The four pairs of XIS and XRT are
co-aligned together, and have the same field of view (FOV) of
$\timeform{17.8^{\prime}}\times\timeform{17.8^{\prime}}$. Since one of three
front-side illuminated (FI) CCD chips, XIS2, was not operational since 2006
November, we utilized the data from the remaining two FI sensors (XIS0 and 3)
and a back-side illuminated (BI) one (XIS1). In the present observation, the
XIS was operated in the normal mode without any window or burst option, but
incorporating the spaced-row charge injection method (Nakajima et al., 2008)
to restore the energy resolution of the CCDs.
After removing periods of the Earth elevation angle less than $5^{\circ}$
(ELV$<5^{\circ}$), the day Earth elevation angle less than $20^{\circ}$
(DYE$\\_$ELV$<20^{\circ}$), and the South Atlantic Anomaly, we achieved a net
exposure of 132 ks. Flickering pixels were removed from the data by using
`cleansis` version 1.7. Then, cleaned event files were generated employing the
same event extraction criteria as in the Suzaku pipe line processing (version
2).
The present data reduction and analysis were performed using HEADAS package
version 6.4.1 and `XSPEC` version 11.3.2. In spectral fitting, redistribution
matrix files and ancillary response files (ARFs) for the XIS/XRT were
generated using `xisrmfgen` version 2007-05-14 and `xissimarfgen` (Ishisaki et
al., 2007) version 2008-04-05, respectively, with the calibration files which
are provided by the calibration database (CALDB) version 2008-04-01. In the
spectral fitting described below, we ignored data in the $1.8-1.9$ keV band so
as to avoid calibration uncertainties around the Si-K edge.
Events with energies above 10 keV, taken with the Hard X-ray Detector (HXD;
Takahashi et al. (2007)), were not utilized in the present analysis. This is
because the HXD lacks imaging capability, and hence we cannot exclude
contamination by X-rays from a number of point sources associated with 47 Tuc
(eg. Verbunt & Hasinger (1998); Grindlay et al. (2001); Heinke et al. (2005)).
Since the spatial resolution of the XIS/XRT is $\sim\timeform{2^{\prime}}$, we
cannot exclude, using the XIS data alone, X-ray point sources that overlap
with the EE. To determine their spectral shapes and fluxes, we also utilized
archived Chandra ACIS data of 47 Tuc acquired in 2000 March for a total
exposure of about 70 ks (obsid=953 and 955). We used `CIAO` (Chandra
Interactive Analysis of Observations) version 4.0.2 and CALDB version 3.4.5 to
extract point source spectra. Like in the Suzaku data analysis, we also used
`XSPEC` when performing model fitting to the ACIS spectra.
## 3 Image Analysis
### 3.1 Soft and hard band images
In Fgure 1, we present images obtained with the FI cameras (XIS0 and 3) in the
soft ($0.5-1.5$ keV) and hard ($1.5-6.0$ keV) bands, after subtracting the non
X-ray background (NXB) and correcting for vignetting and exposure. We
estimated the NXB of the XIS using dark (night) Earth data taken within $\pm
150$ days of our observation of 47 Tuc. The night Earth data were summed up,
with weights according to geomagnetic cut-off rigidity which the spacecraft
experienced at the data acquisition. This was performed by `xisnxbgen` (Tawa
et al., 2008). Then, we created NXB images in the soft and hard bands, and
subtracted them from the raw images. After subtracting the NXB, we smoothed
each image with a two-dimensional Gaussian of
$\sigma=\timeform{6^{\prime\prime}}$. The diffuse X-ray backgrounds, namely
the cosmic X-ray background (CXB) and Galactic diffuse emission, are still
included in the images.
In figure 1, we see several point sources, and the very bright 47 Tuc core
region which consists of numerous X-ray point sources (e.g., Heinke et al.
(2005)). At the center of the two images, we also observe a clear
concentration of X-ray events as Krockenberger & Grindlay (1995) and Okada et
al. (2007) reported. Thus, we reconfirm the EE phenomenon with the Suzaku
data. To extract photons from the EE region, we define a circular region
(white circle in figure 1) with a radius of $\timeform{150^{\prime\prime}}$,
centered on
$(\timeform{00h24m44.2s},\timeform{-71D59^{\prime}33.5^{\prime\prime}})$ where
Okada et al. (2007) found the maximal surface brightness.
As indicated with a black solid circle and a label “PS” in figure 1, a faint
point source is recognized at the north west rim of the event extracting
region. Although the EE is still apparent in the hard band image, the point
source is no longer visible therein. At a consistent position, we find a point
source also in the ACIS image. Therefore, we consider that the XIS source is
not a brightness enhancement associated with the EE, and hereafter exclude it
using a circular region with a radius of $\timeform{1^{\prime}}$. This region
is expected to enclose 50% of X-ray photons from the point source, while the
remaining half will fall out of the region; a half of those photons (25% of
the total flux from the source) are in turn estimated to fall inside the event
extracting region around the EE, and contaminate the EE spectrum. This effect
is considered later in section 4.
(170mm,70mm)figure1.eps
Figure 1: Soft ($0.5-1.5$ keV; left) and hard ($1.5-6.0$ keV; right) band
images of 47 Tuc taken with XIS FI (XIS0 plus XIS3), shown after removing the
two corner regions irradiated with the calibration source. The images are
scaled in units of $4\times 10^{-5}{\rm counts}\ {\rm s}^{-1}~{}{\rm
pixel}^{-1}$. The non X-ray background is subtracted using the night Earth
image (see text), followed by vignetting and exposure correction, although the
diffuse X-ray background is included. The white circle with a radius of
$\timeform{150^{\prime\prime}}$ is the event extracting region for the
extended emission (EE), and the black small one shows a soft point source (PS)
which is excluded from the spectral analysis.
### 3.2 Radial profile of the extended emission
In the previous studies by Krockenberger & Grindlay (1995) and Okada et al.
(2007), the EE was concluded to be extended over an angular radius of
$\sim\timeform{2^{\prime}}$ (2.7 pc assuming a 4.6 kpc distance to 47 Tuc). To
examine the spatial extent of the emission, we calculated its azimuthly
averaged radial profile using the NXB-subtracted and vignetting-corrected
$0.5-6.0$ keV XIS FI image. This was done utilizing a series of annular
extracting regions, each with $\timeform{30^{\prime\prime}}$ width, which are
concentric with the original event extraction region. The result is shown in
figure 2 after subtracting the CXB and Galactic diffuse background rate of
$9.4\times 10^{-7}~{}{\rm counts}\ {\rm s}^{-1}~{}{\rm pixel}^{-1}$, which we
estimated using another region of the CCDs with no evident point sources.
In the same figure, we also plot a radial profile of the point spread function
(PSF) of the XIS/XRT, calculated at the center of the EE, and averaged over
XIS0 and XIS3. Thus, the EE is clearly more extended than the PSF even though
the latter is much broader than those of ROSAT and Chandra.
(80mm,80mm)figure2.eps
Figure 2: Azimuthly averaged radial profiles of the EE (filled circles) shown
in units of counts s-1 pixel-1, after subtracting the CXB and Galactic diffuse
components (see text). The PSF of XIS FI (open squares) is also plotted, being
normalized to have the same maximum value as that of the EE at the innermost
annulus.
## 4 Spectral Analysis
### 4.1 Extraction of spectra
We extracted XIS FI and BI spectra of the EE using the event extracting region
shown in figure 1 (the white circular region but excluding the black circle).
The $0.5-6$ keV band count rates measured with XIS FI and BI are $28.3\pm
0.05\times 10^{-3}~{}{\rm counts}\ {\rm s}^{-1}$ and $22.7\pm 0.04\times
10^{-3}~{}{\rm counts}\ {\rm s}^{-1}$, respectively, with $1\sigma$
statistical errors.
In addition to the EE which is the subject of the present analysis, the
spectra also contain events from the NXB, the Galactic and extragalactic X-ray
backgrounds (altogether, diffuse X-ray background), and X-ray events from
several contaminating X-ray point sources that cannot be resolved with the
XIS/XRT spatial resolution. We assume the Galactic diffuse emission to have a
uniform brightness across the XIS FOV. Although its hard component (Worrall et
al., 1982; Koyama et al., 1986; Ebisawa et al., 2001; Revnivtsev et al., 2006;
Krivonos et al., 2007) has a strong concentration toward the Galactic plane
(with a scale height of $\timeform{1.5D}-\timeform{3D}$ ; e.g., Revnivtsev et
al. (2006), Krivonos et al. (2007)), and hence a steep brightness gradient, it
can be neglected at this high Galactic latitude of $\sim\timeform{45D}$ of 47
Tuc.
### 4.2 Background Spectra
In the following subsections, we estimate the spectral shapes and fluxes of
these individual background components, and create XIS spectral data for each
component. By summing all these components, we construct background spectra,
and then subtract them from the raw XIS spectra of the EE.
#### 4.2.1 The non X-ray Background
We derived the NXB spectrum from the same stacked night-Earth data as
described in section 3.1. Since this component depends on the CCD location, we
examined spectral differences among several circular extracting regions on the
night-Earth image, with the radius ranging from
$\timeform{150^{\prime\prime}}$ to $\timeform{300^{\prime\prime}}$. Each
region is concentric with the EE extracting region (white circle in figure 1).
The derived $0.5-10$ keV NXB spectra were consistent with one another within
$\sim 3$%. Therefore, to minimize the statistical errors of the estimated NXB,
we adopted the largest extracting region ($\timeform{300^{\prime\prime}}$
radius) for both XIS FI and BI.
The constructed NXB spectrum is shown in figure 6 in green. The count rate has
been scaled to the ratio ($\sim 4.8$) of the NXB and signal extracting areas.
The $0.5-6$ keV band count rates with $1\sigma$ statistical errors are $4.4\pm
0.1\times 10^{-3}~{}{\rm counts}\ {\rm s}^{-1}$ (XIS FI) and $5.3\pm 0.1\times
10^{-3}~{}{\rm counts}\ {\rm s}^{-1}$ (XIS BI).
#### 4.2.2 The diffuse X-ray background
The diffuse X-ray background consists of two components; the Galactic and the
extragalactic emissions. The former component is thought to originate from the
Galactic halo and the Local Hot Bubble (eg. Cox & Reynolds (1987)), and
expected to appear at energies below $\sim 2$ keV with its surface brightness
depending considerably on the sky direction. The latter, the extragalactic
component, has been understood as a superposition of numerous extragalactic
active Galactic nuclei. The spectrum is known to be expressed by a power-law
model with a photon index of $\Gamma=1.4$ (e.g. Parmar et al. (1999); Lumb et
al. (2002); Kushino et al. (2002)) at least over the $2-10$ keV band.
In order to determine the local diffuse X-ray background in the present XIS
FOV, we extracted another set of XIS FI and BI spectra from the same
observation data set of 47 Tuc, but applying a mask which excludes point
sources, the core region of 47 Tuc, and the EE itself. The masked image of XIS
FI is shown in figure 3, and the NXB-subtracted (as described in section
4.2.1) spectra of the diffuse X-ray background are plotted in figure 4. We
fitted these spectra jointly with a model which consists of three diffuse
X-ray background components; a thermal emission from the Local Hot Bubble
plasma (`mekal` model in `XSPEC`; Liedahl et al. (1995)); a thermal emission
from the Galactic halo plasma (`mekal`); and a power-law model with a fixed
photon index of 1.4 to account for the extragalactic component (`powerlaw`).
The latter two components were assumed to suffer the line-of-sight Galactic
absorption, with the absorbing column density fixed at $5\times 10^{20}$ atoms
cm-2 (Dickey & Lockman, 1990) which is a typical value toward the present
field. The photoelectric absorption coefficient by Morrison & McCammon (1983),
`wabs` model in XSPEC, was employed. We assumed that the diffuse background
has a uniform surface brightness over the XIS FOV, and utilized an ARF which
was calculated using `xissimarfgen` with the `UNIFORM` option and
`r_max`$=\timeform{20^{\prime}}$. We left free the temperatures, metal
abundances, and surface brightnesses of the two thermal models, as well as the
photon index and surface brightness of the power-law model. The model gave an
acceptable fit with $\chi^{2}_{\nu}=1.10~{}(\nu=175)$; the best fit parameters
are listed in table 1.
As to the power-law component, the best fit model gave the $2-10$ keV flux of
$4.4\times 10^{-8}~{}{\rm erg}\ {\rm cm}^{-2}\ {\rm s}^{-1}\ {\rm sr}^{-1}$,
which is $\sim 20$% lower than the previously reported values; $5.4\times
10^{-8}~{}{\rm erg}\ {\rm cm}^{-2}\ {\rm s}^{-1}\ {\rm sr}^{-1}$ by Lumb et
al. (2002), and $5.7\times 10^{-8}~{}{\rm erg}\ {\rm cm}^{-2}\ {\rm s}^{-1}\
{\rm sr}^{-1}$ by Kushino et al. (2002). The deviation can reasonably be
explained by the spatial fluctuation of the extragalactic emission which can
vary by about 20% (Kushino et al., 2002) depending on XIS pointings.
We might directly subtract the diffuse background spectrum of figure 4 from
that of the EE region. However, this introduces some systematic bias because
the energy-dependent vignetting effect of the XRT (figure 11 of Serlemitsos et
al. (2007)) will cause not only the observed background brightness but also
its spectral shape to differ between the two regions; in the present case, the
two ARFs, one for the masked region (figure 3) while the other for the EE,
differ by $10-20$% (due to energy dependent vignetting effect) in the $2-6$
keV range if we normalize them at 2 keV. Hence, to avoid this problem, we
simulated the expected contribution of the diffuse background to the EE
extracting region using the best fit model explained above and the
corresponding ARF. In producing the fake spectra, we assumed a sufficiently
long exposure ($10^{7}$ s), to suppress statistical errors. This is allowed
because the background components are understood from previous observations.
The $0.5-6$ keV band count rates of the faked spectra are $8.3\times
10^{-3}~{}{\rm counts}\ {\rm s}^{-1}$ (XIS FI) and $5.9\times 10^{-3}~{}{\rm
counts}\ {\rm s}^{-1}$ (XIS BI).
(65mm,65mm)figure3.eps
Figure 3: The XIS FI image after filtering out point sources, the 47 Tuc core
region, and the EE. The image is not corrected for the vignetting or exposure.
The events plotted in the image were used in the modeling of the diffuse X-ray
background in the field of 47 Tuc.
(80mm,80mm)figure4.eps
Figure 4: The NXB-subtracted diffuse X-ray background spectra of XIS FI (black) and BI (red), extracted from the image in figure 3. The solid lines represent the best fit model, while their components are individually plotted in dashed (thermal), dot-dashed (thermal with absorption), and dotted (power law with absorption) lines. Table 1: The best fit model parameters for the diffuse background spectra. Model | Parameter | Value
---|---|---
Thermal 1 | $kT$ | $0.17\pm^{+0.01}_{-0.02}~{}{\rm keV}$
| $Z$**footnotemark: $*$ | $0.1\pm^{+0.9}_{-0.03}$
| $\Sigma$\dagger\daggerfootnotemark: $\dagger$ | $1.07\pm^{+0.51}_{-0.98}$
Absorption | $N_{\rm H}$\ddagger\ddaggerfootnotemark: $\ddagger$ | $5~{}({\rm fixed})$
Thermal 2 | $kT$ | $0.78^{+0.30}_{-0.39}~{}{\rm keV}$
| $Z$**footnotemark: $*$ | $0.03^{+0.04}_{-0.03}$
| $\Sigma$\dagger\daggerfootnotemark: $\dagger$ | $0.75^{+2.28}_{-0.35}$
Power law | $\Gamma$ | $1.4~{}({\rm fixed})$
| $\Sigma$\dagger\daggerfootnotemark: $\dagger$ | $0.38^{+0.03}_{-0.03}$
| $\chi^{2}_{\nu}$ | 1.10 (175)
**footnotemark: $*$ Abundance in terms of the solar value (Anders & Grevesse,
1989).
\dagger\daggerfootnotemark: $\dagger$The $0.5-6$ keV band model surface
brightness in units of $10^{-8}~{}{\rm erg}\ {\rm cm}^{-2}\ {\rm s}^{-1}\ {\rm
sr}^{-1}$. Absorption is not corrected.
\ddagger\ddaggerfootnotemark: $\ddagger$Line-of-sight hydrogen column density
in units of $10^{20}~{}{\rm cm}^{-2}$.
#### 4.2.3 Contamination from point sources
In the previous study using Chandra (Okada et al., 2007), six faint X-ray
point sources were found within $\timeform{2.5^{\prime}}$ of the EE region. In
table 2, we list their positions. Although they were successfully removed in
the Chandra case, we cannot do so from the present XIS data except for the
brightest one described in section 3, because of the broader PSF of the XRT
than that of Chandra. Therefore, we must model and subtract their
contributions, like the diffuse background. As explained below, we estimate
the contribution from the soft point source (Source 1; figure 1) using the
Suzaku XIS data themselves, and those of the remaining five point sources
(Source 2–5) using the Chandra ACIS data assuming that they are not variable.
Table 2: The coordinates of the contaminating point sources. Source # | Coordinate
---|---
1 | $(\timeform{00h24m14.51s},\timeform{-71D58^{\prime}50.4^{\prime\prime}})$
2 | $(\timeform{00h24m38.71s},\timeform{-72D00^{\prime}46.3^{\prime\prime}})$
3 | $(\timeform{00h24m34.83s},\timeform{-72D00^{\prime}40.2^{\prime\prime}})$
4 | $(\timeform{00h24m30.26s},\timeform{-72D00^{\prime}33.8^{\prime\prime}})$
5 | $(\timeform{00h25m00.70s},\timeform{-71D59^{\prime}59.9^{\prime\prime}})$
6 | $(\timeform{00h24m42.63s},\timeform{-71D59^{\prime}22.3^{\prime\prime}})$
Figure 5 shows XIS FI spectrum of Source 1, extracted from the black circle
(figure 1), shown after subtracting the NXB and the diffuse X-ray background.
The FI and BI spectra were fitted with an absorbed single power-law model in
the $0.5-5$ keV band. As listed in table 3, this gave an acceptable fit with
$\chi^{2}_{\nu}=1.10~{}(\nu=30)$. The summed spectrum of the remaining 5 point
sources was extracted from the ACIS data (section 2), using circular regions
each $\timeform{5^{\prime\prime}}$ in radius. The NXB was extracted from
another region of the same ACIS CCD with no evident point sources. Then, we
fitted the summed spectrum with a single power-law model in the $0.8-6$ keV
band. The best-fit ($\chi^{2}_{\nu}=2.54$ and $\nu=4$) model gave a null
hypothesis probability of $0.041$, and the parameters as listed in table 3.
To obtain a summed contribution of all the point sources to the EE, we then
faked the summed spectrum of the 5 point sources and the soft source
separately, by applying appropriate ARFs to the best fit models described
above. In calculating the ARF for the 5 point sources, we took an average of
individual ARFs weighted by their $0.5-5$ keV ACIS count rates. The ARF for
Source 1 was calculated referring to the XIS/XRT effective area for X-ray
photons which leak into the EE event extracting region; the source position
was set to be that of the soft source (the first row in table 2), whilst it is
located outside the EE region (white circle in figure 1)111A ratio of the
number of photons which leak into the EE region to that of photons falling
inside the Source 1 region (black circle in figure 1) is 23% in the 0.5-5 keV
band, which is close to with the rough estimation ($\sim 25$%) in section 3.1.
. Based on the faked spectrum, the implied $0.5-6$ keV band count rates are
$2.9\times 10^{-3}~{}{\rm counts}\ {\rm s}^{-1}$ (XIS FI) and $2.3\times
10^{-3}~{}{\rm counts}\ {\rm s}^{-1}$ (XIS BI).
(80mm,80mm)figure5.eps
Figure 5: The XIS FI spectrum (black crosses) and the best fit power-law model (solid line) of the soft point source at the north west of the EE. The NXB and diffuse X-ray background are subtracted. Data from XIS BI are excluded from the plot for clarity, although they were incorporated in the fitting. Table 3: The best fit model parameters for the contaminating soft point source and five faint sources. Model | Parameter | Source 1 | Source $2-5$
---|---|---|---
Absorption | $N_{\rm H}$**footnotemark: $*$ | $2.2^{+2.3}_{-1.3}$ | 0 (fixed)
Power law | $\Gamma$ | $5.1^{+2.3}_{-1.1}$ | $1.7\pm 0.2$
| ${\rm flux}$\dagger\daggerfootnotemark: $\dagger$ | $3.9^{+6.1}_{-1.6}$ | $3.2\pm 0.3$
| $\chi^{2}_{\nu}$ | 1.10 (30) | 2.54 (4)
**footnotemark: $*$ Hydrogen column density in units of $10^{21}~{}{\rm
cm}^{{}^{2}}$.
\dagger\daggerfootnotemark: $\dagger$The $0.5-6$ keV band model flux in units
of $10^{-14}~{}{\rm erg}\ {\rm cm}^{-2}\ {\rm s}^{-1}$. Not corrected for the
absorption.
(80mm,80mm)figure6a.eps (80mm,80mm)figure6b.eps
Figure 6: The raw (black) and the background-subtracted (cyan) spectra of the EE obtained with XIS FI (panel a) and BI (panel b). The long-accumulated non X-ray background and the faked diffuse X-ray background are plotted in green and red respectively. The blue line represents the simulated contamination from the six point sources. Table 4: The $0.5-6$ keV count rates of individual spectral components. Component | Rate ($10^{-3}~{}{\rm counts}\ {\rm s}^{-1}$)
---|---
| XIS FI | XIS BI
Raw | 28.3 | 22.7
NXB | 4.4 | 5.3
DXB**footnotemark: $*$ | 8.3 | 5.9
PS\dagger\daggerfootnotemark: $\dagger$ | 2.9 | 2.3
BGD\ddagger\ddaggerfootnotemark: $\ddagger$ | 15.6 | 13.5
EE\S\Sfootnotemark: $\S$ | 12.7 (45%) | 9.2 (40%)
**footnotemark: $*$ Diffuse X-ray background.
\dagger\daggerfootnotemark: $\dagger$Six point sources.
\ddagger\ddaggerfootnotemark: $\ddagger$Sum of the NXB, diffuse X-ray
background, and six point sources.
\S\Sfootnotemark: $\S$Derived from Raw$-$BGD. Ratios to the Raw count rates
are also shown.
### 4.3 Model fitting to the Extended Emission Spectrum
Figure 6 shows the raw EE spectra, in comparison with the background
components estimated so far. Table 4 summerizes the estimated $0.5-6$ keV
count rate of each background component. As a whole, the background amounts to
about $50\%$ of the raw counts in each detector. In figure 6, cyan data points
indicate the EE spectra obtained after subtracting the three background
components. Below, we fit them with several models which give different
physical interpretations. An ARF for the EE was calculated assuming a uniform
circular emitting region with a radius of $\timeform{50^{\prime\prime}}$ based
on the Chandra ACIS imaging result (Okada, 2005; Okada et al., 2007). The FI
and BI spectra were fitted simultaneously, with the overall model
normalization fixed to be the same between them.
First, we fitted the spectra with a single power-law model and a single
temperature optically-thin thermal model (`apec` in `XSPEC`; Smith et al.
(2001)), each subjected to the interstellar absorption (`wabs`) as Okada et
al. (2007) did. The fitting results are shown in figure 7 and listed in table
5. However, neither the power-law nor optically-thin thermal model reproduced
the spectra well, with $\chi^{2}_{\nu}=1.31~{}(\nu=100)$ and $1.33~{}(\nu=99)$
respectively. The obtained photon index ($\Gamma=2.9\pm 0.2$) or the plasma
temperature ($kT=1.7\pm 0.3$ keV) implies a considerably softer spectral shape
than the previous report (Okada (2005); $\Gamma=2.1\pm 0.3$ or
$kT=3.7\pm^{2.7}_{1.3}$ keV). In section 5, we discuss this discrepancy.
(80mm,80mm)figure7a.eps (80mm,80mm)figure7b.eps
Figure 7: Spectral fitting to the XIS FI (black) and BI (red) spectra of the
EE, with (a) a power-law and (b) a single-temperature thermal models.
In figure 7, we notice some spectral structures around 0.85 keV, 1.5 keV, and
5 keV that cannot be explained by the employed models. Suspecting that these
structures originate from redshifted atomic emission lines, we next fitted the
spectra with a thermal model that has a free redshift $z$. The fit was
improved significantly to $\chi^{2}_{\nu}=1.10~{}(\nu=98)$, and yielded the
metal abundance and redshift of $0.38^{+0.25}_{-0.13}$ times solar and
$z=0.34\pm 0.02$, respectively. Especially the spectral features at $\sim
0.85$ keV, $\sim 1.5$ keV and $\sim 5$ keV have been reproduced successfully
by redshifted Fe-L, Si-K and Fe-K lines, respectively. Incorporating $z$ thus
determined, the observed flux can be converted to the intrinsic luminosity of
$L_{0.5-6~{}{\rm keV}}=5.5\times 10^{43}~{}{\rm erg}\ {\rm s}^{-1}$,
$L_{2-10~{}{\rm keV}}=2.8\times 10^{43}~{}{\rm erg}\ {\rm s}^{-1}$, and
$L_{0.1-200~{}{\rm keV}}=1.0\times 10^{44}~{}{\rm erg}\ {\rm s}^{-1}$ in the
$0.5-6~{}{\rm keV}$, $2-10~{}{\rm keV}$, and $0.1-200~{}{\rm keV}$ band
respectively.
Since the XIS background spectrum contains K$\alpha$ emission line from
aluminum used in, for example, the XIS housing and substrate of the CCD, the
line feature at 1.5 keV could be due to residual Al-K lines caused by a wrong
NXB subtraction. To examine this possibility, we also tried spectral fittings
with NXB spectra rescaled by $5-10\%$. However, the feature at $\sim 1.5$ keV
can be seen even after subtracting an NXB spectrum that is rescaled up by
$+10$%. Since the systematic error (or reproducibility) of the NXB estimation
is reported to be 5% (Tawa et al., 2008), we consider that the structure to be
real rather than instrumental. For reference, the fit results remain unchanged
within the errors even if we ignore the $1.4-1.6$ keV range in the fitting.
In figure 8, we notice fitting residuals both in the XIS FI and BI spectra at
3.5 keV. However, they have no corresponding background features (figure 6) or
redshifted major atomic lines. No such features are present in the Chandra
spectrum, either (Okada et al., 2007). They are hence considered as
statistical fluctuations.
(80mm,80mm)figure8.eps
Figure 8: The same EE spectra as presented in figure 7, fitted with a redshifted thermal emission model. Table 5: The best fit parameters of the EE spectra. Model | $N_{\rm H}$**footnotemark: $*$ | $\Gamma$ | $kT$\dagger\daggerfootnotemark: $\dagger$ | $Z$\ddagger\ddaggerfootnotemark: $\ddagger$ | $z$\S\Sfootnotemark: $\S$ | $\Sigma$\|\|footnotemark: $\|$ | $\chi^{2}_{\nu}~{}(\nu)$
---|---|---|---|---|---|---|---
Power law | $20^{+6}_{-5}$ | $2.9\pm 0.2$ | $-$ | $-$ | $-$ | $7.5^{+1.2}_{-0.9}$ | 1.31 (100)
Theraml | $3.3^{+4.4}_{-3.3}$ | $-$ | $1.7\pm 0.3$ | $0.02^{+0.06}_{-0.02}$ | $-$ | $7.5^{+1.6}_{-1.3}$ | 1.33 (99)
Redshifted thermal | $6.9^{+5.7}_{-4.8}$ | $-$ | $2.2^{+0.2}_{-0.3}$ | $0.38^{+0.25}_{-0.13}$ | $0.34\pm 0.02$ | $7.6^{+1.3}_{-1.2}$ | 1.10 (98)
**footnotemark: $*$ Line-of-sight hydrogen column density in units of
$10^{20}~{}{\rm cm}^{-2}$.
\dagger\daggerfootnotemark: $\dagger$Thermal plasma temperature in units of
keV.
\ddagger\ddaggerfootnotemark: $\ddagger$Abundance in terms of the solar value.
\S\Sfootnotemark: $\S$Redshift.
\|\|footnotemark: $\|$The $0.5-6$ keV band model surface brightness in units
of $10^{-7}~{}{\rm erg}\ {\rm cm}^{-2}\ {\rm s}^{-1}\ {\rm sr}^{-1}$.
The above results strongly suggest that the EE is an extragalactic object,
rather than a source associated with 47 Tuc. Further considering the extended
nature and the thermal spectrum, it is most likely a background cluster of
galaxies at $z=0.34$. In the following section, we examine the galaxy cluster
interpretation of the EE based on the Suzaku results.
## 5 Discussion
### 5.1 $kT-L_{\rm{X}}$ relation and the counterpart in other wavelength
As shown so far, the spectra of the EE are well described by thermal plasma
emission with a rest-frame temperature of $kT=2.2$ keV and a redshift of
$z=0.34$. Furthermore, as plotted in figure 9, its luminosity and temperature
are consistent with the known temperature-luminosity relation ($kT-L_{\rm{X}}$
relation) of cluster of galaxies. Therefore, the EE is most naturally
interpreted as a background cluster of galaxies at a moderate redshift.
We find no counterpart in the optical (Digital Sky Survey; e.g. McLean et al.
(2000)) or near infrared (Two Micron All Sky Survey; Skrutskie et al. (2006))
surveys. Using Chandra deep survey data, Boschin (2002) however reported more
than 20 candidates of clusters of galaxies that have no optical counterpart.
The present background galaxy cluster is perhaps a member of those clusters. A
deeper optical imagery will reveal the expected galaxy clustering.
(80mm,80mm)figure9.eps
Figure 9: A $kT-L_{\rm{X}}$ relation of clusters of galaxies. Crosses
represent temperatures and bolometric luminosities of individual galaxy
clusters determined by ASCA observations (data taken from Fukazawa et al.
(2004)). Luminosities were obtained by integrating fluxes over the $0.1-200$
keV band. Filled triangle shows the EE of 47 Tuc.
### 5.2 Comparison with previous reports on the EE
The galaxy cluster interpretation has been examined as an origin of the EE in
previous papers as well. Using the $\log N-\log S$ relation of galaxy
clusters, Krockenberger & Grindlay (1995) and Okada et al. (2007) estimated
probabilities of a chance coincidence of the EE and a background galaxy
cluster emission to be less than 0.5% and 0.6%, respectively. Based on such
low probabilities, these authors argued that the EE cannot be a background
galaxy cluster.
In addition to the above probability estimation, Okada et al. (2007) used the
following argument to rule out the background cluster interpretation. First,
they determined the EE temperature as $kT=3.7~{}$keV from the Chandra ACIS
spectrum. They hence assigned a luminosity of $L_{\rm{X}}=1.1\times
10^{44}~{}{\rm erg}\ {\rm s}^{-1}$ to this putative cluster, using the $kT-
L_{\rm{X}}$ relation of clusters (e.g. Ikebe et al. (2002); Fukazawa et al.
(2004)). Comparing this $L_{\rm{X}}$ with the measured flux, the source
redshift was estimated as $z>0.5$, and hence the observed angular core radius
of the EE, $r_{\rm{c}}\sim\timeform{0.6^{\prime}}$, was converted to a
physical size of $r_{\rm{c}}>360~{}$kpc. Finally, they concluded this
$r_{\rm{c}}$ to be too large for a cluster.
In the present study, the use of the Suzaku XIS has enabled us to achieve two
major improvements (or revisions) over Okada et al. (2007). One is that we
clearly detected redshifted emission lines, which indicate $z=0.34\pm 0.02$;
the Chandra data gave no constraint on $z$. The other is that we measured a
significantly lower temperature, $kT=1.7~{}$keV if assuming $z=0$, or
$kT=2.2~{}$keV at the rest frame if adopting $z=0.34$; the latter now
satisfies the $kT-L_{\rm{X}}$ relation of clusters of galaxies (figure 9). In
addition, using the redshift, the physical core radius is now calculated to be
$\sim 160$ kpc, which is reasonable for galaxy clusters.
As reviewed so far, the difference of our conclusion from that of Okada et al.
(2007) comes mainly from the discrepant EE temperatures, $kT=1.7\pm 0.3~{}$keV
measured with Suzaku (without correction for the redshift) and
$kT=3.7^{+2.7}_{-1.3}~{}$keV with the Chandra ACIS. Possible causes of this
difference include an over estimation of the temperature with Chandra, or an
under estimation with Suzaku, or both. As the former possibility, the most
likely cause is systematic errors in the NXB subtraction. As the latter
possibility, we may presume that during the Suzaku observation, some soft
sources became brighter than in the Chandra observation.
Although the Suzaku data could thus be under-estimating the EE temperature,
significantly higher values of $kT$ would be still consistent with the $kT-
L_{\rm{X}}$ relation. Furthermore, the value of $z=0.34$ is not affected,
since it is determined by the redshifted atomic emission lines. We conclude
that the close spatial coincidence between the EE and 47 Tuc is accidental,
and they are not physically associated with each other.
T.Y. is financially supported by the Japan Society for the Promotion of
Science. This research has made use of data and softwares obtained from the
Data Archive and Transmission System at JAXA/ISAS and the High Energy
Astrophysics Science Archive Research Center, provided by NASA’s Goddard Space
Flight Center respectively. We obtained the Chandra data from the Chandra Data
Archive, and analyzed them with softwares provided by the Chandra X-ray
Center. The present research is supported in part by the Grant-in-Aid for
Scientific Research (S), No. 18104004.
## References
* Anders & Grevesse (1989) Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197
* Bock et al. (1999) Bock, D. C.-J., Large, M. I., & Sadler, E. M. 1999, AJ, 117, 1578
* Boschin (2002) Boschin, W. 2002, A&A, 396, 397
* Cox & Reynolds (1987) Cox, D. P., & Reynolds, R. J. 1987, ARA&A, 25, 303
* Dickey & Lockman (1990) Dickey, J. M., & Lockman, F. J. 1990, ARA&A, 28, 215
* Dogiel et al. (2002) Dogiel, V. A., Inoue, H., Masai, K., Schönfelder, V., & Strong, A. W. 2002, ApJ, 581, 1061
* Ebisawa et al. (2001) Ebisawa, K., Maeda, Y., Kaneda, H., & Yamauchi, S. 2001, Sci., 293, 1633
* Fukazawa et al. (2004) Fukazawa, Y., Makishima, K., & Ohashi, T. 2004, PASJ, 56, 965
* Gendre, Barret, & Webb (2003) Gendre, B., Barret, D., & Webb, N. A. 2008, A&A, 400, 521
* Grindlay et al. (2001) Grindlay, J. E., Heinke, C., Edmonds, P. D., & Murray, S. S. 2008, Sci., 292, 2290
* Hartwick, Cowley, & Grindlay (1982) Hartwick, F. D. A., Cowley, A. P., & Grindlay, J. E. 1982, ApJ, 254, L11
* Heinke et al. (2005) Heinke, C. O., Grindlay, J. E., Edmonds, P. D., Cohn, H. N., Lugger, P. M., Camilo, F., Bogdanov, S., & Freire, P. C. 2005, ApJ, 625, 796
* Hopwood et al. (2000) Hopwood, M. E. L., et al. 2000, MNRAS, 316, L5
* Ikebe et al. (2002) Ikebe, Y., Reiprich, T. H., Böhringer, H., Tanaka, Y., & Kitayama, T. 2002, A&A, 383, 773
* Ishisaki et al. (2007) Ishisaki, Y., et al. 2007, PASJ, 59, S113
* Koch-Miramond & Auriére (1987) Koch-Miramond, L., & Auriére, M. 1987, A&A, 183, 1
* Kokubun et al. (2007) Kokubun, M., et al. 2007, PASJ, 59, S53
* Koyama et al. (1986) Koyama, K., Makishima, K., Tanaka, Y., & Tsunemi, H. 1986, PASJ, 38, 121
* Koyama et al. (2007) Koyama, K., et al. 2007, PASJ, 59, S245
* Krivonos et al. (2007) Krivonos, R., Revnivtsev, M., Churazov, E., Sazonov, S., Grebenev, S., & Sunyaev, R. 2007, A&A, 463, 957
* Krockenberger & Grindlay (1995) Krockenberger, M., & Grindlay, J. E. 1995, ApJ, 451, 200
* Kushino et al. (2002) Kushino, A., Ishisaki, Y., Morita, U., Yamasaki, N. Y., Ishida, M., Ohashi, T., & Ueda, Y. 2002, PASJ, 54, 327
* Kuntz & Snowden (2000) Kuntz, K. D., & Snowden, S. L. 2000, ApJ, 543, 195
* Liedahl et al. (1995) Liedahl, D. A., Osterheld, A. L., & Goldstein, W. H. 1995, ApJ, 438, L115
* McLean et al. (2000) McLean, B. J., Greene, G. R., Lattanzi, M. G., & Pirenne, B. 2000, ASPC, 216, 145
* Lumb et al. (2002) Lumb, D. H., Warwick, R. S., Page, M., & De Luca, A. 2002, A&A, 389, 93L
* Morrison & McCammon (1983) Morrison, R. & McCammon, D. 1983, ApJ, 270, 119
* Mitsuda et al. (2007) Mitsuda, K., et al. 2007, PASJ, 59, S1
* Nakajima et al. (2008) Nakajima, H., et al. 2008, PASJ, 60, S1
* Okada (2005) Okada, Y. 2005, PhD thesis, The University of Tokyo
* Okada et al. (2007) Okada, Y., Kokubun, M., Yuasa, T., & Makishima, K. 2007, PASJ, 59, 727
* Ruderman & Spiegel (1971) Ruderman, M. A., & Spiegel, E. A. 1971, ApJ, 165, 1
* Parmar et al. (1999) Parmar, A. N., Guainazzi, M., Oosterbroek, T., Orr, A., Favata, F., Lumb, D., & Malizia, A. 1999, A&A, 345, 611
* Pietz et al. (1998) Pietz, J., Kerp, J., Kalberla, P. M., Burton, W. B., Hartmann, D., & Mebold, U. 1998, A&A, 332, 55
* Revnivtsev et al. (2006) Revnivtsev, M., Sazonov, S., Gilfanov, M., Churazov, E., & Sunyaev, R. 2006, A&A, 452, 169
* Savage & de Boer (1981) Savage, B. D. & de Boer, K. 1981, ApJ, 243, 460
* Serlemitsos et al. (2007) Serlemitsos, P. J. et al. 2007, PASJ, 59, S9
* Skrutskie et al. (2006) Skrutskie, M. F. et al. 2006, AJ, 131, 1163
* Smith et al. (2001) Smith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, ApJ, 556, L91
* Takahashi et al. (2007) Takahashi, T., et al. PASJ, 59, S35
* Tawa et al. (2008) Tawa, N., et al. 2008, PASJ, 60, S11
* Verbunt & Hasinger (1998) Verbunt, F., & Hasinger, G. 1998, A&A, 336, 895
* Worrall et al. (1982) Worrall, D. M., Marshall, F. E, Boldt, E. A., & Swank, J. H. 1982, ApJ, 255, 111
|
arxiv-papers
| 2009-06-19T06:46:14 |
2024-09-04T02:49:03.425904
|
{
"license": "Public Domain",
"authors": "Takayuki Yuasa, Kazuhiro Nakazawa, and Kazuo Makishima",
"submitter": "Takayuki Yuasa",
"url": "https://arxiv.org/abs/0906.3583"
}
|
0906.3825
|
# Novel solution of Wheeler-DeWitt theory
Łukasz Andrzej Glinka
[email protected]
_International Institute for Applicable_
_Mathematics & Information Sciences,_
_Hyderabad (India) & Udine (Italy),_
_B.M. Birla Science Centre,_
_Adarsh Nagar, 500 063 Hyderabad, India_
###### Abstract
We present a novel solution of the Wheeler–DeWitt equation based on the model
resulting due to application of the generalized one-dimensional (1D)
conjecture. The conjecture extends the global 1D one on wave functions
dependent on both matter fields and a generalized dimension which is a
functional of the global one. The residual singularity in the effective
potential is eliminating by an appropriate choice of the dimension.
Application of the dimensional reduction within the obtained two-component 1D
model yields the Dirac equation which is solved in an exact way. By use of the
inverted change of variables in this solution we construct a general solution.
Keywords quantum gravity models ; Wheeler–DeWitt equation ; Schrödinger
equation ; Dirac equation ; one-dimensionality conjecture
PACS 04.60.-m ; 03.65.-w ; 98.80.Qc
## 1 Introduction
The Wheeler–DeWitt theory, well known also as quantum geometrodynamics, is
both the historically first and the basic model of quantum gravity considered
in modern theoretical physics (See _e.g._ Ref. [1]). Understanding of its
physical content, however, is still a great theoretical riddle. Applications
to physical phenomena in high energy physics seems to be the mostly
interesting. The problem with the model has a mathematical nature, _i.e._
model is given by a functional differential equation with respect to a wave
function determined on the Wheeler superspace of on 3-dimensional metrics.
Actually, the Wheeler–DeWitt equation was solved for some highly symmetrical
classical solutions, and its experimental side is studied [2].
Quantum general relativity arises by employing of the $3+1$ splitting of
spacetime metric within the Einstein–Hilbert action supplemented by the
York–Gibbons–Hawking boundary term. It leads to the Hamiltonian form of the
action, and definition of primary and secondary constraints. One of the
secondary constraints, the Hamiltonian contraint, is canonically quantized
according to the Dirac–Faddeev method. In result, there is obtained second
order functional differential equation on superspace of 3-dimensional
embeddings, where the solution is a wave function in general depending on an
induced metric and matter fields. The problem, however, is an establishing of
any solution of the equation. In spite that there is known a formal path
integral solution, the Hartle–Hawking wave function, in general a physical
meaning of the solution is not well defined.
This paper reconsiders the Wheeler–DeWitt equation by using of the generalized
1D conjecture, discussed in some aspect in [3], and having sources in generic
cosmology [4]. The conjecture is based on reduction of the equation into the
Wheeler superspace subset, called DeWitt minisuperspace. The global dimension
is an embedding’s volume form, and obtained potential is the Wheeler–DeWitt
one, with exchange of $\sqrt{h}$ for $2/3h$. The our idea is an application of
the change of variables which could regularize the singular character of the
potential. The regularization is the generalized dimension being a special
functional of the global one. After solution of the received theory, we apply
inverted change of variables within the solution, and in result the solution
of the Wheeler–DeWitt equation is constructed by a novel method.
Paper is organized as follows. In Section 2 basic established facts are
referred. Section 3 presents the conjecture and the change of variables.
Dimensional reduction of the model is done in Section 4, and 1D wave function
is constructed in Section 5. In Section 6 the general solution is received,
and Section 7 briefly discusses all results.
## 2 Canonical Quantum Gravity
Let us recall the basics of Wheeler–DeWitt theory. General relativity [5],
governed by the Einstein field equations (in units $8\pi G/3=1$, $c=1$)
$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}{{}^{(4)}\\!}R+\Lambda
g_{\mu\nu}=3T_{\mu\nu},$ (1)
where $\Lambda$ is cosmological constant and $T_{\mu\nu}$ is stress-energy
tensor
$T_{\mu\nu}=-\dfrac{2}{\sqrt{-g}}\dfrac{\delta S_{\phi}[g]}{\delta
g^{\mu\nu}}\quad,\quad S_{\phi}[g]\equiv\int_{M}d^{4}x\sqrt{-g}L_{\phi},$ (2)
and $L_{\phi}$ is Matter fields Lagrangian, models spacetime by a
4-dimensional pseudo–Riemannian manifold $(M,g)$ with a metric $g_{\mu\nu}$,
connections $\Gamma^{\rho}_{\mu\nu}$, curvature tensor
$R^{\lambda}_{\mu\alpha\nu}$, second fundamental form
$R_{\mu\nu}=R^{\lambda}_{\mu\lambda\nu}$, and scalar curvature
${{}^{(4)}\\!}R=g^{\kappa\lambda}R_{\kappa\lambda}$. If $M$ is closed and has
an induced spacelike boundary $(\partial M,h)$ with a metric $h_{ij}$, second
fundamental form $K_{ij}$, and an extrinsic curvature $K=h^{ij}K_{ij}$ then
(1) arise by variational principle used to the Hilbert action with the
York–Gibbons–Hawking term [6]
$S[g]\\!=\\!\int_{M}d^{4}x\sqrt{-g}\left\\{-\dfrac{1}{6}{{}^{(4)}\\!}R+\dfrac{\Lambda}{3}\right\\}+S_{\phi}[g]-\dfrac{1}{3}\int_{\partial
M}d^{3}x\sqrt{h}K\quad.$ (3)
The Nash embedding theorem [7] allows using $3+1$ splitting [8]
$\displaystyle g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}+N^{i}N_{i}&N_{j}\\\
N_{i}&h_{ij}\end{array}\right]\quad,\quad
h_{ik}h^{kj}=\delta_{i}^{j}\quad,\quad N^{i}=h^{ij}N_{j},$ (6)
for which the action (3) takes the canonical form $S[g]=\int dtL$ with
$\displaystyle L=\int_{\partial
M}d^{3}x\left\\{\pi_{\phi}\dot{\phi}+\pi\dot{N}+\pi^{i}\dot{N_{i}}+\pi^{ij}\dot{h}_{ij}-NH-
N_{i}H^{i}\right\\},$ (7)
where $\pi$’s are canonical conjugate momenta, and $H$, $H^{i}$ are [9]
$\displaystyle\pi_{\phi}=\frac{\partial
L_{\phi}}{\partial\dot{\phi}}\quad,\quad\pi=\frac{\partial
L}{\partial\dot{N}}\quad,\quad\pi^{i}=\frac{\partial
L}{\partial\dot{N_{i}}}\quad,\quad\pi^{ij}=\sqrt{h}\left(K^{ij}-Kh^{ij}\right),$
(8) $\displaystyle H^{i}=2\pi^{ij}_{\leavevmode\nobreak\ ;j}\quad,\quad
H=\sqrt{h}\left\\{{{}^{(3)}\\!R}[h]+K^{2}-K_{ij}K^{ij}-2\Lambda-6\varrho[\phi]\right\\},$
(9)
with ${{}^{(3)}\\!R}\equiv h^{ij}R_{ij}$,
$\varrho[\phi]=n^{\mu}n^{\nu}T_{\mu\nu}$,
$n^{\mu}=(1/N)\left[1,-N^{i}\right]$, and holds
$\dot{h}_{ij}=2NK_{ij}+N_{i|j}+N_{j|i}.$ (10)
where $N_{i|j}$ is an intrinsic covariant derivative of $N_{i}$. DeWitt [10]
showed that $H^{i}$ are generators of the spatial diffeomorphisms
$\widetilde{x}^{i}=x^{i}+\xi^{i}$, _i.e._
$\displaystyle i\left[h_{ij},\int_{\partial M}H_{a}\xi^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle-
h_{ij,k}\xi^{k}-h_{kj}\xi^{k}_{\leavevmode\nobreak\
,i}-h_{ik}\xi^{k}_{\leavevmode\nobreak\ ,j}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (11) $\displaystyle i\left[\pi^{ij},\int_{\partial
M}H_{a}\xi^{a}d^{3}x\right]$ $\displaystyle=$
$\displaystyle-\left(\pi^{ij}\xi^{k}\right)_{,k}+\pi^{kj}\xi^{i}_{\leavevmode\nobreak\
,k}+\pi^{ik}\xi^{j}_{\leavevmode\nobreak\ ,k}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (12)
where $H_{i}=h_{ij}H^{j}$, and that the first-class algebra is satisfied
$\displaystyle i\left[H_{i}(x),H_{j}(y)\right]=\int_{\partial
M}H_{a}c^{a}_{ij}d^{3}z\quad,\quad
i\left[H(x),H_{i}(y)\right]=H\delta^{(3)}_{,i}(x,y),$ (13) $\displaystyle
i\left[\int_{\partial M}H\xi_{1}d^{3}x,\int_{\partial
M}H\xi_{2}d^{3}x\right]=\int_{\partial
M}H^{a}\left(\xi_{1,a}\xi_{2}-\xi_{1}\xi_{2,a}\right)d^{3}x.$ (14)
where
$c^{a}_{ij}=\delta^{a}_{i}\delta^{b}_{j}\delta^{(3)}_{,b}(x,z)\delta^{(3)}(y,z)-(i\leftrightarrow
j,x\leftrightarrow y)$ are structure constants of the diffeomorphism group,
and all Lie brackets of $\pi$’s and $H$’s vanish. Time-preservation [11] of
the primary constraints, _i.e._ $\pi\approx 0$, $\pi^{i}\approx 0$, leads to
the secondary constraints - scalar (Hamiltonian) and vector respectively
$\displaystyle H\approx 0\quad,\quad H^{i}\approx 0\quad,$ (15)
Scalar constraint yields dynamics, vector one merely reflects
diffeoinvariance. Using the canonical momentum $\pi^{ij}$ within the scalar
constraint yield the Einstein–Hamilton–Jacobi equation (See [12] and some
modern studies [13])
$H=G_{ijkl}\pi^{ij}\pi^{kl}-\sqrt{h}\left({}^{(3)}R[h]-2\Lambda-6\varrho[\phi]\right)\approx
0\quad,$ (16)
where
$G_{ijkl}\equiv(2\sqrt{h})^{-1}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right)$
is the metric on superspace, a factor space of all $C^{\infty}$ Riemannian
metrics on $\partial M$, and a group of all $C^{\infty}$ diffeomorphisms of
$\partial M$ that preserve orientation [14]. The Dirac–Faddeev primary
canonical quantization method [11, 15]
$\displaystyle i\left[\pi^{ij}(x),h_{kl}(y)\right]$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)\delta^{(3)}(x,y)\quad,$
(17) $\displaystyle i\left[\pi^{i}(x),N_{j}(y)\right]$ $\displaystyle=$
$\displaystyle\delta^{i}_{j}\delta^{(3)}(x,y)\quad,\quad
i\left[\pi(x),N(y)\right]=\delta^{(3)}(x,y)\quad,$ (18)
used for the constraint (16) yields the Wheeler–DeWitt equation [12, 10]
$\left\\{G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}+\sqrt{h}\left({{}^{(3)}\\!R}[h]-2\Lambda-6\varrho[\phi]\right)\right\\}\Psi[h_{ij},\phi]=0\quad,$
(19)
and other first class constraints merely reflect diffeoinvariance
$\pi\Psi[h_{ij},\phi]=0\quad,\quad\pi^{i}\Psi[h_{ij},\phi]=0\quad,\quad
H^{i}\Psi[h_{ij},\phi]=0\quad,$ (20)
and are not important in this model, called quantum geometrodynamics.
## 3 1D conjecture
### 3.1 Global dimension
Global one–dimensionality within the quantum General Relativity (19)
considered in [3], arises from the change of variables in the Wheeler–DeWitt
equation
$h_{ij}\rightarrow h=\det
h_{ij}=\dfrac{1}{3}\varepsilon^{ijk}\varepsilon^{abc}h_{ia}h_{jb}h_{kc}\quad,$
(21)
where $\varepsilon^{ijk}$ is the Levi-Civita density. Using of the Jacobi rule
for differentiation of a determinant of a metric $g_{\mu\nu}$ in the 3+1
splitting one obtains
$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}\longrightarrow N^{2}\delta
h=N^{2}hh^{ij}\delta h_{ij},$ (22)
and consequently one establishes the differentiation
$\dfrac{\delta}{\delta h_{ij}}=hh^{ij}\dfrac{\delta}{\delta h}\quad.$ (23)
Applying (23) within the quantum geometrodynamics (19) and doing double
contraction of the superspace metric with an embedding metric one receives
$G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}=-\dfrac{3}{2}h^{3/2}\dfrac{\delta^{2}}{\delta h^{2}},$ (24)
so that finally the Wheeler–DeWitt equation (19) can be rewritten as
$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+V_{eff}[h,\phi]\right)\Psi[h,\phi]=0.$
(25)
Here $V_{eff}[h,\phi]$ is the effective potential
$V_{eff}[h,\phi]\equiv\dfrac{2}{3}\dfrac{{{}^{(3)}\\!R}}{h}-\dfrac{4}{3}\dfrac{\Lambda}{h}-\dfrac{4}{h}\varrho[\phi].$
(26)
First term of the potential (26) describes contribution due to an embedding
geometry only, the second one is mix of the cosmological constant and an
embedding geometry, and the third component is due to Matter fields and an
embedding geometry. In result we have to deal with the wave function of a type
$\Psi[h,\phi]$, and the basic Wheeler–DeWitt wave function $\Psi[h_{ij},\phi]$
can be reconstructed by inverse change of variables $h\rightarrow h_{ij}$.
### 3.2 Generalized dimensions
The potential (26) is singular type, which can be eliminated by the general
change of variables
$\displaystyle h\rightarrow\xi=\xi[h],$ (27)
$\displaystyle\delta\xi=\left(\dfrac{\delta\xi}{\delta h}\right)hh^{ij}\delta
h_{ij},$ (28)
where a generalized dimension $\xi[h]$ is any functional in the global
dimension $h$. With (27) the one-dimensional equation (25) becomes
$\left\\{\left(\dfrac{\delta\xi}{\delta
h}\right)^{2}\dfrac{\delta^{2}}{\delta{\xi^{2}}}+V_{eff}\left[\xi,\phi\right]\right\\}\Psi\left[\xi,\phi\right]=0,$
(29)
so that for all nonsingular cases $\dfrac{\delta\xi}{\delta h}\neq 0$ one
writes
$\left\\{\dfrac{\delta^{2}}{\delta{\xi^{2}}}+V[\xi,\phi]\right\\}\Psi\left[\xi,\phi\right]=0,$
(30)
where $V[\xi,\phi]$ is given by
$V[\xi,\phi]=\left(\dfrac{\delta\xi}{\delta
h}\right)^{-2}V_{eff}\left[\xi,\phi\right].$ (31)
In fact the choice of $\xi$ is a kind of the choice of a ”gauge”, naturally
$\xi[h]\equiv h$ is the generic gauge, _i.e._ the case when a generalized
dimension becomes the global dimension. Other choices can be generated
directly from this case. Note that the following choice
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{8}{3}}\sqrt{{h}},$ (32) $\displaystyle\delta\xi$
$\displaystyle=$ $\displaystyle\sqrt{\dfrac{2}{3}}\sqrt{{h}}h^{ij}\delta
h_{ij},$ (33)
cancels the singularity $1/h$ present in the effective potential
$V_{eff}\left[h,\phi\right]$ (26), and the equation (30) reads
$\left\\{\dfrac{\delta^{2}}{\delta{\xi^{2}}}+{{}^{(3)}\\!R[\xi]}-2\Lambda-6\varrho[\phi]\right\\}\Psi\left[\xi,\phi\right]=0,$
(34)
Solving (34) and applying inverse change of variables $\xi\rightarrow h_{ij}$
the basic Wheeler–DeWitt wave function $\Psi\left[h_{ij},\phi\right]$ can be
reconstructed. The appropriate normalization condition should be chosen as
$\int\left|\Psi\left[\xi,\phi\right]\right|^{2}\delta\mu(\xi,\phi)=1,$ (35)
where $\mu(\xi,\phi)$ is an invariant measure.
## 4 Dimensional reduction
Let us chose the product measure $\mu(\xi,\phi)=\delta\xi\delta\phi$. Eq. (30)
can be derived as the Euler-Lagrange equation of motion by variational
principle $\delta S[\Psi]=0$ applied to the action
$\displaystyle
S[\Psi]=-\dfrac{1}{2}\int\delta\xi\delta\phi\Psi[\xi,\phi]\left(\dfrac{\delta^{2}}{\delta{\xi^{2}}}+V[\xi,\phi]\right)\Psi[\xi,\phi]=$
(36)
$\displaystyle=-\dfrac{1}{2}\int\delta\phi\Psi[\xi,\phi]\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}+\dfrac{1}{2}\int\delta\xi\delta\phi\left\\{\left(\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}\right)^{2}+V[\xi,\phi]\Psi^{2}[\xi,\phi]\right\\},$
(37)
where partial differentiation was used. Choosing the coordinate system so that
the boundary term vanishes
$-\dfrac{1}{2}\int\delta\phi\Psi[\xi,\phi]\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}=0,$
(38)
and using the fact that
$S[\Psi]\equiv\int\delta\xi\delta\phi
L\left[\Psi[\xi,\phi],\delta\Psi[\xi,\phi]/\delta\xi\right],$ (39)
one obtains the Lagrangian characteristic for Euclidean field theory
$L\left[\Psi[\xi,\phi],\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}\right]=\dfrac{1}{2}\left(\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi}\right)^{2}+\dfrac{V[\xi,\phi]}{2}\Psi^{2}[\xi,\phi],$
(40)
for which the corresponding canonical conjugate momentum is
$\Pi_{\Psi}[\xi,\phi]=\dfrac{\partial
L}{\partial\left(\delta\Psi[\xi,\phi]/\delta\xi\right)}=\dfrac{\delta\Psi[\xi,\phi]}{\delta\xi},$
(41)
so that the choice (38) actually means orthogonal coordinates
$\Psi[\xi,\phi]\Pi_{\Psi}[\xi,\phi]=0,$ (42)
for any values of $\xi$ and $\phi$. With using of the relation (41) the Eq.
(30) can be rewritten in the form
$\dfrac{\delta\Pi_{\Psi}[\xi,\phi]}{\delta\xi}+V[\xi,\phi]\Psi[\xi,\phi]=0,$
(43)
and its combination with the Eq. (41) yield the appropriate Dirac equation
$\left(i\gamma\dfrac{\delta}{\delta\xi}-M[\xi,\phi]\right)\Phi[\xi,\phi]=0,$
(44)
where we have employed the notation
$\Phi[\xi,\phi]=\left[\begin{array}[]{c}\Pi_{\Psi}[\xi,\phi]\\\
\Psi[\xi,\phi]\end{array}\right]\quad,\quad
M[\xi,\phi]=\left[\begin{array}[]{cc}1&0\\\ 0&V[\xi,\phi]\end{array}\right],$
(45)
and the $\gamma$-matrices algebra consists only one element - the Pauli matrix
$\sigma_{y}$
$\gamma=\left[\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right]\equiv\sigma_{y}\quad,\quad\gamma^{2}=I,$ (46)
where $I$ is the identity matrix, that in itself obey the algebra
$\left\\{\gamma,\gamma\right\\}=2\delta_{E}\quad,\quad\delta_{E}=\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right].$ (47)
Dimensional reduction of the one component second order theory (30) yields the
two component first order one (44) possessing the Clifford algebra of
Euclidean type [16] $\mathcal{C}\ell_{1,1}(\mathbb{R})$ that is the matrix
algebra possessing a complex $2$-dimensional representation. Restricting to
$Pin_{1,1}(\mathbb{R})$ yield a 2D spin representations; restricting to
$Spin_{1,1}(\mathbb{R})$ splits it onto a sum of two 1D Weyl representations;
$\mathcal{C}\ell_{1,1}(\mathbb{R})$ decomposes into a direct sum of two
isomorphic central simple algebras or a tensor product
$\displaystyle\mathcal{C}\ell_{1,1}(\mathbb{R})=\mathcal{C}\ell^{+}_{1,1}(\mathbb{R})\oplus\mathcal{C}\ell^{-}_{1,1}(\mathbb{R})=\mathcal{C}\ell_{2,0}(\mathbb{R})\otimes\mathcal{C}\ell_{0,0}(\mathbb{R}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (48)
$\displaystyle\mathcal{C}\ell_{1,1}(\mathbb{R})\cong\mathbb{R}(2)\quad,\quad\mathcal{C}\ell^{\pm}_{1,1}(\mathbb{R})=\dfrac{1\pm\gamma}{2}\mathcal{C}\ell_{1,1}(\mathbb{R})\cong\mathbb{R}\quad,\quad\mathcal{C}\ell_{0,0}(\mathbb{R})\cong\mathbb{R}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (49)
## 5 1D wave function
The Dirac equation (44) can be rewritten in the Schrödinger form
$i\dfrac{\delta\Phi[\xi,\phi]}{\delta\xi}=H[\xi,\phi]\Phi[\xi,\phi]\quad,\quad
H[\xi,\phi]=i\left[\begin{array}[]{cc}0&-V[\xi,\phi]\\\
1&0\end{array}\right].$ (50)
Solution of the evolution (50) can be written as
$\Phi[\xi,\phi]=U[\xi,\phi]\Phi[\xi^{I},\phi],$ (51)
where $\Phi[\xi^{I},\phi]$ is an initial data vector with respect to $\xi$
only, and $U[\xi,\phi]$ is unitary evolution operator
$\displaystyle
U=\exp\left\\{-i\int_{\Sigma(\xi)}\delta\xi^{\prime}H[\xi^{\prime},\phi]\right\\}=\exp\left\\{-i\Omega(\xi,\phi)\langle
H\rangle(\xi,\phi)\right\\},$ (52)
where $\Sigma(\xi)$ is finite integration area in $\xi$-space, $\Omega$ is the
volume of full configuration space, and $\langle H\rangle(\phi)$ is an
averaged energy given by the formulas
$\Omega(\xi,\phi)=\int_{\Sigma(\xi,\phi)}\delta\xi^{\prime}\delta\phi^{\prime}\quad,\quad\langle
H\rangle(\xi,\phi)=\dfrac{1}{\Omega(\xi,\phi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}H[\xi^{\prime},\phi].$
(53)
where $\Sigma(\xi,\phi)=\Sigma(\xi)\times\Sigma(\phi)$ is finite integration
region of full configurational space. Explicitly one obtains
$\displaystyle
U[\xi,\phi]=\mathbf{1}_{2}\cosh\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]+$ (54)
$\displaystyle+\left[\begin{array}[]{cc}0&\sqrt{{\langle
V\rangle(\xi,\phi)}}\\\ \left(\sqrt{{\langle
V\rangle(\xi,\phi)}}\right)^{-1}&0\end{array}\right]\sinh\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],$ (57)
with
$\langle
V\rangle(\xi,\phi)=\dfrac{1}{\Omega(\xi,\phi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}V[\xi^{\prime},\phi].$
(58)
Elementary algebraic manipulations yield the generalized one-dimensional wave
function as
$\displaystyle\Psi[\xi,\phi]$ $\displaystyle=$
$\displaystyle\Psi[\xi^{I},\phi]\cosh\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]+$ (59) $\displaystyle+$
$\displaystyle\Pi_{\Psi}[\xi^{I},\phi]\left(\sqrt{{\langle
V\rangle(\xi,\phi)}}\right)^{-1}\sinh\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],$
and the canonical conjugate momentum as the solution is
$\displaystyle\Pi_{\Psi}[\xi,\phi]$ $\displaystyle=$
$\displaystyle\Pi_{\Psi}[\xi^{I},\phi]\cosh\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right]+$ (60) $\displaystyle+$
$\displaystyle\Psi[\xi^{I},\phi]\sqrt{{\langle
V\rangle(\xi,\phi)}}\sinh\left[\Omega(\xi,\phi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],$
where $\Psi[\xi^{I},\phi]$ and $\Pi_{\Psi}[\xi^{I},\phi]$ are initial data
with respect to $\xi$ only. Applying, however, the equation (41) for (60) one
obtains the relation
$\displaystyle\Pi_{\Psi}[\xi,\phi]=\dfrac{\Pi_{\Psi}[\xi^{I},\phi]}{\sqrt{{\langle
V\rangle}}}\dfrac{\delta}{\delta\xi}\left[\Omega\sqrt{{\langle
V\rangle}}\right]\cosh\left[\Omega\sqrt{{\langle V\rangle}}\right]+$
$\displaystyle+\left[\Psi[\xi^{I},\phi]\dfrac{\delta}{\delta\xi}\left[\Omega\sqrt{{\langle
V\rangle}}\right]+\Pi_{\Psi}[\xi^{I},\phi]\dfrac{\delta}{\delta\xi}\left[\left(\sqrt{{\langle
V\rangle}}\right)^{-1}\right]\right]\sinh\left[\Omega\sqrt{{\langle
V\rangle}}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (61)
where for shorten notation $\Omega\equiv\Omega(\xi,\phi)$ and $\langle
V\rangle\equiv\langle V\rangle(\xi,\phi)$, which after calculation of the
functional derivatives
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\dfrac{\delta}{\delta\xi}\left[\Omega\sqrt{{\langle
V\rangle}}\right]$ $\displaystyle=$ $\displaystyle\dfrac{1}{2}\sqrt{{\langle
V\rangle}}\left(\dfrac{\delta\Omega}{\delta\xi}+1\right),$ (62)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\dfrac{\delta}{\delta\xi}\left[\left(\sqrt{{\langle
V\rangle}}\right)^{-1}\right]$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left[\Omega\sqrt{{\langle
V\rangle}}\right]^{-1}\left(\dfrac{\delta\Omega}{\delta\xi}-1\right),$ (63)
and using it within the formula (5) yields
$\displaystyle\Pi_{\Psi}[\xi,\phi]=\Pi_{\Psi}[\xi^{I},\phi]\dfrac{1}{2}\left(\dfrac{\delta\Omega}{\delta\xi}+1\right)\cosh\left[\Omega\sqrt{{\langle
V\rangle}}\right]+$
$\displaystyle+\left[\Psi[\xi^{I},\phi]\dfrac{\sqrt{{\langle
V\rangle}}}{2}\left(\dfrac{\delta\Omega}{\delta\xi}+1\right)+\dfrac{\Pi_{\Psi}[\xi^{I},\phi]}{2\Omega\sqrt{{\langle
V\rangle}}}\left(\dfrac{\delta\Omega}{\delta\xi}-1\right)\right]\sinh\left[\Omega\sqrt{{\langle
V\rangle}}\right].\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (64)
After comparison with (60) one obtains the system of equations
$\displaystyle\left\\{\begin{array}[]{cc}\dfrac{1}{2}\left(\dfrac{\delta\Omega}{\delta\xi}+1\right)=1,\vspace*{10pt}\\\
\Psi[\xi^{I},\phi]\dfrac{1}{2}\left(\dfrac{\delta\Omega}{\delta\xi}+1\right)+\dfrac{\Pi_{\Psi}[\xi^{I},\phi]}{\Omega\langle
V\rangle}\dfrac{1}{2}\left(\dfrac{\delta\Omega}{\delta\xi}-1\right)=\Psi[\xi^{I},\phi]\end{array}\right..$
(67)
The first equation of the system (67) yields the relation
$\dfrac{\delta\Omega}{\delta\xi}=1=\int_{\Sigma(\phi)}\delta\phi^{\prime},$
(68)
where the last integral arises by the first formula in (53), which after
application to the second equation gives the self-consistent identity
$\Psi[\xi^{I},\phi]=\Psi[\xi^{I},\phi]$. It means also that the volume
$\Omega(\xi,\phi)$ is $\phi$-invariant, _i.e._
$\Omega(\xi,\phi)=\int_{\Sigma(\xi)}\delta\xi^{\prime}=\Omega(\xi).$ (69)
Directly from (59) the probability density can be deduced easily as
$\displaystyle|\Psi[\xi,\phi]|^{2}$ $\displaystyle=$
$\displaystyle(\Psi[\xi^{I},\phi])^{2}\cosh^{2}\left[\Omega\sqrt{\langle
V\rangle}\right]+$ (70) $\displaystyle+$
$\displaystyle(\Pi_{\Psi}[\xi^{I},\phi])^{2}\left(\langle
V\rangle\right)^{-1}\sinh^{2}\left[\Omega\sqrt{\langle V\rangle}\right]+$
$\displaystyle+$
$\displaystyle\Psi[\xi^{I},\phi]\Pi_{\Psi}[\xi^{I},\phi]\left(\sqrt{\langle
V\rangle}\right)^{-1}\sinh\left[2\Omega\sqrt{\langle V\rangle}\right],$
and in the light of the relation (42) it simplifies to
$|\Psi[\xi,\phi]|^{2}=(\Psi[\xi^{I},\phi])^{2}\cosh^{2}\left[\Omega\sqrt{\langle
V\rangle}\right]+(\Pi_{\Psi}[\xi^{I},\phi])^{2}\left(\langle
V\rangle\right)^{-1}\sinh^{2}\left[\Omega\sqrt{\langle V\rangle}\right].$ (71)
Putting by hands the following separation conditions
$\displaystyle\Psi[\xi^{I},\phi]=\Psi[\xi^{I}]\Gamma_{\Psi}[\phi]\quad,\quad\Pi_{\Psi}[\xi^{I},\phi]=\Pi_{\Psi}[\xi^{I}]\Gamma_{\Pi}[\phi],$
(72)
where $\Gamma_{\Psi}$ and $\Gamma_{\Pi}$ are functionals of $\phi$ only and
$\Psi[\xi^{I}]$, and $\Pi_{\Psi}[\xi^{I}]$ are constant functionals, and
applying the usual normalization one obtains the simple constraint
$\int_{\Sigma(\xi,\phi)}|\Psi[\xi^{\prime},\phi^{\prime}]|^{2}\delta\xi^{\prime}\delta\phi^{\prime}=1\longrightarrow
A(\Pi_{\Psi}[\xi^{I}])^{2}+B(\Psi[\xi^{I}])^{2}-1=0,$ (73)
where the constants $A$ and $B$ are given by the integrals
$\displaystyle A$ $\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\Gamma_{\Pi}[\phi^{\prime}]\left(\langle
V^{\prime}\rangle\right)^{-1}\sinh^{2}\left[\Omega^{\prime}\sqrt{\langle
V^{\prime}\rangle}\right]\delta\xi^{\prime}\delta\phi^{\prime},$ (74)
$\displaystyle B$ $\displaystyle=$
$\displaystyle\int_{\Sigma(\xi,\phi)}\Gamma_{\Psi}[\phi^{\prime}]\cosh^{2}\left[\Omega^{\prime}\sqrt{\langle
V^{\prime}\rangle}\right]\delta\xi^{\prime}\delta\phi^{\prime},$ (75)
which in our assumption are convergent and finite. The equation (73), however,
can be solved straightforwardly. In result one obtains the relation
$\displaystyle\Pi_{\Psi}[\xi^{I}]=\pm\sqrt{{\dfrac{1}{A}-\dfrac{B}{A}(\Psi[\xi^{I}])^{2}}},$
(76)
which together with
$\Pi_{\Psi}[\xi^{I},\phi]=\dfrac{\delta\Psi[\xi^{I},\phi]}{\delta\xi^{I}}$ and
(72 yields differential equation
$\dfrac{1}{\Gamma[\phi]}\dfrac{\delta\Psi[\xi^{I}]}{\delta\xi^{I}}=\pm\sqrt{{\dfrac{1}{A}-\dfrac{B}{A}(\Psi[\xi^{I}])^{2}}},$
(77)
where $\Gamma[\phi]\equiv\dfrac{\Gamma_{\Pi}[\phi]}{\Gamma_{\Psi}[\phi]}$,
which can be integrated
$\sqrt{A}\int\dfrac{\delta\Psi[\xi^{I}]}{\sqrt{{1-B(\Psi[\xi^{I}])^{2}}}}=\pm\Gamma[\phi]\xi^{I}+C,$
(78)
where $C$ is a constant of integration, with the result
$\sqrt{{A/B}}\arcsin\left\\{\sqrt{{B/A}}\Psi[\xi^{I}]\right\\}=\pm\Gamma[\phi]\xi^{I}+C,$
(79)
so that after elementary algebraic manipulations one obtains
$\Psi[\xi^{I}]=\sqrt{{A/B}}\sin\theta(\xi^{I},\phi),$ (80)
where
$\theta(\xi^{I},\phi)=\sqrt{{B/A}}\left(\pm\Gamma[\phi]\xi^{I}+C\right),$ (81)
However, because $\Psi[\xi^{I}]$ must be a functional of $\xi^{I}$ only, must
holds $\Gamma[\phi]=\Gamma_{0}$, where $\Gamma_{0}$ is a constant, for which
$\theta(\xi^{I},\phi)=\theta(\xi^{I})$. Taking into account the relation (76)
one obtains finally
$\displaystyle\Psi[\xi^{I}]=\sqrt{{A/B}}\sin\theta(\xi^{I})\quad,\quad\Pi_{\Psi}[\xi^{I}]=\pm\sqrt{{\dfrac{1}{A}-\sin^{2}\theta(\xi^{I})}}.$
(82)
In the light of the equation (42), however, must holds one of the relations
$\displaystyle\sin\theta(\xi^{I})\equiv
0\quad,\quad\sin\theta(\xi^{I})=\pm\sqrt{{1/A}}.$ (83)
The first relation in (83) means that
$\sqrt{{B/A}}\left(\pm\Gamma_{0}\xi^{I}+C\right)=k\pi\longrightarrow\xi^{I}=\pm\dfrac{1}{\Gamma_{0}}\left(\sqrt{{A/B}}k\pi-C\right),$
(84)
where $k\in\mathbb{Z}$ is an integer. Similarly the second relation in (83)
gives
$\xi^{I}=\pm\dfrac{1}{\Gamma_{0}}\left(\pm\sqrt{{A/B}}\arcsin\sqrt{{1/A}}-C\right).$
(85)
For the first case one has
$\Psi[\xi^{I}]=0\quad,\quad\Pi_{\Psi}[\xi^{I}]=\pm\sqrt{{1/A}},$ (86)
and for the second one hold
$\displaystyle\Psi[\xi^{I}]=\pm\sqrt{{1/B}}\quad,\quad\Pi_{\Psi}[\xi^{I}]=0.$
(87)
Finally we see that the generalized one-dimensional wave function (59) is
$\Psi[\xi,\phi]=\pm\Gamma_{\Psi}[\phi]\Gamma_{0}\sqrt{{\dfrac{1}{A}}}\left(\sqrt{{\langle
V\rangle(\xi,\phi)}}\right)^{-1}\sinh\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right],$ (88)
in the first case of (83), and for the second one
$\Psi[\xi,\phi]=\pm\Gamma_{\Psi}[\phi]\sqrt{{\dfrac{1}{B}}}\cosh\left[\Omega(\xi)\sqrt{{\langle
V\rangle(\xi,\phi)}}\right].$ (89)
## 6 General solution
The general solutions of the Wheeler–DeWitt equation (19) can be now
constructed immediately from the generalized one-dimensional solutions (88)
and (89) by putting in the integrals
$\Omega(\xi)=\int_{\Sigma(\xi,\phi)}\delta\xi^{\prime}\quad,\quad\langle
V\rangle(\xi,\phi)=\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}V[\xi^{\prime},\phi],$
(90)
the $\xi$-measure following form combination of the relations (32) and (33)
$\displaystyle\delta\xi=\sqrt{\dfrac{2}{3}}\sqrt{{h}}h^{ij}\delta h_{ij}.$
(91)
Because, however, the potential $V[\xi,\phi]$ has a form
$V[\xi,\phi]={{}^{(3)}\\!R[\xi]}-2\Lambda-6\varrho[\phi],$ (92)
one has nice separability
$\langle
V\rangle(\xi,\phi)=\dfrac{1}{\Omega(\xi)}\int_{\Sigma(\xi)}\delta\xi^{\prime}\leavevmode\nobreak\
{{}^{(3)}\\!R[\xi^{\prime}]}-2\Lambda-6\rho[\phi],$ (93)
so that in fact for a concrete 3-dimensional embedding we should estimate the
functional average of the 3-dimensional Ricci scalar
$\displaystyle\langle{{}^{(3)}\\!R[h]}\rangle=\dfrac{1}{\Omega(h_{ij})}\int_{\Sigma(h_{ij})}\delta
h_{ij}^{\prime}\sqrt{\dfrac{2}{3}}\sqrt{{h^{\prime}}}{h^{ij}}^{\prime}\leavevmode\nobreak\
{{}^{(3)}\\!R[h^{\prime}]},$ (94)
where
$\Omega(h_{ij})=\int_{\Sigma(h_{ij})}\delta
h_{ij}^{\prime}\sqrt{\dfrac{2}{3}}\sqrt{{h^{\prime}}}{h^{ij}}^{\prime},$ (95)
which yields the functional average of the potential
$\langle
V\rangle(h_{ij},\phi)=\langle{{}^{(3)}\\!R[h]}\rangle-2\Lambda-6\rho[\phi].$
(96)
Using the formula (96) within the solutions (88) and (89) one obtains the
general solutions of the Wheeler–DeWitt equation due to the 1D conjecture
$\displaystyle\Psi[h_{ij},\phi]=\pm\Gamma_{\Psi}[\phi]\Gamma_{0}\sqrt{{\dfrac{1}{A}}}\left(\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right)^{-1}\sinh\left[\Omega(h_{ij})\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (97)
$\displaystyle\Psi[h_{ij},\phi]=\pm\Gamma_{\Psi}[\phi]\sqrt{{\dfrac{1}{B}}}\cosh\left[\Omega(h_{ij})\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right].\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (98)
Here the constants $A$ and $B$ are defined as the integrals
$\displaystyle
A=\sqrt{{\dfrac{2}{3}}}\Gamma_{0}\int_{\Sigma(h_{ij},\phi)}\Gamma_{\Psi}[\phi^{\prime}]\dfrac{\sinh^{2}\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]}{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\sqrt{{h^{\prime}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (99) $\displaystyle
B=\sqrt{{\dfrac{2}{3}}}\int_{\Sigma(h_{ij},\phi)}\Gamma_{\Psi}[\phi^{\prime}]\cosh^{2}\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]\sqrt{{h^{\prime}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (100)
and assumed to be convergent and finite. Using for the solutions (97) and (98)
the usual normalization condition
$\int_{\Sigma(h_{ij},\phi)}|\Psi[h_{ij},\phi]|^{2}\sqrt{\dfrac{2}{3}}\sqrt{{h^{\prime}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi=1,$ (101)
leads to the relations
$|\Gamma_{\Psi}[\phi]\Gamma_{0}|^{2}=1\quad,\quad\Gamma_{\Psi}[\phi]\Gamma_{0}=1,$
(102)
which yield $\Gamma_{\Psi}[\phi]=1/\Gamma_{0}$, $\Gamma_{0}=1$, so that
finally one obtains
$\displaystyle\Psi_{1}[h_{ij},\phi]=\pm\sqrt{{\dfrac{1}{A}}}\left(\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right)^{-1}\sinh\left[\Omega(h_{ij})\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (103)
$\displaystyle\Psi_{2}[h_{ij},\phi]=\pm\sqrt{{\dfrac{1}{B}}}\cosh\left[\Omega(h_{ij})\sqrt{{\langle
V\rangle(h_{ij},\phi)}}\right],\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (104)
where now
$\displaystyle
A=\sqrt{{\dfrac{2}{3}}}\int_{\Sigma(h_{ij},\phi)}\dfrac{\sinh^{2}\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]}{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\sqrt{{h^{\prime}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (105) $\displaystyle
B=\sqrt{{\dfrac{2}{3}}}\int_{\Sigma(h_{ij},\phi)}\cosh^{2}\left[\Omega(h_{ij}^{\prime})\sqrt{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}\right]\sqrt{{h^{\prime}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}.\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (106)
The solutions (103) and (107) describe two independent states in the quantum
gravity model given by the Wheeler–DeWitt equation (19).
Because, however, the equation (19) is linear, in general the superposition
$\displaystyle\Psi[h_{ij},\phi]=\sum_{i=1,2}\alpha_{i}\Psi_{i}[h_{ij},\phi]$
(107)
where $\alpha_{i}$ are arbitrary constants, is its a general solution. In the
light of the normalization condition (101), it means that the constraint holds
$|\alpha_{1}|^{2}+|\alpha_{2}|^{2}+(\alpha^{\star}_{1}\alpha_{2}+\alpha_{1}\alpha^{\star}_{2})I=1,$
(108)
where
$I=\sqrt{{\dfrac{1}{AB}}}\int_{\Sigma(h_{ij},\phi)}\dfrac{\sinh\left[2\Omega(h_{ij}^{\prime})\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}\right]}{2\sqrt{{\langle
V\rangle(h_{ij}^{\prime},\phi^{\prime})}}}\sqrt{\dfrac{2}{3}}\sqrt{{h^{\prime}}}{h^{ij}}^{\prime}\delta
h_{ij}^{\prime}\delta\phi^{\prime}.$ (109)
For vanishing $I=0$ one obtains form (108) simply
$|\alpha_{2}|=\sqrt{{1-|\alpha_{1}|^{2}}}\quad,\quad|\alpha_{1}|\geqslant 1.$
(110)
The case of $I\neq 0$ is more complicated. Note that (108) can be rewritten in
form
$(\alpha_{1}+\alpha_{2}I)\alpha^{\star}_{1}+(\alpha_{2}+\alpha_{1}I)\alpha_{2}^{\star}=0\longrightarrow\dfrac{\alpha^{\star}_{1}}{\alpha_{2}^{\star}}=\dfrac{-\alpha_{1}I+\alpha_{2}}{\alpha_{1}+\alpha_{2}I},$
(111)
or in the equivalent form
$C\alpha^{\star}_{1}=-\alpha_{1}I+\alpha_{2}\quad,\quad
C\alpha_{2}^{\star}=\alpha_{1}+\alpha_{2}I,$ (112)
where $0\neq C\in\mathbb{R}$. The relations (112) establish the absolute
values on
$C|\alpha_{1}|^{2}=-\alpha^{2}_{1}I+\alpha_{2}\alpha_{1}\quad,\quad
C|\alpha_{2}|^{2}=\alpha_{1}\alpha_{2}+\alpha_{2}^{2}I,$ (113)
which after mutual adding and using of (108) yields the equation
$CI[(\alpha^{\star}_{1}-\alpha_{2})\alpha_{2}+(\alpha_{2}^{\star}+\alpha_{1})\alpha_{1}]=\alpha_{1}\alpha_{2}+\alpha_{2}\alpha_{1},$
(114)
which gives the relations
$CI(\alpha^{\star}_{1}-\alpha_{2})=\alpha_{1}\quad,\quad
CI(\alpha_{2}^{\star}+\alpha_{1})=\alpha_{2}.$ (115)
Using of the complex decomposition for $\alpha$ and $\alpha_{2}$ within (115)
leads to
$\Re\alpha_{2}=(CI-1)\Re\alpha_{1}\quad,\quad\Im\alpha_{2}=(CI-1)\Im\alpha_{1},$
(116)
or equivalently
$\alpha_{2}=(CI-1)\alpha_{1}\quad,\quad|\alpha_{2}|^{2}=(CI-1)^{2}|\alpha_{1}|^{2}.$
(117)
Employing (117) within the constraint (108) yields to
$|\alpha_{1}|^{-2}=IC^{2}+(I^{2}-2I)C-I+2.$ (118)
Because, however, both $|\alpha_{i}|^{2}\in\mathbb{R}$ as squares of absolute
values, one obtains the region of values of the constant $C$ in dependence on
the integral $I$
$C\in[-\infty,C_{-}]\cup[C_{+},\infty]\quad,\quad
C_{\pm}=\dfrac{I-2}{2}\left[1\pm\sqrt{{1-\dfrac{4}{I(I-2)}}}\right],$ (119)
where for $C_{\pm}\in\mathbb{R}$ the condition
$I\in[-\infty,1-\sqrt{{5}}]\cup[1+\sqrt{{5}},\infty]$ holds.
## 7 Outlook
This paper has discussed the selected consequence arising due to application
of the generalized one-dimensional conjecture within the Wheeler–DeWitt
quantum geometrodynamics. We have shown that employing the conjecture
immediately yield construction of a general solution. The obtained formulation
in general uses the Lebesgue–Stieltjes 1D integrals. There are open problems
related to the novel wave functions. Especially, black holes exploration by
the presented method seems to be intriguing. Similarly discussion of non
vanishing cosmological constant, and conformal flat classical solutions are
interesting. The other problem is generalization of the results for the case
of D-branes.
## Acknowledgements
Author thanks Profs. I. Ya. Aref’eva, K. A. Bronnikov, I. L. Buchbinder and V.
N. Pervushin for many valuable discussions.
## References
* [1] I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Action in Quantum Gravity. Institute of Physics Publishing (1992);
D. J. Gross, T. Piran, and S. Weinberg (eds.), Two Dimensional Quantum Gravity
and Random Surfaces. World Scientific (1992);
M. C. Bento, O. Bertolami, J. M. Mourão, and R. F. Picken (eds.), Classical
and quantum gravity. World Scientific (1993);
G. Esposito, Quantum Gravity, Quantum Cosmology and Lorentzian Geometries.
Springer (1994);
E. Prugovečki, Principles of Quantum General Relativity. World Scientific
(1995);
R. Gambini and J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity.
Cambridge University Press (1996);
G. Esposito, A. Yu. Kamenshchik, and G. Pollifrone, Euclidean Quantum Gravity
on Manifolds with Boundary. Springer (1997);
I. G. Avramidi, Heat Kernel and Quantum Gravity. Springer (2000);
S. Carlip, Quantum Gravity in 2+1 Dimensions. Cambridge University Press
(2003);
C. Rovelli, Quantum Gravity. Cambridge University Press (2004);
A. Gomberoff and D. Marolf (eds.), Lectures on Quantum Gravity. Springer
(2005);
D. Rickles, S. French, and J. Saatsi (eds.), The Structural Foundations of
Quantum Gravity. Clarendon Press (2006);
D. Gross, M. Henneaux, and A. Sevrin (eds.), The Quantum Structure of Space
and Time. World Scientific (2007);
C. Kiefer, Quantum Gravity. 2nd ed., Oxford University Press (2007);
T. Thiemann, Modern Canonical Quantum General Relativity. Cambridge University
Press (2007).
* [2] D. Giulini, C. Kiefer and C. Lämmerzahl (eds.), Quantum Gravity. From Theory To Experimental Search. Springer (2003);
G. Amelino-Camelia and J. Kowalski-Glikman (eds.), Planck Scale Effects in
Astrophysics and Cosmology. Springer (2005);
B. Fauser, J. Tolksdorf, and E. Zeidler (eds.) Quantum Gravity. Mathematical
Models and Experimental Bounds. Birkhäuser (2007);
D. Oriti, Approaches to Quantum Gravity. Toward a New Understanding of Space,
Time, and Matter. Cambridge University Press (2009).
* [3] L. A. Glinka, arXiv:0808.1035[gr-qc] to appear in Gravitation and Cosmology; Concepts Phys. 6, 19 (2009) arXiv:0809.5216[gr-qc]; New Adv. Phys. 2, 1 (2008) arXiv:0803.1533[gr-qc]
* [4] L. A. Glinka, arXiv:gr-qc/0612079 to appear in Gravitation and Cosmology; SIGMA 3, 087 (2007) arXiv:0707.3341 [gr-qc]; AIP Conf. Proc. 1018, 94 (2008) arXiv:0801.4157 [gr-qc]
* [5] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. Freeman (1973); R. M. Wald, General Relativity. University of Chicago (1984); L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics Vol. 2. The Classical Theory of Fields. 4th ed., Butterworth (2000);
S. Carroll, Spacetime and Geometry. An introduction to General Relativity.
Addison–Wesley (2004).
* [6] J. W. York, Phys. Rev. Lett. 28, 1082 (1972); G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
* [7] J. F. Nash, Ann. Math. 56, 405 (1952); _ibid._ 63, 20 (1956); S. Masahiro, Nash Manifolds. Lect. Notes Math. 1269, Springer (1987); M. Günther, Ann. Global Anal. Geom. 7, 69 (1989); Math. Nachr. 144, 165 (1989).
* [8] R. Arnowitt, S. Deser and Ch.W. Misner, in Gravitation: An Introduction to Current Research, ed. by L. Witten, p. 227, Wiley (1962). arXiv:gr-qc/0405109 and references therein.
* [9] A. Hanson, T. Regge, and C. Teitelboim, Constrained Hamiltonian Systems. Accademia Nazionale dei Lincei (1976).
* [10] B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
* [11] P. A. M. Dirac, Lectures on Quantum Mechanics. Belfer Graduate School of Science, Yeshiva University (1964).
* [12] J. A. Wheeler, Geometrodynamics. Academic Press (1962); Einsteins Vision. Springer (1968).
* [13] A. O. Barvinsky and D. V. Nesterov, Nucl. Phys. B 608, 333 (2001); M. J. W. Hall, K. Kumar, and M. Reginatto, J. Phys. A: Math. Gen. 36, 9779 (2003); T. Kubota, T. Ueno, and N. Yokoi, Phys. Lett. B 579, 200 (2004); N. Pinto-Neto, Found. Phys. 35, 577 (2005); C. Kiefer, T. Lück, and P. Moniz, Phys. Rev. D 72, 045006 (2005); B. M. Barbashov, V. N. Pervushin, A. F. Zakharov, and V. A. Zinchuk, AIP Conf. Proc. 841, 362 (2006); A. B. Henriques, Gen. Rel. Grav. 38, 1645 (2006); M. P. Dabrowski, C. Kiefer, and B. Sandhöfer, Phys. Rev. D 74, 044022 (2006); V. N. Pervushin and V. A. Zinchuk, Phys. Atom. Nucl. 70, 593 (2007); R. Carroll, Theor. Math. Phys. 152, 904 (2007). Ch. Soo, Class. Quantum Grav. 24, 1547 (2007); P. Gusin, Phys. Rev. D 77, 066017 (2008); B. S. DeWitt and G. Esposito, Int. J. Geom. Meth. Mod. Phys. 5, 101 (2008); I. Ya. Aref’eva and I. Volovich, Int. J. Geom. Meth. Mod. Phys. 5, 641 (2008).
* [14] A. E. Fischer, Gen. Rel. Grav. 15, 1191 (1983); J. Math. Phys. 27, 718 (1986).
* [15] L. D. Faddeev, Usp. Fiz. Nauk 136, 435 (1982).
* [16] V. V. Fernández, A. M. Moya, and W. A. Rodrigues Jr, Adv. Appl. Clifford Alg. 11, 1 (2001).
|
arxiv-papers
| 2009-06-20T20:47:26 |
2024-09-04T02:49:03.436246
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lukasz Andrzej Glinka",
"submitter": "Lukasz Glinka",
"url": "https://arxiv.org/abs/0906.3825"
}
|
0906.3827
|
# Thermodynamics of space quanta
models quantum gravity
Łukasz Andrzej Glinka
E-mail: [email protected]
_International Institute for Applicable_
_Mathematics & Information Sciences,_
_Hyderabad (India) & Udine (Italy),_
_B.M. Birla Science Centre,_
_Adarsh Nagar, 500 063 Hyderabad, India_
###### Abstract
Canonically quantized $3+1$ general relativity with the global one
dimensionality (1D) conjecture defines the model, which dimensionally reduced
and secondary quantized yields the 1D quantum field theory wherein generic
one-point correlations create physical scales.
This simple quantum gravity model, however, can be developed in a wider sense.
In this paper we propose to consider _ab initio_ thermodynamics of space
quanta as the quantum gravity phenomenology. The thermodynamics is constructed
in the entropic formalism.
Keywords quantum gravity models ; $3+1$ general relativity ; low dimensional
quantum field theories ; global one-dimensionality ; thermodynamics of space
quanta.
PACS 04.60.-m ; 05.30.Jp ; 05.70.Ce; 11.10.Kk ; 98.80.Qc
## 1 Introduction
Both the theory and phenomenology of quantum gravity possess the most
fundamental meaning for the contemporary theoretical physics. Possibly the
theory of quantized gravitational fields will able to predict unknown facts
and open way for new physics. The efforts of many generations of physicists
and mathematicians working on quantum gravity unquestionably have given
significant contribution to science. In this a great success, however,
understanding the physical role of quantum gravity seems to be still very
distant and intriguing perspective (For some proposals see _e.g._ Ref. [1]).
In this paper we discuss the next implication following form the simple model
of quantum gravity [2] having strict roots in the generic quantum cosmology
[3]. The model was constructed within the Wheeler–DeWitt theory, called
quantum geometrodynamics, with taking into account the global one-dimensional
conjecture. The conjecture states that geometrodynamical wave functions are
dependent on the one dimension only, that is an embedding volume form. It
follows from the assumption that matter fields are functional of a volume form
only. It reduces the Wheeler–DeWitt theory to the superspace strata, called
minisuperspace. By application of the dimensional reduction the resulting
model can be presented in the Dirac equation form with the Euclidean Clifford
algebra $\mathcal{C}\ell_{1,1}(\mathbb{R})$, and by appropriate
diagonalization procedure the equation can be quantized in the Fock space.
Obtained 1D quantum field theory defines quantum gravity model, wherein
quantum correlations yield physical scales.
However, the investigated simple model of quantum gravity can be developed and
applied. This paper gives one of its possible physical implications, that is
thermodynamics of space quanta. By space quanta we understand quantum states
of a 3-dimensional embedding. The Fock space formulation gives a possibility
to consider density matrix related to the model, and build formal
thermodynamics. As the example we are discussing the one-particle
approximation. Entropy and energy are calculated, and their appropriate
renormalization is done. In result we obtain the 2nd order Eulerian system,
and all thermodynamic quantities are calculated in frames of the standard
statistical mechanics by application of first principles only, _i.e._
thermodynamics is done _ab initio_. In this way we receive the model of
quantum gravity strictly related to phenomenology.
Structurally the paper is organized as follows. The preliminary section 2
briefly presents the simple model of quantum gravity. Section 3 is devoted to
the development of the model, that is the thermodynamics of space quanta. In
the final section 4 the new results are discussed.
## 2 The simple quantum gravity model
Let us summarize the quantum gravity model [2]. Regarding general relativity
[4] spacetime is a 4-dimensional pseudo-Riemannian manifold $(M,g)$ with a
metric $g_{\mu\nu}$ satisfying the Einstein field equations
$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}{{}^{(4)}}\\!R+g_{\mu\nu}\Lambda=3T_{\mu\nu},$
(1)
where units $c=8\pi G/3=1$ were used, $R_{\mu\nu}$ is the second fundamental
form, ${{}^{(4)}}\\!R$ is the scalar curvature, $\Lambda$ is cosmological
constant, and $T_{\mu\nu}$ is the stress-energy tensor arising from a Matter
fields Lagrangian $\mathcal{L}_{\psi}$ by
$T_{\mu\nu}=-\dfrac{2}{\sqrt{{-g}}}\dfrac{\delta S_{\phi}}{\delta
g^{\mu\nu}}\quad,\quad S_{\phi}=\int d^{4}x\sqrt{{-g}}\mathcal{L}_{\psi}.$ (2)
For $M$ closed, having a boundary $(\partial M,h)$ with an induced metric
$h_{ij}$ and the Gauss curvature tensor $K_{ij}$, (1) are equations of motion
for the Einstein–Hilbert action supplemented by the York–Gibbons–Hawking term
[5]
$S[g]=\int_{M}d^{4}x\sqrt{-g}\left\\{-\dfrac{1}{6}{{}^{(4)}}R+\dfrac{\Lambda}{3}\right\\}+S_{\phi}-\dfrac{1}{3}\int_{\partial
M}d^{3}x\sqrt{h}K,$ (3)
where $K=h^{ij}K_{ij}$. One can parameterize a metric by the $3+1$ splitting
[7]
$g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}+N_{i}N^{i}&N_{j}\\\
N_{i}&h_{ij}\end{array}\right]\quad,\quad N^{i}=h^{ij}N_{j}\quad,\quad
h_{ik}h^{kj}=\delta_{i}^{j},$ (4)
which for stationary $\phi$ arises by a timelike Killing vector field and
global spacelike foliation $t=\mathit{const}$ on $M$, and satisfies the Nash
embedding theorem [6]. With this (3) takes the Hamilton form $S=\int dtL$ with
the Lagrangian
$\displaystyle L=\int_{\partial
M}d^{3}x\left\\{\pi\dot{N}+\pi^{i}\dot{N_{i}}+\pi^{ij}\dot{h}_{ij}+\pi_{\phi}\dot{\phi}-NH-
N_{i}H^{i}\right\\},$ (5) $\displaystyle\pi_{\phi}=\frac{\partial
L_{\phi}}{\partial\dot{\phi}}\quad,\quad\pi=\frac{\partial
L}{\partial\dot{N}}\quad,\quad\pi^{i}=\frac{\partial L}{\partial\dot{N_{i}}},$
(6) $\displaystyle\pi^{ij}=\frac{\partial
L}{\partial\dot{h}_{ij}}=\sqrt{h}\left(h^{ij}K-K^{ij}\right)\quad,\quad\dot{h}_{ij}=N_{i|j}+N_{j|i}-2NK_{ij},$
(7) $\displaystyle
H=\sqrt{h}\left\\{K^{2}-K_{ij}K^{ij}+{{}^{(3)}}R-2\Lambda-6\varrho\right\\}\quad,\quad
H^{i}=-2\pi^{ij}_{\leavevmode\nobreak\ ;j},$ (8)
where $\varrho=n^{\mu}n^{\nu}T_{\mu\nu}$,
$n^{\mu}=(1/N)\left[1,-N^{i}\right]$, ${{}^{(3)}}R=h^{ij}R_{ij}$. Time-
preservation of the primary constraints [8, 9] yields the secondary ones
$\pi\approx 0\quad,\quad\pi^{i}\approx 0\longrightarrow H\approx 0\quad,\quad
H^{i}\approx 0$ (9)
called the Hamiltonian (scalar) and the diffeomorphism (vector) constraint.
Vector constraint reflects spatial diffeoinvariance, scalar one is dynamical.
Regarding DeWitt [9] $H^{i}$ generates the diffeomorphisms
$\widetilde{x}^{i}=x^{i}+\xi^{i}$
$\displaystyle i\left[h_{ij},\int_{\partial M}H_{a}\xi^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle-
h_{ij,k}\xi^{k}-h_{kj}\xi^{k}_{\leavevmode\nobreak\
,i}-h_{ik}\xi^{k}_{\leavevmode\nobreak\ ,j}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (10) $\displaystyle i\left[\pi^{ij},\int_{\partial
M}H_{a}\xi^{a}d^{3}x\right]$ $\displaystyle=$
$\displaystyle-\left(\pi^{ij}\xi^{k}\right)_{,k}+\pi^{kj}\xi^{i}_{\leavevmode\nobreak\
,k}+\pi^{ik}\xi^{j}_{\leavevmode\nobreak\ ,k}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (11)
and the first-class constraints algebra holds
$\displaystyle i\left[H_{i}(x),H_{j}(y)\right]=\int_{\partial
M}H_{a}c^{a}_{ij}d^{3}z\quad,\quad
i\left[H(x),H_{i}(y)\right]=H\delta^{(3)}_{,i}(x,y),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (12) $\displaystyle
i\left[\int_{\partial M}H\xi_{1}d^{3}x,\int_{\partial
M}H\xi_{2}d^{3}x\right]=\int_{\partial
M}H^{a}\left(\xi_{1,a}\xi_{2}-\xi_{1}\xi_{2,a}\right)d^{3}x.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (13)
Here $H_{i}=h_{ij}H^{j}$, and
$c^{a}_{ij}=\delta^{a}_{i}\delta^{b}_{j}\delta^{(3)}_{,b}(x,z)\delta^{(3)}(y,z)-(i\leftrightarrow
j,x\leftrightarrow y)$ are the structure constants of the spatial
diffeomorphism group. The canonical quantization [8, 10]
$\displaystyle
i\left[\pi^{ij}(x),h_{kl}(y)\right]=\dfrac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)\delta^{(3)}(x,y),$
(14) $\displaystyle
i\left[\pi^{i}(x),N_{j}(y)\right]=\delta^{i}_{j}\delta^{(3)}(x,y)\quad,\quad
i\left[\pi(x),N(y)\right]=\delta^{(3)}(x,y),$ (15)
applied to the Hamiltonian constraint into the Hamilton–Jacobi form [11]
$G_{ijkl}\pi^{ij}\pi^{kl}-\sqrt{h}\left({{}^{(3)}}R-2\Lambda-6\varrho\right)=0,$
(16)
where $G_{ijkl}$ is the Wheeler metric on superspace [12]
$G_{ijkl}\equiv\dfrac{1}{2\sqrt{h}}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right),$
(17)
yields the Wheeler–DeWitt equation [9, 13, 14] modeling quantum gravity
$\left\\{G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}+h^{1/2}\left({{}^{(3)}}R-2\Lambda-6\varrho\right)\right\\}\Psi[h_{ij},\phi]=0.$
(18)
Other first-class reflect diffeoinvariance of a wave function
$\Psi[h_{ij},\phi]$
$\pi\Psi[h_{ij},\phi]=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\pi^{i}\Psi[h_{ij},\phi]=0,\leavevmode\nobreak\ \leavevmode\nobreak\
H^{i}\Psi[h_{ij},\phi]=0.$ (19)
The Wheeler–DeWitt equation (18) is independent on time quantum mechanics on
the superspace of 3-dimensional embeddings. The simple model reduces the
superspace to its strata, called minisuperspace.
The simple model assumes that Matter fields are one-dimensional (1D)
functionals
$\phi=\phi[h]\quad,\quad
h=\dfrac{1}{3}\varepsilon^{ijk}\varepsilon^{abc}h_{ia}h_{jb}h_{kc}\quad,$ (20)
where $\varepsilon$ is the Levi-Civita tensor, so that the conjectured 1D wave
functions
$\Psi[h_{ij},\phi]\rightarrow\Psi[h],$ (21)
satisfy the global one-dimensional evolution
$\left\\{G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}+h^{1/2}\left({{}^{(3)}}R-2\Lambda-6\varrho[h]\right)\right\\}\Psi[h]=0.$
(22)
Assumption (21) is analogous to the generic model [3], but the 1D theory (22)
holds for nonhomogeneous isotropic quantum cosmologies.
Considering the Jacobi rule for differentiation of a determinant [4] together
with the $3+1$ splitting (4) one obtains
$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}\longrightarrow N^{2}\delta
h=N^{2}hh^{ij}\delta h_{ij},$ (23)
which reduces the differential operator in (22)
$\dfrac{\delta}{\delta h_{ij}}\Psi[h]=hh^{ij}\dfrac{\delta}{\delta
h}\Psi[h]\longrightarrow G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}\Psi[h]=-\dfrac{3}{2}h^{3/2}\dfrac{\delta^{2}}{\delta h^{2}}\Psi[h],$
(24)
and yields the simple 1D quantum gravity model
$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}-m^{2}\right)\Psi=0\quad,\quad
m^{2}=\dfrac{2}{3h}\left({{}^{(3)}}R-2\Lambda-6\varrho[h]\right).$ (25)
where $m$ is the mass of the classical field $\Psi[h]$. In fact (25) is a
field-theoretic equation of motion $\delta S[\Psi]/\delta\Psi=0$ for the
Euclidean action
$S[\Psi]=\int\delta hL[\Psi,\Pi_{\Psi}]\quad,\quad
L=\dfrac{1}{2}\Pi_{\Psi}^{2}+\dfrac{m^{2}}{2}\Psi^{2},$ (26)
where $\Pi_{\Psi}=\dfrac{\delta\Psi}{\delta h}$ is conjugate momentum which
allows rewrite (25) in two-component model in the Dirac equation form
$\left(i\gamma\dfrac{\delta}{\delta
h}-M\right)\Phi=0\quad,\quad\Phi=\left[\begin{array}[]{c}\Pi_{\Psi}\\\
\Psi\end{array}\right]\quad,\quad M=\left[\begin{array}[]{cc}1&0\\\
0&-m^{2}\end{array}\right],$ (27)
with the Euclidean Clifford algebra $\mathcal{C}\ell_{1,1}(\mathbb{R})$ [15]
$\gamma=\left[\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right]\quad,\quad\gamma^{2}=I\quad,\quad\left\\{\gamma,\gamma\right\\}=2\delta_{E}\quad,\quad\delta_{E}=\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right],$ (28)
having a $2D$ complex representation. Restricting to $Pin_{1,1}(\mathbb{R})$
yield a 2D spin representations; restricting to $Spin_{1,1}(\mathbb{R})$
splits it onto a sum of two 1D Weyl representations;
$\mathcal{C}\ell_{1,1}(\mathbb{R})$ decomposes into a direct sum of two
isomorphic central simple algebras or a tensor product
$\displaystyle\mathcal{C}\ell_{1,1}(\mathbb{R})=\mathcal{C}\ell^{+}_{1,1}(\mathbb{R})\oplus\mathcal{C}\ell^{-}_{1,1}(\mathbb{R})=\mathcal{C}\ell_{2,0}(\mathbb{R})\otimes\mathcal{C}\ell_{0,0}(\mathbb{R}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (29)
$\displaystyle\mathcal{C}\ell_{1,1}(\mathbb{R})\cong\mathbb{R}(2)\quad,\quad\mathcal{C}\ell^{\pm}_{1,1}(\mathbb{R})=\dfrac{1\pm\gamma}{2}\mathcal{C}\ell_{1,1}(\mathbb{R})\cong\mathbb{R}\quad,\quad\mathcal{C}\ell_{0,0}(\mathbb{R})\cong\mathbb{R}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (30)
The Dirac equation (27) can be rewritten in the dynamical Fock reper
$\mathfrak{B}$
$\displaystyle\mathbf{\Phi}=\mathbb{Q}\mathfrak{B},\leavevmode\nobreak\
\leavevmode\nobreak\
\mathbb{Q}=\left[\begin{array}[]{cc}1/\sqrt{2|m|}&1/\sqrt{2|m|}\\\
-i\sqrt{|m|/2}&i\sqrt{|m|/2}\end{array}\right],$ (33)
$\displaystyle\mathfrak{B}=\left\\{\left[\begin{array}[]{c}\mathsf{G}[h]\\\
\mathsf{G}^{\dagger}[h]\end{array}\right]:\left[\mathsf{G}[h^{\prime}],\mathsf{G}^{\dagger}[h]\right]=\delta\left(h^{\prime}-h\right),\left[\mathsf{G}[h^{\prime}],\mathsf{G}[h]\right]=0\right\\}.$
(36)
Determining a reper $\mathfrak{F}$ by the diagonalization due to the
Bogoliubov transformation and the Heisenberg equations of motion
$\displaystyle\mathfrak{F}=\left[\begin{array}[]{cc}u&v\\\
v^{\ast}&u^{\ast}\end{array}\right]\mathfrak{B}\quad,\quad\dfrac{\delta\mathfrak{F}}{\delta
h}=\left[\begin{array}[]{cc}-i\Omega&0\\\
0&i\Omega\end{array}\right]\mathfrak{F},$ (41)
where $|u|^{2}-|v|^{2}=1$, $u$, $v$, $\Omega$ are functionals of $h$, one
obtains
$\dfrac{\delta\mathbf{b}}{\delta
h}=\mathbb{X}\mathbf{b}\quad,\quad\mathbf{b}=\left[\begin{array}[]{c}u\\\
v\end{array}\right]\quad,\quad\Omega\equiv 0,$ (42)
so that $\mathfrak{F}$ is the Fock the initial data static reper ($I$) with
correct vacuum
$\mathfrak{F}=\left\\{\left[\begin{array}[]{c}\mathsf{G}_{I}\\\
\mathsf{G}^{\dagger}_{I}\end{array}\right]:\left[\mathsf{G}_{I},\mathsf{G}^{\dagger}_{I}\right]=1,\left[\mathsf{G}_{I},\mathsf{G}_{I}\right]=0\right\\}\quad,\quad\mathsf{G}_{I}\left|\mathrm{VAC}\right\rangle=0,$
(43)
and integrability of (42) can be done in the superfluid parametrization
$\displaystyle u=\dfrac{\mu+1}{2\sqrt{\mu}}e^{i\theta}\quad,\quad
v=\dfrac{\mu-1}{2\sqrt{\mu}}e^{-i\theta}\quad,\quad\theta=m_{I}\int_{h_{I}}^{h}\mu^{\prime}\delta
h^{\prime},$ (44)
where $\mu\equiv\mu[h]$, $\mu^{\prime}=\mu[h^{\prime}]$ is a mass scale. In
result one obtains the solution
$\mathbf{\Phi}=\mathbb{Q}\mathbb{G}\mathfrak{F}\quad,\quad\mathbb{G}=\left[\begin{array}[]{cc}u^{\star}&-v^{\star}\\\
-v&u\end{array}\right],$ (45)
and particulary one establishes the field operator and the generic one-point
correlator
$\displaystyle\mathbf{\Psi}=\frac{1}{\sqrt{2m_{I}}}\left(\dfrac{e^{-i\theta}}{2\mu}\mathsf{G}_{I}+\dfrac{e^{i\theta}}{2\mu}\mathsf{G}_{I}^{\dagger}\right)\quad,\quad\left\langle\mathrm{VAC}\right|\mathbf{\Psi}^{\dagger}[h]\mathbf{\Psi}[h]\left|\mathrm{VAC}\right\rangle=\dfrac{1}{\mu^{2}},$
(46)
where the quantum correlator was normalized to unity in $h_{I}$. The static
reper formulation defines the concept of space quanta - the quantized fields
associated with an 3-dimensional embedding.
## 3 The thermodynamics
Thermodynamic equilibrium corresponding to quantum field theory in the static
Fock reper, allows using of first principles of statistical mechanics [16],
and formulate _ab initio_ thermodynamics of space quanta. Let us test the one-
particle density matrix approximation.
### 3.1 One-particle density matrix. Entropy and energy
In the one-particle approximation the density operator $\mathsf{D}$ is
equivalent to an occupation number operator. Thermodynamic equilibrium is
determined with respect to the static reper, so that the one-particle density
matrix in equilibrium $\mathbb{D}$ is given by the Von Neumann–Heisenberg
picture
$\displaystyle\mathsf{D}$ $\displaystyle=$
$\displaystyle{\mathsf{G}}^{\dagger}{\mathsf{G}}=\mathfrak{F}^{\dagger}\mathbb{D}\mathfrak{F},$
(47) $\displaystyle\mathbb{D}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4\mu}\left[\begin{array}[]{cc}(\mu+1)^{2}&1-\mu^{2}\\\
1-\mu^{2}&(\mu-1)^{2}\end{array}\right].$ (50)
Note that $\det\mathbb{D}=0$, that means in the one-particle approximation the
corresponding thermodynamics is not invertible. Employing (50) one can
establish the occupation number value
$N=\dfrac{\mathrm{Tr}\left(\mathbb{D}^{2}\right)}{\mathrm{Tr}\mathbb{D}}=\dfrac{\mu^{2}+1}{2\mu},$
(51)
and the entropy can be derived from its basic definition
$\displaystyle
S=-\dfrac{\mathrm{Tr}(\mathbb{D}\ln\mathbb{D})}{\mathrm{Tr}\mathbb{D}}=\sum_{n=1}^{\infty}\sum_{k=1}^{n}\dfrac{(-1)^{k}}{n}\binom{n}{k-1}S_{k},$
(52)
where $\binom{n}{m}$ are the Newton binomial symbols, and
$S_{k}=\dfrac{\mathrm{Tr}(\mathbb{D}^{k})}{\mathrm{Tr}\mathbb{D}}=N^{k-1},$
(53)
are cluster entropies. The series (52) converges for the spectral radius
values
$\rho(\mathbb{D}-\mathbb{I})<1\Longrightarrow\mu\in(1;2+\sqrt{3}),$ (54)
or equivalently for $m\in(1;2+\sqrt{3})m_{I}$, with the result
$S=-\dfrac{\zeta(1)}{2}\left(\dfrac{\mu^{2}-1}{\mu^{2}+1}\right)^{2}-\dfrac{\mu^{4}+6\mu^{2}+1}{(\mu^{2}+1)^{2}}\ln\dfrac{(\mu-1)^{2}}{2\mu},$
(55)
where $\zeta(s)=\sum_{n=1}^{\infty}\dfrac{1}{n^{s}}$ is the Riemann zeta
function; $\zeta(1)$ is formally infinite.
Note that by straightforward application of the Hagedorn hadronization formula
$m\sim T_{H}$ [17], where $m$ is the mass of the system, one can establishes
the hadronized temperature as
$\dfrac{T_{H}}{T_{I}}=\mu,$ (56)
where $k_{B}T_{I}=m_{I}c^{2}$. By the relation $\langle m\rangle c^{2}\sim
k_{B}\langle T_{H}\rangle$ one obtains averaged hadronized temperature
normalized to $T_{I}$ value
$\left\langle\dfrac{T_{H}}{T_{I}}\right\rangle=\langle\mu\rangle=\dfrac{1+\sqrt{3}}{2}\approx
2.732,$ (57)
so that one can establish the ratio
$\dfrac{\left\langle
T_{H}\right\rangle}{T_{H}}\in\left(\dfrac{\sqrt{3}-1}{2},\dfrac{\sqrt{3}+1}{2}\right)\approx\left(0.366,1.366\right).$
(58)
Defining anisotropy as $\Delta T_{H}=\left\langle T_{H}\right\rangle-T_{H}$
one derives
$\dfrac{\Delta
T_{H}}{T_{H}}\in\left(\dfrac{\sqrt{3}-3}{2},\dfrac{\sqrt{3}-1}{2}\right)\approx(-0.633;0.366),$
(59)
so that averaged anisotropy is
$\left\langle\dfrac{\Delta T_{H}}{T_{H}}\right\rangle=0.5.$ (60)
By the first approximation, (57) can be identified with background
temperature, _e.g._ for $T_{I}\sim 1K$ it is exactly averaged CMB radiation
temperature. Next approximations of the density matrix or fuzzing of the
interval $\mu\in(1;2+\sqrt{3})$ will give next orders of the numbers.
In the static reper Hamiltonian matrix $\mathbb{H}$ of the system equals
$\displaystyle\mathsf{H}$ $\displaystyle=$
$\displaystyle\dfrac{m}{2}\left(\mathsf{G}^{\dagger}\mathsf{G}+\mathsf{G}\mathsf{G}^{\dagger}\right)=\mathfrak{F}^{\dagger}\mathbb{H}\mathfrak{F},$
(61) $\displaystyle\mathbb{H}$ $\displaystyle=$
$\displaystyle\dfrac{m_{I}}{4}\left[\begin{array}[]{cc}1+\mu^{2}&1-\mu^{2}\\\
1-\mu^{2}&1+\mu^{2}\end{array}\right],$ (64)
and has discrete spectrum for fixed mass scale
$\mathrm{Spec}\leavevmode\nobreak\
\mathbb{H}=\left\\{\dfrac{m_{I}}{2}\mu^{2},\dfrac{m_{I}}{2}\right\\}.$ (65)
The internal energy calculated from the Hamiltonian matrix (64) is
$U=\dfrac{\mathrm{Tr}(\mathbb{D}\mathbb{H})}{\mathrm{Tr}\mathbb{D}}=\dfrac{m_{I}}{4}(\mu^{2}+1).$
(66)
The Hamiltonian matrix $\mathbb{H}$, however, consists constant term
$\mathbb{H}_{I}$
$\mathbb{H}_{I}=\dfrac{m_{I}}{4}\left[\begin{array}[]{cc}1&1\\\
1&1\end{array}\right]$ (67)
which can be eliminated by simple renormalization
$\mathbb{H}\rightarrow\mathbb{H}^{\prime}=\mathbb{H}-\mathbb{H}_{I}=\dfrac{m_{I}}{4}\left[\begin{array}[]{cc}\mu^{2}&-\mu^{2}\\\
-\mu^{2}&\mu^{2}\end{array}\right].$ (68)
The renormalized Hemiltonian spectrum is
$\mathrm{Spec}\leavevmode\nobreak\
\mathbb{H}^{\prime}=\left\\{\dfrac{m_{I}}{2}\mu^{2},0\right\\},$ (69)
and straightforward computation of the renormalized internal energy yields the
following result
$U^{\prime}=\dfrac{\mathrm{Tr}(\mathbb{D}\mathbb{H}^{\prime})}{\mathrm{Tr}\mathbb{D}}=\dfrac{m_{I}}{4}\mu^{2}\equiv
U-U_{I},$ (70)
where $U_{I}=\dfrac{m_{I}}{4}$, which has the Eulerian homogeneity of degree 2
$U^{\prime}[\alpha\mu]=\alpha^{2}U^{\prime}[\mu].$ (71)
In this manner thermodynamics describing space quanta behavior can be
formulated in the way typical for the Eulerian systems.
### 3.2 _Ab initio_ thermodynamics of space quanta
Three elementary physical quantities – occupation number $N$, internal energy
$U$, and entropy $S$ – was just derived, so that one can conclude formal
thermodynamics. Actually the entropy (55) is infinite by the presence of
formal infinity $\zeta(1)$. Straightforward calculation shows that temperature
$T=\delta U/\delta S$ arising from the entropy (55) is dependent on $\zeta(1)$
and initial data mass $m_{I}$. Obtained quantity has the finite limit, if and
only if we scale initial data mass $m_{I}\rightarrow m_{I}\zeta(1)$. Because
mass $m$ is related to length $l$ by $m\sim 1/l$, the limit
$m_{I}\rightarrow\infty$ corresponds with a point object $l_{I}\rightarrow 0$.
However, scaling of initial data is not good physical procedure, _i.e._ has
not well-defined physical meaning. It can be shown that the entropy
renormalization $S\rightarrow-S/\zeta(1)$ in the formal limit
$\zeta(1)\rightarrow\infty$ gives the equivalent result for the thermodynamics
with no using initial data scaling. The renormalization corresponds to an
initial quantum state of an embedding being a point, and yields perfect
accordance with the second law of thermodynamics
$S\longrightarrow
S^{\prime}=\lim_{\zeta(1)\rightarrow\infty}\dfrac{-S}{\zeta(1)}=\dfrac{1}{2}\left(\dfrac{\mu^{2}-1}{\mu^{2}+1}\right)^{2}\geqslant
0.$ (72)
Calculating temperature $T$ of space quanta one obtains
$T=\dfrac{\delta U^{\prime}}{\delta
S^{\prime}}=m_{I}\dfrac{(\mu^{2}+1)^{3}}{8(\mu^{2}-1)},$ (73)
and one sees that initially, _i.e._ for $\mu=1$, temperature is infinite. It
is the Hot Big Bang (HBB) phenomenon. After HBB system is cooled right up
until mass scale value $\mu_{PT}=\sqrt{2}\approx 1.414$ and then is warmed. In
fact $\mu_{PT}$ is the phase transition point, namely, the energetic heat
capacity $C_{U}$ having the form
$C_{U}=T\dfrac{\delta S^{\prime}}{\delta T}=\dfrac{\delta U^{\prime}}{\delta
T}=\dfrac{(\mu^{2}-1)^{2}}{(\mu^{2}-2)(\mu^{2}+1)^{2}},$ (74)
possesses the singularity in the point $\mu_{PT}$. The generalized law of
equipartition $\delta U/\delta T=f/2$ establishes degrees of freedom $f$
number
$f=2C_{U}.$ (75)
The Helmholtz free energy $F=U^{\prime}-TS^{\prime}$ that is
$F=-\dfrac{m_{I}}{16}(\mu^{4}-4\mu^{2}-1),$ (76)
is finite for finite $m_{I}$, increases since $\mu=1$ till $\mu_{PT}$, and
then decreases. So, the thermal equilibrium point is the HBB point
$\mu_{eq}=1$. In the region of mass scales $1\leqslant\mu<\mu_{PT}$ mechanical
isolation is absent, but it is after phase transition $\mu>\mu_{PT}$.
Calculating the chemical potential
$\omega=\dfrac{\delta F}{\delta
N}=-m_{I}\dfrac{\mu^{3}(\mu^{2}-2)}{2(\mu^{2}-1)},$ (77)
one sees that in $\mu_{eq}$ it diverges and in $\mu_{PT}$ it vanishes. Using
of (77) together with the occupation number $N$ and the Helmholtz free energy
$F$ yields appropriate free energy defined by the Landau grand potential
$\Omega$
$\Omega=F-\omega N=m_{I}\dfrac{3\mu^{6}+\mu^{4}-11\mu^{2}-1}{16(\mu^{2}-1)},$
(78)
so that the corresponding Massieu–Planck free entropy $\Xi$ can be also
derived
$\Xi=-\dfrac{\Omega}{T}=-\dfrac{3\mu^{6}+\mu^{4}-11\mu^{2}-1}{2(\mu^{2}+1)^{3}},$
(79)
and consequently the grand partition function $Z$ is established as
$\displaystyle
Z=e^{\Xi}=\exp\left\\{-\dfrac{3\mu^{6}+\mu^{4}-11\mu^{2}-1}{2(\mu^{2}+1)^{3}}\right\\}.$
(80)
The 2nd order Eulerian homogeneity yields the equation of state $PV/T=\ln Z$
and determines the product of pressure $P$ and volume $V$ as
$PV=-\Omega,$ (81)
so together with the Gibbs–Duhem equation $V\delta P=S^{\prime}\delta
T+N\delta\omega$ allows to establish the pressure
$|P|=\exp\left\\{-\int\left(S+N\dfrac{\delta\omega}{\delta
T}\right)\dfrac{\delta T}{\Omega}\right\\}.$ (82)
Similarly, the first law of thermodynamics, $-\delta\Omega=S^{\prime}\delta
T+P\delta V+N\delta\omega$, and the equation of state (81) determine the
volume $|V|=|\Omega|/|P|$, which by positiveness is $V=|V|$. Regarding (81)
the pressure $P=|P|$ for $\Omega=-|\Omega|<0$, and $P=-|P|$ for
$\Omega=|\Omega|>0$, so that
$\displaystyle
P=\left\\{\begin{array}[]{ll}\dfrac{m_{I}^{7}a_{0}}{\mu^{2}-1}\dfrac{(\mu^{2}+a_{2})^{b_{2}+1}}{(\mu^{2}+a_{3})^{b_{3}-1}}|\mu^{2}-a_{1}|^{b_{1}+1}&,\leavevmode\nobreak\
\mathrm{iff}\leavevmode\nobreak\
1\leqslant\mu\leqslant\sqrt{a_{1}}\vspace*{10pt}\\\
\dfrac{-m_{I}^{7}a_{0}}{\mu^{2}-1}\dfrac{(\mu^{2}+a_{2})^{b_{2}+1}}{(\mu^{2}+a_{3})^{b_{3}-1}}|\mu^{2}-a_{1}|^{b_{1}+1}&,\leavevmode\nobreak\
\mathrm{iff}\leavevmode\nobreak\ \sqrt{a_{1}}\leqslant\mu\leqslant
2+\sqrt{3}\end{array}\right.$ (85)
where $a_{0}\approx 6.676\cdot 10^{6}$ and
$\displaystyle a_{1}\approx 1.802\leavevmode\nobreak\ ,\leavevmode\nobreak\
a_{2}\approx 0.090\leavevmode\nobreak\ ,\leavevmode\nobreak\ a_{3}\approx
2.046\leavevmode\nobreak\ ,$ (86) $\displaystyle b_{1}\approx
0.014\leavevmode\nobreak\ ,\leavevmode\nobreak\ b_{2}\approx
0.410\leavevmode\nobreak\ ,\leavevmode\nobreak\ b_{3}\approx
1.092\leavevmode\nobreak\ .$ (87)
For the mass scales $1\leqslant\mu<\sqrt{a_{1}}$ $P$ decreases from positive
infinity to zero, vanishes in the point $\mu=\sqrt{a_{1}}\approx 1.343$, and
decreases from zero to negative infinity for $\sqrt{a_{1}}<\mu\leqslant
2+\sqrt{3}$. Regarding the relation $V=|\Omega|/|P|$, $V$ is a fixed parameter
and can be established as
$V=\dfrac{1}{16a_{0}m_{I}^{6}}\dfrac{1}{|\mu^{2}-a_{1}|^{b_{1}}}\dfrac{(\mu^{2}+a_{3})^{b_{3}}}{(\mu^{2}+a_{2})^{b_{2}}}.$
(88)
Equivalently the thermodynamics of space quanta can be expressed by the size
scale $\lambda=\dfrac{1}{\mu}$. There are the relations relating both the
scales with an occupation number
$\displaystyle\lambda=N\left(1\mp\sqrt{{1-\dfrac{1}{N^{2}}}}\right)\quad,\quad\mu=N\left(1\pm\sqrt{{1-\dfrac{1}{N^{2}}}}\right),$
(89)
that in the limit of infinite $N$ are equal
$\displaystyle\lambda_{N=\infty}=\left\\{0,\infty\right\\}\quad,\quad\mu_{N=\infty}=\left\\{\infty,0\right\\},$
(90)
so there are two possible asymptotic behaviors. The first case, _i.e._
$\lambda=0$, $\mu=\infty$, can be interpreted with a black hole as well as
with HBB, the second one, _i.e._ $\lambda=\infty$, $\mu=0$, as stable
classical physical object.
## 4 Discussion
In this paper we have presented the next implication of the simple model of
quantum gravity [2]. This algorithm has yielded constructive and plausible
phenomenology, that is thermodynamics, in the discussed case describing space
quanta behavior. The model applies to all $3+1$ splitted general relativistic
spacetimes which satisfy the Mach principle, _i.e._ are isotropic. Their
importance for elementary particle physics, cosmology and high energy
astrophysics is experimentally confirmed; one can say that these are
phenomenological spacetimes.
As the example of _ab initio_ formulation of thermodynamics we have employed
the one-particle approximation of density matrix. The renormalization method
was applied for entropy and the Hamiltonian matrix, and has yielded the second
order Eulerian homogeneity property. The Landau grand potential $\Omega$ and
the Massieu–Planck free entropy $\Xi$ was used in the thermodynamic
description. Grand partition function $Z$ was established. Thermodynamic
volume $V$ was determined as fixed parameter. Other thermodynamical potentials
was derived in frames of the entropic formalism, that accords with the first
and the second principles of thermodynamics. Physical information following
from the thermodynamics of space quanta is the crucial point of the presented
construction. Actually the approach of this paper differ from other ones (Cf.
_e.g._ [18]) by _ab initio_ quantum gravity phenomenology.
In our opinion studying special physical phenomena by the proposed approach
seems to be the most important prospective arising from the thermodynamics of
space quanta. From experimental point of view the presented considerations
possess possible usefulness, because of bosonic systems are common in high
energy physics.
## Acknowledgements
Author thanks Prof. I. L. Buchbinder for valuable suggestions in correction of
the primary notes, and is grateful to Profs. A. B. Arbuzov, I. Ya. Aref’eva,
K. B. Bronnikov, and V. N. Pervushin for discussions.
## References
* [1] I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Action in Quantum Gravity. Institute of Physics Publishing (1992);
D. J. Gross, T. Piran, and S. Weinberg (eds.), Two Dimensional Quantum Gravity
and Random Surfaces. World Scientific (1992);
G. W. Gibbons and S. W. Hawking (eds.), Euclidean Quantum Gravity. World
Scientific (1993);
G. Esposito, Quantum Gravity, Quantum Cosmology and Lorentzian Geometries.
Springer (1994);
J. Ehlers and H. Friedrich (eds.), Canonical Gravity: From Classical to
Quantum. Springer (1994);
E. Prugovečki, Principles of Quantum General Relativity. World Scientific
(1995);
R. Gambini and J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity.
Cambridge University Press (1996);
G. Esposito, A. Yu. Kamenshchik, and G. Pollifrone, Euclidean Quantum Gravity
on Manifolds with Boundary. Springer (1997);
P. Fré, V. Gorini, G. Magli, and U. Moschella, Classical and Quantum Black
Holes. Institute of Physics Publishing (1999);
I. G. Avramidi, Heat Kernel and Quantum Gravity. Springer (2000);
B. N. Kursunoglu, S. L. Mintz, and A. Perlmutter (eds.), Quantum Gravity,
Generalized Theory of Gravitation and Superstring Theory-Based Unification.
Kluwer (2002);
S. Carlip, Quantum Gravity in 2+1 Dimensions. Cambridge University Press
(2003);
D. Giulini, C. Kiefer and C. Lämmerzahl (eds.), Quantum Gravity. From Theory
To Experimental Search. Springer (2003);
C. Rovelli, Quantum Gravity. Cambridge University Press (2004);
G. Amelino-Camelia and J. Kowalski-Glikman (eds.), Planck Scale Effects in
Astrophysics and Cosmology. Springer (2005);
A. Gomberoff and D. Marolf (eds.), Lectures on Quantum Gravity. Springer
(2005);
D. Gross, M. Henneaux, and A. Sevrin (eds.), The Quantum Structure of Space
and Time. World Scientific (2007);
C. Kiefer, Quantum Gravity. 2nd ed., Oxford University Press (2007);
T. Thiemann, Modern Canonical Quantum General Relativity. Cambridge University
Press (2007);
D. Oriti, Approaches to Quantum Gravity. Toward a New Understanding of Space,
Time, and Matter. Cambridge University Press (2009).
* [2] L. A. Glinka, [arXiv:0808.1035 [gr-qc]] submitted to Gravitation and Cosmology.
* [3] L. A. Glinka, AIP Conf. Proc. 1018, 94 (2008) arXiv:0801.4157 [gr-qc]; SIGMA 3, 087 (2007) arXiv:0707.3341 [gr-qc]; arXiv:gr-qc/0612079
* [4] Ch. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. Freeman (1973); R. M. Wald, General Relativity. University of Chicago (1984); L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics Vol. 2. The Classical Theory of Fields. 4th ed., Butterworth (2000);
S. Carroll, Spacetime and Geometry. An introduction to General Relativity.
Addison–Wesley (2004).
* [5] J. W. York, Phys. Rev. Lett. 28, 1082 (1972); G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977).
* [6] J. F. Nash, Ann. Math. 56, 405 (1952); _ibid._ 63, 20 (1956); S. Masahiro, Nash Manifolds. Lect. Notes Math. 1269, Springer (1987); M. Günther, Ann. Global Anal. Geom. 7, 69 (1989); Math. Nachr. 144, 165 (1989).
* [7] R. Arnowitt, S. Deser and Ch.W. Misner, in Gravitation: An Introduction to Current Research, ed. by L. Witten, p. 227, Wiley (1962). arXiv:gr-qc/0405109 and all references of the authors’ papers therein;
B. DeWitt, The Global Approach to Quantum Field Theory, Vol. 1, 2. Int. Ser.
Monogr. Phys. 114, Clarendon Press (2003).
* [8] P. A. M. Dirac, Lectures on Quantum Mechanics. Belfer Graduate School of Science, Yeshiva University (1964).
* [9] B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
* [10] L. D. Faddeev, Usp. Fiz. Nauk 136, 435 (1982).
* [11] M. J. W. Hall, K. Kumar, and M. Reginatto, J. Phys. A: Math. Gen. 36, 9779 (2003); N. Pinto-Neto, Found. Phys. 35, 577 (2005); B. M. Barbashov, V. N. Pervushin, A. F. Zakharov, and V. A. Zinchuk, AIP Conf. Proc. 841, 362 (2006); V. N. Pervushin and V. A. Zinchuk, Phys. Atom. Nucl. 70, 593 (2007); R. Carroll, Theor. Math. Phys. 152, 904 (2007).
* [12] J. A. Wheeler, in Relativity, Groups, and Topology. ed. by C. DeWitt and B. S. DeWitt, Gordon and Breach (1964); in Analytic Methods in Mathematical Physics. ed. by R. P. Gilbert and R. Newton, Gordon and Breach (1970); A. E. Fischer, in Relativity. ed. by M. Carmeli, S. I. Fickler, and L. Witten, Plenum (1970); Gen. Rel. Grav. 15, 1191 (1983); J. Math. Phys. 27, 718 (1986).
* [13] J. A. Wheeler, Geometrodynamics. Academic Press (1962); Einsteins Vision. Springer (1968).
* [14] B. S. DeWitt and G. Esposito, Int. J. Geom. Meth. Mod. Phys. 5, 101 (2008); P. Gusin, Phys. Rev. D 77, 066017 (2008); I. Ya. Aref’eva and I. Volovich Int. J. Geom. Meth. Mod. Phys. 05, 641 (2008); Ch. Soo, Class. Quantum Grav. 24, 1547 (2007); A. B. Henriques, Gen. Rel. Grav. 38, 1645 (2006); M. P. Dabrowski, C. Kiefer, and B. Sandhöfer, Phys. Rev. D 74, 044022 (2006); C. Kiefer, T. Lück, and P. Moniz, Phys. Rev. D 72, 045006 (2005); T. Kubota, T. Ueno, and N. Yokoi, Phys. Lett. B 579, 200 (2004); K. Meissner, Class. Quantum Grav. 21, 5245 (2004); A. Ashtekar, M. Bojowald, and J. Lewandowski, Adv. Theor. Math. Phys. 7, 233 (2003); A. O. Barvinsky and D. V. Nesterov, Nucl. Phys. B 608, 333 (2001).
* [15] V. V. Fernández, A. M. Moya, and W. A. Rodrigues Jr, Adv. Appl. Clifford Alg. 11, 1 (2001).
* [16] K. Huang, Statistical Mechanics. 2nd ed., Wiley, New York (1987).
* [17] R. Hagedorn, Suppl. Nuovo Cim. 3, 147 (1965); _ibid._ 6, 169 (1968).
* [18] J. Ambjørn, J. Jurkiewicz, and R. Loll, Phys. Rev. Lett. 93, 131 (2004); F. Markopoulou and L. Smolin, Class. Quantum Grav. 24, 3813 (2007); B. F. L. Ward, Int. J. Mod. Phys. D 17, 627 (2008). M. Bojowald, Class. Quantum Grav. 26, 075020 (2009); A. Ashtekar, W. Kaminski, and J. Lewandowski. Phys. Rev. D 79, 064030 (2009);
|
arxiv-papers
| 2009-06-20T20:34:19 |
2024-09-04T02:49:03.442721
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lukasz Andrzej Glinka",
"submitter": "Lukasz Glinka",
"url": "https://arxiv.org/abs/0906.3827"
}
|
0906.4096
|
# An Event Based Approach to Situational Representation
Naveen Ashish Dmitri V. Kalashnikov Sharad Mehrotra Nalini Venkatasubramanian
Calit2 and ICS Department
University of California, Irvine
Irvine, CA 92707, USA
[email protected]
###### Abstract
Many application domains require representing interrelated real-world
activities and/or evolving physical phenomena. In the crisis response domain,
for instance, one may be interested in representing the state of the unfolding
crisis (e.g., forest fire), the progress of the response activities such as
evacuation and traffic control, and the state of the crisis site(s). Such a
situation representation can then be used to support a multitude of
applications including situation monitoring, analysis, and planning. In this
paper, we make a case for an event based representation of situations where
events are defined to be domain-specific significant occurrences in space and
time. We argue that events offer a unifying and powerful abstraction to
building situational awareness applications. We identify challenges in
building an Event Management System (EMS) for which traditional data and
knowledge management systems prove to be limited and suggest possible
directions and technologies to address the challenges.
## 1 Introduction
A large number of applications in different domains, including the emergency
response and disaster management domain that motivates our work, require
capturing and representing information about real-world situations as they
unfold. Such situations may correspond to interrelated real-world activities
or evolving physical phenomena. Such situational data is usually extracted
from multi-sensory data of different (and possibly mixed) modalities (such as
text, audio, video, sensor-data etc) and applications ranging from situation
monitoring to planning are built on top of the captured state or
representation of the real-world. These applications involve querying and
analyzing about events and entities that constitute a situation as well as
about relationships amongst or between them. In the crisis response domain,
for example, various types of sensors at the crisis site, field reports,
communication amongst first responders, news reports, eye witness accounts,
etc. can be used to extract a representation of the crisis situation, the
state of the crisis site, as well as the progress of response activities. Such
a representation can then be utilized to build a system to support customized
crisis monitoring capabilities. For instance, it could be used to provide a
big picture overview to the officials at the Emergency Operations Center (EOC)
to enable response planning and resource scheduling. Likewise, it could be
used to provide localized information of immediate interest to the first-
responders on the field in support of search and rescue activities, and to
provide up-to-date information to concerned or affected citizens via a
community portal.
Today such situational awareness systems are built in a relatively ad-hoc
fashion as applications on top of existing data and knowledge management
systems. Such system, designed as general purpose tools for data (knowledge)
management do not support an abstraction suited for representing situations
and building situation awareness applications. With the view of overcoming the
limitations of existing data management systems, in this paper, we promote an
event-centric approach to modeling and representing situational information.
In very general terms, an event is an occurrence of something of interest of a
certain type at a certain place at/over a certain period of time.111Our view
of event is similar in spirit to its definition in the Webster Dictionary
where it is defined as a significant happening or an occurrence . It is a
fundamental entity of observed physical reality represented by a point
designated by three coordinates of place and one of time in the space-time
continuum . An event is a semantic, domain-dependent concept, where an event
(depending upon the domain) may have associated with it a set of entities that
play different roles, and may bear relationships to other events and/or to
entities in the real-world. Events, in our view, provide a natural abstraction
for modeling, representing, and reasoning about situations. Not only does the
event abstraction provide a natural mechanism/interface for users to
query/reason/analyze situational data, it also provides a natural framework to
incorporate prior domain or context knowledge in a seamless manner when
reasoning about situations (illustrated in more detail later).
While events are domain and application specific concepts, and their types,
properties, and associated entities and relationships may vary from situation
to situation, it is also true that despite their differences, events, in
general, have certain common properties and relationships (e.g., spatial, and
temporal properties), and they support a few common operations and analyses
(e.g., spatial and temporal analysis). Exploiting these commonalities, it
becomes possible to design a general-purpose event management system (EMS)
that serves as a framework for representing and reasoning about situations. In
such an EMS, an event would be treated as a first-class object much in the
same way objects and entities are treated in traditional data management
systems.
Our goal in this paper is to explore the viability of a general event
management system and to identify challenges in developing such an event
management system. We envision such a system (or a system of systems) to
support all the components necessary to build event-based situational
representations. Specifically, it would support mechanisms to specify domain-
specific events, entities, and relationships of interest, provide tools to
incorporate domain semantics in reasoning, support languages for querying and
analyzing events, as well as mechanisms to indexing and other capabilities to
enable efficient data processing. With the above in mind, we enumerate the
following requirements for a general purpose Event Management System (EMS).
1. 1.
Situation Modeling Capabilities. The system should lend itself naturally to
modeling events and relationships at an appropriate level of abstraction. Just
as schemas or ER diagrams are successful modeling primitives for enterprise
structured data, the EMS must provide for appropriate and natural modeling of
events and relationships.
2. 2.
Desired Data Management Capabilities. An event management system must also
have capabilities desired of any data management system, such as simplicity of
use, appropriate query language (for events and event relationship oriented
queries in this case), efficient querying and storage, and also
interoperability with legacy data sources.
3. 3.
Semantic Representation and Reasoning Capabilities An EMS also needs to
incorporate domain and context knowledge when extracting, representing or
reasoning about events. Thus capabilities must be provided to represent such
domain and context knowledge (i.e., semantics) as well as utilize it when
answering queries about events or relationships between them.
We note that the event management system we seek does not need to be designed
from scratch, nor does it need to be designed in isolation from other existing
technologies. Many of the required capabilities for EMS have been studied and
developed in related areas such as GIS, multimedia and spatio-temporal
databases, and data management at large, which can be leveraged. We will
discuss the existing literature in this context in Section 2. Designing EMS,
however, opens many significant research challenges the foremost of which is
identifying an abstraction that captures commonalities across events that can
be refined to meet the needs of a large class of situational awareness
applications and can also be efficiently realized (potentially using existing
data management and knowledge management tools). The key challenge is to
develop a generic yet useful model of an event that can be used as a basis of
the system design. Besides the above challenge, there are numerous other
technical challenges that arise when representing situational data using
events. These include the challenge of disambiguating events as well as
representing their spatial and temporal properties. While prior work exists on
disambiguation, and on space and time representation, as will become evident
in the paper, many of the solutions/approaches need to be redesigned when
representing data at the level of events.
In the remainder of this paper, we address some of the challenges in
developing an EMS. Specifically, we describe a model for events that is
guiding our design of an EMS (Section 3), and discuss some of the technical
challenges (and solutions we designed) for representing spatial properties of
events as well as techniques for event disambiguation (Section 5).
## 2 Related Work
Formal methods for reasoning about events based on explicit representation of
events date back at least to work on situation calculus [12]. Situation
calculus treats situations as snapshots of the state of the world at some time
instants. Actions change one situation to another. These actions are
instantaneous, have no duration, and have immediate and permanent effects upon
situations. Another formal method is the event calculus [11] which explicitly
represents events (including actions) that belong to an event type and
generate new situations from old ones. Predicates in event calculus are
defined over fluents which are time-varying property of a domain, expressed by
a proposition that evaluates to true or false, depending on the time and the
occurrence of relevant events. Predicates on fluents include:
Occurs(event,time), HoldsAt(fluent,time), Initiates (event,fluent,time),
Terminates (event,fluent,time).
Reactive systems (including active databases and large system monitoring
applications) also explicitly store and reason about events. Here, an event is
defined as a system generated message about an activity of interest and it
belongs to an event type (situation). In addition to representing events,
these applications are also interested in detecting the occurrence of events
(e.g. [21, 1, 14]). Besides detecting and storing primitive events, these
applications detect and represent composite events which are defined as some
sequence of primitive events using an event algebra (e.g. [24]). As in
situation calculus, an event is considered to occur at a precisely determined
point in time and has no duration. Although composite events span a time
interval they are typically associated with the time-point of the last
component event. However, events can be mapped to time intervals to apply
queries over the duration of the sequence that a “compound composite” event
matches. All event types consist of core attributes like time point of
occurrence, event identifier, event type label, event source identity, and so
on. There are a number of commercial products that support such applications
including IBM’s Tivoli Enterprise Console (and its Common Base Event
Infrastructure) and iSpheres EPL Server.
Our focus is different from this body of work in several ways, namely: (1)
Events in reactive systems are well-defined structured messages with
restricted variations. In contrast, real-world events are communicated in
diverse formats like text, video, audio, etc., (2) Since we deal with real
world events, we consider spatial aspects of events which are not dealt with
in reactive systems, (3) Information about real-world events can be much more
imprecise (as it is derived from potentially noisy source like human reports)
and much more complex. (4) Relationships between events (e.g. causation) in
reactive systems are typically strong and easier to detect due to the static
nature of the environment (system configuration) in which the events occur.
Real-world events have weaker relationships and include temporal, spatial and
domain relationships.
Recent work in video content representation has also considered events as
foundations of an ontology-driven representation [16, 18, 3]. The goal of this
body of work thus far has been on producing a video event mark-up language
that can facilitate data exchange and event recognition. As in situation
calculus, an event is defined as a change in the state of an object. A state
is a spatio-temporal property valid at a given instant or stable within a time
interval. Events can be primitive or composite. Primitive events are state
changes directly inferred from the observables in the video data. Primitive
events are more abstract than states but they represent the finest granularity
of events. As in situation calculus [12], time is the critical distinguishing
factor between states and events. For example, two identical states with
different time values represent two different events. A composite event is a
combination of states and events. Specifically, a composite event is defined
by sequencing primitive events in a certain manner. This sequencing can be
single-threaded (single-agent based) or multi-threaded (multi-agent based).
Events, states and entities can be related to each other using predicates.
Spatial and temporal relationships are defined as predicates on members of the
time and space domains linked to events. In general this body of work is
object-centric, i.e. assumes knowledge of objects precedes knowledge of the
event as it defines events as changes in object states. As discussed in
Section 3, we adopt an event-centric approach. Besides, the constructs in this
body of work are tailored to automatic recognition of events from video while
we focus on facilitating queries on event data.
Event-oriented approaches have also been studied in spatio-temporal data
management. The goal here is to represent events associated with
geographical/spatial objects. As noted in [22], the effort on spatio-temporal
event representation has evolved in three stages: (1) Temporal snapshot of
spatial configurations of events, (2) Object change (captured in terms of
change primitives such as creation, destruction, appearance, disappearance,
transmission, fission, and fusion) stored as a sequence of past states, and
recently (3) Full-fledged representation of changes in terms of events,
actions (initiated occurrences), and processes. An example of stage 1
representation is [11] where, starting with an initial state (base map),
events are recorded in a chain-like fashion in increasing temporal order, with
each event associated with a list of all changes that occurred since the last
update of the event vector. The Event-based Spatio-Temporal Data Model (ESTDM)
[17] is an example of stage 2. ESTDM groups time-stamped layers to show
observations of a single event in a temporal sequence. The ESTDM stores
changes in relation to previous state rather than a snapshot of an instance.
An event component shows changes to a predefined location (a raster cell) at a
particular point in time. The SPAN ontology [6] that defines an
event/action/process view and the process calculus based approach of [22] that
can also represent event-event relationships are examples of stage 3.
Basically, in stage 3, rather than the sequence of past states of each object,
the events that caused the state changes are modeled resulting in a a more
richer representation. As such, stage 3 can: (1) tell us “why” a state exists,
and (2) enable us to represent which event caused a state change when multiple
events (or sets of events) can potentially cause the same state change.
## 3 Towards an EMS System
This section discusses the issues involved in building an EMS system. Our
vision of an EMS is a system that manages events just as a DMBS system manages
structured enterprise information. Thus for an EMS system we are concerned
with the (DBMS like) issues of representing event information (modeling),
querying and analyzing events, and finally ingesting event information from
information sources about situations and events. Unlike the concept of records
stored in traditional DBMSs, events are a semantically richer concept that
lead to some unique challenges. For instance, unlike enterprise systems, where
the information to be managed is structured and is available in that form as
is, event information is embedded in reports (for instance a text (news)
report about a situation or a video (news) coverage) describing or covering
situations related to those events. Events need to be extracted from such
reports. Given the extracted event information we may be left with uncertainty
about what the information is referring to; for instance (as elaborated on
later) there may be ambiguity about some entity referred to in an event and
also ambiguity about locations mentioned or referred to. Such uncertainty and
ambiguity must be adequately resolved or represented. Furthermore, given that
events are a semantic notion, interpreting events, as well as interpreting
queries about events requires mechanisms to incorporate domain knowledge and
context with both event extraction and querying/reasoning. Querying/analysis
techniques on top of EMSs must be able to deal with uncertainty in event
descriptions and the corresponding query languages must support constructs to
support spatial and temporal reasoning.
A high-level schematic overview of an EMS system is illustrated in Figure 1.
Crucial to the design of an EMS is to develop a model of events. In the
following, we discuss some of the key considerations and issues in modeling
events, and in developing techniques for querying, analyzing, and extracting
and disambiguating events.
Figure 1: Event Management System.
### 3.1 Information Modeling
In a DBMS system we start with capturing the real world using design models
such as the ER model. We then create application specific schemas, in a
particular database model, such as the relational model. Finally there is a
physical realization of each database (see Figure 2). In EMS systems too we
need to capture the real world (situations and events) in an appropriate
design model222 The work in [20] proposes some thoughts on an extended ER
model for modeling event information. We also need to pay attention to domain
knowledge, i.e., prior world knowledge that may be related to the events. As
we shall explain later, domain knowledge plays a critical role in various
facets of an EMS system. Such domain knowledge may be represented in
ontologies which we elaborate on in later sub-sections. We then move to
application specific event schemas and application specific domain knowledge
(as instantiated ontologies) and finally a physical realization of the
information. These levels of abstraction are schematically shown in Figure 1.
Figure 2: DBMS and EMS Systems
We now discuss the various elements and relationships for modeling events.
### 3.2 Building Blocks of the Event Model
Report: A report is the fundamental information source containing event
information. A report could be of any modality, for instance an (audio) phone
call reporting the event, information in text reports such as text alerts or
news stories, or audio-visual information from say a live TV coverage of a
situation. A report is defined then as a physical atomic unit that describes
one or more events.
Event: An event is an instance of an event type in space and time. So an
instance of a vehicle having overturned on a road, is an example of an event.
A situation comprises of a number of events. Events are extracted from
reports.
Entity: An entity is an object that occupies space and exists for an extended
period of time. Events generally have entities, such as people, objects (such
as say cars or planes), etc. associated with them. For instance a vehicle
overturning event will have the particular vehicle overturned as one of the
entities associated with that event.
Milieu: A milieu is the spatial, temporal or spatio-temporal context in which
an event, an object or report is situated. Continuing with the vehicle
overturning example, the time and place where the incident occurred are
milieus associated with the event.
As an example, a model for a vehicle overturning event is illustrated in 3.
The figure illustrates various aspects of the event model introduced so far.
It is shown using an ER diagram in the spirit of [20]. The model captures a
VEHICLE OVERTURN event, which has associated entities such as VEHICLE (the
vehicle which overturned) and also one or more REPORTER entities which are the
person(s) and/or organization(s) reporting that event.
Figure 3: Vehicle Overturning Event
Notice that space and time in the event model should be represented separately
from events and objects. This approach is similar to others like VERL [16] and
OLAP (where time and space are dimensions). This approach (1) allows us to
simultaneously use various forms of space and time values (i.e.
interval/point, region/point, and so on), (2) allows us to associate
attributes to space (e.g. name, geocode, shape, population) and time
instances, (3) explicitly represent and reason about relationships between
space and time instances (for instance using spatial and temporal
hierarchies).
Events and entities in events may be related amongst or across each other in
different ways. Such relationships may be extracted from the reports or
inferred from domain knowledge (as described later).
Relationships Each event may be associated with entities as described above,
creating the notion of an Event-Entity Relationship. The relationships between
events and entities can be described as that of participation (an entity
participates in an event) or, conversely, involvement (an event involves an
entity). an object). Entities participate in events with a given role.
Examples of kinds of these roles include: agentive (object produces,
perpetuates, terminates a particular event), influencing (facilitate,
hindrance), mediating (indirect influence) [6, 23].
Entities may be related to each other. Consider another event, that of a
report of a foul smell in the neighborhood with is reported by someone and for
which there is a known victim (affected by the smell or the associated gas). A
model for this event is illustrated in 4. The REPORTER and VICTIM are two
entities associated with this event. However the two entities could themselves
be related, for instance if the VICTIM is, say, a colleague of the REPORTER.
We then have the notion of an Entity-Entity Relationship.
Figure 4: Foul Smell Report Event
Events are related to time and space instances. For instance the vehicle
overturning event(referred to as VO) could have occurred at 10:15 am on May 1,
2005 at the intersection of 1rst and Main in Irvine, CA. This gives rise to
the notion of an Event-Milieu Relationship, denoted by types such as:
$\text{\em at-time}(e,t)$ or $\text{\em during}(e,T)$, $\text{\em at-
location}(e,L)$, $\text{\em near-location}(e,L)$, and so on. In these
examples, $e$ stands for an event, $t$ and $T$ stand for a time point and
interval, respectively, and $L$ stands for a location. For instance we could
instantiate: $\text{\em at-time}(VO,May12005:10-15am)$
Locations and times are related amongst themselves as well. Thus we need to
capture spatio-temporal or Milieu-Milieu Relationships. Relationships between
temporal milieu include point-point relationships like $\text{\em
before}(t_{1},t_{2})$, point-interval relationship like $\text{\em
begins}(t,T)$, and so on. Temporal relationships can also be cyclic (e.g.
calendar months) and hierarchical (e.g. containment) relationship between
intervals or periods. Relationships between spatial milieu include part-whole
(subsumption) relationships, region-region relationships (e.g. touch,
disjoint), proximity, and so on. The spatial milieu can also be hierarchical
where each level has its own sets of regions and topological relationships
that can be used in spatial reasoning. Many temporal relationships or known or
can be determined a-priori (for instance a relationship that year 2004 is
before year 2005), also spatial relationships, especially between explicitly
specified geographic locations can be known a-priori.
Events can be related to other events. For instance one event could cause
another event (the vehicle overturning could cause another event, namely that
of a road block); an event could hinder other events (the roadblock could
hinder traffic movement in the area). Further, complex events may be composed
of multiple constituent events. Composition of events is important since such
composition relationships form hierarchies resulting in composite events.
Finally, in the situational awareness domain, each event is associated with
one or more reports. Thus we have an Event-Report relationship capturing what
report(s) are associated with an event.
These relationships are illustrated in 5. For instance for the 2 events, the
vehicle overturn occurred before the foul smell report so a milieu-milieu
relationship (before) between the event times illustrates that.
Figure 5: Event Relationships
#### 3.2.1 Domain Knowledge
A data model and instance data can suffice for developing applications, as is
typically the case in enterprise information systems. However in situational
awareness applications, (prior) domain or context knowledge can be valuable as
well. For instance in the above example of a foul smell report, it may help to
have knowledge about various smells, what chemicals or hazards they may be
perpetrated by etc. This holds true in general, for instance for a fire
situation it can be valuable to have (precompiled) knowledge about fires and
fire-spreads in addition to the information about that particular situation.
The same is true for spatial information, as detailed information about places
associated with an event is available before the event.
Thus capturing and representing domain knowledge is an important issue.
Ontologies are a suitable framework for representing such knowledge. An
ontology is defined as a specification of a conceptualization [7]. Such
ontologies can capture knowledge for a particular domain of interest as well
as capture geospatial knowledge of various areas. The Semantic-Web 333
http://www.w3.org/2001/sw/ community has devoted significant efforts to
developing standard, universally accepted, and machine processable ontology
formalisms in recent years. Common ontology representation formalisms,
endorsed by the W3C444 http://www.w3.org/ include the Resource Description
Framework (RDF) [15] and its companion RDF Schema(RDFS) [4], DAML+OIL, OWL
[13], etc.
### 3.3 Querying and Analysis
Once the event model is determined, an EMS needs to support mechanisms to
support retrieval and analysis of events based on their properties and
relationships. A natural way to view events is as a network or a graph. We
refer to such an event graph as an EventWeb. The EventWeb is an attributed
graph where nodes corresponds to events, entities or reports and edges
correspond to a variety of relationships among events. Both nodes and edges
could have associated types that determine the associated attributes.
Given a graph view, a graph-based query language can be used for querying and
analysis for events. We have developed one such graph language named Graph
Analytic Language (GAAL) that extends previous graph languages by supporting
aggregation and grouping operators [dawit]. Besides supporting navigational
queries (based on path expressions), and selection queries (based on
attributes associated with nodes and edges), GAAL also supports the concepts
of aggregation and grouping. Using these operators, GAAL can be used to
support analytical queries similar in spirit to how OLAP operators are used to
support analytical queries over relational data. Such operators allow
analytical queries such as queries such as centrality of a node in a graph ,
degree of connectivity between specified nodes etc. Using GAAL over eventWeb,
numerous types of analysis such as causal analysis, dependency analysis,
impact analysis etc. can be performed. The aggregation and grouping features
of GAAL can also be used to perform such analysis over events at multiple
levels of composition/resolution.
While GAAL as a basis of an event language has a certain appeal for EMS, there
are numerous directions in which it will need to be extended to make it
suitable for event based systems. First, it needs to be extended to support
spatial and temporal reasoning. Space and time (milieu) are integral
components of any event based system. Space and time associated with events
usually correspond to locations or regions (point /intervals). Since such
locations can induce an infinite number of possible spatial and temporal
relationships, any one of which could be of interest to the user, such
relationships are best not represented as edges in an EventWeb. Instead, an
event-based language should support projection of events from and to spatial
and temporal dimensions and should seamlessly combine spatial and temporal
reasoning along with graph based queries.
Another challenge arises due to the semantic nature of events. Unlike
languages designed for structured/semi-structured data, where the primary
concern is to develop a mechanism to navigate through the structure, semantic
associations can play a vital role in expressing and interpreting queries in
event based systems. For instance we may be interested in knowing whether
there is any relationship between the vehicle overturn event and the foul
smell report event. Just the situational information per-se cannot help us in
uncovering such relationships. However incorporation of domain knowledge helps
us uncover such possible relationships. For instance the situational
information in conjunction with domain knowledge (represented as various
ontologies and relationships between the ontologies) enables us to infer that
OVERTURNED VEHICLES which-can-be OVERTURNED-CHEMICAL-TRUCKS which-can-cause
CHEMICAL-DISPERSIONS which-can-create FOUL SMELLs. This is illustrated in
Figure 6. The language designed for event based system must enable such
semantic associations. The work in [2] defines the concept of a semantic
association over semantically related entities and presents algorithms for
extracting such semantic associations between entities.
Figure 6: Domain Knowledge and Event Relationships
Finally, event information is inherently imprecise/uncertain. Events are
extracted from reports and extractors might or might not be able to precisely
determine the properties of events. Furthermore, there may be ambiguity in the
values as well as relationships associated with events (see discussion below).
Such uncertainties must be represented in the basic event model and query
language as well as associated query semantics must accommodate for such
uncertainties.
### 3.4 Data Ingestion
Unlike enterprise information systems, where the information to be stored is
available or entered in required format (i.e., tuples), event information is
present in reports of different modalities that talk about or cover a
situation containing that event. Thus events have to be extracted from such
reports. As mentioned earlier, the extracted event information can have
elements of uncertainty and ambiguity. For instance a reference to an entity
such as the vehicle may be ambiguous and we may need to determine (if
possible) which vehicle is the reference to. Finally, uncertainty might be
inherent in the description and not resolvable. For instance, spatial location
of an event may be specified as near the station . In such a case, query
processing techniques to handle such uncertainties must be developed.
Our progress so far in working towards an event management system has been on
techniques of event ingestion. Specifically, we have developed approaches for
(1) extraction and representation of spatial properties of events from textual
reports and techniques for answering spatial queries using the representation,
and (2) domain-independent techniques for disambiguating references and
entities associated with events. We are currently developing techniques that
exploit domain knowledge (expressed as ontologies) as well as context for
information extraction.
We discuss our work on representing and reasoning about spatial properties of
events described in textual reports and also our techniques for disambiguation
in the following two section. Such techniques form the building block of an
EMS which is our eventual goal.
## 4 Handling spatial uncertainty
As mentioned above, analyzing spatial properties of events is an inevitable
part of many decision making and analysis tasks on event data. In our work we
have addressed the problem of representing and querying uncertain location
information about real-world events that are described using free text. As a
motivating example, consider (again) the excerpts from two (fictional)
transcripts of 911 calls in Orange County (OC):
* •
…a massive accident involving a hazmat truck on N-I5 between Sand Canyon and
Alton Pkwy. …
* •
…a strange chemical smell on Rt133 between I405 and Irvine Blvd. …
These reports talk about the same event (an accident in this case) that
happened at some point-location in Laguna Niguel, CA. However, neither the
reports specify the exact location of the accident, nor do they mention Laguna
Niguel explicitly. We would like to represent such reports in a manner that
enables efficient evaluation of spatial queries and analyses. For instance,
the representation must enable us to retrieve accident reports in a given
geographical region (e.g., Irvine, Laguna Niguel, which are cities in OC).
Likewise, it should enable us to determine similarity between reports based on
their spatial properties; e.g., we should be able to determine that the above
reports might refer to the same location.
Before we describe our technical approach, we briefly discuss our motivation
for studying the afore-mentioned problem. We have already alluded to the
usefulness of spatial reasoning over free text for 911 dispatch in the example
above. We further note that such solutions are useful in a variety of other
application scenarios in emergency response. For instance, such a system could
support real-time triaging and filtering of relevant communications and
reports among first responders (and the public) during a crisis. In our
project, we are building Situational Awareness (SA) tools to enable social
scientists and disaster researchers to perform spatial analysis over two such
datasets: (1) the transcribed communication logs and reports filed by the
first responders after the 9/11 disaster, and (2) newspaper articles and blog
reports covering the S.E. Asia Tsunami disaster. We believe that techniques
such as ours can benefit a very broad class of applications where free text is
used to describe events.
Our goal is to represent and store uncertain locations specified in reports in
the database so as to allow for efficient execution of analytical queries.
Clearly, merely storing location in the database as free text is not
sufficient to answer either spatial queries or to disambiguate reports based
on spatial locations. For example, a spatial query such as ‘retrieve accident
reports in the city of Laguna Niguel’ will not retrieve either of the reports
mentioned earlier.
To support spatial analysis on free text reports, we need to project the
spatial properties of the event described in the report onto the domain
$\Omega$. For that, we model uncertain event locations as random variables
that have certain probability density functions (pdfs) associated with them.
We develop techniques to map free text onto the corresponding pdf defined over
the domain.
Our approach is based on the assumption555We have validated this claim through
a careful study of a variety of crisis related data sets we have collected in
the past. that people report event locations based on certain landmarks. Let
$\Omega\subset R^{2}$ be a 2-dimensional physical space in which the events
described in the reports are immersed. Landmarks correspond to significant
spatial objects such as buildings, streets, intersections, regions, cities,
areas, etc. embedded in the space. Spatial location of events specified in
those reports can be mapped into spatial expressions (s-expressions) that are,
in turn, composed of a set of spatial descriptors (s-descriptors) (such as
near, behind, infrontof, etc) described relative to landmarks. Usually, the
set of landmarks, the ontology of spatial descriptors, as well as, their
interpretation are domain and context dependent. Figure 7 shows excerpts of
free text referring to event locations and the corresponding spatial
expressions.
free text | s-expression
---|---
‘between buildings $A$ and $B$’ | $\text{\tt between}(A,B)$
‘near building A’ | $\text{\tt near}(A)$
‘on interstate I5, near L.A.’ | $\text{\tt within}(\text{I5})\land\text{\tt near}(L.A.)$
Figure 7: Examples of s-expressions.
These expressions use $A$ and $B$ as landmarks. While the spatial locations of
landmarks are precise, spatial expressions are inherently uncertain: they
usually do not provide enough information to identify the exact point-
locations of the events.
Figure 8: Free text location $\mapsto$ pdf $\in\Omega$.
Our approach to representing uncertain locations described in free text
consists of a two-step process, illustrated in Figure 8. First, a location
specified as a free-text is mapped into the corresponding s-expression. That
in turn is mapped to its corresponding pdf representation. Given such a model,
we develop techniques to represent, store and index pdfs so as to support
spatial analysis and efficient query execution over the pdf representations.
Our primary contributions in this direction are:
* •
An approach to mapping uncertain location information from free text into the
corresponding pdfs in the domain $\Omega$.
* •
Methods for representation and efficient storage of complex pdfs in database.
* •
Identification of queries relevant to SA applications.
* •
Indexing techniques and algorithms for efficient spatial query processing.
### 4.1 Modeling location uncertainty
We model each uncertain location $\ell$, as a continuous random variable
(r.v.) which takes values $(x,y)\in\Omega$ and has a certain probability
density function (pdf) $f(x,y)$ associated with it. Interpreted this way, for
any spatial region $R$, the probability that $\ell$ is inside $R$ is computed
as $\int_{R}f_{\ell}(x,y)dxdy$. The set of points for which $f(x,y)\not=0$ is
called uncertainty region $U_{\ell}$ of $\ell$. More specifically, we are
interested in conditional density $f(x,y|report)$ which describes possible
locations of the event given a particular report. While a report might contain
many types of information that can influence $f(x,y|report)$, we concentrate
primarily on direct references to event locations, such as ‘near building A’.
To map locations specified as free text into the corresponding density
functions, we employ a divide-and-conquer approach. We first map a free text
location into the corresponding s-expression which is a composition of
s-descriptors. S-descriptors are less complex than s-expressions, and can be
mapped into the corresponding pdfs. The desired pdf for the s-expression is
computed by combining the pdfs for s-descriptors. The last step of this
process incorporates the prior-distribution into the solution.
Mapping free text onto s-expression. Mapping of free-text locations into the
s-expressions is achieved by employing spatial ontologies. The development of
spatial ontologies is not a focus of our work on spatial uncertainty handling,
but we will summarize some of the related concepts in order to explain our
approach.
behind | totheleftof | infrontof | near
---|---|---|---
between | totherightof | withindist | within
indoor | outdoor | |
Figure 9: Examples of s-descriptors.
The basic idea is that each application domain $\mathcal{A}$ has, in general,
its own spatial ontology $\mathcal{D}(\mathcal{A})$. The ontology defines what
constitutes the landmarks in $\mathcal{A}$, and the right way of specifying
them in the ontology. It also defines the set of basic s-descriptors
$\\{\mathcal{D}_{1},\mathcal{D}_{2},\ldots,\mathcal{D}_{n}\\}$, such that any
free-text location from $\mathcal{A}$ can be mapped onto a composition of
s-descriptors. Examples of s-descriptors are provided in Figure 9. Each
s-descriptor is of the form
$\mathcal{D}_{i}(\mathcal{L}_{1},\mathcal{L}_{2},\ldots,\mathcal{L}_{m})$: it
takes as input $m\in N$ landmarks, where $m$ is determined by the type of
s-descriptor and can be zero. For instance, Figure 7 shows some free text
referring to event locations and the corresponding spatial expressions. Some
s-descriptors may not take any parameters. For instance, an ontology may use
the concept of indoor and outdoor, to mean ‘in some building’ and ‘not in any
building’ respectively.
We have addressed the most common type of s-expression: AND-expressions.
Another type of an s-expression is an OR-expression, but it rarely arises in
practice. An expression of type AND arises when the same location $\ell$ is
described using $n$ different descriptions $s_{1},s_{2},\ldots,s_{n}$, which
we denote as $s=s_{1}\land s_{2}\land\cdots\land s_{n}$. Here
$s_{1},s_{2},\ldots,s_{n}$ are subexpressions of $s$. As an example of an AND-
expression, assume a person is asked ‘where are you?’ to which he replies ‘I
am near building $A$ and near building $B$’, which corresponds to the
s-expression $\text{\tt near}(A)\land\text{\tt near}(B)$.
Pdf for a single s-descriptor. We observe that merely representing locations
as spatial expressions is not sufficient. We also need to be able to project
the meaning of each s-expression onto the domain $\Omega$. We achieve that by
(a) being able to compute the projection (i.e., the pdf) of each individual
s-descriptor in the s-expression; and (b) being able to combine the
projections. This process is illustrated in Figure 10.
Figure 10: Combination of s-descriptors $\mapsto$ pdf $\in\Omega$.
We first describe how a basic s-descriptor can be projected into $\Omega$ in
an automated fashion. Then, we will demonstrate how to compose those
projections to determine the pdfs for s-expressions. It is important to note
that our overall approach is independent from the algorithms for mapping basic
s-descriptors into pdfs.
To illustrate the steps of the algorithm more vividly, consider a simple
(a) Part of campus. (b) $\text{\tt outdoor}\land\text{\tt near}(A)$.
Figure 11: Buildings on a campus and various pdfs.
scenario demonstrated in Figure 11(c). This figure shows a portion of a
university campus with three buildings $A$, $B$, and $C$. Assume a person
reports an event that happened at location $\ell=\text{\tt
outdoor}\land\text{\tt near}(A)$.
The method we use for modeling the pdf
$f(x,y|\mathcal{D}(\mathcal{L}_{1},\mathcal{L}_{2},\ldots,\mathcal{L}_{m}))$
for any s-descriptor
$\mathcal{D}(\mathcal{L}_{1},\mathcal{L}_{2},\ldots,\mathcal{L}_{m})$ requires
making reasonable assumptions about the functional form of
$f(x,y|\mathcal{D}(\mathcal{L}_{1},\mathcal{L}_{2},\ldots,\mathcal{L}_{m}))$.
The model depends on the nature of each descriptor, and the spatial properties
of the landmarks it takes as input, such as the size landmark footprints,
their heights. The model is calibrated by learning the parameters from data.
The modeling assumptions can be refined or rejected later on, e.g. using
Bayesian framework.
For instance, for s-descriptor outdoor we can define the pdf $f(x,y|{\tt
outdoor})$ as having the uniform distribution everywhere inside the domain
$\Omega$ except for the footprints of the buildings that belong to $\Omega$.
That is $f(x,y)=c$ for any point $(x,y)\in\Omega$ except when $(x,y)$ is
inside the footprint of a building, in which case $f(x,y)=0$. The real-valued
constant $c$ is such that $\int_{\Omega}f(x,y)dxdy=1$.
Another example is an s-desriptor of $\text{\tt near}(A)$ which means
somewhere close to the landmark $A$ (the closer the better), but not inside
$A$. Let us note that, unlike the density for outdoor, the real density for
$\text{\tt near}(A)$ clearly is not uniform. Rather, a more reasonable pdf can
be a variation of the truncated-Gaussian density, centered at the center of
the landmark, with variance determined by the spatial properties of the
landmark $A$ (its height, the size of its footprint). Also, since the location
cannot be inside $A$, the values of that density should be zero for each point
inside the footprint of the landmark. This way we can determine the pdf for
each instantiated s-descriptor in an automated, non-manual fashion.
The pdf of a spatial expression. We have developed formulas for computing the
pdfs for AND-expressions, by being able to combine the pdfs of the underlying
basic s-descriptors. Figure 11(b) illustrates an example of a pdf for the
s-expression $\text{\tt outdoor}\land\text{\tt near}(A)$, evaluated in the
context of the scenario in Figure 11(a). Note that in SA domains pdfs might
have very complex shapes, significantly more involved than those traditionally
used. Thus we devise special methods for representing and storing pdfs.
### 4.2 Spatial Queries
SA applications require support of certain types of queries, the choice of
which is motivated by several factors. The three salient factors are: the
necessity, triaging capabilities, and quick response time. The necessity
factor means determining which core types of spatial queries (e.g., range, NN,
etc) are necessary to support common analytical tasks in such applications. In
crisis situations triaging capabilities can play a decisive role by reducing
the amount of information the analyst should process. Those capabilities
operate by restricting the size of query result sets and filtering out, or,
triaging, only most important results, possibly in a ranked order. Similarly,
the solutions that achieve quick query response time, perhaps by sacrificing
other (less important) qualities of the system, are required to be able to
cope with large amounts of data.
Figure 12: Examples of RQ$(R)$ and SQ$(q)$.
We have studied extensively two fundamental types of queries, which must be
supported by SA applications: region and similarity queries, illustrated in
Figure 12. We have designed and evaluated several modifications of those basic
types of queries, which support triaging capabilities and allow for more
efficient query processing. Specifically, we have developed algorithms for
efficient evaluation of the threshold and top-$k$ versions of those queries.
### 4.3 Representing and indexing pdfs
Histogram representation of pdfs. In order to represent and manipulate pdfs
with complex shapes, we first quantize the space by viewing the domain
$\Omega$ as a fine uniform grid $G$ with cells of size $\delta\times\delta$.
The grid $G$ is virtual and is never materialized. Using this grid we then
convert the pdfs into the corresponding histograms. That is, for the pdf
$f_{\ell}(x,y)$ of a location $\ell$ we compute the probability
$p_{ij}^{\ell}$ of $\ell$ to be inside cell $G_{ij}$, i.e.
$p_{ij}^{\ell}=\int_{G_{ij}}f_{\ell}(x,y)dxdy$. The set of all
$p_{ij}^{\ell}\not=0$ defines a histogram for $\ell$.
(a) Before compression. (b) After compression.
Figure 13: Quad-tree representation of pdf.
When we approximate a pdf with a histogram representation, we loose the
information about the precise shape of the pdf, but in return we gain several
advantages. The main advantage is that manipulations with pmfs are less
computationally expensive than the corresponding operations with pdfs, which
involve costly integrations. The latter is essential, given that SA
applications require quick query response time, especially in crisis
scenarios. Therefore, the loss due to approximation and the advantages of
using pmf should be balanced to achieve a reasonable trade-off.
Quad-tree representation of pdfs. We further improve the histogram
representations of pdfs by indexing histograms with quad-trees. First we build
a complete quad-tree $\mathcal{T}_{\ell}$ for each histogram $H_{\ell}$. Each
node $\mathcal{N}$ in the resulting quad-tree $\mathcal{T}_{\ell}$ stores
certain aggregate information that allows for efficient query processing. We
have explored several quad-tree (lossy) compression algorithms that trade
accuracy of representation for efficiency of query processing.
Indexing quad-trees with a grid. Assume the goal is to evaluate a $\tau$-RQ
with some threshold. The quad-tree representation of pdfs might help to
evaluate this query over each individual location $\ell\sim f_{\ell}(x,y)$
stored in the database faster. However, if nothing else is done, answering
this query will first require a sequential scan over all the locations stored
in the database, which is undesirable both in terms of disk I/O as well as CPU
cost. To solve this problem we can create a directory index on top of
$U_{\ell}$ (or, MBR of the histogram) for each location $\ell$ in the
database. We have designed a specific grid-based index for this goal and
demonstrated its superiority over traditional techniques by extensive
empirical evaluation.
We have extensively studied the proposed approach empirically. In our
experiments, we use a real geographic dataset, which covers $4\times 4$ km2
the New York, Manhattan area. The uncertain location data was derived based on
the 164 reports filed by Police Officers who participated in the events of
September 11, 2001. The number of the reports is rather small for testing
database solutions; hence we have generated synthetic datasets of
s-expressions, based on our analysis of the reports. The experimental study
showed the feasibility and the efficiency of the proposed approach as well as
its superiority over existing techniques.
## 5 Event disambiguation
The area of information quality studies various problems that arise when raw
datasets must be converted to a normalized representation so that they can be
analyzed by various applications. The same problems are unavoidable when the
information from raw reports, especially from textual information created by
humans, must be processed to create event representations.. The problem with
data can arise at all levels of event representation: (1) at the attribute
level, the values of the attributes of events/relationships can be incomplete,
uncertain, erroneous, or missing; (2) at the event level, duplicate events
might exist in the database; (3) at the relationship level, due to uncertain
description of events, there might be uncertainty in how a relationship/edge
should be created in the EventWeb, i.e. which entities this edge should
connect.
Event disambiguation is the task of creating accurate event representations
from raw datasets, possibly collected from multi-modal data sources. In the
following discussion we will focus primarily on the event disambiguation
challenges that are related to deduplication. Those challenges arise mainly
because the information might be compiled from different data sources, which
may describe the same events. A good example is news reports, which often
describe the same event. Another example is people calling in a 911 center to
report an accident: a major accident is typically reported multiple times by
different people, putting a strain on 911 centers. Detecting duplicates is
important in this context for proper resource planning and response.
In fact, removing duplicates in datasets is one of the key driving forces
behind the whole research area of information quality. The reason is that (a)
the problem is common in datasets; and (b) duplicate data items often
negatively affect data mining algorithms, which produce wrong results on non-
deduplicated data. The approaches for solving such problems with data are
classified into domain-specific or domain-independent categories. Since we are
interested in applying our algorithm to a variety of SA domains, we will be
looking into domain-independent techniques.
The disambiguation problem is challenging because the same event can be
described very differently in different data sources, and even in a single
data source. Traditional domain-independent cleaning techniques rely on
analyzing event features for disambiguation, hence we refer to them as the
feature-based similarity (FBS) methods. They measure the degree of similarity
of two events by first computing the similarity of their attributes and then
combining those attribute similarities into overall similarity of the two
events. However, there is additional information often available in datasets,
which is not explored by traditional techniques. This information is in the
relationships that exist among entities stored in the dataset. An analysis of
the connection strength $c(u,v)$ between two entities $u$ and $v$, stored in
the relationship chains between them, can help to decide whether $u$ and $v$
refer to the same entity or not.
### 5.1 Disambiguation problems
Figure 14: Event disambiguation.
The generic framework we are currently developing, called Relationship-based
Event Disambiguation (RED), can help solve various disambiguation challenges.
One of those challenges is illustrated in Figure 14, where the problem is
formulated as follows. When processing an incoming event, the application may
determine that the event is already stored in the database. However, the
description of the event might be such that it matches the descriptions of
multiple stored events, instead of a single one. The goal is, for the event
being processed, to identify the right matching event, stored in the database.
This problem is fairly generic, and it arises not only for event data. In
Figure 14 this point is illustrated by showing that the goal is, for a
description “J. Smith”, to determine to which particular “J. Smith” it refers
to, in the given context.
Another challenge that RED can help to solve is to deduplicate the same events
from the dataset. That is, given that events in the dataset are represented
via descriptions, the goal is to consolidate all the descriptions into groups.
The ideal consolidation is such that each resulting group is composed of the
event descriptions of just one event, and all the descriptions of one event
are assigned to just one group.
### 5.2 RED approach
Figure 15: Traditional methods vs. RED approach
Figure 15 illustrates the difference between the traditional feature-based
domain-independent data cleaning techniques and RED. The traditional
techniques, at the core, rely on analyzing object features. Those features are
either standard/regular features of the objects, or the “context attributes” –
the features derived from the context, which a few the recent techniques might
be able to employ. RED however proposes to enhance the core of those
approaches, by considering relationships that exists among entities.
To analyze relationships, the approach views the underlying dataset as an
attributed relational graph (ARG). The nodes in this graph represent entities
and the edges represent relationships. The analysis is based on what we refer
to as the Context Attraction Principle (CAP). The CAP is a hypothesis, which
has been proven empirically for various datasets. Simply put, it states that
if two entities $u$ and $v$ refer to the same object, then the connection
strength between their context is strong, compared to the case where $u$ and
$v$ refer to different objects.
#### 5.2.1 Reference disambiguation
To solve the first disambiguation challenge identified above, known as
reference disambiguation, RED introduces the new concept of a choice node,
illustrated
Figure 16: Choice node.
in Figure 16. A choice node is created to represent an uncertain relationship.
In Figure 16 the choice node $v^{*}_{u}$ is created for a relationship between
objects $u$ and $v$. However, the description of object $v$ is such that it
matches objects $v_{1},v_{2},\ldots,v_{n}$ and there is also a possibility
that the described object is not in the database, denoted by a virtual object
$z$. The choice node $v^{*}_{u}$ represents the fact that $u$ is connected to
either one $v_{1},v_{2},\ldots,v_{n}$ or $z$. That is if there were no
uncertainty and we knew the ground truth, only one edge would exist i.e.,
between $u$ and $v$. The goal is to decide which $v_{i}$ is $v$. The edges
between $v^{*}_{u}$ and $v_{1},v_{2},\ldots,v_{n},z$ have weights associated
with them. The weight $p_{i}$ is a real number between zero and one, which
reflects the degree of confidence that $v$ is $v_{i}$. All weights sum to 1.
The goal is then to assign those weights. As the final step, the algorithm
picks $v_{i}$ with the largest $p_{i}$ as $v$.
To assign those weights the algorithm discovers relationships that exist
between $u$ and each $v_{i}$. It then uses a $c(u,v)$ model to compute the
connection strength $c(u,v_{i})$ stored in the discovered relationships, for
each $v_{i}$. Since in general the discovered relationships can go via choice
nodes, $c(u,v_{i})$ returns an equation that relates $c(u,v_{i})$ to the edge
weights of other choice nodes (rather than a scalar value). The algorithm then
constructs another equation that relates $p_{i}$ and $c(u,v_{k})$ for all
$v_{k}$. This procedure is repeated for each choice node in the ARG. In the
end the algorithm maps the disambiguation problem into the corresponding
optimization problem, which can be solved either using an of the shelf math
solver or iteratively.
Note that the algorithm does not process entities sequentially, but rather
solves the problem for all the objects simultaneously, by resolving the
corresponding optimization problem.
#### 5.2.2 Connection strength models
Recently there has been a spike of interest by various research communities in
the measures directly related to the $c(u,v)$ measure. Since the $c(u,v)$
measure is at the core of the proposed RED approach, we next analyze several
principal models for computing $c(u,v)$. For brevity, we limit the discussion
to the models that have been employed in our work. For some of these models,
we use only their semantic aspects, while procedurally we compute $c(u,v)$
differently.
Diffusion Kernels. The earliest work in this direction that we can trace is in
the area of kernel-based pattern analysis [19], specifically the work on the
diffusion kernels, which are defined below.
A base similarity graph $G=(S,E)$ for a dataset $S$ is considered. The
vertices in the graph are the data items in $S$. The undirected edges in this
graph are labeled with a ‘base’ similarity $\tau({\bf x},{\bf y})$ measure.
That measure is also denoted as $\tau_{1}({\bf x},{\bf y})$, because only the
direct links (of size 1) between nodes are utilized to derive this similarity.
The base similarity matrix ${\bf B}={\bf B}_{1}$ is then defined as the matrix
whose elements ${\bf B}_{\bf xy}$, indexed by data items, are computed as
${\bf B}_{\bf xy}=\tau({\bf x},{\bf y})=\tau_{1}({\bf x},{\bf y})$. Next, the
concept of base similarity is naturally extended to paths of arbitrary length
$k$. To define $\tau_{k}({\bf x},{\bf y})$, the set of all paths $P_{\bf
xy}^{k}$ of length $k$ between the data items ${\bf x}$ and ${\bf y}$ is
considered. The similarity is defined as the sum over all these paths of the
products of the base similarities of their edges:
$\tau_{k}({\bf x},{\bf y})=\sum_{({\bf x}_{1}{\bf x}_{2}\ldots{\bf x}_{k})\in
P_{\bf xy}^{k}}\prod_{i=1}^{k}\tau_{1}({\bf x}_{i-1},{\bf x}_{i})$
Given such $\tau_{k}({\bf x},{\bf y})$ measure, the corresponding similarity
matrix ${\bf B}_{k}$ is defined. It can be shown that ${\bf B}_{k}={\bf
B}^{k}$. The idea behind this process is to enhance the base similarity by
those indirect similarities. For example, the base similarity ${\bf B}_{1}$
can be enhanced with similarity ${\bf B}_{2}$, e.g by considering a
combination of the two matrices: ${\bf B}_{1}+{\bf B}_{2}$. The idea
generalizes to more then two matrices. For instance, by observing that in
practice the relevance of longer paths should decay, it was proposed to
introduce a decay factor $\lambda$ and define what is known as the exponential
diffusion kernel: ${\bf K}=\sum_{k=0}^{\infty}\frac{1}{k!}\lambda^{k}{\bf
B}^{k}=\exp(\lambda{\bf B}).$ The von Neumann diffusion kernel is defined
similarly: ${\bf K}=\sum_{k=0}^{\infty}\lambda^{k}{\bf B}^{k}=({\bf
I}-\lambda{\bf B})^{-1}.$ The diffusion kernels can be computed efficiently by
performing eigen-decomposition of ${\bf B}$, that is ${\bf B}={\bf
V}^{\prime}{\bf\Lambda}{\bf V}$, where the diagonal matrix ${\bf\Lambda}$
contains the eigenvalues of B, and by making an observation that for any
polynomial $p(x)$, the following holds $p({\bf V}^{\prime}{\bf\Lambda}{\bf
V})={\bf V}^{\prime}p({\bf\Lambda}){\bf V}$. The elements of the matrix ${\bf
K}$ exactly define what we refer to as the connection strength: $c({\bf
x},{\bf y})={\bf K}_{\bf xy}$.
The solutions proposed for the diffusion kernels work well, if the goal is to
compute $c(u,v)$ for all the elements in the dataset. They are also very
useful for illustration purposes. However in data cleaning the task is
frequently to compute only some of $c(u,v)$’s, thus more efficient solutions
are possible. Also, often after computing one $c(u,v)$, the graph is adjusted
in some way, which affects the values of the rest of $c(u,v)$’s, computed
after that.
Random walks in graphs. Another common model used for computing $c(u,v)$ is to
compute it as the probability to reach node $v$ from node $u$ via random walks
in the graph. That model has been studied extensively, including in our work
on reference disambiguation [10, 8].
Parameterizable models. In the context of data cleaning the existing
techniques have several disadvantages. One disadvantage is that the true
‘base’ similarity is rarely known in real-world datasets. Some existing
techniques try to mitigate that by imposing a similarity model. However, the
CAP principle implies its own similarity measure, and any imposed model,
created for its own sake in isolation from the specific application, might
have little to do with it. Ideally, the similarity measure should be derived
directly from data for the specific application at hand that employs it. One
step toward achieving this, is to consider parameterizable models and then try
to learn an optimal combination of parameters directly from data. We have
explored such an approach in [9] for the problem of reference disambiguation.
The model is somewhat similar to that of the diffusion kernels but where
certain base similarities $\tau({\bf x},{\bf y})$ are initially specified a as
weight-variables, which are learned later directly from data.
#### 5.2.3 Object consolidation
The second challenge that RED solves is known as object consolidation. The
goal of object consolidation is to accurately group the representations of the
same objects together. Our ongoing work solves this challenge by combining
feature based similarity with analysis of relationships, in a similar manner
to the solution proposed for reference disambiguation. However, the overall
problem is different from that of reference disambiguation and solved by
employing graph partitioning algorithms [5].
## 6 Conclusion
In this paper we proposed our vision of an Event Management System (EMS) as a
platform for building Situational Awareness (SA) applications, with events as
the fundamental abstraction that comprise situations. We discussed several
aspects of an EMS, such as information modeling, querying and analysis, and
data ingestion, and also presented our work on location uncertainty reasoning
and event disambiguation in more detail.
## References
* [1] A. Adi and O. Etzion. Amit - the situation manager. The VLDB Journal, 13.
* [2] K. Anyanwu and A. Sheth. ?-queries: Enabling querying for semantic associations on the semantic web. 2003\.
* [3] F. Bremond and et al. Ontologies for video events. Technical Report INRIA Rapport de recherche no 5189, Univ. of Mass. Amherst, 2004.
* [4] D. Brickely and R.V.Guha. Rdf schema. Technical report, W3C: http://www.w3.org/TR/rdf-schema/, 2004.
* [5] Z. Chen, D. V. Kalashnikov, and S. Mehrotra. Exploiting relationships for object consolidation. Submitted for Publication, Mar. 2005.
* [6] P. Grenon and B. Smith. Snap and span: Towards dynamic spatial ontology. Journal of Spatial Cognition and Computation, 4(1):69–103, 2004\.
* [7] T. R. Gruber. A translation approach to portable ontology specifications. Knowledge Acquisition, 5(2):199–220, 1993.
* [8] D. Kalashnikov and S. Mehrotra. Exploiting relationships for domain-independent data cleaning. SIAM SDM, 2005. ext. ver., http://www.ics.uci.edu/~dvk/pub/sdm05.pdf.
* [9] D. V. Kalashnikov and S. Mehrotra. Learning importance of relationships for reference disambiguation. Submitted for Publication, Dec. 2004. http://www.ics.uci.edu/~dvk/RelDC/TR/TR-RESCUE-04-23.pdf.
* [10] D. V. Kalashnikov, S. Mehrotra, and Z. Chen. Exploiting relationships for domain-independent data cleaning. In SIAM International Conference on Data Mining (SIAM SDM 2005), Newport Beach, CA, USA, April 21–23 2005.
* [11] R. A. Kowalski and M. J. Sergot. A logic-based calculus of events. New Generation Computing, 4.
* [12] J. M. McCarthy and P. J. Hayes. Some philosophical problems form the stand point of artificaial intelligence. Reading in Artificial Intelligence.
* [13] D. McGuinness and F. van Harmelen. Owl: Web ontology language. Technical report, W3C: http://www.w3.org/TR/owl-features/, 2004.
* [14] J. Mihaeli and O. Etzion. Event database processing. In ADBIS, 2004.
* [15] E. Miller, R. Swick, and D. Brickely. Resource description framework (rdf). Technical report, W3C: http://www.w3.org/RDF/, 2004.
* [16] R. Nevatia, J. Hobs, and B. Bolles. An ontology for video event representation. In The 2004 Conference on Computer Vision and Pattern Recognition Workshop, pages 119 – 128, 2004.
* [17] D. Peuquet and N. Duan. An event-based spatio-temporal data model (estdm) for temporal analysis of geographical data. Int. Journal of GIS, 9(1):7–24, 1995.
* [18] T. Z. R. Nevatia and S. Hongeng. Hierarchical language-based representation of events in video streams. In The 2003 Conference on Computer Vision and Pattern Recognition Workshop, 2003.
* [19] J. Shawe-Taylor and N. Cristianni. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004.
* [20] R. Singh, Z. Li, P. Kim, D. Pack, and R. Jain. Event-based modeling and processing of digital media. In Workshop on Computer Vision Meets Databases (CVDB), 2004.
* [21] M. D. Spiteri. An Architecture for the notification, storage and retrieval of events. PhD thesis, 2000.
* [22] M. Worboys. Event-oriented approaches to geographic phenomena. International Journal of Geographic Information Systems, 19(1):1–28, 2005.
* [23] M. F. Worboys and K. Hornsby. From objects to events: Gem, the geospatial event model. In Third International Conference on GIScience, 2004.
* [24] D. Zimmer and R. Unland. On the semantics of complex events in active database management systems. In ICDE, pages 392–399, 1999.
|
arxiv-papers
| 2009-06-22T19:40:12 |
2024-09-04T02:49:03.453357
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Naveen Ashish, Dmitri Kalashnikov, Sharad Mehrotra and Nalini\n Venkatasubramanian",
"submitter": "Naveen Ashish",
"url": "https://arxiv.org/abs/0906.4096"
}
|
0906.4108
|
# Cosmological Constraints from Gravitational Lens Time Delays
Dan Coe and Leonidas A. Moustakas Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Dr, MS 169-327, Pasadena, CA 91109
###### Abstract
Future large ensembles of time delay lenses have the potential to provide
interesting cosmological constraints complementary to those of other methods.
In a flat universe with constant ${\rm w}$ including a Planck prior, LSST time
delay measurements for $\sim 4,000$ lenses should constrain the local Hubble
constant $h$ to $\sim 0.007$ ($\sim 1\%$), $\Omega_{de}$ to $\sim 0.005$, and
${\rm w}$ to $\sim 0.026$ (all 1-$\sigma$ precisions). Similar constraints
could be obtained by a dedicated gravitational lens observatory (OMEGA) which
would obtain precise time delay and mass model measurements for $\sim 100$
well-studied lenses. We compare these constraints (as well as those for a more
general cosmology) to the “optimistic Stage IV” constraints expected from weak
lensing, supernovae, baryon acoustic oscillations, and cluster counts, as
calculated by the Dark Energy Task Force. Time delays yield a modest
constraint on a time-varying ${\rm w}(z)$, with the best constraint on ${\rm
w}(z)$ at the “pivot redshift” of $z\approx 0.31$. Our Fisher matrix
calculation is provided to allow time delay constraints to be easily compared
to and combined with constraints from other experiments. We also show how
cosmological constraining power varies as a function of numbers of lenses,
lens model uncertainty, time delay precision, redshift precision, and the
ratio of four-image to two-image lenses.
###### Subject headings:
cosmological parameters – dark matter — distance scale — galaxies: halos —
gravitational lensing — quasars: general
††slugcomment: Accepted for Publication in the Astrophysical Journal
## 1\. Introduction
The HST Key Project relied on 40 Cepheids to constrain Hubble’s constant
$H_{0}$ to 11% (Freedman et al., 2001). The first convincing measurements of
the accelerating expansion rate of the universe (suggesting the existence of
dark energy) by Riess et al. (1998) and Perlmutter et al. (1999) required 50
and 60 supernovae, respectively. So far, time delays have only been reliably
measured for $\sim 16$ gravitational lenses, thanks to dedicated lens
monitoring from campaigns such as COSMOGRAIL (Eigenbrod et al., 2005). Yet
recent analyses of 10–16 time delay lenses already claim to match or surpass
the Key Project’s 11% precision on $H_{0}$ (Saha et al., 2006; Oguri, 2007;
Coles, 2008). Future surveys promise to yield hundreds or even thousands of
lenses with well-measured time delays, which will enable us to obtain much
tighter constraints on $H_{0}$ as well as constraints on other cosmological
parameters.
To date, most efforts have focused on studies of individual time delay lenses.
In theory, one might be able to control all systematics and constrain $H_{0}$
unambiguously given a single “golden lens”. Such a lens would have a
sufficiently simple and well-measured geometry. The closest to a golden lens
may be B1608+656. In Suyu et al. (2009b), the authors claim all systematics
have been controlled to 5%. A new estimate for $H_{0}$ based on this lens is
forthcoming (Suyu et al., 2009a).
Historically, analyses of individual lenses have yielded varying answers for
$H_{0}$ (see the Appendix of Jackson 2007 for a recent review). This can be
attributed to two factors, both of which, it appears, are now being overcome.
The first factor is simple intrinsic variation in lens properties (especially
mass slope) and environment (lensing contributions from neighboring galaxies).
Consider the following estimate from a simple empirical argument. If
statistical uncertainties on $H_{0}$ decrease as $1/\sqrt{N}$ (assuming
systematics can be controlled), and the current uncertainty from 16 lenses is
$\sim 10$%, then the uncertainty on a single lens might be $\sim 40$%. Thus,
assuming $h=0.7$ (where $H_{0}=100h~{}{\rm km}~{}{\rm s}^{-1}~{}{\rm
Mpc}^{-1}$), individual lenses may be expected to yield a wide range of
$h=0.42$ – 0.98 (1-$\sigma$). (We will revisit these assumptions in this
work.)
The second factor in the wide range of reported $H_{0}$ values is that
different analyses have assumed different mass profiles to model the lenses,
including isothermal, de Vaucouleurs, and mass follows light. There is
substantial weight of evidence that galaxy lenses are roughly isothermal on
average, at least within approximately the scale radius (e.g., Koopmans et
al., 2006). Theoretical work supports this idea, showing that a wide range of
plausible luminous plus dark matter profiles all combine to yield roughly an
isothermal profile at the Einstein radius, though the slope may deviate from
isothermal beyond that radius (van de Ven et al., 2009).
In recent years we have witnessed a steady increase in the number of strong
lenses discovered by searches such as CLASS (Myers et al., 2003), SLACS
(Bolton et al., 2006), SL2S (Cabanac et al., 2007), SQLS (Inada et al., 2008),
HAGGLeS (Marshall et al., 2009b), and searches of AEGIS (Moustakas et al.,
2007) and COSMOS (Faure et al., 2008). Based on this experience, we can expect
that future surveys such as Pan-STARRS111The Panoramic Survey Telescope &
Rapid Response System, http://pan-starrs.ifa.hawaii.edu (Kaiser, 2004),
LSST222The Large Synoptic Survey Telescope, http://www.lsst.org (Ivezic et
al., 2008), JDEM / IDECS333The Joint Dark Energy Mission,
http://jdem.gsfc.nasa.gov, and SKA444The Square Kilometer Array,
http://www.skatelescope.org (Lazio, 2008) will yield an explosion in the
number of strong lenses known (e.g., Koopmans et al., 2004; Fassnacht et al.,
2004; Marshall et al., 2005). Prospects for using these lenses to constrain
the nature of dark matter over the course of the next decade were presented in
Moustakas et al. (2009), Koopmans et al. (2009a), and Marshall et al. (2009a).
It is reasonable to expect that time delays will be reliably measured for
large numbers of these lenses, whether through repeated observations in
surveys (Pan-STARRS and LSST), auxiliary monitoring, and/or through tailored
specific missions such as OMEGA (Moustakas et al., 2008). Increased sample
size, improved lens model constraints, and higher precision redshifts and time
delay measurements will all improve constraints on $H_{0}$ and other
cosmological parameters, as we present below.
A more precise measurement of $H_{0}$ will yield tighter constraints on both
the dark energy equation of state parameter (${\rm w}$) and the flatness of
our universe ($\Omega_{k}$), independently of the results of future dark
energy surveys (Blake et al., 2004; Hu, 2005; Albrecht et al., 2006; Olling,
2007). To this end, the SHOES Program (Supernovae and $H_{0}$ for the Equation
of State) has obtained new observations of supernovae and Cepheid variables
with reduced systematics. Recently, Riess et al. (2009) published a
redetermination of $H_{0}=74.2\pm 3.6{\rm km}~{}{\rm s}^{-1}~{}{\rm
Mpc}^{-1}$, or 5% uncertainty including both statistical and systematic
errors. Their $H_{0}$ determination plus WMAP 5-year data alone constrain
${\rm w}=-1.12\pm 0.12$ (assuming constant ${\rm w}$).
Riess et al. (2009) also make the following important point that bears
repeating. The seemingly tight constraints on $H_{0}$ derived from CMB + BAO +
SN experiments are in fact predictions or inferences of $H_{0}$ given those
data and a cosmological model. They are no substitute for direct measurement
of $H_{0}$ such as that presented in their work or the HST Key Project.
Olling (2007) reviews several methods with the potential to directly constrain
$H_{0}$. Water masers, for example, hold much promise (Braatz et al., 2008;
Braatz, 2009). Time delays and water masers both yield direct geometric
measurements of the universe to the redshifts of the observed sources ($z\sim
2$ or greater for time delay lenses), bypassing all distance ladders.
Time delays do not simply constrain $H_{0}$. To first order, each time delay
is proportional to the angular diameter distance to the lensed object and thus
inversely proportional to $H_{0}$. An additional factor involves a ratio of
two other distances – from observer to lens and from lens to source. All three
of these distances have a complex (though weaker) dependence on the other
cosmological parameters ($\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm
w}_{a}$) which contribute to the expansion history of the universe.
Most time delay analyses ignore this weaker dependence on
($\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a}$), in effect
assuming these parameters are known perfectly. In this paper we show how
relaxing this “perfect prior” increases the uncertainties on $H_{0}$. As dark
energy surveys endeavor to place constraints on ${\rm w}$ and the flatness of
our universe $\Omega_{k}$, we must study how time delays can contribute to
these constraints without assuming the very parameters we would like to
constrain. In this work we also study the ability of large time delay
ensembles to constrain ($\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm
w}_{a}$).
The idea to use time delay lenses to measure $H_{0}$ was first proposed by
Refsdal (1964). Strong gravitational lenses are elegant geometric consequences
of how light travels through the universe while grazing massive galaxies. When
the line of sight alignment is very close, light takes multiple paths around
the curved space of the lens. These paths form multiple images, and the light
takes a different amount of time to travel each path. Light passing closer to
the lens is deflected by a larger angle (increasing its path length) and
experiences a greater relativistic time dilation, further delaying its
arrival. If the source flares up, or otherwise varies in intensity (e.g., if
it is an active galactic nucleus, or AGN), we can observe these “time delays”
between or among the images. These time delays are functions of the angular
diameter distances between the source, lens, and observer, as well as the
properties of the lens itself.
The ability of time delays to constrain other cosmological parameters has also
been explored. Lewis & Ibata (2002) explored various combinations of
($\Omega_{m},\Omega_{\Lambda}$) in a flat universe and various (${\rm
w}_{0},{\rm w}_{a}$) for fixed ($\Omega_{m},\Omega_{de}$). Most notably, they
calculated constraints on ($h,{\rm w}$) from ensembles of lenses assuming
constant ${\rm w}$ and ($\Omega_{m},\Omega_{\Lambda}$) = (0.3, 0.7), finding
that $h$ and ${\rm w}$ would not be strongly constrained. We show that the
addition of a Planck prior improves these constraints considerably. Linder
(2004) investigated constraints on the dark energy parameters (${\rm
w}_{0},{\rm w}_{a}$) from various methods, touting the complimentarity of
strong lensing to that of other methods. However, they concede that the unique
positive correlation in strong lensing (${\rm w}_{0},{\rm w}_{a}$) constraints
evaporates when including degeneracies other cosmological parameters. Mörtsell
& Sunesson (2006) and Dobke et al. (2009) examined the constraints that large
ensembles of lenses might place on $H_{0}$ and $\Omega_{\Lambda}=1-\Omega_{m}$
(assuming a flat universe). Below we present the first full treatment of the
cosmological constraints expected on
($h,\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a}$) from ensembles
of time delay lenses including various priors.
Lens statistics from well-controlled searches for strongly-lensed sources have
also been used to constrain cosmology (e.g., Chae, 2007; Oguri et al., 2008).
If time delays can be obtained for the lenses in such a sample, the lens
statistics and time delays might combine to yield tighter cosmological
constraints. This potential is not explored in this work.
Cosmological constraints can also be obtained from symmetric strong lenses for
which velocity dispersions have been measured (e.g., Paczynski & Gorski, 1981;
Futamase & Hamaya, 1999; Yamamoto et al., 2001; Lee & Ng, 2007). Assuming an
isothermal model, the measured velocity dispersion determines the Einstein
radius solely as a function of cosmology (given redshifts measured to the lens
and source). Yamamoto et al. (2001) studied the future potential for this
method to constrain cosmology using a Fisher matrix analysis.
The reader is invited to skip ahead to our results in §5, where cosmological
constraints expected from time delays (according to our calculations) are
compared to those expected from other methods (weak lensing, supernovae,
baryon acoustic oscillations, and cluster counts). Table 2 summarizes the
assumed priors including a guide to specific sections and figures.
The remainder of our paper is organized as follows. In §2 we provide the time
delay equations and discuss how cosmology is derived from observed time
delays. We define the quantity
${\mathcal{T}}_{\mathcal{C}}(h,\Omega_{m},\Omega_{de},\Omega_{k},{\rm
w}_{0},{\rm w}_{a};z_{L},z_{S})$ which time delays are capable of
constraining. In §3 we estimate the constraints on
${\mathcal{T}}_{\mathcal{C}}$ expected from future experiments. (A more
detailed analysis of lensing simulations is presented in a companion paper Coe
& Moustakas 2009a, hereafter Paper I.) In §4 we illustrate the dependence of
${\mathcal{T}}_{\mathcal{C}}$ on cosmological parameters
($h,\Omega_{m},\Omega_{de},{\rm w}_{0},{\rm w}_{a}$). In §5, as highlighted
above, we give projections for time delay constraints on
$(h,\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a})$ and compare to other
methods. Systematic biases are discussed in §6 and their impact on our ability
to constrain cosmology is analyzed in another companion paper (Coe &
Moustakas, 2009c, hereafter Paper III). Finally we present our conclusions in
§7.
We assume all constraints to be centered on the concordance cosmology $h=0.7$,
$\Omega_{m}=0.3$, $\Omega_{de}=0.7$, $\Omega_{k}=0$, ${\rm w}_{0}=-1$, and
${\rm w}_{a}=0$, where $H_{0}=100h~{}{\rm km}~{}{\rm s}^{-1}~{}{\rm
Mpc}^{-1}$.
## 2\. Cosmological Constraints from Time Delays
### 2.1. Time Delay Equations
A galaxy at redshift $z_{L}$ strongly lenses a background galaxy at redshift
$z_{S}$ to produce multiple images. Either two or four images are typically
produced.555An additional central demagnified image is also produced by every
lens with a central mass profile shallower than isothermal. Such images are
rarely bright enough to be detected, thus we ignore them throughout this work.
We refer to these cases as “doubles” and “quads”, respectively. The lensing
effect delays each image in reaching our telescope by a different amount of
time, given by
$\Delta\tau=\frac{(1+z_{L})}{c}{\mathcal{D}}\left[\text@frac{1}{2}\left|{\bm{\theta}}-{\bm{\beta}}\right|^{2}-\phi\right]$
(1)
(e.g., Blandford & Narayan, 1986) with terms defined below. The factors in the
time delay equation can be grouped into a product of two terms:
$\Delta\tau={\mathcal{T}}_{\mathcal{C}}{\mathcal{T}}_{\mathcal{L}}.$ (2)
The first factor,
${\mathcal{T}}_{\mathcal{C}}\equiv\frac{(1+z_{L})}{c}{\mathcal{D}},$ (3)
is a function of cosmology and the lens and source redshifts, $z_{L}$ and
$z_{S}$. The second factor,
${\mathcal{T}}_{\mathcal{L}}\equiv\left[\text@frac{1}{2}\left|{\bm{\theta}}-{\bm{\beta}}\right|^{2}-\phi\right],$
(4)
is a function of the projected lens potential $\phi$, the source galaxy’s
position on the sky $\bm{\beta}$, and the image positions $\bm{\theta}$.
We concentrate on the cosmological dependence of
${\mathcal{T}}_{\mathcal{C}}$. The factor
${\mathcal{D}}\equiv\frac{D_{L}D_{S}}{D_{LS}}$ (5)
is a ratio of the angular-diameter distances from observer to lens
$D_{L}=D_{A}(0,z_{L})$, observer to source $D_{S}=D_{A}(0,z_{S})$, and lens to
source $D_{LS}=D_{A}(z_{L},z_{S})$. Angular-diameter distances are calculated
as follows (Fukugita et al., 1992, filled beam approximation; see also Hogg
1999):
$D_{A}(z_{1},z_{2})=\frac{c}{H_{0}}\frac{E_{A}(z_{1},z_{2})}{1+z_{2}},$ (6)
$E_{A}=\frac{{\rm
sinn}\left[\sqrt{\left|\Omega_{k}\right|}E^{\star}_{A}\right]}{\sqrt{\left|\Omega_{k}\right|}},$
(7)
where ${\rm sinn}(u)=\sin(u)$, $u$, or $\sinh(u)$ for an open, flat, or closed
universe respectively ($\Omega_{k}<0$, $\Omega_{k}=0$, or $\Omega_{k}>0$). The
curvature is given by $\Omega_{k}\equiv 1-(\Omega_{m}+\Omega_{\Lambda})$,
while
$E^{\star}_{A}(z_{1},z_{2})=\int_{z_{1}}^{z_{2}}\frac{dz^{\prime}}{E(z^{\prime})}.$
(8)
The normalized Hubble parameter $E(z)$ can have different expressions
depending on the cosmology assumed:
$\displaystyle E(z)$ $\displaystyle\equiv$ $\displaystyle\frac{H(z)}{H_{0}}$
$\displaystyle=$
$\displaystyle\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}}$
$\displaystyle=$
$\displaystyle\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{de}(1+z)^{3(1+{\rm
w})}}$ $\displaystyle=$ $\displaystyle\sqrt{\cdots+\Omega_{de}(1+z)^{3(1+{\rm
w}_{0}+{\rm w}_{a})}\exp{\left(\frac{-3{\rm w}_{a}z}{1+z}\right)}}.$
Here we have progressed from a universe with a cosmological constant
$\Omega_{\Lambda}$ to one with dark energy with an equation of state $p={\rm
w}\rho$. In the last line, the last term has been rewritten in terms of an
evolving dark energy equation of state
$\displaystyle{\rm w}$ $\displaystyle=$ $\displaystyle{\rm w}_{0}+{\rm
w}_{a}(1-a)$ (10) $\displaystyle=$ $\displaystyle{\rm w}_{0}+{\rm
w}_{a}\left(\frac{z}{1+z}\right),$ (11)
a common parametrization first introduced by Chevallier & Polarski (2001) and
Linder (2003). The universe scale factor $a=(1+z)^{-1}$.
We next define the dimensionless ratio
${\mathcal{E}}\equiv\frac{E_{L}E_{S}}{E_{LS}}$ (12)
with factors defined similarly to those above for $D_{A}$:
$E_{L}=E_{A}(0,z_{L})$, $E_{S}=E_{A}(0,z_{S})$, $E_{LS}=E_{A}(z_{L},z_{S})$.
We find that many factors cancel, and ${\mathcal{T}}_{\mathcal{C}}$ simplifies
to:
${\mathcal{T}}_{\mathcal{C}}=\frac{\mathcal{E}(\Omega_{m},\Omega_{de},\Omega_{k},{\rm
w}_{0},{\rm w}_{a})}{H_{0}}.$ (13)
We see here clearly that time delays
($\Delta\tau={\mathcal{T}}_{\mathcal{C}}{\mathcal{T}}_{\mathcal{L}}$) scale
inversely with $H_{0}$. There is also a complex though weaker dependence on
the other cosmological parameters as embedded in $\mathcal{E}$.
### 2.2. Deriving Cosmology from Time Delays
Given observed time delays $\Delta\tau$ and assuming a lens model (and thus
${\mathcal{T}}_{\mathcal{L}}$), one can obtain measures of
${\mathcal{T}}_{\mathcal{C}}$. These measures will have some scatter due to
both observational uncertainties and deviations of the lens from the assumed
model.
Recent studies suggest that galaxy lenses, on average, have roughly isothermal
profiles within the Einstein radius (see §1). Deviations from this simple
description include variation in lens slope, external shear, mass sheets, and
substructure. Oguri (2007) parametrized the deviations as the “reduced time
delay”, the ratio of the observed time delay to that expected due to an
isothermal potential in a given lens:
$\Xi\equiv\frac{\Delta\tau}{\Delta\tau_{\rm iso}}.$ (14)
In our notation, these observed deviations are due to deviations in the lens
model:
$\Xi_{\mathcal{L}}\equiv\frac{{\mathcal{T}}_{\mathcal{L}}}{{\mathcal{T}}_{{\mathcal{L}}\rm,iso}}.$
(15)
By assuming an isothermal model
(${\mathcal{T}}_{\mathcal{L}}={\mathcal{T}}_{{\mathcal{L}}\rm,iso}$), these
deviations get absorbed into the derived cosmology:
$\Xi_{\mathcal{C}}\equiv\frac{{\mathcal{T}}_{\mathcal{C}}}{{\mathcal{T}}_{{\mathcal{C}}\rm,true}},$
(16)
where ${\mathcal{T}}_{{\mathcal{C}}\rm,true}$ is the true cosmology. For
example, a lens which is steeper than isothermal yields $\Xi_{\mathcal{L}}>1$;
thus when assuming an isothermal model ($\Xi_{\mathcal{L}}=1$), we derive
$\Xi_{\mathcal{C}}>1$ (since $\Xi=\Xi_{\mathcal{C}}\Xi_{\mathcal{L}}$). In
traditional analyses assuming fixed $\mathcal{E}$, $\Xi_{\mathcal{C}}>1$ would
simply yield a low $h$. This approximation is adequate for small samples of
lenses but not for the large samples to come in the near future (§5.4.1).
Similarly, observational uncertainties affecting $\Delta\tau$ are absorbed
into the derived cosmology. In this paper, we study how observational and
intrinsic (lens model) uncertainties combine to yield scatter in the observed
$\Delta\tau$. We will assume these measurements yield
${\mathcal{T}}_{\mathcal{C}}$ with the correct mean but a simple Gaussian
scatter and explore how this propagates to Gaussian uncertainties on
cosmological parameters.
In practice we do not expect $\Xi_{\mathcal{L}}$ and measurements of
$\Delta\tau$ to have Gaussian scatter, but these serve as useful
approximations. The true expected $P(\Xi)$ from time delay measurements and
methods for handling these distributions are studied in Oguri (2007) and Paper
I.
## 3\. Constraints on ${\mathcal{T}}_{\mathcal{C}}$ from Future Experiments
### 3.1. Extrapolating from Current Empirical Results
Recent studies have constrained ${\mathcal{T}}_{\mathcal{C}}$ to $\sim 10\%$
using time delays, where ${\mathcal{T}}_{\mathcal{C}}$ encodes all of the
cosmological dependencies (§2.1). Constraints on ${\mathcal{T}}_{\mathcal{C}}$
have generally been interpreted to be equivalent to direct constraints on $h$.
This assumption is reasonable for current sample sizes, but will need to be
revised in the future (§5.4.1). Using 16 lenses, Oguri (2007) obtain
$h=0.70\pm 0.06{\rm(stat.)}$. Similar studies by Saha et al. (2006) and Coles
(2008) using a different method obtain similar constraints using 10 and 11
lenses, respectively. The latter finds $h=0.71^{+0.06}_{-0.08}$.
We will adopt the Oguri (2007) uncertainty of 8.6% with 16 lenses as the
“current” uncertainty in ${\mathcal{T}}_{\mathcal{C}}$.666The Oguri (2007)
simulations initially suggested an uncertainty of $\sim 4\%$ in
${\mathcal{T}}_{\mathcal{C}}$. However jackknife resampling of the data
revealed the true uncertainty to be twice as much. Under-prescribed shear in
the simulations was cited as a potential cause for the discrepancy. We note
that the time delay uncertainties in this sample are roughly and broadly
scattered about $\Delta(\Delta\tau)=2$ days.777We adopt a notation in which
“$\Delta$” refers to uncertainties with units and “$\delta$” to fractional
uncertainties. Thus a time delay of 20 days measured to 2-day precision has
$\Delta(\Delta\tau)=2$ days and $\delta(\Delta\tau)=0.1$.
We can improve on these ${\mathcal{T}}_{\mathcal{C}}$ constraints in three
ways: obtaining larger samples of lenses, better constraining our lens models,
and obtaining more precise time delay measurements. As we explain below, we
expect future surveys such as Pan-STARRS and LSST to improve on the sample
size while the lens model and time delay uncertainties will remain about the
same. These surveys will have to contend with a lack of spectroscopic
redshifts for most objects, but the gains in sample size will more than
compensate. Similarly tight constraints on ${\mathcal{T}}_{\mathcal{C}}$ could
also be obtained by studying relatively fewer lenses in great detail, as we
discuss below.
Here we consider statistical uncertainties only, with systematics to be
discussed in §6. We will assume that all other things being equal, increasing
our sample size beats down our errors by $\sqrt{N}$ for $N$ lenses. This
assumption is borne out well by our detailed simulations (Paper I), for the
case of no systematic uncertainties.
Based on the current constraint of $\delta{\mathcal{T}}_{\mathcal{C}}\approx
8.6\%$ from 16 lenses (Oguri, 2007), we project that simply increasing the
sample of lenses would produce constraints of
$\delta{\mathcal{T}}_{\mathcal{C}}\approx 34\%/\sqrt{N}$. We will define this
as the uncertainty from lens models and time delay measurements:
$\delta\Xi_{{\mathcal{L}}\tau}\sim 0.344$. Photometric redshifts would degrade
these constraints as estimated below (§3.3).
### 3.2. Future Surveys
Pan-STARRS and LSST will both survey the sky repeatedly, opening the time
domain window for astronomical study over vast solid angles. Pan-STARRS 1
(PS1) has recently begun its $3\pi$ survey, repeatedly observing the entire
visible sky to $\sim$23rd magnitude every week over a 3-year period. LSST
promises similar coverage and depth every 3 nights with first light scheduled
for 2014.
These surveys will reveal many time-variable sources, among them
gravitationally-lensed quasars. The persistent monitoring over many years
should yield time delays “for free” for many strongly-lensed quasars.
Simulations (M. Oguri 2009, private communication) show that Pan-STARRS 1 and
LSST are expected to yield $\sim 1,000$ and $\sim 4,000$ strongly-lensed
quasars with quad fractions of 19% and 14%, respectively.
We will assume that these surveys will measure time delays to about 2-day
precision, or similar to that of our current sample of time delay lenses. This
is consistent with predictions based on detailed simulations by Eigenbrod et
al. (2005) which study factors including survey cadence, object visibility,
and the complicating effects of microlensing. We note this estimate may be a
bit optimistic for PS1 with its slower sampling rate compared to LSST.
The expected redshift distributions of the lenses and sources can be roughly
approximated by the Gaussian distributions $z_{L}=0.5\pm 0.15$ and
$z_{S}=2.0\pm 0.75$ with $z_{S}>z_{L}$ (Fig. 1), as adopted by Dobke et al.
(2009). Obviously the two distributions will be correlated, but we approximate
them as being independent.
Figure 1.— Distributions of lens and source redshifts used in this paper.
These Gaussian distributions ($z_{L}=0.5\pm 0.15$, $z_{S}=2.0\pm 0.75$;
$z_{S}>z_{L}$) were used by Dobke et al. (2009) as reasonable approximations
for near-future missions including LSST.
As surveys attain fainter magnitude limits, it is believed that the
magnification bias enjoyed by quads will be diminished. Future surveys are
thus expected to yield lower quad fractions ($\sim 19\%,14\%$) than the
current sample of time delay lenses (6 / 16 = 37.5%). This might improve the
expected constraints on ${\mathcal{T}}_{\mathcal{C}}$ from future surveys as
quads have been shown to yield time delays with more scatter and thus less
reliable estimates of ${\mathcal{T}}_{\mathcal{C}}$ (Oguri, 2007, Paper
I).888This is believed to be due to the fact that some of the factors
(especially external shear) which cause scatter in $\Xi$ also raise the
likelihood that a lens will produce quad images rather than a double. However,
we find this to be mitigated by the fact that quads yield multiple time delay
measurements (one for each pair of images), while doubles only yield a single
$\Delta\tau$ measurement. Based on our detailed simulations and analysis
(Paper I), we find quads and doubles to have approximately equal power to
constrain ${\mathcal{T}}_{\mathcal{C}}$. This simplifies our analysis; the
quad-to-double ratio need not be considered when estimating
$\delta{\mathcal{T}}_{\mathcal{C}}$ for a given experiment. To allay any
concern, we stress that this assumption actually makes our estimates of
$\delta{\mathcal{T}}_{\mathcal{C}}$ more conservative for future surveys which
have lower quad fractions than the current sample.
For each double or quad, image pairs can be further classified by their
geometry. For example, image pairs with small opening angles are found to
yield larger scatter in $\Delta\tau$ (Oguri, 2007, Paper I). Detailed analyses
in these papers quantify these scatters, enabling a well-informed prior
$P(\Xi)$ to be placed on each image pair as a function of geometry. The
details are unimportant here though we have made use of the constraint this
analysis has put on ${\mathcal{T}}_{\mathcal{C}}$ (Oguri, 2007).
### 3.3. Photometric Redshift Uncertainties
Currently all lenses which have reliable time delay measurements also have
spectroscopic redshifts measured for both lenses and sources (e.g., Oguri,
2007). The telescope time required to obtain spectroscopic redshifts is
generally a small fraction of that required to obtain accurate time delays, so
the extra investment is worthwhile.
Future surveys which repeatedly scan the sky, however, will yield time delays
for many more lenses than may be followed up spectroscopically. For these
lenses we will have to rely on photometric redshift measurements. These
uncertainties will degrade the constraints possible on the cosmological
parameters.
Photometric redshift uncertainties for the lenses (typically elliptical
galaxies at $z_{L}\sim 0.5$) are expected to be $\Delta z_{L}\sim
0.04(1+z_{L})$, similar to that found in the CFHT Legacy Survey (Ilbert et
al., 2006). Redshift uncertainties for the lensed sources (quasars) are
expected to be somewhat higher. We will adopt $\Delta z_{S}\sim
0.10(1+z_{S})$, roughly that found in the analysis of $\sim$one million SDSS
quasars (Richards et al., 2009).
Obtaining photometric redshifts in ground-based images will often be
complicated by cross-contamination of flux among the lens and multiple images.
Yet improved photometric redshift techniques are also being developed with
LSST in mind (Schmidt et al., 2009), so it is perhaps too early to say whether
our estimated redshift uncertainties are too optimistic or pessimistic for a
future ground-based survey. Some of the most common catastrophic redshift
degeneracies can clearly be avoided by considering the observed image
separations, time delays, etc. Most obviously, the common degeneracy between
$z\sim 0.2$ and $z\sim 3$ (e.g., Coe et al., 2006) can be neatly averted since
a lens at $z\sim 3$ or a source at $z\sim 0.2$ would clearly stand out.
Assuming the above redshift uncertainties, we now determine how these
propagate into uncertainties on ${\mathcal{T}}_{\mathcal{C}}$. For simplicity,
let us assume that redshift uncertainties are Gaussian. Let us further assume
that uncertainty in $\Xi$ scales linearly with redshift uncertainty. (This is
approximately true for reasonable uncertainty levels $\Delta z\lesssim 0.2$.)
Using equations 7 – 13, we find for a typical lens-source combination with
$(z_{L},z_{S})=(0.5,2.0)$, that lens and source redshift uncertainties
translate to $\delta\Xi_{\mathcal{Z}_{L}}\sim 2.75\Delta z_{L}$ and
$\delta\Xi_{\mathcal{Z}_{S}}\sim-0.16\Delta z_{S}$, respectively. Given the
above redshift uncertainties, these evaluate to
$\delta\Xi_{\mathcal{Z}_{L}}\sim 0.16$ and $\delta\Xi_{\mathcal{Z}_{S}}\sim
0.05$. These relations are strong functions of redshift and become
catastrophic for sources very close to the lens. We plot this behavior in Fig.
2. If accurate and precise redshifts are not available, we must concentrate
our analysis on systems with high separation in redshift between the lens and
source.
For a lens ensemble with Gaussian redshift distributions $z_{L}=0.5\pm 0.15$
and $z_{S}=2.0\pm 0.75$, we find $\delta\Xi_{\mathcal{Z}_{L}}\sim 0.175$ and
$\delta\Xi_{\mathcal{Z}_{S}}\sim 0.028$. To calculate these uncertainties, we
sum the $\chi^{2}$ of individual lens-source combinations, weighting by the
probability $P_{i}$ of observing that combination:
$\frac{1}{\sigma^{2}}=\sum_{i}\frac{P_{i}}{\sigma_{i}^{2}}.$ (17)
Note that this sum naturally assigns more weight to more confident
measurements.
Assuming the lens and source redshift uncertainties can be added in
quadrature,
$\delta\Xi_{\mathcal{Z}}^{2}=\delta\Xi_{\mathcal{Z}_{L}}^{2}+\delta\Xi_{\mathcal{Z}_{S}}^{2},$
(18)
we find $\delta\Xi_{\mathcal{Z}}\sim 0.177$.
Figure 2.— Photometric redshift uncertainties’ contributions to cosmological
uncertainties in ${\mathcal{T}}_{\mathcal{C}}$. Left: Uncertainty in
${\mathcal{T}}_{\mathcal{C}}$ (grayscale and contours) from lens redshift
uncertainties of $0.04(1+z_{L})$, plotted as a function of lens redshift
$z_{L}$ and the lens-source redshift difference $z_{S}-z_{L}$. The dashed
contours show the redshift distribution (1- and 2-$\sigma$ contours) assumed
in this work. A dot at $(z_{L},z_{S})=(0.5,2.0)$ marks the center of the
distributions. Right: Same for source redshift uncertainties of
$0.10(1+z_{L})$. Note that the plots have different grayscales. For sources
close to the lens (small $z_{S}-z_{L}$), redshift uncertainties become
catastrophic yielding large $\delta{\mathcal{T}}_{\mathcal{C}}$. Lens redshift
uncertainties are also problematic at low $z_{L}$.
Of course, these are just estimates for large ensembles. In practice, redshift
probability distributions $P(z)$ for individual galaxies will be properly
folded into the $P({\mathcal{T}}_{\mathcal{C}})$ determinations. Biased
redshifts would yield biased ${\mathcal{T}}_{\mathcal{C}}$, the effects of
which we study in Paper III.
### 3.4. Projected Constraints from Large Surveys
We now calculate the total uncertainty $\delta{\mathcal{T}}_{\mathcal{C}}$
expected for large surveys with photometric redshifts. The combined lens model
and time delay uncertainties are $\delta\Xi_{{\mathcal{L}}\tau}\sim 0.344$,
based on extrapolation of the current empirical Oguri (2007) finding (§3.1).
We estimate uncertainties of $\delta\Xi_{\mathcal{Z}}\sim 0.177$ due to
redshift uncertainties of $\Delta z_{L}\sim 0.04(1+z_{L})$ and $\Delta
z_{S}\sim 0.10(1+z_{S})$ for the lenses and sources, respectively (§3.3).
The simplest estimate of the total uncertainty is to add these uncertainties
in quadrature:
$\delta\Xi^{2}=\delta\Xi_{{\mathcal{L}}\tau}^{2}+\delta\Xi_{\mathcal{Z}}^{2}.$
(19)
This yields $\delta\Xi\sim 0.387$.
To be more precise, all of the uncertainties should be added in quadrature for
each lens individually before combining them according to Eq. 17. Repeating
the analysis in this way, we find $\delta\Xi\sim 0.402$.
Thus we expect large surveys with photometric uncertainties given above to
yield $\delta{\mathcal{T}}_{\mathcal{C}}\sim 40\%/\sqrt{N}$. We project
$\delta{\mathcal{T}}_{\mathcal{C}}\sim 1.3\%$ for PS1 (1,000 lenses) and
$\delta{\mathcal{T}}_{\mathcal{C}}\sim 0.64\%$ for LSST (4,000 lenses).
Table 1 summarizes the progress we can expect to make in “Stages”
corresponding to those defined by the Dark Energy Task Force (DETF; Albrecht
et al. 2006, 2009): “Stage I” = current, “II” = ongoing, “III” = currently
proposed, “IV” = large new mission. Again, we stress these are estimates of
statistical uncertainties only. Large surveys are compared to dedicated
monitoring and detailed analysis of a smaller sample of lenses.
We might have made our analysis more sophisticated still, calculating
$\delta\Xi_{{\mathcal{L}}\tau}$, $\delta\Xi_{\mathcal{Z}_{L}}$, and
$\delta\Xi_{\mathcal{Z}_{S}}$ individually for each lens-source combination in
our ensemble. Lenses and sources at higher redshift, for example, will be
brighter and higher magnification cases on average, altering their
$\delta\Xi_{{\mathcal{L}}\tau}$ somewhat. The approximations made in our above
analysis should suffice for our purposes here.
Table 1Estimated Current and Future Constraints on ${\mathcal{T}}_{\mathcal{C}}$ Stage | Experiment | $N_{\rm L}$ | quads | $\Delta z$aaSpectroscopic or photometric redshift measurements. For the latter we assume $\Delta z_{L}=0.04(1+z_{L})$ and $\Delta z_{S}=0.10(1+z_{S})$. | $\Delta(\Delta\tau)$ | $\delta\Xi_{\mathcal{L}}$ | $\delta{\mathcal{T}}_{\mathcal{C}}$
---|---|---|---|---|---|---|---
I | current | 16 | 38% | spec | 2 days | $\cdots$ | 8.6%
II | Pan-STARRS 1 | 1,000 | 19% | phot | 2 days | $\cdots$ | 1.27%
IV | LSST | 4,000 | 14% | phot | 2 days | $\cdots$ | 0.64%
IV | OMEGA | 100 | 100% | spec | 0.1 day | 5% | 0.5%
IV | LSST + OMEGA | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.4%
Figure 3.— Constraints on $\delta{\mathcal{T}}_{\mathcal{C}}$ as a function
of ensemble size and observational uncertainties. The current ensemble has
time delays measured to roughly $\Delta(\Delta\tau)=2$ day precision and
spectroscopic redshifts measured for all lenses and sources. Future large
surveys (“quantity”) should have similar time delay precisions but photometric
redshifts measured for lenses ($\Delta z_{L}=0.04(1+z_{L})$) and sources
($\Delta z_{S}=0.10(1+z_{S})$). A dedicated campaign (“quality”) could in
principle obtain tight lens model constraints ($\delta\Xi_{\mathcal{L}}=5\%$)
with high-precision time delays ($\Delta(\Delta\tau)=0.1$ day) and
spectroscopic redshifts.
### 3.5. Quality vs. Quantity
Thus far we have assumed that detailed observations and analysis would not be
performed on the lenses. The alternative is to study fewer lenses in more
detail, reducing the uncertainties for each lens. In practice, we expect both
strategies to be pursued and the combined power of both analyses to place the
tightest possible constraints on ${\mathcal{T}}_{\mathcal{C}}$.
Moustakas et al. (2008) have designed a mission concept that would be
dedicated to monitoring a sample of four-image lenses, with the primary goal
of constraining fundamental properties of dark matter. This space-based
Observatory for Multi-Epoch Gravitational Lens Astrophysics (OMEGA) would
monitor 100 time delay lenses to achieve precise and accurate $\lesssim 0.1$
day time delay measurements. Supporting measurements would aim to reduce the
model uncertainty of each lens to 5% ($\delta\Xi_{\mathcal{L}}=0.05$) and thus
constrain ${\mathcal{T}}_{\mathcal{C}}$ to 5% with each lens, as claimed
recently for B1608+656 (Suyu et al., 2009a). These supporting measurements,
including velocity dispersion in the lens and characterization of the group
environment (see discussion in §6.2), would be carried out either with OMEGA
itself or though coordinated efforts by ground-based telescopes and JWST.
Spectroscopic redshifts would also be obtained for the 100 lens galaxies and
lensed quasars.
Lenses targeted by OMEGA will be quads, enabling measurements of time delay
ratios among the image pairs. This would provide constraints on the dark
matter substructure mass function (Keeton & Moustakas, 2009; Keeton, 2009,
Moustakas et al., in preparation).
Given lens models accurate to 5% for 100 galaxies, we might expect OMEGA to
yield $\delta{\mathcal{T}}_{\mathcal{C}}\sim 5\%/\sqrt{100}=0.5\%$. The time
delays would be measured with sufficient precision so as not to contribute
significantly to the total uncertainty in $\delta{\mathcal{T}}_{\mathcal{C}}$.
The multiple time delay measurements per lens (quad) also help reduce this
contribution. Based on the expected time delay distribution for a sample of
quads (Paper I), we estimate that $\Delta(\Delta\tau)=0.1$-day uncertainties
would inflate the ${\mathcal{T}}_{\mathcal{C}}$ uncertainty only to $\sim
0.515\%$.
If both LSST and OMEGA obtain their measurements of
${\mathcal{T}}_{\mathcal{C}}$ free of significant systematics, their combined
power could further reduce the uncertainty to
$\delta{\mathcal{T}}_{\mathcal{C}}\sim 0.4\%$.
## 4\. Dependence of ${\mathcal{T}}_{\mathcal{C}}$ on Cosmology
We expect LSST time delay lenses to constrain ${\mathcal{T}}_{\mathcal{C}}$ to
$\sim 0.64\%$. In this section we begin to explore how this “Stage IV”
constraint translates to constraints on cosmological parameters. We study the
dependence of ${\mathcal{T}}_{\mathcal{C}}$ on
$(h,\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a})$ for several
cosmologies as outlined in Table 2.
Table 2Cosmologies explored in this work
Cosmology | $h$ | $\Omega_{m}$ | $\Omega_{de}$ / $\Omega_{\Lambda}$aaWhen ${\rm w}_{0}=-1$ and ${\rm w}_{a}=0$, $\Omega_{de}=\Omega_{\Lambda}$, the cosmological constant. | $\Omega_{k}$ | ${\rm w}_{0}$ | ${\rm w}_{a}$ | SectionsbbIn §4 the ${\mathcal{T}}_{\mathcal{C}}$ dependencies are explored. In §5 additional priors are assumed and time delay constraints are compared to those from other methods. | Figures
---|---|---|---|---|---|---|---|---
Flat universe with cosmological constant | Free | $1-\Omega_{\Lambda}$ | Free ($\Omega_{\Lambda}$) | 0 | $-1$ | 0 | §4.1 | 4, 5
Curved universe with cosmological constant | Free | $1-(\Omega_{\Lambda}+\Omega_{k})$ | Free ($\Omega_{\Lambda}$) | Free | $-1$ | 0 | §4.2 | 6, 7
Flat universe with constant ${\rm w}$ccGiven this cosmology, we assume a Planck prior in §5.2. | Free | $1-\Omega_{de}$ | Free | 0 | Free | 0 | §4.3, §5.2 | 8, 9, 12
Flat universe with time-variable ${\rm w}$ | Free | $1-\Omega_{de}$ | Free | 0 | Free | Free | §4.4 | 10
General (curved with time-variable ${\rm w}$)ddGiven a general cosmology, in §5.3 we assume a prior of Planck + “Stage II” WL+SN+CL (see that section for details). | Free | $1-(\Omega_{de}+\Omega_{k})$ | Free | Free | Free | Free | §5.3 | 13, 14, 17
Note. — We consider six cosmological parameters of which five are independent
since $\Omega_{m}+\Omega_{de}+\Omega_{k}=1$.
### 4.1. Flat universe with a cosmological constant ($h$,
$\Omega_{\Lambda}=1-\Omega_{m}$)
First, we add a single free parameter $\Omega_{\Lambda}$ (in addition to $h$)
in considering a flat universe with a cosmological constant (${\rm w}=-1$).
Given $\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ from an ensemble with all
lenses at $z_{L}=0.5$ and all sources at $z_{S}=2.0$, we would obtain
confidence contours shown in Fig. 4.
The shape of these curves shifts somewhat as a function of $z_{L}$ and
$z_{S}$. Given an ensemble of lenses and sources with Gaussian redshift
distributions $z_{L}=0.5\pm 0.15$ and $z_{S}=2.0\pm 0.75$ as discussed above,
we begin to break the ($h,\Omega_{\Lambda}$) degeneracy (Table 5). Assuming a
flat universe, Stage IV time delays could provide independent evidence for
$\Omega_{\Lambda}>0$. Whether this remains interesting by Stage IV remains to
be seen. The constraints on $h$ are certainly tighter and would be improved by
the introduction of a prior on $\Omega_{\Lambda}$, which we defer until §5.
Figure 4.— Confidence contours (1- and 2-$\sigma$ colored bands) for ($h$,
$\Omega_{\Lambda}=1-\Omega_{m}$) given “Stage IV”
$\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ obtained from an ensemble with all
lenses and sources at $z_{L}$, $z_{S}$ = (0.5, 2.0). Here we assume a flat
universe with a cosmological constant (${\rm w}=-1$). Also plotted are
contours of constant
$\Xi_{\mathcal{C}}\equiv{\mathcal{T}}_{\mathcal{C}}/{\mathcal{T}}_{{\mathcal{C}}\rm,true}$,
where ${\mathcal{T}}_{{\mathcal{C}}\rm,true}\approx 0.99$ for the input
redshifts and cosmology. The input cosmology ($h,\Omega_{m},\Omega_{\Lambda}$)
= (0.7, 0.3, 0.7) is marked with dotted lines and a white dot.
Figure 5.— Confidence contours (1- and 2-$\sigma$ colored bands) for ($h$,
$\Omega_{\Lambda}=1-\Omega_{m}$) given
$\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ and assuming a flat universe with a
cosmological constant (${\rm w}=-1$). Each of the three fainter curves
corresponds to all lenses and sources at the same pair of redshifts: $z_{L}$,
$z_{S}$ = (0.65, 2.75), (0.5, 2.0), (0.35, 1.25), as marked. Next we consider
an ensemble of lenses and sources with Gaussian redshift distributions:
$z_{L}$, $z_{S}$ = ($0.5\pm 0.15$, $2.0\pm 0.75$). These yield the tighter
constraints (marked “ensemble”). The input cosmology
($h,\Omega_{m},\Omega_{\Lambda}$) = (0.7, 0.3, 0.7) is marked with a white
dot.
### 4.2. Curved universe with cosmological constant
($h,\Omega_{m},\Omega_{\Lambda},\Omega_{k}$)
If we relax the flatness parameter, adding another free parameter $\Omega_{m}$
(where curvature is determined by
$\Omega_{k}=1-(\Omega_{m}+\Omega_{\Lambda})$), we run into the degeneracy in
Fig. 6. Plotted as colored bands are the ($\Omega_{m}$, $\Omega_{\Lambda}$)
confidence contours assuming constant $h=0.7$ given
$\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ from an ensemble with all lenses
and sources at $z_{L}$, $z_{S}$ = (0.5, 2.0). As $h$ varies, these contours
move as shown.
An ensemble of lenses with a range of redshifts shrinks the confidence
contours somewhat, as we see in Fig. 7, though the strong
($h,\Omega_{m},\Omega_{\Lambda}$) degeneracy remains. Even adopting an
aggressive 3% prior on $h$, we find neither $\Omega_{m}$ nor
$\Omega_{\Lambda}$ can be constrained individually. However, the degeneracy
does exhibit a strong preference toward a flat or nearly flat universe.
Finally, we note the ($h,\Omega_{m},\Omega_{\Lambda}$) degeneracy can be more
cleanly broken if our ensemble includes a significant fraction of lenses at
$z_{L}=1$ and higher.
Figure 6.— Confidence contours (1- and 2-$\sigma$ colored bands) for
($\Omega_{m},\Omega_{\Lambda}$) given
$\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ obtained from an ensemble with all
lenses and sources at $z_{L}$, $z_{S}$ = (0.5, 2.0). The colored bands shift
in ($\Omega_{m},\Omega_{\Lambda}$) space as $h$ varies. A cosmological
constant (${\rm w}=-1$) is assumed. The input cosmology
($h,\Omega_{m},\Omega_{\Lambda}$) = (0.7, 0.3, 0.7) is marked with a white
dot. Flat cosmologies lie along the dotted line, and this line’s intersection
with the colored bands explains the strange shape of the colored bands in the
previous plot.
Figure 7.— Additional confidence contours for
($\Omega_{m},\Omega_{\Lambda}$). The middle set of contours was plotted in the
previous figure. The top set of contours assumes an ensemble of lenses and
sources $z_{L}$, $z_{S}$ = ($0.5\pm 0.15$, $2.0\pm 0.75$). Finally, the bottom
set of contours is for the ensemble and allowing a 3% uncertainty in $h$.
### 4.3. Flat universe with constant dark energy EOS
($h,\Omega_{de}=1-\Omega_{m},{\rm w}$)
Current cosmological constraints are consistent with a flat universe with a
cosmological constant (as explored in §4.1). As a first perturbation to this
model, it is common to explore constraints on ${\rm w}\neq-1$ while
maintaining constant ${\rm w}$ in a flat universe. This cosmology has three
free parameters ($h,\Omega_{de},{\rm w}$) with $\Omega_{m}=1-\Omega_{de}$.
Given enough data and appropriate priors, time delay lenses could place strong
constraints on the dark energy equation of state parameter ${\rm w}$ (see
§5.2). Figs. 8 and 9 explore the dependence of ${\mathcal{T}}_{\mathcal{C}}$
on $({\rm w},\Omega_{de})$ assuming a flat universe and constant ${\rm w}$.
Figure 8.— Confidence contours (1- and 2-$\sigma$ colored bands) for (${\rm
w},\Omega_{de}=1-\Omega_{m}$) assuming a flat universe with constant ${\rm w}$
given $\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ obtained from an ensemble
with all lenses and sources at $z_{L}$, $z_{S}$ = (0.5, 2.0). The colored
bands shift in (${\rm w},\Omega_{de}$) space as $h$ varies. The input
cosmology ($h,\Omega_{de},{\rm w}$) = (0.7, 0.7, -1) is marked with a white
dot. Figure 9.— Confidence contours for (${\rm w}$,
$\Omega_{de}=1-\Omega_{m}$), assuming a flat universe. As in Fig. 7, we plot a
“Stage IV” ensemble of lenses at a range of redshifts, the lenses all at the
same redshift, and the ensemble allowing 3% uncertainty in $h$.
### 4.4. Flat universe with time-variable dark energy EOS
($h,\Omega_{de}=1-\Omega_{m},{\rm w}_{0},{\rm w}_{a}$)
The most interesting constraints we can hope to place on dark energy are to
verify or falsify the following: ${\rm w}=-1$ (cosmological constant) and
${\rm w}_{a}=0$ (constant ${\rm w}$). In Fig. 10 we explore the dependence of
${\mathcal{T}}_{\mathcal{C}}$ on $({\rm w}_{0},{\rm w}_{a})$ (see Eq. 10). The
colored bands are the constraints we could obtain given perfect knowledge of
($h,\Omega_{m},\Omega_{de}$). The solid lines on the left show the curves’
migration as a function of $h$. On the right, we also explore dependence on
$\Omega_{de}$ for a flat universe ($\Omega_{m}+\Omega_{de}=1$).
Figure 10.— Left: Confidence contours (1- and 2-$\sigma$ colored bands) for
(${\rm w}_{0},{\rm w}_{a}$) given $\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$
obtained from an ensemble with all lenses and sources at $z_{L}$, $z_{S}$ =
(0.5, 2.0) and assuming $h=0.7$ and perfect knowledge of
($\Omega_{m},\Omega_{de}$). As shown, these bands shift in (${\rm w}_{0},{\rm
w}_{a}$) space as $h$ varies. The input cosmology ($h,{\rm w}_{0},{\rm
w}_{a}$) = (0.7, -1, 0) is marked with a white dot. Right: Dependence of the
(${\rm w}_{0},{\rm w}_{a}$) contours on $\Omega_{de}$, assuming a flat
cosmology. Dashed lines show the $h$ dependence from the left plot. Solid
lines of increasing thickness show contours of $\Omega_{de}$ decreasing in 0.1
increments.
## 5\. Cosmological Constraints from Future Experiments
We now consider the full parameter space
$(h,\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a})$ and derive the
constraints that may be placed on these parameters given constraints on
${\mathcal{T}}_{\mathcal{C}}$ along with various priors. Stage IV time delay
constraints are compared to those expected from other experiments as estimated
by the Dark Energy Task Force (Albrecht et al., 2006, 2009). To efficiently
explore this parameter space, we perform Fisher matrix analyses.
### 5.1. Fisher Matrix Analysis
The Fisher matrix formalism provides a simple way to study uncertainties of
many correlated parameters. Constraints from various experiments and/or
specific priors may be combined with ease. A “quick-start” instructional guide
and software are provided in a companion paper (Coe, 2009). Fisher matrices
approximate all uncertainties as Gaussians. The true uncertainties may be
somewhat higher and non-Gaussian. The full information of the dependencies as
shown in §4 is not retained. Yet as cosmological parameters are constrained
close to their true values, these approximations should suffice.
As above we consider a “Stage IV” ensemble of time delays which constrains
${\mathcal{T}}_{\mathcal{C}}$ to 0.64% with Gaussian distributions of lens and
source redshifts ($z_{L}=0.5\pm 0.15$; $z_{S}=2.0\pm 0.75$). Assuming such a
Gaussian distribution for ${\mathcal{T}}_{\mathcal{C}}$ and the aforementioned
redshift ensemble, we calculate (numerically) the Fisher matrix for
cosmological parameters of interest. The Fisher matrix consists of partial
derivatives of $\chi^{2}$ with respect to the parameters. For parameters
($p_{i},p_{j}$), element ($i,j$) in the Fisher matrix is given by
$F_{ij}=\frac{1}{2}\frac{\partial\chi^{2}}{\partial p_{i}\partial p_{j}}.$
(20)
The Stage IV ($\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$) time delay Fisher
matrix is given in Table 3 for the cosmological parameters
$(h,\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a})$. The Fisher matrix may be
easily scaled to other $\delta{\mathcal{T}}_{\mathcal{C}}$ values. For
example, to scale from LSST (4,000 lenses;
$\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$) to Pan-STARRS 1 (1,000 lenses;
$\delta{\mathcal{T}}_{\mathcal{C}}=1.27\%$), simply divide all the values in
the Fisher matrix by $4=4,000/1,000=(1.27/0.64)^{2}$. Or multiply them by
$1.6=(0.64/0.4)^{2}$ to explore the LSST + OMEGA constraints
($\delta{\mathcal{T}}_{\mathcal{C}}=0.4\%$). If one is interested in
constraints on $\Omega_{m}=1-(\Omega_{de}+\Omega_{k})$,
$\omega_{m}\equiv\Omega_{m}h^{2}$, or any other related variable, a
transformation of variables can be performed as outlined in Coe (2009).
Table 3Stage IV Fisher matrix expectation for $(h,\Omega_{de},\Omega_{k},{\rm
w}_{0},{\rm w}_{a})$
given $\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ | $h$ | $\Omega_{de}$ | $\Omega_{k}$ | ${\rm w}_{0}$ | ${\rm w}_{a}$
---|---|---|---|---|---
$h$ | 49824.9224 | -1829.7018 | -4434.2995 | 4546.8899 | 122.5319
$\Omega_{de}$ | -1829.7018 | 88.3760 | 200.9795 | -189.2658 | -8.4386
$\Omega_{k}$ | -4434.2995 | 200.9795 | 463.5732 | -445.5690 | -17.9694
${\rm w}_{0}$ | 4546.8899 | -189.2658 | -445.5690 | 441.9725 | 15.2981
${\rm w}_{a}$ | 122.5319 | -8.4386 | -17.9694 | 15.2981 | 1.0394
In Fig. 11 we show the time delay constraints possible on all parameters and
pairs of parameters assuming perfect knowledge of all the other parameters.
These plots can be compared to those presented in §4. Such perfect priors are
unrealistic, but they help to demonstrate the parameter dependencies and
degeneracies.
Figure 11.— Constraints placed on pairs of parameters derived from our Fisher
matrix analysis assuming perfect knowledge of all other parameters given
$\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ obtained from an ensemble with
Gaussian distributions of lens and source redshifts ($z_{L}=0.5\pm 0.15$;
$z_{S}=2.0\pm 0.75$). All off-diagonal plots show 1- and 2-$\sigma$ colored
ellipses. Along the diagonal are constraints on individual parameters assuming
perfect knowledge of all others. The y axes along the diagonal are units of
relative probability, different from the off-diagonal plots.
### 5.2. Flat universe with constant ${\rm w}$
We first consider the simple case of a flat universe with constant ${\rm w}$.
This is a common perturbation to the concordance cosmology. The goal is to
detect deviation from ${\rm w}=-1$, equivalent to the cosmological constant
$\Lambda$. This 3-parameter cosmology ($h,\Omega_{de},{\rm w}$, with
$\Omega_{m}=1-\Omega_{de}$) was explored above in §4.3.
The top row of Fig. 12 shows Stage IV time delay constraints with a Planck
prior in a flat universe with constant ${\rm w}$. Given these priors, we
estimate that time delays will constrain $h$ to 0.007 ($\sim 1\%$),
$\Omega_{de}$ to 0.005, and ${\rm w}$ to 0.026 (all 1-$\sigma$ precisions).
In the bottom row of Fig. 12, we compare these time delay constraints (TD) to
those expected from other methods: weak lensing (WL), baryon acoustic
oscillations (BAO), supernovae (SN), and cluster counts (CL). We consider
“optimistic Stage IV” expectations from these methods as calculated by the
Dark Energy Task Force (DETF; Albrecht et al. 2006, 2009) and made available
in the software DETFast999http://www.physics.ucdavis.edu/DETFast/. A Planck
prior (also calculated by the DETF) is again assumed for all experiments.
In manipulating the DETF Fisher matrices we adopt their cosmology
($\Omega_{m},\Omega_{de},h$) = (0.27, 0.73, 0.72), but we revert to our chosen
cosmology ($\Omega_{m},\Omega_{de},h$) = (0.3, 0.7, 0.7) for the rest of our
analysis. These differences have negligible impact on our results.
Figure 12.— Top row: Cosmological constraints from “Stage IV” time delays
plus a Planck prior in a flat universe with constant ${\rm w}$. We assume an
ensemble of time delays which constrains ${\mathcal{T}}_{\mathcal{C}}$ to
0.64% (see text for details). Time delays plus Planck constrain $h$ to 0.007
(1%), $\Omega_{de}$ to 0.005, and ${\rm w}$ to 0.026 (all 1-$\sigma$
precisions). Bottom row: Comparison of “optimistic Stage IV” constraints
expected from time delays (TD), weak lensing (WL), supernovae (SN), baryon
acoustic oscillations (BAO), and cluster counts (CL). The time delay
constraints are as plotted in the top row. For the other experiments we use
Fisher matrix calculations provided by the Dark Energy Task Force (DETF). For
each parameter pair, experiments are plotted in order of ${\rm
FOM}\propto({\rm Ellipse~{}Area})^{-1}$, with the best experiment on top.
Lewis & Ibata (2002) considered similar constraints from time delay lenses but
found much weaker constraints on ($h,{\rm w}$), even with all other
cosmological parameters fixed. One of the cases they considered was 500 lenses
with 15% uncertainty each, which translates to $15\%/\sqrt{500}=0.66\%$ total
uncertainty, very similar to the 0.64% uncertainty we estimate for LSST given
4,000 lenses with a much higher uncertainty (effectively 40%) assumed per
lens. For this case, they find $0.99\lesssim h\lesssim 1.10$ and
$-1.48\lesssim{\rm w}\lesssim-0.88$ (95% confidence). When we perform a
similar analysis, assuming $\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ and
perfect knowledge of ($\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{a}$), we
obtain similar uncertainties (without biases, by construction): $h=0.7\pm
0.02$ and ${\rm w}=-1\pm 0.21$ (1-$\sigma$). But with the addition of a Planck
prior, even while relaxing the perfect prior on
($\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{a}$), we find improved
constraints of $h=0.7\pm 0.007$ and ${\rm w}=-1\pm 0.026$ (1-$\sigma$). Planck
clearly complements the strong lensing constraints well to produce tight
constraints on ($h$, ${\rm w}$).
### 5.3. General Cosmology
We now assume a general cosmology allowing for curvature and a time-varying
${\rm w}$. To help constrain this larger parameter space
($h,\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a}$, with
$\Omega_{m}=1-(\Omega_{de}+\Omega_{k})$), we add additional priors. In
addition to the Planck prior, we adopt “Stage II” (near-future) constraints
from weak lensing (WL) + supernovae (SN) + cluster counts (CL), all as
calculated by the DETF. The DETF uses this prior (in addition to Planck) in
many of their calculations comparing the performance of Stage III – IV
techniques.
The Stage II DETF WL + SN + CL prior yields the following uncertainties:
$\Delta h=0.031$ (4.4%), $\Delta\Omega_{de}=0.023$, $\Delta\Omega_{k}=0.010$,
$\Delta{\rm w}_{0}=0.128$, $\Delta{\rm w}_{a}=0.767$ (along with various
covariances between parameters). The addition of the Planck prior reduces
these to: $\Delta h=0.017$ (2.4%), $\Delta\Omega_{de}=0.012$,
$\Delta\Omega_{k}=0.003$, $\Delta{\rm w}_{0}=0.115$, $\Delta{\rm
w}_{a}=0.525$. Note that Stage II WL+SN+CL constrains $h$ well enough (to
4.4%) that an HST Key Project prior ($h=0.72\pm 0.08$) appears to be
unnecessary. Even SHOES ($h=0.742\pm 0.036$, or 4.9%) provides a weaker
constraint on $h$. However, as noted in the introduction, these combined
WL+SN+CL experiments yield a prediction of $h$ based on an assumed
cosmological model and are no substitute for local measurements of $h$ (Riess
et al., 2009).
These Stage II constraints are also rather optimistically combined, assuming
that all experiments have converged on the same best fit cosmology without
systematic offsets among them. The true Stage II constraints should be
somewhat weaker.
Plotted in Fig. 13 are time delay constraints assuming a prior of Planck +
Stage II WL+SN+CL. A progression is shown from Stage I (present) time delay
constraints ($\delta{\mathcal{T}}_{\mathcal{C}}=8.6\%$) through Stage II
($\delta{\mathcal{T}}_{\mathcal{C}}=1.27\%$) and on to Stage IV
($\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$). The current constraints barely
improve upon this aggressive prior. While the Stage II – IV constraints
certainly improve upon the prior, note that the outer bounds of the time delay
and prior ellipses nearly intersect. This indicates that the size of the time
delay ellipse is controlled by that of the prior, at least for these
constraints and prior. Were the prior significantly weaker or the time delay
constraints significantly stronger, we have verified that the time delay
ellipses would shrink well within the prior ellipses.
Figure 13.— Cosmological constraints from time delays in a general cosmology
assuming priors of Planck + Stage II (WL+SN+CL) as calculated by the DETF. A
progression is shown from the prior (outermost ellipse, 2-$\sigma$) to Stage I
(current) time delay constraints ($\delta{\mathcal{T}}_{\mathcal{C}}=8.6\%$;
gray ellipse, 2-$\sigma$) to Stage II constraints
($\delta{\mathcal{T}}_{\mathcal{C}}=1.4\%$; black ellipse, 2-$\sigma$) to
Stage IV constraints ($\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$; colored
ellipses, 1-$\sigma$ and 2-$\sigma$). Along the diagonal are plotted
constraints on individual parameters marginalizing over all others. The y axes
along the diagonal are units of relative probability, different from the off-
diagonal plots. For each plot, we marginalize over all other parameters,
unlike Fig. 11 in which unrealistic perfect priors were assumed for
illustrative purposes.
In Fig. 14 we compare Stage IV time delay constraints to those expected from
other methods for various parameters of interest. Plotted are constraints on
($h,\Omega_{k}$), ($h,{\rm w}_{0}$), and (${\rm w}_{0},\Omega_{k}$), and
(${\rm w}_{0},{\rm w}_{a}$). An example of how these constraints combine is
given in §5.4.2.
Figure 14.— Comparisons of “Stage IV” constraints possible from time delays
(TD), weak lensing (WL), supernovae (SN), baryon acoustic oscillations (BAO),
and cluster counts (CL) in a general cosmology (allowing for curvature and a
time-variable ${\rm w}$). For TD, we assume an ensemble which constrains
${\mathcal{T}}_{\mathcal{C}}$ to 0.64% (see text for details). For the rest we
use “optimistic Stage IV” expectations calculated from Fisher matrices
provided by the Dark Energy Task Force (DETF). A prior of Planck + Stage II
(WL+SN+CL) is assumed for all five experiments and is plotted in gray. For
each parameter pair, experiments are plotted in order of ${\rm
FOM}\propto({\rm Ellipse~{}Area})^{-1}$, with the best experiment on top.
We give extra attention to constraints on the dark energy parameters (${\rm
w}_{0},{\rm w}_{a}$). The DETF figure of merit (FOM) for a given experiment is
defined as the inverse of the area of the ellipse in the (${\rm w}_{0},{\rm
w}_{a}$) plane. In Fig. 15 we plot FOM for various experiments versus the
“pivot redshift”, defined as follows. For a time-varying ${\rm w}(z)$, time
delays constrain ${\rm w}$ best at $z\approx 0.31$. This redshift is known as
the pivot redshift (Huterer & Turner, 2001; Hu & Jain, 2004) and can also be
calculated simply from the (${\rm w}_{0},{\rm w}_{a}$) constraints (Coe,
2009). As in the previous plot, we assume a prior of Planck + Stage II
(WL+SN+CL).
Figure 15.— Dark energy figure of merit (${\rm FOM}\propto\left(({\rm
w}_{0},{\rm w}_{a})~{}{\rm Ellipse~{}Area}\right)^{-1}$, normalized relative
to the prior) versus pivot redshift for various “optimistic Stage IV”
experiments with a prior of Planck + Stage II (WL+SN+CL) The pivot redshift is
the redshift at which ${\rm w}(z)$ is best constrained.
### 5.4. Time delays do not simply constrain $h$
#### 5.4.1 Relaxing the “perfect prior” on
$(\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a})$
To date, analyses of time delay lenses have quoted uncertainties on
${\mathcal{T}}_{\mathcal{C}}$ as uncertainties on $h$, assuming $\delta
h=\delta{\mathcal{T}}_{\mathcal{C}}$. This assumption has been valid to date,
but future constraints on $h$ will be weaker than the constraints on
${\mathcal{T}}_{\mathcal{C}}$, that is $\delta
h>\delta{\mathcal{T}}_{\mathcal{C}}$.
This is demonstrated in Fig. 16 left. The dashed line shows $\delta
h=\delta{\mathcal{T}}_{\mathcal{C}}$, or the “perfect prior” on
($\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a}$) generally
assumed in analyses. For future samples (at the left side of the plot), as
this prior is loosened, we find $\delta h>\delta{\mathcal{T}}_{\mathcal{C}}$.
In Fig. 16 right, we plot $\delta h/\delta{\mathcal{T}}_{\mathcal{C}}$. For
example, given a “Stage II” prior on WL+SN+CL, and LSST constraints on time
delays ($\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$), we find $\delta h\sim
2.2\delta{\mathcal{T}}_{\mathcal{C}}\sim 1.4\%$. Alternatively, assuming a
Planck prior in a flat universe with constant ${\rm w}$, we would find $\delta
h\sim 1.4\delta{\mathcal{T}}_{\mathcal{C}}\sim 0.90\%$.
(Note that the Stage II WL+SN+CL prior claims a constraint of $\delta h=0.03$,
such that it outperforms current constraints from time delays $\delta
h=\delta{\mathcal{T}}_{\mathcal{C}}$.)
Figure 16.— Demonstration that $\delta h>\delta{\mathcal{T}}_{\mathcal{C}}$
for future ensembles. Left: Constraints on $h$ versus constraints on
${\mathcal{T}}_{\mathcal{C}}$ for various priors. Along the top horizontal
axis we plot experiments with corresponding
$\delta{\mathcal{T}}_{\mathcal{C}}$: current constraints (8.6%), Pan-STARRS 1
(1.27%), LSST (0.64%), OMEGA (0.5%), and LSST + OMEGA (0.4%). The priors are
different combinations of the following: Planck, a flat universe, constant
${\rm w}$, and a “Stage II” prior from (WL+SN+CL). This Stage II prior
constrains $\Omega_{k}$ to 0.01, so the additional prior of flatness helps it
little here. The bottom line is the “perfect prior”, perfect knowledge of
($\Omega_{de},\Omega_{m},\Omega_{k},{\rm w}_{0},{\rm w}_{a}$) as is generally
assumed, for which $\delta{\mathcal{T}}_{\mathcal{C}}=\delta h$. Right:
Relative constraints on $h$ compared to the perfect prior. For example, given
the Stage II prior, we find $\delta h\sim
2.2\delta{\mathcal{T}}_{\mathcal{C}}$.
#### 5.4.2 Time delays provide more than constraints on $h$
In the introduction we commented on the ability of any experiment to improve
constraints on ${\rm w}$ and $\Omega_{k}$ simply by tightening the constraints
on $h$. Several methods have the potential to further improve the constraints
on $h$ (Olling, 2007). Do time delays offer more than a simple constraint on
$h$ for the purposes of constraining the dark energy equation of state?
In Fig. 17 we compare Stage IV time delays (left) to a simple $h$ constraint
(right) in ability to constrain dark energy. Each is combined with Stage IV
supernovae constraints plus a prior of Planck + Stage II
WL+SN+CL.101010Strictly speaking we have not taken the proper care in
combining constraints from the Stage II and Stage IV supernova experiments, as
their nuisance parameters have been marginalized over in the DETF Fisher
matrices. But this analysis will suffice for illustrative purposes here. We
find time delays are more powerful than the simple $h$ constraint. The (SN +
TD + prior) figure of merit (FOM) on (${\rm w}_{0},{\rm w}_{a}$) is $\sim
19\%$ higher than that from (SN + H + prior).
The “H” constraint $\delta h=0.009$ was chosen such that when combined with
the prior, the resulting $\delta h$ would equal that from TD + prior. Both H +
prior and TD + prior yield $\delta h=0.008$. However we find TD outperforms
even a perfect H prior ($\delta h\sim 0$) by 13%. Simply put, the time delay
constraints on ($\Omega_{m},\Omega_{de},\Omega_{k},{\rm w}_{0},{\rm w}_{a}$)
are clearly making contributions.
When combined with experiments other than SN, TD offers less marked
improvements over H constraints. Replacing SN with BAO, WL, and CL, we find TD
outperforms H by 7%, 5%, and 3%, respectively.
Figure 17.— Left: Combined constraints on (${\rm w}_{0},{\rm w}_{a}$) from
Stage IV time delays (TD) and supernovae (SN). A prior of Planck + Stage II
(WL+SN+CL) is assumed. The TD + prior constraint yields $\delta h=0.008$ (not
shown). Right: Similar plot combining Stage IV SN with a $\delta h=0.009$
constraint on Hubble’s constant (that which also yields $\delta h=0.008$ when
combined with the prior). Time delays yield a 19% improvement in figure of
merit (${\rm FOM}\propto\left(({\rm w}_{0},{\rm w}_{a})~{}{\rm
Ellipse~{}Area}\right)^{-1}$), versus the constraint on $h$ alone. SN + TD
shows the most dramatic such improvement vs. SN + H. Replacing SN with the
other experiments (BAO, WL, CL) we find lesser improvements vs. H of 7%, 5%,
and 3%, respectively.
### 5.5. Lens and Source Redshift Distribution
We have been considering the Gaussian redshift distributions $z_{L}=0.5\pm
0.15$, $z_{S}=2.0\pm 0.75$ introduced by Dobke et al. (2009) as reasonable
approximate assumptions for near-future missions. We find that the
cosmological parameter constraints are not extremely sensitive to variations
in these redshift distributions.
For $\delta{\mathcal{T}}_{\mathcal{C}}=0.64\%$ plus our Planck + Stage II
(WL+SN+CL) prior, we find the following. A lower tighter lens redshift
distribution of $z_{L}=0.2\pm 0.1$ improves the constraint on $h$ by 22% and
on $\Omega_{de}$ by 12% at the expense of the ${\rm w}_{0}$ and ${\rm w}_{a}$
constraints, which degrade by 8% and 10%, respectively. A higher tighter lens
redshift distribution of $z_{L}=1.0\pm 0.1$ has less leverage, as the $h$ and
$\Omega_{de}$ constraints degrade by 15% and 14%, respectively with little
benefit to the other parameters. Neither broader lens redshift distributions
nor variations on the source redshift distribution have much impact on the
parameter constraints.
When time delay constraints are tighter
($\delta{\mathcal{T}}_{\mathcal{C}}<0.64\%$), with the same priors, the lens
redshift distribution begins to have a greater impact. We reserve study of
such “beyond Stage IV” constraints for future work.
## 6\. Systematics
As with any measurement, there are many potential sources of systematic bias,
as alluded to throughout this work. At the risk of putting the cart before the
horse, we have presented systematic-free projections for time delay
cosmological constraints. These should serve to motivate a more considered
look at systematics, in the context of the behavior of random uncertainties in
these studies. Ideally, efforts should be undertaken to reduce systematics on
a timescale comparable to that presented here (e.g., 0.64% by “Stage IV”). If
this cannot be accomplished, we study prospects for estimating cosmological
parameters in spite of large residual systematic biases in Paper III (Coe &
Moustakas, 2009c).
Here we discuss briefly the greatest potential sources of systematic bias. We
should consider which of our main sources of statistical uncertainty (lens
modelling, redshift measurements, and time delay measurements) could also
contribute significant systematic bias. Time delay uncertainties are generally
not expected to be biased in any preferred direction. Redshift biases are
somewhat worrisome but will not be discussed further here. Most daunting are
potential biases due to imprecise lens modeling.
Whether we determine the appropriate lens model for the “typical” (“average”)
lens in an ensemble or we constrain each individual lens model well, we must
use the following tools to measure lens properties. The largest statistical
uncertainties and potential systematic biases involve measurements of the lens
mass density slope and perturbing mass sheets.
### 6.1. Lens Mass Density Slope
Regarding mass slope, this paper has focused on the statistical strategy which
assumes that we know the correct mean of mass slopes. Evidence currently
suggests that lenses are isothermal ($\alpha=1$, $\gamma=2$)111111We use two
definitions common in the literature regarding lens slope: two-dimensional
mass surface density $\kappa\propto r^{2-\alpha}$, and three-dimensional mass
surface density $\rho\propto r^{-\gamma}$. These parameters are related by
$\alpha+\gamma\approx 3$ (see discussion in van de Ven et al., 2009). on
average. Yet a recent analysis of 58 SLACS lenses finds a slightly higher
average slope of $\gamma=2.085^{+0.025}_{-0.018}({\rm stat.})\pm 0.1({\rm
syst.})$ (Koopmans et al., 2009b). If the average proved to be exactly
$\gamma=2.085$, this would result in an 8.5% bias in
${\mathcal{T}}_{\mathcal{C}}$
($\delta{\mathcal{T}}_{\mathcal{C}}=\delta\gamma/2=\delta\alpha$) were we to
assume an average of $\gamma=2$ instead.
Mass profile slopes for individual lenses are determined by measuring mass
within two radii: the Einstein radius (from the positions of multiple images)
and a smaller radius (from velocity dispersions). The latter require detailed
spectroscopy (e.g., Koopmans et al., 2006). It will not be feasible to obtain
the required measurements for all time delay lenses detected in future
surveys, but small samples of these can be selected for such detailed study.
### 6.2. Mass Sheets
Mass sheets can be equally harmful as a source of systematics as
${\mathcal{T}}_{\mathcal{C}}$ bias also scales linearly with projected mass
density, $\delta{\mathcal{T}}_{\mathcal{C}}\sim\kappa$. Mass sheets can result
from both mass within the lens group environment and mass along the line of
sight (over- or under-densities) all the way from source to observer. The
former is the dominant effect. Simulations (Dalal & Watson, 2005) suggest that
group members contribute $\kappa_{\rm env}=0.03\pm 0.6$ dex (i.e.,
$\log_{10}(\kappa_{\rm env})=\log_{10}(0.03)\pm 0.6$) for a 1-$\sigma$ upper
bound of $\kappa_{\rm env}=0.12$, or 12% bias on
${\mathcal{T}}_{\mathcal{C}}$. Mass along the line of sight is generally lower
and more nearly fluctuates about the cosmic average but should also be
accounted for. Hilbert et al. (2007) measured mass along the lines of sight to
strong lenses in the Millennium simulation. For sources at $z_{S}=2$, the
central 68% span $-0.0355<\kappa_{\rm los}<0.0475$ (Paper I).
Efforts are made to measure $\kappa_{\rm env}$ for individual lenses via
spectroscopic (and photometric) studies (e.g., Momcheva et al., 2006; Auger,
2008) and simulations which estimate the effects of nearby neighbors (e.g.,
Keeton & Zabludoff, 2004; Dalal & Watson, 2005). Similar studies also attempt
to identify groups along the line of sight and estimate their mass sheet
contributions (e.g., Fassnacht et al., 2006).
The alternative is a statistical approach. Measurements of $\kappa_{\rm env}$
or $\kappa_{\rm los}$ would not be required for individual lenses if we had
knowledge of the distributions $P(\kappa_{\rm env})$ and $P(\kappa_{\rm los})$
for strong lenses. These distributions could be obtained from simulations, and
one could attempt to correct for the expected bias for lenses to reside in
high density regions (Dalal & Watson, 2005; Oguri et al., 2005). However, one
might wonder whether these distributions and corrections would prove accurate
to the percent level. Any errors would yield residual systematics in our
estimation of ${\mathcal{T}}_{\mathcal{C}}$.
To aid such a statistical approach, lenses in obvious groups can be excluded
from the analysis leaving only those systems with low $\kappa_{\rm env}$. Such
low mass systems would introduce smaller biases, though a detailed exploration
of this approach will await future work.
## 7\. Conclusions
We have presented the first analysis of the potential of gravitational lens
time delays to constrain a broad range of cosmological parameters. The
cosmological constraining power $\delta{\mathcal{T}}_{\mathcal{C}}$ was
calculated for Pan-STARRS 1, LSST, and OMEGA based on expected numbers of
lenses (including the quad-to-double ratio) as well as the expected
uncertainties in lens models, photometric redshifts, and time delays. Our
Fisher matrix results are provided to allow time delay constraints to be
easily combined with and compared to constraints from other methods.
We concentrate on “Stage IV” constraints from LSST. In a flat universe with
constant ${\rm w}$ including a Planck prior, LSST time delay measurements for
$\sim 4,000$ lenses should constrain $h$ to $\sim 0.007$ ($\sim 1\%$),
$\Omega_{de}$ to $\sim 0.005$, and ${\rm w}$ to $\sim 0.026$ (all 1-$\sigma$
precisions). We compare these results as well as those for a general cosmology
to other “optimistic Stage IV” constraints expected from weak lensing,
supernovae, baryon acoustic oscillations, and cluster counts, as calculated by
the Dark Energy Task Force (DETF).
Combined with appropriate priors (those adopted by the DETF), time delays
provide modest constraints on a time-varying ${\rm w}(z)$ that complement the
constraints expected from other methods. Time delays constrain ${\rm w}$ best
at $z\approx 0.31$, the “pivot redshift” for this method.
We find that LSST and OMEGA represent about an even trade in “quantity versus
quality” in terms of constraining cosmology with time delays. LSST could yield
$\delta{\mathcal{T}}_{\mathcal{C}}\sim 0.64\%$ by measuring time delays for
4,000 lenses, while OMEGA could yield $\delta{\mathcal{T}}_{\mathcal{C}}\sim
0.5\%$ by obtaining high-precision time delay measurements and lens model
constraints for 100 lenses with spectroscopic redshifts. The combined
statistical power of these two missions could further improve the cosmological
constraints to $\delta{\mathcal{T}}_{\mathcal{C}}\sim 0.4\%$.
We acknowledge useful conversations with Phil Marshall, Matt Auger, Chuck
Keeton, Chris Kochanek, Ben Dobke, Chris Fassnacht, Lloyd Knox, Jason Dick,
Andreas Albrecht, Tony Tyson, and Jason Rhodes. We are grateful to the DETF
for releasing Fisher matrices detailing their estimates of cosmological
constraints from various experiments. We thank the referee for useful comments
which led us to significantly improve the manuscript. This work was carried
out at Jet Propulsion Laboratory, California Institute of Technology, under a
contract with NASA.
## References
* Albrecht et al. (2009) Albrecht, A., Amendola, L., Bernstein, G., Clowe, D., Eisenstein, D., Guzzo, L., Hirata, C., Huterer, D., et al. 2009, ArXiv e-prints [ADS]
* Albrecht et al. (2006) Albrecht, A., Bernstein, G., Cahn, R., Freedman, W. L., Hewitt, J., Hu, W., Huth, J., Kamionkowski, M., et al. 2006, ArXiv Astrophysics e-prints [ADS]
* Auger (2008) Auger, M. W. 2008, MNRAS, 383, L40 [ADS]
* Blake et al. (2004) Blake, C. A., Abdalla, F. B., Bridle, S. L., & Rawlings, S. 2004, New Astronomy Review, 48, 1063 [ADS]
* Blandford & Narayan (1986) Blandford, R. & Narayan, R. 1986, ApJ, 310, 568 [ADS]
* Bolton et al. (2006) Bolton, A. S., Burles, S., Koopmans, L. V. E., Treu, T., & Moustakas, L. A. 2006, ApJ, 638, 703 [ADS]
* Braatz (2009) Braatz, J. 2009, Astronomy, 2010, 23 [ADS]
* Braatz et al. (2008) Braatz, J. A., Reid, M. J., Greenhill, L. J., Condon, J. J., Lo, K. Y., Henkel, C., Gugliucci, N. E., & Hao, L. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 395, Frontiers of Astrophysics: A Celebration of NRAO’s 50th Anniversary, ed. A. H. Bridle, J. J. Condon, & G. C. Hunt, 103–+ [ADS]
* Cabanac et al. (2007) Cabanac, R. A., Alard, C., Dantel-Fort, M., Fort, B., Gavazzi, R., Gomez, P., Kneib, J. P., Le Fèvre, O., et al. 2007, A&A, 461, 813 [ADS]
* Chae (2007) Chae, K.-H. 2007, ApJ, 658, L71 [ADS]
* Chevallier & Polarski (2001) Chevallier, M. & Polarski, D. 2001, International Journal of Modern Physics D, 10, 213 [ADS]
* Coe (2009) Coe, D. 2009, ArXiv e-prints [ADS]
* Coe et al. (2006) Coe, D., Benítez, N., Sánchez, S. F., Jee, M., Bouwens, R., & Ford, H. 2006, AJ, 132, 926 [ADS]
* Coe & Moustakas (2009a) Coe, D. A. & Moustakas, L. A. 2009a, in prep. (Paper I)
* Coe & Moustakas (2009c) —. 2009c, in prep. (Paper III)
* Coles (2008) Coles, J. 2008, ApJ, 679, 17 [ADS]
* Dalal & Watson (2005) Dalal, N. & Watson, C. R. 2005, in 25 Years After the Discovery: Some Current Topics on Lensed QSOs, ed. L. J. Goicoechea [ADS]
* Dobke et al. (2009) Dobke, B. M., King, L. J., Fassnacht, C. D., & Auger, M. W. 2009, MNRAS, 397, 311 [ADS]
* Eigenbrod et al. (2005) Eigenbrod, A., Courbin, F., Vuissoz, C., Meylan, G., Saha, P., & Dye, S. 2005, A&A, 436, 25 [ADS]
* Fassnacht et al. (2006) Fassnacht, C. D., Gal, R. R., Lubin, L. M., McKean, J. P., Squires, G. K., & Readhead, A. C. S. 2006, ApJ, 642, 30 [ADS]
* Fassnacht et al. (2004) Fassnacht, C. D., Marshall, P. J., Baltz, A. E., Blandford, R. D., Schechter, P. L., & Tyson, J. A. 2004, in Bulletin of the American Astronomical Society, Vol. 36, Bulletin of the American Astronomical Society, 1531–+ [ADS]
* Faure et al. (2008) Faure, C., Kneib, J.-P., Covone, G., Tasca, L., Leauthaud, A., Capak, P., Jahnke, K., Smolcic, V., et al. 2008, ApJS, 176, 19 [ADS]
* Freedman et al. (2001) Freedman, W. L., Madore, B. F., Gibson, B. K., Ferrarese, L., Kelson, D. D., Sakai, S., Mould, J. R., Kennicutt, Jr., R. C., et al. 2001, ApJ, 553, 47 [ADS]
* Fukugita et al. (1992) Fukugita, M., Futamase, T., Kasai, M., & Turner, E. L. 1992, ApJ, 393, 3 [ADS]
* Futamase & Hamaya (1999) Futamase, T. & Hamaya, T. 1999, Progress of Theoretical Physics, 102, 1037 [ADS]
* Hilbert et al. (2007) Hilbert, S., White, S. D. M., Hartlap, J., & Schneider, P. 2007, MNRAS, 382, 121 [ADS]
* Hogg (1999) Hogg, D. W. 1999, ArXiv Astrophysics e-prints [ADS]
* Hu (2005) Hu, W. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 339, Observing Dark Energy, ed. S. C. Wolff & T. R. Lauer, 215–+ [ADS]
* Hu & Jain (2004) Hu, W. & Jain, B. 2004, Phys. Rev. D, 70, 043009 [ADS]
* Huterer & Turner (2001) Huterer, D. & Turner, M. S. 2001, Phys. Rev. D, 64, 123527 [ADS]
* Ilbert et al. (2006) Ilbert, O., Arnouts, S., McCracken, H. J., Bolzonella, M., Bertin, E., Le Fèvre, O., Mellier, Y., Zamorani, G., et al. 2006, A&A, 457, 841 [ADS]
* Inada et al. (2008) Inada, N., Oguri, M., Becker, R. H., Shin, M.-S., Richards, G. T., Hennawi, J. F., White, R. L., Pindor, B., et al. 2008, AJ, 135, 496 [ADS]
* Ivezic et al. (2008) Ivezic, Z., Tyson, J. A., Allsman, R., Andrew, J., Angel, R., & for the LSST Collaboration. 2008, ArXiv e-prints [ADS]
* Jackson (2007) Jackson, N. 2007, Living Reviews in Relativity, 10, 4 [ADS]
* Kaiser (2004) Kaiser, N. 2004, in Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, Vol. 5489, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. J. M. Oschmann, Jr., 11–22 [ADS]
* Keeton (2009) Keeton, C. R. 2009, ArXiv e-prints [ADS]
* Keeton & Moustakas (2009) Keeton, C. R. & Moustakas, L. A. 2009, ApJ, 699, 1720 [ADS]
* Keeton & Zabludoff (2004) Keeton, C. R. & Zabludoff, A. I. 2004, ApJ, 612, 660 [ADS]
* Koopmans et al. (2009a) Koopmans, L. V. E., Auger, M., Barnabe, M., Bolton, A., Bradac, M., Ciotti, L., Congdon, A., Czoske, O., et al. 2009a, ArXiv e-prints [ADS]
* Koopmans et al. (2009b) Koopmans, L. V. E., Bolton, A., Treu, T., Czoske, O., Auger, M. W., Barnabè, M., Vegetti, S., Gavazzi, R., et al. 2009b, ApJ, 703, L51 [ADS]
* Koopmans et al. (2004) Koopmans, L. V. E., Browne, I. W. A., & Jackson, N. J. 2004, New Astronomy Review, 48, 1085 [ADS]
* Koopmans et al. (2006) Koopmans, L. V. E., Treu, T., Bolton, A. S., Burles, S., & Moustakas, L. A. 2006, ApJ, 649, 599 [ADS]
* Lazio (2008) Lazio, J. 2008, in American Institute of Physics Conference Series, Vol. 1035, The Evolution of Galaxies Through the Neutral Hydrogen Window, ed. R. Minchin & E. Momjian, 303–309 [ADS]
* Lee & Ng (2007) Lee, S. & Ng, K.-W. 2007, Phys. Rev. D, 76, 043518 [ADS]
* Lewis & Ibata (2002) Lewis, G. F. & Ibata, R. A. 2002, MNRAS, 337, 26 [ADS]
* Linder (2003) Linder, E. V. 2003, Physical Review Letters, 90, 091301 [ADS]
* Linder (2004) —. 2004, Phys. Rev. D, 70, 043534 [ADS]
* Marshall et al. (2005) Marshall, P., Blandford, R., & Sako, M. 2005, New Astronomy Review, 49, 387 [ADS]
* Marshall et al. (2009a) Marshall, P. J., Auger, M., Bartlett, J. G., Bradac, M., Cooray, A., Dalal, N., Dobler, G., Fassnacht, C. D., et al. 2009a, ArXiv e-prints [ADS]
* Marshall et al. (2009b) Marshall, P. J., Hogg, D. W., Moustakas, L. A., Fassnacht, C. D., Bradač, M., Schrabback, T., & Blandford, R. D. 2009b, ApJ, 694, 924 [ADS]
* Momcheva et al. (2006) Momcheva, I., Williams, K., Keeton, C., & Zabludoff, A. 2006, ApJ, 641, 169 [ADS]
* Mörtsell & Sunesson (2006) Mörtsell, E. & Sunesson, C. 2006, Journal of Cosmology and Astro-Particle Physics, 1, 12 [ADS]
* Moustakas et al. (2009) Moustakas, L. A., Abazajian, K., Benson, A., Bolton, A. S., Bullock, J. S., Chen, J., Cheng, E., Coe, D., et al. 2009, ArXiv e-prints [ADS]
* Moustakas et al. (2008) Moustakas, L. A., Bolton, A. J., Booth, J. T., Bullock, J. S., Cheng, E., Coe, D., Fassnacht, C. D., Gorjian, V., et al. 2008, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7010, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series [ADS]
* Moustakas et al. (2007) Moustakas, L. A., Marshall, P., Newman, J. A., Coil, A. L., Cooper, M. C., Davis, M., Fassnacht, C. D., Guhathakurta, P., et al. 2007, ApJ, 660, L31 [ADS]
* Myers et al. (2003) Myers, S. T., Jackson, N. J., Browne, I. W. A., de Bruyn, A. G., Pearson, T. J., Readhead, A. C. S., Wilkinson, P. N., Biggs, A. D., et al. 2003, MNRAS, 341, 1 [ADS]
* Oguri (2007) Oguri, M. 2007, ApJ, 660, 1 [ADS]
* Oguri et al. (2008) Oguri, M., Inada, N., Strauss, M. A., Kochanek, C. S., Richards, G. T., Schneider, D. P., Becker, R. H., Fukugita, M., et al. 2008, AJ, 135, 512 [ADS]
* Oguri et al. (2005) Oguri, M., Keeton, C. R., & Dalal, N. 2005, MNRAS, 364, 1451 [ADS]
* Olling (2007) Olling, R. P. 2007, MNRAS, 378, 1385 [ADS]
* Paczynski & Gorski (1981) Paczynski, B. & Gorski, K. 1981, ApJ, 248, L101 [ADS]
* Perlmutter et al. (1999) Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R. A., Nugent, P., Castro, P. G., Deustua, S., Fabbro, S., et al. 1999, ApJ, 517, 565 [ADS]
* Refsdal (1964) Refsdal, S. 1964, MNRAS, 128, 307 [ADS]
* Richards et al. (2009) Richards, G. T., Myers, A. D., Gray, A. G., Riegel, R. N., Nichol, R. C., Brunner, R. J., Szalay, A. S., Schneider, D. P., et al. 2009, ApJS, 180, 67 [ADS]
* Riess et al. (1998) Riess, A. G., Filippenko, A. V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P. M., Gilliland, R. L., Hogan, C. J., et al. 1998, AJ, 116, 1009 [ADS]
* Riess et al. (2009) Riess, A. G., Macri, L., Casertano, S., Sosey, M., Lampeitl, H., Ferguson, H. C., Filippenko, A. V., Jha, S. W., et al. 2009, ApJ, 699, 539 [ADS]
* Saha et al. (2006) Saha, P., Coles, J., Macciò, A. V., & Williams, L. L. R. 2006, ApJ, 650, L17 [ADS]
* Schmidt et al. (2009) Schmidt, S., Newman, J., Tyson, J., Connolly, A., Wittman, D., Matthews, D., Margoniner, V., Choi, A., et al. 2009, in Bulletin of the American Astronomical Society, Vol. 41, Bulletin of the American Astronomical Society, 368–+ [ADS]
* Suyu et al. (2009a) Suyu, S. H., Marshall, P. J., Auger, M. W., Hilbert, S., Blandford, R. D., Koopmans, L. V. E., Fassnacht, C. D., & Treu, T. 2009a, ArXiv e-prints [ADS]
* Suyu et al. (2009b) Suyu, S. H., Marshall, P. J., Blandford, R. D., Fassnacht, C. D., Koopmans, L. V. E., McKean, J. P., & Treu, T. 2009b, ApJ, 691, 277 [ADS]
* van de Ven et al. (2009) van de Ven, G., Mandelbaum, R., & Keeton, C. R. 2009, MNRAS, 398, 607 [ADS]
* Yamamoto et al. (2001) Yamamoto, K., Kadoya, Y., Murata, T., & Futamase, T. 2001, Progress of Theoretical Physics, 106, 917 [ADS]
|
arxiv-papers
| 2009-06-22T21:21:15 |
2024-09-04T02:49:03.464101
|
{
"license": "Public Domain",
"authors": "Dan Coe and Leonidas Moustakas",
"submitter": "Dan Coe",
"url": "https://arxiv.org/abs/0906.4108"
}
|
0906.4123
|
# Fisher Matrices and Confidence Ellipses: A Quick-Start Guide and Software
Dan Coe [email protected] Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr, MS 169-327, Pasadena, CA 91109
###### Abstract
Fisher matrices are used frequently in the analysis of combining cosmological
constraints from various data sets. They encode the Gaussian uncertainties of
multiple variables. They are simple to use, and I show how to get up and
running with them quickly. Python software is also provided. I cover how to
obtain confidence ellipses, add data sets, apply priors, marginalize,
transform variables, and even calculate your own Fisher matrices. This
treatment is not new, but I aim to provide a clear and concise reference
guide. I also provide references and links to more sophisticated treatments
and software.
###### Subject headings:
cosmology
††slugcomment: Version 1
## 1\. Outline
I explain how to do/obtain the following with/from Fisher matrices:
§ 2: Confidence Ellipses
§ 3: Manipulation:
Marginalization, Priors, Adding Data Sets
§ 4: How to Calculate your Own Fisher Matrices
§ 5: How to transform variables
§ 6: Dark energy pivot redshift
§ 7: Discussion (brief) about what Fisher matrices are
§ 8: Software I’ve come across (including my own)
§ 9: How you can contribute to this paper
## 2\. Fisher Matrices $\Rightarrow$ Confidence Ellipses
The inverse of the Fisher matrix is the covariance matrix:
$\left[F\right]^{-1}=\left[C\right]=\left[\begin{array}[]{cc}\sigma_{x}^{2}&\sigma_{xy}\vspace{0.07in}\\\
\sigma_{xy}&\sigma_{y}^{2}\end{array}\right]$ (1)
$\sigma_{x}$ and $\sigma_{y}$ are the 1-$\sigma$ uncertainties in your
parameters $x$ and $y$, respectively (marginalizing over the other).
$\sigma_{xy}=\rho\sigma_{x}\sigma_{y}$, where $\rho$ is known as the
correlation coefficient. $\rho$ varies from 0 (independent) to 1 (completely
correlated). Examples are plotted in Fig. 1.
The ellipse parameters are calculated as follows (e.g., Unknown, 2008):
$a^{2}=\frac{\sigma_{x}^{2}+\sigma_{y}^{2}}{2}+\sqrt{\frac{(\sigma_{x}^{2}-\sigma_{y}^{2})^{2}}{4}+\sigma_{xy}^{2}}$
(2)
$b^{2}=\frac{\sigma_{x}^{2}+\sigma_{y}^{2}}{2}-\sqrt{\frac{(\sigma_{x}^{2}-\sigma_{y}^{2})^{2}}{4}+\sigma_{xy}^{2}}$
(3) $\tan 2\theta=\frac{2\sigma_{xy}}{\sigma_{x}^{2}-\sigma_{y}^{2}}$ (4)
We then multiply the axis lengths $a$ and $b$ by a coefficient $\alpha$
depending on the confidence level we are interested in. For 68.3% CL
(1-$\sigma$), $\Delta\chi^{2}\approx 2.3$,
$\alpha=\sqrt{\Delta\chi^{2}}\approx 1.52$. Other values can be found in Table
1. These can be calculated following e.g., Lampton et al. (1976).
The area of the ellipse is given by
$\displaystyle A$ $\displaystyle=$ $\displaystyle\pi(\alpha a)(\alpha b)$ (5)
$\displaystyle=$ $\displaystyle\pi(\Delta\chi^{2})ab$ (6) $\displaystyle=$
$\displaystyle\pi\sigma_{x}\sigma_{y}\sqrt{1-\rho^{2}}$ (7)
The inverse of the area is a good measure of figure of merit. The Dark Energy
Task Force (DETF; Albrecht et al., 2006, 2009) used ${\rm FOM}=\pi/A$ for the
ability of experiments (WL, SN, BAO, CL) to constrain the dark energy equation
of state parameters ($w_{0},w_{a}$).
### 2.1. Probability $P(x,y)$
Interested in the probability that specific values are correct for parameters
$x$ and $y$? The probability function $P(x,y)$ given best fit values
$(x_{0},y_{0})$ and 1-$\sigma$ uncertainties $(\sigma_{x},\sigma_{y})$ is
calculated as follows:
$\chi^{2}=\frac{\left(\displaystyle\frac{\Delta
x}{\sigma_{x}}\right)^{2}+\left(\displaystyle\frac{\Delta
y}{\sigma_{y}}\right)^{2}-2\rho\left(\displaystyle\frac{\Delta
x}{\sigma_{x}}\right)\left(\displaystyle\frac{\Delta
y}{\sigma_{y}}\right)}{1-\rho^{2}}$ (8)
$P(x,y)=\exp\left(-\frac{\chi^{2}}{2}\right)$ (9)
with $\Delta x\equiv x-x_{0}$ and $\Delta y\equiv y-y_{0}$. Note for $\rho=0$
(uncorrelated $x$ and $y$), the $\chi^{2}$ formula looks familiar. For
correlated $x$ and $y$ ($\rho>0$), $\chi^{2}$ is reduced.
Table 1Confidence Ellipses: $\sigma$ | CL | $\Delta\chi^{2}$ | $\alpha$
---|---|---|---
1 | 68.3% | 2.3 | 1.52
2 | 95.4% | 6.17 | 2.48
3 | 99.7% | 11.8 | 3.44
Figure 1.— 68.3% (1-$\sigma$) confidence ellipses for parameters $x$ and $y$
with 1-$\sigma$ uncertainties $\sigma_{x}$ and $\sigma_{y}$ and correlation
coefficient $\rho$. In the first three panels, we plot as dashed lines the
marginalized 1-$\sigma$ uncertainty for each variable: $\alpha\sigma_{x}$ and
$\alpha\sigma_{y}$, where $\alpha\approx\sqrt{2.3}\approx 1.52$. In the
bottom-right panel, we zoom in to show the intersections with the axes:
$\pm\beta\sigma_{x}$ and $\pm\beta\sigma_{y}$, where $\beta\approx
2.13\sqrt{1-\rho}$ (for $\rho\approx 1$).
## 3\. Manipulation: Marginalization, Priors, Adding Data Sets, and More
Consider a Fisher matrix provided by the DETF (Table 2) for optimistic Stage
IV BAO observations for the following variables:
($\omega_{m},\Omega_{\Lambda},\Omega_{k}$), where
$\omega_{m}\equiv\Omega_{m}h^{2}$ and
$\Omega_{m}+\Omega_{\Lambda}+\Omega_{k}=1$. The covariance matrix (inverse of
the Fisher matrix) is given in Table 3. For example, the top-left element
tells us that $\Delta\omega_{m}\approx 0.00566\approx\sqrt{3.20E-5}$.
Table 2Example Fisher Matrix | $\omega_{m}$ | $\Omega_{\Lambda}$ | $\Omega_{k}$
---|---|---|---
$\omega_{m}$ | 2,376,145 | 796,031 | 615,114
$\Omega_{\Lambda}$ | 796,031 | 274,627 | 217,371
$\Omega_{k}$ | 615,114 | 217,371 | 178,014
Table 3Corresponding Covariance Matrix | $\omega_{m}$ | $\Omega_{\Lambda}$ | $\Omega_{k}$
---|---|---|---
$\omega_{m}$ | 3.20E-5 | -1.56E-4 | 8.02E-5
$\Omega_{\Lambda}$ | -1.56E-4 | 8.71E-4 | -5.25E-4
$\Omega_{k}$ | 8.02E-5 | -5.25E-4 | 3.69E-4
### 3.1. Marginalization
When quoting these uncertainties on $\omega_{m}$, the other variables
($\Omega_{\Lambda},\Omega_{k}$) have automatically been marginalized over.
That is, their probabilities have been integrated over: they have been set
free to hold any values while we calculate the range of acceptable
$\omega_{m}$.
To calculate a new Fisher matrix marginalized over any variable, simply remove
that variable’s row and column from the covariance matrix, and take the
inverse of that to yield the new Fisher matrix.
### 3.2. Fixing Parameters
Suppose instead want the opposite: perfect knowledge of a parameter. For
example, we want to consider a flat universe with a fixed value of
$\Omega_{k}=0$. To do this, simply remove $\Omega_{k}$ from the Fisher matrix
(Table 4). The new covariance matrix and parameter uncertainties are
calculated from the revised Fisher matrix.
Alternatively, the on-diagonal element corresponding to that parameter can be
set to a very large value. For example, if we set the bottom-right element in
Table 2 to $10^{12}$, that would correspond to a $10^{-6}$ uncertainty in
$\omega_{m}$, or nearly fixed. Note that higher values in the Fisher matrix
correspond to higher certainty.
Table 4Fisher Matrix with Fixed $\Omega_{k}=0$ | $\omega_{m}$ | $\Omega_{\Lambda}$
---|---|---
$\omega_{m}$ | 2,376,145 | 796,031
$\Omega_{\Lambda}$ | 796,031 | 274,627
### 3.3. Priors
Rather than fixing a parameter to an exact value, we may want to place a prior
such as $\Delta\Omega_{k}=0.01$ (1-$\sigma$). In this case, simply add
$1/\sigma^{2}=10^{4}$ to the on-diagonal element corresponding to that
variable (in this case, the bottom left element).
### 3.4. Adding Data Sets
To combine constraints from multiple experiments, simply add their Fisher
matrices: $F=F_{1}+F_{2}$. Strictly speaking, any marginalization should be
performed after the addition. But if the “nuisance parameters” are
uncorrelated between the two data sets, then marginalization may be performed
before the addition.
## 4\. How to Calculate your Own Fisher Matrices
Given the badness of fit $\chi^{2}(x,y)$, your 2-D Fisher matrix can be
calculated as follows:
$\left[F\right]=\frac{1}{2}\left[\begin{array}[]{cc}\displaystyle\frac{\partial^{2}}{\partial
x^{2}}&\displaystyle\frac{\partial^{2}}{\partial x\partial
y}\vspace{0.07in}\\\ \displaystyle\frac{\partial^{2}}{\partial x\partial
y}&\displaystyle\frac{\partial^{2}}{\partial y^{2}}\end{array}\right]\chi^{2}$
(10)
In other words,
$F_{ij}=\displaystyle\frac{1}{2}\frac{\partial\chi^{2}}{\partial p_{i}\partial
p_{j}}$.
These derivatives are simple to calculate numerically:
$\displaystyle\frac{\partial^{2}\chi^{2}}{\partial
x^{2}}\approx\frac{\chi^{2}(x_{0}+\Delta
x,y_{0})-2\chi^{2}(x_{0},y_{0})+\chi^{2}(x_{0}-\Delta x,y_{0})}{(\Delta
x)^{2}}$ (11)
$\displaystyle\frac{\partial\chi^{2}}{\partial
x}\approx\frac{\chi^{2}(x_{0}+\Delta x,y_{0})-\chi^{2}(x_{0}-\Delta
x,y_{0})}{2\Delta x}\\\ $ (12)
$\displaystyle\frac{\partial^{2}\chi^{2}}{\partial x\partial
y}=\frac{\partial\displaystyle\frac{\partial\chi^{2}}{\partial x}}{\partial
y}$ (13)
## 5\. Transformation of Variables
Suppose we are given a Fisher matrix in terms of variables $p=(x,y,z)$ but we
are interested in constraints on related variables $p^{\prime}=(a,b,c)$. We
can obtain a new Fisher matrix as follows:
$F^{\prime}_{mn}=\sum_{ij}\frac{\partial p_{i}}{\partial
p^{\prime}_{m}}\frac{\partial p_{j}}{\partial p^{\prime}_{n}}F_{ij}$ (14)
Let’s spell this out explicitly. Here is the expression for element $(a,b)$ in
the new Fisher matrix:
$\displaystyle F^{\prime}_{ab}$ $\displaystyle=$ $\displaystyle\frac{\partial
x}{\partial a}\frac{\partial x}{\partial b}F_{xx}+\frac{\partial x}{\partial
a}\frac{\partial y}{\partial b}F_{xy}+\frac{\partial x}{\partial
a}\frac{\partial z}{\partial b}F_{xz}$ (15) $\displaystyle+$
$\displaystyle\frac{\partial y}{\partial a}\frac{\partial x}{\partial
b}F_{yx}+\frac{\partial y}{\partial a}\frac{\partial y}{\partial
b}F_{yy}+\frac{\partial y}{\partial a}\frac{\partial z}{\partial b}F_{yz}$
(16) $\displaystyle+$ $\displaystyle\frac{\partial z}{\partial
a}\frac{\partial x}{\partial b}F_{zx}+\frac{\partial z}{\partial
a}\frac{\partial y}{\partial b}F_{zy}+\frac{\partial z}{\partial
a}\frac{\partial z}{\partial b}F_{zz}$ (17)
This can be calculated using matrices:
$[F^{\prime}]=[M]^{T}[F][M]$ (18)
where $M_{ij}=\displaystyle\frac{\partial p_{i}}{\partial p^{\prime}_{j}}$:
$\left[M\right]=\left[\begin{array}[]{ccc}\displaystyle\frac{\partial
x}{\partial a}&\displaystyle\frac{\partial x}{\partial
b}&\displaystyle\frac{\partial x}{\partial c}\vspace{0.1in}\\\
\displaystyle\frac{\partial y}{\partial a}&\displaystyle\frac{\partial
y}{\partial b}&\displaystyle\frac{\partial y}{\partial c}\vspace{0.1in}\\\
\displaystyle\frac{\partial z}{\partial a}&\displaystyle\frac{\partial
z}{\partial b}&\displaystyle\frac{\partial z}{\partial c}\end{array}\right]$
(19)
and $[M]^{T}$ is the transpose.
All of these partial derivatives should be evaluated numerically, plugging in
best-fit values of the parameters.
### 5.1. Transformation Example
Suppose we are given a Fisher matrix in terms of
($\omega_{m},\Omega_{\Lambda},\Omega_{k}$), but we are interested in
($\Omega_{m},\Omega_{\Lambda},h$). Here $\omega_{m}\equiv\Omega_{m}h^{2}$ and
$\Omega_{k}=1-\Omega_{m}-\Omega_{\Lambda}$. Suppose further that the best-fit
cosmology is $(\Omega_{m},\Omega_{\Lambda},h)=(0.3,0.7,0.7)$. Our
transformation matrix is evaluated as follows:
$\displaystyle\left[M\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\displaystyle\frac{\partial\omega_{m}}{\partial\Omega_{m}}&\displaystyle\frac{\partial\omega_{m}}{\partial\Omega_{\Lambda}}&\displaystyle\frac{\partial\omega_{m}}{\partial
h}\vspace{0.1in}\\\
\displaystyle\frac{\partial\Omega_{\Lambda}}{\partial\Omega_{m}}&\displaystyle\frac{\partial\Omega_{\Lambda}}{\partial\Omega_{\Lambda}}&\displaystyle\frac{\partial\Omega_{\Lambda}}{\partial
h}\vspace{0.1in}\\\
\displaystyle\frac{\partial\Omega_{k}}{\partial\Omega_{m}}&\displaystyle\frac{\partial\Omega_{k}}{\partial\Omega_{\Lambda}}&\displaystyle\frac{\partial\Omega_{k}}{\partial
h}\end{array}\right]$ (23) $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}h^{2}&0&2\Omega_{m}h\vspace{0.1in}\\\
0&1&0\vspace{0.1in}\\\
-1&-1&0\end{array}\right]=\left[\begin{array}[]{ccc}0.49&0&0.42\vspace{0.1in}\\\
0&1&0\vspace{0.1in}\\\ -1&-1&0\end{array}\right]$ (31)
## 6\. Pivot Redshift
Given the dark energy equation of state parameterization
$w=w_{0}+(1-a)w_{a}$ (32)
where $1/a=1+z$, if you have calculated a Fisher Matrix for dark energy
parameters $w_{0}$ and $w_{a}$, go ahead and calculate the pivot redshift,
too:
$z_{p}=\frac{-1}{1+\frac{\displaystyle\Delta w_{a}}{\displaystyle\rho\Delta
w_{0}}}$ (33)
At this redshift, $w(z)$ is best constrained (e.g., Fig. 16 of Huterer &
Turner, 2001). Rather than presenting constraints on ($w_{0},w_{a}$),
constraints on ($w_{p},w_{a}$) can be presented. That is, we constrain the
value of $w$ at $z=z_{p}$ rather than at $z=0$ (along with $w$’s rate of
change with time $w_{a}$).
The ($w_{p},w_{a}$) confidence ellipse has no tilt; there is no correlation
between the two, by definition.111Thus the DETF chooses a more interesting
ellipse to plot: ($w_{p},\Omega_{DE}$). But the area of the ($w_{p},w_{a}$)
ellipse is equal to the area of the ($w_{0},w_{a}$) ellipse. From this and Eq.
7 it follows that
$\Delta w_{p}=\Delta w_{0}\sqrt{1-\rho^{2}}$ (34)
And if $w$ is constant, then $\Delta w_{p}=\Delta w_{0}$.
Derivation of the pivot redshift formula follows from (Albrecht et al., 2006),
calculating the uncertainty of
$w_{p}=w_{0}+(1-a_{p})w_{a}$ (35)
$(\Delta w_{p})^{2}=(\Delta w_{0})^{2}+((1-a_{p})\Delta
w_{a})^{2}+2(1-a_{p})\Delta w_{0,a}$ (36)
where $\Delta w_{0,a}=\rho\Delta w_{0}\Delta w_{a}$, and then minimizing
$\Delta w_{p}$ for $a_{p}$.
## 7\. Discussion
Fisher matrices encode the Gaussian uncertainties in a number of parameters.
Confidence ellipses can be easily calculated over any pair of parameters.
These provide an optimistic approximation to the true probability
distribution. The true uncertainties may be larger and non-Gaussian. Note the
best fit values themselves are not encoded in the Fisher matrices, and must be
provided separately.
Fisher matrices allow one to easily manipulate parameter constraints over many
variables. It is easy to add data sets, add priors, marginalize over
parameters, and transform variables, as shown here.
A more in-depth discussion of Fisher matrices and issues surrounding their use
can be found in (Albrecht et al., 2009).
This is the paper I’d wished I could find when I began my work with Fisher
matrices: projections for cosmological constraints from gravitational lens
time delays (Coe & Moustakas, 2009).
## 8\. Software
Fisher.py222http://www.its.caltech.edu/%7Ecoe/Fisher/ Python – simple
manipulation of Fisher matrices and plotting of ellipses
DETFast333http://www.physics.ucdavis.edu/DETFast/ (Albrecht et al., 2006) JAVA
– Compare expectations of cosmological constraints from different experiments
with your choice of priors with a few clicks!
Fisher4Cast444http://www.cosmology.org.za/ (Bassett et al., 2009) Matlab –
most sophisticated
Your ad here.
## 9\. Contribute
This is meant to be a brief guide, but if I’ve failed to reference another
useful guide or your software or if I’ve neglected some detail (subtle or
otherwise) about Fisher matrices, please e-mail me at coe(at)caltech.edu, and
I’ll be happy to update this document. Also please tell me if any section is
unclear.
If I have not covered a useful topic, it is probably outside my knowledge of
Fisher matrices. For example, I have not covered the analysis of Monte Carlo
Markov Chains (MCMC) as provided, for example, by the WMAP Lambda website.555
http://lambda.gsfc.nasa.gov/ If a generous reader could explain to me (or
point me to an appropriate reference on) how to extract confidence contours
and a Fisher matrix from a MCMC, I would be grateful and include the
explanation here, giving due credit to the contributor.
I thank Olivier Dore for referring me to the DETFast software written by Jason
Dick and Lloyd Knox whom I also thank for answering my questions about their
software. It is a valuable resource. Once I took off these training wheels and
began to produce my own plots, DETFast is still a valuable resource for Fisher
matrices calculated by the DETF encoding their estimates of cosmological
constraints from various future experiments. This work was carried out at Jet
Propulsion Laboratory, California Institute of Technology, under a contract
with NASA.
## References
* Albrecht et al. (2009) Albrecht, A., Amendola, L., Bernstein, G., Clowe, D., Eisenstein, D., Guzzo, L., Hirata, C., Huterer, D., et al. 2009, ArXiv e-prints [ADS]
* Albrecht et al. (2006) Albrecht, A., Bernstein, G., Cahn, R., Freedman, W. L., Hewitt, J., Hu, W., Huth, J., Kamionkowski, M., et al. 2006, ArXiv Astrophysics e-prints [ADS]
* Bassett et al. (2009) Bassett, B. A., Fantaye, Y., Hlozek, R., & Kotze, J. 2009, ArXiv e-prints [ADS]
* Coe & Moustakas (2009) Coe, D. A. & Moustakas, L. A. 2009, ArXiv e-prints [ADS]
* Huterer & Turner (2001) Huterer, D. & Turner, M. S. 2001, Phys. Rev. D, 64, 123527 [ADS]
* Lampton et al. (1976) Lampton, M., Margon, B., & Bowyer, S. 1976, ApJ, 208, 177 [ADS]
* Unknown (2008) Unknown. 2008, Bivariate Normal Distribution and Error Ellipses [LINK]
|
arxiv-papers
| 2009-06-23T06:49:51 |
2024-09-04T02:49:03.473880
|
{
"license": "Public Domain",
"authors": "Dan Coe",
"submitter": "Dan Coe",
"url": "https://arxiv.org/abs/0906.4123"
}
|
0906.4245
|
# On a Generalization of Alexander Polynomial for Long Virtual Knots
Denis Afanasiev
###### Abstract
We construct new invariant polynomial for long virtual knots. It is a
generalization of Alexander polynomial. We designate it by $\zeta$ meaning an
analogy with $\zeta$-polynomial for virtual links. A degree of
$\zeta$-polynomial estimates a virtual crossing number. We describe some
application of $\zeta$-polynomial for the study of minimal long virtual
diagrams with respect number of virtual crossings.
Virtual knot theory was invented by Kauffman around 1996 [Ka1]. Long virtual
knot theory was invented in [GPV] by M. Goussarov, M. Polyak, and O. Viro.
$\zeta$-polynomial for virtual link was introduced independently by several
authors (see [KR],[Saw],[SW],[Ma1]), for the proof of their coincidence, see
[BF]. The idea of two types of classical crossings in a long diagram, which
were called $\circ$ (circle) and $\ast$ (star), was invented by V.O. Manturov
(see [Ma4],[Ma3]). In present paper we called $\circ$ and $\ast$ crossings by
early overcrossing and early undercrossing respectively. To consider early
overcrossings and early undercrossings is the basis idea for a construction of
$\zeta$-polynomial in the case of long virtual knots.
###### Definition 1.1.
By a long virtual knot diagram we mean a smooth immersion
$f:\mathbb{R}\rightarrow\mathbb{R}^{2}$ such that:
1) outside some big circle, we have $f(t)=(t,0)$;
2) each intersection point is double and transverse;
3) each intersection point is endowed with classical (with a choice for
underpass and overpass specified) or virtual crossing structure.
###### Definition 1.2.
A long virtual knot is an equivalence class of long virtual knot diagrams
modulo generalized Reidemeister moves.
###### Definition 1.3.
By an arc of a long virtual knot diagram we mean a connected component of the
set, obtained from the diagram by deleting all virtual crossings (at classical
crossing the undercrossing pair of edges of the diagram is thought to be
disjoint as it is usually illustrated).
###### Definition 1.4.
We say that two arcs $a,a^{\prime}$ belong to the same long arc if there
exists a sequence of arcs $a=a_{1},\dots,a_{n+1}=a^{\prime}$ and virtual
crossings $c_{1},\dots,c_{n}$ such that for $i=1,\dots,n$ the arcs
$a_{i},a_{i+1}$ are incident to $c_{i}$ from opposite sides.
Throughout the paper, we mean that initial and final long arcs, ${\gamma}_{-}$
and ${\gamma}_{+}$, form united long arc
$\gamma={\gamma}_{-}\cup{\gamma}_{+}$. Let $D$ be a long virtual diagram with
$n\geqslant 1$ classical crossings. Hence, there is a natural pairing of all
classical crossings and all long arcs: classical crossing $v$ and long arc
$\gamma$, which emanates from $v$, are paired.
We say that classical crossing $v$ is early overcrossing (early undercrossing)
if we have an arc passing over (under) $v$ at first, in the natural order on
long virtual diagram (see also [KM], p. 139).
###### Definition 1.5.
An incidence coefficient $[v:a]\in
T=\mathbb{Z}[p,p^{-1},q,q^{-1}]/((p-1)(p-q),(q-1)(p-q))$ of classical crossing
$v$ and arc $a$ is defined as a sum of some of three polynomials:
$[v:a]={\varepsilon}_{1}1+{\varepsilon}_{2}(t^{sgn\,v}-1)+{\varepsilon}_{3}(-t^{sgn\,v})$,
where ${\varepsilon}_{i}\in\\{0,1\\},i=1,2,3$; $t=p$ if $v$ is early
overcrossing, $t=q$ if $v$ is early undercrossing; $sgn\,v$ denotes local
writhe number of $v$. We set ${\varepsilon}_{1}=1\Leftrightarrow$ arc $a$ is
emanating from $v$; ${\varepsilon}_{2}=1\Leftrightarrow$ $a$ is passing over
$v$; ${\varepsilon}_{3}=1\Leftrightarrow$ $a$ is coming into $v$. If $v$ and
$a$ are not incident we set $[v:a]=0$.
Let us enumerate all classical crossings of $D$ by numbers $1,...,n$ in
arbitrary way and associate with each classical crossings the emanating long
arc. Our generalization of Alexander polynomial for long virtual knots is
defined as determinant of $n\times n$-matrix $A(D)$ with elements
$A_{ij}:=\sum_{a\subset{\gamma}^{j}}\,[v_{i}:a]s^{deg\,a}\in T[s,s^{-1}]$
The function $deg:\\{$arcs of D$\\}\rightarrow\mathbb{Z}$ is defined according
to the rules:
(1) if arc $a$ is a first at a long arc, $deg\,a=0$;
(2) if arcs $a$ and $b$ are neighbour on a long arc, $a$ precedes $b$, then
$deg\,b=deg\,a+1$, if we pass from the left to the right with respect to the
transversal arc, and $deg\,b=deg\,a-1$ otherwise. In the first case we called
such virtual crossing increasing, in the second case — decreasing.
It easy to see that polynomial $\zeta(D)=det\,A(D)$ does not depend on a
numeration of classical crossings.
By analogy with [AM] we formulate following three theorems.
###### Theorem 1.1.
If virtual diagrams $D,D^{\prime}$ are equivalent then
$\zeta(D^{\prime})=q^{r}\zeta(D)$ for some integer $r$.
###### A sketch of the proof.
The invariance of $\zeta$ for Reidemeister moves
$\Omega_{1}^{\prime},\Omega_{2}^{\prime},\Omega_{3}^{\prime}$ is evident. The
checking of invariance for $\Omega^{\prime}$ and $\Omega_{2}$ is similar to
the case of $\zeta$-polynomial for virtual link (see [Ma2],[Ma3]).
There are two types of the first Reidemeister move $\Omega_{1}$:
${\Omega}_{1}^{p}$, if we have early overcrossing, and ${\Omega}_{1}^{q}$, if
we have early undercrossing. It easy to calculate that
$\zeta({\Omega}_{1}^{p}(D))=\zeta(D)$, $\zeta({\Omega}_{1}^{q}(D))=q^{\pm
1}\zeta(D)$.
It is convenient to use the Laplace theorem (about determinants) to check that
$det\,A(\Omega_{3}(D))=det\,A(D)$. We check equality for $10$ pair of $3\times
3$-minors of matrices $A(\Omega_{3}(D))$ and $det\,A(D)$. Two of these pairs
give equalities only if we set $(p-1)(p-q)=0,(q-1)(p-q)=0$.
∎
###### Theorem 1.2.
Let $k$ be the number of virtual crossings on a long virtual diagram $D$. Then
$deg_{s}\,\zeta(D)\leqslant k$.
From Theorems 1.1 and 1.2 we easily conclude
###### Corollary 1.1.
If $deg_{s}\,\zeta(D)=k$ then $D$ has minimal virtual crossing number.
For checking of minimality by using Corollary 1.1 it is convenient to use
###### Theorem 1.3.
The $s^{k}$-th coefficient of $\zeta(D)$ is equal to $det\,B$, where
$B_{ij}=[v_{i}:a_{j}]$ if $\exists\,a_{j}\subset{\gamma}^{j}$ s.t.
$deg\,a_{j}=$#of increasing virtual crossings on $\gamma^{j}$, and $B_{ij}=0$
otherwise, $i,j=1,...,n$.
Example. In Figure we draw long virtual diagram $D_{r,l}$ which closure is
unknot. Arcs $a_{j}$, $j=1,...,n$, are marked by thick lines. By Theorem 1.3
the $s^{k}$-th coefficient of $\zeta(D)$ is equal to
$|[v_{i}:a_{j}]|_{i,j=1,...,n}=$ $q^{r+l}(qp^{-1}-1)=q-p\neq 0$ in the ring
$T$. Consequently, $D_{r,l}$ is minimal by Corollary 1.1.
Figure 1: Long knot $D_{r,l}$, $r,l\geqslant 0$
By using our $\zeta$-polynomial we can proof following Conjecture in a
particular case. Here symbol $*$ denotes usual product of long knots.
Conjecture. If $D$ is a minimal long virtual diagram with respect number of
virtual crossings, K is a long classical knot diagram, then $D*K$ is also
minimal.
###### Theorem 1.4.
(the particular case of Conjecture)
If $D$ is a minimal long virtual diagram s.t. $deg_{s}\,\zeta(D)$ is equal to
virtual crossing number of $D$, $K$ is a long classical knot diagram, then
$D*K$ is minimal.
For a proof of Theorem 1.4 we use following lemmas. Let $l$ be a number of
long arc $\gamma={\gamma}_{-}\cup{\gamma}_{+}$, where ${\gamma}_{-}$ and
${\gamma}_{+}$ are initial and final long arcs respectively. Then
$A_{il}:=\sum_{a\subset\gamma}\,[v_{i}:a]s^{deg\,a}=$
$\sum_{a\subset{\gamma}_{-}}\,[v_{i}:a]s^{deg\,a}+\sum_{a\subset{\gamma}_{+}}\,[v_{i}:a]s^{deg\,a}$.
Consequently, $det\,A(D)=det\,A^{-}(D)+det\,A^{+}(D)$, where
$A^{\pm}_{il}=\sum_{a\subset{\gamma}_{\pm}}\,[v_{i}:a]s^{deg\,a}$,
$A^{\pm}_{ij}=A_{ij}$ for $j\neq l$. Thus, we have the natural decomposition
of $\zeta$-polynomial: $\zeta(D)=\zeta_{-}(D)+\zeta_{+}(D)$, where
$\zeta_{\pm}(D):=det\,A^{\pm}(D)$.
###### Lemma 1.1.
$\zeta_{-}(D_{1}*D_{2})=-\zeta_{-}(D_{1})\zeta_{-}(D_{2})$;
$\zeta_{+}(D_{1}*D_{2})=\zeta_{+}(D_{1})\zeta_{+}(D_{2})$.
###### Lemma 1.2.
$x\in T=\mathbb{Z}[p,p^{-1},q,q^{-1}]/((p-1)(p-q),(q-1)(p-q))$ is zero divisor
$\Leftrightarrow$ $x|_{p=1,\,q=1}=0$.
###### Proof of Theorem 1.4.
By Lemma 1.1 $\zeta(D*K)=\zeta_{-}(D*K)+\zeta_{+}(D*K)=$
$-\zeta_{-}(D)\zeta_{-}(K)+\zeta_{+}(D)\zeta_{+}(K)=$ $\zeta_{+}(K)\zeta(D)$,
because $\zeta(K)=0$. Consequently, $deg_{s}\,\zeta(D*K)=deg_{s}\,\zeta(D)$ if
$\zeta_{+}(K)\in T$ is not zero divisor.
It easy to check that $\zeta_{+}(K)|_{p=1,\,q=1}=\pm\Delta(K)|_{t=1}$, where
$\Delta$ denotes Alexander polynomial. It is known that $\Delta(K)|_{t=1}=\pm
1$. Hence, by Lemma 1.2 $\zeta_{+}(K)$ is not zero divisor, because
$\zeta_{+}(K)|_{p=1,\,q=1}\neq 0$. ∎
## Acknowledgements
The author is grateful to V.O. Manturov for idea of $\zeta$-polynomial for
long virtual knots and fruitful consultations.
## References
* [AM] D.M. Afanasiev, V.O. Manturov, On Virtual Crossing Number Estimates For Virtual Links, Journal of Knot Theory and Its Ramifications, Vol. 18, No. 6 (2009).
* [BF] A. Bartholemew, R. Fenn (2003), Quaternionic invariants of virtual knots and links, Journal of Knot Theory and Its Ramifications, 17 (2),2008 pp. 231-251
* [GPV] M. Goussarov, M. Polyak, and O. Viro, Finite type invariants of classical and virtual knots, Topology, 2000, V. 39, pp. 1045–1068.
* [Ka1] L. H. Kauffman, Virtual knot theory, Eur. J. Combinatorics. 1999. V. 20, N. 7, pp. 662–690.
* [KM] L.H. Kauffman, V.O. Manturov, Virtual biquandles, Fundamenta Mathematicae 188 (2005), pp. 103-146.
* [KR] L.H.Kauffman, D.Radford (2002), Bi-oriented quantum algebras and a generalized Alexander polynomial for virtual links, AMS Contemp. Math., 318, pp. 113-140.
* [Ma1] V.O. Manturov, An Invariant 2-variable polynomial for virtual links (2002), (Russian Math. Surveys), 57, No.5, P.141-142.
* [Ma2] V.O. Manturov, Knot Theory, Chapman & Hall, London, CRC Press.
* [Ma3] V.O. Manturov, Teoriya Uzlov (Knot Theory), (Moscow-Izhevsk, RCD), 2005 (in Russian).
* [Ma4] Long virtual knots and their invariants, ibid. 13 (2004), 1029-1039.
* [Saw] J. Sawollek (2002), On Alexander-Conway Polynomials for Virtual Knots and Links, J. Knot Theory and Its Ramifications, 12 (6), pp.767-779.
* [SW] D.Silver and S.Williams (2001), Alexander Groups and Virtual Links, J. of Knot Theory and Its Ramifications, 10 (1), pp. 151-160.
|
arxiv-papers
| 2009-06-23T12:53:57 |
2024-09-04T02:49:03.479680
|
{
"license": "Public Domain",
"authors": "Afanasiev Denis",
"submitter": "Denis Afanasiev Michailovich",
"url": "https://arxiv.org/abs/0906.4245"
}
|
0906.4342
|
# The Sizes of the X-ray and Optical Emission Regions of RXJ 1131–1231
X. Dai11affiliation: Department of Astronomy, University of Michigan, 500
Church Street, Ann Arbor MI 48109 , C.S. Kochanek22affiliation: Department of
Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus OH 43210
33affiliation: Center for Cosmology and Astroparticle Physics, The Ohio State
University, 140 West 18th Avenue, Columbus OH 43210 , G. Chartas44affiliation:
Department of Astronomy and Astrophysics, Pennsylvania State University,
University Park, PA 16802 , S. Kozłowski22affiliation: Department of
Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus OH 43210
, C.W. Morgan55affiliation: Department of Physics, United States Naval
Academy, 572C Holloway Road, Annapolis, MD 21402 , G. Garmire44affiliation:
Department of Astronomy and Astrophysics, Pennsylvania State University,
University Park, PA 16802 , E. Agol66affiliation: Department of Astronomy,
University of Washington, 3910 15th Avenue, Seattle WA 98105
###### Abstract
We use gravitational microlensing of the four images of the $z=0.658$ quasar
RXJ 1131–1231 to measure the sizes of the optical and X-ray emission regions
of the quasar. The (face-on) scale length of the optical disk at rest frame
400nm is $R_{\lambda,O}=1.3\times 10^{15}$ cm, while the half-light radius of
the rest frame 0.3–17 keV X-ray emission is $R_{1/2,X}=2.3\times 10^{14}$ cm.
The formal uncertainties are factors of $1.6$ and $2.0$, respectively. With
the exception of the lower limit on the X-ray size, the results are very
stable against any changes in the priors used in the analysis. Based on the
H$\beta$ line-width, we estimate that the black hole mass is $M_{1131}\simeq
10^{8}M_{\odot}$, which corresponds to a gravitational radius of $r_{g}\simeq
2\times 10^{13}$ cm. Thus, the X-ray emission is emerging on scales of $\sim
10r_{g}$ and the 400 nm emission on scales of $\sim 70r_{g}$. A standard thin
disk of this size should be significantly brighter than observed. Possible
solutions are to have a flatter temperature profile or to scatter a large
fraction of the optical flux on larger scales after it is emitted. While our
calculations were not optimized to constrain the dark matter fraction in the
lens galaxy, dark matter dominated models are favored. With well-sampled
optical and X-ray light curves over a broad range of frequencies there will be
no difficulty in extending our analysis to completely map the structure of the
accretion disk as a function of wavelength.
###### Subject headings:
accretion – accretion disks – black hole physics – gravitational
lensing—quasars: individual (RXJ 1131–1231)
## 1\. Introduction
A significant problem for theoretical studies of quasars is that we cannot
spatially resolve their emission regions to test models (e.g. Blaes 2004). For
example, in this paper we study the gravitational lens RXJ 1231–1131 (RXJ1131
hereafter), where we observe four images of a $z_{s}=0.658$ quasar lensed by a
$z_{l}=0.295$ elliptical galaxy (Sluse et al. 2003). Based on the H$\beta$
line-width from Sluse et al. (2003), and a magnification corrected estimate of
the continuum luminosity, we estimate111Using the Bentz et al. (2006) mass
normalizations. For the Kaspi et al. (2005) normalization we obtain
$M_{1131}=(6.9\pm 1.6)\times 10^{7}M_{\odot}$, which is consistent with the
earlier estimate of Peng et al. (2006) of $6\times 10^{7}M_{\odot}$ also using
the Kaspi et al. (2005) normalizations. We use the Peng et al. (2006) masses
in Morgan et al. (2009) because we lacked spectra for the full sample of
objects. that the black hole mass, $M_{BH}$, is $M_{1131}=(1.3\pm 0.3)\times
10^{8}M_{\odot}$. This corresponds to a gravitational radius of
$r_{g}={GM_{BH}\over c^{2}}\simeq\left(1.9\times
10^{13}\right)\left[{M_{BH}\over M_{1131}}\right]\hbox{cm},$ (1)
that subtends only $0.001h^{-1}$ micro-arcseconds.
Gravity, however, has provided us with a natural telescope for studying the
structure of quasars through the microlensing produced by stars in the lens
galaxy (see the review by Wambsganss 2006). Microlensing has a natural outer
length scale corresponding to the Einstein radius of the stars,
$\langle R_{E}\rangle=D_{OS}\left[{4G\langle M\rangle D_{LS}\over
c^{2}D_{OL}D_{OS}}\right]^{1/2}=\left(4.6\times 10^{16}\right)\left[{\langle
M\rangle\over M_{\odot}}\right]^{1/2}\hbox{cm},$ (2)
where $\langle M\rangle$ is the mean mass of the stars, and the distances
$D_{OL}$, $D_{OS}$ and $D_{LS}$ are the angular diameter distances between the
observer, lens and source. The microlenses also generate caustic lines on
which the magnification diverges, which means that our gravitational telescope
can, for all practical purposes, resolve arbitrarily small sources. The size
of the source is encoded in the amplitude of the microlensing variability as
the source, lens, and observer move relative to the caustic patterns – big
sources have smaller variability amplitudes than small sources. The technique
can be applied to any emission arising from scales more compact than a few
$\langle R_{E}\rangle$.
If we model the accretion disk by a thermally radiating thin disk with a
temperature profile of $T\propto R^{-3/4}$ (Shakura & Sunyaev 1973)222In our
present analysis we can neglect the drop in temperature and emission near the
inner edge of the accretion disk as it has little effect on the results., we
can measure the scale $R_{\lambda}$ defined by the point where the photon
energy equals the disk temperature, $kT=hc/\lambda_{rest}$, by two routes
other than microlensing. First, we can estimate it from the observed flux at
some wavelength. For example, at I-band the radius is
$R_{\lambda}\simeq{2.8\times 10^{15}\over\sqrt{\cos i}}{D_{OS}\over
r_{H}}\left[{\lambda_{obs}\over\mu\hbox{m}}\right]^{3/2}10^{-0.2(I-19)}\hbox{cm}.$
(3)
where $r_{H}=c/H_{0}$ is the Hubble radius, and $i$ is the inclination angle
of the disk. Based on HST observations (Sluse et al. 2006; Kozłowski et al.
2009), we estimate that the magnification-corrected flux is $\hbox{I}\simeq
20.7\pm 0.1$ mag ($\lambda_{obs}=0.814\mu$m), which corresponds to an R-band
(400 nm in the quasar rest frame) size of $R_{\lambda,O}=(3.5\pm 0.2)\times
10^{14}(\cos i)^{-1/2}$ cm or about $18r_{g}$. The flux size depends on the
mean magnification of the images as $1/\sqrt{\langle\mu\rangle}$, which can
introduce a $\sim 50\%$ systematic uncertainty into this size estimate.
Second, thin disk theory predicts that
$\displaystyle R_{\lambda}$ $\displaystyle=$
$\displaystyle{1\over\pi^{2}}\left[{45\over 16}{\lambda^{4}r_{g}\dot{M}\over
h_{p}}\right]^{1/3}$ $\displaystyle=$ $\displaystyle(2.5\times
10^{15})\left[{\lambda_{rest}\over\mu m}\right]^{4/3}\left[{M_{BH}\over
M_{1131}}\right]^{2/3}\left[{L\over\eta L_{E}}\right]^{1/3}\hbox{cm},$
which implies an R-band disk size $R_{\lambda,O}=1.6\times 10^{15}$ cm
($82r_{g}$) if the disk is radiating at the Eddington limit $(L/L_{E})=1$ with
an efficiency of $\eta=0.1$. Note that these two size estimates can be
reconciled only if $(L/\eta L_{E})(M_{BH}/M_{1131})^{2}\simeq 0.1(\cos
i)^{-3/2}$, corresponding to a sub-Eddington accretion rate, an overestimated
black hole mass, or a problem in the disk model since there is no evidence for
the 1–2 mags of extinction in the lens galaxy that would be needed raise the
flux size up to that from thin disk theory (Eqn. 1). Adding the inner disk
edge or using a simple relativistic disk model (Novikov & Thorne 1973, Page &
Thorne 1974) changes this problem little.
The expected size of the X-ray emitting regions is more problematic because
there is no comparably simple model for our theoretical expectations. There is
a general consensus that the X-ray continuum emission is due to unsaturated
inverse Compton scattering of soft photons by hot electrons in a corona
surrounding the inner parts of the accretion disk (see the review by Reynolds
& Nowak 2003), but the extent and geometrical configuration of the X-ray
emission region is an open question. The X-ray continuum from the corona
illuminates the disk to produce Fe K$\alpha$ emission lines, whose broad
widths indicate that they are generated close to the inner edge of the
accretion disk (e.g. Fabian et al. 2005).
While there were a number of early attempts at estimating accretion disk sizes
using microlensing (e.g. Wambsganss, Schneider & Paczyński 1990, Rauch &
Blandford 1991, Wyithe et al. 2000b, Wambsganss et al. 2000, Goicoechea et al.
2003), it is only in the last few years that it has become possible to make
large numbers of microlensing size estimates. In particular, Pooley et al.
(2007) argue that the optical sizes estimated from microlensing must be
considerably larger than the optical “flux” sizes of Eqn. 3. This was
confirmed by Morgan et al. (2009) in a more detailed analysis that also found
that the optical sizes agree better with the thin disk size estimate (Eqn. 1)
than the flux size and have a scaling with black hole mass consistent with the
$M_{BH}^{2/3}$ scaling for Eddington-limited thin disks.
Recent studies have started to examine the temperature dependence of disks
through the scaling of disk size with wavelength (Anguita et al. 2008,
Poindexter et al. 2008, Agol et al. 2009, Bate et al. 2009, Floyd et al. 2009,
Mosquera et al. 2009). Studies of the microlensing of the X-ray emission are
more limited, but indicate that the X-ray emission is much more compact than
the optical (Dai et al. 2003, Pooley et al. 2006, 2007, Kochanek et al. 2006,
Morgan et al. 2008, Chartas et al. 2009), tracking much closer to the inner
edge of the accretion disk. In this paper we estimate the sizes of the optical
and X-ray emission regions of RXJ1131 using microlensing. In §2 we describe
the data and the analysis method. In §3 we discuss the results, their
implications and directions for further research. We use an $\Omega_{0}=0.3$
flat cosmological model with $H_{0}=100h$ km s-1 Mpc-1 and $h=0.7$.
## 2\. Data and Analysis
The optical data consist of the five seasons of R-band monitoring data
described in Kozłowski et al. (2009). For our present analysis we simply
shifted the light curves by their measured time delays (Kozłowski et al.
2009). The X-ray data, all ACIS observations from the Chandra Observatory,
consist of the epoch presented by Blackburne et al. (2006) plus the 5 epochs
presented in Chartas et al. (2009). Each of the Chartas et al. (2009) epochs
consisted of a 5 ksec observation using ACIS-S3 in 1/8 sub-array mode from
which we measure the 0.2-10 keV flux. Chartas et al. (2009) also reanalyzed
the Blackburne et al. (2006) data to properly correct for the “pile-up”
effect. We do not use the X-ray fluxes of image D in our analysis because we
cannot presently be certain its flux ratios relative to A–C are unaffected by
source variability given the roughly $3$ month time delay between D and A–C
(Kozłowski et al. 2009). As we can see from Fig. 5, the X-ray source must be
more compact than the optical source because the X-ray flux ratios are
dramatically more variable.
A full description of our microlensing analysis method is presented in
Kochanek (2004) and Kochanek et al. (2006). In essence, we create the
microlensing magnification patterns we would see for a broad range of lens
models and source sizes, then randomly generate light curves to find ones that
fit the data well. We then use Bayes’ theorem to combine the results for the
individual trials to infer probability distributions for physically
interesting variables including the uncertainties created by all the other
variables.
We fit the lens as in Kozłowski et al. (2009), modeling it as a $R_{e}=1\farcs
7$ de Vaucouleurs model for the stellar distribution embedded in an NFW halo.
We consider a sequence of models described by $f_{*}$, the fraction of mass in
the stellar component relative to a constant mass-to-light ratio model with
$f_{*}\equiv 1$ and no halo. We include models with $f_{*}=0.1$ to $1$ in
equal steps, and the time delay measurements favor $f_{*}\simeq 0.2$. These
lead to the values for the convergence $\kappa$, shear $\gamma$ and fraction
of the convergence in stars $\kappa_{*}/\kappa$ reported in Table. 1.
The stars creating the microlensing magnification were drawn from a power law
mass function $dN/dM\propto M^{-1.3}$ with a ratio of 50 between the minimum
and maximum masses that roughly matches the Galactic disk mass function of
Gould (2000). We know from previous theoretical studies that the choice of the
mass function will have little effect on our conclusions given the other
sources of uncertainty (e.g. Paczyński 1986, Wyithe et al. 2000a). The mean
mass $\langle M\rangle$ is left as a variable with a uniform prior over the
mass range $0.1<\langle M/M_{\odot}\rangle<1.0$.
For each model we generated 8 random realizations of the star fields near each
image. The magnification patterns had an outer scale of $10\langle
R_{E}\rangle=4.6\langle M/M_{\odot}\rangle^{1/2}\times 10^{17}$ cm and a pixel
scale of $10\langle R_{E}\rangle/8192=5.6\langle
M/M_{\odot}\rangle^{1/2}\times 10^{13}~{}\hbox{cm}\simeq 3r_{g}$, so we should
be able to model sources as compact as the inner edge of the accretion disk.
We modeled the relative velocities as in Kochanek (2004), where for RXJ1131
the projection of the CMB dipole velocity (Kogut et al. 1993) on the lens
plane is 47 km/s, the lens velocity dispersion estimated from the Einstein
radius is $350$ km/s, and the estimated rms peculiar velocities of the lens
and source galaxies are $180$ and $140$ km/s respectively.
The source model for both the optical and X-ray sources is a face-on disk with
a temperature profile $T\propto R^{-3/4}$ radiating as a black body (Shakura &
Sunyaev 1973), so the surface brightness profile of the disk is
$I(R)\propto\left[\exp((R/R_{\lambda})^{3/4})-1\right]^{-1}$ (5)
with the single parameter being the scale length $R_{\lambda}$. While it is
true that this profile lacks the central drop in emissivity and that it is not
a physical model for the non-thermal X-ray emission, the microlensing analysis
is not sensitive to these details. The estimate of the half-light radius
($R_{1/2}\simeq 2.44R_{\lambda}$) is essentially independent of the assumed
profile (Kochanek 2004, Mortonsen, Schechter & Wambsganss 2005). We used a
$46\times 61$ logarithmic grid of trial source sizes for the X-ray and optical
sources with a spacing of $0.05$ dex.
We do, however, allow for the possibility that fraction $f_{\hbox{no}\mu}=0$
to $40\%$ of the optical emission is generated on scales much larger than the
disk and is unaffected by microlensing. Such large scale emission could have
two physical origins. First, the optical continuum can be significantly
contaminated by emission lines, both the obvious broad lines and the less
obvious Fe and Balmer pseudo-continuum emission ($\sim 30\%$ of the emission
in some Seyferts, Maoz et al. 1993), that are believed to be produced on much
larger scales than the disk. For our R-band light curves, there are no strong
emission lines in the filter band pass, but the blue edge of the Balmer
continuum emission ($\sim 6000$Å) does lie inside the band pass (roughly
$5700$–$7200$Å). Second, even if the observed photons were generated by the
accretion disk, a fraction could be scattered on much larger scales, leading
to an effectively larger source. These two possibilities are not equivalent,
as the line emission is due to reprocessing of shorter wavelength UV photons
rather than the observed R-band continuum.
A basic problem for any microlensing analysis is the degree to which the
“macro” lens model correctly sets the average magnifications. Each light
curve, $m_{i}(t)=s(t)+\mu_{i}+\delta\mu_{i}(t)+\Delta_{i}$ is defined by the
source light curve $s(t)$, the macro model magnification $\mu_{i}$, the
microlensing magnification $\delta\mu_{i}(t)$ and a possible offset
$\Delta_{i}$. These offsets can be non-zero due to problems in the macro model
or the presence of unrecognized substructures that perturb the magnifications
(e.g. Kochanek & Dalal 2004), because of differential absorption due to dust
or gas in the lens galaxy (e.g. Falco et al. 1999, Dai & Kochanek 2009), or
due to contamination of the light curves by flux from the quasar host or lens
galaxy. For the latter two possibilities, the offsets would differ between the
optical and X-ray light curves. Given a sufficiently long light curve, the
offsets can be determined from the data, but they are poorly constrained until
the light curve is a good statistical sampling of the magnification pattern.
We will consider four treatments of this problem to ensure that such
systematic problems do not affect our results. The basic division we will
refer to as Cases I and II. In Case I we allow the magnification offsets
$\Delta_{i}$ to float independently for the two bands constrained by a term
$\Delta_{i}^{2}/2\sigma^{2}$ in the log likelihood with $\sigma=0.5$ mag. In
Case II we allow them to float, but use the same offsets $\Delta_{i}$ for both
the optical and X-ray light curves. These are weak constraints, so the
resulting distributions for the offsets are broad. To make sure we are not
allowing too much freedom, we also examined limiting the range of the offsets
to $|\Delta_{i}|<0.3$ mag in Cases I’ and II’.
The advantage of the less constrained strategies is that they are robust
against the systematic errors that can plague the absolute magnifications of
the images. It is certainly true that analyses using only the “DC” flux ratios
(e.g. Pooley et al. 2006, 2007, Bate et al. 2009, Floyd et al. 2009) require
less data than our “AC” approach, but they can also lead to conclusions
dominated by these systematic errors. The “AC” approach also has the advantage
that including the effects of the velocities allows us to estimate source
sizes in centimeters without simply assuming a mean mass $\langle M\rangle$.
However, when we use loose priors on the DC flux ratios, we lose significant
information on the locations of the images relative to the magnified and
demagnified regions of the patterns. As such, it is a conservative approach.
We consider all four offset treatments in order to explore their consequences
on estimates of the source size and the amount of dark matter in the lens.
We used 8 statistical realizations of the microlensing magnification patterns
for each of the 10 stellar surface densities ($f_{*}$) and for 5 un-
microlensed fractions of optical light ($f_{\hbox{no}\mu}$). We modeled the
data sequentially, making $10^{6}$ trials for each optical source size and
case, and then fitting each trial that was a reasonable statistical fit to the
optical data to the X-ray data for each of the X-ray source sizes. In this
second step we considered both the case where the X-ray and optical share the
same intrinsic flux ratios and where they are allowed to differ.
Figure 1.— The probability distributions for the size of the X-ray (top) and
R-band ($400$ nm in the rest frame) optical (bottom) emission regions for the
log (solid) and linear (dashed) size priors. These sizes are marginalized over
$f_{\hbox{no}\mu}$. The vertical lines mark the gravitational radius $r_{g}$
for a $M_{1131}$ black hole, the Einstein radius for $\langle
M\rangle=M_{\odot}$ and the accretion disk size estimates based on either the
I-band flux (Eqn. 3) or thin disk theory (Eqn. 1). The microlensing sizes and
the I-band flux estimates can also be scaled by a $(\cos i)^{-1/2}$
inclination dependence from the assumed face-on case ($i=0^{\circ}$).
Figure 2.— The correlated probability distributions for the size of the
optical and X-ray source sizes in Einstein units of $\langle
M/M_{\odot}\rangle^{1/2}$ cm. The contours are drawn at the 68%, 90% and 95%
maximumum likelihood contours for one variable for log (solid) and linear
(dashed) size priors.
Figure 3.— Source size dependence on parameters. The optical (top) and X-ray
(bottom) source sizes ($R_{\lambda}$) as a function of the fraction
$f_{\hbox{no}\mu}$ of the optical flux that is not microlensed. The triangles,
squares, pentagons and circles show the results for the Case I (independent
magnitude offsets for both bands), Case II (common magnitude offsets), Case I’
(independent offsets limited to $|\Delta_{i}|<0.3$) and Case II’ (common
offsets limited to $|\Delta_{i}|<0.3$) treatments of the magnitude offsets.
The filled (open) symbols show the results for the logarithmic (linear) priors
on the source sizes. The horizontal lines show the same physical scales as in
Fig. 1 and the dashed curve shows the expected scaling of the optical size
with $f_{\hbox{no}\mu}$ if we keep the half-light radius of the optical
emission fixed. The half light radius of the disk emission is always
$R_{1/2}=2.44R_{\lambda}$, but the half light radius of the disk emission
combined with the unmicrolensed large scale emission grows with
$f_{\hbox{no}\mu}$, reaching $R_{1/2}=5.87R_{\lambda}$ for
$f_{\hbox{no}\mu}=0.4$.
Figure 4.— Dependence on halo structure. The solid/squares and
dotted/triangles show the likelihood of $f_{*}$, the fraction of mass in the
stellar component in the lens model compared to a constant $M/L$ model
($f_{*}\equiv 1$), for weakly constrained (Case I+II) or strongly constrained
(Case I’+II’) treatments of the magnification offsets. Dark matter dominated
models are always favored, but the low $f_{*}$ models implied by the time
delays are only strongly favored when we force the offsets to be small. The
line without points shows the fraction $1-\kappa_{*}/\kappa$ of the local
surface density near image A that is comprised of smoothly distributed dark
matter.
Figure 5.— The X-ray (top) and R-band optical (bottom) flux ratios between the
A$-$B and B$-$C images along with the tracks across the microlensing patterns
for images A (left) and B (right). The large circle shown on each pattern is
the Einstein radius, while the small circles have the half-light radius of the
optical disk and are shown at the positions corresponding to the epochs of the
X-ray observations. The overall length of the line corresponds to one decade
of motion. Darker colors represent logarithmically higher magnifications with
an overall magnification range from $1/30$ to $30$. This is a Case I example
with fairly large differential offsets. It has a high stellar surface density
($f_{*}=0.7$), a large amount of smooth optical emission
($f_{\hbox{no}\mu}=0.4$), and the X-ray source is 14 times smaller than the
optical.
Figure 6.— As in Fig. 5. This is a Case II’ example, so the magnification
offsets are small and the same for both the optical and X-ray data. It has a
very low stellar surface density ($f_{*}=0.1$), a little smooth optical
emission ($f_{\hbox{no}\mu}=0.2$), and the X-ray source is 32 times smaller
than the optical.
Figure 7.— As in Fig. 5. This is a Case I’ example, so the magnification
offsets are small but differ between the optical and X-ray data. It has a low
stellar surface density ($f_{*}=0.3$), a little smooth optical emission
($f_{\hbox{no}\mu}=0.2$), and the X-ray source is 28 times smaller than the
optical. In the left panel we are at the edge of the pattern (although the
Kochanek (2004) pattern creation method here produces periodic patterns that
allow us to wrap the light curves across edges).
## 3\. Results and Discussion
Fig. 1 shows the main result for the estimated size of the X-ray and optical
emission regions. These combine all four treatments of the magnification
offsets. Also note that in order to preserve the meaning of size ratios in
Fig. 1, we used the scale $R_{\lambda}$ of a face-on disk for both.
Physically, the X-ray emission is better characterized by its half light
radius, $R_{1/2}=2.44R_{\lambda}$. The scale length of the thin disk also
scales as $\cos^{-1/2}i$ if not viewed face on. We show the results for two
different priors on the disk sizes, a logarithmic ($P(R_{\lambda})\propto
1/R_{\lambda}$) and a uniform ($P(R_{\lambda})\propto\hbox{constant}$) prior,
and this has minor effects for the optical estimate and significant effects
for the X-ray estimate. For the logarithmic prior we formally find that the
(face on) optical disk scale length is $\log(R_{\lambda,O}/\hbox{cm})=15.11$
($14.89<\log(R_{\lambda,O}/\hbox{cm})<15.32$) and that the X-ray half-light
radius is $\log(R_{1/2,X}/\hbox{cm})=14.36$
($14.04<\log(R_{1/2,X}/\hbox{cm})<14.68$). These estimates use both the prior
on the velocities and a uniform prior for the mass over the range $0.1<\langle
M/M_{\odot}\rangle<1$. We will focus on results including this mass prior, but
note that if we make no assumption about $\langle M\rangle$, the sizes change
little. With only the velocity priors we find
$\log(R_{\lambda,O}/\hbox{cm})=15.02$
($14.75<\log(R_{\lambda,O}/\hbox{cm})<15.27$) and
$\log(R_{1/2,X}/\hbox{cm})=14.02$ ($13.67<\log(R_{1/2,X}/\hbox{cm})<14.38$).
The source sizes become a little bit smaller, but the net effect is very
modest for the reason outlined in Kochanek (2004).333In Einstein units, one
can achieve the observed variability using either a large source moving
rapidly or a small source moving slowly, with a degeneracy of roughly
$\hat{r}\propto\hat{v}$. We always impose a prior on the physical velocity
$v\propto\hat{v}\langle M\rangle^{1/2}$, so the physical source size
$r\propto\hat{r}\langle M\rangle^{1/2}\propto\hat{v}\langle
M\rangle^{1/2}\propto v$ is essentially independent of $\langle M\rangle$
given a prior on the velocity.
The X-ray size is more sensitive to the priors because the convergence of the
probability distributions for small sources is poor when the light curve is
sparsely sampled. Fig. 2 shows likelihoods for the source size in the Einstein
units used for the basic calculations, and we see that they converge for small
X-ray sizes when we use a linear prior but not for a logarithmic prior. The
problem is not due to the pixel scale of the maps, but due to the lack of a
well-sampled peak in the X-ray data. Very small source sizes are constrained
by the magnification peaks observed during a caustic crossing. If the light
curve only samples up to some minimum physical distance from a caustic
crossing, then it will constrain sources sizes significantly smaller than that
distance poorly and the likelihood function will flatten for small source
sizes. A logarithmic size prior then favors these small scales compared to a
linear prior, leading to significant differences. Thus, our lower limits on
the size of the X-ray emission are at a minimum prior dependent. More
conservatively, the results could be interpreted as providing only an upper
bound on the size of the X-ray emitting region.
Fig. 3 shows how the sizes depend on the priors, the treatment of the
magnification offsets and the fraction $f_{\hbox{no}\mu}$ of the optical
emission that is unaffected by microlensing. The X-ray size is affected only
by the choice of the size prior. The optical size is only affected by
$f_{\hbox{no}\mu}$. There are no significant differences between the results
for the four magnification offset cases. In order to produce the same optical
variability with a larger fraction $f_{\hbox{no}\mu}$, we must shrink the disk
scale length $R_{\lambda}$. Mortonson et al. (2005) argue that the effects of
microlensing are largely determined by the half-light radius of the source,
which is $R_{1/2}=2.44R_{\lambda}$ in the limit that $f_{\hbox{no}\mu}=0$. As
we increase $f_{\hbox{no}\mu}$, the disk scale length shrinks roughly by the
amount needed to keep the half light radius constant, with
$R_{1/2}=5.87R_{\lambda}$ when $f_{\hbox{no}\mu}=0.4$. Note, however, that the
scaling for this particular model will break down when $f_{\hbox{no}\mu}=1/2$.
The larger values of $f_{\hbox{no}\mu}$ are favored, with likelihood ratios of
$0.35$, $0.40$, $0.64$, $0.92$ and $1.0$ for $f_{\hbox{no}\mu}=0$, $0.1$,
$0.2$, $0.3$ and $0.4$ respectively. These differences are only marginally
significant, but they are in the sense of favoring (effectively) a flatter
temperature profile. A flatter temperature profile can help to reconcile the
differences between the larger microlensing and thin disk theory sizes as
compared to the smaller flux sizes. Such flatter temperature profiles are
generally consistent with the studies of the optical/infrared wavelength
dependence of microlensing (Anguita et al. 2008, Eigenbrod et al. 2008,
Poindexter et al. 2008, Mosquera et al. 2009, Bate et al. 2009), but are not
required. The one exception is Floyd et al. (2009), who find a limit requiring
a steeper temperature profile. Some of this information is also present in the
overall spectral energy distribution, and it is a long standing problem that
the spectra of quasars do not match the predictions of thin disk theory (see
Koratkar & Blaes 1999, Gaskell 2008).
Whether increasing $f_{\hbox{no}\mu}$ helps to resolve the size discrepancies
depends on the physical model for the contamination. Line emission is
reprocessed shorter wavelength emission, so as we increase $f_{\hbox{no}\mu}$
we are also reducing the fraction of the observed emission due to the disk and
the flux size also shrinks as $(1-f_{\hbox{no}\mu})^{1/2}$. If, however, we
view it as scattering fraction $f_{\hbox{no}\mu}$ of the continuum emission on
some large scale, then the flux size estimate is unchanged and the effect
helps to reduce the discrepancy. Resolving the discrepancy with
$f_{\hbox{no}\mu}$ would require that most of the optical emission does not
reach us directly from the accretion disk.
While the source sizes show little dependence on the treatment of the
magnification offsets, estimates of the amount of dark matter in the lens are
sensitive to how strongly we constrain the models to match the observed macro
model flux ratios, as illustrated in Fig. 4. By leaving the offsets relatively
free, so as to conservatively estimate the source sizes, we have not optimized
the calculation for probing dark matter. The Case I and II models, where we
very loosely constrain the allowed magnification offsets, marginally favor
models with $f_{*}\simeq 0.3$. The Case I’ and II’ models, where we only
accept small offsets, favor the same dark matter dominated model more
strongly. This range for $f_{*}$ is also that favored by the time delays
measurements from Kozłowski et al. (2009). Note, however, that we are not in
the “lagoon and caldera” regime for the microlensing patterns noted by
Schechter & Wambsganss (2002) because of the very high magnifications
($\mu\sim 50$ for image A at low $\kappa_{*}/\kappa$ rather than $\mu\sim
10$).
Figs. 5-7 show several examples of model light curves that fit the data
reasonably well. These were also selected to have velocities consistent with
masses of order $0.1$-$1.0M_{\odot}$. The solutions are not unique, but they
illustrate how simple changes in the source size dramatically alter the
amplitude of the variability. A common theme to the solutions is that the A
and B images are generally required to lie in “active” regions of the patterns
in order to produce such large changes in the X-ray fluxes. This means that we
can expect the dramatic variability observed in this system to continue for an
extended period of time ($1$–$10$ years). It is also interesting to note that
significant changes in the optical fluxes are also likely. The larger optical
source size washes out the effects of the closely spaced caustics that help to
drive the X-ray variability. But the overall changes between the high
magnification ridges and the demagnified valleys are still significant, and we
should see overall changes in the optical fluxes several times those observed
to date.
The implications of these results for theoretical models are mixed. The size
of the disk is grossly similar to that expected from thin disk theory, and as
we have summarized in Morgan et al. (2009), we find disk sizes that scale with
black hole mass and optical wavelength roughly as expected (Eqn. 1). We also
find that the X-ray emission arises from significantly closer to the expected
inner edge of the accretion disk than the optical emission, as we might expect
for a hot corona. The optical size is broadly consistent with the expectations
for an Eddington luminosity black hole with a mass estimated from the emission
line widths (Eqn. 1). But the size is inconsistent with that expected for a
thermally radiating disk with the observed magnitude (Eqn. 3). This is the
discrepancy originally noted by Pooley et al. (2006), which we explore more
quantitatively in Morgan et al. (2009).
Should we conclude that the thin disk model is wrong or simply that we have
oversimplified the optical radiation transfer? We considered contamination by
line emission or scattering of the optical photons, finding that this can
modestly reduce the disk size for the range where up to 40% of the optical
emission does not come directly from the disk. Our simple emission model
neglects the disk atmosphere and heating of the outer disk by radiation from
the inner disk, all processes which would tend to make the optical emission
region larger than the point where the disk has a temperature matching the
photon wavelength without any change in the underlying properties of the disk.
Many of these effects are included in recent disk models such as Hubeny et al.
(2001) or Li et al. (2005) for disk spectra. We examined face-on models with
$M_{BH}=10^{8}M_{\odot}$, $\dot{M}=0.09M_{\odot}$ yr-1 and a BH spin of
$a=0.998$ using the Hubeny et al. (2001) models to compute our definition of
the disk scale ($R_{\lambda}$, where $kT=hc/\lambda$) and the half-light
radius ($R_{1/2}$). The scale $R_{\lambda}$ is the most sensitive to the
assumptions, with $R_{\lambda}/r_{g}=41$ for $\lambda_{obs}=0.814\mu$m in our
simplified disk model but equal to $36$/$34$ for black body non-
relativistic/relativistic disk models (BB NR/REL) and to $28$/$26$ for non-LTE
non-relativistic/relativistic models (NLTE NR/REL). The model dependence is
much reduced if we compare with the half-light radii ($R_{1/2}/r_{g}=100$ for
the simple model, $114$/$117$ for the NLTE NR/REL models, and $99$/$102$ for
the BB NR/REL models). A flatter temperature profile than $T\propto R^{-3/4}$
would help, since at fixed total flux the half light radius increases. For
example if $T\propto R^{-1/2}$, the flux would be only 20% that of our
standard profile for the same half-light radius. Indeed, such a flat
temperature profile would also come much closer to matching the observed
spectra of quasars (e.g. Koratkar & Blaes 1999, Gaskell 2008) and would be
representative of models dominated by irradiation. In general, however,
current microlensing results on temperature profiles do not favor such flat
profiles even if they generally allow somewhat flatter profiles (e.g. Anguita
et al. 2008, Eigenbrod et al. 2008, Poindexter et al. 2008, Mosquera et al.
2009, Bate et al. 2009, Floyd et al. 2009).
The key to disentangling these problems is to expand the measurements over a
broad range of wavelengths, so that we can constrain the temperature profile
of the disk, and over a broad range of black hole masses and accretion rates.
For the particular case of RXJ1131, we have programs to continue the X-ray
monitoring of the system and to use HST to monitor the ultraviolet flux ratios
of the images. Obtaining a robust lower limit to the size of the X-ray
emitting region may require denser sampling of the X-ray light curve.
Measuring the mid-infrared flux ratios of the images would be the most
important step towards rigorously imposing the relative macro magnifications
with no offsets, as these would only be affected by the macro model and any
large scale gravitational perturbations (satellites).
We would like to thank M. Dietrich, P. Osmer, B. Peterson and R. Pogge for
many discussions on quasar structure, and O. Blaes for his comments. We would
like to thank D. Sluse for supplying their spectrum of the system. The
calculations in this paper were carried out on a Beowulf cluster obtained as
part of the Cluster Ohio program of the Ohio Supercomputer Center. Support for
this work was provided by NASA through Chandra Award GTO-07700072 and by the
NSF through grant AST 0708082.
## References
* Agol et al. (2009) Agol, E., Gogarten, S. M., Gorjian, V., & Kimball, A. 2009, ApJ, 697, 1010
* (2) Anguita, T., Schmidt, R.W., Turner, E.L., Wambsganss, J., Webster, R.L., Loomis, K.A., Long, D., & McMillan, R., 2008, A&A, 480, 327
* (3) Bate, N.F., Floyd, D.J.E., Webster, R.L., & Wyithe, J.S.B., 2008, MNRAS, 391, 1955
* (4) Bentz, M.C., Peterson, B.M., Pogge, R.W., Vestergaarrd, M., Onken, C.A., 2006, ApJ, 644, 133
* (5) Blackburne, J.A., Pooley, D., & Rappaport, S., 2006, ApJ, 640, 569
* (6) Blaes, O.M., 2004, in Les Houches Summer School LXXVIII (Springer: Berlin) 137
* (7) Chartas, G., Kochanek, C.S., Dai, X., Poindexter, S., & Garmire, G., 2009, ApJ, 693, 174
* (8) Dai, X., Chartas, G., Agol, E., Bautz, M.W., & Garmire, G.P., 2003, ApJ, 589, 100
* (9) Dai, X., & Kochanek, C.S., 2009, ApJ, 692, 677
* (10) Eigenbrod, A., Courbin, F., Meylan, G., Agol, E., Anguita, T., Schmidt, R.W., & Wambsganss, J., 2008, A&A, 490, 933
* (11) Fabian, A.C., Miniutti, G., Iwasawa, K., & Ross, R.R., 2005, MNRAS, 361, 795
* (12) Falco, E.E., Impey, C.D., Kochanek, C.S., Lehár, J., McLeod, B.A., Rix, H.-W., Keeton, C.R., Muñoz, J.A. & Peng, C.Y., 1999, ApJ, 523, 617
* (13) Floyd, D.J., Bate, N.F., & Webster, R.L., 2009, MNRAS in press [arXiv:0905.2651]
* (14) Gaskell, C.M., 2008, Revista Mexicana de Astronomia Y Astrofisica Conference Series, 32, 1 [arXiv:0711.2113]
* (15) Goicoechea, L.J., Alcaide, D., Mediavilla, E., Munõz, J.A., 2003, A&A, 397, 517
* Gould (2000) Gould, A. 2000, ApJ, 535, 928
* (17) Hubeny, I., Blaes, O., Krolik, J.H., & Agol, E., 2001, ApJ, 559, 680
* (18) Kaspi, S., Maoz, D., Netzer, H., Peterson, B.M., Vestergaard, M., & Jannuzi, B.T., 2006, ApJ, 629, 61
* (19) Kochanek C.S., 2004, ApJ, 605, 58
* (20) Kochanek, C.S., & Dalal, N., 2004, ApJ, 610, 69
* (21) Kochanek, C.S., Dai, X., Morgan, C., Morgan, N., Poindexter, S., & Chartas, G., 2006, Statistical Challenges in Modern Astronomy IV, 2007, G. J. Babu and E. D. Feigelson (eds.), San Francisco:Astron. Soc. Pacific [astro-ph/0609112]
* (22) Koratkar, A., & Blaes, O., 1999, PASP, 111, 1
* (23) Kozłowski, S., Morgan, N.D., Kochanek, C.S., Poindexter, S., Falco, E.E., & Dai, X., 2009, in preparation
* (24) Kogut, A., et al., 1993, ApJ, 419, 1
* (25) Li, L.-X., Zimmerman, E.R., Narayan, R., & McClintock, J.E., 2005, ApJS 157, 355
* (26) Maoz, D., Netzer, H., Peterson, B.M., et al., 1993, ApJ, 404, 576
* (27) Morgan, C.W., Kochanek, C.S., Morgan, N.D., & Falco, E.E., 2009, ApJ submitted [arXiv:0707.0305]
* Morgan et al. (2008) Morgan, C. W., Kochanek, C. S., Dai, X., Morgan, N. D., & Falco, E. E. 2008, ApJ, 689, 755
* (29) Mortonson, M.J., Schechter, P.L., & Wambsganss, J., 2005, ApJ, 628, 594
* Mosquera et al. (2009) Mosquera, A. M., Muñoz, J. A., & Mediavilla, E. 2009, ApJ, 691, 1292
* (31) Novikov, I.D., & Thorne, K.S., 1973, in Black Holes, ed. C. De Witt & B. De Witt (New York: Gordan & Breach) 343
* (32) Paczyński, B., 1986, ApJ, 301, 503
* (33) Page, D.N., & Thorne, K.S., 1974, ApJ, 191, 499
* (34) Peng, C.Y., Impey, C.D., Rix, H.-W., Kochanek, C.S., Keeton, C.R., Falco, E.E., Lehár, J., & McLeod, B.A., 2006 ApJ in press [astro-ph/0603248]
* (35) Poindexter, S, Morgan, N, & Kochanek, C.S., 2008, ApJ, 673, 34
* Pooley et al. (2006) Pooley, D., Blackburne, J. A., Rappaport, S., Schechter, P. L., & Fong, W.-f. 2006, ApJ, 648, 67
* Pooley et al. (2007) Pooley, D., Blackburne, J. A., Rappaport, S., & Schechter, P. L. 2007, ApJ, 661, 19
* (38) Rauch, K.P., & Blandford, R.D., ApJ, 1991, ApJL, 381, 39L
* (39) Reynolds, C.S., & Nowak, M.A., 2003, PhR, 377, 389
* (40) Schechter, P.L., et al., 2003, ApJ, 584, 657
* (41) Schechter, P.L. & Wambsganss, J., 2002, ApJ, 580, 685
* (42) Shakura, N.I., & Sunyaev, R.A., 1973, A&A, 24, 337
* (43) Sluse, D., Surdej, J., Claeskens, J.-F., Hutsemekers, D., Jean, C., Courbin, F., Nakos, T., Billeres, M., & Khmil, S.V., 2003, A&A, 406, L43
* (44) Sluse, D., Claeskens, J.-F., Altieri, B., Cabanac, R.A., Garcet, O., Hutsemekers, D., Jean, C., Smette, A., & Surdej, J., 2006, A&A, 449, 539
* (45) Wambsganss, J., 2006, in Gravitational Lensing: Strong Weak and Micro, Saas-Fee Advanced Course 33, G. Meylan, P. North, P. Jetzer, eds., (Springer: Berlin) 453 [astro-ph/0604278]
* (46) Wambsganss, J., Schmidt, R.W., Colley, W.N., Kundić, T., Turner, E.L., 2000, A&A, 362, L37
* (47) Wambsganss, J., Schneider, P., & Paczyński, B., 1990, ApJ, 358, L33
* (48) Wyithe, J.S.B., Webster, R.L., & Turner, E.L., 2000a, MNRAS, 315, 51
* (49) Wyithe, J.S.B., Webster, R.L., & Turner, E.L., 2000b, MNRAS, 315, 62
Table 1Microlensing Model Parameters
$f_{*}$ | Image | $\kappa$ | $\gamma$ | $\kappa_{*}/\kappa$
---|---|---|---|---
0.1 | A | 0.667 | 0.359 | 0.030
| B | 0.631 | 0.325 | 0.027
| C | 0.644 | 0.306 | 0.028
| D | 1.079 | 0.493 | 0.079
0.2 | A | 0.618 | 0.412 | 0.067
| B | 0.581 | 0.367 | 0.060
| C | 0.595 | 0.346 | 0.062
| D | 1.041 | 0.631 | 0.159
0.3 | A | 0.569 | 0.465 | 0.110
| B | 0.530 | 0.410 | 0.099
| C | 0.546 | 0.387 | 0.103
| D | 1.001 | 0.635 | 0.242
0.4 | A | 0.519 | 0.518 | 0.162
| B | 0.480 | 0.453 | 0.146
| C | 0.496 | 0.427 | 0.153
| D | 0.964 | 0.895 | 0.329
0.5 | A | 0.469 | 0.572 | 0.226
| B | 0.430 | 0.497 | 0.204
| C | 0.447 | 0.469 | 0.214
| D | 0.925 | 1.018 | 0.421
0.6 | A | 0.419 | 0.626 | 0.305
| B | 0.379 | 0.540 | 0.278
| C | 0.397 | 0.511 | 0.290
| D | 0.890 | 1.139 | 0.520
0.7 | A | 0.369 | 0.679 | 0.406
| B | 0.329 | 0.584 | 0.375
| C | 0.348 | 0.552 | 0.390
| D | 0.851 | 1.288 | 0.625
0.8 | A | 0.318 | 0.734 | 0.541
| B | 0.278 | 0.628 | 0.507
| C | 0.297 | 0.595 | 0.524
| D | 0.816 | 1.412 | 0.740
0.9 | A | 0.268 | 0.787 | 0.725
| B | 0.228 | 0.671 | 0.697
| C | 0.247 | 0.637 | 0.711
| D | 0.781 | 1.530 | 0.863
1.0 | A | 0.217 | 0.842 | 1.000
| B | 0.178 | 0.714 | 1.000
| C | 0.196 | 0.679 | 1.000
| D | 0.740 | 1.639 | 1.000
Note. — The macro models are parametrized by $f_{*}$, the fraction of mass in
the de Vaucouleurs component relative to a constant $M/L$ model with
$f_{*}\equiv 1$. The microlensing model parameters are the convergence
$\kappa$, shear $\gamma$ and the fraction of the convergence in stars
$\kappa_{*}/\kappa$.
|
arxiv-papers
| 2009-06-23T20:00:46 |
2024-09-04T02:49:03.485747
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "X. Dai (1), C.S. Kochanek (2), G. Chartas (3), S. Kozlowski (2), C.W.\n Morgan (4), G. Garmire (3), and E. Agol (4) ((1) Department of Astronomy,\n University of Michigan, (2) Department of Astronomy and Center for Cosmology\n and Astroparticle Physics, The Ohio State University, (3) Department of\n Astronomy and Astrophysics, Pennsylvania State University, (4) Department of\n Physics, United States Naval Academy, (5) Department of Astronomy, University\n of Washington)",
"submitter": "Christopher S. Kochanek",
"url": "https://arxiv.org/abs/0906.4342"
}
|
0906.4359
|
# Systematic reduction of sign errors in many-body problems: generalization of
self-healing diffusion Monte Carlo to excited states
Fernando Agustín Reboredo Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831, USA
###### Abstract
A recently developed self-healing diffusion Monte Carlo algorithm [PRB 79,
195117] is extended to the calculation of excited states. The formalism is
based on an excited-state fixed-node approximation and the mixed estimator of
the excited-state probability density. The fixed-node ground state wave-
functions of inequivalent nodal pockets are found simultaneously using a
recursive approach. The decay of the wave-function into lower energy states is
prevented using two methods: i) The projection of the improved trial-wave
function into previously calculated eigenstates is removed; and ii) the
reference energy for each nodal pocket is adjusted in order to create a kink
in the global fixed-node wave-function which, when locally smoothed, increases
the volume of the higher energy pockets at the expense of the lower energy
ones until the energies of every pocket become equal. This reference energy
method is designed to find nodal structures that are local minima for
arbitrary fluctuations of the nodes within a given nodal topology. It is
demonstrated in a model system that the algorithm converges to many-body
eigenstates in bosonic and fermionic cases.
###### pacs:
02.70.Ss,02.70.Tt
## I Introduction
Although several important chemical and physical properties of matter are
determined by the lowest energy electronic configuration (or ground state), a
significant number of physical properties are crucially dependent on the
excitation spectra. These properties range from electronic optical excitations
to transport and thermodynamic behavior.
While elegant theories that take advantage of the variational principle have
been formulated for the ground state, hohenberg ; kohn the theories on the
excitation spectra are far more complex. hedin65 Therefore, although excited
states are extremely important, our understanding of them is limited as
compared with the ground state.
Diffusion quantum Monte Carlo (DMC) is the method of choice to obtain the
ground state energy of systems with more than $\sim\\!20$ electrons. The DMC
algorithm ceperley80 transforms the calculation of an excited state (e.g.,
the fermionic ground state) into a ground state calculation. The accuracy of
the method depends, however, on a previous estimate of the zeros (nodes) of
the wave-function.
The ground state wave-function of most many-body Hamiltonians $\mathcal{H({\bf
R})}$ is a bosonic (symmetric) wave-function without nodes. Any other
eigenstate of a many-body Hamiltonian $\mathcal{H({\bf R})}$ must have nodes
in order to be orthogonal to the bosonic ground state. In the case of fermions
(e.g., electrons), the ground state must be antisymmetric. Therefore, the
electronic ground state is an excited state of the many-body Hamiltonian
$\mathcal{H({\bf R})}$ and must have nodes (hyper-surfaces in $3N_{e}$ space
where the wave-function becomes zero and changes sign, being $N_{e}$ the
number of particles).
The standard diffusion Monte Carlo (DMC) approach ceperley80 finds the lowest
energy $E^{DMC}_{T}$ of all the wave-functions that share the nodes
$S_{T}({\bf R})$ of a trial wave-function $\Psi_{T}({\bf R})$, where ${\bf R}$
is a point in the $3N_{e}$ coordinate space. This lowest energy wave-function
is denoted as the fixed-node ground state $\Psi_{FN}({\bf R})$.
Since “no nodes” is a condition easy to satisfy, the ground state energy of a
bosonic system can be found with a precision limited only by statistical and
time-step errors. For any other eigenstate $\Psi_{n}({\bf R})$, a good
approximation of its nodal surface $S_{n}({\bf R})$ must be provided in order
to avoid systematic errors. Departures in $S_{T}({\bf R})$ from the exact
nodes $S_{n}({\bf R})$ cause, in general, errors of the energy as compared
with the exact eigenstate energy. foulkes99 For the fermionic ground state,
the standard DMC algorithm provides only an upper bound of the ground state
energy. anderson79 ; reynolds82 Moreover, if $\Psi_{n}({\bf R})$ is non
degenerate, any departure of $S_{T}({\bf R})$ from $S_{n}({\bf R})$ creates a
kink in the fixed-node ground state. keystone Accordingly, accurate many-body
calculations require methods to obtain and improve $S_{T}({\bf R})$. The
problem of searching the exact nodes $S_{n}({\bf R})$, the surfaces in
$3N_{e}$ space where the wave-function of an arbitrary eigenstate $n$ changes
sign, is one of the outstanding problems in condensed matter theory.
ceperley91
This paper is the natural conclusion of earlier work. In Ref. rosetta, we
showed that even the exact Kohn-Shamkohn wave-functions cannot be expected to
provide accurate nodal structures for DMC calculations. However, we also
showed that an optimal Kohn-Sham-like nodal potential exists. Subsequently in
Ref. keystone, we demonstrated that the nodes of the fermionic ground state
wave-function can be found in an iterative process by locally smoothing the
kinks of the fixed-node wave-function. We also showed that an effective nodal
potential can be found to obtain a compact representation of an optimized
trial wave-function with good nodes. While some details are rederived here,
reading those papers before this one is highlyfn:highly recommended.
In this paper the self-healing diffusion Monte Carlo method (SHDMC) is
extended to find the nodes, wave-functions, and energies of low-energy eigen-
states of bosonic and fermionic systems.
## II The simple SHDMC algorithm for the ground state
This paper describes how to extend the “simple SHDMC algorithm” (as described
in Section III.C of Ref. keystone, ) to excited states. An extension to
optimize the multi-determinant expansion, (see Section IV in Ref. keystone, )
is clearly possible and will be explained elsewhere.
The ground state SHDMC algorithm builds upon the importance sampling DMC
method. ceperley80 The standard diffusion Monte Carlo approach is based on
the Ceperley-Alderceperley80 equation: units
$\displaystyle\frac{\partial f({\bf R},\tau)}{\partial\tau}\\!$
$\displaystyle=$ $\displaystyle\\!{\bf\nabla_{R}^{2}}f({\bf
R},\tau)-\\!{\bf\nabla_{R}}\left(f({\bf
R},\tau){\bf\nabla_{R}}ln\left|\Psi_{T}({\bf R})\right|^{2}\right)$ (1)
$\displaystyle-\left[E_{L}({\bf R})-E_{T}\right]f({\bf R},\tau)\;,$
where $E_{L}({\bf R})=[\hat{\mathcal{H}}\Psi_{T}({\bf R})]/\Psi_{T}({\bf R})$
is the “local energy,” $\hat{\mathcal{H}}$ is the many-body Hamiltonian
operator, ${\bf R}$ denotes a point in $3N_{e}$ space, and $E_{T}$ is a
reference energy. Equation (1) is often solved numericallyceperley80 using a
large number $N_{c}$ of electron configurations (or walkers) which are points
${{\bf R}_{i}}$ in the $3N_{e}$ space. These walkers i) randomly diffuse
according to the first term in Eq. (1) and ii) drift according to the second
term a time $\delta\tau$. In addition, iii) the walkers branch {or pass on}
with probability $p=1-\exp[(E_{L}({\bf R})-E_{T})\delta\tau]$ {or
$p=\exp[(E_{L}({\bf R}_{i})-E_{T})\delta\tau]-1$ }. To prevent large
fluctuations in the population of walkers and excessive branching or killing,
often a statistical weight is assigned to each walker. A detailed review of
the numerical methods used for minimizing errors and accelerating DMC
calculations is given in Ref. umrigar93, .
In the limit of $\tau\rightarrow\infty$, the distribution function of the
walkers in an importance sampling DMC algorithm is given byceperley80
$\displaystyle f({\bf R},\tau\rightarrow\infty)$ $\displaystyle=$
$\displaystyle\Psi_{T}^{*}({\bf R})\Psi_{FN}({\bf
R})\;e^{-(E^{DMC}_{T}-E_{T})\tau}$ $\displaystyle=$
$\displaystyle\lim_{N_{c}\rightarrow\infty}\lim_{j\rightarrow\infty}\frac{1}{N_{c}}\sum_{i}^{N_{c}}W_{i}^{j}(j)\;\delta\left({\bf
R-R}_{i}^{j}\right)\;.$
The ${\bf R}_{i}^{j}$ in Eq. (II) correspond to the positions of walker $i$ at
the step $j$ for an equilibrated DMC run of $N_{c}$ configurations. The
original SHDMC method for the ground state was implemented in a mixed
branching with weights scheme. For reasons that will be clear below, it is
easier to formulate a method for excited states with a constant number of
walkers with weights $W_{i}^{j}(k)$ which are given by
$W_{i}^{j}(k)=e^{-\left[E_{i}^{j}(k)-E_{T}\right]k\;\delta\tau},$ (3)
with $k$ being a number of steps, $\delta\tau$ the time step, and
$E_{i}^{j}(k)=\frac{1}{k}\sum_{\ell=0}^{k-1}E_{L}({\bf R}_{i}^{j-\ell})\;.$
(4)
The energy reference $E_{T}$ in Eq. (3) is adjusted so that
$\sum_{i}W_{i}^{j}(k)\approx N_{c}$ assuming a constant $E_{T}$ for $k$ steps.
Note that setting all $W_{i}^{j}(k)=1$ in Eq. (II) gives at equilibrium, by
construction, a distribution $f({\bf R})=|\Psi_{T}({\bf R})|^{2}$, because
this is equivalent to setting $E_{L}({\bf R})=E_{T}$ in Eq. (1). If one sets
the initial distribution of walkers as $f({\bf R},0)=|\Psi_{T}({\bf R})|^{2}$,
then the distribution of walkers at imaginary time $\tau=k\delta\tau$ is given
by
$\displaystyle f({\bf R},\tau)$ $\displaystyle=$ $\displaystyle\Psi_{T}({\bf
R})\left[e^{-\tau\hat{\mathcal{H}}_{FN}}\Psi_{T}({\bf R})\right]$
$\displaystyle=$ $\displaystyle\Psi_{T}({\bf R})\Psi_{T}({\bf R},\tau)$
$\displaystyle=$
$\displaystyle\lim_{N_{c}\rightarrow\infty}\frac{1}{N_{c}}\sum_{i}^{N_{c}}W_{i}^{j}(k)\delta\left({\bf
R-R}_{i}^{j}\right)\;.$
Therefore, at equilibrium and in a no branching approach, the weights
$W_{i}^{j}(k)$ contain all the difference between $f({\bf R},\tau)$ and
$|\Psi_{T}({\bf R})|^{2}$ . In Eq. (II) $e^{-\tau\hat{\mathcal{H}}_{FN}}$ is
the fixed-node evolution operator, which is a function of the fixed-node
Hamiltonian operator $\hat{\mathcal{H}}_{FN}$ given by
$\hat{\mathcal{H}}_{FN}=\hat{\mathcal{H}}-E_{T}+\\!\infty\
\lim_{\epsilon\rightarrow 0}\theta\left\\{\epsilon-d_{m}[S_{T}({\bf
R^{\prime}})-{\bf R}]\right\\}\;.$ (6)
The third term in the right-hand side of Eq. (6) adds an infinite potential at
the points ${\bf R}$ with minimum distance to any point of the nodal surface
$d_{m}[S_{T}({\bf R^{\prime}})-{\bf R}]$ smaller than $\epsilon$. fn:nodelta
Using Eq. (II) one can formally obtain
$\langle{\bf R}|\Psi_{T}(\tau)\rangle=\Psi_{T}({\bf
R},\tau)=e^{-\tau\hat{\mathcal{H}}_{FN}}\Psi_{T}({\bf R})=\frac{f({\bf
R},\tau)}{\Psi_{T}({\bf R})}\;,$ (7)
and using Eq. (II) one obtains
$\langle{\bf R}|\Psi_{FN}\rangle=\Psi_{FN}({\bf
R})=\lim_{\tau\rightarrow\infty}\Psi_{T}({\bf
R},\tau)e^{(E^{DMC}_{T}-E_{T})\tau}\;.$ (8)
The trial wave-function $\Psi_{T}({\bf R})$ is commonly a product of an
antisymmetric function $\Phi_{T}({\bf R})$ and a Jastrowfn:Jastrow factor
$e^{J({\bf R})}$. Often $\Phi_{T}({\bf R})$ is a truncated sum of Slater
determinants or pfaffians $\Phi_{n}({\bf R})$:
$\langle{\bf R}|\Psi_{T}\rangle=\Psi_{T}({\bf R})=e^{J({\bf
R})}\sum_{n}^{\sim}\lambda_{n}\Phi_{n}({\bf R})\;.$ (9)
In Ref. keystone, we proved that we can evaluate
$e^{-\tau\hat{\mathcal{H}}}|\Psi_{T}\rangle$ for $\tau\rightarrow\infty$ using
a numerically stable algorithm. The analytical derivation of the
algorithmkeystone can be summarizedfn:highly here as
$\displaystyle|\Psi_{0}\rangle$ $\displaystyle=$
$\displaystyle\lim_{\tau\rightarrow\infty}e^{-\tau\hat{\mathcal{H}}}|\Psi_{T}^{\ell=0}\rangle$
(10) $\displaystyle=$
$\displaystyle\lim_{\stackrel{{\scriptstyle\ell\rightarrow\infty}}{{\tau\rightarrow\infty}}}\prod_{\ell}(e^{-\delta\tau^{\prime}\hat{\mathcal{H}}}e^{-\tau\hat{\mathcal{H}}^{(\ell-1)}_{FN}})|\Psi_{T}^{\ell=0}\rangle$
$\displaystyle=$
$\displaystyle\lim_{\stackrel{{\scriptstyle\ell\rightarrow\infty}}{{\tau\rightarrow\infty}}}\prod_{\ell}(\tilde{D}e^{-\tau\hat{\mathcal{H}}^{(\ell-1)}_{FN}})|\Psi_{T}^{\ell=0}\rangle$
$\displaystyle=$ $\displaystyle|\Psi_{T}^{\ell\rightarrow\infty}\rangle\;.$
The operator $\tilde{D}$ is defined in Eq. (16). Equation (II) means that the
ground state $|\Psi_{0}\rangle$ fn:groundstate can be obtained recursively by
generating a new trial wave-function $|\Psi_{T}^{\ell}\rangle$ from a fixed-
node DMC calculation that uses the previous trial wave-function
$|\Psi_{T}^{\ell-1}\rangle$, which is given by
$\displaystyle|\Psi_{T}^{\ell}\rangle$ $\displaystyle=$
$\displaystyle\tilde{D}\lim_{\tau\rightarrow\infty}e^{-\tau\mathcal{H}^{(\ell-1)}_{FN}}|\Psi_{T}^{\ell-1}\rangle$
$\displaystyle=$ $\displaystyle\tilde{D}|\Psi_{FN}^{\ell}\rangle\;.$
Equation (II) means that new coefficients $\lambda_{n}$ of a truncated
expansion of a trial wave-function of the form given in Eq. (9) are obtained
numerically from the distribution of walkers of a DMC run as
$\langle\lambda_{n}\rangle=\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}W_{i}^{j}(k\gg
1)\;\xi_{n}^{*}({\bf R}_{i}^{j})\;\gamma({\bf R}_{i}^{j})\;,$ (13)
where
$\xi_{n}({\bf R})=e^{-2J({\bf R})}\frac{\Phi_{n}({\bf R})}{\Phi_{T}({\bf R})}$
(14)
and keystone ; umrigar93
$\gamma({\bf R})=\frac{-1+\sqrt{1+2|{\bf v}|^{2}\tau}}{|{\bf
v}|^{2}\tau}\text{ with }{\bf v}=\frac{\nabla\Psi_{T}({\bf R})}{\Psi_{T}({\bf
R})}\;.$ (15)
A complete explanation of our method is given in Ref. keystone, . Briefly
here, our method systematically improves the nodes for three main reasons:
1) The projectors in Eq. (14) include only functions $\Phi_{n}({\bf R})$ that
retain all symmetries of the ground state. In more technical terms, the ground
state is expanded only with functions that belong to the same irreducible
representation. This means that if the $\Phi_{n}({\bf R})$ are determinants,
for example, the bosonic ground state is excluded. Therefore, fluctuations
that depart from the fermionic Hilbert space are filtered and do not propagate
into the trial wave-function from one DMC run to the next SHDMC iteration.
2) The projection of $\Psi_{FN}({\bf R})$ into a finite set of $\Phi_{n}({\bf
R})$ with low non-interacting energy can be shownkeystone to be equivalent to
locally smoothing the kinks at the node of the fixed-node wave-function with a
function of the form
$\langle{\bf R}|\tilde{D}|{\bf R^{\prime}}\rangle=\tilde{\delta}\left({\bf
R,R^{\prime}}\right)=\sum_{n}^{\sim}\Phi_{n}({\bf R})\Phi_{n}^{*}({\bf
R^{\prime}})\;.$ (16)
We proved that a large class of local smoothing functions have the same effect
on the nodes as a Gaussian, under certain conditions, which includes the case
of Eq. (16). In turn, in Ref. keystone, we proved that, to linear order in
$\sqrt{\delta\tau^{\prime}}$, the convolution of a Gaussian with any
continuous function has the same effect on the nodes as the imaginary time
propagator $e^{-\delta\tau^{\prime}\hat{\mathcal{H}}}$ [this allows replacing
Eq. (10) by Eq. (II)].
Thus our method can be viewed as the recursive application of two operators on
the trial wave-function: i) $e^{-\tau\mathcal{H}_{FN}}$ that turns
$|\Psi_{T}\rangle$ into $|\Psi_{FN}\rangle$ and ii) $\tilde{D}$ that samples
and truncates the expansion and changes the nodes as
$e^{-\tau\hat{\mathcal{H}}}$. Accordingly, our method is formally related to
the shadow wave-function shadow and the A-function approach bianchi93 ;
bianchi96 [see Eq. (10)].
3) Finally, we argued that the method is robust against statistical noise,
because the kink should increase with the distance between the exact node
$S({\bf R})$ and the node of the trial wave-function $S_{T}({\bf R})$ [the
kink must disappear for $S_{T}({\bf R})=S({\bf R})$]. In addition, we took the
relative error in $\lambda_{n}$ as truncation criterion for $\tilde{D}$.
## III Extension of the Self-Healing DMC algorithm to excited states
A detailed explanation of the advantages and limitations of the standard
fixed-node approximation for excited states is given in Ref. foulkes99, This
paper explores the possibility of overcoming these limitations in calculating
excited states by excluding the projection of lower energy states from the set
of $\xi_{n}({\bf R})$. However, in to follow this path the problem of
inequivalent nodal pockets has to be addressed.
### III.1 Inequivalent nodal pockets
The expression “nodal pocket” denotes a volume in $3N_{e}$ space enclosed by
the nodal surface $S_{T}({\bf R})$. It has been shown ceperley91 that the
ground state of any fermionic Hamiltonian with a local potential has nodal
pockets that belong to the same class, meaning that the complete $3N_{e}$
space can be covered by applying all symmetry operations (e.g., particle
permutations) to just one nodal pocket. Therefore, if the trial wave-function
is obtained from such a Hamiltonian, all nodal pockets are equivalent by
symmetry. For the ground state, one can obtain the fixed-node wave-function in
just one pocket and map it to the rest of the $3N_{e}$ space using
permutations of the particles and other symmetries of $\hat{\mathcal{H}}$.
In the case of arbitrary excited states, there are inequivalent nodal pockets
that present a challenge to the fixed-node approach. HLRbook Due to this
inequivalent pocket problem, alternatives to the fixed-node method and
variations have been tried. ceperley88 ; barnett91 ; blume97 ; dasilva01 ;
nightingale00 ; luchow03 ; schautz04 ; umrigar07 ; purwanto09 Self-healing
DMCkeystone implicitly takes advantage of the equivalence of nodal pockets in
the fermionic ground state and must be extended to the inequivalent pocket
case. For this reason a nonbranching formulation is used in the excited state
case.
### III.2 Equilibration of walkers in inequivalent nodal pockets
A first complication, which has a simple solution, of the nonbranching fixed-
node approximation is that the number of walkers in each nodal pocket is also
fixed by the nodes. As a consequence of the drift or “quantum force” term
[second term in Eq. (1)], the walkers are repelled from the regions where the
wave-function is zero and they cannot cross the node for
$\delta\tau\rightarrow 0$. The fact that the population in each nodal pocket
is fixed has no consequence for the ground state because all nodal pockets are
equivalent. For the ground state it is not important in which nodal pocket the
walker is trapped because particle permutations can move every walker into the
same nodal pocket and the projectors $\xi_{n}({\bf R})$ in Eq. (14) are
invariant under such permutations.
However, in the case of excited states, which have more nodes than those
required by symmetry, fn:permutations there are inequivalent nodal pockets.
In a nonbranching DMC scheme with weights, the population is locked from the
start in a set of pockets. If the initial distribution of $N_{c}$ walkers is
chosen with a Metropolis algorithm to match $|\Psi_{T}({\bf R})|^{2}$, there
would be random variations in the starting population of the order of
$\sqrt{N_{c}/N_{p}}$, where $N_{p}$ is the number of inequivalent nodal
pockets. This would cause systematic errors if the wave-function coefficients
$\lambda_{n}$ were sampled without taking preventive measures. Moreover, even
if the initial numbers of walkers in each pocket were set “by hand” (to be
proportional to the integral $|\Psi_{T}({\bf R})|^{2}$ in each pocket), the
resolution of the sampling cannot be better than $1/N_{c}$. The importance of
this error grows if $N_{c}$ is small or if the number of inequivalent nodal
pockets is large.
To prevent this error from occurring, some walkers are simply allowed to cross
the node after the wave-function coefficients are sampled. At the end of a
sub-block of $k$ steps, for every walker $i$ at ${\bf R}_{i}$, a random move
${\bf\Delta R}_{i}$ is generated with a Gaussian distribution using
$\sigma^{2}=\delta\tau^{\prime}$, without the drift velocity contribution.
This move is accepted only if the wave function changes sign with a Metropolis
probability $p=\max\left\\{1,[\Psi_{T}({\bf R}_{i}{\bf+\Delta
R})/\Psi_{T}({\bf R}_{i})]^{2}\right\\}$. This ensures that i) the
distribution of walkers remains proportional to $|\Psi_{T}({\bf R})|^{2}$ and
ii) the average number of walkers in each pocket is proportional to the
integral of $|\Psi_{T}({\bf R})|^{2}$ as the number of sub-blocks $M$ tends to
$\infty$.
### III.3 Unequal fixed-node energies in inequivalent nodal pockets
A second complication of the fixed-node approach for the general case of
excited states appears because small departures of $S_{T}({\bf R})$ from the
exact nodes $S_{n}({\bf R})$ often will result in inequivalent nodal pockets
having fixed-node solutions with different fixed-node energies. When nodal
pockets are not equivalent, a standard DMC algorithm will converge to a
“single nodal pocket” population. In this case, the lowest energy pocket will
contain all the walkers in a branching algorithm [or all significant weights
($W_{i}^{j}(k)\neq 0$ )]. Accordingly, the average energy sampled will
correspond to the lowest energy nodal pocket, which will be different from
that of the true excited-state energy (see Chapter 6 in Ref. HLRbook, and
references therein).
If the coefficients of an excited-state fixed-node wave-function are sampled
with the same procedure used for the ground statekeystone [see Eq. (13)],
they would correspond to a function that is different from zero just at the
class of nodal pockets with lowest DMC energy and zero everywhere else. This
function will not be, in general, orthogonal to the lower energy states.
Moreover, this will result in kinks at the nodes in the wave-function sampled
with Eq. (13) between lowest energy nodal pockets and inequivalent ones.
A first preventive measure to avoid a single pocket population is to avoid the
limit $\tau\rightarrow\infty$ in Eqs. (II) and (II) which replaces
$|\Psi_{FN}^{\ell}\rangle$ by
$e^{-k\delta\tau\mathcal{H}^{(\ell-1)}_{FN}}|\Psi_{T}^{\ell-1}\rangle$ in Eq.
(II). As a result $k$ in Eq. (13) is limited to small values, which brings all
values of $W_{i}^{j}(k)$ closer to $1$. Since the approach is recursive, the
limit of $\tau\rightarrow\infty$ is reached as $\ell\rightarrow\infty$ (since
successive applications of the algorithm are accumulated in
$|\Psi_{T}^{\ell}\rangle$). In addition, to prevent the wave-function from
falling into lower energy states, two techniques are used: i) direct
projection and ii) unequal reference energies.
### III.4 Direct projection
While the trial wave-function can be forced to be orthogonal to the ground
state, or any other excited state calculated before, the fixed-node wave-
function can develop a projection into lower energy states, because the DMC
algorithm only requires $\Psi_{FN}({\bf R})$ to be zero at the nodes
$S_{T}({\bf R})$. To prevent excited states from drifting into lower energy
states, let me assume, for a moment, that approximated expressions of the
excited states $\langle{\bf
R}|e^{\hat{J}}|\breve{\Phi}_{n}\rangle=\Psi_{n}({\bf R})=e^{J({\bf
R})}\breve{\Phi}_{n}({\bf R})$ with $n\leq\nu$ can be obtained and used to
build the projector
$\hat{P}=e^{\hat{J}}\left[1-\sum_{n}^{\nu}|\breve{\Phi}_{n}\rangle\langle\breve{\Phi}_{n}^{*}|\right]e^{-\hat{J}}\;\;,$
(17)
where the operator $e^{\hat{J}}$ is the multiplication by a Jastrow. Since the
$|\breve{\Phi}_{n}\rangle$ shall be obtained statistically, they will have
errors and will not form an orthogonal basis in general. Therefore,
$\langle\breve{\Phi}_{n}^{*}|$ are elements of the conjugated basis that
satisfy $\langle\breve{\Phi}_{n}^{*}|\breve{\Phi}_{m}\rangle=\delta_{n,m}$.
They can be constructed inverting the overlap matrix
$S_{n,m}=\langle\breve{\Phi}_{n}|\breve{\Phi}_{m}\rangle$ as
$\langle\breve{\Phi}_{n}^{*}|=\sum_{m}S^{-1}_{n,m}\langle\breve{\Phi}_{m}|\;.$
(18)
Then, the extension of the self-healing algorithm to the next excited
$|\Psi_{\nu+1}\rangle$ can be rederived analytically as follows:
$\displaystyle|\Psi_{\nu+1}\rangle$ $\displaystyle=$
$\displaystyle\lim_{\tau\rightarrow\infty}\hat{P}\;e^{-\tau\hat{\mathcal{H}}}\hat{P}|\Psi_{T,\nu+1}^{\ell=0}\rangle$
$\displaystyle=$
$\displaystyle\lim_{\ell\rightarrow\infty}\hat{P}\;\prod_{\ell}\left(e^{-(\delta\tau^{\prime}+k\delta\tau)\hat{\mathcal{H}}}\hat{P}\right)|\Psi_{T,\nu+1}^{\ell=0}\rangle$
$\displaystyle=$
$\displaystyle\lim_{\ell\rightarrow\infty}\hat{P}\;\prod_{\ell}\left(e^{-\delta\tau^{\prime}\hat{\mathcal{H}}}e^{-k\delta\tau\hat{\mathcal{H}}_{FN}^{(\ell-1)}}\hat{P}\right)|\Psi_{T,\nu+1}^{\ell=0}\rangle$
$\displaystyle\simeq$
$\displaystyle\lim_{\ell\rightarrow\infty}\hat{P}\;\prod_{\ell}\left(\tilde{D}e^{-k\delta\tau\hat{\mathcal{H}}^{(\ell-1)}_{FN}}\hat{P}\right)|\Psi_{T,\nu+1}^{\ell=0}\rangle$
$\displaystyle=$
$\displaystyle|\Psi_{T,\nu+1}^{\ell\rightarrow\infty}\rangle.$
Equation (III.4) means that for any initial trial wave-function
$|\Psi_{T,\nu+1}^{\ell=0}\rangle$ with
$\hat{P}|\Psi_{T,\nu+1}^{\ell=0}\rangle\neq 0$, one can obtain the next
excited state $|\Psi_{\nu+1}\rangle$ recursively. The numerical implementation
of the algorithm for excited states (see Section IV for details) is almost
identical to the ground state versionkeystone with three differences: i)
there is no branching and the product $k\delta\tau$ is chosen so as
$W_{i}^{j}(k)\simeq 1$ [see Eq. (13)], ii) the projection of the vector of
coefficients $\lambda_{n}$ into the ones corresponding to eigenstates
calculated earlier is removed with $\hat{P}$, and iii) some walkers cross the
node after $k$ time steps (see above).
Eq. (III.4) holds in the limit of $N_{c}\rightarrow\infty$,
$\delta\tau\rightarrow 0$, $\delta\tau^{\prime}\rightarrow 0$, $\ell
k\delta\tau\rightarrow\infty$, and $\ell\delta\tau^{\prime}\rightarrow\infty$.
In the derivation of Eq. (III.4), the following properties were used:
$\hat{P}^{2}=\hat{P}$, and $[\hat{\mathcal{H}},\hat{P}]\simeq 0$. In Ref.
keystone, it was shown that, under certain conditions,
$S\left[e^{-\delta\tau^{\prime}\hat{\mathcal{H}}}e^{-k\delta\tau\hat{\mathcal{H}}^{(\ell-1)}_{FN}}\hat{P}|\Psi_{T}^{\ell}\rangle\right]\simeq
S\left[\tilde{D}e^{-k\delta\tau\hat{\mathcal{H}}^{(\ell-1)}_{FN}}\hat{P}|\Psi_{T}^{\ell}\rangle\right]\;;$
(20)
that is, the nodes of the two functions in the brackets are approximately the
same.
Note that the second term in brackets of Eq. (17) has precisely the form given
in Eq. (16). By construction, this term would generate a function with nodes
corresponding to a linear combination of lower energy eigenstates. The
projector $\hat{P}$, instead, excludes any change in the wave-functions
introduced by the projection and sampling operator $\tilde{D}$ or by
$e^{-\tau\mathcal{H}^{(\ell-1)}_{FN}}$ in the direction of lower energy wave-
functions (which includes their nodes).
### III.5 Adjusting the reference energy in each nodal pocket
If walkers at one side of the node have more weight than at the other (because
of inequivalent pockets with different fixed-node energies), the propagated
wave-function obtained by sampling the walkers will be multiplied by a larger
(smaller) factor for the low (high) energy side of the nodal surface. This
generates an additional contribution to the kink at the node that, when
locally smoothed, increases the volume of lower energy pockets at the expense
of the higher energy ones, causing the volume of the lower (higher) energy
pockets to grow (diminish). This, in turn, will have an impact on the kinetic
energy: due to quantum confinement effects, the difference in fixed-node
energies will increase in the next iteration. This very interesting effect in
fact acts to our advantage by helping us to find the ground state even when
starting from a very poor wave-function. keystone For excited states, this
effect is prevented by i) limiting the maximum value of $k$ and ii) the
projector $\hat{P}$ in Eq. (III.4). However, the eigenstates
$|\Psi_{n}\rangle$ will have statistical errors that can create systematic
errors in the higher states. To partially prevent these errors, and to limit
the number of orthogonality constraints, the energy reference can be changed
in order to invert this contribution to the kink to our advantage.
While a single reference energy $E_{T}$ can still be used for the DMC run in
each block, the projectors of Eq. (13) are redefined using a reference energy
dependent on the nodal pocket. In addition, following a suggestion of C.
Umrigar, umrigar_private the change in the coefficients $\delta\lambda_{n}$
is sampled instead of the total value $\lambda_{n}$.
$\displaystyle\lambda_{n}^{\ell}$ $\displaystyle=$
$\displaystyle\lambda_{n}^{\ell-1}+\langle\delta\lambda_{n}\rangle$ (21)
$\displaystyle\langle\delta\lambda_{n}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}(W_{i}^{j}(k)e^{-\beta\left[E_{T}-\bar{E}_{i}^{j}(j_{0})\right]k\;\delta\tau}-1)\;\xi_{n}^{*}({\bf
R}_{i}^{j})\;\gamma({\bf R}_{i}^{j})\;,$
where $\beta$ is an adjustable parameter and
$\bar{E}_{i}^{j}(j_{0})=\frac{\sum_{m=j_{0}}^{j}W_{i}^{m}(k)\gamma({\bf
R}_{i}^{m})E_{L}({\bf R}^{m}_{i})}{\sum_{m=j_{0}}^{j}W_{i}^{m}(k)\gamma({\bf
R}_{i}^{m})}$ (22)
is the weighted average of the local energy during the lifetime of the walker
$i$ since the start of the block or the last time it crossed the node at step
$j_{0}$. If $\beta=1$ is selected in Eq. (21), the factor
$e^{-\beta[E_{T}-\bar{E}_{i}^{j}(j_{0})]}$ just replaces in the definition of
the weights [see Eq. (3)] $E_{T}$ by $\bar{E}_{i}^{j}(j_{0})$. The energy
$\bar{E}_{i}^{j}(j_{0})$ for $j-j_{0}\gg k$ is expected to converge to the
fixed-node energy of the nodal pocket where the walker $i$ is trapped;
however, only the last two-thirds of the block are used to accumulate values
to allow $\bar{E}_{i}^{j}(j_{0})$ to equilibrate.
It was argued before that, for $\beta=0$, the differences in the fixed-node
energies of neighboring nodal pockets create a contribution to the kink that,
when locally smoothed, increases the volume of nodal pockets with low fixed-
node energy. For $\beta>1$, it is likely that this contribution to the kink is
inverted so that the volume of the lower (higher) energy pockets is reduced
(increased) by the smoothing function (16). Therefore, it can be assumed that
a value of $\beta>1$ should stabilize the higher energy nodal pockets,
increasing their volume and, thus, reducing their energy. This process will
stop when the fixed-node energy of all nodal pockets becomes equal.
Note that by introducing this artificial contribution to the kink, one may
stabilize some nodal structures, preventing nodal fluctuations that reduce the
energy of one nodal pocket at the expense of the others. However, fluctuations
that lower the energy of every nodal pocket are not prevented. Therefore, if
several eigenstates have the same nodal topology, higher energy states could
drift into lower energy ones if orthogonality constraints [see Eq. (17)] are
not imposed.
Finally, note that choosing $\beta>1$ can also cause problems if the quality
of the wave-function is not good or if the statistics is poor. For example, a
small statistical fluctuation in the values of $\lambda_{n}$ could create a
new nodal pocket with high energy. In successive blocks (as $\ell$ increases),
this pocket will grow at the expense of the others, causing the total energy
to rise.
## IV SHDMC algorithm for excited states
A basis of $\Phi_{n}({\bf R})$ must be constructed, taking advantage of all
the symmetries of $\hat{\mathcal{H}}$.fn:permutations The $\Phi_{n}({\bf R})$
should be selected to be eigen-functions of a noninteracting many-body system
keystone belonging to the same irreducible representation for every symmetry
group of $\hat{\mathcal{H}}$. The calculation must be repeated for each
irreducible representation. Note that the same algorithm is used for bosons or
fermions: the only difference is the basis used to expand the wave-functions.
The calculation of excited states with SHDMC is composed of a sequence of
blocks. Each block $\ell$ has $M$ sub-blocks with $k$ standard DMC steps.
The basic algorithm is the following:
1. 1.
An initial set of coefficients for the expansion of the trial wave-function is
selected.
2. 2.
The changes $\delta\lambda_{n}$ are accumulated [see Eqs. (14) and (21)] at
the end of each sub-block. Some walkers near the node can cross it at the end
of each sub-block.
3. 3.
At the end of each block $\ell$, the error in $\delta\lambda_{n}$ is
evaluated. If this error is larger than 25% of
$\lambda_{n}+\delta\lambda_{n}$, then $\lambda_{n}$ is set to zero; keystone
otherwise, $\lambda_{n}$ is set to $\lambda_{n}+\delta\lambda_{n}$.
4. 4.
A new trial wave-function is constructed at the end of each block $\ell$ using
the new values of the coefficients sampled after removing with $\hat{P}$ the
projection into eigenstates calculated earlier.
5. 5.
If the scalar product between the vector of new $\delta\lambda_{n}$ with the
one obtained in the previous block ($\ell-1$) is positive, the number of sub-
blocks $M$ is increased by one. Otherwise, $M$ is multiplied by a factor
larger than one (e.g., $1.25$). This factor increases the statistics reducing
the impact of noise. fn:changes
6. 6.
Steps 2-6 are repeated until the variance of the weights $W_{i}^{j}(k)$ is
smaller than a prescribed tolerance (see Fig. 6 in Section V).
7. 7.
The projector $\hat{P}$ is updated to include the new excited state.
8. 8.
Steps 1-7 are repeated until a desired number of excited states is obtained.
### IV.1 Remarks
Some points about the application of the algorithm should be addressed before
discussing the results.
* •
In this paper, to test the method, intentionally poor trial wave-functions
have been selected as a starting point. Good initial wave-functions and a good
Jastrow are advised in real production runs in large systems. Methods to
select good initial trial wave-functions will be discussed elsewhere.
* •
Time-step errors and, in particular, persistent walker configurationsumrigar93
can cause significant problems. When this happens it often results in an
increase in the error bar of every $\lambda_{n}$ which causes a large
reduction in the number of coefficients retained in the trial wave-function.
This problem is avoided in the algorithm by discarding the entire block if a
50% reduction in the number of basis functions retained is detected.
Nevertheless, if the quality of the initial $\Psi_{T}({\bf R})$ is bad, it is
strongly recommended to reduce the time step $\delta\tau$. As the quality of
the wave-function improves with successive iterations, one can increase
$\delta\tau$. For fast convergence $\sqrt{k\;\delta\tau}$ should be of the
order of the interparticle distance.
* •
As a strategy, it is better to run at first using $\beta=0$ in Eq. (21)
including every state calculated before in $\hat{P}$ [see Eq. (17)]. Once the
wave-function $\Psi_{T}({\bf R})$ is converged, one can set $\hat{P}=1$ and
$\beta=1$ and monitor if $\Psi_{T}({\bf R})$ evolves into a subset of lower
energy states. To prevent the propagation of errors of every lower energy
state included in $\hat{P}$ into the next excited state, a run including only
this subset in $\hat{P}$ can be performed.
* •
To obtain accurate total energies, a long run with large $k$ is required (this
is almost a standard DMC run).
* •
SHDMC should not be used blindly as a library routine. The calculation of
excited states with SHDMC is a task that will probably remain limited to
quantum Monte Carlo experts. While, in contrast, density functional
approximated methods have suddenly become very easy to use, it is not quite
clear to the author that requiring expertise and a deep understanding is a
disadvantage. Any new code using SHDMC should be tested in a small system
where analytical solutions or results with an alternative approachumrigar07
are available. The comparison with a soluble model is presented in the next
section.
## V Applications to Model Systems
This section compares the methods described above for the calculation of
excited states with SHDMC, with full configuration interaction (CI)
calculations in the model system used in Refs. rosetta, and keystone, .
Briefly, the lower energy eigenstates are found for two electrons moving in a
two dimensional square with a side length $1$ with a repulsive interaction
potential of the formunits $V({\bf r},{\bf
r^{\prime}})=8\pi^{2}\gamma\cos{[\alpha\pi(x-x^{\prime})]}\cos{[\alpha\pi(y-y^{\prime})]}$
with $\alpha=1/\pi$ and $\gamma=4$. The many-body wave-function is expanded in
functions $\Phi_{n}({\bf R})$ that are eigenstates of the noninteracting
system. The $\Phi_{n}({\bf R})$ are linear combinations of functions of the
form $\prod_{\nu}\sin(m_{\nu}\pi x_{\nu})$ with $m_{\nu}\leq 7$. Full CI
calculations are performed to obtain a nearly exact expression of the lower
energy states of the system $\Psi_{n}({\bf R})=\sum_{m}a_{m}^{n}\Phi_{m}({\bf
R})$.
We solve the problem both for the singlet and the triplet case. The singlet
state of this system is bosonic-like, since the ground state wave-function has
no nodes. The lowest energy excitations of the noninteracting problem
$\Phi_{n}({\bf R})$ that have the same symmetry (that is, that are invariant
under exchange of particles, and under all symmetry operations of the group
$D_{4}$) are selected to expand $\hat{\mathcal{H}}$. For the case of the
triplet, the wave-function must change sign for permutations of the particles.
The ground state is, however, degenerate (belongs to the $E$ representation of
$D_{4}$). The $E$ representation can be described by wave-function even (odd)
for reflections in $x$ and odd (even) for reflections in $y$. We choose the
wave-functions that are odd in the $x$ direction: belonging to a $D_{2}$
subgroup of the $D_{4}$ symmetry. For more details on the triplet ground state
calculations, see Refs. rosetta, and keystone, .
To facilitate the comparison with the full CI results, projectors
$\xi_{n}({\bf R})$ are constructed with the same basis functions used in the
CI expansion. For the same reason, no Jastrow function is used [$J=0$ in Eq.
(14)].
To test the method, poor initial trial wave-functions are intentionally chosen
as follows: For the ground state the lowest energy function of the
noninteracting system is selected. For the $n^{th}$ ($n=\nu+1$) excited state,
the initial trial wave-function $|\Psi_{T,n}^{\ell=0}\rangle$ was constructed
by completing the first $\nu$ columns of a determinant with the first $\nu+1$
coefficients of the $\nu$ eigenstates calculated before. Subsequently, the
vector of cofactors of the last column was calculated. The coefficients of
this vector are used to construct a trial wave-function orthogonal to all the
eigenstates calculated earlier.
Figure 1: (Color online) Self-healed DMC run obtained for successive
eigenstates belonging to the $A_{1}$ (trivial) irreducible representation of
the group $D_{4}$ in the singlet state. Black lines denote the average value
of the local energy. The horizontal blue dashed lines mark the energy of the
corresponding excitation in the full CI calculation.
Figure 1 shows the results of successive SHDMC runs for the singlet ground
state and the next $8$ excitations that belong to the same symmetry (total
spin $S=0$, and irreducible representation $A_{1}$ of the group $D_{4}$). The
SHDMC calculations were done using $N_{c}=200$ walkers with a sub-block length
$k=50$, a time step $\delta\tau=0.0002$, units $\delta\tau^{\prime}=0.002$
(for the ground state $\delta\tau^{\prime}=0$ ) and, $\beta=1$ in Eq. (21).
The lines in Fig. 1 join the values obtained for the weighted average of the
local energy $E_{L}({\bf R})$ for each time step. The horizontal dashed lines
mark the energy of the nearly analytical result obtained with full CI. The
agreement between SHDMC and full CI is extremely good. As higher energy
eigenstates are calculated however, and the number of nodal pockets and nodal
surfaces increases, time step errors start to play a dominant role. In
particular, for the $9^{th}$ excitation (not shown) $\delta\tau$ must be
reduced.
The occasional peaks (or drops) observable in the data are correlated with the
update of $\Psi_{T}({\bf R})$, and their reduction also reflects a systematic
improvement in the trial wave-function. At the end of each block, the trial
wave-function coefficients $\lambda_{n}$ are updated and all weights are reset
to 1. They gradually reach equilibrium values when new energies are sampled,
completing a sub-block of length $k$. As a result, at the beginning of each
block, the energy sampled is the average of the trial wave-function energy,
which is often different than the DMC energy sampled thereafter (but it can be
smaller or higher for a bad trial wave-function with small $N_{c}$).
One interesting result is that some orthogonality constraints are not required
to obtain some excited states. This is the case, for example, of the first
excited state calculated with $\beta=1$. This is presumably due to the fact
that the number of nodal pockets is different for the excited state and the
ground state and the decay path from the first excited state to the ground
state is obstructed by the formation of a kink between inequivalent nodal
pockets if a value of $\beta\approx 1$ is used. This is also the case for
states $6$ and $7$, which were obtained before state 5 despite the fact that
they have higher energy.
A similar effect is observed in some triplet excitations. Due to the choice of
initial trial wave-function and the kink induced by $\beta=1$, the $3^{rd}$
excitation is found before the $2^{nd}$, and the $5^{th}$ is obtained before
the $2^{nd}$ and the $4^{th}$. This interesting effect disappears if $\beta=0$
is chosen.
Table 1 shows the logarithm of the residual projection
$L_{rp}=\log\left(1-|\langle\Psi_{n}^{CI}|\Psi_{n}\rangle|\right)$ (23)
of the excited state wave-function $|\Psi_{n}\rangle$ sampled with SHDMC onto
the corresponding full CI result $|\Psi_{n}^{CI}\rangle$ as a function of the
number of iterations for different eigenstates. The states are ordered as they
first appear in the calculation.
In addition, Table 1 compares the values of the eigen-energies obtained with
CI and SHDMC. The agreement is very good. In some cases the difference is
larger than the error bar. This might signal that small nodal errors remain.
Note that there is no upper bound theorem for excited states but for the
ground state within an abelian irreducible representation.foulkes99
Table 1: Values obtained for $L_{rp}$ [see Eq. (23) ] for a total of (a) $4\times 10^{4}$ (b) $8\times 10^{4}$ and (c) $12\times 10^{4}$ DMC steps and corresponding eigen-energies for two electrons in a square box with a model interaction. The logarithm of the residual projection $L_{rp}$ of the SHDMC wave-function with the corresponding full result CI is given for different eigenstates belonging to the same symmetry of the ground state as a function of the number of steps used to sample the wave-function. The states are included in the order they were obtained. State | Spin | Rep. | $L_{rp}$ | $L_{rp}$ | $L_{rp}$ | CI | SHDMC |
---|---|---|---|---|---|---|---|---
| | | a | b | c | Energy | Energy |
0 | S | A1 | -14.84 | -15.05 | | 328.088 | 328.089 | (2)
1 | S | A1 | -6.80 | -8.85 | | 374.106 | 374.103 | (6)
2 | S | A1 | -7.23 | -8.69 | | 409.960 | 409.954 | (3)
3 | S | A1 | -4.42 | -6.07 | | 418.508 | 418.66 | (2)
4 | S | A1 | -3.65 | -5.01 | | 454.630 | 454.84 | (2)
6 | S | A1 | -.– | -4.85 | -6.22 | 477.019 | 477.100 | (5)
7 | S | A1 | -3.90 | -5.26 | | 492.216 | 491.98 | (1)
5 | S | A1 | -5.60 | -6.17 | | 468.854 | 468.845 | (13)
8 | S | A1 | -5.09 | -6.49 | | 503.805 | 503.92 | (1)
0 | T | E | -8.49 | -8.71 | | 342.137 | 342.191 | (5)
1 | T | E | -4.37 | -4.35 | | 385.908 | 387.80 | (1)
3 | T | E | -3.06 | -3.35 | | 422.670 | 423.60 | (2)
5 | T | E | -4.04 | -5.48 | | 438.791 | 438.70 | (1)
2 | T | E | -2.31 | -2.31 | | 411.887 | 416.07 | (1)
Figure 2: (Color Online) Logarithm of the residual projection [see Eq. (23)]
for the ground (square), first (diamond), second (up triangle) and third (down
triangle) eigenstates with $A_{1}$ symmetry and S=0.
Figure 2 shows $L_{rp}$ at the end of each block for the ground state and low-
lying excitations of the system as a function of the total number of SHDMC
steps. The calculations were done by first running $\sim\\!40\,000$ SHDMC
steps for each eigenstate before starting the calculation of the next.
Subsequently, an additional set of $\sim\\!40\,000$ SHDMC steps was run,
improving the projector $\hat{P}$. The kinks in the data around
$\sim~{}40\,000$ are due to the changes in the coefficients of the lower
energy states involved in $\hat{P}$ [see Eq. (17)].
One important conclusion of Table 1 and Figure 2 is that errors in the
determination of lower energy states calculated earlier only propagate
“locally” because of the orthogonality constraints in Eq. (17). This error
does not have a strong impact on much higher energy excitations. This is
apparently due to the fact that each newly calculated excitation tends to
occupy the Hilbert space left by lower excitations due to statistical error.
This is clear, for example, for the $5^{th}$ and $8^{th}$ excitations, which
have an error much smaller than several excitations calculated earlier (e.g.,
$3^{rd}$ and $4^{th}$). The error in the $3^{rd}$ and $4^{th}$ excitations is
mainly due to mixing among themselves. This result is important because it
means that the present method can be used to calculate several higher
excitations in spite of the errors in lower energy ones.
Figure 3: (Color online) Change in the values of the multi-determinant
expansion as the DMC self-healing algorithm progresses for the $5^{th}$
excited state of the singlet state of $A_{1}$ symmetry. Light gray colors
denote older coefficients, whereas darker ones denote more converged results.
The full CI results are highlighted in small red diamonds.
Figure 3 shows the evolution of the values of the coefficients
$\lambda_{n}^{\ell}$ of $|\Psi_{T}^{\ell}\rangle$ as a function of the
coefficient index $n$ for the $5^{th}$ excited state corresponding to the
singlet configuration of the $A_{1}$ representation of the group $D_{4}$. The
shade of gray is light for the older (small $\ell$) coefficients and deepens
to black for the final results (large $\ell$). The calculation started from a
trial wave-function orthogonal to the states calculated before as described
above.
The coefficients of the wave-function sampled with SHDMC overlap with the ones
obtained with full CI (see Table 1). Similar results are obtained for all the
other excited states calculated. An important observation is that the
coefficients $\lambda_{n}$ evolve continuously towards the exact solution,
which suggests the possibility of accelerated algorithms that extrapolate the
values of $\delta\lambda_{n}$.
Some eigenstates are significantly more difficult to calculate than others.
This is typically the case for eigenstates with similar eigenvalues (e.g., the
$6^{th}$ excitation in the singlet case). A bigger challenge, however, is when
$E_{L}({\bf R})$ is ill behaved, for example, the case of the $2^{nd}$,
$4^{th}$, and $6^{th}$ excitations of the triplet state. Even the full CI
wave-function with 300 basis functions has a large variance for $E_{L}({\bf
R})$. In that case the coefficients obtained with SHDMC and CI are different.
This is due to the fact that the two methods minimize different things: CI
minimizes
$\langle~{}\Psi_{n}~{}|(\hat{\mathcal{H}}-E_{n})^{2}|\Psi_{n}\rangle$ on a
truncated basis, and SHDMC minimizes $\int E_{L}({\bf R})f({\bf R},\tau){\bf
dR}$ with
$\langle\Psi_{T}|\hat{P}|\Psi_{T}\rangle=\langle\Psi_{T}|\Psi_{T}\rangle$.
Accordingly, the fact that the results are different indicates that neither
calculation, CI or SHDMC, is converged with the basis chosen. The $4^{th}$ and
$6^{th}$ excitations with E symmetry in the triplet case obtained with SHDMC
are a linear combination of the corresponding ones in full CI.
Figure 4: (Color online) Average of the local energy $E_{L}({\bf R})$ as a
function to the number of DMC time steps for two SHDMC runs with $\hat{P}=1$
starting from a converged trial wave-function corresponding to the $8^{th}$
singlet excitation of $A_{1}$ symmetry with a) $\beta=1.05$ and b) $\beta=0$
in Eq. (21). The dotted lines mark the beginning of some of the fixed-node DMC
blocks of a SHDMC run for the $\beta=0$ case. Same conventions as in Fig. 1.
Figure 4 shows the effect of $\hat{P}$ and $\beta$ [see Eq. (21)] on a SHDMC
run. The figure shows the average of the local energy $E_{L}({\bf R})$ for two
calculations that start from the final trial wave-function obtained for the
$8^{th}$ singlet excitation with $A_{1}$ symmetry (please compare it with Fig.
1). Both calculations were run with the same parameters as in Fig. 1 with two
exceptions: i) $\hat{P}=1$ was used, which removes the orthogonality
constraints, and ii) one calculation was run with $\beta=1.05$ and the other
with $\beta=0$ in Eq. (21). An initial number of blocks $M=20$ was used.
Both calculations depart from the initial configuration. However, the run with
$\beta=0$ falls very quickly to the singlet ground state. The calculation with
$\beta=1.05$ remains much longer in the vicinity of the $8^{th}$ excitation.
This clearly shows the stabilizing effect unequal energy references on excited
states. Since presumably the $8^{th}$ excitation is not the minimum of its
nodal topology, it finally drifts away. For the $\beta=1.05$ case with
$\delta\tau=0.0002$, the algorithm becomes numerically unstable to noise after
the $\sim\\!50],000$ time step because the variance in the distribution of
weights of the walkers increases and the statistics is dominated by a reduced
number of walkers.
In contrast, the first excitation does not drift with $\beta\simeq 1$ and
$\hat{P}=1$ (not shown).
### V.1 Coulomb interaction results and discussion
Figure 5: (Color online) Average of the local energy $E_{L}({\bf R})$ of 200
walkers as the SHDMC algorithm converges to the ground, first and second
eigenstates with $A_{1}$ symmetry and S=0 of two electrons with Coulomb
interactions in a square box.
The use of a simplified electron-electron interaction facilitates the CI
calculations and the validation of the optimization method. However, it is
also important to test the convergence and stability of the method with a
realistic Coulomb interaction as in the case of the ground state. keystone
The results shown in this section have an interaction potential of the
formunits $V({\bf r},{\bf r^{\prime}})=20\pi^{2}/|{\bf r-r\prime}|$ as in
Ref. keystone, . To mimic the difficulties that the algorithm would have to
overcome in larger or more realistic systems, the Jastrow term is not
included, i.e. $J=0$. Most SHDMC parameters are the same as in the model
interaction case. All calculations with Coulomb interactions were run with
$\beta=0$, the initial number of sub-blocks $M=6$, and the time step reduced
to $\delta\tau=0.0001$. The initial trial wave-functions were selected with
the criteria used for the model case.
Figure 5 shows the average of the local energy $E_{L}({\bf R})$ obtained for
the ground state and the first two excitations with the same symmetry (singlet
$A_{1}$). The results are qualitatively similar to those obtained with the
model potential. It is evident from the data that the variance of $E_{L}({\bf
R})$ and its average are reduced as the wave-function is optimized.
Occasionally, $E_{L}({\bf R})$ might rise when $\hat{P}$ is updated (improving
the description of lower energy states).
The energy of the singlet ground state is 400.749 $\pm$ 0.013, which is only
slightly smaller than the lowest triplet energykeystone 402.718 $\pm$ 0.008
with symmetry $E$. These energies are very close because of the dominance of
the Coulomb repulsion as compared to the kinetic energy, which forces the
particles to be well separated and therefore the cost of a node in the triplet
state is small. This result is consistent with the choice of parameters that
sets the system in the highly correlated regime. The energies obtained for the
first and second excitations areunits $468.56\pm 0.09$ and $515.50\pm 0.08$
respectively.
Figure 6: (Color online) Logarithm of the variance of the weights of the
walkers distribution as a function of the SHDMC block index $\ell$ for the
$2^{nd}$ excitation with $A^{1}$ symmetry with Coulomb interaction (see Fig.
5). The lines are visual guides.
While Figs. 1 and 5 are qualitatively similar, the results shown in Fig. 1 are
more convincing since they are directly compared with full CI calculations and
they are less noisy, as noted by one referee. When the model interaction
potential is replaced by a Coulomb interaction, full CI calculations are still
possible, but they involve the numerical calculation of $16471$ integrals with
Coulomb singularities. CI calculations are typically done using a Gaussian
basis, dupuis which limits the impact of the matrix element integrals of
these singularities. However, as the size of the system increases, CI
calculations become too expansible numerically. Accordingly, self-reliant
methods to validate the quality of the SHDMC wave-functions must be developed.
As noted earlier, in a fixed population scheme, the weights contain all the
difference between $f({\bf R},\tau)$ and $|\Psi_{T}({\bf R})|^{2}$ . Since
$f({\bf R},\tau)$ and $|\Psi_{T}({\bf R})|^{2}$ should be equal if
$\Psi_{T}({\bf R})$ is an eigenstate, the variance of the weights can be used
to measure the quality of the wave-function. Figure 6 shows the evolution of
the logarithm of the variance $L_{var}$ of the weights of the walkers
$W_{i}^{j}(k)$ [see Eq. (3)] as a function of the SHDMC block index $\ell$.
$L_{var}$ is evaluated as
$L_{var}=\log{\sqrt{\frac{1}{N_{c}}\sum_{i,j}(W_{i}^{kj}(k)-1)^{2}}}\;.$ (24)
By using a linear order expansion in $\delta\tau$ in Eq. (3) and using Eq.
(4), it is straightforward to relate Eq. (24) to the variance of
$E_{i}^{j}(k)$. The latter is an average of $E_{L}({\bf R})$. A common measure
of the quality of the ground state wave-function is the variance of
$E_{L}({\bf R})$.
The results shown in Fig. 6 correspond to the $2^{nd}$ singlet excitation with
$A_{1}$ symmetry (see Fig. 5). Similar results are obtained for the ground
state and the first excitation (not shown). The error bar in $L_{var}$ is
smaller than the size of the symbols. The fluctuations in $L_{var}$ result
from the random fluctuations of the coefficients $\lambda_{n}$ that are
obtained statistically. Note that in spite of the noise, a clear trend shows
the improvement of the quality of the wave-function and $E_{T}$ as the SHDMC
algorithm progresses. However, these improvements are not uniform, which is
reflected by the oscillations in $L_{var}$ in Fig. 6 and in the amplitude of
$E_{L}({\bf R})$ in Fig. 5. A careful user of SHDMC should track $L_{var}$ and
use the best quality wave-function to calculate energies and $\hat{P}$.
## VI Summary
An algorithm to obtain the approximate nodes, wave-functions, and energies of
arbitrary low-energy eigenstates of many-body Hamiltonians has been presented.
This algorithm is a generalization of the “simple” self-healing diffusion
Monte Carlo method developed for the calculation of the ground state of
fermionic systems,keystone which in turn is built upon the standard DMC
method. ceperley80
At least in the case of the tested system, wave-functions and energies that
continuously approach fully converged configuration interaction calculations
can be obtained depending only on the computational time. The wave-function,
in turn, allows the calculation of any observable.
It is found that some special eigenstates, presumably the minimum energy
eigenstate for a given nodal topology, can be obtained without calculating the
lower excitations by artificially generating a kink in the propagated function
using unequal energy references in different nodal pockets.
The present method can be implemented easily in existing codes. Ongoing tests
on the ground state methodkeystone in larger systems give serious
hopefn:tests that the current generalization will also be useful.
While there are methods to obtain the excitation spectra of a many-body
Hamiltonian in a variational Monte Carlo context kent98 ; umrigar07 they
require obtaining the Hamiltonian and the overlap matrix elements. This
requirement would present a challenge for very large systems. SHDMC is a
complementary technique that could potentially scale better for larger sizes.
The evaluation and storage of the matrix elements of $\hat{\mathcal{H}}$ is
not required. The number of quantities sampled [the projectors $\xi_{n}({\bf
R})$, Eq. (14)] is equal to the number of basis functions $n_{b}$. In
contrast, energy minimization methods or configuration interaction (CI)
require the evaluation of $n_{b}^{2}$ matrix elements. In addition, the
solution of a generalized eigenvalue problem with statistical noise is
avoided. This can be an advantage in very large systems since algorithms for
eigenvalue problems are difficult to scale to take maximum advantage of large
supercomputers. In contrast, the sampling of a large number of determinants
can be trivially distributed on different processors. Moreover, recent
advances in determinant evaluation could facilitate sampling a very large
number of projectors $\xi_{n}({\bf R})$. nukala09
An apparent disadvantage of SHDMC is that the method is recursive. This
disadvantage is partially removed since i) the number of blocks $M$ used to
collect data is increased only if necessary to improve the wave-function
significantly, fn:changes ii) and, the propagation to large imaginary times
is avoided by using precisely this recursive approach that accumulates the
propagation in successive blocks. In addition, a small value of
$k\;\delta\tau$ limits large fluctuations in the weights, which recently have
been claimed to cause an exponential cost in the convergence of DMC results.
nemec09
The dominant cost of the present algorithm to obtain the wave-functions and
their nodes scales as $N_{e}^{3}\times n_{max}\times n_{b}\times n_{st}$, with
$n_{max}$ being the number of excited states, $n_{b}$ the number of projectors
$\xi_{n}({\bf R})$ sampled, and $n_{st}$ the total number of SHDMC steps. Of
course, the error and the cost depend on the quality of the method used to
construct $\Phi_{n}({\bf R})$ and the quality of the initial trial wave-
functions. Systematic errors decrease when $n_{b}$ is large, and the
statistical error decreases when $n_{st}$ increases. For a fixed absolute
error, $n_{b}$ is expected to increase exponentially with the number of
electrons $N_{e}$. keystone
Note that in order to describe an arbitrary wave-function of a system with
$N_{e}$ electrons and a typical size $L$ in $D>1$ dimensions with a resolution
$R_{s}$, one needs approximately $(L/R_{s})^{(D\;N_{e})}$ basis functions. The
nodal surface alone requires $(L/R_{s})^{(D\;N_{e}-1)}$ degrees of freedom.
Therefore, finding an algorithm to obtain the nodes $S_{n}({\bf R})$ of any
eigenstate $n$ with an arbitrary interaction in a time polynomial in $N_{e}$
is potentially a “Philosopher’s Stone” quest. However, if exponential factors
actually control the accuracy of the DMC approach, as claimed, nemec09 just a
rock solid method to find the nodes which simultaneously improves the wave-
function (reducing the population fluctuations) could be considered a
satisfactory solution. The presented work could be the basis of such a method.
In ongoing work, SHDMC methods are being developed and tested in larger
systems.
## Acknowledgments
The author would like thank C. Umrigar for suggesting the sampling of
$\delta\lambda_{n}$ instead of the absolute value of the coefficients. The
author also thanks R. Q. Hood, M. Bajdich and P. R. C. Kent for a critical
reading of the manuscript and for related discussions. Finally, the author
thanks the anonymous referee who inspired the calculations presented in Figs.
4 and 6.
Research performed at the Materials Science and Technology Division sponsored
by the Department of Energy and the Laboratory Directed Research and
Development Program of Oak Ridge National Laboratory, managed by UT-Battelle,
LLC, for the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.
## References
* (1) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
* (2) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
* (3) L. Hedin, Phys. Rev. 139, A796 (1965).
* (4) D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
* (5) W. M. C. Foulkes, R. Q. Hood, and R. J. Needs, Phys. Rev. B 60, 4558 (1999).
* (6) J. B. Anderson, Int. J. Quantum Chem. 15, 109 (1979).
* (7) P. J. Reynolds, D. M. Ceperley, B. J. Alder, and W. A. Lester, J. Chem. Phys. 77, 5593 (1982).
* (8) F. A. Reboredo, R. Q. Hood, and P. R. C. Kent, Phys. Rev. B 79, 195117 (2009)
* (9) D. M. Ceperley, J. Stat. Phys. 63, 1237 (1991).
* (10) F. A. Reboredo and P. R. C. Kent, Phys. Rev. B 77, 245110 (2008).
* (11) Understanding sections I, II, III, V and VI of Ref. keystone, is required before reading this article.
* (12) The energy unit is $\hbar^{2}/(2m)$.
* (13) C. J. Umrigar, M. P. Nightingale, and K. J. Runge, J. Chem. Phys. 99, 2865 (1993).
* (14) Note that the limit $\epsilon\rightarrow 0$ is taken after the amplitude of the potential tends to $\infty$. Thus, this potential does not have the $\delta({\bf R})$ form, and every eigenstate of $\mathcal{H}_{FN}$ must be zero at $S_{T}({\bf R})$.
* (15) The Jastrow factor does not change the nodes but accelerates convergence and improves the algorithm’s numerical stability.
* (16) The ground state $|\Psi_{0}\rangle$ is formally obtained byapplying the evolution operator $e^{-\tau\hat{\mathcal{H}}}$ to a trial wave-function $|\Psi_{T}^{\ell=0}\rangle$ in the limit $\tau~{}\rightarrow~{}\infty$.
* (17) S. Vitiello, K. Runge, and M. H. Kalos, Phys. Rev. Lett. 60, 1970 (1988).
* (18) R. Bianchi, D. Bressanini, P. Cremaschi, M. Mella, and G. Morosi, J. Chem. Phys. 98, 7204 (1993).
* (19) R. Bianchi, D. Bressanini, P. Cremaschi, M. Mella, and G. Morosi, Int. J. Quant. Chem. 57, 321 (1996).
* (20) B. L. Hammond, W. A. Lester, Jr., and P. J. Reynolds Monte Carlo Methods in Ab Initio Quantum Chemistry (World Scientific, Singapore-New Jersey-London-Hong Kong, 1994).
* (21) D. M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988).
* (22) R. N. Barnett, R. P. Reynolds, and W. A. Lester, J. Chem. Phys. 96, 2141 (1991).
* (23) D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, Phys. Rev. E 55, 3664 (1997).
* (24) W. D. da Silva and P. H. Acioli, J. Chem. Phys. 114, 9720 (2001).
* (25) M. P. Nightingale and V. Melik-Alaverdian, Phys. Rev. Lett. 87, 043401 (2001).
* (26) A. Lüchow, D. Neuhauser, J. Ka, R. Baer, J. Chen, and V. A. Mandelshtam, J. Phys. Chem. A 107, 7175 (2003).
* (27) F. Schautz, F. Buda, and C. Filippi, J. Chem. Phys. 121, 5836 (2004).
* (28) C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G. Hennig, Phys. Rev. Lett. 98, 110201 (2007).
* (29) W. Purwanto, S. Zhang, and H. Krakauer, J. Chem. Phys. 130, 094107 (2009).
* (30) All symmetries of $\hat{\mathcal{H}}$ must be considered, which includes space group symmetries, spin, and particle permutations.
* (31) C. Umrigar, private communication. A description of the benefits of his suggested improvement for the ground state will be published elsewhere.
* (32) If the change in the wave-function coefficients is dominated by random noise, the scalar product between the old and the new $\delta\lambda_{n}$ can be negative and $M$ is multiplied by a factor larger than 1.
* (33) M. Dupuis, and J. A. Montgomery, J. Comput. Chem. 14, 1347-1363 (1993).
* (34) P. K. V. V. Nukala and P. R. C. Kent, J. Chem. Phys. 130, 204105 (2009).
* (35) A SrLi dimer with $13$ electrons has been compared with energy minimization calculations.umrigar07 We have also a proof of principle for C${}_{20}^{+2}$ (78 electrons and 700 determinants).
* (36) P. R. C. Kent, R. Q. Hood, M. D. Towler, R. J. Needs, and G. Rajagopal, Phys. Rev. B. 57, 15293 (1998).
* (37) N. Nemec, in http://arxiv.org/abs/0906.0501
|
arxiv-papers
| 2009-06-24T17:53:01 |
2024-09-04T02:49:03.493313
|
{
"license": "Public Domain",
"authors": "Fernando A. Reboredo",
"submitter": "Fernando Reboredo",
"url": "https://arxiv.org/abs/0906.4359"
}
|
0906.4408
|
## 0.1 The Enigma of the Mass.
V.G. Plekhanov
Computer Science College, Erika Street 7a, Tallinn, 10416, Estonia
Abstract.
The different manifestations of the mass effects in the microphysics (isotope
effect) are presented for the first time. The bright effects observe in all
branches of physics: nuclear, atomic, and molecular as well as solid state
physics. Charge symmetry breaking in the strong interaction occurs because of
the difference between the masses of the up and down quarks. At present the
Standard Model can’t explain the observed mass pattern (Mn, Mp, mu, md etc.)
and their hierarchy. The last one doesn’t permit us to find the origin of the
isotope effect. The origin of the mass of the matter will be clarified when
the mechanism of chiral symmetry breaking in QCD is established.
Mass is a one of the fundamental properties of matter. It relates to classical
as well as modern physics (quantum mechanics or modern theory of gravitation
(see, e.g. [1]). Although the physical meaning of mass was discovered by
Einstein more than a century ago, when he introduced in physics the concept of
rest energy (E0) [2], the concept of mass still doesn’t have strict
mathematical determination. Indeed, according to the notion of the
relativistical physics (see, e.g. [3]) mass is determined by the next
expression
m2 = $\frac{\text{E}^{2}}{\text{c}^{4}}$ \-
$\frac{\overrightarrow{\text{p}^{2}}}{\text{c}^{2}}$ (1).
And in the case of resting body ($\overrightarrow{\text{p}}$ = 0) we have
m = $\frac{\text{E}_{0}}{\text{c}^{2}}$ (2).
From equation (2) it can be seen that the mass is proportioned to the rest
energy. If we put c = 1, in that case we see that the mass of body equals its
rest energy. The mass of a body is not a constant, it varies with changes in
its energy. Namely, rest energy ”slumbering” in massive bodies partly is
released in chemical and especially nuclear reactions. In spite of equivalence
of the mass of the body and rest energy, especially nuclear physics and
physics of elementary particles, the task of mass has not been solved. Until
present time the spectrum of the discrete hierarchy of elementary particles
mass hasn’t had a successful theoretical explantion [4,6]. As is well-known on
the boundary of the 19 and 20 centuries there was an opinion that the mass of
the electron has the electromagnetic origin [1,4]. However, later
investigations showed that the electromagnetic part of the mass of the
electron has a small contribution to its full mass [3]. Nevertheless, the
modern view connects the origin of the mass with nonlocal gravitational
fields, which nature is due to electromagnetic interaction [8 -11].This
conclusion reflects those fact, that the space between separated particles in
essence isn’t empty, it is filled with the material medium - the physical
fields. The space inside the atom is filled with electromagnetic field, and
inside nucleus - more densier and stronger field which is called sometimes
meson one.
The present letter is devoted to the elucidation of the origin of mass, so far
as only its nature closely connected with the origin of the isotope effect,
the experimental manifestation of which more persuasively testified in the
last fifty years in all branches of physics (nuclear, atomic, molecular as
well as solid state (see, e.g. reviews [12-14])). On the other hand it is
necessary to underline that only isotope effect is a direct manifestation of
the mass effect in microphysics. It should be added that the direct
measurments of the energy of zero-point vibrations owing to isotope effect in
solids shows the good agreement of the experimental values with the results of
the calculation of quantum electrodynamics in solids [13, 14].
Below we describe shortly the manifestations of the isotope effect in
molecular as well as solid state physics (more details see [14]). The
discovery [15] of the new fullerene allotropes of carbon, exemplified by C60
and soon followed by an efficient method for their synthesis [14], led to a
burst of theoretical and experimental activity on their physical properties.
Much of this activity concentrated on the vibrational properties of C60 and
their elucidation by Raman scattering [15]. Comparison between theory and
experiment was greatly simplified by the high symmetry (Ih), resulting in only
ten Raman active modes for the isolated molecule and the relative weakness of
solid state effect [15], causing the crystalline C60 (c - C60) Raman spectrum
at low resolution to deviate only slightly from that expected for the isolated
molecule [15]. Since the natural abundance of 13C is 1.11% (see, e.g. [12]),
almost half of all C60 molecules made from natural graphite contain one or
more 13C isotopes. If the squared frequency of a vibrational mode in a C60
molecule with n13C isotopes is written as a series $\ \omega^{2}$ =
$\omega_{(0)}^{2}$ \+ $\omega_{(1)}^{2}$ \+ $\omega_{(2)}^{2}$ \+
$\omega_{(3)}^{2}$ \+ …… in the mass perturbation (where
$\omega_{\left(0\right)}$ is an eigenmode frequency in a C60 molecule with 60
12C atoms), nondegenerate perturbation theory predicts for the two totally
symmetric Ag modes a first - order correction given
$\frac{\omega_{(1)}^{2}}{\omega_{(0)}^{2}}$ = - $\frac{\text{n}}{\text{720}}$.
(3)
This remarkable result, independent of the relative position of the isotopes
within the molecule and equally independent of the unperturbed eigenvector, is
a direct consequence of the equivalence of all carbon atoms in icosahedral
C60. To the same order of accuracy within nondegenerate perturbation theory,
the Raman polarizability derivatives corresponding to the perturbed modes are
equal to their unperturbed counterparts, since the mode eigenvectors remain
unchanged. These results lead to the following conclusion [15]: The Ag Raman
spectrum from a set of noninteracting C60 molecules will mimic their mass
spectrum if the isotope effect on these vibrations can be described in terms
of first - order nondegenerate perturbation theory. It is no means obvious
that C60 will meet the requirements for the validity of this simple theorem. A
nondegenerate perturbation expansion is only valid if the Agmode is
sufficiently isolated in frequency from its neighboring modes. Such isolation
is not, of course, required by symmetry. Even if a perturbation expansion
converges, there is no a priori reason why second - and higher - order
correction to Eq. (3) should be negligible. As was shown in cited paper the
experimental Raman spectrum (see below) of C60 does agree with the prediction
of Eq. (3). Moreover, as was shown in quoted paper, experiments with
isotopically enriched samples display the striking correlation between mass
and Raman spectra predicted by the above simple theorem. Fig. 1 shows a high -
resolution Raman spectrum at 30 K in an energy range close to the high -
energy pentagonal - pinch Ag(2) vibration according to [15]. Three peaks are
resolved, with integrated intensity of 1.00; 0.95; and 0.35 relative to the
strongest peak. The insert of this figure shows the evolution of this spectrum
as the sample is heated. The peaks cannot be resolved beyond the melting
temperature of CS2 at 150 K. The theoretical fit yields a separation of 0.98
$\pm$ 0.01 cm-1 between two main peaks and 1.02 $\pm$ 0.02 cm-1 between the
second and third peaks. The fit also yields full widths at half maximum
(FDWHM) of 0.64; 0.70 and 0.90 cm-1, respectively. The mass spectrum of this
solution shows three strong peaks (Fig. 1b) corresponding to mass numbers 720;
721 and 722, with intensities of 1.00; 0.67 and 0.22 respectively as predicted
from the known isotopic abundance of 13C. The authors [15] assign the highest
- energy peak at 1471 cm-1 to the Ag(2) mode of isotopically pure C60 (60 12C
atoms). The second peak at 1470 cm-1 is assigned to C60 molecules with one 13C
isotope, and the third peak at 1469 cm-1 to C60 molecules with two 13C
isotopes. The separation between the peaks is in excellent agreement with the
prediction from Eq. (3), which gives 1.02 cm-1. In addition, the width of the
Raman peak at 1469 cm-1, assigned to a C60 molecule with two 13C atoms, is
only 30 % larger than the width of the other peaks. This is consistent with
the prediction of Eq. (3) too, that the frequency of the mode will be
independent of the relative position of the 13C isotopes within the molecule.
The relative intensity between two isotope and one isotope Raman lines agrees
well with the mass spectrum ratios. Concluding this part we stress that the
Raman spectra of C60 molecules show remarkable correlation with their mass
spectra. Thus the study of isotope - related shift offers a sensitive means to
probe the vibrational dynamics of C60.
Next examples of the dependence of the exciton spectra in solds on the isotope
effect demonstrate below. Isotopic substitution only affects the wavefunction
of phonons; therefore, the energy values of electron levels in the Schrödinger
equation ought to have remained the same. This, however, is not so, since
isotopic substitution modifies not only the phonon spectrum, but also the
constant of electron-phonon interaction (see [12]). It is for this reason that
the energy values of purely electron transition in molecules of hydride and
deuteride are found to be different. This effect is even more prominent when
we are dealing with a solid [16]. Intercomparison of absorption spectra for
thin films of LiH and LiD at room temperature revealed that the longwave
maximum (as we know now, the exciton peak ) moves 64.5 meV towards the shorter
wavelengths when H is replaced with D [17].
The mirror reflection spectra of mixed and pure LiD crystals cleaved in liquid
helium are presented in Fig. 2. For comparison, on the same diagram we have
also plotted the reflection spectrum of LiH crystals with clean surface. All
spectra have been measured with the same apparatus under the same conditions.
As the deuterium concentration increases, the long-wave maximum broadens and
shifts towards the shorter wavelengths. As can clearly be seen in Fig. 2, all
spectra exhibit a similar long-wave structure. This circumstance allows us to
attribute this structure to the excitation of the ground (Is) and the first
excited (2s) exciton states. The energy values of exciton maxima for pure and
mixed crystals at 2 K are presented in Table 22 of ref. [12]. The binding
energies of excitons E${}_{\text{b}}$, calculated by the hydrogen-like
formula, and the energies of interband transitions E${}_{\text{g}}$ are also
given in Table 22.
Going back to Fig. 2, it is hard to miss the growth of $\Delta_{\text{12}}$,
which in the hydrogen-like model causes an increase of the exciton Rydberg
with the replacement of isotopes . When hydrogen is completely replaced with
deuterium, the exciton Rydberg (in the Wannier-Mott model) increases by 20%
from 40 to 50 meV, whereas E${}_{\text{g}}$ exhibits a 2% increase, and at 2
$\div$ 4.2 K is $\Delta$E${}_{\text{g}}$ = 103 meV. This quantity depends on
the temperature, and at room temperature is 73 meV, which agrees well enough
with $\Delta$E${}_{\text{g}}$ = 64.5 meV as found in the paper of Kapustinsky
et al. [17]. The single-mode nature of exciton reflection spectra of mixed
crystals LiH${}_{\text{x}}$D${}_{\text{1-x}}$ agrees qualitatively with the
results obtained with the virtual crystal model (see e.g. Elliott et al. [18];
Onodera and Toyozawa [19]), being at the same time its extreme realization,
since the difference between ionization potentials ($\Delta\zeta$) for this
compound is zero. According to the virtual crystal model, $\Delta\zeta$ = 0
implies that $\Delta$E${}_{\text{g}}$ = 0, which is in contradiction with the
experimental results for LiH${}_{\text{x}}$D${}_{\text{1}}$-${}_{\text{x}}$
crystals. The change in E${}_{\text{g}}$ caused by isotopic substitution has
been observed for many broad-gap and narrow-gap semiconductor compounds (see
also [12]).
All of these results are documented in Table 22 of Ref.[12], where the
variation of E${}_{\text{g}}$, E${}_{\text{b}}$, are shown at the isotope
effect. We should highlighted here that the most prominent isotope effect is
observed in LiH crystals, where the dependence of E${}_{\text{b}}$ = f
(C${}_{\text{H}}$) is also observed and investigated. To end this section, let
us note that E${}_{\text{g}}$ decreases by 97 cm${}^{\text{-1}}$ when
${}^{\text{7}}$Li is replaced with ${}^{\text{6}}$Li.
Detailed investigations of the exciton reflectance spectrum in CdS crystals
were done by Zhang et al. [20]. Zhang et al. studied only the effects of Cd
substitutions, and were able to explain the observed shifts in the band gap
energies, together with the overall temperature dependence of the band gap
energies in terms of a two-oscillator model provided that they interpreted the
energy shifts of the bound excitons and n = 1 polaritons as a function of
average S mass reported earlier by Kreingol’d et al. [21] as shifts in the
band gap energies. However, Kreingol’d et al. [21] had interpreted these
shifts as resulting from isotopic shifts of the free exciton binding energies
, and not the band gap energies, based on their observation of different
energy shifts of features which they identified as the n = 2 free exciton
states (for details see [21]). The observations and interpretations, according
Meyer at al. [22], presented by Kreingol’d et al. [21] are difficult to
understand, since on the one hand a significant band gap shift as a function
of the S mass is expected , whereas it is difficult to understand the origin
of the relatively huge change in the free exciton binding energies which they
claimed. Very recently Meyer et al. [22] reexamine the optical spectra of CdS
as function of average S mass, using samples grown with natural Cd and either
natural S ($\sim$ 95% 32S), or highly enriched (99% 34S). These author
observed shifts of the bound excitons and the n = 1 free exciton edges
consistent with those reported by Kreingol’d et al. [21], but, contrary to
their results, Meyer et al. observed essentially identical shifts of the free
exciton excited states, as seen in both reflection and luminescence
spectroscopy. The reflectivity and photoluminescence spectra in polarized
light ($\overrightarrow{E}$ $\bot$ $\overrightarrow{C}$) over the A and B
exciton energy regions for the two samples depicted on the Fig. 3. For the
$\overrightarrow{E}$ $\bot$ $\overrightarrow{C}$ polarization used in Fig. 3
both A and B excitons have allowed transitions, and therefore reflectivity
signatures. Fig. 3 reveals both reflectivity signatures of the n = 2 and 3
states of the A exciton as well that of the n = 2 state of the B exciton.
In Table 18 of Ref. [14] the results of Meyer et al. summarized the energy
differences $\Delta$E = E (Cd34S) - E (CdnatS), of a large number of bound
exciton and free exciton transitions, measured using photoluminescence,
absorption, and reflectivity spectroscopy, in CdS made from natural S (CdnatS,
95% 32S) and from highly isotopically enriched 34S (Cd34S, 99% 34S). As we can
see from Fig. 3, all of the observed shifts are consistent with a single
value, 10.8$\pm$0.2 cm-1. Several of the donor bound exciton photoluminescence
transitions, which in paper [22] can be measured with high accuracy, reveal
shifts which differ from each other by more than the relevant uncertainties,
although all agree with the 10.8$\pm$0.2 cm-1 average shift. These small
differences in the shift energies for donor bound exciton transitions may
reflect a small isotopic dependence of the donor binding energy in CdS (see,
also [12]). This value of 10.8$\pm$0.2 cm-1 shift agrees well with the value
of 11.8 cm-1 reported early by Kreingol’d et al. [21] for the Bn=1 transition,
particularly when one takes into account the fact that enriched 32S was used
in that earlier study, whereas Meyer et al. have used natural S in place of an
isotopically enriched Cd32S (for details see [22]).
Authors [21] conclude that all of the observed shifts arise predominantly from
an isotopic dependence of the band gap energies, and that the contribution
from any isotopic dependence of the free exciton binding energies is much
smaller. On the basis of the observed temperature dependencies of the
excitonic transitions energies, together with a simple two-oscillator model,
Zhang et al. [20] earlier calculated such a difference, predicting a shift
with the S isotopic mass of 950 $\mu$eV/amu for the A exciton and 724
$\mu$eV/amu for the B exciton. Reflectivity and photoluminescence study of
natCd32S and natCd34S performed by Kreingol’d et al. [21] shows that for anion
isotope substitution the ground state (n = 1) energies of both A and B
excitons have a positive energy shifts with rate of $\partial$E/$\partial$MS =
740 $\mu$eV/amu. Results of Meyer et al. [22] are consistent with a shift of
$\sim$710 $\mu$eV/amu for both A and B excitons. Finally, it is interesting to
note that the shift of the exciton energies with Cd mass is 56 $\mu$eV/amu
[20], an order of magnitude less than found for the S mass (more details see
[12, 13]).
The brought examples clearly indicate mass dependence of the electron and
phonon states (see more details [14]) but on the other side it simply
underlines the primary importance in microphysics the difference of mass
between neutron (Mn) and proton (Mp). Really small difference in their masses
Mn \- Mp = 1.2333317 MeV leads to the bright effects in microphysics.
According to the last data [9], the experimental neutron-proton mass
difference of Mn \- Mp = 1.2333317 MeV is received as estimated
electromagnetic contribution Mn \- M${}_{p}\mid^{\text{em}}=$ -0.76 $\pm$ 0.30
MeV, and the remaining mass difference is determined to a strong isospin
breaking contribution of Mn \- M${}_{p}\mid^{\text{d-u}}$= 2.05 $\pm$ 0.30
MeV. In other words the last contribution is a result of difference in mass of
d - and u - quarks (see, also [10, 11]).
As we all know, the observed world - stars, planets, galaxy as well as
surrounding objects consist from the nuclei, neutrons, protons and electrons.
The mass of electrons has a small contribution to the total mass ( less than
0.1% (see, e.g. [1]). Therefore, that we knew that the origin of the mass of
the observed worlds needs to be elucidated the origin of nuclear mass. As we
know the nucleon consists from u - and d - quarks. But the mass of u - and d -
quarks is so small, that is their sum is a small part of the nucleon mass (1 -
2 % [6]). In modern physics of elementary particles it is considered that the
mass of nucleon arises from the spontaneous breaking of a chiral symmetry in
quantum chromodynamics (QCD) [23] and may be expressed over vacuum condensate
(see [5] and references therein).This model has an approximate formula which
expresses the mass of nucleon over quarks condensate [5]
m = [-2(2$\pi$)${}^{2}\langle$0$\mid\overline{\text{q}}$q$\mid$0$\rangle$]1/3
(4),
where m is nucleon mass, $\langle$0$\mid\overline{\text{q}}$q$\mid$0$\rangle$
is quarks condensate, q is the field of u - or d - quarks. The chiral symmetry
in QCD tresult in the expression for the quarks condensate (so called Gell -
Mann - Oakes - Renner formula [24])
$\langle$0$\mid\overline{\text{q}}$q$\mid$0$\rangle$ =
-$\frac{\text{1}}{\text{2}}\frac{\text{m}_{\pi}\text{
f}_{\pi}}{\text{m}_{u}\text{ + m}_{\text{d}}}$ (5).
Here mπ and fπ are the mass and decay constant of $\pi$ \- meson. The defined
value of quarks condensate on the ground of $\tau$ \- decay [5,6] equals
$\langle$0$\mid\overline{\text{q}}$q$\mid$0$\rangle$ = - (254 MeV)3 $\pm$ 10
%. (6)
Put the last value into the expression (4) it gives the nuclon’s mass m = 1.08
GeV, when the experimental value of nucleon’s mass equals m = 0.94 MeV. From
comparison of these values we see that the difference between experimental
value of m and theoretical estimation is 0.15 GeV, that surpasses the
experimental value of the difference Mn \- Mp = 1.2333317 MeV much order. The
last one means that in such model (as well as in the model of constituent
quarks) we have neither the mass difference of the nucleons nor its number in
nuclei and, consequently, isotope effect. But the experimental manifestations
of the isotope effect was demonstrated above in the different branches of
microphysics. Considering the quarks structure of nucleon (the wavefunction of
the neutron is udd, and for proton one is uud) that is the quark strucure
indicates the different construction of the neutron and proton, but this model
doesn’t quantative describe the mass of nucleons.
Thus, the origin of the isotope effect is closely connected with the different
origin of u - and d - quarks and with solution the spectrum and hierarhy of
the elementary particles mass and more common with the solution of the nature
of mass (see, also [25]).
Acknowledgements. I deeply thank to Prof. B.L. Ioffe for the enlighting
discussion on the origin of mass, and Prof. P. Sneider for improving my
English.
Figure captions.
Fig. 1. a - unpolarized Raman spectrum in the frequency region of the
pentagonal - pinch mode, for a frozen sample of nonisotopically enriched C60
in CS2 at 30 K. The points are the experimental data, and the solid curve is a
three - Lorentzian fit. The highest - frequency peak is assigned to the
totally symmetric pentagonal - pinch Ag mode in isotopically pure 12C60. The
other two peaks are assigned to the perturbed pentagonal - pinch mode in
molecules having one and two 13C - enriched C60, respectively. The insert
shows the evolution of these peaks as the solution is heated. b - the points
give the measured unpolarized raman spectrum in the pentagonal - pinch region
for a frozen solution of 13C - enriched C60 in CS2 at 30 K. The solid line is
a theoretical spectrum computed using the sample’s mass spectrum, as described
in the text (after [15]).
Fig. 2. Mirror reflection spectra of crystals: 1 - LiH; 2 - LiHxD1-x; 3 - LiD;
at 4.2 K. 4 - source of light without crystal. Spectral resolution of the
instrument is indicated on the diagram (after [12]).
Fig. 3. a - Reflection spectra in the A and B excitonic polaritons region of
CdnatS and Cd34S at 1.3K with incident light in the
$\overrightarrow{\text{E}}$ $\perp$ $\overrightarrow{C}$. The broken vertical
lines connecting peaks indicate measured enrgy shifts reported in Table 18 of
Ref. [14]. In this polarization, the n = 2 and 3 excited states of the A
exciton, and the n = 2 excited state of the B exciton, can be observed. b -
Polarized photoluminescence spectra in the region of the A${}_{\text{n = 2}}$
and A${}_{\text{n = 3}}$ free exciton recombination lines of CdnatS and Cd34S
taken at 1.3 K with the $\overrightarrow{\text{E}}$ $\perp$
$\overrightarrow{C}$. The broken vertical lines connecting peaks indicate
measured enrgy shifts reported in Table 18 of Ref. [14] (after [22]).
References.
1\. M. Jammer, Concepts of mass in classical and modern physics, Harvard
University Press, Cambridge - Massachsets (1961).
2\. A. Einstein, Ann. Phys. (Leipzig) 20, 371 (1906).
3\. L.D. Landau, E.M. Lifshitz, The classical theory of fields, Pergamon, New
York (1958).
4\. L.B. Okun, Physics Today, June 1989; Uspekhi Fiz. Nauk 158, 512 (1989) (in
Russian).
5\. B.L. Ioffe, Uspekhi Fiz. Nauk 171, 1273 (2001) (in Russian); Progr. Part.
Nucl. Phys. 56, 232 (2006).
6\. C.D. Frogatt, Surveys High Energy Physics 18, 77 (2003); The Problem of
Mass , ArX:hep - ph/0312220.
7\. A. Dobado and A.L. Maroto, Phys. Rev. D60, 104045-9 (1999).
8\. J.J. Kelly, Phys. Rev. C70, 068202 (2004).
9\. S.R. Beane, K. Originas and M.J. Savage, Nucl. Phys. B768, 38, (2007).
10\. G.A. Miller, A.K. Opper, E.J. Stephenson, Annual Review of Nuclear
Science 56, 253 (2006).
11\. G.A. Miller, Phys. Rev. Lett. 99, 112001 (2007); The Neutron Negative
Central Charge Density: an Inclusive - Exclusive Connection, ArXiv 0806.3977.
12\. V.G. Plekhanov, Phys. Reports 410, 1 (2005).
13\. M.Cardona, M.L.W. Thewalt, Rev. Mod. Phys. 77, 1173 (2005).
14\. V.G. Plekhanov, will be published.
15\. J. Menendez and J.B. Page, Vibrational spectroscopy of C60, in, M.
Cardona and G. Guntherodt, eds., Light Scattering in Solids VIII, Springer,
Berlin - Heidelberg (2000) (Vol. 76 in Topics in Applied Physics).
16\. V.G. Plekhanov, Isotope effects in solid state physics, Academic Press,
San Diego (2001).
17\. A.F. Kapustinsky, L.M. Shamovsky, K.S. Bayushkina, Acta Physicochim.
(USSR) 7, 799 (1937).
18\. R.J. Elliott, J.AA. Krumhansl, P.L. Leath, Rev. Mod. Phys. 46, 465
(1974).
19\. Y.Onodera and Y. Toyozawa, J. Phys. Soc. Japan 24, 341 (1968).
20\. M. Zhang, M. Ghieler, T. Ruf, Phys. Rev. B57, 9716 (1998).
21\. F.I. Kreingol’d, K.F. Lider, M.B. Shabaeva, Fiz. Tverd. Tela 26, 3940
(1984) (in Russian).
22\. T.A. Meyer, M.L.W. Thewalt and R. Lauck, Phys. Rev. B69, 115214-5 (2004).
23.J. Grasser and H. Leutwyller, Phys. Reports 87, 77 (1982); H. Leutwyller,
Insights and Puzzles in Light Quark Physics, ZrXiv:hep - ph/070063138.
24\. M. Gell-Mann, R.J. Oakes, B. Renner, Phys. Rev. 175, 2195 (1968).
25\. I. F. Ginzburg, Uspekhi Fiz. Nauk (Moscow) 179, 525 (2009) (in Ruassian).
|
arxiv-papers
| 2009-06-24T06:44:41 |
2024-09-04T02:49:03.502301
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.G. Plekhanov",
"submitter": "Vladimir Plekhanov",
"url": "https://arxiv.org/abs/0906.4408"
}
|
0906.4409
|
# Common Borel radius of an algebroid function and its derivative
Wu Nan1 and Xuan Zu-xing1,2 1Department of Mathematical Sciences Tsinghua
University Beijing, 100084 People’s Republic of China [email protected]
2Basic Department Beijing Union University No.97 Bei Si Huan Dong Road
Chaoyang District Beijing, 100101 People’s Republic of China
[email protected]
(Date: , Preliminary version)
###### Abstract.
In this article, by comparing the characteristic functions, we prove that for
any $\nu$-valued algebroid function $w(z)$ defined in the unit disk with
$\limsup_{r\rightarrow 1-}T(r,w)/\log\frac{1}{1-r}=\infty$ and the hyper order
$\rho_{2}(w)=0$, the distribution of the Borel radius of $w(z)$ and
$w^{\prime}(z)$ is the same. This is the extension of G. Valiron’s conjecture
for the meromorphic functions defined in $\widehat{\mathbb{C}}$.
###### Key words and phrases:
Algebroid functions, Borel radius.
###### 2000 Mathematics Subject Classification:
Primary 30D35.
The work is supported by NSF of China (No.10871108)
## 1\. Introduction and Main Results
The value distribution theory of meromorphic functions due to R.
Nevanlinna(see [2] for standard references) was extended to the corresponding
theory of algebroid functions by H. Selberg [3], E. Ullrich [9] and G. Valiron
[10] around 1930. The singular direction for $w(z)$ is one of the main objects
studied in the theory of value distribution of algebroid functions. Several
types of singular directions have been introduced in the literature. Their
existence and some connections between them have also been established [4, 7,
11].
In 1928, G. Valiron [12] asked the following:
_Does there exist a common Borel direction of a meromorphic function and its
derivative?_
This question was investigated by many mathematicians, such as G.Valiron [13],
A.Rauch [5], C.T. Chuang [1]. They proved the existence of common Borel
directions under some conditions. However, it is still an open problem till
now. For the case of the unit disk, Zhang [15] solved the problem, he proved
that the Borel radius of a meromorphic function of finite order is the same as
its derivative. We associated it to the algebroid functions and ask weather
the Borel radius of a $\nu-$valued algebroid function is the same to its
derivative. To state our results clearly, we begin with some basic nations for
algebroid functions.
Let $w=w(z)(z\in\Delta)$ be the $\nu$-valued algebroid function defined by the
irreducible equation
(1.1) $A_{\nu}(z)w^{\nu}+A_{\nu-1}(z)w^{\nu-1}+\cdots+A_{0}(z)=0,$
where $A_{\nu}(z),...,A_{0}(z)$ are analytic functions without any common
zeros. The single-valued domain $\widetilde{R}_{z}$ of definition of $w(z)$ is
a $\nu$-valued covering of the $z$-plane and it is a Riemann surface.
A point in $\widetilde{R}_{z}$ is denoted by $\widetilde{z}$ if its projection
in the $z$-plane is $z$. The open set which lies over $|z|<r$ is denoted by
$|\widetilde{z}|<r$. Let $n(r,a)$ be the number of zeros, counted according to
their multiplicities, of $w(z)-a$ in $|\widetilde{z}|\leq r,$ $n(r,a)$ be the
number of distinct zeros of $w(z)-a$ in $|\widetilde{z}|\leq r.$ Let
$\displaystyle N(r,a)$ $\displaystyle=$
$\displaystyle\frac{1}{\nu}\int_{0}^{r}\frac{n(t,a)-n(0,a)}{t}dt+\frac{n(0,a)}{\nu}\log{r},$
$\displaystyle m(r,a)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\nu}\int_{|\widetilde{z}|=r}\sum\limits_{j=1}^{\nu}\log^{+}|\frac{1}{w_{j}(re^{i\theta})-a}|d\theta,\
\ z=re^{i\theta},$ $\displaystyle T(r,a)$ $\displaystyle=$ $\displaystyle
m(r,a)+N(r,a).$
where $|\widetilde{z}|=r$ is the boundary of $|\widetilde{z}|\leq r$.
Moreover, $S(r,w)$ is a conformal invariant and is called the mean covering
number of $|\widetilde{z}|\leq r$ into $w$-sphere. We call
$T(r,w)=T(r,\infty)$ the characteristic function of $w(z)$. It is known from
[[3], $3^{o}$, p.84] that $T(r,a)=m(r,\infty)+N(r,\infty)+O(1).$ We define the
order and hyper order of a $\nu$-valued algebroid function as
$\rho(w)=\limsup\limits_{r\rightarrow 1-}\frac{\log
T(r,w)}{\log\frac{1}{1-r}},$
and
$\rho_{2}(w)=\limsup\limits_{r\rightarrow 1-}\frac{\log\log
T(r,w)}{\log\frac{1}{1-r}}.$
Given an angular domain
$\Delta(\theta_{0},\varepsilon)=\\{z||\operatorname{arg}z-\theta_{0}|<\varepsilon\\},0<\varepsilon<\frac{\pi}{2},$
we denote $\\{z:|z|<r,|\operatorname{arg}z-\theta|<\varepsilon\\}$ by
$\Omega(r,\theta,\varepsilon)$ and write $\widetilde{\Omega}$ for the part of
$\widetilde{R}_{z}$ on $\Omega(r,\theta,\varepsilon)$.
$\overline{n}(r,\Delta(\theta,\varepsilon),w=a)$ denotes the numbers of
$w(z)-a$ in $\widetilde{\Omega}$(not counting multiplicities).
$\displaystyle\overline{N}(r,\Delta(\theta,\varepsilon),w=a)$ $\displaystyle=$
$\displaystyle\frac{1}{\nu}\int_{0}^{r}\frac{\overline{n}(t,\Delta(\theta,\varepsilon),w=a)-\overline{n}(0,\Delta(\theta,\varepsilon),w=a)}{t}dt$
$\displaystyle+$
$\displaystyle\frac{\overline{n}(0,\Delta(\theta,\varepsilon),w=a)}{\nu}\log
r$
is called the counting function of zeros of $w(z)-a$ in $\Omega$.
Next, we give the definition of the Borel radius of a $\nu$-valued algebroid
function in the unit disk.
###### Definition 1.1.
A radius $L(\theta):\operatorname{arg}z=\theta,0<|z|<1$ is called a Borel
radius of a $\nu$-valued algebroid function $w(z)$ of order $\rho$, if for any
$\varepsilon>0$
$\limsup\limits_{r\rightarrow
1-}\frac{\log\overline{N}(r,\Delta(\theta,\varepsilon),w=a)}{\log\frac{1}{1-r}}=\rho$
holds for any $a\in\mathbb{\widehat{C}}$, except for $2\nu$ exceptions.
In this note, we give a positive answer to the G. Valiron’s conjecture for
algebroid functions defined in the unit disk.
###### Theorem 1.1.
The distribution of the Borel radius of a $\nu$-valued algebroid function
$w(z)$ with the order $0\leq\rho(w)<\infty$ and
$\limsup\limits_{r\rightarrow 1-}\frac{T(r,w)}{\log\frac{1}{1-r}}=\infty$
is the same to that of its derivative.
###### Theorem 1.2.
The distribution of the Borel radius of a $\nu$-valued algebroid function
$w(z)$ with order $\rho(w)=\infty$ and the hyper order $\rho_{2}(w)=0$ is the
same to that of its derivative.
We will prove the above two theorems synchronously.
## 2\. Primary knowledge
###### Lemma 2.1.
Let $w(z)$ be the $\nu$-valued algebroid function defined by (1.1) in the unit
disk, $z=z(\zeta)$ be a conformal mapping from the unit disk $D(\zeta)$ into
$D(z)$. Then $M(\zeta)=w(z(\zeta))$ and $M^{\prime}(\zeta)$ are also
$\nu$-valued algebroid functions. Furthermore, we can see that
$G(\zeta)=w(z(\zeta))$ is determined by
$A_{\nu}(z(\zeta))M^{\nu}(\zeta)+A_{\nu-1}(z(\zeta))M^{\nu-1}(\zeta)+\cdots+A_{0}(z(\zeta))=0,$
and $M^{\prime}(\zeta)=w^{\prime}(z(\zeta))z^{\prime}(\zeta)$.
Lemma 2.1 is apparent and we omit the proof of it. The following lemma is an
analogue of Lemma 2.1 in [15].
###### Lemma 2.2.
Set
$G(r,\theta,\eta)=\\{z:0<|z|<r,|\operatorname{arg}z-\theta|<\eta\\},$
$\alpha=\frac{\pi}{2\eta},$
$\zeta(z)=\frac{(ze^{-i\theta})^{2\alpha}+2(ze^{-i\theta})^{\alpha}-1}{(ze^{-i\theta})^{2\alpha}-2(ze^{-i\theta})^{\alpha}-1}.$
The function $\zeta=\zeta(z)$ defined above maps conformally the unit disk
$D(\zeta)=\\{\zeta:|\zeta|<1\\}$ onto the sector $G(1,\theta,\eta)$. By
$z=z(\zeta)$ we denote the inverse function of the function $\zeta(z)$. Write
$M(\zeta)=w(z(\zeta))$, where $w(z)$ is a $\nu-$valued algebroid function in
the sector $G(1,\theta,\eta)$. Then for any value $a$ on the complex plane, we
have
(1) Set $\beta=2^{-\alpha-\frac{5}{2}}$. Then
$\overline{N}(r,\Delta(\theta,\frac{\eta}{2}),w=a)\leq\frac{2}{\beta}\overline{N}(1-\beta(1-r),M=a)+O(1),$
when $r\rightarrow 1-$.
(2) Set $\delta=\frac{1}{16\alpha}$. Then
$\overline{N}(\gamma,M=a)\leq\frac{2}{\delta}\overline{N}(1-\delta(1-\gamma),\Delta(\theta,\eta),w=a)+O(1),$
when $\gamma\rightarrow 1-$.
(3) For any $0<t<1$, we have
(2.1) $T(t,z^{\prime}(\zeta))\leq 3\log\frac{2}{1-t},\ \
T(t,\frac{1}{z^{\prime}(\zeta)})\leq 3\log\frac{2}{1-t}+\log\frac{\pi}{\eta}.$
Here we generalize the corresponding results of meromorphic functions to
algebroid functions. This lemma for meromorphic functions was first
established by Zhang in [16]. He proved that the function $\zeta=\zeta(z)$
maps the unit disk $D(\zeta)=\\{\zeta:|\zeta|<1\\}$ onto the sector
$G(1,\theta,\eta)$ conformally. Furthermore, after a calculation Zhang found
that this function has the following perfect properties:
(2.2)
$\zeta(\\{z:\frac{1}{2}<|z|<r,|\operatorname{arg}z-\theta|<\frac{\eta}{2}\\})\subset\\{\zeta:|\zeta|<1-2^{-\frac{\pi}{2\eta}-\frac{\pi}{2}}(1-r)\\}$
and
(2.3)
$z(\\{\zeta:|\zeta|<\gamma\\})\subset\\{z:|z|<1-\frac{\eta}{8\pi}(1-\gamma),|\operatorname{arg}z-\theta|<\eta\\}.$
This is important. The number of roots of algebroid functions or meromorphic
functions are conformal invariant consequently he obtained this result.
Remark. As we know that the term $T(r,\Omega,f)$, whose definition can be seen
in Page 233 of [8] is conformal invariant, where $f$ is a meromorphic function
in the angular domain $G(1,\theta,\eta)$. By (2.2) and (2.3) we have the
following
$T(r,\Delta(\theta,\frac{\eta}{2}),f(z))\leq
T(1-2^{-\frac{\pi}{2\eta}-\frac{\pi}{2}}(1-r),f(z(\zeta)))$
and
$T(\gamma,f(z))\leq
T(1-\frac{\eta}{8\pi}(1-\gamma),\Delta(\theta,\eta),f(z(\zeta))).$
From the above we can see that the order of $T(r,f(z(\zeta)))$ is $\rho$ in
the unit disk if and only if there exists a $\varepsilon$ such that
(2.4) $\limsup\limits_{r\rightarrow 1-}\frac{\log
T(r,\Delta(\theta,\varepsilon),f)}{\log\frac{1}{1-r}}=\rho.$
Since $L(\theta)$ is a Borel radius of a meromorphic function $f$ in the unit
disk if and only if there exists a $\varepsilon$ such that (2.4) holds.
Therefore, we can simplify the Zhang’s proof for $L(\theta)$ is a Borel radius
if and only if the order of $T(r,f(z(\zeta)))$ is $\rho$.
###### Lemma 2.3.
Let $h(r)$ is a real non-negative and non-decreasing function defined in
$(0,1)$, $E\subset(0,1)$ is a set with $\int_{E}\frac{1}{1-r}dr<\infty$. If
(2.5) $\limsup\limits_{r\rightarrow 1-}\frac{\log
h(r)}{\log\frac{1}{1-r}}=\rho,$
then we have
(2.6) $\limsup\limits_{r\notin E,r\rightarrow 1-}\frac{\log
h(r)}{\log\frac{1}{1-r}}=\rho.$
Now we give the proof of Lemma 2.3.
###### Proof.
If $\rho=0$, it is easy to see that the conclusion naturally holds. Here we
only consider the case $0<\rho\leq\infty$.
We choose a $0<\lambda<1$ such that
$\log\frac{1}{\lambda}>K_{E},$
where $K_{E}=\int_{E}\frac{dr}{1-r}<\infty$. Suppose (2.6) is not true, then
there exists a number $0<\rho_{1}<\rho$, such that
$\limsup\limits_{r\notin E,r\rightarrow 1^{-}}\frac{\log
h(r)}{\log\frac{1}{1-r}}=\rho_{1}<\rho.$
From (2.5), we can take a sequence $\\{r_{n}\\}\subset(r_{0},1)$ with
$r_{n}\rightarrow 1-$ such that
(2.7) $\limsup\limits_{n\rightarrow\infty}\frac{\log
h(r_{n})}{\log\frac{1}{1-r_{n}}}=\rho.$
Since for each $n$
$\begin{split}\int_{[r_{n},\lambda r_{n}+(1-\lambda)]\backslash
E}\frac{dr}{1-r}&\geq\int_{[r_{n},\lambda
r_{n}+(1-\lambda)]}\frac{dr}{1-r}-\int_{E}\frac{dr}{1-r}\\\
&=\log\frac{1}{\lambda}-K_{E}>0,\end{split}$
there exists a $r_{n}^{\prime}\in[r_{n},\lambda r_{n}+(1-\lambda)]\backslash
E$. By the increasing property of $\log h(r)$, we have
$\frac{\log h(r_{n}^{\prime})}{\log\frac{1}{1-r_{n}^{\prime}}}\geq\frac{\log
h(r_{n})}{\log\frac{1}{\lambda(1-r_{n})}}=\frac{\log
h(r_{n})}{\log\frac{1}{\lambda}+\log\frac{1}{1-r_{n}}},$
and then we have
$\begin{split}\limsup\limits_{n\rightarrow\infty}\frac{\log
h(r_{n})}{\log\frac{1}{1-r_{n}}}&=\limsup\limits_{n\rightarrow\infty}\frac{\log
h(r_{n})}{\log\frac{1}{\lambda}+\log\frac{1}{1-r_{n}}}\\\
&\leq\limsup\limits_{r_{n}^{\prime}\rightarrow 1-}\frac{\log
h(r_{n}^{\prime})}{\log\frac{1}{1-r_{n}^{\prime}}}\\\
&\leq\limsup\limits_{r\rightarrow 1-,\ r\in[r_{0},1]\backslash E}\frac{\log
h(r)}{\log\frac{1}{1-r}}=\rho_{1}<\rho.\end{split}$
This contradicts to (2.7). Our Lemma is confirmed. ∎
In 1988, Zeng [14] established the following lemma which is a classical result
for algebroid functions and is useful for our study.
###### Lemma 2.4.
[14] Let $w(z)$ be the $\nu$-valued algebroid function defined by (1.1), then
$w^{\prime}(z)$ is also a $\nu$-valued algebroid function in the unit disk and
$\rho(w)=\rho(w^{\prime})$.
The following lemma is the second fundamental theorem for algebroid functions
in the unit disk, whose proof can be seen in [3], and we can obtain the error
term $S(r,w)$ by the same method as used in meromorphic functions.
###### Lemma 2.5.
Let $w(z)$ be a $\nu-$valued algebroid function in the unit disk, and
$a_{1},a_{2},\cdots,a_{q}$ be $q$ different values on the complex sphere, then
we have
$(q-2\nu)T(r,w)<\sum\limits_{i=1}^{q}\overline{N}(r,w=a_{i})+S(r,w),$
where
$S(r,w)=\begin{cases}O(\log\frac{1}{1-r})&\text{,if $\lambda(w)<\infty$},\\\
O(\log\frac{1}{1-r}+\log T(r,w)),r\notin E&\text{,if
$\lambda(w)=\infty$}.\end{cases}$
where $E$ is a set such that $E\subset(0,1)$ and
$\int_{E}\frac{1}{1-r}dr<\infty$.
In general, we can write the second fundamental theorem as follows
$(q-2\nu)T(r,w)<\sum\limits_{i=1}^{q}\overline{N}(r,w=a_{i})+O(\log\frac{1}{1-r}+\log
T(r,w)),r\notin E.$
## 3\. Main lemma
Now we are in the position to show our main lemma which is crucial to our
theorems.
###### Lemma 3.1.
Let $w(z)$ be a $\nu-$valued algebroid function of order $\rho(w)=\rho$
($0\leq\rho\leq\infty$) , $\limsup_{r\rightarrow
1-}T(r,w)/\log\frac{1}{1-r}=\infty$ and $\rho_{2}(w)=0$ in the unit disc
$D(z)$. Then a $radius$ $L(\theta)$ is a Borel radius of the algebroid
function $w(z)$ if and only if for any $0<\eta<1$, the function
$M(\zeta)=w(z(\zeta))$ is a $v-$valued algebroid function of order $\rho$ in
the unit disk $D(\zeta)$, where $z=z(\zeta)$ is the function described in
Lemma 2.2, mapping the unit disk $D(\zeta)$ onto the sector
$G(1,\theta,\eta)$.
###### Proof.
$"\Longrightarrow"$
Let $L(\theta)$ be a Borel radius of the function $w(z)$. Then for any fixed
$0<\eta<1$, there exist $2\nu+1$ different values $a_{1},\cdots,a_{2\nu+1}$ on
the complex plane, such that
$\limsup\limits_{r\rightarrow
1-}\frac{\log\overline{N}(r,\Delta(\theta,\varphi),w=a_{i})}{\log\frac{1}{1-r}}=\rho,(i=1,2,\cdots,2\nu+1;\varphi=\eta,\frac{\eta}{2}).$
Applying Lemma 2.2 to the function $w(z)$, we have
$\begin{split}\limsup\limits_{\gamma\rightarrow
1-}\frac{\log\overline{N}(\gamma,M=a_{i})}{\log\frac{1}{1-\gamma}}&=\limsup\limits_{r\rightarrow
1-}\frac{\log\frac{2}{\beta}\overline{N}(1-\beta(1-r),M=a_{i})}{\log\frac{1}{1-(1-\beta(1-r))}}\\\
&\geq\limsup\limits_{r\rightarrow
1-}\frac{\log\overline{N}(r,\Delta(\theta,\frac{\eta}{2}),w=a_{i})}{\log\frac{1}{1-r}}=\rho(i=1,2,\cdots,2\nu+1).\\\
\end{split}$
Therefore the order of the function $M(\zeta)$ is not less than $\rho$. Apply
Lemma 2.2 to the function $w(z)$, we have
$\begin{split}\limsup\limits_{\gamma\rightarrow
1-}\frac{\log\overline{N}(\gamma,M=a_{i})}{\log\frac{1}{1-\gamma}}&\leq\limsup\limits_{\gamma\rightarrow
1-}\frac{\log\frac{2}{\delta}\overline{N}(1-\delta(1-\gamma),\Delta(\theta,\eta),w=a_{i})}{\log\frac{1}{1-(1-\delta(1-\gamma))}}\\\
&=\limsup\limits_{r\rightarrow
1-}\frac{\log\overline{N}(r,\Delta(\theta,\eta),w=a_{i})}{\log\frac{1}{1-r}}=\rho(i=1,2,\cdots,2\nu+1).\\\
\end{split}$
Applying the second fundamental theorem to the function $M(\zeta)$. We obtain
$T(\gamma,M)\leq\sum\limits_{i=1}^{2\nu+1}\overline{N}(\gamma,M=a_{i})+O(\log\frac{1}{1-\gamma}+\log
T(\gamma,M)),\gamma\notin E,$
where $E$ is a set with $\int_{E}\frac{1}{1-\gamma}d\gamma<\infty$. Hence
$\limsup\limits_{\gamma\notin E,\gamma\rightarrow 1-}\frac{\log
T(\gamma,M)}{\log\frac{1}{1-\gamma}}\leq\limsup\limits_{\gamma\notin
E,\gamma\rightarrow
1-}\frac{\log\sum\limits_{i=1}^{2\nu+1}\overline{N}(\gamma,M=a_{i})}{\log\frac{1}{1-\gamma}}=\rho.$
Applying Lemma 2.3, we can see that the order of the function $G(\zeta)$ is
$\rho$.
$"\Longleftarrow"$ Now for any fixed $0<\eta<1$, let $M(\zeta)=w(z(\zeta))$ be
a $\nu-$valued algebroid function of order $\rho$ in the unit disk $D(\zeta)$,
where $z=z(\zeta)$ is the mapping function defined in Lemma 2.2. Then for any
$2\nu+1$ different values $a_{1},a_{2},\cdots,a_{2\nu+1}$, applying the second
fundamental theorem, we have
$\begin{split}T(\gamma,M)&\leq\sum\limits_{i=1}^{2\nu+1}\overline{N}(\gamma,M=a_{i})+O(\log\frac{1}{1-\gamma}+\log
T(\gamma,M))\\\
&\leq\sum\limits_{i=1}^{2\nu+1}\frac{2}{\delta}\overline{N}(1-\delta(1-\gamma),\Delta(\theta,\eta),w=a_{i})+O(\log\frac{1}{1-\gamma}+\log
T(\gamma,M))\\\ \end{split}$
hence by Lemma 2.3
$\begin{split}\rho&=\limsup\limits_{\gamma\notin E,\gamma\rightarrow
1-}\frac{\log
T(\gamma,M)}{\log\frac{1}{1-\gamma}}\leq\limsup\limits_{\gamma\rightarrow
1-}\frac{\log\sum\limits_{i=1}^{2\nu+1}\overline{N}(1-\delta(1-\gamma),\Delta(\theta,\eta),w=a_{i})}{\log\frac{1}{1-(1-\delta(1-\gamma))}}\\\
&=\limsup\limits_{r\rightarrow
1-}\frac{\log\sum\limits_{i=1}^{2\nu+1}\overline{N}(r,\Delta(\theta,\eta),w=a_{i})}{\log\frac{1}{1-r}}\leq\limsup\limits_{r\rightarrow
1-}\frac{\log\sum\limits_{i=1}^{2\nu+1}\overline{N}(r,w=a_{i})}{\log\frac{1}{1-r}}=\rho.\end{split}$
Thus $L(\theta)$ is a Borel radius of the function $w(z)$. ∎
## 4\. Proof of the theorems
Suppose that $w(z)$ is a $v-$valued algebroid function of order $\rho$ in the
unit disk $D(z)$ and $L(\theta)$ be a Borel radius of $w(z)$. For any
$0<\eta<1$, we write $M(\zeta)=w(z(\zeta))$, where $z=z(\zeta)$ is the
function in Lemma 2.2. Since
$M^{\prime}(\zeta)=w^{\prime}(z(\zeta))z^{\prime}(\zeta)$, we have
$T(t,M^{\prime}(\zeta))\leq T(t,w^{\prime}(z(\zeta)))+T(t,z^{\prime}(\zeta))$
$T(t,w^{\prime}(z(\zeta)))\leq
T(t,M^{\prime}(\zeta))+T(t,\frac{1}{z^{\prime}(\zeta)})=T(t,M^{\prime}(\zeta))+T(t,z^{\prime}(\zeta))+O(1).$
Combining the above two inequalities and noting Lemma 2.2, we have
(4.1)
$|T(t,M^{\prime}(\zeta))-T(t,w^{\prime}(z(\zeta))|\leq|T(t,z^{\prime}(\zeta))|\leq
3\log\frac{2}{1-t}+\log\frac{\pi}{\eta}.$
By Lemma 2.4, we can see that $\rho(M^{\prime})=\rho(M)=\rho$. Therefore the
order of the function $w^{\prime}(z(\zeta))$ is also $\rho$. Then by Lemma
3.1, $L(\theta)$ is also a Borel radius of the function $w^{\prime}(z)$.
Next we suppose that $L(\theta)$ is a Borel radius of the function
$w^{\prime}(z)$. By Lemma 3.1, the function $w^{\prime}(z(\zeta))$ is an
algebroid function of order $\rho$ in the unit disk $D(\zeta)$. Then the order
of the function $M(\zeta)$ is also $\rho$. Moreover, we use Lemma 3.1, we
obtain that $L(\theta)$ is a Borel radius of the function $w(z)$.
## 5\. Open question
In some literatures, we have known that a radius $L(\theta)$ is a Borel radius
of a $\rho-$order meromorphic function if and only if there exists a
$\varepsilon>0$ such that
(5.1) $\limsup\limits_{r\rightarrow 1-}\frac{\log
T(r,\Delta(\theta,\varepsilon),f)}{\log\frac{1}{1-r}}=\rho.$
And it is easy to prove that if $L(\theta)$ is a Borel radius of a
$\rho-$order algebroid function $w(z)$, then (5.1) holds. Here we ask if the
converse proposition holds.
## References
* [1] C. T. Chuang, _Un théorème relatif aux directions de Borel des fonctions meromorphes d’ordre fini,_ C.R.Acad.Sci., 204(1937), 951-952.
* [2] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
* [3] Y. Z. He and X. Z. Xiao, Algebroid functions and ordinary differential equations(in Chinese), Science Press, China, 1988.
* [4] A. Rauch, Sur les algébroïdes enti$\grave{e}$res, C. R. Acad. Sci. Paris 202(1936) 2041-043.
* [5] A. Rauch, _Cas où une direction de Borel d’une fonction entière $f(z)$ d’ordre finiest aussi direction de Borel pour $f^{\prime}(z)$,_ C.R.Acad. Sci., 199(1934), 1014-1016.
* [6] H. Selbreg, Algebroide Funktionen und Umkehrfunktionen Abelscher Integrale, Avh. Norske Vid. Akad. Oslo 8(1934), 1-72.
* [7] N. Toda, Sur les directions de Julia et de Borel des fonctions algébroıdes, Nagoya Math. Journal. 34(1969), 1-23.
* [8] M. Tsuji, Potential theory in modern function theory, Maruzen Co. LTD Tokyo., 1959
* [9] E. Ullrich, Über den Einfluss der verzweigtheit einer Algebroide auf ihre Wertverteilung, J.reine ang. Math. 169(1931), 198-220.
* [10] G. Valiron, Sur la derivée des fonctions algébroïdes, Bull. Sci. Math. 59(1931), 17-39.
* [11] G. Valiron, Sur les directions de Borel des fonctions algébroïdes m$\acute{e}$romorphes d’ordre infini, C. R. Acad. Sci. Paris. 206 (1938), 735-737.
* [12] G. Valiron, Recherches sur le theoreme de M.Borel dans la theorie des fonctions meromorphes, Acta Math., 52(1928), 67-92.
* [13] G. Valirion, _Lectures on the general theory of integral functions, Edouard Privat_ , Toulouse, 1923.
* [14] F. F. Zeng, _The order of the drivertive function of an algebroid function._ , J. Jishou Univ., 1, 2(1988), 1-9. (in Chinese)
* [15] Q. D. Zhang, _Common Borel radii of a mermorphic function and its derivative in the unit disc._ , J. Chengdou Univ. Information Tech., 1, 17(2002), 1-4.
* [16] Q. D. Zhang, _Distribution of Borel radii of meromorphic functions in the unit disc._ , Acta Math. Sinica., 2, 42(1999), 351-358 (in Chinese)
|
arxiv-papers
| 2009-06-24T06:47:11 |
2024-09-04T02:49:03.507694
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nan Wu, Zuxing Xuan",
"submitter": "Zuxing Xuan",
"url": "https://arxiv.org/abs/0906.4409"
}
|
0906.4473
|
# Existence et unicité globale pour le système de Navier-Stokes axisymétrique
anisotrope
Hammadi Abidi 111Département de Mathématiques Faculté des Sciences de Tunis
Campus universitaire 2092 Tunis, Tunisia. [email protected] et Marius Paicu
222Laboratoire de Mathématique Université Paris Sud Bâtiment 425, 91405 Orsay
France [email protected]
Abstract : We study in this paper the axisymmetric $3$-D Navier-Stokes system
where the horizontal viscosity is zero. We prove the existence of a unique
global solution to the system with initial data of Yudovitch type.
Résumé : Nous étudions dans ce papier le système de Navier-Stokes $3$-D
axisymétrique avec viscosité horizontale nulle. Nous allons prouver que le
système est globalement bien posé pour des données de type Yudovitch.
AMS Subject Classifications : 35Q30 (35Q35 76D03 76D05 76D09)
Keywords : Navier-Stokes anisotrope; Existence globale; Unicité.
## 1 Introduction
L’écoulement tridimensionnel d’un fluide homogène visqueux incompressible est
régi par les équations de Navier-Stokes que nous rappelons ici :
${\rm(NS)}\;\left\\{\begin{array}[]{rl}&\partial_{t}u+(u\cdot\nabla)u-\nu_{h}(\partial_{x}^{2}+\partial_{y}^{2})u-\nu_{v}\partial_{z}^{2}u=-\nabla
p\\\ &{\mathop{\rm div}}\,u=0\\\ &u_{|t=0}=u_{0}.\end{array}\right.$
Ci-dessus $\nu_{h}$ (resp. $\nu_{v}$) représente la viscosité horizontale
(resp. verticale), la vitesse $u$ est un champ de vecteurs inconnu dépendant
du temps $t$ et de la variable d’espace $x\in{\mathbb{R}}^{3}$ et $\nabla p$
correspond au gradient de la pression et peut être interprété comme le
multiplicateur de Lagrange associé à la contrainte d’incompressibilité
$\mathop{\rm div}\,u=0.$
Dans le cas ou les coefficients de viscosité $\nu_{h}$ et $\nu_{v}$ sont
strictement positives, on sait que le système $(\rm{NS})$ admet une solution
globale dans l’espace d’énergie $L^{2}$ d’après les travaux de J. Leray [13].
Ensuite dans les années soixante H. Fujita et T. Kato [9] ont démontré, par
des techniques de semi-groupe que $(\rm{NS})$ est localement bien posé pour
des données initiales dans l’espace de Sobolev homogène
$\dot{H}^{\frac{1}{2}}.$ L’existence globale est établie pour des données
petites devant $\inf\\{\nu_{h},\nu_{v}\\}.$ D’autres résultats semblables ont
été prouvés dans des espaces fonctionnels qui sont tous invariants par
changement d’échelle de l’équation considérée (voir par exemple [4] et [11]).
Dans le cas ou $\nu_{h}>0$ et $\nu_{v}=0$ le système $(\rm{NS_{h}})$ a été
étudiée pour la première fois par J.-Y. Chemin et al. [5]. Plus exactement ont
démontré l’existence locale en temps d’une solution, lorsque la donnée
initiale est dans l’espace de Sobolev anisotrope $H^{0,{1\over 2}+},$ avec
$H^{0,s}=\big{\\{}u\in
L^{2}\;\big{|}\;(\int_{{\mathbb{R}}^{2}}\|u(x,y,\cdot)\|^{2}_{H^{s}({\mathbb{R}})}dxdy)^{1\over
2}<\infty\big{\\}}.$ L’existence globale est établie pour des données petites
devant la viscosité $\nu_{h}.$ Par contre l’unicité a été prouvé pour des
données dans $H^{0,{3\over 2}+}.$ Notons que l’unicité dans le cas où la
donnée $u_{0}\in H^{0,{1\over 2}+}$ a été obtenue par D. Iftimie [10]. Ensuite
M. Paicu [14], a démontré que le système $(\rm{NS_{h}})$ est localement bien
posé dans l’espace de Besov anisotrope ${\mathscr{B}}^{0,{1\over
2}}=\big{\\{}u\in{\mathcal{S}}^{\prime}\big{|}\displaystyle\sum_{q\in{\mathbb{Z}}}(\int_{2^{q-1}\leq|z|\leq
2^{q}}|z|\|{\mathcal{F}}u(\cdot,\cdot,z)\|_{L^{2}({\mathbb{R}}^{2})}^{2}dz)^{1\over
2}<\infty\big{\\}},$ l’existence globale a été prouvé pour des données petites
devant $\nu_{h}.$ Récemment J.-Y. Chemin et P. Zhang [6] ont obtenu un
résultat similaire en travaillant dans un espace de Besov anisotrope d’indice
négatif.
Dans la suite, on suppose que le fluide est uniquement verticalement visqueux,
c’est-à-dire, que $\nu_{h}=0$ et $\nu_{v}>0.$ Dans cette partie on ne
s’intéressera pas à la dépendance par rapport à la viscosité $\nu_{v}$ des
quantités à mesurer, et l’on supposera donc pour simplifier que $\nu_{v}=1.$
Dans ce cas le système devient :
${\rm(NS_{v})}\;\left\\{\begin{array}[]{rl}&\partial_{t}u+(u\cdot\nabla)u-\partial_{z}^{2}u=-\nabla
p\\\ &{\mathop{\rm div}}\,u=0\\\ &u_{|t=0}=u_{0}.\end{array}\right.$
Rappelons que dans le cas ou $\nu_{h}>0$ et $\nu_{v}=0,$ la condition
d’incompressibilité, c’est-à-dire,
$\partial_{x}u^{1}+\partial_{y}u^{2}+\partial_{z}u^{3}=0,$ a permis aux
auteurs de prouver un effet régularisant pour la troisième composante $u^{3}$
à partir du laplacien horizontal. Par contre dans notre cas on a un seul effet
régularisant qui rend l’étude du système très difficile. Pour cela on
s’intéresse à des solutions particulières, plus exactement des solutions
axisymétriques, puisque dans se cas, on a ${\mathop{\rm
div}}\,u=\partial_{r}u^{r}+{u^{r}\over r}+\partial_{z}u^{z}=0.$ Avant de
donner plus de détails, il convient de préciser ce que nous entendons par
données et solutions axisymétriques.
###### Définition 1.1.
On dit qu’un champ de vecteurs $u$ est axisymétrique si et seulement si il
possède une symétrie cylindrique de réflexion, c’est-à-dire,
$u=u^{r}(r,z)e_{r}+u^{z}(r,z)e_{z}$
où $\big{(}e_{r},e_{\theta},e_{z}\big{)}$ est la base cylindrique.
Une fonction scalaire est dite axisymétrique si elle ne dépend pas de la
variable angulaire $\theta.$
Le système de Navier-Stokes classique (dans le cas $\nu_{h}=\nu_{v}>0$) a déjà
été étudié par plusieurs auteurs, le premiers résultats étant dues à M.
Ukhovskii et V. Youdovitch [17] et O. A. Ladyzhenskaya [12].
Dans ce cas la vorticité de $u$ que est définie par $\omega:=\nabla\times u,$
admet dans le repère cylindrique une seule composante portée par $e_{\theta}$:
$\omega=\omega^{\theta}e_{\theta}\hskip 14.22636pt\mbox{avec}\hskip
14.22636pt\omega^{\theta}=\partial_{z}u^{r}-\partial_{r}u^{z}$
et qui vérifie l’équation suivante:
$\partial_{t}\omega+(u^{r}\partial_{r}+u^{z}\partial_{z})\omega-\frac{u^{r}}{r}\omega-\partial^{2}_{z}\omega=0,$
et par suite $\omega/r$ vérifie l’équation de trasport-diffusion:
$\partial_{t}{\omega\over r}+(u^{r}\partial_{r}+u^{z}\partial_{z}){\omega\over
r}-\partial^{2}_{z}{\omega\over r}=0.$
Il est alors possible de montrer par une méthode d’énergie que pour tout
$p\in[1,\infty]$ (resp. $p\in]1,2]$) la norme de $\omega/r$ (resp.
$r^{-1}\partial_{z}\omega$) dans $L^{p}$ (resp. $L^{2}_{t}(L^{p})$) est
contrôlée par celle de ${\omega_{0}}/r.$ D’après la loi de Biot-Savart, on
démontre (voir Proposition 3.1) que
$|{u^{r}\over r}|\lesssim{1\over|\cdot|}\star|r^{-1}\partial_{z}\omega|.$
Ainsi la condition d’incompressibilité nous permet de contrôler
$\partial_{r}u^{r}$ puisque $\partial_{r}u^{r}=-{u^{r}\over
r}-\partial_{z}u^{z}.$ Notre résultat principal est le suivant (concernant la
définition de l’espace de Lorentz voir la section suivante):
###### Théorème 1.1.
Soit $\omega_{0}\in L^{{3\over 2},1}({\mathbb{R}}^{3})$ tel que
${\omega_{0}\over r}\in L^{{3\over 2},1}({\mathbb{R}}^{3}).$ Soit $u_{0}$ le
champ de vecteurs avec ${\mathop{\rm div}}\,u_{0}=0$ et
$\omega_{0}=\nabla\times u_{0}$ donné par la loi de Biot-Savart :
$u_{0}(X)={1\over
4\pi}\int_{{\mathbb{R}}^{3}}\frac{(X-Y)\times\omega_{0}(Y)}{|X-Y|^{3}}\,dY.$
Alors le système ${\rm(NS_{v})}$ admet une solution globale $u$ tel que la la
vorticité $\omega$ satisfait
$\displaystyle\omega\in L^{\infty}_{loc}\big{(}{\mathbb{R}}_{+};\,L^{{3\over
2},1}({\mathbb{R}}^{3})\big{)},\hskip 28.45274pt\partial_{z}\omega\in
L^{2}_{loc}\big{(}{\mathbb{R}}_{+};\,L^{{3\over
2},1}({\mathbb{R}}^{3})\big{)}$ $\displaystyle{\omega\over r}\in
L^{\infty}_{loc}\big{(}{\mathbb{R}}_{+};\,L^{{3\over
2},1}({\mathbb{R}}^{3})\big{)},\hskip 25.6073pt\partial_{z}{\omega\over r}\in
L^{2}_{loc}\big{(}{\mathbb{R}}_{+};\,L^{{3\over
2},1}({\mathbb{R}}^{3})\big{)}.$
De plus pour tout $t\geq 0,$ on a
$\|\omega(t)\|_{L^{{3\over 2},1}}+\|\partial_{z}\omega\|_{L^{2}_{t}(L^{{3\over
2},1})}\leq C\|\omega_{0}\|_{L^{{3\over 2},1}}\exp\big{(}Ct^{1\over
2}\|r^{-1}\omega_{0}\|_{L^{{3\over 2},1}}\big{)}$
et
$\|r^{-1}\omega(t)\|_{L^{{3\over
2},1}}+\|r^{-1}\partial_{z}\omega\|_{L^{2}_{t}(L^{{3\over 2},1})}\leq
C\|r^{-1}\omega_{0}\|_{L^{{3\over 2},1}}.$
En outre, cette solution est unique si de plus $\partial_{r}\omega_{0}\in
L^{{3\over 2},1}.$
###### Remarque 1.1.
Rappelons que pour des données initiales de type Yudovitch R. Danchin [7] à
démontre que le système d’Euler axisymétrique est globalement bien pose. Plus
exactement il démontre que le système est globalement bien posé lorsque
$\omega_{0}\in L^{3,1}\cap L^{\infty}$ et $\omega_{0}/r\in L^{3,1}.$ Récemment
H. Abidi et al. [2] ont montré que le système d’Euler axisymétrique est
globalement bien pose dans des espace critiques plus précisément lorsque
$u_{0}\in B^{{3\over p}+1}_{p,1}$ pour $p\in[1,\infty]$ et $\omega_{0}/r\in
L^{3,1}.$
###### Remarque 1.2.
On note aussi quand obtient un résultat similaire que H. Abidi [1]. En effet,
dans cet article, l’auteur démontre que le système de Navier-Stokes
axisymétrique (i.e, $\nu_{h}=\nu_{v}>0$) est globalement bien posé lorsque la
donnée initiale vérifie $u_{0}\in W^{2,p}({\mathbb{R}}^{3})$ pour $1<p<2$.
Nous pouvons obtenir l’existence des solutions pour des données initiales de
régularité encore plus faible. L’unicité en revanche semble être beaucoup plus
difficile à obtenir avec cette régularité très faible. Nous avons le résultat
suivant.
###### Théorème 1.2.
Soit $\omega_{0}\in L^{\frac{6}{5}}\cap L^{{6\over 5}+,1}({\mathbb{R}}^{3})$
tel que ${\omega_{0}\over r}\in L^{\frac{6}{5}}\cap L^{{6\over
5}+,1}({\mathbb{R}}^{3}).$ Soit $u_{0}$ le champ de vecteurs avec
${\mathop{\rm div}}\,u_{0}=0$ et $\omega_{0}=\nabla\times u_{0}$ donné par la
loi de Biot-Savart. Alors le système ${\rm(NS_{v})}$ admet une solution
globale $u$ tel que la la vorticité $\omega$ satisfait
$\displaystyle\big{(}\omega,\frac{\omega}{r}\big{)}\in
L^{\infty}_{loc}\big{(}{\mathbb{R}}_{+};\,L^{\frac{6}{5}}\cap L^{{6\over
5}+,1}({\mathbb{R}}^{3})\big{)},\hskip
28.45274pt\big{(}\partial_{z}\omega,\partial_{z}\frac{\omega}{r}\big{)}\in
L^{2}_{loc}\big{(}{\mathbb{R}}_{+};\,L^{\frac{6}{5}}\cap L^{{6\over
5}+,1}({\mathbb{R}}^{3})\big{)}.$
## 2 Notation et préliminaires
On dit que $A\lesssim B$ s’il existe une constante $C$ strictement positive
telle que $A\leq CB.$ La notation $C$ désigne une constante générique qui peut
changer d’une ligne à une autre. Soient $X$ un espace de Banach et
$p\in[1,\infty],$ on désigne par $L^{p}(0,T;\,X)$ l’ensemble des fonctions $f$
mesurables sur $(0,T)$ à valeurs dans $X,$ telles que
$t\longmapsto\|f(t)\|_{X}$ appartient à $L^{p}(0,T).$ On note $C([0,T);\,X)$
l’espace des fonctions continues de $[0,T)$ à valeurs dans $X,$
$C_{b}([0,T);\,X)\overset{d\acute{e}f}{=}C([0,T);\,X)\cap
L^{\infty}(0,T;\,X).$ Enfin on désigne par $p^{\prime}$ l’exposant conjugué de
$p$ défini par $\frac{1}{p}+\frac{1}{p^{\prime}}=1.$
Avant d’introduire la définition de l’espace de Lorentz, on commence par
rappel la réarrangement d’une fonction. Soit $f$ une fonction mesurable, on
définit son réarrangement $f^{*}:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ par la
formule
$f^{*}(\lambda):=\inf\Big{\\{}s\geq
0;\,\big{|}\\{x/\,|f(x)|>s\\}\big{|}\leq\lambda\Big{\\}}.$
###### Définition 2.1.
(espace de Lorentz) Soient $f$ une fonction mesurable et $1\leq
p,q\leq\infty.$ Alors $f$ appartient a l’espace de Lorentz $L^{p,q}$ si
$\|f\|_{L^{p,q}}\overset{d\acute{e}f}{=}\begin{cases}\Big{(}\int^{\infty}_{0}(t^{1\over
p}f^{*}(t))^{q}{dt\over t}\Big{)}^{1\over q}<\infty&\text{si $q<\infty$}\\\
\displaystyle\sup_{t>0}t^{1\over p}f^{*}(t)<\infty&\text{si
$q=\infty$}.\end{cases}$
Nous pouvons également définir les espaces de Lorentz comme interpolation
réelle des espaces de Lebesgue :
$L^{p,q}:=(L^{p_{0}},L^{p_{1}})_{(\theta,q)},$
avec $1\leq p_{0}<p<p_{1}\leq\infty,$ $0<\theta<1$ satisfait ${1\over
p}={1-\theta\over p_{0}}+{\theta\over p_{1}}$ et $1\leq q\leq\infty,$ muni de
la norme
$\|f\|_{L^{p,q}}:=\Big{(}\int_{0}^{\infty}\big{(}t^{-\theta}K(t,f)\big{)}^{q}{dt\over
t}\Big{)}^{1\over q}$
avec
$K(f,t):=\displaystyle\inf_{f=f_{0}+f_{1}}\big{\\{}\|f_{0}\|_{L^{p_{0}}}+t\|f_{1}\|_{L^{p_{1}}}\;\,\big{|}\;f_{0}\in
L^{p_{0}},\,f_{1}\in L^{p_{1}}\big{\\}}.$
L’espace de Lorentz vérifie les propriétés suivantes (pour plus de détails
voir [15]) :
###### Proposition 2.1.
Soient $f\in L^{p_{1},q_{1}},$ $g\in L^{p_{2},q_{2}}$ et $1\leq
p,q,p_{j},q_{j}\leq\infty,$ pour $1\leq j\leq 2.$
* •
Si ${1\over p}={1\over p_{1}}+{1\over p_{2}}$ et ${1\over q}={1\over
q_{1}}+{1\over q_{2}},$ alors
$\|fg\|_{L^{p,q}}\lesssim\|f\|_{L^{p_{1},q_{1}}}\|g\|_{L^{p_{2},q_{2}}}.$
* •
Si $1<p<\infty,$ ${1\over p}+1={1\over p_{1}}+{1\over p_{2}}$ et ${1\over
q}={1\over q_{1}}+{1\over q_{2}},$ alors
$\|f\ast g\|_{L^{p,q}}\lesssim\|f\|_{L^{p_{1},q_{1}}}\|g\|_{L^{p_{2},q_{2}}},$
pour $p=\infty,$ et ${1\over q_{1}}+{1\over q_{2}}=1,$ alors
$\|f\ast
g\|_{L^{\infty}}\lesssim\|f\|_{L^{p_{1},q_{1}}}\|g\|_{L^{p_{2},q_{2}}}.$
* •
Pour $1\leq p\leq\infty$ et $1\leq q_{1}\leq q_{2}\leq\infty,$ on a
$L^{p,q_{1}}\hookrightarrow L^{p,q_{2}}\hskip 28.45274pt\mbox{et}\hskip
28.45274ptL^{p,p}=L^{p}.$
Dans le repère cylindrique $\omega=\nabla\times u$ admet une seule composante
portée par $e_{\theta}$ et dans le repère cartésienne deux composantes:
$\omega=(\omega^{1},\omega^{2},0)$
avec $\omega^{1}=\partial_{y}u^{3}-\partial_{z}u^{2}$ et
$\omega^{2}=\partial_{z}u^{1}-\partial_{x}u^{3},$ $u^{j}$ pour $1\leq j\leq 3$
les composantes de $u$ dans la base cartésienne et $(x,y,z)$ les variables
dans cette base. Le fait que $u^{\theta}=0,$ alors dans le repère cylindrique,
on a:
$\displaystyle u\cdot\nabla=u^{r}\partial_{r}+u^{z}\partial_{z},$
$\displaystyle{\mathop{\rm div}}\,u=\partial_{r}u^{r}+{u^{r}\over
r}+\partial_{z}u^{z}$ $\displaystyle\mbox{et}\hskip
71.13188ptu^{r}=\omega^{\theta}=0\hskip 14.22636pt\mbox{sur la droite}\hskip
14.22636ptr=0.$
Le dernier point on peut le déduire du fait que $u^{\theta}=0:$ en effet,
comme
$u^{\theta}=u\cdot e_{\theta}$
ainsi
$-yu^{1}+xu^{2}=0.$ (2.1)
Et par suite $u^{1}=0$ (resp. $u^{2}=0$) sur le plan $x=0$ (resp. $y=0$). Pour
$\omega^{\theta},$ on utilise le fait que $\omega$ est portée par
$e_{\theta},$ ce qui implique
$x\omega^{1}+y\omega^{2}=0,$
et par suite $\omega^{1}$ (resp. $\omega^{2}$) est nulle sur le plan $x=0$
(resp. $y=0$). D’où le résultat. Rappelons que si $u$ est solution de
$(NS_{v}),$ alors $\omega$ vérifie l’équation suivante
$\partial_{t}\omega+(u^{r}\partial_{r}+u^{z}\partial_{z})\omega-\frac{u^{r}}{r}\omega-\partial^{2}_{z}\omega=0,$
mais comme $u^{\theta}=0,$ alors
$\partial_{t}\omega+(u\cdot\nabla)\omega-\frac{u^{r}}{r}\omega-\partial^{2}_{z}\omega=0.$
(2.2)
Autrement dit, dans le cas axisymétrique, $(NS_{v})$ se ramène à un problème
d’évolution bidimensionnel. Rappelons qu’en dimension 2,
$\omega=\partial_{x}u^{2}-\partial_{y}u^{1},$ vérifie l’équation de transport-
diffusion suivante :
$\partial_{t}\omega+(u\cdot\nabla)\omega-\partial^{2}_{z}\omega=0.$
En dimension 3 dans le cas axisymétrique $\frac{\omega}{r}$ joue un rôle
similaire puisque
$\partial_{t}\frac{\omega}{r}+(u\cdot\nabla)\frac{\omega}{r}-\partial_{z}^{2}\frac{\omega}{r}=0.$
(2.3)
## 3 Démonstration du théorème 1.1
### 3.1 Estimations a priori
D’après l’équation (2.3) et la loi de Biot-Savart, on peut contrôler des
quantités très importantes, qui nous permet de démontrer l’existence globale.
Plus exactement, on a la proposition suivante.
###### Proposition 3.1.
Soient $(p,q,\lambda)\in[1,\infty]^{3},$ alors on a les inégalités suivantes :
* •
Si ${3\over 2}\leq p<\infty$ tel que ${1\over q}={1\over 3}+{1\over p},$ alors
$\displaystyle\|u\|_{L^{p,\lambda}}\lesssim\|\omega\|_{L^{q,\lambda}},\qquad\|{u^{r}\over
r}\|_{L^{p,\lambda}}\lesssim\|{\omega\over
r}\|_{L^{q,\lambda}},\qquad\|\partial_{z}u^{r}\|_{L^{p,\lambda}}\lesssim\|\partial_{z}\omega\|_{L^{q,\lambda}},$
$\displaystyle\|\partial_{z}u^{z}\|_{L^{p,\lambda}}\lesssim\|\partial_{z}\omega\|_{L^{q,\lambda}}\hskip
8.5359pt\mbox{et}\hskip
8.5359pt\|\partial_{z}u^{z}\|_{L^{p,\lambda}}+\|\partial_{r}u^{z}\|_{L^{p,\lambda}}\lesssim\|\partial_{r}\omega\|_{L^{q,\lambda}}+\|{\omega\over
r}\|_{L^{q,\lambda}}.$
* •
Si $3\leq p<\infty$ tel que ${1\over q}={2\over 3}+{1\over p},$ alors
$\displaystyle\|u^{r}\|_{L^{p,\lambda}}\lesssim\|\partial_{z}\omega\|_{L^{q,\lambda}},\hskip
28.45274pt\|{u^{r}\over r}\|_{L^{p,\lambda}}\lesssim\|\partial_{z}{\omega\over
r}\|_{L^{q,\lambda}}$
$\displaystyle\|u^{z}\|_{L^{p,\lambda}}\lesssim\|\partial_{r}\omega\|_{L^{q,\lambda}}+\|{\omega\over
r}\|_{L^{q,\lambda}},\hskip
14.22636pt\|\partial_{z}u^{z}\|_{L^{p,\lambda}}\lesssim\|\partial_{z}\partial_{r}\omega\|_{L^{q,\lambda}}+\|\partial_{z}{\omega\over
r}\|_{L^{q,\lambda}}$
et
$\|\partial_{r}u^{r}\|_{L^{p,\lambda}}\lesssim\|\partial_{z}\partial_{r}\omega\|_{L^{q,\lambda}}+\|\partial_{z}{\omega\over
r}\|_{L^{q,\lambda}}.$
* •
Dans le cas limite, c’est-à-dire, $p=\infty$
$\displaystyle\|u\|_{L^{\infty}}\lesssim\|\omega\|_{L^{3,1}},\qquad\|u^{r}\|_{L^{\infty}}\lesssim\|\partial_{z}\omega\|_{L^{{3\over
2},1}},\qquad\|{u^{r}\over r}\|_{L^{\infty}}\lesssim\|\partial_{z}{\omega\over
r}\|_{L^{{3\over 2},1}}$
$\displaystyle\|u^{z}\|_{L^{\infty}}\lesssim\|\partial_{r}\omega\|_{L^{{3\over
2},1}}+\|{\omega\over r}\|_{L^{{3\over 2},1}},\hskip
14.22636pt\|\partial_{z}u^{z}\|_{L^{\infty}}\lesssim\|\partial_{z}\partial_{r}\omega\|_{L^{{3\over
2},1}}+\|\partial_{z}{\omega\over r}\|_{L^{{3\over 2},1}}$
et
$\|\partial_{r}u^{r}\|_{L^{\infty}}\lesssim\|\partial_{z}\partial_{r}\omega\|_{L^{{3\over
2},1}}+\|\partial_{z}{\omega\over r}\|_{L^{{3\over 2},1}}.$
###### Proof.
D’après la loi de Biot-Savart, on a
$u(X)={1\over{4\pi}}\int_{{\mathbb{R}}^{3}}{{X-X^{\prime}}\over{|X-X^{\prime}|^{3}}}\times\,\omega(X^{\prime})dX^{\prime},$
(3.1)
avec $X=(x,y,z)$ et $X^{\prime}=(x^{\prime},y^{\prime},z^{\prime}),$ et par
suite
$|u|\lesssim{1\over|\cdot|^{2}}\star|\omega|,$
or par définition de l’espace de Lorentz (définition 2.1), on a
${1\over|X|^{2}}\in L^{{3\over 2},\infty}({\mathbb{R}}^{3})$
ainsi grâce à la Proposition 2.1, on en déduit
$\|u\|_{L^{p,\lambda}}\lesssim\|\omega\|_{L^{{3p\over 3+p},\lambda}}\hskip
28.45274pt\mbox{pour ${3\over 2}\leq p<\infty$}\hskip
28.45274pt\mbox{et}\hskip
28.45274pt\|u\|_{L^{\infty}}\lesssim\|\omega\|_{L^{3,1}}.$
D’après l’égalité (3.1), on a
$u^{1}(x)=-{1\over{4\pi}}\int_{{\mathbb{R}}^{3}}{{z-z^{\prime}}\over{|X-X^{\prime}|^{3}}}\,\omega^{2}(X^{\prime})dX^{\prime}$
et
$u^{2}={1\over{4\pi}}\int_{{\mathbb{R}}^{3}}{{z-z^{\prime}}\over{|X-X^{\prime}|^{3}}}\,\omega^{1}(X^{\prime})dX^{\prime}$
avec
$\omega^{1}(X^{\prime})=-\sin\theta^{\prime}\,\omega^{\theta}(X^{\prime})$ et
$\omega^{2}(X^{\prime})=\cos\theta^{\prime}\,\omega^{\theta}(X^{\prime}).$
Ainsi
$\displaystyle u^{r}(X)$
$\displaystyle=\cos\theta\,u^{1}(X)+\sin\theta\,u^{2}(X)$
$\displaystyle={1\over{4\pi}}\int_{{\mathbb{R}}^{3}}{{z-z^{\prime}}\over{|X-X^{\prime}|^{3}}}\big{\\{}-\cos\theta\cos\theta^{\prime}-\sin\theta\sin\theta^{\prime}\big{\\}}\omega^{\theta}(X^{\prime})dX^{\prime}$
où l’on désigne par $(r,\theta,z)$ les variables dans le repère cylindrique,
rappelons que dans ce repère $X=(r\cos\theta,r\sin\theta,z)$ et
$X^{\prime}=(r^{\prime}\cos\theta^{\prime},r^{\prime}\sin\theta^{\prime},z^{\prime}).$
Et par suite
$\displaystyle u^{r}(X)$
$\displaystyle=-{1\over{4\pi}}\int_{{\mathbb{R}}^{3}}{{z-z^{\prime}}\over{|X-X^{\prime}|^{3}}}\big{\\{}\cos\theta\sin\theta^{\prime}+\sin\theta\cos\theta^{\prime}\big{\\}}\omega^{\theta}(X^{\prime})dX^{\prime}$
$\displaystyle=-{1\over{4\pi}}\int_{{\mathbb{R}}^{3}}{{z-z^{\prime}}\over{|X-X^{\prime}|^{3}}}\cos(\theta-\theta^{\prime})\omega^{\theta}(r^{\prime},z^{\prime})r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime},$
or
${{z-z^{\prime}}\over{|X-X^{\prime}|^{3}}}=\partial_{z^{\prime}}{1\over|X-X^{\prime}|},$
ainsi par intégration par parties, on trouve
$u^{r}(X)={1\over{4\pi}}\int_{{\mathbb{R}}^{3}}{1\over{|X-X^{\prime}|}}\cos(\theta-\theta^{\prime})\partial_{z^{\prime}}\omega^{\theta}(r^{\prime},z^{\prime})r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime}.$
Mais comme $u^{r}$ ne dépend pas de $\theta$ (X=(r,0,z)), alors
$u^{r}(t,r,z)={1\over{4\pi}}\int_{{\mathbb{R}}^{3}}{1\over{|X-X^{\prime}|}}\cos\theta^{\prime}\partial_{z^{\prime}}\omega^{\theta}(t,r^{\prime},z^{\prime})r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime},$
(3.2)
ce qui implique que
$|u^{r}|\lesssim{1\over|\cdot|}\star|\partial_{z^{\prime}}\omega|.$
Or par définition de l’espace de Lorentz, on a
${1\over|X|}\in L^{3,\infty}({\mathbb{R}}^{3})$
ainsi grâce à la Proposition 2.1, on obtient l’inégalité souhaitée. Pour la
deuxième inégalité de la proposition, grâce a l’égalité (3.2), on a
$|\partial_{z}u^{r}|\lesssim{1\over|\cdot|^{2}}\star|\partial_{z^{\prime}}\omega|,$
en conséquence la Proposition 2.1, donne l’inégalité désirée. Pour
${u^{r}\over r},$ on utilise l’identité (3.2) et on suit les mêmes calculs de
[16], on trouve
$\displaystyle u^{r}(t,r,z)$
$\displaystyle={1\over{4\pi}}\int_{{\mathbb{R}}_{+}\times[0,2\pi]\times{\mathbb{R}}}{\cos\theta^{\prime}\partial_{z^{\prime}}\omega^{\theta}(t,r^{\prime},z^{\prime})\over{(r^{2}+r^{\prime
2}-2rr^{\prime}\cos\theta^{\prime}+(z-z^{\prime})^{2})^{1\over
2}}}\,r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime}$
$\displaystyle={1\over{4\pi}}\int_{{\mathbb{R}}_{+}\times[-{\pi\over
2},{\pi\over
2}]\times{\mathbb{R}}}{\cos\theta^{\prime}\partial_{z^{\prime}}\omega^{\theta}(t,r^{\prime},z^{\prime})\over{(r^{2}+r^{\prime
2}-2rr^{\prime}\cos\theta^{\prime}+(z-z^{\prime})^{2})^{1\over
2}}}\,r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime}$
$\displaystyle+{1\over{4\pi}}\int_{{\mathbb{R}}_{+}\times[{\pi\over
2},{3\pi\over
2}]\times{\mathbb{R}}}{\cos\theta^{\prime}\partial_{z^{\prime}}\omega^{\theta}(t,r^{\prime},z^{\prime})\over{(r^{2}+r^{\prime
2}-2rr^{\prime}\cos\theta^{\prime}+(z-z^{\prime})^{2})^{1\over
2}}}\,r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime}$
pour la deuxième partie, on effectue le changement de variable suivant
$\theta^{\prime}\to\theta^{\prime}+\pi,$ on aura
$\displaystyle u^{r}(t,r,z)$
$\displaystyle={1\over{4\pi}}\int_{{\mathbb{R}}_{+}}\int_{-{\pi\over
2}}^{{\pi\over
2}}\int_{{\mathbb{R}}}{\cos\theta^{\prime}\partial_{z^{\prime}}\omega^{\theta}(t,r^{\prime},z^{\prime})\over{(r^{2}+r^{\prime
2}-2rr^{\prime}\cos\theta^{\prime}+(z-z^{\prime})^{2})^{1\over
2}}}\,r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime}$ (3.3)
$\displaystyle-{1\over{4\pi}}\int_{{\mathbb{R}}_{+}}\int_{-{\pi\over
2}}^{{\pi\over
2}}\int_{{\mathbb{R}}}{\cos\theta^{\prime}\partial_{z^{\prime}}\omega^{\theta}(t,r^{\prime},z^{\prime})\over{(r^{2}+r^{\prime
2}+2rr^{\prime}\cos\theta^{\prime}+(z-z^{\prime})^{2})^{1\over
2}}}\,r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime}.$
Si $|X-X^{\prime}|\leq r,$ on utilise l’égalité (3.2) et le fait que
$r^{\prime}\leq 2r,$ on trouve
$\Big{|}\int_{|X-X^{\prime}|\leq
r}{\cos\theta^{\prime}\partial_{z^{\prime}}\omega^{\theta}(t,r^{\prime},z^{\prime})\over|X-X^{\prime}|}r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime}\Big{|}\lesssim
r\int_{{\mathbb{R}}^{3}}{1\over{|X-X^{\prime}|}}\big{|}\partial_{z^{\prime}}{\omega(t,X^{\prime})\over
r^{\prime}}\big{|}dX^{\prime}.$
Si $|X-X^{\prime}|\geq r,$ on utilise l’égalité (3.3) et le fait que
$\displaystyle\Big{|}\Big{(}r^{2}+r^{\prime
2}+2rr^{\prime}\cos\theta^{\prime}+(z-z^{\prime})^{2}\Big{)}^{-{1\over 2}}$
$\displaystyle-\Big{(}r^{2}+r^{\prime
2}-2rr^{\prime}\cos\theta^{\prime}+(z-z^{\prime})^{2}\Big{)}^{-{1\over
2}}\Big{|}$ $\displaystyle\leq{2r\over|X-X^{\prime}|^{2}},$
car $-{\pi\over 2}\leq\theta^{\prime}\leq{\pi\over 2}.$ Ainsi dans cette
région, on trouve
$\displaystyle\Big{|}\int_{|X-X^{\prime}|\geq
r}{\cos\theta^{\prime}\partial_{z^{\prime}}\omega^{\theta}(t,r^{\prime},z^{\prime})\over|X-X^{\prime}|}r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime}\Big{|}$
$\displaystyle\lesssim r\int_{|X-X^{\prime}|\geq
r}{1\over{|X-X^{\prime}|^{2}}}|\partial_{z^{\prime}}\omega(t,X^{\prime})|dX^{\prime}$
$\displaystyle\vspace*{2cm}\lesssim r\int_{|X-X^{\prime}|\geq
r}{r^{\prime}\over{|X-X^{\prime}|^{2}}}\big{|}\partial_{z^{\prime}}{\omega(t,X^{\prime})\over
r^{\prime}}\big{|}dX^{\prime},$
après, on utilise le fait que $r^{\prime}=r^{\prime}-r+r$ et
$|r^{\prime}-r|\leq|X-X^{\prime}|,$ on obtient
$\Big{|}\int_{|X-X^{\prime}|\geq
r}{\cos\theta^{\prime}\partial_{z^{\prime}}\omega^{\theta}(t,r^{\prime},z^{\prime})\over|X-X^{\prime}|}r^{\prime}dr^{\prime}d\theta^{\prime}dz^{\prime}\Big{|}\lesssim
r\int_{{\mathbb{R}}^{3}}{\over{|X-X^{\prime}|}}\big{|}\partial_{z^{\prime}}{\omega(t,X^{\prime})\over
r^{\prime}}\big{|}.$
Donc
$|u^{r}(t,X)|\lesssim
r\int_{{\mathbb{R}}^{3}}{1\over{|X-X^{\prime}|}}\big{|}\partial_{z^{\prime}}{\omega(t,X^{\prime})\over
r^{\prime}}\big{|}dX^{\prime},$
aussi on a
$|u^{r}(t,X)|\lesssim
r\int_{{\mathbb{R}}^{3}}{1\over{|X-X^{\prime}|^{2}}}\big{|}{\omega(t,X^{\prime})\over
r^{\prime}}\big{|}dX^{\prime}.$
Pour conclure il suffit d’utiliser les lois de convolutions. Concernant
$u^{z}$ d’après la loi de Biot-Savart, on a
$u^{z}(X)={1\over
4\pi}\int_{{\mathbb{R}}^{3}}{\frac{(x-x^{\prime})\omega^{2}(X^{\prime})-(y-y^{\prime})\omega^{1}(X^{\prime})}{|X-X^{\prime}|^{3}}}\,dX^{\prime}.$
(3.4)
Or
${\frac{x-x^{\prime}}{|X-X^{\prime}|^{3}}}=\partial_{x^{\prime}}{\frac{1}{|X-X^{\prime}|}}\quad\mbox{et}\quad-{\frac{y-y^{\prime}}{|X-X^{\prime}|^{3}}}=-\partial_{y^{\prime}}{\frac{1}{|X-X^{\prime}|}},$
alors par intégration par parties, on obtient
$u^{z}(X)={1\over
4\pi}\int_{{\mathbb{R}}^{3}}{\frac{\partial_{y^{\prime}}\omega^{1}-\partial_{x^{\prime}}\omega^{2}}{|X-X^{\prime}|}}\,dX^{\prime}.$
Mais en coordonnées cylindriques, on a
$\displaystyle\partial_{x^{\prime}}=\cos\theta^{\prime}\partial_{r^{\prime}}-{1\over
r^{\prime}}\sin\theta^{\prime}\partial_{\theta^{\prime}},\quad\partial_{y^{\prime}}=\sin\theta^{\prime}\partial_{r^{\prime}}+{1\over
r^{\prime}}\cos\theta^{\prime}\partial_{\theta^{\prime}},$
$\displaystyle\omega^{1}=-\sin\theta^{\prime}\omega^{\theta}\quad\mbox{et}\quad\omega^{2}=\cos\theta^{\prime}\omega^{\theta},$
et par suite
$\displaystyle\partial_{y^{\prime}}\omega^{1}-\partial_{x^{\prime}}\omega^{2}$
$\displaystyle=-\sin^{2}\theta^{\prime}\partial_{r^{\prime}}\omega^{\theta}-{1\over
r^{\prime}}\cos^{2}\theta^{\prime}\omega^{\theta}-\big{(}\cos^{2}\theta^{\prime}\partial_{r^{\prime}}\omega^{\theta}+{1\over
r^{\prime}}\sin^{2}\theta^{\prime}\omega^{\theta})$
$\displaystyle=-\partial_{r^{\prime}}\omega^{\theta}-{\omega^{\theta}\over
r^{\prime}},$
ainsi
$u^{z}(X)=-{1\over
4\pi}\int_{{\mathbb{R}}^{3}}{1\over|X-X^{\prime}|}\big{(}\partial_{r^{\prime}}\omega^{\theta}+{\omega^{\theta}\over
r^{\prime}}\big{)}dX^{\prime}.$ (3.5)
Donc
$|u^{z}|\lesssim{1\over|\cdot|}\star\big{(}|\partial_{r^{\prime}}\omega|+|{\omega\over
r^{\prime}}|\big{)},$
de même pour la dérivé par rapport à $z,$ on suit les mêmes calculs et grâce
aux égalités (3.4) et (3.5), on trouve
$|\partial_{z}u^{z}|\lesssim\begin{cases}{1\over|\cdot|^{2}}\star|\partial_{z^{\prime}}\omega|\\\
{1\over|\cdot|^{2}}\star\big{(}|\partial_{r^{\prime}}\omega|+|{\omega\over
r^{\prime}}|\big{)}\\\
{1\over|\cdot|}\star\big{(}|\partial_{z^{\prime}}\partial_{r^{\prime}}\omega|+|\partial_{z^{\prime}}{\omega\over
r^{\prime}}|\big{)}.\end{cases}$
Donc d’après les lois de convolutions, on déduit les inégalités souhaitées.
Concernant $\partial_{r}u^{z},$ d’après l’égalité (3.5), on a
$\partial_{r}u^{z}(X)={1\over
4\pi}\int_{{\mathbb{R}}^{3}}{r-r^{\prime}\cos\theta^{\prime}\over|X-X^{\prime}|^{3}}\big{(}\partial_{r^{\prime}}\omega^{\theta}-{\omega^{\theta}\over
r^{\prime}}\big{)}dX^{\prime},$
alors
$|\partial_{r}u^{z}|\lesssim{1\over|\cdot|^{2}}\star\big{(}|\partial_{r^{\prime}}\omega|+|{\omega\over
r^{\prime}}|\big{)}$
car
$\big{|}{r-r^{\prime}\cos\theta^{\prime}\over|X-X^{\prime}|}\big{|}\leq 1.$
Enfin pour $\partial_{r}u^{r},$ il suffit d’utiliser le fait que
${\mathop{\rm div}}\,u=\partial_{r}u^{r}+{u^{r}\over r}+\partial_{z}u^{z}=0.$
D’où la proposition. ∎
D’après la Proposition 3.1, on a besoin de contrôlé $\omega$ dans l’espace de
Lorentz $L^{{3\over 2},1},$ qui est l’objet de la proposition suivante. Plus
exactement on va donner une estimation de la solution de l’équation transport-
diffusion.
###### Proposition 3.2.
Soient $1<p<2,$ $1\leq q\leq\infty,$ $\omega_{0}\in L^{p,q}$ et $u$ un champ
de vecteurs axisymétrique régulière tels que ${u^{r}\over r}\in
L^{1}_{t}(L^{\infty})$ et ${\mathop{\rm div}}\,u=0.$ Soit $\omega\in
L^{\infty}_{t}(L^{p,q})$ et $\partial_{z}\omega\in L^{2}_{t}(L^{p,q})$ une
solution du système suivant
${\rm(TD_{mod})}\;\left\\{\begin{array}[]{rl}&\partial_{t}\omega+(u\cdot\nabla)\omega-\frac{u^{r}}{r}\omega-\partial^{2}_{z}\omega=0\\\
&\omega_{|t=0}=\omega_{0}.\end{array}\right.$
Alors
$\|\omega(t)\|_{L^{p,q}}+\|\partial_{z}\omega\|_{L^{2}_{t}(L^{p,q})}\lesssim\|\omega_{0}\|_{L^{p,q}}e^{\int_{0}^{t}\|{u^{r}\over
r}\|_{L^{\infty}}}.$
###### Proof.
Tout d’abord on va estimer $\omega$ dans les espaces de Lebesgue. Soit
$1<p<\infty,$ on multiplie l’équation vérifiée par $\omega$ par
$|\omega|^{p-1}{\rm sign}\,\omega.$ On obtient après intégrations par parties
combinées avec le fait que ${\mathop{\rm div}\nolimits\,u}=0$
$\frac{1}{p}\frac{d}{dt}\|\omega\|_{L^{p}}^{p}+{4(p-1)\over
p^{2}}\Big{\|}\partial_{z}|\omega|^{\frac{p}{2}}\Big{\|}_{L^{2}}^{2}=\int_{{\mathbb{R}}^{3}}\frac{u^{r}}{r}|\omega|^{p}dx,$
par suite l’inégalité de Hölder plus l’intégration par rapport au temps,
impliquent
$\|\omega(t)\|_{L^{p}}^{p}+{4(p-1)\over
p}\Big{\|}\partial_{z}|\omega|^{\frac{p}{2}}\Big{\|}_{L^{2}_{t}(L^{2})}^{2}\leq\|\omega_{0}\|_{L^{p}}^{p}+p\int_{0}^{t}\|\frac{u^{r}}{r}(\tau)\|_{L^{\infty}}\|\omega(\tau)\|_{L^{p}}^{p}d\tau.$
Ainsi le lemme de Gronwall, implique que
$\|\omega(t)\|_{L^{p}}^{p}+{4(p-1)\over
p}\Big{\|}\partial_{z}|\omega|^{\frac{p}{2}}\Big{\|}_{L^{2}_{t}(L^{2})}^{2}\leq\|\omega_{0}\|_{L^{p}}^{p}\exp\Big{(}p\int_{0}^{t}\|\frac{u^{r}}{r}(\tau)\|_{L^{\infty}}d\tau\Big{)}.$
(3.6)
Pour estimer $\partial_{z}\omega$ dans $L^{p}$ nous allons utiliser le lemme
suivant. Admettons-le pour le moment.
###### Lemme 3.1.
Soient $1\leq p\leq 2$ et $f\in L^{p}({\mathbb{R}}^{N})$ tel que
$\partial_{i}|u|^{\frac{p}{2}}\in L^{2}({\mathbb{R}}^{N}).$ Alors
$\|\partial_{i}f\|_{L^{p}}\lesssim\Big{\|}\partial_{i}|f|^{\frac{p}{2}}\Big{\|}_{L^{2}}\|f\|_{L^{p}}^{\frac{2-p}{2}}.$
Pour $p\leq 2,$ on en déduit grâce au Lemme 3.1 et l’inégalité (3.6), que
$\displaystyle\|\partial_{z}\omega\|_{L^{2}_{t}(L^{p})}$
$\displaystyle\lesssim\Big{(}\int_{0}^{t}\Big{\|}\partial_{z}|\omega|^{\frac{p}{2}}\Big{\|}_{L^{2}}^{2}\|\omega\|_{L^{p}}^{2-p}d\tau\Big{)}^{1\over
2}$ $\displaystyle\lesssim\|\omega\|_{L^{\infty}_{t}(L^{p})}^{2-p\over
2}\Big{\|}\partial_{z}|\omega|^{\frac{p}{2}}\Big{\|}_{L^{2}_{t}(L^{2})}$
$\displaystyle\lesssim\|\omega_{0}\|_{L^{p}}\exp\Big{(}\int_{0}^{t}\|\frac{u^{r}}{r}(\tau)\|_{L^{\infty}}d\tau\Big{)}.$
Donc
$\|\omega(t)\|_{L^{p}}+\|\partial_{z}\omega\|_{L^{2}_{t}(L^{p})}\lesssim\|\omega_{0}\|_{L^{p}}\exp\Big{(}\int_{0}^{t}\|\frac{u^{r}}{r}(\tau)\|_{L^{\infty}}d\tau\Big{)}.$
(3.7)
On désigne par ${\mathcal{T}}$ et ${\mathcal{S}}$ les opérateurs suivants:
$\displaystyle{\mathcal{T}}:$ $\displaystyle L^{p}\longrightarrow L^{p}\hskip
56.9055pt{\mathcal{S}}:\hskip 28.45274ptL^{p}\longrightarrow L^{2}_{t}(L^{p})$
$\displaystyle\omega_{0}\longmapsto\omega\hskip
106.69783pt\omega_{0}\longmapsto\partial_{z}\omega,$
avec $\omega$ solution du système ${\rm(TD_{mod})}.$ Par définition, on a
${\mathcal{T}}$ et ${\mathcal{S}}$ sont linéaires, alors par définition de
l’espace de Lorentz (interpolation réelle) et [3], on obtient
$\|\omega(t)\|_{L^{p,q}}+\|\partial_{z}\omega(\tau)\|_{L^{2}_{t}(L^{p,q})}\lesssim\|\omega_{0}\|_{L^{p,q}}\exp\Big{(}\int_{0}^{t}\|\frac{u^{r}}{r}(\tau)\|_{L^{\infty}}d\tau\Big{)}.$
(3.8)
D’où la proposition. ∎
En suivant les mêmes calculs, on déduit le corollaire suivant.
###### Corollaire 3.1.
Soient $1<p<2,$ $1\leq q\leq\infty,$ $r^{-1}\omega_{0}\in L^{p,q}$ et $u$ un
champ de vecteurs axisymétrique régulière tel que ${\mathop{\rm div}}\,u=0.$
Soit $r^{-1}\omega\in L^{\infty}_{t}(L^{p,q})$ et $r^{-1}\partial_{z}\omega\in
L^{2}_{t}(L^{p,q})$ une solution du système suivant
$\left\\{\begin{array}[]{rl}&\partial_{t}{\omega\over
r}+(u\cdot\nabla){\omega\over r}-\partial^{2}_{z}{\omega\over r}=0\\\
&{\omega\over r}_{|t=0}={\omega_{0}\over r}.\end{array}\right.$
Alors
$\Big{\|}{\omega\over r}(t)\Big{\|}_{L^{p,q}}+\Big{\|}\partial_{z}{\omega\over
r}\Big{\|}_{L^{2}_{t}(L^{p,q})}\lesssim\Big{\|}{\omega_{0}\over
r}\Big{\|}_{L^{p,q}}.$
###### Remarque 3.1.
D’après l’inégalité (3.6) et le fait que
$\|{\omega\over r}(t)\|_{L^{p}}\leq\|{\omega_{0}\over r}\|_{L^{p}},$
on en déduit grâce à [3], que $\forall(p,q)\in]1,\infty[\times[1,\infty]$
$\|\omega(t)\|_{L^{p,q}}\leq\|\omega_{0}\|_{L^{p,q}}e^{\int_{0}^{t}\|\frac{u^{r}}{r}(\tau)\|_{L^{\infty}}d\tau}$
et
$\|{\omega\over r}(t)\|_{L^{p,q}}\leq\|{\omega_{0}\over r}\|_{L^{p,q}}.$
D’après la Proposition 3.1, le Corollaire 3.1 et l’inégalité de Hölder, on a
$\displaystyle\Big{\|}{u^{r}\over
r}\Big{\|}_{L^{1}_{t}(L^{\infty})}\lesssim\Big{\|}\partial_{z}{\omega\over
r}\Big{\|}_{L^{1}_{t}(L^{{3\over 2},1})}$ $\displaystyle\lesssim t^{1\over
2}\Big{\|}\partial_{z}{\omega\over r}\Big{\|}_{L^{2}_{t}(L^{{3\over 2},1})}$
(3.9) $\displaystyle\lesssim t^{1\over 2}\Big{\|}{\omega_{0}\over
r}\Big{\|}_{L^{{3\over 2},1}}.$
Et par suite pour tout $p\in]1,2[$ et $q\in[1,\infty]$ les inégalités (3.8) et
(3.9), impliquent
$\|\omega(t)\|_{L^{p,q}}+\|\partial_{z}\omega\|_{L^{2}_{t}(L^{p,q})}\leq
C\|\omega_{0}\|_{L^{p,q}}e^{Ct^{1\over 2}\|{\omega_{0}\over r}\|_{L^{{3\over
2},1}}}.$ (3.10)
Ainsi la Proposition 3.1, Remarque 3.1 et l’inégalité (3.9), impliquent que
pour tout $(p,q)\in({3\over 2},\infty)\times[1,\infty],$
$\|u(t)\|_{L^{p,q}}\leq C\|\omega_{0}\|_{L^{{3p\over 3+p},q}}e^{Ct^{1\over
2}\|{\omega_{0}\over r}\|_{L^{{3\over 2},1}}}.$
Donc, si $\omega\in L^{{3\over 2},1},$ alors l’inégalité précédente implique
que $u\in L^{3,1},$ qui est inclus dans l’espace dual de $L^{{3\over 2},1}.$
Et par suite grâce à la Proposition II.1 dans [8] et l’équation que vérifie
$\omega$ (2.2), on déduit le résultat d’existence suivant.
###### Corollaire 3.2.
Soit $\omega_{0}^{\theta}\in L^{{3\over 2},1}({\mathbb{R}}^{3})$ une fonction
axisymétrique tel que ${\omega_{0}^{\theta}\over r}\in L^{{3\over
2},1}({\mathbb{R}}^{3}).$ Soit $u_{0}$ le champ de vecteurs axisymétrique tel
que ${\mathop{\rm div}}\,u_{0}=0$ et avec vorticité
$\omega_{0}=\omega_{0}^{\theta}(r,z)e_{\theta}$ donné par la loi de Biot-
Savart :
$u_{0}(X)={1\over
4\pi}\int_{{\mathbb{R}}^{3}}\frac{X-Y}{|X-Y|^{3}}\times\omega_{0}(Y)\,dY.$
Alors le système ${\rm(NS_{v})}$ admet une solution globale $u$ tel que la
vorticité $\omega$ satisfait
$\displaystyle\omega\in{\mathscr{C}}\big{(}{\mathbb{R}}_{+};\,L^{{3\over
2},1}({\mathbb{R}}^{3})\big{)},\hskip 28.45274pt\partial_{z}\omega\in
L^{2}_{loc}\big{(}{\mathbb{R}}_{+};\,L^{{3\over
2},1}({\mathbb{R}}^{3})\big{)}$ $\displaystyle{\omega\over
r}\in{\mathscr{C}}\big{(}{\mathbb{R}}_{+};\,L^{{3\over
2},1}({\mathbb{R}}^{3})\big{)},\hskip 25.6073pt\partial_{z}{\omega\over r}\in
L^{2}_{loc}\big{(}{\mathbb{R}}_{+};\,L^{{3\over
2},1}({\mathbb{R}}^{3})\big{)}.$
De plus pour tout $t\geq 0,$ on a
$\|\omega(t)\|_{L^{{3\over 2},1}}+\|\partial_{z}\omega\|_{L^{2}_{t}(L^{{3\over
2},1})}\leq C\|\omega_{0}\|_{L^{{3\over 2},1}}e^{Ct^{1\over
2}\|r^{-1}\omega_{0}\|_{L^{{3\over 2},1}}}$
et
$\|r^{-1}\omega(t)\|_{L^{{3\over
2},1}}+\|r^{-1}\partial_{z}\omega\|_{L^{2}_{t}(L^{{3\over 2},1})}\leq
C\|r^{-1}\omega_{0}\|_{L^{{3\over 2},1}}.$
Démonstration du Lemme 3.1.
Remarquons tout d’abord que
$\|\partial_{i}f\|_{L^{p}}=\|\partial_{i}|f|\|_{L^{p}}\hskip
28.45274pt\mbox{et}\hskip 28.45274pt|f|=|f|^{{p\over 2}{2\over p}},$
ainsi, on a
$\partial_{i}|f|={p\over 2}\partial_{i}(|f|^{{p\over 2}})|f|^{{2-p\over 2}}.$
Et par suite l’inégalité de Hölder, implique que
$\|\partial_{i}u\|_{L^{p}}\lesssim\Big{\|}\partial_{i}|u|^{\frac{p}{2}}\Big{\|}_{L^{2}}\|u\|_{L^{p}}^{2-p\over
2}.$
D’où le lemme. $\square$
### 3.2 Unicité
Pour démontrer l’unicité de solution pour le système ${\rm(NS_{v})},$ il
suffit de le prouver pour l’équation (2.2). Soient $\omega_{1}$ et
$\omega_{2}$ deux solutions, et on désignons par
$\delta\omega=\omega_{2}-\omega_{1}$ leur différence, qui vérifie le système
suivant :
$\left\\{\begin{array}[]{rl}&\partial_{t}\delta\omega+(u_{2}\cdot\nabla)\delta\omega-\partial_{z}^{2}\delta\omega=-(\delta
u\cdot\nabla)\omega_{1}+{u^{r}_{2}\over r}\delta\omega+{\delta u^{r}\over
r}\omega_{1}\\\ &{\delta\omega}_{|t=0}=0.\end{array}\right.$
L’espace dans lequel on va estimer la différence est $L^{p}$ avec ${6\over
5}\leq p<{3\over 2}.$ Admettons pour le moment le lemme suivant.
###### Lemme 3.2.
Soient $\omega_{i}$ avec $1\leq i\leq 2$ deux solutions de l’équation (2.2)
ayant les mêmes données initiales. Supposons que pour $i=1,2$ on ait
$\omega_{i}\in L^{\infty}_{t}(L^{{3\over 2},1}),\hskip
14.22636pt\partial_{z}\omega_{i}\in L^{2}_{t}(L^{{3\over 2},1})\hskip
14.22636pt\mbox{et}\hskip 14.22636pt\partial_{r}\omega_{i}\in
L^{\infty}_{t}(L^{{3\over 2},1}).$
Alors
$\delta\omega\in L^{\infty}_{t}(L^{p})\hskip 28.45274pt\mbox{et}\hskip
28.45274pt\partial_{z}|\delta\omega|^{p\over 2}\in L^{2}_{t}(L^{2}).$
L’estimation d’énergie implique que
$\displaystyle{1\over p}{d\over dt}\|\delta\omega\|_{L^{p}}^{p}+{4(p-1)\over
p^{2}}\Big{\|}\partial_{z}|\omega|^{\frac{p}{2}}\Big{\|}_{L^{2}}^{2}$
$\displaystyle\leq\|{u^{r}_{2}\over
r}\|_{L^{\infty}}\|\delta\omega\|_{L^{p}}^{p}+\|{\omega_{1}\delta u^{r}\over
r}\|_{L^{p}}\|\delta\omega\|_{L^{p}}^{p-1}$ $\displaystyle+\|(\delta
u\cdot\nabla)\omega_{1}\|_{L^{p}}\|\delta\omega\|_{L^{p}}^{p-1}.$
D’après l’inégalité de Hölder, l’injection de Sobolev, la Proposition 3.1 et
le Lemme 3.1, on a
$\displaystyle\|{\omega_{1}\delta u^{r}\over r}\|_{L^{p}}$
$\displaystyle+\|(\delta
u\cdot\nabla)\omega_{1}\|_{L^{p}}\leq\big{(}\|{\omega_{1}\over r}\|_{L^{3\over
2}}+\|\partial_{r}\omega_{1}\|_{L^{3\over 2}}\big{)}\|\delta
u^{r}\|_{L^{3p\over 3-2p}}$
$\displaystyle+\|\partial_{z}\omega_{1}\|_{L^{6}_{h}(L^{\frac{3}{2}}_{v})}\|\delta
u^{z}\|_{L^{6p\over 6-p}_{h}(L^{\frac{3p}{3-2p}}_{v})}$
$\displaystyle\lesssim\big{(}\|{\omega_{1}\over r}\|_{L^{3\over
2}}+\|\partial_{r}\omega_{1}\|_{L^{3\over
2}}\big{)}\|\partial_{z}\delta\omega\|_{L^{p}}+\|\partial_{z}\partial_{r}\omega_{1}\|_{L^{\frac{3}{2}}}\|\delta
u^{z}\|_{L^{6p\over 6-p}_{h}(L^{\frac{3p}{3-2p}}_{v})}$
$\displaystyle\lesssim\Big{(}\|{\omega_{1}\over r}\|_{L^{3\over
2}}+\|\partial_{r}\omega_{1}\|_{L^{3\over
2}}\Big{)}\|\partial_{z}|\delta\omega|^{p\over
2}\|_{L^{2}}\|\delta\omega\|_{L^{p}}^{2-p\over
2}+\|\partial_{z}\partial_{r}\omega_{1}\|_{L^{\frac{3}{2}}}\|\delta
u^{z}\|_{L^{6p\over 6-p}_{h}(L^{\frac{3p}{3-2p}}_{v})}.$
Concernant $\|\delta u^{z}\|_{L^{6p\over 6-p}_{h}(L^{\frac{3p}{3-2p}}_{v})}$
on utilise le fait que
$\Delta\delta u^{z}=\partial_{r}\delta\omega+\frac{\delta\omega}{r},$
et par suite par intégration par parties, on aura
$|\delta u^{z}|\lesssim\frac{1}{|\cdot|^{2}}\star|\delta\omega|,$
alors d’après les lois de convolution, on obtient
$\displaystyle\|\delta u^{z}\|_{L^{6p\over 6-p}_{h}(L^{\frac{3p}{3-2p}}_{v})}$
$\displaystyle\lesssim\|\delta\omega\|_{L^{{6p\over 6-p},{6p\over
6-p}}_{h}(L^{p}_{v})}$ $\displaystyle\lesssim\|\delta\omega\|_{L^{p}}.$
Et par suite l’inégalité de Young, implique que
$\displaystyle{d\over dt}\|\delta\omega\|_{L^{p}}^{p}$
$\displaystyle\leq\Big{(}\|{u^{r}_{2}\over
r}\|_{L^{\infty}}+\|{\omega_{1}\over r}\|_{L^{3\over
2}}^{2}+\|\partial_{r}\omega_{1}\|_{L^{3\over
2}}^{2}+\|\partial_{z}\partial_{r}\omega_{1}\|_{L^{\frac{3}{2}}}\Big{)}\|\delta\omega\|_{L^{p}}^{p}.$
Donc on a l’unicité si $\partial_{r}\omega_{1}\in L^{2}_{t}(L^{3\over 2})$ et
$\|\partial_{z}\partial_{r}\omega_{1}\|_{L^{\frac{3}{2}}}$ puisque l’inégalité
(3.9) et le Corollaire 3.1, impliquent $\big{(}\|{u^{r}_{2}\over
r}\|_{L^{\infty}}+\|{\omega_{1}\over r}\|_{L^{3\over 2}}^{2})\in L^{1}_{t}.$
Dans un première temps on démontre que $\partial_{r}\omega_{1}\in
L^{2}_{t}(L^{3\over 2}).$ Plus exactement on prouve qu’on a propagation de la
régularité $\partial_{r}\omega$ dans l’espace de Lorentz $L^{{3\over 2},1}$
plus l’effet régularisant.
### 3.3 Propagation de la régularité $\partial_{r}\omega$
###### Proposition 3.3.
Soient $\omega_{0}\in L^{{3\over 2},1}\cap L^{3,1}$ tels que $\omega_{0}/r\in
L^{{3\over 2},1}$ et $\partial_{r}\omega_{0}\in L^{{3\over 2},1}.$ Soit
$\partial_{r}\omega\in L^{\infty}_{t}(L^{{3\over 2},1}),$
$\partial_{z}\partial_{r}\omega\in L^{2}_{t}(L^{{3\over 2},1})$ une solution
du système suivant
$\left\\{\begin{array}[]{rl}&\partial_{t}\partial_{r}\omega+(u\cdot\nabla)\partial_{r}\omega-\partial_{z}^{2}\partial_{r}\omega=-{u^{r}\over
r}{\omega\over r}+\partial_{r}u^{r}{\omega\over r}+{u^{r}\over
r}\partial_{r}\omega-\partial_{r}u^{r}\partial_{r}\omega-\partial_{r}u^{z}\partial_{z}\omega\\\
&{\partial_{r}\omega}_{|t=0}=\partial_{r}\omega_{0}.\end{array}\right.$
Alors
$\|\partial_{r}\omega(t)\|_{L^{{3\over
2},1}}+\|\partial_{z}\partial_{r}\omega\|_{L^{2}_{t}(L^{{3\over
2},1})}\leq\Phi(t,\omega_{0}),$
avec
$\Phi(t,\omega_{0})=e^{C\exp{\sqrt{t}C(\omega_{0})}}$
###### Proof.
En prenant le produit scalaire au sens $L^{p}$ pour $1<p\leq 2$ de l’équation
qui vérifie $\partial_{r}\omega$ combinés avec $\partial_{r}u^{r}=-{u^{r}\over
r}-\partial_{z}u^{z}$ et l’inégalité de Hardy que implique que
$\|r^{-1}\omega\|_{L^{p}}\lesssim\|\partial_{r}\omega\|_{L^{p}},$ on trouve
$\displaystyle{1\over p}{d\over dt}\|\partial_{r}\omega\|_{L^{p}}^{p}$
$\displaystyle+{4(p-1)\over p^{2}}\|\partial_{z}|\partial_{r}\omega|^{p\over
2}\|_{L^{2}}^{2}\leq 4\|{u^{r}\over
r}\|_{L^{\infty}}\|\partial_{r}\omega\|_{L^{p}}^{p}+\|\partial_{z}u^{z}{\omega\over
r}\|_{L^{p}}\|\partial_{r}\omega\|_{L^{p}}^{p-1}$
$\displaystyle+\|\partial_{r}u^{z}\partial_{z}\omega\|_{L^{p}}\|\partial_{r}\omega\|_{L^{p}}^{p-1}+\int\partial_{z}u^{z}|\partial_{r}\omega|^{p}.$
Par intégration par parties plus l’inégalité de Cauchy-Schwartz, on a
$\int\partial_{z}u^{z}|\partial_{r}\omega|^{p}=-2\int
u^{z}|\partial_{r}\omega|^{p\over 2}\partial_{z}|\partial_{r}\omega|^{p\over
2}\leq 2\|u^{z}\|_{L^{\infty}}\|\partial_{z}|\partial_{r}\omega|^{p\over
2}\|_{L^{2}}\|\partial_{r}\omega\|_{L^{p}}^{p\over 2}.$
Grâce à l’inégalité de Young et le fait que
$\partial_{r}u^{z}=\partial_{z}u^{r}-\omega,$ on obtient
$\|\partial_{r}u^{z}\partial_{z}\omega\|_{L^{p}}=\|\partial_{z}u^{r}\partial_{z}\omega-{1\over
2}\partial_{z}\omega^{2}\|_{L^{p}}\leq\|\partial_{z}u^{r}\partial_{z}\omega\|_{L^{p}}+\|\partial_{z}\omega^{2}\|_{L^{p}}.$
Et par suite
$\displaystyle{1\over p}{d\over dt}\|\partial_{r}\omega\|_{L^{p}}^{p}$
$\displaystyle+\|\partial_{z}|\partial_{r}\omega|^{p\over
2}\|_{L^{2}}^{2}\lesssim\Big{(}\|{u^{r}\over
r}\|_{L^{\infty}}+\|u^{z}\|_{L^{\infty}}^{2}\Big{)}\|\partial_{r}\omega\|_{L^{p}}^{p}$
(3.11) $\displaystyle+\Big{(}\|\partial_{z}u^{z}{\omega\over
r}\|_{L^{p}}+\|\partial_{z}u^{r}\partial_{z}\omega\|_{L^{p}}+\|\partial_{z}\omega^{2}\|_{L^{p}}\Big{)}\|\partial_{r}\omega\|_{L^{p}}^{p-1}.$
grâce à l’inégalité de Hölder et par interpolation, on trouve
$\displaystyle\|\partial_{z}u^{z}{\omega\over r}\|_{L^{p}}$
$\displaystyle\leq\|{\omega\over
r}\|_{L^{p}_{h}(L^{\infty}_{v})}\|\partial_{z}u^{z}\|_{L^{\infty}_{h}(L^{p}_{v})}$
$\displaystyle\lesssim\|{\omega\over
r}\|_{L^{p}}^{\frac{p-1}{p}}\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{\frac{1}{p}}\|\partial_{z}u^{z}\|_{L^{\infty}_{h}(L^{p}_{v})}.$
Comme
$\Delta\partial_{z}u^{z}=\partial_{z}\partial_{r}\omega+\partial_{z}\frac{\omega}{r},$
alors par integration par parties, on trouve
$\partial_{z}u^{z}=-{1\over{4\pi}}\int_{{\mathbb{R}}^{3}}\frac{r^{\prime}-r\cos\theta^{\prime}}{\big{(}r^{2}+{r^{\prime}}^{2}-2rr^{\prime}\cos\theta^{\prime}+(z-z^{\prime})^{2}\big{)}^{\frac{3}{2}}}\,\partial_{z^{\prime}}\omega\,r^{\prime}dr^{\prime}dz^{\prime}d\theta^{\prime},$
et par suite
$|\partial_{z}u^{z}|\lesssim\frac{1}{|X|^{2}}\star|\partial_{z}\omega|,$
ainsi
$\|\partial_{z}u^{z}\|_{L^{\infty}_{h}(L^{p}_{v})}\lesssim\|\partial_{z}\omega\|_{L^{2,1}_{h}(L^{p}_{v})}.$
Comme $1<p<2,$ alors par interpolation, on a
$\|f\|_{L^{2,1}({\mathbb{R}}^{2})}\lesssim\|f\|_{L^{p}}^{\frac{2p-2}{p}}\|\nabla
f\|_{L^{p}}^{\frac{2-p}{p}}.$
Ainsi d’après l’inégalité de Hardy, on trouve
$\|\partial_{z}u^{z}{\omega\over r}\|_{L^{p}}\lesssim\|{\omega\over
r}\|_{L^{p}}^{\frac{p-1}{p}}\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{\frac{1}{p}}\|\partial_{z}\omega\|_{L^{p}}^{\frac{2p-2}{p}}\|\partial_{r}\partial_{z}\omega\|_{L^{p}}^{\frac{2-p}{p}},$
et par suite le Lemme 3.1, implique que
$\|\partial_{z}u^{z}{\omega\over r}\|_{L^{p}}\lesssim\|{\omega\over
r}\|_{L^{p}}^{\frac{p-1}{p}}\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{\frac{1}{p}}\|\partial_{z}\omega\|_{L^{p}}^{\frac{2p-2}{p}}\|\partial_{r}\omega\|_{L^{p}}^{\frac{(2-p)^{2}}{2p}}\|\partial_{z}|\partial_{r}\omega|^{\frac{p}{2}}\|_{L^{p}}^{\frac{2-p}{p}},$
ainsi on aura grâce à l’inégalité de Young
$\displaystyle\|\partial_{z}u^{z}{\omega\over
r}\|_{L^{p}}\|\partial_{r}\omega\|_{L^{p}}^{p-1}$ $\displaystyle\leq
c_{\varepsilon}\|{\omega\over
r}\|_{L^{p}}^{\frac{2p-2}{3p-2}}\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{\frac{2}{3p-2}}\|\partial_{z}\omega\|_{L^{p}}^{\frac{4p-4}{3p-2}}\|\partial_{r}\omega\|_{L^{p}}^{\frac{3p^{2}-6p+4}{3p-2}}$
(3.12)
$\displaystyle+\varepsilon\|\partial_{z}|\partial_{r}\omega|^{\frac{p}{2}}\|_{L^{p}}^{2}.$
Mais comme $p-1\leq\frac{3p^{2}-6p+4}{3p-2}\leq p,$ alors
$\displaystyle{1\over p}{d\over dt}$
$\displaystyle\|\partial_{r}\omega\|_{L^{p}}^{p}+\|\partial_{z}|\partial_{r}\omega|^{p\over
2}\|_{L^{2}}^{2}$ $\displaystyle\lesssim\Big{(}\|{\omega\over
r}\|_{L^{p}}^{\frac{2p-2}{3p-2}}\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{\frac{2}{3p-2}}\|\partial_{z}\omega\|_{L^{p}}^{\frac{4p-4}{3p-2}}+\|{u^{r}\over
r}\|_{L^{\infty}}+\|u^{z}\|_{L^{\infty}}^{2}\Big{)}\|\partial_{r}\omega\|_{L^{p}}^{p}$
$\displaystyle+\Big{(}\|{\omega\over
r}\|_{L^{p}}^{\frac{2p-2}{3p-2}}\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{\frac{2}{3p-2}}\|\partial_{z}\omega\|_{L^{p}}^{\frac{4p-4}{3p-2}}+\|\partial_{z}u^{r}\partial_{z}\omega\|_{L^{p}}+\|\partial_{z}\omega^{2}\|_{L^{p}}\Big{)}\|\partial_{r}\omega\|_{L^{p}}^{p-1}.$
Ainsi le lemme de Gronwall, implique que
$\displaystyle\|\partial_{r}\omega(t)\|_{L^{p}}+\|\partial_{z}|\partial_{r}\omega|^{p\over
2}\|_{L^{2}_{t}(L^{2})}^{2\over p}$
$\displaystyle\leq\Big{(}\|\partial_{r}\omega_{0}\|_{L^{p}}+C\int_{0}^{t}\big{(}\|{\omega\over
r}\|_{L^{p}}^{\frac{2p-2}{3p-2}}\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{\frac{2}{3p-2}}\|\partial_{z}\omega\|_{L^{p}}^{\frac{4p-4}{3p-2}}+\|\partial_{z}u^{r}\partial_{z}\omega\|_{L^{p}}+\|\partial_{z}\omega^{2}\|_{L^{p}}\big{)}d\tau\Big{)}$
$\displaystyle\times e^{C\int_{0}^{t}\big{(}\|{\omega\over
r}\|_{L^{p}}^{\frac{2p-2}{3p-2}}\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{\frac{2}{3p-2}}\|\partial_{z}\omega\|_{L^{p}}^{\frac{4p-4}{3p-2}}+\|{u^{r}\over
r}\|_{L^{\infty}}+\|u^{z}\|_{L^{\infty}}^{2}+\|{\omega_{0}\over
r}\|_{L^{3\over 2}}^{2}\big{)}d\tau},$
enfin le Lemme 3.1 et l’inégalité, assurent que
$\displaystyle\|\partial_{r}\omega(t)\|_{L^{p}}+\|\partial_{z}\partial_{r}\omega\|_{L^{2}_{t}(L^{p})}\leq$
$\displaystyle
C\Big{(}\|\partial_{r}\omega_{0}\|_{L^{p}}+\int_{0}^{t}\big{(}\|{\omega\over
r}\|_{L^{p}}^{2}+\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{2}+\|\partial_{z}\omega\|_{L^{p}}^{2}+\|\partial_{z}u^{r}\partial_{z}\omega\|_{L^{p}}+\|\partial_{z}\omega^{2}\|_{L^{p}}\big{)}d\tau\Big{)}$
$\displaystyle\times e^{C\int_{0}^{t}\big{(}\|{\omega\over
r}\|_{L^{p}}^{2}+\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{2}+\|\partial_{z}\omega\|_{L^{p}}^{2}+\|{u^{r}\over
r}\|_{L^{\infty}}+\|u^{z}\|_{L^{\infty}}^{2}+\|{\omega_{0}\over
r}\|_{L^{3\over 2}}^{2}\big{)}d\tau}.$
Par définition de l’espace de l’espace de Lorentz et d’après [3] l’estimation
précédente reste vraie dans $L^{{\frac{3}{2}},1},$ et par suite
$\displaystyle\|\partial_{r}\omega(t)\|_{L^{{\frac{3}{2}},1}}+\|\partial_{z}\partial_{r}\omega\|_{L^{2}_{t}(L^{{\frac{3}{2}},1})}\leq$
$\displaystyle
C\Big{(}\|\partial_{r}\omega_{0}\|_{L^{{\frac{3}{2}},1}}+\int_{0}^{t}\big{(}\|{\omega\over
r}\|_{L^{{\frac{3}{2}},1}}^{2}+\|\partial_{z}{\omega\over
r}\|_{L^{{\frac{3}{2}},1}}^{2}+\|\partial_{z}\omega\|_{L^{{\frac{3}{2}},1}}^{2}+\|\partial_{z}u^{r}\partial_{z}\omega\|_{L^{{\frac{3}{2}},1}}+\|\partial_{z}\omega^{2}\|_{L^{{\frac{3}{2}},1}}\big{)}d\tau\Big{)}$
$\displaystyle\times e^{C\int_{0}^{t}\big{(}\|{\omega\over
r}\|_{L^{{\frac{3}{2}},1}}^{2}+\|\partial_{z}{\omega\over
r}\|_{L^{{\frac{3}{2}},1}}^{2}+\|\partial_{z}\omega\|_{L^{{\frac{3}{2}},1}}^{2}+\|{u^{r}\over
r}\|_{L^{\infty}}+\|u^{z}\|_{L^{\infty}}^{2}+\|{\omega_{0}\over
r}\|_{L^{3\over 2}}^{2}\big{)}d\tau}.$
Rappelons que
$\|\omega(t)\|_{L^{{3\over 2},1}}+\|\partial_{z}\omega\|_{L^{2}_{t}(L^{{3\over
2},1})}\leq C\|\omega_{0}\|_{L^{{3\over 2},1}}\exp\big{(}Ct^{1\over
2}\|r^{-1}\omega_{0}\|_{L^{{3\over 2},1}}\big{)}$
et
$\|\frac{u^{r}}{r}\|_{L^{1}_{t}(L^{\infty})}\lesssim\sqrt{t}\|\frac{\omega_{0}}{r}\|_{L^{{\frac{3}{2}},1}}\qquad\mbox{et}\qquad\|\frac{\omega}{r}\|_{L^{{\frac{3}{2}},1}}+\|\partial_{z}\frac{\omega}{r}\|_{L^{2}_{t}(L^{{\frac{3}{2}},1})}\lesssim\|\frac{\omega_{0}}{r}\|_{L^{{\frac{3}{2}},1}}.$
Donc grâce aux Propositions (2.1) et (3.1), on aura
$\displaystyle\int_{0}^{t}\|\partial_{z}u^{r}\partial_{z}\omega\|_{L^{{\frac{3}{2}},1}}$
$\displaystyle\leq\int_{0}^{t}\|\partial_{z}u^{r}\|_{L^{6,2}}\|\partial_{z}\omega\|_{L^{2,2}}$
$\displaystyle\lesssim\int_{0}^{t}\|\partial_{z}\omega\|_{L^{2}}^{2},$
et par suite les inégalités (3.7) et (3.9), impliquent
$\int_{0}^{t}\|\partial_{z}u^{r}\partial_{z}\omega\|_{L^{{\frac{3}{2}},1}}\lesssim\|\omega_{0}\|_{L^{2}}^{2}e^{Ct^{1\over
2}\|{\omega_{0}\over r}\|_{L^{{3\over 2},1}}}.$ (3.13)
Concernant $\|\partial_{z}\omega^{2}\|_{L^{1}_{t}(L^{{3\over 2},1})}$ la
Proposition 2.1, implique que
$\displaystyle\int_{0}^{t}\|\partial_{z}\omega^{2}\|_{L^{{3\over 2},1}}$
$\displaystyle\lesssim\int_{0}^{t}\||\omega|^{1\over
2}\|_{L^{6,2}}\|\partial_{z}|\omega|^{3\over 2}\|_{L^{2}}$
$\displaystyle\lesssim t^{1\over
2}\|\omega\|_{L^{\infty}_{t}(L^{3,1})}^{{1\over
2}}\|\partial_{z}|\omega|^{3\over 2}\|_{L^{2}_{t}(L^{2})},$
ainsi les inégalités (3.6) et (3.9) et la Remarque 3.1, impliquent
$\displaystyle\int_{0}^{t}\|\partial_{z}\omega^{2}\|_{L^{{3\over
2},1}}\lesssim t^{1\over 2}\|\omega_{0}\|_{L^{3,1}}^{2}e^{Ct^{1\over
2}\|\omega_{0}/r\|_{L^{{3\over 2},1}}}.$ (3.14)
Pour $\|u^{z}\|_{L^{2}_{t}(L^{\infty})}^{2}$ la Proposition 3.1 et la Remarque
3.1 entraînent
$\int_{0}^{t}\|u^{z}\|_{L^{\infty}}^{2}\lesssim\int_{0}^{t}\|\omega\|_{L^{3,1}}^{2}\lesssim
t\|\omega_{0}\|_{L^{3,1}}^{2}e^{Ct^{1\over 2}\|{\omega_{0}\over
r}\|_{L^{{3\over 2},1}}}.$ (3.15)
D’où la proposition. ∎
En fait on peut travailler avec des données moins régulières mais le prix à
payer est l’absence d’un contrôle explicite de la solution en fonction de la
donnée initiale. Ceci est précisé dans la proposition suivante.
###### Proposition 3.4.
Soient $\omega_{0}\in L^{{3\over 2},1}$ tels que $\omega_{0}/r\in L^{{3\over
2},1}$ et $\partial_{r}\omega_{0}\in L^{{3\over 2},1}.$ Soit
$\partial_{r}\omega\in L^{\infty}_{t}(L^{{3\over 2},1}),$
$\partial_{z}\partial_{r}\omega\in L^{2}_{t}(L^{{3\over 2},1})$ une solution
du système suivant
$\left\\{\begin{array}[]{rl}&\partial_{t}\partial_{r}\omega+(u\cdot\nabla)\partial_{r}\omega-\partial_{z}^{2}\partial_{r}\omega=-{u^{r}\over
r}{\omega\over r}+\partial_{r}u^{r}{\omega\over r}+{u^{r}\over
r}\partial_{r}\omega-\partial_{r}u^{r}\partial_{r}\omega-\partial_{r}u^{z}\partial_{z}\omega\\\
&{\partial_{r}\omega}_{|t=0}=\partial_{r}\omega_{0}.\end{array}\right.$
Alors
$\|\partial_{r}\omega(t)\|_{L^{{3\over
2},1}}+\|\partial_{z}\partial_{r}\omega\|_{L^{2}_{t}(L^{{3\over
2},1})}\leq\gamma(t,\omega_{0}).$
###### Proof.
La preuve s’effectue en deux étapes, on démontre premièrement qu’on a
propagation de la régularité localement et après on en déduit globalement
grâce a l’effet régularisant et la Proposition 3.3. En prenant le produit
scalaire au sens $L^{p}$ pour $1<p\leq 2$ de l’équation qui vérifie
$\partial_{r}\omega$ combinés avec $\partial_{r}u^{r}=-{u^{r}\over
r}-\partial_{z}u^{z}$ et l’inégalité de Hardy que implique que
$\|r^{-1}\omega\|_{L^{p}}\lesssim\|\partial_{r}\omega\|_{L^{p}},$ on trouve
$\displaystyle{1\over p}{d\over dt}\|\partial_{r}\omega\|_{L^{p}}^{p}+$
$\displaystyle{4(p-1)\over p^{2}}\|\partial_{z}|\partial_{r}\omega|^{p\over
2}\|_{L^{2}}^{2}\leq 4\|{u^{r}\over
r}\|_{L^{\infty}}\|\partial_{r}\omega\|_{L^{p}}^{p}$ (3.16)
$\displaystyle+\int\partial_{z}u^{z}{\omega\over
r}|\partial_{r}\omega|^{p-1}+\int\partial_{r}u^{z}\partial_{z}\omega\partial_{r}\omega^{p-1}+\int\partial_{z}u^{z}|\partial_{r}\omega|^{p}$
$\displaystyle\lesssim\|{u^{r}\over
r}\|_{L^{\infty}}\|\partial_{r}\omega\|_{L^{p}}^{p}+\|\partial_{z}u^{z}{\omega\over
r}\|_{L^{p}}\|\partial_{r}\omega\|_{L^{p}}^{p-1}$
$\displaystyle+\int\partial_{r}u^{z}\partial_{z}\omega|\partial_{r}\omega|^{p-1}+\int\partial_{z}u^{z}|\partial_{r}\omega|^{p}.$
Rappelons que d’après l’inégalité (3.12), on a
$\displaystyle\|\partial_{z}u^{z}{\omega\over
r}\|_{L^{p}}\|\partial_{r}\omega\|_{L^{p}}^{p-1}$ $\displaystyle\leq
c_{\varepsilon}\|{\omega\over
r}\|_{L^{p}}^{\frac{2p-2}{3p-2}}\|\partial_{z}{\omega\over
r}\|_{L^{p}}^{\frac{2}{3p-2}}\|\partial_{z}\omega\|_{L^{p}}^{\frac{4p-4}{3p-2}}\|\partial_{r}\omega\|_{L^{p}}^{\frac{3p^{2}-6p+4}{3p-2}}$
$\displaystyle+\varepsilon\|\partial_{z}|\partial_{r}\omega|^{\frac{p}{2}}\|_{L^{p}}^{2}.$
Concernant Le terme
$\int\partial_{r}u^{z}\partial_{z}\omega|\partial_{r}\omega|^{p-1},$
on utilise le fait que $\partial_{r}u^{z}=\partial_{z}u^{r}-\omega$ et
l’inégalité de Minkowski, on obtient
$\displaystyle\int\partial_{r}u^{z}\partial_{z}\omega$
$\displaystyle|\partial_{r}\omega|^{p-1}\leq(\|\omega\partial_{z}\omega\|_{L^{p}}+\|\partial_{z}u^{r}\partial_{z}\omega\|_{L^{p}})\|\partial_{r}\omega\|_{L^{p}}^{p-1}$
$\displaystyle\leq\big{(}\|\omega\|_{L^{\infty}_{v}(L^{2p}_{h})}+\|\partial_{z}u^{r}\|_{L^{\infty}_{v}(L^{2p}_{h})}\big{)}\|\partial_{z}\omega\|_{L^{p}_{v}(L^{2p}_{h})}\|\partial_{r}\omega\|_{L^{p}}^{p-1}$
$\displaystyle\leq\big{(}\|\omega\|_{L^{\infty}_{v}(L^{2p}_{h})}+\|\partial_{z}u^{r}\|_{L^{2p}_{h}(L^{\infty}_{v})}\big{)}\|\partial_{z}\omega\|_{L^{p}_{v}(L^{2p}_{h})}\|\partial_{r}\omega\|_{L^{p}}^{p-1}.$
On se rappelle maintenant que
$\partial_{z}\omega,\partial_{z}(\frac{\omega}{r})$ ainsi que
$\partial_{z}\partial_{r}\omega$ sont dans $L^{p}.$ Donc, on a
$\partial_{z}\omega\in L^{p}_{v}(W^{1,p}({\mathbb{R}}^{2}_{h}))$, et comme
$W^{1,p}({\mathbb{R}}^{2}_{h})\subset L^{p}({\mathbb{R}}^{2}_{h})\cap
L^{\frac{2p}{2-p}}({\mathbb{R}}^{2}_{h})\subset L^{2p}({\mathbb{R}}^{2}_{h}).$
Donc par interpolation et grâce à l’inégalité de Sobolev, on trouve
$\displaystyle\|\partial_{z}\omega\|_{L^{p}_{v}L^{2p}_{h}}$
$\displaystyle\lesssim\|\partial_{z}\omega\|_{L^{p}}^{1-\frac{1}{p}}\|\partial_{z}\omega\|_{L^{p}_{v}(L^{\frac{2p}{2-p}}_{h})}^{\frac{1}{p}}$
(3.17)
$\displaystyle\lesssim\|\partial_{z}\omega\|_{L^{p}}^{1-\frac{1}{p}}\big{(}\|\partial_{r}\partial_{z}\omega\|_{L^{p}}+\|\partial_{z}\frac{\omega}{r}\|_{L^{p}}\big{)}^{\frac{1}{p}}.$
D’autre part,
$\displaystyle\|\omega\|_{L^{\infty}_{v}L^{2p}_{h}}$
$\displaystyle\leq\|\omega\|_{L^{p}_{v}(L^{2p}_{h})}^{1-\frac{1}{p}}\|\partial_{z}\omega\|_{L^{p}_{v}L^{2p}_{h}}^{1\over
p}$
$\displaystyle\leq\|\omega\|_{L^{p}}^{(1-\frac{1}{p})^{2}}\big{(}\|\partial_{r}\omega\|_{L^{p}}+\|\frac{\omega}{r}\|_{L^{p}}\big{)}^{\frac{1}{p}(1-\frac{1}{p})}$
$\displaystyle\times\|\partial_{z}\omega\|_{L^{p}}^{\frac{1}{p}(1-\frac{1}{p})}\big{(}\|\partial_{z}\partial_{r}\omega\|_{L^{p}}+\|\partial_{z}\frac{\omega}{r}\|_{L^{p}}\big{)}^{\frac{1}{p^{2}}}.$
Concernant $\partial_{z}u^{r},$ on utilise le fait que
$u^{r}=\frac{1}{|X|}\ast\partial_{z}\omega,$ (3.18)
on trouve
$\|\partial_{z}u^{r}\|_{L^{\infty}_{v}}\lesssim\frac{1}{{\sqrt{x^{2}+y^{2}}}^{1+\frac{1}{p}}}\ast\|\partial_{z}\omega\|_{L^{p}_{v}},$
ainsi pour $p\leq 2,$ on trouve par interpolation
$\displaystyle\|\partial_{z}u^{r}\|_{L^{2p}_{h}(L^{\infty}_{v})}$
$\displaystyle\lesssim\|\partial_{z}\omega\|_{L^{2,1}_{h}(L^{p}_{v})}$ (3.19)
$\displaystyle\lesssim\|\partial_{z}\omega\|_{L^{p}}^{\frac{2(p-1)}{p}}\|\nabla_{h}\partial_{z}\omega\|_{L^{p}}^{\frac{2-p}{p}}$
$\displaystyle\lesssim\|\partial_{z}\omega\|_{L^{p}}^{\frac{2(p-1)}{p}}\big{(}\|\partial_{r}\partial_{z}\omega\|_{L^{p}}+\|\partial_{z}\frac{\omega}{r}\|_{L^{p}}\big{)}^{\frac{2-p}{p}}.$
En tenu compte de l’inégalité de Hardy que implique que
$\|\frac{\omega}{r}\|_{L^{p}}\lesssim\|\partial_{r}\omega\|_{L^{p}}\qquad\hbox{et}\qquad\|\partial_{z}\frac{\omega}{r}\|_{L^{p}}\lesssim\|\partial_{z}\partial_{r}\omega\|_{L^{p}}\qquad\mbox{pour}\quad
p\neq 2,$
on obtient
$\displaystyle\int\partial_{r}u^{z}\partial_{z}\omega|\partial_{r}\omega|^{p-1}$
$\displaystyle\lesssim\Big{(}\|\omega\|_{L^{p}}^{(1-\frac{1}{p})^{2}}\|\partial_{r}\omega\|_{L^{p}}^{\frac{1}{p}(1-\frac{1}{p})}\|\partial_{z}\omega\|_{L^{p}}^{\frac{1}{p}(1-\frac{1}{p})}\|\partial_{z}\partial_{r}\omega\|_{L^{p}}^{\frac{1}{p^{2}}}$
$\displaystyle+\|\partial_{z}\omega\|_{L^{p}}^{\frac{2(p-1)}{p}}\|\partial_{r}\partial_{z}\omega\|_{L^{p}}^{\frac{2-p}{p}}\Big{)}\|\partial_{z}\omega\|_{L^{p}}^{1-\frac{1}{p}}\|\partial_{r}\partial_{z}\omega\|_{L^{p}}^{\frac{1}{p}}\|\partial_{r}\omega\|_{L^{p}}^{p-1}$
$\displaystyle\lesssim\|\omega\|_{L^{p}}^{(1-\frac{1}{p})^{2}}\|\partial_{z}\omega\|_{L^{p}}^{1-\frac{1}{p^{2}}}\|\partial_{r}\omega\|_{L^{p}}^{p-1+\frac{1}{p}(1-\frac{1}{p})}\|\partial_{r}\partial_{z}\omega\|_{L^{p}}^{\frac{1}{p}(1+\frac{1}{p})}$
$\displaystyle+\|\partial_{z}\omega\|_{L^{p}}^{\frac{3(p-1)}{p}}\|\partial_{r}\omega\|_{L^{p}}^{p-1}\|\partial_{r}\partial_{z}\omega\|_{L^{p}}^{\frac{3-p}{p}}$
grâce au Lemme 3.1 et l’inégalité de Young, on aura
$\displaystyle\|\partial_{r}\partial_{z}\omega\|_{L^{p}}^{\frac{1}{p}+\frac{1}{p^{2}}}$
$\displaystyle\|\partial_{r}\omega\|_{L^{p}}^{p+\frac{1}{p}-\frac{1}{p^{2}}-1}\|\omega\|_{L^{p}}^{(1-\frac{1}{p})^{2}}\|\partial_{z}\omega\|_{L^{p}}^{1-\frac{1}{p^{2}}}$
$\displaystyle\lesssim\|\partial_{z}|\partial_{r}\omega|^{\frac{p}{2}}\|_{L^{2}}^{\frac{1}{p}+\frac{1}{p^{2}}}\|\partial_{r}\omega\|_{L^{p}}^{p+\frac{3}{2p}-\frac{3}{2}}\|\omega\|_{L^{p}}^{(1-\frac{1}{p})^{2}}\|\partial_{z}\omega\|_{L^{p}}^{1-\frac{1}{p^{2}}}$
$\displaystyle\leq\varepsilon\|\partial_{z}|\partial_{r}\omega|^{\frac{p}{2}}\|_{L^{2}}^{2}+c_{\varepsilon}\|\omega\|_{L^{p}}^{2(p-1)\over{2p+1}}\|\partial_{z}\omega\|_{L^{p}}^{{2(p+1)}\over{2p+1}}\|\partial_{r}\omega\|_{L^{p}}^{p\,\frac{2p^{2}+3-3p}{2p^{2}-p-1}}$
et
$\displaystyle\|\partial_{z}\omega\|_{L^{p}}^{\frac{3(p-1)}{p}}\|\partial_{r}\omega\|_{L^{p}}^{p-1}\|\partial_{r}\partial_{z}\omega\|_{L^{p}}^{\frac{3-p}{p}}\leq\varepsilon\|\partial_{z}|\partial_{r}\omega|^{\frac{p}{2}}\|_{L^{2}}^{2}+c_{\varepsilon}\|\partial_{z}\omega\|_{L^{p}}^{2}\|\partial_{r}\omega\|_{L^{p}}^{\frac{3p^{2}-7p+6}{3p-3}}.$
Donc pour $1<p\leq 2$
$\displaystyle\int$
$\displaystyle\partial_{r}u^{z}\partial_{z}\omega|\partial_{r}\omega|^{p-1}\leq
2\varepsilon\|\partial_{z}|\partial_{r}\omega|^{\frac{p}{2}}\|_{L^{2}}^{2}$
(3.20)
$\displaystyle+c_{\varepsilon}\|\omega\|_{L^{p}}^{2(p-1)\over{2p+1}}\|\partial_{z}\omega\|_{L^{p}}^{{2(p+1)}\over{2p+1}}\|\partial_{r}\omega\|_{L^{p}}^{p\,\frac{2p^{2}+3-3p}{2p^{2}-p-1}}+c_{\varepsilon}\|\partial_{z}\omega\|_{L^{p}}^{2}\|\partial_{r}\omega\|_{L^{p}}^{\frac{3p^{2}-7p+6}{3p-3}}.$
Enfin concernant le terme $\int\partial_{z}u^{z}|\partial_{r}\omega|^{p},$ par
intégration par parties plus l’inégalité de Cauchy-Schwartz, on a
$\displaystyle\int\partial_{z}u^{z}|\partial_{r}\omega|^{p}=$
$\displaystyle-2\int u^{z}|\partial_{r}\omega|^{p\over
2}\partial_{z}|\partial_{r}\omega|^{p\over 2}.$ (3.21)
$\displaystyle\lesssim\|u^{z}\|_{L^{\infty}}\|\partial_{r}\omega\|_{L^{p}}^{\frac{p}{2}}\big{\|}\partial_{z}|\partial_{r}\omega|^{p\over
2}\big{\|}_{L^{2}}.$
Comme $\Delta u^{z}=\partial_{r}\omega+\frac{\omega}{r},$ alors par
integration par parties, on trouve
$u^{z}=-{1\over{4\pi}}\int_{{\mathbb{R}}^{3}}\frac{r^{\prime}-r\cos\theta^{\prime}}{\big{(}r^{2}+{r^{\prime}}^{2}-2rr^{\prime}\cos\theta^{\prime}+(z-z^{\prime})^{2}\big{)}^{\frac{3}{2}}}\,\omega\,r^{\prime}dr^{\prime}dz^{\prime}d\theta^{\prime},$
et par suite
$|u^{z}|\lesssim\frac{1}{|X|^{2}}\star|\omega|,$
alors
$\|u^{z}\|_{L^{\infty}_{h}}\lesssim\frac{1}{|z|^{\frac{2-p}{p}}}\ast\|\omega\|_{L^{\frac{2p}{2-p}}_{h}},$
ainsi l’injection de Sobolev et l’inégalité de Hardy, impliquent
$\displaystyle\|u^{z}\|_{L^{\infty}_{v}(L^{\infty}_{h})}$
$\displaystyle\lesssim\|\omega\|_{L^{{\frac{p}{p-1}},1}_{v}(L^{\frac{2p}{2-p}}_{h})}$
$\displaystyle\lesssim\|\partial_{r}\omega\|_{L^{{\frac{p}{p-1}},1}_{v}(L^{p}_{h})}.$
Or par interpolation
$\|f\|_{L^{{\frac{p}{p-1}},1}({\mathbb{R}})}\lesssim\|f\|_{L^{p}}^{\frac{2p-2}{p}}\|\nabla
f\|_{L^{p}}^{\frac{2-p}{p}},$
ainsi pour $1<p<2,$ on obtient grâce au Lemme 3.1
$\displaystyle\|u^{z}\|_{L^{\infty}}$
$\displaystyle\lesssim\|\partial_{r}\omega\|_{L^{p}}^{\frac{2p-2}{p}}\|\partial_{z}\partial_{r}\omega\|_{L^{p}}^{\frac{2-p}{p}}$
$\displaystyle\lesssim\|\partial_{r}\omega\|_{L^{p}}^{\frac{p}{2}}\|\partial_{z}|\partial_{r}\omega|^{\frac{p}{2}}\|_{L^{2}}^{\frac{2-p}{p}}$
En injectant l’inégalité précédente dans l’inégalité (3.21) et on utilisant
l’inégalité de Young, on trouve pour $1<p<2$
$\int\partial_{z}u^{z}|\partial_{r}\omega|^{p}\leq
c_{\varepsilon}\|\partial_{r}\omega\|_{L^{p}}^{\frac{p^{2}}{p-1}}+\varepsilon\big{\|}\partial_{z}|\partial_{r}\omega|^{p\over
2}\big{\|}_{L^{2}}^{2}.$ (3.22)
Donc les inégalités (3.16), (3.12), (3.20) et (3.22), impliquent
$\displaystyle{d\over dt}\|\partial_{r}\omega\|_{L^{p}}^{p}+$
$\displaystyle\Big{\|}\partial_{z}|\partial_{r}\omega|^{p\over
2}\Big{\|}_{L^{2}}^{2}\lesssim\|{u^{r}\over
r}\|_{L^{\infty}}\|\partial_{r}\omega\|_{L^{p}}^{p}+\|\partial_{z}\omega\|_{L^{p}}^{2}\|\partial_{r}\omega\|_{L^{p}}^{\frac{3p^{2}-7p+6}{3p-3}}$
(3.23) $\displaystyle+\|{\omega\over
r}\|_{L^{p}}^{\frac{2p-2}{3p-2}}\Big{\|}\partial_{z}{\omega\over
r}\Big{\|}_{L^{p}}^{\frac{2}{3p-2}}\|\partial_{z}\omega\|_{L^{p}}^{\frac{4p-4}{3p-2}}\|\partial_{r}\omega\|_{L^{p}}^{\frac{3p^{2}-4p+2}{3p-2}}$
$\displaystyle+\|\omega\|_{L^{p}}^{{2(p-1)}\over{2p+1}}\|\partial_{z}\omega\|_{L^{p}}^{{2(p+1)}\over{2p+1}}\|\partial_{r}\omega\|_{L^{p}}^{p\,\frac{2p^{2}+3-3p}{2p^{2}-p-1}}+\|\partial_{r}\omega\|_{L^{p}}^{\frac{p^{2}}{p-1}}.$
En integrant l’inégalité précédente et on tenu compte des l’inǵalités (3.9) et
de Hardy, on obtient
$\displaystyle\|\partial_{r}\omega\|_{L^{\infty}_{t}(L^{p})}^{p}+$
$\displaystyle\Big{\|}\partial_{z}|\partial_{r}\omega|^{p\over
2}\Big{\|}_{L^{2}_{t}(L^{2})}^{2}\lesssim\|\partial_{r}\omega_{0}\|_{L^{p}}^{p}+t^{\frac{1}{2}}\|{\omega_{0}\over
r}\|_{L^{{3\over 2},1}}\|\partial_{r}\omega\|_{L^{\infty}_{t}(L^{p})}^{p}$
$\displaystyle+\|\partial_{z}\omega\|_{L^{2}_{t}(L^{p})}^{2}\|\partial_{r}\omega\|_{L^{\infty}_{t}(L^{p})}^{\frac{3p^{2}-7p+6}{3p-3}}$
$\displaystyle+t^{\frac{p-1}{3p-2}}\Big{\|}\partial_{z}{\omega\over
r}\Big{\|}_{L^{2}_{t}(L^{p})}^{\frac{2}{3p-2}}\|\partial_{z}\omega\|_{L^{2}_{t}(L^{p})}^{\frac{2p-2}{3p-2}}\|\partial_{r}\omega\|_{L^{\infty}_{t}(L^{p})}^{\frac{3p^{2}-2p}{3p-2}}$
$\displaystyle+t^{\frac{p}{2p+1}}\|\omega\|_{L^{\infty}_{t}(L^{p})}^{{2(p-1)}\over{2p+1}}\|\partial_{z}\omega\|_{L^{2}_{t}(L^{p})}^{{2(p+1)}\over{2p+1}}\|\partial_{r}\omega\|_{L^{\infty}_{t}(L^{p})}^{p\,\frac{2p^{2}+3-3p}{2p^{2}-p-1}}+t\|\partial_{r}\omega\|_{L^{\infty}_{t}(L^{p})}^{\frac{p^{2}}{p-1}}.$
Et par suite pour $1<p<2,$ il existe $T>0$ tels que $\partial_{r}\omega\in
L^{\infty}_{T}(L^{p})$ et $\partial_{z}\partial_{r}\omega\in
L^{2}_{T}(L^{p}),$ ainsi il existe $t_{0}\in[0,T]$ tels que
$\partial_{r}\omega(t_{0})\in L^{p}$ et
$\partial_{z}\partial_{r}\omega(t_{0})\in L^{2}_{T}(L^{p}).$ Par définition de
l’espace de Lorentz, on en déduire les mêmes résultats. Ainsi il existe
$t_{1}$ tel que $\omega(t_{1})\in L^{{3\over 2},1}\cap L^{3,1}.$ Pour conclure
la démonstration il suffit d’utilisé la Proposition 3.3. D’où la proposition.
∎
Démonstration du Lemme 3.2.
Pour démontrer que $\delta\omega\in L^{\infty}_{t}(L^{p})$ et
$\partial_{z}|\omega|^{p\over 2}\in L^{2}_{t}(L^{2})$ il suffit de prouver que
$(u_{2}\cdot\nabla)\delta\omega+(\delta
u\cdot\nabla)\omega_{1}-{u^{r}_{2}\over r}\delta\omega-{\delta u^{r}\over
r}\omega_{1}\in L^{1}_{t}(L^{p})$ pour $p\leq{3\over 2}.$ D’après l’inégalité
de Hölder, par interpolation et grâce à la Proposition 3.1 et [18], on a
$\displaystyle\|(u_{2}\cdot\nabla)\delta\omega\|_{L^{p}}$
$\displaystyle\leq\|u_{2}\|_{L^{3p\over
3-2p}}\sum_{i=1}^{2}(\|\partial_{r}\omega_{i}\|_{L^{3\over
2}}+(\|\partial_{z}\omega_{i}\|_{L^{3\over 2}})$
$\displaystyle\leq\|u_{2}\|_{L^{3}}^{3-2p\over
p}\|u_{2}\|_{L^{\infty}}^{3(p-1)\over
p}\sum_{i=1}^{2}(\|\partial_{r}\omega_{i}\|_{L^{3\over
2}}+\|\partial_{z}\omega_{i}\|_{L^{3\over 2}})$
$\displaystyle\lesssim\|\omega_{2}\|_{L^{3\over 2}}^{3-2p\over
p}\|\omega_{2}\|_{L^{3,1}}^{3(p-1)\over
p}\sum_{i=1}^{2}(\|\partial_{r}\omega_{i}\|_{L^{3\over
2}}+\|\partial_{z}\omega_{i}\|_{L^{3\over 2}})$
$\displaystyle\lesssim\|\omega_{2}\|_{L^{3\over 2}}^{3-2p\over
p}\big{(}\|\partial_{r}\omega_{2}\|_{L^{{3\over 2},1}}+\|{\omega_{2}\over
r}\|_{L^{{3\over 2},1}}+\|\partial_{z}\omega_{2}\|_{L^{{3\over
2},1}}\big{)}^{3(p-1)\over p}$
$\displaystyle\times\sum_{i=1}^{2}(\|\partial_{r}\omega_{i}\|_{L^{3\over
2}}+\|\partial_{z}\omega_{i}\|_{L^{3\over 2}})$
et par suite les deux propositions précédentes combinée avec le Corollaire
3.2, impliquent $(u_{2}\cdot\nabla)\delta\omega\in L^{1}_{t}(L^{p}),$ les
mêmes calculs donnent $(\delta u\cdot\nabla)\omega_{1}\in L^{1}_{t}(L^{p}).$
Pour ${u^{r}_{2}\over r}\delta\omega$ grâce à l’inégalité de Hölder, par
interpolation et la Proposition 3.1, on obtient
$\displaystyle\|{u^{r}_{2}\over
r}\delta\omega\|_{L^{p}}\leq\|u^{r}_{2}\|_{L^{3p\over
3-2p}}\|{\delta\omega\over r}\|_{L^{3\over 2}}\leq$
$\displaystyle\sum_{i=1}^{2}\|{\omega_{i}\over r}\|_{L^{3\over
2}}\|u^{r}_{2}\|_{L^{3}}^{3-2p\over p}\|u^{r}_{2}\|_{L^{\infty}}^{3(p-1)\over
p}$ $\displaystyle\lesssim\sum_{i=1}^{2}\|{\omega_{i}\over r}\|_{L^{3\over
2}}\|\omega_{2}\|_{L^{3\over 2}}^{3-2p\over
p}\|\partial_{z}\omega_{2}\|_{L^{{3\over 2},1}}^{3(p-1)\over p}.$
Et par suite le Corollaire 3.2 et le fait que ${3(p-1)\over p}\leq 2$
impliquent ${u^{r}_{2}\over r}\delta\omega\in L^{1}_{t}(L^{p})$ les mêmes
calculs donnent ${\delta u^{r}\over r}\omega_{1}\in L^{1}_{t}(L^{p}).$ D’où le
lemme. $\square$
## 4 Existence pour des données moins régulières
Dans cette partie nous démontrons le Théorème 1.2 d’existence des solutions
pour des données initiales moins régulières. Pour cela nous avons besoin de
prendre en compte encore plus des estimations anisotropes sur
$\frac{u^{r}}{r}$. Nous avons, pour tout $1<p\leq\frac{3}{2}$, l’inégalité
suivante
$\|\frac{u^{r}}{r}\|_{L^{\infty}_{h}(L^{\frac{p}{3-2p}}_{v})}\leq
C\|\partial_{z}\frac{\omega}{r}\|_{L^{p,1}}.$
En effet: d’après les estimations de la Proposition 3.1 on a
$|\frac{u^{r}}{r}|\lesssim\frac{1}{|X|}\star|\partial_{z}\frac{\omega}{r}|.$
Donc
$\|\frac{u^{r}}{r}\|_{L^{\infty}_{h}}\lesssim\|\frac{1}{\sqrt{|X_{h}|^{2}+z^{2}}}\|_{L^{p^{\prime}}_{h}}\star\|\partial_{z}\frac{\omega}{r}\|_{L^{p}_{h}}$
En utilisant le fait que la primitive de $r(r^{2}+z^{2})^{-{p^{\prime}\over
2}}$ est $\sqrt{r^{2}+z^{2}}^{2-p^{\prime}}$ a une constante près, on trouve
$\|\frac{u^{r}}{r}\|_{L^{\infty}_{h}}\lesssim\frac{1}{|z|^{\frac{2}{p}-1}}\star\|\partial_{z}\frac{\omega}{r}\|_{L^{p}_{h}}.$
On prend maintenant la norme $L^{\frac{p}{3-2p}}$ en variable verticale pour
obtenir
$\|\frac{u^{r}}{r}\|_{L^{\infty}_{h}(L^{\frac{p}{3-2p}}_{v})}\leq
C\|\partial_{z}\frac{\omega}{r}\|_{L^{p,1}}.$
On peut ainsi contrôler la norme de $\omega$ dans tout $L^{p},$ rappelons que
$\omega$ vérifie l’équation suivante
$\partial_{t}\omega+u\nabla\omega-\frac{u^{r}}{r}\omega-\partial_{z}^{2}\omega=0$
Donc pour $1<p\leq 3/2$, on a
$\displaystyle\frac{1}{2}\frac{d}{dt}\||\omega(t)|^{p/2}\|_{L^{2}}^{2}+\|\partial_{z}(|\omega|^{p/2})\|^{2}_{L^{2}}$
$\displaystyle\leq\int|\frac{u^{r}}{r}||\omega|^{p/2}|\omega|^{p/2}$
$\displaystyle\leq\|\frac{u^{r}}{r}\|_{L^{\infty}_{h}L^{\frac{p}{3-2p}}_{v}}\||\omega|^{p/2}\|_{L^{2}_{h}(L^{\frac{2p}{3(p-1)}}_{v})}^{2}.$
Comme $H^{s}({\mathbb{R}}_{v})\subset L^{\frac{2p}{3p-3}}({\mathbb{R}}_{v})$
pour $s=(3-2p)/(2p)$, alors
$\||\omega|^{p/2}\|_{L^{2}_{h}(L^{\frac{2p}{3p-3}}_{v})}^{2}\leq\||\omega|^{p/2}\|_{L^{2}}^{(4p-3)/p}\|\partial_{z}(|\omega|^{p/2})\|_{L^{2}}^{(3-2p)/p}.$
Donc
$\displaystyle\frac{1}{2}\frac{d}{dt}\||\omega(t)|^{p/2}\|_{L^{2}}^{2}+\|\partial_{z}(|\omega|^{p/2})\|^{2}_{L^{2}}$
$\displaystyle\leq\frac{1}{2}\|\frac{u^{r}}{r}\|_{L^{\infty}_{h}(L^{\frac{p}{3-2p}}_{v})}^{\frac{2p}{4p-3}}\||\omega|^{p/2}\|^{2}_{L^{2}}+\frac{1}{2}\|\partial_{z}(|\omega|^{p/2})\|_{L^{2}}^{2}$
$\displaystyle\leq
C\|\partial_{z}\frac{\omega}{r}\|_{L^{p,1}}^{\frac{2p}{4p-3}}\||\omega|^{p/2}\|^{2}_{L^{2}}+\frac{1}{2}\|\partial_{z}(|\omega|^{p/2})\|_{L^{2}}^{2}.$
Par le lemme de Gronwall et vu que
$\|\partial_{z}\frac{\omega}{r}\|_{L^{\frac{2p}{4p-3}}_{t}(L^{p,1})}\leq
t^{\frac{3(p-1)}{4p-3}}\|\frac{\omega_{0}}{r}\|_{L^{p,1}},$ nous obtenons
$\|\omega\|_{L^{p}}+\|\partial_{z}\omega\|_{L^{2}_{t}(L^{p})}\leq\|\omega_{0}\|_{L^{p}}\exp(Ct^{\frac{3(p-1)}{4p-3}}\|\frac{\omega_{0}}{r}\|_{L^{p,1}}),$
et par interpolation
$\|\omega\|_{L^{p,1}}+\|\partial_{z}\omega\|_{L^{2}_{t}(L^{p,1})}\leq\|\omega_{0}\|_{L^{p,1}}\exp(Ct^{\frac{3(p-1)}{4p-3}}\|\frac{\omega_{0}}{r}\|_{L^{p,1}}).$
En particulier l’inégalité précédente est valable pour $p=6/5,$ ainsi on peut
montrer l’existence globale d’une solution lorsque $\frac{\omega}{r}\in
L^{\frac{6}{5}+,1}$ et $\omega_{0}\in L^{\frac{6}{5}+,1}$. Tout d’abord, on
note que $\omega_{0}\in L^{\frac{6}{5},1}$ implique que $u_{0}\in L^{2}$ et
par l’estimation d’énergie on a
$\|u(t)\|^{2}_{L^{2}}+2\int_{0}^{t}\|\partial_{z}u\|^{2}_{L^{2}}\leq\|u_{0}\|^{2}_{L^{2}}.$
D’autre part, comme $\omega\in L^{\infty}_{t}(L^{\frac{6}{5}+,1})$ et
$\|\omega\|_{L^{p}}\approx\|\nabla u\|_{L^{p}}$ pour $1<p<+\infty,$ alors
$u\in L^{\infty}_{t}(\dot{W}^{1,\frac{6}{5}+})$ et donc finalement $u\in
L^{\infty}_{t}(W^{1,\frac{6}{5}+}({\mathbb{R}}^{3}))$ qui est un sous espace
de $L^{\infty}_{t}(L^{2}({\mathbb{R}}^{3}))$ avec l’inclusion compacte dans la
topologie de $L^{2}_{loc}({\mathbb{R}}^{3})$ à $t$ fixé. Donc, on peut
construire la solution en utilisant uniquement $\omega\in L^{\frac{6}{5}}\cap
L^{\frac{6}{5}+,1}$ et $\frac{\omega}{r}\in L^{\frac{6}{5}}\cap
L^{\frac{6}{5}+,1}$ par passage à la limite dans une suite des solutions
approchées, axisymétriques et régulières de l’équation
$\partial_{t}u+\text{div\,}(u\otimes u)-\partial_{z}^{2}u=-\nabla p.$
Plus précisement, soit $u_{0}\in L^{2}({\mathbb{R}}^{3})$ de sorte que
$\omega_{0}\in L^{\frac{6}{5}}\cap L^{\frac{6}{5}+,1}$ et
$\frac{\omega_{0}}{r}\in L^{\frac{6}{5}}\cap L^{\frac{6}{5}+,1}$. Soit $J_{n}$
l’opérateur de troncature sur les basses fréquences défini par
$J_{n}u={\mathcal{F}}^{-1}(\chi(\xi 2^{-n}){\mathcal{F}}u(\xi))$, où
$\mathcal{F}$ dénote la transformée de Fourier et $\chi$ est une fonction
radiale régulière qui vaut $1$ sur une boule autour de zéro. On sait que pour
$u_{0}$ champ axisymétrique sans swirl on a $J_{n}u_{0}$ est aussi
axysimétrique sans swirl et régulièr (voir [2]). Donc, il existe un unique
solution globale régulière, axisymétrique et sans swirl $u^{n}$ solution du
problème
$(NS_{n})\begin{cases}\partial_{t}u_{n}+\text{div\,}(u_{n}\otimes
u_{n})-n^{-1}\Delta_{h}u_{n}-\partial_{3}^{2}u_{n}=-\nabla p_{n}\\\
\text{div\,}u_{n}=0\\\ u_{n}|_{t=0}=J_{n}u_{0}.\end{cases}$
En tenant compte du fait que $J_{n}\omega_{0}$ et $\frac{J_{n}\omega_{0}}{r}$
sont uniformément bornés dans $L^{\frac{6}{5}}\cap L^{\frac{6}{5}+,1}$ (voir
[7]) nous obtenons que $u_{n}$ est une suite uniformément bornée dans
$L^{\infty}_{t}(W^{1,\frac{6}{5}+}({\mathbb{R}}^{3}))$. En utilisant
l’équation vérifiée par $u_{n}$ on trouve aisément que $\partial_{t}u_{n}$ est
bornée dans $L^{\infty}_{t}(H^{-N})$ pour $N$ assez grand. En tenant compte du
fait que l’inclusion $W^{1,\frac{6}{5}+}({\mathbb{R}}^{3})$ dans
$L^{2}_{loc}({\mathbb{R}}^{3})$ est compacte et comme $u_{n}$ est bornée dans
$C_{loc}(H^{-N})$ nous obtenons par le lemme de Arzela-Ascoli, quitte à
extraire une sous suite, que $u_{n}$ converge fortement vers un $u$ dans
$C_{loc}(H^{-N}_{loc})$. En interpolant avec le fait que $u_{n}$ est suite
bornée dans $L^{\infty}(W^{1,\frac{6}{5}+})$ on trouve que $u_{N}\to u$ dans
$L^{\infty}_{loc}(L^{2}({\mathbb{R}}^{3}))$. Cela suffit pour passer à la
limite dans les termes non-linéaires et on trouve que $u_{n}\otimes u_{n}\to
u\otimes u$ dans ${\mathcal{D}}^{\prime}.$ Finalement, par passage à la limite
dans $(NS_{n})$ nous obtenons une solution globale axisymétrique sans swirl
$u$ de $(NS_{v})$.
## References
* [1] H. Abidi: Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes, Bull. Sci. Math. 132 (2008), no. 7, 592–624.
* [2] H. Abidi, T. Hmidi et S. Keraani: On the global well-posedness for the axisymmetric Euler equations, à paraître dans Mathematische Annalen.
* [3] J. Bergh and J. Löfström, Interpolation spaces, Springer-Verlag, 1976.
* [4] M. Cannone, Y. Meyer et F. Planchon: Solutions auto-similaires des équations de Navier-Stokes, Séminaire sur les équations aux dérivées partielles, 1993 -1994, exp. No 12 pp. $\acute{E}$cole polytech, palaiseau, 1994.
* [5] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier: Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal. 34 (2000), no. 2, 315–335.
* [6] J.-Y. Chemin and P. Zhang: On the global wellposedness to the $3$-D incompressible anisotropic Navier-Stokes equations, Commun. Math. Phys. 272 (2007), 529–566.
* [7] R. Danchin: Axisymmetric incompressible flows with bounded vorticity, Russian Math. Surveys 62 (2007), no. 3, 73–94.
* [8] R. Di Perna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511–547.
* [9] H. Fujita and T. Kato: On the Navier-Stokes initial value problem I, Archive for rational mechanics and analysis 16 (1964), 269–315.
* [10] D. Iftimie: A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity, SIAM J. Math. Anal. 33 (2002), no. 6, 1483–1493.
* [11] H. Koch and D. Tataru: Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), no 1, 22–35.
* [12] O. A. Ladyzhenskaya: Unique solvability in large of a three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zapisky Nauchnych Sem. LOMI 7 (1968), 155–177.
* [13] J. Leray: Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, Journal de mathématique pures et appliquées 12 (1933), 1–82.
* [14] M. Paicu: $\acute{E}$quation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana 21 (2005), no. 1, 179–235.
* [15] R. O’Neil, Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), 129–142.
* [16] T. Shirota and T. Yanagisawa, Note on global existence for axial ly symmetric solutions of the Euler system, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), no. 10, 299–304.
* [17] M. Ukhovskii and V. Yudovitch, Axially symmetric flows of ideal and viscous fluids filling the whole space, Journal of applied mathematics and mechanics, 32 (1968), 52–69.
* [18] L. Tartar, Imbedding theorems of Sobolev spaces into Lorentz spaces, Bollettino U. M. I. (8) 1-B (1998), 479–500.
|
arxiv-papers
| 2009-06-24T13:34:54 |
2024-09-04T02:49:03.514419
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hammadi Abidi, Marius Paicu",
"submitter": "Hammadi Abidi",
"url": "https://arxiv.org/abs/0906.4473"
}
|
0906.4675
|
# Competition for Popularity in Bipartite Networks
Mariano Beguerisse Díaz [email protected] Centre for
Integrative Systems Biology, Imperial College London, South Kensington Campus,
London, SW7 2AZ, U.K. Division of Biology, Imperial College London, South
Kensington Campus, London, SW7 2AZ, U.K. Mason A. Porter
[email protected] Oxford Centre for Industrial and Applied Mathematics,
Mathematical Institute, University of Oxford, OX1 3LB, U.K. CABDyN Complexity
Centre, University of Oxford, OX1 1HP, U.K. Jukka-Pekka Onnela
[email protected] CABDyN Complexity Centre, University of Oxford, OX1
1HP, U.K. Department of Physics, University of Oxford, OX1 3PU, U.K.
Department of Biomedical Engineering and Computational Science, Helsinki
University of Technology, FIN-02015 HUT, Finland Harvard Kennedy School,
Harvard University, Cambridge, MA 02138, U.S.A.
###### Abstract
We present a dynamical model for rewiring and attachment in bipartite networks
in which edges are added between nodes that belong to catalogs that can either
be fixed in size or growing in size. The model is motivated by an empirical
study of data from the video rental service Netflix, which invites its users
to give ratings to the videos available in its catalog. We find that the
distribution of the number of ratings given by users and that of the number of
ratings received by videos both follow a power law with an exponential cutoff.
We also examine the activity patterns of Netflix users and find bursts of
intense video-rating activity followed by long periods of inactivity. We
derive ordinary differential equations to model the acquisition of edges by
the nodes over time and obtain the corresponding time-dependent degree
distributions. We then compare our results with the Netflix data and find good
agreement. We conclude with a discussion of how catalog models can be used to
study systems in which agents are forced to choose, rate, or prioritize their
interactions from a very large set of options.
Bipartite Networks, Human Dynamics, Catalog Networks, Bursts, Rate Equations
###### pacs:
89.75.Hc, 89.65.-s, 05.90.+m
Human dynamics, which is concerned with the characterization of human activity
in time, has been the subject of intense and exciting research over the last
few years barabasi-2005-435 ; evans-2008-3 ; J.P.Onnela05012007 . In one
typical problem setting, individuals are endowed with limited resources, and
there are numerous activities, behaviors, and/or products that compete against
each other for those resources. Although such situations admit a natural
formulation using bipartite (two-mode) networks that connect individuals to
activities, human dynamics has surprisingly seldom been studied from this
perspective. In the present paper, we analyze bipartite networks constructed
from a large data set of video ratings by the users of a video rental company
over a period of six years. To analyze the time evolution of these networks,
we introduce the concept of a catalog network, and we use this approach to
explore the driving forces behind the video rating behavior of individuals. We
believe that such a framework can be used to study many other phenomena in
human dynamics that involve the allocation of and competition for scarce
resources.
## I Introduction
Numerous natural and man-made systems involve interactions between large
numbers of entities. The structural configuration of interactions is typically
rather complicated, so the study of such systems often benefits greatly from
network representations albert-2002-74 ; Newman:2003 ; guido . A network is
usually abstracted mathematically as a graph whose nodes represent the
entities and whose edges represent the interactions between the entities
HandbookGraphTheory . In many cases, edges can be weighted or directed, and
more complicated frameworks such as hypergraphs can also be employed. The
number of edges connected to a node in an unweighted network is known as its
degree, and the degree distribution of a network is given by the collection of
numbers that give the fraction of nodes that have degree $k$ (for all values
of $k$) Newman:2003 . In weighted networks, one considers the weight of an
edge rather than simply whether or not it exists.
Because networked systems are not static, the last decade has witnessed a
particular interest in models that attempt to address their growth and
evolution guido . Perhaps the best-known model of network growth was
formulated by Barabási and Albert BarabasiAlbert1999 ; albert-2002-74 .
Similar models were also constructed decades earlier by Simon
HERBERTA.SIMON12011955 and Price Price:1965 . Barabási and Albert examined
networks arising from diverse settings and found that their degree
distributions often seemed to follow power laws, which are functions of the
form $f(x)\sim x^{-\alpha}$ (with $\alpha>0$). They proposed a growth
mechanism, which they called preferential attachment (Price had called it
cumulative advantage) to try to explain their observations. One starts with a
small seed network and—in the simplest form of the mechanism—iteratively adds
individual nodes that each possess exactly one edge. One connects each new
node to an existing one chosen at random with probability proportional to its
degree. That is, the probability to choose node $m_{i}$ with degree $k_{i}$ is
$P(m_{i})=\frac{k_{i}}{\sum_{j=1}^{N}k_{j}}\,,$
where the total number of nodes $N$ indicates the size of the network. Because
nodes with higher degrees have correspondingly higher probabilities to receive
new edges, the preferential attachment growth mechanism leads naturally to a
power-law degree distribution BarabasiAlbert1999 ; PhysRevLett.85.4629 .
Because of ideas like preferential attachment and the resulting insights on
the origin of heavy-tailed degree distributions that one sees, e.g., in the
World Wide Web or scientific collaboration networks, the study of networks has
grown immensely during the last ten years Newman:2003 ; NewmanSciCol ; guido .
However, most of this research has concentrated on one-mode (unipartite)
networks, in which all of the nodes are of the same type. It is perhaps under-
appreciated that other graph structures are also very important in many
applications Latapy200831 . Even the simplest generalization, known as a two-
mode or bipartite network, has been studied much more sparingly than
unipartite networks. Bipartite networks contain two categories (partite sets)
of nodes: $\mathcal{U}=\\{u_{1},u_{2},\dots,u_{U}\\}$ (with $U$ members) and
$\mathcal{M}=\\{m_{1},m_{2},\dots,m_{M}\\}$ (with $M$ members). As shown in
Fig. 1, each (undirected) edge connects a node in $\mathcal{U}$ to one in
$\mathcal{M}$ HandbookGraphTheory . Bipartite networks abound in applications:
They can represent affiliation networks in which people are connected to
organizations or committees MasonPorter05172005 , ecological networks with
links between cooperating species in an ecosystem Saavedra07532 , and more
Zhang20086869 ; Guillaume04bipartitegraphs ; PhysRevE.72.036120 ; evans:056101
; baseball .
Figure 1: (Color online) A bipartite network with nodes in the partite sets
$\mathcal{U}=\\{1,2,3,4\\}$ and $\mathcal{M}=\\{A,B,C\\}$. Each edge connects
a number to a letter.
A bipartite network possesses a degree distribution for each of the two node
types. We denote the adjacency matrix of a weighted bipartite network by
$\mathbf{G}\in\mathbb{R}^{U\times M}$. Each matrix element $\mathbf{G}_{ij}$
has a nonzero value if and only if an edge exists between nodes $u_{i}$ and
$m_{j}$. We denote the matrices that result from the two unipartite
projections as
$\mathbf{G}_{\mathcal{U}}=\mathbf{G}\mathbf{G}^{T}\in\mathbb{R}^{U\times U}$
and $\mathbf{G}_{\mathcal{M}}=\mathbf{G}^{T}\mathbf{G}\in\mathbb{R}^{M\times
M}$. The degree of a node in a unipartite projection network is then the
number of nodes of the same type with which the node shares at least one
neighbor in the original bipartite network. The node strengths similarly
incorporate connection strengths from the original bipartite network. (Recall
that the “strength” of a node is the sum of the strengths of the edges
connected to it.) For example, in an unweighted affiliation network, the two
projections give the weighted connection strength (the number of common
affiliations) among individuals and the interlock (the number of common
people) among organizations ceo ; MasonPorter05172005 .
Many of the real-life systems that can be represented by bipartite networks
are dynamic, as the existence and connectivity of both nodes and edges can
change in time. For example, a person might retire or leave one organization
to join another. One of the simplest types of changes is edge rewiring, in
which one end of an edge is fixed to a node and the other end moves from one
node to another (such as in the aforementioned change of affiliation). Because
of the important insights they can offer, network rewiring models have
received increasing attention WattsStrogatz ; PhysRevE.72.036120 ;
evans:056101 ; Dorogovtsev2003396 ; PhysRevE.72.026131 ; fan026103 ; lind .
They are closely related to abstract urn models from probability theory polya
; feller ; Godreche0953 , models of language competition Stauffer2007835 , and
models of transmission of cultural artifacts Bentley2003 . More generally,
they can help lead to a better understanding of any system in which the nature
or existence of an interaction among agents changes over time evans-2008-3 .
The rest of our presentation is organized as follows. In Section II, we
analyze a large data set of time-stamped video ratings from the video rental
service Netflix that we model as a bipartite network of people and videos. In
Section III, we examine the bursty behavior of individual users. In Section
IV, we develop a catalog model of bipartite network growth and evolution. We
then study the Netflix data using this model in Section V. Finally, we discuss
our results and present directions for future research in Section VI.
## II Netflix Video Ratings
Netflix is an online video rental service that encourages its users to rate
the videos they rent in order to improve their personalized recommendations.
As part of the Netflix Prize competition Netflixprize , in which the company
challenged the public to improve their video recommendation algorithm, Netflix
released a large, anonymized collection of user-assigned ratings of videos in
its catalog. In this paper, we use the Netflix data to study human dynamics in
the form of video ratings from a limited catalog. One can also examine the
dynamics of the ratings themselves, which would complement a recent empirical
study of video ratings that used data from the Internet Movie Database (IMDB)
imdb . The Netflix data consist of 100,480,507 ratings of 17,770 videos. The
ratings, which were given by 480,189 Netflix users between October 1998 and
December 2005, were sampled uniformly at random by Netflix from the set of
users who had rated at least 20 videos netflixPaper . Each entry in the data
includes the video ID, user ID, rating score (an integer from 1 to 5), and
submission date. To illustrate some of the temporal dynamics in the data, we
show in Fig. 2 the total number of ratings for each day from July to August
2003. The number of daily ratings exhibits a weekly pattern in which Mondays
and Tuesdays have the highest activity and Saturdays and Sundays have the
lowest. This reflects the weekly patterns in human work–leisure habits.
Figure 2: Number of daily ratings for each day in July and August 2003. The
mean number of ratings per day over this period is 30,449. The dashed vertical
lines indicate Tuesdays. Figure 3: (Color online) Number of ratings in the
Netflix data versus time from the beginning of 2000 to the end of 2005.
Circles indicate data from Netflix and the dashed red curve is a fit to
equation (1).
Figure 3 shows the total number of ratings from 2000 to the end of 2005. These
ratings seems to grow exponentially, which we confirm by fitting the data to
the function
$r(t)=a_{r}\left(e^{b_{r}t}-1\right)$ (1)
using nonlinear least squares. We obtain the parameter values $a_{r}\approx
6.3656\times 10^{5}$ and $b_{r}\approx 0.0024$.
Figure 4: (Color online) Number of users (top) and videos (bottom) in the
Netflix data versus time from the beginning of 2000 to the end of 2005.
Circles indicate data from Netflix and the dashed red curves are fits to
equations (2) and (3) for users and videos, respectively.
The number of users also grows exponentially, as shown on the top panel of Fig
4. The dashed curve in the plot is the fit to
$u(t)=a_{u}(e^{b_{u}t}-1)\,,$ (2)
where we obtain $a_{u}\approx 1.0018\times 10^{4}$ and $b_{u}\approx 0.0018$.
We will need to take the exponential growth of the system into account when
comparing data from dates that are far apart from each other.
In the bottom panel of Fig. 4, we show the number of videos from 2000 to 2005.
The number of videos appears to grow roughly linearly as a function of time,
but in fact it is better described by the relation
$m(t)=a_{m}+b_{m}t^{c_{m}}\,,$ (3)
where fitting yields $a_{m}\approx 2780.00$, $b_{m}\approx 0.6705$, and
$c_{m}\approx 1.3097$.
### II.1 Bipartite Network Formulation
The Netflix data can be represented as a bipartite network. The two types of
nodes in this network are users and videos. We use $\mathcal{U}$ to denote the
set of users and $\mathcal{M}$ to denote the set of videos. We ignore the
rating values and consider only the presence or absence of a rating event,
which corresponds to an edge between a user and a video in the unweighted
bipartite network. The large size and longitudinal nature of the data provides
a valuable opportunity to study video rating in the context of human dynamics,
as has been done previously with mobile telephone networks J.P.Onnela05012007
; nature06958 , book sale rankings Sornette.93.228701 , and electronic and
postal mail usage patterns barabasi-2005-435 ; oliveira2005437 .
### II.2 Degree Distributions
The bipartite video-rating network has one degree distribution for the user
nodes and another one for the video nodes. Keeping in mind the observations in
Fig. 2, we examine the cumulative degree distributions of individual days. The
distributions have a similar functional form for each day in the data set. We
fit them to a power law with an exponential cutoff,
$F(k)\sim k^{-a}e^{-bk}\,,$ (4)
using a modification of the method discussed by Clauset et al.
Clauset:2007p5520 . As an example, we show in Fig. 5 the cumulative degree
distributions for one day. Table 1 gives the parameter values that we found in
our fits of the data to equation (4). Despite the weekly pattern of the
ratings shown in Fig. 2, we did not find any significant differences between
the values of $a$ and $b$ for different days of the week. Hence, although the
number of daily ratings does differ significantly among weekdays, such
differences seem to not have much effect on the aggregate structure of the
network.
The problem setting sheds some insight into the observed functional form of
the degree distribution. Users select which videos to rate from a large set of
possibilities and possess time limitations on the number of videos that they
are able to watch and rate. As in any market, videos must compete against each
other for users’ attention. One can also anticipate that certain videos
saturate their market, especially in the case of niche videos whose audience
is small to begin with. Once the demand for a niche video has been met, it
virtually ceases to receive further ratings. On the other hand, blockbusters
might continue receiving numerous ratings for a long period of time.
Figure 5: (Color online) Cumulative degree distributions of user (top) and
video (bottom) nodes for August 26, 2003 (a Tuesday). The dashed curves are
the fits to equation (4) with parameters $a\approx 0.9828$, $b\approx 0.0057$
for the users and $a\approx 0.6622$, $b\approx 0.0070$ for the videos.
$a$ $b$ Mean Var Mean Var Videos 0.6580 0.0200 0.0686 0.0100 Users 0.8381
0.0573 0.0116 0.0007
Table 1: Fitting parameters of the daily video and user degree distributions
from 2000 to 2005 for the power law with exponential cutoff in (4).
### II.3 Clustering coefficients
To investigate the local connectivity of nodes and examine the impact of
highly-connected nodes, we calculate bipartite clustering coefficients
martacluster ; Zhang20086869 . In bipartite networks, a clustering coefficient
for a node can be calculated by counting the number of cycles of length 4
(i.e., the number of “squares”) that include the node and dividing the result
by the total possible number of squares that could include the node. As stated
by Zhang et al. in Zhang20086869 , the possible (or underlying) number of
squares is calculated by adding the potential links (including existing ones)
between a particular node and the neighbors of its neighbors. In Fig. 6 we
show how a square occurs in a bipartite network when two neighbors of a node
have another neighbor in common. Bipartite networks cannot have triangles
(three mutually-connected nodes) because two nodes of the same type cannot be
neighbors, so a square is the shortest possible cycle.
Figure 6: (Color online) Examples of how to calculate clustering coefficients
for bipartite (top) and unipartite (bottom) networks. In the bipartite
network, solid lines indicate edges that form the square that includes node
$B$, whose bipartite clustering coefficient calculated according to equation
(5) is $C_{4}=1/5$. One obtains this result because there are five possible
squares for this node ({$1A2B$, $1C2B$, $1A4B$, $1C4B$, $2C4B$}) but only one
of them ($2C4B$) actually exists. In the unipartite network, the solid lines
indicate edges that form the triangles that include node $1$. If this were an
unweighted network, for which $G_{ij}\in\\{0,1\\}$ for all $i$ and $j$, then
one would obtain an unweighted clustering coefficient of $C_{3}(1)=2/3$. To
calculate the value of its weighted clustering coefficient $\tilde{C}_{3}$, we
use equation (6).
The definition of a clustering coefficient of node $m_{i}$ in an unweighted
bipartite network is Zhang20086869 :
$C_{4}(m_{i})=\frac{\sum_{h,j}{q_{i_{jh}}}}{\sum_{j,h}{\left[(k_{j}-\eta_{i_{jh}})+(k_{h}-\eta_{i_{jh}})+q_{i_{jh}}\right]}}\,,$
(5)
where $q_{i_{jh}}$ is the observed number of squares containing $m_{i}$ and
any two neighbors $u_{h}$ and $u_{j}$. The degrees of the neighbors are
$k_{h}$ and $k_{j}$, respectively, and $\eta_{i_{jh}}=q_{i_{jh}}+1$. The
possible number of squares is calculated adding the degrees of the nodes
$u_{h}$ and $u_{j}$ minus the link that each shares with $m_{j}$ if the three
nodes are not part of a square to avoid double-counting. If the three nodes
are part of a square, then the square represented by the deleted link must be
added again, hence $(k_{j}-\eta_{i_{jh}})+(k_{h}-\eta_{i_{jh}})+q_{i_{jh}}$ in
the denominator of equation (5).
Figure 7: (Color online) Bipartite clustering coefficients $C_{4}(m_{i})$ for
video (blue) and user nodes (inset, green) for August 12, 2003 (a Tuesday).
The mean values for this day are $\langle
C_{4}\rangle=\frac{1}{M}\sum_{i=1}^{M}C_{4}(m_{i})\approx 0.02606$ for the
videos and $\langle C_{4}\rangle=\frac{1}{U}\sum_{i=1}^{U}C_{4}(u_{i})\approx
0.03144$ for the users.
In Fig. 7, we show the values of $C_{4}(m_{i})$ for the video and user nodes
for a single day (Tuesday, August 12, 2003). In Table 2, we show the mean
values of the bipartite clustering coefficient of all one-day snapshots of
Netflix in 2003. In spite of the weekday-dependent variation in the number of
daily ratings, the values of the bipartite clustering coefficient do not vary
significantly across weekdays. However, the values of $\langle C_{4}\rangle$
increase almost by 80% for both node-types on weekends. For a network
constructed from a single day’s data, only about 2% of the possible squares
typically exist; this is comparable to what would occur in a random network
with the same degree distributions. To investigate whether the presence of
blockbuster nodes (which have high degrees and increase considerably the
number of possible squares) has any effect on the value of $\langle
C_{4}\rangle$, we calculated the clustering coefficient after removing the top
ten most rated videos. We did not find any conclusive evidence of blockbusters
driving the value of the clustering coefficient; some of them caused the value
of $\langle C_{4}\rangle$ to go down and others caused it to go up.
One can also examine clustering coefficients in the weighted unipartite
networks given by the projected adjacency matrices $\mathbf{G}_{\mathcal{U}}$
and $\mathbf{G}_{\mathcal{M}}$. We calculate the weighted clustering
coefficient for each projection using the formula PhysRevE.71.065103
$\tilde{C}_{3}(m_{i})=\frac{2}{k_{i}(k_{i}-1)}\left[\frac{1}{G_{M}}\sum_{j,h}\left(G_{ij}G_{ih}G_{hj}\right)^{1/3}\right]\,,$
(6)
where $k_{i}$ is again the degree of node $m_{i}$, $G_{ij}$ is the weight of
the edge between $m_{i}$ and $m_{j}$, and $G_{M}=\max(G_{ij})$ denotes the
maximum edge weight in the network. The geometric mean
$(G_{ij}G_{ih}G_{hj})^{1/3}$ of the edge weights give the “intensity” of the
$(i,j,h)$-triangle. When the network is unweighted,
$(G_{ij}G_{ih}G_{hj})^{1/3}$ is $1$ if and only if all edges in the
$(i,j,h)$-triangle exist and $0$ if they do not, reducing the equation to the
unweighted unipartite clustering coefficient
$C_{3}(m_{i})=\frac{2t_{i}}{k_{i}(k_{i}-1)}\,,$ (7)
where $t_{i}$ is the number of triangles that include node $m_{i}$.
Figure 8: Weighted clustering coefficient $\tilde{C}_{3}(u_{i})$ for nodes in
the unipartite projection onto users for August 4, 2003. The $x$-axis
represents node degrees, and the $y$-axis represents $\tilde{C}_{3}(u_{i})$.
The mean values for this day are
$\langle\tilde{C}_{3}\rangle=\frac{1}{U}\sum_{i=1}^{U}\tilde{C}_{3}(u_{i})\approx
0.0013$ for the projection onto users and $\tilde{C}_{3}\approx 0.0086$ for
the projection onto videos (not shown). The inset shows values of the
unweighted coefficient $C_{3}(u_{i})$ from the same data.
$\langle C_{4}\rangle$ $\langle\tilde{C}_{3}\rangle$ mean var mean var Videos
0.02039 0.0007 0.0056 $10^{-6}$ Users 0.02092 0.0012 0.0044 $10^{-6}$
Table 2: Means and variances of $\langle C_{4}\rangle$ (for the bipartite
network) and $\langle\tilde{C}3\rangle$ (for the projections) of videos and
users on single-day snapshots of 2003, calculated using equations (5) and (6).
In Fig. 8, we show the $\tilde{C}_{3}(u_{i})$ values for the user projection
$\mathbf{G}_{\mathcal{U}}$ (with 10,228 nodes and 814,667 edges) from Tuesday,
August 4, 2003. In Table 2, we show the mean clustering-coefficient values for
the projected user and video networks for all single-day snapshots of 2003.
The values of $\langle\tilde{C}3\rangle$ did not vary much among weekdays,
except for the videos’ $\langle\tilde{C}3\rangle$ that almost doubled its
value on the weekends from an average of 0.0045 from Monday to Friday to
0.0086 on Saturday and Sunday.
Given the values of $\langle C_{4}\rangle$ in Table 2, it is unsurprising that
the values of $\langle\tilde{C}_{3}\rangle$ are also typically low. In the
inset of Fig. 8, we show the values of the users’ unweighted clustering
coefficient $C_{3}$, which are naturally much higher. For example, about 4000
users have $C_{3}=1.0$, indicating that all potential triangles exist among
these users. This differentiates one set of nodes from the rest. This feature,
which we observe often in the data, arises from the dominant video of the day.
For August 4, 2003, this video (which is typically a blockbuster) was
Daredevil, which had 396 ratings and created many edges in the user projection
among the users who rated it. Removing Daredevil from the bipartite network
also removes these deviant nodes. This feature is not apparent if one
calculates only the unweighted unipartite clustering coefficient ${C}_{3}$.
Just as we did with $\langle C_{4}\rangle$, and given the dramatic effect
observed by removing Daredevil, we calculated $\langle C_{4}\rangle$ for the
projected network of users removing the ten most rated videos. We found that
for every additional video removed, the value of $\langle C_{3}\rangle$
increased by 0.2%, while for $\langle\tilde{C}_{3}\rangle$ the increment was
slightly larger.
## III User Bursts
Figure 9: (Color online) Cumulative distribution of the inter-event time
between the ratings of one Netflix user. The user signed up on April 4, 2000,
and has a degree of 940 based on ratings cast over a period of almost five
years. The dashed curve indicates the fit to the function $F(x)\sim
x^{-\alpha}$, which yields $\alpha\approx 2.27$ in this case. The inset shows
the number of days between consecutive video ratings.
A close examination of the rating habits of individual users can also yield
rich and informative insights. Recent research has shown that people tend to
have bursts of e-mail and postal correspondence, in which they send and
receive numerous messages within short periods of time, followed by long
periods of inactivity barabasi-2005-435 ; oliveira2005437 ; burstbook . We
find similar features in the Netflix data, as about 70% of the users exhibit
bursty behavior by rating several videos in one go after several days of no
activity. We illustrate this phenomenon in Fig. 9 by plotting the cumulative
distribution of inter-event times between the ratings of one user over a
period of almost five years. We fit this distribution to a power law $F(x)\sim
x^{-\alpha}$ using the method discussed in Ref. Clauset:2007p5520 to
determine the value of the exponent $\alpha$. We can similarly provide
estimates for possible power laws (with actual power laws over roughly two
decades of data) among the other bursty users, though the value of $\alpha$
depends on the final degree (i.e., the total number of rated videos) of the
user. For example, bursty users with final degrees between 100 and 1000 have a
mean exponent of $\alpha\approx 2.54$, whereas those with final degrees of at
least 4000 have a mean exponent of $\alpha\approx 3.17$. Additionally, there
are several types of users among those who do not exhibit bursty dynamics. In
particular, some users rated only a very small number of videos (which may be
due to the sampling done by Netflix) and others exhibit seemingly unrealistic
levels of rating activity. (For example, there are 47 users who signed up in
January 2004 or later and who have rated more than 4000 videos each.)
## IV Catalog Networks
The above empirical investigation of the Netflix data motivates the
development of an evolution model for bipartite catalog networks, which arise
in a diverse set of applications. Such networks have two sets of nodes whose
numbers can be fixed or dynamic, and edges are placed one at a time between
previously unconnected edges that are chosen according to predefined rules.
One continues to add edges until a predefined final time has been reached or
the system has become saturated, at which point every node in one partite set
is connected to every node in the other partite set. The Netflix network can
be studied using such a catalog network framework; it starts completely
disconnected (nobody has rated any videos), and the users start choosing and
rating videos from the catalog. Depending on the way the data set is sampled,
the catalogs can be static (e.g. a one-day snapshot) or dynamic (e.g. the full
data set). Catalog models of network evolution are closely related to the
network rewiring problem studied by Plato and Evans evans:056101 ;
evans-2008-3 that features fixed sets of artifacts and individuals. Every
individual has one affiliation (a connection) with an artifact and can
reassign this connection to another node as the network evolves. In contrast,
in a catalog network, any edge that has been placed between two nodes in the
network is permanent. Consequently, catalog networks are suited to describing
records of interactions that are assigned dynamically and then remain
permanently in the system.
As before, $\mathcal{U}$ denotes the set of users and $\mathcal{M}$ denotes
the set of videos. The size of $\mathcal{U}$ is $u(r)$ and the size of
$\mathcal{M}$ is $m(r)$, where $r$ denotes a discrete time that is indexed by
the ratings. That is, we take every rating event as a time step, so when we
discuss time in this context, we are referring to “rating time” and not
physical time unless we indicate otherwise. Because $m(r)$ and $u(r)$ are not
always integers, we define $U(r)=\left\lfloor u(r)\right\rfloor$ and
$M(r)=\left\lfloor m(r)\right\rfloor$ as the (nonnegative integer) numbers of
user and video nodes, respectively. The associated time-dependent catalog
vectors, $D_{\mathcal{U}}$ and $D_{\mathcal{M}}$, have components given by the
degrees of each node in the catalog:
$D_{\mathcal{U}}(r)=\begin{bmatrix}k_{u_{1}}(r)\\\ k_{u_{2}}(r)\\\ \vdots\\\
k_{u_{U}(r)}(r)\end{bmatrix},\qquad
D_{\mathcal{M}}(r)=\begin{bmatrix}k_{m_{1}}(r)\\\ k_{m_{2}}(r)\\\ \vdots\\\
k_{m_{M}(r)}(r)\end{bmatrix}\,.$ (8)
These vectors have size $U(r)$ and $M(r)$, respectively. We denote by
$N_{\mathcal{U}}(r,k)$ (with $k\in\\{0,1,\ldots,M(r)\\}$) and
$N_{\mathcal{M}}(r,k)$ (with $k\in\\{0,1,\ldots,U(r)\\}$) the numbers of users
and videos, respectively, that have degree $k$ at rating time $r$. One can
normalize $N_{\mathcal{U}}(r,k)$ to obtain the proportion of nodes with degree
$k$ given by $\hat{N}_{\mathcal{U}}(r,k)=\frac{1}{U(r)}N_{\mathcal{U}}(r,k)$.
An analogous relation holds for $\hat{N}_{\mathcal{M}}(r,k)$.
Based on our intuition about the choosing and rating of videos, we add edges
to the network using a combination of linear preferential attachment and
uniform attachment. On one hand, one expects the choice of a user to be driven
in part by the choices made by others, as popular videos are more likely to
attract further viewings and hence ratings. On the other hand, one also
expects an element of idiosyncrasy on the part of each user, allowing him or
her to choose any video from the catalog regardless of the choices of others.
This results in two time-dependent probabilities—one for users and one for
videos—each of which consists of a convex combination of preferential and
uniform attachment. More specifically, each time an edge is added to the
network, we select a user and a video to be connected by this new edge. The
video (user) node is chosen using uniform attachment with probability $1-q$
(respectively, $1-p$) and linear preferential attachment with probability $q$
(respectively, $p$). The addition of an edge occurs during a single discrete
(rating) time step, as is common in models of network evolution. Combining
these ideas, a video node with degree $k_{i}$ is chosen with probability
$\displaystyle P_{\mathcal{M}}(r,k_{i})$
$\displaystyle=\frac{1-q}{M(r)-N_{\mathcal{M}}(r,U(r))}$
$\displaystyle\quad+\frac{qk_{i}}{\|D_{\mathcal{M}}(r)\|_{1}-U(r)N_{\mathcal{M}}(r,U(r))}\,,$
(9)
and a user node with degree $h_{i}$ is chosen with probability
$\displaystyle P_{\mathcal{U}}(r,h_{i})$
$\displaystyle=\frac{1-p}{U(r)-N_{\mathcal{U}}(r,M(r))}$
$\displaystyle\quad+\frac{ph_{i}}{\|D_{\mathcal{U}}(r)\|_{1}-M(r)N_{\mathcal{U}}(r,M(r))}\,,$
(10)
where the values of the parameters $p,q\in[0,1]$ are fixed,
$\|D_{\mathcal{U}}(r)\|_{1}=\sum_{i=1}^{U(r)}k_{i}(r)$, and
$\|D_{\mathcal{M}}(r)\|_{1}=\sum_{i=1}^{M(r)}h_{i}(r)$. The probabilities
$P_{\mathcal{U}}(r,h_{i})$ and $P_{\mathcal{M}}(r,k_{i})$ change over time as
the degrees of the nodes change when edges are added to the network.
The denominators in equations (9-10) contain the terms
$N_{\mathcal{M}}(r,U(r))$ and $N_{\mathcal{U}}(r,M(r))$ because once a node of
either type is fully connected, it is no longer eligible to receive any new
connections and is effectively no longer in the catalog until a new node of
the other type arrives. When $r=0$, one obtains
$\|D_{m}(0)\|_{1}=\|D_{u}(0)\|_{1}=0$ and
$N_{\mathcal{M}}(0,U(r))=N_{\mathcal{U}}(0,M(r))=0$, which would result in
division by zero. To overcome this problem, we follow the standard procedure
employed in network growth models albert-2002-74 by seeding the algorithm
with an edge that connects two randomly-chosen nodes (one from each of the
partite sets). This is equivalent to shifting the rating-time variable and
changing the initial conditions to $\|D_{m}(0)\|_{1}=\|D_{u}(0)\|_{1}=1$.
### IV.1 Rate Equations
One can use rate equations (i.e., master equations) to investigate the
dynamics of the degree distributions of a catalog network. This type of
approach has been used successfully to study a variety of other networks
PhysRevLett.85.4629 ; evans:056101 ; evans-2008-3 ; PhysRevLett.86.3200 ;
PhysRevE.71.036127 ; Newman:2003 . The analysis of the degree distribution for
videos in the catalog model is identical to the one for users, as only the
constants and sizes of the catalogs are different. Accordingly, we present our
results for the degree distributions of the videos; one obtains the results
for user distributions by changing $q$ to $p$, $M(r)$ to $U(r)$, and
$P_{\mathcal{M}}(r,k)$ to $P_{\mathcal{U}}(r,k)$. For notational convenience,
we also drop the subscripts in this subsection, so $N(r,k)$ denotes the number
of nodes with degree $k$ at time $r$. To construct the rate equations, one
must consider how many nodes pass through $N(r,k)$ (i.e. turn into nodes of
degree $k$ and $k+1$) for $k\in\\{0,1,2,\ldots,U(r)\\}$. This yields
$\displaystyle\frac{\mathrm{d}N(r,0)}{\mathrm{d}r}$
$\displaystyle=m^{\prime}(r)-P_{\mathcal{M}}(r,0)N(r,0)\,,$
$\displaystyle\frac{\mathrm{d}N(r,k)}{\mathrm{d}r}$
$\displaystyle=P_{\mathcal{M}}(r,k-1)N(r,k-1)$ (11) $\displaystyle-
P_{\mathcal{M}}(r,k)N(r,k)\,,\quad k>0,$
where $m^{\prime}(r)=\frac{\mathrm{d}m(r)}{\mathrm{d}r}$. The initial
conditions are
$\displaystyle N(0,0)$ $\displaystyle=M(0)-1\,,$ $\displaystyle N(0,1)$
$\displaystyle=1\,,$ (12) $\displaystyle N(0,k)$ $\displaystyle=0\,,\quad
k>1\,.$
Equation (11) is a system of coupled nonlinear ordinary differential equations
(ODEs). The positive and negative terms account, respectively, for an increase
and decrease in the number of nodes of a given degree as nodes receive new
edges. The equation for $N(r,0)$ has $m^{\prime}(r)$ as a positive term to
indicate the entry of new nodes (with degree $0$) to the network. The time-
dependent probabilities $P_{\mathcal{M}}(r,k)$ are defined in equation (9). In
the case of fixed catalogs, there is a maximum value of $k$, so the final
equation in (11) takes a slightly different form (see below).
#### IV.1.1 Fixed Catalogs
We begin by analyzing the evolution of the network with fixed catalog sizes,
so $U(r)=U$, $M(r)=M$, and $m^{\prime}(r)=0$ for all $r$. Because a finite,
fixed number of users and videos are available in the catalogs, the network
can only evolve until time $r=UM$. At this point, the system becomes saturated
(i.e., $N_{\mathcal{U}}(MU,M)=U$ and $N_{\mathcal{M}}(MU,U)=M$), and no
additional edges can be added to the network. Note additionally that the
equations in (11) change slightly for fixed catalogs. In particular, the last
equation for nodes with degree $U$ changes to
$\frac{\mathrm{d}N(r,U)}{\mathrm{d}r}=P_{\mathcal{M}}(r,U-1)N(r,U-1)\,,$ (13)
which only has one positive term because nodes with degree $U$ stay that way
until the end of the process.
Additionally, while the degree distribution of a network generated using the
catalog model with static node sets is time-dependent, the long-time
asymptotic behavior is always the same:
$\lim_{r\to UM}N(r,k)=\left\\{\begin{array}[]{lll}M\,,&\mathrm{if}&k=U\,,\\\
0\,,&\mathrm{if}&k<U\,,\end{array}\right.$
which gives a de facto final condition to the system in (11-13). Accordingly,
we examine degree distributions for $r\leq UM-1$.
Figure 10: (Color online) Degree distributions of video nodes averaged over
500 simulations of a fixed catalog network with $U=100$ users, $M=30$ videos,
and $q=0.8$ at rating times $r=500$ (red diamonds) and $r=1000$ (blue
squares). The solid curves are the solutions to the differential equation
(11). Figure 11: (Color online) Numbers of nodes $N(r,0)$ with degree $0$ (red
triangles) and $N(r,100)$ (blue circles) with degree $100$ from 500
simulations of a fixed catalog network with $U=100$, $M=30$, and $p=0.8$.
Inset: Decrease of $N(t,0)$ on a semi-logarithmic scale, which appears to
decrease exponentially. The solid curves come from the solutions of (11).
In Fig. 10, we show the degree distribution of the video nodes averaged over
500 simulations of a fixed catalog network with $U=100$ and $M=30$ at
different times during its evolution. As the discrete time $r$ increases, the
peaks of the functions travels towards higher values of $k$ and decrease as if
they were diffusing. We also observe a jump in $N(r,k)$ at $k=U$. This occurs
because there are nodes in the network that become fully connected during the
edge-assignment process (see Fig. 11). Interestingly, Johnson et al. showed
recently that the time-dependent degree distributions observed in some
networks that undergo edge rewiring with preferential attachment follow
nonlinear diffusion processes PhysRevE.79.050104 .
Figure 12: (Color online) Mean of $N(r,k)$ for user nodes in 500 simulations
of a fixed catalog network with $U=100$, $M=30$, and $p=0.5$. The axes are
(rating) time $r$ and degree $k$, and the color indicates the value of
$\log(N(r,k)+1)$. The horizontal line at the top of the image is the
discontinuity (as seen with the video nodes in Fig. 11) that corresponds to
the value of $N(r,M)$ and reflects the appearance of fully-connected user
nodes.
Figure 12 reveals how the user nodes achieve full connectivity between $r=0$
and $r=UM-1$. The image shows the “paths” that user nodes follow in the
$(r,k)$-plane between $(0,0)$ and $(UM-1,M)$. For example, the nodes that
follow a steep (high $k$ for early $r$) trajectory are the ones that receive
many links early on. Their degree grows mostly from preferential attachment in
the edge-assignment mechanism, and they accordingly achieve full connectivity
early in the process. The nodes that acquire edges more slowly initially begin
to receive edges very fast as $r$ approaches $UM$ (because other nodes have
already saturated), explaining the steep climb in the upper right corner of
the figure.
Figure 13: (Color online) Numerical solution of $N(r,k)$ for video nodes from
equation (11) with a fixed catalog and $q=0.8$, $M=30$, and $U=100$. (We again
plot $\log(N(r,k)+1)$.) The horizontal line at $k=100$ corresponds to the
saturated nodes $N(r,U)$. The inset shows a plot of $N(r,U-1)$ for the same
network.
The “final” condition that $N(UM-1,U)=M$ makes the system in (11) very stiff
for high values of $k$ and $r$. Fig. 13 shows the path that the video nodes
follow in the $(r,k)$-plane (i.e., the same information as in Fig. 12 but for
video nodes) but for the numerical solutions of (11) instead of direct network
simulations. In the inset of the Fig. , we show the profile of $N(r,U-1)$
which evinces the aforementioned stiffness. Because all nodes must be fully
connected at $r=UM-1$, nodes with low degrees begin to receive many edges for
high values of $r$. This causes $N(r,k)$ for high $k$ to peak late in the
process, and the nodes “travel” through values of $k$ rather quickly, which
explains the incredibly steep slope of $N(r,U-1)$ as $r$ approaches $UM-1$.
The value of $q$ affects the width of the region (light colored) in the
$(r,k)$ plane. For lower values of $q$ (e.g., $q=0.3$), uniform random
attachment dominates and the region of activity becomes narrower. The nodes
attain edges at roughly the same pace. For larger values of $q$, the first
nodes to receive edges become more likely to continue receiving more nodes
until they saturate, and the area of activity of the nodes becomes wider (see
Fig. 13).
#### IV.1.2 Growing Catalogs
In the previous section, we described the dynamics of catalog networks when
the sizes of the catalogs are fixed. While this provides a good baseline
investigation, catalogs can grow in many applications—for example, Netflix
gains both new subscribers and new videos almost every day. Accordingly, in
this section we study the dynamics of (11) for growing catalogs for which
$m^{\prime}(r)>0$.
Figure 14: (Color online) Numerical solution of $N(r,k)$ for video nodes from
equation (11) with $q=0.8$, $m(r)=30+0.007r$, and $U=100+0.05r$. (We again
plot $\log(N(r,k)+1)$.) The increasing diagonal line gives $U(r)$ and
represents the temporarily saturated nodes. In the inset, we show a plot of
$N(r,0)$ on a semilogarithmic scale. We observe a rapid initial decrease
followed by a slower increase as the catalog grows.
The system no longer has an obligatory final time, and the saturation level of
nodes is now time-dependent. For example, a user that has degree $M(r)$ is
saturated temporarily until a new video “arrives”—i.e., until time $r+\Delta
r$ so that $M(r+\Delta r)-M(r)>0$ and there is a new video to rate.
In Fig. 14, we show a numerical solution to equation (11) where $m(r)$ and
$u(r)$ are linear functions of $r$. Instead of the horizontal line of fully
connected nodes along $k=100$ in Fig. 13, the saturation of the nodes follows
the growth of $U(r)$. In the inset of Fig. 14, we show the time profile of
$N(r,0)$. Initially, it has what appears to be exponential descent before it
starts to grow slowly as the catalog size increases, in contrast to what we
observed in Fig. 11. The early rapid decay is explained by the absence of many
nodes with high degrees, so nodes with lower degrees receive edges. As $r$
increases, the better-connected nodes receive more edges (because for $q=0.8$
the dominant mechanism is linear preferential attachment) and the population
of nodes with fewer edges increases slowly. In Section V, we discuss how the
Netflix data displays some of these features.
## V Netflix as a Catalog Network
We now investigate how well our catalog model captures the human dynamics
revealed by the Netflix data. To do this, we sample the data set while keeping
in mind the following considerations:
* •
Because of the way we have defined our catalog network growth model, we must
consider the evolution of the Netflix data in “rating time”, in which every
new rating (which adds an edge in the network) constitutes a time step.
* •
Although there might be a (physical) time difference between a node (either
user or video) joining Netflix and the node receiving its first edge, this
information is not included in the data. Many videos receive more than one
rating on their first day, so their entry to the network is reflected by
increases in the value of $N(r,k)$ for several values of $k$. We will have to
take this into account when comparing our model to the data.
### V.1 Growth and Dynamics
To compare our results to the data, we express the growth of the numbers of
videos and users as a function of rating time $r$. Solving for $t$ in equation
(1) gives
$t=\frac{1}{b_{r}}\log{\left(\frac{r}{a_{r}}+1\right)}\,.$ (14)
We substitute (14) into (2) to obtain the new expressionfor the users as a
function of ratings:
$u(r)=a_{u}\left[\left(\frac{r}{a_{r}}+1\right)^{b_{u}/b_{r}}-1\right]\,.$
(15)
We follow the same procedure for the videos to obtain
$m(r)=a_{m}+b_{m}\left\\{\frac{1}{b_{r}}\log{\left(\frac{r}{a_{r}}+1\right)}\right\\}^{c_{m}}\,.$
(16)
Figure 15: (Color online) Users (top) and videos (bottom) as a function of
ratings. We use circles to show the data from Netflix and dashed curves to
show the predictions from equations (15) and (16). We use the parameter values
obtained in Sec. II.
In Fig. 15, we show the numbers of users and videos versus the number of
ratings in the network. Observe that the predictions from equations (15-16)
agree very well with the data.
Figure 16: (Color online) Video degree distribution $N_{\mathrm{data}}(r,k)$
in the Netflix data set in 2000. (We again plot
$\log(N_{\mathrm{data}}(r,k)+1)$.) We show data for videos with degrees
ranging from 1 to 4794.
Figure 16 shows the time-dependent degree distribution of videos in the
Netflix data set for the year 2000. The sample in the plot consists of 365
measurements (one for each day) of $r$ and $N(r,k)$. The highest degree in
this sample is 4794; this is well below the theoretical maximum of 9289
according to the expression for $u(r)$ in equation (15), so the network is not
experiencing node saturation. We can rewrite the probability that a video node
receives an edge as
$P_{\mathcal{M}}(r,k_{i})=\frac{1-q}{M(r)}+\frac{qk_{i}}{\|D_{\mathcal{M}}(r)\|_{1}}\,.$
The rate equation for the evolution of the degree distribution is
$\displaystyle\frac{\mathrm{d}N(r,1)}{\mathrm{d}r}=$
$\displaystyle\delta_{1}m^{\prime}(r)-P_{\mathcal{M}}(r,1)N(r,1)\,,$
$\displaystyle\frac{\mathrm{d}N(r,k)}{\mathrm{d}r}=$
$\displaystyle\delta_{k}m^{\prime}(r)+P_{\mathcal{M}}(r,k-1)N(r,k-1)$ (17)
$\displaystyle\quad-P_{\mathcal{M}}(r,k)N(r,k),\quad k>1.$
The initial conditions are $N(0,1)=m(0)$ and $N(0,k)=0$ for $k>1$. As noted
earlier, the lowest degree a node can have in the data is $1$ and the entry
degree of the nodes can have any value of $k$. We denote by $\delta_{k}$ the
proportion of new nodes whose entry degree is $k$, such that
$\sum_{k}\delta_{k}=1$. We investigated how many ratings do videos receive on
the day that they entered the system and found that over $97\%$ of the new
nodes receive $3$ or fewer ratings. Consequently, we have set
$\delta_{1}=0.8$, $\delta_{2}=0.15$, and $\delta_{3}=0.05$.
To see how well our model describes the Netflix video data in the year 2000,
we define $N_{k}(q)$ as the $4794\times 365$ matrix obtained solving the
system in (17) and $N_{\mathrm{data}}$ obtained from the data sample. These
two matrices contain the values of $N(r,k)$ from the sample and from the
equations for all values of $k$ and $r$. The matrices are of the given size
because we sample the degree distribution once per day and the maximum degree
observed is 4794. We define the error function
$E(q)=\|N_{k}(q)-N_{\mathrm{data}}\|\,,$ (18)
where $\|\cdot\|$ is the Euclidean matrix norm. To find the optimum value
$q^{*}$, we minimize $E(q)$ using the Nelder-Mead derivative-free simplex
method lagarias:112 . We found that the value of $q$ that minimizes (18) is
$q^{*}\approx 0.9795$, meaning that according to the model about $98\%$ of the
decisions to rate a video by users are guided by its popularity (i.e.,
preferential attachment).
Figure 17: (Color online) Values of $N(r,10)$ (videos with degree 10) obtained
by solving (17) using $q=0.9795$ (red curve) and the data from Netflix that we
report in Fig. 16 (blue dots). Figure 18: (Color online) Values of $N(r,50)$
(videos with degree 50) obtained by solving (17) using $q=0.9795$ (red curve)
and the data from Netflix that we report in Fig. 16 (blue dots).
In Figs. 17 and 18, we compare the values of $N(r,k)$ that we obtained in our
model to those in the data. In spite of the noise in the data, our model is
able to reproduce the temporal dynamics of $N(r,k)$.
Figure 19: (Color online) Cumulative degree distribution of video nodes on the
last day (915628 ratings) of the sample from year 2000. We obtained this by
solving (17) using $q=0.9795$ (red curve) and directly from the data (blue
dots).
In Fig. 19, we show the approximation of our model to the cumulative degree
distribution of the videos on the last day of the sample (i.e., for all values
of $k$ and $r=915628$, the number of ratings at the end of year 2000), which
agrees very well with the data.
Although $q^{*}\approx 0.9795$ suggests that the way the users choose to rate
videos is dominated by the popularity of the films, we should stress that the
model developed here is a very simple one. There are probably many other
processes influencing the decisions of the users, including different external
(to the user) factors, such as advertisements, press, and the underlying
social network the users are embedded in.
## VI Conclusions
We have analyzed a large network of video ratings given by the users of the
Netflix video rental service. We studied the system using a bipartite network
of videos and users and employed this perspective to reveal interesting
features in the dynamics of video rating, such as weekly patterns in video
ratings and bursts of activity followed by long idle periods. We calculated
clustering coefficients for one-day snapshots, concluding that their low
values arise from the presence of high-degree nodes (i.e., videos with a large
number of ratings and users who rate many videos). We also showed that the
degree distributions of both the user and video nodes resemble power laws with
exponential cutoffs.
Motivated by the structural and dynamical features we observed in the Netflix
data, we formulated a mechanism of network evolution in the form of “catalog
networks” for bipartite systems. Such networks are initially empty (aside from
a seed), and edges are created between two types of nodes based on some
predefined rules. New nodes can also be added to the network during the wiring
process. In our model, we considered a combination of uniform random
attachment and linear preferential attachment. We derived a set of coupled
ordinary differential equations that describe the time-evolution of the degree
distributions of such catalog networks. Presupposing this mechanism and
employing the Netflix data, we found that users seem to choose videos
according to preferential attachment about 98% of the time and uniform
attachment about 2% of the time. This suggests that the number of ratings for
a given video is driven almost completely by its popularity (preferential
attachment) and only in very small measure by the intrinsic preferences of
users. While interesting, the extreme dominance of a preferential-attachment
mechanism might be due in part to the simplicity of our model and the absence
of information about the underlying social network of the users, which can
have considerable influence over the video choices.Additionally, our model
does not incorporate external influences such as media coverage and promotion
campaigns that can certainly affect the popularity of videos. One can refine
such insights by considering more sophisticated attachment mechanisms that
incorporate the actual scores of the video ratings (not just their existence),
the age of the videos, user social networks (see Refs. Asur2010 and
Ratkiewicz2010 for recent interesting study), interactions among users, media
presence of videos, and more. Our simple catalog model thereby serves as a
good starting point for an abundance of interesting generalizations.
The Netflix data, which is both large and publicly available, provides an
excellent vehicle to study many of the features that have been observed in
network representations of systems in which agents exercise preferences or
choices, such as citation, collaboration, and social networks Price:1965 ;
albert-2002-74 ; Sornette.93.228701 ; Salganik02102006 ; Lambiotte2005 ;
oliveira2005437 . In this paper, we formulated a catalog model to understand
the human dynamics of video rating. In our view, catalog models are suitable
in many other contexts, including studying certain electoral systems (such as
the preamble to preferential voting elections) AustralianPolitics.com ,
professional sports drafts HockeyDraft , and retail shopping. To achieve
insights in such a diverse array of settings, the catalog model presented
herein can be generalized in numerous interesting ways to incorporate external
agents, underlying networks or cliques of individuals, and more.
## Acknowledgements
We thank M. Barahona, R. Desikan, T. Evans, P. Ingram, N. Jones, R. Lambiotte,
S. Lanning, D. Lazer, P. Mucha, D. Plato, S. Saavedra, D. Smith, and J. Stark
for useful comments and suggestions. We also acknowledge Netflix Inc. for
providing the data, which was released publicly as part of their prize
competition. This work was in part done as a dissertation for the MSc in
Mathematical Modelling and Scientific Computing at the University of Oxford.
MBD was supported by a Chevening Scholarship and a BBSRC–Microsoft Dorothy
Hodgkin Postgraduate Award. MAP acknowledges a research award (#220020177)
from the James S. McDonnell Foundation. JPO is supported by the Fulbright
Program.
## References
* (1) A.-L. Barabasi, Nature 435, 207 (2005).
* (2) T. S. Evans and A. D. K. Plato, Networks and Heterogeneous Media 3, 221 (2008).
* (3) J.-P. Onnela et al., Proceedings of the National Academy of Sciences 104, 7332 (2007).
* (4) R. Albert and A.-L. Barabási, Reviews of Modern Physics 74, 47 (2002).
* (5) M. E. J. Newman, SIAM Review 45, 167 (2003).
* (6) M. E. J. Newman, Proceedings of the National Academy of Sciences 98, 404 (2001)
* (7) G. Calderelli, Scale-Free Networks, Oxford University Press, 2007.
* (8) J. L. Gross and J. Yellen, editors, Handbook of Graph Theory, CRC Press, 2004.
* (9) A.-L. Barabási and R. Albert, Science 286, 509 (1999).
* (10) H. A. Simon, Biometrika 42, 425 (1955).
* (11) D. J. d. S. Price, Science 149, 510 (1965).
* (12) P. L. Krapivsky, S. Redner, and F. Leyvraz, Physical Review Letters 85, 4629 (2000).
* (13) M. Latapy, C. Magnien, and N. D. Vecchio, Social Networks 30, 31 (2008).
* (14) M. A. Porter, P. J. Mucha, M. E. J. Newman, and C. M. Warmbrand, Proceedings of the National Academy of Sciences 102, 7057 (2005).
* (15) S. Saavedra, F. Reed-Tsochas, and B. Uzzi, Nature 457, 463 (2009).
* (16) P. Zhang et al., Physica A 387, 6869 (2008).
* (17) J.-L. Guillaume and M. Latapy, Bipartite graphs as models of complex networks, in Aspects of Networking, pages 127–139, Springer, 2004.
* (18) J. Ohkubo, K. Tanaka, and T. Horiguchi, Physical Review E 72, 036120 (2005).
* (19) T. S. Evans and A. D. K. Plato, Physical Review E 75, 056101 (2007).
* (20) S. Saavedra et al., Physica A 389, 1131 (2010).
* (21) G. F. Davis, M. Yoo, and W. E. Baker, Strategic Organization 1, 301 (2003).
* (22) D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998).
* (23) S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, Nuclear Physics B 666, 396 (2003).
* (24) K. Park, Y.-C. Lai, and N. Ye, Physical Review E 72, 026131 (2005).
* (25) H. Fan, Z. Wang, T. Ohnishi, H. Saito, and K. Aihara, Physical Review E 78, 026103 (2008).
* (26) J. Lindquist, J. Ma, P. van den Driessche, and F. H. Willeboordse, Physica D 238, 370 (2009).
* (27) G. Polya, Annalea de 1’Institut Henri Poincaré 1, 117 (1931).
* (28) W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, John Wiley & Sons, Inc., 1957.
* (29) C. Godrèche and J. M. Luck, Journal of Physics: Condensed Matter 14, 1601 (2002).
* (30) D. Stauffer, X. Castelló, V. M. Eguíluz, and M. S. Miguel, Physica A 374, 835 (2007).
* (31) R. A. Bentley and S. J. Shennan, American Antiquity 68, 459 (2003).
* (32) Netflix-Prize, http://www.netflixprize.com.
* (33) J. Lorenz, European Physical Journal B 71, 251 (2009).
* (34) J. Bennett and S. Lanning, Proceedings of KDD Cup and Workshop 2007 (2007).
* (35) M. C. González, C. A. Hidalgo, and A.-L. Barabási, Nature 453, 779 (2008).
* (36) D. Sornette, F. Deschâtres, T. Gilbert, and Y. Ageon, Physical Review Letters 93, 228701 (2004).
* (37) J. G. Oliveira and A.-L. Barabási, Nature 437, 1251 (2005).
* (38) A.-L. Barabási, Bursts: The Hidden Pattern Behind Everything We Do, Dutton Adult, 2010.
* (39) A. Clauset, C. R. Shalizi, and M. E. J. Newman, SIAM Review 51, 661 (2009).
* (40) P. G. Lind, M. C. González, and H. J. Herrmann, Physical Review E 72, 056127 (2005).
* (41) J.-P. Onnela, J. Saramäki, J. Kertész, and K. Kaski, Physical Review E 71, 065103 (2005).
* (42) australianpolitics.com, History & features of the australian electoral system, 2008, Consulted on Feb, 10, 2009.
* (43) R. Pastor-Satorras and A. Vespignani, Physical Review Letters 86, 3200 (2001).
* (44) A. Barrat and R. Pastor-Satorras, Physical Review E 71, 036127 (2005).
* (45) S. Johnson, J. J. Torres, and J. Marro, Physical Review E 79, 050104 (2009).
* (46) J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, SIAM Journal on Optimization 9, 112 (1998).
* (47) S. Asur and B. A. Huberman, (2010), arXiv:1003.5699.
* (48) J. Ratkiewicz, F. Menczer, S. Fortunato, A. Flammini, and A. Vespignani, (2010), arXiv:1005.2704v1.
* (49) M. J. Salganik, P. S. Dodds, and D. J. Watts, Science 311, 854 (2006).
* (50) R. Lambiotte and M. Ausloos, Phys. Rev. E 72, 066107 (2005).
* (51) A. Summers, T. Swartz, and R. Lockhart, In Statistical Thinking in Sports, chapter 15. Optimal drafting in hockey pools, pages 249–262, Chapman & Hall/CRC, 2007.
|
arxiv-papers
| 2009-06-25T12:40:19 |
2024-09-04T02:49:03.526921
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mariano Beguerisse-Diaz, Mason A. Porter, Jukka-Pekka Onnela",
"submitter": "Mariano Beguerisse D\\'iaz",
"url": "https://arxiv.org/abs/0906.4675"
}
|
0906.4784
|
# The Orbital Evolution Induced by Baryonic Condensation in Triaxial Halos
Monica Valluri1111E-mail:[email protected] (MV); ; [email protected]
(VPD), Victor P. Debattista2, Thomas Quinn3, Ben Moore4
1 Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
2 RCUK Fellow; Jeremiah Horrocks Institute, University of Central Lancashire,
Preston, PR1 2HE, UK
3 Astronomy Department, University of Washington, Box 351580, Seattle, WA
98195-1580, USA
4 Department of Theoretical Physics, University of Zürich, Winterthurerstrasse
190, CH-8057, Zürich, Switzerland
(Accepted - Dec, 11, 2009)
###### Abstract
Using spectral methods, we analyse the orbital structure of prolate/triaxial
dark matter (DM) halos in $N$-body simulations in an effort to understand the
physical processes that drive the evolution of shapes of dark matter halos and
elliptical galaxies in which central masses are grown. A longstanding issue is
whether the change in the shapes of DM halos is the result of chaotic
scattering of the major family of box orbits that serves as the back-bone of a
triaxial system, or whether they change shape adiabatically in response to the
evolving galactic potential. We use the characteristic orbital frequencies to
classify orbits into major orbital families, to quantify orbital shapes, and
to identify resonant orbits and chaotic orbits. The use of a frequency-based
method for distinguishing between regular and chaotic $N$-body orbits
overcomes the limitations of Lyapunov exponents which are sensitive to
numerical discreteness effects. We show that regardless of the distribution of
the baryonic component, the shape of a DM halo changes primarily due to
changes in the shapes of individual orbits within a given family. Orbits with
small pericentric radii are more likely to change both their orbital type and
shape than orbits with large pericentric radii. Whether the evolution is
regular (and reversible) or chaotic (and irreversible), depends primarily on
the radial distribution of the baryonic component. The growth of an extended
baryonic component of any shape results in aregular and reversible change in
orbital populations and shapes, features that are not expected for chaotic
evolution. In contrast the growth of a massive and compact central component
results in chaotic scattering of a significant fraction of both box and long-
axis tube orbits, even those with pericenter distances much larger than the
size of the central component. Frequency maps show that the growth of a disk
causes a significant fraction of halo particles to become trapped by major
global orbital resonances. We find that despite the fact that shape of a DM
halo is always quite oblate following the growth of a central baryonic
component, a significant fraction of its orbit population has the
characteristics of its triaxial or prolate progenitor.
††pagerange: The Orbital Evolution Induced by Baryonic Condensation in
Triaxial Halos–References††pubyear: 2009
## 1 Introduction
The condensation of baryons to the centres of dark matter halos is known to
make them more spherical or axisymmetric (Debattista et al., 2008, hereafter
D08). D08 found that the halo shape changes by $\Delta(b/a)\ga 0.2$ out to at
least half the virial radius. This shape change reconciles the strongly
prolate-triaxial shapes found in collisionless $N$-body simulations of the
hierarchal growth of halos (Bardeen et al., 1986; Barnes & Efstathiou, 1987;
Frenk et al., 1988; Dubinski & Carlberg, 1991; Jing & Suto, 2002; Bailin &
Steinmetz, 2005; Allgood et al., 2006) with observations, which generally find
much rounder halos (Schweizer et al., 1983; Sackett & Sparke, 1990; Franx & de
Zeeuw, 1992; Huizinga & van Albada, 1992; Kuijken & Tremaine, 1994; Franx et
al., 1994; Buote & Canizares, 1994; Bartelmann et al., 1995; Kochanek, 1995;
Olling, 1995, 1996; Schoenmakers et al., 1997; Koopmans et al., 1998; Olling &
Merrifield, 2000; Andersen et al., 2001; Buote et al., 2002; Oguri et al.,
2003; Barnes & Sellwood, 2003; Debattista, 2003; Iodice et al., 2003; Diehl &
Statler, 2007; Banerjee & Jog, 2008).
What is the physical mechanism driving shape change? Options suggested in the
literature include two possibilities. The first is that the presence of a
central mass concentration scatters box orbits that serve as the backbone of a
triaxial potential, rendering them chaotic (Gerhard & Binney, 1985; Merritt &
Valluri, 1996). Chaotic orbits in a stationary potential do not conserve any
integrals of motion other than the energy $E$ and consequently are free to
uniformly fill their allowed equipotential surface. Since the potential is, in
general, rounder than the density distribution chaotic diffusion results in
evolution to a more oblate or even a spherical shape (Merritt & Quinlan, 1998;
Kalapotharakos, 2008). The second possibility is that the change of the
central potential occurs because the growth of the baryonic component causes
orbits of collisionless particles in the halo to respond by changing their
shapes in a regular (and therefore reversible) manner (Holley-Bockelmann et
al., 2002).
Time dependence in a potential is also believed to result in chaotic mixing
(Terzić & Kandrup, 2004; Kandrup & Novotny, 2004) and has been invoked as the
mechanism that drives violent relaxation. However, a more recent analysis of
mixing during a major merger showed that the rate and degree of mixing in
energy and angular momentum are not consistent with chaotic mixing, but rather
that particles retain strong memory of their initial energies and angular
momenta even in strongly time dependent potentials (Valluri et al., 2007).
One of the principal features of chaotic evolution is irreversibility. This
irreversibility arises from two properties of chaotic orbits. First, chaotic
orbits are exponentially sensitive to small changes in initial conditions even
in a collisionless system. Second, chaotic systems display the property of
chaotic mixing (Lichtenberg & Lieberman, 1992). Using this principle of
irreversibility, D08 argued that if chaotic evolution is the primary driver of
shape change, then if, subsequently, the central mass concentration is
artificially “evaporated”, the system would not be able to revert to its
original triaxial distribution. D08 showed that growing baryonic components
inside prolate/triaxial halos led to a large change in the shape of the halo.
Despite these large changes, by artificially evaporating the baryons, they
showed that the underlying halo phase space distribution is not grossly
altered unless the baryonic component is too massive or centrally
concentrated, or transfers significant angular momentum to the halo. This led
them to argue that chaotic evolution alone cannot explain the shape change
since such a process is irreversible. They speculated that at most only slowly
diffusive chaos occurred in their simulations. Using test particle orbit
integrations they also showed that box orbits largely become deformed,
possibly changing into tube orbits, during disk growth, but do not become
strongly chaotic.
D08 employed irreversibility as a convenient proxy for the presence of chaos.
In this paper we undertake an orbital analysis of some of the models studied
by D08 to better understand the mechanism that drives shape change. Our goal
is to understand whether chaotic orbits are an important driver of shape
change and if so under what conditions they are important. We also wish to
understand how the orbital populations in halos change when a centrally
concentrated baryonic component grows inside a triaxial dark matter halo.
Finally we would like to understand under what circumstances orbits change
their classification.
This paper is organised as follows. In § 2 we describe the simulations used in
this paper and briefly describe three models from D08 as well as two
additional simulations. In § 3 we describe the principal technique: Numerical
Analysis of Fundamental Frequencies (NAFF) that we use to obtain frequency
spectra and fundamental frequencies and describe how these frequencies are
used to characterise orbits. In § 4 we describe the results of our analysis of
five different simulations. In § 5 we summarise our results and discuss their
implications.
## 2 Numerical Simulations
Run Number | Run | Halo | $r_{200}$ | $M_{200}$ | $M_{b}$ | $f_{b}$ | $R_{b}$ | $t_{g}$ | $t_{e}$
---|---|---|---|---|---|---|---|---|---
(from D08) | Description | | [kpc] | [$10^{12}{{\mathrm{M}}_{\odot}}$] | [$10^{11}{{\mathrm{M}}_{\odot}}$] | | [kpc] | [Gyr] | [Gyr]
SA1 | Triax+Disk | A | 215 | 4.5 | 1.75 | 0.039 | 3.0 | 5 | 2.5
PlA3 | Triax+Bulg | A | 215 | 4.5 | 1.75 | 0.039 | 1.0 | 5 | 2.5
PfB2 | Prolt+Ellip | B | 106 | 0.65 | 0.7 | 0.108 | 3.0 | 10 | 4
PlA4 | Triax+hardpt | A | 215 | 4.5 | 1.75 | 0.039 | 0.1 | 5 | 2.5
PlB3 | Prolt+hardpt | B | 106 | 0.65 | 0.35 | 0.054 | 0.1 | 5 | 5
Table 1: The simulations in this paper. $M_{b}$ is the mass in baryons and
$f_{b}$ is the baryonic mass fraction. For the particle simulations (PfB2,
PlB3, PlA3, PlA4), $R_{\rm b}$ refers to the softening of the spherical
baryonic distribution particle(s). For simulation SA1, $R_{\rm b}$ refers to
the scale length of the baryonic disk.
We formed prolate/triaxial halos via mergers of systems, as described in Moore
et al. (2004). The initially spherical NFW (Navarro et al., 1996) halos were
generated from a distribution function using the method described in
Kazantzidis et al. (2004) with each halo composed of two mass species arranged
on shells. The outer shell has more massive particles than the inner one,
similar to the method described by Zemp et al. (2008), which allows for higher
mass resolution at small radii. Our model halo A was generated by the head-on
merger of two prolate halos, themselves the product of a binary merger of
spherical systems. The first merger placed the concentration $c=10$ halos 800
kpc apart approaching each other at 50 ${\mathrm{km\,s^{-1}}}$, while the
second merger starts with the remnant at rest, 400 kpc from an identical copy.
The resulting halo is highly prolate with a mild triaxiality. Halo model B was
produced by the merger of two spherical halos starting at rest, 800 kpc apart
and is prolate, with $\left<b/a\right>=\left<c/a\right>\simeq 0.58$. Halo A
has $\left<b/a\right>\simeq 0.45$ and $\left<c/a\right>\simeq 0.35$ while halo
B has $\left<b/a\right>=\left<c/a\right>\simeq 0.58$ (see Figure 3 of D08 for
more details). Both halos A and B consist of $4\times 10^{6}$ particles. The
outer particles are $\sim 18$ times more massive in halo A and $\sim 5$ times
more massive in halo B. A large part of the segregation by particle mass
persists after the mergers and the small radius regions are dominated by low
mass particles (cf. Dehnen, 2005). We used a softening parameter
$\epsilon=0.1$ kpc for all halo particles. The radius, $r_{200}$, at which the
halo density is 200 times the mean density of the Universe and the total mass
within this radius, $M_{200}$, are given in Table 1.
Once we produced the prolate/triaxial halos, we inserted a baryonic component,
either a disk of particles that remains rigid throughout the experiments or
softened point particles. The parameters that describe the distribution of the
baryonic components are given in Table1. In four of the models (PlA3, PlA4,
PfB2 and PlB3) the baryonic component is simply a softened point mass with
softening scale length given by $R_{b}$. In model SA1, the density
distribution of the disk was exponential with scale length of the baryonic
component $R_{\rm b}$ and Gaussian scale-height $z_{\rm b}/R_{\rm b}=0.05$.
The disk was placed with its symmetry axis along the triaxial halo’s short
axis in model SA1 (additional orientations of the disk relative to the
principal axes were also simulated but their discussion is deferred to a
future paper). Initially, the disk has negligible mass, but it grows
adiabatically and linearly with time to a mass $M_{b}$ during a time $t_{g}$.
After this time, we slowly evaporated it during a time $t_{e}$. We stress that
this evaporation is a numerical convenience for testing the effect of chaos on
the system, and should not be mistaken for a physical evolution. The disk is
composed of $300K$ equal-mass particles each with a softening $\epsilon=100$
pc. From $t=0$ to $t_{g}+t_{e}$ the halo particles are free to move and
achieve equilibrium with the baryons as their mass changes, but all disk
particles are frozen in place. The masses of models with single softened
particles are also grown in the same way; these are models PfB2, PlB3 from
D08, and PlA3 and PlA4 which are new to this paper. D08’s naming convention
for these experiments used “P” subscripted by “f” for particles frozen in
place and by “l” for live particles free to move.
Three different baryonic components are grown in the triaxial halo A: in model
SA1 the baryons are in the form of a disk grown perpendicular to the short
axis and the model is referred to as Triax+Disk; in model PlA3 the baryonic
component is a softened central point mass resembling a bulge and the model is
referred to as Triax+Bulg; finally in model PlA4 the baryonic component is a
hard central point mass with a softening of 0.1 kpc and the model is referred
to as Triax+hardpt. Two different baryonic components are grown in the prolate
halo B: in model PfB2 the baryonic component loosely resembles an elliptical
galaxy so this run is referred to as Prolt+Ellip; in model PlB3 the baryonic
component is a hard central point mass with a softening of 0.1 kpc and is
referred to as Prolt+hardpt. For model PfB2, D08 showed that there is no
significant difference in the evolution if the central particle is live
instead of frozen, all other things being equal. PlA3 was constructed
specifically for this paper in order to have a triaxial halo model with a
moderately soft spherical baryonic distribution which can be contrasted with
the prolate halo model PfB2, while PlA4 is a triaxial halo model which can be
contrasted with the prolate halo model PlB3.
For each model there were 5 phases in evolution. The initial triaxial or
prolate halo without the baryonic component is referred to as phase a. There
is then a phase (of duration $t_{g}$) during which the halo’s shape is
evolving as the baryonic component is grown adiabatically. We do not study
orbits in this phase since the potential is evolving with time. When the
baryonic component has finished growing to full strength, and the halo has
settled to a new equilibrium the model is referred to as being in phase b. The
baryonic component is “adiabatically evaporated” over a timescale of duration
$t_{e}$ listed in Table 1. Again we do not study orbits during this period
when the potential is evolving with time. After the baryonic component has
been adiabatically evaporated completely and the halo has returned to an
equilibrium configuration the halo is referred to as being in phase c. We only
study the orbits in the halo during the three phases when the halo is in
equilibrium and is stationary (not evolving with time).
The growth of the baryonic component induces several changes in the
distributions of the DM halo particles: first is an increase in central
density relative to the original NFW halo due to increase in the depth of the
central potential (an effect commonly referred to as “baryonic compression”);
second, the halos become more oblate especially within 0.3$r_{vir}$. The
details of the changes in the density and velocity distributions of DM
particles differ slightly depending on the nature of the baryonic component
(D08).
All the simulations in this paper, which are listed in Table 1, were evolved
with pkdgrav an efficient, multi-stepping, parallel tree code (Stadel, 2001).
We used cell opening angle for the tree code of $\theta=0.7$
throughout222Opening angle $\theta$ is used in tree codes to determine how
long-range forces from particles acting at a point are accumulated (Barnes &
Hut, 1986). . Additional details of the simulations can be found in D08.
### 2.1 Computing Orbits
In each of the halos studied we selected a subsample of between 1000-6000
particles and followed their orbits in each of the three stationary phases of
the evolution described in the previous section. The particles were randomly
chosen in the halos at $t=0$ such that they were inside a fixed outer radius
(either 100 or 200 kpc). Since the particles were selected at random from the
distribution function, they have the same overall distribution as the entire
distribution function within the outer radius selected. We integrated the
motion of each a test particle while holding all the other particles fixed in
place. We used a fixed timestep of 0.1 Myr and integrated for 50 Gyr, storing
the phase space coordinates of each test particle every 1 Myr. We used such
long integration times to ensure we are able to obtain accurate measurements
of frequencies (as described in the next section). We carried out this
operation for the same subset of particles at phases a, b and c. In model SA1
(Triax+Disk) we integrated the orbits of 6000 particles which in phase a were
within $r=200$ kpc. In model PlA3 (Triax+Bulg) and PlA4 (Triax+hardpt) we
considered a subsample of 5000 particles starting within $r=100$ kpc. In
models PfB2 and PlB3 we considered orbits of 1000 particles within $r=200$
kpc. We integrated their orbits as above but we used a smaller timestep
$\delta t=10^{4}$ years in the case of PlA4 and PlB3, which had harder central
point masses. The orbit code computes forces in a frozen potential using an
integration scheme that uses forces calculated from the PKDGRAV tree; we used
the orbit integration parameters identical to those used for the evolution of
the self-consistent models.
## 3 Frequency Analysis
In a 3-dimensional galactic potential that is close to integrable, all orbits
are quasi-periodic. If an orbit is quasi-periodic (or regular), then any of
its coordinates can be described explicitly as a series,
$\displaystyle x(t)=\sum_{k=1}^{\infty}A_{k}e^{i\omega_{k}t},$ (1)
where the $\omega_{k}$’s are the oscillation frequencies and the $A_{k}$’s are
the corresponding amplitudes. In a three dimensional potential, each
$\omega_{k}$ can be written as an integer linear combination of three
fundamental frequencies $\omega_{1},\omega_{2},\omega_{3}$ (one for each
degree of freedom). If each component of the motion of a particle in the
system (e.g. $x(t)$) is followed for several ($\sim 100$) dynamical times, a
Fourier transform of the trajectory yields a spectrum with discrete peaks. The
locations of the peaks in the spectrum correspond to the frequencies
$\omega_{k}$ and their amplitudes $A_{k}$ can be used to compute the linearly
independent fundamental frequencies (Boozer, 1982; Kuo-Petravic et al., 1983;
Binney & Tremaine, 2008).
Binney & Spergel (1982, 1984) applied this method to galactic potentials and
obtained the frequency spectra using a least squares technique to measure the
frequencies $\omega_{k}$. Laskar (1990, 1993) developed a significantly
improved numerical technique (Numerical Analysis of Fundamental Frequencies,
hereafter NAFF) to decompose a complex time series of the phase space
trajectory of an orbit of the form $x(t)+iv_{x}(t)$, (where $v_{x}$ is the
velocity along the $x$ coordinate). Valluri & Merritt (1998) developed their
own implementation of this algorithm that uses integer programming to obtain
the fundamental frequencies from the frequency spectrum. In this paper we use
this latter implementation of the NAFF method.
We refer readers to the above papers and to Section 3.7 of Binney & Tremaine
(2008) for a detailed discussion of the main idea behind the recovery of
fundamental frequencies. For completeness we provide a brief summary here. The
NAFF algorithm for frequency analysis allows one to quickly and accurately
compute the fundamental frequencies that characterise the quasi-periodic
motion of regular orbits. The entire phase space at a given energy can then be
represented by a frequency map which is a plot of ratios of the fundamental
frequencies of motion. A frequency map is one of the easiest ways to identify
families of orbits that correspond to resonances between the three degrees of
freedom.
The structure of phase space in 3-dimensional galactic potentials is quite
complex and we summarize some of its properties here to enable the reader to
more fully appreciate the results of the analysis that follows. When an
integrable potential is perturbed, its phase space structure is altered,
resulting in the appearance of resonances (Lichtenberg & Lieberman, 1992).
Resonances are regions of phase space where the three fundamental frequencies
are not linearly independent of each other, but two or more of them are
related to each other via integer linear relations. As the perturbation in the
potential increases, the potential deviates further and further from
integrability, and a larger and larger fraction of the phase space becomes
associated with resonances.
In a three dimensional potential, orbits that satisfy one resonance condition
such as $l\omega_{x}+m\omega_{y}+n\omega_{z}=0$ are referred to as “thin
orbit” resonances since they cover the surface of a two dimensional surface in
phase space (Merritt & Valluri, 1999). If two independent resonance conditions
between the fundamental frequencies exist, then the orbit is a closed periodic
orbit. Orbits that have frequencies close to the resonant orbit frequencies
are said to be resonantly trapped. Such orbits tend to have properties similar
to that of the parent resonance, but get “thicker” as their frequencies move
away from the resonance. At the boundary of the region of phase space occupied
by a resonant family is a region called the “separatrix”. The separatrix is
the boundary separating orbits with different orbital characteristics. In this
case it is the region between orbits that have frequencies that are similar to
the resonant orbits and orbits that are not resonant. Chaotic orbits often
occur in a “stochastic layer” close to resonances and at the intersections of
resonances. In fact one of the primary factors leading to an increase in the
fraction of chaotic orbits is the overlap of resonances (Chirikov, 1979).
Chaotic orbits that are close to a resonant family are referred to as
“resonantly trapped” or “sticky orbits” (Habib et al., 1997) and are often
only weakly chaotic. Orbits that are “sticky” behave like the resonant parent
orbit for extremely long times and therefore do not diffuse freely over their
energy surface or undergo significant chaotic mixing.
The frequency analysis method allows one to map the phase space structure of a
distribution function and to easily identify the most important resonances by
plotting ratios of pairs of frequencies (e.g. $\omega_{x}/\omega_{z}$ vs.
$\omega_{y}/\omega_{z}$) for many thousands of orbits in the potential. In
such a frequency map, resonances appear as straight lines. Stable resonances
appear as filled lines about which many points cluster, and unstable
resonances appear as “blank” or depopulated lines. The strength of the
resonances can be determined by the number of orbits that are associated with
them.
### 3.1 Overcoming microchaos in $N$-body simulations
In $N$-body systems like those considered in this paper, the galactic mass
distribution is realised as a discrete set of point masses. The discretization
of the potential is known to result in exponential deviation of nearby orbits,
even in systems where all orbits are expected to be regular (Miller, 1964;
Goodman et al., 1993; Kandrup & Smith, 1991; Valluri & Merritt, 2000;
Hemsendorf & Merritt, 2002). However as the number of particles in a
simulation is increased, and when point masses are softened, the majority of
orbits begin to appear regular despite the fact that their non-zero Lyapunov
exponent implies that they are chaotic. Hemsendorf & Merritt (2002) showed
that this Lyapunov exponent saturates at a finite value beyond a few hundred
particles and corresponds to an $e$-folding timescale of 1/20 of a system
crossing time (for systems with $N\sim 10^{5}$ particles). Despite having
large Lyapunov exponents (i.e. short e-folding times) these orbits behave and
look much like regular orbits (Kandrup & Sideris, 2001; Kandrup & Siopis,
2003). This property of $N$-body orbits to have non-zero Lyapunov exponents
has been referred to as “microchaos” (Kandrup & Sideris, 2003) or the “Miller
Instability” (Hemsendorf & Merritt, 2002; Valluri et al., 2007) and suggests
that Lyapunov exponents, while useful in continuous potentials, are not a good
measure of chaotic behavior resulting from the global potential when applied
to $N$-body systems. This is a strong motivation for our use of a frequency
based method which, as we demonstrate, is extremely effective at
distinguishing between regular and chaotic orbits and is apparently largely
unaffected by microchaos.
We now discuss how frequency analysis can be used to distinguish between
regular and chaotic orbits. In realistic galactic potentials most chaotic
orbits are expected to be weakly chaotic and lie close to regular orbits
mimicking their behaviour for long times. The rate at which weakly chaotic
orbits change their orbital frequencies can be used as a measure of chaos.
Laskar (1993) showed that the change in the fundamental frequencies over two
consecutive time intervals can be used as a measure of the stochasticity of an
orbit. This method has been used to study the phase space structure in
galactic potentials (Papaphilippou & Laskar, 1996, 1998; Valluri & Merritt,
1998). Examples of frequency spectra for each component of motion, and their
resolution into 3 fundamental frequencies are given by Papaphilippou & Laskar
(1998) for different types of orbits. For each time series the spectrum is
analysed and the three fundamental frequencies are obtained. In Cartesian
coordinates the frequencies would be $\omega_{x},\omega_{y},\omega_{z}$.
For each orbit we therefore divide the integration time of 50 Gyr time into
two consecutive segments and use NAFF to compute the fundamental frequencies
$\omega_{x},\omega_{y},\omega_{z}$ (note that all frequencies in this paper
are in units of Gyr-1, therefore units are not explicitly specified
everywhere). We compute the three fundamental frequencies
$\omega_{x}(t_{1}),\omega_{y}(t_{1}),\omega_{z}(t_{1})$ and
$\omega_{x}(t_{2}),\omega_{y}(t_{2}),\omega_{z}(t_{2})$ in each of the two
intervals $t_{1}$ and $t_{2}$ respectively. We compute the “frequency drift”
for each frequency component as:
$\displaystyle\log(\Delta
f_{x})=\log{|{\frac{\omega_{x}(t_{1})-\omega_{x}(t_{2})}{\omega_{x}(t_{1})}}|},$
(2) $\displaystyle\log(\Delta
f_{y})=\log{|{\frac{\omega_{y}(t_{1})-\omega_{y}(t_{2})}{\omega_{y}(t_{1})}}|},$
(3) $\displaystyle\log(\Delta
f_{z})=\log{|{\frac{\omega_{z}(t_{1})-\omega_{z}(t_{2})}{\omega_{z}(t_{1})}}|}.$
(4)
We define the frequency drift parameter $\log(\Delta f)$ (logarithm to base
10) to be the value associated with the largest of the three frequencies
$f_{x},f_{y},f_{z}$ . The larger the value of the frequency drift parameter,
the more chaotic the orbit.
Identifying truly chaotic behavior however also requires that we properly
account for numerical noise. In previous studies orbits were integrated with
high numerical precision for at least 100 orbital periods, resulting in highly
accurate frequency determination. For instance, Valluri & Merritt (1998) found
that orbital frequencies in a triaxial potential could be recovered with an
accuracy of $10^{-10}$ for regular orbits and $10^{-4}-10^{-6}$ for stochastic
orbits using integration times of at least 50 orbital periods per orbit.
In order to use frequency analysis to characterise orbits as regular or
chaotic in $N$-body systems, it is necessary to assess the numerical accuracy
of orbital frequencies obtained by the NAFF code. To quantify the magnitude of
frequency drift that arises purely from discretization effects (the microchaos
discussed above) we select a system that is spherically symmetric and in
dynamical equilibrium. All orbits in a smooth spherically symmetric potential
are rosettes confined to a single plane (Binney & Tremaine, 2008) and are
regular. Hence any drift in orbital frequencies can be attributed entirely to
discretization errors (including minute deviations of the $N$-body potential
from perfect sphericity). As a test of our application of the NAFF code to
$N$-body potentials we analyse orbits in spherical NFW halos of two different
concentrations ($c=10$ and $c=20$). The halos are represented by $10^{6}$
particles and have mass $\sim 2\times 10^{12}$ M⊙. Particles in both cases
come in two species with softening of 0.1 kpc and 0.5 kpc. We carried out the
frequency analysis of 1000 randomly selected orbits which were integrated for
50 Gyr in the frozen $N$-body realisations of each of the NFW halos.
Figure 1 shows the distribution of values of $\log(\Delta f)$ for both halos.
In both cases the distribution has a mean value of $\log(\Delta f)=-2.29$,
with standard deviations of 0.58 (for the $c=10$ halo) and 0.54 (for the
$c=20$ halo). Both distributions are significantly skewed toward small values
of $\log(\Delta f)$ (skewness = -0.85) and are more peaky than Gaussian
(kurtosis = 1.95). Despite the fact that the two NFW halos have different
concentrations, the distributions of $\log(\Delta f)$ are almost identical,
indicating that our chaotic measure is largely independent of the central
concentration.
Figure 1: Distributions of frequency drift parameter $\log(\Delta f)$ for 1000
orbits in two different spherical NFW halos. Despite the difference in
concentration $c=10$ and $c=20$ the distributions are almost identical having
a mean of $\log(\Delta f)=-2.29$ and a standard deviation $\sigma\simeq 0.56$,
with a significant skewness toward small values of $\log(\Delta f)$.
To define a threshold value of $\log(\Delta f)$ at which orbits are classified
as chaotic we note that 99.5% of the orbits have values of $\log(\Delta
f)<-1.0$. Since all orbits in a stationary spherical halo are expected to be
regular, we attribute all larger values of $\log(\Delta f)$ to numerical noise
arising from the discretization of the potential. Henceforth, we classify an
orbit in our $N$-body simulations to be regular if it has $\log(\Delta
f)<-1.0$.
Figure 2: $\log(\Delta f)$ versus $\omega$ (in units of Gyr-1) for orbits with
$n_{p}>20$ in the two spherical NFW halos. Stars are for the halo with $c=10$
and the open circles are for the halo with $c=20$. The 1000 particles are
binned in $\omega$ so that each bin contains the same number of particles. The
vertical error bars represent the standard deviation in each bin. The straight
lines are fits to the data, the slopes of both lines are consistent with zero.
To accurately measure the frequency of an orbit it is necessary to sample a
significant part of its phase-space structure (i.e. the surface of a 2-torus
in a spherical potential or the surface of the 3-torus in a triaxial
potential). Valluri & Merritt (1998) showed that the accuracy of the frequency
analysis decreases significantly when orbits were integrated for less than 20
oscillation periods. Inaccurate frequency determination could result in
misclassifying orbits as chaotic (since inaccurate frequency measurement can
also lead to larger frequency drifts). We test the dependence of $\log(\Delta
f)$ on the number of orbital periods $n_{p}$ by plotting the frequency drift
parameter against the largest orbital frequency (for orbits with $n_{p}>20$)
in both NFW halos in Figure 2333We use the fractional change in the largest of
the three fundamental frequencies measured over two contiguous time intervals
(frequency drift) as a measure of chaos (Laskar, 1990). For situations where a
large fraction of orbits is resonant, it may be more appropriate to use the
smallest of the three frequencies or the component with the largest
amplitude.. We use $\omega$ instead of $n_{p}$ since $n_{p}\propto\omega$ but
is harder to compute accurately. Particles are binned in equal intervals in
$\omega$ and the error bars represent the standard deviation in each bin. The
straight-lines are best fits to the data-points. The slopes of the correlation
for the $c=10$ halo (solid line) and for the $c=20$ halo (dot-dashed line) are
both consistent with zero, indicating that $\log(\Delta f)$ is largely
independent of $\omega$ (and hence of $n_{p}$). Henceforth we only use orbits
which execute more than 20 orbital periods in the 50 Gyr over which they are
integrated. The excluded orbits lie predominantly at large radii and are not
significantly influenced by the changes in the inner halo that are
investigated here. This rejection criterion affects about 25% of the orbits in
the triaxial dark matter halos that we consider later.
Figure 3: left: comparison of frequencies computed from the low and high time-
sampling runs $\omega_{H}$ versus $\omega_{L}$ respectively; middle:
comparison of diffusion parameter $\log(\Delta f)$ measured in low and high
sampling runs; right: histograms of diffusion parameter $\log(\Delta f)$.
The effect of central concentration on the accuracy of frequency estimation is
of particular concern during phase b, when the potential is deepened due to
the growth of a baryonic component. In this phase, frequencies of those orbits
which are strongly influenced by the deepened potential are increased.
Consequently some orbits execute many more orbital periods during phase b than
they do in phase a or phase c. However we have fixed the orbital sampling time
period (not integration timestep) to 1 Myr in all phases. In principle coarse
time sampling should not be a concern since the long integration time can
still ensure a proper coverage of the phase-space torus. To ensure that the
sampling frequency per orbital period does not significantly alter the
frequency estimation we re-simulated one model (SA1) in phase b and stored the
orbits 5 times more often (i.e. at time intervals of 0.2 Myr). We compared the
frequencies of orbits computed for the low ($\omega_{L}$) and high
($\omega_{H}$) time-sampling runs. Figure 3 (left) shows that there is a
strong correspondence between frequencies obtained with the two different
samplings. We also found (Fig. 3 middle) that the frequency drift parameter
$\log(\Delta f)$ obtained from the two runs are highly correlated although
there is some increase in scatter for orbits with values of $\log(\Delta
f)>-2.$ Since the scattered points lie roughly uniformly above and below the
1:1 correlation line, there is no evidence that the higher sampling rate gives
more accurate frequencies. The right panel shows that the overall distribution
of $\log(\Delta f)$ is identical for the two runs. We find that 95% of the
particles showed a frequency difference $<0.1$% between the two different
sampling rates. From these tests we conclude that our choice of sampling rate
in the phase b is unlikely to significantly affect the frequency measurements
of the majority of orbits. We therefore adopt the lower orbit sampling
frequency for all the analysis that follows.
### 3.2 Orbit classification
Carpintero & Aguilar (1998, hereafter CA98) showed that once a frequency
spectrum of an orbit is decomposed into its fundamental frequencies, the
relationships between the values of the frequencies
($\omega_{x},\omega_{y},\omega_{z}$) can be used to classify the orbits in a
triaxial potential into the major orbit families as boxes, long ($x$) axis
tubes and short ($z$) axis tubes. (CA98 point out that it is difficult to
distinguish between the inner long-axis tubes and outer long-axis tubes from
their frequencies alone. Therefore we do not attempt to distinguish between
these two families with our automatic classification scheme.) In addition to
classifying orbits into these three broad categories, they showed that if one
or more of the fundamental frequencies is an integer linear combination of the
other frequencies, the orbit can be shown to be resonant (either a periodic
orbit or an open resonance). We followed the scheme outlined by CA98 to
develop our own algorithm to classify orbits as boxes, long-axis tubes
(abbreviated as L-tubes) and short-axis-tubes (abbreviated as S-tubes) and to
also identify orbits that are associated with low-order resonances. We do not
describe the classification scheme here since it is essentially identical to
that described by CA98, the main difference lies in that we use NAFF to obtain
the fundamental frequencies of orbits in the $N$-body model, whereas they used
a method based on that of Binney & Spergel (1984). We tested our automated
classification by visually classifying 60 orbits that were randomly selected
from the different models. We then ran our automated orbit classifier on this
sample, and compared our visual classification with that resulting from the
automated classifier. The two methods agreed for 58/60 orbits (a 96% accuracy
rate assuming that the visual classification is perfectly accurate). Hereafter
we assume that our automated classification is accurate 96% of the time and
therefore any orbit fractions quoted have an error of $\pm 4$%.
### 3.3 Quantifying orbital shapes
In any self-consistent potential the distribution of shapes of the majority of
the orbits match the overall shape of the density profile. The elongation
along the major axis is provided either by box orbits or by inner L-tubes. The
ratios of the fundamental frequencies of orbits can be used to characterise
their overall shape. Consider a triaxial potential in which the semi-major
axis (along the $x$-axis) has a length $a_{x}$, the semi-intermediate axis has
length $a_{y}$, and the semi-minor axis has a length $a_{z}$. The fact that
$a_{x}>a_{y}>a_{z}$ implies that the oscillation frequencies along these axes
are $|\omega_{x}|<|\omega_{y}|<|\omega_{z}|$ for any (non resonant) orbit with
the same over-all shape as the density distribution (we consider only the
absolute values of the frequencies since their signs only signify the sense of
oscillation). We can use this property to define an average “orbit shape
parameter” ($\chi_{s}$) for any orbit. For an orbit whose overall shape
matches the shape of the potential,
$\displaystyle|\omega_{z}|>|\omega_{y}|>|\omega_{x}|$
$\displaystyle\Rightarrow$
$\displaystyle{\frac{|\omega_{y}|}{|\omega_{z}|}}>{\frac{|\omega_{x}|}{|\omega_{z}|}}$
$\displaystyle\chi_{s}$ $\displaystyle\equiv$
$\displaystyle{\frac{|\omega_{y}|}{|\omega_{z}|}}-{\frac{|\omega_{x}|}{|\omega_{z}|}}>0.$
(5)
The orbit shape parameter $\chi_{s}$ is positive for orbits with elongation
along the figure. The larger the value of $\chi_{s}$, the greater the degree
of elongation along the major axis. Very close to the centre of the potential
it is possible for orbits to have greater extent along the $y$ axis than along
the $x$ axis, as is sometimes the case with outer L-tubes. For such orbits
$\chi_{s}$ is slightly negative. An orbit for which all frequencies are almost
equal would enclose a volume that is almost spherical. For such an orbit,
$\chi_{s}\sim 0$ (which we refer to as “round”). Note that orbits which are
close to axisymmetric about the short ($z$) axis (i.e. the S-tubes) also have
$\chi_{s}\sim 0$ because $\omega_{x}\sim\omega_{y}$ regardless of the value of
$\omega_{z}$. Our definition of shape parameter does not permit us to
distinguish between truly round orbits for which
$\omega_{x}\sim\omega_{y}\sim\omega_{z}$ and S-tubes, but both contribute to a
more oblate axisymmetric potential.
## 4 Results
For every model in Table 1 the three fundamental frequencies
$\omega_{x},\omega_{y},\omega_{z}$ of each of the orbits in a selected
subsample were computed separately in each of the three phases. For each orbit
the largest of the three fundamental frequencies is assumed to represent the
dominant frequency of motion. The absolute value of this quantity is referred
to as the largest fundamental frequency: we use
$\omega_{a},\omega_{b},\omega_{c}$ to refer to the largest fundamental
frequencies of an orbit in each of the three phases a, b, c. In addition to
computing the fundamental frequencies over the entire 50 Gyr interval, we
split the interval into two equal halves and computed the frequencies in each
to compute the frequency drift parameter $\log(\Delta f)$ defined in §3. All
orbits with $\log(\Delta f)<-1.0$ are identified as regular and the rest are
identified as chaotic. For each orbit we also compute the total energy ($E$),
the absolute value of the total specific angular momentum ($|j_{\rm tot}|$),
the number of orbital periods ($n_{p}$), and the pericenter and apocenter
distance from the centre of the potential ($r_{\rm peri}$, $r_{\rm apo}$).
In this section we consider results of five simulations, SA1 (Triax+Disk),
PlA3 (Triax+Bulg), PlA4 (Triax+hardpt), PfB2 (Prolt+Ellip), and PlB3
(Prolt+hardpt). We shall show that halos A and B have very different initial
and final orbital properties despite the fact that their shapes in the
presence of baryonic components are very similar (D08).
### 4.1 Distributions of orbital frequencies
The frequency distribution of randomly selected orbits in a triaxial halo can
be used to characterise the orbital structure of phase space. It is useful to
begin by discussing our expectations for how orbital frequencies change in
response to growth of a central baryonic component. The potential is
significantly deeper in phase b compared to phase a, consequently the most
tightly bound orbits in the initial potential increase their frequencies. In
contrast orbits which largely lie outside the central mass concentration do
not experience much deformation or much change in their frequencies. The
higher the initial frequency the greater will be the frequency increase. Hence
we expect a faster-than-linear increase in frequency in phase b relative to
the frequency in phase a.
When the baryonic component is evaporated, the halo expands once more and the
halos regain their triaxiality in models SA1, PlA3 and PfB2 but are
irreversibly deformed in runs PlA4 and PlB3. One way to investigate the cause
of the difference in these behaviours is to look for correlations between
largest fundamental frequencies of each orbit in each of the three phases.
When the growth of the baryonic component causes an adiabatic change in
orbits, one expects that their frequencies $\omega_{b}$ will change in a
regular (i.e. monotonic) way so that the particles which are deepest in the
potential experience the greatest frequency increase. In Figures 4 and 5 we
plot correlations between the frequencies $\omega_{a}$, $\omega_{b}$ and
$\omega_{c}$.
In Figures 4 we show results for the three models whose baryonic scale length
is greater than 1 kpc: the left panels show that $\omega_{b}$ increases
faster-than-linearly with $\omega_{a}$, as expected with fairly small scatter.
The right hand panels show that $\omega_{c}$ is quite tightly correlated with
$\omega_{a}$ in all three models with the tightest correlation for simulation
SA1 (the dashed line shows the 1:1 correlation between the two frequencies).
The deviation from the dashed line and the scatter is only slightly larger in
simulations PlA3 and PfB2. The strong correlation between $\omega_{c}$ and
$\omega_{a}$ in these models supports the argument by D08 that the growth of
the baryonic component resulted in regular rather than chaotic evolution. In
all three models only a small fraction of points deviate from the dashed line
for the highest frequencies.
Figure 4: For the three models with extended baryonic components the left
panels show $\omega_{b}$ versus $\omega_{a}$ and right panels show
$\omega_{c}$ versus $\omega_{a}$ (frequencies in Gyr-1). Dashed lines in each
panel show the 1:1 correlation between each pair of frequencies. From top to
bottom the models contain a baryonic disk (SA1), a spherical bulge (PlA3) and
a spherical elliptical (PfB2).
Figure 5: Same as Fig. 4, but for the two models with hard point masses of 0.1
kpc softening. Top panels show the effect in the triaxial halo (dominated by
box orbits) and bottom panel shows the effect on the prolate halo (dominated
by L-tube orbits).
In contrast, the two models with a hard central point mass, PlB3 and PlA4,
(Fig. 5) show a non-monotonic change in $\omega_{b}$ in response to the growth
of the central point mass as well as a higher degree of scattering in
frequency space. In particular we note that orbits with small values of
$\omega_{a}$ (i.e. those which are most weakly bound and have large
apocenters) have the largest values of $\omega_{b}$, which is in striking
contrast to the situation in Figure 4. We also see that $\omega_{b}$ sometimes
decreased instead of increasing - again evidence for a scattering in frequency
space rather than an adiabatic change. There is also greater scatter in the
right-hand panels pointing to a less complete recovery in the frequencies
$\omega_{c}$ after the baryonic component is evaporated. Thus we see that when
the central point mass is hard and compact there is significant orbit
scattering.
It is clear from a comparison of Figures 4 and 5 that a baryonic component
with a scale length of $R_{b}\sim 1$ kpc or larger generally causes a regular
adiabatic change in the potential while a hard point mass ($R_{b}\sim 0.1$
kpc) can produce significant chaotic scattering.
In both Figures 4 and 5 we see in the right hand panels that
$\omega_{c}<\omega_{a}$ especially at large values of $\omega_{a}$ (i.e. all
points in the figures lie systematically below the line, indicating a decrease
in the frequency $\omega_{c}$). Thus particles must have gained some energy
implying that there has been a slight expansion in the DM distribution
following the evaporation of the baryonic component.
Figure 6: Kernel density distributions of $\Delta\omega_{ab}$ (left panel) and
of $\Delta\omega_{ac}$ (right panel) for particles in the halos SA1, PfB2,
PlA3, PlA4, and PlB3 and as indicated by line legends. Each curve normalised
to unit integral. The inset shows the full histogram for PlA4 plotted on a
different scale.
What fraction of orbits experience a large fractional change in frequency of
an orbit from phase a to phase b, and from phase a to phase c? To investigate
this we define:
$\displaystyle\Delta\omega_{ab}$ $\displaystyle=$
$\displaystyle|(\omega_{a}-\omega_{b})/\omega_{a}|$ (6)
$\displaystyle\Delta\omega_{ac}$ $\displaystyle=$
$\displaystyle|(\omega_{a}-\omega_{c})/\omega_{a}|.$ (7)
The first quantity is a measure of the change in frequency distribution of
orbits induced by the presence of the baryonic component, while the latter
quantity measures the irreversibility of the evolution following the
“evaporation of the baryonic component”. In Figure 6 we plot kernel density
histograms of the distribution of the frequency change $\Delta\omega_{ab}$
(left panel) and $\Delta\omega_{ac}$ (right panel) for orbits in all five
models as indicated by the line-legends. Each curve is normalized so that the
area under it is unity.
The distribution of $\Delta\omega_{ab}$ is much wider for models PlA3, PlA4
and PlB3 than for the other two models. In these three models the scale-length
of the baryonic component is $\leq 1$ kpc and results in a broad distribution
of $\Delta\omega_{ab}$, indicating that orbits over a wide range of
frequencies experience significant frequency change. For model PlA4 (dotted
line) the histogram of values of $\Delta\omega_{ab}$ appears almost flat on
the scale of this figure because it is spread out over a much larger range of
abscissa values indicating that many more particles are significantly
scattered in phase b. The full distribution for PlA4 (see inset panel) is
similar in form to PlA3 and PlB3.
The right panels show that only a small number of orbits in models SA1, PlA3
and PfB2 experience an irreversible frequency change $\Delta\omega_{ac}>20\%$,
with the majority of particles experiencing less than 10%. In contrast in
models PlA4 and PlB3, the distribution of $\Delta\omega_{ac}$ is much broader:
a significant fraction of particles have experienced a large (20-50%)
permanent change in their frequencies, reflecting the fact that the models
with a hard-compact point mass are the only ones which experience irreversible
chaotic scattering.
Figure 7: For four models small dots show $\Delta\omega_{ac}$ versus
$\omega_{a}$ (left) and versus $\omega_{b}$ (right) for all particles
analysed. Large solid dots with error bars show mean and standard deviation of
particles in 15 bins in frequency. Top two panels are for models with an
extended baryonic component (SA1, PfB2), lower two panels show models (PlA4,
PlB3) with a compact baryonic component.
Figure 8: For model PlA4 (top panels) and PlB3 (bottom panels) left:
$\Delta\omega_{ac}$ versus $r_{\rm peri}$; right: $\Delta\omega_{ac}$ versus
$|j_{\rm tot}|$.
Are there specific orbital characteristics that contribute to a large
permanent frequency change, $\Delta\omega_{ac}$, between the two triaxial
phases? We address this by determining how this quantity relates to other
orbital properties. In Figure 7 we plot $\Delta\omega_{ac}$ versus
$\omega_{a}$ (left panels) and versus $\omega_{b}$ (right panels) for four of
our models. In the top two panels (SA1, and PfB2 \- models with an extended
baryonic component) there is no evidence of a dependence of frequency change
on $\omega_{b}$ and only a slight increase in $\Delta\omega_{ac}$ at the
highest values of $\omega_{a}$ (results for PlA3 are not shown but are very
similar to those for SA1).
On the other hand, the lower two panels (PlA4 and PlB3 - models with a compact
hard baryonic component) show that there is a strong correlation between
$\Delta\omega_{ac}$ and orbital frequency $\omega_{a}$ indicating that the
orbits with the highest frequencies ($\omega_{a}$) experience the largest
frequency change, $\Delta\omega_{ac}$. This is evidence that scattering by the
hard central point mass is greatest for particles that are most tightly bound
and therefore closest to the central potential, confirming previous
expectations (Gerhard & Binney, 1985; Merritt & Valluri, 1996). The absence of
an appreciable correlation with $\omega_{b}$ is the consequence of scattering
of orbits in frequency.
In Figure 8 we plot $\Delta\omega_{ac}$ versus $r_{\rm peri}$ (left panels)
and versus $|j_{\rm tot}|$ (the total specific angular momentum of an orbit
averaged over its entire orbit in phase a) (right panels) for orbits in the
two models with compact central point mass (PlA4 and PlB3). (We do not show
plots for models SA1, PlA3 and PfB2, because they show no correlation between
$\Delta\omega_{ac}$ and either $|j_{\rm tot}|$ or $r_{\rm peri}$.) The left
panels of Figure 8 shows that orbits which pass closest to the central point
mass experience the most significant scattering. The absence of a correlation
with $|j_{\rm tot}|$ however indicates that scattering is independent of the
angular momentum of the orbit. In the next section we show that halo A (the
initial triaxial halo for model PlA4) is dominated by box orbits while halo B
is initially prolate, and is dominated by L-tubes which circulate about the
long axis. Contrary to the prevailing view that centrophilic box orbits are
more strongly scattered by a central point mass than centrophobic tube orbits,
these figures provide striking evidence that chaotic scattering is equally
strong for the centrophobic L-tubes that dominate model PlB3 as it is for the
centrophilic box orbits that dominate model PlA4. We return to a fuller
discussion of the cause of this scattering in § 5.
### 4.2 Changes in Orbital Classification
Table 2: Orbit composition of the models. The numbers represent the fraction of orbits in each family. Type | Run SA1 | Run PlA3 | Run PfB2 | Run PlA4 | Run PlB3
---|---|---|---|---|---
Phase | a | b | c | a | b | c | a | b | c | a | b | c | a | b | c
Boxes | 0.86 | 0.43 | 0.83 | 0.84 | 0.16 | 0.76 | 0.15 | 0.09 | 0.29 | 0.84 | 0.17 | 0.80 | 0.15 | 0.03 | 0.21
L-tubes | 0.11 | 0.09 | 0.12 | 0.12 | 0.43 | 0.15 | 0.78 | 0.75 | 0.54 | 0.12 | 0.35 | 0.11 | 0.78 | 0.78 | 0.59
S-Tubes | 0.02 | 0.27 | 0.03 | 0.02 | 0.33 | 0.06 | 0.07 | 0.09 | 0.16 | 0.02 | 0.26 | 0.04 | 0.07 | 0.11 | 0.14
Chaotic | 0.01 | 0.21 | 0.02 | 0.02 | 0.08 | 0.03 | 0.00 | 0.07 | 0.01 | 0.02 | 0.21 | 0.05 | 0.00 | 0.08 | 0.06
As we discussed in § 3.2, relationships between the fundamental frequencies of
a regular orbit can be used to classify it as a box orbit, a L-tube or a
S-tube orbit. Quantifying the orbital composition of the two different halos A
and B and how their compositions change in response to the growth of a
baryonic component yields further insight into the factors that lead to halo
shape change. Orbits were first classified as regular or chaotic based on
their drift parameter $\log(\Delta f)$ as described in § 3.1. Regular orbits
were then classified into each of three orbital families using the
classification scheme outlined in § 3.2. The results of this orbit
classification for each model, in each of the three phases, are given in Table
2.
The most striking difference between the initial triaxial models is that halo
A (phase a of models SA1, PlA3 and PlA4) is dominated by box orbits (84-86%)
while halo B (phase a of models PfB2 and PlB3) is dominated by L-tubes (78%).
(The small differences between models SA1, PlA3 and PlA4 in phase a is purely
a consequence of the selection of different subsets of orbits from halo A.)
None of the initial models has a significant fraction of S-tubes or of chaotic
orbits.
The very different orbit compositions of halos A and B in phase a results in
rather different evolutions of their orbital populations in response to the
growth of a central baryonic component. Although the growth of the disk
results in a significant decrease in the box orbit fraction (from 86% to 43%)
with boxes being converted to either S-tubes or becoming chaotic in phase b,
model SA1 is highly reversible suggesting an adiabatic change in the
potential. In model PlA3 and PlA4, the more compact spherical baryonic
components decrease the fraction of box orbits even more dramatically (from
84% down to 16-17%), pointing to the vulnerability of box orbits to
perturbation by a central component. Despite the similar changes in the
orbital populations of the two models, PlA3 is much more reversible than model
PlA4, indicating that both the shape of the central potential and its
compactness play a role in converting box orbits to other families and that
the change in orbit type is not evidence for chaotic scattering. It is
striking that the more compact point mass in model PlA4 results in
significantly more chaotic orbits (21%) compared to 8% in PlA3.
Figure 9: Distributions of $r_{\rm peri}$ for different orbit types.
Distributions of each of the four different orbital types as indicated by the
line-legends. Distribution in phase a is given by black curves and
distribution in phase b is shown by red curves. The integral under each curve
is proportional to the number of orbits of that orbital type.
While halo A is initial dominated by box orbits, halo B is initially dominated
by L-tubes, which dominate the orbit population in halo B in all three phases.
The growth of the baryonic component in phase b causes the box orbit fraction
to decrease (especially in model PlB3) while the fraction of chaotic orbits
increases slightly. The more extended point mass in PfB2 causes a larger
fraction of L-tubes to transform to orbits of another type than does the
harder point mass in PlB3, despite the fact that there is much greater
scattering in the latter model. A comparison between the model PlA4 and PlB3
show that their orbit populations in the presence of a baryonic component
differ significantly due to the different original orbit populations, while
their degree of irreversibility is identical (e.g. Fig. 6) since the point
mass in the two models is identical.
A significant fraction (21%) of the orbits in phase b of model SA1 and PlA4
are classified as chaotic (orbits with drift rate $\log(\Delta f)>-1.0$), in
comparison with 9%, 7% and 8% in models PlA3, PfB2 and PlB3 respectively.
While the presence of such a large fraction of chaotic orbits in phase b of
model PlA4 may be anticipated from previous work, the high fraction of chaotic
orbits in SA1 (Triax+Disk) is puzzling. To address concerns about
classification error that could arise from errors in the accuracy of our
frequency computation, we showed, in Figure 3, that changing the frequency at
which orbits were sampled by a factor of five did not result in any change in
the overall distribution of $\log(\Delta f)$, and hence should not affect our
classification of orbits as regular or chaotic. Another puzzling fact is that,
although model SA1 in phase b has such a significant fraction of chaotic
orbits, the orbit fractions essentially revert almost exactly to their
original ratios once the disk is evaporated in phase c. Hence, the large
fraction of chaotic orbits in phase b do not appear to cause much chaotic
mixing. We will return to a more complete investigation of this issue in §
4.4.
In Figure 9 we investigate how orbits of different types (boxes, L-tubes,
S-tubes, chaotic) are distributed with $r_{\rm peri}$, and how this
distribution changes from phase a (black curves) to phase b (red curves). In
phase a the initially triaxial halo A models (black curves) are dominated by
box orbits. The fraction of box orbits is significantly decreased in phase b.
In particular box orbits with large $r_{\rm peri}$ are transformed equally to
short axis tubes and chaotic orbits, while some box orbits at small $r_{\rm
peri}$ are converted to long-axis tubes. In contrast halo B models are
dominated by L-tubes in both phases. Rather striking is how little the
fraction of L-tubes in the halo B models changes, despite the fact that the
halos are significantly more oblate axisymmetric in phase b than in phase a.
We saw in Figure 6 that a significant fraction of orbits experience strong
scattering that manifests as a change in their orbital frequencies, and in
Figure 8 we noted that the orbits with the smallest pericenter radii
experience the largest change in frequency. In both models PlA4 and PlB3 the
compact central point mass significantly reduced the box orbits. However model
PlB3 only has a small fraction (15%) of box orbits and it seems unlikely that
the chaotic scattering of this small fraction of orbits off the central point
mass is entirely responsible for driving the evolution of halo shape. Also
PfB2 which has a much more extended baryonic component shows a change in orbit
population which closely parallels PlB3 and we saw that PfB2 is quite
reversible and shows little evidence for chaotic scattering. It is clear (from
Fig. 9) that in the prolate models (halo B) the majority of the orbits are
L-tubes with large pericenter radii ($\left<r_{\rm peri}\right>\sim 3$ kpc)
and these remain L-tubes in phase b. How then do these prolate models evolve
to more spherical models while retaining their dominant orbit populations? To
address this question we will now investigate the distribution of orbital
shapes in each model at each phase of the evolution.
### 4.3 Changes in orbital shape
Figure 10: Kernel density histograms of the distribution of orbital shape
parameter $\chi_{s}$ for each of the four models: SA1 (top left), PlA4 bottom
left , PfB2 (top right ) and PlB3 (bottom right) (PlA3 is not shown since it
is very similar to PlA4.) Distributions of $\chi_{s}$ in phase a are shown by
solid curves, in phase b by dot-dashed curves, and in phase c by dashed
curves. In all models, a large fraction of orbits in phase b are “round”
($\chi_{s}\simeq 0.$.)
A parameter, $\chi_{s}$, to quantify the shape of an orbit was defined in
Equation 3 of § 3.3. Recall that this quantity is positive when the orbit is
elongated along the major axis of the triaxial figure, is negative when
elongated along the intermediate axis, and almost zero when the orbit is
“round” ($\omega_{x}\sim\omega_{y}\sim\omega_{z}$) or roughly axisymmetric
about the minor axis ($\omega_{x}\sim\omega_{y}$). In Figure 10 we show the
shape distributions for the orbits in four of our five models. For each model
we show kernel density histograms for models in phase a (solid curves), phase
b (dot-dashed curve), and phase c (dashed curves). In each plot the curves are
normalized such that the integral under each curve is unity. We define orbits
to be elongated if $\chi_{s}\ga 0.25$, and to be “round” if $|\chi_{s}|\leq
0.1$.
Before the growth of the baryonic component (phase a: solid curves) the halo A
models (left panels SA1, PlA4) have a distribution of orbital shapes that has
a large peak at $\chi_{s}\sim 0.35$, arising from elongated orbits and a very
small peak at $\chi_{s}\sim 0$ due to round orbits (model PlA3 is not shown
but is similar to PlA4.) In halo B models, (right panels PfB2, PlB3) on the
other hand, the distribution of shapes is double peaked with about one third
of all orbits contributing to the peak at $\chi_{s}\sim 0$. This implies one
third of its orbits in the initially prolate halo B are “round”. In both halo
A and B however, the larger of the two peaks has a value of $\chi_{s}\sim
0.35$ corresponding to quite elongated orbits. Despite the quite different
underlying orbital distributions (halo A models dominated by box orbits while
the halo B models are dominated by L-tubes). This illustrates that despite
having different orbital compositions, a significant fraction of their orbits
are similarly elongated.
The dot-dashed curves in all the panels show the distribution of orbital
shapes in phase b. In all four models there is a dramatic increase in the peak
at $|\chi_{s}|\sim 0$, pointing to a large increase in the fraction of round
(or S-tubes) at the expense of the elongated (L-tube or box) orbits. In the
halo B models the elongated orbits are significantly diminished indicating
that the elongated L-tubes in phase a are easily deformed to “round” orbits in
phase b (most likely squat inner-L-tubes). However, in model SA1 there is a
large fraction of orbits with intermediate values of elongation
$0.1\leq\chi_{s}\leq 0.4$.
In phase c (dashed curves) all models show the dominant peak shifting back to
quite high elongation values of $\chi_{s}\sim 0.3$ (although this is slightly
lower than $\chi_{s}\sim 0.35$ in phase a). The downward shift in the peak is
most evident in model PlB3 (Prolt+hardpt), which as we saw before, exhibits
the greatest irreversibility in shape. The scattering of a large fraction of
the orbits by the hard central potential in model PlB3 seen in Figures 7 and 8
is the major factor limiting reversibility of the potential. The smallest
shift is for model SA1 (Triax+Disk), which exhibited the greatest
reversibility.
We can also investigate how the shapes of orbits vary with pericentric radius.
We expect that orbits closer to the central potential should become rounder
($\chi_{s}\rightarrow 0$) than orbits further out. We see that this
expectation is borne out in Figure 11 where we plot orbital shape parameter
$\chi_{s}$ versus $r_{\rm peri}$ in both phase a (left hand plots) and phase b
(right hand plots). In each plot the dots show values for individual orbits.
The solid curves show the mean of the distribution of points in each of 15
bins in $r_{\rm peri}$. Curves are only plotted if there are more than 30
particles in a particular orbital family (PlA4 is not shown since it is
similar to PlA3). For models SA1 (Triax+Disk) and PlA3 (Triax+Bulg) the figure
confirms that elongated orbits in the initial halo A were box orbits (black
dots and curves) and L-tubes (red dots and curves). The S-tubes (blue dots and
curves) are primarily responsible for the “round” population at $\chi_{s}\sim
0$. In phase b (right-hand panels) of both SA1 and PlA3 there is a clear
tendency for the elongated orbits (boxes, L-tubes and chaotic) to become
rounder at small pericenter distances, but they continue to be somewhat
elongated at intermediate to large radii. Chaotic orbits in phase b of model
SA1 appear to span the full range of pericentric radii and are not confined to
small radii. (Note that the density of dots of a given colour is indicative of
the number of orbits of a given type but the relative fractions are better
judged from Fig. 9 and Table 2.)
For phase a in the models PfB2 (Prolt+Ellip) and PlB3 (Prolt+hardpt) (left
panels of each plot), boxes and L-tubes are elongated ($\chi_{s}\geq 0.25$),
except at large $r_{\rm peri}\geq 8$ kpc where they become rounder. We see a
trend for the average orbital shape (as indicated by the curves) in phase b to
become round at small pericenter radii.
Note that in all the plots the curves only show the average shape of orbits of
a given type at any radius. The points show that in the case of the L-tubes in
particular, the red dots tend to be distributed in two “clouds”: one with
large elongations $\chi_{s}>0.3$ and one with small elongation $\chi_{s}\sim
0.1$.
Thus in all four models it is clear that orbits that are elongated along the
major axis of the triaxial potential in phase a become preferentially rounder
at small pericenter radii in phase b. It is this change in orbital shape that
plays the most significant role in causing the overall change in the shape of
the density in the baryonic phase444Due to our chosen definition of shape
parameter, S-tubes generally have $\chi_{s}\sim 0$ regardless of radius,
because $\omega_{x}\sim\omega_{y}$..
Figure 11: For models SA1, PlA3, PfB2 and PlB3, the orbital shape parameter
$\chi_{s}$ for each orbit is plotted against its pericentric radius $r_{\rm
peri}$ as a small dot. The orbits of each of the four major orbital families
are colour coded as in the figure legends. Left hand panels are for phase a
and right hand panels are for phase b. The solid curves show the mean value of
$\chi_{s}$ for all particles of that particular family, in 15 bins in $r_{\rm
peri}$. Curves are not plotted if there are fewer than 30 orbits in a given
orbital family. We used a kernel regression algorithm to smooth the curves.
### 4.4 Frequency maps and chaotic orbits
We saw in Table 2 that phase b of model SA1 (Triax+Disk) and of model PlA4
(Triax+hardpt) have a significant fraction (21%) of chaotic orbits (i.e.
orbits with $\log(\Delta f)>-1$). While PlA4 shows significant lack of
reversibility, which we can attribute to the presence of this high fraction of
chaotic orbits, model SA1 does not show evidence for irreversibility.
Figure 12 shows kernel density histograms of the chaotic drift parameter
$\log(\Delta f)$ for orbits in each of the three phases in model SA1. It is
obvious that in phases a and c there is only a small fraction of chaotic
orbits (i.e. orbits with $\log(\Delta f)>-1$), whereas a much more significant
fraction of orbits lie to the right of this value in phase b. Even the peak of
the distribution in phase b is quite significantly shifted to higher drift
values.
In this section we investigate the surprising evidence that the chaotic orbits
in phase b of model SA1 do not appear to mix. One possible reason for the lack
of diffusion of the chaotic orbits is that the timescale for evolution is not
long enough. Indeed D08 report that evolving run SA1 with the disk at full
mass for an additional 5 Gyr after the growth of the disk is complete, leads
to a larger irreversible evolution (see their Figure 3a). Nonetheless, even in
that case the irreversible evolution was only marginally larger than when the
disk was evaporated right after it grew to full mass. Moreover the growth time
was 5 Gyr which means that the halo was exposed to a massive disk for a
cosmologically long time.
A second possible reason for the lack of chaotic diffusion is that most of the
chaotic orbits in this phase of the simulation are “sticky”. The properties of
“sticky chaotic orbits” were discussed in § 3. In a series of experiments
designed to measure the rate of chaotic mixing, Merritt & Valluri (1996)
showed that while ensembles of strongly chaotic orbits diffused and filled an
equipotential surface on timescales between 30-100 dynamical times, similar
ensembles of “sticky” or resonantly trapped orbits diffused much less quickly
and only filled a small fraction of the allowed surface after very long times.
Figure 12: Histograms of frequency drift parameter $\log(\Delta f)$ for the
three phases of model SA1 as indicated by the line-legends.
Laskar (1990) showed that frequency maps are a powerful way to identify
resonances in dynamical systems. Frequency maps are obtained by plotting
ratios of the 3 fundamental frequencies for each individual orbit. If a large
and representative orbit population is selected, they can provide a map of the
phase space structure of the potential including all the resonances.
Resonances appear as straight lines on the frequency map since their
fundamental frequencies satisfy a condition like
$l\omega_{x}+m\omega_{y}+n\omega_{z}=0$. This method of mapping the phase
space has the advantage that since it only depends on the ratios of the
frequencies and not on the frequencies themselves, it can be used to map phase
space for large ensembles of particles without requiring them to be iso-
energetic. This is a significant advantage over mapping schemes like Poincaré
surfaces-of-section, when applied to an $N$-body simulation where particles,
by design, are initialised to be smoothly distributed in energy. Thus one can
use the method to identify global resonances spanning a large range of orbital
energies in $N$-body simulations.
Figure 13: Frequency maps of particles in phase a and phase b for four models.
For each particle the ratio of the fundamental frequencies
$\omega_{y}/\omega_{z}$ is plotted versus $\omega_{x}/\omega_{z}$ is plotted
by a single dot. The dots are colour coded by the energy of the particle in
phase a. The most tightly bound particles are coloured blue, and the least
bound particles are coloured red. Model SA1 has 6000 particles, model PlA4 and
PlA3 have 5000 particles, while PlB3 has 1000 particles each.
In Figure 13 we present frequency maps for four of the five models in phase a
(left panels) and phase b (right panels) (PfB2 is not shown since the
frequency maps for this model are indistinguishable from those for PlB3). For
each orbit the ratios of the fundamental frequencies $\omega_{y}/\omega_{z}$
and $\omega_{x}/\omega_{z}$ are plotted against each other. Particles are
colour coded by their energy in phase a. The energy range in phase a was
divided into three broad energy bins, with equal numbers of particles per bin.
The most tightly bound particles are coloured blue, the least bound particles
are coloured red and the intermediate energy range is coloured green.
Resonance lines are seen in the clustering of particles in all the maps. The
most striking of the frequency maps is that for phase b of model SA1
(Triax+Disk). This map has significantly more prominent resonance lines,
around which many points cluster, than any of the other maps. Three strong
resonances and several weak resonances are clearly seen as prominent straight
lines. The horizontal line at $\omega_{y}/\omega_{z}=1$ corresponds to the
family of orbits associated with the 1:1 closed (planar) orbit that circulates
around the $x$-axis, namely the family of “thin shell” L-tubes. The diagonal
line running from the bottom left corner to the top right corner with a slope
of unity ($\omega_{y}/\omega_{z}=\omega_{x}/\omega_{z}$) corresponds to the
family of orbits that circulates about the $z$-axis: the family engendered by
the “thin shell” S-tubes. Since this latter family shares the symmetry axis of
the disk, it is significantly strengthened in phase b by the growth of the
disk. In addition to having many more orbits associated with it, this
resonance extends over a much wider range in energy as evidenced by the color
segregation along the resonance line (blue points to the bottom left and red
points at the top right). This segregation is the result of an increase in the
gradient of the potential along the $z$-axis due to the growth of the disk,
which results in an increase in $\omega_{z}$. The more tightly bound a
particle, the greater the increase in $\omega_{z}$, and the greater the
decrease in both its ordinate and abscissa. The most bound particles (blue
points) therefore move away from their original positions towards the bottom
left hand corner of the plot. The least bound particles (red points) are
furthest from the center of the potential and these points experience the
least displacement - although these points also shift slightly toward the
resonance lines.
Figure 14: Several chaotic orbits in phase b of model SA1. Orbits are plotted
in two Cartesian projections over two different time segments of 10 Gyr (top
two plots in each panel show the first time segment, while bottom panels show
the second time segment). See text for details.
A third prominent resonance is the vertical line at
$\omega_{x}/\omega_{z}=0.5$ that corresponds to orbits associated with the
family of banana (1:2 resonant) orbits. This banana (boxlet) resonance is also
enhanced by the growth of the disk since this family of orbits, while not
axisymmetric, is characterised by large excursions along the $x$-axis and
smaller excursions in the $z$ direction. Several shorter resonance lines are
seen but are too sparsely populated in this plot to properly identify.
The frequency map for model PlA3 in phase b shows that a spherical baryonic
component produces a rather different phase space structure than that produced
by the disk. In particular it is striking that the most tightly bound (blue)
points are now clustered at the intersection of the horizontal and diagonal
resonances namely around the closed period orbits 1:1:1. This may be
understood as the consequence of the growth of the spherical baryonic point
mass around which all orbits are rosettes and since no direction is preferred
all orbits are “round”. The 1:2 banana resonance is also less prominent in
this model (largely because the deep central potential destabilises this
boxlet family).
The frequency map for model PlA4 phase b shows the greatest degree of
scattering, as evidenced by the thickest resonance lines. We attribute this to
the large number of chaotic orbits in this model. Apart from the broad
clustering of points around the diagonal (S-tube) and horizontal (L-tube)
resonances there are no strong resonance lines seen in this map. Unlike the
map for PlA3 which shows a clustering of tightly bound (blue) points at the
1:1:1 periodic orbit resonances, the blue points are widely scattered in the
frequency map of PlA4.
The frequency maps for model PlB3 shows that most of the orbits in this model
are associated with the (1:1) L-tube family (horizontal line). A smaller
number of orbits is associated with the 1:1 S-tube resonance (diagonal line).
We saw previously that the growth of the baryonic components in this prolate
halo caused little change in the orbit families. This is confirmed by the fact
that the frequency maps in both phases are remarkably similar except for an
increase in the clustering of points at the intersection of the horizontal
(L-tube) and the diagonal (S-tube) resonance, which occurs for the same reason
as in PlA3. Since halo B is initially a highly prolate model, it has (as we
saw previously) only a small fraction of box (and boxlet) orbits and in
particular no banana orbits.
It is quite striking that in phase a the frequency maps show significantly
less segregation by energy, and only a few resonances. This is because the
initial triaxial models were generated out of mergers of spherical NFW halos
which were initially constructed so that orbits were smoothly distributed in
phase space. The increase in the number of resonances following the growth of
a baryonic component is one of the anticipated consequences of resonant
trapping that occurs during the adiabatic change in a potential (e.g. Tremaine
& Yu, 2000; Binney & Tremaine, 2008).
To test the conjecture that the majority of chaotic orbits in model SA1 (phase
b) are resonantly trapped, we compute the number of chaotic orbits that lie
close to a major resonance line. We define “closeness” to the resonance by
identifying those orbits whose frequency ratios lie $\pm\alpha$ of the
resonant frequency ratio. For example we consider an orbit to be close to the
(1:1) L-tube resonance (horizontal line in map), if
$|\omega_{y}/\omega_{z}-1|\leq\alpha$. We find that the fraction of chaotic
orbits in phase b, that lie close to one of the three major resonances
identified above, is 51% when $\alpha=0.01$ and 62% when $\alpha=0.03$. Weaker
resonances lines (which are hard to recognise due to the sparseness of the
data points) may also trap some of the chaotic orbits. This supports our
conjecture that the main reason that model SA1 does not evolve in phase b,
despite the presence of a significant fraction of chaotic orbits, is that the
majority of the chaotic orbits are trapped around resonances and therefore
behave like regular orbits for very long times.
In Figure 14 we plot four examples of chaotic orbits in phase b of SA1, which
illustrate how resonantly trapped or “sticky” chaotic orbits look. Each panel
of four sub-plots shows a single orbit plotted in two Cartesian projections
(side-by-side). The top pair of subplots show the orbit over the first 10 Gyr
long time segment, while the bottom pair shows the same orbit over a second 10
Gyr time segment. The two time segments were separated by 10 Gyr. For
illustration we selected orbits with a range of drift parameters. The orbit in
the top-left panel is an example of an orbit that conforms to our notion of a
chaotic orbit that explores more phase space as time progresses, and has a
large drift parameter of $\log(\Delta f)=-0.48$. The top-right panel shows a
S-tube orbit that suddenly migrates to a box orbit (this orbit has a
$\log(\Delta f)=-0.54$) and was probably in the separatrix region between the
S-tube and box families. The bottom-left panel shows an orbit that is
originally a S-tube that becomes trapped around a resonant boxlet (“fish”)
family (with $\log(\Delta f)=-0.66$), while the bottom right-hand panel shows
a weakly chaotic box orbit (with $\log(\Delta f)=-0.94$). Of the 21% of orbits
in phase b that are chaotic ($\log(\Delta f)\geq-1$), only $\sim 5$% have
$\log(\Delta f)\geq-0.5$. This fraction is small enough that one does not
expect it to result in significant chaotic mixing.
## 5 Summary and Discussion
Since it was first proposed, the idea that a central black hole would scatter
centrophilic box orbits in triaxial galaxies resulting in more axisymmetric
potentials (Gerhard & Binney, 1985) has frequently been used to explain the
shape change in a variety of systems from the destruction of bars by central
black holes (Norman et al., 1996) to the formation of more oblate galaxy
clusters in simulations with gas (Kazantzidis et al., 2004).
Experiments on chaotic mixing indicated that the timescales for such mixing is
about 30 - 100 dynamical times (Merritt & Valluri, 1996), which is much longer
than the timescales for evolution of dark matter halos in simulations with gas
(Kazantzidis et al., 2004). In addition recent detailed studies of $N$-body
simulations with controlled experiments have shown that the role of chaotic
mixing may be less dramatic than conjectured by these previous studies. A
study of relaxation of collisionless systems following the merger of two
spherical galaxies showed that despite the fact that a large fraction of the
orbits in a system undergoing violent relaxation are chaotic, the timescales
for chaotic diffusion and mixing are too long for this process to play a
significant role (Valluri et al., 2007). In fact, even after violent
relaxation, orbits retain strong memories of their initial energies and
angular momenta.
D08 argued that since chaos introduces an irreversible mixing, numerical
experiments in which evolution is driven by chaotic orbits should not be
reversible. These authors studied the macroscopic shapes of triaxial dark
matter halos in response to the growth of a baryonic component. Unless the
baryons were too centrally concentrated, or transported angular momentum to
the halo, the evolution they saw was reversible, from which they concluded
that much of the shape change arises from deformations in the shape of
individual orbits rather than significant chaotic scattering. In this paper we
investigated this issue in significantly greater detail by applying the
Numerical Analysis of Fundamental Frequencies (NAFF) technique that allows us
to quantify the degree to which chaotic diffusion drives evolution and to
identify the primary physical processes that cause halo shape change. The
frequency based method is able to distinguish between regular and chaotic
orbits, making it more useful than Lyapunov exponents which are known to be
sensitive to discretization effects in $N$-body systems (Hemsendorf & Merritt,
2002). We use the method to quantify the drift in frequencies of large
representative samples of orbits, thereby quantifying the degree of chaos in
the systems we study. It also allowed us to map the phase space structure of
the initial and final halos and to quantify the relationship between the
change in the shapes of individual orbits and the shape of the halo as a
whole.
Applying various analysis methods to orbits in five systems we demonstrated
that the conclusion reached by D08 that chaos is not an important driver of
shape evolution when the baryonic component is extended is indeed valid. As
did D08, we also found that significant chaotic scattering does occur when the
baryonic component is in the form of a hard central point mass (of scale
length $\sim 0.1$ kpc). It is interesting that regardless of the original
orbital composition of the triaxial or prolate halo, and regardless of the
shape of radial scale length of the baryonic component, halos become more
oblate following the growth of a baryonic component. Thus two quite different
processes (chaotic scattering and adiabatic deformation) result in similar
final halo shapes even in halos with very different orbital compositions.
We explored two different initial halos, one in which box orbits were the
dominant elongated population (halo A) and the other in which L-tubes
dominated the initial halo (halo B). Despite the different orbit compositions
both models exhibit similar overall evolution with regard to the shapes of
orbits. In the halo A models, the box orbits were much more likely to change
to either L-axis or S-tubes, whereas in the halo B models, the dominant family
of L-tubes largely retained their orbital classification while deforming their
shapes.
Below we list the main results of this paper:
1. 1.
Correlations between the orbital frequencies in the three different phases
$\omega_{a}$, $\omega_{b}$ and $\omega_{c}$ are a useful way to search for
regular versus chaotic evolution of orbits. The orbital frequencies in the
three phases are found to be strongly correlated with each other when the
baryonic component is extended (Fig. 4), but show significant scattering when
the baryonic component is a compact point mass (of scale length about 0.1 kpc)
(Fig. 5). In the more extended distributions, only a small fraction of the
orbits experience significant change in their original orbital frequencies
when the baryons are evaporated, while both the magnitude of scattering in
orbital frequency as well as the fraction of orbits experiencing scattering,
increases as the baryonic component becomes more compact (Fig. 6).
2. 2.
In the three models with relatively extended baryonic components, the change
in orbital frequency between phase c and phase a ($\Delta\omega_{ac}$) is not
correlated with orbital frequency (Fig. 7), pericenter distance or orbital
angular momentum. When the baryonic component is a hard point mass, however,
the frequency change is greater for orbits that are deeper in the potential
and therefore have both a higher initial orbital frequency (Fig. 7) and
smaller pericentric radius (Fig. 8). Scattering in frequency affects both the
centrophilic box orbits as well as centrophobic L-tubes.
3. 3.
The growth of a baryonic component in halo A (either disk or softened point
mass) causes box orbits with large pericentre radii to be converted to
S-tubes, L-tubes or become chaotic (Fig.9 top panel). While this change is
almost completely reversible in the case of the disk or a diffuse point mass,
it is less so when the baryonic component is a hard point mass. In halo B,
which is dominated by L-tubes, the growth of the baryonic component causes
almost no change in the orbital composition of the halo, indicating that the
L-tubes are not destroyed but deformed (Fig.9 bottom panel). Even though PlB3
(Prolt+hardpt) is a model with significant orbit scattering by the hard
central point mass, the process appears to mainly convert elongated inner
L-tube orbits to somewhat rounder outer L-tubes. In model PlA4 (Triax+hardpt)
the box orbits are scattered onto S-tubes or chaotic orbits. The significant
amount of scattering seen for even centrophobic L-tube orbits shows that the
evolution is not due to direct scattering by a central point mass as sometimes
assumed. Two alternative possibilities are more likely to account for the
significant scattering in frequency. First the change in the symmetry and
depth of the central potential is a perturbation to the potential that gives
rise to an increase in the region of phase space occupied by resonances
(Kandrup 1998 - private communication). As the resonances overlap there is an
increase in the degree of chaotic behavior (Chirikov, 1979). The second option
is that the point mass attains equipartition with the background mass
distribution, resulting in Brownian motion (Merritt, 2005). The Brownian
motion can cause the centre of the point mass to wander within a region of
radius $\sim 0.1-1$ kpc which can result in a significant change in the
maximum gravitational force experienced by an orbit from one pericentre
encounter to the next. This change in the maximum gravitational force
manifests as scattering of the orbit which is equally effective for both box
orbits and long-axis tubes. Indeed a small wandering of the central massive
point is seen in the $N$-body simulations of PlA4; this motion is not included
in our orbit calculations since all particles are frozen in place when
calculating orbits. While it is beyond the scope of this paper to explore this
issue further, we caution that the motion of the point mass in our $N$-body
simulations is likely to over-estimate the magnitude of the Brownian motion,
since this depends on the mass resolution of the background particles. This
suggests that in a real galaxy the evolution of the shape is much more likely
to be driven by smooth adiabatic deformation of orbits than chaotic
scattering.
4. 4.
In triaxial halos, the orbital shapes sharply peaked distribution with the
most elongated orbits ($\chi_{s}>0.25$) are either boxes or L-tubes. In the
prolate halos the second peak at $\chi_{s}\sim 0$ contains a third of the
orbits and is composed of squat outer L-tubes and some box orbits. The growth
of a baryonic component of any kind causes orbits of all types to become
“rounder”, especially at small pericenter radii. This change in orbital shape
distribution with radius is the primary cause of the change of halo shapes in
response to the growth of a baryonic component. This is consistent with the
findings of D08 who also found that the orbits in the models became quite
round.
5. 5.
The growth of a disk causes a large fraction of halo orbits to become
resonantly trapped around major resonances. The three most important
resonances are those associated with the 1:1 tube (thin shell) orbit that
circulates about the short axis in the $x-y$-plane, the 1:1 tube (thin shell)
orbit that circulates about the long axis in the $y-z$-plane and the 1:2
banana resonance in the $x-z$-plane. We saw from the frequency maps that the
resonant trapping of the halo particles depends both on the form of the
baryonic component grown in the halo as well as on the initial orbital
population of the halo.
Thus we conclude that the evolution of galaxy and halo shapes following the
growth of a central component occurs primarily due to regular adiabatic
deformation of orbital shapes in response to the changing central potential.
Chaotic scattering of orbits may be important particularly for orbits with
small pericentre radii but only when the central point mass is extremely
compact. Contrary to previous expectations, chaotic scattering is only
slightly more effective for centrophilic box orbits than it is for
centrophobic L-tubes. Boxes can be scattered onto both L- and S-tube orbits
and a significant fraction become chaotic. When the compact central point mass
scatters L-tubes as it does in a prolate halo, they are scattered onto other
L-tube orbits rather than onto S-tube orbits. The strong chaotic scattering
that we see on centrophobic L-tube orbits has not been previously anticipated.
An important implication of our analysis is that while the shapes of halos
(and by extension elliptical galaxies) become more oblate (especially at small
radii), following the growth of a baryonic component, the majority of their
orbits are not S-tubes as might be predicted from their shapes. Instead our
analysis shows that orbits prefer to maintain their orbital characteristics,
and the majority of the orbits are those which would be generally found in
triaxial galaxies. This is particularly important for studies of the internal
dynamics of elliptical galaxies since the fact that their shapes appear nearly
axisymmetric need not imply that their orbital structure is as simple as the
structure of oblate elliptical galaxies. Modifying the shapes to slightly
triaxial could result in significant changes in their orbit populations and
consequently could affect both the inferred dynamical structure as well as the
estimates of the masses components such as the supermassive black holes in
these galaxies (van den Bosch & de Zeeuw, 2009).
Finally, our finding that the growth of a stellar disk can result in a large
fraction of halo orbits becoming trapped in resonances could have important
implications for observational studies of the Milky Way’s stellar halo. The
computation expense of the orbit calculations forced us to restrict the size
of the frequency map for model SA1 to 6000 particles. This is only a tiny
fraction of the particles in the original simulation. Despite the smallness of
the sample, the frequency maps (Fig. 13) shows a rich resonant structure which
implies that the particles (either stars or dark matter) in the stellar and
dark matter halos of our Galaxy, particularly those close to the plane of the
disk, are likely to be associated with resonances, rather than being smoothly
distributed in phase space (this is in addition to structures arising due to
tidal destruction of dwarf satellites). Although significantly greater
resolution is required to resolve such resonances than is currently available,
this could have significant implications for detection of structures in
current and upcoming surveys of the Milky Way such as SDSS-III (Segue) and
Gaia (Perryman et al., 2001; Wilkinson & et al., 2005) and in on-going direct
detection experiments which search for dark matter candidates.
## Acknowledgments
M.V. is supported by the University of Michigan. V.P.D. thanks the University
of Zürich for hospitality during part of this project. Support for one of
these visits by Short Visit Grant # 2442 within the framework of the ESF
Research Networking Programme entitled ’Computational Astrophysics and
Cosmology’ is gratefully acknowledged. Support for a visit by M.V. to the
University of Central Lancashire at an early stage of this project was made
possible by a Livesey Grant held by V.P.D. All simulations in this paper were
carried out at the Arctic Region Supercomputing Center. We would like to thank
the referee Fred Adams for his very thoughtful and constructive report which
helped improve the paper.
## References
* Allgood et al. (2006) Allgood B., Flores R. A., Primack J. R., Kravtsov A. V., Wechsler R. H., Faltenbacher A., Bullock J. S., 2006, MNRAS, 367, 1781
* Andersen et al. (2001) Andersen D. R., Bershady M. A., Sparke L. S., Gallagher J. S., Wilcots E. M., 2001, ApJ, 551, L131
* Bailin & Steinmetz (2005) Bailin J., Steinmetz M., 2005, ApJ, 627, 647
* Banerjee & Jog (2008) Banerjee A., Jog C. J., 2008, ArXiv e-prints, 806
* Bardeen et al. (1986) Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ, 304, 15
* Barnes & Sellwood (2003) Barnes E. I., Sellwood J. A., 2003, AJ, 125, 1164
* Barnes & Efstathiou (1987) Barnes J., Efstathiou G., 1987, ApJ, 319, 575
* Barnes & Hut (1986) Barnes J., Hut P., 1986, Nature, 324, 446
* Bartelmann et al. (1995) Bartelmann M., Steinmetz M., Weiss A., 1995, A&A, 297, 1
* Binney & Spergel (1982) Binney J., Spergel D., 1982, ApJ, 252, 308
* Binney & Spergel (1984) Binney J., Spergel D., 1984, MNRAS, 206, 159
* Binney & Tremaine (2008) Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Princeton University Press
* Boozer (1982) Boozer A. H., 1982, Physics of Fluids, 25, 520
* Buote & Canizares (1994) Buote D. A., Canizares C. R., 1994, ApJ, 427, 86
* Buote et al. (2002) Buote D. A., Jeltema T. E., Canizares C. R., Garmire G. P., 2002, ApJ, 577, 183
* Carpintero & Aguilar (1998) Carpintero D. D., Aguilar L. A., 1998, MNRAS, 298, 1
* Chirikov (1979) Chirikov B. V., 1979, Phys. Rep., 52, 263
* Debattista (2003) Debattista V. P., 2003, MNRAS, 342, 1194
* Debattista et al. (2008) Debattista V. P., Moore B., Quinn T., Kazantzidis S., Maas R., Mayer L., Read J., Stadel J., 2008, ApJ, 681, 1076
* Dehnen (2005) Dehnen W., 2005, MNRAS, 360, 892
* Diehl & Statler (2007) Diehl S., Statler T. S., 2007, ApJ, 668, 150
* Dubinski (1994) Dubinski J., 1994, ApJ, 431, 617
* Dubinski & Carlberg (1991) Dubinski J., Carlberg R. G., 1991, ApJ, 378, 496
* Franx & de Zeeuw (1992) Franx M., de Zeeuw T., 1992, ApJ, 392, L47
* Franx et al. (1994) Franx M., van Gorkom J. H., de Zeeuw T., 1994, ApJ, 436, 642
* Frenk et al. (1988) Frenk C. S., White S. D. M., Davis M., Efstathiou G., 1988, ApJ, 327, 507
* Gerhard & Binney (1985) Gerhard O. E., Binney J., 1985, MNRAS, 216, 467
* Goodman et al. (1993) Goodman J., Heggie D. C., Hut P., 1993, ApJ, 415, 715
* Habib et al. (1997) Habib S., Kandrup H. E., Mahon M. E., 1997, ApJ, 480, 155
* Hemsendorf & Merritt (2002) Hemsendorf M., Merritt D., 2002, ApJ, 580, 606
* Holley-Bockelmann et al. (2002) Holley-Bockelmann K., Mihos J. C., Sigurdsson S., Hernquist L., Norman C., 2002, ApJ, 567, 817
* Huizinga & van Albada (1992) Huizinga J. E., van Albada T. S., 1992, MNRAS, 254, 677
* Iodice et al. (2003) Iodice E., Arnaboldi M., Bournaud F., Combes F., Sparke L. S., van Driel W., Capaccioli M., 2003, ApJ, 585, 730
* Jing & Suto (2002) Jing Y. P., Suto Y., 2002, ApJ, 574, 538
* Kalapotharakos (2008) Kalapotharakos C., 2008, MNRAS, 389, 1709
* Kandrup & Novotny (2004) Kandrup H. E., Novotny S. J., 2004, Celestial Mechanics and Dynamical Astronomy, 88, 1
* Kandrup & Sideris (2001) Kandrup H. E., Sideris I. V., 2001, Phys. Rev. E, 64, 056209
* Kandrup & Sideris (2003) Kandrup H. E., Sideris I. V., 2003, ApJ, 585, 244
* Kandrup & Siopis (2003) Kandrup H. E., Siopis C., 2003, MNRAS, 345, 727
* Kandrup & Smith (1991) Kandrup H. E., Smith H. J., 1991, ApJ, 374, 255
* Kazantzidis et al. (2004) Kazantzidis S., Kravtsov A. V., Zentner A. R., Allgood B., Nagai D., Moore B., 2004, ApJ, 611, L73
* Kazantzidis et al. (2004) Kazantzidis S., Magorrian J., Moore B., 2004, ApJ, 601, 37
* Kochanek (1995) Kochanek C. S., 1995, ApJ, 445, 559
* Koopmans et al. (1998) Koopmans L. V. E., de Bruyn A. G., Jackson N., 1998, MNRAS, 295, 534
* Kuijken & Tremaine (1994) Kuijken K., Tremaine S., 1994, ApJ, 421, 178
* Kuo-Petravic et al. (1983) Kuo-Petravic G., Boozer A. H., Rome J. A., Fowler R. H., 1983, Journal of Computational Physics, 51, 261
* Laskar (1990) Laskar J., 1990, Icarus, 88, 266
* Laskar (1993) Laskar J., 1993, Celestial Mechanics and Dynamical Astronomy, 56, 191
* Lichtenberg & Lieberman (1992) Lichtenberg A. J., Lieberman M. A., 1992, Regular and chaotic motion. Applied Mathematical Sciences, New York: Springer, 1992
* Merritt (2005) Merritt D., 2005, ApJ, 628, 673
* Merritt & Quinlan (1998) Merritt D., Quinlan G. D., 1998, ApJ, 498, 625
* Merritt & Valluri (1996) Merritt D., Valluri M., 1996, ApJ, 471, 82
* Merritt & Valluri (1999) Merritt D., Valluri M., 1999, AJ, 118, 1177
* Miller (1964) Miller R. H., 1964, ApJ, 140, 250
* Moore et al. (2004) Moore B., Kazantzidis S., Diemand J., Stadel J., 2004, MNRAS, 354, 522
* Navarro et al. (1996) Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563
* Norman et al. (1996) Norman C. A., Sellwood J. A., Hasan H., 1996, ApJ, 462, 114
* Oguri et al. (2003) Oguri M., Lee J., Suto Y., 2003, ApJ, 599, 7
* Olling (1995) Olling R. P., 1995, AJ, 110, 591
* Olling (1996) Olling R. P., 1996, AJ, 112, 481
* Olling & Merrifield (2000) Olling R. P., Merrifield M. R., 2000, MNRAS, 311, 361
* Papaphilippou & Laskar (1996) Papaphilippou Y., Laskar J., 1996, A&A, 307, 427
* Papaphilippou & Laskar (1998) Papaphilippou Y., Laskar J., 1998, A&A, 329, 451
* Perryman et al. (2001) Perryman M. A. C., de Boer K. S., Gilmore G., Høg E., Lattanzi M. G., Lindegren L., Luri X., Mignard F., Pace O., de Zeeuw P. T., 2001, A&A, 369, 339
* Sackett & Sparke (1990) Sackett P. D., Sparke L. S., 1990, ApJ, 361, 408
* Schoenmakers et al. (1997) Schoenmakers R. H. M., Franx M., de Zeeuw P. T., 1997, MNRAS, 292, 349
* Schweizer et al. (1983) Schweizer F., Whitmore B. C., Rubin V. C., 1983, AJ, 88, 909
* Stadel (2001) Stadel J. G., 2001, Ph.D. Thesis, University of Washington
* Terzić & Kandrup (2004) Terzić B., Kandrup H. E., 2004, MNRAS, 347, 957
* Tremaine & Yu (2000) Tremaine S., Yu Q., 2000, MNRAS, 319, 1
* Valluri & Merritt (1998) Valluri M., Merritt D., 1998, ApJ, 506, 686
* Valluri & Merritt (2000) Valluri M., Merritt D., 2000, in Gurzadyan V. G., Ruffini R., eds, The Chaotic Universe, Proceedings of the Second ICRA Network Workshop, Advanced Series in Astrophysics and Cosmology, vol.10, Edited by V. G. Gurzadyan and R. Ruffini, World Scientific, 2000, p.229 Orbital Instability and Relaxation in Stellar Systems. pp 229–+
* Valluri et al. (2007) Valluri M., Vass I. M., Kazantzidis S., Kravtsov A. V., Bohn C. L., 2007, ApJ, 658, 731
* van den Bosch & de Zeeuw (2009) van den Bosch R. C. E., de Zeeuw P. T., 2009, MNRAS(in press)
* Wilkinson & et al. (2005) Wilkinson M. I., et al. 2005, MNRAS, 359, 1306
* Zemp et al. (2008) Zemp M., Moore B., Stadel J., Carollo C. M., Madau P., 2008, MNRAS, 386, 1543
|
arxiv-papers
| 2009-06-25T20:24:53 |
2024-09-04T02:49:03.538027
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Valluri (Univ. of Michigan), V.P. Debattista (Univ. of Central\n Lancashire), T. Quinn (Univ. of Washington), B. Moore (Univ. of Zurich)",
"submitter": "Monica Valluri",
"url": "https://arxiv.org/abs/0906.4784"
}
|
0906.4839
|
# An Ultraviolet Study of Star-Forming Regions in M31
Yongbeom Kang11affiliation: Department of Astronomy and Space Science,
Chungnam National University, Daejeon, 305-764, Korea; [email protected],
[email protected] 22affiliation: Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, MD 21218, USA; [email protected] , Luciana
Bianchi22affiliation: Department of Physics and Astronomy, Johns Hopkins
University, Baltimore, MD 21218, USA; [email protected] 33affiliation:
Corresponding authors , and Soo-Chang Rey11affiliation: Department of
Astronomy and Space Science, Chungnam National University, Daejeon, 305-764,
Korea; [email protected], [email protected] 33affiliation: Corresponding authors
###### Abstract
We present a comprehensive study of star-forming (SF) regions in the nearest
large spiral galaxy M31. We use $GALEX$ far-UV (1344-1786 Å, FUV) and near-UV
(1771-2831 Å, NUV) imaging to detect young massive stars and trace the recent
star formation across the galaxy. The FUV and NUV flux measurements of the SF
regions, combined with ground-based data for estimating the reddening by
interstellar dust from the massive stars they contain, are used to derive
their ages and masses. The $GALEX$ imaging, combining deep sensitivity and
coverage of the entire galaxy, provides a complete picture of the recent star
formation in M31 and its variation with environment throughout the galaxy. The
FUV and NUV measurements are sensitive to detect stellar populations younger
than a few hundred Myrs. We detected 894 SF regions, with size $\geq$ 1600 pc2
above an average FUV flux limit of $\sim$26 ABmag arcsecond-2, over the whole
26 kpc (radius) galaxy disk. We derive the star-formation history of M31
within this time span. The star formation rate (SFR) from the youngest UV
sources (age $\leq$ 10 Myr) is comparable to that derived from H$\alpha$, as
expected. We show the dependence of the results on the assumed metallicity.
When star formation detected from IR measurements of the heated dust is added
to the UV-measured star formation (from the unobscured populations) in the
recent few Myrs , we find the SFR has slightly decreased in recent epochs,
with a possible peak between 10 and 100 Myrs, and an average value of SFR
$\sim$0.6 or 0.7 M☉ yr-1 (for metallicity Z=0.02 or 0.05 respectively) over
the last 400 Myrs.
stars: early-type — galaxies: evolution — galaxies: individual (M31) —
galaxies: star clusters — galaxies: structure — ultraviolet: galaxies —
ultraviolet: stars
††slugcomment: accepted, 25 June 2009
## 1 Introduction
In the cold dark matter framework, large spiral galaxies are built
hierachically, and there is much observational evidence of galaxy interactions
to support this. However, we still face many challenges in understanding the
details of galaxy evolution. In this regard, it is interesting to study the
hot, massive stellar populations in nearby galaxies which can be a robust
tracer of recent star formation activity due to their short lives. Young
massive stars contribute to the global characteristics at the current epoch of
their host galaxy and have a major role in the galaxy evolution.
Young massive stars emit powerful ultraviolet (UV) radiation, therefore UV
imaging is ideal to detect and characterize these stars, which are confused
with older stellar populations in observations at longer wavelengths (Bianchi,
2006, and references therein). In most cases, young massive stars are formed
in associations in the galactic disk. We define these stellar associations
from UV imaging, in order to study the spatial structure and intensity of the
recent star formation. Galaxy Evolution Explorer (GALEX, Martin et al., 2005)
imaging in far-UV (FUV) and near-UV (NUV) passbands is particularly useful to
study massive stellar populations. Specifically, the integrated FUV $-$ NUV
color of young stellar populations is very sensitive to the age of the
population, owing to the rapid evolution of the most massive stars (e.g.
Bianchi 2007, 2009: Fig. 9). Consequently, UV imaging data allow us to
unambiguously identify the young stellar populations and to estimate their
ages and masses from colors and extinction-corrected UV luminosities,
respectively. These results provide the means of probing the history and
modality of recent star formation in galaxies.
M31 is one of the two largest spiral galaxies in the Local Group, along with
the Milky Way. However, there has been growing evidence that the Milky Way and
M31 have different properties (Ibata et al., 2007, and references therein).
M31 shows a lower star formation rate (SFR) than the Milky Way (Kennicutt,
1998; Massey et al., 2007; Hou et al., 2009; Fuchs et al., 2009). Furthermore,
there are suggestions that M31 appears to be a more typical spiral galaxy than
the Milky Way (Hammer et al., 2007). As with the bulk of local spirals, M31
shows evidences for a formation and evolution history affected by merging
and/or accretion events, including substructures in its halo (Hammer et al.,
2007, and references therein). In this respect, it is important to investigate
the star-forming (SF) regions in M31. We present the first extensive study of
the young stellar populations from UV imaging, covering the entire M31 disk
within 26 kpc radius, and extending beyond this radius in some fields. By
studying the SF regions in M31 traced by UV imaging, we investigate the recent
star formation history of this galaxy.
In section 2, we describe the GALEX UV data used. We describe the detection
and photometry of SF regions, from UV imaging, in section 3, and estimate
their interstellar extinction in section 4. In section 5, we select OB
associations from ground-based photometry of stellar sources and compare them
with our UV-defined SF regions. The age and mass of the SF regions are derived
in section 6, the star formation history in section 7, and the conclusions are
presented in section 8.
## 2 Data
We focus on the disk region of M31 where most of the recent SF regions are
located. We adopt the disk semi-major axis value of R = 26 kpc from Walterbos
& Kennicutt (1988). We considered 20 fields from the GALEX fourth data release
(GR4). GALEX observed the disk region of M31 as part of the Nearby Galaxies
Survey (NGS, Bianchi et al., 2003). We rejected the fields which have only FUV
observations and exposure time shorter than 2,000 s (except for
“PS_M31_MOS03”), then selected the fields closer to the major axis when
different pointings are located along the galaxy’s major axis. The
“PS_M31_MOS03” field has a shorter exposure than 2,000 s but it is the only
GALEX field observed in the south outermost disk region of M31. As a result,
we selected 7 GALEX fields covering the entire disk region (see Fig. 1 and
Table 1). Each selected GALEX field has FUV (1344-1786 Å) and NUV (1771-2831
Å) imaging with the same exposure time. GALEX FUV and NUV imaging has 4.2 and
5.3 arcsec resolution (FWHM) or about $\sim$19 pc in M31, and the field of
view (FOV) is about 1.27 and 1.25 degree respectively (Morrissey et al.,
2007). We used in our analysis only the central 1.1 degree diameter portion of
the field, for best photometry. GALEX images have a sampling of 1.5 arcsec
pixel-1 which corresponds to 5.67 pc, assuming a distance of 785 kpc
(McConnachie et al., 2005).
## 3 Photometry of Star-Forming Regions
The FUV images provide a direct measure of the flux from young massive stars
not heavily embedded in interstellar dust. Most SF regions are gravitationally
unbound systems and have irregular shapes. Rather than using aperture
photometry, we constructed contours of the SF regions to trace their
morphology, and measure their UV flux and flux density. The procedure was
originally developed by Tolea (2009) in his dissertation, and we have modified
the procedure for a more precise detection and photometry of the SF regions.
Our procedure consisted of three steps. The first step was to detect all image
pixels which have flux above a certain threshold in each FUV image. An
important factor in detecting and defining SF regions is the brightness limit.
The second step was to define the contours of each SF region from contiguous
pixels detected above the threshold over a minimum area. In this way, we can
define contours of SF regions even if they have a complicated shape, and
reject isolated stars which are smaller than the minimum size. The third step
was to estimate the background and to measure the flux of the defined SF
regions. Even though the background in the UV image is much lower than in the
optical, it is important to correctly subtract its contribution from the
source photometry. We performed various tests for determining the optimum flux
threshold, minimum size of a SF region, and best background subtraction
method.
We used the field “NGA_M31_MOS0” which has the longest exposure time (6,811 s)
in our selected GALEX data, to test and refine our procedures, which were then
applied to all our selected fields. This field is good for testing various
types of SF regions because it contains portions from innermost to outermost
spiral arms and the large OB association NGC 206. First of all, we compared
various thresholds for the detection of source pixels. We used the background-
subtracted image (“-intbgsub”) provided by the GR4 pipeline. We estimated the
mean background value by the sigma clipping method. We examined the results
using thresholds of two, three, and five sigma above the mean background value
(see Fig. 2). We detect fainter objects if we use the lower thresholds,
however the SF regions in the spiral arms merge together and the contamination
by the background (including older, diffuse populations) is larger. We can
easily define the bright regions if we use higher thresholds, however we
cannot detect the faint ones. The FUV magnitude limits of our detected SF
regions from each threshold are shown in Fig. 3, they are $\sim$21.5 (low),
$\sim$21.0 (mid), and $\sim$20.4 (high threshold) mag in the AB magnitude
system. We adopted a threshold of three sigma above the mean background value,
which showed in our analysis less contamination by background and marginally
detects the faint regions. This results in an average FUV flux threshold of
$\sim$0.0032 c s-1 pixel-1, or $\sim$25.9 mag arcsec-2. Then, we considered
the minimum acceptable size of the SF regions, in order to eliminate
contamination by background objects, artificial sources, foreground stars, and
isolated bright stars in M31. We considered 3$\times$FWHM of GALEX ($\sim$13
arcsec or $\sim$8.5 pixels) as minimum diameter of a SF region, therefore we
adopted a requirement of a minimum of 50 contiguous pixels ($\sim$1,600 pc2 or
$\sim$40 pc) for the smallest SF region. We contoured the contiguous pixels
selected for measuring the defined regions and we estimated the center of each
SF region by the mid-point of the maximum diameter of its contour.
These adopted threshold and minimum size of the SF regions, were then applied
to all the selected fields. However, the exposure time differences among
fields induces variation of the detection limit. Therefore, we compared the
detection fraction in overlapping image regions, one by one against the
“NGA_M31_MOS0” field. We first tried a fixed threshold ($\sim$0.0032 c s-1
pixel-1) for source detection in the “NGA_M31_MOS0” image. In this case, in
images with shorter exposure time sources are over-detected and include noise.
A second method used the variable threshold of three sigma above the mean
background value from each image. This produced similar contours in the
overlapping regions. However, some sources in the shorter exposure time images
were undetected because the background has larger sigma values. Finally, we
adopted the ratio between detected and undetected pixels of overlapping
regions. This case produced slight over-detection in the shorter exposure time
images. Therefore, we used the last method, manually varying the thresholds in
each field, such as to obtain similar detections in overlapping regions. The
final selected FUV pixel maps are shown in Fig. 4. As a last step, we compared
the detected SF regions within overlapping image portions by visual inspection
to select a final catalog of unique sources. At the end, we obtained a catalog
of 894 SF regions cleaned of artifacts, isolated stars, and overlapping
objects.
We measured the flux within the contours of the SF regions as defined above,
and subtracted the local background, estimated in a circular annulus
surrounding the source. We initially used an inner/outer size of the annulus
of 1.5/3 times the size of each SF region. However, for the largest sources,
which are mostly found along the spiral arms, the annulus for background
measurement scaled in this way becomes too large and includes unrelated
stellar populations. Therefore, we tested three different procedures to
estimate the background. One was to measure the local background (“Ap. sky” in
Fig. 5), adjusting the radius of the background annulus according to the size
of the source. We adopted a variable scaling of inner/outer radii for the
background annulus, adjusted as 2/4, 2/3, 1.5/2.5, 1.1/1.3, and 1.05/1.2 times
the source size (RMAX) for the following ranges of source size respectively:
RMAX $<$ 10, 10 $\leq$ RMAX $<$ 20, 20 $\leq$ RMAX $<$ 50, 50 $\leq$ RMAX $<$
100, and 100 $\leq$ RMAX ([pixels]). Such ranges were found adequate to ensure
a large enough area for the background measurement for the smallest sources,
while preventing excessively large areas to be included in the calculations
for large sources. The second background estimate was obtained from the GR4
pipeline background image (“-skybg”) , and the third measurement was performed
by applying a median filter to the image. In the case of the GR4 pipeline
background (“Pipe. sky” in Fig. 5), a sky background image is produced by a
5$\times$5 median filter size which is good for the case of Poisson
distribution (Morrissey et al., 2007). The background measured using this
image, underestimates the local contribution by the galaxy’s background light,
especially from surrounding stellar populations in the spiral arms. In the
case of the median filtered background (“Med. sky” in Fig. 5), a smaller
filter size (3$\times$3 pixels) than the standard pipeline was used to produce
the background images. The choice of a smaller filter size was driven by the
consideration of spiral arm regions which host most of the SF regions. The
results from this background estimate are similar to those from the local
background and reflect well the brightness of spiral arms. The results from
the three methods are compared in Fig. 5. The background from the pipeline
’sky’ image is always underestimated because it measures the lowest sky level
and not the local diffuse stellar population surrounding the source. In the
right-side panels in Fig. 5, we see that this estimate is not sensitive to the
spiral arm enhancements, as the local-background estimates are. The local sky
estimate is similar to the results from the median-filtered measurements but
shows less scatter for sources with multiple observations in overlapping
regions. In Fig. 6, we compare the photometry results using the three
different methods for background subtraction in color-magnitude diagrams. Most
SF regions have FUV $-$ NUV color between $-0.5$ and $1.0$ in ABmag. The
measurements from the pipeline and median-filtered backgrounds induce brighter
NUV than FUV, especially for sources fainter than 20 mag. We finally adopted
the result from the local background. The resulting catalog of SF regions and
their GALEX photometry is given in Table 2.
## 4 OB Stars and Interstellar Extinction
In order to derive the physical parameters of the SF regions from the
integrated photometry, we must take into account the interstellar extinction.
We estimated the reddening of each UV SF region from the reddening of the
massive stars included within its contour. For massive stars we used the
reddening-free parameter Q (Massey et al., 1995; Bianchi & Efremova, 2006). We
used the optical point-source measurements of M31 sources from P. Massey
(priv. comm., 2009), which is a revised M31 catalog from the NOAO survey data
described by Massey et al. (2006, 2007). We selected sources with $UBV$
measurements having photometric errors lower than 0.1 mag in all bands
(108,089 objects out of their 371,781 total sources catalog), which results in
a magnitude limit of about 23rd mag in $V$-band. This data is deep enough to
select OB type stars which we used to estimate the interstellar reddening of
our SF regions. We selected OB type stars by comparing colors and brightness
of the Galactic OB type main sequence stars from Aller et al. (1982). We
selected stars from O3 to B2V, because a B2V star has $M_{V}$ = $-2.45$ which
is $V$ = $22.02$ in M31 ($m-M$ = 24.47) in absence of extinction. We don’t
know the internal reddening of M31, therefore we used the reddening-free
parameter $Q_{UBV}$ to select the OB stars.
$Q_{UBV}$ = $(U-B)$ \- ${E(U-B)}\over{E(B-V)}$$(B-V)$
The $E(U-B)/E(B-V)$ ratio is a constant (0.72) for Milky Way dust type (Massey
et al., 1995) and does not vary much for other dust types, within a reasonably
small range of $E(B-V)$ (e.g. Bianchi et al., 2007; Bianchi & Efremova, 2006,
and references therein). We selected the OB stars which have $Q_{UBV}$ between
$-0.97$ (O3V) and $-0.67$ (B2V). In order to reduce contamination, we also
adopted magnitude and color limits; $-6.0$ $\leq$ $M_{V}$ $\leq$ $-0.28$,
$-0.34$ $\leq$ $B-V$ $\leq$ 0.46, and $-1.22$ $\leq$ $U-B$ $\leq$ $-0.336$
(Aller et al., 1982). The color and magnitude limits are derived assuming a
maximum reddening value of $E(B-V)$ $\leq$ 0.7. The location of the selected
OB stars in color-magnitude diagrams are presented in Fig. 7. With these
restrictions, we finally selected 22,655 O-B2 stars from the data of Massey et
al. (2006). The spatial distribution of these stars, shown in Fig. 1,
represents well the spiral structure of M31.
The interstellar extinction of the selected OB stars was estimated from the
reddening free parameter QUBV, using the empirical relationship for giant and
main-sequence stars by Massey et al. (1995).
$(B-V)_{0}=-0.013+0.325Q_{UBV}=(B-V)-E(B-V)$
The estimated $E(B-V)$ has mean, median, and mode value of about 0.34, 0.32,
and 0.29, respectively, from our selected OB stars. The mean reddening value
of OB stars is larger than Massey et al. (2007)’s typical value $E(B-V)$ =
0.13 which is estimated visually from the location of the “blue plume” in the
color-magnitude diagram. This difference may be caused by our selection of
blue stars for the reddening estimate. We also explore in this paper more
reddened regions than the average entire stellar population. We estimated the
extinction by interstellar dust for each SF region from the average reddening
of the OB stars within the SF region contour. For the SF regions outside of
Massey et al. (2006) survey, we assumed $E(B-V)$ = 0.20. The spatial
distributions of estimated interstellar reddening of OB stars and SF regions
are shown in Fig. 8. The interstellar extinction decreases from the inner disk
region outwards. In particular, inner-most and south-west areas where we could
not detect SF regions, have high interstellar extinction. We will return to
this point in the next section. Our detection method is based on the FUV flux,
which could vanish in high interstellar extinction regions.
## 5 OB Associations Defined from Stellar Photometry
For comparison with our FUV-defined SF regions, we also used a Path Linkage
Criterion (PLC: Battinelli, 1991) method (explored by Ivanov, 1996, 1998;
Magnier et al., 1993; Tolea, 2009) to detect OB associations using O-B2 stars.
We applied the PLC method varying the minimum number of stars (Nmin) and
maximum link distance (ds) (see Fig. 9). The best choice of Nmin and ds was
found to be Nmin = 5 stars and ds = 10.4 arcsec ($\sim$40 pc). Magnier et al.
(1993) found 174 OB associations and estimated a total number of $\sim$420
associations in M31 from a similar method but different optical photometry.
With this method, we found 650 OB associations in M31 from the O-B2 stars
selected by us from Massey et al. (2006) photometry, which is $\sim$ 2 mag
deeper than what Magnier et al. (1993) used. We compared these OB associations
with the 894 SF regions selected from FUV imaging. They mostly overlap with
the FUV-detected regions (see Fig. 10), however some of FUV-selected SF
regions have a larger area than OB associations defined from stellar
photometry, and some additional OB associations are found from stellar
photometry in high interstellar extinction regions. The observation fields of
Massey et al. (2006) cover a smaller area than our GALEX imaging, therefore we
compared our results within 17 kpc de-projected distance (about 1 deg2; dashed
ellipse in Fig. 10) from the center of M31. The total area of the SF regions
derived from the FUV contours (Section 3), and of the OB associations derived
from stellar photometry (above), are 4.1 % and 3.5 % of the area of the 17 kpc
disk, respectively. The numbers of O-B2 stars are 9094/11350 and 8670/11774
inside/outside of the UV-defined SF regions and of OB associations. The
average projected density of OB stars in the associations is 0.017 and 0.018
stars arcsec-2 from UV-selected SF regions and OB associations selected from
stellar photometry, respectively. The projected density of stars inside SF
regions is about a factor of 20 higher than in general field. The comparison
between the two methods is interesting, because the FUV-selected contours are
more affected by interstellar reddening than the optical stellar photometry,
on the other hand optical bands are not as sensitive as the FUV is to the Teff
of the hottest stars (e.g. Bianchi, 2006, and references therein). The
similarity of the total number of OB stars detected inside young associations
and in the field, by the two methods is remarkable. The slight difference in
the estimated area of the associations may be due, at least partly, to the low
($\approx$5 arcsec) spatial resolution of the GALEX imaging.
## 6 Ages and Masses of Star-Forming Region
We estimated the ages of the SF regions by comparing the measured (FUV $-$
NUV) colour with synthetic Simple Stellar Population (SSP) models, reddened by
the extinction amount estimated for each region. We explored effects of
metallicity and dust type on the results (e.g. Bianchi, 2006, 2009). Then we
estimated the masses of the SF regions from the reddening-corrected UV
luminosity and the derived ages. We used two sets of SSP models, one from
Bruzual & Charlot (2003) (BC03) and the other from Padua (PD: A. Bressan,
priv. comm., 2007). The ages derived from the two grids of models do not
differ significantly (see lower left panel of Fig. 11). Ages estimated using
the BC03 models tend to be slightly younger than those derived using the PD
models, below 30 Myrs, and do not differ at all for older populations (Fig.
11, lower left panel). The small age difference propagates to the derived
masses, as shown in the lower-right panel of Fig. 11. The differences between
results from the two model grids is not significant. We used the PD models in
our analysis.
Most SF regions have UV colour between $-0.5$ and 1.0 (see upper left panel of
Fig. 11). The estimated ages of most SF regions in our sample are younger than
400 Myrs, reflecting our FUV-based selection. Our detection limit, plotted
with a line in Fig. 11, indicates quantitatively how the flux-detection limit
translates into mass detection limit, as a function of age, and shows that we
cannot detect low mass SF regions at older ages, as expected. However, we also
notice a lack of massive SF regions at younger ages. This will be discussed
later.
We explored three metallicity values: subsolar (Z=0.008), solar (Z=0.02), and
supersolar (Z=0.05) metallicity, although M31 is believed to have a typical
metallicity about twice higher than the MW (e.g. Massey, 2003, and references
therein). Our census of young stellar populations based on wide-field FUV
imaging has an unprecedented extent, while direct metallicity measurements
from spectroscopy are confined to limited samples. Therefore, we wanted to
assess in general the dependence of our results on metallicity, which may vary
in some environments. We also examined the effect of four types of
interstellar dust: Milky Way ($R_{V}$ = 3.1; MW, Cardelli et al. (1989)),
average Large Magellanic Cloud (AvgLMC), 30 Doradus (LMC2) (Misselt et al.,
1999), and Small Magellanic Cloud (SMC, Gordon & Clayton, 1998) dust
extinction. The resulting ages and masses of the SF regions are plotted in
Figs 11, 12, and 13.
The differences in derived ages and masses for three metallicity values and
different types of interstellar dust are presented in Fig. 12. We considered
metallicity values of no less than Z=0.008 because we expect the young SF
regions not to be as metal poor as old globular clusters. In Fig. 12 ages and
masses derived from models with solar metallicity, and assuming MW-type
interstellar reddening (RV = 3.1), are compared to results from subsolar
(Z=0.008) and supersolar (Z=0.05) metallicities (top four panels). The derived
ages are older for subsolar metallicity, and younger for supersolar
metallicity, with respect to solar metallicity results. The differences are
most significant for ages younger than 100 Myrs (see also Fig. 9 of Bianchi,
2009) and are up to a factor of $\sim$3 at most. Because the effect is
stronger at certain ages, the number distribution of SF regions with ages also
differs, as shown in Fig. 12. This will be taken into account in the following
analysis, where we derive the global SF in M31 as a function of time. The
difference in the derived masses is not conspicuous, considering the
uncertainties. The uncertainty of the ages derived by comparing the photometry
to synthetic population models, reported in Table 2, is derived by propagating
only the photometric errors, because we investigated and showed separately the
effects of different metallicity values and dust types. The uncertainty on the
derived masses reflects the uncertainty on the photometry and the age.
Reddening corrections, as derived in Section 5, are applied.
The lower four panels of Fig. 12 show the effects of the correction for
reddening. If the selective extinction by dust is steeper in the UV than the
MW dust, as observed for example in the LMC (average) or the extreme starburst
regions 30 Dor (labelled as “AvgLMC” and “LMC2”, respectively, in Fig. 12),
the dereddened UV luminosity will be higher but the dereddened FUV $-$ NUV
color bluer, resulting in much younger ages and consequently lower masses. An
LMC-type dust is however not likely in M31. Bianchi et al. (1996) report UV
extinction curves in M31 similar to the average MW extinction, from UV spectra
of OB stars. Moreover, if we apply UV extinction steeper than MW dust, most
measured FUV $-$ NUV colors appear to be over-corrected, when compared to SSP
model predictions. Out of 847 SF regions whose (FUV $-$ NUV)0 is within the
model color range when dereddened with MW dust type, only 569/227/26 SF
regions have (FUV $-$ NUV)0 colors within the model range if the progressively
steeper dust types AvgLMC/LMC2/SMC are applied. Therefore, UV-steep dust
extinction seems to be not realistic in most cases.
## 7 Results
### 7.1 Spatial distribution of the Star-Forming regions
The FUV-selected SF regions follow the disk structure of M31 and their spatial
distribution traces the recent star formation in M31 (Fig. 13). As can be seen
in Fig. 13, most SF regions have de-projected distances between 40 and 75
arcmin (9 and 17 kpc) from the center of M31, with two peaks in this region.
This region is well-known as the ring of fire or the star-formation ring
(Block et al., 2006, and references therein). In this star formation ring
between 40 and 75 arcmin from the galaxy center, the number of young ($<$ 10
Myrs) SF regions is similar to that of older ($>$ 10 Myrs) ones. One thing of
interest is that the number of younger SF regions (82) is larger than the
number of older SF regions (40) outside of this ring ($d_{de-projected}$ $>$
75 arcmin). Inside of the star formation ring, the number of younger SF
regions (14), however, is less than that of older ones (89). This suggests
that the M31 disk formed stars continuously during the last few hundreds Myrs
at least and, furthermore, the outer disk shows more recent star formation.
The size of some younger SF regions is larger than that of older SF regions,
however most of them are less massive (as derived from the UV flux) than older
regions. Young, large SF regions may be broken into several SF regions with
time, while some of the dense, small SF regions may survive longer than the
others.
### 7.2 Recent Star Formation History in M31
We noticed a lack of massive SF regions younger than 50 Myrs in Fig. 11. We
would expect to find some massive young SF regions if star formation was
constant. We estimated the total SFR in M31 using the ages and masses of the
SF regions derived in Section 6. We added the estimated initial masses of the
SF regions separated in four time bins, to investigate the SFR time evolution.
The results are shown in Fig. 14, and reported in Table 3, for three
metallicity values. Although in each age interval the derived SFR depends on
the assumed metallicity, in all cases Fig.14 shows an apparent decrease of SFR
in the recent epoch ($<$ 10 Myrs) with respect to the average value in the
interval 10-100 Myrs ago, the difference being smallest (and probably not
significant) for supersolar metallicity, which is currently believed to be the
most probable value for M31. The dashed line across the whole time interval is
the average from the SF regions of all ages. When interpreting this diagram,
we must first of all recall that the masses are estimated from the UV flux
above a certain threshold, and the corresponding limit for mass detection
increases at older ages (Fig. 11, continuous line). Therefore, the total
stellar mass formed over 100 Myrs ago may be relatively underestimated
compared to that of younger populations, due to our FUV selection. This bias
makes the apparent decrease in SFR at young ages more robust. Most of the
time-binned and mean SFRs are lower than one solar mass per year. These values
can be compared to the M31’s SFRs from IR and H$\alpha$. Barmby et al. (2006)
estimated a SFR of 0.4 $M_{\sun}$ yr-1 from 8 $\micron$m non-stellar emission,
higher than our UV-derived SFR in the $<$4 Myrs bin, if we assume solar
metallicity, but comparable to our value for supersolar metallicity. This
shows that each indicator alone, UV or IR, may miss about half of the most
recently formed stellar mass. Massey et al. (2007) estimated 0.05 $M_{\sun}$
yr-1 from H$\alpha$ luminosities, lower (within a factor of two) than our
estimate for the $<$10 Myrs bin (solar metallicity).
## 8 Conclusion and Discussion
We used 7 GALEX fields covering the entire disk of M31 (out to $\sim$ 26 kpc
radius) to study its young stellar population. We detected 894 SF regions from
the FUV imaging, and measured their integrated FUV and NUV fluxes. We
estimated the interstellar extinction in each SF region from the OB stars
within its contour, using the ground-based stellar photometry of Massey et al.
(2006). We estimated ages and masses of our SF regions in M31, detected from
FUV imaging, by comparing the FUV and NUV measurements to population synthesis
models. Most are younger than 400 Myrs (UV is not sensitive to older ages).
Interestingly, there are no massive SF regions at young ages (age $<$ 50 Myrs
in Fig. 11), suggesting an apparent decrease of the SFR, as detected from the
FUV imaging.
There may be a potential bias, due to our definition of the SF contours from
the FUV image, concerning the size distribution and average surface
brightness. Younger SF complexes tend to be more compact, and older stellar
associations are less dense, in general (Bianchi, 2006; Efremova & Bianchi,
2009). Therefore, the same flux detection algorithm may break a young SF area
into several peaks, but these contiguous regions may appear connected if they
spread out at older ages. This bias, however, does not appear significant from
visual inspection of the complex SF areas along spiral arms, and younger
associations tend to have larger size in our selection. In any case, the bias
would only affect the apparent distribution [with age] of sizes and masses of
the individual SF regions, and not affect the results when we add all
individual masses in a wide age interval, in order to derive the total SFR in
M31. Because we have estimated ages and masses of the individual SF complexes,
and the UV flux is sensitive to a much broader age range than e.g. H$\alpha$
or IR, we were also able to derive the recent SFR in M31 as a function of
time. We estimated the SFR in four time intervals adding the masses of the SF
regions of corresponding ages. The resulting masses are restricted by our
detection limit which is shown in Fig. 11. The plot in Fig. 14 shows a recent
apparent decrease of the SFR, as derived from the UV flux. However, we know
that the youngest and compact SF regions are often still embedded in dust
(e.g. Bianchi, 2006, and references therein). These escape detection from UV
imaging, but are instead revealed by IR emission from heated dust. Therefore,
for a complete account of star formation in the very young age bins ($<$ 10
Myrs), we added the IR-measured star formation from Barmby et al. (2006). The
total SFR at young ages (UV + IR estimates) is shown as thick lines at the top
of vertical arrows in Fig. 14. This more realistic and complete estimate of
the star formation in the recent few million years significantly reduces the
apparent recent decrease in SFR, as derived from UV flux only.
We also show the average stellar mass formed in a $\leq$ 10 Myrs bin, from our
UV measurements, for comparison to H$\alpha$ estimates. Because H$\alpha$ is
an indirect measurement of the ionizing photons from the short-lived O-type
stars, the SFR derived from this method must be compared to our age bin of
$\leq$ 10 Myrs. The UV and H$\alpha$ SFR estimates agree within a factor of
two, for solar metallicity, as shown in Fig. 14. If we assume supersolar
metallicity for all sources (Z=0.05), then the H$\alpha$ underestimates the
SFR by a factor of ten with respect to the SFR derived from the UV sources in
the recent 10 Myrs. We note that the results shown in Fig. 14 are derived by
dereddening the UV fluxes with MW-type dust extinction. If UV-steeper
reddening applies to the intrinsic dust extinction in some SF regions, their
derived ages would be younger (see Fig. 12), resulting in a SFR lower for
older ages and higher for younger ages than the values shown in Fig.14, and
increasing the discreapancy with the H$\alpha$ estimate.
Even when the IR- and UV-derived SFRs are added at the youngest epochs, there
seem to be a recent decrease of SFR, or a peak of SFR between $\sim$ 10-100
Myrs (although the UV estimate of SF at older ages is a lower limit, as
previously explained). This seems to suggest that M31 had a starburst during
this interval. A possible scenario is that galaxy interaction may have induced
violent star formation around this time interval. Gordon et al. (2006)
considered a collisional event with M32 around 20 Myrs ago to explain the ring
structure by dynamical models. Block et al. (2006) also postulated a head-on
collision event with M32 and estimated it happened around 210 Myrs ago. We now
have a possible evidence of a recent starburst in M31, which may have
constructed the ring structure.
This work presented the first estimate of the recent SFR based on measurements
of individual SF regions across the entire galaxy, from UV imaging. Massey et
al. (2007) provide a comparison of SF, based on H$\alpha$ measurements, among
Local Group galaxies, and several other authors give estimates of SF in nearby
galaxies. However, their results are mostly derived under the assumption of
continuous star formation, translating global fluxes into SF. Our study
provides a time-resolved SFR over the past few hundred million years. We will
perform a similar analysis on other Local Group galaxies, for a consistent
comparison of results from our method within a range of physical environments.
Such comparison should also clarify the relative calibration of UV, IR, and
H$\alpha$ as SF indicators as a function of galaxy physical conditions.
In future works, we will also compare the parameters describing the properties
of SF regions from the integrated measurements with resolved studies of their
stellar populations (from ground-based and Bianchi’s HST programs data). We
will estimate ages and masses from deeper GALEX images to test the limit of
our current detection threshold.
YBK was supported by the Korea Research Foundation Grant funded by the Korean
Government(MOEHRD) (KRF-2007-612-C00047). SCR acknowledges support from the
KOSEF through the Astrophysical Research Center for the Structure and
Evolution of the Cosmos (ARCSEC). We are grateful to D. Thilker for
discussions about the background subtraction, to K. Kuntz for a careful
reading of the manuscript and useful comments, and A. Tolea for providing some
of the procedures he developed for his dissertation with Luciana Bianchi and
K. Kuntz. We are also grateful to P. Massey for discussions about the
reddening value and for providing the revised photometry catalog. This work is
based on archival data from the NASA Galaxy Evolution Explorer (GALEX) which
is operated for NASA by the California Institute of Technology under NASA
contract NAS5-98034. The GALEX data presented in this paper were obtained from
the Multimission Archive at the Space Telescope Science Institute (MAST).
Support for MAST for non-HST data is provided by the NASA Office of Space
Science via grant NAG5-7584 and by other grants and contracts. GALEX (Galaxy
Evolution Explorer) is a NASA Small Explorer, launched in April 2003. We
gratefully acknowledge NASA’s support for construction, operation, and science
analysis of the GALEX mission, developed in cooperation with the Centre
National d’Etudes Spatiales of France and the Korean Ministry of Science and
Technology.
## References
* Aller et al. (1982) Aller, L. H., et al. 1982, Landolt-Bornstein: Numerical Data and Functional Relationships in Science and Technology
* Barmby et al. (2006) Barmby, P., et al. 2006, ApJ, 650, L45
* Battinelli (1991) Battinelli, P. 1991, A&A, 244, 69
* Bianchi (2006) Bianchi, L., & et al. 2006, The Ultraviolet Universe: Stars from Birth to Death, 26th meeting of the IAU, Joint Discussion 4, 16-17 August 2006, Prague, Czech Republic, JD04, 37, 4
* Bianchi (2009) Bianchi, L. 2009, Ap&SS, 320, 11
* Bianchi & Efremova (2006) Bianchi, L., & Efremova, B. V. 2006, AJ, 132, 378
* Bianchi et al. (1996) Bianchi, L., Clayton, G. C., Bohlin, R. C., Hutchings, J. B., & Massey, P. 1996, ApJ, 471, 203
* Bianchi et al. (2003) Bianchi, L., Madore, B., Thilker, D., Gil de Paz, A., Martin, C., & The GALEX Team 2003, The Local Group as an Astrophysical Laboratory, 10
* Bianchi et al. (2007) Bianchi, L., et al. 2007, ApJS, 173, 659
* Block et al. (2006) Block, D. L., et al. 2006, Nature, 443, 832
* Bruzual & Charlot (2003) Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000
* Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
* Efremova & Bianchi (2009) Efremova, B. V. & Bianchi, L. 2009, in preparation
* Fuchs et al. (2009) Fuchs, B., Jahreiß, H., & Flynn, C. 2009, AJ, 137, 266
* Gordon & Clayton (1998) Gordon, K. D., & Clayton, G. C. 1998, ApJ, 500, 816
* Gordon et al. (2006) Gordon, K. D., et al. 2006, ApJ, 638, L87
* Hammer et al. (2007) Hammer, F., Puech, M., Chemin, L., Flores, H., & Lehnert, M. D. 2007, ApJ, 662, 322
* Hou et al. (2009) Hou, J., Yin, J., Boissier, S., Prantzos, N., Chang, R. X., & Chen, L. 2009, IAU Symposium, 254, 27
* Ibata et al. (2007) Ibata, R., Martin, N. F., Irwin, M., Chapman, S., Ferguson, A. M. N., Lewis, G. F., & McConnachie, A. W. 2007, ApJ, 671, 1591
* Ivanov (1996) Ivanov, G. R. 1996, A&A, 305, 708
* Ivanov (1998) ——. 1998, A&A, 337, 39
* Kennicutt (1998) Kennicutt, R. C., Jr. 1998, ApJ, 498, 541
* Magnier et al. (1993) Magnier, E. A., et al. 1993, A&A, 278, 36
* Martin et al. (2005) Martin, D. C., et al. 2005, ApJ, 619, L1
* Massey (2003) Massey, P. 2003, ARA&A, 41, 15
* Massey et al. (1995) Massey, P., Armandroff, T. E., Pyke, R., Patel, K., & Wilson, C. D. 1995, AJ, 110, 2715
* Massey et al. (2006) Massey, P., Olsen, K. A. G., Hodge, P. W., Strong, S. B., Jacoby, G. H., Schlingman, W., & Smith, R. C. 2006, AJ, 131, 2478
* Massey et al. (2007) Massey, P., Olsen, K. A. G., Hodge, P. W., Jacoby, G. H., McNeill, R. T., Smith, R. C., & Strong, S. B. 2007, AJ, 133, 2393
* McConnachie et al. (2005) McConnachie, A. W., Irwin, M. J., Ferguson, A. M. N., Ibata, R. A., Lewis, G. F., & Tanvir, N. 2005, MNRAS, 356, 979
* Misselt et al. (1999) Misselt, K. A., Clayton, G. C., & Gordon, K. D. 1999, ApJ, 515, 128
* Morrissey et al. (2007) Morrissey, P., et al. 2007, ApJS, 173, 682
* Tolea (2009) Tolea, A. C. 2009, Ph.D. Thesis
* Walterbos & Kennicutt (1988) Walterbos, R. A. M., & Kennicutt, R. C., Jr. 1988, A&A, 198, 61
Figure 1: $GALEX$ fields covering the M31 disk region from the GR4 data
release. Black circles are our selected $GALEX$ fields and grey circles are
additional $GALEX$ observations, not selected. Black dots are OB type stars
selected from the ground-based photometric catalog of Massey et al. (2006), as
described in Section 4. Figure 2: Differences of source detection in the
“NGA_M31_MOS0” field using three different flux thresholds. Red contours are
derived using the low threshold (mean background + 2$\sigma$), blue contours
using the mid threshold (mean background + 3$\sigma$), and orange contours
using the high threshold (mean background + 5$\sigma$). The solid circle is
the FOV used in our analysis and the dashed circle is the whole $GALEX$ FOV.
Enlargements of sample regions are shown in the rectangular panels. Background
images are the grey scaled FUV images of the “NGA_M31_MOS0” field. Figure 3:
The color-magnitude diagram of SF regions in the “NGA_M31_MOS0” field. Red
dots are detections using the low flux threshold (mean background +
2$\sigma$), blue dots are for the mid threshold (mean background + 3$\sigma$),
and orange dots are for the high threshold (mean background + 5$\sigma$).
Dashed lines are the magnitude limit of each threshold. Figure 4: The detected
FUV pixel map of the selected 7 $GALEX$ fields. The solid circles are the FOV
of our selection (1.1 degree diameter) and the dashed circles are the whole
FOV of $GALEX$. Figure 5: Comparison of three different methods of sky
background estimations. The sky fraction in our photometry (“NGA_M31_MOS0”
field) is displayed as the fraction of sky flux over object’s original flux
(sky un-subtracted flux). In the first and second column, it is plotted with
the measured magnitude of objects and the size of the objects, respectively.
In the third column, it is displayed with de-projected distance from the
center of M31. The upper 9 panels are NUV measurements and the lower 9 panels
are FUV. The background is always higher in NUV because this filter includes
light from older, more diffuse populations. Therefore, the background
subtraction is more critical for NUV. Figure 6: Comparison of photometry of
the UV sources obtained from three different types of sky subtraction for the
“NGA_M31_MOS0” field. Black dots are obtained subtracting the sky flux
measured from the local background. The end points of the blue arrows (left
panel) are the photometry results using the GR4 pipeline sky. The end points
of the red arrows (right panel) are from our median (3$\times$3 pixel)
filtered sky. Figure 7: The selection of OB type stars from the ground-based
photometry of Massey et al. (2006). The plotted points have magnitude error in
$UBV$ bands lower than 0.1 mag. Blue points are our selected OB stars. The
lines represent the intrinsic color as a function of Teff for luminosity
classes main-sequence (V, red), giant (III, green), and supergiant (Ib and Ia,
blue and purple) from Aller et al. (1982). The red arrow in each panel is the
direction of reddening with $E(B-V)=0.7$. Figure 8: Number distributions of
reddening for the selected OB stars and for the detected SF regions (left
panels). Spatial distributions of OB stars and SF regions are presented color-
coded by three ranges of reddening in the right panels. Figure 9: The mean
normalized fluctuation function changes with the minimum number of stars of an
association (Nmin) and the maximum link distance (ds) between stars. The black
arrow indicates the maximum peak of this function, which defined the
parameters adopted for our selection of OB associations. Figure 10: Spatial
distribution of SF regions detected from the $GALEX$ imaging (blue contours)
and OB associations from optical stellar photometry (red contours). Black dots
are the selected OB stars. The solid ellipse has a 26 kpc de-projected radius
and the dashed ellipse has a 17 kpc de-projected radius.
Figure 11: Upper left panel: the colour-magnitude diagram of the SF regions.
Upper right panel: the estimated ages and masses assuming solar metallicity
(Z=0.02) with interstellar extinction of MW dust type using the PD model grid.
The line below the data points is the model-estimate of our detection limit,
based on the limiting magnitudes. Lower left/right panels: age/mass
differences between two sets of models, assuming solar metallicity with MW
dust type. Figure 12: Ages and masses of the SF regions, derived assuming
various metallicities and dust types. Results from the PD models are shown.
Figure 13: Top panel: the spatial distribution of SF regions (yellow contours)
on a colour composite image (blue: FUV, green: FUV+NUV, red: NUV). The blue
and bright blob on the outermost ellipse is a bright Galactic foreground star
(HD 3431). Bottom panels: distributions of ages, masses, size, and mass-per-
unit-area of SF regions against the de-projected distance from the center of
M31. Different symbol colors indicate different age bins (red: older than 100
Myrs, green: 10 - 100 Myrs, blue: younger than 10 Myrs). The dashed vertical
lines correspond to the ellipses drawn on the top image at 40, 75, and 120
arcmin, corresponding to deprojected distances in M31 of 9, 17, and 27 kpc,
respectively. The ages and masses shown here were derived using metallicity
Z=0.05 and MW (RV=3.1) dust type. Figure 14: SFRs at recent epochs in M31,
from our SF regions. Black lines are values derived for solar metallicity,
blue lines are for Z=0.05, and green lines are for Z=0.008. In all cases MW-
type dust was assumed to correct the UV luminosities for interstellar
extinction, as discussed in Section 6. Solid horizontal lines represent the
SFR in four time bins: $<$4, $<$10, 10-100, and 100-400 Myrs, and the dashed
lines are mean values over the past 400 Myrs. The red line is the SFR from
H$\alpha$ (Massey et al., 2007), shown in the $<$10 Myrs age bin which traces
only the youngest stars capable of ionizing the ISM. The thick horizontal
lines above vertical arrows on the $<$4 Myrs age bin and $<$10 Myrs indicate
the total SFRs obtained by adding the SFR from IR measurements (Barmby et al.,
2006) to our UV-based estimates.
Table 1: Details of the selected $GALEX$ fields of M31’s disk region No | Field Name | R.A. | Dec. | Exp. time | Adopted thresh. | Detected pixels
---|---|---|---|---|---|---
| | [deg] | [deg] | [s] | [c/s] | above thresh.
1 | NGA_M31_MOS11 | 12.204914 | 42.956084 | 3340.35 | 0.00397 | 5533
2 | NGA_M31_MOS8 | 12.164794 | 42.030947 | 2702.75 | 0.00384 | 36791
3 | NGA_M31_MOS18 | 11.403102 | 42.370208 | 3244.45 | 0.00365 | 77217
4 | NGA_M31_MOS4 | 11.253077 | 41.863775 | 3589.70 | 0.00373 | 209395
5 | PS_M31_MOS00 | 10.683594 | 41.277741 | 3760.10 | 0.00353 | 188512
6 | NGA_M31_MOS0 | 10.173990 | 40.836523 | 6811.30 | 0.00324 | 188847
7 | PS_M31_MOS03 | 9.957439 | 40.358496 | 1182.20 | 0.00446 | 152669
Table 2: The UV detected SF regions in M31.aaFull catalog available in
electronic version.
Id | R.A.J2000 | Dec.J2000 | FUV | FUVerr | NUV | NUVerr | $E(B-V)$ | Area | Ageb,cb,cfootnotemark: | Agemin/maxb,cb,cfootnotemark: | Massb,cb,cfootnotemark: | Massmin/maxb,cb,cfootnotemark: | Ageb,db,dfootnotemark: | Agemin/maxb,db,dfootnotemark: | Massb,db,dfootnotemark: | Massmin/maxb,db,dfootnotemark:
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [ABmag] | [ABmag] | [ABmag] | [ABmag] | [mag] | [arcsec2] | [Myrs] | [Myrs]/[Myrs] | [M☉] | [M☉]/[M☉] | [Myrs] | [Myrs]/[Myrs] | [M☉] | [M☉]/[M☉]
1003 | 12.013985 | 42.690975 | 18.870 | 0.020 | 19.158 | 0.013 | 0.20 | 112.5 | 1.6 | 1.3/ 1.9 | 3.7e+02 | 4.4e+02/ 3.7e+02 | -99.0 | -99.0/ 1.2 | -9.9e+01 | -9.9e+01/ 3.8e+02
2040 | 11.951906 | 42.073174 | 19.124 | 0.025 | 19.331 | 0.016 | 0.06 | 126.0 | 4.1 | 2.9/ 5.7 | 1.4e+02 | 9.0e+01/ 2.2e+02 | 2.3 | 1.8/ 2.7 | 8.2e+01 | 8.2e+01/ 1.1e+02
3069 | 11.725769 | 41.968216 | 19.770 | 0.034 | 19.986 | 0.023 | 0.05 | 155.2 | 3.7 | 2.1/ 5.8 | 6.8e+01 | 5.4e+01/ 1.1e+02 | 2.1 | 1.4/ 2.7 | 4.2e+01 | 5.6e+01/ 5.6e+01
3071 | 11.715304 | 41.971684 | 18.494 | 0.018 | 18.420 | 0.010 | 0.13 | 328.5 | 58.7 | 49.9/ 66.5 | 1.5e+04 | 1.2e+04/ 1.9e+04 | 22.4 | 17.6/ 30.8 | 5.0e+03 | 5.0e+03/ 8.2e+03
3074 | 11.641906 | 41.971554 | 20.070 | 0.039 | 20.057 | 0.023 | 0.19 | 117.0 | 40.5 | 24.5/ 49.7 | 3.2e+03 | 1.4e+03/ 4.4e+03 | 10.1 | 6.8/ 17.5 | 7.4e+02 | 4.5e+02/ 1.8e+03
3077 | 11.618323 | 41.983509 | 18.498 | 0.019 | 18.633 | 0.012 | 0.30 | 452.2 | 7.1 | 5.7/ 8.8 | 3.3e+03 | 2.5e+03/ 4.9e+03 | 3.1 | 2.8/ 3.6 | 1.2e+03 | 1.2e+03/ 1.8e+03
3081 | 11.635375 | 41.991177 | 16.531 | 0.007 | 16.393 | 0.004 | 0.19 | 2072.2 | 79.4 | 76.8/ 82.0 | 2.2e+05 | 2.2e+05/ 2.2e+05 | 48.0 | 45.3/ 50.4 | 1.6e+05 | 1.6e+05/ 1.6e+05
3082 | 11.658252 | 41.985649 | 18.267 | 0.016 | 18.131 | 0.008 | 0.26 | 229.5 | 74.8 | 68.9/ 80.6 | 6.4e+04 | 6.4e+04/ 7.6e+04 | 43.3 | 34.6/ 49.3 | 4.2e+04 | 2.8e+04/ 5.6e+04
3083 | 11.625877 | 41.988590 | 19.501 | 0.032 | 19.476 | 0.019 | 0.46 | 207.0 | 19.9 | 14.8/ 29.8 | 1.8e+04 | 7.8e+03/ 3.0e+04 | 6.2 | 4.8/ 8.1 | 4.0e+03 | 2.6e+03/ 6.8e+03
3085 | 11.648593 | 41.992626 | 20.509 | 0.055 | 20.507 | 0.033 | 0.30 | 112.5 | 28.4 | 15.2/ 47.0 | 3.6e+03 | 2.2e+03/ 6.8e+03 | 7.7 | 4.8/ 15.5 | 8.1e+02 | 3.1e+02/ 2.7e+03
3086 | 11.654299 | 41.999249 | 17.305 | 0.011 | 17.136 | 0.006 | 0.20 | 1055.2 | 89.3 | 85.2/ 94.1 | 1.4e+05 | 1.4e+05/ 1.4e+05 | 54.9 | 52.2/ 57.6 | 8.9e+04 | 8.9e+04/ 1.2e+05
… | … | … | … | … | … | … | … | … | … | …/… | … | …/… | … | …/… | … | …/…
bbfootnotetext: “-99.0” and “-9.9e+01” indicate the cases where the observed
color is outside the range of model colors at all ages.
ccfootnotetext: Metallicity Z=0.02 and average MW ($R_{V}$ = 3.1) dust type.
ddfootnotetext: Metallicity Z=0.05 and average MW ($R_{V}$ = 3.1) dust type.
Table 3: M31 SFR derived from UV flux in different age intervals. Metallicity | SFR[$M_{\sun}$/yr] from UV-detected SF regions
---|---
(Z) | $<$4 Myrs**The value inside parentheses is obtained by adding the SFR from IR measurements (Barmby et al., 2006) to the SFR derived from our UV measurements. | $<$10 Myrs**The value inside parentheses is obtained by adding the SFR from IR measurements (Barmby et al., 2006) to the SFR derived from our UV measurements. | 10-100 Myrs | 100-400 Myrs | $<$400 Myrs
0.008 | 0.006(0.406) | 0.014(0.414) | 0.325 | 0.610 | 0.532
0.020 | 0.033(0.433) | 0.062(0.462) | 0.826 | 0.569 | 0.616
0.050 | 0.192(0.592) | 0.434(0.834) | 1.811 | 0.361 | 0.690
|
arxiv-papers
| 2009-06-26T04:22:28 |
2024-09-04T02:49:03.551083
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yongbeom Kang, Luciana Bianchi, Soo-Chang Rey",
"submitter": "Yongbeom Kang",
"url": "https://arxiv.org/abs/0906.4839"
}
|
0906.4843
|
Loop Groups, Higgs Fields and Generalised String Classes
Raymond Vozzo
Thesis submitted for the degree of
Doctor of Philosophy
in
Pure Mathematics
at
The University of Adelaide
School of Mathematical Sciences
June 22, 2009
###### Contents
1. Abstract
2. Signed Statement
3. Acknowledgements
4. 1 Introduction
5. 2 String structures, bundle gerbes and Higgs fields
1. 2.1 String structures
1. 2.1.1 Spin structures
2. 2.1.2 String structures
2. 2.2 Bundle gerbes
1. 2.2.1 $U(1)$-bundles
2. 2.2.2 Bundle gerbes
3. 2.3 Central extensions and the lifting bundle gerbe
1. 2.3.1 Simplicial line bundles and central extensions
2. 2.3.2 The lifting bundle gerbe
4. 2.4 The string class of an $LG$-bundle
5. 2.5 Higgs fields, $LG$-bundles and the string class
1. 2.5.1 Higgs fields and $LG$-bundles
2. 2.5.2 The string class and the first Pontrjagyn class
6. 3 Higgs fields and characteristic classes for $\Omega G$-bundles
1. 3.1 String structures and the path fibration
1. 3.1.1 Classifying maps and characteristic classes
2. 3.1.2 String structures and the path fibration
2. 3.2 Higher string classes for $\Omega G$-bundles
3. 3.3 The universal string class for $L^{\scriptscriptstyle{\vee}}G$-bundles
1. 3.3.1 Classification of $L^{\scriptscriptstyle{\vee}}G$-bundles
2. 3.3.2 The universal string class
7. 4 String structures for $LG\rtimes S^{1}$-bundles
1. 4.1 The string class of an $LG\rtimes S^{1}$-bundle
1. 4.1.1 The string class via lifting bundle gerbes
2. 4.1.2 Reduced splittings for lifting bundle gerbes
2. 4.2 Higgs fields, $LG\rtimes S^{1}$-bundles and the string class
1. 4.2.1 Higgs fields and $LG\rtimes S^{1}$-bundles
2. 4.2.2 The string class and the first Pontrjagyn class
3. 4.3 String structures for $LG\rtimes\operatorname{Diff}(S^{1})$-bundles
8. A Infinite-dimensional manifolds and Lie groups
1. A.1 Fréchet spaces
2. A.2 Groups of maps
3. A.3 The path fibration
9. B Classification of semi-direct product bundles
1. B.1 Classification of semi-direct product bundles
2. B.2 $LG\rtimes S^{1}$-bundles
## Abstract
We consider various generalisations of the string class of a loop group
bundle. The string class is the obstruction to lifting a bundle whose
structure group is the loop group $LG$ to one whose structure group is the
Kac-Moody central extension of the loop group.
We develop a notion of higher string classes for bundles whose structure group
is the group of based loops, $\Omega G$. In particular, we give a formula for
characteristic classes in odd dimensions for such bundles which are associated
to characteristic classes for $G$-bundles in the same way that the string
class is related to the first Pontrjagyn class of a certain $G$-bundle
associated to the loop group bundle in question. This provides us with a
theory of characteristic classes for $\Omega G$-bundles analogous to Chern-
Weil theory in finite dimensions. This also gives us a geometric
interpretation of the well-known transgression map $H^{2k}(BG)\to
H^{2k-1}(G).$
We also consider the obstruction to lifting a bundle whose structure group is
not the loop group but the semi-direct product of the loop group with the
circle, $LG\rtimes S^{1}$. We review the theory of bundle gerbes and their
application to central extensions and lifting problems and use these methods
to obtain an explicit expression for the de Rham representative of the
obstruction to lifting such a bundle. We also relate this to a generalisation
of the so-called ‘caloron correspondence’ (which relates $LG$-bundles over $M$
to $G$-bundles over $M\times S^{1}$) to a correspondence which relates
$LG\rtimes S^{1}$-bundles over $M$ to $G$-bundles over $S^{1}$-bundles over
$M$.
## Signed Statement
This work contains no material which has been accepted for the award of any
other degree or diploma in any university or other tertiary institution and,
to the best of my knowledge and belief, contains no material previously
published or written by another person, except where due reference has been
made in the text.
I consent to this copy of my thesis, when deposited in the University Library,
being available for loan and photocopying, subject to the provisions of the
Copyright Act 1968.
SIGNED: ....................... DATE: .......................
## Acknowledgements
I would like to thank my supervisors, Michael and Mathai, for their invaluable
assistance and guidance, without which this project surely would never have
finished (or, at least, would have taken a significantly longer time). I have
learned more about how Mathematics is done in the last three years from
talking to them than I thought there even was to know. Thanks are also due to
Danny Stevenson for many useful discussions (and possibly even more so for his
willingness to help me).
Thanks are also due to all the staff in the admin office, whose hard work has
made my life much easier over the past few years.
I would also like to thank my fellow postgrads, all of whom have made the last
four years of my life more than bearable (to say the least!). I shall not
endeavour to name everyone here (since I want to fit this whole thing on one
page!), however special thanks are due in particular to Ric Green, David
Roberts, David Butler, Jonathan Tuke, Rongmin Lu, Glenis Crane and Jessica
Kasza. Their readiness to place their personal health at risk by ingesting
dangerous amounts of caffeine (with the possible exception of Rongmin and
David of course) just so we can take a break and discuss something other than
Mathematics for 15 (or 45 as the case may be) minutes is greatly appreciated.
By the same token, special thanks are due to David R and Ric for many, many
helpful discussions on not only my research topic but Mathematics and Physics
in general. Over the past three or four years we have discovered that other
people’s (Mathematical) problems are much more interesting than one’s own and
thanks to David and Ric I believe I have profited immensely from this fact.
Finally, I would like to thank my family for their continual love and support.
Without them this PhD would have remained a dream. I am immensely grateful to
my parents, Armando and Lucy, for their constant encouragement and to my
brother Jonathon and my sister Nicola, who have helped me keep some semblance
of my childhood (despite being nearly 25) and have never questioned why their
brother doesn’t have a job or a home of his own. And to my beautiful fiancée
Emily, for being my beautiful fiancée, and for putting up with so much
incomprehensible Maths over the years and whose love and patience has made all
this possible.
## Chapter 1 Introduction
String structures first appeared in Killingback’s paper [22] as a string
theory version of the well-known spin structures that are important in quantum
field theory. The results came out of a study of global anomalies in the
worldsheet of a string and the idea was motivated by an observation of Witten
[45] that the Dirac-Ramond operator in string theory can be considered as
Dirac-type operator on the loop space.
Recall that if one is given a principal $SO(n)$-bundle (for example the frame
bundle of a manifold), a spin structure is given by a lifting of the structure
group of this bundle to its simply connected double cover
$\operatorname{Spin}(n).$ Killingback’s idea then, is to replace the bundles
which appear in the definition of a spin structure with an infinite-
dimensional bundle whose structure group is the loop group of
$\operatorname{Spin}(n)$ and consider a lifting of this bundle. More
generally, if $G$ is a compact Lie group and $LG$ is its loop group, we could
consider lifting any $LG$-bundle $P\to M$ to a bundle whose structure group is
the central extension of $LG.$ It turns out that the obstruction $s(P)$ to the
existence of such a lift is an element of the degree three cohomology of the
base, $H^{3}(M,{\mathbb{Z}}).$ Killingback proved that, in the case where the
$LG$-bundle $P$ is in fact given by taking loops in a principal $G$-bundle
$Q\to X,$ this obstruction class is the transgression of the first Pontrjagyn
class of $Q.$ That is,
$s(P)=\int_{S^{1}}\operatorname{ev}^{*}p_{1}(Q),$
where $\operatorname{ev}\colon S^{1}\times LX\to X$ is the evaluation map. The
class $s(P)\in H^{3}(LX,{\mathbb{Z}})$ is called the string class of $P.$ The
link with spin structures and Witten’s observation regarding the Dirac-Ramond
operator is given by noticing that in quantum field theory the Dirac operator
can only be defined if spacetime is spin and correspondingly in string theory
the Dirac-Ramond operator can only be defined if spacetime is string (i.e.
has a string structure).
The present work grew out of an attempt to answer some questions naturally
arising from some papers concerning string structures and loop group bundles.
In [35] Murray and Stevenson use techniques from the theory of bundle gerbes
to give an explicit formula for a representative in de Rham cohomology of the
string class of a general $LG$-bundle and provide a link with previous work on
monopoles. The theory of gerbes was first introduced by Giraud [17] and
studied extensively in Brylinski’s book [4]. Gerbes provide a geometric
realisation for degree three cohomology in an analogous way to the way in
which line bundles (or $U(1)$-bundles) provide a geometric realisation of
degree two cohomology. Gerbes are essentially sheaves of groupoids satisfying
certain descent conditions but can be tricky to work with in practice. A much
more appealing (at least from a differential geometric point of view) approach
to the theory of gerbes, called bundle gerbes, was introduced by Murray [32].
These have been studied further (see for example [10, 18, 28, 30, 33]) and
have found applications in physics as well as differential geometry (see for
example [3, 7, 8, 15, 40]). Insofar as a gerbe can be considered a sheaf of
groupoids, bundle gerbes can be viewed as bundles of groupoids. They have a
degree three characteristic class associated with them, called the Dixmier-
Douady class, which can be described in terms of cocycles. However, one can
also define a notion of connection and curvature (more precisely, 3-curvature)
for a bundle gerbe and, using differential geometric methods, obtain a
differential form representative for the image in real cohomology of the
Dixmier-Douady class in analogy with the way the Chern class of a
$U(1)$-bundle is represented in real cohomology by the curvature of the
bundle. Bundle gerbes arise very naturally in lifting problems such as the
string structure example. This is the approach taken in [35] where a de Rham
representative of the string class for a loop group bundle $P$ is given in
terms of data on the bundle. Namely, the authors find that the string class is
given by
$s(P)=-\frac{1}{4\pi^{2}}\int_{S^{1}}\langle F,\nabla\Phi\rangle\,d\theta$
where $F$ is the curvature of $P,$ $\nabla\Phi$ is the covariant derivative of
a Higgs field for $P$ and the bracket is the Killing form suitably normalised.
They also extend Killingback’s result – that is, giving the string class in
terms of the Pontrjagyn class for some $G$-bundle – by using the so-called
‘caloron correspondence’ (which first appeared in [16]) which relates
$LG$-bundles over $M$ to $G$-bundles over $M\times S^{1}.$ In particular,
there is a bijective correspondence between isomorphism classes of principal
$LG$-bundles over $M$ and isomorphism classes of principal $G$-bundles over
$M\times S^{1}$ and if $P\to M$ is an $LG$-bundle and $\widetilde{P}\to
M\times S^{1}$ is its corresponding $G$-bundle, then the authors find that the
string class of $P$ is given by integrating the first Pontrjagyn class of
$\widetilde{P}:$
$s(P)=\int_{S^{1}}p_{1}(\widetilde{P}).$
The first formula above can be used to recover the result from [11] in which
the authors calculate the string class for the universal $\Omega
G$-bundle111Actually, in [11] the authors work with the group of smooth maps
from the interval into $G$ whose endpoints agree. In this thesis we extend
their work to the group of smooth maps from the circle into $G$. (where
$\Omega G$ is the based loop group) and show that the string class is a
characteristic class for loop bundles (that is, $\Omega G$-bundles of the form
$\Omega Q\to\Omega X$ for some $G$-bundle $Q\to X$). A model for the
classifying space of $\Omega G$ is given by the group $G$ itself and
$H^{3}(G,{\mathbb{Z}})={\mathbb{Z}}$ so it is not unreasonable to expect the
string class in this case to be the generator of this group. This is in fact
true and it is shown that the string class for any loop bundle is given by the
pull-back of this class by a classifying map for the bundle.
This thesis deals with two natural questions which arise when one considers
these results. The first concerns the relationship between the string class
and the Pontrjagyn class and the fact that the string class is a
characteristic class for loop bundles. It is natural, firstly, to look for a
way to generalise this to $\Omega G$-bundles which are not necessarily loop
bundles but, also, it seems possible that there is a more general theory of
characteristic classes for loop group bundles which is related to
characteristic class theory for $G$-bundles (i.e. Chern-Weil theory). In the
first part of this thesis we provide answers to these problems. We give a
generalisation of the result from [11] to $\Omega G$-bundles which are not
loop bundles, that is, we show that the string class is a characteristic
class. We then develop a notion of higher string classes for $\Omega
G$-bundles which are also characteristic classes and are related to
characteristic classes for $G$-bundles. In particular, we develop a kind of
Chern-Weil theory for $\Omega G$-bundles which gives characteristic classes
from invariant polynomials on the Lie algebra ${\mathfrak{g}}$ of $G$ and data
on the $\Omega G$-bundle. This theory side-steps the complications which arise
when trying to define the Chern-Weil map directly for bundles with infinite-
dimensional structure group (for example, see [38]). It also provides a
geometric interpretation of the well-known transgression map $\tau\colon
H^{2k}(BG)\to H^{2k-1}(G).$
The next question which it is natural to ask concerns the caloron
correspondence described above (i.e. the correspondence between $LG$-bundles
over $M$ and $G$-bundles over $M\times S^{1}$). In trying to find a formula
for the string class in terms of the Pontrjagyn class of a $G$-bundle (as in
[35]) one finds that it is necessary to make use of the caloron
correspondence. So it is natural then to ask what kind of correspondence
exists in the case where the $G$-bundle is not over $M\times S^{1}$ but over a
non-trivial principal $S^{1}$-bundle over $M$ and, further, whether the
methods of bundle gerbes can be applied to the lifting problem in this case.
In fact, the first part of this question has been answered in [1] in
connection with the Kaluza-Klein reduction of M-theory to type IIA
supergravity. It turns out that there is a bijective correspondence between
isomorphism classes of $G$-bundles over $S^{1}$-bundles and classes of bundles
whose structure group is not the loop group, but the semi-direct product
$LG\rtimes S^{1}.$ In the latter part of this thesis we prove that this
correspondence also holds on the level of connections (as in the case of a
trivial circle bundle) and consider the lifting problem for an $LG\rtimes
S^{1}$-bundle. We use the methods of [35] to find a de Rham representative for
the image in real cohomology of the class which is the obstruction to the
existence of this lift. We also provide a calculation of this class using a
different method introduced by Gomi [18], that of reduced splittings.
The outline of this thesis is as follows: In chapter 2 we describe the
necessary background. We recall some important facts about spin structures and
give an overview of Killingback’s results on string structures. We also review
the theory of bundle gerbes and their application to lifting problems. We then
present, in some detail, the theory and results from Murray and Stevenson’s
paper [35], including the calculation of the string class for a general
$LG$-bundle and the correspondence between $LG$-bundles over $M$ and
$G$-bundles over $M\times S^{1}.$ We also include the extension of
Killingback’s result from this paper.
In chapter 3 we show that the string class is a characteristic class for
$\Omega G$-bundles (Theorem 3.1.3) and generalise some of the results from
chapter 2 (albeit, only in the case of the based loop group) to higher
dimensions. That is, we define cohomology classes in any odd dimension which
are related to characteristic classes for $G$-bundles (in the same way that
the string class is related to the Pontrjagyn class) and we prove that these
are themselves characteristic classes. This gives a method of finding
characteristic classes for an $\Omega G$-bundle given a universal
characteristic class for $G$-bundles (that is, an element of $H^{*}(BG)$).
This is detailed in Theorem 3.2.8. We also provide a partial generalisation to
the case of the free loop group (although here we work with the group of
smooth maps from the interval into $G$ whose endpoints agree). We give a model
for the universal bundle and calculate its string class.
In chapter 4 we present the calculation of the string class of an $LG\rtimes
S^{1}$-bundle (that is, the obstruction to lifting the structure group of an
$LG\rtimes S^{1}$-bundle to its central extension). This is given in Theorem
4.1.3. We also give the generalisation of the caloron correspondence from [1]
which relates $G$-bundles over $S^{1}$-bundles to $LG\rtimes S^{1}$-bundles.
We show that this correspondence holds on the level of connections as well
(Proposition 4.2.2). This allows us to prove a generalisation of the result
from [35] relating the string class to the Pontrjagyn class of the
corresponding $G$-bundle (Theorem 4.2.3). Finally, we briefly outline how
these results can be used to gain information about the more general case of
lifting a bundle whose structure group is
$LG\rtimes\operatorname{Diff}(S^{1}),$ that is, where the loops in $LG$ are
acted upon by general (orientation preserving) diffeomorphisms of the circle.
We make a final comment on terminology and conventions. Throughout this thesis
we will work with many variations of the loop group. We give these here for
convenience. The group of smooth maps $\operatorname{Map}(S^{1},G)$ is denoted
by $LG$ and the subgroup of based loops which start at the identity by $\Omega
G.$ In chapter 3 we consider slightly more general variants of these groups
which consist of smooth maps from the interval $[0,2\pi]$ into $G$ whose
endpoints agree. These are denoted by $L^{\scriptscriptstyle{\vee}}G$ in the
free case and $\Omega^{\scriptscriptstyle{\vee}}G$ in the based case. Finally,
the terms principal $G$-bundle and $G$-bundle are used interchangeably and all
bundles are assumed to be principal bundles unless specifically stated
otherwise. Also, the circle group is denoted by either $U(1)$ or $S^{1}$ – we
make no distinction between the two.
## Chapter 2 String structures, bundle gerbes and Higgs fields
In this chapter we shall present the relevant background required for the rest
of the thesis. Namely, we describe the existing results on string structures
and develop the theory of bundle gerbes, which will feature quite heavily in
the sequel.
### 2.1 String structures
The existence of spinors and the Dirac operator is an essential aspect of
quantum field theory. It is well known that in order to define these objects
the underlying spacetime $M$ must be a spin manifold. In [45], in a study of
global anomalies, Witten shows that there occurs a global anomaly in the
worldline of a supersymmetric point particle in quantum mechanics unless $M$
admits a spin structure. The analogue of this in string theory, that is, a
global anomaly in the worldsheet of a string, was also studied in some detail.
Killingback, in [22], uses these results to determine topological conditions
on the spacetime $M.$ These conditions led to the definition of a so-called
_string structure_ on $M.$ Let us first recall, then, what we mean by a _spin
structure_ and show how to find the analogue of this in string theory.
#### 2.1.1 Spin structures
Let $M$ be an orientable manifold and $F\to M$ its frame bundle. Then $F$ is a
principal $SO(n)$-bundle. There is a simply connected double cover of $SO(n),$
called $\operatorname{Spin}(n)$ that fits into the exact sequence
$0\to{\mathbb{Z}}_{2}\to\operatorname{Spin}(n)\to SO(n)\to 0.$
Thus we can consider lifting the frame bundle of $M$ to a principal
$\operatorname{Spin}(n)$-bundle where by a _lift_ of $F\to M$ we mean a
principal $\operatorname{Spin}(n)$-bundle $\hat{F}\to M$ such that there is a
bundle map $\hat{F}\to F$ that commutes with the homomorphism
$\operatorname{Spin}(n)\to SO(n).$ If such a lift exists, we say $M$ has a
_spin structure_ , or simply that $M$ is _spin_. More generally, we can
consider any principal $SO(n)$-bundle $P\to M$ and ask for a lift of $P$ to a
principal $\operatorname{Spin}(n)$-bundle. If a lift exists in this case we
say that $P$ has a spin structure. It can be shown (see for example [25]) that
a spin structure exists for $P$ if and only if the second Stiefel-Whitney
class, $w_{2}(P),$ vanishes.
#### 2.1.2 String structures
As mentioned above, the Dirac operator, an integral element of quantum field
theory, cannot be defined unless $M$ is a spin manifold. The analogue of this
operator in string theory is the Dirac-Ramond operator. In [45] Witten argued
that the Dirac-Ramond operator can be considered as a Dirac-like operator on
$LM,$ the loop space of $M.$ Thus, in searching for an analogous result for
string theory, one is led to study principal bundles over $LM.$ This is the
subject of [22]. We shall briefly outline Killingback’s argument here. Denote
by $LX$ the loop space of $X,$ that is, the set of smooth maps from the circle
into $X,$ $\operatorname{Map}(S^{1},X).$ Consider a principal $G$-bundle $Q\to
M$ (for $G$ a compact, simple, simply-connected Lie group). Then by
considering the associated loop spaces, we obtain a principal
$LG$-bundle111For the proof that this in in fact a Fréchet principal bundle,
see [11] $LQ\to LM,$ We shall call such a bundle a _loop bundle_. In the case
that $X=G,$ we have the loop group of $G$ which has been extensively studied
(see for example [39]). There is an extension of this group by the circle
$S^{1},$
$0\to S^{1}\to\widehat{LG}\to LG\to 0.$
This extension is central in the sense that the image of $S^{1}$ in
$\widehat{LG}$ is in the centre of $\widehat{LG}.$ We shall look more closely
at this central extension later. For now, let us just outline Killingback’s
result. Killingback considers, as the analogue of a spin structure for string
theory, a lifting of the $LG$-bundle $LQ$ to a principal $\widehat{LG}$-bundle
$\widehat{LQ}.$ The exact sequence above leads to an exact sequence of sheaves
of groups over $LM.$ That is,
$\underline{S}^{1}\to\underline{\widehat{LG}}\to\underline{LG},$
where $\underline{{\mathcal{G}}}$ is the sheaf of ${\mathcal{G}}$-valued
functions over $LM.$ In general, if we have a short exact sequence of sheaves
of abelian groups over $X$
$\underline{A}\to\underline{B}\to\underline{C},$
then this leads to a long exact sequence of sheaf cohomology groups (see [4])
$\cdots\to H^{n}(X,\underline{A})\to H^{n}(X,\underline{B})\to
H^{n}(X,\underline{C})\to H^{n+1}(X,\underline{A})\to\cdots$
The same is not true, however, in the nonabelian case since we cannot define
the cohomology groups $H^{j}(X,\underline{A})$ for $j>1.$ Indeed, if $A,B$ and
$C$ are nonabelian, then $H^{1}(X,\underline{A}),$ $H^{1}(X,\underline{B})$
and $H^{1}(X,\underline{C})$ are not groups but pointed sets. In this case, we
can write down an exact sequence of pointed sets
$0\to H^{0}(X,\underline{A})\to H^{0}(X,\underline{B})\to
H^{0}(X,\underline{C})\to H^{1}(X,\underline{A})\to H^{1}(X,\underline{B})\to
H^{1}(X,\underline{C}),$
where by exactness here we mean the image of any map is exactly the pre-image
of the basepoint in the next set in the sequence. There is no connecting
homomorphism $H^{1}(X,\underline{C})\to H^{2}(X,\underline{A})$ and so the
sequence terminates. If we assume that $A$ is central in $B,$ however, then
$H^{j}(X,\underline{A})$ is an abelian group for all $j$ and it is possible to
extend the sequence above one more step to the right ([4], Theorem 4.1.4)
$0\to H^{0}(X,\underline{A})\to H^{0}(X,\underline{B})\to
H^{0}(X,\underline{C})\\\ \to H^{1}(X,\underline{A})\to
H^{1}(X,\underline{B})\to H^{1}(X,\underline{C})\to H^{2}(X,\underline{A}).$
The short exact sequence above therefore leads to an exact sequence in sheaf
cohomology
$\ldots\to H^{1}(LM,\underline{S}^{1})\to
H^{1}(LM,\underline{\widehat{LG}})\to H^{1}(LM,\underline{LG})\to
H^{2}(LM,\underline{S}^{1}),$
where, since $\widehat{LG}$ and $LG$ are in general nonabelian,
$H^{1}(LM,{\widehat{LG}})$ and $H^{1}(LM,\underline{LG})$ are just pointed
sets, whereas $H^{1}(LM,\underline{S}^{1})$ and $H^{2}(LM,\underline{S}^{1})$
are abelian groups. Now, since the set of isomorphism classes of principal
${\mathcal{G}}$-bundles over $LM$ is in bijective correspondence with the set
$H^{1}(LM,\underline{{\mathcal{G}}})$ we see that the $LG$-bundle $LQ\in
H^{1}(LM,\underline{LG})$ has a lift to an $\widehat{LG}$-bundle exactly when
$LQ$ is the image of an element in $H^{1}(LM,\underline{\widehat{LG}}).$ That
is, when the image of $LQ$ in $H^{2}(LM,\underline{S}^{1})$ is zero.
Therefore, the obstruction to lifting a loop bundle $LQ\to LM$ is a class in
$H^{2}(LM,\underline{S}^{1}).$ Now recall that the short exact sequence of
groups
$0\to{\mathbb{Z}}\to{\mathbb{R}}\to S^{1}\to 0,$
leads to an exact sequence of sheaves (as above)
$\underline{{\mathbb{Z}}}\to\underline{{\mathbb{R}}}\to\underline{S}^{1},$
which in turn leads to a long exact sequence of sheaf cohomology groups
$\ldots\to H^{2}(LM,\underline{{\mathbb{Z}}})\to
H^{2}(LM,\underline{{\mathbb{R}}})\to H^{2}(LM,\underline{S}^{1})\to
H^{3}(LM,\underline{{\mathbb{Z}}})\to\ldots$
(since ${\mathbb{Z}},{\mathbb{R}}$ and $S^{1}$ are all abelian). However,
because $\underline{{\mathbb{R}}}$ is a soft sheaf,
$H^{*}(LM,\underline{{\mathbb{R}}})=0$ and we have the following well known
result (see for example [4])
$H^{2}(LM,\underline{S}^{1})\simeq H^{3}(LM,{\mathbb{Z}}).$
So we see that the obstruction to lifting the $LG$-bundle $LQ$ to an
$\widehat{LG}$-bundle is a class in $H^{3}(LM,{\mathbb{Z}}).$ Since this
lifting is the analogue in string theory of a spin structure for $M,$ we call
it a _string structure_ for $M$ and we call the obstruction class $s(LQ)\in
H^{3}(LM,{\mathbb{Z}})$ the _string class_. Killingback’s main result, then,
is a characterisation of this class in terms of the first Pontrjagyn class of
the $G$-bundle $Q\to M.$ In particular, if $p_{1}(Q)\in H^{4}(M,{\mathbb{Z}})$
is the first Pontrjagyn class of $Q$, then Killingback shows that the
transgression of this is the string class of $LQ.$ That is, the string class
is given by pulling-back $p_{1}(Q)$ by the evaluation map
$\operatorname{ev}\colon LM\times S^{1}\to M$ to give a class on $LM\times
S^{1}$ and integrating over $S^{1}:$
$s(LQ)=\int_{S^{1}}\operatorname{ev}^{*}p_{1}(Q).$
We shall give a proof of this formula later (in section 2.5) following the
methods in [35].
### 2.2 Bundle gerbes
In order to perform calculations involving the string class and to extend
Killingback’s result, we shall use the theory of bundle gerbes [32], in
particular, the lifting bundle gerbe (see section 2.3). In this section we
briefly outline the theory (developed largely in [32] and [33]) behind these
objects. Bundle gerbes can be considered, in some sense, as ‘higher’ versions
of $U(1)$-bundles. Therefore, we start with some basic results on these
bundles before describing the theory of bundle gerbes.
#### 2.2.1 $U(1)$-bundles
As mentioned, we shall begin by recalling some facts about $U(1)$-bundles and
some constructions involving these bundles. Firstly, note that if $P\to M$ is
a $U(1)$-bundle with right action given by $(p,z)\mapsto pz$ (for $p\in P$ and
$z\in U(1)$) then there is a _dual_ bundle, denoted $P^{*},$ which is the same
as $P$ but with the action given by $(p,z)\mapsto pz^{-1}.$ Of course this is
only a right action because $U(1)$ is abelian. Further, if $Q$ is another
$U(1)$-bundle over $M,$ we can form the fibre product over $M,$
$P\times_{M}Q,$ which is a principal $U(1)\times U(1)$-bundle over $M$ whose
fibres are the product of the fibres of $P$ and $Q$ (i.e.
$(P\times_{M}Q)_{m}=P_{m}\times Q_{m}$). By factoring out by the ‘anti-
diagonal’ inside $U(1)\times U(1),$ that is, the set $\\{(z,z^{-1})\\},$ we
obtain a principal $U(1)$-bundle called the _contracted product_ of $P$ and
$Q$ and denoted $P\otimes Q.$ It is easy to see that $P\otimes P^{*}$ is
canonically trivialised by the section $s\colon m\mapsto[p,p^{*}],$ where $p$
is any point in the fibre of $P$ above $m$ and $p^{*}$ is the same point
considered as an element of $P^{*}.$ For if $s_{\alpha}$ and $s_{\beta}$ are
two such local sections then suppose $s_{\alpha}(m)=[p,p^{*}]$ and
$s_{\beta}(m)=[q,q^{*}],$ then we have that $[q,q^{*}]=[pz,p^{*}z^{-1}]$ for
some $z\in U(1)$ and so $s_{\alpha}=s_{\beta}.$
Note that if instead of considering $U(1)$-bundles we equivalently considered
complex hermitian line bundles then the dual would correspond to the linear
dual of a line bundle (i.e. the bundle whose fibres are the dual of those of
the original bundle) and the contracted product would correspond to the tensor
product of line bundles (the bundle whose fibres are the tensor product of the
fibres of the original two bundles). Note also that if $P$ and $Q$ have
transition functions $g_{\alpha\beta}$ and $h_{\alpha\beta}$ respectively
relative to some open cover of $M$ then $P^{*}$ has transition functions
$g_{\alpha\beta}^{-1}$ and $P\otimes Q$ has transition functions
$g_{\alpha\beta}h_{\alpha\beta}.$
Another important property of $U(1)$-bundles on $M$ is the way in which they
relate to $H^{2}(M,{\mathbb{Z}}).$ If a $U(1)$-bundle $P$ has transition
functions $g_{\alpha\beta}$ then on triple overlaps these satisfy the cocycle
condition
$g_{\beta\gamma}^{\phantom{-1}}g_{\alpha\gamma}^{-1}g_{\alpha\beta}^{\phantom{-1}}=1$
and thus form a class in $H^{1}(M,\underline{U(1)}).$ Thus, from the argument
in the previous section we have that a $U(1)$-bundle defines a class in
$H^{2}(M,{\mathbb{Z}}).$ This class is called the _Chern class_ of the bundle
$P.$ It is a standard result (see for example [4]) that the Chern class
classifies $U(1)$-bundles up to isomorphism and, further, that given any class
in $H^{2}(M,{\mathbb{Z}})$ one can construct a $U(1)$-bundle. So we see that
isomorphism classes of $U(1)$-bundles are in bijective correspondence with
$H^{2}(M,{\mathbb{Z}}).$ The Chern class is additive in the sense that if
$c(P)$ and $c(Q)$ are the Chern classes of $P$ and $Q$ respectively, then
$c(P\otimes Q)=c(P)+c(Q)$ and $c(P^{*})=-c(P).$ It is natural in the sense
that if we pull-back the bundle $P\to M$ by a map $f\colon N\to M$ to give a
$U(1)$-bundle $f^{*}P\to N$ then $c(f^{*}P)=f^{*}c(P).$
We can actually represent the image of the Chern class in real cohomology
using differential forms quite easily. If $A$ is a connection on $P$ whose
curvature is $F,$ then $F/2\pi i$ is a closed integral form and its class in
the de Rham cohomology group $H^{2}(M)$ is the image in real cohomology of the
Chern class of $P.$
#### 2.2.2 Bundle gerbes
##### Definitions and basic constructions
Having reviewed some of the basic properties of $U(1)$-bundles in the previous
section, we would now like to present another object, first introduced in [32]
and studied further in [33], which is in some sense a higher dimensional
version of a $U(1)$-bundle as we shall see shortly.
Consider a surjective submersion $Y\xrightarrow{\pi}M.$ We can form the fibre
product of $Y$ with itself, which we denote $Y^{[2]},$ and we have (as before)
$Y^{[2]}=\\{(y_{1},y_{2})\in Y\times Y\mid\pi(y_{1})=\pi(y_{2})\\}.$
Note that since $\pi$ is a submersion $Y^{[2]}$ is a submanifold of $Y^{2}$.
In general we have the _p-fold fibre product_ $Y^{[p]}$ defined similarly. We
define the maps $\pi_{i}\colon Y^{[p+1]}\to Y^{[p]}(i=1,\ldots,p+1)$ to be
omission of the $i^{\text{th}}$ factor,
$\pi_{i}(y_{1},\ldots,y_{p+1})=(y_{1},\ldots,y_{i-1},y_{i+1},\ldots,y_{p+1}).$
We have, then, the following definition:
###### Definition 2.2.1 ([32]).
A _bundle gerbe_ over a manifold $M$ is a pair $(P,Y)$ where $Y\to M$ is a
surjective submersion and $P\to Y^{[2]}$ is a $U(1)$-bundle and such that
there is a _bundle gerbe multiplication_ , which is a smooth isomorphism
$m\colon\pi_{3}^{*}P\otimes\pi_{1}^{*}P\xrightarrow{\sim}\pi_{2}^{*}P$
of $U(1)$-bundles over $Y^{[3]}.$ Further, this multiplication is required to
be associative whenever triple products are defined. That is, if
$P_{(y_{1},y_{2})}$ denotes the fibre of $P$ over $(y_{1},y_{2})\in Y^{[2]}$
then the following diagram commutes for all $(y_{1},y_{2},y_{3},y_{4})\in
Y^{[4]}$:
$\textstyle{P_{(y_{1},y_{2})}\otimes P_{(y_{2},y_{3})}\otimes
P_{(y_{3},y_{4})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{id}\otimes
m}$$\scriptstyle{m\otimes\text{id}}$$\textstyle{P_{(y_{1},y_{3})}\otimes
P_{(y_{3},y_{4})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{P_{(y_{1},y_{2})}\otimes
P_{(y_{2},y_{4})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{P_{(y_{1},y_{4})}}$
We sometimes denote a bundle gerbe simply by $P.$
We typically depict a bundle gerbe thusly:
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{[2]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\vphantom{[2]}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M}$
We can characterise the bundle gerbe multiplication and its associativity in a
different way using sections of bundles related to $P$ as follows. If $Q\to
Y^{[p-1]}$ is a $U(1)$-bundle, define the bundle $\delta Q\to Y^{[p]}$ as
$\delta
Q=\pi_{1}^{*}Q\otimes(\pi_{2}^{*}Q)^{*}\otimes\pi_{3}^{*}Q\otimes\ldots$
Then it is easy to show that $\delta\delta Q$ is canonically trivial. One can
show that the bundle gerbe multiplication is equivalent to a section $s$ of
$\delta P\to Y^{[3]}$ and that the associativity condition is equivalent to
the condition that $\delta s=1$ as a section of $\delta\delta P$ (where $1$
denoted the canonical section of $\delta\delta P$). Indeed if $p$ and $q$ are
elements of $P_{(y_{1},y_{2})}$ and $P_{(y_{2},y_{3})}$ respectively, we can
define a section $s$ of $\delta P$ by
$s(y_{1},y_{2},y_{3})=p\otimes m(p,q)^{*}\otimes q,$
then the associativity of $m$ forces the condition $\delta s=1.$ Note that
these conditions reflect the definition of a _simplicial line bundle_ from
[5]. So we see that a bundle gerbe is the same as a simplicial line bundle
over the simplicial space defined by the fibre products $Y^{[p]}.$ We shall
discuss simplicial spaces and this relationship more in section 2.3.
In [32] Murray claimed that bundle gerbes were essentially bundles of
groupoids. Although it is not essential for our purposes let us briefly
explain what is meant by this. Recall (see [26]) that a groupoid is a small
category with all arrows invertible. Consider then a bundle gerbe $(P,Y)$ over
$M.$ If we consider the elements of the fibre over $m,$ $Y_{m},$ as the
objects of a category, then the elements of the fibre $P_{(y_{1},y_{2})}$ are
the morphisms from $y_{1}$ to $y_{2}$ and the bundle gerbe multiplication
gives a way of composing these morphisms. Since
$P_{(y_{1},y_{2})}^{\vphantom{*}}\simeq P_{(y_{2},y_{1})}^{*}$ and
$P_{(y,y)}\simeq Y^{[2]}\times U(1)$ (which can be shown using the bundle
gerbe multiplication), this category is a groupoid. In [32] the theory of
$U(1)$-groupoids is presented in more detail as a prelude to the introduction
of bundle gerbes.
Just as for $U(1)$-bundles, various constructions are possible with bundle
gerbes [32]. Consider a map $f\colon N\to M.$ We can pull-back the submersion
$Y\to M$ to a submersion $f^{*}Y\to N.$ This gives a map $\hat{f}\colon
f^{*}Y\to Y$ covering $f$ which induces a map (also called $\hat{f}$)
$(f^{*}Y)^{[2]}\to Y^{[2]}.$ Thus we can pull-back the $U(1)$-bundle $P\to
Y^{[2]}$ by $\hat{f}$ to give a bundle $\hat{f}^{*}P\to(f^{*}Y)^{[2]}.$ So we
have a bundle gerbe over $N$ called the _pull-back_ and which we will denote
$f^{*}P.$ We can also define the _dual_ of $(P,Y)$ by taking the dual of the
$U(1)$-bundle $P$ over $Y^{[2]}.$ We denote this by $P^{*}.$ We can form the
_product_ of two bundle gerbes $(P,Y)$ and $(Q,X)$ over $M,$ denoted $P\otimes
Q,$ by taking the fibre product $Y\times_{M}X$ over $M$ and the $U(1)$-bundle
$P\otimes Q$ over $(Y\times_{M}X)^{[2]}.$
We say two bundle gerbes $(P,Y)$ and $(Q,X)$ over $M$ are _isomorphic_ if
there is an isomorphism $Y\to X$ covering the identity on $M$ and a bundle
isomorphism $P\to Q$ covering the induced map $Y^{[2]}\to X^{[2]}$ and which
commutes with the bundle gerbe multiplication.
A particular example of a bundle gerbe is given by taking a $U(1)$-bundle $P$
over $Y$ and defining $\delta P$ over $Y^{[2]}$ as above. That is, $\delta
P=\pi_{1}^{*}P\otimes(\pi_{2}^{*}P)^{*}.$ Since $\delta\delta P$ is
canonically trivial over $Y^{[3]},$ it has a canonical section $s$ which
defines the bundle gerbe multiplication. This is called the _trivial bundle
gerbe_ and in general we say a bundle gerbe is _trivial_ if it is isomorphic
to one of this form.
As was pointed out in [33] there is another notion of equivalence, in addition
to isomorphism, for bundle gerbes. This is the notion of _stable isomorphism_
, first introduced in [7] and studied in detail in [33]. Two bundle gerbes
$(P,Y)$ and $(Q,X)$ are called _stably isomorphic_ if there are trivial bundle
gerbes $T_{1}$ and $T_{2}$ such that $P\otimes T_{1}\simeq Q\otimes T_{2}$ or,
equivalently, if $P\otimes Q^{*}$ is trivial. It turns out that stable
isomorphism is in some sense the correct notion of equivalence for bundle
gerbes because, as we shall see next, all bundle gerbes have a characteristic
class associated to them and this class classifies them up to stable
isomorphism. That is, two bundle gerbes have the same associated class exactly
when they are stably isomorphic. This class is called the _Dixmier-Douady
class_ and it is to this which we now turn our attention.
##### Bundle gerbes and degree three cohomology
As mentioned earlier, bundle gerbes can be considered as higher dimensional
$U(1)$-bundles. We now explain why this is the case and describe how to
construct a characteristic class for bundle gerbes which is analogous to the
Chern class for $U(1)$-bundles.
Let $(P,Y)$ be a bundle gerbe over $M$ and choose a good cover
$\\{U_{\alpha}\\}$ of $M$ over which $Y\to M$ admits local sections. This is
always possible (see [2]). Suppose that $s_{\alpha}\colon U_{\alpha}\to Y$ is
a local section. We have a section of $Y^{[2]}$ over double overlaps given by
$(s_{\alpha},s_{\beta})\colon U_{\alpha\beta}\to Y^{[2]},$
where $U_{\alpha\beta}=U_{\alpha}\cap U_{\beta}.$ As $U_{\alpha\beta}$ is
contractible, the pull-back $P_{\alpha\beta}=(s_{\alpha},s_{\beta})^{*}P$ of
$P$ by this section is trivial. The fibres of $P_{\alpha\beta}$ are given by
$(P_{\alpha\beta})_{m}=P_{(s_{\alpha}(m),s_{\beta}(m))}.$ Choose a section
$\sigma_{\alpha\beta}$ of this bundle. That is, a map
$\sigma_{\alpha\beta}\colon U_{\alpha\beta}\to P$
such that $\sigma_{\alpha\beta}(m)\in P_{(s_{\alpha}(m),s_{\beta}(m))}.$ On
triple overlaps $U_{\alpha\beta\gamma}$ the bundle gerbe multiplication gives
$m(\sigma_{\alpha\beta},\sigma_{\beta\gamma})=g_{\alpha\beta\gamma}\sigma_{\alpha\gamma}$
for some $g_{\alpha\beta\gamma}\colon U_{\alpha\beta\gamma}\to U(1).$ On
overlaps $U_{\alpha\beta\gamma\delta}$ the associativity of this
multiplication gives the cocycle condition
$g_{\beta\gamma\delta}^{\vphantom{-1}}g_{\alpha\gamma\delta}^{-1}g_{\alpha\beta\delta}^{\vphantom{-1}}g_{\alpha\beta\gamma}^{-1}=1.$
Thus the functions $g_{\alpha\beta\gamma}$ define a class in
$H^{2}(M,\underline{U(1)})\simeq H^{3}(M,{\mathbb{Z}}).$ This class is
independent of any choices and is called the _Dixmier-Douady class_ of $P$ and
denoted $\textsl{DD}(P).$ In [32] it is proven that this class is precisely
the obstruction to the bundle gerbe being trivial. We also have the following
results regarding the Dixmier-Douady class for the constructions presented
earlier: If $(P,Y)$ and $(Q,X)$ are bundle gerbes over $M$ then
$\textsl{DD}(P\otimes Q)=\textsl{DD}(P)+\textsl{DD}(Q)$ and
$\textsl{DD}(P^{*})=-\textsl{DD}(P).$ The Dixmier-Douady class is natural with
respect to pull-backs, that is, $\textsl{DD}(f^{*}P)=f^{*}\textsl{DD}(P).$
As mentioned at the end of the previous section, the Dixmier-Douady class
classifies bundle gerbes up to stable isomorphism. This is clear because $P$
and $Q$ are stably isomorphic exactly when $P\otimes Q^{*}$ is trivial and so
the result follows from the fact that $\textsl{DD}(P\otimes
Q^{*})=\textsl{DD}(P)-\textsl{DD}(Q)$ and that trivial bundle gerbes have zero
Dixmier-Douady class.
In [32] it is also shown that every class in $H^{3}(M,{\mathbb{Z}})$ is the
Dixmier-Douady class of some bundle gerbe. This means that there is a
bijection between $H^{3}(M,{\mathbb{Z}})$ and stable isomorphism classes of
bundle gerbes. Thus bundle gerbes provide a geometric realisation of elements
in $H^{3}(M,{\mathbb{Z}})$ in an analogous way to that of $U(1)$-bundles and
$H^{2}(M,{\mathbb{Z}}).$
##### Connective structures on bundle gerbes
We have seen now the way in which bundle gerbes play a role for degree three
cohomology analogous to that of $U(1)$-bundles and degree two cohomology. As
we saw in section 2.2.1 $U(1)$-bundles have the nice property that the image
of their Chern class in real cohomology is represented by the form $F/2\pi i,$
where $F$ is the curvature of the bundle. We would now like to study
connective structures on bundle gerbes and, as we shall see, a similar result
is true in this case.
Consider first the $p$-fold fibre product $Y^{[p]}$ as before. Let
$\Omega^{q}(Y^{[p]})$ denote the space of differential $q$-forms on $Y^{[p]}.$
Then we can define a map
$\delta\colon\Omega^{q}(Y^{[p]})\to\Omega^{q}(Y^{[p+1]})$ as the alternating
sum of pull-backs by the projections $\pi_{i}:$
$\delta=\sum_{i=1}^{p+1}(-1)^{i-1}\pi_{i}^{*}.$
Then $\delta^{2}=0$ and so we have a complex
$0\to\Omega^{q}(M)\xrightarrow{\pi^{*}}\Omega^{q}(Y)\xrightarrow{\,\delta\,}\Omega^{q}(Y^{[2]})\xrightarrow{\,\delta\,}\Omega^{q}(Y^{[3]})\xrightarrow{\,\delta\,}\ldots$
In [32] it is proven that this complex has no cohomology. That is, the above
sequence is exact for all $q\geq 0.$ We shall use this result shortly.
A bundle gerbe connection is a connection $A$ for the $U(1)$-bundle $P$ that
respects the bundle gerbe product in the sense that the induced connection on
$\pi_{2}^{*}P$ is the same as the image of the induced connection on
$\pi_{3}^{*}P\otimes\pi_{1}^{*}P$ under the bundle gerbe multiplication. Note
that if $s\colon Y^{[3]}\to\delta P$ is the section defining this
multiplication, then this means that a bundle gerbe connection satisfies
$s^{*}(\delta A)=0.$ That is, $\delta A$ is flat with respect to $s.$ Using
this observation, it is easy to see that bundle gerbe connections always
exist. For consider a connection $A$ on $P$ that does not necessarily commute
with the product. We cannot say that $s^{*}(\delta A)=0$ but note that
$\delta(s^{*}(\delta A))=(\delta s)^{*}(\delta\delta A),$ which is zero since
$\delta s=1$ as a section of $\delta\delta P$ and $\delta\delta A$ is flat
with respect to the canonical trivialisation of $\delta\delta P.$ Therefore,
by the exact sequence above there is some $a\in\Omega^{1}(Y^{[2]})$ such that
$\delta a=s^{*}(\delta A)$ and so $s^{*}(\delta(A-\pi^{*}a))=0$ (where
$\pi\colon P\to Y^{[2]}$ is the projection). Therefore, $A-\pi^{*}a$ is a
bundle gerbe connection.
If $F$ is the curvature of a bundle gerbe connection $A$ viewed as a 2-form on
$Y^{[2]},$ then $\delta F=s^{*}(\delta dA)=d(s^{*}(\delta A))=0.$ This means
that there is some $B\in\Omega^{2}(Y)$ satisfying $F=\delta B.$ A choice of
such a $B$ is called a _curving_ for $P.$ Note that if $B^{\prime}$ is another
choice of curving then $B$ and $B^{\prime}$ differ by a $\delta$-closed (and
hence $\delta$-exact) 2-form on $Y$. As $\delta$ and $d$ commute, we have that
$\delta(dB)=d(\delta B)=dF=0.$ Therefore there is a 3-form $H$ on $M$ such
that $dB=\pi^{*}H$ (for $\pi$ the projection $Y\to M$). $H$ is called the
_3-curvature_ of $P.$ It is closed and a different choice of $B$ or $H$ would
result in a difference of an exact form. So $H$ defines a cohomology class in
$H^{3}(M).$ It turns out that the 3-form $H/2\pi i$ is integral and that
$H/2\pi i$ is a representative of the Dixmier-Douady class of $P$ in real
cohomology.
### 2.3 Central extensions and the lifting bundle gerbe
In this thesis, we wish to apply the theory of bundle gerbes to the study of
central extensions of Lie groups and, in particular, to lifting problems as in
section 2.1. For this purpose we use a particular bundle gerbe called the
_lifting bundle gerbe_ and in this section we review the basic definitions and
results required to develop the theory. We shall start by outlining the theory
of central extensions, following [5].
#### 2.3.1 Simplicial line bundles and central extensions
We begin by recalling some simplicial techniques. Recall (see [14]) that a
_simplicial space_ is a collection of spaces $\\{X_{p}\\}\,(p=0,1,2,\ldots)$
together with maps $d_{i}\colon X_{p}\to X_{p-1}$ and $s_{j}\colon X_{p}\to
X_{p+1}$ for $i,j=0,\ldots,p$, called _face_ and _degeneracy_ maps
respectively, which satisfy the simplicial identities
$\displaystyle d_{i}d_{j}$ $\displaystyle=d_{j-1}d_{i},\qquad i<j,$
$\displaystyle s_{i}s_{j}$ $\displaystyle=s_{j+1}s_{i},\qquad i\leq j,$
$\displaystyle d_{i}s_{j}$ $\displaystyle=\begin{cases}s_{j-1}d_{i},&i<j\\\
\text{id},&i=j,\,j+1\\\ s_{j}d_{i-1},&i>j+1.\end{cases}$
If we are working in the category of manifolds and smooth maps we say that
$\\{X_{p}\\}$ is a _simplicial manifold_. For example, consider the
collection222Note that here $X_{0}=Y,X_{1}=Y^{[2]},X_{2}=Y^{[3]},\ldots$ and
so on $\\{Y^{[p+1]}\\}$ of fibre products as in the previous section. These
form a simplicial manifold with the obvious face and degeneracy maps. Note
that for a general simplicial manifold $\\{X_{p}\\}$ we can define a complex
similar to the one described in section 2.2 by using the pull-backs of the
face maps $d_{i}.$ That is, we define
$\delta\colon\Omega^{q}(X_{p})\to\Omega^{q}(X_{p+1})$ by
$\delta=\sum_{i=0}^{p}(-1)^{i}d_{i}^{*}.$
Also, as before, if $Q$ is a $U(1)$-bundle (or an hermitian line bundle) over
$X_{p}$ then we can define a bundle over $X_{p+1}$ by
$\delta Q=d_{0}^{*}Q\otimes(d_{1}^{*}Q)^{*}\otimes d_{2}^{*}Q\otimes\ldots$
The particular example of interest to us is a certain simplicial manifold
associated to a Lie group which we describe presently. Let ${\mathcal{G}}$ be
a Lie group. There is a simplicial manifold called
$N{\mathcal{G}}=\\{N{\mathcal{G}}_{p}\\}$ given by the manifolds
$\\{{\mathcal{G}}^{p}\\}$ and face and degeneracy maps $d_{i}$ and $s_{j}$
where
$d_{i}(g_{1},\ldots,g_{p+1})=\begin{cases}(g_{2},\ldots,g_{p+1}),&i=0\\\
(g_{1},\ldots,g_{i-1}g_{i},g_{i+1},\ldots,g_{p+1}),&1\leq i\leq p-1\\\
(g_{1},\ldots,g_{p}),&i=p\end{cases}$
and
$s_{j}(g_{1},\ldots,g_{p+1})=(g_{1},\ldots,g_{j-1},1,g_{j},\ldots,g_{p+1}).$
We would like to consider central extensions of ${\mathcal{G}}$ by the circle
and show how they are related to $N{\mathcal{G}}.$ For this, we follow
Brylinski and McLaughlin [5] where the result is phrased in terms of
simplicial line bundles. We have the following definition
###### Definition 2.3.1 ([5]).
Let $\\{X_{p}\\}$ be a simplicial manifold. A _simplicial line bundle_ over
$\\{X_{p}\\}$ is a line bundle $L$ on $X_{1}$ together with a section $s$ of
the bundle $\delta L\to X_{2}$ such that $\delta s=1$ as a section of
$\delta\delta L.$
Notice the similarity with the definition of a bundle gerbe. In fact, instead
of using $U(1)$-bundles, we can rephrase everything about bundle gerbes in
terms of line bundles and we see that a bundle gerbe is the same thing as a
simplicial line bundle over the simplicial space $\\{Y^{[p]}\\}.$
Now consider a central extension of ${\mathcal{G}}$ by the circle
$U(1)\to\widehat{{\mathcal{G}}}\xrightarrow{p}{\mathcal{G}}.$
If we think of this as a $U(1)$-bundle
$\widehat{{\mathcal{G}}}\to{\mathcal{G}}$ then we must have a multiplication
$M\colon\widehat{{\mathcal{G}}}\times\widehat{{\mathcal{G}}}\to\widehat{{\mathcal{G}}}$
which covers the multiplication on ${\mathcal{G}},$ that is,
$m=d_{1}\colon{\mathcal{G}}\times{\mathcal{G}}\to{\mathcal{G}}.$ Because
$\widehat{{\mathcal{G}}}$ is a central extension we must have
$M(\hat{g}z,\hat{h}w)=M(\hat{g},\hat{h})(zw)$ for any
$\hat{g},\hat{h}\in\widehat{{\mathcal{G}}}$ and $z,w\in U(1).$ In a similar
way to that in which the bundle gerbe multiplication on a bundle gerbe $P$
gave rise to a section of $\delta P,$ this gives a section of
$\delta\widehat{{\mathcal{G}}},$
$s(g,h)=\hat{g}\otimes M(\hat{g},\hat{h})^{*}\otimes\hat{h},$
where $\hat{g}$ and $\hat{h}$ are points in the fibres over $g$ and $h$
respectively. The associativity of this multiplication is equivalent to the
condition $\delta s=1$ as before and hence a central extension gives rise to a
simplicial line bundle. In fact it can be shown that they are equivalent and
we have the result from [5]:
###### Theorem 2.3.2 ([5]).
A simplicial line bundle over the simplicial manifold $N{\mathcal{G}}$ is a
central extension of ${\mathcal{G}}$ by the circle.
We wish to perform explicit calculations using differential forms so,
following [34] and [35], we shall rephrase this result in terms of
differential forms on ${\mathcal{G}}^{p}$ and give a method of constructing
central extensions using these forms. Consider then, a connection $\nu$ for
$\widehat{{\mathcal{G}}}$ thought of as a $U(1)$-bundle over ${\mathcal{G}}.$
As in the treatment of bundle gerbe connections in section 2.2 we can consider
the induced connection $\delta\nu$ on the bundle
$\delta\widehat{{\mathcal{G}}}\to{\mathcal{G}}\times{\mathcal{G}}$ and then,
as this bundle is trivial, we can pull-back $\delta\nu$ by the section $s.$
Let $\alpha=s^{*}(\delta\nu).$ In general $\alpha$ is non-zero. However, we
have that $\delta\alpha=\delta(s^{*}(\delta\nu))=(\delta
s)^{*}(\delta\delta\nu)=0.$ Furthermore, we also have
$d\alpha=s^{*}(d\delta\nu)=\delta R,$ where $R$ is the curvature of $\nu$
viewed as a form on ${\mathcal{G}}.$ Therefore we have constructed from the
central extension a pair of forms $(R,\alpha),$ where
$R\in\Omega^{2}({\mathcal{G}})$ is closed and integral and
$\alpha\in\Omega^{1}({\mathcal{G}}\times{\mathcal{G}})$ is such that $\delta
R=d\alpha$ and $\delta\alpha=0.$ In fact, as we shall now show, this pair is
sufficient to reconstruct the central extension. Recall (see for example [4])
that given an integral 2-form $R\in\Omega^{2}({\mathcal{G}})$ there exists a
principal $U(1)$-bundle $P\to{\mathcal{G}}$ with a connection $a$ whose
curvature is $R.$ Also, it is a standard result (see [23]) that if $Q$ is a
bundle over a simply connected base which admits a flat connection $A,$ then
$Q$ is trivial and there is a section $s$ of $Q$ such that $s^{*}A=0.$ In
terms of the construction here, this means that we can find a bundle
$P\to{\mathcal{G}}$ with curvature $R$ and because $d\alpha=\delta R,$ we have
that $\delta a-\pi^{*}\alpha$ is a flat connection on $\delta
P\to{\mathcal{G}}\times{\mathcal{G}}.$ Therefore, there is a section $s$ of
$\delta P$ satisfying $s^{*}(\delta a)=\alpha.$ As before, this section
defines a multiplication and we can calculate $\delta s$ which we want to be
equal to $1.$ Now, $(\delta s)^{*}(\delta\delta a)=\delta(s^{*}(\delta
a))=\delta\alpha=0$ and for the canonical section $1$ we also have
$1^{*}(\delta\delta a)=0.$ This means that they differ by an element of $U(1)$
and so rather than associativity of the multiplication $M$ defined by $s$ we
have
$M(M(\hat{g},\hat{h}),\hat{k})=zM(\hat{g},M(\hat{h},\hat{k}))$
for some $z\in U(1).$ However, if we choose some $\hat{g}$ in the fibre above
the identity $e$ in ${\mathcal{G}}$ then $M(\hat{g},\hat{g})$ is also in the
fibre above $e$ and so $\hat{g}$ and $M(\hat{g},\hat{g})$ differ by some $w\in
U(1).$ That is, $M(\hat{g},\hat{g})=\hat{g}w.$ Let $\hat{h}$ and $\hat{k}$
both be equal to $\hat{g}\in\pi^{-1}(e).$ Then the formula above reads
$M(M(\hat{g},\hat{g}),\hat{g})=zM(\hat{g},M(\hat{g},\hat{g}))$
and so $\hat{g}w^{2}=\hat{g}w^{2}z$ and we see that in fact $z=1.$
Thus we have constructed a central extension from the pair $(R,\alpha)$ and
this construction recovers the original extension (which follows from the fact
that $P$ has curvature $R$ and the definition of $\alpha$ above). Note that
isomorphic central extensions (where by isomorphic, we mean isomorphic as
$U(1)$-bundles and as groups) give rise to the same $R$ and $\alpha$ and that
in constructing the pair $(R,\alpha)$ if we had chosen a different connection,
by adding on the pull-back of a 1-form $\eta$ on ${\mathcal{G}},$ then we
would have the pair $(R+d\eta,\alpha+\delta\eta).$ Also, note that the section
constructed above from the flat connection is not unique but changing this by
multiplying by a constant $z$ in $U(1)$ would change $M$ to $Mz$ and, as the
extension is central, this would give an isomorphic central extension. So, as
in [35], we have a bijection between isomorphism classes of central extensions
with connection and pairs of forms satisfying the conditions above.
#### 2.3.2 The lifting bundle gerbe
Having reviewed a method for constructing central extensions, we would like
now to link the theory of central extensions with that presented earlier on
bundle gerbes. We present a particular example of a bundle gerbe related to
central extensions, first introduced in [32], called the _lifting bundle
gerbe_ whose Dixmier-Douady class is precisely the obstruction to lifting a
${\mathcal{G}}$-bundle $P$ to a $\widehat{{\mathcal{G}}}$-bundle
$\widehat{P}.$
Consider then a principal ${\mathcal{G}}$-bundle $P\to M.$ Choose a good cover
of $M$ and consider the transition functions $g_{\alpha\beta}$ of $P$ relative
to this cover. We can choose lifts of these functions $\hat{g}_{\alpha\beta}$
which take values in $\widehat{{\mathcal{G}}}$ and these are candidates for
the transition functions of the lift $\widehat{P}.$ However, transition
functions are required to satisfy the cocycle condition
$g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}$ on triple overlaps but the
lifts $\hat{g}_{\alpha\beta}$ only satisfy
$\hat{g}_{\alpha\beta}\hat{g}_{\beta\gamma}=\epsilon_{\alpha\beta\gamma}\hat{g}_{\alpha\gamma}$
for some $U(1)$-valued function $\epsilon_{\alpha\beta\gamma}.$ This means
that the $\hat{g}_{\alpha\beta}$’s are not necessarily transition functions.
However, due to the fact that $\widehat{{\mathcal{G}}}$ is a central
extension, it can be shown that the functions $\epsilon_{\alpha\beta\gamma}$
satisfy the cocycle condition
$\epsilon_{\beta\gamma\delta}^{\vphantom{-1}}\epsilon_{\alpha\gamma\delta}^{{-1}}\epsilon_{\alpha\beta\delta}^{\vphantom{-1}}\epsilon_{\alpha\beta\gamma}^{{-1}}=1.$
Therefore, $\epsilon_{\alpha\beta\gamma}$ defines a class in
$H^{2}(M,\underline{U(1)})\simeq H^{3}(M,{\mathbb{Z}}).$ As per the discussion
in section 2.1, this class is the obstruction to lifting the transition
functions $g_{\alpha\beta}$ to transition functions $\hat{g}_{\alpha\beta}$
and hence the obstruction to lifting $P$ to $\widehat{P}.$
If we take the principal ${\mathcal{G}}$-bundle $P\to M$ and consider the
fibre product $P^{[2]}\rightrightarrows P$ then there is a natural map
$\tau\colon P^{[2]}\to{\mathcal{G}},$ called the _difference map_ , given by
$p_{1}\tau(p_{1},p_{2})=p_{2}.$ If we view $\widehat{{\mathcal{G}}}$ as a
$U(1)$-bundle over ${\mathcal{G}}$ then we can pull-back
$\widehat{{\mathcal{G}}}$ by this map to obtain a $U(1)$-bundle over
$P^{[2]}:$
$\textstyle{\tau^{*}\widehat{{\mathcal{G}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{{\mathcal{G}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{[2]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{{\mathcal{G}}^{\vphantom{[2]}}}$
where
$\tau^{*}\widehat{{\mathcal{G}}}=\left\\{(p_{1},p_{2},\hat{g})\mid
p(\hat{g})=\tau(p_{1},p_{2})\right\\}.$
Note that $\tau(p_{1},p_{2})\tau(p_{2},p_{3})=\tau(p_{1},p_{3})$ and so,
because the multiplication in $\widehat{{\mathcal{G}}}$ covers that in
${\mathcal{G}},$ we have an induced map
$\tau^{*}\widehat{{\mathcal{G}}}_{(p_{1},p_{2})}\otimes\tau^{*}\widehat{{\mathcal{G}}}_{(p_{2},p_{3})}\to\tau^{*}\widehat{{\mathcal{G}}}_{(p_{1},p_{3})}$
which serves as a bundle gerbe multiplication for the bundle gerbe
$(\tau^{*}\widehat{{\mathcal{G}}},P)$ over $M.$ This bundle gerbe is called
the _lifting bundle gerbe_. We would now like to examine its Dixmier-Douady
class. Recall from section 2.2 the construction of the Dixmier-Douady class of
a bundle gerbe. This involves taking sections $s_{\alpha}$ and $s_{\beta}$ of
$P$ to give a section $(s_{\alpha},s_{\beta})$ of $P^{[2]}$ over
$U_{\alpha\beta}.$ We then pull-back the bundle
$\tau^{*}\widehat{{\mathcal{G}}}$ by $(s_{\alpha},s_{\beta})$ to give a bundle
$(s_{\alpha},s_{\beta})^{*}(\tau^{*}\widehat{{\mathcal{G}}})\to
U_{\alpha\beta}.$ The Dixmier-Douady class of
$\tau^{*}\widehat{{\mathcal{G}}}$ is related to sections of this bundle, that
is, maps $\sigma_{\alpha\beta}\colon
U_{\alpha\beta}\to\tau^{*}\widehat{{\mathcal{G}}}$ such that
$\sigma(m)\in\tau^{*}\widehat{{\mathcal{G}}}_{(s_{\alpha}(m),s_{\beta}(m))}$.
The bundle gerbe multiplication (which in this case is given by the
multiplication in $\widehat{{\mathcal{G}}}$) gives
$\sigma_{\alpha\beta}\sigma_{\beta\gamma}=g_{\alpha\beta\gamma}\sigma_{\alpha\gamma}$
for some $U(1)$-valued function $g_{\alpha\beta\gamma}$ and the image of this
in $H^{3}(M,{\mathbb{Z}})$ is a representative for the Dimier-Douady class of
$\tau^{*}\widehat{{\mathcal{G}}}.$ Note at this point, however, that as $P$ is
a principal ${\mathcal{G}}$-bundle, the sections $s_{\alpha}$ and $s_{\beta}$
are related by the transition functions $g_{\alpha\beta}.$ That is,
$s_{\beta}=s_{\alpha}g_{\alpha\beta}.$ This means that
$(s_{\alpha},s_{\beta})^{*}(\tau^{*}\widehat{{\mathcal{G}}})$ is given by
triples $(s_{\alpha},s_{\beta},\hat{g})$ where $p(\hat{g})=g_{\alpha\beta}.$
So in fact a section $\sigma_{\alpha\beta}$ is given by the candidate
transition functions $\hat{g}_{\alpha\beta}.$ Therefore, the sections
$\sigma_{\alpha\beta}$ satisfy
$\hat{g}_{\alpha\beta}\hat{g}_{\beta\gamma}=\epsilon_{\alpha\beta\gamma}\hat{g}_{\alpha\gamma},$
or
$\hat{g}_{\beta\gamma}^{\vphantom{-1}}\hat{g}_{\alpha\gamma}^{-1}\hat{g}_{\alpha\beta}^{\vphantom{-1}}=\epsilon_{\alpha\beta\gamma},$
which is precisely the relation above for the obstruction to the existence of
a lift. Thus the Dixmier-Douady class of the lifting bundle gerbe
$(\tau^{*}\widehat{{\mathcal{G}}},P)$ measures the obstruction to lifting the
${\mathcal{G}}$-bundle $P$ to a $\widehat{{\mathcal{G}}}$-bundle
$\widehat{P}.$ So the lifting bundle gerbe is trivial exactly when $P$ lifts
to a $\widehat{{\mathcal{G}}}$ bundle.
In the next section we shall demonstrate how to find a representative for the
obstruction class of a particular lifting problem using the methods outlined
already from the theory of bundle gerbes.
### 2.4 The string class of an $LG$-bundle
Having outlined the theory of central extensions and bundle gerbes we are now
in a position to extend Killingback’s result to general $LG$-bundles. In this
section we will review the calculations from [35] which give an explicit
expression for (the image in real cohomology of) the string class of an
$LG$-bundle $P\to M,$ where here we do not require $P$ to be a loop bundle as
in section 2.1.
##### The central extension of the loop group
In the previous section we showed how to classify isomorphism classes of
central extensions of a Lie group ${\mathcal{G}}$ using a 2-form $R$ on
${\mathcal{G}}$ and a 1-form $\alpha$ on ${\mathcal{G}}\times{\mathcal{G}}.$
Now suppose that ${\mathcal{G}}=LG,$ the loop group of a compact, simple,
simply connected Lie group. In this case we can give these forms explicitly,
thus making it possible to perform calculations involving the central
extension $\widehat{LG}$ of $LG.$
In [39] Pressley and Segal give a well known expression for the curvature of a
connection on the central extension $\widehat{LG}.$ Namely,
$R=\frac{i}{4\pi}\int_{S^{1}}\langle\Theta,\partial\Theta\rangle\,d\theta,$
where $\Theta$ is the (left-invariant) Maurer-Cartan form on $LG,$ which is
defined pointwise, $\partial$ denotes the derivative in the loop direction,
that is, the derivative with respect to $\theta$ and $\langle\,\,,\,\rangle$
is an invariant inner product333We shall refer to this as the Killing form
since all invariant, bilinear, symmetric forms on ${\mathfrak{g}}$ are
proportional and so this is just the Killing form with a suitable
normalisation. on $L{\mathfrak{g}}$ (defined pointwise) normalised so the
longest root has length squared equal to 2. To construct the central extension
we also need a 1-form $\alpha$ satisfying $\delta R=d\alpha$ and
$\delta\alpha=0.$ In this case it is easy to find such an $\alpha.$ First note
that $\delta R=\pi_{1}^{*}R-m^{*}R+\pi_{2}^{*}R$ where $m$ is the
multiplication in $LG$ and $\pi_{i}$ is the projection $LG\times LG\to LG$
which omits the $i^{\text{th}}$ factor. Then $\pi_{i}^{*}R$ is given by
$\frac{i}{4\pi}\int_{S^{1}}\langle\pi_{i}^{*}\Theta,\partial\pi_{i}^{*}\Theta\rangle\,d\theta.$
and using the identities
$\partial\Theta=ad(\gamma^{-1})d(\partial\gamma^{\vphantom{-1}}\gamma^{-1}),$
at the point $\gamma\in LG,$ and
$\partial\left(ad(\gamma^{-1})X\right)=ad(\gamma^{-1})[X,\partial\gamma\gamma^{-1}]+ad(\gamma^{-1})\partial
X,$
for a vector $X\in L{\mathfrak{g}},$ we can calculate $m^{*}R$ to be
$\frac{i}{4\pi}\int_{S^{1}}\langle\Theta_{1},\partial\Theta_{1}\rangle+\langle[\Theta_{1},\Theta_{1}],\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1}\rangle+\langle\Theta_{1},d(\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1})\rangle\\\
+\langle\Theta_{2},\partial(ad(\gamma_{2}^{-1})\Theta_{1})\rangle+\langle\Theta_{2},\partial\Theta_{2}\rangle\,d\theta,$
where we have written $\Theta_{1}$ for $\pi_{2}^{*}\Theta$ and so on. So
$\delta
R=-\frac{i}{4\pi}\int_{S^{1}}\langle[\Theta_{1},\Theta_{1}],\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1}\rangle+\langle\Theta_{1},d(\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1})\rangle+\langle\Theta_{2},\partial(ad(\gamma_{2}^{-1})\Theta_{1})\rangle\,d\theta,$
and using the identities above and integration by parts, we have
$\delta R=\frac{i}{2\pi}\int_{S^{1}}\langle
d\Theta_{1},\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1}\rangle-\langle\Theta_{1},d(\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1})\rangle\,d\theta.$
Therefore, if we define
$\alpha=\frac{i}{2\pi}\int_{S^{1}}\langle\pi_{2}^{*}\Theta,\pi_{1}^{*}Z\rangle\,d\theta,$
for $Z\colon LG\to L{\mathfrak{g}}$ the function
$\gamma\mapsto\partial\gamma\gamma^{-1},$ then we see that $d\alpha=\delta R.$
Also, one can check that $\delta\alpha=0.$
##### A connection for the lifting bundle gerbe
Now that we have a construction of $\widehat{LG}$ in terms of the differential
forms $R$ and $\alpha,$ we can consider the problem of lifting the $LG$-bundle
$P\to M$ to an $\widehat{LG}$-bundle $\widehat{P}\to M.$ We can write down the
lifting bundle gerbe for this problem, that is, the bundle gerbe
$(\tau^{*}\widehat{LG},P)$ over $M,$ and we would like a connection on this
bundle gerbe so we can calculate its Dixmier-Douady class.
Consider, then, the map $\tau\colon P^{[2]}\to LG$ above. We can extend this
to a map $\tau\colon P^{[k+1]}\to LG^{k}$ by defining
$\tau(p_{1},\ldots,p_{k+1})=(\tau(p_{1},p_{2}),\ldots,\tau(p_{k},p_{k+1})).$
This is a _simplicial map_. That is, it commutes with the face and degeneracy
maps for the simplicial manifolds $\\{P^{[k]}\\}$ and $\\{LG^{k}\\}.$ This
means that for differential forms on these manifolds, $\delta$ commutes with
pull-back by $\tau.$ Now consider the connection $\nu$ on $\widehat{LG}$
(whose curvature is the form $R$). The natural choice for a bundle gerbe
connection would be the pull-back, $\tau^{*}\nu,$ of this form to
$\tau^{*}\widehat{LG}.$ However, $\tau^{*}\nu$ is not a bundle gerbe
connection because it does not respect the product. That is,
$s^{*}(\delta\tau^{*}\nu)$ is non-zero. We know from the discussion on bundle
gerbe connections in section 2.2 that $\delta(s^{*}(\delta\tau^{*}\nu))=0$ and
so there is some form $\epsilon$ on $P^{[2]}$ such that
$\delta\epsilon=s^{*}(\delta\tau^{*}\nu).$ Then $\tau^{*}\nu-\epsilon$ will be
a bundle gerbe connection on $\tau^{*}\widehat{LG}.$ In fact, in this case,
since $\alpha=s^{*}(\delta\nu)$ by definition, we have
$s^{*}(\delta\tau^{*}\nu)=\tau^{*}\alpha.$ So
$\delta(s^{*}(\delta\tau^{*}\nu))=\delta\tau^{*}\alpha=\tau^{*}\delta\alpha=0$
as $\delta\alpha=0$ and so $\epsilon$ satisfies
$\delta\epsilon=\tau^{*}\alpha.$ Thus it suffices to find a 1-form $\epsilon$
on $P^{[2]}$ satisfying $\delta\epsilon=\tau^{*}\alpha.$
The form $\tau^{*}\alpha$ is given by
$\frac{i}{2\pi}\int_{S^{1}}\langle\tau_{12}^{*}\Theta,\tau_{23}^{*}Z\rangle\,d\theta$
where we have written $\tau_{ij}$ for $\tau(p_{i},p_{j}).$ In order to solve
for $\epsilon,$ we need to choose a connection $A$ on $P.$ Then using the
equation $p_{1}\tau(p_{1},p_{2})=p_{2}$ and the Leibnitz rule (see [23]), we
find the identity
$\pi_{1}^{*}A=ad(\tau_{12}^{-1})\pi_{2}^{*}A+\tau_{12}^{*}\Theta.$
Therefore we have
$\tau^{*}\alpha=\frac{i}{2\pi}\int_{S^{1}}\langle\pi_{13}^{*}A-ad(\tau_{12}^{-1})\pi_{23}^{*}A,\partial\tau_{23}^{\vphantom{-1}}\tau_{23}^{-1}\rangle\,d\theta,$
where $\pi_{23}(p_{1},p_{2},p_{3})=p_{1},$ etc. Now define
$\epsilon=\frac{i}{2\pi}\int_{S^{1}}\langle\pi_{2}^{*}A,\tau^{*}Z\rangle\,d\theta.$
Then, using the simplicial identities and the fact that
$\tau_{ij}\tau_{jk}=\tau_{ik},$ we have
$\displaystyle\delta\epsilon$
$\displaystyle=\pi_{1}^{*}\epsilon-\pi_{2}^{*}\epsilon+\pi_{3}^{*}\epsilon$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\langle\pi_{13}^{*}A,\tau_{23}^{*}Z\rangle-\langle\pi_{23}^{*}A,\tau_{13}^{*}Z\rangle+\langle\pi_{23}^{*}A,\tau_{12}^{*}Z\rangle\,d\theta$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\langle\pi_{13}^{*}A,\tau_{23}^{*}Z\rangle-\langle\pi_{23}^{*}A,ad(\tau_{12}^{\vphantom{{}^{*}}})\tau_{23}^{*}Z\rangle\,d\theta$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\langle\pi_{13}^{*}A-ad(\tau_{12}^{-1})\pi_{23}^{*}A,\partial\tau_{23}^{\vphantom{-1}}\tau_{23}^{-1}\rangle\,d\theta.$
It turns out [43] that in general, $\epsilon$ can be written in terms of
$\alpha$ and $A.$ We shall demonstrate in section 4.1 how to find $\epsilon$
in general.
Since we want to calculate the 3-curvature of the lifting bundle gerbe, we are
really interested in the curvature of the connection $\tau^{*}\nu-\epsilon.$
This is given by $\tau^{*}R-d\epsilon.$ Using the identities given above, we
have
$\displaystyle\tau^{*}R$
$\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\langle\tau^{*}\Theta,\partial\tau^{*}\Theta\rangle\,d\theta$
$\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\langle
A_{2}-ad(\tau^{-1})A_{1},\partial(A_{2}-ad(\tau^{-1})A_{1})\rangle\,d\theta$
$\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\langle A_{2},\partial
A_{2}\rangle+\langle A_{1},\partial
A_{1}\rangle+\langle[A_{1},A_{1}],\tau^{*}Z\rangle-2\langle
ad(\tau^{-1})A_{1},\partial A_{2}\rangle\,d\theta,$ and $\displaystyle
d\epsilon$ $\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\langle
dA_{1},\tau^{*}Z\rangle-\langle A_{1},d(\tau^{*}Z)\rangle\,d\theta$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\langle
dA_{1},\tau^{*}Z\rangle-\langle A_{1},\partial
A_{1}\rangle+\langle[A_{1},A_{1}],\tau^{*}Z\rangle-\langle
ad(\tau^{-1})A_{1},\partial A_{2}\rangle\,d\theta.$
Therefore
$\tau^{*}R-d\epsilon=\frac{i}{4\pi}\int_{S^{1}}\langle\pi_{1}^{*}A,\partial\pi_{1}^{*}A\rangle-\langle\pi_{2}^{*}A,\partial\pi_{2}^{*}A\rangle-2\langle\pi_{2}^{*}F,\tau^{*}Z\rangle\,d\theta,$
where $F=dA+\frac{1}{2}[A,A]$ is the curvature of $A.$
##### A curving for the lifting bundle gerbe
The next step is to find a curving for $\tau^{*}\widehat{LG}.$ That is, we
wish to find some 2-form $B$ on $P$ such that $\delta B=\tau^{*}R-d\epsilon.$
Note that $\delta\colon\Omega^{2}(P)\to\Omega^{2}(P^{[2]})$ is given by
$\delta=\pi_{1}^{*}-\pi_{2}^{*},$ so we can write $\tau^{*}R-d\epsilon$ as
$\delta\left(\frac{i}{4\pi}\int_{S^{1}}\langle A,\partial
A\rangle\,d\theta\right)-\frac{i}{2\pi}\int_{S^{1}}\langle\pi_{2}^{*}F,\tau^{*}Z\rangle\,d\theta.$
Thus we just need to find some $B_{2}\in\Omega^{2}(P)$ such that
$\delta
B_{2}=\frac{i}{2\pi}\int_{S^{1}}\langle\pi_{2}^{*}F,\tau^{*}Z\rangle\,d\theta.$
To solve this equation, we use a _Higgs field_ for the bundle $P.$ A Higgs
field is a map $\Phi\colon P\to L{\mathfrak{g}}$ satisfying
$\Phi(p\gamma)=ad(\gamma^{-1})\Phi(p)+\gamma^{-1}\partial\gamma.$
It is clear that Higgs fields exist. Since they exist when $P$ is trivial and
convex combinations of Higgs fields are also Higgs fields, we can use a
partition of unity to construct a Higgs field in general. We shall explain the
geometric significance of this map in the next section. For now, note that if
we pull back $\Phi$ to $P^{[2]}$ it satisfies
$ad(\tau)\pi_{1}^{*}\Phi=\pi_{2}^{*}\Phi+\tau^{*}Z.$
This just comes from the condition above and the definition of $\tau.$ Then we
see that
$\displaystyle\langle\pi_{2}^{*}F,\tau^{*}Z\rangle$
$\displaystyle=\langle\pi_{2}^{*}F,ad(\tau)\pi_{1}^{*}\Phi\rangle-\langle\pi_{2}^{*}F,\pi_{2}^{*}\Phi\rangle$
$\displaystyle=\langle
ad(\tau^{-1})\pi_{2}^{*}F,\pi_{1}^{*}\Phi\rangle-\langle\pi_{2}^{*}F,\pi_{2}^{*}\Phi\rangle.$
But one can demonstrate (in a similar manner to the proof of the equation
above relating $\pi_{1}^{*}A$ and $\pi_{2}^{*}A$) that the curvature $F$
satisfies
$\pi_{1}^{*}F=ad(\tau^{-1})\pi_{2}^{*}F$
and so we have
$\langle\pi_{2}^{*}F,\tau^{*}Z\rangle=\langle\pi_{1}^{*}F,\pi_{1}^{*}\Phi\rangle-\langle\pi_{2}^{*}F,\pi_{2}^{*}\Phi\rangle.$
Therefore, a curving is given by
$B=\frac{i}{2\pi}\int_{S^{1}}\tfrac{1}{2}\langle A,\partial A\rangle-\langle
F,\Phi\rangle\,d\theta.$
##### The string class of an $LG$-bundle
Now that we have a curving for the lifting bundle gerbe we can find a
representative for the string class $s(P)$ by calculating the 3-curvature
$H=dB.$ We have
$dB=\frac{i}{2\pi}\int_{S^{1}}\tfrac{1}{2}\langle dA,\partial
A\rangle-\tfrac{1}{2}\langle A,\partial dA\rangle-\langle
dF,\Phi\rangle-\langle F,d\Phi\rangle\,d\theta.$
Integration by parts and the Bianchi identity $dF=[F,A]$ yields
$dB=\frac{i}{2\pi}\int_{S^{1}}\langle dA,\partial A\rangle-\langle
F,[A,\Phi]\rangle-\langle F,d\Phi\rangle\,d\theta$
and since the integral over the circle of $\langle[A,A],\partial A\rangle$
vanishes, we find
$dB=\frac{i}{2\pi}\int_{S^{1}}\langle F,\partial A\rangle-\langle
F,[A,\Phi]\rangle-\langle F,d\Phi\rangle\,d\theta.$
This descends to a form on $M$ and so
$H=-\frac{i}{2\pi}\int_{S^{1}}\langle F,\nabla\Phi\rangle\,d\theta,$
where
$\nabla\Phi=d\Phi+[A,\Phi]-\partial A.$
Thus we have the result from [35]
###### Theorem 2.4.1 ([35]).
Let $P\to M$ be a principal $LG$-bundle. Let $A$ be a connection on $P$ with
curvature $F$ and let $\Phi$ be a Higgs field for $P.$ Then the string class
of $P$ is represented in de Rham cohomology by the form
$-\frac{1}{4\pi^{2}}\int_{S^{1}}\langle F,\nabla\Phi\rangle\,d\theta,$
where $\nabla\Phi$ is the covariant derivative above.
### 2.5 Higgs fields, $LG$-bundles and the string class
Recall Killingback’s result from section 2.1 regarding string structures of a
loop bundle. That is, if $Q\to M$ is a principal $G$-bundle and $LQ\to LM$ is
the $LG$-bundle obtained by taking loops, then the string class of $LQ$ is the
transgression of the first Pontrjagyn class of $Q,$ i.e.
$s(LQ)=\int_{S^{1}}\operatorname{ev}^{*}p_{1}(Q).$
In the last section we obtained, following the methods of [35], a general
expression for the string class of a principal $LG$-bundle $P\to M$ which is
not necessarily a loop bundle. In this case we can prove a result analogous to
Killingback’s by using a correspondence between $LG$-bundles and certain
$G$-bundles. This will also enable us to provide an easy proof of
Killingback’s result.
#### 2.5.1 Higgs fields and $LG$-bundles
The following correspondence first appeared in [16] in a study of calorons
(monopoles for the loop group) and, in the context in which we are interested,
in [35]. We shall present the construction here in some detail since we will
generalise this result in section 4.2 to $LG\rtimes S^{1}$-bundles and it will
be instructive to see the introductory case in depth.
We wish to set up a bijective correspondence between $LG$-bundles over $M$ and
$G$-bundles over $M\times S^{1}.$ Consider the $LG$-bundle $P\times S^{1}\to
M\times S^{1}$ where the $LG$ action is trivial on the $S^{1}$ factor. Then
use the evaluation map $\operatorname{ev}\colon LG\times S^{1}\to G$ to form
the associated $G$-bundle $\widetilde{P}\to M\times S^{1}$. That is, define
$\widetilde{P}$ by
$\widetilde{P}=(P\times G\times S^{1})/LG$
where $LG$ acts on $P\times G\times S^{1}$ by
$(p,g,\theta)\gamma=(p\gamma,\gamma(\theta)^{-1}g,\theta).$ Then there is a
right $G$ action on $\widetilde{P}$ given by $[p,g,\theta]h=[p,gh,\theta]$
(where square brackets denote equivalence classes) and a projection
$\tilde{\pi}\colon\widetilde{P}\to M\times S^{1}$ given by
$\tilde{\pi}([p,g,\theta])=(\pi(p),\theta).$ This action is free and
transitive on the fibres (which are the orbits of the $G$ action) and hence
$\widetilde{P}\to M\times S^{1}$ is a principal $G$-bundle.
Conversely, given a $G$-bundle $\widetilde{P}\to M\times S^{1}$ we can define
fibrewise an $LG$-bundle $P\to M$ by taking sections of $\widetilde{P}$
restricted to a point in $M.$ That is, the fibre of $P$ over $m$ is
$P_{m}=\Gamma(\widetilde{P}_{|\\{m\\}\times S^{1}})$
or
$P_{m}=\\{f\colon
S^{1}\to\widetilde{P}\,|\,\tilde{\pi}(f(\theta))=(m,\theta)\\}.$
The $LG$ action here is the obvious one derived from the $G$ action on
$\widetilde{P}.$ The transition functions of this bundle are simply the
transition functions of $\widetilde{P}$ considered as functions from an open
set of $M$ to $LG,$ for if $\\{U_{\alpha}\times S^{1}\\}$ is an open cover of
$M\times S^{1}$ and $\tilde{s}_{\alpha}$ is a section of $\widetilde{P}$ then
since elements of $P$ are loops in $\widetilde{P},$ a section of $P$ is given
by $s_{\alpha}(m)(\theta)=\tilde{s}_{\alpha}(m,\theta).$ If $s_{\beta}$ is
another such section, then the transition functions of $P,$
$g_{\alpha\beta}\colon U_{\alpha}\cap U_{\beta}\to LG,$ are given by
$s_{\beta}(m)=s_{\alpha}(m)g_{\alpha\beta}(m).$
Evaluating at $\theta$ gives
$s_{\beta}(m)(\theta)=s_{\alpha}(m)(\theta)g_{\alpha\beta}(m)(\theta).$
But $s_{\beta}(m)(\theta)=\tilde{s}_{\beta}(m,\theta)$ (and similarly for
$\alpha$), so we have
$g_{\alpha\beta}(m)(\theta)=\tilde{g}_{\alpha\beta}(m,\theta)$
where $\tilde{g}_{\alpha\beta}$ are the transition functions for
$\widetilde{P}.$ We can actually give a global description of this bundle
quite easily by considering the map
$\eta\colon M\to L(M\times S^{1});\quad m\mapsto(\theta\mapsto(m,\theta)).$
That is, $\eta(m)(\theta)=(m,\theta).$ Then the bundle $P$ is the pullback of
the $LG$-bundle $L\widetilde{P}\to L(M\times S^{1}):$
$\textstyle{\eta^{*}L\widetilde{P}=P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\widetilde{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{L(M\times
S^{1}).}$
Thus we have a way of constructing a $G$-bundle given an $LG$-bundle and vice
versa. It remains to be shown that this is a bijection on the set of
isomorphism classes of these bundles. That is, if we start with a $G$-bundle
$\widetilde{P}$ and construct $P$ and then form the $G$-bundle corresponding
to that bundle, say $\widetilde{P}^{\prime},$ we have that
$\widetilde{P}^{\prime}$ is isomorphic to $\widetilde{P}.$ And similarly, if
we start with $P$ and construct $\widetilde{P}$ and then construct the
$LG$-bundle corresponding to that, say $P^{\prime},$ then these are
isomorphic. To see this, first consider a $G$-bundle $\widetilde{P}$ and
construct $P$ as above. Then $\widetilde{P}^{\prime}$ is given by
$\widetilde{P}^{\prime}=(P\times G\times S^{1})/LG$
where for $[p,g,\theta]\in(P\times G\times S^{1})/LG,\,p$ is a map
$S^{1}\to\widetilde{P}$ as above. Define a bundle map by
$f\colon\widetilde{P}^{\prime}\to\widetilde{P};\quad[p,g,\theta]\mapsto
p(\theta)g.$
This is well-defined, since
$[p\gamma,\gamma(\theta)^{-1}g,\theta]\stackrel{{\scriptstyle
f}}{{\mapsto}}(p\gamma)(\theta)\gamma(\theta)^{-1}g=p(\theta)g$ and commutes
with the $G$ action, since $[p,g,\theta]h=[p,gh,\theta]\stackrel{{\scriptstyle
f}}{{\mapsto}}p(\theta)gh=(p(\theta)g)h.$ Hence $f$ is a bundle isomorphism.
On the other hand, if we consider an $LG$-bundle $P$ and construct
$\widetilde{P}=(P\times G\times S^{1})/LG$ then $P^{\prime}$ is given by the
pull-back above. Notice that if we define the map $\hat{\eta}\colon P\to
L\widetilde{P}$ by
$\hat{\eta}(p)(\theta)=[p,1,\theta]$
then $\hat{\eta}$ covers $\eta\colon M\to L(M\times S^{1}),$ that is,
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\eta}}$$\textstyle{L\widetilde{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{L(M\times
S^{1})}$
commutes, and so $P$ is isomorphic to the pull-back $P^{\prime}.$ Thus we have
proven
###### Proposition 2.5.1 ([16, 35]).
There is a bijective correspondence between isomorphism classes of principal
$G$-bundles over $M\times S^{1}$ and isomorphism classes of principal
$LG$-bundles over $M.$
Importantly for our purposes, this correspondence holds on the level of
connections as well. More specifically, if we have a $G$-bundle with
connection we can construct an $LG$-bundle with connection and Higgs field
and, conversely, given an $LG$-bundle with connection and Higgs field we can
construct a $G$-bundle with connection. We shall see that the Higgs field is
essentially the $S^{1}$ component of the connection on $\widetilde{P}.$
Suppose we have a connection $\tilde{A}$ on $\widetilde{P}.$ We can define a
connection on $P$ (which is an $L{\mathfrak{g}}$-valued 1-form) by
$A_{p}(X)(\theta)=\tilde{A}_{p(\theta)}(X_{\theta}),$ where $X$ is a vector in
$T_{p}P$ (i.e. a vector field along $p$ in $\widetilde{P}$), and so
$X_{\theta}\in T_{p(\theta)}\widetilde{P}.$ This is a connection by virtue of
the fact that $\tilde{A}$ is. If we view $\tilde{A}$ as a splitting of the
tangent space at each point in $\widetilde{P},$ then we can easily see that
$A$ is given by essentially the same splitting since for each $\theta\in
S^{1},$ $T_{p}P$ splits as
$T_{p(\theta)}\widetilde{P}\simeq V_{p(\theta)}\widetilde{P}\oplus
H_{p(\theta)}\widetilde{P}$
where $V_{p(\theta)}\widetilde{P}$ is the vertical subspace at $p(\theta)$ and
$H_{p(\theta)}\widetilde{P}$ is the horizontal subspace.
Suppose instead we are given an $LG$-bundle $P$ with connection $A$ and Higgs
field $\Phi.$ Then we can define a form on $P\times G\times S^{1}$ by
$\tilde{A}=ad(g^{-1})A(\theta)+\Theta+ad(g^{-1})\Phi\,d\theta.$
This form descends to a form on $\widetilde{P}$ and the connection (also
called $\tilde{A}$) is given by this equation considered as a form on
$(P\times G\times S^{1})/LG.$ To show that this is well defined, we need to
check that it is independent of the lift of a vector in $\widetilde{P}.$ That
is, if $\hat{X}$ and $\hat{X}^{\prime}$ are two lifts of the vector $X\in
T_{[p,g,\theta]}\widetilde{P}$ to the fibre in $P\times G\times S^{1}$ above
$[p,g,\theta],$ then $\tilde{A}(\hat{X})=\tilde{A}(\hat{X}^{\prime}).$ Suppose
then, that $\hat{X}\in T_{(p,g,\theta)}(P\times G\times S^{1})$ and
$\hat{X}^{\prime}\in T_{(p,g,\theta)\gamma}(P\times G\times S^{1}).$ Then
$\hat{X}\gamma\in T_{(p,g,\theta)\gamma}(P\times G\times S^{1}),$ and
$\hat{X}^{\prime}$ and $\hat{X}\gamma$ differ by a vertical vector (with
respect to the $LG$ action) at
$(p,g,\theta)\gamma=(p\gamma,\gamma(\theta)^{-1}g,\theta)$ and so it is
sufficient to show that $\tilde{A}$ is zero on vertical vectors and invariant
under the $LG$ action (since then
$\tilde{A}(\hat{X}^{\prime})=\tilde{A}(\hat{X}\gamma+\text{vertical})=\tilde{A}(\hat{X})$).
Because any compact Lie group has a faithful representation as matrix group
[39], we can expand the exponential map as $\exp(t\xi)=1+t\xi+\ldots.$
Therefore, the vertical vector at $(p,g,\theta)$ generated by $\xi\in
L{\mathfrak{g}}$ is
$\displaystyle V$
$\displaystyle=\frac{d}{dt}{\bigg{|}_{0}}(p,g,\theta)\exp(t\xi)$
$\displaystyle=\frac{d}{dt}{\bigg{|}_{0}}(p\exp(t\xi),\exp(-t\xi(\theta))g,\theta)$
$\displaystyle=(\iota_{p}(\xi),-\xi(\theta)g,0),$
(where we have written $\left.\frac{d}{dt}\right|_{0}$ for the derivative
evaluated at $t=0$), and so
$\displaystyle\tilde{A}(V)$
$\displaystyle=ad(g^{-1})A(\iota_{p}(\xi))(\theta)-g^{-1}\xi(\theta)g$
$\displaystyle=g^{-1}\xi(\theta)g-g^{-1}\xi(\theta)g$ $\displaystyle=0.$
So $\tilde{A}$ is zero on vertical vectors. Now, suppose
$\hat{X}=(X,g\zeta,x_{\theta})$ is given by
$\frac{d}{dt}{\bigg{|}_{0}}(\gamma_{X}(t),g\exp(t\zeta),\theta+tx),$
where $\gamma_{X}(t)$ is a path in $P$ whose tangent vector at $0$ is $X$ and
where $\zeta$ and $x$ are elements of the Lie algebras of $G$ and $S^{1}$
respectively. Then
$\displaystyle\hat{X}\gamma$
$\displaystyle=\frac{d}{dt}{\bigg{|}_{0}}(\gamma_{X}(t)\gamma,\gamma(\theta+tx)g\exp(t\zeta),\theta+tx)$
$\displaystyle=\frac{d}{dt}{\bigg{|}_{0}}(\gamma_{X}(t)\gamma,\gamma(\theta)gt\zeta+tx\partial\gamma(\theta)g,\theta+tx)$
$\displaystyle=(X\gamma,\gamma(\theta)g(\zeta+xad(g^{-1})\gamma(\theta)^{-1}\partial\gamma(\theta)),x).$
So
$\tilde{A}_{(p\gamma,\gamma(\theta)^{-1}g,\theta)}(\hat{X}\gamma)=\tilde{A}_{(p\gamma,\gamma(\theta)^{-1}g,\theta)}(X\gamma,\gamma(\theta)g(\zeta+xad(g^{-1})\gamma(\theta)^{-1}\partial\gamma(\theta)),x)\\\
\phantom{\tilde{A}_{(p\gamma,\gamma(\theta)^{-1}g,\theta)}(\hat{X}\gamma)}=ad((\gamma(\theta)^{-1}g)^{-1})A(X\gamma)+\zeta+xad(g^{-1})\gamma(\theta)^{-1}\partial\gamma(\theta)\\\
+ad((\gamma(\theta)^{-1}g)^{-1})x\Phi(p\gamma)\\\
\phantom{\tilde{A}_{(p\gamma,\gamma(\theta)^{-1}g,\theta)}(\hat{X}\gamma)}=ad(g^{-1})ad(\gamma)ad(\gamma^{-1})A(X)(\theta)+\zeta+xad(g^{-1})\gamma(\theta)^{-1}\partial\gamma(\theta)\\\
+ad(g^{-1})xad(\gamma)(ad(\gamma^{-1})\Phi(p)+\gamma^{-1}\partial\gamma)\\\
\phantom{\tilde{A}_{(p\gamma,\gamma(\theta)^{-1}g,\theta)}(\hat{X}\gamma)}=ad(g^{-1})A(X)(\theta)+\zeta+ad(g^{-1})x\Phi(p).\\\
$
Therefore $\tilde{A}$ is invariant under the $LG$ action and so defines a form
on $\widetilde{P}.$ This form is a connection form since if
$[X,g\zeta,x_{\theta}]$ is a vector at $[p,g,\theta],$ then
$[X,g\zeta,x_{\theta}]h=[X,gh\,ad(h^{-1})\zeta,x_{\theta}]$ and so
$\displaystyle\tilde{A}([X,g\zeta,x_{\theta}]h)$
$\displaystyle=ad(h^{-1}g^{-1})A(X)(\theta)+ad(h^{-1})\zeta+ad(h^{-1}g^{-1})x\Phi(p)$
$\displaystyle=ad(h^{-1})\tilde{A}([X,g\zeta,x_{\theta}])$
and further, the vertical vector at $[p,g,\theta]$ generated by
$\zeta\in{\mathfrak{g}}$ is given by
$\displaystyle V_{\zeta}$
$\displaystyle=\frac{d}{dt}_{|_{0}}[p,g\exp(t\zeta),\theta]$
$\displaystyle=[0,g\zeta,0]$
and so $\tilde{A}(V_{\zeta})=\zeta.$
We have shown already that the correspondence outlined above is a bijection
between isomorphism classes of bundles. Now we will show that in fact it is a
bijection between isomorphism classes of bundles with connection. So given a
$G$-bundle $\widetilde{P}$ with connection $\tilde{A},$ we construct the
$LG$-bundle $P$ with the connection $A$ as above. Then construct the
$G$-bundle $\widetilde{P}^{\prime}$ (which is isomorphic to $\widetilde{P}$)
and give it the connection $\tilde{A}^{\prime}$ which we just outlined. Of
course, to do this we’ll need a Higgs field for $P.$ Recalling that elements
of $P$ are essentially loops in $\widetilde{P},$ we can define a Higgs field
by
$\Phi(p)=\tilde{A}(\partial p).$
This is a Higgs field since if we calculate $\Phi(p\gamma)$ we get
$\displaystyle\tilde{A}(\partial(p\gamma))$
$\displaystyle=\tilde{A}((p\gamma)_{*}\frac{\partial}{\partial\theta})$
$\displaystyle=\tilde{A}((\partial
p)\gamma+\iota_{p\gamma}(\gamma^{-1}\partial\gamma))$
$\displaystyle=ad(\gamma^{-1})\tilde{A}(\partial
p)+\gamma^{-1}\partial\gamma.$
(Note that this is essentially the $S^{1}$ part of $\tilde{A}.$ That is, if we
take a section $\tilde{s}$ of $\widetilde{P}\to M\times S^{1}$ we can get a
section $s$ of $P\to M$ by $s(m)(\theta):=\tilde{s}(m,\theta).$ Then if we
pull-back $\Phi$ by $s$ we get
$\displaystyle(s^{*}\Phi)(m)(\theta)$
$\displaystyle=(\tilde{s}^{*}\tilde{A})(m,\theta)\left(\frac{\partial}{\partial\theta}\right)$
$\displaystyle=(\tilde{s}^{*}\tilde{A})_{\theta}(m,\theta)$
where $(\tilde{s}^{*}\tilde{A})_{\theta}$ is the $S^{1}$ part of
$(\tilde{s}^{*}\tilde{A})$ – i.e. the coefficient of $d\theta$ – and since the
$\frac{\partial}{\partial\theta}$ kills all but the $d\theta$ part.)
Therefore, the connection $\tilde{A}^{\prime}$ is given in terms of
$\tilde{A}$ as
$\tilde{A}^{\prime}_{[p,g,\theta]}=ad(g^{-1})\tilde{A}_{p(\theta)}+\Theta+ad(g^{-1})\tilde{A}(\partial
p)d\theta.$
Recall that $\widetilde{P}^{\prime}$ is isomorphic to $\widetilde{P}$ via the
map
$f\colon\widetilde{P}^{\prime}\to\widetilde{P};\quad[p,g,\theta]\mapsto
p(\theta)g,$
so we would like to have $f^{*}\tilde{A}=\tilde{A}^{\prime}.$ Now,
$f^{*}\tilde{A}([X,g\zeta,x_{\theta}])=\tilde{A}(f_{*}[X,g\zeta,x_{\theta}])$
and, as before, if $\gamma_{X}(t)$ is a path in $P$ whose tangent vector at
$0$ is $X$ and if $\zeta$ and $x$ are elements of the Lie algebras of $G$ and
$S^{1}$ respectively, then
$f_{*}[X,g\zeta,x_{\theta}]=\frac{d}{dt}{\bigg{|}_{0}}(\gamma_{X}(t)(\theta+tx)g\exp(t\zeta))\\\
\phantom{f_{*}[X,g\zeta,x_{\theta}]}=\left(\frac{d}{dt}(\gamma_{X}(t))(\theta+tx)g\exp(t\zeta)+\gamma_{X}(t)(\theta+tx)g\frac{d}{dt}\exp(t\zeta)\right.\\\
+\partial\gamma_{X}(t)(\theta+tx)xg\exp(t\zeta)\left.\vphantom{\frac{d}{dt}}\right){\bigg{|}_{0}}\\\
\phantom{f_{*}[X,g\zeta,x_{\theta}]}=X(\theta)g+\iota_{p(\theta)g}(\zeta)+\partial
p(\theta)xg\\\ $
and so
$\displaystyle f^{*}\tilde{A}([X,g\zeta,x_{\theta}])$
$\displaystyle=ad(g^{-1})A(X)+\zeta+ad(g^{-1})A(\partial p(\theta))x$
$\displaystyle=\tilde{A}^{\prime}([X,g\zeta,x_{\theta}]).$
If, on the other hand, we had started with the $LG$-bundle $P$ with connection
$A$ (and Higgs field $\Phi$), then $A^{\prime}$ would be given by
$A^{\prime}_{p}(X)(\theta)=\tilde{A}_{p(\theta)}(X_{\theta})$
and recalling that the isomorphism between $P$ and $P^{\prime}$ is essentially
given by $f(p)=(\theta\mapsto[p,1,\theta]),$ we have
$f^{*}A^{\prime}_{p}(X)(\theta)=A_{p}(X)(\theta).$
Hence, we have
###### Proposition 2.5.2 ([35]).
The correspondence from Proposition 2.5.1 extends to a bijection between
$G$-bundles on $M\times S^{1}$ with connection and $LG$-bundles on $M$ with
connection and Higgs field.
#### 2.5.2 The string class and the first Pontrjagyn class
As mentioned previously, the correspondence above provides us with a result
analogous to Killingback’s. We have
###### Theorem 2.5.3 ([35]).
Let $P\to M$ be an $LG$-bundle and $\widetilde{P}\to M\times S^{1}$ the
corresponding $G$-bundle. Then the string class of $P$ is given by integrating
over the circle the first Pontrjagyn class of $\widetilde{P}.$ That is,
$s(P)=\int_{S^{1}}p_{1}(\widetilde{P}).$
###### Proof.
If $\tilde{F}$ is the curvature of a connection on $\widetilde{P}$ then the
Pontrjagyn form is given by
$p_{1}(\widetilde{P})=-\frac{1}{8\pi^{2}}\langle\tilde{F},\tilde{F}\rangle.$
In this case we know that $\tilde{A}$ is given as in the previous section.
That is,
$\tilde{A}=ad(g^{-1})A+\Theta+ad(g^{-1})\Phi\,d\theta,$
so we can calculate its curvature using
$\tilde{F}=d\tilde{A}+\frac{1}{2}[\tilde{A},\tilde{A}].$ Now,
$\tfrac{1}{2}[\tilde{A},\tilde{A}]=\tfrac{1}{2}[ad(g^{-1})A+\Theta+ad(g^{-1})\Phi\,d\theta,ad(g^{-1})A+\Theta+ad(g^{-1})\Phi\,d\theta]\\\
\phantom{\tfrac{1}{2}[\tilde{A},\tilde{A}]}=\tfrac{1}{2}ad(g^{-1})[A,A]+\tfrac{1}{2}[\Theta,\Theta]+[\Theta,ad(g^{-1})A]\\\
+ad(g^{-1})[A,\Phi]d\theta+[\Theta,ad(g^{-1})\Phi]d\theta.$
So we just need to calculate
$d\tilde{A}=d(ad(g^{-1})A)+d\Theta+d(ad(g^{-1})\Phi)d\theta.$ Now, if $\omega$
is a 1-form then for tangent vectors $X$ and $Y$ we have
$d\omega(X,Y)=\tfrac{1}{2}\left\\{X(\omega(Y))-Y(\omega(X))-\omega([X,Y])\right\\},$
so let $(X,g\xi,x_{\theta})$ and $(Y,g\zeta,y_{\theta})$ be two tangent
vectors to $\widetilde{P}$ at the point $[p,g,\theta].$ Then for
$d(ad(g^{-1})A),$ first calculate
$\displaystyle(X,g\xi,x_{\theta})(ad(g^{-1})$ $\displaystyle
A_{p}(Y)_{\theta})$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(1-t\xi)g^{-1}A_{\gamma_{X}(t)}(Y)_{(\theta+tx)}g(1+t\xi)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}\left(ad(g^{-1})A_{\gamma_{X}(t)}(Y)_{\theta}\right)+ad(g^{-1})\partial
A_{p}(Y)x-[\xi,ad(g^{-1})A_{p}(Y)_{\theta}].$
This yields
$d(ad(g^{-1})A)=ad(g^{-1})dA-ad(g^{-1})\partial A\wedge
d\theta-[\Theta,ad(g^{-1})A].$
Similarly, for $d(ad(g^{-1})\Phi)d\theta$ we have
$\displaystyle(X,g\xi,x_{\theta})(ad(g^{-1})$ $\displaystyle\Phi(p)_{\theta})$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(1-t\xi)g^{-1}\Phi(\gamma_{X}(t))_{(\theta+tx)}g(1+t\xi)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}\left(ad(g^{-1})\Phi(\gamma_{X}(t))\right)+ad(g^{-1})\partial\Phi
x-[\xi,ad(g^{-1})\Phi(p)_{\theta}],$
and so
$d(ad(g^{-1})\Phi)d\theta=ad(g^{-1})d\Phi\wedge
d\theta-[\Theta,ad(g^{-1})\Phi]d\theta.$
Putting these together gives
$\tilde{F}=ad(g^{-1})dA-ad(g^{-1})\partial A\wedge
d\theta-[\Theta,ad(g^{-1})A]+d\Theta\\\ +ad(g^{-1})d\Phi\wedge
d\theta-[\Theta,ad(g^{-1})\Phi]d\theta+\tfrac{1}{2}ad(g^{-1})[A,A]\\\
+\tfrac{1}{2}[\Theta,\Theta]+[\Theta,ad(g^{-1})A]+ad(g^{-1})[A,\Phi]d\theta+[\Theta,ad(g^{-1})\Phi]d\theta\\\
\phantom{\tilde{F}}=ad(g^{-1})\left(dA+\tfrac{1}{2}[A,A]+d\Phi\wedge
d\theta+[A,\Phi]d\theta-\partial A\wedge d\theta\right)\\\ $
That is,
$\tilde{F}=ad(g^{-1})\left(F+\nabla\Phi\,d\theta\right).$
Then the Pontrjagyn form is given by
$p_{1}(\widetilde{P})=-\frac{1}{8\pi^{2}}\left(\langle F,F\rangle+2\langle
F,\nabla\Phi\rangle\,d\theta\right),$
and integrating over the circle gives the required result. ∎
##### A proof of Killingback’s result
We now have a result which is more general than Killingback’s result since it
can be applied to a general $LG$-bundle, not just a loop bundle. We now show
how Theorem 2.5.3 gives a method for proving Killingback’s result.
###### Corollary 2.5.4.
Let $LQ\to LM$ be a loop bundle, that is, a principal $LG$-bundle obtained by
taking loops in a $G$-bundle $Q\to M.$ Then
$s(LQ)=\int_{S^{1}}\operatorname{ev}^{*}p_{1}(Q).$
###### Proof.
We know that the string class of $LQ$ is given by the integral over the circle
of the first Pontrjagyn class of the corresponding $G$-bundle over $LM\times
S^{1}.$ We show that this bundle is isomorphic to the pull-back of $Q$ by the
evaluation map, then the result follows. The $G$-bundle $\widetilde{LQ}$ is
given by $(LQ\times G\times S^{1})/LG.$ Define the map $\widetilde{LQ}\to Q$
by $[q,g,\theta]\mapsto q(\theta)g.$ As in section 2.5.1 above, this map is
well-defined and commutes with the $G$-action. Furthermore, it covers the
evaluation map $LM\times S^{1}\to M$ and so $\widetilde{LQ}$ is isomorphic to
$\operatorname{ev}^{*}Q$ and hence
$p_{1}(\widetilde{LQ})=\operatorname{ev}^{*}p_{1}(Q).$ ∎
## Chapter 3 Higgs fields and characteristic classes for $\Omega G$-bundles
In our discussion of string structures in chapter 2 we were concerned mainly
with the loop group $LG$ and its central extension $\widehat{LG}.$ In this
chapter we shall, for the most part, be considering the subgroup of $LG$ given
by those loops which begin at the identity in $G,$ that is, the _based_ loop
group, which we shall denote $\Omega G.$ We will return to the discussion of
free loops in section 3.3.
### 3.1 String structures and the path fibration
In this section we will outline the result from [11] concerning string
structures for certain $\Omega G$-bundles.111Actually, in [11] Carey and
Murray work with the group of smooth maps from the interval $[0,2\pi]$ into
the group $G$ whose endpoints agree. We shall look more closely at this group
in section 3.3. Here we will be extending their results to the subgroup of
based smooth maps $S^{1}\to G.$ In particular, we shall see that if $Q\to M$
is a principal $G$-bundle, then the string class for the $\Omega G$-bundle
$\Omega Q\to\Omega M$ is a characteristic class for such bundles. To be
precise, what we mean here is that we have chosen a base point $m_{0}$ in $M$
and a base point $q_{0}$ in the fibre above $m_{0}$ and then $\Omega
Q\to\Omega M$ is an $\Omega G$-bundle. By ‘string class’ we mean the
obstruction to lifting $\Omega Q$ to an $\widehat{\Omega G}$-bundle, where
$\widehat{\Omega G}$ is the central extension of $\Omega G.$ (Actually, since
we are working with differential forms, we are really concerned with the image
in real cohomology of the string class – however, we make no distinction
between the terms here.) We will also generalise this to the case of a general
$\Omega G$-bundle, that is, one which is not necessarily a loop bundle.
#### 3.1.1 Classifying maps and characteristic classes
In the interests of being self-contained we shall begin by giving a short
overview of the theory of classifying maps and characteristic classes before
moving on to the specific case we are interested in. Recall that
${\mathcal{G}}$-bundles over $M$ are classified by (homotopy classes of) maps
to the classifying space $B{\mathcal{G}}.$ A ${\mathcal{G}}$-bundle is then
(isomorphic to) the pull-back by this map of the universal bundle
$E{\mathcal{G}}\to B{\mathcal{G}}.$ This bundle is characterised by the fact
that it is a principal ${\mathcal{G}}$-bundle and that $E{\mathcal{G}}$ is a
contractible space. If $P\to M$ is a ${\mathcal{G}}$-bundle, a map $f\colon
M\to B{\mathcal{G}}$ such that $P$ is isomorphic to the pull-back
$f^{*}E{\mathcal{G}}$ is called a classifying map for $P.$
A characteristic class associates to a ${\mathcal{G}}$-bundle $P\to M$ a class
$c(P)$ in $H^{*}(M).$ It must be natural with respect to pull-backs in the
sense that if $g\colon N\to M$ is a smooth map then $c$ must associate to the
pull-back bundle $g^{*}P\to N$ the class given by the pull-back of $c(P).$
That is,
$c(g^{*}P)=g^{*}c(P).$
Note that since all ${\mathcal{G}}$-bundles are pulled-back from the universal
bundle, then if $P\to M$ is a ${\mathcal{G}}$-bundle with classifying map $f,$
all its characteristic classes are of the form $f^{*}c(E{\mathcal{G}})$ for
some characteristic class $c.$ That is, the set of characteristic classes for
${\mathcal{G}}$-bundles is in bijective correspondence with the cohomology
group $H^{*}(B{\mathcal{G}}).$
#### 3.1.2 String structures and the path fibration
In general, both the classifying space and the universal bundle for a group
can be difficult to describe. For the based loop group $\Omega G,$ however, we
have the following construction [6]: Let $PG$ be the space of paths in $G,$
$p\colon{\mathbb{R}}\to G$ such that $p(0)$ is the identity and
$p^{-1}\partial p$ is periodic. Then this is acted on by $\Omega G$ and
$\textstyle{\Omega
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{PG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G}$
is an $\Omega G$-bundle called the _path fibration_ , where the projection
$\pi$ sends a path $p$ to its value at $2\pi.$ $PG$ is contractible and so the
path fibration is a model for the universal $\Omega G$-bundle and we have
$B\Omega G=G.$ (See Appendix A for details.)
Since we are assuming that $G$ is compact, simple and simply connected, we
know that $H^{3}(G,{\mathbb{Z}})={\mathbb{Z}}$ and there is an expression for
the generator of this group. Namely, the 3-form on $G$ given by
$\omega=\frac{1}{48\pi^{2}}\langle\Theta,[\Theta,\Theta]\rangle.$
In [11] Carey and Murray show the string class of the path fibration (for the
case of loops which are smooth on $(0,2\pi)$) is given by the 3-form $\omega$
by giving an explicit construction of the lift of $PG$ which exists precisely
when this class vanishes. We will use Theorem 2.4.1 to calculate the string
class of the path fibration. Firstly we need a connection on $PG.$ This is
given in [9]: Let $\alpha$ be a smooth real-valued function on $[0,2\pi]$ such
that $\alpha(0)=0,\alpha(2\pi)=1$ and all the derivatives of $\alpha$ vanish
at the endpoints. Then $\alpha$ can be extended to a function on
${\mathbb{R}}$ and a connection in $PG$ is given by
$A=\Theta-\alpha\,ad(p^{-1})\pi^{*}\widehat{\Theta},$
where $\widehat{\Theta}$ is the _right_ invariant Maurer-Cartan form. The
horizontal projection of a tangent vector $X$ using this connection is
$hX=\alpha\,X(2\pi)p(2\pi)^{-1}p.$
We can calculate the curvature of $A$ using the covariant derivative $F=DA.$
For tangent vectors $X$ and $Y,$ we have
$\displaystyle F(X,Y)$ $\displaystyle=\frac{1}{2}A([hX,hY])$
$\displaystyle=\frac{1}{2}A\left(\alpha^{2}\left[X(2\pi)p(2\pi)^{-1},Y(2\pi)p(2\pi)^{-1}\right]p\right)$
$\displaystyle=\frac{1}{2}\left(\Theta-\alpha\,ad(p^{-1})\pi^{*}\widehat{\Theta}\right)\left(\alpha^{2}\left[X(2\pi)p(2\pi)^{-1},Y(2\pi)p(2\pi)^{-1}\right]p\right)$
$\displaystyle=\frac{1}{2}\left(\alpha^{2}-\alpha\right)ad(p^{-1})\left[X(2\pi)p(2\pi)^{-1},Y(2\pi)p(2\pi)^{-1}\right].$
So
$F=\frac{1}{2}\left(\alpha^{2}-\alpha\right)ad(p^{-1})[\pi^{*}\widehat{\Theta},\pi^{*}\widehat{\Theta}].$
In order to use Theorem 2.4.1 we also need a Higgs field for $PG.$ Define the
map $\Phi\colon PG\to L{\mathfrak{g}}$ by
$\Phi(p)=p^{-1}\partial p.$
Then $\Phi$ is a Higgs field, since for $\gamma\in\Omega G$ we have
$\displaystyle\Phi(p\gamma)$ $\displaystyle=(p\gamma)^{-1}\partial(p\gamma)$
$\displaystyle=ad(\gamma^{-1})p^{-1}\partial p+\gamma^{-1}\partial\gamma.$
The formula for the string class uses $\nabla\Phi=d\Phi+[A,\Phi]-\partial A.$
We can calculate
$d\Phi=\partial\Theta+[\Phi,\Theta],$
$[A,\Phi]=[\Theta,\Phi]-\alpha\,[ad(p^{-1})\pi^{*}\widehat{\Theta},\Phi]$
and
$\partial
A=\partial\Theta-\partial\alpha\,ad(p^{-1})\pi^{*}\widehat{\Theta}-\alpha\,[ad(p^{-1})\pi^{*}\widehat{\Theta},\Phi].$
So we have
$\nabla\Phi=\partial\alpha\,ad(p^{-1})\pi^{*}\widehat{\Theta}.$
Therefore, by Theorem 2.4.1 we have
$\displaystyle s(PG)$
$\displaystyle=-\frac{1}{8\pi^{2}}\int_{S^{1}}\left\langle\left(\alpha^{2}-\alpha\right)ad(p^{-1})[\pi^{*}\widehat{\Theta},\pi^{*}\widehat{\Theta}],\partial\alpha\,ad(p^{-1})\pi^{*}\widehat{\Theta}\right\rangle\,d\theta$
$\displaystyle=-\frac{1}{8\pi^{2}}\langle[\widehat{\Theta},\widehat{\Theta}],\widehat{\Theta}\rangle\int_{S^{1}}\left(\alpha^{2}-\alpha\right)\partial\alpha\,d\theta$
$\displaystyle=\frac{1}{48\pi^{2}}\langle\Theta,[\Theta,\Theta]\rangle,$
where the last line follows from the $ad$-invariance of the Killing form. Thus
we see that the string class of the path fibration is the generator of the
degree three cohomology of $G.$
Now, consider again a based loop bundle $\Omega Q\xrightarrow{\,\Omega
G\,}\Omega M.$ In [11] Carey and Murray write down the classifying map for
such bundles and then show, by explicitly calculating the integral of the
(pull-back by the evaluation map of the) first Pontrjagyn class of $Q,$ that
the string class is the pull-back by this map of the 3-form $\omega.$ To write
down the classifying map of the bundle $\Omega Q\to\Omega M$ choose a
connection for it. Then take a loop $\gamma\in\Omega Q$ and project it down to
$\pi\circ\gamma\in\Omega M.$ Lift this back up horizontally to
$\gamma_{h}\in\Omega Q,$ so that $\pi\circ\gamma=\pi\circ\gamma_{h}.$ Then the
_holonomy_ , $\operatorname{hol}(\gamma)\in PG$ is given by
$\gamma=\gamma_{h}\operatorname{hol}(\gamma).$ This covers the usual
holonomy222Note that we can define the holonomy since we have chosen
basepoints in $M$ and $Q.$ $\operatorname{hol}\colon\Omega M\to G,$ so we
have:
$\textstyle{\Omega
Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{hol}}$$\textstyle{PG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{hol}}$$\textstyle{G}$
Thus $\operatorname{hol}$ is a classifying map for the bundle $\Omega
Q\to\Omega M.$ Now, using Corollary 2.5.4 and by calculating explicitly
$\int_{S^{1}}\operatorname{ev}^{*}p_{1}(Q),$ we can show that
$s(\Omega Q)=\operatorname{hol}^{*}\omega+\text{\emph{exact}}.$
We shall show this in more detail in the next section when we generalise this
result to the case of higher classes for general $\Omega G$-bundles (that is,
an $\Omega G$-bundle which is not necessarily a loop bundle). For now let us
assume this result and show how it leads us to a more general statement.
To generalise the result above to a general $\Omega G$-bundle
$P\xrightarrow{\,\Omega G\,}M,$ we need a classifying map for such bundles.
Consider the $\Omega G$-bundle $P\to M.$ Choose a Higgs field $\Phi\colon P\to
L{\mathfrak{g}}$ for $P.$ It is possible to solve the equation
$\Phi(p)=g^{-1}\partial g$ for $g\in PG.$ We define the _Higgs field holonomy_
, $\operatorname{hol}_{\Phi},$ to be the solution to this equation satisfying
the initial condition $g(0)=1$. Note that if $\operatorname{hol}_{\Phi}(p)=g$
then since
$\Phi(ph)=ad(h^{-1})\Phi(p)+h^{-1}\partial h$
and
$(gh)^{-1}\partial(gh)=ad(h^{-1})g^{-1}\partial g+h^{-1}\partial h,$
we see that $\operatorname{hol}_{\Phi}(p\cdot
h)=\operatorname{hol}_{\Phi}(p)h$ and hence $\operatorname{hol}_{\Phi}$
descends to a map (also called $\operatorname{hol}_{\Phi})\,M\to G$ and is a
classifying map for $P\to M.$
A natural question arises at this point: If $Q\to M$ is a $G$-bundle with
connection $A$ then we can define the holonomy of a loop $\gamma\in\Omega Q.$
However, since the loop bundle $\Omega Q\to\Omega M$ is an $\Omega G$-bundle,
we can also choose a Higgs field for it and define the Higgs field holonomy of
a loop $\gamma$ in this bundle. Can we find the Higgs field $\Phi$ such that
$\operatorname{hol}_{\Phi}=\operatorname{hol}$? Define $\Phi$ in terms of $A$
as in section 2.5, that is,
$\Phi(\gamma)=A(\partial\gamma).$
Then using $\gamma=\gamma_{h}\operatorname{hol}(\gamma),$ we find
$\partial\gamma=\partial\gamma_{h}\cdot\operatorname{hol}(\gamma)+\iota_{\gamma_{h}}(\operatorname{hol}(\gamma)^{-1}\partial\operatorname{hol}(\gamma)).$
Since $\gamma_{h}$ is horizontal (in the sense that all its tangent vectors
are horizontal), applying the connection form $A$ gives
$A(\partial\gamma)=\operatorname{hol}(\gamma)^{-1}\partial\operatorname{hol}(\gamma).$
Therefore, $\operatorname{hol}_{\Phi}=\operatorname{hol}.$
We can extend the result from [11] by finding a relationship between
$\operatorname{hol}_{\Phi}$ and $\operatorname{hol}$ in general:
We can modify the correspondence in section 2.5, which relates $LG$-bundles
over $M$ and $G$-bundles over $M\times S^{1},$ to one which applies to $\Omega
G$-bundles. We say a $G$-bundle over $M\times S^{1}$ is _framed_ over
$M\times\\{0\\}$ if it is trivial over $M\times\\{0\\}$. A particular
trivialisation is called a _framing_. Given this, then, $\Omega G$-bundles
correspond to $G$-bundles over $M\times S^{1}$ which are framed over
$M\times\\{0\\}.$ This means we take a $G$-bundle $\widetilde{P}\to M\times
S^{1}$ and a section (i.e. a framing) $s\colon
M\times\\{0\\}\to\widetilde{P}$ and the fibre of $P$ over $m$ has a base point
given by $s(m,0).$ Using this correspondence, define a bundle map
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\Omega\widetilde{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\Omega(M\times
S^{1})}$
by $\eta(m)=\theta\mapsto(m,\theta),$ or, on the total space,
$\eta(p)=\theta\mapsto[p,1,\theta].$ Then we have:
###### Lemma 3.1.1.
Let $P\to M$ be an $\Omega G$-bundle with connection and Higgs field $\Phi,$
$\widetilde{P}\to M\times S^{1}$ its corresponding $G$-bundle and $\eta$ as
above. Then
$\displaystyle\operatorname{hol}_{\Phi}=\operatorname{hol}\circ\eta.$
###### Proof.
If $\tilde{A}$ is the connection form on $\widetilde{P}$ then
$\tilde{\Phi}\colon\Omega\widetilde{P}\to L{\mathfrak{g}}$ defined by
$\tilde{\Phi}(\gamma)=\tilde{A}(\partial\gamma)$
gives us that
$\operatorname{hol}_{\tilde{\Phi}}=\operatorname{hol}$
as above. Therefore we need only show that
$\operatorname{hol}_{\Phi}=\operatorname{hol}_{\tilde{\Phi}}\circ\eta.$
Let $p\in P.$ Consider the unique horizontal path $\eta(p)_{h}$ such that
$\tilde{\pi}(\eta(p))=\tilde{\pi}(\eta(p)_{h})$
given by projecting $\eta(p)$ to $\Omega(M\times S^{1})$ and lifting
horizontally back to $\Omega\widetilde{P}.$ The tangent vector to the loop
$\eta(p)$ at the point $\theta$ is given by the derivative
$\partial\eta(p)_{\theta}$ and since $\eta(p)_{h}$ is horizontal we have that
$\tilde{A}(\eta(p)_{h,\theta})=0.$
Now, $\eta(p)_{\theta}=[p,1,\theta]$, so we can explicitly calculate
$\partial\eta(p)_{\theta}:$
$\frac{\partial}{\partial\theta}\eta(p)_{\theta}=[0,0,1].$
Recall that the connection $\tilde{A}$ is given in terms of the connection $A$
and Higgs field $\Phi$ for $P$ as
$\tilde{A}=ad(g^{-1})A+\Theta+ad(g^{-1})\Phi\,d\theta.$
Therefore, we have $\tilde{A}(\partial\eta(p))=\Phi(p).$ Or, in terms of the
Higgs field for $\Omega\widetilde{P},$
$\Phi=\tilde{\Phi}\circ\eta.$
As above, we have
$\tilde{\Phi}(\eta(p))=\operatorname{hol}(\eta(p))^{-1}\partial\operatorname{hol}(\eta(p)),$
and therefore
$\operatorname{hol}_{\Phi}=\operatorname{hol}_{\tilde{\Phi}}\circ\eta.$
∎
We see that $\operatorname{hol}_{\Phi}$ factors through $\operatorname{hol}.$
In order to use this we need the following result:
###### Lemma 3.1.2.
In the situation of Lemma 3.1.1, for degree 4 differential forms on $M\times
S^{1}$ we have
$\displaystyle\eta^{*}\int_{S^{1}}\operatorname{ev}^{*}=\int_{S^{1}}.$
###### Proof.
Note first that we have
$\displaystyle M$ $\displaystyle\times S^{1}\,$
$\displaystyle\xrightarrow{\,\eta\times 1\,}\,\,$ $\displaystyle\Omega(M\times
S^{1})\times S^{1}\,$ $\displaystyle\xrightarrow{\,\operatorname{ev}\,}\,\,$
$\displaystyle M\times S^{1}$ $\displaystyle(m$ $\displaystyle,\phi)$
$\displaystyle\longmapsto$ $\displaystyle(\theta\mapsto(m,\theta),\phi)$
$\displaystyle\longmapsto$ $\displaystyle(m,\phi)$
so, $\operatorname{ev}\circ(\eta\times 1)$ is the identity. Therefore, we have
$\int_{S^{1}}=\int_{S^{1}}(\eta\times 1)^{*}\operatorname{ev}^{*},$
so it suffices to show that
$\int_{S^{1}}(\eta\times 1)^{*}=\eta^{*}\int_{S^{1}}.$
That is, that the following diagram commutes
$\textstyle{\Omega^{4}(\Omega(M\times S^{1})\times
S^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\eta\times
1)^{*}}$$\scriptstyle{\int_{S^{1}}}$$\textstyle{\Omega^{4}(M\times
S^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\int_{S^{1}}}$$\textstyle{\Omega^{3}(\Omega(M\times
S^{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{*}}$$\textstyle{\Omega^{3}(M)}$
Consider $\omega\in\Omega^{4}(\Omega(M\times S^{1})\times S^{1}).$ Then if
$X_{1},X_{2}$ and $X_{3}$ are tangent vectors to $M$ we have
$\displaystyle\left(\int_{S^{1}}(\eta\times
1)^{*}\omega\right)(X_{1},X_{2},X_{3})$ $\displaystyle=\int_{S^{1}}(\eta\times
1)^{*}\omega(\widehat{X}_{1},\widehat{X}_{2},\widehat{X}_{3})$
$\displaystyle=\int_{S^{1}}\omega((\eta\times
1)_{*}\widehat{X}_{1},(\eta\times 1)_{*}\widehat{X}_{2},(\eta\times
1)_{*}\widehat{X}_{3}),$
where $\widehat{X}_{i}$ ($i=1,2,3$) is a lift of $X_{i}$ to $M\times S^{1}$.
On the other hand, if $\widehat{\eta_{*}X}_{i}$ is a lift of $\eta_{*}X_{i}$
to $\Omega(M\times S^{1})\times S^{1}$, then
$\displaystyle\eta^{*}\left(\int_{S^{1}}\omega\right)(X_{1},X_{2},X_{3})$
$\displaystyle=\left(\int_{S^{1}}\omega\right)(\eta_{*}X_{1},\eta_{*}X_{2},\eta_{*}X_{3})$
$\displaystyle=\int_{S^{1}}\omega(\widehat{\eta_{*}X}_{1},\widehat{\eta_{*}X}_{2},\widehat{\eta_{*}X}_{3}).$
Since the expressions above are independent of the lift chosen, we can use the
natural splitting of the tangent bundles to $M\times S^{1}$ and
$\Omega(M\times S^{1})\times S^{1}$ to define $\widehat{X}_{i}=(X_{i},0)$ and
$\widehat{\eta_{*}X}_{i}=(\eta_{*}X_{i},0)$ and so we have
$\displaystyle\left(\int_{S^{1}}(\eta\times
1)^{*}\omega\right)(X_{1},X_{2},X_{3})$
$\displaystyle=\int_{S^{1}}\omega((\eta_{*}\widehat{X}_{1},0),(\eta_{*}\widehat{X}_{2},0),(\eta_{*}\widehat{X}_{3},0))$
$\displaystyle=\eta^{*}\left(\int_{S^{1}}\omega\right)(X_{1},X_{2},X_{3}).$
∎
Combining Lemmas 3.1.1 and 3.1.2, we have:
###### Theorem 3.1.3.
The string class of an $\Omega G$-bundle $P\to M$ is the characteristic class
corresponding to $\omega\in H^{3}(G)$.
###### Proof.
On the level of cohomology we have
$\displaystyle s(P)$ $\displaystyle=\int_{S^{1}}p_{1}(\widetilde{P})$
$\displaystyle=\eta^{*}\int_{S^{1}}\operatorname{ev}^{*}p_{1}(\widetilde{P})$
$\displaystyle=\eta^{*}s(\Omega\widetilde{P})$
$\displaystyle=\eta^{*}\operatorname{hol}^{*}\omega$
$\displaystyle=\operatorname{hol}_{\Phi}^{*}\omega.$
∎
### 3.2 Higher string classes for $\Omega G$-bundles
We have seen in the last section that the string class is a characteristic
class for $\Omega G$-bundles and we know from section 2.5 (Theorem 2.5.3) that
it is naturally associated to the first Pontrjagyn class of the corresponding
$G$-bundle. Indeed, the fact that the string class is given by integrating the
first Pontrjagyn class was used to show that it is natural. In this section we
will generalise these ideas to higher degree classes for $\Omega G$-bundles.
These classes will be naturally associated to a characteristic class for
$G$-bundles in the same way the string class is related to the Pontrjagyn
class.
We can summarise the results from the previous section with the following
diagram
$\textstyle{H^{4}(BG)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C\text{-}W_{\widetilde{P}}}$$\scriptstyle{\tau}$$\textstyle{H^{4}(M\times
S^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\int_{S^{1}}}$$\textstyle{H^{3}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{hol}_{\Phi}^{*}}$$\textstyle{H^{3}(M)}$
The top arrow here is the usual Chern-Weil map (see below). The map $\tau$ is
the _transgression_ (see for example [13] or [21]) which we shall describe
presently. As long as $G$ is compact and connected, $H^{2k}(BG)$ is isomorphic
to the set of multilinear, symmetric, $ad$-invariant functions on
${\mathfrak{g}}\times\ldots\times{\mathfrak{g}}$ ($k$ times). Let $f$ be such
a function and let $Q\to M$ be a $G$-bundle with connection. Then the Chern-
Weil map, $C\text{-}W_{Q},$ takes $f$ to the class on $M$ given by
$f(F,\ldots,F),$ where $F$ is the curvature of the connection on $Q.$ This is
well-defined and independent of choice of connection. (For details we refer
the reader to [24].) In this case the transgression map $\tau$ is given by
$\tau(f)=\left(-\frac{1}{2}\right)^{k-1}\frac{k!(k-1)!}{(2k-1)!}\,f(\Theta,[\Theta,\Theta],\ldots,[\Theta,\Theta]),$
where, as usual, $\Theta$ is the Maurer-Cartan form on $G.$ In terms of the
result above, we have seen that in the case where the polynomial $f$ is given
by $f(X,Y)=-\frac{1}{8\pi^{2}}\langle X,Y\rangle$ and the $G$-bundle is
$\widetilde{P}\to M\times S^{1},$ then the Chern-Weil map gives the Pontrjagyn
class of $\widetilde{P}$ and the diagram commutes. Furthermore, the element
that fits in the bottom right hand corner is the string class of the
corresponding $\Omega G$-bundle $P\to M.$ That is,
$\displaystyle\int_{S^{1}}p_{1}(\widetilde{P})$ $\displaystyle=s(P)$
$\displaystyle=-\frac{1}{4\pi^{2}}\int_{S^{1}}\langle F,\nabla\Phi\rangle
d\theta$
$\displaystyle=\frac{1}{48\pi^{2}}\operatorname{hol}_{\Phi}^{*}\langle\Theta,[\Theta,\Theta]\rangle.$
It is natural to ask now whether there is a similar theory for general and
higher degree characteristic classes. That is, whether we can set up the
following diagram
$\textstyle{H^{2k}(BG)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C\text{-}W_{\widetilde{P}}}$$\scriptstyle{\tau}$$\textstyle{H^{2k}(M\times
S^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\int_{S^{1}}}$$\textstyle{H^{2k-1}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{hol}_{\Phi}}$$\textstyle{H^{2k-1}(M)}$
and give a formula for the element that ends up in the bottom right-hand
corner given a general polynomial in the top left.
As above, the usual Chern-Weil theory tells us that if we start with an
invariant polynomial $f\in H^{2k}(BG)$ then the element in $H^{2k}(M\times
S^{1})$ that we end up with is $f(\tilde{F},\ldots,\tilde{F})$ where
$\tilde{F}$ is the curvature of the $G$-bundle $\widetilde{P}$ on $M\times
S^{1}.$ Note that if we write out $f(\tilde{F},\ldots,\tilde{F})$ in terms of
the curvature and Higgs field on the corresponding $\Omega G$-bundle $P\to M,$
we get
$\displaystyle f(\tilde{F},\ldots,\tilde{F})$
$\displaystyle=f(F+\nabla\Phi\,d\theta,\ldots,F+\nabla\Phi\,d\theta)$
$\displaystyle=f(F,\ldots,F)+kf(\nabla\Phi\,d\theta,F,\ldots,F)$
since $f$ is multilinear and symmetric and all terms with more than one
$d\theta$ will vanish. From now on we will adopt the convention that whenever
$f$ has repeated entries they will be ordered at the end and we will write
them only once. That is, whatever appears as the last entry in $f$ is repeated
however many times required to fill the remaining slots. (For example,
$f(F)=f(F,\ldots,F)$ and
$f(\nabla\Phi,F)d\theta=f(\nabla\Phi,F,\ldots,F)d\theta.$) So integrating this
over the circle gives
$\int_{S^{1}}f(\tilde{F})=k\int_{S^{1}}f(\nabla\Phi,F)\,d\theta.$
So $k\int_{S^{1}}f(\nabla\Phi,F)d\theta$ is our candidate for the element in
$H^{2k-1}(M)$ which corresponds to $f\in H^{2k}(BG)$ and makes the diagram
commute.
Note that if we evaluate this expression for the path fibration we have
$\displaystyle k\int_{S^{1}}f(\nabla\Phi,F)\,d\theta$
$\displaystyle=f(\Theta,[\Theta,\Theta])\left(\frac{1}{2}\right)^{k-1}k\int_{S^{1}}\left(\alpha^{2}-\alpha\right)^{k-1}\partial\alpha\,d\theta$
$\displaystyle=f(\Theta,[\Theta,\Theta])\left(\frac{1}{2}\right)^{k-1}k\int_{S^{1}}\sum_{i=0}^{k-1}\binom{k-1}{i}(-1)^{k-1-i}\alpha^{2i}\alpha^{k-1-i}\partial\alpha\,d\theta$
$\displaystyle=f(\Theta,[\Theta,\Theta])\left(-\frac{1}{2}\right)^{k-1}k\sum_{i=0}^{k-1}\binom{k-1}{i}(-1)^{i}\frac{1}{k+i}.$
It turns out [44] that the coefficient above is equal to the coefficient in
the definition of the transgression map $\tau$. That is,
$k\sum_{i=0}^{k-1}\binom{k-1}{i}\frac{(-1)^{i}}{k+i}=\frac{k!(k-1)!}{(2k-1)!}.$
Therefore, we have for the path fibration
$k\int_{S^{1}}f(\nabla\Phi,F)\,d\theta=\tau(f).$
So what we are really asking for is a theory which associates to any
characteristic class for $G$-bundles (that is, any polynomial in $H^{2k}(BG)$)
a characteristic class for an $\Omega G$-bundle over $M.$ That is a map
$H^{2k}(BG)\to H^{2k-1}(M)$ which gives characteristic classes for $\Omega
G$-bundles over $M.$ Thus we need to show firstly that
$k\int_{S^{1}}f(\nabla\Phi,F)d\theta$ is closed and independent of choice of
connection and Higgs field. Also, we need to show that it is cohomologous to
the pull-back by the classifying map $\operatorname{hol}_{\Phi}$ of the
$(2k-1)$-form $\tau(f)$ defined above. We shall call
$k\int_{S^{1}}f(\nabla\Phi,F)d\theta$ the _string $(2k-1)$-class associated to
$f$_ and write $s_{2k-1}^{P}(f).$ To be more precise
###### Definition 3.2.1.
Let $\widetilde{P}$ be a framed $G$-bundle over $M\times S^{1}$ and $P$ its
corresponding $\Omega G$-bundle over $M.$ Suppose that $f\in H^{2k}(BG)$ is an
invariant polynomial representing the characteristic class $f(\tilde{F})\in
H^{2k}(M\times S^{1}).$ Then its associated _string $(2k-1)$-class_ is the
class in $H^{2k-1}(M)$ given by
$s_{2k-1}^{P}(f)=k\int_{S^{1}}f(\nabla\Phi,F)\,d\theta,$
where $\Phi$ is a Higgs field for $P$ and $F$ is the curvature of a connection
on $P.$
Note that we still have to show that $s_{2k-1}^{P}(f)$ is closed and well-
defined. We have
###### Proposition 3.2.2.
The string $(2k-1)$-class is closed.
###### Proof.
To show that $s_{2k-1}^{P}(f)$ is closed we use the following result which
follows from Lemmas 1 and 2 on pages 294–295 of [24]:
###### Lemma 3.2.3.
Let $\psi$ be an $ad$-invariant, vertical form on the total space of a
principal bundle. Then $\psi$ projects to a form on the base space. For such a
form, the exterior derivative is equal to the covariant exterior derivative.
That is, $d\psi=D\psi.$
Thus we only need to show that $Ds_{2k-1}^{P}(f)=0.$ Now,
$\displaystyle Dk\int_{S^{1}}f(\nabla\Phi,F)\,d\theta$
$\displaystyle=k\int_{S^{1}}f(D(\nabla\Phi),F)\,d\theta+k(k-1)\int_{S^{1}}f(\nabla\Phi,DF,F)\,d\theta$
$\displaystyle=k\int_{S^{1}}f(D(\nabla\Phi),F)\,d\theta$
using the Bianchi identity. We can calculate $D(\nabla\Phi).$ For tangent
vectors $X$ and $Y,$
$\displaystyle D(\nabla\Phi)(X,Y)$ $\displaystyle=d(\nabla\Phi)(hX,hY)$
$\displaystyle=(d^{2}\Phi+[dA,\Phi]-[A,d\Phi]-\partial(dA))(hX,hY)$
where $(hX,hY)$ is the projection of $(X,Y)$ onto the horizontal subspace at
that point. Using the fact that $dA(hX,hY)=F(X,Y)$ and $A(hX)=A(hY)=0,$ we
have:
$D(\nabla\Phi)(X,Y)=[F(X,Y),\Phi]-\partial F(X,Y)$
That is,
$D(\nabla\Phi)=[F,\Phi]-\partial F.$
So we have,
$\displaystyle Dk\int_{S^{1}}f(\nabla\Phi,F)\,d\theta$
$\displaystyle=k\int_{S^{1}}f([F,\Phi],F)\,d\theta-k\int_{S^{1}}f(\partial
F,F)\,d\theta$
and $ad$-invariance of $f$ (which we will discuss in more detail later)
implies the first term on the right hand side vanishes while integration by
parts implies the second term vanishes. Therefore, $s_{2k-1}^{P}(f)$ is
closed.
∎
We also have
###### Proposition 3.2.4.
The string $(2k-1)$-class is independent of choice of connection and Higgs
field.
###### Proof.
In order to see that $s_{2k-1}^{P}(f)$ is independent of choice of connection
and Higgs field consider 2 different connection forms, $A_{0}$ and $A_{1},$ on
$P$ and 2 different Higgs fields, $\Phi_{0}$ and $\Phi_{1}.$ Since the space
of connections is an affine space and the same is true for Higgs fields, we
can consider lines joining the 2 connections and Higgs fields respectively.
Define:
$\alpha:=A_{1}-A_{0},\qquad\qquad\varphi:=\Phi_{1}-\Phi_{0}$
and
$A_{t}:=A_{0}+t\alpha,\qquad\qquad\Phi_{t}:=\Phi_{0}+t\varphi$
for $t\in[0,1].$ Now consider the corresponding connection form on
$\widetilde{P}$
$\tilde{A}_{t}=\tilde{A}_{0}+t(\tilde{A}_{1}-\tilde{A}_{0})\\\
\phantom{\tilde{A}_{t}}=ad(g^{-1})A_{0}+\Theta+ad(g^{-1})\Phi_{0}d\theta+t(ad(g^{-1})A_{1}+ad(g^{-1})\Phi_{1}d\theta\\\
-ad(g^{-1})A_{0}-ad(g^{-1})\Phi_{0}d\theta)\\\
\phantom{\tilde{A}_{t}}=ad(g^{-1})A_{0}+\Theta+ad(g^{-1})\Phi_{0}d\theta+t\tilde{\alpha}\\\
$
where
$\tilde{\alpha}=ad(g^{-1})\alpha+ad(g^{-1})\varphi d\theta.$
Note that
$\tilde{A}_{t}=ad(g^{-1})A_{t}+\Theta+ad(g^{-1})\Phi_{t}d\theta.$
Recall that $f(\tilde{F})=f(F)+kf(\nabla\Phi,F)d\theta.$ We shall show
$f(\tilde{F}_{0})$ and $f(\tilde{F}_{1})$ differ by an exact form, (where
$\tilde{F}_{0}$ and $\tilde{F}_{1}$ are the curvature forms of $\tilde{A}_{0}$
and $\tilde{A}_{1}$ respectively) so that the class defined by
$k\int_{S^{1}}f(\nabla\Phi_{1},F_{1})d\theta=\int_{S^{1}}f(\tilde{F})$ is
independent of $A$ and $\Phi.$ For this we will need the following lemma:
###### Lemma 3.2.5.
$\displaystyle D_{t}\tilde{\alpha}=\frac{d}{dt}\tilde{F}_{t}.$
###### Proof.
Firstly, we calculate $\tilde{F}_{t}$:
$\displaystyle\tilde{F}_{t}$
$\displaystyle=d\tilde{A}_{t}+\tfrac{1}{2}[\tilde{A}_{t},\tilde{A}_{t}]$
$\displaystyle=ad(g^{-1})\left(F_{t}+\nabla\Phi_{t}\wedge d\theta\right)$
$\displaystyle=ad(g^{-1})\left(dA_{t}+\tfrac{1}{2}[A_{t},A_{t}]+(d\Phi_{t}+[A_{t},\Phi_{t}]-\partial
A_{t})\wedge d\theta\right)$
$\displaystyle=ad(g^{-1})\left(dA_{0}+td\alpha+\tfrac{1}{2}[A_{t},A_{t}]+(d\Phi_{0}+td\varphi+[A_{t},\Phi_{t}]-\partial
A_{0}-t\partial\alpha)\wedge d\theta\right).$
Therefore $\frac{d}{dt}\tilde{F}_{t}$ is given by
$\displaystyle\frac{d}{dt}\tilde{F}_{t}$
$\displaystyle=ad(g^{-1})\left(d\alpha+\frac{1}{2}\frac{d}{dt}[A_{t},A_{t}]+(d\varphi+\frac{d}{dt}[A_{t},\Phi_{t}]-\partial\alpha)\wedge
d\theta\right)$
$\displaystyle=ad(g^{-1})\left(d\alpha+\frac{1}{2}[\alpha,A_{t}]+\frac{1}{2}[A_{t},\alpha]+(d\varphi+[\alpha,\Phi_{t}]+[A_{t},\varphi]-\partial\alpha)\wedge
d\theta\right)$
$\displaystyle=ad(g^{-1})\left(d\alpha+[\alpha,A_{t}]+(d\varphi+[\alpha,\Phi_{t}]+[A_{t},\varphi]-\partial\alpha)\wedge
d\theta\right),$
since $\displaystyle\frac{d}{dt}A_{t}=\alpha$ and
$\displaystyle\frac{d}{dt}\Phi_{t}=\varphi.$ Next we calculate
$D_{t}\tilde{\alpha}$ by calculating $d\tilde{\alpha}$ and evaluating it on
horizontal (with respect to $\tilde{A}_{t}$) vectors. At a point
$(p,g,\theta)$ in $\widetilde{P}$ and for vectors $(X,g\xi,x_{\theta})$ and
$(Y,g\zeta,y_{\theta})$ at $(p,g,\theta)$ we have:
$d\tilde{\alpha}_{(p,g,\theta)}(X,g\xi,x_{\theta},Y,g\zeta,y_{\theta})\\\
=\tfrac{1}{2}\left\\{(X,g\xi,x_{\theta})(\tilde{\alpha}_{(p,g,\theta)}(Y,g\zeta,y_{\theta}))-(Y,g\zeta,y)(\tilde{\alpha}_{(p,g,\theta)}(X,g\xi,x_{\theta}))\right.\\\
\left.-\tilde{\alpha}_{(p,g,\theta)}([(X,g\xi,x_{\theta}),(Y,g\zeta,y_{\theta})])\right\\}.$
So we need to calculate
1. 1.
$(X,g\xi,x_{\theta})(\tilde{\alpha}_{(p,g,\theta)}(Y,g\zeta,y_{\theta}))$, and
2. 2.
$\tilde{\alpha}_{(p,g,\theta)}([(X,g\xi,x_{\theta}),(Y,g\zeta,y_{\theta})]).$
If $\gamma_{X}(t)$ is a curve whose tangent vector is $X,$ we have:
$(X,g\xi,x_{\theta})(\tilde{\alpha}_{(p,g,\theta)}(Y,g\zeta,y_{\theta}))\\\
=\frac{d}{dt}\bigg{|}_{0}\left\\{(1-t\xi)g^{-1}\alpha_{\gamma_{X}(t)}(Y)_{(\theta+tx)}g(1+t\xi)+(1-t\xi)g^{-1}\varphi_{\gamma_{X}(t),(\theta+tx)}g(1+t\xi)y\right\\}\\\
=\frac{d}{dt}\bigg{|}_{0}\left\\{-t\xi
g^{-1}\alpha_{\gamma_{X}(t)}(Y)_{(\theta+tx)}g+g^{-1}\alpha_{\gamma_{p}(t)}(Y)_{(\theta+tx)}gt\xi+g^{-1}\alpha_{\gamma_{X}(t)}(Y)_{\theta}g\right.\\\
+g^{-1}\partial\alpha_{\gamma_{X}(0)}(Y)_{\theta}xtg+-t\xi
g^{-1}\varphi_{\gamma_{X}(t),(\theta+tx)}gy+g^{-1}\varphi_{\gamma_{X}(t),(\theta+tx)}gt\xi
y\\\
\left.+g^{-1}\varphi_{\gamma_{X}(t),\theta}gy+g^{-1}\partial\varphi_{\gamma_{X}(0),\theta}gtxy\right\\}\\\
=-\xi
g^{-1}\alpha_{p}(Y)_{\theta}g+g^{-1}\alpha_{p}(Y)_{\theta}g\xi+g^{-1}\frac{d}{dt}\bigg{|}_{0}\alpha_{\gamma_{X}(t)}(Y)_{\theta}g+g^{-1}\partial\alpha_{p}(Y)_{\theta}gx\\\
-\xi g^{-1}\varphi_{p,\theta}gy+g^{-1}\varphi_{p,\theta}g\xi
y+g^{-1}\frac{d}{dt}\bigg{|}_{0}\varphi_{\gamma_{X}(t),\theta}gy+g^{-1}\partial\varphi_{p,\theta}gxy.$
Also,
$\displaystyle\tilde{\alpha}_{(p,g,\theta)}([(X,g\xi,x_{\theta}),$
$\displaystyle(Y,g\zeta,y_{\theta})]$
$\displaystyle=ad(g^{-1})\alpha_{p}([X,Y])+ad(g^{-1})\varphi_{p}d\theta([x,y])$
$\displaystyle=ad(g^{-1})\alpha_{p}([X,Y])$
Therefore,
$d\tilde{\alpha}_{(p,g,\theta)}(X,g\xi,x_{\theta},Y,g\zeta,y_{\theta})\\\
=\frac{1}{2}\left\\{[ad(g^{-1})\alpha_{p}(Y),\xi]+ad(g^{-1})\left(\frac{d}{dt}\bigg{|}_{0}\alpha_{\gamma_{X}(t)}(Y)_{\theta}\right)+ad(g^{-1})\partial\alpha_{p}(Y)x\right.\\\
+[ad(g^{-1})\varphi_{p,\theta},\xi]y+ad(g^{-1})\left(\frac{d}{dt}\bigg{|}_{0}\varphi_{\gamma_{p}(t),\theta}\right)y\\\
-[ad(g^{-1})\alpha_{p}(X),\zeta]-ad(g^{-1})\left(\frac{d}{dt}\bigg{|}_{0}\alpha_{\gamma_{X}(t)}(X)_{\theta}\right)-ad(g^{-1})\partial\alpha_{p}(X)y\\\
-[ad(g^{-1})\varphi_{p,\theta},\zeta]x-ad(g^{-1})\left(\frac{d}{dt}\bigg{|}_{0}\varphi_{\gamma_{X}(t),\theta}\right)x\\\
\left.-ad(g^{-1})\alpha_{p}([X,Y])\vphantom{\frac{d}{dt}\bigg{|}_{0}}\right\\}$
That is,
$d\tilde{\alpha}=-[ad(g^{-1})\alpha,\Theta]+ad(g^{-1})d\alpha-
ad(g^{-1})\partial\alpha\wedge d\theta\\\ +[ad(g^{-1})\varphi,\Theta]\wedge
d\theta+ad(g^{-1})d\varphi\wedge d\theta$
$\phantom{d\tilde{\alpha}}=ad(g^{-1})\left(d\alpha+d\varphi\wedge
d\theta-\partial\alpha\wedge
d\theta\right)-[ad(g^{-1})\alpha+ad(g^{-1})\varphi d\theta,\Theta]\\\ {}$
To calculate $D_{t}\tilde{\alpha}$ we need to know what the horizontal
projection (with respect to $\tilde{A}_{t}$) of a vector looks like. If $X$ is
a tangent vector at $p$ we can calculate its horizontal projection as
$hX=X-\iota_{p}(A(X)),$ where $\iota_{p}(A(X))$ is the vector at $p$ generated
by the Lie algebra element $A(X).$ So for the vector $(X,g\xi,x_{\theta})$ we
have
$h(X,g\xi,x_{\theta})=(X,g\xi,x_{\theta})-\iota_{(p,g,\theta)}(\tilde{A}_{t}(X,g\xi,x_{\theta})).$
Now,
$\displaystyle\iota_{(p,g,\theta)}(\tilde{A}_{t}(X,g\xi,x_{\theta}))$
$\displaystyle=\frac{d}{ds}\bigg{|}_{0}(p,g(1+s\tilde{A}_{t}(X,g\xi,\theta+x)),\theta)$
$\displaystyle=\frac{d}{ds}\bigg{|}_{0}(p,gs\tilde{A}_{t}(X,g\xi,\theta+x),\theta)$
$\displaystyle=(0,g\tilde{A}_{t}(X,g\xi,x_{\theta}),0),$
and therefore,
$h(X,g\xi,x_{\theta})=(X,g(\xi-\tilde{A}_{t}(X,g\xi,x_{\theta})),x_{\theta}).$
Putting this into the formula above for $d\tilde{\alpha},$ we obtain
$D_{t}\tilde{\alpha}=ad(g^{-1})\left(d\alpha+d\varphi\wedge
d\theta-\partial\alpha\wedge
d\theta\right)-[ad(g^{-1})\alpha+ad(g^{-1})\varphi
d\theta,\Theta-\tilde{A}_{t}]$
and inserting the formula for $\tilde{A}_{t}$ in terms of $A_{t}$ and
$\Phi_{t}$ in the second term we obtain
$\displaystyle-[ad(g^{-1})\alpha+$ $\displaystyle ad(g^{-1})\varphi
d\theta,\Theta-\tilde{A}_{t}]$
$\displaystyle=-[ad(g^{-1})\alpha+ad(g^{-1})\varphi d\theta,\Theta-
ad(g^{-1})A_{t}-\Theta-ad(g^{-1})\Phi_{t}d\theta]$
$\displaystyle=-[ad(g^{-1})\alpha+ad(g^{-1})\varphi
d\theta,-ad(g^{-1})A_{t}-ad(g^{-1})\Phi_{t}d\theta]$
$\displaystyle=ad(g^{-1})[\alpha+\varphi d\theta,A_{t}+\Phi_{t}d\theta]$
and therefore
$D_{t}\tilde{\alpha}=ad(g^{-1})\left(d\alpha+d\varphi\wedge
d\theta-\partial\alpha\wedge
d\theta+[\alpha,A_{t}]+[\alpha,\Phi_{t}]d\theta+[A_{t},\varphi]d\theta\right)$
which is equal to $\dfrac{d}{dt}\tilde{F}_{t}.$ This completes the proof of
Lemma 3.2.5. ∎
Now, if we set
$\psi=k\int_{0}^{1}f(\tilde{\alpha},\tilde{F}_{t})dt$
then
$\displaystyle d\psi$ $\displaystyle=D\psi\qquad\text{ (by Lemma
\ref{L:dpsi=Dpsi}) }$
$\displaystyle=k\int_{0}^{1}f(D_{t}\tilde{\alpha},\tilde{F}_{t})dt$
$\displaystyle=k\int_{0}^{1}f(\frac{d}{dt}\tilde{F}_{t},\tilde{F}_{t})dt$
$\displaystyle=\int_{0}^{1}\frac{d}{dt}f(\tilde{F}_{t})dt$
$\displaystyle=f(\tilde{F}_{1})-f(\tilde{F}_{0}).$
So $s_{2k-1}^{P}(f)$ is independent of choice of connection and Higgs field.
∎
It remains only to prove that $s_{2k-1}^{P}(f)$ is the pull-back of $\tau(f)$
by $\operatorname{hol}_{\Phi}.$ For this we follow the argument in [11] that
will give us a formula for $f(\tilde{F})$ that we can use to calculate
$s_{2k-1}^{\Omega\widetilde{P}}(f)$ for a loop bundle
$\Omega\widetilde{P}\xrightarrow{\Omega G}\Omega(M\times S^{1})$ and then we
can use Lemma 3.1.1 to generalise to a general $\Omega G$-bundle.
If we start with the $G$-bundle $\widetilde{P}\to M\times S^{1}$ we can pull-
back by the evaluation map $\operatorname{ev}\colon[0,1]\times\Omega(M\times
S^{1})\to(M\times S^{1})$ to get a trivial bundle
$\operatorname{ev}^{*}\widetilde{P}$ over $[0,1]\times\Omega(M\times S^{1}).$
A section is given by
$h\colon[0,1]\times\Omega(M\times
S^{1})\to\operatorname{ev}^{*}\widetilde{P};\quad(t,\gamma)\mapsto\hat{\gamma}(t),$
where $\hat{\gamma}$ is the horizontal lift of $\gamma.$ If $\tilde{A}$ is the
connection in $\widetilde{P}$ we can pull it back to
$\operatorname{ev}^{*}\widetilde{P}$ and then back to
$[0,1]\times\Omega(M\times S^{1})$ to obtain
$\tilde{A}^{\prime}:=h^{*}\operatorname{ev}^{*}\tilde{A}.$
We can calculate the curvature $\tilde{F}$ of $\tilde{A}$ and pull it back by
$\operatorname{ev}$ to $[0,1]\times\Omega(M\times S^{1})$ and because this is
a product manifold we can decompose it into parts with a $dt$ and parts
without a $dt.$ Under this decomposition, we have
$\operatorname{ev}^{*}\tilde{F}=-\frac{\partial}{\partial
t}\tilde{A}^{\prime}\wedge dt+\tilde{F}^{\prime},$
where we call the component without a $dt$ $\tilde{F}^{\prime}$ since if we
view the form $\tilde{A}^{\prime}$ for fixed $t_{0}$ as a connection form on
$\Omega(M\times S^{1})$ then its curvature is $\tilde{F}^{\prime}$ evaluated
at $t_{0}.$
Now, we want to calculate $\int_{S^{1}}f(\tilde{F})$ and using Lemma 3.1.2 we
have for a general $\Omega G$-bundle $P\to M,$
$\displaystyle\int_{S^{1}}f(\tilde{F})$
$\displaystyle=\eta^{*}\int_{S^{1}}\operatorname{ev}^{*}f(\tilde{F})$
$\displaystyle=\eta^{*}\int_{S^{1}}f(\operatorname{ev}^{*}\tilde{F}).$
So we wish to calculate explicitly
$\int_{S^{1}}f(\operatorname{ev}^{*}\tilde{F}).$ If we view the circle as the
interval $[0,1]$ with endpoints identified, then we can write
$\int_{S^{1}}f(\operatorname{ev}^{*}\tilde{F})=\int_{[0,1]}f(\operatorname{ev}^{*}\tilde{F})$
and so we have
$\displaystyle k\int_{S^{1}}f(\nabla\Phi,F)d\theta$
$\displaystyle=\eta^{*}\int_{S^{1}}f(\operatorname{ev}^{*}\tilde{F})$
$\displaystyle=\eta^{*}\int_{[0,1]}f(-\frac{\partial}{\partial
t}\tilde{A}^{\prime}\wedge dt+\tilde{F}^{\prime})$
$\displaystyle=\eta^{*}\int_{[0,1]}f(\tilde{F}^{\prime})-k\eta^{*}\int_{[0,1]}f(-\frac{\partial}{\partial
t}\tilde{A}^{\prime},\tilde{F}^{\prime})dt$
$\displaystyle=-k\eta^{*}\int_{[0,1]}f(-\frac{\partial}{\partial
t}\tilde{A}^{\prime},\tilde{F}^{\prime})dt.$
Using the formula
$\tilde{F}^{\prime}=d\tilde{A}^{\prime}+\frac{1}{2}[\tilde{A}^{\prime},\tilde{A}^{\prime}],$
we can write this as:
$-k\eta^{*}\left\\{\int_{[0,1]}f(\partial\tilde{A}^{\prime},d\tilde{A}^{\prime})dt\right.\\\
\left.+(k-1)\frac{1}{2}\int_{[0,1]}f(\partial\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt+\ldots\right.\\\
...+\binom{k-1}{k-2}\left(\frac{1}{2}\right)^{k-2}\int_{[0,1]}f(\partial\tilde{A}^{\prime},d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
\left.+\left(\frac{1}{2}\right)^{k-1}\int_{[0,1]}f(\partial\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\right\\}$
where we have written $\partial\tilde{A}^{\prime}$ for
$\partial\tilde{A}^{\prime}/\partial t.$ Thus we need to work with the general
term
$\binom{k-1}{i}\left(\frac{1}{2}\right)^{i}\int_{[0,1]}f(\partial\tilde{A}^{\prime},\underbrace{d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime}}_{k-i-1},\underbrace{[\tilde{A}^{\prime},\tilde{A}^{\prime}],\ldots,[\tilde{A}^{\prime},\tilde{A}^{\prime}]}_{i})dt.$
To deal with these terms we shall use integration by parts and the
$ad$-invariance of $f.$ Thus we need to know in detail how $ad$-invariance
works.
###### Lemma 3.2.6.
Let $\varphi_{1},\ldots,\varphi_{k}$ be ${\mathfrak{g}}$-valued forms of
degree $q_{1},\ldots,q_{k}$ respectively. Then if $A$ is a
${\mathfrak{g}}$-valued $p$-form, we have
$f([\varphi_{1},A],\varphi_{2},\ldots,\varphi_{k})\\\
=f(\varphi_{1},[A,\varphi_{2}],\ldots,\varphi_{k})+(-1)^{pq_{2}}f(\varphi_{1},\varphi_{2},[A,\varphi_{3}],\ldots,\varphi_{k})+\ldots\\\
\ldots+(-1)^{p(q_{2}+\ldots
q_{k-1})}f(\varphi_{1},\ldots,\varphi_{k-1},[A,\varphi_{k}]).$
###### Proof.
We can expand $\varphi_{i}$ as $\varphi_{i}=\varphi_{i,j}\omega_{i}^{j}$ for
$\varphi_{i,j}\in{\mathfrak{g}}$ and $\omega_{i}^{j}$ a $q_{i}$-form. Then we
have
$f(\varphi_{1},\ldots,\varphi_{k})=f(\varphi_{1,j_{1}},\ldots,\varphi_{k,j_{k}})\omega_{1}^{j_{1}}\wedge\ldots\wedge\omega_{k}^{j_{k}}.$
Now if $A$ is a ${\mathfrak{g}}$ valued $p$-form and we write
$A=A_{i}\alpha^{i}$ as above, then
$f([A,\varphi_{1}],\varphi_{2},\ldots,\varphi_{k})\\\
\phantom{f}=f([A_{i},\varphi_{1,j_{1}}],\varphi_{2,j_{2}},\ldots,\varphi_{k,j_{k}})\alpha^{i}\wedge\omega_{1}^{j_{1}}\wedge\ldots\wedge\omega_{k}^{j_{k}}\\\
\phantom{f}=f(\varphi_{1,j_{1}},[\varphi_{2,j_{2}},A_{i}],\ldots,\varphi_{k,j_{k}})(-1)^{p(q_{1}+q_{2})}\omega_{1}^{j_{1}}\wedge\omega_{2}^{j_{2}}\alpha^{i}\wedge\ldots\wedge\omega_{k}^{j_{k}}\\\
+f(\varphi_{1,j_{1}},\varphi_{2,j_{2}},[\varphi_{3,j_{3}},A_{i}],\ldots,\varphi_{k,j_{k}})(-1)^{p(q_{1}+q_{2}+q_{3})}\omega_{1}^{j_{1}}\wedge\omega_{2}^{j_{2}}\wedge\omega_{3}^{j_{3}}\wedge\alpha^{i}\wedge\ldots\wedge\omega_{k}^{j_{k}}\\\
\ldots+f(\varphi_{1,j_{1}},\varphi_{2,j_{2}},\ldots,[\varphi_{k,j_{k}},A_{i}])(-1)^{p(q_{1}+q_{2}+\ldots+q_{k})}\omega_{1}^{j_{1}}\wedge\omega_{2}^{j_{2}}\wedge\ldots\wedge\omega_{k}^{j_{k}}\wedge\alpha^{i}$
That is,
$f([A,\varphi_{1}],\varphi_{2},\ldots,\varphi_{k})\\\
=(-1)^{pq_{1}}f(\varphi_{1},[\varphi_{2},A],\ldots,\varphi_{k})+(-1)^{p(q_{1}+q_{2})}f(\varphi_{1},\varphi_{2},[\varphi_{3},A],\ldots,\varphi_{k})+\ldots\\\
\ldots+(-1)^{p(q_{1}+\ldots+q_{k})}f(\varphi_{1},\ldots,\varphi_{k-1},[\varphi_{k},A]),$
which we can write as:
$f([\varphi_{1},A],\varphi_{2},\ldots,\varphi_{k})\\\
=f(\varphi_{1},[A,\varphi_{2}],\ldots,\varphi_{k})+(-1)^{pq_{2}}f(\varphi_{1},\varphi_{2},[A,\varphi_{3}],\ldots,\varphi_{k})+\ldots\\\
\ldots+(-1)^{p(q_{2}+\ldots
q_{k-1})}f(\varphi_{1},\ldots,\varphi_{k-1},[A,\varphi_{k}]).$
∎
We are now in a position to prove
###### Proposition 3.2.7.
$s_{2k-1}^{P}(f)=\operatorname{hol}_{\Phi}^{*}\tau(f).$
###### Proof.
To calculate the general term given above, we integrate by parts in the
$\Omega(M\times~{}S^{1})$ and $t$ directions giving
$\int_{[0,1]}f_{i}dt=\int_{[0,1]}f(d\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
+i\int_{[0,1]}f(\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},d[\tilde{A}^{\prime},\tilde{A}^{\prime}],[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
-d\int_{[0,1]}f(\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt$
and
$\int_{[0,1]}f_{i}dt=f(\tilde{A}^{\prime}_{1},d\tilde{A}^{\prime}_{1},\ldots,d\tilde{A}^{\prime}_{1},[\tilde{A}^{\prime}_{1},\tilde{A}^{\prime}_{1}])-f(\tilde{A}^{\prime}_{0},d\tilde{A}^{\prime}_{0},\ldots,d\tilde{A}^{\prime}_{0},[\tilde{A}^{\prime}_{0},\tilde{A}^{\prime}_{0}])\\\
-(k-1-i)\int_{[0,1]}f(\tilde{A}^{\prime},\partial
d\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
-i\int_{[0,1]}f(\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},\partial[\tilde{A}^{\prime},\tilde{A}^{\prime}],[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt$
where we have written $f_{i}$ for the integrand of the general term given
earlier. Combining these gives
$(k-i)\int_{[0,1]}f_{i}dt=f_{i,1}-f_{i,0}-i\int_{[0,1]}f(\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},\partial[\tilde{A}^{\prime},\tilde{A}^{\prime}],[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
+i(k-1-i)\int_{[0,1]}f(\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},d[\tilde{A}^{\prime},\tilde{A}^{\prime}],[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
-(k-1-i)d\int_{[0,1]}f(\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt$
where we have written $f_{i,1}$ and $f_{i,0}$ for $f_{i}$ evaluated at $t=1$
and $0$ respectively. Using $ad$-invariance, the term on the middle line
simplifies as follows:
$\int_{[0,1]}f(\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},d[\tilde{A}^{\prime},\tilde{A}^{\prime}],[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
=2\int_{[0,1]}f([d\tilde{A}^{\prime},\tilde{A}^{\prime}],\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
=2\int_{[0,1]}f(d\tilde{A}^{\prime},[\tilde{A}^{\prime},\partial\tilde{A}^{\prime}],\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
-2\int_{[0,1]}f(d\tilde{A}^{\prime},\partial\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}],d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
+2(k-2-i)\int_{[0,1]}f(d\tilde{A}^{\prime},\partial\tilde{A}^{\prime},\tilde{A}^{\prime},[\tilde{A}^{\prime},d\tilde{A}^{\prime}],d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
=\int_{[0,1]}f(d\tilde{A}^{\prime},\partial[\tilde{A}^{\prime},\tilde{A}^{\prime}],\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
-2\int_{[0,1]}f(\partial\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
-(k-2-i)\int_{[0,1]}f(\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},d[\tilde{A}^{\prime},\tilde{A}^{\prime}],[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt$
and so
$(k-1-i)\int_{[0,1]}f(\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},d[\tilde{A}^{\prime},\tilde{A}^{\prime}],[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
=\int_{[0,1]}f(\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},\partial[\tilde{A}^{\prime},\tilde{A}^{\prime}],[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt\\\
-2\int_{[0,1]}f(\partial\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt.$
Inserting this into the formula for $\int f_{i}dt$ gives
$(k-i)\int_{[0,1]}f_{i}dt=f_{i,1}-f_{i,0}-2i\int_{[0,1]}f_{i}dt\\\
-(k-1-i)d\int_{[0,1]}f(\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt$
and hence
$(k+i)\int_{[0,1]}f_{i}dt\\\
=f_{i,1}-f_{i,0}-(k-1-i)d\int_{[0,1]}f(\partial\tilde{A}^{\prime},\tilde{A}^{\prime},d\tilde{A}^{\prime},\ldots,d\tilde{A}^{\prime},[\tilde{A}^{\prime},\tilde{A}^{\prime}])dt.$
So we have the following expression for $s_{2k-1}^{P}(f):$
$k\int_{S^{1}}f(\nabla\Phi,F)d\theta\\\
=-k\eta^{*}\left\\{\sum_{i=0}^{k-1}\binom{k-1}{i}\left(\frac{1}{2}\right)^{i}\frac{1}{k+i}\left(f_{i,1}-f_{i,0}-(k-i-1)dc_{i}\vphantom{\tilde{f}}\right)\right\\}$
where $c_{i}$ is the last integral in the equation above (with $i$
$[\tilde{A}^{\prime},\tilde{A}^{\prime}]$’s).
Now since $\tilde{A}^{\prime}_{0}=0$ and
$h(0,\gamma)=h(1,\gamma)\operatorname{hol}(\gamma)$ (where $h$ is the section
from earlier), we have that
$\tilde{A}^{\prime}_{0}=ad(\operatorname{hol}^{-1})\tilde{A}^{\prime}_{1}+\operatorname{hol}^{-1}d\operatorname{hol}$
and so
$\tilde{A}^{\prime}_{1}=-d\operatorname{hol}\operatorname{hol}^{-1}.$
Therefore we have that $f_{i,0}=0$ and we can calculate $f_{i,1}$ in terms of
$f_{0,1}$ as follows:
$\displaystyle f_{0,1}$
$\displaystyle=f(\tilde{A}^{\prime}_{1},d\tilde{A}^{\prime}_{1})$
$\displaystyle=f(-d\operatorname{hol}\operatorname{hol}^{-1},d(-d\operatorname{hol}\operatorname{hol}^{-1}))$
$\displaystyle=(-1)^{k}\left(\frac{1}{2}\right)^{k-1}\operatorname{hol}^{*}f(\Theta,[\Theta,\Theta])$
and in general,
$\displaystyle f_{i,1}$
$\displaystyle=f(\tilde{A}^{\prime}_{1},d\tilde{A}^{\prime}_{1},\ldots,d\tilde{A}^{\prime}_{1},[\tilde{A}^{\prime}_{1},\tilde{A}^{\prime}_{1}])$
$\displaystyle=(-1)^{k-i}\left(\frac{1}{2}\right)^{k-1-i}\operatorname{hol}^{*}f(\Theta,[\Theta,\Theta])$
$\displaystyle=(-1)^{i}2^{i}f_{0,1}$
using the fact that
$d(-d\operatorname{hol}\operatorname{hol}^{-1})=-\frac{1}{2}[d\operatorname{hol}\operatorname{hol}^{-1},d\operatorname{hol}\operatorname{hol}^{-1}].$
Therefore we have
$k\int_{S^{1}}f(\nabla\Phi,F)d\theta\\\
=\left(-\frac{1}{2}\right)^{k-1}k\sum_{i=0}^{k-1}\binom{k-1}{i}\frac{(-1)^{i}}{k+i}\operatorname{hol}_{\Phi}^{*}f(\Theta,[\Theta,\Theta])\\\
+k\sum_{i=0}^{k-i}\binom{k-1}{i}\left(\frac{1}{2}\right)^{i}\frac{1}{k+i}(k-i-1)dc_{i}.$
We have seen already that the coefficient above is equal to the coefficient in
the definition of the transgression map:
$k\sum_{i=0}^{k-1}\binom{k-1}{i}\frac{(-1)^{i}}{k+i}=\frac{k!(k-1)!}{(2k-1)!}.$
So we see that the pull-back of the transgression of $f$ is cohomologous to
the string $(2k-1)$-class.
∎
Combining Propositions 3.2.2, 3.2.4 and 3.2.7, we have the following Theorem
###### Theorem 3.2.8.
The diagram
$\textstyle{H^{2k}(BG)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C\text{-}W_{\widetilde{P}}}$$\scriptstyle{\tau}$$\textstyle{H^{2k}(M\times
S^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\int_{S^{1}}}$$\textstyle{H^{2k-1}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{hol}_{\Phi}^{*}}$$\textstyle{H^{2k-1}(M)}$
commutes. Furthermore, the composition map
$H^{2k}(BG)\to H^{2k-1}(M)$
associates to any invariant polynomial its string $(2k-1)$-class, which is a
characteristic class.
### 3.3 The universal string class for
$L^{\scriptscriptstyle{\vee}}G$-bundles
We would now like to return to the study of the free loop group. In this
section, we shall give a partial generalisation of the results in the previous
section. However, we shall be working with a slightly different group than in
the rest of this thesis. For the remainder of this chapter we shall be
considering the group of smooth maps from the interval $[0,2\pi]$ into $G$
whose endpoints are coincident. This group shall be denoted
$L^{\scriptscriptstyle{\vee}}G.$ Note that $LG\subseteq
L^{\scriptscriptstyle{\vee}}G.$ We also have the based version
$\Omega^{\scriptscriptstyle{\vee}}G$ of this group consisting of maps
$[0,2\pi]\to G$ such that the endpoints are mapped to the identity in $G.$
We will give a classifying theory for $L^{\scriptscriptstyle{\vee}}G$ bundles
and present a calculation for the string class of the universal
$L^{\scriptscriptstyle{\vee}}G$-bundle.
#### 3.3.1 Classification of $L^{\scriptscriptstyle{\vee}}G$-bundles
In order to extend the ideas from the previous section (namely, calculating
the string class of the universal $L^{\scriptscriptstyle{\vee}}G$-bundle) we
need a model for $EL^{\scriptscriptstyle{\vee}}G.$ To construct this we view
$L^{\scriptscriptstyle{\vee}}G$ as the semi-direct product
$\Omega^{\scriptscriptstyle{\vee}}G\rtimes G.$ The group multiplication is
given by
$(\gamma_{1},g_{1})(\gamma_{2},g_{2})=(g_{2}^{-1}\gamma_{1}g_{2}\gamma_{2},g_{1}g_{2})$
and the isomorphism between $\Omega^{\scriptscriptstyle{\vee}}G\rtimes G$ and
$L^{\scriptscriptstyle{\vee}}G$ is
$\displaystyle\Omega^{\scriptscriptstyle{\vee}}G\rtimes G$
$\displaystyle\xrightarrow{\sim}L^{\scriptscriptstyle{\vee}}G;\quad(\gamma,g)\mapsto
g\gamma.$
On the level of Lie algebras, the isomorphism is
$\displaystyle\Omega^{\scriptscriptstyle{\vee}}{\mathfrak{g}}\rtimes{\mathfrak{g}}$
$\displaystyle\xrightarrow{\sim}L^{\scriptscriptstyle{\vee}}{\mathfrak{g}};\quad(\xi,X)\mapsto
X+\xi.$
We therefore need a model for the universal
$\Omega^{\scriptscriptstyle{\vee}}G\rtimes G$-bundle. For this, we shall take
the product of the universal $\Omega^{\scriptscriptstyle{\vee}}G$-bundle and
the universal $G$-bundle. A model for the universal
$\Omega^{\scriptscriptstyle{\vee}}G$-bundle is given by the space of maps from
the interval $[0,2\pi]$ into $G,$ denoted $P^{\scriptscriptstyle{\vee}}G.$ The
based loop group $\Omega^{\scriptscriptstyle{\vee}}G$ acts on this space by
right multiplication and evaluation at the endpoint of a path gives a locally
trivial $\Omega^{\scriptscriptstyle{\vee}}G$-bundle
$P^{\scriptscriptstyle{\vee}}G\to G.$ As our study of
$\Omega^{\scriptscriptstyle{\vee}}G$ will be confined to this section, we
shall refer to $P^{\scriptscriptstyle{\vee}}G$ as the _path fibration_ without
any risk of confusion. $P^{\scriptscriptstyle{\vee}}G$ is contractible since
any path $p$ can be homotopied to the identity path by the map
$h\colon I\times P^{\scriptscriptstyle{\vee}}G\to
P^{\scriptscriptstyle{\vee}}G;\quad(t,p)\mapsto(\theta\mapsto p(t\theta)).$
Therefore the path fibration is a model for the universal
$\Omega^{\scriptscriptstyle{\vee}}G$-bundle. So, for our model for
$EL^{\scriptscriptstyle{\vee}}G$ we shall take the space
$P^{\scriptscriptstyle{\vee}}G\times EG$ which is contractible since
$P^{\scriptscriptstyle{\vee}}G$ and $EG$ are both contractible. This is acted
on by $\Omega^{\scriptscriptstyle{\vee}}G\rtimes G:$
$(p,x)(\gamma,g)=(g^{-1}pg\gamma,xg)$
where $xg$ is the right action of $G$ on $EG.$ This action is free (since $G$
acts on $EG$ freely) and transitive on fibres (since the action on $EG$ is
transitive and the equation $g^{-1}p_{1}g\gamma=p_{2}$ can always be solved)
and so $P^{\scriptscriptstyle{\vee}}G\times EG$ is a model for
$EL^{\scriptscriptstyle{\vee}}G$ and $BL^{\scriptscriptstyle{\vee}}G$ is equal
to $(P^{\scriptscriptstyle{\vee}}G\times
EG)/(\Omega^{\scriptscriptstyle{\vee}}G\rtimes G).$ In fact, if we consider
the map
$(P^{\scriptscriptstyle{\vee}}G\times
EG)/(\Omega^{\scriptscriptstyle{\vee}}G\rtimes G)\to(G\times
EG)/G;\quad[p,x]\mapsto[p(2\pi),x],$
where $[h,x]=[g^{-1}hg,xg],$ we can see this is well-defined, since
$[p,x]=[g^{-1}pg\gamma,xg]\mapsto[g^{-1}p(2\pi)g\gamma(2\pi),xg]=[p(2\pi),x].$
Furthermore, this is onto, as the projection $P^{\scriptscriptstyle{\vee}}G\to
G$ is onto, and 1–1, for if we consider two elements
$[p,x],\,[q,y]\in(P^{\scriptscriptstyle{\vee}}G\times
EG)/(\Omega^{\scriptscriptstyle{\vee}}G\rtimes G)$ such that
$[p(2\pi),x]=[q(2\pi),y]$ we have $y=xg$ and $q(2\pi)=g^{-1}p(2\pi)g.$ That
is, the paths $q$ and $g^{-1}pg$ have the same endpoint. Therefore, the path
$g^{-1}p^{-1}gq$ is actually a (based) loop. And since
$q=g^{-1}pg(g^{-1}p^{-1}gq),$ we have
$\displaystyle[q,y]$ $\displaystyle=[g^{-1}pg\gamma,xg]$
$\displaystyle=[p,x],$
where $\gamma=g^{-1}p^{-1}gq\in\Omega^{\scriptscriptstyle{\vee}}G.$ Thus we
have a diffeomorphism between $BL^{\scriptscriptstyle{\vee}}G$ and $(G\times
EG)/G$ (or simply $G\times_{G}EG$). Note that this allows us to calculate the
cohomology of $BL^{\scriptscriptstyle{\vee}}G$ as the equivariant cohomology
of $G$ (with its adjoint action). That is,
$H(BL^{\scriptscriptstyle{\vee}}G)=H_{G}(G).$
Given an $L^{\scriptscriptstyle{\vee}}G$-bundle $P\to M$ we can write down the
classifying map of this bundle as follows. Choose a Higgs field, $\Phi,$ for
$P.$ Then define the map $f\colon P\to P^{\scriptscriptstyle{\vee}}G\times EG$
by
$f(q)=(\operatorname{hol}_{\Phi}(q),f_{G}(q)),$
where $\operatorname{hol}_{\Phi}$ is the Higgs field holonomy and $f_{G}$ is
the classifying map for the $G$-bundle associated to $P$ by the projection
$L^{\scriptscriptstyle{\vee}}G\to G$ given by mapping a loop to its
start/endpoint (or equivalently, the projection
$\Omega^{\scriptscriptstyle{\vee}}G\rtimes G\to G$). That is, $f(q)=(p,x)$
where $p^{-1}\partial p=\Phi(q)$ and $x$ is $f_{G}$ applied to the image of
$q$ in $P\times_{L^{\scriptscriptstyle{\vee}}G}G.$ It is easy to see that this
is equivariant with respect to the $L^{\scriptscriptstyle{\vee}}G$ action and
hence descends to a map $M\to BL^{\scriptscriptstyle{\vee}}G$ since if
$(\gamma,g)\in\Omega^{\scriptscriptstyle{\vee}}G\rtimes G$ then
$\displaystyle f(q(g\gamma))$
$\displaystyle=(\operatorname{hol}_{\Phi}(q(g\gamma)),f_{G}(q)g)$
and so $f$ is equivariant in the $EG$ slot (by virtue of the fact that $f_{G}$
is a classifying map) and also in the $P^{\scriptscriptstyle{\vee}}G$ slot
since if $\operatorname{hol}_{\Phi}(q)=p$ then
$\displaystyle\Phi(q(g\gamma))$
$\displaystyle=ad((g\gamma)^{-1})\Phi(q)+(g\gamma)^{-1}\partial(g\gamma)$
$\displaystyle=ad((g\gamma)^{-1})\Phi(q)+\gamma^{-1}\partial\gamma$
and
$\displaystyle(p(\gamma,g))^{-1}\partial(p(\gamma,g))$
$\displaystyle=(g^{-1}pg\gamma)^{-1}\partial(g^{-1}pg\gamma)$
$\displaystyle=\gamma^{-1}g^{-1}p^{-1}g(g^{-1}\partial
pg\gamma+g^{-1}pg\partial\gamma)$
$\displaystyle=ad((g\gamma)^{-1})p^{-1}\partial p+\gamma^{-1}\partial\gamma$
and so
$\operatorname{hol}_{\Phi}(q(g\gamma))=p(\gamma,g)=\operatorname{hol}_{\Phi}(q)(g\gamma).$
#### 3.3.2 The universal string class
Now that we have a model for the universal
$L^{\scriptscriptstyle{\vee}}G$-bundle we would like to calculate its string
class according to Theorem 2.4.1. So far everything we have said works on the
topological level. In order to use Theorem 2.4.1 however, the first thing we
need is a connection on $P^{\scriptscriptstyle{\vee}}G\times EG.$ Now,
$P^{\scriptscriptstyle{\vee}}G$ is already a smooth manifold. In order to
define a smooth structure and find a connection on $EG$ we use the results in
[36, 37]. As long as the dimension of the base of the $G$-bundle $P\to M$ is
less than or equal to $n$ this gives a construction of a smooth bundle
$EG_{n}\to BG_{n}$ with connection which is a model for the universal
$G$-bundle. From now on we assume therefore that the dimension of the base of
our $L^{\scriptscriptstyle{\vee}}G$-bundle is fixed (and less than or equal to
$n$ for some $n$).
To define a connection we need to know what a vertical vector looks like.
Consider the vector in $T_{(p,x)}(P^{\scriptscriptstyle{\vee}}G\times
EG_{n})=T_{p}P^{\scriptscriptstyle{\vee}}G\times T_{x}EG_{n}$ generated by the
Lie algebra element
$(\xi,X)\in\Omega^{\scriptscriptstyle{\vee}}{\mathfrak{g}}\rtimes{\mathfrak{g}}:$
$\displaystyle\iota_{(p,x)}(\xi,X)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}((1-tX)p(1+tX)(1+t\xi),xe^{tX})$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(t(-Xp+pX+p\xi),xe^{tX})$
$\displaystyle=(p(X-ad(p^{-1})X+\xi),\iota_{x}(X)).$
Note that the $P^{\scriptscriptstyle{\vee}}G$ part of a vertical vector is a
vector field along $p$ that ends at $p(2\pi)(X-ad(p(2\pi)^{-1})X)$ (since
$\xi$ is a based loop). We will assume that we have a connection in $EG_{n}$
since this is always possible by the discussion above. Call this connection
$a.$ So to find the horizontal part of a vector $(V,W)\in
T_{p}P^{\scriptscriptstyle{\vee}}G\times T_{x}EG_{n}$ we need a vector field
along $p$ that ends at $V(2\pi)-p(2\pi)(X-ad(p(2\pi)^{-1})X)$ (since then
$V-\\{\text{this vector}\\}$ will end at the right point to be vertical).
Consider the vector field
$\left(\frac{\theta}{2\pi}\right)p\left\\{ad(p^{-1})\left(V(2\pi)p(2\pi)^{-1}-ad(p(2\pi))a(W)+a(W)\right)\right\\}.$
If we define the horizontal projection of $(V,W),h(V,W),$ to be the vector
field above together with the horizontal component of $W$ (that is,
$hW=W-\iota_{x}(a(W))$), then we have an invariant splitting of the tangent
space at each point in $P^{\scriptscriptstyle{\vee}}G\times EG_{n}.$ This is
easily verified: Since the $EG_{n}$ part has a connection, we need only check
the $P^{\scriptscriptstyle{\vee}}G$ part. First calculate the right action on
the vector above (which we will call $hV$ even though technically the part of
the connection on $P^{\scriptscriptstyle{\vee}}G$ is not actually a connection
itself):
$\left(hV(\gamma,g)\right)_{(g^{-1}pg\gamma,xg)}\\\
=\left(\frac{\theta}{2\pi}\right)g^{-1}p\left\\{ad(p^{-1})\left(V(2\pi)p(2\pi)^{-1}-ad(p(2\pi))a(W)+a(W)\right)\right\\}g\gamma.$
Compare this with the horizontal projection of a vector $V^{\prime}$ at
$(p,x)(\gamma,g)=(g^{-1}pg\gamma,xg):$
$hV^{\prime}_{(g^{-1}pg\gamma,xg)}\\\
\phantom{hV^{\prime}_{(g^{-1}pg\gamma}}=\left(\frac{\theta}{2\pi}\right)g^{-1}pg\gamma\left\\{ad(g^{-1}p^{-1}g\gamma)^{-1}\left(V^{\prime}(2\pi)g^{-1}p(2\pi)^{-1}g\right.\right.\\\
\left.\left.-ad(g^{-1}p(2\pi)g)a(W^{\prime})+a(W^{\prime})\right)\right\\}\\\
\phantom{hV^{\prime}_{(g^{-1}pg\gamma}}=\left(\frac{\theta}{2\pi}\right)g^{-1}p\left\\{ad(p^{-1})g\left(V^{\prime}(2\pi)g^{-1}p(2\pi)^{-1}g\right.\right.\\\
\left.\left.-ad(g^{-1})ad(p(2\pi))ad(g)a(W^{\prime})+a(W^{\prime})\right)g^{-1}\right\\}g\gamma\\\
\phantom{hV^{\prime}_{(g^{-1}pg\gamma}}=\left(\frac{\theta}{2\pi}\right)g^{-1}p\left\\{ad(p^{-1})g\left(V^{\prime}(2\pi)g^{-1}p(2\pi)^{-1}g\right.\right.\\\
\left.\left.-ad(g^{-1})ad(p(2\pi))ad(g)ad(g^{-1})a(W)+ad(g^{-1})a(W)\right)g^{-1}\right\\}g\gamma$
(for $W=W^{\prime}g^{-1}$)
$\phantom{hV^{\prime}_{(g^{-1}pg\gamma}}=\left(\frac{\theta}{2\pi}\right)g^{-1}p\left\\{ad(p^{-1})\left(gV^{\prime}(2\pi)g^{-1}p(2\pi)^{-1}-ad(p(2\pi))a(W)+a(W)\right)\right\\}g\gamma\\\
\phantom{hV^{\prime}_{(g^{-1}pg\gamma}}=\left(\frac{\theta}{2\pi}\right)g^{-1}p\left\\{ad(p^{-1})\left(V(2\pi)p(2\pi)^{-1}-ad(p(2\pi))a(W)+a(W)\right)\right\\}g\gamma\\\
$
(for $V=V^{\prime}(\gamma,g)^{-1},$ so that
$V^{\prime}(2\pi)=g^{-1}V(2\pi)g)$).
So we see that the push forward of the vector $hV$ is horizontal (at
$(g^{-1}pg\gamma,xg)$) and conversely the vector $hV^{\prime}$ is the push
forward of a horizontal vector at $(p,x).$ Thus we have defined a horizontal
splitting of $T_{(p,x)}(P^{\scriptscriptstyle{\vee}}G\times EG_{n})$ for each
$(p,x).$ To find the connection form for this connection we need to recover
the Lie algebra element $(\xi,X)$ from the vector $(V,W).$ We know that the
vector
$v(V,W)\\\
=\left(V-\left(\frac{\theta}{2\pi}\right)p\left\\{ad(p^{-1})\left(V(2\pi)p(2\pi)^{-1}-ad(p(2\pi))a(W)+a(W)\right)\right\\},a(W)\right)$
is the vertical component of $(V,W)$ and that the vertical vector generated by
$(\xi,X)\in\Omega^{\scriptscriptstyle{\vee}}{\mathfrak{g}}\rtimes{\mathfrak{g}}$
looks like
$(p(X-ad(p^{-1})X+\xi),\iota_{x}(X)).$
Thus to recover $\xi$ from $v(V,W)$ we just subtract $p(a(W)-ad(p^{-1})a(W))$
and, writing $A$ for the part of the connection on
$P^{\scriptscriptstyle{\vee}}G,$ we have
$A(V,W)=\\\
p^{-1}V-\left(\frac{\theta}{2\pi}\right)ad(p^{-1})\left\\{V(2\pi)p(2\pi)^{-1}-ad(p(2\pi))a(W)+a(W)\right\\}\\\
-(a(W)-ad(p^{-1})a(W)).$
Therefore, the connection form $(A,a)$ is given by
$(A,a)=\left(\Theta-\left(\frac{\theta}{2\pi}\right)ad(p^{-1})\left\\{\operatorname{ev}_{2\pi}^{*}\hat{\Theta}-ad(p(2\pi))a+a\right\\}-\left(a-ad(p^{-1})a\right),a\right)$
where $\Theta$ is the Maurer-Cartan form, $\hat{\Theta}$ is the right Maurer-
Cartan form and $\operatorname{ev}_{2\pi}\colon
P^{\scriptscriptstyle{\vee}}G\to G$ is evaluation at the endpoint of a path.
It can be easily checked that this form satisfies the conditions for a
connection. It will be useful later on to write this as a form valued in
$L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}.$ To do this we use the
isomorphism of Lie algebras given in section 3.3.1. The connection form
becomes
$A_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}=\Theta-\left(\frac{\theta}{2\pi}\right)ad(p^{-1})\left\\{\operatorname{ev}_{2\pi}^{*}\hat{\Theta}-ad(p(2\pi))a+a\right\\}+ad(p^{-1})a.$
To calculate the string class we will need the curvature of this connection
and a Higgs field. As usual, the curvature (as an
$L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}$-valued form) is given by the
formula
$F_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}=DA_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}$
where $D$ is the covariant exterior derivative. So we have
$F_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}((V,W),(V^{\prime},W^{\prime}))=\tfrac{1}{2}A_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}([h(V,W),h(V^{\prime},W^{\prime})]).$
Now,
$[h(V,W),h(V^{\prime},W^{\prime})]=([hV,hV^{\prime}],[hW,hW^{\prime}])\\\
=\left(\left[\left(\frac{\theta}{2\pi}\right)p\left\\{ad(p^{-1})\left(V(2\pi)p(2\pi)^{-1}-ad(p(2\pi))a(W)+a(W)\right)\right\\},\right.\right.\\\
\left(\left.\frac{\theta}{2\pi}\right)p\left\\{ad(p^{-1})\left(V^{\prime}(2\pi)p(2\pi)^{-1}-ad(p(2\pi))a(W^{\prime})+a(W^{\prime})\right)\right\\}\right],\\\
\left.\vphantom{\frac{\theta}{2\pi}}[hW,hW^{\prime}]\right)$
and calculating just the first slot gives
$p\left(\frac{\theta}{2\pi}\right)^{2}ad(p^{-1})\left\\{[V(2\pi)p(2\pi)^{-1},V^{\prime}(2\pi)p(2\pi)^{-1}]\right.\\\
-[V(2\pi)p(2\pi)^{-1},ad(p(2\pi))a(W^{\prime})]+[V(2\pi)p(2\pi)^{-1},a(W^{\prime})]\\\
-[ad(p(2\pi))a(W),V^{\prime}(2\pi)p(2\pi)^{-1}]+ad(p(2\pi))[a(W),a(W^{\prime})]\\\
-[ad(p(2\pi))a(W),a(W^{\prime})]+[a(W),V^{\prime}(2\pi)p(2\pi)^{-1}]\\\
\left.-[a(W),ad(p(2\pi))a(W^{\prime})]+[a(W),a(W^{\prime})]\right\\}.$
This yields
$F_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}=\\\
\left(\left(\frac{\theta}{2\pi}\right)^{2}-\left(\frac{\theta}{2\pi}\right)\right)ad(p^{-1})\left\\{\tfrac{1}{2}[\operatorname{ev}_{2\pi}^{*}\hat{\Theta},\operatorname{ev}_{2\pi}^{*}\hat{\Theta}]-[\operatorname{ev}_{2\pi}^{*}\hat{\Theta},ad(p(2\pi)^{-1})a]+\tfrac{1}{2}[a,a]\right.\\\
\left.+[\operatorname{ev}_{2\pi}^{*}\hat{\Theta},a]-[ad(p(2\pi))a,a]+[a,a]\right\\}-\left(\frac{\theta}{2\pi}\right)ad(p^{-1})(f-ad(p(2\pi))f)+ad(p^{-1})f$
where $f$ is the curvature of $a.$
The other piece of data we need to calculate the string class is a Higgs field
for $EL^{\scriptscriptstyle{\vee}}G.$ Define the map $\Phi\colon
P^{\scriptscriptstyle{\vee}}G\times
EG_{n}\to\Omega^{\scriptscriptstyle{\vee}}{\mathfrak{g}}\rtimes{\mathfrak{g}}$
by
$\Phi(p,x)=(p^{-1}\partial p,0).$
Or, as a map to $L^{\scriptscriptstyle{\vee}}{\mathfrak{g}},$
$\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}(p,x)=p^{-1}\partial p.$
Then by the calculation at the end of section 3.3.1 we see that
$\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}$ is a Higgs field for
$P^{\scriptscriptstyle{\vee}}G\times EG_{n}.$ Next we need to calculate
$\nabla\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}=d\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}+[A_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}},\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}]-\partial
A_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}.$
We have
$\displaystyle d\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}(V,W)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}(pe^{t\xi})$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(e^{-t\xi}p^{-1}\partial(pe^{t\xi}))$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(e^{-t\xi}p^{-1}\partial
pe^{t\xi}+e^{-t\xi}\partial e^{t\xi})$ $\displaystyle=p^{-1}\partial p\xi-\xi
p^{-1}\partial p+\partial\xi,$
for $V=\frac{d}{dt}\big{|}_{0}\,p\exp(t\xi).$ That is,
$d\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}=[\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}},\Theta]+\partial\Theta.$
So
$\nabla\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}=[\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}},\Theta]+\partial\Theta\\\
+\left[\Theta-\left(\frac{\theta}{2\pi}\right)ad(p^{-1})\left\\{\operatorname{ev}_{2\pi}^{*}\hat{\Theta}-ad(p(2\pi))a+a\right\\}+ad(p^{-1})a,\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}\right]\\\
-\partial\left(\Theta-\left(\frac{\theta}{2\pi}\right)ad(p^{-1})\left\\{\operatorname{ev}_{2\pi}^{*}\hat{\Theta}-ad(p(2\pi))a+a\right\\}+ad(p^{-1})a\right)$
$\phantom{\nabla\Phi_{L^{\scriptscriptstyle{\vee}}{\mathfrak{g}}}}=\frac{1}{2\pi}ad(p^{-1})\left\\{\operatorname{ev}_{2\pi}^{*}\hat{\Theta}-ad(p(2\pi))a+a\right\\}.\\\
$
So the string class for $P^{\scriptscriptstyle{\vee}}G\times EG_{n}$ is
$-\frac{1}{4\pi^{2}}\int_{S^{1}}\left\langle\left(\frac{\theta^{2}}{4\pi^{2}}-\frac{\theta}{2\pi}\right)\left(\tfrac{1}{2}[\operatorname{ev}_{2\pi}^{*}\hat{\Theta},\operatorname{ev}_{2\pi}^{*}\hat{\Theta}]-[\operatorname{ev}_{2\pi}^{*}\hat{\Theta},ad(p(2\pi)^{-1})a]\right.\right.\\\
+\tfrac{1}{2}[a,a]\left.+[\operatorname{ev}_{2\pi}^{*}\hat{\Theta},a]-[ad(p(2\pi))a,a]+[a,a]\right)\\\
-\left(\frac{\theta}{2\pi}\right)(f-ad(p(2\pi))f)+f,\left.\frac{1}{2\pi}\left(\operatorname{ev}_{2\pi}^{*}\hat{\Theta}-ad(p(2\pi))a+a\right)\right\rangle$
$=-\frac{1}{8\pi^{2}}\left\langle-\tfrac{1}{3}\left(\tfrac{1}{2}[\hat{\Theta},\hat{\Theta}]-[\hat{\Theta},ad(p(2\pi)^{-1})a]\right.\right.\\\
+\tfrac{3}{2}[a,a]\left.+[\hat{\Theta},a]-[ad(p(2\pi))a,a]\right)\\\
+ad(p(2\pi))f+f,\left.\left(\hat{\Theta}-ad(p(2\pi))a+a\right)\right\rangle.$
## Chapter 4 String structures for $LG\rtimes S^{1}$-bundles
Thus far we have discussed central extensions of both the loop group (in
chapter 2) and the based loop group (in chapter 3). The loop group $LG$ has a
natural action of the circle given by rotating loops. In this chapter, we
shall consider the more general case where we allow rotations of the loops in
$LG.$ That is, we shall be working with the semi-direct product $LG\rtimes
S^{1}.$ This group arises when we consider a natural generalisation of the
caloron correspondence from section 2.5. There we showed that a $G$-bundle
over $M\times S^{1}$ corresponds to an $LG$-bundle over $M$. If we allow the
base space of the $G$-bundle to be a non-trivial $S^{1}$-bundle (rather than
$M\times S^{1}$) we obtain not an $LG$-bundle but an $LG\rtimes S^{1}$-bundle.
If, further, we consider a non-trivial $S^{1}$ fibre bundle (instead of a
principal bundle), we obtain an $LG\rtimes\operatorname{Diff}(S^{1})$-bundle.
In this chapter then, we will calculate the obstruction to lifting a principal
$LG\rtimes S^{1}$-bundle $P$ to a principal $\widehat{LG\rtimes S^{1}}$-bundle
$\widehat{P}.$ In section 4.2 we will construct a correspondence for
$LG\rtimes S^{1}$-bundles in analogy with the caloron correspondence from
chapter 2. This will be used to prove a theorem which extends Theorem 2.5.3
relating the string class and the first Pontrjagyn class. In section 4.3 we
shall consider the lifting problem for the more general case where we allow
general (orientation preserving) diffeomorphisms of the loops in $LG,$ that
is, principal bundles with structure group
$LG\rtimes\operatorname{Diff}(S^{1}).$
### 4.1 The string class of an $LG\rtimes S^{1}$-bundle
In this section we present a formula for the obstruction to lifting a
principal $LG\rtimes S^{1}$-bundle $P$ to a principal $\widehat{LG\rtimes
S^{1}}$-bundle $\widehat{P},$ which we call the _string class_ of $P.$ We
shall follow the methods of [35], outlined in section 2.4. In section 4.1.2 we
will give another method for calculating the 3-curvature of a lifting bundle
gerbe, first presented in [18], and apply this to the problem of the string
class of an $LG\rtimes S^{1}$-bundle.
#### 4.1.1 The string class via lifting bundle gerbes
Let $LG\rtimes S^{1}$ be the semi-direct product, whose multiplication is
given by
$(\gamma_{1},\phi_{1})(\gamma_{2},\phi_{2})=(\gamma_{1}\rho_{\phi_{1}}(\gamma_{2}),\phi_{1}+\phi_{2}),$
where $\rho_{\phi}(\gamma)(\theta)=\gamma(\theta-\phi).$ For convenience, let
us record some facts about the Lie algebra of $LG\rtimes S^{1}$ here. The
bracket on the Lie algebra $L{\mathfrak{g}}\rtimes i{\mathbb{R}}$ is given by
$[(\xi,x),(\zeta,y)]=([\xi,\zeta]-x\partial\zeta+y\partial\xi,0)$
and the adjoint action of $LG\rtimes S^{1}$ on $L{\mathfrak{g}}\rtimes
i{\mathbb{R}}$ is
$ad(\gamma,\phi)(\xi,x)=\left(ad(\gamma)\rho_{\phi}(\xi)+x\,\partial\gamma\gamma^{-1},x\right).$
##### The central extension of $LG\rtimes S^{1}$
Recall from section 2.4 that in order to perform calculations involving the
lifting bundle gerbe, we needed an explicit construction of the central
extension of $LG.$ This was given following the construction in section 2.3 in
terms of a pair of differential forms satisfying a certain compatibility
condition. Namely, a pair $(R,\alpha),$ where $R$ is a closed, integral 2-form
on $LG$ and $\alpha$ is a 1-form on $LG\times LG,$ satisfying the conditions
$\delta R=d\alpha$ and $\delta\alpha=0.$ In a similar manner, for what follows
we will require an explicit construction of the central extension of
$LG\rtimes S^{1}.$ Note, however, that the construction in section 2.3 only
works for ${\mathcal{G}}$ a simply connected Lie group. This is because in
order to construct the extension given the pair $(R,\alpha)$ we used the fact
that a flat bundle over a simply connected base has a section satisfying
certain conditions. This allowed us to find a $U(1)$-bundle $P$ over
${\mathcal{G}}$ such that $\delta P\to{\mathcal{G}}\times{\mathcal{G}}$ was
trivial and had a section which defined the multiplication on the central
extension.111See the discussion in section 2.3.1. However, even though the
semi-direct product $LG\rtimes S^{1}$ is not simply connected we can modify
the construction from section 2.3 slightly to cover this case [35]. This
involves replacing the 2-form $R$ with a differential character [12] for the
bundle $\widehat{{\mathcal{G}}}\to{\mathcal{G}}$. That is, we add to our pair
$(R,\alpha)$ a homomorphism $h\colon Z_{1}({\mathcal{G}})\to U(1)$ satisfying
$h(\partial\sigma)=\exp\left(\int_{\sigma}R\right)$
for every two-cycle $\sigma$ in ${\mathcal{G}}.$ We also require the
compatibility condition
$(\delta h)(\gamma)=\exp\left(\int_{\gamma}\alpha\right)$
for every closed one-cycle $\gamma$ in ${\mathcal{G}}\times{\mathcal{G}}.$
Therefore, we need to find a triple of objects $(R,\alpha,h)$ as above. Note
first that
$H^{2}(LG\rtimes S^{1})\simeq H^{2}(LG).$
To see this, we observe that as $LG\rtimes S^{1}=LG\times S^{1}$ as a space,
the Künneth formula (see [2]) gives
$H^{2}(LG\rtimes S^{1})\simeq H^{2}(LG)\otimes H^{0}(S^{1})\oplus
H^{1}(LG)\otimes H^{1}(S^{1}),$
since $H^{2}(S^{1})=0.$ Now, $H^{0}(S^{1})\simeq
H^{1}(S^{1})\simeq{\mathbb{R}},$ so we have
$H^{2}(LG\rtimes S^{1})\simeq H^{2}(LG)\oplus H^{1}(LG).$
Recall, however, that $LG\simeq\Omega G\times G$ as a space, and so
$\pi_{1}(LG)=\pi_{2}(G)\times\pi_{1}(G).$ Therefore, as $G$ is simply
connected, so is $LG,$ and thus $H_{1}(LG,{\mathbb{R}})=0$ by the Hurewicz
Theorem (see for example [20]). Therefore, by the Universal Coefficient
Theorem (see for example [27]) $H^{1}(LG)=0,$ and so $H^{2}(LG\rtimes
S^{1})\simeq H^{2}(LG).$ Thus, we take as the 2-form $R,$ the pull-back of the
form from section 2.4 to $LG\rtimes S^{1}.$ That is,
$R=\frac{i}{4\pi}\int_{S^{1}}\langle\Theta,\partial\Theta\rangle\,d\theta.$
Note that since we are integrating over the circle, this expression is
unchanged when each term is rotated by a fixed angle. That is,
$\frac{i}{4\pi}\int_{S^{1}}\langle\rho_{\phi}(\Theta),\partial\rho_{\phi}(\Theta)\rangle\,d\theta=\frac{i}{4\pi}\int_{S^{1}}\langle\Theta,\partial\Theta\rangle\,d\theta$
Now, to find $\alpha$ we need to calculate $\delta
R=\pi_{1}^{*}R-m^{*}R+\pi_{2}^{*}R,$ where as before, $\pi_{i}$ is the
projection $LG\rtimes S^{1}\times LG\rtimes S^{1}\to LG\rtimes S^{1}$ which
omits the $i^{\text{th}}$ factor and $m$ is the multiplication defined above.
As in chapter 2, $\pi_{i}^{*}R$ is given by
$\frac{i}{4\pi}\int_{S^{1}}\langle\pi_{i}^{*}\Theta,\partial\pi_{i}^{*}\Theta\rangle\,d\theta$
and so it remains to calculate $m^{*}R.$ For this, note that a tangent vector
to $LG\rtimes S^{1}$ at the point $(\gamma,\phi)$ can be written as
$(\gamma,\phi)(\xi,x)=(\gamma\rho_{\phi}(\xi),x_{\phi})$ for some $(\xi,x)\in
L{\mathfrak{g}}\rtimes i{\mathbb{R}}$ by using the left multiplication to
transport elements of the Lie algebra to the point $(\gamma,\phi).$ Therefore,
we can calculate $m^{*}R$ by noting that
$m^{*}R((\gamma_{1}\rho_{\phi_{1}}(\xi_{1}),x_{1\phi_{1}}),(\gamma_{2}\rho_{\phi_{2}}(\xi_{2}),x_{2\phi_{2}}))=R(m_{*}((\gamma_{1}\rho_{\phi_{1}}(\xi_{1}),x_{1\phi_{1}}),(\gamma_{2}\rho_{\phi_{2}}(\xi_{2}),x_{2\phi_{2}})))$
and calculating the push-forward of $m.$ We have
$m_{*}((\gamma_{1}\rho_{\phi_{1}}(\xi_{1}),x_{1\phi_{1}}),(\gamma_{2}\rho_{\phi_{2}}(\xi_{2}),x_{2\phi_{2}}))\\\
=\frac{d}{dt}\bigg{|}_{0}\left(\gamma_{1}(1+t\xi_{1}^{\rho_{1}})\rho_{(\phi_{1}+tx_{1})}(\gamma_{2})\rho_{(\phi_{1}+tx_{1})}(1+t\xi_{2}^{\rho_{2}})),\phi_{1}+\phi_{2}+t(x_{1}+x_{2})\right),$
where we have written (for example) $\xi_{1}^{\rho_{1}}$ for
$\rho_{\phi_{1}}(\xi_{1})$. As the multiplication on the $S^{1}$ factor is not
twisted, the second slot above will give $x_{1}+x_{2}.$ Thus it suffices to
calculate the first slot only. Using the fact that
$\frac{d}{dt}\bigg{|}_{0}\rho_{(\phi_{1}+tx_{1})}(\gamma_{2})=-x_{1}\rho_{\phi_{1}}(\partial\gamma_{2}),$
we have
$\displaystyle\frac{d}{dt}\bigg{|}_{0}$
$\displaystyle\left(\gamma_{1}(1+t\xi_{1}^{\rho_{1}})\rho_{(\phi_{1}+tx_{1})}(\gamma_{2})\rho_{(\phi_{1}+tx_{1})}(1+t\xi_{2}^{\rho_{2}})\right)$
$\displaystyle=\gamma_{1}\xi_{1}^{\rho_{1}}\gamma_{2}^{\rho_{1}}+\gamma_{1}\gamma_{2}^{\rho_{1}}\xi_{2}^{\rho_{2}}-x_{1}\gamma_{1}\partial\gamma_{2}^{\rho_{1}}$
$\displaystyle=\gamma_{1}\gamma_{2}^{\rho_{1}}\rho_{(\phi_{1}+\phi_{2})}\left((\gamma_{2}^{-1}\xi_{1}\gamma_{2}^{\vphantom{-1}})^{\rho_{2}^{-1}}+\xi_{2}-x_{1}(\gamma_{2}^{-1}\partial\gamma_{2}^{\vphantom{-1}})^{\rho_{2}^{-1}}\right).$
Therefore, $m^{*}R$ evaluated on the pairs of tangent vectors
$\left((\gamma_{1},\phi_{1})(\xi_{1},x_{1}),(\gamma_{2},\phi_{2})(\xi_{2},x_{2})\right)$
and
$\left((\gamma_{1},\phi_{1})(\zeta_{1},y_{1}),(\gamma_{2},\phi_{2})(\zeta_{2},y_{2})\right)$
is given by
$\frac{i}{4\pi}\int_{S^{1}}\left\langle(ad(\gamma_{2}^{-1})\xi_{1})^{\rho_{2}^{-1}}+\xi_{2}-x_{1}(\gamma_{2}^{-1}\partial\gamma_{2}^{\vphantom{-1}})^{\rho_{2}^{-1}},\partial\left((ad(\gamma_{2}^{-1})\zeta_{1})^{\rho_{2}^{-1}}+\zeta_{2}-y_{1}(\gamma_{2}^{-1}\partial\gamma_{2}^{\vphantom{-1}})^{\rho_{2}^{-1}}\right)\right\rangle\,d\theta,$
where we have used the fact that the integral is unchanged by rotating
everything by $\rho_{(\phi_{1}+\phi_{2})}^{-1}.$ Expanding this, we have
$\frac{i}{4\pi}\int_{S^{1}}\left\langle(ad(\gamma_{2}^{-1})\xi_{1}),\partial(ad(\gamma_{2}^{-1})\zeta_{1})\right\rangle+\left\langle\xi_{2},\partial\zeta_{2}\right\rangle\\\
+x_{1}y_{1}\left\langle(ad(\gamma_{2}^{-1})Z_{2}),\partial(ad(\gamma_{2}^{-1})Z_{2})\right\rangle\\\
+\left\langle(ad(\gamma_{2}^{-1})\xi_{1}),\partial\zeta_{2}^{\rho_{2}}\right\rangle+\left\langle\xi_{2}^{\rho_{2}},\partial(ad(\gamma_{2}^{-1})\zeta_{1})\right\rangle\\\
-y_{1}\left\langle\xi_{2}^{\rho_{2}},\partial(ad(\gamma_{2}^{-1})Z_{2})\right\rangle-
x_{1}\left\langle(ad(\gamma_{2}^{-1})Z_{2}),\partial\zeta_{2}^{\rho_{2}}\right\rangle\\\
-y_{1}\left\langle(ad(\gamma_{2}^{-1})\xi_{1}),\partial(ad(\gamma_{2}^{-1})Z_{2})\right\rangle\\\
-x_{1}\left\langle(ad(\gamma_{2}^{-1})Z_{2}),\partial(ad(\gamma_{2}^{-1})\zeta_{2})\right\rangle
d\theta,$
where as before $Z$ is the function $\gamma\mapsto\partial\gamma\gamma^{-1}$
and, again, we have used the rotation invariance of the integral. Using the
$ad$-invariance of the Killing form and integration by parts, along with the
identity from section 2.4,
$\partial\left(ad(\gamma^{-1})X\right)=ad(\gamma^{-1})[X,Z]+ad(\gamma^{-1})\partial
X$
for a vector $X\in L{\mathfrak{g}},$ this simplifies to
$\frac{i}{4\pi}\int_{S^{1}}\left\langle[\xi_{1},\zeta_{1}],Z_{2}\right\rangle+\left\langle\xi_{1},\partial\zeta_{1}\right\rangle+\left\langle\xi_{2},\partial\zeta_{2}\right\rangle\\\
+\left\langle
ad(\gamma_{2}^{-1})\xi_{1},\partial\zeta_{2}^{\rho_{2}}\right\rangle-\left\langle\partial\xi_{2}^{\rho_{2}},ad(\gamma_{2}^{-1})\zeta_{1}\right\rangle-
x_{1}\left<Z_{2},\partial\zeta_{1}\right\rangle+y_{1}\left\langle\partial\xi_{1},Z_{2}\right\rangle\\\
-x_{1}\left\langle
ad(\gamma_{2}^{-1})Z_{2},\partial\zeta_{2}^{\rho_{2}}\right\rangle+y_{1}\left\langle\partial\xi_{2}^{\rho_{2}},ad(\gamma_{2}^{-1})Z_{2}\right\rangle
d\theta,$
or simply
$m^{*}R=\frac{i}{4\pi}\int_{S^{1}}\left\langle[\Theta_{1},\Theta_{1}],Z_{2}\right\rangle+\left\langle\Theta_{1},\partial\Theta_{1}\right\rangle+\left\langle\Theta_{2},\partial\Theta_{2}\right\rangle\\\
+2\left\langle
ad(\gamma_{2}^{-1})\Theta_{1},\partial\Theta_{2}^{\rho_{2}}\right\rangle-2\left\langle\mu_{1}ad(\gamma_{2}^{-1})Z_{2},\partial\Theta_{2}^{\rho_{2}}\right\rangle-2\left\langle\mu_{1}Z_{2},\partial\Theta_{1}\right\rangle
d\theta,$
where $\mu$ represents the Maurer-Cartan form on $S^{1}$. Therefore, we have
$\delta
R=\frac{i}{2\pi}\int_{S^{1}}-\tfrac{1}{2}\left\langle[\Theta_{1},\Theta_{1}],Z_{2}\right\rangle-\left\langle
ad(\gamma_{2}^{-1})\Theta_{1},\partial\Theta_{2}^{\rho_{2}}\right\rangle\\\
+\left\langle\mu_{1}ad(\gamma_{2}^{-1})Z_{2},\partial\Theta_{2}^{\rho_{2}}\right\rangle+\left\langle\mu_{1}Z_{2},\partial\Theta_{1}\right\rangle
d\theta.$
Recall from section 2.4 that for the loop group case, the form $\alpha$ such
that $d\alpha=\delta R$ is given by
$\alpha=\frac{i}{2\pi}\int_{S^{1}}\left\langle\Theta_{1},Z_{2}\right\rangle
d\theta.$
When evaluated on the vector $(\gamma_{1}\xi_{1},\gamma_{2}\xi_{2})$ tangent
to the point $(\gamma_{1},\gamma_{2})\in LG\times LG,$ $\alpha$ is given by
$\frac{i}{2\pi}\int_{S^{1}}\left\langle\xi_{1},\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1}\right\rangle
d\theta.$
Consider the generalisation of this form to $LG\rtimes S^{1}\times LG\rtimes
S^{1}.$ That is, define $\alpha_{1}$ as
$\alpha_{1}(\gamma_{1}\xi^{\rho_{1}},x_{1\phi_{1}},\gamma_{2}\xi^{\rho_{2}},x_{2\phi_{2}})=\frac{i}{2\pi}\int_{S^{1}}\left\langle\xi_{1},\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1}\right\rangle,$
or
$\alpha_{1}=\frac{i}{2\pi}\int_{S^{1}}\big{\langle}\Theta_{1}^{\rho_{1}^{-1}},Z_{2}\big{\rangle}\,d\theta.$
We can calculate the derivative of this form via
$d\alpha_{1}(X,Y)=\tfrac{1}{2}\left\\{X(\alpha_{1}(Y))-Y(\alpha_{1}(X))-\alpha_{1}([X,Y])\right\\},$
for tangent vectors $X$ and $Y.$ Thus we need to calculate
$\displaystyle(\gamma_{1}\xi_{1}^{\rho_{1}},$ $\displaystyle
x_{1\phi_{1}},\gamma_{2}\xi_{2}^{\rho_{2}},x_{2\phi_{2}})\left(\alpha_{1}(\gamma_{1}\zeta_{1}^{\rho_{1}},y_{1\phi_{1}},\gamma_{2}\zeta_{2}^{\rho_{2}},y_{2\phi_{2}})\right)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}\frac{i}{2\pi}\int_{S^{1}}\left\langle\zeta_{1},\partial(\gamma_{2}(1+t\xi_{2}^{\rho_{2}}))(1-t\xi_{2}^{\rho_{2}})\gamma_{2}^{-1}\right\rangle
d\theta$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\left\langle\zeta_{1},ad(\gamma_{2})\partial\xi_{2}^{\rho_{2}}\right\rangle
d\theta,$
and
$\displaystyle\alpha_{1}([(\gamma_{1}\xi_{1}^{\rho_{1}},x_{1\phi_{1}}),$
$\displaystyle(\gamma_{1}\zeta_{1}^{\rho_{1}},y_{1\phi_{1}})],[(\gamma_{2}\xi_{2}^{\rho_{2}},x_{2\phi_{2}}),(\gamma_{2}\zeta_{2}^{\rho_{2}},y_{2\phi_{2}})])$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\left\langle[(\xi_{1},x_{1}),(\zeta_{1},y_{1})],\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1}\right\rangle
d\theta$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\left\langle[\xi_{1},\zeta_{1}],\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1}\right\rangle-\left\langle
x_{1}\partial\zeta_{1}-y_{1}\partial\xi_{1},\partial\gamma_{2}^{\vphantom{-1}}\gamma_{2}^{-1}\right\rangle
d\theta.$
Therefore, we have
$d\alpha_{1}=\frac{1}{2\pi}\int_{S^{1}}-\tfrac{1}{2}\left\langle[\Theta,\Theta],Z_{2}^{\vphantom{-1}}\right\rangle-\left\langle
ad(\gamma_{2}^{-1})\Theta_{1},\partial\Theta_{2}^{\rho_{2}}\right\rangle+\left\langle\mu_{1}Z_{2},\partial\Theta_{1}^{\vphantom{-1}}\right\rangle
d\theta.$
Note that $\delta R$ does not equal $d\alpha_{1}.$ However,
$\delta
R-d\alpha_{1}=\frac{i}{2\pi}\int_{S^{1}}\left\langle\mu_{1}ad(\gamma_{2}^{-1})Z_{2},\partial\Theta_{2}^{\rho_{2}}\right\rangle
d\theta.$
Using the identity
$ad(\gamma)\partial\Theta^{\rho}=dZ,$
we see that
$\delta
R-d\alpha_{1}=\frac{i}{2\pi}\int_{S^{1}}\left\langle\mu_{1}Z_{2},dZ_{2}\right\rangle
d\theta.$
Now, if we define
$\alpha_{2}=-\frac{i}{4\pi}\int_{S^{1}}\left\langle\mu_{1}Z_{2},Z_{2}\right\rangle
d\theta,$
then
$\displaystyle d\alpha_{2}$
$\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\left\langle\mu_{1}dZ_{2},Z_{2}\right\rangle+\left\langle\mu_{1}Z_{2},dZ_{2}\right\rangle
d\theta$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\left\langle\mu_{1}Z_{2},dZ_{2}\right\rangle
d\theta$ $\displaystyle=\delta R-d\alpha_{1}.$
Thus $\alpha$ is given by
$\alpha=\frac{i}{2\pi}\int_{S^{1}}\big{\langle}\pi_{2}^{*}\Theta^{\rho^{-1}}\\!-\tfrac{1}{2}\pi_{2}^{*}\mu\,\pi_{1}^{*}Z,\pi_{1}^{*}Z\big{\rangle}\,d\theta,$
and $\delta R=d\alpha.$ One can also easily check that $\delta\alpha=0.$
Notice that the $2$-form $R$ is left invariant and the $1$-form $\alpha$ is
left invariant in the first slot. To find the homomorphism $h\colon
Z_{1}(LG\rtimes S^{1})\to U(1)$ we note that since $\pi_{1}(LG\rtimes
S^{1})={\mathbb{Z}}$ any cycle $a\in Z_{1}(LG\rtimes S^{1})$ can be written as
$n\gamma+\partial\sigma,$ for some two-cycle $\sigma,$ where $\gamma$ is the
generator of $H_{1}(LG\rtimes S^{1}),$ a loop around the $S^{1}$ factor. It is
easy to see that the integral of $\alpha$ over the generators of
$H_{1}(LG\rtimes S^{1}\times LG\rtimes S^{1})$ vanishes, that is,
$\int_{\gamma_{1}}\alpha=0=\int_{\gamma_{2}}\alpha$
for $\gamma_{1},\gamma_{2}$ loops around the first and second $S^{1}$ factors
respectively. This suggests that we define
$h(a)=h(\partial\sigma)=\exp\left(\int_{\sigma}R\right).$
This is well defined since if
$a=n\gamma+\partial\sigma=n\gamma+\partial\sigma^{\prime}$ then
$\partial(\sigma-\sigma^{\prime})=0$ and so
$\displaystyle\int_{\sigma-\sigma^{\prime}}R\in 2\pi i{\mathbb{Z}}$ (since $R$
is integral). Because the integral of $\alpha$ over the generators of
$H_{1}(LG\rtimes S^{1}\times LG\rtimes S^{1})$ vanishes, it is easy to check
that for any one-cycle $\gamma$ we have
$(\delta h)(\gamma)=\exp\left(\int_{\gamma}\alpha\right).$
Thus we have proven
###### Proposition 4.1.1.
The triple $(R,\alpha,h)$ as above determines a central extension of the semi-
direct product $LG\rtimes S^{1}.$
##### A connection for the lifting bundle gerbe
Now that we have a construction of the central extension of $LG\rtimes S^{1},$
the next step is to write down a bundle gerbe connection for the lifting
bundle gerbe. Recall from section 2.4 that if $P$ is an $LG\rtimes
S^{1}$-bundle and $\nu$ is a connection on the central extension
$\widehat{LG\rtimes S^{1}}$ thought of as a bundle over $LG\rtimes S^{1}$ then
a bundle gerbe connection is given by $\tau^{*}\nu-\epsilon,$ where $\epsilon$
is some $1$-form on $P^{[2]}$ satisfying $\delta\epsilon=\tau^{*}\alpha.$ In
the $LG$ case, this form was given by
$\epsilon=\frac{i}{2\pi}\int_{S^{1}}\langle\pi_{2}^{*}A,\tau^{*}Z\rangle\,d\theta,$
where $A$ is a connection on $P.$ As mentioned in section 2.4, it is possible
to write $\epsilon$ in general in terms of $\alpha$ [43]. We shall now
demonstrate how to do this. Let $P$ be a ${\mathcal{G}}$-bundle with
connection $A.$ Recall that $A$ satisfies
$\pi_{1}^{*}A=ad(\tau_{12}^{-1})\pi_{2}^{*}A+\tau_{12}^{*}\Theta.$
For tangent vectors $(X_{1},X_{2},X_{3})$ at $(p_{1},p_{2},p_{3})\in P^{[3]},$
we can calculate
$(\delta\alpha)_{(1,\tau_{12},\tau_{23})}(A(X_{1}),\tau_{12}(X_{1},X_{2}),\tau_{23}(X_{2},X_{3}))=\\\
\phantom{(\delta\alpha)}\alpha_{(\tau_{12},\tau_{23})}(\tau_{12}(X_{1},X_{2}),\tau_{23}(X_{2},X_{3}))\\\
-\alpha_{(\tau_{12},\tau_{23})}(m_{*}(A(X_{1}),\tau_{12}(X_{1},X_{2})),\tau_{23}(X_{2},X_{3}))\\\
+\alpha_{(1,\tau_{12}\tau_{23})}(A(X_{1}),m_{*}(\tau_{12}(X_{1},X_{2}),\tau_{23}(X_{2},X_{3})))\\\
-\alpha_{(1,\tau_{12})}(A(X_{1}),\tau_{12}(X_{1},X_{2})).$
Notice that the first term above is actually $\tau^{*}\alpha.$ Since
$\delta\alpha=0,$ we have
$(\tau^{*}\alpha)_{(p_{1},p_{2},p_{3})}(X_{1},X_{2},X_{3})=\\\
\alpha_{(\tau_{12},\tau_{23})}(m_{*}(A(X_{1}),\tau_{12}(X_{1},X_{2})),\tau_{23}(X_{2},X_{3}))\\\
-\alpha_{(1,\tau_{12}\tau_{23})}(A(X_{1}),m_{*}(\tau_{12}(X_{1},X_{2}),\tau_{23}(X_{2},X_{3})))\\\
+\alpha_{(1,\tau_{12})}(A(X_{1}),\tau_{12}(X_{1},X_{2})).$
Now, if we define $\epsilon$ in terms of $\alpha$ and $A$ as
$\epsilon_{(p_{1},p_{2})}(X_{1},X_{2})=\alpha_{(1,\tau_{12})}(A(X_{1}),\tau_{12}(X_{1},X_{2}))$
then we have
$(\delta\epsilon)_{(p_{1},p_{2},p_{3})}(X_{1},X_{2},X_{3})=\\\
\alpha_{(1,\tau_{23})}(A(X_{2}),\tau_{23}(X_{2},X_{3}))-\alpha_{(1,\tau_{13})}(A(X_{1}),\tau_{13}(X_{1},X_{3}))\\\
+\alpha_{(1,\tau_{12})}(A(X_{1}),\tau_{12}(X_{1},X_{2})).$
Using the fact that $\tau_{13}=\tau_{12}\tau_{23},$ we see
$\alpha_{(1,\tau_{13})}(A(X_{1}),\tau_{13}(X_{1},X_{2}))=\alpha_{(1,\tau_{12}\tau_{23})}(A(X_{1}),m_{*}(\tau_{12}(X_{1},X_{2}),\tau_{23}(X_{2},X_{3})))$
and since $\alpha$ is left invariant in the first slot, and using the equation
above relating $A(X_{1})$ and $A(X_{2}),$ we have
$\displaystyle\alpha_{(1,\tau_{23})}($ $\displaystyle
A(X_{2}),\tau_{23}(X_{2},X_{3}))$
$\displaystyle=\alpha_{(\tau_{12},\tau_{23})}(\tau_{12}A(X_{2}),\tau_{23}(X_{2},X_{3}))$
$\displaystyle=\alpha_{(\tau_{12},\tau_{23})}(\tau_{12}(ad(\tau_{12}^{-1})A(X_{1})+\tau_{12}^{-1}(\tau_{12}(X_{1},X_{2}))),\tau_{23}(X_{2},X_{3}))$
$\displaystyle=\alpha_{(\tau_{12},\tau_{23})}(\tau_{12}ad(\tau_{12}^{-1})A(X_{1})+\tau_{12}(X_{1},X_{2}),\tau_{23}(X_{2},X_{3})),$
which equals
$\alpha_{(\tau_{12},\tau_{23})}(m_{*}(A(X_{1}),\tau_{12}(X_{1},X_{2})),\tau_{23}(X_{2},X_{3})).$
Thus we have $\delta\epsilon=\tau^{*}\alpha.$
Consider now the $LG\rtimes S^{1}$-bundle $P.$ Choose a connection $(A,a)$ for
$P,$ where $A$ and $a$ are $1$-forms on $P$ with values in $L{\mathfrak{g}}$
and $i{\mathbb{R}}$ respectively. Note that $a$ is a connection for the
associated $S^{1}$-bundle $P/LG$ whereas $A$ is not a connection form. In
fact, if $X$ is a tangent vector to $P,$ we have
$\displaystyle(A,a)(X(\gamma,\phi))$
$\displaystyle=ad(\gamma,\phi)^{-1}(A(X),a(X))$
$\displaystyle=\left(\rho_{\phi^{-1}}\left(ad(\gamma^{-1})A(X)-a(X)\gamma^{-1}\partial\gamma\right),a(X)\right),$
and so $A$ does not have the correct transformation properties to be a
connection.222Notice the similarity with the treatment of connections for the
universal $\Omega G\rtimes G$-bundle in section 3.3. Given $(A,a)$ then, we
can write down the $1$-form $\epsilon\in\Omega^{1}(P^{[2]})$ as above:
$\epsilon=\frac{i}{2\pi}\int_{S^{1}}\left\langle\pi_{2}^{*}A-\tfrac{1}{2}\pi_{2}^{*}a\,\tau^{*}Z,\tau^{*}Z\right\rangle
d\theta.$
It is easy to check that $\delta\epsilon=\tau^{*}\alpha$ and so we have that
$\tau^{*}\nu-\epsilon$ is a connection for the lifting bundle gerbe. Of
course, as in section 2.4 we are concerned with finding a curving for this
bundle gerbe and so we are really interested in calculating the curvature of
this connection, given by $\tau^{*}R-d\epsilon.$ Recall that for a connection
$A$ on a ${\mathcal{G}}$-bundle, we have the formula
$\pi_{1}^{*}A=ad(\tau^{-1})\pi_{2}^{*}A+\tau^{*}\Theta.$
In the case where ${\mathcal{G}}=LG\rtimes S^{1},$ the formula relating
$\pi_{1}^{*}(A,a)=(A_{2},a_{2})$ and $\pi_{2}^{*}(A,a)=(A_{1},a_{1})$ is
$(A_{2},a_{2})=\left(\rho_{\tau_{S^{1}}}^{-1}\left(ad(\tau_{LG}^{-1})A_{1}-a_{1}\tau_{LG}^{-1}\partial\tau_{LG}^{\vphantom{-1}}\right)+\tau_{LG}^{*}(\rho_{\tau_{S^{1}}}^{-1}(\Theta)),a_{1}+\tau_{S^{1}}^{*}\mu\right)$
where we have written the difference map $\tau$ as $(\tau_{LG},\tau_{S^{1}}).$
That is, $\tau_{LG}$ is the $LG$ part of $\tau$ and $\tau_{S^{1}}$ is the
circle part. From now on, we will simply write $\tau$ and assume that it is
clear from the context which part we mean. In particular, then, we have
$\tau^{*}\rho_{\tau}^{-1}(\Theta)=A_{2}-\rho_{\tau}^{-1}\left(ad(\tau^{-1})A_{1}+a_{1}\tau^{-1}\partial\tau\right).$
Note that here we have used the fact that the Maurer-Cartan form on $LG\rtimes
S^{1}$ is not the pair $(\Theta,\mu)$ but in fact includes a rotation of
$\Theta.$ So at the point $(\gamma,\phi),$ it is given by
$(\rho_{\phi^{-1}}(\Theta),\mu).$
We can use this to calculate $\tau^{*}R-d\epsilon.$ Writing $A^{\rho}$ for
$\rho(A)$ and so on, as before, we have
$\tau^{*}R=\frac{i}{4\pi}\int_{S^{1}}\langle\tau^{*}\Theta,\partial\tau^{*}\Theta\rangle
d\theta\\\ \phantom{\tau^{*}R}=\frac{i}{4\pi}\int_{S^{1}}\langle
A_{2}^{\rho}-ad(\tau^{-1})A_{1}+a_{1}\tau^{-1}\partial\tau,\partial(A_{2}^{\rho}-ad(\tau^{-1})A_{1}+a_{1}\tau^{-1}\partial\tau)\rangle
d\theta\\\ \phantom{\tau^{*}R}=\frac{i}{4\pi}\int_{S^{1}}\langle
A_{2},\partial A_{2}\rangle-2\langle
A_{2}^{\rho},\partial(ad(\tau^{-1})A_{1})\rangle+2\langle
A_{2}^{\rho},a_{1}\partial(\tau^{-1}\partial\tau)\rangle\\\ +\langle
ad(\tau^{-1})A_{1},\partial(ad(\tau^{-1})A_{1})\rangle-2\langle
ad(\tau^{-1})A_{1},a_{1}\partial(\tau^{-1}\partial\tau)\rangle\\\ +\langle
a_{1}\tau^{-1}\partial\tau,a_{1}\partial(\tau^{-1}\partial\tau)\rangle
d\theta\\\ \phantom{\tau^{*}R}=\frac{i}{4\pi}\int_{S^{1}}\langle
A_{2},\partial A_{2}\rangle-2\langle
A_{2}^{\rho},\partial(ad(\tau^{-1})A_{1})\rangle+2\langle
A_{2}^{\rho},a_{1}\partial(\tau^{-1}\partial\tau)\rangle\\\ +\langle
ad(\tau^{-1})A_{1},\partial(ad(\tau^{-1})A_{1})\rangle-2\langle
ad(\tau^{-1})A_{1},a_{1}\partial(\tau^{-1}\partial\tau)\rangle d\theta.$
The last term vanishes since $a_{1}\wedge a_{1}=0.$ For $d\epsilon$ we have:
$d\epsilon=\frac{i}{2\pi}d\int_{S^{1}}\langle
A_{1}-\tfrac{1}{2}a_{1}\tau^{*}Z,\tau^{*}Z\rangle d\theta\\\
\phantom{d\epsilon}=\frac{i}{2\pi}\int_{S^{1}}\langle
dA_{1},\tau^{*}Z\rangle-\langle A_{1},d(\tau^{*}Z)\rangle-\tfrac{1}{2}\langle
da_{1}\tau^{*}Z,\tau^{*}Z\rangle+\langle a_{1}\tau^{*}Z,d(\tau^{*}Z)\rangle
d\theta\\\ $
and using the fact that
$d(\tau^{*}Z)=ad(\tau)\partial(\tau^{*}\Theta^{\rho}),$
$\phantom{d\epsilon}=\frac{i}{2\pi}\int_{S^{1}}\langle
dA_{1},\tau^{*}Z\rangle-\langle
A_{1},ad(\tau)\partial(\tau^{*}\Theta^{\rho})\rangle\\\ -\tfrac{1}{2}\langle
da_{1}\tau^{*}Z,\tau^{*}Z\rangle+\langle
a_{1}\tau^{*}Z,ad(\tau)\partial(\tau^{*}\Theta^{\rho})\rangle d\theta\\\
\phantom{d\epsilon}=\frac{i}{2\pi}\int_{S^{1}}\langle
dA_{1},\tau^{*}Z\rangle\\\ -\langle
A_{1},ad(\tau)\partial(A_{2}^{\rho}-ad(\tau^{-1})A_{1}+a_{1}\tau^{-1}\partial\tau)\rangle-\tfrac{1}{2}\langle
da_{1}\tau^{*}Z,\tau^{*}Z\rangle\\\ +\langle
a_{1}\tau^{*}Z,ad(\tau)\partial(A_{2}^{\rho}-ad(\tau^{-1})A_{1}+a_{1}\tau^{-1}\partial\tau)\rangle
d\theta\\\ \phantom{d\epsilon}=\frac{i}{2\pi}\int_{S^{1}}\langle
dA_{1},\tau^{*}Z\rangle-\langle A_{1},ad(\tau)\partial
A_{2}^{\rho}\rangle+\langle
A_{1},ad(\tau)\partial(ad(\tau^{-1})A_{1})\rangle\\\ -\langle
A_{1},a_{1}ad(\tau)\partial(\tau^{-1}\partial\tau)\rangle-\tfrac{1}{2}\langle
da_{1}\tau^{*}Z,\tau^{*}Z\rangle\\\ +\langle a_{1}\tau^{*}Z,ad(\tau)\partial
A_{2}^{\rho}\rangle-\langle
a_{1}\tau^{*}Z,ad(\tau)\partial(ad(\tau^{-1})A_{1})\rangle d\theta.$
Therefore,
$\tau^{*}R-d\epsilon=\frac{i}{4\pi}\int_{S^{1}}\langle A_{2},\partial
A_{2}\rangle-2\langle dA_{1},\tau^{*}Z\rangle-\langle
A_{1},ad(\tau)\partial(ad(\tau^{-1})A_{1})\rangle\\\ +2\langle
a_{1}\tau^{-1}\partial\tau,\partial(ad(\tau^{-1})A_{1})\rangle+\langle
da_{1}\tau^{*}Z,\tau^{*}Z\rangle d\theta,$
using the $ad$ invariance of the Killing form and integration by parts. Then,
using the identity from before,
$\partial(ad(\tau^{-1})A)=ad(\tau^{-1})[A,\tau^{*}Z]+ad(\tau^{-1})\partial A,$
yields
$\tau^{*}R-d\epsilon=\frac{i}{4\pi}\int_{S^{1}}\langle A_{2},\partial
A_{2}\rangle-2\langle dA_{1},\tau^{*}Z\rangle-\langle
A_{1},[A_{1},\tau^{*}Z]\rangle-\langle A_{1},\partial A_{1}\rangle\\\
+2\langle\tau^{*}Za_{1},[A_{1},\tau^{*}Z]\rangle+2\langle
a_{1}\tau^{*}Z,\partial A_{1}\rangle+\langle da_{1}\tau^{*}Z,\tau^{*}Z\rangle
d\theta\\\ \phantom{\tau^{*}R-d\epsilon}=\frac{i}{4\pi}\int_{S^{1}}\langle
A_{2},\partial A_{2}\rangle-\langle A_{1},\partial A_{1}\rangle-2\langle
dA_{1},\tau^{*}Z\rangle-\langle[A_{1},A_{1}],\tau^{*}Z\rangle\\\
+2\langle\tau^{*}Za_{1},\partial A_{1}\rangle+\langle
da_{1}\tau^{*}Z,\tau^{*}Z\rangle d\theta.$
Note now that if $(F,f)$ is the curvature of the connection $(A,a)$ then we
have
$\displaystyle(F,f)(X,Y)$
$\displaystyle=(dA(X,Y)+\tfrac{1}{2}[(A,a)(X),(A,a)(Y)],da(X,Y))$
$\displaystyle=(dA(X,Y)+\tfrac{1}{2}([A(X),A(Y)]-a(X)\partial
A(Y)+a(Y)\partial A(X)),da(X,Y)).$
That is,
$(F,f)=(dA+\tfrac{1}{2}[A,A]-a\wedge\partial A,da).$
Therefore, the formula above for $\tau^{*}R-d\epsilon$ reads
$\tau^{*}R-d\epsilon=\frac{i}{4\pi}\int_{S^{1}}\left\langle\pi_{1}^{*}A,\partial\pi_{1}^{*}A\right\rangle-\left\langle\pi_{2}^{*}A,\partial\pi_{2}^{*}A\right\rangle-2\left\langle\pi_{2}^{*}F-\tfrac{1}{2}\pi_{2}^{*}f\,\tau^{*}Z,\tau^{*}Z\right\rangle
d\theta.$
##### A curving for the lifting bundle gerbe
Recall that in order to find the 3-curvature of the lifting bundle gerbe, and
hence a representative for the image in real cohomology of the Dixmier-Douady
class, we need a curving for $\tau^{*}\widehat{LG\rtimes S^{1}}.$ That is,
some 2-form $B$ on $P$ such that $\delta B=\tau^{*}R-d\epsilon.$ Note that
$\delta=\pi_{1}^{*}-\pi_{2}^{*}$ and
$\tau^{*}R-d\epsilon=\delta\left(\frac{i}{4\pi}\int_{S^{1}}\left\langle
A,\partial A\right\rangle
d\theta\right)-\frac{i}{2\pi}\int_{S^{1}}\left\langle\pi_{2}^{*}F-\tfrac{1}{2}\pi_{2}^{*}f\,\tau^{*}Z,\tau^{*}Z\right\rangle
d\theta.$
To deal with the second term above, we use a similar method to the one in
section 2.4. Namely, we will need a Higgs field for the $LG\rtimes
S^{1}$-bundle $P.$
###### Definition 4.1.2.
A _Higgs field_ for $P$ is a map $\Phi\colon P\to L{\mathfrak{g}}$ satisfying
$\Phi(p(\gamma,\phi))=\rho_{\phi}^{-1}\left(ad(\gamma^{-1})\Phi(p)+\gamma^{-1}\partial\gamma\right).$
We shall explain the geometric significance of this map in section 4.2. As in
the $LG$ case, Higgs fields exist for $LG\rtimes S^{1}$-bundles. Note that the
condition above implies that a Higgs field $\Phi$ satisfies
$\pi_{1}^{*}\Phi=\rho_{\tau}^{-1}\left(ad(\tau^{-1})\pi_{2}^{*}\Phi+\tau^{-1}\partial\tau\right)$
or simply,
$ad(\tau)\Phi_{2}^{\rho}=\Phi_{1}+\tau^{*}Z.$
Using this, the second term in $\tau^{*}R-d\epsilon$ becomes
$\frac{i}{2\pi}\int_{S^{1}}\left\langle
F_{1}-\tfrac{1}{2}f_{1}\,\tau^{*}Z,ad(\tau)\Phi_{2}^{\rho}-\Phi_{1}\right\rangle
d\theta.$
Since $(F,f)$ is a curvature, it satisfies
$\pi_{1}^{*}(F,f)=ad(\tau^{-1})\pi_{2}^{*}(F,f).$
That is, $f_{2}=f_{1}$ and
$F_{2}=\rho_{\tau}^{-1}\left(ad(\tau^{-1})F_{1}-f_{1}\tau^{-1}\partial\tau\right),$
or
$ad(\tau)F_{2}^{\rho}=F_{1}-f_{1}\tau^{*}Z.$
Using this, we have
$\displaystyle\frac{i}{2\pi}\int_{S^{1}}$ $\displaystyle\left\langle
F_{1}-\tfrac{1}{2}f_{1}\,\tau^{*}Z,ad(\tau)\Phi_{2}^{\rho}-\Phi_{1}\right\rangle
d\theta$ $\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\left\langle
F_{1}+ad(\tau)F_{2}^{\rho},ad(\tau)\Phi_{2}^{\rho}-\Phi_{1}\right\rangle
d\theta$ $\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\left\langle
F_{1},ad(\tau)\Phi_{2}^{\rho}\right\rangle-\left\langle
F_{1},\Phi_{1}\right\rangle+\left\langle
F_{2},\Phi_{2}\right\rangle-\left\langle
ad(\tau)F_{2}^{\rho},\Phi_{1}\right\rangle d\theta$
$\displaystyle=\delta\left(\frac{i}{4\pi}\int_{S^{1}}\left<F,\Phi\right\rangle
d\theta\right)+\frac{i}{4\pi}\int_{S^{1}}\left\langle
F_{1},ad(\tau)\Phi_{2}^{\rho}\right\rangle-\left\langle
ad(\tau)F_{2}^{\rho},\Phi_{1}\right\rangle d\theta.$
Note, however, that the second integral above simplifies further
$\displaystyle\frac{i}{4\pi}\int_{S^{1}}$ $\displaystyle\left\langle
F_{1},ad(\tau)\Phi_{2}^{\rho}\right\rangle-\left\langle
ad(\tau)F_{2}^{\rho},\Phi_{1}\right\rangle d\theta$
$\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\left\langle
ad(\tau)F_{2}+f_{1}\tau^{*}Z,ad(\tau)\Phi_{2}^{\rho}\right\rangle-\left\langle
F_{1}-f_{1}\tau^{*}Z,\Phi_{1}\right\rangle d\theta$
$\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\langle F_{2},\Phi_{2}\rangle-\langle
F_{1},\Phi_{1}\rangle+\langle
f_{1}\tau^{*}Z,ad(\tau)\Phi_{2}^{\rho}+\Phi_{1}\rangle d\theta$
$\displaystyle=\delta\left(\frac{i}{4\pi}\int_{S^{1}}\left\langle
F,\Phi\right\rangle d\theta\right)+\frac{i}{4\pi}\int_{S^{1}}\left\langle
f_{1}\tau^{*}Z,2\Phi_{1}+\tau^{*}Z\right\rangle d\theta$
$\displaystyle=\delta\left(\frac{i}{4\pi}\int_{S^{1}}\left\langle
F,\Phi\right\rangle d\theta\right)+\frac{i}{4\pi}\int_{S^{1}}2\left\langle
f_{1}\tau^{*}Z,\Phi_{1}\right\rangle+\left\langle
f_{1}\tau^{*}Z,\tau^{*}Z\right\rangle d\theta.$
Therefore, $\tau^{*}R-d\epsilon$ is equal to
$\delta\left(\frac{i}{4\pi}\int_{S^{1}}\left\langle A,\partial
A\right\rangle-2\left\langle F,\Phi\right\rangle
d\theta\right)-\frac{i}{4\pi}\int_{S^{1}}2\left\langle
f_{1}\tau^{*}Z,\Phi_{1}\right\rangle+\left\langle
f_{1}\tau^{*}Z,\tau^{*}Z\right\rangle d\theta.$
So it is enough to find a $B_{2}\in\Omega^{2}(P)$ such that
$\delta B_{2}=\frac{i}{4\pi}\int_{S^{1}}2\left\langle
f_{1}\tau^{*}Z,\Phi_{1}\right\rangle+\left\langle
f_{1}\tau^{*}Z,\tau^{*}Z\right\rangle d\theta.$
Consider, then, the form
$\frac{i}{4\pi}\int_{S^{1}}\left\langle\Phi,f\Phi\right\rangle d\theta.$
We have
$\delta\left(\frac{i}{4\pi}\int_{S^{1}}\left\langle\Phi,f\Phi\right\rangle
d\theta\right)\\\
\phantom{\delta(\int_{S^{1}}\langle\Phi}=\frac{i}{4\pi}\int_{S^{1}}\left\langle\Phi_{2},f_{2}\Phi_{2}\right\rangle-\left\langle\Phi_{1},f_{1}\Phi_{1}\right\rangle
d\theta\\\
\phantom{\delta(\int_{S^{1}}\langle\Phi}=\frac{i}{4\pi}\int_{S^{1}}\left\langle
ad(\tau^{-1})(\Phi_{1}+\tau^{*}Z),f_{1}ad(\tau^{-1})(\Phi_{1}+\tau^{*}Z)\right\rangle-\left\langle\Phi_{1},f_{1}\Phi_{1}\right\rangle
d\theta\\\
\phantom{\delta(\int_{S^{1}}\langle\Phi}=\frac{i}{4\pi}\int_{S^{1}}\left\langle\Phi_{1},f_{1}\Phi_{1}\right\rangle+\left\langle\tau^{*}Z,f_{1}\Phi_{1}\right\rangle+\left\langle\Phi_{1},f_{1}\tau^{*}Z\right\rangle\\\
+\langle\tau^{*}Z,f_{1}\tau^{*}Z\rangle-\left\langle\Phi_{1},f_{1}\Phi_{1}\right\rangle
d\theta\\\
\phantom{\delta(\int_{S^{1}}\langle\Phi}=\frac{i}{4\pi}\int_{S^{1}}2\left\langle
f_{1}\tau^{*}Z,\Phi_{1}\right\rangle+\left\langle
f_{1}\tau^{*}Z,\tau^{*}Z\right\rangle d\theta.\\\ $
Therefore, a curving for the lifting bundle gerbe is given by
$B=\frac{i}{4\pi}\int_{S^{1}}\left\langle A,\partial A\right\rangle-2\langle
F+\tfrac{1}{2}f\Phi,\Phi\rangle\,d\theta.$
##### The string class of an $LG\rtimes S^{1}$-bundle
The last step now that we have found a curving for the lifting bundle gerbe is
to calculate the $3$-curvature $H=dB.$ Then $H/2\pi i$ is integral and
represents the real image of the Dixmier-Douady class of
$\tau^{*}\widehat{LG\rtimes S^{1}}$ (and hence the obstruction to lifting
$P$). We have
$\displaystyle dB$ $\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\left\langle
dA,\partial A\right\rangle-\left\langle A,d(\partial
A)\right\rangle-2\left\langle dF,\Phi\right\rangle-2\left\langle
F,d\Phi\right\rangle-\left\langle
d\Phi,f\Phi\right\rangle-\left\langle\Phi,fd\Phi\right\rangle d\theta$
$\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\left\langle dA,\partial
A\right\rangle+\left\langle\partial A,dA\right\rangle-2\left\langle
dF,\Phi\right\rangle-2\left\langle F,d\Phi\right\rangle-2\left\langle
d\Phi,f\Phi\right\rangle d\theta$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\left\langle dA,\partial
A\right\rangle-\left\langle dF,\Phi\right\rangle-\left\langle
F,d\Phi\right\rangle-\left\langle d\Phi,f\Phi\right\rangle d\theta.$
To proceed further, we require the Bianchi identity for $(F,f).$ Note that
$d(F,f)=([dA,A]-f\wedge\partial A+a\wedge\partial(dA),d^{2}a).$
In particular, this means that
$dF=[F,A]-f\wedge\partial A+a\wedge\partial F,$
since
$\displaystyle[F,A]$
$\displaystyle=[dA,A]+\tfrac{1}{2}[[A,A],A]-[a\wedge\partial A,A]$
$\displaystyle=[dA,A]-[a\wedge\partial A,A],$
and
$\displaystyle a\wedge\partial F$
$\displaystyle=a\wedge\partial(dA)+\tfrac{1}{2}a\wedge\partial[A,A]-a\wedge\partial(a\wedge\partial
A)$ $\displaystyle=a\wedge\partial(dA)+[a\wedge\partial A,A].$
Using this, and the fact that $\int_{S^{1}}\langle[A,A],\partial A\rangle
d\theta$ and $\langle a\wedge\partial A,\partial A\rangle$ both vanish (so
that $\int_{S^{1}}\langle dA,\partial A\rangle d\theta=\int_{S^{1}}\langle
F,\partial A\rangle d\theta$), the expression for $dB$ becomes
$dB=\frac{i}{2\pi}\int_{S^{1}}\left\langle F,\partial
A\right\rangle-\left\langle[F,A]-f\wedge\partial A+a\wedge\partial
F,\Phi\right\rangle-\left\langle F,d\Phi\right\rangle-\left\langle
d\Phi,f\Phi\right\rangle d\theta\\\
\phantom{dB}=\frac{i}{2\pi}\int_{S^{1}}\left\langle F,\partial
A\right\rangle-\left\langle[F,A],\Phi\right\rangle+\left\langle
f\wedge\partial A,\Phi\right\rangle-\left\langle a\wedge\partial
F,\Phi\right\rangle-\left\langle F,d\Phi\right\rangle-\left\langle
d\Phi,f\Phi\right\rangle d\theta\\\
\phantom{dB}=\frac{i}{2\pi}\int_{S^{1}}\left\langle F+f\Phi,\partial
A\right\rangle-\left\langle F,[A,\Phi]\right\rangle-\left\langle
a\wedge\partial F,\Phi\right\rangle-\left\langle F+f\Phi,d\Phi\right\rangle
d\theta\\\ \phantom{dB}=\frac{i}{2\pi}\int_{S^{1}}\left\langle
F+f\Phi,\partial A\right\rangle-\left\langle
F,[A,\Phi]\right\rangle+\left\langle F,a\partial\Phi\right\rangle-\left\langle
F+f\Phi,d\Phi\right\rangle d\theta\\\
\phantom{dB}=\frac{i}{2\pi}\int_{S^{1}}\left\langle F+f\Phi,\partial
A-[A,\Phi]+a\partial\Phi-d\Phi\right\rangle d\theta.\\\ $
Where the last line follows from the fact that $\int_{S^{1}}\langle
f\Phi,a\partial\Phi\rangle d\theta$ and $\langle f\Phi,[A,\Phi]\rangle$ both
vanish. If we define the covariant derivative of $\Phi$ by
$\nabla\Phi=d\Phi+[A,\Phi]-\partial A-a\partial\Phi,$
then one can easily check that it is (twisted) equivariant for the adjoint
action. That is,
$\nabla\Phi(X(\gamma,\phi))=\rho_{\phi}^{-1}\left(ad(\gamma^{-1})\nabla\Phi(X)\right),$
for any tangent vector $X.$ The same is true for the quantity $F+f\Phi,$ and
so using the $ad$-invariance of the Killing form and the rotation invariance
of the integral, Lemma 3.2.3 implies that $H=dB$ descends to a form on $M.$
Thus we have proven
###### Theorem 4.1.3.
Let $P\to M$ be a principal $LG\rtimes S^{1}$-bundle and let $\Phi$ be a Higgs
field for $P$ and $(A,a)$ be a connection for $P$ with curvature $(F,f).$ Then
the string class of $P,$ that is, the obstruction to lifting $P$ to an
$\widehat{LG\rtimes S^{1}}$-bundle, is represented in de Rham cohomology by
$-\frac{1}{4\pi^{2}}\int_{S^{1}}\langle F+f\Phi,\nabla\Phi\rangle d\theta,$
where
$\nabla\Phi=d\Phi+[A,\Phi]-\partial A-a\partial\Phi.$
#### 4.1.2 Reduced splittings for lifting bundle gerbes
In this section we shall present an alternative method for finding the curving
of a lifting bundle gerbe and show how to apply this to the problem above.
This method uses _reduced splittings_ and was first introduced by Gomi [18].
In [4] Brylinski considers the problem of lifting a principal
${\mathcal{G}}$-bundle $P$ to a $\widehat{{\mathcal{G}}}$-bundle
$\widehat{P},$ for which he uses a _bundle splitting_. He relates the
obstruction class to the _scalar curvature_ of a certain connection on
$\widehat{P}.$ In [18] Gomi phrases this in such a way that he can use the
theory of lifting bundle gerbes in order to calculate the obstruction class.
We shall begin by briefly outlining Brylinski’s results before describing the
reduced splittings of Gomi.
Let ${\mathcal{G}}$ be a Lie group with central extension
$\widehat{{\mathcal{G}}}.$ If $\mathfrak{G}$ and $\widehat{\mathfrak{G}}$ are
the Lie algebras of ${\mathcal{G}}$ and $\widehat{{\mathcal{G}}}$ respectively
then we have an extension of Lie algebras
$0\to i{\mathbb{R}}\to\widehat{\mathfrak{G}}\to\mathfrak{G}\to 0.$
We can define an action of ${\mathcal{G}}$ on $\widehat{\mathfrak{G}}$ by
lifting the adjoint action of ${\mathcal{G}}$ on its Lie algebra. That is, we
define
$ad\colon{\mathcal{G}}\times\widehat{\mathfrak{G}}\to\widehat{\mathfrak{G}}$
by
$ad(g)\hat{\xi}=ad(\hat{g})\hat{\xi},$
where $\hat{\xi}\in\widehat{\mathfrak{G}}$ and
$\hat{g}\in\widehat{{\mathcal{G}}}$ projects to $g\in{\mathcal{G}}.$ This is
well-defined since $U(1)$ acts trivially on $\widehat{\mathfrak{G}}$ and any
two lifts of $g$ differ by an element of $U(1).$ Consider now a principal
${\mathcal{G}}$-bundle $P.$ We can write down an exact sequence of vector
bundles associated to $P$ as follows. Let
$\operatorname{Ad}_{{\mathfrak{g}}}(P)$ denote the adjoint bundle of $P$ where
${\mathcal{G}}$ acts on the Lie algebra ${\mathfrak{g}}.$ For example,
$\operatorname{Ad}_{\mathfrak{G}}(P)$ is the usual adjoint bundle of $P$ and
$\operatorname{Ad}_{i{\mathbb{R}}}(P)=P\times_{ad}i{\mathbb{R}}\simeq M\times
i{\mathbb{R}}.$ Since ${\mathcal{G}}$ acts via the adjoint action on the exact
sequence above, we have an exact sequence of vector bundles
$0\to\operatorname{Ad}_{i{\mathbb{R}}}(P)\to\operatorname{Ad}_{\mathfrak{G}}(P)\to\operatorname{Ad}_{\widehat{\mathfrak{G}}}(P)\to
0.$
This means that $\operatorname{Ad}_{\mathfrak{G}}(P)$ is isomorphic to the
direct sum of $M\times i{\mathbb{R}}$ and
$\operatorname{Ad}_{\widehat{\mathfrak{G}}}(P).$ A choice of isomorphism is
called a _bundle splitting_. That is,
###### Definition 4.1.4 ([4]).
A _bundle splitting_ of $P$ is a vector bundle map
$L\colon\operatorname{Ad}_{\widehat{\mathfrak{G}}}(P)\to\operatorname{Ad}_{i{\mathbb{R}}}(P)$
which is the identity on the (trivial) subbundle
$\operatorname{Ad}_{i{\mathbb{R}}}(P).$
As mentioned above, Brylinski uses the notion of scalar curvature to calculate
the obstruction to the existence of a lift of $P.$ This is essentially the
$i{\mathbb{R}}$ part of the curvature of a connection on $\widehat{P}.$ More
precisely,
###### Definition 4.1.5 ([4]).
Let $\hat{A}$ be a connection on $\widehat{P}$ with curvature $\hat{F},$
viewed as a $2$-form on $M$ with values in
$\operatorname{Ad}_{\widehat{\mathfrak{G}}}(\widehat{P})\simeq\operatorname{Ad}_{\widehat{\mathfrak{G}}}(P).$
Let $L$ be a bundle splitting of $P.$ The _scalar curvature_ of $\hat{A}$ is
the $i{\mathbb{R}}$-valued $2$-form
$K=L\circ\hat{F}.$
To see how this is related to the obstruction class, let $\\{U_{\alpha}\\}$ be
a good cover of $M$ over which $P$ is trivial. Then there exists a lift
$\widehat{P}_{\alpha}$ of $P|_{U_{\alpha}}\to U_{\alpha}.$ Choose a connection
$A_{\alpha}$ on $P|_{U_{\alpha}}$ and let $K_{\alpha}$ be the scalar curvature
of a connection $\hat{A}_{\alpha}$ on $\widehat{P}_{\alpha}$ which is
compatible with $A_{\alpha}$ in the sense that the pull-back of $A_{\alpha}$
to $\widehat{P}_{\alpha}$ coincides with the image of $\hat{A}_{\alpha}$ in
$\mathfrak{G}.$ That is,
$f^{*}A_{\alpha}=p(\hat{A}_{\alpha}),$
where $f$ is the bundle map $\widehat{P}_{\alpha}\to P|_{U_{\alpha}}$ and $p$
is the projection $\widehat{{\mathcal{G}}}\to{\mathcal{G}}.$ Brylinski’s
result, then, is that the (real image of the) obstruction class restricted to
$U_{\alpha}$ coincides with the derivative of the scalar curvature,
$dK_{\alpha}.$
As mentioned, Gomi’s results interpolate between the method described above
and the theory of lifting bundle gerbes which we have used extensively. He
utilises so-called reduced splittings to write down a formula for the curving
of the lifting bundle gerbe associated to a lifting problem and relates the
curving to the scalar curvature. In the case where a splitting of the Lie
algebra of $\widehat{\mathfrak{G}}$ has been specified, reduced splittings are
equivalent to bundle splittings. To describe Gomi’s results, let us assume we
have chosen a splitting of the Lie algebra $\widehat{\mathfrak{G}}$ as
$\mathfrak{G}\oplus i{\mathbb{R}}.$
###### Definition 4.1.6 ([18]).
The _group cocycle_ for the central extension $\widehat{{\mathcal{G}}}$ is the
map $\sigma\colon{\mathcal{G}}\times\mathfrak{G}\to i{\mathbb{R}}$ defined by
$\sigma(g,\xi)=ad(g)(\xi,0)-(ad(g)\xi,0),$
where $ad(g)$ acts on $\widehat{\mathfrak{G}}$ as described above.
The group cocycle gives information about the multiplication in
$\widehat{{\mathcal{G}}}$ in the same way as the $1$-form $\alpha$ which we
used. In fact, as we shall see, to apply Gomi’s results to the case where
${\mathcal{G}}$ is either the loop group $LG$ or the semi-direct product
$LG\rtimes S^{1},$ we shall give $\sigma$ in terms of $\alpha.$
###### Definition 4.1.7 ([18]).
A _reduced splitting_ for a principal ${\mathcal{G}}$-bundle $P$ is a map
$\ell\colon P\times\mathfrak{G}\to i{\mathbb{R}}$ which is linear in the
second factor and satisfies
$\ell(p,\xi)=\ell(pg,ad(g^{-1})\xi)+\sigma(g^{-1},\xi).$
The relation between reduced splittings and bundle gerbe curvings is given by
the following theorem.
###### Theorem 4.1.8 ([18]).
Let $F$ be the curvature of a connection $A$ on $P$ and $\ell$ be a reduced
splitting for $P.$ Define a $2$-form $\kappa$ on $P$ by
$\kappa_{p}=\ell(p,F).$ Then a curving for the lifting bundle gerbe associated
to the lifting problem for $P$ is given by
$B=\frac{1}{2}\omega(A,A)+\kappa,$
where
$\omega(\xi,\zeta)=[(\xi,0),(\zeta,0)]_{\widehat{\mathfrak{G}}}-([\xi,\zeta]_{\mathfrak{G}},0)$
is the cocycle classifying the central extension.
To connect this with Brylinski’s work, Gomi proves the following theorem
relating the curving and the scalar curvature.
###### Theorem 4.1.9 ([18]).
Let $P$ be a principal ${\mathcal{G}}$-bundle and $\widehat{P}$ be a lift of
$P.$ Let $A$ be a connection on $P$ and $\hat{A}$ be a compatible connection
on $\widehat{P}.$ Then the curving can be written as
$B=\pi^{*}K-\tilde{F},$
where $\tilde{F}$ is the curvature of the connection $\hat{A}-f^{*}A$ on
$\widehat{P}$ (for $f\colon\widehat{P}\to P$ the bundle map defining the lift)
and $K$ is the scalar curvature of $\hat{A}.$
We would now like to consider the case where ${\mathcal{G}}=LG\rtimes S^{1}.$
We shall define a reduced splitting for $P$ so we can use Theorem 4.1.8 to
calculate a curving and show that it is in agreement with the results from
section 4.1.1. The group cocycle in this case is given by
$\displaystyle\sigma((\gamma,\phi)^{-1},(\xi,x))$
$\displaystyle=\alpha_{((1,1),(\gamma,\phi))}((\xi,x),(0,0))$
$\displaystyle=\frac{i}{2\pi}\int_{S^{1}}\left\langle\xi-\tfrac{1}{2}x\partial\gamma\gamma^{-1},\partial\gamma\gamma^{-1}\right\rangle
d\theta,$
and we have
###### Proposition 4.1.10.
A reduced splitting for the $LG\rtimes S^{1}$-bundle $P$ is given by
$\ell(p,(\xi,x))=-\frac{i}{2\pi}\int_{S^{1}}\left\langle\xi+\tfrac{1}{2}x\,\Phi(p),\Phi(p)\right\rangle
d\theta,$
where $\Phi$ is a Higgs field for $P.$
###### Proof.
We need only show that it satisfies the transformation property above. We can
calculate
$\ell(p(\gamma,\phi),ad(\gamma,\phi)^{-1}(\xi,x))\\\
\phantom{\ell(p}=-\frac{i}{2\pi}\int_{S^{1}}\left\langle ad(\gamma^{-1})(\xi-
xZ)+\tfrac{1}{2}x\,ad(\gamma^{-1})(\Phi(p)+Z),ad(\gamma^{-1})(\Phi(p)+Z)\right\rangle
d\theta\\\
\phantom{\ell(p}=-\frac{i}{2\pi}\int_{S^{1}}\left\langle\xi,\Phi(p)\vphantom{\tfrac{1}{2}}\right\rangle-\left\langle
xZ,\Phi(p)\vphantom{\tfrac{1}{2}}\right\rangle+\left\langle\tfrac{1}{2}x\,\Phi(p),\Phi(p)\right\rangle+\left\langle\tfrac{1}{2}xZ,\Phi(p)\right\rangle\\\
+\left\langle\xi,Z\vphantom{\tfrac{1}{2}}\right\rangle-\left\langle
xZ,Z\vphantom{\tfrac{1}{2}}\right\rangle+\left\langle\tfrac{1}{2}x\,\Phi(p),Z\right\rangle+\left\langle\tfrac{1}{2}x\,Z,Z\right\rangle
d\theta\\\
\phantom{\ell(p}=-\frac{i}{2\pi}\int_{S^{1}}\left\langle\xi+\tfrac{1}{2}x\,\Phi(p),\Phi(p)\right\rangle+\left\langle
X-\tfrac{1}{2}xZ,Z\right\rangle d\theta\\\
\phantom{\ell(p}=\ell(p,(\xi,x))-\sigma((\gamma,\phi)^{-1},(\xi,x))\\\ $
as required. ∎
Note that in order to use Theorem 4.1.8, we need the cocycle $\omega.$ This is
simply given by the form $R$ which defines the central extension. In
particular,
$\omega((\xi,x),(\zeta,y))=\frac{i}{2\pi}\int_{S^{1}}\left\langle\xi,\partial\zeta\right\rangle
d\theta.$
Therefore, for the curving of the lifting bundle gerbe, Theorem 4.1.8 gives
$\displaystyle B$
$\displaystyle=\frac{1}{2}R((A,a),(A,a))-\frac{i}{2\pi}\int_{S^{1}}\left\langle
F+\tfrac{1}{2}f\,\Phi,\Phi\right\rangle d\theta$
$\displaystyle=\frac{i}{4\pi}\int_{S^{1}}\left\langle A,\partial
A\right\rangle-2\langle F+\tfrac{1}{2}f\,\Phi,\Phi\rangle\,d\theta,$
where as before, $(A,a)$ is a connection on $P$ and $(F,f)$ is its curvature.
### 4.2 Higgs fields, $LG\rtimes S^{1}$-bundles and the string class
Now that we have an explicit formula for the string class of an $LG\rtimes
S^{1}$-bundle $P,$ it is natural to ask whether there is some relation with
the Pontrjagyn class of a $G$-bundle related to $P$ in some way, as was the
case with the string class of an $LG$-bundle presented in chapter 2. In
particular, in section 2.5, following [35], we saw that there was a
correspondence between $LG$-bundles over $M$ (with connection and Higgs field)
and $G$-bundles over $M\times S^{1}$ (with connection) (Propositions 2.5.1 and
2.5.2) and we used this to prove that the string class of $P$ is given by
integrating over the circle the first Pontrjagyn class of the corresponding
$G$-bundle (Theorem 2.5.3). In this section, we shall show there is a
correspondence between $LG\rtimes S^{1}$-bundles over $M$ and $G$-bundles over
$S^{1}$-bundles over $M,$ which holds on the level of connections as well. As
in section 2.5 we shall use this correspondence to prove that the string class
of $P$ is given in terms of the Pontrjagyn class of some $G$-bundle.
#### 4.2.1 Higgs fields and $LG\rtimes S^{1}$-bundles
The following correspondence first appeared in [1]. We will present it here in
detail and also extend it to the level of connections.
Suppose that we have a principal $G$-bundle over a principal $S^{1}$-bundle:
$\textstyle{\widetilde{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{1}}$$\textstyle{M}$
We would like to mimic the construction of the $LG$-bundle in section 2.5
where we essentially took loops in $\widetilde{P}$ such that their image in
$M\times S^{1}$ commuted with the obvious $S^{1}$ action on this space. That
is, for a loop $f\in L\widetilde{P}_{m}$ in the fibre above $\\{m\\}\times
S^{1}$ we required that $\tilde{\pi}(f(\theta))=(m,\theta).$ The difference
here is that we cannot choose a global section $M\to Y$ and thus there is no
way of choosing a ‘starting point’ for the loop $\tilde{\pi}(f)\colon S^{1}\to
Y.$ We can, however, still require that the map $\tilde{\pi}(f)$ commutes with
the $S^{1}$ action on $Y$ (which we will write as addition). That is, we can
define
$P=\\{f\colon
S^{1}\to\widetilde{P}\mid\tilde{\pi}(f(\theta+\phi))=\tilde{\pi}(f(\theta))+\phi\\}$
and there is a canonical map $P\to M.$ $P$ is acted on by $LG\rtimes S^{1}:$
$(f(\gamma,\phi))(\theta)=f(\theta+\phi)\gamma(\theta+\phi),$
i.e.
$f(\gamma,\phi)=\rho_{\phi}^{-1}(f\gamma).$
It is a right action since
$\displaystyle f(\gamma_{1},\phi_{1})(\gamma_{2},\phi_{2})$
$\displaystyle=\rho_{(\phi_{1}+\phi_{2})}^{-1}f\rho_{(\phi_{1}+\phi_{2})}^{-1}\gamma_{1}\rho_{\phi_{2}}^{-1}(\gamma_{2})$
$\displaystyle=\rho_{(\phi_{1}+\phi_{2})}^{-1}(f\gamma_{1}\rho_{\phi_{1}}(\gamma_{2}))$
$\displaystyle=f(\gamma_{1}\rho_{\phi_{1}}(\gamma_{2}),\phi_{1}+\phi_{2}).$
It preserves the fibres of $P$ since the $G$ action on $\widetilde{P}$
preserves fibres and the $S^{1}$ action on $Y$ preserves fibres. It is also
free and transitive on fibres and therefore $P\to M$ is a principal $LG\rtimes
S^{1}$-bundle. Note that local triviality of this bundle follows from the
local triviality of $Y$ as follows: Choose a good cover of $M$ and let $U$ be
an open set such that we can find a local section $s\colon U\to Y_{|_{U}}.$
There is a map $P\to Y$ given by $f\mapsto\tilde{\pi}(f(0)).$ If we pull-back
$P$ by $s$ then $s^{*}P\to U$ is trivial (since $U$ is contractible).
Conversely, suppose we are given a principal $LG\rtimes S^{1}$-bundle $P\to
M.$ Following the construction in section 2.5, define
$\widetilde{P}=(P\times G\times S^{1})/LG\rtimes S^{1},$
where $[p,g,\theta]=[p(\gamma,\phi),\gamma(\theta)^{-1}g,\theta-\phi].$ A $G$
action on $\widetilde{P}$ is given by $[p,g,\theta]h=[p,gh,\theta].$ There is
a natural projection from $\widetilde{P}$ to the $S^{1}$-bundle associated to
$P$ via the homomorphism $LG\rtimes S^{1}\to S^{1},$ that is,
$\widetilde{P}\to(P\times S^{1})/LG\rtimes S^{1}\simeq P/LG,$ given by
$\tilde{\pi}([p,g,\theta])=[p,\theta].$ This makes $\widetilde{P}$ into a
principal $G$-bundle. Thus, given the $LG\rtimes S^{1}$-bundle $P\to M$ we can
construct a $G$-bundle over an $S^{1}$-bundle:
$\textstyle{\dfrac{P\times G\times S^{1}}{LG\rtimes
S^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{\dfrac{P\times
S^{1}}{LG\rtimes
S^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{1}}$$\textstyle{M}$
We would like to show that both constructions above are invertible (as we did
for the constructions in the $LG$ case). Assume, then, that we are given an
$LG\rtimes S^{1}$-bundle $P\to M$ and have constructed the $G$-bundle
$\widetilde{P}$ over the $S^{1}$-bundle $P/LG\to M$ as above. Then use the
first correspondence above to form the $LG\rtimes S^{1}$-bundle $P^{\prime}\to
M$ (by taking certain loops in $\widetilde{P}$). So we have
$P^{\prime}=\\{f\colon S^{1}\to(P\times G\times S^{1})/LG\rtimes
S^{1}\mid\tilde{\pi}(f(\theta+\phi))=\tilde{\pi}(f(\theta))+\phi\\}$
and a bundle isomorphism is given by
$P\to P^{\prime};\quad p\mapsto f_{p}=(\theta\mapsto[p,1,\theta]).$
It is easily checked that this map commutes with the $LG\rtimes S^{1}$ action,
for
$\displaystyle p(\gamma,\phi)\mapsto$ $\displaystyle~{}f_{p(\gamma,\phi)}$
$\displaystyle=$ $\displaystyle~{}[p(\gamma,\phi),1,\theta]$ and on the other
hand, $\displaystyle f_{p}(\gamma,\phi)=$
$\displaystyle~{}f_{p}(\theta+\phi)\gamma(\theta+\phi)$ $\displaystyle=$
$\displaystyle~{}[p,1,\theta+\phi]\gamma(\theta+\phi)$ $\displaystyle=$
$\displaystyle~{}[p,\gamma(\theta+\phi),\theta+\phi]$ $\displaystyle=$
$\displaystyle~{}[p(\gamma,\phi),1,\theta].$
So we have that $P\simeq P^{\prime}.$ If, on the other hand, we are given the
$G$-bundle over the $S^{1}$-bundle $\widetilde{P}\to Y\to M$ and have
constructed $P\to M,$ then we can construct $\widetilde{P}^{\prime}\to P/LG\to
M$ and we would like for these bundles to be isomorphic. That is,
$\widetilde{P}^{\prime}\simeq\widetilde{P}$ and $P/LG\simeq Y.$ Firstly,
consider the map $P/LG\simeq P\times_{LG\rtimes S^{1}}S^{1}\to Y$ defined by
$[f,\theta]\mapsto\tilde{\pi}(f(\theta)).$ This is well-defined on equivalence
classes:
$[f,\theta]=[\rho_{\phi}^{-1}(f\gamma),\theta-\phi]\mapsto\tilde{\pi}(f(\theta-\phi+\phi)\gamma(\theta-\phi+\phi))=\tilde{\pi}(f(\theta)).$
It commutes with the $S^{1}$ action on $P\times_{LG\rtimes S^{1}}S^{1}:$
$[f,\theta+\alpha]\mapsto\tilde{\pi}(f(\theta+\alpha))=\tilde{\pi}(f(\theta))+\alpha$
by the definition of $P$ in terms of $\widetilde{P}.$ Thus $P\times_{LG\rtimes
S^{1}}S^{1}\simeq Y.$ For $\widetilde{P}^{\prime}$ and $\widetilde{P}$
consider the bundle map
$\widetilde{P}^{\prime}\to\widetilde{P};\quad[f,g,\theta]\mapsto f(\theta)g.$
This is well-defined:
$[f,g,\theta]=[f(\gamma,\phi),\gamma(\theta)^{-1}g,\theta-\phi]\mapsto
f(\theta-\phi-\phi)\gamma(\theta-\phi+\phi)\gamma(\theta)^{-1}g=f(\theta)g,$
and commutes with the $G$ action:
$[f,g,\theta]h=[f,gh,\theta]\mapsto f(\theta)gh=(f(\theta)g)h.$
Therefore, it is a bundle isomorphism and
$\widetilde{P}^{\prime}\simeq\widetilde{P}.$ Thus we have proven
###### Proposition 4.2.1 ([1]).
There is a bijective correspondence between isomorphism classes of principal
$LG\rtimes S^{1}$-bundles over $M$ and isomorphism classes of principal
$G$-bundles over principal $S^{1}$-bundles over $M.$
As in section 2.5 the correspondences here hold on the level of connections as
well. We shall now describe how to derive the connections corresponding to one
another.
Suppose we are given a connection $\tilde{A}$ on $\widetilde{P}\to Y$ and a
connection $\tilde{a}$ on $Y\to M.$ This amounts to a splitting of the tangent
spaces $T_{\tilde{p}}\widetilde{P}\simeq V_{\tilde{p}}\widetilde{P}\oplus
H_{\tilde{p}}\widetilde{P}$ at each point $\tilde{p}\in\widetilde{P}$ and also
$T_{y}Y\simeq V_{y}Y\oplus H_{y}Y$ at each point $y\in Y.$ Since $P$ is given
by certain loops in $\widetilde{P},$ a vector $X\in T_{f}P$ is really a vector
field along $f$ in $\widetilde{P}.$ So, $X_{\theta}\in
T_{f(\theta)}\widetilde{P}.$ Thus we can use the splittings of the tangent
spaces of $\widetilde{P}$ and $Y$ to define a splitting for the tangent space
to $P$ at $f$ for each $\theta.$ So we have
$\displaystyle T_{f(\theta)}\widetilde{P}$ $\displaystyle\simeq
V_{f(\theta)}\widetilde{P}\oplus H_{f(\theta)}\widetilde{P}$
$\displaystyle\simeq V_{f(\theta)}\widetilde{P}\oplus
V_{\tilde{\pi}(f(\theta))}Y\oplus H_{\tilde{\pi}(f(\theta))}Y$
$\displaystyle\simeq V_{f(\theta)}\widetilde{P}\oplus
V_{\tilde{\pi}(f(\theta))}Y\oplus T_{(\pi_{Y}\circ\tilde{\pi})(f(\theta))}M,$
using the isomorphisms $H_{f(\theta)}\widetilde{P}\simeq
T_{\tilde{\pi}(f(\theta))}Y$ and $H_{\tilde{\pi}(f(\theta))}Y\simeq
T_{(\pi_{Y}\circ\tilde{\pi})(f(\theta))}M.$ We can find the 1-form for this
connection by calculating
$X_{\theta}-\widehat{\widehat{\pi_{*}X_{\theta}}}$
which equals $\iota_{f(\theta)}(A_{f}(X)_{\theta}),$ where
$\pi=\pi_{Y}\circ\tilde{\pi}$ and $\widehat{\widehat{V}}$ is the horizontal
lift of a vector on $M$ first to $Y,$ then to $\widetilde{P}.$ Note that using
the connection on $Y$ we have
$\iota_{\tilde{\pi}(f(\theta))}(\tilde{a}(\tilde{\pi}_{*}X_{\theta}))=\tilde{\pi}_{*}X_{\theta}-\widehat{\pi_{*}X_{\theta}},$
and so
$\widehat{\pi_{*}X_{\theta}}=\tilde{\pi}_{*}X_{\theta}-\iota_{\tilde{\pi}(f(\theta))}(\tilde{a}(\tilde{\pi}_{*}X_{\theta})).$
Lifting everything, we have
$\widehat{\widehat{{\pi_{*}X_{\theta}}}}=\widehat{\tilde{\pi}_{*}X_{\theta}}-\widehat{\iota_{\tilde{\pi}(f(\theta))}(\tilde{a}(\tilde{\pi}_{*}X_{\theta}))},$
and thus
$\iota_{f(\theta)}(A_{f}(X)_{\theta})=X_{\theta}-\widehat{\tilde{\pi}_{*}X_{\theta}}+\widehat{\iota_{\tilde{\pi}(f(\theta))}(\tilde{a}(\tilde{\pi}_{*}X_{\theta}))}.$
But
$X_{\theta}-\widehat{\tilde{\pi}_{*}X_{\theta}}=\iota_{f(\theta)}(\tilde{A}(X_{\theta}))$
and so we have
$\iota_{f(\theta)}(A_{f}(X)_{\theta})=\iota_{f(\theta)}(\tilde{A}(X_{\theta}))+\widehat{\iota_{\tilde{\pi}(f(\theta))}(\tilde{a}(\tilde{\pi}_{*}X_{\theta}))}.$
To make use of this we need to be able to write $A$ as an
$L{\mathfrak{g}}$-valued 1-form and an $i{\mathbb{R}}$-valued 1-form. That is,
$A(X)_{\theta}=(\xi(\theta),x)$ for $\xi\in L{\mathfrak{g}}$ and $x\in
i{\mathbb{R}}.$ To that end, consider the vertical vector $V$ in $T_{f}P$
generated by the Lie algebra element $(\xi,x):$
$\displaystyle V_{\theta}$
$\displaystyle=\frac{d}{dt}_{|_{0}}f(\exp(t\xi),tx)(\theta)$
$\displaystyle=\frac{d}{dt}_{|_{0}}f(\theta+tx)\exp(t\xi(\theta+tx))$
$\displaystyle=\frac{d}{dt}_{|_{0}}\left(f(\theta)+f^{\prime}(\theta)tx\vphantom{{}^{2}}\right)\left(1+t\xi(\theta)+O(t^{2})\right)$
$\displaystyle=\iota_{f(\theta)}(\xi(\theta))+xf^{\prime}(\theta).$
Since $A$ is a connection, it returns the Lie algebra element corresponding to
the vertical part of a vector $X.$ Therefore, we must solve the following
equation for $\xi$ and $x:$
$\iota_{f(\theta)}(\tilde{A}(X_{\theta}))+\widehat{\iota_{\tilde{\pi}(f(\theta))}(\tilde{a}(\tilde{\pi}_{*}X_{\theta}))}=\iota_{f(\theta)}(\xi(\theta))+xf^{\prime}(\theta).$
Applying $\tilde{A}$ to both sides gives
$\tilde{A}(X_{\theta})=\xi(\theta)+x\tilde{A}(f^{\prime}(\theta)),$
since
$\widehat{\iota_{\tilde{\pi}(f(\theta))}(\tilde{a}(\tilde{\pi}_{*}X_{\theta}))}$
is horizontal with respect to $\tilde{A}.$ Thus, we have
$\xi(\theta)=\tilde{A}(X_{\theta}-xf^{\prime}(\theta)).$
Taking instead, $\tilde{\pi}_{*}$ of both sides gives
$\iota_{\tilde{\pi}(f(\theta))}(\tilde{a}(\tilde{\pi}_{*}X_{\theta}))=x\,\tilde{\pi}_{*}f^{\prime}(\theta),$
since the vectors $\iota_{f(\theta)}(\tilde{A}(X_{\theta}))$ and
$\iota_{f(\theta)}(\xi(\theta))$ are vertical in $\widetilde{P}$. Then
applying $\tilde{a}$ to both sides yields
$\tilde{a}(\tilde{\pi}_{*}X_{\theta})=x\,\tilde{a}(\tilde{\pi}_{*}f^{\prime}(\theta)).$
So (with a slight abuse of notation) we can write the connection form on $P$
as
$(A,a)_{f}(X)_{\theta}=(\tilde{A}(X_{\theta}-a(X)f^{\prime}(\theta)),a(X)),$
where $\tilde{A}$ and $\tilde{a}$ are connection forms on $\widetilde{P}$ and
$Y$ respectively and $a(X)$ is given by the formula for $x$ above. Now that we
have the connection on $P$ in this form we can check explicitly that it
satisfies the conditions for a connection. By construction, it satisfies
$(A,a)(\iota_{f}(\xi,x))=(\xi,x)$ and so we just need to check that
$(A,a)(X(\gamma,\phi))=ad(\gamma,\phi)^{-1}(A,a)(X).$ Recall that the adjoint
action of $LG\rtimes S^{1}$ on its Lie algebra is given by
$ad(\gamma,\phi)^{-1}(\xi,x)=\left(\rho_{\phi}^{-1}\left(ad(\gamma^{-1})\xi-\gamma^{-1}\partial\gamma\,x\right),x\right)$
and so
$ad(\gamma,\phi)^{-1}(A,a)(X)_{\theta}=\left(\rho_{\phi}^{-1}(ad(\gamma^{-1})\tilde{A}(X_{\theta}-a(X)f^{\prime}(\theta))-\gamma^{-1}\partial\gamma\,a(X)),a(X)\right).$
On the other hand, the action of $LG\rtimes S^{1}$ on the tangent vector $X$
is
$X(\gamma,\phi)=\rho_{\phi}^{-1}(X\gamma).$
Therefore,
$\displaystyle(A,a)(X(\gamma,\phi))_{\theta}$
$\displaystyle=\left(\tilde{A}(X(\gamma,\phi)_{\theta}-a(X(\gamma,\phi))\partial(f(\theta+\phi)\gamma(\theta+\phi))),a(X(\gamma,\phi))\right)$
$\displaystyle=\left(\tilde{A}(\rho_{\phi}^{-1}(X\gamma)_{\theta}-a(\rho_{\phi}^{-1}(X\gamma)_{\theta})\partial(f(\theta+\phi)\gamma(\theta+\phi))),a(\rho_{\phi}^{-1}(X\gamma)_{\theta})\right)$
$\displaystyle\begin{split}&=\left(\tilde{A}(\rho_{\phi}(X\gamma)_{\theta}-a(\rho_{\phi}^{-1}(X\gamma)_{\theta})\\{\partial
f(\theta+\phi)\gamma(\theta+\phi)\right.\\\
&\left.\phantom{(\tilde{A}(\rho_{\phi}^{-1}(X\gamma)_{\theta}-a(\rho_{\phi}^{-1}(X\gamma}+f(\theta+\phi)\partial\gamma(\theta+\phi)\\}),a(\rho_{\phi}^{-1}(X\gamma)_{\theta})\right).\end{split}$
Since $\tilde{A}$ is a connection, we have
$\tilde{A}(\rho_{\phi}^{-1}(X\gamma))_{\theta}=\rho_{\phi}^{-1}(ad(\gamma^{-1})\tilde{A}(X))_{\theta}$
and $\tilde{A}(\partial
f(\theta+\phi)\gamma(\theta+\phi))=\rho_{\phi}^{-1}(ad(\gamma^{-1})\tilde{A}(\partial
f(\theta))).$ Also, since $a$ is $i{\mathbb{R}}$-valued, we have
$a(\rho_{\phi}^{-1}(X\gamma)_{\theta})=a(X).$ Therefore,
$(A,a)(X(\gamma,\phi))_{\theta}=\left(\rho_{\phi}^{-1}(ad(\gamma^{-1})\tilde{A}(X_{\theta}-a(X)\partial
f(\theta)))\right.\\\
\left.-a(X)\tilde{A}(f(\theta+\phi)\partial\gamma(\theta+\phi))),a(X)\vphantom{\tilde{A}}\right).$
But, $f(\theta+\phi)\partial\gamma(\theta+\phi)$ is really shorthand for
$\iota_{f(\theta+\phi)}(\rho_{\phi}^{-1}(\gamma^{-1}\partial\gamma))$ and so
$\tilde{A}(f(\theta+\phi)\partial\gamma(\theta+\phi))=\tilde{A}(\iota_{f(\theta+\phi)}(\rho_{\phi}^{-1}(\gamma^{-1}\partial\gamma)))=\rho_{\phi}^{-1}(\gamma^{-1}\partial\gamma).$
Thus, we have
$(A,a)(X(\gamma,\phi))_{\theta}=\left(\rho_{\phi}^{-1}(ad(\gamma^{-1})\tilde{A}(X_{\theta}-a(X)\partial
f(\theta))-a(X)\gamma^{-1}\partial\gamma),a(X)\vphantom{\tilde{A}}\right),$
as required.
As for the $LG$-bundle case in section 2.5, to define a connection333Of
course, here we need to define a connection on $Y$ as well as on
$\widetilde{P}.$ on $\widetilde{P}$ given the bundle $P$ we need a connection
and Higgs field for $P$. Unlike the case in the previous section, however, in
order to define a connection we require a Higgs field to satisfy a slightly
different condition. Recall that a Higgs field for an $LG\rtimes S^{1}$-bundle
$P$ satisfies
$\Phi(p(\gamma,\phi))=\rho_{\phi}^{-1}\left(ad(\gamma^{-1})\Phi(p)+\gamma^{-1}\partial\gamma\right).$
It will be instructive to define a Higgs field for $P$ given the bundles
$\widetilde{P}\to Y\to M$ now since we will need this later to show that the
construction is invertible. Define then, the map $\Phi\colon P\to
L{\mathfrak{g}}$ by
$\Phi(f)=\tilde{A}(\partial f).$
This is a Higgs field since
$\displaystyle\Phi(f(\gamma,\phi))$
$\displaystyle=\tilde{A}(\rho_{\phi}^{-1}(\partial
f\gamma)+\rho_{\phi}^{-1}(\gamma^{-1}\partial\gamma)$
$\displaystyle=\tilde{A}(\rho_{\phi}^{-1}(\partial
f\gamma))+\iota_{\rho_{\phi}(f)}(\rho_{\phi}^{-1}(\gamma^{-1}\partial\gamma)$
$\displaystyle=ad(\rho_{\phi}^{-1}(\gamma^{-1}))\tilde{A}(\rho_{\phi}^{-1}(\partial
f))+\rho_{\phi}^{-1}(\gamma^{-1}\partial\gamma)$
$\displaystyle=\rho_{\phi}^{-1}\left(ad(\gamma^{-1})\tilde{A}(\partial
f)+\gamma^{-1}\partial\gamma\right)$
$\displaystyle=\rho_{\phi}^{-1}\left(ad(\gamma^{-1})\Phi(f)+\gamma^{-1}\partial\gamma\right).$
To define a connection on $\widetilde{P}=(P\times G\times S^{1})/LG\rtimes
S^{1}$ we need to be able to write a form on $P\times G\times S^{1}$ which is
zero on vertical vectors (with respect to the $LG\rtimes S^{1}$ action) and
invariant under the $LG\rtimes S^{1}$ action (so as to ensure that it is well-
defined). Thus we need to calculate the action of $LG\rtimes S^{1}$ on: the
connection, $(A,a),$ on $P,$ the Higgs field, $\Phi,$ on $P$ and the Maurer-
Cartan forms $\Theta$ and $d\theta$ on $G$ and $S^{1}$ respectively. Then we
can combine these in an invariant way. We can calculate the action of
$(\gamma,\phi)$ on the connection on $P$:
$\displaystyle(\gamma,\phi)^{*}(A,a)(X)$ $\displaystyle=(A,a)(X(\gamma,\phi))$
$\displaystyle=ad(\gamma,\phi)^{-1}(A,a)(X)$
$\displaystyle=\left(\rho_{\phi}^{-1}\left(ad(\gamma^{-1})A(X)-\gamma^{-1}\partial\gamma\,a(X)\right),a(X)\right),$
and we know that the Higgs field satisfies
$\Phi(p(\gamma,\phi))=\rho_{\phi}^{-1}\left(ad(\gamma^{-1})\Phi(p)+\gamma^{-1}\partial\gamma\right),$
and the Maurer-Cartan form on $S^{1}$ is unchanged. To calculate the action on
the Maurer-Cartan form on $G,$ consider a vector $(X,g\zeta,x_{\theta})\in
T_{(p,g,\theta)}(P\times G\times S^{1}).$ We have:
$\displaystyle(X,g\zeta,x_{\theta})(\gamma,\phi)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(\gamma_{X}(t)(\gamma,\phi),\gamma(\theta+tx)^{-1}g\exp(t\zeta),\theta+tx-\phi)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(\gamma_{X}(t)(\gamma,\phi),(\gamma(\theta)^{-1}-\gamma(\theta)^{-1}\partial\gamma(\theta)\gamma(\theta)^{-1}tx)g(1+t\zeta),\theta+tx-\phi)$
$\displaystyle=(X(\gamma,\phi),\gamma(\theta)^{-1}g\zeta-\gamma(\theta)^{-1}\partial\gamma(\theta)\gamma(\theta)^{-1}gx,x)$
$\displaystyle=(X(\gamma,\phi),\gamma(\theta)^{-1}g\\{\zeta-x\,ad(g^{-1})\partial\gamma(\theta)\gamma(\theta)^{-1}\\},x),$
and so
$\displaystyle(\gamma,\phi)^{*}\Theta(g\zeta)$
$\displaystyle=\Theta_{\gamma(\theta)^{-1}g}(\gamma(\theta)^{-1}g\\{\zeta-x\,ad(g^{-1})\partial\gamma(\theta)\gamma(\theta)^{-1}\\})$
$\displaystyle=\zeta-x\,ad(g^{-1})\partial\gamma(\theta)\gamma(\theta)^{-1}.$
Now consider the form on $P\times G\times S^{1}$ given by
$\tilde{A}=ad(g^{-1})A+\Theta+ad(g^{-1})\Phi(a+d\theta).$
This is invariant under the $LG\rtimes S^{1}$ action, for
$\displaystyle(\gamma,\phi)^{*}\tilde{A}_{(p,g,\theta)}$
$\displaystyle(X,g\zeta,x_{\theta})$
$\displaystyle=\tilde{A}_{(p(\gamma,\phi),\gamma(\theta)^{-1}g,\theta+\phi)}(X(\gamma,\phi),\gamma(\theta)^{-1}g\\{\zeta-x\,ad(g^{-1})\partial\gamma(\theta)\gamma(\theta)^{-1}\\},x)$
$\displaystyle=ad(g^{-1}\gamma(\theta))\rho_{\phi}^{-1}\left(ad(\gamma^{-1})A(X)_{\theta-\phi}-\gamma^{-1}\partial\gamma_{\theta-\phi}\,a(X)\right)$
$\displaystyle\qquad\qquad+\zeta-x\,ad(g^{-1})\partial\gamma(\theta)\gamma(\theta)^{-1}$
$\displaystyle\qquad\qquad+ad(g^{-1}\gamma(\theta))\rho_{\phi}^{-1}\left(ad(\gamma^{-1})\Phi(p)_{\theta-\phi}+\gamma^{-1}\partial\gamma_{\theta-\phi}\right)\left(a(X)+x\right)$
$\displaystyle=ad(g^{-1}\gamma(\theta))\left(ad(\gamma^{-1})A(X)_{\theta}-\gamma^{-1}\partial\gamma_{\theta}\,a(X)\right)$
$\displaystyle\qquad\qquad+\zeta-x\,ad(g^{-1})\partial\gamma(\theta)\gamma(\theta)^{-1}$
$\displaystyle\qquad\qquad+ad(g^{-1}\gamma(\theta))\left(ad(\gamma^{-1})\Phi(p)_{\theta}+\gamma^{-1}\partial\gamma_{\theta}\right)\left(a(X)+x\right)$
$\displaystyle=ad(g^{-1})A(X)_{\theta}+\zeta+ad(g^{-1})\Phi(p)_{\theta}\left(a(X)+x\right)$
$\displaystyle=\tilde{A}_{(p,g,\theta)}(X,g\zeta,x_{\theta}),$
by the calculations above. So for it to be well-defined on the quotient space
we just need to check that it vanishes on vertical vectors. The vertical
vector at the point $(p,g,\theta)$ generated by the vector $(\xi,x)$ is
$\displaystyle V$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(p,g,\theta)(\exp(t\xi),tx)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(p(\exp(t\xi),tx),(1-t\xi(\theta))g,\phi-
tx)$ $\displaystyle=(\iota_{p}(\xi,x),-g\,ad(g^{-1})\xi(\theta),-x),$
and so
$\displaystyle\tilde{A}(V)$
$\displaystyle=ad(g^{-1})A(\iota_{p}(\xi,x))_{\theta}-ad(g^{-1})\xi(\theta)+ad(g^{-1})\Phi(p)(a(\iota_{p}(\xi.x))-x)$
$\displaystyle=ad(g^{-1})\xi(\theta)-ad(g^{-1})\xi(\theta)+ad(g^{-1})\Phi(p)(x-x)$
$\displaystyle=0.$
Thus we have defined a $G$-valued 1-form on $\widetilde{P}.$ $\tilde{A}$ is in
fact a connection form, since if we evaluate it on the vertical vector
generated by $\zeta\in{\mathfrak{g}},$ that is,
$\iota_{[p,g,\theta]}(\zeta)=(0,g\zeta,0),$ we get $\tilde{A}(g\zeta)=\zeta$
and further,
$\displaystyle\tilde{A}((X,g\zeta,x_{\theta})h)$
$\displaystyle=\tilde{A}(X,ghh^{-1}\zeta h,x_{\theta})$
$\displaystyle=\left(ad(gh)^{-1}A+ad(h^{-1})\Theta+ad(gh)^{-1}\Phi(a+d\theta)\right)(X,g\zeta,x_{\theta})$
$\displaystyle=ad(h^{-1})\tilde{A}(X,g\zeta,x_{\theta}).$
To define a connection on the $S^{1}$-bundle $P/LG$ we just take the
projection of the $i{\mathbb{R}}$-valued 1-form $a$ which is a connection
form.
What remains to be shown now is that the constructions presented here for
connections on $P,$ $\widetilde{P}$ and $Y$ are invertible. In particular,
suppose we have the $LG\rtimes S^{1}$-bundle $P\to M$ with connection $(A,a)$
and Higgs field $\Phi$ and have constructed $\widetilde{P}\to Y\to M$ with
connections $\tilde{A}$ and $\tilde{a}.$ Then if we construct the
corresponding $LG\rtimes S^{1}$-bundle $P^{\prime}$ (which is isomorphic to
$P$ via the map $f\colon P\to P^{\prime};\,p\mapsto
f_{p}=(\theta\mapsto[p,1,\theta])$) and the connection
$(A^{\prime},a^{\prime})$ for $P^{\prime},$ we would like to show that
$f^{*}(A^{\prime},a^{\prime})=(A,a).$ Note that for the vector $X\in T_{p}P$
we have
$f_{*}X=(X,0,0)\in T_{f_{p}}P^{\prime}.$
Therefore,
$\displaystyle f^{*}(A^{\prime},a^{\prime})(X)$
$\displaystyle=(A^{\prime},a^{\prime})(X,0,0)$
$\displaystyle=(\tilde{A}(X),a^{\prime}(X))$
$\displaystyle=(A(X),a^{\prime}(X))$
by the definition of $A^{\prime}$ in terms of $\tilde{A}$ and $\tilde{A}$ in
terms of $A$ and also $a^{\prime}(X)=\tilde{a}(\tilde{\pi}_{*}X)=a(X).$ On the
other hand, suppose we had the bundles $\widetilde{P}\to Y\to M$ with
connections $\tilde{A}$ and $\tilde{a}$ and constructed $P\to M$ with
connection $(A,a)$ and Higgs field $\Phi(f)=\tilde{A}(\partial f).$ Then we
would like to show that if we construct the bundles $\widetilde{P}\to Y\to M$
with connections $\tilde{A}^{\prime}$ and $\tilde{a}^{\prime},$ we have
$\tilde{A}^{\prime}=f^{*}\tilde{A}$ where
$f\colon\widetilde{P}^{\prime}\xrightarrow{\sim}\widetilde{P}$ is the
isomorphism given by $[f,g,\theta]\mapsto f(\theta)g.$ Note that at the point
$[p,g,\theta]$ we have
$\displaystyle f_{*}(X,g\zeta,x_{\theta})$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}\gamma_{X(\theta+tx)}(t)g\exp(t\zeta)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}\gamma_{X(\theta)}(t)g+\partial\gamma_{X(\theta)}(0)xg+\gamma_{X(\theta)}(0)g\zeta$
$\displaystyle=X(\theta)g+\partial p(\theta)xg+p(\theta)g\zeta$
and therefore,
$\displaystyle f^{*}\tilde{A}(X,g\zeta,x_{\theta})$
$\displaystyle=\tilde{A}(X(\theta)g+\partial p(\theta)xg+p(\theta)g\zeta)$
$\displaystyle=\tilde{A}(X(\theta)g)+x\tilde{A}(\partial p(\theta)g)+\zeta$
$\displaystyle=ad(g^{-1})\tilde{A}(X(\theta))+x\,ad(g^{-1})\tilde{A}(\partial
p(\theta))+\zeta$
while for $\tilde{A}^{\prime}$ we have
$\displaystyle\tilde{A}^{\prime}(X,g\zeta,x_{\theta})$
$\displaystyle=ad(g^{-1})A(X)+\zeta+ad(g^{-1})\Phi(p)(a(X)+x)$
$\displaystyle=ad(g^{-1})\left(\tilde{A}(X)-a(X)\tilde{A}(\partial
p)\right)+\zeta+ad(g^{-1})\tilde{A}(\partial p)(a(X)+x)$
$\displaystyle=f^{*}\tilde{A}(X,g\zeta,x_{\theta}).$
Thus we have proven the analogue of Proposition 2.5.2
###### Proposition 4.2.2.
The correspondence from Proposition 4.2.1 extends to a bijection between
$G$-bundles with connection over $S^{1}$-bundles with connection and
$LG\rtimes S^{1}$-bundles with connection and Higgs field.
#### 4.2.2 The string class and the first Pontrjagyn class
Now that we have extended the correspondence from section 2.5, we are in a
position to extend the result concerning the string class and the Pontrjagyn
class (Theorem 2.5.3). Recall that Theorem 2.5.3 extended Killingback’s result
to a general $LG$-bundle $P\to M$ by relating the string class of $P$ to the
first Pontrjagyn class of the corresponding $G$-bundle $\widetilde{P}\to
M\times S^{1}.$ In particular, the string class of $P$ is given by integrating
$p_{1}(\widetilde{P})$ over the circle. We would like now to extend this
further to the case where $P\to M$ is an $LG\rtimes S^{1}$-bundle and
$\widetilde{P}$ is the corresponding $G$-bundle over a circle bundle $Y$ over
$M.$ In this case we find that the string class is given by integrating the
first Pontrjagyn class of $\widetilde{P}$ over the fibre of the circle bundle
$Y$. In particular, we have the following theorem
###### Theorem 4.2.3.
Let $P\to M$ be a principal $LG\rtimes S^{1}$-bundle and $\widetilde{P}\to
Y\to M$ be the corresponding $G$-bundle over an $S^{1}$-bundle. Then the
string class of $P$ is given by the integration over the fibre of the first
Pontrjagyn class of $\widetilde{P}.$ That is,
$s(P)=\int_{S^{1}}p_{1}(\widetilde{P}).$
###### Proof.
We prove this in analogy with the proof of Theorem 2.5.3, that is, by
calculating the integral of the first Pontrjagyn class of $\widetilde{P}.$
Recall that the first Pontrjagyn class is given by
$p_{1}=-\frac{1}{8\pi^{2}}\langle\tilde{F},\tilde{F}\rangle,$
where $\tilde{F}=d\tilde{A}+\tfrac{1}{2}[\tilde{A},\tilde{A}]$ is the
curvature of the connection $\tilde{A}$ corresponding to the pair $(A,\Phi)$
on $P.$ We have
$\tilde{F}=d(ad(g^{-1})A+\Theta+ad(g^{-1})\Phi(a+d\theta))\\\
+\tfrac{1}{2}[ad(g^{-1})A+\Theta+ad(g^{-1})\Phi(a+d\theta),ad(g^{-1})A+\Theta+ad(g^{-1})\Phi(a+d\theta)]\\\
\phantom{\tilde{F}}=d(ad(g^{-1})A+\Theta+ad(g^{-1})\Phi(a+d\theta))\\\
+\tfrac{1}{2}[ad(g^{-1})A,ad(g^{-1})A]+[ad(g^{-1})A,\Theta]+[ad(g^{-1})A,ad(g^{-1})\Phi(a+d\theta)]\\\
+\tfrac{1}{2}[\Theta,\Theta]+[\Theta,ad(g^{-1})\Phi(a+d\theta)]+\tfrac{1}{2}[ad(g^{-1})\Phi(a+d\theta),ad(g^{-1})\Phi(a+d\theta)].$
To calculate $d(ad(g^{-1})A+\Theta+ad(g^{-1})\Phi(a+d\theta))$ we use
$d(ad(g^{-1})A+\Theta+ad(g^{-1})\Phi(a+d\theta))((X,g\xi,x_{\theta}),(Y,g\zeta,y_{\theta}))\\\
\phantom{d(ad}=\tfrac{1}{2}\left\\{(X,g\xi,x_{\theta})(ad(g^{-1})A(Y)_{\theta})-(Y,g\zeta,y_{\theta})(ad(g^{-1})A(Y)_{\theta})\right.\\\
\left.-ad(g^{-1})A([X,Y])_{\theta}\right\\}\\\ +d\Theta\\\
\phantom{d(ad}+\tfrac{1}{2}\left\\{(X,g\xi,x_{\theta})(ad(g^{-1})(a(Y)+y)\Phi(p)_{\theta})\right.\\\
\left.-(Y,g\zeta,y_{\theta})(ad(g^{-1})(a(X)+x)\Phi(p)_{\theta})-ad(g^{-1})[x,y]\Phi(p)_{\theta}\right\\},\\\
$
for tangent vectors $(X,g\xi,x_{\theta})$ and $(Y,g\zeta,y_{\theta})$ at the
point $[p,g,\theta]\in\widetilde{P}.$ For the first term, calculate
$\displaystyle(X,g$ $\displaystyle\xi,x_{\theta})(ad(g^{-1})A(Y)_{\theta})$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}(1-t\xi)g^{-1}A_{\gamma_{p}(t)}(Y)_{(\theta+tx)}g(1+t\xi)$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}g^{-1}A_{\gamma_{p}(t)}(Y)_{(\theta+tx)}g-t\xi
g^{-1}A_{\gamma_{p}(t)}(Y)_{(\theta+tx)}g+g^{-1}A_{\gamma_{p}(t)}(Y)_{(\theta+tx)}gt\xi$
$\displaystyle=\frac{d}{dt}\bigg{|}_{0}g^{-1}A_{\gamma_{p}(t)}(Y)_{\theta}g+g^{-1}\partial
A(Y)_{\theta}xg-\xi g^{-1}A(Y)_{\theta}g+g^{-1}A(Y)_{\theta}g\xi.$
Combining this with the other terms for the first derivative above, we have
$d(ad(g^{-1})A)=ad(g^{-1})dA-ad(g^{-1})\partial A\wedge
d\theta-[\Theta,ad(g^{-1})A].$
For the last term, calculate
$(X,g\xi,x_{1\theta})(ad(g^{-1})(a(Y)+y)\Phi(p)_{\theta})\\\
=\frac{d}{dt}\bigg{|}_{0}(1-t\xi)g^{-1}(a_{\gamma_{p}(t)}(Y)+y)\Phi(\gamma_{p}(t))_{(\theta+tx)}g(1+t\xi)\\\
=\frac{d}{dt}\bigg{|}_{0}g^{-1}(a_{\gamma_{p}(t)}(Y)+y)\Phi(\gamma_{p}(t))_{(\theta+tx)}g\\\
-t\xi g^{-1}(a_{\gamma_{p}(t)}(Y)+y)\Phi(\gamma_{p}(t))_{(\theta+tx)}g\\\
+g^{-1}(a_{\gamma_{p}(t)}(Y)+y)\Phi(\gamma_{p}(t))_{(\theta+tx)}gt\xi\\\
=\frac{d}{dt}\bigg{|}_{0}g^{-1}(a_{\gamma_{p}(t)}(Y)+y)\Phi(p)_{\theta}g+\frac{d}{dt}_{|_{0}}g^{-1}(a(Y)+y)\Phi(\gamma_{p}(t))_{\theta}g\\\
+g^{-1}(a(Y)+y)\partial\Phi(p)_{\theta}xg-\xi
g^{-1}(a(Y)+y)\Phi(p)_{\theta}g\\\ +g^{-1}(a(Y)+y)\Phi(p)_{\theta}g\xi.\\\ $
Subtracting $(Y,g\zeta,y_{\theta})(ad(g^{-1})(a(X)+x)\Phi(p)_{\theta})$ from
this gives
$d(ad(g^{-1})\Phi(a+d\theta))=ad(g^{-1})f\Phi+ad(g^{-1})d\Phi\wedge(a+d\theta)\\\
-[\Theta,ad(g^{-1})(a+d\theta)\Phi]-ad(g^{-1})a\partial\Phi\wedge d\theta.$
We also have $d\Theta=-\tfrac{1}{2}[\Theta,\Theta].$ Therefore, the curvature
of $\widetilde{P}$ is given by
$\tilde{F}=d(ad(g^{-1})A+\Theta+ad(g^{-1})\Phi(a+d\theta))\\\
+\tfrac{1}{2}[ad(g^{-1})A,ad(g^{-1})A]+[ad(g^{-1})A,\Theta]+[ad(g^{-1})A,ad(g^{-1})\Phi(a+d\theta)]\\\
+\tfrac{1}{2}[\Theta,\Theta]+[\Theta,ad(g^{-1})\Phi(a+d\theta)]+\tfrac{1}{2}[ad(g^{-1})\Phi(a+d\theta),ad(g^{-1})\Phi(a+d\theta)]\\\
\phantom{\tilde{F}}=ad(g^{-1})dA-ad(g^{-1})\partial A\wedge
d\theta+ad(g^{-1})f\Phi\\\
+ad(g^{-1})d\Phi\wedge(a+d\theta)-ad(g^{-1})a\partial\Phi\wedge d\theta\\\
+\tfrac{1}{2}[ad(g^{-1})A,ad(g^{-1})A]+[ad(g^{-1})A,ad(g^{-1})\Phi(a+d\theta)].\\\
$
That is,
$\tilde{F}=ad(g^{-1})\left(F+f\Phi+\nabla\Phi\wedge(a+d\theta)\right).$
So the first Pontrjagyn class is
$p_{1}=-\frac{1}{8\pi^{2}}\langle\tilde{F},\tilde{F}\rangle\\\
\phantom{p_{1}}=-\frac{1}{8\pi^{2}}\left\langle
F+f\Phi+\nabla\Phi\wedge(a+d\theta),F+f\Phi+\nabla\Phi\wedge(a+d\theta)\right\rangle\\\
\phantom{p_{1}}=-\frac{1}{8\pi^{2}}\Big{(}\left\langle
F+f\Phi,F+f\Phi\right\rangle-2\left\langle
F+f\Phi,\nabla\Phi\wedge(a+d\theta)\right\rangle\\\
-\left\langle\nabla\Phi\wedge(a+d\theta),\nabla\Phi\wedge(a+d\theta)\right\rangle\Big{)}\\\
\phantom{p_{1}}=-\frac{1}{8\pi^{2}}\Big{(}\left\langle
F+f\Phi,F+f\Phi\right\rangle-2\left\langle F+f\Phi,\nabla\Phi\wedge
a\right\rangle-2\left\langle F+f\Phi,\nabla\Phi\right\rangle
d\theta\Big{)}.\\\ $
Thus, integrating $p_{1}$ over the fibre, we get
$-\frac{1}{4\pi^{2}}\int_{S^{1}}\langle F+f\Phi,\nabla\Phi\rangle d\theta,$
which is the expression from Theorem 4.1.3.
∎
### 4.3 String structures for $LG\rtimes\operatorname{Diff}(S^{1})$-bundles
So far in this chapter we have generalised the results from [35] to include
the possibility of rotating loops. That is, we have worked with the semi-
direct product $LG\rtimes S^{1}.$ We would like to conclude now with a brief
outline of one way in which the results we have seen regarding $LG\rtimes
S^{1}$ lead us to information about a more general situation. Namely, we shall
consider the problem of lifting a bundle whose structure group is the semi-
direct product $LG\rtimes\operatorname{Diff}(S^{1}).$ That is, we shall allow
an action of the orientation preserving diffeomorphisms of the circle on the
loops in $LG.$
The group $\operatorname{Diff}(S^{1})$ has a well known central extension. In
particular, the Lie algebra of this extension is the Virasoro algebra (see for
example [29]). In this section, we would like to consider the central
extension of the semi-direct product above
$U(1)\to\widehat{LG\rtimes\operatorname{Diff}(S^{1})}\to
LG\rtimes\operatorname{Diff}(S^{1}).$
Thus far, we have seen that principal $LG$-bundles over $M$ correspond to
principal $G$-bundles over $M\times S^{1}$ (via the caloron correspondence)
and in the previous section we showed that isomorphism classes of principal
$LG\rtimes S^{1}$-bundles are in bijective correspondence with isomorphism
classes of principal $G$-bundles over principal $S^{1}$-bundles. If instead we
considered a principal $G$-bundle over a general $S^{1}$ fibre bundle444Such
bundles have structure group $\operatorname{Diff}(S^{1})$ and give rise to
principal $\operatorname{Diff}(S^{1})$-bundles in a natural way. we would find
that these bundles correspond to principal
$LG\rtimes\operatorname{Diff}(S^{1})$-bundles.
Now let $R\to M$ be a principal $LG\rtimes\operatorname{Diff}(S^{1})$-bundle.
We are interested in finding the obstruction to lifting this bundle to an
$\widehat{LG\rtimes\operatorname{Diff}(S^{1})}$-bundle $\widehat{R}.$ The
following result, due to Smale, gives us a way of using our previous results
to solve this problem. Namely, we have
###### Theorem 4.3.1 ([42]).
$\operatorname{Diff}(S^{1})$ is homotopy equivalent to $S^{1}.$
This means that if $Y\to M$ is a $\operatorname{Diff}(S^{1})$-bundle then its
transition functions can be chosen to be valued in $S^{1}$ and so $Y$ actually
admits an action of the circle (by identifying $Y$ locally with $S^{1}\times
U$ (for some open subset $U\subseteq M$) and rotating the $S^{1}$ factor).
This makes $Y$ into a principal $S^{1}$-bundle. In particular, then, if we
have a $G$-bundle $\widetilde{P}$ over an $S^{1}$ fibre bundle $Y\to M$ we can
replace the $LG\rtimes\operatorname{Diff}(S^{1})$-bundle in question with an
$LG\rtimes S^{1}$-bundle. That is, $R$ has a reduction to a principal
$LG\rtimes S^{1}$-bundle $P,$ so $R=P\times_{LG\rtimes
S^{1}}LG\rtimes\operatorname{Diff}(S^{1}).$ We can thus give the lift of $R$
in terms of the central extension of $LG\rtimes\operatorname{Diff}(S^{1})$ and
the lift $\widehat{P}$ of $P.$ In particular, we have a bundle map
$\widehat{P}\times_{\widehat{LG\rtimes
S^{1}}}\widehat{LG\rtimes\operatorname{Diff}(S^{1})}\to P\times_{LG\rtimes
S^{1}}LG\rtimes\operatorname{Diff}(S^{1})$
given by
$[\hat{p},\widehat{(\gamma,\varphi)}]\mapsto[p,(\gamma,\varphi)],$
where $\hat{p}$ is a lift of $p$ to $\widehat{P}$ and
$\widehat{(\gamma,\varphi)}$ is a lift of $(\gamma,\varphi)$ to the central
extension of $LG\rtimes\operatorname{Diff}(S^{1}).$ This map commutes with the
homomorphism $\widehat{LG\rtimes\operatorname{Diff}(S^{1})}\to
LG\rtimes\operatorname{Diff}(S^{1})$ and so
$\widehat{P}\times_{\widehat{LG\rtimes
S^{1}}}\widehat{LG\rtimes\operatorname{Diff}(S^{1})}$ is a lift of $R.$
## Appendix A Infinite-dimensional manifolds and Lie groups
In this thesis we have largely been concerned with the loop group of a compact
group. This is an example of an infinite-dimensional Lie group – specifically,
it is a _Fréchet_ Lie group. In this Appendix we collect some of the basic
results on Fréchet manifolds and Lie groups. We follow closely the expositions
presented in [19], [31] and [39]
### A.1 Fréchet spaces
We will begin with some basic definitions and examples of the sorts of spaces
we shall be dealing with. An infinite-dimensional manifold, like any manifold,
is a topological space modelled on some sort of Euclidean space. In the case
we are considering, this is a locally convex topological vector space called a
_Fréchet_ space.
###### Definition A.1.1.
A _Fréchet_ space is a complete metrisable Hausdorff locally convex
topological vector space, where by _locally convex_ we mean a space whose
topology is generated from some family of seminorms.111An equivalent
definition of local convexity for a topological vector space is that every
neighbourhood of $0$ contains a neighbourhood which is convex. This is the
definition used in [31].
Perhaps the most immediate example of a Fréchet space is given by any Banach
space. In general, however, there are examples of Fréchet spaces which are not
Banach spaces. The particular example we will consider is the space of all
smooth maps222More generally, the space of smooth sections of a vector bundle
over a compact manifold is also a Fréchet space. We shall restrict our
interest however, to the case of a trivial bundle (that is, the space of all
maps as above) since this covers the case we are really interested in – the
Lie algebra of the loop group, $\operatorname{Map}(S^{1},{\mathfrak{g}}).$
from a compact manifold $X$ into a vector space $V,$ that is, the space
$\operatorname{Map}(X,V).$ We define the topology on this space in terms of a
collection of neighbourhoods of the zero map. (Since this is a topological
vector space this will give the topology completely.) To do this, choose a
small neighbourhood $E$ of $0\in V.$ Then consider an open coordinate chart
$U\subseteq X$ with local coordinates $x_{1},\ldots,x_{m}$ and a compact set
$K\subseteq U.$ We define a family of sub-basic neighbourhoods (for each
choice of coordinate chart, compact set, neighbourhood of $0\in V$ and non-
negative integer $n$)
$N=\\{f\colon X\to V\mid\partial^{k}f/\partial x_{i_{1}}\ldots x_{i_{k}}\in E\
\forall\,x\in K,0\leq k\leq n,i_{j}\in\\{1,\ldots,m\\}\\}.$
Finite intersections of sets of this form give the basic neighbourhoods for
the topology on $\operatorname{Map}(X,V).$
The above example is important for our purposes since the special case of maps
from the circle into the Lie algebra of a compact group $G$ will be the
Fréchet space on which the loop group $LG$ is modelled.
### A.2 Groups of maps
Now that we have seen an example of a Fréchet space, we can give an example of
an infinite-dimensional manifold modelled on this space. This is the space
$\operatorname{Map}(X,G)$ of smooth maps from a compact manifold $X$ into a
compact Lie group $G$ and it is in fact an example of an infinite-dimensional
Lie group.
To define the coordinate charts for this manifold consider an open
neighbourhood $U$ of the identity in $G$. Using the exponential map, this is
homeomorphic to an open neighbourhood of the identity in ${\mathfrak{g}},$ say
$\tilde{U}.$ The set $\tilde{\mathcal{U}}:=\operatorname{Map}(X,\tilde{U})$ is
then an open neighbourhood of the identity in
$\operatorname{Map}(X,{\mathfrak{g}})$ and an atlas for
$\operatorname{Map}(X,G)$ is given by the open sets $\mathcal{U}f$ (where
$\mathcal{U}:=\operatorname{Map}(X,U)$), which are also homeomorphic to
$\tilde{\mathcal{U}}.$ The case where $X$ is the circle is the loop group
$LG.$
Note that there is a slightly more general example given by taking sections of
a fibre bundle over $X.$ Recall from the previous section that sections of a
vector bundle form a Fréchet space. Given a fibre bundle $Y\xrightarrow{\pi}X$
we can associate to any section $f\colon X\to Y$ a vector bundle over $X,$
called the vertical tangent bundle to $f$ and denoted $T_{\text{vert}}Y_{f},$
whose fibre at $x\in X$ is given by all vertical tangent vectors to $Y$ at
$f(x).$ That is, $T_{\text{vert}}Y_{f}=\\{V\in T_{f(x)}Y\mid\pi_{*}V=0\\}.$
Then the sections of $T_{\text{vert}}Y_{f}\to X$ form a Fréchet space and
there is a diffeomorphism from a neighbourhood of the zero section to a
neighbourhood of the image of $f$ in $Y$ which serves as a coordinate chart.
### A.3 The path fibration
In chapter 3 we made extensive use of a particular $\Omega G$-bundle called
the _path fibration_. This is a model for the universal $\Omega G$-bundle. In
this section we shall explain why this is in fact a locally trivial $\Omega
G$-bundle. Recall that the total space of the path fibration is defined as
$PG=\\{p:{\mathbb{R}}\to G\mid p(0)=1\text{ and }p^{-1}dp\text{ is
periodic}\\}.$
We can equivalently view this as the space of connections on the trivial
$G$-bundle over the circle, since if $p$ is a path in $G$ as above then
$p^{-1}dp$ is a ${\mathfrak{g}}$-valued 1-form on $S^{1}$ and conversely, each
connection form $A$ on the trivial $G$-bundle over $S^{1}$ uniquely determines
a periodic path by solving the ordinary differential equation $A=p^{-1}dp$
subject to the initial condition $p(0)=1.$ This means that $PG$ is
contractible. Note that when viewed as the space of connections $\Omega G$
acts freely on the right of this space by gauge transformations. Notice also
that if $p$ and $q$ are two paths in the same fibre of the projection
$PG\xrightarrow{\pi}G$ (so $p(2\pi)=q(2\pi)$) then $p^{-1}q$ is a smooth based
loop, since if $f(t)=(p^{-1}q)(t+2\pi)$ then $f$ satisfies the same
differential equation as $p^{-1}q$ and $f(0)=1$ so $f=p^{-1}q$ and thus
$p^{-1}q$ is periodic. This means that $q=p\gamma$ for some $\gamma\in\Omega
G$ and so $PG/\Omega G=G.$
For the local triviality of this bundle, consider an open neighbourhood $U$ of
the identity in $G.$ We can define a map
$U\times\Omega G\xrightarrow{\sim}\pi^{-1}(U);\quad(g,\gamma)\mapsto p,$
where $p(t)=\exp(t\xi)\gamma(t)$ and $\exp(2\pi\xi)=g.$ The inverse of this
map is given by
$p\mapsto(\pi(p),\exp(t\pi(p))^{-1}p).$
This gives us a trivialisation near the identity. To extend this to a local
trivialisation for the entire bundle we consider the open cover $\\{Uh\\}$ for
$h\in G.$ Let $\tilde{h}$ be a path ending at $h$ (that is,
$\pi(\tilde{h})=h$). Then the maps
$Uh\times\Omega G\xrightarrow{\sim}\pi^{-1}(Uh);\quad(g,\gamma)\mapsto p_{h},$
for $p_{h}(t)=\tilde{h}(t)\exp(t\xi)\gamma(t),$ give a local trivialisation
for the path fibration. So we have that the path fibration is a model for the
universal $\Omega G$-bundle.
## Appendix B Classification of semi-direct product bundles
### B.1 Classification of semi-direct product bundles
In section 3.3 we gave a model for the universal $L^{\vee}G$-bundle (where
$L^{\vee}G$ is the group of smooth maps $[0,2\pi]\to G$ with coincident
endpoints) by utilising its description as the semi-direct product
$\Omega^{\vee}G\rtimes G.$ Following those ideas we can actually give a
classification theory for general $K\rtimes H$-bundles.
Suppose $K$ and $H$ are Lie groups and we have an action $\varphi\colon
H\to\operatorname{Aut}(K).$ Then we can form the semi-direct product $K\rtimes
H,$ where the multiplication is defined by
$(k_{1},h_{1})(k_{2},h_{2})=(k_{1}\varphi_{h_{1}}(k_{2}),h_{1}h_{2}).$
We can give a model for the classifying space $E(K\rtimes H)$ as follows.
Consider the space $EK\times EH.$ This is contractible, since both $EK$ and
$EH$ are. Suppose we can find a left action of $H$ on $EK.$ That is, some
$\tilde{\varphi}\colon H\to\operatorname{Diff}(EK)$ such that
$\tilde{\varphi}_{h_{1}}\tilde{\varphi}_{h_{2}}=\tilde{\varphi}_{h_{1}h_{2}}.$
Suppose also that this action satisfies
$\tilde{\varphi}_{h}(xk)=\tilde{\varphi}_{h}(x)\varphi_{h}(k)$
for all $x\in EK.$ Then we can define a right action of $K\rtimes H$ on
$EK\times EH$ by
$(x,y)(k,h)=(\tilde{\varphi}_{h^{-1}}(xk),yh),$
where $(x,y)\in EK\times EH.$ This is clearly a right action since
$\displaystyle(\tilde{\varphi}_{h_{1}^{-1}}(xk_{1}),yh_{1})(k_{2},h_{2})$
$\displaystyle=(\tilde{\varphi}_{h_{2}^{-1}}(\tilde{\varphi}_{h_{1}^{-1}}(xk_{1})k_{2}),yh_{1}h_{2})$
$\displaystyle=(\tilde{\varphi}_{(h_{1}h_{2})^{-1}}(xk_{1}\varphi_{h_{1}}(k_{2})),yh_{1}h_{2})$
$\displaystyle=(x,y)(k_{1}\varphi_{h_{1}}(k_{2}),h_{1}h_{2}).$
It is also free and transitive on fibres and so
$\textstyle{EK\times
EH\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(EK\times
EH)/(K\rtimes H)}$
is a model for the universal bundle. To see that $\tilde{\varphi}$ exists,
consider the following construction of $EK$ [41] (see also [14]). Let
$\Delta^{n}$ be the standard $n$-simplex in ${\mathbb{R}}^{n+1}.$ That is,
$\Delta^{n}=\\{(t_{0},\ldots,t_{n})\mid t_{i}\geq 0,\textstyle\sum
t_{i}=1\\}.$
Then
$EK=\bigsqcup_{n\geq 0}\Delta^{n}\times K^{n+1}/\sim,$
where we make the identifications
$\left((t_{0},\ldots,t_{i-1},0,t_{i+1},\ldots,t_{n}),(k_{0},\ldots,k_{n})\right)\sim\left((t_{0},\ldots,t_{n}),(k_{0},\ldots,k_{i-1},1,k_{i+1},\ldots,k_{n})\right).$
Equivalently, we can think of $EK$ as the set of formal linear combinations of
elements of $K:$
$EK=\left\\{\textstyle\sum t_{i}k_{i}\mid t_{i}\geq 0,\textstyle\sum
t_{i}=1,k_{i}\in K\right\\}$
where in any given sum, only finitely many of the $t_{i}$’s are non-zero. Then
$\tilde{\varphi}$ is given by
$\tilde{\varphi}_{h}\left(\textstyle\sum t_{i}k_{i}\right)=\textstyle\sum
t_{i}\varphi_{h}(k_{i}).$
Using this construction, we can also write down a classifying map for any
$K\rtimes H$-bundle $P\xrightarrow{\pi}M.$ For this we will need a
correspondence between these bundles and certain pairs of $K$-bundles and
$H$-bundles. Let us briefly outline this correspondence now. First note that
there is a homomorphism $K\rtimes H\to H$ and so we can form the associated
$H$-bundle $P\times_{K\rtimes H}H\xrightarrow{\pi_{H}}M,$ where
$[p,h]=[p(k^{\prime},h^{\prime}),h^{\prime-1}h],\,$
$[p,h]h^{\prime}=[phh^{\prime}]$ and $\pi_{H}([p,h])=\pi(p).$ Further, there’s
a free action of $K$ on $P$ that identifies $P\times_{K\rtimes H}H$ with
$P/K.$ Namely, $pk=p(k,1).$ Then we have that
$P\xrightarrow{\pi_{K}}P\times_{K\rtimes H}H$ is a principal $K$-bundle.111For
the proof of the local triviality of this bundle, see [23], Proposition 5.5, p
57. Thus, we have constructed a $K$-bundle over an $H$-bundle out of the
$K\rtimes H$-bundle $P$ that we started with. In addition, we have an action
of $H$ on $P$ that covers the $H$ action on $P/K.$ That is, define $ph=p(1,h)$
and then $\pi_{K}(ph)=[p(1,h),1]=[p,h]=[p,1]h=\pi_{K}(p)h.$ This $H$ action
also has the property that
$(ph)k=p(1,h)(k,1)=p(\varphi_{h}(k),h)=(p\varphi_{h}(k))h.$ Therefore, we have
constructed a $K$-bundle with a twisted $H$-equivariant action as above over
an $H$-bundle:
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K,H}$$\textstyle{P/K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H}$$\textstyle{M}$
In fact, this construction is invertible. That is, given a $K$-bundle over an
$H$-bundle that satisfies the properties above, we can construct a $K\rtimes
H$-bundle. Suppose, then, that we have two Lie groups $K$ and $H$ with an
action $\varphi\colon H\to\operatorname{Aut}(K)$ as above. Suppose also that
we have a principal $K$-bundle $P\xrightarrow{\pi_{K}}P/K$ and a principal
$H$-bundle $P/K\xrightarrow{\pi_{H}}M$ and that there’s an $H$ action on $P$
covering that on $P/K$ and such that $(ph)k=(p\varphi_{h}(k))h.$ We can define
an action of $K\rtimes H$ on $P$ by $p(k,h)=(pk)h.$ This is a right action
since
$\displaystyle p(k_{1},h_{1})(k_{2},h_{2})$
$\displaystyle=(((pk_{1})h_{1})k_{2})h_{2}$
$\displaystyle=(((pk_{1})\varphi_{h_{1}}(k_{2}))h_{1})h_{2}$
$\displaystyle=p(k_{1}\varphi_{h_{1}}(k_{2}),h_{1}h_{2}).$
It is a free action, for suppose that $p(k,h)=p.$ Then $(pk)h=p$ and so
$\pi_{K}((pk)h)=\pi_{K}(p).$ But $\pi_{K}((pk)h)=\pi_{K}(pk)h$ and
$\pi_{K}(pk)=\pi_{K}(p),$ so we have $\pi_{K}(p)h=\pi_{K}(p)$ and therefore
$h=1$ since the $H$ action is free. But if $h=1$ we have that $pk=p$ and so
$k=1.$ We also have that $P/(K\rtimes H)=(P/K)/H=M.$ To see that $P\to M$ is
locally trivial, consider an open set $U\subset M$ over which $P/K$ is
trivial. Then there exists a section $s\colon U\to P/K.$ Since $U$ is
contractible, the pull-back $s^{*}P$ over $U$ is trivial and so there exists a
section $s^{\prime}\colon U\to s^{*}P.$ But a choice of section
$s^{\prime}\colon U\to s^{*}P$ is equivalent to a map $\sigma\colon U\to P$
such that $\pi_{K}(\sigma(x))=s(x).$ That is, such that $\pi(\sigma(x))=x.$ So
$\sigma$ is a local section of $P\to M.$ Therefore, we have that $P\to M$ is a
principal $K\rtimes H$-bundle.
Using this correspondence, we can write down a classifying map for $P.$ That
is, a map $f\colon P\to EK\times EH$ such that $f(p(k,h))=f(p)(k,h).$ Firstly,
note that if $P\xrightarrow{\pi}M$ is a ${\mathcal{G}}$-bundle then we can
write the classifying map as follows: Let $\\{U_{\alpha}\\}$ be an open cover
of $M$ over which $P$ is trivial. Then $\pi^{-1}(U_{\alpha})$ is isomorphic to
$U_{\alpha}\times{\mathcal{G}}.$ Now choose local sections $s_{\alpha}\colon
U_{\alpha}\to\pi^{-1}(U_{\alpha})$ and define the functions
$g_{\alpha}\colon\pi^{-1}(U_{\alpha})\to{\mathcal{G}}$ by
$s_{\alpha}(m)=(m,g_{\alpha}(s_{\alpha}(m))),$ where we have used the
isomorphism to identify $\pi^{-1}(U_{\alpha})$ with
$U_{\alpha}\times{\mathcal{G}}.$ Now, let $\\{\psi_{\alpha}\\}$ be a partition
of unity subordinate to $\\{U_{\alpha}\\}.$ Then define the map
$f_{{\mathcal{G}}}\colon P\to E{\mathcal{G}}$ by
$f_{{\mathcal{G}}}(p)=\sum\psi_{\alpha}(\pi(p))g_{\alpha}(p).$
This is clearly ${\mathcal{G}}$-equivariant and so defines the classifying map
for $P.$
Now consider again the case where ${\mathcal{G}}=K\rtimes H.$ Write the
classifying map $f$ as a pair of functions $(f_{K},f_{H}).$ Then we require
that
$(f_{K}(p(k,h)),f_{H}(p(k,h)))=(\tilde{\varphi}_{h^{-1}}(f_{K}(p)k),f_{H}(p)h).$
Using the correspondence above, we can construct a pair of bundles
$P\xrightarrow{\pi_{K}}P/K\xrightarrow{\pi_{H}}M.$ Define $f_{H}$ to be the
classifying map of the $H$-bundle $P/K.$ To define $f_{K},$ consider an open
cover $\\{U_{\alpha}\\}$ of $M$ as above. Consider the cover
$\\{V_{\alpha}\\}$ of $P/K$ where $V_{\alpha}=\pi_{H}^{-1}(U_{\alpha}).$ $P$
is trivial over $V_{\alpha}$ since we can construct a local section as
follows. Identify $V_{\alpha}$ with $U_{\alpha}\times H.$ Then over the subset
$U_{\alpha}\times\\{1\\},$ $P$ has a section, say $\sigma_{\alpha}.$ We can
define a section of $P$ over $U_{\alpha}\times H$ by forcing $H$-equivariance.
That is, by defining
$\chi_{\alpha}(s_{\alpha}(m)h)\colon=\sigma_{\alpha}(m)h,$ where $s_{\alpha}$
is a local section of $P/K.$ So $\pi_{K}^{-1}(V_{\alpha})\simeq
U_{\alpha}\times H\times K.$ Then we can define the functions $k_{\alpha}$ as
above and we see that $k_{\alpha}(ph)=\varphi_{h^{-1}}(k_{\alpha}(p))$ (which
follows from the fact that $(pk)h=(ph)\varphi_{h^{-1}}(k)$). Therefore, if we
choose partitions of unity $\\{\psi_{a}\\}$ subordinate to $\\{U_{\alpha}\\}$
and $\\{\chi_{\alpha}\\}$ subordinate to $\\{V_{\alpha}\\},$ we can define
$f(p)=\left(\sum\chi_{\alpha}(\pi(p))k_{\alpha}(p),\sum\psi_{\alpha}(\pi(p))h_{\alpha}(\pi_{K}(p))\right),$
which is $K\rtimes H$-equivariant because
$\displaystyle f(p(k,h))$
$\displaystyle=\left(\sum\chi_{\alpha}(\pi(p))k_{\alpha}((pk)h),\sum\psi_{\alpha}(\pi(p))h_{\alpha}(\pi_{K}(ph))\right)$
$\displaystyle=\left(\sum\chi_{\alpha}(\pi(p))\varphi_{h^{-1}}(k_{\alpha}(p)k),\sum\psi_{\alpha}(\pi(p))h_{\alpha}(\pi_{K}(p))h\right)$
$\displaystyle=f(p)(k,h).$
Thus $f$ is a classifying map for $P.$
### B.2 $LG\rtimes S^{1}$-bundles
We have shown in the previous section that principal $K\rtimes H$-bundles are
equivalent to $K$-bundles with a twisted equivariant $H$ action over
$H$-bundles. Consider now the case where $K=LG$ and $H=S^{1},$ as in chapter
4. We have already seen (see section 4.2) that there is a bijective
correspondence between isomorphism classes of principal $LG\rtimes
S^{1}$-bundles and isomorphism classes of principal $G$-bundles over
$S^{1}$-bundles. The result from section B.1, however, implies that we could
construct a principal $LG$-bundle over a circle bundle. Namely, the bundle
$P\to P/LG=(P\times S^{1})/LG\rtimes S^{1}$ is a principal $LG$ bundle. We
would like to understand the relationship between the $LG$-bundle we have
constructed and the $G$-bundle we have constructed in section 4.2. Consider
the map
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\widetilde{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P/LG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{P}/G}$
given by $f(p)=[p,1,1]$ (and where the induced map $LG\to G$ is the
homomorphism $\gamma\mapsto\gamma(1)$). This is a bundle map since
$\displaystyle f(p\gamma)$ $\displaystyle=[p(\gamma,1),1,1]$
$\displaystyle=[p,\gamma(1),1]$ $\displaystyle=[p,1,1]\gamma(1)$
$\displaystyle=f(p)\gamma(1).$
Therefore, we see that $\widetilde{P}\simeq P\times_{LG}G$ (via the
isomorphism $[p,g]\mapsto[p,g,1]$). So $\widetilde{P}$ is given by extending
the structure group of $P$ from $LG$ to $G.$
## Bibliography
* [1] A. Bergman and U. Varadarajan. Loop groups, Kaluza-Klein reduction and M-theory. J. High Energy Phys., (6):043, 28 pp. (electronic), 2005.
* [2] R. Bott and L. W. Tu. Differential forms in algebraic topology, volume 82 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982.
* [3] P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray, and D. Stevenson. Twisted $K$-theory and $K$-theory of bundle gerbes. Comm. Math. Phys., 228(1):17–45, 2002.
* [4] J.-L. Brylinski. Loop spaces, characteristic classes and geometric quantization, volume 107 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1993.
* [5] J.-L. Brylinski and D. A. McLaughlin. The geometry of degree four characteristic classes of line bundles on loop spaces I. Duke Math. J., 75(3):603–638, 1994.
* [6] A. Carey and J. Mickelsson. A gerbe obstruction to quantization of fermions on odd-dimensional manifolds with boundary. Lett. Math. Phys., 51(2):145–160, 2000.
* [7] A. Carey, J. Mickelsson, and M. Murray. Index theory, gerbes, and Hamiltonian quantization. Comm. Math. Phys., 183(3):707–722, 1997.
* [8] A. L. Carey, S. Johnson, M. K. Murray, D. Stevenson, and B.-L. Wang. Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories. Comm. Math. Phys., 259(3):577–613, 2005.
* [9] A. L. Carey and J. Mickelsson. The universal gerbe, Dixmier-Douady class, and gauge theory. Lett. Math. Phys., 59(1):47–60, 2002.
* [10] A. L. Carey, J. Mickelsson, and M. K. Murray. Bundle gerbes applied to quantum field theory. Rev. Math. Phys., 12(1):65–90, 2000.
* [11] A. L. Carey and M. K. Murray. String structures and the path fibration of a group. Comm. Math. Phys., 141(3):441–452, 1991.
* [12] J. Cheeger and J. Simons. Differential characters and geometric invariants. In Geometry and topology (College Park, Md., 1983/84), volume 1167 of Lecture Notes in Math., pages 50–80. Springer, Berlin, 1985\.
* [13] S. S. Chern and J. Simons. Characteristic forms and geometric invariants. Ann. of Math. (2), 99:48–69, 1974.
* [14] J. L. Dupont. Curvature and characteristic classes. Lecture Notes in Mathematics, Vol. 640. Springer-Verlag, Berlin, 1978\.
* [15] C. Ekstrand and J. Mickelsson. Gravitational anomalies, gerbes, and Hamiltonian quantization. Comm. Math. Phys., 212(3):613–624, 2000.
* [16] H. Garland and M. K. Murray. Kac-Moody monopoles and periodic instantons. Comm. Math. Phys., 120(2):335–351, 1988.
* [17] J. Giraud. Cohomologie non abélienne. Springer-Verlag, Berlin, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 179.
* [18] K. Gomi. Connections and curvings on lifting bundle gerbes. J. London Math. Soc. (2), 67(2):510–526, 2003.
* [19] R. S. Hamilton. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.), 7(1):65–222, 1982.
* [20] A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.
* [21] J. L. Heitsch and H. B. Lawson, Jr. Transgressions, Chern-Simons invariants and the classical groups. J. Differential Geometry, 9:423–434, 1974.
* [22] T. P. Killingback. World-sheet anomalies and loop geometry. Nuclear Phys. B, 288(3-4):578–588, 1987.
* [23] S. Kobayashi and K. Nomizu. Foundations of differential geometry. Vol I. Interscience Publishers, a division of John Wiley & Sons, New York-Lond on, 1963.
* [24] S. Kobayashi and K. Nomizu. Foundations of differential geometry. Vol. II. Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969\.
* [25] H. B. Lawson, Jr. and M.-L. Michelsohn. Spin geometry, volume 38 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1989.
* [26] S. MacLane. Categories for the working mathematician. Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5.
* [27] W. S. Massey. A basic course in algebraic topology, volume 127 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991.
* [28] E. Meinrenken. The basic gerbe over a compact simple Lie group. Enseign. Math. (2), 49(3-4):307–333, 2003.
* [29] J. Mickelsson. Current algebras and groups. Plenum Monographs in Nonlinear Physics. Plenum Press, New York, 1989.
* [30] J. Mickelsson. From gauge anomalies to gerbes and gerbal actions. math-ph/0812.1640, 2008.
* [31] J. Milnor. Remarks on infinite-dimensional Lie groups. In Relativity, groups and topology, II (Les Houches, 1983), pages 1007–1057. North-Holland, Amsterdam, 1984.
* [32] M. K. Murray. Bundle gerbes. J. London Math. Soc. (2), 54(2):403–416, 1996.
* [33] M. K. Murray and D. Stevenson. Bundle gerbes: stable isomorphism and local theory. J. London Math. Soc. (2), 62(3):925–937, 2000.
* [34] M. K. Murray and D. Stevenson. Yet another construction of the central extension of the loop group. In Geometric analysis and applications (Canberra, 2000), volume 39 of Proc. Centre Math. Appl. Austral. Nat. Univ., pages 194–200. Austral. Nat. Univ., Canberra, 2001.
* [35] M. K. Murray and D. Stevenson. Higgs fields, bundle gerbes and string structures. Comm. Math. Phys., 243(3):541–555, 2003.
* [36] M. S. Narasimhan and S. Ramanan. Existence of universal connections. Amer. J. Math., 83:563–572, 1961.
* [37] M. S. Narasimhan and S. Ramanan. Existence of universal connections. II. Amer. J. Math., 85:223–231, 1963.
* [38] S. Paycha and S. Rosenberg. Traces and characteristic classes on loop spaces. In Infinite dimensional groups and manifolds, volume 5 of IRMA Lect. Math. Theor. Phys., pages 185–212. de Gruyter, Berlin, 2004.
* [39] A. Pressley and G. Segal. Loop groups. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1986. Oxford Science Publications.
* [40] U. Schreiber, C. Schweigert, and K. Waldorf. Unoriented WZW models and holonomy of bundle gerbes. Comm. Math. Phys., 274(1):31–64, 2007.
* [41] G. Segal. Classifying spaces and spectral sequences. Inst. Hautes Études Sci. Publ. Math., (34):105–112, 1968.
* [42] S. Smale. Diffeomorphisms of the $2$-sphere. Proc. Amer. Math. Soc., 10:621–626, 1959.
* [43] D. Stevenson. Private communication.
* [44] B. Sury, T. Wang, and F.-Z. Zhao. Identities involving reciprocals of binomial coefficients. J. Integer Seq., 7(2):Article 04.2.8, 12 pp. (electronic), 2004\.
* [45] E. Witten. Global anomalies in string theory. In Symposium on anomalies, geometry, topology (Chicago, Ill., 1985), pages 61–99. World Sci. Publishing, Singapore, 1985.
|
arxiv-papers
| 2009-06-26T05:24:06 |
2024-09-04T02:49:03.564471
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Raymond Vozzo",
"submitter": "Raymond F. Vozzo",
"url": "https://arxiv.org/abs/0906.4843"
}
|
0906.5018
|
# Radiation from relativistic shocks with turbulent magnetic fields
K.-I. Nishikawa J. Niemiec M. Medvedev B. Zhang P. Hardee Å. Nordlund J.
Frederiksen Y. Mizuno H. Sol M. Pohl D. H. Hartmann M. Oka G. J. Fishman
National Space Science and Technology Center, Huntsville, AL 35805, USA
Institute of Nuclear Physics PAN, ul. Radzikowskiego 152, 31-342 Kraków,
Poland Department of Physics and Astronomy, University of Kansas, KS 66045,
USA Department of Physics, University of Nevada, Las Vegas, NV 89154, USA
Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL
35487, USA Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej
30, 2100 Copenhagen Ø, Denmark LUTH, Observatore de Paris-Meudon, 5 place
Jules Jansen, 92195 Meudon Cedex, France Department of Physics and Astronomy,
Iowa State University, Ames, IA 50011, USA Department of Physics and
Astronomy, Clemson University, Clemson, SC 29634, USA 1Space Sciences
Laboratory, University of California, Berkeley, California 94720, USA
NASA/MSFC, Huntsville, AL 35805, USA
###### Abstract
Using our new 3-D relativistic electromagnetic particle (REMP) code
parallelized with MPI, we investigated long-term particle acceleration
associated with a relativistic electron-positron jet propagating in an
unmagnetized ambient electron-positron plasma. The simulations were performed
using a much longer simulation system than our previous simulations in order
to investigate the full nonlinear stage of the Weibel instability and its
particle acceleration mechanism. Cold jet electrons are thermalized and
ambient electrons are accelerated in the resulting shocks. Acceleration of
ambient electrons leads to a maximum ambient electron density three times
larger than the original value. Behind the bow shock in the jet shock strong
electromagnetic fields are generated. These fields may lead to time dependent
afterglow emission. We calculated radiation from electrons propagating in a
uniform parallel magnetic field to verify the technique. We also used the new
technique to calculate emission from electrons based on simulations with a
small system. We obtained spectra which are consistent with those generated
from electrons propagating in turbulent magnetic fields with red noise. This
turbulent magnetic field is similar to the magnetic field generated at an
early nonlinear stage of the Weibel instability. A fully developed shock
within a larger system generates a jitter/synchrotron spectrum.
###### keywords:
acceleration of particles, galaxies, jets, gamma rays bursts, magnetic fields,
plasmas, shock waves, radiation
, , , , , , , , , , , , and
## 1 RPIC Simulations
Particle-in-cell (PIC) simulations can shed light on the physical mechanism of
particle acceleration that occurs in the complicated dynamics within
relativistic shocks. Recent PIC simulations of relativistic electron-ion and
electron-positron jets injected into an ambient plasma show that acceleration
occurs within the downstream jet nishi03 ; nishi05 ; Hededal & Nishikawa
(2005); nishi06 ; ram07 ; chang08 ; anat08a ; anat08b ; sironi09m . In
general, these simulations have confirmed that relativistic jets excite the
Weibel instability, which generates current filaments and associated magnetic
fields medv99 , and accelerates electrons Hededal & Nishikawa (2005); nishi06
; ram07 ; chang08 ; anat08a ; anat08b ; sironi09m .
Therefore, the investigation of radiation resulting from accelerated particles
(mainly electrons and positrons) in turbulent magnetic fields is essential for
understanding radiation mechanisms and their observable spectral properties.
In this report we present a new numerical method to obtain spectra from
particles self-consistently traced in our PIC simulations.
Figure 1: The averaged values of electron density (a) and field energy (b)
along the $x$ at $t=3750\omega_{\rm pe}^{-1}$. Fig. 1a shows jet electrons
(red), ambient electrons (blue), and the total electron density (black). Fig.
1b shows electric field energy (red) and magnetic field energy (blue) divided
by the total kinetic energy.
Pair Jets Injected into Unmagnetized Pair Plasmas using a Large System
We have performed simulations using a system with ($L_{\rm x},L_{\rm y},L_{\rm
z})=(4005\Delta,$ $131\Delta,131\Delta)$ ($\Delta=1$: grid size) and a total
of $\sim 1$ billion particles (12 particles$/$cell$/$species for the ambient
plasma) in the active grid zones nishi09 . In the simulations the electron
skin depth, $\lambda_{\rm ce}=c/\omega_{\rm pe}=10.0\Delta$, where
$\omega_{\rm pe}=(4\pi e^{2}n_{\rm e}/m_{\rm e})^{1/2}$ is the electron plasma
frequency and the electron Debye length $\lambda_{\rm e}$ is half of the grid
size. Here the computational domain is six times longer than in our previous
simulations nishi06 ; ram07 . The electron number density of the jet is
$0.676n_{\rm e}$, where $n_{\rm e}$ is the ambient electron density and
$\gamma=15$. The electron/positron thermal velocity of the jet is $v^{\rm
e}_{\rm j,th}=0.014c$, where $c=1$ is the speed of light.
Figure 2: The case with a strong magnetic field ($B_{\rm x}=3.7$) and larger
perpendicular velocity ($v_{\perp 1}=0.1c,v_{\perp 2}=0.12c$). The paths of
two electrons moving helically along the $x-$direction in a homogenous
magnetic field shown in the $x-y$-plane (a). The two electrons radiate a time
dependent electric field. An observer situated at great distance along the
n-vector sees the retarded electric field from the moving electrons at the
rest frame (b). The observed power spectrum at different viewing angles from
the two electrons (c). Frequency is in units of $\omega_{\rm pe}^{-1}$.
Figure 1 shows the averaged (in the $y-z$ plane) electron density and
electromagnetic field energy along the jet at $3750\omega_{\rm pe}^{-1}$. The
resulting profiles of jet (red), ambient (blue), and total (black) electron
density are shown in Fig. 1a. The ambient electrons are accelerated by the jet
electrons and pile up towards the front part of jet. At the earlier time the
ambient plasma density increases linearly behind the jet front. At the later
time the ambient plasma shows a rapid increase to a plateau behind the jet
front, with additional increase to a higher plateau farther behind the jet
front. The jet density remains approximately constant except near the jet
front.
Figure 3: Two-dimensional images in the $x-z$ plane at $y/\Delta=65$ for
$t=450\omega_{\rm pe}^{-1}.$ The colors indicate the x-component of current
density generated by the Weibel instability, with the x- and z-components of
magnetic field represented by arrows (a). Phase space distributions as a
function of $x/\Delta-\gamma v_{\rm x}$ plotted for the jet (red) and ambient
(blue) electrons at the same time.
The Weibel instability remains excited by continuously injected jet particles
and the electromagnetic fields are maintained at a high level, about four
times that seen in a previous, much shorter grid simulation system ($L_{\rm
x}=640\Delta$). At the earlier simulation time a large electromagnetic
structure is generated and accelerates the ambient plasma. As shown in Fig.
1b, at the later simulation time the strong magnetic field extends up to
$x/\Delta=2,000$. These strong fields become very small beyond $x/\Delta=2000$
in the shocked ambient region nishi06 ; ram07 .
Figure 4: Two-dimensional images in the $x-z$ plane at $y/\Delta=65$ for
$t=525\omega_{\rm pe}^{-1}.$ The colors indicate the x-component of current
density generated by the Weibel instability, with the x- and z-components of
magnetic field represented by arrows (a). Phase space distributions as a
function of $x/\Delta-\gamma v_{\rm x}$ plotted for the jet (red) and ambient
(blue) electrons at the same time.
The acceleration of ambient electrons becomes visible when jet electrons pass
about $x/\Delta=500$. The maximum density of accelerated ambient electrons is
attained at $t=1750\omega_{\rm pe}^{-1}$. The maximum density gradually
reaches a plateau as seen in Fig. 1a. The maximum electromagnetic field energy
is located at $x/\Delta=1,700$ as shown in Fig. 1b.
### 1.1 New Numerical Method for Calculating Synchrotron Emission
Let a particle be at position ${\bf{r}_{0}}(t)$ at time $t$ nishi08 ; Hededal
(2005); Hededal & Nordlund (2005). At the same time, we observe the electric
field from the particle from position $\bf{r}$. However, because of the finite
velocity of light, we observe the particle at an earlier position
$\bf{r}_{0}(\rm{t}^{{}^{\prime}})$ where it was at the retarded time
$t^{{}^{\prime}}=t-\delta t^{{}^{\prime}}=t-\bf{R}(\rm{t}^{{}^{\prime}})/c$.
Here $\bf{R}(\rm{t}^{{}^{\prime}})=|\bf{r}-\bf{r}_{0}(\rm{t}^{{}^{\prime}})|$
is the distance from the charge (at the retarded time $t^{{}^{\prime}}$) to
the observer.
After some calculation and simplifying assumptions the total energy $W$
radiated per unit solid angle per unit frequency from a charged particle
moving with instantaneous velocity $\boldsymbol{\beta}$ under acceleration
$\boldsymbol{\dot{\beta}}$ can be expressed as Rybicki and Lightman (1979);
Jackson (1999)
$\displaystyle\frac{d^{2}W}{d\Omega d\omega}$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}cq^{2}}{16\pi^{3}}\left|\int^{\infty}_{-\infty}\frac{\bf{n}\times[(\bf{n}-\boldsymbol{\beta})\times\boldsymbol{\dot{\beta}}]}{(1-\boldsymbol{\beta}\cdot\bf{n})^{2}}e^{i\omega(t^{{}^{\prime}}-\bf{n}\cdot\bf{r}_{0}({\rm
t}^{{}^{\prime}})/{\rm c})}dt^{{}^{\prime}}\right|^{2}$ (1)
Here,
$\bf{n}\equiv\bf{R}(\rm{t}^{{}^{\prime}})/|\bf{R}(\rm{t}^{{}^{\prime}})|$ is a
unit vector that points from the particle’s retarded position towards the
observer.
The observer’s viewing angle is set by the choice of $\bf{n}$ ($n_{\rm
x}^{2}+n_{\rm y}^{2}+n_{\rm z}^{2}=1$). The choice of unit vector $\bf{n}$
along the direction of propagation of the jet (hereafter taken to be the
$x$-axis) corresponds to head-on emission. For any other choice of $\bf{n}$
(e.g., $\theta_{\gamma}=1/\gamma$), off-axis emission is seen by the observer.
In order to calculate radiation from relativistic jets propagating along the
$x$ direction nishi08 we consider a test case which includes a parallel
magnetic field ($B_{\rm x}$), and jet velocity of $v_{\rm j1,2}=0.99c$. Two
electrons are injected with different perpendicular velocities ($v_{\perp
1}=0.1c,v_{\perp 2}=0.12c$). A maximum Lorenz factor of
$\gamma_{\max}=\\{(1-(v_{\rm j2}^{2}+v_{\perp 2}^{2})/c^{2}\\}^{-1/2}=13.48$
is calculated with the larger perpendicular velocity.
Figure 2 shows electron trajectories in the $x-y$ plane (red: $v_{\perp
2}=0.12c$, blue: $v_{\perp 1}=0.1c$) (a: left panel), the radiation (retarded)
electric field (b: middle panel), and spectra (right panel) for the case
$B_{\rm x}=3.70$. The two electrons are propagating left to right with
gyration in the $y-z$ plane (not shown). The gyroradius is about $0.44\Delta$
for the electron with the larger perpendicular velocity. The seven curves show
the power spectrum at viewing angles of 0∘ (red), 10∘ (orange), 20∘ (yellow),
30∘ (moss green), 45∘ (green), 70∘ (light blue), and 90∘ (blue). The higher
frequencies become stronger at the $10^{\circ}$ viewing angle. The critical
angle for off-axis radiation $\theta_{\gamma}=180^{\circ}/(\pi\gamma_{\max})$
for this case is 4.25∘. As shown in this panel, the spectrum at a larger
viewing angle ($>20^{\circ}$) has smaller amplitude.
Since the jet plasma has a large velocity $x$-component in the simulation
frame, the radiation from the particles (electrons and positrons) is strongly
beamed along the $x$-axis (jitter radiation) Medvedev (2000, 2006).
Equations 6.30a and 6.30b show that the radiation with the viewing angle
$\alpha=0$ disappears (see Fig. 6.5 in the textbook of Rybicki and Lightman
Rybicki and Lightman (1979)). However, based on two other textbooks, radiation
at the viewing angle $0^{\circ}$ should not vanish beke66 ; land80 . This
aspect is shown in Fig. 2c, and at the higher frequency the amplitude at the
viewing angle $10^{\circ}$ is stronger than that with viewing angle
$0^{\circ}$.
### 1.2 The Standard Synchrotron Radiation Model
A synchrotron shock model is widely adopted to describe the radiation
mechanism in the external shock thought to be responsible for observed broad-
band GRB afterglows Zhang & Meszaros (2004); Piran (2005a, b); Zhang (2007);
Nakar (2007). Associated with this model are three major assumptions that are
adopted in almost all current GRB afterglow models. Firstly, electrons are
assumed to be “Fermi” accelerated at the relativistic shocks and to have a
power-law distribution with a power-law index $p$ upon acceleration, i.e.
$N(E_{\rm e})dE_{\rm e}\propto E^{-p}dE_{\rm e}$. This is consistent with
recent PIC simulations of the shock formation and particle acceleration
anat08b and also some Monte Carlo models Achterberg et al. (2001); Ellison &
Double (2002); Lemoine & Pelletier (2003), but see Niemiec, & Ostrowski
(2006); Niemiec, Ostrowski, & Pohl (2006). Secondly, a fraction $\epsilon_{\rm
e}$ (generally taken to be $\leq 1$) of the total electrons associated with
ISM baryons are accelerated, and the total electron energy is a fraction
$\epsilon_{\rm e}$ of the total internal energy in the shocked region.
Thirdly, the strength of the magnetic fields in the shocked region is unknown,
but its energy density ($B^{2}/8\pi$) is assumed to be a fraction
$\epsilon_{B}$ of the internal energy. These assumed “micro-physics”
parameters, $p,\epsilon_{\rm e}$ and $\epsilon_{\rm B}$, whose values are
obtained from spectral fits Panaitescu, A., Kumar (2001); Yost et al. (2003)
reflect a lack of knowledge of the underlying microphysics Waxman (2006).
The typical observed emission frequency from an electron with (comoving)
energy $\gamma_{\rm e}m_{\rm e}c^{2}$ in a frame with a bulk Lorentz factor
$\Gamma$ is $\nu=\Gamma\gamma_{\rm e}^{2}(eB/2\pi m_{\rm e}c)$. Three critical
frequencies are defined by three characteristic electron energies. These are
$\nu_{\rm m}$ (the injection frequency), $\nu_{\rm c}$ (the cooling
frequency), and $\nu_{\rm M}$ (the maximum synchrotron frequency). In our
simulations of GRB afterglows, there is one additional relevant frequency,
$\nu_{\rm a}$, due to synchrotron self-absorption at lower frequencies
Meszaros, Rees, & Wijer (1998); Sari, Piran, & Narayan (1998); Zhang (2007);
Nakar (2007).
The general agreement between the blast wave dynamics and the direct
measurements of the fireball size argue for the validity of this model’s
dynamics Zhang (2007); Nakar (2007). The shock is most likely collisionless,
i.e. mediated by plasma instabilities Waxman (2006). The electromagnetic
instabilities mediating the afterglow shock are expected to generate magnetic
fields. Afterglow radiation was therefore predicted to result from synchrotron
emission of shock accelerated electrons Meszaros & Rees (1997). The observed
spectrum of afterglow radiation is indeed remarkably consistent with
synchrotron emission of electrons accelerated to a power-law distribution,
providing support for the standard afterglow model based on synchrotron
emission of shock accelerated electrons Piran (1999, 2000, 2005a); Zhang &
Meszaros (2004); Meszaros (2002, 2006); Zhang (2007); Nakar (2007).
In order to determine the luminosity and spectrum of synchrotron radiation,
the strength of the magnetic field ($\epsilon_{\rm B}$) and the energy
distribution of the electrons ($p$) must be determined. Due to the lack of a
first principles theory of collisionless shocks, a purely phenomenological
approach to the model of afterglow radiation was ascribed without
investigating in detail the processes responsible for particle acceleration
and magnetic field generation Waxman (2006). Rather, one simply assumes that a
fraction $\epsilon_{\rm B}$ of the post-shock thermal energy density is
carried by the magnetic field, that a fraction $\epsilon_{\rm e}$ is carried
by electrons, and that the energy distribution of the electrons is a power-
law, $d\log n_{\rm e}/d\log\varepsilon=p$ (above some minimum energy
$\varepsilon_{0}$ which is determined by $\epsilon_{\rm e}$ and $p$),
$\epsilon_{\rm B}$, $\epsilon_{\rm e}$ and $p$ are treated as free parameters,
determined by observations. It is important to clarify here that the
constraints implied on these parameters by the observations are independent of
any assumptions regarding the nature of the afterglow shock and the processes
responsible for particle acceleration or magnetic field generation. Any model
should satisfy these observational constraints.
The properties of synchrotron (or “jitter”) emission from relativistic shocks
will be determined by the magnetic field strength and structure and the
electron energy distribution behind the shock. The characteristics of jitter
radiation may be important to understanding the complex time evolution and/or
spectral structure in gamma-ray bursts Preece et al. (1998). For example,
jitter radiation has been proposed as a means to explain GRB spectra below the
peak frequency that are harder than the “line of death” spectral index
associated with synchrotron emission Medvedev (2000, 2006), i.e., the observed
spectral power scales as $F_{\nu}\propto\nu^{2/3}$, whereas synchrotron
spectra are $F_{\nu}\propto\nu^{1/3}$ or softer Medvedev (2006). Thus, it is
essential to calculate radiation production by tracing electrons (positrons)
in self-consistently treated small-scale electromagnetic fields.
### 1.3 Calculating Synchrotron and Jitter Emission from Electron
Trajectories in Self-consistently Generated Magnetic Fields
In order to obtain the spectrum of synchrotron (jitter) emission, we consider
an ensemble of electrons selected in the region where the Weibel instability
has fully grown and electrons are accelerated in the generated magnetic
fields. In order to validate our numerical method we performed simulations
using a small system with ($L_{\rm x},L_{\rm y},L_{\rm
z})=(645\Delta,131\Delta,131\Delta)$ ($\Delta=1$: grid size) and a total of
$\sim 0.5$ billion particles (12 particles$/$cell$/$species for the ambient
plasma) in the active grid zones nishi06 . First we performed simulations
without calculating radiation up to $t=450\omega_{\rm pe}^{-1}$. The jet front
is located around about $x/\Delta=480$. We selected 12,150 electrons for each
jet and ambient electrons randomly. Recently, a similar calculation has been
carried out for the radiation from accelerated electrons in laser-wakefield
acceleration Martins et al. (2009) and in shocks Sironi & Spitkovsky (2009).
Figure 3 shows (a) the current filaments generated by the Weibel instability
and (b) the phase space of $x/\Delta-\gamma V_{\rm x}$ for jet electrons (red)
and ambient electrons (blue) at $t=450\omega_{\rm pe}^{-1}$.
Figure 5: Spectra obtained from jet and ambient electrons for the two viewing
angles. Spectra with jet electrons are shown in red ($0^{\circ}$) and orange
($5^{\circ}$). Spectra from ambient electrons show the lowest levels by blue
($0^{\circ}$) and light blue ($5^{\circ}$).
Figure 4 shows (a) the $x$-component of current density generated by the
Weibel instability and (b) the phase space of jet electrons and ambient
electrons after $t_{\rm s}=75\omega_{\rm pe}^{-1}$ (at $t=525\omega_{\rm
pe}^{-1}$).
Figure 6: The case with a weak magnetic field ($B_{\rm x}=0.37$) and small
perpendicular velocity ($v_{\perp 1}=0.01c,v_{\perp 2}=0.012c$). The paths of
two electrons moving helically along the $x-$direction in a homogenous
magnetic field shown in the $x-y$-plane (a). The two electrons radiate a time
dependent electric field. An observer situated at great distance along the
n-vector sees the retarded electric field from the moving electrons (b). The
observed power spectrum at different viewing angles from the two electrons
(c). Frequency is in units of $\omega_{\rm pe}^{-1}$.
We calculated the emission from 12,150 electrons during the sampling time
$t_{\rm s}=t_{\rm 2}-t_{\rm 1}=75\omega_{\rm pe}^{-1}$ with Nyquist frequency
$\omega_{\rm N}=1/2\Delta t=200\omega_{\rm pe}$ where $\Delta
t=0.005\omega_{\rm pe}^{-1}$ is the simulation time step and the frequency
resolution $\Delta\omega=1/t_{\rm s}=0.0133\omega_{\rm pe}$.
The spectra shown in Fig. 5 are obtained for emission from jet electrons and
ambient electrons separately. In this case the spectra are calculated for
head-on radiation ($0^{\circ}$) and $5^{\circ}$. The radiation from jet
electrons show Bremsstrahlung-like spectra as a red line ($0^{\circ}$) and
orange line ($5^{\circ}$) Hededal (2005). The spectra with jet electrons are
different from the spectra shown in Fig. 2c. Since the magnetic fields
generated by the Weibel instability are rather weak and the jet electrons are
not much accelerated, the trajectories of jet electrons are almost straight
with only a slight bent.
We compare these spectra with our known spectra obtained from two (jet)
electrons, the case with a parallel magnetic field ($B_{\rm x}=0.37$), and jet
velocity of $v_{\rm j1,2}=0.99c$. Two electrons are injected with different
perpendicular velocities ($v_{\perp 1}=0.01c,v_{\perp 2}=0.012c$). A maximum
Lorenz factor of $\gamma_{\max}=\\{(1-(v_{\rm j2}^{2}+v_{\perp
2}^{2})/c^{2}\\}^{-1/2}=7.114$ accompanies the larger perpendicular velocity.
The critical angle for off-axis radiation
$\theta_{\gamma}=180^{\circ}/(\pi\gamma_{\max})$ for this case is 8.05∘.
Comparing the spectra with Figs. 5 and 6c we find similarities. The lower
frequencies have flat spectra and the higher frequencies decrease
monotonically. The slope in Fig. 5 is less steep than that in Fig. 6c. This is
due to the fact that the spread of Lorenz factors of jet electrons is larger
and the average Lorenz factor is larger as well. Furthermore, even the
magnetic field strength is not so large, however the slope of the spectra
seems to be consistent with Fig. 7.16 (left) with the turbulent magnetic field
with the red noise ($\mu=-3$) in Hededal’s Ph. D. thesis Hededal (2005). We
obtained several different parameters with jet electrons and ambient magnetic
field. However, the strength of the magnetic fields generated by the Weibel
instability is small, therefore the spectra for these cases are very similar
to Fig. 5. As shown in Fig. 7.12 in Hededal’s Ph. D. thesis Hededal (2005),
the trajectories of jet electrons have to be chaotic to produce a jitter-like
spectrum Fig. 7.22 Hededal (2005).
As shown in Fig. 1b, the magnetic field energy in the region $x/\Delta<500$ is
small ($\epsilon_{\rm B}<0.07$), therefore, as expected, the spectra look like
those emitted from electrons propagating in a turbulent magnetic field with
red noise (see also Figs. 3a and 4a).
### 1.4 Discussions
Emission obtained with the method described above is obtained self-
consistently, and automatically accounts for magnetic field structures on
small scales responsible for jitter emission. By performing such calculations
for simulations with different parameters, we can investigate and compare the
different regimes of jitter- and synchrotron-type emission Medvedev (2000,
2006). The feasibility of this approach has already been demonstrated Hededal
(2005); Hededal & Nordlund (2005), and its implementation is straightforward.
Thus, we should be able to address the low frequency GRB spectral index
violation of the synchrotron spectrum line of death Medvedev (2006).
Medvedev and Spitkovsky recently showed that electrons may cool efficiently at
or near the shock jump and are capable of emitting a large fraction of the
shock energy Medvedev & Spitkovsky (2009). Such shocks are well-resolved in
existing PIC simulations; therefore, the microscopic structure can be studied
in detail. Since most of the emission in such shocks would originate from the
vicinity of the shock, the spectral power of the emitted radiation can be
directly obtained from finite-length simulations and compared with
observational data.
As shown in Fig. 1, behind the trailing shock the electrons are accelerated
and strong magnetic fields are generated. Therefore, this region seems to
produce the emission that is observed by satellites. We will calculate more
spectra based on our RPIC simulations and compare in detail with Fermi data.
### 1.5 Achknowledgments
This work is supported by NSF-AST-0506719, AST-0506666, AST-0908040,
AST-0908010, NASA-NNG05GK73G, NNX07AJ88G, NNX08AG83G, NNX08AL39G, and
NNX09AD16G. JN was supported by MNiSW research projects 1 P03D 003 29 and N
N203 393034, and The Foundation for Polish Science through the HOMING program,
which is supported through the EEA Financial Mechanism.Simulations were
performed at the Columbia facility at the NASA Advanced Supercomputing (NAS).
and IBM p690 (Copper) at the National Center for Supercomputing Applications
(NCSA) which is supported by the NSF. Part of this work was done while K.-I.
N. was visiting the Niels Bohr Institute. Support from the Danish Natural
Science Research Council is gratefully acknowledged. This report was finalized
during the program “Particle Acceleration in Astrophysical Plasmas” at the
Kavli Institute for Theoretical Physics which is supported by the National
Science Foundation under Grant No. PHY05-51164.
## References
* (1)
* (2) K.-I. Nishikawa, P. Hardee, G. Richardson, R. Preece, H. Sol, H., and G. J. Fishman, _ApJ_ , 595, 555–563 (2003)
* (3) K.-I. Nishikawa, P. Hardee, G. Richardson, R. Preece, H. Sol, H., and G. J. Fishman, _ApJ_ , 623, 927–937 (2005)
* Hededal & Nishikawa (2005) C. B. Hededal, and K.-I. Nishikawa, 2005, _ApJ_ , 623, L89–L92, (2005)
* (5) K.-I. Nishikawa, P. Hardee, C. B. Hededal, and G. J. Fishman, _ApJ_ , 642, 1267–1274 (2006)
* (6) E. Ramirez-Ruiz, K.-I. Nishikawa, and C. B. Hededal, _ApJ_ , 671, 1877–1885 (2007)
* (7) P. Chang, A. Spitkovsky, and J. Arons, _ApJ_ 674 378–387 (2008)
* (8) A. Spitkovsky, _ApJ_ 673 L39–L42 (2008a)
* (9) A. Spitkovsky, _ApJ_ 682 L5–L8 (2008b)
* (10) L. Sironi, and A. Spitkovsky, _ApJ_ 698 1523–1549 (2009)
* (11) M. V. Medvedev, and A. Loeb, _ApJ_ , 526 697–706 (1999)
* (12) K.-I. Nishikawa, J. Niemiec, P. E. Hardee, M. Medvedev, H. Sol, Y. Mizuno, B. Zhang, M. Pohl, M. Oka, and D. H. Hartmann, _ApJ_ , 689 L10–L13 (2009)
* (13) K.-I. Nishikawa, J. Niemiec, H. Sol, M. Medvedev, et al. in Proceedings of The 4th Heidelberg International Symposium on High Energy Gamma-Ray Astronomy, eds, F. A. Aharonian, W. Hofmann, F. Rieger 1085 589–593 (2009) (arXiv:0809.5067)
* Hededal (2005) C.B. Hededal, Ph.D. thesis (2005) (arXiv:astro-ph/0506559)
* Hededal & Nordlund (2005) C.B. Hededal, and Å. Nordlund, _ApJL_ , submitted (2005) (arXiv:astro-ph/0511662)
* Rybicki and Lightman (1979) G. B. Rybicki and A. P. Lightman, Radiative Processes in Astrophysics, John Wiley & Sons, New York, 1979
* Jackson (1999) J. D. Jackson, Classical Electrodynamics, Interscience, 1999
* Medvedev (2000) M. V. Medvedev, _ApJ_ , 540 704–714 (2000)
* Medvedev (2006) M. V. Medvedev, _ApJ_ , 637 869–872 (2006)
* (20) G. Bekefi, Radiative Processes in Plasmas, John Wiley & Sons, New York, 1966
* (21) L. D. Landau, and E. M. Lifshitz, The Classical Theory of Fields, Elsevier Science & Technology Books, 1980
* Zhang & Meszaros (2004) B. Zhang, and P. Meszaros, _Int. J. Mod. Phys._ , A19 2385–2472 (2004)
* Piran (2005a) T. Piran, _Rev. Mod. Phys._ , 76 1143–1210 (2005a)
* Piran (2005b) T. Piran, in the proceedings of Magnetic Fields in the Universe, Angra dos Reis, Brazil, Nov. 29-Dec 3, 2004, Ed. E. de Gouveia del Pino, _AIPC_ , 784, 164–174, (2005b)
* Zhang (2007) B. Zhang, _Chin. J. Astron. Astrophys._ , 7, 1–50 (2007)
* Nakar (2007) E. Nakar, _Phys. Reports_ , 442, 166–236 (2007)
* Achterberg et al. (2001) A. Achterberg, Y. A. Gallant, J. G. Kirk, and A. X. Guthmann, _MNRAS_ , 328, 393–408 (2001)
* Ellison & Double (2002) D. C. Ellison, and G. P. Double, _Astroparticle Phys._ , 18 213–228 (2002)
* Lemoine & Pelletier (2003) M. Lemoine, and G. Pelletier, _ApJ_ , 589 L73–L76 (2003)
* Niemiec, & Ostrowski (2006) J. Niemiec, and M. Ostrowski, _ApJ_ 641 984–992 (2006)
* Niemiec, Ostrowski, & Pohl (2006) J. Niemiec, M. Ostrowski, and M. Pohl, _ApJ_ , 650 1020–1027 (2006)
* Panaitescu, A., Kumar (2001) A. Panaitescu, and P. Kumar, _ApJ_ , 560 L49–L52 (2001)
* Yost et al. (2003) S. A. Yost, F. A. Harrison, R. Sari, D. A. Frail, _ApJ_ , 597 459–473 (2003)
* Waxman (2006) E. Waxman, _Plasma Phys. Control. Fusion_ , 48 B137–B151 (2006)
* Meszaros, Rees, & Wijer (1998) P. Meszaros, M. J. Rees, R. A. M. J., Wijers, _ApJ_ , 499 301–308 (1998)
* Sari, Piran, & Narayan (1998) R. Sari, T. Piran, and R. Narayan, _ApJ_ , 497 L17–L20 (1998)
* Meszaros & Rees (1997) P. Meszaros, and M. J. Rees, _ApJ_ , 482 L29–L32 (1997)
* Meszaros (2002) P. Meszaros, _ARAA_ , 40 137–169 (2002)
* Meszaros (2006) P. Meszaros, _Rept. Prog. Phys._ , 69 2259–21 (2006)
* Piran (1999) T. Piran, _Phys. Rep._ , 314 575–667 (1999)
* Piran (2000) T. Piran, _Phys. Rep._ , 333 529–553 (2000)
* Preece et al. (1998) R. D., Preece, M. S. Briggs, R. S. Mallozzi, G. N. Pendleton, W. S. Paciesas, and D. L. Band, _ApJ_ , 506 L23–L26 (1998)
* Martins et al. (2009) J.L. Martins, S.F. Martins, R.A. Fonseca and L.O. Silva, _Proc. of SPIE_ , 7359 73590V-1–8 (2009)
* Sironi & Spitkovsky (2009) L. Sironi and A. Spitkovsky, _ApJ_ , submitted (2009) (arXiv:0908.3193)
* Medvedev & Spitkovsky (2009) M.V. Medvedev, and A. Spitkovsky, _ApJ_ , 700 956–964 (2009)
|
arxiv-papers
| 2009-06-26T23:05:43 |
2024-09-04T02:49:03.593107
|
{
"license": "Public Domain",
"authors": "K.-I. Nishikawa, J. Niemiec, M. Medvedev, B. Zhang, P. Hardee, A.\n Nordlund, J. Frederiksen, Y. Mizuno, H. Sol, M. Pohl, D. H. Hartmann, M. Oka,\n and G. J. Fishman",
"submitter": "Ken-Ichi Nishikawa",
"url": "https://arxiv.org/abs/0906.5018"
}
|
0906.5055
|
# Spectra of Upper-triangular Operator Matrix111This work is supported by the
NSF of China (Grant Nos. 10771034, 10771191 and 10471124) and the NSF of
Fujian Province of China (Grant Nos. Z0511019, S0650009).
Shifang Zhang1, Huaijie Zhong2, Junde Wu1222Corresponding author: E-mail:
[email protected]
1Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
2Department of Mathematics, Fujian Normal University, Fuzhou 350007, P. R.
China
Abstract Let $X$ and $Y$ be Banach spaces, $A\in B(X)$, $B\in B(Y)$, $C\in
B(Y,X)$, $M_{C}=\left(\begin{array}[]{cc}A&C\\\ 0&B\\\ \end{array}\right)$ be
the operator matrix acting on the Banach space $X\oplus Y$. In this paper, we
give out 20 kind spectra structure of $M_{C}$, decide 18 kind spectra filling-
in-hole properties of $M_{C}$, and present 10 examples to show that some
conclusions about the spectra structure or filling-in-hole properties of
$M_{C}$ are not true.
Keywords: Banach spaces, Upper-triangular operator matrix, Spectra, filling-
in-hole.
## 1 Introduction and basic concepts
It is well known that if $H$ is a Hilbert space and $T$ is a bounded linear
operator defined on $H$ and $H_{1}$ is an invariant closed subspace of $T$,
then $T$ can be represented for the form of
$T=\left(\begin{array}[]{cc}*&*\\\ 0&*\\\ \end{array}\right):H_{1}\oplus
H_{1}^{\perp}\rightarrow H_{1}\oplus H_{1}^{\perp},$
which motivated the interest in $2\times 2$ upper-triangular operator matrices
(see [2], [3], [6], [8-25], [28-32]). Throughout this paper, let $X$ and $Y$
be complex infinite dimensional Banach spaces and $B(X,Y)$ be the set of all
bounded linear operators from $X$ into $Y$, for simplicity, we write $B(X,X)$
as $B(X)$. Let $X^{*}$ be the dual space of $X$. If $T\in B(X,Y)$, then
$T^{*}\in B(Y^{*},X^{*})$ denotes the dual operator of $T$.
For $T\in B(X,Y)$, let $R(T)$ and $N(T)$ denote the range and kernel of $T$,
respectively, and denote $\alpha(T)=\dim N(T)$, $\beta(T)=\dim Y/R(T)$. If
$T\in B(X)$, the ascent $asc(T)$ of $T$ is defined to be the smallest
nonnegative integer $k$ (if it exists) which satisfies that
$N(T^{k})=N(T^{k+1})$. If such $k$ does not exist, then the ascent of $T$ is
defined as infinity. Similarly, the descent $des(T)$ of $T$ is defined as the
smallest nonnegative integer $k$ (if it exists) for which
$R(T^{k})=R(T^{k+1})$ holds. If such $k$ does not exist, then $des(T)$ is
defined as infinity, too. If the ascent and the descent of $T$ are finite,
then they are equal (see [13]). For $T\in B(X)$, if $R(T)$ is closed and
$\alpha(T)<\infty$, then $T$ is said to be an upper semi-Fredholm operator, if
$\beta(T)<\infty$, then $T$ is said to be a lower semi-Fredholm operator. If
$T\in B(X)$ is either upper or lower semi-Fredholm operator, then $T$ is said
to be a semi-Fredholm operator. For semi-Fredholm operator $T$, its index ind
$(T)$ is defined as ind $(T)=\alpha(T)-\beta(T).$
Now, we introduce the following important operator classes:
The sets of all invertible operators, bounded below operators, surjective
operators, left invertible operators, right invertible operators on $X$ are
defined, respectively, by
$\displaystyle G(X):=\\{T\in B(X):T\makebox{ is invertible}\\},$
$\displaystyle G_{+}(X):=\\{T\in B(X):T\makebox{ is injective and}\
R(T)\makebox{ is closed}\\},$ $\displaystyle G_{-}(X):=\\{T\in B(X):T\makebox{
is surjective}\\},$ $\displaystyle G_{l}(X):=\\{T\in B(X):T\makebox{ is left
invertible}\\},$ $\displaystyle G_{r}(X):=\\{T\in B(X):T\makebox{ is right
invertible}\\}.$
The sets of all Fredholm operators, upper semi-Fredholm operators, lower semi-
Fredholm operators, left semi-Fredholm operators, right semi-Fredholm
operators on $X$ are defined, respectively, by
$\displaystyle\Phi(X):=\\{T\in B(X):\alpha(T)<\infty\makebox{ and
}\beta(T)<\infty\\},$ $\displaystyle\Phi_{+}(X):=\\{T\in
B(X):\alpha(T)<\infty\makebox{ and }R(T)\makebox{ is closed}\\},$
$\displaystyle\Phi_{-}(X):=\\{T\in B(X):\beta(T)<\infty\\},$
$\displaystyle\Phi_{l}(X):=\\{T\in B(X):R(T)\makebox{ is a closed and
complemented subspace of }X\makebox{ and }\,\,\alpha(T)<\infty\\},$
$\displaystyle\Phi_{r}(X):=\\{T\in B(X):N(T)\makebox{ is a closed and
complemented subspace of }X\makebox{ and }\,\,\beta(T)<\infty\\}.$
The sets of all Weyl operators, upper semi-Weyl operators, lower semi-Weyl
operators, left semi-Weyl operators, right semi-Weyl operators on $X$ are
defined, respectively, by
$\displaystyle\Phi_{0}(X):=\\{T\in\Phi(X):\makebox{ind}(T)=0\\},$
$\displaystyle\Phi_{+}^{-}(X):=\\{T\in\Phi_{+}(X):\makebox{ind}(T)\leq 0\\},$
$\displaystyle\Phi_{-}^{+}(X):=\\{T\in\Phi_{-}(X):\makebox{ind}(T)\geq 0\\},$
$\displaystyle\Phi_{lw}(X):=\\{T\in\Phi_{l}(X):\makebox{ind}(T)\leq 0\\},$
$\displaystyle\Phi_{rw}(X):=\\{T\in\Phi_{r}(X):\makebox{ind}(T)\geq 0\\}.$
The sets of all Browder operators, upper semi-Browder operators, lower semi-
Browder operators, left semi-Browder operators, right semi-Browder operators
on $X$ are defined, respectively, by
$\displaystyle\Phi_{b}(X):=\\{T\in\Phi(X):asc(T)=des(T)<\infty\\},$
$\displaystyle\Phi_{ab}(X):=\\{T\in\Phi_{+}(X):asc(T)<\infty\\},$
$\displaystyle\Phi_{sb}(X):=\\{T\in\Phi_{-}(X):des(T)<\infty\\},$
$\displaystyle\Phi_{lb}(X):=\\{T\in\Phi_{l}(X):asc(T)<\infty\\},$
$\displaystyle\Phi_{rb}(X):=\\{T\in\Phi_{r}(X):des(T)<\infty\\}.$
By the help of above set classes, for $T\in B(X)$, we can define its
corresponding 22 kind spectra, respectively, as following:
the spectrum: $\sigma(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in
G(X)\\}$,
the approximate point spectrum:
$\sigma_{a}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in G_{-}(X)\\}$,
the defect spectrum: $\sigma_{su}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in G_{+}(X)\\}$,
the left spectrum: $\sigma_{l}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in
G_{l}(X)\\}$,
the right spectrum: $\sigma_{r}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in G_{r}(X)\\}$,
the essential spectrum: $\sigma_{e}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi(X)\\}$,
the upper semi-Fredholm spectrum:
$\sigma_{SF+}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{+}(X)\\},$
the lower semi-Fredholm spectrum:
$\sigma_{SF-}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{-}(X)\\},$
the left semi-Fredholm spectrum:
$\sigma_{le}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{l}(X)\\},$
the right semi-Fredholm spectrum:
$\sigma_{re}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{r}(X)\\},$
the Weyl spectrum: $\sigma_{w}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{0}(X)\\},$
the upper semi-Weyl spectrum:
$\sigma_{aw}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{+}^{-}(X)\\},$
the lower semi-Weyl spectrum:
$\sigma_{sw}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{-}^{+}(X)\\},$
the left semi-Weyl spectrum:
$\sigma_{lw}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{lw}(X)\\},$
the right semi-Weyl spectrum:
$\sigma_{rw}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{rw}(X)\\},$
the Browder spectrum: $\sigma_{b}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{b}(X)\\},$
the Browder essential approximate point spectrum:
$\sigma_{ab}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{ab}(X)\\},$
the lower semi-Browder spectrum:
$\sigma_{sb}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{sb}(X)\\},$
the left semi-Browder spectrum:
$\sigma_{lb}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{lb}(X)\\},$
the right semi-Browder spectrum:
$\sigma_{rb}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{rb}(X)\\},$
the Kato spectrum: $\sigma_{K}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{+}(X)\cup\Phi_{-}(X)\\},$
the third Kato spectrum:
$\sigma_{K_{3}}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{l}(X)\cup\Phi_{r}(X)\\}.$
It is well known that all of these spectra are compact nonempty subsets of
complex plane ${\mathbb{C}}$ and have the following relationship:
(1)
$\sigma_{K}(T)\subseteq\sigma_{SF+}(T)\subseteq\sigma_{aw}(T)\subseteq\sigma_{ab}(T)\subseteq\sigma_{b}(T),$
(2)
$\sigma_{K}(T)\subseteq\sigma_{SF-}(T)\subseteq\sigma_{sw}(T)\subseteq\sigma_{sb}(T)\subseteq\sigma_{b}(T),$
(3)
$\sigma_{K_{3}}(T)\subseteq\sigma_{le}(T)\subseteq\sigma_{lw}(T)\subseteq\sigma_{lb}(T)\subseteq\sigma_{b}(T),$
(4)
$\sigma_{K_{3}}(T)\subseteq\sigma_{re}(T)\subseteq\sigma_{rw}(T)\subseteq\sigma_{rb}(T)\subseteq\sigma_{b}(T),$
(5)
$\partial(\sigma_{b}(T))\subseteq\partial(\sigma_{w}(T))\subseteq\partial(\sigma_{e}(T))\subseteq\sigma_{K}(T)\subseteq\sigma_{e}(T)\subseteq\sigma_{w}(T)\subseteq\sigma_{b}(T)\subseteq\sigma(T),$
(6)
$\partial(\sigma(T))\subseteq\sigma_{a}(T)\cap\sigma_{su}(T)\subseteq\sigma_{l}(T)\subseteq\sigma_{r}(T)\subseteq\sigma(T).$
For a compact subset $M$ of $\mathbb{C}$, we use $accM$, $intM$ and $isoM$,
respectively, to denote all the points of accumulation of $M$, the interior of
$M$ and all the isolated points of $M$.
An operator $T\in B(X)$ is said to be Drazin invertible if there exists an
operator $T^{D}\in B(X)$ such that
$TT^{D}=T^{D}T,\quad\quad T^{D}TT^{D}=T^{D},\quad\quad T^{k+1}T^{D}=T^{k}$
for some nonnegative integer $k$ ([13], [31]).
The operator $T^{D}$ is said to be a Drazin inverse of $T$. It follows from
[13] that $T^{D}$ is unique. The smallest $k$ in the previous definition is
called as the Drazin index of $T$ and denoted by $i(T)$. Now, we can define
the Drazin spectrum, the ascent spectrum and the descent spectrum of $T$,
respectively, as following:
$\sigma_{D}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\makebox{ is not Drazin
invertible}\\},$ $\sigma_{asc}(T)=\\{\lambda\in{\mathbb{C}}:asc(T-\lambda
I)=\infty\\},$ $\sigma_{des}(T)=\\{\lambda\in{\mathbb{C}}:des(T-\lambda
I)=\infty\\}.$
The sets $\sigma_{D}(T)$, $\sigma_{asc}(T)$ and $\sigma_{des}(T)$ are closed
but may be empty ([7], [31]).
Now, we continue to introduce the following operator classes which were
discussed in [1], [3-5] and [26-27]:
$\displaystyle BF(X)=\\{T\in B(X):T=T_{1}\oplus T_{2},\,\makebox{
where}\,T_{1}\,\makebox{is a Fredholm operator and
}\,T_{2}\,\makebox{nilpotent}\\},$ $\displaystyle BW(X)=\\{T\in
B(X):T=T_{1}\oplus T_{2},\,\makebox{ where}\,T_{1}\,\makebox{is a Weyl
operator and }\,T_{2}\,\makebox{nilpotent}\\},$ $\displaystyle
R_{4}(X)=\\{T\in B(X):des(T)<\infty,\,\,R(T^{des(T)})\makebox{ is closed}\\},$
$\displaystyle R_{9}(X)=\\{T\in
B(X):asc(T)<\infty,\,\,R(T^{asc(T)+1})\makebox{ is closed}\\},$ $\displaystyle
SF_{0}(X)=\\{T\in\Phi_{+}(X)\cup\Phi_{-}(X):\alpha(T)=0\,\,\makebox{or}\,\,\beta(T)=0\\},$
$\displaystyle M(X)=\\{T\in B(X):T\,\,\makebox{is left or right
invertible}\\},$ $\displaystyle D(X)=\\{T\in B(X):R(T)\,\,\makebox{is closed
and}N(T)\subseteq\cap_{n=1}^{\infty}R(T^{n})\\}.$
Their corresponding spectra can be defined, respectively, by
The B-Fredholm spectrum: $\sigma_{BF}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in BF(X)\\},$
The B-Weyl spectrum: $\sigma_{BW}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in BW(X)\\},$
The right Drazin spectrum: $\sigma_{rD}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in R_{4}(X)\\},$
The left Drazin spectrum: $\sigma_{lD}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in R_{9}(X)\\},$
$\sigma_{SF_{0}}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in
SF_{0}(X)\\},$
$\sigma_{lr}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in M(X)\\},$
The semi-regular spectrum: $\sigma_{se}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in D(X)\\}$.
In [3], the semi-regular spectrum $\sigma_{se}(T)$ of $T$ is also called as
the regular spectrum and denoted by $\sigma_{g}(T)$.
The spectra
$\sigma_{BF},\sigma_{BW},\sigma_{rD},\sigma_{lD},\sigma_{SF_{0}},\sigma_{lr}$
are closed and $\sigma_{SF_{0}},\sigma_{lr},\sigma_{se}$ are nonempty ([1],
[3-5], [26-27]).
Let $D(\lambda,r)$ be the open disc centered at $\lambda\in\mathbb{C}$ with
radius $r>0$, $\overline{D}(\lambda,r)$ be the corresponding closed disc.
$T\in B(X)$ is said to have the Single Valued Extension Property (SVEP, for
short) ([28]) at $\lambda$ if there exists $r>0$ such that for each open
subset $V\subseteq D(\lambda,r)$, the constant function $f\equiv 0$ is the
only analytic solution of the equation
$(T-\mu)f(\mu)=0,\forall\mu\in V.$
For $T\in B(X)$, we denote
$S(T)=\\{\lambda\in{\mathbb{C}}:T\makebox{ fails to have the SVEP at
}\lambda\\}.$
If $S(T)=\emptyset$, then $T$ is said to have SVEP.
Henceforth, for $A\in B(X)$, $B\in B(Y)$ and $C\in B(Y,X)$, we put
$M_{C}=\left(\begin{array}[]{cc}A&C\\\ 0&B\\\ \end{array}\right)$. It is clear
that $M_{C}\in B(X\oplus Y)$.
For a given bounded linear operator $T\in B(X)$, as showed above, we have
defined 32 kinds spectra. Now, we are interesting in deciding the spectra
structure of operator matrix $M_{C}$. [28] and [16] have told us that for
spectra $\sigma,\sigma_{e},\sigma_{w},\sigma_{a},\sigma_{su}$ and
$\sigma_{aw}$, we have
$\sigma(M_{C})\cup(S(A^{*})\cap
S(B))=\sigma(A)\cup\sigma(B),\qquad\qquad\qquad\qquad$ $None$
$\sigma_{e}(M_{C})\cup(S(A^{*})\cap
S(B))=\sigma_{e}(A)\cup\sigma_{e}(B)\cup(S(A^{*})\cap S(B)),$ $None$
$\sigma_{w}(M_{C})\cup[S(A)\cap S(B^{*})]\cup[S(A^{*})\cap
S(B)]=\qquad\qquad\qquad$ $\sigma_{w}(A)\cup\sigma_{w}(B)\cup[S(A)\cap
S(B^{*})]\cup[S(A^{*})\cap S(B)],\qquad\qquad$ $None$ $\sigma_{su}(M_{C})\cup
S(B)=\sigma_{su}(A)\cup\sigma_{su}(B)\cup S(B),\qquad\qquad\qquad\qquad$
$None$ $\sigma_{a}(M_{C})\cup S(A^{*})=\sigma_{a}(A)\cup\sigma_{a}(B)\cup
S(A^{*}),\qquad\qquad\qquad\qquad$ $None$ $\sigma_{aw}(M_{C})\cup(S(A)\cap
S(B^{*}))\cup S(A^{*})=\sigma_{aw}(A)\cup\sigma_{aw}(B)\cup(S(A)\cap
S(B^{*}))\cup S(A^{*}).$ $None$
In this paper, first, in Section 3, we decide another 20 kind spectra
structure of $M_{C}$, that is:
Spectra $\sigma_{D}$ and $\sigma_{b}$ have equation (1) form, spectra
$\sigma_{des},\sigma_{r},\sigma_{rb},$ $\sigma_{sb},\sigma_{re}$ and
$\sigma_{SF-}$ have equation (4) form, spectra
$\sigma_{l},\sigma_{lb},\sigma_{ab},$ $\sigma_{le}$ and $\sigma_{SF+}$ have
equation (5) form, spectrum $\sigma_{lw}$ has equation (6) form. Moreover, if
spectrum $\sigma_{*}=\sigma_{sw}$ or $\sigma_{rw}$, then it has the following
form
$\sigma_{*}(M_{C})\cup(S(A)\cap S(B^{*}))\cup
S(B)=\sigma_{*}(A)\cup\sigma_{*}(B)\cup(S(A)\cap S(B^{*}))\cup S(B).$ $None$
If spectrum $\sigma_{*}=\sigma_{K},\sigma_{K_{3}},\sigma_{SF0}$ or
$\sigma_{lr}$, then it has the following forms
$\sigma_{*}(M_{C})\cup S(A^{*})\cup S(A)\cup
S(B)=\sigma_{*}(A)\cup\sigma_{*}(B)\cup S(A^{*})\cup S(A)\cup S(B)$ $None$
and
$\sigma_{*}(M_{C})\cup S(A)\cup S(B)\cup
S(B^{*})=\sigma_{*}(A)\cup\sigma_{*}(B)\cup S(A)\cup S(B)\cup S(B^{*}).$
$None$
On the other hand, Han and Lee in [19] studied the so-call filling-in-hole
problem of operator matrix, their result is:
$\sigma(A)\cup\sigma(B)=\sigma(M_{C})\cup W_{\sigma}(A,B,C),$ $None$
where $W_{\sigma}(A,B,C)$ is the union of some holes in $\sigma(M_{C})$ and
$W_{\sigma}(A,B,C)\subseteq\sigma(A)\cap\sigma(B)$.
That is, the passage from $\sigma(A)\cup\sigma(B)$ to $\sigma(M_{C})$ is the
punching of some open sets in $\sigma(A)\cap\sigma(B)$.
Moreover, in [12, 22, 31-32], the authors showed that for the spectra
$\sigma_{b},\sigma_{w},\sigma_{e}$ and $\sigma_{D}$, the equation (10) is also
true.
In [12, 20], the authors showed that if spectrum
$\sigma_{*}=\sigma_{a},\sigma_{SF+},\sigma_{SF-},\sigma_{ab}$ or
$\sigma_{sb}$, then
$\sigma_{*}(A)\cup\sigma_{*}(B)=(M_{C})\cup W_{\sigma_{*}}(A,B,C),$ $None$
where $W_{\sigma_{*}}(A,B,C)$ is contained in the union of some holes in
$\sigma_{*}(M_{C})$.
Let $M$ be a compact subset of ${\mathbb{C}}$. The set
$\eta{(M)}=\\{w:\ |P(w)|\leq max\\{|P(z)|:\ z\in M\\}\mbox{ for every
polynomial}\,P\\}$
is called to be the polynomially convex hull of $M$.
In [3], the authors proved that
$\eta(\sigma_{se}(A)\cup\sigma_{se}(B))=\eta(\sigma_{se}(M_{C})).$ $None$
Those spectra of $M_{C}$ which satisfy equation (10), (11), (12),
respectively, are called to have the filling-in-hole property, generalized
filling-in-hole property, convex filling-in-hole property, respectively.
In Section 4, we continue to study the filling-in-hole problem of another 18
kind spectra of $M_{C}$, we show that each spectrum of the 18 kind spectra of
$M_{C}$ has one of the above 3 kind filling-in-hole properties.
In Section 5, we present some interesting examples to show that some
conclusions about the spectra structure or filling-in-hole problem of $M_{C}$
are not true.
## 2 Main Lemmas and Proofs
Lemma 2.1 ([31]). If $T\in B(X)$, then the following statements are
equivalent:
(i). $T$ is Drazin invertible.
(ii). $T=T_{1}\oplus T_{2},$ where $T_{1}$ is invertible and $T_{2}$ is
nilpotent.
(iii). There exists a nonnegative integer $k$ such that
$des(T)=asc(T)=k<\infty.$
(iv). $T^{*}$ is Drazin invertible.
Lemma 2.2 ([31]). For $(A,B)\in B(X)\times B(Y)$, if $M_{C}$ is Drazin
invertible for some $C\in B(Y,\,X)$, then
(i). $des(B)<\infty$ and $asc(A)<\infty,$
(ii). $des(A^{*})<\infty$ and $asc(B^{*})<\infty$.
Lemma 2.3 ([1] Theorem 3.81). Let $T\in B(X)$ and $des(T-\lambda_{0})<\infty$.
Then the following statements are equivalent:
(i). $T$ has the SVEP at $\lambda_{0}$.
(ii). $asc(T-\lambda_{0})<\infty$.
(iii). $\lambda_{0}$ is a pole of the resolvent.
(iv). $\lambda_{0}$ is an isolated point of $\sigma(T)$.
Lemma 2.4 ([ 22, 31-32]). Let $(A,B)\in B(X)\times B(Y)$ and $C\in B(Y,X)$. We
have:
(i). if any two of operators $A,B$ and $M_{C}$ are invertible, then so is the
third,
(ii). if any two of operators $A,B$ and $M_{C}$ are Fredholms, then so is the
third,
(iii). if any two of operators $A,B$ and $M_{C}$ are Weyl, then so is the
third,
(iv). if any two of operators $A,B$ and $M_{C}$ are Drazin invertible, then so
is the third,
(v). if any two of operators $A,B$ and $M_{C}$ are Browder, then so is the
third.
Lemma 2.5 ([16]). Let $(A,B)\in B(X)\times B(Y)$ and $C\in B(Y,X)$. If $A$ has
infinite ascent, then $M_{C}$ has infinite ascent, if $B$ has infinite
descent, then $M_{C}$ has infinite descent.
Lemma 2.6. Let $(A,B)\in B(X)\times B(Y)$ and $C\in B(Y,X)$. We have:
(i). if $B$ is invertible, then $des(M_{C})<\infty$ if and only if
$des(A)<\infty$,
(ii). if $B=0$, then $des(M_{C})<\infty$ if and only if $des(A)<\infty$,
(iii). if $A$ is invertible, then $asc(M_{C})<\infty$ if and only if
$asc(B)<\infty$,
(iv). if $A=0$, then $asc(M_{C})<\infty$ if and only if $asc(B)<\infty$.
Proof. (i). If $B$ is invertible, then $des(B)<\infty$, thus, it follows from
$des(A)<\infty$ that $des(M_{C})<\infty$. In fact, if $des(A)=p$ and
$des(B)=q$, then we can claim that $R(M_{C}^{2n+1})=R(M_{C}^{2n})$ for each
$C\in B(Y,X)$, where $n=max\\{p,q\\}$. For this, note that
$R(M_{C}^{2n+1})\subseteq R(M_{C}^{2n})$ is obvious, it is sufficient to show
that $R(M_{C}^{2n})\subseteq R(M_{C}^{2n+1})$.
Suppose that $u_{0}=(u_{1},u_{2})\in R(M_{C}^{2n}).$ Then there exists
$(x_{0},y_{0})\in X\oplus Y$ such that
$(u_{1},u_{2})=M_{C}^{2n}(x_{0},y_{0})=(A^{2n}x_{0}+\sum_{i=0}^{n-1}A^{2n-i-1}CB^{i}y_{0}+\sum_{i=n}^{2n-1}A^{2n-i-1}CB^{i}y_{0},\
B^{2n}y_{0}),$
where $A^{0}=I$ and $B^{0}=I$. That $B^{2n}y_{0}=u_{2}$ is clear. In view of
$R(B^{n})=R(B^{n+1}),$ there exists $y_{1}\in Y,$ such that
$B^{n}y_{0}=B^{n+1}y_{1}.$
Therefore,
$\displaystyle u_{1}$
$\displaystyle=A^{2n}x_{0}+\sum_{i=0}^{n-1}A^{2n-i-1}CB^{i}y_{0}+\sum_{i=n}^{2n-1}A^{2n-i-1}CB^{i}y_{0}$
$\displaystyle=A^{2n}x_{0}+\sum_{i=0}^{n-1}A^{2n-i-1}CB^{i}y_{0}+\sum_{i=n}^{2n-1}A^{2n-i-1}CB^{i+1}y_{1}$
$\displaystyle=A^{2n}x_{0}+\sum_{i=0}^{n-1}A^{2n-i-1}CB^{i}y_{0}+\sum_{i=n+1}^{2n}A^{2n-i}CB^{i}y_{1}$
$\displaystyle=A^{n}(A^{n}x_{0}+\sum_{i=0}^{n-1}A^{n-i-1}CB^{i}y_{0})+\sum_{i=n+1}^{2n}A^{2n-i}CB^{i}y_{1}$
$\displaystyle=A^{n}(A^{n}x_{0}+\sum_{i=0}^{n-1}A^{n-i-1}CB^{i}y_{0}-\sum_{i=0}^{n}A^{n-i}CB^{i}y_{1})+\sum_{i=0}^{2n}A^{2n-i}CB^{i}y_{1}.$
Moreover, it follows from $des(A)=p\leq n<\infty$ that
$R(A^{n})=R(A^{n+1})=\cdots=R(A^{2n+1})$, hence, there exists $x_{1}\in X$
such that
$A^{n}(A^{n}x_{0}+\sum_{i=0}^{n-1}A^{n-i-1}CB^{i}y_{0}-\sum_{i=1}^{n-1}A^{n-i-1}CB^{i+1}y_{1})=A^{2n+1}x_{1}.$
Note that $(u_{1},u_{2})=M_{C}^{2n+1}(x_{1},y_{1}),$ so
$R(M_{C}^{2n})\subseteq R(M_{C}^{2n+1}).$
If $B$ is invertible and $des(M_{C})<\infty$, now we consider two cases to
show that $des(A)<\infty$.
Case I. If $R(M_{C})=R(M_{C}^{2})$, we claim that $R(A)=R(A^{2})$. In fact,
$R(A)\supseteq R(A^{2})$ is obvious. If $y\in R(A)$, there exists $x\in X$
such that $y=Ax.$ Thus, $\left(\begin{array}[]{c}y\\\
0\end{array}\right)=\left(\begin{array}[]{cc}A&C\\\ 0&B\\\
\end{array}\right)\left(\begin{array}[]{c}x\\\ 0\end{array}\right).$ That is
$\left(\begin{array}[]{c}y\\\ 0\end{array}\right)\in R(M_{C})=R(M_{C}^{2}).$
Hence, there exists $\left(\begin{array}[]{c}x_{1}\\\
y_{1}\end{array}\right)\in X\oplus Y$ such that
$\left(\begin{array}[]{c}y\\\
0\end{array}\right)={\left(\begin{array}[]{cc}A&C\\\ 0&B\\\
\end{array}\right)}^{2}\left(\begin{array}[]{c}x_{1}\\\
y_{1}\end{array}\right)=\left(\begin{array}[]{cc}A^{2}&AC+CB\\\ 0&B^{2}\\\
\end{array}\right)\left(\begin{array}[]{c}x_{1}\\\
y_{1}\end{array}\right)=\left(\begin{array}[]{c}A^{2}x_{1}+(AC+CB)y_{1}\\\
B^{2}y_{1}\end{array}\right).$
Moreover, since $B$ is invertible, it is easy to show that
$y_{1}=0,y=A^{2}x_{1}.$
Thus $y\in R(A^{2}),$ and so $R(A)\subseteq R(A^{2}).$
Case II. If $1<des(M_{C})=p<\infty$, put $M_{p}=M_{C}^{p}$. Then
$M_{p}=\left(\begin{array}[]{cc}A^{p}&\sum_{i=1}^{p}A^{p-i}CB^{i-1}\\\
0&B^{p}\\\ \end{array}\right)$ and $R(M_{p})=R(M_{p}^{2})$, where
$A^{0}=I,B^{0}=I.$ By using the same methods as in case I, we can prove that
$R(A^{p})\subseteq R(A^{2p})$. So $R(A^{p})=R(A^{2p})$. It follows from the
conclusion that $R(A^{p})=R(A^{p+1}).$
Combining Case I with Case II, we complete the proof of (i).
(ii). It follows from the above argument methods that we only need to show
that if $B=0$ and $R(M_{C})=R(M_{C}^{2})$, then $R(A)=R(A^{2})$. In fact, if
$y\in R(A)$, then there exists $x\in X$ such that $y=Ax,$ so
$\left(\begin{array}[]{c}y\\\
0\end{array}\right)=\left(\begin{array}[]{cc}A&C\\\ 0&0\\\
\end{array}\right)\left(\begin{array}[]{c}x\\\ 0\end{array}\right)$. Since
$\left(\begin{array}[]{c}y\\\ 0\end{array}\right)\in{R(M_{C})=R(M_{C}^{3})},$
there exists $\left(\begin{array}[]{c}x_{1}\\\ y_{1}\end{array}\right)\in
X\oplus Y$ such that
$\left(\begin{array}[]{c}y\\\
0\end{array}\right)={\left(\begin{array}[]{cc}A&C\\\ 0&0\\\
\end{array}\right)}^{3}\left(\begin{array}[]{c}x_{1}\\\
y_{1}\end{array}\right)=\left(\begin{array}[]{cc}A^{2}&0\\\ 0&0\\\
\end{array}\right)\left(\begin{array}[]{c}Ax_{1}+Cy_{1}\\\
0\end{array}\right).$
Thus we have $y=A^{2}(Ax_{1}+Cy_{1})$, so $y\in R(A^{2}),$ this showed that
$R(A)\subseteq R(A^{2})$. Note that $R(A^{2})\subseteq R(A)$ is obvious,
therefore $R(A)=R(A^{2}).$
(iii). If $A$ is invertible, then $asc(A)<\infty$, thus, it follows easily
from $asc(B)<\infty$ that $asc(M_{C})<\infty$ (see Lemma 2.2 in [11]).
If $A$ is invertible and $asc(M_{C})<\infty$, now we consider two cases to
show that $asc(B)<\infty$.
Case I. $N(M_{C})=N(M_{C}^{2})$. We claim that $N(B)=N(B^{2}).$ If $y\in
N(B^{2})$, then $B^{2}y=0$. Note that $A$ is invertible, it can be proved that
$M_{C}^{2}\left(\begin{array}[]{c}-A^{-2}(AC+CB)y\\\
y\end{array}\right)=\left(\begin{array}[]{c}0\\\
B^{2}y\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\end{array}\right).$
Thus,
$\left(\begin{array}[]{c}-A^{-2}(AC+CB)y\\\ y\end{array}\right)\in
N(M_{C}^{2})=N(M_{C}).$
Therefore
$\left(\begin{array}[]{cc}A&C\\\ 0&B\\\
\end{array}\right)\left(\begin{array}[]{c}-A^{-2}(AC+CB)y\\\
y\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\end{array}\right),$
which implies $By=0$, thus $N(B^{2})\subseteq N(B)$. Note that $N(B)\subseteq
N(B^{2})$ is obvious, so $N(B^{2})=N(B)$.
Case II. If $1<asc(M_{C})=p<\infty$, put $M_{p}=M_{C}^{p}$. Then
$M_{p}=\left(\begin{array}[]{cc}A^{p}&\sum_{i=1}^{p}A^{p-i}CB^{i-1}\\\
0&B^{p}\\\ \end{array}\right)\,\,\makebox{ and}\,\,N(M_{p})=N(M_{p}^{2}),$
where $A^{0}=I,B^{0}=I.$ By the same methods as in case I, we can prove that
$N(B^{p})\subseteq N(B^{2p})$ and so it is easy to obtain that
$N(B^{p})=N(B^{p+1}).$
Combining case I with case II, we prove (iii).
(iv). We only prove that if $A=0$ and $N(M_{C})=N(M_{C}^{2})$, then
$N(B)=N(B^{2}).$ Let $y\in N(B^{2})$. Then $B^{2}y=0$. So we have
$M_{C}^{3}\left(\begin{array}[]{c}0\\\
y\end{array}\right)=\left(\begin{array}[]{cc}0&CB^{2}\\\ 0&B^{3}\\\
\end{array}\right)\left(\begin{array}[]{c}0\\\
y\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\end{array}\right),$
which implies that
$\left(\begin{array}[]{c}0\\\ y\end{array}\right)\in N(M_{C}^{3})=N(M_{C}).$
Thus
$\left(\begin{array}[]{cc}0&C\\\ 0&B\\\
\end{array}\right)\left(\begin{array}[]{c}0\\\
y\end{array}\right)=\left(\begin{array}[]{c}Cy\\\
By\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\end{array}\right).$
This showes that $By=0,$ so $N(B^{2})\subseteq N(B)$. It follows easily from
the conclusion that $N(B)=N(B^{2}).$ The lemma is proved.
The following lemma is important and it is widely used in the proofs of our
main theorems in Section 3 and Section 4.
Lemma 2.7. For $(A,B)\in B(X)\times B(Y)$ and $C\in B(Y,X)$, we have:
(i). if $A$ is Drazin invertible, then $asc(M_{C})<\infty$
($des(M_{C})<\infty$) if and only if $asc(B)<\infty$ ($des(B)<\infty$),
(ii). if $B$ is Drazin invertible, then $des(M_{C})<\infty$
($asc(M_{C})<\infty$ ) if and only if $des(A)<\infty$ ($asc(A)<\infty$),
(iii). if $A$ is Browder operator, then $M_{C}$ is left (right, upper, lower)
semi-Browder operator if and only if $B$ is left (right, upper, lower) semi-
Browder operator,
(iv). if $B$ is Browder operator, then $M_{C}$ is left (right, upper, lower)
semi-Browder operator if and only if $A$ is left (right, upper, lower) semi-
Browder operator,
(v). if $A$ is Fredholm operator, then $M_{C}$ is left (right, upper, lower)
semi-Fredholm operator if and only if $B$ is left (right, upper, lower) semi-
Fredholm operator,
(vi). if $B$ is Fredholm operator, then $M_{C}$ is left (right, upper, lower)
semi-Fredholm operator if and only if $A$ is left (right, upper, lower) semi-
Fredholm operator,
(vii). if $A$ is Weyl operator, then $M_{C}$ is left (right, upper, lower)
semi-Weyl operator if and only if $B$ is left (right, upper, lower) Weyl
operator,
(viii). if $B$ is Weyl operator, then $M_{C}$ is left (right, upper, lower)
semi-Weyl operator if and only if $A$ is left (right, upper, lower) semi-Weyl
operator,
(viiii). if $A$ is invertible, then $M_{C}$ is left (right) invertible if and
only if $B$ is left (right)invertible,
(vv). if $B$ is invertible, then $M_{C}$ is left (right) invertible if and
only if $A$ is left (right) invertible,
(vvi). if $A$ is invertible, then $M_{C}$ bounded below if and only if $B$
bounded below,
(vvii). if $B$ is invertible, then $M_{C}$ is surjective if and only if $A$ is
surjective.
Proof. That from (v) to (vv) are well known and from (i) to (iv) are new. Here
we only prove (i).
If $A$ is Drazin invertible and $i(A)=k<\infty$, then
$A=\left(\begin{array}[]{cc}A_{1}&0\\\ 0&A_{2}\\\
\end{array}\right):R(A^{k})\oplus N(A^{k})\longrightarrow R(A^{k})\oplus
N(A^{k}),$
where $A_{1}$ is invertible and $A_{2}^{k}=0.$ Thus,
$M_{C}=\left(\begin{array}[]{ccc}A_{1}&0&C_{1}\\\ 0&A_{2}&C_{2}\\\ 0&0&B\\\
\end{array}\right):R(A^{k})\oplus N(A^{k})\oplus Y\longrightarrow
R(A^{k})\oplus N(A^{k})\oplus Y.$
So
$M_{C}^{k}=\left(\begin{array}[]{ccc}A_{1}^{k}&0&\sum_{i=1}^{k}A_{1}^{k-i}C_{1}B^{i-1}\\\
0&0&\sum_{i=1}^{k}A_{2}^{k-i}C_{2}B^{i-1}\\\ 0&0&B^{k}\\\
\end{array}\right):R(A^{k})\oplus N(A^{k})\oplus Y\longrightarrow
R(A^{k})\oplus N(A^{k})\oplus Y.$
It follows from Lemma 2.6 that
$\displaystyle asc(M_{C})<\infty$ $\displaystyle\Longleftrightarrow
asc(M_{C}^{k})<\infty$ $\displaystyle\Longleftrightarrow
asc(\left(\begin{array}[]{cc}0&\sum_{i=1}^{k}A_{2}^{k-i}C_{2}B^{i-1}\\\
0&B^{k}\\\ \end{array}\right))<\infty$ $\displaystyle\Longleftrightarrow
asc(B^{k})<\infty$ $\displaystyle\Longleftrightarrow asc(B)<\infty.$
Moreover, if $A$ is Drazin invertible, then it follows from Lemma 2.5 and the
proof of Lemma 2.6 that
$des(M_{C})<\infty\Leftrightarrow des(B)<\infty.$
Thus (i) is proved.
Lemma 2.8 ([1] Theorem 3.4). For $T\in B(X)$, the following properties hold:
(i). if $asc(T)<\infty$, then $\alpha(T)\leq\beta(T)$,
(ii). if $des(T)<\infty$, then $\beta(T)\leq\alpha(T)$,
(iii). if $asc(T)=des(T)<\infty$, then $\beta(T)=\alpha(T)$,
(iv). if $\beta(T)=\alpha(T)<\infty$ and either $asc(T)$ or $des(T)$ is
finite, then $asc(T)=des(T)$.
Lemma 2.9 ([1] Corollary 3.19). Let $T\in B(X)$ and
$\lambda_{0}-T\in\Phi_{\pm}(X)$, where
$\Phi_{\pm}(X)=\Phi_{+}(X)\cup\Phi_{-}(X)$. We have:
(i). if $T$ has the SVEP at $\lambda_{0}$, then ind $(\lambda_{0}I-T)\leq 0$,
(ii). if $T^{*}$ has the SVEP at $\lambda_{0}$, then ind $(\lambda_{0}I-T)\geq
0$.
Lemma 2.10 ([31]). If each neighborhood of $\lambda$ contains a point that is
not an eigenvalue of operator $T$ and $\lambda-T$ has a finite descent, then
$\lambda-T$ is Drazin invertible.
## 3 Spectra structure of operator matrix $M_{C}$
The following theorems in this section tell us 20 kind spectra structure of
$M_{C}$, that is:
Theorem 3.1.
$\sigma_{D}(M_{C})\cup(S(A^{*})\cap S(B))=\sigma_{D}(A)\cup\sigma_{D}(B).$
Proof. First, Theorem 2.9 in [31] told us that
$\sigma_{D}(M_{C})\subseteq\sigma_{D}(A)\cup\sigma_{D}(B)$. Note that
$S(A^{*})\cap S(B)\subseteq int(\sigma(A^{*}))\cap
int(\sigma(B))=int(\sigma(A))\cap
int(\sigma(B))\subseteq\sigma_{D}(A)\cup\sigma_{D}(B).$
It follows that
$\sigma_{D}(M_{C})\cup(S(A^{*})\cap
S(B))\subseteq\sigma_{D}(A)\cup\sigma_{D}(B).$
On the other hand, if
$\lambda\in(\sigma_{D}(A)\cup\sigma_{D}(B))\setminus\sigma_{D}(M_{C})$, then
$M_{C}-\lambda$ is Drazin invertible. It follows from Lemma 2.2 that
$des(B-\lambda)<\infty\,\,\makebox{ and}\,\,des(A^{*}-\lambda)<\infty.$
Now we claim that $\lambda\in S(A^{*})\cap S(B)$. In fact, if $\lambda\not\in
S(A^{*})\cap S(B)$. There are two cases:
If $\lambda\not\in S(A^{*})$, note that $des(A^{*}-\lambda)<\infty$, it
follows from Lemmas 2.1 and Lemma 2.3 that $asc(A^{*}-\lambda)<\infty$ and
$A-\lambda$ is Drazin invertible. Furthermore, Lemma 2.4 tells us that
$B-\lambda$ is also Drazin invertible, this is a contradiction.
Similarly, we can prove that $\lambda\not\in S(B)$ is also impossible. Thus,
we have
$\sigma_{D}(A)\cup\sigma_{D}(B)\subseteq\sigma_{D}(M_{C})\cup(S(A^{*})\cap
S(B)).$
The theorem is proved.
Theorem 3.2.
$\sigma_{b}(M_{C})\cup(S(A^{*})\cap S(B))=\sigma_{b}(A)\cup\sigma_{b}(B).$
Proof. It follows from Lemma 2.4 that
$\sigma_{b}(M_{C})\subseteq\sigma_{b}(A)\cup\sigma_{b}(B).$ Moreover, it is
easy to know that
$\sigma_{D}(A)\cup\sigma_{D}(B)\subseteq\sigma_{b}(A)\cup\sigma_{b}(B)$, thus,
it follows from the proof of Theorem 3.1 that $S(A^{*})\cap
S(B)\subseteq\sigma_{b}(A)\cup\sigma_{b}(B).$ Hence
$\sigma_{b}(M_{C})\cup(S(A^{*})\cap
S(B))\subseteq\sigma_{b}(A)\cup\sigma_{b}(B).$
For the contrary inclusion, it is sufficient to prove that
$(\sigma_{b}(A)\cup\sigma_{b}(B))\setminus\sigma_{b}(M_{C})\subseteq(S(A^{*})\cap
S(B)).$
Let $\lambda\in(\sigma_{b}(A)\cup\sigma_{b}(B))\setminus\sigma_{b}(M_{C})$.
Then $M_{C}-\lambda$ is a Drazin invertible Fredholm operator, and so
$A-\lambda\in\Phi_{+}(X),B-\lambda\in\Phi_{-}(Y)$. Moreover, It follows from
Lemma 2.2 that
$des(B-\lambda)<\infty\,\,\makebox{ and}\,\,asc(A-\lambda)<\infty.$
Now, we show that $\lambda\in S(A^{*})\cap S(B)$. In fact, if $\lambda\not\in
S(A^{*})\cap S(B)$, then when $\lambda\not\in S(A^{*})$, we can prove as in
Theorem 3.1 that $A-\lambda$ is Drazin invertible. Note that
$A-\lambda\in\Phi_{+}(X)$, it is easy to show from Lemma 2.8 that
$A-\lambda\in\Phi_{b}(X).$ Thus, by Lemma 2.4 we get that
$B-\lambda\in\Phi_{b}(Y)$, and hence
$\lambda\not\in\sigma_{b}(A)\cup\sigma_{b}(B)$, which is a contradiction. So,
$\lambda\in S(A^{*})$.
Similarly, we can show that $\lambda\in S(B)$. The theorem is proved.
Theorem 3.3. If
$\sigma_{*}=\sigma_{des},\sigma_{r},\sigma_{rb},\sigma_{sb},\sigma_{re}$ or
$\sigma_{SF-}$, then
$\sigma_{*}(M_{C})\cup S{(B)}=\sigma_{*}(A)\cup\sigma_{*}(B)\cup S{(B)}.$
Proof. Observe that
$M_{C}=\left(\begin{array}[]{cc}I&0\\\ 0&B\\\
\end{array}\right)\left(\begin{array}[]{cc}I&C\\\ 0&I\\\
\end{array}\right)\left(\begin{array}[]{cc}A&0\\\ 0&I\\\ \end{array}\right).$
It is easy to prove that
$\sigma_{*}(B)\subseteq\sigma_{*}(M_{C})\subseteq\sigma_{*}(A)\cup\sigma_{*}(B)$
and
$\sigma_{*}(M_{C})\cup S{(B)}\subseteq\sigma_{*}(A)\cup\sigma_{*}(B)\cup
S{(B)}.$
for each
$\sigma_{*}=\sigma_{des},\sigma_{r},\sigma_{rb},\sigma_{sb},\sigma_{re}$ or
$\sigma_{SF-}$. So, in order to prove the theorem, we only need to prove that
$(\sigma_{*}(A)\cup\sigma_{*}(B))\setminus\sigma_{*}(M_{C})\subseteq S(B).$
If $\lambda\in(\sigma_{*}(A)\cup\sigma_{*}(B))\setminus\sigma_{*}(M_{C})$ and
$\sigma_{*}=\sigma_{des},\sigma_{r},\sigma_{rb},\sigma_{sb},\sigma_{re}$ or
$\sigma_{SF-}$, we show that $\lambda\in S(B).$ In fact, if $\lambda\not\in
S(B)$, we consider the following four cases:
Case (I). Let $\sigma_{*}=\sigma_{des}$. Since
$\lambda\in(\sigma_{des}(A)\cup\sigma_{des}(B))\setminus\sigma_{des}(M_{C})$
and $\lambda\not\in S(B)$, it follows from Lemmas 2.5 and 2.3 that
$des(B-\lambda)<\infty$ and $asc(B-\lambda)<\infty$, so $B-\lambda$ is Drazin
invertible. Thus, Lemma 2.7 tells us that $des(A-\lambda)<\infty,$ this is a
contradiction. Therefore, it follows that
$(\sigma_{des}(A)\cup\sigma_{des}(B))\setminus\sigma_{des}(M_{C})\subseteq
S(B)$.
Case (II). Let $\sigma_{*}=\sigma_{r}$. Note that
$\lambda\in(\sigma_{r}(A)\cup\sigma_{r}(B))\setminus\sigma_{r}(M_{C})$, so
$B-\lambda\in G_{r}(Y).$ That is $B-\lambda\in\Phi_{rb}(Y)$ and
$R(B-\lambda)=Y$. Since $\lambda\not\in S(B)$, it follows from Lemma 2.9 that
ind$(B-\lambda)=\alpha(B-\lambda)-\beta(B-\lambda)\leq 0$. Thus
$\alpha(B-\lambda)=0$, and so $B-\lambda\in\Phi_{0}(Y)$. Also since
$R(B-\lambda)=Y$, $B-\lambda$ is invertible. It follows from Lemma 2.7 that
$A-\lambda\in G_{r}(X)$, this contradicts
$\lambda\in(\sigma_{r}(A)\cup\sigma_{r}(B))$. Therefore
$(\sigma_{r}(A)\cup\sigma_{r}(B))\setminus\sigma_{r}(M_{C})\subseteq S(B)$.
Case (III). Let $\sigma_{*}=\sigma_{rb}$. Since
$\lambda\in(\sigma_{rb}(A)\cup\sigma_{rb}(B))\setminus\sigma_{rb}(M_{C})$,
$B-\lambda\in\Phi_{rb}(Y).$ In particular, $B-\lambda\in\Phi_{r}(Y)$,
ind$(B-\lambda)\geq 0$. Note that $\lambda\not\in S(B)$, it follows from Lemma
2.9 that ind$(B-\lambda)\leq 0$. Thus $B-\lambda\in\Phi_{0}(Y)$ and
$des(B-\lambda)<\infty$, and then Lemma 2.8 shows us that
$B-\lambda\in\Phi_{b}(Y)$. Lemma 2.7 tells us that $A-\lambda\in\Phi_{rb}(X),$
this is a contradiction. So we get that
$(\sigma_{rb}(A)\cup\sigma_{rb}(B))\setminus\sigma_{rb}(M_{C})\subseteq S(B)$.
Case (IV). Let $\sigma_{*}=\sigma_{re}$. Since
$\lambda\in(\sigma_{re}(A)\cup\sigma_{re}(B))\setminus\sigma_{re}(M_{C})$, it
is easy to show that $B-\lambda\in\Phi_{r}(Y)$. Note that $\lambda\not\in
S(B)$, it follows from Lemma 2.9 that ind$(B-\lambda)\leq 0$, so
$\alpha(B-\lambda)\leq\beta(B-\lambda)<\infty.$ Thus $B-\lambda\in\Phi(X)$. By
using Lemma 2.7, we have $A-\lambda\in\Phi_{r}(X),$ this is a contradiction
and so we can prove that
$(\sigma_{re}(A)\cup\sigma_{re}(B))\setminus\sigma_{re}(M_{C})\subseteq S(B)$.
For $\sigma_{*}=\sigma_{sb}$ or $\sigma_{SF-}$, the proof methods are similar,
we omit them.
Similarly, we can prove also the following theorem:
Theorem 3.4. If $\sigma_{*}=\sigma_{l},\sigma_{lb},\sigma_{ab},\sigma_{le}$ or
$\sigma_{SF+}$, then
$\sigma_{*}(M_{C})\cup S{(A^{*})}=\sigma_{*}(A)\cup\sigma_{*}(B)\cup
S{(A^{*})}.$
Theorem 3.5.
$\sigma_{lw}(M_{C})\cup(S(A)\cap S(B^{*}))\cup
S(A^{*})=\sigma_{lw}(A)\cup\sigma_{lw}(B)\cup(S(A)\cap S(B^{*}))\cup
S(A^{*}).$
Proof. Note that
$\sigma_{lw}(M_{C})\subseteq\sigma_{lw}(A)\cup\sigma_{lw}(B),$ it is obvious
that
$\sigma_{lw}(M_{C})\cup(S(A)\cap S(B^{*}))\cup
S(A^{*})\subseteq\sigma_{lw}(A)\cup\sigma_{lw}(B)\cup(S(A)\cap S(B^{*}))\cup
S(A^{*}).$
For the contrary inclusion, let $\lambda\not\in\sigma_{lw}(M_{C})\cup(S(A)\cap
S(B^{*}))\cup S(A^{*}).$ Then $M_{C}-\lambda$ is left semi-Weyl operator, and
it is easy to prove that $A-\lambda\in\Phi_{l}(X)$. From the assumption we
also get that either $A$ and $A^{*}$ or $A^{*}$ and $B^{*}$ have the SVEP at
$\lambda$. If $A$ and $A^{*}$ have the SVEP at $\lambda$, it follows from
Lemma 2.9 that $A-\lambda\in\Phi_{0}(X)$. By using Lemma 2.7, it follows that
$B-\lambda\in\Phi_{l}(Y)$ and ind$(M_{C}-\lambda)=$
ind$(A-\lambda)+$ind$(B-\lambda)\leq 0$. Thus $B-\lambda\in\Phi_{l}(Y)$ with
ind$(B-\lambda)=$ind$(M_{C}-\lambda)$-ind$(A-\lambda)$=ind$(M_{C}-\lambda)\leq
0.$ Hence $\lambda\not\in\sigma_{lw}(A)\cup\sigma_{lw}(B).$ On the other hand,
if $A^{*}$ and $B^{*}$ have the SVEP at $\lambda$, it is obvious that
$M_{C}^{*}$ have the SVEP at $\lambda$. It follows from Lemma 2.9 that
$M_{C}-\lambda\in\Phi_{0}(X\oplus Y)$. Thus, it is easy to show that
$A-\lambda\in\Phi_{l}(X)$ and $B-\lambda\in\Phi_{r}(Y)$. Also, note that
$B^{*}$ and $A^{*}$ have the SVEP at $\lambda$, it follows from Lemma 2.9 that
ind$(A-\lambda)\geq 0$ and ind$(B-\lambda)\geq 0$. Hence it follows from
$A-\lambda\in\Phi_{l}(X)$ proved above that $A-\lambda\in\Phi(X)$, so Lemma
2.4 tells us that $B-\lambda\in\Phi(Y)$. Moreover, in view of
$0=$ind$(M_{C}-\lambda)=$ind$(A-\lambda)+$ind$(B-\lambda)$,
ind$(A-\lambda)\geq 0$ and ind$(B-\lambda)\geq 0$, it is clear that
ind$(A-\lambda)$=ind$(B-\lambda)=0$. Thus $A-\lambda$ and $B-\lambda$ are both
Weyl operators, which implies that
$\lambda\not\in\sigma_{lw}(A)\cup\sigma_{lw}(B).$ It follows that
$\lambda\not\in\sigma_{lw}(A)\cup\sigma_{lw}(B)$ when
$\lambda\not\in\sigma_{lw}(M_{C})\cup(S(A)\cap S(B^{*}))\cup S(A^{*}).$ Thus,
the contrary inclusion is clear. The theorem is proved.
Similarly, we can prove also the following theorem:
Theorem 3.6. If $\sigma_{*}=\sigma_{sw}$ or $\sigma_{rw}$, then
$\sigma_{*}(M_{C})\cup(S(A)\cap S(B^{*}))\cup
S(B)=\sigma_{*}(A)\cup\sigma_{*}(B)\cup(S(A)\cap S(B^{*}))\cup S(B).$
Theorem 3.7. If $\sigma_{*}=\sigma_{K},\sigma_{K_{3}},\sigma_{SF0}$ or
$\sigma_{lr}$, then
$\sigma_{*}(M_{C})\cup S(A^{*})\cup S(A)\cup
S(B)=\sigma_{*}(A)\cup\sigma_{*}(B)\cup S(A^{*})\cup S(A)\cup S(B)$
and
$\sigma_{*}(M_{C})\cup S(A^{*})\cup S(B)\cup
S(B^{*})=\sigma_{*}(A)\cup\sigma_{*}(B)\cup S(A^{*})\cup S(B)\cup S(B^{*}).$
Proof. We only prove when $\sigma_{*}=\sigma_{K}$, equation (8) holds. First,
we prove that
$\sigma_{K}(M_{C})\cup S(A^{*})\cup S(A)\cup
S(B)\subseteq\sigma_{K}(A)\cup\sigma_{K}(B)\cup S(A^{*})\cup S(A)\cup S(B).$
In fact, let $\lambda\not\in\sigma_{K}(A)\cup\sigma_{K}(B)\cup S(A^{*})\cup
S(A)\cup S(B).$ Then $A-\lambda$ is a semi-Fredholm operator with $A^{*}$, $A$
have the SVEP at $\lambda$, this implies $A-\lambda$ is a Weyl operator. Note
that $B-\lambda$ is also a semi-Fredholm operator, it follows from Lemma 2.7
that $M_{C}-\lambda$ is a semi-Fredholm operator. Thus
$\lambda\not\in\sigma_{K}(M_{C})\cup S(A^{*})\cup S(A)\cup S(B).$ So it is
clear that $\sigma_{K}(M_{C})\cup S(A^{*})\cup S(A)\cup
S(B)\subseteq\sigma_{K}(A)\cup\sigma_{K}(B)\cup S(A^{*})\cup S(A)\cup S(B).$
For the contrary inclusion, let $\lambda\not\in\sigma_{K}(M_{C})\cup
S(A^{*})\cup S(A)\cup S(B).$ Then $M_{C}-\lambda$ is a semi-Fredholm operator,
that is, $M_{C}-\lambda$ is either a upper semi-Fredholm operator or a lower
semi-Fredholm operator. If $M_{C}-\lambda\in\Phi_{+}(X\oplus Y)$, then it is
easy to prove that $A-\lambda\in\Phi_{+}(X)$. Since $A^{*}$ has the SVEP at
$\lambda$, it follows from lemma 2.9 that $A-\lambda\in\Phi(X)$. Hence it
follows Lemma 2.7 that $B-\lambda\in\Phi_{+}(Y)$. Thus
$\lambda\not\in\sigma_{K}(A)\cup\sigma_{K}(B)$, and so
$\lambda\not\in\sigma_{K}(A)\cup\sigma_{K}(B)\cup S(A^{*})\cup S(A)\cup S(B).$
On the other hand, if $M_{C}-\lambda\in\Phi_{-}(X\oplus Y)$, then
$B-\lambda\in\Phi_{-}(Y)$. Since $B$ has the SVEP at $\lambda$, it follows
from lemma 2.9 that $B-\lambda\in\Phi(Y)$. Similar to the above arguments, we
can also obtain that $\lambda\not\in\sigma_{K}(A)\cup\sigma_{K}(B)\cup
S(A^{*})\cup S(A)\cup S(B)$. This proves equation (8).
We are interesting in the following question, it is perhaps difficult:
Open question 3.8. Do other spectra of $M_{C}$ have the equations (1) to (9)
forms ?
## 4 Filling-in-hole Problem of $M_{C}$
In Section 1, we pointed out that if spectrum
$\sigma_{*}=\sigma,\sigma_{b},\sigma_{w},\sigma_{e}$ or $\sigma_{D}$, then
$\sigma_{*}(A)\cup\sigma_{*}(B)=\sigma_{*}(M_{C})\cup W_{\sigma_{*}}(A,B,C),$
where $W_{\sigma_{*}}(A,B,C)$ is the union of some holes in
$\sigma_{*}(M_{C})$ and
$W_{\sigma_{*}}(A,B,C)\subseteq\sigma_{*}(A)\cap\sigma_{*}(B)$. The following
theorem shows the relationship among
$W_{\sigma}(A,B,C),W_{\sigma_{b}}(A,B,C)$, $W_{\sigma_{D}}(A,B,C)$ and
$W_{\sigma_{w}}(A,B,C)$:
Theorem 4.1. For $(A,B)\in B(X)\times B(Y)$ and $C\in B(Y,X)$, we have
(i). $W_{\sigma}(A,B,C)\subseteq W_{\sigma_{b}}(A,B,C)\subseteq
W_{\sigma_{D}}(A,B,C),$
(ii). $W_{\sigma_{b}}(A,B,C)\subseteq W_{\sigma_{w}}(A,B,C).$
In particular, the following states are equivalent:
(a). $W_{\sigma}(A,B,C)=\emptyset,$
(b). $W_{\sigma_{b}}(A,B,C)=\emptyset,$
(c). $W_{\sigma_{D}}(A,B,C)=\emptyset.$
Proof. (i). Let $\lambda\in W_{\sigma}(A,B,C)$. It follow from equation (1)
that $\lambda\in(\sigma(A)\cup\sigma(B))\setminus\sigma(M_{C})$. Thus
$A-\lambda$ is left invertible and $B-\lambda$ is right invertible. That
$\lambda\not\in\sigma_{b}(M_{C})$ is obvious. Now we claim that
$\lambda\in\sigma_{b}(A)\cup\sigma_{b}(B)$. If not, by Lemma 2.4, we have that
both $A-\lambda$ and $B-\lambda$ are Browder operators. This implies that
$\lambda\in\Phi_{0}(A)\cap\Phi_{0}(B)$. Moreover, since $A-\lambda$ is left
invertible and $B-\lambda$ is right invertible, $A-\lambda$ and $B-\lambda$
are invertible, this contradicts $\lambda\in\sigma(A)\cup\sigma(B)$, so
$\lambda\in\sigma_{b}(A)\cup\sigma_{b}(B)$, thus $W_{\sigma}(A,B,C)\subseteq
W_{\sigma_{b}}(A,B,C)$ is proved.
For the inclusion $W_{\sigma_{b}}(A,B,C)\subseteq W_{\sigma_{D}}(A,B,C)$, note
that $\sigma_{b}(M_{C})\supseteq\sigma_{D}(M_{C})$, so it is sufficient to
show that if
$\lambda\in(\sigma_{b}(A)\cup\sigma_{b}(B))\setminus\sigma_{b}(M_{C})$, then
$\lambda\in(\sigma_{D}(A)\cup\sigma_{D}(B))$. Let
$\lambda\in(\sigma_{b}(A)\cup\sigma_{b}(B))\setminus\sigma_{b}(M_{C})$. Then
$M_{C}-\lambda\in\Phi_{b}(X\oplus Y)$, so $A-\lambda\in\Phi_{l}(X)$ and
$B-\lambda\in\Phi_{r}(Y)$. We claim
$\lambda\in\sigma_{D}(A)\cup\sigma_{D}(B)$. If not, it follows from Lemma 2.4
that $\lambda\in\rho_{D}(A)\cap\rho_{D}(B)$. Since $B-\lambda\in\Phi_{r}(Y)$,
Lemma 2.1 and Lemma 2.8 tell us that $B-\lambda$ is a Fredholm operator. This
implies that $B-\lambda$ is a Drazin invertible Fredholm operator, that is,
$B-\lambda$ is a Browder operator. By Lemma 2.4, we get that $A-\lambda$ is
also a Browder operator. Thus $\lambda\in(\Phi_{b}(A)\cap\Phi_{b}(B))$, which
contradicts with the assumption that
$\lambda\in(\sigma_{b}(A)\cup\sigma_{b}(B))$, so we have
$\lambda\in\sigma_{D}(A)\cup\sigma_{D}(B)$, thus
$W_{\sigma_{b}}(A,B,C)\subseteq W_{\sigma_{D}}(A,B,C)$ is also proved.
(ii). To prove $W_{\sigma_{b}}(A,B,C)\subseteq W_{\sigma_{w}}(A,B,C)$, by
Lemma 2.4 and the fact that $\sigma_{w}(M_{C})\subseteq\sigma_{b}(M_{C})$, it
sufficient to show that if
$\lambda\in(\sigma_{b}(A)\cup\sigma_{b}(B))\setminus\sigma_{b}(M_{C})$, then
$\lambda\in\sigma_{w}(A)\cup\sigma_{w}(B)$. Indeed, if
$\lambda\in(\sigma_{b}(A)\cup\sigma_{b}(B))\setminus\sigma_{b}(M_{C})$, then
$M_{C}-\lambda\in\Phi_{b}(X\oplus Y)$. It follows Lemma 2.2 that there exist
some nonnegative integer $k$ and $l$ such that $asc(A-\lambda)=k<\infty$ and
$des(B-\lambda)=l<\infty$. Now we claim
$\lambda\in\sigma_{w}(A)\cup\sigma_{w}(B)$. Otherwise,
$\lambda\in\Phi_{0}(A)\cap\Phi_{0}(B)$. Moreover, Since
$asc(A-\lambda)=k<\infty$, by Lemma 2.8 we have $des(A-\lambda)<\infty.$ That
is $A-\lambda$ is a Drazin invertible Fredholm operator, so
$A-\lambda\in\Phi_{b}(X).$ Using Lemma 2.4, we get that
$B-\lambda\in\Phi_{b}(Y)$ and so $\lambda\in\Phi_{b}(A)\cap\Phi_{b}(B)$, this
contradicts with the assumption that
$\lambda\in(\sigma_{b}(A)\cup\sigma_{b}(B))$. Thus we have
$\lambda\in\sigma_{w}(A)\cup\sigma_{w}(B)$ and (ii) is proved.
Finally, it follows from Corollary 2.12 in [31] that if $\
W_{\sigma}(A,B,C)=\emptyset$, then $W_{\sigma_{D}}(A,B,C)=\emptyset.$ This
completed the proof of theorem.
The following result which generalizes Lemma 3.2 of [12] is useful in studying
the filling-in-hole problem of $M_{C}$.
Proposition 4.2. For each $T\in B(X)$, we have
(i). If
$\sigma_{*}=\sigma_{lr},\sigma_{SF0},\sigma_{su},\sigma_{r},\sigma_{a},\sigma_{l}$
or $\sigma_{se}$, then $\eta(\sigma_{*}(T))=\eta(\sigma(T))$.
(ii). If
$\sigma_{*}=\sigma_{ab},\sigma_{lb},\sigma_{sb},\sigma_{rb},\sigma_{aw},\sigma_{lw},\sigma_{sw},\sigma_{rw},\sigma_{w},\sigma_{SF+},\sigma_{SF-},\sigma_{le},\sigma_{re},\sigma_{e},\sigma_{K}$
or $\sigma_{K_{3}}$, then $\eta(\sigma_{*}(T))=\eta(\sigma_{b}(T))$.
(iii).
$\eta(\sigma_{D}(T))=\eta(\sigma_{des}(T))=\eta(\sigma_{rD}(T))=\eta(\sigma_{lD}(T))$.
Proof. (i). Note that if
$\sigma_{*}=\sigma_{lr},\sigma_{SF0},\sigma_{su},\sigma_{r},\sigma_{a}$ or
$\sigma_{l}$, then
$\partial\sigma\subseteq\sigma_{su}\cap\sigma_{a}\subseteq\sigma_{*}\subseteq\sigma$,
$\partial\sigma\subseteq\sigma_{se}\subseteq\sigma$ ([1]), so (i) is proved.
(ii). If
$\sigma_{*}=\sigma_{ab},\sigma_{lb},\sigma_{sb},\sigma_{rb},\sigma_{aw},\sigma_{sw},\sigma_{sw},\sigma_{rw},\sigma_{w},\sigma_{SF+},\sigma_{SF-},\sigma_{le},\sigma_{re},\sigma_{e},\sigma_{K},\sigma_{K_{3}}$
or $\sigma_{SF+}$, it is well known that
$\partial\sigma_{b}\subseteq\sigma_{K}\subseteq\sigma_{*}\subseteq\sigma_{b}$,
so we have $\eta(\sigma_{*}(T))=\eta(\sigma_{b}(T))$.
(iii). First, we prove that $\eta(\sigma_{D}(T))=\eta(\sigma_{des}(T)).$ That
$\sigma_{des}(T)\subseteq\sigma_{D}(T)$ is clear. If
$\lambda\in\partial(\sigma_{D}(T))$, there exist $\\{\lambda_{n}\\}$ such that
$\\{\lambda_{n}\\}\cap\sigma(T)=\emptyset\,\,\makebox{and}\,\,\lambda_{n}\rightarrow\lambda.$
If $\lambda\not\in\sigma_{des}(T),$ then $\sigma_{des}(T-\lambda)<\infty$ and
hence by Lemma 2.10 that $\lambda\not\in\sigma_{D}(T)$, which contradicts with
$\lambda\in\partial(\sigma_{D}(T))\subseteq\sigma_{D}(T)$. Thus it follows
that $\lambda\in\sigma_{des}(T)$ when $\lambda\in\partial(\sigma_{D}(T))$, so
$\partial(\sigma_{D}(T))\subseteq\sigma_{des}(T).$ Note that
$\sigma_{des}(T)\subseteq\sigma_{rD}(T)\subseteq\sigma_{D}(T)$, thus,
$\eta(\sigma_{D}(T))=\eta(\sigma_{rD}(T)).$ Moreover, since $\sigma_{rD}$ and
$\sigma_{lD}$ are dual, we have
$\eta(\sigma_{lD}(T))=\eta(\sigma_{rD}(T^{*}))=\eta(\sigma_{D}(T^{*}))=\eta(\sigma_{D}(T)).$
Therefore,
$\eta(\sigma_{D}(T))=\eta(\sigma_{des}(T))=\eta(\sigma_{rD}(T))=\eta(\sigma_{lD}(T))$.
This proved (iii).
The following theorems decide 18 kind spectra filling-in-hole properties of
$M_{C}$:
Theorem 4.3. If
$\sigma_{*}=\sigma_{des},\sigma_{su},\sigma_{r},\sigma_{rb},\sigma_{sw},\sigma_{rw}$
or $\sigma_{re}$, then $\sigma_{*}$ has the generalized filling-in-hole
property. That is
$\sigma_{*}(A)\cup\sigma_{*}(B)=\sigma_{*}(M_{C})\cup W_{\sigma_{*}}(A,B,C),$
where $W_{\sigma_{*}}(A,B,C)$ is contained in the union of some holes in
$\sigma_{*}(M_{C})$. In particular,
(i).
$W_{\sigma_{des}}(A,B,C)\subseteq[(\sigma_{des}(A)\cap\sigma_{asc}(B))\setminus\sigma_{des}(B)]$
is contained in the union of all holes in $\sigma_{des}(B).$
(ii). If $\sigma_{*}=\sigma_{su},\sigma_{r},\sigma_{rb}$ or $\sigma_{re}$,
then
$W_{\sigma_{*}}(A,B,C)\subseteq[(\sigma_{*}(A)\cap\sigma_{*}(B^{*}))\setminus\sigma_{*}(B)]$
is contained in the union of all holes in $\sigma_{*}(B).$
(iii). If $\sigma_{*}=\sigma_{rw}$ or $\sigma_{sw}$, then
$W_{\sigma_{*}}(A,B,C)\subseteq[(\sigma_{*}(A)\cup\sigma_{*}(B^{*}))\setminus\sigma_{SF-}(B)]$
is contained in the union of all holes in $\sigma_{*}(B).$
Proof. (i). It follows from Lemma 2.5 and the proof of Lemma 2.6 (i) that
$\sigma_{des}(A)\cup\sigma_{des}(B)\supseteq\sigma_{des}(M_{C})\supseteq\sigma_{des}(B).$
This implies that $\sigma_{des}$ has the generalized filling-in-holes property
and
$W_{\sigma_{des}}(A,B,C)\subseteq\sigma_{des}(A)\setminus\sigma_{des}(B).$
Moreover, note that Lemma 2.7, we can prove that
$W_{\sigma_{des}}(A,B,C)\subseteq[(\sigma_{des}(A)\cap\sigma_{asc}(B))\setminus\sigma_{\sigma_{des}}(B)].$
To see this, let $\lambda\in W_{\sigma_{des}}(A,B,C).$ Then
$\sigma_{des}(A-\lambda)=\infty$ and $\sigma_{des}(B-\lambda)<\infty.$ If
$\sigma_{asc}(B-\lambda)<\infty,$ then by Lemma 2.7 (ii) we know that
$\sigma_{des}(A-\lambda)<\infty$, which is a contradiction. Thus
$\lambda\in\sigma_{asc}(B)$. Next, we can claim that $W_{\sigma_{des}}(A,B,C)$
is contained in the union of all holes in $\sigma_{des}(B),$ that is,
$W_{\sigma_{des}}(A,B,C)\subseteq\eta(\sigma_{des}(B)).$ Otherwise, there
exists $\lambda\in W_{\sigma_{des}}(A,B,C)\setminus\eta(\sigma_{des}(B)).$ By
Proposition 4.2 we have that $\eta(\sigma_{des}(B))=\eta(\sigma_{D}(B))$. Thus
$\lambda\not\in\eta(\sigma_{D}(B)).$ Furthermore, Lemma 2.7 tells us that
$des(A-\lambda)<\infty\Leftrightarrow des(M_{C}-\lambda)<\infty,$ which is a
contradiction with the assumption that $\lambda\in W_{\sigma_{des}}(A,B,C).$
Thus, it follows that $W_{\sigma_{des}}(A,B,C)$ is contained in the union of
all holes in $\sigma_{des}(B).$
(ii). We only prove $\sigma_{*}=\sigma_{su}$ case. Note that $A$ and $B$ are
surjective imply that $M_{C}$ is surjective, $M_{C}$ is surjective implies
that $B$ is also surjective. So we have
$\sigma_{su}(B)\subseteq\sigma_{su}(M_{C})\subseteq\sigma_{su}(A)\cup\sigma_{su}(B).$
This shows that $\sigma_{su}$ has the generalized filling-in-hole property and
$W_{\sigma_{su}}(A,B,C)\subseteq\sigma_{su}(A)\setminus\sigma_{su}(B).$
Next we claim that $W_{\sigma_{su}}(A,B,C)\subseteq\sigma_{a}(B).$ If not,
there exists $\lambda\in W_{\sigma_{su}}(A,B,C)\setminus\sigma_{a}(B).$
Combine this fact with the inclusion
$W_{\sigma_{su}}(A,B,C)\subseteq\sigma_{su}(A)\setminus\sigma_{su}(B)$ proved
above, we have that $B-\lambda$ is invertible. By Lemma 2.7 it follows that
$A-\lambda$ is surjective, this is a contradiction. Thus
$W_{\sigma_{su}}(A,B,C)\subseteq\sigma_{a}(B)$, and so
$W_{\sigma_{su}}(A,B,C)\subseteq(\sigma_{su}(A)\cap\sigma_{a}(B))\setminus\sigma_{su}(B)=(\sigma_{su}(A)\cap\sigma_{su}(B^{*}))\setminus\sigma_{su}(B).$
Similar to the proof of (1), we can get that $W_{\sigma_{su}}(A,B,C)$ is
contained in the union of all holes in $\sigma_{su}(B).$
(iii) is obvious by Proposition 4.2.
Duality, we have the following:
Theorem 4.4. If $\sigma_{*}=\sigma_{l},\sigma_{lb},\sigma_{aw},\sigma_{lw}$ or
$\sigma_{le}$, then $\sigma_{*}$ has the generalized filling-in-holes
property. That is
$\sigma_{*}(A)\cup\sigma_{*}(B)=\sigma_{*}(M_{C})\cup W_{\sigma_{*}}(A,B,C),$
where $W_{\sigma_{*}}(A,B,C)$ is contained in the union of some holes in
$\sigma_{*}(M_{C})$. In particular,
(i). If $\sigma_{*}=\sigma_{l},\sigma_{lb}$ or $\sigma_{le}$, then
$W_{\sigma_{*}}(A,B,C)\subseteq[(\sigma_{*}(B)\cap\sigma_{*}(A^{*}))\setminus\sigma_{*}(A)]$
is contained in the union of all holes in $\sigma_{*}(A).$
(ii). If $\sigma_{*}=\sigma_{aw}$ or $\sigma_{lw}$, then
$W_{\sigma_{*}}(A,B,C)\subseteq[(\sigma_{*}(B)\cup\sigma_{*}(A^{*}))\setminus\sigma_{SF+}(A)]$
is contained in the union of all holes in $\sigma_{*}(A).$
Theorem 4.5. If
$\sigma_{*}=\sigma_{lD},\sigma_{rD},\sigma_{lr},\sigma_{K_{3}},\sigma_{K}$ or
$\sigma_{SF_{0}}$, then $\sigma_{*}$ has the convex filling-in-hole property.
That is
$\eta(\sigma_{*}(A)\cup\sigma_{*}(B))=\eta(\sigma_{*}(M_{C})).$
Proof. It follows from Lemma 3.1 of [12] that
$\eta(\sigma(A)\cup\sigma(B))=\eta(\sigma(M_{C})),\eta(\sigma_{b}(A)\cup\sigma_{b}(B))=\eta(\sigma_{b}(M_{C})),$
$\eta(\sigma_{D}(A)\cup\sigma_{D}(B))=\eta(\sigma_{D}(M_{C})).$ Combine those
facts with Proposition 4.2 that it is easy to prove the theorem.
We are also interesting in the following question:
Open question 4.6. Do other spectra of $M_{C}$ have the filling-in-hole
properties ?
## 5 Examples
Now, we present examples to show that some conclusions about the spectra
structure or the filling-in-hole properties of $M_{C}$ are not true.
The following example shows that for spectrum
$\sigma_{*}=\sigma_{a},\sigma_{l},\sigma_{SF+},\sigma_{le},\sigma_{aw}$,
$\sigma_{lw},\sigma_{ab},\sigma_{lb},\sigma_{su},$
$\sigma_{r},\sigma_{SF-},\sigma_{re}$,$\sigma_{sw},\sigma_{rw},\sigma_{sb}$ or
$\sigma_{rb}$, it not only has not the equation (1) form, but also has not the
filling-in-hole property.
Example 5.1 ([12]). Let $X$ be the direct sum of countably many copies of
$\ell^{2}:=\ell^{2}(N)$. Thus, the elements of $X$ are the sequences
$\\{x_{j}\\}_{j=1}^{\infty}$ with $x_{j}\in\ell^{2}$ and
$\sum_{j=1}^{\infty}\|x_{j}\|^{2}<\infty$. Put $Y=\ell^{2}$. Let $V$ be the
forward shift on $\ell^{2}$:
$V:\ell^{2}\to\ell^{2},\quad\\{z_{1},z_{2},\ldots\\}\mapsto\\{0,z_{1},z_{2},\ldots\\},$
define the operators $A$ and $C$ by
$A:X\to X,\quad\\{x_{1},x_{2},\ldots\\}\mapsto\\{Vx_{1},Vx_{2},\ldots\\},$
$C:Y\to
X,\quad\\{y_{1},y_{2},\ldots\\}\mapsto\\{y_{1}e_{1},y_{2}e_{1},\ldots\\}.$
where $e_{1}=\\{1,0,0,\ldots\\}$. Let $B=0$ and consider the operator
$M_{C}=\left(\begin{array}[]{ll}A&C\\\ 0&B\end{array}\right):X\oplus Y\to
X\oplus Y.$
If
$\sigma_{*}=\sigma_{a},\sigma_{l},\sigma_{SF+},\sigma_{le},\sigma_{aw},\sigma_{lw},\sigma_{ab}$
or $\sigma_{lb}$, then
(i). $\sigma_{*}(A)\cup\sigma_{*}(B)\cup(S(A^{*})\cap
S(B))=\sigma_{*}(A)\cup\sigma_{*}(B)=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid=1\\}\cup\\{0\\}.$
(ii). $\sigma_{*}(M_{C})\cup(S(A^{*})\cap
S(B))=\sigma_{*}(M_{C})=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid=1\\}.$
(iii). $(\sigma_{*}(A)\cup\sigma_{*}(B))\setminus\sigma_{*}(M_{C})=\\{0\\}.$
Thus, it follows from (i) and (ii) that equation (1) does not hold for
spectrum
$\sigma_{*}=\sigma_{a},\sigma_{l},\sigma_{SF+},\sigma_{le},\sigma_{aw}$,
$\sigma_{lw},\sigma_{ab}$ or $\sigma_{lb}$. By duality, we can also show that
equation (1) does not hold for spectrum
$\sigma_{*}=\sigma_{su},\sigma_{r},\sigma_{SF-},\sigma_{re}$,
$\sigma_{sw},\sigma_{rw},\sigma_{sb}$ or $\sigma_{rb}$. Moreover, from (iii)
we knew that none of the above 16 kind spectra has the filling-in-hole
property.
This following example shows that ascent spectrum $\sigma_{asc}$ has not
equations (1) to (9) form.
Example 5.2. Let $X=Y=\ell^{2}$ and $\\{e_{n}\\}_{n\geq 1}$ be a basis of
$\ell^{2}$. Define $A,B,C\in B(\ell^{2})$ by
$Ae_{i}=\frac{1}{i}e_{2i}\,\,\,\makebox{for}\,\,i=1,2,\cdots,$
$Be_{1}=0,Be_{i}=\frac{1}{i}e_{i-1}\,\,\,\,\,\makebox{for}\,\,i=2,3,\cdots,$
$Ce_{i}=e_{2i-1}\,\,\,\,\makebox{for}\,\,i=1,2,3,\cdots.$
Then we have
$\displaystyle\\{0\\}=\sigma_{asc}(A)\cup\sigma_{asc}(B)$
$\displaystyle=\sigma_{asc}(A)\cup\sigma_{asc}(B)\cup S(A)\cup S(A^{*})\cup
S(B)\cup S(B^{*})$ $\displaystyle\not=\sigma_{asc}(M_{C})\cup S(A)\cup
S(A^{*})\cup S(B)\cup S(B^{*})$ $\displaystyle=\sigma_{asc}(M_{C})=\emptyset.$
In [16], the authors claimed that
$(\sigma_{ab}(A)\cup\sigma_{ab}(B))\setminus\sigma_{ab}(M_{C})\subseteq
S(A^{*})\cap\sigma_{D}(B),$
where $\sigma_{D}(B)$ was denoted by $F(B)$, which implies that
$(\sigma_{ab}(A)\cup\sigma_{ab}(B))\cup(S(A^{*})\cap\sigma_{D}(B))=\sigma_{ab}(M_{C})\cup(S(A^{*})\cap\sigma_{D}(B))$
$None$
The following example shows that neither the claim nor equation (13) is true.
Example 5.3. Let $X=Y=\ell^{2}$ and
$A\\{x_{1},x_{2},\ldots\\}\mapsto\\{x_{1},0,x_{2},0,\ldots\\},$
$B\\{x_{1},x_{2},\ldots\\}\mapsto\\{0,0,0,\ldots\\},$
$C\\{x_{1},x_{2},\ldots\\}\mapsto\\{0,x_{1},0,x_{2},\ldots\\}.$
Then
$\sigma_{ab}(A)=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid=1\\},\sigma_{ab}(B)=\\{0\\},\sigma_{ab}(M_{C})=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid=1\\},$
$S(A^{*})\cap\sigma_{D}(B)=\emptyset,\sigma_{ab}(M_{C})\not=\sigma_{ab}(A)\cup\sigma_{ab}(B).$
So
$(\sigma_{ab}(A)\cup\sigma_{ab}(B))\setminus\sigma_{ab}(M_{C})\not\subseteq
S(A^{*})\cap\sigma_{D}(B).$
The following example shows that for spectrum
$\sigma_{*}=\sigma_{aw},\sigma_{lw},\sigma_{sw}$ or $\sigma_{rw}$, it has not
equations (4) and (5) form.
Example 5.4. Let $X=Y=\ell^{2}$ and define $T,S,C\in B(\ell^{2})$ by
$T\\{x_{1},x_{2},x_{3},\ldots\\}\mapsto\\{0,x_{1},x_{2},\ldots\\},$
$S\\{x_{1},x_{2},x_{3},\ldots\\}\mapsto\\{x_{2},x_{4},x_{6},\ldots,\\},$
$C\\{x_{1},x_{2},x_{3},\ldots\\}\mapsto\\{0,0,0,\ldots\\}.$
(i). Put $A=S,B=T^{2}$. Then $A^{*}$ and $B$ have the SVEP and
$\sigma_{aw}(A)\cup\sigma_{aw}(B)\cup S(A^{*})\cup
S(B)=\sigma_{lw}(A)\cup\sigma_{lw}(B)\cup S(A^{*})\cup
S(B)=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid\leq 1\\},$
and
$\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid=1\\}=\sigma_{aw}(M_{C})\cup
S(A^{*})\cup S(B)=\sigma_{lw}(M_{C})\cup S(A^{*})\cup S(B).$
So, when $\sigma_{*}=\sigma_{aw}$ or $\sigma_{lw}$, we have
$\sigma_{*}(A)\cup\sigma_{*}(B)=\sigma_{*}(A)\cup\sigma_{*}(B)\cup
S(A^{*})\cup S(B)\not=\sigma_{*}(M_{C})\cup S(A^{*})\cup
S(B)=\sigma_{*}(M_{C}).$
(ii). Put $A=S^{2},B=T$. Then $A^{*}$ and $B$ have the SVEP and
$\sigma_{sw}(A)\cup\sigma_{sw}(B)\cup S(A^{*})\cup
S(B)=\sigma_{rw}(A)\cup\sigma_{rw}(B)\cup S(A^{*})\cup
S(B)=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid\leq 1\\}$
and
$\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid=1\\}=\sigma_{sw}(M_{C})\cup
S(A^{*})\cup S(B)=\sigma_{rw}(M_{C})\cup S(A^{*})\cup S(B).$
Thus, when $\sigma_{*}=\sigma_{sw}$ or $\sigma_{rw}$, we have
$\sigma_{*}(A)\cup\sigma_{*}(B)=\sigma_{*}(A)\cup\sigma_{*}(B)\cup
S(A^{*})\cup S(B)\not=\sigma_{*}(M_{C})\cup S(A^{*})\cup
S(B)=\sigma_{*}(M_{C}).$
The following example shows that for spectrum
$\sigma_{*}=\sigma_{K},\sigma_{K_{3}},\sigma_{SF0}$ or $\sigma_{lr}$, it not
only has not equations (4) and (5) form, but also has not generalized filling-
in-hole property.
Example 5.5. Let $X=Y=\ell^{2}$ and define $A,B,C\in B(\ell^{2})$ by
$A\\{x_{1},x_{2},x_{3},\ldots\\}\mapsto\\{x_{2},x_{4},x_{6},\ldots\\},$
$B\\{x_{1},x_{2},x_{3},\ldots\\}\mapsto\\{0,x_{1},0,x_{2},\ldots,\\},$
$C\\{x_{1},x_{2},x_{3},\ldots\\}\mapsto\\{0,0,0,\ldots\\}.$
It is easy to show that $A^{*}$ and $B$ have the SVEP and if
$\sigma_{*}=\sigma_{K},\sigma_{K_{3}},\sigma_{SF0}$ or $\sigma_{lr}$, then
$\sigma_{*}(A)\cup\sigma_{*}(B)=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid=1\\}$,
$\sigma_{*}(M_{C})=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid\leq 1\\}.$ Thus,
when $\sigma_{*}=\sigma_{K},\sigma_{K_{3}},\sigma_{SF0}$ or $\sigma_{lr}$, we
have
$\sigma_{*}(A)\cup\sigma_{*}(B)\cup S(A^{*})\cup
S(B)\not=\sigma_{*}(M_{C})\cup S(A^{*})\cup S(B)$
and
$\sigma_{*}(A)\cup\sigma_{*}(B)\not\supseteq\sigma_{*}(M_{C}).$
The following example shows that the inclusions in Theorem 4.1 may be strict
in general.
Example 5.6. Let $X_{n}$ be a complex $n$ dimensional Hilbert space. Define
$T,S,C_{3}\in B(\ell^{2})$ by
$T\\{x_{1},x_{2},x_{3},\cdots\\}=\\{0,x_{1},x_{2},x_{3},\cdots\\},$
$S\\{x_{1},x_{2},x_{3},\cdots\\}=\\{x_{2},x_{3},x_{4},\cdots\\},$
$C_{3}:=I-TS.$
(i). Put $A=T$, $C=\left(\begin{array}[]{cc}C_{3}&0\end{array}\right)$
$:\ell^{2}\oplus X_{n}\longrightarrow\ell^{2}$,
$B=\left(\begin{array}[]{cc}S&0\\\ 0&0\end{array}\right)$ $:\ell^{2}\oplus
X_{n}\longrightarrow\ell^{2}\oplus X_{n}$. Then
$W_{\sigma}(A,B,C)=\\{\lambda\in{\mathbb{C}}:0<\mid\lambda\mid<1$ $\\},$
$W_{\sigma_{b}}(A,B,C)=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid<1$ $\\}$,
thus $W_{\sigma}(A,B,C)\not=W_{\sigma_{b}}(A,B,C)$.
(ii). Put $A=T$, $C=\left(\begin{array}[]{cc}C_{3}&0\end{array}\right)$
$:\ell^{2}\oplus\ell^{2}\longrightarrow\ell^{2}$,
$B=\left(\begin{array}[]{cc}S&0\\\ 0&0\end{array}\right)$
$:\ell^{2}\oplus\ell^{2}\longrightarrow\ell^{2}\oplus\ell^{2}$. Then
$W_{\sigma_{b}}(A,B,C)=\\{\lambda\in{\mathbb{C}}:0<\mid\lambda\mid<1$ $\\},$
$W_{\sigma_{D}}(A,B,C)=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid<1$ $\\}$,
thus $W_{\sigma_{b}}(A,B,C)\not=W_{\sigma_{D}}(A,B,C)$.
(iii). Put $A=T$, $C=0$, $B=S$. Then $W_{\sigma_{b}}(A,B,C)=\emptyset,$
$W_{\sigma_{w}}(A,B,C)=\\{\lambda\in{\mathbb{C}}:\mid\lambda\mid<1\\}$, thus
$W_{\sigma_{b}}(A,B,C)\not=W_{\sigma_{w}}(A,B,C)$.
The following example shows that for $W_{\sigma_{e}}(A,B,C)$,
$W_{\sigma_{w}}(A,B,C)$ and $W_{\sigma_{D}}(A,B,C)$, there are no inclusion
relationship among them.
Example 5.7. Let $T,S,T_{1},S_{1},C_{1},C_{2},C_{3}\in B(\ell^{2})$ be defined
by
$T\\{x_{1},x_{2},x_{3},\cdots\\}=\\{0,x_{1},x_{2},x_{3},\cdots\\},$
$S\\{x_{1},x_{2},x_{3},\cdots\\}=\\{x_{2},x_{3},x_{4},\cdots\\},$
$T_{1}\\{x_{1},x_{2},x_{3},\cdots\\}=\\{0,x_{1},0,x_{2},0,x_{3},\cdots\\},$
$S_{1}\\{x_{1},x_{2},x_{3},\cdots\\}=\\{x_{2},x_{4},x_{6},\cdots\\},$
$C_{1}\\{x_{1},x_{2},x_{3},\cdots\\}=\\{x_{1},0,x_{3},0,x_{5},\cdots\\},$
$C_{2}\\{x_{1},x_{2},x_{3},\cdots\\}=\\{0,0,x_{1},0,x_{3},0,x_{5},\cdots\\},$
$C_{3}=I-TS.$
(i). If $A=T,B=S,C=C_{3}$, then $W_{\sigma_{w}}(A,B,C)\not\subseteq
W_{\sigma_{e}}(A,B,C).$
(ii). If $A=T,B=S,C=0$, then $W_{\sigma_{w}}(A,B,{C})\not\subseteq
W_{\sigma_{D}}(A,B,{C}).$
(iii). Let $A=T_{1},$ $B=\left(\begin{array}[]{cc}S_{1}&0\\\ 0&S\\\
\end{array}\right)$$:\ell^{2}\oplus\ell^{2}\longrightarrow\ell^{2}\oplus\ell^{2}$
and
$C=\left(\begin{array}[]{cc}C_{1}&0\end{array}\right)$$:\ell^{2}\oplus\ell^{2}\longrightarrow\ell^{2}.$
Then $W_{\sigma_{e}}(A,B,{C})\not\subseteq W_{\sigma_{D}}(A,B,{C}).$
(iv). If $A=T_{1},B=S_{1},C=C_{2}$, then $W_{\sigma_{e}}(A,B,{C})\not\subseteq
W_{\sigma_{w}}(A,B,{C}).$
(v). If $A=T$, $B=\left(\begin{array}[]{cc}S&0\\\ 0&0\\\
\end{array}\right)$$:\ell^{2}\oplus\ell^{2}\longrightarrow\ell^{2}\oplus\ell^{2}$
and
$C=\left(\begin{array}[]{cc}C_{3}&0\end{array}\right)$$:\ell^{2}\oplus\ell^{2}\longrightarrow\ell^{2},$
then $W_{\sigma_{D}}(A,B,{C})\not\subseteq
W_{\sigma_{e}}(A,B,{C}),W_{\sigma_{D}}(A,B,{C})\not\subseteq
W_{\sigma_{w}}(A,B,{C}).$
Note that $\sigma(T)\supseteq\sigma_{b}(T)\supseteq\sigma_{D}(T)\supseteq
acc\sigma(T)$ is well known, so we have
$\eta(\sigma(T))\supseteq\eta(\sigma_{b}(T))\supseteq\eta(\sigma_{D}(T))\supseteq\eta(acc\sigma(T)).$
The following example shows that the above inclusions may be strict.
Example 5.8. Let $X_{n}$ be a $n$ dimensional complex Hilbert space. Define
operators $A\in B(\ell^{2})$ by
$A\\{x_{1},x_{2},x_{3},\cdots\\}=\\{0,\frac{1}{2}x_{1},\frac{1}{3}x_{2},\frac{1}{4}x_{3},\cdots\\}.$
Then $\sigma(A)=\sigma_{D}(A)=\sigma_{des}(A)=\\{0\\}$ and
$acc\sigma(A)=\emptyset$. If consider operator
$T=\left(\begin{array}[]{ccc}0&0&0\\\ 0&3I&0\\\
0&0&5+A\end{array}\right):X_{n}\oplus\ell^{2}\oplus\ell^{2}\longrightarrow
X_{n}\oplus\ell^{2}\oplus\ell^{2},$
we have $\eta(\sigma(T))=\sigma(T)=\\{0,3,5\\}$,
$\eta(\sigma_{b}(T))=\sigma_{b}(T)=\\{3,5\\}$,
$\eta(\sigma_{D}(T))=\sigma_{D}(T)=\\{5\\}$,
$\eta(acc\sigma(T))=acc\sigma(T)=\emptyset$. Thus,
$\eta(\sigma(T))\not=\eta(\sigma_{b}(T))\not=\eta(\sigma_{D}(T))\not=\eta(acc\sigma(T)).$
The following example shows that spectra $\sigma_{rD}$ and $\sigma_{lD}$ have
not the generalized filling-in-hole property.
Example 5.9. Let $X=Y=\ell^{2}$. Define $A=0$ and $B,C$ by
$B\\{x_{1},x_{2},x_{3},\cdots\\}=\\{x_{2},x_{4},x_{6},\cdots\\},$
$C\\{x_{1},x_{2},x_{3},\cdots\\}=\\{x_{1},\frac{1}{\sqrt{2}}x_{3},\frac{1}{\sqrt{3}}x_{5},\cdots\\}.$
Then $0\not\in\sigma_{rD}(A)\cup\sigma_{rD}(B)$ but $0\in\sigma_{rD}(M_{C})$.
So $\sigma_{rD}(M_{C})\not\subseteq\sigma_{rD}(A)\cup\sigma_{rD}(B).$ Since
$\sigma_{rD}$ and $\sigma_{lD}$ are dual (see [26]), thus, neither
$\sigma_{rD}$ nor $\sigma_{lD}$ has the generalized filling-in-holes property.
Let $H,K$ be Hilbert spaces, $(A,B)\in B(H)\times B(K),C\in B(K,H)$. If $A\in
B(H)$, let $A^{*}$ denote the adjoint operator of $A$ and $\sigma_{p}(A)$
denote the point spectrum of $A$. In [3], the authors claimed that
$\eta(\sigma_{se}(A)\cup\sigma_{se}(B))=\eta(\sigma_{se}(M_{C})),$
More precisely,
$\sigma_{se}(A)\cup\sigma_{se}(B)\cup(\overline{\sigma_{p}(A^{*})}\cap\sigma_{p}(B))=\sigma_{se}(M_{C})\cup
W,$ $None$
where $W$ is the union of some holes in $\sigma_{se}(M_{C})$ which happen to
be subsets of $\overline{\sigma_{p}(A^{*})}\cup\sigma_{p}(B)$. The following
example shows that equation (14) is not true.
Example 5.10. Let $X,Y,A,C$ be defined as in Example 5.1. and $B\in B(Y)$ be
defined by
$B:\\{y_{1},y_{2},\ldots\\}\mapsto\\{0,\frac{1}{2}y_{1},\frac{1}{3}y_{2},\frac{1}{4}y_{3},\cdots\\}.$
Consider the operator
$M_{C}=\left(\begin{array}[]{ll}A&C\\\ 0&B\end{array}\right):X\oplus Y\to
X\oplus Y.$
Then we have
(i). $\sigma_{se}(M_{C})=\sigma_{se}(A)=\\{\lambda:\mid\lambda\mid=1\\}$,
(ii). $\sigma(B)=\sigma_{{se}}(B)=\\{0\\},$ $\sigma_{p}(B)=\emptyset,$
(iii). $\overline{\sigma_{p}(A^{*})}\cap\sigma_{p}(B)=\emptyset.$
Thus
$W=(\sigma_{{se}}(A)\cup\sigma_{{se}}(B)\cup(\overline{\sigma_{p}(A^{*})}\cap\sigma_{p}(B))\setminus\sigma_{{se}}(M_{C})=\\{0\\},$
so $W$ is just a point but not an open set. This showed that the above
conclusion is not true.
Acknowledgment
The authors are grateful to Doctor Qiaofen Jiang for the valuable suggestions
on Lemma 2.7 and Theorem 3.1.
## References
* [1] P. Aiena. Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, 2004.
* [2] M. Barraa and M. Boumazgour. A note on the spectrum of an upper triangular operator matrix, Proc. Amer. Math. Soc., 131(2003), 3083-3088.
* [3] M. Barraa, M. Boumazgour. On the perturbations of spectra of upper triangular operator matrices, J. Math. Anal. Appl., 347(2008), 315-322.
* [4] M. Berkani. Index of B-Fredholm operators and generalization of a Weyl theorem, Proc. Amer. Math. Soc., 130(2002), 1717-1723.
* [5] M. Berkani. On a class of quasi-Fredholm operators, Integr. Equ. Oper. Theory, 34(1999), 244-249.
* [6] C. Benhida, E. H. Zerouali and H. Zguitti. Spectral properties of upper-triangular block operators, Acta Sci. Math. (Szeged), 71(2005), 607-616.
* [7] M. Burgos, A. Kaidi, M. Mbekhta and M. Oudghiri. The descent spectrum and perturbations, J. operator Theory, 56(2006), 259-271.
* [8] X. H. Cao. Browder spectra for upper triangular operator matrices, J. Math. Anal. Appl., 342(2008), 477-484.
* [9] X. H. Cao, M.Z. Guo, B. Meng. Drazin spectrum and Wely’s theorem for operator matrices, J. Math. Res. Exposition, 26( 2006), 413-422.
* [10] X. H. Cao, M.Z. Guo, B. Meng. Semi-Fredholm spectrum and Weyl’s theory for operator matrices, Acta Math. Sinica, 22(2006), 169-178.
* [11] X. H. Cao, M.Z. Guo, B. Meng. Weyl’s theorem for upper triangular operator matrices, Linear Algebra Appl., 402(2005), 61-73.
* [12] X. L. Chen, S. F. Zhang, H. J. Zhong. On the filling in holes problem of operator matrices, Linear Algebra Appl., (2008), doi:10.1016/j.laa.2008.08.022.
* [13] D. S. Djordjević. Perturbations of spectra of operator matrices, J. Operator Theory, 48(2002), 467-486.
* [14] S. V. Djordjević, Y. M. Han. A note on Weyl’s theorem for operator matrices, Proc. Amer. Math. Soc., 131(2002), 2543-2547.
* [15] S. V. Djordjević, Y. M. Han. spectral continuity for operator matrices, Glasg. Math. J., 43(2001), 487-490.
* [16] S. V. Djordjević, H. Zguitti. Essential point spectra of operator matrices though local spectral theory, J. Math. Anal. Appl., 338(2008), 285-291.
* [17] H. K. Du, J. Pan. Perturbation of spectrums of $2\times 2$ operator matrices, Proc. Amer. Math. Soc., 121(1994), 761-766.
* [18] H. Elbjaoui, E. H. Zerouali. Local spectral theory for $2\times 2$ operator matrices, Int. J. Math. Math. Sci., 42(2003), 2667-2672.
* [19] J. K. Han, H. Y. Lee, W. Y. Lee. Invertible completions of $2\times 2$ upper triangular operator matrices. Proc. Amer. Math. Soc., 128(1999), 119-123.
* [20] I. S. Hwang, W. Y. Lee. The boundedness below of $2\times 2$ upper triangular operator matrices, Integr. Equ. Oper. Theory, 39(2001), 267-276.
* [21] W. Y. Lee. Weyl’s theorem for operator matrices, Integr. equ. oper. theory, 32(1998), 319-331.
* [22] W. Y. Lee. Weyl spectra of operator matrices, Proc. Amer. Math. Soc., 129(2000), 131-138.
* [23] Y. Li, X. H. Sun, H. K. Du. A note on the left essential spectra of operator matrices, Acta Math. Sinica, 23(2007), 2235-2240.
* [24] Y. Li, X. H. Sun, H. K. Du. The intersection of left(right) spectra of 2 $\times$ 2 upper triangular operator matrices, Linear Algebra Appl., 418(2006), 112-121.
* [25] Y. Li, H. K. Du. The intersection of essential approximate point spectra of operator matrices, J. Math. Anal. Appl., 323(2006), 1171-1183.
* [26] M. Merkhta, V. Müller. On the axiomatic theory of spectrum II., Studia Math., 119(1996), 129-147.
* [27] C. Schmoeger. Perturbation properties of some classes of operators, Rendiconti di Matematica, Serie VII Volume 14 Roma, (1994), 533-541.
* [28] E. H. Zerouali, H. Zguitti. Perturbation of spectra of operator matrices and local spectral theory, J. Math. Anal. Appl., 324(2006), 992-1005.
* [29] H. Y. Zhang, H. K. Du. Browder spectra of upper-triangular operator matrices, J. Math. Anal. Appl., 323(2006), 700-707.
* [30] S. F. Zhang, H. J. Zhong. A note of Browder spectrum of operator matrices, J. Math. Anal. Appl., 344(2008), 927-931 .
* [31] S. F. Zhang, H. J. Zhong, Q. F. Jiang. Drazin spectrum of operator matrices on the Banach space, Linear Algebra Appl., 429(2008), 2067-2075.
* [32] Y. N. Zhang, H. J. Zhong, L. Q. Lin. Browder spectra and essential spectra of operator matrices, Acta Math. Sinica, 24 (2008), 947-954.
|
arxiv-papers
| 2009-06-27T09:25:45 |
2024-09-04T02:49:03.600335
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhang Shifang, Zhong Huaijie, Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0906.5055"
}
|
0906.5056
|
# Fredholm Perturbation of Spectra of $2\times 2$ Upper Triangular Matrix
111This work is supported by the NSF of China (Grant Nos. 10771034, 10771191
and 10471124) and the NSF of Fujian Province of China (Grant Nos. Z0511019,
S0650009).
Shifang Zhang1,2, Huaijie Zhong2, Junde Wu1222Corresponding author: Junde Wu,
E-mail: [email protected]
1Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
2Department of Mathematics, Fujian Normal University, Fuzhou 350007, P. R.
China,
[email protected], [email protected]
Abstract As we knew, study the perturbation theory of spectra of operator is a
very important project in mathematics physics, in particular, in quantum
mechanics. In this paper, we characterize the Fredholm perturbation for the
Weyl spectrum, essential spectrum, spectrum, left spectrum, right spectrum,
lower semi-Fredholm spectrum, upper semi-Weyl spectrum and lower semi-Weyl
spectrum of upper triangular operator matrix
$M_{C}=\left(\begin{array}[]{cc}A&C\\\ 0&B\\\ \end{array}\right)$.
Keywords Operator matrix; spectra; perturbation.
AMS classifications 47A10
## 1 Introduction
Let $H$ and $K$ be the complex infinite dimensional separable Hilbert spaces,
$B(H,K)$ be the set of all bounded linear operators from $H$ into $K$. For
simplicity, we write $B(H,H)$ as $B(H).$ If $T\in B(H,K)$, we use $R(T)$ and
$N(T)$ to denote the range and kernel of $T$, respectively, and define
$\alpha(T)=\dim N(T)$ and $\beta(T)=\dim(K/R(T))$. For $T\in B(H,K)$, if
$R(T)$ is closed and $\alpha(T)<\infty$, we call $T$ an upper semi-Fredholm
operator; if $\beta(T)<\infty$, then $T$ is called a lower semi-Fredholm
operator. If $T$ is either an upper or lower semi-Fredholm operator, then $T$
is called a semi-Fredholm operator. In this case, the index of $T$ is defined
as ind$(T)=\alpha(T)-\beta(T).$ If $T$ is a semi-Fredholm operator with
$\alpha(T)<\infty$ and $\beta(T)<\infty$, then $T$ is called a Fredholm
operator. For $T\in B(H)$, the ascent asc$(T)$ and the descent des$(T)$ are
given by asc$(T)=\inf\\{k\geqslant 0:N(T^{k})=N(T^{k+1})\\}$ and
des$(T)=\inf\\{k\geqslant 0:R(T^{k})=R(T^{k+1})\\}$, respectively; the infimum
over the empty set is taken to be $\infty$.
Let $G(H,K),G_{l}(H,K)$, $G_{r}(H,K),\Phi(H,K),\Phi_{+}(H,K)$ and
$\Phi_{-}(H,K)$, respectively, denote the sets of all invertible operators,
left invertible operators, right invertible operators, Fredholm operators,
upper semi-Fredholm operators and lower semi-Fredholm operators from $H$ into
$K$. The sets of all Weyl operators, upper semi-Weyl operators and lower semi-
Weyl operators from $H$ into $K$ are defined, respectively, by
$\Phi_{0}(H,K):=\\{T\in\Phi(H,K):$ ind$(T)=0\\},$
$\Phi_{+}^{-}(H,K):=\\{T\in\Phi_{+}(H,K):$ ind$(T)\leq 0\\},$
$\Phi_{-}^{+}(H,K):=\\{T\in\Phi_{-}(H,K):$ ind$(T)\geq 0\\}.$
When $H=K$, the above 9 kind operator classes are also abbreviated as
$G(H),G_{l}(H)$, $G_{r}(H),\Phi(H),$ $\Phi_{+}(H)$, $\Phi_{-}(H),$
$\Phi_{0}(H)$,$\Phi_{+}^{-}(H)$ and $\Phi_{-}^{+}(H),$ respectively.
For $T\in B(H)$, its corresponding spectra are, respectively, defined by
the spectrum: $\sigma(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\makebox{ is not
invertible}\\},$
the left spectrum: $\sigma_{l}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I$ is
not left invertible$\\},$
the right spectrum: $\sigma_{r}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I$ is
not right invertible$\\},$
the essential spectrum: $\sigma_{e}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi(H)\\},$
the upper semi-Fredholm spectrum:
$\sigma_{SF+}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{+}(H)\\},$
the lower semi-Fredholm spectrum:
$\sigma_{SF-}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda I\not\in\Phi_{-}(H)\\},$
the Weyl spectrum: $\sigma_{w}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{0}(H)\\},$
the upper semi-Weyl spectrum:
$\sigma_{aw}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{+}^{-}(H)\\},$
the lower semi-Weyl spectrum:
$\sigma_{sw}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{-}^{+}(H)\\},$
the Browder spectrum: $\sigma_{b}(T)=\\{\lambda\in{\mathbb{C}}:T-\lambda
I\not\in\Phi_{b}(H)\\},$ where $\Phi_{b}(H):=\\{T\in\Phi(H):$ asc$(T)<\infty$
and des$(T)<\infty\\}.$
It is well known that all the above spectra are compact nonempty subsets of
complex plane ${\mathbb{C}}$.
Let $H$ be a Hilbert space and $T$ be a bounded linear operator defined on $H$
and $H_{1}$ be an invariant closed subspace of $T$. Then $T$ can be
represented by the form of
$T=\left(\begin{array}[]{cc}*&*\\\ 0&*\\\ \end{array}\right):H_{1}\oplus
H_{1}^{\perp}\rightarrow H_{1}\oplus H_{1}^{\perp},$
which motivated the interest in $2\times 2$ upper-triangular operator matrices
(see [1-19]).
Henceforth, for $A\in B(H)$, $B\in B(K)$ and $C\in B(K,H)$, we put
$M_{C}=\left(\begin{array}[]{cc}A&C\\\ 0&B\\\ \end{array}\right)$. It is clear
that $M_{C}\in B(H\oplus K)$. Recent, people studied the perturbation theory
of some spectra of $M_{C}$, for example, in [8], for the spectrum
$\sigma(M_{C})$, the perturbation result is
$\bigcap_{C\in
B(K,\,H)}\sigma(M_{C})=\sigma_{l}(A)\cup\sigma_{r}(B)\cup\\{\lambda\in{\mathbb{C}}:\alpha(B-\lambda)\not=\beta(A-\lambda)\\}.$
$None$
In [5], for the Weyl spectrum $\sigma_{w}(M_{C})$ and the essential spectrum
$\sigma_{e}(M_{C})$, the perturbation results are
$\bigcap_{C\in
B(K,\,H)}\sigma_{w}(M_{C})=\sigma_{SF+}(A)\cup\sigma_{SF-}(B)\cup\\{\lambda\in{\mathbb{C}}:\alpha(A-\lambda)+\alpha(B-\lambda)\not=\beta(A-\lambda)+\beta(B-\lambda)\\}$
$None$
and
$\bigcap_{C\in
B(K,\,H)}\sigma_{e}(M_{C})=\sigma_{SF+}(A)\cup\sigma_{SF-}(B)\cup$
$\\{\lambda\in{\mathbb{C}}:\min(\beta(A-\lambda),\alpha(B-\lambda))<\max(\beta(A-\lambda),\alpha(B-\lambda))=\infty\\}.$
$None$
In [1-3, 10], the authors also characterize completely sets $\bigcap_{C\in
B(K,\,H)}\sigma_{*}(M_{C})$, where $\sigma_{*}(M_{C})$ may be the Browder
spectrum, left spectrum, right spectrum, lower semi-Fredholm spectrum, upper
semi-Fredholm spectrum, lower semi-Weyl spectrum or upper semi-Weyl spectrum
of $M_{C}$, respectively.
Moreover, in [13-15], for the spectra $\sigma_{*}(M_{C})$, where
$\sigma_{*}=\sigma_{r},\sigma_{SF-}$ or $\sigma_{sw}$, its perturbation result
is
$\bigcap_{C\in G(K,\,H)}\sigma_{*}(M_{C})=(\bigcap_{C\in
B(K,\,H)}\sigma_{*}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{is
compact}\\};$ $None$
for the spectra $\sigma_{*}(M_{C})$, where
$\sigma_{*}=\sigma_{l},\sigma_{SF+}$ or $\sigma_{aw}$, its perturbation result
is
$\bigcap_{C\in G(K,\,H)}\sigma_{*}(M_{C})=(\bigcap_{C\in
B(K,\,H)}\sigma_{*}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:B-\lambda\,\makebox{is
compact}\\};$ $None$
for the spectra $\sigma_{*}(M_{C})$, where $\sigma_{*}=\sigma,\sigma_{e}$ or
$\sigma_{w}$, its perturbation result is
$\bigcap_{C\in G(K,\,H)}\sigma_{*}(M_{C})=(\bigcap_{C\in
B(K,\,H)}\sigma_{*}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{or}\,B-\lambda\,\makebox{is
compact}\\}.$ $None$
Note that equations (1) to (3) showed the perturbation of all bounded linear
operator $C$ in $B(K,H)$, and equations (4) to (6) showed the perturbation of
all bounded invertible linear operator $C$ in $G(K,H)$.
In this paper, we characterize the Fredholm perturbation for the Weyl
spectrum, essential spectrum, spectrum, left spectrum, right spectrum, lower
semi-Fredholm spectrum, upper semi-Weyl spectrum and lower semi-Weyl spectrum
of $M_{C}$.
## 2 Main results and proofs
At first, in order to characterize the perturbation of Weyl spectrum of
$M_{C}$, we need the following:
Lemma 1. For a given pair $(A,B)\in B(H)\times B(K)$, the following statements
are equivalent:
(i). there exists some $C\in B(K,H)$ such that $M_{C}\in\Phi_{0}(H\oplus K),$
(ii). $A\in\Phi_{+}(H)$, $B\in\Phi_{-}(K)$ and
$\alpha(A)+\alpha(B)=\beta(A)+\beta(B),$
(iii). there exists some $Q\in G(K,H)$ such that $M_{Q}\in\Phi_{0}(H\oplus
K),$
(iv). there exists some $Q\in\Phi(K,H)$ such that $M_{Q}\in\Phi_{0}(H\oplus
K).$
Proof. (i)$\Leftrightarrow$(ii) was proved in [5, Theorem 3.6].
(ii)$\Rightarrow$(iii). It is sufficient to prove that if $A\in\Phi_{+}(H)$,
$B\in\Phi_{-}(K)$ and $\beta(A)=\alpha(B)=\infty,$ then there exists $Q\in
G(K,H)$ such that $M_{Q}\in\Phi_{0}(H\oplus K).$ To show this, there are three
cases to consider:
Case 1. Suppose $\alpha(A)=\beta(B)<\infty$. Define an operator
$Q:K\rightarrow H\,\makebox{ by}\,Q=\left(\begin{array}[]{cc}T_{1}&0\\\
0&T_{2}\\\ \end{array}\right):N(B)\oplus N(B)^{\perp}\rightarrow
R(A)^{\perp}\oplus R(A),$ where $T_{1}$ and $T_{2}$ are invertible operators.
Obviously, $Q\in G(K,H)$ and $M_{Q}\in\Phi(H\oplus K)$. Also, it is evident
that $N(M_{Q})=N(A)\oplus\\{0\\}$ and $R(M_{Q})^{\perp}=\\{0\\}\oplus
R(B)^{\perp}$. Thus $\alpha(M_{Q})=\beta(M_{Q})=\alpha(A)=\beta(B)<\infty$,
and hence $M_{Q}\in\Phi_{0}(H\oplus K)$ is clear.
Case 2. Suppose $\beta(B)<\alpha(A)<\infty$ and put $l=\alpha(A)-\beta(B).$
Note that $\beta(A)=\dim N(B)^{\perp}=\infty$, let $R(A)^{\perp}=H_{1}\oplus
H_{2}$ and $\dim H_{2}=l$, $N(B)^{\perp}=K_{1}\oplus K_{2}$ and
$\dim(K_{1})=l.$
Define an operator $Q:K\rightarrow H\,\makebox{
by}\,Q=\left(\begin{array}[]{ccc}T_{1}&0&0\\\ 0&T_{2}&0\\\ 0&0&T_{3}\\\
\end{array}\right):N(B)\oplus K_{1}\oplus K_{2}\rightarrow H_{1}\oplus
H_{2}\oplus R(A),$ where $T_{1},T_{2}$ and $T_{3}$ are invertible operators.
Obviously, $Q\in B(K,H)$ is invertible. Now we claim that
$M_{Q}\in\Phi_{0}(H\oplus K)$. In fact, $M_{Q}$ has the following form:
$M_{Q}=\left(\begin{array}[]{ccccc}0&0&T_{1}&0&0\\\ 0&0&0&T_{2}&0\\\
0&A_{1}&0&0&T_{3}\\\ 0&0&0&B_{1}&B_{2}\\\ 0&0&0&0&0\\\
\end{array}\right):{N(A)}\oplus{N(A)}^{\perp}\oplus{N(B)}\oplus{K_{1}}\oplus{K_{2}}\longrightarrow
H_{1}\oplus{H_{2}}\oplus{R(A)}\oplus{R(B)}\oplus{R(B)}^{\perp},$ where
$A_{1}\in B(N(B)^{\perp},R(A))$ and $(B_{1}\,\,B_{2})\in B((K_{1}\oplus
K_{2}),R(B))$ are invertible operators. Moreover, observe that $\dim
K_{1}<\infty,$ we have $B_{1}\in G(K_{1},R(B_{1}))$, $B_{2}\in
G(K_{2},R(B_{2}))$ and $\dim K_{1}=\dim R(B_{1})=\dim(R(B)\ominus R(B_{2})).$
Now let $W_{1}=\left(\begin{array}[]{ccccc}I&0&0&0&0\\\ 0&I&0&0&0\\\
0&0&I&0&0\\\ 0&-B_{1}T_{2}^{-1}&0&I&0\\\ 0&0&0&0&I\\\
\end{array}\right):{N(A)}\oplus{N(A)}^{\perp}\oplus{N(B)}\oplus{K_{1}}\oplus{K_{2}}\longrightarrow{N(A)}\oplus{N(A)}^{\perp}\oplus{N(B)}\oplus{K_{1}}\oplus{K_{2}},$
and $W_{2}=\left(\begin{array}[]{ccccc}I&0&0&0&0\\\
0&I&0&0&-A_{1}^{-1}T_{3}\\\ 0&0&I&0&0\\\ 0&0&0&I&0\\\ 0&0&0&0&I\\\
\end{array}\right):H_{1}\oplus{H_{2}}\oplus{R(A)}\oplus{R(B)}\oplus{R(B)}^{\perp}\longrightarrow
H_{1}\oplus{H_{2}}\oplus{R(A)}\oplus{R(B)}\oplus{R(B)}^{\perp}.$
Then $W_{1}M_{Q}W_{2}=\left(\begin{array}[]{ccccc}0&0&T_{1}&0&0\\\
0&0&0&T_{2}&0\\\ 0&A_{1}&0&0&0\\\ 0&0&0&0&B_{2}\\\ 0&0&0&0&0\\\
\end{array}\right):{N(A)}\oplus{N(A)}^{\perp}\oplus{N(B)}\oplus{K_{1}}\oplus{K_{2}}\longrightarrow
H_{1}\oplus{H_{2}}\oplus{R(A)}\oplus{R(B)}\oplus{R(B)}^{\perp}.$ Since
$A_{1},T_{1}$ and $T_{2}$ are invertible, we get that
$R(W_{1}M_{Q}W_{2})=H_{1}\oplus{H_{2}}\oplus{R(A)}\oplus{R(B_{2})}\oplus\\{0\\}$
and
$N(W_{1}M_{Q}W_{2})=N(A)\oplus\\{0\\}\oplus\\{0\\}\oplus\\{0\\}\oplus\\{0\\},$
and
$R(W_{1}M_{Q}W_{2})^{\perp}=\\{0\\}\oplus\\{0\\}\oplus\\{0\\}\oplus({R(B)}\ominus{R(B_{2})})\oplus{R(B)^{\perp}}.$
Thus $W_{1}M_{Q}W_{2}\in\Phi(H\oplus K)$ and
$\displaystyle\alpha(W_{1}M_{Q}W_{2})$ $\displaystyle=\alpha(A)=l+\beta(B)$
$\displaystyle=\dim K_{1}+\beta(B)$
$\displaystyle=\dim({R(B)}\ominus{R(B_{2})})+\beta(B)$
$\displaystyle=\beta(W_{1}M_{Q}W_{2})<\infty.$
So $W_{1}M_{Q}W_{2}\in\Phi_{0}(H\oplus K).$ Also since $W_{1}$ and $W_{2}$ are
invertible, it follows that $M_{Q}\in\Phi_{0}(H\oplus K).$
Case 3. Suppose $\alpha(A)<\beta(B)<\infty$, put $l=\beta(B)-\alpha(A).$ Since
$\dim R(A)=\dim N(B)=\infty$, let R(A)$=H_{1}\oplus H_{2}$ and $\dim H_{1}=l$,
$N(B)=K_{1}\oplus K_{2}$ and $\dim(K_{2})=l.$ That $\dim
H_{2}=\dim(K_{1})=\infty$ is clear. Define an operator $Q:K\rightarrow
H\,\makebox{ by}\,Q=\left(\begin{array}[]{ccc}T_{1}&0&0\\\ 0&T_{2}&0\\\
0&0&T_{3}\\\ \end{array}\right):K_{1}\oplus K_{2}\oplus
N(B)^{\perp}\rightarrow R(A)^{\perp}\oplus H_{1}\oplus H_{2},$ where $T_{1},$
$T_{2}$ and $T_{3}$ are invertible operators. Obviously, $Q\in G(K,H)$.
Similar to the proof of Case 2, we can also show that
$M_{Q}\in\Phi_{0}(H\oplus K)$.
It follows from Case 1 to Case 3 that (ii)$\Rightarrow$(iii).
Finally, (iii)$\Rightarrow$(iv) and (iv)$\Rightarrow$(i) are clear. The lemma
is proved.
From Lemma 1 and Equation (2), we have the following:
Theorem 1. For a given pair $(A,B)\in B(H)\times B(K)$, we have
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{w}(M_{C})=\bigcap_{C\in
G(K,\,H)}\sigma_{w}(M_{C})=\bigcap_{C\in B(K,\,H)}\sigma_{w}(M_{C})$
$=\sigma_{SF+}(A)\cup\sigma_{SF-}(B)\cup\\{\lambda\in{\mathbb{C}}:\alpha(A-\lambda)+\alpha(B-\lambda)\not=\beta(A-\lambda)+\beta(B-\lambda)\\}.$
In order to characterize the perturbation of essential spectrum of $M_{C}$, we
need the following:
Lemma 2. For a given pair $(A,B)\in B(H)\times B(K)$, the following statements
are equivalent:
(i). there exists some $C\in B(K,H)$ such that $M_{C}\in\Phi(H\oplus K),$
(ii).
$\left\\{\begin{array}[]{l}A\in\Phi(H)\,\,\makebox{and}\,\,B\in\Phi(K)\\\
\makebox{or}\,\,A\in\Phi_{+}(H),B\in\Phi_{-}(K)\,\,\makebox{and}\,\,\beta(A)=\alpha(B)=\infty,\end{array}\right.$
(iii). there exists some $Q\in G(K,H)$ such that $M_{Q}\in\Phi(H\oplus K),$
(iv). there exists some $Q\in\Phi(K,H)$ such that $M_{Q}\in\Phi(H\oplus K).$
Proof. (i)$\Rightarrow$(ii). Suppose that $M_{C}\in\Phi(H\oplus K)$ for some
$C\in B(K,H)$. It follows from [5, Theorem 3.2] that
$A\in\Phi_{+}(H),B\in\Phi_{-}(K)$. Moreover, by [19, Lemma 2.2] we have that
either both $A$ and $B$ are Fredholm operators or neither $A$ nor $B$ is a
Fredholm operator. Thus $\beta(A)=\alpha(B)=\infty$ when neither $A$ nor $B$
is a Fredholm operator.
(ii)$\Rightarrow$(iii). To do this, if $A\in\Phi(H)$ and $B\in\Phi(K)$, then
$M_{C}\in\Phi(H\oplus K)$ for every $C\in B(K,H)$. On the other hand, if
$A\in\Phi_{+}(H)$, $B\in\Phi_{-}(K)$ and $\beta(A)=\alpha(B)=\infty.$ Define
an operator $Q:K\rightarrow H\,\makebox{
by}\,Q=\left(\begin{array}[]{cc}T_{1}&0\\\ 0&T_{2}\\\
\end{array}\right):N(B)\oplus N(B)^{\perp}\rightarrow R(A)^{\perp}\oplus
R(A)$, where $T_{1}$ and $T_{2}$ are invertible operators. Obviously, $Q\in
G(K,H)$, and it is easy to show that $M_{Q}\in\Phi(H\oplus K)$.
(iii) $\Rightarrow$ (iv) and (iv) $\Rightarrow$ (i) are obvious. The lemma is
proved.
From Lemma 2 and Equation (3) we have the following immediately:
Theorem 2. For a given pair $(A,B)\in B(H)\times B(K)$, we have
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{e}(M_{C})=\bigcap_{C\in
G(K,\,H)}\sigma_{e}(M_{C})=\bigcap_{C\in B(K,\,H)}\sigma_{e}(M_{C})$
$=\sigma_{SF+}(A)\cup\sigma_{SF-}(B)\cup\\{\lambda\in{\mathbb{C}}:\min(\beta(A-\lambda),\alpha(B-\lambda))<\max(\beta(A-\lambda),\alpha(B-\lambda))=\infty\\}.$
In order to characterize the perturbation of spectrum of $M_{C}$, we need the
following lemma which is a generalization in [9, Theorem 2] in the case of
Hilbert spaces:
Lemma 3. For a given pair $(A,B)\in B(H)\times B(K)$, the following statements
are equivalent:
(i). there exists some $C\in B(K,H)$ such that $M_{C}$ is invertible,
(ii). $A$ is left invertible, $B$ is right invertible and
$\beta(A)=\alpha(B),$
(iii). there exists some $Q\in G(K,H)$ such that $M_{Q}$ is invertible,
(iv). there exists some $Q\in\Phi(K,H)$ such that $M_{Q}$ is invertible.
Proof. (i)$\Rightarrow$(ii) is prove in [9, Theorem 2]. In fact, if $M_{C}$ is
invertible, it is easy to show that $A$ is left invertible and $B$ is right
invertible, which implies that $\alpha(A)=\beta(B)=0$. Moreover, it follows
from Lemma 1 that $\alpha(A)+\alpha(B)=\beta(A)+\beta(B),$ thus
$\beta(A)=\alpha(B)$.
(ii)$\Rightarrow$(iii). Suppose $A$ is left invertible, $B$ is right
invertible and $\beta(A)=\alpha(B)$. Define an operator $Q:K\rightarrow
H\,\makebox{ by}\,Q=\left(\begin{array}[]{cc}T_{1}&0\\\ 0&T_{2}\\\
\end{array}\right):N(B)\oplus N(B)^{\perp}\rightarrow R(A)^{\perp}\oplus
R(A),$ where $T_{1}$ and $T_{2}$ are invertible operators. it is evident that
$Q\in G(K,H)$ and $M_{Q}\in G(H\oplus K)$.
(iii) $\Rightarrow$ (iv) and (iv) $\Rightarrow$ (i) are obvious. The lemma is
proved.
From Lemma 3 and Equation (1), the following theorem is immediate:
Theorem 3. For a given pair $(A,B)\in B(H)\times B(K)$, We have
$\bigcap_{C\in\Phi(K,\,H)}\sigma(M_{C})=\bigcap_{C\in
G(K,\,H)}\sigma(M_{C})=\bigcap_{C\in B(K,\,H)}\sigma(M_{C})$
$=\sigma_{l}(A)\cup\sigma_{r}(B)\cup\\{\lambda\in{\mathbb{C}}:\alpha(B-\lambda)\not=\beta(A-\lambda)\\}.$
In order to characterize the perturbation for left spectrum, right spectrum,
lower semi-Weyl spectrum, upper semi-Weyl spectrum and lower semi-Fredholm
spectrum of $M_{C}$, we need the following three lemmas:
Lemma 4. For a given pair $(A,B)\in B(H)\times B(K)$, if either $A$ or $B$ is
a compact operator, then for each $C\in\Phi(K,H)$, $M_{C}$ is not a semi-
Fredholm operator.
Proof. If $B$ is a compact operator, then we can claim that $M_{C}$ is not a
semi-Fredholm operator for each $C\in\Phi(K,H)$. If not, assume that
$C_{0}\in\Phi(K,H)$ such that $M_{C_{0}}$ is a semi-Fredholm operator. Since
$C_{0}\in\Phi(K,H)$, there exists $C_{1}\in\Phi(H,K)$ such that
$C_{0}C_{1}=I+K$, where $K\in B(H)$ is a compact operator. Note that
$\left(\begin{array}[]{cc}A&C_{0}\\\ 0&B\\\
\end{array}\right)\left(\begin{array}[]{cc}I&0\\\ -C_{1}A&I\\\
\end{array}\right)=\left(\begin{array}[]{cc}A-C_{0}C_{1}A&C_{0}\\\
-BC_{1}A&B\\\ \end{array}\right)=\left(\begin{array}[]{cc}-KA&C_{0}\\\
-BC_{1}A&B\\\ \end{array}\right),$
we have that $\left(\begin{array}[]{cc}-KA&C_{0}\\\ -BC_{1}A&B\\\
\end{array}\right)$ is a semi-Fredholm operator. Also since $K$ and $B$ are
compact operators, both $\left(\begin{array}[]{cc}0&0\\\ -BC_{1}A&0\\\
\end{array}\right)$ and $\left(\begin{array}[]{cc}-KA&0\\\ 0&B\\\
\end{array}\right)$ are also compact. Thus
$\left(\begin{array}[]{cc}0&C_{0}\\\ 0&0\\\ \end{array}\right)$ is a semi-
Fredholm operator, which is impossible. So $M_{C}$ is not a semi-Fredholm
operator for each $C\in\Phi(K,H)$.
Similarly, we can prove when $A$ is a compact operator, $M_{C}$ is not a semi-
Fredholm operator for each $C\in\Phi(K,H)$. The lemma is proved.
Lemma 5. The following statements are equivalent:
(i). $B$ is not compact,
(ii). for each given $A\in\Phi_{+}(H)$, if $\beta(A)=\infty$, then there
exists an operator $C\in G(K,H)$ such that $M_{C}$ is an upper semi-Weyl
operator and $\alpha(M_{C})=\alpha(A)$,
(iii). for each given $A\in\Phi_{+}(H)$, if $\beta(A)=\infty$, then there
exists an operator $C\in\Phi(K,H)$ such that $M_{C}$ is an upper semi-Weyl
operator and $\alpha(M_{C})=\alpha(A)$,
(iv). for each given $A\in\Phi_{+}(H)$, if $\beta(A)=\infty$, then there
exists an operator $C\in G(K,H)$ such that $M_{C}$ is an upper semi-Weyl
operator,
(v). for each given $A\in\Phi_{+}(H)$, if $\beta(A)=\infty$, then there exists
an operator $C\in\Phi(K,H)$ such that $M_{C}$ is an upper semi-Weyl operator,
(vi). for each given $A\in\Phi_{+}(H)$, if h $\beta(A)=\infty$, then there
exists an operator $C\in G(K,H)$ such that $M_{C}$ is an upper semi-Fredholm
operator,
(vii). for each given $A\in\Phi_{+}(H)$, if $\beta(A)=\infty$, then there
exists an operator $C\in\Phi(K,H)$ such that $M_{C}$ is an upper semi-Fredholm
operator.
Proof. Obviously, we only need to prove the implications (i) $\Rightarrow$
(ii) and (vii) $\Rightarrow$ (i).
(vii) $\Rightarrow$ (i). If $B$ is compact, then it follows from Lemma 4 that
$M_{C}$ is not a semi-Fredholm operator for each $C\in\Phi(K,H)$, which
contradicts with (vii). Thus $B$ is not compact.
(i) $\Rightarrow$ (ii). Suppose that $B$ is not compact. Then we consider the
following two cases:
Case 1. Assume that $R(B)$ is closed. Since the assumption that $B$ is not
compact, we have that $\dim{N(B)}^{\perp}=\infty.$ Also since
$\beta(A)=\infty$, let ${R(A)}^{\perp}=H_{1}\oplus H_{2}$ with $\dim
H_{1}=\dim N(B)$ and $\dim H_{2}=\infty.$ Define an operator $C:K\rightarrow
H$ by
$C=\left(\begin{array}[]{cc}C_{1}&0\\\ 0&C_{2}\\\
\end{array}\right):{N(B)}\oplus{N(B)}^{\perp}\longrightarrow{H_{1}}\oplus(H_{2}\oplus
R(A)),$
where $C_{1}\in B(N(B),H_{1})$ and $C_{2}\in B({N(B)}^{\perp},H_{2}\oplus
R(A))$ are invertible operators. Obviously, operator $C$ is invertible. By
[12, Lemma 2], $M_{C}$ is an upper semi-Fredholm operator. Moreover, it is
easy to prove that $N(M_{C})=N(A)\oplus\\{0\\}$ and
$\dim{R(M_{C})}^{\perp}\geq\dim H_{2}=\infty.$ Thus, $M_{C}$ is an upper semi-
Weyl operator and $\alpha(M_{C})=\alpha(A)$.
Case 2. Assume that $R(B)$ is not closed. By [13, Lemma 3.6] and its proof, we
can obtain an operator $C\in G(K,H)$ such that $M_{C}$ is an upper semi-Weyl
operator and $\alpha(M_{C})=\alpha(A)$. The lemma is proved.
Duality, we have:
Lemma 6. The following statements are equivalent:
(i). $A$ is not compact,
(ii). for each given $B\in\Phi_{-}(K)$, if $\alpha(B)=\infty$, then there
exists an operator $C\in G(K,H)$ such that $M_{C}$ is a lower semi-Weyl
operator and $\beta(M_{C})=\beta(B)$,
(iii). for each given $B\in\Phi_{-}(K)$, if $\alpha(B)=\infty$, then there
exists an operator $C\in\Phi(K,H)$ such that $M_{C}$ is a lower semi-Weyl
operator and $\beta(M_{C})=\beta(B)$,
(iv). for each given $B\in\Phi_{-}(K)$, if $\alpha(B)=\infty$, then there
exists an operator $C\in G(K,H)$ such that $M_{C}$ is a lower semi-Weyl
operator,
(v). for each given $B\in\Phi_{-}(K)$, if $\alpha(B)=\infty$, then there
exists an operator $C\in\Phi(K,H)$ such that $M_{C}$ is a lower semi-Weyl
operator,
(vi). for each given $B\in\Phi_{-}(K)$, if $\alpha(B)=\infty$, then there
exists an operator $C\in G(K,H)$ such that $M_{C}$ is a lower semi-Fredholm
operator,
(vii). for each given $B\in\Phi_{-}(K)$, if $\alpha(B)=\infty$, then there
exists an operator $C\in\Phi(K,H)$ such that $M_{C}$ is a lower semi-Fredholm
operator.
Our Theorem 4 and Theorem 5 following show the similar conclusions as Equation
(4)-(5).
Theorem 4. For a given pair $(A,B)\in B(H)\times B(K)$, we have
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{*}(M_{C})=(\bigcap_{C\in
B(K,\,H)}\sigma_{*}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{is
compact}\\},$
where $\sigma_{*}\in\\{\sigma_{r},\sigma_{SF-},\sigma_{sw}\\}.$
Proof. According to Lemma 4, it is clear that
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{*}(M_{C})\supseteq(\bigcap_{C\in
B(K,\,H)}\sigma_{*}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$
In order to show the theorem, we only need to prove that
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{*}(M_{C})\subseteq(\bigcap_{C\in
B(K,\,H)}\sigma_{*}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$
(i). Suppose that $\sigma_{*}(\cdot)=\sigma_{SF-}(\cdot)$ and
$\lambda\not\in(\bigcap_{C\in
B(K,\,H)}\sigma_{SF-}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$ Then $A-\lambda$ is not compact and there exists $C\in
B(K,\,H)$ such that $M_{C}-\lambda\in\Phi_{-}(H\oplus K),$ and hence
$B-\lambda\in\Phi_{-}(K).$
Case 1. $\alpha(B-\lambda)=\infty$. It follows from Lemma 6 that there exists
$C\in\Phi(K,H)$ such that $M_{C}-\lambda$ is a lower semi-Fredholm operator.
This implies that
$\lambda\not\in\bigcap_{C\in\Phi(K,\,H)}\sigma_{SF-}(M_{C})$. It is clear that
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{SF-}(M_{C})\subseteq(\bigcap_{C\in
B(K,\,H)}\sigma_{SF-}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$
Case 2. $\alpha(B-\lambda)<\infty$. This implies that $B-\lambda\in\Phi(K),$
and so $A-\lambda\in\Phi_{-}(H)$ since $M_{C}-\lambda\in\Phi_{-}(H\oplus K).$
Thus, we have that $M_{C}-\lambda$ is a lower semi-Fredholm operator for each
$C\in B(K,\,H)$, which means
$\lambda\not\in\bigcap_{C\in\Phi(K,\,H)}\sigma_{SF-}(M_{C})$. Thus
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{SF-}(M_{C})\subseteq\bigcap_{C\in
B(K,\,H)}\sigma_{SF-}(M_{C})\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$
Together Case 1 with Case 2, we have
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{SF-}(M_{C})=(\bigcap_{C\in
B(K,\,H)}\sigma_{SF-}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$
(ii). Suppose that $\sigma_{*}(\cdot)=\sigma_{r}(\cdot)$ and
$\lambda\not\in(\bigcap_{C\in
B(K,\,H)}\sigma_{r}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$ Then $A-\lambda$ is not compact and there exists $C\in
B(K,\,H)$ such that $M_{C}-\lambda\in G_{r}(H\oplus K),$ and hence
$B-\lambda\in G_{r}(K).$
Case 1. $\alpha(B-\lambda)=\infty$. It follows from Lemma 6 that there exists
$C\in\Phi(K,H)$ such that $M_{C}-\lambda$ is a lower semi-Weyl operator and
$\beta(M_{C}-\lambda)=\beta(B-\lambda)$. Note that $B-\lambda$ is surjective,
then $M_{C}-\lambda$ is also surjective. This implies that
$\lambda\not\in\bigcap_{C\in\Phi(K,\,H)}\sigma_{r}(M_{C})$. It is clear that
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{r}(M_{C})\subseteq\bigcap_{C\in
B(K,\,H)}\sigma_{r}(M_{C})\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$
Case 2. $\alpha(B-\lambda)<\infty$. This means that $B-\lambda\in\Phi(K),$ so
it is easy to prove that $A-\lambda\in\Phi_{-}(H).$ Moreover, it follows from
[10, Corollary 2] that $\alpha(B-\lambda)\geq\beta(A-\lambda).$ Next we claim
that there exists some $C\in\Phi(K,\,H)$ such that
$\lambda\not\in\bigcap_{C\in\Phi(K,\,H)}\sigma_{r}(M_{C})$. For this, let
${N(B-\lambda)}^{\perp}=K_{1}\oplus K_{2}$ with $\dim
K_{2}=\dim\beta(A-\lambda)$. Define an operator $Q:K\rightarrow H$ by
$Q=\left(\begin{array}[]{cc}C_{1}&0\\\ 0&C_{2}\\\
\end{array}\right):({N(B-\lambda)}\oplus K_{1})\oplus K_{2}\longrightarrow
R(A-\lambda)\oplus R(A-\lambda)^{\perp},$
where $C_{1}\in B({N(B-\lambda)}\oplus K_{1},R(A-\lambda))$ and $C_{2}\in
B(K_{2},R(A-\lambda)^{\perp})$ are invertible operators. Obviously, operator
$Q\in G(K,\,H)$ and $M_{C}-\lambda$ is surjective. Thus
$\bigcap_{C\in G(K,\,H)}\sigma_{r}(M_{C})\subseteq\bigcap_{C\in
B(K,\,H)}\sigma_{r}(M_{C})\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$
Together Case 1 with Case 2, we have
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{r}(M_{C})=\bigcap_{C\in
B(K,\,H)}\sigma_{r}(M_{C})\cup\\{\lambda\in{\mathbb{C}}:A-\lambda\,\makebox{
is compact}\\}.$
Similarly, when $\sigma_{*}=\sigma_{sw},$ we can prove the conclusion is also
true.
By the proof methods of Theorem 4, we can prove the following result:
Theorem 5. For a given pair $(A,B)\in B(H)\times B(K)$, we have
$\bigcap_{C\in\Phi(K,\,H)}\sigma_{*}(M_{C})=(\bigcap_{C\in
B(K,\,H)}\sigma_{*}(M_{C}))\cup\\{\lambda\in{\mathbb{C}}:B-\lambda\,\makebox{is
compact}\\},$
where $\sigma_{*}\in\\{\sigma_{l},\sigma_{aw}\\}.$
## References
* [1] X. H. Cao. Browder spectra for upper triangular operator matrices, J. Math. Anal. Appl., 342(2008), 477-484.
* [2] X. H. Cao, M. Z. Guo, B. Meng. Semi-Fredholm spectrum and Weyl’s theory for operator matrices, Acta Math. Sinica, 22(2006), 169-178.
* [3] X. H. Cao, B. Meng. Essential approximate point spectra and Weyl’s theorem for upper triangular operator matrices, J. Math. Anal. Appl., 304(2005), 759-771.
* [4] X. L. Chen, S. F. Zhang, H. J. Zhong. On the filling in holes problem for operator matrices, Linear Algebra Appl., 430(2009),558-563
* [5] D. S. Djordjević. Perturbations of spectra of operator matrices, J. Operator Theory, 48(2002), 467-486.
* [6] S. V. Djordjević, Y. M. Han. spectral continuity for operator matrices, Glasg. Math. J., 43(2001), 487-490.
* [7] S. V. Djordjević, H. Zguitti. Essential point spectra of operator matrices though local spectral theory, J. Math. Anal. Appl., 338(2008), 285-291.
* [8] H. K. Du, J. Pan. Perturbation of spectrums of $2\times 2$ operator matrices, Proc. Amer. Math. Soc., 121(1994), 761-766.
* [9] J. K. Han, H. Y. Lee, W. Y. Lee. Invertible completions of $2\times 2$ upper triangular operator matrices. Proc. Amer. Math. Soc., 128(1999), 119-123.
* [10] I. S. Hwang, W. Y. Lee. The boundedness below of $2\times 2$ upper triangular operator matrices, Integr. Equ. Oper. Theory, 39(2001), 267-276.
* [11] W. Y. Lee. Weyl spectra of operator matrices, Proc. Amer. Math. Soc., 129(2000), 131-138.
* [12] Y. Li, H. K. Du. The intersection of left and right essential spectra of 2 $\times$ 2 operator matrices, Bull. Lond. Math. Soc. , 36(2004), 811-819.
* [13] Y. Li, H. K. Du. The intersection of essential approximate point spectra of operator matrices, J. Math. Anal. Appl., 323(2006), 1171-1183.
* [14] Y. Li, X. H. Sun, H. K. Du. The intersection of left(right) spectra of 2 $\times$ 2 upper triangular operator matrices, Linear Algebra Appl., 418(2006), 112-121.
* [15] Y. Li, X. H. Sun, H. K. Du. A note on the left essential approximate point spectra of operator matrices, Acta Math. Sinica, 23(2007), 2235-2240.
* [16] E. H. Zerouali, H. Zguitti. Perturbation of spectra of operator matrices and local spectral theory, J. Math. Anal. Appl., 324(2006), 992-1005.
* [17] S. F. Zhang, H. J. Zhong. A note of Browder spectrum of operator matrices, J. Math. Anal. Appl., 344(2008), 927-931.
* [18] S. F. Zhang, H. J. Zhong, Q. F. Jiang. Drazin spectrum of operator matrices on the Banach space, Linear Algebra Appl., 429(2008), 2067-2075.
* [19] Y. N. Zhang, H. J. Zhong, L. Q. Lin. Browder spectra and essential spectra of operator matrices, Acta Math. Sinica, 24(2008), 947-954.
|
arxiv-papers
| 2009-06-27T09:31:42 |
2024-09-04T02:49:03.608760
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhang Shifang, Zhong Huaijie, Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0906.5056"
}
|
0906.5268
|
Veech groups of Loch Ness monsters
Piotr Przytyckia111Partially supported by MNiSW grant N201 012 32/0718 and the
Foundation for Polish Science. , Gabriela Schmithüsenb222Partially supported
by Landesstiftung Baden-Württemberg. & Ferrán Valdezc333Partially supported by
Sonderforschungsbereich/Transregio 45 and ANR Symplexe.
a Institute of Mathematics, Polish Academy of Sciences,
Śniadeckich 8, 00-956 Warsaw, Poland
_e-mail:_ [email protected]
b Institute of Algebra and Geometry, Faculty of Mathematics,
University of Karlsruhe, D-76128 Karlsruhe, Germany
_e-mail:_ [email protected]
c Max-Planck Institut für Mathematik,
Vivatsgasse 7, 53111 Bonn, Germany
_e-mail:_ [email protected]
###### Abstract
We classify Veech groups of tame non-compact flat surfaces. In particular we
prove that all countable subgroups of $\mathbf{GL}_{+}(2,\mathbb{R})$ avoiding
the set of mappings of norm less than $1$ appear as Veech groups of tame non-
compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group
of any tame flat surface is either countable, or one of three specific types.
## 1 Introduction
For a compact flat surface $S$, the _Veech group_ of $S$ is the subgroup of
$\rm\mathbf{SL}(2,\mathbb{R})$ formed by the differentials of the orientation
preserving affine homeomorphisms of $S$. Veech groups of compact flat surfaces
are related to the dynamics of the geodesic flow [Ve].
Our goal is to describe all possible Veech groups one can obtain for tame non-
compact flat surfaces (see Definition 2.2), introduced in [V2]. An example
_par excellence_ of a tame non-compact flat surface is the surface associated
to the billiard game on an irrational angled polygonal table. This surface is
of infinite genus and has only one end [V1]. A surface with those properties
is called a _Loch Ness monster_ (see [G]). We distinguish the role of this
”monster” in our main result.
To state it, we need the following notation. We denote by
$\mathcal{U}\subset\mathbf{GL}_{+}(2,\mathbb{R})$ the set of matrices $M$ such
that $||Mv||<||v||$ for all $v\in\mathbb{R}^{2}$, where $||\cdot||$ is the
Euclidean norm on $\mathbb{R}^{2}$. We denote
* •
by $P\subset\mathbf{GL}_{+}(2,\mathbb{R})$ the group of matrices
$\begin{pmatrix}1&t\\\ 0&s\end{pmatrix},\mathrm{where}\
t\in\mathbb{R},s\in\mathbb{R}_{+},$
* •
by $P^{\prime}\subset\mathbf{GL}_{+}(2,\mathbb{R})$ the group of matrices
generated by $P$ and $-\mathrm{Id}$.
Note that $P$ has index $2$ in $P^{\prime}$.
We prove the following.
###### Theorem 1.1.
Let $G\subset\mathbf{GL}_{+}(2,\mathbb{R})$ be a Veech group of a tame flat
surface. Then one of the following holds.
1. (i)
$G$ is countable and disjoint from $\mathcal{U}$.
2. (ii)
$G$ is conjugate to $P$.
3. (iii)
$G$ is conjugate to $P^{\prime}$.
4. (iv)
$G=\mathbf{GL}_{+}(2,\mathbb{R})$.
Conversely, we prove the following.
###### Theorem 1.2.
Any subgroup $G$ of $\mathbf{GL}_{+}(2,\mathbb{R})$ satisfying assertion (i),
(ii) or (iii) of Theorem 1.1 can be realized as a Veech group of a tame flat
surface $X$ which is a Loch Ness monster.
In particular, every cyclic subgroup of $\mathbf{SL}(2,\mathbb{R})$ or every
Fuchsian group can be realized as the Veech group of a tame flat surface which
is a Loch Ness monster. For compact flat surfaces, such questions are still
open (see [HZ, Problems 5, 6]). Furthermore, observe that a cocompact Fuchsian
group cannot be the Veech group of a compact flat surface [Ve], but occurs as
the Veech group of a tame flat surface, which is a Loch Ness monster.
We will see that the only tame flat surfaces with Veech group
$\mathbf{GL}_{+}(2,\mathbb{R})$, as in (iv) of Theorem 1.1, are cyclic
branched coverings of the flat plane (see Lemmas 3.2 and 3.3). In particular
$\mathbf{GL}_{+}(2,\mathbb{R})$ cannot be realized as a Veech group of a tame
Loch Ness monster.
In our article we restrict in Definition 2.3 of the Veech group to affine
homeomorphisms which preserve the orientation. If we allow orientation
reversing ones, substituting $\mathbf{GL}(2,\mathbb{R})$ in place of
$\mathbf{GL}_{+}(2,\mathbb{R})$ in the statements of our theorems, they remain
valid, except that we need to add three more ’’parabolic‘‘ subgroups to the
pair $P$ and $P^{\prime}$. No new ideas appear in the proofs. Thus we restrict
to the orientation preserving case to simplify the formulation and the
arguments.
The article is organized as follows. In Section 2 we recall the definition of
a tame non-compact flat surface and its Veech group.
We divide the proofs of Theorems 1.1 and 1.2 into two parts. In Section 3 we
treat the case where the group $G$ is uncountable. More precisely, we prove
that if in the hypothesis of Theorem 1.1 we assume that $G$ is uncountable,
then it satisfies assertion (ii), (iii) or (iv) (Proposition 3.1). Conversely,
we prove that any group satisfying assertion (ii) or (iii) can be realized as
a Veech group of a tame flat surface which is a Loch Ness monster (Lemmas 3.7
and 3.8).
In Section 4 we study the remaining case, where $G$ is countable. In other
words, we prove that any group satisfying assertion (i) of Theorem 1.1 can be
realized as a Veech group of a tame flat surface which is a Loch Ness monster
(Proposition 4.1). This construction is the main point of the article.
Conversely, we prove that if we assume in the hypothesis of Theorem 1.1 that
$G$ is countable, then it satisfies assertion (i) (Lemma 4.15).
Acknowledgments. We thank the faculty and staff of Max-Planck Institut in
Bonn, where part of this work was carried out. We furthermore thank the
Landesstiftung Baden–Württemberg and the Department of Mathematics of the
University of Karlsruhe that enabled the authors to meet and work together.
## 2 Preliminaries
In this section we briefly recall the definition and features of non-compact
flat surfaces. For more details, we refer the reader to [V2].
Let $(S,\omega)$ be a pair formed by a connected Riemann surface $S$ and a
non-zero holomorphic $1$–form $\omega$ on $S$. Denote by $Z(\omega)\subset S$
the zero locus of the form $\omega$. Local integration of $\omega$ endows
$S\setminus Z(\omega)$ with an atlas whose transition functions are
translations of $\mathbb{C}$. The pullback of the standard translation
invariant flat metric on the complex plane defines a flat metric on
$S\setminus Z(\omega)$. Let $\widehat{S}$ be the metric completion of
$S\setminus Z(\omega)$. Each point in $Z(\omega)$ has a neighborhood isometric
to the neighborhood of $0\in\mathbb{C}$ with the metric coming from the 1–form
$z^{k}dz$ for some $k>1$ (which is the metric induced via a cyclic branched
covering of $\mathbb{C}$). The points in $Z(\omega)$ are called _finite angle
singularities_.
###### Definition 2.1.
A point $p\in\widehat{S}$ is called an _infinite angle singularity_ of $S$, if
there exists a neighborhood of $p$ isometric to the neighborhood of the
branching point of the infinite cyclic branched covering of $\mathbb{C}$. We
denote the set of infinite angle singularities of $\widehat{S}$ by
$Y_{\infty}(\omega)$.
###### Definition 2.2.
The pair $(S,\omega)$ is called a _tame flat surface_ , if
$\widehat{S}\setminus S$ equals $Y_{\infty}(\omega)$.
Let $\mathrm{Aff}_{+}(S)$ be the group of affine orientation preserving
homeomorphisms of a tame flat surface $S$ (we assume that $S$ comes with a
preferred $1$–form $\omega$). Consider the differential
$\mathrm{Aff}_{+}(S)\overset{D}{\longrightarrow}\mathbf{GL}_{+}(2,\mathbb{R})$
that associates to every $\phi\in\mathrm{Aff}_{+}(S)$ its (constant) Jacobian
derivative $D\phi$.
###### Definition 2.3.
Let $S$ be a tame flat surface. We call $G(S)=D(\mathrm{Aff}_{+}(S))$ the
_Veech group_ of $S$.
We define _saddle connections_ and _holonomy vectors_ in the context of tame
non-compact flat surfaces exactly in the same way as for compact ones, see
[V2].
We refer the reader to [HS, Ve] for more details on Veech groups of compact
flat surfaces, and to [HW, HSc, V2, H] for explicit examples of Veech groups
of tame flat surfaces which are Loch Ness monsters.
## 3 Uncountable Veech groups
In this section we prove Theorems 1.1 and 1.2 in the case where $G$ is
uncountable. Under this assumption we restate Theorem 1.1 in the following
way.
###### Proposition 3.1.
If the Veech group of a tame flat surface is uncountable, then it is conjugate
to $P$, conjugate to $P^{\prime}$ or equals the whole
$\mathbf{GL}_{+}(2,\mathbb{R})$.
We begin the proof with the following.
###### Lemma 3.2.
If a tame flat surface $S$ has no saddle connections and its Veech group $G$
is uncountable, then $G$ equals $P^{\prime}$ or
$\mathbf{GL}_{+}(2,\mathbb{R})$. In the latter case $S$ is a cyclic branched
covering of the flat plane.
Proof. First assume that $S$ has no singularities. Then the universal cover of
$S$ is the flat plane and $S$ is either (i) the plane itself, or (ii) a flat
cylinder which is a quotient of the plane by a cyclic group, or (iii) it is
compact. Since $G$ is uncountable, $S$ is not compact. In case (i) we have
that $G=\mathbf{GL}_{+}(2,\mathbb{R})$. In case (ii) we have that $G$ is
conjugate to $P^{\prime}$ by a rotation.
Now assume that $S$ has a singularity $x_{0}$ (which might be of finite or
infinite angle). Since there are no saddle connections issuing from $x_{0}$,
we have that $\widehat{S}$ is isometric to a (possibly infinite) cyclic
branched covering of $\mathbb{R}^{2}$. Hence
$G=\mathbf{GL}_{+}(2,\mathbb{R})$. $\square$
To complete the proof of Proposition 3.1 it remains to prove the following.
###### Lemma 3.3.
If the Veech group $G$ of a tame flat surface $S$ carrying saddle connections
is uncountable, then $G$ is conjugate to $P$ or $P^{\prime}$.
Proof. Step 1. _All saddle connections of $S$ are parallel_.
Since there are only countably many homotopy classes of arcs joining
singularities of $\widehat{S}$, the set of saddle connections of $S$, and thus
the set $V\subset\mathbb{R}^{2}$ of holonomy vectors, is countable. If $s_{1}$
and $s_{2}$ are two non-parallel saddle connections, then let $v_{1}$, $v_{2}$
be their holonomy vectors. For each $g\in G$ we define
$\eta(g)=(g(v_{1}),g(v_{2}))\in V\times V$. Since $\\{v_{1},v_{2}\\}$ is a
basis of $\mathbb{R}^{2}$, we have that $\eta$ is an embedding. But $V\times
V$ is countable. Contradiction. This concludes Step 1.
Without loss of generality we may assume that all saddle connections are
horizontal. Let ${\rm Spine}(S)\subset\widehat{S}$ be the union of the set of
singularities together with all singular horizontal geodesics (this includes
saddle connections). We claim that ${\rm Spine}(S)$ is connected and complete
w.r.t. its intrinsic path metric. The latter follows from the completeness of
$\widehat{S}$. The former follows from the fact that any two singularities of
$\widehat{S}$ are connected by a concatenation of saddle connections, which
are horizontal by Step 1.
Step 2. _We have that $P\subset G$._
Let $C$ be the closure of a component of $\widehat{S}\setminus{\rm Spine}(S)$.
It is a complete Riemann surface with nonvanishing holomorphic $1$–form and
horizontal boundary. The boundary of $C$ is connected, since otherwise there
would be a non-horizontal saddle connection joining singularities in different
boundary components. Hence $C$ is either a half-plane or a half-cylinder with
horizontal boundary. In particular, for any $g\in P$ we have that $C$ admits
an orientation preserving affine homeomorphism with differential $g$, which
fixes its boundary. Hence for any $g\in P$, there is an orientation preserving
affine homeomorphism $\overline{g}\in{\rm Aff_{+}}(S)$, with
$D\overline{g}=g$, which fixes ${\rm Spine}(S)$ and is prescribed
independently on each component of the complement.
Step 3. _We have that $G\subset P^{\prime}$._
Let $\vec{\mathbf{e}}$ denote the unit horizontal vector in $\mathbb{R}^{2}$.
We prove that for every $g\in G$ we have
$g(\vec{\mathbf{e}})=\pm\vec{\mathbf{e}}$. Otherwise, assume that there is an
orientation preserving affine homeomorphism $\overline{g}\in{\rm Aff_{+}}(S)$
with differential $g$ for which $g(\vec{\mathbf{e}})=\lambda\vec{\mathbf{e}}$,
with $|\lambda|\neq 1$. Then $\overline{g}$ or its inverse acts as a
contraction on ${\rm Sing}(S)$. By the Banach fixed point theorem, the
iterates of any singularity under $\overline{g}$ or its inverse accumulate on
the fixed point of $\overline{g}$. Since the set of singularities is invariant
under the action of $\overline{g}$, this implies that it has an accumulation
point. Contradiction.
We summarize. By Steps 2 and 3 we have that $P\subset G\subset P^{\prime}$.
Since $P$ is of index $2$ in $P^{\prime}$, we have that $G=P$ or
$G=P^{\prime}$. $\square$
We now provide examples of Loch Ness monsters with Veech groups $P$ and
$P^{\prime}$. First we introduce the following vocabulary, which will become
particularly useful in Section 4.
###### Definition 3.4.
Let $S$ be a tame flat surface. A _mark_ on $S$ is an oriented finite length
geodesic (with endpoints) on $S$ which does not meet singularities. If $S$ is
simply connected, a mark is determined by its endpoints. The _slope_ of a mark
is its holonomy vector, which lies in $\mathbb{R}^{2}$.
If $m,m^{\prime}$ are two disjoint marks on $S$ with equal slopes, we can
perform the following operation. We cut $S$ along $m$ and $m^{\prime}$, which
turns $S$ into a surface with boundary consisting of four straight segments.
Then we reglue these segments to obtain a tame flat surface $S^{\prime}$
different from the one we started from. We say that $S^{\prime}$ is obtained
from $S$ by _regluing along $m$ and $m^{\prime}$_.
Let $S_{0}=S\setminus(m\cup m^{\prime})$. Then $S^{\prime}$ admits a natural
embedding $i$ of $S_{0}$. If $A\subset S_{0}$, then we say that $i(A)$ is
_inherited_ by $S^{\prime}$ from $A$.
###### Remark 3.5.
If $S^{\prime}$ is obtained from $S$ by regluing, then the number of
singularities of $S^{\prime}$ of a fixed angle equals the one of $S$, except
for $4\pi$–angle singularities, whose number is greater by $2$ in $S^{\prime}$
(we put $\infty+2=\infty$). The Euler characteristic of $S$ is greater by $2$
than the Euler characteristic of $S^{\prime}$.
We can extend the notion of regluing to families of marks.
###### Definition 3.6.
Let $S$ be a tame flat surface. Assume that
$\mathcal{M}=(m_{n})_{n=1}^{\infty}$ and
$\mathcal{M^{\prime}}=(m^{\prime}_{n})_{n=1}^{\infty}$ are ordered families of
disjoint marks, which do not accumulate in $\widehat{S}$, and such that the
slope of $m_{n}$ equals the slope of $m^{\prime}_{n}$, for each $n$. Let
$S_{0}=S$ and let $S_{n}$ be obtained from $S_{n-1}$ by regluing along $m_{n}$
and $m^{\prime}_{n}$. Let $S^{\prime}$ be the Riemann surface equipped with a
holomorphic $1$–form which is the limit of $S_{n}$. The limit exists since the
marks do not accumulate, but might not be a tame flat surface. We say that
$S^{\prime}$ is obtained from $S$ by _regluing along $\mathcal{M}$ and
$\mathcal{M}^{\prime}$_. If $A\subset
S\setminus(\mathcal{M}\cup\mathcal{M^{\prime}})$, then we define the subset of
$S^{\prime}$ _inherited_ from $A$ as before.
We are ready to perform the following constructions.
###### Lemma 3.7.
There is a tame Loch Ness monster with Veech group $P$.
Proof. Let $A$ and $A^{\prime}$ be two oriented flat planes, equipped with
origins that allow us to identify them with $\mathbb{R}^{2}$. Let
$\mathcal{C},\mathcal{C}^{\prime}$ be families of marks with endpoints
$(4n+1)\vec{\mathbf{e}},(4n+3)\vec{\mathbf{e}}$, for $n\geq 1$, on
$A,A^{\prime}$, respectively, where $\vec{\mathbf{e}}$ denotes, as before, the
horizontal unit vector in $\mathbb{R}^{2}$. Let $\hat{A}$ be the tame flat
surface obtained from $A\cup A^{\prime}$ by regluing along $\mathcal{C}$ and
$\mathcal{C^{\prime}}$.
The group $P$ acts on $A$ and $A^{\prime}$ under identification with
$\mathbb{R}^{2}$. This action carries over to $\hat{A}$. Hence the Veech group
$G$ of $\hat{A}$ contains $P$. By Lemma 3.3, we have that $G=P$ or
$G=P^{\prime}$. But in the latter case, the affine homeomorphism with
differential $-\mathrm{Id}$ must act on ${\rm Sing}(\hat{A})$ (defined in the
proof of Lemma 3.3) by an orientation reversing isometry. Since there is no
such isometry, we conclude that $G=P$.
By Remark 3.5, we have that $\hat{A}$ has infinite genus. It has one end (this
follows in particular from Lemma 4.3). Hence $\hat{A}$ is a Loch Ness monster
with Veech group $P$. $\square$
###### Lemma 3.8.
There is a tame Loch Ness monster with Veech group $P^{\prime}$.
Proof. Similarly as in the proof of Lemma 3.7, let $A$ and $A^{\prime}$ be two
oriented flat planes, equipped with origins that allow us to identify them
with $\mathbb{R}^{2}$. Let $\mathcal{C},\mathcal{C}^{\prime}$ be families of
marks with endpoints $(4n+1)\vec{\mathbf{e}},(4n+3)\vec{\mathbf{e}}$, on
$A,A^{\prime}$, respectively, where this time we take $n\in\mathbb{Z}$, and we
order the marks into sequences. Let $\hat{A}$ be the tame flat surface
obtained from $A\cup A^{\prime}$ by regluing along $\mathcal{C}$ and
$\mathcal{C^{\prime}}$.
This time the action of the whole group $P^{\prime}$ carries over to
$\hat{A}$. Hence the Veech group $G$ of $\hat{A}$ contains $P^{\prime}$. By
Lemma 3.3 we have that $G=P^{\prime}$. The surface $\hat{A}$ is a Loch Ness
monster by the same argument as in the proof of Lemma 3.7. $\square$
Lemmas 3.7 and 3.8 prove Theorem 1.2 in the case where $G$ is uncountable.
## 4 Countable Veech groups
The main part of this section is devoted to the proof of Theorem 1.2 in the
case where the group $G\subset\mathbf{GL}_{+}(2,\mathbb{R})$ is countable. In
other words, we prove the following.
###### Proposition 4.1.
For any countable subgroup $G$ of $\mathbf{GL}_{+}(2,\mathbb{R})$ disjoint
from $\mathcal{U}=\\{g\in\mathbf{GL}_{+}(2,\mathbb{R})\colon||g||<1\\}$ there
exists a tame flat surface $S=S(G)$, which is a Loch Ness monster, with Veech
group $G$.
In fact the group $\mathrm{Aff}_{+}(S)$ will map isomorphically onto $G$ under
the differential map. This means that the group $G$ will act on $S$ via affine
homeomorphisms with appropriate differentials. Here we adopt the convention
that an action of a group $G$ on a set $X$ is a mapping $(g,x)\rightarrow
g\cdot x$ such that $(gh)\cdot x=g\cdot(h\cdot x)$ and $\mathrm{Id}\cdot x=x$.
We begin with an outline of the proof of Proposition 4.1. We make use of the
fact that any group $G$ acts on its Cayley graph $\Gamma$. We turn $\Gamma$
equivariantly into a flat surface. With each vertex $g$ of $\Gamma$ we
associate a flat surface $V_{g}$ which can be cut into a flat plane $A_{g}$
and a _decorated surface_ $\widetilde{L}^{\prime}_{g}$, whose role is
explained later.
To guarantee tameness, we do not want the singularities of different $V_{g}$
to accumulate. Let $(g,g^{\prime})$ be an edge of $\Gamma$ such that
$g^{-1}g^{\prime}$ is the $i$‘th generator of $G$. We associate to this edge a
_buffer surface_ $\hat{E}^{i}_{g}$ which connects $V_{g}$ to $V_{g^{\prime}}$,
but separates them by a definite distance.
We keep track of the end in the following way. First we provide that each
$V_{g}$ and $\hat{E}^{i}_{g}$ is one-ended. Then we provide that after gluing
all $V_{g}$ and $\hat{E}^{i}_{g}$, their ends actually merge into one end.
In this way we construct a one-ended flat surface with a faithful affine
action of $G$. The role of the decorated surface $\widetilde{L}^{\prime}_{g}$
is to prevent the group of orientation preserving affine homeomorphisms of the
surface from being richer than $G$. To achieve this,
$\widetilde{L}^{\prime}_{g}$ is decorated with special singularities. This
guarantees that every orientation preserving affine homeomorphism of the
surface permutes this set of singularities and with some more care we
establish that it actually acts as one of the elements of $G$.
We begin by explaining how to obtain a nice action of
$\mathbf{GL}_{+}(2,\mathbb{R})$ on a disjoint union of affine copies of any
flat surface.
###### Definition 4.2.
Let $S_{\mathrm{Id}}$ be a tame flat surface. For each
$g\in\mathbf{GL}_{+}(2,\mathbb{R})$, we denote by $S_{g}$ the affine copy of
$S_{\mathrm{Id}}$, whose atlas differs from the one of $S_{\mathrm{Id}}$ by
post-composing each chart with $g$. In other words, $S_{g}$ comes with a
canonical affine homeomorphism $\overline{g}\colon S_{\mathrm{Id}}\rightarrow
S_{g}$ with differential $g$. Moreover, $\mathbf{GL}_{+}(2,\mathbb{R})$ acts
on the union of all $S_{g^{\prime}}$ so that $\overline{g}$ maps each
$S_{g^{\prime}}$ onto $S_{gg^{\prime}}$, with differential $g$.
We provide the following criterion for $1$–endedness. Let $\Gamma$ be a
connected graph. Let $A$ be the union, over $v\in\Gamma^{(0)}$, of $1$–ended
tame flat surfaces $A_{v}$ without infinite angle singularities. Assume that
each $A_{v}$ is equipped with infinite families of marks
$\mathcal{C}^{e}_{v}$, for each edge $e$ issuing from $v$, and additional,
possibly finite, two families of marks
$\mathcal{C}_{v},\mathcal{C}^{\prime}_{v}$, of the same cardinality. Assume
that all these marks are disjoint and do not accumulate. In particular this
implies that $\Gamma^{(0)}$ is countable. Moreover, assume that for each edge
$e=(v,v^{\prime})$ the slopes of the marks in $\mathcal{C}^{e}_{v}$ and
$\mathcal{C}^{e}_{v^{\prime}}$ agree. Additionally, assume that the slopes of
the marks in $\mathcal{C}_{v}$ and $\mathcal{C}^{\prime}_{v}$ agree.
###### Lemma 4.3.
Let $S$ be the surface obtained from $A$ by regluing along
$\mathcal{C}^{e}_{v}$ and $\mathcal{C}^{e}_{v^{\prime}}$, for all edges
$e=(v,v^{\prime})$ in $\Gamma^{(1)}$, and along $\mathcal{C}_{v}$ and
$\mathcal{C}^{\prime}_{v}$, for all vertices $v$ in $\Gamma^{(0)}$. Then $S$
is $1$–ended. If $\Gamma$ has an edge or if it has only one vertex $v$ but
with infinite $C_{v}$ (or if $A_{v}$ has infinite genus), then $S$ has
infinite genus.
Unless $\Gamma$ has no edges (it has then only one vertex $v$) and
additionally $\mathcal{C}_{v}$ is finite and $A_{v}$ has finite genus, we have
that $S$ has infinite genus.
Proof. For each vertex $v$ in $\Gamma^{(0)}$, choose a basepoint $O_{v}$ in
$A_{v}$. Let $B_{v}(r)$ be the closure in $S$ of the subset inherited from the
ball of radius $r$ around $O_{v}$ with appropriate marks removed.
We order all vertices of $\Gamma$ into a sequence $(v_{j})_{j=1}^{\infty}$.
For $l\geq 1$, let
$K_{l}=\bigcup_{j=1}^{l}B_{v_{j}}(l).$
Then $K_{l}$ is a family of compact sets which has the property that each
compact set in $S$ is contained in $K_{l}$, for some $l\geq 1$.
Now we prove that the complement of each $K_{l}$ is connected. Since the
$A_{v}$ are complete non-positively curved and $1$–ended, since balls and the
marks we consider are convex, and since those marks are disjoint, we have that
all
$A_{v_{j}}^{\prime}=A_{v_{j}}\setminus(B_{v_{j}}(l)\cup_{e}\mathcal{C}_{v_{j}}^{e}\cup\mathcal{C}_{v_{j}}\cup\mathcal{C}^{\prime}_{v_{j}})$
are connected. Since $\Gamma$ is connected, all $\mathcal{C}^{e}_{v}$ are
infinite, and $K_{l}$ intersects only a finite number of marks, we have that
all $A^{\prime}_{v_{j}}$ are in the same connected component of $S\setminus
K_{l}$. Since the union of $A^{\prime}_{v_{j}}$ is dense in $S\setminus
K_{l}$, this implies that $S\setminus K_{l}$ is connected.
Thus $S$ is $1$–ended. If $\Gamma$ has at least one edge or $C_{v}$ is
infinite, then $S$ has infinite genus by Remark 3.5. $\square$
We describe the construction of the _buffer surfaces_ , which will correspond
to the edges of the Cayley graph $\Gamma$ of $G$. We denote the base vectors
$(1,0),(0,1)$ of $\mathbb{R}^{2}$ by $\vec{\mathbf{e}}$ and
$\vec{\mathbf{f}}$, respectively.
###### Construction 4.4.
Let $E_{\mathrm{Id}},E^{\prime}_{\mathrm{Id}}$ be two oriented flat planes,
equipped with origins that allow us to identify them with $\mathbb{R}^{2}$. We
define the following families of slope $\vec{\mathbf{e}}$ marks on
$E_{\mathrm{Id}}\cup E^{\prime}_{\mathrm{Id}}$. Let $\mathcal{S}$ be the
family of marks on $E_{\mathrm{Id}}$ with endpoints
$4n\vec{\mathbf{e}},(4n+1)\vec{\mathbf{e}}$, for $n\geq 1$, and let
$\mathcal{S}_{\mathrm{glue}}$ be the family of marks on $E_{\mathrm{Id}}$ with
endpoints $(4n+2)\vec{\mathbf{e}},(4n+3)\vec{\mathbf{e}}$, for $n\geq 1$. Let
$\mathcal{S}^{\prime}$ be the family of marks on $E^{\prime}_{\mathrm{Id}}$
with endpoints $2n\vec{\mathbf{f}},2n\vec{\mathbf{f}}+\vec{\mathbf{e}}$, for
$n\geq 1$, and let $\mathcal{S}^{\prime}_{\mathrm{glue}}$ be the family of
marks on $E^{\prime}_{\mathrm{Id}}$ with endpoints
$(2n+1)\vec{\mathbf{f}},(2n+1)\vec{\mathbf{f}}+\vec{\mathbf{e}}$, for $n\geq
1$. Let $\hat{E}_{\mathrm{Id}}$ be the tame flat surface obtained from
$E_{\mathrm{Id}}$ and $E^{\prime}_{\mathrm{Id}}$ by regluing along
$\mathcal{S}_{\mathrm{glue}}$ and $\mathcal{S}^{\prime}_{\mathrm{glue}}$. We
call $\hat{E}_{\mathrm{Id}}$ the _buffer surface_. We record that
$\hat{E}_{\mathrm{Id}}$ comes with distinguished families of marks inherited
from $\mathcal{S},\mathcal{S}^{\prime}$, for which we retain the same
notation.
###### Lemma 4.5.
Let $\hat{E}_{\mathrm{Id}}$ be the buffer surface and let
$g\in\mathbf{GL}_{+}(2,\mathbb{R})\setminus\mathcal{U}$. Then the distance in
$\hat{E}_{g}$ (see Definition 4.2) between $\overline{g}\mathcal{S}$ and
$\overline{g}\mathcal{S}^{\prime}$ is at least $\frac{1}{\sqrt{2}}$.
Proof. Denote by $\hat{d}$ the distance in $\hat{E}_{g}$ between
$\overline{g}\mathcal{S}$ and $\overline{g}\mathcal{S}^{\prime}$. Let $d$ be
the distance in $E_{g}$ between $\overline{g}\mathcal{S}$ and
$\overline{g}\mathcal{S}_{\mathrm{glue}}$ and let $d^{\prime}$ be the distance
in $E^{\prime}_{g}$ between $\overline{g}\mathcal{S}^{\prime}_{\mathrm{glue}}$
and $\overline{g}\mathcal{S}^{\prime}$. Then we have that $\hat{d}\geq
d+d^{\prime}$. Moreover, $d=|g(\vec{\mathbf{e}})|$ and
$d^{\prime}=\min_{|s|\leq 1}|g(\vec{\mathbf{f}}+s\vec{\mathbf{e}})|.$
Let $s\in[-1,1]$ be such that the minimum is attained, that is
$d^{\prime}=|g(\vec{\mathbf{f}}+s\vec{\mathbf{e}})|$. If
$d+d^{\prime}<\frac{1}{\sqrt{2}}$, then
$|g(\vec{\mathbf{f}})|\leq|g(\vec{\mathbf{f}}+s\vec{\mathbf{e}})|+|s||g(\vec{\mathbf{e}})|<\frac{1}{\sqrt{2}}.$
Hence for any $v=x\vec{\mathbf{e}}+y\vec{\mathbf{f}}\in\mathbb{R}^{2}$ we have
that
$|g(v)|\leq|x||g(\vec{\mathbf{e}})|+|y||g(\vec{\mathbf{f}})|<\frac{1}{\sqrt{2}}(|x|+|y|)\leq\sqrt{x^{2}+y^{2}}=|v|.$
Thus $||g||<1$. Contradiction. $\square$
Now we construct the _decorated surface_ which will force rigidity of the
affine homeomorphism group.
###### Construction 4.6.
Let $L_{\mathrm{Id}}$ be an oriented flat plane, equipped with an origin. Let
$\widetilde{L}_{\mathrm{Id}}$ be the threefold cyclic branched covering of
$L_{\mathrm{Id}}$, which is branched over the origin. Denote the projection
map from $\widetilde{L}_{\mathrm{Id}}$ onto $L_{\mathrm{Id}}$ by $\pi$. Denote
by $R$ the closure in $\widetilde{L}_{\mathrm{Id}}$ of one connected component
of the pre-image under $\pi$ of the open right half-plane in
$L_{\mathrm{Id}}$. On $R$ consider coordinates induced from $L_{\mathrm{Id}}$
via $\pi$. Denote by $\mathcal{C}^{\prime}$ the family of marks in $R$ with
endpoints $(2n-1)\vec{\mathbf{e}},2n\vec{\mathbf{e}}$, for $n\geq 1$, and
denote by $t$ and $b$ the two marks in $\widetilde{L}_{\mathrm{Id}}$ with
endpoints in $R$ with coordinates $\vec{\mathbf{f}},2\vec{\mathbf{f}}$ and
$-\vec{\mathbf{f}},-2\vec{\mathbf{f}}$, respectively. Let
$\widetilde{L}^{\prime}_{\mathrm{Id}}$ be the tame flat surface obtained from
$\widetilde{L}_{\mathrm{Id}}$ by regluing along $t$ and $b$. We call
$\widetilde{L}^{\prime}_{\mathrm{Id}}$ the _decorated surface_.
###### Remark 4.7.
We keep the notation $\mathcal{C}^{\prime}$ for the family of marks inherited
by $\widetilde{L}^{\prime}_{\mathrm{Id}}$. We denote the point inherited from
the origin by $O$. Then $O$ is a $6\pi$–angle singularity outside
$\mathcal{C}^{\prime}$.
###### Remark 4.8.
Let $S$ be a tame flat surface with a non-accumulating (in $\widehat{S}$)
family $\mathcal{C}$ of marks with slopes $\vec{\mathbf{e}}$. Assume that
$S^{\prime}$ is obtained from $\widetilde{L}^{\prime}_{\mathrm{Id}}\cup S$ by
regluing along $\mathcal{C}^{\prime}$ and $\mathcal{C}$. Then there are only
three saddle connections issuing from the point inherited from $O$ by
$S^{\prime}$. Their interiors are all contained in the subset inherited from
$R\setminus(t\cup b\cup\mathcal{C}^{\prime})$ and their holonomy vectors equal
$-\vec{\mathbf{f}},\vec{\mathbf{e}}$, and $\vec{\mathbf{f}}$. Hence the angles
between these saddle connections are $\frac{\pi}{2},\frac{\pi}{2}$ and $5\pi$.
We are now ready for our main construction. Recall that $\mathcal{U}$ denotes
the set of linear mappings of norm less than one.
###### Construction 4.9.
Let $G$ be a nontrivial countable subgroup of
$\mathbf{GL}_{+}(2,\mathbb{R})\setminus\mathcal{U}$. Denote the generators of
$G$ by $a_{i}$, where $i\geq 1$. If $G$ is trivial, we consider a single
generator $a_{1}=\mathrm{Id}$. Let $A_{\mathrm{Id}}$ be an oriented flat
plane, equipped with an origin. Let $A$ be the union of $A_{g}$ over $g\in G$
(see Definition 4.2).
For $i\geq 0$ let $\mathcal{C}^{i}$ be the family of marks on
$A_{\mathrm{Id}}$ with endpoints $i\vec{\mathbf{f}}+(2n-1)\vec{\mathbf{e}},\
i\vec{\mathbf{f}}+2n\vec{\mathbf{e}}$, for $n\geq 1$. All these marks are
pairwise disjoint. Now, given $x_{1},y_{1}\in\mathbb{R}$, consider the family
$\mathcal{C}^{-1}$ of marks on $A_{\mathrm{Id}}$ with endpoints
$(nx_{1},y_{1}),\ (nx_{1},y_{1})+a_{1}^{-1}(\vec{\mathbf{e}})$, for $n\geq 1$.
Choose $x_{1}>0$ sufficiently large and $y_{1}<0$ sufficiently small (i.e.
$-y_{1}>0$ sufficiently large) so that all these marks are pairwise disjoint
and disjoint from the ones in $\mathcal{C}^{i}$ for $i\geq 0$.
Observe that a translate of the lower half-plane in $A_{\mathrm{Id}}$ is
avoided by all already constructed marks. In this way we can inductively, for
all $i\geq 2$, choose $x_{i},-y_{i}\in\mathbb{R}$ sufficiently large so that
the marks with endpoints $(nx_{i},y_{i}),\
(nx_{i},y_{i})+a_{i}^{-1}(\vec{\mathbf{e}})$, for $n\geq 1$, are pairwise
disjoint and disjoint with the previously constructed marks. We denote these
families by $\mathcal{C}^{-i}$. None of the described marks accumulate.
Let $\widetilde{L}^{\prime}_{\mathrm{Id}}$ be the decorated surface from
Construction 4.6 and let $\widetilde{L}^{\prime}$ be the union of
$\widetilde{L}^{\prime}_{g}$ over $g\in G$ (see Definition 4.2). For each
$g\in G$ let $V_{g}$ be the flat surface obtained from
$A_{g}\cup\widetilde{L}^{\prime}_{g}$ by regluing along the families of marks
$\overline{g}\mathcal{C}^{0}$ and $\overline{g}\mathcal{C}^{\prime}$. The
regluing is allowed, since all the slopes equal $g(\vec{\mathbf{e}})$. The
surface $V_{g}$ is complete, in particular it is tame. Let $V$ be the union of
the $V_{g}$ over $g\in G$. The action of $G$ on $A$ and on
$\widetilde{L}^{\prime}$ carries over to an action on $V$, and we retain the
same notation for this action. It still has the property that the differential
of $\overline{g}$ equals $g$, for each $g\in G$. We keep the notation
$\mathcal{C}^{i}$, for $i\neq 0$, for the families of marks that are inherited
from the families of marks on $A_{\mathrm{Id}}$ by $V_{\mathrm{Id}}$.
For each $i\geq 1$ we consider a copy $\hat{E}^{i}_{Id}$ of the buffer surface
$\hat{E}_{Id}$ defined in Construction 4.4. We denote the copies of
$\mathcal{S},\mathcal{S}^{\prime}$ in $\hat{E}^{i}_{Id}$ by
$\mathcal{S}^{i},\mathcal{S}^{\prime i}$. Let $E$ be the union of all
$\hat{E}^{i}_{g}$, over $g\in G$ and all $i\geq 1$. Let $S=S(G)$ be the
Riemann surface equipped with the holomorphic $1$–form obtained from $V\cup E$
by regluing along the following pairs of families of marks. For each $i\geq 1$
and $g\in G$, we reglue the family $\overline{g}\mathcal{C}^{i}$ with
$\overline{g}\mathcal{S}^{i}$ and the family $\overline{g}\mathcal{S}^{\prime
i}$ with $\overline{g}\overline{a}_{i}\mathcal{C}^{-i}$. Note that this is
allowed since all slopes of these marks equal $g(\vec{\mathbf{e}})$. Moreover,
the action of $G$ carries over to $S$, and we retain the same notation for
this action.
###### Remark 4.10.
By Remarks 3.5 and 4.7 the set of singularities of $S$ with angle $6\pi$ is
the set of the $G$–translates of the point inherited by $S$ from $O$ (for
which we retain the same notation). By Remark 4.7 the translates
$\overline{g}O$ of $O$ in $S$ are pairwise different, for different $g\in G$.
###### Lemma 4.11.
$S$ is a Loch Ness Monster.
Proof. This follows from Lemma 4.3 applied to the graph $\Gamma^{\prime}$
obtained from the Cayley graph $\Gamma$ of $G=\langle a_{i}\rangle_{i\geq 1}$.
We get $\Gamma^{\prime}$ from $\Gamma$ by subdividing each edge of $\Gamma$
into three parts and by adding for each original vertex $v$ of $\Gamma$ an
additional vertex $v^{\prime}$ and an edge joining $v^{\prime}$ to $v$.
$\square$
###### Lemma 4.12.
$S$ is a tame flat surface.
Proof. Let $\bar{V}_{g}$, respectively $\bar{E}_{g}^{i}$, denote the closures
in $S$ of the subsets inherited from $V_{g}\setminus\overline{g}(\cup_{i\neq
0}\mathcal{C}^{i})$, respectively
$\hat{E}_{g}^{i}\setminus\overline{g}(\mathcal{S}^{i}\cup\mathcal{S}^{\prime
i})$.
It is enough to prove that $S$ is complete. Let $(x_{k})$ be a Cauchy sequence
on $S$. By Lemma 4.5 we may assume that there is some $g\in G$ such that all
$x_{k}$ lie in the union of $\bar{V}_{g}$ and the adjacent affine buffer
surfaces $\bar{E}^{i}_{g}$ and $\bar{E}^{i}_{ga^{-1}_{i}}$. Since the
components of
$\bar{V}_{g}\cap\left(\bigcup_{i}(\bar{E}^{i}_{g}\cup\bar{E}^{i}_{ga^{-1}_{i}})\right)$
form a discrete subset in $\bar{V}_{g}$, we may assume that all $x_{k}$ lie in
$\bar{V}_{g}$ and in a single adjacent buffer surface. Since both
$\bar{V}_{g}$ and the buffer surface are complete, $(x_{k})$ converges, as
required. $\square$
###### Lemma 4.13.
Any orientation preserving affine homeomorphism of $S$ is equal to
$\overline{g}$ for some $g\in G$.
Proof. Let $\psi$ be an orientation preserving affine homeomorphism of $S$. By
Remark 4.10, $\psi$ must permute the set of the $G$–translates of $O$. Hence
$\psi(O)=\overline{g}(O)$, for some $g\in G$. We are going to prove that
$\psi=\overline{g}$, which means that $\varphi=\overline{g}^{-1}\circ\psi$
equals the identity. For the time being we know only that $\varphi(O)=O$.
By Remark 4.7, there are only three saddle connections issuing from $O$.
Exactly one angle formed by them at $O$ exceeds $\pi$. Hence $\varphi$, which
is an orientation preserving affine homeomorphism fixing $O$, must fix all
these saddle connections. Therefore $\varphi$ is equal to the identity in the
neighborhood of $O$, which implies that $\varphi$ is the identity. $\square$
We summarize with the following.
Proof of Proposition 4.1. If
$G\subset\mathbf{GL}_{+}(2,\mathbb{R})\setminus\mathcal{U}$ is countable, and
nontrivial, then Construction 4.9 provides a Riemann surface $S=S(G)$ with a
holomorphic $1$–form. Moreover, $G$ acts on $S$ by affine homeomorphisms with
appropriate differentials. By Lemma 4.12 the flat surface $S$ is tame. By
Lemma 4.11 it is a Loch Ness monster. By Lemma 4.13 the Veech group of $S$
does not exceed $G$. $\square$
This establishes Theorem 1.2 in the case where the group $G$ is countable.
###### Remark 4.14.
If we do not require in Proposition 4.1 that our flat surface is a Loch Ness
monster, then it suffices to take only one mark from each infinite family of
marks, instead of the whole family, in Construction 4.9.
If in Construction 4.9 we take, for positive odd $i$, the marks in
$\mathcal{C}^{i}$ to have endpoints
$i\vec{\mathbf{f}}+(2n-1-\frac{1}{2^{i}})\vec{\mathbf{e}},\
i\vec{\mathbf{f}}+(2n-\frac{1}{2^{i}})\vec{\mathbf{e}}$, then there are
Euclidean triangles of arbitrarily small area, with vertices in singularities,
embedded in $S$. This is unlike in the case of compact flat surfaces, where
small triangles appear only if the Veech group is not a lattice [SW].
Conversely, we have the following.
###### Lemma 4.15.
If the Veech group $G$ of a flat surface $S$ is countable, then $G$ is
disjoint from $\mathcal{U}$.
Proof. First consider the case, where $S$ has a singularity $x$. Recall that
$\widehat{S}$ denotes the metric completion of $S$ and that the action of the
group of orientation preserving affine homeomorphisms of $S$ extends to an
action on $\widehat{S}$. Suppose that there is an orientation preserving
affine homeomorphism $\phi$ of $S$ with $D\phi\in\mathcal{U}$. Then $\phi$
extends to a contraction on $\widehat{S}$. By the Banach fixed point theorem,
the sequence $\phi^{k}(x)$ converges in $\widehat{S}$. If $x$ is not the fixed
point of $\phi$, then this contradicts tameness.
Assume now that $x$ is the fixed point of $\phi$ and the only singularity of
$S$. Then $S$ is simply connected. Otherwise by pushing a homotopically
nontrivial loop going through $x$ by the iterates of $\phi$ we obtain
arbitrarily short homotopically nontrivial loops through $x$, which
contradicts tameness. Hence $S$ is a cyclic branched covering of $\mathbb{C}$
and thus $G=\mathbf{GL}_{+}(2,\mathbb{R})$ which is not countable,
contradiction.
If $S$ does not have singularities, its universal cover is the flat plane.
Since $G$ is countable, $S$ must be a flat torus and we have that
$G\subset\mathbf{SL}(2,\mathbb{R})$ which is disjoint from $\mathcal{U}$.
$\square$
This proves Theorem 1.1 in the case where $G$ is countable.
## References
|
arxiv-papers
| 2009-06-29T14:00:13 |
2024-09-04T02:49:03.617808
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Piotr Przytycki, Gabriela Schmithuesen, Ferran Valdez",
"submitter": "Ferran Valdez",
"url": "https://arxiv.org/abs/0906.5268"
}
|
0906.5360
|
# A Remark on Kac-Wakimoto Hierarchies of D-type
Chao-Zhong Wu
Department of Mathematical Sciences, Tsinghua University,
Beijing 100084, P. R. China [email protected]
###### Abstract
For the Kac-Wakimoto hierarchy constructed from the principal vertex operator
realization of the basic representation of the affine Lie algebra
$D_{n}^{(1)}$, we compute the coefficients of the corresponding Hirota
bilinear equations, and verify the coincidence of these bilinear equations
with the ones that are satisfied by Givental’s total descendant potential of
the $D_{n}$ singularity, as conjectured by Givental and Milanov in [13].
Keywords: Kac-Wakimoto hierarchy, bilinear equation, principal realization,
total descendant potential
## 1 Introduction
The theory on representation theoretical aspects of soliton equations
developed by Date, Jimbo, Kashiwara, Miwa [1]-[4] and Kac, Wakimoto [17, 18]
plays a significant role in several research areas of modern mathematical
physics. For each affine Lie algebra $\mathfrak{g}$ together with an
integrable highest weight representation $V$ of $\mathfrak{g}$ and a vertex
operator construction $R$ of $V$, Kac and Wakimoto formulated a hierarchy of
soliton equations. These equations can be written down in terms of Hirota
bilinear equations and their super analogue [18]. When $\mathfrak{g}$ is the
untwisted affinization of a simply laced finite Lie algebra, the Kac-Wakimoto
hierarchy coincides with the corresponding generalized Drinfeld-Sokolov
hierarchy defined by Groot, Hollowood and Miramontes [14, 15]. In particular,
if the highest weight representation is the basic one, and the vertex operator
realization is constructed from the principal Heisenberg subalgebra, then the
Kac-Wakimoto hierarchy is equivalent to the Drinfeld-Sokolov hierarchy
associated to $\mathfrak{g}$ and the vertex $c_{0}$ of its Dynkin diagram [5].
In [10, 11], Givental constructed the total descendant potential for any
semisimple Frobenius manifold [6]. This potential is supposed to satisfy the
axioms dictated by Gromov-Witten theory, such as the string equation, dilaton
equation, topological recursion relations, and Virasoro constraints. Recently
Givental and Milanov [12, 13] showed that the total descendant potentials for
semisimple Frobenius manifolds associated to simple singularities satisfy
certain Hirota bilinear (quadratic) equations, and proved that for the
$A_{n}$, $D_{4}$ and $E_{6}$ singularities these equations are equivalent to
the corresponding Kac-Wakimoto hierarchies. They also conjectured that this
fact is true for all simple singularities.
In this note we compute explicitly the coefficients of the Kac-Wakimoto
hierarchy constructed from the principal vertex operator realization of the
basic representation of the affine Lie algebra $D_{n}^{(1)}$, while these
coefficients are implicitly defined in [18] except for the case $n=4$. This
computation verifies Givental and Milanov’s conjecture for the $D_{n}$
singularity.
## 2 Kac-Wakimoto hierarchies of ADE-type
Let $\mathfrak{g}$ be an untwisted affine Lie algebra of ADE-type, with rank
$n$, Coxeter number $h$, and normalized invariant bilinear form
$(\cdot\mid\cdot)$. The set of simple roots and simple coroots are denoted by
$\\{\alpha_{i}\\}_{i=0}^{n}$ and $\\{\alpha_{i}^{\vee}\\}_{i=0}^{n}$
respectively.
We denote the principal gradation of $\mathfrak{g}$ as
$\mathfrak{g}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{j}$. The Cartan
subalgebra of $\mathfrak{g}$, i.e., the $0$-component $\mathfrak{g}_{0}$, has
the following two decompositions
$\mathfrak{g}_{0}=\mathring{\mathfrak{h}}\oplus\mathbb{C}c\oplus\mathbb{C}d=\bar{\mathfrak{h}}\oplus\mathbb{C}c\oplus\mathbb{C}d.$
Here on the one hand
$\mathring{\mathfrak{h}}=\sum_{i=1}^{n}\mathbb{C}\alpha_{i}^{\vee}$, $c$ is
the central element and $d$ is determined by the constraint
$(\mathring{\mathfrak{h}}|d)=0,~{}~{}(c|d)=1,~{}~{}(d|d)=0;$
on the other hand, the subspace $\bar{\mathfrak{h}}$ is so chosen that the
difference of the projections of any $x\in\mathfrak{g}_{0}$ onto
$\mathring{\mathfrak{h}}$ and $\bar{\mathfrak{h}}$ is given by
$\mathring{x}-\bar{x}=h^{-1}(\mathring{\rho}^{\vee}|\mathring{x})c$, where
$\mathring{\rho}^{\vee}$ is an element of $\mathring{\mathfrak{h}}$ defined by
the condition
$\langle\alpha_{i},\mathring{\rho}^{\vee}\rangle=1,\quad i=1,\dots,n.$ (2.1)
Let $E$ be the set of exponents of $\mathfrak{g}$. For each $j\in E$ there
exists $H_{j}\in\mathfrak{g}_{j}$ satisfying
$(H_{i}|H_{j})=h\,\delta_{i,-j},\quad[H_{i},H_{j}]=i\,\delta_{i,-j}\,c.$ (2.2)
They generate the principal Heisenberg subalgebra
$\mathfrak{s}=\mathbb{C}c+\sum_{j\in E}\mathbb{C}H_{j}$.
In Kac and Wakimoto’s construction of their hierarchies, it is essential to
choose two bases $\\{v_{i}\\}$, $\\{v^{i}\\}$ of $\mathfrak{g}$ that are dual
to each other. These two bases read
$\displaystyle\\{v_{i}\\}~{}:~{}$
$\displaystyle\frac{1}{\sqrt{h}}H_{j}~{}(j\in E),~{}X^{(r)}_{m}~{}(1\leq r\leq
n;m\in\mathbb{Z}),~{}c,~{}d;$ (2.3) $\displaystyle\\{v^{i}\\}~{}:~{}$
$\displaystyle\frac{1}{\sqrt{h}}H_{-j}~{}(j\in E),~{}Y^{(r)}_{-m}~{}(1\leq
r\leq n;m\in\mathbb{Z}),~{}d,~{}c$ (2.4)
such that
$\displaystyle\\{X^{(r)}_{0}\\}_{r=1}^{n},\\{Y^{(r)}_{0}\\}_{r=1}^{n}\hbox{
are two bases of }\bar{\mathfrak{h}},$ (2.5)
$\displaystyle[H_{j},X^{(r)}_{m}]=\beta_{r,\bar{j}}X^{(r)}_{m+j},~{}~{}[H_{j},Y^{(r)}_{-m}]=-\beta_{r,\bar{j}}Y^{(r)}_{-m+j},$
(2.6) $\displaystyle(X^{(r)}_{l}|Y^{(s)}_{-m})=\delta_{r,s}\delta_{l,m}$ (2.7)
where $0<\bar{j}<h$ is the remainder of $j$ modulo $h$, and
$\beta_{r,\bar{j}}$ are some complex numbers which depend on the choice of the
two bases of $\mathfrak{g}$ .
Let $E_{+}$ be the set of positive exponents. A representation of the
Heisenberg subalgebra $\mathring{\mathfrak{s}}$ on the Fock space
$\mathbb{C}[t_{j};\,j\in E_{+}]$ is given by
$c\mapsto 1,~{}~{}H_{j}\mapsto\frac{\partial}{\partial
t_{j}},~{}~{}H_{-j}\mapsto j\,t_{j},~{}~{}j\in E_{+}.$
This can be lifted to a basic representation $L(\Lambda_{0})$ of
$\mathfrak{g}$ as follows:
$\displaystyle\sum_{m\in\mathbb{Z}}X^{(r)}_{m}z^{-m}\mapsto-h^{-1}(\mathring{\rho}^{\vee}|\mathring{X}^{(r)}_{0})X^{(r)}(t;z),$
$\displaystyle\sum_{m\in\mathbb{Z}}Y^{(r)}_{-m}z^{m}\mapsto-h^{-1}(\mathring{\rho}^{\vee}|\mathring{Y}^{(r)}_{0})X^{(r)}(-t;z),$
$\displaystyle d_{0}:=hd+\mathring{\rho}^{\vee}\mapsto-\sum_{j\in
E_{+}}j\,t_{j}\frac{\partial}{\partial t_{j}},$
where $X^{(r)}(t;z)$ $(1\leq r\leq n)$ are the vertex operators
$X^{(r)}(t;z)=\Big{(}\exp\sum_{j\in
E_{+}}\beta_{r,\bar{j}}\,t_{j}z^{j}\Big{)}\Big{(}\exp-\sum_{j\in
E_{+}}\frac{\beta_{r,\overline{-j}}}{jz^{j}}\frac{\partial}{\partial
t_{j}}\Big{)}.$
Such a realization of the basic representation $L(\Lambda_{0})$ is called the
_principal vertex operator construction_ , see [17, 18] for details.
###### Theorem 2.1 ([18])
Consider the basic representation of a simply laced affine Lie algebra
$\mathfrak{g}$ on the Fock space $L(\Lambda_{0})=\mathbb{C}[t_{j};\,j\in
E_{+}]$ constructed as above. Denote by ${G}$ the Lie group of the derived
algebra $\mathfrak{g}^{\prime}$ of $\mathfrak{g}$. A nonzero $\tau\in
L(\Lambda_{0})$ lies in the orbit $G\cdot 1$ if and only if $\tau$ satisfies
the following hierarchy of Hirota bilinear equations:
$\begin{split}&\Big{(}-2h\sum_{j\in
E_{+}}j\,y_{j}D_{j}+\sum_{r=1}^{n}g_{r}\sum_{m\geq
1}S_{m}^{E}(2\beta_{r,\bar{j}}\,y_{j})S_{m}^{E}(-\frac{\beta_{r,\overline{-j}}}{j}D_{j})\Big{)}\times\\\
&~{}~{}\times\Big{(}\exp\sum_{j\in
E_{+}}y_{j}D_{j}\Big{)}\tau\cdot\tau=0.\end{split}$ (2.8)
Here
$g_{r}=(\mathring{\rho}^{\vee}|\mathring{X}^{(r)}_{0})(\mathring{\rho}^{\vee}|\mathring{Y}^{(r)}_{0})$,
$S_{m}^{E}$ are the elementary Schur polynomials of $\mathfrak{g}$ defined by
$\exp\sum_{j\in E_{+}}y_{j}z^{j}=\sum_{m\geq 0}S_{m}^{E}(y_{j})z^{m}$, and
$D_{j}$ are the Hirota bilinear operators defined by $D_{j}\,f\cdot
g=\left.\frac{\partial}{\partial u}\right|_{u=0}f(t_{j}+u)g(t_{j}-u)$.
Kac and Wakimoto gave explicitly the coefficients $g_{r},\beta_{r,j}$ for the
affine Lie algebras $A_{n}^{(1)}$, $D_{4}^{(1)}$ and $E_{6}^{(1)}$ in [18],
however, these coefficients remain implicit for other affine Lie algebras. We
proceed to compute them for the affine Lie algebra $D^{(1)}_{n}$ in the next
section.
## 3 Bilinear equations for $D_{n}^{(1)}$
Let $\mathfrak{g}$ be an affine Lie algebra of type $D_{n}^{(1)}$. In this
section we want to construct the two bases (2.3), (2.4) of $\mathfrak{g}$, and
then write down the Kac-Wakimoto bilinear equations (2.8). Our result implies
that Givental and Milanov’s conjecture on the total descendant potential of
$D_{n}$ singularity is true.
Let us consider the corresponding simple Lie algebra first. The simple Lie
algebra $\mathring{\mathfrak{g}}$ of type $D_{n}$ possesses the following
$2n$-dimensional matrix realization [5]:
$\mathring{\mathfrak{g}}=\left\\{A\in\mathbb{C}^{2n\times 2n}\mid
A=-SA^{T}S\right\\},\ S=\sum_{i=1}^{n}(-1)^{i-1}(e_{ii}+e_{2n+1-i,2n+1-i}).$
(3.1)
Here $e_{i,j}$ is the $2n\times 2n$ matrix that takes value $1$ at the
$(i,j)$-entry and zero elsewhere, and $A^{T}=(a_{l+1-j,k+1-i})$ for any
$k\times l$ matrix $A=(a_{ij})$. In this matrix realization, a set of Weyl
generators can be chosen as
$\displaystyle e_{i}=e_{i+1,i}+e_{2n+1-i,2n-i}~{}(1\leq i\leq
n-1),~{}e_{n}=\frac{1}{2}(e_{n+1,n-1}+e_{n+2,n}),$ (3.2) $\displaystyle
f_{i}=e_{i,i+1}+e_{2n-i,2n+1-i}~{}(1\leq i\leq
n-1),~{}f_{n}={2}(e_{n-1,n+1}+e_{n,n+2}),$ (3.3) $\displaystyle
h_{i}=-e_{i,i}+e_{i+1,i+1}-e_{2n-i,2n-i}+e_{2n+1-i,2n+1-i}~{}(1\leq i\leq
n-1),$ (3.4) $\displaystyle
h_{n}=-e_{n-1,n-1}-e_{n,n}+e_{n+1,n+1}+e_{n+2,n+2}.$ (3.5)
Besides them we also need the following elements in $\mathring{\mathfrak{g}}$:
$\displaystyle
e_{0}=\frac{1}{2}(e_{1,2n-1}+e_{2,2n}),~{}~{}f_{0}=2(e_{2n-1,1}+e_{2n,2}),$
(3.6) $\displaystyle h_{0}=e_{1,1}+e_{2,2}-e_{2n-1,2n-1}-e_{2n,2n}.$ (3.7)
Recall the normalized Killing form $(A|B)=\frac{1}{2}\mathrm{tr}\,(AB)$ and
the Coxeter number $h=2n-2$ of $\mathring{\mathfrak{g}}$. We denote the
$\mathbb{Z}/h\mathbb{Z}\,$-principal gradation of $\mathring{\mathfrak{g}}$ as
$\mathring{\mathfrak{g}}=\bigoplus_{j\in\mathbb{Z}/h\mathbb{Z}}\mathring{\mathfrak{g}}_{j},$
then we have $e_{i}\in\mathring{\mathfrak{g}}_{\bar{1}}$,
$f_{i}\in\mathring{\mathfrak{g}}_{\overline{-1}}$,
$h_{i}\in\mathring{\mathfrak{g}}_{\bar{0}}$ for $i=0,\dots,n$.
Let $\Lambda=\sum_{i=0}^{n}{e}_{i}$ and $\mathring{\mathfrak{s}}$ be the
centralizer of $\Lambda$ in $\mathring{\mathfrak{g}}$. Then
$\mathring{\mathfrak{s}}$ is a Cartan subalgebra of $\mathring{\mathfrak{g}}$.
We fix a basis $\\{T_{j}|\,j\in I\\}$ of $\mathring{\mathfrak{s}}$ as
$\displaystyle T_{j}=$ $\displaystyle\Lambda^{j},\qquad j=1,3,\ldots,2n-3,$
$\displaystyle T_{(n-1)^{\prime}}=$
$\displaystyle\sqrt{n-1}\,\kappa\Big{(}e_{n,1}-\frac{1}{2}e_{n+1,1}-\frac{1}{2}e_{n,2n}+\frac{1}{4}e_{n+1,2n}$
$\displaystyle\quad+(-1)^{n}\big{(}e_{2n,n+1}-\frac{1}{2}e_{2n,n}-\frac{1}{2}e_{1,n+1}+\frac{1}{4}e_{1,n}\big{)}\Big{)}$
where $\kappa=1$ (resp. $\sqrt{-1}$) when $n$ is even (resp. odd), and $I$ is
the set of exponents of $\mathring{\mathfrak{g}}$ given by
$I=\\{1,3,5,\ldots,2n-3\\}\cup\\{(n-1)^{\prime}\\}.$
Here $(n-1)^{\prime}$ indicates that when $n$ is even the multiplicity of the
exponent $n-1$ is $2$. These matrices $T_{j}$ belong to
$\mathring{\mathfrak{g}}_{j}$ respectively, and satisfy
$(T_{i}|T_{h-j})=(n-1)\delta_{i,j}.$
To construct the desired bases, we need the root space decomposition of
$\mathring{\mathfrak{g}}$ with respect to $\mathring{\mathfrak{s}}$. Note that
the set of eigenvalues of $\Lambda$ is
$\\{\omega\in\mathbb{C}\mid\omega^{h}=1\\}\cup\\{0\\},$
in which the multiplicity of $0$ is $2$. We choose the eigenvectors
$\eta_{\omega},\eta_{0},\eta_{0^{\prime}}$ associated to eigenvalues $\omega$,
$0$ respectively as follows
$\displaystyle\eta_{\omega}=(\frac{1}{2},\omega^{-1},\ldots,\omega^{-(n-1)},\
\frac{1}{2}\omega^{n-1},\omega^{n-2},\ldots,\omega,1)^{t},$
$\displaystyle\eta_{0}=(-\frac{1}{2}\psi_{1}+\psi_{2n})+\kappa^{-1}(\psi_{n}-\frac{1}{2}\psi_{n+1}),$
$\displaystyle\eta_{0^{\prime}}=(-\frac{1}{2}\psi_{1}+\psi_{2n})-\kappa^{-1}(\psi_{n}-\frac{1}{2}\psi_{n+1}),$
where $\psi_{i}$ is the $2n$-dimensional column vector with the $i$-th entry
being $1$ and all other entries being zero, and $\cdot^{t}$ is the usual
transposition of matrices. These eigenvectors give a common eigenspace
decomposition for $T_{j}\ (j\in I)$:
$\displaystyle T_{j}\,\eta_{\alpha}=\alpha^{j}\,\eta_{\alpha},\quad
j=1,3,\dots,2n-2,$ $\displaystyle
T_{(n-1)^{\prime}}\,\eta_{\alpha}=\big{(}(-1)^{n-1}\delta_{\alpha,0}+(-1)^{n}\delta_{\alpha,0^{\prime}}\big{)}\sqrt{n-1}\,\eta_{\alpha}.$
Introduce a map $\sigma:\,\mathbb{C}^{2n\times 2n}\to\mathring{\mathfrak{g}}$,
$A\mapsto A-SA^{T}S$, and define the $2n\times 2n$ matrices
$A_{(\alpha,\beta)}=\sigma(\eta_{\alpha}\eta_{-\beta}^{T}),$
where $\alpha,\beta$ are eigenvalues of $\Lambda$. These matrices satisfy
$\displaystyle[T_{j},A_{(\alpha,\beta)}]=(\alpha^{j}+\beta^{j})A_{(\alpha,\beta)},\quad
j=1,3,\ldots,2n-3,$
$\displaystyle[T_{(n-1)^{\prime}},A_{(\alpha,\beta)}]=(\delta_{\alpha,0}-\delta_{\alpha,0^{\prime}}+\delta_{\beta,0}-\delta_{\beta,0^{\prime}})\sqrt{n-1}\,A_{(\alpha,\beta)},$
from which one can obtain the root space decomposition of
$\mathring{\mathfrak{g}}$ with respect to $\mathring{\mathfrak{s}}$.
Now denote by $A_{(\alpha,\beta),j}$ the homogeneous components of
$A_{(\alpha,\beta)}$ in $\mathring{\mathfrak{g}}_{j}$, and fix
$\omega=\exp\big{(}2\pi i/h\big{)}$. One can verify the following relations
$\displaystyle(A_{(1,\omega^{r}),0}|A_{(-1,-\omega^{s}),0})$
$\displaystyle=-h\delta_{r,s},$
$\displaystyle(A_{(1,\omega^{r}),0}|A_{(-1,\alpha),0})$ $\displaystyle=0,$
$\displaystyle(A_{(1,\alpha),0}|A_{(-1,\beta),0})$
$\displaystyle=2(1-\delta_{\alpha,\beta}),$
where $1\leq r,s\leq n-2$ and $\alpha,\beta\in\\{0,0^{\prime}\\}$. According
to these relations, we choose two bases of $\mathring{\mathfrak{g}}$:
$\displaystyle\\{T_{j}\mid j\in I\\}\cup\\{\tilde{X}^{(r)}_{m}\mid
r=1,\dots,n;\ m\in\mathbb{Z}/h\mathbb{Z}\\},$ $\displaystyle\\{T_{j}\mid j\in
I\\}\cup\\{\tilde{Y}^{(r)}_{m}\mid r=1,\dots,n;\
m\in\mathbb{Z}/h\mathbb{Z}\\},$ $\begin{array}[]{cccc}\hline\cr&1\leq r\leq
n-2&r=n-1&r=n\\\
\hline\cr\tilde{X}^{(r)}_{m}:&\frac{1}{\sqrt{h}}{A}_{(1,\omega^{r}),m}&\frac{1}{\sqrt{2}}{A}_{(1,0),m}&\frac{1}{\sqrt{2}}{A}_{(1,0^{\prime}),m}\\\
\tilde{Y}^{(r)}_{m}:&-\frac{1}{\sqrt{h}}{A}_{(-1,-\omega^{r}),m}&\frac{1}{\sqrt{2}}{A}_{(-1,0^{\prime}),m}&\frac{1}{\sqrt{2}}{A}_{(-1,0),m}\\\
\hline\cr\end{array}$ (3.8)
The above two bases of $\mathring{\mathfrak{g}}$ help us to construct a pair
of dual bases (2.3), (2.4) of the affine Lie algebra $\mathfrak{g}$ that
satisfy (2.5)-(2.7). We use the principal realization of $\mathfrak{g}$ [17]
$\mathfrak{g}=\bigoplus_{m\in\mathbb{Z}}\lambda^{m}\mathring{\mathfrak{g}}_{\bar{m}}\oplus\mathbb{C}c\oplus\mathbb{C}d.$
Note that the set of exponents of $\mathfrak{g}$ is $E=I+h\,\mathbb{Z}$, and
the principal Heisenberg subalgebra is generated by
$H_{j}=\sqrt{2}\,\lambda^{j}\,T_{\bar{j}},\ j\in E.$
The two bases (2.3), (2.4) of $\mathfrak{g}$ can be chosen as
$\displaystyle\frac{1}{\sqrt{h}}H_{j},\
X^{(r)}_{m}=\lambda^{m}\tilde{X}^{(r)}_{\bar{m}},\ c,\ d;$
$\displaystyle\frac{1}{\sqrt{h}}H_{-j},\
Y^{(r)}_{-m}=\lambda^{-m}\tilde{Y}^{(r)}_{\overline{-m}},\ d,\ c$
with the coefficients $\beta_{r,j}$ that appear in (2.6) given by
$\displaystyle\beta_{r,j}=$
$\displaystyle\left\\{\begin{array}[]{ll}\sqrt{2}(1+\omega^{rj}),&r=1,2,\ldots,n-2,\
j\neq(n-1)^{\prime},\\\ \sqrt{2},&r=n-1,n,\ j\neq(n-1)^{\prime},\\\
\sqrt{2n-2}(\delta_{r,n-1}-\delta_{r,n}),&j=(n-1)^{\prime}.\end{array}\right.$
(3.12)
To write down the Kac-Wakimoto bilinear equations (2.8), we still need to
compute the constants
$g_{r}=(\mathring{\rho}^{\vee}|\mathring{X}^{(r)}_{0})(\mathring{\rho}^{\vee}|\mathring{Y}^{(r)}_{0})$.
Note that in the principal realization of $\mathfrak{g}$, the Weyl generators
are given by
$\tilde{e}_{i}=\lambda\,e_{i},\ \tilde{f}_{i}=\lambda^{-1}f_{i},\
\alpha_{i}^{\vee}=h_{i}+\frac{c}{h},~{}~{}i=0,\dots,n,$
so we have
$(\mathring{\rho}^{\vee}|\mathring{X}^{(r)}_{0})=\left(\mathring{\rho}^{\vee}\left|X^{(r)}_{0}+\frac{c}{h}\sum_{i=1}^{n}a_{i}\right.\right)=\sum_{i=1}^{n}a_{i},$
where $a_{i}$ are the coefficients in the following linear expansion
$X^{(r)}_{0}=\sum_{i=1}^{n}a_{i}\,h_{i}=\sum_{i=1}^{n}a_{i}\,\left(\alpha_{i}^{\vee}-\frac{c}{h}\right)\in\mathring{\mathfrak{g}}_{0}.$
According to the realization (3.2)-(3.5), given any
$\mathrm{diag}(b_{1},b_{2},\dots,b_{2n})=\sum_{i=1}^{n}a_{i}\,h_{i}\in\mathring{\mathfrak{g}}_{0},$
the summation $\sum_{i=1}^{n}a_{i}$ reads
$\sum_{i=1}^{n}a_{i}=-\sum_{i=1}^{n-1}(n-i)b_{i}.$
By using this formula, we obtain
$g_{r}=\left\\{\begin{array}[]{ll}\frac{n-1}{2}\frac{2-\omega^{r}-\omega^{-r}}{2+\omega^{r}+\omega^{-r}},&r=1,\ldots,n-2,\\\
\frac{(n-1)^{2}}{2}&r=n-1,n.\end{array}\right.$ (3.13)
###### Proposition 3.1
The constants $g_{r}$ and $\beta_{r,j}$ in the Kac-Wakimoto hierarchy of
bilinear equations (2.8) for $D_{n}^{(1)}$ are given by (3.12) and (3.13).
Note that the values $\beta_{r,j}$ depend on the choice of the dual bases
(2.3), (2.4). However, it is easy to see that the constants $g_{r}$ are
independent of the choice of such bases.
In [13], Givental and Milanov proved that the total descendant potential for
semisimple Frobenius manifolds associated to a simple singularity satisfies
the following hierarchy of Hirota bilinear equations:
$\begin{split}\mathrm{res}_{z=0}&z^{-1}\sum_{r=1}^{n}g_{r}e^{\sum_{j\in
E_{+}}2\beta_{r,\bar{j}}\,z^{j}y_{j}}e^{-\sum_{j\in
E_{+}}\,\beta_{r,\overline{-j}}\,z^{-j}\partial_{y_{j}}/j}\tau(t+y)\tau(t-y)\\\
&=\Big{(}2h\sum_{j\in
E_{+}}j\,y_{j}\partial_{y_{j}}+\frac{nh(h+1)}{12}\Big{)}\tau(t+y)\tau(t-y),\end{split}$
(3.14)
where the coefficients $\beta_{r,j}$ are the same as in (2.8), and $g_{r}$ are
given explicitly in [13]. By comparing the constants $g_{r}$ (3.13) with those
in [13], we obtain the following corollary.
###### Corollary 3.2
The hierarchy (3.14) for the $D_{n}$ singularity coincides with the Kac-
Wakimoto hierarchy of type $D_{n}^{(1)}$ associated to the basic
representation and its principal vertex operator construction.
Namely, we conform Givental and Milanov’s conjecture [13] for the case
$D_{n}$.
## 4 Concluding remarks
We study in [19] the tau structure of the Drinfeld-Sokolov hierarchy
associated to $D_{n}^{(1)}$ and the zeroth vertex of its Dynkin diagram
following the approach of [7]. So we can define the tau function by using the
tau symmetry of the Hamiltonian structures, and establish the equivalence
between this definition of the tau function for this hierarchy and that given
by Hollowood and Miramontes [15]. Basing on the tau structure, we plan to show
that this Drinfeld-Sokolov hierarchy coincides with the bihamiltonian
integrable hierarchy constructed according to the axiomatic scheme developed
by Dubrovin and Zhang [7] on the formal loop space of the semisimple Frobenius
manifold associated to the $D_{n}$-type Weyl group. This assertion together
with the result of this note would imply that Givental’s total descendant
potential associated to the $D_{n}$ singularity is a tau function of Dubrovin
and Zhang’s hierarchy.
While we prepared to do an analogous computation for the cases $E_{7}$,
$E_{8}$ of Givental and Milanov’s conjecture [13], we learned from [9] that
Frenkel, Givental and Milanov have obtained a proof of this conjecture in
general. We hope however that this short note might be helpful to a better
understanding of the relationship between Givental’s total descendant
potentials and integrable systems.
Acknowledgments. The author would like to thank Boris Dubrovin, Si-Qi Liu and
Youjin Zhang for advises, he would also like to thank Todor Milanov for
helpful comments. This work is partially supported by the National Basic
Research Program of China (973 Program) No.2007CB814800.
## References
* [1] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T. Transformation groups for soliton equations. III. Operator approach to the Kadomtsev-Petviashvili equation. J. Phys. Soc. Japan 50 (1981), no. 11, 3806–3812.
* [2] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa T. Transformation groups for soliton equations. VI. KP hierarchies of orthogonal and symplectic type. J. Phys. Soc. Japan 50 (1981), no. 11, 3813–3818.
* [3] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T. Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type. Phys. D 4 (1981/82), no. 3, 343–365.
* [4] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T. Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy. Publ. Res. Inst. Math. Sci. 18 (1982), no. 3, 1077–1110.
* [5] Drinfeld, V.G.; Sokolov, V.V. Lie algebras and equations of Korteweg-de Vries type. (Russian) Current problems in mathematics, Vol. 24, 81–180, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.
* [6] Dubrovin, B. Geometry of $2$D topological field theories. Integrable systems and quantum groups (Montecatini Terme, 1993), 120–348, Lecture Notes in Math., 1620, Springer, Berlin, 1996.
* [7] Dubrovin, B.; Zhang, Y. Normal forms of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, preprint arXiv: math.DG/0108160, 2001.
* [8] Dubrovin, B.; Zhang, Y. Universal integrable hierarchy of the topological type, in preparation.
* [9] Fan, H.; Jarvis, T.J.; Ruan, Y. The Witten equation, mirror symmetry and quantum singularity theory, preprint arXiv: math.AG/0712.4021v3.
* [10] Givental, A. Semi-simple Frobenius structures at higher genus. International Mathematics Research Notices 2001, no. 23 (2001): 1265-1286.
* [11] Givental, A. Gromov-Witten invariants and quantization of quadratic Hamiltonians. Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. Mosc. Math. J. 1 (2001), no. 4, 551–568, 645.
* [12] Givental, A. $A_{n-1}$-singularities and $n$KdV hierarchies. Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday. Mosc. Math. J. 3 (2003), no. 2, 475–505, 743.
* [13] Givental, A.; Milanov, T.E. Simple singularities and integrable hierarchies. The breadth of symplectic and Poisson geometry, 173–201, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005.
* [14] de Groot, M.F.; Hollowood, T.J.; Miramontes, J.L. Generalized Drinfeld-Sokolov hierarchies. Comm. Math. Phys. 145 (1992), no. 1, 57–84.
* [15] Hollowood, T.J.; Miramontes, J.L. Tau-functions and generalized integrable hierarchies. Comm. Math. Phys. 157 (1993), no. 1, 99–117.
* [16] Jimbo, M.; Miwa, T. Solitons and infinite-dimensional Lie algebras. Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 943–1001.
* [17] Kac, V.G. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990.
* [18] Kac, V.G.; Wakimoto, M. Exceptional hierarchies of soliton equations. Theta functions—Bowdoin 1987, Part 1 (Brunswick, ME, 1987), 191–237, Proc. Sympos. Pure Math., 49, Part 1, Amer. Math. Soc., Providence, RI, 1989.
* [19] Liu, S.Q.; Wu, C.Z.; Zhang, Y. Tau structures of Drinfeld-Sokolov hierarchies of D-type, in preparation.
|
arxiv-papers
| 2009-06-29T20:06:10 |
2024-09-04T02:49:03.625786
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chao-Zhong Wu",
"submitter": "Chaozhong Wu",
"url": "https://arxiv.org/abs/0906.5360"
}
|
0907.0005
|
# Exploring Dark Matter with Milky Way substructure
Michael Kuhlen1111To whom correspondence should be addressed; E-mail:
[email protected]., Piero Madau2, Joseph Silk3
1School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540
2Department of Astronomy & Astrophysics, University of California, Santa Cruz,
CA 95064
3Department of Physics, University of Oxford, Oxford, OX1 3RH, UK
> The unambiguous detection of Galactic dark matter annihilation would unravel
> one of the most outstanding puzzles in particle physics and cosmology.
> Recent observations have motivated models in which the annihilation rate is
> boosted by the Sommerfeld effect, a non-perturbative enhancement arising
> from a long range attractive force. Here we apply the Sommerfeld correction
> to Via Lactea II, a high resolution N-body simulation of a Milky-Way-size
> galaxy, to investigate the phase-space structure of the Galactic halo. We
> show that the annihilation luminosity from kinematically cold substructure
> can be enhanced by orders of magnitude relative to previous calculations,
> leading to the prediction of $\gamma$-ray fluxes from up to hundreds of dark
> clumps that should be detectable by the Fermi satellite.
In the standard cold dark matter (CDM) paradigm of structure formation, a
weakly interacting massive particle (WIMP) of mass $m_{\chi}\sim 100$ GeV-10
TeV ceases to annihilate when the universe cools to a temperature of
$T_{f}\sim m_{\chi}/20$, about one nano-second after the Big Bang. A
thermally-averaged cross-section at freeze-out of $\langle\sigma
v\rangle_{0}\approx 3\times 10^{-26}$ cm3 s-1 results in a relic abundance
consistent with observations (?). Perturbations in the dark matter density are
amplified by gravity after the universe becomes matter dominated, around ten
thousand years after the Big Bang: the smallest structures (“halos”) collapse
earlier when the universe is very dense and merge to form larger and larger
systems over time. Today, galaxies like our own Milky Way are embedded in
massive, extended halos of dark matter that are very lumpy, teeming with self-
bound substructure (“subhalos”) that survived this hierarchically assembly
process (?, ?, ?). Indirect detection of high energy antiparticles and
$\gamma$-rays from dark matter halos provides a potential “smoking gun”
signature of WIMP annihilation (?). The usual assumption that WIMP
annihilation proceeds at a rate that does not depend, in the non-relativistic
$v/c\ll 1$ limit, on the particle relative velocities implies that the primary
astrophysical quantity determining the annihilation luminosity today is the
local density squared. WIMP annihilations still occur in the cores of
individual substructures, but with fluxes that are expected to be dauntingly
small. The latest calculations show that only a handful of the most massive
Galactic subhalos may, in the best case, be detectable in $\gamma$-rays by the
Fermi satellite (?, ?).
The Sommerfeld enhancement, a velocity-dependent mechanism that boosts the
dark matter annihilation cross-section over the standard $\langle\sigma
v\rangle_{0}$ value (?, ?, ?, ?), may provide an explanation for the
experimental results of the PAMELA satellite reporting an increasing positron
fraction in the local cosmic ray flux at energies between 10 and 100 GeV (?),
as well as for the surprisingly large total electron and positron flux
measured by the ATIC and PPB-BETS balloon-borne experiments (?, ?). Very
recent Fermi (?) and H.E.S.S. (?) data appear to be inconsistent with the ATIC
and PPB-BETS measurements, but still exhibit departures with respect to
standard expectations from cosmic ray propagation models. Although
conventional astrophysical sources of high energy cosmic rays, such as nearby
pulsars or supernova remnants, may provide a viable explanation (?, ?, ?), the
possibility of Galactic DM annihilation as a source remains intriguing (?, ?,
?). In this case, cross-sections a few orders of magnitude above what is
expected for a thermal WIMP are required (?).
Figure 1: A: The distribution of velocity dispersion for Via Lactea II
particles within 400 kpc. The dispersions are calculated from the nearest 32
neighbors of each particle. B: Sommerfeld enhancement factor as a function of
velocity for four representative models, exhibiting $S\sim 1/v$ (cyan and
orange curves) and $\sim 1/v^{2}$ behavior (magenta and brown), and high (cyan
and magenta) versus low (orange and brown) saturation velocities. C and D: The
corresponding distributions of S-factors for Via Lactea II particles.
The Sommerfeld non-perturbative increase in the annihilation cross-section at
low velocities is the result of a generic attractive force between the
incident dark matter particles that effectively focuses incident plane-wave
wavefunctions. The force carrier may be the $W$ or $Z$ boson of the weak
interaction (?), $m_{\phi}=80-90$ GeV/$c^{2}$, or a lighter boson,
$m_{\phi}\sim$ GeV/$c^{2}$, mediating a new interaction in the dark sector (?,
?). Upon introduction of a force with coupling strength $\alpha$, the
annihilation cross-section is shifted to $\langle\sigma
v\rangle=S\langle\sigma v\rangle_{0}$, where the Sommerfeld correction $S$
disappears ($S=1$) in the limit $v/c\rightarrow 1$ (thus leaving unchanged the
weak scale annihilation cross-section during WIMP freeze-out in the early
universe). When $v/c\ll\alpha$, $S\approx\pi\alpha c/v$ (“$1/v$” enhancement),
but it levels off to $S_{\rm max}\approx 6\alpha m_{\chi}/m_{\phi}$ at
$v/c\approx 0.5m_{\phi}/m_{\chi}$ because of the finite range of the
interaction. For specific parameter combinations, i.e. when
$m_{\chi}/m_{\phi}\approx n^{2}/\alpha$ where $n$ is an integer, the (Yukawa)
potential develops bound states, and these give rise to large, resonant cross-
section enhancements where $S$ grows approximately as $1/v^{2}$ before
saturating (see Supporting Online Material).
Figure 2: All-sky maps (in a Mollweide projection) of the Sommerfeld-enhanced
annihilation surface brightness ($\int_{\rm los}\rho^{2}S\;d\ell$) from all
Via Lactea II dark matter particles within 400 kpc. The observer is located at
8 kpc from the halo center along the host halo’s intermediate principal axis.
A: No Sommerfeld enhancement. B: $S\sim 1/v$, saturated at $\sim 1$ km s-1. C:
$S\sim 1/v^{2}$ saturated at $\sim 5$ km s-1. The maps have been normalized to
give the same total smooth host halo flux.
The Sommerfeld effect connects dynamically the dark and the astrophysics
sectors. Because the typical velocities of dark matter particles in the Milky
Way today are of the order of $v/c\sim 10^{-3}$, the resulting boost in the
annihilation rate may provide an explanation to the puzzling Galactic signals.
Compared to particles in the smooth halo component, the Sommerfeld correction
preferentially enhances the annihilation luminosity of cold, lower velocity
dispersion substructure, as emphasized previously by (?, ?, ?). Detailed
knowledge of the full phase-space density of dark matter particles in the
Milky Way is thus necessary to reliably compute the expected signals. Here we
use the Via Lactea II cosmological simulation, a high precision calculation of
the assembly of the Galactic CDM halo, for a systematic investigation of the
impact of Sommerfeld-corrected models on present and future indirect dark
matter detection efforts. Via Lactea II employs just over one billion
$4,100\,\,\rm M_{\odot}$ particles to follow, with a force resolution of 40
pc, the formation of a $1.9\times 10^{12}\,\,\rm M_{\odot}$ Milky-Way size
halo and its substructure from redshift $z=104$ to the present (?, ?, ?).
(Fig. 1 A) The smooth halo particles, whose velocity dispersions are set by
the global potential, typically have three-dimensional velocity dispersion
$\sigma>100\,\,{\rm km\,s^{-1}}$. Particles in self-bound subhalos dominate at
lower velocity dispersions. The total mass fraction of particles with
$\sigma<5\,\,{\rm km\,s^{-1}}$ is 1%. We calculated the Sommerfeld enhancement
factors $S$ on a particle-by-particle basis by averaging $S(v)$ over a
Maxwell-Boltzmann distribution of relative velocities with one-dimensional
velocity dispersion given by $\sqrt{2/3}\;\sigma$ (see the Supporting Online
Materials for details) (Fig. 1 C, D).
The large Sommerfeld boost expected for $v/c\sim 10^{-4}-10^{-5}$ make cold
subhalos more promising sources of annihilation $\gamma$-rays than the higher
density but much hotter region around the Galactic Center (Fig. 2). In
Sommerfeld-enhanced models, substructures are much more clearly visible, and
can even outshine the Galactic Center when the cross-section is close to
resonance and saturates at low velocities. Furthermore, baryonic processes
will tend to heat up the Galactic Center and dim its Sommerfeld boost, and
thereby increase the relative detectability of subhalos. Dark matter halos are
not isothermal and have smaller velocity dispersions in the center (see
Supporting Online Material). In addition to an overall increase in the
annihilation rate, this “temperature inversion” leads to a relative
brightening of the center at the expense of the diffuse flux from the
surrounding region (Fig. 3). The subhalo exhibits its own population of
subclumps, also Sommerfeld-enhanced.
Figure 3: Annihilation rate maps (projections of $\rho^{2}S$ out to the tidal
radius) of one of the most massive ($M\sim 2\times 10^{9}\,\,\rm M_{\odot}$)
subhalos in Via Lactea II, for the same models as in Fig. 2. The images have
not been normalized. Compared to the $S=1$ case (A), the total luminosity is
2,200 and 160,000 higher for $S\sim 1/v$ (B) and $S\sim 1/v^{2}$ (C),
respectively.
To address quantitatively the detectability of Sommerfeld-enhanced subhalos by
the Fermi Space Telescope, we have converted the annihilation flux calculated
from our simulation (?) into a predicted $\gamma$-ray flux and compared it to
the expected backgrounds. We investigated two different classes of particle
physics models (Table 1):
1. i)
those motivated by (?), in which the force carrier is the conventional weak
force gauge boson, the W or Z particle, and the mass of the dark matter
particle is $\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}4$ TeV. We have
chosen four representative values of $m_{\chi}$ and $\alpha$, which lie
increasingly close to a $S\sim 1/v^{2}$ resonance. In these models the main
source of $\gamma$-rays is the decay of neutral pions that are produced in the
hadronization of the annihilation products;
2. ii)
and those in which the annihilation is mediated by a new dark sector force
carrier $\phi$ (?). The choice of parameters ($m_{\chi},m_{\phi},\alpha$)
follows Meade, Papucci, & Volansky (MPV) (?) and satisfies constraints from
recent H.E.S.S. measurements of the Galactic Center (?) and the Galactic Ridge
(?), as well as the PAMELA measurement of the local positron fraction above 10
GeV (?) and the ATIC (?) and PPB-BETS (?) measurements of the total
$(e^{+}+e^{-})$ flux above 100 GeV. Models MPV-1 incorporate all three
constraints and models MPV-2 only the H.E.S.S. and PAMELA data. We considered
models away from (a) and close to (b) resonance, thereby covering both $S\sim
1/v$ and $\sim 1/v^{2}$ behavior. The data favor a light force carrier,
$m_{\phi}\approx 200$ MeV, and the $\gamma$-rays originate then as final state
radiation (internal bremsstrahlung) accompanying the decay of the $\phi$’s
into $e^{+}e^{-}$ pairs.
The magnitude of the relativistic cross section was fixed to the standard
value of $\langle\sigma v\rangle_{0}=3\times 10^{-26}$ cm3 s-1. $\gamma$-ray
spectra are shown in Figure S2 in the Supporting Online Material.
Table 1: Summary of the models used to assess subhalo detectability with Fermi. Particle physics parameters are: $m_{\chi}$, the mass of the dark matter particle, $m_{\phi}$, the mass of the force carrier, and $\alpha$, the coupling constant. In the two right-most columns we give $S_{\rm max}$, the maximum Sommerfeld enhancement obtained, and the saturation velocity $v_{\rm sat}$, defined as the velocity at which $S$ reaches 90% of $S_{\rm max}$. Model | $m_{\chi}$ | $m_{\phi}$ | $\alpha\times 100$ | $S_{\rm max}$ | $v_{\rm sat}$
---|---|---|---|---|---
| (TeV) | (GeV) | | | (km s-1)
LS-1 | 4.30 | 90 | $3.307$ | 1,500 | 80
LS-2 | 4.45 | 90 | $3.297$ | 12,000 | 28
LS-3 | 4.50 | 90 | $3.288$ | 70,000 | 12
LS-4 | 4.55 | 90 | $3.281$ | 430,000 | 4.7
MPV-1a | 1.0 | 0.2 | $4.000$ | 3,000 | 7.4
MPV-1b | 1.0 | 0.2 | $3.739$ | 16,000 | 2.4
MPV-2a | 0.25 | 0.2 | $4.000$ | 480 | 40
MPV-2b | 0.25 | 0.2 | $4.500$ | 40,000 | 3.3
We determined the Fermi detection significance by summing the annihilation
photons from all the pixels in our all-sky maps covering a given subhalo, and
compared this to the square root of the number of background photons from the
same area. We counted a subhalo as “detectable” if it had a total signal-to-
noise greater than 5 (Table 2). The numbers are quite large, implying that
individual subhalos should easily be detected by Fermi if Sommerfeld
enhancements are important. Even in the most conservative cases (MPV-1a and
MPV-2a) around ten or more subhalos should be discovered after 5 years of
observation. In fact, on the basis of all models considered here it is
predicted that Fermi should be able to accumulate enough flux in its first
year of observations to detect several dark matter subhalos at more than
$5\sigma$ significance, a prediction that will soon be tested, and that may
open up the door to studies of non-gravitational dark matter interactions and
new particle physics. The central brightening discussed above results in a
smaller angular extent of a given subhalo’s detectable region: the stronger
the Sommerfeld enhancement the fewer pixels exceed the detection threshold.
Nevertheless, for all models considered here the majority of detectable
subhalos would be resolved sources for Fermi.
Table 2: Detectable Subhalos. The number of subhalos that would be detected with $>5\sigma$ significance by Fermi after 1, 2, 5, and 10 years in orbit, for different Sommerfeld-enhanced dark matter particle models. In the two right-most columns we give the median distance and mass of the detectable clumps after 5 years in orbit. Model | 1 yr | 2 yr | 5 yr | 10 yr | $\tilde{D}$ | $\tilde{M}_{\rm sub}$
---|---|---|---|---|---|---
| | | | | (kpc) | ($\,\rm M_{\odot}$)
LS-1 | 12 | 19 | 29 | 38 | 24 | $1.4\times 10^{7}$
LS-2 | 72 | 99 | 167 | 244 | 42 | $9.5\times 10^{6}$
LS-3 | 225 | 311 | 457 | 583 | 56 | $6.2\times 10^{6}$
LS-4 | 410 | 528 | 730 | 919 | 66 | $4.9\times 10^{6}$
MPV-1a | 5 | 7 | 12 | 15 | 16 | $9.8\times 10^{7}$
MPV-1b | 9 | 14 | 25 | 36 | 25 | $4.4\times 10^{6}$
MPV-2a | 12 | 18 | 29 | 38 | 24 | $1.4\times 10^{7}$
MPV-2b | 187 | 254 | 397 | 518 | 55 | $4.5\times 10^{6}$
Another question of interest is whether Sommerfeld-corrected substructure
would lead to a significant boost in the local production of high energy
positrons, arising from dark matter annihilation in subhalos within a
diffusion region of a few thousand parsecs from Earth, as well as of
antiprotons within a correspondingly larger diffusion region. The local dark
matter distribution at the Sun’s location appears quite smooth in the highest-
resolution numerical simulations to date (?, ?, ?). Tidal forces efficiently
strip matter from subhalos passing close to the Galactic Center and often
completely destroy them. Further substructure depletion may be expected from
interactions with the stellar disk and bulge. In the Via Lactea II simulation,
the mean number of $>10^{5}\,\,\rm M_{\odot}$ subhalos within 1 kpc of the Sun
is only 0.04, and one must reach three times farther to find one clump on
average. Without the Sommerfeld effect, this dearth of nearby substructure
leads to a local annihilation boost of less than 1%, and at most 20% in the
rare case of a nearby clump, as found in a statistical approach (?). The
picture changes with Sommerfeld enhancement. The low velocity dispersion of
cold substructure leads to a greatly increased luminosity compared to the
hotter smooth component. For typical $1/v$ models, subhalos resolved in our
simulation within 2 kpc contribute on average about half as much luminosity as
the smooth component, and up to 5 times as much in rare cases. If the
Sommerfeld enhancement is resonant ($S\sim 1/v^{2}$), then these subhalos
dominate by a factor of 20 on average and by as much as 200 in rare cases.
## References and Notes
* 1. G. Jungman, M. Kamionkowski, K. Griest, Phys. Rep. 267, 195 (1996)
* 2. A. Klypin, A. V. Kravtsov, O. Valenzuela, F. Prada, Astrophys. J. 522, 82 (1999)
* 3. B. Moore, S. Ghigna, F. Governato, G. Lake, T. Quinn, J. Stadel, P. Tozzi, Astrophys. J. Lett. 524, 19 (1999)
* 4. J. Diemand, M. Kuhlen, P. Madau, Astrophys. J. 657, 262 (2007)
* 5. J. Silk, M. Srednicki, Phys. Rev. Lett. 53, 624 (1984)
* 6. M. Kuhlen, J. Diemand, P. Madau, Astrophys. J. 686, 262 (2008)
* 7. L. Pieri, G. Bertone, E. Branchini, Mon. Not. R. Astron. Soc. 384, 1627 (2008)
* 8. J. Hisano, S. Matsumoto, M. M. Nojiri, Phys. Rev. Lett. 92, 031303 (2004)
* 9. J. March-Russell, S. M. West, Phys. Lett. B 676, 133 (2009)
* 10. M. Lattanzi, J. Silk, Phys. Rev. D 79, 083523 (2009)
* 11. N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, N. Weiner, Phys. Rev. D 79, 015014 (2009)
* 12. O. Adriani, et al., submitted to Nature (available at http://arxiv.org/abs/0810.4995) (2008)
* 13. J. Chang, et al., Nature 456, 362 (2008)
* 14. S. Torii, et al., submitted to Astropart. Physics (available at http://arxiv.org/abs/0809.0760) (2008)
* 15. A. A. Abdo, et al., Phys. Rev. Lett., 102, 181101 (2009)
* 16. H.E.S.S. Collaboration: F.Aharonian, et al., preprint (available at http://arxiv.org/abs/0905.0105) (2009)
* 17. S. Profumo, preprint (available at http://arxiv.org/abs/0812.4457) (2009)
* 18. N. J. Shaviv, E. Nakar, T. Piran, preprint (available at http://arxiv.org/abs/0902.0376) (2009)
* 19. D. Malyshev, I. Cholis, J. Gelfand, submitted to Phys. Rev. D (available at http://arxiv.org/abs/0903.1310) (2009)
* 20. L. Bergström, J. Edsjö, G. Zaharijas, preprint (available at http://arxiv.org/abs/0905.0333) (2009)
* 21. P. Meade, M. Papucci, A. Strumia, T. Volansky, preprint (available at http://arxiv.org/abs/0905.0480) (2009)
* 22. M. Kuhlen, D. Malyshev, Phys. Rev. D 79, 123517 (2009)
* 23. M. Cirelli, M. Kadastik, M. Raidal, A. Strumia, Nucl. Phys. B 813, 1 (2009)
* 24. The “dark sector” of our universe consists of dark matter, dark energy, and possibly additional dark forces.
* 25. B. E. Robertson, A. R. Zentner, Phys. Rev. D, 79, 083525 (2009)
* 26. J. Bovy, Phys. Rev. D 79, 083539 (2009)
* 27. J. Diemand, M. Kuhlen, P. Madau, M. Zemp, B. Moore, D. Potter, J. Stadel, Nature 454, 735 (2008)
* 28. P. Madau, M. Kuhlen, J. Diemand, B. Moore, M. Zemp, D. Potter, J. Stadel, Astrophys. J. Lett. 689, 41 (2008)
* 29. M. Zemp, J. Diemand, P. Madau, M. Kuhlen, B. Moore, D. Potter, J. Stadel, L. Widrow, Mon. Not. R. Astron. Soc. 394, 641 (2009)
* 30. Even at the extremely high resolution of Via Lactea II the cuspy subhalo centers are often not sufficiently well resolved to give accurate central surface brightnesses. We corrected the centers according to an analytical model that has been calibrated to our numerical simulations (see Supporting Online Material).
* 31. P. Meade, M. Papucci, T. Volansky, preprint (available at http://arxiv.org/abs/0901.2925) (2009)
* 32. F. Aharonian, et al., Phys. Rev. Lett. 97, 221102 (2006)
* 33. F. Aharonian, et al., Nature 439, 695 (2006)
* 34. M. Vogelsberger, et al., Mon. Not. R. Astron. Soc. 395, 797 (2009)
* 35. J. Lavalle, Q. Yuan, D. Maurin, X. Bi, Astron. Astrophys. 479, 427 (2008)
* 1.
Support for this work was provided by NASA through grant NNX08AV68G (P.M.) and
by the William L. Loughlin Fellowship at the Institute for Advanced Study
(M.K.). This work would not have been possible without the expertise and
invaluable contributions of all the members of the Via Lactea Project team. We
thank M. Lattanzi, S. Profumo, N. Arkani-Hamed, N. Weiner, P. Meade and T.
Volansky for enlightening discussions, D. Shih for help with the Sommerfeld
enhancement calculations, and M. Papucci for providing $\gamma$-ray spectra.
Supporting Online Material
www.sciencemag.org
Supporting text
Figs. S1, S2, S3, S4, S5, S6
Supporting Online Material
“Exploring Dark Matter with Milky Way substructure”
Kuhlen, Madau, & Silk
## S1 Sommerfeld enhancement
Figure S1: A: The Sommerfeld enhancement factor $S$ as a function of
$m_{\chi}/m_{\phi}$ (the mass ratio of the dark matter particle to the force
carrier particle) at a fixed coupling ($\alpha=0.30$) for different
velocities. B and C: $S$ as a function of velocity for the models for which we
calculate subhalo detectability with Fermi. The parameters of the models are
given in Table 1 in the main text.
A Sommerfeld enhancement to the annihilation cross-section arises when the
dark matter (DM) particle is heavy compared to the gauge boson mediating the
interaction: $m_{\chi}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}m_{\phi}/\alpha$,
where $\alpha=\lambda^{2}/4\pi$, and $\lambda$ is the coupling between the
dark matter particle $\chi$ and the force carrier $\phi$. The magnitude of the
enhancement can be determined by solving the two-body radial Schrödinger
equation, as shown in (?, ?). Defining the additional dimensionless parameter
$\beta^{*}=\sqrt{\frac{\alpha m_{\phi}}{m_{\chi}}},$ (1)
one can distinguish three regimes for the dependence of the enhancement $S$ as
a function of the velocity (?):
* i)
for large velocities, $\beta\equiv v/c\gg\alpha$, there is no Sommerfeld
enhancement, $S\sim 1$. This ensures that the relic abundance of dark matter
is not affected, since the velocities were close to relativistic at freeze-
out;
* ii)
at intermediate velocities, $\beta^{*}\ll\beta\ll\alpha$, the Sommerfeld
enhancement follows a $1/v$ behavior, $S\approx\pi\alpha/\beta$;
* iii)
at low velocities, $\beta\ll\beta^{*}$, resonances appear for certain values
of $m_{\chi}$, due to the presence of bound states. In this case $S$ grows as
$1/v^{2}$ before saturating at a velocity that depends on how close to
resonance $m_{\chi}$ lies. In the non-resonant case the Sommerfeld enhancement
saturates when the deBroglie wavelength of the particle $\sim(m_{\chi}v)^{-1}$
becomes comparable to the range of the interaction $\sim m_{\phi}^{-1}$), i.e.
when $v\sim m_{\phi}/m_{\chi}$.
Figure S1 shows the rich behavior of the Sommerfeld enhancement, obtained by
numerical integration of the Schrödinger equation with a Yukawa potential.
The magnitude of the Sommerfeld enhancement at a given location depends on the
details of the distribution of relative velocities of the DM particles.
Accurately determining the phase-space structure as a function of position is
a notoriously difficult task even with the highest resolution N-body
simulations. Some recent investigations based on cosmological N-body
simulations have found significant structure in the coarse-grained velocity
distribution of the host halo and departures from a simple,
single-”temperature” Maxwell-Boltzmann distribution (?, ?). These features are
remnants of the accretion history of the host halo due to incomplete phase
mixing. The centers of subhalos, however, are likely more well-mixed and may
show less significant departures from a Maxwellian. At any rate, a full
characterization of the phase-space distribution at all locations in our
simulation is beyond the scope of this paper, and hence we make the
simplifying assumption that the distribution of relative velocities is of the
Maxwell-Boltzmann form
$f(v_{\rm rel};\sigma_{\mu,\rm 1D})=4\pi\,\frac{1}{(2\pi\sigma_{\mu,\rm
1D})^{3/2}}\,v_{\rm rel}^{2}\,\exp\left[-\frac{1}{2}\left(\frac{v_{\rm
rel}}{\sigma_{\mu,\rm 1D}}\right)^{2}\right],$ (2)
with a one-dimensional velocity dispersion
$\sigma_{\mu,\rm
1D}^{2}\equiv\frac{k\,T}{\mu}=2\frac{k\,T}{m}=2\,\sigma_{m,\rm 1D}^{2},$ (3)
where $\mu=m_{1}m_{2}/(m_{1}+m_{2})=m/2$ is the reduced mass of a two particle
system. From the simulated particles we determine a three-dimensional velocity
dispersion, $\sigma^{2}\equiv\sigma_{m,\rm 3D}^{2}=3\,\sigma_{m,\rm
1D}^{2}=3/2\,\sigma_{\mu,\rm 1D}^{2}$ (for systems with zero velocity
anisotropy). The Maxwell-Boltzmann-weighted Sommerfeld enhancement is then
given by
$S(\sigma)=\int_{0}^{\infty}f(v;\sqrt{2/3}\,\sigma)\;S(v)\;dv.$ (4)
## S2 Gamma-rays from dark matter annihilation
Figure S2: The $\gamma$-ray spectrum per annihilation for the models under
consideration here. In the LS models ($\chi\chi\rightarrow ZZ\;{\rm
or}\;W^{+}W^{-}$) the $\gamma$-rays come from the decay of pions produced in
the decay of the bosons. The dotted line indicates the internal bremsstrahlung
contribution in the case of $W^{+}W^{-}$ (?). In the MPV models
($\chi\chi\rightarrow\phi\phi$) the $\gamma$-rays originate as final state
radiation associated with the decay of the $\phi$ carriers into $e^{+}e^{-}$
pairs.
Since the dark matter particle is neutral it does not couple directly to the
electromagnetic field, and hence annihilations straight into two monochromatic
photons (or a photon and a Z boson) are typically strongly suppressed.
Nevertheless $\gamma$-rays can be a significant by-product of dark matter
annihilations, since they can arise either from the decay of neutral pions
produced in the hadronization of the annihilation products, or through
internal bremsstrahlung associated with annihilations into charged particles,
or from interactions of energetic leptons with the surrounding interstellar
photons (inverse Compton scattering). We do not consider the latter process
here, since we are focusing on the annihilation signal from dark matter
subhalos which are unlikely to harbor a sufficiently high stellar radiation
field.
The $\gamma$-ray spectra per annihilation that we use in our detectability
calculation are shown in Figure S2. In the Lattanzi & Silk models the
annihilation results in two neutral $Z$ bosons or a pair of $W^{+}$ and
$W^{-}$ bosons, and the dominant source of $\gamma$-rays is neutral pion
decay. For $m_{\chi}=4.5$ TeV, every annihilation results in $\sim 26$ photons
with energies between 3 and 300 GeV. In the MPV models, the mass of the $\phi$
particle is so low (by design), that only decays into $e^{+}e^{-}$ pairs are
kinematically allowed, and we must rely on final state radiation (internal
bremsstrahlung) for the $\gamma$-ray signal. This results in fewer 3-300 GeV
$\gamma$-rays per annihilation (0.39 for $m_{\chi}=1$ TeV, 0.30 for
$m_{\chi}=250$ GeV), but this is partially compensated by the smaller dark
matter particle mass and hence higher number density at fixed mass density.
## S3 The velocity structure of the Via Lactea II host and its subhalos
Figure S3: Left panel: The radial dependence of the density (A), velocity
dispersion (B), and the velocity anisotropy
$\beta=1-\frac{1}{2}\frac{\sigma_{\theta}^{2}+\sigma_{\phi}^{2}}{\sigma_{r}^{2}}$
(C) for the smooth host halo component. The solid red line shows the values
calculated from all particles in spherical shells. For the $\rho$ and $\sigma$
profiles we also show the median (dark green dashed line) and the 68% region
(light green shaded region) of the particle-by-particle quantities determined
from the nearest 32 neighbors. The vertical dashed line indicates our estimate
for the convergence radius of the density profile (380 pc). The shape of the
$\sigma$ profile implies that Sommerfeld enhancement will preferentially
brighten the very central region of the host halo at the expense of the
surrounding region. The anisotropy profile clearly shows that the host halo is
not isotropic, with a slight radial anisotropy persisting down to the
convergence radius. Right panel (D-F): The same quantities averaged over the
100 most massive subhalos. The bullets indicate the median and the error bars
the 68% scatter around the median. In D and E we also plot the best-fitting
NFW (solid) and Einasto ($\alpha$ fixed at 0.17, dotted) profiles. Note that
the subhalo profiles should not be considered converged below $\sim
0.1\;r_{\rm Vmax}$.
In Figure S3 we present radial profiles of the density $\rho$, velocity
dispersion $\sigma$, and velocity anisotropy parameter $\beta$ for the smooth
host halo and averaged over the 100 most massive subhalos. We determine these
profiles by first binning all particles into equally spaced logarithmic radial
shells, and then calculating
$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}m_{i}}{4\pi r^{2}dr},$ (5)
$\displaystyle\sigma_{j}^{2}(r)$ $\displaystyle=$
$\displaystyle\langle(v_{j}-\langle v_{j}\rangle_{i})^{2}\rangle_{i},$ (6)
$\displaystyle\sigma^{2}(r)$ $\displaystyle=$
$\displaystyle\sigma_{x}^{2}(r)+\sigma_{y}^{2}(r)+\sigma_{z}^{2}(r),$ (7)
$\displaystyle\beta(r)$ $\displaystyle=$ $\displaystyle
1-\frac{1}{2}\frac{\sigma_{\theta}^{2}(r)+\sigma_{\phi}^{2}(r)}{\sigma_{r}^{2}(r)},$
(8)
where the sum and averages (denoted by $\left<\right>_{i}$) are over all
particles in a given spherical shell. These profiles are indicated by the
solid red line for the host halo in the left panels of Figure S3. For the
$\rho(r)$ and $\sigma(r)$ profiles we also show the median and 68% interval of
the distribution of particle density $\rho_{i}$ and velocity dispersion
$\sigma_{i}$, both calculated from the 32 nearest neighboring particles. The
two estimates agree quite well with each other, with the slightly higher
(lower) median $\rho_{i}$ ($\sigma_{i}$) indicating a negative (positive) skew
of the distribution, presumably due to spherical averaging of a triaxial mass
distribution. Since 32 particles are not sufficient for a good estimate of
$\beta$, we don’t show the distributions for the velocity anisotropy.
The density profile of the Via Lactea II host has been discussed in (?) and we
present it here merely for completeness. Down to our convergence radius of
$380$ pc it is well fit by a generalized NFW profile,
$\rho(r)=\frac{\rho_{s}\;2^{3-\gamma}}{(r/r_{s})^{\gamma}(1+r/r_{s})^{3-\gamma}},$
(9)
with a central slope of $\gamma=1.2$, but an Einasto profile,
$\rho(r)=\rho_{s}\exp{\left[-\frac{\alpha}{2}\left((r/r_{s})^{\alpha}-1\right)\right]},$
(10)
with $\alpha=0.167$ fits almost as well. The velocity dispersion profile
exhibits the central “temperature inversion” typical of cold dark matter
halos: $\sigma(r)$ peaks at about the scale radius and decreases with
decreasing radius (?, ?, ?). The $\beta(r)$ profile shows the well established
trend of a considerable amount of radial anisotropy in the outskirts of the
halo and decreasing towards the center (?, ?, ?). We find that even at the
convergence radius a slight amount of radial anisotropy remains.
In the right panels of Figure S3 we show the $\rho$, $\sigma$, and $\beta$
profiles averaged over the 100 most massive subhalos in our simulations. We
scaled the radius of each subhalo by its $r_{\rm Vmax}$ and calculated radial
profiles using 30 equally spaced logarithmic bins from $r/r_{\rm Vmax}=0.01$
to 2. In each radial bin we then determined the median and 68% region of the
distribution of values over all 100 subhalos, rejecting bins containing less
than 100 particles. We only plotted bins containing values from more than 10
subhalos. It is difficult to estimate a convergence radius for these subhalos;
the host halo convergence radius of 380 pc corresponds to (0.05 - 0.5) $r_{\rm
Vmax}$ for these 100 subhalos, so these average profiles should not be
considered converged below $\sim 0.1r_{\rm Vmax}$.
The median density profile nicely follows the anticipated NFW-like profile. We
have overplotted the best-fitting NFW (solid line) and Einasto (with fixed
$\alpha=0.17$, dotted line) profiles. Clearly the resolution is not good
enough to allow a quantitative analysis of the asymptotic central slope of the
density profile, but it remains cuspy as far down as we can resolve. The
velocity dispersion profile looks qualitatively the same as the host halo’s,
with a peak around $0.25r_{\rm Vmax}$ and decreasing towards smaller radii.
Not surprisingly, subhalos are not isothermal. The strong increase in the
$84^{\rm th}$ percentile of $\sigma$ at large radii is due to contamination by
host halo particles which artificially inflate $\sigma$. Although the $\beta$
profile is quite noisy, it is clear that subhalos too exhibit a slight radial
anisotropy even down at the smallest radii that we can access.
An important caveat to these findings is that our simulations completely
neglect the effects of baryons on the DM distribution. Gas cooling, star
formation, supernova feedback, and stellar dynamical processes might alter
both the DM density and velocity dispersion profiles. In fact, not even the
sign of these effects is clear at the moment: adiabatic contraction generally
leads to a steepening of the central density profile (?, ?), but dynamical
friction acting on baryonic condensations tends to remove the central cusp (?,
?). The velocity dispersion of DM particles, on the other hand, is more likely
to increase in regions affected by baryonic processes. These complications
will be most important for the Galactic Center. The high mass-to-light ratios
observed in Galactic dwarf satellites (?) indicate that they are completely DM
dominated and hence likely much less affected by baryonic physics.
Interactions with the Milky Way’s stellar and gaseous disk probably strip a
significant fraction of mass from some DM subhalos, but the dense central
regions responsible for most of the annihilation luminosity are relatively
well protected.
## S4 Detectability Calculation
We calculated the annihilation flux directly from our simulations following
the procedure detailed in (?), with one important modification to account for
the Sommerfeld enhancement. The intensity in a given direction $(\theta,\phi)$
is now given by
$\mathcal{I}(\theta,\phi)=\mathcal{G}\int_{\rm
los}\rho^{2}S(\sigma;m_{\chi},m_{\phi},\alpha)d\ell,$ (11)
where $\mathcal{G}$ contains most of the particle physics dependence (the
particle mass, the high velocity annihilation cross section, and the
$\gamma$-ray spectrum per annihilation event) and
$S(\sigma;m_{\chi},m_{\phi},\alpha)$ is the Maxwell-Boltzmann-weighted
Sommerfeld enhancement factor at a velocity dispersion $\sigma$ for a given
particle physics model. For discrete particles, with masses $m_{i}$, distances
$d_{i}$, densities $\rho_{i}$, and velocity dispersions $\sigma_{i}$, this
integral becomes a discrete sum over all particles in a given map pixel,
$\sum_{i}\frac{\rho_{i}S(\sigma_{i})m_{i}}{4\pi d_{i}^{2}}.$ (12)
We determined $\rho_{i}$ and $\sigma_{i}$ from the nearest 32 neighbors of the
$i$th particle, but have checked that our results do not change significantly
for 64 neighbors. We have implemented several additional improvements over
(?): (a) we corrected a calculation error in the conversion from simulation
fluxes to gamma-ray counts (the subhalo fluxes were a factor of two too large
in (?)), (b) we use a 15% lower effective area ($\sim$ 7,300 cm2), as
suggested by the performance of the LAT instrument measured in orbit (?), and
(c) we switched to the HEALPix222http://healpix.jpl.nasa.gov/ equal area
pixelization scheme, setting $N_{\rm side}=512$, which corresponds to a solid
angle per pixel of $\Delta\Omega=4\times 10^{-6}$ sterad, comparable to the
angular resolution of Fermi’s LAT detector above 3 GeV. We assumed a LAT
exposure time equal to 0.153 of the time in orbit, a combination of the $\sim
4\pi/5$ sr field of view, 90% trigger live time, and 15% data acquisition
outage during South Atlantic Anomaly passages (?). Our analysis was restricted
to one fiducial observer located at 8 kpc from the host halo center along the
intermediate axis of its density ellipsoid, and we refer the reader to (?) for
a discussion of the signal variance arising from different observer locations.
Due to the finite resolution of our simulation the centers of all our halos
are artificially heated and less dense than they would be at higher
resolution. This results in central brightnesses that are lower than would be
expected for an NFW or Einasto density profile. We have corrected the central
flux from the host halo and all subhalos using an analytical extrapolation of
the density and velocity dispersion profiles. We considered both an NFW
($\gamma=1$) profile and an Einasto profile with $\alpha=0.17$, which has been
shown to fit Galactic-scale dark matter halos (?). These analytical profiles
are matched to the measured values of $V_{\rm max}$ and $r_{\rm Vmax}$, which
are robustly determined for subhalos with more than 200 particles ($M>8\times
10^{5}\,\rm M_{\odot}$). The relations between $(V_{\rm max},r_{\rm Vmax})$
and $(\rho_{s},r_{s})$ are
$r_{\rm Vmax}=f_{r}\;r_{s}\qquad V_{\rm max}^{2}=f_{V}\;4\pi
G\rho_{s}r_{s}^{2},$ (13)
with $f_{r}\approx 2.163\;(2.212)$ and $f_{V}\approx 0.865\;(0.897)$ for the
NFW (Einasto) profile. We use the spherical Jeans equation to solve for the
corresponding velocity dispersion profile, assuming $\beta=0$. For a halo at a
given distance we can then solve for a Sommerfeld-enhanced surface brightness
profile as a function of angle from the halo center, average it over the
angular resolution of our maps ($\Delta\Omega=4\times 10^{-6}$ sterad), and
use this to correct our simulated maps. For the host halo we only correct
pixels within $\sim 2.7^{\circ}$ from the center, corresponding to the density
profile convergence radius of 380 pc. For the subhalos we ensure that all
pixels within the projected scale radius $r_{s}$ have a surface brightness at
least as high as the expectation from the analytical extrapolation. These
correction factors are typically not very large: over all subhalos and all
Sommerfeld models, the median (root mean square) correction factor for the
central pixel is 2.2 (5.0).
Note that we neglect the possible enhancement of a subhalo’s luminosity
arising from additional clumpy substructure below our simulation’s resolution
limit. The magnitude of this so-called “substructure boost factor” depends on
uncertain extrapolations of the abundance, distribution, and internal
properties of the low mass subhalos, and will not be significantly increased
by the Sommerfeld effect due to its saturation at low velocities. While this
boost typically doesn’t affect the central surface brightness very much, it
may somewhat increase the angular extent of a given subhalo’s signal.
The annihilation signal from individual subhalos must compete with a number of
diffuse $\gamma$-ray backgrounds, of both astrophysical and DM annihilation
origin. At low Galactic latitudes the dominant astrophysical background arises
arises from the interaction of high energy cosmic rays with interstellar gas
(pion decay and bremsstrahlung) and radiation fields (inverse Compton). This
background has been measured by the EGRET instrument aboard the Compton
satellite at 0.5 degree resolution out to $\sim 30$ GeV (?). An improved
measurement of the spectral and angular properties of this background is one
of the goals of the Fermi mission, and preliminary data at intermediate
Galactic latitudes have already been presented by the Fermi collaboration (?).
Here we employ a theoretical model of this background (the GALPROP
“conventional” model (?, ?)), which matches the EGRET and preliminary Fermi
measurements with sufficient accuracy for our purposes. At high Galactic
latitudes ($|b|\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}20^{\circ}$), an
isotropic extragalactic $\gamma$-ray background from unresolved blazars may
become dominant. We include such a background with an intensity and spectrum
as measured by EGRET (?). Note that this is probably an overestimate, since
Fermi will likely resolve some fraction of this background into point sources.
We do not include the contributions from astrophysical foreground and
background point sources, but we have checked that none of the Sommerfeld-
enhanced subhalos would have been detected by EGRET, assuming a point-source
sensitivity of $2\times 10^{-7}\gamma$ cm2 s-1 from 0.1 to 30 GeV (?).
In addition to these astrophysical backgrounds, individually detectable
subhalos must compete with the diffuse background from DM annihilation in the
smooth host halo and from the population of individually undetectable subhalos
(see Section S5) (?, ?). In models without Sommerfeld enhancement and for
$S\sim 1/v$ models, these DM annihilation backgrounds are negligible compared
to the astrophysical backgrounds (although they themselves constitute a signal
worth searching for). In resonant $S\sim 1/v^{2}$ models with low saturation
velocity (e.g. LS-3, LS-4, MPV-1b, MPV-2b), however, the unresolved subhalo
flux becomes the dominant background and must be accounted for. For
completeness we have included all four components in the detectability
calculation in all cases.
Figure S4: Cumulative distribution of the angular size of the detectable
subhalos. We plot the fraction of S/N$>5$ subhalos with more than $N_{\rm
pix}$ pixels exceeding the Fermi detection threshold after 5 years in orbit,
for the LS models in panel A, and for the MPV models in panel B.
Subhalo detectability for Fermi is assessed as follows. First we calculate a
signal-to-noise ratio ($S/N$) per pixel by dividing the number of source
photons arising in the subhalo map by the square root of the number of photons
in the background map. Next we select all pixels with $S/N>1$ and identify
contiguous regions, which we associate with individual subhalos based on
proximity of the brightest pixel with a subhalo center. If more than one
subhalo center coincides with a given brightest pixel, we pick the subhalo
with the larger expected surface brightness ($L/r_{s}^{2}\sim V_{\rm
max}^{4}/r_{\rm Vmax}^{3}$). For each of these contiguous regions we then
calculate a subhalo detection significance
$S/N=\frac{N_{s}}{\sqrt{N_{b}}},$ (14)
where $N_{s}$ and $N_{b}$ are the total number of source and background
$\gamma$-rays over the contiguous region. This definition is a good proxy for
detection significance under the assumption that an estimate of the background
can be subtracted out and only Poisson fluctuations remain. Note that it is
not the uncertainty of the flux itself (which would be
$N_{s}/\sqrt{N_{s}+N_{b}}$), but instead an estimate of the significance of
having detected a departure from a smooth background.
In Figure S4 we show the cumulative distribution of the angular size (the
number of pixels exceeding the Fermi detection threshold after 5 years in
orbit) of the detectable subhalos. Although models with stronger Sommerfeld
enhancement result in smaller detectable regions, the majority of all subhalos
would still be resolved sources for Fermi. This effect raises the possibility
of future observations being able to discriminate between different
Sommerfeld-enhanced models.
## S5 Diffuse Flux from Unresolved Subhalos
Figure S5: The annihilation intensity as a function of angle $\psi$ from the
Galactic Center, for the host halo (red), individual subhalos (blue), and
unresolved subhalos (magenta dashed). A: No Sommerfeld enhancement. B: Model
MPV-2a ($S\sim 1/v$, high $v_{\rm sat}$). C: Model LS-4 ($S\sim 1/v^{2}$, low
$v_{\rm sat}$). For the host halo and unresolved components we plot the mean,
for the individual subhalo profile the maximum intensity over all pixels in a
given $\psi$ bin. Individual subhalos outshine the diffuse unresolved subhalo
background, even in the strongly Sommerfeld-enhanced case.
For a typical CDM power spectrum of density fluctuations one would expect dark
matter clumps on scales all the way down to a cutoff set by collisional
damping and free streaming in the early universe (?, ?). For WIMP dark matter,
typical cut-off masses are $m_{0}=10^{-12}$ to $10^{-4}\,\rm M_{\odot}$ (?,
?), some 10 to 20 orders of magnitude below Via Lactea II’s mass resolution.
In this case the Galactic dark matter halo might host an enormous number
($10^{10}-10^{22}$, see (?)) of small mass subhalos, whose combined
annihilation signal could result in a sizeable $\gamma$-ray background. If the
Sommerfeld enhancement didn’t saturate at a finite velocity, this background
would easily outshine any other Galactic $\gamma$-ray signal. Even with
saturation one must ask whether the Sommerfeld-enhanced annihilation
background from this population of unresolved subhalos would outshine
individual subhalos.
In order to address this question, we have extended the analytical model of
(?) to allow for Sommerfeld enhancement. The overall intensity of this
background depends sensitively on a number of uncertain parameters governing
the subhalo population as a whole (slope and normalization of the mass
function, their dependence on distance to the host center) and the mass
dependent properties of individual subhalos (density and velocity dispersion
profiles, concentrations). Although the model is calibrated to numerical
simulations at the high mass end, it relies on an extrapolation over many
orders of magnitude in mass below the simulation’s resolution limit that is
very uncertain.
Our model employs a radially anti-biased subhalo mass function
$\frac{dn(M,r)}{dM}=1.05\times 10^{-8}\,\rm M_{\odot}^{-1}\;{\rm
kpc}^{-3}\left(\frac{M}{10^{6}\,\rm
M_{\odot}}\right)^{-2}\left(1+\frac{r}{18.5\;{\rm kpc}}\right)^{-2}$ (15)
with a low mass cutoff of $m_{0}=10^{-6}\,\rm M_{\odot}$, and assumes an
Einasto density and velocity dispersion profile with a concentration-mass
relation according to (?) and a radial dependence of
$c(M,r)=c^{\rm B01}(M)\left(\frac{r}{400\;{\rm kpc}}\right)^{-0.286}.$ (16)
We refer the reader to the Appendix of (?) for details about the calculation.
In Figure S5 we show the resulting azimuthally averaged intensity as a
function of angle from the Galactic Center, and compare it to the smooth host
halo signal and the peaks due to individual subhalos. While the unresolved
subhalo background is brighter than the smooth halo component everywhere but
in the very center, individual subhalos have higher central surface
brightnesses and can easily outshine it. This holds true in all Sommerfeld
models that we have considered.
## S6 Local Luminosity Boost From Substructure
Figure S6: The effect of substructure on the local annihilation rate. A: The
mean and maximum number of Via Lactea II subhalos inside 100 randomly placed
spheres 8 kpc from the halo center versus the radius of these sample spheres.
The mean subhalo occupancy becomes unity at $r=3$ kpc. B: The density
“clumping factor” $\langle\rho^{2}\rangle/\langle\rho\rangle^{2}$ over the 100
sample spheres. C: The ratio of the subhalo to host halo contributions to the
annihilation luminosity for three representative Sommerfeld models (none,
$1/v$, and $1/v^{2}$). Only subhalos resolved in our simulation are accounted
for.
Here we assess the role that nearby subhalos play for the Sommerfeld-enhanced
production of high energy electrons and positrons. Because these energetic
particles lose energy as they diffuse through the Galactic magnetic field,
only those that are produced within a few kpc of Earth are of interest.
We considered 100 spheres of radius 20 kpc, with randomly positioned centers 8
kpc from the host halo center. For each of these spheres we determined the
cumulative number of subhalos $N_{\rm sub}$, a “clumping factor”, defined as
$\langle\rho^{2}\rangle/\langle\rho\rangle^{2}$, and the ratio of total
subhalo to host halo luminosity as a function of enclosed radius in the
sphere. For the subhalo luminosity we used the analytical NFW estimate, as
explained in Section S4. Figure S6 shows the mean and maximum values of these
quantities over all 100 sample spheres. The low local subhalo abundance is
reflected in a small mean $N_{\rm sub}$ within a few kpc of the Sun. Only 3 of
the 100 sample spheres have any subhalos within 1 kpc of their center. The
mean subhalo occupancy becomes unity at 3 kpc, but one has to go out to 7 kpc
before every single sphere contains at least one subhalo. The clumping factor
captures the enhancement of the annihilation luminosity compared to a
homogeneous density background. It has contributions from the overall density
stratification, from the Poisson noise of the density estimator, and from
unbound and bound substructure. The sharp rise towards 8kpc is due to the
cuspy nature of the host halo density profile. The bottom panel of Figure S6
shows that without Sommerfeld enhancements the subhalos do not contribute
significantly to the local annihilation luminosity. In a typical $S\sim 1/v$
model, however, subhalos contribute on average about half as much as the host
halo, and in rare cases 5 times more. For models on resonance ($S\sim
1/v^{2}$), the subhalos completely dominate the host halo, and provide on
average 20 times as much luminosity as the host halo. Again we have neglected
the possible additional contribution from subhalos below our simulation’s
resolution limit.
## References and Notes
* 1. N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, N. Weiner, Phys. Rev. D 79, 015014 (2009)
* 2. M. Lattanzi, J. Silk, Phys. Rev. D 79, 083523 (2009)
* 3. M. Vogelsberger, et al., Mon. Not. R. Astron. Soc. 395, 797 (2009)
* 4. M. Zemp, J. Diemand, P. Madau, M. Kuhlen, B. Moore, D. Potter, J. Stadel, L. Widrow, Mon. Not. R. Astron. Soc. 394, 641 (2009)
* 5. T. Bringmann, L. Bergström, J. Edsjö, J. High Energy Phys. 1, 49 (2008)
* 6. J. Diemand, M. Kuhlen, P. Madau, M. Zemp, B. Moore, D. Potter, J. Stadel, Nature 454, 735 (2008)
* 7. J. Diemand, B. Moore, J. Stadel, Mon. Not. R. Astron. Soc. 352, 535 (2004)
* 8. M. Hoeft, J. P. Mücket, S. Gottlöber, Astrophys. J. 602, 162 (2004)
* 9. W. Dehnen, D. E. McLaughlin, Mon. Not. R. Astron. Soc. 363, 1057 (2005)
* 10. G. R. Blumenthal, S. M. Faber, R. Flores, J. R. Primack, Astrophys. J., 301, 27 (1986)
* 11. O. Y. Gnedin, A. V. Kravtsov, A. A. Klypin, & D. Nagai, Astrophys. J., 616, 16 (2004)
* 12. A. A. El-Zant, Y. Hoffman, J. R. Primack, F. Combes, I. Shlosman, Astrophys. J. Lett., 607, 75 (2004)
* 13. E. Romano-Díaz, I. Shlosman, Y. Hoffman, C. Heller, Astrophys. J. Lett. 685, 105 (2008)
* 14. J. D. Simon, M. Geha, Astrophys. J., 670, 313 (2007)
* 15. M. Kuhlen, J. Diemand, P. Madau, Astrophys. J. 671, 1135 (2007)
* 16. M. Kuhlen, J. Diemand, P. Madau, Astrophys. J. 686, 262 (2008)
* 17. R. Johnson, private communication
* 18. J. F. Navarro, et al., Mon. Not. R. Astron. Soc. 349, 1039 (2004)
* 19. S. D. Hunter, et al., Astrophys. J. 481, 205 (1997)
* 20. I. V. Moskalenko, A. W. Strong, talk at the ENTApP DM workshop 2009 (slides available at http://indico.cern.ch/materialDisplay.py?materialId=slides&confId=44160) (2009)
* 21. A. W. Strong, I. V. Moskalenko, Astrophys. J. 509, 212 (1998)
* 22. I. V. Moskalenko, A. W. Strong, J. F. Ormes, M. S. Potgieter, Astrophys. J. 565, 280 (2002)
* 23. P. Sreekumar, et al., Astrophys. J. 494, 523 (1998)
* 24. R. C. Hartman, et al., Astrophys. J. Suppl. 123, 79 (1999)
* 25. L. Pieri, G. Bertone, E. Branchini, Mon. Not. R. Astron. Soc. 384, 1627 (2008)
* 26. A. M. Green, S. Hofmann, D. J. Schwarz, J. Cosmol. Astropart. Phys. 8, 3 (2005)
* 27. A. Loeb, M. Zaldarriaga, Phys. Rev. D 71, 103520 (2005)
* 28. S. Profumo, K. Sigurdson, M. Kamionkowski, Phys. Rev. Lett. 97, 031301 (2006)
* 29. T. Bringmann, submitted to New Journal of Physics (available at http://arxiv.org/abs/0903.0189) (2009)
* 30. J. S. Bullock, et al., Mon. Not. R. Astron. Soc. 321, 559 (2001)
|
arxiv-papers
| 2009-06-30T20:00:07 |
2024-09-04T02:49:03.642557
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Kuhlen (IAS, Princeton), P. Madau (UC Santa Cruz), J. Silk (U. of\n Oxford)",
"submitter": "Michael Kuhlen",
"url": "https://arxiv.org/abs/0907.0005"
}
|
0907.0256
|
arxiv-papers
| 2009-07-01T22:50:30 |
2024-09-04T02:49:03.653275
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Scott Morrison",
"submitter": "Scott Morrison",
"url": "https://arxiv.org/abs/0907.0256"
}
|
|
0907.0357
|
# Coexistence of the antiferromagnetic and superconducting order and its
effect on spin dynamics in electron-doped high-$T_{c}$ cuprates
Cui-Ping Chen National Laboratory of Solid State of Microstructure and
Department of Physics, Nanjing University, Nanjing 210093, China Hong-Min
Jiang National Laboratory of Solid State of Microstructure and Department of
Physics, Nanjing University, Nanjing 210093, China Jian-Xin Li National
Laboratory of Solid State of Microstructure and Department of Physics, Nanjing
University, Nanjing 210093, China
###### Abstract
In the framework of the slave-boson approach to the
$t-t^{\prime}-t^{\prime\prime}-J$ model, it is found that for electron-doped
high-$T_{c}$ cuprates, the staggered antiferromagnetic (AF) order coexists
with superconducting (SC) order in a wide doping level ranged from underdoped
to nearly optimal doping at the mean-field level. In the coexisting phase, it
is revealed that the spin response is commensurate in a substantial frequency
range below a crossover frequency $\omega_{c}$ for all dopings considered, and
it switches to the incommensurate structure when the frequency is higher than
$\omega_{c}$. This result is in agreement with the experimental measurements.
Comparison of the spin response between the coexisting phase and the pure SC
phase with a $d_{x^{2}-y^{2}}$-wave pairing plus a higher harmonics term
(DP+HH) suggests that the inclusion of the two-band effect is important to
consistently account for both the dispersion of the spin response and the non-
monotonic gap behavior in the electron-doped cuprates.
###### pacs:
74.20.Mn, 74.25.Ha, 75.40.Gb
## I Introduction
The pairing symmetry of the hole-doped high-$T_{c}$ superconductors is
generally believed to have the dominant $d_{x^{2}-y^{2}}$-wave pairing.
However, the pairing symmetry of the electron-doped high-$T_{c}$
superconductors is still under debate. 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 While no
consensus has been reached yet, more and more recent experimental results have
suggested that the order parameter of electron-doped cuprates is likely to
have a dominant $d_{x^{2}-y^{2}}$-wave pairing symmetry, 3 ; 4 ; 5 ; 6 ; 7 ; 8
and with an unusual non-monotonic gap function.
Although various explanations have been proposed to account for the non-
monotonic behavior, they can generally be categorized into two scenarios. 5 ;
7 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 One is to extend the
superconducting (SC) gap out of the simple $d_{x^{2}-y^{2}}$-wave via the
inclusion of higher harmonics term. 5 ; 7 ; 9 ; 10 ; 11 ; 12 ; 13 From
theoretical perspective, the non-monotonic $d_{x^{2}-y^{2}}$-wave gap appears
under the assumption that the $d_{x^{2}-y^{2}}$-wave pairing is caused by the
interaction with the continuum of overdamped antiferromagnetic (AF) spin
fluctuations. In this scenario, the non-monotonic gap behavior is described by
the combination effect of a $d_{x^{2}-y^{2}}$-wave paring plus a higher
harmonics term (DP+HH). Therefore, it is an intrinsic property of the SC state
regardless of the presence of the AF order, and a simple one-band model is
used to reproduce the non-monotonic gap behavior. The other argues that the
non-monotonic behavior is the outcome of the coexistence of the AF and the SC
orders. 14 ; 15 ; 16 ; 17 ; 18 This scenario assumes that the AF order
disguises the $d_{x^{2}-y^{2}}$-wave character of SC gap. When the AF order is
introduced, the resulting quasiparticle (QP) excitation can be gapped by both
orders and behaves to be non-monotonic, although the SC gap itself is
monotonic. The scenario gained support from the angle-resolved photoemission
spectra (ARPES) measurements, where two inequivalent Fermi pockets around
$(\pi,0)$ and $(\pi/2,\pi/2)$ have been detected. 19 ; 20 This phenomena is
well explained in terms of the $\mathbf{k}$-dependent band-folding effect
associated with an AF order which splits the band into upper and lower
branches, 14 ; 20 ; 21 ; 22 leading to the two-band and/or two-gap model.
Recently, neutron scattering experiments in electron-doped cuprates have
revealed that the spin response is commensurate in a substantial frequency
range below a crossover frequency $\omega_{c}$ ,23 ; 24 ; 25 ; 26 ; 27 ; 28
which constitutes a distinct difference from the widely studied hour-glass
dispersion in the hole-doped cuprates.29 Although, both scenarios mentioned
above can account for the non-monotonic gap behavior of the electron-doped
cuprates, a comparative study on the spin dynamics between the two scenarios
is deserved to demonstrate the possible differences and therefore serve to
select the reasonable model for electron-doped high-$T_{c}$ cuprates.
In this paper, we investigate the spin dynamics in the coexisting phase of the
AF and the $d_{x^{2}-y^{2}}$-wave SC orders. The calculation is based on a
self-consistent determination of the QP dispersion, the AF order and the SC
gap at the slave-boson mean-field level of the
$t-t^{\prime}-t^{\prime\prime}-J$ model. It is shown that the AF and SC orders
compete and coexist in a substantial doping range in the underdoped regime.
The spin response is commensurate below a crossover frequency $\omega_{c}$ for
all dopings considered, and it becomes incommensurate when the frequency is
higher than $\omega_{c}$. This result is qualitatively consistent with
experiments. 23 ; 24 ; 25 ; 26 ; 27 ; 28 While in the framework of the pure
SC state with $d_{x^{2}-y^{2}}$-wave and/or DP+HH, 30 ; 31 ; 32 , though an
extended region of a commensurate spin fluctuation also exists, it evolves
into an incommensurate spin fluctuation at low frequencies, which is not
consistent experiments. Therefore, our result suggests that the inclusion of
the two-band effect resulting from the coexisting AF and SC orders is
important to consistently account for both the spin dynamics and the non-
monotonic gap behavior in the electron-doped cuprates.
The paper is organized as follows. In Sec. II, we introduce the theoretical
model and carry out the analytical calculations. In Sec. III, we present the
numerical results with some discussions. Finally, we present the conclusion in
Sec. IV.
## II THEORETICAL MODEL
The Hamiltonian of the two dimensional $t-t^{\prime}-t^{\prime\prime}-J$ model
on a square lattice is written in the form,
$\displaystyle H$ $\displaystyle=$
$\displaystyle-t\sum_{<ij>,\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+H.c.)-t_{1}\sum_{<ij>_{2},\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+H.c.)-t_{2}\sum_{<ij>_{3},\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+H.c.)$
(1)
$\displaystyle+J\sum_{<ij>}(\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\frac{1}{4}n_{i}n_{j})-\mu_{0}\sum_{<i>,\sigma}c^{\dagger}_{i\sigma}c_{i\sigma}.$
Where the summations $<ij>$, $<ij>_{2}$, $<ij>_{3}$ run over the nearest-
neighbor(n$\cdot$n), the next-n$\cdot$n, and the third-n$\cdot$n pairs
respectively, $\mathbf{S}_{i}$ is the spin on site $i$. This Hamiltonian can
be used to model both hole-doped and electron-doped systems after a particle-
hole transformation. For electron doping, one has $t$$<$$0$, $t_{1}$$>$$0$ and
$t_{2}$$<$$0$. The slave-boson mean-field theory (SBMFT) is used to decouple
the electron operators $c_{i\sigma}$ to bosons $b_{i}$ carrying the charge and
fermions $f_{i\sigma}$ representing the spin. Then, the local constraint
$b^{\dagger}_{i}b_{i}+\sum_{i\sigma}f^{\dagger}_{i\sigma}f_{i\sigma}$=$1$ is
satisfied averagely at the mean-field (MF) level. We choose the spinon pairing
order
$\Delta_{ij}$=$<f_{i\uparrow}f_{j\downarrow}-f_{i\downarrow}f_{j\uparrow}>$=$\pm\Delta$
, where $\Delta_{ij}=\Delta(-\Delta)$ for bond $<ij>$ along $x(y)$ direction,
the uniform bond order
$\chi_{ij}$=$\sum_{\sigma}<f^{\dagger}_{i\sigma}f_{j\sigma}>$=$\chi$, the AF
order
$<f_{i\uparrow}^{{\dagger}}f_{i\uparrow}-f_{i\downarrow}^{{\dagger}}f_{i\downarrow}>/2=(-1)^{i}m$,
and replace $b_{i}$ by $<b_{i}>$=$\sqrt{x}$ due to boson condensation. After
the Fourier transformation, the mean-field (MF) Hamiltonian can be written in
the Nambu representation,
$H=\sum_{\mathbf{k}}C^{{\dagger}}(\mathbf{k})\hat{A}(\mathbf{k})C(\mathbf{k})+2NJ(\chi^{2}+m^{2}+\Delta^{2}/2)-N\mu,$
(2)
here, the Nambu operator
$C^{{\dagger}}(\mathbf{k})=(f_{\mathbf{k}\uparrow}^{{\dagger}},f_{\mathbf{k+Q}\uparrow}^{{\dagger}},f_{\mathbf{-k}\downarrow},f_{\mathbf{-k-Q}\downarrow})$,
and
$\displaystyle\hat{A}(\mathbf{k})=\left(\begin{array}[]{c c c
c}{\epsilon_{\mathbf{k}}}&{-2Jm}&{-J\Delta_{\mathbf{k}}}&0\\\
{-2Jm}&{\epsilon_{\mathbf{k+Q}}}&0&{J\Delta_{\mathbf{k}}}\\\
{-J\Delta_{\mathbf{k}}}&0&{-\epsilon_{\mathbf{k}}}&{-2Jm}\\\
0&{J\Delta_{\mathbf{k}}}&{-2Jm}&{-\epsilon_{\mathbf{k+Q}}}\end{array}\right),$
(7)
where, $\epsilon_{\mathbf{k}}=(-2tx-J\chi)(\cos k_{x}+\cos k_{y})-4t_{1}x\cos
k_{x}\cos k_{y}-2t_{2}x(\cos 2k_{x}+\cos 2k_{y})-\mu$ and
$\Delta_{\mathbf{k}}=\Delta(\cos k_{x}-\cos k_{y})$. $\mu$ is the renormalized
chemical potential, $N$ is the total number of lattice sites, and
$\mathbf{Q}=(\pi,\pi)$ is the AF momentum. Note that the wave vector
$\mathbf{k}$ is restricted to the magnetic Brillouin zone (MBZ) in all
follows.
Diagonalizing of the Hamiltonian (2) by an unitary matrix
$\hat{U}(\mathbf{k})$ leads to four energy bands
$E_{1}(\mathbf{k})=E^{+}_{\mathbf{k}}$,
$E_{2}(\mathbf{k})=E^{-}_{\mathbf{k}}$,
$E_{3}(\mathbf{k})=-E^{-}_{\mathbf{k}}$,
$E_{4}(\mathbf{k})=-E^{+}_{\mathbf{k}}$, with
$E^{\pm}_{\mathbf{k}}=\sqrt{(\xi_{\mathbf{k}}^{\pm})^{2}+(J\Delta_{\mathbf{k}})^{2}},$
(8)
where
$\xi_{\mathbf{k}}^{\pm}=\epsilon^{+}_{\mathbf{k}}\pm\sqrt{(\epsilon^{-}_{\mathbf{k}})^{2}+4J^{2}m^{2}}$
with
$\epsilon^{\pm}_{\mathbf{k}}=(\epsilon_{\mathbf{k}}\pm\epsilon_{\mathbf{k+Q}})/2$.
And the free energy is written down (Boltzmann constant $k_{B}=1$),
$F=-2T\sum_{\mathbf{k},\nu=\pm}\ln[2\cosh(\frac{E_{\mathbf{k}}^{\nu}}{2T})]-\mu
N+2NJ(\chi^{2}+m^{2}+\Delta^{2}/2).$ (9)
The MF order parameters $\chi$, $\Delta$, $m$ and the chemical potential $\mu$
for different dopings $x$ can be calculated from the self-consistent equations
obtained by $\partial F/\partial\chi=0$, $\partial F/\partial\Delta=0$,
$\partial F/\partial m=0$, and $\partial F/\partial\mu=-N(1-x)$, respectively.
The magnitudes of the parameters are chosen as $t$=$-3.0J$, $t_{1}$=$1.02J$,
$t_{2}$=$-0.51J$ and $J$=$100$ meV to model the Fermi surface observed in
ARPES experiment.19 ; 20
Then, the bare spin susceptibility (transverse) is given by,
$\chi^{\pm}_{0}(\mathbf{q},\mathbf{q}^{{}^{\prime}},\tau)=\frac{1}{N}<S^{+}_{\mathbf{q}}(\tau)S^{-}_{-\mathbf{q}^{\prime}}(0)>_{(0)},$
(10)
where $<\cdots>_{(0)}$ means thermal average over the eigenstates of $H$,
$S^{+}_{\mathbf{q}}=\sum_{k}f^{+}_{\mathbf{k+q}\uparrow}f_{\mathbf{k}\downarrow}$
is the spin operator. Considering that $\mathbf{k}$ is restricted to the MBZ,
an explicit calculation shows that the spin susceptibility should be expressed
in the following matrix form,
$\displaystyle\
\hat{\chi}_{0}^{\pm}(\mathbf{q},\omega)=\left(\begin{array}[]{c
c}{\chi_{0}^{\pm}(\mathbf{q},\mathbf{q},\omega)}&{\chi_{0}^{\pm}(\mathbf{q},\mathbf{q+Q},\omega)}\\\
{\chi_{0}^{\pm}(\mathbf{q+Q},\mathbf{q},\omega)}&{\chi_{0}^{\pm}(\mathbf{q+Q},\mathbf{q+Q},\omega)}\end{array}\right),$
(13)
where, the nondiagonal correlation function $\chi_{0}^{\pm}$ with
$\mathbf{q}^{\prime}=\mathbf{q+Q}$ arises due to the umklapp process. The
matrix elements of the bare spin susceptibility, which come from the particle-
hole $(p-h)$ excitations, are given by,
$\displaystyle\chi_{0}^{\pm}(\mathbf{q},\omega)_{\eta\eta^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{i,j=1}^{2}\sum_{m,n=1}^{2}\sum_{\mathbf{k}}[a_{1}\frac{f(E_{m}(\mathbf{k}))-f(E_{n}(\mathbf{k+q}))}{\omega+E_{n}(\mathbf{k+q})-E_{m}(\mathbf{k})+i\Gamma}+a_{2}\frac{f(E_{n}(\mathbf{k+q}))-f(E_{m}(\mathbf{k}))}{\omega-
E_{n}(\mathbf{k+q})+E_{m}(\mathbf{k})+i\Gamma}$ (14)
$\displaystyle+b_{1}\frac{1-f(E_{m}(\mathbf{k}))-f(E_{n}(\mathbf{k+q}))}{\omega+E_{n}(\mathbf{k+q})+E_{m}(\mathbf{k})+i\Gamma}+b_{2}\frac{f(E_{m}(\mathbf{k}))+f(E_{n}(\mathbf{k+q}))-1}{\omega-
E_{n}(\mathbf{k+q})-E_{m}(\mathbf{k})+i\Gamma}],$
where, $f(E_{\mathbf{\mathbf{k}}})$ is the Fermi function and
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle
U_{in}^{*}(\mathbf{k+q})U_{(j+\eta^{\prime}-\eta)n}(\mathbf{k+q})U_{im}(\mathbf{k})U_{jm}^{*}(\mathbf{k})+U_{in}^{*}(\mathbf{k+q})U_{(j+2)n}(\mathbf{k+q})U_{im}(\mathbf{k})U_{(j+2+\eta^{\prime}-\eta)m}^{*}(\mathbf{k}),$
$\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle
U_{(i+2)n}(\mathbf{k+q})U_{(j+2+\eta^{\prime}-\eta)n}^{*}(\mathbf{k+q})U_{(i+2)m}^{*}(k)U_{(j+2)m}(\mathbf{k})+U_{(i+2)n}(\mathbf{k+q})U_{jn}^{*}(\mathbf{k+q})U_{(i+2)m}^{*}(\mathbf{k})U_{(j+\eta^{\prime}-\eta)m}(\mathbf{k}),$
$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle
U_{in}^{*}(\mathbf{k+q})U_{(j+\eta^{\prime}-\eta)n}(\mathbf{k+q})U_{(i+2)m}^{*}(\mathbf{k})U_{(j+2)m}(\mathbf{k})-U_{in}^{*}(\mathbf{k+q})U_{(j+2)n}(\mathbf{k+q})U_{(i+2)m}^{*}(\mathbf{k})U_{(j+\eta^{\prime}-\eta)m}(\mathbf{k}),$
$\displaystyle b_{2}$ $\displaystyle=$ $\displaystyle
U_{(i+2)n}(\mathbf{k+q})U_{(j+2+\eta^{\prime}-\eta)n}^{*}(\mathbf{k+q})U_{im}(\mathbf{k})U_{jm}^{*}(\mathbf{k})-U_{(i+2)n}(\mathbf{k+q})U_{jn}^{*}(\mathbf{k+q})U_{im}(\mathbf{k})U_{(j+2+\eta^{\prime}-\eta)m}^{*}(\mathbf{k}).$
(15)
The renormalized spin susceptibility due to the spin fluctuations is obtained
via the random-phase approximation (RPA),
$\hat{\chi}^{\pm}(\mathbf{q},\omega)=\frac{\hat{\chi}_{0}^{\pm}(\mathbf{q},\omega)}{\hat{1}+\alpha\hat{J}_{q}\hat{\chi}_{0}^{\pm}(\mathbf{q},\omega)},$
(16)
where,
$\displaystyle\hat{J}_{q}=\left(\begin{array}[]{c c}{J(\mathbf{q})}&{0}\\\
{0}&{J(\mathbf{q+Q})}\end{array}\right)$ (19)
with $J(\mathbf{q})=J(\cos q_{x}+\cos q_{y})$. In the coexisting phase of the
AF order and SC order, $\alpha$ is taken as 1. As for the pure SC state with
DP+HH, we choose a slightly small $\alpha=0.72$, the criteria for choosing
$\alpha$ is to set the AF instability at $x=0.12$. The parameter
$\Gamma$=$0.04J$ is introduced to account for the QP damping rate which comes
from the scattering off other fluctuations that are not included here.
## III NUMERICAL RESULTS AND DISCUSSION
In Fig. 1, we show the MF parameters $\chi$, $m$ and $\Delta$ as a function of
doping $x$. For a comparison, we also show the doping dependence of the MF SC
gap $\Delta_{1}$ obtained without considering the AF order by setting $m=0$.
It is seen that the staggered magnetization $m$ decreases with increasing
doping $x$, and goes sharply to zero at $x\approx 0.16$, which implies a phase
transition from the antiferromagnetism (AFM) phase to the paramagnetic phase.
The SC order parameter, on the other hand, increases its value initially up to
an optimal doping level, and then decreases upon further doping, forming a
generic SC dome. 33 However, the SC order parameter $\Delta_{1}$ without the
inclusion of the AF order exhibits a monotonic decrease with doping, which
deviates obviously from the experimental observations. Furthermore, the SC
order parameter $\Delta$ with an AF order shows a noticeable suppression
compared to $\Delta_{1}$, exhibits a competitive character with the AF order.
But, they also coexist in a substantial doping range. The MF phase diagram
also shows that the optimal doping is rather low compared to that deduced from
the experiments. This may be due to the fact the SBMFT includes only the MF
value of the order parameters and treats the no-double occupancy on the
average. However, the similarity of the phase diagram obtained by the SBMFT to
that of the variational quantum-cluster theory 16 ; 22 validates the SBMFT as
a low energy effective theory. Here our aim is to use the MF theory as an
effective model to study the effect of the AF order on the spin dynamics, and
then to compare the two-band and/or two-gap model with the simple one-band
model. Therefore, the relatively simple SBMFT is qualitatively enough for our
purpose. We note that a similar phase diagram has been obtained before. 15
The doping dependence of the renormalized spin susceptibility
Im$\chi(\mathbf{q},\omega)$ at a low frequency $\omega=0.04J$ in the
coexisting phase is presented in Fig. 2. In this figure, it is found that the
low-energy excitations exhibit commensurate peaks for all $x$, which consists
with the experiments well 27 . The inset shows the spin susceptibility
Im$\chi(\mathbf{q},\omega)$ at doping $x=0.15$ in the pure SC state with DP+HH
which is used to reproduce a non-monotonic SC gap behavior. One can see that
the spin response is incommensurate at low frequency without considering the
AF order.
Detailed frequency dependence of the spin response in the coexisting phase and
the pure SC phase with DP+HH at doping $x=0.15$ are shown in Figs. 3(a) and
3(b), and Figs. 3(c) and 3(d), respectively. The difference in the low
frequency regime of the two phases is more evident here. The spin fluctuation
is commensurate in a substantial frequency range below a crossover frequency
$\omega_{c}\approx 0.52J$ and down to the lowest frequency considered in the
coexisting phase, and it switches to be incommensurate when the frequency is
higher than $\omega_{c}\approx 0.52J$ [Figs. 3(a) and 3(b)]. This feature
agrees with the neutron-scattering measurements on electron-doped cuprates
that have been reported recently. 23 While for the pure SC phase with DP+HH,
the spin response is incommensurate at low frequency, then it switches to be
commensurate within the intermediate frequency range, and becomes
incommensurate again at higher frequency. 31 These results can be summarized
in the intensity plot of the imaginary part of the renormalized spin
susceptibility Im$\chi(\mathbf{q},\omega)$ as a function of frequency and
momentum along $(\pi,q_{y})$ direction, as shown in Fig. 4. In the figure, the
solid line indicating the peak position is the dispersion of spin excitations.
The commensurate spin fluctuation prevails below $\omega_{c}$ for the
coexisting system [Fig. 4(a)]. For the pure SC phase with DP+HH, the
dispersion shows a hourglass-like behavior [Fig. 4(c)], which is similar to
the hole-doped one, and does not consistent with the experiments on electron-
doped cuprates. 23
In the presence of the AF order, the energy band of quasiparticles is split
into two bands. Therefore, the particle-hole excitations that contributed to
the spin susceptibility are composed of two kinds of excitations, the intra-
band and the inter-band excitations. In Fig. 5, we present the results for the
bare spin susceptibility $\chi_{0}(\mathbf{q},\omega)$ (without the RPA
correction) coming from the intra-band and the inter-band contributions,
respectively. Figs. 5(a1) and 5(a2) denote the imaginary part of
$\chi_{0}(\mathbf{q},\omega)$, Figs. 5(b1) and 5(b2) the real part. One
obvious feature is that, the intra-band contribution is zero at the AF
momentum $\mathbf{Q}=(\pi,\pi)$, leading to the incommensurate spin response.
It results from the fact that the coherence factor in the spin susceptibility
due to the intra-band excitations,
$1-[(2Jm)^{2}-\varepsilon_{\mathbf{k+q}}\varepsilon_{\mathbf{k}}]/[{\sqrt{\varepsilon_{\mathbf{k+q}}^{2}+(2Jm)^{2}}\sqrt{\varepsilon_{\mathbf{k}}^{2}+(2Jm)^{2}}}]$
[where, $\varepsilon_{\mathbf{k}}=(-2tx-J\chi)(\cos k_{x}+\cos k_{y})$] is
zero at $\mathbf{Q}$. While, the inter-band contribution is commensurate for
all frequencies. At low frequencies, the inter-band excitations have a larger
contribution to the spin susceptibility than the intra-band excitations, so
the spin fluctuation is commensurate. However, with the increase of frequency,
the intensity of Im$\chi_{0}(\mathbf{q},\omega)$ due to the intra-band
contributions increases more rapidly than the inter-band contribution. As a
result, the spin fluctuation switches from a commensurate to an incommensurate
structure.
## IV conclusion
In this paper, we have investigated the spin dynamics in the electron-doped
cuprates in the coexisting phase of the $d_{x^{2}-y^{2}}$-wave SC and AF
orders, and compared the results with that in the dominant
$d_{x^{2}-y^{2}}$-wave phase with a higher harmonics term. In the coexisting
phase, we found that the spin response is commensurate in a substantial
frequency range below a crossover frequency $\omega_{c}$ for all dopings
considered, and it switches to be incommensurate when the frequency is higher
than $\omega_{c}$. The theoretical calculations are shown to be in good
agreement with the experimental measurements. However, in the dominant
$d_{x^{2}-y^{2}}$-wave phase with a higher harmonics term, the dispersion is
just like that of the hole-doped one, namely exhibits a hourglass-like
dispersion, which is not consistent with experiments. Thus, our result
suggests that the inclusion of the two-band effect is important to
consistently account for both the dispersion of the spin response and the non-
monotonic gap behavior in the electron-doped cuprates.
###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China
(10525415), the Ministry of Science and Technology of China (973 project
Grants Nos.2006CB601002,2006CB921800), and the China Postdoctoral Science
Foundation (Grant No. 20080441039).
Figure 1: (Color online) Mean-field phase diagram for
$t-t^{\prime}-t^{\prime\prime}-J$ model, where $\Delta_{1}$ is the
superconducting order parameter without considering the AF order. The model
parameters are taken as: $t=-3.0J$, $t^{\prime}=1.02J$,
$t^{\prime\prime}=-0.51J$. Figure 2: (Color online) Doping dependence of
Im$\chi(\mathbf{q},\omega)$ in the coexisting phase of the AF and SC order at
low frequency $\omega=0.04J$. The momentum is scanned along $(\pi,q_{y})$. The
inset shows Im$\chi(q,\omega)$ at $\omega=0.04J$ for the pure SC state with
DP+HH at doping $x=0.15$. Figure 3: (Color online) Frequency dependence of
Im$\chi(\mathbf{q},\omega)$ at doping $x$=$0.15$. The momentum is scanned
along $(\pi,q_{y})$. (a) and (b) are in the coexistence of AF and SC state.
(c) and (d) are in the pure SC state with DP+HH. Figure 4: (Color online)
Intensity plot of Im$\chi(\mathbf{q},\omega)$ as a function of frequency
$\omega$ and momentum $\mathbf{q}$ at doping $x$=$0.15$. The momentum is
scanned along $(\pi,q_{y})$. The solid line is the peak position. (a) is in
the coexistence of AF and SC state and (b) in the pure SC state with DP+HH.
Figure 5: (Color online) Frequency dependence of the intra- and inter-band
contributions to the bare spin susceptibility $\chi_{0}(\mathbf{q},\omega)$
[(a1) and (a2) denote the imaginary part, (b1) and (b2) the real part] in the
coexistence of AF and SC state at doping $x$=$0.15$. (a1) and (b1) show the
intra-band contributions , and (a2) and (b2) the inter-band contribution.
## References
* (1) C. C. Tsuei and J. R. Kirtley, Phys. Rev. Lett. 85, 182 (2000); A. D. Darminto, H. J. H. Smilde, V. Leca, D. H. A. Blank, H. Rogalla, and H. Hilgenkamp, Phys. Rev. Lett. 94, 167001 (2005); A. Biswas, P. Fournier, M. M. Qazilbash, V. N. Smolyaninova, H. Balci, and R. L. Greene, Phys. Rev. Lett. 88, 207004 (2002).
* (2) L. Shan, Y. Huang, Y. L. Wang, S. Li, J. Zhao, P. Dai, Y. Z. Zhang, C. Ren, and H. H. Wen, Phys. Rev. B 77, 014526 (2008); A. Zimmers, Y. Noat, T. Cren, W. Sacks, D. Roditchev, B. Liang, and R. L. Greene, Phys. Rev. B 76, 132505 (2007); G.-Q. Zheng, T. Sato, Y. Kitaoka, M. Fujita, and K. Yamada, Phys. Rev. Lett. 90, 197005 (2003); J. A. Skinta, M.-S. Kim, T. R. Lemberger, T. Greibe and M. Naito, Phys. Rev. Lett. 88, 207005 (2002).
* (3) T. Sato, T. Kamiyama, T. Takahashi, K. Kurahashi, and K. Yamada, Science 291, 1517 (2001).
* (4) N. P. Armitage, D. H. Lu, D. L. Feng, C. Kim, A. Damascelli, K. M. Shen, F. Ronning, Z. X. Shen, Y. Onose, Y. Taguchi, and Y. Tokura, Phys. Rev. Lett. 86, 1126 (2001).
* (5) H. Matsui, K. Terashima, T. Sato, T. Takahashi, M. Fujita, and K. Yamada, Phys. Rev. Lett. 95, 017003 (2005).
* (6) S. R. Park, Y. S. Roh, Y. K. Yoon, C. S. Leem, J. H. Kim, B. J. Kim, H. Koh, H. Eisaki, N. P. Armitage, and C. Kim, Phys. Rev. B 75, 060501(R) (2007).
* (7) G. Blumberg, A. Koitzsch, A. Gozar, B. S. Dennis, C. A. Kendziora, P. Fournier, and R. L. Greene, Phys. Rev. Lett. 88, 107002 (2002).
* (8) Y. Dagan, R. Beck, and R. L. Greene, Phys. Rev. Lett. 99, 147004 (2007).
* (9) P. Krotkov and A. V. Chubukov, Phys. Rev. Lett. 96, 107002 (2006).
* (10) H. Yoshimura and D. S. Hirashima, J. Phys. Soc. Jpn. 73, 2057 (2004); 74, 712 (2005).
* (11) V. A. Khodel, V. M. Yakovenko, M. V. Zverev, and H. Kang, Phys. Rev. B 69, 144501 (2004).
* (12) T. Watanabe, T. Miyata, H. Yokoyama, Y. Tanaka, and J.-I. Inoue, J. Phys. Soc. Jpn. 74, 1942 (2005).
* (13) P. Krotkov and A. V. Chubukov, Phys. Rev. B 74, 014509 (2006).
* (14) Q. S. Yuan, F. Yuan, and C. S. Ting, Phys. Rev. B 73, 054501 (2006); Q. S. Yuan, X. Z. Yan, and C. S. Ting, $ibid$. 74, 214503 (2006).
* (15) W. Yuan, H.-D. Lü, H.-Y. Lu, and Q.-H. Wang, Phys. Rev. B 77, 064515 (2008).
* (16) T. Das, R. S. Markiewicz, and A. Bansil, Phys. Rev. B 74, 020506(R) (2006); M. Aichhorn, E. Arrigoni, M. Potthoff, and W. Hanke, $ibid$. 74, 024508 (2006).
* (17) H. G. Luo and T. Xiang, Phys. Rev. Lett. 94, 027001 (2005).
* (18) C. S. Liu, H. G. Luo, W. C. Wu, and T. Xiang, Phys. Rev. B 73, 174517 (2006).
* (19) N. P. Armitage, F. Ronning, D. H. Lu, C. Kim, A. Damascelli, K. M. Shen, D. L. Feng, H. Eisaki, Z.-X. Shen, P. K. Mang, N. Kaneko, M. Greven, Y. Onose, Y. Taguchi, and Y. Tokura, Phys. Rev. Lett. 88, 257001 (2002).
* (20) H. Matsui, K. Terashima, T. Sato, T. Takahashi, S.-C. Wang, H.-B. Yang, H. Ding, T. Uefuji, and K. Yamada, Phys. Rev. Lett. 94, 047005 (2005).
* (21) C. Kusko, R. S. Markiewicz, M. Lindroos, and A. Bansil, Phys. Rev. B 66, 140513(R) (2002).
* (22) D. S$\acute{e}$n$\acute{e}$chal, P.-L. Lavertu, M.-A. Marois, and A.-M. S. Tremblay, Phys. Rev. Lett. 94, 156404 (2005).
* (23) S. D. Wilson, S. Li, H. Woo, P. Dai, H. A. Mook, C. D. Frost, S. Komiya, and Y. Ando, Phys. Rev. Lett. 96, 157001 (2006)
* (24) K. Yamada, K. Kurahashi, T. Uefuji, S. Park, S.-H. Lee, and Y. Endoh, Phys. Rev. Lett. 90, 137004 (2003)
* (25) H. J. Kang, P. Dai, H. A. Mook, D. N. Argyriou, V. Sikolenko, J. W. Lynn, Y. Kurita, S. Komiya, and Y. Ando, Phys. Rev. B 71, 214512 (2005)
* (26) S. D. Wilson, S. Li, P. Dai, W. Bao, J. Chung, H. J. Kang, S. H. Lee, S. Komiya, Y. Ando, and Q. Si, Phys. Rev. B 74, 144514 (2006)
* (27) M. Fujita, M. Matsuda, S.-H. Lee, M. Nakagawa, and K. Yamada, Phys. Rev. Lett. 101, 107003 (2008)
* (28) E. M. Motoyama, G. Yu, I. M. Vishik, O. P. Vajk, P. K. Mang, and M. Greven, Nature 445, 186 (2007).
* (29) P. Bourges, Y. Sidis, H. F. Fong, L. P. Renault, J. Bossy, A. Ivanov, and B. Keimer, Science 288, 1234 (2000); P. Dai, H. A. Mook, R. D. Hunt, and F. Dogan, Phys. Rev. B 63, 054525 (2001); S. M. Hayden, H. A. Mook, P. Dai, T. G. Perring, and F. Dogan, Nature 429, 531 (2004); J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, Nature 429, 534 (2004).
* (30) J. X. Li, J. Zhang, and J. Luo, Phys. Rev. B 68, 224503 (2003).
* (31) C. P. Chen, T. Zhou and J. X. Li, Physica C 469, 234 (2009).
* (32) K. Krüger, S. D. Wilson, L. Shan, S. Li, Y. Huang, H.-H. Wen, S.-C Zhang, P. Dai, J. Zaanen, Phys. Rev. B 76, 094506 (2007).
* (33) P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 000017 (2006).
|
arxiv-papers
| 2009-07-02T12:14:23 |
2024-09-04T02:49:03.666015
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Cui-Ping Chen, Hong-Min Jiang, and Jian-Xin Li",
"submitter": "Hong-Min Jiang",
"url": "https://arxiv.org/abs/0907.0357"
}
|
0907.0467
|
§ NON-ARCHIMEDEAN ANALYSIS ON THE EXTENDED HYPERREAL LINE $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
_{\MATHBF{D}}$ AND SOME TRANSCENDENCE CONJECTURES OVER FIELD $%
%TCIMACRO{\U{211A} }%
\MATHBB{Q}
$ AND $^{\AST }%
%TCIMACRO{\U{211A} }%
\MATHBB{Q}
_{\PROTECT\OMEGA }.$
Jaykov Foukzon
Israel Institute of Technology
Abstract. In this paper possible completion of the
Robinson non-archimedean field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ constructed by Dedekind sections. As interesting example I show how, a few
simple ideas from non-archimedean analysis on the pseudo-ring $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ gives a short clear nonstandard reconstruction for the
Euler's original proof of the Goldbach-Euler theorem. Given an analytic
function of one complex variable $f\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\left[ z\right] ,$we investigate the arithmetic nature of the values of $f$
at transcendental points.
1.Some transcendence conjectures over field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
2.Modern nonstandard analysis and non-archimedean analysis on
the extended hyperreal line $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Chapter I.The classical hyperreals numbers.
I.1.1. The construction non-archimedean field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I.1.2. The brief nonstandard vocabulary.
I.2. The higher orders of hyper-method.Second
order transfer principle.
I.2.1. What are the higher orders of hyper-method?
I.2.2. The higher orders of hyper-method by using countable
I.2.3. Divisibility of hyperintegers.
I.3. The construction non-archimedean pseudo-ring $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I.3.1. Generalized pseudo-ring of Wattenberg-Dedekind hyperreals $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
and hyperintegers $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
I.3.1.1. Strong and weak Dedekind cuts.Wattenberg-Dedekind
and hyperintegers.
I.3.2. The topology of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I.3.3. Absorption numbers in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I.3.3.1. Absorption function and numbers in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I.3.3.2. Special Kinds of Idempotents in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I.3.3.3. Types of $\alpha $ with a given $\mathbf{ab.p.}(\alpha ).$
I.3.3.4. $\varepsilon $-Part of $\alpha $ with $\mathbf{ab.p.}%
(\alpha )\neq 0.$
I.3.3.5. Multiplicative idempotents.
I.3.3.6. Additive monoid of Dedekind hyperreal integers $^{\ast }%
\breve{%
\mathbb{Z}%
I.3.5. Pseudo-ring of Wattenberg hyperreal integers $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
I.3.6. External summation of countable and hyperfinite
sequences in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I.3.7. The construction non-archimedean field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}^{\omega }$ as Dedekind
completion of countable non-standard models of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I.4. The construction non-archimedean field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I.4.1. Completion of ordered group and fields in general by
"Cauchy pregaps".
I.4.1.1. Totally ordered group and fields
I.4.1.2. Cauchy completion of ordered group and fields.
I.4.2.1. The construction non-archimedean field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}$ by using Cauchy
hypersequence in ancountable field $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
I.4.2.2. The construction non-archimedean field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}^{\omega }$ as Cauchy
completion of countable non-standard models of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
Chapter II.Euler's proofs by using non-archimedean analysis on the
pseudo-ring $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ revisited.
II.1.Euler's original proof of the Goldbach-Euler Theorem revisited.
III. Non-archimedean analysis on the extended hyperreal line $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ and
transcendence conjectures over field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
.$Proof that $\ e+\pi $ and $e\cdot \pi $ is
Chapter III.Non-archimedean analysis on the extended hyperreal line
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
and transcendence conjectures over field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
III.1. Proof that $e$ is $\#$-transcendental and that $\ e+\pi $
and $e\cdot \pi $ is irrational.
III.2. Nonstandard generalization of the Lindeman Theorem.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
§ LIST OF NOTATION.
$^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$..................................................... the set of
infinite natural numbers
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\infty }$.................................................the set of
infinite hyper real numbers
$\mathbf{L}_{\ast }\mathbf{=L}\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $.........................................the set of the limited
members of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\mathbf{I}_{\ast }=\mathbf{I}\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $..................................the set of the infinitesimal
members of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
halo$\left( x\right) =\mu \left( x\right) =x+\mathbf{I}_{\ast }$
............................................halo (monad) of $x\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\mathbf{st}\left( a\right) $
......................................................Robinson standard part
of $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\left[ z_{1},...,z_{\mathbf{n}}\right] $.....................internal
polynomials over $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ in $\ \mathbf{n}\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ variables
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\left[ z_{1},...,z_{\mathbf{n}}\right] $.....................internal
polynomials over $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ in $\mathbf{n}\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ variables
$^{\ast }%
%TCIMACRO{\U{2102} }%
\mathbb{C}
\left[ z_{1},...,z_{\mathbf{n}}\right] $.....................internal
polynomials over $^{\ast }%
%TCIMACRO{\U{2102} }%
\mathbb{C}
$ in $\mathbf{n}\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ variables
$^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
completion of the ring $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
completion of the field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
.........................................................Cauchy completion
of the field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\varepsilon _{\mathbf{d}}=\sup \left[ x|x\in \mu \left( 0\right) \right]
=\inf \left[ x|x\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}\right] $..............................$\mu \left( 0\right) $ $\subset $
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}$ $\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\Delta _{\mathbf{d}}=\sup \left(
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}\right) =\inf \left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+\infty }\right) $
$WST\left( \alpha \right) $.........Wattenberg standard part of $\ \alpha
\in \left( -\Delta _{\mathbf{d}},\Delta _{\mathbf{d}}\right) _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}$ (Def.1.3.2.3)
$\mathbf{ab.p.}\left( \alpha \right) $
....................................absorption part of $\alpha \in $ $^{\ast
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ (Def 1.3.3.1.1)
$\left[ \alpha \right] _{\varepsilon }$
\varepsilon $-part of $-\Delta _{\mathbf{d}}<\alpha <\Delta _{\mathbf{d}}$
$\left[ \alpha |b^{\#}\right] _{\varepsilon }$
...................................................$\varepsilon $-part of $%
\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ for a given $b\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
"Arthur stopped at the steep descent into the quarry, froze in his
steps,straining to look down and into the distance, extending his long
neck.Redrick joined him. But he did not look where Arthur was looking. Right
at their feet the road into the quarry began, torn up many years ago by the
treads and wheels of heavy vehicles.To the right was a white steep
slope,cracked by the heat; the next slope was half excavated, and among the
rocks and rubble stood a dredge, its lowered bucket jammed impotently
against the side of the road. And,as was to be expected, there was nothing
else to be seen on the road..."
Arkady and Boris
"Roadside Picnic"
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
§ INTRODUCTION.
1.Some transcendence conjectures over field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
.$In 1873 French mathematician, Charles Hermite, proved that $e$ is
transcendental. Coming as it did 100 years after Euler had established the
significance of $e,$ this meant that the issue of transcendence was one
mathematicians could not afford to ignore.Within 10 years of Hermite's
breakthrough,his techniques had been extended by Lindemann and used to add $%
\pi $ to the list of known transcendental numbers. Mathematician then tried
to prove that other numbers such as $e+\pi $ and $e\times \pi $ are
transcendental too,but these questions were too difficult and so no further
examples emerged till today's time. The transcendence of $e^{\pi }$had been
proved in1929 by A.O.Gel'fond.
Conjecture 1. The numbers $e+\pi $ and $e\times \pi $ are
Conjecture 2. The numbers $e$ and $\pi $ are
algebraically independent.
However, the same question with $e^{\pi }$ and $\pi $ has been answered:
Theorem.1.(Nesterenko,1996 [22]) The numbers $e^{\pi }$ and $\pi $
are algebraically
During of XX th century,a typical question: is whether $f(\alpha )$ is a
dental number for each algebraic number $\alpha $ has been investigated and
answered many authors.Modern result in the case of entire functions
satisfying a linear differential equation provides the strongest results,
related with Siegel's $E$-functions [22],[27].Ref. [22] contains references
to the subject before 1998, including Siegel $E$ and $G$ functions.
Theorem.2.(Siegel C.L.) Suppose that $\lambda \in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,\lambda \neq -1,-2,...,\alpha \neq 0.$
\begin{array}{cc}
\begin{array}{c}
\\
\varphi _{\lambda }\left( z\right) =\sum_{n=0}^{\infty }\dfrac{z^{n}}{\left(
\lambda +1\right) \left( \lambda +2\right) \cdot \cdot \cdot \left( \lambda
+n\right) }. \\
\end{array}
& \text{ }\left( 1.1\right) \text{\ \ }%
\end{array}%
Then $\varphi _{\lambda }\left( \alpha \right) $ is a transcendental number
for each algebraic number $\alpha \neq 0.$
Given an analytic function of one complex variable $f\left( z\right) \in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\left[ z\right] ,$we
investigate the arithmetic nature of the values of $f\left( z\right) $ at
Conjecture 3.Is whether $f(\alpha )$ is a irrational
number for given
transcendental number $\alpha .$
Conjecture 4.Is whether $f(\alpha )$ is a transcendental
number for given
transcendental number $\alpha .$
In particular we investigate the arithmetic nature of the values of
classical polylogarithms $Li_{s}\left( z\right) $ at transcendental
points.The classical polylogarithms
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
Li_{s}\left( z\right) =\sum_{n\geq 1}\dfrac{z^{n}}{n^{s}} \\
\end{array}
& \left( 1.2\right)%
\end{array}%
for $s=1,2,...$ and $|z|\leq 1$ with $(s;z)=(1;1),$ are ubiquitous. The study
of the arithmetic nature of their special values is a fascinating subject
[35] very few is known.Several recent investigations concern the values
of these functions at $z=1:$ these are the values at the positive integers
of Riemann zeta function
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\zeta \left( s\right) =Li_{s}\left( z=1\right) =\sum_{n\geq 1}\dfrac{1}{n^{s}%
} \\
\end{array}
& \left( 1.3\right)%
\end{array}%
One knows that $\zeta (3)$ is irrational [36],and that inInitely many values
$\zeta (2n+1)$ of the zeta function at odd integers are irrational.
Conjecture 4.Is whether $Li_{s}\left( \alpha \right) $ is
a irrational number for given
transcendental number $\alpha .$
Conjecture 5.Is whether $Li_{s}\left( \alpha \right) $ is
a transcendental number for given
transcendental number $\alpha .$
2.Modern nonstandard analysis and non-archimedean analysis
on the extended hyperreal line $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$Nonstandard analysis, in its early period of development,
shortly after having been established by A. Robinson [1],[4],[5] dealt
mainly with nonstandard extensions of some traditional mathematical
structures. The system of its foundations, referred to as "model-theoretic
foundations" was proposed by Robinson and E. Zakon [12]. Their approach was
based on the type-theoretic concept of superstructure $V(S)$ over some set
of individuals $S$ and its nonstandard extension (enlargement) $^{\ast
}V(S), $ usually constructed as a (bounded) ultrapower of the "standard"
superstructure $V(S).$They formulated few principles concerning the
elementary embedding $V(S)\longmapsto $ $^{\ast }V(S),$ enabling the use of
methods of nonstandard analysis without paying much attention to details of
construction of the particular nonstandard extension.
In classical Robinsonian nonstandard analysis we usualy deal only with
completely internal objects which can defined by internal set theory $%
\mathbf{IST}$ introduced by E.Nelson [11]. It is known that $\mathbf{IST}$
is a conservative extension of $ZFC.$ In $\mathbf{IST}$ all the classical
infinite sets, e.g., $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2124} }%
\mathbb{Z}
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ or $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ acquire new, nonstandard elements (like "infinite" natural numbers
or "infinitesimal" reals). At the same time, the families $%
^{\sigma }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ $=$ $\left\{ x\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
:\mathbf{st}\left( x\right) \right\} $ or $^{\sigma }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ $=$ $\left\{ x\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\mathbf{st}\left( x\right) \right\} $ of all standard,i.e., "true," natural
numbers or reals, respectively, are not sets in $\mathbf{IST}$ at all. Thus,
for a traditional mathematician inclined to ascribe to mathematical objects
a certain kind of objective existence or reality, accepting $\mathbf{IST}$
would mean confessing that everybody has lived in confusion, mistakenly
having regarded as, e.g., the set $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ just its tiny part $^{\sigma }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ (which is not even a set) and overlooked the rest. Edvard Nelson and Karel
Hrbc̆ek have improved this lack by introducing several "nonstandard" set
theories dealing with standard, internal and external sets [13].
Note that in contrast with early period of development of the nonstandard
analysis in latest period many mathematicians dealing with external and
internal set simultaneously,for example see [14],[15],[16],[17].
Many properties of the standard reals $x\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ suitably reinterpreted, can be transfered to the internal hyperreal number
system. For example, we have seen that $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$like $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$is a totally ordered field. Also, jast $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ contain the natural number $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ as a discrete subset with its own characteristic properties,$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ contains the hypernaturals $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ as the corresponding discrete subset with analogous properties.For
example, the standard archimedean property
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\forall x_{x\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\forall y_{y\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\exists n_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left[ \left( \left\vert x\right\vert <\left\vert y\right\vert \right)
\dashrightarrow n\left\vert x\right\vert \geq \left\vert y\right\vert \right]
\\
\end{array}
& \left( 1.4\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
is preserved in non-archimedean field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ in respect hypernaturals $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
,$i.e. the next property is satisfied
$\ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \forall x_{x\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\forall y_{y\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\exists n_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left[ \left( \left\vert x\right\vert <\left\vert y\right\vert \right)
\dashrightarrow n\left\vert x\right\vert \geq \left\vert y\right\vert \right]
. \\
\end{array}
& \left( 1.5\right)%
\end{array}%
\ \ \ \ $
$\bigskip $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
However, there are many fundamental properties of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ do not transfered to $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
I. This is the case one of the fundamental supremum
property of the standard totally ordered field $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$It is easy to see that it apper bound property does not necesarily holds
by considering, for example, the (external) set $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ itself which we ragard as canonically imbedded into hyperreals $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$ This is a non-empty set which is bounded above (by any of the infinite
member in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$) but does not have a least apper bound in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$ However by using transfer one obtain the next statement [18] :
Weak supremum property for $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:$Every non-empty internal subset
$A\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ which has an apper bound in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ has a least apper bound in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
This is a problem, because any advanced variant of the analysis on the field
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is needed more strong fundamental supremum property. At first sight one
can improve this lack by using corresponding external constructions which
known as Dedekind sections and Dedekind completion (see section I.3.
).We denote corresponding Dedekind completion by symbol $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$. It is clear that $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is completely external object. But unfortunately $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is not iven a non-archimedean ring but non-archimedean
pseudo-ring only. However this lack does not make greater
difficulties because non-archimedean pseudo-ring $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ contains non-archimedean subfield $\mathbf{\Re }_{\mathbf{c}%
}\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that $\mathbf{\Re }_{\mathbf{c}}\approx $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}.$Here $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}$ this is a Cauchy completion of the non-archimedean
field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ (see section I.4.).
II. This is the case two of the fundamental Peano's
induction property:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\forall B\left[ \left[ \left( 1\in B\right) \wedge \forall x\left( x\in
B\implies x+1\in B\right) \right] \implies B=%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right] \\
\end{array}
& \text{ \ \ \ \ \ }\left( 2.1\right)%
\end{array}%
does not necesarily holds for arbitrary subset $B\subset $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
.$ Therefore (2.1)
is true for $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ when interpreted in $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ i.e.,
$\ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\forall ^{\mathbf{int}}B\left[ \left[ \left( 1\in B\right) \wedge \forall
x\left( x\in B\implies x+1\in B\right) \right] \implies B=\text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right] \\
\end{array}
& \text{ \ \ \ \ \ \ }\left( 2.2\right)%
\end{array}%
is true for $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ provided that we read "$\forall B$" as "for each internal subset $B$ of $%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$", i.e. as $\forall ^{\mathbf{int}}B.$ In general the importance of
internal versus external entities rests on the fact that each statement that
is true for $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is true for $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ provided its quantifiers are restricted to the internal entities (subset)
of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ only [18].This is a problem, because any advanced variant of the analysis
on the field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is needed more strong induction property than property
(2).In this paper I have improved this lack by using external construction
two different types for operation of exteral summation:
$\bigskip $
$\ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}q_{n}, \\
\\
\#Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}q_{n}^{\#} \\
\end{array}
& \left( 2.3\right)%
\end{array}%
\ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
and two different types for operation of exteral multiplication:
$\bigskip \ \
\begin{array}{cc}
\begin{array}{c}
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
}q_{n}, \\
\\
\#Ext-\dprod\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}q_{n}^{\#} \\
\end{array}
& \text{ \ \ \ \ \ \ }\left( 2.4\right)%
\end{array}%
for arbitrary countable sequences such as $q_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and $q_{n}^{\#}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
As interesting example I show how, this external constructions from
non-archimedean analysis on the pseudo-ring $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ gives a short and clear nonstandard reconstruction for the
Euler's original proof of the Goldbach-Euler theorem.
§ I.THE CLASSICAL HYPERREALS NUMBERS.
§ I.1.1.THE CONSTRUCTION NON-ARCHIMEDEAN FIELD $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
Let $\Re $ denote the ring of real valued sequences with the usual pointwise
operations.If $x$ is a real number we let $s_{x}$ denote the constant
sequence,$\mathbf{s}_{x}=x$ for all $n.$ The function sending $x$ to $%
\mathbf{s}_{x}$ is a one-to-one ring homomorphism,providing an embedding of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ into $\Re $. In the following, wherever it is not too confusing we will
not distinguish between $x\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and the constant function $\mathbf{s}_{x},$leaving the reader to derive
intent from context. The ring $\Re $ has additive identity $0$ and
multiplicative identity $1.$ $\Re $ is not a field because if $r$ is any
sequence having $0$ in its range it can have no multiplicative inverse.
There are lots of zero divisors in $\Re $.
We need several definitions now. Generally, for any set $S,$ $\mathbf{P}(S)$
denotes the set of all subsets of $S.$ It is called the power set of $S.$
Also, a subset of $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ will be called cofinite if it contains all but finitely many members of $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$. The symbol $\varnothing $ denotes the empty set. A partition of a set $S$
is a decomposition of $S$ into a union of sets, any pair of which have no
elements in common.
Definition.1.1.1. An ultrafilter $\mathbf{H}$ over $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ is a family of sets for which:
(i) $\varnothing \notin \mathbf{H\subset P(}%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\mathbf{),%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\in H.}$
(ii) Any intersection of finitely many members of $\mathbf{H}$ is
in $\mathbf{H.}$
(iii) $A\subset
%TCIMACRO{\U{2115} }%
\mathbb{N}
,B\in \mathbf{H}\Rightarrow A\cup B\in \mathbf{H.}$
(iv) If $V_{1},...,V_{n}$ is any finite partition of $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ then $\mathbf{H}$ contains exactly
one of the $V_{i}.$
If, further,
(v) $\mathbf{H}$ contains every cofinite subset of $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
the ultrafilter is called free.
If an ultrafilter on $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ contains a finite set then it contains a one-point set,
and is nothing more than the family of all subsets of $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ containing that
point. So if an ultrafilter is not free it must be of this type, and is
a principal ultrafilter.
The existence of a free ultrafilter containing any given infinite subset of $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
is implied by the Axiom of Choice.
Remark 1.1.1. Suppose that $x\in X.$ An ultrafilter denoted
$\mathbf{prin}_{X}\left( x\right) \subseteq X$ consisting of all subsets $%
S\subseteq X$ which contain $x,$ and called
the principal ultrafilter generated by $x.$
Proposition 1.1.1. If an ultrafilter $\mathbf{\tciFourier }
$ on $X$ contains a finite set $S\subseteq X,$
then $\mathbf{\tciFourier }$ is principal.
Proof: It is enough to show $\mathbf{\tciFourier }$ contains $%
\left\{ x\right\} $ for some $x\in S.$ If not, then
$\mathbf{\tciFourier }$ contains the complement $X\backslash \left\{
x\right\} $ for every $x\in S$, and therefore also
the finite intersection $\mathbf{\tciFourier }\ni \dbigcap\limits_{x\in
S}X\backslash \left\{ x\right\} =X\backslash S,$ which contradicts the fact
that $S\in \tciFourier .$It follows that nonprincipal ultrafilters can exist
only on infinite
sets $X$, and that every cofinite subset of $X$ (complement of a finite set)
belongs to such an ultrafilter.
$\mathbf{Remark}$ $\mathbf{1.1.2.}$Our construction below depends on the use
of a free-not a
We are going to be using conditions on sequences and sets to define subsets
of $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$. We introduce a convenient shorthand for the usual “set
builder” notation. If $P$ is a property that can be true
or false for natural numbers we use $[[P]]$ to denote $\{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
|P(n)$ is true $\}.$ This notation will only be employed during a discussion
to decide if the set of natural numbers defined by $P$ is in $\mathbf{H,}$
or not. For example, if $s,t$ is a pair of sequences in $\Re $ we define
three sets of integers For example, if $s,t$ is a pair of sequences in $S$
we define three sets of integers
$\ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ [[s<t]],\ \ [[s=t]],\ [[s>t]]. \\
\end{array}
& \text{ \ \ \ \ \ \ \ \ \ }\left( 1.1.1\right)%
\end{array}%
\ \ $
Since these three sets partition $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$, exactly one of them is in $\mathbf{H,}$ and we
declare $s\equiv t$ when $[[s=t]]\in \mathbf{H.}$
Lemma 1.1.1. $\equiv $ is an equivalence relation on $\Re $. We
denote the
equivalence class of any sequence $s$ under this relation by $[s].$
Define for each $r\in \Re $ the sequence $\tilde{r}$ by
$\ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \ \tilde{r}=\left\{
\begin{array}{c}
0\text{ \ \ }\mathbf{iff}\text{ }r_{n}=0 \\
r_{n}^{-1}\text{ }\mathbf{iff}\text{ }r_{n}\neq 0%
\end{array}%
\right\} . \\
\end{array}
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(
\end{array}%
Lemma 1.1.2. (a) There is at most one constant
sequence in any class $[r].$
(b) $[0]$ is an ideal in $\Re $ so $\Re /[0]$ is a commutative ring
with identity [1].
(c) Consequently $[r]=r+[0]=\{r+t|t\in \lbrack 0]\}$ for all $r\in
\Re $.
(d) If $[r]\neq \lbrack 0]$ then $[\tilde{r}]\cdot \lbrack r]=[1].$
So $[r]^{-1}=[\tilde{r}].$
From Lemma 1.1.2., we conclude that $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ defined to be $\Re /[0],$ is a field
containing an embedded image of as a subfield. $[0]$ is a maximal ideal in
$\Re $.
Definition.1.1.2.This quotient ring is called the field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ of classical
hyperreal numbers.
We declare $[s]<[t]$ provided $[[s<t]]\in \mathbf{H.}$
Recall that any field with a linear order $<$ is called an ordered field
(i) $x+y>0$ whenever $x,y>0$
(ii) $x\cdot y>0$ whenever $x,y>0$
(iii) $x+z>y+z$ whenever $x>y$
Theorem 1.1.3. (a) The relation given above is a linear
order on $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ and
makes $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ into an ordered field. As with any ordered field, we define $|x|$
for $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ to be $x$ or $-x,$ whichever is nonnegative.
(b) If $x,y$ are real then $x\leq y$ if and only if $[x]\leq
\lbrack y].$ So the ring morphism
of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ into $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is also an order isomorphism onto its image in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Because of this last theorem and the essential uniqueness of the real
numbers it is common to identify the embedded image of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ with $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ itself. Though obviously circular, one does something similar when
identifying $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ with its isomorphic image in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$, and $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ itself with the corresponding subset of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$. This kind of notational simplification usually does not cause problems.
Now we get to the ideas that prompted the construction. Define the sequence $%
r$ by $r_{n}=\left( n+1\right) ^{-1}$ . For every positive integer $%
k,[[r<k^{-1}]]\in \mathbf{H.}$So $0<[r]<1/k.$ We have found a positive
hyperreal smaller than (the embedded image of) any real number. This is our
first nontrivial infinitesimal number. The sequence $\tilde{r}$ is given by $%
\tilde{r}_{n}=n+1.$So $[r]^{-1}=[\tilde{r}]>k$ for every positive integer $%
k.[r]^{-1}$ is a hyperreal larger than any real number.
§ I.1.2.THE BRIEF NONSTANDARD VOCABULARY.
Definition.1.1.2.1. We call a member $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$-limited if there are members $\ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\ \
\ \ \ \ a,b\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ with $a<x<b.$
We will use $\mathbf{L}_{\ast }\mathbf{=L}\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $ to indicate the limited members of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$. $x$ is called
%TCIMACRO{\U{211d} }%
\mathbb{R}
$-unlimited if it not $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
These terms are preferred to “finite” and
which are reserved for concepts related to cardinality.
Definition.1.1.2.2. If $x,y\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and $x<y$ we use $^{\ast }[x,y]$ to denote
$\{t\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
|x\leq t\leq y\}.$
This set is called a closed hyperinterval. Open and half-open
hyperintervals are defined and denoted similarly.
Definition.1.1.2.3. A set $S\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is called hyperbounded if there are
members $x,y$ of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ for which $S$ is a subset of the hyperinterval
$^{\ast }[x,y].$Abusing standard vocabulary for ordered sets, $S$ is called
if $x$ and $y$ can be chosen to be limited members of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ $x$ and $y$ could, in
fact, be chosen to be real if $S$ is bounded.
The vocabulary of bounded or hyperbounded above and below can be
Definition.1.1.2.3. We call a member $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ infinitesimal if $|x|<a$ for
every positive $a\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$ We write $x\approx 0$ iff $x$ is infinitesimal.
The only real infinitesimal is obviously $0.$
We will use $\mathbf{I}_{\ast }=\mathbf{I}\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $ to indicate the infinitesimal members of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Definition.1.1.2.4. A member $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is called appreciable if it is limited
but not infinitesimal.
Definition.1.1.2.5. Hyperreals $x$ and $y$ are said to have
appreciable separation if $|x-y|$ is appreciable.
We will be working with various subsets $S$ of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and adopt the
following convention: $S_{\infty }=S\backslash \mathbf{L}_{\ast }=\{x\in
S|x\notin \mathbf{L}_{\ast }\}.$ These are the
unlimited members of $S,$if any.
Definition.1.1.2.6. (a) We say two hyperreals $x,y$ are
infinitesimally close
or have infinitesimal separation if $|x-y|\in \mathbf{I}_{\ast }.$
We use the notation $x\approx y$ to indicate that $x$ and $y$ are
infinitesimally close.
(b) They have limited separation if $|x-y|\in \mathbf{L}_{\ast }%
\mathbf{.}$
(c) Otherwise they are said to have unlimited separation.
We define the halo of $x$ by $\mathbf{halo}(x)=x+\mathbf{I}_{\ast }.$ There
can be at most one real
number in any halo. Whenever $\mathbf{halo}(x)\cap
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is nonempty we define the
shadow of $x,$ denoted $\mathbf{shad}(x),$ to be that unique real number.
The galaxy of $x$ is defined to be $\mathbf{gal}(x)=x+\mathbf{L}_{\ast }%
\mathbf{.}$ $\mathbf{gal}(x)$ is the set of
hyperreal numbers $a$ limited distance away from $x.$ So if $x$ is limited
$\mathbf{gal}(x)=\mathbf{L}_{\ast }\mathbf{.}$
If n is any fixed positive integer we define $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
^{n}$ to be the set of
equivalence classes of sequences in $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
^{n}$ under the equivalence
relation $x\equiv y$ exactly when $[[x=y]]\in \mathbf{H.}$
Definition.1.1.2.7. We call $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ the set of classical hypernatural or
A. Robinson's hypernatural numbers, $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ the set of classical infinite
hypernatural or A. Robinson's infinite hypernatural numbers,$%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\infty }$ the
set of classical infinite hyperreal or A. Robinson's infinite
numbers,$^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ the set of classical hyperintegers or A. Robinson's
hyperintegers, and $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ the set of classical hyperrational numbers or
A. Robinson's hyperrational numbers.
Theorem 1.1.2.1. $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ $\mathbf{is}$ $\mathbf{not}$ $\mathbf{Dedekind}$
(hint: $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ is bounded above by the member $[\mathbf{t}]\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty },$ where $\mathbf{t}$ is the sequence
given by $t_{n}=n$ for all $n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
.$ But $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ can have no least upper bound: if $n\leq c$
for all $n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ then $n\leq c-1$ for all $n\in $ $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
As another example consider $\mathbf{I}_{\ast }\mathbf{.}$ This set is
(very) bounded, but has no
least upper bound.)
Theorem 1.1.2.2. For every $r\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ there is unique $n\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ with
$n\leq r<n+1.$
§ I.2.THE HIGHER ORDERS OF HYPER-METHOD.SECOND ORDER
TRANSFER PRINCIPLE.
§ I.2.1.WHAT ARE THE HIGHER ORDERS OF HYPER-METHOD?
Usual nonstandard analysis essentially consists only of two fundamental
tools:the (first order) star-map $\ast _{1}\triangleq \ast
$ and the (first order) transfer principle. In most
applications, a third fundamental tool is also considered, namely the
saturation property.
Definition.1.2.1.1. Any universe $\mathbf{U}$ is a nonempty
collection of
"standard mathematical objects" that is closed under
i.e. $a\subseteqq A\in \mathbf{U}$ $\implies a\in \mathbf{U}$ and closed
under the basic mathematical
operations.Precisely, whenever $A,B\in \mathbf{U},$ we require that also the
union $A\cup B$, the intersection $A\cap B,$the set-difference $A\backslash
B $ the
ordered pair $\left\{ A,B\right\} ,$the Cartesian product $A\times B,$ the
powerset $P(A)=\left\{ a|a\subseteqq A\right\} ,$the function-set $%
B^{A}=\left\{ f\text{ }|\text{ }f:A\rightarrow B\right\} $ all
belong to $\mathbf{U.}$A universe $\mathbf{U}$ is also assumed to contain
(copies of) all
sets of numbers $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
,$ $%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
%TCIMACRO{\U{211a} }%
\mathbb{Q}
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{2102} }%
\mathbb{C}
$ $\in \mathbf{U,}$ and to be transitive, i.e. members
of members of $\mathbf{U}$ belong to $\mathbf{U}$ or in formulae: $a\in A\in
\mathbf{U}$ $\implies a\in \mathbf{U}$.
The notion of "standard mathematical object" includes all objects
in the ordinary practice of mathematics, namely: numbers, sets,
functions, relations, ordered tuples, Cartesian products, etc. It is well-
known that all these notions can be defined as sets and formalized in
the foundational framework of Zermelo-Fraenkel axiomatic set theory
From standard assumption: $Con\left( \mathbf{ZFC}\right) $ and Gödel's
completeness theorem
one obtain that $\mathbf{ZFC}$ has a model $M$. A model $M$ of set theory is
standard if the element relation $\in _{M}$of the model $M$ is the actual
relation $\in $ restricted to the model $M,$ i.e.$\in _{M}\triangleq $ $\in
|_{M}$ . A model is called
transitive when it is standard and the base class is a transitive class of
sets. A model of set theory is often assumed to be transitive unless it is
explicitly stated that it is non-standard. Inner models are transitive,
transitive models are standard, and standard models are well-founded.
The assumption that there exists a standard model of $\mathbf{ZFC}$ (in a
universe) is stronger than the assumption that there exists a model. In
fact, if there is a standard model, then there is a smallest standard
model called the minimal model contained in all standard models. The
minimal model contains no standard model (as it is minimal) but
(assuming the consistency of $\mathbf{ZFC}$) it contains some model of $%
\mathbf{ZFC}$ by
the Gödel completeness theorem. This model is necessarily not well
founded otherwise its Mostowski collapse would be a standard model.
(It is not well founded as a relation in the universe, though it
satisfies the axiom of foundation so is "internally" well founded.
Being well founded is not an absolute property[2].) In particular in the
minimal model there is a model of $\mathbf{ZFC}$ but there is no standard
of $\mathbf{ZFC.}$
By the theorem of Löwenheim-Skolem, we can choose transitive models
$M_{\omega }$ of $\mathbf{ZFC}$ of countable cardinality.
Remark 1.2.1.1.In $\mathbf{ZFC,}$ an ordered pair $\left\{
a,b\right\} $ is defined as the
Kuratowski pair $\left\{ \left\{ a\right\} ,\left\{ a,b\right\} \right\} ;$
an $n$-tuple is inductively defined by
$\left\{ a_{1},...,a_{n},a_{n+1}\right\} $ $=\left\{ \left\{
a_{1},...,a_{n}\right\} ,a_{n+1}\right\} ;$ an $n$-place relation $R$ on $A$
is identified with the set $R\subseteq A^{n}$ of $n$-tuples that satisfy it;
a function
$f:A\rightarrow B$ is identi407ed with its graph $\left[ \left\{
a,b\right\} \in A\times B|b=f(a)\right] $.
As for numbers, complex numbers $%
%TCIMACRO{\U{2102} }%
\mathbb{C}
=$ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\times
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ are defined as ordered pairs of
real numbers, and the real numbers $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ are defined as equivalence classes
of suitable sets of rational numbers namely, Dedekind cuts or Cauchy
The rational numbers $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ are a suitable quotient $%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\times
%TCIMACRO{\U{2124} }%
\mathbb{Z}
/_{\approx },$ and the integers
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ are in turn a suitable quotient $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\times
%TCIMACRO{\U{2115} }%
\mathbb{N}
/_{\approx }$. The natural numbers of $\mathbf{ZFC}$ are
defined as the set of von Neumann naturals: $0=\NEG{0}$ and $n+1=\left\{
n\right\} $
(so that each natural number $\left\{ n=0,1,...,n-1\right\} $ is identified
with the set
of its predecessors.)
Each countable model $M_{\omega }$ of $\mathbf{ZFC}$ contains countable
model $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }$ of the
real numbers $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$Every element $x\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }$ defines a Dedekind cut:
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{x}\triangleq \left\{ q\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|q\leq x\right\} \cup \left\{ q\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|q>x\right\} .$We therefore get a order preserving
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }\rightarrow
%TCIMACRO{\U{211d} }%
\mathbb{R}
\\
\end{array}
& \text{ \ \ }\left( 1.2.1\right)%
\end{array}%
and which respects addition and multiplication.
We address the question what is the possible range of $f_{p}?$
Proposition 1.2.1.1. Choose an arbitrary subset $\Theta
\subset
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$Then there is a
model $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\left( \Theta \right) $ such that $f_{p}\left[
%TCIMACRO{\U{211d} }%
\mathbb{R}
\left( \Theta \right) \right] \supset \Theta .$Moreover, the cardinality of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\left( \Theta \right) $ can
be chosen to coincide with $\Theta $, if $\Theta $ is infinite.
Proof. Choose $\Theta \subset
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$For each $\alpha \in \Theta $ choose
$q_{1}\left( \alpha \right) <q_{2}\left( \alpha \right) <...<p_{1}\left(
\alpha \right) <p_{2}\left( \alpha \right) $ with $\underset{n\rightarrow
\infty }{\lim }q_{n}\left( \alpha \right) =$ $\underset{n\rightarrow \infty }%
{\lim }p_{n}\left( \alpha \right) =\alpha .$
We add to the axioms of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ the following axioms:
$\ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \forall \alpha \left( \alpha \in \Theta \right) \exists e_{\alpha
}\forall k\left( k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right) \left[ q_{k}\left( \alpha \right) <e_{\alpha }<p_{k}\left( \alpha
\right) \right] \\
\end{array}
& \text{ \ \ \ \ \ \ \ }\left( 1.2.2\right)%
\end{array}%
Again $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ is a model for each finite subset of these axioms,so that the
compactness theorem implies the existence of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\left( \Theta \right) $ as required,where
the cardinality of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\left( \Theta \right) $ can be chosen to be the cardinality of the set of
axioms, i.e. of $\Theta $, if $\Theta $ is infinite. Note that by
construction $f_{p}\left( e_{\alpha }\right) =\alpha .$
$\bigskip $
Remark 1.2.1.2. It follows by the theorem of Lö
for each countable subset $\Theta _{\omega }\subset
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ we can find a countable model
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }=%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\left( \Theta _{\omega }\right) $ of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that the image of $f_{p}$ contains this subset.Note,
on the other hand, that the image will only be countable, so that the
different models $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\left( \Theta _{\omega }\right) $ will have very different ranges.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
Hyper-Tool # 1: FIRST ORDER STAR-MAP.
Definition.1.2.1.2. The first order star-map is a function
$\ast _{1}:\mathbf{U\rightarrow V}_{1}$
between two universes that associates to each object $A$ $\in $ $\mathbf{U}$
its first order
hyper-extension (or first order non-standard extension) $^{\ast
_{1}}A$ $\in $ $\mathbf{V}_{1}\mathbf{.}$ It is also
assumed that $^{\ast _{1}}n=n$ for all natural numbers $n$ $\in $ $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$, and that the properness
condition $^{\ast _{1}}%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\neq
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ holds.
Remark 1.2.1.3. It is customary to call standard any object $A$ $%
\in $ $\mathbf{U}$ in the
domain of the first order star-map $\ast _{1},$ and first order nonstandard
any object
$B$ $\in $ $\mathbf{V}_{1}$ in the codomain. The adjective standard is also
often used in the
literature for first order hyper-extensions $^{\ast _{1}}A\in \mathbf{V}%
_{1}. $
Hyper-Tool # 2: SECOND ORDER STAR-MAP.
Definition.1.2.1.3. The second order star-map is a function
$\ast _{2}:\mathbf{V}_{1}$ $\mathbf{\rightarrow V}_{2}$ between two
universes that associates to each
object $A$ $\in $ $\mathbf{V}_{1}$ its second order hyper-extension
(or second
order non-standard extension) $^{\ast _{2}}A$ $\in $ $\mathbf{V}_{2}%
\mathbf{.}$
It is also assumed that $^{\ast _{2}}N=N$ for all hyper natural numbers
$N$ $\in $ $^{\ast _{1}}%
%TCIMACRO{\U{2115} }%
\mathbb{N}
,$and that the properness condition $^{\ast _{2}}%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\neq $ $^{\ast _{1}}%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ holds.
Hyper-Tool # 3: FIRST ORDER TRANSFER PRINCIPLE.
Definition.1.2.1.4. Let $P(a_{1},...,a_{n})$ be a property of the
standard objects
$a_{1},...,a_{n}\in $ $\mathbf{U}$ expressed as an "elementary sentence".
Then $P(a_{1},...,a_{n})$ is
true if and only if corresponding sentence $^{\ast _{1}}P\left(
c_{1},...,c_{n}\right) $ is true about the
corresponding hyper-extensions $^{\ast _{1}}a_{1},...,^{\ast _{1}}a_{n}\in
\mathbf{V}_{1}$. That is:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
P(a_{1},...,a_{n})\iff \text{ }^{\ast _{1}}P\left( ^{\ast
_{1}}a_{1},...,^{\ast _{1}}a_{n}\right) . \\
\end{array}
& \left( 1.2.3\right)%
\end{array}%
In particular $P(a_{1},...,a_{n})$ is true if and only if the same sentence $%
P\left( c_{1},...,c_{n}\right) $
is true about the corresponding hyper-extensions $^{\ast
_{1}}a_{1},...,^{\ast _{1}}a_{n}\in \mathbf{V}_{1}$. That
is: $P(a_{1},...,a_{n})\iff $ $P\left( ^{\ast _{1}}a_{1},...,^{\ast
_{1}}a_{n}\right) .$
Hyper-Tool # 4: SECOND ORDER TRANSFER PRINCIPLE.
Definition.1.2.1.5.Let $^{\ast _{1}}P\left( ^{\ast
_{1}}a_{1},...,^{\ast _{1}}a_{n}\right) $ be a property of the first
non-standard objects $^{\ast _{1}}a_{1},...,^{\ast _{1}}a_{n}\in \mathbf{V}%
_{1}$ expressed as an "elementary
§ I.2.2.THE HIGHER ORDERS OF HYPER-METHOD BY USING COUNTABLE UNIVERSES.
Definition.1.2.2.1. Any countable universe $\mathbf{U}_{\omega }$
is a nonempty countable
collection of "standard mathematical objects" that is
closed under subsets,
i.e. $a\subseteqq A\in \mathbf{U}$ $\implies a\in \mathbf{U}$ and closed
under the basic mathematical
operations. Precisely, whenever
$A,B\in \mathbf{U},$ we require that also the union $A\cup B$, the
intersection $A\cap B,$
the set-difference $A\backslash B$ the ordered pair $\left\{ A,B\right\} ,$
the Cartesian product
$A\times B,$ the powerset $P(A)=\left\{ a|a\subseteqq A\right\} ,$the
$B^{A}=\left\{ f\text{ }|\text{ }f:A\rightarrow B\right\} $ all belong to $%
\mathbf{U}_{\omega }\mathbf{.}$A countable universe $\mathbf{U}_{\omega }$
is also
assumed to contain (copies of) all sets of numbers $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
,$ $%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\in \mathbf{U}_{\omega },%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega },$
%TCIMACRO{\U{2102} }%
\mathbb{C}
_{\omega }$ $\in \mathbf{U}_{\omega }\mathbf{,}$ and to be transitive, i.e.
members of members of $\mathbf{U}_{\omega }$ belong
to $\mathbf{U}_{\omega }$ or in formulae: $a\in A\in \mathbf{U}_{\omega }$ $%
\implies a\in \mathbf{U}_{\omega }$.
Remark 1.2.2.1.In any countable model $M_{\omega }$ of $\mathbf{ZFC,%
}$an ordered pair $\left\{ a,b\right\} $
is defined as the Kuratowski pair
$\left\{ \left\{ a\right\} ,\left\{ a,b\right\} \right\} ;$an $n$-tuple is
inductively defined by $\left\{ a_{1},...,a_{n},a_{n+1}\right\} $ $=$
$\left\{ \left\{ a_{1},...,a_{n}\right\} ,a_{n+1}\right\} ;$ an $n$-place
relation $R$ on $A$ is identified with the
countable set $R\subseteq A^{n}$ of $n$-tuples that satisfy it; a function $%
f:A\rightarrow B$
is identified with its graph $\left[ \left\{ a,b\right\} \in A\times B|b=f(a)%
\right] .$
As for numbers, complex numbers $%
%TCIMACRO{\U{2102} }%
\mathbb{C}
_{\omega }=$ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }\times
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }$ are defined as ordered
pairs of real numbers, and the real numbers $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }$ are defined as
countable set of countable equivalence classes of suitable sets of
rational numbers namely,Dedekind cuts or Cauchy sequences.
The rational numbers $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ are a suitable quotient $%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\times
%TCIMACRO{\U{2124} }%
\mathbb{Z}
/_{\approx },$ and the
integers $%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ are in turn a suitable quotient $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\times
%TCIMACRO{\U{2115} }%
\mathbb{N}
/_{\approx }$. The natural
numbers of $\mathbf{ZFC}$ are defined as the set of von Neumann naturals:
$0=\NEG{0}$ and $n+1=\left\{ n\right\} $ (so that each natural number
$\left\{ n=0,1,...,n-1\right\} $ is identified with the set of its
§ I.2.3.DIVISIBILITY OF HYPERINTEGERS.
Definition.1.2.3.1.If $n$ and $d$ are hypernaturals,i.e. $n,d\in $ $%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ or hyperintegers,
i.e. $n,d\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ and $d\neq 0,$then $n$ is divisible by $d$ provided $n=d\cdot k$ for some
hyperinteger $k.$Alternatively, we say:
1.$n$ is a multiple of $d,$
2.$d$ is a factor of $n,$
3.$d$ is a divisor of $n,$
4.$d$ divides $n$ (denoted with $d$ $|$ $n$).
Theorem 1.2.3.1.Transitivity of Divisibility.
For all $a,b,c\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
,$ if $a|b$ and $b|c,$ then $a|c.$
Theorem 1.2.3.2.Every positive hyperinteger greater than $1$
is divisible by a hyperprime number.
Definition.1.2.3.2.Given any integer $n>1,$ the
standard factored form of $n\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ is an expression of $n=$ $^{\ast }\prod_{k=1}^{m}p_{k}^{e_{k}},$
where $m$ is a positive hyperinteger, $p_{1},p_{2},...,p_{m}$ are
hyperprime numbers with $p_{1}<p_{2}<...<p_{m}$ and
$e_{1},e_{2},...,e_{m}$ are positive hyperintegers.
Theorem 1.2.3.3.Given any hyperinteger $n>1,$ there exist
positive hyperinteger $m,$hyperprime numbers $p_{1},p_{2},...,p_{m}$
and positive hyperintegers $e_{1},e_{2},...,e_{m}$ with $n=$ $%
^{\ast }\prod_{k=1}^{m}p_{k}^{e_{k}}.$
Theorem 1.2.3.1. (i) Every pair of elements $%
m,n\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ has a highest
common factor $d=s\times m+t\times n$ for some $s,t\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(ii) For every pair of elements $a,d\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ dividend $a$ and divisor $d,$ with $d\neq 0$
there exist unique integers $q$ and $r$ such that $a=q\times d+r$ and $0\leq
r<\left\vert d\right\vert .$
Definition.1.2.3.2. Suppose that $a=q\times d+r$ and $%
0\leq r<\left\vert d\right\vert .$We call $d$
the quotient and $r$ the remainder.
Redrick, squinting his swollen eyes against the blinding light, silently
him go. He was cool and calm, he knew what was about to happen, and he
knew that he would not watch,but it was still all right to watch, and he did,
feeling nothing in particular,except that deep inside a little worm started
wriggling around and twisting its sharp head in his gut.
Arkady and Boris Strugatsky
"Roadside Picnic"
§ I.3.THE CONSTRUCTION NON-ARCHIMEDEAN PSEUDO-RING$^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
§ I.3.1.GENERALIZED PSEUDO-RINGS WATTENBERG-DEDEKIND HYPERREALS$\ $ $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
_{\MATHBF{D}}$ AND HYPERINTEGERS $^{\AST }%
%TCIMACRO{\U{2124} }%
\MATHBB{Z}
§ I.3.1.1.STRONG AND WEAK DEDEKIND CUTS. WATTENBERG-DEDEKIND
HYPERREALS AND HYPERINTEGERS.
From Theorem 1.2.1.1 above we knov that: $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ $\mathbf{is}$ $\mathbf{not}$ $\mathbf{Dedekind}$
For example, $\mu (0)$ and $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ are bounded subsets of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ which have no
suprema or infima in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Possible standard completion of the field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ can be constructed by
Dedekind sections [23],[24]. In [24] Wattenberg constructed the Dedekind
completion of a nonstandard model of the real numbers and applied the
construction to obtain certain kinds of special measures on the set of
Thus was established that the Dedekind completion $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ of the field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is a
structure of interest not for its own sake only and we establish further
importent applications here. Importent concept was introduce Gonshor [23]
is that of the absorption number of an element $\mathbf{a\in }%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which, roughly
speaking,measures the degree to which the cancellation law
$\mathbf{a}+b=\mathbf{a}+c\implies b=c$ fails for $\mathbf{a}$.
More general construction well known from topoi theory [10].
Definition 1.3.1.1.1. A Dedekind hyperreal $\alpha \in $ $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is a pair
$(U,V)\in \mathbf{P}\left( ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\right) \times $ $\mathbf{P}\left( ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\right) $ satisfying the next conditions:
1.$\exists x\exists y\left( x\in U\wedge y\in V\right) .$
2. $U\cap V=\varnothing .$
3.$\forall x\left( x\in U\iff \exists y\left( y\in V\wedge
x<y\right) \right) .$
4. $\forall x\left( x\in V\iff \exists y\left( y\in V\wedge
y<x\right) \right) .$
5. $\forall x\forall y\left( x<y\implies x\in U\vee y\in V\right) .$
Remark. The monad of $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ the set $\left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
|\text{ }x\approx \alpha \right\} $ is
denoted: $\mu \left( \alpha \right) .$
Monad $\mu \left( 0\right) $ is denoted: $\mathbf{I}_{\ast }.$Supremum of $%
\mathbf{I}_{\ast }$ is denoted: $\varepsilon _{\mathbf{d}}.$
Let $A$ be a subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is bounded or hyperbounded above
then $\sup \left( A\right) $ exists in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Example. (i) $\Delta _{\mathbf{d}}=\sup \left(
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}\right) \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\left\backslash ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right. ,$(ii) $\varepsilon _{\mathbf{d}}=\sup \left( \ \mathbf{I}%
_{\ast }\right) \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\left\backslash ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right. .$
Remark. Anfortunately the set $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ inherits some but by no means all
of the algebraic structure on $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$For example,$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is not a group with
respect to addition since if $x+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}y$ denotes the addition in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ then:
\begin{array}{cc}
\begin{array}{c}
\\
\varepsilon _{\mathbf{d}}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\varepsilon _{\mathbf{d}}=\varepsilon _{\mathbf{d}}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}=\varepsilon _{\mathbf{d}} \\
\end{array}
& \left( 1.3.1.1.1\right)%
\end{array}%
Thus $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is not iven a ring but pseudo-ring only. Thus, one must
proceed somewhat cautiously. In this section more details than is
customary will be included in proofs because some standard properties
which at first glance appear clear often at second glance reveal themselves
to be false in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
We shall briefly remind a way Dedekind's constructions of a pseudo-field
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Definition 1.3.1.2.a.(Strong and weak Dedekind cuts)
(1) Suppose $\preceq $ is a total ordering on $X.$ We write $%
x\preceq y$ if $x$ is
less than or equal to $y$ and we write $x\prec y$ if $x\preceq y$ and $%
x\neq y.$
Then $\left\{ A,B\right\} $ is said to be a strong Dedekind
cut of $\left\langle X,\preceq \right\rangle ,$
if and only if:
1. $A$ and $B$ are nonempty subsets of $X.$
2. $A\cup B=X.$
3. For each $x$ in $A$ and each $y$ in $B,x\preceq y.$
(2) Suppose $\left\{ A,B\right\} $ is a strong Dedekind cut of $%
\left\langle X,\preceq \right\rangle $
Then $\left\{ A,\widetilde{A};B,\widetilde{B}\right\} $ is said to be a
weak Dedekind cut of $\left\langle X,\preceq \right\rangle
if and only if:
1. $A\subsetneqq \widetilde{A},B\subsetneqq \widetilde{B}.$
2. For each $x$ in $A$ there is exist $\widetilde{x}_{1}\in
\widetilde{A}$ such that $x\prec \widetilde{x}$ and
$\widetilde{x}_{2}\in \widetilde{A}$ such that $\widetilde{x}_{2}\prec x.$
3. For each $y$ in $B$ there is exist $\widetilde{y}_{1}\in
\widetilde{B}$ such that $\widetilde{y}_{1}\prec y$ and
$\widetilde{y}_{2}\in \widetilde{B}$ such that $y\prec \widetilde{y}_{2}.$
(3) $A$ is the left-hand part of the strong cut $\left\{
A,B\right\} $ and $B$ is the
riht-hand part of the strong cut $\left\{ A,B\right\} $.
We denote the strong cut as $x=A|B$ or simple $x=A.$
The strong cut $x=A|B$ is less than or equal to the strong cut
$y=C|D$ if $A\subseteqq C.$
(4) $\widetilde{A}$ is the left-hand part of the weak cut $%
\left\{ A,\widetilde{A};B,\widetilde{B}\right\} $ and $\widetilde{B}$ is the
riht-hand part of the weak cut $\left\{ A,\widetilde{A};B,%
\widetilde{B}\right\} $.
We denote the weak cut as $x=\widetilde{A}|\widetilde{B}$ or simple $x=%
\widetilde{A}.$
The weak cut $x=\widetilde{A}|\widetilde{B}$ is less than or equal to the
weak cut
$y=\widetilde{C}|\widetilde{D}$ iff $A\subseteqq C.$
Definition 1.3.2.b.
(1) $c\in X$ is said to be a cut element of $\left\{ A,B\right\} $
if and only if either:
(i) $c$ is in $A$ and $x\preceq c\preceq y$ for each $x$ in $A$ and
each $y$ in $B,$ or
(ii) $c$ is in $B$ and $x\preceq c\preceq y$ for each $x$ in $A$
and each $y$ in $B.$
(2) $c\in X$ is said to be a cut element of $\left\{ A,\widetilde{A}%
;B,\widetilde{B}\right\} $
if and only if either:
(i) $c$ is in $\widetilde{A}$ and $x\preceq c\preceq y$ for each $x$
in $\widetilde{A}$ and each $y$ in $\widetilde{B},$ or
(ii) $c$ is in $\widetilde{B}$ and $x\preceq c\preceq y$ for each $%
x $ in $\widetilde{A}$ and each $y$ in $\widetilde{B}.$
Definition 1.3.2.c.$\left\langle X,\preceq \right\rangle $ is said
to be Dedekind complete if
and only if each strong Dedekind cut of $\left\langle X,\preceq
\right\rangle $,has a cut
Equivalently $\left\langle X,\preceq \right\rangle $ is said to be
Dedekind complete if and
only if each weak Dedekind cut $\widetilde{A}|\widetilde{B}$ of $%
\left\langle X,\preceq \right\rangle $,has a cut
Example. The following theorem is well-known.
Theorem. $\left\langle
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\leqslant \right\rangle $ is Dedekind complete, and for each Dedekind cut
$\left\{ A,B\right\} $,of $\left\langle
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\leqslant \right\rangle $ if $r$ and $s$ are cut elements of $\left\{
A,B\right\} $, then $r=s.$
Making a semantic leap, we now answer the question "what is a
Wattenberg-Dedekind hyperreal number ?"
Definition 1.3.2.d. A Wattenberg-Dedekind hyperreal number
a cut in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is the class of all Dedekind hyperreal numbers $x=A|B$ ($x=A$
We will show that in a natural way $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is a complete ordered
generalized pseudo-ring containing $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Before spelling out what this means, here are some examples of cuts.
$A|B=\left. \left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\text{ }r<1\right\} \right\vert \left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\text{ }r\geq 1\right\} .$
(ii) $\ $
$\ A|B=\left. \left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\left( r\leq 0\right) \vee \left( \text{ }r^{2}<2\right) \right\}
\right\vert \left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\left( r>0\right) \wedge \text{ }\left( r^{2}\geq 2\right) \right\}
$A|B=\left. \left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\text{ }r<\omega \right\} \right\vert \left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\text{ }r\geq \omega \right\} ,$where $\omega \in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$\left. \left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\left( r\leq 0\right) \vee \left( r\in \mathbf{I}_{\ast }\right)
\vee \left( \text{ }r\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{+}\right) \right\} \right\vert $
$\left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\left( r>0\right) \wedge \text{ }\left( r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{+}\backslash \left(
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{+}\cup \mathbf{I}_{\ast }\right) \right) \right\} .$
$\left. \left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\left( r\leq 0\right) \vee \left( r\in \mathbf{I}_{\ast }\right)
\right\} \right\vert \left\{ r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\text{ }|\left( r>0\right) \wedge \text{ }\left( r\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{+}\backslash \mathbf{I}_{\ast }\right) \right\} .$
Remark. It is convenient to say that $A|B\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is a rational (hyperrational)
cut in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ if it is like the cut in examples (i),(iii):
fore some fixed rational
(hyperrational) number $c\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,A$ is the set of all hyperrational $r$ such
that $r<c$ while $B$ is the rest of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
The $B$-set of a rational (hyperrational) cut contains a smollest $c\in $ $%
^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
and conversaly if $A|B$ is a cut in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ and $B$ contains a smollest element
$c$ then $A|B$ is a rational or hyperrational cut at $c.$We write $\breve{c}$
for the rational
hyperrational cut at $c.$This lets us think of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ by identifying $c$
with $\breve{c}.$
Remark. It is convenient to say that:
(1) $A|B\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is an standard cut in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ if it is like the cut in examples
(i)-(ii):fore some cut $A^{\prime }|B^{\prime }\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ the next equality is satisfied:
$A|B=$ $^{\ast }\left( A^{\prime }\right) |^{\ast }\left( B^{\prime }\right)
,$i.e. $A$-set of a cut is an standard set.
(2) $A|B\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is an internal cut or nonstandard
cut in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ if it is like
the cut in example (iii), i.e. $A$-set of a cut is an
internal nonstandard
(3) $A|B\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is an external cut in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ if it is like the cut in
examples (iv)-(v),i.e. $A$-set of a cut is an
external set.
There is an order relation $\left( \cdot \leq \cdot \right) $ on cuts that
fairly cries out for
Definition 1.3.2.e. The cut $x=A|B$ is less than or equal to the
$y=C|D$ if $A\subseteqq C.$
We write $x\leq y$ if $x$ is less than or equal to $y$ and we write $x<y$ if
$x\leq y$ and $x\neq y.$If $x=A|B$ is less than $y=C|D$ then $A\subset C$
$A\neq C,$so there is some $c_{0}\in C\backslash A.$Sinse the $A$-set of a
cut contains
no largest element, there is also a $c_{1}\in C$ with $c_{0}<c_{1}.$All the
hyperrational numbers $c$ with $c_{0}\leq c\leq c_{1}$ belong to $%
C\backslash A.$
Remark. The property distinguishing $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ from $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ and from $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and
which is the bottom of every significant theorem about $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ involves
upper bounds and least upper bounds or equivalently,lower bounds
and gretest lower bounds.
Definition 1.3.2.f. $M\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is an upper bound for a set $S\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ if
each $s\in S$ satisfies $s\leq M.$ We also say that the set $S$ is
above by $M$ iff $M\in $ $\mathbf{L}\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $ We also say that the set $S$ is
hyperbounded above iff $M\notin $ $\mathbf{L}\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) ,$i.e.$\left\vert M\right\vert \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{211d} }%
\mathbb{R}
Definition 1.3.2.g. An upper bound for $S$ that is less
than all other
upper bound for $S$ is a least upper bound for $S.$
The concept of a pseudo-ring originally was introduced by
E. M.Patterson [21].Briefly,Patterson's pseudo-ring is an algebraic
system consisting of an additive abelian group $\mathbf{A}$, a distinguished
subgroup $\mathbf{A}^{\mathbf{\ast }}$ of $\mathbf{A,}$and a multiplication
operation $\mathbf{A}^{\ast }\mathbf{\times A}\rightarrow \mathbf{A}$ under
which $\mathbf{A}^{\ast }$ is a ring and $\mathbf{A}$ a left $\mathbf{A}%
^{\ast }$-module.For convenience, we
denote the pseudo-ring by $\Re =(A^{\ast },A).$
Definition 1.3.1.2.h.Generalized pseudo-ring
is an algebraic system consisting of an abelian
semigroup $\mathbf{A}_{\mathbf{s}}$ (or
abelian monoid $\mathbf{A}_{\mathbf{m}}$),a distinguished subgroup $\mathbf{A%
}_{\mathbf{s}}^{\mathbf{\ast }}$ of $\mathbf{A}_{\mathbf{s}}$ (or a
distinguished subgroup $\mathbf{A}_{\mathbf{m}}^{\mathbf{\ast }}$ of $%
\mathbf{A}_{\mathbf{m}}$),and a multiplication operation
$\mathbf{A}_{\mathbf{s}}^{\ast }\mathbf{\times A}_{\mathbf{s}}\rightarrow
\mathbf{A}_{\mathbf{s}}$ ($\mathbf{A}_{\mathbf{m}}^{\ast }\mathbf{\times A}_{%
\mathbf{m}}\rightarrow \mathbf{A}_{\mathbf{m}}$) under which $\mathbf{A}_{%
\mathbf{s}}^{\ast }$ ($\mathbf{A}_{\mathbf{m}}^{\ast }$) is a ring and
$\mathbf{A}_{\mathbf{s}}$ ($\mathbf{A}_{\mathbf{m}}$) a left $\mathbf{A}_{%
\mathbf{s}}^{\ast }$-module ($\mathbf{A}_{\mathbf{m}}^{\ast }$-module).
For convenience,we denote the generalized pseudo-ring by
$\Re _{\mathbf{s}}=(A_{\mathbf{s}}^{\ast },A_{\mathbf{s}}).$
Pseudo-field is an algebraic system consisting of an
semigroup $\mathbf{A}_{\mathbf{s}}$, a distinguished subgroups $%
\mathbf{A}_{\mathbf{s}}^{\mathbf{\ast }}\subsetneqq \mathbf{A}_{\mathbf{s}}^{%
\mathbf{\#}}$ of $\mathbf{A}_{\mathbf{s}}$ and a
multiplication operations $\mathbf{A}_{\mathbf{s}}^{\ast }\mathbf{\times A}_{%
\mathbf{s}}\rightarrow \mathbf{A}_{\mathbf{s}}$and $\mathbf{A}_{\mathbf{s}}%
\mathbf{\times A}_{\mathbf{s}}^{\ast }\rightarrow \mathbf{A}_{\mathbf{s}}$
which $\mathbf{A}_{\mathbf{s}}^{\ast }$ is a ring,$\mathbf{A}_{\mathbf{s}}^{%
\mathbf{\#}}$ is a field and $\mathbf{A}_{\mathbf{s}}$ is a vector spase over
field $\mathbf{A}_{\mathbf{s}}^{\#}.$
Definition 1.3.1.2$^{\prime }$.I.(Strong and Weak Dedekind
cut tipe
I in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(1) A strong Dedekind cut tipe I $\alpha _{%
\mathbf{s}}=\alpha _{\mathbf{s}}^{\mathbf{I}}$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is a subset
$\alpha _{\mathbf{s}}^{\mathbf{I}}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ of the hyperreals $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ that satisfies these properties:
1. $\alpha _{\mathbf{s}}^{\mathbf{I}}$ is not empty.
2. $\beta _{\mathbf{s}}^{\mathbf{I}}=$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\backslash \alpha _{\mathbf{s}}^{\mathbf{I}}$ is not empty.
3. $\alpha _{\mathbf{s}}^{\mathbf{I}}$ contains no greatest element
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ if $x\in \alpha _{\mathbf{s}}^{\mathbf{I}}$ and $y<x,$ then $y\in \alpha
_{\mathbf{s}}^{\mathbf{I}}$ as well.
(2) A weak Dedekind cut tipe I $\alpha _{w}$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is a subset
$\alpha _{w}^{\mathbf{I}}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ of the hyperreals $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ that satisfies these properties:
1. $\alpha _{w}^{\mathbf{I}}$ is not empty.
2. $\beta _{w}^{\mathbf{I}}=$ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \alpha _{w}^{\mathbf{I}}$ is not empty.
3. $\alpha _{\mathbf{s}}^{\mathbf{I}}$ contains no greatest element.
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ if $x\in \alpha _{w}^{\mathbf{I}}$ and $y<x,$ then there is exists
$z\in \alpha _{w}^{\mathbf{II}}$ such that $z<y$ as well.
Remark. Note that for every weak Dedekind cut $\alpha _{w}^{\mathbf{%
I}}$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ there is
exists unique strong Dedekind cut $\alpha _{\mathbf{s}}^{\mathbf{I}}\left(
\alpha _{w}^{\mathbf{I}}\right) $ in $^{\ast }\mathbf{%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}$ such that:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha _{\mathbf{s}}^{\mathbf{I}}\left( \alpha _{w}^{\mathbf{I}}\right) =%
\left[ \alpha _{w}^{\mathbf{I}}\right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}=\bigcap \left\{ \alpha _{\mathbf{s}}^{\mathbf{I}}|\alpha _{w}^{\mathbf{I}%
}\subset \text{ }\alpha _{\mathbf{s}}^{\mathbf{I}}\right\} . \\
\end{array}
& \left( 1.3.2\right)%
\end{array}%
Example. (1) Let $\Delta $ denotes the set $%
\left\{ x|x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\backslash ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+\infty }\right\} .$ It is
easy to see that $\Delta $ is a strong Dedekind cut in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(2) Let $\Delta \upharpoonright $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ denotes the set $\Delta \cap $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
=\left\{ q|\left( q\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{+\infty }\right) \right\} .$
It is easy to see that $\Delta \upharpoonright $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ is a weak Dedekind cut in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and
$\left[ \Omega _{k}\right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}=\Delta .$
(3) Let $1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}$ denotes the set $\left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
|x<1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\right\} ,1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\triangleq $ $^{\ast }1$ and $0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
denotes the set $\left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
|x<0\right\} ,0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\triangleq $ $^{\ast }0.$It is easy to see that $1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
and $0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}$ is a strong Dedekind cuts in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(3) Let $1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\upharpoonright $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
=\left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
|x<1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\right\} \cap ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ and
$0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\upharpoonright $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
=\left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
|x<0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\right\} \cap ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
.$It is easy to see that $1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\upharpoonright $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
and $0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\upharpoonright $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ is a strong Dedekind cuts in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and
$\left[ 1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\upharpoonright ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}=1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}},\left[ 0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\upharpoonright ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}=0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Definition 1.3.1.2$^{\prime }$.II.(Strong and Weak Dedekind
tipe II in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(1) A strong Dedekind cut tipe II $\alpha _{%
\mathbf{s}}^{\mathbf{II}}$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is a subset $\alpha _{\mathbf{s}}^{\mathbf{II}}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
of the hyperrational numbers $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ that satisfies these
1. $\alpha _{\mathbf{s}}^{\mathbf{II}}$ is not empty.
2. $\beta _{\mathbf{s}}^{\mathbf{II}}=$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\backslash \alpha _{\mathbf{s}}^{\mathbf{II}}$ is not empty.
3. $\alpha _{\mathbf{s}}^{\mathbf{II}}$ contains a greatest element
or $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\backslash \alpha _{\mathbf{s}}^{\mathbf{II}}$ contains no least
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ if $x\in \alpha _{\mathbf{s}}^{\mathbf{II}}$ and $y<x,$ then $y\in \alpha
_{\mathbf{s}}^{\mathbf{II}}$ as well.
(2) A weak Dedekind cut tipe II $\alpha _{w}^{\mathbf{II}}$
in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is a subset $\alpha _{w}^{\mathbf{II}}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
of the hyperrational numbers $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ that satisfies these properties:
1. $\alpha _{w}^{\mathbf{II}}$ is not empty.
2. $\beta _{w}^{\mathbf{II}}=$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\backslash \alpha _{w}^{\mathbf{II}}$ is not empty.
3. $\alpha _{\mathbf{s}}^{\mathbf{II}}$ contains a greatest element
or $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\backslash \alpha _{\mathbf{s}}^{\mathbf{II}}$ contains no least
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ if $x\in \alpha _{w}^{\mathbf{II}}$ and $y<x,$ then there is exists
$z\in \alpha _{w}^{\mathbf{II}}$ such that $z<y$ as well.
Remark. Note that for every weak Dedekind cut $\alpha _{w}^{\mathbf{%
II}}$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
there is exists unique strong Dedekind cut $\alpha _{\mathbf{s}}^{\mathbf{II}%
}\left( \alpha _{w}^{\mathbf{II}}\right) $ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such
that: $\alpha _{\mathbf{s}}^{\mathbf{II}}\left( \alpha _{w}^{\mathbf{II}%
}\right) =\left[ \alpha _{w}^{\mathbf{II}}\right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}=$ $\bigcap \left\{ \alpha _{\mathbf{s}}^{\mathbf{II}}|\alpha _{w}^{\mathbf{%
II}}\subset \text{ }\alpha _{\mathbf{s}}^{\mathbf{II}}\right\} .$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha _{\mathbf{s}}^{\mathbf{II}}\left( \alpha _{w}^{\mathbf{II}}\right) =%
\left[ \alpha _{w}^{\mathbf{II}}\right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}=\bigcap \left\{ \alpha _{\mathbf{s}}^{\mathbf{II}}|\alpha _{w}^{\mathbf{II}%
}\subset \text{ }\alpha _{\mathbf{s}}^{\mathbf{II}}\right\} . \\
\end{array}
& \left( 1.3.3\right)%
\end{array}%
Definition 1.3.1.2$^{\prime }$.a.I. (Strong and Weak
Dedekind cut
tipe I in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
(1) A strong Dedekind cut tipe I $\alpha _{%
\mathbf{s}}=\alpha _{\mathbf{s}}^{\mathbf{I}}$ in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ is a subset
$\alpha _{\mathbf{s}}^{\mathbf{I}}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ of the hyperrational numbers $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ that satisfies
these properties:
1. $\alpha _{\mathbf{s}}^{\mathbf{I}}$ is not empty.
2. $\beta _{\mathbf{s}}^{\mathbf{I}}=$ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \alpha _{\mathbf{s}}^{\mathbf{I}}$ is not empty.
3. $\alpha _{\mathbf{s}}^{\mathbf{I}}$ contains no greatest element
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$ if $x\in \alpha _{\mathbf{s}}^{\mathbf{I}}$ and $y<x,$ then $y\in \alpha
as well.
(2) A weak Dedekind cut tipe I $\alpha _{w}$ in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ is a subset
$\alpha _{w}^{\mathbf{I}}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
of the hyperrational numbers $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ that satisfies these
1. $\alpha _{w}^{\mathbf{I}}$ is not empty.
2. $\beta _{w}^{\mathbf{I}}=$ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \alpha _{w}^{\mathbf{I}}$ is not empty.
3. $\alpha _{\mathbf{s}}^{\mathbf{I}}$ contains no greatest element.
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$ if $x\in \alpha _{w}^{\mathbf{I}}$ and $y<x,$ then there
is exists $z\in \alpha _{w}^{\mathbf{II}}$ such that $z<y$ as well.
Remark. Note that for every weak Dedekind cut $\alpha _{w}^{\mathbf{%
I}}$ in
$^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ there is exists unique strong Dedekind cut $\alpha _{\mathbf{s}}^{\mathbf{I%
}}\left( \alpha _{w}^{\mathbf{I}}\right) $
in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ such that:
$\ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha _{\mathbf{s}}^{\mathbf{I}}\left( \alpha _{w}^{\mathbf{I}}\right) =%
\left[ \alpha _{w}^{\mathbf{I}}\right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}=\bigcap \left\{ \alpha _{\mathbf{s}}^{\mathbf{I}}|\alpha _{w}^{\mathbf{I}%
}\subset \text{ }\alpha _{\mathbf{s}}^{\mathbf{I}}\right\} . \\
\end{array}
& \left( 1.3.4\right)%
\end{array}%
Definition 1.3.1.2$^{\prime }$.a.II. (Strong and Weak
Dedekind cut
tipe II in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
(1) A strong Dedekind cut tipe II $\alpha _{%
\mathbf{s}}^{\mathbf{II}}$ in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ is a subset $\alpha _{\mathbf{s}}^{\mathbf{II}}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
of the hyperrational numbers $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ that satisfies these
1. $\alpha _{\mathbf{s}}^{\mathbf{II}}$ is not empty.
2. $\beta _{\mathbf{s}}^{\mathbf{II}}=$ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \alpha _{\mathbf{s}}^{\mathbf{II}}$ is not empty.
3. $\alpha _{\mathbf{s}}^{\mathbf{II}}$ contains a greatest element
or $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \alpha _{\mathbf{s}}^{\mathbf{II}}$ contains no least
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$ if $x\in \alpha _{\mathbf{s}}^{\mathbf{II}}$ and $y<x,$ then $y\in \alpha
_{\mathbf{s}}^{\mathbf{II}}$ as well.
(2) A weak Dedekind cut tipe II $\alpha _{w}^{\mathbf{II}}$
in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ is a subset $\alpha _{w}^{\mathbf{II}}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
of the hyperrational numbers $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ that satisfies these
1. $\alpha _{w}^{\mathbf{II}}$ is not empty.
2. $\beta _{w}^{\mathbf{II}}=$ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \alpha _{w}^{\mathbf{II}}$ is not empty.
3. $\alpha _{\mathbf{s}}^{\mathbf{II}}$ contains a greatest element
or $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \alpha _{\mathbf{s}}^{\mathbf{II}}$ contains no least
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$ if $x\in \alpha _{w}^{\mathbf{II}}$ and $y<x,$ then there is exists
$z\in \alpha _{w}^{\mathbf{II}}$ such that $z<y$ as well.
Remark. Note that for every weak Dedekind cut $\alpha _{w}^{\mathbf{%
II}}$ in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
there is exists unique strong Dedekind cut $\alpha _{\mathbf{s}}^{\mathbf{II}%
}\left( \alpha _{w}^{\mathbf{II}}\right) $ in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ such
that: $\alpha _{\mathbf{s}}^{\mathbf{II}}\left( \alpha _{w}^{\mathbf{II}%
}\right) =\left[ \alpha _{w}^{\mathbf{II}}\right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}=$ $\bigcap \left\{ \alpha _{\mathbf{s}}^{\mathbf{II}}|\alpha _{w}^{\mathbf{%
II}}\subset \text{ }\alpha _{\mathbf{s}}^{\mathbf{II}}\right\} .$
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha _{\mathbf{s}}^{\mathbf{II}}\left( \alpha _{w}^{\mathbf{II}}\right) =%
\left[ \alpha _{w}^{\mathbf{II}}\right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}=\bigcap \left\{ \alpha _{\mathbf{s}}^{\mathbf{II}}|\alpha _{w}^{\mathbf{II}%
}\subset \text{ }\alpha _{\mathbf{s}}^{\mathbf{II}}\right\} . \\
\end{array}
& \left( 1.3.5\right)%
\end{array}%
Definition 1.3.1.2$^{\prime }$.b. (Strong and Weak Dedekind
cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(1) A strong Dedekind cut $\alpha _{%
\mathbf{s}}=\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{s}}$ in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ is a subset
$\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{s}}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ of the hyperintegers $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ that satisfies these
1. $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ is not empty.
2.$\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{s}}=$ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\backslash \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{s}}$ is not empty.
3. $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\backslash \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{s}}$ contains no least element.
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
,$ if $x\in \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{s}}$ and $y<x,y\notin \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{s}}$ then
there is exists $z\in \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{s}}$ such that $z<y$ as well.
(2) A weak Dedekind cut $\alpha _{w}=\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{w}$ in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ is a subset
$\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{w}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ of the hyperintegers $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ that satisfies these
1. $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{w}$ is not empty.
2.$\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{w}=$ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\backslash \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{w}$ is not empty.
3. $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\backslash \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{w}$ contains no least element.
4. For every $x,y\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
,$ if $x\in \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{w}$ and $y<x,y\notin \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{w}$ then
there is exists $z\in \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{w}$ such that $z<y$ as well.
Remark. Note that for every weak Dedekind cut $\alpha _{w}$ in $%
^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
there is exists unique strong Dedekind cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ such that:
$\alpha _{\mathbf{s}}\left( \alpha _{w}\right) =\left[ \alpha _{w}\right]
_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}=$ $\bigcap \left\{ \alpha _{\mathbf{s}}|\alpha _{w}\subset \text{ }\alpha
_{\mathbf{s}}\right\} .$
Example. (1) Let $\Omega $ denotes the set $%
\left\{ n|n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\backslash ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{+\infty }\right\} .$
It is easy to see that $\Omega $ is a strong Dedekind cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(2) Let $\Omega _{k},k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
,k\neq 0$ denotes the set
$\left\{ n|\left( n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\backslash ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{+\infty }\right) \wedge \left( n|k\right) \right\} .$
It is easy to see that $\Omega $ is a weak Dedekind cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
and $\left[ \Omega _{k}\right] _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}=\Omega .$
Definition 1.3.1.3.a.(Wattenberg-Dedekind hyperreal
(1) A Wattenberg-Dedekind hyperreal number
$\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is a strong Dedekind cut $\alpha =\alpha _{\mathbf{s}}$ in $%
^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
(2) A Wattenberg-Dedekind hyperreal number
$\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is a weak Dedekind cut $\alpha =\alpha _{w}$ in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
(3) We denote the set of all Wattenberg-Dedekind
hyperreal numbers by $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ and we order them by
set-theoretic inclusion, that is to say, for any
$\alpha ,\beta \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},$ $\alpha <\beta $ if and only if $\alpha \subsetneqq \beta $
where the
inclusion is strict.
We further define $\alpha =\beta $ as real numbers
if and are equal as sets. As usual, we write
$\alpha \leqslant \beta $ if $\alpha <\beta $ or $\alpha =\beta $.
Definition 1.3.1.3.b. (Wattenberg-Dedekind hyperrationals $%
^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
(1) A Wattenberg-Dedekind hyperrational is a weak Dedekind cut
$\alpha =\alpha _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}$ in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
Definition 1.3.1.3.c. (Wattenberg-Dedekind hyperintegers $%
^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(1) A Wattenberg-Dedekind hyperinteger is a weak Dedekind cut
$\alpha =\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(2) We denote the set of all Wattenberg-Dedekind
hyperintegers by $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ and we order them by suitable
set-theoretic inclusion, that is to say, for any $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
},\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}<\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ if and only if $\left[ \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\subsetneqq \left[ \beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}$ where the inclusion is
strict. We further define:
(3) weak equality:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}=_{w}\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
} \\
\end{array}
& \left( 1.3.6\right)%
\end{array}%
as Wattenberg-Dedekind hyperintegers iff Dedekind cut $\left[ \alpha \right]
_{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
and $\left[ \beta \right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}$ are equal as sets,i.e.
\begin{array}{cc}
\begin{array}{c}
\\
\forall x\left\{ x\in \left[ \alpha \right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\iff x\in \left[ \beta \right] _{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\right\} \\
\end{array}
& \left( 1.3.7\right)%
\end{array}%
As usual, we write $\alpha \leqslant _{w}\beta $ if $\alpha <\beta $ or $%
\alpha =_{w}\beta $.
(4) strong equality:
\begin{array}{cc}
\begin{array}{c}
\\
\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}=_{s}\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
} \\
\end{array}
& \left( 1.3.8\right)%
\end{array}%
as Wattenberg-Dedekind hyperintegers iff Dedekind cut $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
and $\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ are equal as sets,i.e.
\begin{array}{cc}
\begin{array}{c}
\\
\forall x\left\{ x\in \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\iff x\in \beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\right\} \\
\end{array}
& \left( 1.3.9\right)%
\end{array}%
As usual, we write $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\leqslant _{s}\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ if $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}<\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ or $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}=_{s}\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
Remark. Note that we often write formula $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}=_{s}\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ as
$\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}=\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
Definition 1.3.1.4. Dedekind hyperreal $\alpha $ is said to be
Dedekind hyperirrational if $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \alpha $ contains no least element.
Theorem 1.3.1.1. Every nonempty subset $A\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ of Dedekind
hyperreal numbers that is bounded (hyperbounded) above has a least
upper bound.
Proof. Let $A$ be a nonempty set of hyperreal numbers, such that
for every $\alpha \in A$ we have that $\alpha \leqslant \gamma $ for some
real number $\gamma \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$Now define the set $\sup A=\dbigcup\limits_{\alpha \in
A}\alpha .$ We must show that this set is a Wattenberg-Dedekind hyperreal
number. This amounts to checking the four conditions of a Dedekind cut. $%
\sup A$ is clearly not empty, for it is the nonempty union of nonempty sets.
Because $\gamma $ is a Wattenberg-Dedekind hyperreal number, there is some
hyperrational $x\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ that is not in $\gamma .$ Since every $\alpha \in A$ is a subset of $%
\gamma ,x$ is not in any $\alpha ,$ so $%
x\notin \sup A$ either. Thus, $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \sup A$ is nonempty. If $\sup A$ had a greatest element $g\in $ $%
^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$ then $g\in \alpha $ for some $\alpha \in A.$ Then $g$ would be a greatest
element of $\alpha ,$ but $\alpha $ is a Wattenberg-Dedekind hyperreal
number, so by contrapositive law,$\sup A$ has no greatest element. Lastly,
if $x\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ and $x\in \sup A,$ then $x\in \alpha $ for some $\alpha ,$ so given any $%
y\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$ $y<x$ because $\alpha $ is a Dedekind hyperreal number $y\in \alpha $
whence $y\in \sup A.$Thus $\sup A,$ is a Wattenberg-Dedekind
hyperreal number.Trivially,$\sup A\ $is an upper bound of $A,$ for every $%
\alpha \in A,$ $\alpha \subseteqq \sup A.$ It now suffices to prove that $%
\sup A\leqslant \gamma ,$because was an arbitrary upper bound. But this is
easy, because every $x\in \sup A,x\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ is an element of $\alpha $ for some $\alpha \in A,$ so because $\alpha
\subseteq \gamma ,$ $x\in \gamma .$ Thus, $\sup A$ is the least upper bound
of $A$.
Definition 1.3.1.5.a. Given two Wattenberg-Dedekind hyperreal
numbers $\alpha $ and $\beta $ we define:
1.The additive identity (zero cut) $0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}},$ denoted $0,$is
$0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|\text{ }x<0\right\} .$
2.The multiplicative identity $1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}},$ denoted $1,$is
$1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|\text{ }x<_{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}1_{^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\right\} .$
3. Addition $\alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta $ of $\alpha $ and $\beta $ denoted $\alpha +\beta $ is
$\alpha +\beta \triangleq \left\{ x+y|\text{ }x\in \alpha ,y\in \beta
\right\} .$
It is easy to see that $\alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}=\alpha $ for all $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
It is easy to see that $\alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta $ is a cut in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ and $\alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta =\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha .$
Another fundamental property of cut addition is associativity:
$\left( \alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma =\alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) .$
This follows from the corresponding property of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
4.The opposite $-_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha $ of $\alpha ,$ denoted $-\alpha ,$ is
$-\alpha \triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|\text{ }-x\notin \alpha ,-x\text{ is not the least element of }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\backslash \alpha \right\} .$
5.Remark. We also say that the opposite $-\alpha $ of $\alpha $ is
the additive
inverse of $\alpha $ denoted $\div \alpha $ iff the next equality
is satisfied:
$\alpha +\left( \div \alpha \right) =0.$
6.Remark. It is easy to see that for all internal cut $\alpha ^{%
\mathbf{Int}}$ the opposite
$-\alpha ^{\mathbf{Int}}$ is the additive inverse of $\alpha ^{\mathbf{Int}%
}, $i.e. $\alpha ^{\mathbf{Int}}+\left( \div \alpha ^{\mathbf{Int}}\right)
7.Example. (External cut $X$ without additive inverse $\div X$) For
$x,y\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ we denote:
$x\ll \infty \triangleq \exists r\left[ \left( r\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) \wedge \left( x<\text{ }^{\ast }r\right) \right] ,$ $y\approx
-\infty \triangleq \forall r\left[ r\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
\implies x<\text{ }^{\ast }r\right] .$
Let us consider two Dedekind hyperreal numbers $X$ and $Y$ defined as:
$X=\left\{ x|\left( x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) \wedge \left( x\ll \infty \right) \right\} ,Z=\left\{ x|z<0\right\}
. $
It is easy to see that is no exist cut $Y$ such that: $X+Y=Z.$
Proof. Suppose that cut $Y$ such that: $X+Y=Z$ exist. It is easy to
check that $\forall y\left[ y\in Y\implies y\approx -\infty \right] .$
Suppose that $y\in Y,$then
$\forall x\left[ x\in X\implies x+y\in Z\right] ,$i.e.$\forall \left( x\ll
\infty \right) \left[ x+y<0\right] .$Hence
$\forall \left( x\ll \infty \right) \left[ y<-x\right] ,$i.e. $y\approx
-\infty .$It is easy to check that $Z\nsubseteqq X+Y.$
If $x\in X$ and $y\in Y$ then $x\ll \infty $ and $y\approx -\infty ,$hence $%
x+y\neq -1,$i.e.
$-1\notin X+Y.$Thus $Z\nsubseteqq X+Y.$This is a contradiction.
8.We say that the cut $\alpha $ is positive if $0<\alpha $ or
negative if $\alpha <0.$
The absolute value of $\alpha ,$denoted $\left\vert \alpha \right\vert ,$is $%
\left\vert \alpha \right\vert \triangleq \alpha ,$if $\alpha \geq 0$ and $%
\left\vert \alpha \right\vert \triangleq -\alpha ,$
if $\alpha \leq 0$
9.If $\alpha ,\beta >0$ then multiplication $\alpha \times _{^{\ast
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta $ of $\alpha $ and $\beta $ denoted $\alpha \times \beta
$ is
$\alpha \times \beta \triangleq \left\{ z\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|\text{ }z=x\times y\text{ for some }x\in \alpha ,y\in \beta \text{ with }%
x,y>0\right\} .$
In general, $\alpha \times \beta =0$ if $\alpha =\mathbf{0}$ or $\beta =0%
\mathbf{,}$
$\alpha \times \beta \triangleq \left\vert \alpha \right\vert \times
\left\vert \beta \right\vert $ if $\alpha >0,\beta >0$ or $\alpha <0,\beta <0%
\mathbf{,}$
$\alpha \times \beta \triangleq -\left( \left\vert \alpha \right\vert \cdot
\left\vert \beta \right\vert \right) $ if $\alpha >0,\beta <0\mathbf{,}$or $%
\alpha <0,\beta >0\mathbf{.}$
10. The cut order enjois on $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ the standard additional properties of:
(i) transitivity: $\alpha \leq \beta \leq \gamma
\implies \alpha \leq \gamma .$
(ii) trichotomy: eizer $\alpha <\beta ,\beta <\alpha $
or $\alpha =\beta $ but only one
of the three things is true.
(iii) translation: $\alpha \leq \beta \implies \alpha
+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \leq \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma .$
11.By definition above, this is what we mean when we say that $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is an
ordered pseudo-ring or ordered pseudo-field.
Remark. We embed $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ in the standard way [24].If $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ the
corresponding element, $\alpha ^{\#},$ of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is
\begin{array}{cc}
\begin{array}{c}
\\
\alpha ^{\#}\triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
|x<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\alpha \right\} \\
\end{array}
& \left( 1.3.10\right)%
\end{array}%
Definition 1.3.1.5.b. Given two Wattenberg-Dedekind
hyperintegers $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ and $\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ we define:
1.The additive identity (zero cut) denoted $0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}$ or $0,$is
$0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|\text{ }x\leq 0\right\} .$
2.The multiplicative identity denoted $1_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}$ or $1,$is
$1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|\text{ }x\leq _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}1_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\right\} .$
3. Addition of $\alpha =\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ and $\beta =\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ denoted $\alpha +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta $
or $\alpha +\beta $ is $\alpha +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta \triangleq \left\{ x+_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}y|\text{ }x\in \alpha ,y\in \beta \right\} .$
It is easy to see that $\alpha +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}=\alpha $ for all $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
It is easy to see that $\alpha +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta $ is a cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ and
$\alpha +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta =\beta +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\alpha .$
Another fundamental property of cut addition is
$\left( \alpha +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta \right) +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\gamma =\alpha +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\gamma \right) .$
This follows from the corresponding property of $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
4.The opposite of $\alpha =\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
},$ denoted $-_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\alpha $ or $-\alpha ,$ is
$-_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\alpha \triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|\text{ }-x\notin \alpha \right\} .$
5.Remark. We also say that the opposite $-_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ of $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
is the additive inverse of $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ denoted $\div _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\alpha $ or $\div \alpha $ iff the
next equality is satisfied: $\alpha +_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\left( \div _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\alpha \right) =0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
6.Remark. It is easy to see that for all internal cut $\alpha
_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}^{\mathbf{Int}}$ the
opposite $-_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{Int}}$ is the additive inverse of $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{Int}}+_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\left( \div _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}^{\mathbf{Int}}\right) =0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
8.We say that the cut $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ is positive if $0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}<\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ or
negative if $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}<0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
The absolute value of $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
},$denoted $\left\vert \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\right\vert ,$is $\left\vert \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\right\vert \triangleq \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
if $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\geq 0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}$ and $\left\vert \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\right\vert \triangleq -_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
},$if $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\leq 0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
9.If $\alpha =\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
},\beta =\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}>0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}$ then multiplication of $\alpha $ and $\beta $
denoted $\alpha \times _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta $ or $\alpha \times \beta $ is
$\alpha \times _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta \triangleq \left\{ z\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|\text{ }z=x\times _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}y\text{ for some }x\in \alpha ,y\in \beta \text{ with }x,y>0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\right\} .$
In general, $\alpha \times _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta =0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}$ if $\alpha =0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}$ or $\beta =0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$\alpha \times _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta \triangleq \left\vert \alpha \right\vert \times _{^{\ast
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\left\vert \beta \right\vert $ if $\alpha >0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}},\beta >0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}$ or $\alpha <0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}},\beta <0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$\alpha \times _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta \triangleq -\left( \left\vert \alpha \right\vert \times
_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\left\vert \beta \right\vert \right) $ if $\alpha >0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}},\beta <0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\mathbf{,}$or $\alpha <0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}},\beta >0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
10. The cut order enjois on $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ the standard additional properties of:
(i) transitivity: $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\leq \beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\leq \gamma _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\implies \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\leq \gamma _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(ii) trichotomy: eizer $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}<\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
},\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}<\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ or $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}=\beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}$ but only one
of the three things is true.
(iii) translation: $\alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\leq \beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\implies \alpha _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}\leq \beta _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
Lemma 1.3.1.1.[24].
(i) Addition $\left( \circ +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\circ \right) $ is commutative and associative in$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(ii) $\forall \alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}=\alpha .$
(iii) $\forall \alpha ,\beta \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\alpha ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\#}=\left( \alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) ^{\#}.$
Proof. (i) Is clear from definitions.
(ii) Suppose $a\in \alpha .$ Since $a$ has no greatest element $%
\exists b\left[ \left( b>a\right) \wedge \left( b\in \alpha \right) \right]
. $
Thus $a-b\in 0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}$ and $a=(a$ $-$ $b)+b\in 0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}+\alpha .$
(iii) (a) $\alpha ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\#}\subseteqq \left( \alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) ^{\#}$ is clear since:
$\left( x<\alpha \right) \wedge \left( y<\beta \right) \implies $ $%
x+y<\alpha +\beta .$
(b) Suppose $x<\alpha +\beta .$ Thus $\alpha -\dfrac{\left( \alpha +\beta
\right) -x}{2}<\alpha $ and
$\beta -\dfrac{\left( \alpha +\beta \right) -x}{2}<\beta .$So one obtain
$x=\left[ \left( \alpha -\dfrac{\left( \alpha +\beta \right) -x}{2}\right)
+\left( \beta -\dfrac{\left( \alpha +\beta \right) -x}{2}\right) \right] \in
\alpha ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\#},$
$\left( \alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) ^{\#}\subseteqq \alpha ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\#}.$
Notice, here again something is lost going from $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ to $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ since $a<\beta $ does
not imply $\alpha +\alpha <\beta +\alpha $ since $0<\varepsilon _{\mathbf{d}%
} $ but $0+\varepsilon _{\mathbf{d}}=\varepsilon _{\mathbf{d}}+\varepsilon _{%
\mathbf{d}}=\varepsilon _{\mathbf{d}}.$
Lemma 1.3.1.2.[24].
(i) $\leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}$a linear ordering on $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},$which extends the usual ordering on $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(ii) $\left( \alpha \leq \alpha ^{\prime }\right) \wedge \left(
\beta \leq \beta ^{\prime }\right) \implies \alpha +\beta \leq \alpha
^{\prime }+\beta ^{\prime }.$
(iii) $\left( \alpha <\alpha ^{\prime }\right) \wedge \left( \beta
<\beta ^{\prime }\right) \implies \alpha +\beta <\alpha ^{\prime }+\beta
^{\prime }.$
(iv) $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is dense in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$That is if $\alpha <\beta $ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ there is an $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ then
$\ \ \ \ \ \ \ \alpha <a^{\#}<\beta .$
Lemma 1.3.1.3.[24].
(i) If $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ then $-_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \alpha ^{\#}\right) =\left( -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\alpha \right) ^{\#}.$
(ii) $\mathbf{-}_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) =\alpha .$
(iii) $\alpha \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \iff -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}-_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha .$
(iv) $\left( \mathbf{-}_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) +_{_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\left( -_{_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\beta \right) \leq _{_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\mathbf{-}_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \alpha +_{_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\beta \right) .$
(v) $\forall a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\left( \mathbf{-}_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}a\right) ^{\#}+_{_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\left( -_{_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\beta \right) =-_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( a^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) .$
(vi) $\alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Lemma 1.3.1.4.[24].
(i) $\forall a,b\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\left( a\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}b\right) ^{\#}=a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(ii) Multiplication $\left( \cdot \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\cdot \right) $ is associative and commutative:
$\alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta =\beta \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha ,\left( \alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma =\alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) .$
(iii) $1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha =\alpha ;$ $-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha =-_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha ,$ where $1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}=\left( 1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\right) ^{\#}.$
(iv) $\left\vert \alpha \right\vert \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left\vert \beta \right\vert =\left\vert \beta \right\vert
\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left\vert \alpha \right\vert .$
(v) $\left[ \left( \alpha \geq 0\right) \wedge \left( \beta
\geq 0\right) \wedge \left( \gamma \geq 0\right) \right] \implies $
$\ \ \ \ \ \ \ \implies \alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) =\alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma .$
(vi) $0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha <_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha ^{\prime },0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta <_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\prime }\implies $
$\ \ \ \ \ \ \ \implies \alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta <_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha ^{\prime }\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\prime }.$
Proof.(v) Clearly $\alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) \leq \alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma .$
Suppose $d\in \alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma .$Hence:
$d=ab+a^{\prime }c,$where $a,a^{\prime }\in \alpha ,b\in \beta ,c\in \gamma
. $
Without loss of generality we may assume $a\leq a^{\prime }.$Hence:
$d=ab+a^{\prime }c\leq a^{\prime }b+a^{\prime }c=a^{\prime }\left(
b+c\right) \in \alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) .$
Definition 1.3.1.6. Suppose $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},0<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha $ then $\alpha ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}$ is defined
as follows:
(i) $0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha :\alpha ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\triangleq \inf \left\{ a^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}|a\in \alpha \right\} ,$
(ii) $_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha <_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}0:\alpha ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\triangleq -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Lemma 1.3.1.5.[24].
(i) $\forall a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\left( a^{\#}\right) ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}=\left( a^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}\right) ^{\#}.$
(ii) $\left( \alpha ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}\right) ^{^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}}=\alpha .$
(iii) $0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \implies \beta ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(iv) $\left[ \left( 0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) \wedge \left( 0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) \right] \implies $
$\ \ \ \ \ \implies \left( \alpha ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}\right) \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}\right) \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(v) $\forall a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:a\neq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}0_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}\implies \left( \alpha ^{\#}\right) ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\right) =\left( \alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(vi) $\alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\mathbf{Lemma}$ $\mathbf{1.3.1.5}^{\ast }$.Suppose that $%
a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,a>0,\beta ,\gamma \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$ Then
$a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) =a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma .$
Proof. Clearly $a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) \leq a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma .$
$\left( a^{\#}\right) ^{^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}}\left( a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) \leq $
$\leq \left( a^{\#}\right) ^{^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}}\left( a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( a^{\#}\right) ^{^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}}\left( a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) =$
$=\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma .$Thus $\left( a^{\#}\right) ^{^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}}}\left( a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) \leq \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma $ and
one obtain $a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \leq a^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) .$
$\mathbf{Lemma}$ $\mathbf{1.3.1.6.}$ ($\mathbf{General}$
$\mathbf{Strong}$ $\mathbf{Approximation}$ $\mathbf{%
If $A$ is a nonempty subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded from above, then
$\sup (A)$ is the unique number such that:
(i) $\sup (A)$ is an upper bound for $A$ and
(ii) for any $\alpha \in \sup (A)$ there exists $x\in A$ such that $%
\alpha <x\leq \sup (A).$
Proof. If not, then $\alpha $ is an upper bound of $A$ less than
the least upper
bound $\sup (A)$, which is a contradiction.
Lemma 1.3.1.7.Let $\mathbf{A}$ and $\mathbf{B}$ be nonempty subsets
of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ and
$\mathbf{C}=$ $\left\{ a+b:a\in \mathbf{A},b\in \mathbf{B}\right\} $.If $%
\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from above,hence $\sup \left( \mathbf{A}\right) $ and $\sup \left( \mathbf{B}%
\right) $ exist, then $\mathbf{s}$-$\sup \left( \mathbf{C}\right) $ exist and
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\sup \left( \mathbf{C}\right) =\sup \left( \mathbf{A}\right) +\sup \left(
\mathbf{B}\right) . \\
\end{array}
& \text{ }\left( 1.3.11\right) \text{\ }%
\end{array}%
Proof.Suppose $c<\sup \left( \mathbf{A}\right) +\sup \left( \mathbf{%
B}\right) .$From Lemma 1.3.1.2.(iv) $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is
dense in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$So there is exists $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that $c<x^{\#}<\sup \left( \mathbf{A}\right) +\sup \left( \mathbf{B}%
\right) .$
Suppose that $\alpha ,\beta \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and $\alpha ^{\#}<\sup \left( \mathbf{A}\right) ,\beta ^{\#}<\sup \left(
\mathbf{B}\right) .$
From Lemma 1.3.1.6 (General Strong Approximation Property
) one obtain
there is exists $a\in \mathbf{A},b\in \mathbf{B}$ such that $\alpha
^{\#}<a<\sup \left( \mathbf{A}\right) ,\beta ^{\#}<b<\sup \left( \mathbf{B}%
\right) .$
Suppose $x^{\#}<\alpha ^{\#}+\beta ^{\#}.$Thus one obtain:
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha ^{\#}-\dfrac{\left( \alpha ^{\#}+\beta ^{\#}\right) -x^{\#}}{2}%
<\alpha ^{\#}<a<\sup \left( \mathbf{A}\right) \\
\end{array}
& \text{\ }\left( 1.3.12\right) \text{\ }%
\end{array}%
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\beta ^{\#}-\dfrac{\left( \alpha ^{\#}+\beta ^{\#}\right) -x^{\#}}{2}<\beta
^{\#}<b<\sup \left( \mathbf{B}\right) .\bigskip \\
\end{array}
& \text{ \ \ }\left( 1.3.13\right) \text{\ }%
\end{array}%
So one obtain
$\bigskip $
\begin{array}{cc}
\begin{array}{c}
\\
x^{\#}=\left[ \left( \alpha ^{\#}-\dfrac{\left( \alpha ^{\#}+\beta
^{\#}\right) -x^{\#}}{2}\right) +\left( \beta ^{\#}-\dfrac{\left( \alpha
^{\#}+\beta ^{\#}\right) -x^{\#}}{2}\right) \right] \\
\\
<\alpha ^{\#}+\beta ^{\#}<a+b<\sup \left( \mathbf{A}\right) +\sup \left(
\mathbf{B}\right) . \\
\end{array}
& \text{ \ }\left( 1.3.14\right) \text{\ \ \ \ \ \ \ \ }%
\end{array}%
But $a+b\in \mathbf{C,}$hence by using Lemma 1.3.1.4 one obtain that
$\sup \left( \mathbf{C}\right) =\sup \left( \mathbf{A}\right) +\sup \left(
\mathbf{B}\right) .$
Theorem 1.3.1.2. Let $\mathbf{A}$ and $\mathbf{B}$ be nonempty
subsets of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ and
$\mathbf{C}=$ $\left\{ a+b:a\in \mathbf{A},b\in \mathbf{B}\right\} $.If $%
\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from above,hence $\sup \left( \mathbf{A}\right) $ and $\sup \left( \mathbf{B}%
\right) $ exist, then $\mathbf{s}$-$\sup \left( \mathbf{C}\right) $ exist
and $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\sup \left( \mathbf{C}\right) =\sup \left( \mathbf{A}\right) +\sup \left(
\mathbf{B}\right) . \\
\end{array}
& \text{ \ \ \ \ \ }\left( 1.3.3.1\right) \text{\ \ \ \ }%
\end{array}%
Proof.Suppose $c<\sup \left( \mathbf{A}\right) +\sup \left( \mathbf{%
B}\right) .$From Lemma 1.3.1.2.(iv) $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is
dense in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$So there is exists $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that $c<x^{\#}<\sup \left( \mathbf{A}\right) +\sup \left( \mathbf{B}%
\right) .$
Suppose that $\alpha ,\beta \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and $\alpha ^{\#}<\sup \left( \mathbf{A}\right) ,\beta ^{\#}<\sup \left(
\mathbf{B}\right) .$From Lemma 1.3.1.4
(General Strong Approximation Property)one obtain there is exists
$a\in \mathbf{A},b\in \mathbf{B}$ such that $\alpha ^{\#}<a<\sup \left(
\mathbf{A}\right) ,\beta ^{\#}<b<\sup \left( \mathbf{B}\right) .$ Suppose
$x^{\#}<\alpha ^{\#}+\beta ^{\#}.$Thus one obtain:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha ^{\#}-\dfrac{\left( \alpha ^{\#}+\beta ^{\#}\right) -x^{\#}}{2}%
<\alpha ^{\#}<a<\sup \left( \mathbf{A}\right) \\
\end{array}
& \text{ \ \ \ \ \ \ \ }%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\beta ^{\#}-\dfrac{\left( \alpha ^{\#}+\beta ^{\#}\right) -x^{\#}}{2}<\beta
^{\#}<b<\sup \left( \mathbf{B}\right) .\bigskip \\
\end{array}
& \text{ \ \ \ \ \ \ \ }%
\end{array}%
So one obtain
$\ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
x^{\#}=\left[ \left( \alpha ^{\#}-\dfrac{\left( \alpha ^{\#}+\beta
^{\#}\right) -x^{\#}}{2}\right) +\left( \beta ^{\#}-\dfrac{\left( \alpha
^{\#}+\beta ^{\#}\right) -x^{\#}}{2}\right) \right] < \\
\\
<\alpha ^{\#}+\beta ^{\#}< \\
\\
<a+b<\sup \left( \mathbf{A}\right) +\sup \left( \mathbf{B}\right) . \\
\end{array}
& \text{ \ \ \ \ \ \ \ }%
\end{array}%
But $a+b\in \mathbf{C,}$hence by using Lemma 1.3.1.4 one obtain that
$\sup \left( \mathbf{C}\right) =\sup \left( \mathbf{A}\right) +\sup \left(
\mathbf{B}\right) .$
Theorem 1.3.1.3.Suppose that $\mathbf{S}$ is a non-empty subset of $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is
bounded or hyperbounded from above,i.e. $\sup \left( \mathbf{S}\right) $
exist and suppose that
$\xi \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\xi >0.$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\underset{x\in \mathbf{S}}{\sup }\left\{ \xi ^{\#}\times x\right\} =\xi
^{\#}\times \left( \underset{x\in \mathbf{S}}{\sup }\left\{ x\right\}
\right) =\xi ^{\#}\times \left( \sup \mathbf{S}\right) \mathbf{.} \\
\end{array}
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ }\left( 1.3.3.2\right) \text{\ \ \ \ \ \ \ }%
\end{array}%
Proof.Let $B=\mathbf{s}$-$\sup \mathbf{S.}$Then $B$ is the smallest
number such that, for
any $x\in \mathbf{S,}x$ $\mathbf{\leq B.}$Let $\mathbf{T}=\left\{ \xi
^{\#}\times x|x\in \mathbf{S}\right\} .$Since $\xi ^{\#}>0,\xi ^{\#}\times
x\leq \xi ^{\#}\times B$ for any
$x\in \mathbf{S.}$Hence $\mathbf{T}$ is bounded or hyperbounded above by $%
\xi ^{\#}\times B.$Hence
$\mathbf{T}$ has a supremum $C_{\mathbf{T}}=\mathbf{s}$-$\sup \mathbf{T.}$
Now we have to pruve that $C_{\mathbf{T}}=\xi ^{\#}\times B=$
$=\xi ^{\#}\times \left( \sup \mathbf{S}\right) .$Since $\xi ^{\#}\times
B=\xi ^{\#}\times \left( \sup \mathbf{S}\right) $ is an apper bound for $%
\mathbf{T}$and $C$ is the
smollest apper bound for $\mathbf{T,}C_{\mathbf{T}}\leq \xi ^{\#}\times B.$
Now we repeat the argument above
with the roles of $\mathbf{S}$ and $\mathbf{T}$ reversed. We know that $C_{%
\mathbf{T}}$ is the smallest number
such that, for any $y\in \mathbf{T,}y\leq C_{\mathbf{T}}.$Since $\xi >0$ it
follows that
$\left( \xi ^{\#}\right) ^{-1}\times y\leq \left( \xi ^{\#}\right)
^{-1}\times C_{\mathbf{T}}$ for any $y\in \mathbf{T.}$But $\mathbf{S=}%
\left\{ \left( \xi ^{\#}\right) ^{-1}\times y|y\in \mathbf{T}\right\} .$Hence
$\left( \xi ^{\#}\right) ^{-1}\times C_{\mathbf{T}}$ is an apper bound for $%
\mathbf{S.}$But $B$ is a supremum for $\mathbf{S.}$Hence
$B\leq \left( \xi ^{\#}\right) ^{-1}\times C_{\mathbf{T}}$ and $\xi
^{\#}\times B\leq C_{\mathbf{T}}.$We have shown that $C_{\mathbf{T}}\leq \xi
^{\#}\times B$ and also
that $\xi ^{\#}\times B\leq C_{\mathbf{T}}.$Thus $\xi ^{\#}\times B=C_{%
\mathbf{T}}.$
Theorem 1.3.1.4. Suppose that $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ $\alpha >0,\beta \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},\gamma \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) =\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma . \\
\end{array}
& \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left( 1.3.3.3\right)%
\end{array}%
$\ \ \ \ \ \ \ \ $
Proof.Let us consider any two sets $S_{\beta }\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and $S_{\gamma }\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that:
$\beta =\sup \left( S_{\beta }\right) ,\gamma =\sup \left( S_{\gamma
}\right) .$Thus by using Theorem 1.3.1.3 and$\ $
Theorem 1.3.1.2 one obtain:
$\bigskip $ $%
\begin{array}{cc}
\begin{array}{c}
\\
\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) =\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\sup \left( S_{\beta }+S_{\gamma }\right) = \\
\\
=\sup \left[ \alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( S_{\beta }+S_{\gamma }\right) \right] =\sup \left[
\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}S_{\beta }+\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}S_{\gamma }\right] = \\
\\
=\sup \left( \alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}S_{\beta }\right) +\sup \left( \alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}S_{\gamma }\right) = \\
\\
\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\sup \left( S_{\beta }\right) +\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\sup \left( S_{\gamma }\right) . \\
\end{array}
\end{array}%
$\bigskip $Theorem 1.3.1.5. Suppose that $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\alpha <0,\beta \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\gamma \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$Then $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) =\left( -1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\right) \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left[ \left\vert \alpha ^{\#}\right\vert \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left\vert \alpha ^{\#}\right\vert \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right] . \\
\end{array}
& \text{ \ }\left( 1.3.3.4\right)%
\end{array}%
Proof.Let us consider any set $S_{\gamma }\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that:$\gamma =\sup \left( S_{\gamma }\right) .$Thus by
using Theorem 1.3.1.3, Theorem 1.3.1.2 and$\ $Lemma 1.3.1.3
(v) one obtain:
\begin{array}{cc}
\begin{array}{c}
\\
\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) =\left\vert \alpha ^{\#}\right\vert \times
_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( -1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\right) \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( \beta ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) = \\
\\
=\left\vert \alpha ^{\#}\right\vert \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left[ \left( -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\#}\right) +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) \right] = \\
\\
=\left\vert \alpha ^{\#}\right\vert \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\#}\right) +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left\vert \alpha ^{\#}\right\vert \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma \right) = \\
\\
=\left\vert \alpha ^{\#}\right\vert \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( -1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\right) \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left\vert \alpha ^{\#}\right\vert \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( -1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\right) \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma = \\
\\
=\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta ^{\#}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha ^{\#}\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\gamma . \\
\end{array}
\end{array}%
§ I.3.2.THE TOPOLOGY OF $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
_{\MATHBF{D}}.$WATTENBERG STANDARD PART.
Fortunately topologically, $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has many properties strongly reminiscent
of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ itself. We proceed as follows [24].
Definition 1.3.2.1.
(i) $\left( \alpha ,\beta \right) _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\triangleq \left\{ u|\alpha <_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}u<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right\} ,$
(ii) $\left[ \alpha ,\beta \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\triangleq \left\{ u|\alpha \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}u\leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right\} .$
Definition 1.3.2.2.[24].Suppose $U\subseteqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$. Then $U$ is open if and only
if for every $u\in U,$ $\exists \alpha _{\alpha \in ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\exists \beta _{\beta \in ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left[ \alpha <_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}u<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right] $ such that
$u\in \left( \alpha ,\beta \right) _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\subseteqq U.$
Remark.1.3.2.1.[24]. Notice this is not equivalent to:
$\forall u_{u\in U}\exists \varepsilon _{\varepsilon >0}\left[ \left(
u-\varepsilon ,u+\varepsilon \right) _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\subseteqq U\right] .$
Lemma 1.3.2.1.[24].
(i) $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is dense in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(ii) $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\backslash ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is dense in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Lemma 1.3.2.2.[24]. Suppose $A\subseteqq $ $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$Then $A$ is closed if and
only if:
(i) $\forall E\left( E\subseteqq A\right) $ $E$ bounded
above implies $\sup \left( E\right) \in A,$ and
(ii) $\forall E\left( E\subseteqq A\right) $ $E$
bounded below implies $\inf \left( E\right) \in A.$
Proposition 1.3.2.1.[24].
(i) $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is connected.
(ii) For $\alpha <_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta $ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ set $\left[ \alpha ,\beta \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}$ is compact.
(iii) Suppose $A\subseteqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$Then $A$ is compact if and
only if $A$ is closed and bounded.
(iv) $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is normal.
(v) The map $\alpha \longmapsto -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha $ is continuous.
(vi) The map $\alpha \longmapsto \alpha ^{^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}}$ is continuous.
(vii) The maps $\left( \alpha ,\beta \right) \longmapsto \left(
\alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) $ and $\left( \alpha ,\beta \right) \longmapsto
\left( \alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) $
are not continuous.
Definition 1.3.2.3.[24].(Wattenberg Standard Part)
(i) Suppose $\alpha \in \left( -\Delta _{\mathbf{d}},\Delta _{%
\mathbf{d}}\right) _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}.$Then there is a unique
standard $x\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ called $WST\left( \alpha \right) ,$ such that $x\in \left[ \alpha
-\varepsilon _{\mathbf{d}},\alpha +\varepsilon _{\mathbf{d}}\right] _{^{\ast
%TCIMACRO{\U{211d} }%
\mathbb{R}
(ii) $\alpha \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta $ implies $WST\left( \alpha \right) \leq WST\left( \beta
\right) ,$
(iii) the map $WST\left( \cdot \right) :$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is continuous,
(iv) $WST\left( \alpha +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) =WST\left( \alpha \right) +WST\left( \beta
\right) ,$
(v) $WST\left( \alpha \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\beta \right) =WST\left( \alpha \right) \times WST\left( \beta
\right) ,$
(vi) $WST\left( -_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) =-WST\left( \alpha \right) ,$
(vii) $WST\left( \alpha ^{-1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}}\right) =\left[ WST\left( \alpha \right) \right] ^{-1}$ if $%
\alpha \notin \left[ -\varepsilon _{\mathbf{d}},\varepsilon _{\mathbf{d}}%
\right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Proposition 1.3.2.2.[24].Suppose $f:$ $\left[ a,b\right]
\rightarrow $ $A\subseteqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is internal,
$\ast $-continuous, and monotonic. Then
(1) $f$ has a unique continuous extension $f^{\#}$ $\left[ a,b%
\right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}$ $\overline{A}$ $\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$, where
$\overline{A}$ denotes the closure of $A$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(2)The conclusion (1) above holds iff is piecewise
(i.e., the domain can be decomposed into a finite (not $\ast $-finite) number
of intervals on each of which $f$ is monotonic).
Proposition 1.3.2.3.[24].Suppose $f,g$ are $\ast $
-continuous, piecewise
monotonic functions then
(i) $f\circ g$ is also and
(ii) $\left( f\circ g\right) ^{\#}=\left( f^{\#}\right) \circ
\left( g^{\#}\right) .$
§ I.3.3.ABSORPTION NUMBERS IN $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
_{\MATHBF{D}}$ AND IDEMPOTENTS.
§ I.3.3.1.ABSORPTION FUNCTION AND NUMBERS IN $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
One of standard ways of defining the completion of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ involves restricting oneself to subsets a which have the following
property $\forall \varepsilon _{\varepsilon >0}\exists x_{x\in \alpha }$ $%
\exists y_{y\in \alpha }\left[ y\text{ }-\text{ }x<\varepsilon \right] $. It
is well known that in this case we obtain a field. In fact the proof is
essentially the same as the one used in the case of ordinary Dedekind cuts
in the development of the standard real numbers, $\varepsilon _{\mathbf{d}},$
of course, does not have the above property because no infinitesimal
works.This suggests the introduction of the concept of absorption part $%
\mathbf{ab.p.}\left( \alpha \right) $ of a number $\alpha $ for an element $%
\alpha $ of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which, roughly speaking, measures how much a departs from
having the above property [23]. We also introduce similar concept of an
absorption number $\alpha \left( \mathbf{ab.n.}\right) \beta \triangleq
\mathbf{ab.n.}\left( \alpha ,\beta \right) $ (cut) for given element $\beta $
of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Definition 1.3.3.1.1.[23].$\mathbf{ab.p.}\left( \alpha
\right) \triangleq \left\{ d\geq 0|\forall x_{x\in \alpha }\left[ x+d\in
\alpha \right] \right\} .$
Example 1.3.3.1.(i) $\forall \alpha \in $ $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\mathbf{ab.p.}\left( \alpha \right) =0,$
(ii) $\mathbf{ab.p.}\left( \varepsilon _{\mathbf{d}}\right)
=\varepsilon _{\mathbf{d}},$ (iii) $\mathbf{ab.p.}\left(
-\varepsilon _{\mathbf{d}}\right) =\varepsilon _{\mathbf{d}},$
(iv) $\forall \alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\mathbf{ab.p.}\left( \alpha +\varepsilon _{\mathbf{d}}\right) =\varepsilon
(v) $\ \ \forall \alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\mathbf{ab.p.}\left( \alpha -\varepsilon _{\mathbf{d}}\right) =\varepsilon
Definition 1.3.3.2. $\mathbf{ab.n.}\left( \alpha ,\beta \right)
\iff \alpha +\beta =\alpha .$
Example 1.3.3.2.(i)$\ \forall \beta \approx 0:$ $\mathbf{%
ab.n.}\left( \varepsilon _{\mathbf{d}},\beta \right) ,$
(ii) $\mathbf{ab.n.}\left( \varepsilon _{\mathbf{d}},\varepsilon _{%
\mathbf{d}}\right) ,\mathbf{ab.n.}\left( -\varepsilon _{\mathbf{d}%
},\varepsilon _{\mathbf{d}}\right) ,\mathbf{ab.n.}\left( -\varepsilon _{%
\mathbf{d}},-\varepsilon _{\mathbf{d}}\right) ,$
(iii) $\forall \alpha \in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\mathbf{ab.n.}\left( \alpha +\varepsilon _{\mathbf{d}},\varepsilon _{%
\mathbf{d}}\right) ,\mathbf{ab.n.}\left( \alpha -\varepsilon _{\mathbf{d}%
},\varepsilon _{\mathbf{d}}\right) ,\mathbf{ab.n.}\left( \alpha -\varepsilon
_{\mathbf{d}},-\varepsilon _{\mathbf{d}}\right) ,$
(iv) $\forall \alpha \in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
:\mathbf{ab.n.}\left( \Delta _{\mathbf{d}},\beta \right) ,$
(v) $\mathbf{ab.n.}\left( \Delta _{\mathbf{d}},\Delta _{%
\mathbf{d}}\right) ,\mathbf{ab.n.}\left( -\Delta _{\mathbf{d}},\Delta _{%
\mathbf{d}}\right) ,\mathbf{ab.n.}\left( -\Delta _{\mathbf{d}},-\Delta _{%
\mathbf{d}}\right) .$
Lemma 1.3.3.1.[23].(i) $c<\mathbf{ab.p.}\left( \alpha
\right) $ and $0\leq d<c\implies d\in \mathbf{ab.p.}\left( \alpha \right) $
(ii) $c\in \mathbf{ab.p.}\left( \alpha \right) $ and $d\in \mathbf{%
ab.p.}\left( \alpha \right) \implies c+d\in \mathbf{ab.p.}\left( \alpha
\right) .$
Remark 1.3.3.1. By Lemma 1.3.2.1 $\mathbf{ab.p.}%
\left( \alpha \right) $ may be regarded as an
element of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ by adding on all negative elements of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ to $\mathbf{ab.p.}\left( \alpha \right) .$
Of course if the condition $d\geq 0$ in the definition of $\mathbf{ab.p.}%
\left( \alpha \right) $ is deleted we
automatically get all the negative elements to be in $\mathbf{ab.p.}\left(
\alpha \right) $ since
$x<y\in \alpha \implies x\in \alpha .$The reason for our definition is that
the real interest
lies in the non-negative numbers. A technicality occurs if $\mathbf{ab.p.}%
\left( \alpha \right) =\left\{ 0\right\} $.
We then identify $\mathbf{ab.p.}\left( \alpha \right) $ with $0.$ [$\mathbf{%
ab.p.}\left( \alpha \right) $ becomes $\{x|x<0\}$ which by
our early convention is not in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Remark 1.3.3.1.2.By Lemma 1.3.2.1(
ii), $\mathbf{ab.p.}\left( \alpha \right) $ is idempotent.
Lemma 1.3.3.1.2.[23].
(i) $\mathbf{ab.p.}(\alpha )$ is the maximum element $\beta \in $ $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that $\alpha +\beta =\alpha .$
(ii) $\mathbf{ab.p.}(\alpha )\leq \alpha $ for $\alpha >0.$
(iii) If $\alpha $ is positive and idempotent then $\mathbf{ab.p.}%
(\alpha )=\alpha .$
Lemma 1.3.3.1.3.[23]. Let $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ satsify $\alpha >0.$ Then the following are
equivalent. In what follows assume $a,b>0.$
(i) $\ \ \alpha $ is idempotent,
(ii) $\ a,b\in \alpha \implies a+b\in \alpha ,$
(iii) $a\in \alpha \implies 2a\in \alpha ,$
(iv) $\forall n_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left[ a\in \alpha \implies n\cdot a\in \alpha \right] ,$
(v) $a\in \alpha \implies r\cdot a\in \alpha ,$ for all finite $%
r\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
§ CONNECTION WITH THE VALUE GROUP.
Definition 1.3.3.1.2. We define an equivalence relation on the
elements of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ as follows: $a$ $\symbol{126}$ $b\iff \dfrac{a}{b}$ and $\dfrac{b}{a}$ are
finite.Then the
equivalence classes from a linear ordered set. We denote the order
relation by $\ll .$
The classes may be regarded as orders of infinity.
The subring of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ consisting of the finite elements is a valuation ring,
and the equivalence classes may also be regarded as elements of the
value group. Condition (v) in Lemma 1.3.3.1.3 essentially says that
$a$ $\in \alpha $
and $b\symbol{126}a$ $\implies $ $b\in \alpha ,$i.e. a may be regarded as a
Dedekind cut in the value
§ PROPERTIES OF THE ABSORPTION FUNCTION.
Theorem 1.3.3.1.1.[23]. $\left( -\alpha
\right) +\alpha =-\left[ \mathbf{ab.p.}(\alpha )\right] .$
Theorem 1.3.3.1.2.[23].$\mathbf{ab.p.}(\alpha
+\beta )\geq \mathbf{ab.p.}(\alpha ).$
Theorem 1.3.3.1.3.[23].
(i) $\alpha +\beta \leq \alpha +\gamma \implies -\left[ \mathbf{%
ab.p.}(\alpha )\right] +\beta \leq \gamma .$
(ii) $\alpha +\beta =\alpha +\gamma \implies -\left[
\mathbf{ab.p.}(\alpha )\right] +\beta =\gamma .$
Theorem 1.3.3.1.4.[23].
(i) $\ \mathbf{ab.p.}(-\alpha )=\mathbf{ab.p.}(\alpha ),$
(ii) $\mathbf{ab.p.}(\alpha +\beta )=\max \left\{ \mathbf{ab.p.}%
(\alpha ),\mathbf{ab.p.}(\beta )\right\} .$
We now classify the elements $\beta $ such that $\alpha +\beta =\alpha $.
For positive $\beta $ we
know by Lemma 1.3.3.1.2.(i) that $\alpha +\beta =\alpha $
iff $\beta \leq \mathbf{ab.p.}(\alpha ).$
Theorem 1.3.3.1.5.[23]. Assume $\beta $ $>0.$ If $%
\alpha $ absorbs $-\beta $ then a abosrbs $\beta $.
Theorem 1.3.3.1.6.[23]. Let $0<\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$ Then the following are
(i) $\ \alpha $ is an idempotent,
(ii) $\left( -\alpha \right) +\left( -\alpha \right) =-\alpha ,$
(iii) $\left( -\alpha \right) +\alpha =-\alpha .$
§ SPECIAL EQUIVALENCE RELATIONS ON $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
Let $\Delta $ be a positive idempotent. We define three equivalence relations
$\left( \circ \text{ }\mathbf{R\circ }\right) \mathbf{,}\left( \circ \text{ }%
\mathbf{S\circ }\right) $ and $\left( \circ \text{ }\mathbf{T\circ }\right) $
on $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Definition 1.3.3.1.3.[23].
(i) $\alpha \mathbf{R}\beta \left( \func{mod}\Delta
\right) \iff \alpha +\Delta =\beta +\Delta ,$
(ii) $\alpha \mathbf{S}\beta \left( \func{mod}\Delta \right) \iff
\alpha +\left( -\Delta \right) =\beta +\left( -\Delta \right) ,$
(iii) $\alpha \mathbf{T}\beta \left( \func{mod}\Delta
\right) \iff \exists d\left( d\in \Delta \right) \left[ \left( \alpha
\subset \beta +d\right) \wedge \left( \beta \subset \alpha +d\right) \right]
Remark 1.3.3.1.3.To simplify the notation $\func{mod}\Delta $ is
omitted when we are
dealing with only one $\Delta $. $\mathbf{R}$ and $\mathbf{S}$ are obviously
equivalence relations.
$\mathbf{T}$ is an equivalence relation since $\Delta $ is idempotent.
Remark 1.3.3.1.4.It is immediate that $\mathbf{R,S}$ and $\mathbf{T}
$ are congruence relations
with respect to addition. Also, if $\symbol{126}$ stands for either $\mathbf{%
R,S}$ or $\mathbf{T}$ then $\alpha <\beta <\gamma $
and $\alpha $ $\symbol{126}$ $\gamma \implies \alpha $ $\symbol{126}$ $\beta
.$ To see this it is convenient to have the following
Lemma 1.3.3.1.4.[23]. Suppose $\alpha <\beta $. Then
(i) $\ \ \alpha \mathbf{R}\beta \left( \func{mod}\Delta \right)
\iff \beta \leq \alpha +\Delta ,$
(ii) $\alpha \mathbf{S}\beta \left( \func{mod}\Delta
\right) \iff \beta +\left( -\Delta \right) \leq \alpha .$
Lemma 1.3.3.1.5.[23]. Let $\Delta $ be a positive
idempotent. Then
$-\left[ \alpha +\left( -\Delta \right) \right] +\left( -\Delta \right) \leq
-\alpha .$
Remark 1.3.3.1.5.This is not immediate since the inequality
$\left( -\alpha \right) +\left( -\beta \right) $ $\leq -\left( \alpha +\beta
\right) $ goes the wrong way. In fact, this seems
surprising at first since the first addend may be bigger than one
intuitively expects, e.g. if $\alpha =\Delta =\varepsilon _{\mathbf{d}}$
then $-\left[ \alpha +\left( -\Delta \right) \right] =$
$-\left[ \varepsilon _{\mathbf{d}}+\left( -\varepsilon _{\mathbf{d}}\right) %
\right] =\varepsilon _{\mathbf{d}}>0.$ However,$\varepsilon _{\mathbf{d}%
}+\left( -\varepsilon _{\mathbf{d}}\right) =-\varepsilon _{\mathbf{d}},$ so
inequality is valid after all.
Theorem 1.3.3.1.7.[23].
(i) $\ \ \mathbf{S}$ is a congruence relation with respect to
(ii) $\ \mathbf{T}$ is a congruence relation with respect to
(iii) $\mathbf{R}$ is not a congruence relation with
respect to negation.
Theorem 1.3.3.1.8.[23]. $\alpha +\Delta $ is the maximum
element $\beta $ satisfying
$\beta \mathbf{R\alpha .}$
Theorem 1.3.3.1.9.[23]. $\alpha +\left( -\Delta
\right) $ is the minimum element $\beta $ satisfying
$\beta \mathbf{S\alpha .}$
Theorem 1.3.3.1.9.[23]. $\mathbf{T}%
\subsetneqq \mathbf{R}\subsetneqq \mathbf{S.}$ Both inclusions are proper.
Theorem 1.3.3.1.10.[23].
(i) Let $\Delta _{1}$ and $\Delta _{2}$ be two positive idempotents
such that $\Delta _{2}>\Delta _{1}.$ Then:
$\Delta _{2}+\left( -\Delta _{1}\right) =\Delta _{2},$
(ii) Let $\Delta _{1}$ and $\Delta _{2}$ be two positive
idempotents such that $\Delta _{2}>\Delta _{1}.$ Then:
$\alpha \mathbf{S}\beta \left( \func{mod}\Delta _{1}\right) \implies \alpha
\mathbf{R}\beta \left( \func{mod}\Delta _{2}\right) .$
Theorem 1.3.3.1.11.[23].Let $\Delta _{1}$ and $\Delta _{2}$
be two positive idempotents such
that $\Delta _{2}>\Delta _{1}.$Then $\alpha \mathbf{S}\beta \left( \func{mod}%
\Delta _{1}\right) \implies \alpha \mathbf{T}\beta \left( \func{mod}\Delta
_{2}\right) $ but not conversely.
Theorem 1.3.3.1.12.[23].$\mathbf{S}$ is the smallest
congruence relation with respect
to addition and negation containing $\mathbf{R.}$
Theorem 1.3.3.1.13.[23].Any convex congruence relation $%
\left( \circ \text{ }\symbol{126}\circ \right) $ containing
$\mathbf{T}$ properly must contain $\mathbf{S.}$
§ I.3.3.2.SPECIAL KINDS OF IDEMPOTENTS IN $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
Let $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that $a>0.$ Then $a$ gives rise to two idempotents in a
natural way.
Definition 1.3.3.2.1.[23].
(i) $\mathbf{A}_{a}$ $\triangleq $ $\left\{ x|\exists n_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left[ x<n\cdot a\right] \right\} .$
(ii) $\mathbf{B}_{a}$ $\triangleq $ $\left\{ x|\forall r_{r\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}}\left[ x<r\cdot a\right] \right\} .$
Then it is immediate that $\mathbf{A}_{a}$ and $\mathbf{B}_{a}$ are
idempotents.The usual "$\epsilon /2$
argument" shows this for $\mathbf{B}_{a}.$It is also clear that $\mathbf{A}%
_{a}$ is the smallest
idempotent containing $a$ and $\mathbf{B}_{a}$ is the largest idempotent not
containing $a.$It follows that $\mathbf{B}_{a}$ and $\mathbf{A}_{a}$ are
consecutive idempotents.
Remark 1.3.3.2.1.Note that $\mathbf{B}_{1}=\varepsilon _{\mathbf{d}%
}=\inf \left(
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}\right) $ (which is the set of all
infinite small positive numbers plus all negative numbers) which we
have already considered above. $\mathbf{A}_{1}=\Delta _{\mathbf{d}%
}\triangleq \sup \left(
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}\right) $ (which is the set
of all finite numbers plus all negative numbers) which we have also
already considered above.
Definition 1.3.3.2.2. Let $a\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
(i) $\mathbf{\omega }_{\mathbf{d}}\left[ a\right] $ $\triangleq $ $%
\left\{ x|\exists n_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left[ x<n\cdot a\right] \right\} .$
(ii) $\mathbf{\Omega }_{\mathbf{d}}\left[ a\right] $ = $\left\{
x|\forall r_{r\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}}\left[ x<r\cdot a\right] \right\} ,a\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$
Remark 1.3.3.2.2.Then it is immediate that $\mathbf{\omega }_{%
\mathbf{d}}\left[ a\right] $ and $\mathbf{\Omega }_{\mathbf{d}}\left[ a%
\right] $ are
idempotents. It is also clear that $\mathbf{\omega }_{\mathbf{d}}\left[ a%
\right] $ is the smallest idempotent
containing hypernatural $a$ and $\mathbf{\omega }_{\mathbf{d}}\left[ a\right]
=a\cdot \mathbf{\omega }_{\mathbf{d}}.$ $\mathbf{\Omega }_{\mathbf{d}}\left[
a\right] =a\cdot \varepsilon _{\mathbf{d}}$ is the
largest idempotent not containing $a.$
It follows that $\mathbf{\Omega }_{\mathbf{d}}\left[ a\right] $ and $\mathbf{%
\omega }_{\mathbf{d}}\left[ a\right] $ are consecutive idempotents.
Remark 1.3.3.2.3. Note that $\mathbf{\omega }_{\mathbf{d}}\left[ 1%
\right] =\mathbf{\omega }_{\mathbf{d}}$ (which is the set of all finite
natural numbers $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ plus all negative numbers) which we have also
already considered above.
Theorem 1.3.3.2.1.[23].
(i) No idempotent of the form $\mathbf{A}_{a}$ has an immediate
(ii) All consecutive pairs of idempotents have the form $\mathbf{A}%
_{a}$ and $\mathbf{B}_{a}$
for some $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
§ I.3.3.3. TYPES OF $\PROTECT\ALPHA $ WITH A GIVEN $\MATHBF{AB.P.}(%
\PROTECT\ALPHA ).$
Among elements of $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that $\mathbf{ab.p.}(\alpha )=\Delta $ we can distinguish
two types.
Definition 1.3.3.3.1.[23]. Assume $\Delta >0.$
(i) $\ \alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has type $1$ if $\exists x\left( x\in \alpha \right) \forall y%
\left[ x+y\in \alpha \implies y\in \Delta \right] ,$
(ii) $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has type $2$ if $\forall x\left( x\in \alpha \right) \exists
y\left( y\notin \Delta \right) \left[ x+y\in \alpha \right] ,$i.e.
$\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has type $2$ iff $\alpha $ does not have type $1.$
A similar classification exists from above.
Definition 1.3.3.3.2.[23]. Assume $\Delta >0.$
(i) $\ \alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has type $1\mathbf{A}$ if $\exists x\left( x\notin \alpha
\right) \forall y\left[ x-y\notin \alpha \implies y\in \Delta \right] ,$
(ii) $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has type $2\mathbf{A}$ if $\forall x\left( x\notin \alpha
\right) \exists y\left( y\notin \alpha \right) \left[ x-y\notin \alpha %
\right] .$
Theorem 1.3.3.3.3.[23].
(i) $\ \alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has type $1$ iff $-\alpha $ has type $1\mathbf{A},$
(ii) $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ cannot have type $1$ and type $1\mathbf{A}$ simultaneously.
Theorem 1.3.3.3.4.[23].Suppose $\mathbf{ab.p.}(\alpha
)=\Delta >0.$ Then $\alpha $ has type $1$
iff $\alpha $ has the form $a+$ $\Delta $ for some $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Theorem 1.3.3.3.5.[23].$\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has type $1\mathbf{A}$ iff $\alpha $ has the form $a+$ $%
\left( -\Delta \right) $
for some $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Theorem 1.3.3.3.6.[23].
(i) If $\mathbf{ab.p.}(\alpha )>\mathbf{ab.p.}(\beta )$ then $%
\alpha +\beta $ has type $1$ iff $\alpha $ has type $1.$
(ii) If $\mathbf{ab.p.}(\alpha )=\mathbf{ab.p.}(\beta )$ then $%
\alpha +\beta $ has type $2$ iff either $\alpha $ or $\beta $
has type $2.$
Theorem 1.3.3.3.7.[23]. If $\mathbf{ab.p.}(\alpha )$ has
the form $\mathbf{B}_{a}$ then $\alpha $ has
type $1$ or type $1\mathbf{A.}$
§ I.3.3.4. $\PROTECT\VAREPSILON $-PART OF $\PROTECT\ALPHA $ WITH $%
\MATHBF{AB.P.}(\PROTECT\ALPHA )\NEQ 0.$
$\mathbf{Theorem}$ $\mathbf{1.3.3.4.8.}$ (i) Suppose:
1) $-\Delta _{\mathbf{d}}<\alpha <\Delta _{\mathbf{d}},$
2) $\mathbf{ab.p.}(\alpha )=\varepsilon _{\mathbf{d}}$ i.e. $\alpha $ has
type $1.$
Then there is exist unique $a\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha =\left( ^{\ast }a\right) ^{\#}+\varepsilon _{\mathbf{d}}, \\
\\
a=WST\left( \alpha \right) . \\
\end{array}
& \text{ \ \ }\left( 1.3.3.5\right)%
\end{array}%
(ii) Suppose:
1) $-\Delta _{\mathbf{d}}<\alpha _{1}<\Delta _{\mathbf{d}},-\Delta _{\mathbf{%
d}}<\alpha _{2}<\Delta _{\mathbf{d}},$
2) $\mathbf{ab.p.}(\alpha _{1})=\varepsilon _{\mathbf{d}},\mathbf{ab.p.}%
(\alpha _{2})=\varepsilon _{\mathbf{d}}$ i.e. $\alpha _{1}$ and $\alpha _{2}$
has type $1.$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha _{1}+\alpha _{2}=WST\left( \alpha _{1}\right) +WST\left( \alpha
_{2}\right) +\varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ }\left( 1.3.3.6\right)%
\end{array}%
(iii) Suppose:
1) $-\Delta _{\mathbf{d}}<\alpha <\Delta _{\mathbf{d}},$
2) $\mathbf{ab.p.}(\alpha )=\varepsilon _{\mathbf{d}}$ i.e. $\alpha $ has
type $1.$
Then $\forall b\left( b\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
b^{\#}\times \alpha =b^{\#}\times \left( ^{\ast }WST\left( \alpha \right)
\right) ^{\#}+b^{\#}\times \varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ \ }\left( 1.3.3.7\right)%
\end{array}%
(iv) Suppose:
1) $-\Delta _{\mathbf{d}}<\alpha _{1}<\Delta _{\mathbf{d}},-\Delta _{\mathbf{%
d}}<\alpha _{2}<\Delta _{\mathbf{d}},$
2) $\mathbf{ab.p.}(\alpha _{1})=\varepsilon _{\mathbf{d}},\mathbf{ab.p.}%
(\alpha _{2})=\varepsilon _{\mathbf{d}}$ i.e. $\alpha _{1}$ and $\alpha _{2}$
has type $1.$
Then $\forall b\left( b\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $:
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
b^{\#}\times \left( \alpha _{1}+\alpha _{2}\right) =b^{\#}\times \left(
^{\ast }WST\left( \alpha \right) \right) ^{\#}+ \\
\\
b^{\#}\times \left( ^{\ast }WST\left( \alpha _{2}\right) \right)
^{\#}+b^{\#}\times \varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ \ \ \ \ }\left( 1.3.3.8\right)%
\end{array}%
$\bigskip $
$\mathbf{Theorem}$ $\mathbf{1.3.3.4.9.}$ (i) Suppose:
1) $-\Delta _{\mathbf{d}}<\alpha <\Delta _{\mathbf{d}},$
2) $\mathbf{ab.p.}(\alpha )=-\varepsilon _{\mathbf{d}}$ i.e. $\alpha $ has
type $1A.$
Then there is exist unique $a\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
\begin{array}{cc}
\begin{array}{c}
\\
\alpha =\left( ^{\ast }a\right) ^{\#}-\varepsilon _{\mathbf{d}}. \\
\\
a=WST\left( \alpha \right) . \\
\end{array}
& \text{ \ }\left( 1.3.3.9\right)%
\end{array}%
(ii) Suppose:
1) $-\Delta _{\mathbf{d}}<\alpha _{1}<\Delta _{\mathbf{d}},-\Delta _{\mathbf{%
d}}<\alpha _{2}<\Delta _{\mathbf{d}},$
2) $\mathbf{ab.p.}(\alpha _{1})=-\varepsilon _{\mathbf{d}},\mathbf{ab.p.}%
(\alpha _{2})=-\varepsilon _{\mathbf{d}}$ i.e. $\alpha _{1}$ and $\alpha
_{2} $ has type $1A$ or
3) $\mathbf{ab.p.}(\alpha _{1})=\varepsilon _{\mathbf{d}},\mathbf{ab.p.}%
(\alpha _{2})=-\varepsilon _{\mathbf{d}}$ i.e. $\alpha _{1}$ has type $1$
and $\alpha _{2}$ has
type $1A.$Then:
\begin{array}{cc}
\begin{array}{c}
\\
\alpha _{1}+\alpha _{2}=WST\left( \alpha _{1}\right) +WST\left( \alpha
_{2}\right) -\varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ \ \ }\left( 1.3.3.10\right)%
\end{array}%
(iii) Suppose:
1) $-\Delta _{\mathbf{d}}<\alpha <\Delta _{\mathbf{d}},$
2) $\mathbf{ab.p.}(\alpha )=-\varepsilon _{\mathbf{d}}$ i.e. $\alpha $ has
type $1A.$
Then $\forall b\left( b\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $:
\begin{array}{cc}
\begin{array}{c}
\\
b^{\#}\times \alpha =b^{\#}\times \left( ^{\ast }WST\left( \alpha \right)
\right) ^{\#}-b^{\#}\times \varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ \ \ }\left( 1.3.3.11\right)%
\end{array}%
(iv) Suppose:
1) $-\Delta _{\mathbf{d}}<\alpha _{1}<\Delta _{\mathbf{d}},-\Delta _{\mathbf{%
d}}<\alpha _{2}<\Delta _{\mathbf{d}},$
2) $\mathbf{ab.p.}(\alpha _{1})=-\varepsilon _{\mathbf{d}},\mathbf{ab.p.}%
(\alpha _{2})=-\varepsilon _{\mathbf{d}}$ i.e. $\alpha _{1}$ and $\alpha
_{2} $ has type $1A$ or
3) $\mathbf{ab.p.}(\alpha _{1})=\varepsilon _{\mathbf{d}},\mathbf{ab.p.}%
(\alpha _{2})=-\varepsilon _{\mathbf{d}}$ i.e. $\alpha _{1}$ has type $1$
and $\alpha _{2}$ has
type $1A.$Then $\forall b\left( b\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $:
\begin{array}{cc}
\begin{array}{c}
\\
b^{\#}\times \left( \alpha _{1}+\alpha _{2}\right) =b^{\#}\times \left(
^{\ast }WST\left( \alpha \right) \right) ^{\#}+ \\
\\
b^{\#}\times \left( ^{\ast }WST\left( \alpha _{2}\right) \right)
^{\#}-b^{\#}\times \varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ \ \ \ }\left( 1.3.3.12\right)%
\end{array}%
$\bigskip $
$\mathbf{Definition}$ $\mathbf{1.3.3.4.3.}$Suppose $\alpha >0$ then
$\alpha ^{+}\triangleq \left[ \alpha \right] ^{+}\triangleq \left[ x|\left(
x\in \alpha \right) \wedge \left( x\geq 0\right) \right] .$
Suppose (1) $\alpha >0$ and (2) $\mathbf{ab.p.}(\alpha )=\varepsilon _{%
\mathbf{d}}$ i.e.
$\alpha $ has type $1,$ i.e. $\alpha =\left( ^{\ast }a\right)
^{\#}+\varepsilon _{\mathbf{d}},a=WST\left( \alpha \right) ,a\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Then $\beta \triangleq \left[ \alpha \right] _{\varepsilon },$($%
\varepsilon \approx 0,\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}$) is an $\varepsilon $-part of $\alpha $ iff:
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \forall y\left( y\geq 0\right) \left[ \left( \left[ \left( ^{\ast
}a\right) ^{\#}\right] ^{+}+y\in \alpha ^{+}\right) \wedge \left( \left[
\left( ^{\ast }a\right) ^{\#}\right] ^{+}+y\in \beta \right) \right. \\
\\
\left. \iff y\in \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}^{+}\right]
. \\
\end{array}
& \text{ }\left( 1.3.3.13\right)%
\end{array}%
$\bigskip $
$\mathbf{Theorem}$ $\mathbf{1.3.3.4.10.}$Suppose $0<\alpha <\Delta _{\mathbf{%
d}},\mathbf{ab.p.}(\alpha )=\varepsilon _{\mathbf{d}}$
i.e. $\alpha $ has type $1,$i.e. $\alpha =\left( ^{\ast }a\right)
^{\#}+\varepsilon _{\mathbf{d}},a=WST\left( \alpha \right) ,a\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$0<b<\Delta _{\mathbf{d}},b\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{211d} }%
\mathbb{R}
(i) $\left[ \alpha \right] _{\varepsilon }=\left[ \left( ^{\ast
}a\right) ^{\#}\right] ^{+}+\varepsilon ^{\#}\times \varepsilon _{\mathbf{d}%
(ii) $\left[ b^{\#}+\alpha \right] _{\varepsilon }=\left[ \left(
^{\ast }\left( \mathbf{st}\left( b\right) \right) \right) ^{\#}\right] ^{+}+%
\left[ \left( ^{\ast }a\right) ^{\#}\right] ^{+}+\varepsilon ^{\#}\times
\varepsilon _{\mathbf{d}}^{+}.$
(iii) $\left[ \left( ^{\ast }c\right) ^{\#}+\alpha \right]
_{\varepsilon }=\left[ \left( ^{\ast }\left( c\right) \right) ^{\#}\right]
^{+}+\left[ \left( ^{\ast }a\right) ^{\#}\right] ^{+}+\varepsilon
^{\#}\times \varepsilon _{\mathbf{d}}^{+}.$
$\mathbf{Theorem}$ $\mathbf{1.3.3.4.11.}$(i) Suppose $0<\alpha
_{1}<\Delta _{\mathbf{d}},0<\alpha _{2}<\Delta _{\mathbf{d}},$
$\mathbf{ab.p.}(\alpha _{1})=\mathbf{ab.p.}(\alpha _{2})=\varepsilon _{%
\mathbf{d}},WST\left( \alpha _{1}\right) =a\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
,WST\left( \alpha _{2}\right) =b\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
(i) $\left[ \alpha _{1}+\alpha _{2}\right] _{\varepsilon }=%
\left[ \left( ^{\ast }a\right) ^{\#}\right] ^{+}+\left[ \left( ^{\ast
}b\right) ^{\#}\right] ^{+}+\varepsilon ^{\#}\times \varepsilon _{\mathbf{d}%
(ii) $\left[ \alpha _{1}-\alpha _{2}\right] _{\varepsilon }=\left[
\left( ^{\ast }a\right) ^{\#}\right] ^{+}-\left[ \left( ^{\ast }b\right)
^{\#}\right] ^{+}-\varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}^{+}.$
$\mathbf{Theorem}$ $\mathbf{1.3.3.4.11.\forall }\varepsilon \left(
\varepsilon \approx 0\right) \left[ \alpha ^{+}=\varepsilon _{\mathbf{d}%
}^{+}\iff \left[ \alpha \right] _{\varepsilon }=\varepsilon ^{\#}\times
\varepsilon _{\mathbf{d}}^{+}\right] .$
$\mathbf{Definition}$ $\mathbf{1.3.3.4.4.}$Suppose $\alpha \geq 0$ $\mathbf{%
ab.p.}(\alpha )=\Delta \geq \varepsilon _{\mathbf{d}},\alpha \neq \Delta $
and $\alpha $ has type $1,$i.e. $\alpha $ has representation $\alpha
=a^{\#}+\Delta $ for some
$a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+},a^{\#}\notin \Delta .$
Then $\beta \triangleq \left[ \alpha |a^{\#}\right] _{\varepsilon
}, $($\varepsilon \approx 0,\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}$) is an $\varepsilon $-part of $\alpha $ for a given
$a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}$ iff:
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \forall y\left( y\geq 0\right) \left[ \left( \left[ a^{\#}\right]
^{+}+y\in \alpha ^{+}\right) \wedge \left( \left[ a^{\#}\right] ^{+}+y\in
\beta \right) \iff y\in \varepsilon ^{\#}\times \Delta ^{+}\right] . \\
\end{array}
& \text{ \ }\left( 1.3.3.14\right)%
\end{array}%
Remark. Suppose $\mathbf{ab.p.}(\alpha )=\Delta \geq \varepsilon _{%
\mathbf{d}},\alpha =\Delta .$Then $\beta \triangleq \left[ \alpha
|\Delta \right] _{\varepsilon },$
($\varepsilon \approx 0,\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}$) is an $\varepsilon $-part of $\alpha $ for a given $%
a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ iff:
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \forall y\left( y\geq 0\right) \left[ \left( y\in \alpha ^{+}\right)
\wedge \left( y\in \beta \right) \iff y\in \varepsilon ^{\#}\times \Delta
^{+}\right] . \\
\end{array}
& \text{ }\left( 1.3.3.15\right)%
\end{array}%
Note if $\mathbf{ab.p.}(\alpha )=\varepsilon _{\mathbf{d}}$ and $\alpha
=\left( ^{\ast }a\right) ^{\#}+\varepsilon _{\mathbf{d}},a\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
then $\beta \triangleq \left[ \alpha |\left( ^{\ast }a\right) ^{\#}%
\right] _{\varepsilon }=\left[ \alpha \right] _{\varepsilon },$ $\left[
\alpha \right] _{\varepsilon }$is an $\varepsilon $-part of $\alpha .$
Definition 1.3.3.4.4. Suppose $\alpha >0,$ $\mathbf{ab.p.}(\alpha
)=\Delta \leq -\varepsilon _{\mathbf{d}}$ and $\alpha $
has type $1A,$i.e. $\alpha $ has representation $\alpha =a^{\#}-\Delta $ for
$a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,a^{\#}\notin \Delta .$Then $\beta \triangleq \left[ \alpha |a^{\#}%
\right] _{\varepsilon }$ is an $\varepsilon $-part of $\alpha $ for a
$a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ iff:
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \forall y\left( y\geq 0\right) \left[ \left( \left[ a^{\#}\right]
^{+}-y\in \alpha ^{+}\right) \wedge \left( \left[ a^{\#}\right] ^{+}-y\in
\beta \right) \right. \\
\\
\left. \iff y\in -\varepsilon ^{\#}\times \left( -\Delta \right) ^{+}\right]
. \\
\end{array}
& \text{ \ }\left( 1.3.3.16\right)%
\end{array}%
Note if $\mathbf{ab.p.}(\alpha )=-\varepsilon _{\mathbf{d}}$ i.e. $\alpha
=\left( ^{\ast }a\right) ^{\#}-\varepsilon _{\mathbf{d}},a\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
then $\beta \triangleq \left[ \alpha |\left( ^{\ast }a\right) ^{\#}%
\right] _{\varepsilon }=\left[ \alpha \right] _{\varepsilon },\left[ \alpha %
\right] _{\varepsilon }$ is an $\varepsilon $-part of $-\alpha $.$\ \ $
$\ \ \ $
Theorem 1.3.3.4.10.
(1) Suppose $\alpha >0,$ $\mathbf{ab.p.}(\alpha )=\Delta \geq
\varepsilon _{\mathbf{d}}$ and $\alpha $ has type $1,$
i.e. $\alpha =a^{\#}+\Delta $ for some $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Then $\left[ \alpha |a^{\#}\right] _{\varepsilon }$ has the form
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \alpha |a^{\#}\right] _{\varepsilon }=a^{\#}+\varepsilon ^{\#}\times
\Delta ^{+} \\
\end{array}
& \text{ \ }\left( 1.3.3.17\right)%
\end{array}%
for a given $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(2) Suppose $\alpha >0,$ $\mathbf{ab.p.}(\alpha )=\Delta \leq
-\varepsilon _{\mathbf{d}}$ and $\alpha $ has type $1A,$
i.e. $\alpha =a^{\#}-\Delta $ for some $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Then $\left[ \alpha |a^{\#}\right] _{\varepsilon }$ has the form
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \alpha |a^{\#}\right] _{\varepsilon }=\left[ a^{\#}\right]
^{+}-\varepsilon ^{\#}\times \left( -\Delta \right) ^{+} \\
\end{array}
& \text{\ }\left( 1.3.3.18\right)%
\end{array}%
for a given $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\mathbf{Theorem}$ $\mathbf{1.3.3.4.11.}$
(1) Suppose $\alpha >0,$ $\mathbf{ab.p.}(\alpha )=\varepsilon _{%
\mathbf{d}}$ i.e. $\alpha $ has type $1$ and
$\alpha $ has representation $\alpha =\left( ^{\ast }a\right)
^{\#}+\varepsilon _{\mathbf{d}},$ for some unique $a\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Then $\left[ \alpha \right] _{\varepsilon }$ has the unique form:
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \alpha \right] _{\varepsilon }=\left[ \left( ^{\ast }a\right) ^{\#}%
\right] ^{+}+\varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}^{+}. \\
\end{array}
& \text{ }\left( 1.3.3.19\right)%
\end{array}%
(2) Suppose $\alpha >0,$ $\mathbf{ab.p.}(\alpha )=-\varepsilon _{%
\mathbf{d}}$ i.e. $\alpha $ has type $1A$ and
$\alpha $ has representation $\alpha =\left( ^{\ast }a\right)
^{\#}-\varepsilon _{\mathbf{d}},$ for some unique $a\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Then $\left[ \alpha \right] _{\varepsilon }$ has the unique form:
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \alpha \right] _{\varepsilon }=\left[ \left( ^{\ast }a\right) ^{\#}%
\right] ^{+}-\varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}^{+}. \\
\end{array}
& \text{ }\left( 1.3.3.20\right)%
\end{array}%
$\mathbf{Theorem}$ $\mathbf{1.3.3.4.12.}$ (1) Suppose $%
\mathbf{ab.p.}(\alpha )=\varepsilon _{\mathbf{d}},WST\left( \alpha \right)
\geq 0$
i.e. $\alpha $ has type $1$ and $\alpha $ has representation $\alpha =\left(
^{\ast }a\right) ^{\#}+\varepsilon _{\mathbf{d}},$
for some unique $a\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}.$Then for every $M\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
M\times \left[ \alpha \right] _{\varepsilon }=\left[ M\times \alpha
\left\vert M\times \left( ^{\ast }a\right) ^{\#}\right. \right]
_{\varepsilon }= \\
\\
=M\times \left[ \left( ^{\ast }a\right) ^{\#}\right] ^{+}+\left( \varepsilon
^{\#}\times M\right) \times \varepsilon _{\mathbf{d}}^{+}. \\
\end{array}
& \text{ \ }\left( 1.3.3.21\right)%
\end{array}%
(2) Suppose $\mathbf{ab.p.}(\alpha )=-\varepsilon _{\mathbf{d}%
},WST\left( \alpha \right) \geq 0$
and $\alpha $ has type $1A$ i.e. $\alpha =\left( ^{\ast }a\right)
^{\#}-\varepsilon _{\mathbf{d}},$ for some unique $a\in $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Then for every $M\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
M\times \left[ \alpha \right] _{\varepsilon }=\left[ M\times \alpha
\left\vert M\times \left( ^{\ast }a\right) ^{\#}\right. \right]
_{\varepsilon }= \\
\\
=M\times \left[ \left( ^{\ast }a\right) ^{\#}\right] ^{+}-\left( \varepsilon
^{\#}\times M\right) \times \varepsilon _{\mathbf{d}}^{+}. \\
\end{array}
& \text{ \ }\left( 1.3.3.22\right)%
\end{array}%
Theorem 1.3.3.4.13. (i) Suppose $\mathbf{ab.p.}(\alpha
)=\varepsilon _{\mathbf{d}}$ i.e. $\alpha $ has type $1.$
$\alpha =\varepsilon _{\mathbf{d}}\iff \forall y\left( y\geq 0\right)
\forall \varepsilon \left( \varepsilon \approx 0\right) \left[ \left( y\in
\alpha \right) \wedge \left( y\in \left[ \alpha \right] _{\varepsilon
}\right) \iff y\in \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}^{+}%
\right] .$
(ii) Suppose $\mathbf{ab.p.}(\alpha )=-\varepsilon _{\mathbf{d}}$
i.e. $\alpha $ has type $1A.$
$\alpha =-\varepsilon _{\mathbf{d}}\iff \forall y\left( y\geq 0\right)
\forall \varepsilon \left( \varepsilon \approx 0\right) \left[ \left( y\in
\alpha \right) \wedge \left( y\in \left[ \alpha \right] _{\varepsilon
}\right) \iff y\in -\varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}^{+}%
\right] .$
§ I.3.3.5.MULTIPLICATIVE IDEMPOTENTS.
Definition 1.3.3.5.1.[23]. Let $\left[ S\right] _{\mathbf{d}%
}=\left\{ x|\exists y\left( y\in S\right) \left[ x\leq y\right] \right\} $
.Then $\left[ S\right] _{\mathbf{d}}$
satisfies the usual axioms for a closure operation.
Let $f$ be a continuous strictly increasing function in each variable from a
subset of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
^{n}$ into $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$. Specifically, we want the domain to be the cartesian
product $\prod_{i=1}^{n}A_{i},$ where $A_{i}$ = $\left\{ x|x>a_{i}\right\} $
for some $a_{i}\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$By transfer $f$
extends to a function $^{\ast }f$ from the corresponding subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
^{n}$ into $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
which is also strictly increasing in each variable and continuous in the $Q$
topology (i.e. $\varepsilon $ and $\delta $ range over arbitrary positive
elements in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Definition 1.3.3.5.2.[23]. Let $\alpha _{i}\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},b_{i}\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ then
\begin{array}{cc}
\begin{array}{c}
\\
\left[ f\right] _{\mathbf{d}}\left( \alpha _{1},\alpha _{2},...,\alpha
_{n}\right) = \\
\\
\left[ \left\{ ^{\ast }f\left( b_{1},b_{2},...,b_{n}\right) |\text{ }%
b_{i}\in \alpha _{i}\right\} \right] _{\mathbf{d}} \\
\end{array}
& 1.3.3.23%
\end{array}%
Theorem 1.3.3.5.1.[23]. If $f$ and $g$ are functions of
one variable
then $\left[ f\cdot g\right] _{\mathbf{d}}\left( \alpha \right) =\left( %
\left[ f\right] _{\mathbf{d}}\left( \alpha \right) \right) \cdot \left( %
\left[ g\right] _{\mathbf{d}}\left( \alpha \right) \right) .$
Theorem 1.3.3.5.2.[23].Let $f$ and $g$ be any two terms
obtained by
compositions of strictly increasing continuous functions possibly
containing parameters in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$. Then any relation $^{\ast }f=$ $^{\ast }g$ or $^{\ast }f<$ $^{\ast }g$
valid in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ extends to $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},$i.e. $\left[ f\right] _{\mathbf{d}}\left( \alpha \right) =$ $%
\left[ g\right] _{\mathbf{d}}\left( \alpha \right) $ or $\left[ f\right] _{%
\mathbf{d}}\left( \alpha \right) <$ $\left[ g\right] _{\mathbf{d}}\left(
\alpha \right) .$
Theorem 1.3.3.5.3.[23].The map $\alpha \longmapsto $ $%
\left[ \exp \right] _{\mathbf{d}}\left( \alpha \right) $ maps the set of
additive idempotents onto the set of all multiplicative idempotents
other than $0.$
Similarly, multiplicative absorption can be defined and reduced to
the study of additive absorption. Incidentally the map $\alpha \longmapsto $
$\left[ \exp \right] _{\mathbf{d}}\left( \alpha \right) $
is essentially the same as the map in [34, Theorem 6] which is the
map from the set of ideals onto the set of all prime ideals of the
valuation ring consisting of the finite elements of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
§ I.3.3.6. ADDITIVE MONOID OF DEDEKIND HYPERREAL INTEGERS $^{\AST }\BREVE{%
\MATHBB{Z}%
Well-order relation $\left( \cdot \preceq _{\mathbf{s}}\cdot \right) $ (or
strong well-ordering) on a set $S$ is a total order
on $S$ with the property: that every non-empty subset $S^{\prime }$ of $S$
has a least element in this ordering.The set $S$ together with the
well-order relation $\preceq _{\mathbf{s}}$ is then called a (strong) well-
ordered set. The natural numbers of $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ with the well-order relation $\left( \cdot \leq _{^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\cdot \right) $ are not strong well-ordered set,for there is no smallest
infinite one.
Definition 1.3.3.6.1. Weak well-order relation $\left( \cdot
\preceq _{w}\cdot \right) $ (or weak well-ordering) on
a set $S$ is a total order on $S$ with the property:every non-empty subset $%
S^{\prime }\subseteqq $ $S$ has
a least element in this ordering or $S^{\prime }$ has a greatest lower bound
($\inf \left( S^{\prime }\right) $) in this
ordering.The set $S$ together with the weak well-order relation $\preceq
_{w} $is then called a
weak well-ordered set.
The natural numbers of $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ with the well-order relation $\left( \cdot \leq _{^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\cdot \right) $ are not iven
weak well-ordered set,for there is no $\inf \left( S^{\prime }\right) $ in $%
Let us considered completion of the ring $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$.Possible standard completion
of the ring $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ can be constructed by Dedekind sections. Making a semantic
leap, we now answer the question:"what is a Dedekind hyperintegers
Definition 1.3.3.6.2. A Dedekind hyperinteger is a cut in $%
^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}=\left( ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}},+\right) $ is the class of all Dedekind hyperintegers
$x=A|B,A\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
,B\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$(x=A,A\subsetneqq ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
We will show that in a natural way $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ is a complete ordered additive
monoid containing $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
Before spelling out what this means, here are some examples of cuts in $%
^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$A|B=\left. \left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\text{ }|\text{ }n<1\right\} \right\vert \left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\text{ }|\text{ }n\geq 1\right\} .$
$A|B=\left. \left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\text{ }|\text{ }n<\omega \right\} \right\vert \left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\text{ }|\text{ }n\geq \omega \right\} ,$where $\omega \in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }.$
$A|B=\left. \left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\text{ }|\left( n\leq 0\right) \vee \left( \text{ }n\in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{+}\right) \right\} \right\vert \left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\text{ }|\left( n\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }\right) \right\} .$
$\left. \left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\text{ }|\left( n\leq 0\right) \vee \left[ \left( n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{+}\right) \wedge \left( \underset{i\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\bigwedge }\left( n\leq \omega +i\right) \right) \right] \right\}
\right\vert $
$\left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\text{ }|\left( n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{+}\right) \wedge \text{ }\left( \underset{i\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\bigwedge }\left( n>\omega +i\right) \right) \right\} ,$
where $\omega \in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }.$
Remark. 1.3.3.6.1. It is convenient to say that $A|B\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ is an integer
(hyperinteger) cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ if it is like the cut in examples (i),(ii):
fore some
fixed integer (hyperinteger) number $c\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
,A$ is the set of all integer $n\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
such that $n<c$ while $B$ is the rest of $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
The $B$-set of an integer (hyperinteger) cut contains a smollest $c,$ and
conversaly if $A|B$ is a cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ and $B$ contains a smollest element $c$ then
$A|B$ is an integer (hyperinteger) cut at $c.$We write $\breve{c}$ for the
integer and
hyperinteger cut at $c.$This lets us think of $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\subset $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ by identifying $c$ with $\breve{c}.$
Remark.1.3.3.6.2. It is convenient to say that:
(1) $A|B\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ is an standard cut in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ if it is like the cut in example (i):
fore some cut $A^{\prime }|B^{\prime }\in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ the next equality is satisfied:$A|B=$ $^{\ast }\left( A^{\prime }\right)
|^{\ast }\left( B^{\prime }\right) ,$
i.e. $A$-set of a cut is an standard set.
(2) $A|B\in $ $^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}$ is an internal cut or nonstandard
cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ if it is like the cut
in example (ii), i.e. $A$-set of a cut is an internal
nonstandard set.
(3) $A|B\in $ $^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}$ is an external cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ if it is like the cut in examples (iii)-(iv),
i.e. $A$-set of a cut is an external set.
Definition 1.3.3.6.3. A Dedekind cut $\alpha $ in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ is a subset $\alpha \subset $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ of the
hyperinteger numbers $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ that satisfies these properties:
1. $\alpha $ is not empty.
2. $\beta =$ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\backslash \alpha $ is not empty.
3. $\alpha $ contains no greatest element.
4. For $x,y\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
,$ if $x\in \alpha $ and $y<x,$ then $y\in \alpha $ as well.
Definition 1.3.3.6.4. A Dedekind hyperinteger $\alpha \in $ $^{\ast
\mathbb{Z}%
}_{\mathbf{d}}$ is a
Dedekind cut $\alpha $ in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
.$ We denote the set of all Dedekind hyperinteger
by $^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}$ and we order them by set-theoretic inclusion, that is to
say, for any $\alpha ,\beta \in $ $^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}},$ $\alpha <_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta $ (or $\alpha <\beta $) if and only if $\alpha
\subsetneqq \beta $ where the
inclusion is strict. We further define $\alpha =_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta $ (or $\alpha =\beta $) as hyperinteger
if and are equal as sets. As usual, we write $\alpha \leqslant _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta $ if $\alpha <_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta $
or $\alpha =_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta $.
Definition 1.3.3.6.5. $M\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ is an upper bound for a set $S\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ if each
$s\in S$ satisfies $s\leq _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}M.$ We also say that the set $S$ is bounded above by
$M$ iff $M\in $ $\mathbf{L}\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) .$We also say that the set $S$ is hyperbounded above iff
$M\notin $ $\mathbf{L}\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) ,$i.e. $\left\vert M\right\vert \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{211d} }%
\mathbb{R}
Definition 1.3.3.6.6. An upper bound for $S$ that is less
than all other upper
bound for $S$ is a least upper bound for $S.$
Theorem 1.3.3.6.1 Every nonempty subset $A\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ of Dedekind
hyperinteger that is bounded (hyperbounded) above has a least
upper bound.
Definition 1.3.3.6.7. Given two Dedekind hyperinteger $\alpha $ and
$\beta $ we
1.The additive identity (zero cut) denoted $0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}$ or $\mathbf{0},$is
$0_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|\text{ }x<0\right\} .$
2.The multiplicative identity denoted $1_{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}$ or $1,$is
$1_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|\text{ }x<1\right\} .$
3. Addition $\alpha +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta $ of $\alpha $ and $\beta $ also denoted $\alpha +\beta
$ is
$\alpha +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta \triangleq \left\{ x+y|\text{ }x\in \alpha ,y\in \beta
\right\} .$
It is easy to see that $\alpha +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}0_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}=0_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}$ for all $\alpha \in $ $^{\ast }\breve{%
\mathbb{Z}%
It is easy to see that $\alpha +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta $ is a cut in $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ and $\alpha +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta =\beta +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\alpha .$
Another fundamental property of cut addition is associativity:
$\left( \alpha +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta \right) +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\gamma =\alpha +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\left( \beta +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\gamma \right) .$
This follows from the corresponding property of $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
4.The opposite $-_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\alpha $ of $\alpha ,$ also denoted $-\alpha ,$is
$-\alpha \triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|\text{ }-x\notin \alpha ,-x\text{ is not the least element of }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\backslash \alpha \right\} .$
5.Remark 1.3.3.6.3. We also say that the opposite $-\alpha $ of $%
\alpha $ is the additive
inverse of $\alpha $ denoted $\div \alpha $ iff the next equality
is satisfied: $\alpha +\left( \div \alpha \right) =0.$
6.Remark 1.3.3.6.4. It is easy to see that for all standard and
internal cut $\alpha ^{\mathbf{Int}}$ the
opposite $-\alpha ^{\mathbf{Int}}$ is the additive inverse of $\alpha ^{%
\mathbf{Int}},$i.e. $\alpha ^{\mathbf{Int}}+\left( \div \alpha ^{\mathbf{Int}%
}\right) =0.$
7.We say that the cut $\alpha $ is positive if $0<\alpha $ or
negative if $\alpha <0.$
The absolute value of $\alpha ,$denoted $\left\vert \alpha \right\vert ,$is $%
\left\vert \alpha \right\vert \triangleq \alpha ,$if $\alpha \geq 0$ and $%
\left\vert \alpha \right\vert \triangleq -\alpha ,$if $\alpha \leq 0.$
8. The cut order enjois on $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ the standard additional properties of:
(i) transitivity: $\alpha \leq _{^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}}\beta \leq _{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\gamma \implies \alpha \leq _{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\gamma .$
(ii) trichotomy: eizer $\alpha <_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta ,\beta <_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\alpha $ or $\alpha =_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta $ but only one of the
three things is true.
(iii) translation: $\alpha \leq _{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta \implies \alpha +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\gamma \leq _{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\beta +_{^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}}\gamma .$
9.By definition above, this is what we mean when we say that
$^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}$ is an complete ordered additive monoid.
Remark 1.3.3.6.5. Let us considered Dedekind integer cut $c\in $ $%
^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}$ as subset of
$c\subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$We write $\widetilde{c}=$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$-$\sup \left( c\right) =\underset{x}{\sup }\left\{ x|x\in
c\subset \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\right\} $ for the cut $c\in $ $^{\ast }\breve{%
\mathbb{Z}%
This lets us think of canonical imbeding $^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}$ $\underset{\mathbf{j}_{\mathbf{d}}}{\longmapsto }$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ monoid
$^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}$ into generalized pseudo-field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}\subset _{\longrightarrow }\text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}} \\
\end{array}
\end{array}%
by identifying $c$ with it image $\widetilde{c}=j_{\mathbf{d}}\left(
c\right) .$
Remark 1.3.3.6.6. It is convenient to identify monoid $^{\ast }%
\breve{%
\mathbb{Z}%
}_{\mathbf{d}}$ with it
image $j_{\mathbf{d}}\left( ^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}\right) \subset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
§ I.3.5.PSEUDO-RING OF WATTENBERG HYPERREAL INTEGERS $^{\AST }%
%TCIMACRO{\U{2124} }%
\MATHBB{Z}
The set $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has within it a set $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}\subsetneqq $ $^{\ast }\breve{%
\mathbb{Z}%
}_{\mathbf{d}}$ of Wattenberg hyperreal integers
which behave very much like hyperreals $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ inside $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$In particular the
greatest integer function $^{\ast }\left[ \cdot \right] :$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ extends in a natural way
to $\left[ \alpha \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}:$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\rightarrow $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
Lemma 1.3.5.1.[24].Suppose $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$. Then the following two
conditions on $\alpha $ are equivalent:
(i) $\alpha =\sup \left\{ \nu ^{\#}|\left( \nu \in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) \wedge \left( \nu \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) \right\} ,$
(ii) $\alpha =\inf \left\{ \nu ^{\#}|\left( \nu \in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) \wedge \left( \alpha \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\nu \right) \right\} .$
Definition 1.3.5.7.If $\alpha $ satisfies conditions (i)
or (ii) from lemma 1.3.5.1
$\alpha $ is said to be a $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$-integer or Wattenberg (hyperreal) integer.
Lemma 1.3.5.2.[24].(i) $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ is the closure in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ of $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(ii) $\ ^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\mathbf{d}}$ is the closure in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ of $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
(iii) both $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ and $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\mathbf{d}}$ are closed with respect to taking $\sup $
and $\inf $.
(iv) $\ ^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\mathbf{d}}$ is a weak well-ordered set.
Lemma 1.3.5.3.[24].Suppose that $\lambda ,\nu \in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}.$ Then,
(i) $\ \ \ \ \lambda +_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\nu \in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(ii) $-_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\lambda \in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(iii) $\ \ \lambda \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\nu \in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
Definition 1.3.5.8. Suppose $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}.$ Then, we define
$\left[ \cdot \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}:$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\rightarrow $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ by: $\left[ \alpha \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\triangleq \sup \left\{ \nu |\left( \nu \in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) \wedge \left( \nu \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) \right\} .$
Remark 1.3.5.7.There are two possibilities:
(i) collection $\left\{ \nu |\left( \nu \in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) \wedge \left( \nu \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) \right\} $ has no greatest element. In this
case $\left[ \alpha \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}=\alpha $ since $\left[ \alpha \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha $ implies $\exists a\left( a\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) \left[ \left[ \alpha \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}a<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right] .$
But then $[a]_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha $ which implies $[a]_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha $ contradicting
with $\left[ \alpha \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}<_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}a\leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}[a]_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}+_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}1_{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(ii) collection $\left\{ \nu |\left( \nu \in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) \wedge \left( \nu \leq _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\alpha \right) \right\} $ has a greatest element,$\nu .$In this
case $\left[ \alpha \right] _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}=\nu \in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
Definition 1.3.5.9. $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$-integer $\sup \left(
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right) =\inf \left( ^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }\right) $ we denote $\omega _{\mathbf{d}}.$
Definition 1.3.5.10. Suppose $\nu \in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }.$ Then, we define $\nu $-block $\mathbf{bk}\left[ \nu %
\right] $
as a set of hyper integers such that $\mathbf{bk}\left[ \nu \right] =\left\{
\nu \pm n|n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right\} .$
For $\nu ,\lambda \in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ there are two possibilities:
(i) $\nu -\lambda \in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
.$ In this case $\mathbf{bk}\left[ \nu \right] =\mathbf{bk}\left[ \lambda %
\right] $ and we write $\mathbf{bk}\left[ \nu \right] =\mathbf{bk}\left[
\widetilde{\nu }\right] $ where
$\widetilde{\nu }\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
%TCIMACRO{\U{2124} }%
\mathbb{Z}
(ii) $\left\vert \nu -\lambda \right\vert \in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }.$In this case $\mathbf{bk}\left[ \nu \right] \neq \mathbf{bk}%
\left[ \lambda \right] $ and $\mathbf{bk}\left[ \widetilde{\nu }\right] \neq
\mathbf{bk}\left[ \widetilde{\lambda }\right] .$
Lemma 1.3.5.11. $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
\cup \left( \bigcup_{\widetilde{\nu }}\mathbf{bk}\left[ \widetilde{\nu }%
\right] \right) .$
Proof. Clear by using
[25,Chapt.1,section 9].
§ I.3.6.EXTERNAL SUMMATION OF COUNTABLE AND HYPERFINITE SEQUENCES IN $%
^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
Definition 1.3.6.1. Let $\mathbf{S}_{X}$ denote the group of
permutations of the set $X$
and $\mathbf{H}_{X}$ denote ultrafilter on the set $X.$ Permutation $\sigma
\in \mathbf{S}_{X}$ is admissible
iff $\sigma $ preserv $\mathbf{H}_{X},$i.e. for any $A\in \mathbf{H}_{X}$
the next condition is satisfied:
$\sigma \left( A\right) \in \mathbf{H}_{X}.$
Below we denote by $\widehat{\mathbf{S}}_{X,\mathbf{H}_{X}}$ the subgroup $%
\widehat{\mathbf{S}}_{X,\mathbf{H}_{X}}\subsetneqq \mathbf{S}_{X}$ of the all
admissible permutations.
Definition 1.3.6.2. Let us consider countable sequence $\mathbf{s}%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that:
(a) $\forall n\left( \mathbf{s}_{n}\geq 0\right) $ or
(b) $\forall n\left( \mathbf{s}_{n}<0\right) $ and hyperreal number
denoted $\left[ \mathbf{s}_{n}\right] $ which
formed from sequence $\ \left\{ \mathbf{s}_{n}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ by the law
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \mathbf{s}_{n}\right] = \\
\\
\left( \mathbf{s}_{0},\mathbf{s}_{0}+\mathbf{s}_{1},\mathbf{s}_{0}+\mathbf{s}%
_{1}+\mathbf{s}_{2},...,\sum_{0}^{i}\mathbf{s}_{i},...\right) \in \text{ }%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
. \\
\end{array}
& \text{\ }\left( 1.3.6.1\right)
\end{array}%
Then external sum of the countable sequence $\mathbf{s}_{n}$ denoted
\begin{array}{cc}
\begin{array}{c}
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n} \\
\end{array}
& \text{ \ }\left( 1.3.6.2\right)
\end{array}%
\begin{array}{cc}
\begin{array}{c}
\\
\left( a\right) :Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}\triangleq \inf \left\{ \left[ \mathbf{s}_{\sigma \left(
n\right) }\right] |\text{ }\sigma \in \widehat{\mathbf{S}}_{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
}}\right\} , \\
\\
\left( b\right) :Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}\triangleq \sup \left\{ \left[ \mathbf{s}_{\sigma \left(
n\right) }\right] |\text{ }\sigma \in \widehat{\mathbf{S}}_{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
}}\right\} \\
\end{array}
& \text{\ }\left( 1.3.6.3\right)%
\end{array}%
Example 1.3.6.1. Let us consider countable sequence $\left\{
\mathbf{1}_{n}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ such
that: $\forall n\left( \mathbf{1}_{n}=1\right) .$Hence $\left[ \mathbf{1}_{n}%
\right] =\left( 1,2,3,....,i,...\right) =\varpi \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and using
Eq.(1.3.3) one obtain $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ $
$\bigskip $
\begin{array}{cc}
\begin{array}{c}
\\
Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{1}_{n}=\varpi \in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
. \\
\end{array}
& \text{ \ \ }\left( 1.3.6.4\right)%
\end{array}%
Example 1.3.6.2. Let us consider countable sequence $\left\{
\mathbf{1}_{n}^{\blacktriangledown }\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ such that: $\ \ \ \ \ \ \ $
$\left\{ n|\mathbf{1}_{n}^{\blacktriangledown }=1\right\} \in \mathbf{H}_{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}.$Hence $\left[ \mathbf{1}_{n}^{\blacktriangledown }\right] =\left(
1,2,3,....,i,...\right) \left( \text{mod}\mathbf{H}_{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right) $ $=\varpi \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and
using Eq.(1.3.3) one obtain
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{1}_{n}^{\blacktriangledown }=\varpi \in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
. \\
\end{array}
& \text{ \ }\left( 1.3.6.5\right)%
\end{array}%
Example 1.3.6.3. (Euler's infinite number $E^{\#}$). Let
us consider
countable sequence $\mathbf{h}_{n}=n^{-1}$. Hence
$\ \left[ \mathbf{h}_{n}\right] =\left( 1,1+\dfrac{1}{2},1+\dfrac{1}{2}+%
\dfrac{1}{3},....,1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{i},...\right)
\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
and using Eq.(1.3.3) one obtain
\begin{array}{cc}
\begin{array}{c}
\\
Ext-\dsum\limits_{n=1}^{\infty }\mathbf{h}_{n}=E^{\#}\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}. \\
\end{array}
& \text{ \ \ }\left( 1.3.6.6\right)%
\end{array}%
Definition 1.3.6.8. Let us consider countable sequence $\mathbf{s}%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and
two subsequences denoted $\mathbf{s}_{n}^{+}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ which formed from
sequence $\left\{ \mathbf{s}_{n}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ by the law
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{s}_{n}^{+}=\mathbf{s}_{n}\iff \mathbf{s}_{n}\geq 0, \\
\\
\mathbf{s}_{n}^{+}=0\iff \mathbf{s}_{n}<0 \\
\end{array}
& \text{ }\left( 1.3.6.7\right)%
\end{array}%
and accordingly by the law
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{s}_{n}^{-}=\mathbf{s}_{n}\iff \mathbf{s}_{n}<0, \\
\\
\mathbf{s}_{n}^{-}=0\iff \mathbf{s}_{n}\geq 0 \\
\end{array}
& \text{ \ }\left( 1.3.6.8\right)%
\end{array}%
Hence $\left\{ \mathbf{s}_{n}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}=\left\{ \mathbf{s}_{n}^{+}+\mathbf{s}_{n}^{-}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
Example 1.3.6.4. Let us consider countable sequence$\ $
$\bigskip \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
\begin{array}{cc}
\begin{array}{c}
\\
\ \left\{ \mathbf{1}_{n}^{\pm }\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}=\left\{ 1,-1,1,-1,...,1,-1,...\right\} . \\
\end{array}
& \text{ \ \ }\left( 1.3.6.9\right)%
\end{array}%
Hence $\left\{ \mathbf{1}_{n}^{\pm }\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}=$ $\left\{ \mathbf{1}_{n}^{+}+\mathbf{1}_{n}^{-}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\ \ $where$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\{ \mathbf{1}_{n}^{+}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}=\left\{ 1,0,1,0,...,1,0,...\right\} \\
\\
\left\{ \mathbf{1}_{n}^{-}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}=\left\{ 0,-1,0,-1,...,0,-1,...\right\} . \\
\end{array}
& \text{ \ \ }\left( 1.3.6.10\right)
\end{array}%
Definition 1.3.6.9.The external sum of the arbitrary
sequence $\left\{ \mathbf{s}_{n}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ denoted
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n} \\
\end{array}
& \text{ \ }\left( 1.3.6.12\right)%
\end{array}%
is $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ $
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}\triangleq \\
\\
\left( Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{+}\right) +\left( Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{-}\right) . \\
\end{array}
& \text{ }\left( 1.3.6.13\right)%
\end{array}%
Example 1.3.6.5. Let us consider countable sequence (1.3.9)$\ \ $
Eq.(1.3.3),Eq.(1.3.13) and Eq.(1.3.5) one obtain$\ $
$\bigskip $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\ \mathbf{1}_{nn\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}^{\pm }= \\
\\
\left( Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{1}_{n}^{+}\right) +\left( Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{1}_{n}^{-}\right) = \\
\\
=\varpi -\varpi =0. \\
\end{array}
& \text{ \ \ \ \ \ \ \ \ }\left( 1.3.6.14\right)%
\end{array}%
Definition 1.3.6.10. Let us consider countable sequence $\mathbf{s}%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that: (a) $\forall n\left( \mathbf{s}_{n}^{\#}\geq 0\right) $
(b) $\forall n\left( \mathbf{s}_{n}^{\#}<0\right) .$
Then external sum of the countable sequence $\mathbf{s}_{n}^{\#}$
\begin{array}{cc}
\begin{array}{c}
\\
\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{\#} \\
\end{array}
& \text{ }\left( 1.3.6.15\right)
\end{array}%
$\ $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( a\right) :\#Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{\#}\triangleq \\
\\
\text{ }\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left\{ \left. \dsum\limits_{n\leq k}\mathbf{s}_{\sigma \left(
n\right) }^{\#}\right\vert \text{ }\sigma \in \mathbf{S}_{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} , \\
\\
\left( b\right) :\#Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{\#}\triangleq \\
\\
\text{ }\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left\{ \left. \dsum\limits_{n\leq k}\mathbf{s}_{\sigma \left(
n\right) }^{\#}|\text{ }\right\vert \sigma \in \mathbf{S}_{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} . \\
\end{array}
& \text{ \ }\left( 1.3.6.16\right)%
\end{array}%
Definition 1.3.6.11. Let us consider countable sequence $\mathbf{s}%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
and two subsequences denoted $^{\#}\mathbf{s}_{n}^{+}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}$ which
formed from sequence $\left\{ \mathbf{s}_{n}^{\#}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ by the law
\begin{array}{cc}
\begin{array}{c}
\\
^{\#}\mathbf{s}_{n}^{+}=\mathbf{s}_{n}\iff \mathbf{s}_{n}^{\#}\geq 0, \\
\\
^{\#}\mathbf{s}_{n}^{+}=0\iff \mathbf{s}_{n}^{\#}<0 \\
\end{array}
& \text{ \ \ }\left( 1.3.6.17\right)%
\end{array}%
and accordingly by the law
\begin{array}{cc}
\begin{array}{c}
\\
^{\#}\mathbf{s}_{n}^{-}=\mathbf{s}_{n}^{\#}\iff \mathbf{s}_{n}^{\#}<0, \\
\\
^{\#}\mathbf{s}_{n}^{-}=0\iff \mathbf{s}_{n}^{\#}\geq 0 \\
\end{array}
& \text{ }\left( 1.3.6.18\right)%
\end{array}%
Hence $\left\{ \mathbf{s}_{n}^{\#}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}=\left\{ ^{\#}\mathbf{s}_{n}^{+}+\text{ }^{\#}\mathbf{s}_{n}^{-}\right\}
%TCIMACRO{\U{2115} }%
\mathbb{N}
Definition 1.3.6.12.The external sum of the arbitrary
sequence $\left\{ \mathbf{s}_{n}\right\} _{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ denoted $\ \ \ \ $
$\bigskip $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\#Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{\#} \\
\end{array}
& \text{ \ \ }\left( 1.3.6.19\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
\begin{array}{cc}
\begin{array}{c}
\\
\#Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{\#}\triangleq \\
\\
\left( \#Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\#}\mathbf{s}_{n}^{+}\right) \right) +\left( \#Ext\text{ -}%
\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\#}\mathbf{s}_{n}^{-}\right) \right) . \\
\end{array}
& \text{ \ \ }\left( 1.3.6.20\right)%
\end{array}%
Definition 1.3.6.13. Let us consider an nonempty subset $\mathbf{A}%
\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which
is bounded or hyperbounded from above and such that:
$\sup \left( \mathbf{A}\right) \pm \varepsilon \neq \sup \left( \mathbf{A}%
\right) $ for any $\varepsilon \approx 0.$We call this least upper bound $%
\sup \left( \mathbf{A}\right) $
the strong least upper bound or strong supremum, written
as $\mathbf{s}$-$\sup \left( \mathbf{A}\right) $.
Proposition 1.3.6.1. If $\mathbf{A}$ is a nonempty subset of $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded
from above and strong supremum $\mathbf{s}$-$\sup \left( \mathbf{A}\right) $
exist, then:
(1) $\mathbf{s}$-$\sup \left( \mathbf{A}\right) $ is the unique
number such that $\mathbf{s}$-$\sup \left( \mathbf{A}\right) $ is an upper
for $\mathbf{A}$ and $\mathbf{s}$-$\sup \left( \mathbf{A}\right)
-\varepsilon $ is not a upper bound for $\mathbf{A}$ for any $\varepsilon
\approx 0,\varepsilon >0;$
(2) (The Strong Approximation Property) let $\varepsilon
\approx 0,\varepsilon >0$ there exist $x\in \mathbf{A}$
such that $\mathbf{s}$-$\sup \left( \mathbf{A}\right) -\varepsilon <x\leq
\mathbf{s}$-$\sup \left( \mathbf{A}\right) .$
Proof.(2) If not, then $\mathbf{s}$-$\sup \left(
\mathbf{A}\right) -\varepsilon $ is an upper bound of $\mathbf{A}$ less than
the least
upper bound, which is a contradiction.
Corollary 1.3.6.1. Let $\mathbf{A}$ be bounded or
hyperbounded from above and
non-empty set such that $\mathbf{s}$-$\sup \left( \mathbf{A}\right) $ exist.
There is a function
$\alpha \left( \circ \right) :$ $^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that for all $\mathbf{n\in }$ $^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }$ we have
$\mathbf{s}$-$\sup \left( \mathbf{A}\right) -\mathbf{n}^{-1}<\alpha \left(
\mathbf{n}\right) \leq \mathbf{s}$-$\sup \left( \mathbf{A}\right) $
Theorem 1.3.6.2. Let $\mathbf{A}$ be a non-empty set which is
bounded or
hyperbounded from below. Then the set of lower bounds of $\mathbf{A}$ has a
greatest element.
Proof. Let $-\mathbf{A\triangleq }\left\{ -x|x\in \mathbf{A}%
\right\} .$We know that (i) $\forall x_{x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\forall y_{y\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\left( x\leq y\iff -y\leq -x\right) .$
Let $l_{\mathbf{A}}$ be a lower bound of $\mathbf{A.}$Then $l_{\mathbf{A}%
}\leq x$ for all $x\in \mathbf{A.}$So $-x\leq -l_{\mathbf{A}}$ for all
$x\in \mathbf{A,}$that is $y\leq l_{\mathbf{A}}$ for all $y\in -\mathbf{A.}$
So $-\mathbf{A}$ is bounded above, and non-empty,
so by the Theorem 1.3.1 $\sup \left( -\mathbf{A}\right) $ exists.
We shall prove now that: (ii) $-\sup \left( -\mathbf{A}\right) $ is
a lower bound of $\mathbf{A,}$(iii) if $l_{\mathbf{A}}$ is a lower
bound of $\mathbf{A}$ then $l_{\mathbf{A}}\leq -\sup \left( -\mathbf{A}%
\right) .$ (ii) if $x\in \mathbf{A}$ then $-x\in -\mathbf{A}$ and
so $-x\leq \sup \left( -\mathbf{A}\right) $
Hence by statement (i) $x\geq -\sup \left( -\mathbf{A}\right) $ and
we see that $-\sup \left( -\mathbf{A}\right) $ is a lower
bound of $\mathbf{A.}$(iii) If $l_{\mathbf{A}}\leq x$ for all $x\in
\mathbf{A}$ then $-l_{\mathbf{A}}\geq y$ for all $y\in -\mathbf{A.}$Hence
$-l_{\mathbf{A}}\geq \sup \left( -\mathbf{A}\right) $ by virtue of $\sup
\left( -\mathbf{A}\right) $ being the least upper bound of $-\mathbf{A.}$
Finally we obtain: $l_{\mathbf{A}}\leq -\sup \left( -\mathbf{A}\right) .$
Definition 1.3.6.14. We call this greatest element a
greatest lower bound $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
or infinum of $\mathbf{A}$,written is $\inf
\left( \mathbf{A}\right) .$
Definition 1.3.6.15.Let us consider an nonempty subset $\mathbf{A}%
\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which
is bounded or hyperbounded from below and such that:
$\inf \left( \mathbf{A}\right) \pm \varepsilon \neq \inf \left( \mathbf{A}%
\right) $ for any $\varepsilon >0,$ $\varepsilon \approx 0.$
We call this greatest lower bound $\inf \left( \mathbf{A}\right) $ a
strong greatest lower bound or
strong infinum, written is $\mathbf{s}$-$\inf \left( \mathbf{A}%
\right) $.
Definition 1.3.6.16.Let us consider an nonempty subset $\mathbf{A}%
\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which
is bounded or hyperbounded from below and such that:
(1) there exist $\varepsilon _{0}\approx 0$ such that $\inf \left(
\mathbf{A}\right) \pm \varepsilon _{0}=\inf \left( \mathbf{A}\right) ,$
(2) $\inf \left( \mathbf{A}\right) \pm \varepsilon \neq \inf \left(
\mathbf{A}\right) $ for any $\varepsilon >0$ such that $\varepsilon \geq
\varepsilon _{0}\approx 0.$
We call this greatest lower bound $\inf \left( \mathbf{A}\right) $
almost strong greatest lower bound
or almost strong infinum, written is $o\mathbf{s}$-$\inf
\left( \mathbf{A}\right) $.
Definition 1.3.6.17. Let us consider an nonempty subset $\mathbf{A}%
\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which
is bounded or hyperbounded from below and such that:
(1) $\inf \left( \mathbf{A}\right) \pm \varepsilon =\inf \left(
\mathbf{A}\right) $ for any $\varepsilon >0,$ $\varepsilon \approx 0,$
(2) $\inf \left( \mathbf{A}\right) \pm \varepsilon \neq \inf \left(
\mathbf{A}\right) $ for any $\varepsilon >0$ such that $\varepsilon
\not\approx 0.$
We call this greatest lower bound $\inf \left( \mathbf{A}\right) $
weak greatest lower bound
or weak infinum, written is $w$-$\inf \left( \mathbf{A}\right) $.
Definition 1.3.6.18. Let us consider an nonempty subset $\mathbf{A}%
\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which
is hyperbounded from below and such that:
(1) $\inf \left( \mathbf{A}\right) \pm \alpha =\inf \left( \mathbf{A%
}\right) $ for any $\alpha >0,$ $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
(2) $\inf \left( \mathbf{A}\right) \pm \Gamma \neq \inf \left(
\mathbf{A}\right) $ for any $\Gamma >0$ such that $\Gamma \in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }.$
We call this greatest lower bound $\inf \left( \mathbf{A}\right) $
ultra weak greatest lower bound
or ultra weak infinum, written is $uw$-$\inf \left(
\mathbf{A}\right) $.
Proposition 1.3.6.2. (1) If $\mathbf{A}$ is a nonempty
subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded
from below and strong infinum $\mathbf{s}$-$\inf \left( \mathbf{A}\right) $
exist, then:
$\mathbf{s}$-$\inf \left( \mathbf{A}\right) $ is the unique number such that
$\mathbf{s}$-$\inf \left( \mathbf{A}\right) $ is an upper bound
for $\mathbf{A}$ and $\mathbf{s}$-$\inf \left( \mathbf{A}\right)
+\varepsilon $ is not a lower bound for $\mathbf{A}$ for any $\varepsilon
\approx 0,\varepsilon >0;$
(2) If $\mathbf{A}$ is a nonempty subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded from above, then:
$\mathbf{s}$-$\sup \left( \mathbf{A}\right) $ is the unique number such that
$\mathbf{s}$-$\sup \left( \mathbf{A}\right) $ is an upper bound for $\mathbf{%
and $\mathbf{s}$-$\sup \left( \mathbf{A}\right) -\varepsilon $ is not an
upper bound for $\mathbf{A}$ for any $\varepsilon \approx 0,\varepsilon >0.$
Proposition 1.3.6.3.(a). (Strong Approximation Property.)
(1) If $\mathbf{A}$ is a nonempty subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded (hyperbounded)
from above and such that strong supremum $\mathbf{s}$-$\sup \left( \mathbf{A}%
\right) $ exist, and let
$\varepsilon \approx 0,\varepsilon >0$ there exist $x\in \mathbf{A}$ such
that $\mathbf{s}$-$\sup \left( \mathbf{A}\right) -\varepsilon <x\leq \mathbf{%
s}$-$\sup \left( \mathbf{A}\right) .$
(2) If $\mathbf{A}$ is a nonempty subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded (hyperbounded)
from below and such that strong infinum $\mathbf{s}$-$\inf \left( \mathbf{A}%
\right) $ exist, and let
$\varepsilon \approx 0,\varepsilon >0$ there exist $x\in \mathbf{A}$ such
that $\mathbf{s}$-$\inf \left( \mathbf{A}\right) \leq x<\mathbf{s}$-$\inf
\left( \mathbf{A}\right) +\varepsilon .$
Proof. (1) If not, then $\mathbf{s}$-$\sup \left( \mathbf{A%
}\right) -\varepsilon $ is an upper bound of $\mathbf{A}$ less than the
strong upper bound $\mathbf{s}$-$\sup \left( \mathbf{A}\right) $,
which is a contradiction.
(2) If not, then $\mathbf{s}$-$\inf \left( \mathbf{A}\right)
+\varepsilon $ is an lower bound of $\mathbf{A}$ bigger than the
strong lower bound $\mathbf{s}$-$\inf \left( \mathbf{A}\right) $, which is a
Proposition 1.3.6.3.(b) (The Almost Strong Approximation Property.)
(1) If $\mathbf{A}$ is a nonempty subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded (hyperbounded)
from above and such that almost strong supremum $o\mathbf{s}$-$\sup \left(
\mathbf{A}\right) $ exist, and
let $\varepsilon \approx 0,\varepsilon >0,o\mathbf{s}$-$\sup \left( \mathbf{A%
}\right) \pm \varepsilon \neq o\mathbf{s}$-$\sup \left( \mathbf{A}\right) $
there exist $x\in \mathbf{A}$ such that
$o\mathbf{s}$-$\sup \left( \mathbf{A}\right) -\varepsilon <x\leq o\mathbf{s}$
-$\sup \left( \mathbf{A}\right) .$
(2) If $\mathbf{A}$ is a nonempty subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded (hyperbounded)
from below and such that almost strong infinum $o\mathbf{s}$-$\inf \left(
\mathbf{A}\right) $ exist, and let
$\varepsilon \approx 0,\varepsilon >0,o\mathbf{s}$-$\inf \left( \mathbf{A}%
\right) \pm \varepsilon \neq o\mathbf{s}$-$\inf \left( \mathbf{A}\right) $
there exist $x\in \mathbf{A}$ such that
$o\mathbf{s}$-$\inf \left( \mathbf{A}\right) \leq x<o\mathbf{s}$-$\inf
\left( \mathbf{A}\right) +\varepsilon .$
Proof. (1) If not, then $o\mathbf{s}$-$\sup \left( \mathbf{%
A}\right) -\varepsilon $ is an upper bound of $\mathbf{A}$ less than the
almost strong upper bound $o\mathbf{s}$-$\sup \left( \mathbf{A}\right) $,
which is a contradiction.
(2) If not, then $o\mathbf{s}$-$\inf \left( \mathbf{A}\right)
+\varepsilon $ is an lower bound of $\mathbf{A}$ bigger than the
almost strong lower bound $o\mathbf{s}$-$\inf \left( \mathbf{A}%
\right) $, which is a contradiction.
Proposition 1.3.6.3.(c) (The Weak Approximation Property.)
(1) If $\mathbf{A}$ is a nonempty subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded (hyperbounded)
from above and such that weak supremum $w$-$\sup \left( \mathbf{A}\right) $
exist, and let
$\varepsilon \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\varepsilon >0,w$-$\sup \left( \mathbf{A}\right) \pm \varepsilon \neq w$-$%
\sup \left( \mathbf{A}\right) $ there exist $x\in \mathbf{A}$ such that
$w$-$\sup \left( \mathbf{A}\right) -\varepsilon <x\leq w$-$\sup \left(
\mathbf{A}\right) .$
(2) If $\mathbf{A}$ is a nonempty subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded (hyperbounded)
from below and such that weak infinum $w$-$\inf \left( \mathbf{A}\right) $
exist, and let
$\varepsilon \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\varepsilon >0,w$-$\inf \left( \mathbf{A}\right) \pm \varepsilon \neq w$-$%
\inf \left( \mathbf{A}\right) $ there exist $x\in \mathbf{A}$ such that
$w$-$\inf \left( \mathbf{A}\right) \leq x<w$-$\inf \left( \mathbf{A}\right)
+\varepsilon .$
Proof. (1) If not, then $w$-$\sup \left( \mathbf{A}\right)
-\varepsilon $ is an upper bound of $\mathbf{A}$ less than the
weak upper bound $w$-$\sup \left( \mathbf{A}\right) $, which is a
(2) If not, then
Corollary 1.3.6.2. (1) Let $\mathbf{A}$ be
bounded or hyperbounded from below and
non-empty set such that $\mathbf{s}$-$\inf \left( \mathbf{A}\right) $ exist.
There is a function
$\beta \left( \circ \right) :$ $^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that for all $\mathbf{n\in }$ $^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }$ we have
$\mathbf{s}$-$\inf \left( \mathbf{A}\right) \leq \beta \left( \mathbf{n}%
\right) <\mathbf{s}$-$\inf \left( \mathbf{A}\right) +\mathbf{n}^{-1}.$
(2) Let $\mathbf{A}$ be bounded or hyperbounded from above and
non-empty set such that $\mathbf{s}$-$\sup \left( \mathbf{A}\right) $ exist.
There is a function
$\alpha \left( \circ \right) :$ $^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that for all $\mathbf{n\in }$ $^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }$ we have
$\mathbf{s}$-$\sup \left( \mathbf{A}\right) -\mathbf{n}^{-1}<\alpha \left(
\mathbf{n}\right) \leq \mathbf{s}$-$\sup \left( \mathbf{A}\right) .$
Example 1.3.6.6. (a) The subset $\left\{ \mathbf{%
n}^{-1}\right\} _{\mathbf{n\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }}\triangleq \left\{ \left. \dfrac{1}{\mathbf{n}}\right\vert
\mathbf{n\in }\text{ }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }\right\} \subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
has a strong greatest lower bound $\mathbf{s}$-$\inf \left( \left\{
\mathbf{n}^{-1}\right\} _{\mathbf{n\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }}\right) =0$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(b) The subset $\left\{ n^{-1}\right\} _{\mathbf{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}}\triangleq \left\{ \left. \dfrac{1}{n}\right\vert n\mathbf{\in }\text{ }%
\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has a greatest lower bound
$\inf \left( \left\{ n^{-1}\right\} _{n\mathbf{\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}}\right) $ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ but has not a strong greatest lower bound in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Example 1.3.6.7. The subset $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has the least upper bound $\sup \left(
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) $
in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ but has not strong least upper bound in $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Example 1.3.6.8. The subset $\mathbf{I}_{\ast }$ the all
infinitesimal members of the $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\mathbf{I}_{\ast }\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has least upper bound $\sup \left( \mathbf{I}_{\ast }\right) $
in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ but has not strong least
upper bound in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Example 1.3.6.9. The subset $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has lower bound $\inf \left(
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}\right) $
in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ but has not strong lower bound in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Example 1.3.6.10. The subset $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}\subsetneqq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has lower bound $\inf \left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}\right) $
in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ but has not strong lower bound in $^{\ast
%TCIMACRO{\U{211d} }%
\mathbb{R}
Proposition 1.3.6.4.Let $\mathbf{A}$ and $\mathbf{B}$ be nonempty
subsets of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Theorem 1.3.6.3.A. Let $\mathbf{A}$ and $\mathbf{B}$ be nonempty
subsets of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ and
$\mathbf{C}=$ $\left\{ a+b:a\in \mathbf{A},b\in \mathbf{B}\right\} $.
(1.a) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from above
(hence $\mathbf{s}$-$\sup \left( \mathbf{A}\right) $ and $\mathbf{s}$-$\sup
\left( \mathbf{B}\right) $ exist) then $\mathbf{s}$-$\sup \left( \mathbf{C}%
\right) $ exist
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{s}\text{-}\sup \left( \mathbf{C}\right) =\mathbf{s}\text{-}\sup
\left( \mathbf{A}\right) +\mathbf{s}\text{-}\sup \left( \mathbf{B}\right) \\
\end{array}
& \text{ \ }\left( 1.3.6.21.\mathbf{a}\right)%
\end{array}%
(2.a) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from below
(hence $\mathbf{s}$-$\inf \left( \mathbf{A}\right) $ and $\mathbf{s}$-$\inf
\left( \mathbf{B}\right) $ exist) then $\mathbf{s}$-$\inf \left( \mathbf{C}%
\right) $ exist and
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{s}\text{-}\inf \left( \mathbf{C}\right) =\mathbf{s}\text{-}\inf
\left( \mathbf{A}\right) +\mathbf{s}\text{-}\inf \left( \mathbf{B}\right) \\
\end{array}
& \text{ \ }\left( 1.3.6.22.\mathbf{a}\right)%
\end{array}%
(1.b) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from above,
(hence $o\mathbf{s}$-$\sup \left( \mathbf{A}\right) $ and $o\mathbf{s}$-$%
\sup \left( \mathbf{B}\right) $ exist) then $o\mathbf{s}$-$\sup \left(
\mathbf{C}\right) $ exist
and $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ $
$\bigskip $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
o\mathbf{s}\text{-}\sup \left( \mathbf{C}\right) =o\mathbf{s}\text{-}\sup
\left( \mathbf{A}\right) +o\mathbf{s}\text{-}\sup \left( \mathbf{B}\right)
\\
\end{array}
& \text{ \ }\left( 1.3.6.21.\mathbf{b}\right)%
\end{array}%
(2.b) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from below
(hence $o\mathbf{s}$-$\inf \left( \mathbf{A}\right) $ and $o\mathbf{s}$-$%
\inf \left( \mathbf{B}\right) $ exist) then $o\mathbf{s}$-$\inf \left(
\mathbf{C}\right) $ exist
\begin{array}{cc}
\begin{array}{c}
\\
o\mathbf{s}\text{-}\inf \left( \mathbf{C}\right) =o\mathbf{s}\text{-}\inf
\left( \mathbf{A}\right) +o\mathbf{s}\text{-}\inf \left( \mathbf{B}\right)
\\
\end{array}
& \text{ \ }\left( 1.3.6.22.\mathbf{b}\right)%
\end{array}%
(1.c) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from above,
hence and $w$-$\sup \left( \mathbf{A}\right) $ and $w$-$\sup \left( \mathbf{B%
}\right) $ exist, then $w$-$\sup \left( \mathbf{C}\right) $ exist
$\bigskip \
\begin{array}{cc}
\begin{array}{c}
\\
w\text{-}\sup \left( \mathbf{C}\right) =w\text{-}\sup \left( \mathbf{A}%
\right) +w\text{-}\sup \left( \mathbf{B}\right) \\
\end{array}
& \text{ \ }\left( 1.3.6.21.\mathbf{c}\right)%
\end{array}%
(2.c) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from below
(hence $w$-$\inf \left( \mathbf{A}\right) $ and $w$-$\inf \left( \mathbf{B}%
\right) $ exist) then $w$-$\inf \left( \mathbf{C}\right) $ exist and
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
w\text{-}\inf \left( \mathbf{C}\right) =w\text{-}\inf \left( \mathbf{A}%
\right) +w\text{-}\inf \left( \mathbf{B}\right) \\
\end{array}
& \text{ \ }\left( 1.3.6.22.\mathbf{c}\right)%
\end{array}%
(1.d) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from above
(then $uw$-$\sup \left( \mathbf{A}\right) $ and $uw$-$\sup \left( \mathbf{B}%
\right) $ exist) then $uw$-$\sup \left( \mathbf{C}\right) $ exist
and $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ $
$\bigskip $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
uw\text{-}\sup \left( \mathbf{C}\right) =uw\text{-}\sup \left( \mathbf{A}%
\right) +uw\text{-}\sup \left( \mathbf{B}\right) \\
\end{array}
& \text{\ }\left( 1.3.6.21.\mathbf{d}\right)%
\end{array}%
(2.d) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from below
(hence $uw$-$\inf \left( \mathbf{A}\right) $ and $uw$-$\inf \left( \mathbf{B}%
\right) $ exist) then $uw$-$\inf \left( \mathbf{C}\right) $ exist
and $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ $
$\bigskip $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
uw\text{-}\inf \left( \mathbf{C}\right) =uw\text{-}\inf \left( \mathbf{A}%
\right) +uw\text{-}\inf \left( \mathbf{B}\right) \\
\end{array}
& \text{ \ \ }\left( 1.3.6.22.\mathbf{d}\right)%
\end{array}%
Proof. (1.a) Suppose that $\mathbf{A}$ and $\mathbf{B}$
are bounded or hyperbounded from
above,hence $\mathbf{s}$-$\sup (\mathbf{A})$ and $\mathbf{s}$-$\sup (\mathbf{%
B})$ exist. Let $c$ $\in $ $\mathbf{C}$. Then $c=a+b$ for
numbers $a\in $ $\mathbf{A}$ and $b$ $\in $ $\mathbf{B}$. Since $a\leq
\mathbf{s}$-$\sup (\mathbf{A})$ and $b\leq \mathbf{s}$-$\sup (\mathbf{B})$,
$c=a+b\leq \mathbf{s}$-$\sup (\mathbf{A})$ $+$ $\mathbf{s}$-$\sup (\mathbf{B}%
)$. This shows that $\mathbf{s}$-$\sup (\mathbf{A})+\mathbf{s}$-$\sup (%
\mathbf{B})$ is an
upper bound for $\mathbf{C}$, in particular, $\mathbf{C}$ is bounded or
hyperbounded from above.
Given $\varepsilon >0,$ $\mathbf{s}$-$\sup (\mathbf{A})-\varepsilon /2$ is
not an a strong upper bound for $\mathbf{A}$ hence there
exists $a^{\prime }\in \mathbf{A}$ such that $\mathbf{s}$-$\sup (\mathbf{A}%
)-\varepsilon /2<a^{\prime }.$ Similarly, $\mathbf{s}$-$\sup (\mathbf{B})$ $%
-\varepsilon /2$ is not an
upper bound for $\mathbf{B}$ and there exists $b^{\prime }$ $\in $ $\mathbf{B%
}$ such that $\mathbf{s}$-$\sup (\mathbf{B})-\varepsilon /2<b^{\prime }.$ So
for $c^{\prime }=a^{\prime }+b^{\prime }$ $\in $ $\mathbf{C}$ we have $%
\mathbf{s}$-$\sup (\mathbf{A})$ $+$ $\mathbf{s}$-$\sup (\mathbf{B}%
)-\varepsilon <c^{\prime }.$ This shows that
$\mathbf{s}$-$\sup (\mathbf{A})$ $+$ $\mathbf{s}$-$\sup (\mathbf{B}%
)-\varepsilon $ is not an a strong upper bound for $\mathbf{C}$ for any $%
\varepsilon >0.$
Hence by the Proposition 1.3.1 one obtain: $\mathbf{s}$-$\sup (\mathbf{C})=$
$\mathbf{s}$-$\sup (\mathbf{A})$ $+$ $\mathbf{s}$-$\sup (\mathbf{B})$.
By using Theorem 1.3.6.3.A one obtain:
Theorem 1.3.6.3.B. Let $\mathbf{A}$ and $\mathbf{B}$ be nonempty
subsets of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ and
$\mathbf{C}=$ $\left\{ a+b:a\in \mathbf{A},b\in \mathbf{B}\right\} $.
(1) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from above
(hence $\sup \left( \mathbf{A}\right) $ and $\sup \left( \mathbf{B}\right) $
exist) then $\sup \left( \mathbf{C}\right) $ exist and
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\sup \left( \mathbf{C}\right) =\sup \left( \mathbf{A}\right) +\sup \left(
\mathbf{B}\right) \\
\end{array}
& \text{ \ \ }\left( 1.3.6.23\right) \text{\ }%
\end{array}%
(2) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from below
(hence $\inf \left( \mathbf{A}\right) $ and $\inf \left( \mathbf{B}\right) $
exist) then $\inf \left( \mathbf{C}\right) $ exist and $\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\bigskip $
\begin{array}{cc}
\begin{array}{c}
\\
\inf \left( \mathbf{C}\right) =\inf \left( \mathbf{A}\right) +\inf \left(
\mathbf{B}\right) . \\
\end{array}
& \text{ \ \ }\left( 1.3.6.24\right) \text{\ \ }%
\end{array}%
Theorem 1.3.6.3.C. Setting (1). Suppose that $\mathbf{S}$
is a non-empty subset
of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is bounded or hyperbounded from above and $\mathbf{s}$-$%
\sup \mathbf{S}$ exist
and $\xi \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\xi >0.$Then
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{s}\text{-}\underset{x\in \mathbf{S}}{\sup }\left\{ \xi \times
x\right\} = \\
\\
\xi \times \left( \mathbf{s}\text{-}\underset{x\in \mathbf{S}}{\sup }\left\{
x\right\} \right) =\xi \times \left( \mathbf{s}\text{-}\sup \mathbf{S}%
\right) \mathbf{.} \\
\end{array}
& \text{ \ }\left( 1.3.6.25\right) \text{\ }%
\end{array}%
Setting (2).Suppose that $\mathbf{S}$ is a non-empty subset of $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ which is
bounded or hyperbounded from above and $o\mathbf{s}$-$\sup \mathbf{S}$ exist
and $\xi \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\xi >0.$Then
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
o\mathbf{s}\text{-}\underset{x\in \mathbf{S}}{\sup }\left\{ \xi \times
x\right\} = \\
\\
\xi \times \left( o\mathbf{s}\text{-}\underset{x\in \mathbf{S}}{\sup }%
\left\{ x\right\} \right) =\xi \times \left( o\mathbf{s}\text{-}\sup \mathbf{%
S}\right) \mathbf{.} \\
\end{array}
& \text{ \ }\left( 1.3.6.26\right) \text{\ }%
\end{array}%
Proof. (1) Let $B=\mathbf{s}$-$\sup \mathbf{S.}$Then $B$
is the smallest number such that, for
any $x\in \mathbf{S,}x$ $\mathbf{\leq }B\mathbf{.}$Let $\mathbf{T}=\left\{
\xi \times x|x\in \mathbf{S}\right\} .$Since $\xi >0,\xi \times x\leq \xi
\times B$ for any
$x\in \mathbf{S.}$Hence $\mathbf{T}$ is bounded or hyperbounded above by $%
\xi \times B.$By the
Theorem 1 and setting (1), $\mathbf{T}$ has a strong supremum $C,C=\mathbf{s}
$-$\sup \mathbf{T.}$
Now we have to pruve that $C=\xi \times B.$Since $\xi \times B$ is an apper
bound for $\mathbf{T}$
and $C$ is the smollest apper bound for $\mathbf{T,}C\leq \xi \times B.$Now
we repeat the
argument above with the roles of $\mathbf{S}$ and $\mathbf{T}$ reversed. We
know that $C$ is the
smallest number such that, for any $y\in \mathbf{T,}y\leq C.$Since $\xi >0$
it follows that
$\xi ^{-1}\times y\leq \xi ^{-1}\times C$ for any $y\in \mathbf{T.}$But $%
\mathbf{S=}\left\{ \xi ^{-1}\times y|y\in \mathbf{T}\right\} .$Hence $\xi
^{-1}\times C$ is an
apper bound for $\mathbf{S.}$But $B$ is a strong supremum for $\mathbf{S.}$
Hence $B\leq \xi ^{-1}\times C$
and $\xi \times B\leq C.$We have shown that $C\leq \xi \times B$ and also
that $\xi \times B\leq C.$Thus
$\xi \times B=C.$
Theorem 1.3.6.3.D. Let $\mathbf{A}$ and $\mathbf{B}$ be nonempty
subsets of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that
$0\leq \mathbf{A,}0\leq \mathbf{B}$ and $\mathbf{C}=$ $\left\{ a\times
b:a\in \mathbf{A},b\in \mathbf{B}\right\} $.
(1.a) If $\mathbf{A}$ and $\mathbf{B}$ are bounded or hyperbounded
from above,hence $\mathbf{s}$-$\sup \left( \mathbf{A}\right) $
and $\mathbf{s}$-$\sup \left( \mathbf{B}\right) $ exist, then $\mathbf{s}$-$%
\sup \left( \mathbf{C}\right) $ exist and
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{s}\text{-}\sup \left( \mathbf{C}\right) =\left[ \mathbf{s}\text{-}%
\sup \left( \mathbf{A}\right) \right] \times \left[ \mathbf{s}\text{-}\sup
\left( \mathbf{B}\right) \right] \\
\end{array}
& \text{ \ \ }\left( 1.3.6.21^{\prime }.\mathbf{a}\right)
\end{array}%
Proposition 1.3.6.5. Let $\mathbf{A}$ and $\mathbf{B}$ be nonempty
subsets of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(i) If for every $a\in \mathbf{A}$ there exists $b\in \mathbf{B}$
with $a\leq b$ and $\mathbf{B}$ is bounded
from above,then so is $\mathbf{A}$ and $\sup \left( \mathbf{A}\right) \leq
\sup \left( \mathbf{B}\right) .$
(ii) If for every $b\in \mathbf{B}$ there exists $a\in \mathbf{A}$
with $a\leq b$ and $\mathbf{A}$ is bounded
from below,then so is $\mathbf{B}$ and $\inf \left( \mathbf{A}\right) \leq
\inf \left( \mathbf{B}\right) .$
Proof. (ii) Suppose that for every $b\in \mathbf{%
B}$ there exists $a\in \mathbf{A}$ with $a\leq b$
and $\mathbf{A}$ is bounded from below. Then $\inf (\mathbf{A})$ exists. For
every $b\in \mathbf{B}$ there
is a $\in $ $\mathbf{A}$ such that $a\leq b.$ So $\inf \left( \mathbf{A}%
\right) $ $\leq a\leq b.$Therefore $\inf \left( \mathbf{A}\right) $ is a
bound for $\mathbf{B.}$Hence $\mathbf{B}$ is bounded from below and $\inf
\left( \mathbf{B}\right) $ exists. By
definition of the infimum (greatest lower bound) one obtain:
$\inf \left( \mathbf{B}\right) \geq \inf \left( \mathbf{A}\right) .$
Lemma 1.3.6.1. (a) $\mathbf{s}$-
$\inf \left( ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) $ and $\mathbf{s}$-$\sup \left( ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) $ is not exists in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(b) $\mathbf{s}$-$\sup \left( ^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right) $ is not exists in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(c) $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ is bounded neither from below nor from above in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(d) $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ is not bounded from above in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Proof. (a) Assume that $\mathbf{s}$-$%
\sup \left( ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) $ exists in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$.Then $\mathbf{s}$-$\sup \left( ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) -1$
is not an upper bound and hence there exists $n\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ such that
$\mathbf{s}$-$\sup \left( ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) -1<n$ hence $\mathbf{s}$-$\sup \left( ^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) <n+1.$But since $n+1\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
this is a contradiction. Therefore $\mathbf{s}$-$\sup \left( ^{\ast
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right) $ is not exists in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(d) Assume that $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ has an upper bound, call it $\mathbf{\Theta .}$Hence $\mathbf{\Theta }%
^{-1} $
is a lower bound for the set $\left\{ \mathbf{n}^{-1}\right\} _{\mathbf{n\in
}^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }}$ and consequently
$\inf \left( \left\{ \mathbf{n}^{-1}\right\} _{\mathbf{n\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }}\right) \geq \mathbf{\Theta }^{-1}\neq 0.$But we know that $\ast
$-$\lim_{\mathbf{n}\rightarrow \text{ }^{\ast }\infty }\mathbf{n}^{-1}=0$
which is a contradiction.
Theorem 1.3.6.4. (Generalyzed Archimedean Property of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
For any $\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},\varepsilon \approx 0,\varepsilon >0$ there exists $\mathbf{%
n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ such that $\mathbf{n}^{-1}<\varepsilon .$
Proof. Since $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ is not hyperbounded from above $\varepsilon ^{-1}$ is not an upper
bound for $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
.$ Hence there exists $\mathbf{n}\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ such that $\mathbf{n>}$ $\varepsilon ^{-1}$ and
consequently $\mathbf{n}^{-1}<\varepsilon .$
Theorem 1.3.6.5. For every $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that for the set
$\left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|n\leq x\right\} $one of the next conditions is satisfied:
(i) strong supremum $\mathbf{s}$-$\sup \left( \left\{ n\in \text{ }%
^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|n\leq x\right\} \right) $ exists in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ or
(ii) almost strong supremum $\mathbf{os}$-$\sup \left( \left\{ n\in
\text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|n\leq x\right\} \right) $ exists in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ or
(iii) weak supremum $w$-$\sup \left( \left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|n\leq x\right\} \right) $ exists in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
there exists a unique $m\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ such that $m\leq x<m+1.$
Proof. Let $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Existence: Since $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ is not hyperbounded from below $x$ is not a lower
bound for $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ hence the set $\mathbf{A=}\left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|n\leq x\right\} $is not empty. Moreover,
$x$ is an upper bound for $\mathbf{A}$ by definition of $\mathbf{A.}$Hence,
as a subset of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\mathbf{A}$ has a supremum $\sup \left( \left\{ n\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
|n\leq x\right\} \right) $,call it $\Delta \left( x\right) .$ $\Delta \left(
x\right) -1$ is not an
upper bound for $\mathbf{A}$ hence there exists $m\in \mathbf{A\subset }$ $%
^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ such that $\Delta \left( x\right) -1<m$
and consequently $\Delta \left( x\right) <m+1.$So $m+1\notin \mathbf{A,}$so $%
$m\leq x<m+1.$
Uniqueness: Suppose that $m^{\prime }\leq x<m^{\prime }+1$ for $%
m^{\prime }\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
.$If $m^{\prime }<m,$ then
$m^{\prime }+1\leq m$ implying $m^{\prime }\leq x<m^{\prime }+1\leq m\leq x,$
a contradiction. $m<m^{\prime }$
leads to a similar contradiction. So $m=m^{\prime }.$
Let $E=\left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}|x^{2}=x\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}x<_{\text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}2\right\} .$Note that $1^{2}=1<2,$ so that
$1\in E$ and in particular $E$ is non-empty. Further if $x>2$ then:
$x^{2}=x\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Hence $2$ is an upper bound for $E$ and so we may define $\zeta \triangleq
\sup E.$
Theorem 1.3.6.6. Suppose that $\zeta =\mathbf{s}$-$\sup E$
exist.There exists a unique
positive number $\zeta \triangleq Ext$-$\sqrt{2}\triangleq \#$-$\sqrt{2}\in $
$^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that $\zeta ^{2}=\zeta \times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}}\zeta =_{\text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Proof. Note further that $\zeta $ $>1>0$ is positive. We split the
remainder of
the proof into showing that $\zeta ^{2}<2$ and $\zeta ^{2}>2$ both lead to
Suppose for a contradiction that $\zeta ^{2}<2.$Let $h=\dfrac{1}{2}\min
\left( \zeta ,\dfrac{2-\zeta ^{2}}{3\zeta }\right) >0.\ $
Then $(\zeta +h)^{2}=\zeta ^{2}+2h\times \zeta +h^{2}<\zeta ^{2}+3h\times
\zeta <\zeta ^{2}+(2-\zeta ^{2})=2.$Since $h<\zeta $
and $h<\dfrac{2-\zeta ^{2}}{3\zeta }.$Hence $\zeta +h\in E$ and since $\zeta
=\mathbf{s}$-$\sup E$ we get $\zeta +h<\zeta ,$ a
Suppose instead that $\zeta ^{2}>2.$ Let $h=\dfrac{1}{2}\left( \dfrac{\zeta
^{2}-2}{2\zeta }\right) >0.\ $As $\zeta -h<\zeta $ there
exists $\epsilon \in E$ with $\zeta -h<\epsilon $ by the Strong
Approximation Property; then
$(\zeta -h)^{2}<\epsilon ^{2}<2\Rightarrow \zeta ^{2}-2h\times \zeta
+h^{2}<2.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ $
As $h^{2}>0$ this gives $\zeta ^{2}-2h\times \zeta <2,$ and so, since $\zeta
>0,$ we have
$h>\left( \zeta ^{2}-2\right) /2\zeta $ which contradicts our choice of $h.$
Finally, by trichotomy,
$\zeta ^{2}=2$ follows as the only remaining possibility.
Let $E_{<}=\left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|x^{2}=x\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}x<_{\text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}2\right\} $ and
$E_{>}=\left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|x^{2}=x\times _{^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}x>_{\text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
}2\right\} .$
Hence a Dedekind hyperreal $\#$-$\sqrt{2}\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is a pair $(U,V)\in \mathbf{P}\left( ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\right) \times $ $\mathbf{P}\left( ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\right) $
where $U=E_{<},V=E_{>}.$
Theorem 1.3.6.7.Let $a\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ be any positive hyperreal number. Then for any
$n\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ there exists a unique Dedekind hyperreal number $\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(denoted by$\left( \sqrt[n]{a}\right) _{\mathbf{d}}$) such that $\alpha
Theorem 1.3.6.8.($\ast $-Density of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$). Let $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ be a Dedekind
hyperreal number such that $x\pm \varepsilon \neq x$ for any $\varepsilon
>0, $ $\varepsilon \approx 0.$. For every $\epsilon >0$
there exists a hyperrational number $r\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ such that $x-\epsilon <r<x+\epsilon .$
Proof. Let $\epsilon >0$ be given. By the Generalyzed
Archimedean Property of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
we can pick $n\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ with $n^{-1}<\epsilon .$Let $q=\left[ \left\vert nx\right\vert \right] \in
$ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
.$Since $q\leq nx<q+1,$we
have $\dfrac{q}{n}\leq x<\dfrac{q}{n}+\dfrac{1}{n}<\dfrac{q}{n}+\epsilon .$
Now let $r=\dfrac{q}{n}\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
.$Then $r\leq x<r+\epsilon $ and
hence $x-\epsilon <r<x+\epsilon .$
§ REARRANGEMENTS OF COUNTABLE INFINITE SERIES.
Definition 1.3.6.19.(i) Let be $\left\{ \mathbf{s%
}_{n}\right\} _{n=1}^{\infty }$ countable sequence $\mathbf{s}_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that: (a) $\forall n\left( \mathbf{s}_{n}\geq 0\right) $ or (
b) $\forall n\left( \mathbf{s}_{n}<0\right) $ or
(c) $\left\{ \mathbf{s}_{n}\right\} _{n=1}^{\infty }=\left\{
\mathbf{s}_{n_{1}}\right\} _{n_{1}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{1}}^{\infty }\cup \left\{ \mathbf{s}_{n_{2}}\right\} _{n_{2}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{2}}^{\infty },\forall n_{1}\left( n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\right) \left[ \mathbf{s}_{n_{1}}\geq 0\right] ,$
$\forall n_{2}\left( n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}\right) \left[ \mathbf{s}_{n_{2}}<0\right] ,%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\cup \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
Then external $\flat $-sum of the countable sequence $\mathbf{s}_{n}$ denoted
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\ast }\mathbf{s}_{n}\right) \right) ^{\flat } \\
\end{array}
& \text{\ }\left( 1.3.6.23\right)
\end{array}%
is$\ \ \ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\left( \mathbf{a}\right) \text{ \ \ \ \ }\forall n\left( \mathbf{s}_{n}\geq
0\right) : \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\ast }\mathbf{s}_{n}\right) \right) ^{\flat }\triangleq \\
\\
\triangleq \text{ }\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left\{ \dsum\limits_{n\leq k}\left( ^{\ast }\mathbf{s}_{n}\right)
^{\#}\right\} , \\
\\
\left( \mathbf{b}\right) \text{ \ \ \ \ \ \ }\forall n\left( \mathbf{s}%
_{n}<0\right) : \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\ast }\mathbf{s}_{n}\right) \right) ^{\flat }\triangleq \\
\\
\triangleq \text{ }\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left\{ \dsum\limits_{n\leq k}\left( ^{\ast }\mathbf{s}_{n}\right)
^{\#}\right\} . \\
\\
\left( \mathbf{c}\right) \text{ \ \ }\forall n_{1}\left( n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\right) \left[ \mathbf{s}_{n_{1}}\geq 0\right] , \\
\\
\forall n_{2}\left( n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}\right) \left[ \mathbf{s}_{n_{2}}<0\right] ,%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\cup \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}: \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\ast }\mathbf{s}_{n}\right) \right) ^{\flat }\triangleq \\
\\
\triangleq \left( \#Ext\text{-}\dsum\limits_{n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}}\left( ^{\ast }\mathbf{s}_{n_{1}}\right) \right) ^{\flat }+\left( \#Ext%
\text{-}\dsum\limits_{n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}}\left( ^{\ast }\mathbf{s}_{n_{2}}\right) \right) ^{\flat }. \\
\end{array}
& \text{ \ }\left( 1.3.6.24\right)%
\end{array}%
Definition 1.3.6.20.(i) Let be $\left\{ \mathbf{s%
}_{n}\right\} _{n=1}^{\infty }$ countable sequence $\mathbf{s}_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that: (a) $\forall n\left( \mathbf{s}_{n}\geq 0\right) $ or (
b) $\forall n\left( \mathbf{s}_{n}<0\right) $ or
(c) $\left\{ \mathbf{s}_{n}\right\} _{n=1}^{\infty }=\left\{
\mathbf{s}_{n_{1}}\right\} _{n_{1}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{1}}^{\infty }\cup \left\{ \mathbf{s}_{n_{2}}\right\} _{n_{2}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{2}}^{\infty },\forall n_{1}\left( n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\right) \left[ \mathbf{s}_{n_{1}}\geq 0\right] ,$
$\forall n_{2}\left( n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}\right) \left[ \mathbf{s}_{n_{2}}<0\right] ,%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\cup \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
Then external $\flat $-sum of the countable sequence $\mathbf{s}_{n}$ denoted
$\ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{\#}\right) ^{\flat } \\
\end{array}
& \text{ \ }\left( 1.3.6.23^{\prime }\right)
\end{array}%
is$\ \ $
$\ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\left( \mathbf{a}\right) \text{ \ \ \ \ }\forall n\left( \mathbf{s}_{n}\geq
0\right) : \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{\#}\right) ^{\flat }\triangleq \\
\\
\triangleq \text{ }\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left\{ \dsum\limits_{n\leq k}s_{n}^{\#}\right\} , \\
\\
\left( \mathbf{b}\right) \text{ \ \ \ \ \ \ }\forall n\left( \mathbf{s}%
_{n}<0\right) : \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{\#}\right) ^{\flat }\triangleq \\
\\
\triangleq \text{ }\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left\{ \dsum\limits_{n\leq k}s_{n}^{\#}\right\} . \\
\\
\left( \mathbf{c}\right) \text{ \ \ }\forall n_{1}\left( n_{1}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{1}\right) \left[ \mathbf{s}_{n_{1}}\geq 0\right] , \\
\\
\forall n_{2}\left( n_{2}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{2}\right) \left[ \mathbf{s}_{n_{2}}<0\right] ,%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{2}: \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}^{\#}\right) ^{\flat }\triangleq \\
\\
\triangleq \left( \#Ext\text{-}\dsum\limits_{n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}}\mathbf{s}_{n_{1}}^{\#}\right) ^{\flat }+\left( \#Ext\text{-}%
\dsum\limits_{n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}}\mathbf{s}_{n_{2}}^{\#}\right) ^{\flat }. \\
\end{array}
& \text{ \ }\left( 1.3.6.24^{\prime }\right)%
\end{array}%
(ii) Let be $\left\{ \mathbf{s}_{n}\right\} _{n=1}^{\infty }$
countable sequence $\mathbf{s}_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that (a) $\forall n\left( \mathbf{s}_{n}\geq 0\right) $ or (
b) $\forall n\left( \mathbf{s}_{n}<0\right) $ or
(c) $\left\{ \mathbf{s}_{n}\right\} _{n=1}^{\infty }=\left\{
\mathbf{s}_{n_{1}}\right\} _{n_{1}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{1}}^{\infty }\cup \left\{ \mathbf{s}_{n_{2}}\right\} _{n_{2}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{2}}^{\infty },\forall n_{1}\left( n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\right) \left[ \mathbf{s}_{n_{1}}\geq 0\right] ,$
$\forall n_{2}\left( n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}\right) \left[ \mathbf{s}_{n_{2}}<0\right] ,%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\cup \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
Then external $\flat $-sum of the countable sequence $\mathbf{s}_{n}
$ denoted
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}\right) ^{\flat } \\
\end{array}
& \text{ \ \ \ }\left( 1.3.6.23^{\prime \prime }\right)
\end{array}%
is$\ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\left( \mathbf{a}\right) \text{ \ \ \ \ }\forall n\left( \mathbf{s}_{n}\geq
0\right) : \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}\right) ^{\flat }\triangleq \\
\\
\triangleq \text{ }\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left\{ \dsum\limits_{n\leq k}\mathbf{s}_{n}\right\} , \\
\\
\left( \mathbf{b}\right) \text{ \ \ \ \ \ \ }\forall n\left( \mathbf{s}%
_{n}<0\right) : \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}\right) ^{\flat }\triangleq \\
\\
\triangleq \text{ }\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left\{ \dsum\limits_{n\leq k}\mathbf{s}_{n}\right\} . \\
\\
\left( \mathbf{c}\right) \text{ \ }\forall n_{1}\left( n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\right) \left[ \mathbf{s}_{n_{1}}\geq 0\right] , \\
\\
\forall n_{2}\left( n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}\right) \left[ \mathbf{s}_{n_{2}}<0\right] ,%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\cup \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}: \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}\right) ^{\flat }\triangleq \\
\\
\triangleq \left( \#Ext\text{-}\dsum\limits_{n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}}\mathbf{s}_{n_{1}}\right) ^{\flat }+\left( \#Ext\text{-}%
\dsum\limits_{n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}}\mathbf{s}_{n_{2}}\right) ^{\flat }.%
\end{array}
& \text{ \ \ }\left( 1.3.6.24^{\prime \prime }\right)%
\end{array}%
Theorem 1.3.6.9.(i) Let be $\left\{ \mathbf{s}%
_{n}\right\} _{n=1}^{\infty }$ countable sequence $\mathbf{s}_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that $\forall n\left( n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right) \left[ \mathbf{s}_{n}\geq 0\right] ,$ $\sum_{n=1}^{\infty }\mathbf{s}%
_{n}=\eta <\infty ,$ i.e. infinite series
$\sum_{n=1}^{\infty }\mathbf{s}_{n}$ converges to $\eta $ in $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\ast }\mathbf{s}_{n}\right) \right) ^{\flat }\triangleq \text{ }%
\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left\{ \dsum\limits_{n\leq k}\left( ^{\ast }\mathbf{s}_{n}\right)
^{\#}\right\} = \\
\\
=\text{ }\left( ^{\ast }\eta \right) ^{\#}-\varepsilon _{\mathbf{d}}\in
\text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}, \\
\end{array}
& \left( 1.3.6.25.a\right)%
\end{array}%
(ii) Let be $\left\{ \mathbf{s}_{n}\right\} _{n=1}^{\infty }$
countable sequence $\mathbf{s}_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that $\forall n\left( n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right) \left[ \mathbf{s}_{n}<0\right] ,$ $\sum_{n=1}^{\infty }\mathbf{s}%
_{n}=\eta <\infty ,$ i.e. infinite series
$\sum_{n=1}^{\infty }\mathbf{s}_{n}$ converges to $\eta $ in $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\begin{array}{cc}
\begin{array}{c}
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\ast }\mathbf{s}_{n}\right) \right) ^{\flat }\triangleq \text{ }%
\underset{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left\{ \dsum\limits_{n\leq k}\left( ^{\ast }\mathbf{s}_{n}\right)
^{\#}\right\} = \\
\\
=\text{ }\left( ^{\ast }\eta \right) ^{\#}+\varepsilon _{\mathbf{d}}\in
\text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}, \\
\end{array}
& \left( 1.3.6.25.b\right)%
\end{array}%
(iii) Let be $\left\{ \mathbf{s}_{n}\right\} _{n=1}^{\infty }$
countable sequence $\mathbf{s}_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that
(1) $\left\{ \mathbf{s}_{n}\right\} _{n=1}^{\infty }=\left\{ \mathbf{s}%
_{n_{1}}\right\} _{n_{1}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{1}}^{\infty }\cup \left\{ \mathbf{s}_{n_{2}}\right\} _{n_{2}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{2}}^{\infty },\forall n_{1}\left( n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\right) \left[ \mathbf{s}_{n_{1}}\geq 0\right] ,$
$\forall n_{2}\left( n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}\right) \left[ \mathbf{s}_{n_{2}}<0\right] ,%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\cup \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}$ and
(2) $\dsum\limits_{n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}}\mathbf{s}_{n_{1}}=\eta _{1}<\infty ,\dsum\limits_{n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}}\mathbf{s}_{n_{2}}=\eta _{2}>-\infty .$
$\ \ \ \ \ \ \ \ \ $
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\ast }\mathbf{s}_{n}\right) \right) ^{\flat }\triangleq \\
\\
\triangleq \left( \#Ext\text{-}\dsum\limits_{n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}}\left( ^{\ast }\mathbf{s}_{n_{1}}\right) \right) ^{\flat }\text{ }%
+\left( \#Ext\text{-}\dsum\limits_{n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}}\left( ^{\ast }\mathbf{s}_{n_{2}}\right) \right) ^{\flat }= \\
\\
=\left( ^{\ast }\eta _{1}\right) ^{\#}-\varepsilon _{\mathbf{d}}+\text{ }%
\left( ^{\ast }\eta _{2}\right) ^{\#}+\varepsilon _{\mathbf{d}}= \\
\\
=\left( ^{\ast }\eta _{1}\right) ^{\#}+\text{ }\left( ^{\ast }\eta
_{2}\right) ^{\#}-\varepsilon _{\mathbf{d}}\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}. \\
\end{array}
& \left( 1.3.6.25.c\right)%
\end{array}%
Theorem 1.3.6.10.Let be $\left\{ a_{n}\right\}
_{n=1}^{\infty }$ countable sequence $a_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that $\forall n\left( a_{n}\geq 0\right) $ and $\left( \#Ext\text{-}%
\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}\right) ^{\flat }$ external sum of the
countable sequence $\left\{ a_{n}\right\} _{n=1}^{\infty }$ denoted by $%
\mathbf{s}.$
Let be $\left\{ b_{n}\right\} _{n=1}^{\infty }$ countable sequence where $%
b_{n}=a_{m\left( n\right) }$ any
rearrangement of terms of the sequence $\left\{ a_{n}\right\}
_{n=1}^{\infty }.$
Then external sum $\mathbf{\sigma }=\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}b_{n}\right) ^{\flat }$ of the countable
sequence $\left\{ b_{n}\right\} _{m=1}^{\infty }$ has the same value $%
\mathbf{s}$ as external sum of the
countable sequence $\left\{ a_{n}\right\} \mathbf{,}$i.e.
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{\sigma =s.} \\
\end{array}
& \text{ \ }\left( 1.3.6.25\right)%
\end{array}%
Proof.Let be $\mathbf{\sigma }_{n}=b_{1}+b_{2}+...+b_{n}$
the $n$-th partial sum of the
sequence $\left\{ b_{n}\right\} _{n=1}^{\infty }$ and $\mathbf{s}%
_{m}=a_{1}+a_{2}+...+a_{m}$ the $m$-th partial
sum of the sequence $\left\{ a_{n}\right\} _{n=1}^{\infty }$.
It is easy to see that for any given $n$-th partial sum
$\mathbf{\sigma }_{n}=b_{1}+b_{2}+...+b_{n}$ there is exist $\ m$-th partial
$\mathbf{s}_{m\left( n\right) }=a_{1}+a_{2}+...+a_{m\left( n\right) }$ such
\begin{array}{cc}
\begin{array}{c}
\\
\left\{ a_{m}\right\} _{m=1}^{m\left( n\right) }\supseteqq \left\{
b_{i}\right\} _{i=1}^{n}, \\
\end{array}
& \text{ }\left( 1.3.6.26\right)%
\end{array}%
and there is exist $\ N$-th partial sum
$\mathbf{\sigma }_{N\left( m\right) }=b_{1}+b_{2}+...+b_{n}+...+b_{N\left(
m\right) }$ such that:
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\{ b_{j}\right\} _{j=1}^{N\left( m\right) }\supseteqq \left\{
a_{i}\right\} _{i=1}^{m\left( n\right) }, \\
\end{array}
& \text{ \ }\left( 1.3.6.27\right)%
\end{array}%
By using setting and Eqs.(1.3.26)-(1.3.27) one
obtain inequality
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{\sigma }_{n}\leq \mathbf{s}_{m\left( n\right) }\leq \mathbf{\sigma }%
_{N\left( m\right) }. \\
\end{array}
& \text{ \ \ }\left( 1.3.6.28\right)%
\end{array}%
By using Proposition 1.3.6.5. one obtain
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\underset{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left\{ \mathbf{\sigma }_{n}\right\} \leq \text{ }\underset{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left\{ \mathbf{s}_{m\left( n\right) }\right\} \leq \text{ }\underset%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left\{ \mathbf{\sigma }_{N\left( m\right) }\right\} . \\
\end{array}
& \text{ \ \ }\left( 1.3.6.29\right)%
\end{array}%
Hence $\mathbf{\sigma }\leq \mathbf{s}\leq \mathbf{\sigma }$ and finally we
obtain $\mathbf{\sigma =s.}$
Theorem 1.3.21.
Theorem 1.3.22.(i) Let be $\left\{ a_{n}\right\}
_{n=1}^{\infty }$ countable sequence $a_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that $\forall n\left( a_{n}\geq 0\right) $ and $\#Ext$-$%
\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}$ external sum of the sequence $\left\{ a_{n}\right\} _{n=1}^{\infty
}. $
Then for any $c\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}$ the next equality is satisfied:
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
c^{\#}\times \left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}\right) =\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}c^{\#}\times a_{n}\right) \\
\end{array}
& \text{\ }\left( 1.3.6.30\right)%
\end{array}%
(ii) Let be $\left\{ a_{n}\right\} _{n=1}^{\infty }$ countable
sequence $a_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that $\forall n\left( a_{n}<0\right) $ and $\#Ext$-$\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}$ external sum of the
sequence $\left\{ a_{n}\right\} _{n=1}^{\infty }.$Then for any $c\in $ $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{+}$ the next equality is
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
c^{\#}\times \left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}\right) =\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}c^{\#}\times a_{n}\right) \\
\end{array}
& \text{ \ }\left( 1.3.6.30^{\prime }\right)%
\end{array}%
(iii) Let be $\left\{ \mathbf{s}_{n}\right\} _{n=1}^{\infty }$
countable sequence $\mathbf{s}_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that
$\left\{ \mathbf{s}_{n}\right\} _{n=1}^{\infty }=\left\{ \mathbf{s}%
_{n_{1}}\right\} _{n_{1}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{1}}^{\infty }\cup \left\{ \mathbf{s}_{n_{2}}\right\} _{n_{2}\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{2}}^{\infty },\forall n_{1}\left( n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\right) \left[ \mathbf{s}_{n_{1}}\geq 0\right] ,$
$\forall n_{2}\left( n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}\right) \left[ \mathbf{s}_{n_{2}}<0\right] ,%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}\cup \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
Then the next equality is satisfied:
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \ \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\mathbf{s}_{n}= \\
\\
\#Ext\text{-}\dsum\limits_{n_{1}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{1}}\mathbf{s}_{n_{1}}+\#Ext\text{-}\dsum\limits_{n_{2}\in \widehat{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{2}}\mathbf{s}_{n_{2}} \\
\end{array}
& \text{ \ \ }\left( 1.3.6.30^{\prime \prime }\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
Proof.(i) By using Definition 1.3.20 (ii) and
Theorem 1.3.1.3
one obtain
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
c\times \left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}\right) =c\times \sup \left\{ \left. \dsum\limits_{n\leq
m}a_{n}\right\vert m\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right\} = \\
\\
\sup \left[ c\times \left\{ \left. \dsum\limits_{n\leq m}a_{n}\right\vert
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right\} \right] = \\
\\
\sup \left[ \left\{ \left. c\times \dsum\limits_{n\leq m}a_{n}\right\vert
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right\} \right] = \\
\\
\sup \left[ \left\{ \left. \dsum\limits_{n\leq m}c\times a_{n}\right\vert
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right\} \right] = \\
\\
\left( \#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}c\times a_{n}\right) . \\
\end{array}
& \text{ \ }\left( 1.3.6.31\right)%
\end{array}%
Theorem 1.3.6.13.Let be $\left\{ a_{n}\right\}
_{n=1}^{\infty }$ countable sequence $a_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
and $\#Ext$-$\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}$ external sum of the sequence $\left\{ a_{n}\right\} _{n=1}^{\infty
}. $
Definition 1.3.6.20. Let be $\left\{ a_{n}\right\} _{n=1}^{\infty }$
arbitrary countable Cauchy sequence
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$The upper limit in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ of the countable sequence $\left\{ a_{n}\right\}
_{n=1}^{\infty }$
denoted $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$-$\overline{\overline{\lim a_{n}}}$ is
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\text{-}\overline{\overline{\lim a_{n}}}=\text{ }\underset{m\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left( \underset{n\geq m}{\sup }\left( a_{n}^{\#}\right) \right) .
\\
\end{array}
& \text{ \ \ }\left( 1.3.6.32\right)%
\end{array}%
The lower limit in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ of the countable sequence $\left\{ a_{n}^{\#}\right\}
_{n=1}^{\infty }$
denoted $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$-$\underline{\underline{\lim a_{n}}}$ is
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\text{-}\underline{\underline{\lim a_{n}}}=\text{ }\underset{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left( \underset{n\geq m}{\inf }\left( a_{n}^{\#}\right) \right) .
\\
\end{array}
& \text{ \ \ }\left( 1.3.6.33\right)%
\end{array}%
Theorem 1.3.6.14. Suppose that $%
\lim_{n\rightarrow \infty }a_{n}=\zeta \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$ Then
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\text{-}\overline{\overline{\lim a_{n}}}=\text{ }\left( ^{\ast
}\zeta \right) ^{\#}+\varepsilon _{\mathbf{d}}, \\
\\
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\text{-}\underline{\underline{\lim a_{n}}}=\text{ }\left(
^{\ast }\zeta \right) ^{\#}-\varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ \ }\left( 1.3.6.34\right)
\end{array}%
Definition 1.3.6.21. Let be $\left\{ b_{n}\right\} _{n=1}^{\infty }$
countable sequence $b_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
such that $\sum_{n=1}^{\infty }b_{n}<\infty ,$ i.e. infinite series $%
\sum_{n=1}^{\infty }b_{n}$ converges in $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
The upper sum in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ of the infinite series $\sum_{n=1}^{\infty }b_{n}$ denoted
\begin{array}{cc}
\begin{array}{c}
\\
\overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}b_{n}}} \\
\end{array}
& \text{ \ \ }\left( 1.3.6.32\right)%
\end{array}%
is$\ $ $\ \ \ \ \ \ \ \ \ \ \ \ $
$\bigskip \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}b_{n}}}\text{ }\triangleq \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\text{-}\overline{\overline{\lim \left(
\dsum\limits_{i=1}^{n}b_{i}^{\#}\right) }}= \\
\\
\text{ }\underset{m\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left( \underset{n\geq m}{\sup }\left(
\dsum\limits_{i=1}^{n}b_{i}^{\#}\right) \right) . \\
\end{array}
& \text{ \ \ \ \ \ \ }\left( 1.3.6.33\right)%
\end{array}%
The lower sum in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ of the infinite series $\sum_{n=1}^{\infty }b_{n}$ denoted
$\ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\underline{\underline{\#Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}b_{n}}} \\
\end{array}
& \text{ \ }\left( 1.3.6.34\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\underline{\underline{\#Ext\text{ -}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}b_{n}}}\triangleq \text{ } \\
\\
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\text{-}\underline{\underline{\lim \left(
\dsum\limits_{i=1}^{n}b_{i}\right) }}=\text{ }\underset{m\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left( \underset{n\geq m}{\inf }\left(
\dsum\limits_{i=1}^{n}b_{i}\right) \right) . \\
\end{array}
& \text{\ }\left( 1.3.6.35\right)
\end{array}%
Theorem 1.3.6.15. Suppose that $%
\lim_{n\rightarrow \infty }\dsum\limits_{i=1}^{n}b_{i}=\zeta \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$ Then
$\ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}b_{n}}}=\left( ^{\ast }\zeta \right) ^{\#}+\varepsilon _{\mathbf{d}}, \\
\\
\underline{\underline{\#Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}b_{n}}}=\left( ^{\ast }\zeta \right) ^{\#}-\varepsilon _{\mathbf{d}}, \\
\end{array}
& \text{ \ }\left( 1.3.6.36\right)%
\end{array}%
Definition 1.3.6.22. Let be $\left\{ a_{n}\right\} _{n=1}^{\infty }$
arbitrary countable sequence $a_{n}:%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
The upper sum of the countable sequence $\left\{
a_{n}\right\} _{n=1}^{\infty }$ denoted
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}}} \\
\end{array}
& \text{ \ }\left( 1.3.6.37\right)%
\end{array}%
is$\ $
$\bigskip $
$\ \ \ \ \ \ \ $
$\bigskip \ \
\begin{array}{cc}
\begin{array}{c}
\\
\overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}}}\text{ }\triangleq \text{ }\underset{m\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left( \underset{n\geq m}{\sup }\left(
\dsum\limits_{i=1}^{n}a_{i}\right) \right) . \\
\end{array}
& \text{ \ }\left( 1.3.6.38\right)%
\end{array}%
The lower sum of the countable sequence $a_{n}$ denoted
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\underline{\underline{\#Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}}} \\
\end{array}
& \text{ \ \ }\left( 1.3.6.39\right)%
\end{array}%
is$\ $ $\ \ \ \ \ \ \ \ \ $
$\bigskip $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\underline{\underline{\#Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}a_{n}}}\triangleq \text{ }\underset{m\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left( \underset{n\geq m}{\inf }\left(
\dsum\limits_{i=1}^{n}b_{i}\right) \right) . \\
\end{array}
& \text{ \ }\left( 1.3.6.40\right)%
\end{array}%
Theorem 1.3.6.16. Let be $\left\{ a_{n}\right\}
_{n=1}^{\infty }$ arbitrary countable sequence
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$Then for every $b$ such that $b\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\begin{array}{cc}
\begin{array}{c}
\\
\left( b\right) ^{\#}\times \left( \overline{\overline{\#Ext\text{-}%
\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( a_{n}\right) ^{\#}}}\right) = \\
\\
\overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( b\right) ^{\#}\times \left( a_{n}\right) ^{\#}}}, \\
\\
\left( b\right) ^{\#}\times \left( \underline{\underline{\#Ext-\dsum%
\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( a_{n}\right) ^{\#}}}\right) = \\
\\
\underline{\underline{\#Ext-\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( b\right) ^{\#}\times \left( a_{n}\right) ^{\#}}}. \\
\end{array}
& \text{ }\left( 1.3.6.41\right)%
\end{array}%
Theorem 1.3.6.17. Suppose that $\lim_{n\rightarrow \infty
}\dsum\limits_{i=1}^{n}b_{i}=\zeta \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$Then for every $b$
such that $b\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,b>0:$ $\ \ \ $
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( b\right) ^{\#}\times \left( \overline{\overline{\#Ext\text{-}%
\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\ast }a_{n}\right) ^{\#}}}\right) = \\
\\
\overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( b\right) ^{\#}\times \left( ^{\ast }a_{n}\right) ^{\#}}}= \\
\\
=\left( b\right) ^{\#}\times \left( ^{\ast }\zeta \right) ^{\#}+\left(
b\right) ^{\#}\times \varepsilon _{\mathbf{d}}, \\
\\
\left( b\right) ^{\#}\times \left( \underline{\underline{\#Ext\text{-}%
\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( ^{\ast }a_{n}\right) ^{\#}}}\right) = \\
\\
\underline{\underline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left( b\right) ^{\#}\times \left( ^{\ast }a_{n}\right) ^{\#}}}= \\
\\
=\left( b\right) ^{\#}\times \left[ \left( ^{\ast }\zeta \right)
^{\#}-\varepsilon _{\mathbf{d}}\right] . \\
\end{array}
& \text{ \ \ \ \ \ }\left( 1.3.6.42\right)%
\end{array}%
§ I.3.7.THE CONSTRUCTION NON-ARCHIMEDEAN FIELD $\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ $ $\ ^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
_{\MATHBF{D}}^{\PROTECT\OMEGA }$ AS DEDEKIND COMPLETION OF COUNTABLE NON-STANDARD MODELS OF FIELD $%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
Let $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }$ be a countable field which is elementary equivalent, but not
isomorphic to $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Remark.1.3.7.1. The “elementary
equivalence” means that an (arithmetic)
expression of first order is true in field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }$ if and only if it is true in field $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Note that any non-standard model of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ contains an element $\mathbf{\upsilon \in }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }$ such
that $\mathbf{\upsilon }>x$ for each $x\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
The canonical way to construct a model for $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }$ uses model theory [31]. We
simply take as axioms all axioms of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and additionally the following countable
number of axioms: the existence of an element $\mathbf{\upsilon }$ with $%
\mathbf{\upsilon }>1,\mathbf{\upsilon }>2,...,\mathbf{\upsilon }>n,...%
\mathbf{.}$
Each finite subset of this axioms is satisfied by the standard $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$.By the
compactness theorem in first order model theory, there exists a model which
also satisfies the given infinite set of axioms. By the theorem of Lö
Skolem, we can choose such models of countable cardinality.
Each non-standard model $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ contains the (externally defined) subset
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{fin}}\triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
|\exists n_{n\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left[ -n\leq x\leq n\right] \right\} . \\
\end{array}
& \text{ \ \ }\left( 1.3.7.1\right)%
\end{array}%
Every element $x\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{fin}}$ defines a Dedekind cut:
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
%TCIMACRO{\U{211d} }%
\mathbb{R}
=\left\{ y\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
|\text{ }y\leq x\right\} \cup \left\{ y\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
|y>x\right\} . \\
\end{array}
& \text{ \ \ \ }\left( 1.3.7.2\right)%
\end{array}%
We therefore get a order preserving map $\mathbf{j}_{p}\mathbf{:}^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ which restricts to
the standard inclusion of the standard irrationals and which
respects addition and multiplication. An element of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{fin}}$ is called
infinitesimal,if it is mapped to $0$ under the map $\mathbf{j}_{p}.$
Proposition [30].1.3.7.1.Choose an arbitrary subset $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(i) there is a model $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
^{M}$ such that $\mathbf{j}_{p}\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{fin}}^{M}\right) \supset M.$
(ii) the cardinality of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
^{M}$ can be chosen to coincide with $\mathbf{card}\left( M\right) $,if $M$
Proof. Choose $M\subset
%TCIMACRO{\U{211d} }%
\mathbb{R}
$. For each $m\in M$ choose $q_{1}^{m}<q_{2}^{m}<...<...<p_{2}^{m}<p_{1}^{m}$
with $\lim_{k\rightarrow \infty }q_{k}^{m}=\lim_{k\rightarrow \infty
We add to the axioms of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ the following axioms:$\forall m\in M$ $\exists e_{m}$ such that
$q_{k}^{m}<e_{m}<p_{k}^{m}$ for all $k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
Again, the standard $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is a model for each finite subset of these axioms,
so that the compactness theorem implies the existence of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
^{M}$ as required,
where the cardinality of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
^{M}$ can be chosen to be the cardinality of the set of
axioms, i.e. of $M,$ if $M$ is infinite. Note that by construction $\mathbf{j%
}_{p}\left( e_{m}\right) =e_{m}.$
$\mathbf{Remark.}$1.4.2.2.3.It follows in particular that for each
countable subset of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
we can find a countable model of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that the image of fp contains this
subset. Note, on the other hand, that the image will only be countable, so
that the different models will have very different ranges.
Definition 1.4.2.2.1.[30]. A Cauchy sequence in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }$ is a sequence $\left( a_{k}\right) _{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
such that for every $\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega },$ $\varepsilon >0$ there is an $n_{\varepsilon }\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ such that:
$\forall m_{m\text{ }>\text{ }n_{\varepsilon }}\forall n_{n\text{ }>\text{ }%
n_{\varepsilon }}\left[ \left\vert \text{ }a_{m}-a_{n}\right\vert
<\varepsilon \right] .$
Definition 1.4.2.2.2. We define Cauchy completion $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}^{\omega }\triangleq \left[ ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\omega }\right] _{\mathbf{c}}$ in the
canonical way as equivalence classes of Cauchy sequences.
$\mathbf{Remark.}$1.4.2.2.4. This is a standard construction and
works for all ordered
fields.The result is again a field, extending the original field. Note that,
our case,each point in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}^{\omega }$ is infinitesimally close to a point in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$\mathbf{Remark.}$1.4.2.2.5. In many non-standard models of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$, there are no countable
sequences $\left( a_{k}\right) _{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ tending to zero which are not eventually zero.
Proposition [30].1.3.7.2. Assume that $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is countable.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
§ I.4.THE CONSTRUCTION NON-ARCHIMEDEAN FIELD $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
§ I.4.1.COMPLETION OF ORDERED GROUP AND FIELDS IN GENERAL BY USING
'CAUCHY PREGAPS'.
We cketch here the aspects of the general theory that is concerned with
completion ordered group and fields, to be constructed by using 'Cauchy
pregaps' [32].
Throughout in this section we shall only consider fields which are
linear algebra over ground field $\Bbbk ,$ where $\Bbbk =%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
%TCIMACRO{\U{211d} }%
\mathbb{R}
,^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
§ I.4.1.1.TOTALLY ORDERED GROUP AND FIELDS
Definition 1.4.1.1.1. Let $\left( K,+,\cdot \right) $ be a field
and let $\left( \circ \leq \circ \right) $ be a binary
relation on $K.$Then $\left( K,+,\cdot ,\leq \right) $ is an ordered field if
(i) $\ \ \left( K,\leq \right) $ is totally ordered set,
(ii)$\ \ \left( K,+,\leq \right) $ is an ordered group and
(iii)$\ a,b\in K^{+}\implies a\cdot b\in K^{+}.$
Note the standard convention that the order $\leq $ on an ordered field $K$
is necessarily a total order.
Let $K$ be an ordered field. It is easy to see from Definition 1.4.1.1. that
$\left\vert a\right\vert \cdot \left\vert b\right\vert =\left\vert a\cdot
b\right\vert .$
Let $K,L$ be an ordered fields. An imbeding of $K$ in $L$ is an algebra
monomorphism from $K$ into $L$ which is isotonic. A surjective embedding
is an isomorphism. In the case where exist an isomorphism $K$ onto $L$ then
$K$ and $L$ are isomorphic, and we write $K\cong L.$
Definition 1.4.1.1.2. Let $A$ be an algebra. Then:
1. $A\left[ X\right] $ denotes the algebra of polinomials $p\left(
X\right) $ with coefficients in $A;$
2. $^{\ast }A\left[ X\right] $ denotes the algebra of
hyperpolinomials $P\left( X\right) $ with coefficients in
$^{\ast }A;$
3. in the case where $A$ is a subalgebra of an algebra $B$ and $%
b\in B,$
$\ \ \ A\left[ b\right] \triangleq \left\{ p\left( b\right) |p\in A\left[ X%
\right] \right\} ;$
4. in the case where $A$ is a subalgebra of an algebra $B$ and
$\ \ \ b\in $ $^{\ast }B,^{\ast }A\left[ b\right] \triangleq \left\{ P\left(
b\right) |P\in \text{ }^{\ast }A\left[ X\right] \right\} ;$
Definition 1.4.1.1.3.Let $A$ be a subalgebra of an
algebra $B$ and $b\in B.$
1. $b\in B$ is algebraic over $A$ if there exist $p\in A\left[ X%
\right] \backslash \left\{ 0\right\} $ with $p\left( b\right) =0;$
2. $b\in B$ is transcendental over $A$ if is not algebraic over $A;$
3. $b\in $ $^{\ast }B$ is hyperalgebraic over $A$ if there exist $%
P\in $ $^{\ast }A\left[ X\right] \backslash \left\{ 0\right\} $ with
$P\left( ^{\ast }b\right) =0;$
4. $b\in $ $^{\ast }B$ is hypertranscendental over $A$ if is not
hyperalgebraic over $A;$
Definition 1.4.1.1.4.Let $A$ be a subalgebra of an
algebra $B$ and $b\in B.$
1.$b\in B$ is $w$-$\mathbf{transcendental}$ $\mathbf{over}$ $A$ if:
(a) $b\in B$ is transcendental over $A$ and
(b) there exist $P\in $ $^{\ast }A\left[ X\right] \backslash \left\{
0\right\} $ with $P\left( ^{\ast }b\right) =0$
or with $P\left( ^{\ast }b\right) \approx 0;$
2. $b\in B$ is $\#$-$\mathbf{transcendental}$ $\mathbf{over}$ $A$
(a) $b\in B$ is transcendental over $A$ and
(b) there is no exist $P\in $ $^{\ast }A\left[ X\right] \backslash
\left\{ 0\right\} $ with $P\left( ^{\ast }b\right) \approx 0.$
Example 1.4.1.1.1. Number $\pi \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is $w$-transcendental over$\ \
%TCIMACRO{\U{211a} }%
\mathbb{Q}
There exist $P_{\pi }\left( X\right) \in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\left[ X\right] \backslash \left\{ 0\right\} $ with $P_{\pi }\left( ^{\ast
}\pi \right) \approx 0$ where
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
P_{\pi }\left( X\right) =\left[ \sin \left( X\right) \right] _{N\in \text{ }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }}= \\
\\
\left( ^{\ast }\sum_{m=1}^{N\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }}\dfrac{\left( -1\right) ^{m-1}X^{2m-1}}{\left( 2m-1\right) !}%
\right) . \\
\end{array}
& \left( 1.4.1.1.1\right)%
\end{array}%
Example 1.4.1.1.2. Number $\ln 2\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is $w$-transcendental over$\ \
%TCIMACRO{\U{211a} }%
\mathbb{Q}
There exist $P_{\ln 2}\left( X\right) \in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\left[ X\right] \backslash \left\{ 0\right\} $ with $P_{\ln 2}\left( \ln
2\right) -2\approx 0$ where
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
P_{\ln 2}\left( X\right) =\left[ \exp \left( X\right) \right] _{N\in \text{ }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }} \\
\\
=\left( ^{\ast }\sum_{m=1}^{N\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }}\dfrac{X^{m}}{m!}\right) . \\
\end{array}
& \text{ \ \ }\left( 1.4.1.1.2\right)%
\end{array}%
Definition 1.4.1.1.4.Let $A$ be a subalgebra of an
algebra $B.$Then:
(i) The algebra $B$ is an algebraic extension of $A$ if each
$\ b\in B$ is algebraic over $A;$otherwise $B$ is transcendental extension
of $A.$
(ii) The algebra $^{\ast }B$ is an hyperalgebraic extension of $A$
if each
$\ b\in $ $^{\ast }B$ is hyperalgebraic over $A;$otherwise $B$ is
extension of $A.$
(iii) The algebra $^{\ast }B$ is an $w$-transcendental extension of
$A$ if each
$\ b\in $ $^{\ast }B$ is $w$-transcendental over $A.$
(iv) The algebra $^{\ast }B$ is an $\#$-transcendental extension
of $A$ if each
$\ b\in $ $^{\ast }B$ is $\#$-transcendental over $A.$
Definition 1.4.1.1.5. Let $K$ be a field.
(i) A field $K$ is algebraically closed iff there is no
field $L$ which is a proper algebraic extension of $K,$or,equivalently,
$\ \ \ K$ is algebraically closed iff each non constant $p\in K\left[ X%
\right] $ has a root in $K.$
(ii) A field $K$ is hyperalgebraically closed iff there is no
field $L$ which is a proper hyperalgebraic extension of $K,$
$\ \ \ K$ is hyperalgebraically closed iff each non constant $P\in $ $^{\ast
}K\left[ X\right] $ has a root
in $K,$ i.e. $P\left( ^{\ast }b\right) =0$ fore some $b\in K.$
Definition 1.4.1.1.6. An ordered field $K$ is real-closed if
(a) it has no proper algebraic extension to an ordered field,
or, equivalently,if
(b) the complexification $K_{%
%TCIMACRO{\U{2102} }%
\mathbb{C}
}$ of $K$ is algebraically closed,
or, equivalently,if
(c) every positive element in $K$ is a square and every polinomial over $K$
of odd degree has a root in $K.$
Let $K$ be a real-closed ordered field and take some $c\in K^{+}\backslash
\left\{ 0\right\} $ and
%TCIMACRO{\U{2115} }%
\mathbb{N}
.$ There is a anique element $b\in K^{+}\backslash \left\{ 0\right\} $ such
that $b^{n}=c,$ and
so there is a map $\psi :%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{+}\rightarrow K^{+}\backslash \left\{ 0\right\} $ where $\psi
:\alpha \longmapsto c^{\alpha }.$ Thus
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\psi \left( 0\right) =1,\psi \left( 1\right) =c, \\
\\
\psi \left( \alpha +\beta \right) =\psi \left( \alpha \right) \cdot \psi
\left( \beta \right) ,\left( \alpha ,\beta \in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{+}\right) . \\
\end{array}
& \text{ \ }\left( 1.4.1.1.3\right)%
\end{array}%
§ I.4.1.2.CAUCHY COMPLETION OF ORDERED GROUP AND FIELDS.
Definition 1.4.1.2.1.Let $\left\langle A,B\right\rangle $ be a
pregap in totally ordered group $G$. Then
$\left\langle A,B\right\rangle $ is a Cauchy pregap if $A$ has no maximum,$B$
has no minimum, and,for
each $\varepsilon >0,\varepsilon \in G,$there exist $a\in A$ and $b\in B$
with $b<a+\varepsilon .$
Definition 1.4.1.2.2. The group $G$ is Cauchy complete if, for each
pregap there exists $x\in G$ with $a<<x<<b.$
Remark.1.4.1.1.Thus totally ordered group $G$ is Cauchy
complete iff there
are no Cauchy gaps.
Remark.1.4.1.2.The element $x$ arising in the above definition is
Example 1.4.1.1. (i) The group $\left(
%TCIMACRO{\U{211d} }%
\mathbb{R}
,+\right) $ is certainly Cauchy complete.
(ii) The group $\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,+\right) $ is Cauchy complete.
(iii) The monoid $\left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},+\right) $ is certainly Cauchy complete.
Definition 1.4.1.3. The set of the all Cauchy pregaps in totally
group $G$ we denote by $C\left( G\right) .$
Definition 1.4.1.4. A totally ordered group $G$ is
discrete if the set $G^{+}\backslash \left\{ 0\right\} $
is empty or has a minimum element and is non-discrete otherwise.
For any $\left\langle A_{1},B_{1}\right\rangle \in C\left( G\right) $ and $%
\left\langle A_{2},B_{2}\right\rangle \in Cp\left( G\right) $ we hawe
$\left\langle A_{1}+A_{2},B_{1}+B_{2}\right\rangle \in Cp\left( G\right) .$
Let us define sum of the classes
$\left[ \left\langle A_{1},B_{1}\right\rangle \right] \in Cl\left[ Cp\left(
G\right) \right] $ and $\left[ \left\langle A_{2},B_{2}\right\rangle \right]
\in Cl\left[ Cp\left( G\right) \right] $ by formula
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \left\langle A_{1},B_{1}\right\rangle \right] +\left[ \left\langle
A_{2},B_{2}\right\rangle \right] =\left[ \left\langle
A_{1}+A_{2},B_{1}+B_{2}\right\rangle \right] . \\
\end{array}
& \text{ \ \ }\left( 1.4.1.2.1\right)%
\end{array}%
Then $+$ is well defined in $H_{G}=Cl\left[ Cp\left( G\right) \right] $ and $%
\left\{ Cl\left[ Cp\left( G\right) \right] ,+\right\} =$
$\left\{ H_{G},+\right\} $ is an abelian group.The map
\begin{array}{cc}
\begin{array}{c}
\\
\iota :\left\{ G,+\right\} \hookrightarrow \left\{ H_{G},+\right\} \\
\end{array}
& \text{ \ \ \ }\left( 1.4.1.2.2\right)%
\end{array}%
is a canonical group morphism. It is easy to see that $\left\{
H_{G},+\right\} $ is a
totally ordered group. Let $\left\langle \hat{A},\hat{B}\right\rangle $ be a
Cauchy pregap in $H_{G},$and
$\ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{A}=\bigcup \left\{ A_{\alpha }|\left[ \left( A_{\alpha },B_{\alpha
}\right) \right] \in \hat{A}\text{ for some }B_{\alpha }\right\} \\
\\
\mathbf{B}=\bigcup \left\{ B_{\alpha }|\left[ \left( A_{\alpha },B_{\alpha
}\right) \right] \in \hat{B}\text{ for some }A_{\alpha }\right\} , \\
\end{array}
& \text{ \ \ }\left( 1.4.1.2.3\right)%
\end{array}%
hence $\left\langle \mathbf{A,B}\right\rangle \in Cp\left( G\right) $ and $%
\hat{A}<<\left[ \left( \mathbf{A,B}\right) \right] <<\hat{B}$ and $\left\{
H_{G},+\right\} $ is
Cauchy complete. On it we have the following result:
Theorem 1.4.1.1.[32].Let $G$ be a totally ordered
non-discrete group.
Then the group $\left\{ H_{G},+\right\} $ is defined by formula (1.4.1.1) is
a Cauchy
completion of $G.$
Definition 1.4.1.5. An ordered field $K$ is Cauchy complete if the
totally ordered group $\left\{ K,+\right\} $ is Cauchy complete.
Theorem 1.4.1.2.[32].Let $K$ be an ordered field.Then the
completion $\widetilde{K}$ of the group $\left\{ K,+\right\} $ can be made
into an ordered field
in such a way that $K$ is a subfield of $\widetilde{K}.$ If $K$ is
real-closed, then so is $\widetilde{K}.$
Proof. Suppose $a_{1},a_{2}\in \widetilde{K},a_{1}>0,a_{2}>0$ and $%
a_{1}=\left[ \left( A_{1},B_{1}\right) \right] ,$
$a_{2}=\left[ \left( A_{2},B_{2}\right) \right] .$We may suppose that $%
A_{1},A_{2}\subset K^{+}\backslash \left\{ 0\right\} .$Set
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
A=\left\{ x_{1}\cdot x_{2}|x_{1}\in A_{1},x_{2}\in A_{2}\right\} , \\
\\
B=\left\{ y_{1}\cdot y_{2}|y_{1}\in B_{1},y_{2}\in B_{2}\right\} . \\
\end{array}
& \text{ \ \ }\left( 1.4.1.2.4\right)%
\end{array}%
Then $\left\langle A,B\right\rangle \in Cp\left( \left\{
K,+\right\} \right) .$Define
$\bigskip $
\begin{array}{cc}
\begin{array}{c}
\\
a_{1}\cdot a_{2}=\left[ \left\langle A_{1},B_{1}\right\rangle \right] \cdot %
\left[ \left\langle A_{2},B_{2}\right\rangle \right] \\
\end{array}
& \text{ \ \ \ }\left( 1.4.1.2.5\right)%
\end{array}%
The operation $\left( \circ \cdot \circ \right) $ is well defined in $%
\widetilde{K}$ and $a_{1}\cdot a_{2}>0.$If $a_{1}<0,$
$a_{2}>0$ set $a_{1}\cdot a_{2}=-\left( \left( -a_{1}\right) \cdot
a_{2}\right) ,$ etc.
It is simple to check that the group $\left\{ \widetilde{K},+\right\} $
together with product
$\left( \circ \cdot \circ \right) $ is an ordered field $\widetilde{K}%
\triangleq \left\{ \widetilde{K},+,\cdot \right\} $ with the required
Let $K^{\prime }$ be any ordered field containing $K$ as an order-dense
subfield. Then there is an isotonic morphism from $K^{\prime }$ into $%
\widetilde{K},$ and so
$\widetilde{K}$ is the maximum ordered field containing $K$ as an proper
order-dense subfield. On it we have the following result:
Theorem 1.4.1.3.[32].Let $K$ be an ordered field.Then $K$
is Cauchy
complete iff no ordered field $L$ containing $K$ as an proper
order-dense subfield.
Standard main tool for anderstanding structure of the totally ordered
external group well be Hahn's embedding theorem. We shall associate to
a totally ordered external and internal group $\left( G,+,\leq \right) $ a
'value set' $\Gamma ^{\#},$
define a collections $\tciFourier \left(
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\Gamma \right) ,\tciFourier \left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\Gamma ^{\#}\right) $ of 'formal power series' over $\Gamma $ and
$\Gamma ^{\#}$ and imbed $G$ into $\tciFourier \left(
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\Gamma \right) $ or $\tciFourier \left( ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\Gamma ^{\#}\right) $ correspondingly.
Let us define the value set of external group $G.$
Definition 1.4.1.6.[32]. Let $\left( G,+,\leq \right) $
be a totally ordered external group,
i.e. $\left( G,+,\leq \right) \in V^{Ext}$ and let $x,y\in G.$Set:
(i) $x=o\left( y\right) $ iff $\forall n_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left[ n\cdot \left\vert x\right\vert \leq \left\vert y\right\vert \right]
; $
(ii) $x=O\left( y\right) $ iff $\exists m_{m\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\left[ \left\vert x\right\vert \leq m\cdot \left\vert y\right\vert \right]
; $
(iii) $x$ $\symbol{126}$ $y$ iff $\left[ x=O\left(
y\right) \right] \wedge \left[ y=O\left( x\right) \right] .$
For each $y\in G$ the sets $\left\{ x|x=o\left( y\right) \right\} $ and $%
\left\{ x|x=O\left( y\right) \right\} $ are absolutely
convex subsets of $G.$Is clear that $\left( \circ \text{ }\symbol{126}\circ
\right) $ is an equivalence relation on $G.$
Each $\symbol{126}$ -equivalence class (other than $\left\{ 0\right\} $) is
the union of an interval
contained in $G^{+}\backslash \left\{ 0\right\} $ and an interval contained
in $G^{-}\backslash \left\{ 0\right\} .$
Definition 1.4.1.7.Let $\left( G,+,\leq \right) $ be a totally
ordered external group.
The set $\Gamma =\Gamma _{G}=\left( G\backslash \left\{ 0\right\} \right) /%
\symbol{126}$ of equivalence classes in the value set
of $G$ and the elements of $\Gamma $ are the archimedian classes of $G.$
The quotient map from $G\backslash \left\{ 0\right\} $ onto $\Gamma $ is
denoted by $v:$ $G\backslash \left\{ 0\right\} \rightarrow \Gamma .$It is
the archimedian valuation on $G.$Set $v\left( x\right) \leq v\left( y\right)
$ for $x,y\in G\backslash \left\{ 0\right\} $ iff
$y=O\left( x\right) .$
It is easy checked that $\leq $ is well defined on $\Gamma ,$hence $\left(
\Gamma ,\leq \right) $ is a totally
ordered set, such that
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
v\left( x+y\right) \geq \min \left\{ v\left( x\right) ,v\left( y\right)
\right\} , \\
\\
v\left( x+y\right) =\min \left\{ v\left( x\right) ,v\left( y\right) \right\}
\text{ if } \\
\\
\left[ v\left( x\right) \neq v\left( y\right) \right] \vee \left[ \left(
x\in G^{+}\right) \wedge \left( y\in G^{+}\right) \right] . \\
\end{array}
& \text{\ }\left( 1.4.1.2.6\right)%
\end{array}%
Definition 1.4.1.8. Let $\left( \breve{G},+,\leq \right) $ a
totally ordered internal group,i.e.
$\left( \breve{G},+,\leq \right) \in V^{Int}$ and let $x,y\in G.$In
particular $\breve{G}=$ $^{\ast }\left( G,+,\leq \right) $ for some
$\left( G,+,\leq \right) ,$i.e. in particular $\breve{G}$ is an standard
§ I.4.2.1.THE CONSTRUCTION NON-ARCHIMEDEAN FIELD $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
_{\MATHBF{C}}$ BY USING CAUCHY HYPERSEQUENCE IN ANCOUNTABLE FIELD $^{\AST }%
%TCIMACRO{\U{211A} }%
\MATHBB{Q}
Let $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega _{\alpha }}\triangleq $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,\omega <\omega _{\alpha }$ be a ancountable field which is elementary
equivalent, to $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
.$The “elementary equivalence” means that
(arithmetic) expression of first order is true in field $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega _{\alpha }}$ if and only
if it is true in field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
.$Note that any non-standard model of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ contains
an element $\mathbf{e\in }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega }$ such that $\mathbf{e}>q$ for each $q\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
We define Cauchy completion $\left[ ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega _{\alpha }}\right] _{\mathbf{c}}\triangleq $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}$ in the canonical
way as equivalence classes of Cauchy hypersequences.
$\mathbf{Remark.}$1.4.2.1.1.This is a general construction and
works for all
nonstandard ordered fields $^{\ast }\Bbbk $.The result is again a field $%
\left[ ^{\ast }\Bbbk \right] _{\mathbf{c}}$ which
is potentially different from extending the original field $\Bbbk $, and we
actually see that $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}$ is different from $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$\mathbf{Remark.}$1.4.2.1.2.In many non-standard ancountable models
of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
there are no countable sequences tending to zero which are not
eventually zero.Thus dealing with analysis over field $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ we are
compelled to enter into consideration hypersequences of various
classes: $\mathbf{s}_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}:$ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,\mathbf{s}_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}:$ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and $\mathbf{s}_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}:$ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Definition 1.4.2.1.1. A hypersequence
$\mathbf{s}_{\mathbf{n}}:$ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\supset $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\supset $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ tends
to a $\ast $-limit $\alpha $ ($\alpha \in ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ or $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$) in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ or $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ iff
\begin{array}{cc}
\begin{array}{c}
\\
\exists \alpha \left( \alpha \in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\right) \left[ \forall \varepsilon _{\varepsilon >0}\left(
\varepsilon \in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\right) \exists \mathbf{n}_{0}\left( \mathbf{n}_{0}\in \text{ }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }\right) \right. \\
\\
\left. \forall \mathbf{n}\left[ \mathbf{n\geqslant n}_{0}\implies \left\vert
\alpha -\mathbf{s}_{\mathbf{n}_{0}}\right\vert <\varepsilon \right] \right] .
\\
\end{array}
& \left( 1.4.2.1.1\right)%
\end{array}%
We write $\ast $-$\lim_{\mathbf{n}\text{ }\longrightarrow \text{ \ }^{\ast
}\infty }\mathbf{s}_{\mathbf{n}_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}}=\alpha $ $\ $or$\ \underset{\mathbf{n}\text{ }\longrightarrow \text{ \ }%
^{\ast }\infty }{\ast \text{-}\lim }\mathbf{s}_{\mathbf{n}}=\alpha \ \ $iff
(1.4.1.1.1) is satisfied.
Definition 1.4.2.1.2. A hypersequence $\mathbf{s}_{\mathbf{n}}:$ $%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\supseteqq $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ is
divergent in $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},$ or tends to $^{\ast }\infty $ iff
$\bigskip $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\forall r_{r>0}\left( r\in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\right) \exists \mathbf{n}_{0}\left( \mathbf{n}_{0}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }\right) \forall \mathbf{n}\left[ \mathbf{n}\geqslant \mathbf{n}%
_{0}\right. \\
\\
\left. \implies \left\vert \mathbf{s}_{\mathbf{n}}\right\vert >r\right] .\ \
\\
\end{array}
& \text{\ }\left( 1.4.2.1.2\right)%
\end{array}%
Lemma 1.4.2.1.1. Suppose that $\mathbf{s}_{\mathbf{n}}:$ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\supset $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
(a) If $\underset{\mathbf{n}\text{ }\longrightarrow \text{ \ }%
^{\ast }\infty }{\ast \text{-}\lim }\mathbf{s}_{\mathbf{n}}$ exists in $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,$ then it is unique.
(b) If $\underset{\mathbf{n}\text{ }\longrightarrow \text{ \ }%
^{\ast }\infty }{\ast \text{-}\lim }\mathbf{s}_{\mathbf{n}}$ exists in $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},$ then it is unique.
That is if $\ast $-$\lim_{\mathbf{n}\text{ }\longrightarrow \text{ \ }^{\ast
}\infty }\mathbf{s}_{\mathbf{n}}=\alpha _{1},\ast -\lim_{\mathbf{n}\text{ }%
\longrightarrow \text{ \ }^{\ast }\infty }\mathbf{s}_{\mathbf{n}}=\alpha
_{2} $ then $\alpha _{1}=\alpha _{2}.$
Proof. (a) Let $\varepsilon $ be any positive number $\varepsilon
>0,\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\supset $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$.Then, by
definition, we must be able to find a number $N_{1}$ so that $\left\vert
\mathbf{s}_{\mathbf{n}}-\alpha _{1}\right\vert <\varepsilon $
whenever $\mathbf{n}\geq N_{1}.$
We must also be able to find a number $N_{2}$ so that $\left\vert \mathbf{s}%
_{\mathbf{n}}-\alpha _{2}\right\vert <\varepsilon $
whenever $\mathbf{n}\geq N_{2}.$ Take $\mathbf{m}$ to be the maximum of $%
N_{1}$ and $N_{2}.$ Then both
assertions $\left\vert \mathbf{s}_{\mathbf{m}}-\alpha _{1}\right\vert
<\varepsilon $ and $\left\vert \mathbf{s}_{\mathbf{m}}-\alpha
_{2}\right\vert <\varepsilon $ are true.
This by using triangle inequality allows us to conclude that
$\left\vert \alpha _{1}-\alpha _{2}\right\vert =\left\vert \left( \alpha
_{1}-\mathbf{s}_{\mathbf{m}}\right) +\left( \mathbf{s}_{\mathbf{m}}-\alpha
_{2}\right) \right\vert \leq \left\vert \alpha _{1}-\mathbf{s}_{\mathbf{m}%
}\right\vert +\left\vert \mathbf{s}_{\mathbf{m}}-\alpha _{2}\right\vert
<2\varepsilon .$ So that
$\left\vert \alpha _{1}-\alpha _{2}\right\vert <2\varepsilon .$ But $%
\varepsilon $ can be any positive infinite small number whatsoever.
This could only be true if $\alpha _{1}=\alpha _{2}$, which is what we
wished to show.
Definition 1.4.2.1.3. A Cauchy hypersequence in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ is a sequence $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \mathbf{s}_{\mathbf{n}}:$ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\rightarrow $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}\supseteqq $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ with the following property: for every $\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ such that $\ \ \ \ \ \ \ \ \ \ \varepsilon >0,$there exists
an $\mathbf{n}_{0}\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ such that $\mathbf{m,n}\geqslant n_{0}$ implies $|\mathbf{s}_{%
\mathbf{m}}-\mathbf{s}_{\mathbf{n}}|$ $<\varepsilon ,$
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\forall \varepsilon _{(\varepsilon \in ^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}})}\left( \varepsilon >0\right) \exists \mathbf{n}_{0}\left(
\mathbf{n}_{0}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }\right) \left[ \mathbf{m,n}\geqslant n_{0}\right. \\
\\
\left. \implies |\mathbf{s}_{\mathbf{m}}-\mathbf{s}_{\mathbf{n}%
}|<\varepsilon \right] . \\
\end{array}
& \text{ \ \ }\left( 1.4.2.1.3\right)%
\end{array}%
Lemma 1.4.2.1.2. A hypersequence of numbers $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},$ that
converges is Cauchy hypersequence.
Lemma 1.4.2.1.3. A Cauchy hypersequence $\left( \mathbf{s}_{\mathbf{%
n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ and $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
bounded or hyperbounded.
Proof.Choose in (1.4.2.1) $\varepsilon =1.$ Since the sequence $%
\left( \mathbf{s}_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ is Cauchy,
there exists a positive hyperinteger $N\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ such that $\left\vert \mathbf{s}_{\mathbf{i}}-\mathbf{s}_{\mathbf{j}%
}\right\vert <1$whenever
$\mathbf{i,j\geq }N.$In particular,$\left\vert \mathbf{s}_{\mathbf{i}}-%
\mathbf{s}_{N}\right\vert <1$ whenever $\mathbf{i\geq }N$ By the triangle
$\left\vert \mathbf{s}_{\mathbf{i}}\right\vert -\left\vert \mathbf{s%
}_{N}\right\vert \leq \left\vert \mathbf{s}_{\mathbf{i}}-\mathbf{s}%
_{N}\right\vert $ and therefore,$\left\vert \mathbf{s}_{\mathbf{i}%
}\right\vert <\left\vert \mathbf{s}_{N}\right\vert +1$ for all $\mathbf{%
i\geq }$ $N.$
Definition 1.4.2.1.4. Cauchy hypersequences $(x_{\mathbf{n}})_{%
\mathbf{n}\in ^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ and $(y_{\mathbf{n}})_{\mathbf{n}\in ^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
can be added, multiplied and compared as follows:
(a) $(x_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}+(y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}=(x_{\mathbf{n}}+y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
(b) $(x_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\times (y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}=(x_{\mathbf{n}}\times y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
(c) $\dfrac{(x_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}}{(y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}}=\left( \dfrac{x_{\mathbf{n}}}{y_{\mathbf{n}}}\right) _{\mathbf{n}\in
\text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ iff $\forall \mathbf{n}\left( \mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right) $ $\left[ y_{n}\neq 0\right] ,$
(d) $(x_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}^{-1}=(x_{\mathbf{n}}^{-1})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ iff $\ \forall \mathbf{n}\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
(y_{\mathbf{n}}\neq 0),$
(e) $(x_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\geq (y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ if and only if for every $\epsilon >0,\epsilon \in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
there exists an integer $\mathbf{n}_{0}$ such that $x_{\mathbf{n}%
}\geq y_{\mathbf{n}}-\epsilon $ for all $\mathbf{n>n}_{0}\mathbf{.}$
Definition 1.4.2.1.5.Two Cauchy hypersequences $(x_{\mathbf{n}})_{%
\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ and $(y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
are called equivalent: $(x_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ $\approx _{\mathbf{c}}(y_{n})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ if the hypersequence $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\ $
$(x_{\mathbf{n}}-y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$has $\ast $-limit zero$,$i.e. $\ast $-$\lim_{\mathbf{n\rightarrow }^{\ast
}\infty }(x_{\mathbf{n}}-y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
Lemma 1.4.2.1.4. If $(x_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\approx _{\mathbf{c}}(x_{\mathbf{n}}^{\prime })_{\mathbf{n}\in \text{ }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ and $(y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\approx _{\mathbf{c}}(y_{\mathbf{n}}^{\prime })_{\mathbf{n}\in \text{ }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
are two pairs of equivalent Cauchy hypersequences, then:
(a) hypersequence $(x_{\mathbf{n}}+y_{\mathbf{n}})_{\mathbf{n}\in
\text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ is Cauchy and
\begin{array}{cc}
\begin{array}{c}
\\
(x_{\mathbf{n}}+y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\approx _{\mathbf{c}}(x_{\mathbf{n}}^{\prime }+y_{\mathbf{n}}^{\prime })_{%
\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}, \\
\end{array}
& \text{ \ }\left( 1.4.2.1.4\right)%
\end{array}%
\ \ $
(b) hypersequence $(x_{\mathbf{n}}-y_{\mathbf{n}})_{\mathbf{n}\in
\text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ is Cauchy and
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ (x_{\mathbf{n}}-y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\approx _{\mathbf{c}}(x_{\mathbf{n}}^{\prime }-y_{\mathbf{n}}^{\prime })_{%
\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}, \\
\end{array}
& \text{ \ }\left( 1.4.2.1.5\right)%
\end{array}%
$\ \ \ \ \ \ $
(c) hypersequence $(x_{\mathbf{n}}\times y_{\mathbf{n}})_{\mathbf{n}%
\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ is Cauchy and
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\ (x_{\mathbf{n}}\times y_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\approx _{\mathbf{c}}(x_{\mathbf{n}}^{\prime }\times y_{\mathbf{n}}^{\prime
})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}, \\
\end{array}
& \text{ \ \ }\left( 1.4.2.1.6\right)%
\end{array}%
\ \ \ \ \ \ $
(d) hypersequence $\left( \dfrac{x_{\mathbf{n}}}{y_{\mathbf{n}}}%
\right) _{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ is Cauchy and
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \left( \dfrac{x_{\mathbf{n}}}{y_{\mathbf{n}}}\right) _{\mathbf{n}\in \text{
}^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\approx _{\mathbf{c}}\left( \dfrac{x_{\mathbf{n}}^{\prime }}{y_{\mathbf{n}%
}^{\prime }}\right) _{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
} \\
\end{array}
& \text{\ }\left( 1.4.2.1.7\right)
\end{array}%
iff $\forall \mathbf{n}_{(\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
)}\left[ (y_{\mathbf{n}}\not\equiv 0)\wedge \left( y_{\mathbf{n}}^{\prime
}\not\equiv 0\right) \wedge \left( y_{\mathbf{n}}\not\approx _{\mathbf{c}%
}0\right) \right] ,$
(e) hypersequence $(x_{\mathbf{n}}+0_{\mathbf{n}})_{\mathbf{n}\in
\text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ where $\forall \mathbf{n}_{(\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
)}\left[ 0_{\mathbf{n}}=0\right] $
is Cauchy and
$\ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
(x_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}+(0_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\approx _{\mathbf{c}}(x_{\mathbf{n}})_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}, \\
\end{array}
& \text{ \ \ }\left( 1.4.2.1.8\right)%
\end{array}%
here $(0_{\mathbf{n}})_{\mathbf{n}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ is a null hypersequence,
(f) hypersequence $(x_{\mathbf{n}}\times 1_{\mathbf{n}})_{\mathbf{%
n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ where $\forall \mathbf{n}_{(\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
)}\left[ 1_{\mathbf{n}}=1\right] $ is
Cauchy and
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\times (1_{\mathbf{n}})_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\approx _{\mathbf{c}}(x_{\mathbf{n}})_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}, \\
\end{array}
& \text{ \ \ }\left( 1.4.2.1.9\right)%
\end{array}%
here $(1_{\mathbf{n}})_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ is a unit hypersequence.
(g) hypersequence $(x_{\mathbf{n}})_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\times (x_{\mathbf{n}})_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}^{-1}$ is Cauchy and
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
(x_{\mathbf{n}})_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\times (x_{\mathbf{n}})_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}^{-1}\approx _{\mathbf{c}}(1_{\mathbf{n}})_{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
} \\
\end{array}
& \text{ \ \ }\left( 1.4.2.1.10\right)%
\end{array}%
iff $\forall \mathbf{n}_{(\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
)}\left[ (x_{\mathbf{n}}\not\equiv 0)\wedge \left( x_{\mathbf{n}}\not\approx
_{\mathbf{c}}\left( 0_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right) \right] .$
Proof. (a) From definition of the Cauchy hypersequences
one obtain:
\begin{array}{cc}
\begin{array}{c}
\\
\exists \varepsilon _{1}\exists \mathbf{m}_{\left( \mathbf{m\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }\right) }\mathbf{\forall k}\left( \mathbf{k\geqslant m}\right)
\mathbf{\forall l}\left( \mathbf{l\geqslant m}\right) \left[ \left(
\left\vert x_{\mathbf{k}}-x_{\mathbf{l}}\right\vert <\varepsilon _{1}\right)
\right. \\
\\
\left. \wedge \left( \left\vert y_{\mathbf{k}}-y_{\mathbf{l}}\right\vert
<\varepsilon _{1}\right) \right] . \\
\end{array}
& \text{ }\left( 1.4.2.1.11\right)%
\end{array}%
Suppose $\varepsilon _{1}=\varepsilon /2,$ then from formula above we can to
choose $\mathbf{m=m}\left( \varepsilon _{1}\right) $
such that for all $\mathbf{k\geqslant m,l\geqslant m}$ valid the next
\begin{array}{cc}
\begin{array}{c}
\\
\left\vert (x_{\mathbf{k}}+y_{\mathbf{k}})-(x_{\mathbf{l}}+y_{\mathbf{l}%
})\right\vert =\left\vert (x_{\mathbf{k}}-x_{\mathbf{l}})+\left( y_{\mathbf{k%
}}-y_{\mathbf{l}}\right) \right\vert \leqslant \\
\\
\leqslant \left\vert (x_{\mathbf{k}}-x_{\mathbf{l}})\right\vert +\left\vert
\left( y_{\mathbf{k}}-y_{\mathbf{l}}\right) \right\vert <\varepsilon
/2+\varepsilon /2=\varepsilon , \\
\\
\left\vert (x_{\mathbf{k}}^{\prime }+y_{\mathbf{k}}^{\prime })-(x_{\mathbf{l}%
}^{\prime }+y_{\mathbf{l}}^{\prime })\right\vert =\left\vert (x_{\mathbf{k}%
}^{\prime }-x_{\mathbf{l}}^{\prime })+\left( y_{\mathbf{k}}^{\prime }-y_{%
\mathbf{l}}^{\prime }\right) \right\vert \leqslant \\
\\
\leqslant \left\vert (x_{\mathbf{k}}^{\prime }-x_{\mathbf{l}}^{\prime
})\right\vert +\left\vert \left( y_{\mathbf{k}}^{\prime }-y_{\mathbf{l}%
}^{\prime }\right) \right\vert <\varepsilon /2+\varepsilon /2=\varepsilon .
\\
\end{array}
& \text{ \ }\left( 1.4.2.1.12\right)%
\end{array}%
From Definition 1.4.2.1.5. and inequalities (1.4.12) we have
the statement (a).
(b) Similarly proof the statement (a) we have the next
inequalities: $\ \ \ \ $
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\vert (x_{\mathbf{k}}-y_{\mathbf{k}})-(x_{\mathbf{l}}-y_{\mathbf{l}%
})\right\vert =\left\vert (x_{\mathbf{k}}-x_{\mathbf{l}})+\left( y_{\mathbf{k%
}}-y_{\mathbf{l}}\right) \right\vert \leqslant \\
\\
\leqslant \left\vert (x_{\mathbf{k}}-x_{\mathbf{l}})\right\vert +\left\vert
\left( y_{\mathbf{k}}-y_{\mathbf{l}}\right) \right\vert <\varepsilon
/2+\varepsilon /2=\varepsilon , \\
\\
\left\vert (x_{\mathbf{k}}^{\prime }-y_{\mathbf{k}}^{\prime })-(x_{\mathbf{l}%
}^{\prime }-y_{\mathbf{l}}^{\prime })\right\vert =\left\vert (x_{\mathbf{k}%
}^{\prime }-x_{\mathbf{l}}^{\prime })-\left( y_{\mathbf{k}}^{\prime }-y_{%
\mathbf{l}}^{\prime }\right) \right\vert \leqslant \\
\\
\leqslant \left\vert (x_{\mathbf{k}}^{\prime }-x_{\mathbf{l}}^{\prime
})\right\vert +\left\vert \left( y_{\mathbf{k}}^{\prime }-y_{\mathbf{l}%
}^{\prime }\right) \right\vert <\varepsilon /2+\varepsilon /2=\varepsilon .\
\ \ \ \\
\end{array}
& \text{ \ }\left( 1.4.2.1.13\right)%
\end{array}%
From Definition 1.4.2.1.5 and inequalities (1.4.13) we have the
statement (b).
(c) $\mathbf{\forall k}\left( \mathbf{k\geqslant m}\right)
$ and $\mathbf{\forall l}\left( \mathbf{l\geqslant m}\right) $ we have the
next inequalities:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\vert x_{\mathbf{k}}\cdot y_{\mathbf{k}}-x_{\mathbf{l}}\cdot y_{\mathbf{%
l}}\right\vert =\left\vert \left( x_{\mathbf{k}}\cdot y_{\mathbf{k}}-x_{%
\mathbf{l}}\cdot y_{\mathbf{k}}\right) +\left( x_{\mathbf{l}}\cdot y_{%
\mathbf{k}}-x_{\mathbf{l}}\cdot y_{\mathbf{l}}\right) \right\vert \leqslant
\\
\\
\leqslant \left\vert x_{\mathbf{k}}-x_{\mathbf{l}}\right\vert \cdot
\left\vert y_{\mathbf{l}}\right\vert +\left\vert y_{\mathbf{k}}-y_{\mathbf{l}%
}\right\vert \cdot \left\vert x_{\mathbf{l}}\right\vert , \\
\\
\left\vert x_{\mathbf{k}}^{\prime }\cdot y_{\mathbf{k}}^{\prime }-x_{\mathbf{%
l}}^{\prime }\cdot y_{\mathbf{l}}^{\prime }\right\vert =\left\vert \left( x_{%
\mathbf{k}}^{\prime }\cdot y_{\mathbf{k}}^{\prime }-x_{\mathbf{l}}^{\prime
}\cdot y_{\mathbf{k}}^{\prime }\right) +\left( x_{\mathbf{l}}^{\prime }\cdot
y_{\mathbf{k}}^{\prime }-x_{\mathbf{l}}^{\prime }\cdot y_{\mathbf{l}%
}^{\prime }\right) \right\vert \leqslant \\
\\
\leqslant \left\vert x_{\mathbf{k}}^{\prime }-x_{\mathbf{l}}^{\prime
}\right\vert \cdot \left\vert y_{\mathbf{l}}^{\prime }\right\vert
+\left\vert y_{\mathbf{k}}^{\prime }-y_{\mathbf{l}}^{\prime }\right\vert
\cdot \left\vert x_{\mathbf{l}}^{\prime }\right\vert , \\
\\
\left\vert x_{\mathbf{k}}\cdot y_{\mathbf{k}}-x_{\mathbf{k}}^{\prime }\cdot
y_{\mathbf{k}}^{\prime }\right\vert =\left\vert \left( x_{\mathbf{k}}\cdot
y_{\mathbf{k}}-x_{\mathbf{k}}\cdot y_{\mathbf{k}}^{\prime }\right) +\left(
x_{\mathbf{k}}\cdot y_{\mathbf{k}}^{\prime }-x_{\mathbf{k}}^{\prime }\cdot
y_{\mathbf{k}}^{\prime }\right) \right\vert \leqslant \\
\\
\leqslant \left\vert x_{\mathbf{k}}-x_{\mathbf{k}}^{\prime }\right\vert
\cdot \left\vert y_{\mathbf{k}}^{\prime }\right\vert +\left\vert y_{\mathbf{k%
}}-y_{\mathbf{k}}^{\prime }\right\vert \cdot \left\vert x_{\mathbf{k}%
}\right\vert . \\
\end{array}
& \text{ }\left( 1.4.2.1.14\right)%
\end{array}%
From definition Cauchy hypersequences one obtain $\exists c\forall \mathbf{k:%
}\left\vert x_{\mathbf{k}}\right\vert \leqslant c,\left\vert y_{\mathbf{k}%
}\right\vert \leqslant c,$
$\left\vert x_{\mathbf{k}}^{\prime }\right\vert \leqslant c,\left\vert y_{%
\mathbf{k}}^{\prime }\right\vert \leqslant c.$ From Definition
1.4.2.1.5 and inequalities (1.4.14) we
have the statement (c).
Let $\Re _{\mathbf{c}}^{\ast }$ denote the set of the all equivalence
classes $\left\{ \left( x_{n}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \in \Re _{\mathbf{c}}^{\ast }$
Using Lemma 1.4.1. one can define an equivalence relation $\approx _{\mathbf{%
c}},$which is
compatible with the operations defined above, and the set $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}=\Re _{\mathbf{c}}^{\ast }/\approx _{\mathbf{c}}$
is satisfy of the all usual field axioms of the hyperreal numbers.
Lemma 1.4.2.1.5. Suppose that
$\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} ,\left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} ,\left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}},$ then:$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\bigskip \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\text{(\textbf{a}) } \\
\\
\text{(\textbf{b})} \\
\\
\\
\\
\text{(\textbf{c}) } \\
\\
\\
\\
\text{(\textbf{d})} \\
\\
\\
\\
\text{(\textbf{e})} \\
\\
\\
\text{(\textbf{f})} \\
\\
\text{(\textbf{g})} \\
\\
\text{(\textbf{i})} \\
\\
\text{(\textbf{j})} \\
\\
\text{(\textbf{k})} \\
\\
\\
\end{array}%
\begin{array}{c}
\\
\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} +\left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} =\left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} +\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} , \\
\\
\left[ \left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} +\left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \right] +\left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} = \\
\\
=\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} +\left[ \left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} +\left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \right] , \\
\\
\left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left[ \left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} +\left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \right] = \\
\\
=\left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} +\left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} , \\
\\
\left[ \left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \right] \times \left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in
}^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} = \\
\\
=\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left[ \left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \right] , \\
\\
\left[ \left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \right] \times \left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in
}^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} = \\
=\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left[ \left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \right] , \\
\\
\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} +\left\{ \left( 0_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} =\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} , \\
\\
\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \cdot \left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} ^{-1}=\left\{ \left( 1_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} , \\
\\
\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( 0_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} =\left\{ \left( 0_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} , \\
\\
\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( 1_{\mathbf{n}}\right) _{{}}\right\} =\left\{
\left( x_{\mathbf{n}}\right) _{{}}\right\} , \\
\\
\left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} <\left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \wedge \left\{ \left( 0_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} <\left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \implies \\
\\
\implies \left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( x_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} <\left\{ \left( z_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} \times \left\{ \left( y_{\mathbf{n}}\right) _{\mathbf{n\in }^{\ast
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\right\} . \\
\end{array}
& \text{ \ \ }\left( 1.4.2.1.15\right) \text{\ }%
\end{array}%
Proof. Statements (a),(b),(c),(d),(e),(f),(g),(i) (j) and
(k) is evidently from
Lemma.1.4.1 and definition of the equivalence relation $\approx _{%
\mathbf{c}}.$
§ I.4.2.2.THE CONSTRUCTION NON-ARCHIMEDEAN FIELD $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
_{\MATHBF{C}}^{\PROTECT\OMEGA }$ AS CAUCHY COMPLETION OF COUNTABLE
NON-STANDARD MODELS OF FIELD $%
%TCIMACRO{\U{211A} }%
\MATHBB{Q}
Let $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega }$ be a countable field which is elementary equivalent, but not
isomorphic to $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
Remark.1.4.2.2.1. The “elementary
equivalence” means that an
(arithmetic) expression of first order is true in field $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega }$ if and only if it
is true in field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
Note that any non-standard model of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ contains an element $\mathbf{e\in }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega }$ such
that $\mathbf{e}>q$ for each $q\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
The canonical way to construct a model for $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega }$ uses model theory [30],[31].
We simply take as axioms all axioms of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ and additionally the following
countable number of axioms: the existence of an element $\mathbf{e}$ with
$\mathbf{e}>1,\mathbf{e}>2,...,\mathbf{e}>n,...\mathbf{.}$Each finite subset
of this axioms is satisfied by the
standard $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$.By the compactness theorem in first order model theory, there
exists a model which also satisfies the given infinite set of axioms. By the
theorem of Löwenheim-Skolem, we can choose such models of countable
Each non-standard model $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ contains the (externally defined) subset
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\mathbf{fin}}\triangleq \left\{ x\in \text{ }^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|\exists n_{n\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left[ -n\leq x\leq n\right] \right\} . \\
\end{array}
& \text{ }\left( 1.4.2.2.1\right)%
\end{array}%
Every element $x\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\mathbf{fin}}$ defines a Dedekind cut:
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
%TCIMACRO{\U{211a} }%
\mathbb{Q}
=\left\{ q\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|\text{ }q\leq x\right\} \cup \left\{ q\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
|q>x\right\} . \\
\end{array}
& \text{\ }\left( 1.4.2.2.2\right)%
\end{array}%
We therefore get a order preserving map $\mathbf{j}_{op}\mathbf{:}^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ which restricts to
the standard inclusion of the standard rationals and which
respects addition and multiplication. An element of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\mathbf{fin}}$ is called
infinitesimal,if it is mapped to $0$ under the map $\mathbf{j}_{op}.$
Proposition [30].1.4.2.2.1.Choose an arbitrary subset $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(i) there is a model $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
^{M}$ such that $\mathbf{j}_{op}\left( ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\mathbf{fin}}^{M}\right) \supset M.$
(ii) the cardinality of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
^{M}$ can be chosen to coincide with $\mathbf{card}\left( M\right) $,if $M$
Proof. Choose $M\subset
%TCIMACRO{\U{211d} }%
\mathbb{R}
$. For each $m\in M$ choose $q_{1}^{m}<q_{2}^{m}<...<...$
$<p_{2}^{m}<p_{1}^{m}$ with $\lim_{k\rightarrow \infty
}q_{k}^{m}=\lim_{k\rightarrow \infty }p_{k}^{m}=m.$
We add to the axioms of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ the following axioms:$\forall m\in M$ $\exists e_{m}$ such that
$q_{k}^{m}<e_{m}<p_{k}^{m}$ for all $k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
Again, the standard $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is a model for each finite subset of these axioms,
so that the compactness theorem implies the existence of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
^{M}$ as required,
where the cardinality of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
^{M}$ can be chosen to be the cardinality of the set
of axioms, i.e. of $M,$ if $M$ is infinite. Note that by construction $%
\mathbf{j}_{op}\left( e_{m}\right) =e_{m}.$
Remark.1.4.2.2.2. It follows in particular that for each countable
of $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ we can find a countable model $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega }$ of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ such that the image of
$\mathbf{j}_{op}\left( \circ \right) $ contains this subset.Note, on the
other hand, that the image will
only be countable, so that the different models will have very different
Definition 1.4.2.2.1.[30]. A Cauchy sequence in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega }$ is a sequence $\left( a_{k}\right) _{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
such that for every $\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega },$ $\varepsilon >0$ there is an $n_{\varepsilon }\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ such that:
$\forall m_{m\text{ }>\text{ }n_{\varepsilon }}\forall n_{n\text{ }>\text{ }%
n_{\varepsilon }}\left[ \left\vert \text{ }a_{m}-a_{n}\right\vert
<\varepsilon \right] .$
Definition 1.4.2.2.2. We define Cauchy completion $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{c}}^{\omega }\triangleq \left[ ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega }\right] _{\mathbf{c}}$ in the
canonical way as equivalence classes of Cauchy sequences.
$\mathbf{Remark.}$1.4.2.2.3. This is a standard construction and
works for all
ordered fields.The result is again a field, extending the original field.
Note that, in our case,each point in $\left[ ^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\omega }\right] _{\mathbf{c}}$ is infinitesimally close to a
point in $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$\mathbf{Remark.}$1.4.2.2.4. In many non-standard models of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$, there are no
countable sequences $\left( a_{k}\right) _{k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}$ tending to zero which are not eventually
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
§ II.EULER'S PROOFS BY USING NON-ARCHIMEDEAN ANALYSIS ON THE
PSEUDO-RING $^{\AST }%
%TCIMACRO{\U{211D} }%
\MATHBB{R}
_{\MATHBF{D}}$ REVISITED. II.1.EULER'S
ORIGINAL PROOF OF THE GOLDBACH-EULER THEOREM REVISITED.
That's what he's infected me with,he thought. His madness.That's
why I've come here.That's what I want here. A strange and very new
feeling overwhelmed him. He was aware that the feeling was really not
new at all, that it had been hidden in him for a long time, but that he was
acknowledging it only now, and everything was falling into place.
Arkady and
Boris Strugatsky
"Roadside Picnic"
Euler's paper of 1737 “Variae Observationes Circa Series
Infinitas,” is Euler's first paper that closely follows
the modern Theorem-Proof format. There are no definitions in the paper, or
it would probably follow the Definition-Theorem-Proof format. After an
introductory paragraph in which Euler tells part of the story of the
problem, Euler gives us a theorem and a "proof". Euler's "proof" begins with
an 18-th century step that treats infinity as a number. Such steps
became unpopular among rigorous mathematicians about a hundred years later.
He takes $x$ to be the "sum" of the harmonic series:
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
x=1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+ \\
\\
+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+...+\dfrac{1}{n}+... \\
\end{array}
& \text{ \ \ \ \ \ \ \ }\left( 2.4\right)%
\end{array}%
The Euler's original proofs is one of those examples of completely
misuse of
divergent series to obtain completely correct results so frequent
during the seventeenth and eighteenth centuries.The acceptance of Euler's
proofs seems to lie in the fact that,at the time,Euler (and most of his
contemporaries) actually manipulated a model of real numbers which included
infinitely large and infinitely small numbers.A model that much later
Bolzano would try to build on solid grounds and that today is called
“nonstandard” after A.Robinson definitely
established it in the 1960's [1],[2],[3],[4],[5]. This last approach,
though, is completely in tune with Euler's proof [7] Nevertheless using
ideas borrowed from modern nonstandard analysis the same reconstruction
rigorous by modern Robinsonian standards is not found. In
particular "nonstandard" proof proposed in paper [7] is not completely
nonstandard becourse authors use the solution Catalan's conjecture [9]
Unfortunately completely correct proofs of the Goldbach-Euler Theorem, was
presented many authors as rational reconstruction only in terms which could
be considered rigorous by modern Weierstrassian standards.
In this last section we show how, a few simple ideas from non-archimedean
analysis on the pseudoring $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}},$ vindicate Euler's work.
Theorem 2.1.1. (Euler [6],[8]) Consider the following series,
\begin{array}{cc}
\begin{array}{c}
\\
\dfrac{1}{3}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{15}+\dfrac{1}{24}+ \\
\\
+\dfrac{1}{26}+\dfrac{1}{31}+\dfrac{1}{35}+... \\
\end{array}
& \text{\ }\left( 2.4.1\right)
\end{array}%
whose denominators, increased by one, are all the numbers which
are powers of the integers, either squares or any other higher
degree.Thus each term may be expressed by the formula
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\dfrac{1}{m^{n}-1} \\
\end{array}
& \text{ \ }\left( 2.4.2\right)%
\end{array}%
where $m$ and $n$ are integers greater than one. The sum of this series is $%
Proof. Let
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \mathbf{h}=\mathbf{cl}\left( 1+\dfrac{1}{2},1+\dfrac{1}{2}+\dfrac{1}{3},1+%
\dfrac{1}{2}+\dfrac{1}{3}+\right. \\
\\
\left. \dfrac{1}{4},1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}%
,...\right) \\
\end{array}
& \text{ \ \ }\left( 2.4.3\right)%
\end{array}%
from Eq.(2.4.3), we obtain
$\ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
1=\mathbf{cl}\left( \dfrac{1}{2},\dfrac{1}{2}+\dfrac{1}{4},\dfrac{1}{2}+%
\dfrac{1}{4}+\dfrac{1}{8},\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16%
},\right. \\
\\
\left. \dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\dfrac{1}{32}%
\dfrac{1}{2^{i}},...\right) -\varepsilon _{1}, \\
\\
\varepsilon _{1}\approx 0, \\
\\
\varepsilon _{1}=\mathbf{cl}\left( \dfrac{1}{2^{M}},\dfrac{1}{2^{M+1}},...,%
\dfrac{1}{2^{M+i}},...\right) \\
\end{array}
& \left( 2.4.4\right)%
\end{array}%
Thus we obtain
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \mathbf{h}-1= \\
\\
\mathbf{cl}\left( 1,1+\dfrac{1}{3},1+\dfrac{1}{3}+\dfrac{1}{5},1+\dfrac{1}{3}%
}{7},\right. \\
\\
\left. 1+\dfrac{1}{3}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{9}+%
\dfrac{1}{10},...\right) -\varepsilon _{1}. \\
\end{array}
& \text{ }\left( 1.4.5\right)%
\end{array}%
From Eq.(2.4.5), we obtain
$\ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\dfrac{1}{2}=\mathbf{cl}\left( \dfrac{1}{3},\dfrac{1}{3}+\dfrac{1}{9},\dfrac{%
1}{3}+\dfrac{1}{9}+\dfrac{1}{27},...,\right. \\
\\
\left. \dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+...+\dfrac{1}{3^{i}}%
,...\right) -\varepsilon _{2}, \\
\\
\varepsilon _{2}\approx 0, \\
\\
\varepsilon _{2}=\dfrac{1}{2}\mathbf{cl}\left( \dfrac{1}{3^{M}},\dfrac{1}{%
3^{M+1}},...,\dfrac{1}{3^{M+i}},...\right) \\
\end{array}
& \text{ \ \ \ \ }\left( 2.4.6\right)%
\end{array}%
we obtain
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \mathbf{h}-\left( 1+\dfrac{1}{2}\right) =\mathbf{cl}\left( 1,1+\dfrac{1}{5}%
\\
\\
\left. 1+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{10}+\dfrac{1}{11}%
,...\right) -\left( \varepsilon _{1}+\varepsilon _{2}\right) . \\
\end{array}
& \text{ \ \ }\left( 2.4.7\right)%
\end{array}%
$\ \ \ $
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\dfrac{1}{4}=\mathbf{cl}\left( \dfrac{1}{5},\dfrac{1}{5}+\dfrac{1}{25},%
\dfrac{1}{5}+\dfrac{1}{25}+\dfrac{1}{125},...,\right. \\
\\
\left. \dfrac{1}{5}+\dfrac{1}{25}+\dfrac{1}{125}+...+\dfrac{1}{5^{i}}%
,...\right) -\varepsilon _{3}, \\
\\
\varepsilon _{3}\approx 0, \\
\\
\varepsilon _{3}=\dfrac{1}{4}\mathbf{cl}\left( \dfrac{1}{5^{M}},\dfrac{1}{%
5^{M+1}},...,\dfrac{1}{5^{M+i}},...\right) \\
\end{array}
& \text{ }\left( 2.4.8\right)
\end{array}%
Finally we obtain
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \mathbf{h}-\left( 1+\dfrac{1}{2}+\dfrac{1}{4}\right) = \\
\mathbf{cl}\left( 1+\dfrac{1}{6},1+\dfrac{1}{6}+\dfrac{1}{7},1+\dfrac{1}{6}+%
\dfrac{1}{7}+\dfrac{1}{10},...\right) - \\
\\
-\left( \varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3}\right) . \\
\end{array}
& \text{ \ \ }\left( 2.4.9\right)%
\end{array}%
Proceeding similarly, i.e. deleting all the all terms that remain,we get
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \mathbf{h}-\left[ \mathbf{\Im }_{n}\right] = \\
\\
=\mathbf{cl}\left( 1+\dfrac{1}{5},...,1+\dfrac{1}{m\left( n^{\prime }\right)
},1+\dfrac{1}{m\left( n^{\prime }\right) }+...,...\right) - \\
\\
-\left( \#Ext-\sum_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\varepsilon _{n}\right) , \\
\\
m>n^{\prime }\left( n\right)
\end{array}
& \text{ \ }\left( 2.4.10\right)
\end{array}%
where $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ $
$\bigskip $
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{\Im }_{n}=\left( 1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{5},...,\dfrac{%
1}{n^{\prime }\left( n\right) },...\right) \\
\end{array}
& \text{ \ \ \ }\left( 2.4.11\right)%
\end{array}%
whose denominators, increased by one, are all the numbers
which are not powers. From Eq.(2.4.10) we obtain
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{h}-\left[ \mathbf{\Im }_{n}\right] = \\
\\
1+\left( \#Ext-\sum_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\varepsilon _{n}\right) = \\
=1+\epsilon \\
\\
\epsilon =\text{ }\#Ext-\sum_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}\varepsilon _{n}\approx 0.%
\end{array}
& \text{ \ \ }\left( 2.4.12\right)%
\end{array}%
Thus we obtain
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{h}-\left[ \mathbf{\Im }_{n}\right] =1+\epsilon , \\
\end{array}
& \text{ }\left( 2.4.13\right)%
\end{array}%
Substitution Eq.(2.4.3) into Eq.(2.4.13) gives
\begin{array}{cc}
\begin{array}{c}
\\
1+\epsilon =\dfrac{1}{3}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{15}+\dfrac{1}{24%
}+\dfrac{1}{26}+... \\
\\
\epsilon \approx 0 \\
\end{array}
& \text{ \ }\left( 2.4.14\right)%
\end{array}%
series whose denominators, increased by one, are all the powers
of the integers and whose sum is one.
Time passed, and more or less coherent thoughts came to him.
Well,that's it,he thought unwillingly. The road is open. He could
go down right now, but it was better, of course, to wait a while.
The meatgrinders can be tricky.
He got up,automatically brushed off his pants, and started down into
the quarry.The sun was broiling hot, red spots floated before his eyes,
the air was quivering on the floor of the quarry, and in the shimmer it
seemed that the ball was dancing in place like a buoy on the waves.
He went past the bucket, superstitiously picking up his feet higher and
making sure not to step on the splotches. And then, sinking into the rubble,
he dragged himself across the quarry to the dancing, winking ball.
Arkady and Boris Strugatsky
"Roadside Picnic"
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
§ III.NON-ARCHIMEDEAN ANALYSIS ON THE EXTENDED HYPERREAL LINE $^{\AST
%TCIMACRO{\U{211D} }%
\MATHBB{R}
_{\MATHBF{D}}$ AND TRANSCENDENCE CONJECTURES OVER FIELD $%
%TCIMACRO{\U{211A} }%
\MATHBB{Q}
.$PROOF THAT $\ E+\PROTECT\PI $ AND $E\CDOT \PROTECT\PI $ IS IRRATIONAL.
$\ \ \ \ \ \ \ \ $
$\ \ \ \ \ \ \ \ \ \ \ $
§ III.1.HYPERRATIONAL APPROXIMATION OF THE IRRATIONAL NUMBERS.
The next simple result shows that in a way the hyperrationals already
"incorporate" the real numbers (see e.g. [25] Ch.II Thm. 2).
Theorem 3.1.1.Let $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\mathbf{fin}}$ be the ring of finite hyperrationals, and let $\Im $ be
the maximal ideal of its infinitesimals. Then $%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ and $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\mathbf{fin}}/\Im $ are isomorphic
as ordered fields.
Theorem 3.1.2.(Standard form of Dirichlet's Approximation
Theorem).Let be $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ positive real number and $n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ a positive
integer. Then there is an integer $k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ and an integer $b\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ with
$0<k<n,$ for which
\begin{array}{cc}
\begin{array}{c}
\\
-\dfrac{1}{n}<k\cdot \alpha -b<\dfrac{1}{n}. \\
\end{array}
& (\mathbf{DAP})\text{\ \ }\left( 3.1.1\right) \text{\ }%
\end{array}%
Definition 3.1.1. A “$\mathbf{D}$
-approximation” to $\alpha $ is a rational
number $\dfrac{p}{q}\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$, whose denominator is a positive integer
%TCIMACRO{\U{2115} }%
\mathbb{N}
,$ with
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\vert \alpha -\dfrac{p}{q}\right\vert <\dfrac{1}{q^{2}}. \\
\end{array}
& \text{ }\ (\mathbf{DAP1})\text{\ \ }\left( 3.1.2\right) \text{\ }%
\end{array}%
Theorem 3.1.3. If $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is irrational it has infinitely many
Remark 3.1.1.[37]. Let sequence $\dfrac{p_{n}}{q_{n}}$ be
a convergent to $\alpha $ in the
sense such that:
$\ \ \
\ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha =\dfrac{p_{n}}{q_{n}}+\dfrac{\theta _{q_{n}}}{q_{n}^{2}}, \\
\\
\left( p_{n},q_{n}\right) =1,\left\vert \theta _{q_{n}}\right\vert
<1,n=0,1,2,... \\
\end{array}
& \text{ }(\mathbf{DAP3})\text{\ \ }\left( 3.1.3\right) \text{\ }%
\end{array}%
i.e. there is exist infinite sequence $\left( p_{n},q_{n}\right) \in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\times
%TCIMACRO{\U{2115} }%
\mathbb{N}
such that $q_{n+1}>q_{n}$ and
$\ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\alpha =\dfrac{p_{n}}{q_{n}}+\dfrac{\theta _{q_{n}}}{q_{n}^{2}}, \\
\\
\left( p_{n},q_{n}\right) =1,\left\vert \theta _{q_{n}}\right\vert <1. \\
\end{array}
& \text{ \ \ \ \ }(\mathbf{DAP4})\text{\ \ }\left( 3.1.4\right) \text{\ }%
\end{array}%
Theorem 3.1.3 shows that each irrational number $\alpha $ has infinitely many
convergents of the form $\mathbf{DAP4.}$
Definition 3.1.2. (i) Let $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is irrational number. A $\ast $-$\mathbf{D}$-approximation
to $^{\ast }\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is a number $\dfrac{P}{Q}\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$ $P\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty }$ whose denominator is
a positive hyperinteger $Q\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty },$with
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\vert ^{\ast }\alpha -\dfrac{P}{Q}\right\vert <\dfrac{1}{Q^{2}}, \\
\\
\left( P,Q\right) =1. \\
\end{array}
& \text{ \ }(\ast \text{-}\mathbf{DAP})\text{\ \ }\left( 3.1.5\right) \text{%
\ }%
\end{array}%
(ii) Let $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is irrational number. A “$\#$-$\mathbf{D}$
-approximation” to $\left( ^{\ast }\alpha \right) ^{\#}\in
$ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
is a Wattenberg hyperrational number $\dfrac{P^{\#}}{Q^{\#}}\in $ $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
_{\mathbf{d}},$ $P^{\#}\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty ,\mathbf{d}}$ whose
denominator is a positive hyperinteger $Q^{\#}\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty ,\mathbf{d}},$with
$\ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\vert \left( ^{\ast }\alpha \right) ^{\#}-\dfrac{P^{\#}}{Q^{\#}}%
\right\vert <\dfrac{1^{\#}}{Q^{\#2}}, \\
\\
\left( P^{\#},Q^{\#}\right) =1^{\#}. \\
\end{array}
& \text{ \ }(\#\text{-}\mathbf{DAP})\text{\ \ }\left( 3.1.6\right) \text{\ }%
\end{array}%
Definition 3.1.3. (i) Let $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is irrational number. A hyperrational
approximation to $\alpha $ is a $\ast $-$\mathbf{D}$-approximation to $%
^{\ast }\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
(ii) Let $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is irrational number. A Wattenberg hyperrational
approximation to $\alpha $ is a $\#$-$\mathbf{D}$-approximation
” to $\left( ^{\ast }\alpha \right) ^{\#}\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Theorem 3.1.3.(Nonstandard form of Dirichlet's Approximation
(1) If $\alpha $ is irrational it has infinitely many $\ast $-$%
\mathbf{D}$-approximations such that
for any two $\ast $-$\mathbf{D}$-approximations $P_{1}/Q_{1}$ and $%
P_{2}/Q_{2}$ the next equality is
$\ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\dfrac{P_{1}}{Q_{1}}\approx \dfrac{P_{2}}{Q_{2}}, \\
\end{array}
& \text{ \ }\left( 3.1.7\right) \text{\ \ }%
\end{array}%
$\ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( ^{\ast }\alpha \right) ^{\#}=\left( \dfrac{P_{1}}{Q_{1}}\right) ^{\#}%
\text{ }\left( \func{mod}\varepsilon _{\mathbf{d}}\right) . \\
\end{array}
& \text{ \ \ }\left( 3.1.8\right) \text{\ \ \ }%
\end{array}%
(2) If $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is irrational then $^{\ast }\alpha \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ has representation
$\ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }\alpha =\dfrac{P}{Q}+\dfrac{\theta _{Q}}{Q^{2}}, \\
\\
P\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty },Q\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty },\left( P,Q\right) =1, \\
\\
\left( P,Q\right) =1,\left\vert \theta _{Q}\right\vert <1. \\
\end{array}
& \text{ \ }(\ast \text{-}\mathbf{DAP1})\text{\ \ }\left( 3.1.9\right) \text{%
\ }%
\end{array}%
(3) If $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is irrational then $\left( ^{\ast }\alpha \right) ^{\#}\in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
_{\mathbf{d}}$ has representation
$\ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( ^{\ast }\alpha \right) ^{\#}=\dfrac{P^{\#}}{Q^{\#}}+\dfrac{\left(
\theta _{Q}\right) ^{\#}}{Q^{\#2}}, \\
\\
P^{\#}\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty ,\mathbf{d}},Q^{\#}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty ,\mathbf{d}},\left( P^{\#},Q^{\#}\right) =1, \\
\\
\left( P^{\#},Q^{\#}\right) =1,\left\vert \left( \theta _{Q}\right)
^{\#}\right\vert <1^{\#}. \\
\end{array}
& \text{ \ \ }(\#\text{-}\mathbf{DAP1})\text{\ \ }\left( 3.1.10\right) \text{%
\ }%
\end{array}%
Definition 3.1.4. A real number $\alpha \in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is a Liouville number if for
every positive integer $m\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$, there is exist infinite sequence
$\left( p_{n},q_{n}\right) \in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\times
%TCIMACRO{\U{2115} }%
\mathbb{N}
,n=0,1,2,...$ such that
$\ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
0<\left\vert \alpha -\dfrac{p_{n}}{q_{n}}\right\vert <\dfrac{1}{q^{m}}. \\
\end{array}
& \text{ \ \ }\left( 3.1.11\right) \text{\ \ }%
\end{array}%
Remark 3.1.2.This is well known that all Liouville numbers
transcendental. From the inequality (3.1.11) one obtain directly:
Theorem 3.1.4. (i) Any Liouville number $\alpha _{l}\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ for every positive
hyperinteger $N\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ has a hyperrational approximation such that
$\bigskip $
$\ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
0<\left\vert ^{\ast }\alpha _{l}-\dfrac{P}{Q}\right\vert <\dfrac{1}{Q^{N}},
\\
\\
P\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty },N,Q\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty },\left( P,Q\right) =1. \\
\end{array}
& \text{ }\left( 3.1.12\right) \text{\ \ }%
\end{array}%
(ii) Any Liouville number $\alpha _{l}\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ for every positive hyperinteger
$N\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ has a Wattenberg hyperrational approximation such that
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
0<\left\vert \left( ^{\ast }\alpha _{l}\right) ^{\#}-\dfrac{P^{\#}}{Q^{\#}}%
\right\vert <\dfrac{1^{\#}}{\left( Q^{N}\right) ^{\#}}, \\
\\
P^{\#}\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty ,\mathbf{d}},Q^{\#}\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty ,\mathbf{d}}, \\
\\
N\in \text{ }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty },\left( P^{\#},Q^{\#}\right) =1^{\#}. \\
\end{array}
& \text{ }\left( 3.1.13\right) \text{\ \ }%
\end{array}%
Theorem 3.1.5. Every Liouville number $\alpha _{l}\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ are $\#$-transcendental
over field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$ i.e., there is no real $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$-analytic function
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( x\right) =\dsum\limits_{n=0}^{\infty }a_{n}x^{n}<\infty ,0\leq
\left\vert x\right\vert \leq r\leq e$ with rational coefficients
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ such that $g_{%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( \alpha _{l}\right) .$
$\ \ \ \ \ \ \ \ \ \ \ \ $
§ III.2.PROOF THAT $E$ IS #-TRANSCENDENTAL.
Definition 3.2.1. Let $g\left( x\right) :%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\rightarrow
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ be any real analytic function
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( x\right) =\dsum\limits_{n=0}^{\infty }a_{n}x^{n},\left\vert
x\right\vert <r, \\
\\
\forall n\left[ a_{n}\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
\right] \\
\end{array}
& \text{\ }\left( 3.2.1\right)%
\end{array}%
defined on an open interval $I\subset
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ such that $0\in I.$
We call this function given by Eq.(3.2.1) $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$-analytic function and denote $g_{%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( x\right) .$
Definition 3.2.2. Arbitrary transcendental number $z\in
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ is called $\ \ \ \ \ $
$\#$-transcendental number over field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$, if no exist $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$-analytic function $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( x\right) $ such that $g_{%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( z\right) =0,$i.e. for every $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$-analytic function $g_{%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( x\right) $ the
inequality $g_{%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( z\right) \neq 0$ is satisfies.
Definition 3.2.3.Arbitrary transcendental number $z$ called $w$
number over field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$,if $z$ is not $\#$-transcendental number over field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
exist $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$-analytic function $g_{%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( x\right) $ such that $g_{%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( z\right) =0.$
Example 3.2.1. Number $\pi $ is transcendental but number $\pi $ is
not $\ \ \ \ \ \ \ $
$\ \#$-transcendental number over field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ as
(1) function $\sin x$ is a $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$-analytic and
(2)$\ \sin \left( \dfrac{\pi }{2}\right) =1,$i.e.$\ \ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
-1+\dfrac{\pi }{2}-\dfrac{\pi ^{3}}{2^{3}3!}+\dfrac{\pi ^{5}}{2^{5}5!}- \\
\\
-\dfrac{\pi ^{7}}{2^{7}7!}+...+\dfrac{\left( -1\right) ^{2n+1}\pi ^{2n+1}}{%
2^{2n+1}\left( 2n+1\right) !}+...=0. \\
\end{array}
& \text{ }\left( 3.2.2\right)%
\end{array}%
Theorem 3.2.1.Number $e$ is $\#$-transcendental over
field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
Proof I.To prove $e$ is $\#$-transcendental number we must show
that $e$ it is
not $w$-transcendental, i.e., there is no exist real $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$-analytic function
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( x\right) =\dsum\limits_{n=0}^{\infty }a_{n}x^{n},0\leq \left\vert
x\right\vert \leq r\leq e$ with rational coefficients
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ such that $\ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\dsum\limits_{n=0}^{\infty }a_{n}e^{n}=0. \\
\end{array}
& \text{ \ \ \ \ \ \ \ \ \ }\left( 3.2.3\right)%
\end{array}%
Suppose that $e$ is $w$-transcendental, i.e., there is exist an $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
function $\breve{g}_{%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( x\right) =\dsum\limits_{n=0}^{\infty }\breve{a}_{n}x^{n},$with
rational coefficients:
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \breve{a}_{0}=\dfrac{k_{0}}{m_{0}},\breve{a}_{1}=\dfrac{k_{1}}{m_{1}}%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
, \\
\\
\ \breve{a}_{0}>0 \\
\end{array}
& \text{ \ }\left( 3.2.4\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
such that the next equality is satisfied:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\dsum\limits_{n=0}^{\infty }\breve{a}_{n}e^{n}=0. \\
\end{array}
& \text{ \ }\left( 3.2.5\right)%
\end{array}%
Hence there is exist sequences$\ \ \left\{ n_{i}\right\} _{i=0}^{\infty }$
and $\ \left\{ n_{j}\right\} _{j=1}^{\infty }$ such that$\ $
$\bigskip $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\forall k\left[ \dsum\limits_{n=0}^{n_{i}\leq k}\breve{a}_{n}e^{n}>0\right] ,%
\underset{i\rightarrow \infty }{\lim }\dsum\limits_{n=0}^{n_{i}}\breve{a}%
_{n}e^{n}=0, \\
\\
\forall k\left[ \dsum\limits_{n=1}^{n_{j}\leq k}\breve{a}_{n}e^{n}<0\right] ,%
\underset{j\rightarrow \infty }{\lim }\dsum\limits_{n=1}^{n_{j}}\breve{a}%
_{n}e^{n}=0. \\
\end{array}
& \text{ \ \ \ }\left( 3.2.6\right)%
\end{array}%
From Eqs.(3.2.6) by using definitions one obtain the next
$\ \ \ \ \ \ \ \ \ \ \
\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( ^{\ast }\breve{a}_{0}\right) ^{\#}+\left[ \overline{\overline{\#Ext%
\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\backslash \left\{ 0\right\} }\left( ^{\ast }\breve{a}_{n}\right)
^{\#}\times \left( ^{\ast }e^{n}\right) ^{\#}}}\right] _{\varepsilon }= \\
\\
=\left[ \left( ^{\ast }\left( \underset{i\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left( \dsum\limits_{n=0}^{n_{i}}\breve{a}_{n}e^{n}\right) \right)
\right) ^{\#}+\varepsilon _{\mathbf{d}}\right] _{\varepsilon }= \\
\\
\text{ }=\left( ^{\ast }\left( \underset{i\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\inf }\left( \dsum\limits_{n=0}^{n_{i}}\breve{a}_{n}e^{n}\right) \right)
\right) ^{\#}+\varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}=\varepsilon
^{\#}\times \varepsilon _{\mathbf{d}}, \\
\\
\varepsilon \approx 0,\varepsilon \in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\\
\end{array}
& \text{ \ \ }\left( 3.2.7\right)%
\end{array}%
$\bigskip $and by similar way
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( ^{\ast }\breve{a}_{0}\right) ^{\#}+\left[ \underline{\underline{\#Ext%
\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\backslash \left\{ 0\right\} }\left( ^{\ast }\breve{a}_{n}\right)
^{\#}\times \left( ^{\ast }e^{n}\right) ^{\#}}}\right] _{\varepsilon }= \\
\\
\text{ }=\text{ }\underset{j\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
}{\sup }\left( \dsum\limits_{n=1}^{n_{j}}\breve{a}_{n}e^{n}\right)
-\varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}=-\varepsilon ^{\#}\times
\varepsilon _{\mathbf{d}}, \\
\\
\varepsilon \approx 0,\varepsilon \in \text{ }^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
. \\
\end{array}
& \text{ \ }\left( 3.2.8\right)%
\end{array}%
Let us considered hypernatural number $\Im \in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ defined by
countable sequence
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\Im =\left( m_{0},m_{0}\times m_{1},...,m_{0}\times m_{1}\times ...\times
m_{n},...\right) \\
\end{array}
& \text{ \ }\left( 3.2.9\right)
\end{array}%
By using Eq.(3.2.7) and Eq.(3.2.9) one obtain
$\ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\Im ^{\#}\times \left( ^{\ast }\breve{a}_{0}\right) ^{\#}+\Im ^{\#}\times %
\left[ \overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\backslash \left\{ 0\right\} }\left( ^{\ast }\breve{a}_{n}\right)
^{\#}\times \left( ^{\ast }e^{n}\right) ^{\#}}}\right] _{\varepsilon }= \\
\\
\Im ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}, \\
\\
\Im ^{\#}\times \left( ^{\ast }\breve{a}_{0}\right) ^{\#}+ \\
\\
+\left[ \overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\backslash \left\{ 0\right\} }\Im ^{\#}\times \left( ^{\ast }\breve{a}%
_{n}\right) ^{\#}\times \left( ^{\ast }e^{n}\right) ^{\#}}}\left\vert \Im
^{\#}\times \left( ^{\ast }c\right) ^{\#}\right. \right] _{\varepsilon }= \\
\\
=\Im _{0}^{\#}+\left[ \overline{\overline{\#\text{-}Ext\text{-}%
\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\backslash \left\{ 0\right\} }\Im _{n}^{\#}\times \left( ^{\ast
}e^{n}\right) ^{\#}}}\left\vert \Im ^{\#}\times \left( ^{\ast }c\right)
^{\#}\right. \right] = \\
\\
=\Im ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}, \\
\\
%TCIMACRO{\U{211d} }%
\mathbb{R}
, \\
\\
\Im _{n}^{\#}\triangleq \Im ^{\#}\times \left( ^{\ast }\breve{a}_{n}\right)
%TCIMACRO{\U{2115} }%
\mathbb{N}
. \\
\end{array}
& \text{ }\left( 3.2.10\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
Now we have to pruve that Eq.(3.2.10) leads to contradiction.
$\mathbf{Proof}$ $\mathbf{I.}$
Part I. Let be
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
M_{0}\left( n,p\right) =\dint\limits_{0}^{+\infty }\left[ \dfrac{x^{p-1}%
\left[ \left( x-1\right) ...\left( x-n\right) \right] ^{p}e^{-x}}{\left(
p-1\right) !}\right] dx\neq 0, \\
\end{array}
& \text{ }\left( 3.2.11\right)%
\end{array}%
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
M_{k}\left( n,p\right) =e^{k}\dint\limits_{k}^{+\infty }\left[ \dfrac{x\left[
^{p-1}\left( x-1\right) ...\left( x-n\right) \right] ^{p}e^{-x}}{\left(
p-1\right) !}\right] dx, \\
\\
k=1,2,... \\
\end{array}
& \text{ \ }\left( 3.2.12\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ $
\begin{array}{cc}
\begin{array}{c}
\\
\varepsilon _{k}\left( n,p\right) =e^{k}\dint\limits_{0}^{k}\left[ \dfrac{%
x^{p-1}\left[ \left( x-1\right) ...\left( x-n\right) ^{p}\right] e^{-x}}{%
\left( p-1\right) !}\right] dx, \\
\\
k=1,2,..., \\
\end{array}
& \text{ }\left( 3.2.13\right)%
\end{array}%
where $p\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ this is any prime number.Using Eq.(3.2.9)-Eq.(3.2.13) by simple calculation one obtain:$\ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
M_{k}\left( n,p\right) +\varepsilon _{k}\left( n,p\right) =e^{k}M_{0}\neq 0,
\\
\\
k=1,2,... \\
\end{array}
& \text{ \ \ }\left( 3.2.14\right)%
\end{array}%
and consequently
$\ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
e^{k}=\dfrac{M_{k}\left( n,p\right) +\varepsilon _{k}\left( n,p\right) }{%
M_{0}} \\
\\
k=1,2,... \\
\end{array}
& \text{ \ }\left( 3.2.15\right)%
\end{array}%
By using equality
$\bigskip $
\begin{array}{cc}
\begin{array}{c}
\\
x^{p-1}\left[ \left( x-1\right) ...\left( x-n\right) \right] ^{p}= \\
\\
\left( -1\right) ^{n}\left( n!\right) ^{n}x^{n-1}+\dsum\limits_{\mu
=p+1}^{\left( n+1\right) \times p}c_{\mu -1}x^{\mu -1}, \\
\\
c_{\mu }\in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
,\mu =p,p+1,...,\left[ (n+1)\times p\right] -1, \\
\end{array}
& \text{ \ }\left( 3.2.16\right)%
\end{array}%
from Eq.(3.2.11) one obtain
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
M_{0}\left( n,p\right) =\left( -1\right) ^{n}\left( n!\right) ^{p}\dfrac{%
\Gamma \left( p\right) }{\left( p-1\right) !}+ \\
\\
\dsum\limits_{\mu =p+1}^{\left( n+1\right) \times p}c_{\mu -1}\dfrac{\Gamma
\left( \mu \right) }{\left( p-1\right) !}= \\
\\
=\left( -1\right) ^{n}\left( n!\right) ^{p}+c_{p}p+c_{n+1}p\left( p+1\right)
+...= \\
\\
=\left( -1\right) ^{n}\left( n!\right) ^{p}+p\times \Theta _{1},\Theta
%TCIMACRO{\U{2124} }%
\mathbb{Z}
, \\
\\
\Gamma \left( \mu \right) =\dint\limits_{0}^{+\infty }x^{\mu -1}e^{-x}dx. \\
\\
M_{0}\left( n,p\right) =\left( -1\right) ^{n}\left( n!\right) ^{p}+p\times
\Theta _{1},\Theta _{1}\in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\\
\end{array}
& \text{ \ }\left( 3.2.17\right)%
\end{array}%
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
M_{0}\left( n,p\right) =\left( -1\right) ^{n}\left( n!\right) ^{p}+ \\
\\
p\cdot \Theta _{1}\left( n,p\right) ,\Theta _{1}\left( n,p\right) \in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
. \\
\end{array}
& \text{ \ }\left( 3.2.18\right)%
\end{array}%
By subsitution $x=k+u\implies dx=du$ from Eq.(3.2.13.) one obtain
\begin{array}{cc}
\begin{array}{c}
\\
M_{k}\left( n,p\right) = \\
\dint\limits_{0}^{+\infty }\left[ \dfrac{\left( u+k\right) ^{p-1}\left[
\left( u+k-1\right) \times ...\times u\times ...\times \left( u+k-n\right) %
\right] ^{p}e^{-u}}{\left( p-1\right) !}\right] du \\
\\
k=1,2,3,... \\
\end{array}
& \text{ }\left( 3.2.19\right)%
\end{array}%
By using equality
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left( u+k\right) ^{p-1}\left[ \left( u+k-1\right) \times ...\times u\times
...\times \left( u+k-n\right) \right] ^{p}= \\
\\
=\dsum\limits_{\mu =p+1}^{\left( n+1\right) \times p}d_{\mu -1}u^{\mu -1},
\\
\\
d_{\mu }\in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
,\mu =p,p+1,...,\left[ (n+1)\times p\right] -1, \\
\end{array}
& \text{ \ }\left( 3.2.20\right)%
\end{array}%
and by subsitution Eq.(3.2.20) into Eq.(3.2.19) one obtain
$\ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
M_{k}\left( n,p\right) =\dfrac{1}{\left( p-1\right) !}\dint\limits_{0}^{+%
\infty }\dsum\limits_{\mu =p+1}^{\left( n+1\right) \times p}d_{\mu -1}u^{\mu
-1}du= \\
\\
p\cdot \Theta _{2}\left( n,p\right) , \\
\\
\Theta _{2}\left( n,p\right) \in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
, \\
\\
k=1,2,...\text{ }. \\
\end{array}
& \text{ \ }\left( 3.2.21\right)%
\end{array}%
There is exists sequences $a\left( n\right) ,n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ and $g_{k}\left( n\right) ,k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ such that
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\vert x\left( x-1\right) ...\left( x-n\right) \right\vert <a\left(
n\right) , \\
\\
0\leq x\leq n, \\
\\
\left\vert x\left( x-1\right) ...\left( x-n\right) e^{-x+k}\right\vert
<g_{k}\left( n\right) , \\
\\
0\leq x\leq n,k=1,2,...\text{ }. \\
\end{array}
& \text{ \ \ }\left( 3.2.22\right)%
\end{array}%
Substitution the inequalities (3.2.22.) into Eq.(3.2.13.) gives
\begin{array}{cc}
\begin{array}{c}
\\
\varepsilon _{k}\left( n,p\right) \leq g_{k}\left( n\right) \dfrac{\left[
a\left( n\right) \right] ^{p-1}}{\left( p-1\right) !}\dint\limits_{0}^{k}dx%
\leq \\
\\
\leq \dfrac{n\cdot g_{k}\left( n\right) \cdot \left[ a\left( n\right) \right]
^{p-1}}{\left( p-1\right) !}. \\
\end{array}
& \text{ }\left( 3.2.23\right)%
\end{array}%
By using transfer, from Eq.(3.2.11.) and Eq.(3.2.18.) one obtain
$\ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }M_{0}\left( \mathbf{n,p}\right) = \\
\\
\text{ }^{\ast }\left( \dint\limits_{0}^{+\infty }\left[ \dfrac{x^{p-1}\left[
\left( x-1\right) ...\left( x-n\right) \right] ^{p}e^{-x}}{\left( p-1\right)
!}\right] dx\right) = \\
\\
=\left( -1\right) ^{\mathbf{n}}\left( \mathbf{n}!\right) ^{\mathbf{p}}+%
\mathbf{p}\times \text{ }^{\ast }\Theta _{1}\left( \mathbf{n},\mathbf{p}%
\right) , \\
\\
^{\ast }\Theta _{1}\left( \mathbf{n},\mathbf{p}\right) \in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty }, \\
\\
\mathbf{n},\mathbf{p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{ }\left( 3.2.24\right)%
\end{array}%
From Eq.(3.2.12.) and Eq.(3.2.21) one obtain
$\ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
M_{k}\left( n,p\right) =e^{k}\dint\limits_{k}^{+\infty }\left[ \dfrac{x\left[
^{p-1}\left( x-1\right) ...\left( x-n\right) \right] ^{p}e^{-x}}{\left(
p-1\right) !}\right] dx= \\
\\
\dint\limits_{0}^{+\infty }\left[ \dfrac{\left( u+k\right) ^{p-1}\left[
\left( u+k-1\right) \times ...\times u\times ...\times \left( u+k-n\right) %
\right] ^{p}e^{-u}}{\left( p-1\right) !}\right] du= \\
\\
=p\cdot \Theta _{2}\left( n,p\right) , \\
\\
\Theta _{2}\left( n,p\right) \in
%TCIMACRO{\U{2124} }%
\mathbb{Z}
, \\
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
. \\
\end{array}
& \text{ }\left( 3.2.25\right)%
\end{array}%
Using transfer, from Eq.(3.2.25.) one obtain $\forall k\left( k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right) :$
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }M_{k}\left( \mathbf{n,p}\right) =\left( ^{\ast }e^{k}\right) \times
\text{ }^{\ast }\left( \dint\limits_{k}^{+\infty }\left[ \dfrac{x\left[ ^{%
\mathbf{p}-1}\left( x-1\right) ...\left( x-\mathbf{n}\right) \right] ^{%
\mathbf{p}}e^{-x}}{\left( \mathbf{p}-1\right) !}\right] dx\right) = \\
\\
=\text{ }^{\ast }\left( \dint\limits_{0}^{+\infty }\left[ \dfrac{\left(
u+k\right) ^{\mathbf{p}-1}\left[ \left( u+k-1\right) \times ...\times
u\times ...\times \left( u+k-\mathbf{n}\right) \right] ^{\mathbf{p}}e^{-u}}{%
\left( \mathbf{p}-1\right) !}\right] du\right) = \\
\\
=\mathbf{p}\times \text{ }^{\ast }\Theta _{2}\left( \mathbf{n,p}\right) , \\
\\
^{\ast }\Theta _{2}\left( \mathbf{n,p}\right) \in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty }, \\
\\
k=1,2,3,... \\
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
, \\
\\
\mathbf{n},\mathbf{p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{ }\left( 3.2.26\right)%
\end{array}%
Using transfer, from inequality (3.2.23.) one obtain $\forall k\left( k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right) :$
$\ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }\varepsilon _{k}\left( \mathbf{n,p}\right) \leq \text{ }^{\ast
}g_{k}\left( \mathbf{n}\right) \times \dfrac{\left[ ^{\ast }a\left( \mathbf{n%
}\right) \right] ^{\mathbf{p}-1}}{\left( \mathbf{p}-1\right) !}\times \left[
\text{ }^{\ast }\left( \dint\limits_{0}^{k}dx\right) \right] \leq \\
\\
\leq \dfrac{\mathbf{n}\cdot \left[ ^{\ast }g_{k}\left( \mathbf{n}\right) %
\right] \cdot \left[ ^{\ast }a\left( \mathbf{n}\right) \right] ^{\mathbf{p}%
-1}}{\left( \mathbf{p}-1\right) !}, \\
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
, \\
\\
\mathbf{n},\mathbf{p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{\ }\left( 3.2.27\right)%
\end{array}%
By using transfer again, from Eq.(3.2.15.) one obtain $\forall k\left( k\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\right) :$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\text{ }^{\ast }\left( e^{k}\right) =\left( ^{\ast }e\right) ^{k}=\dfrac{%
^{\ast }M_{k}\left( \mathbf{n,p}\right) +\text{ }^{\ast }\varepsilon
_{k}\left( \mathbf{n,p}\right) }{^{\ast }M_{0}\left( \mathbf{n,p}\right) },
\\
\\
k=1,2,..., \\
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
, \\
\\
\mathbf{n},\mathbf{p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{ \ }\left( 3.2.28\right)%
\end{array}%
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\bigskip $
(Part II) By using Eq.(3.2.28.) one obtain
$\ \ \ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\text{ }\left[ ^{\ast }\left( e^{k}\right) \right] ^{\#}=\left[ \left(
^{\ast }e\right) ^{\#}\right] ^{k}= \\
\\
\dfrac{\left[ ^{\ast }M_{k}\left( \mathbf{n,p}\right) \right] ^{\#}+\text{ }%
\left[ ^{\ast }\varepsilon _{k}\left( \mathbf{n,p}\right) \right] ^{\#}}{%
\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) \right] ^{\#}}, \\
\\
k=1,2,..., \\
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
, \\
\\
\mathbf{n},\mathbf{p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{ }\left( 3.2.29\right)%
\end{array}%
By using Eq.(3.2.24.) one obtain
\begin{array}{cc}
\begin{array}{c}
\\
\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) \right] ^{\#}=\left[ \left(
-1\right) ^{\mathbf{n}}\right] ^{\#}\left[ \left( \mathbf{n}!\right) ^{%
\mathbf{p}}\right] ^{\#}+ \\
\\
\mathbf{p}^{\#}\times \text{ }\left[ ^{\ast }\Theta _{1}\left( \mathbf{n},%
\mathbf{p}\right) \right] ^{\#}, \\
\\
\left[ ^{\ast }\Theta _{1}\left( \mathbf{n},\mathbf{p}\right) \right]
^{\#}\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty ,\mathbf{d}}, \\
\\
\mathbf{n},\mathbf{p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{ \ }\left( 3.2.30\right)%
\end{array}%
By using Eq.(3.2.26.) one obtain
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left[ ^{\ast }M_{k}\left( \mathbf{n,p}\right) \right] ^{\#}=\mathbf{p}%
^{\#}\times \left[ \text{ }^{\ast }\Theta _{2}\left( \mathbf{n,p}\right) %
\right] ^{\#}, \\
\\
\left[ ^{\ast }\Theta _{2}\left( \mathbf{n,p}\right) \right] ^{\#}\in \text{
}^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\infty ,\mathbf{d}}, \\
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
, \\
\\
\mathbf{n},\mathbf{p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{ \ }\left( 3.2.31\right)%
\end{array}%
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ $
By using inequality (3.2.27) one obtain
\begin{array}{cc}
\begin{array}{c}
\\
\left[ ^{\ast }\varepsilon _{k}\left( \mathbf{n,p}\right) \right] ^{\#}\leq
\dfrac{\mathbf{n}^{\#}\cdot \left[ g_{k}\left( \mathbf{n}\right) \right]
^{\#}\cdot \left[ \left[ a\left( \mathbf{n}\right) \right] ^{\mathbf{p}-1}%
\right] ^{\#}}{\left[ \left( \mathbf{p}-1\right) !\right] ^{\#}}, \\
\\
%TCIMACRO{\U{2115} }%
\mathbb{N}
, \\
\\
\mathbf{n},\mathbf{p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{ }\left( 3.2.32\right)%
\end{array}%
Substitution Eq.(3.2.28) into Eq.(3.2.10) gives
\begin{array}{cc}
\begin{array}{c}
\\
\Im _{0}^{\#}+\left[ \overline{\overline{\#Ext\text{-}\dsum\limits_{n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\backslash \left\{ 0\right\} }\Im _{n}^{\#}\times \left( ^{\ast
}e^{n}\right) ^{\#}}}\left\vert \Im ^{\#}\times \left( ^{\ast }c\right)
^{\#}\right. \right] _{\varepsilon }= \\
\\
\Im _{0}^{\#}+\left[ \overline{\overline{\#Ext\text{-}\dsum\limits_{k=1}^{%
\infty }\Im _{k}^{\#}\times \dfrac{\left[ ^{\ast }M_{k}\left( \mathbf{n,p}%
\right) \right] ^{\#}+\text{ }\left[ ^{\ast }\varepsilon _{k}\left( \mathbf{%
n,p}\right) \right] ^{\#}}{\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}}}}\left\vert \Im ^{\#}\times \left( ^{\ast }c\right)
^{\#}\right. \right] _{\varepsilon } \\
\\
=\Im ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}},. \\
\end{array}
& \left( 3.2.33\right)%
\end{array}%
Multiplying Eq.(3.2.33) by number $\left[ ^{\ast }M_{0}\left( \mathbf{n,p}%
\right) \right] ^{\#}\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
_{\mathbf{d}}$ one obtain
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\Im _{0}^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}+ \\
\\
\left[ \overline{\overline{\#Ext\text{-}\dsum\limits_{k=1}^{\infty }\text{ }%
\left\{ \Im _{k}^{\#}\times \left[ ^{\ast }M_{k}\left( \mathbf{n,p}\right) %
\right] ^{\#}+\Im _{k}^{\#}\times \left[ ^{\ast }\varepsilon _{k}\left(
\mathbf{n,p}\right) \right] ^{\#}\right\} }}\right. \\
\\
\left. \left\vert \Im ^{\#}\times \left[ ^{\ast }M_{0}\left( \mathbf{n,p}%
\right) \right] ^{\#}\times \left( ^{\ast }c\right) ^{\#}\right. \right]
_{\varepsilon }= \\
\\
=\Im ^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ }\left( 3.2.34\right)%
\end{array}%
By using inequality (3.2.32) for we will choose prime hyper number $\mathbf{%
p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }$
given $\varepsilon $ such that:$\ $
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \overline{\overline{\#Ext\text{-}\dsum\limits_{k=1}^{\infty }\text{ }%
\Im _{k}^{\#}\times \left[ ^{\ast }\varepsilon _{k}\left( \mathbf{n,p}%
\right) \right] ^{\#}}}\left\vert \Im ^{\#}\times \left( ^{\ast }c\right)
^{\#}\right. \right] _{\varepsilon }\in \\
\\
\in \Im ^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}\times \varepsilon \times \varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{\ }\left( 3.2.35\right)%
\end{array}%
Hence from Eq.(3.2.34) and Eq.(3.2.35) one obtain
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\Im _{0}^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}+ \\
\\
+\left[ \overline{\overline{\#Ext\text{-}\dsum\limits_{k=1}^{\infty }\text{ }%
\Im _{k}^{\#}\times \left[ ^{\ast }M_{k}\left( \mathbf{n,p}\right) \right]
^{\#}}}\left\vert \Im ^{\#}\times \left[ ^{\ast }M_{0}\left( \mathbf{n,p}%
\right) \right] ^{\#}\times \left( ^{\ast }c\right) ^{\#}\right. \right]
_{\varepsilon }= \\
\\
=\Im ^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ }\left( 3.2.36\right)%
\end{array}%
We will choose prime hyper number $\mathbf{p\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }$ such that
\begin{array}{cc}
\begin{array}{c}
\\
\mathbf{p}^{\#}\mathbf{>\max }\left( \Im ^{\#},\left\vert \Im
_{0}^{\#}\right\vert ,\mathbf{n}^{\#}\mathbf{.}\right) \\
\end{array}
& \text{ \ }\left( 3.2.37\right)%
\end{array}%
Hence by using Eq.(3.2.20) one obtain:
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) \right] ^{\#}\nmid \mathbf{p}%
^{\#} \\
\end{array}
& \text{ \ }\left( 3.2.38\right)%
\end{array}%
and consequently $\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) \right]
^{\#}\neq 0^{\#}.$And by using (3.2.20),(3.2.28)
one obtain:
\begin{array}{cc}
\begin{array}{c}
\\
\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) \right] ^{\#}\times \Im
_{0}^{\#}\nmid \mathbf{p}^{\#}\mathbf{.} \\
\end{array}
& \text{ \ }\left( 3.2.39\right)
\end{array}%
By using Eq.(3.2.22) one obtain
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left[ ^{\ast }M_{k}\left( \mathbf{n,p}\right) \right] ^{\#}\mid \mathbf{p}%
^{\#}\mathbf{,} \\
\\
k=1,2,...\mathbf{.} \\
\end{array}
& \text{ \ \ \ \ }\left( 3.2.40\right)%
\end{array}%
By using Eq.(3.2.36) one obtain
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\Xi \left( \mathbf{n,p,}\varepsilon \right) =\mathbf{a.p.}\left\{ \left[ \Xi
\left( \mathbf{n,p}\right) \right] _{\varepsilon }\right\} = \\
\\
\mathbf{ab.p.}\left\{ \left[ \overline{\overline{\#Ext\text{-}%
\dsum\limits_{k=1}^{\infty }\text{ }\Im _{k}^{\#}\times \left[ ^{\ast
}M_{k}\left( \mathbf{n,p}\right) \right] ^{\#}}}\left\vert \Im ^{\#}\times %
\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) \right] ^{\#}\times \left(
^{\ast }c\right) ^{\#}\right. \right] _{\varepsilon }\right\} \\
\\
=\Im ^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}. \\
\\
\Xi \left( \mathbf{n,p}\right) =\overline{\overline{\#Ext\text{-}%
\dsum\limits_{k=1}^{\infty }\text{ }\Im _{k}^{\#}\times \left[ ^{\ast
}M_{k}\left( \mathbf{n,p}\right) \right] ^{\#}}} \\
\end{array}
& \left( 3.2.41\right)%
\end{array}%
It is easy to see that Wattenberg hypernatural number $\Xi \left( \mathbf{n,p%
}\right) $ has
tipe $1\mathbf{.}$Hence Wattenberg hypernatural number $\Xi \left( \mathbf{%
n,p}\right) $ has
represantation: $\ \ \ \ \ \ \ \ \ $
$\bigskip $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\Xi \left( \mathbf{n,p}\right) =\mathbf{p}^{\#}\mathbf{\times m+}\Im
^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) \right]
^{\#}\times \varepsilon _{\mathbf{d}}, \\
\\
\mathbf{m\in }\text{ }^{\ast }\mathbf{%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}_{\mathbf{d}}. \\
\end{array}
& \text{ }\left( 3.2.42\right)%
\end{array}%
By using (3.2.42) one obtain represantation
\begin{array}{cc}
\begin{array}{c}
\\
\Xi \left( \mathbf{n,p,\varepsilon }\right) =\left[ \Xi \left( \mathbf{n,p}%
\right) \right] _{\varepsilon }= \\
\\
\mathbf{p}^{\#}\mathbf{\times m+}\Im ^{\#}\times \text{ }\left[ ^{\ast
}M_{0}\left( \mathbf{n,p}\right) \right] ^{\#}\times \varepsilon ^{\#}\times
\varepsilon _{\mathbf{d}}, \\
\\
\mathbf{m\in }\text{ }^{\ast }\mathbf{%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
}_{\mathbf{d}}. \\
\end{array}
& \text{ \ \ \ }\left( 3.2.43\right)%
\end{array}%
Substitution Eq.(3.2.43) into Eq.(3.2.36) gives
\begin{array}{cc}
\begin{array}{c}
\\
\left\{ \Im _{0}^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}%
\right) \right] ^{\#}+\mathbf{p}^{\#}\mathbf{\times m}\right\} \mathbf{+} \\
\\
+\Im ^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}= \\
\\
=\Im ^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ \ }\left( 3.2.45\right)%
\end{array}%
By using Eq.(3.2.39)-Eq.(3.2.40) one obtain:
\begin{array}{cc}
\begin{array}{c}
\\
\left\{ \Im _{0}^{\#}\times \left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}+\mathbf{p}^{\#}\mathbf{\times m}\right\} \nmid \mathbf{p}^{\#}
\\
\end{array}
& \text{ \ }\left( 3.2.46\right)%
\end{array}%
and consequently $\left\{ \Im _{0}^{\#}\times \left[ ^{\ast }M_{0}\left(
\mathbf{n,p}\right) \right] ^{\#}+\mathbf{p}^{\#}\mathbf{\times m}\right\}
\neq 0^{\#}.$But on the
other hand, for sufficiently infinite smoll $\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ idempotent
$\Im ^{\#}\times $ $\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) \right]
^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}$ does not
absorbs Wattenberg
hypernatural number $\left\{ \Im _{0}^{\#}\times \left[ ^{\ast }M_{0}\left(
\mathbf{n,p}\right) \right] ^{\#}+\mathbf{p}^{\#}\mathbf{\times m}\right\} $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\{ \Im _{0}^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}%
\right) \right] ^{\#}+\mathbf{p}^{\#}\mathbf{\times m}\right\} \mathbf{+} \\
\\
+\Im ^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}\neq \\
\\
\neq \Im ^{\#}\times \text{ }\left[ ^{\ast }M_{0}\left( \mathbf{n,p}\right) %
\right] ^{\#}\times \varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}. \\
\end{array}
& \text{ \ \ }\left( 3.2.47\right)%
\end{array}%
Thus for sufficiently infinite small $\varepsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ inequality (3.2.47) in a
contradiction with Eq.(3.2.45).This contradiction proves that $e$ is not
$w$-transcendental. Hence $e$ is $\#$-transcendental.
Proof II. (Part I)
To prove $e$ is #-transcendental we must show it is not $w$-transcendental,
i.e., there is no real analytic function $g_{%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
}\left( x\right) =\dsum\limits_{n=0}^{\infty }b_{n}x^{n},e\leq \left\vert
x\right\vert \leq r$ with
rational coefficients $b_{0},b_{1},...,b_{n},...\in
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$ such that
\begin{array}{cc}
\begin{array}{c}
\\
\dsum\limits_{n=0}^{\infty }b_{n}e^{n}=0. \\
\\
b_{0}=\dfrac{k_{0}}{m_{0}},b_{n}=\dfrac{k_{n}}{m_{n}}. \\
\end{array}
& \text{ \ }\left( 3.2.48\right)%
\end{array}%
1. Assume that $b_{0},b_{1},...,b_{n},...\in $ $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,b_{0}\neq 0.$
Let $f\left( x\right) :%
%TCIMACRO{\U{211d} }%
\mathbb{R}
\rightarrow
%TCIMACRO{\U{211d} }%
\mathbb{R}
$ be a polynomial of degree $m\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
.$ Then (repeated) integrations by
parts gives
\begin{array}{cc}
\begin{array}{c}
\\
\dint\limits_{0}^{k}f\left( x\right) e^{-x}dx=\left. -f\left( x\right)
e^{-x}\right\vert _{0}^{k}+\dint\limits_{0}^{k}f^{\text{ }\prime }\left(
x\right) e^{-x}dx= \\
\\
=\left. -\left( f\left( x\right) +f^{\text{ }\prime }\left( x\right) +...+f^{%
\text{ }\left( m\right) }\left( x\right) \right) e^{-x}\right\vert _{0}^{k}
\\
\end{array}
& \text{ }\left( 3.2.49\right)%
\end{array}%
Multiply by $b_{k}e^{k}\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ and add up: Then
\begin{array}{cc}
\begin{array}{c}
\\
\dsum\limits_{k=0}^{n}b_{k}e^{k}\dint\limits_{0}^{k}f\left( x\right)
e^{-x}dx= \\
=\left( f\left( 0\right) +f^{\text{ }\prime }\left( 0\right) +...+f^{\text{ }%
\left( m\right) }\left( 0\right) \right) \dsum\limits_{k=0}^{n}b_{k}e^{k}-
\\
\\
-\dsum\limits_{k=0}^{n}b_{k}\left( f\left( k\right) +f^{\text{ }\prime
}\left( k\right) +...+f^{\text{ }\left( m\right) }\left( k\right) \right) .
\\
\end{array}
& \text{ \ }\left( 3.2.50\right)%
\end{array}%
By using transfer from Eq.(3.2.50) one obtain
\begin{array}{cc}
\begin{array}{c}
\\
\dsum\limits_{k=0}^{n}\left( ^{\ast }b_{k}\left( ^{\ast }e^{k}\right)
\right) \times \text{ }^{\ast }\left( \dint\limits_{0}^{k}f\left( x\right)
e^{-x}dx\right) - \\
\\
-\left( ^{\ast }f\left( 0\right) +\text{ }^{\ast }f^{\text{ }\prime }\left(
0\right) +...+\text{ }^{\ast }f^{\text{ }\left( \mathbf{m}\right) }\left(
0\right) \right) \dsum\limits_{k=0}^{n}\left( ^{\ast }b_{k}^{\ast
}e^{k}\right) \\
\\
=-\dsum\limits_{k=0}^{n}\left( ^{\ast }b_{k}\right) \left( ^{\ast }f\left(
k\right) +\text{ }^{\ast }f^{\text{ }\prime }\left( k\right) +...+\text{ }%
^{\ast }f^{\text{ }\left( \mathbf{m}\right) }\left( k\right) \right) , \\
\\
\mathbf{m\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{ \ }\left( 3.2.51\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\dsum\limits_{k=0}^{n}\left( ^{\ast }b_{k}\left( ^{\ast }e^{k}\right)
\right) =\dsum\limits_{k=0}^{n}\left( ^{\ast }b_{k}\right) \left( \dfrac{%
\Delta _{k}}{\Delta _{0}}-\dfrac{\gamma _{k}}{\Delta _{0}}\right) , \\
\\
\gamma _{k}=\left( ^{\ast }b_{k}\left( ^{\ast }e^{k}\right) \right) \times
\text{ }^{\ast }\left( \dint\limits_{0}^{k}f\left( x\right) e^{-x}dx\right) ,
\\
\\
\Delta _{0}=\left( ^{\ast }f\left( 0\right) +\text{ }^{\ast }f^{\text{ }%
\prime }\left( 0\right) +...+\text{ }^{\ast }f^{\text{ }\left( \mathbf{m}%
\right) }\left( 0\right) \right) , \\
\\
\Delta _{k}=\left( ^{\ast }f\left( k\right) +\text{ }^{\ast }f^{\text{ }%
\prime }\left( k\right) +...+\text{ }^{\ast }f^{\text{ }\left( \mathbf{m}%
\right) }\left( k\right) \right) . \\
\end{array}
& \text{ }\left( 3.2.52\right)
\end{array}%
From Eq.(3.2.52) one obtain
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\dsum\limits_{k=0}^{n}\left( ^{\ast }b_{k}\right) ^{\#}\times \left( ^{\ast
}e^{k}\right) ^{\#}=\dsum\limits_{k=0}^{n}\left( ^{\ast }b_{k}\right)
^{\#}\times \left( \dfrac{\Delta _{k}^{\#}}{\Delta _{0}^{\#}}-\dfrac{\gamma
_{k}^{\#}}{\Delta _{0}^{\#}}\right) , \\
\\
\gamma _{k}^{\#}=\left( \left( ^{\ast }b_{k}\left( ^{\ast }e^{k}\right)
\right) \times \text{ }^{\ast }\left( \dint\limits_{0}^{k}f\left( x\right)
e^{-x}dx\right) \right) ^{\#}, \\
\\
\Delta _{0}^{\#}=\left( ^{\ast }f\left( 0\right) +\text{ }^{\ast }f^{\text{ }%
\prime }\left( 0\right) +...+\text{ }^{\ast }f^{\text{ }\left( \mathbf{m}%
\right) }\left( 0\right) \right) ^{\#}, \\
\\
\Delta _{k}^{\#}=\left( ^{\ast }f\left( k\right) +\text{ }^{\ast }f^{\text{ }%
\prime }\left( k\right) +...+\text{ }^{\ast }f^{\text{ }\left( \mathbf{m}%
\right) }\left( k\right) \right) ^{\#}. \\
\end{array}
& \text{ \ }\left( 3.2.53\right)
\end{array}%
2. We will choose $f\left( x\right) $ of the form
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
f\left( x\right) =\dfrac{1}{\left( P-1\right) !}x^{P-1}\cdot \left(
x-1\right) ^{P}\cdot \left( x-2\right) ^{P}\cdot ...\cdot \left( x-n\right)
^{P} \\
\end{array}
& \text{ \ }\left( 3.2.54\right)
\end{array}%
where $P\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ is a prime number. Note that for $0\leq x\leq n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ we have$\bigskip \ \ \ \ \ $
$\bigskip $
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\left\vert f\left( x\right) \right\vert \leq \dfrac{n^{\left( n+1\right)
\cdot P}}{\left( P-1\right) !}=\dfrac{\left[ A\left( n\right) \right] ^{P}}{%
\left( P-1\right) !}, \\
\\
A\left( n\right) =n^{n+1}. \\
\end{array}
& \text{ \ \ }\left( 3.2.55\right)%
\end{array}%
By using transfer from Eq.(3.2.54)-Eq.(3.2.55) one obtain
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }\left( \text{ }f\left( x\right) \right) =\dfrac{1}{\left( \mathbf{P}%
-1\right) !}x^{\mathbf{P}-1}\cdot \left( x-1\right) ^{\mathbf{P}}\cdot
\left( x-2\right) ^{\mathbf{P}}\cdot ...\cdot \left( x-\mathbf{n}\right) ^{%
\mathbf{P}}, \\
\\
\left\vert ^{\ast }\left( \text{ }f\left( x\right) \right) \right\vert \leq
\dfrac{\mathbf{n}^{\left( \mathbf{n}+1\right) \cdot \mathbf{P}}}{\left(
\mathbf{P}-1\right) !}=\dfrac{\left[ A\left( \mathbf{n}\right) \right] ^{%
\mathbf{P}}}{\left( \mathbf{P}-1\right) !},A\left( \mathbf{n}\right) =%
\mathbf{n}^{\mathbf{n}+1} \\
\\
\mathbf{P,n\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }. \\
\end{array}
& \text{ \ }\left( 3.2.56\right)
\end{array}%
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\overline{\overline{\#\text{-}\dsum\limits_{k=0}^{\infty }\left\vert \left(
\left( ^{\ast }b_{k}\right) ^{\#}\times \left( ^{\ast }e^{k}\right)
^{\#}\right) \times \text{ }\left[ ^{\ast }\left(
\dint\limits_{0}^{k}f\left( x\right) e^{-x}dx\right) \right]
^{\#}\right\vert }}\leq \\
\\
\leq \left( \overline{\overline{\#\text{-}\dsum\limits_{k=0}^{\infty
}\left\vert \left( ^{\ast }b_{k}\right) ^{\#}\right\vert \times \left(
^{\ast }e^{k}\right) ^{\#}}}\right) \times \dfrac{\left( \left[ A\left(
\mathbf{n}\right) \right] ^{\mathbf{P}}\right) ^{\#}}{\left[ \left( \mathbf{P%
}-1\right) !\right] ^{\#}}\leq \\
\\
\leq \dfrac{\Delta _{\mathbf{d}}\mathbf{\times }\left( \left[ A\left(
\mathbf{n}\right) \right] ^{\mathbf{P}}\right) ^{\#}}{\left[ \left( \mathbf{P%
}-1\right) !\right] ^{\#}}, \\
\end{array}
& \text{ \ }\left( 3.2.57\right)%
\end{array}%
It is easy to see that for $\mathbf{P\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }.$ large enough, this is less than $\epsilon ^{\#}$
for a given $\epsilon \approx 0,\epsilon \in $ $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
Thus for $\mathbf{P\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }$ large enough by using Eq.(3.2.53) one obtain
\begin{array}{cc}
\begin{array}{c}
\\
\overline{\overline{\#\text{-}\dsum\limits_{k=0}^{\infty }\left( ^{\ast
}b_{k}\right) ^{\#}\times \left( ^{\ast }e^{k}\right) ^{\#}}}= \\
\\
\left( ^{\ast }b_{0}\right) ^{\#}+\overline{\overline{\#\text{-}%
\dsum\limits_{k=1}^{\infty }\left( ^{\ast }b_{k}\right) ^{\#}\times \left(
\dfrac{\Delta _{k}^{\#}}{\Delta _{0}^{\#}}-\dfrac{\gamma _{k}^{\#}}{\Delta
_{0}^{\#}}\right) }}, \\
\\
\overline{\overline{\#-\dsum\limits_{k=0}^{\infty }\left\vert \gamma
_{k}^{\#}\right\vert }}<\epsilon ^{\#},\epsilon \approx 0, \\
\\
\Delta _{0}^{\#}=\left( ^{\ast }f\left( 0\right) +\text{ }^{\ast }f^{\text{ }%
\prime }\left( 0\right) +...+\text{ }^{\ast }f^{\text{ }\left( \mathbf{m}%
\right) }\left( 0\right) \right) ^{\#}, \\
\\
\Delta _{k}^{\#}=\left( ^{\ast }f\left( k\right) +\text{ }^{\ast }f^{\text{ }%
\prime }\left( k\right) +...+\text{ }^{\ast }f^{\text{ }\left( \mathbf{m}%
\right) }\left( k\right) \right) ^{\#}. \\
\end{array}
& \text{ }\left( 3.2.58\right)%
\end{array}%
$\ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \overline{\overline{\#\text{-}\dsum\limits_{k=0}^{\infty }\left(
^{\ast }b_{k}\right) ^{\#}\times \left( ^{\ast }e^{k}\right) ^{\#}}}\right]
_{\varepsilon }= \\
\\
\left( ^{\ast }b_{0}\right) ^{\#}+\left[ \overline{\overline{\#\text{-}%
\dsum\limits_{k=1}^{\infty }\left( ^{\ast }b_{k}\right) ^{\#}\times \left(
\dfrac{\Delta _{k}^{\#}}{\Delta _{0}^{\#}}-\dfrac{\gamma _{k}^{\#}}{\Delta
_{0}^{\#}}\right) }}\right] _{\varepsilon }, \\
\\
\overline{\overline{\#-\dsum\limits_{k=0}^{\infty }\left\vert \gamma
_{k}^{\#}\right\vert }}<\epsilon ^{\#},\epsilon \approx 0, \\
\end{array}
& \text{\ }\left( 3.2.59\right)%
\end{array}%
By using Theorem 1. and Eq.(3.2.48) one obtain
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\overline{\overline{\#\text{-}\dsum\limits_{k=0}^{\infty }\left( ^{\ast
}b_{k}\right) ^{\#}\times \left( ^{\ast }e^{k}\right) ^{\#}}}=\varepsilon _{%
\mathbf{d}}. \\
\end{array}
& \text{ \ }\left( 3.2.60\right)%
\end{array}%
By using Theorem 1.3.4 and Eq.(3.2.60) one obtain
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \overline{\overline{\#\text{-}\dsum\limits_{k=0}^{\infty }\left(
^{\ast }b_{k}\right) ^{\#}\times \left( ^{\ast }e^{k}\right) ^{\#}}}\right]
_{\varepsilon }=\left[ \varepsilon _{\mathbf{d}}\right] _{\varepsilon }. \\
\end{array}
& \text{ \ }\left( 3.2.61\right)%
\end{array}%
By using Eq.(3.2.59) and Eq.(3.2.60) one obtain
\begin{array}{cc}
\begin{array}{c}
\\
\left[ \varepsilon _{\mathbf{d}}\right] _{\varepsilon }=\left( ^{\ast
}b_{0}\right) ^{\#}+\left[ \overline{\overline{\#\text{-}\dsum%
\limits_{k=1}^{\infty }\left( ^{\ast }b_{k}\right) ^{\#}\times \left( \dfrac{%
\Delta _{k}^{\#}}{\Delta _{0}^{\#}}-\dfrac{\gamma _{k}^{\#}}{\Delta _{0}^{\#}%
}\right) }}\right] _{\varepsilon }, \\
\\
\overline{\overline{\#-\dsum\limits_{k=0}^{\infty }\left\vert \gamma
_{k}^{\#}\right\vert }}<\epsilon ^{\#}\left( \varepsilon \right) , \\
\\
\epsilon ,\varepsilon \approx 0. \\
\end{array}
& \text{ }\left( 3.2.62\right)%
\end{array}%
\ \ \ \ \ \ \ \ \ $
\begin{array}{cc}
\begin{array}{c}
\\
\Delta _{0}^{\#}\times \left[ \varepsilon _{\mathbf{d}}\right] _{\varepsilon
}=\Delta _{0}^{\#}\times \left( ^{\ast }b_{0}\right) ^{\#}+ \\
\\
\left[ \overline{\overline{\#\text{-}\dsum\limits_{k=1}^{\infty }\left(
^{\ast }b_{k}\right) ^{\#}\times \left( \Delta _{k}^{\#}-\gamma
_{k}^{\#}\right) }}\left\vert \Delta _{0}^{\#}\times \left( ^{\ast }c\right)
^{\#}\right. \right] _{\varepsilon }, \\
\\
%TCIMACRO{\U{211d} }%
\mathbb{R}
. \\
\end{array}
& \text{ \ }\left( 3.2.63\right)%
\end{array}%
Multiplying Eq.(3.2.63) by number$\ \ \Im ^{\#},$ where
$\Im =\left( m_{0},m_{0}\times m_{1},...,m_{0}\times m_{1}\times ...\times
m_{n},...\right) \ \ $gives
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\Im ^{\#}\times \Delta _{0}^{\#}\times \varepsilon ^{\#}\times \varepsilon _{%
\mathbf{d}}= \\
\\
=\Im ^{\#}\times \Delta _{0}^{\#}\times \left( ^{\ast }b_{0}\right) ^{\#}+
\\
\\
+\Im ^{\#}\times \left[ \overline{\overline{\#\text{-}\dsum\limits_{k=1}^{n}%
\left( ^{\ast }b_{k}\right) ^{\#}\times \left( \Delta _{k}^{\#}-\gamma
_{k}^{\#}\right) }}\left\vert \Delta _{0}^{\#}\times \left( ^{\ast }c\right)
^{\#}\right. \right] _{\varepsilon }, \\
\end{array}
& \text{ \ }\left( 3.2.64\right)%
\end{array}%
$\bigskip $thus
\begin{array}{cc}
\begin{array}{c}
\\
\Im ^{\#}\times \Delta _{0}^{\#}\times \varepsilon ^{\#}\times \varepsilon _{%
\mathbf{d}}= \\
\\
=\Im ^{\#}\times \Delta _{0}^{\#}\times \left( ^{\ast }b_{0}\right) ^{\#}+
\\
\\
\left[ \overline{\overline{\#\text{-}\dsum\limits_{k=1}^{n}\left( \Im
_{k}^{\#}\times \Delta _{k}^{\#}-\Im ^{\#}\gamma _{k}^{\#}\right) }}%
\left\vert \Im ^{\#}\times \Delta _{0}^{\#}\times \left( ^{\ast }c\right)
^{\#}\right. \right] _{\varepsilon }= \\
\\
=\Im ^{\#}\times \Delta _{0}^{\#}\times \left( ^{\ast }b_{0}\right) ^{\#}+
\\
\\
\left[ \overline{\overline{\#\text{-}\dsum\limits_{k=1}^{n}\left( \Im
_{k}^{\#}\times \Delta _{k}^{\#}-\Im ^{\#}\times \gamma _{k}^{\#}\right) }}%
\left\vert \Im ^{\#}\times \Delta _{0}^{\#}\times \left( ^{\ast }c\right)
^{\#}\right. \right] _{\varepsilon }, \\
\\
\Im ^{\#}\times \left( \overline{\overline{\#-\dsum\limits_{k=0}^{\infty
}\left\vert \gamma _{k}^{\#}\right\vert }}\right) <\epsilon ^{\#}\left(
\varepsilon \right) <\varepsilon ^{\#},\varepsilon \approx 0, \\
\\
\Im _{0}^{\#}=\Im ^{\#}\times \left( ^{\ast }b_{0}\right) ^{\#},\Im
_{k}^{\#}=\Im ^{\#}\times \left( ^{\ast }b_{k}\right) ^{\#},k=1,2,... \\
\end{array}
& \text{ \ \ }\left( 3.2.65\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ $
3. If
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
h\left( x\right) =\dfrac{g\left( x\right) \left( x-a\right) ^{\mathbf{P}}}{%
\mathbf{P}!} \\
\\
\mathbf{P\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }, \\
\end{array}
& \text{ \ \ \ }\left( 3.2.66\right)%
\end{array}%
where $g\left( x\right) $ is any hyper polynomial with hyper integer
and $a\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ then the derivatives $h^{\left( j\right) }\left( a\right) =0$ for $0\leq j<%
\mathbf{P}$ and in
general $h^{\left( j\right) }\left( a\right) \in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ for all $j\geq 0.$ Since $f\left( x\right) /\mathbf{P}$ has this form with
$a\in \left\{ 1,2,...,\mathbf{n}\right\} $ it follows that $f^{\text{ }%
\left( j\right) }\left( k\right) $ is an integer and is divisible
by $\mathbf{P},$ for all $j\geq 0$ and for $k\in \left\{ 1,2,...,\mathbf{n}%
\right\} .$ Thus $\mathbf{P}$ divides all terms
on the RHS of Eq.(3.2.65) having $k\neq 0.$
4. It remains to consider the terms with $k=0.$Note that $^{\ast
}f\left( x\right) $ has the form
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
\begin{array}{cc}
\begin{array}{c}
\\
^{\ast }f\left( x\right) =\dsum\limits_{\mathbf{j}=\mathbf{P}-1}^{\mathbf{m}}%
\dfrac{c_{\mathbf{j}}x^{\mathbf{j}}}{\left( \mathbf{P}-1\right) !} \\
\end{array}
& \text{ \ }\left( 3.2.67\right)%
\end{array}%
$\ $
where $c_{\mathbf{P}-1}=\left( \pm \text{ }\mathbf{n}!\right) ^{\mathbf{P}}$
and $c_{\mathbf{j}}\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
$ for all $\mathbf{j}\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
.$Then $^{\ast }f^{\text{ }\left( \mathbf{j}\right) }\left( 0\right) =0$
for $\mathbf{j}<\mathbf{P}-1,$ $^{\ast }f^{\text{ }\left( \mathbf{P}%
-1\right) }\left( 0\right) =c_{\mathbf{P}-1}$ and $^{\ast }f^{\text{ }\left(
\mathbf{j}\right) }\left( 0\right) =c_{\mathbf{j}}\cdot \mathbf{j}!/\left(
\mathbf{P}-1\right) !$ for $\mathbf{j}\geq \mathbf{P}$ so
$\mathbf{P}$ divides $^{\ast }f^{\text{ }\left( \mathbf{j}\right) }\left(
0\right) $ if $\mathbf{j}\neq \mathbf{P}-1.$
5. The only term remaining on the RHS of Eq.(3.2.65) is
$\ \
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \Im ^{\#}\times \Delta _{0}^{\#}\times \left( ^{\ast }b_{0}\right)
^{\#}\times \left( ^{\ast }f^{\text{ }\left( \mathbf{P}-1\right) }\left(
0\right) \right) ^{\#} \\
\\
=\left( \left( \pm \text{ }\mathbf{n}!\right) ^{\mathbf{P}}\right) ^{\#}. \\
\end{array}
& \text{ }\left( 3.2.68\right)%
\end{array}%
This term is not divisible by $\mathbf{P}^{\#}\mathbf{\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty ,\mathbf{d}}$ if $\mathbf{P\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
}_{\infty }$ is prime with
$\mathbf{P}>|^{\ast }b_{0}|\times \mathbf{n}.$Thus, we may choose $\mathbf{P}
$ so that
$\Im ^{\#}\times \left( \overline{\overline{\#-\dsum\limits_{k=0}^{\infty
}\left\vert \gamma _{k}^{\#}\right\vert }}\right) <\varepsilon ^{\#}$ and so
that in the RHS of Eq.(3.2.65),
$\mathbf{P}$ divides every term
\begin{array}{cc}
\begin{array}{c}
\\
\ \ \Im ^{\#}\times \Delta _{0}^{\#}\times \left( ^{\#}b_{k}\right) \times
\left( ^{\ast }f^{\text{ }\left( \mathbf{j}\right) }\left( k\right) \right)
^{\#},\mathbf{j\in }^{\ast }\mathbf{%
%TCIMACRO{\U{2115} }%
\mathbb{N}
} \\
\end{array}
& \text{ \ }\left( 3.2.69\right)%
\end{array}%
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
except for $\ \Im ^{\#}\times \Delta _{0}^{\#}\times \left(
^{\#}b_{0}\right) \times \left( ^{\ast }f^{\text{ }\left( \mathbf{P}%
-1\right) }\left( 0\right) \right) ^{\#}.$ Therefore the RHS has
representation $\Gamma ^{\#}+\Im ^{\#}\times \Delta _{0}^{\#}\times
\varepsilon ^{\#}\times \varepsilon _{\mathbf{d}}$ such that $\Gamma \in $ $%
^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,\ \Gamma \geq 1.$
Thus one obtain
\begin{array}{cc}
\begin{array}{c}
\\
\Im ^{\#}\times \Delta _{0}^{\#}\times \varepsilon ^{\#}\times \varepsilon _{%
\mathbf{d}}=\Gamma ^{\#}+\Im ^{\#}\times \Delta _{0}^{\#}\times \varepsilon
^{\#}\times \varepsilon _{\mathbf{d}} \\
\end{array}
& \text{ }\left( 3.2.70\right)%
\end{array}%
This is a contradiction.This contradiction proves that $e$ is not
$w$-transcendental. Hence $e$ is $\#$-transcendental.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
§ III.3.NONSTANDARD GENERALIZATION OF THE LINDEMAN THEOREM.
Theorem 3.3.1.(Nonstandard Lindeman Theorem)The number $^{\ast }e$
cannot satisfy an equation of the next form:
\begin{array}{cc}
\begin{array}{c}
\\
a_{1}\cdot \left( ^{\ast }e\right) ^{\alpha _{1}}+a_{2}\cdot \left( ^{\ast
}e\right) ^{\alpha _{2}}+...+a_{N}\cdot \left( ^{\ast }e\right) ^{\alpha
_{N}}\approx 0, \\
\end{array}
& \text{ \ }\left( 3.3.1\right)%
\end{array}%
in which at least one coefficient $a_{n},n=1,2,...,N\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ is different from
zero, no two exponents $\alpha _{n},n=1,2,...,N\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ are equal, and all numbers
$\alpha _{n},n=1,2,...,N\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ are hyperalgebraic.
Proposition 3.3.1.Let $\rho _{1},\rho _{2},...,\rho _{m},$ $m\in $ $%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ be the roots of the
hyperpolynomial equation $a\cdot z^{m}+b\cdot z^{m-1}+c\cdot z^{m-2}+...=0$
with integral
coefficients $a,b,c,...\in $ $^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
.$ Then any symmetric hyperpolynomial in the
quantities $a\cdot \rho _{1},a\cdot \rho _{2},...,a\cdot \rho _{m}$ with
integral coefficients, is an hyperinteger.
Proposition 3.3.2.Suppose given a hyperpolynomial in $m\in $ $%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ variables $\alpha _{i_{1}},$
in $n\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ variables $\beta _{i_{2}},...,$and in $k\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ variables $\sigma _{i_{l}},l\in $ $^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ which is
symmetric in the $\alpha $'s in the $\beta $'s,...,and in the $\sigma $'s,
and which has hyperrational
coefficients. If the $\alpha $'s are chosen to be all the roots of a
equation with rational coefficients, and similarly for the $\beta $'s, , and
for the $\sigma $'s
then the value of the polynomial is hyperrational.
Definition 3.3.1.A hyperpolynomial is said to be irreducible over
the rationale
if it cannot be factored into hyperpolynomials of lower degree with
Definition 3.3.2.If $\alpha _{1}$ is a root of an irreducible
hyperpolynomial equation with
hyperrational coefficients, whose other roots are $\alpha _{2},\alpha
_{3},,...,\alpha _{n},n\in ^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ then the
hyperalgebraic numbers $\alpha _{1},\alpha _{2},\alpha _{3},,...,\alpha _{n}$
are said to be the conjugates of $\alpha _{1}.$
Proposition 3.3.3.Any hyperpolynomial with hyperrational
coefficients can be
factored into irreducible polynomials with hyperrational coefficients.
Proposition 3.3.4.Over the field $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
$, an hyperalgebraic number is a root of a
unique irreducible hyperpolynomial with hyperrational coefficients and
coefficient unity. Such an equation has no multiple roots.
Proposition 3.3.5.The Van der Monde determinant $\det \left\vert
\left( \rho _{k}\right) ^{i-1}\right\vert $ vanishes only
if two or more of the $\rho $'s are equal.
§ III.4. THE NUMBERS $E$ AND $\PROTECT\PI $ ARE ANALYTICALLY
First of all, recall that is an entire function,in $2$ variables, with
coefficients in field $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$is a function $f(z_{1},z_{2})$ which is analytic in $G\subseteqq $ $%
%TCIMACRO{\U{2102} }%
\mathbb{C}
\times
%TCIMACRO{\U{2102} }%
\mathbb{C}
$\ \ $
$\ \ \ \ \ \
\begin{array}{cc}
\begin{array}{c}
\\
f(z_{1},z_{2})=\sum_{i=0}^{\infty }\sum_{j=0}^{\infty
}c_{i,j}z_{1}^{i}z_{2}^{j}, \\
\\
c_{i,j}\in \text{ }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
. \\
\end{array}
& \text{ \ \ \ \ }\left( 3.4.1\right)%
\end{array}%
Definition 3.3.1.Two complex numbers $\alpha \in $ $%
%TCIMACRO{\U{2102} }%
\mathbb{C}
$ and $\beta \in $ $%
%TCIMACRO{\U{2102} }%
\mathbb{C}
$ are said to be
analytically dependent if there is a nonzero entire function $f(z_{1},z_{2})$
in $2$
variables, with hyperinteger coefficients $c_{i,j}\in $ $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,$ such that $f(\alpha ,\beta )=0.$
Otherwise, $\alpha \in $ $%
%TCIMACRO{\U{2102} }%
\mathbb{C}
$ and $\beta \in $ $%
%TCIMACRO{\U{2102} }%
\mathbb{C}
$ are said to be analytically independent.
§ APPENDIX A. HYPER ALGEBRAIC NUMBERS.
§ 1.DEFINITIONS OF SYMMETRIC POLYNOMIALS
AND SYMMETRIC FUNCTIONALS.
Consider a monic polynomial $P\left( z\right) $ in $z\in
%TCIMACRO{\U{2102} }%
\mathbb{C}
$ of degree $n\in
%TCIMACRO{\U{2115} }%
\mathbb{N}
\begin{array}{cc}
\begin{array}{c}
\\
P\left( z\right) =1+a_{1}z+a_{2}z^{2}+...+a_{n-1}z^{n-1}+a_{n}z^{n} \\
\end{array}
& \text{ \ \ }\left( A.1.1\right)%
\end{array}%
There exist $n$ roots $z_{1},\ldots ,z_{n}$ of $P$ and that one is expressed
by the relation
\begin{array}{cc}
\begin{array}{c}
\\
P\left( z\right) =1+a_{1}z+a_{2}z^{2}+...+a_{n-1}z^{n-1}+a_{n}z^{n}= \\
\\
=\left( 1-\dfrac{z}{z_{1}}\right) \left( 1-\dfrac{z}{z_{2}}\right) \cdot
\cdot \cdot \left( 1-\dfrac{z}{z_{n}}\right) = \\
\\
z_{1}^{-1}\cdot z_{2}^{-1}\cdot \cdot \cdot z_{n}^{-1}\left( z_{1}-z\right)
\cdot \left( z_{2}-z\right) \cdot \cdot \cdot \left( z_{n}-z\right) = \\
\\
=\left( -1\right) ^{n}z_{1}^{-1}\cdot z_{2}^{-1}\cdot \cdot \cdot
z_{n}^{-1}\left( z-z_{1}\right) \cdot \left( z-z_{2}\right) \cdot \cdot
\cdot \left( z-z_{n}\right) \\
\\
\hat{P}\left( z\right) =P\left( z\right) /\left( -1\right)
^{n}z_{1}^{-1}\cdot z_{2}^{-1}\cdot \cdot \cdot z_{n}^{-1}= \\
\\
\hat{a}_{0} \\
\end{array}
& \text{ \ }\left( A.1.2\right)
\end{array}%
Thus by comparison of the coefficients one finds
$\ \ \
\begin{array}{cc}
\begin{array}{c}
\\
a_{1}=-\sum_{1\leq i\leq n}\dfrac{1}{z_{i}}, \\
\\
a_{2}=\sum_{1\leq i<j\leq n}\dfrac{1}{z_{i}z_{j}}, \\
\\
\cdot \cdot \cdot \cdot \cdot \\
\\
a_{m}= \\
\\
\cdot \cdot \cdot \cdot \cdot \\
\\
a_{n-1}=\left( -1\right) ^{n-1}\sum_{1\leq i_{1}<i_{2}<...<i_{n-1}\leq
n}\prod_{i\neq j}\dfrac{1}{z_{j}} \\
\\
a_{n}=\left( -1\right) ^{n}\prod_{1\leq i\leq n}\dfrac{1}{z_{i}}. \\
\end{array}
& \text{ \ }\left( A.1.3\right)
\end{array}%
Definition. Let us defined $n$ polynomials expressed by the
$\bigskip $
$\ $
\begin{array}{cc}
\begin{array}{c}
\\
e_{1}(z_{1},...,z_{n})=\sum_{1\leq i\leq n}\dfrac{1}{z_{i}}=-a_{1}, \\
\\
e_{2}(z_{1},...,z_{n})=\sum_{1\leq i<j\leq n}\dfrac{1}{z_{i}z_{j}}=a_{2}, \\
\\
\cdot \cdot \cdot \cdot \cdot \\
\\
e_{m}(z_{1},...,z_{n})= \\
\\
\cdot \cdot \cdot \cdot \cdot \\
\\
e_{n-1}(z_{1},...,z_{n})=a_{n-1}= \\
\\
e_{n}(z_{1},...,z_{n})=a_{n}=\left( -1\right) ^{n}\prod_{1\leq i\leq n}%
\dfrac{1}{z_{i}}. \\
\end{array}
& \text{ \ \ \ }\left( A.1.4\right)%
\end{array}%
The polynomial $e_{m}(z_{1},...,z_{n})$ is called the $m$-th symmetric
It has the following property:$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\bigskip $
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ $
§ 2.NONSTANDARD POLYNOMIALS.
The set of natural,integer,rational, real, complex or any algebraic numbers
is denoted by $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
%TCIMACRO{\U{2124} }%
\mathbb{Z}
%TCIMACRO{\U{211a} }%
\mathbb{Q}
%TCIMACRO{\U{211d} }%
\mathbb{R}
%TCIMACRO{\U{2102} }%
\mathbb{C}
,\Bbbk $ respectively, and their nonstandard extensions
$^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
,^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
,^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
,^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
,^{\ast }%
%TCIMACRO{\U{2102} }%
\mathbb{C}
,^{\ast }\Bbbk .$
Definition A.2.1.(Nonstandard polynomials) Nonstandard
polynomial of
hyper degree $\mathbf{d\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ in $x$ with coefficients in nonstandard field $^{\ast }\Bbbk $
is an
expression defined by internal hyper finite sum of the form
\begin{array}{cc}
\begin{array}{c}
\\
f\left( x\right) =\sum_{j=0}^{\mathbf{d}}a_{j}x^{j}=a_{0}+a_{1}x+...+a_{%
\mathbf{d-}1}x^{\mathbf{d-}1} \\
\\
+a_{\mathbf{d}}x^{\mathbf{d}}\in \text{ }^{\ast }\Bbbk \left[ x\right] , \\
\\
\text{ \ }\forall j\left[ a_{j}\in \text{ }^{\ast }\Bbbk \right] . \\
\end{array}
& \text{ \ }\left( A.2.1\right)
\end{array}%
Definition A.2.2. (Algebraic hyper integers) If $\alpha
\in $ $^{\ast }%
%TCIMACRO{\U{2102} }%
\mathbb{C}
$ is a root of a
monic nonstandard polynomial of hyper degree $\mathbf{d\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty },$namely a root of
a polynomial of the form
\begin{array}{cc}
\begin{array}{c}
\\
f\left( x\right) =\sum_{j=0}^{\mathbf{d}}a_{j}x^{j}=a_{0}+a_{1}x+...+a_{%
\mathbf{d-}1}x^{\mathbf{d-}1} \\
\\
+a_{\mathbf{d}}x^{\mathbf{d}}\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\left[ x\right] , \\
\\
\text{\ }\forall j\left[ a_{j}\in \text{ }^{\ast }%
%TCIMACRO{\U{2124} }%
\mathbb{Z}
\right] \\
\end{array}
& \text{ \ \ \ \ \ }\left( A.2.2\right)
\end{array}%
and $\alpha $ is not the root of such a polynomial of hyper degree less then
then $\alpha $ is colled an algebraic hyper integer of hyper degree
$\mathbf{d\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }.$
Definition A.2.3. (Nonstandard algebraic numbers) An
algebraic number $\alpha $ of hyper degree $\mathbf{d\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ is a root of a monic
nonstandard polynomial of hyper degree $\mathbf{d\in }^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty },$ and not be the root of
an the nonstandard polynomial of hyper degree $\mathbf{d}_{1}\mathbf{\in }%
^{\ast }%
%TCIMACRO{\U{2115} }%
\mathbb{N}
_{\infty }$ less then $\mathbf{d.}$
Remark. We have to mach examles standard real numbers that are not
standard algebraic numbers, such as $\ln 2$ and $\pi .$These are
examples of
standard transcendental numbers, which are not standard
We will establish that every hyper finite extension of $^{\ast }%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
Definition A.2.4. (Simple hyper finite extentions
and nonstandard
polynomials) If $\alpha \in $ $^{\ast }E$ an hyper finite extention
field of a given
nonstandard field $^{\ast }F,$then $\alpha $ is colled hyper
algebraic over $^{\ast }F$ if $f\left( \alpha \right) =0$
for some nonzero $f\left( x\right) \in F\left[ x\right] .$If $\alpha $
§ REFERENCES
[1] Goldblatt,R.,Lectures on the Hyperreals. Springer-Verlag, New
NY, 1998
[2] Henle,J. and Kleinberg, E., Infinitesimal Calculus. Dover
Publications, Mineola,NY, 2003.
[3] Loeb, P. and Wolff, M.,Nonstandard Aalysis for the Working
Mathematician.Kluwer Academic Publishers, Dordrecht,
The Netherlands, 2000.
[4] Roberts, A. M.,Nonstandard Analysis. Dover Publications,
Mineola, NY, 2003.
[5] Robinson, A.,Non-Standard Analysis (Rev. Ed.). Princeton
University Press, Princeton, NJ,
[6] Euler L.,Variae observationes circa series infinitas. 1737. 29p.
[7] Viader P.,Bibiloni L., Jaume P. On a series of Goldbach and
[8] Edward C., (Author) How Euler Did it. Hardcover - Jul 3,2007.
[9] Metsnkyl T.,Catalan's conjecture: another old Diophantine
problem solved, Bull. Amer. Math. Soc. (N.S.) 41 (2004),
no. 1, 43–57.
[10] Goldblatt,R.,Topoi 2-ed.,NH,1984.
[11] Nelson E. Internal set theory: a new approach to nonstandard
analysis. Bull. Amer. Math. Soc, 1977.
[12] Robinson A., Zakon E., Applications of Model Theory to Algebra,
Analysis, and Probability.C.I.T.,Holt, Rinehart and
p.109 -122,1967.
[13] Hrbc̆ek K. Axiomatic foundations for nonstandard analysis,
Fundamenta Mathematicae, vol. 98 (1978),pp.1-19.
[14] Jin R.,The sumset phenomenon.Proceedings of the AMS Volume
130,Number 3, Pages 855-861.
[15] Foukzon J., Nonstandard analysis and structural theorems of a
general nonlocallycompact Hausdorf Abelian group.
International Workshop on Topological Groups.Pamplona,
August 31st - September 2nd.
[16] Foukzon J., A definition of topological invariants for wild knots and
links by using non standard internal S. Albeverio
The 22-nd Annual Geometric Topology Workshop.Colorado June
9th-11th, 2005.
[17] Foukzon J., Generalized Pontryagian's duality theorem.
2006 International Conference on Topology and its
Applications,June 23-26, 2006, Aegion,Greece.
[18] Albeverio S., Fenstad J.E., Hoegh-Krohn R., Lindstrem T.
Nonstandard methods in stochastic analysis and mathematical
physics. Academic Press, Inc.1986, 590p.
[19] Kusraev A. G., Kutateladze S. S. Nonstandard Methods of Analysis.
Novosibirsk: Nauka, 1990; Dordrecht: Kluwer,1995.
[20] Laczkovich M. Conjecture and proof. 2001.
[21] Patterson E. M.The Jacobson radical of a pseudo-ring.
Math. Zeitschr. 89, 348–364 (1965).
[22] Nesterenko,Y.V., Philippon.Introduction to Algebraic Independence
Theory. Series: Lecture Notes in Mathematics,Vol.1752
(Eds.) 2001, XIII, 256 pp.,Softcover ISBN: 3-540-41496-7
[23] Gonshor, H., Remarks on the Dedekind completion of a nonstandard
model of the reals.Pacific J. Math. Volume 118, Number1
[24] Wattenberg, F. $\left[ 0,\infty \right] $-valued, translation
invariant measures on $%
%TCIMACRO{\U{2115} }%
\mathbb{N}
$ and
the Dedekind completion of $^{\ast }%
%TCIMACRO{\U{211d} }%
\mathbb{R}
.$Pacific J. Math. Volume 90,
Number 1 (1980), 223-247.
[25] Davis M. Applied Nonstandard Analysis.Wiley,New York,London,
Sydney, Toronto, 1977, xii + 181 pp.,
[26] Foukzon J.2006 Spring Central Sectional Meeting Notre Dame,IN,
April 8-9,2006 Meeting #1016 The solution of one very old
problem in
transcendental numbers theory. Preliminary report.
[27] Waldschmidt M., Algebraic values of analytic functions.Journal of
Computational and Applied Mathematics 160 (2003)
[28]
[29]
[30] Laubenheimer P.,Schick T.,Stuhler U. Completions of countable
non-standard models of $%
%TCIMACRO{\U{211a} }%
\mathbb{Q}
.$ http://arxiv.org/abs/math/0604466v3
[31] Robinson A.,Nonstandard Arithmetic.Bull.Amer.Math.Soc.Volume 73,
Number 6 (1967),818-843.
[32] Harold G. Dales, W. Hugh Woodin Super-real fields: totally ordered
fields with additional structure.
[33] Arkeryd Leif O. Cutland, N.J. Henson C. Nonstandard analysis: theory
and applications.Ward (Eds.) 1997, 384 p., Hardcover
ISBN: 978-0-7923-4586-2
[34] Gonshor H., The ring of finite elements in a non-standard model of the
reals, J.London Math. Soc, (2) 3 (1971), 493-500.
[35] Cartier P., Functions polylogarithmes, nombres polyzëta et groupes
prounipotents, Sém. Bourbaki, 53$^{\acute{e}me}$ anné
e, 2000–2001, no 884,
Mars 2001, 36 pp.
[36] Apéry R., Irrationalité de $\zeta (2)$ et $\zeta (3),$ Ast
érisque 61 (1979) 11–13.
[37] $\ $Balog,A. Perelli,A.Diophantine approximation by square-free
Annali della Scuola Normale Superiore di Pisa - Classe di
Sér.4, 11 no. 3 (1984), p. 353-359. $\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ $
|
arxiv-papers
| 2009-07-02T19:35:27 |
2024-09-04T02:49:03.679938
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Jaykov Foukzon",
"submitter": "Jaykov Foukzon",
"url": "https://arxiv.org/abs/0907.0467"
}
|
0907.0499
|
# Agent-Oriented Approach for Detecting and Managing Risks in Emergency
Situations
Fahem Kebair and Frédéric Serin F. Kebair is PhD student in computer science
with LITIS–Laboratoire d’Informatique de Traitement de l’Information et des
Systsème, University of Le Havre, 25 rue Philippe Lebon, 76058, Le Havre,
Cedex, France, e-mail: [email protected]. Serin is professor
assistant in computer science with LITIS, e-mail: frederic.serin@univ-
lehavre.fr
###### Abstract
This paper presents an agent-oriented approach to build a decision support
system aimed at helping emergency managers to detect and to manage risks. We
stress the flexibility and the adaptivity characteristics that are crucial to
build a robust and efficient system, able to resolve complex problems. The
system should be independent as much as possible from the subject of study.
Thereby, an original approach based on a mechanism of perception,
representation, characterisation and assessment is proposed. The work
described here is applied on the RoboCupRescue application. Experimentations
and results are provided.
###### Index Terms:
Assessment agents, clusters, decision support system, factual agents.
## I Introduction
The use of Decision Support Systems (DSSs) has considerably increased, during
the last decade, due to the complexity of the problems faced by the decision
makers. Indeed, the need for decision support tools should be, if anything,
increasing [10]. In some domains or circumstances, making a decision is an
arduous task that requires some abilities exceeding the human capacities. We
can think decision-making in Simon’s decision making model, which consists in
intelligence, design and choice [11]. Based on this model, the complexity of
decision making lies in the difficulty to get a clear insight into the problem
to resolve, to process the vast amount of collected information, to make the
right choice in time and to harmonise finally the set of decisions made by the
decision makers or the organisations. Therefore, computer-based systems may be
very helpful to support decision making, especially when the environment
problem is complex, dynamic and partially known. Processing and managing
information issued from such an environment represents a challenge to the DSS
developers. However, DSS are well known to be customized for a specific
purpose and can rarely be reused. Moreover, DSSs only support circumstances
which lie in the known and knowable spaces and do not support complex
situations sufficiently [4]. This led us to think DSSs must be flexible and
adaptive to be effective in solving complex problems as the risk and crisis
management. Flexibility allows the use of the system in different subject of
studies with minor changes. In other words, the system operates in a generic
manner and relies on specific knowledge that are defined by experts of the
domain. Adaptivity is an essential characteristic to build intelligent
information systems which draws increasingly the attention of the scientists
in computer science and in artificial intelligence. Thanks to the adaptivity,
the system may adapt its behaviour autonomously by altering its internal
structure and changing its behaviour to better respond to the change of its
environment. The multiagent systems technology is an appropriate solution to
achieve these two objectives. Intelligent agents [13] are able to self-perform
actions and to interact with other agents and their environment in order to
carry out some objectives and to react to changes they perceive by adapting
their behaviours.
In this paper we propose an agent-oriented approach aimed at building a DSS
that has as role to help emergency managers to detect and to manage risks in
emergency situations. The system perceives facts occurred in the environment,
represents them and analyses them to assess the current situation. To evaluate
the situation, the system uses an analogical reasoning based on the following
postulate: if a given situation A seems like a situation B, then it is likely
that the consequences of the situation A will be similar to those of B.
Consequently, the risk appeared in B become a potential risk of A. An internal
multi-level kernel is used to insure the whole decision-support process. We
utilise an earthquake scenario using the RoboCupRescue Simulation System
(RCRSS) [7] [9] in order to illustrate our approach. Experimentations and
results are provided and discussed.
## II Decision Support System for Risk Detection and Management
### II-A Definitions and Approaches
The Risk is a concept that denotes a potential negative impact to an asset or
some characteristic of value that may arise from some present process or
future event. There are many more and less precise definitions of risk. They
do depend on specific applications and situational contexts. It can be
assessed qualitatively or quantitatively. In our context, we are interested in
natural and technological risks. The management of these risks often
represented a large-scale challenge for the individuals and the organisations,
since they are hard to predict and their occurrences are much sudden. The risk
management may be defined as the systematic application of management
policies, procedures and practices to the tasks of establishing the context,
identifying, analysing, evaluating, treating, monitoring and communicating
risk [1]. This process is complex and exceeds widely the human abilities. The
use of the DSS in this case is indispensable. Indeed, DSSs are interactive,
computer-based systems that aid users in judgement and choice activities. They
provide data storage and retrieval but enhance the traditional information
access and retrieval functions with support for model building and model-based
reasoning. They support framing, modeling, and problem solving [2]. In the
context of the risks and crisis management, the DSS must insure the following
functionalities:
* •
Evaluation of the current situation, the system must detect/recognize an
abnormal event;
* •
Evaluation/Prediction of the consequences, the system must assess the event by
identifying the possible consequences;
* •
Intervention planning, the system must help the emergency responders in
planning their interventions thanks to an actions plan (or procedures) that
must be the most appropriate to the situation.
Figure 1: Whole DSS architecture
### II-B DSS Architecture
The kernel is the main part of the DSS and has as role to manage all the
decision-support process. The environment includes essentially the actors and
Distributed Information Systems (DIS) and feeds permanently the system with
information describing the state of the current situation. In order to
apprehend and to deal with these information, specific knowledge related to
the domain as ontologies and proximity measures are required. The final goal
of the DSS is to provide an evaluation of the situation by comparing it with
past experimented situations stored as scenarios in a Scenario Base (SB).
The kernel is a MAS operating on three levels. It intends to detect
significant organisations that give a meaning to data in order to support
finally the decision making. We aim, from such a structure, to equip the
system with an adaptable and a partially generic architecture that may be
easily adjusted to new cases of studies. Moreover, its suppleness makes the
system able to operate autonomously and to change its behaviour according to
the evolution of the problem environment. As follows a description of each
level:
* •
Situation representation: One fundamental step of the system is to represent
the current situation and its evolution over time. Indeed, the system
perceives the facts that occur in the environment and creates its own
representation of the situation thanks to a factual agents organisation. This
approach has as purpose to let emerge subsets of agents.
* •
Situation assessment: A set of assessment agents are related to scenarios
stored in a SB. These agents scrutinise permanently the factual agents
organisation to find agents clusters enough close to their scenarios. This
mechanism is studied “manually” by an expert of the domain and is similar to a
Case-Based Reasoning (CBR) [8], except it is dynamic and incremental.
According to the application, one or more most pertinent scenarios are
selected to inform decision-makers about the state of the current situation
and its probable evolution, or even to generate a warning in case of detecting
a risk of crisis. The evaluation of the situation will be then reinjected in
the perception level in order to confirm the position of the system about the
current situation. This characteristic is inspired from the feedbacks of the
natural systems. In that manner, the system learns from its successes or from
its failures.
* •
Automating decisions: Outcomes generated by the assessment agents are captured
by a set of performative agents and are transformed in decisions that may be
used directly by the final users.
### II-C RoboCupRescue Case Study
The RCRSS is an agent-based simulator which intends to reenact the rescue
mission problem in real world. An earthquake scenario is reproduced including
various kinds of incidents as the traffic after earthquake, buried civilians,
road blockage, fire accidents, etc. A set of heterogeneous agents (RCR agents)
coexist in the disaster space: rescue agents that are fire brigades, ambulance
teams and police forces, and civilians agents. We focus, in this application,
on the development of the rescue agents behaviours. Our final goal is to use
the DSS in order to improve their decision-making ability and to support them
during their rescue operations.
A model of the RoboCupRescue disaster space and the properties of its
components, and the RCR agents are detailled in [12]. We use this model in
order to extract knowledge and to formalise information.
## III Dynamic Representation of the Situation: Factual Agents
The system perceives and represents the facts occurred in the situation in an
original manner using factual agents. Factual agents are reactive and
proactive agents according to the agents definition given in [13]. Each agent
carries an elementary datum that represents an observed fact and that aims to
manage it over time. This information is presented in the shape of a Factual
Semantic Feature (FSF), more details about this structure and how it is
formalised and managed by a factual agent is provided in [6].
The objective by using factual agents in the representation situation level is
to reflect the dynamic change of the situation and to let emerge, from this
view, agents subsets. These subsets may be representative of some situations
that are close to some others encountered in the past. The analysis of these
agents groups is based on geometric criteria, insuring thus the independence
of the treatment from the subject of study. Each factual agent exposes
behavioural activities that are characterised thanks to numerical indicators.
The latter form a behavioural vector that draws, by its variations, the
dynamics of the agent during its live. This gives a meaning to the state of
the agent inside its organisation and consequently to the prominence of the
semantic character that it carries.
The goal of our approach is to characterise the factual agents organisation by
forming dynamically agents clusters and comparing them with stored scenarios.
The clustering algorithms seem appropriate to this objective, since they are
able to create objects groups in an unsupervised way. However, these methods
present some deficiencies in our case. The main ones are the need to specify
some parameters as the minimal distance between two objects, required by
density-based algorithms [3]; or the minimal length of a cluster, required by
Kmeans algorithms [5]. Moreover, the experimentations we led using these
methods showed us that we are unable to analyse instantaneously the obtained
clusters neither to reproduce them. We changed therefore our way for
proceeding by confiding this task to the assessment agents. These agents will
search through the factual agents in order to form clusters, that should be
the closest to the scenarios to which they are linked. We think this approach
is more suitable for our problem, since it does not require specific knowledge
and we are certain that the obtained clusters have probably a meaning and may
be easily interpreted. In addition we may exploit the assets of the agents,
especially their adaptivity and their communication abilities.
## IV Situation Assessment
### IV-A Assessment Agents
Each assessment agent is linked to a scenario stored in the SB (see Figure2).
Each scenario is composed of one or more factual agents clusters, this depends
on the treated application. A cluster is made up of a set of elements, each
one includes an FSF, the indicators values of the factual agent associated to
this FSF and the size of its Acquaintances Network (AN). Thus, a cluster
element has the following structure: $FSF:V_{I1}\dots V_{In}:S_{AN}$, with
$V_{I}$ a value of indicator $I$, and an example of an FSF is (fire,
intensity, strong, location, $2^{nd}$ street, time, 10:00 pm).
The role of the assessment agents is to scrutinise permanently the
organisation of the factual agents in order to extract agents clusters that
should be similar as much as possible to their scenarios. A relevance, which
is the sum average of all the similarities values of a created cluster
elements, is attributed to each cluster to indicate its proximity to a stored
scenario. This value is included in a range of [0,1]. The more the relevance
is near to 1, the more the cluster is close to its scenario maker and vice
versa. The clusters, and consequently the assessment agents, are sorted
according to their relevances and the selected agents depend on their rank and
the size of their clusters .i.e. the first agents covering the bulk of the
situation are selected.
*
Figure 2: Role of the assessment agents in the DSS
To find close elements in the factual agents organisation, the assessment
agents look only at the numeric properties of the agents and disregard the
semantic characters that they carry. This insures the genericity of the
mechanism. The assessment agents compare the elements of their scenarios with
those carried by the factual agents by computing distances between them. The
compared data are vectors defined by the $n$ indicators of the factual agent
and its AN size. The cosine similarity measure (see equation 1) is used in
order to compute the similarity between these vectors. The similarity value is
included in a range of [0,1]. A value of 1 means the perfect equality between
the two vectors, whereas 0 means their total divergence.
$CS(V_{1},V_{2})=\dfrac{x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}}{\sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}}\sqrt{x_{2}^{2}+y_{2}^{2}+z_{2}^{2}}}$
(1)
With $V_{1}$ and $V_{2}$ two vectors, and $x_{i}$, $y_{i}$ and $z_{i}$ are
their respective coordinates.
### IV-B Experimentations
We have made experimentations on the RCR application dealing with fires
situations. We have developed a prototype allowing the representation and the
assessment of risks. The perceived facts in the disaster space are related to
the fires propagation and to the fire brigades activities that try to
extinguish these fires. The system includes a factual agents organisation for
the perception and the representation of the situation and a set of assessment
agents to deal with the facts evolution. At this progression stage of our
work, the assessment situation is limited to the recognition of factual agents
clusters according to past ones defined and experimented beforehand. We have
defined therefore, from a starting scenario, a clusters set that we intend to
regain in other similar scenarios by forming similar clusters. To modify an
RCR scenario, we change the strategy applied by the fire brigades. This allows
to have a different perception of the environment and different behaviours of
the agents.
Figure 3: First test example at the beginning of the RCR simulation
Figure 3 shows two views of the disaster space state at the beginning of the
simulation–at the $6^{th}$ second. The left view belongs to the starting
scenario, the right one belongs to a scenario test. What interests us in these
views are the fire brigades agents represented by black ellipses and the fires
represented by black rectangles. Both objects have white identifiers (IDs), we
note that the RCRSS gives randomly new IDs for all the RCR objects in each new
simulation. These two elements are represented in the system by two different
kinds of factual agents. We have identified two factual agents clusters at
this step. Cluster-1 includes starting fires and the first fire brigades
having perceived these fires and which are the most able to put out them.
Cluster-2 contains however the rest of the fire brigades that are in a passive
state.
Table I presents a test example. For this example we have four assessment
agents, each one is associated to one cluster in the base. The table shows
both the stored clusters elements and those created by the assessment agents.
As we see, the two first agents (Agent-2 and Agent-1) regained two analogous
clusters with relatively high relevances ($r$) in the test scenario and cover
all the perceived facts of the situation. These two agents are therefore
selected as the best candidates to provide the final decisions.
TABLE I: Created clusters at the $6^{th}$ second of the RCR simulation Stored clusters | Assessment agents | Similar clusters
---|---|---
Cluster-2: | Agent-2 | Cluster-1, $r$=0.99
fireBrigade#267864071 | | fireBrigade#267888188
fireBrigade#130020552 | | fireBrigade#264158650
fireBrigade#129970323 | | fireBrigade#201310913
fireBrigade#255666267 | | fireBrigade#134192215
fireBrigade#199205638 | | fireBrigade#234821930
fireBrigade#20884048 | | fireBrigade#232695827
fireBrigade#133635968 | | fireBrigade#258896960
Cluster-1: | Agent-1 | Cluster-2, $r$=0.89
fireBrigade#200188078 | | fireBrigade#64866967
fireBrigade#250079625 | | fireBrigade#268275018
fireBrigade#263968700 | | fireBrigade#33546030
fire#238713057 | | fire#265210206
fire#222263253 | | fire#262626275
fire#256855677 | | fire#217816816
Cluster-4: | Agent-4 | Cluster-3, $r$=0.80
Cluster-3: | Agent-3 | Cluster-4, $r$=0.67
The second example (see Figure 4) concerns another scenario in an advanced
stage of the RCR simulation–at the $13^{th}$ second of the simulation–in which
fires are more important and the fire brigades are more active. At this step,
two starting clusters have been identified and stored. Cluster-3 includes fire
brigades in full fight with fires and other important starting fires.
Cluster-4 presents some isolated fire brigades blocked by debris and that are
unable to move. A similar situation is perceived at the $11^{th}$ second of
the test scenario. The most relevant assessment agents are Agent-3 and Agent-4
that succeed in creating two similar clusters, whereas Agent-1 and Agent-2
have retrogressed in the relevances rank.
Figure 4: Second test example in the middle of the RCR simulation TABLE II: Created clusters at the $11^{th}$ second of the RCR simulation Stored clusters | Assessment agents | Similar clusters
---|---|---
Cluster-3: | Agent-3 | Cluster-1, $r$=0.83
fireBrigade#200188078 | | fireBrigade#201310913
fireBrigade#263968700 | | fireBrigade#134192215
fireBrigade#133635968 | | fireBrigade#234821930
fireBrigade#20884048 | | fireBrigade#268275018
fireBrigade#130020552 | | fireBrigade#64866967
fireBrigade#250079625 | | fireBrigade#258896960
fire#222263253 | | fire#265210206
fire#263966785 | | fire#217816816
fire#267173025 | | fire#134174462
fire#150719037 | | fire#165395197
| | fire#115811948
Cluster-4: | Agent-4 | Cluster-2, $r$=0.80
fireBrigade#199205638 | | fireBrigade#264158650
fireBrigade#267864071 | | fireBrigade#267888188
fireBrigade#255666267 | | fireBrigade#232695827
fireBrigade#129970323 | |
Cluster-1 | Agent-1 | Cluster-3, $r$=0.78
Cluster-2 | Agent-2 | Cluster-4, $r$=0.44
## V Conclusion
We have described in this paper an agent-based approach that aims to build a
DSS. The system intends to help emergency planners to detect risks and to
manage crisis situations by perceiving, representing and assessing a current
situation. We think this approach may be adjusted easilly to different
problems types and enables the system to have an adaptive behaviour thanks to
a multiagent multilevel kernel. We are working currently on the assessment
level of the system mechanism. We have presented here first results applied on
the RoboCupRescue. We intend to apply this approach on different subjects of
studies in order to better improve its generic aspect. We aim also to
generalise this approach by setting up a generic modelling of factual agents
clusters that will enhance their formalisation and their management.
## References
* [1] Australian Standard, _AS/NZS 4360:2004: Risk management_ , 2004.
* [2] M. J. Druzdzel and R. R. Flynn, _Decision Support Systems_. In Encyclopedia of Library and Information Science, vol. 67, pp. 120–133, 2000.
* [3] M. Ester, H. P. Kriegel, J. Sander, and X. Xu, _A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise_. Proceedings of 2nd International Conference on Knowledge Discovery and Data Mining (KDD-96), Varna, Bulgaria, pp. 226–231, 1996).
* [4] S. French and C. Niculae, _Believe in the Model: Mishandle the Emergency_. Journal of Homeland Security and Emergency Management, Springer-Verlag New York, Inc.,Secaucus, NJ, USA, vol. 2, pp. 1–18, 2005.
* [5] J. A. Hartigan and M. A. Wong, _A k-means clustering algorithm_. Applied Statistics, vol. 28, pp. 100–108, 1979.
* [6] F. Kebair and F. Serin, _Information Modeling for a Dynamic Representation of an Emergency Situation_. Proceedings of the 4th IEEE International Conference on Intelligent Systems IS’08, Varna, Bulgaria, vol. 1, pp. 2–7, 2008.
* [7] H. Kitano, S. Tadokor, H. Noda, I. Matsubara, T. Takahashi, A. Shinjou, and S. Shimada, _RoboCup Rescue: search and rescue in large-scale disasters as a domain for autonomous agents research_. Proceedings of the IEEE Conference on Systems, Man, and Cybernetics (SMC-99), vol. 6, pp. 739–743, 1999.
* [8] K. Kolodner, _Case-based reasoning_. Morgan Kaufmann, Boston, 1993.
* [9] _RoboCupRescue Official Web Site_. http://www.robocuprescue.org/.
* [10] M. J. Shawn, D. M. Gardner, and H. Thomas, _Research Opportunities in Electronic Commerce_. Decis. Support Syst, 1997, vol. 21, pp. 149–156.
* [11] H. A. Simon, _The New Science of Management Decision_. Prentice Hall PTR, 1977.
* [12] T. Takahashi, _RoboCupRescue Simulation Manual_. Available: http://sakura.meijo-u.ac.jp/ttakaHP/kiyosu/robocup/Rescue/manual-English-v0r4/index.html.
* [13] M. Wooldridge, _An Introduction to MultiAgent Systems_. John Wiley & Sons, 2002.
|
arxiv-papers
| 2009-07-03T14:24:23 |
2024-09-04T02:49:03.716795
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fahem Kebair and Frederic Serin",
"submitter": "Fahem Kebair",
"url": "https://arxiv.org/abs/0907.0499"
}
|
0907.0655
|
# Non linear transport theory for negative-differential resistance states of
two dimensional electron systems in strong magnetic fields.
A. Kunold [email protected], [email protected] Université de
Toulouse; INSA-CNRS-UPS, LPCNO, 135, Av. de Rangueil, 31077 Toulouse, France
Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-
Azcapotzalco, Av. San Pablo 180, México D. F. 02200, México M. Torres
[email protected] Instituto de Física, Universidad Nacional Autónoma de
México, Apartado Postal 20-364, México Distrito Federal 01000, México
###### Abstract
We present a model to describe the nonlinear response to a direct dc current
applied to a two-dimensional electron system in a strong magnetic field. The
model is based on the solution of the von Neumann equation incorporating the
exact dynamics of two-dimensional damped electrons in the presence of
arbitrarily strong magnetic and dc electric fields, while the effects of
randomly distributed impurities are perturbatively added. From the analysis of
the differential resistivity and the longitudinal voltage we observe the
formation of negative differential resistivity states (NDRS) that are the
precursors of the zero differential resistivity states (ZDRS). The theoretical
predictions correctly reproduce the main experimental features provided that
the inelastic scattering rate obey a $T^{2}$ temperature dependence,
consistent with electron-electron interaction effects.
###### pacs:
73.43.Qt,71.70.Di,73.43.Cd,73.50.Bk,73.50.Fq
## I Introduction
In the past few years the study of non-equilibrium magneto-transport in high
mobility two-dimensional electron systems (2DES) has received much attention
due to the experimental finding of intense oscillations of the magneto-
resistivity and zero resistance states (ZRS). Microwave-induced resistance
oscillations (MIRO) were discovered Zudov et al. (2001, 2003); Mani et al.
(2002); Ye et al. (2001) in 2DES samples subjected to microwave irradiation
and moderate magnetic fields. For the MIRO the photoresistance is a function
of the ratio $\epsilon^{ac}=\omega/\omega_{c}$ where $\omega$ and $\omega_{c}$
are microwave and cyclotron frequencies. This outstanding discovery triggered
a great amount of theoretical work Ryzhii (1970); Ryzhii and Vyurkov (2003);
Shi and Xie (2003); Dorozhkin (2003); Durst et al. (2003); Lei and Liu (2003);
Vavilov and Aleiner (2004); Torres and Kunold (2005); Iñarrea and Platero
(2007); Dmitriev et al. (2003, 2005, 2004); Robinson et al. (2004). Our
current understanding of this phenomenon rests upon models that predict the
existence of negative-resistance states (NRS) yielding an instability that
rapidly drive the system into a ZRS Andreev et al. (2003). Two distinct
mechanisms for the generation of NRS are known, one is based in the microwave-
induced impurity scattering Ryzhii (1970); Durst et al. (2003); Lei and Liu
(2003); Shi and Xie (2003); Vavilov and Aleiner (2004); Torres and Kunold
(2005); Iñarrea and Platero (2007), while the second is linked to inelastic
processes leading to a non-trivial distribution function Dorozhkin (2003);
Dmitriev et al. (2003, 2005); Robinson et al. (2004).
An analogous effect, Hall field-induced resistance oscillations (HIRO) has
been observed in high mobility samples in response to a dc-current excitation
Yang et al. (2002); Zhang et al. (2007, 2007). Although MIRO and HIRO are
basically different phenomena both rely on the commensurability of the
cyclotron frequency with a characteristic parameter; in both cases
oscillations are periodic in $1/B$. In HIRO the oscillation peaks, observed in
differential resistance, appear at integer values of the dimensionless
parameter $\epsilon^{dc}=\omega_{H}/\omega$. Here, $\hbar\omega_{H}\approx
eE_{H}(2R_{C})$ is the energy associated with the Hall voltage drop across the
cyclotron diameter; $E_{H}$ is the Hall field and $R_{C}$ the cyclotron radius
of the electron at the Fermi level. It has been found that there are two main
contributions to the HIRO: the inelastic one is related to the formation of a
non-equilibrium distribution function component that oscillates as a function
of the energyVavilov et al. (2007) and the elastic contribution is related to
electron transitions between different LLs due to impurity scatteringLei
(2007). The first one was shown to be dominant at relatively weak electric
fields, and the latter prevails in the strong-field regime.
More recently it has been demonstrated that the effects of a direct dc current
on electron transport can be quite dramatic leading to zero differential
resistance states (ZDRS)Bykov et al. (2007); Romero et al. (2008). As compared
with the HIRO conditions, the ZDRS are observed under dc bias at higher
magnetic fields ($0.5-1.0\,T$) and lower mobilities ($70-85\,m^{2}/Vs$). At
low temperature and above a threshold bias current the differential
resistivity vanishes and the longitudinal dc voltage becomes constant.
Positive values for the differential resistance are recovered at higher bias
as the longitudinal dc voltage slope becomes positive. Bykov et al. analyzed
the results following an approach similar to that of Andreev et al. Andreev et
al. (2003); the presence of the ZDRS is attributed to the formation of
negative differential resistance states (NDRS) that yields an instability that
drives the system into a ZDRS. Similar results where obtained by Chen et al.
Chen et al. (2009)
In this paper we present a model to explain the formation of NDRS. According
to our formalism both the effects of elastic impurity scattering as well as
those related to inelastic processes play an important role. The model is
based on the solution of the von Neumann equation for 2D damped electrons,
subjected to arbitrarily strong magnetic and dc electric fields, in addition
to the weak effects of randomly distributed impurities. This procedures yields
a Kubo formula that includes the non-linear response with respect to the dc
electric field. Considering a current controlled scheme, we obtain a set of
nonlinear self-consistent relations that allow us to determine the
longitudinal and Hall electric fields in terms of the imposed external
current. It is shown that in order to correctly reproduce the main
experimental results the inelastic scattering rate must obey a $T^{2}$
temperature dependence, consistent with electron-electron Coulomb interaction
as the dominant inelastic process.
Figure 1: Differential resistance $r_{xx}$ as a function of the dc bias
$J_{x}$ for $B=0.784T$ temperatures from $T=1K$ to $T=10K$. Figure 2: Electric
field $E_{x}$ as a function of the dc bias $J_{x}$ for $B=0.784T$ and for
fixed temperatures ranging from $T=1K$ to $T=10K$.
## II Model
We start with the Hamiltonian for an electron in the effective mass
approximation in two dimensions subject to a uniform perpendicular magnetic
field $\boldsymbol{B}=\left(0,0,B\right)$, an in-plane electric field
$\boldsymbol{E}=\left(E_{x},E_{y},0\right)$, and the impurity scattering
potential $V$. Hence the dynamics is governed by the total Hamiltonian
$H=H_{e}+V$, with
$H_{e}=H_{0}+e\boldsymbol{E}\cdot\boldsymbol{x}\,,$ (1)
here $H_{0}=\boldsymbol{\Pi}^{2}/2m$, $m$ is the effective mass of the
electron, $e$ is the electron’s charge,
$\boldsymbol{\Pi}=\boldsymbol{p}+e\boldsymbol{A}$ is the velocity operator and
the vector potential in the symmetric gauge is given as
$\boldsymbol{A}=\left(-By,Bx\right)/2$. The impurity scattering potential is
expressed in terms of its Fourier components
$V\left(\boldsymbol{r}\right)={\rm
e}^{-\eta\left|t\right|}\sum_{i}^{N_{i}}\int\frac{d^{2}q}{\left(2\pi\right)^{2}}V\left(q\right)\exp\left[i\boldsymbol{q}\cdot\left(\boldsymbol{r}-\boldsymbol{r}_{i}\right)\right]\,,$
(2)
where $\boldsymbol{r}_{i}$ is the position of the $i$th impurity and $N_{i}$
is the number of impurities. The explicit form of $V\left(q\right)$ depends on
the nature of the scatterersTorres and Kunold (2005), for simplicity we assume
short-range uncorrelated scatterers. The factor $\exp\left(-\eta|t|\right)$
takes care of the adiabatic switching of the impurity potential at the initial
time $t_{0}\to-\infty$.
The motion of a planar electron in magnetic and electric fields can be
decomposed into the guiding center coordinates $\boldsymbol{Q}$ and the
relative coordinates $\boldsymbol{R}=\left(-\Pi_{y},\Pi_{x}\right)/eB$, such
that the position of the electron is given by
$\boldsymbol{r}=\boldsymbol{Q}+\boldsymbol{R}$. The guiding center coordinates
is written as $\boldsymbol{Q}=\left({\cal Q}_{x},{\cal Q}_{y}\right)/eB$, in
the symmetric gauge $\left({\cal Q}_{x},{\cal
Q}_{y}\right)=\left(p_{x}+eBy/2,p_{y}-eBx/2\right)$. The commutation relations
for velocity and guiding center operators are
$\left[\Pi_{x},\Pi_{y}\right]=\left[{\cal Q}_{x},{\cal Q}_{y}\right]=-i\hbar
eB$, with all the other commutators being zero.
Our aim now is to compute the electric current density. In order to calculate
the expectation value of the current density we need the time-dependent matrix
$\rho(t)$ which obeys the von Neumman’s equation $i\hbar\partial\rho/\partial
t=\left[H,\rho\right]$. We assume that in the absence of the impurity
potential the density matrix reduces to the equilibrium density matrix given
by $\rho_{0}=f(H_{0})$, with $f(E)$ given by the Fermi distribution function.
In order to solve the von Neumman’s equation we apply three unitary
transformations: the first two transformations exactly take into account the
effects of the electric and magnetic fields, whereas the third transformation
incorporates the impurity scattering effects to second order in time dependent
perturbation theory. First we consider the unitary transformation
${\cal W}\left(t\right)=e^{\frac{i}{\hbar}\int{\cal
L}dt}e^{-i\frac{v_{x}\Pi_{y}}{\hbar\omega_{c}}}e^{i\frac{v_{y}\Pi_{x}}{\hbar\omega_{c}}}e^{i\frac{X{\cal
Q}_{x}}{\hbar}}e^{i\frac{Y{\cal Q}_{y}}{\hbar}}$ (3)
where $v_{x}\left(t\right)$, $v_{y}\left(t\right)$, $X\left(t\right)$ and
$Y\left(t\right)$ are solutions of the dynamical equations
$\displaystyle\dot{v}_{x}+\frac{1}{\tau_{i}}v_{x}+\omega_{c}v_{y}+\frac{e}{m}E_{x}=0,$
$\displaystyle\dot{X}-\frac{Ey}{B}=0,$ (4)
$\displaystyle\dot{v}_{y}+\frac{1}{\tau_{i}}v_{y}-\omega_{c}v_{x}+\frac{e}{m}E_{y}=0,$
$\displaystyle\dot{Y}+\frac{Ex}{B}=0.$ (5)
Except for the damping terms, these equations follow from the variation of the
classical Lagrangian ${\cal L}$Torres and Kunold (2005). The variables $v_{x}$
and $v_{y}$ correspond to the electron velocity components and $X$ and $Y$ are
the coordinates that follow the drift of the electron’s orbit. In order to
incorporate dissipative effect we added the damping term
$\boldsymbol{v}/\tau_{i}$ the dynamical equations. This procedure yields a
simple scheme to incorporate dissipation to the quantum system. Recent
magnetoresistance experimentsHatke et al. (2009a, b) and theory Vavilov et al.
(2007) suggest, that in 2DES, electron-electron interaction provide an
important contribution to the inelastic scattering rate, giving rise to
$1/\tau_{i}\propto T^{2}$ temperature dependance. Consequently, in what
follows we shall assume that the inelastic scattering rate is given by
$1/\tau_{i}\approx(k_{B}T)^{2}/\hbar E_{F}$ Chaplik (1971); Giuliani and Quinn
(1982); Hatke et al. (2009a, b), where $E_{F}$ is the Fermi energy.
The transformation (3) renders von Neumann equation into the following form
$i\hbar\frac{\partial\left({\cal W}\rho{\cal W}^{{\dagger}}\right)}{\partial
t}\\\ =\left[H_{0}+V\left(t\right),{\cal W}\rho{\cal W}^{{\dagger}}\right].$
The electric field term is conveniently removed from the Hamiltonian to
produce a time-dependent impurity potential
$V\left(t\right)=V\left(x+X\left(t\right)+\frac{v_{y}\left(t\right)}{\omega_{c}},y+Y\left(t\right)-\frac{v_{x}\left(t\right)}{\omega_{c}}\right).$
(6)
We proceed to switch to the interaction picture through the unitary operator
${\cal U}_{0}=\exp\left(iH_{0}t/\hbar\right)$ and solve the remaining equation
up to second order in time dependent perturbation theory obtaining yet another
simplified version of von Neumann equation
$i\hbar\frac{\partial}{\partial t}\left({\cal U}{\cal U}_{0}{\cal W}\rho{\cal
W}^{{\dagger}}{\cal U}_{0}^{{\dagger}}{\cal U}^{{\dagger}}\right)=0,$ (7)
where the time evolution operator is given by
$\displaystyle{\cal U}=$ $\displaystyle
1-\frac{i}{\hbar}\int_{t_{0}}^{t}V_{I}\left(s_{1}\right)ds_{1}$
$\displaystyle-\frac{1}{\hbar^{2}}\int_{t_{0}}^{t}\int_{t_{0}}^{s_{1}}V_{I}\left(s_{1}\right)V_{I}\left(s_{2}\right)ds_{1}ds_{2}\,,$
(8)
here $V_{I}\left(t\right)={\cal U}_{0}V\left(t\right){\cal U}_{0}^{{\dagger}}$
is the impurity potential in the interaction picture. The formal solution to
(7) is given by $\rho\left(t\right)={\cal W}^{{\dagger}}{\cal
U}_{0}^{{\dagger}}{\cal U}^{{\dagger}}\rho\left(t_{0}\right){\cal U}{\cal
U}_{0}{\cal W}$ where $\rho\left(t_{0}\right)=\rho_{0}=f(H_{0})$ is the
equilibrium density matrix at the initial time $t_{0}\to-\infty$.
The density current is proportional to the thermal and time average of the
velocity operator
$\displaystyle\boldsymbol{J}=\frac{e}{S}\int_{-\infty}^{\infty}dt{\rm
Tr}\left[\rho\left(t\right)\boldsymbol{\Pi}\right],$ (9)
where $S$ is the surface of the sample, and the limit $S\to\infty$ is
understood. By performing a cyclic permutation in the trace we obtain
$\boldsymbol{J}=\frac{e}{S}{\rm Tr}\left[\rho\left(t_{0}\right){\cal U}{\cal
U}_{0}{\cal W}\boldsymbol{\Pi}{\cal W}^{{\dagger}}{\cal
U}_{0}^{{\dagger}}{\cal U}^{{\dagger}}\right].$ (10)
After lengthy calculations the components of the density current is worked out
as
$J_{i}=\frac{ne^{2}\tau_{i}}{m}\frac{E_{i}-\omega_{c}\tau_{i}\epsilon_{ij}E_{j}}{1+\omega_{c}^{2}\tau_{i}^{2}}\\\
+\frac{e^{2}}{h}\sum_{\mu\mu^{\prime}}\int
d^{2}q\left(f_{\mu}-f_{\mu^{\prime}}\right)G^{i}_{\mu\mu^{\prime}}\left(q\right)$
(11)
where $i,j=x,y$ and $\epsilon_{i,j}$ is the antisymmetric tensor
($\epsilon_{12}=-\epsilon_{21}=1$ and $\epsilon_{11}=\epsilon_{22}=0$),
$G^{i}_{\mu\mu^{\prime}}=\frac{N_{i}B\left|V\left(q\right)\right|^{2}}{Sm\hbar}\left|D_{\mu\mu^{\prime}}\left(z_{q}\right)\right|^{2}\\\
\frac{q_{i}\Delta_{\mu\mu^{\prime}}+2\left|\epsilon_{ij}\right|q_{j}\omega_{c}\eta}{\Delta^{2}_{\mu\mu^{\prime}}+4\omega_{c}^{2}\eta^{2}}$
(12)
and
$\Delta_{\mu\mu^{\prime}}=\left[\omega_{q}+\omega_{c}\left(\mu-\mu^{\prime}\right)\right]^{2}-\omega_{c}^{2}+\eta^{2}$,
$\omega_{q}=\omega_{x}E_{x}+\omega_{y}E_{y}$,
$\omega_{x}=-\tau_{i}\omega_{c}(q_{x}+q_{y}\tau_{i}\omega_{c})/B(1+\tau_{i}^{2}\omega_{c}^{2})$,
$\omega_{y}=\tau_{i}\omega_{c}(-q_{y}+q_{x}\tau_{i}\omega_{c})/B(1+\tau_{i}^{2}\omega_{c}^{2})$
and $f_{\mu}=f\left(\hbar\omega_{c}\left(\mu+1/2\right)\right)$. The matrix
elements $D_{\mu,\nu}$ are given by
$D_{\mu\mu^{\prime}}\left(z_{q}\right)=\exp\left(-\frac{\left|z_{q}\right|^{2}}{2}\right)\\\
\times\left\\{\begin{array}[]{ll}z_{q}^{\mu-\mu^{\prime}}\sqrt{\frac{\mu^{\prime}!}{\mu!}}L_{\mu^{\prime}}^{\mu-\mu^{\prime}}\left(\left|z_{q}\right|^{2}\right),&\mu\geq\mu^{\prime},\\\
\left(-{z_{q}}^{*}\right)^{\mu^{\prime}-\mu}\sqrt{\frac{\mu!}{\mu^{\prime}!}}L_{\mu^{\prime}}^{\mu^{\prime}-\mu}\left(\left|z_{q}\right|^{2}\right),&\mu\leq\mu^{\prime},\\\
\end{array}\right.$ (13)
where $z_{q}=(q_{x}-iq_{y})/\sqrt{2}$ and $L_{\nu}^{\mu-\nu}$ denotes the
associated Laguerre polynomial.
Retaining a finite value of the switching parameter $\eta$ yields a density of
states for the Landau levels with the Lorentzian form given in Eq. (12); it is
distorted by the electric field through the $\omega_{q}$ term. Henceforth we
will consider $\eta=\Gamma\omega_{c}$. The differential conductivity tensor is
calculated from Eq. (11) as $\sigma_{ij}=\partial J_{i}/\partial E_{j}$.
Finally the differential resistivity tensor is obtained from the inverse of
the conductivity: that is $r_{ij}=\sigma^{-1}_{ij}$.
In the limit of small bias and small magnetic field the expression for the
density current reduces to $J_{x}=ne^{2}\tau_{i}E_{x}\left(1-\alpha\right)/m$
where
$\alpha=\frac{2\pi}{k_{B}T}\frac{e^{-E_{F}/k_{B}T}}{\left(e^{E_{F}/k_{B}T}+1\right)^{2}}\frac{\left|V\right|^{2}N_{i}m}{S\hbar\Gamma^{2}}.$
(14)
Hence the quantum scattering time and the inelastic scattering time can be
related by $\tau=\tau_{i}(1-\alpha)$ or similarly the elastic scattering time
is given by $\tau_{e}=\tau_{i}(1-\alpha)/\alpha$. The factor
$N_{i}\left|V\right|^{2}/S\Gamma^{2}$ present in the expressions for the
density current can be estimated from the sample’s mobility and the inelastic
scattering time.
In a current controled scheme: the longitudinal density current is fixed to a
constant value $J_{0}$ while $J_{y}$ should vanish. This leads to a set of two
implicit equations for the density current
$\displaystyle J_{x}\left(E_{x},E_{y}\right)=J_{0},$ $\displaystyle
J_{y}\left(E_{x},Ey\right)=0,$ (15)
where the explicit form of the functions $J_{i}$ is given in Eq. (11). To
obtain the components of the electric field $E_{x}$ and $E_{y}$, we start
assigning initial values $E_{x}=E_{x_{0}}$ and $E_{y}=E_{y_{0}}$ that solve
these relations in the absence of impurities ($i.e.$ using only the first term
on the R.H.S. of Eq. (11)), then the accuracy of the solution is improved by a
recursive application of Newton’s method.
Figure 3: Electric field $E_{x}$ as a function of the dc bias $J_{x}$ for
$T=2K$ and for fixed magnetic fields ranging from $B=0.5T$ to $B=1.085T$. The
thin lines indicate that $r_{xx}<0$. Figure 4: Electric field $E_{x}$ as a
function of the dc bias $J_{x}$ for $T=2K$ and for magnetic field $B=0.5T$.
The thin lines indicate differential resistivity $r_{xx}<0$. The inset shows a
possible non uniform configuration for the density current.
## III Results
Fig. 1 shows the differential resistivity $r_{xx}=\partial E_{x}/\partial
J_{x}$ as a function of the longitudinal dc density current $J_{x}$ for a
magnetic field $B=0.784T$ and various values of the temperature. We use a
sample mobility $\mu=100m^{2}V/s$, electron density $n=8.2\times
10^{15}m^{-2}$ and a broadening parameter $\Gamma=0.04$. As the value of the
temperature is reduced the differential resistance decrease approaching zero.
We can observe that at low temperature ($T<2K$) and above a threshold bias
current ($J_{x}>0.4A/m$) the differential resistivity becomes negative.
Positive values for the differential resistance are recovered at higher bias
or higher temperatures. The strong temperature dependence observed in this
plots, consistent with the experiments, is originated mainly on the $T^{2}$
dependence of the inelastic scattering rate.
The electric field $E_{x}$ is plotted as a function of the longitudinal
current $J_{x}$ in Fig. 2. It is important to notice that $E_{x}$ differs from
the longitudinal voltage by a geometrical factor. DNRS are observed below
$T=4K$ and above the current threshold $J_{x}>0.4A/m$ in the form of negative
slope curves (see inset of Fig. 2) in accordance with the $r_{xx}$ negative
values observed in Fig. 1. According to Bykov et al. Bykov et al. (2007) the
stability condition is simply expressed as $r_{xx}\geq 0$. Thus the regions in
Figs. 1 and 2 that display a negative differential resistivity are unstable,
and they should rapidly evolve into ZDRS to insure stability. Accordingly in
Fig. 1 we should replace the NDRS by $r_{xx}=0$ and maintain a constant slope
in Fig. 2 instead of the negative slope. At higher values of $J_{x}$ the
differential resistivity becomes positive (Fig. 1) as well as the longitudinal
voltage slope as a result of an increase in the impurity scattering prevalent
at high electric fields. In this regime the large electric field components,
necessary to maintain the strong dc bias and $J_{y}=0$, cause the impurity
terms to strongly participateVavilov et al. (2007).
Fig. 3 display a series of plots of $E_{x}$ field as a function of the
longitudinal density current $J_{x}$ at $T=2K$ for various fixed values of the
magnetic field that correspond to Shubnikov-de Haas oscillations maxima. The
thin lines indicate negative values of $r_{xx}$ that violate the stability
condition. As the magnetic field increases the width of the electric field
plateaus increase and the positive slope is recovered for higher onset density
currents. An isolated plot of the longitudinal electric field $E_{x}$ as a
function of the dc current $J_{x}$ is shown in Fig. 4. In the inset of Fig. 4
we show a nonuniform distribution current similar to the one proposed by Bykov
et al.Bykov et al. (2007). With this configuration not only the stability
condition $r_{xx}>0$ is fulfilled but the electric field is uniform throughout
the sample given that $E_{x}=E_{\rm min}$ for $J_{x1}$ and $J_{x2}$. The
average current density $J_{x}=(J_{x1}y_{1}+J_{x2}y_{2})/(y_{1}+y_{2})$ may be
modulated by varying the sizes $y_{1}$ and $y_{2}$ of the different density
current domains with the restriction that $y_{1}+y_{2}=w$. Notice that more
complicated schemes with more density current modulations also fulfill this
conditions.
## IV Conclusions
We have presented a model for the nonlinear transport of a 2DES placed in a
strong perpendicular magnetic field. The model is based on the solution of the
von Neumann equation for 2D damped electrons, subjected to arbitrarily strong
magnetic and dc electric fields, in addition to the weak effects of randomly
distributed impurities. This procedures yields a Kubo formula that includes
the non-linear response with respect to the dc electric field. Considering a
current controlled scheme, we obtain a set of nonlinear self-consistent
relations that allow us to determine the longitudinal and Hall electric fields
in terms of the imposed external current. NDRS are found in the low
temperature ($T\leq 2$) and moderate bias regime $0.4A/m<J_{x}<1.6A/m$. In low
dc bias (low electric field regime) the dominant mechanism is the inelastic
one. The longitudinal electric field (and voltage) recover they positive slope
in the high bias (high electric field regime). It is shown that in order to
correctly reproduce the main experimental results the inelastic scattering
rate must obey a $T^{2}$ temperature dependence, consistent with electron-
electron Coulomb interaction as the dominant inelastic process.
###### Acknowledgements.
A. Kunold is receiving financial support from “Estancias sabáticas al
extranjero” CONACyT and “Acuerdo 02/06” Rectoría UAM-A. A. Kunold wishes to
thank INSA-Toulouse for his hospitality.
## References
* Zudov et al. (2001) M. A. Zudov, R. R. Du, J. A. Simmons, and J. L. Reno, Phys. Rev. B 64, 201311(R) (2001).
* Zudov et al. (2003) M. A. Zudov, R. R. Du, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 90, 046807 (2003).
* Mani et al. (2002) R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. Johnson, and V. Umansky, Nature 420, 646 (2002).
* Ye et al. (2001) P. D. Ye, L. W. Engel, D. C. Tsui, J. A. Simmons, J. R. Wendt, G. A. Vawter, and J. L. Reno, Appl. Phys. Lett. 79, 2193 (2001).
* Ryzhii (1970) V. I. Ryzhii, Sov. Phys. Solid State 11, 2078 (1970).
* Ryzhii and Vyurkov (2003) V. Ryzhii and V. Vyurkov, Phys. Rev. B 68, 165406 (2003).
* Shi and Xie (2003) J. Shi and X. C. Xie, Phys. Rev. Lett. 91, 086801 (2003).
* Dorozhkin (2003) S. I. Dorozhkin, JETP Lett. 77, 577 (2003).
* Durst et al. (2003) A. C. Durst, S. Sachdev, N. Read, and S. M. Girvin, Phys. Rev. Lett. 91, 086803 (2003).
* Lei and Liu (2003) X. L. Lei and S. Y. Liu, Phys. Rev. Lett. 91, 226805 (2003).
* Vavilov and Aleiner (2004) M. G. Vavilov and I. L. Aleiner, Phys. Rev. B 69, 035303 (2004).
* Torres and Kunold (2005) M. Torres and A. Kunold, Phys. Rev. B 71, 115313 (2005).
* Iñarrea and Platero (2007) J. Iñarrea and G. Platero, Physical Review B (Condensed Matter and Materials Physics) 76, 073311 (2007).
* Dmitriev et al. (2003) I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. Lett. 91, 226802 (2003).
* Dmitriev et al. (2005) I. A. Dmitriev, M. G. Vavilov, I. L. Aleiner, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. B 71, 115316 (2005).
* Dmitriev et al. (2004) I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. B 70, 165305 (2004).
* Robinson et al. (2004) J. P. Robinson, M. P. Kennett, N. R. Cooper, and V. I. Fal’ko, Phys. Rev. Lett. 93, 036804 (2004).
* Andreev et al. (2003) A. V. Andreev, I. L. Aleiner, and A. J. Millis, Phys. Rev. Lett. 91, 056803 (2003).
* Yang et al. (2002) C. L. Yang, J. Zhang, R. R. Du, J. A. Simmons, and J. L. Reno, Phys. Rev. Lett. 89, 076801 (2002).
* Zhang et al. (2007) Jing-qiao Zhang, S. Vitkalov, A. A. Bykov, A. K. Kalagin, and A. K. Bakarov, Physical Review B (Condensed Matter and Materials Physics) 75, 081305(R) (2007).
* Zhang et al. (2007) W. Zhang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Physical Review Letters 98, 106804 (2007).
* Vavilov et al. (2007) M. G. Vavilov, I. L. Aleiner, and L. I. Glazman, Physical Review B (Condensed Matter and Materials Physics) 76, 115331 (2007).
* Lei (2007) X. L. Lei, Applied Physics Letters 90, 132119 (2007).
* Bykov et al. (2007) A. A. Bykov, Jing-qiao Zhang, S. Vitkalov, A. K. Kalagin, and A. K. Bakarov, Physical Review Letters 99, 116801 (2007).
* Romero et al. (2008) N. Romero, S. McHugh, M. P. Sarachik, S. A. Vitkalov, and A. A. Bykov, Physical Review B 78, 153311 (2008).
* Chen et al. (2009) J. C. Chen, Y. Tsai, Y. Lin, T. Ueda, and S. Komiyama, Physical Review B (Condensed Matter and Materials Physics) 79, 075308 (2009).
* Hatke et al. (2009a) A. T. Hatke, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 102, 066804 (2009a).
* Hatke et al. (2009b) A. T. Hatke, M. A. Zudov, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 79, 161308(R) (2009b).
* Chaplik (1971) A. V. Chaplik, Sov. Phys. JETP 33, 997 (1971).
* Giuliani and Quinn (1982) G. F. Giuliani and J. J. Quinn, Phys. Rev. B 26, 4421 (1982).
|
arxiv-papers
| 2009-07-03T15:38:36 |
2024-09-04T02:49:03.724758
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Kunold, M. Torres",
"submitter": "Alejandro Kunold",
"url": "https://arxiv.org/abs/0907.0655"
}
|
0907.0673
|
# Can time-dependent density functional theory predict the excitation energies
of conjugated polymers?
Jianmin Tao Theoretical Division and CNLS, Los Alamos National Laboratory,
Los Alamos, New Mexico 87545 Sergei Tretiak Theoretical Division and CNLS,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Center for
Integrated Nanotechnology, Los Alamos National Laboratory, Los Alamos, New
Maxico 87545 Jian-Xin Zhu Theoretical Division and CNLS, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545
###### Abstract
Excitation energies of light-emitting organic conjugated polymers have been
investigated with time-dependent density functional theory (TDDFT) within the
adiabatic approximation for the dynamical exchange-correlation potential. Our
calculations show that the accuracy of the calculated TDDFT excitation
energies largely depends upon the accuracy of the dihedral angle obtained by
the geometry optimization on ground-state DFT methods. We find that, when the
DFT torsional dihedral angles between two adjacent phenyl rings are close to
the experimental dihedral angles, the TDDFT excitation energies agree fairly
well with experimental values. Further study shows that, while hybrid density
functionals can correctly respect the thumb rule between singlet-singlet and
singlet-triplet excitation energies, semilocal functionals do not, suggesting
inadequacy of the semilocal functionals in predicting triplet excitation
energies of conjugated polymers.
###### pacs:
71.15.Mb, 31.15.ee, 71.45.Gm
The most important progress made in the development of molecular electronics
is the discovery of electroluminescent conjugated polymers thomas98 – that
is, fluorescent polymers that emit light when these polymers in the excited
states are stimulated by, say, electric current. Conjugated polymers are
organic semiconductors with delocalized $\pi$-molecular orbitals along the
polymeric chain. These materials are a major challenge to inorganic materials
which have been dominating the commercial market in light-emitting diodes for
display and other purpose pmay . The attraction of conjugated polymers lies at
their versatility, because their physical properties such as color purity and
emission efficiency can be fine-tuned by manipulation of their chemical
structures. The systematic modification of the properties of emissive polymers
by synthetic design has become a vital component in the optimization of light-
emitting devices.
Theoretical investigation of their optical absorption plays a significant role
in computer-aided design and optimization of the electroluminescent polymers.
The method of choice for the simulation of the optical absorption of
electronic materials is time-dependent density functional theory (TDDFT)
grossbook , owing to its high computational efficiency and comparable
accuracy. TDDFT is the most important extension of Kohn-Sham ground-state DFT,
the standard method in electronic structure calculations. The only
approximation made in TDDFT is the dynamical exchange-correlation (XC)
potential, which includes all unknown many-body effects. The simplest
construction is called adiabatic (ad) approximation zs , which takes the same
form of the static XC potential but replaces the ground-state density
$n_{0}({\bf r})$ with the instantaneous time-dependent density $n({\bf r},t)$:
$v_{\rm xc}^{ad}([n];{\bf r},t)=\delta E_{\rm xc}[n_{0}]/\delta n_{0}({\bf
r})|_{n_{0}({\bf r})=n({\bf r},t)}~{}.$ The advantage of this approach is its
simplicity in both theoretical construction and numerical implementation.
Although the adiabatic TDDFT cannot properly describe multiple excitations, it
has become the most popular approach in the study of low-lying single-particle
excitations (i.e., only one electron in the excitated states).
Figure 1: Chemical structures of the computationally studied light-emitting
congugated polymers. Table 1: Excitation energies of singlet-singlet
($S_{0}-S_{1}$) and singlet-triplet ($S_{0}-T_{1}$) gaps (in units of eV) of
polymers of length of $\sim 10~{}{\rm nm}$ in gas phase calculated using the
adiabatic TDDFT methods with the ground-state geometries optimized on the
respective density functionals. Basis set 6-31G is used in all calculations.
The number in parentheses is the number of rings included in our calculations.
1 hartree = 27.21 eV.
| $S_{0}-S_{1}$ | | $S_{0}-T_{1}^{a}$
---|---|---|---
Polymer | Expta | LSDA | TPSS | TPSSh | B3LYP | PBE0 | | Expta | LSDA | TPSS | TPSSh | B3LYP | PBE0
P3OT(28) | 2.8-3.8 | $0.99$ | $0.99$ | $1.35$ | $1.59$ | $1.76$ | | 1.7-2.2 | $0.90$ | $0.80$ | $0.88$ | $0.96$ | $0.95$
PBOPT(32) | 2.52 | $1.49$ | $1.55$ | $1.96$ | $2.26$ | $2.39$ | | 1.60 | $1.37$ | $1.31$ | $1.42$ | $1.57$ | $1.54$
MEHPPV(16) | 2.48 | $1.14$ | $1.27$ | $1.66$ | $1.94$ | $2.07$ | | 1.30 | $1.04$ | $1.08$ | $1.18$ | $1.31$ | $1.24$
PFO(36) | 3.22 | $2.30$ | $2.45$ | $2.89$ | $3.13$ | $3.30$ | | 2.30 | $2.22$ | $2.23$ | $2.34$ | $2.45$ | $2.43$
DHOPPV(16) | 2.58 | $1.14$ | $1.27$ | $1.67$ | $1.95$ | $2.07$ | | 1.50 | $1.04$ | $1.08$ | $1.18$ | $1.32$ | $1.24$
PPY(24) | 3.4-3.9 | $1.82$ | $2.10$ | $2.61$ | $2.87$ | $3.03$ | | 2.4-2.5 | $1.82$ | $1.99$ | $2.11$ | $2.23$ | $2.20$
CN-MEHPPV(16) | 2.72 | $1.10$ | $1.34$ | $1.84$ | $2.16$ | $2.27$ | | N/A | $1.06$ | $1.22$ | $1.34$ | $1.48$ | $1.43$
PANi(20) | 2.00 | $2.34$ | $2.53$ | $3.05$ | $3.30$ | $3.44$ | | $<0.9$ | $2.31$ | $2.43$ | $2.63$ | $2.75$ | $2.73$
aFrom Ref. monkman01 , in which there is a small red shift in gas phase,
compared to those in solvent (see discussion in the context). bNotation of
Ref. birks is used. Note that all the groups of -(CH2)nCH3 in polymers have
been replaced with the hydrogen (-H).
Our previous studies of small molecules ttz082 and molecular materials tt09
show that the excitation energies obtained with the adiabatic TDDFT agree
fairly well with experiments. In the present work, we calculate the lowest
singlet-singlet ($S_{0}-S_{1}$) and singlet-triplet ($S_{0}-T_{1}$) excitation
energies of a series of light-emitting organic conjugated polymers (see Fig. 1
for their chemical structures). The singlet-singlet excitation is responsible
for the strong ultraviolet (UV) or near-UV optical absorption, while the
singlet-triplet excitation is responsible for weak fluorescence. Our
calculations show that, when the dihedral angles note1 between two adjacent
phenyl rings obtained by the geometry optimization on ground-state DFT methods
are close to experimental dihedral angles, the calculated TDDFT excitation
energies agree well with experiments, regardless of whether the excitations
arise from singlet-singlet excitations or singlet-triplet excitations. This
suggests that in TDDFT calculations, there are two sources of error. One is
from the adiabatic approximation itself note2 , and the other, much larger
than the first one, arises from inaccuracy of the ground-state DFT geometries.
In order to identify these errors, here we employ five commonly-used density
functionals. Two of them, the local spin density approximation (LSDA) and the
meta-generalized gradient approximation (meta-GGA) of Tao, Perdew, Staroverov,
and Scuseria (TPSS) tpss , are pure density functionals, while the other
three, TPSSh sstp1 (a hybrid of the TPSS meta-GGA with $10\%$ exact
exchange), B3LYP b3lyp (a hybrid with $20\%$ exact exchange), and PBE0 gus99
(a hybrid of the Perdew-Burke-Ernzerhof (PBE) pbe96 GGA with $25\%$ exact
exchange) are hybrid functionals with increasing amount of exact exchange from
TPSSh, B3LYP to PBE0.
Moreover, in the simulation of electronic excitations of small molecules and
molecular materials, the most effort has been devoted to the study of the
absorption arising from singlet-singlet excitation, leaving the singlet-
triplet excitation less investigated perun . An important reason for this
omission is that triplet-state energies are not easy to measure through direct
optical absorption due to very low singlet-triplet ($S_{0}-T_{1}$) absorption
coefficient walters and low phosphorescence quantum yield roman
($<10^{-6}$). The major approaches to probe triplet states in conjugated
polymers are the charge recombination or energy transfer, and singlet-triplet
($T_{1}-S_{0}$ or $S_{1}-T_{1}$) intersystem crossing reindl ; parker ; birks
. The observation of $T_{1}-S_{0}$ phosphorescence from molecules initially
excited into $S_{1}$ is clear evidence for a radiationless transition from
$S_{1}$ to an isoenergetic level of the triplet manifold, corresponding to
singlet-triplet intersystem crossing. Singlet-triplet intersystem crossing can
occur either from the zero-point vibrational level of $S_{1}$ or from
thermally-populated vibrational level of $S_{1}$ into an excited vibrational
level of $T_{1}$, or more probably into a higher excited triplet state
$T_{2}$, which is closer in energy to $S_{1}$. It has been found burrows01 ;
burrows02 that the properties of the triplet states directly impact device
performance. For example, the formation of triplet states may cause the loss
of the device efficiency in these materials and thus can limit device
performance and operational life span. Therefore, investigation of triplet
excitations is crucial for a full understanding of electroluminescence
behavior of conjugated organic polymers and for the improvement of new
materials.
Monkman and collaborators burrows02 ; monkman01 investigated the photophysics
of triplet states in a series of conjugated polymers and measured the
excitation energies of the lowest singlet- and triplet-excitated states. Their
measurements show that the excitation energies in general respect the well-
known rule of thumb found for small molecules:
$\displaystyle E_{T}\approx 2E_{S}/3,$ (1)
where $E_{T}$ is the triplet excitation energy and $E_{S}$ is the singlet-
singlet excitation energy. As a second part of our work, we calculate the
singlet-triplet excitation energies of the polymers with the adiabatic TDDFT.
We find that, without exact exchange mixing, a pure semilocal density
functional cannot satisfy the thumb rule of Eq. (1), suggesting inadequacy of
the adiabatic semilocal functionals in predicting the triplet excitation
energies for polymers.
Table 2: Torsions of the conjugated polymers Polymer | Expt | PBE0 | Energy
---|---|---|---
P3OT | $\sim 24\,^{\circ}$ | $\sim 0\,^{\circ}$ | red shift
PBOPT | $\sim 35\,^{\circ}$ | $\sim 40\,^{\circ}$ | On experiment
MEHPPV | $\sim 20\,^{\circ}$ | $\sim 1\,^{\circ}$ | red shift
PFO | $\sim 40\,^{\circ}$ | $\sim 38\,^{\circ}$ | On experiment
DHOPPV | $\sim 20\,^{\circ}$ | $\sim 0\,^{\circ}$ | red shift
PPY | $\gtrsim 0\,^{\circ}$ | $\sim 0-1\,^{\circ}$ | slightly red shift
CN-MEHPPV | $\sim 20\,^{\circ}$ | $\sim 0\,^{\circ}$ | red shift
PANi | $\sim 0\,^{\circ}$ | $\sim 18-26\,^{\circ}$ | too blue shift
Table 3: Excitation energies of singlet-singlet ($S_{0}-S_{1}$) and singlet-
triplet ($S_{0}-T_{1}$) gaps (in units of eV) of polymers of length of $\sim
10~{}{\rm nm}$ in benzene solution calculated using the adiabatic TDDFT
methods with the ground-state geometries optimized on the respective density
functionals. The solvent effects are taken into account through PCM
(polarizable continuum model) method. Basis set 6-31G is used in all
calculations. The number in parentheses is the number of rings included in our
calculations. 1 hartree = 27.21 eV.
| $S_{0}-S_{1}$ | | $S_{0}-T_{1}^{b}$
---|---|---|---
Polymer | Expta | LSD | TPSS | TPSSh | B3LYP | PBE0 | | Expta | LSD | TPSS | TPSSh | B3LYP | PBE0
P3OT(28) | 2.8-3.8 | $0.97$ | $0.97$ | $1.32$ | $1.56$ | $1.73$ | | 1.7-2.2 | $0.89$ | $0.80$ | $0.87$ | $0.95$ | $0.94$
PBOPT(32) | 2.52 | | | | | | | 1.60 | | | | |
MEHPPV(16) | 2.48 | $1.12$ | $1.25$ | $1.64$ | $1.91$ | $2.04$ | | 1.30 | $1.03$ | $1.07$ | $1.18$ | $1.32$ | $1.25$
PFO(36) | 3.22 | $2.30$ | $2.45$ | $2.88$ | $3.12$ | $3.29$ | | 2.30 | $2.22$ | $2.24$ | $2.35$ | $2.46$ | $2.43$
DHOPPV(16) | 2.58 | $1.12$ | $1.25$ | $1.64$ | $1.92$ | $2.04$ | | 1.50 | $1.03$ | $1.07$ | $1.18$ | $1.32$ | $1.25$
PPY(24) | 3.4-3.9 | $2.08$ | $2.16$ | $2.61$ | $2.85$ | $3.01$ | | 2.4-2.5 | $2.02$ | $1.99$ | $2.11$ | $2.23$ | $2.20$
CN-MEHPPV(16) | 2.72 | $1.10$ | $1.32$ | $1.80$ | $2.10$ | $2.21$ | | N/A | $1.05$ | $1.21$ | $1.34$ | $1.48$ | $1.43$
PANi(20) | 2.00 | $2.33$ | $2.53$ | $3.03$ | $3.27$ | $3.41$ | | $<0.9$ | $2.30$ | $2.42$ | $2.62$ | $2.75$ | $2.73$
aFrom Ref. monkman01 . bNotation of Ref. birks is used. Note that all the
groups of -(CH2)nCH3 in polymers have been replaced with the hydrogen (-H).
Computational method: All our calculations were performed on the molecular-
structure code Gaussian 03 g03 . The initial geometries are prepared with
GaussView 4, while the dihedral angles are manually adjusted to be $\sim
30\,^{\circ}$. Then we optimize the geometries on respective ground-state DFT
methods. Finally we calculate the excitation energies from the optimized
ground-state geometries with the adiabatic TDDFT density functionals. For
consistency, basis set 6-31G was used in both ground-state and time-dependent
DFT calculations. In order to check whether our conclusion is affected by the
choice of basis set, we repeat our calculations for polymer P3OT using a
larger basis set 6-31G(d) that has diffusion functions. Our calculations show
that the excitation energy obtained with 6-31G(d) is larger only by $<0.2$ eV
than that obtained with 6-31G basis set. The excitation energies of the
polymers in benzene solvent are calculated with PCM (polarizable continuum
model) cmt97 . The polymers we study here have chain length of $\sim 10$ nm.
Since the groups of -(CH2)nCH3 only has little effect upon the properties of
the polymers ttz082 , these groups have been removed from the backbone of a
polymer and are, therefore, excluded in all calculations.
Table 1 shows the first singlet and triplet excitation energies of the
polymers in gas phase calculated with the adiabatic TDDFT. The experimental
results are also listed for comparison. Usually a polymer is of infinite chain
length. In practical calculations, we only choose several repeating monomeric
units. The number of “molecular” rings included in our calculations for each
polymer is given in the parentheses in Tables 1 and 3. These numbers are
chosen so that the lengths of the polymers are about 10 nm. This size effect
will be reduced by increasing the repeating units. However, adding the
repeating units will simultaneously increase the computational time. On the
other hand, high accuracy usually can be achieved by using large basis set,
which will result in significant increase in computational time. In the
present calculations, we use a basis set which is relatively smaller than
those used in small molecular calculations, and prepare the polymers with
moderate length of chain. This is a balanced choice between the size effect
and the accuracy we can tolerate.
From Table 1 we observe that, among the five adiabatic TDDFT methods, the
adiabatic PBE0 functional yields the most accurate excitation energies. This
is consistent with our previous studies ttz082 ; tt09 . We can see from Table
1 that the difference between the singlet and the triplet excitation energies,
$E_{S}-E_{T}$, is $\sim 0-0.1$ eV for LSDA, $\sim 0.1-0.2$ eV for meta-GGA,
$\sim 0.5$ eV for TPSSh, $\sim 0.6$ eV for B3LYP, and $\sim 0.8$ eV for PBE0.
The difference increases as the amount of exact exchange increases. However,
some studies suggest itc05 ; itc07 that for semilocal density functionals
(LSDA, GGA, and meta-GGA), this difference may vanish in the limit of infinite
chain length, a result similar to the performance of semilocal functionals for
solids. Mixing exact exchange into a semilocal functional will partly correct
the errors from self interaction, improve the asymptotic behavior of the XC
potential, and build in other many-body properties such as excitonic effects
itc05 ; itc07 which have not been taken into account properly in pure density
functional approximations and thus will lead to a finite difference in this
limit.
Interestingly, we find that when the theoretical dihedral angle is smaller
than the experimental dihedral angle, the TDDFT methods tend to underestimate
the excitation energies regardless of whether the excitation is singlet or
triplet. When the theoretical dihedral angle is close to the experimental one,
the TDDFT excitation energies are in good agreement with experiments. Our
calculations show that in rare cases, theoretical dihedral angles can be
greater that experimental estimates. In this case, the excitation energies are
overestimated by the TDDFT methods. A comparison of the dihedral angles
between theoretical and experimental estimates is displayed in Table 2. The
origin of torsional angles (or generally tortional disorder) of polymers is
complicated. It may arise from interchain interaction in amorphous polymeric
materials sergei1 ; sergei2 or from van der Waals interaction dion ;
scheffler between phenyl rings. These effects have not been properly taken
into account in current DFT methods.
The excitation energies of the polymers in benzene solvent are summarized in
Table 3. From Table 3, we can see that the lowest singlet-singlet excitation
energies in solution have a red shift of $\sim 0.01-0.05$ eV, compared to
those in gas phase (Table 1). This is consistent with what we have observed
for oligomers ttz082 ; tt09 . However, this trend does not apply to the
triplet excitation. Triplet excitation energies are nearly the same whether
the polymer is in gas phase or in solution.
In conclusion, we have investigated the lowest excitation energies of several
light-emitting conjugated polymers from the adiabatic TDDFT methods. Our
calculations show that the calculated excitation energies are in good
aggrement with experiments only when the theoretical torsions agree with
experimental estimates. If the theoretical dihedral angles are smaller than
the experiments, the TDDFT excitation energies tend to be underestimated. If
the theoretical dihedral angles are greater than the experiments, as in rare
case, the TDDFT excitation energies tend to be overestimated. Furthermore, we
find that, a semilocal functional without exact exchange mixing does not
satisfy the well-known “two-third” thumb rule relation between the singlet-
singlet and singlet-triplet excitation energies. For semilocal functionals,
the difference in energy between singlet state and triplet state is less than
0.1 eV for polymers with chain length of 10 nm and may vanish in the limit of
infinite chain length. Compared to semilocal functionals, hybrid functionals
yield much larger difference between singlet-singlet and singlet-triplet
excitation energies for polymers with finite chain length as well as with
infinite chain length. This difference increases with more exact exchange
mixed in semilocal functionals, and is nonzero even in the limit of infinite
chain length.
###### Acknowledgements.
The authors thank Richard Martin and John Perdew for valuable discussion and
suggestions. This work was carried out under the auspices of the National
Nuclear Security Administration of the U.S. Department of Energy at Los Alamos
National Laboratory under Contract No. DE-AC52-06NA25396, and was supported by
the LANL LDRD program.
## References
* (1) A. Kraft, A.C. Grimsdale, and A.B. Holmes, Angew. Chem. Int. Ed. 37, 402 (1998); T. Jüstel, H. Nikol, and C. Ronda, Angew. Chem. Int. Ed. 37, 3084 (1998); U. Mitschke and P. Bäuerle, J. Mater. Chem. 10, 1471 (2000); D.Y. Kim, H.N. Cho, and C.Y. Kim, in Progress in Polymer Science, v. 25 (Elsevier Science Ltd. 2000).
* (2) P. May, Phys. World 8, 52 (1995).
* (3) Time-Dependent Density Functional Theory, Lecture Notes in Physics, Vol. 706, edited by M.A.L. Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross (Springer, Berlin, 2006).
* (4) A. Zangwill and P. Soven, Phys. Rev. Lett. 45, 204 (1980).
* (5) J. Tao, S. Tretiak, and J.-X. Zhu, J. Phys. Chem. 112, 13701 (2008).
* (6) J. Tao and S. Tretiak, J. Chem. Theory Comput. 5, 866 (2009).
* (7) Dihedral angle is defined as the angle formed by two planes containing two adjacent phenyl or other “molecular” rings.
* (8) It has been shown that the largest source of error in the prediction of low-lying excitation energies arises from the approximation to the static XC potential [S.J.A. van Gisbergen, F. Kootstra, P.R.T. Schipper, O.V. Gritsenko, J.G. Snijders, and E.J. Baerends, J. Chem. Phys. 109, Phys. Rev. A 57, 2556 (1998)].
* (9) J. Tao, J.P. Perdew, V.N. Staroverov, and G.E. Scuseria, Phys. Rev. Lett. 91, 146401 (2003).
* (10) V.N. Staroverov, G.E. Scuseria, J.Tao, and J.P. Perdew, J. Chem. Phys. 119, 12129 (2003).
* (11) P.J. Stephens, F.J. Devlin, C.F. Chabalowski, and M.J. Frisch, J. Phys. Chem. 1994, 98, 11623.
* (12) C. Adamo, G.E. Scuseria, and V. Barone, J. Chem. Phys. 111, 2889 (1999); M. Ernzerhof and G.E. Scuseria, J. Chem. Phys. 1999, 110, 5029.
* (13) J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
* (14) S. Perun, J. Tatchen, and C.M. Marian, ChemPhysChem 9, 282 (2008).
* (15) K.A. Walters, K.D. Ley, and K.S. Shanze, Chem. Commun. 10, 1115 (1998).
* (16) Yu. V Romanovskii, A. Gerhard, B. Schweitzer, U. Scherf, R.I. Personov, and H. Bässler, Phys. Rev. Lett. 84, 1027 (2000).
* (17) S. Reindl and A. Penzkofer, Chem. Letts. 211, 431 (1996).
* (18) C.A. Parker, Photoluminescence of solutions (Elsevier, Amsterdam, 1968).
* (19) J.B. Birks, Photophysics of aromatic molecules (Wiley-Interscience, London, 1970).
* (20) H.D. Burrows, J. Seixas de Melo, C. Serpa, L.G. Arnaut, A.P. Monkman, I. Hamblett, and S. Navaratnam, J. Chem. Phys. 115, 9601 (2001).
* (21) H.D. Burrows, J. Seixas de Melo, C. Serpa, L.G. Arnaut, M. da G. Miguel, A.P. Monkman, I. Hamblett, and S. Navaratnam, Chem. Letts. 285, 3 (2002).
* (22) A.P. Monkman, H.D. Burrows, L.J. Hartwell, L.E. Horsburgh, I. Hamblett, and S. Navaratnam, Phys. Rev. Lett. 86, 1358 (2001).
* (23) M.J. Frisch et al., Gaussian, Inc., Pittsburgh PA (2003).
* (24) E. Cancès, B. Mennucci, and J. Tomasi, J. Chem. Phys. 107, 3032 (1997).
* (25) S. Tretiak, K. Igumenshchev and V. Chernyak, Phys. Rev. B 71, 033201 (2005).
* (26) K. Igumenshchev, S. Tretiak, and V. Chernyak, J. Chem. Phys. 127, 114902 (2007).
* (27) P. Yang, E.R. Batista, S. Tretiak, A. Saxena, R.L. Martin, and D.L. Smith, Phys. Rev. B 76, 241201(R) (2007).
* (28) S. Kilina, E.R. Batista, P. Yang, S. Tretiak, A. Saxena, R.L. Martin, and D.L. Smith, ACS Nano 2, 1381 (2008).
* (29) M. Dion, H. Rydberg, E. Schroder, D.C. Langreth, and B.I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004).
* (30) A. Tkatchenko and M. Scheffler, Phys. Rev. Lett. 102, 073005 (2009).
|
arxiv-papers
| 2009-07-03T16:50:09 |
2024-09-04T02:49:03.730384
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianmin Tao, Sergei Tretiak, and Jian-Xin Zhu",
"submitter": "Jianmin Tao",
"url": "https://arxiv.org/abs/0907.0673"
}
|
0907.0773
|
WHITTAKER MODULES FOR A LIE ALGEBRA OF BLOCK TYPE
Bin Wang, Xinyun Zhu
###### Abstract
In this paper, we study Whittaker modules for a Lie algebras of Block type. We
define Whittaker modules and under some conditions, obtain a one to one
correspondence between the set of isomorphic classes of Whittaker modules over
this algebra and the set of ideals of a polynomial ring, parallel to a result
from the classical setting and the case of the Virasoro algebra.
Keywords: Whittaker modules, Whittaker vectors.
MR(2000) Subject Classification: 17B10, 17B65, 17B68.
§1. Introduction
Let $\mathfrak{g}$ be a Lie algebra that admits a decomposition
$\mathfrak{g}=\mathfrak{b}_{-}\oplus\mathfrak{n}$
where $\mathfrak{b}_{-},\mathfrak{n}$ are two Lie subalgebras. Let
$\varphi:\mathfrak{n}\rightarrow\mathbb{C}$
be a homomorphism of Lie algebras. For a $\mathfrak{g}$ module $V$ and $v\in
V$, one says that $v$ is a Whittaker vector of type $\varphi$ if
$\mathfrak{n}$ acts on $v$ through $\varphi$. A Whittaker module is then
defined to be a module generated by a Whittaker vector.
The category of Whittaker modules for a given algebra (say, $\mathfrak{g}$)
admits an initial object and we call it a universal Whittaker module (say,
$M$). Then $M$ is isomorphic to $\mathcal{U}(\mathfrak{b}_{-})$ as
$\mathfrak{b}_{-}$ modules. Let $Z$ stand for the center of $\mathfrak{g}$,
and $Z^{\prime\prime}=Z\cap\mathfrak{b}_{-}$. Suppose $\mathfrak{g}$ possesses
the following properties:
1) for each ideal $I$ of $S(Z^{\prime\prime})$, then every Whittaker vector of
the Whittaker module $M/IM$ is of form $p\bar{w}$, with $p\in
S(Z^{\prime\prime})$, where $w$ is a Whittaker generator of M;
2) for each $I\subset S(Z^{\prime\prime})$, then any nontrivial submodule of
$M/IM$ admits a nonzero Whittaker vector.
Then it is not hard to set up a correspondence between the set of isomorphic
classes of Whittaker modules and the one of all the ideals of
$S(Z^{\prime\prime})$.
In this paper, we consider a Lie algebras of Block type, $\mathcal{B}$, which
is an infinite-dimensional Lie algebra with a basis
$\\{x_{a,i}\,|\,a\in\mathbb{Z},i\in\mathbb{N}\\}$ and brackets
$\displaystyle[x_{(a,i)},x_{(b,j)}]=\big{(}(b-1)i-(a-1)j\big{)}x_{(a+b,i+j-1)}.$
(1.1)
and it has the following decomposition
$\displaystyle\mathcal{B}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n},$
(1.2)
where
$\displaystyle\mathfrak{h}=\mbox{Span}_{\mathbb{C}}\\{x_{(a,i)},\,|\,a+i=1\\},$
$\displaystyle\mathfrak{n}=\mbox{Span}_{\mathbb{C}}\\{x_{(a,i)}\,|\,a+i>1\\},$
$\displaystyle\mathfrak{n}_{-}=\mbox{Span}_{\mathbb{C}}\\{x_{(a,i)}\,|\,a+i<1\\}.$
Suppose $\varphi$ is a given good character ( for its definition, see 2.3.2).
Our main result is to show that $\mathcal{B}$ satisfies the above properties
1) and 2) for such $\varphi$ (see §3), and hence obtain a correspondence
between Whittaker modules and ideals of a polynomial ring (of $x_{1,0}$). This
is treated in §4. Note that for general characters, property 1) may not hold.
Whittaker modules were first discovered for $\mathfrak{sl}_{2}{(\mathbb{C})}$
by Arnal and Pinzcon in [1]. Block showed, in [3] that the simple modules for
$\mathfrak{sl}_{2}(\mathbb{C})$ consist of highest (lowest) weight modules,
Whittaker modules and a third family obtained by localization. This
illustrates the prominent role played by Whittaker modules.
Kostant defined Whittaker modules for an arbitrary finite-dimensional complex
semi-simple Lie algebra $\mathfrak{g}$ in [5], and showed that these modules,
up to isomorphism, are in bijective correspondence with ideals of the center
$Z(\mathfrak{g})$. In particular, irreducible Whittaker modules correspond to
maximal ideals of $Z(\mathfrak{g})$. In the quantum setting, Whittaker modules
have been studied by Sevoystanov for $\mathcal{U}_{h}(\mathfrak{g})$ [9] and
by M. Ondrus for $U_{q}(\mathfrak{sl}_{2})$ in [7]. Recently Whittaker modules
have also been studied by M. Ondrus and E. Wiesner for the Virasoro algebra in
[8], X. Zhang and S. Tan for Schrödinger-Virasoro algebra in [12], K.
Christodoulopoulou for Heisenberg algebras in [4], and by G. Benkart and M.
Ondrus for generalized Weyl algebras in [2].
We note that our proofs differ from the ones in the classical setting in the
use of the center of the universal enveloping algebra. The reasoning for this
is similar to the one explained in [8]. Also, our approach to obtaining
property 2) is same as in [10], different from [8].
The paper is organized in the following way. In section 2, we define Whittaker
vectors and Whittaker modules for a class of Lie algebras, and also construct
a universal Whittaker module for them. Then the Whittaker vectors in a
universal Whittaker module are examined in section 3 and the irreducible
Whittaker modules are classified in section 4. In the last section we discuss
some examples.
§2. Preliminaries
2.1. Q-graded Lie algebras
2.1.1. Let $V$ be a vector space over $\mathbb{C}$ and $Q$ a free abelian
additive semigroup. By a $Q$-grading of $V$ we will understand a family
$\\{V_{\alpha}|\alpha\in Q\\}$ of subspaces of $V$ such that
$V=\oplus_{\alpha\in Q}V_{\alpha}$. For a nonzero vector $v\in V_{\alpha}$, we
say $v$ is a homogeneous vector of degree $\alpha$. Let $\mathfrak{g}$ be a
Lie algebra over $\mathbb{C}$ and let $\\{\mathfrak{g}_{\alpha}\,|\,\alpha\in
Q\\}$ be a grading of $\mathfrak{g}$ (as a vector space). Call $\mathfrak{g}$
a $Q$-graded Lie algebra if
$[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]\subset\mathfrak{g}_{\alpha+\beta}$,
for all $\alpha,\beta\in Q$.
Now suppose $Q$ is totally ordered abelian group by the ordering $\leq$ that
is compatible with its additive group structure. Given a $Q$-graded Lie
algebra, $\mathfrak{g}=\oplus_{\alpha\in Q}\mathfrak{g}_{\alpha}$, and a
homomorphism of abelian groups $\pi:Q\rightarrow\mathbb{Z}$ that preserves the
ordering, write
$\mathfrak{g}_{m}=\sum\limits_{\pi(\alpha)=m}\mathfrak{g}_{\alpha}$. Then the
Lie algebra $\mathfrak{g}=\oplus_{i\in\mathbb{Z}}\mathfrak{g}_{i}$ can be
viewed as a $\mathbb{Z}$-graded Lie algebra too.
2.1.2. We now consider the algebra $\mathcal{B}$ defined by $\eqref{b1}$. Let
$Q=\mathbb{Z}\times\mathbb{Z}$, and $\pi:Q\rightarrow\mathbb{Z}$ by
$\pi((a,i))=a+i-1$. Equip $Q$ with a group structure by
$(a,i)*(b,j)=(a+b,i+j-1)$
and a ordering by
$(a,i)<(b,j)\,\,\mbox{if either}\,\,a+i<b+j\,\,\mbox{or}\,\,a+i=b+j,i<j.$
Then $Q$ becomes a totally ordered abelian group, and $\pi$ preserves the
ordering and group structure. Let $Q^{\prime}=\\{\alpha\in
Q\,|\,\pi(\alpha)>0\\},Q^{\prime\prime}=\\{\alpha\in Q\,|\,\pi(\alpha)\leq
0\\}$ and $K_{n}=\\{(a,i)\in Q\,|\,i\geq n\\},n\geq 0$. Set
$K^{\prime}_{n}=K_{n}\cap Q^{\prime},K^{\prime\prime}_{n}=K_{n}\cap
Q^{\prime\prime}$ and $R=\\{(1,0)\\}$. So $\mathcal{B}$ (resp. $\mathfrak{n}$,
resp. $\mathfrak{b}_{-}$) is a $Q$-graded (resp. $Q^{\prime}$-graded, resp.
$Q^{\prime\prime}$-graded) Lie algebra. Write
$K=K_{0},K^{\prime}=K^{\prime}_{0},K^{\prime\prime}=K^{\prime\prime}_{0}.$
2.2. Partitions.
2.2.1. Let $\Lambda$ be a totally ordered set. We define a partition of
$\Lambda$ to be a non-decreasing sequence of elements of $\Lambda$,
$\mu=(\mu_{1},\mu_{2},\cdots,\mu_{r}),\,\,\mu_{1}\leqslant\mu_{2}\leqslant\cdots\leqslant\mu_{r}.$
Denote by $\mathcal{P}(\Lambda)$ the set of all partitions. For
$\lambda=(\lambda_{1},\cdots,\lambda_{r})\in\mathcal{P}(\Lambda)$, we define
the length of $\lambda$ to be $r$, denoted by $\ell(\lambda)$, and for
$\alpha\in\Lambda$, let $\lambda(\alpha)$ denote the number of times $\alpha$
appears in the partition. Clearly any partition $\lambda$ is completely
determined by the values $\lambda(\alpha),\alpha\in\Lambda$. If all
$\lambda(\alpha)=0$, call $\lambda$ the null partition, denoted by $\bar{0}$.
Note that $\bar{0}$ is the only partition of length $=0$. We consider
$\bar{0}$ an element of $\mathcal{P}(\Lambda)$.
Back to the situation of a Lie algebra $\mathfrak{g}$. Define the symbols
$x_{\lambda}$, for all partitions. For $\bar{0}\neq\lambda$, define
$x_{\lambda}$ to be an element of $\mathcal{U}(\mathfrak{g})$, the universal
enveloping algebra of $\mathfrak{g}$, by
$x_{\lambda}=x_{\lambda_{1}}x_{\lambda_{2}}\cdots
x_{\lambda_{r}}=\prod\limits_{\alpha\in
K}x_{\alpha}^{\lambda(\alpha)}\in\mathcal{U}(\mathfrak{g})$
whenever each $x_{\lambda_{i}}$ is well understood as an element of
$\mathfrak{g}$. And let $x_{\bar{0}}=1\in\mathcal{U}{(\mathfrak{g})}$.
By PBW theorem, we know that
$\\{x_{\lambda}\,|\,\lambda\in\mathcal{P}(K\setminus R)\\}$ form a basis of
$\mathcal{U}(\mathcal{B})$ over $S(Z)$ where $Z=\mathbb{C}x_{(1,0)}$ and
$S(Z)$ is the polynomial ring of $x_{(1,0)}$.
2.2.2. $\mathcal{U}{(\mathfrak{g})}$ (denoted by $\mathcal{U}$) naturally
inherits a grading from the one of $\mathfrak{g}$. Namely, for any $\alpha\in
Q$, set $\mathcal{U}_{\alpha}=Span_{\mathbb{C}}\\{x_{1}x_{2}\cdots
x_{k}\,|\,x_{i}\in\mathfrak{g}_{\alpha_{i}},1\leq i\leq
k,\sum\limits_{i=1}^{k}\alpha_{i}=\alpha\\}$, and then
$\mathcal{U}=\bigoplus\limits_{\alpha\in Q}\mathcal{U}_{\alpha}$ is a
$Q$-graded algebra, i.e.
$\mathcal{U}_{\alpha}\mathcal{U}_{\beta}\subseteq\mathcal{U}_{\alpha+\beta}$.
Similarly, $\mathcal{U}(\mathfrak{n})$ (resp.
$\mathcal{U}({\mathfrak{b}_{-}})$) inherit a grading from $\mathfrak{n}$,
(resp. $\mathfrak{b}_{-}$ ). If $x\in\mathcal{U}_{\alpha}$, then we say $x$ is
a homogeneous element of degree $\alpha$. Set $|\bar{0}|=0$ and
$|\lambda|=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{{\ell}(\lambda)},\,\forall\lambda\neq\bar{0}$.
Then $x_{\lambda}$ is a homogeneous element of degree $|\lambda|$. If $u(\neq
0)$ is not homogeneous but a sum of finitely many nonzero homogeneous
elements, then denote by $mindeg(u)$ the minimum degree of its nonzero
homogeneous components.
Now let us, for convenience, call any product of elements $x_{\alpha}^{s}$ (
$\alpha\in K\setminus R,s\geq 0$) in $\mathcal{U}$ and elements of $S(Z)$ a
monomial, of height equal to the sum of the various $s$’s occurring. Then we
have, by PBW theorem,
Lemma. For $\alpha,\beta\in K\setminus R,t,k\geq 0$ ,
$x_{\beta}^{t}x_{\alpha}^{k}$ is a $S(Z)$-linear combination of
$x_{\alpha}^{k}x_{\beta}^{t}$ along with other monomials of height $<t+k$.
$\Box$
This allows us to make the following definition. If $x\in\mathcal{U}$ is a sum
of monomials of height $\leq l$, we say $ht(x)\leq l$.
2.2.3. We need some more notation. For
$\lambda=(\lambda_{1},\lambda_{2},\cdots\lambda_{r})\in\mathcal{P}(K),0<i\leq
r,0\leq j<r,$write
$\displaystyle\lambda\\{i\\}=(\lambda_{1},\cdots,\lambda_{i}),\,\lambda\\{0\\}=\bar{0}$
$\displaystyle\lambda[j]=(\lambda_{j+1},\cdots,\lambda_{r}),\,\lambda[r]=\bar{0}$
$\displaystyle\lambda<i>=(\lambda_{1},\cdots,\lambda_{i-1},\hat{\lambda_{i}},\lambda_{i+1},\cdots\lambda_{r}).$
Lemma Write $\mathcal{U}^{\prime\prime}$ for $\mathcal{U}(\mathfrak{b}_{-})$.
Let $0\neq x\in\mathfrak{n}_{\beta},0\neq
y\in\mathcal{U}^{\prime\prime}_{\gamma}$ with $\pi(\beta)>0,\pi(\gamma)\leq
0$.
1) if $s=\pi(\beta+\gamma)>0$, then
$[x,y]=\sum\limits_{s\leq\pi(\alpha)\leq\pi(\beta)}u_{\alpha}$ with
$u_{\alpha}=\sum\limits_{i}v^{(\alpha,i)}w^{(\alpha,i)}$ where
$w^{(\alpha,i)}\in\mathfrak{n}_{\alpha},v^{(\alpha,i)}\in\mathcal{U}^{\prime\prime}_{\beta+\gamma-\alpha}$
and $ht(v^{(\alpha,i)})<ht(y)$ if $v^{(\alpha,i)}\neq 0$;
2)if $\pi(\beta+\gamma)\leq 0$, then
$[x,y]=\sum\limits_{0<\pi(\alpha)\leq\pi(\beta)}u_{\alpha}+u$ with
$u\in\mathcal{U}^{\prime\prime}_{\beta+\gamma}$ and
$u_{\alpha}=\sum\limits_{i}v^{(\alpha,i)}w^{(\alpha,i)}$ where
$w^{(\alpha,i)}\in\mathfrak{n}_{\alpha},v^{(\alpha,i)}\in\mathcal{U}^{\prime\prime}_{\beta+\gamma-\alpha}$
and $ht(v^{(\alpha,i)})<ht(y)$ if $v^{(\alpha,i)}\neq 0$.
Proof Write $y=\sum_{\lambda}f_{\lambda}x_{\lambda}$ where $f_{\lambda}\in
S(Z),\lambda\in\mathcal{P}(K^{\prime\prime}\setminus R)$.
But for any $x\in\mathfrak{n}_{\beta}$, one has,
$\displaystyle[x,x_{\lambda}]$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\ell(\lambda)}x_{\lambda\\{i-1\\}}[x,x_{\lambda_{i}}]x_{\lambda[i]}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{\ell(\lambda)}x_{\lambda<i-1>}[x,x_{\lambda_{i}}]+\sum_{i=1}^{\ell(\lambda)}x_{\lambda\\{i-1\\}}[[x,x_{\lambda_{i}}],x_{\lambda[i]}]$
Then one can easily deduce that the lemma follows. $\Box$
2.3 Whittaker Module
2.3.1. Definition Given a Lie algebra homomorphism
$\varphi:\mathfrak{n}\rightarrow\mathbb{C}$, for a $\mathcal{B}$-module $V$, a
vector $v\in V$ is said to be a Whittaker vector of type $\varphi$ if
$xv=\varphi(x)v$ for all $x\in\mathfrak{n}$. Furthermore, if $v$ generates
$V$, then we call $V$ a Whittaker module of type $\varphi$ and $v$ a cyclic
Whittaker vector of $V$.
2.3.2. One says a Lie algebra homomorphism
$\varphi:\mathfrak{n}\rightarrow\mathbb{C}$ is nonsingular, if
$\varphi(x_{(a,i)})\neq 0$, for all $(a,i)\in Q$ with $a+i=2$. For $n\geq
1,s\geq 1$, let $\varphi_{m,j}^{(n,s)}=\varphi(x_{(2-a,a)})$, where
$a=m+j+s-1$, for $m\geq 1,0\leq j\leq n$. Denote by $H^{(n,s)}$ the
$\infty\times(n+1)$ matrix whose $(m,j)$ entry is $\varphi_{m,j}^{(n,s)}$.
Definition A Lie algebra homomorphism
$\varphi:\mathfrak{n}\rightarrow\mathbb{C}$ is said to be a good character if
$\varphi$ is nonsingular and for all $n\geq 1,s\geq 1$, rank$(H^{(n,s)})=n+1$.
For a given $\varphi:\mathfrak{n}\rightarrow\mathbb{C}$, define
$\mathbb{C}_{\varphi}$ to be the one-dimensional $\mathfrak{n}$-module given
by the action $xa=\varphi(x)a$ for all $x\in\mathfrak{n}$ and
$a\in\mathbb{C}$. Then the induced $\mathcal{B}$-module,
$\displaystyle
M_{\varphi}=\mathcal{U}(\mathcal{B})\otimes_{\mathcal{U}(\mathfrak{n})}\mathbb{C}_{\varphi},$
is a Whittaker module of type $\varphi$ with the cyclic Whittaker vector
$w={\bf 1}\otimes 1$. By PBW theorem, it’s easy to see that
$\\{x_{\lambda}w\,|\,\lambda\in\mathcal{P}(K_{0}^{\prime\prime}\setminus
R)\\}$ is a basis of $M_{\varphi}$ over $S(Z)$ where $Z=\mathbb{C}x_{(1,0)}.$
Besides, for any ideal $I$ of $S(Z)$, define
$L_{\varphi,I}=M_{\varphi}/IM_{\varphi}$ and denote by $p_{I}$ the canonical
homomorphism. Then $L_{\varphi,I}$ is a Whittaker module for $\mathcal{B}$.
The following lemma makes $M_{\varphi}$ become a universal Whittaker module.
Lemma Fix $\varphi$ and $M_{\varphi}$ as above. Let $V$ be a Whittaker module
of type $\varphi$ generated by a Whittaker vector $w^{\prime}$. Then there is
a unique map $\phi:M_{\varphi}\rightarrow V$ taking $w=1\otimes 1$ to
$w^{\prime}$.
Proof. Uniqueness is obvious. Consider $u\in\mathcal{U}(\mathcal{B})$. One can
write, by PBW,
$u=\sum\limits_{\alpha}b_{\alpha}n_{\alpha},\,\,b_{\alpha}\in\mathcal{U}(\mathfrak{b}_{-}),n_{\alpha}\in\mathcal{U}(\mathfrak{n})$
If $uw=0$, then $uw=\sum\limits_{\alpha}b_{\alpha}\varphi(n_{\alpha})w=0$, and
therefore $\sum\limits_{\alpha}b_{\alpha}\varphi(n_{\alpha})=0$. Now it’s easy
to see that the map $\phi:M_{\varphi}\rightarrow V$, defined by
$\phi(uw)=uw^{\prime}$, is well defined. $\Box$
2.3.3. Let $A=S(Z),Z=\mathbb{C}x_{(1,0)}$ and $I$ be an ideal of $A$. Write
$M=M_{\varphi},\mathcal{P}_{\leq 0}=\mathcal{P}(K^{\prime\prime}\setminus R)$
and $w^{\prime}=p_{I}w\in M/IM.$
Lemma $M/IM$ admits a basis,
$\\{x_{\lambda}w^{\prime}\,|\,\lambda\in\mathcal{P}_{\leq 0}\\},$ over $A/I$.
Proof Note that $M=\bigoplus\limits_{\lambda\in\mathcal{P}_{\leq
0}}Ax_{\lambda}w$. Hence,
$M/IM=A/I\otimes_{A}M=A/I\otimes_{A}(\bigoplus_{\lambda\in\mathcal{P}_{\leq
0}}Ax_{\lambda}w)=\bigoplus_{\lambda\in\mathcal{P}_{\leq
0}}(A/I)x_{\lambda}w^{\prime}.$
Then the lemma follows immediately. $\Box$
2.3.4. Assume now that $I$ is an ideal of $A=S(Z)$. Write
$\mathcal{U}^{\prime}$ for $\mathcal{U}(\mathfrak{n})$, and
$\mathcal{U}^{\prime\prime}$ for $\mathcal{U}(\mathfrak{b}_{-})$. Then
$V=M/IM$ has a natural grading as a vector space. Namely, based on Lemma
2.3.2, let, for any $\alpha\in Q^{\prime\prime}$,
$V_{\alpha}=\\{x=\sum\limits_{\lambda\in\mathcal{P}_{\leq
0}}a_{\lambda}x_{\lambda}w\,|\,a_{\lambda}\in(A/I),a_{\lambda}=0\,\,\mbox{if}\,\,|\lambda|\neq\alpha\\}$
and then clearly $V=\bigoplus\limits_{\alpha\in Q^{\prime\prime}}V_{\alpha}$.
We say that a nonzero homogeneous vector $v$ in $M$ is of degree $\alpha$ if
$v\in V_{\alpha}$. If $v\,(\neq 0)$ is not homogeneous but a sum of finitely
many nonzero homogeneous vectors, then define $mindeg(v)$ to be the minimum
degree of its nonzero homogeneous components. Meanwhile, for any nonzero
vector $v\in V$, let $d(v)=mindeg(v)$ and then there uniquely exist
$v_{i}\in\mathcal{U}^{\prime\prime}_{\alpha},\alpha\geq d(v)$ such that
$v=\sum\limits_{d(v)\leq\alpha}v_{\alpha}w$ with $v_{d(v)}\neq 0$. Then define
$\ell(v)=ht(v_{d(v)})$. Note $V$ can also be equipped with a
$\mathbb{Z}$-grading through $\pi:Q\rightarrow\mathbb{Z}$. With this grading,
we can introduce notation $mindeg_{1}(v)$ and $\ell_{1}(v)$, for each $0\neq
v\in V$, parallel to $mindeg(v)$ and $\ell(v)$ respectively.
§3. Whittaker Vectors in $M_{\varphi}$ and $L_{\varphi,I}$
In this section, we characterize the Whittaker vectors in Whittaker modules
for $\mathcal{B}$, where $\varphi$ is a fixed nonsingular Lie algebra
homomorphism from $\mathfrak{n}\rightarrow\mathbb{C}$. Let
$M=M_{\varphi},w={\bf 1}\otimes 1$, and
$Z=\mathfrak{Z}(\mathcal{B})=\mathbb{C}x_{(1,0)}$, the center of
$\mathcal{B}$. Notation as in 2.1.2.
3.1. Assume that $I$ is an ideal of $A=S(Z)$. Set $V=M/IM$ and
$w^{\prime}=p_{I}(w)$. Write
$\mathcal{P}_{1}=\mathcal{P}(K_{1}^{\prime\prime}),\mathcal{P}_{2}=\mathcal{P}(\bar{K}\setminus
R)$ where $\bar{K}=K^{\prime\prime}\setminus K^{\prime\prime}_{1}$. Then we
have the following lemma.
Lemma Assume $\varphi$ is a good character, then every Whittaker vector of $V$
is of form $pw^{\prime}$ with $p\in A=S(Z)$.
Proof Suppose $w^{\prime\prime}$ is a Whittaker vector of $V$. We can write,
by Lemma 2.3.3,
$\displaystyle
w^{\prime\prime}=\sum_{\lambda\in\mathcal{P}_{1},\mu\in\mathcal{P}_{2}}p_{\lambda,\mu}x_{\lambda}x_{\mu}w^{\prime}$
(3.1)
where $p_{\lambda,\mu}\in A/I$. Obviously it is enough to show that
$p_{\lambda,\mu}=0$ if either $\lambda\neq\bar{0}$ or $\mu\neq\bar{0}$.
Case a), suppose there exists a $\mu\neq\bar{0}$ such that
$p_{\lambda,\mu}\neq 0$ for some $\lambda\in\mathcal{P}_{1}.$
Let $\Lambda_{1}=\\{(a,i)\,|\,a+i\leq 0,i\geq
1\\},\Lambda_{2}=\\{(a,i)\,|\,a+i\leq 1,i\geq 2\\}$ and
$\Lambda=\Lambda_{1}\cup\Lambda_{2}.$ Then
$K^{\prime\prime}_{1}=\Lambda\cup\\{(0,1)\\}$. Set
$\mathcal{P}^{\prime}_{1}=\mathcal{P}(\Lambda)$. Then obviously one can
rewrite equation (3.1) as
$\displaystyle
w^{\prime\prime}=\sum_{\lambda\in\mathcal{P}^{\prime}_{1},\mu\in\mathcal{P}_{2}}\sum_{s\geq
0}q_{\lambda,\mu,s}x_{\lambda}y^{s}x_{\mu}w^{\prime}$ (3.2)
where $y=x_{(0,1)},q_{\lambda,\mu,s}\in A/I$. Then there exists a $\mu\neq 0$
such that $q_{\lambda,\mu,s}\neq 0$ for some $\lambda,s$ by our assumption.
Now take a $N>2$ so that for any $\lambda\in\mathcal{P}^{\prime}_{1}$, if
$\exists\,\lambda_{i}=(a,j)$ with $j>N-2$, then $q_{\lambda,\mu,s}=0.$
Obviously this can be achieved since there are only finitely many nonzero
$q_{\lambda,\mu,s}$. Put $u=x_{(2-N,N)}.$ Consider
$\displaystyle(u-\varphi(u))w^{\prime\prime}=$ $\displaystyle\sum$
$\displaystyle q_{\lambda,\mu,s}[u,x_{\lambda}y^{s}x_{\mu}]w^{\prime}$
$\displaystyle=$ $\displaystyle\sum$ $\displaystyle
q_{\lambda,\mu,s}[u,x_{\lambda}]y^{s}x_{\mu}w^{\prime}$ $\displaystyle+$
$\displaystyle\sum q_{\lambda,\mu,s}x_{\lambda}[u,y^{s}]x_{\mu}w^{\prime}$
$\displaystyle+$ $\displaystyle\sum
q_{\lambda,\mu,s}x_{\lambda}y^{s}[u,x_{\mu}]w^{\prime}.$
But note that it can be easily showed by induction that $[u,y^{s}]=f_{s}(y)u$,
where $f_{s}(y)$ is a polynomial of $y$ with $deg(f_{s}(y))\leq s-1.$
Therefore,
$\displaystyle(u-\varphi(u))w^{\prime\prime}=$ $\displaystyle\sum$
$\displaystyle q_{\lambda,\mu,s}[u,x_{\lambda}]y^{s}x_{\mu}w^{\prime}$ (3.3)
$\displaystyle+$ $\displaystyle\sum
q_{\lambda,\mu,s}x_{\lambda}f_{s}(y)x_{\mu}\varphi(u)w^{\prime}+\sum
q_{\lambda,\mu,s}x_{\lambda}f_{s}(y)[u,x_{\mu}]w^{\prime}$ (3.4)
$\displaystyle+$ $\displaystyle\sum
q_{\lambda,\mu,s}x_{\lambda}y^{s}[u,x_{\mu}]w^{\prime}.$ (3.5)
Let $\pi_{1}:Q\rightarrow\mathbb{Z}$ by $\pi_{1}(a,i)=a.$ Set
$t=min\\{\pi_{1}(\mu_{1})\,|\,\mu\in\mathcal{P}_{2},\mbox{and}\,\,q_{\lambda,\mu,s}\neq
0,\,\mbox{for some}\,\,\lambda,s.\\}$
Then $t\leq 0.$ Take a $\tau\in\mathcal{P}_{2}$ such that $\tau_{1}=(t,0)$ and
$q_{\lambda,\mu,s}\neq 0$ for some $\lambda,s.$ Let $r$ be the maximum integer
such that there exists a $\lambda$, s.t. $q_{\lambda,\tau,r}\neq 0$. Set
$\tau^{\prime}=\tau<1>=(\tau_{2},\cdot\cdot\cdot\tau_{\ell(\tau)})$ and
$\alpha=(t-N+2,N-1).$
Note that
$\\{x_{\lambda}y^{k}x_{\mu}\,|\,\lambda\in\mathcal{P}^{\prime}_{1},\mu\in\mathcal{P}_{2},s\geq
0\\}$ form a basis of $V=M/IM$ over $A/I$. Clearly, under this basis, the
representation of the formula $\eqref{f3}$ contains nonzero terms involving
$x_{\alpha}y^{r}x_{\tau^{\prime}}$ which are linearly independent from each
other. However, it is easy to see that for $\eqref{f1}$ and $\eqref{f2}$,
there are no terms involving $x_{\alpha}y^{r}x_{\tau^{\prime}}$. Hence,
$(u-\varphi(u))w^{\prime\prime}\neq 0.$ This contradicts with the assumption
that $w^{\prime\prime}$ is a Whittaker vector.
Case b), suppose $p_{\lambda,\mu}=0$ whenever $\mu\neq\bar{0}$, and there
exists at least a $\lambda\neq\bar{0}$ such that $p_{\lambda,\bar{0}}\neq 0.$
Let $L=\\{(a,i)\,|\,a+i=1,i\geq 1\\}$ and $L^{\prime}=\\{(a,i)\,|\,a+i<1,i\geq
1\\}$. Set $\mathcal{Q}=\mathcal{P}(L)$ and
$\mathcal{Q}^{\prime}=\mathcal{P}(L^{\prime})$. Then one can rewrite equation
(3.1) as
$\displaystyle
w^{\prime\prime}=\sum_{\lambda\in\mathcal{Q}^{\prime},\mu\in\mathcal{Q}}f_{\lambda,\mu}x_{\lambda}x_{\mu}w^{\prime}$
where $f_{\lambda,\mu}\in A/I$.
i). Assume that there exists a $\lambda(\neq\bar{0})\in\mathcal{Q}^{\prime}$
such that $f_{\lambda,\mu}\neq 0$ for some $\mu\in\mathcal{Q}$.
Define $\pi_{2}:Q\rightarrow\mathbb{Z}$ by $\pi_{2}(a,i)=i$. Take a $n_{0}>0$
such that $f_{\lambda,\mu}=0$ if either there is a $i$ such that
$n_{0}\leq\pi_{2}(\lambda_{i})$ or there is a $j$ such that
$n_{0}\leq\pi_{2}(\mu_{j})$. Let
$\alpha_{0}=max\\{\lambda_{\ell(\lambda)}\,|\,\exists\,\mu
s.t.f_{\lambda,\mu}\neq 0\\},\tau_{0}=(2-n_{0},n_{0})$, and $y=x_{\tau_{0}}$.
Consider
$\displaystyle(y-\varphi(y))w^{\prime\prime}=$
$\displaystyle\sum_{\lambda,\mu}f_{\lambda,\mu}[y,x_{\lambda}x_{\mu}]w^{\prime}$
$\displaystyle=$ $\displaystyle\sum
f_{\lambda,\mu}[y,x_{\lambda}]x_{\mu}w^{\prime}$ (3.7) $\displaystyle+\sum
f_{\lambda,\mu}x_{\lambda}[y,x_{\mu}]w^{\prime}.$
Note that
$\triangle=\\{x_{\lambda}x_{\mu}x_{\gamma}\,|\,\lambda\in\mathcal{Q}^{\prime},\mu\in\mathcal{Q},\gamma\in
pp_{2}\\}$ form a basis of $V=M/IM$ over $A/I$. Clearly, under this basis, the
representation of the formula $\eqref{g1}$ contains nonzero terms involving
$x_{\tau_{0}*\alpha_{0}}$ which are linearly independent from each other.
However, it is easy to see that for $\eqref{g2}$, there are no terms involving
$x_{\tau_{0}*\alpha_{0}}$. Hence, $(u-\varphi(u))w^{\prime\prime}\neq 0.$ This
contradicts with the assumption that $w^{\prime\prime}$ is a Whittaker vector.
ii). Assume $f_{\lambda,\mu}=0$ if $\lambda\neq\bar{0}$, and there exists at
least a $\mu\neq\bar{0}$ such that $f_{\bar{0},\mu}\neq 0$.
In this case, we write
$\displaystyle
w^{\prime\prime}=\sum_{\mu\in\mathcal{Q}}b_{\mu}x_{\mu}w^{\prime}$
where $b_{\mu}=f_{\bar{0},\mu}$.
Let $\sigma_{0}=(1-s,s)$ for some $s\geq 1$, and
$\sigma_{0}^{\prime}=(1-s-n,s+n)$ for some $n\geq 1$ be such that for all
$\mu\in\mathcal{Q}$, if either $\mu_{1}<\sigma_{0}$ or
$\mu_{\ell(\mu)}>\sigma_{0}^{\prime}$, then $b_{\mu}=0$. Set
$y_{m}=x_{(2-m,m)},m\geq 1$. Consider
$\displaystyle(y_{m}-\varphi(y_{m}))w^{\prime\prime}=$
$\displaystyle\sum_{\mu}b_{\mu}[y_{m},x_{\mu}]w^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{\mu}\sum_{i}^{\ell(\mu)}b_{\mu}x_{\mu\\{i-1\\}}[y_{m},x_{\mu_{i}}]x_{\mu[i]}w^{\prime}$
$\displaystyle=$
$\displaystyle\sum_{\mu}\sum_{i}^{\ell(\mu)}b_{\mu}x_{\mu<i>}\varphi([y_{m},x_{\mu_{i}}])w^{\prime}$
(3.9)
$\displaystyle+\sum_{\mu}\sum_{i=1}^{\ell(\mu)}b_{\mu}x_{\mu\\{i-1\\}}[[y_{m},x_{\mu_{i}}],x_{\mu[i]}]w^{\prime}.$
Let $l=max\\{\ell(\mu)\,|\,b_{\mu}\neq 0\\}$ and take a
$\lambda\in\mathcal{Q}$ such that $\ell(\lambda)=l-1$, and $\exists\,\mu$ s.t.
$b_{\mu}\neq 0,\lambda=\mu<i>$ for some $i$. Now set
$\sigma_{i}=(1-s-i,s+i),0\leq i\leq n$ and let $0\leq i_{1}\leq
i_{2}\leq\cdots\leq i_{k}$ be such that $\\{\sigma_{i_{j}}\,|\,1\leq j\leq
k\\}=\\{\alpha\,|\,\lambda(\alpha)\neq 0,\alpha\in L\\}$ and
$t_{j}=\lambda(\sigma_{i_{j}})$. So,
$\lambda=(\sigma_{i_{1}},\cdots,\sigma_{i_{1}},\sigma_{i_{2}},\cdots,\sigma_{i_{2}},\cdots,\sigma_{i_{k}},\cdots,\sigma_{i_{k}})$
where $\sigma_{i_{j}}$ appears $t_{j}$ times. Note also that
$t_{1}+t_{2}+\cdots t_{k}=l-1$. Moreover, for $0\leq a\leq n$, if $a\neq
i_{j},\forall j$, then define $\lambda^{(a)}$ to be the partition such that
$\lambda^{(a)}(\sigma_{a})=1,\lambda^{(a)}(\sigma_{i_{j}})=t_{j}$; if
$a=i_{v}$ for some $v$, then define $\lambda^{(a)}$ to be the partition such
that $\lambda^{(a)}(\sigma_{i_{v}})=t_{v}+1$, and
$\lambda^{(a)}(\sigma_{i_{j}})=t_{j},j\neq v$. Clearly
$\ell(\lambda^{(a)})=l$. Observe that for each $\mu\in\mathcal{Q}$ with
$b_{\mu}\neq 0$ and $\mu<i>=\lambda$ for some $i$, we have $\mu=\lambda^{(a)}$
for some $a$.
Now it is easy to see that the coefficient $c_{\lambda}$ of
$x_{\lambda}w^{\prime}$ in the representation of the formula $\eqref{h1}$,
under the basis $\triangle$, is
$\displaystyle
c_{\lambda}=\sum_{j=0}^{n}b_{\lambda^{(j)}}(-s-j)d_{j}\varphi_{m,j}$
where $d_{i_{v}}=t_{v}+1,1\leq v\leq k$, $d_{j}=1$ if $j\neq
i_{1},i_{2},...,i_{k}$ and $\varphi_{m,j}=\varphi_{m,j}^{(n,s)}$ that is
defined in 2.3.2. Meanwhile, one immediately deduces that
$x_{\lambda}w^{\prime}$ does not appear in the representation of the formula
$\eqref{h2}$. Hence $c_{\lambda}=0$, since $w^{\prime\prime}$ is a Whittaker
vector, that is, for all $m\geq 1$,
$\displaystyle\sum_{j=0}^{n}b_{\lambda^{(j)}}(-s-j)d_{j}\varphi_{m,j}=0.$
(3.10)
Then the assumption that $\varphi$ is a good character implies that
$b_{\lambda^{(j)}}=0,0\leq j\leq n.$ But this contradicts with the choice of
$\lambda$. $\Box$
3.2. Lemma Any nontrivial submodule of $V=M/IM$ contains a nonzero Whittaker
vector.
Proof Let $V_{1}$ be a submodule of $V$. Suppose $V_{1}$ contains no nonzero
Whittaker vector. Use the notation in 2.3.4. Let
$t=max\\{mindeg_{1}(v)\,|\,v\neq 0,v\in
V_{1}\\},l=min\\{\ell_{1}(v)\,|\,mindeg_{1}(v)=t,v\in V_{1},v\neq 0\\}$. Take
a $u\in V_{1}$ such that $mindeg_{1}(u)=t,\ell_{1}(u)=l$ (clearly, $l>0$ ).
Write $u=\sum\limits_{0\geq a\geq t}u_{a}w^{\prime}$, where
$u_{a}=\sum\limits_{\pi(|\lambda|)=a}p_{\lambda,a}x_{\lambda}$, with
$p_{\lambda,a}\in A/I$. Since $u$ is not a Whittaker vector, there exists a
$x\in\mathfrak{n}_{\sigma}$, for some $\pi(\sigma)>0$ such that
$u^{\prime}:=xu-\varphi(x)u=\sum\limits_{0\geq a\geq t}[x,u_{a}]w^{\prime}\neq
0$, where $[x,u_{\alpha}]$ stands for
$\sum\limits_{\pi(|\lambda|)=a}p_{\lambda,a}[x,x_{\lambda}]$. Note that
$u^{\prime}$ is contained in $V_{1}$. Then, it’s easy to see
$mindeg_{1}([x,u_{\alpha}]w^{\prime})\geq a\geq t$, by Lemma 2.2.3, if
$[x,u_{\alpha}]\neq 0$. So we have $mindeg_{1}(u^{\prime})\geq t$ and hence
$mindeg_{1}(u^{\prime})=t$ for the definition of $t$. In this case,
$[x,u_{t}]w^{\prime}\neq 0$ and $mindeg_{1}([x,u_{t}]w^{\prime})=t$. But this
forces $\ell_{1}([x,u_{\tau_{0}}]w^{\prime})<ht(u_{t})=\ell_{1}(u)$ (c.f.
Lemma 2.2.3). Thus, $\ell_{1}(u^{\prime})<l$, which contradicts with the
definition of $l$. $\Box$
3.3. Remark 1). Lemma 3.1 and 3.2 suggest that $\mathcal{B}$ satisfies the
properties 1) and 2), therefore with the technique developed in [10], one can
set a correspondence of the set of Whittaker modules and the set of ideals of
$S(Z)$. This is treated in the next section.
2). If $\varphi$ is nonsingular but not a good character, then Lemma 3,1 may
not hold. For example, if $\varphi(x_{(2-i,i)})=1,i\geq 0$, then
$(4x_{\alpha}^{2}+x_{\beta}^{2}-4x_{\alpha}x_{\beta})w^{\prime}$ is a
Whittaker vector, where $\alpha=(0,1),\beta=(-1,2)$. However,
$(4x_{\alpha}^{2}+x_{\beta}^{2}-4x_{\alpha}x_{\beta})w^{\prime}$ is not
contained in $A/Iw^{\prime}$.
3). Almost all nonsingular characters are good characters. To see this, denote
by $G^{(n,s)}$ the matrix formed by the first $n+1$ rows and columns of
$H^{(n.s)},n\geq 1,s\geq 1$. Then obviously, the set
$\\{det(G^{(n,s)})\,|\,n\geq 1,s\geq 1\\}$ consists of countable polynomials
of $\varphi(x_{2-a,a}),a\geq 0$. Hence the statement follows from the fact
that a nonsingular character $\varphi$ is good if all $det(G^{(n,s)})\neq
0,n\geq 1,s\geq 1$.
§4. Whittaker Modules for $\mathcal{B}$
The results and their proofs are exactly parallel to [10]. Notation as in
2.1.2. Fix a nonsingular homomorphism
$\varphi:\mathfrak{n}\rightarrow\mathbb{C}$, and let $M=M_{\varphi},w={\bf
1}\otimes 1$. Let $A=S(Z)$.
4.1.1 Proposition Let $N$ be a submodule of $M=M_{\varphi}$. Then $N=IM$ for
some ideal $I$ of $A=S(Z)$.
Proof Set $I=\\{x\in A\,|\,xw\in N\\}$. One immediately sees that $I$ is an
ideal of $A$ and $IM\subseteq N$. So we can view $N/IM$ as a submodule of
$M/IM$. If $N\neq IM$, then there exists $pw^{\prime}\in N/IM$, with
$pw^{\prime}\neq 0,p\in A$, $(w^{\prime}=p_{I}(w))$ by Lemma 3.1 and 3.2. So
$pw\in N$ and hence $p\in I$. Therefore $pw\in IM$, which contradicts with the
fact that $pw^{\prime}\neq 0$ in $N/IM$. Thus, $N=IM$. $\Box$
4.1.2 proposition Then any nontrivial submodule of a Whittaker module of type
$\varphi$ contains a nontrivial Whittaker submodule of type $\varphi$.
Proof It follows immediately from Proposition 4.1.1 and Lemma 3.2. $\Box$
4.2. The character $\varphi:\mathfrak{n}\rightarrow\mathbb{C}$ naturally
extends to a character of $\mathcal{U}(\mathfrak{n})$. Let
$\mathcal{U}_{\varphi}(\mathfrak{n})$ be the kernel of this extension so that
$\mathcal{U}(\mathfrak{n})=\mathbb{C}\oplus\mathcal{U}_{\varphi}(\mathfrak{n})$.
Hence,
$\mathcal{U}(\mathcal{B})=\mathcal{U}(\mathfrak{b}_{-})\otimes\mathcal{U}(\mathfrak{n})=\mathcal{U}(\mathfrak{b}_{-})\oplus
I_{\varphi}$
where
$I_{\varphi}=\mathcal{U}(\mathcal{B})\mathcal{U}_{\varphi}(\mathfrak{n})$. For
any $u\in\mathcal{U}(\mathcal{B})$, let
$u^{\varphi}\in\mathcal{U}(\mathfrak{b}_{-})$ be its component in
$\mathcal{U}(\mathfrak{b}_{-})$ relative to the above decomposition of
$\mathcal{U}(\mathcal{B})$.
If $V$ is a Whittaker module generated by a Whittaker vector $v$, let
$\mathcal{U}_{v}(\mathcal{B})$ (resp. $\mathcal{U}_{V}(\mathcal{B})$) be the
annihilator of $v$ (resp. $V$).Then we have, immediately,
$V\simeq\mathcal{U}(\mathcal{B})/\mathcal{U}_{v}(\mathcal{B})$. Set
$A_{V}=A\cap\mathcal{U}_{V}(\mathcal{B})$.
4.2.1. Proposition Let $V$ be any $\mathcal{B}$ module that admits a cyclic
Whittaker vector $v$. Then
$\mathcal{U}_{v}(\mathcal{B})=\mathcal{U}(\mathcal{B})A_{V}+I_{\varphi}.$
Proof Obviously the right hand side of the equation is contained in the left
hand side. So it is enough to show the other way around. Using the universal
property of $M$, we can choose a surjective homomorphism $\psi:M\rightarrow V$
that sends $w={\bf 1}\otimes 1$ to $v$. Let $Y=ker(\psi)$. Then $Y=IM$ for
some $I\subseteq A$, by 4.1.1.
But for any $x\in\mathcal{U}_{v}(\mathcal{B})$, i.e. $xv=0$, one has
$x^{\varphi}v=0$ and hence $x^{\varphi}w\in Y$. Then
$x^{\varphi}w=\sum_{i}p_{i}x_{i}w,\,x_{i}\in\mathcal{U}^{\prime\prime},p_{i}\in
I$. Thus, $x^{\varphi}=\sum\limits_{i}p_{i}x_{i}\subset\mathcal{U}I$. But
clearly $I\subseteq A_{V}$, therefore $x^{\varphi}\in\mathcal{U}A_{V}$. So,
$x\in\mathcal{U}A_{V}+I_{\varphi}$. $\Box$
4.2.2. Theorem The correspondence
$V\rightarrow A_{V}$
sets up a bijection between the set of all the isomorphic classes of Whittaker
modules for $\mathcal{B}$ and the set of all the ideals of $A=S(Z)$.
Proof Note that for any $I\subseteq A$, if let $V=M/IM$, then $A_{V}=I$. Now
the theorem follows immediately.
4.2.3. Corollary For any maximal ideal $\mathfrak{m}\in S(Z)$,
$L_{\varphi,\mathfrak{m}}=M/\mathfrak{m}M$ is simple and any simple Whittaker
module of type $\varphi$ is of this form.
Proof Observe that if $I\subseteq J\subseteq S(Z)$, then $M/JM$ is a quotient
of $M/IM$. The corollary now follows from Theorem 4.2.2 immediately. $\Box$
## References
* [1] D. Arnal and G. Pinczon, On algebraically irreducible representations of he Lie algebra $\mathfrak{sl}_{2}$ J. Math. Phys. 15 (1974), 350–359.
* [2] G. Benkart and M. Ondrus, Whittaker modules for Generalized Weyl Algebras, arXiv:0803. 3570.
* [3] R. Block, The irreducible representations of the Lie algebra $\mathfrak{sl}_{2}$ and of the Weyl algebra, Adv. Math. 39 (1981), 69–110.
* [4] K. Christodoulopoulou, Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras, J. Alg. 320 (2008) 2871–2890.
* [5] B. Kostant, On Whittaker vectors and representation theory, Invent. Math., 48 (1978), 101–184.
* [6] D. Liu, Y. Wu , L. Zhu, Whittaker Modules for, preprint, arXiv:0801.2603v2.
* [7] M. Ondrus, Whittaker modules for $U_{q}({\mbox{sl}}_{2})$, J. Alg. 289 (2005), 192–213.
* [8] M. Ondrus and E. Wiesenr, Whittaker modules for the Virasoro algebra, arXiv:0805.2686.
* [9] A. Sevostyanov, Quantum deformation of Whittaker modules and Toda lattice, Duke Math. J., (2000), 204 211–238.
* [10] B. Wang, Whittaker Modules for graded Lie algebras, arXiv: 0902.3801
* [11] B. Wang, J. Li, Whittaker Modules for $W$-algebra $W(2,2)$, arXiv: 0902.1592.
* [12] X. Zhang and S. Tan, Whittaker modules and a class of new modules similar as Whittaker modules for the Schrödinger-Virasoro algebra, arXiv:0812.3245v1. Department of Mathematics, Changshu Institute of Technology, Changshu 215500, China, Email: [email protected] Department of Mathematics, University of Texas of the Permian Basin, Odessa, TX, 79762, Email: [email protected]
|
arxiv-papers
| 2009-07-04T16:29:12 |
2024-09-04T02:49:03.737700
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bin Wang, Xinyun Zhu",
"submitter": "Xinyun Zhu",
"url": "https://arxiv.org/abs/0907.0773"
}
|
0907.0842
|
# the automorphism groups of quasi-galois closed arithmetic schemes
Feng-Wen An School of Mathematics and Statistics, Wuhan University, Wuhan,
Hubei 430072, People’s Republic of China [email protected]
###### Abstract.
Assume that $X$ and $Y$ are arithmetic schemes, i.e., integral schemes of
finite types over $Spec(\mathbb{Z})$. Then $X$ is said to be quasi-galois
closed over $Y$ if $X$ has a unique conjugate over $Y$ in some certain
algebraically closed field, where the conjugate of $X$ over $Y$ is defined in
an evident manner. Now suppose that $\phi:X\rightarrow Y$ is a surjective
morphism of finite type such that $X$ is quasi-galois closed over $Y$. In this
paper the main theorem says that the function field $k\left(X\right)$ is
canonically a Galois extension of $k\left(Y\right)$ and the automorphism group
${Aut}(X/Y)$ is isomorphic to the Galois group $Gal(k(X)/k\left(Y\right))$; in
particular, $\phi$ must be affine. Moreover, let $\dim X=\dim Y$. Then $X$ is
a pseudo-galois cover of $Y$ in the sense of Suslin-Voevodsky.
###### Key words and phrases:
arithmetic scheme, automorphism group, pseudo-galois, Galois group, geometric
class field
###### 2000 Mathematics Subject Classification:
Primary 14J50; Secondary 14G40, 14G45, 14H30, 14H37, 12F10
Contents
Introduction
1\. Notation and Definition
1.1. Convention
1.2. Affine covering with values in a given field
1.3. Quasi-galois closed varieties
2\. Statement of The Main Theorem
3\. Proof of The Main Theorem
3.1. Affine structures
3.2. A quasi-galois closed variety has only one maximal affine structure among
others
with values in a fixed field
3.3. Definition for conjugations of a given field
3.4. A quasi-galois field has only one conjugation
3.5. Definition for conjugations of an open set
3.6. Quasi-galois closed varieties and conjugations of open sets
3.7. Automorphism groups of quasi-galois closed varieties and Galois groups of
the
function fields
3.8. Proof of the main theorem
References
## Introduction
Let $k_{1}$ be an algebraic extension of a field $k$ and let $X$ be an
algebraic variety defined over $k_{1}$. Then by a $k-$automorphism $\sigma$ of
$\overline{k}$, we get a conjugate $X^{\sigma}$ of $X$ over $k$. $X$ is said
to be _normally algebraic_ over $k$, defined by Weil, if $X$ coincides with
each of the conjugates of $X$ over $k$ (see [20]). It is well-known that
algebraic varieties and their conjugates behave like conjugates of fields and
have almost all of the algebraic properties (for example, see [7, 20]). But
their complex topological properties are very different from each other (for
example, see [13, 17]). On the other hand, in the geometric version of class
field theory, following Weil’s [21] and [22], Lang uses algebraic varieties to
describe unramified class fields over function fields in several variables
(see [12]); then in virtue of Bloch’s [5] and others’ foundations, Kato and
Saito use algebraic fundamental groups to obtain unramified class field theory
(see [10, 16]). Here, the main feature is to use the abelianized fundamental
groups of algebraic or arithmetic schemes to describe abelian class fields
(for example, see [11, 15, 23, 24]).
Motivated by those works, in this paper we will suggest a definition that an
arithmetic variety is said to be _quasi-galois closed_ if it has a unique
conjugate in an algebraically closed field (see _Definition 1.1_), which can
be regarded as a generalization from the notion that an algebraic variety is
normally algebraic over a number field to the one that an arithmetic variety
is quasi-galois closed over a fixed arithmetic one. Here, an _arithmetic
variety_ is an integral scheme of finite type over
$Spec\left(\mathbb{Z}\right)$. Then we will try to use these relevant data of
such arithmetic varieties to obtain some information of Galois extensions of
function fields in several variables.
The following is the _Main Theorem_ of the paper (i.e., _Theorem 2.1_).
###### Main Theorem 0.1.
_(Theorem 2.1)_ Let $X$ and $Y$ be two arithmetic varieties. Assume that $X$
is quasi-galois closed over $Y$ by a surjective morphism $\phi$ of finite
type. Then there are the following statements.
* •
$f$ is affine.
* •
$k\left(X\right)$ is canonically a Galois extension of $k(Y)$.
* •
There is a group isomorphism
${Aut}\left(X/Y\right)\cong Gal(k\left(X\right)/k(Y)).$
* •
Particularly, let $\dim X=\dim Y$. Then $X$ is a pseudo-galois cover of $Y$ in
the sense of Suslin-Voevodsky.
See [18] for the definition of _pseudo-galois_ covers of schemes. Note that
here $k(X)$ is not necessarily algebraic over $k(Y)$ by $\phi$ in the first
property above. That is, the morphism $\phi$ is not necessarily finite.
Hence, the _Main Theorem_ of the paper shows us some evidence that there
exists a nice relationship between quasi-galois closed arithmetic varieties
and Galois extensions of functions fields in several variables.
For the case that $\phi$ is finite, it can be seen that quasi-galois closed
arithmetic varieties behave like Galois extensions of number fields and their
automorphism groups can be regarded as the Galois groups of the field
extensions.
In deed, one has been attempted to use the data of such varieties $X/Y$ to
describe a given Galois extension $E/F$ for a long time and one says that
$X/Y$ are a _model_ for $E/F$ if the Galois group $Gal\left(E/F\right)$ is
isomorphic to the automorphism group ${Aut}\left(X/Y\right)$ (for example, see
[7, 14, 15, 18, 19]). The _Main Theorem_ gives us such a model for function
fields in several variables.
In [18, 19], Suslin and Voevodsky obtain several good properties for pseudo-
galois covers of varieties for the case that the morphism $\phi$ is finite,
where they also give the existence of pseudo-galois covers. If the arithmetic
varieties are of the same dimensions, it is seen that there is no essential
difference between our “quasi-galois closed” and “pseudo-galois cover”.
However, there is a main difference between the two types of covers if the
structure morphism is not finite. For example, let $t$ be a variable over
$\mathbb{Q}$. Then $Spec(\mathbb{Z}[t])/Spec(\mathbb{Z})$ is quasi-galois
closed but not pseudo-galois. Hence, to some degree, the _Main Theorem_ of the
paper gives us a sufficient condition for the existence of such a pseudo-
galois cover in a more generalized case in the category of arithmetic
varieties, where the function fields are in several variables.
The _Main Theorem_ of the paper can be regarded as a generalization of
_Proposition 1.1_ in [7], _Page 106_ , for the case of function fields in
several variables.
Now let us give some applications of quasi-galois closed covers such as the
following.
In [2] we will prove the existence of quasi-galois closed covers of arithmetic
schemes and then by these covers we will give an explicit construction of the
geometric model for a prescribed Galois extension of a function field in
several variables over a number field.
In [4] we will use quasi-galois closed covers to define and compute a qc
fundamental group for an arithmetic scheme. Then we will prove that the étale
fundamental group of an arithmetic scheme is a normal subgroup in our qc
fundamental group. Hence, our group gives us a prior estimate of the étale
fundamental group. The quotient group reflects the topological properties of
the arithmetic scheme.
Particularly, in [3] we will use quasi-galois closed covers to give the
computation of the étale fundamental group of an arithmetic scheme.
### 0.1. Outline of the Proof for the Main Theorem
The whole of §3 will be devoted to the proof of the Main Theorem of the
present paper, where we will proceed in several subsections.
In §3.1 we will recall some preliminary facts on affine structures on
arithmetic schemes (see [1]). Here, affine structures on a scheme behave like
differential structures on a differential manifold. In §3.2 we will prove that
a quasi-galois closed arithmetic variety has one and only one maximal affine
structure among others with values in a fixed algebraically closed field (see
_Proposition 3.9_).
In §3.3 we will define conjugations of a given field and a quasi-galois
extension of a field in an evident manner. For the case of algebraic
extensions, “conjugation” is exactly “conjugate” and “quasi-galois” is exactly
“normal”.
Let $K$ be a finitely generated extension of a fixed field $k$. In §3.4 we
will demonstrate that $K$ is quasi-galois over $k$ if and only if $K$ has only
one conjugation over $k$ (see _Corollary 3.14_). Moreover, $K$ is a Galois
extension of $k$ if $K$ is quasi-galois and separably generated over $k$ (see
the proof of _Theorem 3.26_).
Then conjugations and quasi-galois extensions for fields will be geometrically
realized in arithmetic varieties. In §3.5 we will define conjugations of an
open subset in an arithmetic variety in an evident manner. An open subset of
an arithmetic variety is said to have a quasi-galois set of conjugations if
all of its conjugations can be affinely realized in the variety.
Now let $\phi:X\rightarrow Y$ be a surjective morphism of finite type between
arithmetic varieties. Suppose that $X$ is quasi-galois closed over $Y$ by the
structure morphism $\phi$.
In §3.6 we will establish a relationship between the conjugations of fields
and the conjugations of open subsets in arithmetic varieties. In deed, the
discussions on fields and schemes are parallel. It will be proved that affine
open sets in $X$ have quasi-galois sets of conjugations (see _Theorem 3.23_)
and that the function field $k(X)$ is canonically a quasi-galois extension of
the function field $k(Y)$ (see _Theorem 3.24_).
In §3.7 we will prove that the automorphism group of $X$ over $Y$ is
isomorphic the Galois group of the function field $k(X)$ over $k(Y)$ (see
_Theorem 3.26_), which is the dominant part of the Main Theorem in the paper.
Finally in §3.8 we will complete the proof for the Main Theorem of the paper.
### Acknowledgements
The author would like to express his sincere gratitude to Professor Li Banghe
for his advice and instructions on algebraic geometry and topology. Thanks for
an anonymous referee’s comments.
## 1\. Notation and Definitions
### 1.1. Convention
In this paper, an arithmetic variety is an integral scheme of finite type over
$Spec\left(\mathbb{Z}\right)$. A $k-$variety is an integral scheme of finite
type over a field $k$. By a variety, we will understand an arithmetic variety
or a $k-$variety. Let $k(X)\triangleq\mathcal{O}_{X,\xi}$ denote the function
field of a variety $X$ (with generic point $\xi$).
Let $X$ and $Y$ be varieties over a fixed variety $Z$. $Y$ is said to be a
conjugate of $X$ over $Z$ if there is an isomorphism $\sigma:X\rightarrow Y$
over $Z$. Let $Aut\left(X/Z\right)$ denote the group of automorphisms of $X$
over $Z$.
Let $D$ be an integral domain. Denote by $Fr(D)$ the field of fractions on
$D$. If $D$ is a subring of a field $\Omega$, $Fr(D)$ will be assumed to be
contained in $\Omega$.
Let $E$ be an extension of a field $F$. Note that here $E$ is not necessarily
algebraic over $F$. Recall that $E$ is a Galois extension of $F$ if $F$ is the
invariant subfield of the Galois group $Gal(E/F)$.
### 1.2. Affine covering with values in a given field
Let $(X,\mathcal{O}_{X})$ be a scheme. As usual, an affine covering of the
scheme $(X,\mathcal{O}_{X})$ is a family
$\mathcal{C}_{X}=\\{(U_{\alpha},\phi_{\alpha};A_{\alpha})\\}_{\alpha\in\Delta}$
such that for each $\alpha\in\Delta$, $\phi_{\alpha}$ is an isomorphism from
an open set $U_{\alpha}$ of $X$ onto the spectrum $Spec{A_{\alpha}}$ of a
commutative ring $A_{\alpha}$. Each
$(U_{\alpha},\phi_{\alpha};A_{\alpha})\in\mathcal{C}_{X}$ is called a local
chart. An affine covering $\mathcal{C}_{X}$ of $(X,\mathcal{O}_{X})$ is said
to be reduced if $U_{\alpha}\neq U_{\beta}$ holds for any $\alpha\neq\beta$ in
$\Delta$.
Sometimes, we will denote by $(X,\mathcal{O}_{X};\mathcal{C}_{X})$ a scheme
$(X,\mathcal{O}_{X})$ with a given affine covering $\mathcal{C}_{X}$. For the
sake of brevity, a local chart $(U_{\alpha},\phi_{\alpha};A_{\alpha})$ will be
denoted by $U_{\alpha}$ or $(U_{\alpha},\phi_{\alpha})$.
Let $\mathfrak{Comm}$ be the category of commutative rings with identity.
Fixed a subcategory $\mathfrak{Comm}_{0}$ of $\mathfrak{Comm}$. An affine
covering $\\{(U_{\alpha},\phi_{\alpha};A_{\alpha})\\}_{\alpha\in\Delta}$ of
$(X,\mathcal{O}_{X})$ is said to be with values in $\mathfrak{Comm}_{0}$ if
$\mathcal{O}_{X}(U_{\alpha})=A_{\alpha}$ holds and $A_{\alpha}$ is contained
in $\mathfrak{Comm}_{0}$ for each $\alpha\in\Delta$.
In particular, let $\Omega$ be a field (large enough) and let
$\mathfrak{Comm}(\Omega)$ be the category consisting of the subrings of
$\Omega$ and their isomorphisms. An affine covering $\mathcal{C}_{X}$ of
$(X,\mathcal{O}_{X})$ with values in $\mathfrak{Comm}(\Omega)$ is said to be
with values in the field $\Omega$.
### 1.3. Quasi-galois closed varieties
Let $X$ and $Y$ be two varieties and let $f:X\rightarrow Y$ be a surjective
morphism of finite type.
###### Definition 1.1.
The variety $X$ is said to be quasi-galois closed over $Y$ by $f$ if there is
an algebraic closed field $\Omega$ and a reduced affine covering
$\mathcal{C}_{X}$ of $X$ with values in $\Omega$ such that for any conjugate
$Z$ of $X$ over $Y$ the following conditions are satisfied:
$(i)$ $(X,\mathcal{O}_{X})=(Z,\mathcal{O}_{Z})$ holds if $(Z,\mathcal{O}_{Z})$
has a reduced affine coverings with values in $\Omega$.
$(ii)$ Each local chart contained in $\mathcal{C}_{Z}$ is contained in
$\mathcal{C}_{X}$ for any reduced affine covering $\mathcal{C}_{Z}$ of
$(Z,\mathcal{O}_{Z})$ with values in $\Omega$.
In particular, if $Y$ is $Spec(\mathbb{Z})$ or $Spec(k)$, such a variety $X$
is said to be a quasi-galois closed variety.
###### Remark 1.2.
The existence of quasi-galois closed varieties.
$(i)$ For the case of varieties, the finite group actions on varieties can
produce quasi-galois closed varieties (For example, see [6, 7, 14, 18, 19]).
$(ii)$ For the case of schemes, there is another way to obtain quasi-galois
closed schemes. Let $X$ be a scheme with a finite number of conjugates. Then
the disjoint union of the conjugates of $X$ will be quasi-galois closed over
$X$.
$(iii)$ For a general case, in [2] we will prove the existence of quasi-galois
closed schemes over arithmetic schemes.
## 2\. Statement of The Main Theorem
Here is the _Main Theorem_ of the present paper, which will be proved in §3.
###### Theorem 2.1.
_(Main Theorem)._ Let $X$ and $Y$ be two arithmetic varieties. Assume that $X$
is quasi-galois closed over $Y$ by a surjective morphism $\phi$ of finite
type. Then there are the following statements.
* •
$f$ is affine.
* •
$k\left(X\right)$ is canonically a Galois extension of $k(Y)$.
* •
There is a group isomorphism
${Aut}\left(X/Y\right)\cong Gal(k\left(X\right)/k(Y)).$
* •
Particularly, let $\dim X=\dim Y$. Then $X$ is a pseudo-galois cover of $Y$ in
the sense of Suslin-Voevodsky.
###### Remark 2.2.
By the first property in _Theorem 2.1_ it is seen that there exists a nice
relationship between quasi-galois closed arithmetic varieties and Galois
extensions of functions fields in several variables.
###### Remark 2.3.
If $\dim X=\dim Y$, it is seen that quasi-galois closed arithmetic varieties
behave like Galois extensions of number fields and their automorphism groups
can be regarded as the Galois groups of the field extensions. If $\dim X>\dim
Y$, _Theorem 2.1_ can be regarded as a generalization of that in _Proposition
1.1_ in [7], _Page 106_ for function fields in several variables.
###### Remark 2.4.
We have attempted to use the data of such varieties $X/Y$ to describe a given
finite Galois extension $E/F$ in such a manner that $X/Y$ are said to be a
_model_ for $E/F$ if the Galois group $Gal\left(E/F\right)$ is isomorphic to
the automorphism group ${Aut}\left(X/Y\right)$ (for example, see [7, 14, 15,
18, 19]). Hence, _Theorem 2.1_ afford us such a model for function fields in
several variables.
###### Remark 2.5.
If $\dim X=\dim Y$, we have pseudo-galois covers of arithmetic varieties in
the sense of Suslin-Voevodsky (see [18, 19]); it is seen that there is no
essential difference between our “quasi-galois closed”and “pseudo-galois
cover”. However, suppose $\dim X>\dim Y$. Then it is seen that there is a main
difference between the two types of covers. For example,
$Spec(\mathbb{Z}[t])/Spec(\mathbb{Z})$ is quasi-galois closed but not pseudo-
galois, where $t$ is a variable over $\mathbb{Q}$. Hence, _Theorem 2.1_ gives
us a sufficient condition for the existence of such a pseudo-galois cover in a
more generalized case in the category of arithmetic varieties, where the
function fields are in several variables.
## 3\. Proof of the Main Theorem
In this section we will proceed in several subsections to prove the main
theorem of the paper.
### 3.1. Affine structures
Let us recall some preliminary results on affine structures (see [An]) which
will be used in the following subsections. Here, affine structures on a
schemes can be regarded as a counterpart of differential structures on a
manifold in topology (for example, see [8]).
Let $\mathfrak{Comm}$ be the category of commutative rings with identity.
Fixed a subcategory $\mathfrak{Comm}_{0}$ of $\mathfrak{Comm}$.
###### Definition 3.1.
A pseudogroup $\Gamma$ of affine transformations (with values in
$\mathfrak{Comm}_{0}$) is a subcategory of $\mathfrak{Comm}_{0}$ such that the
algebra isomorphisms contained in $\Gamma$ satisfying the conditions
$(i)-(v)$:
$\left(i\right)$ Each $\sigma\in\Gamma$ is an isomorphism between algebras
$dom\left(\sigma\right)$ and $rang\left(\sigma\right)$ contained in
$\mathfrak{Comm}_{0}$, called the domain and range of $\sigma$, respectively.
$\left(ii\right)$ Let $\sigma\in\Gamma$. Then the inverse $\sigma^{-1}$ is
contained in $\Gamma.$
$\left(iii\right)$ The identity map $id_{A}$ on $A$ is contained in $\Gamma$
if there is some $\delta\in\Gamma$ with $dom\left(\delta\right)=A.$
$\left(iv\right)$ Let $\sigma\in\Gamma$. Then the isomorphism induced by
$\sigma$ defined on the localization $dom\left(\sigma\right)_{f}$ of the
algebra $dom\left(\sigma\right)$ at any nonzero $f\in dom\left(\sigma\right)$
is contained in $\Gamma.$
$\left(v\right)$ Let $\sigma,\delta\in\Gamma$. Assume for some $\tau\in\Gamma$
there are isomorphisms $dom\left(\tau\right)\cong dom\left(\sigma\right)_{f}$
and $dom\left(\tau\right)\cong rang\left(\delta\right)_{g}$ with $0\not=f\in
dom\left(\sigma\right)$ and $0\not=g\in rang\left(\delta\right).$ Then the
isomorphism factorized by $dom\left(\tau\right)$ from
$dom\left(\sigma\right)_{f}$ onto $rang\left(\delta\right)_{g}$ is contained
in $\Gamma$.
Let $X$ be a topological space and let $\Gamma$ be a pseudogroup of affine
transformations with values in $\mathfrak{Comm}_{0}$.
###### Definition 3.2.
An affine $\Gamma-$atlas $\mathcal{A}$ on $X$ (with values in
$\mathfrak{Comm}_{0}$) is a collection of triples
$\left(U_{j},\varphi_{j};A_{j}\right)$ with $j\in\Delta$, called local charts,
satisfying the conditions $(i)-(iii)$:
$\left(i\right)$ For every $\left(U_{j},\phi_{j};A_{j}\right)\in\mathcal{A}$,
$U_{j}$ is an open subset of $X$ and $\phi_{j}$ is an homeomorphism of $U_{j}$
onto $Spec\left(A_{j}\right)$ with $A_{j}\in\Gamma$ such that
$U_{i}\not=U_{j}$ holds for any $i\not=j$ in $\Delta$.
For the sake of brevity, such a triple $\left(U_{j},\phi_{j};A_{j}\right)$
will be denoted sometimes by $U_{j}$ or by a pair
$\left(U_{j},\phi_{j}\right)$.
$\left(ii\right)$ $\bigcup_{j\in\Delta}U_{j}$ is an open covering of $X.$
$\left(iii\right)$ Take any
$\left(U_{i},\phi_{i},A_{i}\right),\left(U_{j},\phi_{j},A_{j}\right)\in\mathcal{A}$
with $U_{i}\cap U_{j}\not=\varnothing$. Then there is a local chart
$\left(W_{ij},\phi_{ij}\right)\in\mathcal{A}$ with $W_{ij}\subseteq U_{i}\cap
U_{j}$ such that the isomorphism between the localizations
$\left(A_{j}\right)_{f_{j}}$ and $\left(A_{i}\right)_{f_{i}}$ induced by the
map
$\phi_{j}\circ\phi_{i}^{-1}\mid_{W_{ij}}:\phi_{i}(W_{ij})\rightarrow\phi_{j}(W_{ij})$
is contained in $\Gamma$, where $\phi_{i}\left(W_{ij}\right)\cong
Spec\left(A_{i}\right)_{f_{i}}$ and $\phi_{j}\left(W_{ij}\right)\cong
Spec\left(A_{j}\right)_{f_{j}}$ are homeomorphic for some $f_{i}\in A_{i}$ and
$f_{j}\in A_{j}.$
###### Definition 3.3.
Two affine $\Gamma-$atlases $\mathcal{A}$ and $\mathcal{A}^{\prime}$ on $X$
are said to be $\Gamma-$compatible if the condition below is satisfied:
Take any $\left(U,\phi,A\right)\in\mathcal{A}$ and
$\left(U^{\prime},\phi^{\prime},A^{\prime}\right)\in\mathcal{A}^{\prime}$ with
$U\cap U^{\prime}\not=\varnothing.$ Then there is a local chart
$\left(W,\phi^{\prime\prime}\right)\in\mathcal{A}\bigcap\mathcal{A}^{\prime}$
with $W\subseteq U\cap U^{\prime}$ such that the isomorphism between the
localizations $A_{f}$ and $\left(A^{\prime}\right)_{f^{\prime}}$ induced by
the map
$\phi^{\prime}\circ\phi^{-1}\mid_{W}:\phi(W)\rightarrow\phi^{\prime}(W)$ is
contained in $\Gamma$, where $\phi\left(W\right)\cong SpecA_{f}$ and
$\phi^{\prime}\left(W\right)\cong Spec\left(A^{\prime}\right)_{f^{\prime}}$
are homeomorphic for some $f\in A$ and $f^{\prime}\in A^{\prime}.$
By an affine $\Gamma-$structure on $X$ (with values in $\mathfrak{Comm}_{0}$)
we understand a maximal affine $\Gamma-$atlas $\mathcal{A}\left(\Gamma\right)$
on $X$. Here, an affine $\Gamma-$atlas $\mathcal{A}$ on $X$ is said to be
maximal (or complete) if it can not be contained properly in any other affine
$\Gamma-$atlas of $X.$
###### Remark 3.4.
Fixed a pseudogroup $\Gamma$ of affine transformations. By Zorn’s Lemma it is
seen that for any given affine $\Gamma-$atlas $\mathcal{A}$ on $X$ there is a
unique affine $\Gamma-$structure $\mathcal{A}_{m}$ on $X$ satisfying
$\left(i\right)$ $\mathcal{A}\subseteq\mathcal{A}_{m};$
$\left(ii\right)$ $\mathcal{A}$ and $\mathcal{A}_{m}$ are $\Gamma-$compatible.
In such a case, $\mathcal{A}$ is said to be a base for $\mathcal{A}_{m}$ and
$\mathcal{A}_{m}$ is the affine $\Gamma-$structure defined by $\mathcal{A}.$
###### Definition 3.5.
Let $\mathcal{A}\left(\Gamma\right)$ be a affine $\Gamma-$structure on $X$.
Assume that there is a sheaf $\mathcal{F}$ of rings on $X$ such that
$\left(X,\mathcal{F}\right)$ is a locally ringed space and that
$\phi_{\alpha\ast}\mathcal{F}\mid_{U_{\alpha}}\left(SpecA_{\alpha}\right)=A_{\alpha}$
holds for each
$\left(U_{\alpha},\phi_{\alpha};A_{\alpha}\right)\in\mathcal{A}\left(\Gamma\right)$.
Then $\mathcal{A}\left(\Gamma\right)$ is said to be admissible on $X$ and
$\mathcal{F}$ is said to be an extension of $\mathcal{A}\left(\Gamma\right)$.
It is evident that such a sheaf $\mathcal{F}$ on $X$ affords us a scheme
$(X,\mathcal{F})$. That is, an extension of an affine structure on a space is
a scheme.
Let $(X,\mathcal{O}_{X})$ be a scheme and $U_{\alpha}$ an affine open set of
$X$. Take an isomorphism
$(\phi_{\alpha},{\phi_{\alpha}}^{\sharp}):(U_{\alpha},{\mathcal{O}_{X}}_{\mid
U_{\alpha}})\rightarrow(SpecA_{\alpha},\mathcal{O}_{SpecA_{\alpha}})$. In
general, the ring ${\mathcal{O}_{X}}(U)$ is isomorphic to $A_{\alpha}$ by
${\phi_{\alpha}}^{\sharp}$. Here, we choose the ring $A_{\alpha}$ to be such
that ${\mathcal{F}}(U)=A_{\alpha}$ in the definition above. This can be done
according to the preliminary facts on affine schemes (see [6]).
###### Remark 3.6.
It is easily seen that all extensions of a fixed admissible affine structure
on a space are isomorphic schemes (see [An]).
### 3.2. A quasi-galois closed variety has only one maximal affine structure
among others with values in a fixed field
Let $\mathfrak{Comm}_{/k}$ be the category of finitely generated algebras
(with identities) over a given field $k$. We will consider the pseudogroup of
affine transformations with values in $\mathfrak{Comm}_{/k}$ in this
subsection.
Fixed a $k-$variety $\left(X,\mathcal{O}_{X}\right)$ with a given reduced
affine covering $\mathcal{C}_{X}$. That is, each reduced affine covering gives
us a pseudogroup of affine transformations in a natural manner.
In deed, define $\Gamma(\mathcal{C}_{X})$ to be the set of identities
$1_{A_{\alpha}}:A_{\alpha}\rightarrow A_{\alpha}$ and isomorphisms
$\sigma_{\alpha\beta}:\left(A_{\alpha}\right)_{f_{\alpha}}\rightarrow\left(A_{\beta}\right)_{f_{\beta}}$
of $k-$algebras for any $0\not=f_{\alpha}\in A_{\alpha}$ and
$0\not=f_{\beta}\in A_{\beta}$, where $A_{\alpha}$ and $A_{\beta}$ are
contained in $\mathfrak{Comm}/k$ such that there are some affine open subsets
$U_{\alpha}$ and $U_{\beta}$ of $X$ with
$(U_{\alpha},\phi_{\alpha}),(U_{\beta},\phi_{\beta})\in\mathcal{C}_{X}$
satisfying
$\phi_{\alpha}\left(U_{\alpha}\right)=SpecA_{\alpha},\phi_{\beta}\left(U_{\beta}\right)=SpecA_{\beta}.$
Then $\Gamma(\mathcal{C}_{X})$ is a pseudogroup in $\mathfrak{Comm}_{/k}$,
called the (maximal) pseudogroup of affine transformations in
$\left(X,\mathcal{O}_{X};\mathcal{C}_{X}\right)$.
It is seen that $\mathcal{C}_{X})$ is an affine
$\Gamma(\mathcal{C}_{X})-$atlas on $X$. Denote by
$\mathcal{A}(\mathcal{C}_{X})$ the affine $\Gamma(\mathcal{C}_{X})-$structure
on $X$ defined by an affine $\Gamma(\mathcal{C}_{X})-$atlas $\mathcal{C}_{X}$
on $X$.
###### Remark 3.7.
Let $\mathfrak{Comm}_{/\mathbb{Z}}$ be the category of finitely generated
algebras (with identities) over $\mathbb{Z}$. We can similarly define a
pseudogroup $\Gamma$ of affine transformations with values in
$\mathfrak{Comm}_{/\mathbb{Z}}$ and then discuss affine $\Gamma-$structures
such as the above.
In particular, let $\Omega$ be a field and let $\mathfrak{Comm}(\Omega)$ be
the category consisting of the subrings of $\Omega$ and their isomorphisms. An
affine structure of a variety $X$ with values in $\mathfrak{Comm}(\Omega)$ is
said to be with values in the field $\Omega$.
###### Remark 3.8.
Let $\left(X,\mathcal{O}_{X}\right)$ be a variety (i.e, an arithmetic variety
or a $k-$variety).
$(i)$ Different reduced affine coverings $\mathcal{C}_{X}$ have different
pseudogroups $\Gamma(\mathcal{C}_{X})$ of affine transformations.
$(ii)$ Each reduced affine covering $\mathcal{C}_{X}$ is an admissible affine
atlas. In particular, $\mathcal{O}_{X}$ is an extension of $\mathcal{C}_{X}$
on the underlying space $X$.
$(iii)$ As an admissible affine atlas, each reduced affine covering
$\mathcal{C}_{X}$ can have many extensions $\mathcal{F}_{X}$ on the space $X$.
With such an extension we have a scheme $(X,\mathcal{F}_{X})$, called an
associate scheme of $\left(X,\mathcal{O}_{X}\right)$.
$(iv)$ By _Remark 3.6_ it is seen that all associate schemes of
$\left(X,\mathcal{O}_{X}\right)$ are isomorphic.
###### Proposition 3.9.
Let $X$ and $Y$ be varieties with $X$ quasi-galois closed over $Y$. Then the
variety $X$ has one and only one affine structure
$\mathcal{A}(\mathcal{O}_{X})$ with values in an algebraic closed field
$\Omega$ satisfying the below properties:
$(i)$ The structure sheaf $\mathcal{O}_{X}$ is an extension of
$\mathcal{A}(\mathcal{O}_{X})$.
$(ii)$ By set inclusion, $\mathcal{A}(\mathcal{O}_{X})$ is maximal among the
whole of the affine structures on the underlying space of $X$ with values in
$\Omega$. That is, take any affine $\Gamma-$structure $\mathcal{B}$ on the
space of $X$ with values in $\Omega$. Then we must have
$\mathcal{B}\subseteq\mathcal{A}(\mathcal{O}_{X})$ and
$\Gamma\subseteq\Gamma(\mathcal{A}(\mathcal{O}_{X}))$, where
$\Gamma(\mathcal{A}(\mathcal{O}_{X}))$ is the maximal pseudogroup of affine
transformations of $(X,\mathcal{O}_{X};\mathcal{A}(\mathcal{O}_{X}))$.
In particular, we can choose $\Omega$ to be an fixed algebraic closure of the
function field $k(X)$ of $X$.
Here, $\mathcal{A}(\mathcal{O}_{X})$ will be called the natural affine
structure of $(X,\mathcal{O}_{X})$ with values in $\Omega$.
###### Proof.
Let $\mathcal{C}_{X}$ and $\mathcal{C}_{X}^{\prime}$ be two affine structures
on the underlying space $X$ with respect to pseudogroups $\Gamma$ and
$\Gamma^{\prime}$ respectively, which are both with values in some field
$\Omega$. As an affine structure is a maximal affine atlas, it is clear that
$\mathcal{C}_{X}$ and $\mathcal{C}_{X}^{\prime}$ are reduced affine coverings
on the space $X$. By _Definition 1.1_ , we must have either
$\mathcal{C}_{X}\subseteq\mathcal{C}_{X}^{\prime},\Gamma\subseteq\Gamma^{\prime}$
or
$\mathcal{C}_{X}\supseteq\mathcal{C}_{X}^{\prime},\Gamma\supseteq\Gamma^{\prime}.$
Let $\Sigma$ be the set of affine structures on the underlying space $X$ with
values in $\Omega$. By set inclusion, $\Sigma$ is a partially ordered set
since any two affine structures are compatible with the pseudogroups of affine
transformations.
Hence, $\Sigma$ is totally ordered. The unique maximal element in $\Sigma$ is
the desired affine structure, where we choose the field $\Omega$ to be an
algebraic closure of the functional field $\mathcal{O}_{X,\xi}=k(X)$ of $X$. ∎
### 3.3. Definition for conjugations of a given field
Let $K$ be an extension of a field $k$. Here $K/k$ is not necessarily
algebraic. $K$ is said to be $k-$quasi-galois (or, quasi-galois over $k$) if
each irreducible polynomial $f(X)\in F[X]$ that has a root in $K$ factors
completely in $K\left[X\right]$ into linear factors for any intermediate field
$k\subseteq F\subseteq K$.
Let $E$ be a finitely generated extension of $k$. The elements
$w_{1},w_{2},\cdots,w_{n}\in E\setminus k$
are said to be a $(r,n)-$nice $k-$basis of $E$ (or simply, a nice $k-$basis)
if the following conditions are satisfied:
$E=k(w_{1},w_{2},\cdots,w_{n})$;
$w_{1},w_{2},\cdots,w_{r}$ constitute a transcendental basis of $E$ over $k$;
$w_{r+1},w_{r+2},\cdots,w_{n}$ are linearly independent over
$k(w_{1},w_{2},\cdots,w_{r})$, where $0\leq r\leq n$.
###### Definition 3.10.
Let $E$ and $F$ be two finitely generated extensions of a field $k$. $F$ is
said to be a $k-$conjugation of $E$ (or, a conjugation of $E$ over $k$) if
there is a $(r,n)-$nice $k-$basis $w_{1},w_{2},\cdots,w_{n}$ of $E$ such that
$F$ is a conjugate of $E$ over $k(w_{1},w_{2},\cdots,w_{r})$.
We will denote by $\tau_{(r,n)}$ such an isomorphism from $F$ onto $E$ over
$k(w_{1},w_{2},\cdots,w_{r})$ with respect to the $(r,n)-$nice $k-$basis.
###### Remark 3.11.
Let $F$ be a $k-$conjugation of $E$. Then $F$ is contained in the algebraic
closure $\overline{E}$ of $E$.
It will be proved that a finitely generated field is quasi-galois if and only
if it has only one conjugation (see _Corollary 3.14_). For the case of
algebraic extensions, this is exactly to say that a field is normal if and
only if it has only one conjugate field.
### 3.4. A quasi-galois field has only one conjugation
We give the below criterion for a quasi-galois field by conjugations, which
behaves like a normal field and its conjugate field for the case of algebraic
extensions.
###### Theorem 3.12.
Let $K$ be a finitely generated extension of a field $k$. The following
statements are equivalent.
$\left(i\right)$ $K$ is a quasi-galois field over $k$.
$\left(ii\right)$ Take any $x\in K$ and any subfield $k\subseteq F\subseteq
K$. Then each conjugation of $F\left(x\right)$ over $F$ is contained in $K$.
$\left(iii\right)$ Each $k-$conjugation of $K$ is contained in $K$.
###### Proof.
$\left(i\right)\implies\left(ii\right).$ Take any $x\in K$ and any $k\subseteq
F\subseteq K.$ If $x$ is a variable over $F$, the field $F\left(x\right)$ is
the unique $k-$conjugation of $F\left(x\right)$ in
$\overline{F\left(x\right)}$ ($\subseteq\overline{K}$). If $x$ is algebraic
over $F$, a $F-$conjugation of $F\left(x\right)$ which is exactly a
$F-$conjugate of $F\left(x\right)$ is contained in $K$ by the assumption that
$K$ is $k-$quasi-galois; then all $F-$conjugates of $F\left(x\right)$ in
$\overline{F\left(x\right)}$ ($\subseteq\overline{K}$) is contained in $K$.
$\left(ii\right)\implies\left(i\right).$ Let $F$ be a field with $k\subseteq
F\subseteq K$. Take any irreducible polynomial $f\left(X\right)$ over $F.$
Suppose that $x\in K$ satisfies the equation $f\left(x\right)=0$. Then such an
$F-$conjugation of $F(x)$ is an $F-$conjugate. By $\left(ii\right)$ it is seen
that every $F-$conjugate $z\in\overline{F}$ of $x$ is contained in $K$; hence,
$K$ is quasi-galois over $k.$
$\left(ii\right)\implies\left(iii\right).$ Hypothesize that there is a
$k-$conjugation $H$ of $K$ in $\overline{K}$ is not contained in $K,$ that is,
$H\setminus K$ is a nonempty set. Take any $x_{0}\in H\setminus K$.
Choose a $(r,n)-$nice $k-$basis $w_{1},w_{2},\cdots,w_{n}$ of $K$ which make
$H$ be a $k-$conjugation of $K$. By _Remark 3.11_ it is seen that $H$ is
contained in the algebraic closure of $k(w_{1},w_{2},\cdots,w_{n})$. As
$w_{1},w_{2},\cdots,w_{r}$ are all variables over $k$, it is seen that
$w_{1},w_{2},\cdots,w_{r}$ are all contained in the intersection of $H$ and
$K$. By _Definition 3.10_ it is seen that there is an isomorphism
$\sigma:H\rightarrow K$ of fields over $k(w_{1},w_{2},\cdots,w_{r})$.
It is evident that the specified element $x_{0}$ must be algebraic over
$k(w_{1},w_{2},\cdots,w_{r})$. Then the field
$k(w_{1},w_{2},\cdots,w_{r},x_{0})$ is a conjugate of the field
$k(w_{1},w_{2},\cdots,w_{r},\sigma(x_{0}))$ over
$k(w_{1},w_{2},\cdots,w_{r})$.
From $\left(ii\right)$ we have $k(w_{1},w_{2},\cdots,w_{r},x_{0})\subseteq K$.
In particular, $x_{0}$ is contained in $K$, which is in contradiction with the
hypothesis above. Therefore, every $k-$conjugation of $K$ is in $K.$
$\left(iii\right)\implies\left(ii\right).$ Take any $x\in K$ and any field $F$
such that $k\subseteq F\subseteq K$. If $x$ is a variable over $F,$
$F\left(x\right)$ is the unique $F-$conjugation in $\overline{K}$ of
$F\left(x\right)$ itself by _Remark 3.11_ again; hence, $F\left(x\right)$ is
contained in $K.$
Now suppose that $x$ is algebraic over $F.$ Let $z\in\overline{K}$ be an
$F-$conjugate of $x$. If $F=K,$ we have $\sigma_{x}=id_{K};$ then $z=x\in K$.
If $F\not=K$, from _Lemma 3.13_ below we have a field
$F\left(z,v_{1},v_{2},\cdots,v_{s},w_{s+1},\cdots,w_{m}\right)$ that is an
$F-$conjugation of $K$; it is seen that the element $z$ is contained in an
$F-$conjugation of $K$; as $k\subseteq K$, an $F-$conjugation of $K$ must be
an $k-$conjugation of $K$; by $(iii)$ we must have $z\in K$. This proves
$(ii)$. ∎
###### Lemma 3.13.
Fixed a finitely generated extension $K$ of a field $k$ and a field $F$ with
$k\subseteq F\subsetneqq K$. Let $x\in K$ be algebraic over $F$ and let $z$ be
a conjugate of $x$ over $F$. Then there is a $(s,m)-$nice
$F\left(x\right)-$basis $v_{1},v_{2},\cdots,v_{m}$ of $K$ and an
$F-$isomorphism $\tau$ from the field
$K=F\left(x,v_{1},v_{2},\cdots,v_{s},v_{s+1},\cdots,v_{m}\right)$
onto a field of the form
$F\left(z,v_{1},v_{2},\cdots,v_{s},w_{s+1},\cdots,w_{m}\right)$
such that
$\tau(x)=z,\tau(v_{1})=v_{1},\cdots,\tau(v_{s})=v_{s}$
where $w_{s+1},w_{s+2},\cdots,w_{m}$ are elements contained in an extension of
$F$. In particular, we have
$w_{s+1}=v_{s+1},w_{s+2}=v_{s+2},\cdots,w_{m}=v_{m}$
if $z$ is not contained in $F(v_{1},v_{2},\cdots,v_{m})$.
###### Proof.
We will proceed in two steps according to the assumption that $s=0$ or
$s\not=0$.
_Step 1_. Let $s\not=0$. That is, $v_{1}$ is a variable over
$F\left(x\right)$.
Let $\sigma_{x}$ be the $F-$isomorphism between fields $F(x)$ and $F(z)$ with
$\sigma_{x}(x)=z$. From the isomorphism $\sigma_{x}$ we obtain an isomorphism
$\sigma_{1}$ of $F\left(x,v_{1}\right)$ onto $F\left(z,v_{1}\right)$ defined
by
$\sigma_{1}:\frac{f(v_{1})}{g(v_{1})}\mapsto\frac{\sigma_{x}(f)(v_{1})}{\sigma_{x}(g)(v_{1})}$
for any polynomials $f[X_{1}],g[X_{1}]\in F\left(x\right)[X_{1}]$ with
$g[X_{1}]\neq 0$.
It is easily seen that $g(v_{1})=0$ if and only if ${\sigma_{x}(g)(v_{1})}=0$.
Hence, the map $\sigma_{1}$ is well-defined.
Similarly, for the elements $v_{1},v_{2},\cdots,v_{s}\in K$ that are variables
over $F(x)$, there is an isomorphism
$\sigma_{s}:F\left(x,v_{1},v_{2},\cdots,v_{s}\right)\longrightarrow
F\left(z,v_{1},v_{2},\cdots,v_{s}\right)$
of fields defined by
$x\longmapsto z\text{ and }v_{i}\longmapsto v_{i}$
for $1\leq i\leq s$, where we have the restrictions
$\sigma_{i+1}|_{F\left(x,v_{1},v_{2},\cdots,v_{i}\right)}=\sigma_{i}.$
If $s=m$, we have $K=F\left(x,v_{1},v_{2},\cdots,v_{s}\right)$ and it follows
that the field $F\left(z,v_{1},v_{2},\cdots,v_{s}\right)$ is an
$F-$conjugation of $K$. We put $s\leqslant m-1$.
_Step 2_. Let $s=0$. That is exactly to consider the case $v_{s+1}\in K$ since
$v_{s+1}$ is algebraic over the field $F(v_{1},v_{2},\cdots,v_{s})\subseteq
K$. We have two cases for the element $v_{s+1}$.
_Case (i)_. Suppose that $z$ is not contained in
$F(v_{1},v_{2},\cdots,v_{s+1})$.
We have an isomorphism $\sigma_{s+1}$ between the fields
$F\left(x,v_{1},v_{2},\cdots,v_{s+1}\right)$ and
$F\left(z,v_{1},v_{2},\cdots,v_{s+1}\right)$ given by
$x\longmapsto z\text{ and }v_{i}\longmapsto v_{i}$
with $1\leq i\leq s+1.$
The map $\sigma_{s+1}$ is well-defined. In deed, by the below Claim† it is
seen that $f\left(v_{s+1}\right)=0$ holds if and only if
$\sigma_{s}\left(f\right)\left(v_{s+1}\right)=0$ holds for any polynomial
$f\left(X_{s+1}\right)\in
F\left(x,v_{1},v_{2},\cdots,v_{s}\right)\left[X_{s+1}\right]$.
_Case (ii)_. Suppose that $z$ is contained in the field
$F(v_{1},v_{2},\cdots,v_{s+1})$.
By the below Claim†† we have an element $v_{s+1}^{\prime}$ contained in an
extension of $F$ such that the fields $F(x,v_{s+1})$ and
$F(z,v_{s+1}^{\prime})$ are isomorphic over $F$.
Then by the same procedure as in _Case (i)_ of Claim† it is seen that the
fields $F(x,v_{s+1},v_{1},v_{2},\cdots,v_{s})$ and
$F(z,v_{s+1}^{\prime},v_{1},v_{2},\cdots,v_{s})$ are isomorphic over $F$.
Hence, in such a manner we have an $F-$isomorphism $\tau$ from the field
$F\left(x,v_{1},v_{2},\cdots,v_{s},v_{s+1},\cdots,v_{m}\right)$
onto the field of the form
$F\left(z,v_{1},v_{2},\cdots,v_{s},w_{s+1},\cdots,w_{m}\right)$
such that
$\tau(x)=z,\tau(v_{1})=v_{1},\cdots,\tau(v_{s})=v_{s},$
where $w_{s+1},w_{s+2},\cdots,w_{m}$ are elements contained in an extension of
$F$. This completes the proof of the lemma. ∎
Claim†. Given any $f\left(X,X_{1},X_{2},\cdots,X_{s+1}\right)$ in the
polynomial ring $F\left[X,X_{1},X_{2},\cdots,X_{s+1}\right]$. Suppose that $z$
is not contained in the field $F(v_{1},v_{2},\cdots,v_{s+1})$. Then
$f\left(x,v_{1},v_{2},\cdots,v_{s+1}\right)=0$ holds if and only if
$f\left(z,v_{1},v_{2},\cdots,v_{s+1}\right)=0$ holds.
###### Proof.
Here we use Weil’s algebraic theory of specializations (See [20]) to prove the
claim. For $v_{s+1}$ there are two cases:
$v_{s+1}\in\overline{F};$
$v_{s+1}\in\overline{F(v_{1},v_{2},\cdots,v_{s+1})}\setminus\overline{F},$
where $\overline{F}$ denotes the algebraic closure of the field $F$.
_Case (i)_. Let
$v_{s+1}\in\overline{F(v_{1},v_{2},\cdots,v_{s+1})}\setminus\overline{F}$.
By _Theorem 1_ in [20], _Page 28_ , it is clear that $\left(z\right)$ is a
(generic) specialization of $\left(x\right)$ over $F$ since $z$ and $x$ are
conjugates over $F$. From _Proposition 1_ in [20], _Page 3_ , it is seen that
$F\left(v_{1},v_{2},\cdots,v_{s+1}\right)$ and the field $F(x)$ are free with
respect to each other over $F$ since $x$ is algebraic over $F$. That is,
$F\left(v_{1},v_{2},\cdots,v_{s+1}\right)$ is a free field over $F$ with
respect to $(x)$. By _Proposition 3_ in [20], _Page 4_ , it is seen that
$F\left(v_{1},v_{2},\cdots,v_{s+1}\right)$ and the algebraic closure
$\overline{F}$ are linearly disjoint over $F$. That is,
$F\left(v_{1},v_{2},\cdots,v_{s+1}\right)$ is a regular extension of $F$ (For
detail, see [20], _Page 18_).
Then by _Theorem 5_ in [20], _Page 29_ , it is seen that
$\left(z,v_{1},v_{2},\cdots,v_{s+1}\right)$ is a (generic) specialization of
$\left(x,v_{1},v_{2},\cdots,v_{s+1}\right)$ over $F$ since $\left(z\right)$ is
a (generic) specialization of $\left(x\right)$ over $F$ and
$\left(v_{1},v_{2},\cdots,v_{s+1}\right)$ is a (generic) specialization of
$\left(v_{1},v_{2},\cdots,v_{s+1}\right)$ itself over $F$.
_Case (ii)_. Let $v_{s+1}\in\overline{F}$.
By the above assumption for $z$ it is seen that $z$ is not contained in the
field $F(v_{s+1})$. It is easily seen that there is an isomorphism between the
fields $F(x,v_{s+1})$ and $F(z,v_{s+1})$. It follows that $(z,v_{s+1})$ is a
(generic) specialization of $(x,v_{s+1})$ over $F$. By the same procedure as
in the above _Case (i)_ it is seen that
$\left(z,v_{s+1},v_{1},v_{2},\cdots,v_{s}\right)$ is a (generic)
specialization of $\left(x,v_{s+1},v_{1},v_{2},\cdots,v_{s}\right)$ over $F$.
Now take any polynomial $f\left(X,X_{1},X_{2},\cdots,X_{s+1}\right)$ over $F$.
According to _Cases (i)-(ii)_ , it is seen that
$f\left(x,v_{1},v_{2},\cdots,v_{s+1}\right)=0$ holds if and only if
$f\left(z,v_{1},v_{2},\cdots,v_{s+1}\right)=0$ holds by the theory for generic
specializations. This completes the proof. ∎
Claim††. Assume that $F(u)$ and $F(u^{\prime})$ are isomorphic over $F$ given
by $u\mapsto u^{\prime}$. Let $w$ be an element contained in an extension of
$F$. Then there is an element $w^{\prime}$ contained in some extension of $F$
such that the fields $F(u,w)$ and $F(u^{\prime},w^{\prime})$ are isomorphic
over $F$.
###### Proof.
It is immediate from _Proposition 4_ in [20], _Page 30_. ∎
###### Corollary 3.14.
Let $K$ be a finitely generated extension of a field $k$. Then $K$ is a quasi-
galois field over $k$ if and only if $K$ has one and only one conjugation over
$k$.
###### Proof.
Prove $\Leftarrow$. Let $K$ have only one $k-$conjugation $H$. We must have
$H=K$ and then each $k-$conjugation of $K$ is contained in $K$. By _Theorem
3.12_ it is seen that $K$ is a quasi-galois field over $k$.
Prove $\Rightarrow$. Let $K$ be a $k-$quasi-galois field and $H$ a
$k-$conjugation of $K$. Choose a $k-$isomorphism $\tau$ of $H$ onto $K$ and a
$(s,m)-$nice $k-$basis $v_{1},v_{2},\cdots,v_{m}$ of $K$ such that $H$ is a
conjugate of $K$ over $F$ by $\tau$, where $F\triangleq
k(v_{1},v_{2},\cdots,v_{s})$. We have $F\subseteq H\subseteq K$.
Hypothesize $H\subsetneqq K$. Fixed any $x_{0}\in K\setminus H$. For the
element $x_{0}$ there are two cases.
_Case (i)_. Let $x_{0}$ be a variable over $H$. We have
$\dim_{k}H=\dim_{k}K=s<\infty$
since $H$ and $K$ are conjugations over $k$. But from $x_{0}\in K\setminus H$,
it is seen that
$1+\dim_{k}H=\dim_{k}H(x_{0})\leq\dim_{k}K$
hold; from it we will obtain a contradiction.
_Case (ii)_. Let $x_{0}$ be algebraic over $H$. As
$\overline{H}\subseteq\overline{F}$, we have $x_{0}\in\overline{F}$; it
follows that $x_{0}$ is algebraic over $F$. It is clear that we have
$[H:F]=[K:F]<\infty$
since $H$ is a conjugate of $K$ over $F$ by $\tau$. But from $x_{0}\in
K\setminus H$, it is seen that
$2+[H:F]\leq[H(x_{0}):F]\leq[K:F]$
hold; from it we will obtain a contradiction.
Therefore, the set $K\setminus H$ is empty and we must have $K=H$. ∎
### 3.5. Definition for conjugations of an open set
The notion on conjugations of an open set in a given variety that will be
defined in this subsection can be regarded as a geometric counterpart to that
for the case of fields in §3.3.
Let us first consider the case for integral domains. Here we let
$Fr\left(D\right)$ denote the fractional field of an integral domain $D$.
###### Definition 3.15.
Let $D\subseteq D_{1}\cap D_{2}$ be three integral domains.
$(i)$ The ring $D_{1}$ is said to be $D-$quasi-galois (or, quasi-galois over
$D$) if the field $Fr\left(D_{1}\right)$ is a quasi-galois extension of
$Fr\left(D\right)$.
$(ii)$ Assume that there is a $(r,n)-$nice $k-$basis
$w_{1},w_{2},\cdots,w_{n}$ of the field $Fr(D_{1})$ and an $F-$isomorphism
$\tau_{(r,n)}:Fr(D_{1})\rightarrow Fr(D_{2})$ of fields such that
$\tau_{(r,n)}(D_{1})=D_{2}$, where $k=Fr(D)$ and we set $F\triangleq
k(w_{1},w_{2},\cdots,w_{r})$ to be contained in the intersection
$Fr(D_{1})\cap Fr(D_{2})$.
Then the ring $D_{1}$ is said to be a $D-$conjugation of the ring $D_{2}$ (or,
a conjugation of $D_{2}$ over $D$).
Then consider an integral scheme $Z$. Let $z\in Z$. By the structure sheaf
$\mathcal{O}_{Z}$ on $Z$, we have the canonical embeddings
$i^{Z}_{U}:\mathcal{O}_{Z}(U)\rightarrow k(Z);$
$i^{Z}_{z}:\mathcal{O}_{Z,z}\rightarrow k(Z);$
$i^{z}_{U}:\mathcal{O}_{Z}(U)\rightarrow\mathcal{O}_{Z,z}$
for every open set $U$ of $Z$ containing $z$, where
$k\left(Z\right)=\mathcal{O}_{Z,\xi}$ is the function field of $X$ and $\xi$
is the generic point of $Z$.
We will identify these integral domains with their images, that is, we will
take the rings
$\mathcal{O}_{Z}(U)\subseteq\mathcal{O}_{Z,z}\subseteq k\left(Z\right)$
as subrings of the function field $k\left(Z\right)$. This leads us to obtain
the following definitions.
Now fixed any two $k-$varieties (or, arithmetic varieties) $X$ and $Y$ and let
$\phi:X\rightarrow Y$ be a morphism of finite type. Take a point $y\in\phi(X)$
and an open set $V$ in $Y$ with $V\cap\phi(X)\neq\emptyset$.
###### Definition 3.16.
Assume that $U_{1}$ and $U_{2}$ are open sets of $X$ such that either $U_{1}$
or $U_{2}$ is contained in $\phi^{-1}(V)$. The open set $U_{1}$ is said to be
a $V-$conjugation of the open set $U_{2}$ if the ring
$i^{X}_{U_{1}}(\mathcal{O}_{X}(U_{1}))$ ($\subseteq k(X)$) is a conjugation of
the ring $i^{X}_{U_{2}}(\mathcal{O}_{X}(U_{2}))$ ($\subseteq k(X)$) over the
ring $i^{X}_{\phi^{-1}(V)}(\phi^{\sharp}(\mathcal{O}_{Y}(V)))$ ($\subseteq
k(X)$), where
$\phi^{\sharp}:\mathcal{O}_{Y}(V)\rightarrow\phi_{\ast}\mathcal{O}_{X}(V)=\mathcal{O}_{X}(\phi^{-1}(V))$
is the ring homomorphism.
If $U_{1}$ and $U_{2}$ are both contained in $\phi^{-1}(V)$, such a
$V-$conjugation is said to be geometric.
###### Remark 3.17.
It is seen that the above conjugation of an open set is well-defined since we
have
$\begin{array}[]{l}i^{X}_{\phi^{-1}(V)}(\phi^{\sharp}(\mathcal{O}_{Y}(V)))\\\
=i^{X}_{U_{1}}(i_{\phi^{-1}(V)}^{U_{1}}(\phi^{\sharp}(\mathcal{O}_{Y}(V))))\\\
=i^{X}_{U_{1}\cap U_{2}}(i_{\phi^{-1}(V)}^{U_{1}\cap
U_{2}}(\phi^{\sharp}(\mathcal{O}_{Y}(V))))\\\
=i^{X}_{U_{2}}(i_{\phi^{-1}(V)}^{U_{2}}(\phi^{\sharp}(\mathcal{O}_{Y}(V)))).\end{array}$
In particular, if $\phi$ is surjective, we have
$\phi^{\sharp}(k(Y))\subseteq k(X);$
$\phi^{\sharp}(i^{Y}_{V}(\mathcal{O}_{Y}(V))=i^{X}_{\phi^{-1}(V)}(\phi^{\sharp}(\mathcal{O}_{Y}(V))).$
###### Definition 3.18.
Assume that either $x_{1}\in X$ or $x_{2}\in X$ is contained in
$\phi^{-1}\left(y\right)$. The point $x_{1}$ is said to be a $y-$conjugation
of the point $x_{2}$ if the ring
$i^{X}_{x_{1}}\left(\mathcal{O}_{X,x_{1}}\right)$ ($\subseteq k(X)$) is a
conjugation of the ring $i^{X}_{x_{2}}\left(\mathcal{O}_{X,x_{2}}\right)$
($\subseteq k(X)$) over the ring
$i^{X}_{x_{1}}(\phi^{\sharp}(\mathcal{O}_{Y,y}))$ ($\subseteq k(X)$), where
$\phi^{\sharp}:\mathcal{O}_{Y,y}\rightarrow\mathcal{O}_{X,x_{1}}$ is the ring
homomorphism.
If $x_{1}$ and $x_{2}$ are both contained in $\phi^{-1}(y)$, such a
$y-$conjugation is said to be geometric.
###### Remark 3.19.
The above conjugation of a point is well-defined. In deed, by _Remark 3.17_ we
have
$i^{X}_{x_{1}}(\phi^{\sharp}(\mathcal{O}_{Y,y}))=i^{X}_{x_{2}}(\phi^{\sharp}(\mathcal{O}_{Y,y}))$
as subrings of $k(X)$ according to the preliminary facts on direct systems of
rings.
Let $A$ be a commutative ring with identity. $A$ is said to be affinely
realized in $X$ by an open set $U$ of $X$ if we have $A=\mathcal{O}_{X}(U)$.
$A$ is said to be affinely realized in $X$ by a point $x$ of $X$ if we have
$A=\mathcal{O}_{X,x}$. This is a hint of the following notion for the case of
varieties.
###### Definition 3.20.
An open set $U\subseteq\phi^{-1}(V)$ in the variety $X$ is said to have a
quasi-galois set of $V-$conjugations in $X$ if each conjugation $A$ of the
ring $i^{X}_{U}(\mathcal{O}_{X}(U))$ over the ring
$i^{X}_{\phi^{-1}(V)}(\phi^{\sharp}(\mathcal{O}_{Y}(V)))$ can be affinely
realized canonically by an open set $U_{A}$ of $X$ such that
$A=i^{X}_{U_{A}}(\mathcal{O}_{X}(U_{A})).$
It is easily seen that such an open set $U_{A}$ can be contained in the set
$\phi^{-1}(V)$.
###### Definition 3.21.
A point $x\in\phi^{-1}\left(y\right)$ in the variety $X$ is said to have a
quasi-galois set of $y-$conjugations in $X$ if each conjugation $A$ of the
ring $i^{X}_{x}\left(\mathcal{O}_{X,x}\right)$ over the ring
$i^{X}_{x}(\phi^{\sharp}(\mathcal{O}_{Y,y}))$ can be affinely realized
canonically by a point $x_{A}$ of $X$ such that
$A=i^{X}_{x_{A}}\left(\mathcal{O}_{X,x_{A}}\right)$.
In particular, the fiber $\phi^{-1}\left(y\right)$ is said to be quasi-galois
over $y$ if each point of the fiber $\phi^{-1}\left(y\right)$ has a quasi-
galois set of $y-$conjugations in $X$.
###### Remark 3.22.
Let $y\in V$. By _Theorem 3.23_ below it is easily seen that each point
$x_{0}\in\phi^{-1}\left(y\right)$ has a quasi-galois set of $y-$conjugations
implies that each affine open set $U\subseteq\phi^{-1}(V)$ containing $x_{0}$
has a quasi-galois set of $V-$conjugations in $X$.
### 3.6. Quasi-galois closed varieties and conjugations of open sets
In this subsection we will obtain some properties of quasi-galois closed
varieties by virtue of conjugations of open sets.
###### Theorem 3.23.
Let $X$ and $Y$ be two $k-$varieties (or, two arithmetic varieties) such that
$X$ is quasi-galois closed over $Y$ by a surjective morphism $\phi$ of finite
type.
$(i)$ Fixed any affine open set $V$ of $Y$. Then each affine open set
$U\subseteq\phi^{-1}(V)$ has a quasi-galois set of $V-$conjugations in $X$.
$(ii)$ Let $\Omega$ be an fixed algebraic closure of the functional field
$k(X)$. Then we have
$\mathcal{O}_{X}(U)\subseteq\mathcal{O}_{X,x_{0}}\subseteq\Omega$
exactly as subsets for any point $x\in X$ and any affine open set $U$ of $X$
containing $x$.
###### Proof.
Let $\Omega$ be an fixed algebraic closure of the functional field $k(X)$ of
$X$. By _Proposition 3.9_ we have the natural affine structure
$\mathcal{A}(\mathcal{O}_{X})$ of the variety $(X,\mathcal{O}_{X})$ such that
$\mathcal{A}(\mathcal{O}_{X})$ is with values in $\Omega$ and that
$\mathcal{O}_{X}$ is an extension of $\mathcal{A}(\mathcal{O}_{X})$.
$(i)$ Hypothesize that there is an affine open set
$U_{0}\subseteq\phi^{-1}(V)$ such that a conjugation $H$ of
$i^{X}_{U_{0}}(\mathcal{O}_{X}(U_{0}))$ over
$i^{X}_{\phi^{-1}(V)}(\phi^{\sharp}(\mathcal{O}_{Y}(V)))$ can not be affinely
realized canonically by any open set $U^{\prime}$ of $X$ with
$H=i^{X}_{U^{\prime}}(\mathcal{O}_{X}(U^{\prime})).$
Evidently, $H\not=i^{X}_{U_{0}}(\mathcal{O}_{X}(U_{0}))$. From the field
$\Omega$ we have
$\mathcal{O}_{X}(U_{0})=i^{X}_{U_{0}}(\mathcal{O}_{X}(U_{0}))$ and then
$H\not=\mathcal{O}_{X}(U_{0})$.
Put
$\mathcal{C}^{\prime}_{X}=\\{(U_{0},\phi^{\prime}_{0};H)\\}\bigcup(\mathcal{A}(\mathcal{O}_{X})\setminus\\{(U_{0},\phi_{0};A_{0})\\})$
where $(U_{0},\phi_{0};A_{0})\in\mathcal{A}(\mathcal{O}_{X})$ and
$\phi^{\prime}_{0}(U_{0})=Spec(H)$ is an isomorphism.
Let $\Gamma(\mathcal{C}^{\prime}_{X})$ be the maximal pseudogroup of affine
transformations in $(X,\mathcal{O}_{X};\mathcal{C}^{\prime}_{X})$ and let
$\mathcal{A}^{\prime}(\mathcal{O}_{X})$ be the affine
$\Gamma(\mathcal{C}^{\prime}_{X})-$structure defined by the reduced affine
covering $\mathcal{C}^{\prime}_{X}$.
By gluing schemes (see [9]), it is easily seen that
$\mathcal{A}^{\prime}(\mathcal{O}_{X})$ is admissible and there is a sheaf
$\mathcal{O}^{\prime}_{X}$ on $X$ such that $\mathcal{O}^{\prime}_{X}$ is an
extension of $\mathcal{A}^{\prime}(\mathcal{O}_{X})$. Then
$(X,\mathcal{O}^{\prime}_{X})$ is a scheme such that
$\mathcal{O}^{\prime}_{X}(U_{0})$ is exactly equal to the ring $H$ since they
are both subrings of $\Omega$.
As $\mathcal{A}(\mathcal{O}_{X})$ and $\mathcal{A}^{\prime}(\mathcal{O}_{X})$
are both with values in $\Omega$, in virtue of $(ii)$ of _Proposition 3.9_ we
have
$\mathcal{A}(\mathcal{O}_{X})\supseteq\mathcal{A}^{\prime}(\mathcal{O}_{X})$;
as affine structures are reduced coverings of $X$, we must have
$\mathcal{O}_{X}(U_{0})=\mathcal{O}^{\prime}_{X}(U_{0})=H$
since $(U_{0},\phi^{\prime}_{0};H)\in\mathcal{A}^{\prime}(\mathcal{O}_{X})$,
which will be in contradiction with the hypothesis above. Therefore, each
affine open set $U\subseteq\phi^{-1}(V)$ has a quasi-galois set of
$V-$conjugations in $X$.
$(ii)$ Fixed a point $x_{0}\in X$. Let $I_{x_{0}}$ (respectively, $J_{x_{0}}$)
be the index family of open sets (respectively, affine open sets) of $X$
containing $x_{0}$. By set inclusion, $I_{x_{0}}$ and $J_{x_{0}}$ are
partially ordered sets and then are directed sets. It is easily seen that
$J_{x_{0}}$ and $I_{x_{0}}$ are cofinal since affine open sets for a base for
the topology on the space of $X$. Hence, the stalk $\mathcal{O}_{X,x_{0}}$ at
$x_{0}$ is the direct limit of the system of rings $\mathcal{O}_{X}(U)$ with
$U\in J_{x_{0}}$.
Now consider the natural affine structure $\mathcal{A}(\mathcal{O}_{X})$ with
values in $\Omega$. Take each local chart
$(U,\phi;A)\in\mathcal{A}(\mathcal{O}_{X})$ with $U\in J_{x_{0}}$. It is seen
that we have
$i^{X}_{U}(\mathcal{O}_{X}(U))=\mathcal{O}_{X}(U)=A\subseteq\Omega$
since $\mathcal{O}_{X}$ is an extension of $\mathcal{A}(\mathcal{O}_{X})$.
Similarly, take any affine open sets $U_{1},U_{2}$ of $X$ containing $x_{0}$
such that $U_{1}\subseteq U_{2}$. We have
$i^{U_{2}}_{U_{1}}(\mathcal{O}_{X}(U_{2}))=\mathcal{O}_{X}(U_{2})\subseteq\Omega.$
It follows that for the stalk of $\mathcal{O}_{X}$ at $x_{0}$ we have
$\mathcal{O}_{X,x_{0}}=\bigcup_{U\in
J_{x_{0}}}\mathcal{O}_{X}(U)\subseteq\Omega.$
(Please notice that all isomorphisms $i^{X}_{U}$ and $i^{U_{2}}_{U_{1}}$ here
are exactly identity maps only for affine open sets!) ∎
###### Theorem 3.24.
Let $X$ and $Y$ be two $k-$varieties (or, two arithmetic varieties) such that
$X$ is quasi-galois closed over $Y$ by a surjective morphism $\phi$ of finite
type. Then the function field $k\left(X\right)$ is a quasi-galois extension
over the image $\phi^{\sharp}(k\left(Y\right))$ of the function field $f(Y)$.
###### Proof.
By _Proposition 3.9_ it is seen that there is the natural affine structure
$\mathcal{A}(\mathcal{O}_{X})$ of the variety $(X,\mathcal{O}_{X})$ with
values in $\Omega$ and $\mathcal{O}_{X}$ is an extension of
$\mathcal{A}(\mathcal{O}_{X})$, where $\Omega$ is an fixed algebraic closure
of the functional field $k(X)$ of $X$.
Now fixed any conjugation $H$ of $k\left(X\right)$ over $\phi^{\sharp}(k(Y))$.
Take any element $w_{0}\in H$. Let $\sigma:H\rightarrow k\left(X\right)$ be an
isomorphism over $\phi^{\sharp}(k(Y))$. Put $u_{0}=\sigma\left(w_{0}\right).$
In virtue of _Theorem 3.23_ we have
$\mathcal{O}_{X}(U)\subseteq k(X)=\mathcal{O}_{X,\xi}\subseteq\Omega$
exactly as subsets of $\Omega$ for any affine open set $U$ of $X$, where $\xi$
is the generic point of $X$. Then we have
$\bigcup_{U}\mathcal{O}_{X}(U)=\mathcal{O}_{X,\xi}$
since affine open sets $U$ of $X$ form a base for the topology of $X$.
It follows that there are affine open subsets $V_{0}$ of $Y$ and
$U_{0}\subseteq\phi^{-1}\left(V_{)}\right)$ of $X$ such that $u_{0}$ is
contained in $\mathcal{O}_{X}\left(U_{0}\right)$. By _Theorem 3.23_ again it
is seen that there is some affine open set $W_{0}$ of $X$ such that $W_{0}$ is
a $V-$conjugation of $U_{0}$ and that the element $w_{0}$ is contained in
$\mathcal{O}_{X}(W_{0})$.
Hence, $w_{0}$ is contained in $k(X)=\mathcal{O}_{X,\xi}\subseteq\Omega$. This
proves any given conjugation $H$ of $k\left(X\right)$ over
$\phi^{\sharp}(k(Y))$ is contained in $k(X)$. From _Theorem 3.12_ it is seen
that the function field $k(X)$ is a quasi-galois extension of the field
$\phi^{\sharp}(k(Y))$. ∎
At last we have the following corollary.
###### Corollary 3.25.
Let $X$ and $Y$ be two $k-$varieties (or, two arithmetic varieties) such that
$X$ is quasi-galois closed over $Y$ by a surjective morphism $\phi$ of finite
type. Suppose that each $V-$conjugation of $U$ is geometric for any affine
open sets $V\subseteq Y$ and $U\subseteq\phi^{-1}(V)$.
Then each point $x_{0}\in\phi^{-1}(y_{0})$ has a quasi-galois set of geometric
$y-$conjugations in $X$ for any point $y_{0}\in Y$.
###### Proof.
Fixed a point $y_{0}\in Y$ and a point $x_{0}\in\phi^{-1}(y_{0})$. Take any
affine open sets $V\subseteq Y$ and $U\subseteq\phi^{-1}(V)$ such that
$x_{0}\in U$ and $y_{0}\in V$. By _Theorem 3.23_ it is seen that $U$ has a
quasi-galois set of geometric $V-$conjugations; from _Theorem 3.24_ it is seen
that each point $x_{0}\in\phi^{-1}(y_{0})$ has a quasi-galois set of geometric
$y-$conjugations in $X$ for any point $y_{0}\in Y$. ∎
### 3.7. Automorphism groups of quasi-galois closed varieties and Galois
groups of the function fields
For automorphism groups of quasi-galois closed varieties, we have the
following result.
###### Theorem 3.26.
Let $X$ and $Y$ be two $k-$varieties (or, two arithmetic varieties) such that
$X$ is quasi-galois closed over $Y$ by a surjective morphism $\phi$ of finite
type. Suppose that $k\left(X\right)/\phi^{\sharp}(k(Y))$ is separably
generated. Then the function field $k\left(X\right)$ is a Galois extension of
$\phi^{\sharp}(k(Y))$ and there is a group isomorphism
${Aut}\left(X/Y\right)\cong Gal(k\left(X\right)/\phi^{\sharp}(k(Y))).$
###### Proof.
(i). Prove that the function field $k\left(X\right)$ is a finitely generated
Galois extension of $\phi^{\sharp}(k(Y))$.
Without loss of generality, assume that $k\left(X\right)$ is a transcendental
extension over $\phi^{\sharp}(k(Y))$. By _Corollary 3.14_ and _Theorem 3.24_
it is seen that every conjugation of $k\left(X\right)$ over
$\phi^{\sharp}(k(Y))$ is $k\left(X\right)$ itself. It needs to prove that
there exists a $\sigma_{0}\in Gal(k\left(X\right)/\phi^{\sharp}(k(Y)))$ such
that $\phi^{\sharp}(k(Y))$ is the invariant subfield of $\sigma_{0}$.
In deed, take a $(r,n)-$nice $F-$basis $v_{1},v_{2},\cdots,v_{n}$ of
$k\left(X\right)$. By the assumption above it is seen that $r\geqslant 1$
holds and $k\left(X\right)$ is an algebraic Galois extension of the field
$F_{0}\triangleq\phi^{\sharp}(k(Y))(v_{1},v_{2},\cdots,v_{r})$. Fixed any
$\tau_{0}\in Gal(k\left(X\right)/F_{0})$ with
$\tau_{0}\not=id_{k\left(X\right)}$. Let $\tau_{1}\in
Gal(F_{0}/\phi^{\sharp}(k(Y)))$ be given by
$v_{1}\mapsto\frac{1}{v_{1}},v_{2}\mapsto\frac{1}{v_{2}},\cdots,v_{r}\mapsto\frac{1}{v_{r}}.$
We have a $\sigma_{0}\in Gal(k\left(X\right)/\phi^{\sharp}(k(Y)))$ defined by
$\tau_{0}$ and $\tau_{1}$ in such a manner
$\frac{f(v_{1},v_{2},\cdots,v_{n})}{g(v_{1},v_{2},\cdots,v_{n})}\in k(X)$
$\mapsto\frac{f(\tau_{1}(v_{1}),\tau_{1}(v_{2}),\cdots,\tau_{1}(v_{r}),\tau_{0}(v_{r+1}),\cdots,\tau_{0}(v_{n}))}{g(\tau_{1}(v_{1}),\tau_{1}(v_{2}),\cdots,\tau_{1}(v_{r}),\tau_{0}(v_{r+1}),\cdots,\tau_{0}(v_{n}))}\in
k(X)$
for any polynomials $f(X_{1},X_{2},\cdots,X_{n})$ and
$g(X_{1},X_{2},\cdots,X_{n})\not=0$ over the field $\phi^{\sharp}(k(Y))$ such
that $g(v_{1},v_{2},\cdots,v_{n})\not=0$. By $\tau_{0}$ it is seen that we
have
$g(v_{1},v_{2},\cdots,v_{n})=0$
if and only if
$g(\tau_{1}(v_{1}),\tau_{1}(v_{2}),\cdots,\tau_{1}(v_{r}),\tau_{0}(v_{r+1}),\cdots,\tau_{0}(v_{n}))=0$
holds. Hence, $\sigma_{0}$ is well-defined.
It is seen that $\phi^{\sharp}(k(Y))$ is the invariant subfield of
$\sigma_{0}$ and then $\phi^{\sharp}(k(Y))$ is the invariant subfield of
$Gal(k\left(X\right)/\phi^{\sharp}(k(Y)))$. Therefore, $k\left(X\right)$ is a
Galois extension of $\phi^{\sharp}(k(Y))$.
(ii). Now let $\mathcal{A}(\mathcal{O}_{X})$ be the natural affine structure
of the variety $(X,\mathcal{O}_{X})$ with values in an fixed algebraic closure
$\Omega$ of the functional field $k(X)$.
For an open set $H$ in $X$, we have an isomorphism
$\tau_{H}:\Gamma(H,\mathcal{O}_{X})\cong\mathcal{O}_{X}(H)$ of algebras and an
embedding
$i^{X}_{H}:\mathcal{O}_{X}(H)\rightarrow\mathcal{O}_{X,\xi}\subseteq\Omega$,
where $\xi$ is the generic point of $x$.
For the sake of convenience, all such rings $\Gamma(H,\mathcal{O}_{X})$ and
$\mathcal{O}_{X}(H)$ are regarded as the subrings of the function field $k(X)$
by the maps $i^{X}_{H}\circ\tau_{H}$ and $i^{X}_{H}$, respectively.
The function field $k(X)=\mathcal{O}_{X,\xi}$ is regarded as the set of the
elements of the forms $(U,f)$, where $U$ is an open set of $X$ and $f$ is an
element of $\mathcal{O}_{X}(U)$ ($\subseteq\Omega$). That is,
$k(X)=\\{(U,f):f\in\mathcal{O}_{X}(U)\text{ and }U\subseteq X\text{ is
open}\\}.$
In the following we will proceed in several steps to demonstrate that there
exists an isomorphism
$t:{Aut}\left(X/Y\right)\cong{Gal}\left(k\left(X\right)/\phi^{\sharp}\left(k\left(Y\right)\right)\right)$
of groups.
_Step 1._ Fixed any automorphism
$\sigma=\left(\sigma,\sigma^{\sharp}\right)\in Aut\left(X/Y\right).$ That is,
$\sigma:X\longrightarrow X$ is a homeomorphism and
$\sigma^{\sharp}:\mathcal{O}_{X}\rightarrow\sigma_{\ast}\mathcal{O}_{X}$ is an
isomorphism of sheaves of rings on $X$. As $\dim X<\infty$, we have
$\sigma(\xi)=\xi$. It follows that
$\sigma^{\sharp}:k\left(X\right)=\mathcal{O}_{X,\xi}\rightarrow\sigma_{\ast}\mathcal{O}_{X,\xi}=k\left(X\right)$
is an automorphism of $k(X)$. Let $\sigma^{\sharp-1}$ denote the inverse of
$\sigma^{\sharp}$.
Take any open subset $U$ of $X$. We have the restriction
$\sigma=(\sigma,\sigma^{\sharp}):(U,\mathcal{O}_{X}|_{U})\longrightarrow(\sigma(U),\mathcal{O}_{X}|_{\sigma(U)})$
of open subschemes. That is,
$\sigma^{\sharp}:\mathcal{O}_{X}|_{\sigma(U)}\rightarrow\sigma_{\ast}\mathcal{O}_{X}|_{U}$
is an isomorphism of sheaves on $\sigma(U)$. In particular,
$\sigma^{\sharp}:\mathcal{O}_{X}(\sigma(U))=\mathcal{O}_{X}|_{\sigma(U)}(\sigma(U))\rightarrow\mathcal{O}_{X}(U)=\sigma_{\ast}\mathcal{O}_{X}|_{U}(\sigma(U))$
is an isomorphism of rings.
For every $f\in\mathcal{O}_{X}|_{U}(U)$, we have
$f\in\sigma_{\ast}\mathcal{O}_{X}|_{U}(\sigma(U));$
hence
$\sigma^{\sharp-1}(f)\in\mathcal{O}_{X}(\sigma(U)).$
Now define a mapping
$t:Aut\left(X/Y\right)\longrightarrow
Gal\left(k\left(X\right)/\phi^{\sharp}(k\left(Y\right))\right)$
given by
$\sigma=(\sigma,\sigma^{\sharp})\longmapsto
t(\sigma)=\left\langle\sigma,\sigma^{\sharp-1}\right\rangle$
such that
$\left\langle\sigma,\sigma^{\sharp-1}\right\rangle:\left(U,f\right)\in\mathcal{O}_{X}(U)\longmapsto\left(\sigma\left(U\right),\sigma^{\sharp-1}\left(f\right)\right)\in\mathcal{O}_{X}(\sigma(U))$
is a mapping of $k(X)$ into $k(X)$.
_Step 2._ Prove that $t$ is well-defined. In deed, given any
$\sigma=\left(\sigma,\sigma^{\sharp}\right)\in Aut\left(X/Y\right).$
For any $(U,f),(V,g)\in k(X)$, we have
$(U,f)+(V,g)=(U\cap V,f+g)$
and
$(U,f)\cdot(V,g)=(U\cap V,f\cdot g);$
then we have
$\begin{array}[]{l}\left\langle\sigma,\sigma^{\sharp-1}\right\rangle((U,f)+(V,g))\\\
=\left\langle\sigma,\sigma^{\sharp-1}\right\rangle((U\cap V,f+g))\\\
=(\sigma(U\cap V),\sigma^{\sharp-1}(f+g))\\\ =(\sigma(U\cap
V),\sigma^{\sharp-1}(f))+(\sigma(U\cap V),\sigma^{\sharp-1}(g))\\\
=(\sigma(U),\sigma^{\sharp-1}(f))+(\sigma(V),\sigma^{\sharp-1}(g))\\\
=\left\langle\sigma,\sigma^{\sharp-1}\right\rangle((U,f))+\left\langle\sigma,\sigma^{\sharp-1}\right\rangle((V,g))\end{array}$
and
$\begin{array}[]{l}\left\langle\sigma,\sigma^{\sharp-1}\right\rangle((U,f)\cdot(V,g))\\\
=\left\langle\sigma,\sigma^{\sharp-1}\right\rangle((U\cap V,f\cdot g))\\\
=(\sigma(U\cap V),\sigma^{\sharp-1}(f\cdot g))\\\ =(\sigma(U\cap
V),\sigma^{\sharp-1}(f))\cdot(\sigma(U\cap V),\sigma^{\sharp-1}(g))\\\
=(\sigma(U),\sigma^{\sharp-1}(f))\cdot(\sigma(V),\sigma^{\sharp-1}(g))\\\
=\left\langle\sigma,\sigma^{\sharp-1}\right\rangle((U,f))\cdot\left\langle\sigma,\sigma^{\sharp-1}\right\rangle((V,g)).\end{array}$
It follows that $\left\langle\sigma,\sigma^{\sharp-1}\right\rangle$ is an
automorphism of $k\left(X\right).$
It needs to prove that $\left\langle\sigma,\sigma^{\sharp-1}\right\rangle$ is
an isomorphism over $\phi^{\sharp}(k(Y))$. In deed, consider the given
morphism
$\phi=(\phi,\phi^{\sharp}):(X,\mathcal{O}_{X})\rightarrow(Y,\mathcal{O}_{Y})$
of schemes. Evidently, $\phi(\xi)$ is the generic point of $Y$ and $\xi$ is
invariant under any automorphism $\sigma\in Aut\left(X/Y\right)$. Then
$\sigma^{\sharp}:\mathcal{O}_{X,\xi}\rightarrow\mathcal{O}_{X,\xi}$ is an
isomorphism of algebras over
$\phi^{\sharp}(\mathcal{O}_{Y,\phi(\xi)})=\phi^{\sharp}(k(Y))$. Hence,
$\left\langle\sigma,\sigma^{\sharp-1}\right\rangle|_{\phi^{\sharp}(k(Y))}=id_{\phi^{\sharp}(k(Y))}.$
This proves
$\left\langle\sigma,\sigma^{\sharp-1}\right\rangle\in
Gal\left(k\left(X\right)/\phi^{\sharp}(k\left(Y\right)\right)).$
That is, $t$ is a well-defined map.
Prove that $t$ is a homomorphism between groups. In fact, take any
$\sigma=\left(\sigma,\sigma^{\sharp}\right),\delta=\left(\delta,\delta^{\sharp}\right)\in
Aut\left(X/Y\right).$
By preliminary facts on schemes (see [6]) we have
$\delta^{\sharp-1}\circ\sigma^{\sharp-1}=(\delta\circ\sigma)^{\sharp-1};$
then
$\left\langle\delta,\delta^{\sharp-1}\right\rangle\circ\left\langle\sigma,\sigma^{\sharp-1}\right\rangle=\left\langle\delta\circ\sigma,\delta^{\sharp-1}\circ\sigma^{\sharp-1}\right\rangle.$
Hence, the map
$t:Aut\left(X/Y\right)\rightarrow
Gal\left(k\left(X\right)/\phi^{\sharp}\left(k\left(Y\right)\right)\right)$
is a homomorphism of groups.
_Step 3._ Prove that ${t}$ is injective. Assume
$\sigma,\sigma^{\prime}\in{Aut}\left(X/Y\right)$ such that
$t\left(\sigma\right)=t\left(\sigma^{\prime}\right).$ We have
$\left(\sigma\left(U\right),\sigma^{\sharp-1}\left(f\right)\right)=\left(\sigma^{\prime}\left(U\right),\sigma^{\prime\sharp-1}\left(f\right)\right)$
for any $\left(U,f\right)\in k\left(X\right).$ In particular, we have
$\left(\sigma\left(U_{0}\right),\sigma^{\sharp-1}\left(f\right)\right)=\left(\sigma^{\prime}\left(U_{0}\right),\sigma^{\prime\sharp-1}\left(f\right)\right)$
for any $f\in\mathcal{O}_{X}(U_{0})$ and any affine open subset $U_{0}$ of $X$
such that $\sigma\left(U_{0}\right)$ and $\sigma^{\prime}\left(U_{0}\right)$
are both contained in $\sigma\left(U\right)\cap\sigma^{\prime}\left(U\right)$.
As $\mathcal{O}_{X}$ is an extension of $\mathcal{A}(\mathcal{O}_{X})$, there
are three subrings
$A_{0}=\mathcal{O}_{X}(U_{0}),B_{0}=\mathcal{O}_{X}(\sigma(U_{0})),\text{ and
}{B_{0}}^{\prime}=\mathcal{O}_{X}(\sigma^{\prime}(U_{0}))$
of $k(X)$ such that
$B_{0}=\sigma^{\sharp-1}(A_{0})=\sigma^{\prime\sharp-1}(A_{0})=B_{0}^{\prime}.$
By preliminary facts on affine schemes (see [6]) again, it is seen that
$\sigma|_{U_{0}}=\sigma^{\prime}|_{U_{0}}$
holds as isomorphisms of schemes. As $U_{0}$ is dense in $X$, we have
$\sigma=\sigma|_{\overline{U_{0}}}=\sigma^{\prime}|_{\overline{U_{0}}}=\sigma^{\prime}$
on the whole of $X$. This proves that $t$ is an injection.
_Step 4._ Prove that ${t}$ is surjective. Fixed any element $\rho$ of the
group
$Gal\left(k\left(X\right)/\phi^{\sharp}\left(k\left(Y\right)\right)\right)$.
As $k(X)=\\{(U_{f},f):f\in\mathcal{O}_{X}(U_{f})\text{ and }U_{f}\subseteq
X\text{ is open}\\}$, we have
$\rho:\left(U_{f},f\right)\in
k\left(X\right)\longmapsto\left(U_{\rho\left(f\right)},\rho\left(f\right)\right)\in
k\left(X\right),$
where $U_{f}$ and $U_{\rho(f)}$ are open sets in $X$, $f$ is contained in
$\mathcal{O}_{X}(U_{f})$, and $\rho(f)$ is contained in
$\mathcal{O}_{X}(U_{\rho(f)})$.
We will proceed in the following several sub-steps to prove that each element
of $Gal\left(k\left(X\right)/\phi^{\sharp}\left(k\left(Y\right)\right)\right)$
give us a unique element of ${Aut}(X/Y)$.
(a) Fixed any affine open set $V$ of $Y$. Prove that for each affine open set
$U\subseteq\phi^{-1}(V)$ there is an affine open set $U_{\rho}$ in $X$ such
that $\rho$ determines an isomorphism between affine schemes
$(U,\mathcal{O}_{X}|_{U})$ and $(U_{\rho},\mathcal{O}_{X}|_{U_{\rho}})$.
In fact, take any local chart
$\left(U,\phi;A_{U}\right)\in\mathcal{A}(\mathcal{O}_{X})$ with
$U\subseteq\phi^{-1}(V)$ for some affine open set $V$ of $Y$. Here
$\mathcal{A}(\mathcal{O}_{X})$ is the natural affine structure of the variety
$X$ with values in $\Omega$. As $\mathcal{O}_{X}$ is an extension of
$\mathcal{A}(\mathcal{O}_{X})$, by _Theorem 3.23_ we have
$A=\mathcal{O}_{X}(U)=\\{\left(U_{f},f\right)\in
k\left(X\right):U_{f}\supseteq U\\}$
since $U$ is an affine open set of $X$. Put
$B=\\{\left(U_{\rho\left(f\right)},\rho\left(f\right)\right)\in
k\left(X\right):\left(U_{f},f\right)\in A\\}.$
Then $B$ is a subring of $k(X)$. As $\rho$ is an isomorphism over
$\phi^{\sharp}(k(Y))$, it is seen that by $\rho$ the rings $A$ and $B$ are
isomorphic algebras over $\phi^{\sharp}(k(Y))$. It follows that $A$ and $B$
are conjugations over $\phi^{\sharp}(\mathcal{O}_{Y}(V))$.
By _Theorem 3.23_ again it is seen that $U$ has a quasi-galois set of
$V-$conjugations in $X$. Then there is an open set $U_{\rho}$ that is a
$V-$conjugation of $U$ such that $B=\mathcal{O}_{X}(U_{\rho})$. As $U$ is
affine open, it is clear that $U_{\rho}$ is affine open.
Hence, by $\rho$ we have a unique isomorphism
$\lambda_{U}=\left(\lambda_{U},\lambda_{U}^{\sharp}\right):(U,\mathcal{O}_{X}|_{U})\rightarrow(U_{\rho},\mathcal{O}_{X}|_{U_{\rho}})$
of the affine open subscheme in $X$ such that
$\rho|_{\mathcal{O}_{X}(U)}=\lambda_{U}^{\sharp-1}:\mathcal{O}_{X}(U)\rightarrow\mathcal{O}_{X}(U_{\rho}).$
(b) Take any affine open sets $V\subseteq Y$ and
$U,U^{\prime}\subseteq\phi^{-1}(V)$. Prove that
$\lambda_{U}|_{U\cap U^{\prime}}=\lambda_{U^{\prime}}|_{U\cap U^{\prime}}$
holds as morphisms of schemes.
In fact, by the above construction for each $\lambda_{U}$ it is seen that
$\lambda_{U}^{\sharp}$ and $\lambda^{\sharp}_{U^{\prime}}$ coincide on the
intersection $U\cap U^{\prime}$ since we have
$\rho|_{\mathcal{O}_{X}(U\cap U^{\prime})}=\lambda_{U}|_{U\cap
U^{\prime}}^{\sharp-1}:\mathcal{O}_{X}(U\cap
U^{\prime})\rightarrow\mathcal{O}_{X}({(U\cap U^{\prime})}_{\rho});$
$\rho|_{\mathcal{O}_{X}(U\cap U^{\prime})}=\lambda_{U^{\prime}}|_{U\cap
U^{\prime}}^{\sharp-1}:\mathcal{O}_{X}(U\cap
U^{\prime})\rightarrow\mathcal{O}_{X}({(U\cap U^{\prime})}_{\rho}).$
For any point $x\in U\cap U^{\prime}$, we must have
$\lambda_{U}(x)=\lambda_{U^{\prime}}(x)$. Otherwise, if
$\lambda_{U}(x)\not=\lambda_{U^{\prime}}(x)$, will have an affine open subset
$X_{0}$ of $X$ that contains one of the two points $\lambda_{U}(x)$ and
$\lambda_{U^{\prime}}(x)$ but does not contain the other since the underlying
space of $X$ is a Kolmogrov space. Assume $\lambda_{U}(x)\in X_{0}$ and
$\lambda_{U^{\prime}}(x)\not\in X_{0}$. We choose an affine open subset
$U_{0}$ of $X$ such that $x\in U_{0}\subseteq U\cap U^{\prime}$ and
$\lambda_{U}(U_{0})\subseteq X_{0}$ since we have
$\lambda_{U}(U\cap U^{\prime})={(U\cap U^{\prime})}_{\rho}\subseteq U_{\rho};$
$\lambda_{U^{\prime}}(U\cap U^{\prime})={(U\cap U^{\prime})}_{\rho}\subseteq
U^{\prime}_{\rho}.$
However, by the definition for each $\lambda_{U}$, we have
$\lambda_{U}(U_{0})=(U_{0})_{\rho}=\lambda_{U^{\prime}}(U_{0});$
then
$\lambda_{U^{\prime}}(x)\in(U_{0})_{\rho}\subseteq X_{0},$
where there will be a contradiction. Hence, $\lambda_{U}$ and
$\lambda_{U^{\prime}}$ coincide on $U\cap U^{\prime}$ as mappings of spaces.
(c) By gluing $\lambda_{U}$ along all such affine open subsets $U$, we have a
homeomorphism $\lambda$ of $X$ onto $X$ as a topological space given by
$\lambda:x\in X\mapsto\lambda_{U}(x)\in X$
where $x$ belongs to $U$ and $U$ is an affine open subset of $X$ such that
$\phi(U)$ is contained in some affine open subset $V$ of $Y$. That is,
$\lambda|_{U}=\lambda_{U}.$ By (b) it is seen that $\lambda$ is well-defined.
It is clear that $\lambda$ is also an automorphism of the scheme
$(X,\mathcal{O}_{X})$.
Show that $\lambda$ is contained in $Aut\left(X/Y\right)$ such that
$t\left(\lambda\right)=\rho$. In deed, as $\rho$ is an isomorphism of $k(X)$
over $\phi^{\sharp}\left(k(Y)\right)$, it is seen that the isomorphism
$\lambda_{U}$ is over $Y$ by $\phi$ for any affine open subset $U$ of $X$;
then $\lambda$ is an automorphism of $X$ over $Y$ by $\phi$ such that
$t\left(\lambda\right)=\rho$ holds.
This proves that there exists $\lambda\in Aut\left(X/Y\right)$ such that
$t(\lambda)=\rho$ for each $\rho\in
Gal\left(k\left(X\right)/\phi^{\sharp}\left(k\left(Y\right)\right)\right)$.
Hence, ${t}$ is surjective. This completes the proof. ∎
###### Corollary 3.27.
Let $X$ and $Y$ be arithmetic varieties and let $X$ be quasi-galois closed
over $Y$ by a surjective morphism $\phi$ of finite type. Then there is a
natural isomorphism
$\mathcal{O}_{Y}\cong\phi_{\ast}(\mathcal{O}_{X})^{{Aut}\left(X/Y\right)}$
where $(\mathcal{O}_{X})^{{Aut}\left(X/Y\right)}(U)$ denotes the invariant
subring of $\mathcal{O}_{X}(U)$ under the natural action of
${Aut}\left(X/Y\right)$ for any open subset $U$ of $X$.
###### Proof.
Fixed any affine open sets $U_{0}$ of $X$ and $V_{0}$ of $Y$ with
$U_{0}\subseteq\phi^{-1}(V_{0})$. By _Theorem 3.26_ we have
$\phi^{\sharp}(\mathcal{O}_{Y}(V_{0}))=(\mathcal{O}_{X})^{{Aut}\left(X/Y\right)}(U_{0})=\phi_{\ast}(\mathcal{O}_{X})^{{Aut}\left(X/Y\right)}(V_{0})$
since $k(X)=Fr(\mathcal{O}_{X}(U_{0}))$ and $k(Y)=Fr(\mathcal{O}_{X}(V_{0}))$.
Now take any open set $V$ of $Y$ and put $U=\phi^{-1}(V)$. We must have
$\phi^{\sharp}(\mathcal{O}_{Y}(V))=(\mathcal{O}_{X})^{{Aut}\left(X/Y\right)}(U)=\phi_{\ast}(\mathcal{O}_{X})^{{Aut}\left(X/Y\right)}(V).$
Otherwise, if there is some element $w$ contained in the difference set
$(\mathcal{O}_{X})^{{Aut}\left(X/Y\right)}(U)\setminus\phi^{\sharp}(\mathcal{O}_{Y}(V))$,
we will have
$w\in\mathcal{O}_{X}^{{Aut}\left(X/Y\right)}(U_{1})\setminus\phi^{\sharp}(\mathcal{O}_{Y}(V_{1}))$
and then we will obtain a contradiction, where $U_{1}\subseteq U$ and
$V_{1}\subseteq V$ are affine open sets such that
$U_{1}\subseteq\phi^{-1}(V_{1})$. This completes the proof. ∎
###### Remark 3.28.
Let $X$ and $Y$ be arithmetic varieties and let $X$ be quasi-galois closed
over $Y$ by a surjective morphism $\phi$ of finite type. By _Corollary 3.27_
it is easily seen that the morphism $f$ must be affine.
### 3.8. Proof of the main theorem
Now we are ready to prove the main theorem of the paper, _Theorem 2.1_ in §2.
###### Proof.
(Proof of Theorem 2.1.) By _Theorem 3.26_ and _Remark 3.28_ it needs only to
prove that $X$ is a pseudo-galois cover of $Y$ if $X$ and $Y$ have the same
dimensions.
Let $\dim X=\dim Y$ and $G={Aut}\left(X/Y\right)$. It is clear that $\phi$ is
invariant under the natural action of ${Aut}\left(X/Y\right)$ on $X$. By
_Corollary 3.27_ , we have
$\mathcal{O}_{Y}\cong\phi_{\ast}(\mathcal{O}_{X})^{G}$. By _Theorem 3.26_
again it is seen that $G$ is a finite group. By §5 of [18], it is immediate
that $\phi$ is finite and then $X$ is a pseudo-galois cover of $Y$ by $\phi$.
∎
###### Remark 3.29.
Fixed any two arithmetic varieties $X$ and $Y$ such that $X$ is quasi-galois
closed over $Y$ by a surjective morphism $\phi$ of finite type. By _Theorem
2.1_ it is seen that the natural action of automorphism group
${Aut}\left(X/Y\right)$ on the fiber $\phi^{-1}(y)$ is transitive at each
$y\in Y$.
It follows that each point $x_{0}\in\phi^{-1}(y_{0})$ has a quasi-galois set
of geometric $y_{0}-$conjugations in $X$ for any point $y_{0}$ of $Y$.
## References
* [1] An, F-W. The affine structures on a ringed space and schemes. eprint arXiv:0706.0579.
* [2] An, F-W. on the existence of geometric models for function fields in several variables. eprint arXiv:0909.1993.
* [3] An, F-W. on the étale fundamental groups of arithmetic schemes. eprint arXiv:0910.0157.
* [4] An, F-W. On the arithmetic fundamental groups. eprint arXiv:0910.0605.
* [5] Bloch, S. Algebraic $K-$Theory and Classfield Theory for Arithmetic Surfaces. Annals of Math, 2nd Ser., Vol 114, No. 2 (1981), 229-265.
* [6] Grothendieck, A; Dieudonné, J. Éléments de Géoemétrie Algébrique. vols I-IV, Pub. Math. de l’IHES, 1960-1967.
* [7] Grothendieck, A; Raynaud, M. Rev$\hat{e}$tements $\acute{E}$tales et Groupe Fondamental (SGA1). Springer, New York, 1971.
* [8] Guillemin, V; Sternberg, S. Deformation Theory of Pseudogroup Structures. Memoirs of the Amer Math Soc, Vol 1, No. 64, 1966.
* [9] Hartshorne, R. Algebraic Geometry. Springer, New York, 1977.
* [10] Kato, K; Saito, S. Unramified Class Field Theory of Arithmetical Surfaces. Annals of Math, 2nd Ser., Vol 118, No. 2 (1983), 241-275.
* [11] Kerz, M; Schmidt, A. Covering Data and Higher Dimensional Global Class Field Theory. eprint arXiv:0804.3419.
* [12] Lang, S. Unramified Class Field Theory Over Function Fields in Several Variables. Annals of Math, 2nd Ser., Vol 64, No. 2 (1956), 285-325.
* [13] Milne, J and Suh, J. Nonhomeomorphic Conjugates of Connected Shimura Varieties. eprint arXiv:0804.1953.
* [14] Mumford, D; Fogarty, J; Kirwan, F. Geometric Invariant Theory. Third Enlarged Ed. Springer, Berlin, 1994.
* [15] Raskind, W. Abelian Calss Field Theory of Arithmetic Schemes. K-theory and Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, Vol 58, Part 1 (1995), 85-187.
* [16] Saito, S. Unramified Class Field Theory of Arithmetical Schemes. Annals of Math, 2nd Ser., Vol 121, No. 2 (1985), 251-281.
* [17] Serre, J-P. Exemples de variétés projectives conjuguées non homéomorphes, C. R. Acad. Sc. Paris,, Vol 258 (1964), 4194-4196.
* [18] Suslin, A; Voevodsky, V. Singular homology of abstract algebraic varieties. Invent. Math. 123 (1996), 61-94.
* [19] Suslin, A; Voevodsky, V. Relative Cycles and Chow Sheaves, in _Cycles, Transfers, and Motivic Homology Theories_ , Voevodsky, V; Suslin, A; Friedlander, E M. Annals of Math Studies, Vol 143. Princeton University Press, Princeton, NJ, 2000.
* [20] Weil, A. Foundations of Algebraic Geometry. Amer Math Soc, New York, 1946.
* [21] Weil, A. Variétés Abeliennes et Courbes Algébriques. Hermann, Paris, 1948.
* [22] Weil, A. Numbers of Solutions of Equations in Finite Fields. Bull of the Amer Math Soc, Vol 55 (1949), 497-508.
* [23] Wiesend, G. A Construction of Covers of Arithmetic Schemes. J. Number Theory, Vol 121 (2006), No. 1, 118-131.
* [24] Wiesend, G. Class Field Theory for Arithmetic Schemes. Math Zeit, Vol 256 (2007), No. 4, 717-729.
|
arxiv-papers
| 2009-07-05T10:29:49 |
2024-09-04T02:49:03.745370
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feng-Wen An",
"submitter": "Feng-Wen An",
"url": "https://arxiv.org/abs/0907.0842"
}
|
0907.0855
|
11institutetext: School of Mathematics, University of Leeds, Leeds LS2 9JT,
UK. Web: http://maths.leeds.ac.uk/~kisilv/
Molecular dynamics and particle methods Semiclassical theories and
applications Molecular dynamics and other numerical methods
# Comment on ‘Do we have a consistent non-adiabatic quantum-classical
mechanics?’
Vladimir V. Kisil On leave from Odessa University [email protected]
###### Abstract
We argue with claims of the paper [1] that the quantum-classic bracket
introduced in [4] produces “artificial coupling” and has “genuinely classical
nature”.
###### pacs:
02.70.Ns
###### pacs:
03.65.Sq
###### pacs:
31.15.Qg
## 1 Introduction
This is a comment on the paper [1], which evaluates the quantum-classical (QC)
bracket:
$\displaystyle{}[K_{1},K_{2}]_{qc}$ $\displaystyle=$
$\displaystyle\frac{1}{\mathrm{i}h}[K_{1},K_{2}]+\frac{1}{2}\left(\\{K_{1},K_{2}\\}-\\{K_{2},K_{1}\\}\right)$
(1)
$\displaystyle-\left.\mathrm{i}\partial_{h_{2}}[K_{1},K_{2}]\right|_{h_{2}=0},$
introduced in paper [4, (26)]. The authors in [1] claimed that the QC bracket
(1) exhibits:
* •
an artificial coupling property (i.e., coupling between the subsystems in the
absence of an interaction);
* •
a genuinely classical nature (i.e., the apparent mixed quantum classical form
reduces to a purely classical form for both subsystems).
The assessment in [1] oversaw the following points:
1. 1.
QC bracket (1) is the image of the universal bracket [4, (22)]:
$\left\\{\\!\left[k_{1},k_{2}\right]\\!\right\\}=(k_{1}*k_{2}-k_{2}*k_{1})(\mathcal{A}_{1}+\mathcal{A}_{2}),$
(2)
under QC representation [4, (20)]. The universal bracket consists of
convolution commutator and antiderivative operators [4, (12)]. QC bracket
requires consideration of the first jet space [4]: the bracket is determined
not only by their values of observables at $h_{2}=0$ but also by the values of
their first derivative with respect to $h_{2}$ at zero (see the last term in
(1)).
2. 2.
The derivation QC bracket (1) is independent of p-mechanisation procedure
introduced in [4, (23)]:
$\displaystyle q_{j}$ $\displaystyle\mapsto$ $\displaystyle
Q_{j}=\delta^{\prime}_{x_{j}}(g_{1};g_{2}),$ (3) $\displaystyle p_{1}$
$\displaystyle\mapsto$ $\displaystyle
P_{j}=\chi^{\prime}_{s_{k}}(s_{1}+s_{2})*\delta^{\prime}_{y_{j}}(g_{1};g_{2}),\quad$
(4)
where $j=1$, $2$ and $k=3-j$.
Here _p-mechanisation_ [3, § 3.3], as an analog of quantisation, is a
prescription how to build p-mechanical observables out of classical ones. It
may not be very explicit in [4], but the deduction of the bracket (1) is
compatible with different choices of p-mechanisation, however the value of the
bracket will be different, see Exs. 3 and 4 below. To illustrate this in the
present comment we use p-mechanisation given by the Weyl (symmetric) calculus
based on the following correspondence, cf. [4, (23)], [1, (19)] and (3)–(4):
$q_{j}\mapsto Q_{j}=\delta^{\prime}_{x_{j}}(g_{1};g_{2}),\quad p_{j}\mapsto
P_{j}=\delta^{\prime}_{y_{j}}(g_{1};g_{2}),\quad$ (5)
Then the quantum-quantum image of the universal bracket (2) of the respective
coordinate and momentum observables is:
$[Q_{j},P_{j}]_{qq}=\frac{h_{1}+h_{2}}{h_{k}}I,\qquad k=3-j.$ (6)
Now we review the above two claims from the paper [1].
## 2 Artificial coupling property
There is the following claim in [1, 3001-p3]: “It must be underlined that eq.
(16) describes an artificial interaction even if the two systems are not
coupled by the Hamiltonian.” This coupling property is attributed to the fact
that quantum-quantum bracket in [4, (25)] and [1, (16)] always contains both
Planck’s constants $h_{1}$ and $h_{2}$, which are generated by the presence of
both antiderivative operators in the definition of universal bracket (2).
###### Example 1.
In order to exam the claim let us consider an uncoupled Hamiltonian
$H(q_{1},p_{1},q_{2},p_{2})=H_{1}(q_{1},p_{1})+H_{2}(q_{2},p_{2})$. The
p-mechanisation (as well as quantisation) is a linear map [3, § 3.3], thus
this uncoupled structure will be preserved. Let $\hat{B}$ be an observable
depending only from $\hat{X}_{2}$ and $\hat{D}_{2}$, thus it will commute with
$H_{1}$. Therefore the commutator of $B$ and $H$ will be the same as $B$ and
$H_{2}$. The QC bracket is an image under a representation of the usual
commutator, thus the universal bracket (2) of $B$ and $H$ will be the same as
$B$ and $H_{2}$. Consequently the $\hat{H}_{1}$ will not affect the dynamics
of such an observable $\hat{B}$.
Therefore there is no coupling in the following meaning: arbitrary change of
the Hamiltonian $H_{1}$ of the first subsystem will not affect dynamics of any
observable build from coordinates and momenta of the second system only.
## 3 Genuinely classical nature
The paper [1, 3001-p3] said “In ref. [8] it was suggested that the dynamical
equation (16), in the limit $h_{1}=h$ and $h_{2}\rightarrow 0$, yields a QC
dynamics.” However the derivation in [4] of the QC bracket intentionally
avoids any kind of semiclassic limits due to its potential danger, see such an
attempt in [2] and Example 2 below. The actual method evaluates the image of
the universal bracket (2) under the QC representation [4, (20)] of the group
$\mathbb{D}^{m}$.
The paper [1, 3001-p4] “corrected” the original derivation of QC bracket
replacing the initial set of Planck constants $h_{1}$ and to $h_{2}$ by the
new one $h_{\mathrm{eff}}$ defined by the expression:
$\frac{1}{h_{\mathrm{eff}}}=\frac{1}{h_{1}}+\frac{1}{h_{2}}.$ (7)
However this transformation is singular for $h_{1}h_{2}=0$ and needs special
clarifications how to proceed for such values.
###### Example 2.
Let us consider the transformation $U_{h}:f(x,y)\mapsto f(hx,\frac{1}{h}y)$,
which is a unitary operator $L_{2}(\mathbb{R}^{2})\rightarrow
L_{2}(\mathbb{R}^{2})$ for any $h>0$. However this does not allow us “to take
the limit $h\rightarrow 0$” through the straightforward substitution $h=0$.
Furthermore the paper [1, 3001-p3] claims that “we have shown that the
equation of motion (16) does not lead to a non-trivial QC limit”. However,
this is caused by p-mechanisation (3)–(4), cf. the next two examples.
###### Example 3.
Let $B_{1}$ and $B_{2}$ are squares of coordinate $Q$ and momentum $P$
observables (of the quantum subsystem) respectively. Under p-mechanisation
(3)–(4) they are represented by squares of corresponding convolutions. Then
the commutator (first term of bracket (1)) of their QC representations is
zero, the second term in (1)) vanishes since no classical observables present,
and the third termin (1)) is equal to QC image of the observable $4QP$. Thus
the total bracket is indeed the same as the Poisson bracket for those
observables.
Let us examine the above claim for the p-mechanisation (5) and assume that two
p-mechanical observables $B_{1}$ and $B_{2}$, that is two convolutions on the
group $\mathbb{D}^{n}{}$ [4, p. 876], for any fixed $g_{1}$ are multiples of
the delta function in $g_{2}$, e.g. as in Ex. 3. Under the QC representation
$\rho_{(h;q,p)}$ [4, (20)] those observables become operators
$\rho_{(h;q,p)}(B_{1})$ and $\rho_{(h;q,p)}(B_{2})$ on the state space for the
quantum subsystem without any dependence from classical coordinates $p$, $q$
and the respective Planck constant $h_{2}$. Correspondingly the second and the
third terms of the bracket (1) vanish and this bracket is equal to the
(quantum) commutator
$\frac{1}{\mathrm{i}h}[\rho_{(h;q,p)}(B_{1}),\rho_{(h;q,p)}(B_{2})]$.
Therefore if we admit the claim [1, 3001-p3] that QC bracket (1) always
coincides with the purely classic Poisson bracket, then we have to accept that
any quantum commutator is always equal to the Poisson bracket.
###### Example 4.
Under p-mechanisation (5) the squares of momentum and coordinates from Ex. 3
are represented by convolutions with kernels
$\delta^{\prime\prime}_{x_{1}x_{1}}(g_{1};g_{2})$ and
$\delta^{\prime\prime}_{y_{1}y_{1}}(g_{1};g_{2})$. Their commutator on the
group $\mathbb{D}^{1}{}$ is
$4\delta^{\prime\prime\prime}_{x_{1}y_{1}s_{1}}+2\delta^{\prime\prime}_{s_{1}s_{1}}$.
Thus the universal bracket (2) is
$\left\\{\\!\left[B_{1},B_{2}\right]\\!\right\\}=4\delta^{\prime\prime}_{x_{1}y_{1}}+2\delta^{\prime}_{s_{1}}+(4\delta^{\prime\prime\prime}_{x_{1}y_{1}s_{1}}+2\delta^{\prime\prime}_{s_{1}s_{1}})\mathcal{A}_{2}.$
(8)
In the QC representation of $\mathbb{D}^{1}{}$ the last term of the sum
vanishes and two first terms produce $4QP+2\mathrm{i}hI$. This is the quantum
commutator of $Q^{2}$ and $P^{2}$ times $\frac{1}{\mathrm{i}h}$. There is no
unitary representation to get rid of the purely imaginary term $2\mathrm{i}hI$
in order to reduce the QC bracket of $B_{1}$ and $B_{2}$ to the value $4QP$ of
their Poisson bracket.
## 4 Conclusion
In this paper we demonstrated that the QC bracket (1) does not possess itself
two properties of “artificial coupling” and “genuinely classical nature” as
claimed in [1]. Unfortunately the claims [1] were uncritically translated by
some other authors, see [5, 6].
We showed that for a decoupled Hamiltonian the dynamics of observables
localised in one subsystem is unaffected by the Hamiltonian of the other
subsystem. The “classical nature” described in [1] is rooted in the
p-mechanisation used in [4] and does not appear with other choice of
p-mechanical observables.
The main conclusion of the commented paper [1] is: “We suggest that a
different Ansatz for the equations of motion, could indeed produce non-trivial
QC equations”. This comment is aimed to clarify possible directions for such a
search.
## 5 Acknowledgements
I am grateful to Dr. Frederica Agostini for useful discussions and providing
me with a part of her unpublished thesis. Prof. O.V. Prezhdo and an anonymous
referee made suggestions, which improved presentation in this comment.
## References
* [1] Agostini F., Caprara S. Ciccotti G. Europhys. Lett. EPL 782007Art. 30001, 6 10.1209/0295-5075/78/30001.
* [2] Prezhdo O. V. Kisil V. V. Phys. Rev. A (3) 561997162 arXiv:quant-ph/9610016.
* [3] Kisil V. V. J. Phys. A 372004183 arXiv:quant-ph/0212101, On-line. .
* [4] Kisil V. V. Europhys. Lett. 722005873 arXiv:quant-ph/0506122, On-line.
* [5] Hall M. J. W. Physical Review A782008042104.
* [6] Zhan F., Lin Y. Wu B. Journal of Chemical Physics 1282008315204\.
|
arxiv-papers
| 2009-07-05T13:55:49 |
2024-09-04T02:49:03.755024
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vladimir V. Kisil",
"submitter": "Vladimir V Kisil",
"url": "https://arxiv.org/abs/0907.0855"
}
|
0907.1028
|
# Hubbard-U band-structure methods
R C Albers1, N E Christensen2, and A Svane2 1Theoretical Division, Los Alamos
National Laboratory, Los Alamos, NM 87545, USA 2Department of Physics and
Astronomy, Aarhus University, Denmark [email protected]
###### Abstract
The last decade has seen a large increase in the number of electronic-
structure calculations that involve adding a Hubbard term to the local density
approximation band-structure Hamiltonian. The Hubbard term is then solved
either at the mean-field level or with sophisticated many-body techniques such
as dynamical mean field theory. We review the physics underlying these
approaches and discuss their strengths and weaknesses in terms of the larger
issues of electronic structure that they involve. In particular, we argue that
the common assumptions made to justify such calculations are inconsistent with
what the calculations actually do. Although many of these calculations are
often treated as essentially first-principles calculations, in fact, we argue
that they should be viewed from an entirely different point of view, viz., as
phenomenological many-body corrections to band-structure theory.
Alternatively, they may also be considered to be just a more complex Hubbard
model than the simple one- or few-band models traditionally used in many-body
theories of solids.
###### pacs:
71.10.-w,71.15.-m,71.27.+a,71.28.+d
††: J. Phys.: Condens. Matter
## 1 Introduction
Since the very early days of quantum mechanics, the exact electronic
Hamiltonian, written in terms of the kinetic energy and electrostatic
interactions between the electrons and with the nucleus, has been known. In
non-relativistic second-quantized form, for example, this can be written as
$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\sum_{\sigma}\int
d\bm{r}\psi^{\dagger}_{\sigma}(\bm{r})[\frac{-\hbar^{2}}{2m}\nabla^{2}+V_{N}(\bm{r})-\mu]\psi_{\sigma}(\bm{r})$
(1) $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{\sigma\sigma^{\prime}}\int
d\bm{r}d\bm{r^{\prime}}\psi^{\dagger}_{\sigma}(\bm{r})\psi^{\dagger}_{\sigma^{\prime}}(\bm{r^{\prime}})V_{C}(\bm{r}-\bm{r^{\prime}})\psi_{\sigma^{\prime}}(\bm{r^{\prime}})\psi_{\sigma}(\bm{r}),$
where $V_{N}(\bm{r})$ is the Coulomb interaction between the electrons and the
nuclei, $V_{C}(\bm{r}-\bm{r^{\prime}})$ is the Coulomb interaction
($e^{2}/|\bm{r}-\bm{r^{\prime}}|$) between the electrons,
$\psi^{\dagger}_{\sigma}(\bm{r})$ and $\psi_{\sigma}(\bm{r})$ are creation and
destruction operators for an electron at $\bm{r}$ with spin $\sigma$, and
$\mu$ is the chemical potential. Since this Hamiltonian has strong
interactions between all the electrons in a solid, it is fundamentally a many-
body problem by nature, and is too intractable to be capable of exact
solution. Density functional theory and its application to calculations of the
electronic band structure (BS) of materials[1, 2, 3] has had a profound impact
on Condensed Matter Physics. With fast computers, excellent numerical
algorithms, and good basis sets for expanding the resulting one-particle
equations, it has been possible to reliably predict many properties of
materials from first-principles, such as the correct ground-state crystal
structure and good values for the internal atomic positions and lattice
parameters by minimizing the total energy of the band structure.
Nonetheless, despite the excellent success of the local density approximation
(LDA) band-structure (LDA-BS) theory in this respect (note that by the term
LDA we also include the gradient corrected or GGA versions of this theory),
there have been many well known examples of its failures as well, such as band
gaps in semiconductors that are about 50–80% smaller than the experimentally
measured values and an incorrect description of many aspects of the electronic
structure of strongly correlated electron systems such as mixed-valence,
transition-metal oxides, heavy fermions, and high-temperature superconductors.
These failures can usually either be attributed to the use of a local
potential to represent exchange, which is actually non-local, or more
importantly to not adequately treating the many-body electronic correlations.
Most involve experiments aimed at some type of spectroscopic or excited-state
properties of the electronic quasiparticles such as photoemission or the
presence or absence of band gaps.
To remedy these problems, many new techniques have been developed,
particularly with respect to strongly correlated electron systems. These
methods usually add a model Hubbard-Hamiltonian term to the band-structure
Hamiltonian, and hence have the generic form:
$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{\bm{k},ilm_{l}\sigma,i^{\prime}l^{\prime}m_{l^{\prime}}\sigma^{\prime}}[H^{0}(\bm{k})]_{ilm_{l}\sigma,i^{\prime}l^{\prime}m_{l^{\prime}}\sigma^{\prime}}\hat{c}_{\bm{k}ilm_{l}\sigma}^{\dagger}\hat{c}_{\bm{k}i^{\prime}l^{\prime}m_{l^{\prime}}\sigma^{\prime}}$
(2) $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{i,m\sigma\neq
m^{\prime}\sigma^{\prime}}U_{im\sigma,im^{\prime}\sigma^{\prime}}\hat{n}_{im\sigma}\hat{n}_{im^{\prime}\sigma^{\prime}}+\hat{V}_{DC}.$
The first term is an LDA one-electron band-structure Hamiltonian summed over
Bloch vectors $\bm{k}$ and with orbitals at lattice sites $i$, orbital
momentum $l$, azimuthal quantum number $m_{l}$, and spin $\sigma$. The second
term is a Hubbard-$U$ term that only applies to the $f$ orbitals, which we
will use as a prototype for the strongly correlated orbitals (in many of the
materials mentioned above, these orbitals are actually $d$ orbitals). To
simplify the discussion we use an orthogonalized form of the Hamiltonian that
ignores any potential overlap integrals between the orbitals. The term
$V_{DC}$ is usually called the “double-counting” correction term (for a fairly
complete discussion of this term in the literature, see Ref. [4], and
references therein). We will refer to this Hamiltonian as a Hubbard-$U$ band-
structure (HU-BS) Hamiltonian. Different methods solve this Hubbard-$U$ term
to various degrees of sophistication.
The techniques involving the HU-BS Hamiltonian have been considered by many to
be a revolutionary new devlopment in electronic structure theory, especially
for strongly correlated electronic systems. A recent review article by Held on
dynamical mean-field theory (DMFT)[5], for example, ends with a claim that is
commonly echoed in many places, viz., that “the advances in electronic
structure calculations through DMFT put our ability to predict physical
quantities of such strongly correlated materials onto a similar level as
conventional electronic structure calculations for weakly correlated
materials—at last.”
Given such optimism in the field, we feel that it is timely to review these
new HU-BS methods from the perspective of basic electronic structure theory.
Because the mathematical details of the methods have already been heavily
reviewed many times recently, we will focus this review primarily on more
fundamental aspects. Given the history of electronic structure methods and
what we know about the underlying theory, what is the role and usefullness of
HU-BS approaches? How predictive are they in practice? How much can we trust
the results of such theories and how optimistic can we be that they represent
the revolutionary breakthrough ascribed to them? In addition, how seriously
should we view the development of even more sophisticated methods based on
this approach, especially in light of considerations involving the underlying
foundations of the starting Hamiltonian upon which these sophisticated
mathematical tools are employed?
Finally, to be clear about the focus of this review, we note that we will
ignore electron-phonon and other vibrational aspects of electronic structure
in this paper, as well as pairing and superconductivity, and will only
consider the case where all of the atoms are at static positions within the
unit cell of a periodic solid.
## 2 Failures of Band Structure
Before turning to specific aspects of the HU-BS methods, it is useful to begin
by reviewing the failures of conventional LDA-BS methods that motivate the
search for improvements. By understanding what has gone wrong, we will gain
insight into what the HU-BS methods are attempting to achieve.
A brief catalog of typical failures include: (1) BS predictions of metallic
materials that are experimentally known to be insulating (e.g., CoO and FeO),
(2) absence of magnetism for materials that are magnetic (e.g., for many
undoped high-temperature superconducting oxides) and vice versa (such as Pu),
(3) band gaps that are much too small compared with experiment (e.g., for many
semiconductors), (4) electronic specific heats that are drastically too small
(e.g., for heavy fermion materials), (5) missing peaks at the Fermi energy
(e.g., Kondo-like peaks), and (6) missing satellite spectra (e.g., as occurs
in Ni). More examples can no doubt be found, but this list suffices for our
purposes.
When examining this list, it becomes clear that many of the problems listed
have to do with the spectral properties of the electronic structure of a
material. However, as explained carefully in the early classic papers[1, 2] on
LDA, this type of theory is designed to minimize the total ground-state energy
of the electrons in a material as a functional of the spatial distribution of
the number density of electrons. Thus, the eigenvalues of the Kohn-Sham
equations[2] were never supposed to represent the actual quasiparticle
spectrum of electrons. Nonetheless, because the eigenvalues often, in fact,
are a reasonably good representation of the spectral properties measured in
experiments, this identification is commonly made in practice. Hence, although
everyone admits that this has no justification, most attempts to improve BS
theory are actually attempts to make corrections to the eigenvalue spectra to
bring it into agreement with various spectroscopies that probe the
quasiparticle properties of the materials, such as optical and photoemission
spectra. Even the metal versus insulator problem involves this issue, since
this distinction depends upon knowing the quasiparticle spectral distribution
as a function of energy.
From this very basic point of view, one could strongly question why band-
structure theory should be used for any spectral property, since there is no
formal justification for such an application! So, in this respect, a correct
starting point for a HU-BS description should actually begin with an
explanation of what spectral features an LDA-BS description can be expected to
accurately predict, and what many-body modifications need to be made to
improve this description. If LDA bands are to be used for the quasiparticle
description of the non-$f$ electrons in this approach, it is important that
this part of the theory should be placed on a firmer foundation. As far as we
know, this has never been done in any satisfactory way.
In order to examine this question, the best approach is probably to consider
the GW approximation[6, 7]. Such a theory is developed in a Green’s function
formalism, which is necessary in order to calculate spectral properties. The
one-shot GW approximation can be written as an RPA-like correction to any one-
particle Hamiltonian, such as, for example, an LDA band-structure Hamiltonian.
GW theory has a formal derivation, and it is very clear what physics it
includes and what it does not include.
In this type of approach, one could ignore the original derivation of LDA
theory, and simply treat the Kohn-Sham equations as an approximate one-
electron representation of the electronic Hamiltonian. The Green’s function
for this Hamiltonian can be used to calculate spectral properties, and, if
desired, to lowest-order in the screened Coulomb interaction, these results
can then be approximately corrected to provide a better many-body theory of
the electronic structure. Framed in this way, one can ask if better one-
electron Hamiltonians than LDA would provide a better starting point for
spectral properties. In fact, for example, ideas based on GW theory for such
an improved Hamiltonian has been proposed by van Schilfgaarde, Kotani, and
coworkers[8]. Of course, better many-body corrections would then need to be
added to any such one-electron approach.
Historically, electronic-structure methods have forked into two paths. The
beginnings of this division were seen even 40 years ago[7]: “on the one hand,
we have had the enormous wealth of energy band calculations which have had
tremendous success in explaining the properties of specific solids, but in
which the connection with first principles is not always apparent. On the
other hand, we have seen the spectacular progress of many-body theory applied
to the solid state, which has given a number of new results, although often of
a rather general and formal nature, such as to provide the justification and a
formal basis for a one-electron theory.” In today’s perspective, a very large
effort has gone into improving the first-principles local-density
approximations in order to provide the best possible one-electron theory of
electronic structure, with the main focus on the accuracy of the total energy
functional. The advantage of this “fork” is that such theories are usually
first-principles (i.e., parameter free) and provide a detailed calculation of
specific wave functions and their spatial distribution with respect to the
actual crystal structure of the material, and also include the atomic number
and core electrons of the relevant atoms. It is also usually possible to find
the optimal atomic locations by energy minimizations. The weakness of these
types of theories is their poor treatment of the many-body and quasiparticle
aspects of the electron-electron interaction. On the other hand, approaches in
the second fork have attempted to focus on this many-body character, albeit in
the form of simplified model Hamiltonians such as the Hubbard or Anderson
Hamiltonians, which can then be solved by a variety of sophisticated many-body
techniques involving various levels of approximation.
More recently, these two “forks” have been merged into unified approaches,
e.g., LDA+U (see, for example, Ref. [9] and references therein) or dynamical
mean-field theory, DMFT (see, for example, Refs. [10, 11, 5] and references
therein), that are believed to include the best aspects of both types of
approaches. These are the HU-BS methods mentioned above. In order for theory
to provide a proper guidance or context for various experiments on different
types of materials, it is essential to retain the details about the types of
atoms, their orbital character, and the atomic locations of the atoms in the
unit cell. Otherwise, the calculations often become generic and less useful.
This is included in the BS part of the theory. On the other hand, many
materials clearly exhibit important many-body effects that must be treated
with more sophistication than LDA-BS methods. This is treated by many-body
methods applied to the model Hubbard term in the theory. Because these HU-BS
theories have been “built” on the BS Hamiltonian, in the literature and at
scientific conferences, one commonly finds that many of these calculations
have been de facto considered as quasi first-principles methods. It is the
purpose of this paper to counter this prevailing assumption and to provide a
proper context for these new techniques.
As mentioned above, the types of electronic-structure calculations that we
will consider, in particular, are all based upon adding an additional
Hubbard-U term (or one of its variants) to the band-structure Hamiltonian and
then solving the resulting many-body problem to some level of treatment. When
examining such approaches, the critical question to ask is what such “hybrid”
approaches mean, or how one should understand them. How first principles are
such approaches and do they provide an adequate treatment of the electronic
structure? This type of discussion rarely occurs in the literature, but yet is
crucial if the field is to properly advance.
## 3 What is a good band structure; what does a band structure measure?
Since HU-BS methods are designed to correct band-structure calculations and to
make them more realistic (i.e., to have better agreement with experiment), it
is useful to review what we may mean by a good band structure or what a band
structure actually measures. In this regard, we can begin by reminding
ourselves what goes into a band-structure and what quantities result. The
input to a band structure are the atomic positions and types of atoms (e.g.,
Cu or Si) within a unit cell. The band-structure method then involves
generating a one-electron Hamiltonian and calculating the electronic wave
functions and energy eigenvalues. It also provides a total energy, and number
density (or charge density) and spin density as a function of position. From
the energy eigenvalues the density of states can be calculated.
The total energy as a function of unit cell dimensions and atomic positions is
very useful since changes in the total energy can provide forces on atoms that
can be used in molecular dynamics programs, for example, and can provide
energy differences between different crystal structures. A good band structure
could be defined in terms of how well it calculates this total energy.
However, this is not the main focus of this article. We are more concerned
with quasiparticles, spectral properties, and the energy distribution of
electrons. These come from the energy eigenvalues or dispersion relations
(energy eigenvalues as a function of the k-vector in the Brillouin zone).
To answer the question of what a good band structure is and how we can
experimentally confirm such a band structure, one has to first ask first what
are the fundamental intrinsic mathematical formulations of the electronic
structure of a solid and secondly how various experimental spectroscopies are
related to this formulation. About 40 years ago, Hedin and Lundqvist wrote a
very significant review article[7] that very clearly delineated the answer to
these questions.
With respect to the first question, the answer surely must be that fundamental
theoretical functions that must be calculated are the one-particle and, more
generally, $n$-particle Green’s functions. The one-particle Green’s function,
for example, provides information on the energy needed to add or remove one
electron from the solid as well as the energy dependent spectral density,
which can be written in terms of the imaginary part of the Green’s function.
The two-particle Green’s function arises in a simple and direct calculation of
the total energy (although this can be reformulated in terms of the one-
particle Green’s function as well), as well as the dielectric and other
response and correlation functions. Many of these are needed to evaluate
neutron scattering and magnetic response functions, for example, such as
magnetic susceptibilities or superconducting pairing. The value of the Green’s
function approach is that it can incorporate simple approximations like one-
electron approaches and yet can be generalized to contain the full many-body
physics of the electronic structure, also including, for example, plasmons or
other collective excitations.
When the one-electron band structure is put into a Green’s function form, the
results are very simple. The imaginary part of the Green’s function is just a
sum over delta functions at the energies of the different eigenvalues. Because
the band-structure is a one-electron theory each electron acts independently
and excitations involve only differences between the various energy
eigenvalues with no correction effects. Because the quasiparticle spectra are
just a series of delta functions, the lifetime of each quasiparticle is
infinite (there is no broadening of the spectral function by lifetime
effects). Also, because of the independent particle approximation, there are
no collective excitations.
For this reason, the predicted spectrum of the band-structure is simply a
series of sharp quasiparticle excitations (band energies as a function of
$\mathbf{k}$). Corrections to this spectrum could come in two possible forms:
(1) single-electron corrections that would shift the energy eigenvalues as a
function of $\mathbf{k}$, and (2) addition of a frequency (and
$\mathbf{k}$-dependent) self-energy that could also shift the effective
quasiparticle energies (through the real part of the self energy evaluated at
the quasiparticle energy) as well as provide quasiparticle lifetimes (through
the imaginary part of the self energy evaluated at the quasiparticle energy),
and other-excited state effects such as satellite features at other energies.
As we will see below, mean-field Hubbard model theories are examples of the
first type of correction, and dynamic many-body theor es like DMFT the latter
type.
To understand how “good” this band structure is requires experimental
verification of the spectral properties or other ways of evaluating the
Green’s function that band-structure predicts. This is far from an easy task
and in general involves correcting raw experimental data for a variety of
matrix elements, and other surface and experimental effects (for example,
secondary electrons and experimental resolution, etc.). These issues are
discussed later in this article (see Section 8). Here, it should only be noted
that many of these correction effects are often not carefully taken into
account and our knowledge of the “experimental” spectral functions are
probably not very good for most materials.
Finally, since HU-BS methods only correct the “strongly correlated” orbitals
and leave the other (usually $s$, $p$, and some $d$) orbitals unchanged, the
question of how well conventional band-structure theory applies to the
spectroscopic properties of these more extended orbitals is actually very
important and should be studied much more systematically than has been done up
to now.
## 4 The Hubbard term
There are several important features about the HU-BS Hamiltonian that must be
emphasized. First, the band-structure part of the Hamiltonian identifies
specific orbitals and various hybridizations that provide a realistic
description of the underlying electronic structure and take into account the
correct underlying crystal structure. In addition, such calculations are
first-principles and involve no adjustable parameters. Secondly, the Hubbard
term requires knowing the occupation numbers of the “correlated” $f$ orbitals
(as mentioned above, we use the convention that $f$ orbitals will be the
correlated orbitals in this paper).
It should be pointed out that this “hybrid” Hamiltonian has no derivation. It
is written down based on an intuitive understanding of the electronic
structure. The two terms are simply added together with no formal
justification. The connection between the two terms comes through the $f$
occupation numbers in the Hubbard term, which are assumed to be the same
orbitals as the $f$ orbitals of the underlying band-structure (the first term
of the Hamiltonian). Hence the many-body treatment, which will only be applied
to the second or Hubbard term involves a projection of the Bloch states onto
the $f$ orbitals.
The main assumption made by theories that involve adding a Hubbard $U$ term is
that band-structure calculations treat the Coulomb repulsion between electrons
at the mean-field level and that a more sophisticated many-body treatment is
necessary to handle strong electronic correlation effects. Thus the Hubbard
$U$ term is reintroduced in a simplified way so that a proper many-body
treatment can be performed on this term. To avoid double counting, the mean-
field evaluation of this term is subtracted out in the belief that this
removes the same amount of Coulomb repulsion from the band-structure part of
the Hamiltonian. Hence, many-body effects are then included at some level of
sophistication while mean-field effects are cancelled out.
It is important to consider whether these assumptions make any sense. We
believe that in fact they are seriously flawed. For example, the Hubbard $U$
term is strongly screened and appears nowhere in the original Hamiltonian,
which directly treats the explicit unscreened Coulomb repulsion. In addition,
LDA calculations include a local exchange-correlation potential and hence
involve more than a mean-field (or Hartree) treatment of Coulomb repulsion. A
more straightforward approach would be to do a Hartree calculation of the
electronic structure and then add an unscreened Hubbard $U$ term upon which to
do the many-body treatment (including screening). This would be a disaster and
such a theoretical approach would lead to enormous errors in the electronic
structure.
One useful way to assess the validity of adding this term to the BS
Hamiltonian is to take the local limit of this theory. It is often asserted
that HU-BS methods become more exact as the correlated orbital becomes more
localized. An extreme version of localization is to consider the isolated
atom. For example, it should be possible to do both LDA+$U$ and DMFT for an
isolated atom. In addition, the constrained LDA methods for calculating the
effective $U$ should be very easy!
If this were to be done, the results would probably be very poor indeed.
Certainly the $U$ would not have the large screening of the solid and would
most likely revert back to the 20–30 eV characteristic of the unscreened
Coulomb integrals. In addition, most of the multiplet structure of the atom
would be missing, unless it involved only direct $f$-$f$ multiplets and hence
perhaps was specifically taken into account by the Hubbard-$U$ term.
This example is actually a very useful illustration of how dangerous it is to
assume that the HU-BS Hamiltonian is a good Hamiltonian to describe the
overall electronic structure of a system. It directly demonstrates how
strongly the HU-BS method truncates the original exact Hamiltonian and
therefore how severe this approximation is. Is it possible to really assume
that the screening of a solid can kill off so many aspects of the electronic
structure that such a simplified Hamiltonian as in the HU-BS method is
justified? Thus, it shows very clearly how drastically we have reduced the
actual complexity of the electronic structure when applying HU-BS methods.
Obviously, one must take the results from such theories with many misgivings.
In fact, as we argue elsewhere in this review, it only makes sense to give up
on the notion that these types of calculations are first principles in any
sense of this word, and that they can only reflect a convenient way to modify
the spectral weight of the band-structure predictions so as to better fit and
interpret experimental data.
## 5 Mean-Field Corrections
Mean-field corrections to the original band-structure through the use of the
Hubbard term (i.e., LDA+$U$; see, for example, Ref. [9] and references
therein) provides an important illustration of how the addition of model
Hamiltonian terms modifies and affects the original band structure. These
applications are especially simple in that they do not change the one-electron
character of the Hamiltonian and hence can be solved simply and accurately. In
effect, they are simply a slightly different band-structure than the LDA
starting point.
In a Hubbard framework, these modifications are written in terms of the
occupation operator of specific orbitals (the “strongly correlated” orbitals).
Hence they all have the same generic form:
$H_{MF}=\sum_{im\sigma}V_{im\sigma}\hat{n}_{im\sigma}$
where $V_{im\sigma}$ is a function of the occupation numbers of the $f$
orbitals on the same site $i$, and $\hat{n}_{im\sigma}$ is the number operator
for the $f$ orbital $im\sigma$. In the mean-field approximation this is
usually a linear function of the occupation numbers
$V_{im\sigma}=V_{im\sigma}^{0}+\sum_{m^{\prime}\sigma^{\prime}}U_{im\sigma,im^{\prime}\sigma^{\prime}}n_{im^{\prime}\sigma^{\prime}},$
where $V_{im\sigma}^{0}$ and $U_{im\sigma,im^{\prime}\sigma^{\prime}}$ are
numerical constants, and $n_{im^{\prime}\sigma^{\prime}}$ are the occupation
numbers of the $f$ orbitals (which have to be solved self-consistently in the
course of the calculation). This approach can, of course, also be generalized
for nearest-neighbor or more distant Coulomb-like interactions.
The first point to note about these relationships is that they depend on the
number operator of the correlated orbitals. Hence they are orbital-dependent
interactions and require a projection of the electronic states onto the number
density on these orbitals in order to specify the interaction. Thus they
depend specifically on the basis set that is used. Intuitively, they are meant
to be intra-atomic corrections, so that one prefers that these orbitals look
as atomic-like as possible. There are actually two different choices that can
be made in this regard. Since many BS methods involve muffin-tin basis sets
that have specific numerical wave functions for each type of orbital angular
momentum, one could project these occupation numbers onto an occupation number
for only these parts of the wavefunctions. As the wavefunctions in a solid
extend both into the interstitial region and other atomic spheres, such
occupation numbers would always be less than one when projected onto the
radial wave function of any specific sphere. Alternatively, one can view these
atom-centered basis functions to be Wannier functions centered on each site,
and to use maximally localized Wannier functions so that the portion of each
Wannier function has as much atomic-like character as possible on the relevant
atomic center. Each of these choices has some drawback. The first choice is
somewhat ill-determined since it involves the way the wavefunctions are
normalized, and the second choice puts parts of the Wannier function into the
interstitial region and onto the $s$, $p$, and $d$ orbitals of other atoms and
hence loses some of the intra-atomic character that is being corrected for.
Both choices depend on the size of the muffin-tin radii and how much of the
wavefunctions are localized within a given sphere. Either choice is only
likely to be somewhat satisfactory for very localized or atomic-like
wavefunctions and to become less well defined as the wave functions become
more diffuse. In practice, there is some interplay between the value of the
projection used and the parameters of the model Hamiltonian term that is used.
For example, if the method chosen for the projections leads to smaller
occupation numbers, one can correct for this by increasing the values used for
the mean-field (i.e., the Hubbard-$U$) parameters.
The first correction factor ($V_{im\sigma}^{0}$) is usually described as a
“double counting” term and simply shifts the bare energy level of that
specific $f$ orbital up or down in energy. It can be used to precisely place
the energy of any given orbital wherever it needs to be in order to achieve
good agreement with experiment. If it is independent of the z-projection of
the orbital ($m$), it simply shifts all the $f$ orbitals up or down. An
alternative way of viewing this correction is as a way of modifying the
occupation number of any orbital. By shifting their energy up or down, one
controls how much of the orbital is occupied. For example, this type of
interaction is identical to that which is used as the Lagrange parameter in
constrained calculations of the Hubbard $U$.
In the literature, different choices are made for this first correction factor
for different “flavors” of mean-field theories. Since the “double counting” is
actually an illusion (one is actually not adding and substracting terms from
the original starting Hamiltonian), the only satisfactory way to choose which
method one wants to employ (or perhaps to add even a different constant shift
of the orbitals) is to compare with experiment. Otherwise there is no
fundamental physics argument to choose one method in preference to the other.
The second term in mean-field theories (involving
$U_{im\sigma,im^{\prime}\sigma^{\prime}}$) is an orbital polarization term. It
causes different $f$ orbitals to shift in terms of their relative energies
depending on the specific occupations of each orbital and on the values of the
coefficients $U$ that are chosen (especially if they are positive or
negative). Given this functional form, any polarization that is desired in
order to fit experiment can be forced upon the band-structure solution if the
proper coefficients are chosen to do this.
In addition to these effects, the orbital dependence of these Hubbard terms
also makes it possible to include non-local exchange effects, since orbital-
dependent interactions can be used to represent a non-local function. For
example, in pseudopotential theories an $l$-dependent potential is often added
to represent the non-local character of the pseudopotentials. Since the
starting LDA potentials use a local exchange potential, the Hubbard terms can
be a way of correcting the LDA band structure for non-local exchange. In the
quasi-particle self-consistent method screened non-local exchange interactions
coming from the GW approximation are included as orbital-dependent potentials
to correct the one-electron band structure[8]. Similarly, it has long been
known that the LDA-BS method suffers from self-interaction errors, and the
LDA+U may be viewed as a method to remove self-interactions. In Hartree-Fock
theory, for example, the self-Coulomb and self-exchange interactions exactly
cancel, but once the exchange interaction is described in terms of a local
potential, this cancellation is no longer exact. The screening of the Hubbard
U parameters may then be justified by the fact that part of the self-exchange
is being removed by the local potential.
## 6 Dynamical Corrections
The mean-field treatments considered in the previous sections are basically
different variations on band-structure calculations. All are one-electron
theories and, at best, simply modify the disperson relations of the bands
(energy versus $\mathbf{k}$). However, as was well explained in the early
classic paper by Hedin and Lundqvist[7], the exact Green’s function for the
electronic structure contains a significant frequency dependence in its self
energy. Since a band-structure calculation is a static approximation for the
electronic structure, it has no frequency dependence, and completely misses
this structure. Hence, if one is going to correct band-structure theory in
order to provide a more realistic electronic structure, it is essential to
consider how to incorporate self-energy effects.
The GW approximation, as its name suggests, automatically generates a self-
energy that is proportional to a Green’s function times a screened Coulomb
energy. Although this self energy is the lowest order term in an expansion in
the screened Coulomb energy, it still incorporates some important features
that more sophisticated treatments will need to include. For example, for
weakly correlated systems it maintains the quasiparticle structure inherent in
the band structure. Hence the spectral function often has a strong peak at the
quasiparticle energy. The energy of this peak can be considered to represent
the corrected band-dispersion relations and the width provides a lifetime for
the quasiparticle. It can also correct the size of band-gaps in semiconductors
and open gaps in systems that otherwise would be metallic within LDA band-
structure theory (although sometimes this requires using LDA+U or other
theories to first create a gapped electronic structure as the starting point
for a one-shot GW calculation). Because it incorporates a non-local screened
exchange term, it can also provide the type of corrections that traditionally
have come from Hartree-Fock-like theories, for example, such as are often
added by LDA+U approaches. Hence it can account for some of the modifications
discussed in the previous section on mean-field approaches.
Although simpler many-body approaches can be incorporated into HU-BS
approaches[12, 13, 14], the state-of-the-art methods are now almost
exclusively DMFT. This involves a non-perturbative many-body solution of the
Hubbard term that is performed by mapping the original problem onto a single-
impurity Anderson model (SIAM) and solving the SIAM as exactly as possible. It
requires a projection onto the strongly correlated $f$ orbitals. The method
produces a self-energy coming from the $f$ orbitals only. These can generate
satellite spectra (lower Hubbard bands) as well as Kondo-like peaks at the
Fermi energy and large specific heat enhancements. They will also provide an
electronic lifetime for states that have a significant $f$ character. However,
these lifetimes only come from the $f$-electron self energy, while the other
$s$, $p$, and $d$ electrons have no self-energy or lifetimes and are the
original starting band-structure dispersions. Such theories are useful if
electronic correlations dominate the “interesting” parts of the electronic
structure. The Hubbard-$U$ parameter must either be estimated from constrained
Hubbard-$U$ calculations or fit to experiment. Although much success has been
claimed for these types of theories, the experimental verification is often
qualitative. The critical assumption of the single-site DMFT is that the self
energy of the correlated electron states is $\mathbf{k}$ independent.
One very serious issue with DMFT approaches is the “solver” for the SIAM
equation in the theory. At the present time, many different solvers are used.
Most SIAM solvers, whether from iterative perturbation theory or non-crossing
approaches, etc., have large uncertainties in the correctness of the many-body
solutions they provide. Only the quantum Monte Carlo and numerical
renormalization group solvers are exact. However, even for these, despite new
algorithmic advances such as continuous time quantum Monte Carlo techniques,
there is considerable uncertainty about the quality of their results. For
example, the quantum Monte Carlo methods requires an analytic continuation
from the imaginary frequencies that are calculated by the method to the real
frequencies needed for physical properties. Even with new and sophisticated
techniques for accomplishing this analytic continuation such as those
involving maximum entropy, the real frequency results are very sensitive to
small changes in the imaginary frequency results leading to concerns of large
errors. Also, the Monte Carlo solvers are most accurate at high temperatures
and become increasingly untrustworthy at low temperatures where most of the
interesting correlation physics lies. Overall, from the point of view of the
SIAM solvers, at this time one has to strongly question the accuracy of most
DMFT calculations. In particular, it is clear that different solvers will give
different results, as emphasized at the beginning of Sec. 5 of Held’s DMFT
review.[5]
In general many-body theories must be added to band-structure methods if the
correct electronic structure is to be produced. However, self-energy effects
need to be generalized to correct the non-correlated orbitals as well as the
correlated orbitals. All of the quasiparticles (except those exactly at the
Fermi surface) have finite lifetimes and are likely to require corrections to
their dispersion relations relative to the LDA starting point. In addition,
plasmon, lower Hubbard band, and other non-quasi-particle features will in
general be present in the electronic structure. Such effects are not included
in the LDA band structures.
## 7 Is the HU-BS approach a real electronic structure method?
At this point, based on the previous discussion, it is useful to summarize our
review of the content of the electronic structure implicit in the HU-BS
methods. The most striking comment that can be made on this method is the
starting Hamiltonian itself, Eq. (2). Compared with the exact Hamiltonian, Eq.
(1), it is clear that such a drastic simplification has been made that the
credibility of the HU-BS Hamiltonian cannot be taken at face value but must be
carefully assessed. Exactly what has been done?
From the form of Eq. (2) and the fact that the Hubbard term is a model term
whereas the first term is an attempt at a first-principles description of the
electronic structure, it is reasonable to interpret this Hamiltonian in terms
of its most fundamental part, the band-structure Hamiltonian, and a correction
term, the Hubbard term. In addition it is commonly assumed that the double-
counting term is just the same term but treated in the same mean-field way as
the local-density approximation, and thus one is essentially adding or
subtracting the same effect in order to do a more exact treatment of the most
difficult part of the physics. However, this is not really credible. Neither
the Hubbard term nor the double-counting term exist in the original
Hamiltonian. They simply represent an “ansatz” that has been inserted by hand.
Thus, they can only be sensibly understood as a “correction” to the band-
structure Hamiltonian. These terms are a means by which to include additional
many-body physics that was left out when the rather drastic approximations
needed to formulate the LDA Hamiltonian were made. Hence they make it possible
to build in new features such as satellite peaks and to adjust the
quasiparticle spectra of the band structure.
To approach Hubbard $U$ theories in this spirit provides new flexibility and
should make it easier to resolve certain controversies that often arise, such
as which LDA+U theory is the best approximation. For example, once it is
realized that double-counting is not an issue, one can focus more on what
electronic-structure effects have been left out of the band-structure approach
and what “model” terms could best correct for these effects with a better
many-body treatment. In fact, one could question, for example, whether other
expressions that are different from the Hubbard $U$ term would lead to better
corrections or whether one should instead add corrections to the self-energy
of the electronic Green’s function instead of adding additional terms to the
Hamiltonian.
An important consideration is whether the HU-BS approach can actually work.
How do we know what physics is left out, and why do we believe that the model
term is the right correction factor? Finally, can we actually do the many-body
physics in a sufficiently correct way to believe that we have significantly
improved our understanding of the electronic structure? A bad treatment might
actually lead us to a worse description. Also, an important aspect of this
approach is that we need to include parameters in the theory in order to mask
our ignorance of the real many-body microscopic theory that we are at present
unable to successfully attack. What are the implications of being forced into
a parameterized approach?
Before delving into such matters, however, it is useful to examine more
closely the Hubbard-$U$ term again. If $U$ is treated as a matrix and allowed
to depend on the $m$ projection of the $f$ orbitals, this term is exactly of
the same form as the original Coulomb integrals for a fixed basis of $f$
orbitals. In this sense it has the same physics as the Coulomb term for a
fixed (or frozen) atomic basis, although the basis functions are limited to
one type ($f$ orbitals only) and these functions are extremely limited in
scope (a minimal $f$ basis). If used for an atomic calculation, such a limited
basis would give very poor results for treating the electron-electron Coulomb
terms. So, why should we expect an accurate treatment in the solid? We believe
that, in fact, this term does not provide an accurate treatment. The $U$
matrix that is used in the HU-BS approximation is heavily screened. What is
actually going on is that the many-body treatment of the Hubbard-$U$ term is
being used on something that looks like the original Coulomb term. Hence, the
form of the results (the types of peaks and excitations in the Green’s
function) has a frequency dependence and quasiparticle spectra similar to what
a real Coulomb term would generate. By scaling down the Hubbard $U$ one
reduces the strength of this effect while retaining the same functional form
(freqency or spectral dependence). Thus, if the original electronic structure
is missing peaks or features, this is a way of reintroducing them. At the same
time one has a tuning parameter that can be used to fit the peaks in an
experimental spectra. Thus, such a theory provides realistic spectra that can
be fit to experiment, and with which to correct the LDA band structure for
these types of missing features. Because it is not the electronic structure
calculation of any actual electronic-structure Hamiltonian but of a model or
pseudo-Hamiltonian, the accuracy of the results really doesn’t matter. As long
as the right types of peaks or other features that are seen in experiment are
present in the many-body results, the Hubbard-$U$ parameter can be scaled up
or down to fit the experimental peaks or features. Essentially, the HU-BS
approach is just a model solution of a Coulomb-like term, with the final
results scaled and then mixed with some LDA band structure.
In practice, as discussed above, mean-field treatments of the Hubbard-$U$ term
are used to add in Hartree-Fock like atomistic structure into the one-particle
spectra. These can essentially orbitally polarize the correlated $f$ shell of
electrons. They can also account for SIC-like corrections. For dynamical
theories like DMFT, two effects are commonly introduced: (1) an additional
peak in the photoemission (the lower Hubbard-$U$ band), and a narrowing of the
correlated quasi-particle bands. These are all that are required to fit the
experimental data. Besides the Hubbard-$U$ parameter itself, additional
parameters such as the Hubbard-$J$, etc, can be added if the single $U$
parameter is too crude to fit the experimental data. Hence, since there are
plenty of available parameters and such limited data set with which to fit to,
the HU-BS approach is almost certain to be in good agreement with experiment.
This argument could be turned upside down, of course. Is there any
experimental data that show the essential correctness of the HU-BS approach
other than being a simple fitting procedure? We have been unable to find any
such examples, which leads us to the conclusion that HU-BS approaches are
simply ways to add in some crude many-body effects that are left out of the
original band-structure calculations. Similar questions could be framed in
another way. For example, are there any surprises from these types of
calculations that could not have been guessed from the model calculations
alone? How much physics does the band-structure piece of the Hamiltonian add
that is not included in the Hubbard-U term? What new physics has really come
from the merging of these two Hamiltonians? Is there any rigorous confirmation
of DMFT or any other HU-BS approach that goes beyond a fit to some
experimental data?
The ideal approach would, of course, be to start from the exact microscopic
theory of electronic structure and then to make various approximations that
then lead to different levels of sophistication in the solution. This is
similar to the line of theories starting with Hartree and Hartree-Fock
solutions, through various flavors of local-density approximations, and up
through GW-type theories. However, at this point our abilities to calculate
true many-body effects from first principles appears to have hit a dead end,
in the sense that it is unclear how to go further with a tractable theory.
This is, in fact, the driving force for developing Hubbard-$U$-like
approaches. Modern many-body theory has heavily focused on solving simplified
electronic-structure Hamiltonians based on a simple nearest-neighbor tight-
binding treatment of the Hubbard Hamiltonian, often based on a single orbital
per unit cell. By simplifying the electronic structure, it was possible to
focus on the complex mathematical manipulations that are necessary to treat
the many-body aspects of the theory. The price that was paid for this approach
was to lose the connection to the specific types of orbitals, atoms, and their
geometries possessed by real materials. Hence one ended up with “spherical
cow” approximations to the electronic structure of materials that could not
well describe Fermi surface or photoemission details of materials of interest.
To include these material-dependent properties the LDA band-structure was then
added back into the various approaches, with the treatment of the Hubbard $U$
term projected onto the most localized or strongly correlated orbitals. Since
the model calculations were viewed as simplifications of the real electronic
structure, this lead to the conclusion that one had to add and subtract terms
from the band-structure Hamiltonian, leading to the notion of double-counting,
etc.
We believe that the correct many-body treatment of the microscopic electronic
Hamiltonian is still too difficult for current levels of theory. Hence some
simplifications of the many-body effects will require the introduction of
model terms or expression that parameterize corrections of the first-principle
theory. What these additional terms do in practice is to push spectral weight
of the electronic structure away from that calculated by the original band-
structure theory. For example, in some materials, remnants of the original
atomic structure show up as satellite features (often described as lower
Hubbard bands) below the conventional valence band structure or as additional
peaks in the density of states at the Fermi energy (often described in terms
of Kondo effects in many theories). These effects cannot naturally arise in
the one-electron-type approach of band-structure theory. Since they cannot be
calculated from first-principles, one has to add in parameters to the theory
to force the electronic-structure theory to agree with available experimental
data. One can then question how best to correct the original band-structure
theory to force this agreement, and what understanding such a theory provides
about real electronic structure that an exact theory would predict. There are
also questions about the robustness of such corrections. For example, is each
correction materials specific, or can trends be determined for classes of
materials that continuously tune these parameters. Also, is there enough
physics in the “correction terms” to allow one to understand the correct
mechanisms controlling the functionality and many-body properties described by
such theories?
## 8 Experimental Verification
The key to making progress is good experimental data. Since we cannot trust
the current level of theory to accurately predict materials properties,
especially when part of the electronic structure depends upon unknown
parameters, experimental data is necessary to guide theory.
The chief obstacle with respect to experimental data is that most experiments
do not directly measure the fundamental mathematical properties of the
electronic structure, viz., the various Green’s functions and spectral
densities of states. If we need completely different electronic-structure
theories for each specific type of spectroscopy, we will only attain many
random bits of information that do not form a coherent whole. To be useful
there has to be a common ground where all the experimental data converge. This
common ground has to be a fundamental property of the electronic structure and
it must be amenable to theoretical techniques. Hence it makes sense to focus
on spectral and other fundamental properties of the electronic structure.
From this point of view, one has to ask what each type of spectroscopy
measures and how each one can shed light on the various Green’s functions or
their spectral representations. This will depend upon a very clear theoretical
understanding of all of the physical processes that are involved in each
spectroscopy and how to account for these in order to pull out of the raw data
the fundamental information about spectral functions. At the present time,
very little emphasis has been placed on this. Most interpretations of
experimental data are very simple minded and have not changed much in the last
30 years. Spectrometers and the physical and electronic equipment used in the
various techniques has undergone enormous improvements, but this is not the
case for the fundamental theory needed to interpret the data. This is a
situation that desperately cries out for improvement. The best way to advance
our understanding of strongly correlated electron materials, for example, is
to improve our understanding of what each spectroscopy accurately measures
about their properties. This can only be achieved if we have a proper
understanding of the fundamental physics of each spectroscopic method.
While one could discuss many different types of spectroscopy here, probably
the most heavily used spectroscopies that most directly measure electronic
spectral densities involve interactions of photons with matter, such as
optical spectroscopy and photoelectron spectroscopy. In each case, the
fundamental process involves the absorption of a photon by exciting an
electron from an occupied state of the solid to an unoccupied state. It is
useful to do at least a brief exploratory discussion of how these types of
experiments can be related to the fundamental electronic stucture. Here we
will only discuss photoelectron spectoscopy as a prototype for the types of
discussion that need to be more generally employed.
The simplest theories of photoemission (see, for example, the recent review
paper on the cuprate superconductors, Ref. [15], that cites many earlier
review papers, as well as the standard book on the subject, Ref. [16]) use the
three-step model developed in the early days by Spicer and coworkers. This
treats the photoemission process as involving: (1) an electronic excitation of
the system by a photoelectron, (2) the transport of the photoelectron to the
surface, and (3) the escape of the photoelectron through the surface to the
vacuum where it is detected. Even this very simplified model already hints at
how complicated photoelectron spectroscopy really is. Not only excitation
processes need to be described, but electron transport and detection as well.
Also, the surface clearly must play an important role.
If we just focus on the first process, the excitation of the electrons by the
photon, even this is not simply related to the spectral function of the
desired one-particle Green’s function. Clearly, this must involve the
transition from an occupied electronic level to an unoccupied. In a band
picture, one would calculate this from the Golden Rule, and this would involve
occupied electrons of a given band index and $\mathbf{k}$ being excited to a
higher lying band with some matrix element squared (which would also, of
course, have selection rules). This involves a convolution of occupied and
excited states. From this, how is the spectral function $A(\mathbf{k},\omega)$
to be determined?
In practice, the cuprate review article[15] (see their Eq. (12)) recommends
using the sudden approximation and the formula for the observed electron
intensity:
$I(\mathbf{k},\omega)=I_{0}(\mathbf{k},\nu,\mathbf{A})f(\omega)A(\mathbf{k},\omega)$
where $\mathbf{k}=\mathbf{k}_{\parallel}$ is the in-plane momentum, $\omega$
is the electron energy with respect to the Fermi level,
$I_{0}(\mathbf{k},\nu,\mathbf{A})$ is proportional to a squared one-electron
matrix element and therefore depends on the electron momentum as well as the
energy and polarization of the incoming photon, and $f(\omega)$ is the Fermi
function. The function $A(\mathbf{k},\omega)$ is the electron spectral
function.
With even this additional level of simplification, analysis is still not
completely simple. In most experimental papers, it appears as if the angle-
resolved experiments simply track peaks in the observed energy distribution
curves as a function of angle and energy. These peak energies are then plotted
relative to an estimated Fermi energy to produce an “experimental” band
structure. However, what about the matrix elements
$I_{0}(\mathbf{k},\nu,\mathbf{A})$. If these are strongly k or energy
dependent, they could certainly drastically distort apparent band positions.
One also has to question how valid the sudden approximation is (see, for
example, the classic discussion in Ref. [17]). The chief argument for such a
simplified analysis of photoemission is that the results appear somewhat
similar to one-electron band-structure calculations!
In the above formula, one has to question what happened to the unoccupied
states that the photoelectron is excited to? Hüfner’s book suggests using
free-electron band-structure expressions for accounting for this quantity,
which would involve a more complex analysis than given by the above formula.
However, electronic band-structure calculations suggest that there is
significant band-structure effects that strongly distort even fairly high
energy unoccupied electrons away from their free-electron energies. This is
likely to be the case for the relatively low energies used in the UV
photoelectron range, which has the highest precision. What are these
corrections and how much do they change the determination of
$A(\mathbf{k},\omega)$?
Finally, one should consider the effects of the surface and a large number of
other physical processes such as secondary electrons that complicate the
experiment (see, for example, the Hüfner book). Even 35 years ago, this
complexity was recognized as important for understanding the comparison
between band theory and photoemission experiments[18, 19]. However, today,
most of this complexity seems to have been swept under the rug, with simple
peak evaluations trusted as reliable estimates of the spectral functions! It
is certainly incumbent upon the experimentalists to correct their data as
carefully as possible in order to provide the best experimentally determined
spectral function as possible.
To us, one of the most problematical aspects of photoemission is its high
surface sensitive. Besides possible effects in shifting the peak positions in
which quasiparticle energies are based or on the appearance of new surface
electronic-structure peaks, another example of problems with surface sensitive
spectroscopies is the danger of artifical narrowing of strongly correlated
electron bands. Tight-binding theory suggests that the order of magnitude of
the band width is proportional to the number of near neighbors times the
nearest-neighbor hopping matrix element (see, for example, the analysis in
Ref. [18]). At a surface, there are fewer nearest neighbors and bands should
narrow. This is actually observed in LDA-like calculations of surfaces. See,
for example, Ref. [20]. Correlation effects may artificially enhance such
effects, leading to a significant narrowing of bands that is purely a surface
effect. In the HU-BS methods, band narrowing can be caused by increasing the
value of the Hubbard $U$. If this is fit to photoemission that is really
measuring the surface band width, significant errors may be introduced and
misleading conclusions drawn. Perhaps many strongly correlated materials are
far less correlated than they appear, and the band narrowing observed in
experiment is just a measure of enhanced “surface” electronic structure? It is
unknown how surfaces may modify satellite, Kondo, and other many-body
features. Again, it is possible that they greatly enhance such effects.
While it is clear that these types of experiments desperately need a good
theoretical underpinning to aid in the interpretation of the data, on the
other hand, the very fact that such experiments are surface sensitive makes it
very difficult to develop the precise theory needed to interpret them.
Surfaces introduce changes that depend on how they are prepared and are a much
less intrinsic property of a material than bulk electronic structure. Often
the presence of oxygen, hydrogen, or other impurities can significantly modify
the nature of the surface. In addition, there is the possibility of
preferential segregation of bulk impurities to surfaces, particularly if some
heat treatment or annealing has been performed. For a good theory to be
developed, it is necessary to have a very precise knowledge of all of the
atomic positions and types of atoms at a surface, before attempting to account
for the excitation process of the photon, and the transport of the resulting
photoelectron to the surface, emission through the surface, and its
collection. This involves very complex physics, and is quite difficult.
However, until we have a better understanding of the theory of photoemission,
and how the nature of these types of measurements affect the resulting
electronic properties that are measured, it will always be somewhat dangerous
to rely upon such experimental data to tune the parameters of a strongly
correlated material.
In addition to surface sensitivity, lifetime effects can also be problematic,
and may limit precise measurements to energy regions around the Fermi energy.
Usually the lifetime of an occupied electron state increases rapidly as its
energy moves farther below the Fermi energy. In photoemission, this rapidly
washes out the experimentally determined dispersion relations and it becomes
difficult to know what the quasiparticle energies are deep (or even
moderately) below the Fermi energy. Since the electronic lifetimes are an
intrinisic bulk effect, this effect cannot be reduced in any type of
spectroscopy. This can, for example, make it difficult to measure shifts
between the bottom of the $s$ band and the position of the bottom of the $f$
band in actinides, which might be useful to know if one wants to understand
how nonlocal exchange potentials shift localized electronic states relative to
delocalized.
Viewed more broadly, excitation spectra can have also additional effects that
are unrelated to ground-state electronic structure, making it difficult to
know what the intrinsic electronic structure is in the absence of the specific
experimental probe used to measure the electronic structure. A well known
example of this is exciton effects in semiconductors. In an optical probe, the
incoming photon excites an occupied electron to an unoccupied state, creating
an electron-hole pair. The electron and hole repeatedly scatter off of each
other (this is usually calculated by a Bethe-Salpeter equation) and the
resulting excitation lies in the energy gap of the semiconductor. If not
accounted for, this would erroneously lead to a conclusion that the intrinsic
band gap is smaller than it actually is. Other issues are explicit excitation
processes that are different from ground-state electronic structure, such as
shake-up, shake-off, and other multiplet or other intra-atomic processes
involving electronic excitations that occur nearly simultaneoulsy with the
one-elecron process of interest.
Given these experimental difficulties, one can question how well we know the
experimental electronic structure, and whether the many-body corrections that
we are including by fitting to such data is really correct. There is thus a
real need to develop a better theoretical underpinning for the various
experimental techniques that are being used so that we can more reliably
interpret such data.
## 9 Summary
HU-BS methods involve adding a Hubbard model term to an LDA band-structure
Hamiltonian. The Hubbard model term is a static Coulomb interaction for frozen
orbitals with matrix elements that are scaled to fit experiment. A double-
counting term is actually just a way of preventing the average energy of the
correlated orbitals from being pushed too high in energy and should be
considered just another parameter of the theory that fits the correct average
occupation of the correlated orbitals. The LDA band-structure is a model for
the non-correlated orbitals. It replaces the small number of nearest-neighbor
hopping terms of traditional model Hamiltonians with the full complexity of
all of the relevant orbitals of the various atoms in the solid. However, it
suffers from the defect that LDA does not correctly treat the spectral
properties of these orbitals. In particular, non-local exchange and self-
interaction correction effects are improperly treated. The accuracy of the HU-
BS methods cannot be determined very well, because it is difficult to correct
any current spectroscopy sufficiently to accurately measure the intrinsic
spectral functions of the electrons in the solid. In practice, the HU-BS
methods add lower Hubbard band peaks and narrow the band-width of the
correlated states. The linear term can also be used to introduce Hartree-Fock
like structure to open band gaps and orbitally polarize the electrons. Because
these methods use parameters, they are fits to the experimentally observed
spectra (whether these are an accurate measurement of the actual spectral
functions or not) and are not first-principles methods. They should be viewed
as simply more elaborate model calculations to include more orbitals than
traditional Hubbard models, which often only have one or a very small number
of orbitals. Because the LDA term takes care of the non-correlated orbital
interactions, the number of fitting parameters of a traditional Hubbard model
is reduced for these extra orbitals at the price of the loss in accuracy
entailed by the LDA method.
Progress in the future has to involve two aspects. The first is better first-
principles starting points that incorporate more and more of the correct
physics. The better these are and the more physics they incorporate, the fewer
the corrections that need to be made to compare with experiments. Secondly,
better solutions to a variety of strongly interacting models are needed. What
does the frequency dependence of the exact self-energy for these various
models look like? Will they show any surprises, such as additional features in
their frequency dependence? If, for example, the exact theory simply shifts
the lower Hubbard sattelite away from the position of a less accurate theory,
this could be corrected for by simply modifying the strength of the Hubbard
$U$ parameter. What is particularly important here, and which has not been
carefully examined in the past, is the generic functional forms that the many-
body solutions involve. For example, the Hubbard Hamiltonian usually causes
lower and upper Hubbard-band satellite features. What is the functional form
of the frequency response of the self energy that causes such satellite
structure to appear in the spectral response of the many-body system? Could
one parameterize this in such a way as to compare different many-body
solutions and understand how to include model frequency dependent functional
forms in self-energy corrections to the starting band-structure solutions?
Could one model these in a way similar to what is done in Fermi liquid theory?
For example, if one knows that some quadratic or perhaps exponential frequency
dependence must show up in the exact theory, could one simply parameterize
this response to correct the band structure?
Finally, much more work must go into the interpretation of various
spectroscopies if these are to be accurately related to the bulk spectral
functions predicted by theory. Without good experimental bulk spectral
functions, it is impossible to tell how good or poor our current models of
electronic structure are. Without a true first-principles method, the
parameters of the HU-BS models must be fit to experiment. The resulting
electronic structure will heavily depend on the quality of the experimental
data that is fit to.
This work was carried out under the auspices of the National Nuclear Security
Administration of the U.S. Department of Energy at Los Alamos National
Laboratory under Contract No. DE-AC52-06NA25396, and the Center for
Theoretical Natural Science at Aarhus University.
## References
## References
* [1] P. Hohenberg and W. Kohn. Phys. Rev., 136:B864, 1965.
* [2] W. Kohn and L. J. Sham. Phys. Rev., 140:A1133, 1965.
* [3] R. O. Jones and O. Gunnarsson. Rev. Mod. Phys., 61:689, 1989.
* [4] A. G. Petukhov, I. I. Mazin, L. Chioncel, and A. I. Lichtenstein. Phys. Rev. B, 67:153106, 2003.
* [5] K. Held. Adv. in Physics, 56:829, 2007.
* [6] L. Hedin. Phys. Rev., 139:A796, 1965.
* [7] Lars Hedin and Stig Lundqvist. Effects of electron-electron and electron-phonon interactions on the one-electron states of solids. In Frederick Seitz, David Turnbull, and Henry Ehrenreich, editors, Solid State Physics, volume 23, pages 1–181. Academic Press, New York, 1969\.
* [8] Takao Kotani, Mark van Shilfgaarde, and Sergey V. Faleev. Phys. Rev. B, 76:165106, 2007.
* [9] V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein. J. Phys.: Condens. Matter, 9:767, 1997.
* [10] A. George, G. Kotliar, W. Krauth, and M. J. Rozenburg. Rev. Mod. Phys., 68:13, 1996.
* [11] G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcoullet, and C. A. Marianetti. Rev. Mod. Phys., 78:865, 2006.
* [12] M. M. Steiner, R. C. Albers, D. J. Scalapino, and L. J. Sham. Phys. Rev. B, 43:1637, 1991.
* [13] M. M. Steiner, R. C. Albers, and L. J. Sham. Phys. Rev. B, 45:13272, 1992.
* [14] M. M. Steiner, R. C. Albers, and L. J. Sham. Phys. Rev. Lett., 72:2923, 1991.
* [15] Andrea Damascelli, Zahid Hussain, and Zhi-Xun Shen. Rev. Mod. Phys., 75:473, 2003.
* [16] Stefan Hüffner. Photoelectron Spectroscopy. Springer, Berlin, second edition, 1996.
* [17] Lars Hedin. J. Phys.: Condens. Matter, 11:R489, 1999.
* [18] N. Egede Christensen and B. Feuerbacher. Phys. Rev. B, 10:2349, 1974.
* [19] B. Feuerbacher and N. Egede Christensen. Phys. Rev. B, 10:2373, 1974.
* [20] O. Eriksson, R. C. Albers, and A. M. Boring. Phys. Rev. Lett., 66:1350, 1991.
|
arxiv-papers
| 2009-07-06T16:03:58 |
2024-09-04T02:49:03.762934
|
{
"license": "Public Domain",
"authors": "R. C. Albers, N. E. Christensen, and A. Svane",
"submitter": "Robert Albers",
"url": "https://arxiv.org/abs/0907.1028"
}
|
0907.1436
|
# Attaining mean square boundedness of a marginally stable stochastic linear
system with a bounded control input
Federico Ramponi , Debasish Chatterjee , Andreas Milias-Argeitis , Peter
Hokayem and John Lygeros Automatic Control Laboratory, ETL I28, ETH Zürich,
Physikstrasse 3, 8092 Zürich, Switzerland http://control.ee.ethz.ch
{ramponif,chatterjee,milias,hokayem,lygeros}@control.ee.ethz.ch
###### Abstract.
In this article we construct control policies that ensure bounded variance of
a noisy marginally stable linear system in closed-loop. It is assumed that the
noise sequence is a mutually independent sequence of random vectors, enters
the dynamics affinely, and has bounded fourth moment. The magnitude of the
control is required to be of the order of the first moment of the noise, and
the policies we obtain are simple and computable.
This research was partially supported by the Swiss National Science
Foundation, grant 200021-122072.
## 1\. Introduction
Stabilization of stochastic linear systems with bounded control inputs has
attracted considerable attention over the years. This is due to the fact that
incorporating bounds on the control is of paramount importance in practical
applications; suboptimal control strategies such as receding-horizon control
[Chatterjee et al., 2009; Hokayem et al., 2009], and rollout algorithms
[Bertsekas, 2000], among others, were designed to incorporate such constraints
with relative ease, and have become widespread in applications. However, the
following question remains open: _when is a linear system with possibly
unbounded additive stochastic noise globally stabilizable with bounded
inputs?_ In this article we shall provide sufficient conditions that give a
positive answer to this question with minimal hypotheses.
Bounded input control has a rich and important history in the control
literature [Yang et al., 1992; Sussmann et al., 1994; Yang et al., 1997; Lin
et al., 1996; Stoorvogel et al., 2007]. The deterministic version of the
bounded input stabilization problem was solved completely in a series of
articles [Yang et al., 1992; Sussmann et al., 1994] culminating in [Yang et
al., 1997]. It was demonstrated in [Yang et al., 1997] that global asymptotic
stabilization of a discrete-time linear system
($(\star)$) $x_{t+1}=Ax_{t}+Bu_{t}$
with bounded feedback inputs is possible if and only if the transition matrix
has spectral radius at most $1$, and the pair $(A,B)$ is stabilizable with
arbitrary controls. Moreover, extensions to the output feedback case have
appeared in [Bao et al., 2000; Chitour and Lin, 2003].
In the presence of affine stochastic noise the linear system ($(\star)$ ‣ 1)
becomes $x_{t+1}=Ax_{t}+Bu_{t}+w_{t}$, where $(w_{t})_{t\in\mathbb{N}_{0}}$ is
a collection of independent (but not necessarily identically distributed)
random vectors in $\mathbb{R}^{d}$ with possibly inter-dependent components at
each time $t$. With an arbitrary noise it is clearly not possible to ensure
mean-square boundedness; for instance, if the noise has a spherically
symmetric Cauchy distribution on $\mathbb{R}^{d}$, then given any initial
condition $x_{0}\in\mathbb{R}^{d}$, the second moment of $x_{1}$ does not even
exist. Similarly, if the second moment of the noise becomes unbounded with
time, it is not possible to control the second moment of the process
$(x_{t})_{t\in\mathbb{N}_{0}}$. It is necessary to assume, at least, that the
noise has bounded variance.
Going beyond this necessary condition, it is not difficult to establish mean-
square boundedness of such a system with bounded controls under the assumption
that $A$ is Schur stable, i.e., all eigenvalues of $A$ are contained in the
interior of the unit disk (the proof of this fact relies on standard Foster-
Lyapunov techniques [Meyn and Tweedie, 1993]). However, to the best of our
knowledge, there is no proof that the same can be ensured for a marginally
stable linear system. Results in this direction were reported in [Stoorvogel
et al., 2007], but to the best of our understanding _conclusive_ proofs of the
facts reported in the present article are still missing in the literature.
In this article, we develop easily computable bounded control policies for the
case when $A$ is marginally stable and $(A,B)$ is stabilizable. Our policy is
not anyway stationary and is in general chosen from the class of finite
$k$-history-dependent and/or non-stationary policies. With respect to the case
when $A$ is orthogonal, it turns out that if the system is reachable in one
step (i.e., $\operatorname{rank}B=$ the dimension of the state space), we do
get stationary feedback policies. In the more general case when the system
($(\star)$ ‣ 1) is reachable in $k$ steps (with arbitrary controls), we
propose a feedback policy for a sub-sampled system derived from the original
one, which, for the actual system, turns out to be a $k$-history-dependent
policy. In fact, in this case we realize our policy as successive
concatenations of a fixed $k$-length policy. In the most general situation we
propose a $k$-history-dependent policy, where $k$ is now the reachability
index of the particular subsystem of $(A,B)$ for which the dynamics matrix is
orthogonal. In all the mentioned cases, the length of the policy is at most
equal to the dimension of the state space; memory requirements for even the
most general case are, therefore, modest.
Note that in our setting we do _not_ assume that the noise is white. For our
purposes the requirements on the noise are rather general, namely, the fourth
moment of the noise should be uniformly bounded, and the noise vectors should
be independent of each other (identical distribution at each time is not
assumed). In particular, we do _not_ assume Gaussian structure of the noise.
It turns out that to ensure stabilization we need the controller to be
sufficiently strong, in the sense that the control input norm bound should be
bigger than a uniform bound on the first moment of the noise.
Section 2 contains a precise statement of our result in the most general
hypotheses ($A$ marginally stable and $(A,B)$ stabilizable), and a brief
sketch of the proof. In Section 3, after some preliminary material, we prove
the attainability of bounded second moment for a random walk, then we
generalize the result under weaker and weaker hypotheses, finally culminating
in the proof of the main theorem of Section 2. Section 4 presents a numerical
example illustrating our results, and Section 5 concludes the article with a
conjecture.
## 2\. Main result
### 2.1. Statement of the theorem
Consider the discrete-time linear system
((2.1)) $x_{t+1}=Ax_{t}+Bu_{t}+w_{t},\qquad x_{0}=x,\quad t\in\mathbb{N}_{0},$
where the following hold: $x\in\mathbb{R}^{d}$ is given; the state $x_{t}$ at
time $t$ takes values in $\mathbb{R}^{d}$; $A\in\mathbb{R}^{d\times d}$, all
the eigenvalues of $A$ lie in the closed unit circle, and those eigenvalues
$\lambda$ such that $|\lambda|=1$ have equal algebraic and geometric
multiplicities; $B\in\mathbb{R}^{d\times m}$, and the control $u_{t}$ at time
$t$ takes values in $\mathbb{R}^{m}$; $(w_{t})_{t\in\mathbb{N}_{0}}$ is an
$\mathbb{R}^{d}$-valued random process with mean zero and
$\mathsf{E}\bigl{[}w_{t}w_{t}^{\scriptscriptstyle{\mathrm{T}}}\bigr{]}=Q_{t}$.
Our objective is to synthesize a $k$-history-dependent control policy111See
§3.1 for definitions of policies. $\pi=(\pi_{t})_{t\in\mathbb{N}_{0}}$,
consisting of successive concatenations of $k$-length sequence
$\tilde{\pi}_{0:k-1}\coloneqq\bigl{[}\tilde{\pi}_{0},\cdots,\tilde{\pi}_{k-1}\bigr{]}$
of maps, $\tilde{\pi}_{i}:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{m}$ for
$i=0,\ldots,k-1$, such that $\pi_{t}:\mathbb{R}^{d\times
k}\longrightarrow\mathbb{R}^{m}$ is measurable,
$u_{t}\coloneqq\pi_{t}\bigl{(}x_{t},x_{t-1},\ldots,x_{t-k+1}\bigr{)}$, the
sequence $(u_{t})_{t\in\mathbb{N}_{0}}$ is bounded, and the state of the
closed-loop system
((2.2))
$x_{t+1}=Ax_{t}+B\pi_{t}\bigl{(}x_{t},x_{t-1},\ldots,x_{t-k+1}\bigr{)}+w_{t},\qquad
x_{0}=x,\quad t\in\mathbb{N}_{0},$
has bounded second-order moment. (To simplify the notation, we fix
$x_{-k+1}=\cdots=x_{-1}=x_{0}$.) The following is our main result:
###### (2.3) Theorem.
Consider the system ((2.1)). Suppose that the pair $(A,B)$ is stabilizable,
and that $\sup_{t\in\mathbb{N}_{0}}\mathsf{E}\bigl{[}\left\lVert
w_{t}\right\rVert^{4}\bigr{]}<\infty$. Then there exist an $R>0$ and a
deterministic $k$-history-dependent policy $(\pi_{t})_{t\in\mathbb{N}_{0}}$,
with $k\leq d$ and $\left\lVert\pi_{t}(\cdot)\right\rVert\leqslant R$ for
every $t$, such that
1. (P1)
for every fixed $x\in\mathbb{R}^{d}$ the process
$(x_{t})_{t\in\mathbb{N}_{0}}$ that solves the recursion ((2.2)) satisfies
$\sup_{t\in\mathbb{N}_{0}}\mathsf{E}_{x}\bigl{[}\left\lVert
x_{t}\right\rVert^{2}\bigr{]}<\infty$, and
2. (P2)
in the absence of the random noise the origin is asymptotically stable for the
closed-loop system.
### 2.2. Sketch of the proof
Our proof is built in a series of steps, moving from simpler to progressively
more complex systems. The starting point is the $d$-dimensional random walk
$x_{t+1}=x_{t}+u_{t}+w_{t}$. In this case we employ the main result of
[Pemantle and Rosenthal, 1999] to design a policy that guarantees mean-square
boundedness of the closed-loop system. We then consider the system
$x_{t+1}=Ax_{t}+Bu_{t}+w_{t}$, where $u_{t}$ is a $d$-dimensional control
input, $\operatorname{rank}B=d$, and $A$ is orthogonal. With the help of a
time-varying injective linear transformation this case is reduced to the
$d$-dimensional random walk. The third case that we consider is that of the
system $x_{t+1}=Ax_{t}+Bu_{t}+w_{t}$, where $u_{t}\in\mathbb{R}^{m}$ and $A$
is orthogonal. This is reduced to the second case above with the aid of an
injective linear transformation derived from the reachability matrix of the
pair $(A,B)$ (recall that by assumption the reachability matrix has rank $d$).
Finally, the general case when $A$ is just stable and $(A,B)$ stabilizable is
reduced to the third case with the observation that, in view of the stability
hypothesis, $A$ acts as an orthogonal map on its invariant subspace that
corresponds to the eigenvalues that lie on the unit circle.
Arguments for establishing mean-square boundedness of stochastic dynamical
systems typically rely on $L_{1}$-bounded-ness of a Lyapunov-like functional
of the system. The latter can be established in at least three different ways:
The first is via the classical Foster-Lyapunov drift-conditions [Foss and
Konstantopoulos, 2004; Meyn and Tweedie, 1993] and its various refinements;
the second is via excursion-theoretic analysis [Chatterjee and Pal, 2008] that
relies primarily on the existence of certain supermartingales as long as the
process is outside some bounded set; the third is via martingale inequalities
[Pemantle and Rosenthal, 1999], which applies to more general scalar-valued
processes than Markov processes, and in the presence of bounded controls,
provides the basic machinery for establishing our Theorem (2.3).
## 3\. Proof of the main result
### 3.1. Preliminaries
Let $\mathbb{N}_{0}$ be the set of nonnegative integers $\\{0,1,2,\ldots\\}$.
The standard $2$-norm on Euclidean spaces is denoted by
$\left\lVert\cdot\right\rVert$ and the absolute value on $\mathbb{R}$ by
$\left\lvert{\cdot}\right\rvert$. In a Euclidean space we denote by
${\mathcal{B}}_{r}$ the closed Euclidean ball of radius $r$ centered at the
origin. If $(y_{t})_{t\in\mathbb{N}_{0}}$ is a random process on a probability
space $(\Omega,\mathfrak{F},\mathsf{P})$, taking values in some Euclidean
space, we let $\mathsf{E}_{x}[\varphi(y_{s};s=0,1,\ldots,t)]$ denote the
conditional expectation of a measurable mapping $\varphi$ of the process up to
time $t$, given the initial condition $y_{0}=x$; in particular we define the
$n$-th moment of $y_{t}$ as $\mathsf{E}_{x}[\left\lVert
y_{t}\right\rVert^{n}]$. We denote conditional expectation given a
sub-$\sigma$-algebra $\mathfrak{F}^{\prime}$ of $\mathfrak{F}$ as
$\mathsf{E}[\cdot\,|\,\mathfrak{F}^{\prime}]$. For $r>0$ let
$\operatorname{sat}_{r}:\mathbb{R}^{d}\longrightarrow{\mathcal{B}}_{r}$ be
defined by $\operatorname{sat}_{r}(y)\coloneqq y$ if $y\in{\mathcal{B}}_{r}$
and $\operatorname{sat}_{r}(y)\coloneqq ry/\left\lVert y\right\rVert$
otherwise. Note that $\operatorname{sat}_{r}(\cdot)$ is _not_ the component-
wise saturation function. Given matrices $A\in\mathbb{R}^{d\times d}$ and
$B\in\mathbb{R}^{d\times m}$ we define the $k$-step reachability matrix
${\mathcal{R}}_{k}\coloneqq\left[\begin{array}[]{cccc}B&AB&\cdots&A^{k-1}B\end{array}\right]$.
We specialize the general definition of a policy [Hernández-Lerma and
Lasserre, 1996, Chapter 2] to our setting. A policy
$\pi\coloneqq(\pi_{t})_{t\in\mathbb{N}_{0}}$ is a sequence of measurable maps
$\pi_{t}:\mathbb{R}^{d\times k}\longrightarrow\mathbb{R}^{m}$ for some
$k\in\mathbb{N}$, such that the control at time $t$ is
$\pi_{t}\bigl{(}x_{t},x_{t-1},\ldots,x_{t-k+1}\bigr{)}$. The policy
$\pi=(\pi_{t})_{t\in\mathbb{N}_{0}}$ we have defined is also known as a
deterministic $k$-history-dependent policy in the literature. A special case
of these policies is a deterministic feedback policy or simply a feedback if
$k=1$ in the definition of a deterministic history-dependent policy. Under
deterministic feedback policies the closed-loop system is Markovian
[Hernández-Lerma and Lasserre, 1996, Proposition 2.3.5]. A further special
case is when $\pi_{t}=f$, a fixed measurable mapping
$f:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{m}$ for $t\in\mathbb{N}_{0}$; this
is known as a stationary feedback policy.
###### (3.1) Lemma.
Let $B_{1},\cdots,B_{k}$ be $d\times m$ matrices,
$M\coloneqq\left[\begin{array}[]{ccc}B_{1}&\cdots&B_{k}\end{array}\right]$,
and $\sigma_{d}$ denote the minimum singular value of $M$. If
$\operatorname{rank}M=d$, then for all $r>0$ every vector $v\in\mathbb{R}^{d}$
belonging to ${\mathcal{B}}_{r}$ can be expressed as
$v=\sum_{i=1}^{k}B_{i}u_{i}$, with $u_{i}\in\mathbb{R}^{m}$ and $\left\lVert
u_{i}\right\rVert\leq r\sigma_{d}^{-1}$. In particular, if
$B\in\mathbb{R}^{d\times d}$ and $\operatorname{rank}B=d$, then every vector
$v\in\mathbb{R}^{d}$ belonging to ${\mathcal{B}}_{r}$ can be expressed as
$v=Bu$, where $u\in\mathbb{R}^{d}$, $\left\lVert u\right\rVert\leq
r\sigma_{d}^{-1}$.
###### Proof.
$\operatorname{rank}M=d$ implies that $km\geq d$. Hence,
$M=\left[\begin{array}[]{ccc}B_{1}&\cdots&B_{k}\end{array}\right]\in\mathbb{R}^{d\times
km}$ is a “flat” matrix. Let
$M=USV^{\scriptscriptstyle{\mathrm{T}}}=U\left[\begin{array}[]{cc}\Sigma&0\\\
\end{array}\right]V^{\scriptscriptstyle{\mathrm{T}}}$ be a singular value
decomposition of $M$, where $\Sigma=\mathop{\rm
diag}(\sigma_{1},...,\sigma_{d})$. Since $M$ has full rank, the matrix
$\Sigma$ is invertible. Hence every vector $v\in\mathbb{R}^{d}$ can be
expressed as $v=Mu$, where $u=M^{+}v$ and
$M^{+}=V\left[\begin{array}[]{c}\Sigma^{-1}\\\ 0\\\
\end{array}\right]U^{\scriptscriptstyle{\mathrm{T}}}\in\mathbb{R}^{km\times
d}$ is the Moore-Penrose pseudoinverse of $M$. Since $U,V$ are orthogonal, for
any $\rho>0$ we have $\inf_{\left\lVert u\right\rVert=\rho}\left\lVert
Mu\right\rVert=\inf_{\left\lVert
V^{\scriptscriptstyle{\mathrm{T}}}u\right\rVert=\rho}\left\lVert
U\left[\begin{array}[]{cc}\Sigma&0\\\
\end{array}\right]V^{\scriptscriptstyle{\mathrm{T}}}u\right\rVert=\inf_{\left\lVert\upsilon\right\rVert=\rho}\left\lVert\Sigma\upsilon\right\rVert=\rho\sigma_{d}.$
Hence, the image of ${\mathcal{B}}_{\rho}$ under $M$ contains
${\mathcal{B}}_{\rho\sigma_{d}}$, and if we choose $\rho=r\sigma_{d}^{-1}$,
then the image of ${\mathcal{B}}_{\rho}$ under $M$ contains
${\mathcal{B}}_{r}$. Notice that $\sigma_{d}^{-1}$ is also the greatest
singular value of $M^{+}$, and indeed we have $\sup_{\left\lVert
v\right\rVert=r}\left\lVert M^{+}v\right\rVert=\sup_{\left\lVert
U^{\scriptscriptstyle{\mathrm{T}}}v\right\rVert=r}\left\lVert
V\left[\begin{array}[]{c}\Sigma^{-1}\\\ 0\\\
\end{array}\right]U^{\scriptscriptstyle{\mathrm{T}}}v\right\rVert=\sup_{\left\lVert\nu\right\rVert=r}\bigl{\|}\Sigma^{-1}\nu\bigr{\|}=r\sigma_{d}^{-1}.$
Summing up, every $v\in{\mathcal{B}}_{r}$ can be expressed as $v=Mu$, where
$u\in\mathbb{R}^{km}$ and $\left\lVert u\right\rVert\leq r\sigma_{d}^{-1}$. It
remains to notice that $u$ can be partitioned according to the partition of
$M$, that is
$v=Mu=\left[\begin{array}[]{cccc}B_{1}&B_{2}&\cdots&B_{k}\end{array}\right]\begin{bmatrix}u_{1}^{\scriptscriptstyle{\mathrm{T}}}&\cdots&u_{k}^{\scriptscriptstyle{\mathrm{T}}}\end{bmatrix}^{\scriptscriptstyle{\mathrm{T}}}=\sum_{i=1}^{k}B_{i}u_{i}$
and the bound $\left\lVert u\right\rVert\leq r\sigma_{d}^{-1}$ implies
$\left\lVert u_{i}\right\rVert\leq r\sigma_{d}^{-1}$ for all $i=1\cdots k$. ∎
### 3.2. The $d$-dimensional random walk
At the core of our proof is the $d$-dimensional random walk:
((3.2)) $x_{t+1}=x_{t}+u_{t}+w_{t},\qquad x_{0}=x,\quad t\in\mathbb{N}_{0},$
with the state $x_{t}\in\mathbb{R}^{d}$, the control $u_{t}\in\mathbb{R}^{d}$
with $\left\lVert u_{t}\right\rVert\leqslant r$ for some $r>0$, the noise
process $(w_{t})_{t\in\mathbb{N}_{0}}$ satisfies the following assumption:
###### (3.3) Assumption.
* $\diamond$
$(w_{t})_{t\in\mathbb{N}_{0}}$ are mutually independent $d$-dimensional random
vectors (not necessarily identically distributed),
* $\diamond$
$\mathsf{E}[w_{t}]=0$,
$\mathsf{E}\bigl{[}w_{t}w_{t}^{\scriptscriptstyle{\mathrm{T}}}\bigr{]}=Q_{t}$
for all $t\in\mathbb{N}_{0}$,
* $\diamond$
there exist $C_{4}>0$ such that $\mathsf{E}\bigl{[}\left\lVert
w_{t}\right\rVert^{4}\bigr{]}\leqslant C_{4}$ for all
$t\in\mathbb{N}_{0}$.$\diamondsuit$
Let $C_{1}\coloneqq\sup_{t\in\mathbb{N}_{0}}\mathsf{E}\bigl{[}\left\lVert
w_{t}\right\rVert\bigr{]}$; this is well-defined because by Jensen’s
inequality we have $C_{1}\leqslant\sqrt[4]{C_{4}}$. Let
$(\mathfrak{F}_{t})_{t\in\mathbb{N}_{0}}$ be the natural filtration of the
system ((3.2)). Our proof of Theorem (2.3) relies on the following (immediate)
adaptation of the fundamental result [Pemantle and Rosenthal, 1999, Theorem
1].
###### (3.4) Proposition.
Let $(\xi_{t})_{t\in\mathbb{N}_{0}}$ be a sequence of nonnegative random
variables on some probability space $(\Omega,\mathfrak{F},\mathsf{P})$, and
let $(\mathfrak{F}_{t})_{t\in\mathbb{N}_{0}}$ be any filtration to which
$(\xi_{t})_{t\in\mathbb{N}_{0}}$ is adapted. Suppose that there exist
constants $b>0$, and $J,M<\infty$, such that $\xi_{0}\leqslant J$, and for all
$t$:
((3.5))
$\displaystyle\mathsf{E}\bigl{[}\xi_{t+1}-\xi_{t}\big{|}\mathfrak{F}_{t}\bigr{]}\leqslant-b\quad\text{on
the event }\\{\xi_{t}>J\\},\quad\text{and}$ ((3.6))
$\displaystyle\mathsf{E}\bigl{[}\left\lvert{\xi_{t+1}-\xi_{t}}\right\rvert^{4}\big{|}\xi_{0},\ldots,\xi_{t}\bigr{]}\leqslant
M.$
Then there exists a constant $c=c(b,J,M)>0$ such that
$\displaystyle{\sup_{t\in\mathbb{N}_{0}}\mathsf{E}\bigl{[}\xi_{t}^{2}\bigr{]}\leqslant
c}$.
###### (3.7) Lemma.
Consider the system ((3.2)), and define $\xi_{t}\coloneqq\left\lVert
x_{t}\right\rVert$, $t\in\mathbb{N}_{0}$. There exists a constant $b>0$, such
that for any $r>C_{1}$ condition ((3.5)) holds in closed-loop with the control
$u_{t}=-\operatorname{sat}_{r}(x_{t})$.
###### Proof.
Fix $t\in\mathbb{N}_{0}$ and $r>C_{1}$. We have
$\displaystyle\mathsf{E}\bigl{[}\xi_{t+1}-\xi_{t}\big{|}\mathfrak{F}_{t}\bigr{]}$
$\displaystyle=\mathsf{E}\bigl{[}\left\lVert x_{t+1}\right\rVert-\left\lVert
x_{t}\right\rVert\big{|}\mathfrak{F}_{t}\bigr{]}=\mathsf{E}\bigl{[}\bigl{\|}x_{t}+u_{t}+w_{t}\bigr{\|}-\left\lVert
x_{t}\right\rVert\big{|}\mathfrak{F}_{t}\bigr{]}$
$\displaystyle=\mathsf{E}\bigl{[}\left\lVert
x_{t}-\operatorname{sat}_{r}(x_{t})+w_{t}\right\rVert-\left\lVert
x_{t}\right\rVert\big{|}\mathfrak{F}_{t}\bigr{]}$
$\displaystyle\leqslant\mathsf{E}\bigl{[}\left\lVert
x_{t}-\operatorname{sat}_{r}(x_{t})\right\rVert+\left\lVert
w_{t}\right\rVert-\left\lVert
x_{t}\right\rVert\big{|}\mathfrak{F}_{t}\bigr{]}.$
Let $J=r$ and $b\coloneqq r-C_{1}$. On the set $\\{\left\lVert
x_{t}\right\rVert>J\\}$ we have $\left\lVert
x_{t}-\operatorname{sat}_{r}(x_{t})\right\rVert-\left\lVert
x_{t}\right\rVert=-r$. From the above we get, on the set $\\{\left\lVert
x_{t}\right\rVert>J\\}$,
$\displaystyle\mathsf{E}\bigl{[}\xi_{t+1}-\xi_{t}\big{|}\mathfrak{F}_{t}\bigr{]}$
$\displaystyle\leqslant\mathsf{E}\bigl{[}\left\lVert
x_{t}-\operatorname{sat}_{r}(x_{t})\right\rVert+\left\lVert
w_{t}\right\rVert-\left\lVert
x_{t}\right\rVert\big{|}\mathfrak{F}_{t}\bigr{]}$
$\displaystyle=-r+\mathsf{E}\bigl{[}\left\lVert w_{t}\right\rVert\bigr{]}$
$\displaystyle\leqslant-b,$
where $b$ is positive by our hypothesis. The assertion follows. ∎
###### (3.8) Lemma.
Consider the system ((3.2)) and define $\xi_{t}\coloneqq\left\lVert
x_{t}\right\rVert$, $t\in\mathbb{N}_{0}$. Then for the closed-loop system with
$u_{t}=-\operatorname{sat}_{r}(x_{t})$ there exists a constant $M=M(C_{4})>0$
such that ((3.6)) holds.
###### Proof.
Fix $r>C_{1}$. Applying the triangle inequality successively, we have
$\left\lvert{\xi_{t+1}-\xi_{t}}\right\rvert^{4}=\left\lvert{\left\lVert
x_{t+1}\right\rVert-\left\lVert
x_{t}\right\rVert}\right\rvert^{4}\leqslant\left\lVert
x_{t+1}-x_{t}\right\rVert^{4}=\left\lVert
u_{t}+w_{t}\right\rVert^{4}\leqslant\bigl{(}r+\left\lVert
w_{t}\right\rVert\bigr{)}^{4},$
which leads to
$\mathsf{E}\bigl{[}\left\lvert{\xi_{t+1}-\xi_{t}}\right\rvert^{4}\,\big{|}\,\xi_{0},\ldots,\xi_{t}\bigr{]}\leqslant\mathsf{E}\bigl{[}\bigl{(}r+\left\lVert
w_{t}\right\rVert\bigr{)}^{4}\big{|}\xi_{0},\ldots,\xi_{t}\bigr{]}=\mathsf{E}\bigl{[}\bigl{(}r+\left\lVert
w_{t}\right\rVert\bigr{)}^{4}\bigr{]}.$
Since the fourth moment of $w_{t}$ is uniformly bounded, expanding the right-
hand side above and applying Jensen’s inequality shows that there exists some
$M=M(C_{4})>0$ such that $\mathsf{E}\bigl{[}\bigl{(}r+\left\lVert
w_{t}\right\rVert\bigr{)}^{4}\bigr{]}\leqslant M$. The assertion follows. ∎
###### (3.9) Proposition.
For $r>0$ consider the system ((3.2)) under the deterministic stationary
feedback policy $u_{t}=-\operatorname{sat}_{r}(x_{t})$:
((3.10)) $x_{t+1}=x_{t}-\operatorname{sat}_{r}(x_{t})+w_{t},\qquad
x_{0}=x,\quad t\in\mathbb{N}_{0}.$
Then for every $r>C_{1}$ the system ((3.10)) satisfies
$\sup_{t\in\mathbb{N}_{0}}\mathsf{E}_{x}\bigl{[}\left\lVert
x_{t}\right\rVert^{2}\bigr{]}\leqslant c$ for some $c=c(x,C_{1})<\infty$.
###### Proof.
Let $r=C_{1}+b$ for some $b>0$ and $J\coloneqq\max\bigl{\\{}r,\left\lVert
x\right\rVert\bigr{\\}}$. Lemma (3.7) guarantees that ((3.5)) holds, and Lemma
(3.8) shows that there exists an $M>0$ such that ((3.6)) holds. The assertion
now is an immediate consequence of Proposition (3.4). ∎
### 3.3. The case of $A$ orthogonal
Next we establish part (P1) of the main theorem in the particular case of $A$
being orthogonal.
###### (3.11) Lemma.
Consider the system $y_{t+1}=Ay_{t}+u_{t}+w_{t}$, where $y_{t}$ and $u_{t}$
take values in $\mathbb{R}^{d}$, $A$ is orthogonal, and
$(w_{t})_{t\in\mathbb{N}_{0}}$ satisfies Assumption (3.3). There exist a
constant $r>0$ and a deterministic stationary policy $\pi=(f,f,\cdots)$ such
that $\left\lVert f(y)\right\rVert\leqslant r$ for all $y\in\mathbb{R}^{d}$
and $t\in\mathbb{N}_{0}$, and the closed-loop system
((3.12)) $y_{t+1}=Ay_{t}+f(y_{t})+w_{t}$
under this policy satisfies
$\sup_{t\in\mathbb{N}_{0}}\mathsf{E}_{x}\bigl{[}\left\lVert
y_{t}\right\rVert^{2}\bigr{]}<\infty$.
###### Proof.
Consider the process $(z_{t})_{t\in\mathbb{N}_{0}}$ defined by
$z_{t}\coloneqq(A^{\scriptscriptstyle{\mathrm{T}}})^{t}\ y_{t}$. The second
moment of $z_{t}$ is the same as that of $y_{t}$ due to orthogonality of $A$:
$\mathsf{E}_{x}\bigl{[}\left\lVert
z_{t}\right\rVert^{2}\bigr{]}=\mathsf{E}_{x}\bigl{[}\left\lVert(A^{\scriptscriptstyle{\mathrm{T}}})^{t}\
y_{t}\right\rVert^{2}\bigr{]}=\mathsf{E}_{x}\bigl{[}y_{t}^{\scriptscriptstyle{\mathrm{T}}}A^{t}(A^{\scriptscriptstyle{\mathrm{T}}})^{t}y_{t}\bigr{]}=\mathsf{E}_{x}\bigl{[}y_{t}^{\scriptscriptstyle{\mathrm{T}}}y_{t}\bigr{]}=\mathsf{E}_{x}\bigl{[}\left\lVert
y_{t}\right\rVert^{2}\bigr{]}.$
Now we have
((3.13)) $\begin{split}z_{t+1}&=(A^{\scriptscriptstyle{\mathrm{T}}})^{t+1}\
y_{t+1}=(A^{\scriptscriptstyle{\mathrm{T}}})^{t}\
y_{t}+(A^{\scriptscriptstyle{\mathrm{T}}})^{t+1}\
u_{t}+(A^{\scriptscriptstyle{\mathrm{T}}})^{t+1}\
w_{t}=z_{t}+\bar{u}_{t}+\bar{w}_{t},\end{split}$
where the mapping
$u_{t}\longmapsto\bar{u}_{t}\coloneqq(A^{\scriptscriptstyle{\mathrm{T}}})^{t+1}\
u_{t}$ is isometric and invertible, and $(\bar{w}_{t})_{t\in\mathbb{N}_{0}}$
defined by $\bar{w}_{t}\coloneqq(A^{\scriptscriptstyle{\mathrm{T}}})^{t+1}\
w_{t}$, is a sequence of zero-mean, independent (although in general not
identically distributed) random vectors, with fourth moment given by
$\mathsf{E}\bigl{[}\left\lVert\bar{w}_{t}\right\rVert^{4}\bigr{]}=\mathsf{E}\bigl{[}\bigl{\|}(A^{\scriptscriptstyle{\mathrm{T}}})^{t+1}\
w_{t}\bigr{\|}^{4}\bigr{]}=\mathsf{E}\bigl{[}\left\lVert
w_{t}\right\rVert^{4}\bigr{]}\leqslant C_{4}.$ Due to Proposition (3.9), there
exists a constant $r$ such that the closed-loop system ((3.13)) under the
policy $\bar{u}_{t}=-\operatorname{sat}_{r}(z_{t})\eqqcolon\bar{f}(z_{t})$ has
bounded second moment. Consequently, the original system ((3.12)) has bounded
second moment under the policy
$u_{t}=A^{t+1}\bar{u}_{t}=A^{t+1}\bar{f}(z_{t})=-A^{t+1}\operatorname{sat}_{r}\left((A^{\scriptscriptstyle{\mathrm{T}}})^{t}\
y_{t}\right)\eqqcolon f_{t}(y_{t}).$
Noting that for any orthogonal matrix $A$ we have
$\operatorname{sat}_{r}(Ay)=A\operatorname{sat}_{r}(y)$, we arrive at
$u_{t}=f_{t}(y_{t})=-A\operatorname{sat}_{r}(y_{t})\eqqcolon f(y_{t}),$ which
is indeed a stationary feedback. Moreover, since $\left\lVert
A\operatorname{sat}_{r}(y_{t})\right\rVert\leq r$, we have $\left\lVert
f(y_{t})\right\rVert\leq r$. ∎
In the following we will consider a nonstationary policy obtained by
successive concatenations of a $k$-length policy $(f_{0},f_{1},\cdots
f_{k-1})$ acting on the “sub-sampled” process $(x_{nk})_{n\in\mathbb{N}_{0}}$.
More precisely, our policy has the form $u_{t}=Bf_{t\ {\bf mod}\ k}(x_{(t\div
k)k})$ where the “$\div$” symbol denotes integer division and “${\bf mod}$”
its remainder. In words, we break the time line into segments of length $k$,
and within each segment we let the controls be given by $f_{0},f_{1},\cdots
f_{k-1}$, applied in this order always to the first state observed in the
segment. For example, $x_{1}=x_{0}+Bf_{0}(x_{0})+w_{0}$,
$x_{2}=x_{1}+Bf_{1}(x_{0})+w_{1}$, …, $x_{k}=x_{k-1}+Bf_{k-1}(x_{0})+w_{k-1}$,
$x_{k+1}=x_{k}+Bf_{0}(x_{k})+w_{k}$, $x_{k+2}=x_{k+1}+Bf_{1}(x_{k})+w_{k+1}$,
and so on.
###### (3.14) Lemma.
Consider the system
((3.15)) $x_{t+1}=Ax_{t}+Bu_{t}+w_{t},$
where $x_{t}$ takes values in $\mathbb{R}^{d}$, $u_{t}$ takes values in
$\mathbb{R}^{m}$, $A$ is orthogonal, the pair $(A,B)$ is reachable in $k$
steps (i.e., $\operatorname{rank}{\mathcal{R}}_{k}=d$, where
${\mathcal{R}}_{k}=\left[\begin{array}[]{cccc}B&AB&\cdots&A^{k-1}B\end{array}\right]$),
and $(w_{t})_{t\in\mathbb{N}_{0}}$ satisfies Assumption (3.3). Then there
exist a constant $\rho>0$ and a policy $\pi=(f_{0},f_{1},\cdots
f_{k-1},f_{0},f_{1},\cdots)$ such that $\left\lVert
f_{i}(x)\right\rVert\leq\rho$ for all $x\in\mathbb{R}^{d}$, and the closed-
loop system
((3.16)) $x_{t+1}=Ax_{t}+Bf_{t\ {\bf mod}\ k}(x_{(t\div k)k})+w_{t}$
under this policy satisfies
$\sup_{t\in\mathbb{N}_{0}}\mathsf{E}_{x}\bigl{[}\left\lVert
x_{t}\right\rVert^{2}\bigr{]}<\infty$.
###### Proof.
Let $\tau\in\mathbb{N}_{0}$ and consider the evolution of ((3.15)) from time
$\tau k$ to time $(\tau+1)k$:
((3.17)) $\begin{split}x_{(\tau+1)k}&=A^{k}\ x_{\tau
k}+{\mathcal{R}}_{k}\begin{bmatrix}u_{(\tau+1)k-1}\\\ \vdots\\\ u_{\tau
k}\end{bmatrix}+\sum_{i=0}^{k-1}A^{k-1-i}w_{\tau k+i}=\bar{A}x_{\tau
k}+\bar{u}_{\tau}+\tilde{w}_{\tau},\end{split}$
where $\tilde{w}_{\tau}\coloneqq\sum_{i=0}^{k-1}A^{k-1-i}w_{\tau k+i}$ is a
random vector with mean zero and bounded fourth moment. Since
${\mathcal{R}}_{k}$ has full rank, Lemma (3.1) implies that for arbitrary
$r>0$, any $\bar{u}_{\tau}$ in ${\mathcal{B}}_{r}$ can be expressed as
$\bar{u}_{\tau}=\sum_{i=0}^{k-1}A^{k-1-i}Bu_{\tau k+i}$, where $\left\lVert
u_{\tau k+i}\right\rVert\leq r\sigma_{d}^{-1}$ and $\sigma_{d}$ is the
smallest singular value of ${\mathcal{R}}_{k}$. But from Lemma (3.11) we know
that there exists a particular $r>0$ such that, under the stationary policy
$\bar{u}_{\tau}=f(x_{\tau k})=-\bar{A}\operatorname{sat}_{r}(x_{\tau k})$, the
“sub-sampled” system ((3.17)) has bounded second moment, and
$\left\lVert\bar{u}_{\tau}\right\rVert\leq r$. Therefore, if we choose
$\rho=r\sigma_{d}^{-1}$, there exists a constant $c=c(x,C_{1},C_{4})>0$ such
that $\sup_{\tau\in\mathbb{N}_{0}}\mathsf{E}_{x}\bigl{[}\left\lVert x_{\tau
k}\right\rVert^{2}\bigr{]}\leqslant c$. It follows from the system dynamics
that for $n=0,\ldots,k-1$,
$\displaystyle\mathsf{E}_{x}\bigl{[}\left\lVert x_{\tau
k+n}\right\rVert^{2}\bigr{]}$ $\displaystyle\leqslant
2\bigl{(}c+n^{2}r^{2}\sigma_{1}(B)^{2}\bigl{)}+k\max_{n=0,\ldots,k-1}\operatorname{tr}Q_{\tau
k+n}$ $\displaystyle\leqslant
2\bigl{(}c+n^{2}r^{2}\sigma_{1}(B)^{2}\bigl{)}+k\sqrt{C_{4}},$
where the last step follows from Jensen’s inequality. Since the right-hand
side above constitutes a uniform bound, this proves the assertion. ∎
###### (3.18) Remark.
The actual policy for ((3.15)) is $\begin{bmatrix}u_{(\tau+1)k-1}\\\ \vdots\\\
u_{\tau
k}\end{bmatrix}=-{\mathcal{R}}_{k}^{+}\bar{A}\operatorname{sat}_{r}(x_{\tau
k}).$ The proof above shows that all the inputs
$u_{(\tau+1)k-1},\cdots,u_{\tau k}$ can be computed at time $\tau k$ in order
to counteract the future effect of the current state, i.e. $\bar{A}x_{\tau
k}$, and ignoring the effect of the noise for the following $k$ steps. In the
particular case when $B\in\mathbb{R}^{d\times d}$ has full rank, $m=d$, and
obviously $k=1$, the above policy is stationary, and in particular it has the
form: $u_{t}=f(x_{t})=-B^{-1}A\operatorname{sat}_{r}(x_{t}).$ Once again we
have $\left\lVert u_{t}\right\rVert\leq r\sigma_{d}^{-1}$, where this time
$\sigma_{d}$ is the smallest singular value of $B$. $\vartriangleleft$
### 3.4. Proof of Theorem (2.3)
###### Proof.
Consider the system ((2.1)), with $(A,B)$ stabilizable and
$(w_{t})_{t\in\mathbb{N}_{0}}$ with bounded fourth moment. If $A$ is Schur
stable (that is, all the eigenvalues of $A$ belong to the interior of the unit
disk), the system with zero input has bounded second moment and is
asymptotically stable, and there is nothing to prove. Otherwise, there exists
a change of base in the state-space that brings the original pair $(A,B)$ to a
new pair $\bigl{(}\tilde{A},\tilde{B}\bigr{)}$, where $\tilde{A}$ is in real
Jordan form [Horn and Johnson, 1990, p. 150]. In particular, choosing a
suitable ordering of the Jordan blocks, we can ensure that the pair
$\bigl{(}\tilde{A},\tilde{B}\bigr{)}$ has the form
$\left(\bigl{[}\begin{smallmatrix}A_{11}&0\\\
0&A_{22}\end{smallmatrix}\bigr{]},\bigl{[}\begin{smallmatrix}B_{1}\\\
B_{2}\end{smallmatrix}\bigr{]}\right)$, where $A_{11}$ is Schur stable, and
$A_{22}$ has its eigenvalues on the unit circle. Due to the stability
hypothesis (the algebraic and geometric multiplicities of the eigenvalues of
$A_{22}$ are equal), $A_{22}$ is therefore block-diagonal with elements on the
diagonal being either $\pm 1$ or $2\times 2$ rotation matrices. As a
consequence, $A_{22}$ is orthogonal. Moreover, since $(A,B)$ is stabilizable,
the pair $(A_{22},B_{2})$ must be reachable in a number of steps $k\leq d$
which depends on the dimension of $A_{22}$ and the structure of
$(A_{22},B_{2})$, since it contains precisely the modes of $A$ which are not
asymptotically stable. Summing up, we can reduce the original system
$x_{t+1}=Ax_{t}+Bu_{t}+w_{t}$ to the form
$\Bigl{[}\begin{smallmatrix}x^{(1)}_{t+1}\\\
x^{(2)}_{t+1}\end{smallmatrix}\Bigr{]}=\Bigl{[}\begin{smallmatrix}A_{11}x^{(1)}_{t}\\\
A_{22}x^{(2)}_{t}\end{smallmatrix}\Bigr{]}+\Bigl{[}\begin{smallmatrix}B_{1}\\\
B_{2}\end{smallmatrix}\Bigr{]}u_{t}+\Bigl{[}\begin{smallmatrix}w^{(1)}_{t}\\\
w^{(2)}_{t}\end{smallmatrix}\Bigr{]},$ where $A_{11}$ is Schur stable,
$A_{22}$ is orthogonal, $(A_{22},B_{2})$ is reachable, and
$\Bigl{(}\Bigl{[}\begin{smallmatrix}w^{(1)}_{t}\\\
w^{(2)}_{t}\end{smallmatrix}\Bigr{]}\Bigr{)}_{t\in\mathbb{N}_{0}}$ is derived
from $(w_{t})_{t\in\mathbb{N}_{0}}$ by means of linear transformations. We
know that since $A_{11}$ is Schur stable, the noise
$\bigl{(}w^{(1)}_{t}\bigr{)}_{t\in\mathbb{N}_{0}}$ has bounded second moment,
and the control inputs $(u_{t})_{t\in\mathbb{N}_{0}}$ are bounded, then the
$x^{(1)}$ sub-system is mean-square bounded under any Markovian control
[Chatterjee et al., 2009, §4]. Therefore, if under some bounded policy the
$x^{(2)}$ sub-system is mean-square bounded, the original system will also be
mean-square bounded under the same policy. Thus, at least for the proof of
(P1), it suffices to restrict our attention to the subsystem described by the
pair $\bigl{(}A_{22},B_{2}\bigr{)}$. Suppose that this subsystem is reachable
in a certain number $k\leq d$ of steps.
The proof of (P1) coincides with the proof of Lemma (3.14), where we obtain
$\rho=r\sigma_{d}^{-1}$ for $r>C_{1}$ and $\sigma_{d}$ is the smallest
singular value of ${\mathcal{R}}_{k}$. (Here,
${\mathcal{R}}_{k}=\left[\begin{array}[]{cccc}B_{2}&A_{22}B_{2}&\cdots&A_{22}^{k-1}B_{2}\end{array}\right]$.)
As the control authority required in the claim of the theorem, we choose
precisely $R=\rho$.
To prove (P2), notice that for the closed-loop “sub-sampled” system without
noise under the policy
$u_{t}=-{\mathcal{R}}_{k}^{+}\bar{A}\operatorname{sat}_{r}\bigl{(}x_{t}^{(2)}\bigr{)}$,
where $\bar{A}=A_{22}^{k}$, it holds:
((3.19)) $x_{(\tau+1)k}^{(2)}=\bar{A}x_{\tau
k}^{(2)}-\bar{A}\operatorname{sat}_{r}\bigl{(}x_{\tau k}^{(2)}\bigr{)}.$
As long as $x_{\tau k}^{(2)}$ is outside ${\mathcal{B}}_{r}$, $\left\lVert
x_{(\tau+1)k}^{(2)}\right\rVert=\left\lVert x_{\tau k}^{(2)}\right\rVert-r$.
Hence, in a finite number of steps it must hold $\left\lVert x_{\tau
k}^{(2)}\right\rVert<r$. When for some $\bar{\tau}$ we have $\left\lVert
x_{(\bar{\tau}-1)k}^{(2)}\right\rVert<r$, by the definition of
$\operatorname{sat}_{r}(\cdot)$ we have $x_{\bar{\tau}k}^{(2)}=0$, and
consequently $x_{\tau k}^{(2)}=0$ for all $\tau\geq\bar{\tau}$. Hence, the
state of the closed-loop “sub-sampled” system converges to zero in finite time
for any initial condition. Then, according to the chosen policy, for all
$\tau\geq\bar{\tau}$ we have $\begin{bmatrix}u_{(\tau+1)k-1}\\\ \vdots\\\
u_{\tau k}\end{bmatrix}=-{\mathcal{R}}_{k}^{+}\bar{A}x_{\tau k}^{(2)}=0$ and
$\bar{u}_{\tau}={\mathcal{R}}_{k}\begin{bmatrix}u_{(\tau+1)k-1}\\\ \vdots\\\
u_{\tau k}\end{bmatrix}=0,$ and consequently, for $\tau\geq\bar{\tau}$ and
$\tau k\leq t<(\tau+1)k$ we also have $x_{t}^{(2)}=0$, that is,
$x_{t}^{(2)}=0\;\;\forall\,t\geq\bar{\tau}k$, which proves (P2) for the
subsystem $\bigl{(}A_{22},B_{2}\bigr{)}$ of our system ((2.1)).
Finally, to extend the result (P2) to the general case (where $A=\mathop{\rm
diag}(A_{11},A_{22})$), it suffices to note that, since for $t\geq\bar{\tau}k$
it also holds $u_{t}=0$, from the time $\bar{\tau}k$ onwards the subsystem
$(A_{11},B_{1})$ is in open loop. Since we imposed $A_{11}$ to be Schur
stable, the state $x^{(1)}_{t}$ of the latter converges to zero as
$t\rightarrow\infty$. This proves the theorem. ∎
## 4\. Numerical Example
An example follows, which shows that our nonlinear policy is readily
computable, and effective in bounding the state of a stable linear system in
the mean square. We executed $1000$ runs of simulation of the system
$x_{t+1}=Ax_{t}+Bu_{t}+w_{t}$, where
$A=\left[\begin{smallmatrix}\cos{\varphi_{1}}&-\sin{\varphi_{1}}&0&0\\\
\sin{\varphi_{1}}&\cos{\varphi_{1}}&0&0\\\ 0&0&0.5&0\\\
0&0&0&0.9\end{smallmatrix}\right]$, $B=\left[\begin{smallmatrix}1\\\ 0\\\ 0\\\
0\end{smallmatrix}\right]$, with $\varphi_{1}=0.8$,
$x_{0}=\left[\begin{array}[]{cccc}10&20&30&40\end{array}\right]^{\top}$, and
where $w_{t}$ is a Gaussian white noise with variance $I_{4}$. This system is
marginally stable and, as is easily seen, the $2$-dimensional subsystem with
eigenvalues on the unit circle is reachable in $2$ steps, whereas the
$2$-dimensional Schur-stable subsystem is not reachable at all. The control
authority $R$ was chosen approximately equal to $3.6$ according to a rough
estimate of $C_{1}=\mathsf{E}_{x}\bigl{[}\left\lVert
w_{t}\right\rVert\bigr{]}$. It should be noticed that smaller values of $R$
are also sufficient to stabilize the system.
Figure 4.1. Empirical average of $||x_{t}||^{2}$ over $1000$ runs.
Figure 4.1 shows the empirical average of $||x_{t}||^{2}$ over the $1000$
runs, respectively with disabled control, with the chosen control authority,
and with one tenth of the chosen control authority.
## 5\. A Conjecture
We conjecture that if the noise has bounded variance, then _given any
arbitrary positive uniform upper-bound_ on the norm of the control, there
exists a _stationary feedback policy_ such that the closed-loop system is
mean-square bounded. It appears to us that a proof of this conjecture will
require substantially new and nontrivial techniques.
## References
* [1]
* Bao et al. [2000] Bao, X., Lin, Z. and Sontag, E. D. [2000], ‘Finite gain stabilization of discrete-time linear systems subject to actuator saturation’, Automatica 36(2), 269–277.
* Bertsekas [2000] Bertsekas, D. P. [2000], Dynamic Programming and Optimal Control, Vol. 1, 2 edn, Athena Scientific.
* Chatterjee et al. [2009] Chatterjee, D., Hokayem, P. and Lygeros, J. [2009], ‘Stochastic receding horizon control with bounded control inputs—a vector-space approach’, IEEE Transactions on Automatic Control . Under review. http://arxiv.org/abs/0903.5444.
* Chatterjee and Pal [2008] Chatterjee, D. and Pal, S. [2008], ‘An excursion-theoretic approach to stability of stochastic hybrid systems’, http://arxiv.org/abs/0901.2269.
* Chitour and Lin [2003] Chitour, Y. and Lin, Z. [2003], ‘Finite gain $l_{p}$ stabilization of discrete-time systems subject to actuator saturation: The case of $p=1$’, IEEE Transactions on Automatic Control 48(12), 2196–2198.
* Foss and Konstantopoulos [2004] Foss, S. and Konstantopoulos, T. [2004], ‘An overview of some stochastic stability methods’, Journal of Operations Research Society of Japan 47(4), 275–303.
* Hernández-Lerma and Lasserre [1996] Hernández-Lerma, O. and Lasserre, J. B. [1996], Discrete-Time Markov Control Processes: Basic Optimality Criteria, Vol. 30 of Applications of Mathematics, Springer-Verlag, New York.
* Hokayem et al. [2009] Hokayem, P., Chatterjee, D. and Lygeros, J. [2009], On stochastic receding horizon control with bounded control inputs, in ‘IEEE Conference on Decision and Control and Chinese Control Conference’, Shanghai, China. http://arxiv.org/abs/0902.3944.
* Horn and Johnson [1990] Horn, R. A. and Johnson, C. R. [1990], Matrix Analysis, Cambridge University Press, Cambridge.
* Lin et al. [1996] Lin, Z., Saberi, A. and Stoorvogel, A. A. [1996], ‘Semi-global stabilization of linear discrete-time systems subject to input saturation via linear feedback—an ARE-based approach’, IEEE Transactions on Automatic Control 41(8), 1203–1207.
* Meyn and Tweedie [1993] Meyn, S. P. and Tweedie, R. L. [1993], Markov Chains and Stochastic Stability, Springer-Verlag, London.
* Pemantle and Rosenthal [1999] Pemantle, R. and Rosenthal, J. S. [1999], ‘Moment conditions for a sequence with negative drift to be uniformly bounded in $L^{r}$’, Stochastic Processes and their Applications 82(1), 143–155.
* Stoorvogel et al. [2007] Stoorvogel, A. A., Saberi, A. and Weiland, S. [2007], On external semi-global stochastic stabilization of linear systems with input saturation, in ‘American Control Conference’, pp. 5845–5850.
* Sussmann et al. [1994] Sussmann, H. J., Sontag, E. D. and Yang, Y. [1994], ‘A general result on the stabilization of linear systems using bounded controls’, IEEE Transactions on Automatic Control 39(12), 2411–2425.
* Yang et al. [1997] Yang, Y., Sontag, E. D. and Sussmann, H. J. [1997], ‘Global stabilization of linear discrete-time systems with bounded feedback’, Systems & Control Letters 30(5), 273–281.
* Yang et al. [1992] Yang, Y., Sussmann, H. J. and Sontag, E. D. [1992], Stabilization of linear systems with bounded controls, in M. Fliess, ed., ‘Proceedings of the Nonlinear Control Systems Design Symposium’, IFAC Publications, pp. 15–20.
|
arxiv-papers
| 2009-07-09T13:41:21 |
2024-09-04T02:49:03.779322
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Federico Ramponi, Debasish Chatterjee, Andreas Milias-Argeitis, Peter\n Hokayem, John Lygeros",
"submitter": "Debasish Chatterjee",
"url": "https://arxiv.org/abs/0907.1436"
}
|
0907.1483
|
Radiative corrections to the Higgs potential in the LH model
Antonio Dobado, Lourdes Tabares-Cheluci
Departamento de Física Teórica I,
Universidad Complutense de Madrid, E-28040 Madrid, Spain
Siannah Peñaranda
Departamento de Física Teórica, Universidad de Zaragoza, E-50009, Zaragoza,
Spain
Javier Rodriguez-Laguna
Departamento de Matemáticas, Universidad Carlos III de Madrid, E-28911,
Madrid, Spain
ABSTRACT
In this work we compute the radiative corrections to the Higgs mass and the
Higgs quartic couplings coming from the Higgs sector itself and the scalar
fields $\phi$ in the Littlest Higgs (LH) model. The restrictions that the new
contributions set on the parameter space of the models are also discussed.
Finally this work, together with our three previous papers, complete our
program addressed to compute the relevant contributions to the Higgs low-
energy effective potential in the LH model and the analysis of their
phenomenological consequences.
## 1 Introduction
The discovery of a Higgs boson and the elucidation of the mechanism
responsible for the electroweak symmetry breaking are some of the major goals
of present and future searches in particle physics. Because of the precise
data obtained for a long time to test the Standard Model (SM) of particle
interactions, and the recent measurements of the $W$ and the top masses at the
Fermilab Tevatron [1], the SM has been confirmed as the right model describing
the electroweak phenomena at the current experimental energy scale. However,
the origin of the electroweak symmetry breaking, for which the Higgs boson is
responsible in the SM, remains elusive. The quadratically divergent
contributions to the Higgs mass and the electroweak precision observables
imply different scales for physics beyond the SM, being the first one below
$1$ TeV and the second one above $10$ TeV. This is the so called little
hierarchy problem. As it is well known the mass of the Higgs boson receives
one-loop corrections that are quadratic in the loop momenta. The largest
contributions come from the top quark loop, with smaller corrections coming
from loops of the electroweak gauge bosons and of the Higgs boson itself.
Cancellations between the top sector and other sectors must occur in order to
have the Higgs mass lighter than $200$ GeV as expected from the electroweak
precision test of the SM, which requires a fine-tuning of one part in 100. As
this situation is quite unnatural various theories and models have been
designed to solve this problem.
An interesting attempt to deal with it is the so called Littlest Higgs model
(LH) [2], inspired in an old suggestion by Georgi and Pais [3], which tries to
solve the little hierarchy problem by adding new particles with masses O(TeV)
and symmetries which protect the Higgs mass from those dangerous quadratically
divergent contributions (see [4] and [5] for reviews). These particles include
the Goldstone bosons (GB) corresponding to a global spontaneous symmetry
breaking (SSB) from the $SU(5)$ to the $SO(5)$ group, a new third generation
vector quark called $T$ and the gauge bosons corresponding to an additional
gauge group which contains at least a $SU(2)_{R}$ and eventually a new
hypercharge $U(1)$. In this case, and contrary to the supersymmetric theories,
cancellation occurs between same-statistics particles. However, LH models
typically leave uncanceled logarithmic divergencies which requires additional
new contributions at some higher scale to preserve a small Higgs boson mass.
Many of such models with different theory space have been constructed [2, 6],
and electroweak precision constraints on various little Higgs models have been
investigated by performing global fits to the precision data [7, 8, 9, 10,
11]. The existence of the different new states in these models could give rise
to a very rich phenomenology, which could be probed at the CERN Large Hadron
Collider (LHC) [12, 13].
Nevertheless, it is clear that any viable model has to fulfill the basic
requirement of reproducing the SM model at low energies. In particular, from
the LH model it is possible in principle to compute the Higgs low-energy
effective potential and then, by comparing with the SM potential, to obtain
their phenomenological consequences including new restrictions on the
parameter space of the LH model itself. For example, one can obtain the one-
loop contribution to the parameters of the standard Higgs potential,
$V=-\mu^{2}HH^{{\dagger}}+\lambda(HH^{{\dagger}})^{2};$ (1.1)
where $\mu^{2}$ and $\lambda$ denote the well known Higgs mass and Higgs self-
couplings parameters. Then it is possible to set restrictions over the LH
parameters space by imposing the condition $\mu^{2}=\lambda v^{2}$, where $v$
is the SM vacuum expectation value ($H=(0,v)/\sqrt{2}$). The $\mu^{2}$ sign
and value are well known [2, 13], and effectively they are the right ones to
produce the electroweak symmetry breaking, giving a Higgs mass
$m_{H}^{2}=2\mu^{2}$. However, the full expression for the radiative
corrections to $\lambda$ has not been analyzed in detail so far. In principle
both $\mu^{2}$ and $\lambda$ receive contributions from fermion, gauge boson
and scalar loops, besides others that could come from the ultraviolet
completion of the LH model. We have previously computed the contributions to
the Higgs effective potential in the LH model coming from the fermion sector
and the gauge boson sector [14, 15]. On the other hand, several relations for
the threshold corrections to the $\lambda$ parameter in the presence of a $10$
TeV cut-off, depending on the UV-completion of the theory, have been reported
(see, for example [17]). Besides, we have computed the effective potential for
the doublet Higgs and the triplet $\phi$ [16], coming from the fermionic and
gauge boson one-loop contributions and from the higher order effective
operators needed for the ultraviolet completion of the model.
In [14] and [15] we computed and analyzed the fermion contributions to the low
energy Higgs effective potential together with the effects of virtual heavy
and electroweak gauge bosons present in the LH model. We have illustrated in
these works the kind of constraints on the possible values of the LH
parameters that can be set by requiring the complete LH effective potential to
reproduce exactly the SM potential. The radiative corrections to $\lambda$, at
the one-loop level, had not been previously computed. The computation of
$\lambda$ is important for several reasons: First, it must be positive, for
the low energy effective action to make sense. In addition, from the effective
potential (1.1), one gets the simple formula $m^{2}_{H}=2\lambda v^{2}$ or,
equivalently, $\mu^{2}=\lambda v^{2}$, where $v$ is set by the experiment (for
instance from the muon lifetime) to be $v\simeq 245$ GeV. In our
phenomenological discussion in [14, 15] we have shown that the one-loop
effective potential of the LH model cannot reproduce the SM potential with a
low enough Higgs mass, $m^{2}_{H}=2\lambda v^{2}=2\mu^{2}$, in agrement with
the present experimental constraints.
In order to solve this problem we computed in [16] the effective potential for
the doublet Higgs and the triplet $\phi$; coming from the fermionic and gauge
boson one-loop contributions and also from the higher order effective
operators, as defined in [12]. The relevant terms of this effective potential
can be read as,
$\displaystyle V_{eff}(H,\phi)$ $\displaystyle=$
$\displaystyle-\mu_{fg}^{2}HH^{{\dagger}}+\lambda_{fg}(HH^{{\dagger}})^{2}$
(1.2)
$\displaystyle+\lambda_{\phi^{2}}f^{2}\mbox{tr}(\phi\phi^{{\dagger}})+i\lambda_{H^{2}\phi}f(H\phi^{{\dagger}}H^{T}-H^{*}\phi
H^{{\dagger}})\,,$
where $\mu_{fg}^{2}>0$ and $\lambda_{fg}>0$.
With this potential we studied the regions of the LH parameter space giving
rise to the SM electroweak symmetry breaking. Although radiative corrections
from fermion and gauge boson loops were discussed in [14, 15], the radiative
contributions to $\lambda_{\phi^{2}}$ and $\lambda_{H^{2}\phi}$ have not been
computed so far. New constraints over the LH parameter space emerge once we
impose the new relation between coefficients of the effective Higgs potential
namely;
$v^{2}={\mu_{fg}^{2}}/{\lambda_{fg}-\lambda_{H^{2}\phi}^{2}/\lambda_{\phi^{2}}}$.
In particular, the lowest value found for the $\mu$ parameter was $390$ GeV
[16], which implied a Higgs boson mass of about $m_{H}\simeq 550$ GeV, still
not compatible with the present experimental constraints.
On the other hand it is well known that the radiative corrections coming from
the Higgs itself and the $\phi$ fields could also provide relevant
contributions to the effective potential. Thus the main goal of the present
work is to check wether these corrections could really reduce the Higgs mass
to solve the above mentioned problem, making the LH model compatible with the
present phenomenology.
This work is organized as follows: In Section 2 we briefly explain the LH
model. A summary on the SSB and the mass eigenstates is presented in Section
3. We set the notation in the two aforementioned sections. Section 4 is
devoted to the computation of the radiative corrections contributions to the
Higgs mass and quartic coupling coming from the scalar sector loops. In
Section 5 we analyze the constraints that our computation establishes on the
LH parameters and, finally, in Section 6 we present the conclusions. The
expressions of the coefficients of the effective potential (1.2) coming from
the radiative corrections and the effective operators are listed in the
Appendix.
## 2 The model
The LH model is based on the assumption that there is a physical system with a
global $SU(5)$ symmetry that is spontaneously broken to a $SO(5)$ symmetry at
a high scale $\Lambda$ through a vacuum expectation value (v.e.v) of order
$f$. Thus, 14 Goldstone bosons (GB) are obtained as a consequence of this
breaking. In this work we will consider two different versions of the LH
model. In the first one the $SU(5)$ subgroup $[SU(2)\times U(1)]^{2}$ is
gauged. We refer to this version as _Model I_. In the second one the gauge
group is $[SU(2)^{2}\times U(1)]$ (_Model II_) [14, 15]. In both cases some of
the GB acquire masses through radiative corrections coming from the gauge
bosons and the $t$, $b$ and $T$ fermions loops.
The starting Lagrangian of the LH model is given by [2, 12, 13]:
$\textit{L}=\textit{L}_{\Sigma}+\textit{L}_{YK}$ (2.3)
where $\textit{L}_{\Sigma}$ is the Non Linear Sigma Model (NLSM) lagrangian:
$\textit{L}_{\Sigma}=\frac{f^{2}}{8}\mbox{tr}[(D_{\mu}\Sigma)(D^{\mu}\Sigma)^{\dagger}]\,;$
(2.4)
and $\textit{L}_{YK}$ the Yukawa couplings for fermions and scalars:
$\textit{L}_{YK}=-\frac{\lambda_{1}}{2}f\overline{u}_{R}\epsilon_{mn}\epsilon_{ijk}\Sigma_{im}\Sigma_{jn}\chi_{Lk}-\lambda_{2}f\overline{U}_{R}U_{L}+\mbox{h.c.}\,.$
(2.5)
In the above Lagrangians $\Sigma$ is the GB matrix given by:
$\Sigma=e^{2i\Pi/f}\Sigma_{0}$ (2.6)
where $\Sigma_{0}$ can be chosen to be:
$\Sigma_{0}=\left(\begin{array}[]{ccc}0&0&\textbf{1}\\\ 0&1&0\\\
\textbf{1}&0&0\\\ \end{array}\right)\,,$ (2.7)
with 1 being the $2\times 2$ unit matrix, and the $\Pi$ matrix can be
parameterized as:
$\displaystyle\Pi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\frac{-i}{\sqrt{2}}H^{{\dagger}}&\phi^{{\dagger}}\\\
\frac{i}{\sqrt{2}}H&0&\frac{-i}{\sqrt{2}}H^{*}\\\
\phi&\frac{i}{\sqrt{2}}H^{T}&0\\\ \end{array}\right),$ (2.11)
where$H=(H^{0},H^{+})$ is the SM Higgs doublet and $\phi$ is the triplet given
by:
$\phi=\left(\begin{array}[]{cc}\phi^{0}&\frac{1}{\sqrt{2}}\phi^{+}\\\
\frac{1}{\sqrt{2}}\phi^{+}&\phi^{++}\end{array}\right)\,.$ (2.12)
The covariant derivative $D_{\mu}$ is defined as:
Model I $\displaystyle D_{\mu}\Sigma$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Sigma-i\sum_{k=1}^{2}g_{k}W^{a}_{k}(Q_{k}^{a}\Sigma+\Sigma
Q_{k}^{aT})-i\sum_{k=1}^{2}g^{\prime}_{k}B_{k}(Y_{k}\Sigma+\Sigma Y_{k}^{T})$
Model II $\displaystyle D_{\mu}\Sigma$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Sigma-i\sum_{k=1}^{2}g_{k}W^{a}_{k}(Q_{k}^{a}\Sigma+\Sigma
Q_{k}^{aT})-ig^{\prime}B(Y\Sigma+\Sigma Y^{T})\,,$ (2.13)
where $g$ and $g^{\prime}$ are the gauge couplings, $W_{k}^{a}$ $(a=1,2,3)$
and $B_{k}\,,B$ are the $SU(2)$ and $U(1)$ gauge fields respectively,
$Q_{1ij}^{a}=\sigma_{ij}^{a}/2$ for $i,j=1,2$ and zero otherwise,
$Q_{2ij}^{a}=\sigma_{i-3,j-3}^{a*}/2$ for $i,j=4,5$ and zero otherwise,
$Y_{1}=diag(-3,-3,2,2,2)/10$, $Y_{2}=diag(-2,-2,-2,3,3)/10$ and
$Y=diag(-1,-1,0,1,1)/2$. The Yukawa Lagrangian in (2.5) describes the
interactions between GB and fermions, more exactly, the third generations of
quarks plus the extra $T$ quark appearing in the LH model. The indices in
$\textit{L}_{YK}$ are defined such that $m,n=4,5$, $i,j=1,2,3$, and
$\displaystyle\overline{u}_{R}$ $\displaystyle=$ $\displaystyle
c\,\overline{t}_{R}+s\,\overline{T}_{R}\,,$ $\displaystyle\overline{U}_{R}$
$\displaystyle=$ $\displaystyle-s\,\overline{t}_{R}+c\,\overline{T}_{R},$
(2.14)
with:
$\displaystyle c$ $\displaystyle=$
$\displaystyle\cos\theta=\frac{\lambda_{2}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}},$
$\displaystyle s$ $\displaystyle=$
$\displaystyle\sin\theta=\frac{\lambda_{1}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}}\,,$
(2.15)
and
$\chi_{L}=\left(\begin{array}[]{c}u\\\ b\\\ U\\\
\end{array}\right)_{L}=\left(\begin{array}[]{c}t\\\ b\\\ T\\\
\end{array}\right)_{L}.$ (2.16)
In addition to the above terms it is needed to add to the LH Lagrangian the
Yang-Mills terms corresponding to the various gauge fields, and also the gauge
fixing and Faddeev-Popov terms. Some of the gauge fields get massive at the
tree level through the Higgs mechanism associated to the $SU(5)/SO(5)$
symmetry breaking. By using the Landau gauge, which is the most appropriate
for the kind of computations we are presenting here (see [15] for further
details), the quadratic part of the complete gauge boson Lagrangian can be
written as:
$\textit{L}_{\Omega}=\frac{1}{2}\Omega^{\mu}((\Box+M_{\Omega}^{2})g_{\mu\nu}-\partial_{\mu}\partial_{\nu}+2\tilde{I}\,g_{\mu\nu})\Omega^{\nu}\,,$
(2.17)
where $\Omega$ stands for any of the gauge bosons,
Model I $\displaystyle\Omega^{\mu}=({W^{\prime}}^{\mu a},W^{\mu
a},{B^{\prime}}^{\mu},B^{\mu}),$ Model II
$\displaystyle\Omega^{\mu}=({W^{\prime}}^{\mu a},W^{\mu a},B^{\mu})\,,$ (2.18)
being the mass matrix eigenstates,
Model I $\displaystyle M_{\Omega}=(M_{W^{\prime}}1_{3\times 3},0_{3\times
3},M_{B^{\prime}},0),$ Model II $\displaystyle
M_{\Omega}=(M_{W^{\prime}}1_{3\times 3},0_{3\times 3},0)\,,$ (2.19)
with $M_{W^{\prime}}=f\sqrt{g_{1}^{2}+g_{2}^{2}}/2$ and
$M_{B^{\prime}}=f\sqrt{g_{1}^{{}^{\prime}2}+g_{2}^{{}^{\prime}2}}/\sqrt{20}$.
The gauge boson mass eigenstates are defined such as:
$\displaystyle W^{a}$ $\displaystyle=$ $\displaystyle
c_{\psi}W_{1}^{a}+s_{\psi}W_{2}^{a},$ $\displaystyle W^{{}^{\prime}a}$
$\displaystyle=$ $\displaystyle s_{\psi}W_{1}^{a}-c_{\psi}W_{2}^{a},$ (2.20)
where
$\displaystyle s_{\psi}$ $\displaystyle=$
$\displaystyle\sin\psi=\frac{g_{1}}{\sqrt{g_{1}^{2}+g_{2}^{2}}},$
$\displaystyle c_{\psi}$ $\displaystyle=$
$\displaystyle\cos\psi=\frac{g_{2}}{\sqrt{g_{1}^{2}+g_{2}^{2}}},$ (2.21)
and
$\displaystyle B$ $\displaystyle=$ $\displaystyle
c^{\prime}_{\psi}B_{1}+s^{\prime}_{\psi}B_{2},$ $\displaystyle B^{\prime}$
$\displaystyle=$ $\displaystyle
s^{\prime}_{\psi}B_{1}-c^{\prime}_{\psi}B_{2},$ (2.22)
with
$\displaystyle
s^{\prime}_{\psi}=\sin\psi^{\prime}=\frac{g^{\prime}_{1}}{\sqrt{{g^{\prime}}_{1}^{\,2}+{g^{\prime}}_{2}^{\,2}}},$
$\displaystyle
c^{\prime}_{\psi}=\cos\psi^{\prime}=\frac{{g^{\prime}}_{2}}{\sqrt{{g^{\prime}}_{1}^{\,2}+{g^{\prime}}_{2}^{\,2}}}\,.$
(2.23)
$\tilde{I}$ is the interaction matrix between the gauge bosons and the $H$ and
$\phi$ scalars which can be found in our previous works [15, 16].
By adding the appropriate kinetic terms, the complete Lagrangian for the
quarks becomes:
$\displaystyle\textit{L}_{\chi}=\overline{\chi}_{R}(i{\partial\mkern-9.0mu/}-M+\hat{I})\chi_{L}+\mbox{h.c.}\,,$
(2.24)
where
$\chi_{R}=\left(\begin{array}[]{c}t\\\ b\\\ T\\\ \end{array}\right)_{R}\,,$
$M=$diag$(0,0,m_{T})$ with $m_{T}=f\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}$ and
$\hat{I}$ is the scalar-quark interaction matrix. The elements of this matrix
can be found in [14, 16]. For more details about the model, including Feynman
rules and also some phenomenological results see for example [12].
## 3 Effective operators
It is well known that the effective Higgs potential receive also contributions
from additional operators coming from the ultraviolet completion of the LH
model. Obviously these operators must be consistent with the symmetries of the
LH model [2, 12, 18]. At the lowest order they can be parameterized by two
unknown coefficients $a$ and $a^{\prime}$ $\sim O(1)$. The form of these
effective operators is, for the fermion sector [12]:
$\textit{O}_{f}=-a^{\prime}\frac{1}{4}\lambda_{1}^{2}f^{4}\epsilon^{wx}\epsilon_{yz}\epsilon^{ijk}\epsilon_{kmn}\Sigma_{iw}\Sigma_{jx}\Sigma^{*my}\Sigma^{*nz}\,,$
(3.25)
where $i,j,k,m,n$ run over 1,2,3 and $w,x,y,z$ run over 4,5 and for the gauge
sector we have for _Model I_ :
$\displaystyle\textit{O}_{gb}=\frac{1}{2}af^{4}\left\\{g_{j}^{2}\sum_{a=1}^{3}\mbox{Tr}\left[(Q_{j}^{a}\Sigma)(Q_{j}^{a}\Sigma)^{*}\right]+g_{j}^{{}^{\prime}2}\mbox{Tr}\left[(Y_{j}\Sigma)(Y_{j}\Sigma)^{*}\right]\right\\}\,,$
(3.26)
with $j=1,2$ and $Q_{j}^{a}$ and $Y_{j}$ being the generators of the
$SU(2)_{j}$ and $U(1)_{j}$ groups, respectively. In the case of _Model II_ :
$\displaystyle\textit{O}_{gb}=\frac{1}{2}cf^{4}\left\\{g_{j}^{2}\sum_{a=1}^{3}\mbox{Tr}\left[(Q_{j}^{a}\Sigma)(Q_{j}^{a}\Sigma)^{*}\right]+g^{{}^{\prime}2}\mbox{Tr}\left[(Y\Sigma)(Y\Sigma)^{*}\right]\right\\}\,,$
(3.27)
where $j=1,2$ and $Y$ is the generator of the unique $U(1)$ group.
By expanding the GB field matrix $\Sigma$ in these effective operators, we
obtain their different contributions to the coefficients of the effective
potential (1.2). The results are presented in the Appendix.
The complete result for the coefficients of the Higgs potential is given by
the sum of the contributions coming from the effective operators, as given
above, and the radiative contributions coming from all sectors of the model,
as will be discussed in the following.
## 4 SSB and mass eigenstates
In the LH model the electroweak symmetry breaking is triggered, in principle,
by the Higgs potential generated by one-loop radiative corrections, including
both, fermion and gauge boson loops, and the effective operators introduced in
the previous section. Obviously, this potential is invariant under the
electroweak gauge group $SU(2)\times U(1)$ and also should have the correct
form to break this symmetry spontaneously to $U(1)_{em}$. The relevant terms
for this work are given in (1.2). Quartic terms involving $\phi^{4}$ and
$H^{2}\phi^{2}$ are not included since they give subleading contributions to
the Higgs mass. These parameters were computed in our previous works [14, 15,
16] and are given in the Appendix for completeness.
The scalar potential, as given in (1.2), reaches its minimum at:
$\langle HH^{{\dagger}}\rangle={v^{2}}/{2}$ and
$\,\,\langle\phi\phi^{{\dagger}}\rangle={v^{\prime}}^{2}$ with:
$v^{2}=\frac{\mu_{fg}^{2}}{\lambda_{fg}-\lambda_{H^{2}\phi}^{2}/\lambda_{\phi^{2}}},\hskip
56.9055pt{v^{\prime}}^{2}=\frac{\lambda_{H^{2}\phi}}{\sqrt{2}\lambda_{\phi^{2}}}\frac{v^{2}}{f}.$
(4.28)
Note that both, the doublet and triplet scalars, get a v.e.v., $v$ and
$v^{\prime}$ respectively. A standard choice for the components of these
fields at the vacuum is:
$\displaystyle H^{+}=0,\hskip 28.45274ptH_{0}=\frac{v}{\sqrt{2}},\hskip
28.45274pt\phi_{0}=-v^{\prime},\hskip 28.45274pt\phi^{+}=\phi^{++}=0.$ (4.29)
Then $H$ and $\phi$ can be parameterized as:
$\displaystyle H=(w^{+},\frac{1}{\sqrt{2}}(v+h+iw_{0}))\hskip
8.5359pt\mbox{and}\hskip
8.5359pt\phi=\left(\begin{array}[]{cc}-v^{\prime}+\frac{1}{\sqrt{2}}(\xi+i\rho)&\frac{1}{\sqrt{2}}\phi^{+}\\\
\frac{1}{\sqrt{2}}\phi^{+}&\phi^{++}\end{array}\right).$ (4.32)
Obviously the new fields describe fluctuations around the vacuum and the
potential written in terms of them can be split in four sectors, namely, the
scalar, the pseudoscalar, the charged and the doubly charged. For the first
three sectors we find that the new fields are not mass eigenstates. By
diagonalizing the corresponding mass matrices we obtain the mass eigenstates
in each case. I.e., for the scalar sector:
$\displaystyle h$ $\displaystyle=$ $\displaystyle
c_{0}\mathcal{H}+s_{0}\Phi_{0},\hskip 56.9055ptm^{2}_{\mathcal{H}}\equiv
m^{2}_{fg}=2\,\mu_{fg}^{2},$ $\displaystyle\xi$ $\displaystyle=$
$\displaystyle c_{0}\Phi_{0}-s_{0}\mathcal{H},\hskip
56.9055ptm^{2}_{\Phi_{0}}=M_{\phi}^{2}+2\,m^{2},$ (4.33)
the pseudoscalar sector:
$\displaystyle w_{0}$ $\displaystyle=$ $\displaystyle
c_{P}G^{0}+s_{P}\Phi^{P},\hskip 56.9055ptm^{2}_{G^{0}}=0,$ $\displaystyle\rho$
$\displaystyle=$ $\displaystyle c_{P}\Phi^{P}-s_{P}G^{0},\hskip
56.9055ptm^{2}_{\Phi^{P}}=M_{\phi}^{2}+2m^{2},$ (4.34)
and the charged sector:
$\displaystyle w^{+}$ $\displaystyle=$ $\displaystyle
c_{+}G^{+}+s_{+}\Phi^{+},\hskip 42.67912ptm^{2}_{G^{+}}=0,$
$\displaystyle\phi^{+}$ $\displaystyle=$ $\displaystyle
c_{+}\Phi^{+}+s_{+}G^{+},\hskip
42.67912ptm^{2}_{\Phi^{+}}=M_{\phi}^{2}+m^{2},$ (4.35)
with $M_{\phi}^{2}=\lambda_{\phi^{2}}f^{2}$,
$m^{2}=v^{2}\lambda_{H^{2}\phi}^{2}/\lambda_{\phi^{2}}$. The doubly charged
sector remains unchanged with a mass $M_{\phi}$.
Where the notation introduced for the mass eigenstates is the following:
$\mathcal{H}$ and $\Phi_{0}$ are neutral scalars, $\Phi^{P}$ is a neutral
pseudoscalar, $\Phi^{+}$ and $\Phi^{++}$ are the charged and doubly charged
scalars, and $G^{+}$ and $G^{0}$ are the would-be Goldstone bosons
corresponding to the SM $W$ and $Z$.
In terms of the mass eigenstates the leading order in the
$\mathcal{O}(v^{2}/f^{2})$ expansion of the potential is given by:
$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{fg}^{2}\mathcal{H}^{2}+\frac{1}{2}m_{\Phi_{0}}^{2}\Phi_{0}^{2}+\frac{1}{2}m_{\Phi^{p}}^{2}{\Phi^{P}}^{2}$
$\displaystyle+$ $\displaystyle
m_{\Phi^{+}}^{2}\Phi^{+}\Phi^{-}+v\lambda_{fg}\mathcal{H}^{3}+v\lambda_{fg}{G^{0}}^{2}\mathcal{H}+2v\lambda_{fg}{G^{+}}{G^{-}}\mathcal{H}$
$\displaystyle+$
$\displaystyle\frac{\lambda_{fg}}{4}\mathcal{H}^{4}+\frac{\lambda_{fg}}{2}\mathcal{H}^{2}{G^{0}}^{2}+\lambda_{fg}\mathcal{H}^{2}{G^{+}}{G^{-}}$
$\displaystyle-$
$\displaystyle\frac{\lambda_{H^{2}\phi}}{\sqrt{2}}f\mathcal{H}^{2}\Phi_{0}-\sqrt{2}\lambda_{H^{2}\phi}f\mathcal{H}{G^{0}}\Phi^{P}-\lambda_{H^{2}\phi}f(\mathcal{H}{G^{-}}\Phi^{+}+\mathcal{H}{G^{+}}\Phi^{-})+...$
## 5 Goldstone boson sector contributions
The objective of this section is the computation of the radiative
contributions to the Higgs mass and the Higgs quartic coupling coming from the
GB sector. The relevant Lagrangian is given by:
$\displaystyle\textit{L}_{GB}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\Pi)(\partial^{\mu}\Pi)+\frac{1}{f^{2}}\left((\partial_{\mu}\Pi)(\partial^{\mu}\Pi)\Pi\Pi+\Pi(\partial_{\mu}\Pi)\Pi(\partial^{\mu}\Pi)\right)-V_{eff}.$
In order to calculate the radiative contributions we write this Lagrangian in
terms of the mass eigenstates and we split the Higgs field as
$\mathcal{H}=\mathcal{\overline{H}}+\mathcal{\tilde{H}}$ where
$\mathcal{\overline{H}}$ is the vacuum field and $\mathcal{\tilde{H}}$
describes the field fluctuations around this point. Then the first two terms
of the Lagrangian above become:
$\displaystyle\textit{L}_{Kin}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1+2\frac{\mathcal{\overline{H}}^{2}}{f^{2}}\right)(\partial_{\mu}\mathcal{\tilde{H}})(\partial^{\mu}\mathcal{\tilde{H}})+\frac{1}{2}\left(1+\frac{\mathcal{\overline{H}}^{2}}{2f^{2}}\right)(\partial_{\mu}\Phi_{0})(\partial^{\mu}\Phi_{0})$
(5.38) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(1+\frac{\mathcal{\overline{H}}^{2}}{2f^{2}}\right)(\partial_{\mu}{G^{0}})(\partial^{\mu}{G^{0}})+\frac{1}{2}\left(1+\frac{\mathcal{\overline{H}}^{2}}{2f^{2}}\right)(\partial_{\mu}\Phi^{P})(\partial^{\mu}\Phi^{P})$
$\displaystyle+$
$\displaystyle\left(1+\frac{\mathcal{\overline{H}}^{2}}{4f^{2}}\right)(\partial_{\mu}\Phi^{+})(\partial^{\mu}\Phi^{-})+\left(1+\frac{\mathcal{\overline{H}}^{2}}{2f^{2}}\right)(\partial_{\mu}{G^{+}})(\partial^{\mu}{G^{-}})$
$\displaystyle+$
$\displaystyle(\partial_{\mu}\Phi^{++})(\partial^{\mu}\Phi^{--}).$
Obviously, all the kinetic terms in this formula, but the last one, are not
properly normalized. Therefore we write the fields in terms of a new set of
properly normalized fields up to order $1/f^{2}$ as:
$\displaystyle\Upsilon$ $\displaystyle=$
$\displaystyle\left(1-\frac{\mathcal{\overline{H}}^{2}}{4f^{2}}\right)\Upsilon^{\prime}\hskip
28.45274pt\mbox{with}\hskip
28.45274pt\Upsilon^{(^{\prime})}={G^{0(^{\prime})}},G^{\pm(^{\prime})},\Phi_{0}^{(^{\prime})},\Phi^{P(^{\prime})},$
(5.39) $\displaystyle\mathcal{\tilde{H}}$ $\displaystyle=$
$\displaystyle\left(1-\frac{\mathcal{\overline{H}}^{2}}{f^{2}}\right)\mathcal{H}^{\prime},$
(5.40) $\displaystyle\Phi^{\pm}$ $\displaystyle=$
$\displaystyle\left(1-\frac{\mathcal{\overline{H}}^{2}}{8f^{2}}\right)\Phi^{\pm^{\prime}},$
(5.41)
so that the Lagrangian is just:
$\displaystyle\textit{L}_{Kin}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\mathcal{H}^{\prime}\partial^{\mu}\mathcal{H}^{\prime}+\frac{1}{2}\partial_{\mu}\Phi_{0}^{\prime}\partial^{\mu}\Phi_{0}^{\prime}+\frac{1}{2}\partial_{\mu}{G^{0}}^{\prime}\partial^{\mu}{G^{0}}^{\prime}+\frac{1}{2}\partial_{\mu}{\Phi^{P}}^{\prime}\partial^{\mu}{\Phi^{P}}^{\prime}$
(5.42) $\displaystyle+$
$\displaystyle\partial_{\mu}G^{+^{\prime}}\partial^{\mu}G^{-^{\prime}}+\partial_{\mu}\Phi^{+^{\prime}}\partial^{\mu}\Phi^{-^{\prime}}+\partial_{\mu}\Phi^{++}\partial^{\mu}\Phi^{--},$
Then, the effective potential $V_{eff}$ is given by:
$\displaystyle
V_{eff}=\frac{1}{2}m_{fg}^{2}\mathcal{\overline{H}}^{2}+\frac{\lambda_{fg}}{4}\mathcal{\overline{H}}^{4}+V_{eff}^{ss}+V_{eff}^{ps}+V_{eff}^{cs}+...$
(5.43)
where
$\displaystyle V_{eff}^{ss}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{\Phi_{0}}^{2}\Phi_{0}^{{}^{\prime}2}+\frac{1}{2}m_{fg}^{2}\mathcal{H}^{{}^{\prime}2}+\frac{3}{2}\lambda_{fg}\overline{\mathcal{H}}^{2}\mathcal{H}^{{}^{\prime}2}-\frac{\lambda_{\phi^{2}}}{4}\overline{\mathcal{H}}^{2}\Phi_{0}^{{}^{\prime}2}$
(5.44) $\displaystyle-$
$\displaystyle\sqrt{2}\lambda_{H^{2}\phi}f\overline{\mathcal{H}}\mathcal{H}^{\prime}\Phi_{0}^{\prime}-\frac{\lambda_{H^{2}\phi}}{\sqrt{2}}f\overline{\mathcal{H}}^{2^{\prime}}\Phi_{0}^{\prime},$
$\displaystyle V_{eff}^{ps}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{\Phi^{p}}^{2}{\Phi^{P}}^{{}^{\prime}2}+\frac{\lambda_{fg}}{2}\overline{\mathcal{H}}^{2}{G^{0}}^{{}^{\prime}2}-\frac{\lambda_{\phi^{2}}}{4}\overline{\mathcal{H}}^{2}{\Phi^{P}}^{{}^{\prime}2}$
(5.45) $\displaystyle-$
$\displaystyle\sqrt{2}\lambda_{H^{2}\phi}f\overline{\mathcal{H}}{G^{0}}^{\prime}{\Phi^{P}}^{\prime},$
$\displaystyle V_{eff}^{cs}$ $\displaystyle=$ $\displaystyle
m_{\Phi^{+}}^{2}\Phi^{{}^{\prime}+}\Phi^{{}^{\prime}-}+\lambda_{fg}\overline{\mathcal{H}}^{2}G^{{}^{\prime}+}G^{{}^{\prime}-}-\frac{\lambda_{\phi^{2}}}{4}\overline{\mathcal{H}}^{2}\Phi^{{}^{\prime}+}\Phi^{{}^{\prime}-}$
(5.46) $\displaystyle-$
$\displaystyle\lambda_{H^{2}\phi}f\overline{\mathcal{H}}(G^{{}^{\prime}-}\Phi^{{}^{\prime}+}+G^{{}^{\prime}+}\Phi^{{}^{\prime}-}),$
Observe that the third terms in (5.44), (5.45) and (5.46) describe the new
interactions which come from the new normalization of the fields and the fact
that the triplet boson mass is $\mathcal{O}(f^{2})$. These interactions play a
decisive role to cancel the quadratic divergences that come from the GB loops.
Finally, we can see that the split into different scalar sectors is maintained
after diagonalization and normalization. This fact is important in order to
simplify the computation of the radiative contributions coming from the GB.
Thus we can deal with each scalar sector in an independent way being the
computations in all cases similar. We illustrate this by computing the
($\mathcal{H^{\prime}},\Phi_{0}^{\prime}$) contribution and then we apply the
same method to the other scalars.
### 5.1 Scalar sector contribution
The Lagrangian for the scalar sector
($\mathcal{H^{\prime}},\Phi_{0}^{\prime}$) is given by:
$\displaystyle\mathcal{L}^{ss}(\overline{\mathcal{H}},\mathcal{H}^{\prime},\Phi_{0}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\mathcal{H}^{\prime}\partial^{\mu}\mathcal{H}^{\prime}+\frac{1}{2}\partial_{\mu}\Phi_{0}^{\prime}\partial^{\mu}\Phi_{0}^{\prime}-V_{eff}^{ss}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\mathcal{H}^{\prime}\partial^{\mu}\mathcal{H}^{\prime}+\frac{1}{2}\partial_{\mu}\Phi_{0}^{\prime}\partial^{\mu}\Phi_{0}^{\prime}-\frac{1}{2}m_{fg}^{2}\mathcal{H}^{{}^{\prime}2}-\frac{1}{2}m_{\Phi_{0}}^{2}\Phi_{0}^{{}^{\prime}2}$
$\displaystyle-$
$\displaystyle\frac{3}{2}\lambda_{fg}\overline{\mathcal{H}}^{2}\mathcal{H}^{{}^{\prime}2}+\frac{\lambda_{\phi^{2}}}{4}\overline{\mathcal{H}}^{2}\Phi_{0}^{{}^{\prime}2}+\sqrt{2}f\lambda_{H^{2}\phi}\overline{\mathcal{H}}\mathcal{H}^{\prime}\Phi_{0}^{\prime}+\frac{\lambda_{H^{2}\phi}}{\sqrt{2}}\overline{\mathcal{H}}^{2}\Phi_{0}^{\prime}.$
The effective action for the $\overline{\mathcal{H}}$ is:
$e^{iS_{eff}[\overline{\mathcal{H}}]}=\int[d\mathcal{H}^{\prime}][d\Phi_{0}^{\prime}]e^{i\int
dx\mathcal{L}^{ss}},$ (5.48)
From the (5.1) we observe that the integration can be computed in two steps:
First we concentrate on the $\Phi_{0}^{\prime}$ field and then we integrate
the $\mathcal{H^{\prime}}$ field. After integrating $\Phi_{0}^{\prime}$ we get
the $\mathcal{H}$ effective action:
$\displaystyle S_{eff}^{ss}[\overline{\mathcal{H}},\mathcal{H^{\prime}}]$
$\displaystyle=$
$\displaystyle-\frac{i}{2}\mbox{Tr}\log\left[1+G_{\Phi_{0}}\frac{\lambda_{\phi^{2}}}{2}\overline{\mathcal{H}}^{2}\right]$
(5.49) $\displaystyle-$ $\displaystyle f^{2}\lambda_{H^{2}\phi}^{2}\int
dxdy\overline{\mathcal{H}}^{2}\mathcal{H}^{\prime}_{x}G_{\Phi_{0}xy}\mathcal{H}^{\prime}_{y}-\frac{\lambda_{H^{2}\phi}^{2}}{4}f^{2}\int
dxdyG_{\Phi_{0}xy}\overline{\mathcal{H}}^{4}\delta_{yx}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\mbox{Tr}\left(G_{\Phi_{0}}\frac{\lambda_{\phi^{2}}}{2}\overline{\mathcal{H}}^{2}\right)^{k}+\tilde{I}_{2}+\tilde{I}_{4}\,,$
where the $\Phi_{0}^{\prime}$ propagator is given by:
$G_{\Phi_{0}}(x,y)=\int
d\tilde{k}e^{ik(x-y)}\frac{1}{k^{2}-m_{\Phi_{0}}^{2}},$ (5.50)
here $\tilde{k}\equiv d^{4}k/(2\pi)^{4}$, and
$\displaystyle\tilde{I}_{2}$ $\displaystyle=$
$\displaystyle-f^{2}\lambda_{H^{2}\phi}^{2}\int
dxdy\overline{\mathcal{H}}^{2}\mathcal{H}^{\prime}_{x}G_{\Phi_{0}xy}\mathcal{H}^{\prime}_{y}\,,$
(5.51) $\displaystyle\tilde{I}_{4}$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{H^{2}\phi}^{2}}{4}f^{2}\int
dxdy\overline{\mathcal{H}}^{4}\delta_{xy}G_{\Phi_{0}xy}\,.$ (5.52)
Observe that we have obtained three terms. The first and the third ones are
$\mathcal{H}^{\prime}$ independent and they will give the $\Phi_{0}^{\prime}$
radiative contributions to the Higgs mass and the quartic coupling.
Now integrating out $\mathcal{H}^{\prime}$ we find its contribution to the
$\overline{\mathcal{H}}$ effective action:
$\displaystyle S^{ss}[\overline{\mathcal{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\mbox{Tr}\log\left[1+G_{\mathcal{H}^{\prime}}(-3\lambda_{fg}\overline{\mathcal{H}}^{2}+2\tilde{I}_{2})\right]$
(5.53) $\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\mbox{Tr}\left(G_{\mathcal{H}}(3\lambda_{fg}\overline{\mathcal{H}}^{2}-2\tilde{I}_{2})\right)+...\,,$
where $G_{\mathcal{H}^{\prime}}$ is the $\mathcal{H}^{\prime}$ propagator,
$G_{\mathcal{H}^{\prime}}(x,y)=\int
d\tilde{k}e^{ik(x-y)}\frac{1}{k^{2}-m_{fg}^{2}}\,.$ (5.54)
Finally, by taking into account (5.49) and (5.53), we obtain the
$\overline{{\mathcal{H}}}$ effective action which reads:
$\displaystyle S^{ss}[\overline{\mathcal{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\mbox{Tr}\left(G_{\Phi_{0}}\frac{\lambda_{\phi^{2}}}{2}\overline{\mathcal{H}}^{2}\right)^{k}$
$\displaystyle+\frac{i}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\mbox{Tr}(G_{\mathcal{H}^{\prime}}(3\lambda_{fg}\overline{\mathcal{H}}^{2}+\tilde{I}_{2}))^{k}+\tilde{I}_{4}.$
In order to obtain the scalar contribution to the Higgs mass we only need to
consider the $k=1$ term in the expansion (5.1). The generic loop diagrams are
shown in Fig. 1. Then, for $k=1$,
$\displaystyle S^{(1)ss}[\mathcal{\overline{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\lambda_{\phi^{2}}\int
dxdy(G_{\Phi_{0}xy}\overline{\mathcal{H}}^{2}\delta_{yx})+\frac{i}{2}\int
dxdyG_{\mathcal{H}^{\prime}xy}(3\lambda_{fg}\overline{\mathcal{H}}^{2}\delta_{yx}+\tilde{I}_{2}\delta_{yx})$
(5.56) $\displaystyle=$ $\displaystyle-\frac{i}{4}\lambda_{\phi^{2}}\int
dx\overline{\mathcal{H}}^{2}I_{0}(m_{\Phi_{0}}^{2})+\frac{3}{2}i\lambda_{fg}\int
dx\overline{\mathcal{H}}^{2}I_{0}(m_{fg}^{2})$
$\displaystyle+i\lambda_{H^{2}\phi}^{2}f^{2}\int
dx\overline{\mathcal{H}}^{2}I_{3}(m_{\Phi_{0}}^{2},m_{fg}^{2})\,,$
with
$\displaystyle I_{0}(M^{2})$ $\displaystyle\equiv$ $\displaystyle\int
d\tilde{k}\frac{i}{(k^{2}-M^{2})}=\frac{1}{(4\pi)^{2}}\left[\Lambda^{2}-M^{2}\log\left(1+\frac{\Lambda^{2}}{M^{2}}\right)\right],$
(5.57) $\displaystyle I_{3}(M_{a}^{2},M_{b}^{2})$ $\displaystyle\equiv$
$\displaystyle\int d\tilde{p}\frac{i}{(p^{2}-M_{a}^{2})(p^{2}-M_{b}^{2})}$
$\displaystyle=$
$\displaystyle-\frac{1}{(4\pi)^{2}}\frac{1}{M_{a}^{2}-M_{b}^{2}}\left[M_{a}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{a}^{2}}\right)-M_{b}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{b}^{2}}\right)\right].$
$\Gamma_{L},~{}\Gamma_{H}$$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$$\Gamma_{L}$$\Gamma_{H}$$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$(a)$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$(b)
Figure 1: (a) Scalar sector loops contributing to the Higgs mass.
$\Gamma_{L}=\mathcal{H}^{\prime},{G^{0}}^{\prime}$ or $G^{\pm^{\prime}}$ and
$\Gamma_{H}=\Phi_{0}^{\prime},\Phi^{P}$ or $\Phi^{\pm^{\prime}}$. (b)
Contribution to the Higgs quartic coupling from the $\Phi_{0}^{\prime}$
propagator.
For the quartic coupling Higgs correction coming from $\tilde{I}_{4}$ we have:
$\tilde{I}_{4}=\frac{\lambda_{H^{2}\phi}^{2}}{4\lambda_{\phi^{2}}}\int
dx\overline{\mathcal{H}}^{4}+...$ (5.59)
where we have expanded the $\Phi_{0}^{\prime}$ propagator in powers of
$k^{2}/m_{\Phi_{0}^{\prime}}^{2}$ and kept just the first term.
### 5.2 Pseudoscalar sector and charged sector contributions
The computation of the contributions from the pseudoscalar and charged sectors
is similar to the previous ones with only one difference, i.e.: these sectors
do not give a contribution to the Higgs quartic coupling. They just contribute
to the Higgs mass. Then the results for the pseudoscalar sector are:
$\displaystyle S^{(1)ps}[\mathcal{\overline{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{4}\lambda_{\phi^{2}}\int
dx\overline{\mathcal{H}}^{2}I_{0}(m_{\Phi^{p}}^{2})+\frac{1}{2}i\lambda_{fg}\int
dx\overline{\mathcal{H}}^{2}I_{0}(0)$ (5.60) $\displaystyle+$ $\displaystyle
i\lambda_{H^{2}\phi}^{2}f^{2}\int
dx\overline{\mathcal{H}}^{2}I_{3}(m_{\Phi^{p}}^{2},0).$
and for the charged sector:
$\displaystyle S^{(1)cs}[\mathcal{\overline{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{4}\lambda_{\phi^{2}}\int
dx\overline{\mathcal{H}}^{2}I_{0}(m_{\Phi^{+}}^{2})+i\lambda_{fg}\int
dx\overline{\mathcal{H}}^{2}I_{0}(0)$ (5.61) $\displaystyle+$ $\displaystyle
i\lambda_{H^{2}\phi}^{2}f^{2}\int
dx\overline{\mathcal{H}}^{2}I_{3}(m_{\Phi^{+}}^{2},0).$
Notice the there is no contribution coming from the doubly charged scalar
sector.
### 5.3 Analytical results
Now by adding (5.56), (5.60), (5.61) we obtain the total radiative corrections
to the Higgs mass from the GB sector up to order $\mathcal{O}(v^{2}/f^{2})$
which reads:
$\displaystyle\Delta m_{GB}^{2}$ $\displaystyle=$
$\displaystyle\frac{3}{(4\pi)^{2}}\left\\{\left(-\frac{\lambda_{\phi^{2}}}{4}+\lambda_{fg}\right)\Lambda^{2}+\left(\frac{\lambda_{\phi^{2}}}{4}+\frac{\lambda_{H^{2}\phi}^{2}}{\lambda_{\phi^{2}}}\right)M_{\phi}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{\phi}^{2}}\right)\right.$
(5.62)
$\displaystyle\left.-\frac{1}{2}\lambda_{fg}m_{fg}^{2}\log\left(1+\frac{\Lambda^{2}}{m_{fg}^{2}}\right)\right\\}\,,$
where, in order to simplify the computations, we have considered the heavy
scalar fields as degenerate since $m^{2}/M_{\phi}^{2}$ is of the order of
$\mathcal{O}(v^{2}/f^{2})$ (see eq. (4)).
The coefficients of the Higgs potential $\lambda_{fg},\lambda_{\phi^{2}}$ and
$\lambda_{H^{2}\phi}^{2}$ appearing in eq. (5.62) receive contributions from
both the radiative corrections and the effective operators (see Appendix).
Since the contributions to $\lambda_{fg}$ and $\lambda_{\phi^{2}}$ contain
terms of the order of $\Lambda^{2}$, divergencies $\mathcal{O}(\Lambda^{4})$
and $\mathcal{O}(\Lambda^{2})$ emerge from the first term in (5.62). However,
these divergencies cancel due to the relationship between $\lambda_{\phi^{2}}$
and $\lambda_{fg}$, namely:
$\displaystyle\lambda_{fg}^{\Lambda^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\lambda_{\phi^{2}}^{\Lambda^{2}},$
$\displaystyle\lambda_{fg}^{EO}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\lambda_{\phi^{2}}^{EO},$ (5.63)
where the index $\Lambda^{2}$ refers to the quadratically divergent terms and
$EO$ represents the part of these coefficients coming from the effective
operators. This fact occurs in the fermionic and gauge boson sectors, where
the quadratic divergences coming from light and heavy modes of the same
statistics cancel [2]. Then the corrections summarized in $\Delta m_{GB}^{2}$
(eq. 5.62) are at most of the order
$\mathcal{O}(\Lambda^{2}\log(\Lambda^{2}/M^{2}))$. It is important to stress
that the above cancellations occur exactly only in _Model I_ (as you can
easily check from the results given in the Appendix). However, in _Model II_
(where only the $SU(2)\times SU(2)\times U(1)$ is gauged), there are
$\mathcal{O}(\Lambda^{2})$ terms coming from the U(1) sector which do not
cancel. However, such terms appear always with a squared gauge coupling
$g^{\prime}$ factor which is very small ($g^{\prime 2}/g^{2}\sim 0.3$ in the
SM) and then their contribution is not expected to be too large.
Finally, from (5.59), the radiative correction to the quartic coupling is:
$\displaystyle\tilde{I}_{4}=\frac{1}{4}\Delta\lambda_{GB}\int
dx\overline{\mathcal{H}}^{4}\,,$
being
$\displaystyle\Delta\lambda_{GB}=\frac{\lambda_{H^{2}\phi}^{2}}{\lambda_{\phi^{2}}}.$
(5.64)
In summary, taking into account (5.59) and (5.62), the Higgs boson potential
can be written as:
$V=\frac{1}{2}m_{\overline{\mathcal{H}}}^{2}{\overline{\mathcal{H}}}^{2}+\frac{1}{4}\lambda_{\overline{\mathcal{H}}}{\overline{\mathcal{H}}}^{4},$
(5.65)
where the Higgs mass is given by,
$m_{\overline{\mathcal{H}}}^{2}=2(\mu_{fg}^{2}-\Delta m_{GB}^{2}),$ (5.66)
and the quartic Higgs couplings is,
$\lambda_{\overline{\mathcal{H}}}=\lambda_{fg}-\frac{\lambda_{H^{2}\phi}^{2}}{\lambda_{\phi^{2}}}.$
(5.67)
It is important to note that we have obtained the GB contributions after
having broken the SM symmetry through the fermion and gauge boson radiative
corrections. In this fact we differ from other analysis performed in the
literature (see for example [2, 18]), where these scalar contributions are
computed at the tree level from the effective operators only. Moreover, in our
case, the coefficients of the potential (1.2) do not depend only on the two
unknown coefficients $a$ and $a^{\prime}$, but also on the scale $f$ and the
cutoff $\Lambda$, thus setting more restrictions on the space parameter as we
will see in the following.
## 6 Numerical Results and Phenomenological Discussion
In this section we continue our study about the allowed region of the
parameter space of the LH model started in our previous papers [14, 15, 16].
In the present one we complete this phenomenological study, taking into
account also the contributions from the Goldstone boson sector to the Higgs
mass and quartic coupling obtained above. The LH parameters different
relationships and their relevant ranges considered are the following:
First, we impose the minimum condition for the complete effective potential
(1.2):
$v^{2}=\frac{\mu_{fg}^{2}}{\lambda_{fg}-\lambda_{H^{2}\phi}^{2}/\lambda_{\phi^{2}}}\,.$
(6.68)
This condition is crucial in order to reproduce the electroweak symmetry
breaking.
If we want to study the allowed region of the parameter space in these models,
we should also take into account other constraints imposed by requiring the
consistency of the LH models with the electroweak precision data. There exist
several studies of the corrections to electroweak precision observables in the
Little Higgs models, exploring whether there are regions of the parameter
space in which the model is consistent with the available data [12, 13, 7, 8,
4, 5, 9, 10, 11]. In _Model I_ with a gauge group $SU(2)\times SU(2)\times
U(1)\times U(1)$ we have a multiplet of heavy $SU(2)$ gauge bosons and a heavy
$U(1)$ gauge boson. The last one leads to large electroweak corrections and
some problems with the direct observational bounds on the $Z^{\prime}$ boson
from Tevatron [7, 8]. Then, a very strong bound on the symmetry breaking scale
$f$, $f>4$ TeV at $95\%$ C.L, is found [7]. However, it is known that this
bound is lowered to $1-2$ TeV for some region of the parameter space [8] by
gauging only $SU(2)\times SU(2)\times U(1)$ (_Model II_). For this reason, in
the following we will concentrate only on this model.
On the other hand, in order to avoid small values for the $W^{\prime}$ mass
and a very strong coupling constant, we set the range of the $\psi$ mixing
angle (for the $SU(2)$ group) to be $0.1<c{{}_{\psi}}<0.9$ [15]. In addition,
the condition
$\lambda_{T}\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$\,}$
0.5 is established from the top mass [12], setting the bounds on the couplings
$\lambda_{1},\lambda_{2}\geq m_{t}/v$ or $\lambda_{1}\lambda_{2}\geq
2(m_{t}/v)^{2}$. In order to avoid a large fine-tuning in the Higgs potential
[2, 13] we set the condition
$m_{T}\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}$\,}$
2.5 TeV. Then, since $m_{T}$ grows linearly with $f$, $f$ should be less than
about one TeV [14]. Following the restrictions on the parameters given in
[15], we take $0.8$ TeV $<f<1$ TeV. Finally the usual condition
$\Lambda\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}$\,}4\pi
f$ is also imposed.
By using the constraints on the LH parameters given above, taking into account
also that the Higgs mass is experimentally restricted to the range $114$ GeV
$<m_{\overline{\mathcal{H}}}<200$ GeV, and by imposing the minimum condition
(6.68), we analyze the available regions for the remaining LH parameters. To
do that we include the contributions of both radiative corrections and
effective operators. In fact, in order to see the role played for each of
them, we consider three different cases: having just radiative corrections
(RC), just effective operators (EO) and the most general case including both
of them (RC+EO).
Figure 2: (a) Values of $\lambda_{T}$, $\Lambda$ and $c_{\psi}$ which are
possible solutions for the LH model. Here $f$ vary between $0.8$ and $1$ TeV,
and $a$ and $a^{\prime}$ are $\mathcal{O}(1)$. The three separate surfaces
correspond with the three different cases analyzed in this section. (b) Values
of $a$ and $a^{\prime}$ which are possible solutions for the LH model. The
$\lambda_{T}$, $\Lambda$, $c_{\psi}$ and $f$ ranges are described in the text.
In Fig. 2 we show the allowed regions of the parameter space for the three
different cases analyzed; RC (red region), EO (blue region) and RC+EO (green
region). In Fig. 2.a we show the possible solutions to the LH model in the
$(\Lambda,c_{\psi},\lambda_{T})$ space varying $f$ between $0.8$ TeV and $1$
TeV and by assuming that the $a$ and $a^{\prime}$ parameters are of the order
of $\mathcal{O}(1)$. From these results there are two important issues to
remark. First, when only radiative corrections are included we do not find any
solution for the LH model if $\Lambda>6$ TeV. Unfortunately, precision
electroweak data rule out new strong interactions at scales below about $10$
TeV. On the contrary, in the other two cases, RC+EO and EO, the possible
values for the cut-off are larger. This fact implies also that the mass of the
$\phi$ fields must be about $2$ TeV when the model includes only radiative
corrections unlike in the other two cases where it is about $5$ TeV (see Fig.
3). In Fig. 2.b we show the possible values for the unknown $a$ and
$a^{\prime}$ parameters. Here, the other parameters have been varied in the
ranges set above. The two cases considered are RC+EO and EO only. We find that
the set of possible solutions include in both cases positive values for $a$.
In the RC+EO case we obtain large and negative values for $a^{\prime}$,
whereas in the EO case $a^{\prime}$ takes small and positive values. Notice
also that $a$ is always positive. This is important since it is known that
$a<0$ leads to a large v.e.v for the scalar triplet.
The reason for the differences of the parameter solutions for the three cases
come from the $\Delta m_{GB}^{2}$ cutoff dependence when the radiative
contributions are included. For example, in the case where only the radiative
corrections are taken into account, a cut-off $\Lambda$ bigger than $6$ TeV
produces GB contributions resulting in a negative Higgs mass. However, by
dropping the value of $\Lambda$ we get a LH parameter space where the
condition (6.68) is satisfied and the Higgs mass is well inside the
experimental constraints. In the RC+EO case, the $a^{\prime}$ parameter can
take values which help to compensate the big effect of the GB radiative
contributions (see also Fig. 4) thus allowing larger cutoff values.
Figure 3: (a) $m_{T}$ as a function of $\lambda_{T}$, (b) $M_{W^{\prime}}$ as
a function of $\cos\psi$ and (c) $M_{\phi}$ as a function of $\lambda_{T}$ and
$\cos\psi$, where the $\Lambda$, $f$, $a$ and $a^{\prime}$ parameters vary
between ranges described in the text.
For completeness, Fig. 3 shows the mass values for the heavy particles in the
three different cases analyzed. Each point of the figures is a possible
solution of the LH model. In this way, these regions represent the possible
values for the masses of the heavy particles predicted by the LH model, which
are compatible with electroweak symmetry breaking and precision data. The
region of possible values for the masses coming from EO contributions is
clearly larger than in the case of considering RC alone. Notice that the
theoretical lower bounds in the heavy states masses,
$M_{\phi},M_{W^{\prime}}\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$\,}$
1 TeV, and the condition
$m_{T}\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}$\,}$
2.5 TeV are fulfilled.
Figure 4: These figures show the average and standard deviation of both the
fermionic and gauge boson contribution $\mu_{fg}$ and the Goldstone boson
contribution $\Delta m_{GB}$ to the Higgs mass as a function of $\lambda_{T}$.
(a) RC case, (b) RC+EO case and (c) EO case.
To complete our study, we compare the contributions to the Higgs mass coming
from the different sectors i.e. fermionic and gauge bosons ($\mu_{fg}$) and on
the other hand the GB contribution ($\Delta m_{GB}$), as a function of
$\lambda_{T}$. We show the average and standard deviation for each
contribution (Fig. 4). In all physical cases it can be seen that
$\mu_{fg}>\Delta m_{GB}$, thus yielding a real value for the Higgs mass (eq.
5.66). It is also remarkable the higher variability of $\Delta m_{GB}$
compared with $\mu_{fg}$. The reason is that both the parameters appearing in
the radiative corrections, i.e. $f,\Lambda,\lambda_{T},\cos\phi$, and the two
EO parameters $a$ and $a^{\prime}$, play an important role in the final
results of $\Delta m_{GB}$ (see the discussion above).
Finally, as an example, we give in the Table.1 the lowest Higgs mass values
found for the three cases considered in this work.
Parameters | RC | RC+EO | EO
---|---|---|---
$m_{\overline{\mathcal{H}}}$ | $156.66$ GeV | $114.69$ GeV | $116.94$ GeV
$\mu_{fg}$ | $359.54$ GeV | $236.87$ GeV | $288.70$ GeV
$\Delta_{Gb}$ | $342.04$ GeV | $222.55$ GeV | $275.53$ GeV
$\lambda_{\overline{\mathcal{H}}}$ | $0.97$ | $0.90$ | $1.42$
$f$ | $0.86$ TeV | $0.96$ TeV | $0.82$ TeV
$\Lambda$ | $5$ TeV | $11.64$ TeV | $10.01$ TeV
$\lambda_{T}$ | $0.6$ | $0.61$ | $0.53$
$c_{\psi}$ | $0.18$ | $0.16$ | $0.3$
$a$ | $0$ | $0.98$ | $1.06$
$a^{\prime}$ | $0$ | $-1.25$ | $0.5$
Table 1: The lowest values for the Higgs mass found for the three cases: RC,
RC+EO and EO.
## 7 Conclusion
In this work we have completed our program of computing the relevant
contributions to the Higgs low-energy effective potential in the context of
the Littlest Higgs models based on the $SU(5)/SO(5)$ coset. To the radiative
corrections coming from the fermions and the gauge bosons considered so far,
we have added here the effect the scalar loops and also the effective
operators emerging from the ultraviolet completion of the model.
In particular we have computed in detail the main contributions to the Higgs
mass and its quartic coupling. From our previous works, in which only
fermionic and gauge boson radiative corrections were included, it was clear
that the effect of the scalar sector could be decisive in order to have the
appropriate cancellations between the different sectors of the model to give a
Higgs mass within the present experimental limits. We have performed our
analytical computations for two different versions of the model called _Model
I_ and _Model II_ having as gauge groups $[SU(2)\times U(1)]^{2}$ and
$SU(2)^{2}\times U(1)$ respectively.
In order to complete our analysis, we have concentrated on studying those
regions of the parameter space where the model could give rise to an
acceptable phenomenology. In particular we have done a detailed numerical
search for _Model II_ since _Model I_ seems to be incompatible with the
present experimental data [7, 8]. We have analyzed three cases: 1) radiative
corrections only (RC), 2) radiative corrections and effective operators
(RC+EO) and 3) effective operator only (EO). From this analysis we get that
this model is compatible with the expected Higgs mass provided that the
contribution of the effective operators is included. We also conclude that the
Goldstone boson contributions are fundamental to obtain a low enough Higgs
particle mass. For example a Higgs mass $m_{H}\simeq 115GeV$ can be obtained
when radiative and effective operator contributions are both taken into
account.
Summarizing, we have arrived to the conclusion that the $SU(5)/S(5)$ Littlest
Higgs model with gauge group $[SU(2)\times U(1)]^{2}$ is phenomenologically
viable through some tuning in the parameter space, assuming a careful
inclusion of fermions, gauge bosons, scalar loops and effective operators.
In any case it will be the LHC, whose main goal is to disentangle the
mechanism of the electroweak symmetry breaking, which will decide if Littlest
Higgs models are appropriate for describing mechanism or not.
Acknowledgments: This work is supported by DGICYT (Spain) under project number
FPA2008-00592 and by the Universidad Complutense/CAM: UCM-BSCH GR58/08 910309.
The work of S.P. is supported by a Ramón y Cajal contract from MEC (Spain) and
partially by CICYT (grant FPA2006-2315) and DGIID-DGA (grant 2008-E24/2).The
work of J.R.L. is supported by project number FIS2006-04885. We would like to
thank J.R.Espinosa for useful discussions.
Appendix
a. Coefficients coming from loops computation
_Model I_
$\displaystyle\mu_{fg}^{2}$ $\displaystyle=$
$\displaystyle\mu^{2}_{f}+\mu^{2}_{g}$ $\displaystyle=$ $\displaystyle
N_{c}\frac{m_{T}^{2}\lambda_{t}^{2}}{4\pi^{2}}\log\left(1+\frac{\Lambda^{2}}{m_{T}^{2}}\right)$
$\displaystyle-$
$\displaystyle\frac{3}{64\pi^{2}}\left[3g^{2}M_{W^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)+g^{{}^{\prime}2}M_{B^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}\right)\right]$
$\displaystyle\lambda_{f}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{(4\pi)^{2}}\left[2(\lambda_{t}^{2}+\lambda_{T}^{2})\frac{\Lambda^{2}}{f^{2}}\right.$
$\displaystyle-$
$\displaystyle\log\left(1+\frac{\Lambda^{2}}{m_{T}^{2}}\right)\left(-\frac{2m_{T}^{2}}{f^{2}}\left(\frac{5}{3}\lambda_{t}^{2}+\lambda_{T}^{2}\right)+4\lambda_{t}^{4}+4(\lambda_{T}^{2}+\lambda_{t}^{2})^{2}\right)$
$\displaystyle-$
$\displaystyle\left.4\lambda_{T}^{2}\frac{1}{1+\frac{m_{T}^{2}}{\Lambda^{2}}}\left(\frac{m_{T}^{2}}{f^{2}}-2\lambda_{t}^{2}-\lambda_{T}^{2}\right)-4\lambda_{t}^{4}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)\right]$
$\displaystyle-$ $\displaystyle\frac{3}{(16\pi
f)^{2}}\left[-\left(\frac{g^{2}}{c_{\psi}^{2}s_{\psi}^{2}}+\frac{g^{{}^{\prime}2}}{c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}\right)\Lambda^{2}\right.$
$\displaystyle+$
$\displaystyle\left.g^{2}M_{W^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)\left(4+\frac{1}{c_{\psi}^{2}s_{\psi}^{2}}+2g^{{}^{\prime}2}\frac{(c_{\psi}^{2}s_{\psi}^{{}^{\prime}2}+s_{\psi}^{2}c_{\psi}^{{}^{\prime}2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}\frac{f^{2}}{M_{W^{\prime}}^{2}-M_{B^{\prime}}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.g^{{}^{\prime}2}M_{B^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}\right)\left(\frac{4}{3}+\frac{1}{c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}+2g^{2}\frac{(c_{\psi}^{2}s_{\psi}^{{}^{\prime}2}+s_{\psi}^{2}c_{\psi}^{{}^{\prime}2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}\frac{f^{2}}{M_{B^{\prime}}^{2}-M_{W^{\prime}}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)\left(3g^{4}+2(3g^{2}+g^{{}^{\prime}2})g^{2}\frac{(s_{\psi}^{2}-c_{\psi}^{2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(1+\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}\right)\left(g^{{}^{\prime}4}+2(g^{2}+g^{{}^{\prime}2})g^{{}^{\prime}2}\frac{(s_{\psi}^{{}^{\prime}2}-c_{\psi}^{{}^{\prime}2})^{2}}{c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)\left(3g^{4}+g^{{}^{\prime}4}+8g^{2}g^{{}^{\prime}2}\right)-3f^{2}\frac{g^{4}}{1-\frac{M_{W^{\prime}}^{2}}{\Lambda^{2}}}-f^{2}\frac{g^{{}^{\prime}4}}{1-\frac{M_{B^{\prime}}^{2}}{\Lambda^{2}}}\right]$
$\displaystyle\lambda_{\phi^{2}f}$ $\displaystyle=$
$\displaystyle\frac{8N_{c}}{(4\pi
f)^{2}}(\lambda_{t}^{2}+\lambda_{T}^{2})\left(\Lambda^{2}-m_{T}^{2}\log\left(\frac{\Lambda^{2}}{m_{T}^{2}}+1\right)\right)$
$\displaystyle+$ $\displaystyle\frac{3}{4(4\pi
f)^{2}}\left[\frac{g^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\Lambda^{2}-g^{2}M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\left(\frac{(s_{\psi}^{2}-c_{\psi}^{2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}}-4\right)\right.$
$\displaystyle+$
$\displaystyle\left.\frac{g^{{}^{\prime}2}}{c_{\psi^{\prime}}^{2}s_{\psi^{\prime}}^{2}}\Lambda^{2}-g^{{}^{\prime}2}M_{B^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}+1\right)\frac{(s_{\psi^{\prime}}^{2}-c_{\psi^{\prime}}^{2})^{2}}{c_{\psi^{\prime}}^{2}s_{\psi^{\prime}}^{2}}\right]$
$\displaystyle\lambda_{H^{2}\phi}$ $\displaystyle=$
$\displaystyle-\frac{4N_{c}}{(4\pi
f)^{2}}\left[(\lambda_{t}^{2}+\lambda_{T}^{2})\Lambda^{2}-\lambda_{T}^{2}m_{T}^{2}\log\left(\frac{\Lambda^{2}}{m_{T}^{2}}+1\right)\right]$
$\displaystyle+$ $\displaystyle\frac{3}{8(4\pi
f)^{2}}\left[g^{2}\frac{s_{\psi}^{2}-c_{\psi}^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\left(\Lambda^{2}-M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\right)\right.$
$\displaystyle+$
$\displaystyle\left.g^{{}^{\prime}2}\frac{s_{\psi^{\prime}}^{2}-c_{\psi^{\prime}}^{2}}{c_{\psi^{\prime}}^{2}s_{\psi^{\prime}}^{2}}\left(\Lambda^{2}-M_{B^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}+1\right)\right)\right]\,,$
_Model II_
$\displaystyle\mu_{fg}^{2}$ $\displaystyle=$
$\displaystyle\mu^{2}_{f}+\mu^{2}_{g}$ $\displaystyle=$ $\displaystyle
N_{c}\frac{m_{T}^{2}\lambda_{t}^{2}}{4\pi^{2}}\log\left(1+\frac{\Lambda^{2}}{m_{T}^{2}}\right)-\frac{3}{64\pi^{2}}\left(3g^{2}M_{W^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)+g^{{}^{\prime}2}\Lambda^{2}\right)$
$\displaystyle\lambda_{fg}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{(4\pi)^{2}}\left[2(\lambda_{t}^{2}+\lambda_{T}^{2})\frac{\Lambda^{2}}{f^{2}}\right.$
$\displaystyle-$
$\displaystyle\log\left(1+\frac{\Lambda^{2}}{m_{T}^{2}}\right)\left(-\frac{2m_{T}^{2}}{f^{2}}\left(\frac{5}{3}\lambda_{t}^{2}+\lambda_{T}^{2}\right)+4\lambda_{t}^{4}+4(\lambda_{T}^{2}+\lambda_{t}^{2})^{2}\right)$
$\displaystyle-$
$\displaystyle\left.4\lambda_{T}^{2}\frac{1}{1+\frac{m_{T}^{2}}{\Lambda^{2}}}\left(\frac{m_{T}^{2}}{f^{2}}-2\lambda_{t}^{2}-\lambda_{T}^{2}\right)-4\lambda_{t}^{4}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)\right]$
$\displaystyle-$ $\displaystyle\frac{3}{(16\pi
f)^{2}}\left[-\frac{g^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\Lambda^{2}+\frac{4}{3}{g^{\prime}}^{2}\Lambda^{2}+g^{2}M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\left(4+\frac{1}{c_{\psi}^{2}s_{\psi}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)\left(3g^{4}+2(3g^{2}+{g^{\prime}}^{2})g^{2}\frac{(s_{\psi}^{2}-c_{\psi}^{2})^{2}}{s_{\psi}^{2}c_{\psi}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)(3g^{4}+{g^{\prime}}^{4}+8g^{2}{g^{\prime}}^{2})-3f^{2}\frac{g^{4}}{1-\frac{M_{W^{\prime}}^{2}}{\Lambda^{2}}}\right]$
$\displaystyle\lambda_{\phi^{2}}$ $\displaystyle=$
$\displaystyle\frac{8N_{c}}{(4\pi
f)^{2}}(\lambda_{t}^{2}+\lambda_{T}^{2})\left(\Lambda^{2}-m_{T}^{2}\log\left(\frac{\Lambda^{2}}{m_{T}^{2}}+1\right)\right)$
$\displaystyle+$
$\displaystyle\frac{3}{64\pi^{2}f^{2}}\left[\frac{g^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\Lambda^{2}-g^{2}M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\left(\frac{(s_{\psi}^{2}-c_{\phi}^{2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}}-4\right)\right]$
$\displaystyle+$ $\displaystyle\frac{3g^{{}^{\prime}2}}{(4\pi
f)^{2}}\Lambda^{2}$
$\displaystyle\lambda_{H^{2}\phi}$ $\displaystyle=$
$\displaystyle-\frac{4N_{c}}{(4\pi
f)^{2}}\left[(\lambda_{t}^{2}+\lambda_{T}^{2})\Lambda^{2}-\lambda_{T}^{2}m_{T}^{2}\log\left(\frac{\Lambda^{2}}{m_{T}^{2}}+1\right)\right]$
$\displaystyle+$
$\displaystyle\frac{3g^{2}}{8(4f\pi)^{2}}\frac{s_{\psi}^{2}-c_{\psi}^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\left(\Lambda^{2}-M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\right)\,,$
b. Coefficients coming from effective operators
_Modelo I_
$\displaystyle\lambda_{fg}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{8}\left(\frac{g^{2}}{s_{\psi}^{2}c_{\psi}^{2}}+\frac{g^{{}^{\prime}2}}{s_{\psi}^{{}^{\prime}2}c_{\psi}^{{}^{\prime}2}}\right)+2a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle{\lambda_{\phi^{2}}}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{2}\left(\frac{g^{2}}{s_{\psi}^{2}c_{\psi}^{2}}+\frac{g^{{}^{\prime}2}}{s_{\psi}^{{}^{\prime}2}c_{\psi}^{{}^{\prime}2}}\right)+8a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle{\lambda_{H^{2}\phi}}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{4}\left(g^{2}\frac{c_{\psi}^{2}-s_{\psi}^{2}}{s_{\psi}^{2}c_{\psi}^{2}}+g^{{}^{\prime}2}\frac{c_{\psi}^{{}^{\prime}2}-s_{\psi}^{{}^{\prime}2}}{s_{\psi}^{{}^{\prime}2}c_{\psi}^{{}^{\prime}2}}\right)+4a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
_Modelo II_
$\displaystyle\lambda_{fg}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{8}\left(\frac{g^{2}}{s_{\psi}^{2}c_{\psi}^{2}}\right)-\frac{a}{3}g^{{}^{\prime}2}+2a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle{\lambda_{\phi^{2}}}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{2}\left(\frac{g^{2}}{s_{\psi}^{2}c_{\psi}^{2}}\right)+4a{g^{{}^{\prime}2}}+8a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle{\lambda_{H^{2}\phi}}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{4}g^{2}\frac{c_{\psi}^{2}-s_{\psi}^{2}}{s_{\psi}^{2}c_{\psi}^{2}}+4a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle\mu^{2\,\rm EO}$ $\displaystyle=$ $\displaystyle
af^{2}g^{{}^{\prime}2}$
## References
* [1] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 99 (2007) 151801 [arXiv:0707.0085 [hep-ex]]; T.T.E. Group et al. [Tevatron Electroweak Working Group and CDF Collaboration and D0 Collab], arXiv:0803.1683 [hep-ex].
* [2] N. Arkani-Hamed, A.G. Cohen, E.Katz, A.E.Nelson, JHEP 0207:034 (2002), hep-ph/0206021.
* [3] H. Georgi and A. Pais, Phys. Rev. D10, 539 (1974) H. Georgi and A. Pais, Phys. Rev. D12, 508 (1975).
* [4] M. Schmaltz, D. Tucker-Smith, Ann. Rev. Nucl. Part. Sci. 55 (2005) 229, hep-ph/0502182.
* [5] M. Perelstein, Prog. Part. Nucl. Phys. 58 (2007) 247, hep-ph/0512128.
* [6] M. Schmaltz, JHEP 0408 (2004) 056, hep-ph/0407143; H. C. Cheng and I. Low, JHEP 0408, 061 (2004), hep-ph/0405243; D. E. Kaplan and M. Schmaltz, JHEP 0310 (2003) 039, hep-ph/0302049; H. C. Cheng and I. Low, JHEP 0309, 051 (2003), hep-ph/0308199; I. Low, W. Skiba and D. Tucker-Smith, Phys. Rev. D 66, 072001 (2002), hep-ph/0207243.
* [7] C. Csaki, J. Hubisz, G. D. Kribs, P. Meade, J. Terning, Phys. Rev. D67 (2003) 115002, hep-ph/0211124.
* [8] C. Csaki, J. Hubisz, G. D. Kribs, P. Meade, J. Terning, Phys. Rev. D68, 035009 (2003), hep-ph/0303236.
* [9] Z. Han, W. Skiba, Phys. Rev. D72, 035005 (2005), hep-ph/0506206; H.E. Logan, Phys. Rev. D70, 115003 (2004), hep-ph/0405072; T. Han, H. E. Logan, B. McElrath, L. T. Wang, Phys. Lett. B563, 191 (2003) [Erratum-ibid. B603, 257 (2004)], hep-ph/0302188.
* [10] J. L. Hewett, F. J. Petriello, T. G. Rizzo, JHEP 0310, 062 (2003), hep-ph/0211218. T. Gregoire, D. R. Smith, J. G. Wacker, Phys. Rev. D69, 115008 (2004), hep-ph/0305275; M. C. Chen and S. Dawson, Phys. Rev. D70, 015003 (2004), hep-ph/0311032; W. Kilian, J. Reuter, Phys. Rev. D70, 015004 (2004), hep-ph/0311095; G. Marandella, C. Schappacher, A. Strumia, Phys. Rev. D72, 035014 (2005), hep-ph/0502096.
* [11] S. K. Kang, C. S. Kim and J. Park, Phys. Lett. B 666 (2008) 38, arXiv:0805.0179 [hep-ph]; M.C. Chen, Mod. Phys. Lett. A21, 621 (2006), hep-ph/0601126; S.R. Choudhury, A.S. Cornell, N. Gaur, A. Goyal, hep-ph/0604162; J.A. Conley, J. Hewett, M. P. Le, Phys. Rev. D72, 115014 (2005), hep-ph/0507198; C.O. Dib, R. Rosenfeld, A. Zerwekh, AIP Conf. Proc. 815, 296 (2006), hep-ph/0509013; Z. Berezhiani, P.H. Chankowski, A. Falkowski, S. Pokorski, Phys. Rev. Lett. 96, 031801 (2006), hep-ph/0509311.
* [12] T. Han, Heather E. Logan, B. McElrath, Lian-Tao Wang, Phys. Rev. D67, 095005 (2003), hep-ph/0301040.
* [13] M. Perelstein, M. E. Peskin, A. Pierce, Phys. Rev. D69, 075002 (2004), hep-ph/0310039.
* [14] A. Dobado, L. Tabares and S. Peñaranda, Eur. Phys. J. C50 (2007) 647, hep-ph/0606031.
* [15] A. Dobado, L. Tabares, S. Peñaranda, Phys. Rev. D75 (2007) 083527, hep-ph/0612131.
* [16] A. Dobado, L. Tabares, S. Peñaranda, Eur. Phys. J. C 58 (2008) 471, arXiv:0711.4471 [hep-ph].
* [17] F. Bazzocchi, M. Fabbrichesi, M. Piai, Phys. Rev. D72 (2005) 095019, hep-ph/0506175.
* [18] J. Alberto Casas, Jose Ramón Espinosa, Irene Hidalgo, JHEP 0503:038.
* [19] A. Dobado, A. Gómez-Nicola, A.L. Maroto and J.R. Peláez, Effective Lagrangians for the Standard Model, Springer-Verlag, Heidelberg, (1997).
* [20] A. Manohar, H. Georgi, Nucl. Phys. B234, 189 (1984); M. A. Luty, Phys. Rev. D57, 1531 (1998), hep-ph/9706235; A. G. Cohen, D. B. Kaplan, A. E. Nelson, Phys. Lett. B412, 301 (1997).
* [21] J. R. Espinosa and J. M. No, JHEP 0701 (2007) 006, hep-ph/0610255.
|
arxiv-papers
| 2009-07-09T10:06:24 |
2024-09-04T02:49:03.787253
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Antonio Dobado, Lourdes Tabares-Cheluci (Madrid U.), Siannah Penaranda\n (Zaragoza U.), Javier Rodriguez-Laguna (Carlos III U.)",
"submitter": "Lourdes Tabares",
"url": "https://arxiv.org/abs/0907.1483"
}
|
0907.1632
|
# Incorporating Integrity Constraints in Uncertain Databases
Naveen Ashish$~{}^{\ 1}$, Sharad Mehrotra$~{}^{2}$, Pouria Pirzadeh$~{}^{\ 3}$
$~{}^{\ }$Calit2 and ICS Department, UC Irvine
Irvine CA 92697 USA
$~{}^{1}[email protected]
$~{}^{2}[email protected]
###### Abstract
We develop an approach to incorporate additional knowledge, in the form of
general-purpose integrity constraints (ICs), to reduce uncertainty in
probabilistic databases. While incorporating ICs improves data quality (and
hence quality of answers to a query), it significantly complicates query
processing. To overcome the additional complexity, we develop an approach to
map an uncertain relation $U$ with ICs to another uncertain relation
$U^{\prime}$ that approximates the set of consistent worlds represented by
$U$. Queries over $U$ can instead be evaluated over $U^{\prime}$ achieving
higher quality (due to reduced uncertainty in $U^{\prime}$) without additional
complexity in query processing due to ICs. We demonstrate the effectiveness
and scalability of our approach to large datasets with complex constraints. We
also present experimental results demonstrating the utility of incorporating
integrity constraints in uncertain relations, in the context of an information
extraction application.
## I Introduction
Recent advances in probabilistic models for information extraction, document
classification, and automated tagging has revived significant interest in
probabilistic data management. Extraction techniques based on models such as
conditional random fields (CRFs) [1] create a database wherein tuples and/or
attribute values have associated explicit estimates of probability. Multiple
probabilistic models [2, 3, 4], of varying expressivity, have been developed
to represent such uncertain data along with efficient query processing
approaches [5, 3, 2] to support search and analysis capability on such
uncertain databases. In this paper we develop an approach to incorporating
additional semantics, in the form of database integrity constraints, that can
reduce uncertainty in data, which, in turn, could positively impact
applications such as query answering and retrieval
Consider the example in Fig 1 where we have an Employee relation with
uncertainty, represented using ”or-sets” [2] for each attribute . For instance
in tuple 1 the job-title of the employee ”jim” is either instructor (with
probability $0.7$) or a manager (with probability $0.3$). This uncertain
relation represents 4 possible worlds which are the 4 possibilities of this
relation based on different attribute value choices in each attribute. A query
$Q$ over such a probabilistic database returns tuples that satisfy $Q$ in one
of the possible worlds along with their corresponding probabilities. Now
consider additional semantics in the form of say a functional dependency (FD)
that states that a person cannot hold the same job title in 2 different
divisions, i.e., (name, job-title) $\rightarrow$ division. Given this
knowledge, we know that two out of the four possible worlds, where the first
tuple has ”Jim” as a ”manager” (in ”training”), are impossible as they violate
the functional dependency. A natural extension to the query semantics is to
return only those tuples that satisfy the query $Q$ in one of the consistent
possible worlds. As a result, the answer to the query about ”jim” above should
not include the tuple $\\{jim,manager,training\\}$ since such a tuple is not
part of any consistent instance of the relation.
Employee
---
name | job-title | division | degree
jim | instructor(0.7) | training | MBA
| manager(0.3) | |
jim | manager | marketing | MBA
jim (0.5) | consultant | innovation | PhD
jill (0.5) | | |
Constraint (C): (name, job-title) $\rightarrow$ division
Figure 1: Uncertain Relation with Constraints
Incorporating additional knowledge such as constraints to reduce uncertainty
in query results has indeed been explored to various degrees in different
uncertain database models and systems [3, 6, 7]. For instance Trio [3] and [6]
permit the specification of constraints, but at the data instance level i.e.,
between individual attribute or tuple instances. For instance using the
notation T1(2) to represent the second tuple instance (possibility) in the
first tuple in Fig 1 i.e., (jim, manager, training, MBA) and T2(1) to
represent the first (and only) tuple instance in the second tuple, we could
specify a constraint such as T1(2) XOR T2(1) which states that only one of
these tuple instances can exist together in a possible world. Query answering
approaches for such models [6] can address only a small number ($<$ 20) of
such constraint instances. [8] considers very restricted forms of FD and IND
constraints in addition to database statistics, to address a different problem
- that of determining a maximum likelihood estimate of the probability of a
query answer in a data integration setting. MayBMS [7] has considered more
general integrity constraints at the level of individual tuples as well as
functional dependencies in their probabilistic model based on representing
uncertain databases using world set decompositions. Their approach for
factoring FDs however can be shown to be exponential - as we illustrate in the
related work section. This is not surprising as it can be shown that answering
even simple selection queries exactly over uncertain relation in presence of
integrity constraints (e.g., a single functional dependency), is NP-hard. We
state the following:
Statement 1: Given a U-relation, U, and a functional dependency (FD) F defined
over U, identifying a possible world instance $pw_{q}\in PW_{U}$ such that
$pw_{q}\models F$ or determining that no such instance exists is NP-Hard. We
refer to this problem as the The FD consistency problem.
Proof: This follows by a reduction from the 3-SAT problem which is known to be
NP-Hard. The proof is as follows:
1\. Given an instance of 3-SAT i.e., a CNF expression:
$(x_{11}\vee x_{12}\vee x_{13})$ $\wedge$ $(x_{21}\vee x_{22}\vee x_{23})$
…$\wedge$ $(x_{n1}\vee x_{n2}\vee x_{n3})$.
We will now create a corresponding uncertain relation U as follows.
2\. Consider any one conjunct, each conjunct is of the form of one of
$(x_{1}\vee x_{2}\vee x_{3})$, $(x_{11}\vee x_{2}\vee\neg x_{3})$, $(\neg
x_{11}\vee\neg x_{2}\vee x_{3})$, or $(\neg x_{11}\vee\neg x_{2}\vee\neg
x_{3})$
3\. Take the following actions based on the type of the conjunct:
(i) Type is: $(x_{1}\vee x_{2}\vee x_{3})$
Create the following 3 uncertain tuples, each tuple with 3 tuple instances
(choices), in U:
{$T_{x1},T_{x2},T_{x3}$}
{$T_{x1},T_{x2},T_{x3}$}
{$T_{x1},T_{x2},T_{x3}$}
Let the tuple have some attributes (which are at least 3 in number). Let the
first attribute be a tuple instance identifier (ID), tuple instance $T_{x1}$
has ID 1, tuple instance $T_{x2}$ has ID 2, etc.
(ii) Type is: $(x_{1}\vee x_{2}\vee\neg x_{3})$
Note that we can also treat this as $x_{3}$ $\rightarrow$ ($x_{1}\vee x_{2}$)
Create the following tuple instances in U:
{$T_{x1},T_{x2},$-$T_{x3}$}
{$T_{x1},T_{x2},$-$T_{x3}$}
-$T_{xi}$ is a tuple instance created such that $T_{xi}$ and -$T_{xi}$ violate a functional dependency (FD). Pick 2 attributes A and B in the tuple. To inject an FD violation assign $T_{xi}$ as [i,..,a1,..,b1..] and -$T_{xi}$ as [i,..,a1,..,b2..] where a1 is a value of the A attribute and b1 and b2 are values of the B attribute. The instances $T_{xi}$ and -$T_{xi}$ thus violate the FD: A $\rightarrow$ B.
(iii) Type is: $(\neg x_{1}\vee\neg x_{2}\vee x_{3})$
We can also treat this as $(x_{1}\wedge x_{2})$ $\rightarrow$ $x_{3}$ Create
the following tuple instances in U:
{-$T_{x1},$-$T_{x2},T_{x3}$}
(iv) Type is: $(\neg x_{1}\neg\vee x_{2}\neg\vee x_{3})$
We can also treat this as $(x_{1}\wedge x_{2})$ $\rightarrow$ $\neg x_{3}$
Create the following tuple instances in U:
{-$T_{x1},$-$T_{x2},T_{y}$}
$T_{y}$ is made such that $T_{y}$ and $T_{x3}$ violate the FD: A $\rightarrow$
B.
At this point we have an instance of an FD consistency problem i.e., we have
an uncertain relation U with uncertain tuples and a single FD: A $\rightarrow$
B on this relation. This reduction, from the original 3-SAT problem has been
done in time polynomial in the size of the original 3-SAT problem.
Our claim is that a solution to the 3-SAT problem exists iff there is a
solution to the FD consistency problem we have derived. Say we have a solution
to the FD consistency problem. Each tuple is some $T_{xi}$. For each i, we can
have only one of $T_{xi}$ or -$T_{xi}$ in the consistent relation obtained (so
as to not violate the FD). For any $T_{xi}$ in the solution we set xi to 1 in
the 3-SAT problem, for any -$T_{xi}$ in the solution we set xi to 0. With this
assignment we will necessarily have a truth assignment for the xi s for which
the 3-SAT formula is true. Also if there is a truth assignment that makes the
3-SAT formual true then there necessarily exists a solution to the FD
consistency problem (for each xi assigned to 1 we retain $T_{xi}$ in the
solution and for each xi assigned to 0 we retain -$T_{xi}$). Conversely if
there is no solution to the FD consistency problem then there is no solution
to the 3-SAT problem.
Our claim above is thus valid that a solution to the 3-SAT problem exists iff
there is a solution to the translated FD consistency problem. As 3-SAT is NP-
complete it follows that the FD-consistency problem is NP-Hard.
Given the intractibility of answering queries exactly in presence of ICs, we
take a different approach that attempts to replace a given uncertain relation
$U$ by another sub-relation that is also (a special case of) an uncertain
relation $U^{\prime}$ into which any constraints provided over $U$ have been
factored in. Ideally, $U^{\prime}$ represents all the possible worlds of $U$
that are consistent w.r.t. $C$ and eliminates possible worlds that are
inconsistent. The uncertain relation shown in Fig 2 is such a $U^{\prime}$ for
the relation $U$ in Fig1. Answers to queries over $U^{\prime}$ ,which can be
efficiently evaluated using independence semantics, would thus be exactly the
answers were we to execute the query over consistent possible worlds of $U$.
In general, such a $U^{\prime}$ that exactly captures the set of consistent
possible worlds of $U$ might not exist. For instance if we modify the second
tuple in the relation in Fig 1 to be
jim | manager | marketing (0.5) | MBA
---|---|---|---
| | training(0.5) |
we can see that no sub-relation (in the or-set based model we use) can exactly
represent the consistent possible worlds of $U$. Our goal, thus, is to
identify a “good” sub-relation that mirrors/approximates the original
uncertain relation (and constraints) closely. Such a ”good” approximation
would eliminate as many of the inconsistent worlds of the original relation as
possible while at the same time minimizing the number of consistent worlds
that would invariably be eliminated as a by product. The paper devises
mechanisms to computing such a good approximation for the original uncertain
relation given a set of integrity constraints (IC). We consider a large class
of attribute, tuple, and relation level ICs - including FDs, aggregation
constraints and other kinds of ICs that other approaches have not considered.
Note that queries over a sub relation into which constraints have been
factored can be answered efficiently. While simple selection queries over a
single relation can be answered efficiently in a straightforward mechanism,
techniques developed in [9] can be used to answer more complex single as well
as multi relation queries. Our specific contributions can be summarized as:
(i) We present a more general approach for factoring a large class of ICs into
uncertain databases that other systems have not considered, (ii) By using
approximations our approach can scalably handle uncertain databases with a
high degree of data ”dirtiness” (fraction of fields that are uncertain).
name | job-title | division | degree
---|---|---|---
jim | instructor (0.7) | training | MBA
jim | manager | marketing | MBA
jill (0.5) | consultant | innovation | PhD
jim (0.5) | | |
Figure 2: Alternate Representation
The rest of the paper is organized as follows. In Section 2, we formally
define our notion of uncertain relations, state our problem of generating
(tractable) sub-relations of uncertain relations as part of our approach to
providing efficient retrieval over uncertain relations with constraints.
Section 3 and 4 together develops our approach where we borrow from and build
upon techniques from areas such as database repair [10] and work in compact
representation of probabilistic distributions [1]. In Section 5, we
demonstrate both the scalability and efficiency of our approach as well as
impact of considering ICs on quality of the information extraction task.
Section 6 gives an overview of related works and the last section concludes
the paper.
## II Formalization
In this section we formally define uncertain relations with constraints and
postulate the problem of generating approximations of such uncertain relations
that facilitate efficient query answering.
Uncertain Relation An uncertain relation, $U$, is defined as:
* •
$U$ = $\\{t_{1},t_{2},\cdots,t_{n}\\}$; i.e., $U$ is a relation which is a set
of $n$ tuples.
* •
$t_{i}=(a_{i1},a_{i2},\cdots,a_{is})$ ; each tuple is a sequence of $s$
attributes.
* •
$a_{ij}=\\{(a_{ij}^{1},c_{ij}^{1}),...,(a_{ij}^{k_{ij}},c_{ij}^{k_{ij}})\\}$ ;
Each attribute is a set of possible attribute values with an associated
probability distribution. The set is referred to as the attribute world.
$k_{ij}$ is the number of choices in the attribute world $a_{ij}$, and
$\sum_{p=1}^{k_{ij}}c_{ij}^{p}=1$ .
Each uncertain relation $U$ represents a set of possible worlds, $PW_{U}$.
Each possible world corresponds to choosing a value for each attribute
$a_{ij}$, a specific value from its attribute world. Let $pw$ be a variable
over the possible worlds. A possible world $pw$ = $pw_{q}$ $\in$ $PW_{U}$ is
defined by a function, $f_{q}$ : $f_{q}(x,y)\rightarrow I$; where
$x\in\\{1,2,..,n\\}$, $y\in\\{1,2,..,s\\}$ and $I\in\\{1,2,\cdots,k_{xy}\\}$.
The number of such unique functions is $\prod_{i=1}^{n}\prod_{j=1}^{s}k_{ij}$
which is the number of possible worlds. The probability distribution $P_{I}$
defined over $PW_{U}$ under the assumption of independence is :
$\begin{split}\forall pw_{q}\in
PW_{U},P_{I}(pw=pw_{q})=\prod_{i=1}^{n}\prod_{j=1}^{s}c_{ij}^{f_{q}(i,j)}\end{split}$
(1)
Note that $\sum_{all\,worlds\,q}P_{I}(pw=pw_{q})=1$.
The above model for representing database uncertainty is based on the or-set
relations [11] where an attribute value is essentially a set of possible
values with an associated probability distribution.
Uncertain Relation with Constraints We now associate constraints with
uncertain relations, defining an uncertain relation with constraints denoted
as $U+C$, where $U$ is an uncertain relation as defined above and $C$ is a set
of integrity constraints over $U$. Let $PW_{U}^{C}$ denote the subset of
possible worlds in $PW_{U}$ that are consistent w.r.t (all) the constraints,
C. i.e., $PW_{U}^{C}=\\{pw_{q}|pw_{q}\in PW_{U}$ and $pw_{q}\models C\\}$. The
set of possible worlds not consistent w.r.t. C is denoted as $PW_{U}^{\neg
C}=\\{pw_{q}|pw_{q}\in PW_{U}$ and $pw_{q}\not\models C\\}$. The uncertain
relation with constraints, $U+C$, is interpreted as a set of possible worlds
of $U$ with the probability distribution redefined as follows:
$\begin{split}P(pw=pw_{q})&=0,ifpw_{q}\not\models C\\\
P(pw=pw_{q})&=P(pw=pw_{q}|pw\in PW_{U}^{C}),ifpw_{q}\models C\\\
&=\frac{P(pw\in PW_{U}^{C}\mid pw_{q})P_{I}(pw=pw_{q})}{P(pw\in
PW_{U}^{C})}\\\ &(Bayes^{\prime}theorem)\\\ &=\gamma
P_{I}(pw=pw_{q})\end{split}$ (2)
As $P(pw\in PW_{U}^{C}\mid pw_{q})P(pw_{q})=1$ since $pw\models C.$ Also
$\gamma$ = 1/(1 - $\lambda$) where
$\lambda=\Sigma_{pw_{q}}P_{I}(pw_{q}),pw_{q}\in PW_{U}^{\neg C}$.
Sub-relations Consider an uncertain relation $U$. If we replace the possible
values of each attribute $a_{ij}$ in each tuple $t_{i}$ in $U$ with a subset
of the possible values for that attribute in $U$, we arrive at what we call a
sub-relation of $U$. We denote the sub-relation of $U$ by $U^{\prime}$.
Strictly speaking $U^{\prime}$ is not an uncertain relation as it does not
necessarily provide a complete probabilistic distribution over possible
relations. It is used however to represent a subset of the possible worlds for
an uncertain relation. A sub-relation $U^{\prime}$ is additionally associated
with a constant factor $\gamma_{U^{\prime}}$ and the probability of any world
$pw=pw_{q}\in PW_{U^{\prime}}$ is given by
$p(pw=pw_{q})=\gamma_{U^{\prime}}\prod_{i=1}^{n}\prod_{j=1}^{s}cij^{f_{q}(i,j)}$
i.e., the probability of any world is recalibrated with the
$\gamma_{U^{\prime}}$ factor. The factor $\gamma_{U^{\prime}}$ is derived from
Equation 2 which ensures that the probability of any consistent world in
$U^{\prime}$ is exactly the same as in $U+C$. Note however that $U^{\prime}$
may represent some inconsistent worlds as well and assign a non-zero
probability to such worlds.
As an example, Fig 3 represents a sub-relation of the uncertain relation in
Fig 1 (and with the second tuple modified). $\lambda$ for the uncertain
relation is 0.15. Thus $\gamma_{U^{\prime}}$ = 1/0.85 = 1.17 which is how the
$\gamma_{U^{\prime}}$ factor for the sub-relation in Fig 3 is derived. We
define a sub-tuple of an uncertain tuple (any tuple in an uncertain relation
is an uncertain tuple) analogously, where replacing the set of attribute
values in each attribute in the tuple with one of its subsets provides us with
a sub-tuple of that uncertain tuple.
We use sub-relations to approximate an uncertain relation $U$ with constraints
$C$. Ideally, we would like the sub-relation $U^{\prime}$ to represent the
exact set of consistent possible worlds of $U$ and to eliminate all of the
inconsistent possible worlds. However, as discussed in the introduction, such
a $U^{\prime}$ might not exist and, as a result, our goal will be to identify
the ”best” approximation of $U+C$. In order to define a concept of ”best” we
need to define a metric to evaluate how well does a specific sub-relation
capture $U+C$.
name | job-title | division | degree
---|---|---|---
jim | instructor(0.7) | training | BA
jim | manager | marketing (0.5) | MBA
| | training(0.5) |
jim (0.5) | consultant | innovation | PhD
jill (0.5) | | |
$\gamma$ = 1.17
Figure 3: Sub-relation
Quality of Approximation:. Let $U$ be an uncertain relation with associated
integrity constraints $C$ and let $U^{\prime}$ be a sub-relation approximation
of $U$. Let $P_{c}$ be the (total) consistent mass in $U+C$ (i.e., the sum of
the probabilities of the possible worlds of $U$ that are consistent). Also,
let $C_{r}$ ($I_{r}$) be the consistent (inconsistent) mass of $U$ retained in
$U^{\prime}$ respectively. The quality of $U^{\prime}$, denoted by
$Q_{U^{\prime}}$ is defined as:
$Q_{U^{\prime}}=\frac{C_{r}}{P_{c}}-I_{r}$
Note that this metric considers the absolute inconsistent mass retained and
the relative consistent mass retained because it is the fraction of consistent
mass retained that we would like to be high (as opposed to its absolute value
which may be low). A quality value of 1 is the best achievable. We also note
that since the approximate representation $U^{\prime}$ might eliminate
consistent possible worlds (in addition to eliminating inconsistent worlds),
the results of a query $Q$ over $U^{\prime}$ might include false negatives
(i.e., tuples that should be part of the answer since they satisfy the query
in some consistent world, but are not part of the result over $U^{\prime}$).
While introducing false negatives might be unacceptable for certain
applications, for applications of probabilistic databases that motivate our
work such as information extraction and query answering, we believe that a
modest reduction in one of precision or recall in exchange for a significant
increase in the other is a desirable tradeoff.
Problem Formalization Given the above definition of quality, we can now
formally state our objective as that of generating a sub-relation $U^{\prime}$
of $U$ that has the highest quality. That is, $\forall Y\in U^{\prime}_{M}$:
$Q_{U^{\prime}}\geq Q_{Y}$, where $U^{\prime}_{M}$ is the set of ”all” sub-
relations. Unfortunately, identifying such an ”optimal” sub-relation is NP-
hard even when we consider a single functional dependency or a tuple level
constraint as we will see in the next section [12]. We will therefore restrict
ourselves to heuristic techniques to finding $U^{\prime}$ that attempt to
maximize $Q_{U^{\prime}}$.
## III Incorporating ICs in an Uncertain Relation
In this section, we describe our approach to generating the approximate sub-
relation $U^{\prime}$ given an uncertain relation $U$ and a set of constraints
$C$ that hold over $U$. Our approach starts with the original relation $U$,
selects a constraint $C_{i}\in C$, and attempts to resolve $C_{i}$ by dropping
(a minimal number) of attribute values from tuples in $U$ such that the
resulting sub-relation does not violate $C_{i}$. The process of resolving
constraints (or ”fixing” the relation $U$) is iteratively carried out until
the algorithm deems that the benefit of further removing inconsistency no
longer outweigh the loss of the consistent worlds that results as a by-product
of ”fixing” the uncertain relation. Before we discuss the details of the
algorithm, we first need to specify the nature of integrity constraints (IC)
that we consider in the paper. The approach we use to fix the uncertain
relation depends upon the nature of the integrity constraint.
We classify ICs into the following three different categories:
(i) Attribute level ICs: Constraints that depend on the values of a specific
attribute in a tuple, and not on other attributes in the same tuple or other
tuples. An example can be CHECK degreelevel(degree) that states, through a
user defined function (UDF), that the value for the degree must be at least a
4-year college degree. We will assume that attribute level ICs can be checked
efficiently (in polynomial time).
(ii) Tuple level ICs: Constraints that are dependent on the values of two or
more attributes within a specific tuple, and not on the values of attributes
of different tuples. As an example: CHECK compatible(job-title,degree) may
represent a tuple level IC that enforces, also through a UDF, some
compatibility between a person’s job title and his degree (e.g., that a
”manager” must have at least an ”MBA” degree, etc.). We will assume that each
instance of a tuple-level IC can be checked for constraint violation
efficiently (in polynomial time.) In addition, we will assume that the arity
of the constraint, i.e. number of attributes associated with the constraint is
small enough such that enumerating all tuple instances that could be potential
constraint violations is tractable.
(iii) Relation level ICs: Constraints that exist across different attributes
from different tuples. For instance a constraint:
CREATE ASSERTION no-multiple-divisions
CHECK (SELECT COUNT division FROM employees GROUP BY (name, job-title) == 1)
states that the same person cannot have the same job-title in two different
divisions. This constraint is essentially the FD (name, job-title)
$\rightarrow$ (division). Note that "Check" constraints at the attribute level
(or at the tuple level) that depend upon other tuples will also be classified
as relation level constraints.
The set of constraints, $C$, is a union of attribute level, tuple level, and
relational level constraints, represented as $C_{a}$, $C_{t}$, and $C_{r}$
respectively i.e., $C=Ca\cup Ct\cup Cr$.
We next discuss our strategies to resolve attribute, tuple, and relation level
constraints independently. After describing our strategies to resolve single
constraints, we will describe our algorithm to resolve the set of constraints
$C$. In the remainder of the section, we will use the example uncertain
relation with constraints in Figure 4 as an example for illustration.
Relation: U
name job-title division deg jim instructor (0.7) training BA (0.2) manager
(0.3) MBA (0.8) jim manager marketing MBA jill (0.5) consultant innovation AAB
(0.4) jim (0.5) PhD (0.6)
Constraints: C
Attribute level ICs (Ca)
1\. CHECK degreelevel(deg)
All employees have at least a 4 year college degree.
Tuple level ICs (Ct)
1\. CHECK compatible(division,deg)
All ”training” division employees have at least an ”MBA” degree.
Relation level ICs (Cr)
1\. CHECK (name, job-title) $\rightarrow$ division
An employee does not hold the same title in 2 different divisions
Figure 4: Uncertain Relation with Constraints
jim | instructor (0.7) | training | BA (0.2)
---|---|---|---
| manager (0.3) | | MBA (0.8)
jim | manager | marketing | MBA
jill (0.5) | consultant | innovation | PhD (0.6)
jim (0.5) | | |
(a) $U_{1}$: Factored attribute levels ICs
jim instructor (0.7) training MBA (0.8) manager (0.3) jim manager marketing
MBA jill (0.5) consultant innovation PhD (0.6) jim (0.5)
(b) $U_{2}$: Factored tuple level ICs
jim instructor (0.7) training MBA (0.8) jim manager marketing MBA jill (0.5)
consultant innovation PhD (0.6) jim (0.5)
(c) $U^{\prime}$: Factored relation level ICs, final approximation
TABLE I: Generating Approximations
Resolving attribute level ICs is actually tivial as in any attribute world we
simply eliminate any attribute instance that is not consistent with an IC in
$C_{a}$. This is illustrated in table I (a) where the AAB value in tuple 3 is
dropped. We note that the sub-relation that results from resolving $C_{a}$
removes only the inconsistent worlds but does not remove any consistent ones.
### III-A Resolving A Tuple Level IC
To resolve a tuple level constraint $C_{tup}\in C$, we can consider each tuple
$T$ of the uncertain relation independently. Given an uncertain tuple $T$ and
a specific tuple level constraint $C_{tup}$, we would ideally like to arrive
at a sub-tuple $T^{\prime}$ (of $T$) that is equivalent to $T+C_{tup}$, i.e.
it satisfies $C_{tup}$, while allowing the same set of possible consistent
instances as $T$. Unlike the case of attribute level IC, dropping attribute
values from tuples in $U$ that violate $C_{tup}$ might result in one or more
consistent instances to be eliminated as well. As a result, the resulting sub-
relation $U^{\prime}$ might not exactly represent the set of consistent
possible instances in $U+C_{tup}$. Fig 5 illustrates such an example with a
constraint that all training division employees have at least an ”MBA” level
degree. Dropping any attribute value from the tuple results in a loss of a
consistent instance. For instance, removing ”BA” from the attribute world of
degree attribute results in a sub-tuple that satisfies the considered tuple
level IC, but it eliminates the consistent possible instance in which ”jim”
works in ”marketing” division with a ”BA” degree. Furthermore, the problem of
identifying the sub-relation that optimally approximates (in terms of quality)
the set of possible worlds of the uncertain relation $U$ consistent w.r.t. a
single tuple level constraint $C_{tup}$ remains NP-hard. We state the
following:
Statement 2: Determining an optimal approximation T$\prime$ of an uncertain
tuple T is NP-Hard
Proof: This follows by a reduction from the functional dependency (FD)
consistency problem. The proof is as follows:
1) Consider any given instance of an FD consistency problem (U,F) where U is a
U-relation and F is an FD over U.
2) Create a new tuple, T, as follows. For every tuple $t_{i}\in$ U create a
new attribute $A_{t_{i}}$ in T. For each tuple instance $t_{i}^{k}$ in every
tuple $t_{i}$ in U, create a corresponding attribute value instance in
$A_{t_{i}}$. Finally, provide a uniform probability distribution in all
attribute worlds in T.
3) For every instance of a pair of tuple instances $t_{i}^{m}$ and $t_{i}^{n}$
(i$\neq$j) that violate F, create an instance of a constraint violation
between the corresponding attribute value instances in T.
4) T is an uncertain tuple that possibly also has some constraint violations
across attribute values. Note that the reduction from the FD consistency
problem to this uncertain tuple T is done in time polynomial in the size of
the original problem.
5) Generate an optimal approximation T$\prime$ of T. If there is any tuple
instance that is consistent in T then at least one such consistent instance
must appear in T$\prime$. This is because all consistent tuple instances in T
have the same probability and all inconsistent instances have a probability of
0. Also if T$\prime$ is empty then this implies that there are no consistent
tuple instances whatsoever in T.
6) The tuple instances in T directly correspond to relation instances in the
original FD consistency problem as there is a 1-1 mapping from the attribute
values instances in attributes in T to tuple instances in tuples in U. Any
consistent tuple instance in T directly corresponds to a consistent relation
instance in U.
7) The original problem of determining a consistent relation instance in U (or
determining that none exists) is however NP-Hard. This implies that the
problem of optimal tuple approximation, that this was reduced, to is also NP-
Hard.
Given the intractability of identifying the optimal sub-relation, we focus on
developing a heuristic approach to find a ”good” approximation that preserves
as much of consistent mass as possible (see Sec. 2) which we describe next.
name | job-title | division | degree
---|---|---|---
jim | instructor | training (0.6) | BA (0.7)
| | marketing (0.4) | MBA (0.3)
Figure 5: Sample U-tuple, for which no proper sub-tuple exists Figure 6: Graph
Representation of Uncertain Tuple
Algorithm: APPLY_TUPLE_IC
---
Input: Uncertain relation $U_{0}$, Tuple level IC $C_{tup}$
Output: Sub-relation $U_{1}$
1: APPLY_TUPLE_IC_SR ($U_{0},C_{tup}$)
2: $t_{new}$ $\leftarrow$ $\o$
3: for | (each tuple t in $U_{0}$)
4: | $t_{new}$ $\leftarrow$ $t_{new}$ $\cup$ APPLY_TUPLE_IC_SR_TUPLE(t, $C_{tup}$)
5: $U_{1}$ $\leftarrow$ form_relation($t_{new}$)
6: return $U_{1}$
1: APPLY_TUPLE_IC_SR_TUPLE (T, $C_{tup}$)
2: | ATTRIBUTE_MARGINALS(T,S)
3: | G $\leftarrow$ graph_representation(T, $C_{tup}$)
4: | I $\leftarrow$ independent_nodes(G)
5: | N $\leftarrow$ $G-I$
6: | Nb $\leftarrow$ best_candidate_to_delete(N)
7: | G $\leftarrow$ delete(G,Nb)
8: T $\leftarrow$ tuple_representation(G,$\gamma$)
9: return(T)
form_relation: Construct a new relation.
graph_representation: Convert uncertain tuple to graph.
independent_nodes: Find nodes without any edge.
best_candidate_to_delete: Find proper node to remove.
tuple_representation: Convert graph to tuple.
For a given tuple $T$ of $U$ and a tuple level constraint $C_{tup}$, we start
with constructing a graph representation of $T$ in which nodes correspond to
attribute value instances in each attribute, and edges and hyper-edges
represent sets of attribute value instances (across attributes) that violate
the tuple level constraints. The graph representation of the first tuple in
Fig 4 is shown in Fig 6. We now delete nodes in this graph till all the
(hyper) edges disappear, the resulting graph represents the attribute value
instances that are consistent w.r.t. $C_{tup}$ and can hence be retained in
the approximation. For choosing nodes to drop, recall that we are interested
in approximations with high quality i.e., where any consistent mass dropped is
minimal. The consistent mass associated with any individual node (attribute
value) is given by its marginal probability in the tuple. The marginal
probability of an attribute value instance $a_{ij}^{k}$, denoted as
$p_{MARG}(a_{ij}^{k})$ is defined as the sum of the probabilities of all the
tuple instances implied by the uncertain tuple that include that attribute
world instance.
$p_{MARG}(a_{ij}^{k})=\sum_{all\,instances\,t\in T\,\wedge\,a_{ij}^{k}\in
t}p(t)$ (3)
We adorn the graph nodes with their associated marginal probabilities. We then
choose the nodes to drop in a greedy fashion biasing towards dropping nodes
with low marginal probabilities, till all (hyper) edges have been eliminated.
As an example, consider again the graph in figure 6, and its corresponding
sub-tuple. Having just one tuple level IC, and a pair of violating possible
attribute values, we can eliminate the only existing inconsistency, shown as
the dashed edge in the graph, by dropping one of its corresponding nodes. In
this case, we decide to drop $a_{4}^{2}$, the ”BA” value, according to its
marginal probability, which is 0 comparing to the marginal probability of
$a_{3}^{1}$, ”training” value, which is 0.8.The complete algorithm is
described in Algorithm APPLY _TUPLE_IC.
Note that our approach requires that the marginal probability value for each
attribute value instance in the tuple be known. Unfortunately, computing the
marginal probabilities of attribute values instances in an uncertain tuple can
be shown to be NP-Hard. Instead, we can estimate such marginals using
statistical sampling. We employ naive-MC (Monte-Carlo) sampling. The procedure
for estimating marginal probabilities of attribute value instances in a tuple,
based on sampling randomly generated tuple instances, is described in
algorithm ATTRIBUTE_MARGINALS.
Algorithm: ATTRIBUTE_MARGINALS
---
Input: Uncertain tuple T, Number of samples S
1: ATTRIBUTE_MARGINALS(T,S) {
2: for | (all attribute value instances $a_{ij}^{k}$ in all attributes in T)
3: | $p_{MARG}(a_{ij}^{k}$) $\leftarrow$ 0
5: for | (i = 1 through S)
6: | $t_{samp}$ $\leftarrow$ random_sample(T)
7: | for | (all attribute value instances $a_{ij}^{k}$ $\in$ $t_{samp}$)
8: | | $p_{MARG}(a_{ij}^{k}$) $\leftarrow$ $p_{MARG}(a_{ij}^{k}$) + $p(t_{samp})$
10: for | (all attribute value instances $a_{ij}^{k}$ $\in$ T)
11: | $p_{MARG}(a_{ij}^{k})$ $\leftarrow$ $p_{MARG}(a_{ij}^{k}$)/S
12: return($\\{p_{MARG}(a_{ij}^{k}\\}$ )
random_sample(T): random tuple instance.
Statement 3: The derivation of marginal probabilities of attribute value
instances in an uncertain tuple, or of tuple instances in an uncertain
relation with constraints, is NP-Hard.
Proof: Given an instance (U,F) of the FD consistency problem we determine the
marginal probabilities of the tuple instances in each tuple in U. A consistent
relation instance in U exists iff the marginal probability of at least one of
the tuple instances (in any tuple in U) is $>$0\. The original FD consistency
problem is however NP-Hard. This implies that determining the marginal
probabilities of tuple instances in tuples in a U-relation is also NP-Hard
For determining the complexity of determining marginal probabilities of
attribute values in an uncertain tuple we make a reduction from the FD
consistency problem. Given an instance of the FD consistency problem (U,F) we
create an uncertain tuple, T, exactly as in the proof for Statement 2 above. A
consistent instance in the FD consistency problem is present iff the marginal
probability of (at least) one of the attribute value instances in T is $>$ 0\.
The original FD consistency problem is however NP-Hard. This implies that
determining the marginal probabilities of attribute value instance in
attributes in an uncertain tuple is also NP-Hard.
### III-B Resolving A Relation Level IC
For a relation level IC the instances of violations of that IC could be
exponential in the number of tuples. The approach we used for resolving tuple
level ICs - which involves exhaustively enumerating and imprinting all
instances of violations, is thus not practical for relation level ICs. Also in
the context of a relation level IC, we will use the term ”tuple instance” to
refer to the projection of the tuple instances onto the attributes that are
part of the IC. Resolving a specific relational level IC, $C_{rel}$, in an
uncertain relation $U$, comprises the following two steps:
a) Within $U$ we identify sets of tuple instances where each set can
potentially violate $C_{rel}$. For instance for a functional dependency IC,
$A\rightarrow B$, where $A$ and $B$ are two sets of attributes according to
the schema of $U$, any set of tuple instances which agree on the value of $A$,
form a set of tuple instances that could potentially violate the FD. A
possible relation of $U$, where tuple instances are drawn from such a set,
could be inconsistent with $C_{rel}$. Given any $C_{rel}$, all such sets of
tuple instances can be determined exhaustively (the number of such sets is
proportional to the number of distinct attribute values of $A$). We refer to
any such a set as a ”NEED-FIX” class for that $C_{rel}$. As an example, a
NEED-FIX class for the FD constraint over the uncertain relation in Fig 7 (a)
is illustrated in Fig 7 (b). The tuple instances are denoted by first
specifying the tuple number in the uncertain relation and then the tuple
instance number within each tuple in ().
b) We eliminate the inconsistencies in any NEED-FIX class considering each
class individually. This is achieved by dropping tuple instances in the class
till consistency is achieved. We refer to this as ”fixing” a class. For
instance for a NEED-FIX class corresponding to a functional dependency
$A\rightarrow B$, we would drop tuple instances until all the tuple instances
in that class agree on the value of the attribute(s) in $B$. Note that in
general there may be many different combinations of tuple instances that can
be dropped that will achieve consistency. For instance, the NEED-FIX class in
Fig 7 (b) can be fixed by dropping either the 1st tuple instance, or the 2nd
and 3rd tuple instances in the class.
CONSTRAINT TYPE | Generating NEED-FIX class(NF) | Fixing NF | Complexity*
---|---|---|---
Type: Functional Dependency (FD) | 1) For each tuple instance t in each tuple T in U. | 1) Group the tuple instances by the value of B | $O(N_{t})$
Format: A $\rightarrow$ B where A,B are subsets of columns in U | 2) Initialize a new NEED-FIX class, NF, with the single member t.
3) For any tuple $T\prime$ in U that contains
a tuple instance $t\prime$ such that $t\prime$.A=t.A,
add $t\prime$ to NF.
4) Add NF to the pool of NEED-FIX
classes. | 2) Select the value for B for which the sum of the marginals (of the tuple instances) in that group is the highest.
3) Drop all tuple instances with
values for B other than the above
selected value. |
Type: Inclusion Dependency(IND) | 1) Initialize a new NEED-FIX class,NF, to NULL. | 1) Drop all tuple instances in NF. | $O(N_{t})$
Format: U.A $\in$ E.B where E is a fixed relation and A, B are subsets of columns in U and E respectively | 2) For any tuple instance t in tuple T, if t.A $\neg\in$ E.B then add T to NF.
3) Add NF to the pool of NEED-FIX classes. | |
Type: Aggregation
Format: GROUP BY A COUNT $<$ G ;where A is a subset of columns in U and G is an integer. | 1) For each tuple instance t in each tuple T in U.
2) Initialize a new NEED-FIX class,NF, with the single member t.
3) For any tuple $T\prime$ in U that contains a tuple instance $t\prime$ such
that $t\prime$.A=t.A, add $t\prime$ to NF.
4) Add NF to the pool of NEED-FIX classes. | 1) Let Nnf be the number if tuple instances in NF.
2) If Nnf $<$ G then we are done.
3) Else Nnf $-$ G tuple instances have to be dropped. Drop those Nnf $-$ G tuple instances from NF for which the sum of the marginal values is minimum. | $O(^{N_{T}}C_{N_{T}\\-G})$
Type: Aggregation
Format: GROUP BY A EXP(B) $\theta$ val; where EXP is one of {AVERAGE, SUM, COUNT}, A is a subset of columns in U, and B is a (numeric) column in U, and $\theta$ is one of $\\{=,\leq,<\\}$ | Same as above. | 1) Exhaustively search all combinations of tuple instances that can be dropped to make NF consistent wrt this constraint.
2) Determine the combination with the minimum total marginal value and drop the tuple instances in that combination. | $O((2^{P})^{N_{T}})$
Type: SET Constraint
Format: Q $\theta$ E.B ; where Q = (SELECT A FROM U WHERE CND), CND is a query condition, and $\theta$ is one of $\\{=,\leq,<\\}$ | 1) Initialize a new NEED-FIX class,NF, to NULL.
2) For any tuple instance t in tuple T in the result of Q, if t.A $\in$ E.B
then add T to NF.
3) Add NF to the pool of NEED-FIX classes. | 1) Drop all tuple instances in NF. | $O(N_{t})$
TABLE II: Addressing Relation Level Constraints ($N_{t}$: Total number of
tuple instances in NF; $N_{T}$: Total number of tuples represented in NF; $P$:
Maximum number of tuple instances in any tuple in NF.)
name | job-title | division
---|---|---
jim | instructor (0.5) | marketing
| consultant (0.5) |
jim | instructor (0.3) | training
| manager (0.7) |
jim | instructor | training
ICs:
1\. CHECK (name, job-title) $\rightarrow$ division
2\. CHECK GROUP BY (name, job-title) COUNT * $<$ 2
(a) Example uncertain relation with constraints
| 1(1) jim instructor marketing
---
2(1) jim instructor training
3(1) jim instructor training
| 2(1) jim instructor training
---
3(1) jim instructor training
(b) NEED-FIX class: FD IC | (c) NEED-FIX class: Aggregation IC
Figure 7: An Example
The process of determining a NEED-FIX class, fixing it, and also the
computational complexity of the fix operation are dependent on the type of the
relational constraint that is being addressed. We described the generation and
fixing of NEED-FIX classes for FD type ICs above. For aggregation constraints,
such as the 2nd IC in the example in Fig 7 (a), NEED-FIX classes are
determined by grouping together tuple instances that agree on the attributes
that we must group by according to the aggregation constraint. One such NEED-
FIX class is shown in Fig 7 (c) where we have grouped together tuple instances
by (name, job-title). The fix is a process of eliminating tuple instances such
that the aggregation constraint condition is satisfied, in this example
dropping either of the tuple instances in the NEED-FIX class will ensure this.
Table II presents the specific kinds of relation ICs addressed and the
associated complexity. The procedures for generating and fixing NEED-FIX
classes for IND and SET constraints are straightforward and we do not present
them here for lack of space. As we have seen there can be multiple sets of
tuple instances that can be dropped to fix a NEED-FIX class. The choice of an
optimal set of tuple instances to drop is made based on the marginal
probabilities of each tuple instance. Formally, the marginal probability,
$p_{MARG}(t)$, of a tuple instance, t in an uncertain relation U is defined
as:
$p_{MARG}(t)$ = $\Sigma_{all\,instances\,u\in U}$ $p(u)$ ; $t\in u$. Like
attribute marginals, the derivation of tuple instance marginals in a relation
is also NP- Hard[12]. We employ naive-MC sampling for estimating these
marginals in a manner analogous to the attribute marginals estimation, and
here we sample randomly generated relation instances.
For any NEED-FIX class we can determine the combination of tuple instances
with lowest (total) marginal probability, that if dropped will eliminate the
inconsistencies in that class. The complexity of resolving a NEED-FIX class is
polynomial in the size of the NEED-FIX class for (the permitted) FDs, INDs and
SET constraints and is exponential (in the size of the NEED-FIX class) for the
aggregation constraints. For aggregation constraints we use a simple hill-
climbing procedure to find a set of tuple instances to drop that will remove
the inconsistency in the NEED-FIX class and also have a low (total) marginal
probability.
Algorithm: APPLY_RELATION_IC
---
Input: Uncertain relation $U_{0}$, Relation level IC $C_{rel}$
Output: Sub-relation $U_{1}$
1: APPLY_RELATION_IC($U_{0},C_{rel}$)
2: $NF$ $\leftarrow$ $\o$
3: $\gamma$ $\leftarrow$ estimate_gamma($U_{0},C_{rel}$)
4: for | (each tuple instance t in each tuple T in $U_{0}$) {
5: | $NF_{t}$ $\leftarrow$ generate_need_fix_class(t,$U_{0}$, $C_{rel}$)
6: | NF $\leftarrow$ NF $\cup$ $NF_{t}$
7: }
8: $NF$ $\leftarrow$ FIX(NF)
9: $U_{1}$ $\leftarrow$ form_relation($NF$, $\gamma$)
10: return $U_{1}$
generate_need_fix_class: generates a new NEED-FIX class
given a tuple instance and a relation level IC.
fix: fix a particular NEED-FIX class
form_relation: form a new relation.
## IV Using a Multi-Row Representation
Revisiting the example in Fig 5 we realized that in order to achieve
consistency (by dropping some instances) some loss of consistent instances was
invariable. This is because of the model simplicity and we have been using
what is called a single-row model [1]. A representation model that permits
multiple rows for each tuple, known as a multi-row model, can overcome this
limitation as illustrated in Fig 8 where the approximation now exactly
captures the uncertain tuple in Fig 5.
jim | instructor (0.7) | training (0.6) | MBA (0.8) 1
---|---|---|---
| manager (0.3) | marketing (0.4) |
jim | instructor (0.7) | | MBA (0.8) 1
| manager (0.3) | marketing (0.4) | BA (0.2)
Figure 8: Uncertain Tuple Approximation Figure 9: Multi-Row Example
While in the above example the multi-row representation exactly captured the
uncertain tuple with a small number of rows (2), this is not the case in
general. We present the following:
Statement: The number of rows in a multi-row representation required to
exactly capture an uncertain tuple with constraints can be exponential (in the
size of the largest attribute world in the tuple) in the worst case.
Proof: Given a tuple and a set of constraint violations (let us consider only
binary constraints violations across pairs of attribute values wlog) assume
that there is a multi-row representation with $M$ rows. Consider any row, $r$,
where we have at least one attribute that has at least 2 attribute values. We
insert a new violation between any of these (multiple) attribute values and
any attribute value in any other attribute in the tuple. Now $r$ must
necessarily be split into at least 2 rows to exactly capture the consistent
tuple instances. We can continue inserting violations in rows in this manner
with an upper bound of $KC_{2}A^{2}$ violations that we will insert where K is
the number of attributes. The number of rows that we will form in the multi-
row model can however be as much as $O(A_{K})$ i.e., exponential in the
(maximum) size of the attribute worlds in the tuple.
An approximation that takes exponential space is not tractable to reason with
and we are interested in multi-row approximations where the number of rows is
bounded by a constant or at least a factor that is polynomial in the size of
the original uncertain tuple. With such a restriction we can at best achieve
an optimal approximation as opposed to an exact one in the general case. Any
multi-row approximation is defined by 2 kinds of parameters, one is the number
of rows in the representation and the other is the assignment of probabilistic
values to attribute value instances within each attribute within each row. The
complexity of deriving an optimal approximation is an issue however, we
present the following:
Statement: For a multi-row representation with a bounded number of rows,
determining multi-row model parameters that result in an optimal approximation
of a tuple is NP-Hard.
Proof: This too follows from a reduction from the FD consistency problem, and
the proof is analogous to as for the single row model.
Algorithm: APPLY_TUPLE_IC_MR
---
Input: Uncertain relation $U_{0}$, Tuple level IC $C_{tup}$
Output: Sub-relation $U_{1}$
1: APPLY_TUPLE_IC_MR ($U_{0},C_{tup}$)
2: $t_{new}$ $\leftarrow$ $\o$
3: for | (each tuple t in $U_{0}$)
4: | $t_{new}$ $\leftarrow$ $t_{new}$ $\cup$ APPLY_TUPLE_IC_MR_TUPLE(t,$C_{tup}$)
5: $U_{1}$ $\leftarrow$ form_relation($t_{new}$)
6: return $U_{1}$
1: APPLY_TUPLE_IC_MR_TUPLE(t,$C_{tup}$,M)
2: $V\leftarrow$ determine_violation_sets(T)
3: m=0, F=0 4: while | (m $<$M and inconsistent(T))
6: | $TopV\leftarrow top\\_violation\\_set(V)$
7: | $T\leftarrow split(T,TopV)$
8: end while
9: for | (each row R $\in$ T)
10: | $R\leftarrow$ APPLY_TUPLE_IC_SR(R)
11: end for
12: return T
What we employ is a heuristic approach to generating a multi-row approximation
for a given uncertain tuple. We describe our approach using the example of
binary tuple level constraints although the basic approach is valid for
general (k-ary) tuple level constraints. Continuing with the graph
representation of a tuple as described earlier, we recall that our aim was to
eliminate (hyper) edges in the graph by dropping nodes. In the multi-row model
our aim is to instead split the graph into multiple sub-graphs such that the
(hyper) edges are eliminated - this is illustrated in Fig 9 where the original
tuple graph is split into two sub-graphs neither of wich contains the edge.The
idea is to split a tuple graph recursively in this manner till (i) No sub-
graph contains any edges, or (ii) The number of sub-graphs exceeds the number
of available rows per tuple - whichever is earlier. Each sub-graph then
corresponds to a row in the multi-row representation of the tuple. Consider an
uncertain tuple T and three of its attributes $A_{i}$, $A_{j}$, and Am with
attribute value instances as shown in fig 10. Focusing on attributes $A_{i}$
and $A_{j}$, certain attribute value instances in $A_{i}$ may be inconsistent
with certain instances in $A_{j}$, based on 1 or more (binary) tuple level
constraints. For an attribute value instance $a_{i_{k}}$ $\in$ $A_{i}$, define
$cons(a_{i_{k}},A_{j})$ as the set of those attribute value instances in
$A_{j}$ that are consistent with $a_{i_{k}}$ i.e., wrt the tuple level
constraints. Now consider a particular row in a multi-row representation for
T. A row is said to be consistent wrt attributes $A_{i}$ and $A_{j}$ iff all
attribute value instances for attribute $A_{i}$ in that row are consistent
with all attribute value instances for $A_{j}$ in that row. A row is said to
be completely consistent if it is consistent wrt all pairs of attributes in
the uncertain tuple. A row is inconsistent if it is not consistent wrt at
least one pair of attributes $A_{i}$ and $A_{j}$. It follows that any row will
be inconsistent iff there are 2 attributes $A_{i}$ and $A_{j}$ such that there
are two instances in $A_{i}$, $a_{i_{k1}}$ and $a_{i_{k2}}$ and
$cons(a_{i_{k1}},Aj)\neq cons(a_{i_{k2}},Aj)$. We denote any such pair of
attribute value instances and pair of attributes where this an inconsistency
as a violation set $v=<a_{i_{k1}},a_{i_{k2}},Aj>$.
Figure 10: Split to Multi-Row
To eliminate the inconsistencies across $A_{i}$ and $A_{j}$, the strategy we
follow, in the single-row model, is to eliminate certain attribute value
instances from attributes $A_{i}$ and/or $A_{j}$ till consistency is achieved.
This comes at a cost of possibly eliminating certain consistent instances as
well and we provided a mechanism to estimate this loss using marginal values
in the previous section. We denote as $loss(v)$ the estimate of the consistent
mass loss associated with making violation set $v$ consistent by eliminating
attribute value instances. With the luxury of multiple row, we can instead
split a row with a violation set into 2 rows as shown in fig 10. The resulting
2 rows are necessarily consistent wrt $A_{i}$ and $A_{j}$. Denote this
operation as that of splitting on a violation set. Note that the
inconsistencies are eliminated but no consistent mass is lost in the process.
We can perform such splitting on all violation sets for the uncertain tuple,
the number of violation sets is polynomial in the (maximum) number of
attribute value instances in each attribute and the number of attributes.
While this will ensure that we end up in a multi-row representation that
exactly captures all the consistent tuple instances in the original uncertain
tuple, the number of rows created can be exponential. The number of rows
however is bounded. Prioritizing and considering violation sets, based on
decreasing order of $loss(v)$, we split them till either all inconsistencies
are eliminated or we reach the limit of the number of rows, whichever is
earlier. Should the limit on number of rows be reached first there will be
rows that do have inconsistencies (still) present. We employ the single-row
approximation on each of these rows. The heuristic rationale is that the
additional row created due to splitting is in a sense saving us the associated
loss value of consistent mass.
## V Integrity Constraint Selection
In the previous section we studied how individual ICs of different kinds can
be applied to remove inconsistency in an uncertain relation with constraints.
Strictly speaking, when we state we are resolving a tuple (relation) IC we
mean we are resolving that tuple (relation) IC in a particular tuple (NEED-
FIX) class that is inconsistent with that IC. This is what the term ”resolving
an IC” will imply now on. In this section we describe how a set of ICs can be
applied so as to achieve an approximation $U^{\prime}$ of good quality. Note
that if our goal was to simple eliminate all the inconsistency we could apply
all the ICs and achieve this, however we realize that a significant amount of
consistent instances can be lost this way. The challenge is to find an optimal
subset of ICs to apply such that the quality of the approximation achieved is
maximized.
### V-A Utility of Each IC
For each individual IC we can determine whether resolving it will cause the
overall quality to increase or decrease. Assume that for any IC we have an
estimate of the inconsistent mass lost, $IC_{L}$, and the consistent mass
lost, $CM_{L}$, as a result of applying that IC. We define the utility of an
IC, $UT$, as $UT=IC_{L}-CM_{L}/Pc$, where $Pc$ is the total consistent mass in
the uncertain relation. The reader can verify, given the quality measure
definition in Section 2, that the overall quality will necessarily increase
after resolving that IC if its utility $UT$ is $>0$.
We need to be able to determine the utility for any IC. Recall that a tuple
inconsistent with a tuple level IC can be resolved by dropping a single
attribute value in some attribute (involved in the IC violation). $IC_{L}$ in
this case can be determined by computing the probability of the tuple
instances in the tuple that are indeed inconsistent w.r.t. that IC. $CM_{L}$
on the other hand is nothing but the marginal probability of the attribute
value instance that we will drop. Determining $UT$ for a relation level IC is
relatively more complicated. Recall that resolving a tuple IC in each NEED-FIX
class is a process of eliminating attribute values in possibly multiple
tuples. $IC_{L}$ is determined by statistical sampling within a NEED-FIX class
i.e., by randomly generating relation possibilities from the NEED-FIX class
and estimating what is inconsistent . Now to fix the class if the attribute
values to be dropped (across different tuples) are $av_{1},\cdots,av_{n}$ then
$P(av_{1}\cup\cdots\cup av_{n})$ is a measure of the consistent mass lost by
dropping these attributes. This is essentially the estimation of a DNF formula
which can be also be done using Monte-Carlo sampling and applying the Luby-
Karp-Madras estimation algorithm [13].
Algorithm: Greedy_IC_Resolution
---
Input: Uncertain relation $U$, Set of ICs $C$, Threshold $B$
Output: Sub-relation $U^{\prime}$
1: $C_{t}$ $\leftarrow$ $C$
2: $U_{t}$ $\leftarrow$ $U$
3: $c_{m}$ $\leftarrow$ $null$
4: initialize_utilities($C_{t}$)
5: while | ($C_{r}(U_{t})$ $>$ $B$ and $C_{t}$ $\neq$ $\o$)
6: do
7: | UPDATE_UTILITIES($U_{t}$,$C_{t}$)
8: | $c_{m}$ $\leftarrow$ select_best_IC($C_{t}$)
9: | $U_{t}$ $\leftarrow$ resolve($U_{t}$, $c_{m}$)
10: | $C_{t}$ $\leftarrow$ $C_{t}$ \- $c_{m}$
11: end
12: return $U_{t}$
1: UPDATE_UTILITIES($U_{t}$,$C_{t}$)
2: for | (each IC $c_{i}$ in $C_{t}$)
3: | $benefit(c_{i})$ $\leftarrow$ calculate_benefit($c_{i}$, $U_{t}$)
4: | $cost(c_{i})$ $\leftarrow$ calculate_cost($c_{i}$, $U_{t}$)
5: | $utility(c_{i})$ $\leftarrow$ benefit($c_{i}$) - cost($c_{i}$)
initialize_utilities: Define a utility value for each IC
initialized with unknown
select_best_IC: Select the IC with the maximum utility
resolve: Resolve given IC in the sub-relation according
to its type
calculate_benefit: Calculate benefit of given IC, if
resolved in the sub-relation
calculate_cost: Calculate cost of given IC, if resolved
in the sub-relation
### V-B IC Selection
Based on the utility, we need to determine an optimal set of ICs to choose to
arrive at a maximum quality approximation. The complexity in this problem is
caused by the fact that there can be shared dependencies amongst the
resolution for certain ICs, specifically this happens if some of the attribute
values to be dropped are common across multiple ICs. The utility of applying a
set of multiple ICs thus cannot be determined from the utilities of the
individual ICs alone. The problem of determining a subset of ICs that
maximizes the resulting approximation quality, where costs and benefits may be
shared across the ICs can in fact be restated as the Budgeted Maximum Coverage
(BMC) problem [14], which unfortunately is NP-Hard. We thus provide a
heuristic algorithm that attempts to find a subset of ICs to apply such that
we achieve an approximation of high (although not necessarily the highest)
quality.
Our approach is to first consider all tuple level ICs and associated tuples
and resolve them (or not). We then move on to considering relation ICs and
associated NEED-FIX classes. Within each of the two categories of ICs we
consider and resolve an IC and an associated tuple or NEED-FIX class in a
greedy fashion. The algorithm selects ICs (and tuples or NEED-FIX classes) in
descending order of the associated utility. After each iteration, the
utilities of each of the ICs (and tuples or NEED-FIX classes) are
recalibrated. This is to factor in the attribute value instances that have
already been dropped as a result of the ICs that have so far been applied. The
algorithm applies ICs sequentially in this manner, recomputes utilities at
each iteration, and terminates when we have no more ICs with an associated
utility that is $>0$. Greedy_IC_Resolution describes this algorithm.
Estimation of Key Quantities In the above approximation and IC selection
algorithms we require the value $Pc$ \- total consistent mass in an uncertain
relation, $Cr$ \- consistent mass retained and $Ir$ \- inconsistent mass
retained for any approximation $U^{\prime}$. Determining any of these values
is also NP-Hard. We state:
Statement 4: The derivation of the total consistent mass $\delta$ (or $\gamma$
= 1/$\delta$) factor for a U-relation with constraints is NP-Hard
Proof: Given an instance of the FD consistency problem, consider the $\delta$
factor for the uncertain relation U in that problem. A consistent relation
instance in the original FD consistency problem is present iff $\delta$, the
total consistent mass is $>$0\. This implies that if $\delta$ (or $\gamma$)
can be determined in polynomial time, then the FD consistency problem can be
addressed in polynomial time as well. As the original FD consistency problem
is NP-Hard, it follows that determining the $\delta$ (or $\gamma$) factor for
a U-relation is also NP-Hard.
We thus resort to statistical sampling to estimate these values instead. A
naive approach however is not applicable in this case. Consider estimating
$Pc$ given an uncertain relation U. We can estimate the average consistent
mass per world instance, $Pc_{AVG}$, and then multiply this by the number of
world instances (which we can compute directly). We recall Hoeffding’s
inequality [15] from basic probability theory which states:
Hoeffding’s Inequality: Let $X_{1},X_{2},...,X_{n}$ be iid random variables,
while for all $i$ we have $a_{i}\leq X_{i}\leq b_{i}$, and also let
$S=\sum_{i=1}^{n}X_{i}$. Then we have:
$\begin{split}Pr(S-E[S]\geq nt)\leq
e^{(-2nt^{2})/\sum_{i=1}^{n}(b_{i}-a_{i})^{2}}\end{split}$ (4)
Or: $Pr(SAvgX-EAvgX\geq t)\leq e^{(-2nt^{2})/\sum_{i=1}^{n}(b_{i}-a_{i})^{2}}$
where $SAvgX=(\sum_{i=1}^{n}X_{i})/n$ is the sample average and EAvgX is the
expected average of the $X_{i}$s.
Treating the mass of a single world instance as a random variable $X_{i}$
above the sample average $SAvgX$ is an estimate of $Pc_{AVG}$. The value $t$
is a measure of the error. To estimate a small quantity such as $Pc_{AVG}$
which for an uncertain relation with 3 attributes, 2 attribute values per
attribute, and 100 tuples is itself of the order of $2^{-300}$, to within say
a 10% error requires $t$ to be accordingly small as well. Plugging such a
small value of t and using 0 and 1 as lower and upper bounds for $X_{i}$ we
see that we require an extremely large number of samples (order of $10^{30}$)
$n$ to achieve a probability of 0.9 that the estimation error is within 10%.
Instead of the average consistent mass we estimate the ratio, R, of the
consistent mass to the total mass. We define a block in U (or $U^{\prime}$) to
be any subset of relation instances from the possible world of U (or
$U^{\prime}$). For any such block $B_{i}$ define the quantity:
$R_{B_{i}}$ = Total consistent mass in $B_{i}$/Total mass in $B_{i}$
We choose block size for a block $B_{i}$ such that $R_{B_{i}}$ can be computed
by exhaustively enumerating through all instances in that block. The average
value of $R_{B_{i}}$, referred to as AvgR, is simply
$\sum_{i=1}^{N}R_{B_{i}}/N$. Unlike $Pc_{AVG}$ or $Cr_{AVG}$, AvgR is in
general not such an infinitesimally small quantity (for instance 0.3 could be
a value of AvgR). Thus the number of samples required to estimate AvgR to
within a reasonable accuracy is significantly smaller, for instance a
confidence of 0.9 of estimating this to within 10% error would require
sampling just a few hundred such blocks (Equation (4)). Now for both $U$ or an
approximation $U^{\prime}$ we can determine the total mass. For $U$ it is
simply 1, and for $U^{\prime}$ we can just compute it given $U^{\prime}$.
Having the total mass, and a reasonable estimate of AvgR we can derive
reasonably accurate estimates of $Pc$, or $Cr$ and $Ir$.
## VI Other Issues
While we have described the basic approach to resolving various kinds of ICs
we would like to discuss some additional issues related to the representation
model and the IC resolution algorithms.
Model Expressivity The or-set model we have used is simple and efficient but
also limited in expressivity. With more expressive models we will achieve
better quality approximations as this will mean having to drop less consistent
mass. We have begun exploring more expressive models with using a mutli-row
representation for tuples where a tuple can be represented as multiple rows of
or-sets of attributes. This is illustrated in Fig 11 where we note that we can
now exactly represent the consistent instances of tuple of Fig 5. Our
experimental results also show that we achieve better quality approximations
using multiple rows. Developing an approach for approximating an uncertain
relation with constraints to a more complex model such as that based on world
set decompositions and ”ws-sets” [5] is indeed an interesting direction for
future work.
jim | instructor | training (0.6) | MBA (0.3) 1
---|---|---|---
jim | instructor (0.3) | marketing (0.4) | BA (0.7) 1
| | | MBA (0.3)
Figure 11: Multi-Row Representation
Incrementality While in most applications we expect the complete uncertain
relation and set of ICs to be provided upfront, we can also envision scenarios
where the additions to the ICs, to the uncertain relation itself (i.e., new
tuples), or both, are provided incrementally. Rather than recompute
$U^{\prime}$ from scratch in such cases, we present an incremental approach.
Consider first the case where we have approximated an uncertain relation $U$
to $U^{\prime}$ given a set of ICs, and are now given a new set off additional
ICs $C_{Na}\cup C_{Nt}\cup C_{Nr}$, where $C_{Na}$, $C_{Nt}$, $C_{Nr}$ are the
additional attribute, tuple, and relation level ICs respectively. Our approach
is to start with $U^{\prime}$, apply the additional attribute ICs, and then
apply the additional tuple and relation ICs in a greedy fashion using the
algorithm GREEDY_IC_Resolution. The steps are as follows:
1\. Resolve $C_{Na}$ in $U^{\prime}$ resulting in $U_{1}\prime$
2\. Resolve $C_{Nt}\cup C_{Nr}$ in a greedy fashion on $U_{1}\prime$,
resulting in $U_{2}\prime$ which is now the new approximation of $U$.
The other case is when new tuples are provided for $U$. Let the set of new
tuples be $U_{N}$. As attribute and tuple level ICs are local to individual
tuples we need resolve the (existing) attribute and tuple level ICs only in
$U_{N}$. New violations of relation level ICs however can occur within the
tuples in $U_{N}$ or across the tuple in $U_{N}$ and $U^{\prime}$. We thus
proceed as follows.
1\. Resolve $C_{Na}$ in $U_{N}$ resulting in $U_{N1}$.
2\. Resolve $C_{Nt}$ in a greedy fashion on $U_{N1}$ resulting in $U_{N2}$.
3\. Resolve $C_{Nr}$ in a greedy fashion on $U\prime\cup U_{N2}$, resulting in
$U_{1}\prime$ which is now the new approximation of $U$.
Operations and ICs We achieve consistency with the ICs by essentially deleting
tuples (the deletion of an attribute value instance can be viewed as deleting
the tuple instances that get dropped as a consequence). In database repair one
can in general consider any of tuple deletion, addition, or modification to
repair a database to make it consistent with a set of given ICs. Our model is
to start with a complete uncertain relation i.e., one where we know of all the
possible relations that that uncertain relation implies. Starting with the
complete space of possibilities, the only meaningful operation to ensure
consistency given ICs, is to eliminate possible relations that are
inconsistent with any of the ICs. Coming back to a repair perspective, the
deletion of tuples is the only viable option in this framework. Another
related aspect is that we permit only particular subtypes of ICs within the
classes of relation ICs addressed as shown in Table II. This is to ensure that
a NEED-FIX class wrt these kinds of constraints can always be fixed using
tuple deletion.
## VII Experimental Evaluation
We present experimental evaluation results in two different experimental set-
ups. The first set-up is to assess the impact of incorporating constraints on
applications that use uncertain relations - specifically we choose the
application of information extraction, and assess an eventual improvement in
extraction accuracy with the use of constraints. This experiment is over a
real dataset of free text bios of researchers collected from their homepages
on the open Web. The second set-up is to evaluate the effectiveness of our
approach for approximating an uncertain relation with constraints and our
primary goal is to assess the quality of the approximations achieved. We
employ a synthetic dataset in this case. We describe below the two sets of
experiments and results.
### VII-A Application Impact
We consider the application of information extraction (IE), in particular the
task of ”slot-filling” or extracting relations from text. Our goal is to
assess any improvement in extraction accuracy that can be achieved with the
use of ICs. We store the extracted data provided by a given extractor in an
uncertain relation. We further define a set of ICs that are meaningful for the
particular relation that is being extracted. We then compare the accuracy of
retrieval done over the original uncertain relation, with the uncertain
relation refined incorporating the ICs.
#### VII-A1 Dataset
We have chosen the extraction task of extracting details of a researcher, such
as her job-title, employer, academic degrees and their associated dates and
alma-maters from free text bios on their Web pages. We have collected around
500 such Web pages of bios from the homepages of researchers in the field of
computer science. We identified 48 different items or slots to be extracted
from each Web page which correspond to the above mentioned data items such as
degrees, dates, employers etc.
#### VII-A2 Uncertain Relation Representation
We then trained and employed the TIES [16] information extraction system to
extract these slots from the collection of Web pages. The extracted data is
first represented in an uncertain relation. We consider each Web page as
providing the data for a single tuple in this relation. State-of-the-art
extraction systems such as TIES now provide a space of multiple possibilities
for an extracted value for a slot, typically having each possible value
associated with a confidence score. The extracted values provided by the
extractor for a particular slot are part of the space of attribute values for
the corresponding attribute and tuple in the uncertain relation. Also other
possible values for that slot, identified through a tokenization process are
included in the space of possible attribute values, realizing a complete space
of attribute value possibilities. As an example, for a particular page (tuple)
say the extractor returns the set of values [(2005 9.2) (2001 1.3)] for the
PhD Date attribute i.e., two possible values and associated confidence scores.
Also suppose that through tokenization we know that one other token, (2003),
which is also of the type date (which is the domain for the PhD Date
attribute), could also be a value for that slot. The attribute world formed
based on this information is $\\{$(2005 0.6), (2001, 0.1), (2003, 0.3) $\\}$.
(The details such as the determination of the probabilistic distribution in
each attribute world are important in general, but not to this discussion).
The set of attribute worlds corresponding to all slots for a page forms an
uncertain tuple and the set of all such tuples (corresponding to all pages)
corresponds to the extracted uncertain relation that we will call $U_{bios}$.
#### VII-A3 Integrity Constraints
Next, we author a set of integrity constraints that capture the semantics of
the bios relation. For instance we know that people receive their PhD degrees
only after their bachelors degrees (in the same major at least), or we know
that a person who received his PhD in 1978 is not likely to have a current job
title of an Assistant Professor. For this domain we were able to specify a
total of over 40 ICs spanning the attribute, tuple, and relation levels. A
subset of such constraints are: 1) All computer science degrees were awarded
after 1959. 2) A person receives his doctoral degree only after his bachelors
degree (same major). 3) A NULL value for a degree implies NULL values for the
associated alma-mater and degree date. 4) The PhD degree alma mater and
employer of a person are different. The first constraint above can be
expressed as an attribute level IC while the other 3 can be expressed as tuple
ICs over $U_{bios}$. Strictly speaking, some of the above constraints (such as
4) are ”soft” constraints in that they hold mostly but not necessarily always.
For our purpose we treat them as hard constraints.
#### VII-A4 Results
We evaluated the precision and recall of retrieval over several different
slots in $U_{bios}$. We compare the accuracy of retrieval over the original
extracted uncertain relation $U_{bios}$, with that over $U_{bios}$ augmented
with the domain ICs. We consider precision and recall on a per-slot basis,
where:
Precision for a slot s, PR(s), is defined as:
$PR(S)=\sum_{all\,tuples\,t}p(v)_{s,t}/N$ (5)
where v is the correct value for the slot s in tuple t, $p(v)_{s,t}$ is the
probability associated with value v for slot s in tuple t, and N is the number
of tuples returned.
Recall for a slot s, RE(s), is defined as:
$PR(S)=\sum_{all\,tuples\,retrieved\,r}p(v)_{s,r}/\sum_{all\,tuples\,t}p(v)_{s,t}$
(6)
where v is the value for slot s in tuple t.
Slot | $p_{i}$ $\mid$ $p_{c}$ | $r_{i}$ $\mid$ $r_{c}$ | $f_{i}$ $\mid$ $f_{c}$
---|---|---|---
Title | $0.95\mid 0.8$ | $0.78\mid 0.94$ | $0.85\mid 0.82$
Employer | $0.79\mid 0.82$ | $0.65\mid 0.69$ | $0.71\mid 0.75$
PhD Degree | $0.98\mid 0.98$ | $1\mid 1$ | $0.98\mid 0.98$
PhD School | $0.69\mid 0.76$ | $0.36\mid 0.58$ | $0.47\mid 0.66$
PhD Date | $0.69\mid 0.86$ | $0.46\mid 0.83$ | $0.55\mid 0.84$
Bach School | $0.93\mid 0.9$ | $0.3\mid 0.49$ | $0.45\mid 0.63$
Bach Date | $0.88\mid 1$ | $0.62\mid 0.96$ | $0.73\mid 0.98$
TABLE III: Extraction Accuracy with Constraints
Given $U_{bios}$ and the set of ICs specified over this relation we generate
an approximation of $U_{bios}$ plus the ICs, $U_{bios}\prime$ using our
approach. Table 3 provides the retrieval accuracy, in terms of precision,
recall, and f-measure, for a subset (for brevity) of the slots over both
$U_{bios}$ and $U_{bios}\prime$. Here $p_{I}$, $r_{I}$, and $f_{I}$ are
precision, recall and f-measure respectively over $U_{bios}$, and $p_{c}$,
$r_{c}$, and $f_{c}$ are the corresponding values over $U_{bios}\prime$ . We
observe that both precision and recall for many slots are significantly
improved in $U_{bios}\prime$ compared to $U_{bios}$, thus demonstrating the
effectiveness of employing ICs. Albeit in some cases we see a (minor) drop in
precision which is due to treating what should be soft constraints as hard.
Note that these are extraction accuracy improvements over the output of
extraction systems that are representative of the state of the art and also
have been provided extensive training data in the application domain. These
results demonstrate the utility of employing constraints in the context of an
actual application of information extraction where the use of constraints
significantly improves the retrieval quality. We also demonstrate (in Fig 12)
the increase in overall extraction accuracy (aggregated over all the slots in
the relation) as a function of the number of ICs incorporated.
Figure 12: Extraction Accuracy
### VII-B Assessing Effectiveness of Approximation Approach
Our aim is to arrive at a good quality approximation of an uncertain relation
with constraints. For more detailed analysis of scalability, sensitivity, and
robustness of our algorithm we performed empirical evaluation on synthetic
data. We evaluate the quality of approximation that we can achieve with our
greedy algorithm. We also compare our results with the alternative algorithm
of removing all inconsistent instances.
#### VII-B1 Synthetic Dataset Generation
We implemented an uncertain relation generator which lets us generate
uncertain relations under pre-specified settings for different parameters. The
key parameters are described in Fig 13 below. The generation parameters allow
us to control and configure various factors such as the size of the uncertain
relation, uncertainty in the uncertain relation, kinds and number of ICs, the
”dirtiness” of the relation i.e., the degree of inconsistency in the original
relation etc.
Param | Description
---|---
A | Number of attributes in relation.
$MAX$ | Maximum number of choices in one attribute
C | Total Number of ICs
D | Maximum arity of a (tuple) IC
R | Number of tuples
$\alpha$ | Degree of data dirtiness (% fields uncertain)
Figure 13: Synthetic Data Generator Parameters
The generation of an uncertain relation with constraints comprises of the
following basic steps: 1) Generate an initial (clean) uncertain relation
according to the relation size, schema size, and relation uncertainty degree
parameters. This includes the definition of a probability distribution over
the uncertain relation. 2)Generate specific ICs at attribute, tuple, and
relation levels based on the number of ICs parameter. 3) Inject instances of
violations for the attribute, tuple, and relation level ICs in randomly chosen
attributes, tuples, and sets of tuples (respectively) according to the degree
of dirtiness parameters.
#### VII-B2 Experimental Results
On a synthetic dataset of 1000 tuples with 25 ICs (of different types) Figure
14 demonstrates the consistent (Cr) and inconsistent mass (Ir) in the
approximation as a function of the number of ICs (iterations) applied. Figure
15 illustrates (for 2 cases of different initial consistency) the
approximation quality as a function of the IC iterations applied. We applied
the ICs in order that the greedy algorithm selects them, the greedy algorithm
terminates according to the utility based criterion whereas the brute force
algorithm of resolving all ICs runs on. These results are typical of the many
traces we conducted. We clearly see the superiority of our greedy IC selection
algorithm ($U^{\prime}$) which terminates when resolving ICs is no longer
beneficial, as opposed to the brute force approach ($U_{all}$) of resolving
all ICs that can cause the quality to significantly degenerate.
In Figure 16 we illustrate the sensitivity of approximation quality (shown
averaged over several traces) to (a) the initial consistent mass in the
relation, and (b) the degree of uncertainty in the original uncertain relation
- which is controlled by the $MAX$ parameter. We observe (a) that uncertain
relations of higher original consistency result in better quality
approximations, whereas (b) quality depends on other factors such as the
degree of inconsistency, constraint distribution etc., as opposed to relation
uncertainty defined in terms of the number of attribute value choices $MAX$.
(a) Consistent Mass Retained
(b) Inconsistent Mass Retained
Figure 14: Cr and Ir in each iteration
(a) Quality
(b) Quality
Figure 15: Quality after resolving each IC
(a)
(b)
Figure 16: Sensitivity Figure 17: Quality in Multi-row Model # Tuples | # ICs | Marginals (ms) | IC Resolution (ms)
---|---|---|---
100 | 5 | $<$ 1 | 703
1000 | 50 | $<$ 1 | 4922
10000 | 500 | 48 | 99167 ( 2 min)
50000 | 2500 | 1078 | 2591384( 53 min)
TABLE IV: Time vs Relation Size
Figure 17 shows the advantage of using a richer multirow model where we can
see that the appxroximation quality increases as more rows are provided for a
multirow representation of each tuple. Finally in Table IV we present the time
required for approximation generation with increasing tuples and IC
violations, where we show the time for marginals computation and the (total)
IC resolution time. Note that once the approximation has been generated we can
answer queries very efficiently on the resulting approximation as the
constraints have been factored in. The approximation generation times show
that our approach is scalable to large datasets. The experients were conducted
on an IBM XSeries_445 machine with 4 Intel Xeon 3 GHz processors, 17GB Ram
running Windows Server 2003. We must also mention that we have been unable to
provide comparative experimental results with a related system such as MayBMS
(in particular) as the ”assert” operation meant to materialize a database
recalibrated given an IC is not provided in the current system.
## VIII Related Work
Probabilistic databases have been an area of activity since the 1980s with
foundational works such as [17, 18] extending the relational model and algebra
to represent and support uncertainty in databases. Current active projects -
MystiQ[9], MayBMS, Trio, or Orion [2] employ different underlying uncertain
database representation formalisms that either vary subtly, or in some cases
significantly across each other, for instance MystiQ using ”or-tuples”, Trio
using or-sets but with additional ”lineage” information, and MayBMS using more
expressive world set decompositions (WSDs). MayBMS has considers conditioning
probabilistic databases with ICs which is motivated from a data cleaning
perspective, dealing with ”equality generating dependencies” (equivalent to
the tuple level ICs) and just functional dependencies (FDs) from amongst
relation level ICs (as opposed to the larger class of relation ICs that we
address). Their approach to resolving ICs is quite different from ours.
Instead of applying ICs to an uncertain database as we do, they augment
queries with the ICs so that the ICs are resolved at query time. The approach
to factor in FDs using a chase based procedure [5] can result in an
exponential blow up even with a single FD. Each relation is represented as
decomposed into multiple ”components” the product of which yields the entire
relation. Each component essentially contains the values of an attribute or a
set of attributes. Their algorithm is to consider pairs of tuples violating
the FD, take each attribute in the FD and merge the components containing
those attributes for the pair of tuples into a new component, and then clean
the new component by eliminating attribute value combinations that are
inconsistent with the FD. In the case of a relation R, with FD $A\rightarrow
B$, and pair-wise violations $(t_{1},t_{2}),(t_{2},t_{3}),,(t_{K-1},t_{K})$
with this FD, we will end up with a component that has as columns
($t_{1}.A,t_{2}.A,..,t_{K}.A,t_{1}.B,..,t_{K}.B$) and in the rows of this
component have all consistent combinations of attribute A and B values. The
size of this component is $O(M^{K})$ where M is the degree of uncertainty
(choices) in the attributes. Further, the chase based procedure must select
the consistent combinations only and its compelxity is also $O(M^{K})$. Even
with modest values of say M=2 and K= 30, $M^{K}$ is extremely large. While we
observe that their approach is exponential, we note that the authors
essentially meant the technique to be used in the context of data that has
only very few violations, in which case their approach will work fine. This is
substantiated by their experiments which have been done with a degree of data
dirtiness as low as 0.001% - 0.005% and also stated as a valid assumption by
them given the focus on data cleaning applications. In contrast, our approach
is applicable to databases with a much higher degree of data dirtiness, for
applications such as information extraction where literally all fields in the
data can be uncertain i.e., with a degree of dirtiness of 100% ! Also in our
synthetic data evaluation we have used an $\alpha$ (dirtiness) factor of at
least 5% (Table IV). To the best of our knowledge our work is the first to a)
Provide an approach to factoring a large class of ICs, including many kinds of
relation level ICs such as FDs, aggregation constraints, inclusion
dependencies, and set constraints in a correct manner into an uncertain
database, b) Provide an approach to incorporating ICs that makes no
assumptions on factors such as the degree of data dirtiness and is thus
applicable to applications where the degree of data dirtiness can in practice
be quite high.
In information extraction, the approach developed in [1] is to approximate a
complex CRF distribution that represents text segmentation possibilities into
a probabilistic relational model. This work however does not consider any
dependencies across different extracted segments, where each extracted segment
is treated as a tuple. We address such dependencies as relation level ICs. In
[1] the probability distribution being approximated is known to be generated
from a CRF and an efficient forward-backward-message-passing algorithm is
employed for marginal computation, vs our setting where marginal probabilities
must be estimated. We compared with database repair [19, 10, 20] earlier and
further note that most prior work on repair has considered only a limited set
of constraints, such as [10] which deals with only functional (FD) and
inclusion (IND) dependencies whereas our paper addresses a large class of
attribute, tuple, and relation level ICs. Work on consistent query answering
(CQA) deals with a related but different problem of answering queries over a
dirty database considering constraints over the database - this is established
as a hard problem in general [20] with practical approaches [21] provided
considering only primary key constraints.
## IX Conclusion
We have developed an approach for incorporating integrity constraints into
uncertain relations by approximating the uncertain relations. There are
several interesting directions for future work, including considering more
expressive uncertain database representation models, that we are working on.
## References
* [1] R. Gupta and S. Sarawagi, “Creating probabilistic databases from information extraction models,” in _VLDB_ , 2006, pp. 965–976.
* [2] D. Suciu and N. Dalvi, “Foundations of probabilistic answers to queries,” in _Tutorial at ACM SIGMOD_ , 2005.
* [3] P. Agrawal, O. Benjelloun, A. D. Sarma, C. Hayworth, S. U. Nabar, T. Sugihara, and J. Widom, “Trio: A system for data, uncertainty, and lineage,” in _VLDB_ , 2006, pp. 1151–1154.
* [4] S. Abiteboul, R. Hull, and V. Vianu, _Foundations of Databases_. Addison-Wesley, 1995.
* [5] L. Antova, “Efficient representation and processing of incomplete information,” Master’s thesis, Saarland University, Feb 2006, http://www.cs.cornell.edu/$\sim$lantova/.
* [6] M. A. Soliman, I. F. Ilyas, and K. C.-C. Chang, “Urank: formulation and efficient evaluation of top-k queries in uncertain databases,” in _ACM SIGMOD_ , 2007.
* [7] C. Koch and D. Olteanu, “Conditioning probabilistic databases,” in _PVLDB 1(1)_ , 2008.
* [8] N. N. Dalvi and D. Suciu, “Answering queries from statistics and probabilistic views,” in _VLDB_ , 2005.
* [9] ——, “Management of probabilistic data: foundations and challenges,” in _PODS_ , 2007, pp. 1–12.
* [10] P. Bohannon, W. Fan, M. Flaster, and R. Rastogi, “A cost-based model and effective heuristic for repairing constraints by value modification,” in _ACM SIGMOD_ , 2005.
* [11] T. Imielinski, S. Naqvi, and K. Vadaparty, “Incomplete object a data model for design and planning applications,” in _ACM SIGMOD_ , 1991, pp. 288–297.
* [12] N. Ashish, S. Mehrotra, and P. Pirzadeh, “Incorporating integrity constraints in uncertain databases (extended),” http://www.ics.uci.edu/$\sim$ashish/techrep, Tech. Rep.
* [13] R. Karp, M. Luby, and N. Madras, “Monte-carlo approximation algorithms for enumeration problems,” _Journal of Algorithms_ , vol. 10, pp. 429–448, 1989\.
* [14] S. Khuller, A. Moss, and J. S. Naor, “The budgeted maximum coverage problem,” _Information Processing Letters_ , vol. 70, no. 1, pp. 39–45, 1999.
* [15] A. Papoulis, _Probability, Random Variables and Stochastic Processes_. McGraw-Hill Companies, 1991\.
* [16] “Ties: Trainable information extraction system,” http://tcc.itc.it/research/textec/tools-resources/ties.html.
* [17] D. Barbara, H. Garcia-Molina, and D. Porter, “The management of probabilistic data,” _IEEE TKDE_ , vol. 4, no. 5, pp. 487–502, 1992.
* [18] R. Cavallo and M. Pittarelli, “The theory of probabilistic databases,” in _Proc of VLDB_ , 1987.
* [19] A. Lopatenko and L. Bravo, “Efficient approximation algorithms for repairing inconsistent databases,” _ICDE_ , pp. 216–225, 2007.
* [20] J. Chomicki, “Consistent query answering: Five easy pieces,” in _ICDT_ , 2007, pp. 1–17.
* [21] A. Fuxman and R. J. Miller, “First-order query rewriting for inconsistent databases,” in _ICDT_ , 2005.
|
arxiv-papers
| 2009-07-09T18:45:29 |
2024-09-04T02:49:03.798737
|
{
"license": "Public Domain",
"authors": "Naveen Ashish, Sharad Mehrotra, Pouria Pirzadeh",
"submitter": "Naveen Ashish",
"url": "https://arxiv.org/abs/0907.1632"
}
|
0907.1855
|
Strain-Induced Alignment in Collagen Gels
D. Vader1, A. Kabla2, D. Weitz1,3, L. Mahadevan1,∗
1 School of Engineering and Applied Sciences, Harvard University, Cambridge,
MA, USA
2 Department of Engineering, University of Cambridge, Cambridge, UK
3 Department of Physics, Harvard University, Cambridge, MA, USA
## Abstract
Collagen is the most abundant extracellular-network-forming protein in animal
biology and is important in both natural and artificial tissues, where it
serves as a material of great mechanical versatility. This versatility arises
from its almost unique ability to remodel under applied loads into anisotropic
and inhomogeneous structures. To explore the origins of this property, we
develop a set of analysis tools and a novel experimental setup that probes the
mechanical response of fibrous networks in a geometry that mimics a typical
deformation profile imposed by cells in vivo. We observe strong fiber
alignment and densification as a function of applied strain for both
uncrosslinked and crosslinked collagenous networks. This alignment is found to
be irreversibly imprinted in uncrosslinked collagen networks, suggesting a
simple mechanism for tissue organization at the microscale. However,
crosslinked networks display similar fiber alignment and the same geometrical
properties as uncrosslinked gels, but with full reversibility. Plasticity is
therefore not required to align fibers. On the contrary, our data show that
this effect is part of the fundamental non-linear properties of fibrous
biological networks.
Citation: Vader D, Kabla A, Weitz D, Mahadevan L (2009) Strain-Induced
Alignment in Collagen Gels. PLoS ONE 4(6): e5902.
doi:10.1371/journal.pone.0005902
Received March 6, 2009; Accepted April 21, 2009; Published June 16, 2009
Copyright: © 2009 Vader et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.
Funding: This work was supported by NIH Bioengineering Research Partnership
grant R01 CA085139-01A2 and by NSF IGERT program in biomechanics. The funders
had no role in study design, data collection and analysis, decision to
publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests
exist.
$\ast$ E-mail: [email protected]
## Introduction
Fiber networks, which arise in a range of natural and technological
situations, are prime candidates for a wide spectrum of applications requiring
tunable mechanical, transport and chemical properties [1]. In nature, these
networks self-assemble to generate important structural and functional
elements at various length scales: actin, intermediate filaments and
microtubules are the main components of the cytoskeleton [2]; spectrin confers
versatile qualities to red blood cell membranes [3]; fibrin is an essential
element in hemostasis [4]; collagen is the main component of the extracellular
matrix (ECM) in the animal kingdom [5] and cellulose is used by plants to
build cell walls [6].
The mechanical function of biological fiber networks is essentially two-fold:
(i) at the subcellular (actin, spectrin) and supracellular (collagen, fibrin)
scales, the material offers little resistance and high sensitivity to small
deformations, allowing it to be easily remodeled locally ; (ii) at larger
strains it stiffens strongly to ensure cell and tissue integrity [7]. The non-
linear stiffening, while observed in many biological systems [8, 9], is not
fully understood yet, with theories focusing on one of two broad mechanisms:
(i) microstructural nonlinearities of individual filaments [8], and (ii)
collective non-affine deformations of multiple filaments [10, 11]. To unravel
the relative importance of these mechanisms, a range of experimental tools
have been developed to quantify the network’s mechanical non-linearity in
systematic ways and relate the material micro-structure (network density and
morphology, fiber behavior) to the mesoscopic stress-strain laws. These tools
fall into two broad categories: simple shear in cone-plate or parallel plate
geometries, and uniaxial/biaxial stretch.
Simple shear deformations are commonly used to study purified protein
networks. This technique requires low sample volumes and provides a consistent
set of experimental tools and generic protocols to probe the visco-elastic
properties of soft gels in both the small-strain (linear) and large-strain
(non-linear) regimes, and in addition, normal stresses can be measured. Recent
data collected by Janmey et al. [9] show in particular that sheared
biopolymers exert negative normal forces, a fact that is in contradiction with
the hyperelastic behavior of other well studied elastomers. The broad
availability of experimental data in that geometry has encouraged a large
number of related theoretical and numerical studies [12, 13, 14], focused
primarily on the linear response of the material. However, since simple shear
rheology assumes that the material undergoes purely isochoric deformations in
the limit of small strains, it only allows for partial exploration of material
behavior. In particular, these experiments do not allow one to study
completely the non-linear regime (strain typically larger than 10%) that is
most relevant in many biological situations (single cell or tissue
deformation). And furthermore, it does not allow for a probe of the
dilatational rheology of the networks.
In contrast, at mesoscopic scales, uniaxial and biaxial testing are most
common for tissue mechanical characterization [15, 7, 16, 17] and have been
used to study reconstituted collagen networks [18, 19, 20, 21, 22], the
simplest tissue equivalents [23]. In contrast with simple shear, uniaxial
stretch generically leads to non-isochoric deformations, and hence allows one
to measure quantities such as the material Poisson ratio which can have values
as large as $\nu\approx 3$ for strongly deformed collagen gels. These values
arise in highly anisotropic materials, as reported for instance for solid
foams [24], and it is somewhat surprising to see similar behavior in in vitro
collagen gels which display little or no anisotropy in their undeformed state.
To understand this, we recall that early studies [25] on cell/matrix
interactions show that cells or groups of cells tend to generate tensile
forces on the extracellular environment . When cell colonies were plated on
fibrous materials such as collagenous gels, Harris and Stopak reported the
formation of anisotropic and denser regions connecting these cellular
assemblies, and showed that the matrix structure has a strong influence on
cell motility. Although these observations are well accepted, little is known
about the mechanical response of a fibrous matrix subject to an internal local
strain. Neither of the mechanical characterizations described previously focus
on how deformation changes the microstructure at the fiber scale, an issue of
particular importance in the large strain regime, that is all too easy to
observe (Figure 1).
In this paper, we use collagen type I gels as a model system to address this
question and shed light on the morphological evolution of both the fiber
111Although the expression ”collagen fiber” traditionally refers to large
bundles of collagen fibrils, we will use the words ”fiber” and ”fibril”
interchangeably in this paper to refer to the 0.5 micron diameter bundles. and
the network on an externally imposed stretching strain. Collagen is a
convenient biomaterial for biomechanical studies for a number of reasons: a)
it is readily available in large amounts, which makes it suitable for
milliliter-size gels; b) in vitro reconstituted networks have fibers that are
easily identifiable using confocal microscopy; c) many of its properties have
been extensively studied [26, 27, 28, 29, 30, 18]; d) the large diameter of
the fibers ($\approx 0.5\mu m$ for collagen fibrils [21]) and the stability of
the network [31, 32] make it easy to handle and image over a range of spatial
and temporal scales; e) fibrillogenesis is conveniently controlled in vitro by
pH, temperature and concentration [28, 26].
Figure 1: Collagen gel morphological changes induced by presence of cells.
(A) Single U87 glioblastoma cell in a collagen network 10 hours after gel
polymerization. bar=50 $\mu$m. (B) Several U87 cells on the surface of a
collagen gel 10 hours after gel polymerization. bar=200 $\mu$m. (C) Two cell
colonies embedded in a collagen matrix 48 hours after gel polymerization.
bar=200 $\mu$m. Fibers (artificial red color) are imaged through confocal
reflectance; cell nuclei (green) are labeled with a GFP-histone heterodimer.
We first verified the presence of cell-induced alignments and densification
with our experimental system. As shown in figure 1A, an isolated human
glioblastoma cell (see Methods) in a collagen network induces stress
variations and modifies the network texture in its vicinity [33, 34]. Several
isolated cells on the surface of a collagen gel produce fiber alignment and
network densification along lines connecting individual cells (figure 1B).
Following Stopak and Harris, we also observed fiber alignment on macroscopic
length scales when we introduce large cell assemblies in the same extra-
cellular environment (figure 1C). Since active matrix remodeling is restricted
to the vicinity of living cells, such an effect can only be accounted for by
the mechanical properties of the network.
With these observations in mind, we employ a specific experimental approach
and develop a set of tools to quantitatively study the coupling between strain
and the morphology of fibrous networks in a range of strain and strain rates
that are typical of many biomechanical situations. Experiments on cell
colonies suggest that such a process can be conveniently studied at the
millimeter scale, and over a time-scale of a few hours. However, instead of
using cells to deform the extra-cellular matrix, we use an external imposed
displacement to stretch collagenous samples and monitor the gel response. In
particular, this experiment allows us to probe a range of dynamical regimes
and independently tune the biochemistry (crosslinking) to study the coupling
of tensile strain to network density and fiber orientation in a controlled
setting and investigate the origin and generality of these mechanical
processes. This also allows us to address the outstanding question of the
mechanical reversibility of these patterns in an extracellular environment.
## Methods
### Network synthesis
In vitro collagen networks are prepared according to a previously described
cell culture-compatible protocol [35], with a final collagen concentration
ranging from 0.5 to 4.0mg/mL. Solutions consist of 10% 10X minimum essential
medium (Invitrogen, Carlsbad, CA), 10% fetal bovine serum (JRH Biosciences,
Lenexa, KS), 1% penicillin-streptomycin (Invitrogen), bovine collagen diluted
to desired concentration (from 3.1mg/mL or 6.4mg/mL batch, Inamed
Biomaterials, Fremont, CA), a few $\mu$L of 1M sodium hydroxide (NaOH, Sigma,
St. Louis, MO) to bring pH to neutral, 50mM sodium bicarbonate (NaHCO3, Sigma)
buffer and deionized water. 800$\mu$L of solution are pipetted onto glass-
bottom Petri dishes (MatTek, Ashland, MA) Samples then polymerize for 30-60
minutes in a cell incubator at 37∘C, 5% CO2. After polymerization, samples are
20mm in diameter and 1-2mm in height.
In addition to untreated in vitro collagen gels, we also prepare polymerized
samples, to which we add glutaraldehyde (GA) - a common cell and tissue
fixative. The effect of this is an increase in the overall stiffness of the
gel by at least an order of magnitude (from a few tens of Pascals to over 1kPa
for a 1mg/mL gel, as evaluted using a cone-plate rheometer), without
noticeably changing the structure of the collagen network (fiber width and
length, mesh size). 2mL of 4% v/v GA (Sigma, St. Louis, MO) in water is
pipetted onto the sample, which is then incubated once more for at least 2
hours. It is subsequently rinsed twice with deionized water. Before use, all
samples are immersed into 2mL of deionized water, to allow the gel to swell,
and to reduce friction.
### Cell experiments
U87 human glioblastoma cells are cultured as described in [35]. After
passaging the cells, tissue equivalents are generated by diluting the cells to
approximately $10^{5}$/mL in an unpolymerized collagen solution at 1.5mg/mL.
After polymerization in a 37∘C temperature- and humidity-controlled incubator,
the spacing between individual cells, as seen in figures 1a and 1b, is on the
order of 100$\mu$m. Cell colonies (or spheroids), with an estimated $10^{3}$
cells, are generated with the hanging droplet method [36] and subsequently
seeded in 500$\mu$L of collagen solution at 1.5mg/mL shortly before it
polymerizes.
### Bulk rheology
We use an AR-G2 (TA Instruments, New Castle, DE) rheometer with a 4∘, 40mm
cone-plate geometry with a 109$\mu$m gap. 1.2mL of collagen solution is
pipetted onto the 37∘C preheated bottom plate of the rheometer and the cone is
lowered onto the sample. We use a solvent trap to prevent the sample from
drying during the measurement. During polymerization, the increase in G′ and
G′′ is probed by continuously oscillating the sample at a fixed 0.5% strain
amplitude and at a frequency of 0.2Hz. The oscillatory strain sweep is
performed at the same frequency and temperature, after the gel has polymerized
for 2-3hrs. The strain amplitude is increased logarithmically until the sample
breaks.
During oscillatory strain sweeps, we simultaneously record the maximum stress
and strain of the sample for each oscillatory cycle. To characterize the onset
of stiffening from the stress-strain data, we define the critical strain as
the value at which the stress $\sigma$ exceeds the product
$G^{\prime}_{0}\gamma$ by more than 10%, where $G^{\prime}_{0}$ is the elastic
modulus in the small-strain linear regime.
### Mechanical setup
We place a polymerized collagen sample onto a glass cover slip and perforate
it with two rough-ended 1mm-diameter glass cylinders (capillaries) (see
figures 2A,B), which are gently pushed all the way to the glass bottom to
prevent the collagen from slipping beneath them. Each cylinder is attached to
two secondary transverse elastic capillary rods, themselves attached to two
linear transducers (Newport, Irvine, CA) controlled by the ESP300 controller
(Newport). The transverse capillaries act as springs that allow to maintain
contact with the bottom cover slip of the dish with constant pressure. The
tips, initially 1cm from each other, can then be moved apart at speeds ranging
from 0.125 to 12.5 $\mu m/s$, corresponding to strain rates of 2.5$\cdot
10^{-5}$ to 2.5$\cdot 10^{-3}$ per second; this range includes measured rates
of cell-induced contraction [37]. The movement of the tips results in the
local stretching of the gel sitting between them. For imaging purposes, the
whole mechanical setup (motors and tips) is clamped to the microscope sample
holder plate.
Figure 2: Mechanical setup and sample imaging. (A) Side and (B) top views of
the mechanical setup used to deform the network; the collagen gel has a
pancake-like shape, typically 1mm in thickness and 20mm in diameter. As
defined in our experiments, the stretch axis is $x$. Drawn to scale, the two
squares represent the fields-of-view of the wide-field fluorescence images
(5x) and confocal reflectance images (60x). (C) Correlation of multiple slices
over time gives an estimate of the interslice distances, and hence vertical
strain. (D) Collagen network (blue) obtained with confocal reflectance.
Fluorescent tracers (pink) are embedded in the network. Scale bar 20$\mu m$.
(E) Wide-field fluorescence image of the embedded beads. Scale bar 500$\mu m$.
### Confocal imaging
We use a Zeiss LSM 510 Meta (Carl Zeiss Microimaging Inc., Thornwood, NY)
equipped with a 488nm Argon laser line and several photomutliplier tubes. We
set the Meta channel of the microscope (which allows for selection of specific
wavelengths) to detect wavelengths between 474 and 494nm to allow for the fact
that we work in reflectance mode [38], which has the significant advantage of
avoiding the use of fluorescent dyes in the samples. A 60X 1.2-NA Olympus
water immersion objective (Olympus America Inc., Center Valley, PA) is mounted
onto the microscope.
While deforming the sample, we acquire timelapse 2D confocal images at various
heights between 50 and 150$\mu$m from the bottom surface. Tracking in-plane
deformations in multiple slices improves the statistics of our analysis;
moreover, tracking out-of-plane motion via image correlation allows us to
estimate the vertical deformation of the sample (figure 2C). The timelapse
interval is 10s, which corresponds to 0.25% imposed deformation at the typical
strain rate.
### Image segmentation and fiber detection
Fiber orientation is calculated for each 2D confocal slice by several image
processing steps: the raw images are filtered using a 2D Gaussian blur and
subsequently thresholded so that at 0% stretch, 10% of the pixels are above
that threshold; this threshold value is applied to all subsequent images of
the same experiment. A circular window of diameter 15px moves across the
thresholded image, and the 2nd order moment tensor $M$, defined below, is
locally calculated using binary pixel weights: below the threshold level, the
pixel weight $A_{ij}$ is 0, above that level, $A_{ij}$ is 1. Typically, the
moment tensor quantifies the spatial distribution of weight around a center of
mass and its eigenvalues give an indication as to whether weights are
distributed isotropically around the center or not. Similarly here, $M$
quantifies the distribution of pixel intensities around the intensity-weighted
center of mass, with $X_{ij}$ and $Y_{ij}$ the pixel coordinates in the local
circular window:
$M\\!=\\!\\!\\!\\\ \left[\\!\\!\\!\\!\begin{array}[]{cc}\
\\!\\!\\!\scriptstyle\sum A_{ij}(X_{ij}-\bar{X})^{2}&\
\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\scriptstyle\sum
A_{ij}(X_{ij}-\bar{X})(Y_{ij}-\bar{Y})\\\ \ \\!\scriptstyle\sum
A_{ij}(X_{ij}-\bar{X})(Y_{ij}-\bar{Y})&\
\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\scriptstyle\sum
A_{ij}(Y_{ij}-\bar{Y})^{2}\end{array}\\!\\!\\!\\!\right]$ (1)
$r=\frac{\hbox{Max}\left\\{\hbox{Eigenvalues}(M)\right\\}}{\hbox{Min}\left\\{\hbox{Eigenvalues}(M)\right\\}}$
(2)
The ratio $r$ characterizes the aspect ratio of binarized image fragment
enclosed in the sliding window and can be used to detect fibers: a single
fiber passing through the middle of the window, with a diameter less than half
the window size will yield a high aspect ratio; a single fiber in the corner
of the window or multiple fibers in the window will give a low aspect ratio.
We only keep the regions with $r>2$ for which the eigenvector corresponding to
the higher eigenvalue of $M$ provides the local fiber orientation $\phi$.
The choice of window size (15 pixels) is a compromise between increasing
angular resolution at low strains and avoiding multiple fibers in the window.
As shown later on, the fiber density strongly increases at high strain,
resulting in less accuracy in the orientation analysis. In practice, this sets
the limits of the method. See Methods S1, Figure S1 and Figure S2 for
validation of this image processing algorithm.
### Orientational order parameter
To quantify the network anisotropy as a function of the applied deformation,
we calculate an orientation tensor $\Omega$ and an associated nematic order
parameter $\mu$ from the distribution of the fiber orientation $\phi$ at each
time step, defined by:
$\Omega=\left(\begin{array}[]{cc}\left<\cos^{2}(\phi)\right>&\left<\cos\phi\sin\phi\right>\\\
\left<\cos\phi\sin\phi\right>&\left<\sin^{2}\phi\right>\end{array}\right)$ (3)
$\mu=\hbox{Max}\left\\{\hbox{Eigenvalues}(2\;\\!\Omega-\hbox{\bf
Id})\right\\}$ (4)
where $\bf Id$ is the identity matrix, and $<\cdot>$ denotes spatial averaging
over the domain of interest. The order parameter ranges from $\mu=0$ for a
uniform angular (isotropic) distribution to $\mu=1$ for a perfectly aligned
system. Although this parameter, based on a 2D image analysis, only
characterizes the order in the horizontal plane, it does provide a suitable
signature of the microstructure evolution, and in particular its non-linear
behavior.
### Deformation field at the mesoscale
We also characterize, at the mesoscopic millimeter scale, the strain field
$\varepsilon_{xx}$, $\varepsilon_{yy}$ and $\varepsilon_{zz}$ induced by the
imposed displacement of the glass tips. Vertical strain is estimated by
following the individual displacement of confocal z-stacks. For each slice of
a stack taken at a time $t$, this is done by calculating the correlation with
the slices obtained at a neighboring time $t+\Delta t$ (see figure 2C). The
height of the slice which provides the largest correlation value indicates the
new location of the material layer and this information is used to calculate
the vertical component of the strain.
The deformation in the $xy$ plane is measured by a PIV (particle imaging
velocimetry) method. 1$\mu m$ diameter rhodamine Fluospheres (Invitrogen,
Carlsbad, CA), at a volume ratio of 1:100000, are used as tracers to measure
the deformation of the gel at millimeter length-scales. Most of the carboxy-
coated particles stick to the network, as seen in figure 2D. Using a 5x lens
on a Zeiss wide-field microscope and focusing on the middle of the sample, we
image the local density of these particles, which displays heterogeneities as
seen in figure 2E. An image cross-correlation technique is then used to track
these heterogeneities at a scale of $10-50\mu m$ as the network is
progressively deformed. To identify the local displacement of a mesoscopic
region of the gel located at $(x,y)$ from time $t$ to $t+\Delta t$, we extract
a domain of 48x48 pixels surrounding $(x,y)$ at $t$ and look for the best
matching region - i.e. the one that maximizes the cross-correlation function -
in the image obtained at $t+\Delta t$. The cross-correlation function is
defined as:
$\rho_{AB}=\sum{(A_{ij}-A_{avg})(B_{ij}-B_{avg})/(\sigma_{A}\sigma_{B})}$ (5)
where A and B are the pixel intensity values associated with the two regions
of interest, $A_{avg}$ and $B_{avg}$ are the local average pixel intensities,
and $\sigma_{A}$ and $\sigma_{B}$ are the standard deviations of intensity
values of those regions.
This tracking through cross-correlation process is iterated over time to
extract the full trajectory of the material point and corresponds to a
Lagrangian description of the material. For a grid deformation that is fairly
homogeneous in the field of view of the microscope (millimeter scale), we use
a least-squares planar fit of the nodal displacement to quantify the material
deformation at the mesoscale. The deviation from the fit provides a measure of
the error on the deformation field.
These measurements allow us to extract a number of strain and stress
characteristics. In particular, in our geometry, the normal stresses
$\sigma_{yy}$ and $\sigma_{zz}$ are negligible as the gel is not attached on
the lateral sides, and this allows us to estimate the incremental Poisson
ratio, defined as:
$\nu_{xy}=-\partial\varepsilon_{yy}/\partial\varepsilon_{xx}$ (6)
which characterizes the coupling of incremental deformations in orthogonal
directions. In two dimensions and in the small deformation limit, the area of
the grid $A(\epsilon_{xx})$ is related to the Poisson ratio via the following
relationship:
$\frac{1}{A(0)}\frac{dA}{d\epsilon_{xx}}=1-\nu_{xy}$ (7)
Most materials respond to tension with a slight increase in their area (volume
in three dimensions), which for an isotropic material translates into the
condition:
$\nu_{xy}\leq 1$ (8)
The analogous condition in 3 dimensions is $\nu_{xy}\leq 0.5$. At large
deformations for an isotropic material, the criterion is slightly more
complex, but the critical strain, beyond which the change of area (or volume
in 3D) with respect to elongational strain becomes negative, remains of the
same order in practice.
## Results
### Rheological characterization of the samples
The range of collagen concentrations we work with (0.5-4.0 mg/mL) display mesh
sizes from 1-5 $\mu$m (measured through analysis of confocal reflectance
slices as in [35]) and span over two orders of magnitude in shear modulus (see
figure 3A), with values in close agreement with previously reported data [35,
39, 40]. The linear shear modulus $G^{\prime}$ (measured at small deformation)
has a strong dependence on the concentration $c$, with a behavior well
approximated by $G^{\prime}\sim c^{3}$ over the probed range. The onset of
nonlinear strain-stiffening typically occurs at strains of the order of 5%
(see figure 3B and [28]) and has only a weak dependence on collagen
concentration (see figure 3A inset).
Figure 3: Bulk rheology measurements. (A) Linear elastic modulus as a
function of collagen concentration. A power-law of 3 is shown for comparison.
Inset: the onset of non-linear mechanical behavior, as defined in Methods. (B)
Elastic modulus $G^{\prime}$ during oscillatory strain sweeps and as a
function of collagen concentration. Figure 4: Typical results of a sample
stretching experiment at micro- and meso-scale. The two montages of 5 images
each show, for two different 1mg/mL samples and at various strains, (A) wide-
field fluorescence images of beads embedded within the network - the super-
imposed grid is the result of the tracking of the deformation field (scale bar
500$\mu$m); (B) direct imaging of the fibers through CRLSM (scale bar
50$\mu$m). In inset, each corresponding $\phi$ histogram, with angle values
going from 0 to $\pi$. For the same samples depicted above: (C) represents the
evolution of the orientation statistics; the color at each point corresponds
to the relative count of fibers oriented along a specific direction at a given
strain $\varepsilon_{xx}$. (D) shows the order parameter $\mu$ resulting from
the data in (C); the curve beyond 15% stretch is grayed out due to the lack of
confidence of the order parameter when the high value of the density prevents
a proper detection of the fibers (see Methods). (E) gives the deformations
$\varepsilon_{yy}$ and $\varepsilon_{zz}$ as a function of the local strain
$\varepsilon_{xx}$. In inset, the incremental Poisson ratio $\nu_{xy}$ as a
function of the imposed deformation.
### Fiber alignment and non-linear Poisson effect
Figures 4A and 4B illustrate the evolution of the network microstucture over
the domain as it is stretched. At low strains ($<5\%$), no particular
alignment can be observed; however, above this threshold both fiber alignment
and network density increase. This is quantified in the figures 4C and 4D
where we show, as a function of the applied strain, the probability
distribution of local in-plane fiber orientation and the resulting order
parameter.
For a prescribed displacement of the capillary tips, we also characterized the
gel microstructure as we move away from the axis connecting the two
capillaries. Both the alignment and fiber density (expressed as fraction of
pixels above a given threshold) decrease (figure 5). This picture is, as
expected, in direct agreement with the alignment pattern induced by cell
colonies pulling on extra-cellular matrix shown in figure 1, where fiber
alignment and density are maximum along the axis joining the colonies, and
decay away from it.
Figure 5: Characterization of the spatial variations of collagen fiber
alignment and densification. (A-C) Confocal reflectance images of 1mg/mL
collagen network stretched up to 15%; images are located at (A) 0, (B) 0.5 and
(C) 2 millimeters from the stretching axis. bars = 50$\mu m$. (D) Order
parameter and density as a function of distance from the stretching axis, for
the same sample. Letters A, B and C on figures (D) and (E) refer to
corresponding images above. (E) Drawn to scale, locations of images (A), (B)
and (C) with respect to stretching axis.
In figure 4E we show the average induced strain components $\varepsilon_{yy}$
and $\varepsilon_{zz}$ as a function of the externally applied strain
$\varepsilon_{xx}$. When $\varepsilon_{xx}\leq 2\%$ there is little
contraction in the transverse direction so that the Poisson ratio
$\nu_{xy}\sim 0$. As the applied strain increases, the material first thins by
contracting in the $z$ direction when $2\%\leq\varepsilon_{xx}\leq 5\%$, and
only when $\varepsilon_{xx}\geq 5\%$ does it also contract in the transverse
$y$ direction, with observed values of $\nu_{xy}$ as high as 5 (figure 4E
inset). The lag in response between these two directions can be attributed to
the sample geometry as well as a slight initial anisotropy of fiber
orientations in the $yz$ plane [21]; here we consider only the properties of
the planar projection of the network. The large in-plane incremental Poisson
ratio $\nu_{xy}$ quantifies the change in local fiber density and is
consistent with the confocal observations of densification. In order to
quantify and compare these changes with an independent measure of the geometry
of deformation, we define a critical strain $\epsilon_{crit}$ for which
$dA(\epsilon_{crit})/d\varepsilon_{xx}=0$, beyond which the areal strain
(yellow frame on figure 4A) starts to decrease with the applied strain. We
find that $\epsilon_{crit}\sim 5\%$, consistent with the critical strain
observed for fiber alignment.
The critical deformation, as defined above from the kinematic behavior in
local stretching tests, can be directly compared with the strain associated
with the mechanical stiffening measured in rheological experiments (see figure
3). These two quantities, measured independently, show good correlations in
their values and trends. For unfixed collagen samples, critical strain values,
measured either from rheological meansurements or from the kinematics, range
from a few percents at high collagen concentration to 15 % at low
concentration (see figure 6). The critical strain is therefore very weakly
dependant on the concentration, in particular compared with the variation of
the elastic modulus at small deformation that varies over more than two orders
of magnitude in the same concentration range. GA-fixed samples, whose
stiffness is estimated at an order of magnitude higher than their non-fixed
counterparts, also show similar values for $\epsilon_{crit}$. Taken together,
these results show that the strains above which the gel behavior becomes non-
linear as evidenced i) from the elastic modulus for a simple shear geometry
and ii) from the Poisson ratio in the local stretching experiments are related
with each other and only weakly sensitive to physical scales such as the
actual value of the elastic modulus.
Figure 6: Comparison of critical strains measured from bulk rheology and
2-point stretching. Comparison of critical strain measured from the PIV method
($\epsilon_{crit}$ such that $dA(\epsilon_{crit})/d\varepsilon_{xx}=0$, see
Results) on locally stretched samples with the onset of strain stiffening
($\gamma_{crit}$ such that $\sigma>1.1G^{\prime}_{0}\gamma$, see Results)
obtained from rheological measurements.
### Orientational ordering is an elastic effect
Strain-induced alignment arises a priori from a combination of reversible
elastic effects and irreversible inelastic effects. To disentangle these two
contributions, we apply repeated strain cycles to the sample. All pure type-I
collagen samples display very little reversibility regardless of their
concentration once the imposed strains exceed about 10%; the gel never
recovers its initial configuration once the capillary tips return to their
initial location. In figures 7A,B we show the evolution of the fiber
orientation histograms as the material is cyclically stretched up to strains
of 15%: fiber aligment is permanently imprinted (7C-E).
Figure 7: Characterization of the reversibility of alignment and
densification in crosslinked and uncrosslinked collagen samples. Response to
cycles of deformation for untreated collagen gels (A-E) and gels treated with
glutaraldehyde (F-K). (A) Histograms of the fiber orientation as a function of
the imposed strain for untreated sample; color at each point corresponds to
the relative count of fibers oriented along a specific direction at a given
strain $\varepsilon_{xx}$. (B) Resulting order parameter for untreated sample.
(C-E) Confocal reflectance images of a 1mg/mL collagen sample cycled 4 times
up to 15% stretch and back to 0%, showing the extent of the reversibility at
the microstructural level: (C) beginning of 1st stretch cycle; (D) middle of
1st stretch cycle; (E) end of 1st stretch cycle; bars = 50$\mu m$. (F)
Histograms of the fiber orientation as a function of imposed strain for
glutaraldehyde-treated sample; color-coding as in (A). (G) Resulting order
parameter and density for glutaraldehyde-treated sample. (H-J) Confocal images
of a 1mg/mL collagen samples with glutaraldehyde, cycled 4 times up to 15%
stretch and back to 0%: (H) beginning of 3rd stretch cycle; (I) middle of 3rd
stretch cycle; (J) end of 3rd stretch cycle; bars = 25$\mu m$. (K) Mesoscopic
response to 15% strain cycles for various strain rates at a constant imposed
strain amplitude. The bold line corresponds to the strain rate used for all
other experiments.
With the addition, post-polymerization, of GA to our samples (see Methods), we
change collagen’s material properties (elastic modulus, plasticity threshold)
without changing the microstructure of the network. For small deformations,
the initial response is similar to that of the unfixed sample, indicating that
fiber alignment and the anomalous Poisson effect are only weakly sensitive to
the fiber material properties. Furthermore, during cycles of applied
deformation - i.e. ramping up the applied strain to 15% and returning back to
0% at the same rate - GA-crosslinked samples exhibit near-reversibility at the
microscale, with fiber images of successive cycles being almost identical
(figures 7H-J). Figures 7F and 7G quantify this reversibility in terms of the
orientation histogram and the order parameter over multiple cycles. Consistent
with this behavior, we find (see figure 7K) that the local deformation field
at the mesoscale evolves along a reversible path for the same strain rate
($2.5\cdot 10^{-4}$/s). Taken together, these results show that fiber
plasticity, though observed for pure collagen samples, is unimportant in
determining alignment at microscopic scales and the large Poisson ratio at
larger scales. That is, strain-induced alignment is primarily an elastic
effect.
The reversible behavior characterized above might then serve as a baseline to
study more subtle effects (e.g. time-dependent and/or irreversible processes)
that can be observed after many cycles or different strain rates. We see, in
particular, a slight decrease in the amplitude of the alignment with the
number of cycles, indicating that fiber plasticity still occurs, although on
much larger timescales. In figure 7K we show the effects of strain rate on the
response of the system for a fixed strain amplitude. At lower strain rates the
system does not recover completely after a full cycle. This offset in the
response can be attributed to a slow creeping process occurring over a time-
scale of a few hours. At larger strain rates, one expects to see dynamic
effects related to viscous dissipation. Previous studies on the poroelasticity
of collagen networks [41, 18, 20, 42, 43], have reported equilibration times
ranging from a few seconds to a few hours, probably due to the diversity of
geometries and setups used for all these measurements. We find that dynamic
effects associated with higher strain rates induce an asymmetry in the
response between loading and unloading, influencing the unloading curve more
than the loading curve. This shows that the material responds with different
time-scales in extension and compression; this, in turn, suggests that
different physical processes are involved during the loading which is
dominated by fiber stretching and unloading which is dominated by fiber
bending.
## Discussion
In this study, we used an experimental and computational approach to quantify
the emergence of fiber alignment as a collagen sample is stretched. This
behavior is consistent with observations of cell-induced morphological changes
in tissue equivalents and sheds light on a biologically relevant material non-
linearity that arises from stress heterogeneities in fiber networks. Fiber
alignment at the microscale results in tissue densification when boundary
conditions allow it; this leads to high values of the Poisson ratio at large
deformations, and is observed for all studied collagen concentrations, with or
without addition of glutaraldehyde.
The reversibility of fiber alignment and gel densification, seen in
crosslinked collagen samples, show that these effects are primarily elastic.
Experiments on a piece of synthetic felt [44] have demonstrated that geometry
alone can account for such a behavior based on the generic non-linearity of
individual fibers, stiff in extension, but soft in compression
(bending/buckling). Under uniaxial loading, a tensile stress is necessarily
balanced in the microstructure by a compressive load on fibers normal to the
stretch direction, leading to a collapse of the material in the normal
direction and a strong enghancement of fiber alignment along the load
direction, as observed here. This argument also explains the correlation
between the moment where stress builds up (onset of non-linearity in
rheological measurements) and the critical strain associated with the Poisson
effect. This is also consistent with the bulk rheology experiments performed
by Janmey et al. [9] who report large negative normal stresses in all tested
fibrous materials. The normal stress in volume-constrained geometries (as in
simple shear flows) is the counterpart of high Poisson ratios in unconstrained
tests such as the local stretch performed in the present study.
Our findings seem to contrast with the experimental results of Tower et al.
[19] on stretched collagen samples, where either irreversible alignment (pure
collagen sample) or early fracture with negligible alignment (GA crosslinked)
occurs, suggesting that fiber plasticity is a key player in the alignment
process. The difference with the present study can be readily explained from a
geometrical standpoint: Tower et al. used a geometry in which the width of the
stretched region is comparable to its length; this might significantly
constrain transverse motion of the material and prevent local volume changes
to their full extent. By contrast, the 2-point stretching device we use
ensures that the material is free to move along the direction normal to the
load.
Our experiments have demonstrated the importance of sample geometry and
boundary conditions on the microstructure and mechanical response of
reconstituted biopolymer gels. For functional tissues, it is known that
mechanical properties are often finely controlled by the texture of the
underlying extracellular matrix as well [45]. Understanding the mechanisms
leading to such organization is an important step in learning how it happens
in the formation of natural tissues and for developing strategies to engineer
suitable tissue equivalents. We have shown here that ECM texturization can be
brought about simply by applying a deformation with a purely mechanical
device, without any intervention due to active modeling by cells. This
external perturbation is applied post-polymerization, in contrast with
previous reports of fiber alignment induced by a flow [46] or a magnetic field
[47, 48] during collagen polymerization. This provides direct support to the
in vivo studies of post-polymerization collagen texturization in developing
tissues [49]. However, the microscopic origin of the permanent texturization
occurring in our uncrosslinked samples remains unclear. Fibril-fibril
junctions are likely to be where plastic deformation occurs, allowing fibrils
to slide with respect to one another and thus inducing irreversible changes in
the network topology. One of the effects of glutaraldehyde crosslinking is to
strengthen these junctions and reduce the amount of plastic reorganization
allowed at the network level.
Whatever their origin, geometrical changes in the network structure are known
to influence, in turn, cell behavior. Our mechanical setup allows for the
dynamical control of network texture in a passive system, but clearly can be
used to study how active cells respond to externally-induced anisotropy. Such
experiments will provide more insights into specific mechanisms of
mechanotransduction and cell behavior, which are crucial to processes such as
morphogenesis, stem cell differentiation, metastasis and wound healing. More
generally, the external control of fiber alignment post polymerization not
only offers a convenient way to design anisotropic tissue equivalents with
collagen, but can also, when applied to other biopolymer systems, shed light
on a range of analogous phenomena, such as actin gel contraction [50] or
platelet-fibrin interactions [51], where microscopic agents interact through
the network and lead to large scale evolution and reorganization of matter.
## References
* 1. Star A, Tu E, Niemann J, Gabriel JCP, Joiner CS, et al. (2006) Label-free detection of dna hybridization using carbon nanotube network field-effect transistors. Proc Natl Acad Sci U S A 103: 921–926.
* 2. Alberts B (2002) Molecular Biology of the Cell. Garland Science.
* 3. Gov N, Zilman AG, Safran S (2003) Cytoskeleton confinement and tension of red blood cell membranes. Phys Rev Lett 90: 228101.
* 4. Weisel JW (2004) The mechanical properties of fibrin for basic scientists and clinicians. Biophys Chem 112: 267–276.
* 5. Palsson B, Bhatia S (2003) Tissue Engineering. Prentice Hall.
* 6. Taiz L, Zeiger E (2002) Plant Physiology. Sinauer Associates.
* 7. Sacks MS (2000) Biaxial mechanical evaluation of planar biological materials. Journal of Elasticity 61: 199-246.
* 8. Storm C, Pastore JJ, MacKintosh FC, Lubensky TC, Janmey PA (2005) Nonlinear elasticity in biological gels. Nature 435: 191–194.
* 9. Janmey PA, Georges PC, Hvidt S (2007) Basic rheology for biologists. Methods Cell Biol 83: 3–27.
* 10. Heussinger C, Frey E (2006) Floppy modes and nonaffine deformations in random fiber networks. Phys Rev Lett 97: 105501.
* 11. Wyart M, Liang H, Kabla A, Mahadevan L (2008) Elasticity of floppy and stiff random networks. Phys Rev Lett 101: 215501.
* 12. Head DA, Levine AJ, MacKintosh FC (2003) Deformation of cross-linked semiflexible polymer networks. Phys Rev Lett 91: 108102.
* 13. Wilhelm J, Frey E (2003) Elasticity of stiff polymer networks. Phys Rev Lett 91: 108103.
* 14. Onck PR, Koeman T, van Dillen T, van der Giessen E (2005) Alternative explanation of stiffening in cross-linked semiflexible networks. Phys Rev Lett 95: 178102.
* 15. Billiar KL, Sacks MS (1997) A method to quantify the fiber kinematics of planar tissues under biaxial stretch. J Biomech 30: 753–756.
* 16. Ivancic PC, Coe MP, Ndu AB, Tominaga Y, Carlson EJ, et al. (2007) Dynamic mechanical properties of intact human cervical spine ligaments. Spine J 7: 659–665.
* 17. Cheng T, Gan RZ (2007) Mechanical properties of stapedial tendon in human middle ear. J Biomech Eng 129: 913–918.
* 18. Knapp D, Barocas V, Moon A (1997) Rheology of reconstituted type i collagen gel in confined compression. Journal of Rheology 41: 971-993.
* 19. Tower TT, Neidert MR, Tranquillo RT (2002) Fiber alignment imaging during mechanical testing of soft tissues. Ann Biomed Eng 30: 1221–1233.
* 20. Girton TS, Barocas VH, Tranquillo RT (2002) Confined compression of a tissue-equivalent: collagen fibril and cell alignment in response to anisotropic strain. J Biomech Eng 124: 568–575.
* 21. Roeder BA, Kokini K, Robinson JP, Voytik-Harbin SL (2004) Local, three-dimensional strain measurements within largely deformed extracellular matrix constructs. J Biomech Eng 126: 699–708.
* 22. Krishnan L, Weiss JA, Wessman MD, Hoying JB (2004) Design and application of a test system for viscoelastic characterization of collagen gels. Tissue Eng 10: 241–252.
* 23. Tranquillo RT (1999) Self-organization of tissue-equivalents: the nature and role of contact guidance. Biochem Soc Symp 65: 27–42.
* 24. Lee T, Lakes R (1997) Anisotropic polyurethane foam with poisson s ratio greater than 1. Journal of Materials Science 32: 2397-2401.
* 25. Harris AK, Wild P, Stopak D (1980) Silicone rubber substrata: a new wrinkle in the study of cell locomotion. Science 208: 177–179.
* 26. Raub CB, Suresh V, Krasieva T, Lyubovitsky J, Mih JD, et al. (2007) Noninvasive assessment of collagen gel microstructure and mechanics using multiphoton microscopy. Biophys J 92: 2212–2222.
* 27. Gentleman E, Lay AN, Dickerson DA, Nauman EA, Livesay GA, et al. (2003) Mechanical characterization of collagen fibers and scaffolds for tissue engineering. Biomaterials 24: 3805–3813.
* 28. Roeder BA, Kokini K, Sturgis JE, Robinson JP, Voytik-Harbin SL (2002) Tensile mechanical properties of three-dimensional type i collagen extracellular matrices with varied microstructure. J Biomech Eng 124: 214–222.
* 29. Brightman AO, Rajwa BP, Sturgis JE, McCallister ME, Robinson JP, et al. (2000) Time-lapse confocal reflection microscopy of collagen fibrillogenesis and extracellular matrix assembly in vitro. Biopolymers 54: 222–234.
* 30. Bozec L, Horton M (2005) Topography and mechanical properties of single molecules of type i collagen using atomic force microscopy. Biophys J 88: 4223–4231.
* 31. Leikina E, Mertts MV, Kuznetsova N, Leikin S (2002) Type i collagen is thermally unstable at body temperature. Proc Natl Acad Sci U S A 99: 1314–1318.
* 32. Persikov AV, Brodsky B (2002) Unstable molecules form stable tissues. Proc Natl Acad Sci U S A 99: 1101–1103.
* 33. Friedl P, Z nker KS, Br cker EB (1998) Cell migration strategies in 3-d extracellular matrix: differences in morphology, cell matrix interactions, and integrin function. Microsc Res Tech 43: 369–378.
* 34. Kim A, Lakshman N, Petroll WM (2006) Quantitative assessment of local collagen matrix remodeling in 3-d culture: the role of rho kinase. Exp Cell Res 312: 3683–3692.
* 35. Kaufman LJ, Brangwynne CP, Kasza KE, Filippidi E, Gordon VD, et al. (2005) Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns. Biophys J 89: 635–650.
* 36. Kelm JM, Timmins NE, Brown CJ, Fussenegger M, Nielsen LK (2003) Method for generation of homogeneous multicellular tumor spheroids applicable to a wide variety of cell types. Biotechnol Bioeng 83: 173–180.
* 37. Pizzo AM, Kokini K, Vaughn LC, Waisner BZ, Voytik-Harbin SL (2005) Extracellular matrix (ecm) microstructural composition regulates local cell-ecm biomechanics and fundamental fibroblast behavior: a multidimensional perspective. J Appl Physiol 98: 1909–1921.
* 38. Gunzer M, K mpgen E, Br cker EB, Z nker KS, Friedl P (1997) Migration of dendritic cells in 3d-collagen lattices. visualisation of dynamic interactions with the substratum and the distribution of surface structures via a novel confocal reflection imaging technique. Adv Exp Med Biol 417: 97–103.
* 39. Parsons JW, Coger RN (2002) A new device for measuring the viscoelastic properties of hydrated matrix gels. J Biomech Eng 124: 145–154.
* 40. Velegol D, Lanni F (2001) Cell traction forces on soft biomaterials. i. microrheology of type i collagen gels. Biophys J 81: 1786–1792.
* 41. Wille JJ, Elson EL, Okamoto RJ (2006) Cellular and matrix mechanics of bioartificial tissues during continuous cyclic stretch. Ann Biomed Eng 34: 1678–1690.
* 42. Chandran PL, Barocas VH (2004) Microstructural mechanics of collagen gels in confined compression: poroelasticity, viscoelasticity, and collapse. J Biomech Eng 126: 152–166.
* 43. Ozerdem B, T zeren A (1995) Physical response of collagen gels to tensile strain. J Biomech Eng 117: 397–401.
* 44. Kabla A, Mahadevan L (2007) Nonlinear mechanics of soft fibrous networks. J R Soc Interface 4: 99–106.
* 45. Shadwick RE (1999) Mechanical design in arteries. J Exp Biol 202: 3305–3313.
* 46. Lee P, Lin R, Moon J, Lee LP (2006) Microfluidic alignment of collagen fibers for in vitro cell culture. Biomed Microdevices 8: 35–41.
* 47. Barocas VH, Girton TS, Tranquillo RT (1998) Engineered alignment in media equivalents: magnetic prealignment and mandrel compaction. J Biomech Eng 120: 660–666.
* 48. Guo C, Kaufman LJ (2007) Flow and magnetic field induced collagen alignment. Biomaterials 28: 1105–1114.
* 49. Stopak D, Wessells NK, Harris AK (1985) Morphogenetic rearrangement of injected collagen in developing chicken limb buds. Proc Natl Acad Sci U S A 82: 2804–2808.
* 50. Bendix PM, Koenderink GH, Cuvelier D, Dogic Z, Koeleman BN, et al. (2008) A quantitative analysis of contractility in active cytoskeletal protein networks. Biophys J 94: 3126–3136.
* 51. Shah JV, Janmey PA (1997) Strain hardening of fibrin gels and plasma clots. Rheological Acta 36: 262-268.
|
arxiv-papers
| 2009-07-10T15:56:36 |
2024-09-04T02:49:03.813051
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Vader, A. Kabla, D. Weitz, L. Mahadevan",
"submitter": "David Vader",
"url": "https://arxiv.org/abs/0907.1855"
}
|
0907.1908
|
# Comment on “On the Crooks fluctuation theorem and the Jarzynski equality”
[J. Chem. Phys. 129, 091101 (2008)]
Artur B. Adib [email protected] Laboratory of Chemical Physics, NIDDK,
National Institutes of Health, Bethesda, Maryland 20892-0520, USA
###### Abstract
It has recently been argued that a self-consistency condition involving the
Jarzynski equality (JE) and the Crooks fluctuation theorem (CFT) is violated
for a simple Brownian process [L. Y. Chen, J. Chem. Phys. 129, 091101 (2008)].
This note adopts the definitions in the original formulation of the JE and CFT
and demonstrates the contrary.
Consider the problem of an overdamped Brownian particle moving under the
action of a time-dependent potential $U(x,t)$ over the whole line
$-\infty<x<\infty$ in the time interval $0\leq t\leq\tau$. Let us define the
following objects of interest (for simplicity of notation, I will set
$k_{B}T=1$):
(a) The free energy difference
$\Delta F:=-\ln\frac{Z_{\tau}}{Z_{0}},$ (1)
where the partition function $Z_{t}$ corresponding to the instantaneous energy
surface $U(x,t)$ is defined as
$Z_{t}:=\int_{-\infty}^{\infty}\\!dx\,e^{-U(x,t)};$ (2)
(b) The work functional defined along a given trajectory $x(t)$,
$W[x(t)]:=\int_{0}^{\tau}\\!dt\,\frac{\partial U(x(t),t)}{\partial t},$ (3)
where the partial derivative $\partial/\partial t$ keeps $x(t)$ constant; and
lastly
(c) The Jarzynski average, defined as the average $\langle\cdot\rangle$ over
all possible trajectories $x(t)$ propagated under $U(x,t)$ from $t=0$ to
$t=\tau$, where $x(0)$ is sampled from the initial Boltzmann distribution
$p(x(0))=e^{-U(x(0),0)}/Z_{0}$. (An expression for such an average in terms of
e.g. path integrals can be immediately written down, but this is not relevant
for the present discussion).
Making explicit and unambiguous use of definitions (a)-(c), the Jarzynski
equality (JE) Jarzynski (1997)
$\left\langle e^{-W}\right\rangle=e^{-\Delta F},$ (4)
has been proven through several different approaches in the literature (see
Jarzynski (2008) for a review).
Previous derivations notwithstanding, a skeptic might question the validity of
Eq. (4) at two rather distinct levels; namely, the appropriateness of the
definitions (a)-(c), or—once such definitions are agreed upon—the correctness
of the ensuing equality itself. The first is mostly a matter of historical
context and semantics, and will not be the subject of the present paper (see
e.g. Vilar and Rubi (2008); Horowitz and Jarzynski (2008) for a discussion on
definitions (a)-(b)). The second, on the other hand, would imply some type of
mathematical inconsistency in the steps leading up to Eq. (4).
In Ref. Chen (2008), the two levels of inquiry above have been blurred
together, leading its author to—among other misconceptions—claim that Eq. (4)
is “violated for a simple model system driven far from equilibrium.” What is
missing in such claim is the necessary qualification that the original
underpinnings of the equality, i.e. definitions (a)-(c), have been tampered
with. The object of this note is to point out that precisely the opposite
conclusion is reached if one uses the original definitions.
To proceed, consider the forward and reverse energy functions, $U_{f}(x,t)$
and $U_{r}(x,t)$ respectively, satisfying the condition
$U_{f}(x,t)=U_{r}(x,\tau-t),$ (5)
for $0\leq t\leq\tau$. This mapping between energy functions is the continuum
analog of Crooks’s forward and reverse processes Crooks (1998), and was
presumably implicit in the treatment of Ref. Chen (2008). (When such energy
functions are used instead of $U(x,t)$ in the definitions (a)-(b), the
appropriate subscripts $f$ and $r$ will be added to the corresponding
quantities; for (c), i.e. the average $\left\langle\cdot\right\rangle$, the
energy function is specified by the subscript of the quantity inside the
brackets).
Since the forward and reverse energy functions coincide at the opposite ends
of the time interval, i.e. $U_{f}(x,0)=U_{r}(x,\tau)$ and
$U_{f}(x,\tau)=U_{r}(x,0)$, we have trivially $\Delta F_{r}=-\Delta F_{f}$,
and thus the JE leads directly to Eq. (4) of Ref. Chen (2008):
$\left\langle e^{-W_{f}}\right\rangle\,\left\langle
e^{-W_{r}}\right\rangle=1.$ (6)
It is worth emphasizing that this result exists independently of the Crooks
fluctuation theorem (CFT) or its generalizations, such as Eq. (1) of Ref. Chen
(2008); indeed, as we have just seen, it is merely an application of the
Jarzynski equality twice for energy functions that coincide at opposite ends
of the time interval, as expressed above. Nonetheless, for problems satisfying
Eq. (5) and microscopic reversibility, its breakdown would imply that both the
CFT and the JE are violated, as for such problems the former also requires Eq.
(6) to be true Chen (2008).
The central exercise of Ref. Chen (2008) was to consider a simple choice of
$U(x,t)$ that allows for analytical computations, and test the validity of the
JE and CFT through Eq. (6). This program was carried out indirectly by
considering systems whose work distributions are known to be Gaussian, in
which case the exponential averages of work in Eq. (6) can be
straightforwardly reduced to their first and second moments. This Gaussian
assumption leads finally to Chen’s self-consistency condition (Eq. (6) of Ref.
Chen (2008)), namely
$\left\langle W_{f}\right\rangle+\left\langle
W_{r}\right\rangle=\frac{1}{2}\left(\sigma_{f}^{2}+\sigma_{r}^{2}\right),$ (7)
where $\sigma_{f}^{2}\equiv\langle(W_{f}-\langle W_{f}\rangle)^{2}\rangle$ and
similarly for $\sigma_{r}^{2}$. In the following, I will consider the same
model system of Ref. Chen (2008), and compute such moments analytically using
the original definitions (a)-(c).
Chen’s system can be described by the forward energy function
$U_{f}(x,t)=\frac{k}{2}x^{2}-f_{0}\theta(t)\,x.$ (8)
This is a simple harmonic potential whose center $z(t)=f_{0}\theta(t)/k$ is
instantaneously displaced, at $t=0$, from the initial position $z(0)=0$ to the
final position $z(\tau)=f_{0}/k$. [I am using the convention that the step
function $\theta(t)$ vanishes at $t=0$, which is consistent with Chen’s choice
of initial conditions, viz. $\langle x(0)\rangle=0$. This convention implies,
in particular, that $\int_{0}^{\tau}\\!dt\,x(t)\dot{\theta}(t)=x(0)$ for any
finite $\tau>0$, a result that is used in Eqs. (9) and (12)]. Accordingly,
$U_{f}(x,0)=\frac{k}{2}x^{2}$, and with definition (b) we have
$W_{f}[x(t)]=-f_{0}\int_{0}^{\tau}\\!dt\,x(t)\delta(t)=-f_{0}\,x(0).$ (9)
Thus, the moments of $W_{f}$ reduce to moments of the initial coordinate
$x(0)$, which give
$\left\langle
W_{f}\right\rangle=0,\quad\left\langle\sigma_{f}^{2}\right\rangle=\frac{f_{0}^{2}}{k}.$
(10)
Similarly, with the reverse energy function (cf. Eq. (5))
$U_{r}(x,t)=\frac{k}{2}x^{2}-f_{0}\theta(\tau-t)\,x,$ (11)
we have
$W_{r}[x(t)]=f_{0}\int_{0}^{\tau}\\!dt\,x(t)\delta(\tau-t)=f_{0}\,x(\tau).$
(12)
Since the original equilibrium state specified by $U_{r}(x(0),0)$ is not
perturbed until $t=\tau$, the coordinate $x(\tau)$ is distributed like $x(0)$,
and thus
$\left\langle
W_{r}\right\rangle=\frac{f_{0}^{2}}{k},\quad\left\langle\sigma_{r}^{2}\right\rangle=\frac{f_{0}^{2}}{k}.$
(13)
With the results given by Eqs. (10) and (13), Chen’s self-consistency
condition (Eq. (7)) is immediately satisfied, q.e.d.
Some final remarks are in order. Though never unambiguously stated in Ref.
Chen (2008), it seems like the analysis offered by Chen departed from the
original formulation of the Jarzynski equality and the Crooks fluctuation
theorem in the definition of work, i.e. in definition (b) of the present note.
Using the original definition, the free energy difference $\Delta F$ obtained
via the Jarzynski equality and Eqs. (10) and (13) (cf. Eq. (5) of Ref. Chen
(2008)) is $-f_{0}^{2}/(2k)$ for the forward, and $+f_{0}^{2}/(2k)$ for the
reverse process, in agreement with the different approach of Ref. Horowitz and
Jarzynski (2008), and consistent with the simple observation that the free
energy difference between two states changes sign upon a change of direction
in the process. Had Chen adhered to the original definitions, no controversy
would have arisen.
The author would like to thank Attila Szabo for bringing Chen’s manuscript to
his attention. This research was supported by the Intramural Research Program
of the NIH, NIDDK.
## References
* Jarzynski (1997) C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
* Jarzynski (2008) C. Jarzynski, Eur. Phys. J. B 64, 331 (2008).
* Vilar and Rubi (2008) J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 100, 020601 (2008).
* Horowitz and Jarzynski (2008) J. Horowitz and C. Jarzynski, Phys. Rev. Lett. 101, 098901 (2008).
* Chen (2008) L. Y. Chen, J. Chem. Phys. 129, 091101 (2008).
* Crooks (1998) G. E. Crooks, J. Stat. Phys. 90, 1481 (1998).
|
arxiv-papers
| 2009-07-10T20:46:37 |
2024-09-04T02:49:03.821903
|
{
"license": "Public Domain",
"authors": "Artur B. Adib",
"submitter": "Artur Adib",
"url": "https://arxiv.org/abs/0907.1908"
}
|
0907.1938
|
Fluctuations Destroying Long-Range Order
in SU(2) Yang-Mills Theory
Tohru Koma
Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo
171-8588, JAPAN
e-mail: [email protected]
We study lattice SU(2) Yang-Mills theory with dimension $d\geq 4$. The model
can be expressed as a $(d-1)$-dimensional O(4) non-linear $\sigma$-model in a
$d$-dimensional heat bath. As is well known, the non-linear $\sigma$-model
alone shows a phase transition. If the quark confinement is a consequence of
absence of a phase transition for the Yang-Mills theory, then the fluctuations
of the heat bath must destroy the long-range order of the non-linear
$\sigma$-model. In order to clarify whether this is true, we replace the
fluctuations of the heat bath with Gaussian random variables, and obtain a
Langevin equation which yields the effective action of the non-linear
$\sigma$-model through analyzing the Fokker-Planck equation. It turns out that
the fluctuations indeed destroy the long-range order of the non-linear
$\sigma$-model within a mean field approximation estimating a critical point,
whereas for the corresponding U(1) gauge theory, the phase transition to the
massless phase remains against the fluctuations.
## 1 Introduction
We study Euclidean SU(2) Yang-Mills theory on the hypercubic lattice
$\mathbb{Z}^{d}$ with dimension $d\geq 4$. It is widely believed that111See,
for example, the book [1]. the gauge theory shows a quark confinement phase
with a mass gap for all the values of the coupling in dimensions $d=4$. On the
other hand, the corresponding U(1) gauge theory in dimensions $d=4$ is proven
to show the existence of a deconfining transition to a massless phase [2, 3].
Thus it is expected that there exists a crucial difference between SU(2) and
U(1) gauge theories.
In this paper, we explore the origin of this difference. For this purpose, we
go back to the paper by Durhuus and Fröhlich [4]. They showed that the
$d$-dimensional Yang-Mills system can be interpreted as many
$(d-1)$-dimensional non-linear $\sigma$-models which are stacked up in the
$d$-th direction and coupled through $(d-1)$-dimensional external Yang-Mills
fields.222See also related articles [5, 6]. When we give our eye to one of the
$(d-1)$-dimensional non-linear $\sigma$-models, the system can be interpreted
as a $(d-1)$-dimensional non-linear $\sigma$-model in a $d$-dimensional heat
bath. When we turn off the interaction between the non-linear $\sigma$-model
and the heat bath, the non-linear $\sigma$-model becomes the standard O(4)
non-linear $\sigma$-model because the gauge group SU(2) is homeomorphic to
$3$-sphere $\mathbb{S}^{3}$. As is well known, the O(4) non-linear
$\sigma$-model is proven to show a phase transition [7] in dimensions greater
than or equal to three. This implies that, if the quark confinement is a
consequence of absence of a phase transition for the Yang-Mills theory, then
the fluctuations of the external Yang-Mills fields must destroy the long-range
order of the O(4) non-linear $\sigma$-model.
The effective action of the $(d-1)$-dimensional non-linear $\sigma$-model can
be derived by integrating out the degrees of freedom of the heat bath.
However, carrying out the integration is very difficult. Instead of doing so,
we replace the fluctuations of the external Yang-Mills fields with Gaussian
random variables. Within this approximation, the spins of the non-linear
$\sigma$-model can be interpreted as “particles” which move on
$\mathbb{S}^{3}$, acted by the two-body interaction and the random forces.
Namely the dynamics of the “particles” obeys a Langevin equation [8]. As is
well known, a Langevin dynamics yields a Fokker-Planck equation which
describes the time evolution of the distribution of the “particles”. In the
present system, the effective action of the non-linear $\sigma$-model can be
derived from the steady state solution to the corresponding Fokker-Planck
equation. In the effective action so obtained, the attractive potential
between the two “particles” is modified by the fluctuations of the external
Yang-Mills fields.
We show that the height and the width of the barrier of the attractive
potential depend on the coupling constant of the Yang-Mills theory. Roughly
speaking, the critical value of the coupling constant for the phase transition
to a massless phase can be estimated by the height and the width of the
barrier of the attractive potential. Therefore the critical value becomes a
function of the coupling constant. In consequence, we obtain that within a
certain mean field approximation, the critical value is always strictly less
than the value of the coupling constant itself for weak couplings. This
implies that the critical value must be equal to zero, i.e., there is no phase
transition to a massless phase for non-zero coupling constants.
On the other hand, the corresponding U(1) gauge theory shows that the
attractive potential does not depend on the coupling constant for weak
coupling constants within the same approximation. Namely the fluctuations of
the external Yang-Mills fields does not affect the critical behavior of the
O(2) non-linear $\sigma$-model.
This paper is organized as follows. In the next section, we express SU(2)
Yang-Mills theory in the form of the O(4) non-linear $\sigma$-model with a
large heat bath, following Durhuus and Fröhlich [4]. In Section 3, we obtain
the Langevin equation for the “particles” moving on $\mathbb{S}^{3}$, by
replacing the fluctuations of the heat bath with Gaussian random variables. In
the standard procedure, the Langevin equation yields the Fokker-Planck
equation for the distribution of the “particles”. In Section 4, a steady state
solution to the Fokker-Planck equation is obtained. The result immediately
yields the effective action of the non-linear $\sigma$-model. Further, we show
that the phase transition of the O(4) non-linear $\sigma$-model disappears,
owing to the fluctuations, within a mean field approximation for the effective
action so obtained. In Section 5, we apply the same method to the
corresponding U(1) gauge theory, and show that the phase transition to the
massless phase remains against the fluctuations.
## 2 Yang-Mills theory as a $\sigma$-model in a heat bath
Let $\Lambda$ be a sublattice of $\mathbb{Z}^{d}$. The SU(2) gauge field on
$\Lambda$ is a map from the oriented links or nearest neighbour pairs
$\langle{\bf q},{\bf q}^{\prime}\rangle$ of sites, ${\bf q},{\bf q}^{\prime}$,
of the lattice $\Lambda$ into the Lie group $G=$SU(2),
$\langle{\bf q},{\bf q}^{\prime}\rangle\longmapsto U_{{\bf q}{\bf
q}^{\prime}}\in G,$ (2.1)
obeying
$U_{{\bf q}^{\prime}{\bf q}}=\left(U_{{\bf q}{\bf q}^{\prime}}\right)^{-1}.$
(2.2)
Let $\gamma$ be an oriented path which is written $\gamma=\langle{\bf
q}_{1},{\bf q}_{2}\rangle\langle{\bf q}_{2},{\bf
q}_{3}\rangle\cdots\langle{\bf q}_{n-1},{\bf q}_{n}\rangle$ with the oriented
links, $\langle{\bf q}_{i},{\bf q}_{i+1}\rangle$ of the neighboring sites,
${\bf q}_{i},{\bf q}_{i+1}$, for $i=1,2,\ldots,n-1$. When ${\bf q}_{1}={\bf
q}_{n}$, the path $\gamma$ is a loop. For an oriented path $\gamma$, we write
$U_{\gamma}=U_{{\bf q}_{1}{\bf q}_{2}}U_{{\bf q}_{2}{\bf q}_{3}}\cdots U_{{\bf
q}_{n-1}{\bf q}_{n}}.$ (2.3)
The Euclidean action of pure Yang-Mills theory on the lattice
$\Lambda\subset\mathbb{Z}^{d}$ is given by
${\cal A}_{d}^{\rm YM}(\Lambda):=-\frac{1}{2}\sum_{p\subset\Lambda}{\rm
Re}\,{\rm Tr}\,U_{\partial p},$ (2.4)
where $p$ denotes an oriented plaquette(unit square) of $\Lambda$, and
$\partial p$ is the oriented loop formed by the four sides of $p$. The
orientation of the loop $\partial p$ obeys the orientation of the plaquette
$p$. The expectation value is given by
$\left\langle\cdots\right\rangle_{\Lambda}:=Z_{\Lambda}^{-1}\int\prod_{b\subset\Lambda}dU_{b}(\cdots)\exp\left[-\beta{\cal
A}_{d}^{\rm YM}(\Lambda)\right]$ (2.5)
with the inverse temperature $\beta$ and the normalization $Z_{\Lambda}$,
where $b$ is a link in $\Lambda$ and $dU_{b}$ is the Haar measure of the gauge
group $G=$SU(2).
Following Durhuus and Fröhlich [4], we use the relation between the
$d$-dimensional Yang-Mills action and a $(d-1)$-dimensional non-linear
$\sigma$-model. The coordinates of a lattice site ${\bf q}$ are denoted
$(x^{(1)},x^{(2)},\ldots,x^{(d-1)},x^{(d)})=({\bf i},x^{(d)})$ with ${\bf
i}=(x^{(1)},\ldots,x^{(d-1)})\in\mathbb{Z}^{d-1}$. Write
$\Lambda_{\tau}=\Lambda\cap\\{{\bf q}:x^{(d)}=\tau\\}$ for the
$(d-1)$-dimensional hyperplane, and
$\Lambda^{0}=\Lambda\cap\mathbb{Z}^{d-1}\times\\{0\\}$ for the projection onto
$\mathbb{Z}^{d-1}$ lattice. Let $U_{{\bf i}{\bf j}}^{h}(\tau)$ denote the
gauge field $U_{{\bf q}{\bf q}^{\prime}}$ assigned to the link $\langle{\bf
q},{\bf q}^{\prime}\rangle$ in $\Lambda_{\tau}$ with ${\bf q}=({\bf i},\tau)$
and ${\bf q}^{\prime}=({\bf j},\tau)$, and $U_{\bf i}^{v}(\tau)$ the gauge
field $U_{{\bf q}{\bf q}^{\prime}}$ with ${\bf q}=({\bf i},\tau)$ and ${\bf
q}^{\prime}=({\bf i},\tau+1)$. The former are called horizontal gauge fields
localized at $x^{(d)}=\tau$, and the latter are called vertical gauge fields
localized in the slice $[\tau,\tau+1]$. Now the Yang-Mills action can be
rewritten as
${\cal A}_{d}^{\rm
YM}(\Lambda)=-\frac{1}{2}\sum_{\tau}\sum_{p\subset\Lambda_{\tau}}{\rm
Re}\,{\rm Tr}\,U_{\partial p}^{h}-\frac{1}{2}\sum_{\tau}\sum_{\langle{\bf
i},{\bf j}\rangle\subset\Lambda^{0}}{\rm Re}\,{\rm Tr}\,{U_{\bf
i}^{v}(\tau)}^{-1}U_{{\bf i}{\bf j}}^{h}(\tau)U_{\bf j}^{v}(\tau)U_{{\bf
j}{\bf i}}^{h}(\tau+1).$ (2.6)
The first term in the right-hand side is a sum of Yang-Mills actions which
depend on the horizontal gauge fields in $(d-1)$-dimensional hyperplane at
$x^{(d)}=\tau$. As to the second term, the vertical gauge fields in different
slices are not coupled to each other. Therefore the summand about $\tau$ in
the second term is written in an action of a $(d-1)$-dimensional non-linear
$\sigma$-model for the vertical gauge fields as
${\cal
A}_{d-1}^{\sigma}(\Lambda^{0};U^{h}(\tau),U^{h}(\tau+1))=-\frac{1}{2}\sum_{\langle{\bf
i},{\bf j}\rangle\subset\Lambda^{0}}{\rm Re}\,{\rm Tr}\,{U_{\bf
i}^{v}(\tau)}^{-1}U_{{\bf i}{\bf j}}^{h}(\tau)U_{\bf j}^{v}(\tau)U_{{\bf
j}{\bf i}}^{h}(\tau+1)$ (2.7)
in the external gauge fields, $U^{h}(\tau)=\\{U_{{\bf i}{\bf j}}^{h}(\tau)\\}$
and $U^{h}(\tau+1)=\\{U_{{\bf i}{\bf j}}^{h}(\tau+1)\\}$.
Let $\mathbb{S}^{3}$ denote the $3$-sphere. In order to express the gauge
fields in terms of spins ${\bf S}\in\mathbb{S}^{3}$, we use the homeomorphism
$\varphi:\mathbb{S}^{3}\rightarrow{\rm SU(2)}$ which is defined by [4]
$\varphi({\bf
S})=\varphi\left(S^{(0)},S^{(1)},S^{(2)},S^{(3)}\right)=\left(\matrix{S^{(0)}+iS^{(3)}&-S^{(1)}+iS^{(2)}\cr
S^{(1)}+iS^{(2)}&S^{(0)}-iS^{(3)}\cr}\right)$ (2.8)
with the radius $(S^{(0)})^{2}+(S^{(1)})^{2}+(S^{(2)})^{2}+(S^{(3)})^{2}=1$.
Then the interaction potential $V_{12}$ between two spins ${\bf S}_{1}$ and
${\bf S}_{2}$ in the non-linear $\sigma$-model (2.7) can be written
$V_{12}=-\frac{1}{2}\,{\rm Re}\,{\rm Tr}\,\varphi\left({\bf
S}_{1}\right)^{-1}\varphi\left(\mbox{\boldmath$\sigma$}_{1}\right)\varphi\left({\bf
S}_{2}\right)\varphi\left(\mbox{\boldmath$\sigma$}_{2}\right)^{-1},$ (2.9)
where we have written $\mbox{\boldmath$\sigma$}_{1}$ and
$\mbox{\boldmath$\sigma$}_{2}$ for the external horizontal gauge fields. When
the external gauge fields, $\mbox{\boldmath$\sigma$}_{\ell}$, take the vacuum
configurations,
$\mbox{\boldmath$\sigma$}_{1}=\mbox{\boldmath$\sigma$}_{2}=(1,0,0,0)$, the
interaction becomes that of the O(4) non-linear $\sigma$-model in $(d-1)$
dimensions as
$V_{12}=-\frac{1}{2}\,{\rm Re}\,{\rm Tr}\,\varphi\left({\bf
S}_{1}\right)^{-1}\varphi\left({\bf S}_{2}\right)=-{\bf S}_{1}\cdot{\bf
S}_{2}=-\sum_{k=0}^{3}S_{1}^{(k)}S_{2}^{(k)}.$ (2.10)
As is well known, the O(4) non-linear $\sigma$-model shows a long-range order
of spins at low temperatures in three or higher dimensions [7]. The long-range
order leads to the perimeter law of the decay of the Wilson loop [4]. The
perimeter law implies deconfinement of quarks. If the confinement of quarks
indeed occurs in the SU(2) gauge theory, the fluctuations of the external
gauge fields around the vacuum must destroy the long-range order of the O(4)
non-linear $\sigma$-model.
In order to take account of the fluctuations around the vacuum configuration
of the external gauge fields, we approximate $\mbox{\boldmath$\sigma$}_{\ell}$
as
$\mbox{\boldmath$\sigma$}_{\ell}=\left(\sqrt{1-\left|\hat{\mbox{\boldmath$\sigma$}}_{\ell}\right|^{2}},\hat{\mbox{\boldmath$\sigma$}}_{\ell}\right)\approx\left(1,\hat{\mbox{\boldmath$\sigma$}}_{\ell}\right)$
(2.11)
with small fluctuations,
$\hat{\mbox{\boldmath$\sigma$}}_{\ell}=\left(\sigma_{\ell}^{(1)},\sigma_{\ell}^{(2)},\sigma_{\ell}^{(3)}\right),\quad\mbox{for}\
\ell=1,2.$ (2.12)
We write
$\delta\mbox{\boldmath$\sigma$}_{\ell}=(0,\hat{\mbox{\boldmath$\sigma$}}_{\ell})$.
Then the two-body potential is written
$V_{12}\approx-{\bf S}_{1}\cdot{\bf S}_{2}-\frac{1}{2}\,{\rm Re}\,{\rm
Tr}\,\varphi({\bf
S}_{1})^{-1}\varphi^{\prime}(\delta\mbox{\boldmath$\sigma$}_{1})\varphi({\bf
S}_{2})-\frac{1}{2}\ {\rm Re}\ {\rm Tr}\ \varphi({\bf S}_{1})^{-1}\varphi({\bf
S}_{2})\varphi^{\prime}(-\delta\mbox{\boldmath$\sigma$}_{2}),$ (2.13)
dropping the second order333The contributions of the second order of the
fluctuations $\delta\mbox{\boldmath$\sigma$}_{\ell}$ give order of temperature
$T=\beta^{-1}$ in the potential $V_{12}$. Therefore one can expect that the
contributions of the second order slightly modifies the coupling constants of
the interaction potentials at low temperatures. in the fluctuations
$\delta\mbox{\boldmath$\sigma$}_{\ell}$. Here we have written
$\varphi^{\prime}(\delta\mbox{\boldmath$\sigma$})=\left(\matrix{i\sigma^{(3)}&-\sigma^{(1)}+i\sigma^{(2)}\cr\sigma^{(1)}+i\sigma^{(2)}&-i\sigma^{(3)}\cr}\right).$
(2.14)
The right-hand side of (2.13) can be written
$V_{12}\approx V_{0}+V_{\rm R}$ (2.15)
with
$V_{0}=-{\bf S}_{1}\cdot{\bf S}_{2}$ (2.16)
and
$V_{\rm R}=-\sqrt{2}\,\hat{\mbox{\boldmath$\sigma$}}_{+}\cdot\left(\hat{\bf
S}_{1}\times\hat{\bf
S}_{2}\right)-\sqrt{2}\,\hat{\mbox{\boldmath$\sigma$}}_{-}\cdot\left(S_{1}^{(0)}\hat{\bf
S}_{2}-S_{2}^{(0)}\hat{\bf S}_{1}\right),$ (2.17)
where
$\hat{\mbox{\boldmath$\sigma$}}_{\pm}=\frac{1}{\sqrt{2}}\left(\hat{\mbox{\boldmath$\sigma$}}_{2}\pm\hat{\mbox{\boldmath$\sigma$}}_{1}\right),$
(2.18)
and
$\hat{\bf
S}_{\ell}=\left(S_{\ell}^{(1)},S_{\ell}^{(2)},S_{\ell}^{(3)}\right),\quad\ell=1,2.$
(2.19)
Thus the present system can be expressed as the O(4) non-linear $\sigma$-model
in the heat bath. The interaction between the non-linear $\sigma$-model and
the heat bath is given by $V_{\rm R}$.
## 3 Langevin dynamics for two particles on $\mathbb{S}^{3}$.
If we can integrate out the degrees of freedom of the heat bath, then we can
obtain the effective action of the non-linear $\sigma$-model. However, it is
very difficult problem. Instead of this way, we replace the fluctuations of
the external gauge fields with Gaussian random variables. Then, the spins of
the $\sigma$-model can be interpreted as the “particles” which move on
$\mathbb{S}^{3}$, acted by the two-body interaction and the random forces.
In order to derive the effective two-body interaction between two spins of the
$\sigma$-model within this approximation, we first introduce the Langevin
equation for the two “particles”. We write
${\hat{x}}_{\ell}=(x_{\ell}^{(1)},x_{\ell}^{(2)},x_{\ell}^{(3)})$, $\ell=1,2$,
for the local coordinates of the two 3-spheres $\mathbb{S}^{3}$. Then the
Langevin equation [8] is given by
$\frac{d}{dt}x_{\ell}^{(i)}=F_{0,\ell}^{(i)}+F_{{\rm
R},\ell}^{(i)},\quad\ell=1,2;\ \ i=1,2,3.$ (3.1)
with the forces, $F_{0,\ell}^{(i)},F_{{\rm R},\ell}^{(i)}$, which are given by
the gradient444See, for example, the book [9]. of the potentials as
$F_{0,\ell}^{(i)}=-g^{ij}_{\ \ell}\partial_{j,\ell}V_{0}$ (3.2)
and
$F_{{\rm R},\ell}^{(i)}=-g^{ij}_{\ \ell}\partial_{j,\ell}V_{\rm R},$ (3.3)
where $g^{ij}_{\ \ell}$ is the matrix inverse of the metric tensor
$g_{ij,\ell}$ for the “particle” $\ell$, and we have used the Einstein
summation convention and written
$\partial_{i,\ell}=\frac{\partial}{\partial x_{\ell}^{(i)}}.$ (3.4)
Let $\rho_{t}({\hat{x}}_{1},{\hat{x}}_{2})$ be the distribution of the two
“particles” on $\mathbb{S}^{3}\times\mathbb{S}^{3}$. The expectation value of
the function $f({\hat{x}}_{1},{\hat{x}}_{2})$ on
$\mathbb{S}^{3}\times\mathbb{S}^{3}$ at time $t$ is given by
$\left\langle
f\right\rangle_{t}:=\int_{\mathbb{S}^{3}\times\mathbb{S}^{3}}f({\hat{x}}_{1},{\hat{x}}_{2})\rho_{t}({\hat{x}}_{1},{\hat{x}}_{2})d\mu_{1}d\mu_{2},$
(3.5)
where we have written
$d\mu_{\ell}=\sqrt{{\rm
det}\,g_{\ell}}\,dx_{\ell}^{(1)}dx_{\ell}^{(2)}dx_{\ell}^{(3)}\quad\mbox{for \
\ }\ell=1,2.$ (3.6)
For a small $\Delta t>0$, the following relation must hold:
$\left\langle f\right\rangle_{t+\Delta
t}=\mathbb{E}\int_{\mathbb{S}^{3}\times\mathbb{S}^{3}}f({\hat{x}}_{1}(t+\Delta
t),{\hat{x}}_{2}(t+\Delta
t))\rho_{t}({\hat{x}}_{1},{\hat{x}}_{2})d\mu_{1}d\mu_{2}+{\cal O}((\Delta
t)^{2}),$ (3.7)
where $\mathbb{E}$ stands for the average over the fluctuations
$\hat{\mbox{\boldmath$\sigma$}}_{\ell}$, $\ell=1,2$, and
${\hat{x}}_{\ell}(t+\Delta t)$ is the solution of the Langevin equation (3.1)
with the initial conditions ${\hat{x}}_{\ell}(t)={\hat{x}}_{\ell}$ at time
$t$. As usual, we assume that, for the short interval $[t,t+\Delta t]$, the
fluctuations ${\hat{\sigma}}_{\ell}^{(i)}$ are constant, and satisfy
$\mathbb{E}\left[\sigma_{\ell}^{(i)}\right]=0,\quad\mathbb{E}\left[\sigma_{\ell}^{(i)}\sigma_{\ell}^{(j)}\right]=\frac{\alpha}{\Delta
t}\delta^{ij}\quad\mbox{and}\quad\mathbb{E}\left[\sigma_{1}^{(i)}\sigma_{2}^{(j)}\right]=\frac{\alpha^{\prime}}{\Delta
t}\delta^{ij},$ (3.8)
where $\alpha$ and $\alpha^{\prime}$ are a nonnegative constant, and
$\delta^{ij}$ is the Kronecker delta. Physically, a natural assumption is that
$\alpha$ and $\alpha^{\prime}$ satisfy the condition
$\alpha>\alpha^{\prime}>0$. From the relation between the fluctuations and the
temperature of the heat bath, both of $\alpha$ and $\alpha^{\prime}$ are
proportional to the temperature $\beta^{-1}$ of the heat bath.
From the Langevin equation (3.1), we have
$x_{\ell}^{(i)}(s)-x_{\ell}^{(i)}(t)=\int_{t}^{s}dt^{\prime}\frac{dx_{\ell}^{(i)}(t^{\prime})}{dt}=\int_{t}^{s}dt^{\prime}F_{\ell}^{(i)}({\tilde{x}}(t^{\prime})),$
(3.9)
where we have written $F_{\ell}^{(i)}=F_{0,\ell}^{(i)}+F_{R,\ell}^{(i)}$ and
${\tilde{x}}(t)=({\hat{x}}_{1}(t),{\hat{x}}_{2}(t))$. Using this relation, we
obtain
$F_{\ell}^{(i)}({\tilde{x}}(t^{\prime}))=F_{\ell}^{(i)}({\tilde{x}}(t))+\sum_{m,k}\frac{\partial
F_{\ell}^{(i)}({\tilde{x}}(t))}{\partial
x_{m}^{(k)}}\int_{t}^{t^{\prime}}dt^{\prime\prime}F_{m}^{(k)}({\tilde{x}}(t^{\prime\prime}))+\cdots.$
(3.10)
Combining these, the expansion with respect to $\Delta t$ is derived as
$x_{\ell}^{(i)}(t+\Delta
t)=x_{\ell}^{(i)}(t)+F_{\ell}^{(i)}({\tilde{x}}(t))\Delta
t+\frac{1}{2}\sum_{m,k}\frac{\partial F_{\ell}^{(i)}({\tilde{x}}(t))}{\partial
x_{m}^{(k)}}F_{m}^{(k)}({\tilde{x}}(t))(\Delta t)^{2}+\cdots.$ (3.11)
Substituting this into (3.7) and using (3.8), the order of $\Delta t$ yields
$\displaystyle\int_{M}d\mu
f({\tilde{x}})\frac{\partial\rho_{t}({\tilde{x}})}{\partial t}$
$\displaystyle=$ $\displaystyle\int_{M}d\mu\sum_{\ell,i}\frac{\partial
f({\tilde{x}})}{\partial
x_{\ell}^{(i)}}F_{0,\ell}^{(i)}({\tilde{x}})\rho_{t}({\tilde{x}})$ (3.12)
$\displaystyle+$ $\displaystyle\frac{\Delta
t}{2}\int_{M}d\mu\sum_{\ell,i;m,j}\frac{\partial^{2}f({\tilde{x}})}{\partial
x_{\ell}^{(i)}\partial x_{m}^{(j)}}\mathbb{E}\left[F_{{\rm
R},\ell}^{(i)}({\tilde{x}})F_{{\rm
R},m}^{(j)}({\tilde{x}})\right]\rho_{t}({\tilde{x}})$ $\displaystyle+$
$\displaystyle\frac{\Delta t}{2}\int_{M}d\mu\sum_{\ell,i;n,k}\frac{\partial
f({\tilde{x}})}{\partial x_{\ell}^{(i)}}\mathbb{E}\left[\frac{\partial F_{{\rm
R},\ell}^{(i)}({\tilde{x}})}{\partial x_{n}^{(k)}}F_{{\rm
R},n}^{(k)}({\tilde{x}})\right]\rho_{t}({\tilde{x}}),$
where we have written $M=\mathbb{S}^{3}\times\mathbb{S}^{3}$ and
$d\mu=d\mu_{1}d\mu_{2}$. Since this equation holds for any function $f$, we
can derive the equation of the time evolution for the distribution $\rho_{t}$,
i.e., the Fokker-Planck equation.
To this end, consider first the first term in the right-hand side of (3.12).
Note that
$\displaystyle\sum_{i}\frac{\partial f({\tilde{x}})}{\partial
x_{\ell}^{(i)}}F_{0,\ell}^{(i)}({\tilde{x}})\rho_{t}({\tilde{x}})$
$\displaystyle=$ $\displaystyle\sum_{i}\frac{1}{\sqrt{{\rm
det}\,g_{\ell}}}\frac{\partial}{\partial x_{\ell}^{(i)}}\sqrt{{\rm
det}\,g_{\ell}}\,F_{0,\ell}^{(i)}({\tilde{x}})f({\tilde{x}})\rho_{t}({\tilde{x}})$
(3.13) $\displaystyle-$
$\displaystyle\sum_{i}f({\tilde{x}})\frac{1}{\sqrt{{\rm
det}\,g_{\ell}}}\frac{\partial}{\partial x_{\ell}^{(i)}}\sqrt{{\rm
det}\,g_{\ell}}\,F_{0,\ell}^{(i)}({\tilde{x}})\rho_{t}({\tilde{x}})$
$\displaystyle=$ $\displaystyle{\rm
div}_{\ell}\left[F_{0,\ell}({\tilde{x}})f({\tilde{x}})\rho_{t}({\tilde{x}})\right]-f({\tilde{x}})\,{\rm
div}_{\ell}\left[F_{0,\ell}({\tilde{x}})\rho_{t}({\tilde{x}})\right],$
where ${\rm div}_{\ell}$ stands for the divergence for the “particle” $\ell$.
Combining this with the divergence theorem,555See, for example, Theorem 5.11
in Chap. II of the book [9].
$\int_{\mathbb{S}^{3}}d\mu_{\ell}\;{\rm div}_{\ell}\,v_{\ell}=0,$ (3.14)
for a vector field $v_{\ell}$ on $\mathbb{S}^{3}$, the first term in the
right-hand side of (3.12) is written as
$\sum_{\ell,i}\int_{M}d\mu\left(\partial_{i,\ell}f\right)F_{0,\ell}^{(i)}\rho_{t}=-\sum_{\ell}\int_{M}d\mu\,f\,{\rm
div}_{\ell}\left(F_{0,\ell}\rho_{t}\right).$ (3.15)
As to the second and third terms in the right-hand side of (3.12), we must
compute the second moments of the random forces. But one can treat these terms
in the same way as in the above. The detail is given in Appendix A. As a
result, the Fokker-Planck equation is given by
$\displaystyle\frac{\partial\rho_{t}}{\partial t}$ $\displaystyle=$
$\displaystyle-\sum_{\ell}{\rm
div}_{\ell}\left(F_{0,\ell}\rho_{t}\right)+(\alpha+\alpha^{\prime})\sum_{\ell}\left\\{\Delta_{\ell}\rho_{t}-{\rm
div}_{\ell}\left[\xi_{\ell}\,{\rm
div}_{\ell}(\xi_{\ell}\rho_{t})\right]\right\\}$ (3.16)
$\displaystyle-(\alpha+\alpha^{\prime})\left\\{{\rm
div}_{1}\left[\mbox{\boldmath$\eta$}_{1}W\cdot{\rm
div}_{2}(\mbox{\boldmath$\eta$}_{2}\rho_{t})\right]+{\rm
div}_{2}\left[\mbox{\boldmath$\eta$}_{2}W\cdot{\rm
div}_{1}(\mbox{\boldmath$\eta$}_{1}\rho_{t})\right]\right\\}$
$\displaystyle-2\alpha^{\prime}\sum_{m,n}{\rm
div}_{m}\left[\hat{\mbox{\boldmath$\zeta$}}_{m}\cdot{\rm
div}_{n}(\hat{\mbox{\boldmath$\zeta$}}_{n}\rho_{t})\right],$
where $\Delta_{\ell}$ is the Laplacian for the “particle” $\ell$, and we have
written $W={\bf S}_{1}\cdot{\bf S}_{2}$; the vector fields, $\xi_{\ell}$,
$\mbox{\boldmath$\eta$}_{\ell}$ and $\hat{\mbox{\boldmath$\zeta$}}_{\ell}$,
are given by
$\xi_{\ell}^{i}:=g^{ij}_{\ \ell}\partial_{j,\ell}W,$ (3.17)
$\mbox{\boldmath$\eta$}_{\ell}^{i}:=g^{ij}_{\ \ell}\partial_{j,\ell}{\bf
S}_{\ell}$ (3.18)
and
$\hat{\mbox{\boldmath$\zeta$}}_{\ell}^{i}:=g^{ij}_{\
\ell}\partial_{j,\ell}\left(S_{1}^{(0)}{\hat{\bf S}}_{2}-S_{2}^{(0)}{\hat{\bf
S}}_{1}\right)$ (3.19)
for $i=1,2,3$ and $\ell=1,2$. Here the vectors
$\mbox{\boldmath$\eta$}_{\ell}^{i}$ have four components like ${\bf
S}_{\ell}$, and $\hat{\mbox{\boldmath$\zeta$}}_{\ell}^{i}$ have three
components like ${\hat{\bf S}}_{\ell}$. This Fokker-Planck equation can be
written
$\frac{\partial\rho_{t}}{\partial t}=-{\rm div}\,J\quad\mbox{with}\ \ {\rm
div}\,J={\rm div}_{1}J_{1}+{\rm div}_{2}J_{2}$ (3.20)
in terms of the current $J=(J_{1},J_{2})$ which is given by
$J_{\ \ell}^{i}=g^{ij}_{\ \ell}J_{j,\ell}$ (3.21)
with
$\displaystyle J_{j,1}$ $\displaystyle=$
$\displaystyle-(\partial_{j,1}V_{0})\rho_{t}-(\alpha+\alpha^{\prime})\left\\{\partial_{j,1}\rho_{t}-\left[(\partial_{j,1}W){\rm
div}_{1}(\xi_{1}\rho_{t})+W(\partial_{j,1}{\bf S}_{1})\cdot{\rm
div}_{2}(\mbox{\boldmath$\eta$}_{2}\rho_{t})\right]\right\\}$ (3.22)
$\displaystyle+2\alpha^{\prime}\hat{\mbox{\boldmath$\zeta$}}_{j,1}\cdot\left[{\rm
div}_{1}(\hat{\mbox{\boldmath$\zeta$}}_{1}\rho_{t})+{\rm
div}_{2}(\hat{\mbox{\boldmath$\zeta$}}_{2}\rho_{t})\right]$
and with $J_{j,2}$ given by exchanging the subscripts 1 and 2 in $J_{j,1}$.
Here we have written
$\hat{\mbox{\boldmath$\zeta$}}_{i,\ell}:=\partial_{i,\ell}\left(S_{1}^{(0)}{\hat{\bf
S}}_{2}-S_{2}^{(0)}{\hat{\bf S}}_{1}\right).$ (3.23)
## 4 A steady state for the Fokker-Planck dynamics
The effective potential $V_{\rm eff}$ between the two “particles” is derived
from a steady distribution $\rho_{t}=\rho$ for the Fokker-Planck equation
(3.20), as in (4.7) below. For a steady distribution $\rho_{t}=\rho$, the
Fokker-Planck equation (3.20) becomes ${\rm div}\,J=0$. In order to obtain the
solution near the north pole, ${\bf S}_{\ell}=(1,0,0,0)$, for $\ell=1,2$, we
introduce the local coordinates, $(x_{\ell},y_{\ell},z_{\ell})$ for
$\ell=1,2$, as
${\bf
S}_{\ell}=\left(\sqrt{1-x_{\ell}^{2}-y_{\ell}^{2}-z_{\ell}^{2}},x_{\ell},y_{\ell},z_{\ell}\right).$
(4.1)
We write
${\bf r}=(x,y,z)=(x_{1}-x_{2},y_{1}-y_{2},z_{1}-z_{2})$ (4.2)
and
${\bf R}=(X,Y,Z)=(x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2}).$ (4.3)
We also write $r=|{\bf r}|$ and $R=|{\bf R}|$. In order to solve the partial
differential equation ${\rm div}\,J=0$, we employ the Cauchy-Kowalevski type
expansion666See, for example, Sec. D of Chap. 1 in the book [11]. with respect
to small $x_{\ell},y_{\ell},z_{\ell}$.
Let us compute the $x$-component $J_{x,1}$ of the current $J_{1}$ for the
particle 1. Note that
$V_{0}=-{\bf S}_{1}\cdot{\bf S}_{2}=-1+\frac{1}{2}r^{2}+\frac{1}{8}({\bf
r}\cdot{\bf R})^{2}+\cdots.$ (4.4)
Immediately,
$\frac{\partial V_{0}}{\partial x_{1}}=x+\frac{1}{4}({\bf r}\cdot{\bf
R})x+\frac{1}{4}({\bf r}\cdot{\bf R})X+\cdots.$ (4.5)
Therefore, the first term of $J_{x,1}$ of (3.22) becomes
$-(\partial_{x,1}V_{0})\rho=\left[-x-\frac{1}{4}({\bf r}\cdot{\bf
R})x-\frac{1}{4}({\bf r}\cdot{\bf R})X+\cdots\right]\rho.$ (4.6)
In order to treat the rest of the terms of $J_{x,1}$, we assume that the
steady state solution $\rho_{t}=\rho$ of ${\rm div}\,J=0$ has the form,
$\rho=\exp[-\beta V_{\rm eff}],$ (4.7)
where $V_{\rm eff}$ is the effective potential to be determined, and $\beta$
is the inverse temperature of the heat bath. Both of $\alpha$ and
$\alpha^{\prime}$ are proportional to the temperature $\beta^{-1}$ as
mentioned in the preceding section. The effective potential $V_{\rm eff}$ must
be vanishing for ${\bf r}=0$ because the two-body potential (2.13) becomes
constant irrespective of the external fluctuations for ${\bf S}_{1}={\bf
S}_{2}$. From this and taking account of the spherical and exchange
symmetries, we assume that the effective potential $V_{\rm eff}$ can be
expended as
$V_{\rm eff}=C_{20}r^{2}+C_{40}r^{4}+C_{22}r^{2}R^{2}+C_{22}^{\prime}({\bf
r}\cdot{\bf R})^{2}+\cdots,$ (4.8)
where $C_{20},C_{40},C_{22}$ and $C_{22}^{\prime}$ are the coefficients to be
determined. In the following, we take $\alpha$ and $\alpha^{\prime}$ to be
small, and ignore the order of $\alpha$ and $\alpha^{\prime}$.
For small $x_{\ell},y_{\ell},z_{\ell}$, the current $J_{x,1}$ is written
$\displaystyle J_{x,1}$ $\displaystyle=$
$\displaystyle\left[-x-\frac{1}{4}({\bf r}\cdot{\bf R})x-\frac{1}{4}({\bf
r}\cdot{\bf
R})X\right]\rho-(\alpha-\alpha^{\prime})\left(\frac{\partial}{\partial
x_{1}}-\frac{\partial}{\partial x_{2}}\right)\rho$ $\displaystyle+$
$\displaystyle(\alpha+\alpha^{\prime})\left[x\left(x\frac{\partial\rho}{\partial
x_{1}}+y\frac{\partial\rho}{\partial y_{1}}+z\frac{\partial\rho}{\partial
z_{1}}\right)+x\left(x_{2}\frac{\partial\rho}{\partial
x_{2}}+y_{2}\frac{\partial\rho}{\partial
y_{2}}+z_{2}\frac{\partial\rho}{\partial
z_{2}}\right)-\frac{r^{2}}{2}\frac{\partial\rho}{\partial x_{2}}\right]$
$\displaystyle+$ $\displaystyle
2\alpha^{\prime}\left[-x_{1}\left(x\frac{\partial\rho}{\partial
x_{1}}+y\frac{\partial\rho}{\partial y_{1}}+z\frac{\partial\rho}{\partial
z_{1}}\right)+x_{2}\left(x_{1}\frac{\partial\rho}{\partial
x_{1}}+y_{1}\frac{\partial\rho}{\partial
y_{1}}+z_{1}\frac{\partial\rho}{\partial z_{1}}\right)\right.$
$\displaystyle\left.-\left(\frac{3}{2}x+\frac{1}{2}X\right)\left(x_{2}\frac{\partial\rho}{\partial
x_{2}}+y_{2}\frac{\partial\rho}{\partial
y_{2}}+z_{2}\frac{\partial\rho}{\partial
z_{2}}\right)-r_{2}^{2}\frac{\partial\rho}{\partial
x_{1}}+\frac{1}{2}(r_{1}^{2}+r_{2}^{2})\frac{\partial\rho}{\partial
x_{2}}\right]+\cdots.$
The derivation is given in Appendix B. Let us substitute $\rho$ of (4.7) with
the effective potential (4.8) into this right-hand side. First of all, since
the leading order which is proportional to $x\exp[-\beta V_{\rm eff}]$ must be
vanishing, we have
$4\beta(\alpha-\alpha^{\prime})C_{20}=1.$ (4.10)
Since we can choose
$\beta=\frac{1}{\alpha-\alpha^{\prime}}$ (4.11)
without loss of generality, we have
$C_{20}=\frac{1}{4}.$ (4.12)
Using these, we get
$-(\alpha-\alpha^{\prime})\left(\frac{\partial}{\partial
x_{1}}-\frac{\partial}{\partial x_{2}}\right)\exp\left[-\beta V_{\rm
eff}\right]=\left(\frac{\partial V_{\rm eff}}{\partial x_{1}}-\frac{\partial
V_{\rm eff}}{\partial x_{2}}\right)\exp\left[-\beta V_{\rm eff}\right]$ (4.13)
with
$\left(\frac{\partial}{\partial x_{1}}-\frac{\partial}{\partial
x_{2}}\right)V_{\rm eff}=x+8C_{40}r^{2}x+4C_{22}R^{2}x+4C_{22}^{\prime}({\bf
r}\cdot{\bf R})X+\cdots.$ (4.14)
Moreover we have
$\left(x\frac{\partial}{\partial x_{1}}+y\frac{\partial}{\partial
y_{1}}+z\frac{\partial}{\partial z_{1}}\right)\rho=\left(-\frac{1}{2}\beta
r^{2}+\cdots\right)\exp[-\beta V_{\rm eff}],$ (4.15)
$\left(x_{1}\frac{\partial}{\partial x_{1}}+y_{1}\frac{\partial}{\partial
y_{1}}+z_{1}\frac{\partial}{\partial z_{1}}\right)\rho=\left[-\frac{1}{4}\beta
r^{2}-\frac{1}{4}\beta({\bf r}\cdot{\bf R})+\cdots\right]\exp[-\beta V_{\rm
eff}],$ (4.16) $\left(x_{2}\frac{\partial}{\partial
x_{2}}+y_{2}\frac{\partial}{\partial y_{2}}+z_{2}\frac{\partial}{\partial
z_{2}}\right)\rho=\left[-\frac{1}{4}\beta r^{2}+\frac{1}{4}\beta({\bf
r}\cdot{\bf R})+\cdots\right]\exp[-\beta V_{\rm eff}],$ (4.17)
$-\frac{r^{2}}{2}\frac{\partial\rho}{\partial
x_{2}}=\left[-\frac{\beta}{4}xr^{2}+\cdots\right]\exp[-\beta V_{\rm eff}]$
(4.18)
and
$-r_{2}^{2}\frac{\partial\rho}{\partial
x_{1}}+\frac{1}{2}(r_{1}^{2}+r_{2}^{2})\frac{\partial\rho}{\partial
x_{2}}=\frac{\beta}{4}x\left[r^{2}+R^{2}-({\bf r}\cdot{\bf
R})\right]\exp[-\beta V_{\rm eff}]+\cdots.$ (4.19)
Substituting these into (LABEL:Jx1expand), we obtain
$\displaystyle J_{x,1}\exp[\beta V_{\rm eff}]$ $\displaystyle=$
$\displaystyle\left[8C_{40}-1\right]r^{2}x+\frac{\alpha^{\prime}\beta}{2}\left[r^{2}X-({\bf
r}\cdot{\bf R})x\right]$ (4.20) $\displaystyle+$
$\displaystyle\left[4C_{22}+\frac{\alpha^{\prime}\beta}{2}\right]R^{2}x+\left[4C_{22}^{\prime}-\frac{(\alpha+\alpha^{\prime})\beta}{4}\right]({\bf
r}\cdot{\bf R})X+\cdots.$
From ${\rm div}\,J=0$, the coefficients must satisfy the relations,
$5(8C_{40}-1)+\alpha^{\prime}\beta=0$ (4.21)
and
$3\left[4C_{22}+\frac{\alpha^{\prime}\beta}{2}\right]+\left[4C_{22}^{\prime}-\frac{(\alpha+\alpha^{\prime})\beta}{4}\right]=0.$
(4.22)
Using these relations, the current $J_{x,1}$ can be written
$J_{x,1}=\left\\{-\frac{\alpha^{\prime}\beta}{5}r^{2}x+\frac{\alpha^{\prime}\beta}{2}\left[r^{2}X-({\bf
r}\cdot{\bf R})x\right]+A\left[R^{2}x-3({\bf r}\cdot{\bf
R})X\right]\right\\}\exp[-\beta V_{\rm eff}]+\cdots$ (4.23)
with the constant,
$A=4C_{22}+\frac{\alpha^{\prime}\beta}{2},$ (4.24)
which we cannot determine in the present method. Clearly one notices that in
${\rm div}\,J$, there appear the other terms,
$\frac{1}{5}\alpha^{\prime}\beta^{2}r^{4}\quad\mbox{and}\quad-A\beta[r^{2}R^{2}-3({\bf
r}\cdot{\bf R})^{2}].$ (4.25)
These are higher order in powers of the local coordinates but order of
$\beta$. Since the equation ${\rm div}\,J=0$ must hold, this implies that
there must exist some terms of order of $\beta$ in the effective potential
$V_{\rm eff}$ so as to cancel the above terms of (4.25).
When both of the coefficients $C_{22}$ and $C_{22}^{\prime}$ depend on
$\beta$, the corresponding terms may appear in the expansion. In this case,
from (4.22), we have
$C_{22}\sim C\beta\quad\mbox{and}\quad C_{22}^{\prime}\sim-3C\beta$ (4.26)
with some constant $C$ for a large $\beta$. Substituting these into $V_{\rm
eff}$, we have
$V_{\rm eff}\sim\frac{1}{4}r^{2}+C_{40}r^{4}+C\beta[r^{2}R^{2}-3({\bf
r}\cdot{\bf R})^{2}].$ (4.27)
This leads to instability of binding of the two particles because the value of
$R^{2}$ is expected to become larger than order of $\beta^{-1}$ in the thermal
equilibrium. Thus we require that both of $C_{22}$ and $C_{22}^{\prime}$ are
order of 1.
In consequence, we need the following terms in the effective potential $V_{\rm
eff}$:
$C_{60}r^{6},\quad C_{42}r^{4}R^{2},\quad C_{42}^{\prime}r^{2}({\bf
r}\cdot{\bf R})^{2}.$ (4.28)
Here all the coefficients, $C_{60},C_{42},C_{42}^{\prime}$, are proportional
to $\beta$ for a large $\beta$. In the same way as in the above, we can
determine these coefficients as
$C_{60}=-\frac{3!}{7!}\alpha^{\prime}\beta^{2},\quad
C_{42}=\frac{1}{56}A\beta\quad\mbox{and}\quad
C_{42}^{\prime}=-\frac{3}{56}A\beta$ (4.29)
so as to cancel the above terms (4.25) which appear in ${\rm div}\,J$. As a
result, the dominant contributions in the effective potential $V_{\rm eff}$
are given by
$V_{\rm
eff}\sim\frac{1}{4}r^{2}-\frac{3!}{7!}\alpha^{\prime}\beta^{2}r^{6}+\frac{1}{56}A\beta
r^{2}[r^{2}R^{2}-3({\bf r}\cdot{\bf R})^{2}]$ (4.30)
for a large $\beta$ because the second, third and fourth terms in the right-
hand side of (4.8) do not affect the critical behavior.
Now we discuss the critical behavior of the $(d-1)$-dimensional $\sigma$ model
with the above two-body interaction $V_{\rm eff}$. Consider first the case of
$A=0$. Namely the effective potential is given by
$V_{\rm eff}\sim\frac{1}{4}r^{2}-\frac{3!}{7!}\alpha^{\prime}\beta^{2}r^{6}$
(4.31)
for small $r$ and large $\beta$. The second term lowers the potential barrier.
Within a mean-field approximation [12], the critical temperature $T_{\rm C}$
can be estimated by the volume and the height of the potential well. More
precisely, $T_{\rm C}\sim({\rm volume})\times({\rm height})$. In the present
case, the width $w$ and the height $h$ of the effective potential $V_{\rm
eff}$ are estimated as
$w\sim(\lambda\beta)^{-1/4},\quad h\sim(\lambda\beta)^{-1/2},$ (4.32)
where we have written
$\lambda=12\cdot\frac{3!}{7!}\alpha^{\prime}\beta.$ (4.33)
Therefore the critical temperature $T_{\rm C}$ is estimated as
$T_{\rm C}\sim w^{3}\times h\sim(\lambda\beta)^{-5/4}.$ (4.34)
This is lower than $\beta^{-1}$ for small temperature $T=\beta^{-1}$. This
implies that the true critical temperature must be equal to zero.
In the case of $A\neq 0$, the third term in the right-hand side of (4.30) may
heighten the potential barrier if $R^{2}$ does not take a small value. But it
is impossible that the term heightens the potential barrier in all the
directions of ${\bf r}$. Thus we reach the same conclusion, $T_{\rm C}=0$.
Let us make the following two remarks:
1. 1.
Our argument can be applied to the systems in arbitrary dimensions. Therefore
a reader might think that our method suggests no phase transition for non-
Abelian lattice gauge theory also in five or higher dimensions. On this point,
we should remark the following: We used the two-body approximation,
considering only a single plaquette. When dealing with two plaquettes within
our method, three- and four-body interactions would appear in the effective
potential for the non-linear $\sigma$-model. The resulting interactions may
change the conclusion of this section. Namely a high-dimensional system may
exhibit a phase transition. Actually, in five or higher dimensions, the effect
of the three- or four-body interactions may not be ignored because the number
of the neighboring plaquettes for a fixed plaquette becomes large, compared to
low-dimensional systems. However, taking account of such interactions is not
so easy.
2. 2.
Consider the O(4) non-linear $\sigma$-model on the three-dimensional lattice
with the effective two-body interaction which we obtained. Then the
correlation length of the model leads to an estimate of the string tension [4,
5]. Does the scaling limit so obtained give the standard continuum? This
problem must be very important. But it is very difficult to compute the low
temperature asymptotics of the correlation length for such a weakly attractive
potential.
## 5 Difference between U(1) and SU(2) gauge theories
Let us see difference between U(1) and SU(2) gauge theories.
For this purpose, we apply the present method to the abelian case $G=$U(1). In
the case, the gauge field $U_{b}$ on a link $b$ is written
$U_{b}=\exp[i\theta_{b}]$ (5.1)
in terms of the angle variable $\theta_{b}\in[0,2\pi)$. Therefore the two-body
interaction $V_{12}$ between $\theta_{1}$ and $\theta_{2}$ is written
$V_{12}=-\cos(\theta_{1}-\theta_{2}+\sigma_{1}-\sigma_{2}),$ (5.2)
where $\sigma_{1}$ and $\sigma_{2}$ are the angle variables of the external
fields. We write $\theta=\theta_{1}-\theta_{2}$ and
$\delta\sigma=\sigma_{1}-\sigma_{2}$, and assume that $\delta\sigma$ is a
small fluctuation. Under this assumption, the potential can be approximated as
$V_{12}\approx-\cos\theta+\delta\sigma\sin\theta.$ (5.3)
Then the Langevin equation is given by
$\frac{d\theta}{dt}=-\sin\theta-\delta\sigma\cos\theta.$ (5.4)
As usual, we assume
$\mathbb{E}[(\delta\sigma)^{2}]=\frac{\alpha}{\Delta t}$ (5.5)
for a small $\Delta t$. In the same way as in the SU(2) case, we obtain the
Fokker-Planck equation,
$\frac{\partial\rho_{t}}{\partial
t}=\left[\frac{\partial}{\partial\theta}\sin\theta+\frac{\alpha}{2}\frac{\partial}{\partial\theta}\sin\theta\cos\theta+\frac{\alpha}{2}\frac{\partial^{2}}{\partial\theta^{2}}\cos^{2}\theta\right]\rho_{t}.$
(5.6)
For a steady state $\rho_{t}=\rho$, we have
$\left[\sin\theta+\frac{\alpha}{2}\sin\theta\cos\theta+\frac{\alpha}{2}\frac{\partial}{\partial\theta}\cos^{2}\theta\right]\rho=0.$
(5.7)
One can easily find the solution,
$\rho=\cases{\displaystyle{(\cos\theta)^{-1}\exp\left[-2\alpha^{-1}/{\cos\theta}\right]},&for
$-\pi/2<\theta<\pi/2$;\cr\quad 0,&otherwise.}$ (5.8)
Since the diffusion disappears at $\theta=\pm\pi/2$ in the right-hand side of
(5.4), the “particle” cannot move beyond the points. Clearly, we have
$\rho\sim{\rm const.}\exp[-\alpha^{-1}\theta^{2}]$ (5.9)
for a small $\theta$. Thus there is no term which is proportional to
$\alpha^{-1}$ or higher powers of $\alpha^{-1}$ in the effective potential,
and the critical behavior can be expected to be the same as the standard O(2)
nonlinear-$\sigma$ model. This is consistent with the rigorous result of [2,
3].
## Appendix A Derivation of the Fokker-Planck equation
Consider first the case with $\alpha^{\prime}=0$ in (3.8). We introduce
$\sigma^{ij}$ satisfying $\sigma^{ji}=-\sigma^{ij}$ with
$(\sigma^{01},\sigma^{02},\sigma^{03})=(\sigma_{+}^{(1)},\sigma_{+}^{(2)},\sigma_{+}^{(3)}),\quad\mbox{and}\quad(\sigma^{23},\sigma^{31},\sigma^{12})=(\sigma_{-}^{(1)},\sigma_{-}^{(2)},\sigma_{-}^{(3)}).$
(A.1)
Then the random potential $V_{\rm R}$ of (2.17) can be written
$V_{\rm
R}=-\frac{1}{\sqrt{2}}\varepsilon_{ijk\ell}\,\sigma^{ij}\,S_{1}^{(k)}S_{2}^{(\ell)},$
(A.2)
where $\varepsilon_{ijk\ell}$ is completely antisymmetric and satisfies
$\varepsilon_{0123}=+1$, and we have used the Einstein summation convention.
From $\alpha^{\prime}=0$, we have
$\mathbb{E}\left[\sigma^{\alpha\beta}\sigma^{mn}\right]=\frac{\alpha}{\Delta
t}\left(\delta^{\alpha m}\delta^{\beta n}-\delta^{\alpha n}\delta^{\beta
m}\right).$ (A.3)
Using (A.2) and (A.3), we obtain
$\displaystyle\mathbb{E}\left[\left(\partial_{\ell,1}V_{\rm
R}\right)\left(\partial_{k,1}V_{\rm R}\right)\right]$ (A.4) $\displaystyle=$
$\displaystyle\frac{1}{2}\mathbb{E}\left[\varepsilon_{\alpha\beta\gamma\delta}\sigma^{\alpha\beta}\left(\partial_{\ell,1}S_{1}^{(\gamma)}\right)S_{2}^{(\delta)}\varepsilon_{mnst}\sigma^{mn}\left(\partial_{k,1}S_{1}^{(s)}\right)S_{2}^{(t)}\right]$
$\displaystyle=$ $\displaystyle\frac{\alpha}{2\Delta
t}\varepsilon_{\alpha\beta\gamma\delta}\varepsilon_{mnst}(\delta^{\alpha
m}\delta^{\beta n}-\delta^{\alpha n}\delta^{\beta
m})\left(\partial_{\ell,1}S_{1}^{(\gamma)}\right)S_{2}^{(\delta)}\left(\partial_{k,1}S_{1}^{(s)}\right)S_{2}^{(t)}$
$\displaystyle=$ $\displaystyle\frac{2\alpha}{\Delta
t}\sum_{\gamma,\delta}\left[\left(\partial_{\ell,1}S_{1}^{(\gamma)}\right)\left(\partial_{k,1}S_{1}^{(\gamma)}\right)S_{2}^{(\delta)}S_{2}^{(\delta)}-\left(\partial_{\ell,1}S_{1}^{(\gamma)}\right)S_{2}^{(\gamma)}\left(\partial_{k,1}S_{1}^{(\delta)}\right)S_{2}^{(\delta)}\right].$
Using the metric
$g_{ij,\ell}=\frac{\partial{\bf S}_{\ell}}{\partial
x_{\ell}^{(i)}}\cdot\frac{\partial{\bf S}_{\ell}}{\partial x_{\ell}^{(j)}}$
(A.5)
of $\mathbb{S}^{3}$ for the “particle” $\ell$, the above result is written
$\mathbb{E}\left[\left(\partial_{\ell,1}V_{\rm
R}\right)\left(\partial_{k,1}V_{\rm R}\right)\right]=\frac{2\alpha}{\Delta
t}\left[g_{\ell k,1}-(\partial_{\ell,1}W)(\partial_{k,1}W)\right]$ (A.6)
and
$\mathbb{E}\left[\left(\partial_{\ell,2}V_{\rm
R}\right)\left(\partial_{k,2}V_{\rm R}\right)\right]=\frac{2\alpha}{\Delta
t}\left[g_{\ell k,2}-(\partial_{\ell,2}W)(\partial_{k,2}W)\right],$ (A.7)
where we have written $W={\bf S}_{1}\cdot{\bf S}_{2}$. Similarly, we have
$\mathbb{E}\left[\left(\partial_{k,1}\partial_{j,1}V_{\rm
R}\right)\left(\partial_{\ell,1}V_{\rm R}\right)\right]=\frac{2\alpha}{\Delta
t}\sum_{\gamma,\delta}\left[\frac{\partial^{2}S_{1}^{(\gamma)}}{\partial
x_{1}^{(k)}\partial x_{1}^{(j)}}\frac{\partial S_{1}^{(\gamma)}}{\partial
x_{1}^{(\ell)}}S_{2}^{(\delta)}S_{2}^{(\delta)}-\frac{\partial^{2}S_{1}^{(\gamma)}}{\partial
x_{1}^{(k)}\partial x_{1}^{(j)}}S_{2}^{(\gamma)}\frac{\partial
S_{1}^{(\delta)}}{\partial x_{1}^{(\ell)}}S_{2}^{(\delta)}\right].$ (A.8)
Combining this with
$\sum_{\gamma}\frac{\partial^{2}S_{1}^{(\gamma)}}{\partial x_{1}^{(k)}\partial
x_{1}^{(j)}}\frac{\partial S_{1}^{(\gamma)}}{\partial
x_{1}^{(\ell)}}=\Gamma_{kj,1}^{m}g_{m\ell,1},$ (A.9)
we obtain
$\mathbb{E}\left[\left(\partial_{k,1}\partial_{j,1}V_{\rm
R}\right)\left(\partial_{\ell,1}V_{\rm R}\right)\right]=\frac{2\alpha}{\Delta
t}\left[\Gamma_{kj,1}^{m}g_{m\ell,1}-\left(\partial_{k,1}\partial_{j,1}W\right)\left(\partial_{\ell,1}W\right)\right],$
(A.10)
where $\Gamma_{k\ell,1}^{m}$ are the Christoffel symbols [9]. In the same way,
we get
$\mathbb{E}\left[\left(\partial_{\ell,1}V_{\rm
R}\right)\left(\partial_{k,2}V_{\rm R}\right)\right]=-\frac{2\alpha}{\Delta
t}W\left(\partial_{\ell,1}\partial_{k,2}W\right)$ (A.11)
and
$\mathbb{E}\left[\left(\partial_{k,2}\partial_{j,1}V_{\rm
R}\right)\left(\partial_{\ell,2}V_{\rm R}\right)\right]=-\frac{2\alpha}{\Delta
t}\left(\partial_{k,2}W\right)\left(\partial_{j,1}\partial_{\ell,2}W\right).$
(A.12)
Using (A.6), we have
$\displaystyle\mathbb{E}\left[F_{{\rm R},1}^{(i)}F_{{\rm R},1}^{(j)}\right]$
$\displaystyle=$ $\displaystyle\mathbb{E}\left[g^{i\ell}_{\
1}\left(\partial_{\ell,1}V_{\rm R}\right)g^{jk}_{\
1}\left(\partial_{k,1}V_{\rm R}\right)\right]$ (A.13) $\displaystyle=$
$\displaystyle\frac{2\alpha}{\Delta t}g^{i\ell}_{\ 1}g^{jk}_{\ 1}\left[g_{\ell
k,1}-(\partial_{\ell,1}W)(\partial_{k,1}W)\right]$ $\displaystyle=$
$\displaystyle\frac{2\alpha}{\Delta t}\left(g^{ij}_{\
1}-\xi_{1}^{i}\xi_{1}^{j}\right),$
where $\xi_{\ell}^{i}$ is the vector field which is given by (3.17). From
(A.6) and (A.10), we obtain
$\displaystyle\sum_{k}\mathbb{E}\left[\frac{\partial F_{{\rm
R},1}^{(i)}}{\partial x_{1}^{k}}F_{{\rm R},1}^{(k)}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(\partial_{k,1}g^{ij}_{\
1}\partial_{j,1}V_{\rm R}\right)\left(g^{k\ell}_{\ 1}\partial_{\ell,1}V_{\rm
R}\right)\right]$ (A.14) $\displaystyle=$
$\displaystyle\left(\partial_{k,1}g^{ij}_{\ 1}\right)g^{k\ell}_{\
1}\,\mathbb{E}\left[\left(\partial_{j,1}V_{\rm
R}\right)\left(\partial_{\ell,1}V_{\rm R}\right)\right]+g^{ij}_{\
1}g^{k\ell}_{\ 1}\,\mathbb{E}\left[\left(\partial_{k,1}\partial_{j,1}V_{\rm
R}\right)\left(\partial_{\ell,1}V_{\rm R}\right)\right]$ $\displaystyle=$
$\displaystyle\frac{2\alpha}{\Delta t}\left(\partial_{k,1}g^{ij}_{\
1}\right)g^{k\ell}_{\
1}\left[g_{j\ell,1}-\left(\partial_{j,1}W\right)\left(\partial_{\ell,1}W\right)\right]$
$\displaystyle+$ $\displaystyle\frac{2\alpha}{\Delta t}g^{ij}_{\
1}g^{k\ell}_{\
1}\left[\Gamma_{kj,1}^{m}g_{m\ell,1}-\left(\partial_{k,1}\partial_{j,1}W\right)\left(\partial_{\ell,1}W\right)\right]$
$\displaystyle=$ $\displaystyle\frac{2\alpha}{\Delta
t}\left[\partial_{j,1}g^{ij}_{\ 1}+g^{ij}_{\
1}\Gamma_{kj,1}^{k}-\left(\partial_{k,1}\xi_{1}^{i}\right)\xi_{1}^{k}\right]$
$\displaystyle=$ $\displaystyle\frac{2\alpha}{\Delta
t}\left[\frac{1}{\sqrt{{\rm det}\,g_{1}}}\partial_{j,1}g^{ij}_{\ 1}\sqrt{{\rm
det}\,g_{1}}-\left(\partial_{k,1}\xi_{1}^{i}\right)\xi_{1}^{k}\right]$
where we have used777See, for example, Sec.7 of Chap. I of the book [10].
$\Gamma_{kj,1}^{k}=\frac{1}{\sqrt{{\rm det}\,g_{1}}}\partial_{j,1}\sqrt{{\rm
det}\,g_{1}}.$ (A.15)
In the same way, the relations (A.11) and (A.12) yield
$\displaystyle\mathbb{E}\left[F_{{\rm R},1}^{(i)}F_{{\rm R},2}^{(j)}\right]$
$\displaystyle=$ $\displaystyle g^{i\ell}_{\ 1}g^{jk}_{\
2}\,\mathbb{E}\left[\left(\partial_{\ell,1}V_{\rm
R}\right)\left(\partial_{k,2}V_{\rm R}\right)\right]$ (A.16) $\displaystyle=$
$\displaystyle-\frac{2\alpha}{\Delta t}g^{i\ell}_{\ 1}g^{jk}_{\
2}\,W\left(\partial_{\ell,1}\partial_{k,2}W\right)$
and
$\displaystyle\sum_{k}\mathbb{E}\left[\frac{\partial F_{{\rm
R},1}^{(i)}}{\partial x_{2}^{k}}F_{{\rm R},2}^{(k)}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(\partial_{k,2}g^{ij}_{\
1}\partial_{j,1}V_{\rm R}\right)\left(g^{k\ell}_{\ 2}\partial_{\ell,2}V_{\rm
R}\right)\right]$ (A.17) $\displaystyle=$ $\displaystyle g^{ij}_{\
1}g^{k\ell}_{\ 2}\,\mathbb{E}\left[\left(\partial_{k,2}\partial_{j,1}V_{\rm
R}\right)\left(\partial_{\ell,2}V_{\rm R}\right)\right]$ $\displaystyle=$
$\displaystyle-\frac{2\alpha}{\Delta t}g^{ij}_{\ 1}g^{k\ell}_{\
2}\left(\partial_{k,2}W\right)\left(\partial_{j,1}\partial_{\ell,2}W\right),$
respectively. The contribution from the two random forces $F_{{\rm R},\ell}$
with the same indexes $\ell=1$ in the right-hand side of (3.12) becomes
$\displaystyle I_{11}$ $\displaystyle:=$ $\displaystyle\frac{\Delta
t}{2}\left\\{\sum_{i,j}\int_{M}d\mu\,\frac{\partial^{2}f}{\partial
x_{1}^{(i)}\partial x_{1}^{(j)}}\mathbb{E}\left[F_{{\rm R},1}^{(i)}F_{{\rm
R},1}^{(j)}\right]+\sum_{i,k}\int_{M}d\mu\,\frac{\partial f}{\partial
x_{1}^{(i)}}\mathbb{E}\left[\frac{\partial F_{{\rm R},1}^{(i)}}{\partial
x_{1}^{(k)}}F_{{\rm R},1}^{(k)}\right]\right\\}\rho_{t}$ (A.18)
$\displaystyle=$
$\displaystyle\alpha\sum_{i,j}\int_{M}d\mu\,\frac{\partial^{2}f}{\partial
x_{1}^{(i)}\partial x_{1}^{(j)}}\left(g^{ij}_{\
1}-\xi_{1}^{i}\xi_{1}^{j}\right)\rho_{t}$ $\displaystyle+$
$\displaystyle\alpha\sum_{i}\int_{M}d\mu\,\frac{\partial f}{\partial
x_{1}^{(i)}}\left[\frac{1}{\sqrt{{\rm det}\,g_{1}}}\partial_{j,1}g^{ij}_{\
1}\sqrt{{\rm
det}\,g_{1}}-\left(\partial_{k,1}\xi_{1}^{i}\right)\xi_{1}^{k}\right]\rho_{t},$
where we have used (A.13) and (A.14). Note that
$\displaystyle\int_{M}d\mu\,\left[\sum_{i,j}g^{ij}_{\
1}\frac{\partial^{2}f}{\partial x_{1}^{(i)}\partial
x_{1}^{(j)}}+\sum_{i}\left(\frac{1}{\sqrt{{\rm
det}\,g_{1}}}\partial_{j,1}g^{ij}_{\ 1}\sqrt{{\rm
det}\,g_{1}}\right)\frac{\partial f}{\partial x_{1}^{(i)}}\right]\rho_{t}$
(A.19) $\displaystyle=$ $\displaystyle\int_{M}d\mu\,\left(\frac{1}{\sqrt{{\rm
det}\,g_{1}}}\partial_{j,1}g^{ij}_{\ 1}\sqrt{{\rm
det}\,g_{1}}{\partial_{i,1}f}\right)\rho_{t}$ $\displaystyle=$
$\displaystyle\int_{M}d\mu\,(\Delta_{1}f)\rho_{t}=\int_{M}d\mu\,f\left(\Delta_{1}\rho_{t}\right),$
where the second equality follows from the property888See, for example,
Corollary 5.13 in Chap. II of the book [9]. of the Laplacian $\Delta_{\ell}$.
The rest of the contributions in the right-hand side of (A.18) are computed as
$\displaystyle\int_{M}d\mu\,\left[\sum_{i,j}\frac{\partial^{2}f}{\partial
x_{1}^{(i)}\partial x_{1}^{(j)}}\xi_{1}^{i}\xi_{1}^{j}+\sum_{i}\frac{\partial
f}{\partial
x_{1}^{(i)}}\left(\partial_{k,1}\xi_{1}^{i}\right)\xi_{1}^{k}\right]\rho_{t}$
(A.20) $\displaystyle=$ $\displaystyle\int_{M}d\mu\,\left[\frac{1}{\sqrt{{\rm
det}\,g_{1}}}\partial_{i,1}\sqrt{{\rm
det}\,g_{1}}(\partial_{j,1}f)\xi_{1}^{i}\xi_{1}^{j}\rho_{t}-(\partial_{j,1}f)\frac{1}{\sqrt{{\rm
det}\,g_{1}}}\partial_{i,1}\sqrt{{\rm
det}\,g_{1}}\xi_{1}^{i}\xi_{1}^{j}\rho_{t}\right]$ $\displaystyle+$
$\displaystyle\int_{M}d\mu\,(\partial_{j,1}f)(\partial_{i,1}\xi_{1}^{j})\xi_{1}^{i}\rho_{t}$
$\displaystyle=$
$\displaystyle-\int_{M}d\mu\,(\partial_{j,1}f)\xi_{1}^{j}\frac{1}{\sqrt{{\rm
det}\,g_{1}}}\partial_{i,1}\sqrt{{\rm det}\,g_{1}}\xi_{1}^{i}\rho_{t}$
$\displaystyle=$
$\displaystyle-\int_{M}d\mu\,(\partial_{j,1}f)\xi_{1}^{j}\,{\rm
div}_{1}\left[\xi_{1}\rho_{t}\right]$ $\displaystyle=$
$\displaystyle-\int_{M}d\mu\,\frac{1}{\sqrt{{\rm
det}\,g_{1}}}\partial_{j,1}\sqrt{{\rm det}\,g_{1}}\xi_{1}^{j}f\,{\rm
div}_{1}\left(\xi_{1}\rho_{t}\right)$ $\displaystyle+$
$\displaystyle\int_{M}d\mu\,f\frac{1}{\sqrt{{\rm
det}\,g_{1}}}\partial_{j,1}\sqrt{{\rm det}\,g_{1}}\xi_{1}^{j}\,{\rm
div}_{1}\left(\xi_{1}\rho_{t}\right)$ $\displaystyle=$
$\displaystyle\int_{M}d\mu\,f\,{\rm div}_{1}\left[\xi_{1}\,{\rm
div}_{1}\left(\xi_{1}\rho_{t}\right)\right],$
where we have used the divergence theorem (3.14). Substituting this and (A.19)
into (A.18), we obtain
$I_{11}=\alpha\int_{M}d\mu\,f\left\\{\Delta_{1}\rho_{t}-{\rm
div}_{1}\left[\xi_{1}{\rm div}_{1}(\xi_{1}\rho_{t})\right]\right\\}.$ (A.21)
Next consider the contribution from the two random forces $F_{{\rm R},\ell}$
with different indexes, $\ell=1$ and $\ell=2$, in the right-hand side of
(3.12). Using (A.16) and (A.17), we obtain
$\displaystyle I_{12}$ $\displaystyle:=$ $\displaystyle\frac{\Delta
t}{2}\left\\{\sum_{i,j}\int_{M}d\mu\,\frac{\partial^{2}f}{\partial
x_{1}^{(i)}\partial x_{2}^{(j)}}\mathbb{E}\left[F_{{\rm R},1}^{(i)}F_{{\rm
R},2}^{(j)}\right]+\sum_{i,k}\int_{M}d\mu\,\frac{\partial f}{\partial
x_{1}^{(i)}}\mathbb{E}\left[\frac{\partial F_{{\rm R},1}^{(i)}}{\partial
x_{2}^{(k)}}F_{{\rm R},2}^{(k)}\right]\right\\}\rho_{t}$ (A.22)
$\displaystyle=$
$\displaystyle-\alpha\int_{M}d\mu\,\sum_{i,j}\frac{\partial^{2}f}{\partial
x_{1}^{(i)}\partial x_{2}^{(j)}}g^{i\ell}_{\ 1}g^{jk}_{\
2}W(\partial_{\ell,1}\partial_{k,2}W)\rho_{t}$
$\displaystyle-\alpha\int_{M}d\mu\,\sum_{i}\frac{\partial f}{\partial
x_{1}^{(i)}}g^{ij}_{\ 1}g^{k\ell}_{\
2}(\partial_{k,2}W)(\partial_{j,1}\partial_{\ell,2}W)\rho_{t}$
$\displaystyle=$ $\displaystyle-\alpha\int_{M}d\mu\,\frac{1}{\sqrt{{\rm
det}\,g_{2}}}\partial_{j,2}\sqrt{{\rm det}\,g_{2}}g^{jk}_{\
2}(\partial_{i,1}f)g^{i\ell}_{\ 1}(\partial_{\ell,1}\partial_{k,2}W)W\rho_{t}$
$\displaystyle+\alpha\int_{M}d\mu\,(\partial_{i,1}f)\frac{1}{\sqrt{{\rm
det}\,g_{2}}}\partial_{j,2}\sqrt{{\rm det}\,g_{2}}g^{jk}_{\ 2}g^{i\ell}_{\
1}(\partial_{\ell,1}\partial_{k,2}W)W\rho_{t}$
$\displaystyle-\alpha\int_{M}d\mu\,(\partial_{i,1}f)g^{ij}_{\ 1}g^{k\ell}_{\
2}(\partial_{k,2}W)(\partial_{j,1}\partial_{\ell,2}W)\rho_{t}$
$\displaystyle=$
$\displaystyle\alpha\int_{M}d\mu\,(\partial_{i,1}f)g^{i\ell}_{\
1}W\frac{1}{\sqrt{{\rm det}\,g_{2}}}\partial_{j,2}\sqrt{{\rm
det}\,g_{2}}g^{jk}_{\ 2}(\partial_{\ell,1}\partial_{k,2}W)\rho_{t},$
where we have used the divergence theorem (3.14). Recalling $W={\bf
S}_{1}\cdot{\bf S}_{2}$, we have
$\partial_{\ell,1}\partial_{k,2}W=\left(\partial_{\ell,1}{\bf
S}_{1}\right)\cdot\left(\partial_{k,2}{\bf S}_{2}\right).$ (A.23)
Substituting this into the above result, we get
$\displaystyle I_{12}$ $\displaystyle=$
$\displaystyle\alpha\int_{M}d\mu\,(\partial_{i,1}f)g^{i\ell}_{\
1}(\partial_{\ell,1}{\bf S}_{1})W\cdot\frac{1}{\sqrt{{\rm
det}\,g_{2}}}\partial_{j,2}\sqrt{{\rm det}\,g_{2}}g^{jk}_{\
2}\left(\partial_{k,2}{\bf S}_{2}\right)\rho_{t}$ (A.24) $\displaystyle=$
$\displaystyle\alpha\int_{M}d\mu\,(\partial_{i,1}f)\mbox{\boldmath$\eta$}_{1}^{i}W\cdot{\rm
div}_{2}\left(\mbox{\boldmath$\eta$}_{2}\rho_{t}\right)$ $\displaystyle=$
$\displaystyle\alpha\int_{M}d\mu\,\frac{1}{\sqrt{{\rm
det}\,g_{1}}}\partial_{i,1}\sqrt{{\rm
det}\,g_{1}}\mbox{\boldmath$\eta$}_{1}^{i}fW\cdot{\rm
div}_{2}\left(\mbox{\boldmath$\eta$}_{2}\rho_{t}\right)$
$\displaystyle-\alpha\int_{M}d\mu\,f\frac{1}{\sqrt{{\rm
det}\,g_{1}}}\partial_{i,1}\sqrt{{\rm
det}\,g_{1}}\mbox{\boldmath$\eta$}_{1}^{i}W\cdot{\rm
div}_{2}\left(\mbox{\boldmath$\eta$}_{2}\rho_{t}\right)$ $\displaystyle=$
$\displaystyle-\alpha\int_{M}d\mu\,f\,{\rm
div}_{1}\left[\mbox{\boldmath$\eta$}_{1}W\cdot{\rm
div}_{2}\left(\mbox{\boldmath$\eta$}_{2}\rho_{t}\right)\right],$
where $\mbox{\boldmath$\eta$}_{\ell}^{i}$ is given by (3.18). From (3.12),
(3.15), (A.18), (A.21), (A.22) and (A.24), we obtain the Fokker-Planck
equation,
$\displaystyle\frac{\partial\rho_{t}}{\partial t}$ $\displaystyle=$
$\displaystyle-\sum_{\ell}{\rm
div}_{\ell}\left(F_{0,\ell}\rho_{t}\right)+\alpha\sum_{\ell}\left\\{\Delta_{\ell}\rho_{t}-{\rm
div}_{\ell}\left[\xi_{\ell}\,{\rm
div}_{\ell}(\xi_{\ell}\rho_{t})\right]\right\\}$ (A.25)
$\displaystyle-\alpha\left\\{{\rm
div}_{1}\left[\mbox{\boldmath$\eta$}_{1}W\cdot{\rm
div}_{2}(\mbox{\boldmath$\eta$}_{2}\rho_{t})\right]+{\rm
div}_{2}\left[\mbox{\boldmath$\eta$}_{2}W\cdot{\rm
div}_{1}(\mbox{\boldmath$\eta$}_{1}\rho_{t})\right]\right\\},$
for $\alpha^{\prime}=0$.
Next consider the case with $\alpha^{\prime}\neq 0$. To begin with, we note
that
$\displaystyle\mathbb{E}\left[\sigma_{+}^{(i)}\sigma_{+}^{(j)}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbb{E}\left[\left(\sigma_{2}^{(i)}+\sigma_{1}^{(i)}\right)\left(\sigma_{2}^{(j)}+\sigma_{1}^{(j)}\right)\right]$
(A.26) $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\mathbb{E}[\sigma_{2}^{(i)}\sigma_{2}^{(j)}]+\mathbb{E}[\sigma_{1}^{(i)}\sigma_{1}^{(j)}]+\mathbb{E}[\sigma_{2}^{(i)}\sigma_{1}^{(j)}]+\mathbb{E}[\sigma_{1}^{(i)}\sigma_{2}^{(j)}]\right\\}$
$\displaystyle=$ $\displaystyle\frac{\alpha+\alpha^{\prime}}{\Delta
t}\delta^{ij}.$
Similarly,
$\mathbb{E}\left[\sigma_{-}^{(i)}\sigma_{-}^{(j)}\right]=\frac{\alpha-\alpha^{\prime}}{\Delta
t}\delta^{ij}.$ (A.27)
Further, we have
$\displaystyle\mathbb{E}\left[\sigma_{+}^{(i)}\sigma_{-}^{(j)}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbb{E}\left[\left(\sigma_{2}^{(i)}+\sigma_{1}^{(i)}\right)\left(\sigma_{2}^{(j)}-\sigma_{1}^{(j)}\right)\right]$
(A.28) $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\mathbb{E}[\sigma_{2}^{(i)}\sigma_{2}^{(j)}]-\mathbb{E}[\sigma_{1}^{(i)}\sigma_{1}^{(j)}]-\mathbb{E}[\sigma_{2}^{(i)}\sigma_{1}^{(j)}]+\mathbb{E}[\sigma_{1}^{(i)}\sigma_{2}^{(j)}]\right\\}$
$\displaystyle=$ $\displaystyle 0.$
Since we can write
$\mathbb{E}\left[\sigma_{-}^{(i)}\sigma_{-}^{(j)}\right]=\frac{\alpha+\alpha^{\prime}}{\Delta
t}\delta^{ij}-\frac{2\alpha^{\prime}}{\Delta t}\delta^{ij},$ (A.29)
it is sufficient to calculate the corrections from the second term in this
right-hand side, with replacing $\alpha$ with $\alpha+\alpha^{\prime}$ in the
above result (A.25).
In (A.13), the correction to $\mathbb{E}\left[g^{i\ell}_{\
1}(\partial_{\ell,1}V_{\rm R})g^{jk}_{\ 1}(\partial_{k,1}V_{\rm R})\right]$ is
given by
$-\frac{4\alpha^{\prime}}{\Delta
t}\hat{\mbox{\boldmath$\zeta$}}_{1}^{i}\cdot\hat{\mbox{\boldmath$\zeta$}}_{1}^{j},$
(A.30)
where $\hat{\mbox{\boldmath$\zeta$}}_{\ell}^{i}$ is given by (3.19).
Similarly, the correction to $\mathbb{E}\left[(\partial_{k,1}g^{ij}_{\
1}\partial_{j,1}V_{\rm R})(g^{k\ell}_{\ 1}\partial_{\ell,1}V_{\rm R})\right]$
in (A.14) is given by
$-\frac{4\alpha^{\prime}}{\Delta
t}\left(\partial_{k,1}\hat{\mbox{\boldmath$\zeta$}}_{1}^{i}\right)\cdot\hat{\mbox{\boldmath$\zeta$}}_{1}^{k}.$
(A.31)
Therefore the same calculations as those from (A.18) to (A.21) yield the
correction,
$-2\alpha^{\prime}{\rm
div}_{1}\left[\hat{\mbox{\boldmath$\zeta$}}_{1}\cdot{\rm
div}_{1}(\hat{\mbox{\boldmath$\zeta$}}_{1}\rho_{t})\right],$ (A.32)
in the right-hand side of the Fokker-Planck equation (A.25).
In (A.16), the correction to $\mathbb{E}\left[g^{i\ell}_{\
1}(\partial_{\ell,1}V_{\rm R})g^{jk}_{\ 2}(\partial_{k,2}V_{\rm R})\right]$ is
given by
$-\frac{4\alpha^{\prime}}{\Delta
t}\hat{\mbox{\boldmath$\zeta$}}_{1}^{i}\cdot\hat{\mbox{\boldmath$\zeta$}}_{2}^{j}.$
(A.33)
Further, the correction to $\mathbb{E}\left[(\partial_{k,2}g^{ij}_{\
1}\partial_{j,1}V_{\rm R})(g^{k\ell}_{\ 2}\partial_{\ell,2}V_{\rm R})\right]$
in (A.17) is given by
$-\frac{4\alpha^{\prime}}{\Delta
t}\left(\partial_{k,2}\hat{\mbox{\boldmath$\zeta$}}_{1}^{i}\right)\cdot\hat{\mbox{\boldmath$\zeta$}}_{2}^{k}.$
(A.34)
Therefore similar calculations to those from (A.22) to (A.24) yield the
correction,
$-2\alpha^{\prime}{\rm
div}_{1}\left[\hat{\mbox{\boldmath$\zeta$}}_{1}\cdot{\rm
div}_{2}(\hat{\mbox{\boldmath$\zeta$}}_{2}\rho_{t})\right],$ (A.35)
in the right-hand side of the Fokker-Planck equation (A.25). In consequence,
the Fokker-Planck equation is given by
$\displaystyle\frac{\partial\rho_{t}}{\partial t}$ $\displaystyle=$
$\displaystyle-\sum_{\ell}{\rm
div}_{\ell}\left(F_{0,\ell}\rho_{t}\right)+(\alpha+\alpha^{\prime})\sum_{\ell}\left\\{\Delta_{\ell}\rho_{t}-{\rm
div}_{\ell}\left[\xi_{\ell}\,{\rm
div}_{\ell}(\xi_{\ell}\rho_{t})\right]\right\\}$ (A.36)
$\displaystyle-(\alpha+\alpha^{\prime})\left\\{{\rm
div}_{1}\left[\mbox{\boldmath$\eta$}_{1}W\cdot{\rm
div}_{2}(\mbox{\boldmath$\eta$}_{2}\rho_{t})\right]+{\rm
div}_{2}\left[\mbox{\boldmath$\eta$}_{2}W\cdot{\rm
div}_{1}(\mbox{\boldmath$\eta$}_{1}\rho_{t})\right]\right\\}$
$\displaystyle-2\alpha^{\prime}\sum_{m,n}{\rm
div}_{m}\left[\hat{\mbox{\boldmath$\zeta$}}_{m}\cdot{\rm
div}_{n}(\hat{\mbox{\boldmath$\zeta$}}_{n}\rho_{t})\right].$
## Appendix B Derivation of the expansion (LABEL:Jx1expand)
The metric $g_{ij,\ell}$ of $\mathbb{S}^{3}$ is computed as
$g_{ij,\ell}=\left(\matrix{1+\gamma_{\ell}{x_{\ell}^{2}}&\gamma_{\ell}{x_{\ell}y_{\ell}}&\gamma_{\ell}{x_{\ell}z_{\ell}}\cr\gamma_{\ell}{y_{\ell}x_{\ell}}&1+\gamma_{\ell}{y_{\ell}^{2}}&\gamma_{\ell}{y_{\ell}z_{\ell}}\cr\gamma_{\ell}{z_{\ell}x_{\ell}}&\gamma_{\ell}{z_{\ell}y_{\ell}}&1+\gamma_{\ell}{z_{\ell}^{2}}\cr}\right)=\left(\matrix{1+x_{\ell}^{2}&x_{\ell}y_{\ell}&x_{\ell}z_{\ell}\cr
y_{\ell}x_{\ell}&1+y_{\ell}^{2}&y_{\ell}z_{\ell}\cr
z_{\ell}x_{\ell}&z_{\ell}y_{\ell}&1+z_{\ell}^{2}\cr}\right)+\cdots,$ (B.1)
where we have written
$\gamma_{\ell}=\frac{1}{\sqrt{1-r_{\ell}^{2}}}\quad\mbox{with}\ \
r_{\ell}=\sqrt{x_{\ell}^{2}+y_{\ell}^{2}+z_{\ell}^{2}}.$ (B.2)
Therefore, the inverse $g^{ij}_{\ \ell}$ is given by
$g^{ij}_{\
\ell}=\left(\matrix{1-x_{\ell}^{2}&-x_{\ell}y_{\ell}&-x_{\ell}z_{\ell}\cr-
y_{\ell}x_{\ell}&1-y_{\ell}^{2}&-y_{\ell}z_{\ell}\cr-
z_{\ell}x_{\ell}&-z_{\ell}y_{\ell}&1-z_{\ell}^{2}\cr}\right)+\cdots.$ (B.3)
Using this, we have
$\displaystyle(\partial_{x,1}W){\rm div}_{1}(\xi_{1}\rho)$ $\displaystyle=$
$\displaystyle\frac{\partial{\bf S}_{1}\cdot{\bf S}_{2}}{\partial
x_{1}}\frac{1}{\sqrt{{\rm det}\,g_{1}}}\partial_{i,1}\sqrt{{\rm
det}\,g_{1}}g^{ij}_{\ 1}(\partial_{j,1}{\bf S}_{1}\cdot{\bf S}_{2})\rho$ (B.4)
$\displaystyle=$ $\displaystyle-xg^{ij}_{\ 1}(\partial_{j,1}{\bf
S}_{1}\cdot{\bf S}_{2})\partial_{i,1}\rho+\cdots$ $\displaystyle=$
$\displaystyle x\left(x\frac{\partial\rho}{\partial
x_{1}}+y\frac{\partial\rho}{\partial y_{1}}+z\frac{\partial\rho}{\partial
z_{1}}\right)+\cdots.$
Similarly,
$\displaystyle W(\partial_{x,1}{\bf S}_{1})\cdot{\rm
div}_{2}(\mbox{\boldmath$\eta$}_{2}\rho)$ $\displaystyle=$ $\displaystyle
W(\partial_{x,1}{\bf S}_{1})\cdot g^{ij}_{\ 2}(\partial_{j,2}{\bf
S}_{2})\partial_{i,2}\rho+\cdots$ (B.5) $\displaystyle=$ $\displaystyle
W(\partial_{x,1}\partial_{j,2}{\bf S}_{1}\cdot{\bf S}_{2})g^{ij}_{\
2}\partial_{i,2}\rho+\cdots$ $\displaystyle=$ $\displaystyle
W\left\\{\partial_{j,2}\left[-x-\frac{1}{2}({\bf r}\cdot{\bf
R})x_{1}+\cdots\right]\right\\}g^{ij}_{\ 2}\partial_{i,2}\rho+\cdots$
$\displaystyle=$ $\displaystyle Wg^{i1}_{\
2}\partial_{i,2}\rho+W\left[x_{1}x_{2}\frac{\partial\rho}{\partial
x_{2}}+x_{1}y_{2}\frac{\partial\rho}{\partial
y_{2}}+x_{1}z_{2}\frac{\partial\rho}{\partial z_{2}}\right]+\cdots$
$\displaystyle=$ $\displaystyle\left(1-\frac{1}{2}r^{2}\right)\left[g^{11}_{\
2}\frac{\partial\rho}{\partial x_{2}}+g^{21}_{\ 2}\frac{\partial\rho}{\partial
y_{2}}+g^{31}_{\ 2}\frac{\partial\rho}{\partial z_{2}}\right]$
$\displaystyle+\left[x_{1}x_{2}\frac{\partial\rho}{\partial
x_{2}}+x_{1}y_{2}\frac{\partial\rho}{\partial
y_{2}}+x_{1}z_{2}\frac{\partial\rho}{\partial z_{2}}\right]+\cdots$
$\displaystyle=$ $\displaystyle\frac{\partial\rho}{\partial
x_{2}}-\frac{1}{2}r^{2}\frac{\partial\rho}{\partial
x_{2}}-\left[x_{2}^{2}\frac{\partial\rho}{\partial
x_{2}}+x_{2}y_{2}\frac{\partial\rho}{\partial
y_{2}}+x_{2}z_{2}\frac{\partial\rho}{\partial z_{2}}\right]$
$\displaystyle+\left[x_{1}x_{2}\frac{\partial\rho}{\partial
x_{2}}+x_{1}y_{2}\frac{\partial\rho}{\partial
y_{2}}+x_{1}z_{2}\frac{\partial\rho}{\partial z_{2}}\right]+\cdots$
$\displaystyle=$ $\displaystyle\frac{\partial\rho}{\partial
x_{2}}-\frac{1}{2}r^{2}\frac{\partial\rho}{\partial
x_{2}}+x\left[x_{2}\frac{\partial\rho}{\partial
x_{2}}+y_{2}\frac{\partial\rho}{\partial
y_{2}}+z_{2}\frac{\partial\rho}{\partial z_{2}}\right]+\cdots.$
We write
$\hat{\mbox{\boldmath$\zeta$}}_{i,\ell}=\left(\zeta_{i,\ell}^{(1)},\zeta_{i,\ell}^{(2)},\zeta_{i,\ell}^{(3)}\right).$
(B.6)
Note that
$\displaystyle\zeta_{x,1}^{(a)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
x_{1}}\left(S_{1}^{(0)}S_{2}^{(a)}-S_{2}^{(0)}S_{1}^{(a)}\right)$ (B.7)
$\displaystyle=$
$\displaystyle\frac{-x_{1}}{\sqrt{1-r_{1}^{2}}}S_{2}^{(a)}-\sqrt{1-r_{2}^{2}}\frac{\partial
S_{1}^{(a)}}{\partial x_{1}}.$
Therefore, we have
$\displaystyle\hat{\mbox{\boldmath$\zeta$}}_{x,1}$ $\displaystyle=$
$\displaystyle\left(\frac{-x_{1}x_{2}}{\sqrt{1-r_{1}^{2}}}-\sqrt{1-r_{2}^{2}},\frac{-x_{1}y_{2}}{\sqrt{1-r_{1}^{2}}},\frac{-x_{1}z_{2}}{\sqrt{1-r_{1}^{2}}}\right)$
(B.8) $\displaystyle=$
$\displaystyle\left({-x_{1}x_{2}}-\sqrt{1-r_{2}^{2}},{-x_{1}y_{2}},{-x_{1}z_{2}}\right)+\cdots.$
In the same way,
$\hat{\mbox{\boldmath$\zeta$}}_{y,1}=\left(-y_{1}x_{2},-y_{1}y_{2}-\sqrt{1-r_{2}^{2}},-y_{1}z_{2}\right)+\cdots$
(B.9)
and
$\hat{\mbox{\boldmath$\zeta$}}_{z,1}=\left(-z_{1}x_{2},-z_{1}y_{2},-z_{1}z_{2}-\sqrt{1-r_{2}^{2}}\right)+\cdots.$
(B.10)
From these results, we obtain
$\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot\hat{\mbox{\boldmath$\zeta$}}_{x,1}=1-r_{2}^{2}+2x_{1}x_{2}+\cdots,$
(B.11)
$\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot\hat{\mbox{\boldmath$\zeta$}}_{y,1}=y_{1}x_{2}+x_{1}y_{2}+\cdots$
(B.12)
and
$\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot\hat{\mbox{\boldmath$\zeta$}}_{z,1}=z_{1}x_{2}+x_{1}z_{2}+\cdots.$
(B.13)
Using these, we have
$\displaystyle\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot{\rm
div}_{1}(\hat{\mbox{\boldmath$\zeta$}}_{1}\rho)$ $\displaystyle=$
$\displaystyle\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot g^{ij}_{\
1}\hat{\mbox{\boldmath$\zeta$}}_{j,1}\partial_{i,1}\rho+\cdots$ (B.14)
$\displaystyle=$ $\displaystyle(1-r_{2}^{2}+2x_{1}x_{2})g^{i1}_{\
1}\partial_{i,1}\rho$ $\displaystyle+$
$\displaystyle(y_{1}x_{2}+x_{1}y_{2})g^{i2}_{\
1}\partial_{i,1}\rho+(z_{1}x_{2}+x_{1}z_{2})g^{i3}_{\
1}\partial_{i,1}\rho+\cdots$ $\displaystyle=$
$\displaystyle(1-r_{2}^{2}+2x_{1}x_{2})\left[(1-x_{1}^{2})\frac{\partial\rho}{\partial
x_{1}}-x_{1}y_{1}\frac{\partial\rho}{\partial
y_{1}}-x_{1}z_{1}\frac{\partial\rho}{\partial z_{1}}\right]$ $\displaystyle+$
$\displaystyle(y_{1}x_{2}+x_{1}y_{2})\frac{\partial\rho}{\partial
y_{1}}+(z_{1}x_{2}+x_{1}z_{2})\frac{\partial\rho}{\partial z_{1}}+\cdots$
$\displaystyle=$ $\displaystyle\frac{\partial\rho}{\partial
x_{1}}-r_{2}^{2}\frac{\partial\rho}{\partial
x_{1}}-x_{1}\left(x\frac{\partial\rho}{\partial
x_{1}}+y\frac{\partial\rho}{\partial y_{1}}+z\frac{\partial\rho}{\partial
z_{1}}\right)$ $\displaystyle+$ $\displaystyle
x_{2}\left(x_{1}\frac{\partial\rho}{\partial
x_{1}}+y_{1}\frac{\partial\rho}{\partial
y_{1}}+z_{1}\frac{\partial\rho}{\partial z_{1}}\right)+\cdots.$
In the same way,
$\hat{\mbox{\boldmath$\zeta$}}_{x,2}=\left(x_{1}x_{2}+\sqrt{1-r_{1}^{2}},x_{2}y_{1},x_{2}z_{1}\right)+\cdots,$
(B.15)
$\hat{\mbox{\boldmath$\zeta$}}_{y,2}=\left(y_{2}x_{1},y_{1}y_{2}+\sqrt{1-r_{1}^{2}},y_{2}z_{1}\right)+\cdots$
(B.16)
and
$\hat{\mbox{\boldmath$\zeta$}}_{z,2}=\left(z_{2}x_{1},z_{2}y_{1},z_{1}z_{2}+\sqrt{1-r_{1}^{2}}\right)+\cdots.$
(B.17)
Combining these, (B.8), (B.9) and (B.10), we obtain
$\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot\hat{\mbox{\boldmath$\zeta$}}_{x,2}=-\left(1-\frac{1}{2}r_{1}^{2}-\frac{1}{2}r_{2}^{2}+2x_{1}x_{2}\right)+\cdots,$
(B.18)
$\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot\hat{\mbox{\boldmath$\zeta$}}_{y,2}=-2x_{1}y_{2}+\cdots$
(B.19)
and
$\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot\hat{\mbox{\boldmath$\zeta$}}_{z,2}=-2x_{1}z_{2}+\cdots.$
(B.20)
Using these, we have
$\displaystyle\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot{\rm
div}_{2}(\hat{\mbox{\boldmath$\zeta$}}_{2}\rho)$ $\displaystyle=$
$\displaystyle\hat{\mbox{\boldmath$\zeta$}}_{x,1}\cdot g^{ij}_{\
2}\hat{\mbox{\boldmath$\zeta$}}_{j,2}\partial_{i,2}\rho+\cdots$ (B.21)
$\displaystyle=$
$\displaystyle-\left(1-\frac{1}{2}r_{1}^{2}-\frac{1}{2}r_{2}^{2}+2x_{1}x_{2}\right)g^{i1}_{\
2}\partial_{i,2}\rho$ $\displaystyle-2x_{1}y_{2}g^{i2}_{\
2}\partial_{i,2}\rho-2x_{1}z_{2}g^{i3}_{\ 2}\partial_{i,2}\rho+\cdots$
$\displaystyle=$ $\displaystyle-g^{i1}_{\
2}\partial_{i,2}\rho+\frac{1}{2}(r_{1}^{2}+r_{2}^{2})\frac{\partial\rho}{\partial
x_{2}}$ $\displaystyle-2x_{1}\left(x_{2}\frac{\partial\rho}{\partial
x_{2}}+y_{2}\frac{\partial\rho}{\partial
y_{2}}+z_{2}\frac{\partial\rho}{\partial z_{2}}\right)+\cdots$
$\displaystyle=$ $\displaystyle-\frac{\partial\rho}{\partial
x_{2}}+\frac{1}{2}(r_{1}^{2}+r_{2}^{2})\frac{\partial\rho}{\partial x_{2}}$
$\displaystyle-$
$\displaystyle\left(\frac{3}{2}x+\frac{1}{2}X\right)\left(x_{2}\frac{\partial\rho}{\partial
x_{2}}+y_{2}\frac{\partial\rho}{\partial
y_{2}}+z_{2}\frac{\partial\rho}{\partial z_{2}}\right)+\cdots.$
Substituting (4.6), (B.4), (B.5), (B.14) and (B.21) into (3.22), we obtain the
expansion (LABEL:Jx1expand).
## References
* [1] I. Montvay and G. Münster, Quantum Fields on a Lattice, Cambridge University Press, 1994.
* [2] A. H. Guth, Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory, Phys. Rev. D, 21: 2291–2307 (1980).
* [3] J. Fröhlich and T. Spencer, Massless phase and symmetry restoration in Abelian gauge theories and spin systems, Commun. Math. Phys. 83: 411–454 (1982).
* [4] B. Durhuus and J. Fröhlich, A connection between $\nu$-dimensional Yang-Mills theory and $(\nu-1)$-dimensional, non-linear $\sigma$-models, Commun. Math. Phys. 75: 103–151 (1980).
* [5] P. Orland, (2+1)-dimensional lattice QCD, Phys. Rev. D 71: 054503 (2005); Integrable models and confinement in (2+1)-dimensional weakly-coupled Yang-Mills theory, Phys. Rev. D 74: 085001 (2006).
* [6] P. Orland, String tensions and representations in anisotropic (2+1)-dimensional weakly-coupled Yang-Mills theory, Phys. Rev. D 75: 025001 (2007); Glueball masses in (2+1)-dimensional anisotropic weakly-coupled Yang-Mills theory, Phys. Rev. D 75: 101702(R) (2007); Composite strings in (2+1)-dimensional anisotropic weakly-coupled Yang-Mills theory, Phys. Rev. D 77: 025035 (2008); Near-integrability and confinement for high-energy hadron-hadron collisions, Phys. Rev. D 77: 056004 (2008).
* [7] J. Fröhlich, B. Simon and T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking, Commun. Math. Phys. 50: 79–95 (1976).
* [8] G. Parigi and Y.-S. Wu, Perturbation theory without gauge fixing, Sci. Sin. 24: 483–496 (1981).
* [9] T. Sakai, Riemannian Geometry, Amer. Math. Soc., Providence, R. I., 1996.
* [10] L. P. Eisenhart, Riemannian Geometry, Princeton University Press, 1966.
* [11] G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1995.
* [12] N. Martzel and C. Aslangul, Mean-field treatment of the many-body Fokker-Planck equation, J. Phys. A: Math. Gen. 34: 11225–11240 (2001).
|
arxiv-papers
| 2009-07-11T04:41:51 |
2024-09-04T02:49:03.827706
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tohru Koma",
"submitter": "Tohru Koma",
"url": "https://arxiv.org/abs/0907.1938"
}
|
0907.1959
|
# The triangle and the open triangle
Gady Kozma
###### Abstract.
We show that for percolation on any transitive graph, the triangle condition
implies the open triangle condition.
## 1\. Introduction
Let $G$ be a vertex-transitive111A vertex-transitive graph, and any other
notion not specifically defined, may be found in Wikipedia. connected graph,
and let $p$ be some number in $[0,1]$. We say that $p$-percolation on $G$
satisfies the triangle condition if for some $v\in G$
$\sum_{x,y\in G}\mathbb{P}(v\leftrightarrow x)\mathbb{P}(x\leftrightarrow
y)\mathbb{P}(y\leftrightarrow v)<\infty.$ (1)
where $x\leftrightarrow y$ implies that there exists an open path between $x$
and $y$. Here and below we abuse notations by denoting “$v$ is a vertex of
$G$” by $v\in G$. Of course, by transitivity, the sum is in fact independent
of $v$. This note is far too short to explain the importance of the triangle
condition. Suffices to say that it the triangle condition holds at the
_critical_ $p$, then many exponents take their _mean-field_ values. See [AN84,
N87, BA91, KN09] for corollaries of the triangle condition. On the other hand,
the triangle condition holds in many interesting cases, see [HS90, HHS08] for
the graphs $\mathbb{Z}^{d}$ with $d$ sufficiently large, and [S01, S02, K] for
various other transitive graphs. See [G99] or [BR06] for a general
introduction to percolation.
In many applications the triangle condition (1) is not so convenient to use.
One instead uses the _open_ triangle condition, which states that
$\lim_{R\to\infty}\max_{w\not\in B(v,R)}\sum_{x,y\in
G}\mathbb{P}(v\leftrightarrow x)\mathbb{P}(x\leftrightarrow
y)\mathbb{P}(y\leftrightarrow w)=0,$
where $B(v,R)$ stands for the ball around $v$ with radius $R$ in the graph (or
shortest path) distance. Clearly, the open triangle condition implies the
(closed) triangle condition (recall that if $y$ and $y^{\prime}$ are neighbors
in the graph then $\mathbb{P}(x\leftrightarrow y)\geq
c\mathbb{P}(x\leftrightarrow y^{\prime})$ for some constant $c$ independent of
$x$, $y$ and $y^{\prime}$). The contents of lemma 2.1 of Barsky & Aizenman
[BA91] is the reverse implication. The proof in [BA91] is specific to the
graph $\mathbb{Z}^{d}$ as it uses the Fourier transform of the function
$f(x)=\mathbb{P}(\vec{0}\leftrightarrow x)$. The purpose of this note is to
generalize this to any transitive graph, namely
###### Theorem.
Let $G$ be a vertex-transitive graph and let $p\in[0,1]$. Assume $G$ satisfies
the triangle condition at $p$. Then $G$ satisfies the open triangle condition
at $p$.
This result is not particularly important. For example, in [S01, S02] the
author simply circumvents the problem by working directly with the open
triangle condition. The advantage of making the triangle condition “the”
marker for mean-field behavior is mostly aesthetic. The real reason for the
existance of this note is to demonstrate an application of operator theory,
specifically of spectral theory, to percolation. Operator theory is a
fantastically powerful tool whose absence from the percolation scene is behind
many of the difficulties one encounters. I aim to remedy this situation, even
if by very little.
I wish to thank Asaf Nachmias for pointing out some omissions in a draft
version of the paper, and Michael Aizenman for an intersting discussion of
alternative proof approaches.
## 2\. The proof
Before starting the proof proper, let us make a short heuristic argument.
Define the infinite matrix
$B(v,w)=\mathbb{P}(v\leftrightarrow w)$ (2)
where in the notation we assume that $v\leftrightarrow v$ always so
$B(v,v)=1$. By [AN84] $B$, considered as an (unbounded) operator on $l^{2}(G)$
is a positive operator. Hence the same holds for
$Q(v,w)=\sum_{x,y}B(v,x)B(x,y)B(y,w)$ (3)
which is just $B^{3}$ (as an infinite matrix or as an unbounded operator). It
is possible to take the square root of any positive operator, so denote
$S=\sqrt{Q}$. We get
$Q(v,w)=\langle Q\mathbf{1}_{v},\mathbf{1}_{w}\rangle=\langle
S\mathbf{1}_{v},S\mathbf{1}_{w}\rangle$
where $\mathbf{1}_{v}$ is the element of $l^{2}(G)$ defined by
$\mathbf{1}_{v}(x)=\begin{cases}1&v=x\\\ 0&v\neq x.\end{cases}$
Hence the triangle condition $Q(v,v)<\infty$ implies that $||Sv||<\infty$. But
$S$ is invariant to the automorphisms of $G$ (as a root of $Q$ which is
invariant to them) so $S\mathbf{1}_{w}$ is a map of $S\mathbf{1}_{v}$ under an
automorphism $\varphi$ taking $v$ to $w$. But any vector in $l^{2}$ is almost
orthogonal to sufficiently far away “translations” (namely, the automorphisms
of $G$), so $\langle S\mathbf{1}_{v},S\mathbf{1}_{w}\rangle\to 0$ as the graph
distance of $v$ and $w$ goes to $\infty$, as required.
Why is this even a heuristic and not a full proof? Because of the benign
looking expression $\langle Q\mathbf{1}_{v},\mathbf{1}_{w}\rangle$ which is in
fact meaningless. $Q$ is an unbounded operator and hence it cannot be applied
to any vector in $l^{2}(G)$, and there is nothing guaranteeing that
$\mathbf{1}_{v}$ will be in its domain. For example, in a sufficiently spread-
out lattice in $\mathbb{R}^{d}$ one has that $\mathbb{P}(x\leftrightarrow
y)\approx|x-y|^{2-d}$ [HHS03] which gives with a simple calculation that the
triangle condition holds whenever $d>6$ while $Q\mathbf{1}_{v}\in l^{2}$ only
when $d>12$.
The proof below circumvents this problem by decomposing $B$ into a sum of
positive bounded operators using specific properties of $B$. Somebody more
versed in the theory of unbounded operators might have constructed a more
direct proof.
We start the proof proper with
###### Definition.
Let $\varphi$ be an automorphism of the graph $G$. We define the isometry
$\Phi=\Phi_{\varphi}$ of $l^{2}(G)$ corresponding to $\varphi$ by
$(\Phi(f))(v)=f(\varphi^{-1}(v)).$ (4)
It is easy to check that $\Phi\mathbf{1}_{v}=\mathbf{1}_{\varphi(v)}$ and that
the support of $\Phi f$ is $\varphi($the support of $f)$.
###### Lemma.
Let $f\in l^{2}(G)$, let $v\in G$ and let $\delta>0$. Then there exists an
$R=R(f,\delta,v)$ such that for any $w\not\in B(v,R)$ and any automorphism
$\varphi$ of $G$ taking $v$ to $w$ one has
$|\langle\Phi_{\varphi}f,f\rangle|<\delta$ (5)
###### Proof.
Let $A\subset G$ be some finite set of vertices such that
$\sqrt{\sum_{v\not\in A}|f(v)|^{2}}<\frac{1}{3||f||}\delta.$
Write now
$f=f_{\mathrm{loc}}+f_{\mathrm{glob}}\mbox{ where
}f_{\mathrm{loc}}=f\cdot\mathbf{1}_{A}.$
By the definition of $A$, $||f_{\mathrm{glob}}||<\frac{1}{3||f||}\delta$, and
so by Cauchy-Schwarz,
$|\langle\Phi f,f\rangle|\leq|\langle\Phi
f_{\mathrm{loc}},f_{\mathrm{loc}}\rangle|+2||f_{\mathrm{glob}}||\cdot||f_{\mathrm{loc}}||+||f_{\mathrm{glob}}||^{2}<|\langle\Phi
f_{\mathrm{loc}},f_{\mathrm{loc}}\rangle|+\delta.$ (6)
Define now
$R=2\max_{x\in A}d(v,x)+1.$
To see (5), let $w$ and $\varphi$ be as above. We get, for any $x\in A$,
$d(\varphi(x),v)\geq d(v,w)-d(\varphi(x),w).$
Now, $d(\varphi(x),w)=d(\varphi(x),\varphi(v))=d(x,v)<\frac{1}{2}R$ because
$\varphi$ is an automorphism of $G$. Hence we get
$d(\varphi(x),v)>R-{\textstyle\frac{1}{2}}R$
implying that $\varphi(x)\not\in A$ as it is too far. In other words,
$A\cap\varphi(A)=\emptyset$ which implies that
$\langle\Phi_{\varphi}f_{\mathrm{loc}},f_{\mathrm{loc}}\rangle=0$. With (6),
the lemma is proved. ∎
###### Proof of the theorem.
We will not keep $p$ in the notations as it does not change throughout the
proof. For every $n\in\mathbb{N}$ and every $v,w\in G$, let $B_{n}(v,w)$ be
defined by
$B_{n}(v,w)=\mathbb{P}(v\leftrightarrow w,\,|\mathcal{C}(v)|=n)$
where $\mathcal{C}(v)$ is the cluster of $v$ i.e. the set of vertices
connected to $v$ by open paths, and $|\mathcal{C}(v)|$ is the number of
vertices in $\mathcal{C}(v)$. Clearly $B_{n}(v,w)\geq 0$ and
$B(v,w)=\sum_{n=1}^{\infty}B_{n}(v,w)$ (7)
where $B$ is as above (2). Therefore we may write
$\displaystyle Q(v,w)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:defQ})}}{{=}}\sum_{x,y}B(v,x)B(x,y)B(y,w)\stackrel{{\scriptstyle(\ref{eq:BBn})}}{{=}}\sum_{x,y}B(v,x)\left(\sum_{n=1}^{\infty}B_{n}(x,y)\right)B(y,w)=$
$\displaystyle\stackrel{{\scriptstyle\hphantom{(\ref{eq:defQ})}}}{{=}}\sum_{n=1}^{\infty}\sum_{x,y}B(v,x)B_{n}(x,y)B(y,w)$
(8)
where the change of order of summation in the last equality is justified since
all terms are positive. Now, the vector
$B\mathbf{1}_{w}=\left(B(y,w)\right)_{y\in G}$
is in $l^{2}(G)$ because
$\sum_{y}B(y,w)^{2}\leq\sum_{y,x}B(w,y)B(y,x)B(x,w)<\infty.$
Further, each $B_{n}$, considered as an operator on $l^{2}(G)$ is bounded,
because the sum of the (absolute values of the) entries in each row and each
column is finite. From this we conclude that $B_{n}B\mathbf{1}_{w}\in
l^{2}(G)$ and we may present the sum in (8) in an $l^{2}$ notation as
$Q(v,w)=\sum_{n=1}^{\infty}\langle
B_{n}B\mathbf{1}_{v},B\mathbf{1}_{w}\rangle.$ (9)
Next we employ the argument of Aizenman & Newman [AN84] to show that $B_{n}$
is a positive operator. This means that $B_{n}(v,w)=B_{n}(w,v)$ (which is
obvious) and that $\langle B_{n}f,f\rangle\geq 0$ for any (real-valued) $f\in
l^{2}$. It is enough to verify this for $f$ with finite support. But in this
case we can write
$\displaystyle\langle B_{n}f,f\rangle$
$\displaystyle=\sum_{v,w}f(v)f(w)\mathbb{P}(v\leftrightarrow
w,\,|\mathcal{C}(v)|=n)=$ $\displaystyle(*)\qquad$
$\displaystyle=\mathbb{E}\Big{(}\sum_{v,w}f(v)f(w)\mathbf{1}_{\\{v\leftrightarrow
w,|\mathcal{C}(v)|=n\\}}\Big{)}=$
$\displaystyle=\mathbb{E}\Big{(}\sum_{\mathcal{C}\>\mathrm{s.t.}\>|\mathcal{C}|=n}\;\sum_{v,w\in\mathcal{C}}f(v)f(w)\Big{)}=\mathbb{E}\Big{(}\sum_{\mathcal{C}\>\mathrm{s.t.}\>|\mathcal{C}|=n}\Big{(}\sum_{v\in\mathcal{C}}f(v)\Big{)}^{2}\Big{)}\geq
0.$
where $(*)$ is where we used the fact that $f$ has finite support to justify
taking the expectation out of the sum. The notation $\mathbf{1}_{E}$ here is
for the indicator of the event $E$. Thus $B_{n}$ is positive.
We now apply the spectral theorem for _bounded_ positive operators to take the
square root of $B_{n}$. See [EMT04], lemma 6.3.5 for the specific case of
taking the root of a positive operator and chapter 7 for general spectral
theory. Denote $S_{n}=\sqrt{B_{n}}$. This implies, of course, that
$S_{n}^{2}=B_{n}$ but also that $S_{n}$ is positive and that it commutes with
any operator $\Phi$ that commutes with $B_{n}$.
Returning to (9) we now write
$Q(v,w)=\sum_{n=1}^{\infty}\langle
S_{n}^{2}B\mathbf{1}_{v},B\mathbf{1}_{w}\rangle=\sum_{n=1}^{\infty}\langle
S_{n}B\mathbf{1}_{v},S_{n}B\mathbf{1}_{w}\rangle.$ (10)
The fact that $Q(v,v)<\infty$ therefore implies that
$\sum_{n=1}^{\infty}||S_{n}B\mathbf{1}_{v}||^{2}<\infty.$ (11)
Our only use of the triangle condition.
Fix now some $\epsilon>0$. By (11) we can find some $N$ such that
$\sum_{n=N+1}^{\infty}||S_{n}B\mathbf{1}_{v}||^{2}<\tfrac{1}{2}\epsilon.$ (12)
Since $S_{n}B\mathbf{1}_{v}\in l^{2}(G)$, we can use the lemma, and we use it
with
$f_{\mathrm{lemma}}=S_{n}B\mathbf{1}_{v}\quad
v_{\mathrm{lemma}}=v\qquad\delta_{\mathrm{lemma}}=\frac{\epsilon}{2N}\>.$
We get some $R_{n}$ such that for any $\varphi$ taking $v$ outside of
$B(v,R_{n})$,
$|\langle\Phi_{\varphi}S_{n}B\mathbf{1}_{v},S_{n}B\mathbf{1}_{v}\rangle|\leq\frac{\epsilon}{2N}\>.$
Some standard abstract nonsense shows that the invariance of $B_{n}$ i.e. the
fact that $B_{n}(x,y)=B_{n}(\varphi(x),\varphi(y))$ implies that
$B_{n}\Phi=\Phi B_{n}$. Hence also $S_{n}\Phi=\Phi S_{n}$ so
$\langle\Phi S_{n}B\mathbf{1}_{v},S_{n}B\mathbf{1}_{v}\rangle=\langle
S_{n}B\Phi\mathbf{1}_{v},S_{n}B\mathbf{1}_{v}\rangle=\langle
S_{n}B\mathbf{1}_{\varphi(v)},S_{n}B\mathbf{1}_{v}\rangle.$
Define $R=\max\\{R_{1},\dotsc,R_{N}\\}$. We get, for every $w\not\in B(v,R)$,
$\sum_{n=1}^{N}\langle S_{n}B\mathbf{1}_{v},S_{n}B\mathbf{1}_{w}\rangle\leq
N\delta=\tfrac{1}{2}\epsilon.$ (13)
(12) takes care of the other sum,
$\displaystyle\sum_{n=N+1}^{\infty}\langle
S_{n}B\mathbf{1}_{v},S_{n}B\mathbf{1}_{w}\rangle$
$\displaystyle\leq\sum_{n=N+1}^{\infty}||S_{n}B\mathbf{1}_{v}||\cdot||S_{n}B\mathbf{1}_{w}||=$
$\displaystyle=\sum_{n=N+1}^{\infty}||S_{n}B\mathbf{1}_{v}||^{2}<\tfrac{1}{2}\epsilon.$
(14)
We are done. We get that for any $w\not\in B(v,R)$,
$Q(v,w)\stackrel{{\scriptstyle(\ref{eq:QSBSB})}}{{=}}\sum_{n=1}^{\infty}\langle
S_{n}B\mathbf{1}_{v},S_{n}B\mathbf{1}_{w}\rangle\stackrel{{\scriptstyle(\ref{eq:sum1N},\ref{eq:sumNinf})}}{{\leq}}\epsilon$
as required. ∎
_Closing remark._ Comparing the proof here to that of Barsky & Aizenman
[BA91], it seems as if there is something missing in their argument. This is
not true. Justifying the change of order of summation in [BA91] is completely
standard — for example, by examining Cesàro sums — and does not deserve any
special remark.
## References
* [AN84] Michael Aizenman and Charles M. Newman, _Tree graph inequalities and critical behavior in percolation models_. J. Statist. Phys. 36:1-2 (1984), 107–143.
* [BA91] David J. Barsky and Michael Aizenman, _Percolation critical exponents under the triangle condition_. Ann. Probab. 19:4 (1991), 1520–1536.
* [BR06] Béla Bollobás and Oliver Riordan, _Percolation_. Cambridge University Press, New York, 2006.
* [EMT04] Yuli Eidelman, Vitali Milman and Antonis Tsolomitis, _Functional analysis. An introduction_. Graduate Studies in Mathematics, 66\. American Mathematical Society, Providence, RI, 2004.
* [G99] Geoffrey Grimmett, _Percolation._ Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999.
* [HHS03] Takashi Hara, Remco van der Hofstad and Gordon Slade, _Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models_. Ann. Probab. 31:1 (2003), 349–408.
* [HS90] Takashi Hara and Gordon Slade, _Mean-field critical behaviour for percolation in high dimensions_. Commun. Math. Phys. 128:2 (1990), 333–391.
* [HHS08] Markus Heydenreich, Remco van der Hofstad R. and Akira Sakai, _Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk_. J. Statist. Phys., 132:6 (2008), 1001–-1049.
* [K] Gady Kozma, _Percolation on a product of two trees_ , in preparation
* [KN09] Gady Kozma and Asaf Nachmias, _The Alexander-Orbach conjecture holds in high dimensions_. To appear in Invent. Math., preprint available from
http://arxiv.org/abs/0806.1442
* [N87] Bao Gia Nguyen, _Gap exponents for percolation processes with triangle condition_. J. Statist. Phys. 49:1-2 (1987), 235–243.
* [S01] Roberto H. Schonmann, _Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs_. Commun. Math. Phys. 219:2 (2001) 271-322.
* [S02] Roberto H. Schonmann, _Mean-field criticality for percolation on planar non-amenable graphs_. Commun. Math. Phys. 225:3 (2002), 453-463.
|
arxiv-papers
| 2009-07-11T11:53:42 |
2024-09-04T02:49:03.837013
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gady Kozma",
"submitter": "Gady Kozma",
"url": "https://arxiv.org/abs/0907.1959"
}
|
0907.1965
|
11institutetext: Centre for Mathematical Modelling, University of Leicester,
Leicester, LE1 7RH, UK, {ag153,tt51}@le.ac.uk 22institutetext: Siberian
Federal University, Krasnoyarsk, 660041, Russia
# Law of the Minimum Paradoxes
Alexander N. Gorban 11 Lyudmila I. Pokidysheva 22 Elena V. Smirnova 22 Tatiana
A. Tyukina 11
###### Abstract
Keywords: Liebig s Law, Adaptation, Fitness, Stress
Keywords: Liebig s Law, Adaptation, Fitness, Stress
## 1 Introduction
### 1.1 The Law of the Minimum
The “law of the minimum” states that growth is controlled by the scarcest
resource (limiting factor) Liebig1 . This law is usually believed to be the
result of Justus von Liebig’s research (1840) but the agronomist and chemist
Carl Sprengel published in 1828 an article that contained in essence the Law
of the Minimum and this law can be called the Sprengel–Liebig Law of the
Minimum. van der Ploeg1999 .
This concept is illustrated on Fig. 1.
Figure 1: The Law of the Minimum. Coordinates $c_{1}$, $c_{2}$ are normalized
values of factors. For a given state $s=(c_{1}(s),c_{2}(s))$, the bold solid
line $\min\\{c_{1},c_{2}\\}=\min\\{c_{1}(s),c_{2}(s)\\}$ separates the states
with better conditions (higher productivity) from the states with worse
conditions. On this line the conditions do not differ significantly from $s$
because of the same value of the limiting factor. The dot dash line shows the
border of survival. On the dashed line the factors are equally important
($c_{1}=c_{2}$).
This concept was originally applied to plant or crop growth. Many times it was
criticized, rejected, and then returned to and demonstrated quantitative
agreement with experiments Liebig1 ; Liebig2+ ; Liebig3+ .
The law of the minimum was extended then to more a general conception of
factors, not only for the elementary physical description of available
chemical substances and energy. Any environmental factor essential for life
that is below the critical minimum, or that exceeds the maximum tolerable
level could be considered as a limiting one.
There were several attempts to create a general theory of factors and
limitation in ecology, physiology and evolutionary biology. Tilman Tilman1980
proposed an equilibrium theory of resource competition based on classification
of interaction in pairs of resources. They may be: (1) essential, (2) hemi-
essential, (3) complementary, (4) perfectly substitutable, (5) antagonistic,
or (6) switching. This interaction depends on spatial heterogeneity of
resource distributions. For various resource types, the general criterion for
stable coexistence of species was developed.
Bloom, Chapin and Mooney BloomChapinMooney1985 ; ChapinSchulzeMooney1990
elaborated the economical metaphor of ecological concurrency. This analogy
allowed them to merge the optimality and the limiting approach and to
formulate four “theorems”. In particular, Theorem 3 states that a plant should
adjust allocation so that, for a given expenditure in acquiring each resource,
it achieves the same growth response: Growth is equally limited by all
resources. This is a result of adjustment: adaptation makes the limiting
factors equally important. They also studied the possibility for resources to
substitute for one another (Theorem 4) and introduced the concept of “exchange
rate”.
For human physiology the observation that adaptation makes the limiting
factors equally important was supported by many data of human adaptation to
the Far North conditions (or, which is the same, disadaptation causes
inequality of factors and leads to appearance of single limiting factor)
GorSmiCorAd1st . The theory of factors – resource interaction was developed
and supported by experimental data. The results are used for monitoring of
human populations in Far North Sedov .
In their perspectives paper, Sih and Gleeson SihGleeson1995 considered three
inter-related issues which form the core of evolutionary ecology: (1) key
environmental factors; (2) organismal traits that are responses to the key
factors; (3) the evolution of these key traits. They suggested to focus on
’limiting traits’ rather than optimal traits. Adaptation leads to optimality
and equality of traits as well as of factors but under variations some traits
should be more limiting than others. From the Sih and Gleeson point of view,
there is a growing awareness of the potential value of the limiting traits
approach as a guide for studies in both basic and applied ecology.
The critics of the Law of the Minimum is usually based on the “colimitation”
phenomenon: limitation of growth and surviving by a group of equally important
factors and traits. For example, analysis of species-specific growth and
mortality of juvenile trees at several contrasting sites suggests that light
and other resources can be simultaneously limiting, and challenges the
application of the Law of the Minimum to tree sapling growth Kobe1996 .
The concept of multiple limitation was proposed for unicellular organisms
based on the idea of the nutritional status of an organism expressed in terms
of state variables vandenBerg1998 . The property of being limiting was defined
in terms of the reserve surplus variables. This approach was illustrated by
numerical experiments.
In the world ocean there are High Nutrient–Low Chlorophyll regions where
chlorophyll concentrations are lower than expected concentrations given the
ambient phosphate and nitrate levels. In these regions, limitations of
phytoplankton growth by other nutrients like silicate or iron have been
hypothesized and supported by experiments. This colimitation was studied using
a nine-component ecosystem model embedded in the HAMOCC5 model of the oceanic
carbon cycle Aumontatal2003 .
The double–nutrient–limited growth appears also as a transition regime between
two regimes with single limiting factor. For bacteria and yeasts at a constant
dilution rate in the chemostat, three distinct growth regimes were recognized:
(1) a clearly carbon-limited regime with the nitrogen source in excess, (2) a
double–nutrient–limited growth regime where both the carbon and the nitrogen
source were below the detection limit, and (3) a clearly nitrogen-limited
growth regime with the carbon source in excess. The position of the
double–nutrient–limited zone is very narrow at high growth rates and becomes
broader during slow growth EgliZinn2003 ; ZinnWitholtEgli2004 .
Decomposition of soil organic matter is limited by both the available
substrate and the active decomposer community. The colimitation effects
strongly affect the feedbacks of soil carbon to global warming and its
consequences WutzlerReichstein2008 .
Dynamics of communities leads to colimitation on community level even if
organisms and populations remain limited by single factors. Communities are
likely to adjust their stoichiometry by competitive exclusion and coexistence
mechanisms. It guaranties simultaneous limitation by many resources and
optimal use of them at the community scale. This conclusion was supported by a
simple resource ratio model and an experimental test carried out in microcosms
with bacteria Dangeratal2008 .
In spite of the long previous discussion of colimitation, in 2008 Saito and
Goepfert stressed that this notion is “an important yet often misunderstood
concept” SaitoGoepfert2008 . They describe the potential nutrient colimitation
pairs in the marine environment and define three types of colimitation:
1. I.
Independent nutrient colimitation concerns two elements that are generally
biochemically mutually exclusive, but are also both found in such low
concentrations as to be potentially limiting. Example: nitrogen–phosphorus
colimitation.
2. II.
Biochemical substitution colimitation involves two elements that can
substitute for the same biochemical role within the organism. Example:
zinc–cobalt colimitation.
3. III.
Biochemically dependent colimitation refers to the limitation of one element
that manifests itself in an inability to acquire another element. Example:
zinc–carbon colimitation.
The experimental colimitation examples of the first type do not refute the Law
of the Minimum completely but rather support the following statement: the
ecological systems of various levels, from an organism to a community, may
avoid the monolimitation regime either by the natural adjustment of their
consumption structure BloomChapinMooney1985 ; SemSem or just by living in the
transition zone between the monolimitation regimes. From the general point of
view SihGleeson1995 , such a transition zone is expected to be quite narrow
(as a vicinity of a surface where factors are equal) but in some specific
situations it may be broad, for example, for slow growth regimes in the
chemostat EgliZinn2003 ; ZinnWitholtEgli2004 .
The type II and type III colimitations should be carefully separated from the
usual discussion of the Law of the Minimum limitation. For these types of
colimitation, two (or more) nutrients limit growth rates simultaneously,
either through the effect of biochemical substitution (type II) or by
depressing the ability for the uptake of another nutrient (type III)
SaitoGoepfert2008 . The type II and type III colimitations give us examples of
the “non-Liebig” organization of the system of factors.
The Law of the Minimum is one of the most important tools for mathematical
modeling of ecological systems. It gives a clue for making of the first model
for multi-component and multi-factor systems. This clue sounds rather simple:
first of all, we have to take into account the most important factors which
are, probably, limiting factors. Everything else should be excluded and return
back only in a case when a ’sufficient reason’ is proved (following the famous
“Principle of Sufficient Reason” by Leibnitz, one of the four recognized laws
of thought).
It is suggested to consider the Liebig production function as the “archetype”
for ecological modeling Nijlandatal2008 . The generalizations of the Law of
the Minimum were supported by the biochemical idea of limiting reaction steps
(see, for example, Brown or recent review GorRadLim ). Three classical
production functions, the Liebig, Mitscherlich and Liebscher relations between
nutrient supply and crop production, are limiting cases of an integrated model
based on the Michaelis–Menten kinetic equation Nijlandatal2008 .
Applications of the Law of the Minimum to the ecological modeling are very
broad. The quantitative theories of the bottom–up control of the phytoplankton
dynamics is based on the influence of limiting nutrients on growth and
reproduction. The most used is the Droop model and its generalizations
Droop1973 ; LegovicCruzado1997 ; Ballantyneatal2008 .
The Law of the Minimum was combined with the evolutionary dynamics to analyze
the “Paradox of the plankton” Shoreshatal2008 formulated by Hutchinson
Hutchinson1961 in 1961: How it is possible for a number of species to coexist
in a relatively isotropic or unstructured environment all competing for the
same sorts of materials… According to the principle of competitive exclusion…
we should expect that one species alone would outcompete all of the others. It
was shown that evolution exacerbates the paradox and it is now very far from
the resolution.
The theory of evolution from monolimitation toward colimitation was developed
that takes into account the viruses attacks on the phytoplankton receptors
Menge2009 . In the classic theory Tilman1982 , evolution toward colimitation
decreases equilibrium resource concentrations and increases equilibrium
population density. In contrary, under influence of viruses, evolution toward
co-limitation may have no effect on equilibrium resource concentrations and
may decreases the equilibrium population density Menge2009 .
The Law of the Minimum was used for modeling of microcolonial fungi growth on
rock surfaces Chertovatal2004 . The analysis demonstrated, that a continued
lack of organic nutrition is a dominating environmental factor limiting growth
on stone monuments and other exposed rock surfaces in European temperate and
Mediterranean climate.
McGill McGill2005 developed a model of coevolution of mutualisms where one
resource is traded for another resource. The mechanism is based on the Law of
the Minimum in combination with Tilman’s approach to resource competition
Tilman1980 ; Tilman1982 . It was shown that resource limitations cause
mutualisms to have stable population dynamics.
The Law of the Minimum produces the piecewise linear growth functions which
are non-smooth and very far from being linear. This nonlinearity transforms
normal or uniform distributions of resource availabilities into skewed crop
yield distribution and no natural satisfactory motivation exists in favor of
any simple crop yield distribution Hennessy2009 . With independent, identical,
uniform resource availability distributions the yield skew is positive, and it
is negative for normal distributions.
The standard linear tools of statistics such as generalized linear models do
not work satisfactory for systems with limiting factors. Conventional
correlation analysis conflicts with the concept of limiting factors. This was
demonstrated in a study of the spatial distribution of Glacier lily in
relation to soil properties and gopher disturbance Thomsonatal1996 . For
systems with limiting factors, quantile regression performs much better. It
has strong theoretical justification in Law of the Minimum Austin2007 .
Some of the generalizations of the Law of the Minimum went quite far from
agriculture and ecology. The law of the minimum was applied to economics
EcolEcon and to education, for example EcolEdu .
Recently, a strong mathematical background was created for the Law of the
Minimum. Now the limiting factors theory together with static and dynamic
limitation in chemical kinetics GorRadLim ; GorbanRadZin2010 are considered
as the realization of the Maslov dequantization KoMa97 ; LitvinovMaslov2005 ;
Litvinov2007 and idempotent analysis. Roughly speaking, the limiting factor
formalism means that we should handle any two quantities $c_{1},c_{2}$ either
as equal numbers or as numbers connected by the relation $\gg$: either
$c_{1}\gg c_{2}$ or $c_{1}\ll c_{2}$. Such a hard non-linearity can arise in
the smooth dynamic models because of the time-scale separation vandenBerg1998
.
Dequantization of the traditional mathematics leads to a mathematics over
tropical algebras like the max-plus algebra. Since the classical work of
Kleene Kleene these algebras are intensively used in mathematics and computer
science, and the concept of dequantization and idempotent analysis opened new
applications in physics and other natural sciences (see the comprehensive
introduction in Litvinov2007 ). Liebig’s and anti-Liebig’s (see Definition 1
below) systems of factors may be considered as realizations of max-plus or
min-plus asymptotics correspondingly.
### 1.2 Fitness Convexity, Concavity and Various Interactions Between Factors
There exist an opposite type of organization of the system of factors, which,
from the first glance, seems to be symmetric to Liebig’s type of interaction
between them. In Liebig’s systems, the factor with the worst value determines
the growth and surviving. The completely opposite situation is: the factor
with the best value determines everything. We call such a system “anti-
Liebig’s” one. Of course, it seems improbable that all the possible factors
interact following the Law of the Minimum or the fully opposite anti-Liebig’s
rule. Interactions between factors in real systems are much more complicated
SaitoGoepfert2008 . Nevertheless, we can state a question about hierarchical
decomposition of the system of factors in elementary groups with simple
interactions inside, then these elementary groups can be clustered into
super–factors with simple interactions between them, and so on.
Let us introduce some notions and notations. We consider organisms that are
under the influence of several factors $F_{1},...F_{q}$. Each factor has its
intensity $f_{i}$ ($i=1,...q$). For convenience, we consider all these factors
as negative or harmful. This is just a convention about the choice of axes
directions: a wholesome factor is just a “minus harmful” factor.
At this stage, we do not specify the nature of these factors. Formally, they
are just inputs in the adaptation dynamics, the arguments of the fitness
functions.
The fitness function is the central notion of the evolutionary and ecological
dynamics. This is a function that maps the environmental factors and traits of
the organism into the reproduction coefficient, that is, its contribution, in
offspring to its population. Fisher proposed to construct fitness as a
combination of independent individual contribution of various traits
Fisher1930 . Haldane Haldane1932 criticized the approach based traits on
independent actions of traits. Modern definitions of fitness function are
based on adaptation dynamics. For the structured populations, the fitness
should be defined through the dominant Lyapunov exponents G1984 ;
MetzNisbetGeritz1992 . In the evolutionary game theory Maynard-Smith1982 ,
payoff represents Darwinian fitness and describes how the use of the strategy
improves an animal’s prospects for survival and reproduction. Recently, the
Fisher and Haldane approaches are combined WaxmanWelch2005 : Haldane’s concern
is incorporated into Fisher’s model by allowing the intensity of selection to
vary between traits.
It is a nontrivial task to measure the fitness functions and action of
selection in nature, but now it was done for many populations and phenotypical
traits KingsolverPfennig2007 . Special statistical methods for life-history
analysis for inference of fitness and population growth are developed and
tested Shawatal2008 .
In our further analysis we do not need exact values of fitness but rather it
existence and some qualitative features.
First of all, let us consider an oversimplified situation with identical
organisms. Given phenotypical treats, fitness $W$ is a function of factors
loads: $W=W(f_{1},\ldots,f_{q})$. This assumption does not take into account
physiological adaptation that works as a protection system and modifies the
factor loads. This modification is in the focus of our analysis in the follow-
up sections, and now we neglect adaptation. The convention about axes
direction means that all the partial derivatives of $W$ are non-positive
$\partial W/\partial f_{i}\leq 0$.
By definition, for Liebig’s system of factors $W$ is a function of the worst
(maximal) factor intensity: $W=W(\max\\{f_{1},\ldots,f_{q}\\})$ (Fig. 2a) and
for anti-Liebig’s system it is the function of the best (minimal) factor
intensity $W=W(\min\\{f_{1},\ldots,f_{q}\\})$ (Fig. 2c). Such representations
as well as the usual formulation of the Law of the Minimum require special
normalization of factor intensities to compare the loads of different factors.
For Liebig’s systems of factors the superlevel sets of $W$ given by
inequalities $W\geq w_{0}$ are convex for any level $w_{0}$ in a convex domain
(Fig. 2a). For anti-Liebig’s systems of factors the sublevel sets of $W$ given
by inequalities $W\leq w_{0}$ are convex for any level $w_{0}$ in a convex
domain (Fig. 2c).
(a) Liebig’s system
(b) Generalized Liebig’s system
(c) Anti-Liebig’s system
(d) Synergistic system
Figure 2: Various types of organization of the system of factors. For a given
state $s$ the bold solid line is given by the equation $W(f_{1},f_{2})=W(s)$.
This line separates the area with higher fitness (“better conditions”) from
the line with lower fitness (“worse conditions”). In Liebig’s (a) and
generalized Liebig’s systems (b) the area of better conditions is convex, in
“anti-Liebig’s” systems (c) and the general synergistic systems (d) the area
of worse conditions is convex. The dot dash line shows the border of survival.
On the dashed line the factors are equally important ($f_{1}=f_{2}$).
These convexity properties are essential for optimization problems which arise
in the modeling of adaptation and evolution. Let us take them as definitions
of the generalized Liebig and anti-Liebig systems of factors:
Definition 1.
* •
A system of factors is the generalized Liebig system in a convex domain $U$,
if for any level $w_{0}$ the superlevel set $\\{f\in U\ |\ W(f)\geq w_{0}\\}$
is convex (Fig. 2b).
* •
A system of factors is the generalized anti-Liebig system in a convex domain
$U$, if for any level $w_{0}$ the sublevel set $\\{f\in U\ |\ W(f)\leq
w_{0}\\}$ is convex (Fig. 2d).
We call the generalized anti-Liebig systems of factors the synergistic systems
because this definition formalizes the idea of synergy: in the synergistic
systems harmful factors superlinear amplify each other.
(a) Liebig’s system
(b) Generalized Liebig’s system
(c) Anti-Liebig’s system
(d) Synergistic system
Figure 3: Conditional optimisation for various systems of factors. Because of
convexity conditions, fitness achieves its maximum on an interval $L$ for
Liebig’s system (a) on the diagonal (the factors are equally important), for
generalized Liebig’s systems (b) near the diagonal, for anti-Liebig’s system
(c) and for the general synergistic system (d) this maximum is one of the ends
of the interval $L$.
Conditional maximization of fitness destroys the symmetry between Liebig’s and
anti-Liebig’s systems as well as between generalized Liebig’s systems and
synergistic ones. Following the geometric approach of Tilman1980 ; Tilman1982
we illustrate this optimization on Fig. 3. The picture may be quite different
from the conditional maximization of a convex function near its minima point
(compare, for example, Figs. 3c,3d to Fig. from SihGleeson1995 ).
Individual adaptation changes the picture. In the next subsection we discuss
possible mechanism of these changes.
### 1.3 Adaptation Energy and Factor–Resource Models
The reaction of an organism to the load of a single factor may have plateaus
(intervals of tolerance considered in Shelford’s “law of tolerance”, Odum ,
Chapter 5). The dose–response curves may be nonmonotonic Colborn or even
oscillating. Nevertheless, we start from a very simple abstract model that is
close to the usual factor analysis.
We consider organisms that are under the influence of several harmful factors
$F_{1},...F_{q}$ with intensities $f_{i}$ ($i=1,...q$). Each organism has its
adaptation systems, a “shield” that can decrease the influence of external
factors. In the simplest case, it means that each system has an available
adaptation resource, $R$, which can be distributed for the neutralization of
factors: instead of factor intensities $f_{i}$ the system is under pressure
from factor values $f_{i}-a_{i}r_{i}$ (where $a_{i}>0$ is the coefficient of
efficiency of factor $F_{i}$ neutralization by the adaptation system and
$r_{i}$ is the share of the adaptation resource assigned for the
neutralization of factor $F_{i}$, $\sum_{i}r_{i}\leq R$). The zero value
$f_{i}-a_{i}r_{i}=0$ is optimal (the fully compensated factor), and further
compensation is impossible and senseless.
For unambiguity of terminology, we use the term “factor” for all factors
including any deficit of available external resource or even some illnesses.
We keep the term “resource” for internal resources, mostly for the
hypothetical Selye’s “adaptation energy”.
We represent the organisms, which are adapting to stress, as the systems which
optimize distribution of available amount of a special adaptation resource for
neutralization of different aggressive factors (we consider the deficit of
anything needful as a negative factor too). These factor–resource models with
optimization are very convenient for the modeling of adaptation. We use a
class of models many factors – one resource.
Interaction of each system with a factor $F_{i}$ is described by two
quantities: the factor $F_{i}$ pressure $\psi_{i}=f_{i}-a_{i}r_{i}$ and the
resource $r_{i}$ assigned to the factor $F_{i}$ neutralization. The first
quantity characterizes, how big the uncompensated harm is from that factor,
the second quantity measures, how intensive is the adaptation answer to the
factor (or how far the system was modified to answer the factor $F_{i}$
pressure).
Already one factor–one resource models of adaptation produce the tolerance
law. We demonstrate below that it predicts the separation of groups of
organism into two subgroups: the less correlated well–adapted organisms and
highly correlated organisms with a deficit of the adaptation resource. The
variance is also higher in the highly correlated group of organisms with a
deficit of the adaptation resource.
This result has a clear geometric interpretation. Let us represent each
organism as a data point in an $n$-dimensional vector space. Assume that they
fall roughly within an ellipsoid. The well-adapted organisms are not highly
correlated and after normalization of scales to unit variance the
corresponding cloud of points looks roughly as a sphere. The organisms with a
deficit of the adaptation resource are highly correlated, hence in the same
coordinates their cloud looks like an ellipsoid with remarkable eccentricity.
Moreover, the largest diameter of this ellipsoid is larger than for the
well–adapted organisms and the variance increases together with the
correlations.
This increase of variance together with correlations may seem counterintuitive
because it has no formal backgrounds in definitions of the correlation
coefficients and variance. This is an empirical founding that under stress
correlations and variance increase together, supported by many observations
both for physiological and financial systems. The factor-resource models give
a plausible explanation of this phenomena.
The crucial question is: what is the resource of adaptation? This question
arose for the first time when Selye published the concept of adaptation energy
and experimental evidence supporting this idea SelyeAEN ; SelyeAE1 . Selye
found that the organisms (rats) which demonstrate no differences in normal
environment may differ significantly in adaptation to an increasing load of
environmental factors. Moreover, when he repeated the experiments, he found
that adaptation ability decreases after stress. All the observations could be
explained by existence of an universal adaptation resource that is being spent
during all adaptation processes.
Selye’s ideas allow the following interpretation: the aggressive influence of
the environment on the organism may be represented as action of independent
factors. The system of adaptation consists of subsystems, which protect the
organism from different factors. These subsystems consume the same resource,
the adaptation energy. The distribution of this resource between the
subsystems depends on environmental conditions.
Later the concept of adaptation energy was significantly improved GP_AE1952 ,
plenty of indirect evidence supporting this concept were found, but this
elusive adaptation energy is still a theoretical concept, and in the modern
“Encyclopedia of Stress” we read: “As for adaptation energy, Selye was never
able to measure it…” AEencicl . Nevertheless, the notion of adaptation energy
is very useful in the analysis of adaptation and is now in wide use (see, for
example, BreznitzAEappl ; SchkadeOccAdAE2003 ). It should be specially
stressed that the adaptation energy is neither physical energy nor a
substance. This idealization describes the experimental results: in many
experiments it was demonstrated, that organisms under load of various factors
behave as if they spend a recourse, which is the same for different factors.
This recourse may be exhausted and then the organism dies.
The idea of exchange can help in the understanding of adaptation energy: there
are many resources, but any resource can be exchanged for another one. To
study such an exchange an analogy with the currency exchange is useful.
Following this analogy, we have to specify, what is the exchange rate, how
fast this exchange could be done (what is the exchange time), what is the
margin, how the margin depends on the exchange time. There may appear various
limitations of the amount of the exchangeable resource, and so on. The
economic metaphor for ecological concurrency and adaptation was elaborated in
1985 BloomChapinMooney1985 ; ChapinSchulzeMooney1990 but much earlier, in
1952, it was developed for physiological adaptation GP_AE1952 .
Market economics seems closer to the idea of resource universalization than
biology is, but for biology this exchange idea also seems useful. Of course
there exist some limits on the possible exchanges of different resources. It
is possible to include the exchange processes into models, but many questions
appear immediately about unknown coefficients. Nevertheless, we can follow
Selye’s arguments and postulate the adaptation energy as a universal
adaptation resource.
The adaptation energy is neither physical energy nor a substance. This is a
theoretical construction, which may be considered as a pool of various
exchangeable resources. When an organism achieves the limits of resource
exchangeability, the universal non-specific stress and adaptation syndrome
transforms (disintegrates) into specific diseases. Near this limit we have to
expect the critical retardation of exchange processes.
Adaptation optimizes the state of the system for given available amounts of
the adaptation resource. This idea seems very natural, but it may be a
difficult task to find the objective function that is hidden behind the
adaptation process. Nevertheless, even an assumption about the existence of an
objective function and about its general properties helps in analysis of
adaptation process.
Assume that adaptation should maximize a fitness function $W$ which depends on
the compensated values of factors, $\psi_{i}=f_{i}-a_{i}r_{i}$ for the given
amount of available resource:
$\left\\{\begin{array}[]{l}W(f_{1}-a_{1}r_{1},f_{2}-a_{2}r_{2},...f_{q}-a_{q}r_{q})\
\to\ \max\ ;\\\ r_{i}\geq 0$, $f_{i}-a_{i}r_{i}\geq 0$,
$\sum_{i=1}^{q}r_{i}\leq R\ .\end{array}\right.$ (1)
The only question is: how can we be sure that adaptation follows any
optimality principle? Existence of optimality is proven for microevolution
processes and ecological succession. The mathematical backgrounds for the
notion of “natural selection” in these situations are well–established after
work by Haldane (1932) Haldane1932 and Gause (1934) Gause . Now this
direction with various concepts of fitness (or “generalized fitness”)
optimization is elaborated in many details (see, for example, review papers
Bom02 ; Oechssler02 ; GorbanSelTth ).
The foundation of optimization is not so clear for such processes as
modifications of a phenotype, and for adaptation in various time scales. The
idea of genocopy–phenocopy interchangeability was formulated long ago by
biologists to explain many experimental effects: the phenotype modifications
simulate the optimal genotype (West-Eberhardgenocopy-phenocopy , p. 117). The
idea of convergence of genetic and environmental effects was supported by an
analysis of genome regulation ZuckerkandlConvergGenEnv (the principle of
concentration–affinity equivalence). The phenotype modifications produce the
same change, as evolution of the genotype does, but faster and in a smaller
range of conditions (the proper evolution can go further, but slower). It is
natural to assume that adaptation in different time scales also follows the
same direction, as evolution and phenotype modifications, but faster and for
smaller changes. This hypothesis could be supported by many biological data
and plausible reasoning. (See, for example, the case studies of relation
between evolution of physiological adaptation Hoffman1978 ; Greene1999 , a
book about various mechanisms of plants responses to environmental stresses
Lerner1999 , a precise quantitative study of the relationship between
evolutionary and physiological variation in hemoglobin Miloatal2007 and a
modern review with case studies FuscoMinelli2010 .)
It may be a difficult task to find an explicit form of the fitness function
$W$, but for our qualitative analysis we need only a qualitative assumption
about general properties of $W$. First, we assume monotonicity with respect to
each coordinate:
$\frac{\partial W(\psi_{1},\ldots\psi_{q})}{\partial\psi_{i}}\leq 0\,.$ (2)
A system of factors is Liebig’s system, if
$W=W\left(\max_{1\leq i\leq q}\\{f_{i}-a_{i}r_{i}\\}\right)\ .$ (3)
This means that fitness depends on the worst factor pressure.
A system of factors is generalized Liebig’s system, if for any two different
vectors of factor pressures $\mathbf{\psi}=(\psi_{1},...\psi_{q})$ and
$\mathbf{\phi}=(\phi_{1},...\phi_{q})$ ($\mathbf{\psi}\neq\mathbf{\phi}$) the
value of fitness at the average point $(\mathbf{\psi}+\mathbf{\phi})/2$ is
greater, than at the worst of points $\mathbf{\psi}$, $\mathbf{\phi}$:
$W\left(\frac{\mathbf{\psi}+\mathbf{\phi}}{2}\right)>\min\\{W(\mathbf{\psi}),W(\mathbf{\phi})\\}\
.$ (4)
Any Liebig’s system is, at the same time, generalized Liebig’s system because
for such a system the fitness $W$ is a decreasing function of the maximal
factor pressure, the minimum of $W$ corresponds to the maximal value of the
limiting factor and
$\max\left\\{\frac{\psi_{1}+\phi_{1}}{2},\ldots,\frac{\psi_{q}+\phi_{q}}{2}\right\\}\leq\max\\{\max\\{\psi_{1},\ldots,\psi_{q}\\},\max\\{\phi_{1},\ldots,\phi_{q}\\}\\}\
.$
The opposite principle of factor organization is synergy: the superlinear
mutual amplification of factors. The system of factors is a synergistic one,
if for any two different vectors of factor pressures
$\mathbf{\psi}=(\psi_{1},...\psi_{q})$ and
$\mathbf{\phi}=(\phi_{1},...\phi_{q})$ ($\mathbf{\psi}\neq\mathbf{\phi}$) the
value of fitness at the average point $(\mathbf{\psi}+\mathbf{\phi})/2$ is
less, than at the best of points $\mathbf{\psi}$, $\mathbf{\phi}$:
$W\left(\frac{\mathbf{\psi}+\mathbf{\phi}}{2}\right)<\max\\{W(\mathbf{\psi}),W(\mathbf{\phi})\\}\
.$ (5)
A system of factors is anti-Liebig’s system, if
$W=W\left(\min_{1\leq i\leq q}\\{f_{i}-a_{i}r_{i}\\}\right)\ .$ (6)
This means that fitness depends on the best factor pressure. Any anti-Liebig
system is, at the same time a synergistic one because for such a system the
fitness $W$ is a decreasing function of the minimal factor pressure, the
maximum of $W$ corresponds to the minimal value of the factor with minimal
pressure and
$\min\left\\{\frac{\psi_{1}+\phi_{1}}{2},\ldots,\frac{\psi_{q}+\phi_{q}}{2}\right\\}\geq\min\\{\min\\{\psi_{1},\ldots,\psi_{q}\\},\min\\{\phi_{1},\ldots,\phi_{q}\\}\\}$
We prove that adaptation of an organism to Liebig’s system of factors, or to
any synergistic system, leads to two paradoxes of adaptation:
* •
Law of the Minimum paradox: If for a randomly selected pair, ( State of
environment – State of organism ), the Law of the Minimum is valid (everything
is limited by the factor with the worst value) then, after adaptation, many
factors (the maximally possible amount of them) are equally important.
* •
Law of the Minimum inverse paradox: If for a randomly selected pair, ( State
of environment – State of organism ), many factors are equally important and
superlinearly amplify each other then, after adaptation, a smaller amount of
factors is important (everything is limited by the factors with the worst non-
compensated values, the system approaches the Law of the Minimum).
In this paper, we discuss the individual adaptation. Other types of
adaptations, such as changes of the ecosystem structure, ecological succession
or microevolution lead to the same paradoxes if the factor–resource models are
applicable to these processes.
## 2 One-Factor Models, the Law of Tolerance, and the Order–Disorder
Transition
The question about interaction of various factors is very important, but,
first of all, let us study the one-factor models. Each organism is
characterized by measurable attributes $x_{1},\ldots x_{m}$ and the value of
adaptation resource, $R$.
### 2.1 Tension–Driven Models
In these models, observable properties of interest $x_{k}$ $(k=1,...m)$ can be
modeled as functions of factor pressure $\psi$ plus some noise $\epsilon_{k}$.
Let us consider one-factor systems and linear functions (the simplest case).
For the tension–driven model the attributes $x_{k}$ are linear functions of
tension $\psi$ plus noise:
$x_{k}=\mu_{k}+l_{k}\psi+\epsilon_{k}\ ,$ (7)
where $\mu_{k}$ is the expectation of $x_{k}$ for fully compensated factor,
$l_{k}$ is a coefficient, $\psi=f-ar_{f}\geq 0$, and $r_{f}\leq R$ is amount
of available resource assigned for the factor neutralization. The values of
$\mu_{k}$ could be considered as “normal” (in the sense opposite to
“pathology”), and noise $\epsilon_{k}$ reflects variability of norm.
If systems compensate as much of factor value, as it is possible, then
$r_{f}=\min\\{R,f/a\\}$, and we can write:
$\psi=\left\\{\begin{array}[]{ll}&f-aR\ ,\ \ {\rm if}\ \ f>aR\ ;\\\ &0,\ \ \
{\rm else.}\end{array}\right.$ (8)
Individual systems may be different by the value of factor intensity (the
local intensity variability), by amount of available resource $R$ and, of
course, by the random values of $\epsilon_{k}$. If all systems have enough
resource for the factor neutralization ($aR>f$) then all the difference
between them is in the noise variables $\epsilon_{k}$. Nobody will observe any
change under increase of the factor intensity, until violation of inequality
$F<r$.
Let us define the dose–response curve as
$M_{k}(f)=\mathbf{E}(x_{k}|f).$
Due to (7)
$M_{k}(f)=\mu_{k}+l_{k}\mathbf{P}(aR<f)(f-a\mathbf{E}(R|aR<f))\ ,$ (9)
where $\mathbf{P}(aR<f)$ is the probability of organism to have insufficient
amount of resource for neutralization of the factor load and
$\mathbf{E}(R|aR<f)$ is the conditional expectation of the amount of resource
if it is insufficient.
The slope $\mathrm{d}M_{k}(f)/\mathrm{d}f$ of the dose–response curve (9) for
big values of $f$ tends to $l_{k}$, and for small $f$ it could be much
smaller. This plateau at the beginning of the dose-response curve corresponds
to the law of tolerance (Victor E. Shelford, 1913, Odum , Chapter 5).
If the factor value increases, and for some of the systems the factor
intensity $f$ exceeds the available compensation $aR$ then for these systems
$\psi>0$ and the term $l_{k}\psi$ in Eq. (7) becomes important. If the noise
of the norm $\epsilon_{k}$ is independent of $\psi$ then the correlation
between different $x_{k}$ increases monotonically with $f$.
With increase of the factor intensity $f$ the dominant eigenvector of the
correlation matrix between $x_{k}$ becomes more uniform in the coordinates,
which tend asymptotically to $\pm\frac{1}{\sqrt{m}}$.
For a given value of the factor intensity $f$ there are two groups of
organisms: the well–adapted group with $R\geq f$ and $\psi=0$, and the group
of organisms with deficit of adaptation energy and $\psi>0$. If the
fluctuations of norm $\epsilon_{k}$ are independent for different $k$ (or just
have small correlation coefficients), then in the group with deficit of
adaptation energy the correlations between attributes is much bigger than in
the well–adapted group. If we use metaphor from physics, we can call this two
groups two phases: the highly correlated phase with deficit of adaptation
energy and the less correlated phase of well-adapted organisms.
In this simple model (7) we just formalize Selye’s observations and
theoretical argumentation. One can call it Selye’s model. There are two other
clear possibilities for one factor–one resource models.
### 2.2 Response–Driven Models
What is more important for values of the observable quantities $x_{k}$: the
current pressure of the factors, or the adaptation to this factor which
modified some of parameters? Perhaps, both, but let us introduce now the
second simplest model.
In the response–driven model of adaptation, the quantities $x_{k}$ are modeled
as functions of adaptive response $ar_{f}$ plus some noise $\epsilon_{k}$:
$x_{k}=\mu_{k}+q_{k}ar_{f}+\epsilon_{k}\ .$ (10)
When $f$ increases then, after threshold $f=aR$, the term $l_{k}ar_{f}$
transforms into $l_{k}aR$ and does not change further. The observable
quantities $x_{k}$ are not sensitive to changes in the factor intensity $f$
when $f$ is sufficiently large. This is the significant difference from the
behavior of the tension-driven model (7), which is not sensitive to change of
$f$ when $f$ is sufficiently small.
### 2.3 Tension–and–Response Driven 2D One–Factor Models.
This model is just a linear combination of Eqs. (7) and (10)
$x_{k}=\mu_{k}+l_{k}\psi+q_{k}ar_{f}+\epsilon_{k}\ .$ (11)
For small $f$ (comfort zone) $\psi=0$, the term $l_{k}\psi$ vanishes,
$ar_{f}=f$ and the model has the form $x_{k}=\mu_{k}+q_{k}f+\epsilon_{k}$. For
intermediate level of $f$, if systems with both signs of inequality
$f\gtreqless aR$ are present, the model imitates 2D (two-factor) behavior.
After the threshold $f\geq aR$ is passed for all systems, the model
demonstrates 1D behavior again:
$x_{k}=\mu_{k}+l_{k}f+(q_{k}-l_{k})aR+\epsilon_{k}\ .$ For small $f$ the
motion under change of $f$ goes along direction $q_{k}$, for large $f$ it goes
along direction $l_{k}$.
Already the first model of adaptation (7) gives us the law of tolerance and
practically important effect of order–disorder transition under stress. Now we
have no arguments for decision which of these models is better, but the second
model (10) has no tolerance plateau for small factor values, and the third
model has almost two times more fitting parameters. Perhaps, the first choice
should be the first model (7), with generalization to (11), if the described
two-dimensional behaviour is observed.
## 3 Law of the Minimum Paradox
Liebig used the image of a barrel – now called Liebig’s barrel – to explain
his law. Just as the capacity of a barrel with staves of unequal length is
limited by the shortest stave, so a plant’s growth is limited by the nutrient
in shortest supply.
Adaptation system acts as a cooper and repairs the shortest stave to improve
the barrel capacity. Indeed, in well-adapted systems the limiting factor
should be compensated as far as this is possible. It seems obvious because of
very natural idea of optimality, but arguments of this type in biology should
be considered with care.
Assume that adaptation should maximize a objective function $W$ (1), which
satisfies the Law of the Minimum (3 and the monotonicity requirement (2) under
conditions $r_{i}\geq 0$, $f_{i}-a_{i}r_{i}\geq 0$, $\sum_{i=1}^{q}r_{i}\leq
R$. (Let us remind that $f_{i}\geq 0$ for all $i$.)
Description of the maximizers of $W$ gives the following theorem (the proof is
a straightforward consequence of the Law of the Minimum and monotonicity of
$W$).
Theorem 1. For any objective function $W$ that satisfies conditions (3) the
optimizers $r_{i}$ are defined by the following algorithm.
1. 1.
Re-enumerate factors in the order of their intensities: $f_{1}\geq
f_{2}\geq...f_{q}$.
2. 2.
Calculate differences $\Delta_{j}=f_{j}-f_{j+1}$ (take formally
$\Delta_{0}=\Delta_{q+1}=0$).
3. 3.
Find such $k$ ($0\leq k\leq q$) that
$\sum_{j=1}^{k}\left(\sum_{p=1}^{j}\frac{1}{a_{p}}\right)\Delta_{j}\leq
R\leq\sum_{j=1}^{k+1}\left(\sum_{p=1}^{j}\frac{1}{a_{p}}\right)\Delta_{j}\ .$
For $R<\Delta_{1}$ we put $k=0$, for $R>\sum_{j=1}^{k+1}j\Delta_{j}$ we take
$k=q$.
4. 4.
If $k<q$ then the optimal amount of resource $r_{j_{l}}$ is
$r_{l}=\left\\{\begin{array}[]{ll}&\frac{\Delta_{l}}{a_{l}}+\frac{1}{a_{l}\sum_{p=1}^{k}\frac{1}{a_{p}}}\left(R-\sum_{j=1}^{k}\left(\sum_{p=1}^{j}\frac{1}{a_{p}}\right)\Delta_{j}\right)\
,\ \ {\rm if}\ \ l\leq k+1\ ;\\\ &0\ ,\ \ \ \ \ {\rm if}\ \ l>k+1\
.\end{array}\right.$ (12)
If $k=q$ then $r_{i}=f_{i}/a_{i}$ for all $i$. $\square$
After all we have to restore the initial enumeration $f_{i_{1}}\geq
f_{i_{2}}\geq\ldots f_{i_{q}}$. This optimization is illustrated in Fig. 4.
Figure 4: Optimal distribution of resource for neutralization of factors
under the Law of the Minimum. (a) histogram of factors intensity (the
compensated parts of factors are highlighted, $k=3$), (b) distribution of
tensions $\psi_{i}$ after adaptation becomes more uniform, (c) the sum of
distributed resources. For simplicity of the picture, we take here all
$a_{i}=1$.
Hence, if the system satisfies the law of the minimum then the adaptation
process makes more uniform the tension produced by different factors
$\psi_{i}=f_{i}-ar_{i}$ (Fig. 4). Thus adaptation decreases the effect from
the limiting factor and hides manifestations of the Law of the Minimum.
Under the assumption of optimality (1) the law of the minimum paradox becomes
a theorem: if the Law of the Minimum is true then microevolution, ecological
succession, phenotype modifications and adaptation decrease the role of the
limiting factors and bring the tension produced by different factors together.
The cooper starts to repair Liebig’s barrel from the shortest stave and after
reparation the staves are more uniform, than they were before. This cooper may
be microevolution, ecological succession, phenotype modifications, or
adaptation. For the ecological succession this effect (the Law of the Minimum
leads to its violation by succession) was described in Ref. SemSem . For
adaptation (and in general settings too) it was demonstrated in Ref.
GorSmiCorAd1st .
## 4 Law of the Minimum Inverse Paradox
The simplest formal example of “anti–Liebig’s” organization of interaction
between factors gives us the following dependence of fitness from two factors:
$W=-f_{1}f_{2}$: each of factors is neutral in the absence of another factor,
but together they are harmful. This is an example of synergy: the whole is
greater than the sum of its parts. (For our selection of axes direction,
“greater” means “more harm”.)
In according to Definition 1, the system of factors $F_{1},...F_{q}$ is
synergistic, in a convex domain $U$ of the admissible vectors of factor
pressure if for any level $w_{0}$ the sublevel set $\\{\psi\in U\ |\
W(\psi)\leq w_{0}\\}$ is convex. Another definition gives us the synergy
inequality (5). These definitions are equivalent. This proposition follows
from the definition of convexity and standard facts about convex sets (see,
for example, Rockafellar )
Proposition 1. The synergy inequality (5) holds if and only if all the
sublevel sets $\\{\mathbf{f}\ |\ W(\mathbf{f})\leq\alpha\\}$ are strictly
convex.$\square$
(The fitness itself may be a non-convex function.)
This proposition immediately implies that the synergy inequality is invariant
with respect to increasing monotonic transformations of $W$. This invariance
with respect to nonlinear change of scale is very important, because usually
we don’t know the values of function $W$.
Proposition 2. If the synergy inequality (5) holds for a function $W$, then it
holds for a function $W_{\theta}=\theta(W)$, where $\theta(x)$ is an arbitrary
strictly monotonic function of one variable.$\square$
Already this property allows us to study the problem about optimal
distribution of the adaptation resource without further knowledge about the
fitness function.
Assume that adaptation should maximize an objective function
$W(f_{1}-r_{1},...f_{q}-r_{q})$ (1) which satisfies the synergy inequality (5)
under conditions $r_{i}\geq 0$, $f_{i}-a_{i}r_{i}\geq 0$,
$\sum_{i=1}^{q}r_{i}\leq R$. (Let us remind that $f_{i}\geq 0$ for all $i$.)
Following our previous convention about axes directions all factors are
harmful and $W$ is monotonically decreasing function
$\frac{\partial W(f_{1},...f_{q})}{\partial f_{i}}<0\ .$
We need also a technical assumption that $W$ is defined on a convex set in
$\mathbb{R}^{q}_{+}$ and if it is defined for a nonnegative point
$\mathbf{f}$, then it is also defined at any nonnegative point
$\mathbf{g}\leq\mathbf{f}$ (this inequality means that $g_{i}\leq f_{i}$ for
all $i=1,...q$).
The set of possible maximizers is finite. For every group of factors
$F_{i_{1}},...F_{i_{j+1}}$, ($1\leq j+1<q$) with the property
$\sum_{k=1}^{j}\frac{f_{i_{k}}}{a_{i_{k}}}<R\leq\sum_{k=1}^{j+1}\frac{f_{i_{k}}}{a_{i_{k}}}$
(13)
we find a distribution of resource
$\mathbf{r}_{\\{{i_{1}},...{i_{j+1}}\\}}=(r_{i_{1}},...r_{i_{j+1}})$:
$r_{i_{k}}=\frac{f_{i_{k}}}{a_{i_{k}}}\ \ (k=1,...j)\ ,\ \
r_{i_{j+1}}=R-\sum_{k=1}^{j}\frac{f_{i_{k}}}{a_{i_{k}}}\ ,\ \ r_{i}=0\ \ {\rm
for}\ \ i\notin\\{{i_{1}},...{i_{j+1}}\\}\ .$ (14)
This distribution (13) means that the pressure of $j$ factors are completely
compensated and one factor is partially compensated. For $j=0$, Eq. (13) gives
$0<R\leq f_{i_{1}}$ and there exists only one nonzero component in the
distribution (14), $r_{i_{1}}=R$. For $j=q$ all $r_{i}=f_{i}/a_{i}$,
$\sum_{i}r_{i}<R$ and all factors are fully compensated.
We get the following theorem as an application of standard results about
extreme points of convex sets Rockafellar .
Theorem 2. Any maximizer for $W(f_{1}-a_{1}r_{1},...f_{q}-a_{q}r_{q})$ under
given conditions has the form $\mathbf{r}_{\\{{i_{1}},...{i_{j+1}}\\}}$
(14).$\square$
To find the optimal distribution we have to analyze which distribution of the
form (13) gives the highest fitness.
If the initial distribution of factors intensities,
$\mathbf{f}=(f_{1},...f_{q})$, is almost uniform and all factors are
significant then, after adaptation, the distribution of effective tensions,
$\mathbf{\psi}=(\psi_{1},...\psi_{q})$ ($\psi_{i}=f_{i}-a_{i}r_{i}$), is less
uniform. Following Theorem 2, some of factors may be completely neutralized
and one additional factor may be neutralized partially. This situation is
opposite to adaptation to Liebig’s system of factors, where amount of
significant factors increases and the distribution of tensions becomes more
uniform because of adaptation. For Liebig’s system, adaptation transforms low
dimensional picture (one limiting factor) into high dimensional one, and we
expect the well-adapted systems have less correlations than in stress. For
synergistic systems, adaptation transforms high dimensional picture into low
dimensional one (less factors), and our expectations are inverse: we expect
the well-adapted systems have more correlations than in stress (this situation
is illustrated in Fig. 5; compare to Fig. 4). We call this property of
adaptation to synergistic system of factors the law of the minimum inverse
paradox.
Figure 5: Typical optimal distribution of resource for neutralization of
synergistic factors. (a) Factors intensity (the compensated parts of factors
are highlighted, $j=2$), (b) distribution of tensions $\psi_{i}$ after
adaptation becomes less uniform (compare to Fig. 4), (c) the sum of
distributed resources. For simplicity of the picture, we take here all
$a_{i}=1$.
The fitness by itself is a theoretical construction based on the average
reproduction coefficient (instant fitness). It is impossible to measure this
quantity in time intervals that are much shorter than the life length and even
for the life–long analysis it is a non-trivial problem Shawatal2008 .
In order to understand which system of factors we deal with, Liebig’s or
synergistic one, we have to compare theoretical consequences of their
properties and compare them to empirical data. First of all, we can measure
results of adaptation, and use for analysis properties of optimal adaptation
in ensembles of systems for analysis (Fig. 4, Fig. 5).
## 5 Empirical data
In many areas of practice, from physiology to economics, psychology, and
engineering we have to analyze behavior of groups of many similar systems,
which are adapting to the same or similar environment. Groups of humans in
hard living conditions (Far North city, polar expedition, or a hospital, for
example), trees under influence of anthropogenic air pollution, rats under
poisoning, banks in financial crisis, enterprizes in recession, and many other
situations of that type provide us with plenty of important problems, problems
of diagnostics and prediction.
For many such situations it was found that the correlations between individual
systems are better indicators than the value of attributes. More specifically,
in thousands of experiments it was shown that in crisis, typically, even
before obvious symptoms of crisis appear, the correlations increase, and, at
the same time, variance. After the crisis achieves its bottom, it can develop
into two directions: recovering (both correlations and variance decrease) or
fatal catastrophe (correlations decrease, but variance continue to increase).
In this Sec. we review several sets of empirical results which demonstrate
this effect. Now, after 21 years of studying of this effect GorSmiCorAd1st ;
Sedov , we maintain that it is universal for groups of similar systems that
are sustaining a stress and have an adaptation ability. On the other hand,
situations with inverse behavior were predicted theoretically and found
experimentally Mansurov . This makes the problem more intriguing.
Below, to collect information about strong correlations between many
attributes in one indicator, we evaluate the non-diagonal part of the
correlation matrix and delete terms with values below a threshold 0.5 from the
sum:
$G=\sum_{j>k,\ |r_{jk}|>\alpha}|r_{jk}|.$ (15)
This quantity $G$ is a weight of the correlation graph. The vertices of this
graph correspond to variables, and these vertices are connected by edges, if
the absolute value of the correspondent sample correlation coefficient exceeds
$alpha$: $|r_{jk}|>alpha$. Usually, we take $\alpha=0.5$ (a half of the
maximum) if there is no reason to select another value.
### 5.1 Adaptation of Adults for Change of Climatic Zone
Activity of enzymes in human leukocytes was studied Bul1Limf ; Bul2Limf . We
analyzed the short-term adaptation (20 days) of groups of healthy 20-30 year
old men who change their climate zone:
* •
From Far North to the South resort (Sochi, Black Sea) in summer;
* •
From the temperate belt of Russia to the South resort (Sochi, Black Sea) in
summer.
Results are represented in Fig. 6. This analysis supports the basic hypothesis
and, on the other hand, could be used for prediction of the most dangerous
periods in adaptation, which need special care.
Figure 6: Weight of the correlation graphs of activity of enzymes in
leucocytes during urgent adaptation at a resort. For people from Far North,
the adaptation crisis occurs near the 15th day.
We selected the group of 54 people who moved to Far North, that had any
illness during the period of short-term adaptation. After 6 months at Far
North, this test group demonstrates much higher correlations between activity
of enzymes than the control group (98 people without illness during the
adaptation period). We analyzed the activity of enzymes (alkaline phosphatase,
acid phosphatase, succinate dehydrogenase, glyceraldehyde-3-phosphate
dehydrogenase, glycerol- 3-phosphate dehydrogenase, and glucose-6-phosphate
dehydrogenase) in leucocytes: $G=5.81$ in the test group versus $G=1.36$ in
the control group. To compare the dimensionless variance for these groups, we
normalize the activity of enzymes to unite sample means (it is senseless to
use the trace of the covariance matrix without normalization because normal
activities of enzymes differ in order of magnitude). For the test group, the
sum of the enzyme variances is 1.204, and for the control group it is 0.388.
### 5.2 Destroying of Correlations “on the Other Side of Crisis”: Acute
Hemolytic Anemia in Mice
It is very important to understand where the system is going: (i) to the
bottom of the crisis with possibility to recover after that bottom, (ii) to
the normal state, from the bottom, or (iii) to the “no return” point, after
which it cannot recover.
This problem was studied in many situation with analysis of fatal outcomes in
oncological MansurOnco and cardiological Strygina clinics, and also in
special experiments with acute hemolytic anemia caused by phenylhydrazine in
mice mice . The main result here is: when approaching the no-return point,
correlations destroy ($G$ decreases), and variance typically does continue to
increase.
There exist no formal criterion to recognize the situation “on the other side
of crisis”. Nevertheless, it is necessary to select situations for testing of
our hypothesis. Here can help the “general practitioner point of view”
GP_AE1952 based on practical experience. From such a point of view, the
situation described below is on the other side of crisis: the acute hemolytic
anemia caused by phenylhydrazine in mice with lethal outcome.
Figure 7: Adaptation and disadaptation dynamics for mice after phenylhydrazine
injection.
This effect was demonstrated in special experiments mice . Acute hemolytic
anemia caused by phenylhydrazine was studied in CBAxlac mice. After
phenylhydrazine injections (60 mg/kg, twice a day, with interval 12 hours)
during first 5-6 days the amount of red cells decreased (Fig. 7), but at the
7th and 8th days this amount increased because of spleen activity. After 8
days most of the mice died. Dynamics of correlation between hematocrit,
reticulocytes, erythrocytes, and leukocytes in blood is presented in Fig. 7.
Weight of the correlation graph increase precedeed the active adaptation
response, but $G$ decreased to zero before death (Fig. 7), while amount of red
cells increased also at the last day.
### 5.3 Grassy Plants Under Trampling Load
Table 1: Weight $G$ of the correlation graph for different grassy plants under various trampling load Grassy Plant | Group 1 | Group 2 | Group 3
---|---|---|---
Lamiastrum | 1.4 | 5.2 | 6.2
Paris (quadrifolia) | 4.1 | 7.6 | 14.8
Convallaria | 5.4 | 7.9 | 10.1
Anemone | 8.1 | 12.5 | 15.8
Pulmonaria | 8.8 | 11.9 | 15.1
Asarum | 10.3 | 15.4 | 19.5
The effect exists for plants too. The grassy plants in oak tree-plants are
studied RazzhevaikinTrava1996 . For analysis the fragments of forests are
selected, where the densities of trees and bushes were the same. The
difference between those fragments was in damaging of the soil surface by
trampling. Tree groups of fragments are studied:
* •
Group 1 – no fully destroyed soil surface;
* •
Group 2 – 25% of soil surface are destroyed by trampling;
* •
Group 3 – 70% of soil surface are destroyed by trampling.
The studied physiological attributes were: the height of sprouts, the length
of roots, the diameter of roots, the amount of roots, the area of leafs, the
area of roots. Results are presented in Table 1.
### 5.4 Scots Pines Near a Coal Power Station
The impact of emissions from a heat power station on Scots pine was studied
KofmantREES . For diagnostic purposes the secondary metabolites of phenolic
nature were used. They are much more stable than the primary products and hold
the information about past impact of environment on the plant organism for
longer time.
The test group consisted of Scots pines (Pinus sylvestric L) in a 40 year old
stand of the II class in the emission tongue 10 km from the power station. The
station had been operating on brown coal for 45 years. The control group of
Scots pines was from a stand of the same age and forest type, growing outside
the industrial emission area. The needles for analysis were one year old from
the shoots in the middle part of the crown. The samples were taken in spring
in bud swelling period. Individual composition of the alcohol extract of
needles was studied by high efficiency liquid chromatography. 26 individual
phenolic compounds were identified for all samples and used in analysis.
No reliable difference was found in the test group and control group average
compositions. For example, the results for Proantocyanidin content (mg/g dry
weight) were as follows:
* •
Total 37.4$\pm$3.2 (test) versus 36.8$\pm$2.0 (control);
Nevertheless, the variance of compositions of individual compounds in the test
group was significantly higher, and the difference in correlations was huge:
$G=17.29$ for the test group versus $G=3.79$ in the control group.
## 6 Comparison to Econometrics
The simplest Selye’s model (7) seems very similar to the classical one-factor
econometrics models Campbell which assume that the returns of stocks
($\rho_{i}$) are controlled by one factor, the “market” return $M(t)$. In this
model, for any stock
$\rho_{i}(t)=a_{i}+b_{i}M(t)+\epsilon_{i}(t)$ (16)
where $\rho_{i}(t)$ is the return of the $i$th stock at time $t$, $a_{i}$ and
$b_{i}$ are real parameters, and $\epsilon_{i}(t)$ is a zero mean noise. In
our models, the factor pressure characterizes the time window and is slower
variable than the return.
The main difference between models (7) and (16) could be found in the
nonlinear coupling (8) between the environmental property (the factor value
$f$) and the property of individuals (the resource amount $R$). Exactly this
coupling causes separation of a population into two groups: the well-adapted
less correlated group and the highly correlated group with larger variances of
individual properties and amount of resource which is not sufficient for
compensation of the factor load. Let us check whether such a separation is
valid for financial data.
### 6.1 Data Description
For the analysis of correlations in financial systems we used the daily
closing values for companies that are registered in the FTSE 100 index
(Financial Times Stock Exchange Index). The FTSE 100 is a market-
capitalization weighted index representing the performance of the 100 largest
UK-domiciled blue chip companies which pass screening for size and liquidity.
The index represents approximately 88.03% of the UK s market capitalization.
FTSE 100 constituents are all traded on the London Stock Exchange s SETS
trading system. We selected 30 companies that had the highest value of the
capital (on the 1st of January 2007) and stand for different types of business
as well. The list of the companies and business types is displayed in Table 2.
Table 2: Thirty largest companies for analysis from the FTSE 100 index Number | Business type | Company | Abbreviation
---|---|---|---
1 | Mining | Anglo American plc | AAL
2 | | BHP Billiton | BHP
3 | Energy (oil/gas) | BG Group | BG
4 | | BP | BP
5 | | Royal Dutch Shell | RDSB
6 | Energy (distribution) | Centrica | CNA
7 | | National Grid | NG
8 | Finance (bank) | Barclays plc | BARC
9 | | HBOS | HBOS
10 | | HSBC HLDG | HSBC
11 | | Lloyds | LLOY
12 | Finance (insurance) | Admiral | ADM
13 | | Aviva | AV
14 | | LandSecurities | LAND
15 | | Prudential | PRU
16 | | Standard Chartered | STAN
17 | Food production | Unilever | ULVR
18 | Consumer | Diageo | DGE
19 | goods/food/drinks | SABMiller | SAB
20 | | TESCO | TSCO
21 | Tobacco | British American Tobacco | BATS
22 | | Imperial Tobacco | IMT
23 | Pharmaceuticals | AstraZeneca | AZN
24 | (inc. research) | GlaxoSmithKline | GSK
25 | Telecommunications | BT Group | BTA
26 | | Vodafone | VOD
27 | Travel/leasure | Compass Group | CPG
28 | Media (broadcasting) | British Sky Broadcasting | BSY
29 | Aerospace/ | BAE System | BA
30 | defence | Rolls-Royce | RR
a)b)
c)
d)
Figure 8: Correlation graphs for six positions of sliding time window on
interval 10/04/2006 - 21/07/2006. a) Dynamics of FTSE100 (dashed line) and of
$G$ (solid line) over the interval, vertical lines correspond to the points
that were used for the correlation graphs. b) Thirty companies for analysis
and their distributions over various sectors of economics. c) The correlation
graphs for the first three points, FTSE100 decreases, the correlation graph
becomes more connective. d) The correlation graphs for the last three points,
FTSE100 increases, the correlation graph becomes less connective.
Data for these companies are available form the Yahoo!Finance web-site. For
data cleaning we use also information for the selected period available at the
London Stock Exchange web-site. Let $x_{i}(t)$ denote the closing stock price
for the $i$th company at the moment $t$, where $i=\overline{1,30}$, $t$ is the
discrete time (the number of the trading day). We analyze the correlations of
logarithmic returns: $x^{l}_{i}(t)=\ln\frac{x_{i}(t)}{x_{i}(t-1)}$, in sliding
time windows of length $p=20$, this corresponds approximately to 4 weeks of 5
trading days. The correlation coefficients $r_{ij}(t)$ for time moment $t$ are
calculated in the time window $[t-p,t-1]$, which strongly precedes $t$. Here
we calculate correlations between individuals (stocks), and for biological
data we calculated correlations between attributes. This corresponds to
transposed data matrix.
### 6.2 Who Belongs to the Highly Correlated Group in Crisis
For analysis we selected the time interval 10/04/2006 - 21/07/2006 that
represents the FTSE index decrease and restoration in spring and summer 2006
(more data are analyzed in our e-print GSTarXiv ). In Fig. 8 the correlation
graphs are presented for three time moments during the crisis development and
three moments of the restoration. The vertices of this graph correspond to
stocks. These vertices are connected by solid lines is the correspondent
correlation coefficient $|r_{jk}|\geq\sqrt{0.5}$
$(\sqrt{0.5}=\cos(\pi/4)\approx 0.707)$, and by dashed lines if
$\sqrt{0.5}>|r_{jk}|>0.5$.
The correlation graphs from Fig. 8 show that in the development of this crisis
(10/04/2006 - 21/07/2006) the correlated group was formed mostly by two
clusters: a financial cluster (banks and insurance companies) and an energy
(oil/gas) – mining – aerospace/defence and travel cluster. At the bottom of
crisis the correlated phase included almost all stocks. The recovery followed
a significantly different trajectory: the correlated phase in the recovery
seems absolutely different from that phase in the crisis development: there
appeared the strong correlation between financial sector and industry. This is
a sign that after the crisis bottom the simplest Selye’s model is not valid
for a financial market. Perhaps, interaction between enterprizes and
redistribution of resource between them should be taken into account. We need
additional equations for dynamics of the available amounts of resource $R_{i}$
for $i$th stock. Nevertheless, appearance of the highly correlated phase in
the development of the crisis in the financial world followed the predictions
of Selye’s model, at least, qualitatively.
Asymmetry between the drawups and the drawdowns of the financial market was
noticed also in the analysis of the financial empirical correlation matrix of
the 30 companies which compose the Deutsche Aktienindex (DAX)
DrozdComCollNoise2000 .
The market mode was studied by principal component analysis Stanley2002 .
During periods of high market volatility values of the largest eigenvalue of
the correlation matrix are large. This fact was commented as a strong
collective behavior in regimes of high volatility. For this largest
eigenvalue, the distribution of coordinates of the correspondent eigenvector
has very remarkable properties:
* •
It is much more uniform than the prediction of the random matrix theory
(authors of Ref. Stanley2002 described this vector as “approximately
uniform”, suggesting that all stocks participate in this “market mode”);
* •
Almost all components of that eigenvector have the same sign.
* •
A large degree of cross correlations between stocks can be attributed to the
influence of the largest eigenvalue and its corresponding eigenvector
Two interpretations of this eigenvector were proposed Stanley2002 : it
corresponds either to the common strong factor that affects all stocks, or it
represents the “collective response” of the entire market to stimuli. Our
observation supports this conclusion at the bottom of the crisis. At the
beginning of the crisis the correlated group includes stocks which are
sensitive to the factor load, and other stocks are tolerant and form the less
correlated group with the smaller variance. Following Selye’s model we can
conclude that the effect is the result of nonlinear coupling of the
environmental factor load and the individual adaptation response.
## 7 Functional Decomposition and Integration of Subsystems
In the simple factor–resource Selye models the adaptation response has no
structure: the organism just distributes the adaptation resource to
neutralization of various harmful factors. It is possible to make this model
more realistic by decomposition. The resource is assigned not directly
“against factors” but is used for activation and intensification of some
subsystems.
We need to define the hierarchical structure of the organism to link the
behavior in across multiple scales. In integrative and computational
physiology it is necessary to go both bottom–up and top–up approaches. The
bottom–up approach goes from proteins to cells, tissues, organs and organ
systems, and finally to a whole organism Cramlinatal2004 .
The top-down approach starts from a bird’s eye view of the behavior of the
system – from the top or the whole and aims to discover and characterize
biological mechanisms closer to the bottom – that is, the parts and their
interactions Cramlinatal2004 .
There is a long history of discussion of functional structure of the organism,
and many approaches are developed: from the Anokhin theory of functional
systems Sudakov2004 to the inspired by the General Systems approach theory of
“Formal Biological Systems” Chauvet1999 .
The notion of functional systems represents a special type of integration of
physiological functions. Individual organs and tissue elements, are
selectively combined into self-regulating systems organizations to achieve the
necessary adaptive results important for the whole organism. The self-
organization process is ruled by the adaptation needs.
For decomposition of the models of physiological systems, the concept of
principal dynamic modes was developed Marmarelis1997 ; Marmarelis2004 .
In this section, we demonstrate how to decompose the factor–resource models of
the adaptation of the organism to subsystems.
In general, the analysis of interaction of factors is decomposed to
interaction of factors and subsystems. Compensation of the harm from each
factor $F_{i}$ requires activity of various systems. For every system $S_{j}$
a variable, activation level $I_{j}$ is defined. Level 0 corresponds to a
fully disabled subsystem (and for most of essentially important subsystems it
implies death). For each factor $F_{i}$ and every subsystem $S_{j}$ a
“standard level” of activity $\varsigma_{ij}$ is defined. Roughly speaking,
this level of activation of the subsystem $S_{j}$ is necessary for
neutralization of the unit value of the pressure of the factor $F_{i}$. If
$\varsigma_{ij}=0$ then the subsystem $S_{j}$ is not involved in the
neutralization of the factor $F_{i}$.
The compensated value of the factor pressure $F_{i}$ is
$\psi_{i}=f_{i}-\min_{j,\ \varsigma_{ij}\neq
0}\left\\{\frac{I_{j}}{\varsigma_{ij}}\right\\}\ .$ (17)
In this model resources are assigned not to neutralization of factors but for
activation of subsystems. The activation intensity of the subsystem $S_{j}$
depends on the adaptation resource $r_{j}$, assigned to this subsystem:
$I_{j}=\alpha_{j}r_{j}\ .$ (18)
For any given organization of the system of factors, optimization of fitness
together with definitions (17) and (18) lead to a clearly stated optimization
problem. For example, for Liebig’s system of factors we have to find
distributions of $r_{j}$ that either are maximizers in a problem:
$\min_{i}\left\\{f_{i}-\min_{j,\ \varsigma_{ij}\neq
0}\left\\{\frac{\alpha_{j}r_{j}}{\varsigma_{ij}}\right\\}\right\\}\ \ {\rm
for}\ \ r_{j}\geq 0,\ \sum_{j}r_{j}\leq R\ $ (19)
if this minimum is nonnegative, or give a solution to the system of
inequalities
$f_{i}-\min_{j,\ \varsigma_{ij}\neq
0}\left\\{\frac{\alpha_{j}r_{j}}{\varsigma_{ij}}\right\\}\leq 0;\ r_{j}\geq
0,\ \sum_{j}r_{j}\leq R$ (20)
if the minimum in (19) is negative.
For the study of integration in experiment we use principal component analysis
and find, parameters of which systems give significant inputs in the first
principal components. Under the stress, the configuration of the subsystems,
which are significantly involved in the first principal components, changes
Svetlichnaia1997 .
We analyzed interaction of cardiovascular and respiratory subsystems under
exercise tolerance tests at various levels of load. Typically, we observe the
following dynamics of the first factor composition. With increase of the load,
coordinates both the correlations of the subsystems attributes with the first
factor increase up to some maximal load which depend on the age and the health
in the group of patients. After this maximum of integration, if the load
continues to increase then the level of integration decreases Svetlichnaia1997
.
Generalization of Selye’s models by decomposition creates a rich and flexible
system of models for adaptation of hierarchically organized systems. Principal
component analysis Jolliffe2002 with its various nonlinear generalizations
Gorbanatal2008 ; GorbanZinovyev2009 gives a system of tools for extracting
the information about integration of subsystems from the empirical data.
## 8 Conclusion
Due to the law of the minimum paradoxes, if we observe the Law of the Minimum
in artificial systems, then under natural conditions adaptation will equalize
the load of different factors and we can expect a violation of the law of the
minimum. Inversely, if an artificial systems demonstrate significant violation
of the law of the minimum, then we can expect that under natural conditions
adaptation will compensate this violation.
This effect follows from the factor–resource models of adaptation and the idea
of optimality applied to these models. We don’t need an explicit form of
generalized fitness (which may be difficult to find), but use only the general
properties that follow from the Law of the Minimum (or, oppositely, from the
assumption of synergy).
Another consequence of the factor–resource models is the prediction of the
appearance of strongly correlated groups of individuals under an increase of
the load of environmental factors. Higher correlations in those groups do not
mean that individuals become more similar, because the variance in those
groups is also higher. This effect is observed for financial market too and
seems to be very general in ensembles of systems which are adapting to
environmental factors load.
Decomposition of the factor–resource models for the hierarchy of subsystems
allows us to discuss integration of the subsystems in adaptation. For the
explorative analysis of this integration in empirical data the principal
component analysis is the first choice: for the high level of integration
different subsystems join in the main factors.
The most important shortcoming of the factor–resource models is the lack of
dynamics. In the present form it describes adaptation as a single action, the
distribution of the adaptation resource. We avoid any kinetic modeling.
Nevertheless, adaptation is a process in time. We have to create a system of
dynamical models.
## References
* (1) F. Salisbury (1992) Plant Physiology, Plant physiology (4th ed.), Wadsworth, Belmont, CA.
* (2) R.R. van der Ploeg, W. Böhm, M.B. Kirkham, (1999), On the origin of the theory of mineral nutrition of plants and the law of the minimum, Soil Science Society of America Journal 63, 1055–1062.
* (3) Q. Paris, (1992), The Return of von Liebig’s “Law of the Minimum”, Agron. J., 84, 1040–1046
* (4) B.S. Cade, J.W. Terrell, R.L. Schroeder, (1999), Estimating effects of limiting factors with regression quantiles, Ecology 80 (1), 311–323.
* (5) D. Tilman, (1980), Resources: a graphical-mechanistic approach to competition and predation, Am. Nat. 116 (3), 362–393.
* (6) D. Tilman (1982) Resource Competition and Community Structure, Princeton University Press, Princeton, NJ.
* (7) A.J. Bloom, F.S. Chapin III, H.A. Mooney, (1985), Resource limitation in plants – an economic analogy. Annu. Rev. Ecol. Syst. 16, 363–392.
* (8) F.S. Chapin, III, E. Schulze, H.A. Mooney, (1990), The Ecology and Economics of Storage in Plants, Annu. Rev. Ecol. Syst. 21, 423–447.
* (9) A.N. Gorban, V.T. Manchuk, E.V. Petushkova (Smirnova) (1987) Dynamics of physiological paramethers correlations and the ecological-evolutionary principle of polyfactoriality, In: Problemy Ekologicheskogo Monitoringa i Modelirovaniya Ekosistem [The Problems of Ecological Monitoring and Ecosystem Modelling], Vol. 10. Gidrometeoizdat, Leningrad, pp. 187–198.
* (10) K.R. Sedov, A.N. Gorban’, E.V. Petushkova (Smirnova), V.T. Manchuk, E.N. Shalamova, (1988), Correlation adaptometry as a method of screening of the population, Vestn Akad Med Nauk SSSR (10), 69–75. PMID: 3223045
* (11) A. Sih, S.K. Gleeson, (1995), A limits-oriented approach to evolutionary ecology, Trends in Ecology and Evolution 10 (9), 378–382.
* (12) R.K. Kobe (1996) Intraspecific Variation in Sapling Mortality and Growth Predicts Geographic Variation in Forest Composition, In: Ecological Monographs: Vol. 66 (2), pp. 181–201.
* (13) H.A. van den Berg, (1998), Multiple nutrient limitation in unicellulars: reconstructing Liebig’s law, Mathematical Biosciences, 149 (1), 1–22.
* (14) O. Aumont, E. Maier-Reimer, S. Blain, P. Monfray, (2003), An ecosystem model of the global ocean including Fe, Si, P colimitations, Global Biogeochemical Cycles 17 (2), 1060, doi:10.1029/2001GB001745.
* (15) T. Egli, M. Zinn, (2003), The concept of multiple-nutrient-limited growth of microorganisms and its application in biotechnological processes, Biotechnology Advances 22 (1-2), 35–43
* (16) M. Zinn, B. Witholt, T. Egli, (2004), Dual nutrient limited growth: models, experimental observations, and applications, Journal of Biotechnology 113 (1-3), 263–279.
* (17) T. Wutzler and M. Reichstein, (2008), Colimitation of decomposition by substrate and decomposers – a comparison of model formulations, Biogeosciences Discuss. 5, 163–190.
* (18) M. Danger, T. Daufresne, F. Lucas, S. Pissard, G. Lacroix, (2008), Does Liebig’s law of the minimum scale up from species to communities? Oikos 117 (11), 1741–1751
* (19) F.N. Semevsky, S.M. Semenov (1982) Mathematical modeling of ecological processes, Gidrometeoizdat, Leningrad.
* (20) M.A. Saito, T.J. Goepfert, (2008), Some thoughts on the concept of colimitation: Three definitions and the importance of bioavailability, Limnol. Oceanogr. 53 (1), 276–290
* (21) G.O. Nijland, J. Schouls, J. Goudriaan, (2008), Integrating the production functions of Liebig, Michaelis-Menten, Mitscherlich and Liebscher into one system dynamics model, NJAS - Wageningen Journal of Life Sciences 55 (2), 199–224.
* (22) G. C. Brown and C. E. Cooper, (1993). Control analysis applied to a single enzymes: can an isolated enzyme have a unique rate–limiting step? Biochem. J. 294, 87–94.
* (23) A.N. Gorban, O. Radulescu, (2008), Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited, Advances in Chemical Engineering 34, 103–173.
* (24) M.R. Droop, (1973), Some thoughts on nutrient limitation in algae, J. Phycol. 9, 264–272.
* (25) T. Legović, A. Cruzado, (1997), A model of phytoplankton growth on multiple nutrients based on the Michaelis-Menten-Monod uptake, Droop’s growth and Liebig’s law, Ecol. Modelling 99 (1), 19–31.
* (26) F. Ballantyne IV., D.N.L. Menge, A. Ostling and P. Hosseini, (2008), Nutrient recycling affects autotroph and ecosystem stoichiometry, Am. Nat. 171 (4), 511–523.
* (27) N. Shoresh, M. Hegreness, R. Kishony, (2008), Evolution exacerbates the paradox of the plankton, PNAS USA 105 (34), 12365–12369.
* (28) G.E. Hutchinson, (1961), The paradox of the plankton, Am. Nat. 95, 137–145.
* (29) D.N.L. Menge, J.S. Weitz, (2009), Dangerous nutrients: Evolution of phytoplankton resource uptake subject to virus attack, Journal of Theoretical Biology, 257 (1), 104–115.
* (30) O. Chertov, A. Gorbushina, B. Deventer, (2004), A model for microcolonial fungi growth on rock surfaces, Ecological Modelling 177 (3-4), 415–426.
* (31) B. McGill, (2005), A mechanistic model of a mutualism and its ecological and evolutionary dynamics, Ecological Modelling 187 (4), Pages 413–425.
* (32) D.A. Hennessy, (2009), Crop yield skewness under law of the minimum technology, American Journal of Agricultural Economics 91 (1), 197–208.
* (33) M. Austin, (2007), Species distribution models and ecological theory: A critical assessment and some possible new approaches, Ecological Modelling 200 (1-2), 1–19.
* (34) J.D. Thomson, G. Weiblen, B.A. Thomson, S. Alfaro, P. Legendre, (1996), Untangling multiple factors in spatial distributions: lilies, gophers and rocks, Ecology 77, 1698 1715.
* (35) H.E. Daly, (1991), Population and Economics – A Bioeconomic Analysis, Population and Environment 12 (3), 257–263.
* (36) M.Y. Ozden, (2004), Law of the Minimum in Learning, Educational Technology & Society 7 (3), 5–8.
* (37) V. Kolokoltsov, V. Maslov (1997) Idempotent analysis and applications, Kluwer Acad. Publ., Dordrecht.
* (38) G.L. Litvinov, V.P. Maslov (Eds.) (2005) Idempotent mathematics and mathematical physics, Contemporary Mathematics, Vol. 377, AMS, Providence, RI.
* (39) G.L. Litvinov, (2007), The Maslov dequantization, idempotent and tropical mathematics: a brief introduction, Journal of Mathematical Sciences 140 (3), 426-444. E-print: arXiv:math/0507014 [math.GM].
* (40) S.C. Kleene (1956) Representation of events in nerve sets and finite automata, In: J. McCarthy and C. Shannon (Eds), Automata Studies, Princeton University Press, Princeton, pp. 3–40.
* (41) A.N. Gorban, O. Radulescu, A.Y. Zinovyev, (2010), Asymptotology of chemical reaction networks, Chem. Eng. Sci. 65, 2310–2324.
* (42) R.A. Fisher (1930) The genetical theory of natural selection, Oxford University Press, Oxford.
* (43) J.B.S. Haldane (1932) The causes of evolution, Longmans Green, London.
* (44) J.A.J. Metz, R.M. Nisbet, S.A.H. Geritz, (1992), How should we define fitness for general ecological scenarios, Trends Ecol. Evol. 7, 198–202.
* (45) A.N. Gorban (1984) Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis, Nauka, Novosibirsk.
* (46) J. Maynard-Smith (1982) Evolution and the Theory of Games, Cambridge University Press, Cambridge.
* (47) D. Waxman, J.J. Welch, (2005), Fisher’s Microscope and Haldane’s Ellipse, Am. Nat. 166, 447–457.
* (48) J.G. Kingsolver, D.W. Pfennig, (2007), Patterns and Power of Phenotypic Selection in Nature, BioScience 57 (7), 561–572.
* (49) R.G. Shaw, C.J. Geyer, S. Wagenius, H.H. Hangelbroek, J.R. Etterson, (2008), Unifying Life-History Analyses for Inference of Fitness and Population Growth, Am. Nat. 172, E35–E47.
* (50) T. Colborn, D. Dumanoski, J.P. Meyers (1996) Our Stolen Future: Are We Threatening Our Fertility, Intelligence, and Survival? – A Scientific Detective Story, Dutton, Peguin Books, NY.
* (51) H. Selye, (1938), Adaptation Energy, Nature 141 (3577), 926.
* (52) H. Selye, (1938), Experimental evidence supporting the conception of “adaptation energy”, Am. J. Physiol. 123, 758–765.
* (53) B. Goldstone, (1952), The general practitioner and the general adaptation syndrome, S. Afr. Med. J. 26, 88–92, 106–109 PMID: 14901129, 14913266\.
* (54) R. McCarty, K. Pasak (2000) Alarm phase and general adaptation syndrome, in: Encyclopedia of Stress, George Fink (ed.), Vol. 1, Academic Press, pp. 126–130.
* (55) S. Breznitz (Ed.) (1983) The Denial of Stress, International Universities Press, Inc., New York.
* (56) J.K. Schkade, S. Schultz (2003) Occupational Adaptation in Perspectives, Ch. 7 in: Perspectives in Human Occupation: Participation in Life, By Paula Kramer, Jim Hinojosa, Charlotte Brasic Royeen (eds), Lippincott Williams & Wilkins, Baltimore, MD, pp. 181–221.
* (57) G.F. Gause (1934) The struggle for existence, Williams and Wilkins, Baltimore. Online: http://www.ggause.com/Contgau.htm.
* (58) I.M. Bomze, (2002), Regularity vs. degeneracy in dynamics, games, and optimization: a unified approach to different aspects. SIAM Review 44, 394–414.
* (59) J. Oechssler, F. Riedel, (2002), On the Dynamic Foundation of Evolutionary Stability in Continuous Models. Journal of Economic Theory 107, 223–252.
* (60) A.N. Gorban, (2007), Selection Theorem for Systems with Inheritance, Math. Model. Nat. Phenom. 2 (4), 1–45. E-print: arXiv:cond-mat/0405451 [cond-mat.stat-mech].
* (61) E. Zuckerkandl, R. Villet, (1988), Concentration-affinity equivalence in gene regulation: Convergence of genetic and environmental effects. PNAS USA, 85, 4784–4788.
* (62) M.J. West-Eberhard, (2003), Developmental Plasticity and Evolution, Oxford University Press, US.
* (63) R.J. Hoffman, (1978), Environmental Uncertainty and Evolution of Physiological Adaptation in Colias Butterflies, Am. Nat. 112 (988), 999–1015.
* (64) E. Greene (1999) Phenotypic variation in larval development and evolution: polymorphism, polyphenism, and developmental reaction norms, In: The origin and evolution of larval forms, M. Wake, B. Hall (Eds.), Academic Press, New York, pp. 379 410.
* (65) H.R. Lerner (ed.) (1999) Plant responses to environmental stresses: from phytohormones to genome reorganization, Marcel Dekker, Inc., New York.
* (66) R. Milo, J.H. Hou, M. Springer, M.P. Brenner, M.W. Kirschner, (2007), The relationship between evolutionary and physiological variation in hemoglobin, PNAS USA 104 (43), 16998–17003.
* (67) G. Fusco, A. Minelli, (2010), Phenotypic plasticity in development and evolution: facts and concepts, Phil. Trans. R. Soc. B 365 (1540), 547–556.
* (68) E.P. Odum, (1971), Fundamentals of ecology (Third Edition), W. B. Saunders, Comp., Philadelphia – London – Toronto.
* (69) R.T. Rockafellar, (1997), Convex analysis, Princeton University Press, Princeton, NJ.
* (70) A.S. Mansurov, T.P. Mansurova, E.V. Smirnova, L.S. Mikitin, A.V. Pershin (1994) How do correlations between physiological parameters depend on the influence of different systems of stress factors? In: Global & Regional Ecological Problems, R.G. Khlebopros (Ed.), Krasnoyarsk State Technical University Publ., Krasnoyarsk, pp. 499–516.
* (71) G.V. Bulygin, A.S. Mansurov, T.P. Mansurova, E.V. Smirnova (1992) Dynamics of parameters of human metabolic system during the short-term adaptation, Institute of Biophysics, Russian Academy of Sciences, Preprint 180B, Krasnoyarsk.
* (72) G.V. Bulygin, A.S. Mansurov, T.P. Mansurova, A.A. Mashanov, E.V. Smirnova (1992) Impact of health on the ecological stress dynamics. Institute of Biophysics, Russian Academy of Sciences, Preprint 185B, Krasnoyarsk.
* (73) A.S. Mansurov, T.P. Mansurova, E.V. Smirnova, L.S. Mikitin, A.V. Pershin (1995) Human adaptation under influence of synergic system of factors (treatment of oncological patients after operation), Institute of Biophysics Russian Academy of Sciences, Preprint 212B Krasnoyarsk.
* (74) S.O. Strygina, S.N. Dement’ev, V.M. Uskov, G.I. Chernyshova (2000) Dynamics of the system of correlations between physiological parameters in patients after myocardial infarction, In: Mathematics, Computer, Education, Proceedings of conference, Issue 7, Moscow, pp. 685–689.
* (75) L.D. Ponomarenko, E.V. Smirnova, (1998), Dynamical characteristics of blood system in mice with phenilhydrazin anemiya, In: Proceeding of 9th International Symposium “Reconstruction of homeostasis”, Krasnoyarsk, Russia, March 15-20, vol. 1, pp. 42–45.
* (76) I.V. Karmanova, V.N. Razzhevaikin, M.I. Shpitonkov, (1996), Application of correlation adaptometry for estimating a response of herbaceous species to stress loadings, Doklady Botanical Sciences, Vols. 346–348, 4–7. [Translated from Doklady Akademii Nauk SSSR, 346, 1996.]
* (77) P.G. Shumeiko, V.I. Osipov, G.B. Kofman, (1994), Early detection of industrial emission impact on Scots Pine needles by composition of phenolic compounds, In: Global & Regional Ecological Problems, R.G. Khlebopros (Ed.), Krasnoyarsk State Technical University Publ., Krasnoyarsk, pp. 536–543.
* (78) J. Y. Campbell, A.-W. Lo, and A. C. MacKinlay (1997) The Econometrics of Financial Markets, Princeton Unviersity Press, Princeton.
* (79) A. N. Gorban, E. V. Smirnova, T. A. Tyukina, (2009), Correlations, risk and crisis: from physiology to finance, arXiv:0905.0129 [physics.bio-ph].
* (80) S. Drożdż, F. Grümmer , A.Z. Górski, F. Ruf, J. Speth, (2000), Dynamics of competition between collectivity and noise in the stock market, Physica A 287, 440–449.
* (81) V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, T. Guhr, H.E. Stanley, (2002), Random matrix approach to cross correlations in financial data, Phys. Rev. E 65, 066126.
* (82) E.J. Crampin, M. Halstead, P. Hunter, P. Nielsen, D. Noble, N. Smith, M. Tawhai, (2004), Computational physiology and the physiome project, Experimental Physiology 89, 1-26.
* (83) K.V. Sudakov, (2004), Functional Systems Theory: A New Approach to the Question of the Integration of Physiological Processes in the Body, Neuroscience and Behavioral Physiology 34(5), 495–500.
* (84) G.A. Chauvet, (1999), S-propagators: a formalism for the hierarchical organization of physiological systems. Application to the nervous and the respiratory systems, International Journal of General Systems 28 (1), 53–96.
* (85) V.Z. Marmarelis, (1997), Modeling methodology for nonlinear physiological systems, Annals of Biomedical Engineering 25, (2), 239–251.
* (86) V.Z. Marmarelis (2004) Nonlinear Dynamic Modeling of Physiological Systems, Wiley, New York.
* (87) G.N. Svetlichnaia, E.V. Smirnova, L.I. Pokidysheva, (1997), Correlational adaptometry as a method for evaluating cardiovascular and respiratory interaction, Fiziol. Cheloveka 23 (3), 58–62. PMID: 9264951\.
* (88) I.T. Jolliffe (2002) Principal component analysis, series: Springer series in statistics, 2nd ed., XXIX, Springer, New York.
* (89) A.N. Gorban, B. Kegl, D. Wunsch, A. Zinovyev (Eds.) (2008) Principal Manifolds for Data Visualisation and Dimension Reduction, Lecture Notes in Computational Science and Engineering, Vol. 58, Springer, Berlin – Heidelberg – New York.
* (90) A.N. Gorban, A.Y. Zinovyev (2009) Principal Graphs and Manifolds, Ch. 2 in: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods, and Techniques, Emilio Soria Olivas et al. (eds), IGI Global, Hershey, PA, pp. 28–59. E-print: arXiv:0809.0490 [cs.LG].
|
arxiv-papers
| 2009-07-11T13:10:53 |
2024-09-04T02:49:03.842741
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "A.N. Gorban, L.I. Pokidysheva, E.V. Smirnova, T.A. Tyukina",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/0907.1965"
}
|
0907.1968
|
# Quantum Biology
Alessandro Sergi [email protected] School of Physics, University of KwaZulu-
Natal, Pietermaritzburg, Private Bag X01 Scottsville, 3209 Pietermaritzburg,
South Africa
###### Abstract
A critical assessment of the recent developments of molecular biology is
presented. The thesis that they do not lead to a conceptual understanding of
life and biological systems is defended. Maturana and Varela’s concept of
autopoiesis is briefly sketched and its logical circularity avoided by
postulating the existence of underlying living processes, entailing
amplification from the microscopic to the macroscopic scale, with increasing
complexity in the passage from one scale to the other. Following such a line
of thought, the currently accepted model of condensed matter, which is based
on electrostatics and short-ranged forces, is criticized. It is suggested that
the correct interpretation of quantum dispersion forces (van der Waals,
hydrogen bonding, and so on) as quantum coherence effects hints at the
necessity of including long-ranged forces (or mechanisms for them) in
condensed matter theories of biological processes. Some quantum effects in
biology are reviewed and quantum mechanics is acknowledged as conceptually
important to biology since without it most (if not all) of the biological
structures and signalling processes would not even exist. Moreover, it is
suggested that long-range quantum coherent dynamics, including electron
polarization, may be invoked to explain signal amplification process in
biological systems in general.
Published on line in:
Atti della Accademia Peloritana dei Pericolanti Vol. LXXXVII, C1C0901001
(2009).
DOI: 10.1478/C1C0901001
## I Introduction
Biology offers to scientists the most complex systems to study in the
universe. Since scientists themselves are biological systems, such a study is
not just the most interesting that one can think of but effectively introduces
a circularity in the process of knowledge, as was noted by Maturana and Varela
tree : Life systems (the scientists) who try to know life systems (possibly
themselves); in other words, life that tries to know life. It is a platitude
to assert that studying biology from the point of view of physics (i.e., the
point of view of the fundamental laws of the universe) is very difficult st .
From a physicist’s perspective, there are universal laws (and, perhaps,
building blocks) of reality and one should be able to predict the emergence
and the characteristics of biological systems from these very fundamental
laws. Such a gigantic endeavour has not been successful to date.
In this contribution, the thesis that quantum mechanics is a powerful tool for
explaining the characteristics of biological systems will be defended and some
(speculative, at the moment) lines of research will be suggested. From a
certain point of view, the use of quantum mechanics in biology might seem
logical since it is the fundamental theory describing microscopic phenomena in
physics. Within a fully reductionist philosophy (which it is not invoked here)
chemistry, biochemistry, and biology would be only epiphenomena of the
fundamental microscopic laws of physics, i.e., they would be secondary
manifestations of the main microscopic reality with its laws. From another
point of view, it is very strange that one would invoke such a controversial
theory, as quantum mechanics actually is, in order to explain the most complex
phenomena in the universe. As a matter of fact, while everybody agrees on the
main technical points of quantum mechanics, almost nobody agrees on its
interpretation, which seems to depend oddly on the area of research the theory
is applied to. It is not entirely wrong to write that there are almost as many
interpretations of quantum mechanics as there are theoretical physicists: It
is sufficient to search in the contemporary literature of physics journals to
be convinced of this. The ideas underlying this contribution are that quantum
mechanics is the fundamental physical theory in the microscopic world, that
important concepts can arise from its application to biological systems, and
that biological systems, while requiring a multi-level approach, must also be
studied from a microscopic point of view (in order to unfold their universal
characteristics).
The main-stream scientific discipline that currently undertakes the endeavour
of a microscopic explanation of life is molecular biology. Its method is based
on enumerating and elucidating the role of molecules in the life process.
While such a work is of fundamental importance for medicine and applied
biochemistry, it does not seem to lead to a better understanding of the
universal properties of life itself. Such a critic opinion, which has been
defended among others by Kaneko kaneko , will be the starting point of this
contribution. However, while Kaneko invokes dynamical system theory kaneko ,
here quantum mechanics is considered necessary in order to unveil the
universal properties of living systems.
This contribution is organized as follows. In Section II the arguments of
Kaneko kaneko , trying to characterize life and to criticize molecular biology
(for failing to provide an understanding), are followed. In Section II.3 the
universal logic of Maturana and Varela, which is founded on the concept of
autopoiesis, is summarized. Autopoiesis is further analyzed and the more
fundamental concept of living process (based on amplification mechanisms) is
introduced. We feel that _living processes_ can be directly linked to quantum
phenomena and, as such, are more suited to physical modeling. The current
paradigm of condensed matter physics is illustrated and criticized in Section
III. The features of quantum mechanics, with a particular emphasis on those of
interest to biological processes, are sketched in the same Section. In Section
IV a certain number of biological phenomena, where quantum mechanics is
necessary, are reviewed. Van der Waals interactions and quantum mechanical
dispersion forces are presented (in a speculative way) as the main candidates
for the amplification processes necessary to living systems. Finally,
conclusions and perspectives are elucidated in Section V.
## II What is life?
The questions that will be addressed in this Section are: “What kind of system
is life?” and “What does understanding life really means?”. The main thesis is
that such basic questions on life systems are not answered by the main-stream
approach of current biology, which enumerates molecules and genes.
As for understanding life as a process, the molecular paradigm embraced by
contemporary biology has a fundamental flaw: There is no particular molecule,
including DNA, whose presence by itself implies life. In his book kaneko ,
Kaneko presents the semi-serious example of an omelette, which possesses DNA
but which is clearly not alive. Moreover, until one does not gain a general
understanding of what life is, speculations about the possibility that the
molecules, used by living creatures on Earth, could not be the only thing
playing an important role for life are not entirely unreasonable. Hence, if
not the molecules, the specific conditions for life remain to be clarified.
One may attempt to compile a list of the characteristics of living systems.
These are the ability of reproduction, the potentiality to undergo evolution,
the existence of some kind of structure separating an individual’s body from
the external world, some kind of metabolic capacity through which an
individual body is maintained, the existence of some degree of autonomy, and
so on. Despite the various attempts at listing such characteristics, no list
has ever been completely satisfactory or agreed upon. However, there must be a
solution since in many cases human beings have an intuitive ability to
distinguish between living and non-living creatures (excluding limit cases
such as viruses and so on). Acknowledging that certain properties are common
to all living systems, a theoretical physicist would like to search for the
universal properties of living systems. In other words, one would like to
understand the universal logic of life (its logos underlying it as a process)
instead of understanding the specific functioning of a definite organism.
Indeed, since its birth biology has attempted to escape mere enumerationism
through Darwin’s theory of evolution: A universal logic based on the three
processes of variation, reproduction, and selection. Such an evolutionary
logic has been mimicked by computer scientists in order to devise the so-
called genetic algorithms. However, Darwin’s theory alone does not allow one
to determine what kind of properties (or functions) of organisms are possible
in general, nor does it allow one to determine whether any specific property
can be realized in practice. Hence, a universally applicable logic explaining
the emergence of the fundamental properties of living systems is yet to be
found.
### II.1 Molecular biology
Physicists introduced a major trend in biology more than half a century ago.
It is worth mentioning Delbruck and collaborators, who were strongly
influenced by the lectures of Niels Bohr. Perhaps, the most far-reaching
speculation is due to the father of (quantum) wave mechanics, Schrödinger
himself, who suggested that an aperiodic solid could be the “storing device”
for biological information, in his famous book “What is life?” whatislife .
This steered the search leading to the discovery of the DNA molecule by Watson
and Crick.
Since the discovery of DNA, molecular biology has attempted to describe the
universal properties of the phenomena exhibited by living systems in terms of
molecules. The goal was, and still is, to trace down chemical processes from
the level of cells to that of the composing molecules, and to understand the
functioning of each molecule in biological processes (e.g., heredity,
metabolism, motility, and so on). The methodology of molecular biology can be
sketched in the following way. First, one has to consider a system at the
macroscopic level and identify the molecules and genes that are important in
some function under study. The role of each molecule must be clarified and the
interactions of such molecules with other molecules must be found. Then, one
has to devise how the macroscopic functions of the organism arise from the
cooperativity of the microscopic relevant molecules. In order to bring such a
program to completion, one has to cope with the enormous combinatorial
complexity that is due to the great number of different types of molecules
involved and, nevertheless, devise the network/circuit of chemical reactions
and back-reactions entailing life as a process. Hence, molecular biology
originally started as the pursuit of universality, rejecting the
“enumerationism” that preceded it. However, the present days witness a re-
emergence of the enumerative doctrine of the past, even if in different and
more subtle forms. Indeed, under the push of gigantic funding from medical
(and perhaps, army) research companies, molecular biology has now become an
enumerative science again. One can classify the genome project (the listing of
all the human genes), the proteome project (the listing of all proteins), and
the metabolome project (the listing of all the molecules involved in
metabolism) as mere enumerative science. Of course, such projects are of
utmost importance for practical reasons such as the health care of human
beings. The point is that such projects alone do not lead scientists one inch
further in the understanding of the universal logic of life.
Essentially, reductionism is the philosophy underlying such enumerative
projects. However, there are some hidden assumptions behind reductionism that
need to be brought to the foreground. Typically, reductionism is based on the
premise that the properties of individual elements change little in response
to their interaction with the other elements composing the whole. Here one
faces a first problem because interactions are often not small in biological
systems. Kaneko illustrates the example of proteins in the crowded environment
of a living cell, where they effectively constitute a gel kaneko . In some
cases, the distance between two neighbouring atoms in a single protein
molecule is greater than that between either of these atoms and the closest
atoms of other protein molecules. In reality, it is not unreasonable to
believe that a hard-core version of reductionism is bound to fail in the
search for the universal logic of life simply because living systems are not
machines. Typically, in living systems the behaviour of the parts/molecules
alone is different from that of the parts/molecules acting collectively. In
other words, the dynamics of the parts composing the whole is determined by
the whole; an example is given by the process of morphogenesis. In addition,
living systems display no fixed response to a specific stimulus: A given
stimulus can be associated to various possible responses (a mathematical
analogy is provided by many-valued functions). Technically, such a variety of
responses to a fixed stimulus is referred to as absence of stiffness. Such an
absence of stiffness can be further analyzed in terms of softness, or the
dependence of the response on the environmental conditions, and of
spontaneity, or the possibility of associating different outputs to the same
input, depending on the internal state of the living system and its
fluctuations. Softness and spontaneity, together with some form of memory,
give rise to (perhaps) the most striking feature of living systems: Autonomy
(flexibility and adaptability). In other words, living systems do not always
behave in strict accordance with the rules applied to them and, depending on
the situation, the rules they perceive will change (or the living systems are
able to change the rules they abide). Linking the autonomy of living systems
to mere molecular processes seems impossible to the present author. Other
universal properties of living systems that cannot be traced back to molecules
alone kaneko are the stability, the irreversibility in the development
process (i.e., the loss of multipotency from embryonic stem cells to stem
cells, and to committed cells capable of reproducing their own kind only), and
the compatibility of the two faces of the reproduction: The ability to produce
nearly identical offspring and the capacity to generate variations leading to
diversity through evolution.
### II.2 The information paradigm
Since this is the computer age, it is not strange to witness another more
subtle reductionist approach to the logic of life by means of the the
information (computer) paradigm. Such an attempt is subtle because, in
principle, it promises to explain even the autonomy of living systems.
Typically, within an algorithmic approach, the process of development
exhibited by living creatures would be represented as some kind of computer
program (a logical expression in the form of a chain of if-then statements).
For example, if $X$ denotes the concentration of a species of molecules and
$Y$ the concentration of another species, a fundamental bio-chemical process
could be understood by the code:
${\rm If}~{}X~{}\rangle~{}X^{0}~{}{\rm then~{}express}~{}Y\;,$
where $X^{0}$ is a concentration threshold. One serious problem that such an
algorithmic approach must face is that, for continuous values of the
concentration $X$, the fluctuations of $X$ can cause big errors and one should
then explain the algorithmic robustness (stability) of such “living”
computers. The problem of stability is not trivial even when one considers
discrete values of $X$. In fact, life processes involve an enormous variety of
molecules. But, the number of molecules of a given type is often small. This
means that the fluctuations of $X$ are always big and one is in the presence
of a very large chemical noise. An example is given by the process of
development of a multicellular organism kaneko : It seems miraculous that
systems with such a variety of molecules, participating in such a large number
of processes, can result again and again in an almost identical macroscopic
pattern. The situation is analogous to that of a person attempting to stack
many irregularly shaped blocks into a particular form during an intense
earthquake kaneko . A possible solution to the stability problem might be
given by the existence of negative feedback processes in living systems (i.e.,
by some form of error-correction). However, there must also be positive
feedback (amplification) in living systems. Indeed, in the following, the
thesis that the amplification processes themselves are a key to the
understanding of life will be proposed. It is not clear to the author how
error-correction schemes might work in the presence of amplification. However,
for the sake of scientific fairness, one should acknowledge that the above
arguments alone are not enough to rule out completely the information
paradigm. Perhaps, it is the author’s dislike of such a paradigm that leads
him to reject it. Typically, if the universe is some kind of supercomputer and
the phenomena we observe are just the results of its calculations, then
everything should be computable by the computer-universe. However, according
to the laws we know, many non-linear problems show an extreme sensitivity to
the initial conditions, and thus require an infinite precision representation
of the initial information for an exact solution. There are even phenomena
which are not computable in a finite time with a finite memory. In classical
mechanics the three-body problem is not exactly solvable, in quantum mechanics
the two-body problem (e.g., electron dynamics in the helium atom) is not
exactly solvable, and in quantum field theory the zero-body problem (i.e., the
vacuum) is not exactly solvable. Scientists are able to calculate realistic
processes only at the expense of tremendous simplifications and
approximations. The author is not able to imagine what kind of computer-
universe would be able to solve all these problems in the time they take to
occur in reality.
### II.3 Autopoiesis: The universal logic of Maturana and Varela
Molecular biology is not the only attempt to provide a universal logic for
life. In recent times, Maturana and Varela proposed a theory of life that is
not based on molecules and that tries to explain the general features of
living systems, including those features that are not observed but which are
nevertheless possible, in abstract terms tree . These authors state that
understanding a living system means understanding the network of relations
that must occur so that it can exist as a unit. The set of such relations is
called “organization”. The particular and concrete realization of the
organization of a living system (molecules, network of specific chemical
reactions, and so on) is called “structure”. From this distinction, it follows
that the same organization can, in principle, be realized through various
structures.
These authors make another very important distinction. In their analysis tree
it emerges that living systems are closed (e.g., isolated) from the point of
view of their organization. In other words, whatever the logic of life may be,
it must specify the living system as an independent unit, knowing and reacting
to nothing else than its internal state. However, Maturana and Varela also
clarified that, on a different level, viz., the level of thermodynamics,
living systems are open systems: They are not isolated but (from the point of
view of their structure) interact continuously with their own environment. At
this stage, Maturana and Varela explain the process of life in terms of an
unavoidable circularity: It is a peculiar feature of living systems that the
product of their organization is themselves. In other words, their
organization is such that it maintains their structure which, in turn,
implements in practice their organization. Such an unavoidable circularity is
called autopoiesis, and Maturana and Varela proceeds with the identification
of living systems with autopoietic units.
Clearly, the logic of autopoiesis is a universal logic, based on the
circularity of the abstract level of the organization maintaining itself by
means of the concrete (and physical) level of the structure of living systems.
A platonic philosopher might, perhaps, use the word “form” in place of
organization, hence clarifying from the very begining that Maturana and
Varela’s approach is not entirely reductionist. Nevertheless, reductionism is
to be found at the level of the structure, the concrete physical realization
of the living system which must be invoked in order to maintain the
organization itself. Such a circularity is fascinating and problematic at the
same time since, in general, physics and human logic do not like circularity.
It seems difficult to devise concrete mathematical models implementing such
general ideas.
### II.4 Living processes
One way to escape the circularity of autopoiesis would be to decompose it in
terms of more fundamental processes. Indeed, it appears to the present author
that autopoiesis is necessary to explain the persistence in time of a given
living system: The system can stay alive because its actions mantain its
existence. The following question naturally arises: Would it be possible to
speak of transient living systems? In other words, if living systems would not
produce themselves they would disappear almost instantaneously; however, it is
altogether tempting to call “life” such an ephemeral existence. I propose to
consider living processes (although the meaning of this concept is at the
moment unclear) as the fundamental building block of life. Such living
processes would be something more fundamental than autopoiesis for
characterizing life. From such a point of view, autopoiesis would explain the
stable existence of a living system over an extended time interval. it is
intuitive that when autopoiesis stops, a living system dies. Hence, an
autopoietic unit would then be defined as a network of self-sustaining living
processes.
Although the living process has been defined as the fundamental (transient)
building block of life, its characteristics have been left as yet mysterious.
The working hypothesis that is introduced here as a postulate (to be verified
by further analysis) is that the living process is an amplification process,
from the microscopic to the macroscopic scale, which builds up complexity in
structure and organization. Of course, the autopoiesis of Maturana and Varela
would require a feedback process from the macro to the micro scale. However,
according to the above discussion, such a feedback mechanism would be required
for mantaining life, not for life itself. The introduction of the concept of
living process breaks the circularity of autopoiesis. The identification of
the living process with a particular type of amplification process,
transferring information from the microscopic level to the macroscopic one,
which also leads to an increase of complexity in organization, permits, at
least in principle, to devise structural (i.e., physical or mathematical)
models. At this stage, it is also necessary to explain what we mean when
saying that complexity increases in going from smaller to larger scales.
Complexity is the number of constraints (or laws) that the process must
fulfill as the scale increases. The nature of such constraints can be static
or dynamic, and one thus considers a fixed structure or to the time evolution
of the system. It is not difficult to understand that an increasing number of
constraints leads to the generation of forms: What is form if not something
that is specified by boundaries and constraints? Therefore, one can also
define the living process as an amplification process that builds up forms.
The above reflection is, at the moment of writing, only a working hypothesis
and here the author is not going to provide the reader with any specific
mathematical or computer model of a living process. Perhaps, the best that can
be presently done is leaving the reader with a metaphoric image that tries to
convey the idea of amplification building complexity. Hence, one can imagine a
lightning flash which, from the shape of a flux-tube, enlarges
(amplification), not unlike the delta of a river, in order to build up an
intricate tree (augmented complexity) of smaller lightning flashes. Keeping
the poetic spirit of the above example, one can draw the main conclusion of
this Section and state that biological systems, far from being mere machines,
are matter that dances (i.e., matter that moves with an incredible level of
coordination among its constituents). Theoretical physicists are left with the
question whether the standard theories of condensed matter physics can explain
biological systems.
## III The current paradigm of condensed matter physics
The physical description of condensed matter systems is currently dominated by
electrostatics. The most sophisticated, and state-of-the-art, molecular
dynamics simulations of protein molecules in water, see Refs. sergi-rome ;
pellicane as an example, are based on semi-phenomenological force fields,
describing interactions arising from fixed electric charges, Lennard-Jones and
harmonic potentials plus bond constraints that mimic covalent bonding. In
practice, charge shielding causes the existence of short-ranged forces in such
models, which effectively treat matter as an erector set (or meccano). Even
first-principles theories, such as the electron Density Functional Theory
gross , are currently based on electrostatics alone. As a result, they
describe hydrogen bonding, van der Waals, and charge polarization effects only
with difficulty and, on the whole, with unsatisfactory results. Clearly, such
a condensed matter paradigm tries to build long-ranged correlations from
statistical fluctuations of short-ranged interactions.
We surmise that such a paradigm might be flawed at a fundamental level. For
example, it is clear that both the structure and the function of biological
macromolecules largely depend on hydrogen bonding as well as on hydrophobic
and hydrophilic interactions. These are determined by dispersion or van der
Waals forces, precisely those interactions that are not properly described by
the current paradigm. Such dispersion forces depend on the temperature and on
the molecular environment (i.e., they are not additive) milonni ; vanderwaals
. They lead to the existence of long-ranged networks of structural and
dynamical correlations. Van der Waals forces, also known as induced-dipole-
induced-dipole forces, arise from a highly correlated motion of the electronic
clouds of otherwise neutral atoms. Such a correlated motion, which in quantum
mechanical terms is called coherent, takes place even at room temperature and
in densely packed matter. Poetically, one could say that such forces in matter
arise from “dancing” electronic clouds. The explanation of such a dance is
provided by quantum mechanics milonni ; vanderwaals ; ballentine .
Quantum mechanics is widely believed to be the fundamental theory underling
the phenomenological reality. Although the majority of physicists agrees on
its mathematical formulations, its interpretation is highly controversial.
However, on some points there is a wide consensus. For example, there is
almost no dispute on the issue that quantum mechanics has some form of non-
locality built inside bell . The most advanced mathematical formulation of
quantum mechanics takes the form of a field theory zee . Notwithstanding
infinities, field phenomenology is perhaps more soundly funded than
conventionalism or the spooky attitude arising from “particle” interpretations
(see, for example, the discussion of the Einstein-Podolski-Rosen paradox from
the point of view of field theory in Ref. preparata ). In simple terms, the
field is “something” that is extended in space and time by its very definition
and, from a conceptual point of view, this aspect can be accorded more easily
with the non-locality of quantum mechanics, which appears rather puzzling when
interpreted in terms of localized particles. What is important, both for
physics and for the present discussion, is that the quantum phenomenology
(therein including discreteness, diffraction, and coherence ballentine ) does
not substantially raise any dispute.
Discreteness in quantum mechanics is usually associated with the appearance of
stationary energy levels separated by “quanta” of energy: Transitions between
these levels can only take place through the transfer of the required amount
of energy. Such a discreteness arises from the boundary conditions imposed on
the wave function (or functional) and is considered to be well understood. The
existence of discrete values for the magnetic moments, charges, and masses of
fundamental particles is less clearly understood but is easily embedded in the
current formalism of quantum mechanics.
Quantum diffraction takes its name from an analogy with the wave propagation
of light. Ensembles of microscopic particles exhibit wave-like motion arising
from the correlation and the spatial and time memory of single events in an
ensemble: One single particle is able to influence the “whole” so that the
entire ensemble appears to “move” like a wave, thus also displaying
interference effects.
Coherence also takes its name from an analogy with the wave motion of light:
It immediately brings to mind the condition of phase stability that is
necessary to observe interference and, thus, diffraction. In quantum mechanics
one would consider the phase stability of the wave function (or functional in
field theory). However, this would be confined within a formulation of quantum
mechanics in terms of wave functions. Quantum mechanics can also be formulated
in terms of path integrals, charge and mass densities, distributions in phase
space and so on nine . Hence, one needs a definition of quantum coherence less
bound to the mathematical formulation of quantum mechanics itself. Here it is
proposed to define quantum coherence as the property underlying the typical
and highly correlated motion which takes place in an unperturbed, isolated
quantum system. In condensed matter physics, striking examples of quantum
coherent motion are provided by superfluids and superconductive materials zee
. In superfluids the atomic motion is so highly correlated that friction
disappears and the fluid moves as a whole without dissipation. In many
respects, a superconductive material can be considered as a charged superfluid
of paired electrons (Cooper’s pairs) moving in a frictionless way in the
lattice of positively charged ions (making up the solid material). Both
phenomena take place at very low temperatures. Thermal fluctuations are
incoherent by their very essence and destroy coherent quantum fluctuations
with a surprisingly high efficiency: This is the phenomenon of decoherence
decoherence that is displayed by open quantum systems petruccione . Hence, in
real systems some kind of shielding from thermal fluctuations is necessary in
order to observe quantum coherent motions. It is worth remembering that in
both cases of superfluidity and superconductivity, physicists do not possess a
widely agreed microscopic dynamical explanation. In superfluids, a microscopic
picture of rotons, which are the typical many-body excitations of such
systems, has not yet emerged. In other words, although one can approximately
calculate the roton energy spectrum, it is not yet known what a roton is on
the microscopic scale: In other words, nobody actually knows what the rotonic
motion of atoms in a superfluid is. The situation is somewhat better for
superconductivity, where, at least, Cooper’s pairs have been postulated.
However, there is no first-principle explanation of the formation of Cooper’s
pairs. Typically, the celebrated Bardeen-Cooper-Schrieffer theory of
superconductivity ballentine ; zee works only after assuming the phenomenon
of electron pairing. In analogy with mechanics, one can call such theories
kinematical, since they describe the time evolution of the system without
considering the microscopic causes of the motion. This state of affairs should
be contrasted with that of dynamical theories which try to explain the time
evolution of the system under study starting from the underlying microscopic
causes (in analogy with dynamics, the branch of mechanics which explains
motion in relation to forces).
As for its applications in condensed matter systems, quantum mechanics might
provide a synthesis of reductionism and holism. The big fight of the $19^{\rm
th}$ century, between Mach and Ostwald on one side (the champions of
thermodynamics and of the holistic vision of matter) and Maxwell and Boltzmann
(the champions of atomism) on the other, has been resolved in favor of the
latter: The regularity of the chemical laws is nowadays commonly seen as the
realization of the ancient atomistic dream of Democritus and Epicurus.
However, the importance of the specification of the boundary conditions in
quantum mechanics, arising from its non-local or global (holistic) features,
renders it conceptually similar to thermodynamics, where the ensemble must be
specified in terms of the macroscopic conserved quantities.
The main conclusion of this Section is that condensed matter systems can be
dynamically understood only by resorting to quantum mechanics and to the
coherent motion of the electrons, which gives rise to chemical bonding:
Quantum mechanical electronic clouds are matter that dances, the poet would
say. The main question that is left to theoretical physicists is the
following: Condensed matter requires the coherent motion of electrons while
biological systems seem to require a highly correlated motion of heavy atoms;
is quantum mechanics necessary to understand biological systems as dancing
matter? In other words, may coherent electronic motion be the cause of
correlated heavy atomic motion?
## IV Quantum phenomena in biology
From a certain point of view, it should not be a surprise that quantum
mechanics is relevant to biology. After all, quantum mechanics is definitely
relevant to chemistry. However, there is the serious possibility that all
quantum effects in biology are, in practice, trivial nontrivial . By this it
is meant that, although necessary to explain the details of a given biological
phenomenon, quantum mechanics does not need to be understood mathematically by
a biologist who wants to study life processes. Up to a certain extent, such a
position is also held in the present contribution and phantasmal concepts such
as “quantum consciousness” penrose are not even discussed. Because of
decoherence decoherence , it appears highly implausible to the present author
that extended coherent states of heavy atoms, such as those of superfluids and
superconductors, may exist at room temperature inside a biological system
(such as a cell). Is this the end of the story? Since electronic coherence
seems to be fundamental for condensed matter systems (in practice all
electromagnetic forces can be seen as a manifestation of quantum coherent
behaviour collective ), there is still the possibility that such an electronic
coherence is really fundamental biological systems. Such a thesis is here
defended and a working hypothesis is also proposed.
There are some quantum phenomena in biology that definitely need quantum
mechanics for their very existence and whose detailed description is
challenging for the theoretical physicist. However, once their existence is
postulated, they can be used by the biologist without almost any reference to
quantum mechanics itself (in the jargon used in this contribution, one can say
that such phenomena are kinematically described by the biologist). This
occurrence is not very dissimilar from the understanding of the stability of
matter: Without quantum mechanical laws atoms could not exist. However, in a
kinematical way, one can postulate the stability of atoms, disregarding its
cause, and study, for example, the physics of noble gases (at temperatures far
from the absolute zero). On a second thought, perhaps it is not intellectually
fair to classify effects like these as trivial. Typically, the fact that some
fundamental concept can be used as a “black box”, within a more approximate
level of description, should not be used to deem the concept itself as
trivial. Therefore, in disagreement with the definition of triviality adopted
in Ref. nontrivial , here it is defended the thesis that when quantum
mechanics is necessary for the existence of a given biological phenomenon,
that phenomenon is nontrivial on a quantum mechanical basis, even if
biologists may choose to describe it kinematically as a black box. Electric
charge and exciton transfer processes in proteins and photoactive complexes
are examples of such pseudo-trivial quantum effects. The exciton is a many-
body excitation of an interacting system: A charge is excited from its ground
state and undergoes a transition to the excited state leaving a hole (a
missing charge) in the ground state; afterwards, because of many-body
interactions, the charge and the hole move in a coherent way thus giving life
to the exciton. Without quantum mechanics, one would not have ground and
excited energy states and would not have the coherent motion of the charge and
the hole. In biological systems, there are cases in which the motion of the
exciton can be approximated by classical mechanics (and there are cases when
this is not possible) but without quantum mechanics the exciton itself would
not exist. Single charge transfers are intrinsically quantum mechanical only
when tunneling through energy barriers takes place. As a matter of fact, some
charge transfer processes can be modeled classically. However, quantum
coherence could be fundamental also in non-tunneling transfers in order to
establish the right degree of correlation with the environment rearrangement
before, during, and after the charge transfer. An example could be provided by
the process of molecular recognition taking place in odor sensing by human
beings. Brookes and coworkers odor proposed a quantum model to explain odor
selectivity. According to this model, molecules are recognized not only in
terms of their shape (allowing them to dock at the right recognition site) but
also in terms of their phonon frequencies: The molecular phonons provide the
necessary energy to realize an inelastic electron transfer process which takes
place at the recognition site. Such a mechanism would explain why molecules
with the same shape may have different odors and why molecules with different
shape may have the same odor. If this is true, quantum mechanics would be
fundamental even in odor sensing. Of course, a biologist could just assume the
existence of phonons in molecules and their coupling to charges in proteins to
kinematically explain the “mechanics” of odor sensing. But the possibility of
such a mechanism would come from quantum effects anyway and this would be
conceptually very important for the understanding of life.
There are other types of phenomena in biology where quantum coherence is also
fundamental in the kinematics itself. One such phenomenon is the wave-like
motion of massive molecules. In a different context, the possibility that an
ensemble of massive molecules could behave as waves and display diffraction
has been experimentally proven by Zeilinger and coworkers zeilinger . They
have shown that slow beams of fullerene molecules (${\rm C}_{60}$) can give
rise to diffraction effects as much as lighter particles (e.g., electrons and
neutrons) do. The key to understand such a phenomenon is that the de Broglie
wavelength $\lambda$ of any object is inversely proportional to its momentum
$p$: $\lambda=h/p$, where $h$ is the celebrated Plank’s constant. Now, the
momentum is equal to the product of the mass times the velocity of the object
($p=mv$). Therefore, even if $m$ is large, as is the case of a fullerene
molecule, a small velocity $v$ of translation of the centre of mass can be
associated with an appreciable de Broglie wavelength: Indeed, Zeilinger has
been able to measure it. A second crucial step, necessary for the experimental
measurement of $\lambda$ in fullerenes, is that decoherence decoherence
(which is a universal and fast process, taking place on the scales of
femtosecons) must be inhibited in some way, otherwise the wave nature of the
beam would persist for too short times with no observable effects. In ${\rm
C}_{60}$ it turns out that the centre of mass, displaying the wave properties,
is effectively decoupled from the relative motion of the other sixty atoms in
the molecule: The sixty carbon atoms constitute the environment of the centre
of mass of the molecule and without a coupling to the environment (or in the
presence of an important energy gap) there is no decoherence. Can all this be
relevant to biological systems? In biological systems one finds long and heavy
carbon chains. They typically constitute the backbone of any protein. The
motion of such long chains can be represented collectively in terms of modes,
viz., they can be expressed in Fourier frequencies describing the motion of
all the atoms in the chain globally. The dispersion relation links the smaller
frequencies to the motion of the greater number of atoms in the chain, so that
one can say that the massive modes of the chain are slow. As a result they can
have an appreciable de Broglie wavelength. If such slow massive modes become
very weakly coupled to the environment (i.e., the water molecules and/or other
proteins around) they can exhibit interference and diffraction effects in
their motion. This has been been theoretically proven by Tuckerman in a study
of an inter-molecular proton transfer in Malonaldehyde. Such a study was
carried out by means of a very sophisticated first-principle simulation
technique which combines a path integral representation of the heavy carbon
atoms of the molecule and a density functional representation of the valence
electrons tuck . By means of this technique Tuckerman could alternatively
represent the motion of the heavy carbon atoms by means of quantum mechanics
and classical mechanics while always representing the transferring proton
quantum mechanically. Upon calculating the free energy barriers of the
transfer process in different cases, he was able to show the importance of the
quantum motion of the heavy atoms at $T=300$ K for a quantitatively correct
description of the phenomenon.
Another quantum effect in biological systems has been experimentally revealed
quite recently. It has to do with the exciton propagation in photosynthentic
systems such as the Fenna-Matthews-Olson complex photo-1 and in
photosynthetic proteins photo-2 . The charge propagation in such systems
experimentally appears to display a wave-like character even at relatively
high temperatures, of the order of $100\sim 200$ K. It has been suggested that
the protein environment might, in some way, shield the coherence of the
exciton transfer process but the detailed mechanism is not yet understood.
The last example that will be discussed here is similar to the coherent
dynamics in photosynthesis: We refer to the coherent dynamics in chromophore
molecules and to the subsequent process of vision. Such an example will also
be used to introduce and discuss a general characteristic of biological
processes and to propose the working hypothesis pointing to the necessity of
quantum mechanical coherence in living processes (which has already been
discussed in Section II.4). Chromophores are small molecules which are tightly
binded in protein pockets. Examples are given by the p-coumaric acid, the
chromophore of the Photoactive Yellow Protein sergi , and by the retinal in
bacteriorhodopsin duane . Chromophores inside a protein can catch light. After
catching light, the energy is transformed into small atomic rearrangements of
the atoms of the chromophore: Typically, a double bond is twisted and two
hydrogen atoms make a transition from the trans configuration (staying on
opposite sites of the double bond) to the cis configuration (staying on the
same site of the double bond) or vice versa, depending if one speaks of the
p-coumaric acid or retinal, respectively. It has been experimentally
demonstrated in the case of the retinal that the chromophore dynamics inside
the protein is a coherent process duane . Hence, also in this case, one finds
that the protein allows, in some way, the coherent dynamics of extended and
massive systems to take place even at room temperature. An aspect that is not
yet understood is that, while the chromophores freely flip between the trans
and cis configurations in the vacuum, inside the protein this only happens
through a photocycle composed of various steps (of which the trans-cis
transition is only the first stage). The work reported in sergi tried to
elucidate this point in the case of the p-coumaric acid in the Photoactive
Yellow Protein, but no final conclusion could be drawn. Proton pumping from
the chromophore to the protein is also involved. What is of interest to us is
that there is currently no real explanation for the signalling state of the
whole protein after light catching. In other words, the microscopic dynamics,
photon absorption plus atomic motion of the small chromophore molecule, must
be amplified at the level of the protein in order to cause the macroscopic
signalling state (or, at least, its first stage). Therefore, in this
phenomenon one encounters the amplification process that has been postulated
to be the key for living processes in Section II.4. Indeed, such an
amplification step could have been discussed for all the previous examples.
The question in all cases would be: What is the physical mechanism that gives
rise to the amplification process?
The hypothesis that is proposed here is that the amplification process could
be explained by a long-range, coherent polarization dynamics of electronic
degrees of freedom. These are in fact shielded from decoherence because the
interaction with the thermal environment may only take place by overcoming a
significant energy gap: In other words, the energy associated with thermal
fluctuations at room temperature is usually much less than the energy
necessary for a transition to the first electronic excited state. If this were
not the case, van der Waals forces and the hydrogen bonding, for example,
would not be present: The incoherent transitions of different atoms to and
from the excited state (caused by thermal fluctuations) would destroy the
coherence needed by dispersion forces. Moreover, if the dynamics of the
electron clouds is coherent, considering that the forces on the nuclei arise
from the electrons, can one really think of the dynamics of the nuclei as
incoherent? Indeed, there are theoretical reasons that suggest that when
classical degrees of freedom interact consistently with quantum ones, they
also acquire some quantum features kapral . In such a case, atomic
correlations of biological functions in thermally disordered and crowded
environments could also be interpreted in term of quantum electron coherence.
The hypothesis that quantum coherent electron dynamics/polarization could
explain in general the amplification mechanism in biological process is bold
and, at the moment, not substantiated by scientific evidence. As already noted
before, at this stage, it should be interpreted just as a working hypothesis
that needs to be tested.
## V Conclusions and perspectives
Following Kaneko’s book, a critical assessment of the usefulness of the
current trends in molecular biology has been presented. The criticism is that
the elucidation of molecules does not lead to an understanding of life as a
process. The general idea of Maturana and Varela’s autopoiesis, explaining
living systems, has been briefly sketched. Its intrinsic circularity has been
superseded by postulating the existence of a living process, proceeding from
the microscopic scale to the macroscopic scale and building complexity. Both
the microscopic amplification and the increasing complexity have been assumed
to be of fundamental importance for living processes. The amplification
process, in particular, seems easier to be modeled in a mathematical way than
autopoiesis. The concepts behind condensed matter theory and quantum mechanics
have been reviewed, emphasizing that van der Waals interactions and chemical
bonding require, in general, quantum electronic coherence even at room
temperature. Hence, meccano-like (electrostatically founded) condensed matter
theory has been deemed inadequate for the understanding of biological
phenomena. Some quantum effects in biology have also been discussed and the
suggestion that signal amplification may be explained in terms of coherent
quantum electron dynamics has been proposed: Coherent charge distribution
dynamics might be a key to the understanding of biological matter.
Does all this mean that another paradigm of condensed matter theory is needed
in order to understand biological matter? Here, the affirmative answer has
been defended and it has been proposed that long-ranged interactions and
correlations must be included from the start in the theories of biological
processes. This conclusion leads immediately to some working perspectives. One
could try to devise phenomenological computer models that include/postulate
long-ranged correlations in the dynamics and then use them to mimic biological
processes. On a more fundamental level, such correlated states should arise
from first-principle theories like quantum electrodynamics. Hence, one can
embark onto the very ambitious process of finding novel (perhaps, non-
perturbative) solutions of the ground (or first excited) states of quantum
electrodynamics in densely packed (condensed) matter systems.
One last question needs to be clearly answered here. Should a biologist study
quantum mechanics and learn all about Hilbert spaces and linear Hermitian
operators? This is not necessary for describing biological process in a
kinematical way (i.e., disregarding their causes) since quantum mechanics can
be used in many instances as a black-box theory. Nevertheless, one should bear
in mind that most biological structures and signalling process would not even
exist without quantum mechanics. This may not be important for the practice of
biology but is certainly fundamental for understanding its conceptual basis.
## Acknowledgments
This work has been funded through a competitive research grant of the
University of KwaZulu Natal. The travel to the University of Messina (Italy)
has been funded by the National Research Foundation (NRF) of South Africa
through a Knowledge and Interchange (KIC) grant.
Useful discussions with Prof. Giacomo Tripodi and Prof. Owen de Lange (who
also carefully read the manuscript) are acknowledged. I am particularly
indebted to Prof. Paolo Giaquinta, who did not only critically review many of
the ideas here reported (thus helping me shaping them up) but, out of
friendship, also took upon himself the burden of a very meticolous (and for me
very precious) editing of the manuscript.
## References
* (1) H. R. Maturana and F. Varela, _Tree of Knowledge_ (Shambhala, Boston, 1992).
* (2) A. Sergi and G. Tripodi, “Il metodo della fisica e le problematiche della biologia”, _Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat._ 85, C1C0701001 (2007).
* (3) K. Kaneko, _Life: An introduction to Complex Systems Biology_ (Springer, Berlin, 2006).
* (4) E. Schrödinger, _What is life? The Physical Aspect of the Living Cell_ (Cambridge University Press, Cambridge, 1968).
* (5) A. Sergi, M. Falconi, M. Ferrario, A. Desideri and G. Ciccotti, “Effective Binding Force Calculations in a Dimeric Protein by Molecular Dynamics Simulations”, _Journal of Chemical Physics_ 116, 6329 (2002).
* (6) G. Pellicane, G. Smith, and L. Sarkisov, “Molecular Dynamics Characterization of Protein Crystal Contacts in Aqueous Solutions”, _Phys. Rev. Lett._ 101, 248102 (2008).
* (7) R. Dreizler and E. K. U. Gross, _Density Functional Theory_ (Springer-Verlag, Berlin, 1990).
* (8) P. W. Milonni, _The Quantum Vacuum_ (Academic Press, San Diego, 1993).
* (9) V. A. Parsegian, _Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists_ (Cambridge University Press, Cambridge, 2005).
* (10) L. E. Ballentine, _Quantum Mechanics: A Modern Development_ (World Scientific, Singapore, 1998).
* (11) J. S. Bell, _Speakable and Unspeakable in Quantum Mechanics_ (Cambridge University Press, Cambridge, 1987).
* (12) A. Zee, _Quantum Field Theory in a Nutshell_ (Princeton University Press, Princeton, 2003).
* (13) D. F. Styer, M. S. Balkin, K. M. Becker, M. R. Burns, C. E. Dudley, S. T. Forth, J. S. Gaumer, M. A. Kramer, D. C. Oertel, L. H. Park, M. T. Rinkoski, C. T. Smith, and T. D. Wotherspoon, “Nine Formulations of Quantum Mechanics”, _Am. J. Phys._ 70, 288 (2002).
* (14) G. Preparata, _An Introduction to a Realistic Quantum Physics_ (World Scientific, Singapore, 2002).
* (15) _Decoherence and the Appearence of a Classical World in Quantum Physics_ , edited by D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H. D. Zeh (Springer-Verlag, Berlin, 1996).
* (16) H.-P. Breuer and F. Petruccione, _The theory of open quantum systems_ (Oxford University Press, Oxford, 2002).
* (17) H. M. Wiseman, J. Eisert, “Nontrivial quantum effects in biology: A skeptical physicists’ view”, arXiv:0705.1232v2 [physics.gen-ph]. Invited contribution to _Quantum Aspects of Life_ , edited by D. Abbott (World Scientific, Singapore, 2007).
* (18) R. Penrose, _The Emperor’s New Mind_ (Vintage, London, 1990).
* (19) C. A. Mead, _Collective Electrodynamics: Quantum Foundations of Electromagnetism_ (The MIT Press, Boston, 2000)
* (20) M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw and A. Zeilinger, “Wave–particle duality of C molecules”, _Nature_ 401, 680 (1999), L. Hackermüller, S. Uttenthaler, K. Hornberger, E. Reiger, Biörn Brezger, A. Zeilinger and M. Arndt, “Wave Nature of Biomolecules and Fluorofullerenes”, _Phys. Rev. Lett._ 91, 090408 (2003).
* (21) J. C. Brookes, F. Hartoutsiou, A. P. Horsfield, and A. M. Stoneham, “Could Humans Recognize Odor by Phonon Assisted Tunneling?”, _Phys. Rev. Lett._ 98, 038101 (2007).
* (22) M. E. Tuckerman and D. Marx, “Heavy-Atom Skeleton Quantization and Proton Tunneling in Intermediate-Barrier Hydrogen Bonds”, _Phys. Rev. Lett._ 86, 4946 (2001).
* (23) G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mancal, Y.-C. Cheng, R. Blankeship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems”, _Nature_ 446, 782 (2007).
* (24) H. Lee, Y.-C. Cheng, and G. R. Fleming, “Coherence Dynamics in Photosynthesis: Protein Protection of Exciton Coherence”, _Science_ 316, 1462 (2007).
* (25) A. Sergi, M. Grüning, M. Ferrario and F. Buda, “A Density Functional Study of the PYP Chromophore”, _J. Phys. Chem._ 105, 4386 (2001).
* (26) V. I. Prokhorenko, A. M. Nagy, S. A. Waschuk, L. S. Brown, R. R. Birge, and R. J. D. Miller, “Coherent Control of Retinal Isomerization in Bacteriorhodopsin”, _Science_ 313, 1258 (2006).
* (27) R. Kapral and G. Ciccotti, “A statistical mechanical theory of quantum dynamics in classical environments”, in _Lecture Notes in Phys._ 605, pp. 445–472 (Springer-Verlag, Berlin, 2002).
|
arxiv-papers
| 2009-07-11T15:33:53 |
2024-09-04T02:49:03.854154
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alessandro Sergi",
"submitter": "Alessandro Sergi",
"url": "https://arxiv.org/abs/0907.1968"
}
|
0907.2001
|
# Hybrid density functional calculations of the band gap of GaxIn1-xN
Xifan Wu1, Eric J. Walter2, Andrew M. Rappe3, Roberto Car1, and Annabella
Selloni1 1Chemistry Department, Princeton University, Princeton, NJ
08544-0001,USA 2Department of Physics, College of William and Mary,
Williamsburg, Virginia 23187-8795, USA 3The Makineni Theoretical
Laboratories, Department of Chemistry, University of Pennsylvania,
Philadelphia, Pennsylvania 19104-6323, USA
###### Abstract
Recent theoretical work has provided evidence that hybrid functionals, which
include a fraction of exact (Hartree Fock) exchange in the density functional
theory (DFT) exchange and correlation terms, significantly improve the
description of band gaps of semiconductors compared with local and semilocal
approximations. Based on a recently developed order-$N$ method for calculating
the exact exchange in extended insulating systems, we have implemented an
efficient scheme to determine the hybrid functional band gap. We use this
scheme to study the band gap and other electronic properties of the ternary
compound In1-xGaxN using a 64-atom supercell model.
###### pacs:
71.15.DX, 71.15.Mb, 71.20.Nr
The design of novel functional semiconductors with given values of the energy
band gap is an area of intense research Zunger_inverse ; Vasp_alloy ; ZnO_GaN
; Wang ; Bennett . In particular, much attention is focused on the band gap
engineering of group-III nitride semiconductors, whose remarkable optical
properties are important for optoelectronic device applications InGaN_review ;
Zunger_InGaN . To guide the search for compounds with tailored properties
Zunger_inverse , experimental studies are often accompanied by electronic
structure calculations based on density functional theory (DFT) DFT . For
these calculations, the local-density(LDA) or generalized gradient
approximations(GGA) are typically used. Due to the delocalization error of the
LDA and GGA exchange and correlation functionals, however, these approaches
severely underestimate the materials band gaps Yang_Science ; Yang_PRB .
As shown by several recent studies HSE_review , a significant improvement in
the description of semiconductor and insulator band gaps is generally obtained
by using hybrid functionals Hybrid , in which some exact (Hartree-Fock)
exchange is mixed into the exchange and correlation functional. This reduces
the delocalization and derivative discontinuity errors of (semi)local
functionals Yang_Science ; Yang_PRB ; Vasp_alloy ; HSE_review . However,
because of the considerable computational cost of evaluating the non-local
exact exchange term, hybrid functionals have been mostly applied to systems
with small unit cells PBE0_vasp . For the modeling of systems where a large
supercell is needed, an additional screened exchange approximation is usually
made to relieve the computational burden Vasp_alloy ; HSE_review .
Recently Wu et al. (WSC) Xifan introduced an order-$N$ method to calculate
the exact exchange in extended insulating systems. The WSC method is based on
a localized Wannier function representation of the occupied (valence) space,
so that the exchange interaction between two orbitals decays rapidly with the
distance between their centers. A truncation can thus be introduced, which
greatly reduces the computational cost. The effectiveness of the WSC method
was demonstrated by ground state electronic minimizations for crystalline
silicon in supercells with 64 and 216 atoms.
In this paper, we extend the WSC scheme to compute hybrid functional band
gaps. To this end, the system’s first (few) empty conduction state(s) is(are)
determined starting from the ground state calculated via the WSC method. With
hybrid functionals, this requires the computation of the pair exchange between
the empty state and each valence orbital. Even though the empty state is
delocalized, the product between this state and a valence orbital is well
localized, so that the corresponding exchange interaction can be truncated as
in the original WSC method Xifan . We apply our scheme to determine the band
gap of In1-xGaxN, a ternary nitride semiconductor of great technological
interest, and of its parent compounds, InN and GaN, using the PBE0 hybrid
functional PBE0 . Our results show that, compared to the semi-local PBE
functional, PBE0 gives a considerably improved description of the band gap, as
well as of the cation $d$ state binding energy, which is also poorly decribed
by the semilocal functionals.
The PBE0 hybrid functional is constructed by mixing 25% of exact exchange with
the GGA-PBE exchange PBE0 , while the correlation potential is still
represented by the corresponding functional in PBE PBE ,
$E_{xc}^{\rm PBE0}=\frac{1}{4}E_{x}+\frac{3}{4}E_{x}^{\rm PBE}+E_{c}^{\rm
PBE}.$ (1)
Here $E_{x}$ denotes the exact exchange energy, $E_{x}^{\rm PBE}$ is the PBE
exchange, and $E_{c}^{\rm PBE}$ is the PBE correlation functional. $E_{x}$ has
the usual Hartree-Fock form in terms of one-electron orbitals. In the WSC
method, this term is expressed in terms of localized Wannier orbitals
$\\{\widetilde{\varphi}_{i}\\}$. These are obtained through an unitary
transformation of the delocalized Bloch states $\\{\varphi_{i}\\}$
corresponding to occupied bands. In particular, we use maximally localized
Wannier functions (MLWFs) MLWF , which are exponentially localized. In this
way, a significant truncation in both number and size of exchange pairs can be
achieved in real space.
We now turn to the calculation of the band gap. In extended insulating systems
the band gap is simply given by the difference between the eigenvalue of the
highest occupied and the lowest empty state. Once the ground state has been
minimized self-consistently, the eigenvalue of the empty state $\varphi_{e}$
can be obtained through a simple non-selfconsistent calculation. With the
hybrid PBE0 functional, the equation for $\varphi_{e}$ is
$\displaystyle\Bigl{(}$ $\displaystyle-$
$\displaystyle\frac{1}{2}\nabla^{2}+V_{\rm ion}({\bf r})+V_{\rm H}[\,\rho^{\rm
val}({\bf r})\,]+\frac{3}{4}V_{x}^{\rm PBE}[\,\rho^{\rm val}({\bf r})\,]$ (2)
$\displaystyle+$ $\displaystyle V_{c}^{\rm PBE}[\,\rho^{\rm val}({\bf
r})\,]\Bigr{)}\times\varphi_{e}({\bf r})+\frac{1}{4}\int V_{x}^{\rm val}({\bf
r,r^{\prime}})\varphi_{e}({\bf r^{\prime}})d{\bf r^{\prime}}$ (3)
$\displaystyle=$ $\displaystyle\varepsilon_{e}\varphi_{e}({\bf r}),$ (4)
In the above expression we have assumed, for simplicity, a closed-shell system
with $N/2$ doubly occupied one-electron states (extension to spin-polarized
systems is straightforward); $V_{\rm H}$ and $V_{\rm ion}$ are the Hartree and
the ionic (pseudo-)potentials, respectively; $V_{x}^{\rm PBE}$ and $V_{c}^{\rm
PBE}$ are the PBE exchange and correlation potentials. We note that $V_{\rm
H}$, $V_{x}^{\rm PBE}$ and $V_{c}^{\rm PBE}$ are fixed operators as they only
depend on the (fixed) valence charge density $\rho^{\rm val}({\bf
r})=\sum_{j}^{\rm occ}\varphi_{j}^{*}({\bf r})\varphi_{j}({\bf r})$. Finally,
the non-local exact exchange potential ${V}_{x}^{\rm val}({\bf r,r^{\prime}})$
is given by:
${V}_{x}^{\rm val}({\bf r,r^{\prime}})=-2\sum_{j}^{\rm
occ}\frac{\widetilde{\varphi}_{j}^{*}({\bf
r^{\prime}})\widetilde{\varphi}_{j}({\bf r})}{|{\bf r}-{\bf r^{\prime}}|},$
(5)
where the sum runs over all the occupied states. This potential describes the
exchange interaction between the empty state and each of the valence MLWFs
$\\{\widetilde{\varphi}_{j}\\}$.
The action of ${V}_{x}^{\rm val}({\bf r,r^{\prime}})$ on the empty state
$\varphi_{e}$ in Eq. (5) is given by:
$\displaystyle D_{x}^{e}({\bf r})$ $\displaystyle=$
$\displaystyle-2\sum_{j}^{\rm occ}\int d{\bf
r^{\prime}}\frac{\widetilde{\varphi}_{j}^{*}({\bf r^{\prime}})\varphi_{e}({\bf
r^{\prime}})}{|{\bf r}-{\bf r^{\prime}}|}\times\widetilde{\varphi}_{j}({\bf
r})$ (6) $\displaystyle=$ $\displaystyle-2\sum_{j}^{\rm occ}v_{ej}({\bf
r})\widetilde{\varphi}_{j}({\bf r})$ (7)
Here $v_{ej}$ is the Coulomb potential originating from the “exchange charge”
$\rho_{\rm ej}=\widetilde{\varphi}_{j}^{*}({\bf r^{\prime}})\varphi_{e}({\bf
r^{\prime}})$, and satisfies the Poisson equation:
$\nabla^{2}v_{ej}=-4\pi\rho_{ej}$ (8)
It is important to note that, while the empty eigenstate of Eq. (4) is Bloch
like and delocalized in real space, the exchange pair density $\rho_{ej}$ is
confined by the valence MLWFs that are well localized in real space. As a
result, the Poisson equation, Eq. (8), and the action of the exchange
operator, Eq. (7), need only be solved in the region where
$\widetilde{\varphi}_{j}\neq 0$.
We have implemented the above computational procedure for calculating the PBE0
band gap in the CP code of the Quantum-ESPRESSO package. QuantumEspresso The
procedure works as a post processing feature following a PBE0 ground state
calculation by the MLWF-based WSC method. In this work, we use it to calculate
the electronic structure, particularly the band gap, of GaN, InN, and
In1-xGaxN in the zincblende phase. These systems are computationally
challenging because InN and In-rich In1-xGaxN are incorrectly predicted to be
metallic by standard GGA calculations.
The calculations were performed using a 64-atom cubic supercell to model both
In1-xGaxN and its parent compounds, GaN and InN. For each Ga concentration $x$
in the ternary In1-xGaxN compound, only a few selected atomic configurations
were considered, with no specific treatment of disorder effects, as e.g. in
Refs. Zunger_InGaN, ; Wang, ; within our limited sampling, a very weak
dependence of the calculated band gap on the specific cation arrangement was
observed. For direct comparison with experiments and other theoretical
results, the experimental lattice constants of GaN (a = 4.50 Å) and InN (a =
4.98 Å) were used, while the lattice parameter of the alloy was determined by
linear interpolation.
Table 1: Pseudopotential generation parameters. Here “ref.” refers to the reference state occupation, rc refers to the cut-off radius, $q_{c}$ is the cut-off wavevector and $N_{B}$ is the number of Bessel functions used for each channel (see Ref. RRKJ, ). Atom | parameter | $s$ | $p$ | $d$
---|---|---|---|---
N | ref. | $2.0$ | $3.0$ | –
| ${\rm r}_{c}$ | $1.30$ | $1.30$ | –
| $q_{c}$ | $7.50$ | $7.50$ | –
| $N_{B}$ | 10 | 10 | –
Ga | ref. | $2.0$ | $1.0$ | $10.0$
| ${\rm r}_{c}$ | $1.80$ | $2.20$ | $1.80$
| $q_{c}$ | $8.00$ | $8.00$ | $8.36$
| $N_{B}$ | 6 | 8 | 10
In | ref. | $2.0$ | $1.0$ | $10.0$
| ${\rm r}_{c}$ | $1.90$ | $2.30$ | $1.80$
| $q_{c}$ | $8.00$ | $8.00$ | $8.00$
| $N_{B}$ | 8 | 8 | 8
Table 1 shows the reference states and cut-off radii used to construct the
pseudopotentials used in this study. All pseudopotentials were generated using
the OPIUM code OPIUM .
Unlike with traditional density functional theory, Hartree-Fock
pseudopotentials require extra care in their construction. This arises from
the non-local form of the Hartree-Fock exchange potential trail_needs1 ;
bk_exact_xc ; stadele_exact_xc ; engel_exact_xc . The presence of the non-
local exchange potential in Hartree-Fock or Hartree-Fock/DFT hybrids will
often yield pseudopotentials with an unphysical, long-range tail. A correction
procedure is necessary to remove this tail and restore the correct long-range
behavior of the pseudopotential while maintaining the eigenvalue spectrum and
logarithmic derivatives. Recent work trail_needs1 ; trail_needs2 ; AWR has
shown that this approach yields highly accurate Hartre-Fock pseudopotentials.
The pseudopotentials were norm-conserving/RRKJ type RRKJ and were generated
from self-consistent PBE0 all-electron reference states using the approach of
Ref. AWR, . The Ga and In pseudopotentials were obtained from scalar-
relativistic solutions, while the N pseudopotential was non-relativistic. The
local potential was the $s$ channel for all cases. The semi-core $d$ electrons
were treated as valence electrons in In and Ga (this corresponds to 576
valence electrons, i.e. 288 occupied states, in the 64-atom supercell). The
plane-wave energy cutoff was 70 Ry and the Brillouin zone was sampled at the
$\Gamma$ point. Atomic positions in the supercell were relaxed at the GGA-PBE
level.
The PBE0 ground state was determined by the WSC method, using MLWFs to
calculate the exchange interaction among valence electrons Xifan . While the
MLWFs generated from the PBE ground state often give an excellent initial
guess for the PBE0 calculations, for InN and In rich GaxIn1-xN alloy
configurations, the PBE ground state shows an incorrect ordering of the energy
bands. For this reason, instead of PBE Wannier orbitals we used a set of
fictitious localized orbitals at the guess bonding centers as the trial
solutions for Eq. (4). This procedure was essential to obtain the PBE0 ground
state with correct symmetry for InN and In rich GaxIn1-xN. In the empty state
calculations, for each PBE0 ground state MLWF we first defined an orthorhombic
box such that outside this box $\rho_{ej}({\bf r})$ is smaller than a given
cut-off value $\rho^{\rm cut}$; we take this cut-off equal to $2\times
10^{-4}\ {\rm bohr}^{-3}$ in the present work. Then Eq. (8) is solved by the
conjugate gradient method Xifan , and for each pair $\rho_{\rm ej}$ formed by
the empty state and a PBE0 ground state MLWF its action Eq. (7) is applied
only inside the above truncated box. Finally with this $D_{x}^{e}({\bf r})$,
Eq. (4) is solved via a damped second order Car-Parrinello dynamics RC review
.
Figure 1: (Color online.) Isosurfaces of typical $d$-like and $sp^{3}$ like
Wannier orbitals in the InN (on the left) and GaN (on the right) 64-atom
supercell. The Ga, In and N atoms are denoted by the green, red and blue
spheres respectively.
Representative MLWFs for InN in its PBE0 ground state are shown in Fig. 1. Two
types of valence MLWFs are present in our calculations, $d$-like Wannier
orbitals centered at the In sites, and covalent $sp^{3}$-like orbitals
centered between the cations and the anions. As one can see from the figure,
the $d$-like orbitals originating from the cation semi-core states are more
localized than the $sp^{3}$-like ones. The valence MLWFs are qualitatively
similar for GaN, except for a slightly more pronounced localization related to
the larger band gap.
Table 2: Valence band width, band gap and average $d$-band binding energy (eV) of GaN and InN. | | VBW | $E_{g}$ | $E_{d}$
---|---|---|---|---
GaN | PBE0-MLWFs | $17.70$ | $3.52$ | $-16.16$
| PBE | $16.14$ | $1.60$ | $-13.62$
| PBE0, plane waves 111Reciprocal space method in PWSCF (Ref. QuantumEspresso, ) in 2-atom cell and 4$\times$4$\times$4 $k$ points | $17.72$ | $3.61$ |
| GW 222Reference InN_vasp, . | | $3.53$ | $-16.5$
| Experiment 333Reference InN_d, . | | $3.3$ | $-17.7$
InN | PBE0-MLWFs | $17.04$ | $1.09$ | $-15.30$
| PBE results | $15.04$ | $-0.04$ | $-13.48$
| GW 222Reference InN_vasp, . | | $0.78$ | $-15.3$
| Experiment | | $0.61$222Reference InN_vasp, . | $-16.0$333Reference InN_d, .
The band structure properties of GaN and InN that result from our PBE0-MLWFs
calculations are summarized in Table 2. Here we report the valence band width
(VBW), the band gap $E_{g}$ and the average $d$-band binding energy $E_{d}$,
and compare them to PBE calculations (performed with the same 64-atom
supercell used for the PBE0 calculations) and experimental results. For
further comparison, we also report the results of PBE0 calculations performed
using the reciprocal space implementation in Ref. QuantumEspresso, ; we can
see that the agreement between these results and our MLWF-based calculations
is very good. From Table 2 it appears that the GGA-PBE results significantly
overestimate the energetic position of the cation $d$-bands. Because of the
$pd$ repulsion, the overestimated $d$ bands level in turn pushes the $p$ band
upwards, resulting in an underestimated band gap. For InN, this effect leads
to a wrong ordering of the $\Gamma_{1c}$ and $\Gamma_{15v}$ energy levels, and
thus to the incorrect prediction of a metallic ground state. In the PBE0
calculations, the inclusion of exact exchange reduces the delocalization
error. As shown by Table 2, the PBE0 VBW is larger and the $d$-bands level
shifts downwards, in better agreement with the experiment. In turn, this leads
to a considerable improvement of the band gaps of both InN and GaN with
respect to experiment; in particular, the PBE0 band gap becomes 1.09 eV for
InN. It is also worth noticing that calculation of the PBE0 band gap using a
PBE pseudopotential yields a $\sim$ 0.2 eV smaller value than that obtained
with the PBE0 pseudopotential.
Figure 2: (Color online.) (a) PBE0, PBE and experimental band gap of
dependence Ga fraction $x$ (b) Valence band maximum (VBM) and conduction band
minimum as a function of Ga fraction $x$ in In1-xGaxN
Besides confirming the good performance of hybrid functionals for band gap
predictions, the above results for InN and GaN provide evidence of the
reliability of our procedure for calculating the PBE0 band gap. We have thus
applied this procedure to the study of the ternary In1-xGaxN compound, a
system for which the standard reciprocal space approach to calculate the exact
exchange would be extremely cumbersome. Instead, our order-$N$ scheme is well
suited to treat systems for which large supercells are needed. Using a 64-atom
supercell, we then considered In1-xGaxN models with 1(31), 2(30), 3(29),
4(28), 16(16), 28(4), 29(3), 30(2) and 31(1) Ga(In) cations, which correspond
to $x$ = 0.031, 0.063, 0.094, 0.125, 0.5, 0.875, 0.906, 0.938, and 0.969. For
each value of $x$ and a given configuration of Ga(In) atoms, the atomic
positions were relaxed at the PBE level. The computed PBE0 band gap of
In1-xGaxN as a function of the Ga fraction $x$ is shown in Fig. 2(a), together
with experimental bowing and PBE results. We can see that PBE not only
significantly underestimates the band gap but incorrectly shows a metallic
ground state for $x<0.5$. By contrast, a direct band gap at the $\Gamma$ point
is found for all values of $x$ at the PBE0 level. Moreover, PBE0 predicts a
large band gap bowing effect, in qualitative agreement with the experiment
bowing . The band gap can be fitted to the quadratic form
$E_{g}^{\rm alloy}=xE_{g}^{\rm GaN}+(1-x)E_{g}^{\rm InN}-x(1-x)b$ (9)
from which a bowing coefficient $b^{\rm PBE0}$ = 1.63 eV can be extracted,
similar to the value, 1.67 eV, found in previous screened-exchange density
functional ($sx$-LDA) calculations Wang . However, this is somewhat larger
than the experimental value $b^{\rm expt}$ = 1.43 eV bowing , likely because
of the overestimated PBE0 band gap for the In-rich compounds. To gain more
insight into the origin of the large band gap bowing, we have examined how the
valence band maximum (VBM) and conduction band minimum (CBM) depend separately
on $x$, see Fig. 3(a). In this analysis, the average electrostatic potential
was taken as the reference for the band alignment. It can be seen that the VBM
increases almost linearly with $x$, whereas the CBM shows a stronger nonlinear
increase which is responsible for the large bowing coefficient of the alloy.
Figure 3: (Color online.) Isosurfaces of PBE0 eigenstate (a) In3Ga29N where 3
In atoms forms a zigzag chain structure; (b) Ga3In29N where 3 Ga atoms forms a
zigzag chain N atoms are denoted by red, orange and blue spheres respectively
The electronic states in proximity of the VBM are important for the
pholuminescence properties of In1-xGaxN. These states have the character of
$p$ orbitals localized at the N sites. Previous theoretical studies of
In1-xGaxN found that in Ga-rich alloys the amplitude of these states is
enhanced at N sites close to In impuritiesWang ; Zunger_InGaN , suggesting a
localization of photoexcited holes at such sites. This interesting result is
confirmed by our PBE0 hybrid calculations. The enhancement, or hole
localization, is particularly evident when the In impurities are clustered to
form a zigzag In-N-In-N-In chain, as shown in Fig. 3(a). This localization has
been suggested to be the reason of the high efficiency of In1-xGaxN based
emitting devices Wang ; InGaN_review . Interestingly, we found that there is
an opposite effect for the case of Ga impurities in In rich alloys. Here, a
reduction of the $p$ states at the N sites along the Ga-N-Ga-N-Ga-N chain is
observed, see Fig. 3(b).
In conclusion, we have described an efficient procedure to calculate the band
gap of extended insulating systems using hybrid functionals. This procedure is
based on the recently developed WSC order-$N$ method, in which the Hartree
Fock exchange is calculated using MLWFs, and can therefore be used to study
the band gap and other electronic properties of systems with large unit cells.
We have demonstrated the effectiveness of our approach by a study of the band
gap of a ternary compound, In1-xGaxN, that we have modeled using a 64-atom
supercell. Hybrid functional results for this important material are here
reported for the first time, without the approximation of screened exchange,
and show a much better agreement with experiment than conventional DFT-GGA or
LDA calculations. Our approach can be widely used for the band gap engineering
problem in semiconductor alloys.
###### Acknowledgements.
This work has been supported by the Department Of Energy under grant DE-
FG02-06ER-46344, grant DE-FG02-05ER46201 and by AFOSR-MURI F49620-03-1-0330.
A. M. R. was supported by the (US) Department of Energy under grant DE-
FG02-07ER46431
## References
* (1) P. Piquini, P. A. Graf, and A. Zunger, Phys. Rev. Lett. 100, 186403 (2008). S. V. Dudiy and A. Zunger, ibid 97, 046401 (2006).
* (2) A. Grüneis, K. Hummer, M. Marsman, and G. Kresse, Phys. Rev. B 78, 165103 (2008).
* (3) M. N. Huda, Y. Yan, S. -H. Wei, and M. M. Al-Jassim, Phys. Rev. B 78, 195204 (2008).
* (4) B. Lee and L. -W. Wang, J. Appl. Phys. 100, 093717 (2006).
* (5) J. W. Bennett, I. Grinberg, and A. M. Rappe, J. Am. Chem. Soc. 130, 17409 (2008).
* (6) S. F. Chichibu et al., Philos. Mag. 87, 2019 (2007).
* (7) L. Bellaiche, T. Mattila, L. -W. Wang, S. -H. Wei, and A. Zunger, Appl. Phys. Lett. 74, 1842(1999).
* (8) See e.g. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989).
* (9) A. J. Cohen, P. Mori-S anchez, and W. Yang, Science 321, 792 (2008).
* (10) A. J. Cohen, P. Mori-S anchez, and W. Yang, Phys. Rev. B 77, 115123 (2008).
* (11) B. G. Janesko, T. M. Henderson, and G. E. Scuseria, Phys. Chem. Chem. Phys., 11, 443 (2009).
* (12) A. D. Becke, J. Chem. Phys. 98, 1372 (1993).
* (13) M. Marsman, J. Paier, A. Stroppa, and G. Kresse, J. Phys.: Condens. Matter 20, 064201 (2008).
* (14) X. Wu, A. Selloni, and R. Car, Phys. Rev. B 79, 085102 (2009).
* (15) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).
* (16) J. P. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982 (1996).
* (17) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
* (18) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).
* (19) R. Car, in Conceptual Foundations of Materials: A Standard Model for Ground- and Excited-State Properties, Contemporary Concepts of Condensed Matter Science, edited by S. G. Louie and M. L. Cohen (Elsevier, Amsterdam, 2006), Chap. 3, p. 64.
* (20) See http://www.quantum-espresso.org.
* (21) OPIUM pseudopotential package http://opium.sourceforge.net.
* (22) J. R. Trail and R. J. Needs, J. Chem. Phys. 122, 014112 (2005).
* (23) D. M. Bylander and L. Kleinman, Phys. Rev. Lett. 74, 3660 (1995); Phys. Rev. B 52, 14566 (1995); 54, 7891 (1996); 55, 9432 (1997).
* (24) M. Städele, J. A. Majewski, P. Vogl, and A. Görling, Phys. Rev. Lett. 79, 2089 (1997).
* (25) E. Engel, A. Höck, R. N. Schmid, R. M. Dreizler, and N. Chetty, Phys. Rev. B 64, 125111 (2001).
* (26) J. R. Trail and R. J. Needs, J. Chem. Phys. 122, 174109 (2005).
* (27) W. A. Al-Saidi, E. J. Walter, and A. M. Rappe, Phys. Rev. B 77, 075112 (2008).
* (28) A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joannopoulos, Phys. Rev. B 41, R1227 (1990).
* (29) F. Fuchs, J. Furthmüller, F. Bechstedt, M. Shishkin, and G. Kresse, Phys. Rev. B 76, 115109 (2007).
* (30) L. F. J. Piper, T. D. Veal, P. H. Jefferson, C. F. McConville, F. Fuchs, J. Furthmüller,F. Bechstedt, H. Lu and W. J. Schaff, Phys. Rev. B 72, 245319 (2005).
* (31) J. Wu, W. Walukiewicz, K. M. Yu, J. W. Ager III, E. E. Haller, H. Lu, and W. J. Schaff, Appl. Phys. Lett. 80, 4741 (2002).
|
arxiv-papers
| 2009-07-12T03:58:33 |
2024-09-04T02:49:03.863734
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xifan Wu, Eric J. Walter, Andrew M. Rappe, Roberto Car, Annabella\n Selloni",
"submitter": "Xifan Wu",
"url": "https://arxiv.org/abs/0907.2001"
}
|
0907.2003
|
# Generalized quantum operations and almost sharp quantum effects††thanks:
This project is supported by Natural Science Found of China (10771191 and
10471124).
Shen Jun1,2, Wu Junde1 E-mail: [email protected]
###### Abstract
In this paper, we study generalized quantum operations and almost sharp
quantum effects, our results generalize and improve some important conclusions
in [2] and [3].
1Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
2Department of Mathematics, Anhui Normal University, Wuhu 241003, P. R. China
Key Words. Quantum operations, fixed points, almost sharp quantum effects.
This paper is to commemorate my outstanding student Shen Jun, who passed away
accidently on July 1, 2009. Shen Jun made great contributions in sequential
effect algebra theory. He solved four open problems which were presented by
Professor Gudder in International Journal of Theoretical Physics, 44 (2005),
2199-2205.
1\. Introduction
Let $H$ be a Hilbert space, $B(H)$ be the set of bounded linear operators on
$H$, $P(H)$ be the set of projection operators on $H$, $T(H)$ be the set of
trace class operators on $H$, and
$\Gamma=\\{A_{\alpha},A_{\alpha}^{*}\\}_{\alpha\in\Lambda}$ be a set of
operators, where $A_{\alpha}\in B(H)$ satisfy
$\sum\limits_{\alpha}A_{\alpha}A_{\alpha}^{*}\leq I$. A map
$\Phi_{\Gamma}:B(H)\longrightarrow
B(H);B\longmapsto\sum\limits_{\alpha}A_{\alpha}BA_{\alpha}^{*}$ is called a
generalized quantum operation. Each element of
$\Gamma=\\{A_{\alpha},A_{\alpha}^{*}\\}_{\alpha\in\Lambda}$ is said to be a
operation element of $\Phi_{\Gamma}$. If $B\geq 0$, then it is obvious that
$\sum\limits_{\alpha}A_{\alpha}BA_{\alpha}^{*}$ converges in the strong
operator topology, so $\sum\limits_{\alpha}A_{\alpha}BA_{\alpha}^{*}$
converges in the strong operator topology for any $B\in B(H)$. If
$\Phi_{\Gamma}(I)=\sum\limits A_{\alpha}A^{*}_{\alpha}=I$, then
$\Phi_{\Gamma}$ is said to be unital, if
$\sum\limits_{\alpha}A_{\alpha}^{*}A_{\alpha}=I$, then $\Phi_{\Gamma}$ is said
to be trace preserving, if $\sum\limits_{\alpha}A_{\alpha}^{*}A_{\alpha}\leq
I$, then $\Phi_{\Gamma}$ is said to be trace nonincreasing, if
$A_{\alpha}^{*}=A_{\alpha}$ for every $\alpha$, then $\Phi_{\Gamma}$ is said
to be self-adjoint.
The set of fixed points of $\Phi_{\Gamma}$ is $B(H)^{\Phi_{\Gamma}}=\\{B\in
B(H)\mid\Phi_{\Gamma}(B)=B\\}$. Obviously $B(H)^{\Phi_{\Gamma}}$ is closed
under the involution $*$. The commutant ${\Gamma}^{\prime}=\\{B\in B(H)\mid
BA_{\alpha}=A_{\alpha}B,BA_{\alpha}^{*}=A_{\alpha}^{*}B,\alpha\in\Lambda\\}$
of ${\Gamma}$ is a von Neumann algebra.
Quantum operations frequently occur in quantum measurement theory, quantum
probability, quantum computation, and quantum information theory ([1]). If an
operator $A$ is invariant under the quantum operation $\Phi_{\Gamma}$, in
physics, it implies that $A$ is not disturbed by the action of
$\Phi_{\Gamma}$. So, the following problem is interesting and important: if
$A$ is a $\Phi_{\Gamma}$-fixed point, is $A$ commutative with each operation
element of $\Phi_{\Gamma}$? In general, the answer is not and some sufficient
conditions under which the answer is yes were given ([2]).
On the other hand, quantum effects are represented by operators on a Hilbert
space $H$ satisfying that $0\leq A\leq I$, and sharp quantum effects are
represented by projections. An quantum effect $A$ is said to be almost sharp
if $A=PQP$ for projections $P$ and $Q$ ([3]). In [3], some characterizations
of almost sharp quantum effects were obtained.
In this paper, we generalize some theorems in [2] from quantum operations to
generalized quantum operations, from unital to not necessarily unital, and
from trace preserving to trace nonincreasing, we also generalize some results
in [3] and give some more characterizations for almost sharp quantum effects.
2\. Generalized quantum operations
Lemma 2.1. If $\Phi_{\Gamma}$ is a generalized quantum operation, $B,BB^{*}\in
B(H)^{\Phi_{\Gamma}}$, then $BA_{\alpha}=A_{\alpha}B$ for every $\alpha$.
Proof. Since $B\in B(H)^{\Phi_{\Gamma}}$, we have $B^{*}\in
B(H)^{\Phi_{\Gamma}}$. Let we denote $[B,A_{\alpha}]=BA_{\alpha}-A_{\alpha}B$.
Note that
$0\leq[B,A_{\alpha}][B,A_{\alpha}]^{*}=(BA_{\alpha}-A_{\alpha}B)(A_{\alpha}^{*}B^{*}-B^{*}A_{\alpha}^{*})=BA_{\alpha}A_{\alpha}^{*}B^{*}+A_{\alpha}BB^{*}A_{\alpha}^{*}-A_{\alpha}BA_{\alpha}^{*}B^{*}-BA_{\alpha}B^{*}A_{\alpha}^{*}$.
Thus
$0\leq\sum\limits_{\alpha}[B,A_{\alpha}][B,A_{\alpha}]^{*}=B(\sum\limits_{\alpha}A_{\alpha}A_{\alpha}^{*})B^{*}+\Phi_{\Gamma}(BB^{*})-\Phi_{\Gamma}(B)B^{*}-B\Phi_{\Gamma}(B^{*})=B(\sum\limits_{\alpha}A_{\alpha}A_{\alpha}^{*})B^{*}-BB^{*}\leq
0$.
So we conclude that $[B,A_{\alpha}]=0$ for every $\alpha$. That is,
$BA_{\alpha}=A_{\alpha}B$ for every $\alpha$.
Theorem 2.1. If $\Phi_{\Gamma}$ is a generalized quantum operation,
$B,B^{*}B,BB^{*}\in B(H)^{\Phi_{\Gamma}}$, then $B\in{\Gamma}^{\prime}$.
Proof. By Lemma 2.1, $BA_{\alpha}=A_{\alpha}B$ for every $\alpha$. Since $B\in
B(H)^{\Phi_{\Gamma}}$, we have $B^{*}\in B(H)^{\Phi_{\Gamma}}$. Thus by Lemma
2.1 again, $B^{*}A_{\alpha}=A_{\alpha}B^{*}$ for every $\alpha$. Taking
adjoint, we have $BA_{\alpha}^{*}=A_{\alpha}^{*}B$ for every $\alpha$. So we
conclude that $B\in{\Gamma}^{\prime}$.
Theorem 2.2. If $\Phi_{\Gamma}$ is a self-adjoint generalized quantum
operation, $B,BB^{*}\in B(H)^{\Phi_{\Gamma}}$, then $B\in{\Gamma}^{\prime}$.
Proof. By Lemma 2.1, $BA_{\alpha}=A_{\alpha}B$ for every $\alpha$. Since
$A_{\alpha}^{*}=A_{\alpha}$ for every $\alpha$, we conclude that
$B\in{\Gamma}^{\prime}$.
We denote the set of selfadjoint elements in $B(H)^{\Phi_{\Gamma}}$ by
$Re(B(H)^{\Phi_{\Gamma}})$.
Theorem 2.3. If $\Phi_{\Gamma}$ is a generalized quantum operation, then the
following conditions are all equivalent:
(1) $B(H)^{\Phi_{\Gamma}}\subseteq{\Gamma}^{\prime}$;
(2) If $B\in B(H)^{\Phi_{\Gamma}}$, then $B^{*}B\in B(H)^{\Phi_{\Gamma}}$;
(3) If $B\in Re(B(H)^{\Phi_{\Gamma}})$, then $B^{2}\in B(H)^{\Phi_{\Gamma}}$.
Proof. (1)$\Rightarrow$(2): If $B\in B(H)^{\Phi_{\Gamma}}$, then
$B\in{\Gamma}^{\prime}$. Thus $B^{*}\in{\Gamma}^{\prime}$. So
$\Phi_{\Gamma}(B^{*}B)=\sum\limits_{\alpha}A_{\alpha}B^{*}BA_{\alpha}^{*}=B^{*}\sum\limits_{\alpha}A_{\alpha}BA_{\alpha}^{*}=B^{*}\Phi_{\Gamma}(B)=B^{*}B$.
Thus $B^{*}B\in B(H)^{\Phi_{\Gamma}}$.
(2)$\Rightarrow$(3) is obvious.
(3)$\Rightarrow$(1): By Theorem 2.1, If $B\in Re(B(H)^{\Phi_{\Gamma}})$, then
$B\in{\Gamma}^{\prime}$. That is,
$Re(B(H)^{\Phi_{\Gamma}})\subseteq{\Gamma}^{\prime}$. Since
$B(H)^{\Phi_{\Gamma}}$ is closed under the involution $*$, we conclude that
$B(H)^{\Phi_{\Gamma}}\subseteq{\Gamma}^{\prime}$.
Lemma 2.2. If $\\{C_{\beta}\\}_{\beta}\subset B(H)$, $\\{C_{\beta}\\}_{\beta}$
is a nondecreasing net of positive operators converging to some $C_{0}\in
B(H)$ in the strong operator topology, then $tr(C_{\beta})\longrightarrow
tr(C_{0})$, here the trace function $tr(\cdot)$ can take value $+\infty$.
Proof. Since $0\leq C_{\beta}\leq C_{0}$, we have $tr(C_{\beta})\leq
tr(C_{0})$.
For any constant $\xi<tr(C_{0})=\sum\limits_{\gamma\in F}\langle
C_{0}x_{\gamma},x_{\gamma}\rangle$ ( here $\\{x_{\gamma}\\}_{\gamma\in F}$ is
an orthonormal bases of $H$), there exists a finite subset $F_{0}\subseteq F$
such that $\xi<\sum\limits_{\gamma\in F_{0}}\langle
C_{0}x_{\gamma},x_{\gamma}\rangle$. Since $\sum\limits_{\gamma\in
F_{0}}\langle
C_{\beta}x_{\gamma},x_{\gamma}\rangle\longrightarrow\sum\limits_{\gamma\in
F_{0}}\langle C_{0}x_{\gamma},x_{\gamma}\rangle$, we have
$tr(C_{\beta})\geq\sum\limits_{\gamma\in F_{0}}\langle
C_{\beta}x_{\gamma},x_{\gamma}\rangle>\xi$ for all sufficiently large $\beta$.
Thus $tr(C_{\beta})\longrightarrow tr(C_{0})$.
Theorem 2.4. Let $\Phi_{\Gamma}$ be a trace nonincreasing generalized quantum
operation, $B\in T(H)_{+}$, then $\Phi_{\Gamma}(B)\in T(H)_{+}$ and
$tr(\Phi_{\Gamma}(B))\leq tr(B)$.
Proof. Let $F$ be a finite subset of $\Lambda$, then
$tr(\sum\limits_{\alpha\in
F}A_{\alpha}BA_{\alpha}^{*})=tr(\sum\limits_{\alpha\in
F}A_{\alpha}^{*}A_{\alpha}B)\leq\parallel\sum\limits_{\alpha\in
F}A_{\alpha}^{*}A_{\alpha}\parallel tr(B)\leq tr(B)$. Ordering all such $F$ by
including, $\\{\sum\limits_{\alpha\in F}A_{\alpha}BA_{\alpha}^{*}\\}_{F}$ is a
nondecreasing net of positive operators converging to $\Phi_{\Gamma}(B)$ in
the strong operator topology. So by Lemma 2.2 we have
$tr(\sum\limits_{\alpha\in F}A_{\alpha}BA_{\alpha}^{*})\longrightarrow
tr(\Phi_{\Gamma}(B))$. Thus $tr(\Phi_{\Gamma}(B))\leq tr(B)$.
A generalized quantum operation $\Phi_{\Gamma}$ is faithful if for any $B\in
B(H)$, $\Phi_{\Gamma}(B^{*}B)=0$ implies $B=0$.
Theorem 2.5. Let $\Phi_{\Gamma}$ be a trace preserving generalized quantum
operation. We have
(1). $\Phi_{\Gamma}$ is faithful.
(2). If $B\in T(H)$, then $\Phi_{\Gamma}(B)\in T(H)$ and
$tr(\Phi_{\Gamma}(B))=tr(B)$.
Proof. (1). Suppose $B\in B(H)$, $\Phi_{\Gamma}(B^{*}B)=0$. Then
$\sum\limits_{\alpha}A_{\alpha}B^{*}BA_{\alpha}^{*}=0$. So $BA_{\alpha}^{*}=0$
for every $\alpha$. Thus $B=B\sum\limits_{\alpha}A_{\alpha}^{*}A_{\alpha}=0$.
(2). Firstly we suppose $B\in T(H)_{+}$. By Theorem 2.4 we have
$\Phi_{\Gamma}(B)\in T(H)_{+}$. Let $F$ be a finite subset of $\Lambda$,
ordering all such $F$ by including, $\\{\sum\limits_{\alpha\in
F}A_{\alpha}BA_{\alpha}^{*}\\}_{F}$ is a nondecreasing net of positive
operators converging to $\Phi_{\Gamma}(B)$ in the strong operator topology. So
by Lemma 2.2 we have $tr(\sum\limits_{\alpha\in
F}A_{\alpha}BA_{\alpha}^{*})\longrightarrow tr(\Phi_{\Gamma}(B))$.
Since $\Phi_{\Gamma}$ is trace preserving, $\\{\sum\limits_{\alpha\in
F}B^{\frac{1}{2}}A_{\alpha}^{*}A_{\alpha}B^{\frac{1}{2}}\\}_{F}$ is a
nondecreasing net of positive operators converging to $B$ in the strong
operator topology. So by Lemma 2.2 we have $tr(\sum\limits_{\alpha\in
F}B^{\frac{1}{2}}A_{\alpha}^{*}A_{\alpha}B^{\frac{1}{2}})\longrightarrow
tr(B)$. But $tr(\sum\limits_{\alpha\in
F}A_{\alpha}BA_{\alpha}^{*})=tr(\sum\limits_{\alpha\in
F}B^{\frac{1}{2}}A_{\alpha}^{*}A_{\alpha}B^{\frac{1}{2}})$ for every $F$, so
we conclude that $tr(\Phi_{\Gamma}(B))=tr(B)$. By linearity, the result for
arbitrary $B\in T(H)$ now follows.
The next Lemma 2.3 is from [4], it is presumed in [4] that all linear maps on
$C^{*}$-algebras preserve the identity, we modify the proof slightly such that
it suit for our need.
Lemma 2.3. If $\Re_{1}$, $\Re_{2}$ are $C^{*}$-algebras,
$\phi:\Re_{1}\longrightarrow\Re_{2}$ is a 2-positive linear map,
$\|\phi(I)\|\leq 1$, then $\phi(C^{*}C)\geq\phi(C)^{*}\phi(C)$ for every
$C\in\Re_{1}$.
Proof. Let $T=\left(\begin{array}[]{cc}0&C^{*}\\\ C&0\\\ \end{array}\right)\in
M_{2}(\Re_{1})=\Re_{1}\otimes M_{2}$, here $M_{2}$ denote the $C^{*}$-algebra
of $2\times 2$ complex matrices. Then $T=T^{*}$.
Since $\phi\otimes 1_{2}:M_{2}(\Re_{1})\longrightarrow M_{2}(\Re_{2})$ is a
positive linear map and $\|\phi\otimes 1_{2}\|\leq 1$, by [5] Theorem 1 we
have $(\phi\otimes 1_{2})(T^{2})\geq((\phi\otimes 1_{2})(T))^{2}$.
While $T^{2}=\left(\begin{array}[]{cc}C^{*}C&0\\\ 0&CC^{*}\\\
\end{array}\right)$, $(\phi\otimes
1_{2})(T^{2})=\left(\begin{array}[]{cc}\phi(C^{*}C)&0\\\ 0&\phi(CC^{*})\\\
\end{array}\right)$,
$(\phi\otimes 1_{2})(T)=\left(\begin{array}[]{cc}0&\phi(C^{*})\\\ \phi(C)&0\\\
\end{array}\right)$, $((\phi\otimes
1_{2})(T))^{2}=\left(\begin{array}[]{cc}\phi(C)^{*}\phi(C)&0\\\
0&\phi(C)\phi(C)^{*}\\\ \end{array}\right)$.
Thus $\phi(C^{*}C)\geq\phi(C)^{*}\phi(C)$.
It is easy to see that a generalized quantum operation is completely positive
and satisfies the conditions in Lemma 2.3.
An operator $W\in T(H)$ is faithful if for any $A\in B(H)_{+}$,
$tr(W^{*}AW)=0$ implies $A=0$.
Theorem 2.6. Let $\Phi_{\Gamma}$ be a trace nonincreasing generalized quantum
operation. We have
(1). $B(H)^{\Phi_{\Gamma}}\cap T(H)\subseteq{\Gamma}^{\prime}\cap T(H)$;
(2). If $dim(H)<\infty$, then
$B(H)^{\Phi_{\Gamma}}\subseteq{\Gamma}^{\prime}$;
(3). If there exists a faithful operator $W\in T(H)\cap{\Gamma}^{\prime}$,
then $B(H)^{\Phi_{\Gamma}}\subseteq{\Gamma}^{\prime}$.
Proof. (1). Suppose $B\in B(H)^{\Phi_{\Gamma}}\cap T(H)$. Thus $B^{*}B\in
T(H)_{+}$. By Lemma 2.3 we have
$\Phi_{\Gamma}(B^{*}B)\geq\Phi_{\Gamma}(B)^{*}\Phi_{\Gamma}(B)=B^{*}B$. By
Theorem 2.4 we have $\Phi_{\Gamma}(B^{*}B)\in T(H)_{+}$ and
$tr(\Phi_{\Gamma}(B^{*}B))=tr(B^{*}B)$. That is,
$tr(\Phi_{\Gamma}(B^{*}B)-B^{*}B)=0$. So $\Phi_{\Gamma}(B^{*}B)=B^{*}B$. We
conclude that $B^{*}B\in B(H)^{\Phi_{\Gamma}}$. Since $B(H)^{\Phi_{\Gamma}}$
is closed under the involution $*$, we also have $B^{*}\in
B(H)^{\Phi_{\Gamma}}\cap T(H)$. Similarly we have $BB^{*}\in
B(H)^{\Phi_{\Gamma}}$. By Theorem 2.1, We conclude that
$B\in{\Gamma}^{\prime}$. That is, $B(H)^{\Phi_{\Gamma}}\cap
T(H)\subseteq{\Gamma}^{\prime}\cap T(H)$.
(2) follows from (1) immediately.
(3). Suppose $B\in B(H)^{\Phi_{\Gamma}}$. By Lemma 2.3 we have
$\Phi_{\Gamma}(B^{*}B)\geq\Phi_{\Gamma}(B)^{*}\Phi_{\Gamma}(B)=B^{*}B$. Thus
By Theorem 2.4 we have
$0\leq tr(W^{*}(\Phi_{\Gamma}(B^{*}B)-B^{*}B)W)$
$=tr(W^{*}\Phi_{\Gamma}(B^{*}B)W)-tr(W^{*}B^{*}BW)$
$=tr(\Phi_{\Gamma}(W^{*}B^{*}BW))-tr(W^{*}B^{*}BW)\leq 0.$
So $tr(W^{*}(\Phi_{\Gamma}(B^{*}B)-B^{*}B)W)=0$. Since $W$ is faithful, we
conclude that $\Phi_{\Gamma}(B^{*}B)=B^{*}B$. That is, $B^{*}B\in
B(H)^{\Phi_{\Gamma}}$. Since $B(H)^{\Phi_{\Gamma}}$ is closed under the
involution $*$, we also have $B^{*}\in B(H)^{\Phi_{\Gamma}}$. Similarly we
have $BB^{*}\in B(H)^{\Phi_{\Gamma}}$. By Theorem 2.1, we conclude that
$B\in{\Gamma}^{\prime}$. That is,
$B(H)^{\Phi_{\Gamma}}\subseteq{\Gamma}^{\prime}$.
The next theorem is a direct corollary of Theorem 2.6 (2), but we give a
simple elementary proof instead.
Theorem 2.7. Let $\Phi_{\Gamma}$ be a generalized quantum operation,
$\Gamma=\\{A_{\alpha},A_{\alpha}^{*}\\}_{\alpha\in\Lambda}$ is commutative and
$dim(H)<\infty$, then $B(H)^{\Phi_{\Gamma}}\subseteq{\Gamma}^{\prime}$.
Proof. By Theorem 2.5.5 in [6], $\\{A_{\alpha}\\}_{\alpha\in\Lambda}$ can be
diagonalized simultaneously. That is, there exists a set of pairwise
orthogonal nonzero projections $\\{P_{k}\\}_{k}$ such that
$\sum\limits_{k}P_{k}=I$, $A_{\alpha}=\sum\limits_{k}\lambda_{k,\alpha}P_{k}$.
We also can suppose that if $k_{1}\neq k_{2}$, then there exists some $\alpha$
such that $\lambda_{k_{1},\alpha}\neq\lambda_{k_{2},\alpha}$. In fact, if not,
we can combine $P_{k_{1}}$ and $P_{k_{2}}$ into one projection.
Since $\sum\limits_{\alpha}A_{\alpha}A_{\alpha}^{*}\leq I$, we have
$\sum\limits_{\alpha}|\lambda_{k,\alpha}|^{2}\leq 1$ for every $k$. Let
$\xi_{k}=\\{\lambda_{k,\alpha}\\}_{\alpha\in\Lambda}\in l^{2}(\Lambda)$, then
$\|\xi_{k}\|\leq 1$ for every $k$. Thus if
$\langle\xi_{k_{1}},\xi_{k_{2}}\rangle=1$, then by Schwarz inequility we have
$\xi_{k_{1}}=\xi_{k_{2}}$. So by the assumption above, we conclude that
$k_{1}=k_{2}$.
Now we suppose $B\in B(H)^{\Phi_{\Gamma}}$. Then
$B=\sum\limits_{\alpha}A_{\alpha}BA_{\alpha}^{*}$. So
$P_{k}BP_{l}=(\sum\limits_{\alpha}\lambda_{k,\alpha}\overline{\lambda_{l,\alpha}})P_{k}BP_{l}=\langle\xi_{k},\xi_{l}\rangle
P_{k}BP_{l}$ for every $k,l$. Thus we have $P_{k}BP_{l}=0$ for $k\neq l$. So
$B=\sum\limits_{k}P_{k}BP_{k}$. We conclude that $BP_{k}=P_{k}B$ and
$B\in{\Gamma}^{\prime}$. That is,
$B(H)^{\Phi_{\Gamma}}\subseteq{\Gamma}^{\prime}$.
3\. Almost sharp quantum effects
Firstly, let ${\cal E}(H)$ be the set of self-adjoint operators on $H$
satisfying that $0\leq A\leq I$. For $A\in B(H)$, denote $Ker(A)=\\{x\in H\mid
Ax=0\\}$ and $Ran(A)=\\{Ax\mid x\in H\\}$. If $A,B\in{\cal E}(H)$, we call
$A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}$ the sequential product of $A$ and
$B$ (see [7-10]).
Lemma 3.1 ([7-8]). If $A,B\in{\cal E}(H)$, $A\circ B\in P(H)$, then $AB=BA$.
We generalize Corollary 3 in [3] as the following Theorem 3.1.
Theorem 3.1. Suppose $P\in P(H)$, $A\in{\cal E}(H)$, $P\ or\ A\in T(H)$, then
the following conditions are all equivalent:
(1) $P\circ A\in P(H)$;
(2) $tr(PA)=tr(PAPA)$;
(3) $PA\in P(H)$;
(4) $PA$ is idempotent.
Proof. (1)$\Rightarrow$(3). By Lemma 3.1 we have $PA=AP$. Thus $PA=PAP=P\circ
A\in P(H)$.
(3)$\Rightarrow$(4)$\Rightarrow$(2) is obvious.
(2)$\Rightarrow$(1). Since $P\circ A\in T(H)$, we have $(P\circ A)^{2}\in
T(H)$.
$tr(P\circ A)=tr(PAP)=tr(PA)=tr(PAPA)=tr(PAPAP)=tr((PAP)^{2})=tr((P\circ
A)^{2})$.
Since $0\leq P\circ A\leq I$, we have $(P\circ A)^{2}\leq P\circ A$. It
follows from $tr(P\circ A-(P\circ A)^{2})=0$ that $P\circ A=(P\circ A)^{2}$.
So $P\circ A\in P(H)$.
Let $M$ be a von Neumann algebra on $H$. The set of effects in $M$ is ${\cal
E}(M)=\\{A\in M\mid 0\leq A\leq I\\}$. The set of projections or sharp effects
in $M$ is $P(M)=\\{P\in M\mid P=P^{*}=P^{2}\\}$. We denote the usual Murray-
von Neumann relations on $P(M)$ by $\preceq$, $\succeq$ and $\sim$.
For $A\in{\cal E}(M)$, defining the negation of $A$ by $A^{\prime}=I-A$. if
$A=PQP$ for some $P,Q\in P(M)$, we say $A$ is an almost sharp element in $M$.
We say that $A$ is nearly sharp if both $A$ and $A^{\prime}$ are almost sharp
([3]).
We denote the set of almost sharp elements in $M$ by $M_{as}$.
For $A\in{\cal E}(M)$, we denote the projection onto $\overline{Ran(A)}$ and
$Ker(A)$ by $P_{A}$ and $N_{A}$ respectively. It is easy to know that
$P_{A}+N_{A}=I$.
Note that if $A\in\varepsilon(M)$ has the form $A=PQP$ for some $P,Q\in P(M)$,
then $P_{A}\leq P$, thus we also have that $A=P_{A}QP_{A}$ ([3]).
Lemma 3.2 ([3]). Let $A\in{\cal E}(M)$. Then
(1). $A$ is almost sharp iff $P_{AA^{\prime}}\preceq N_{A}$;
(2). $A$ is nearly sharp iff $P_{AA^{\prime}}\preceq N_{A}$ and
$P_{AA^{\prime}}\preceq N_{A^{\prime}}$;
(3). $P_{AA^{\prime}}=P_{A}-N_{A^{\prime}}=I-N_{A}-N_{A^{\prime}}$.
Now, we generalize Theorem 10 in [3] as the following Theorem 3.2 and Theorem
3.3:
Theorem 3.2. Suppose $P\in P(M)$, then the following conditions are all
equivalent:
(1). $P\preceq P^{\prime}$;
(2). $[0,P]\subseteq M_{as}$.
Proof. (1)$\Rightarrow$(2). Suppose $0\leq A\leq P$. Then $P_{A}\leq P$,
$N_{A}\geq P^{\prime}$. Thus $P_{AA^{\prime}}\leq P_{A}\leq P\preceq
P^{\prime}\leq N_{A}$. That is, $P_{AA^{\prime}}\preceq N_{A}$. So by Lemma
3.2 we have $A\in M_{as}$.
(2)$\Rightarrow$(1). Let $A=\frac{1}{2}P$, then $A\in[0,P]\subseteq M_{as}$.
So by Lemma 3.2 we have $P_{AA^{\prime}}\preceq N_{A}$.
It is easy to see that $P_{A}=P$, $N_{A}=P^{\prime}$, $N_{A^{\prime}}=0$. By
Lemma 3.2 we have $P_{AA^{\prime}}=P_{A}-N_{A^{\prime}}=P$. Thus
$P=P_{AA^{\prime}}\preceq N_{A}=P^{\prime}$.
Theorem 3.3. Suppose $P\in P(M)$, then the following conditions are all
equivalent:
(1). $P\sim P^{\prime}$;
(2). $[0,P]\cup[0,P^{\prime}]\subseteq M_{as}$;
(3). If $A\in{\cal E}(M)$, $AP=PA$, then $A=P_{1}Q_{1}P_{1}+P_{2}Q_{2}P_{2}$
with $P_{i},Q_{i}\in P(M)$ and $P_{1}\leq P$, $P_{2}\leq P^{\prime}$.
Proof. (1)$\Longleftrightarrow$(2). By Theorem 3.2.
(2)$\Rightarrow$(3). Suppose $A\in{\cal E}(M)$, $AP=PA$. Then
$A=PAP+P^{\prime}AP^{\prime}$. Since $PAP\in[0,P]$ and
$P^{\prime}AP^{\prime}\in[0,P^{\prime}]$, we have
$PAP,P^{\prime}AP^{\prime}\in M_{as}$. Thus, we can prove the result easily.
(3)$\Rightarrow$(2). Suppose $0\leq A\leq P$. It is easy to see that
$AP=PA=A$. Thus $A=P_{1}Q_{1}P_{1}+P_{2}Q_{2}P_{2}$ with $P_{i},Q_{i}\in P(M)$
and $P_{1}\leq P$, $P_{2}\leq P^{\prime}$. So $A=PAP=P_{1}Q_{1}P_{1}$. That
is, $A\in M_{as}$. We conclude that $[0,P]\subseteq M_{as}$. Similarly
$[0,P^{\prime}]\subseteq M_{as}$.
Let ${\cal B}[0,1]$ be the set of bounded Borel functions on interval $[0,1]$.
Suppose $A\in{\cal E}(M)$, $h\in{\cal B}[0,1]$, $0\leq h\leq 1$, then
$h(A)\in{\cal E}(M)$.
Theorem 3.4. Suppose $A\in{\cal E}(M)$, $h\in{\cal B}[0,1]$, $0\leq h\leq 1$,
$h(0)=0$, $h(1)=1$. We have
(1). $N_{A}\leq N_{h(A)}$, $N_{A^{\prime}}\leq N_{h(A)^{\prime}}$,
$P_{h(A)h(A)^{\prime}}\leq P_{AA^{\prime}}$;
(2). If $A$ is almost sharp, then $h(A)$ is almost sharp;
(3). If $A$ is nearly sharp, then $h(A)$ is nearly sharp.
Proof. (1). If $Ax=0$, then $h(A)(x)=h(0)x=0$. Thus $Ker(A)\subseteq
Ker(h(A))$. That is, $N_{A}\leq N_{h(A)}$.
If $Ax=x$, then $h(A)(x)=h(1)x=x$. Thus $Ker(I-A)\subset Ker(I-h(A))$. That
is, $N_{A^{\prime}}\leq N_{h(A)^{\prime}}$. Thus by Lemma 3.2 we have
$P_{AA^{\prime}}=I-N_{A}-N_{A^{\prime}}\geq
I-N_{h(A)}-N_{h(A)^{\prime}}=P_{h(A)h(A)^{\prime}}$.
(2). If $A$ is almost sharp, by Lemma 3.2 we have $P_{AA^{\prime}}\preceq
N_{A}$. From (1) we have $P_{h(A)h(A)^{\prime}}\leq P_{AA^{\prime}}\preceq
N_{A}\leq N_{h(A)}$. That is, $P_{h(A)h(A)^{\prime}}\preceq N_{h(A)}$. Thus by
Lemma 3.2 again $h(A)$ is almost sharp.
(3). If $A$ is nearly sharp, by Lemma 3.2 we have $P_{AA^{\prime}}\preceq
N_{A}$ and $P_{AA^{\prime}}\preceq N_{A^{\prime}}$. From (1) we have
$P_{h(A)h(A)^{\prime}}\leq P_{AA^{\prime}}\preceq N_{A}\leq N_{h(A)}$ and
$P_{h(A)h(A)^{\prime}}\leq P_{AA^{\prime}}\preceq N_{A^{\prime}}\leq
N_{h(A)^{\prime}}$. That is, $P_{h(A)h(A)^{\prime}}\preceq N_{h(A)}$ and
$P_{h(A)h(A)^{\prime}}\preceq N_{h(A)^{\prime}}$. Thus by Lemma 3.2 again
$h(A)$ is nearly sharp.
Let $C[0,1]$ be the set of continuous functions on interval $[0,1]$. Suppose
$h\in C[0,1]$, we say $h$ satisfy kernel condition if the following three
conditions hold:
(1). $0\leq h\leq 1$;
(2). $h(0)=0$, $h(1)=1$;
(3). $h$ is strictly monotonous.
Suppose $A\in{\cal E}(M)$, $h\in C[0,1]$ satisfies kernel condition, then it
is easy to see that $h(A)\in{\cal E}(M)$, $h^{-1}\in C[0,1]$ also satisfies
kernel condition and $A=h^{-1}(h(A))$.
Theorem 3.5. Suppose $A\in{\cal E}(M)$, $h\in C[0,1]$ satisfy kernel
condition. We have
(1). $N_{A}=N_{h(A)}$, $N_{A^{\prime}}=N_{h(A)^{\prime}}$,
$P_{AA^{\prime}}=P_{h(A)h(A)^{\prime}}$;
(2). $A$ is almost sharp if and only if $h(A)$ is almost sharp;
(3). $A$ is nearly sharp if and only if $h(A)$ is nearly sharp.
Proof. (1). By Theorem 3.4, we have $N_{A}\leq N_{h(A)}$, $N_{A^{\prime}}\leq
N_{h(A)^{\prime}}$, $P_{h(A)h(A)^{\prime}}\leq P_{AA^{\prime}}$. Since
$h(A)\in\varepsilon(M)$, $h^{-1}\in C[0,1]$ satisfy kernel condition, and
$A=h^{-1}(h(A))$, by Theorem 3.4 again, we have $N_{A}\geq N_{h(A)}$,
$N_{A^{\prime}}\geq N_{h(A)^{\prime}}$, $P_{h(A)h(A)^{\prime}}\geq
P_{AA^{\prime}}$. Thus the conclusion follows.
(2) and (3) follow from Lemma 3.2 and (1) immediately.
Corollary 3.1. Suppose $A\in{\cal E}(M)$, $t$ is a positive number. Then
(1). $A$ is almost sharp if and only if $A^{t}$ is almost sharp.
(2). $A$ is nearly sharp if and only if $A^{t}$ is nearly sharp.
References
[1]. Nielsen, M. and Chuang, J. Quantum computation and quantum information,
Cambridge University Press, 2000
[2]. A. Arias, A. Gheondea, S. Gudder. Fixed points of quantum operations. J.
Math. Phys. 43(2002), 5872.
[3]. A. Arias, S. Gudder. Almost sharp quantum effects. J. Math. Phys.
45(2004), 4196.
[4]. M. D. Choi. A Schwarz inequality for positive linear maps on
$C^{*}$-algebras. Illinois J. Math. 18(1974), 565.
[5]. R. V. Kadison. A generalized Schwarz inequality and algebraic invariants
for operator algebras. Ann. of Math. 56(1952), 494.
[6]. R. Horn, C. Johnson. Matrix Analysis. Cambridge University Press,
Cambridge, 1990.
[7]. S. Gudder, G. Nagy. Sequential quantum measurements. J. Math. Phys.
42(2001), 5212.
[8]. S. Gudder, R. Greechie. Sequential products on effect algebras. Rep.
Math. Phys. 49(2002), 87.
[9]. Weihua Liu, Junde Wu. A uniqueness problem of the sequence product on
operator effect algebra ${\cal E}(H)$. J. Phys. A: Math. Theor. 42 (2009),
185206.
[10]. Jun Shen, Junde Wu. Sequential product on standard effect algebra ${\cal
E}(H)$. J. Phys. A: Math. Theor. 44 (2009).
Appendix: Collected papers of Shen Jun
[1]. Jun Shen, Junde Wu. Not each sequential effect algebra is sharply
dominating. Physics Letters A. 373 (2009), 1708-1712.
[2]. Jun Shen, Junde Wu. Sequential product on standard effect algebra ${\cal
E}(H)$. J. Phys. A: Math. Theor. 44 (2009).
[3]. Jun Shen, Junde Wu. Remarks on the sequential effect algebras. Reports on
Math. Phys. 63 (2009), 441-446.
[4]. Jun Shen, Junde Wu. The average value inequality in sequential effect
algebras. Acta Math. Sinica, English Series. Accepted for publishing.
[5]. Jun Shen, Junde Wu. The n-th root of sequential effect algebras.
Submitted.
[6]. Jun Shen, Junde Wu. Spectral representation of infimum of bounded quantum
observables. Submitted.
[7]. Jun Shen, Junde Wu. Generalized quantum operations and almost sharp
quantum effects. Submitted.
|
arxiv-papers
| 2009-07-12T04:03:41 |
2024-09-04T02:49:03.869785
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shen Jun, Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0907.2003"
}
|
0907.2085
|
url]http://power.itp.ac.cn/ suncp/
# One Hair Postulate for Hawking Radiation as Tunneling Process
H. Dong Qing-yu Cai X.F. Liu C. P. Sun [email protected] [ Institute of
Theoretical Physics, Chinese Academy of Science, Beijing 100190, China State
Key Laboratory of Magnetic Resonances and Atomic and Molecular Physics, Wuhan
Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan
430071, China Department of Mathematics, Peking University, Beijing 100871,
China
###### Abstract
For Hawking radiation, treated as a tunneling process, the no-hair theorem of
black hole together with the law of energy conservation is utilized to
postulate that the tunneling rate only depends on the external qualities ( e.g
., the mass for the Schwarzschild black hole ) and the energy of the radiated
particle. This postulate is justified by the WKB approximation for calculating
the tunneling probability. Based on this postulate, a general formula for the
tunneling probability is derived without referring to the concrete form of
black hole metric. This formula implies an intrinsic correlation between the
successive processes of the black hole radiation of two or more particles. It
also suggests a kind of entropy conservation and thus resolves the puzzle of
black hole information loss in some sense.
###### keywords:
tunneling formulism, modified probability, correlation, entropy conservation,
black hole
††journal: Physics Letters B
## 1 Introduction
Hawking discovered that the black hole radiation possesses an exactly thermal
spectrum of temperature depending on the surface gravity of the black hole
[1]. Particularly, the radiation does not depend on the details of the
structure of the object that collapsed to form the black hole. Thus, an
initially pure quantum state will evolve into a mixed thermal state as the
black hole radiates. This phenomenon, known as the paradox of black hole
information loss, obviously violates the quantum unitarity for the closed
system.
Since its appearing, many attempts [2] have been made to resolve this paradox.
In the previous investigations, the radiation is always treated as possessing
the thermal spectrum and the space-time geometry is fixed. Recently, based on
the WKB approximation, the tunneling probability for the Hawking radiation was
derived in the framework of dynamical geometry. It turns out surprisingly that
the radiation spectrum is not exactly thermal [3]. For this reason, it is
found in Ref. [4] that the successively radiated two particles are correlated,
and thus no information is lost in the radiation [4]. Actually, by using the
same approach as that in Ref. [3], the Hawking radiation spectra of various
black holes have been obtained [5, 6, 7, 8, 9, 10] . These results verify the
correlation between the successive radiations and the conservation of the
information in the radiation [11, 12]. We find that the chain rule for the
probability is essential for the information conservation in the black hole
radiation, and we verify case by case that the chain rule indeed holds for
various Hawking radiations coincidentally.
We observe that the above mentioned coincidence can be exactly explained by
the No-hair theorem of black hole together with the law of energy
conservation. In fact, from our “One hair” postulate based on the No-hair
theorem and the law of energy conservation, we are able to derive a general
form of the tunneling probability of Hawking radiation without resorting to
the details of the black hole, such as its geometric structure. We are thus
able to prove that for the tunneling probability obtained from the WKB
approximation, the chain rule is satisfied automatically and the above
mentioned coincidence is of physical necessity. It should be clear that our
results demonstrate the advantage of treating the black hole radiation as a
tunneling process.
This letter is organized as follows. In Sec. 2, Our postulate is stated based
on the No-hair Theorem. In Sec. 3, a general formula for the tunneling
probability is derived from the postulate. In Sec. 4, the tunneling rate for
the Schwarzschild black hole is obtained without referring to its geometry. In
Sec. 5, the case by case verification of our postulate is given for various
black hole radiations.
## 2 “One hair” for Hawking radiation as tunneling
It is well known that all black hole solutions of the Einstein-Maxwell
equations of gravitation and electromagnetism in general relativity can be
completely characterized by only three externally observable classical
parameters: mass, electric charge, and angular momentum. This result is
referred to as No-hair theorem of steady black hole. For our purpose, we
generalize this theorem for the dynamic black hole as follows: the tunneling
probability for the Hawking radiation only depends on the final state of the
steady black hole and the total energy $E_{T}=E_{1}+E_{2}+...+E_{N}$ after
simultaneously radiating N particles with the energies $E_{1},E_{2},...E_{N}$.
Here, there is only “one hair” quantity $E_{T}$ and the tunneling probability
has nothing to do with its partition.
To investigate the above “One-hair ” postulate, let us consider the two
processes in the Hawking radiation, illustrated in Fig. 1:
* 1.
The black hole radiates a single particle with the energy $E_{\mathrm{T}}$, as
illustrated in Fig. 1(a). The mass of the black hole reduces to
$M-E_{\mathrm{T}}$. The tunneling probability is defined as
$p\left(\left\\{E_{\mathrm{T}}\right\\};M\right)$. The black hole can also
simultaneously radiate two particles with the energies $E_{1}$ and $\ E_{2}$
respectively. The probability of this process is denoted by
$p\left(\left\\{E_{1},E_{2}\right\\};M\right).$ Based on the No-hair Theorem
of black hole and the law of energy conservation, we postulate the One-hair
Theorem for black hole radiation: if $E_{\mathrm{T}}=E_{1}+E_{2},$then
$p\left(\left\\{E_{1},E_{2}\right\\};M\right)=p\left(\left\\{E_{T}\right\\};M\right).$
(1)
Actually, we can imagine that after the Hawking radiation the radiated
particle immediately splits into two particles with the energy $E_{x}$ and
$E_{\mathrm{T}}-E_{x}$ respectively, and in the split no particular energy
partition between the two particles is preferred. The One-hair Theorem simply
means that all the splits satisfying the law of energy conservation possess
the same tunneling probability .
* 2.
The black hole firstly radiates a particle with the energy $E_{1}$ and then
radiates another particle with the energy $E_{2}=E_{\mathrm{T}}-E_{1}$, as
illustrated in Fig. 1(b). The mass of the black hole also reduces to
$M-E_{\mathrm{T}}$. The tunneling probability for this process is
$p\left(\left\\{E_{1}:E_{2}\right\\};M\right)=p\left(\left\\{E_{1}\right\\};M\right)p\left(\left\\{E_{2}\right\\};M-E_{1}\right)$
where the conditional probability
$p\left(\left\\{E_{2}\right\\};M-E_{1}\right)$ reflects the fact that the the
mass of the black hole reduces to $M-E_{\mathrm{1}}$ after it radiats the
particle of energy $E_{\mathrm{1}}$.
We remark here that, the first radiated particle is correlated to the second
one, since the conditional tunneling probability of the second one actually
depends on the energy $E_{1}$ of the first one. Most recently, this
correlation is employed to account for the information loss in the black hole
radiation process [4, 11, 12].
In the following we only consider the steady state of the black hole. It will
take a longer time to reach the steady state than the relaxation time of each
radiation. In this case, the One-hair Theorem for black hole radiation can be
re-expressed as
$p\left(\left\\{E_{1},E_{2}\right\\};M\right)=p\left(\left\\{E_{1}:E_{2}\right\\};M\right)$
or
$p\left(\left\\{E_{1},E_{2}\right\\};M\right)=p\left(\left\\{E_{1}\right\\};M\right)p\left(\left\\{E_{2}\right\\};M-E_{1}\right).$
(2)
Here, as only the steady solutions of the black hole radiation are concerned,
we have identified the two processes of simultaneously and successively
radiating two particles. For the multi-particle case, we can recover the chain
rule as
$p\left(\left\\{E_{1}:E_{2}:...:E_{N}\right\\};M\right)=\prod_{p}p\left(E_{p};M-\sum_{j=1}^{p-1}E_{j}\right).$
based on this two-particle case. Thus, to verify the chain rule for various
Hawking radiation, we need only to prove the postulation in Eq. (2).
To justify the above observation, let us briefly review some results derived
from the dynamic calculation based on the generalized WKB approximation. In
reference [3], the tunneling probability for a particle out of the black hole
is defined as
$p\thicksim e^{-2\mathrm{Im}S},$ (3)
where $S$ is the action for an $s$-wave outgoing positive particle. The exact
form of the imaginary part of the action reads
$\mathrm{Im}S=\mathrm{Im}\intop_{M}^{M-E}\intop_{r_{\mathrm{in}}}^{r_{\mathrm{out}}}\frac{dr}{\dot{r}}dH.$
(4)
Here, the Hamiltonian $H$ is defined through the radial null geodesics
equation, and particularly $H=M-E^{\prime}$ for the Schwarzschild black hole.
It is easily seen that $\mathrm{Im}S$ naturally satisfies the above stated
postulate. Then it can be concluded that the conservation of information will
not be broken if Hawking radiation is treated as tunneling process, as has
been proved in many references [4, 11, 12].
Figure 1: Radiation. (a) The black hole radiates a particle with energy
$E_{T}$. (b) The black hole radiates firstly a particle with energy $E_{1}$
and successively another particle with energy $E_{2}$.
## 3 Energy Dependence of Non Thermal Hawking Radiation
In this section, we will present a derivation of the general form of the
tunneling probability based only on the ”One hair ” postulate. Without losing
the generality, we assume
$p\left(\left\\{E\right\\};M\right)=\exp\left[f\left(\left\\{E\right\\};M\right)\right],$
where $f\left(\left\\{E\right\\};M\right)$ is actually the tunneling entropy
for the black hole radiation. It then follows from equation Eq. 2 that
$f\left(\left\\{E_{T}\right\\};M\right)=f\left(\left\\{E_{1}\right\\};M\right)+f\left(\left\\{E_{2}\right\\};M-E_{1}\right).$
(5)
Substituting the Taylor expansion form
$f\left(\left\\{\omega\right\\};M\right)=\sum_{n=0}A_{n}\left(M\right)\omega^{n}$
of the function $f$ into this equation and comparing the coefficients of the
terms with the same orders of $E_{2}$, we obtain the following system of
recursive equations
$\displaystyle 0$ $\displaystyle=$ $\displaystyle A_{0}\left(M-E_{1}\right),$
$\displaystyle\sum_{n=1}A_{n}\left(M\right)C_{n}^{1}E_{1}^{n-1}$
$\displaystyle=$ $\displaystyle A_{1}\left(M-E_{1}\right),$
$\displaystyle\sum_{n=2}A_{n}\left(M\right)C_{n}^{2}E_{1}^{n-2}$
$\displaystyle=$ $\displaystyle A_{2}\left(M-E_{1}\right),$
$\displaystyle\vdots$
$\displaystyle\sum_{n=m}A_{n}\left(M\right)C_{n}^{m}E_{1}^{n-m}$
$\displaystyle=$ $\displaystyle A_{m}\left(M-E_{1},\right)$
$\displaystyle\sum_{n=m+1}A_{n}\left(M\right)C_{n}^{m+1}E_{1}^{n-\left(m+1\right)}$
$\displaystyle=$ $\displaystyle A_{m+1}\left(M-E_{1}\right),$
$\displaystyle\vdots$ .
Differentiating the left hand right hand sides of the above equations with
respect to $E_{1}$ then results in the equation
$\displaystyle(m+1)A_{m+1}(M-E_{1})$ $\displaystyle=$
$\displaystyle\frac{dA_{m}(M-E_{1})}{dE_{1}}$ $\displaystyle=$
$\displaystyle-\frac{dA_{m}(M-E_{1})}{dM}$
for each $m$. Thus we have the recursion formula
$\displaystyle A_{m}\left(M\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{m}\frac{d}{dM}A_{m-1}\left(M\right)$ $\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{m-1}}{m!}\frac{d^{m-1}}{dM^{m-1}}A_{1}\left(M\right).$
and the black hole entropy can be rewritten as
$f\left(\left\\{E\right\\};M\right)=\sum_{m=1}\frac{\left(-1\right)^{m-1}}{m!}\frac{d^{m-1}}{dM^{m-1}}A_{1}\left(M\right)E^{m}.$
(6)
Define the entropy $G\left(M\right)$ for the black hole radiation through
$A_{1}\left(M\right)=-\frac{dG\left(M\right)}{dM},$
the black hole entropy then reads
$f\left(\left\\{E\right\\};M\right)=G\left(M-E\right)-G\left(M\right).$ (7)
This is the main result of this paper. Obviously, $G\left(M\right)$ in Eq. (
7) is a conservation quantity. According to the above result, after a black
hole of mass $M$ radiates a tunneling particle with energy $E$, its entropy
decrease is
$S\left(E,M\right)=-\ln
p\left(\left\\{E\right\\};M\right)=G\left(M\right)-G\left(M-E\right).$ (8)
In deriving the above result, it is tacitly assumed that the black hole does
not carry charge. For charged black hole a similar result can easily be
obtained by the above method. In fact, when a charged black hole with charge
$Q$ radiates a particle with charge $q$, the tunneling probability can be
derived as
$S\left(E,q;M,Q\right)=G\left(M,Q\right)-G\left(M-E,Q-q\right).$ (9)
## 4 Tunneling Probability for Schwarzschild black hole
In this section, we will derive the tunneling probability for the Hawking
radiation of the Schwarzschild black hole without referring to its dynamic
geometry.
We assume that the entropy for black hole radiation is corrected to the second
order of the tunneling energy $E$, namely
$f\left(\left\\{E\right\\};M\right)=A\left(M\right)+B\left(M\right)E+C\left(M\right)E^{2},$
(10)
where $A\left(M\right),B\left(M\right)$ and $C\left(M\right)$ are mass-
dependent functions to be determined. Then equation ( 5) takes the form
$\displaystyle
A\left(M\right)+B\left(M\right)\left(E_{1}+E_{2}\right)+C\left(M\right)\left(E_{1}+E_{2}\right)^{2}$
$\displaystyle=$ $\displaystyle
A\left(M\right)+B\left(M\right)E_{1}+C\left(M\right)E_{1}^{2}$
$\displaystyle+A\left(M-E_{1}\right)+B\left(M-E_{1}\right)E_{2}+C\left(M-E_{1}\right)E_{2}^{2}.$
gives the following equations about $A\left(M\right),B\left(M\right)$ and
$C\left(M\right):$
$\displaystyle A\left(M-E_{1}\right)$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle B\left(M\right)-2C\left(M\right)E_{1}$ $\displaystyle=$
$\displaystyle B\left(M-E_{1}\right),$ $\displaystyle C\left(M\right)$
$\displaystyle=$ $\displaystyle C\left(M-E_{1}\right).$
It then follows that $C\left(M\right)=k$ and $B\left(M\right)=\xi-2kM$, and
the entropy of black hole radiation is obtained as
$f\left(\left\\{E\right\\};M\right)=\left(\xi-2kM\right)E+kE^{2},$ (11)
where $k$ and $\xi$ are constants. If we take $k=4\pi$ and $\xi=0$, then we
recover the well-known result by Parikh and Wilczek:
$f\left(\left\\{E\right\\};M\right)=4\pi\left[\left(M-E\right)^{2}-M^{2}\right].$
(12)
We would like to emphasize again that, in obtaining the above result, we only
make the assumption that the entropy of the black hole is a polynomial of the
radiated energy $E$, and the details of the dynamic geometry do not come into
the derivation. If the entropy is a polynomial of $E$ of degree $1$ , then we
have $f\left(\left\\{E\right\\};M\right)=\xi E$ where $\xi$ is a constant
independent of the mass $M$. Thus, the conventional thermal spectrum
$p^{\prime}\left(E,M\right)=\exp\left(-8\pi EM\right)$ does not satisfy Eq. (
5) about the conditional probability. In that case, $G\left(M\right)=4\pi
M^{2}=A/4$ is the usual entropy for the Schwarzschild black hole, and is
usually called Bekenstein-Hawking entropy of black hole.
According to Ref. [4], the above spectrum function ( 12) indicates that the
two successively radiated particles are actually correlated. Since Hawking
radiation can carry information through this correlation between the radiated
particles, the conservation of total information can be restored by taking
this correlation into account.
## 5 Verification of One-hair Postulate for other black holes
In this section, we will check the radiation spectra of some well known black
holes to see if they satisfy the One-hair postulate expressed by Eq. ( 5).
Reissner-Nordström black hole\- The tunneling probability of a charged
particle with energy $E$ and charge $q$ for the Reissner-Nordström black hole
has been obtained in Ref. [5] as
$p\left(\left\\{E,q\right\\};M,Q\right)=\frac{\exp\left[G_{\mathbf{RN}}\left(M-E,Q-q\right)\right]}{\exp\left[G_{\mathbf{RN}}\left(M,Q\right)\right]},$
(13)
where
$G_{\mathbf{RN}}\left(M,Q\right)=\pi\left(M+\sqrt{M^{2}-Q^{2}}\right)^{2}.$
Clearly, the radiation spectrum for the Reissner-Nordström black hole is not
thermal, and satisfies our One-hair postulate.
Kerr black hole-For the rotating black hole(Kerr black hole), the tunneling
probability is found in Ref. [6] as
$p\left(\left\\{E\right\\},M\right)=\exp\left[G_{\mathbf{K}}\left(M-E\right)-G_{\mathbf{K}}\left(M\right)\right],$
(14)
where
$G_{\mathbf{K}}\left(M\right)=2\pi\left(M^{2}+M\sqrt{M^{2}-a^{2}}\right).$
Obviously, its spectrum structure is in accordance with our One-hair
postulate.
Kerr-Newman black hole\- For the Kerr-Newman black hole, the tunneling
probability for a particle with charge $q$ is obtained in Ref. [6, 7] as
$p\left(\left\\{E,q\right\\};M,Q\right)=\frac{\exp\left[G_{\mathbf{KN}}\left(M-E,Q-q\right)\right],}{\exp\left[G_{\mathbf{KN}}\left(M,Q\right)\right],}$
(15)
where
$G_{\mathbf{KN}}\left(M,Q\right)=\pi\left(M+\sqrt{M^{2}-Q^{2}-a^{2}}\right)^{2}.$
It also satisfies our postulate.
Quantum corrected Hawking radiation-Last, we consider the tunneling with
quantum correction for the Schwarzschild black hole. For the quantum corrected
Hawking radiation, the tunneling probability reads
$\displaystyle p\left(\left\\{E\right\\};M\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{E}{M}\right)^{2\alpha}\exp\left[8\pi
E\left(M-\frac{E}{2}\right)\right]$ (16) $\displaystyle=$
$\displaystyle\exp\left[G\left(M-E\right)-G\left(M\right)\right],$
where
$G\left(M\right)=4\pi M^{2}+2\alpha\ln M.$
This tunneling probability still satisfies our postulate, thus the information
conservation is quite natural. For a detailed discussion about the information
conservation, one can refer to the Refs. [11, 12].
## 6 Summary
In this letter, we suggest the One-hair Postulate to describe Hawking
radiation as tunneling process based on the No-hair theorem and the energy
conservation law. This postulate for tunneling probability naturally leads to
the information conservation for the total system formed by the radiated
particles plus the remnant black hole. Especially, this postulate is used to
determine the tunneling rate by the information (probability) theory method
rather than the dynamic geometry method. Finally, some well known examples are
presented to support the postulate. We expect the viewpoint developed in this
letter will shed light on the parabox of black hole information loss.
## Acknowledgement
We thank Li You and Zhan Xu for useful discussion. The work is supported by
National Natural Science Foundation of China and the National Fundamental
Research Programs of China under Grant No. 10874091 and No. 2006CB921205.
## References
* [1] S.W. Hawking, Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206 (1976)].
* [2] S.W. Hawking, Phys. Rev. D 14, 2460 (1976); Y. Aharonov, A. Casher and S. Nussinov, Phys. Lett. B 191, 51 (1987); L. M. Krauss and F. Wilczek, Phys. Rev. Lett. 62, 1221 (1989); J.Preskill, hep-th/9209058; G. T. Horowitz and J. Maldacena, J. High Energy Phys. 02, 008 (2004); S. W. Hawking, Phys. Rev. D 72, 084013 (2005); S. L. Braunstein and A. K. Pati, Phys. Rev. Lett. 98, 080502 (2007); D. N. Page, Phys. Rev. Lett. 71, 3743 (1993); G. ’t Hooft, Nucl. Phys. B 256, 727 (1985).
* [3] M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85, 5042 (2000).
* [4] B. Zhang, Q. y. Cai, L. You and M. S. Zhan, Phys. Lett. B 675, 98 (2009) arXiv:0903.0893 [hep-th].
* [5] J. Zhang and Z. Zhao, J. High Energy Phys. 10, 055 (2005);
* [6] Q. Q. Jiang, S. Q. Wu, and X. Cai, Phys. Rev. D 73, 064003 (2006);
* [7] J. Zhang and Z. Zhao, Phys. Lett. B 638, 110 (2006);
* [8] M. Arzano, A. J. M. Medved, and E. C. Vagenas, J. High Energy Phys. 0509, 037 (2005). [hep-th/0505266];
* [9] R. Banerjee, B. R. Majhi, and S. Samanta, Phys. Rev. D 77, 124035 (2008);
* [10] K. Nozari and S. H. Mehdipour, Class. Quantum Grav. 25, 175015 (2008).
* [11] Y. Chen and K. Shao, arXiv:0905.0948 [hep-th].
* [12] B. Zhang, Q. y. Cai, L. You and M. S. Zhan, arXiv:0906.5033 [hep-th].
|
arxiv-papers
| 2009-07-13T01:57:37 |
2024-09-04T02:49:03.876377
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H.Dong, Qing-yu Cai, X.F. Liu, C.P.Sun",
"submitter": "H. Dong",
"url": "https://arxiv.org/abs/0907.2085"
}
|
0907.2126
|
# Velocities as a probe of dark sector interactions
Kazuya Koyama, Roy Maartens, Yong-Seon Song Institute of Cosmology &
Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
###### Abstract
Dark energy in General Relativity is typically non-interacting with other
matter. However, it is possible that the dark energy interacts with the dark
matter, and in this case, the dark matter can violate the universality of free
fall (the weak equivalence principle). We show that some forms of the dark
sector interaction do not violate weak equivalence. For those interactions
that do violate weak equivalence, there are no available laboratory
experiments to probe this violation for dark matter. But cosmology provides a
test for violations of the equivalence principle between dark matter and
baryons – via a test for consistency of the observed galaxy velocities with
the Euler equation.
## I Introduction
Dark matter is currently only detected via its gravitational effects, and
there is an unavoidable degeneracy between dark matter and dark energy within
General Relativity. There could be a hidden non-gravitational coupling between
dark matter and dark energy, and thus it is interesting to develop ways of
testing for such an interaction (see Friedman:1991dj ; Gradwohl:1992ue ;
Bean:2008ac for earlier attempts).
One signal of a dark sector interaction could be a violation of the weak
equivalence principle (universality of free fall) by dark matter, under the
non-gravitational drag due to coupled dark energy. Since Galileo shattered the
myth that heavier objects fall faster, the universality of free fall has been
established as a fundamental principle of gravity. Laboratory tests have been
made to show the independence of the acceleration of objects from their masses
and chemical composition. However, these tests apply to baryonic matter, and
no direct probe of dark matter is available.
If the interacting dark sector couples non-gravitationally to baryonic matter,
then existing laboratory tests provide constraints on the dark sector
interaction Bovy:2008gh . Here we assume that there is zero (or negligible)
non-gravitational coupling between the dark sector and standard-model fields.
A difference in the acceleration between dark matter and baryons could show up
in the stellar distribution in tidal trails of satellite galaxies
Kesden:2006vz . This same difference should also show up as an inconsistency
when interpreting the relation between galaxy peculiar velocities and
overdensities, as we explain below.
We assume that gravity on all scales is described by General Relativity. Thus
there is no gravitational mechanism to violate the weak equivalence principle.
Note that this is also true of scalar-tensor theories, since the gravitational
scalar degree of freedom couples equally to all types of matter. Indeed, most
metric theories of modified gravity also respect the weak equivalence
principle (see, e.g., Sotiriou:2007zu ). Various tests have been developed to
discriminate between metric theories of modified gravity, and non-interacting
dark energy models in General Relativity (see, e.g., mg ). But these tests do
not in general apply to dark energy that interacts with dark matter, since a
dark sector interaction can introduce new degeneracies Wei:2008vw . We confine
ourselves to the question of how galaxy peculiar velocities can be used to
detect dark sector interactions within General Relativity.
## II Interacting Dark Energy
We briefly review the necessary background on perturbations of interacting
dark energy models in General Relativity. (For recent work with further
references, see, e.g., Valiviita:2008iv .)
A general dark sector coupling may be described in the background by the
energy balance equations of cold dark matter ($c$) and dark energy ($x$),
$\displaystyle\rho_{c}^{\prime}$ $\displaystyle=$ $\displaystyle-3{\cal
H}\rho_{c}+aQ_{c}\,,$ (1) $\displaystyle\rho_{x}^{\prime}$ $\displaystyle=$
$\displaystyle-3{\cal H}(1+w_{x})\rho_{x}+aQ_{x}\,,~{}~{}Q_{x}=-Q_{c}\,,$ (2)
where $w_{x}=P_{x}/\rho_{x}$, ${\cal H}=d\ln a/d\tau$ and $\tau$ is conformal
time, with $ds^{2}=a^{2}(-d\tau^{2}+d\vec{x}^{\,2}\,)$. Here $Q_{c},Q_{x}$ are
the rates of energy density transfer to dark matter and energy respectively.
In order to avoid stringent “fifth-force” constraints, we assume that baryons
($b$), photons ($\gamma$) and neutrinos ($\nu$) are not coupled to dark energy
and are separately conserved.
In the Newtonian gauge the perturbed metric is given by
$ds^{2}=a^{2}\Big{[}-(1+2\Psi)d\tau^{2}+(1-2\Psi)d\vec{x}\,^{2}\Big{]}\,,$ (3)
where we have neglected anisotropic stress since we are interested in the late
universe. The total (energy-frame) four-velocity is
$u^{\mu}=a^{-1}\Big{(}1-\Psi,\partial^{i}v\Big{)},$ (4)
where the velocity potential $v$ is defined by
$(\rho+P)v=\sum(\rho_{A}+P_{A})v_{A}\,,$ (5)
and $A=c,x,b,\gamma,\nu$. The $A$-fluid four-velocity is
$u^{\mu}_{A}=a^{-1}\Big{(}1-\Psi,\partial^{i}v_{A}\Big{)}.$ (6)
The covariant form of energy-momentum transfer is
$\nabla_{\nu}T^{\mu\nu}_{A}=Q^{\mu}_{A}\,,$ (7)
where $Q^{\mu}_{A}=0$ for $A=b,\gamma,\nu$ in the late universe, while
$Q_{c}^{\mu}=-Q_{x}^{\mu}\neq 0$. The energy-momentum transfer four-vector can
be split relative to the total four-velocity as
$Q_{A}^{\mu}=Q_{A}u^{\mu}+F_{A}^{\mu}\,,~{}~{}Q_{A}=\bar{Q}_{A}+\delta
Q_{A}\,,~{}~{}u_{\mu}F_{A}^{\mu}=0\,,$ (8)
where $Q_{A}$ is the energy density transfer rate and $F_{A}^{\mu}$ is the
momentum density transfer rate, relative to $u^{\mu}$. Then it follows that
$F_{A}^{\mu}=a^{-1}(0,\partial^{i}f_{A})$, where $f_{A}$ is a momentum
transfer potential, and
$\displaystyle Q^{A}_{0}$ $\displaystyle=$
$\displaystyle-a\left[Q_{A}(1+\Psi)+\delta Q_{A}\right],$ (9) $\displaystyle
Q^{A}_{i}$ $\displaystyle=$ $\displaystyle
a\partial_{i}\left(f_{A}+Q_{A}v\right).$ (10)
In the background, the energy-momentum transfer four-vectors have the form
$Q^{\mu}_{c}=a^{-1}(Q_{c},\vec{0}\,)=-Q^{\mu}_{x}\,,$ so that there is no
momentum transfer.
The evolution equations for the dimensionless density perturbation
$\delta_{A}=\delta\rho_{A}/\rho_{A}$ and for the velocity perturbation are:
$\displaystyle\delta_{A}^{\prime}+3{\cal
H}(c_{sA}^{2}-w_{A})\delta_{A}-(1+w_{A})k^{2}v_{A}$
$\displaystyle~{}~{}-3{\cal H}\big{[}3{\cal
H}(1+w_{A})(c_{sA}^{2}-w_{A})+w_{A}^{\prime}\big{]}v_{A}$
$\displaystyle~{}~{}-3(1+w_{A})\Psi^{\prime}={a\over\rho_{A}}\,\delta Q_{A}$
$\displaystyle~{}~{}+{aQ_{A}\over\rho_{A}}\left[\Psi-\delta_{A}-3{\cal
H}(c_{sA}^{2}-w_{A})v_{A}\right]\,,$ (11) $\displaystyle v_{A}^{\prime}+{\cal
H}\big{(}1-3c_{sA}^{2}\big{)}v_{A}+{c_{sA}^{2}\over(1+w_{A})}\,\delta_{A}+\Psi$
$\displaystyle~{}={a\over(1+w_{A})\rho_{A}}\Big{\\{}Q_{A}\big{[}v-(1+c_{sA}^{2})v_{A}\big{]}+f_{A}\Big{\\}}\\!,$
(12)
where $w_{c}=0=c_{sc}^{2}$ and $c_{sx}^{2}=1$.
For our purposes, we are interested in the behaviour of dark matter in the
Newtonian regime on sub-Hubble scales. In this case, the perturbed continuity
and Euler equations reduce to
$\displaystyle\delta_{c}^{\prime}-k^{2}v_{c}$ $\displaystyle=$
$\displaystyle{a\over\rho_{c}}\left(\delta Q_{c}-Q_{c}\delta_{c}\right),$ (13)
$\displaystyle v_{c}^{\prime}+{\cal H}v_{c}+\Psi$ $\displaystyle=$
$\displaystyle{a\over\rho_{c}}\left[Q_{c}(v-v_{c})+f_{c}\right],$ (14)
If the right-hand side of the continuity equation (13) is nonzero, then the
interaction will lead to a bias in the linear regime between dark matter and
baryons Amendola:2001rc , since the baryon overdensities obey
$\delta_{b}^{\prime}-k^{2}v_{b}=0\,.$ (15)
If the right-hand side of the Euler equation (14) is nonzero, then the dark
matter no longer follows geodesics and breaks the weak equivalence principle,
unlike baryons, for which
$v_{b}^{\prime}+{\cal H}v_{b}+\Psi=0\,.$ (16)
In the Newtonian regime, the Poisson equation becomes
$k^{2}\Psi=-4\pi Ga^{2}\left(\rho_{c}\delta_{c}+\rho_{b}\delta_{b}\right)\,.$
(17)
Here we neglect dark energy clustering, assuming that the sound velocity of
dark energy perturbations is $c_{sx}=1$. Dark energy perturbations can be
important on large scales depending on the strength of interactions but they
are not important on sub-horizon scales as long as the sound velocity of dark
energy perturbations is positive – since in that case, the gradient term in
the evolution equation for $\delta_{x}$ [see Eq. (12)] always dominates over
the interaction terms. The evolution equation for $\delta_{c}$ is then given
by
$\displaystyle\delta_{c}^{\prime\prime}+{\cal H}\delta_{c}^{\prime}-4\pi
Ga^{2}\rho_{c}\delta_{c}-{\cal H}\frac{a}{\rho_{c}}(\delta
Q_{c}-Q_{c}\delta_{c})$ $\displaystyle{}-\Big{[}\frac{a}{\rho_{c}}(\delta
Q_{c}-Q_{c}\delta_{c})\Big{]}^{\prime}-\frac{a}{\rho_{c}}k^{2}\Big{[}Q_{c}(v-v_{c})+f_{c}\Big{]}=0.$
## III Different types of interaction
There is no fundamental theory that determines the form of the interaction,
i.e., of $Q_{c}^{\mu}$, so we are forced to use phenomenological models. Here
we consider three types of interaction, each illustrated with a particular
form: interactions that do not change the continuity or Euler equations;
interactions that change only the Euler equation; interactions that change
only the continuity equation. The general case, where both equations are
modified, can be thought of as a linear superposition of the last two cases.
### III.1 Continuity and Euler equations unchanged
A general class of interactions may be defined by requiring that there is no
momentum exchange in dark matter rest frame,
$Q_{c}^{\mu}=Q_{c}u_{c}^{\mu}\,,$ (19)
where $Q_{c}$ remains to be specified. For this class, we find from Eqs. (9)
and (10) that, for any $Q_{c}$, we have $f_{c}=Q_{c}(v_{c}-v)$. Thus Eq. (14)
becomes
$v_{c}^{\prime}+{\cal H}v_{c}+\Psi=0\,.$ (20)
This is the same Euler equation as the non-interacting case, so that the dark
matter velocity is not directly affected by the interaction and there is no
violation of weak equivalence. The dark matter continues to follow geodesics,
and feels no direct drag force from the dark energy.
An example in the form of Eq. (19) is Valiviita:2008iv ; Boehmer:2008av ;
Majerotto:2009np ; Valiviita:2009nu
$Q^{\mu}_{c}=-\Gamma\rho_{c}\,u_{c}^{\mu}\,,$ (21)
where $\Gamma$ is a constant interaction rate. In this case
$Q_{c}=-\Gamma\rho_{c}(1+\delta_{c})$ and Eq. (13) becomes
$\delta_{c}^{\prime}-k^{2}v_{c}=0\,.$ (22)
The continuity equation is therefore the same as in the non-interacting case.
Thus for this form of interaction, there is no violation of the weak
equivalence principle by dark matter, and no bias is induced by the
interaction. In fact, in the Newtonian regime, the only signal of the dark
sector interaction in structure formation to linear order is via the
modification of the background expansion history. The evolution equation (II)
for $\delta_{c}$ becomes
$\delta_{c}^{\prime\prime}+{\cal H}\delta_{c}^{\prime}-4\pi
Ga^{2}(\rho_{c}\delta_{c}+\rho_{b}\delta_{b})=0,$ (23)
which is the same as in the uncoupled case. Thus the only imprint of the dark
sector interaction on $\delta_{c}$ is via the different background evolution
of ${\cal H}$ and $\rho_{c}$.
### III.2 Continuity equation modified
If we keep Eq. (19) but generalize Eq. (21) to CalderaCabral:2008bx ;
CalderaCabral:2009ja
$Q^{\mu}_{c}=-(\Gamma_{c}\rho_{c}+\Gamma_{x}\rho_{x})\,u_{c}^{\mu}\,,$ (24)
then $\delta Q_{c}-Q_{c}\delta_{c}=\Gamma_{x}\rho_{x}(\delta_{c}-\delta_{x})$.
Since dark energy does not cluster on sub-Hubble scales, we can neglect the
$\delta_{x}$ term, and we have
$\delta_{c}^{\prime}-k^{2}v_{c}=a\Gamma_{x}{\rho_{x}\over\rho_{c}}\,\delta_{c}\,.$
(25)
For this interaction, the dark matter continues to follow geodesics by virtue
of Eq. (20), but the continuity equation (25 is modified. As a consequence,
there will be a bias induced by the interaction.
The evolution equation (II) for $\delta_{c}$ becomes
$\displaystyle\delta_{c}^{\prime\prime}+\left({\cal
H}-a\Gamma_{x}\frac{\rho_{x}}{\rho_{c}}\right)\delta_{c}^{\prime}=4\pi
Ga^{2}\rho_{b}\delta_{b}$ $\displaystyle~{}~{}{}+\Big{[}4\pi
Ga^{2}\rho_{c}+2a{\cal
H}\Gamma_{x}\frac{\rho_{x}}{\rho_{c}}+a\Gamma_{x}\Big{(}\frac{\rho_{x}}{\rho_{c}}\Big{)}^{\prime}\Big{]}\delta_{c}\,.$
(26)
(This generalizes CalderaCabral:2009ja , where only the case $\Gamma_{c}=0$ is
considered.)
The modification of the standard evolution for $\delta_{c}$ occurs in 3 ways:
firstly via the modified expansion history in the background ${\cal H}$ and
$\rho_{c}$; secondly by the modified Hubble friction term ${\cal H}\to{\cal
H}[1-a\Gamma_{x}\rho_{x}/{\cal H}\rho_{c}]$; and thirdly by the modified
effective gravitational coupling for dark matter – dark matter particle
interactions,
$G_{\rm eff}=G\Big{[}1+{{\cal H}\rho_{x}\over 2\pi
Ga\rho_{c}^{2}}+{\Gamma_{x}\over 4\pi
Ga\rho_{c}}\Big{(}{\rho_{x}\over\rho_{c}}\Big{)}^{\prime}\Big{]}.$ (27)
### III.3 Euler equation modified
A second general class of interactions has no momentum exchange in the dark
energy frame,
$Q_{c}^{\mu}=Q_{c}u_{x}^{\mu}\,.$ (28)
It follows that $f_{c}=Q_{c}(v_{x}-v)$, and hence
$v_{c}^{\prime}+{\cal H}v_{c}+\Psi={a\over\rho_{c}}Q_{c}(v_{x}-v_{c})\,.$ (29)
In this case, there is an explicit deviation of the dark matter velocity
relative to the non-interacting case. The dark matter no longer follows
geodesics in general. Note that, even though dark energy does not cluster on
sub-Hubble scales, we cannot in general neglect the dark energy velocity
$v_{x}$ relative to the dark matter velocity $v_{c}$ in Eq. (29).
An example of the form of Eq. (28) is Wetterich:1994bg
$Q^{\mu}_{c}=-\alpha\rho_{c}\nabla^{\mu}\varphi\,,$ (30)
where $\varphi$ is the scalar field that describes dark energy and $\alpha$ is
a coupling constant. Note that $\nabla^{\mu}\varphi$ is parallel to the dark
energy four-velocity $u_{x}^{\mu}$:
$u_{x}^{\mu}={1\over
a}\Big{(}1-\Psi,-{\partial^{i}\delta\varphi\over\varphi^{\prime}}\Big{)}\,,~{}~{}v_{x}=-{\delta\varphi\over\varphi^{\prime}}\,.$
(31)
In this case,
$Q_{c}=a^{-1}\alpha(\rho_{c}\varphi^{\prime}+\delta\rho_{c}\varphi^{\prime}+\rho_{c}\delta\varphi^{\prime}-\rho_{c}\varphi^{\prime}\Psi)$.
The perturbed Klein-Gordon equation is Hwang:2001fb
$\displaystyle\delta\varphi^{\prime\prime}+2{\cal
H}\delta\varphi^{\prime}+(k^{2}+a^{2}V_{\varphi\varphi})\delta\varphi$
$\displaystyle~{}~{}{}=2\varphi^{\prime}(\Psi^{\prime}+{\cal
H}\Psi)+2\varphi^{\prime\prime}\Psi-\alpha a^{2}\rho_{c}\delta_{c}\,,$ (32)
where $V(\varphi)$ is the quintessence potential. In the Newtonian regime, the
last term on the right dominates over the other terms, while the $k^{2}$ term
dominates on the left, leading to
$k^{2}\delta\varphi=-\alpha a^{2}\rho_{c}\delta_{c}\,.$ (33)
It follows from Eqs. (29), (30) and (31) that
$v_{c}^{\prime}+{\cal
H}v_{c}+\Psi=-\alpha\varphi^{\prime}\Big{(}v_{c}+{\delta\varphi\over\varphi^{\prime}}\Big{)},$
(34)
confirming the violation of weak equivalence. For the perturbed continuity
equation (13), the right-hand side becomes $-\alpha\delta\varphi^{\prime}$. By
Eq. (33), this term is suppressed by $k^{-2}$ relative to the
$\delta_{c}^{\prime}$ term on the left-hand side, and therefore to a good
approximation we have
$\delta_{c}^{\prime}-k^{2}v_{c}=0\,.$ (35)
Using (33) and (35), the evolution equation (II) for $\delta_{c}$ becomes
$\displaystyle\delta_{c}^{\prime\prime}+({\cal
H}+\alpha\varphi^{\prime})\delta_{c}^{\prime}=4\pi Ga^{2}\rho_{b}\delta_{b}$
$\displaystyle~{}~{}{}+4\pi Ga^{2}\Big{(}1+{\alpha^{2}\over 4\pi
G}\Big{)}\rho_{c}\delta_{c}\,.$ (36)
As in the case of Eq. (26), the modification of the standard evolution for
$\delta_{c}$ occurs in 3 ways Amendola:2001rc : firstly via the modified
expansion history in the background ${\cal H}$ and $\rho_{c}$; secondly by the
modified Hubble friction term ${\cal H}\to{\cal
H}[1+\alpha\varphi^{\prime}/{\cal H}]$; and thirdly by the modified effective
gravitational coupling for dark matter – dark matter particle interactions,
$G_{\rm eff}=G\Big{(}1+{\alpha^{2}\over 4\pi G}\Big{)}.$ (37)
These effects are incorporated in the modified $N$-body simulations for this
form of interacting dark energy Maccio:2003yk .
## IV Testing for dark sector interactions
In this section, we discuss several possible ways to use observations to
constrain the dark sector interactions discussed in the previous section.
### IV.1 Continuity and Euler equations unchanged
We first consider the case where there is no modification to the dynamics of
perturbations in the Newtonian regime. The difference comes purely from the
modified background evolution. If dark matter interacts with dark energy, the
dark matter density no longer decays like $a^{-3}$. This affects the distance
measures in the Universe and thus changes the measurements of CMB, SNe and
Baryon Acoustic Oscillations. By combining these observations, we can measure
today’s matter density and then determine the matter energy density at the
last scattering surface. However, the distance is determined by integrating
over the expansion history and we cannot directly check the deviation at each
redshift from the standard behaviour, $\rho_{c}\propto a^{-3}$.
There is an independent way to measure the dark matter density using structure
formation. From the Poisson equation, the dark matter density can be written
as
$\omega_{m}(a)\equiv\Omega_{m}(a)h^{2}=-\frac{2\Psi(k,a)}{3\delta_{c}(k,a)}\left(\frac{kh}{aH_{0}}\right)^{2},$
(38)
where we neglected the baryon contribution for simplicity (we are only
illustrating the principle, rather than making quantitative predictions). One
way to measure $\delta_{c}$ is to reconstruct $\delta_{c}$ from peculiar
velocities using the continuity equation Eq. (22) because in this case there
is no modification to the continuity equation and no violation of weak
equivalence principle. On the other hand, weak lensing measures directly
$\Psi$ without bias. Thus we can use Eq. (38) to predict the background
evolution of matter density from structure formation.
In Fig. 1, we plot $\omega_{m}/\omega_{m}^{\rm eff}$, where $\omega_{m}$ is
the true matter density measured by weak lensing, and $\omega_{m}^{\rm eff}$
is derived from the background measurement of $\omega_{m}$ at the last
scattering surface, assuming $\rho_{m}\propto a^{-3}$. At late times when the
interaction becomes important, this ratio deviates from 1 due to the non-
adiabatic decay of the dark matter density. In this way, we can check the
modification to the behaviour of the matter density at each redshift, using
tomographic measurements of $\Psi$ from weak lensing.
Figure 1: The ratio between the true matter density obtained from structure
formation and the density estimated from geometrical tests assuming the non-
interacting adiabatic behaviour $\rho_{m}\propto a^{-3}$.
### IV.2 Test of the continuity equation
In the case where the interaction changes only the continuity equation, there
is no difference between the peculiar velocities of baryons and dark matter.
We assume that galaxies can be treated as test particles that are made of
baryons and whose peculiar velocities, $v_{g}$, are determined by baryon
peculiar velocities. Although there is an indication that this assumption is
valid Percival:2008sh , this should be tested by N-body simulations carefully
in the presence of interaction. We leave this for a future work.
With this assumption, we can determine peculiar velocities of baryons,
$v_{b}$, from peculiar velocities of galaxies, $v_{g}$. The latter can be
measured for example by redshift-space distortions (see Song:2008qt ;
White:2008jy for recent work). Then it is possible to determine dark matter
peculiar velocities because $v_{c}=v_{b}$.
On the other hand, density perturbations can be measured from the galaxy
distribution with a knowledge of bias. One possibility to measure bias is to
use weak lensing. Weak lensing measures $\Psi$ without bias and $\delta_{c}$
can be derived from the Poisson equation (17). Note that in order to measure
$\Psi$ from $\delta_{c}$, it is necessary to measure the true evolution of
$\rho_{c}$, which is modified by interactions. However, we found that the
modification to the continuity equation has significant effects even in weak
interactions cases where the effect of interactions on $\rho_{c}$ is
negligible. Thus in the following we only consider the case where we can
neglect the effect of interactions on the evolution of $\rho_{c}$. Another
possibility is to use the peculiar velocity measurements. In the case that we
consider here, the Euler equation is not modified [see Eq. (20)] and it is
possible to reconstruct $\Psi$ from $v_{c}$. Then again using the Poisson
equation, $\delta_{c}$ can be derived Hu:2003pt ; Acquaviva:2008qp ; Song09 .
In this way we can test whether the continuity equation is modified.
Fig. 2 demonstrates the breakdown of the standard continuity equation by an
interacting dark energy model. We used a model where $\Gamma_{x}\neq 0$ and
$\Gamma_{c}=0$.
Figure 2: The breakdown of the continuity equation by an interacting dark
energy model. In this model, $v_{b}=v_{c}$ and
$\delta_{c}^{\prime}-k^{2}v_{c}=a\Gamma_{x}(\rho_{x}/\rho_{c})\delta_{c}$.
### IV.3 Test of weak equivalence principle
The weak equivalence principle is broken when the Euler equation for dark
matter is modified. In this case, there is a difference between the peculiar
velocities of dark matter and baryons. With the assumption that galaxies trace
baryon peculiar velocities, we measure baryon peculiar velocities say from
red-shift distortions. Unlike the previous case, dark matter peculiar
velocities are different. However, without knowing that there is an
interaction between dark matter and dark energy, we estimate dark matter
peculiar velocity as
$v_{c}^{\rm est}=v_{b}=v_{g}\,.$ (39)
The estimated peculiar velocity is different from true peculiar velocity of
dark matter: $v_{c}\neq v_{c}^{\rm est}$.
If the continuity equation is not modified, as happens for the model of Eq.
(30), then the true peculiar velocity satisfies the same continuity equation
as the uncoupled case. Thus, if we use the estimated peculiar velocity, the
continuity equation is apparently broken
$\delta_{c}^{\prime}-k^{2}v_{c}^{\rm est}\neq 0.$ (40)
In this case, the continuity equation is not broken but $v_{c}\neq v_{b}$.
Then we can apply the same analysis as in the previous section. We can measure
$\delta_{c}$ from weak lensing. Then it is possible to prove the breakdown of
the weak equivalence principle through the apparent breakdown of the
continuity equation. Fig. 3 demonstrates this idea.
## V Conclusions
An interaction between dark matter and dark energy could exist in various ways
which are not detectable by any direct probe. We investigated how the Euler
equation and the continuity equation for dark matter could be modified by such
an interaction, taking care to provide a covariant analysis of momentum
transfer. Modification of the Euler equation indicates a deviation of the dark
matter motion from geodesic, under the drag force of dark energy – and a
consequent breaking of the weak equivalence principle for dark matter. Using
three different forms of interaction as examples, we considered interacting
models in which:
(A) neither the Euler nor continuity equations are modified, so that the
effect of the interaction in the Newtonian regime is purely via the different
background evolution;
(B) the Euler equation is unchanged but the continuity equation is modified
(and consequently a new bias is introduced by the interaction);
(C) the continuity equation is unchanged but the Euler equation is modified,
leading to violation of weak equivalence.
We discussed how in principle observations could be used to detect these
different forms of interacting dark energy. In case (A), we used the fact that
the continuity and Euler equations are unchanged to devise a test based on the
non-adiabatic redshifting of the dark matter. This test uses independent
measurements of the Newtonian potential and the density perturbation via the
Poisson equation, to compute the true matter density and show that it deviates
from the non-interacting case.
In cases (B) and (C), the effects of the violation of the continuity equation
or Euler equation are stronger than the non-standard redshifting of the
background matter density. We showed how, given a knowledge of bias from weak
lensing, tests could be constructed for the breakdown of the continuity or the
Euler equation.
A further issue raised by our investigations is how to distinguish interacting
dark energy from modified gravity. This is left for future work.
Figure 3: The breakdown of the weak equivalence principle for dark matter. In
this model, the continuity equation is not broken but $v_{c}\neq v_{b}$.
###### Acknowledgements.
The authors are supported by the UK’s Science & Technology Facilities Council.
KK is supported by the European Research Council and Research Councils UK.
## References
* (1) J. A. Frieman and B. A. Gradwohl, Phys. Rev. Lett. 67, 2926 (1991).
* (2) B. A. Gradwohl and J. A. Frieman, Astrophys. J. 398, 407 (1992).
* (3) R. Bean, E. E. Flanagan, I. Laszlo and M. Trodden, Phys. Rev. D 78, 123514 (2008) [arXiv:0808.1105 [astro-ph]].
* (4) J. Bovy and G. R. Farrar, arXiv:0807.3060 [hep-ph];
S. M. Carroll, S. Mantry, M. J. Ramsey-Musolf and C. W. Stubbs,
arXiv:0807.4363 [hep-ph].
* (5) M. Kesden and M. Kamionkowski, Phys. Rev. D 74, 083007 (2006) [arXiv:astro-ph/0608095];
J. A. Keselman, A. Nusser and P. J. E. Peebles, arXiv:0902.3452 [astro-ph.GA].
* (6) T. P. Sotiriou, V. Faraoni and S. Liberati, Int. J. Mod. Phys. D 17, 399 (2008) [arXiv:0707.2748 [gr-qc]].
* (7) A. Lue, R. Scoccimarro and G. Starkman, Phys. Rev. D 69, 044005 (2004) [arXiv:astro-ph/0307034];
L. Knox, Y. S. Song and J. A. Tyson, Phys. Rev. D 74 (2006) 023512;
M. Ishak, A. Upadhye and D. N. Spergel, Phys. Rev. D 74 (2006) 043513
[arXiv:astro-ph/0507184];
E. V. Linder, Phys. Rev. D 72, 043529 (2005) [arXiv:astro-ph/0507263];
E. Bertschinger, Astrophys. J. 648, 797 (2006) [arXiv:astro-ph/0604485];
S. Wang, L. Hui, M. May and Z. Haiman, Phys. Rev. D 76, 063503 (2007)
[arXiv:0705.0165 [astro-ph]];
W. Hu and I. Sawicki, Phys. Rev. D 76, 104043 (2007) [arXiv:0708.1190 [astro-
ph]];
M. A. Amin, R. V. Wagoner and R. D. Blandford, arXiv:0708.1793 [astro-ph];
B. Jain and P. Zhang, Phys. Rev. D 78, 063503 (2008) [arXiv:0709.2375 [astro-
ph]];
Y. Wang, JCAP 0805, 021 (2008) [arXiv:0710.3885 [astro-ph]];
E. Bertschinger and P. Zukin, arXiv:0801.2431 [astro-ph];
S. F. Daniel, R. R. Caldwell, A. Cooray and A. Melchiorri, arXiv:0802.1068
[astro-ph];
L. Guzzo et al., Nature 451, 541 (2008) [arXiv:0802.1944 [astro-ph]];
Y. S. Song and K. Koyama, JCAP 0901, 048 (2009) [arXiv:0802.3897 [astro-ph]];
Y. S. Song and O. Dore, JCAP 0903 (2009) 025;
G. B. Zhao, L. Pogosian, A. Silvestri and J. Zylberberg, Phys. Rev. D 79,
083513 (2009) [arXiv:0809.3791 [astro-ph]];
G. B. Zhao, L. Pogosian, A. Silvestri and J. Zylberberg, arXiv:0905.1326
[astro-ph.CO];
P. Zhang, R. Bean, M. Liguori and S. Dodelson, arXiv:0809.2836 [astro-ph].
* (8) H. Wei and S. N. Zhang, Phys. Rev. D 78, 023011 (2008) [arXiv:0803.3292 [astro-ph]].
* (9) E. Majerotto, J. Valiviita and R. Maartens, arXiv:0907.4981 [astro-ph.CO].
* (10) J. Valiviita, R. Maartens and E. Majerotto, arXiv:0907.4987 [astro-ph.CO].
* (11) J. Valiviita, E. Majerotto and R. Maartens, JCAP 0807, 020 (2008) [arXiv:0804.0232 [astro-ph]].
* (12) L. Amendola and D. Tocchini-Valentini, Phys. Rev. D 66, 043528 (2002) [arXiv:astro-ph/0111535].
* (13) C. G. Boehmer, G. Caldera-Cabral, R. Lazkoz and R. Maartens, Phys. Rev. D 78, 023505 (2008) [arXiv:0801.1565 [gr-qc]].
* (14) G. Caldera-Cabral, R. Maartens and L. A. Urena-Lopez, Phys. Rev. D 79, 063518 (2009) [arXiv:0812.1827 [gr-qc]].
* (15) G. Caldera-Cabral, R. Maartens and B. M. Schaefer, arXiv:0905.0492 [astro-ph.CO].
* (16) C. Wetterich, Astron. Astrophys. 301, 321 (1995) [arXiv:hep-th/9408025];
L. Amendola, Phys. Rev. D 60, 043501 (1999) [arXiv:astro-ph/9904120];
D. J. Holden and D. Wands, Phys. Rev. D 61, 043506 (2000) [arXiv:gr-
qc/9908026].
* (17) J. c. Hwang and H. Noh, Class. Quant. Grav. 19, 527 (2002) [arXiv:astro-ph/0103244].
* (18) A. V. Maccio, C. Quercellini, R. Mainini, L. Amendola and S. A. Bonometto, Phys. Rev. D 69, 123516 (2004) [arXiv:astro-ph/0309671];
M. Baldi, V. Pettorino, G. Robbers and V. Springel, arXiv:0812.3901 [astro-
ph].
* (19) W. J. Percival and M. White, arXiv:0808.0003 [astro-ph].
* (20) Y. S. Song and W. J. Percival, arXiv:0807.0810 [astro-ph].
* (21) M. White, Y. S. Song and W. J. Percival, arXiv:0810.1518 [astro-ph].
* (22) W. Hu and B. Jain, Phys. Rev. D 70 (2004) 043009 [arXiv:astro-ph/0312395].
* (23) V. Acquaviva, A. Hajian, D. N. Spergel and S. Das, Phys. Rev. D 78 (2008) 043514 [arXiv:0803.2236 [astro-ph]].
* (24) Y-S. Song, C. Sabiu and R. Nichol and C. Miller, submmited to JCAP (2009).
|
arxiv-papers
| 2009-07-13T10:23:09 |
2024-09-04T02:49:03.883006
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kazuya Koyama, Roy Maartens, Yong-Seon Song",
"submitter": "Kazuya Koyama",
"url": "https://arxiv.org/abs/0907.2126"
}
|
0907.2316
|
# Casimir-Lifshitz Interaction between Dielectric Heterostructures
Arash Azari, Himadri S. Samanta, and Ramin Golestanian Department of Physics
and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom
###### Abstract
The interaction between arbitrary dielectric heterostructures is studied
within the framework of a recently developed dielectric contrast perturbation
theory. It is shown that periodically patterned dielectric or metallic
structures lead to oscillatory lateral Casimir-Lifshitz forces, as well as
modulations in the normal force as they are displaced with respect to one
another. The strength of these oscillatory contributions increases with
decreasing gap size and increasing contrast in the dielectric properties of
the materials used in the heterostructures.
###### pacs:
05.40.-a, 81.07.-b, 03.70.+k, 77.22.-d
## I Introduction
In light of the ongoing miniaturization of mechanical devices and the recent
developments in Casimir-Lifshitz interactions Casimir48 ; Lifshitz ; measure ;
lateral-exp ; trench , there has been some recent interest in the effect of
these interactions between the components of small mechanical devices nanomech
. Since these interaction are particularly strong at small distances, it will
be interesting to know how they can be utilized for designing novel mechanical
systems that could work without physical contact and could potentially help
solve the wear problem machine .
In the past few years there have been a surge of interest in developing
techniques that can be used to study the Casimir-Lifshitz interaction in non-
ideal geometries, including geometry perturbation theories GK ; EHGK ;
lambrecht , semiclassical semiclass and classical ray optics Jaffe
approximations, multiple scattering and multipole expansions balian ; klich ;
multipole1 ; multipole2 ; multipole3 , world-line method gies and exact
numerical diagonalization methods Emig-exact ; valery , as well as the
numerical Green’s function calculation method Johnson . These methods have
been used in studying the Casimir force in a variety of different geometries,
which have improved significantly our understanding of the nontrivial geometry
dependence of this effect.
The effect of non-ideal geometry has been shown to lead to a number
interesting effects. For example, it has been suggested that corrugated
surfaces opposite one another can experience an oscillatory lateral Casimir
force GK , which was subsequently observed experimentally lateral-exp . A
recent experiment probing the normal Casimir force between a smooth surface
and surface with tall rectangular corrugations also revealed further evidence
on the non-additive nature of the Casimir force trench . Here, we study the
Casimir-Lifshitz interaction between arbitrary dielectric heterostructures
within the framework of a recently developed formalism ramin ; rg-09 . We
derive a closed form expression for the Casimir-Lifshitz energy between two
dielectric heterostructures (such as the example depicted in Fig. 1) up to the
second order in the perturbation theory and show that a coherent coupling
between the different modes of the spectrum of the dielectric pattern takes
place across the gap. As a special example, we consider unidirectional
periodic heterostructures (see Fig. 1) and calculate the lateral and normal
Casimir-Lifshitz force between them within the same order in the perturbation
theory. We find that coupling between modes with identical wavevectors of the
pattern structures between the different objects can lead to modulations in
the normal force and can give rise to oscillatory later forces, reminiscent of
the lateral Casimir force that appears due to coupling between geometrical
features such as corrugations GK ; lateral-exp .
Figure 1: Schematic representation of two identical semi-infinite and periodic
objects made of intercalated layers of high and low dielectric functions,
occupying the fractions of $f$ and $1-f$, respectively. Here $H$ is the
separation between them, $a$ is a dimensionless lateral displacement, and
$\lambda$ is the wavelength of the periodic structure.
This paper is organized as follows. Section II sketches the dielectric
contrast perturbation theory, and Sec. III elaborates on how it can be used
for dielectric heterostructures giving closed form expressions for the second
order term in the perturbation theory. Section IV gives the results for the
lateral and normal Casimir-Lifshitz force for a number of choices of
materials, and Sec. V contains some discussions and concluding remarks.
## II Theoretical Formulation
To calculate the Casimir-Lifshitz interaction we need to quantize the
electromagnetic field in a background that includes the dielectric or metallic
objects that modify the quantum fluctuations of the field. Describing a
general assortment of dielectric and metallic objects in space via a frequency
dependent dielectric profile $\epsilon(\omega,{\bf r})$, we can write a
general expression for the Casimir-Lifshitz energy as rg-09
$E=\hbar\int_{0}^{\infty}\frac{d\zeta}{2\pi}\;{\rm tr}\ln\left[{\cal
K}_{ij}(\zeta;{\bf r},{\bf r}^{\prime})\right],$ (1)
where
${\cal K}_{ij}=\left[\frac{\zeta^{2}}{c^{2}}\epsilon(i\zeta,{\bf
r})\delta_{ij}+\partial_{i}\partial_{j}-\partial_{k}\partial_{k}\delta_{ij}\right]\delta^{3}({\bf
r}-{\bf r}^{\prime}).$ (2)
We can consider the dielectric function as $\epsilon(i\zeta,{\bf
r})=1+\delta\epsilon(i\zeta,{\bf r})$, and expand Eq. (1) in powers of the
dielectric contrast. A similar approach has been the subject of a few recent
studies barton ; ramin ; buhmann ; rudi ; milton .
The expansion leads to the decomposition of ${\cal K}_{ij}$ into a diagonal
part ${\cal K}_{0,ij}$, corresponding to the empty space, and a perturbation
part $\delta{\cal K}_{ij}$, namely
${\cal K}_{ij}(\zeta;{\bf q},{\bf q}^{\prime})={\cal K}_{0,ij}(\zeta,{\bf
q})(2\pi)^{3}\delta^{3}({\bf q}+{\bf q}^{\prime})+\delta{\cal
K}_{ij}(\zeta;{\bf q},{\bf q}^{\prime}),$ (3)
where
${\cal K}_{0,ij}(\zeta,{\bf
q})=\frac{\zeta^{2}}{c^{2}}\delta_{ij}+q^{2}\delta_{ij}-q_{i}q_{j},$ (4)
and
$\delta{\cal K}_{ij}(\zeta;{\bf q},{\bf
q}^{\prime})=\frac{\zeta^{2}}{c^{2}}\delta_{ij}\delta\tilde{\epsilon}(i\zeta,{\bf
q}+{\bf q}^{\prime}).$ (5)
This yields an expansion
${\rm tr}\ln[{\cal K}]={\rm tr}\ln[{\cal
K}_{0}]+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}\;{\rm tr}[({\cal
K}_{0}^{-1}\delta{\cal K})^{n}],$ (6)
where
$\displaystyle{\cal K}_{0,ij}^{-1}(\zeta,{\bf
q})=\frac{\frac{\zeta^{2}}{c^{2}}\delta_{ij}+q_{i}q_{j}}{\frac{\zeta^{2}}{c^{2}}\left[\frac{\zeta^{2}}{c^{2}}+q^{2}\right]}.$
(7)
The first term is the vacuum energy in the absence of the objects, and the
terms in the series take account of their effect in a perturbative scheme. The
$n$-th order term in Eq. (6) takes on the explicit form
${\rm tr}[({\cal K}_{0}^{-1}\delta{\cal K})^{n}]=\int\frac{d^{3}{\bf
q}^{(1)}}{(2\pi)^{3}}\cdots\frac{d^{3}{\bf
q}^{(n)}}{(2\pi)^{3}}\frac{[\frac{\zeta^{2}}{c^{2}}\delta_{i_{1}i_{2}}+q_{i_{1}}^{(1)}q_{i_{2}}^{(1)}]\cdots[\frac{\zeta^{2}}{c^{2}}\delta_{i_{n}i_{1}}+q_{i_{n}}^{(n)}q_{i_{1}}^{(n)}]}{[\frac{\zeta^{2}}{c^{2}}+q^{(1)2}]\cdots[\frac{\zeta^{2}}{c^{2}}+q^{(n)2}]}\delta\tilde{\epsilon}(i\zeta,-{\bf
q}^{(1)}+{\bf q}^{(2)})\cdots\delta\tilde{\epsilon}(i\zeta,-{\bf q}^{(n)}+{\bf
q}^{(1)}),$ (8)
which involves the Fourier transform of the dielectric contrast profile. Going
to real space, we can rewrite the energy of the system as rg-09
$E=\hbar\int_{0}^{\infty}\frac{d\zeta}{2\pi}\;\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}\int
d^{3}{\bf r}_{1}\cdots d^{3}{\bf r}_{n}{\cal A}_{i_{1}i_{2}}({\bf r}_{1}-{\bf
r}_{2})\cdots{\cal A}_{i_{n}i_{1}}({\bf r}_{n}-{\bf
r}_{1})\left[\frac{\delta\epsilon(i\zeta,{\bf
r}_{1})}{1+\frac{1}{3}\delta\epsilon(i\zeta,{\bf
r}_{1})}\right]\cdots\left[\frac{\delta\epsilon(i\zeta,{\bf
r}_{n})}{1+\frac{1}{3}\delta\epsilon(i\zeta,{\bf r}_{n})}\right],$ (9)
where
${\cal A}_{ij}(\zeta,{\bf r})=\frac{\zeta^{2}}{c^{2}}\frac{{\rm e}^{-\zeta
r/c}}{4\pi r}\left[\delta_{ij}\left(1+\frac{c}{\zeta
r}+\frac{c^{2}}{\zeta^{2}r^{2}}\right)-\frac{r_{i}r_{j}}{r^{2}}\left(1+3\frac{c}{\zeta
r}+3\frac{c^{2}}{\zeta^{2}r^{2}}\right)\right],$ (10)
We now use this formulation to study the Casimir-Lifshitz interaction between
structures with inhomogeneous or patterned dielectric properties.
## III Dielectric Heterostructures
Let us now consider a configuration similar to the one depicted in Fig. 1,
namely two dielectric heterostructures that are placed parallel to each other
at a separation $H$. Using the definition ${\bf r}=({\bf x},z)$, the
dielectric profile can be written as
$\epsilon(i\zeta,{\bf r})=\left\\{\begin{array}[]{ll}\epsilon_{u}(i\zeta,{\bf
x}),&\;\;\;\;\frac{H}{2}\leq z<+\infty,\\\ \\\
1,&\;\;\;\;\frac{-H}{2}<z<\frac{H}{2},\\\ \\\ \epsilon_{d}(i\zeta,{\bf
x}),&\;\;\;\;-\infty<z\leq\frac{-H}{2},\end{array}\right.$ (11)
using the labels u and d for the “up” and “down” bodies respectively.
To keep the calculations tractable, we now focus on the second order term in
the series expansion in Eq. (9). For such two semi-infinite bodies, the second
order interaction term between the bodies can be written as
$E_{2}=-\frac{\hbar}{2\pi^{2}c^{2}}\int_{0}^{\infty}d\zeta\;\zeta^{2}\int
d^{2}{\bf x}d^{2}{\bf x}^{\prime}\int\frac{d^{2}{\bf Q}}{(2\pi)^{2}}\;{\rm
e}^{i{\bf Q}\cdot({\bf x}-{\bf x}^{\prime})}\;{\cal
E}(Q)\left[\frac{\delta\epsilon_{u}(i\zeta,{\bf
x})}{1+\frac{1}{3}\delta\epsilon_{u}(i\zeta,{\bf
x})}\right]\left[\frac{\delta\epsilon_{d}(i\zeta,{\bf
x}^{\prime})}{1+\frac{1}{3}\delta\epsilon_{d}(i\zeta,{\bf
x}^{\prime})}\right],$ (12)
for any lateral dielectric function profile, where
${\cal
E}(Q)=\int_{1}^{\infty}dp\;\frac{[2p^{4}-2p^{2}+1]}{\left[4p^{2}+(cQ/\zeta)^{2}\right]^{3/2}}\;{\rm
e}^{-\frac{\zeta H}{c}\sqrt{4p^{2}+(cQ/\zeta)^{2}}},$ (13)
and $\delta\epsilon_{u,d}(i\zeta,{\bf x})=\epsilon_{u,d}(i\zeta,{\bf x})-1$.
We now focus on the specific example of unidirectional periodic structures as
depicted in Fig. 1, which is made of subsequent layers of materials with
relatively high and low dielectric functions. We can use the periodic
properties of the dielectrics and write them in Fourier series expansion. As
Fig. 1 shows, we can define the dielectric profile of the $d$-object as
$\epsilon_{d}\left(i\zeta,x\right)=\left\\{\begin{array}[]{ll}\epsilon_{l}\left(i\zeta\right),&\;\;\;\;-\frac{\lambda}{2}+s\lambda\leq
x\leq-\frac{f\lambda}{2}+s\lambda,\\\ \\\
\epsilon_{h}\left(i\zeta\right),&\;\;\;\;-\frac{f\lambda}{2}+s\lambda<x<\frac{f\lambda}{2}+s\lambda,\\\
\\\ \epsilon_{l}\left(i\zeta\right),&\;\;\;\;\frac{f\lambda}{2}+s\lambda\leq
x\leq\frac{\lambda}{2}+s\lambda,\end{array}\right.$ (14)
where $s$ is an integer number. We define the Fourier series as
$\frac{\delta\epsilon_{d}\left(i\zeta,x\right)}{1+\frac{1}{3}\delta\epsilon_{d}\left(i\zeta,x\right)}=\sum_{m=-\infty}^{\infty}{\mathcal{C}}_{m}(i\zeta)\;{\rm
e}^{i2\pi mx/\lambda},$ (15)
where
${\mathcal{C}}_{m}(i\zeta)=\frac{\sin m\pi
f}{m\pi}\left[\frac{\delta\epsilon_{h}\left(i\zeta\right)}{1+\frac{1}{3}\delta\epsilon_{h}\left(i\zeta\right)}-\frac{\delta\epsilon_{l}\left(i\zeta\right)}{1+\frac{1}{3}\delta\epsilon_{l}\left(i\zeta\right)}\right],$
(16)
for $m\neq 0$, and
${\mathcal{C}}_{0}(i\zeta)=f\left[\frac{\delta\epsilon_{h}\left(i\zeta\right)}{1+\frac{1}{3}\delta\epsilon_{h}\left(i\zeta\right)}\right]+(1-f)\left[\frac{\delta\epsilon_{l}\left(i\zeta\right)}{1+\frac{1}{3}\delta\epsilon_{l}\left(i\zeta\right)}\right].$
(17)
We can find the corresponding expansion for the $u$-object by changing
$x\rightarrow x+a\lambda$.
Using the Fourier series expansion, one can find the Casimir-Lifshitz energy
between two dielectric heterostructures as depicted in Fig. 1 [up to second
order in the Clausius-Mossotti expansion of Eq. (9)] as
$E_{pp}=-\frac{\hbar
A}{2\pi^{2}c^{2}}{\sum_{m=0}^{\infty}}^{{}^{\prime}}\int_{0}^{\infty}d\zeta\;\zeta^{2}{\cal
E}\left(\frac{2\pi m}{\lambda}\right)\;{\mathcal{C}}^{2}_{m}(i\zeta)\cos(2\pi
ma),$ (18)
where the prime on the summation sign indicates that the $m=0$ term is counted
with half the weight, and the $pp$ index means the energy calculated for the
plate-plate geometry. This result shows that similar to the case of two
corrugated surfaces, two patterned dielectric heterostructures also couple to
each other at the leading order when the two wavelengths of the modulations
are equal GK . Moreover, higher harmonics contribute to the Casimir-Lifshitz
energy with exponentially decaying contributions, such that at large
separations only the fundamental mode (lowest harmonic) will survive Emig-
exact .
## IV The Normal and Lateral Forces
We now use Eq. (18) to calculate the normal and lateral forces between
different types of dielectric and metallic heterostructures. We look at three
different types of materials as examples, namely, gold, silicon, and
air/vacuum, and consider layered materials made of gold-silicon, silicon-air,
and gold-air. We describe the dielectric function of gold using a plasma
model, namely,
$\displaystyle\epsilon(i\zeta)=1+\frac{\omega_{p}^{2}}{\zeta^{2}},$ where
$\omega_{p}$ is the plasma frequency, which is given as $\omega_{p}({\rm
Au})=1.37\times 10^{16}$ rad/s Palik . For silicon we use the Drude-Lorentz
form
$\displaystyle\epsilon(i\zeta)=1+\frac{\omega_{p}^{2}}{\zeta^{2}+\omega_{0}^{2}},$
where $\omega_{p}({\rm Si})=3.3\;\omega_{0}({\rm Si})$ and $\omega_{0}({\rm
Si})=6.6\times 10^{15}$ rad/s Palik . Finally, for air/vacuum we use
$\epsilon(i\zeta)=1$.
Due to difficulties in keeping the surfaces of the objects parallel to each
other, most experiments are performed in plate-sphere geometry. To perform the
calculation of the forces for the plate-sphere configuration, we can use the
Derjaguin Approximation Israelachvili92 , where we replace one of the semi-
infinite objects with a planar surface with a sphere with radius $R$. The
approximation is valid provided that the radius of sphere is much larger than
the distance between the dielectric heterostructures, namely, $R\gg H$. Using
this approximation we can find the normal force between a semi-infinite
dielectric heterostructure and a sphere of the same material composition as
Israelachvili92
$F_{ps}^{nor}=2\pi R\left(\frac{E_{pp}}{A}\right).$ (19)
Using this result, we can find the Casimir-Lifshitz energy for plate-sphere
configuration as
$E_{ps}=-\int_{H}^{\infty}dH^{\prime}\;F_{ps}^{nor}(H^{\prime}),$ (20)
which we can now use to calculate the lateral Casimir force as
$F_{ps}^{lat}=-\frac{1}{\lambda}\;\frac{\partial E_{ps}}{\partial a}.$ (21)
Substituting Eqs. (19) and (20) into Eq. (21), it reads
$F_{ps}^{lat}=\frac{2\pi R}{\lambda}\;\frac{\partial}{\partial
a}\int_{H}^{\infty}dH^{\prime}\;\left(\frac{E_{pp}(H^{\prime})}{A}\right).$
(22)
The above equations are the basis of the results that will be presented below.
Figure 2: Normal Casimir-Lifshitz force between layered dielectric
heterostructures as shown in Fig. 1 in the plate-sphere geometry,
corresponding to (a) gold-silicon, (b) silicon-air, and (c) gold-air, with
$f=0.5$, and to (d) gold-silicon, (e) silicon-air, and (f) gold-air, with
$f=0.2$. The numerical value of corrugation wavelength used is $\lambda=1$
$\mu$m. Different curves correspond to different gap sizes of $H=100$ nm,
$H=300$ nm, and $H=600$ nm. The forces are normalized to $F_{ps}^{0}$ that
corresponds to the normal force when the laterally-averaged dielectric profile
is used.
Figures 2a-c show the normal Casimir-Lifshitz force between two unidirectional
(layered) dielectric heterostructures as shown in Fig. 1 when laterally
displaced with respect to one another by $a\lambda$. It corresponds to the
symmetric case with $f=0.5$, and the corrugation wavelength of $\lambda=1$
$\mu$m. Three different compositions of gold-silicon, silicon-air, and gold-
air are considered each at three different gap sizes of $H=100$ nm, $H=300$
nm, and $H=600$ nm. The normal forces are normalized using the normal force
$F_{ps}^{0}$ that corresponds to the Casimir-Lifshitz force calculated within
the same scheme but with laterally averaged dielectric profile, which
corresponds to the $m=0$ term in the expansion in Eq. (18). The normal force
is found to oscillate as a function of the lateral displacement, having the
maximum value when the regions of high dielectric constant from both sides are
exactly opposite one another, and the minimum value when in the staggered
configuration where regions of higher dielectric constant face regions of
lower dielectric constant. The amplitude of the oscillations increases by
decreasing the gap size, and the effect is progressively stronger when the
contrast between the dielectric properties of the two regions is more
pronounced, with a maximum relative change of 0.7 % for gold-silicon, 7 % for
silicon-air, and 65 % for gold-air, at the closest separation of $H=100$ nm.
In Figs. 2d-f the normal Casimir-Lifshitz forces between the same types of
structures as above are presented, for the asymmetric case of $f=0.2$. One can
see two noticeable differences with the symmetric case. First, the
oscillations are now asymmetric, as enforced by the asymmetry of the
dielectric profile, although the asymmetry weakens as the gaps size increases
and eventually disappears—i.e. the oscillations become symmetric and
harmonic—at sufficiently large separations. This is consistent with the
picture that different harmonics of the dielectric contrast profile in Eq.
(18) couple with each other via an exponential terms that decays with the
corresponding wavelengths of each harmonic and as a result any asymmetry
caused by higher harmonics will die out at large gap sizes. The second new
feature is the significant enhancement of the amplitude of the oscillatory
behavior as a function of the lateral displacement. While it is still the case
that this amplitude increases with increasing contrast between the dielectric
properties of the two materials used in the layered structure, the maximum
relative change is 0.4 % for gold-silicon, 6 % for silicon-air, and 200 % for
gold-air, at the closest separation of $H=100$ nm.
Figure 3: Lateral Casimir-Lifshitz force between layered dielectric
heterostructures as shown in Fig. 1 in the plate-sphere geometry,
corresponding to (a) gold-silicon, (b) silicon-air, and (c) gold-air, with
$f=0.5$, and to (d) gold-silicon, (e) silicon-air, and (f) gold-air, with
$f=0.2$. The numerical values used in these graphs are $\lambda=1$ $\mu$m and
$R=180$ $\mu$m. Different curves correspond to different gap sizes of $H=100$
nm, $H=200$ nm, and $H=400$ nm.
The lateral Casimir-Lifshitz forces for the same layered structures as above
are shown in Figs. 3a-c for the symmetric case with $f=0.5$. In this case, we
have assumed $R=180$ $\mu$m and $\lambda=1$ $\mu$m. Similar to the previous
study, three different compositions of gold-silicon, silicon-air, and gold-air
are considered each at three different gap sizes of $H=100$ nm, $H=200$ nm,
and $H=400$ nm. The lateral force is found to oscillate as a function of the
lateral displacement, reminiscent of the lateral Casimir force that is induced
by geometrical corrugations GK ; lateral-exp . The shape of the oscillatory
function approaches a sinusoidal behavior as the gap size increases,
consistent with the fact that higher harmonics do not contribute to the force
in that limit as also seen in geometrical lateral Casimir effect Emig-exact .
The amplitude of the oscillations increases by decreasing the gap size as well
as the contrast between the dielectric properties of the two regions.
Numerically, we find an amplitude of 0.5 pN for gold-silicon, 8 pN for
silicon-air, and 12 pN for gold-air, at the closest separation of $H=100$ nm.
Figure 3d-f show the lateral Casimir-Lifshitz forces between the same types of
structures as above, for the asymmetric case of $f=0.2$. Similarly, the
profiles of the lateral force are noticeably asymmetric, with the asymmetry
weakening as the gap size is increased and the shape of the profile approaches
that of a sinusoidal function (single harmonic). We also see comparatively
more significant enhancement of the amplitude of the oscillatory behavior as a
function of the lateral displacement. The amplitude of the oscillations is
found as 0.3 pN for gold-silicon, 5 pN for silicon-air, and 7 pN for gold-air,
at the closest separation of $H=100$ nm.
## V Discussion
In this paper, we have proposed a mechanism by which it is possible to create
a lateral Casimir-Lifshitz force as well as controlled modulations in the
normal Casimir-Lifshitz force without geometrical corrugations. A coupling
similar to what exists in the case of corrugated surfaces gives rise to these
oscillatory forces, namely identical modes of the dielectric patterns couple
across the gap to generate a macroscopic coherence in the fluctuations. The
generic features of these oscillatory forces are very similar to those of the
forces caused by corrugations; the effect is stronger and involves more
harmonics at closer separations, while it weakens and only involves the lowest
mode of the pattern in the dielectric contrast at larger separations.
While the difference in the dielectric properties of the materials controls
the general strength of the above results, comparison between Fig. 2 and 3
shows that the modulations in the normal force are more strongly affected by
the contrast in the dielectric properties. The choice of air/vacuum as one
component also allows us to make predictions about geometrical features with
large corrugation amplitudes, which provides an approximation scheme for the
non-perturbative geometrical regime.
In the present calculations we have only used the second order terms in the
dielectric contrast perturbative series. Higher order terms shown in Eq. (8)
will introduce coupling between different modes of the dielectric pattern in a
systematic way, as imposed by the overall conservation of the sum of all
wavevectors (momenta). While the present is aimed at showing in terms of
tractable calculations, one can in principle carry out the calculation of the
Casimir-Lifshitz interaction in such dielectric heterostructures using
numerical diagonalization methods valery .
Controlled interactions between dielectric heterostructures with smooth outer
surfaces could be very useful in practical applications because it will help
avoid the complications of bringing surfaces with geometrical protrusions
close to each other while avoiding contact between them and controlling their
separations. Moreover, it is much easier to pattern dielectric properties of
materials in a controlled way than it is to shape them with the high precision
that is needed for Casimir effect type experiments.
###### Acknowledgements.
The authors thank the ESF Research Network CASIMIR for providing excellent
opportunities for discussion on the Casimir effect and related topics. This
work was supported by EPSRC under Grant EP/E024076/1.
## References
* (1) H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948).
* (2) E.M. Lifshitz, Sov. Phys. JETP 2, 73 (1956); I.E. Dzyaloshinskii, E.M. Lifshitz, and L.P. Pitaevskii, Adv. Phys. 10, 165 (1961).
* (3) S.K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997); U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998); A. Roy and U. Mohideen, Phys. Rev. Lett. 82, 4380 (1999); H.B. Chan et al, Science 291, 1941 (2001); G. Bressi et al, Phys. Rev. Lett. 88, 041804 (2002); R.S. Decca et al Phys. Rev. Lett. 91, 050402 (2003); D.E. Krause, Phys. Rev. Lett. 98, 050403 (2007).
* (4) F. Chen, U. Mohideen, G.L. Klimchitskaya, and V.M. Mostepanenko, Phys. Rev. Lett. 88, 101801 (2002); Phys. Rev. A 66, 032113 (2002);
* (5) H.B. Chan et al, Phys. Rev. Lett. 101, 030401 (2008).
* (6) F.M. Serry et al, J. Microelectromech. Syst. 4, 193 (1995); E. Buks and M.L. Roukes, Phys. Rev. B 63, 033402 (2001); H.B. Chan et al, Phys. Rev. Lett. 87, 211801 (2001); G. Palasantzas and J.Th.M. De Hosson, Phys. Rev. B 72, 121409 (R) (2005).
* (7) A. Ashourvan, M.F. Miri, and R. Golestanian, Phys. Rev. Lett. 98, 140801 (2007); Phys. Rev. E 75, 040103 (R) (2007); T. Emig, Phys. Rev. Lett. 98, 160801 (2007); M. Miri and R. Golestanian, Appl. Phys. Lett. 92, 113103 (2008); F.C. Lombardo et al., J. Phys. A 41 164009 (2008); I. Cavero-Peláez et al., Phys. Rev. D 78 065019 (2008).
* (8) R. Golestanian and M. Kardar, Phys. Rev. Lett. 78, 3421 (1997); Phys. Rev. A 58, 1713 (1998); M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999).
* (9) T. Emig, A. Hanke, R. Golestanian, and M. Kardar, Phys. Rev. Lett. 87, 260402 (2001); Phys. Rev. A 67, 022114 (2003).
* (10) P.A. Maia Neto, A. Lambrecht, and S. Reynaud, Phys. Rev. A 72, 012115 (2005); R.B. Rodrigues, P.A. Maia Neto, A. Lambrecht, and S. Reynaud, Phys. Rev. Lett. 96, 100402 (2006).
* (11) M. Schaden and L. Spruch, Phys. Rev. A 58, 935 (1998).
* (12) R.L. Jaffe and A. Scardicchio, Phys. Rev. Lett. 92, 070402 (2004).
* (13) R. Balian and B. Duplantier, Ann. Phys. (New York) 104, 300 (1977); 112, 165 (1978).
* (14) O. Kenneth and I. Klich, Phys. Rev. Lett. 97, 160401 (2006)
* (15) R. Golestanian, Phys. Rev. E 62, 5242 (2000).
* (16) T. Emig, N. Graham, R.L. Jaffe, and M. Kardar, Phys. Rev. Lett. 99, 170403 (2007).
* (17) T. Emig, N. Graham, R.L. Jaffe, and M. Kardar, Phys. Rev. A 79, 054901 (2009).
* (18) H. Gies and K. Klingmuller, Phys. Rev. Lett. 96, 220401 (2006); Phys. Rev. Lett. 97, 220405 (2006).
* (19) T. Emig, Europhys. Lett. 62, 466 (2003); R. Büscher and T. Emig, Phys. Rev. Lett. 94, 133901 (2005).
* (20) A. Lambrecht and V.N. Marachevsky, Phys. Rev. Lett. 101, 160403 (2008).
* (21) A. Rodriguez, M. Ibanescu, D. Iannuzzi, F. Capasso, J.D. Joannopoulos, and S.G. Johnson, Phys. Rev. Lett. 99, 080401 (2007); A. Rodriguez, M. Ibanescu, D. Iannuzzi, J.D. Joannopoulos, and S.G. Johnson, Phys. Rev. A 76, 032106 (2007); A. Rodriguez, J.D. Joannopoulos, and S.G. Johnson, Phys. Rev. A 77, 062107 (2008).
* (22) G. Barton, J. Phys. A 34, 4083 (2001).
* (23) R. Golestanian, Phys. Rev. Lett. 95, 230601 (2005).
* (24) S.Y. Buhmann and D.-G. Welsch, Appl. Phys. B 82, 2, 189 (2006).
* (25) G. Veble and R. Podgornik, Eur. Phys. J. E 23, 275 279 (2007).
* (26) K.A. Milton, P. Parashar, and J. Wagner, Phys. Rev. Lett. 101, 160402 (2008).
* (27) R. Golestanian, Phys. Rev. A, in press (2009) [arXiv:0905.1046].
* (28) E. D. Palik, Handbook of Optical Constants of Solids, edited by E. D. Palik (Academic Press, New York, 1985).
* (29) J.N. Israelachvili, Intermolecular and Surface Forces (Academic, London, 1992).
|
arxiv-papers
| 2009-07-14T09:53:14 |
2024-09-04T02:49:03.898035
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arash Azari, Himadri S. Samanta, and Ramin Golestanian",
"submitter": "Ramin Golestanian",
"url": "https://arxiv.org/abs/0907.2316"
}
|
0907.2324
|
11institutetext: Institut für Informatik, Ruprecht-Karls-Universität,
Heidelberg, Germany
# Separations of non-monotonic randomness notions
(Preliminary version, 7 July 2009)
Laurent Bienvenu Rupert Hölzl Thorsten Kräling Wolfgang Merkle
###### Abstract
In the theory of algorithmic randomness, several notions of random sequence
are defined via a game-theoretic approach, and the notions that received most
attention are perhaps Martin-Löf randomness and computable randomness. The
latter notion was introduced by Schnorr and is rather natural: an infinite
binary sequence is computably random if no total computable strategy succeeds
on it by betting on bits in order. However, computably random sequences can
have properties that one may consider to be incompatible with being random, in
particular, there are computably random sequences that are highly
compressible. The concept of Martin-Löf randomness is much better behaved in
this and other respects, on the other hand its definition in terms of
martingales is considerably less natural.
Muchnik, elaborating on ideas of Kolmogorov and Loveland, refined Schnorr’s
model by also allowing non-monotonic strategies, i.e. strategies that do not
bet on bits in order. The subsequent “non-monotonic” notion of randomness, now
called Kolmogorov-Loveland-randomness, has been shown to be quite close to
Martin-Löf randomness, but whether these two classes coincide remains a
fundamental open question.
In order to get a better understanding of non-monotonic randomness notions,
Miller and Nies introduced some interesting intermediate concepts, where one
only allows non-adaptive strategies, i.e., strategies that can still bet non-
monotonically, but such that the sequence of betting positions is known in
advance (and computable). Recently, these notions were shown by Kastermans and
Lempp to differ from Martin-Löf randomness. We continue the study of the non-
monotonic randomness notions introduced by Miller and Nies and obtain results
about the Kolmogorov complexities of initial segments that may and may not
occur for such sequences, where these results then imply a complete
classification of these randomness notions by order of strength.
## 1 Introduction
Random sequences are the central object of study in algorithmic randomness and
have been investigated intensively over the last decade, which led to a wealth
of interesting results clarifying the relations between the various notions of
randomness and revealing interesting interactions with notions such as
computational power [2, 5, 11].
Intuitively speaking, a binary sequence is random if the bits of the sequence
do not have effectively detectable regularities. This idea can be formalized
in terms of betting strategies, that is, a sequence will be called random in
case the capital gained by successive bets on the bits of the sequence
according to a fixed betting strategy must remain bounded, with fair payoff
and a fixed set of admissible betting strategies understood.
The notions of random sequences that have received most attention are Martin-
Löf randomness and computable randomness. Here a sequence is called computably
random if no total computable betting strategy can achieve unbounded capital
by betting on the bits of the sequence in the natural order, a definition that
indeed is natural and suggests itself. However, computably random sequences
may lack certain properties associated with the intuitive understanding of
randomness, for example there are such sequences that are highly compressible,
i.e., show a large amount of redundancy, see Theorem 3.1 below. Martin-Löf
randomness behaves much better in this and other respects. Indeed, the Martin-
Löf random sequences can be characterized as the sequences that are
incompressible in the sense that all their initial segments have essentially
maximal Kolmogorov complexity, and in fact this holds for several versions of
Kolmogorov complexity according to celebrated results by Schnorr, by Levin
and, recently, by Miller and Yu [2]. On the other hand, it has been held
against the concept of Martin-Löf randomness that its definition involves
effective approximations, i.e., a very powerful, hence rather unnatural model
of computation, and indeed the usual definition of Martin-Löf randomness in
terms of left-computable martingales, that is, in terms of betting strategies
where the gained capital can be effectively approximated from below, is not
very intuitive.
It can be shown that Martin-Löf randomness strictly implies computable
randomness. According to the preceding discussion the latter notion is too
inclusive while the former may be considered unnatural. Ideally, we would
therefore like to find a more natural characterization of ML-randomness; or,
if that is impossible, we are alternatively interested in a notion that is
close in strength to ML-randomness, but has a more natural definition. One
promising way of achieving such a more natural characterization or definition
could be to use computable betting strategies that are more powerful than
those used to define computable randomness.
Muchnik [10] proposed to consider computable betting strategies that are non-
monotonic in the sense that the bets on the bits need not be done in the
natural order, but such that the bit to bet on next can be computed from the
already scanned bits. The corresponding notion of randomness is called
Kolmogorov-Loveland randomness because Kolmogorov and Loveland independently
had proposed concepts of randomness defined via non-monotonic selecting of
bits.
Kolmogorov-Loveland randomness is implied by and in fact is quite close to
Martin-Löf randomness, see Theorem 4.3 below, but whether the two notions are
distinct is one of the major open problems of algorithmic randomness. In order
to get a better understanding of this open problem and of non-monotonic
randomness in general, Miller and Nies [9] introduced restricted variants of
Kolmogorov-Loveland randomness, where the sequence of betting positions must
be non-adaptive, i.e., can be computed in advance without knowing the sequence
on which one bets.
The randomness notions mentioned so far are determined by two parameters that
correspond to the columns and rows, respectively, of the table in Figure 1.
First, the sequence of places that are scanned and on which bets may be
placed, while always being given effectively, can just be monotonic, can be
equal to $\pi(0),\pi(1),\ldots$ for a permutation or an injection $\pi$ from
${{\mathbb{N}}}$ to ${{\mathbb{N}}}$, or can be adaptive, i.e., the next bit
depends on the bits already scanned. Second, once the sequence of scanned bits
is determined, betting on these bits can be according to a betting strategy
where the corresponding martingale is total or partial computable, or is left-
computable. The known inclusions between the corresponding classes of random
sequences are shown in Figure 1, see Section 2 for technical details and for
the definitions of the class acronyms that occur in the figure.
monotonic permutation injection adaptive total $\mathbf{TMR}$ $=$
$\mathbf{TPR}$ $\supseteq$ $\mathbf{TIR}$ $\supseteq$ KLR $\subseteq$
$\subseteq$ $\subseteq$ $=$ partial $\mathbf{PMR}$ $\supseteq$ $\mathbf{PPR}$
$\supseteq$ $\mathbf{PIR}$ $\supseteq$ KLR $\subseteq$ $\subseteq$ $\subseteq$
$\subseteq$ left-computable MLR = MLR = MLR = MLR
Figure 1: Known class inclusions
The classes in the last row of the table in Figure 1 all coincide with the
class of Martin-Löf random sequences by the folklore result that left-
computable martingales always yield the concept of Martin-Löf randomness, no
matter whether the sequence of bits to bet on is monotonic or is determined
adaptively, because even in the latter, more powerful model one can uniformly
in $k$ enumerate an open cover of measure at most $1/k$ for all the sequences
on which some universal martingale exceeds $k$. Furthermore, the classes in
the first and second row of the last column coincide with the class of
Kolmogorov-Loveland random sequences, because it can be shown that total and
partial adaptive betting strategies yield the same concept of random sequence
[6]. Finally, it follows easily from results of Buhrman et al. [1] that the
class $\mathbf{TMR}$ of computably random sequences coincides with the class
$\mathbf{TPR}$ of sequences that are random with respect to total permutation
martingales, i.e., the ability to scan the bits of a sequence according to a
computable permutation does not increase the power of total martingales.
Concerning non-inclusions, it is well-known that it holds that
$\textnormal{\bf KLR}\subsetneq\mathbf{PMR}\subsetneq\mathbf{TMR}.$
Furthermore, Kastermans and Lempp [3] have recently shown that the Martin-Löf
random sequences form a proper subclass of the class $\mathbf{PIR}$ of partial
injective random sequences, i.e., $\textnormal{\bf
MLR}\subsetneq\mathbf{PIR}$.
Apart from trivial consequences of the definitions and the results just
mentioned, nothing has been known about the relations of the randomness
notions between computable randomness and Martin-Löf randomness in Figure 1.
In what follows, we investigate the six randomness notions that are shown in
Figure 1 in the range between $\mathbf{PIR}$ and $\mathbf{TMR}$, i.e., between
partial injective randomness as introduced below and computable randomness. We
obtain a complete picture of the inclusion structure of these notions, more
precisely we show that the notions are mutually distinct and indeed are
mutually incomparable with respect to set theoretical inclusion, except for
the inclusion relations that follow trivially by definition and by the known
relation $\mathbf{TMR}\subseteq\mathbf{TPR}$, see Figure 2 at the end of this
paper. Interestingly these separation results are obtained by investigating
the possible values of the Kolomogorov complexity of initial segments of
random sequences for the different strategy types, and for some randomness
notions we obtain essentially sharp bounds on how low these complexities can
be.
#### Notation.
We conclude the introduction by fixing some notation. The set of finite
strings (or finite binary sequences, or words) is denoted by $2^{<\omega}$,
$\epsilon$ being the empty word. We denote the set of infinite binary
sequences by $2^{\omega}$. Given two finite strings $w,w^{\prime}$, we write
$w\sqsubseteq w^{\prime}$ if $w$ is a prefix of $w^{\prime}$. Given an element
$x$ of $2^{\omega}$ or $2^{<\omega}$, $x(i)$ denotes the $i$-th bit of $x$
(where by convention there is a $0$-th bit and $x(i)$ is undefined if $x$ is a
word of length less than $i+1$). If $A\in 2^{\omega}$ and
$X=\\{x_{0}<x_{1}<x_{2}<\ldots\\}$ is a subset of ${{\mathbb{N}}}$ then
$A\upharpoonright{X}$ is the finite or infinite binary sequence
$A(x_{0})A(x_{1})\ldots$. We abbreviate $A\upharpoonright{\\{0,\ldots,n-1\\}}$
by $A\upharpoonright{n}$ (i.e., the prefix of $A$ of length $n$).
C and K denote plain and prefix-free Kolmogorov complexity, respectively [2,
5]. The function $\log$ designates the logarithm of base 2. An order is a
function $h:{{\mathbb{N}}}\rightarrow{{\mathbb{N}}}$ that is non-decreasing
and tends to infinity.
## 2 Permutation and injection randomness
We now review the concept of martingale and betting strategy that are central
for the unpredictability approach to define notions of an infinite random
sequence.
###### Definition 1
A martingale is a nonnegative, possibly partial, function
$d:2^{<\omega}\rightarrow{{\mathbb{Q}}}$ such that for all $w\in 2^{<\omega}$,
if $d(w0)$ is defined if and only if $d(w1)$ is, and if these are defined,
then so is $d(w)$, and the relation $2d(w)=d(w0)+d(w1)$ holds. A martingale
succeeds on a sequence $A\in 2^{\omega}$ if $d(A\upharpoonright{n})$ is
defined for all $n$, and $\limsup\,d(A\upharpoonright{n})=+\infty$. We denote
by $\mathrm{Succ}(d)$ the success set of $d$, i.e., the set of sequences on
which $d$ succeeds.
Intuitively, a martingale represents the capital of a player who bets on the
bits of a sequence $A\in 2^{\omega}$ in order, where at every round she bets
some amount of money on the value of the next bit of $A$. If her guess is
correct, she doubles her stake. If not, she loses her stake. The quantity
$d(w)$, with $w$ a string of length $n$, represents the capital of the player
before the $n$-th round of the game (by convention there is a $0$-th round)
when the first $n$ bits revealed so far are those of $w$.
We say that a sequence $A$ is computably random if no total computable
martingale succeeds on it. One can extend this in a natural way to partial
computable martingales: a sequence $A$ is partial computably random if no
partial martingale succeeds on it. No matter whether we consider partial or
total computable martingales, this game model can be seen as too restrictive
by the discussion in the introduction. Indeed, one could allow the player to
bet on bits in any order she likes (as long as she can visit each bit at most
once). This leads us to extend the notion of martingale to the notion of
strategy.
###### Definition 2
A betting strategy is a pair $b=(d,\sigma)$ where $d$ is a martingale and
$\sigma:2^{<\omega}\rightarrow{{\mathbb{N}}}$ is a function.
For a strategy $b=(d,\sigma)$, the term $\sigma$ is called the _scan rule_.
For a string $w$, $\sigma(w)$ represents the position of the next bit to be
visited if the player has read the sequence of bits $w$ during the previous
moves. And as before, $d$ specifies how much money is bet at each move.
Formally, given an $A\in 2^{\omega}$, we define by induction a sequence of
positions $n_{0},n_{1},\ldots$ by
$\left\\{\begin{array}[]{l}n_{0}=\sigma(\epsilon),\\\
n_{k+1}=\sigma\left(A(n_{0})A(n_{1})\ldots A(n_{k})\right)\textnormal{ for all
}k\geq 0\end{array}\right.$
and we say that $b=(d,\sigma)$ succeeds on $A$ if the $n_{i}$ are all defined
and pairwise distinct (i.e., no bit is visited twice) and
$\limsup_{k\rightarrow+\infty}\;d\left(A(n_{0})\ldots A(n_{k})\right)=+\infty$
Here again, a betting strategy $b=(d,\sigma)$ can be total or partial. In
fact, its partiality can be due either to the partiality of $d$ or to the
partiality of $\sigma$. We say that a sequence is Kolmogorov-Loveland random
if no total computable betting strategy succeeds on it. As noted in [8], the
concept of Kolmogorov-Loveland randomness remains the same if one replaces
“total computable” by “partial computable” in the definition.
Kolmogorov-Loveland randomness is implied by Martin-Löf randomness and whether
the two notions can be separated is one of the most important open problems on
algorithmic randomness. As we discussed above, Miller and Nies [9] proposed to
look at intermediate notions of randomness, where the power of non-monotonic
betting strategies is limited. In the definition of a betting strategy, the
scan rule is adaptive, i.e., the position of the next visited bit depends on
the bits previously seen. It is interesting to look at non-adaptive games.
###### Definition 3
In the above definition of a strategy, when $\sigma(w)$ only depends on the
length of $w$ for all $w$ (i.e., the decision of which bit should be chosen at
each move is independent of the values of the bits seen in previous moves), we
identify $\sigma$ with the (injective) function
$\pi:{{\mathbb{N}}}\rightarrow{{\mathbb{N}}}$, where for all $n$ $\pi(n)$ is
the value of $\sigma$ on words of length $n$ ($\pi(n)$ indicates the position
of the bit visited during the $n$-th move), and we say that $b=(d,\pi)$ is an
injection strategy. If moreover $\pi$ is bijective, we say that $b$ is a
permutation strategy. If $\pi$ is the identity, the strategy $b=(d,\pi)$ is
said to be monotonic, and can clearly be identified with the martingale $d$.
All this gives a number of possible non-adaptive, non-monotonic, randomness
notions: one can consider either monotonic, permutation, or injection
strategies, and either total computable or partial computable ones. This gives
a total of six randomness classes, which we denote by
$\mathbf{TMR},\;\mathbf{TPR},\;\mathbf{TIR},\;\mathbf{PMR},\;\mathbf{PPR},\;\mbox{and}\;\mathbf{PIR},$
(1)
where the first letter indicates whether we consider total (T) or partial (P)
strategies, and the second indicates whether we look at monotonic (M),
permutation (P) or injection (I) strategies. For example, the class
$\mathbf{TMR}$ is the class of computably random sequences, while the class
$\mathbf{PIR}$ is the class of sequences $A$ such that no partial injection
strategy succeeds on $A$. Recall in this connection that the known inclusions
between the six classes in (1) and the classes KLR and MLR of Kolmogorov-
Loveland random and Martin-Löf random sequences have been shown in Figure 1
above.
## 3 Randomness notions based on total computable strategies
We begin our study by the randomness notions arising from the game model where
strategies are total computable. As we will see, in this model, it is possible
to construct sequences that are random and yet have very low Kolmogorov
complexity (i.e. all their initial segments are of low Kolmogorov complexity).
We will see in the next section that this is no longer the case when we allow
partial computable strategies in the model.
### 3.1 Building a sequence in $\mathbf{TMR}$ of low complexity
The following theorem is a first illustration of the phenomenon we just
described.
###### Theorem 3.1 (Lathrop and Lutz [4], Muchnik [10])
For every computable order $h$, there is a sequence
$A\in\mathbf{\mathbf{TMR}}$ such that, for all $n\in\mathbb{N}$,
$\mathrm{C}\left({A\upharpoonright{n}}\,\mid\,{n}\right)\leq
h(n)+\textnormal{O}(1).$
###### Proof (Idea)
Defeating one total computable martingale is easy and can be done computably,
i.e., for every total computable martingale $d$ there exists a sequence $A$,
uniformly computable in $d$, such that $A\notin\mathrm{Succ}(d)$. Indeed,
given a martingale $d$. For any given $w$, one has either $d(w0)\leq d(w)$ or
$d(w1)\leq d(w)$. Thus, one can easily construct a computable sequence $A$ by
setting $A\upharpoonright{0}=\epsilon$ and by induction, having defined
$A\upharpoonright{n}$, we choose
$A\upharpoonright{n+1}=(A\upharpoonright{n})i$ where $i\in\\{0,1\\}$ is such
that $d((A\upharpoonright{n})i)\leq d(A\upharpoonright{n})$. This can of
course be done computably since $d$ is total computable, and by construction
of $A$, $d(A\upharpoonright{n})$ is non-increasing, meaning in particular that
$d$ does not succeed against $A$.
Defeating a finite number of total computable martingales is equally easy.
Indeed, given a finite number $d_{1},\ldots,d_{k}$ of such martingales, their
sum $D=d_{1}+\ldots+d_{k}$ is itself a total computable martingale (this
follows directly from the definition). Thus, we can construct as above a
computable sequence $A$ that defeats $D$. And since $D\geq d_{i}$ for all
$1\leq i\leq k$, this implies that $A$ defeats all the $d_{i}$. Note that this
argument would work just as well if we had taken $D$ to be any weighted sum
$\alpha_{1}d_{1}+\ldots+\alpha_{k}d_{k}$, with positive rational constants
$\alpha_{i}$.
We now need to deal with the general case where we have to defeat _all_ total
computable martingales simultaneously. We will again proceed using a
diagonalization technique. Of course, this diagonalization cannot be carried
out effectively, since there are infinitely many such martingales and since we
do not even know whether any one given partial computable martingale is total.
The first problem can easily be overcome by introducing the martingales to
diagonalize against one by one instead of all at the beginning. So at first,
for a number of stages we will only take into account the first computable
martingale $d_{1}$. Then (maybe after a long time) we may introduce the second
martingale $d_{2}$, with a small coefficient $\alpha_{2}$ (to ensure that
introducing $d_{2}$ does not cost us too much) and then consider the
martingale $d_{1}+\alpha_{2}d_{2}$. Much later we can introduce the third
martingale $d_{3}$ with an even smaller coefficient $\alpha_{3}$, and
diagonalize against $d_{1}+\alpha_{2}d_{2}+\alpha_{3}d_{3}$, and so on. So in
each step of the construction we have to consider just a finite number of
martingales.
The non-effectivity of the construction arises from the second problem,
deciding which of our partial computable martingales are total. However, once
we are supplied with this additional information, we can effectively carry out
the construction of $A$. And since for each step we need to consider only
finitely many potentially total martingales, the information we need to
construct the first $n$ bits of $A$ for some fixed $n$ is finite, too. Say,
for example, that for the first $n$ stages of the construction – i.e., to
define $A\upharpoonright{n}$ – we decided on only considering $k$ martingales
$d_{0},\ldots,d_{k}$. Then we need no more than $k$ bits, carrying the
information which martingales among $d_{0},\ldots,d_{k}$ are total, to
describe $A\upharpoonright{n}$. That way, we get
$\mathrm{C}\left({A\upharpoonright{n}}\,\mid\,{n}\right)\leq k+O(1)$.
As can be seen from the above example, the complexity of descriptions of
prefixes of $A$ depends on how fast we introduce the martingales. This is
where our orders come into play. Fix a fast-growing computable function $f$
with $f(0)=0$, to be specified later. We will introduce a new martingale at
every position of type $f(k)$, that is, between positions $[f(k),f(k+1))$, we
will only diagonalize against $k+1$ martingales, hence by the above
discussion, for every $n\in[f(k),f(k+1))$, we have
$\mathrm{C}\left({A\upharpoonright{n}}\,\mid\,{n}\right)\leq k+O(1)$
Thus, if the function $f$ grows faster than the inverse function $h^{-1}$ of a
given order $h$, we get
$\mathrm{C}\left({A\upharpoonright{n}}\,\mid\,{n}\right)\leq h(n)+O(1)$
for all $n$.∎
### 3.2 $\mathbf{TMR}=\mathbf{TPR}$: the averaging technique
It turns out that, perhaps surprisingly, the classes $\mathbf{TMR}$ and
$\mathbf{TPR}$ coincide. This fact was stated explicitely in Merkle et al [8],
but is easily derived from the ideas introduced in Buhrman et al [1]. We
present the main ideas of their proof as we will later need them. We shall
prove:
###### Theorem 3.2
Let $b=(d,\pi)$ be a total computable permutation strategy. There exists a
total computable martingale $d$ such that
$\mathrm{Succ}(b)\subseteq\mathrm{Succ}(d)$.
This theorem states that total permutation strategies are no more powerful
than total monotonic strategies, which obviously entails
$\mathbf{TMR}=\mathbf{TPR}$. Before we can prove it, we first need a
definition.
###### Definition 4
Let $b=(d,\pi)$ be a total injective strategy. Let $w\in 2^{<\omega}$. We can
run the strategy $b$ on $w$ as if it were an element of $2^{\omega}$, stopping
the game when $b$ asks to bet on a bit of position outside $w$. This game is
of course finite (for a given $w$) since at most $|w|$ bets can be made. We
define $\hat{b}(w)$ to be the capital of $b$ at the end of this game.
Formally: $\hat{b}(w)=d\left(w_{\pi(0)}\ldots w_{\pi(N-1)}\right)$ where $N$
is the smallest integer such that $\pi(N)\geq|w|$.
Note that if $b=(d,\pi)$ is a total computable injection martingale, $\hat{b}$
is total computable. If $\hat{b}$ was itself a monotonic martingale, Theorem
3.2 would be proven. This is however not the case in general: suppose
$d(\epsilon)=1$, $d(0)=2$, $d(1)=0$, and $\pi(0)=1$, $\pi(1)=5$ (i.e., $d$
first visits the bit in position $1$, betting everyrhing on the value $0$,
then visits the bit in position $5$). We then have $b(0)=1$ and $b(1)=1$, but
$\hat{b}(00)=2$, $\hat{b}(01)=2$, $\hat{b}(10)=0$ and $\hat{b}(11)=0$, which
shows that $\hat{b}$ is not a martingale.
The trick is, given a betting strategy $b$ and a word $w$, to look at the
_expected value_ of $b$ on $w$, i.e., look at the mathematical expectation of
$b(w^{\prime})$ for large enough extensions $w^{\prime}$ of $w$. Specifically,
given a total betting strategy $b=(d,\pi)$ and a word $w$ of length $n$, we
take an integer $M$ large enough to have
$\pi\left([0,\ldots,M-1]\right)\cap[0,\ldots,n-1]=\pi({{\mathbb{N}}})\cap[0,\ldots,n-1]$
(i.e. the strategy $b$ will never bet on a bit of position less than $n$ after
the $M$-th move), and define:
$\mathrm{Av}_{b}(w)=\frac{1}{2^{M}}\,\sum_{\begin{subarray}{c}w\sqsubseteq
w^{\prime}\\\ |w^{\prime}|=M\end{subarray}}\hat{b}(w^{\prime})$
###### Proposition 1 (Buhrman et al [1], Kastermans-Lempp [3])
* (i)
The quantity $\mathrm{Av}_{b}(w)$ (defined above) is well-defined i.e. does
not depend on $M$ as long as it satisfies the required condition.
* (ii)
For a total injective strategy $b$, $\mathrm{Av}_{b}$ is a martingale.
* (iii)
For a given injective strategy $b$ and a given word $w$ of length $n$,
$\mathrm{Av}_{b}(w)$ can be computed if we know the set
$\pi({{\mathbb{N}}})\cap[0,\ldots,n-1]$. In particular, if $b$ is a total
computable permutation strategy, then $\mathrm{Av}_{b}$ is total computable.
As Buhrman et al. [1] explained, it is not true in general that if a total
computable injective strategy $b$ succeeds against a sequence $A$, then
$\mathrm{Av}_{b}$ also succeeds on $A$. However, this can be dealt with using
the well-known “saving trick”. Suppose we are given a martingale $d$ with
initial capital, say, $1$. Consider the variant $d^{\prime}$ of $d$ that does
the following: when run on a given sequence $A$, $d^{\prime}$ initially plays
exactly as $d$. If at some stage of the game $d^{\prime}$ reaches a capital of
$2$ or more, it then puts half of its capital on a “bank account”, which will
never be used again. From that point on, $d^{\prime}$ bets half of what $d$
does, i.e. start behaving like $d/2$ (plus the saved capital). If later in the
game the “non-saved” part of its capital reaches $2$ or more, then half of it
is placed on the bank account and then $d^{\prime}$ starts behaving like
$d/4$, and so on.
For every martingale $d^{\prime}$ that behaves as above (i.e. saves half of
its capital as soon as it exceeds twice its starting capital), we say that
$d^{\prime}$ has the “saving property”. It is clear from the definition that
if $d$ is computable, then so is $d^{\prime}$, and moreover $d^{\prime}$ can
be uniformly computed given an index for $d$. Moreover, if for some sequence
$A$ one has
$\limsup_{n\rightarrow+\infty}d(A\upharpoonright{n})=+\infty$
then
$\lim_{n\rightarrow+\infty}d^{\prime}(A\upharpoonright{n})=+\infty$
which in particular implies
$\mathrm{Succ}(d)\subseteq\mathrm{Succ}(d^{\prime})$ (it is easy to see that
it is in fact an equality). Thus, whenever one considers a martingale $d$, one
can assume without loss of generality that it has the saving property (as long
as we are only interested in the success set of martingales, not in the growth
rate of their capital). The key property (for our purposes) of saving
martingales is the following.
###### Lemma 1
Let $b=(d,\pi)$ be a total injective strategy such that $d$ has the saving
property. Let $d^{\prime}=\mathrm{Av}_{b}$. Then
$\mathrm{Succ}(b)\subseteq\mathrm{Succ}(d^{\prime})$.
###### Proof
Suppose that $b=(d,\pi)$ succeeds on a sequence $A$. Since $d$ has the saving
property, for arbitrarily large $k$ there exists a finite prefix
$A\upharpoonright{n}$ of $A$ such that a capital of at least $k$ is saved
during the finite game of $b$ against $A$. We then have
$\hat{b}(w^{\prime})\geq k$ for all extensions $w^{\prime}$ of
$A\upharpoonright{n}$ (as a saved capital is never used), which by definition
of $\mathrm{Av}_{b}$ implies $\mathrm{Av}_{b}(A\upharpoonright{m})\geq k$ for
all $m\geq n$. Since $k$ can be chosen arbitrarily large, this finishes the
proof. ∎
Now the proof of Theorem 3.2 is as follows. Let $b=(d,\pi)$ be a total
computable permutation strategy. By the above discussion, let $d^{\prime}$ be
the saving version of $d$, so that
$\mathrm{Succ}(d)\subseteq\mathrm{Succ}(d^{\prime})$. Setting
$b^{\prime}=(d^{\prime},\pi)$, we have
$\mathrm{Succ}(b)\subseteq\mathrm{Succ}(b^{\prime})$. By Proposition 1 and
Lemma 1, $d^{\prime\prime}=\mathrm{Av}_{b^{\prime}}$ is a total computable
martingale, and
$\mathrm{Succ}(b)\subseteq\mathrm{Succ}(b^{\prime})\subseteq\mathrm{Succ}(d^{\prime\prime})$
as wanted.
### 3.3 Understanding the strength of injective strategies: the class
$\mathbf{TIR}$
While the class of computably random sequence (i.e. the class $\mathbf{TMR}$)
is closed under computable permutations of the bits, we now see that this
result does not extend to computable injections. To wit, the following theorem
is true.
###### Theorem 3.3
Let $A\in 2^{\omega}$. Let $\\{n_{k}\\}_{k\in{{\mathbb{N}}}}$ be a computable
sequence of integers such that $n_{k+1}\geq 2n_{k}$ for all $k$. Suppose that
$A$ is such that:
$\mathrm{C}\left({A\upharpoonright{n_{k}}}\,\mid\,{k}\right)\leq\log(n_{k})-3\log(\log(n_{k}))$
for infinitely many $k$. Then $A\notin\mathbf{TIR}$.
###### Proof
Let $A$ be a sequence satisfying the hypothesis of the theorem. Assuming,
without loss of generality, that $n_{0}=0$, we partition ${{\mathbb{N}}}$ into
an increasing sequence of intervals $I_{0},I_{1},I_{2},\ldots$ where
$I_{k}=[n_{k},n_{k+1})$. Notice that we have for all $k$:
$\mathrm{C}\left({A\upharpoonright{I}_{k}}\,\mid\,{k}\right)\leq\mathrm{C}\left({A\upharpoonright{n}_{k+1}}\,\mid\,{k+1}\right)+O(1)$
By the hypothesis of the theorem, the right-hand side of the above inequality
is bounded by $\log(n_{k+1})-3\log(\log(n_{k+1}))$ for infinitely many $k$.
Additionally, we have $|I_{k}|=n_{k+1}-n_{k}$ which by hypothesis on the
sequence $n_{k}$ implies $|I_{k}|\geq n_{k+1}/2$, and hence
$\log(|I_{k}|)=\log(n_{k+1})+O(1)$ and
$\log(\log(|I_{k}|))=\log(\log(n_{k+1}))+O(1)$. It follows that
$\mathrm{C}\left({A\upharpoonright{I}_{k}}\,\mid\,{k}\right)\leq\log(|I_{k}|)-3\log(\log(|I_{k}|))-O(1)$
for infinitely many $k$, hence
$\mathrm{C}\left({A\upharpoonright{I}_{k}}\,\mid\,{k}\right)\leq\log(|I_{k}|)-2\log(\log(|I_{k}|))$
for infinitely many $k$.
Let us call $S_{k}$ the set of strings $w$ of length $|I_{k}|$ such that
$\mathrm{C}\left({w}\,\mid\,{|I_{k}|}\right)\leq\log(|I_{k}|)-2\log(\log(|I_{k}|))$
(to which $A\upharpoonright{I_{k}}$ belongs for infinitely many $k$). By the
standard counting argument, there are at most
$s_{k}=2^{\log(|I_{k}|)-2\log(\log(|I_{k}|))}=\frac{|I_{k}|}{\log^{2}(|I_{k}|)}$
strings in $S_{k}$. For every $k$, we split $I_{k}$ into $s_{k}$ consecutive
disjoint intervals of equal length:
$I_{k}=J^{0}_{k}\cup J^{1}_{k}\cup\ldots\cup J^{s_{k}-1}_{k}$
$\mathbb{N}$$I_{k+1}$$J_{k+1}^{0}$$J_{k}^{s_{k}-1}$$J_{k}^{1}$$J_{k}^{0}$$I_{k}$0
We design a betting strategy as follows. We start with a capital of $2$. We
then reserve for each $k$ an amount $1/(k+1)^{2}$ to be bet on the bits in
positions in $I_{k}$ (this way, the total amount we distribute is smaller than
$2$), and we split this evenly between the $J^{i}_{k}$, i.e. we reserve an
amount $\frac{1}{s_{k}\cdot(k+1)^{2}}$ for every $J^{i}_{k}$. We then
enumerate the sets $S_{k}$ in parallel. Whenever the $e$-th element
$w^{e}_{k}$ of some $S_{k}$ is enumerated, we see $w^{e}_{k}$ as a possible
candidate to be equal to $A\upharpoonright{I_{k}}$, and we bet the reserved
amount $\frac{1}{s_{k}\cdot(k+1)^{2}}$ on the fact that
$A\upharpoonright{I_{k}}$ coincides with $w^{e}_{k}$ on the bits whose
position is in $J^{e}_{k}$. If we are successful (this in particular happens
whenever $w^{e}_{k}=A\upharpoonright{I_{k}}$), our reserved capital for this
$J^{e}_{k}$ is multiplied by $2^{|J^{e}_{k}|}$, i.e. we now have for this
$J^{e}_{k}$, a capital of
$\frac{1}{s_{k}\cdot(k+1)^{2}}\cdot 2^{(|I_{k}|/s_{k})}$
Replacing $s_{k}$ by its value (and remembering that $|I_{k}|\geq
2^{k-O(1)}$), an elementary calculation shows that this quantity is greater
than $1$ for almost all $k$. Thus, our betting strategy succeeds on $A$.
Indeed, for infinitely many $k$, $A\upharpoonright{I_{k}}$ is an element of
$S_{k}$, hence for some $e$ we will be successful in the above sub-strategy,
making an amount of money greater than $1$ for infinitely many $k$, hence our
capital tends to infinity throughout the game. Finally, it is easy to see that
this betting strategy is total: it simply is a succession of doubling
strategies on an infinite c.e. set of words, and it is injective as the
$J^{e}_{k}$ form a partition of ${{\mathbb{N}}}$, and the order of the bits we
bet on is independent of $A$ (in fact, we see our betting strategy succeeds on
_all_ sequences $\alpha$ satisfying the hypothesis of the theorem). ∎
As an immediate corollary, we get the following.
###### Corollary 1
If for a sequence $A$ we have for all $n$
$\mathrm{C}\left({A\upharpoonright{n}}\,\mid\,{n}\right)<\log n-4\log\log
n+\textnormal{O}(1)$, then $A\not\in\mathbf{TIR}$.
Another interesting corollary of our construction is that the class of all
computable sequences can be covered by a single total computable injective
strategy.
###### Corollary 2
There exists a single total computable injective strategy which succeeds
against all computable elements of $2^{\omega}$.
###### Proof
This is because, as we explained above, the strategy we construct in the proof
of Theorem 3.3 succeeds against _every_ sequence $A$ such that
$\mathrm{C}\left({A\upharpoonright{n_{k}}}\,\mid\,{k}\right)\leq\log(n_{k})-3\log(\log(n_{k}))$
for infinitely many $k$. This in particular includes all computable sequences
$A$, for which
$\mathrm{C}\left({A\upharpoonright{n_{k}}}\,\mid\,{k}\right)=O(1)$. ∎
The lower bound on Kolmogorov complexity given in Theorem 3.3 is quite tight,
as witnessed by the following theorem.
###### Theorem 3.4
For every computable order $h$ there is a sequence $A\in\mathbf{TIR}$ such
that $\textnormal{C}(A\upharpoonright n\mid
n)\leq\log(n)+h(n)+\textnormal{O}(1)$. In particular, we have
$\textnormal{C}(A\upharpoonright{n})\leq 2\log(n)+h(n)+\textnormal{O}(1)$.
###### Proof
The proof is a modification of the proof of Theorem 3.1. This time, we want to
diagonalize against all _non-monotonic_ total computable injective betting
strategies. Like in the proof of Theorem 3.1, we add them one by one,
discarding the partial strategies. However, to achieve the construction of $A$
by diagonalization, we will diagonalize against the average martingales of the
strategies we consider. As explained on page 3.2, we can assume that all total
computable injective strategies have the saving property, hence defeating
$\mathrm{Av}_{b}$ is enough to defeat $b$ (by Lemma 1). The proof thus goes as
follows:
Fix a fast growing computable function $f$, to be specified later. We start
with a martingale $D_{0}=1$ (the constant martingale equal to $1$) and
$w_{0}=\epsilon$. For all $k$ we do the following. Assume we have constructed
a prefix $w_{k}$ of $A$ of length $f(k)$, and that we are currently
diagonalizing against a martingale $D_{k}$, so that $D_{k}(w_{k})<2$. We then
enumerate a new partial computable injective betting strategy $b$. If it is
not total, we memorize this fact using one extra bit of information, and we
set $D_{k+1}=D_{k}$. Otherwise, we set $d_{k+1}=\mathrm{Av}_{b}$ and compute a
positive rational $\alpha_{k+1}$ such that
$(D_{k}+\alpha_{k+1}d_{k+1})(w_{k})<2$, and finally set
$D_{k+1}=D_{k}+\alpha_{k+1}d_{k+1}$.
Then, we define $w_{k+1}$ to be the extension of $w_{k}$ of length $f(k+1)$ by
the usual diagonalization against $D_{k+1}$, maintaining the inequality
$D_{k+1}(u)<2$ for all prefixes $u$ of $w_{k+1}$. The infinite sequence $A$
obtained this way defeats all the average martingales of all total computable
injective strategies, hence by Lemma 1, $A\in\mathbf{TIR}$.
It remains to show that $A$ has low Kolmogorov complexity. Suppose we want to
describe $A\upharpoonright{n}$ for some $n\in[f(k),f(k+1))$. This can be done
by giving $n$, the subset of $\\{0,\ldots,k\\}$ (of complexity $k+O(1)$)
corresponding to the indices of the total computable injective strategies
among the first $k$ partial computable ones, and by giving the restriction of
$D_{k+1}$ to words of length at most $n$. From all this, $A\upharpoonright{n}$
can be reconstructed following the above construction. It remains to evaluate
the complexity of the restriction of $D_{k+1}$ to words of length at most $n$.
We already know the total computable injective strategies $b_{0},\ldots,b_{k}$
that are being considered in the definition of $D_{k+1}$. For all $i$, let
$\pi_{i}$ be the injection associated to $b_{i}$. We need to compute, for all
$0\leq i\leq k$, the martingale $d_{i}=\mathrm{Av}_{b_{i}}$ on words of length
at most $n$. By Proposition 1, this can be done knowing
$\pi_{i}({{\mathbb{N}}})\cap[0,\ldots,n-1]$ for all $0\leq i\leq k$. But if
the $\pi_{i}$ are known, this set is uniformly c.e. in $i,n$. Hence, we can
enumerate all the sets $\pi_{i}({{\mathbb{N}}})\cap[0,\ldots,n-1]$ (for $0\leq
i\leq k$) in parallel, and simply give the last couple $(i,l)$ such that $l$
is enumerated in $\pi_{i}({{\mathbb{N}}})\cap[0,\ldots,n-1]$. Since $0\leq
i\leq k$ and $0\leq l<n$, this costs an amount of information
$\textnormal{O}(\log k)+\log n$. To sum up, we get
$\mathrm{C}\left({A\upharpoonright{n}}\,\mid\,{n}\right)\leq
k+\textnormal{O}(\log k)+\log n$
Thus, it suffices to take $f$ growing fast enough to ensure that the term
$\leq k+\textnormal{O}(\log k)$ is smaller than $h(n)+\textnormal{O}(1)$. ∎
## 4 Randomness notions based on partial computable strategies
We now turn our attention to the second line of Figure 1, i.e., to those
randomness notions that are based on partial computable betting strategies.
### 4.1 The class $\mathbf{PMR}$: partial computable martingales are stronger
than total ones
We have seen in the previous section that some sequences in $\mathbf{TIR}$
(and a fortiori $\mathbf{TPR}$ and $\mathbf{TMR}$) may be of very low
complexity, namely logarithmic. This is not the case anymore when one allows
partial computable strategies, even monotonic ones.
###### Theorem 4.1 (Merkle [7])
If $\textnormal{C}(A\upharpoonright{n})=\textnormal{O}(\log n)$ then
$A\not\in\mathbf{PMR}$.
However, the next theorem, proven by An. A. Muchnik, shows that allowing
slightly super-logarithmic growth of the Kolmogorov complexity is enough to
construct a sequence in $\mathbf{PMR}$.
###### Theorem 4.2 (Muchnik et al. [10])
For every computable order $h$ there is a sequence $A\in\mathbf{\mathbf{PMR}}$
such that, for all $n\in\mathbb{N}$,
$\mathrm{C}\left({A\upharpoonright{n}}\,\mid\,{n}\right)\leq
h(n)\log(n)+\textnormal{O}(1).$
###### Proof
The proof is almost identical to the proof of Theorem 3.1. The only difference
is that we insert _all_ partial computable martingales one by one, and
diagonalize against their weighted sum as before. It may happen however, that
at some stage of the construction, one of the martingales becomes undefined.
All we need to do then is to memorize this, and ignore this particular
martingale from that point on. Call $A$ the sequence we obtain by this
construction. We want to describe $A\upharpoonright{n}$. To do so, we need to
specify $n$, and, out of the $k$ partial computable martingales that are
inserted before stage $n$, which ones have diverged, and at what stage, hence
an information of $\textnormal{O}(k\log n)$ (giving the position where a
particular martingale diverges costs $\textnormal{O}(\log n)$ bits, and there
are $k$ martingales. Since we can insert martingales as slowly as we like
(following some computable order), the complexity of $A\upharpoonright{n}$
given $n$ can be taken to be smaller than $h(n)\log n+O(1)$ (where $h$ is a
computable order, fixed before the construction of $A$). ∎
### 4.2 The class $\mathbf{PPR}$
In the case of total strategies, allowing permutation gives no real additional
power, as $\mathbf{TMR}=\mathbf{TPR}$. Very surprisingly, Muchnik showed that
in the case of partial computable strategies, permutation strategies are a
real improvement over monotonic ones. To wit, the following theorem (quite a
contrast to Theorem 4.2!).
###### Theorem 4.3 (Muchnik [10])
If there is a computable order $h$ such that for all $n$ we have
$\textnormal{K}(A\upharpoonright n)\leq n-h(n)-\textnormal{O}(1)$, then
$A\not\in\mathbf{PPR}$.
Note that the proof used by Muchnik in [10] works if we replace K by C in the
above statement.
###### Theorem 4.4
For every computable order $h$ there is a sequence $A\in\mathbf{PPR}$, such
that there are infinitely many $n$ where
$\mathrm{C}\left({A\upharpoonright{n}}\,\mid\,{n}\right)<h(n)$.
Furthermore, if we have an infinite computable set $S\subseteq\mathbb{N}$, we
can choose the infinitely many lengths $n$ such that they all are contained in
$S$.
###### Lemma 2
Let $d$ be a partial computable martingale. Let $\mathcal{C}$ be an
effectively closed subset of $2^{\omega}$. Suppose that $d$ is total on every
element of $\mathcal{C}$. Then there exists a total computable martingale
$d^{\prime}$ such that
$\mathrm{Succ}(d)\cap\mathcal{C}=\mathrm{Succ}(d^{\prime})\cap\mathcal{C}$.
###### Proof
The idea of the proof is simple: the martingale $d^{\prime}$ will try to mimic
$d$ while enumerating the complement $\mathcal{U}$ of $\mathcal{C}$. If at
some stage a cylinder $[w]$ is covered by $\mathcal{U}$, then $d$ will be
passive (i.e. defined but constant) on the sequences extending $w$. As we do
not care about the behavior of $d^{\prime}$ on $\mathcal{U}$ (as long as it is
defined), this will be enough to get the conclusion.
Let $d,\mathcal{C}$ be as above. We build the martingale $d^{\prime}$ on words
by induction. Define $d^{\prime}(\epsilon)=d(\epsilon)$ (here we assume
without loss of generality that $d(\epsilon)$ is defined, otherwise there is
nothing to prove). During the construction, some words will be marked as
inactive, on which the martingale will be passive; initially, there is no
inactive word. On active words $w$, we will have $d(w)=d^{\prime}(w)$.
Suppose for the sake of the induction that $d^{\prime}(w)$ is already defined.
If $w$ is marked as inactive, we mark $w0$ and $w1$ as inactive, and set
$d(w0)=d(w1)=d(w)$. Otherwise, by the induction hypothesis, we have
$d(w)=d^{\prime}(w)$. We then run in parallel the computation of $d(w0)$ and
$d(w1)$, and enumerate the complement $\mathcal{U}$ of $\mathcal{C}$ until one
of the two above events happens:
* (a)
$d(w0)$ and $d(w1)$ become defined. Then set $d^{\prime}(w0)=d(w0)$ and
$d^{\prime}(w1)=d(w1)$
* (b)
The cylinder $[w]$ gets covered by $\mathcal{U}$. In that case, mark $w0$ and
$w1$ as inactive and set $d^{\prime}(w0)=d^{\prime}(w1)=d^{\prime}(w)$
Note that one of these two events _must_ happen: indeed, if $d(w0)$ and
$d(w1)$ are undefined (remember that by the definition of a martingale,
Definition 1, that they are either both defined or both undefined), then this
means that $d$ diverges on _any_ element of $[w0]\cup[w1]=[w]$. Hence, by
assumption, $[w]\cap\mathcal{C}=\emptyset$, i.e. $[w]\subseteq\mathcal{U}$. It
remains to verify that
$\mathrm{Succ}(d)\cap\mathcal{C}=\mathrm{Succ}(d^{\prime})\cap\mathcal{C}$.
Let $A\in\mathcal{C}$. Since $d$ is total on $A$ by assumption, during the
construction of $d^{\prime}$ on $A$, we will always be in case (a), hence we
will have for all $n$,
$d(A\upharpoonright{n})=d^{\prime}(A\upharpoonright{n})$. The result follows
immediately. ∎
###### Corollary 3
Let $b=(d,\pi)$ be a partial computable permutation strategy (resp. injective
strategy). Let $\mathcal{C}$ be an effectively closed subset of $2^{\omega}$.
Suppose that $b$ is total on every element of $\mathcal{C}$. Then there exists
a total computable permutation strategy (resp. injective strategy)
$b^{\prime}$ such that
$\mathrm{Succ}(b)\cap\mathcal{C}=\mathrm{Succ}(b^{\prime})\cap\mathcal{C}$.
###### Proof
This follows from the fact that the image or pre-image of an effectively
closed set under a computable permutation of the bits is itself a closed set:
take $b=(d,\pi)$ and $\mathcal{C}$ as above. Let $\bar{\pi}$ be the map
induced on $2^{\omega}$ by $\pi$, i.e. the map defined for all $A\in
2^{\omega}$ by
$\bar{\pi}(A)=A(\pi(0))A(\pi(1))A(\pi(2))\ldots$
For any given sequence $A\in\mathcal{C}$, $b$ succeeds on $A$ if and only if
$d$ succeeds on $\bar{\pi}(A)$. As $\bar{\pi}(A)\in\bar{\pi}(\mathcal{C})$,
and $\bar{\pi}(\mathcal{C})$ is an effectively closed set, by , there exists a
total martingale $d^{\prime}$ such that
$\mathrm{Succ}(d)\cap\bar{\pi}(\mathcal{C})=\mathrm{Succ}(d^{\prime})\cap\bar{\pi}(\mathcal{C})$.
Thus, $d^{\prime}$ succeeds on $\bar{\pi}(A)$, or equivalently,
$b^{\prime}=(d^{\prime},\pi)$ succeeds on $A$. Thus $b^{\prime}$ is as
desired. ∎
###### Proof (of Theorem 4.4)
Again, this proof is a variant of the proof of Theorem 3.1: we add strategies
one by one, diagonalizing, at each stage, against a finite weighted sum of
total monotonic strategies (i.e. martingales). Of course, not all strategies
have this property, but we can reduce to this case using the techniques we
presented above. Suppose that in the construction of our sequence $A$, we have
already constructed an initial segment $w_{k}$, and that up to this stage we
played against a weighted sum of $k$ total martingales
$D_{k}=\sum_{i=1}^{k}\alpha_{i}\,d_{i}$
where the $d_{i}$ are total computable martingales, ensuring that $D(u)<2$ for
all prefix $u$ of $w$. Suppose we want to introduce a new strategy
$b=(d,\pi)$. There are three cases:
Case 0: the new strategy is not valid, i.e. $\pi$ is not a permutation. In
this case, we just add one bit of extra information to record this, and ignore
$b$ from now on, i.e. we set $w_{k+1}=w_{k}$, $d_{k+1}=0$ (the zero
martingale), and $D_{k+1}=D_{k}+d_{k+1}=D_{k}$.
Case 1: the strategy $b$ is indeed a partial computable permutation strategy,
and there exists an extension $w^{\prime}$ of $w$ such that $D_{k}(u)<2$ for
all prefixes $u$ of $w^{\prime}$, and $b$ diverges on $w^{\prime}$. In this
case, we simply take $w^{\prime}$ as our new prefix of $A$, as it both
diagonalizes against $D$, and defeats $b$ (since $b$ diverges on $w^{\prime}$,
it will not win against _any_ possible extension of $w^{\prime}$). We can thus
ignore $b$ from that point on, so we set $w_{k+1}=w^{\prime}$, $d_{k+1}=0$ and
$D_{k+1}=D_{k}+d_{k+1}=D_{k}$.
Case 2: if we are not in one of the two previous cases, this means that our
strategy $b=(d,\pi)$ is a partial computable permutation strategy, and that
$b$ is total on the whole $\mathrm{\Pi}^{0}_{1}$ class
$\mathcal{C}_{k}=[w_{k}]\cap\\{X\in 2^{\omega}\mid\forall
n\,D_{k}(X\upharpoonright{n})<2\\}$
Thus, by Lemma 3, there exists a total computable permutation strategy
$b^{\prime}$ such that
$\mathrm{Succ}(b)\cap\mathcal{C}_{k}=\mathrm{Succ}(b^{\prime})\cap\mathcal{C}_{k}$.
And by Theorem 3.2, there exists a total computable martingale
$d^{\prime\prime}$ such that
$\mathrm{Succ}(b^{\prime})\subseteq\mathrm{Succ}(d^{\prime\prime})$. Thus, we
can replace $b$ by $d^{\prime\prime}$, and defeating $d^{\prime\prime}$ will
be enough to defeat $b$ as long as the sequence we construct is in
$\mathcal{C}_{k}$. We thus set $d_{k+1}=d^{\prime\prime}$, $w_{k+1}=w_{k}$ and
$D_{k+1}=\sum_{i=1}^{k+1}\alpha_{i}\,d_{i}$
where $\alpha_{k+1}$ is sufficiently small to have $D_{k+1}(w_{k+1})<2$.
Once we have added a new monotonic martingale, we (as usual) computably find
an extension $w^{\prime\prime}$ of $w_{k+1}$, ensuring that $D_{k+1}(u)<2$ for
all prefix $u$ of $w^{\prime\prime}$, taking $w^{\prime\prime}$ long enough to
have
$\mathrm{C}\left({w^{\prime\prime}}\,\mid\,{|w^{\prime\prime}|}\right)\leq
h(|w^{\prime\prime}|)$. We then set $w_{k+1}=w^{\prime\prime}$, then add a
$k+2$-th strategy and so on.
Note that since $w^{\prime\prime}$ can be chosen arbitrarily large, if we have
fixed a computable susbet $S$ of ${{\mathbb{N}}}$, we can also ensure that
$|w^{\prime\prime}|$ belong to $S$ if we like.
It is clear that the infinite sequence $A$ constructed via this process
satisfies
$\mathrm{C}\left({A\upharpoonright{n}}\,\mid\,{n}\right)\leq h(n)$
for infinitely many $n$ (and, since Case 2 happens infinitely often, if we fix
a given computable set $S$, we can ensure that infinitely many of such $n$
belong to $S$). To see that it belongs to $\mathbf{PPR}$, we notice that since
for all $k$, $D_{k+1}\geq D_{k}$ and $w_{k}\sqsubseteq w_{k+1}$, we have
$\mathcal{C}_{k+1}\subseteq\mathcal{C}_{k}$ and thus
$A\in\bigcap_{k}\mathcal{C}_{k}$. Now, given a total computable strategy
$b=(d,\pi)$, let $k$ be the stage where $b$ was considered, and replaced by
the martingale $d_{k}$. Since by construction of $A$, $d_{k+1}$ does not win
against $A$ and by definition of $d_{k}$,
$\mathrm{Succ}(b)\cap\mathcal{C}_{k}\subseteq\mathrm{Succ}(d_{k})\cap\mathcal{C}_{k}$,
it follows that $A\notin\mathrm{Succ}(b)$. ∎
Now that we have assembled all our tools, we can easily prove the desired
results.
###### Theorem 4.5
The following statements hold.
1. 1.
$\mathbf{PPR}\not\subseteq\mathbf{TIR}$
2. 2.
$\mathbf{TIR}\not\subseteq\mathbf{PMR}$
3. 3.
$\mathbf{PMR}\not\subseteq\mathbf{PPR}$
From these results it easily follows that in Figure 2 no inclusion holds
except those indicated and those implied by transitivity.
###### Proof
1. 1.
Choose a computable sequence $\\{n_{k}\\}_{k}$ fulfilling the requirements of
Theorem 3.3 such that $\textnormal{C}(k)\leq\log\log n_{k}$ for all $k$. The
members of this set then form a computable set $S$. Use Theorem 4.4 to
construct a sequence $A\in\mathbf{PPR}$ such that
$\textnormal{C}(A\upharpoonright n\mid n)<\log\log n$ at infinitely many
places in $S$. We then have for infinitely many $k$
$\textnormal{C}(A\upharpoonright n_{k}\mid
k)\leq\textnormal{C}(A\upharpoonright
n_{k})\leq\textnormal{C}(A\upharpoonright n_{k}\mid n_{k})+2\log\log n_{k}\leq
3\log\log n_{k},$
so $A$ cannot be in $\mathbf{TIR}$ according to Theorem 3.3.
2. 2.
Follows immediately from Theorems 3.4 and 4.1.
3. 3.
Follows immediately from Theorems 4.2 and 4.3. ∎
monotonic permutation injection total $\mathbf{TMR}$ $=$ $\mathbf{TPR}$
$\supsetneq$ $\mathbf{TIR}$ $\subsetneq$ $\subsetneq$ $\subsetneq$ partial
$\mathbf{PMR}$ $\supsetneq$ $\mathbf{PPR}$ $\supsetneq$ $\mathbf{PIR}$
Figure 2: Assembled class inclusion results
## References
* [1] Harry Buhrman, Dieter van Melkebeek, Kenneth Regan, D. Sivakumar, and Martin Strauss. A generalization of resource-bounded measure, with application to the BPP vs. EXP problem. SIAM Journal of Computing, 30(2):576–601, 2000.
* [2] Rod Downey and Denis Hirschfeldt. Algorithmic Randomness. Springer, to appear.
* [3] Bart Kastermans and Steffen Lempp. Comparing notions of randomness. Manuscript, 2008.
* [4] James Lathrop and Jack Lutz. Recursive computational depth. Information and Computation, 153(1):139–172, 1999.
* [5] Ming Li and Paul Vitányi. Kolmogorov Complexity and Its Applications. Springer, 2008.
* [6] Wolfgang Merkle. The Kolmogorov-Loveland stochastic sequences are not closed under selecting subsequences. Journal of Symbolic Logic, 68:1362–1376, 2003.
* [7] Wolfgang Merkle. The complexity of stochastic sequences. Journal of Computer and System Sciences, 74(3):350–357, 2008.
* [8] Wolfgang Merkle, Joseph S. Miller, André Nies, Jan Reimann, and Frank Stephan. Kolmogorov-Loveland randomness and stochasticity. Annals of Pure and Applied Logic, 138(1-3):183–210, 2006.
* [9] Joseph Miller and André Nies. Randomness and computability: open questions. Bulletin of Symbolic Logic, 12(3):390–410, 2006.
* [10] Andrei A. Muchnik, Alexei Semenov, and Vladimir Uspensky. Mathematical metaphysics of randomness. Theoretical Computer Science, 207(2):263–317, 1998.
* [11] André Nies. Computability and Randomness. Oxford University Press, 2009.
|
arxiv-papers
| 2009-07-14T10:41:08 |
2024-09-04T02:49:03.903768
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Laurent Bienvenu, Rupert Hoelzl, Thorsten Kraling, Wolfgang Merkle",
"submitter": "Laurent Bienvenu",
"url": "https://arxiv.org/abs/0907.2324"
}
|
0907.2469
|
1–8
# Astrometric Solar-System Anomalies
John D. Anderson1 Michael Martin Nieto2 1Jet Propulsion Laboratory (Retired)
121 S. Wilson Ave., Pasadena, CA 91106-3017 U.S.A.
email: [email protected]
2Theoretical Division (MS-B285), Los Alamos National Laboratory
Los Alamos, New Mexico 87645 U.S.A.
email: [email protected]
(2009)
###### Abstract
There are at least four unexplained anomalies connected with astrometric data.
Perhaps the most disturbing is the fact that when a spacecraft on a flyby
trajectory approaches the Earth within 2000 km or less, it often experiences a
change in total orbital energy per unit mass. Next, a secular change in the
astronomical unit AU is definitely a concern. It is reportedly increasing by
about 15 cm yr-1. The other two anomalies are perhaps less disturbing because
of known sources of nongravitational acceleration. The first is an apparent
slowing of the two Pioneer spacecraft as they exit the solar system in
opposite directions. Some astronomers and physicists, including us, are
convinced this effect is of concern, but many others are convinced it is
produced by a nearly identical thermal emission from both spacecraft, in a
direction away from the Sun, thereby producing acceleration toward the Sun.
The fourth anomaly is a measured increase in the eccentricity of the Moon’s
orbit. Here again, an increase is expected from tidal friction in both the
Earth and Moon. However, there is a reported unexplained increase that is
significant at the three-sigma level. It is prudent to suspect that all four
anomalies have mundane explanations, or that one or more anomalies are a
result of systematic error. Yet they might eventually be explained by new
physics. For example, a slightly modified theory of gravitation is not ruled
out, perhaps analogous to Einstein’s 1916 explanation for the excess
precession of Mercury’s perihelion.
###### keywords:
gravitation, celestial mechanics, astrometry
††volume: 261††journal: Relativity in Fundamental Astronomy:
Dynamics, Reference Frames, and Data Analysis††editors: Sergei Klioner, P.
Kenneth Seidelmann & Michael Soffel
## 1 Earth flyby anomaly
The first of the four anomalies considered here is a change in orbital energy
for spacecraft that fly past the Earth on approximately hyperbolic
trajectories ([Anderson et al. (2008), Anderson et al. 2008]). By means of a
close flyby of a planet, it is possible to increase or decrease a spacecraft’s
heliocentric orbital velocity far beyond the capability of any chemical
propulsion system (see for example [Flandro (1966), Flandro 1966] and [Wiesel
(1989), Wiesel 1989]). It has been known for over a century that when a small
body encounters a planet in the solar system, the orbital parameters of the
small body with respect to the Sun will change. This is related to Tisserand’s
criterion for the identification of comets ([Danby (1988), Danby 1988]).
During a gravity assist, which is now routine for interplanetary missions, the
orbital energy with respect to the planet is conserved. Therefore, if there is
an observed energy increase or decrease with respect to the planet during the
flyby, it is considered anomalous ([Anderson et al. (2007), Anderson et al.
2007]).
Unfortunately, it is practically impossible to detect a small energy change
with planetary flybys, both because an energy change is difficult to separate
from errors in the planet’s gravity field and because of the unfavorable
Doppler tracking geometry of a distant planet. The more favorable geometry of
an Earth flyby is needed. Also the Earth’s gravity field is well known from
the GRACE mission ([Tapley et al. (2004), Tapley et al. 2004]). Earth’s
gravity is not a significant source of systematic error for the flyby orbit
determination ([Anderson et al. (2008), Anderson et al. 2008]).
The flyby anomaly was originally detected in radio Doppler data from the first
of two Earth flybys by the Galileo spacecraft (for a description of the
mission see [Russell (1992), Russell 1992]). After launch on 1989-Oct-18, the
spacecraft made one flyby of Venus on 1990-Feb-10, and subsequently two flybys
of Earth on 1990-Dec-08 and two years later on 1992-Dec-08. The spacecraft
arrived at Jupiter on 1995-Dec-07.
Without these planetary gravity assists, a propulsion maneuver of 9 km s-1
would have been needed to get from low Earth orbit to Jupiter. With them, the
Galileo spacecraft left low Earth orbit with a maneuver of only 4 km s-1. The
first Earth flyby occurred at an altitude of 960 km. The second, which
occurred at an altitude of 303 km, was affected by atmospheric drag, and
therefore it was difficult to obtain an unambiguous measurement of an
anomalous energy change on the order of a few mm s-1.
The anomalistic nature of the flyby is demonstrated by Fig. 1. The pre-perigee
fit produces residuals which are distributed about a zero mean with a standard
error of 0.087 mm s-1. However, when the pre-perigee fit is extrapolated to
the post-perigee data, there is a clear asymptotic bias of 3.78 mm s-1 in the
residuals. Further, the data immediately after perigee indicates that there is
perhaps an anomalous acceleration acting on the spacecraft from perigee plus
2253 s, the first data point after perigee, to about 10 hr, the start of the
asymptotic bias. (A discussion of these residuals and how they were obtained
can be found in [Antreasian & Guinn (1998), Antreasian & Guinn 1998].)
Figure 1: Doppler residuals (observed minus computed) converted to units of
line of sight (LOS) velocity about a fit to the pre-perigee Doppler data, and
the failure of this fit to predict the post perigee data. The mean offset in
the post-perigee data approaches 3.78 mm s-1, as shown by the dashed line. The
solid line connecting the post-perigee data represents an eighth degree
fitting polynomial to data after perigee plus 2.30 hours. The time of perigee
is 1990-Dec-08 20:34:34.40 UTC.
A similar but larger effect was observed during an Earth flyby by the Near
Earth Asteroid Rendezvous (NEAR) spacecraft. The spacecraft took four years
after launch to reach the asteroid (433) Eros in February 2000 ([Dunham et al.
(2005), Dunham et al. 2005]). For the Earth gravity assist in January 1998,
the pre-perigee Doppler data can be fit with a residual standard error of
0.028 mm s-1. Note that the residuals are smaller for NEAR with its Doppler
tracking in the X-Band at about 8.0 GHz, as opposed to Galileo in the S-Band
at about 2.3 GHz. Scattering of the two-way radio signal by free ionospheric
electrons is less at the higher frequency, although systematic and random
effects from atmospheric refraction limit the X-Band tracking accuracy.
Nevertheless, the post-perigee residuals ([Antreasian & Guinn (1998),
Antreasian & Guinn 1998]) show a clear asymptotic bias of 13.51 mm s-1 (see
Fig. 2). There is also some evidence from Fig. 2 that an anomalistic
acceleration might be acting over perhaps plus and minus 10 hours of perigee.
Figure 2: Similar to Fig. 1 but for the NEAR Doppler residuals. The mean
offset in the post-perigee data approaches 13.51 mm s-1, as shown by the
dashed line. The post-perigee data start at perigee plus 2.51 hours. The time
of perigee is 1998-Jan-23 07:22:55.60 UTC.
The anomalistic bias can also be demonstrated for both GLLI and NEAR by
fitting the post-perigee data and using that fit to predict the pre-perigee
residuals ([Anderson et al. (2008), Anderson et al. 2008]). For both
spacecraft, the two pre- and post-perigee fits are consistent with the same
velocity increases shown in Fig. 1 and Fig. 2.
Earth flybys by the Cassini spacecraft on 1999-Aug-18 and the Stardust
spacecraft in January 2001 yielded little or no information on the flyby
anomaly. Both spacecraft were affected by thruster firings which masked any
anomalous velocity change. However, on 2005-Mar-04 the Rosetta spacecraft
swung by Earth on its first flyby and an anomalous energy gain was once again
observed. Rosetta is an ESA mission with space navigation by the European
Space Operations Center (ESOC). As such it provides an independent analysis at
ESOC for both ESA and NASA tracking data for Rosetta ([Morley & Budnik (2006),
Morley & Budnik 2006]). The Rosetta anomaly was confirmed independently at JPL
with an asymptotic velocity increase of (1.80 $\pm$ 0.03) mm s-1
([anderson_etal08, Anderson et al. 2008]). Similar data analysis by [Anderson
et al. (2008)] yielded slightly different velocity changes than indicated by
Fig. 1 and Fig. 2 but with error bars. The best estimates are (3.92 $\pm$
0.03) mm s-1 for GLLI and (13.46 $\pm$ 0.01) mm s-1 for NEAR. Rosetta swung by
the Earth again on 2007-Nov-13 (RosettaII), but this time no anomaly was
reported.
There is most likely a distance dependence to the anomaly. The net velocity
increase is 3.9 mm s-1 for the Galileo spacecraft at a closest approach of 960
km, 13.5 mm s-1 for the NEAR spacecraft at 539 km, and 1.8 mm s-1 for the
Rosetta spacecraft at 1956 km. The altitude of RosettaII is 5322 km, perhaps
too high for a detection of the anomaly. A third Rosetta Earth swing-by
(RosettaIII) is scheduled for 2009-Nov-13 at a more favorable altitude of 2483
km. This third gravity assist, which possibly could reveal the anomaly, will
place Rosetta on a trajectory to rendezvous with Comet
67P/Churyumov–Gerasimenko on 2014-May-22 and a lander will be placed on the
comet on 2014-Nov-10. The spacecraft bus will orbit the comet and escort it
around the Sun until December 2015, when the comet will be at a heliocentric
distance of about one AU.
Indeed there is a distance-independent phenomenological formula that models
the anomaly quite accurately, at least for flybys at an altitude of 2000 km or
less, be that fortuitous or not ([anderson_etal08, Anderson et al. 2008]). The
percentage change in the excess velocity at infinity $v_{\infty}$ is given by
$\displaystyle\frac{\Delta v_{\infty}}{v_{\infty}}$ $\displaystyle=$
$\displaystyle K(\cos\delta_{i}-\cos\delta_{f}),$ (1) $\displaystyle K$
$\displaystyle=$
$\displaystyle\frac{2\omega_{\oplus}R_{\oplus}}{c}=3.099\times 10^{-6},$ (2)
where $\delta_{\\{i,f\\}}$ are the initial (ingoing) and final (outgoing)
declination angles given by
$\sin\delta_{\\{i,f\\}}=\sin I\cos\left(\omega\mp\psi\right).$ (3)
The parameter $\omega_{\oplus}$ is the Earth’s angular velocity of rotation,
$R_{\oplus}$ is the Earth’s mean radius, and $c$ is the velocity of light.
The angle $\psi$ is one half the total bending angle in the flyby trajectory,
$I$ is the osculating orbital inclination to the equator of date, and $\omega$
is the osculating argument of the perigee measured along the orbit from the
equator of date. The angle $\psi$ is related to the osculating eccentricity
$e$ by
$\sin\psi=\frac{1}{e}$ (4)
Alternatively, the total bending angle $2\psi$ can be obtained as the angle
between the asymptotic ingoing and outgoing velocity vectors.
## 2 Increase in the Astronomical Unit
Radar ranging and spacecraft radio ranging to the inner planets result in a
measurement of the AU to an accuracy of 3 m, or a percentage error of $2\times
10^{-11}$, making it the most accurately determined constant in all of
astronomy ([Pitjeva 2007], [Pitjeva & Standish 2009]). In SI units the AU can
be expressed by the constant $A$, or as the number of meters or seconds in one
AU. The two SI units are interchangeable by means of the defining constant
$c$, the speed of light in units m s-1. In this form, and in combination with
the IAU definition of the AU (Resolution No. 10
1976111http://www.iau.org/static/resolutions/IAU1976_French.pdf), there is an
equivalence between the AU and the mass of the Sun $M_{S}$ given by
$GM_{S}\equiv k^{2}A^{3},$ (5)
where $G$ is the gravitational constant and $k$ is Gauss’ constant.
According to IAU Resolution No. 10, $k$ is exactly equal to
$0.01720209895~{}AU^{3/2}~{}d^{-1}$, similar to $c$ exactly equal to
$299792458~{}m~{}s^{-1}$. The value of the AU is connected to the ranging
observations by the time unit used for the time delay of a radar signal or a
modulated spacecraft radio carrier wave, ideally the SI second, or
equivalently the day $d$ of 86400 s. The extraordinary accuracy in the AU is
based on Earth-Mars spacecraft ranging data over an interval from the first
Viking Lander on Mars in 1976 and continuing with Viking from 1976 to 1982,
Pathfinder P (1997), MGS from 1998 to 2003, and Odyssey from 2002 to 2008
([Pitjeva 2009a, Pitjeva 2009b]). In practice the AU is measured in units of
Coordinated Universal Time (UTC), the time scale used by the Deep Space
Network (DSN) in their frequency and timing system. Therefore the AU is given
in SI seconds as determined by International Atomic Time TAI ([Moyer 2003]).
The fitting models for the JPL ephemeris and for the IAA-RAS ephemeris
([Pitjeva & Standish 2009]) are relativistically consistent with ranging
measurements in units of SI seconds. It seems that we really do know the AU to
($149597870700~{}\pm~{}3$) m ([Pitjeva & Standish 2009]).
For purposes of deciding whether a measurement of a change in the AU is
feasible, we simulate Earth-Mars ranging at a 40-day sample interval over a
27-year observing interval starting on 1976-July-01, for a total of 248
simulated normal points. We approximate the tracking geometry by means of a
Newtonian integration of a four-body system consisting of the Sun, the Earth-
Moon barycenter, the Mars barycenter, and the Jupiter barycenter, all treated
as point masses. The initial conditions of the Earth and Mars are adjusted to
give a best fit to the distance between the Earth-Moon barycenter and the Mars
barycenter, as given by DE405. The rms error in this best fit is
$2.6~{}\times~{}10^{-5}$ AU, which is unacceptable as a fitting model, but
sufficient for a covariance analysis. In the real analysis ([Pitjeva 2009a,
Pitjeva 2009b]) the ranging data are represented by hundreds of parameters,
only one of which is the AU.
The parameters for our covariance analysis consist of the 12 state variables
for Earth and Mars, expressed as the Cartesian initial conditions at the July
1976 epoch, plus two parameters ($k_{1},k_{2}$) for $GM_{S}$ as given by
$k^{2}\left[1+k_{1}+k_{2}(t-\bar{t})\right]$ in units of AU${}^{3}d^{-2}$.
This is the most direct way to express a bias in the AU and its secular time
variation as a Newtonian perturbation. 222The AU is not determined in
ephemeris software by means of this physical approach (see Pitjeva (2007),
Pitjeva (2009a) and Pitjeva (2009b) for details). The masses of the three
planetary systems are constant at their DE405 values, and the initial
conditions of the Jupiter system are not included in the covariance matrix,
which makes it a 14$\times$14 matrix. The rank of this matrix is actually 12.
The mean Earth orbit defines the reference plane for the other orbits. Hence
there are only four Earth elements that can be inferred from the data. A
singular value decomposition (SVD) of the 14$\times$14 matrix can be obtained
and its pseudo inverse can be interpreted as the covariance matrix on the 14
parameters ([Lawson & Hanson 1974]). Actually all the information on $k_{1}$
and $k_{2}$ is obtained by the 8th singular value, so a rank 9 pseudo inverse
is more than sufficient for a study of the AU and its time variation. The mean
time $\bar{t}$ is introduced into the secular variation in $GM_{S}$ such that
$k_{1}$ and $k_{2}$ are uncorrelated. This mean time is 13.5 yr for the
simulation, but in the real analysis it should be taken as the mean of all the
observation times.
Taking account of the factor of three in Eq. 5, we normalize the result of the
covariance analysis to a standard error in the AU of 3.0 m, represented by
$k_{1}$ in the rank 12 matrix. The corresponding rank 9 standard error, where
it is assumed that all the remaining five singular values are perfectly known,
is 2.5 m. The corresponding error in the secular variation represented by
$k_{2}$ is 2.9 cm yr-1 for full rank 12 and 2.7 cm yr-1 for rank 9.
We conclude that at least the uncertainty part of the reported increase in the
AU ([Krasinsky & Brumberg 2004]) of (15 $\pm$ 4) cm yr-1 is reasonable. Any
future work should be focused on checking the actual mean value of the secular
increase and perhaps refining it. It is unlikely that its error bar can be
decreased below 3.0 cm yr-1 with existing Earth-Mars ranging data. However, if
the error in the AU can be reduced to $\pm$ 1.0 m with confidence, the error
in its secular variation could perhaps be reduced to $\pm$ 1.0 cm yr-1, with
Earth-Mars ranging alone. Other than that, the Cassini spacecraft carries an
X-Band ranging transponder ([Kliore (2004), Kliore 2004]). Range fixes on
Saturn presumably can be obtained for each Cassini orbital period of roughly
14.3 days over an observing interval from July 2004 to July 2009, or for as
long as the spacecraft is in orbit about Saturn and ranging data are
available. These data are not yet publicly available, but when they are
released, we can expect a standard error in each ranging normal point of about
5 m. Spacecraft ranging to Mercury during the MESSENGER and BepiColombo
missions could also add additional information for the AU and its secular
variation. If the AU is really increasing with time, the planetary orbits by
definition (Eq. 5) are shrinking and their periods are getting shorter, such
that their mean orbital longitudes are increasing quadratically with $t$, the
major effect that can be measured with Earth-planet ranging data.
However, rather than increasing, the AU should be decreasing, mainly as a
result of loss of mass to solar radiation, and to a much lesser extent to the
solar wind. The total solar luminosity is 3.845 $\times~{}10^{26}$ W
([Livingston 1999]). This luminosity divided by c2 gives an estimated mass
loss of 1.350 $\times~{}10^{17}$ kg yr-1. The total mass of the Sun is 1.989
$\times~{}10^{30}$ kg ([Livingston 1999]), so the fractional mass loss is 6.79
$\times~{}10^{-14}$ yr-1. Again with the factor of three from Eq. 5, the
expected fractional decrease in the AU is 2.26 $\times~{}10^{-14}$ yr-1, or a
change in the AU of $-~{}0.338$ cm yr-1. A change this small is not currently
detectable, and it introduces an insignificant bias into the reported
measurement of an AU increase ([Krasinsky & Brumberg 2004]). If the reported
increase is absorbed into a solar mass increase, and not into a changing
gravitational constant G, the inferred solar mass increase is (6.0 $\pm$ 1.6)
$\times~{}10^{18}$ kg yr-1. This is an unacceptable amount of mass accretion
by the Sun each year. It amounts to a fair sized planetary satellite of
diameter 140 km and with a density of 2000 kg m-3, or to about 40,000 comets
with a mean radius of 2000 m. If the reported increase holds up under further
scrutiny and additional data analysis, it is indeed anomalous. Meanwhile it is
prudent to remain skeptical of any real increase. In our opinion the
anomalistic increase lies somewhere in the interval zero to 20 cm yr-1, with a
low probability that the reported increase is a statistical false alarm.
## 3 The Pioneer anomaly
The first missions to fly to deep space were the Pioneers. By using flybys,
heliocentric velocities were obtained that were unfeasible at the time by
using only chemical fuels. Pioneer 10 was launched on 1972-Mar-02 local time.
It was the first craft launched into deep space and was the first to reach an
outer giant planet, Jupiter, on 1973-Dec-04. With the Jupiter flyby, Pioneer
10 reached escape velocity from the solar system. Pioneer 10 has an asymptotic
escape velocity from the Sun of 11.322 km s-1 (2.388 AU yr-1).
Pioneer 11 followed soon after Pioneer 10, with a launch on 1973-Apr-06. It
too cruised to Jupiter on an approximate heliocentric ellipse. This time a
carefully executed flyby of Jupiter put the craft on a trajectory to encounter
Saturn in 1979. So, on 1974-Dec-02, when Pioneer 11 reached Jupiter, it
underwent a Jupiter gravity assist that sent it back inside the solar system
to catch up with Saturn on the far side. It was then still on an ellipse, but
a more energetic one. Pioneer 11 reached Saturn on 1979-Sept-01. Then Pioneer
11 embarked on an escape hyperbolic trajectory with an asymptotic escape
velocity from the Sun of 10.450 km s-1 (2.204 AU yr-1)
The Pioneer navigation was carried out at the Jet Propulsion Laboratory. It
used NASA’s DSN to transmit and obtain the raw radiometric data. An S-band
signal ($\sim$2.11 Ghz) was sent up via a DSN antenna located either at
Goldstone, California, outside Madrid, Spain, or outside Canberra, Australia.
On reaching the craft the signal was transponded back with a (240/221)
frequency ratio ($\sim$2.29 Ghz), and received back at the same station (or at
another station if, during the radio round trip, the original station had
rotated out of view). There the signal was compared with 240/221 times the
recorded transmitted frequency and any Doppler frequency shift was measured
directly by cycle count compared to an atomic clock. The processing of the raw
cycle count produced a data record of Doppler frequency shift as a function of
time, and from this a trajectory was calculated. This procedure was done
iteratively for purposes of converging to a best fit by nonlinear weighted
least squares (minimization of the chi squared statistic, see [Lawson & Hanson
1974]).
However, to obtain the spacecraft velocity as a function of time from this
Doppler shift is not easy. The codes must include all gravitational and time
effects of general relativity to order $(v/c)^{2}$ and some effects to order
$(v/c)^{4}$. The ephemerides of the Sun, planets and their large moons as well
as the lower mass multipole moments are included. The positions of the
receiving stations and the effects of the tides on the exact positions, the
ionosphere, troposphere, and the solar plasma are included.
Given the above tools, precise navigation was possible because, due to a
serendipitous stroke of luck, the Pioneers were spin-stabilized. With spin-
stabilization the craft are rotated at a rate of $\sim$(4-7) rpm about the
principal moment-of-inertia axis. Thus, the craft is a gyroscope and attitude
maneuvers are needed only when the motions of the Earth and the craft move the
Earth from the antenna’s line-of-sight.
The Pioneers were chosen to be spin-stabilized because of other engineering
decisions. As the craft would be so distant from the Sun solar power panels
would not work. Therefore these were the first deep spacecraft to use nuclear
heat from 238Pu as a power source in Radioisotope Thermoelectric Generators
(RTGs). Because of the then unknown effects of long-term radiation damage on
spacecraft hardware, a choice was made to place the RTGs at the end of long
booms. This placed them away from the craft and thereby avoided most of the
radiation that might be transferred to the spacecraft.
Even so, there remained one relatively large effect on this scale that had to
be modeled: the solar radiation pressure of the Sun. This effect is
approximately 1/30,000 that of the Sun’s gravity on the Pioneers. It produced
an acceleration of $\sim 20\times 10^{-8}$ cm s-2 on the Pioneer craft at the
distance of Saturn.
After 1976 small time-samples (approximately 6-month to 1-year averages) of
the data were periodically analyzed. At first nothing significant was found,
But when a similar analysis was done around Pioneer 11 ’s Saturn flyby, things
dramatically changed. (See the first three data points in Fig. 3.) So people
kept following Pioneer 11. They also started looking more closely at the
incoming Pioneer 10 data.
Figure 3: A JPL Orbit Determination Program (ODP) plot of the early unmodeled
accelerations of Pioneer 10 and Pioneer 11, from about 1981 to 1989 and 1977
to 1989, respectively.
By 1987 it was clear that an anomalous acceleration appeared to be acting on
the craft with a magnitude $\sim 8\times 10^{-8}$ cm s-2, directed
approximately towards the Sun. The effect was a concern, but the effect was
small in the scheme of things and did not affect the necessary precision of
the navigation. However, by 1992 it was clear that a more detailed look would
be useful.
An announcement was made at a 1994 conference proceedings. The strongest
immediate reaction was that the anomaly could well be an artifact of JPL’s
Orbit Determination Program (ODP), and could not be taken seriously until an
independent code had tested it. So, a team was gathered that included
colleagues from The Aerospace Corporation and their independent CHASMP
navigation code. Their result was the same as that obtained by JPL’s ODP.
The Pioneer anomaly collaboration’s discovery paper appeared in 1998
([Anderson et al. 1998, Anderson et al. 1998]). and a detailed analysis
appeared in 2002 ([Anderson et al. 2002, Anderson et al. 2002]). The latter
used Pioneer 10 data spanning 1987-Jan-03 to 1998-Jul-22 (when the craft was
40 AU to 70.5 AU from the Sun) and Pioneer 11 data spanning 1987-Jan-05 to
1990-Oct-01 (when Pioneer 11 was 22.4 to 31.7 AU from the Sun). The largest
systematics were, indeed, from heat but the final result for the anomaly, is
that there is an unmodeled acceleration, directed approximately towards the
Sun, of
$a_{P}=(8.74\pm 1.33)\times 10^{-8}~{}\mathrm{cm~{}s^{-2}}.$ (6)
Two later and independent analyses of this data obtained similar results. The
conclusion, then, is that this “Pioneer anomaly” is in the data. The question
is ([Nieto & Anderson 2007]), “What is its origin?”
It is tempting to assume that radiant heat must be the cause of the
acceleration, since only 63 W of directed power could cause the effect (and
much more heat than that is available). The heat on the craft ultimately comes
from the Radioisotope Thermoelectric Generators (RTGs), which yield heat from
the radioactive decay of 238Pu. Before launch, the four RTGs had a total
thermal fuel inventory of 2580 W ($\approx 2070$ W in 2002). Of this heat 165
W was converted at launch into electrical power, which decreased down to $\sim
70$ W. So, heat as a mechanism yielding an approximately constant effect
remains to be clearly resolved, but detailed studies are underway at JPL.
Indeed, from the beginning we observed that a most likely origin is directed
heat radiation ([Anderson et al. 1998, Anderson et al. 1998], [Anderson et al.
2002, Anderson et al. 2002]). However, suspecting this likelihood is different
from proving it. Even so, investigation may well ultimately show that heat was
a larger effect than originally demonstrated by [Anderson et al. 2002,
Anderson et al. (2002)]. Their original estimate of the bias from reflected
heat amounts to only 6.3% of the total anomaly. Nevertheless, a three-sigma
error in the original estimate could amount to a 25% thermal effect. We would
have difficulty accepting anything larger than this three-sigma limit.
On the other hand, if this is a modification of gravity, it is not universal;
i.e., it is not a scale independent force that affects planetary bodies in
bound orbits. The anomaly could, in principle be i) some modification of
gravity, ii) drag from dark matter, or a modification of inertia, or iii) a
light acceleration ([Nieto & Anderson 2007]);
Future study of the anomaly may determine which, if any, of these proposals
are viable.
## 4 Increase in the eccentricity of the Moon’s orbit
A detailed orbital analysis of Lunar Laser Ranging (LLR) data can be found in
[Williams & Boggs 2009, Williams & Boggs (2009)]. A total of 16,941 ranges
were analyzed extending from 1970-Mar-16 to 2008-Nov-22. LLR can measure
evolutionary changes in the geocentric lunar orbit over this interval of 38.7
years. Changes in both the mean orbital motion and eccentricity are observed.
While the mean motion and semi-major axis rates of the lunar orbit are
consistent with physical models for dissipation in Earth and Moon, LLR orbital
solutions consistently reveal an anomalous secular eccentricity variation.
After accounting for tides on the Earth that produce an eccentricity change of
1.3 $\times$ 10-11 yr-1 and tides on the Moon that produce a change of -0.6
$\times$ 10-11 yr-1, there is an anomalous rate of (0.9 $\pm$ 0.3) $\times$
10-11 yr-1, equivalent to an extra 3.5 mm yr-1 in perigee and apogee distance
([Williams & Boggs 2009]). This anomalous eccentricity rate is not understood
and it presents a problem, both for a physical understanding of dissipative
processes in the interiors of Earth and Moon, and for the modeling of
dynamical evolution at the 10-11 yr-1 level.
## References
* [Anderson et al. 1998] Anderson, J.D., Laing, P.A., Lau, E.L., Liu, A.S., Nieto, M.M., & Turyshev, S.G. 1998, Phys. Rev. Lett., 81, 2858
* [Anderson et al. 2002] Anderson, J.D., Laing, P.A., Lau, E.L., Liu, A.S., Nieto, M.M., & Turyshev, S.G. 2002, Phys. Rev. D, 65, 082004
* [Anderson et al. (2007)] Anderson, J.D., Campbell, J.K., & Nieto, M.M. 2007, New Astron., 12, 383
* [Anderson et al. (2008)] Anderson, J.D., Campbell, J.K., Ekelund, J.E., Ellis, J., & Jordan, J.F. 2008, Phys. Rev. Lett., 100, 091102
* [Antreasian & Guinn (1998)] Antreasian, P.G. & Guinn, J.R. 1998, AIAA/AAS Paper No. 98–4287
(http://www2.aiaa.org/citations/mp-search.cfm)
* [Danby (1988)] Danby, J.M.A. 1988, Fundamentals of Celestial Mechanics (Richmond: Willmann-Bell), sec. 8.2
* [Dunham et al. (2005)] Dunham, D.W., Farquhar, R.W., & McAdams, J.V. 2005 Ann. N.Y. Acad. Sci., 1065, 254
* [Flandro (1966)] Flandro, G.A. 1966, Astronaut. Acta, 12, 329
* [Kliore (2004)] Kliore, A.J., Anderson, J.D., Armstrong, J.W. & ten others 2004 Space Science Reviews, 115, 1
* [Krasinsky & Brumberg 2004] Krasinsky, G.A. & Brumberg, V.A. 2004 Celest. Mech. Dynam. Astron., 90, 3
* [Lawson & Hanson 1974] Lawson, C.J. & Hanson, R.J. 1974 Solving Least Squares Problems (Englewood Cliffs: Prentice-Hall)
* [Livingston 1999] Livingston, W.C. 1999, in Allen’s Astrophysical Quantities, Fourth Edition ed. A. N. Cox, (New York, Berlin, Heidelberg: Springer-Verlag), Chap. 14
* [Morley & Budnik (2006)] Morley, T. & Budnik, F. 2006, 19th Int. Symp. on Space Flight Dynamics, Paper No. ISTS 2006-d-52
* [Moyer 2003] Moyer, T.D. 2003, Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation (Print ISBN: 9780471445357, Online ISBN: 9780471728474: John Wiley & Sons), chap. 2
* [Nieto & Anderson 2007] Nieto, M.M. & Anderson, J.D. 2007 Contemp. Phys., 48, 41
* [Pitjeva 2007] Pitjeva, E.V. 2007 in Proceedings of the “Journées Systèmes de Référence Spatio-temporels 2007” (Observatoire de Paris), p. 65.
* [Pitjeva 2009a] Pitjeva, E.V. 2009 This Proceedings
* [Pitjeva 2009b] Pitjeva, E.V. 2009 JOURNEES-2008 Astrometry, Geodynamics and Astronomical Reference Systems ed. M. Soffel & N. Capitaine (Dresden, in press)
* [Pitjeva & Standish 2009] Pitjeva, E.V. & Standish, E.M. 2009, Celest. Mech. Dynam. Astron., 103, 365
* [Russell (1992)] Russell, C.T. 1992, The Galileo Mission (Dordrecht, Boston, London: Kluwer)
* [Tapley et al. (2004)] Tapley, B.D., Bettadpur, S., Watkins, M. & Reigber, C. 2004 Geophys. Res. Lett., 31, L09607
* [Wiesel (1989)] Wiesel, W.E. 1989, Spaceflight Dynamics (New York: McGraw-Hill), sec. 11.5
* [Williams & Boggs 2009] Williams, J.G. & Boggs, D.H. 2009 in Proceedings of 16th International Workshop on Laser Ranging ed. S. Schillak, (Space Research Centre, Polish Academy of Sciences)
|
arxiv-papers
| 2009-07-14T23:33:03 |
2024-09-04T02:49:03.914009
|
{
"license": "Public Domain",
"authors": "John D. Anderson and Michael Martin Nieto",
"submitter": "Michael Martin Nieto",
"url": "https://arxiv.org/abs/0907.2469"
}
|
0907.2492
|
# Towards a Post Reductionist Science:
The Open Universe
Stuart Kauffman
(July 8, 2009)
###### Abstract
We have lived with a world view dominated by reductionism. Yet recently, S.
Hawking has written an article entitled ”Godel and the End of Physics”. His
observations raise the possibility that we should question our foundations.
Core to this is reductionism itself. In turn reductionism finds its roots in
Aristotle’s model of scientific explanation as deductive inference: All men
are mortal, Socrates is a man, therefore Socrates is a mortal. With Newton’s
laws in differential form, reductionsim snaps into place, for given initial
and boundary conditions, integration of those equations is exactly deduction.
Aristotle’s ’efficient cause’ becomes mathematized as deduction. In this paper
I discuss the reality that deductive inference is not the only way we explain
in science. Darwin gave us the Blind Watchmaker, the appearance of design
without a designer. I discuss the role of the opportunity for an adaptation in
the biosphere and claim that such an opportunity is a ’blind final cause’, not
an efficient cause, yet shapes evolution. I also argue that Darwinian
exaptations are not describable by sufficient natural law. Based on an
argument of Sir Karl Popper, I claim that no law, or function, f, maps a
decoherence process in a Special Relativity setting from a specific space-time
slice into its future. If true this suggests there can be no theory of
everything entailing all that happens. I then discuss whether we can view laws
as ’enabling constraints’ and what they enable. Finally, in place of the weak
Anthropic principle in a multiverse, I suggest that we might consider Darwin
all the way down. It is not impossible that a single universe has an abiotic
natural selection process for laws as enabling constraints and that the single
universe that ’wins’ is ours. One possible criterion of winning might be ’most
rapid growth of the Adjacent Possible of the universe’.
Institute for Biocomplexity and Informatics
The University of Calgary
Signal Processing, Tampere University of Technology
External Professor, The Santa Fe Institute
## Introduction
We have lived in a scientific world view dominated by reductionism for at
least 350 years. Reductionism, in S. Weinberg’s view, (1) holds that all that
unfolds in the universe is logically entailed by the fundamental laws of
physics. In the past thirty five years, doubts as to the full adequacy of
reductionism have been increasingly voiced, even in physics. Philip Anderson’s
”More is Different”, (2) and Robert Laughlin’s ”A Different Universe”, (3),
are major examples of this doubt Recently, Stephen Hawking has written an
article entitled ”Godel and the End of Physics”, (4), suggesting that no
finite set of laws may suffice to describe by entailment the evolution of the
universe. If reductionism proves both profoundly useful but ultimately
inadequate, as I think it does, this failure must portend a major change in
our scientific world view. What might follow reductionism as a more fully
adequate approach? I argue that, as this foundation is pulled from under us,
it portends a partially lawless open and creative universe of profound new
interest.
The heart of what I want to explore begins with this: The very laws of physics
may be open to being viewed as enabling constraints \- enabling constraint
laws selected by an abiotic natural selection among a set of possible laws to
yield our extremely complex universe. And our single universe, not the
multiverse and its attending weak Anthropic principle, may be the ’winning’
universe that is enabled by the opportunities afforded by those laws. In
winning, our universe would then have evolved its laws such that the winning
universe is ours. I will discuss initial ideas about what ’winning’ might mean
below.
By appealing to an abiotic natural selection on a set of laws, this view goes
beyond reductionism and explanation purely via logical entailment. As we shall
see, Darwin’s natural selection goes beyond entailment. We have been taught
that science answers only ’how questions’. For example, given Newton’s laws
and the Newtonian world view, Newton’s laws answer how celestial mechanics
occurs. But there is no answer to the question, ’Why Newton’s Laws’?
Scientific enquiry must stop, on the reductionist view, with the ultimate law,
for example Weinberg’s Dreams of a Final Theory, (1). But Darwin’s natural
selection answers ’why’ questions \- why has the vertebrate eye emerged in the
evolution of the universe? Because of a sequence of adaptations achieved by
Natural Selection, thus achieving what philosopher David Depew calls ’blind
teleology’,(5). Darwin reaches beyond reductionism because his ’why’ question
rests on what I shall call ’blind final cause’, as is captured in Richard
Dawkins famous book, ”The Blind Watchmaker”, (6). I will argue that the
opportunity for an adaptation in the biosphere, or for the universe as a
whole, is just such a blind final cause, subsequently achieved by efficient
causes. I will suggest and hope to persuade the reader that blind final cause
is not efficient cause. This issue will prove central to our discussion. In
sharp contrast, since Descartes and Newton, science has been bound to explain
the unfolding of the universe purely in terms of Aristotle’s ’efficient
cause’, mathematized as logical entailment. This assumption is the root of our
long faith in reductionism. Given this assumption and the mathematization of
efficient cause as entailment, the deductive explanatory, and tautological
character, of reductionism is set in place. There is no room for ’lawless
creativity’ in this world view. The logical possibility of blind final cause,
in the evolution of the biosphere, or even the universe as a whole, renders
restriction to our familiar reductionism logically unnecessary, thus goes
beyond our familiar reductionism. Again, reductionism and the consequent faith
in deductive entailment yields a universe barren of creativity, a tautological
realm entailed by the hoped for theory of everything. In contrast, if ’law’ is
enabling constraint, and that enablement enables opportunities that can,
blindly, be seized by the becoming of the universe in its full becoming, then
the universe is open to myriad creativity. The universe is open in ways we
have not dreamed in Western science since Descartes.
This article is organized as follows: In Section 1, I briefly review
Aristotle’s four causes and his model of scientific explanation in the
syllogism. Newton’s laws, plus initial and boundary conditions then sets the
stage for reductionism, with us today, and leads to Hawking’s ”Godel and the
End of Physics”. I end Section 1 by raising the question whether our sole
reliance on efficient cause in science since Newton may be a foundational
problem. In Section 2, I raise the issue of blind final cause in the evolution
of the biosphere in the achievement of adaptations that alter the course of
the biosphere’s evolution, hence that of the universe. Blind final causes are
’opportunities’ for adaptations in a selective niche blindly seized by
evolution. The selective niche is, itself, not an efficient cause, but a blind
final cause of the successful emergence of the adaptation. The selective niche
shapes the course of evolution, but is not an efficient cause of that
evolution. The adaptation is achieved by efficient causes. The universe is
open in ways beyond logical entailment. In Section 3, I describe new grounds
to think that the evolution of the biosphere by Darwinian exaptations, or
’preadaptations’ is not describable by sufficient natural law, where natural
laws are compact descriptions of the regularities of a process. As we will
see, the implications of this evolution by Darwinian preadaptations is that we
cannot make probability statements about such evolution since we do not know
the sample space of possibilities in what I will call the Adjacent Possible of
the biosphere, that in place of sufficient entailing law is a ceaseless
creativity, and that the generation of ’information’ in the biosphere does not
fit Shannon’s theory, where Shannon information requires prior knowledge of
the ensemble of messages. More the becoming of the biosphere is both partially
lawless, yet non-random - a concept we do not yet have in physics. If these
claims are correct, it appears that there can be no entailing Theory of
Everything (TOE). In Section 4, I discuss the issue of whether the co-
evolution of the quantum-classical boundary is describable by sufficient
natural law and suggest that it is not. In its place may be an abiotic natural
selection blind final cause. These ideas seem to have experimental
consequences. In Section 5, I discuss enabling constraints and what they
enable. We have not even the beginning of a theory here, but need to develop
one. I will discuss the fact that in the evolution of the biosphere, evolution
has itself achieved enabling constraints that have improved the very process
of evolution. If we can view law as enabling constraints, then the biosphere
is evolving its own laws such that it evolves better. If so, then it becomes
thinkable that the universe has evolved its laws as well. I will sketch an
initial approach to what I believe is a non-algorithmic process with an
algorithmic set of board games, legal move sets on those board games and the
possibilities enabled by those legal rules. The approach is inadequate, but a
start. In Section 6, I discuss our current cosmological conundrum, the
apparent fine tuning of the 23 constants of nature, which has led to the
suggestion of a multiverse and the weak Anthropic principle. We are driven to
a multiverse hypothesis by reductionism itself. But that reductionism may no
longer be all we need. An alternative reductionist hope is that there really
is a TOE, and we are left to wonder why this, rather than another TOE,
describes the universe. In place of these familiar theories, I raise the
possibility that the laws of physics are enabling constraints that enable a
very complex universe, laws that were selected from some set of possible laws
early in the history of the universe by blind final cause for a ’winning’
persistent and complex universe, and point to some features of physical laws
that are puzzling but interesting hints in this context, including various
conservation laws. To pursue the above agenda, if it has merit, will require
an entire new body of theory concerning enabling constraints and what they
enable. In the final Section 7, I try to discuss the puzzling role and status
of the possible in the origin and history of the universe.
## 1 Aristotle and the Mathematization of
Efficient Causes
Aristotle famously held that there were four causes, formal, final, material
and efficient. In a simple example of a house to be constructed, the formal
cause of the house is the blueprint. The material causes of the house are the
bricks, mortar, beams, and building material. The final cause of the house is
my decision to build the house. The efficient cause is the actual process of
its construction. But Aristotle, as R. Rosen points out in Life Itself (7),
also offered a model of scientific explanation in the syllogism: All men are
mortal. Socrates is a man. Therefore Socrates is a mortal. The logical ’force’
of this logical entailment may play a later role in our sense that natural
efficient cause laws govern rather than describe the unfolding of the
universe.
Newton’s laws, given as differential equations in a state space with initial
and boundary conditions, fulfill Aristotle’s form of scientific explanation as
entailment, for the integration of the differential equations is precisely
deduction. With Newton, Aristotle’s other causes, formal, final, and material,
largely recede from science, which takes itself to explain purely in terms of
efficient cause, mathematized as logical entailment.
That this is the base of reductionism is already evident in Laplace’s famous
claim that a massive computing system, if given the positions and momenta of
all the particles in the universe, could, using Newton’s laws, predict or
retrodict the entire future and past of the universe. I note four features of
Laplace’s reductionism: i. All laws are deterministic, now in doubt given
quantum mechanics and the Copenhagen interpretation including the Born rule.
ii. All that exists ontologically in the universe are particles in motion.
iii. All that happens in the universe is describable by sufficient efficient
cause laws via deductive entailment. iv. There exists at least one language,
here Newton’s, to describe all of reality. None of these four claims will
remain unchallenged below. With the addition of fields, quantum mechanics and
General Relativity, plus the standard model, we have contemporary reductionist
physics and Hawking’s doubts, (4). Indeed, Hawking, in seeing in Godel the
potential ’end of physics’, does so in terms of the pure sufficiency of a
mathematized form of efficient cause law. It is in terms of such mathematized
efficient cause laws that he fears an infinite set of such laws.
When such a crisis as Hawking hints arises, one recourse is to doubt the
fundamental assumptions we make. The use of efficient cause as the sole
explanatory principle may be just that assumption. I now claim that this
assumption is false.
## 2 Blind Final Cause in Biological Evolution
Darwin may rank as the mind who most changed our world view, for with Darwin
we are given, in Richard Dawkin’s fine phrase, ”the Blind Watchmaker”, that
is, the emergence in the biosphere of the appearance of design without a
designer - the teleonomy of which J. Monod speaks so eloquently in Chance and
Necessity,(8). In this 150th year since Darwin’s Origin of Species, we are
still grappling with the implications of his central idea. Philosopher David
Depew at a recent conference on Darwin and Evolution,(5), spoke of the
achievement of an adaptation, say the eye, or even a red light sensitive cell
in the progeny of an organism with no light sensitivity, as a ’blind
teleology’. Depew had in mind just what Darwin told us. This is Monod’s
teleonomy - the appearance of design without a designer. There is no doubt the
eye is an adaptation - indeed the similarity of the vertebrate eye, so
resembling a camera, is stunning. As Monod forcefully points out, only life
appears able to do this. Of course, the eye has evolved multiple time, but
that is beside the point I raise. Other adaptations are unique.
I now raise a central issue. Can we speak of an opportunity for an adaptation
before it occurs? With thanks to G. Kaufman, (9), I translate such an
opportunity for an adaptation, A, as ’A is possible. A might or might not
occur. If A occurs it will tend to be selected and fixed in the population’.
Now it becomes a critical issue to ask what kind of a ’cause’ is the
opportunity for an adaptation, which, if achieved, may change the course of
biological evolution. It is clear that the actual achievement of the
adaptation is via a series of efficient causes. However, the tendency to be
selected is a dispositional term, untranslatable into any finite set of
necessary and sufficient efficient cause conditions, or actual events, for the
achievement of fixation of the adaptation. This means that we cannot state
ahead of time the efficient causes by which a particular adaptation will come
to be achieved. But is the opportunity for the adaptation itself, the very
fact that the eye is an adaptation \- subsequently achieved by non-prestatable
efficient causes - itself an efficient cause? Certainly the opportunity for an
adaptation is not an efficient cause in the straight forward sense of billiard
balls hitting billiard balls. Nor in more sophisticated terms, such as
Maxwell’s equations which are descriptions of efficient causes, is the
opportunity for an adaptation in any clear sense an efficient cause. Further,
for a system with a potential, such as a ball rolling down a warped hill,
where a least action principle can be found, that least action gives a
superficial appearance of a final cause. But there is no hint that the
achievement of an adaptation is a flow on a potential for which a least action
principle might be found. And again the ’tendency’ to become fixed by
selection in the population is not, as noted, reducible to any actual
(efficient cause) events that are necessary and sufficient for fixation to
occur. Thus it does not seem that we can translate the opportunity for an
adaptation into any set of e fficient cause events.
Addy Pross has given a very interesting analysis of this issue in the
biological realm, (10,11). He distinguishes between thermodynamic selection,
which tends toward thermodynamic equilibrium, and ’kinetic selection’ among
replicators for those which maximize a kinetic stability, not a thermodynamic
stability. Pross’s central point is that cells, as open thermodynamic systems,
are unstable thermodynamically, but kinetically stable - they are the winners
of a kinetic race in a ’space of replicators’. I think Pross’s insight is
important, for in the simple case of, say bare replicators, the opportunity
for an adaptation is a means to replicate faster to higher copy number, hence,
as he says, higher kinetic stability. Yet Darwin’s fully biotic selection, and
the economic selection of goods and services which survive in the market
place, both analogous to Pross’s kinetic selection, may go beyond any simple
sense of ’winning the kinetic race’. A butterfly may forego more rapid
reproduction if ’K’ selected for carrying capacity in a nutrient limited
environment, rather than ’R’ selection for replication rate.
I comment that David Deutsch (12) has written extensively on quantum mechanics
and evolution.
I give next four examples, two economic, then two biological, also referred to
below in the section on enabling constraints, to argue that opportunities for
adaptation are blind final causes in the case of the biosphere, and full
Aristotelean final causes in the case of the economy, with the assumption of
responsible free willed economic actors in the latter case. Consider the
following economic facts. In the early 1980s in North America, there were many
television stations, abundant programming, many television sets, and, perhaps
sadly, a multitude of couch potatoes. In the face of this economic niche, was
there an opportunity to invent and successfully market the television remote
channel changer? Yes of course there was and one could obtain venture funding
to do so. Now I ask, was the economic niche mentioned above an efficient cause
of the invention of the television remote? No, it was, rather, as described
below, an enabling condition, or enabling constraint, that a fforded the
opportunity to invent and make money with the television remote. Now consider
the following: In 1943, the computer was invented to calculate shell
trajectories in World War II. Some thirty years later, the invention of the
computer afforded the opportunity to invent and market widely the personal
computer. IBM and Apple made substantial money on the venture. With the
invention of the personal computer and its wide sale, the opportunity arose to
invent and market word processing, and Microsoft made money doing so. But the
invention of word processing afforded the opportunity to store word files. In
turn, stored files afforded the opportunity to share files between CERN
colleagues, which in turn led to the economic-technological niche opportunity
to invent and spread the world wide web. In turn, the web afforded the
opportunity, the niche, where web commerce could find a home and EBay
flourished. In turn, the abundance of information on the web created the oppo
rtunity, the economic niche, for the invention of web portals such as Google.
Now we have achieved the summit of Western civilization with Facebook. Or,
consider the flourishing of ”aps” on cell phones and the growth of text
messaging. Note how each of these opportunities, or enabling conditions,
created a niche into which the next invention made economic sense. But the
opportunities were not efficient causes of the inventions.
It might be thought that the story above relies on human conscious invention,
but the same processes obtain for the evolution of the biosphere. Organisms
occupy niches. As new organisms evolve, new niches are created. But a niche,
for example, that occupied by rabbits, is not an efficient cause of the
evolution of rabbits to fill and persist by existence in that niche. Rather,
the niche is an opportunity which evolution blindly seizes, and adaptations to
fill that niche arise and are selected by efficient cause events as the
adaptations tend to be selected by natural selection. The niche is not an
efficient cause of those adaptations, although the actual steps of adaptation
are themselves achieved by efficient causes. Rather the niche is, as
emphasized below, an enabling constraint that allows rabbits to arise and
’make a living’ in that niche.
A wonderful further set of examples arise in co-evolution. Consider flowers,
insects and birds such as humming birds. Flowers feed the birds and insects
nectar. Pollen rubs off on the insects and birds, is transferred to another
flower and pollinates the latter. Each is the niche of the other, and flowers,
insects and birds have co-evolved their mutual niches for millions of years.
Step by step flowers found new adaptations to attract insects and birds and
manage to be fertilized by insects and birds, and the latter adapted the
stickiness of their hairs and beaks for pollen, and food gathering behavior,
to carry out that fertilization. The adaptation steps were achieved by
efficient causes. The wondrous mutual emergence of the diversity of flowers
and insects and humming birds as mutual co-evolutionary adaptations of ever
creating niches is not efficient cause. Each of the mutualists gradually
builds new opportunities for the other in their evolutionary becoming. The
Buddhists would call this ’co-dependent origination’.
Physicists seeking a theory of everything from which all is entailed by
deduction cannot ignore the biosphere’s becoming, let alone culture,
economics, and history where we become confused about consciousness and free
will. Yet the evolution of the biosphere, say before consciousness evolved, is
squarely in the purported purview of the physicist such as Weinberg. But he
cannot deduce this becoming, for opportunities for adaptations are not
efficient causes, yet, once achieved by efficient causes, alter the course of
evolution of the biosphere.
I conclude that the opportunity for an adaptation is an opportunity for
natural selection to select what will succeed in the current selective
environment, and is a blind final cause, not an efficient cause. It may be
pointed out here that with Darwin and with ourselves, it is essential that we
feel it appropriate to use the phrase ’succeed in the current selective
environment’. The red spotted organism will be a winner in Darwin’s struggle
for existence. But the very phrase ’struggle for existence’ is, as philosopher
Dan Cloud pointed out to me, to place the process of natural selection in a
problem solving framework. But a problem solving framework is not a mere
description of what happens, as is the description of a ball rolling down a
hill. It is, in fact, true that the red spotted organism that is light
sensitive is actually fitter than its non-light sensitive rivals in its
selective environment. This fact that this organism is actually fitter in the
given environment is why - not how, but why - this fitter organism is
selected. ’How’ the selection actually occurs is a sequence of efficient
causes such that the fitter organism dispositionally ’tends’ to win. But we
cannot state what those efficient causes must be. Again, I conclude that the
opportunity for an adaptation is a blind final cause, not an efficient cause
of what merely happens. This is an essential step, for it claims that the
becoming of the biosphere is not sufficiently describable only by efficient
causes. But this will imply that the becoming of the universe including the
biosphere is not describable only by entailment from mathematicized efficient
cause laws. In turn, this means that we are not limited to the tautological
entailments of a final theory of everything, and that an open creativity
beyond entailment is present in the unfolding of the biosphere, economy,
history, and perhaps the universe as a whole.
I remark preliminarily that to speak of an opportunity for an adaptation, we
seem forced to deal with the fact that the adaptation is ’possible’. Already
in Quantum Mechanics and the Schrodinger equation with Copenhagen and the Born
rule, we speak of the Schrodinger wave as a ’possibility wave’ which, when its
modulus is squared, gives the probability of observing possibilities that we
know beforehand. We will soon see that the evolution of the biosphere seems to
force us to a wider possible, where we do not know beforehand what the
possibilities are. All this is puzzling. In General Relativity and the block
spacetime universe, there are only world lines, actuals, and no possibles. We
shall have to begin to inquire about the status of the possible.
## 3 The Evolution of the Biosphere by Darwinian ’Preadaptations’ is
Partially Lawless
Were we to ask Darwin the function of the human heart, he would say it is to
pump blood. Were we to point out that the heart makes heart sounds and moves
water in the pericardial sac, he would say these effects are not the function
of the heart. If we asked why not, he would reply that the heart was selected,
so exists in the universe, because it was of selective advantage to pump blood
in some ancestor and the lineage leading to us.
Already this is interesting because, were the physicist to succeed in deducing
all the causal properties of the heart from its subatomic constituents, she
would have no way to pick out pumping blood as the biological function of the
heart and the putative reason hearts came to exist in the universe. To
describe the function of the heart, she would have to become a paleontologist
and evolutionary biologist, or to simulate the evolution of the biosphere, or
deduce from her theory of everything the emergence of the heart. In two books,
Investigations and Reinventing the Sacred,(13,14), I argue that she cannot
simulate or deduce the emergence of the heart in the unfolding of the
universe. Quantum events matter in evolution, at least by causing mutations.
There is no way to simulate all the quantum processes that have occurred,
including random cosmic rays, or, in accord with Schrodinger, might have
occurred, in the history of the past 5 billion years of the earth, let alone u
niverse. How would one simulate the all the possible consequences of all the
possible temporal instants of a radioactive decay, or a quantum coherent
electron transfer in some protein in some organism in some environment? Now
consider doing so for all the quantum events in the past 5 billion year
history of the Earth and evolving biosphere. More there is no way to confirm
that any such simulation captures the actual quantum history of this
biosphere’s evolution. But can the physicist deduce the becoming of the human
heart in evolution, or evolution more generally. I now argue that the answer
is a resounding ’No’. If I am correct, it appears to have major implications.
Darwin spoke of the fact that a feature of an organism, with some causal
property of no selective significance in the current environment, might be of
selective value in a different selective environment, so be selected.
Typically a new function will arise in the biosphere. These events are called
either ’exaptations’ or Darwinian ’preadaptations’. There is no concept of
evolutionary foresight here. It just happens to turn out that a property that
is of no selective use in one environment is of selective use in another
environment.
I give two examples. Some fish have swim bladders. These are sacs, partially
filled with water, partially filled with air, that adjust neutral buoyancy in
the water column. Paleontologists believe that swim bladders evolved by
exapatation from lung fish. Water got into the lungs of some lung fish, now
there was a sac partially filled with air, partially with water, and so poised
to evolve into a swim bladder. Let us assume the paleontologists are correct.
Now: Did a new function arise in the biosphere? Of course, neutral buoyancy in
the water column. Did the swim bladder affect the further evolution of the
biosphere? Of course, new species, proteins, other molecules, and niches
evolved. Here is a second example. We have three middle ear bones to transmit
sound from our tympanic membrane to our inner ear. These evolved by
preadaptation from three adjacent jaw bones of an early teleost fish. This
case is important because relational ’degrees of freedom’ matter. If the three
bones were not adjacent, but were in the spine, skull and jaw, probably middle
ear bones would not have evolved. Again, did a new function come to exist in
the biosphere? Yes, hearing. Did this new function alter the evolution of the
biosphere? Of course, new species, proteins, niches.
I now come to the critical question: Do you think you could prestate all the
possible Darwinian exaptations of all organisms alive now? You might respond
that we do not know all organisms alive now. I simplify my question: Do you
think you could prestate all possible Darwinian exapatations just for humans?
I have now asked thousands of people. We all agree we cannot carry out this
task. Why not? I think parts of the problem are that we cannot prestate all
possible selective environments, nor know that we had listed them all. Nor can
we prestate all features of one or many organisms, including relational
features, that might turn out to be preadaptations. It is not clear how to
prove this claim. An experiment seems beside the point A theorem seems
impossible at least at present.
I now need to define the ’Adjacent Possible’. Consider a liter of buffer with
1000 different molecular species. Call this set the ’Actual’. Let them react
by a single reaction step. If new species of molecules appear, call these ’The
Adjacent Possible’. Clearly this is well defined in the chemical case, given a
minimal life time of stability for a species. Now let me point to the Adjacent
Possible of the biosphere. Once there were lung fish, swim bladders were in
the Adjacent Possible of the biosphere. Before there were multicelled
organisms, swim bladders were not in the Adjacent Possible of the biosphere.
Admittedly, I use some poorly defined sense of ’adjacent’ here.
Now if we do not know all the possible preadaptations that might arise in the
adjacent possible of the biosphere, then not only do we not know what will
happen, we do not even know what can happen! Can we make probability
statements about the evolution of the biosphere by preadaptations? Consider
flipping a coin 10,000 times. It will come up heads about 5000 times, with a
binomial distribution. But notice that we knew ahead of time all the
possibilities, all heads, all tails, and so forth. We knew the sample space of
the process, so could erect a probability measure on the frequency
interpretation of probabilities for this coin flipping process. But we do not
know the sample space of the evolution of the biosphere by preadaptations, so
can make no probability statements about it. Now Laplace had a different
interpretation of probability. If confronted by N doors, behind one of which
was a treasure, but we had no idea which door, our chances of picking the
right door is 1/N. But notice that we know N, the number of doors. We do not
know N for the evolution of the biosphere, so can make no probability
statements about this process.
If a natural law is a compact description of the regularities of a process,
can we have a sufficient natural law for the emergence of swim bladders? No.
We cannot even state the possibility of the emergence of swim bladders, let
alone their probability. Thus we cannot have a law that is sufficient for
describing the emergence of swim bladders.
This is a major conclusion. The becoming of the biosphere is partially beyond
sufficient natural law. Yet it is also non-random. There is no sufficient law
for the becoming of the swim bladder, yet this new organ does make sense and
is selected in its selective environment, hence its evolutionary emergence is
not random. We have no such concepts in physics of a partially lawless yet
non-random process. But the biosphere appears to be doing just this. The same
is true in the economy, culture and history. But if the emergence of the swim
bladder is not describable by sufficient natural law, it is not entailed by
any theory of everything at the fundamental level of physics. Thus, there can
be no theory of everything! Nor can the evolution of the biosphere be deduced
by mathematized efficient causal law. This failure reinforces the conclusion
that adaptations are blind final causes, and our explanations of the becoming
of the universe are not limited to efficient cause laws. Notice that this
discussion is not that of Hawking about Godel and the End of Physics based on
efficient cause mathematical law and the possible inadequacy of any finite set
of such laws, which may also be valid in its own right.
It is important to pause for a claim about ’the furniture of the universe’.
Are swim bladders ontologically ’real’? Consider proteins length 200 amino
acids. How many are possible with 20 kinds of amino acids? 20 raised to the
200th power, or about 10 to the 260th power. We can make any one of these we
choose. But were the 10 to the 80th particles in the known universe to do
nothing, ignoring space-like separation, on the Planck time scale of 10 to the
-43rd seconds but make proteins length 200 amino acids, it would require 10 to
the 39th power repetitions of the history of the universe to make all these
proteins just once. But this means that, at levels above stable atoms, the
universe is on a unique, utterly non-ergodic trajectory. Most complex things
will never exist, so the existence of the heart is no small matter. But if we
cannot deduce the coming into existence of hearts or swim bladders, and yet
they have causal powers as organized structures and processes, then hearts and
swim bladders are emergent with respect to the fundamental laws of physics and
so are ontologically real parts of the universe. We are not just particles in
motion. Moreover, since most complex things will never exist, the universe is
indefinitely open upward in complexity. And since efficient causes,
mathematized as deductions, do not suffice to describe the unfolding of the
universe including the biosphere, the universe is open and, for the biosphere
and upward, vastly creative.
I also pause to note that the richly interwoven complexity of the biosphere
which has emerged cannot be captured by Shannon information. Shannon assumes
an ensemble of messages, in a prestated alphabet, where all possible messages
are known beforehand, and thus whose entropy can be calculated. But we do not
know the all the possibilities that evolution will unfold. We do not know the
alphabet of processes, entities, and functions that will emerge and integrate
into an evolving biosphere. Whatever information may be, a vexed question,
Shannon information does not seem to apply to the evolution of the biosphere.
Indeed, I do not think that this evolution is even algorithmic, (13,14).
Consider the famous Halting problem, where no compact description of the
behavior of some algorithm may be available. But for the next 11 steps, or any
finite number step of the universal Turing machine, all possible states, in a
prestated alphabet, of tape and head, can be listed. We cannot even get
started on the evolution of the biosphere by preadaptations. So our problem
with the evolution of the biosphere does not seem to be the same as the
problem of there being no compact description for an arbitrary algorithms
behavior. We may well confront the issue that no language describes all of
reality.
I end this section with an economic preadaptation, said to be a true story.
Engineers were trying to invent the tractor, so knew that a massive engine
block would be necessary. This was placed on a succession of chasses, all of
which broke. At last an engineer said, ”You know, the engine block itself is
so big and rigid, we can hang everything off the engine block and use it as
the chasse”. This novel use of the engine block is a Darwinian economic
preadaptations. Economic inventions are rife with similar examples and most
inventions are not used for their initial inventive purpose. This raises the
issue of algorithmicity again. Can we name all uses of a screw driver? No.
This is the ’frame problem’ of computer science, never solved. I think that
the human mind, like the evolution of the biosphere, is not algorithmic, and
the evolution of the economy, culture, and history are not describable by
natural laws, (14). Indeed, historians, who do find out about the real world,
today largely eschew a search for the laws that Marx sought. In part, history
and cultural evolution, like the invention of Google, are instances of
opportunities seized - not merely efficient caused events.
## 4 Is the Coevolution of the Quantum Classical Boundary Lawful?
As we consider the adequacy of reductionism, it becomes of interest to ask if
the boundary between the quantum and classical worlds, their co-evolution, is
lawful. In this section I borrow an argument from Sir Karl Popper in his The
Open Universe, (15), to suggest that this becoming is not describable by
sufficient efficient cause law. The ideas have testable consequences, in
principle.
In Popper’s argument, the setting is Special Relativity. An event A has a past
light cone and a future light cone, with a zone of possible simultaneity
between them. An event B is in the future light cone of A. The past light cone
of B includes the past light cone of A, but includes regions that are space-
like separated so lie outside the past light cone of A. Popper then argues
that, at A, we cannot know the events in the past light cone of B that are
outside the past light cone of A but may influence event B, so we cannot have
a law for the event B before B occurs. If a law is a compact description of
the regularities of a process which an observer at A, and before event B, can
construct, Popper’s argument seems valid. If the observer is not located at A,
then we will be driven to an observer outside the universe, which seems
inadmissible. Popper uses his argument to support indeterminism.
My own setting depends upon the currently popular theory that the transition
from quantum to classical is due to decoherence and loss of phase information
from the system to an environment, quantum, classical or both. The loss of
phase information to the environment means that the system gradually loses the
capacity to exhibit interference patterns like the two slit experiment, the
hallmark of quantum behavior. The transition to classical behavior is often
described as ’for all practical purposes’, (FAPP), since the system’s phase
information continues to exist in the environment. Take a setting like
Popper’s. For example consider a complex organic molecule in a dense solution
of such molecules, and an event A in which two emitted entangled quantum
degrees of freedom that move apart from that molecule and eventually are
absorbed by one or two detectors, event B, say classical, that recede from one
another at constant velocity, the Special Relativity setting. Then Popper’s
argument applies. Before the absorption by classical or other quantum degrees
of freedom, we cannot know what events outside the past light cone of the
complex molecule, event A, and the receding quantum entangled particles, may
impinge upon decoherence upon absorption, event B, and EPR instantaneous
correlation with the quantum decohering molecular system from event A. Then we
do not know how decoherence happens in detail in that molecule. Then there can
be no efficient cause function, F, or law, for the detailed way decoherence of
parts or all of the complex molecule happens. Thus, it appears, we can have no
law for detailed decoherence in this Special Relativity setting. But quantum
mechanics and Special Relativity are consistent, as Dirac’s relativistic
electron equation argues. This claim implies that there is no efficient cause
law, or function, that maps the space time region including event A and the
receding detectors before event B, into a future that includes event B.
If there is no law, what can we say about what happens? I discuss this below.
If there can be no law, then it seems there can be no Theory of Everything
from which all that happens is entailed.
There can, of course, be statistical models of this decoherence process. But
such models are not detailed laws. However, if the above view is correct, it
seems to vitiate full reductionism - the dream that there is an efficient
cause law or set of laws that entails all that happens in the universe.
The situation is even more complex, for the transition from quantum to
classical (for all practical purposes if you wish) and back is thought to be
reversible. Shor’s code for error correction in quantum computers, (16), shows
that in a quantum computer, decohering degrees of freedom can be made to
recohere with addition of information from the outside. H. Briegel has
recently published two papers, (17,18), arguing that a quantum entangled
system can become classical then fully quantum entangled again. Assume Shor
and Briegel are correct. If decoherence is lawless, then even if the classical
to quantum transition is lawful, the total quantum to classical to quantum
reversible process must be lawless. But that means that the coevolution of the
quantum-classical world is not describable by efficient cause mathematical
laws. Again, it seems there can be no Theory of Everything.
Given our interest in Darwin and natural selection, it becomes of considerable
interest that a speculative abiotic natural selection process may arise at the
quantum-classical boundary. Decoherence seems likely to depend upon the local
quantum plus classical environment. The more complex the environment,
presumably the easier and more rapid decoherence of the system will be. Then
quantum degrees of freedom that have decohered to classicity for all practical
purposes, and are more resistant, in that complex ’selective environment’ to
returning to the purely quantum condition, will tend to persist as classical
entities in the universe. This will depend upon the local ’classicity
selective environment’ and is a possible form of abiotic natural selection
with abiotic blind final cause due to the (possibly changing) selective
environment. This argument supplies the start of an answer to ’what happens’
if there is no efficient cause function, or law, at the quantum-classical
boundary. For, as in the case of biological evolution, the selective
environment determines in part how readily a now classical entity tends to
remain classical rather than becoming quantum again by recohering. ’Tends’ is
again a dispositional term. The actual ways that decoherence happens and is
sustained against recoherence will depend upon actual detailed quantum and
classical processes. We cannot prestate those selective environmental
processes that are necessary and sufficient for the now classical (for all
practical purposes) entity to remain classical, FAPP. Thus, there is a process
carrying the system into the future, but no efficient cause law, or function,
describing it.
The above should be experimentally testable. In general, it is now believed
that complex entities decohere more rapidly than simple entities, eg electrons
and photons, which means a bias towards the emergence of classicity in complex
entities. Abiotic natural selection arises here with blind final cause, for we
cannot prestate all the complex environments which may impact decoherence in
any specific way. Anton Zeilinger has recently shown that
Buckmeisterfullerenes interfere in a two slit-like experiment. Presumably, as
the complexity of the objects in this experiment increases, and the complexity
of the surrounding environment increases, decoherence should begin to fail. As
it does, it may be possible to ask whether the decoherence process is fully
lawful or not, for example, by failure of stable statistics in fading
interference bands. More, if the complexity of the environment bears on
decoherence, then at that molecular complexity when interference begins to
fail due to decoherence, one would expect that a dense ’beam’ of the objects
sent through the two slits would behave more classically and show less
interference, than if the objects were sent through the two slits rarely. It
seems that the above are possible new experiments.
Finally, I note that D. d’Lambert commented to me that the above ideas imply
that the quantum measurement problem does not have a solution, (19). Taken
together these ideas, if correct, again seem to imply that there is no Theory
of Everything from which all is logically entailed. I comment that W. Zureck
might strongly disagree, (20).
## 5 Enabling Constraints and What They Enable
In about the year 1200 AD, the Calif of Cairo caused the only hospital in the
Islamic world to be constructed. Because patients were required to be treated
within the hospital, where Maimonides later practiced, it became possible to
train medieval physicians in a new manner. The hospital enabled a new form of
medical education and medical practice. More, the Calif was able, as a sign of
caring, to visit the patients in the hospital and thereby talk to poor people
he could not have met socially. This allowed the Calif to gain different
information about his realm and govern differently.
The hospital acted as an enabling constraint or enabling condition, and
enabled changes in medical education, treatment and governance. We obviously
know this is true, but have virtually no clear ways to think about enabling
constraints or what they enable.
A second example was raised by A. Juarraro in Dynamics in Action, (21). Could
we cash a check 50,000 years ago? No. Think of all the social inventions that
had to occur to allow this bit of human action. Laws, courts, credit,
bankruptcy laws, enforcement procedures, contract law, all had to come into
existence. In the law, the concept of enabling constraints is known. If you
and I enter into a contract, we are thereby constrained, but may be enabled to
form a corporation by that contract, with all the enabled actions of a
corporation in the contemporary world. The stories above of the invention of
the television remote, and the sequence leading from the first computer to
FaceBook, are also examples of situations arising that create new economic
niches and are enabling constraints, but not efficient causes. The enabling
constraints create opportunities seized. We know this is true, but do not
think about it. We have no theory for it.
Enabling constraints arise in biological evolution. A signal case is the
evolution of meiosis, chromosomal recombination and sex. Sex causes a two fold
loss in fitness as two parents are required, not one. But sex allows meiosis
and chromosomal recombination between homologous paternal and maternal
chromosomes that permits two advantageous genes, say A and B, initially with
one on the maternal and one on the paternal chromosome, to recombine so A and
B are on one chromosome and passed via sperm or egg to the offspring. This
process is much faster than waiting for A to arise by mutation on the B
bearing chromosome, so abets more rapid and efficient evolution. In short, sex
is an enabling constraint! The biosphere is not only evolving, it is evolving
the way it is building itself. It is evolving the very way it is evolving. If
sex and recombination yielded the emergence of Mendel’s laws, then life
evolved its own enabling constraint laws by which evolution itself became more
efficient. Then might the universe as a whole evolve its laws so that its
becoming’ was more efficient in a form of a Darwinian race among a set of
candidate enabling constraint laws and some definable notion of ’efficient’?
A second biological example almost certainly arose early in life. Current
cells use DNA, RNA, and encoded protein translation. But the process is very
complex, with transfer RNA and specific protein enzymes each of which charges
the appropriate transfer RNA with the ’right’ amino acid to allow proper
translation of messenger RNA. The entire system is needed for the system to
work. Early in the evolution of life, proto-cells presumably were reproducing,
perhaps did work cycles, but could not have been so complex. Call the
emergence of DNA, RNA, and encoded protein synthesis ’the Darwinian
Transition’. This transition has become an enabling constraint. All of life
since, presumably, the last common ancestor, has used this molecular
machinery: we are constrained to it. Yet this machinery enables the rapid
exploration of protein space by mutations to DNA sequences not needed for core
molecular reproduction. The biosphere, again, is evolving the very way it
evolves. The DNA/RNA/protein translation machinery is a powerful enabling
constraint ’law’, the central Dogma of molecular biology, that has enabled
enhanced evolution.
These biological examples seem deeply important for, unlike human law, no
conscious agency is invoked. The biosphere is building the way it builds
itself by evolving law-like enabling constraints that enable enhanced
biological evolution. This is an existence proof that nature is able to
achieve such a miracle. We broach the universe as a whole below.
I do not believe that these evolutionary processes are algorithmic, (13, 14).
We have no theory of enabling constraints and what they enable, which I also
do not think are algorithmic in general. I have no idea how to study enabling
constraints and what they enable in general, so I now sketch the earliest
stages of an admittedly limited and algorithmic approach to this question that
is now underway.
Consider chess. The rules of chess are the enabling constraints, the laws of
the world of chess. They enable very sophisticated, strategic play, as many of
us more or less know. We do not understand, in general, enabling constraints,
and what poor or superb ’strategies’ can emerge as in the history of chess
play. But a few observations start a discussion. Note that, given the move
rules of chess, the Adjacent Possible for White, or for Black, is fully
determined for each board position. Then I propose to ask, as a game proceeds,
what happens to the ’size’ of the adjacent possible for each side. In the end
game, typically the losing side has almost no adjacent possible, while the
winning side has a very large adjacent possible. How does this happen? How is
it related to the search depth of computer chess programs playing one another,
whether of equal ’strength’ or different strengths? I intend to find out in
this simple case.
Note that in chess the bishop can move along a diagonal that is free,
regardless of the position of the same side’s rook, as long as the rook does
not block the diagonal. The movements of chess pieces are largely independent
of one another except for blocking. But one can imagine chess - like rules in
which all positions of the rooks impacted the legal moves of the bishop. Or in
which all positions of all pieces impacted the allowed moves of the bishop. As
one tinkers with the move rules, and the dependencies of pieces moves on one
another’s positions, what happens to the games that are enabled? What happens
to the adjacent possible? We don’t know. Are the most complex games achieved,
under a to be determined criterion of ’complex’, if the pieces moves are
largely independent of one another? I have no idea.
My colleagues and I are also starting work on board games in which each side
has M pieces, each piece has, for each board condition a set of allowed next
positions. Thus, the adjacent possible for all board positions is perfectly
defined. As we tune the dependency of each piece’s moves on the positions of
is own sides other pieces, what happens to the adjacent possible of each piece
and why? If we start in the same position and step randomly several steps into
the successive adjacent possibles of a piece, and repeat this sequence many
times, do these ’histories’ spread out widely? Do they converge? Are there
some board positions reachable by very many other board positions, and others
that are hardly accessible? If so why? What are the implications of this
possible variation in adjacent possible board positions on the flow of the
games we envision next.
We plan to study games where each side can ’take’ a piece from the other side
by occupying its position via a legal move, as in chess or checkers. We
propose to allow two depth search, so each side can both ’try’ to take an
opponent’s piece and try to avoid having its own pieces taken. A game will be
won when all pieces of one side are taken. We propose to evolve the rules of
moving the pieces, so that winning players (or both winners and losers) can
alter their move rules to a set of ’next move rules’ to evolve toward rules
that allow longer more complex games. One measure of the complexity of a game
is to replay the same game multiple times, treat a board position as a vector,
and concatenate successive board positions of one game until the game is won
into a long vector. Repeated games will give some diversity of these vectors.
The ’normalized compression distance’, (22), between many pairs of games can
then be computed, to gain a measure of how diverse games under a given set of
move rules are. This diversity is one measure of game complexity. As the move
rules evolve toward more complex games, we hope to look at the dependency of
each piece’s adjacent possible move space on the positions of other pieces of
the same side. I hope we find that ’complex games’ evolve relative
independence of one piece’s moves on another’s positions except blocking
positions.
In short, a new body of theory is needed where virtually none exists: What are
enabling constraints and what possibilities do they enable? Board games are
interesting because they are so well defined. They are inadequate because the
move rules enabled by the Cairo hospital were not algorithmic, as are the
board games. An entire new field of research is needed. I believe and feel
sure it is worth exploring. All our legal codes, regulations, the biosphere
and perhaps, as I try to discuss next, the very physical laws of the universe,
are enabling constraints. What do they enable? How?
Notice for further discussion, that the move rules define an Adjacent
Possible. Below I ask where does ’the possible’ come from?
## 6 Might the Laws of Physics Be Abiotically
Selected Enabling Constraints?
Where are we now in fundamental physics and cosmology? We have the Standard
Model and General Relativity, and as yet no clear way to unite them. If the
above argument about lawlessness and abiotic blind final cause at the quantum
classical boundary is right, we may never unite the two. If blind final cause
is present in the evolution of the universe including the biosphere, let alone
human culture, there may be no theory of everything entailing all that occurs.
Meanwhile, we have the well known ’fine tuning’ of the 23 constants of nature.
It is widely believed that without this fine tuning we would not be in a
complex universe with stars, simple and complex atoms, chemistry and life. But
we have no rationale for why the constants have the values they do.
In face of this fine tuning, the current view in physics is of a multiverse,
where each ’pocket’ universe has its own values of the constants, perhaps
randomly distributed, and either the strong or weak Anthropic principle. The
former looks to a Creator God to tune the constants and is held to be outside
science. The latter assumes that only those universes with constants disposed
to allow stars, complex atoms, chemistry and life would have physicists to
puzzle about why the constants of their pocket universe were so tuned as to
allow their existence. Probably the weak Anthropic principle is the dominant
view among physicists today. Leonard Susskind, confronted with 10 to the 500th
string theories, envisions a cosmic landscape, with as many pocket universes,
each with a random choice of string theory from among the 10 to the 500th, and
we are the lucky ones, (23). Lee Smolin, in Life of the Cosmos, (24), imagines
universes born from black holes and emerging with minor variations of the
constants, so a cosmic natural selection among universes for those that are
more fecund because they have many black holes.
It is worth stressing that reductionism itself is what is driving us to the
multiverse. If we cannot account for the fine tuning of the 23 constants of
nature, and if all that arises in any universe is deductively entailed in its
efficient cause Final Theory of Everything, there is no choice but some space
of possible laws or one set of laws but many choices of values of the 23
constants, multiple universes, and some way of distributing the laws, or
constants, among these universes. But, as I note in a moment, Darwin tells us
that we are not limited to efficient causes, and that may change everything.
I now propose ’Darwin all the way down’. Suppose that there was, in the
beginning, or in a ’possible’ before the beginning as I try to discuss below,
an indefinitely or infinitely large set of laws to create universes - I’ll
give a conceivable example in a moment - and a cosmic natural selection
selected, in just one universe, those laws which, as enabling constraints,
enabled our very complex universe precisely because it was able to grow large
and complex, hence by persistent winning ’existence’, won Existence and
persistence are the abiotic analogues of the persistence of saber tooth tigers
existence and persistence in the biosphere. Existence and persistence of a
’winning’ universe that does so ’the best’, is the analogue of Pross’s kinetic
selection in a non-equilibrium chemical replicator system. We will see and are
the winners.
We then are in this universe, because, Darwin-like, it is the universe that
won by blind final cause. We now answer, in principle, a why question and
answer not just with an efficient cause ’how’ answer, but a ’why’ answer. Our
universe won and was able to become a very or most complex universe. That is
why our universe is as it is. For us to be satisfied, what constitutes
’winning’ for a universe must itself be ’natural’. For example, I want to
believe that the biosphere evolves, as a secular trend, to maximize its
Adjacent Possible in the non-ergodic universe: Perhaps as species diversity
and features per species and complexity of features increase, the ease of
forming positive sum games and mutualisms increases, driving further
diversification of organized processes in the biosphere. Perhaps the winning
universe wins by maximizing its Adjacent Possible into which it can ’become’
more rapidly than universes that grow their Adjacent Possibles more slowly.
Like the biosphere, the universe as a whole is vastly non-ergodic. The
metaphor is at least suggestive. Science sometimes starts with a mere
metaphoric image that later crystallizes usefully. The metaphor of the solar
system for atoms is a famous example.
We must note that any effort along these lines is radically unlike familiar
physics, for we are attempting to formulate the question: how are physical
laws enabling constraints and what kinds of universes do they enable? And if
there are a multitude of laws, how might an abiotic natural selection process
with blind final cause work to select among the laws? And what constitutes a
”winning” universe that might, by blind final cause, select the laws that
enable it?
I briefly mention a ’vacuum selection’ principle of which I am the author,
(13). It is only of interest as an example to show that such a vacuum
selection principle might be possible. Smolin and colleagues have explored
loop quantum gravity, (25). Here Planck scale tetrahedra of quantized units of
space build a universe by budding or cloning new tetrahedra on their faces,
via Pachner moves, where the tetrahedra are linked by what are called 15J
symbols. As Louis Crane showed, these 15J symbols, all integers, form a
denumerably infinite series of laws, (26), so can be pictured as a space of
laws with an ordering relation among them. Each 15J symbol implies the way the
discrete analogue of the Schrodinger equation propagates on the space
constructed by the tetrahedra. My idea was to allow uncertainty of the laws
themselves in an early universe, with a universe starting in one state of
geometry, (and ultimately particles), with one 15J symbol, and following all
possible paths to a final state where the 15J laws were different. Thus, if
the particles under these different laws, or geometries themselves, could
interact, quantum interference could arise. I reasoned that some small changes
in the 15J symbols could yield large changes in Schrodinger propagation, hence
yield destructive interference. Other small changes could yield, I hoped, very
small changes in how the Schrodinger equation propagated possibility
amplitudes, so lead to constructive interference. More slowly, Feynmann showed
in his sum over all possible histories formulation of quantum electrodynamics,
that nearly parallel pathways interfered constructively, while radically
twisting pairs of pathways interfered destructively, so near classical
parallel behavior was the most probable. Generalizing to the case where there
is to be uncertainty over the laws themselves, and summing over all histories
from all initial to all final states of tetrahedral space, with the same or
different 15J symbols, I hoped, mere sum over all possible pathways and
constructive interference, as Feynmann showed with a single Schrodinger
equation, would pick out the region(s) in the denumerably infinite space of
laws where constructive interference among a neighboring set of laws would
arise where small changes in 15 J symbols yielded tiny changes in how the
Shrodinger equation propagated, hence a universe whose laws, if initially
fluctuating slightly, showed constructive interference, would arise.
Ultimately, this universe would select out a single law 15J law. This simple
example, merely conceptual, is the start of a possible vacuum selection
principle among an infinite set of laws in a single universe with an infinite
set of possible laws, yet might be able to select the laws and ultimately, I
hoped, the constants, the particles, and all. If even logically possible, this
putative vacuum selection principle suffices as an example of a way a single
universe might evolve its own laws. No multiverse is needed here. If not, we
are not forced, e ven by reductionism in the sum over histories and laws
above, to a unique or very small set of neighboring laws, to posit a
multiverse.
But there may be other vacuum selection principles evolving the laws if we
allow forms of blind final cause for a ’winning universe’ selecting among the
possible laws, all in one universe evolving its laws so it ’becomes’ better.
In the case above of my hoped for vacuum selection principle by constructive
interference over sets of 15J laws, we already have a Feynmann framework to
understand what ’winning’ might mean - constructive interference. As noted, we
need to explore a wider set of what a winning universe that exists and
persists might mean and how that could ”blindly” select among enabling
constraint laws that enable its Adjacent Possible.
Since we do not know what enabling constraints enable what kind of universe,
only hints are available now. What might they be? It would seem, as noted just
above, that ’getting to persist’ - like saber tooth tigers - would be
important. Perhaps relative local independence of classical events, like the
bishop’s moves independent of the rook or the same in complex board games,
might be essential to a winning universe that can become big and complex, or
maximize the growth of its Adjacent Possible, hence win. There are clues to
such ’move independence’. Nother’s theorem, (27), shows that where there are
symmetries, for example of force applied and acceleration achieved, with
temporal, translational and rotational invariances, conservation of energy,
momentum and angular momentum are entailed. Why should these independencies
with respect to these spatial and motion symmetries be a feature of our laws
of physics? Bishops and rooks? Does this enable a universe with a larger
Adjacent Possible? Perhaps, in due course, this still intuitive question can
be formulated precisely.
Our particles form a group, with its symmetries. The group property implies
that the particles transform into one another, hence persist. What if
particles did not do this, but transformed into a spray of ever new particles
such that, even were reversibility allowed, they created an infinite ’jet’ of
particle types that would emerge. Then nothing would persist.
Could such group particle properties emerge from the evolution of random laws?
No one knows. But I now report remarkable results that hint the answer could
be yes.
I describe a wonderful numerical experiment some years ago by Walter Fontana,
(28), at the Santa Fe Institute. Fontana created a ’chemostat’ on his computer
which contained up to 50,000 Lisp expressions. Lisp expressions were chosen
randomly to act on Lisp expressions typically yielding new Lisp expressions.
Selective conditions were maintained by randomly throwing out Lisp expressions
if there were more than 50,000 in the computer chemostat. Fontana found that
at first a stream of unique Lisp expressions were generated. Then one of two
things happened. First, a Lisp expression able to copy itself emerged and took
over the chemostat. If copying was disallowed, collectively autocatalytic sets
of Lisp expressions emerged, in which each was formed by one or more of the
Lisp expression present. Fontana found that these collectively autocatalytic
sets of Lisp expressions formed an algebra, but not a group in that they
lacked an inverse and an identity operator. Nevertheless, his numerical
experiment is a toy example of entities bootstrapping themselves via random
laws co-evolving into self consistent co-creation and stable existence and
persistence. It is a long way to elementary particles forming a group, and
transmitting forces, but perhaps a hint. The transformations among the Lisp
expressions are mediated by Lisp expressions and seem the analogue of forces
carried by particles acting to transform particles into one another in a
group. If models can be explored which include the possibilities of reversible
transformations mediated by the same ’expression’ acting on two
interconverting pairs of ’expressions’, and a ’do nothing’ expression, perhaps
autocatalytic sets of expressions might emerge from a soup of co-evolving
random ’laws’ or ’expressions’ and form algebraic groups, (29).
Why are there conservation laws like that for a perfect harmonic oscillator,
free of friction? In the state space of position and velocity, orbits are
concentric curves. Adjacent curves have 0 Lyapunov exponents. Thus, they are
dynamically critical, and can persist and propagate information without loss
due to convergence in state space nor, in a noisy world, loss due to a
positive Lyapunov exponent and chaos. Why? Electromagnetic waves propagate,
exist and persist therefore, across the universe. They can propagate
information extremely well. Why such conservation laws? What do they enable?
Do they enable a larger Adjacent Possible for an emerging universe?
We do live in an extremely complex universe. If we take the fine tuning
arguments seriously, this is a profound puzzle. Just perhaps abiotic natural
selection provides a radical but ultimately useful new way to think about
this. If so, Godel is not the end of physics, but all this is a possible new
beginning in an open universe.
## 7 The Possible
How can we begin to think about ’the possible’? It seems we have to consider
the possible and its ontological status. We seem to need ’a possible’. I will
proceed in steps based on physical theory and beyond.
First, consider General Relativity and Einstein’s block universe. Here there
are no possibilities at all, only actual geometric world lines. Next consider
Newton, where a state of the system in space and time has a possible future
and past deterministic trajectory. It is not much of a possible, but more than
in General Relativity. Next consider quantum mechanics on the Copenhagen
interpretation and Born rule. Here we have at the fundamental level, a
Schrodinger equation for possibility waves. So we seem forced, on Copenhagen
at least, to consider the ’possible’. But notice an odd fact. We know
beforehand exactly what the quantum degrees of freedom are, spin,
polarization, and so forth, that we will measure. Whitehead, in Process and
Reality, (30), considers a metaphysics of Actuals giving rise to Possibles
that give rise to Actuals. But in quantum mechanics, such as Quantum
Electrodynamics, possibles can give rise to possibles in Feynmann’s sum over
all possible histories and his famous Feynmann diagrams.
Now consider the evolution of the biosphere by Darwinian preadaptations. We
seem to confront an Adjacent Possible of the biosphere where, unlike the
possibles of quantum mechanics, we cannot prestate the relevant degrees of
freedom, eg swim bladders. Unlike familiar quantum mechanics, we do not even
know what the variables might be. This failure may be due to the failure of
human language to describe all of ’relational’ reality in a continuous
spacetime in a denumerably infinite language. In any case, we seem forced to
consider ’the possible’, even one we cannot prestate. The same is true for the
evolution from the computer to FaceBook and history with its Cairo hospital.
Who foresaw the changes in medical training and practice that were enabled,
became possible, then actual?
’The Possible’ produces confusion because, in part, we live among Whitehead’s
Actuals. As with consciousness itself, we don’t know ’where’ the possible is
in space and time.
Consider any physical theory which posits a multiverse with a set of possible
values of the 23 constants, or Susskind’s Cosmic Landscape with pocket
universes having one of the 10 to the 500th string theories. Then it seems we
are forced to consider a set of ’possible’ values of the constants, or string
theories, somehow assigned to, or coming into existence with, universes in the
multiverse. What sense does it make to speak of these possibles before there
are any universes? It seems physicists may have slipped into speaking of these
possibles as, in some sense, real possibilities, outside of any universe(s)
that exist, whatever that means. Can there ’be’ a ’possible’ before’ one or
many universes exist and out of which it or they can become? Can a ’possible’
make any sense without the enabling constraints that seem to define it?
If we can speak of possible values of the 23 constants assigned somehow to
pocket universes, or 10 to the 500th string theories assigned somehow to
pocket universes, then it seems no stranger to consider a space of possible
laws, for example the 10 to the 500th string theories, before there is our one
universe and a very rapid vacuum selection principle, perhaps like mine above,
that, in a single universe, selects by constructive interference among
competing laws, or by blind final cause, that universe that ’wins’. Like the
weak Anthropic principle, such a vacuum selection would answer the question
why the constants have the values they do. And it would answer the question:
why these laws and partial lawlessness. But as noted, to base our thinking on
an abiotic natural selection among a family of laws, perhaps infinite, to
answer this question means understanding what a ’winning’ universe might be,
how the enabling constraint laws enable that winning universe, and how it wins
over other universes in the early evolution of our one universe and thereby
selects its own laws. Like the evolution of sex and reconbination and Mendel’s
laws, are our physical laws ’enabling constraints’ that became hardened and
entrenched as the universe itself evolved, such that the universe was then
constrained to those law? If the discussion above, in whole, is correct, to
pursue this avenue means giving up the reductionist dream of a final theory,
but it may open wide new doors in an open creative universe.
## Conclusion
Reductionism has been a brilliant success. It is built upon the mathmatization
of efficient cause and that cause as deductive entailment. It appears that
this is insufficient to describe the becoming of the biosphere by adaptive
evolution and more the evolution of the biosphere by Darwinian preadaptations.
The considerations above, together with Hawkings Godel and the End of
Physics, suggest we may be approaching a crisis in which 350 years of
reductionistic science will give way not only to emergence, but to an open
universe, partially enabled by enabling constraint laws, partially lawless,
uniting physics with history. There may be lawlessness at the quantum-
classical interface, making a Theory of Everything that explains by entailment
impossible. This one universe may have evolved by abiotic natural selection
among an infinite or vast set of laws to be a winning universe, perhaps by
maximizing the growth of its own Adjacent Possible, hence its own growth.
There might be an approach to the ancient question: Why is there something
rather than nothing? Hawkings Godel and the End of Physics may be only the
beginning of a new physics.
## Acknowledgement
This paper was partially funded by iCORE grant to Kauffman, and a Tekkes grant
to Kauffman as a Finnish Distinguished Professor.
## References
* [1] Weinberg, S., 1992, Dreams of a Final Theory: the Search for the Fundamental Laws of Nature. Pantheon Books, N.Y.
* [2] Anderson, P.W., 1972, More is Different, Science 177: 4047, 393-396.
* [3] Laughlin, R., 2005, A Different Universe, Basic Books, N.Y.
* [4] Hawking, S., 2009, Godel and the End of Physics,
http://www.damtp.cam.ac.uk/strings02/dirac/hawking/
* [5] Depew, D., Lecture, STOQ-Vatican Conference on Darwin and Evolution March 2009, Rome.
* [6] Dawkins, R., 1986, The Blind Watchmaker: Why the Evidence of Evolution Reveals a Universe without Design. W.W. Norton and Co. N.Y.
* [7] Rosen, R.,1980, Life Itself: A Comprehensive Inquiry into the Nature, Origin and Fabrication of Life, Columbia University Press. N.Y.
* [8] Monod, J., 1971, Chance and Necessity, Alfred Knopf, N.Y.
* [9] Kaufman, G., 2009, Personal Communication
* [10] Pross, A. and Khodorkovsky, V., 2004, Extending the concept of kinetic stability. Toward a paradigm for life, J. Phys. Org. Chem, 17: 312-316.
* [11] Pross, A. 2008, How can a chemical system act purposefully? Bridging between life and non-life, J. Phys. Org. Chem. Online, Wiley Interscience.
* [12] Deutsch, D., 1997, The Fabric of Reality, The Penguin Press.
* [13] Kauffman, S. A., 2000, Investigations, Oxford University Press, N.Y.
* [14] Kauffman, S. A., 2008, Reinventing the Sacred, Basic Books, N.Y.
* [15] Popper, K., 1982, The Open Universe: An Argument for Intedetminism. Roman and Littlefield, Lanham, MD.
* [16] Shor, P.W.,1995, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A. 52:4, R2493-R2496.
* [17] Briegel, H. J. and Popescu, S., Entanglement and Intramolecularlar cooling in biological systems? A quantum thermodynamic perspective, arXhiv:0806.4552v1 [quant ph] 27 June 2008
* [18] Cai, J., Popescu. S. and Briegel, H.J., Dynamic entanglement in oscillating molecules. arXiv:0809.4906v1[quant ph] 29 Sept 2008.
* [19] d’Lambert, D. 2009, Personal Communication
* [20] Zurek, W., 2009, Quantum Darwininsm, Nature Physics, vol 5, 181-188.
* [21] Juarraro, A., 1999, Dynamics in Action, Bradford Book, MIT Press, Cambridge MA.
* [22] Nykter, M., Price, N. D., Aldana, M., Ramsey, S. A., Kauffman, S. A., Hood, L., Yli-Harja, O. and Shmulevich, I. (2008). Gene Expression Dynamics in the Macrophage Exhibit Criticality. Proc Natl Acad Sci USA 105: 1897-1900.
* [23] Susskind, L., 2006, The Cosmic Landscape, Little Brown and Co. N.Y.
* [24] Smolin, L., 1997, The Life of the Cosmos, Oxford University Press, N.Y.
* [25] Smolin, L., 2001, Three Roads to Quantum Gravity, Basic Books, N.Y.
* [26] Crane, L. 1993, Personal Communication.
* [27] Nother, E. 1918, Invariante Variationsprobleme, Nachr. D. Konig. Gesellsch. D. Wiss. zu Gottingen, math-physics, 235-257.
* [28] Fontana, W.1991, Algorithmic Chemistry, Artificial Life II, SFI Studies in the Sciences of Complexity vol. X. Ed C.G Langton, C Taylor, J.D. Farmer, and S. Rasmussen, 159-209. Addison-Wesley, N.Y.
* [29] This seems a feasible study.
* [30] Whitehead, A.N., 1929, Process and Reality, An Essay in cosmology, Cambridge University Press. Cambridge, U.K.
|
arxiv-papers
| 2009-07-15T05:28:19 |
2024-09-04T02:49:03.921261
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Stuart Marongwe and Stuart Kauffman",
"submitter": "Stuart Kauffman",
"url": "https://arxiv.org/abs/0907.2492"
}
|
0907.2611
|
# Formation of Sub-millisecond Pulsars and Possibility of Detection
Y. J. Du,1 R. X. Xu,2 G. J. Qiao,2 and J. L. Han1
1National Astronomical Observatories, Chinese Academy of Sciences, Jia-20,
Datun Road, Chaoyang District, Beijing 100012, China
2Department of Astronomy, Peking University, Beijing 100871, China E-mail:
[email protected].
###### Abstract
Pulsars have been recognized as normal neutron stars, but sometimes argued as
quark stars. Sub-millisecond pulsars, if detected, would play an essential and
important role in distinguishing quark stars from neutron stars. We focus on
the formation of such sub-millisecond pulsars in this paper. A new approach to
form a sub-millisecond pulsar (quark star) via accretion induced collapse
(AIC) of a white dwarf is investigated here. Under this AIC process, we found
that: (1) almost all the newborn quark stars could have an initial spin period
of $\sim 0.1$ ms; (2) the nascent quark stars (even with a low mass) have
sufficiently high spin-down luminosity and satisfy the conditions for pair
production and sparking process to be as sub-millisecond radio pulsars; (3) in
most cases, the timescales of newborn quark stars in the phase of spin period
$<1$ (or $<0.5$) ms can be long enough to be detected.
As a comparison, an accretion spin-up process (for both neutron and quark
stars) is also investigated. It is found that, quark stars formed through AIC
process can have shorter periods ($\leq$ 0.5 ms); while the periods of neutron
stars formed in accretion spin-up process must be longer than $0.5$ms. Thus if
a pulsar with a period less than $0.5$ ms can be identified in the future, it
should be a quark star.
###### keywords:
Accretion – Gravitational waves – Stars: Neutron – Pulsars: General
††pagerange: Formation of Sub-millisecond Pulsars and Possibility of
Detection–A
## 1 Introduction
Though it has been more than 40 years since the discovery of radio pulsars,
their real nature is still not yet clear because of the uncertainty about cold
matter at supranuclear density. Both neutron matter and quark matter are two
conjectured states for such compact objects. The objects with the former are
called neutron stars, and with the latter are quark stars. It is an
astrophysical challenge to observationally distinguish real quark stars from
neutron stars (see reviews by, e.g., Madsen 1999; Glendenning 2000; Lattimer &
Prakash 2001; Kapoor & Shukre 2001; Weber 2005; Xu 2008). The most obvious
discrepancy could be the minimal spin period of these two distinct objects.
The minimal periods of these two kinds of objects are related to their
formation process. How fast a neutron star or a quark star can rotate during
the recycling process in low mass X-ray binary (LMXB) has been considered by
several authors (Bulik, Gondek-Rosi$\rm\acute{n}$ska & Klu$\rm\acute{z}$niak
1999; Blaschke et al. 2002; Zdunik, Haensel & Gourgoulhon 2002; Xu 2005; Arras
2005). Friedman, Parker & Ipser (1984) have found that neutron stars with the
softest equation of state can rotate as fast as 0.4 ms. The smallest spin
period for neutron stars computed by Cook, Shapiro & Teukolsky (1994) is about
0.6 ms. Frieman & Olinto (1989) have showed that the maximum rotation rate of
secularly stable quark stars may be less than 0.5 ms. Burderi & D’Amico (1997)
have discussed a possible evolutionary scenario resulting in a sub-millisecond
pulsar and the possibility of detecting a sub-millisecond pulsar with a fine-
tuned pulsar-search survey. Gourgoulhon et al. (1999) have investigated the
maximally rotating configurations of quark stars and showed that the minimal
spin period was between 0.513 ms and 0.640 ms. Burderi et al. (1999) have
predicted that there might exist a yet undetected population of massive sub-
millisecond neutron stars, and the discovery of a sub-millisecond neutron star
would imply a lower limit for its mass of about 1.7 $M_{\odot}$. A detailed
investigation about spin-up of neutron stars to sub-millisecond period,
including a complete statistical analysis of the ratio with respect to normal
millisecond pulsars, was performed by Possenti et al. (1999). The minimal
recycled period was found to be 0.7 ms. Gondek-Rosinska et al. (2001) have
found that the shortest spin period is approximately 0.6 ms through the
maximum orbital frequency of accreting quark stars. Huang & Wu (2003) have
found that the initial periods of pulsars are in the range of 0.6 $\sim$ 2.6
ms using the proper motion data. Zheng et al. (2006) have showed that hybrid
stars instead of neutron or quark stars may lead to sub-millisecond pulsars.
Haensel, Zdunik & Bejger (2008) have discussed the compact stars’ equation of
state (EOS) and the spin-up to sub-millisecond period, via mass accretion from
a disk in a low-mass X-ray binary.
There have been many observational attempts in searching sub-millisecond
pulsars. A possible discovery of a 0.5 millisecond pulsar in Supernova 1987A
is not held true in the follow-up observations (Sasseen 1990; Percival et al.
1995). Bell et al. (1995) reported on optical observation of the low mass
binary millisecond pulsar system PSR J0034-0534, and they used white dwarf
cooling models to speculate that, the limit magnitude of the J0034-0534’s
companion suggested that this millisecond pulsar’s initial spin period was as
short as 0.6 ms. As addressed by D’Amico & Burderi (1999), in particular the
detection of a pulsar with a spin period well below 1 ms could put severe
constraints on the neutron star structure and the absolute ground state for
the baryon matter in nature. They designed an experiment to find sub-
millisecond pulsars with Italian Northern Cross radio telescope near Bologna.
Edwards, van Straten & Bailes (2001) have found none of sub-millisecond
pulsars in a search of 19 globular clusters using the Parks 64 m Radio
telescope at 660 MHz with a time resolution of 25.6 $\rm\mu s$. Han et al.
(2004) did not find any sub-millisecond pulsars from highly polarized radio
source of NVSS (NRAO VLA Sky Survey). Kaaret et al. (2007) have found
oscillations at a frequency of 1122 Hz in an X-ray burst from a transient
source XTE J1739-285 which may contain the fastest rotating neutron star so
far. Significant difficulties do exist in current radio surveys for binary
sub-millisecond pulsars due to strong Doppler modulation and computational
limitations (Burderi et al. 2001).
How do sub-millisecond pulsars form? This is still an open question which we
will explore in this paper. Previously, discussions were concentrated on the
formation of neutron stars or quark stars spun up via accretion in binaries.
We have considered a new approach to create a sub-millisecond pulsar (quark
star) with super-Keplerian spin via accretion induced collapse (AIC) of a
massive white dwarf (WD). The initial spin of the newborn quark star could be
super-Keplerian, and it can have a long lifetime in sub-millisecond phase and
produce enough strong radio emission to be detected.
In $\S 2$, we discuss low mass quark stars formed from AIC of WDs, which can
have an minimal initial period of sub-millisecond. In $\S 3$, the radiation
parameters and the conditions for pair production are estimated in order to
investigate whether the AIC-induced quark stars could be pulsars or not. The
lifetimes of sub-millisecond pulsars are also estimated and the possibility of
detection is discussed. The spin-down evolution diagrams of a newborn quark
star and neutron star are also plotted. In $\S 4$, as a comparison, the sub-
millisecond pulsars formed through accretion acceleration (spin-up) in binary
systems are also considered. In $\S 5$, conclusions and discussions are
presented.
## 2 Sub-millisecond quark stars formed through AIC of white dwarfs
Neutron star’s formation from AIC of a massive white dwarf is widely discussed
by many authors (Nomoto et al. 1979; Nomoto & Kondo 1991; van Paradijs et al.
1997; Fryer et al. 1999; Bravo & García-Senz 1999; Dessart et al. 2006).
Recently, it is pointed out that Galactic core-collapse supernova rate cannot
sustain all the separate neutron star populations (Keane & Kramer 2008), which
implies other mechanisms for forming neutron stars. AIC of a massive WD can be
an important mechanism for pulsar formation, even for isolated pulsars if the
binary systems are destroyed due to strong kicks. We now discuss the
possibility for a low mass quark star formed from AIC of a WD. In a binary
system, when the WD has accreted enough matter from the companion so that its
mass reaches the Chandrasekhar limit, the process of electron capture may
induce gravitational collapse. The detonation waves burn nuclear matter into
strange quark matter which spread out from the inner core of the WD (Lugones,
Benvenuto & Vucetich 1994). A boundary of strange quark matter and nuclear
matter will be found at the radius where the detonation waves stop when
nuclear matter density drops below a critical value. A similar process was
also discussed and calculated by Chen, Yu & Xu (2007). The size of the inner
collapsed core may depend on the chemical composition and accretion history of
the WD (Nomoto & Kondo 1991). Consequently, quark stars with different masses
could be formed.
Figure 1: The relation of mass and radius of WDs. The red line is theoretical
line. The blue triangles and circles are observed WDs’ data which were taken
from Table 1 of Należyty & Madej (2004) and Table 3 & 5 of Provencal et al.
(1998), respectively. Among these data, the WD RE J0317-853 is the most
massive WD, whose mass and radius are 1.34$M_{\odot}$ and 2400 km
respectively. The square is the point of a WD with the Chandrasekhar mass
limit.
Both rigidly and differentially rotating WDs are taken into account. As a
first step, we assume that both the collapsed WD and the newborn quark star
have rigidly rotating configurations for simplicity. The WDs, progenitors of
these quark stars, could have a uniformly rotating configuration due to the
effects of crystallization, as well as an increase of central density may lead
to catastrophic evolution (supernova) (Koester 1974). With these assumptions,
a model of sub-millisecond pulsars’ formation is given below. The initial spin
period of AIC-produced quark stars can be estimated as follows. We assume that
the mass ($M_{\star}$) of the nascent quark star ranges from
$10^{-3}M_{\odot}$ to $1M_{\odot}$, and the white dwarf rotates rigidly at the
Kepler period ($P_{\rm K}$) just before collapsing. The quark star’s rest mass
($M_{\star}$) is approximately equal to the mass ($m_{\rm core}$) of the inner
collapsed core of the white dwarf. If the angular momentum is conserved during
AIC, the newborn quark star can rotate at a much shorter period, $P_{\rm q}$,
then
$I_{\rm core}\frac{2\pi}{P_{K}}=I_{\rm q}\frac{2\pi}{P_{\rm q}}.$ (1)
This is to say,
$P_{\rm q}=\frac{I_{\rm q}}{I_{\rm core}}P_{\rm K},$ (2)
where $I_{\rm q}$ is the quark star’s moment of inertia, and $I_{\rm core}$ is
the moment of inertia of WD’s inner collapsed core, which can be well
approximated by
$I_{\rm core}\simeq\frac{2}{5}M_{\rm core}R_{\rm core}^{2}.$ (3)
The mass and radius of a low mass $(M_{\star}\leqslant 1M_{\odot})$ quark star
could be approximately related by $M_{\star}=(4/3)\pi(4\beta)R^{3}$ (Alcock,
Farhi & Olinto 1986) in the bag model. We have an approximate formula for the
fast rotating quark star’s moment of inertia
$\displaystyle I_{\rm q}$ $\displaystyle=$ $\displaystyle
2\int_{0}^{R}dz\int_{0}^{\sqrt{R^{2}-z^{2}}}\frac{4\beta 2\pi
x^{3}}{\sqrt{1-\frac{4\pi^{2}x^{2}}{c^{2}P_{\rm q}^{2}}}}dx$ (4)
$\displaystyle=$ $\displaystyle\frac{\beta cP_{\rm q}}{16\pi^{4}}[6\pi
c^{3}P_{\rm q}^{3}R-8\pi^{3}cP_{\rm q}R^{3}+(16\pi^{4}R^{4}$ $\displaystyle+$
$\displaystyle 8\pi^{2}c^{2}P_{\rm q}^{2}R^{2}-3c^{4}P_{\rm
q}^{4})\ln\frac{1+\frac{2\pi R}{cP_{\rm
q}}}{\sqrt{1-\frac{4\pi^{2}R^{2}}{c^{2}P_{\rm q}^{2}}}}],$
where z-axis is spin axis; x is integral variable of each disc perpendicular
to the spin axis; $c$ is the speed of light; $R$ is the quark star’s radius;
the bag constant $\beta$ of quark stars is (60–110) MeV fm-3, i.e. (1.07–1.96)
$\times 10^{14}$ g cm-3, $\beta_{14}$ in units of $10^{14}~{}\rm g~{}cm^{-3}$.
For WD, we made a code, using both non-relativistic hydrostatic equilibrium
equation
$\frac{dp}{dr}=-\frac{Gm(r)\rho(r)}{r^{2}},$ (5)
and general equation of state (EOS) for a completely degenerate fermi gas
$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{1}{3\pi^{2}\hbar^{3}}\int_{0}^{p_{F}}\frac{c^{2}p^{4}}{\sqrt{c^{2}p^{2}+m^{2}c^{4}}}dp$
(6) $\displaystyle=$ $\displaystyle 1.42\times 10^{25}\phi(x)~{}\rm
dyn\;cm^{-2},$
where $x\equiv p_{F}/mc$, $\lambda_{e}=\hbar/(mc)$ the electron’s Compton
wavelength, $P_{\rm F}$ the fermi momentum,
$\displaystyle\phi(x)$ $\displaystyle=$
$\displaystyle(8\pi^{2})^{-1}\\{x(1+x^{2})^{1/2}(2x^{2}/3-1)$
$\displaystyle+\ln[x+(1+x^{2})^{1/2}]\\}$
to calculate the mass ($m_{\rm core}$) and moment of inertia ($I_{\rm core}$)
of the collapsed core of a massive WD, where $p$, $\rho$, $G$, $\hbar$ are
pressure, mass density, the gravitational constant and the Planck constant,
respectively.
Using Eqs. (5) and (6), one can make numerical calculation to get the WD’s
theoretical relation of mass and radius (the red line in Figure 1). For
comparison, one can see a figure on line
111http://cococubed.asu.edu/code_pages/coldwd.shtml. Before collapsing, the
mass of a WD is close to the Chandrasekhar mass limit, as high as $M_{\rm
WD}=1.4M_{\odot}$ (Shu 1982), the corresponding radius is much smaller, such
as $R_{\rm WD}=410$ km. We could numerically obtain the initial period,
$P_{\rm q}$, of nascent quark stars with different mass via Eq. (2), and find
that almost all the values of $P_{\rm q}$ are around $\sim 0.1$ ms (See Table
1) if the WD rotates rigidly at an almost Kepler period due to accretion (or
spin-up) in a binary just before collapsing. The newborn quark stars’ surface
spin velocities are well above the Kepler velocities, we regard this as “the
super-Keplerian case”.
WDs may be rotating differentially. The detailed calculations are given in
Appendix A. Therefore, as a follow-up second step, we also use Eqs. (2), (5),
(6) and (A2) to calculate the initial spin period of the nascent quark stars
in the differentially rotating WD model, taking the free parameter $a=0.5$.
The results are also shown in Table 1.
A newborn quark star could certainly rotate differentially, and may be relaxed
to become a rigidly rotating configuration finally. However, the timescale of
the relaxation depends on the viscosity and the state of cold quark matter
(Xu, 2009). Nevertheless, the newborn quark star’s relaxation (from
differentially rotating configuration to rigidly rotating configuration) may
be due to fast solidification after birth. A calculation shows that the
solidification timescale is only $10^{3}-10^{6}$ s (Xu & Liang, 2009).
Therefore, the relaxation timescale could be much shorter than the lifetime of
pulsars within sub-millisecond periods (See the following section 3.3).
The WD RE J0317-853 has the highest observed mass (1.34 $M_{\odot}$ close to
the Chandrasekhar limit) with radius of 2400 km (Należyty & Madej 2004). If a
WD like RE J0317-853 could be in a binary and accreted enough materials to the
Chandrasekhar limit, then it may collapse. Therefore, under this assumption,
we also calculated the initial spin periods $\widehat{P}_{\rm q}$ and
$\widehat{P}_{\rm dif}$ of a nascent quark star. The calculated results are
listed in Table 1\. It is found that, even if a WD has a larger radius such as
2400 km, it can also collapse to a sub-millisecond quark star for either
rigidly or differentially rotating WD models. In the differentially rotating
WD model, it tends to give a rigidly rotating configurations in the limit of
large values of $a$, $P_{\rm dif}$ increases as the parameter $a$ increases.
The conclusions from the rigid rotation model are valid even if differential
rotation is included.
Can a quark star survive even if it rotates at such a high frequency ($\sim
10^{4}$ Hz)? Will it be torn apart by the centrifugal force? There are quite
distinguishing characteristics between neutron stars and quark stars. A low
mass quark star is possible to spin at a super-Keplerian frequency because it
is self-bound by strong interaction. On one hand, as noted by Qiu & Xu (2006),
astrophysical quark matter splitting could be color-charged if color
confinement cannot be held exactly because of causality. On the other hand,
however, rapidly spinning quark matter could hardly split if color confinement
is held exactly. In addition, the recently discovered nature of strongly
coupled quark gluon plasma (sQGP) as realized at Relativistic Heavy Ion
Collider (RHIC) experiment (e.g., Shuryak 2006) may also prevent a super-
Keplerian quark star to split.
The short spin period above is not surprising, and could be verified for a
simplified special case, if both quark star’s density (=4$\beta$) and white
dwarf’s density ($=\rho_{c}$) are uniform. Using Eq. (2) and the mass-radius
relation, we can find the initial period of the quark star to be $P_{\rm
q}=(\rho_{c}/4\beta)^{2/3}P_{\rm WD}\sim 4\times
10^{-3}(\rho_{11}/\beta_{14})^{2/3}P_{\rm WD}$ (with $P_{\rm WD}$ the spin
period of white dwarf, $\rho_{11}=\rho_{c}/10^{11}$g cm3,
$\beta_{14}=\beta/10^{14}\rm g$ cm3), which depends only on the densities of
WD and quark star.
If the WD has not been spun up fully to the Kepler period, i.e., the WD
rotates at a sub-Keplerian period (e.g., several times of $P_{\rm K}$) before
AIC, can the initial period of a newborn quark star formed from such a WD be
of sub-millisecond? We investigated the case of a massive WD (1.4$M_{\odot}$,
410 km) rotating at a period $P_{\rm WD}=5P_{\rm K}\sim 600$ ms. The initial
spin period of quark stars with different mass are as follows: $\hat{P}_{\rm
q}\sim 0.11$ ms for a quark star with mass of $0.001M_{\odot}$; $\sim 0.24$ ms
for $0.01M_{\odot}$; $0.35$ ms for $0.1M_{\odot}$ and $\sim 0.36$ ms for
$1M_{\odot}$. The spin-down feature of such a newborn quark star depends on
its gravitational wave radiation and magnetodipole radiation (see details in
$\S 3$).
Table 1: The minimal initial period ($P_{\rm q}$) and lifetimes ($\tau$) due
to GW and EM radiation in the phase of sub-millisecond period for quark stars
with different masses in the super-Keplerian case. $P_{\rm q}$ and $P_{\rm
dif}$ are calculated via angular momentum conservation using rigidly and
differentially rotating WD model with central density of $10^{11}\rm
g\,cm^{-3}$. $\widehat{P}_{\rm q}$ and $\widehat{P}_{\rm dif}$ are similarly
calculated but using a WD like RE J0317-853 with mass of 1.4$M_{\odot}$ and
radius of 2400 km. $\tau_{1}$ is quark stars’ lifetime in the phase of $<1$
ms, while $\tau_{2}$ is the timescale in the phase of $<0.5$ ms. $P_{\rm q}$
is also used in Table 2 & 3 and Figure 2. $\beta$ is the bag constant,
$\varepsilon_{e}$ is the gravitational ellipticity.
Mass | Radius (km) | $P_{\rm q}$(ms) | $P_{\rm dif}$(ms) | $\widehat{P}_{\rm q}$(ms) | $\widehat{P}_{\rm dif}$(ms)
---|---|---|---|---|---
$(M_{\odot})$ | $\beta=60\rm~{}MeV~{}fm^{-3}$ | $\beta=60\rm~{}MeV~{}fm^{-3}$ | $a=0.5$ | $\beta=60\rm~{}MeV~{}fm^{-3}$ | $a=0.5$
0.001 | 1.04 | 0.0699 | 0.0261 | 0.0481 | 0.0252
0.01 | 2.24 | 0.0751 | 0.0472 | 0.0512 | 0.0470
0.1 | 4.81 | 0.104 | 0.101 | 0.102 | 0.101
1 | 10.37 | 0.221 | 0.218 | 0.218 | 0.218
Mass($M_{\odot}$) | $P_{\rm q}$(ms) | $\tau_{1}$(yr) | $\tau_{2}$(yr)
---|---|---|---
$\varepsilon_{e}=10^{-6}$ | $\varepsilon_{e}=10^{-9}$ | $\varepsilon_{e}=10^{-6}$ | $\varepsilon_{e}=10^{-9}$
0.001 | 0.0699 | $3.4\times 10^{7}$ | $4.5\times 10^{10}$ | $2.1\times 10^{6}$ | $1.1\times 10^{10}$
0.01 | 0.0751 | $7.3\times 10^{5}$ | $2.0\times 10^{10}$ | $4.5\times 10^{4}$ | $4.3\times 10^{9}$
0.1 | 0.104 | $1.6\times 10^{4}$ | $5.4\times 10^{9}$ | $9.0\times 10^{2}$ | $6.5\times 10^{8}$
1 | 0.221 | $3.4\times 10^{2}$ | $3.1\times 10^{8}$ | 2.2$\times 10^{1}$ | $2.0\times 10^{7}$
## 3 Radiation of sub-millisecond quark stars with low masses
The mass of most sub-millisecond quark stars formed from WD’s AIC is so low,
can the quark star produce radiation luminous enough to be observed like
millisecond pulsars? This is related to two aspects. First of all, is the
rotational energy loss rate high enough to power the electromagnetic radiation
as normal neutron stars? Secondly, is the potential drop in the inner gap high
enough for pair production and sparking to take place in the inner gap? These
are necessary conditions for radio emission of pulsars.
### 3.1 The spin-down power of sub-millisecond pulsars
Normal radio pulsars are rotation-powered, and the radiation energy is coming
from the rotational energy loss. Here we neglect gravitational wave radiation
first, then the rate $\dot{E}_{\rm rot}$, is
$\dot{E}_{\rm rot}=\frac{8\pi^{4}R^{6}B^{2}P^{-4}}{3c^{3}}.$ (7)
Comparing the rotational energy loss rate ($\dot{E}_{\rm rot,q}$) of quark
stars with normal neutron stars’ ($\dot{E}_{\rm rot,NS}$), one can have
$\dot{E}_{\rm rot,q}/\dot{E}_{\rm rot,NS}=\frac{R^{6}_{\rm q}B_{\rm
q}^{2}P^{-4}_{\rm q}}{R^{6}_{\rm NS}B_{\rm NS}^{2}P^{-4}_{\rm NS}}.$ (8)
If we take normal parameters, such as the surface magnetic field of polar cap
$B_{\rm q}=10^{8}$ G, $B_{\rm NS}=10^{12}$ G, the rotational period $P_{\rm
q}=0.1$ ms and $P_{\rm NS}=1$ s, the result is $\dot{E}_{\rm
rot,q}/\dot{E}_{\rm rot,NS}=10^{2}$ even for a quark star with a mass of
$0.001M_{\odot}$. This means that the quark stars have enough rotational
energy to radiate, hundred times than normal pulsars, even if the mass is so
low.
Table 2: Gap parameters estimated for sub-millisecond quark stars. $\dot{E}_{\rm rot}$ is the spin-down luminosity; $h_{\rm CR}$ is the curvature radiation (CR) gap height; $\Delta V_{\rm CR}$ is the potential drop of CR gap; $h_{\rm res}$ is the height of resonant ICS gap; $\Delta V_{\rm res}$ is the potential drop of resonant ICS gap; $h_{\rm th}$ is the thermal ICS gap height; $\Delta V_{\rm th}$ is the potential drop of thermal ICS gap. $M(M_{\odot})$ | $\dot{E}_{\rm rot}$(erg s-1) | $h_{\rm CR}$(cm) | $\Delta V_{\rm CR}$(V) | $h_{\rm res}$(cm) | $\Delta V_{\rm res}$(V) | $h_{\rm th}$(cm) | $\Delta V_{\rm th}$(V)
---|---|---|---|---|---|---|---
0.001 | $4.99\times 10^{36}$ | $1.16\times 10^{4}$ | $2.84\times 10^{10}$ | $3.13\times 10^{5}$ | $2.05\times 10^{13}$ | $1.18\times 10^{3}$ | $2.93\times 10^{8}$
0.01 | $3.75\times 10^{38}$ | $1.34\times 10^{4}$ | $3.76\times 10^{10}$ | $3.64\times 10^{5}$ | $2.78\times 10^{13}$ | $1.31\times 10^{3}$ | $3.62\times 10^{8}$
0.1 | $1.02\times 10^{40}$ | $1.72\times 10^{4}$ | $6.19\times 10^{10}$ | $4.61\times 10^{5}$ | $4.46\times 10^{13}$ | $1.62\times 10^{3}$ | $5.47\times 10^{8}$
1 | $5.00\times 10^{40}$ | $2.65\times 10^{4}$ | $1.47\times 10^{11}$ | $6.74\times 10^{5}$ | $9.52\times 10^{13}$ | $2.36\times 10^{3}$ | $1.16\times 10^{9}$
### 3.2 Particle acceleration for sub-millisecond pulsars
In most radio emission models of pulsars, such as RS model (Ruderman &
Sutherland 1975, hereafter RS75), inverse Compton scattering (ICS) model (Qiao
& Lin 1998), the multi-ring sparking model (Gil & Sendyk 2000), the annular
gap model (Qiao et al. 2004) and so on, the potential drop in the inner gap
must be high enough so that the pair production condition can be satisfied.
In the inner vacuum gap model, there is strong electric field parallel to the
magnetic field lines due to the homopolar generator effect. The particles
produced through $\gamma-B$ process in the gap can be accelerated to ultra-
relativistic energy (i.e., the lorentz factor can be $10^{6}$ for normal
pulsars). The potential across the gap is (RS75)
$\bigtriangleup V=\frac{\Omega B}{c}h^{2},$ (9)
where $\Omega$ is the angular frequency of the pulsar; $h$ is the gap height;
$B$ and $c$ represent the magnetic field at the surface of the neutron star
and the speed of light, respectively. As $h$ increases and approaches $r_{p}$,
the potential drop along a field line traversing the gap can not be expressed
by Eq. (9) above. In this case the potential can reach a maximum value
$\bigtriangleup V_{\max}=\frac{\Omega B}{2c}r_{p}^{2},$ (10)
where $r_{p}$ is the radius of the polar cap.
Let us make an estimate about the quark star’s potential drop $\bigtriangleup
V_{\rm q}$ in the polar gap region
$\bigtriangleup V_{\rm q}=\frac{\Omega B_{\rm q}}{2c}r_{\rm p,q}^{2},$ (11)
where $\Omega=2\pi/P_{\rm q}$, $r_{\rm p,q}=R_{\rm q}(2\pi R_{\rm q}/{cP_{\rm
q}})^{1/2}$. For normal neutron stars, $\bigtriangleup V$ can be obtained by
just changing the subscript q to NS. Thus
$\frac{\bigtriangleup V_{\rm q}}{\bigtriangleup V_{\rm NS}}=\frac{B_{\rm
q}R_{\rm q}^{3}P_{\rm q}^{-2}}{B_{\rm NS}R_{\rm NS}^{3}P_{\rm NS}^{-2}}.$ (12)
As one can take $R_{\rm q}=1$ km for a quark star with the mass of
$0.001M_{\odot}$, $B_{\rm q}=10^{8}$ G, $B_{\rm NS}=10^{12}$ G, $R_{\rm
NS}=10$ km, $P_{\rm q}=0.1$ ms and $P_{\rm NS}=1$ s, we find that
$\bigtriangleup V_{\rm q}/\bigtriangleup V_{\rm NS}=10$. This means that the
quark stars can have enough potential drops in the polar cap regions.
In the inner gap model, $\gamma-B$ process plays a very important role, two
conditions should be satisfied at the same time for pair production: (1) to
produce high energy $\gamma$-ray photons, a strong enough potential drop
should be reached; (2) for pair production, the energy component of
$\gamma$-ray photons perpendicular to the magnetic field must satisfy
$E_{\gamma,\perp}\geq 2m_{e}c^{2}$ (Zhang & Qiao 1998).
Particles produced in the gap can be accelerated by the electric field in the
gap and the Lorentz factor of the particles can be written as
$\gamma=\frac{e\bigtriangleup V}{m_{e}c^{2}},$ (13)
where $\gamma$ is the Lorentz factor of the particles accelerated by the
potential $\bigtriangleup V$, $m_{e}$ the mass of an electron or positron, $e$
the charge of an electron.
In $\gamma-B$ process, the conditions for pair production are that the mean
free path of $\gamma$-ray photon in strong magnetic field is equal to the gap
heights, $l\approx h$. The mean free path of $\gamma$-ray photon is given by
(Erber 1966)
$l=\frac{4.4}{e^{2}/\hbar
c}\frac{\hbar}{m_{e}c}\frac{B_{c}}{B_{\perp}}\exp(\frac{4}{3\chi}),$ (14)
where $B_{\rm c}=4.414\times 10^{13}$ G is the critical magnetic field,
$\hbar$ the Planck’s constant,
$\chi=\frac{E_{\gamma}}{2m_{e}c^{2}}\sin\theta\frac{B}{B_{\rm
c}}=\frac{E_{\gamma}}{2m_{e}c^{2}}\frac{B_{\perp}}{B_{\rm c}},$ (15)
and $B_{\perp}$ is the magnetic field perpendicular to the moving direction of
$\gamma$ photons, which can be expressed as (RS75)
$B_{\perp}\approx\frac{h}{\rho}B\approx\frac{l}{\rho}B.$ (16)
Here $l\approx h$ is the condition for sparks (pair production) to take place.
$\rho$ is curvature radius of the magnetic field lines. For dipole magnetic
configuration, it is (Zhang et al. 1997a)
$\rho\approx\frac{4}{3}(\lambda Rc/\Omega)^{1/2}.$ (17)
where $\lambda$ is a parameter to show the field lines, $\lambda=1$
corresponding to the last opening field line. Gamma-ray energy from the
curvature radiation process can be written as
$E_{\gamma,cr}=\hbar\frac{3\gamma^{3}c}{2\rho}.$ (18)
We estimated the gap heights based on Zhang, Qiao & Han (1997b), i.e.
$\displaystyle h_{\rm CR}\simeq 10^{6}P^{3/7}B_{8}^{-4/7}\rho_{6}^{2/7}\rm
cm.$ (19)
When the relevant parameters used are $B=10^{8}$ G , $P=P_{\rm q}$ and
assuming a dipole magnetic configuration, for any mass quark stars, one can
estimate the gap height from curvature radiation (CR) $h_{cr}\approx
10^{4}\rm~{}cm=100~{}\rm m$. This means that even if without multipolar
magnetic field assumption, the quark star can still work well for the CR pair
production.
There are three gap modes for pair production, i.e. resonant ICS mode,
thermal-peak ICS mode and CR mode (Zhang et al. 1997a). Each mode has relevant
gap parameters including gap potential drop $\Delta V$ and the mean free path
$l$ of $\gamma-B$ process. For normal neutron stars, one needs the assumption
of a multipolar magnetic field, $\rho=10^{6}$ cm, as RS75; but for $0.1$ ms
low mass quark stars, the dipole curvature radius is about $10^{6}$ cm. We
estimated gap heights and other parameters based on the work of Zhang, Qiao &
Han (1997b), as shown in Table 2.
One can see from Table 2 that when the high energy gamma-ray photons come from
resonant photon production, the height of the gap is larger. For the thermal-
peak ICS mode, it is one order of magnitude lower than the CR mode, and two
order of magnitude lower than resonant ICS mode. This means that in most
cases, the thermal-peak ICS induced pair production is dominated in the gap.
The newborn sub-millisecond quark stars have enough spin-down luminosities and
gap potential drops (see Table 2), so that they may emit radio or $\gamma$-ray
photons with sufficient luminosities, which can be detected by new facilities,
e.g., FAST and Fermi (formerly GLAST).
### 3.3 Lifetimes of the sub-millisecond pulsars in the phase of a short spin
period
Sub-millisecond pulsars may be very rare, or the timescale for such a pulsar
to stay in the short period phase ($<1$ ms) may not be long enough due to
magnetodipole (EM) radiation and gravitational wave (hereafter GW) radiation
(Andersson 2003). The lowest order GW radiation is bar-mode, which is due to
non-axisymmetric quadrupole moment. Here we consider GW radiation on the bar
mode which exerts a larger braking torque with braking index $n\approx 5$ than
magnetodipole radiation ($n=3$). The rotation frequency drops quickly due to
GW radiation and EM radiation:
$-I\Omega\dot{\Omega}=\frac{32GI^{2}\varepsilon_{e}^{2}\Omega^{6}}{5c^{5}}+\frac{B_{0}^{2}R^{6}\Omega^{4}}{6c^{3}},$
(20)
where $c$ is the speed of light, $\varepsilon_{e}=\Delta a/\bar{a}$ is the
gravitational ellipticity (equatorial ellipticity), $\Delta a$ is the
difference in equatorial radii and $\bar{a}$ is the mean equatorial radii.
To simplify Eq. (20), we introduce the notation
$A=32GI\varepsilon_{e}^{2}/(5c^{5})$ and $D=B_{0}^{2}R^{6}/(6Ic^{3})$, and
integrate the equation in the angular velocity’s domain
$[\Omega_{i}=2\pi/P_{\rm i},\Omega_{0}=2\pi/0.001]$, then
$\tau=\frac{1}{2D}(\frac{1}{\Omega_{0}^{2}}-\frac{1}{\Omega_{i}^{2}})-\frac{A}{2D^{2}}\ln{\frac{\frac{1}{\Omega_{0}^{2}}+\frac{A}{D}}{{\frac{1}{\Omega_{i}^{2}}+\frac{A}{D}}}}.$
(21)
An accurate ellipticity of quark stars is unfortunately uncertain.
Nevertheless, let’s estimate the $\varepsilon_{e}$ to calculate the timescales
in the sub-millisecond period phase for GW and EM radiations. Cutler & Thorne
(2002) suggested $\varepsilon_{e}=(I-I_{0})/I_{0}\leq 10^{-6}$. Regimbau & de
Freitas Pacheco (2003) found from their simulations that
$\varepsilon_{e}=10^{-6}$ is the critical value to have an at least one
detection with interferometers of the first generation (LIGO or VIRGO). It was
shown that direct upper limit was $\varepsilon_{\rm e}\simeq 1.8\times
10^{-4}$ on GW emission from the Crab pulsar using data from the first 9
months of the fifth science run of LIGO (Abbott et al. 2008). In addition,
Owen (2005) showed that the maximum ellipticity of solid quark stars was
$\varepsilon_{\rm e,max}=6\times 10^{-4}$. From the on-line catalogue hosted
by the ATNF 222http://www. atnf.csiro.au/research/pulsar
/catalogue/, the seventh fastest rotating millisecond pulsar is PSR
J0034-0534, which has very low period derivative $\dot{P}\sim 4.96\times
10^{-21}\,\rm s\,s^{-1}$. We thus use such a low $\dot{P}$ and Eq. (20) to
constrain the lower limit of the sub-millisecond pulsars’ ellipticity, which
is $\varepsilon_{\rm e,min}\sim 10^{-9}$ if the stellar mass is one order of
one Solar mass. For quark stars, in order to facilitate to compare with the
neutron stars’ lifetime ($\tau$) in the phase of sub-millisecond period, we
use mean equatorial ellipticities $\varepsilon_{e}=10^{-6}$ and
$\varepsilon_{e}=10^{-9}$ to calculate $\tau$ for both quark stars and neutron
stars through Eq. (21).
In the case of $\varepsilon_{e}=10^{-6}$, if we make the hypothesis that the
rotational energy is lost because of EM radiation, then one can easily derive
$\tau_{\rm EM}=1/(2D)(1/\Omega_{0}^{2}-1/\Omega_{i}^{2})\sim 5.9\times 10^{9}$
yr for a typical compact star with $B_{0}\sim 10^{8}$ G and $M=M_{\odot}$.
While, if we suppose that the rotational energy is lost due to GW radiation,
then $\tau_{\rm GW}=1/(4A)(1/\Omega_{0}^{4}-1/\Omega_{i}^{4})\sim 10^{2}$ yr
for a typical compact star. The energy loss rate of GW & EM radiation in the
phase of sub-millisecond period, for a typical compact star which has a low
magnetic field (108–109 G) either from AIC (Xu 2005) or spun up, are
$\dot{E}_{\rm GW}=32GI^{2}\varepsilon_{e}^{2}\Omega^{6}/(5c^{5})=7.0\times
10^{41}P_{\rm ms}^{-6}\rm\,erg\,s^{-1}$ and $\dot{E}_{\rm
EM}=B_{0}^{2}R^{6}\Omega^{4}/(6c^{3})=9.6\times 10^{34}P_{\rm
ms}^{-4}B_{8}^{2}R_{6}^{6}\rm\,erg\,s^{-1}$, respectively. Even if a quark
star with 1 $M_{\odot}$ formed from WD’s AIC has a high magnetic field such as
$10^{12}$ G, the lifetime $\tau$ in the phase of sub-millisecond is 37 years,
in comparison with $\tau=336$ years for $B_{0}=10^{8}$ G. Then the EM energy
loss is similar to the GW energy loss and becomes very important for
$B_{0}=10^{12}$ G. For $B_{0}$ ranges from $10^{8}$ G to $10^{11}$ G, one
always has $\dot{E}_{\rm GW}\gg\dot{E}_{\rm EM}$ for compact stars with short
spin periods ($<1$ ms). Therefore, in the case of larger ellipticity (e. g.,
$\varepsilon_{e}=10^{-6}$), it is clear that GW radiation dominates the energy
loss in the phase of short period for either recycled or AIC’s compact stars
with low magnetic field. The corresponding lifetime is shorter for a compact
star with higher mass ($\sim M_{\odot}$), but longer for a star with lower
mass ($\sim 0.001M_{\odot}$). However, if the ellipticity is lower, such as
$\varepsilon_{e}=10^{-9}$, EM radiation dominates the rotational energy loss.
The corresponding lifetime of a quark star (even with a high mass $\sim
M_{\odot}$) is long enough for us to detect. Figure 2 shows the relation of
lifetime (in the phase of $<1$ ms) and gravitation ellipticity
$\varepsilon_{\rm e}$ for quark stars.
In the super-Keplerian case, the timescales in the phase of $<$0.5 ms for
quark stars with different mass are also calculated, and listed in Table 1
(See $\tau_{2}$). For a high-mass quark star with larger ellipticity, the
timescale is too small for real detection; but the timescale is $>10^{4}~{}\rm
yr$ for a low mass quark star. Therefore, low mass quark stars with
$\varepsilon_{e}\sim 10^{-6}$ could have much longer lifetime in the phase of
$<0.5$ ms. However, for lower ellipticity, their lifetimes in the phase of
$<0.5$ ms are long enough for quark stars with $\sim 1M_{\odot}$. Once a
pulsar with spin period $<0.5$ ms is ever found, low mass quark stars will be
physically identified.
Figure 2: The relation of lifetime (in the phase of $<1$ ms) and gravitational
ellipticity $\varepsilon_{e}$ for quark stars with masses of 0.001$M_{\odot}$
(solid line), 0.01$M_{\odot}$ (dot-dash line), 0.1$M_{\odot}$ (dashed line),
magnetic field $B=10^{8}$G and the bag constant $\beta=60\rm MeV~{}fm^{-3}$.
The lifetime in the phase of sub-millisecond period is shorter if the quark
star’s mass is higher.
### 3.4 Spin-down rate $\dot{P}$ for newborn quark stars and neutron stars
Figure 3: Spin-down evolution of quark stars due to GW and EM radiations
(period derivative versus spin period), with masses of 0.1$M_{\odot}$,
0.01$M_{\odot}$, 0.001$M_{\odot}$. We choose ellipticity to be $10^{-5}$ (dot-
dash lines), $10^{-7}$ (dashed lines), $10^{-9}$ (solid lines) in the
calculation. It is evident that GW radiation dominates for quark stars with
higher $\varepsilon_{e}$, while EM radiation dominates for lower
$\varepsilon_{e}$. Figure 4: Period derivative versus spin period diagram for
a neutron star with an initial period of 0.5ms, mass of 1.4$M_{\odot}$ and
radius of $10^{6}$km. The neutron star spins down quickly due to high mass
(moment of inertia) for GW radiation.
We also use Eq. (20) to calculate the period derivative ($\dot{P}$) for the
nascent sub-millisecond quark stars and neutron stars.
Figure 3 is a $\dot{P}-P$ diagram that shows the spin-down evolution for quark
stars with different masses. It is found that, for different ellipticity there
are different properties. For high ellipticity such as
$\varepsilon_{e}=10^{-5}$, the $\dot{P}$ can be changing about ten orders of
magnitude for different periods (see the steep slopes of dot-dash lines and
dashed lines). The rotational energy losses in this case are dominated by the
gravitational wave (GW) radiation. For low ellipticity such as
$\varepsilon_{e}=10^{-9}$, in most cases, the rotational energy losses are
dominated by magnetic dipole (EM) radiation and the $\dot{P}$ changes with
periods relatively slow (solid lines).
As a comparison, we also calculate the period derivative ($\dot{P}$) of a
neutron star (with an initial period $0.5$ ms, mass of 1.4$M_{\odot}$ and
radius of $10^{6}$ km). The results are shown in Figure 4. One can see that
the $\dot{P}$ is changing with periods as large as ten orders of magnitude. It
is found that the neutron star spins down much more quickly than low mass
quark stars, because of neutron star’s high mass ($\sim M_{\odot}$) for higher
efficiency of GW radiation.
## 4 Sub-millisecond pulsars formed through accretion in binary systems
Table 3: The minimal equilibrium period for quark stars and lifetimes due to GW and EM radiation in the phase of sub-millisecond period for quark stars with different masses ($10^{-3}M_{\odot}$, $0.1M_{\odot}$, $1.4M_{\odot}$) in the sub-Keplerian case. $\tau_{1}$, $\tau_{2}$, $\tau_{3}$ are calculated by using $\varepsilon_{e}=10^{-6}$, while $\tilde{\tau_{1}}$, $\tilde{\tau_{2}}$, $\tilde{\tau_{3}}$ are calculated by using $\varepsilon_{e}=10^{-9}$. The bag constant $\beta$ is in unit of $\rm Mev\,fm^{-3}$, the accretion ratio $\alpha$ is in unit of the Eddington accretion rate $\dot{M}_{\rm Edd}$. $\beta$ | $\alpha$ | $B_{0}(10^{8}{\rm G})$ | $P_{\rm eqmin}(\rm ms)$ | $\tau_{1}(\rm yr)$ | $\tilde{\tau_{1}}$(yr) | $\tau_{2}(\rm yr)$ | $\tilde{\tau_{2}}$(yr) | $\tau_{3}(\rm yr)$ | $\tilde{\tau_{3}}$(yr)
---|---|---|---|---|---|---|---|---|---
60 | 0.71 | 1.1 | 0.613 | $2.9\times 10^{7}$ | $2.8\times 10^{10}$ | $1.3\times 10^{4}$ | $4.1\times 10^{9}$ | $1.7\times 10^{2}$ | $1.5\times 10^{8}$
110 | 0.85 | 1.4 | 0.453 | $5.1\times 10^{7}$ | $3.6\times 10^{10}$ | $2.3\times 10^{4}$ | $5.6\times 10^{9}$ | $2.3\times 10^{2}$ | $2.6\times 10^{8}$
Table 4: The minimal equilibrium period and lifetimes for GW and EM radiation in the phase of sub-millisecond period of different EOSs of normal neutron stars in the sub-Keplerian case. The mass and radius data of neutron stars are obtained from Figure 2 of Lattimer et al. (2004). $\tau$ and $\tilde{\tau}$ are lifetimes within sub-millisecond period for neutron stars using $\varepsilon_{e}=10^{-6}$ and $\varepsilon_{e}=10^{-9}$ respectively. EOS | $P_{\rm eqmin}$(ms) | Mass$(M_{\odot})$ | Radius(km) | B${}_{0}(10^{8}\rm G)$ | $\dot{M}$($10^{17}\rm\,g\,s^{-1}$) | $\tau$(yr) | $\tilde{\tau}$(yr)
---|---|---|---|---|---|---|---
$\rm AP4$ | 0.55 | 2.21 | 10 | 2 | 6.36 | 103 | $1.02\times 10^{8}$
$\rm GS1$ | 0.52 | 1.38 | 8.27 | 2 | 5 | 255 | $2.37\times 10^{8}$
$\rm PAL6$ | 0.60 | 1.48 | 9.24 | 2 | 6.38 | 177 | $1.65\times 10^{8}$
$\rm MS0$ | 0.76 | 2.76 | 13.31 | 1 | 2.91 | 35 | $3.48\times 10^{7}$
$\rm GM3$ | 0.75 | 1.56 | 10.93 | 2 | 9.47 | 94 | $8.54\times 10^{7}$
$\rm MS1$ | 0.76 | 1.81 | 11.67 | 1 | 2.58 | 90 | $6.83\times 10^{7}$
There is also an important mechanism of “spin-up in binaries” for sub-
millisecond pulsars’ formation, which is widely discussed in the literatures.
We regard this as “sub-Keplerian case” and make a comparison with our proposed
AIC model “super-Keplerian case”. In this section, we will find the minimal
periods for both neutron stars and bare quark stars spun up by accretion in
binary systems. We assume that the initial rotational periods of newborn
pulsars could have an “equilibrium period” with two characteristic parameters:
magnetospheric radius and corotation radius. The magnetospheric radius
$(r_{m})$ is the radius where the ram pressure of particles is equal to the
local magnetic pressure, i.e.
$\displaystyle r_{m}$ $\displaystyle=$ $\displaystyle\phi
R_{A}=\phi(\frac{4\mu_{m}^{2}M^{3/2}}{\dot{M}\sqrt{2G}})^{2/7}=\phi(\frac{B_{0}^{2}R^{6}}{\dot{M}\sqrt{2GM}})^{2/7}$
(24) $\displaystyle=$ $\displaystyle\Big{\\{}\begin{array}[]{l}3.24\times
10^{8}\phi B_{12}^{4/7}M_{1}^{-1/7}R_{6}^{12/7}\dot{M}_{17}^{-2/7}~{}\rm
cm,\\\ 1.857\times 10^{6}\phi
B_{8}^{4/7}M_{1}^{3/7}\beta_{14}^{-4/7}\dot{M}_{17}^{-2/7}~{}\rm
cm,\end{array}$
where $\mu_{m}$ is the magnetic moment of per unit mass of the compact star;
$B_{8}$ is the surface magnetic strength in units of $10^{8}$ G and
$\dot{M}_{17}$ is the accretion rate in units of $10^{17}~{}\rm g~{}s^{-1}$;
$\phi$ is the ratio between the magnetospheric radius and the
Alfv$\rm\acute{e}$n radius (Wang 1997; Burderi & King 1998). Wang (1997)
studied the torque exerted on an oblique rotator and pointed out that $\phi$
decreased from 1.35 to 0.65 as the inclination angle increased from
$0^{\circ}$ to $90^{\circ}$. Here we take $\phi\sim 1$, the influence of
$\phi$ is discussed in $\S 6$.
When $r_{m}$ is very close to the compact star’s radius, we could rewrite the
accretion rate $\dot{M}$ in units of Eddington accretion rate ($\dot{M}_{\rm
Edd}$), with a ratio, $\alpha$, so that
$\dot{M}=\alpha\dot{M}_{\rm Edd}=\alpha\frac{4\pi
cm_{p}R}{\sigma_{T}}=1.0\times 10^{18}\alpha M_{1}^{1/3}\beta^{-1/3}~{}\rm
g~{}s^{-1}.$ (25)
With these equations obtained above, then we can get $r_{m}$ for quark stars,
$r_{m}=9.6\alpha^{-2/7}B_{8}^{4/7}M_{1}^{1/3}\beta_{14}^{-10/21}~{}\rm km.$
(26)
The corotation radius is $r_{c}=1.5\times 10^{8}M_{1}^{1/3}P^{2/3}$ cm. The
spin periods of compact stars cannot exceed the Kepler limit via accretion.
When the compact star was spun up to the Kepler limit by the accreted matter
falling onto the compact star’s surface, for neutron stars, as the equatorial
radius expanded, one can use the simple empirical relation for the maximum
spin frequency
$\Omega_{\rm max}=7700M_{1}^{1/2}R_{6}^{-3/2}~{}\rm s^{-1}$ (27)
(Haensel & Zdunik 1989; Lattimer & Prakash 2004), which leads to
$P_{\rm eq}\geqslant 0.816M_{1}^{-1/2}R_{6}^{3/2}~{}\rm ms,$ (28)
where $M$ and $R$ refer to the neutron star’s mass and radius of nonrotating
configurations.
For quark stars, Gourgoulhon et al. (1999) used a highly precise numerical
code for the 2-D calculations, and found that the $\Omega_{\rm max}$ could be
expressed as $\Omega_{\rm max}=9920\sqrt{\beta_{60}}~{}\rm rad~{}s^{-1}$,
where $\beta_{60}=\beta/(60~{}{\rm MeV~{}fm^{-3}})$, which implied that
$P_{\rm eq}\geqslant 0.633{\beta_{14}^{-1/2}}~{}\rm ms$. These are the so-
called “sub-Keplerian condition”.
The accretion torque, $N$, exerted on the compact star contains two
contributions: one is positive material torque which is carried by the
materials falling onto the star’s surface; the other is magnetic torque which
can be positive or negative, depending on the fastness parameter
$\omega_{s}=\Omega_{\star}/\Omega_{\rm K}=(r_{m}/r_{c})^{3/2}$. It is
suggested that all the torques may cancel one another if the fastness is
$\omega_{s}=(r_{m}/r_{c})^{3/2}\approx 0.884$ (Dai & Li 2006). This implies a
magnetospheric radius of $r_{m}=0.92r_{c}\approx r_{c}$. One can obtain an
equilibrium period of $P_{\rm eq}$ when setting $r_{m}=r_{c}$,
$\displaystyle P_{\rm
eq}=\Big{\\{}\begin{array}[]{lr}0.512B_{8}^{6/7}\beta_{14}^{-5/7}\alpha^{-3/7}~{}\rm
ms,&(a)\\\ 3170B_{12}^{6/7}M_{1}^{-5/7}R_{6}^{18/7}\dot{M}_{17}^{-3/7}~{}\rm
ms.&(b)\end{array}$ (31)
For quark stars, the equilibrium period is independent of mass and radius, and
only dependent on bag constant, surface magnetic field, and accretion rate.
Take $B_{0}$ in the range $[10^{8}~{}\rm G,10^{12}~{}\rm G]$, we may use Eq.
(27a) to calculate the minimal equilibrium period of different EOSs (equation
of state) for quark stars. For $\beta=60~{}\rm MeV~{}fm^{-3}$, when
$\alpha=0.71,B_{8}=1.1$, one can get the minimal period 0.613 ms. For
$\beta=110~{}\rm MeV~{}fm^{-3}$, when $\alpha=0.85,B_{8}=1.4$, the minimal
period is 0.453 ms. (See results in Table 3.)
For neutron stars, data for mass and radius in different EOSs were taken from
Lattimer & Prakash (2004, their Figure 2), $B_{0}$ is in the range
$[10^{8}~{}\rm G,10^{12}~{}\rm G]$. The minimal equilibrium period is
calculated using the Eq. (27b). (See results in Table 4.)
In the sub-Keplerian case, the timescales in the phase of sub-millisecond for
quark stars of different mass and neutron stars of different EOSs are listed
in Table 3 and Table 4, respectively. For typical quark stars as well as
neutron stars with high $\varepsilon_{e}$, their lifetimes in the phase of
sub-millisecond period are about $10^{2}$ years, which result in a too low
detection possibility. However, for low $\varepsilon_{e}$, the lifetime of a
sub-millisecond pulsar (even with a high mass) is long enough.
## 5 Conclusions and Discussions
If a sub-millisecond pulsar is ever found, we have shown that it could be a
quark star based upon plausible scenarios for its origin, the energy available
for radiation and its lifetime. A new possible way to form sub-millisecond
pulsars (quark stars) via AIC of white dwarfs has been discussed in this
paper. In the super-Keplerian case, we derived the initial period $P_{\rm q}$
via angular momentum conservation with consideration of the special and
general relativistic effects, and calculated the lifetime and gap parameters
of a newborn quark star. Quark stars with different masses could have the
minimal rotational period around 0.1 ms. In most cases, quark stars would be
bare (Xu 2002), therefore, a vacuum gap would be formed in the polar cap
region. Based on our rough estimations without considering the effect of frame
dragging (Harding & Muslimov 1998), we found that the basic parameters
(including rotational energy loss) in the gap are suitable for pair (electrons
and positrons) production and sparking. They can be detected as sub-
millisecond radio pulsars.
We also used an approximate formula to calculate nascent quark star’s moment
of inertia, but there are no accurate solutions to fast rotating compact
stars’ configuration until nowadays. It should be investigated precisely in
the future. In the calculation of WD’s mass and radius, we just considered the
non-rotating configuration. But it does not change the conclusions of this
paper. If the central density $\rho_{\rm c}$ of the WD is lower than $10^{11}$
g cm-3 before collapsing, the resulting WD has a larger radius and moment of
inertia, consequently, the newborn quark star could have a smaller spin period
($<$1 ms).
Both the special and general relativistic effects are weak for a low mass
(e.g. Jupiter-like) quark star with a small radius. The rotational energy is
lost via GW and EM radiation. The GW radiation dominates the rotational energy
loss in the phase of sub-millisecond period, if magnetic field of stars is not
so large. Such quark stars therefore have long lifetimes (several million
years if mass $\sim 10^{-3}M_{\odot}$) to maintain their spin periods of sub-
millisecond. We have considered the bar-mode of GW radiation in this paper,
while other GW mode (e.g. r-mode) may be important but not yet considered here
(Xu, 2006b). The subsequent relaxation timescale of a newborn quark star to a
rigidly rotating configuration could be negligible since a quark star may be
solidified soon after birth.
An important constraint for sub-millisecond pulsar’s detection is its lifetime
in the phase of $<1$ ms due to GW and EM radiation. A possible method is
proposed to constrain the lower limit of the pulsars’ equatorial ellipticity,
i. e., $\varepsilon_{\rm e,min}\sim 10^{-9}$, by evaluating millisecond
pulsars’ period derivative via Eq. (20). For larger ellipticity, e. g.,
$\varepsilon_{e}=10^{-6}$, it is clear that GW radiation dominates the energy
loss in the phase of short period for either recycled or AIC’s compact stars.
The corresponding lifetime is shorter for a compact star with higher mass
($\sim M_{\odot}$), but longer for a star with lower mass ($\sim
0.001M_{\odot}$). However, if the ellipticity is lower, e. g.,
$\varepsilon_{e}=10^{-9}$, EM radiation dominates the rotational energy loss.
The corresponding lifetime of a quark star (even with a high mass $\sim
M_{\odot}$) is long enough, and there are no lifetime constraints for sub-
millisecond pulsars’ detection. Solid evidence of quark stars will be obtained
if a pulsar with a period of less than $\sim 0.5$ ms is discovered in the
future.
In the sub-Keplerian case, neutron and “bare” quark stars can be spun up to
sub-millisecond periods (even $\sim 0.5$ ms) through accretion in binary
systems. When neutron stars are spun up to the Kepler limit, the minimal
equilibrium periods depend only on the mass and radius of the nonrotating
configurations. Quark stars’ minimal equilibrium periods depend on the bag
constant.
## Acknowledgments
The authors are very grateful to the referee for valuable comments. We thank
for useful conversations at both the pulsar groups of NAOC and of Peking
University. We are also grateful to Prof. Gao, C. S. for a valuable
discussion. Especially, we appreciate Prof. Chou, Chih Kang for improving our
language. This work is supported by NSFC (10521001, 10573002, 10778611,
10773016 and 10833003) and the Key Grant Project of Chinese Ministry of
Education (305001).
## References
* Abbott et al. (2008) Abbott, B., et al. 2008, ApJ, 683, L45
* Alcock et al. (1986) Alcock, C., Farhi, E., & Olinto, A. 1986, ApJ, 310, 261
* Andersson (2003) Andersson, N. 2003, Classical and Quantum Gravity, 20, 105
* Arras et al. (2005) Arras, P. 2005, ASP Conference Series, Vol. 328
* Basko et al. (1976) Basko, M. M., & Sunyaev, R. A. 1976, 175, 395
* Becker & Truemper (1997) Becker, W., & Truemper, J. 1997, A&A, 326, 682
* Bell et al. (1995) Bell, J. F., Kulkarni, S. R., Bailes, M., Leitch, E. M., & Lyne, A. G. 1995, ApJ, 452, L121
* Blaschke et al. (2002) Blaschke, D., Bombaci, I., Grigorian, H. & Poghosyan, G. 2002, New Astronomy, 107, 112
* Bombaci & Datta (2000) Bombaci, I., & Datta, B. 2000, ApJ, 530, L69
* Bravo & García-Senz (1999) Bravo, E., & García-Senz, D. 1999, MNRAS, 307, 984
* Bulik et al. (1999) Bulik, T., Gondek-Rosi$\acute{n}$ska, D., & Klu$\acute{z}$niak, W. 1999, A&A, 344, L71
* Burderi & D’Amico (1997) Burderi, L., & D’Amico, N. 1997, ApJ, 490, 343
* Burderi & King (1998) Burderi, L., & King, A. R. 1998, ApJ, 505, L135
* Burderi et al. (1999) Burderi, L., Possenti, A., Colpi, M., di Salvo, T., & D’Amico, N. 1999, ApJ, 519, 285
* Burderi et al. (2001) Burderi, L., et al. 2001, ApJ, 560, L71
* Chen et al. (2007) Chen, A., Yu, T. H., & Xu, R. X. 2007, ApJ, 668, L55
* Cook et al. (1994) Cook, G. B., Shapiro, S. L., & Teukolsky, S. A. 1994, ApJ, 424, 823
* Cutler et al. (2002) Cutler, C., & Thorne, K. 2002, arXiv: gr-qc/0204090
* Dai & Li (2006) Dai, H.-L., & Li, X.-D. 2006, A&A, 451, 581
* D’Amico & Burderi (1999) D’Amico, N., & Burderi, L. 1999, Pulsar Timing, General Relativity and the Internal Structure of Neutron Stars, 129
* Dessart et al. (2006) Dessart, L., Burrows, A., Ott, C. D., Livne, E., Yoon, S.-C., & Langer, N. 2006, ApJ, 644, 1063
* Edwards et al. (2001) Edwards, R. T., van Straten, W., & Bailes, M. 2001, ApJ, 560, 365
* Erber (1966) Erber, T. 1966, Reviews of Modern Physics, 38, 626
* Frieman & Olinto (1989) Frieman, J. A., & Olinto, A. V. 1989, Nature, 341, 633
* Friedman et al. (1984) Friedman, J. L., Parker, L., & Ipser, J. R. 1984, Nature, 312, 255
* Fryer et al. (1999) Fryer, C., Benz, W., Herant, M., & Colgate, S. A. 1999, ApJ, 516, 892
* Gil & Sendyk (2000) Gil, J. A., & Sendyk, M. 2000, ApJ, 541, 351
* Glendenning et al. (1995) Glendenning, N. K., Kettner, C., & Weber, F. 1995, ApJ, 450, 253
* Glendenning (2000) Glendenning N. K., _Compact Stars_ (Springer-Verlag, NewYork, 2000), 2nd ed.
* Gondek-Rosińska et al. (2001) Gondek-Rosińska, D., Stergioulas, N., Bulik, T., Kluźniak, W., & Gourgoulhon, E. 2001, A&A, 380, 190
* Gourgoulhon et al. (1999) Gourgoulhon, E., Haensel, P., Livine, R., Paluch, E., Bonazzola, S., & Marck, J.-A. 1999, A&A, 349, 851
* Haensel & Zdunik (1989) Haensel, P., & Zdunik, J. L. 1989, Nature, 340, 617
* Haensel et al. (2008) Haensel, P., Zdunik, J. L., & Bejger, M. 2008, New Astronomy Review, 51, 785
* Han et al. (2004) Han, J. L., Manchester, R. N., Lyne, A. G., & Qiao, G. J. 2004, Young Neutron Stars and Their Environments, 218, 135
* Harding & Muslimov (1998) Harding, A. K., & Muslimov, A. G. 1998, ApJ, 508, 328
* Huang & Wu (2003) Huang, Z.-K., & Wu, X.-J. 2003, Chinese Journal of Astronomy and Astrophysics, 3, 166
* Kaaret et al. (2007) Kaaret, P., et al. 2007, ApJ, 657, L97
* Kapoor & Shukre (2001) Kapoor, R. C., & Shukre, C. S. 2001, A&A, 375, 405
* Keane & Kramer (2008) Keane, E. F., & Kramer, M. 2008, MNRAS, 391, 2009
* Koester (1974) Koester, D. 1974, A&A, 35, 77
* Lattimer & Prakash (2001) Lattimer, J. M., & Prakash, M. 2001, ApJ, 550, 426
* Lattimer et al. (2004) Lattimer, J. M., & Prakash, M. 2004, Science, 304, 536
* Lewin & van den Heuvel (1984) Lewin, W. H. G., & van den Heuvel, E. P. J. 1984, Science, 225, 1143
* Lugones et al. (1994) Lugones, G., Benvenuto, O. G., & Vucetich, H. 1994, Phys. Rev. D, 50, 6100
* Madsen (1999) Madsen J., 1999, Lecture Notes in Physics, 516, 162 (astro-ph/9809032)
* Manchester et al. (2005) Manchester, R. N., Hobbs, G. B., Teoh, A., & Hobbs, M. 2005, AJ, 129, 1993
* Mueller & Eriguchi (1985) Mueller, E., & Eriguchi, Y. 1985, A&A, 152, 325
* Należyty & Madej (2004) Należyty, M., & Madej, J. 2004, A&A, 420, 507
* Nomoto et al. (1979) Nomoto, K., Miyaji, S., Sugimoto, D., & Yokoi, K. 1979, IAU Colloq. 53: White Dwarfs and Variable Degenerate Stars, 56
* Nomoto & Kondo (1991) Nomoto, K., & Kondo, Y. 1991, ApJ, 367, L19
* Owen (2005) Owen, B. J. 2005, Physical Review Letters, 95, 211101
* Percival et al. (1995) Percival, J. W., et al. 1995, ApJ, 446, 832
* Possenti et al. (1999) Possenti, A., Colpi, M., Geppert, U., Burderi, L., & D’Amico, N. 1999, ApJS, 125, 463
* Provencal et al. (1998) Provencal, J. L., Shipman, H. L., Hog, E., & Thejll, P. 1998, ApJ, 494, 759
* Qiao & Lin (1998) Qiao, G. J., & Lin, W. P. 1998, A&A, 333, 172
* Qiao et al. (2004) Qiao, G. J., Lee, K. J., Wang, H. G., Xu, R. X., & Han, J. L. 2004, ApJ, 606, L49
* Qiu & Xu (2006) Qiu, C. X., & Xu, R. X. 2006, Chin. Phys. Lett., 23, 3205
* Regimbau & de Freitas Pacheco (2003) Regimbau, T., & de Freitas Pacheco, J. A. 2003, A&A, 401, 385
* Ruderman & Sutherland (1975) Ruderman, M. A., & Sutherland, P. G. 1975, ApJ, 196, 51
* Sasseen (1990) Sasseen, T. P. 1990, Ph.D. Thesis
* Shu (1982) Shu, F. H. The Physical Universe: An Introduction to Astronomy. Mill Valley, CA: University Science Books, p. 128, 1982.
* Shuryak (2006) Shuryak, E. V. 2006, in: Proceedings of Continuous Advances in QCD (hep-ph/0608177)
* Wang (1997) Wang, Y.-M. 1997, ApJ, 475, L135
* Weber (2005) Weber F. 2005, Prog. Part. Nucl. Phys., 54, 193
* Xu (2002) Xu, R. X. 2002, ApJ, 570, L65
* Xu et al. (2005) Xu, R. X. 2005, MNRAS, 356, 359
* Xu (2006a) Xu, R. X. 2006a, Adv. Space Res., 37, 1992
* Xu (2006b) Xu, R. X. 2006b, Astropart. Phys., 25, 212-219 (astro-ph/0511612)
* Xu (2007) Xu, R. X. 2007, in: Gravitation and Astrophysics: on the occasion of the 90th year of General Relativity, eds. J. M. Nester, C. M. Chen, and J. P. Hsu, World Scientific, p.159 (astro-ph/0605028)
* Xu (2008) Xu, R. X. 2008, in: Proceedings of the international conference “Astrophysics of Compact Objects”, AIP Conference Proceedings, Volume 968, pp.197-202
* Xu (2009) Xu, R. X. 2009, J. Phys. G: Nucl. Part. Phys., in press (arXiv:0812.4491)
* Xu & Liang (2009) Xu, R. X., Liang, E. W. 2009, Science in China Series G: Physics, Mechanics & Astronomy, 52, 315
* van Paradijs et al. (1997) van Paradijs, J., van den Heuvel, E. P. J., Kouveliotou, C., Fishman, G. J., Finger, M. H., & Lewin, W. H. G. 1997, A&A, 317, L9
* Zdunik et al. (2002) Zdunik, J. L., Haensel, P., & Gourgoulhon, E. 2002, A&A, 381, 933
* Zhang et al. (1997) Zhang, B., Qiao, G. J., Lin, W. P., & Han, J. L. 1997a, ApJ, 478, 313
* Zhang et al. (1997) Zhang, B., Qiao, G. J., & Han, J. L. 1997b, ApJ, 491, 891
* Zhang & Qiao (1998) Zhang, B., & Qiao, G. J. 1998, A&A, 338, 62
* Zheng et al. (2006) Zheng, X., Pan, N., Yang, S., Liu, X., Kang, M., & Li, J. 2006, New Astronomy, 12, 165
## Appendix A Differentially Rotating WD Model
WD could be rotating differentially. As stated by (Mueller & Eriguchi 1985),
the WD’s angular velocity $\Omega$ is a function of the distance from the
rotation axis $\widetilde{\omega}$. The angular momentum distribution (so-
called rotation law) is
$\Omega(\widetilde{r})=\Omega_{\rm c}\frac{(aR_{\rm e})^{2}}{(aR_{\rm
e})^{2}+\widetilde{r}^{2}},$ (32)
where $\Omega_{c}$ is the central angular velocity, $R_{e}$ is the equatorial
radius, and $a$ is a free parameter. When differential rotation is taking into
account, we can numerically evaluate the angular momentum of the WD’s inner
collapsed core, i.e.,
$\displaystyle J_{\rm core}$ $\displaystyle=\sum_{\rm i}J_{\rm i}=\sum_{\rm
i}\int_{0}^{\pi}{\sigma 2\pi
r^{4}_{i}\sin^{3}{\theta}\Omega(r_{i}\sin{\theta})}\rm d\theta$
$\displaystyle=\sum_{\rm i}[\frac{m_{\rm core}\Omega_{c}a^{2}R_{\rm
WD}^{2}}{r_{i}^{2}}\times$
$\displaystyle(r^{2}_{i}-0.5\sqrt{\frac{r^{2}_{i}}{a^{2}R_{\rm
WD}^{2}+r^{2}_{i}}}a^{2}R_{\rm
WD}^{2}\ln{\frac{1+\sqrt{\frac{r^{2}_{i}}{a^{2}R_{\rm
WD}^{2}+r^{2}_{i}}}}{1-\sqrt{\frac{r^{2}_{i}}{a^{2}R_{\rm
WD}^{2}+r^{2}_{i}}}}})],$
where $J_{i}$ is the angular momentum of each spherical shell with integral
radius $r_{i}$.
|
arxiv-papers
| 2009-07-15T15:15:37 |
2024-09-04T02:49:03.934772
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. J. Du, R. X. Xu, G. J. Qiao and J. L. Han",
"submitter": "YuanJie Du Mr.",
"url": "https://arxiv.org/abs/0907.2611"
}
|
0907.2640
|
# Towards Hybrid Intensional Programming with JLucid, Objective Lucid, and
General Imperative Compiler Framework in the GIPSY
Serguei A. Mokhov
###### Abstract
Pure Lucid programs are concurrent with very fine granularity. Sequential
Threads (STs) are functions introduced to enlarge the grain size; they are
passed from server to workers by Communication Procedures (CPs) in the General
Intensional Programming System (GIPSY). A JLucid program combines Java code
for the STs with Lucid code for parallel control. Thus first, in this thesis,
we describe the way in which the new JLucid compiler generates STs and CPs.
JLucid also introduces array support.
Further exploration goes through the additional transformations that the Lucid
family of languages has undergone to enable the use of Java objects and their
members, in the Generic Intensional Programming Language (GIPL), and Indexical
Lucid: first, in the form of JLucid allowing the use of pseudo-objects, and
then through the specifically-designed the Objective Lucid language. The
syntax and semantic definitions of Objective Lucid and the meaning of Java
objects within an intensional program are provided with discussions and
examples.
Finally, there are many useful scientific and utility routines written in many
imperative programming languages other than Java, for example in C, C++,
Fortran, Perl, etc. Therefore, it is wise to provide a framework to facilitate
inclusion of these languages into the GIPSY and their use by Lucid programs. A
General Imperative Compiler Framework and its concrete implementation is
proposed to address this issue.
## Acknowledgments
I would like to thank my supervisor Dr. Joey Paquet and Dr. Peter Grogono for
ever lasting patience and caring guidance throughout the variety of learning
experience and their advices and insightful comments to make these
contributions possible. I would also like to thank my friendly team members
with whom we together were lifting the complex GIPSY system off the ground.
Specifically, I would like to mention Chun Lei Ren, Paula Bo Lu, Ai Hua Wu,
Yimin Ding, Lei Tao, Emil Vassev, and Kai Yu Wan for outstanding team work.
Thanks to Dr. Patrice Chalin for an in-depth introduction to semantics of
programming languages. Thanks to Dr. Sabine Bergler and Dr. Leila Kosseim for
the journey through the internals of natural language processing side related
to this work. Thanks to my beloved Irina for helping me to carry through.
This work has been sponsored by NSERC and the Faculty of Engineering and
Computer Science of Concordia University, Montréal, Québec, Canada. This
document was produced in LaTeX with the guidance of Dr. Grogono’s manual in
[Gro01] and Concordia University LaTeX thesis styling maintained by Steve
Malowany, Stan Swiercz, and Patrice Chalin.
###### Contents
1. Acknowledgments
2. 1 Introduction
1. 1.1 Thesis Statement
2. 1.2 Contributions
3. 1.3 Scope of the Thesis
4. 1.4 Structure of the Thesis
3. 2 Background
1. 2.1 Intensional Programming
2. 2.2 The Lucid Programming Language
1. 2.2.1 Brief History and The Family
2. 2.2.2 Indexical Lucid
1. 2.2.2.1 Streams
2. 2.2.2.2 Basic Operators
3. 2.2.2.3 Sequentiality Problem
4. 2.2.2.4 Random Access to Streams
5. 2.2.2.5 Definition of Lucid Operators By Means of @ and #
6. 2.2.2.6 Abstract Syntax of Lucid
7. 2.2.2.7 Concrete GIPL Syntax
8. 2.2.2.8 Semantic Rules
9. 2.2.2.9 Examples of Lucid Programs
3. 2.2.3 Lucid Now
3. 2.3 Hybrid Programming
1. 2.3.1 ML≤
2. 2.3.2 FC++
3. 2.3.3 GLU
4. 2.3.4 GLU#
4. 2.4 Compiler Frameworks
5. 2.5 General Intensional Programming System
1. 2.5.1 Introduction
2. 2.5.2 Goals
3. 2.5.3 General Intensional Programming Compiler
4. 2.5.4 General Eduction Engine
1. 2.5.4.1 Demand Propagation Resources for the GEE
2. 2.5.4.2 Synchronization
5. 2.5.5 Run-time Interactive Programming Environment
6. 2.6 Tools
1. 2.6.1 Java as a Programming Language
1. 2.6.1.1 Java Reflection
2. 2.6.1.2 Java Native Interface (JNI)
3. 2.6.1.3 JUnit
2. 2.6.2 javacc – Java Compiler Compiler
3. 2.6.3 MARF
4. 2.6.4 CVS
5. 2.6.5 Tomcat
6. 2.6.6 Build System
1. 2.6.6.1 Makefiles
2. 2.6.6.2 Eclipse
3. 2.6.6.3 JBuilder
4. 2.6.6.4 Ant
5. 2.6.6.5 NetBeans
7. 2.6.7 readmedir
7. 2.7 Summary
4. 3 Methodology
1. 3.1 JLucid: Lucid with Embedded Java Methods
1. 3.1.1 Rationale
1. 3.1.1.1 Modeling Non-Determinism
2. 3.1.1.2 Loading Existing Java Code with embed()
3. 3.1.1.3 The #JAVA and #JLUCID Code Segments
4. 3.1.1.4 Is JLucid an Intensional Language?
2. 3.1.2 Syntax
3. 3.1.3 Semantics
2. 3.2 Objective Lucid: JLucid with Java Objects
1. 3.2.1 Rationale
1. 3.2.1.1 Pseudo-Objectivism in JLucid
2. 3.2.1.2 Stream of Objects
3. 3.2.1.3 Pure Intensional Object-Oriented Programming
2. 3.2.2 Syntax
3. 3.2.3 Semantics
3. 3.3 General Imperative Compiler Framework
1. 3.3.1 Rationale
2. 3.3.2 Matching Lucid and Java Data Types
3. 3.3.3 Sequential Thread and Communication Procedure Generation
1. 3.3.3.1 Java Sequential Threads
2. 3.3.3.2 Java Communication Procedures
3. 3.3.3.3 C Sequential Threads and Communication Procedures with the JNI
4. 3.3.3.4 Worker Aggregator Definition in the Generator-Worker Architecture
4. 3.4 Summary
1. 3.4.1 Benefits
2. 3.4.2 Limitations
5. 4 Design and Implementation
1. 4.1 Internal Design
1. 4.1.1 General Intensional Programming Compiler Framework
1. 4.1.1.1 General Imperative Compiler Framework
2. 4.1.1.2 Generalization of a Concrete Implementation
3. 4.1.1.3 Resolving Generalization Issues and Binary Compatibility
4. 4.1.1.4 GIPC Preprocessor
5. 4.1.1.5 GIPSY Type System
6. 4.1.1.6 GICF Design
7. 4.1.1.7 Intensional Programming Languages Compiler Framework
8. 4.1.1.8 Sequential Thread and Communication Procedure Interfaces
9. 4.1.1.9 GIPC Design
10. 4.1.1.10 GIPC Class as a Meta Processor
11. 4.1.1.11 Calling Sequence
12. 4.1.1.12 Compiling and Linking
13. 4.1.1.13 Semantic Analyzer
14. 4.1.1.14 Interfacing GIPC and GEE and Compiled GIPSY Program
2. 4.1.2 JLucid
1. 4.1.2.1 Design
2. 4.1.2.2 Grammar Generation
3. 4.1.2.3 Free Java Functions and Java Compiler
4. 4.1.2.4 Arrays
5. 4.1.2.5 Implementing embed()
6. 4.1.2.6 Abstract Syntax Tree and the Dictionary
3. 4.1.3 Objective Lucid
1. 4.1.3.1 Design
2. 4.1.3.2 Grammar Generation
3. 4.1.3.3 Object Instantiation
4. 4.1.3.4 The Dot-Notation
5. 4.1.3.5 Abstract Syntax Tree and the Dictionary
6. 4.1.3.6 Objects as Arrays and Arrays as Objects
2. 4.2 External Design
1. 4.2.1 User Interface
1. 4.2.1.1 WebEditor – A Web Front-End to the GIPSY
2. 4.2.1.2 GIPSY Command-Line Interface
3. 4.2.1.3 RIPE Command-Line Interface
4. 4.2.1.4 GIPC Command-Line Interface
5. 4.2.1.5 GEE Command-Line Interface
6. 4.2.1.6 Regression Testing Application Command-Line Interface
2. 4.2.2 External Software Interfaces
1. 4.2.2.1 JavaCC API
2. 4.2.2.2 MARF Library API
3. 4.2.2.3 Servlets API
3. 4.2.3 Architectural Design and Unit Integration
1. 4.2.3.1 GIPSY
2. 4.2.3.2 GIPSY Exceptions Framework
3. 4.2.3.3 GEE Design
4. 4.2.3.4 RIPE Design
5. 4.2.3.5 Data Flow Graphs Integration
3. 4.3 Summary
6. 5 Testing
1. 5.1 Regression Testing
1. 5.1.1 Introduction
2. 5.1.2 Regression Testing Suite
1. 5.1.2.1 Unit Testing with JUnit
2. 5.1.2.2 Unit Testing with diff
3. 5.1.2.3 Tests
2. 5.2 Portability Testing
3. 5.3 Solving Problems
1. 5.3.1 Prefix Sum
2. 5.3.2 Dining Philosophers
3. 5.3.3 Fast Fourier Transform
1. 5.3.3.1 Fast Fourier Transform in JLucid.
2. 5.3.3.2 Fast Fourier Transform code fragment in Java from MARF.
4. 5.3.4 Moving Car
5. 5.3.5 Game of Life
4. 5.4 Summary
7. 6 Conclusion
1. 6.1 Results
1. 6.1.1 Experiments
2. 6.1.2 Interpretation of Results
2. 6.2 Discussions and Limitations
1. 6.2.1 Lack of Hybrid Intensional-Imperative Semantics Proofs
2. 6.2.2 Genuine Imperative Compilers
3. 6.2.3 Cross-Language Data Type Mapping
4. 6.2.4 Dimension Index Overflow
5. 6.2.5 Hybrid-DFG Integration
6. 6.2.6 Dealing With Side Effects and Abrupt Termination
7. 6.2.7 Imperative Function Overloading
8. 6.2.8 Cross-Imperative Language Calls
9. 6.2.9 Security
8. 7 Future Work
1. 7.1 Formal Verification of Semantic Rules and the GIPSY Type System
2. 7.2 Dealing with Data Flow Graphs in Hybrid Programming
3. 7.3 Security
4. 7.4 Implementation of the C Compiler in GICF
5. 7.5 Fully Explore Array Properties
6. 7.6 Genuine Imperative and Functional Language Compilers
7. 7.7 Visualization and Control of Communication Patterns and Load Balancing
8. 7.8 Target Host Compilation
9. 7.9 The GIPSY Screen Saver
10. 7.10 The GIPSY Server
9. A Definitions and Abbreviations
1. A.1 Abbreviations
10. B Sequential Thread and Communication Procedure Interfaces
1. B.1 Sequential Thread Interface
2. B.2 Communication Procedure Interface
3. B.3 Generated Sequential Thread Wrapper Class
4. B.4 Sample Worker’s Implementation
11. C Architectural Module Layout
1. C.1 GIPSY Java Packages and Directory Structure
2. C.2 GIPSY Modules Packaging
12. D Grammar Generation Scripts for JLucid and Objective Lucid
1. D.1 jlucid.sh
2. D.2 JGIPL.sh
3. D.3 JIndexicalLucid.sh
4. D.4 ObjectiveGIPL.sh
5. D.5 ObjectiveIndexicalLucid.sh
###### List of Figures
1. 1 Concrete Indexical Lucid Syntax
2. 2 GIPL Expressions
3. 3 GIPL where Definitions
4. 4 Concrete GIPL Syntax
5. 5 Operational Semantics of GIPL
6. 6 Natural numbers problem in Indexical Lucid.
7. 7 Natural numbers problem in GIPL.
8. 8 Indexical Lucid program implementing the merge() function.
9. 9 The GIPSY Logo representing the distributed nature of GIPSY.
10. 10 Structure of the GIPSY
11. 11 Initial Conceptual Design of the GIPC
12. 12 Conceptual Design of the GEE
13. 13 Conceptual Design of the RIPE
14. 14 Tomcat Web Applications Manager
15. 1 Indexical Lucid program implementing the merge() function.
16. 2 Indexical Lucid program implementing the merge() function as inline Java method.
17. 3 Indexical Lucid program implementing the merge() function as embed().
18. 4 Illustration of the embed() syntax.
19. 5 Generated corresponding ST to that of Figure 4.
20. 6 Inline Java function declaration.
21. 7 Java method declaration split out from the Lucid part.
22. 8 Natural numbers problem in plain GIPL.
23. 9 Natural numbers problem with two Java methods calling each other.
24. 10 Generated Sequential Thread Class.
25. 11 JLucid Extension to GIPL Syntax
26. 12 JLucid Extension to Indexical Lucid Syntax
27. 13 Additional basic semantic rules to support JLucid
28. 14 Pseudo-objectivism in JLucid.
29. 15 Using pseudo-free Java functions to access object properties in JLucid.
30. 16 Objective Lucid example.
31. 17 Objective Lucid Syntax
32. 18 Additional basic semantic rules to support Objective Lucid
33. 19 Hybrid GIPSY Program Compilation Process
34. 20 Generator-Worker Architecture
35. 1 Example of a hybrid GIPSY program.
36. 2 Another example of a hybrid GIPSY program.
37. 3 Original Framework for the General Intensional Programming Compiler in the GIPSY
38. 4 Modified Framework for the General Intensional Programming Compiler in the GIPSY
39. 5 The FormatTag API.
40. 6 The GIPC Preprocessor.
41. 7 Preprocessor Grammar for a GIPSY program.
42. 8 GIPSY Type System.
43. 9 GICF Design.
44. 10 IPLCF Design.
45. 11 SIPL to GIPL Translator Integration.
46. 12 Sequential Thread and Communication Procedure Class Diagram.
47. 13 All GIPC Compilers.
48. 14 Overall GIPC Design.
49. 15 Sequence Diagram of GIPSY Program Compilation Process.
50. 16 Sequence Diagram of Intensional Compilation Process.
51. 17 Sequence Diagram of Imperative Compilation Process.
52. 18 Semantic Analyzer.
53. 19 Class diagram describing GIPSYProgram.
54. 20 JLucid Design.
55. 21 JLucid Compilation Sequence.
56. 22 Java Compilation Sequence.
57. 23 Objective Lucid Design.
58. 24 Objective Lucid Compilation Sequence.
59. 25 GIPSY WebEditor Interface.
60. 26 JavaCC- and JJTree-generated Modules Used by Several GIPC Modules.
61. 27 MARF Utility Classes used by the GIPSY.
62. 28 Dictionary and DictionaryItem API
63. 29 Dictionary Usage within the GIPSY
64. 30 GIPSY Main Modules.
65. 31 GIPSY Exceptions Framework.
66. 32 GEE Design.
67. 33 The Demand Dispatcher Integrated and Implemented based on Jini.
68. 34 Integration of the Intensional Value Warehouse and Garbage Collection.
69. 35 RIPE Design.
70. 36 DFG Integration Design.
71. 1 Pseudocode of a thread $j$ for the Prefix Sum Problem.
72. 2 The Prefix Sum Problem in JLucid in GIPL Style.
73. 3 The Prefix Sum Problem in JLucid in Indexical Lucid Style.
74. 4 Objective Lucid example of a Car object that changes in time.
75. 5 Eduction Tree for the Natural Numbers Problem.
76. 6 The Natural Numbers Problem in Objective Lucid.
77. 7 Eduction Tree for the Natural Numbers Problem in Objective Lucid.
78. 8 The Life in Haskell.
79. 9 The Life in Indexical Lucid.
80. 1 Sequential Thread Interface.
81. 2 Communication Procedure Interface.
82. 1 GIPSY Java Packages Hierarchy.
###### List of Tables
1. 1 Matching data types between Lucid and Java.
2. 1 Correspondence of the GIPSY .jar files and the modules.
## Chapter 1 Introduction
### 1.1 Thesis Statement
In the previous prototype of the General Intensional Programming System
(GIPSY) there existed limitations to its potential in distributed computing –
lack of sequential threads and communication procedures. Additionally, the
capabilities of Indexical Lucid and GIPL, the primary GIPSY’s languages, were
limited to only computing aspects without input/output, arrays, and some other
essential features (e.g. math, non-determinism, dynamic loading) that exist in
imperative (e.g. Java) languages. We discuss an extension to Generic
Intensional Programming Language (GIPL) and Indexical Lucid with embedded Java
– JLucid. A few problems are solved as an example using the enhanced language.
JLucid brings embedded Java and most of its powers into Indexical Lucid in the
GIPSY by allowing intensional languages to manipulate Java methods as first
class values111The Java methods are not referred to as “functions” as in
functional programming – the Java methods can be passed around as values
inside the Lucid part, but not to or from Java part of a GIPSY program..
However, it is very natural to have objects with Java and manipulate their
members in scientific intensional computation, yet JLucid fails to support
that Java’s capability. Hence, we design Objective Lucid to address this
deficiency. We define the operational semantics of Objective Lucid, and give
some examples of its application.
Existence of JLucid, Objective Lucid, and GLU as well as many useful libraries
written in other imperative languages, such as C/C++, Perl, Python, Fortran
etc. demanded ability to use code written in those languages by intensional
programs, naturally. Thus, we design a first version of the General Imperative
Compiler Framework (GICF) as a part of the GIPSY to allow GIPSY programs to
use virtually any combination of intensional and imperative languages at the
meta level. This is a very ambitious goal; therefore, the proposal is the
first iteration of the framework open for later refinements as it matures
along with the corresponding changes to the run-time system.
### 1.2 Contributions
Primary contributions of this thesis are outlined below:
* •
JLucid
* –
Semantics of pseudo-free Java methods in Lucid programs
* –
Design and implementation of JLucid and its compiler in the GIPSY
* •
Objective Lucid
* –
Semantics of the integration of Java objects in Lucid programs
* –
Design and implementation of the Objective Lucid compiler
* •
General Imperative Compiler Framework
* –
Design and Implementation of the GICF
* –
Embedding of a Java compiler in the GICF
* •
WebEditor to edit, compile, and run GIPSY programs online
* •
System Architecture Issues
* –
Rework and refactoring of most existing system design, both at the
architectural and detailed design levels
* –
Major rework of the architecture and detailed design of GIPC
* –
Java sequential threads generation
* –
Threaded and RMI communication procedures generation
* –
GIPSY Type System222Though the type system may seem not to be related to the
architecture, but it impacted the design most of the main modules in it, so it
was classified as architectural.
* –
GIPSY Exceptions Framework
* –
Regression Testing Infrastructure
* –
Unit Testing Automation with JUnit
The last contributed items touch the rest of the GIPSY, the components and
modules done by other team members. The integration performed (outside of the
main scope of this thesis) demanded extensive testing. Without the integration
and testing work, these other contributions wouldn’t be possible. This also
includes developing and enforcing Coding Conventions and setting up project’s
CVS repository [Mok05b, Mok03a, Mok03b] for the entire project as this work is
to become a manual for the current and future GIPSY developers and
researchers.
### 1.3 Scope of the Thesis
While the Contributions section outlines the major work done, the below
explains what was not done or exhibits some limitations at the time of this
writing:
* •
Integrated imperative compilers aren’t native to the GIPSY, instead we call
external compilers, such as javac, gcc, g++, nmake.exe, bc.exe, perl, etc.
depending on a platform.
* •
Even though the mechanism was designed and implemented to generate CPs and
STs, only two of the concrete implementations of the actual CPs were done: for
local execution and distributed execution by extending the RMI implementation
done by Bo Lu. The other implementations of CPs for Jini, DCOM+, CORBA and
others are being worked on by other team members at the time of this writing.
* •
Semantic rules to have Java objects in Objective Lucid have been developed,
but have not been formally proven to be correct.
* •
When presenting GICF and the Preprocessor syntax, no semantic rules are given
for any of parts of the hybrid programs, except for JLucid and Objective
Lucid, i.e. the semantics of integrated Java itself or C constructs, etc.
* •
JLucid and Objective Lucid are still in their experimental stage of
development and it will take some time before they mature.
### 1.4 Structure of the Thesis
The next chapter provides the necessary background on the Lucid family of
languages, its history, operational semantics, compiler frameworks, and hybrid
programming. Then, it gives the context of this thesis, the GIPSY system, and
the tools and techniques employed to make the contributions possible. The core
of this thesis is based on three publications, namely [MPG05, MP05b, MP05a].
Chapter 3 describes the approach and methodology used to overcome and provide
a solution to the problems stated in Section 1.1. Then, the design
implementation details are presented in Chapter 4. Chapter 5 introduces the
Regression Testing Suite for GIPSY and what kinds of tests were performed and
their limitations. Finally, Chapter 6 and Chapter 7 conclude on the work done,
discuss the results and limitations of the implementation, and lay down some
paths towards enhancing the GIPSY in various areas further. At the end, there
is a list of references, Bibliography, and an Appendix with most common
abbreviations found in this work, CP and ST interfaces, JLucid and Objective
Lucid grammar generation scripts, etc., followed by an overall index.
## Chapter 2 Background
While there is a complete and comprehensive set of references in the
Bibliography chapter that was a great deal of help to the creation of this
work, there are some keynotes that require special mention. The following are
some of the related readings that were sources of inspiration and invaluable
informational food for thought. These include Joey Paquet’s PhD thesis
“Scientific Intensional Programming” [Paq99], related hybrid intensional-
imperative programming in various GLU-related work, such as [JD96, JDA97],
other recent hybrid programming papers, such as [PK04, MS01, SM02], the PhD
thesis of Paula Bo Lu [Lu04] and other theses of the GIPSY group, such as
[Ren02, Din04, Tao04, Wu02], and semantics of programming languages in
[Gro02a, HJ02, Moe04]. Additionally, since this work also deals with compiler
frameworks, a general overview of existing frameworks is presented. An on-line
encyclopedia, Wikipedia [WSoafaotw05], was a valuable resource for the
background and literature review, some of which is summarized in the sections
that follow.
### 2.1 Intensional Programming
Intensional programming is a generalization of unidimensional contextual (also
known as modal logic [Car47, Kri59, Kri63]) programming such as temporal
programming, but where the context is multidimensional and implicit rather
than unidimensional and explicit. Intensional programming is also called
multidimensional programming because the expressions involved are allowed to
vary in an arbitrary number of dimensions, the context of evaluation is thus a
multidimensional context. For example, in intensional programming, one can
very naturally represent complex physical phenomena such as plasma physics
(e.g. in Tensor Lucid in [Paq99]), which are in fact a set of charged
particles placed in a space-time continuum that behaves according to a limited
set of laws of intensional nature. This space-time continuum becomes the
different dimensions of the context of evaluation, and the laws are expressed
naturally using intensional definitions [Paq99]. Joey Paquet’s PhD thesis
discusses the syntax and semantics of the Lucid language, designs GIPL and
Tensor Lucid. While we omit the Tensor Lucid part, the reader is reminded
about the basic properties of the Indexical Lucid and GIPL languages in the
follow up sections in greater detail to provide the necessary context for the
follow up work in Chapter 3 and Chapter 4.
#### Intensional Logic
Intensional programming (IP) is based on intensional (or multidimensional or
modal) logic (where semantics was applied first by [Car47, Kri59, Kri63]),
which, in turn, are based on Natural Language Understanding (aspects, such as,
time, belief, situation, and direction are considered). IP brings in
dimensions and context to programs (e.g. space and time in physics or
chemistry). Intensional logic adds dimensions to logical expressions; thus, a
non-intensional logic can be seen as a constant or a snapshot in all possible
dimensions. Intensions are dimensions at which a certain statement is true or
false (or has some other than a Boolean value). Intensional operators are
operators that allow us to navigate within these dimensions.
#### Temporal Intensional Logic
Temporal intensional logic is an extension of temporal logic that allows to
specify the time in the future or in the past.
(1) $E_{1}$ := it is raining here today
Context: {place:here, time:today}
(2) $E_{2}$ := it was raining here before(today) = yesterday
(3) $E_{3}$ := it is going to rain at(altitude here \+ 500 m) after(today) =
tomorrow
Let’s take $E_{1}$ from (1) above. Then let us fix here to Montreal and assume
it is a constant. In the month of March, 2004, with granularity of day, for
every day, we can evaluate $E_{1}$ to either true or false:
Tags: 1 2 3 4 5 6 7 8 9 ...
Values: F F T T T F F F T ...
If you start varying the here dimension (which could even be broken down into
$X$, $Y$, $Z$), you get a two-dimensional evaluation of $E_{1}$:
City / Day 1 2 3 4 5 6 7 8 9 ...
Montreal F F T T T F F F T ...
Quebec F F F F T T T F F ...
Ottawa F T T T T T F F F ...
The purpose of this example is to remind the reader the basic ideas behind
intensions and intensional programming and what dimensionality is by using
natural language. What follows is formalization of the above in terms of the
Lucid programming language.
### 2.2 The Lucid Programming Language
Let us begin by introducing the Lucid language history and which features of
it came at different stages of its evolution to its present form. This is the
necessary step to further illustrate the purpose of this thesis.
#### 2.2.1 Brief History and The Family
From 1974 to Lucid Today:
1. 1.
Lucid as a Pipelined Dataflow Language through 1974-1977. Lucid was introduced
by Anchroft and Wadge in [AW76, AW77]. Features:
* •
A purely declarative language for natural expression of iterative algorithms.
* •
Goals: semantics and verification of correctness of programming languages (for
details see [AW76, AW77]).
* •
Operators as pipelined streams: one for initial element, and then all for the
successor ones.
2. 2.
Intensions, Indexical Lucid, GRanular Lucid (GLU, [JD96, JDA97]), circa 1996.
More details on these two dialects are provided further in the chapter as they
directly relate to the theme of this thesis. Features:
* •
Random access to streams in Indexical Lucid.
* •
First working hybrid intensional-imperative paradigm (C/Fortran and Indexical
Lucid) in the form of GLU.
* •
Eduction or demand-driven execution (in GLU).
3. 3.
Partial Lucid, Tensor Lucid, 1999 [Paq99].
* •
Partial Lucid is an intermediate experimental language used for demonstrative
purposes in presenting the semantics of Lucid in [Paq99].
* •
Tensor Lucid dialect was developed by Joey Paquet for plasma physics
computations to illustrate advantages and expressiveness of Lucid over an
equivalent solution written in Fortran.
4. 4.
GIPL, 1999 [Paq99].
* •
All Lucid dialects can be translated into this basic form of Lucid, GIPL
through a set of translation rules. (GIPL is in the foundation of the
execution semantics of GIPSY and its GIPC and GEE because its AST is the only
type of AST GEE understands when executing a GIPSY program).
5. 5.
RLucid, 1999, [GP99]
* •
A Lucid dialect for reactive real-time intensional programming.
6. 6.
JLucid, Objective Lucid, 2003 - 2005
* •
These dialects introduce a notion of hybrid and object-oriented programming in
the GIPSY with Java and Indexical Lucid and GIPL, and are discussed great
detail in the follow up chapters of this thesis.
7. 7.
Lucx [WAP05], 2003 - 2005
* •
Kaiyu Wan introduces a notion of contexts as first-class values in Lucid,
thereby making Lucx the true intensional language.
8. 8.
Onyx [Gro04], April 2004.
* •
Peter Grogono makes an experimental derivative of Lucid – Onyx to investigate
on lazy evaluation of arrays.
9. 9.
GLU# [PK04], 2004
* •
GLU# is an evolution of GLU where Lucid is embedded into C++.
#### 2.2.2 Indexical Lucid
When Indexical Lucid came into existence, it allowed accessing context
properties in multiple dimensions. Prior Indexical Lucid, the only implied
dimension was a set of natural numbers. With Indexical Lucid, we can have more
than one dimension, and we can query for a part of the context (any dimensions
of it). Thus, the syntactic definition has been amended to include an ability
to specify which dimensions exactly we are working on.
##### 2.2.2.1 Streams
Lucid variables and expressions are said to be streams of values, through
which one can navigate using some sort of navigational operators. In the
natural language example given earlier the operators were before(), after(),
and at(); here we begin by introducing first() and next() (very much like in
LISP).
If the following equations hold111Note, these are initial conditions of a
definition to illustrate the ideas behind the streams and not an actual
declaration of constructs in the language one would normally write.:
* •
first $X=0$
* •
next $X=X+1$ (like succ in LISP)
where $0$ is a stream of 0’s: $(0,0,0,...,0,...)$. Likewise, $1$ is a stream
of 1’s, and the ‘$+$’ operator performs pair-wise addition of the elements in
the streams according to the implied current dimension index. Thus, $X$ is
defined as a stream, such that:
* •
$x_{0}=0,x_{i+1}=x_{i}+1$, or
* •
$X=(x_{0},x_{1},...,x_{i},...)=(0,1,...,i,...)$
Similarly, if:
* •
first $X=X$
* •
next $Y=Y+$ next $X$
$Y$ here becomes a running sum of $X$:
* •
$y_{0}=x_{0};y_{i+1}=y_{i}+x_{i+1}$
* •
$Y=(y_{0},y_{1},...,y_{i},...)=(0,1,...,i(i+1)/2,...)$
##### 2.2.2.2 Basic Operators
This section defines properties of basic Lucid operators, which were proven by
Paquet in [Paq99].
###### Operator fby.
Operator fby stands for “followed by”. fby allows simply to suppress dimension
index and switch to another stream. As an example the previously shown streams
$X$ and $Y$ can be defined as follows using fby:
* •
$X=0$ fby $X+1=(0,1,2,...,i,...)$
* •
$Y=X$ fby $Y+$ next $X=(0,1,...,i(i+1)/2,...)$
To provide an analogy to lists, we can say that that the following operators
are equivalent:
* •
first and hd
* •
next and tl
* •
fby and cons
###### Informal Definition of first, next, fby.
* •
Definitions:
* –
first $X=(x_{0},x_{0},...,x_{0},...)$
* –
next $X=(x_{1},x_{2},...,x_{i+1},...)$
* –
$X$ fby $Y=(x_{0},y_{0},y_{1},...,y_{i-1},...)$
* •
These are the three operators of the original Lucid.
* •
Indexical Lucid has come into existence with the ability to access an
arbitrary element by some index $i$ in the stream.
###### Operators wvr, asa, and upon.
The other three operators that are slightly more complex informally defined
below:
* •
$X$ wvr $Y=$
if first $Y\neq 0$
then $X$ fby (next $X$ wvr next $Y)$
else (next $X$ wvr next $Y)$
* •
$X$ asa $Y=$ first $(X$ wvr $Y)$
* •
$X$ upon $Y=$
$X$ fby
(if first $Y\neq 0$ then $($next $X$ upon next $Y)$ else $(X$ upon next $Y))$
where wvr stands for whenever, asa stands for as soon as and upon stands for
advances upon. wvr chooses from its left-hand-side operand only values in the
current dimension where the right-hand-side evaluates to true. asa returns the
value of its left-hand-side as a first point in that stream as soon as the
right-hand-side evaluates to true. Unlike asa, upon switches context of its
left-hand-side operand uf the right-hand side is true.
##### 2.2.2.3 Sequentiality Problem
With tagged-token dataflows of the original Lucid operators one could only
define an algorithm with pipelined, or sequential, data flow:
* •
It is wasteful use of computing resources (e.g. to compute an element $i$ we
need $i-1$, but $i-1$ may never be used/needed otherwise).
* •
Sequential access to the stream of values.
##### 2.2.2.4 Random Access to Streams
New intensional operators are introduced to remedy the sequentiality problem:
@ and #. The operators are used as an index # corresponding to the current
position that allows querying the current context, and @ is intensional
navigation to switch the context. With @ and #:
* •
the computation is defined according to a context (here a single integer),
* •
Lucid is no longer a data-flow language and is on the road to intensional
programming, and
* •
the previously introduced intensional operators can be redefined in terms of
the operators # and @.
In terms of the three original operators of first, next, and fby the operators
@ and # are defined as follows:
Definition 1
$\\#=0$ fby $(\\#+1)$
$X@Y=$ if $Y=0$ then first $X$ else (next $X)@(Y-1)$
Both $X$ and $Y$ in the above definition are variable streams, and their
current values are determined by their current context at the time of
evaluation. To redefine the meaning of @ and # Paquet uses the denotational
form, with the following proposition:
Proposition 1
(1) $[\\#]_{i}=i$
(2) $[X@Y]_{i}=[X]_{[Y]_{i}}$
where (1) means the value of # at the current context $i$ is $i$ itself (i.e.
we query the value of our current dimension), and (2) says that evaluate $Y$
at the current context $i$ and then use $Y$ as a new context for $X$.
##### 2.2.2.5 Definition of Lucid Operators By Means of @ and #
First we present the definition of the operators via @ and # denoted in
monospaced font, and then we will provide their equivalence to the original
Lucid operators, denoted as small caps.
Definition 2
(1) first $X$ = $X@0$
(2) next $X=X@(\\#+1)$
(3) $X$ fby $Y=$ if # $=0$ then $X$ else $Y@(\\#-1)$
(4) $X$ wvr $Y=X@T$ where
$T=U$ fby $U@(T+1)$
$U=$ if $Y$ then # else next $U$
end
(5) $X$ asa $Y=$ first $(X$ wvr $Y)$
(6) $X$ upon $Y=X@W$
where $W=0$ fby (if $Y$ then $(W+1)$ else $W)$ end
##### 2.2.2.6 Abstract Syntax of Lucid
Abstract and concrete syntaxes of Lucid for expressions, definitions, and
operators are presented in Figure 2, Figure 3, and Figure 1 for both Indexical
Lucid and GIPL.
op | $::=$ | intensional-op
---|---|---
| $|$ | data-op
intensional-op | $::=$ | i-unary-op
| $|$ | i-binary-op
i-unary-op | $::=$ | first $|$ next $|$ prev
i-binary-op | $::=$ | fby $|$ wvr $|$ asa $|$ upon
data-op | $::=$ | unary-op
| $|$ | binary-op
unary-op | $::=$ | ! $|$ $-$ $|$ iseod
binary-op | $::=$ | arith-op
| $|$ | rel-op
| $|$ | log-op
arith-op | $::=$ | $+$ $|$ $-$ $|$ $*$ $|$ $/$ $|$ %
rel-op | $::=$ | $<$ $|$ $>$ $|$ $<=$ $|$ $>=$ $|$ $==$ $|$ $!=$
log-op | $::=$ | && $|$ ||
Figure 1: Concrete Indexical Lucid Syntax $E$ | $::=$ | $id$
---|---|---
| $|$ | $E(E_{1},...,E_{n})$
| $|$ | if $E$ then $E^{\prime}$ else $E^{\prime\prime}$
| $|$ | $\\#E$
| $|$ | $E@E^{\prime}E^{\prime\prime}$
| $|$ | $E$ where $Q$
Figure 2: GIPL Expressions $Q$ | $::=$ | dimension $id$
---|---|---
| $|$ | $id=E$
| $|$ | $id(id_{1},id_{2},...,id_{n})=E$
| $|$ | $QQ$
Figure 3: GIPL where Definitions
##### 2.2.2.7 Concrete GIPL Syntax
The GIPL is the generic programming language of all intensional languages,
defined by the means of only two intensional operators – @ and #. It has been
proven that other intensional programming languages of the Lucid family can be
translated into the GIPL [Paq99]. The concrete syntax of the GIPL is presented
in Figure 4. It has been amended to support the isoed operator of Indexical
Lucid for completeness and influenced by the productions from Lucx [WAP05] to
allow contexts as first-class values while maintaining backward compatibility
to the GIPL language designed by Paquet in [Paq99].
E ::= id
| E(E,...,E) #LUCX
| E[E,...,E](E,...,E) #GIPL
| if E then E else E fi
| # E
| E @ [E:E] #GIPL
| E @ E #LUCX
| E where Q end;
| [E:E,...,E:E] #LUCX
| iseod E; #INDEXICAL
Q ::= dimension id,...,id;
| id = E;
| id(id,....,id) = E; #LUCX
| id[id,...,id](id,....,id) = E; #GIPL
| QQ
Figure 4: Concrete GIPL Syntax
##### 2.2.2.8 Semantic Rules
Paquet’s PhD thesis [Paq99] presents details of the operational semantics of
GIPL recited here for the unaware reader with a brief description. Figure 5
provides initial operational semantic rules for Indexical Lucid in Hoare Logic
[Moe04, HJ02]. Later on, these rules are extended to support free Java methods
and Java objects in JLucid and Objective Lucid respectively in Chapter 3.
###### Notation
* •
$\mathcal{D}$ represents the definition environment where all symbols are
defined (a dictionary of identifiers).
* •
$\mathcal{D},\mathcal{P}\vdash E:a$ represents current context of evaluation
(a set of dimensions $\mathcal{P}$) and the dictionary that yields a specified
result $a$ under that context given expression $E$.
* •
const, op, dim, func, and var represent what kind of construct types are put
into $\mathcal{D}$ as constants, operators, dimensions, functions, and
variables respectively.
* •
the $\mathbf{E_{Xid}}$ type of rules place different identifier types listed
above into the definition environment $\mathcal{D}$.
* •
the remaining $\mathbf{E_{xyz}}$-style rules correspond to the execution (or
rather application of) of the operators, functions, and conditionals to their
argument expressions given the definition of them in $\mathcal{D}$ and the
current context. Thus, $\mathbf{E_{op}}$ specifies application of a defined
operator function $f$ in the current context to its arguments (usually one for
unary operators and two for binary); $\mathbf{E_{fct}}$ applies the named
function to its arguments translating the formal arguments to actual;
$\mathbf{E_{c_{T}}}$ and $\mathbf{E_{c_{F}}}$ correspond to conditional
evaluation of the then and else branching clauses; $\mathbf{E_{at}}$ and
$\mathbf{E_{tag}}$ correspond to the universal intensional operators @ and #
for switching of and querying for the current context; and $\mathbf{E_{w}}$
corresponds to the scope definition marked by the where clause.
* •
the $\mathbf{Q}$-style rules allow definitions within the scope of the
dimension $\mathbf{Q_{dim}}$ and variable identifier $\mathbf{Q_{id}}$ types
and their composition.
$\displaystyle{\mathbf{E_{cid}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D}(\textit{id})=(\texttt{const},c)}{\mathcal{D},\mathcal{P}\vdash\textit{id}:c}$
$\displaystyle{\mathbf{E_{opid}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D}(\textit{id})=(\texttt{op},f)}{\mathcal{D},\mathcal{P}\vdash\textit{id}:\textit{id}}$
$\displaystyle{\mathbf{E_{did}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D}(\textit{id})=(\texttt{dim})}{\mathcal{D},\mathcal{P}\vdash\textit{id}:\textit{id}}$
$\displaystyle{\mathbf{E_{fid}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D}(\textit{id})=(\texttt{func},\textit{id}_{i},E)}{\mathcal{D},\mathcal{P}\vdash\textit{id}:\textit{id}}$
$\displaystyle{\mathbf{E_{vid}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D}(\textit{id})=(\texttt{var},E)\qquad\mathcal{D},\mathcal{P}\vdash
E:v}{\mathcal{D},\mathcal{P}\vdash\textit{id}:v}$
$\displaystyle{\mathbf{E_{op}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D},\mathcal{P}\vdash
E:\textit{id}\qquad\mathcal{D}(\textit{id})=(\texttt{op},f)\qquad\mathcal{D},\mathcal{P}\vdash
E_{i}:v_{i}}{\mathcal{D},\mathcal{P}\vdash
E(E_{1},\ldots,E_{n}):f(v_{1},\ldots,v_{n})}$
$\displaystyle{\mathbf{E_{fct}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D},\mathcal{P}\vdash
E:\textit{id}\qquad\mathcal{D}(\textit{id})=(\texttt{func},\textit{id}_{i},E^{\prime})\qquad\mathcal{D},\mathcal{P}\vdash
E^{\prime}[\textit{id}_{i}\leftarrow E_{i}]:v}{\mathcal{D},\mathcal{P}\vdash
E(E_{1},\ldots,E_{n}):v}$ $\displaystyle{\mathbf{E_{c_{T}}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D},\mathcal{P}\vdash
E:\textit{true}\qquad\mathcal{D},\mathcal{P}\vdash
E^{\prime}:v^{\prime}}{\mathcal{D},\mathcal{P}\vdash\mathtt{if}\;E\;\mathtt{then}\;E^{\prime}\;\mathtt{else}\;E^{\prime\prime}:v^{\prime}}$
$\displaystyle{\mathbf{E_{c_{F}}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D},\mathcal{P}\vdash
E:\textit{false}\qquad\mathcal{D},\mathcal{P}\vdash
E^{\prime\prime}:v^{\prime\prime}}{\mathcal{D},\mathcal{P}\vdash\mathtt{if}\;E\;\mathtt{then}\;E^{\prime}\;\mathtt{else}\;E^{\prime\prime}:v^{\prime\prime}}$
$\displaystyle{\mathbf{E_{tag}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D},\mathcal{P}\vdash
E:\textit{id}\qquad\mathcal{D}(\textit{id})=(\texttt{dim})}{\mathcal{D},\mathcal{P}\vdash\\#E:\mathcal{P}(\textit{id})}$
$\displaystyle{\mathbf{E_{at}}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D},\mathcal{P}\vdash
E^{\prime}:\textit{id}\qquad\mathcal{D}(\textit{id})=(\texttt{dim})\qquad\mathcal{D},\mathcal{P}\vdash
E^{\prime\prime}:v^{\prime\prime}\qquad\mathcal{D},\mathcal{P}\\!\dagger\\![\textit{id}\mapsto
v^{\prime\prime}]\vdash E:v}{\mathcal{D},\mathcal{P}\vdash
E\;@E^{\prime}\;E^{\prime\prime}:v}$ $\displaystyle{\mathbf{E_{w}}}$
$\displaystyle:$ $\displaystyle\frac{\mathcal{D},\mathcal{P}\vdash
Q\>:\>\mathcal{D}^{\prime},\mathcal{P}^{\prime}\qquad\mathcal{D}^{\prime},\mathcal{P}^{\prime}\vdash
E:v}{\mathcal{D},\mathcal{P}\vdash E\;\mathtt{where}\;Q:v}$
$\displaystyle{\mathbf{Q_{dim}}}$ $\displaystyle:$
$\displaystyle\frac{}{\mathcal{D},\mathcal{P}\vdash\texttt{dimension}\;\textit{id}\>:\>\mathcal{D}\\!\dagger\\![\textit{id}\mapsto(\texttt{dim})],\mathcal{P}\\!\dagger\\![\textit{id}\mapsto
0]}$ $\displaystyle{\mathbf{Q_{id}}}$ $\displaystyle:$
$\displaystyle\frac{}{\mathcal{D},\mathcal{P}\vdash\textit{id}=E\>:\>\mathcal{D}\\!\dagger\\![\textit{id}\mapsto(\texttt{var},E)],\mathcal{P}}$
$\displaystyle{\mathbf{QQ}}$ $\displaystyle:$
$\displaystyle\frac{\mathcal{D},\mathcal{P}\vdash
Q\>:\>\mathcal{D}^{\prime},\mathcal{P}^{\prime}\qquad\mathcal{D}^{\prime},\mathcal{P}^{\prime}\vdash
Q^{\prime}:\mathcal{D}^{\prime\prime},\mathcal{P}^{\prime\prime}}{\mathcal{D},\mathcal{P}\vdash
Q\;Q^{\prime}\>:\>\mathcal{D}^{\prime\prime},\mathcal{P}^{\prime\prime}}$
Figure 5: Operational Semantics of GIPL
##### 2.2.2.9 Examples of Lucid Programs
Two simple examples of Lucid programs are presented. The examples demonstrate
absence of iterative/sequential operation as opposed to the traditional
imperative programming languages.
###### Natural Numbers Problem
An example program in Indexical Lucid that yields 44 as the result is in
Figure 6. The way the program is expanded using the re-definitions of the
Lucid operators, such as fby, employing @ and # in GIPL is shown in Figure 7.
N @.d 2
where
dimension d;
N = 42 fby.d (N + 1);
end;
Figure 6: Natural numbers problem in Indexical Lucid.
N @.d 2
where
dimension d;
N = if (#.d <= 0) then 42 else (N + 1) @.d (#.d - 1) fi;
end;
Figure 7: Natural numbers problem in GIPL.
###### The Hamming Problem
This example (see Figure 8) illustrates the simple use of functions in Lucid.
H
where
H = 1 fby merge(merge(2 * H, 3 * H), 5 * H);
merge(x, y) = if(xx <= yy) then xx else yy
where
xx = x upon(xx <= yy);
yy = y upon(yy <= xx);
end;
end;
Figure 8: Indexical Lucid program implementing the merge() function.
#### 2.2.3 Lucid Now
To summarize, Lucid is a functional programming language where a variable
(stream), a function, a dimension, or even entire context can be a first class
value (i.e. can viewed and manipulated as data). Lucid provides operators,
such as @ and #, to navigate within dimensions and switch contexts. The
language also exhibits the eductive execution model (demand-driven distributed
computation) that augments the semantics with a warehouse (intensional value
cache) and its consistency222Paquet defines the augmented operational
semantics in [Paq99] and Tao implements its first incarnation in GIPSY
[Tao04]. This work has an impact on this aspect by introducing the side
effects with the imperative languages, which will be discussed later..
### 2.3 Hybrid Programming
There have been previous approaches to couple intensional or functional and
imperative and object-oriented paradigms prior to this work. Some recent
related work on the same issue is presented in [BM96, PK04, MS01, SM02] with
the [PK04] being the most relevant. The two major approaches of addressing the
OO issue are – either (1) to extend Lucid to become object-oriented or
objects-aware or (2) make a host imperative language be extended to embed
Lucid. The authors of [PK04] chose the latter by extending GLU-with-C to
GLU#-with-C++, whereas this work approaches the problem from Lucid to Java.
This means a Lucid program is the main one driving the computation. We will
briefly consider the following approaches to the hybrid programming:
* •
ML≤
* •
FC++
* •
GLU
* •
GLU#
#### 2.3.1 ML≤
ML≤ [BM96] is a system introduced in 1996 that proposed to marry OOP and
functional paradigms using their own language and providing the details of the
predicative and decidable typing rules and operational semantics of such a
system. Their main goal is to be able to induce implicit polymorphism of
functional languages in objects. They do not extend an existing functional
language with the OO capabilities, instead they reinterpret all data types as
either abstract or concrete classes and use the dynamic dispatch, a typical OO
feature, on run-time types.
#### 2.3.2 FC++
FC++ [MS01, SM02] tries to promote the functional paradigm in C++. FC++ is a
library add-on to enable higher-order polymorphic functions in a novel use of
C++ type inference that is not very complex and is still expressive. FC++ adds
support for both parametric and subtype polymorphism policies for functions in
order to be able to fit FC++ functions within the C++ object model and pass
higher-order functions as parameters. The FC++ functions are kept as objects
called functoids and use a reference counter machinery for allocation and de-
allocation. Closures in FC++ (operation on a some state and the state itself)
can automatically be created during functoid object creation, but their
“closing” of that state is not automatic and the state values have to be
passed explicitly during the creation process. The library also adds a set of
functional operators from the Haskell Standard Prelude. FC++ comes more from
the OOP-to-functional point of view and conforms with standard software
engineering design patterns and is suitable for the common OO tasks.
#### 2.3.3 GLU
GLU was the most general intensional programming tool recently available
[JD96]. However, experience has shown that, while being very efficient, the
GLU system suffers from a lack of flexibility and adaptability [Paq99]. Given
that Lucid is evolving continually, there is an important need for the
successor to GLU to be able to stand the heat of evolution [Paq99]. The two
major successors of GLU are the GIPSY and GLU# systems.
##### Eduction
The earlier mentioned notion of eduction was first introduced by the GLU
compiler. GLU supports so-called tagged-token demand-driven dataflow where
data elements (tokens) are computed on demand following a dataflow network
defined in Lucid. Data elements flow in the normal flow direction (from
producer to consumer) and demands flow in the reverse order, both being tagged
with their current context of evaluation. This form of lazy computation is
inherited by GIPSY from GLU.
#### 2.3.4 GLU#
GLU# [PK04] is a successor of GLU, which enables Lucid within C++. The authors
argue for the embedding small functional/intensional-language pieces of Lucid
into C++ programs allowing lazy (demand-driven) evaluation of arrays and
functions thereby making Lucid easily accessible within a popular imperative
programming language, such as C++. Because GLU# appeared quite recently (2004)
to when this work was written, its success compared to GLU is yet to be
evaluated; however, it seems to suffer from the same inflexibility GLU did and
targets only C++ as a host language.
### 2.4 Compiler Frameworks
A significant number of compiler frameworks emerged for the past decade. All
try to enable compilation of more than one language, either hybrid or not, in
an uniform manner. Some frameworks or libraries became “frozen” (i.e. non-
extendable) and fixed to a specific set of languages, some other ones were
build with the extension in mind, so it is relatively easy to “plug-in” yet
another compiler into the system (a collection of compilers and the necessary
tools) with minimum integration work required. A brief overview of different
compiler frameworks is given next:
* •
GLU tried to accommodate Fortran, C, and Lucid in one system, but was made so
inflexible [Paq99] that it would take a significant effort to extend it and
add other languages to the system.
* •
GLU# merges Lucid and C++; however, makes no provisions for extension to other
languages on either intensional or imperative side.
* •
Microsoft .NET can also be thought of a commercial heterogeneous compiler
framework (it is more than a compiler framework, but our focus is on
compilers) that allows easy cooperation and application development between
different language models, such as C#, C++, Visual Basic, and Assembly in a
homogeneous environment. However, none of these languages have natively any of
the intensional or functional capabilities, so no native debugging support or
other tools exist, even if one starts using FC++ or GLU# in this environment.
Despite the fact that all programs can be compiled into the common bytecode,
the debugging tools have to be aware of the functional paradigms on a higher
level and they are not (at least at this writing).
* •
The GNU Compiler Collection (GCC) can also be said as a compiler framework
from the free software [CP05]. It supports C, C++, Objective-C, Objective-C++,
Java, Fortran, and Ada. Again, these languages are more of an imperative
nature, but it is far easier to add new language into GCC than to Microsoft
.NET due to its openness.
* •
Finally, the GIPSY presents the GIPC framework that is designed for expansion
and integration of the intensional and imperative (and later functional)
languages. This is presented through the rest of this thesis.
### 2.5 General Intensional Programming System
#### 2.5.1 Introduction
Figure 9: The GIPSY Logo representing the distributed nature of GIPSY.
GIPSY is broadly presented in [WPG03, Lu04, PW05], and others. Please refer to
the online resources [RG05a, PW05, RG05b] to obtain the most current status of
the project. GIPSY is primarily implemented in Java. General GIPSY
architecture is presented in Figure 10. The essence behind GIPSY is demand-
driven computation support for the intensional programming languages, e.g.,
Indexical Lucid, Tensor Lucid [Paq99], etc.
Figure 10: Structure of the GIPSY
The GIPSY consists in three modular sub-systems: the General Intensional
Programming Language Compiler (GIPC); the General Eduction Engine (GEE), and
the Intensional Run-time Interactive Programming Environment (RIPE). The sub-
systems have to be modular so that one implementation of parts of them or the
whole can be replaced by another without having major if any impact on the
other modules. Although the theoretical basis of the language has been
settled, the implementation of an efficient, general and adaptable programming
system for this language raises many questions. The following sections outline
the theoretical basis and architecture of the different components of the
system. All these components are designed in a modular manner to permit the
eventual replacement of each of its components – at compile-time or even at
run-time – to improve the overall efficiency and productivity of the system
[Paq99].
A GIPSY instance sends out little bits of work to others to compute and then
gathers the results in distributed fashion. Of course, synchronization,
latency tolerance, and maximum utilization of resources are primary goals for
the system to be productive. Unlike in most programming language models (see
[ST98]) considered for parallel computation, in GIPSY several key concepts are
considered:
* •
Thread-Level Parallelism (TLP)
* •
Stream-Level Parallelism (SLP)
* •
Cluster-Level Parallelism (CLP)
GIPSY’s parallelism granularity takes into account the amount of TLP, SLP, and
CLP available. TLP determines the maximum number of threads that should or can
be created when a Lucid program is being executed. In other words, TLP defines
on how many pieces of terminal computational work we can chop a big job into.
The goal, as far as programming is concerned, is to program for infinite TLP,
and later adjust (load-balance) at run-time to the actual amount of SLP. SLP
determines the maximum number of streams available to execute the threads.
Here, by “streams” we mean processors but, with the invention of multithreaded
CPUs for a single processor, there may be several thread streams available in
parallel, and hence a more general notion of SLP. The amount of SLP is
machine-dependent and has to be discovered at run-time on remote machines. If
a job is to be run on a single machine, GIPSY tries to maximize SLP
utilization, providing just enough TLP for the machine in question with the
design goal of always assuming infinite TLP. Then load-balancing comes into
play. CLP takes GIPSY to another level — distributed computing, involving
utilization of SLP of the machines across the network nearby or across the
globe over the Internet.
NOTE: the Lucid family of languages has also a notion of streams that refers
to Lucid variables that evaluate in multiple contexts. Every Lucid stream
(e.g. a variable) can potentially be evaluated on any hardware stream
available, but it is important not to confuse the two kinds of streams. The
reason for the existence of the two notions is that both terms were used
independently in each field. Now that parallel architectures and language
models such as Lucid came into proximity, the terms clash.
#### 2.5.2 Goals
The system has to withstand the evolution of the tools, languages, and
underlying platforms, thus be flexible and adaptable to the changes. That is
one of the most important and stringent requirements put on the development of
GIPSY [Paq99]. Other subordinate requirements in compiler design, run-time
system, communication, and user interfaces are presented in detail throughout
the follow up sections.
#### 2.5.3 General Intensional Programming Compiler
Figure 11: Initial Conceptual Design of the GIPC
GIPSY programs are compiled in a two-stage process (see Figure 19, page 19).
First, the intensional part of the GIPSY program is translated in Java, then
the resulting Java program is compiled in the standard way.
The source code consists of two parts: the Lucid part that defines the
intensional data dependencies between variables and the sequential part that
defines the granular sequential computation units (usually written in any
imperative language, e.g. C or Java). The Lucid part is compiled into an
intensional data dependency structure (IDS) describing the dependencies
between each variable involved in the Lucid part. This structure is
interpreted at run-time by the GEE following the demand propagation mechanism.
Data communication procedures used in a distributed evaluation of the program
are also generated by the GIPC according to the data structures definitions
written in the Lucid part, yielding a set of communication procedures (CP).
These are generated following a given communication layer definition such as
provided by RPC (or rather RMI since GIPSY is implemented in Java), CORBA,
Jini, or the WOS [BKU98]. The sequential functions defined in the second part
of the GIPSY program are translated into imperative code using the second
stage imperative compiler syntax, yielding imperative sequential threads (ST).
Intensional function definitions, including higher order functions, will be
flattened using a well-known efficient technique [Ron94, Paq99]. The closures
in the higher order functions case are still applicable because the function
state and the operation on it are correctly passed to the functions by
expanding and using function definitions inline. The insignificant limitation
here is that self-referential closures for such functions cannot be made. The
function elimination in GIPSY pertinent to some of these aspects was
implemented by Wu in [Wu02].
The Figure 11 presents the initial conceptual design of the GIPC. Based on
this design, the GIPSY module integration and the development of the STs and
CPs support has begun. Later on the design was refined in [PGW04, MP05a] and
its latest reincarnation is shown in Figure 4 in Chapter 4, page 4; thus, the
evolution description is delayed until then.
Prior this work, GIPC supported only two Lucid dialects: GIPL and Indexical
Lucid. The initial GIPC compiler was implemented by Chun Lei Ren in [Ren02],
and the translation of the Indexical Lucid into GIPL and the semantic analysis
was implemented by Aihua Wu in [Wu02]. A large integration and re-engineering
effort went into GIPC to approach it to the goals of the GIPSY (see Section
2.5.2) and add more compilers for investigation of the underlying language
models. The results of this effort are presented in the Design and
Implementation chapter (Chapter 4).
#### 2.5.4 General Eduction Engine
Figure 12: Conceptual Design of the GEE
The GIPSY uses a demand-driven model of computation, which is based on the
principle is that certain computation takes effect only if there is an
explicit demand for it. The GIPSY uses eduction, which is demand-driven
computation in conjunction with an intelligent value cache called a warehouse.
Every demand can potentially generate a procedure call, which is either
computed locally or remotely, thus eventually in parallel with other procedure
calls. Every computed value is placed in the warehouse, and every demand for
an already-computed value is extracted from the warehouse rather than computed
again and again (demands that may have side effects, e.g. if we cache results
of STs, shall not be cached). Eduction, thus, reduces the overhead induced by
the procedure calls needed for the computation of demands sequentially. Figure
12 describes the internal conceptual structure and functioning of the GEE.
The GEE itself is composed of three main modules: the executor, the
intensional demand propagator (IDP), and the intensional value warehouse
(IVW). First, the intensional data dependency structure (IDS, which represents
GEER) is fed to the demand generator (DG) by the compiler (GIPC). This data
structure represents the data dependencies between all the variables in the
Lucid part of the GIPSY program. This tells us in what order all demands are
to be generated to compute values from this program. The demand generator
receives the initial demand, that in turn raises the need for other demands to
be generated and computed as the execution progresses. For all non-functional
demands (i.e. demands not associated with the execution of sequential threads
(ST)), the DG makes a request to the warehouse to see if this demand has
already been computed. If so, the previously computed value is extracted from
the warehouse. If not, the demand is propagated further, until the original
demand resolves to a value and is put in the warehouse for further use. This
type of warehousing was introduced by GLU due to its distributed nature to cut
down on communication costs, but it can certainly be applicable to any
functional language, such as LISP, Scheme, Haskell, ML and others to improve
efficiency even on a single machine provided there are no any side effects
whatsoever. The garbage collector can run on the background to clean up old
function-parameters-values tuples periodically, and given that the large
amounts of memory are cheap these days functional languages may gain much more
popularity with the increased performance.
For functional demands (i.e. demands associated with the execution of a
sequential thread), the demands are sent to the demand dispatcher (DD) that
takes care of sending the demand to one of the workers or to resolve it
locally (which normally means that a worker instance is running on the
processor running the generator process). If the demands are sent to a remote
worker, the communication procedures (CP) generated by the compiler are used
to communicate the demand to the worker. The demand dispatcher (DD) receives
some information about the liveness and efficiency of all workers from the
demand monitor (DM), to help it make better decisions in dispatching the
demands.
The demand monitor, after some functional demands are sent to workers, starts
to gather various types of information about each worker, including, but not
limited to:
* •
liveness status (is it still alive, not responding, or dead)
* •
network link performance
* •
response time statistics for all demands sent to it
These data points are accessed by the DD to make better decisions about the
load balancing of the workers, and thus achieving better overall run-time
efficiency.
Bo Lu was the first one to do the original design of the GEE framework [Lu04]
and investigate its performance under threaded and RMI environments. She also
introduced the notion of the Identifier Context (IC) classes – demands
converted into Java code and using Java Reflection [Gre05] to compile, load,
and execute them them at run-time. She also contributed the first version of
the interpreter-based execution engine. Next, Lei Tao contributed the first
incarnation of the intensional value warehouse and garbage collection
mechanisms in [Tao04] based on the popular scientific library called NetCDF.
The author of this thesis put an effort to modularize these all and make them
easier to extend and customize. He also provided the initial GEE application
to start available network services. The GEE was also made aware of the STs
and CPs as well as the new type system, described in Section 4.1.1.5. Further,
Emil Vassev [VP05] produced a very general and functional framework for demand
migration and its implementation, Demand Migration System (DMS) that supports
among other things Jini, CORBA, and .NET Remoting for fault-tolerant demand
transportation system, a part of the Demand Dispatcher. The DMS is still
pending integration as of this writing.
##### 2.5.4.1 Demand Propagation Resources for the GEE
The IDP generates and propagates demands according to the data dependence
structure (DPR, now renamed to GEER in [WPG03]) generated by the GIPC. If a
demand requires some computation, the result can be calculated either locally
or on a remote computing unit. In the latter case, the communication
procedures (CP) generated by the GIPC are used by the GEE to send the demand
to the worker. When a demand is made, it is placed in a demand queue, to be
removed only when the demand has been successfully computed. This way of
working provides a highly fault-tolerant system. One of the weaknesses of GLU
is its inability to optimize the overhead induced by demand-propagation. The
IDP will remedy to this weakness by implementing various optimization
techniques:
* •
Data blocking techniques used to aggregate similar demands at run time, which
will also be used at compile-time in the GIPC for automatic granularization of
data and functions for data-parallel applications
* •
The performance-critical parts (IDP and IVW) are designed as replaceable
modules to enable run-time replacements by more efficient versions adapted to
specific computation-intensive applications
* •
Certain demand paths identified (at compile-time or run-time) as critical will
be compiled to reduce their demand propagation overhead
* •
Extensive compile-time and run-time rank analysis (analysis of the
dimensionality of variables) [Dod96].
##### 2.5.4.2 Synchronization
##### Distributed vs. Parallel
It is important to make a distinction between parallel and distributed
computing. In parallel computing, SLP matters and latency tolerance for memory
references with mostly UMA (uniform memory access) characteristics, whereas in
distributed computing communication is much more expensive (and perhaps even
prohibitive) and CLP matters as well. This setup largely exhibits NUMA (non-
UMA) characteristics (see [Pro03b]) and latency tolerance (and so also fault
tolerance) has a higher significance. This greatly impacts the way we
synchronize in parallel and distributed worlds.
###### Synchronization in Distributed Environment
A distributed environment is a very popular domain these days, so we’ll start
with it first. Typically, the network is the scarce resource and is the
bottleneck for a distributed application because it implies communication
(e.g., MPI), which is often unacceptable. Therefore, many distributed
applications choose not to communicate at all or communicate very little
through message passing. This implies blocking on waiting for the network
requests to propagate, i.e. network latency.
###### Synchronization in Parallel Environment
Synchronization in a parallel environment is more fine-grained, often at the
hardware level (e.g., a full/empty bit in memory cells). Java does not give us
control over such synchronization, so we have to rely on the JVM built for an
architecture that has such synchronization. The JVM has to be developed to
make use of the full/empty bits that are usually represented as future
variables [Pro03b, JA03] in the languages specifically designed for parallel
computing.
##### Secure Synchronization
Secure synchronization is especially pertinent in a distributed environment.
Like any act of communication within worker-generator architecture (see
Section 3.3.3.4) and a warehouse (Figure 19; Section 2.5.4.1), synchronization
has to be secure to avoid (a) over-demanding, (b) incorrect results sent back,
(c) loss of results and demands, and (d) poisoning the warehouse with wrong
data. Secure synchronization implies fault tolerance. In GIPSY, we will rely
on Java’s RMI and Jini over JSSE for secure communication in a distributed
environment, using Java’s synchronization primitives (see Section 2.5.4) to
achieve the goal of secure synchronization. Thus, the reliability and
accountability of the results of a GIPSY program are dependent on these
properties of underlying Java Runtime Environment (JRE) and the communication
protocols used.
##### Implicit vs. Explicit Synchronization
One of the productivity metrics of a software completing its task on time, is
the efficiency of development of (see [Pro03c]) such a software, i.e., the
amount of programmer’s effort required to create and debug the software. This
is essentially a metric, called time-to-solution (TTS) [Pro03c]; from creation
until the end result (e.g. completion of some scientific computation). The
goal is to minimize TTS. One way to achieve this is ease of programming. As
the proportion of the work done by the compiler increases, so does the
reliability of the code, but we target scientific researchers, not just
programmers. Scientific researchers from math and physics should not care
about these issues and, thus, just be concerned mastering the basics of Lucid.
Therefore, the programmer has to be freed from taking care of synchronization
explicitly, which a source of bugs and inefficiency of programming (e.g.,
using Java’s synchronization primitives, such as synchronized, Object.wait(),
Object.notify(), and Object.notifyAll(), [Fla97]). The programmer should
rather focus on the problem being solved and let the compiler/run-time system
deal with the synchronization pain. The GIPSY system, built around the Lucid
family, advocates implicit synchronization either by wrapping around the
Java’s synchronization primitives or through the communication synchronization
and data dependencies (although a complete discussion is beyond the scope of
this thesis, see [Lu04, VP05]).
#### 2.5.5 Run-time Interactive Programming Environment
Figure 13: Conceptual Design of the RIPE
The RIPE is a visual programming aid to the run-time environment (GEE)
enabling the visualization of a dataflow diagram corresponding to the Lucid
part of the GIPSY program, source code editing, launching the compilation and
execution of GIPSY programs. The original conceptual design of RIPE [Paq99] is
illustrated in Figure 13. The user’s points of interaction with the RIPE at
run-time vary in the following ways:
* •
Enable interactive editing of GIPSY programs via a variety of editors
(textual, graphical, web).
* •
Dynamic inspection of the IVW.
* •
Modification of the input/output channels of the program.
* •
Recompilation of the GIPSY programs.
* •
Modification of the communication protocols.
* •
Swapping of the parts of the GIPSY itself (e.g. garbage collection,
optimization, warehouse caching etc. strategies).
Because of the interactive nature of the RIPE, the GIPC is modularly designed
to allow the individual on-the-fly compilation of either the IDS (by changing
the Lucid code), CP (by changing the communication protocol), or ST (by
changing the sequential code). Such a modular design even allows sequential
threads to be programs written in different languages (for now, we are
concentrating on Java sequential threads, but a provision is made for easy
inclusion of other languages with the GICF, Section 4.1.1.1).
The RIPE even enables the graphic development of Lucid programs, translating
the graphic version of the program into a textual version that can then be
compiled into an operational version through a DFG generator of Yimin Ding
[Din04]. However, the development of this facility for graphical programming
posed many problems whose solution is not yet settled, for example
representation of the STs and CPs in the DFG nodes. An extensive and general
requirements analysis will be undertaken, as this interface will have to be
suited to many different types of applications. There is also the possibility
to have a kernel run-time interface on top of which we can plug-in different
types of interfaces adapted to different applications, such as stand-alone,
web-, or server-based.
### 2.6 Tools
This section presents a brief description of a variety of tools that helped
most with the implementation aspects of this work.
#### 2.6.1 Java as a Programming Language
The primary implementation language of GIPSY is Java. This includes using
Java’s Reflection, JNI, and JUnit frameworks and packages. We have chosen to
implement our project using the Java programming language mainly because of
the binary portability of the Java applications as well as its facilities, for
e.g. memory management and communication tasks, so we can concentrate more on
the algorithms instead. Java also provides built-in types and data-structures
to manage collections (build, sort, store/retrieve) efficiently [Fla97,
Mic05b]. There is also source code written in other languages in the main
GIPSY repository. This includes LEFTY code for DFG generation and the code of
the test intensional programs in various Lucid dialects. The Java versions
supported by GIPSY are 1.4 and 1.5. The GIPSY will no longer build on 1.3 and
earlier JDKs.
##### 2.6.1.1 Java Reflection
Java Reflection Framework java.reflect.* [Gre05] allows us to
load/query/discover a given class for all of its API through enumeration of
constructors, fields, methods, etc. at run-time. This is incredibly useful for
dynamic loading and execution of our compilers, identifier context classes,
and sequential threads on local and remote machines.
The basic API from the reflection framework used in the implementation of
GIPSY is the Class class that allows getting arrays of declared Method objects
through the getDeclaredMethods() call that will become the STs at the end,
then for each Method the reflection API allows getting parameter and return
types via getParameterTypes() and getReturnType() calls, which will become the
CPs. The Class.newInstance() method allows instantiating an object off the
newly generated class. Likewise, an enumeration of Constructor objects is
acquired through the Class.getConstructors() call. Constructors in Java are
treated differently from methods because they are not inherited and don’t have
a return type (except that the type of the object they create). We still need
to enumerate them to allow Objective Lucid programs to use the constructors,
default or non-default, directly, so we can get a handle on them similarly to
STs.
##### 2.6.1.2 Java Native Interface (JNI)
The Java Native Interface (JNI) [Ste05] is very useful for the thread
generation component of the GIPC. We rely on JNI to increase the number of
popular imperative languages in which the sequential threads could be written.
Developers use the JNI to handle some specific situations when an application
cannot be written entirely in Java, e.g. when the standard Java classes do not
provide some platform-dependent features an application may require, or use a
library written in another language be accessible to Java applications, or for
performance reasons a small portion of a time-critical code has to be written
say in C or assembly, but still be accessible from a Java application [Ste05].
In GIPSY, the second and third of the listed cases are most applicable (e.g.
to adopt GLU programs). The JNI will allow us to avoid Lucid-to-C or Lucid-
to-C++ type matching as we can do it all through Java and maintain only Lucid-
to-Java type mapping table.
The JNI is made so that the native and Java sides of an application can pass
back and forth objects, strings, arrays and update their state on either end
[Ste05]. The JNI is bi-directional, i.e., allows Java to use the native
libraries and applications and provide access to Java libraries from the
native applications.
The general methodology of creating a JNI application say that interacts with
a C implementation is done in six steps [Ste05]:
1. 1.
Write a Java code with a native method to be implemented in C, the main(), and
the dynamic loading statement for a library (to be compiled in the next
steps).
2. 2.
Compile the Java code with javac and produce a .class file.
3. 3.
Create a C header .h file from the compiled .class file by calling javah. This
header file will provide the necessary #include directives along with the
C-style prototype declaration of the native method.
4. 4.
Next, write the implementation of the function in regular C in a .c file.
5. 5.
Then, create a shared library by compiling the .h and .c files with a C
compiler.
6. 6.
Run the application regularly with the JVM (java).
##### 2.6.1.3 JUnit
JUnit is an open-source Java testing framework used to write and run automated
repeatable unit tests in a hassle-free manner [GB04]. The goal is to sustain
application correctness over time, especially when undergoing a lot of
integration efforts. JUnit is designed with software architecture patterns in
mind and follows best software engineering practices. It encourages developers
to write tests for their applications that withstand time and bit rot.
The main abstract class is TestCase that follows the Command design pattern
that implements the Test interface. This class maintains the name of the tests
(if it fails) and defines the run() method that has to be overridden to do the
actual testing work. The default Template Method run() simply does three
things: setUp(), runTest(), and tearDown(). Their default implementation is to
do nothing, so a developer can override them as necessary. Then, to collect
the test results they apply Collecting Parameter pattern. They use the
TestResult class for that.
JUnit makes a distinction between errors and failures in the following way:
errors to JUnit are mostly unexpected run-time or regular exceptions, whereas
failures are anticipated and are tested for using assertion checks. The errors
and failures are collected for further test failure reporting.
To run tests in a general manner from the point of view of the tester, the
test classes have with a generic interface using the Adapter pattern. JUnit
also offers a pluggable selector capability via the Java Reflection API
[Gre05]. The TestSuite class represents a collection of tests to run. In the
GIPSY, the Regression application (see Section 5.1) comprises concrete
implementation of such a test suite that tests most of the feasible
functionality of the GIPC and GEE modules. See more details of application of
JUnit to the GIPSY in Chapter 5.
#### 2.6.2 javacc – Java Compiler Compiler
JavaCC [VC05], accompanied by JJTree, is the tool the GIPSY project is relying
on since the first implementation [Ren02] to create Java-language parsers and
ASTs off a source grammar files. The Java Compiler Compiler tool implements
the same idea for Java, as do lex/yacc [Lou97] (or flex/bison) for C – reading
a source grammar they produce a parser that complies with this grammar and
gives you a handle on the root of the abstract syntax tree. The GIPL,
Indexical Lucid, JLucid, Objective Lucid, PreprocessorParser, and DFGGenerator
parsers are generated with the JavaCC/JJTree parser generation tools. JavaCC
is a LL(K) [Lou97] parser generator, so the original GIPL and Indexical Lucid
grammars and the new grammars had to be modified to eliminate or avoid the
left recursion.
#### 2.6.3 MARF
Modular Audio Recognition Framework (MARF) library [MCSN05] provides a few
useful utility and storage classes GIPSY is using to manipulate threads,
arrays, option processing, and byte operations. Despite MARF’s belonging to a
voice/speech/natural language recognition and processing library, it contains
a variety of useful utility modules for threading and options processing.
#### 2.6.4 CVS
For managing the source code repository the Concurrent Versions System (CVS)
[BddzzP+05] is used. The CVS allows multiple developers work on the up-to-date
source tree in parallel that keeps tracks of the revision history and works in
an transactional manner. The author produced a mini-tutorial on the CVS
[Mok03a] for the GIPSY Research and Development team, which contains the
necessary summary for the team to work with the project repository.
While CVS has a comprehensive set of commands, the basic set includes:
* •
init to initialize the repository
* •
checkout or co to checkout the source code tree from the repository to a local
directory
* •
update or up to make the local tree up-to-date with the one on the server
* •
add to schedule a new file inside the existing local checkout for addition to
the repository
* •
remove to schedule a new file inside the existing local checkout for removal
from the repository
* •
commit to upload the changes done locally to the server
* •
diff to show the differences between the local and the server versions of the
tree
#### 2.6.5 Tomcat
Apache Jakarta Tomcat [Fou05] is an open-source Java application servlet and
server pages container project from Apache Foundation to run web Java-based
applications written in accordance with the Java Servlet and JavaServer Pages
[Mic05a, Mic05c] specifications developed by Sun Microsystems. Tomcat powers
up the web front end to GIPSY to test intensional programs online. The web
frontend is represented by the WebEditor servlet as of this writing a part of
RIPE which is discussed later in Chapter 4. Tomcat has an easy interface to
deploy Java-based applications and their libraries, e.g. through a manager
presented in Figure 14.
Figure 14: Tomcat Web Applications Manager
Tomcat itself consists from a variety of modules that includes implementation
of the JSP (Jasper engine) and Servlet APIs, a webserver called Coyote, the
application server called Catalina, and many other things for logging,
security, administration, etc.
#### 2.6.6 Build System
The GIPSY’s sources can be built using a variety of ways, using different
compilers and IDEs on different platforms. This includes Linux Makefiles,
IBM’s Eclipse, Borland’s JBuilder, Apache’s Ant, and Sun’s NetBeans.
##### 2.6.6.1 Makefiles
Unix/Linux Makefiles are targeting all Unix systems that support GNU make
(a.k.a gmake) [SMSP00, Mok05a]. Often, to compile all of the GIPSY is just
enough to type in make and the system will be built. All Unix versions support
make, and our system has been tested to build on Red Hat Linux 9, Fedora Core
2, Mac OS X, and Solaris 9. There is a test script make-test.sh that tests
whether we are dealing with the GNU make on Unix systems, as this is the only
make supported.
##### 2.6.6.2 Eclipse
There are project files .project and .classpath that belong to this IDE from
IBM [c+04]. The GIPSY build with this IDE properly and has its library
CLASSPATH set. Eclipse is another open source tool available free of charge
and provides extended tools for Java projects development, refactoring, and
deployment.
##### 2.6.6.3 JBuilder
There is a project file GIPSY.jpx that belongs to this IDE from Borland
[Bor03]. The GIPSY build with this IDE properly and has its library CLASSPATH
set.
##### 2.6.6.4 Ant
There is a project file build.xml that belongs to this build tool from the
Apache Foundation [Con05]. The GIPSY build with this tool properly and has its
library CLASSPATH set. In this case build.xml is a portable way to write a
Makefile in XML.
##### 2.6.6.5 NetBeans
There is a project file nbproject.xml that belongs to this IDE from Sun
[Mic04]. The GIPSY build with this IDE properly and has its library CLASSPATH
set.
#### 2.6.7 readmedir
This script generates a human-readable description of a directory structure
starting from some directory with file listing and possibly descriptions (for
this there should be specially formatted file README.dir in every directory
traversed. The contents of this file will be a part of the output and is a
responsibility of the directory creator/maintainer. The output formats of the
script are LaTeX, HTML, and plain text.
### 2.7 Summary
In this chapter the reader was introduced to the necessary background on the
GIPSY project and how it is being managed starting from the Lucid language
origins to its implementations in the GIPSY and the summary of the tools used
to aid the advancement of the project. In the GIPSY section the three main
modules were introduced, such as GIPC, GEE, and RIPE. While most of the
remaining work has gone into the GIPC in this thesis, the author had to
perform the necessary integration and adjustments to the GEE and RIPE.
## Chapter 3 Methodology
This chapter focuses on the methods and techniques proposed to the solve the
stated problems (see Section 1.1). The approaches described are based on three
publications, namely [MPG05, MP05b, MP05a]. Section 3.1 introduces the JLucid
language and all related considerations including the syntax and semantics.
Next, Objective Lucid is introduced along with its syntax and semantics.
Further, the GICF is introduced by providing the necessary requirements for it
to exist and the way to satisfy them. Lastly, the summary is presented
outlining the benefits and limitations of the proposed solutions.
### 3.1 JLucid: Lucid with Embedded Java Methods
#### 3.1.1 Rationale
The name JLucid comes from the GIPC component known as Java Compiler within
the Sequential Thread (ST) Generator of the GIPSY. It subsumes all of
Indexical Lucid and General Intensional Programming Language (GIPL) [Paq99]
and syntactically allows embedded Java code. In fact, a JLucid program looks
like a partial fusion of the intensional and Java code segments. JLucid gives
a great deal of flexibility to Lucid programs by allowing to use existing
implementations of certain functions in Java, providing I/O facilities and
math routines (that Lucid entirely lacks), and other Java features accessible
to Lucid, arrays, and permits to increase the granularity of computations at
the operator level by allowing the user to define Java operators, i.e.,
functions manipulating objects, thus allowing streams of objects111A more
precise meaning of Java objects within Lucid is explored further in the
Objective Lucid language, including the meaning of an object stream and how
object members are manipulated (see for example Section 3.2 and Section
4.1.3.6). Additionally, since the actual Java objects are flattened into
primitive types, it would be possible to access object members in parallel
manner. in Lucid. JLucid more or less achieves the same goals and mechanisms
as provided by GLU. What we are proposing is a flexible compiler and run-time
system that permits the evolution of languages through a framework approach
[MP05a, PW05].
##### 3.1.1.1 Modeling Non-Determinism
Lucid, by its nature, is deterministic, so introduction of imperative
languages, such as Java, may allow us to model non-determinism in Lucid
programs for example by providing access to random number generators available
to the imperative languages. Non-determinism can also be introduced as a
result of side effects from for example reading a different file each time an
ST is invoked, or making a database query against a table where data regularly
changes, or say by reading the current time of day value. Of course, a special
care should be taken not to cache the results of such STs in the warehouse.
##### 3.1.1.2 Loading Existing Java Code with embed()
In a nutshell, we want to make the following possible for the Indexical Lucid
program in Figure 1 (replicated here from Chapter 2 for convenience) to become
something as in Figure 2 or, alternatively as in Figure 3. The latter form
would allow us to include objects from any types of URLs, local, HTTP, FTP,
etc. The idea behind embed() is to include or to import the code written
already by someone and not to rewrite it in Lucid (which may not be a trivial
task). It is not meant to adjust to URL’s existence at run-time as all embed-
referenced resources are resolved at compile time. We “include” the pointed-to
resource and attempt to compile it where the original program-initiator
resides. If the URL is invalid at compile time, then there will be a compile
error and no computation will be started. embed() by itself does not
necessarily provoke a remote function call.
H
where
H = 1 fby merge(merge(2 * H, 3 * H), 5 * H);
merge(x, y) = if(xx <= yy) then xx else yy
where
xx = x upon(xx <= yy);
yy = y upon(yy <= xx);
end;
end;
Figure 1: Indexical Lucid program implementing the merge() function.
#JAVA
void merge(int x, int y)
{
// java code here
}
#JLUCID
H
where
H = 1 fby merge(merge(2 * H, 3 * H), 5 * H);
end;
Figure 2: Indexical Lucid program implementing the merge() function as inline
Java method.
H
where
H = 1 fby merge(merge(2 * H, 3 * H), 5 * H);
merge(x, y) =
embed("file://path/to/class/Merge.class", "merge", x, y);
end;
Figure 3: Indexical Lucid program implementing the merge() function as
embed().
F
where
dimension d;
F = foo(#d);
where
foo(i) = embed("file://my/classes/Foo.class", "foo", i);
end;
end;
Figure 4: Illustration of the embed() syntax.
Existing Java code, in either .class or .java form, can be loaded with
embed(). Intuitively, we would prefer the approach presented in Figure 4. That
added flexibility requires syntactical extension of Lucid and is not portable.
For the program in Figure 4 to work, foo() has to return a Java type of int,
byte, long, char, String, or boolean, as per Table 1, page 1. A wrapper class
will be created to extend from the Foo and implement the ISequentialThread
interface (see Appendix B.1). General embed() syntax would be defined as
follows:
id(id, id, ...) ::= embed(URI, METHOD, id, id, ...);
where id is the Lucid function name being defined that is mapped to a Java’s
method named METHOD (which may or may not be of the same name as the first
id). The URI is pointing to either .class or .java file. Example URI’s would
be:
foo(a,b) = embed("file://files/Foo.java","bar",a,b); bar(a,b) =
embed("http://www.java.com/Foo.class","foo",a,b); baz(a,b) =
embed("ftp://ftp.file.com/pub/Foo.java","zee",a,b);
These declarations associate Lucid functions with Java implementations. Name
clashes may be avoided, if necessary, by using different function names.
Above, for example, Lucid baz() is implemented by Java zee().
public class <filename>_<machine_name>_<timestamp>
extends my.classes.Foo
implements ISequentialThread
{
// The definition is provided later in the text
}
Figure 5: Generated corresponding ST to that of Figure 4.
There are several ways of making this work. We could extract either a textual
or a bytecode definition of foo(), wrap it in our own class and, (re)compile
it. However, there is an issue here. What about other functions it may use,
like shown in Figure 9 with two methods calling each other? That would mean
extracting those dependencies as well along with the method of interest. This
won’t scale very efficiently. Thus, alternate approaches include: to either
inherit from the desired class as in Figure 5, encapsulate this class
instance, or attempt to wrap the entire class as done for the JAVA segment in
Section 3.1.1.3 below. The former approach would imply having a class variable
instance of the type of that class encapsulated into the wrapper. The latter
approach was chosen as more feasible to implement, although it doesn’t deal
with user-defined classes and subclass and packages the .class or .java file
may require at the moment. Thus, the embed() acts in a way similar to #include
in C/C++ or import in Java of a set of Java definitions to be used in a JLucid
program. Therefore, embed() has to be resolved at compile time. Similar
technique may be taken towards other languages than Java at a later time.
Lucid’s syntax has to be extended to support embed().
##### 3.1.1.3 The #JAVA and #JLUCID Code Segments
This section explores ways of mixing Java and Lucid source code segments in a
single text file and ways of dealing with such a merge.
F
where
dimension d;
F = foo(#d);
where
foo(i) = int foo(int i) { return i + 1; }
end;
end;
Figure 6: Inline Java function declaration.
An attempt to use Java’s methods inline, such as in Figure 6 would be
intuitive, but does not justify the effort spent on syntax analysis.
Therefore, we take the inline definition out of the Lucid part, and make it a
separate outer definition of the same method. Additionally, we explicitly mark
the JLUCID and JAVA code segments to simplify pre-processing of the JLucid
code as presented in Figure 7.
#JAVA
int foo(int i)
{
// Some i + PI
return (int)(java.lang.Math.PI + i);
}
#JLUCID
F
where
dimension d;
F = foo(#d);
end;
Figure 7: Java method declaration split out from the Lucid part.
Given the Natural Numbers Problem (see [Paq99]) in Figure 8 (replicated here
for convenience), one could imagine the function definition for $N$ to be
implemented in Java in two functions. To illustrate the point when two
separate functions can call each other in the JAVA segment or several JAVA
segments. This modified JLucid code along with line numbers is shown in Figure
9. Since we allow one Java method to call another within, we have to wrap them
both into the same class.
N @.d 2
where
dimension d;
N = if (#.d <= 0) then 42 else (N + 1) @.d (#.d - 1) fi;
end;
Figure 8: Natural numbers problem in plain GIPL.
The JLucid code segments after “#JAVA” constructs will be grouped together by
the compiler. For all definitions (functions, classes, variables) in these
segments, their original location in the JLucid source recorded and statically
put in the wrapper class. These definitions will end up in that wrapper class
as well.
It would be possible to have a class defined within a wrapper class or any
other valid Java declaration; even a data member can be included. To
summarize, the Java segments in the JLucid code are a body of a generated
class that implements the ISequentialThread interface.
1 #JAVA
2
3 int getN(int piDimension)
4 {
5 if(piDimension <= 0)
6 return get42();
7 else
8 return getN(piDimension - 1) + 1;
9 }
10
11 int get42()
12 {
13 return 42;
14 }
15
16 #JLUCID
17
18 N @d 2
19 where
20 dimension d;
21 N = getN(#d);
22 end;
Figure 9: Natural numbers problem with two Java methods calling each other.
For the example in Figure 9 the parser would proceed as follows:
* •
In the preprocessing step the source code is split into two parts: the Java
part and the Lucid part. For both parts original source’s line numbers and
length of the definitions are recorded.
* •
Then they both are fed to the respective parsers. Java’s part requires extra
handling: the Java methods (one or more) defined in the code, have to be
wrapped into a class and then JavaCompiler class that takes the Java portion
of the source and feeds it to javac for syntactic and semantic analyses and
byte code generation. They will become parts of a Sequential Thread, ST (see
Section 3.3.3.1) definition fed to Workers (see Section 3.3.3.4).
* •
The Lucid part is processed by the modified Lucid compiler (to include the
syntactical modifications for arrays and embed()) and comes up with the main
AST from that.
* •
The Java STs are then linked into the main AST in place of nodes where the
identifiers of these appear in the Lucid part of the program prior semantic
analysis.
Any method or other definition in the JAVA segment is wrapped into a class.
The generated wrapper class will contain a Hashtable that maps method
signature strings to their starting line in the original JLucid code plus the
length of the definitions in lines of text they occupy statically generated
and initialized. This is needed for the error reporting subsystem in case of
syntax/semantic errors, report back correctly the line in the original JLucid
program and not in the generated class. The class name is created
automatically from the original program name, the machine name it’s being
compiled on, and a timestamp to guarantee enough uniqueness to the generated
class’ name to minimize conflict for multiple such generated classes. Thus,
the JAVA segment in Figure 9 will transform into the generated class as in
Figure 10. This is a short version; for more detailed one please refer to the
Section B.3. In fact, after generating this class (and possibly compiling it)
this situation can be viewed as a special case for embed(), Section 3.1.1.2 or
vice versa. Note, since we have no guarantee the Java methods are side-effects
free in JLucid, their results are not cached in the warehouse.
public class <filename>_<machine_name>_<timestamp>
implements gipsy.interfaces.ISequentialThread {
private OriginalSourceCodeInfo oOriginalSourceCodeInfo;
// Inner class with original source code information
public class OriginalSourceCodeInfo {
// For debugging / monitoring; generated statically
private String strOriginalSource = ...
// Mapping to original source code position for error reporting
private Hashtable oLineNumbers = new Hashtable();
// Body is filled in by the preprocessor statically
public OriginalSourceCodeInfo() {
Vector int_getN_int_piDimension = new Vector();
// Start line and Length in lines
int_getN_int_piDimension.add(new Integer(3));
int_getN_int_piDimension.add(new Integer(7));
oLineNumbers.put("int getN(int piDimension)",
int_getN_int_piDimension);
Vector int_get42 = new Vector();
int_get42.add(new Integer(11));
int_get42.add(new Integer(4));
oLineNumbers.put("int get42()", int_get42);
}
}
// Constructor
public <filename>_<machine_name>_<timestamp>() {
oOriginalSourceCodeInfo = new OriginalSourceCodeInfo();
}
/*
* Implementation of the SequentialThread interface
*/
// Body generated by the compiler
public void run() {
Payload oPayload = new Payload();
oPayload.add("d", new Integer(42));
work(oPayload);
}
// Body generated by the compiler statically
public WorkResult work(Payload poPayload) {
WorkResult oWorkresult = new WorkResult();
oWorkresult.add(getN(poPayload.getVaueOf("d")));
return oWorkResult;
}
/*
* The below are generated off the source file nat2java.ipl
*/
public static int getN(int piDimension) {
if(piDimension <= 0) return get42();
else return getN(piDimension - 1) + 1;
}
public static int get42() {
return 42;
}
}
Figure 10: Generated Sequential Thread Class.
In [MPG05] we required foo() in the previous examples to be static. In fact,
any method or other definition in the JAVA segment were to be transformed to
become static while being wrapped into a class. For example, “int foo()
{return 1;}” would become “public static int foo() {...}”. We insisted on
static declarations only because the sequential threads were not instantiated
by the workers when executed. This restriction has been lifted during
implementation as we instantiate and serialize the sequential thread class as
needed.
##### 3.1.1.4 Is JLucid an Intensional Language?
We treat JLucid as a separate specific intensional programming language (SIPL)
rather than a part of a GIPSY program within existing Indexical Lucid
implementation. Here are some pros and cons of this approach and JLucid as a
separate SIPL approach is the winner. Why extend it as a separate SIPL?
* •
This would serve as an example on how to add other SIPLs.
* •
This would allow us to keep the original Indexical Lucid clean and working.
* •
This would allow functions with Java syntax to be used within a Lucid program
as well as binary Java function calls of pre-compiled classes.
* •
It can be extended to other languages as it turns out to be a successful
approach.
Why not to treat is as a separate SIPL?
* •
We might want to have embedded Java (or other language) in any intensional
language, not just Indexical Lucid. How to make that possible?
* •
It is not truly an SIPL, but a hybrid.
#### 3.1.2 Syntax
In JLucid, we extend the syntax of both GIPL and Indexical Lucid to support
arrays. For example, it is useful to be able to evaluate several array
elements under the same context. This is included by the last $E$ rules of
$E[E,...,E]$ and $[E,...,E]$ in both syntaxes. Arrays are useful to manipulate
a collection Lucid streams under the same context. JLucid arrays are mapped to
Java arrays on the element-by-element basis with the appropriate element type
matching and may only correspond to arrays of primitive types in Java. The
syntax also includes the embed() extension to allow including external Java
code. The JLucid syntax extensions to GIPL and Indexical Lucid are presented
in Figure 11 and Figure 12.
E ::= id
| E(E,...,E)
| if E then E else E fi
| # E
| E @ E
| E where Q end;
| [E:E,...,E:E]
| embed(URI, METHOD, E, E, ...)
| E[E,...,E]
| [E,...,E]
Q ::= dimension id,...,id;
| id = E;
| id(id,...,id) = E;
| QQ
Figure 11: JLucid Extension to GIPL Syntax
E ::= id
| E(E,...,E)
| if E then E else E fi
| # E
| E @ E E
| E where Q end;
| E bin-op E
| un-op E
| embed(URI, METHOD, E, E, ...)
| E[E,...,E]
| [E,...,E]
Q ::= dimension id,...,id;
| id = E;
| id.id,...,id(id,...,id) = E;
| QQ
bin-op ::= fby | upon | asa | wvr
un-op ::= first | next | prev
Figure 12: JLucid Extension to Indexical Lucid Syntax
#### 3.1.3 Semantics
The JLucid extension to the operational semantics of Lucid (see Section
2.2.2.8 on page 2.2.2.8) is defined in Figure 13. As in the original Lucid
semantics, each type of identifier can only be used in the appropriate
situations. Notation:
* •
freefun, ffid, ffdef mean a type of identifier is a hybrid free (i.e. object-
free) function freefun, where ffid is its identifier and ffdef is its
definition (body).
* •
The ${\mathbf{E_{ffid}}}$ rule defines JLucid’s free functions.
* •
The JLucid $\mathbf{\\#JAVA_{ffid}}$ rule add free function definition to the
definition environment.
$\displaystyle{\mathbf{E_{ffid}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{\begin{array}[]{c}\mathcal{D},\mathcal{P}\vdash
E:id\qquad\mathcal{D},\mathcal{P}\vdash
E_{1},\ldots,E_{n}:v_{1},\ldots,v_{n}\\\ \mathcal{D}(id)=({\texttt{freefun,
ffid, {{\text@underline{ffdef}}}}})\\\
\mathcal{D},\mathcal{P}\vdash<\\!\\!{\mathtt{ffid}}(v_{1},\dots,v_{n})\\!\\!>:v\end{array}}{\mathcal{D},\mathcal{P}\vdash
E(E_{1},\ldots,E_{n}):v}$ $\displaystyle{\mathbf{\\#JAVA_{ffid}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{{{\mathtt{\underline{ffdef}}}}=\mathit{frttype}\texttt{
ffid}(\mathit{fargtype_{1}}\;farg_{id_{1}},\dots,\mathit{fargtype_{n}}\;farg_{id_{n}})}{\mathcal{D},\mathcal{P}\vdash{{\mathtt{\underline{ffdef}}}}\>:\>\mathcal{D}\\!\dagger\\![\texttt{ffid}\mapsto(\texttt{freefun,
ffid, {\text@underline{ffdef}}})],\mathcal{P}}$ Figure 13: Additional basic
semantic rules to support JLucid
### 3.2 Objective Lucid: JLucid with Java Objects
#### 3.2.1 Rationale
Objective Lucid is a direct extension of JLucid. The original syntax of
Indexical Lucid (and also for JLucid and GIPL) is augmented to support a so-
called dot-notation. This allows Lucid to manipulate grouped data by using
object’s methods. In fact, the idea is similar to manipulating arrays in
JLucid. The difference with the arrays is that they are manipulated as a
collection of ordered data of elements of the same type, to be evaluated in
the same context. However, an object that varies in some dimension implies
that all its members, possibly of different types, also potentially vary along
this dimension, but across objects, i.e. the objects themselves are not
intensional. An object can be thought of as a heterogeneous collection of
different types of members, which you can access individually using their
name, whereas arrays can be thought of as a homogeneous collection of members
that can be accesses individually using their index.
Just like JLucid [MPG05], Objective Lucid is being developed as a separate
specific intensional programming language (SIPL) within the GIPSY for the same
reasons: keeping the other implementations undisturbed and working while
experimenting on this particular implementation.
##### 3.2.1.1 Pseudo-Objectivism in JLucid
A pseudo-object-oriented approach is already present in JLucid. The program
presented in Figure 14 gives an example of a Java function returning an object
of type Integer.
#JAVA
Integer f()
{
return new Integer("1234");
}
int g()
{
return f().intValue();
}
#JLUCID
A
where
A = g();
end;
Figure 14: Pseudo-objectivism in JLucid.
In JLucid we are not able to manipulate this object directly in intensional
programming as Java does, though we can provide methods, such as g() to access
properties of a particular Java object from within JLucid. However, that
reduces legacy Java code reusability by forcing the programmer to add such
functions in his code to be able to use it in the GIPSY. Another example in
Figure 15 shows how one can make use of objects in JLucid by providing pseudo-
free Java accessors similar to getComputedBar() in the example. They are
pseudo-free because they don’t appear as a part of any Java class to a JLucid
programmer explicitly, but internally they get wrapped into a class when the
code is compiled.
#JAVA
class Foo
{
private int bar;
public Foo()
{
bar = (int)(Math.random() * Integer.MAX_VALUE);
}
public int getBar()
{
return bar;
}
public void computeMod(int piParam)
{
bar = bar % piParam;
}
}
int getComputedBar(int piParam)
{
Foo oFoo = new Foo();
oFoo.computeMod(piParam);
System.out.println("bar = " + bar);
return oFoo.getBar();
}
#JLUCID
Bar
where
Bar = getComputedBar(5);
end;
Figure 15: Using pseudo-free Java functions to access object properties in
JLucid.
In Objective Lucid such explicit workarounds are not necessary anymore, but
this gives us some ideas about how to actually implement some features of
Objective Lucid in practice, i.e., the compiler can generate a number of
pseudo-free accessors to object’s members and use JLucid’s implementation of
Java functions internally.
##### 3.2.1.2 Stream of Objects
An interesting question could be to ask: “What is an object stream?” Is it
that the members of this object vary in the same dimension(s) or they can have
“substreams”? In Objective Lucid we answer this as decomposing public object’s
data members into primitive types and varying them or in simplified manner we
employ object’s effectors. Thus, when there is a demand say for the object’s
state (data members) at some time $t$, there will have to be generated demands
for all of $t$ between $[0,t]$ where at time $0$ an instance of the object is
created. Therefore, the object state changes in the $[0,t]$ interval represent
the object stream in the context of this thesis. There are two possible
outcomes of this evaluation: either a portion of object’s state is altered by
an intensional program or the entire object. In the former case, Lucid only
accesses some object’s members via the dot-notation in the intensional manner,
whereas in the latter case all the members of an object are altered in the
intensional context implicitly. The examples presented in Figure 16, Figure 4,
page 4, and Figure 6, page 6 work on portions of an object, whereas the
examples in Section 4.1.3.6, page 4.1.3.6 work on all the members of an object
at the same time.
##### 3.2.1.3 Pure Intensional Object-Oriented Programming
Objective Lucid has presented a way for Lucid programs to use Java objects.
This may seem rather restrictive and may look like a workaround (though
practical!). An interesting concept would be to extend the Lucid language
itself to create and manipulate pure Lucid objects, not Java objects. This
will allow addressing issues like inheritance and polymorphism and other
attributes of object-oriented programming and will solve the problem of
matching Lucid and Java data types. This is not addressed in this work, but
attempted to be solved in [WP05].
#### 3.2.2 Syntax
The parser is extended to support the `<objectref>.<feature>` dot-notation for
the Lucid part of reference data types. The semantic analysis is augmented to
accommodate objects and user-defined data types. In doing so, Lucid is able to
manipulate Java objects as well as access public variables and methods of
these objects. An example is shown in Figure 16. This example manipulates a
simple object E by evaluating its state at some time “2”. The program begins
with the construction of the object with f1() (or one could call the object
constructor directly), and then the rest of the expressions access public
members x and foo() of the object during expression evaluation.
#JAVA
class ClassXB
{
public int x;
public float b;
public ClassXB()
{
x = 0; b = 1.2;
}
public int foo(int a, float c)
{
return x = (int)(x * a + b * c);
}
ClassXB addx(int b)
{
x += b;
return this;
}
}
ClassXB f1()
{
return new ClassXB();
}
#OBJECTIVELUCID
/*
* The result of this program should be the object E
* to be evaluated at time dimension 2 with its ’x’
* member modified accordingly.
*/
E @time 2
where
dimension time;
E = f1() fby.time A;
A = E.addx(B);
B = E.foo(A @time C, A) + 3;
C = E.x * 2;
end;
Figure 16: Objective Lucid example.
The Objective Lucid syntax is in Figure 17. It is a direct extension of the
JLucid syntax in Figure 12 to support the dot-notation. Essentially, the
extension is the E.id productions. Any E on the left-hand-side can evaluate to
an object type, but the right-hand-side is always an identifier (Java class’
data member or method).
E ::= id
| E(E,...,E)
| if E then E else E fi
| # E
| E @ E E
| E where Q end;
| E bin-op E
| un-op E
| embed(URI, METHOD, E, E, ...)
| E[E,...,E]
| [E,...,E]
| E.id
| E.id(E,...,E)
Q ::= dimension id,...,id;
| id = E;
| E.id = E;
| id.id,...,id(id,...,id) = E;
| QQ
bin-op ::= fby | upon | asa | wvr
un-op ::= first | next | prev
Figure 17: Objective Lucid Syntax
#### 3.2.3 Semantics
To support these extensions to JLucid, the Semantic Analyzer of JLucid
requires more non-trivial changes than the syntax analysis and the dot-
notation implementation due to arbitrary object data types. In order to
perform type checks and apply the semantic rules of Lucid, we place the object
data types into the definition environment $\mathcal{D}$, which is in fact a
semantic equivalent to the data dictionary part of the GEER. This is partly
solved by using the pseudo-free Java functions, which de-objectify the object
members, but in order to be able to do so, we need to have the object types in
the definition environment. The corresponding operational semantic rules from
[Paq99] can be extended as follows.
The Objective Lucid extension to the operational semantics of Lucid is defined
in Figure 18. As in the original Lucid semantics, each type of identifier can
only be used in the appropriate situations. Notation:
* •
class, cid, cdef means it is a Class type of identifier with name cid and a
definition cdef.
* •
classv, cid.cvid, vdef means that the variable is a member variable of a class
classv with identifier cid.cvid given the variable definition vdef within the
class.
* •
$\mathtt{<\\!\\!cid.cvid\\!\\!>}$ means object-member reference within an
intensional program.
* •
classf, cid.cfid, fdef means that the function is a member function of a class
classf with identifier cid.cfid given the variable definition fdef within the
class.
* •
$<\\!\\!{\mathtt{cid.cfid}}(v_{1},\dots,v_{n})\\!\\!>$ represents a object-
function call within an intensional program with actual parameters.
* •
freefun, ffid, ffdef mean a type of identifier is a hybrid free (i.e. object-
free) function freefun, where ffid is its identifier and ffdef is its
definition (body).
* •
By ${{\mathtt{\underline{cdef}}}}={\mathtt{Class\;cid\;\\{\ldots\\}}}$ we
declare a class definition. A class can contain member variable vdef and
member functions definitions fdef.
The rules:
* •
The ${\mathbf{E_{c-vid}}}$ rule defines an object member variable for an
expression for the dot-notation. It is independent from the language in which
we define and express our objects. The rule says that under some context given
two expressions $E$ and $E^{\prime}$ that evaluate to a class-type identifier
$id$ and a variable type identifier $id^{\prime}$ respectively and if the two
together via a dot-notation represent an object-data-member reference, then
the expression $E.E^{\prime}$ evaluates to a value $v$.
* •
Member function calls are resolved by the $\mathbf{E_{c-fct}}$ rule. Similarly
to the ${\mathbf{E_{c-vid}}}$ rule, it defines that given two expressions $E$
and $E^{\prime}$ under some context that evaluate to a class-type identifier
$id$ and a member function type identifier $id^{\prime}$ and a set of
intensional expressions ${E_{1},\ldots,E_{n}}$ evaluates to some values
${v_{1},\ldots,v_{n}}$ and the two identifiers via a dot-notation represent a
member function call with parameters ${v_{1},\ldots,v_{n}}$, then we say the
expression $E.E^{\prime}(E_{1},\ldots,E_{2})$ is a member function call that
under the same context evaluates to some value $v$, i.e. the function always
returns a value. Here we see why it is necessary for Lucid to map a void data
type to implicit Boolean true. This choice may seem a bit arbitrary (for
example, one could pick an integer $1$), but aside from practicality aspect
the mere choice of true may signify a successful termination of a method.
* •
The ${\mathbf{E_{ffid}}}$ rule defines JLucid’s free functions. The rule is a
simpler version of $\mathbf{E_{c-fct}}$ with no class type identifiers
present.
* •
The ${\mathbf{\\#JAVA_{objid}}}$ rule places class definition into the
definition environment.
* •
The $\mathbf{\\#JAVA_{obvjid}}$ and $\mathbf{\\#JAVA_{objfid}}$ rules add
public Java object member variable and function identifiers along with their
definitions to the definition environment.
* •
The JLucid $\mathbf{\\#JAVA_{ffid}}$ rule add free function definition to the
definition environment.
$\displaystyle{\mathbf{E_{c-vid}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{\begin{array}[]{c}\mathcal{D},\mathcal{P}\vdash
E:id\quad\mathcal{D},\mathcal{P}\vdash E^{\prime}:id^{\prime}\\\
\mathcal{D}(id)=({\texttt{class, cid,
{{\text@underline{cdef}}}}})\quad\mathcal{D}(id^{\prime})=({\texttt{classv,
cid.cvid, {{\text@underline{vdef}}}}})\\\
\mathcal{D},\mathcal{P}\vdash<\\!\\!{\mathtt{cid.cvid}}\\!\\!>:v\end{array}}{\mathcal{D},\mathcal{P}\vdash
E.E^{\prime}:v}$ $\displaystyle{\mathbf{E_{c-fct}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{\begin{array}[]{c}\mathcal{D},\mathcal{P}\vdash
E:id\qquad\mathcal{D},\mathcal{P}\vdash
E^{\prime}:id^{\prime}\qquad\mathcal{D},\mathcal{P}\vdash
E_{1},\ldots,E_{n}:v_{1},\ldots,v_{n}\\\ \mathcal{D}(id)=({\texttt{class, cid,
{{\text@underline{cdef}}}}})\qquad\mathcal{D}(id^{\prime})=({\texttt{classf,
cid.cfid, {{\text@underline{fdef}}}}})\\\
\mathcal{D},\mathcal{P}\vdash<\\!\\!{\mathtt{cid.cfid}}(v_{1},\dots,v_{n})\\!\\!>:v\end{array}}{\mathcal{D},\mathcal{P}\vdash
E.E^{\prime}(E_{1},\ldots,E_{n}):v}$ $\displaystyle{\mathbf{E_{ffid}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{\begin{array}[]{c}\mathcal{D},\mathcal{P}\vdash
E:id\qquad\mathcal{D},\mathcal{P}\vdash
E_{1},\ldots,E_{n}:v_{1},\ldots,v_{n}\\\ \mathcal{D}(id)=({\texttt{freefun,
ffid, {{\text@underline{ffdef}}}}})\\\
\mathcal{D},\mathcal{P}\vdash<\\!\\!{\mathtt{ffid}}(v_{1},\dots,v_{n})\\!\\!>:v\end{array}}{\mathcal{D},\mathcal{P}\vdash
E(E_{1},\ldots,E_{n}):v}$ $\displaystyle{\mathbf{\\#JAVA_{objid}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{{{\mathtt{\underline{cdef}}}}={\mathtt{Class\;cid\;\\{\ldots\\}}}}{\mathcal{D},\mathcal{P}\vdash{{\mathtt{\underline{cdef}}}}\>:\>\mathcal{D}\\!\dagger\\![{\mathtt{cid}}\mapsto(\mathtt{class,\;cid,\;{{\underline{cdef}}}})],\;\mathcal{P}}$
$\displaystyle{\mathbf{\\#JAVA_{objvid}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{{{\mathtt{\underline{cdef}}}}={\mathtt{Class\;cid\;\\{\ldots{\mathtt{\underline{vdef}}}\ldots\\}}}\qquad{{\mathtt{\underline{vdef}}}}={{\mathtt{public}\;type\;{\mathtt{vid};}}}}{\mathcal{D},\mathcal{P}\vdash{{\mathtt{\underline{cdef}}}}\>:\>\mathcal{D}\\!\dagger\\![{\mathtt{cid.vid}}\mapsto(\mathtt{classv,\;cid.vid,\;{\underline{vdef}}})],\mathcal{P}}$
$\displaystyle{\mathbf{\\#JAVA_{objfid}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{{{\mathtt{\underline{cdef}}}}={\mathtt{Class\;cid}}\;\\{\ldots{\mathtt{\underline{fdef}}}\ldots\\}\qquad{{\mathtt{\underline{fdef}}}}={\mathtt{public}}\;\mathit{frttype}\texttt{
fid}(\mathit{fargtype_{1}}\;farg_{id_{1}},\dots,\mathit{fargtype_{n}}\;farg_{id_{n}})}{\mathcal{D},\mathcal{P}\vdash{{\mathtt{\underline{cdef}}}}\>:\>\mathcal{D}\\!\dagger\\![\texttt{cid.fid}\mapsto(\texttt{classf,
cid.fid, {\text@underline{fdef}}})],\mathcal{P}}$
$\displaystyle{\mathbf{\\#JAVA_{ffid}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!:\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{{{\mathtt{\underline{ffdef}}}}=\mathit{frttype}\texttt{
ffid}(\mathit{fargtype_{1}}\;farg_{id_{1}},\dots,\mathit{fargtype_{n}}\;farg_{id_{n}})}{\mathcal{D},\mathcal{P}\vdash{{\mathtt{\underline{ffdef}}}}\>:\>\mathcal{D}\\!\dagger\\![\texttt{ffid}\mapsto(\texttt{freefun,
ffid, {\text@underline{ffdef}}})],\mathcal{P}}$ Figure 18: Additional basic
semantic rules to support Objective Lucid
### 3.3 General Imperative Compiler Framework
#### 3.3.1 Rationale
Having to deal with JLucid, Objective Lucid, and Java and a future likely
possibility to include other than Java imperative languages into intensional
ones prompted invention of a general mechanism to handle that and simplify
addition of new languages into the GIPSY for research and experiments. This
generalization touches several critical aspects exposed by the JLucid and
Objective Lucid languages involving such a hybrid programming model. Thus, a
core redesign of the GIPC was necessary to enable this feature. The General
Imperative Compiler Framework (GICF) addresses the generalization issues
(split among this Methodology and Design and Implementation chapters) for the
imperative compilers and suggests later development of a similar framework for
the intensional languages.
The core areas in the hybrid compilation process affect the way an intensional
language program (which now syntactically allows having any number of code
segments written in one or more imperative languages) is compiled. This kind
of program has to be preprocessed first to extract the code segments to be
compiled by the appropriate language compilers and at the same time maintains
syntactic and semantic links between the parts of a hybrid program. This
influences the general intensional compiler instrumentation, such as
generation of sequential threads and communication procedures, function
elimination, GIPL-to-SIPL translation, semantic analysis, and linking (and
later interpreting/executing) of a GIPSY program.
Requirements for any such a framework like GICF imply at least the following
considerations:
* •
having a number of compiler interfaces known to the system that any concrete
compiler implements,
* •
ability to pick such compilers at runtime based on a hybrid program being
compiled,
* •
have a generalized AST that is capable of holding intensional and imperative
nodes,
* •
have the semantic analyzer understand possible data types that any language
may expose (which is a very challenging goal to do correctly), and deal with
function elimination for the imperative parts of the AST,
* •
preprocess by breaking down a hybrid GIPSY program’s source code to be fed to
the appropriate compilers gives us flexibility of allowing to include any
imperative language we want, but complicates maintenance of semantic links
between the intensional and imperative parts for later linking and semantic
analysis. This necessitates development of the two other special segments that
can declare in a uniform manner for GIPSY providing some meta information
about embedded imperative sequential threads, like function and type
identifiers, parameter and return types for communication procedures, and user
data types. Thus, for the former we need a function prototype declaration
segment, that lists all free functions declared within imperative segments to
be used by Lucid and the type declaration segment for the user-defined types
possibly declared in those same imperative segments. The purpose of this meta-
information is two-fold: it will help us maintaining the semantic links via a
dictionary and create so-called “imperative stubs”. The former prompts the
development of the GIPSY Type System (see Section 4.1.1.5, page 4.1.1.5) as
understood by the Lucid language and its incarnation within the GIPSY to
handle types in a more general manner. The latter stubs have to be produced in
order for the intensional language compilers (that stay intact with the
introduced framework) not to choke on “undefined” symbols that really were
defined in the imperative parts, which an existing intensional compiler
running in isolation fails to see.
* •
After all involved compilers are finished doing compilation of their code
segments, they all produce a partial AST. For intensional compilers that means
the main AST with the intensional and stub nodes. For imperative compilers it
is the appropriate imperative AST for each sequential thread. The imperative
AST, in fact, need not to be a real tree and may contain a single imperative
node that would hold a payload of STs (compiled object or byte code), CPs,
type information, and some meta-information (e.g. what language the STs and
CPs are in and for which operating system and native compiler environment).
* •
Then, the imperative stubs have to be replaced by the real imperative nodes at
the linking stage before the semantic analysis.
* •
Once the main tree is formed, the semantic analyzer would use the type system
to verify type information of the intensional-imperative calls within taking
into consideration imperative nodes when doing function elimination and
producing the final “executable” tree, or Demand AST, or DAST, a component of
the GEER.
All this work is motivated by the desire to simplify the addition of new
compilers into the GIPSY environment with minimal integration hassle. The
follow up sections explore some of the issues about primary matching of the
Java and GIPSY data types, followed by the definition of sequential threads
and communication procedures in the GIPSY, and their Worker aggregator. While
the below are sections that lay down a concrete example based on JLucid and
Java, the discussion addressing the generalization of the design and
implementation of these issues are presented in the chapter that follows with
the actual sequence diagram showing implementation details of the above hybrid
compilation process.
#### 3.3.2 Matching Lucid and Java Data Types
Allowing Lucid to call Java functions brings a new set of issues related to
data types. Additional work is required on the semantic analyzer, especially
when it comes to type checks between Lucid and Java parts of a JLucid program.
This is pertinent when Lucid variables or expressions are used as parameters
to Java functions and when a Java function returns a result to be assigned to
a Lucid variable or used in an IP expression. The sets of types in both cases
are not exactly the same. The basic set of Lucid data types as defined by
Grogono [Gro02b] is int, bool, double, string, and dimension. Lucid’s int is
of the same size as Java’s int, and so are double, boolean, and String. Lucid
string and Java String are simply mapped to each other since internally we
implement the former as the latter; thus, one can think of the Lucid string as
a reference when evaluated in the intensional program. Based on this fact, the
lengths of a Lucid string and Java String are the same. Java String is also an
object in Java; however, at this point, a Lucid program has no direct access
to any object properties. We also distinguish the float data type for single-
precision floating point operations. The dimension index type is said to be an
integer for the time being, but might become a float when higher precision of
points in time, for example, will be in demand, or it could even be an
enumerated type of unordered values (though float dimensions will introduce
some very interesting problems). Therefore, we perform data type matching as
presented in Table 1. The return and parameter types matching sets are not the
same because of the size of the types. Additionally, we allow void Java return
type which will always be matched to a Boolean expression true in Lucid as an
expression has to always evaluate to something.
Table 1: Matching data types between Lucid and Java. Return Types of Java Methods | Types of Lucid Expressions
---|---
int, byte, long | int
float | float
double | double
boolean | bool
char, String | string
void | bool::true
Parameter Types Used in Lucid | Corresponding Java Types
string | String
float | float
double | double
int, dimension | int
bool | boolean
The table does not reflect the fact that JLucid is able to manipulate arrays
of values (streams), but these arrays are not Java arrays (Java’s arrays are
objects). In Objective Lucid (see Section 3.2), we also have Java object data
types will also be manipulated by a Lucid program with the Lucid part being
able to access object’s properties and methods and have them as return types
and arguments. As for now our types mapping and restrictions are as per Table
1.
#### 3.3.3 Sequential Thread and Communication Procedure Generation
##### 3.3.3.1 Java Sequential Threads
Sequential threads are imperative functions that can be called in the Lucid
part of a GIPSY program. The data elements of a Lucid program are integers and
the like. Using them as such would result in a very inefficient computation
due to the overhead in generation and propagation of demands. STs overcome
this problem. The notion of sequential thread and granularization of data was
introduced by the GLU (Granular LUcid system [JD96, JDA97].
Figure 19: Hybrid GIPSY Program Compilation Process
Each GIPSY program potentially defines several Java methods that can be called
by the Lucid part of the program. Each of these functions are coded in the
Java part of the GIPSY program; thus, a sequential thread represents by itself
a bit of work to compute split into one or more Java methods. They are
compiled (see Figure 19) to Java byte code by the compiler (GIPC, Figure 10)
and packed into one executable, along with the Communication Procedures (CP)
(see Section 3.3.3.2) needed for the communication between the generator and
worker (Section 3.3.3.4, Figure 20). The notion of worker is thus very close
to the notion of sequential threads, where a worker is basically the
aggregation of the (potentially) several sequential threads that can be
executed by a worker, along with the communications procedures needed for the
generator-worker communication.
Notice that the Generator-Worker Architecture may well be extended so that the
worker and the generator are fused into one; this is under review and is
discussed in [Lu04] and in [VP05]. This gives us distributed generators as
outlined in [Gro02b], but as yet is only a topic for discussion.
##### 3.3.3.2 Java Communication Procedures
The functional demands (i.e., demands that raise the need for a Java function
call) are potentially computed by remote workers, upon demand by the
generator. The demand is sent via the network by the generator to the worker,
along with the data representing the parameters of this Java function call.
Sending this data through the network requires the breaking of the data
structure into packets transmissible via a network. This packing of the
demand’s input data is done by the Communication Procedures, along with some
kind of remote procedure call to the worker using, for example, TCP/IP RPC.
Once the function (the sequential thread) resolves, the worker (Section
3.3.3.4) is responsible for sending back the result to the generator that
called for this demand. That is also done by the CPs.
The CPs are generated by the compiler (GIPC) using the first part of the GIPSY
program: the definition of the data structures sent over the network (i.e.,
the parameter and return types of the Java functions). The GIPC parses these
Java data structures and translates them into an abstract syntax tree. This
tree is then traversed by the CP generator, which generates byte code for the
communication procedures, following the communication protocol that was
selected. Serialization summarizes much of this and Java helps us do it.
The CP generator has to be extremely flexible, as it has to be able to
generate code that uses various kinds of communication schemes. In a nutshell,
CPs determine the way a ST should be delivered to the computing host’s worker
depending on the communication environment. For the localhost, it is plain TLP
(i.e., we create Java threads on a local machine) so
NullCommunicationProcedure (Section B.2) is used. For distributed environment
CPs wrap transport functions over Jini, DCOM+, CORBA, PVM, and RMI (see [Lu04,
VP05]) protocols. Both CP and ST interfaces are presented in Section 4.1.1.8.
##### 3.3.3.3 C Sequential Threads and Communication Procedures with the JNI
This is the methodology of how to extend the Java ST/CP generation concepts to
C (and similarly can be done for C++) with the JNI [Ste05] introduced in
Section 2.6.1.2, page 2.6.1.2. This approach was designed, but not implemented
as of this writing; however, it may serve as a good head start on the
implementation of the CCompiler in GICF.
Much of the ST wrapper class generation code for C will be similar to that of
Java. The main difference is the bodies of the sequential thread functions
will not be present in the generated class as-is, but they will be declared as
native with no Java implementation. The C code chunks will be saved to a .c
file and the corresponding .h fill will be generated declaring all the needed
prototypes with the javah tool provided with the standard distribution of the
JDK. After that, we call an external C compiler to compile the C chunks into a
shared library. Thus, the other modification to the generated wrapper class
the CCompiler has to do, is to add a static initializer with the
System.loadLibrary() call for the newly compiled library with the C
implementation of our ST(s). The generated ST class and the compiled mini-
library can be stored together (e.g. the binary library file can be loaded
into a byte array of the class and deserialized back when about to be
executed) in the imperative node and later be communicated just like Java STs.
A more sophisticated alternative is to do the compilation and dynamic loading
after communication by the engine, but this can be a next step.
As far as type matching concerned, we still can use the same mapping rules
defined in Section 3.3.2 (and subsequently the TypeMap class of the
JavaCompiler presented later on) because with the JNI with still work with
Java and the JVM can do Java-to-native type translation to C or C++ for us,
not only for primitive types, but also for arrays, objects, and strings.
##### 3.3.3.4 Worker Aggregator Definition in the Generator-Worker
Architecture
The GIPSY uses a generator-worker execution architecture as shown in Figure
20. The GEER generated by the GIPC is interpreted (or executed) by the
generator following the eductive model of computation. The low-charge ripe
sequential threads are evaluated locally by the generator. The higher-charge
ripe sequential threads are evaluated on a remote worker. The generator
consists of two systems: the Intensional Demand Propagator (IDP) and the
Intensional Value Warehouse (IVW) [Tao04]. The IDP implements the demand
generation and propagation mechanisms, and the IVW implements the warehouse. A
set of semantic rules that outlines the theoretical aspects of the distributed
demand propagation mechanism has been defined in [Paq99]. The worker simply
consists of a “Ripe Function Executor” (RFE), responsible for the computation
of the ripe sequential threads as demanded by the generator. The sequential
threads are compiled and can be either downloaded/uploaded dynamically by/to
the remote workers. Better efficiency can be achieved by using a shared
network file system.
Figure 20: Generator-Worker Architecture
An example: a GIPSY screen saver would be a sample worker running when the an
ordinary PC is going into an idle mode and normally launches ordinary dancing
bears screensavers, it can actually run our downloaded worker instead and
contribute to computation. When such a worker starts, it has to register it
within a system somehow (see [VP05]), so that the generators are aware of its
presence and can send demands to it. In the event of merging of semantics of a
worker and a generator, such a screensaver would also be able to generate
demands and maintain a local warehouse.
### 3.4 Summary
This chapter presented methodology behind concrete implementations of the
first two hybrid languages in the GIPSY – JLucid and Objective Lucid. Semantic
rules were presented for free Java functions and Java objects to be included
into the Lucid programs and evaluated by the eduction engine in the hybrid
environment. Furthermore, operational semantics of Objective Lucid is clearly
defined and is compatible with the semantics of Lucid. The general
requirements for the GICF, a tool simplifying imperative compiler management
within GIPC, are introduced. The follow up chapter details the architectural
and detailed designs and concrete implementation of the languages as well as
General Intensional Compiler Framework and overall module integration and
their interfaces. Some immediate benefits and limitations are outlined below.
#### 3.4.1 Benefits
* •
JLucid opens the door for STs and CPs and first hybrid programming paradigm in
the GIPSY.
* •
JLucid provides ability to either write Java code alongside the Lucid code or
embed existing one via embed().
* •
Objective Lucid introduces Java objects and their semantics in the GIPSY.
* •
GICF generalizes the embed() mechanism to all languages in the GIPSY.
* •
GICF promotes general type handling in the GIPSY.
* •
GICF promotes general compiler handling in the GIPSY.
* •
GICF generalizes the notion of the STs and CPs for all compilers.
#### 3.4.2 Limitations
* •
JLucid is limited only to GIPL-Java and Indexical Lucid-Java hybrids.
* •
JLucid does not allow Java objects.
* •
JLucid restricts the embed() mechanism only to itself and its derivative –
Objective Lucid.
* •
Objective Lucid is primarily an experimental language to research on Java
objects in the intensional environment.
* •
GICF addresses mostly the imperative compilers, but a similar approach can be
applied to the intensional and functional ones.
## Chapter 4 Design and Implementation
This chapter combines the architectural and detailed designs and integration
of the modules contributed not only by the author of this thesis but also by
the other GIPSY team members. Section 4.1 explores the GIPSY architecture and
implementation of the major components and frameworks. Then, Section 4.2
focuses on the user interface and external library interfaces. User
interfaces, class and sequence diagrams are provided mostly following the top-
down approach. For GIPSY Java packages, directory structure with description
of each package, and .jar file packaging please refer to Appendix C.
### 4.1 Internal Design
The GIPC framework redesign along with the realization of the two children
frameworks of GICF and IPLCF are presented first followed by the design and
implementation of JLucid and Objective Lucid integrated into the new
frameworks.
#### 4.1.1 General Intensional Programming Compiler Framework
The GIPC Framework experienced several iterations of refinements as a result
of this research. Two new frameworks emerged, namely General Imperative
Compiler Framework (GICF) to handle all imperative languages within the GIPSY
and, its counterpart Intensional Programming Languages Compiler Framework
(IPLCF).
##### 4.1.1.1 General Imperative Compiler Framework
GLU [JDA97, JD96], JLucid [MPG05], and later Objective Lucid [MP05b] prompted
the development of a General Imperative Compiler Framework (GICF). The
framework targets integration (embedding of) different imperative languages
into GIPSY (see [RG05a]) programs for portability and extensibility reasons.
GLU promoted C and Fortran functions within; JLucid/Objective Lucid promote
embedded Java. Since GIPSY targets to unite all intensional paradigms in one
research system, we try to be as general as possible and as compatible as
possible and pragmatic at the same time.
For example, if we want to be able to run GLU programs with minimum (if at
all) modifications to the code base, GIPSY has to be extended somehow to
support C- or Fortran-functions just like it does for Java. What if later on
we would need to add C++, Perl, Python, shell scripts, or some other language
for example? The need for a general “pluggable” framework arises to add
imperative code segments within a GIPSY program. We could go even support
multi-segment multi-language (with multiplicity of 3 or more languages) GIPSY
programs. Two examples are presented in Figure 1 and in Figure 2.
#funcdecl
Integer f();
void gee();
void z();
#typedecl
Integer;
#JAVA
Integer f()
{
return new Integer("123");
}
#CPP
#include <iostream>
void gee()
{
cout << "gee" << endl;
}
#PERL
sub z()
{
while(<STDIN>)
{
s/\n//;
print;
}
}
#OBJECTIVELUCID
A @.d 5
where
dimension d;
A = B fby.d (A - 1);
B = C fby.d (B + f().intValue());
C = z() && gee();
end;
Figure 1: Example of a hybrid GIPSY program.
/**
* Language-mix GIPSY program.
*
* $Id: language-mix.ipl,v 1.5 2005/04/25 00:16:30 mokhov Exp $
* $Revision: 1.5 $
* $Date: 2005/04/25 00:16:30 $
*
* @author Serguei Mokhov
*/
#typedecl
myclass;
#funcdecl
myclass foo(int,double);
float bar(int,int):"ftp://newton.cs.concordia.ca/cool.class":baz;
int f1();
#JAVA
myclass foo(int a, double b)
{
return new myclass(new Integer((int)(b + a)));
}
class myclass
{
public myclass(Integer a)
{
System.out.println(a);
}
}
#CPP
#include <iostream>
int f1(void)
{
cout << "hello";
return 0;
}
#OBJECTIVELUCID
A + bar(B, C)
where
A = foo(B, C).intValue();
B = f1();
C = 2.0;
end;
/*
* in theory we could write more than one intensional chunk,
* then those chunks would evaluate as separate possibly
* totally independent expressions in parallel that happened
* to use the same set of imperative functions.
*/
// EOF
Figure 2: Another example of a hybrid GIPSY program.
##### 4.1.1.2 Generalization of a Concrete Implementation
Thus, the JavaCompiler component (see Figure 19), part of GIPC, has to be
generalized, and the JavaCompiler itself be a concrete implementation of this
generalization. The generalization would express itself by having an abstract
class ImperativeCompiler, the generic Preprocessor (vs. JLucidPreprocessor in
Section 4.1.2) should be able to cope with all PLs and know what PLs are
supported through enumerating them. Another thing the GICF buys us is an
ability to have any supported imperative programming language embedded in any
supported intensional programming language. Though this may seem impractical
at the first glance, but the framework is designed such that a lot of syntax,
semantics, and type mapping work is performed by the individual concrete
compiler implementations and not by the generic machinery. The goal here is
that as long as any given compiler within the framework conforms to the
designed interface specification and produces the required data structures,
there should be least possible effort to enable such a compiler in GIPSY.
Thus, the compilation process, semantic checks, linking, and execution at the
meta level of implementation of the GIPC and GEE can be reasonably generalized
without loss of practicality as we shall see. With this great deal of
flexibility, we have several issues:
* •
Binary portability of compiled languages, such as C/C++ on a different host
(this problem theoretically does not exist for Java).
* •
Though some languages, such as Perl, Python, shell scripts, are interpreted, a
version mismatch may happen.
* •
A compiler for interpreted languages other than Java would be rather simple
because should we want to pass the ST code to a remote host, all we need is to
pass the source itself. Of course, in both compiled and interpreted variant
there is a large potential of security vulnerability exploits (e.g. with
malicious code injection), which will have to be dealt with as a part of the
future work. As of this writing, there are no embedded checks in GIPSY for
that; instead a guide of a sandboxed installation of GIPSY will be provided
when the system is released.
* •
Another important issue is having imperative PL nodes in the AST. The issue is
in what such nodes should contain in order for them to be linked back into the
main AST, how to perform semantic analysis of the hybrid code based on the
contents of such nodes, and GEE should go about executing this code.
* •
Various languages define their own set of types and typing rules, gluing them
all together is a very difficult task for semantic analysis and type
inference.
The follow up sections clarify and address most of these issues.
##### 4.1.1.3 Resolving Generalization Issues and Binary Compatibility
In order to fully support GICF, the original GIPC framework in Figure 3
(discussed in detail by Wu and Paquet in [PGW04]) has to be altered in the
following way: the Preprocessor has to be added on top of all the front-end
modules, and new links drawn between the Preprocessor and the other modules
Figure 4. This also changes the data structures flow between the components.
For the unaware reader, what follows is the brief description of the layers,
components, and abbreviations of the conceptual design present in Figure 4:
The front-end and back-end layers are the two bottom ones represent the main
machinery of the GIPC. The front-end compilers and parsers are responsible for
parsing, producing initial syntax trees, STs, and CPs. At this layer, the main
abstract syntax tree AST is always compliant to the one of Generic Intensional
Programming Language (GIPL). If the source code program was written in some
specific intensional programming language (SIPL, e.g. Indexical Lucid or
Tensor Lucid), its AST has to be translated first into GIPL. Both, GIPL and
SIPL type components may translate a Lucid dialect source code into a data
flow (DFG) graph language and back; hence, there is a variety of the DFG
translators. Next, the other two types of conceptual components at the front-
end layer are the data type (DT) and the sequential thread (ST) front-ends.
These correspond to the imperative language compilers and their modules in the
implementation. The DT front-end is responsible for analyzing data-type
definitions in the ST code and producing native (i.e. compiled) representation
of communication procedures (NPCs). The ST front-end is responsible for
compilation an ST code and producing some equivalent of the native compiled
code (NST) as the end result.
The GIPC back-end layer performs finalization of a GIPSY program compilation
by doing semantic analysis and eliminating Lucid functions and producing the
demand AST (DAST) along with linking in the generated STs and CPs from the
imperative side. The GEER generator then produces the final linked version of
a GIPSY program as a resource usable by the GEE (GEER).
The first two layers are meta-level layers that prepare information for the
front-end and back-end layers. The second layer is the GIPC Preprocessor layer
discussed in depth through the rest of this chapter. The top level has to do
with some language specification processing and creating corresponding parsers
and data structures for the front-end layer. SIPL and GIPL front-end
generators have to do with the fact that our SIPL and GIPL parsers are
generated out of a source grammar specification by javacc. Thus, a GIPL
specification corresponds to the GIPL grammar in the GIPL.jjt file and the
GIPL spec processor is the javacc tool. The DT and ST front-end generators
exist for the same idea as the GIPL and SIPL ones do. However, in the current
implementation they are not present either because they are hand-written or we
rely on the external compiler tools (e.g. javac to compile Java STs) to do the
processing for us. The design however implies that these components may
eventually be converted to the genuine imperative compilers within GIPSY
giving greater control and flexibility over the imperative parts than relying
on external tools. Therefore, we may acquire a Java.jjt one day, for example,
and generate a Java parser out of it.
Figure 3: Original Framework for the General Intensional Programming Compiler
in the GIPSY Figure 4: Modified Framework for the General Intensional
Programming Compiler in the GIPSY
###### Format Tag
To address some binary compatibility issues we invent a notion of a format tag
attached to the STs and CPs. The format tag’s purpose is to include meta-
information about STs and CPs such that it includes the programming language,
the object code format, the operating system, compiler, and their versions.
This is important if we are sending platform-dependent compiled code, such as
that of C or C++ from one host to another with different architectural
platforms. The FormatTag API is in Figure 5.
Figure 5: The FormatTag API.
We implement format specifications as a hashtable. We also predefine some
common format tags, such as JAVA, for conveniences as most frequently used.
The class overrides toString() and equals() of Object to define that the two
format tags are only equal if the string representation of all their
specifications are identical.
###### Sending Source Code Text
Not all non-intensional languages require compilation, e.g. Perl, Python, etc.
These can be sent over as plain source code text; thus, the format tag will
indicate the fact. We can go even further with this and send any language as
plain text and compile it on the target host instead prior invocation. For the
task of the source code inclusion we reserved the
SequentialThreadSourceGenerator. Of course, this won’t work for embed-included
binary code via a URI parameter because that code was already compiled by
someone else on some specific platform. As far as current implementation
concerned, the generated ST class does always contain the source code of STs
from the GIPSY program code segments, but it is unused by the GEE except for
debugging as of this writing.
###### Dictionary
The Preprocessor’s dictionary will initially be constructed based on the
#funcdecl and #typedecl program segments. The dictionary will serve as an
input to three other components: the NST generator (for error reporting and
pointers to the nodes in the AST and the compiled code), to the NCP generator
(to analyze the data structures used by STs and generate CPs accordingly), and
to the semantic analyzer, to perform data type matching between the
intensional and imperative parts. Both NCP and NST generators work under the
command of some imperative language compiler and are referred to as
SequentialThreadGenerator and CommunicationProcedureGenerator in their most
general forms, which are subclassed by a concrete language implementation.
##### 4.1.1.4 GIPC Preprocessor
The Preprocessor is something that is invoked first by the GIPC on incoming
GIPSY program’s source code stream. The Preprocessor’s job is to do
preliminary program analysis, processing, and splitting into chunks. Since a
GIPSY program is a hybrid program consisting of different languages in one
source file, there ought to be an interface between all these chunks. Thus,
the Preprocessor after initial parsing and producing the initial parse tree,
constructs a preliminary dictionary of symbols used throughout the program.
This is important for type matching and semantic analysis later on. The
Preprocessor then splits the code segments of the GIPSY program into chunks
preparing them to be fed to the respective concrete compilers for those
chunks. The chunks are represented through the CodeSegment class that the GIPC
collects. The corresponding class diagram of is in Figure 6.
Figure 6: The GIPC Preprocessor.
The Preprocessor can also be told to report certain code segments are invalid
at the preprocessing stage rather delaying the error until the compiler
discovery stage through the addInvalidSegmentName() and addValidSegmentName()
methods and maintaining internal vector of the strings with invalid segment
names. This feature is for example used in Preprocessor’s extensions of
JLucidPreprocessor and ObjectiveLucidPreprocessor later on that filter out
code segments that do not belong to the languages. The filtering logic works
like this:
* •
if no valid and invalid segments are specified, all segments are accepted as
valid at the preprocessing stage. This is the default for general GIPC work.
* •
if some invalid and no valid segments are specified, the Preprocessor will
error out on the invalid segments
* •
if only valid segments are specified, everything else will be treated as
invalid
* •
if both valid and invalid segments are present; the invalid set segments are
ignored and everything that it is not mentioned in the valid set is said to be
invalid.
###### GIPSY Program Segments
Here we define four basic types of segments to be used in a GIPSY program.
These are:
* •
#funcdecl program segment declares function prototypes of imperative-language
functions defined later or externally from this program to be used by the
intensional language part. These prototypes are syntactically universal for
all GIPSY programs and need not resemble the actual function definitions they
describe in their particular programming language.
* •
#typedecl segment lists all user-defined data types that can potentially be
used by the intensional part; usually objects. These are the types that do not
appear in the matching table in Table 1.
* •
#$<$IMPERATIVELANG$>$ segment declares that this is a code segment written in
whatever IMPERATIVELANG may be, for example #JAVA for Java, #CPP for C++,
#PERL for Perl, #PYTHON for Python, etc.
* •
#$<$INTENSIONALLANG$>$ segment declares that this is a code segment written in
whatever INTENSIONALLANG may be, for example #GIPL, #INDEXICALLUCID, #JLUCID,
#OBJECTIVELUCID, #TENSORLUCID, #ONYX111See [Gro04] for details on the Onyx
language., etc. as understood by the GIPSY.
###### Preprocessor Grammar
The initial grammar for the Preprocessor to be able to parse a GIPSY program
is shown in Figure 7. After having parsed a program, we have a Preprocessor
AST (PAST) that will be used further by the compilation process in the GIPC
and its submodules. The grammar and the framework were designed in such a way
so all the previous neat features of JLucid [MP05b]/Objective Lucid [MP05b]
still be present, such as embed() and are accessible to other dialects. In the
GICF, we generalize our function prototype declaration to be able to include
external code of any imperative language.
$\mathtt{<\\!\\!GIPSY\\!\\!>}$ | ::= | $\mathtt{<\\!\\!DECLARATIONS\\!\\!>}$ $\mathtt{<\\!\\!CODESEGMENTS\\!\\!>}$
---|---|---
$\mathtt{<\\!\\!DECLARATIONS\\!\\!>}$ | ::= | $\mathtt{<\\!\\!FUNCDECLS\\!\\!>}$ $\mathtt{<\\!\\!DECLARATIONS\\!\\!>}$
| $|$ | $\mathtt{<\\!\\!TYPEDECLS\\!\\!>}$ $\mathtt{<\\!\\!DECLARATIONS\\!\\!>}$
| $|$ | $\epsilon$
$\mathtt{<\\!\\!FUNCDECLS\\!\\!>}$ | ::= | #funcdecl $\mathtt{<\\!\\!PROTOTYPES\\!\\!>}$
$\mathtt{<\\!\\!TYPEDECLS\\!\\!>}$ | ::= | #typedecl $\mathtt{<\\!\\!TYPES\\!\\!>}$
$\mathtt{<\\!\\!PROTOTYPES\\!\\!>}$ | ::= | $\mathtt{<\\!\\!PROTOTYPE\\!\\!>}$ ; $\mathtt{<\\!\\!PROTOTYPES\\!\\!>}$
| $|$ | $\epsilon$
$\mathtt{<\\!\\!PROTOTYPE\\!\\!>}$ | ::= | $\mathtt{<\\!\\!PSTART\\!\\!>}$ $\mathtt{<\\!\\!EMBED\\!\\!>}$
$\mathtt{<\\!\\!PSTART\\!\\!>}$ | ::= | [ immutable ] $\mathtt{<\\!\\!TYPE\\!\\!>}$ [ [] ] $\mathtt{<\\!\\!ID\\!\\!>}$ ( $\mathtt{<\\!\\!TYPELIST\\!\\!>}$ )
$\mathtt{<\\!\\!EMBED\\!\\!>}$ | ::= | $\epsilon$
| $|$ | : $\mathtt{<\\!\\!LANGID\\!\\!>}$ : $\mathtt{<\\!\\!URI\\!\\!>}$
| $|$ | : $\mathtt{<\\!\\!LANGID\\!\\!>}$ : $\mathtt{<\\!\\!URI\\!\\!>}$ : $\mathtt{<\\!\\!ID\\!\\!>}$
$\mathtt{<\\!\\!TYPES\\!\\!>}$ | ::= | $\mathtt{<\\!\\!TYPE\\!\\!>}$ ; $\mathtt{<\\!\\!TYPES\\!\\!>}$
| $|$ | $\epsilon$
$\mathtt{<\\!\\!TYPELIST\\!\\!>}$ | ::= | $\mathtt{<\\!\\!TYPE\\!\\!>}$ [ [] ]
| $|$ | $\mathtt{<\\!\\!TYPE\\!\\!>}$ [ [] ] , $\mathtt{<\\!\\!TYPELIST\\!\\!>}$
| $|$ | $\epsilon$
$\mathtt{<\\!\\!CODESEGMENT\\!\\!>}$ | ::= | $\mathtt{<\\!\\!LANGDATA\\!\\!>}$ $\mathtt{<\\!\\!LANGID\\!\\!>}$
| $|$ | $\mathtt{<\\!\\!LANGDATA\\!\\!>}$ $\mathtt{<\\!\\!EOF\\!\\!>}$
$\mathtt{<\\!\\!CODESEGMENTS\\!\\!>}$ | ::= | $\mathtt{<\\!\\!CODESEGMENT\\!\\!>}$ $\mathtt{<\\!\\!CODESEGMENTS\\!\\!>}$
| $|$ | $\epsilon$
$\mathtt{<\\!\\!URI\\!\\!>}$ | ::= | $\mathtt{<\\!\\!CHARACTERLITERAL\\!\\!>}$
| $|$ | $\mathtt{<\\!\\!STRINGLITERAL\\!\\!>}$
$\mathtt{<\\!\\!ID\\!\\!>}$ | ::= | $\mathtt{<\\!\\!LETTER\\!\\!>}$ ($\mathtt{<\\!\\!LETTER\\!\\!>}$ $|$ $\mathtt{<\\!\\!DIGIT\\!\\!>}$)*
$\mathtt{<\\!\\!LANGID\\!\\!>}$ | ::= | #$\mathtt{<\\!\\!CAPLETTER\\!\\!>}$ ($\mathtt{<\\!\\!CAPLETTER\\!\\!>}$)*
$\mathtt{<\\!\\!TYPE\\!\\!>}$ | ::= | $\mathtt{<\\!\\!ID\\!\\!>}$
| $|$ | int
| $|$ | double
| $|$ | bool
| $|$ | float
| $|$ | char
| $|$ | string
| $|$ | void
Figure 7: Preprocessor Grammar for a GIPSY program.
The lexical elements, such as LETTER, LANGDATA, DIGIT, CAPLETTER, and
*LITERALs are not listed for brevity as they are merely standard and self-
explanatory lexical tokens except probably LANGDATA – this is character data
allowing any character sequence within except LANGID that serves as a
terminator of a code segment chunk.
Notice, the grammar is not bound to our current set of supported intensional
and imperative languages. Rather, the GIPC attempts to look up appropriate
compiler for each code segment automagically using LANGID for mapping at run-
time. The JavaCC version of the grammar can be found the
PreprocessorParser.jjt file.
The grammar has been amended from what was published in [MP05a] to include
LANGID in the EMBED production, the immutable keyword and arrays subscript
operator [] in the PSTART production. LANGID in EMBED is needed to be able to
pick the appropriate compiler for the included code as it may be written in
any imperative language. The immutable keyword is needed to allow a programmer
to assert that certain STs are immutable meaning given the same parameters
they always return the same result, and, therefore, their result can be safely
cached in the warehouse as such functions are declared side-effects free (e.g.
as the get42() method in Figure 9, page 9 can be marked as immutable). This
marking of methods will allow more efficient caching of the ST results of STs
known not to have side effects and has to be explicitly set by the programmer.
If the programmer by mistake marks a method with side effects as immutable,
then a program may exhibit erroneous execution at run-time by returning a
possibly incorrect value from the warehouse. There is no way to automatically
discover immutability of STs in GIPSY at this time (it may only be possible
when genuine imperative compilers are implemented). The array subscript
operator [] has been added to PSTART and TYPELIST productions to allow GIPSY
arrays (as a generalization of JLucid arrays) that are composed of the
elements of GIPSY types. The concrete imperative compilers implementing the
mapping (if possible) will have to do appropriate conversions from the native
arrays to GIPSY arrays.
##### 4.1.1.5 GIPSY Type System
While the main language of GIPSY, Lucid, is polymorphic and does not have
explicit types, co-existing with other languages necessitates definition of
GIPSY types and their mapping to a particular language being embedded. Figure
8 presents the design aspects of the GIPSY Type System.
Figure 8: GIPSY Type System.
Each class is prefixed with GIPSY to avoid possible confusion with similar
definitions in the java.lang package. The GIPSYVoid type always evaluates to
the Boolean true, as described earlier in Section 3.3.2. The other types wrap
around the corresponding Java object wrapper classes for the primitive types,
such as Integer, Float, etc. Every class keeps a lexeme (a lexical
representation) of the corresponding type in a GIPSY program and overrides
toString() to show the lexeme and the contained value. These types are
extensively used by the Preprocessor, imperative and intensional (for
constants) compilers, the SequentialThreadGenerator,
CommunicationProcedureGenerator, SemanticAnalyzer for the general type of
GIPSY program processing, and by the GEE Executor.
The other special types that have been created are either experimental or do
not correspond to a wrapper of a primitive type. GIPSYIdentifier type case
corresponds to a declaration of some sort of an identifier in a GIPSY program
to be put into the dictionary, be it a variable or a function name with the
reference to their definition. This is an experimental type and may be removed
in the future. Constants and conditionals may be anonymous and thereby not
have a corresponding identifier. GIPSYEmbed is another special transitional
type that encapsulates embedded code via the URL parameter and later is
exploded into multiple types corresponding to STs and their CPs. GIPSYFunction
and its descendant GIPSYOperator correspond to the function types for regular
operators and user defined functions. A GIPSYFunction can either encapsulate
an ordinary Lucid function (as in functional programming an which is
immutable) or an ST function (e.g. a Java method), which may easily be
volatile (i.e. with side effects). These four types are not directly exposed
to a GIPSY programmer and at this point are managed internally. The rest of
the type system is exposed to the GIPSY programmer in the preamble of a GIPSY
program, i.e., the #funcdecl and #typedecl segments, which result in the
embryo of the dictionary for linking, semantic analysis, and execution. Once
ST compilers return, the type data structures (return and parameter types)
declared in the preamble are matched against what was discovered by the
compilers and if the match is successful, the link is made.
##### 4.1.1.6 GICF Design
The GICF is the first generalization framework of hybrid programming in the
GIPSY. Implementation-wise, only Java is implemented as an imperative language
with an external compiler. However, provision was made for C/C++, Perl,
Fortran and Python with stub compilers. The class diagram describing GICF is
shown in Figure 9. On this diagram the interaction between a given imperative
compiler and the SequentialThreadGenerator and CommunicationProcedureGenerator
only shown for JavaCompiler to keep the clearer picture, but the same kind of
association will have to be maintained for all imperative compilers as the
IImperativeCompiler interface mandates. The EImperativeLanguages is a Java
interface enumerating all available imperative language compilers. It is used
by the GIPC to discover a given compiler for a language dynamically. As of
this writing, the enumeration is maintained by hand; however, it is planned to
be generated in the near future with a command-line-driven script or a RIPE
GUI automagically to facilitate addition of new languages.
Figure 9: GICF Design.
##### 4.1.1.7 Intensional Programming Languages Compiler Framework
As a consequence of GICF, a similar approach was applied to the intensional
compilers in the form of IPLCF. See the corresponding class diagram in Figure
10. The IIntensionalCompiler was designed and implemented by all the
intensional compilers we have. An enumeration EIntensionalLanguages of all
supported intensional languages was created, so the GIPC can pick needed
compiler at run-time as determined by the Preprocessor.
Figure 10: IPLCF Design. Figure 11: SIPL to GIPL Translator Integration.
Translation for all intensional compilers is done through the generic
Translator implemented by Aihua Wu in [Wu02]. The Translator has been
integrated into the GIPC.intensional.GenericTranslator package and split and
renamed as in Figure 11. Thus, every SIPL compiler refers to this translator
to acquire a GIPL AST at the end via generic implementation of
IntensionalCompiler.translate(). The Translator was refactored and augmented
to understand GIPSY Types (see Section 4.1.1.5) and ImperativeNode for
imperative languages. The TranslationParser and TranslationLexer collaborate
to compile intensional language translation rules (e.g. IndexicalLucid.rul)
files provided by each SIPL author.
##### 4.1.1.8 Sequential Thread and Communication Procedure Interfaces
This section details Sequential Thread and Communication Procedure interfaces.
The related class diagram is in Figure 12. The ICommunicationProcedure and
ISequentualThread are the core interfaces. Both extend Serializable in order
for us to be able to dump their concrete implementations to disk or
distributed storage using Java’s object serialization machinery. This is
needed for the GIPSYProgram container to be saved to disk or for an ST to be
able to reside in JavaSpaces [Mam05] implementation of the demand space
[VP05]. The ISequentialThread also extends Runnable to be true thread when
materialized, especially for the case of local execution. The Runnable
interface makes it possible for an implementing class to become a thread in
multithreaded environment in Java. The ICommunicationProceduresEnum is an
enumeration of all known to the GIPSY communication procedure types. The
NullCommunicationProcedure and RMICommunicationProcedure represent concrete
implementations for local threaded processing as well as RMI. Therefore, the
SequentialThreadGenerator is an abstract factory for all sequential threads
that has to be overridden by a language-specific sequential thread generator,
e.g. such as JavaSequentialThreadGenerator. Likewise,
CommunicationProcedureGenerator is a factory for CPs. The WorkResult class
represents the result of (computation) work done, which is also has to be
Serializable. Upon various communication needs the CommunicationStats is
returned by the ICommunicationProcedure API or the CommunicationException is
thrown indicating an error. The Worker class represents a collection of STs
and CPs being executed.
Figure 12: Sequential Thread and Communication Procedure Class Diagram.
##### 4.1.1.9 GIPC Design
In Figure 14 there is a hierarchy that all imperative and intensional
compilers should adhere to. The IImperativeCompiler interface is something
every imperative compiler implements to ease up the job of GIPC. A similar
interface has been invented for intensional languages – IIntensionalCompiler
for consistency.
A set of interfaces has been designed for all the present and future compilers
to implement. There are three interfaces so far:
1. 1.
ICompiler is a superinterface for all compiler interfaces. It is implemented
by GIPC itself and by DFGAnalyzer, as shown in Figure 13.
2. 2.
IIntensionalCompiler is a subinterface of ICompiler designated to
differentiate intensional compilers. It is implemented in part by the
IntensionalCompiler abstract class that most (for now all) intensional
compilers implement.
3. 3.
IImperativeCompiler is a counterpart of IIntensionalCompiler. Its purpose is
similar to that of IIntensionalCompiler for imperative languages.
Figure 13: All GIPC Compilers.
The core difference between IIntensionalCompiler and IImperativeCompiler
versus the general ICompiler is that most (except for GIPL) of the intensional
compilers have to perform SIPL-to-GIPL translation; hence, the translate()
method, and all imperative compilers must produce communication procedures and
sequential threads as the result of their work; hence,
generateSequentialThreads() and generateCommunicationProcedures() methods are
provided. The abstract classes IntesionalCompiler and ImperativeCompiler
provide the most common possible implementation for all intensional and
imperative compilers respectively, so the underlying concrete compilers only
have to override some parts specific to the language they are to compile. If
extension of these classes is not possible for some reason (e.g. when writing
external GIPSY plugins when a compiler class already inherits from some other
class), they must implement their corresponding interface. Out of the concrete
classes on the diagram the author of this thesis fully implemented GIPC,
GIPLCompiler, IndexicalLucidCompiler, JLucidCompiler, ObjectiveLucidCompiler,
and JavaCompiler. The DFGAnalyzer of Yimin Ding was made to implement
ICompiler as it in fact compiles the “DFG code” out of GIPL or Indexical
Lucid.
Figure 14: Overall GIPC Design.
The overall design and integration of the GIPC participants is illustrated in
Figure 14. The GIPC class is the main compiler application that drives the
compilation process, so in the general case in invokes the Preprocessor,
intensional and imperative compilers required, the SemanticAnalyzer,
IdentifierContextCodeGenerator, Translator, and the GEERGenerator linker. It
also acts like a facade to other GIPSY modules. The major data structures,
such as AbstractSyntaxTree, Dictionary, CodeSegment, FormatTag,
ImperativeNode, and SimpleNode are created, accessed, or modified throughout
the modules during the compilation process. Out of imperative languages only
JavaCompiler is mentioned as it is the most advanced in this category. The
JLucidCompiler’s JLucidParser underneath invokes both JGIPLParser and
JIndexicalLucidParser as JLucid Section 3.1 provides extensions to both of
these languages. A number of association links have been removed from the
diagram to maintain clarity as these links are intuitive or present in detail
diagrams.
##### 4.1.1.10 GIPC Class as a Meta Processor
The GIPC (a concrete class) acts here as so-called “meta processor” that
drives the entire compilation process and invokes appropriate submodules in
order to come up with a compiled version of a GIPSY program. This involves
calling the Preprocessor, then feeding its output to whatever concrete
compilers for the code segments of the GIPSY program, collecting the output of
them (various ASTs, dictionaries), performing semantic analysis, and linking
all the parts back together in a binary form. This portable binary version of
the GIPSY program is to either be serialized as an executable file for later
execution by the GEE or optionally to be fed directly to the GEE.
##### 4.1.1.11 Calling Sequence
The sequence diagram in Figure 15 illustrates the entire compilation process
and the data structures passed between the modules. This is the roundtrip
description of the implementation efforts. The two followup diagrams detail
the differences in the compilation process between the imperative and
intensional languages. The general compilation process begins by reading the
source GIPSY program and converting it into a meta token stream of types,
declarations, and code segments by the Preprocessor. The Preprocessor takes
that input and with its own parser produces a preprocessor AST and an embryo
of a dictionary with the identifiers and types declared in the imperative code
segments for further semantic linking. The latter is used to produce
imperative stubs for cross-segment type checks. The former contains primarily
code segments written in various languages. The GIPC takes these code segments
and creates appropriate compiler threads, one for each code segment. Then,
each compiler tries to compile its own chunk and produces a portion of a main
AST. Since we treat the IPL part as a main program, its AST is considered to
be the main skeleton tree. The ASTs produced by the imperative compilers
(which really contain a single ImperativeNode) are secondary and should be
merged into the main when appropriate. Once all the compiler threads are
successfully done, the GIPC collects all the ASTs and performs linking via the
GEERGenerator. The combined AST is now a subject to the semantic analysis and
the function elimination. Once semantic analysis is complete, the final post-
linking is performed where all the pieces of the GIPSYProgram are combined
together and its instance is serialized to disk. Optionally, right after
compilation the GEE may be invoked to start the execution of the just compiled
program.
Figure 15: Sequence Diagram of GIPSY Program Compilation Process.
There is no any preference made in GIPC on the number and the order of
intensional and imperative compilers executed. This may result in several main
intensional programs (if the source code contained more than one intensional
code segment) or unused imperative nodes (an imperative segment is declared
but the code from it is unused). For the former we maintain an array of ASTs
in the GIPSYProgram, so that when the actual program is executed, the same
number of the GEE Executor threads are started and all main ASTs are evaluated
in parallel providing the result set of a computation instead of a single
result. Detailed sequence diagrams of the intensional and imperative
compilation processes are in Figure 16 and Figure 17 to illustrate the
differences in compiling intensional and imperative code segments.
Figure 16: Sequence Diagram of Intensional Compilation Process. Figure 17:
Sequence Diagram of Imperative Compilation Process.
##### 4.1.1.12 Compiling and Linking
###### Multiple Intensional Parts
In a GIPSY program we may possibly have multiple intensional parts. For
example, if a GIPSY programmer gave a GIPL expression, an Indexical Lucid
expression and a couple of Java procedures in the same source GIPSY program,
what is the meaning of that setup would be? In this case, we can say that we
evaluate two independent intensional expressions in parallel that happened to
share the same imperative part. Thus, for such a GIPSY program there will be
two instances of GEE running. The GEE is to extended to accept a forest of
ASTs to be processed in parallel.
###### Imperative Stubs
When the Preprocessor completes its job, it has to create some stubs in the
intensional parts of the program for the symbols declared outside of those
parts (e.g. Java functions) so that the appropriate intensional compiler does
not complain about undefined symbols when producing the AST because the
intensional compilers are not aware of anything outside their work scope.
Later on, the corresponding stub nodes in the AST are found and replaced with
the real contents at the linking stage.
###### NCP Generator as a Type Processor
The NCP generator will act very much like a type processor and will have to
look inside the imperative code segments analyzed/compiled by the ST
generator. This kind of type processing is needed to decide on communication
procedures (CPs) to be generated for that ST. It issues warnings if the
compiled version of the data structures to be sent is not portable. The role
of the NCP generators in the GIPSY implementation is played by the imperative
compilers, such as JavaCompiler.
###### GEER Generator as a Linker
The GEER Generator (see GEERGenerator in Figure 19) in the backend acts like a
linker of all parts of a GIPSY program. It gathers all the resources from the
compiler set, such as ASTs, ICs, CPs, STs, and the dictionary. Then, it
replaces the stubs in the intensional part with the nodes from the imperative
ASTs (STs accompanied with their respective CPs) forming a complete composite
AST ready for consumption by the GEE. All this will be serialized as a
GIPSYProgram class instance. The GEERGenerator is invoked two times – first
prior SemanticAnalyzer to assemble a complete AST, and then after semantic
analysis and function elimination to set up the finalized dictionary and
program name.
##### 4.1.1.13 Semantic Analyzer
Figure 18: Semantic Analyzer.
The semantic analyzer detailed design diagram is shown in Figure 18.
Originally implemented by Aihua Wu, the class was renamed from Semantic [Wu02]
to a more complete name of SemanticAnalyzer and placed under the GIPC package.
Relevant changes include integration of storage.Dictionary (previously was
java.util.Vector), storage.DictionaryItem (formerly Item_in_Dict [Wu02]),
storage.FunctionItem (formerly Fun_Item [Wu02], serves for function
description). The SemanticAnalyzer had to be taught to recognize new GIPSY
types (see Section 4.1.1.5) with base GIPSYType class for object, embed, and
array processing, ImperativeNode for sequential threads and communication
procedures, and a general AbstractSyntaxTree.
##### 4.1.1.14 Interfacing GIPC and GEE and Compiled GIPSY Program
Now, let us formally define the notion of a stored compiled GIPSY program, as
a GEER or the interface between the two major modules - GIPC and GEE. Until
this point, the GEE accepted from GIPC as the input AST of an intensional part
and a dictionary of symbols. This suggests having serialized the AST and the
dictionary. With the invent of JLucid, communication procedures (CPs) and
sequential threads (STs) became relevant and should belong to the GIPC-GEE
interface. Thus, a compiled GIPSY program may have several of CPs and STs
serialized along. While STs and CPs are present within imperative AST nodes,
references to them are recorded here for quicker access and decision making by
the GEE. Then, as GEE produces demands (especially over RMI or Jini, [VP05])
for each intensional identifier in the dictionary an Identifier Context (IC)
class created [LGP03, Lu04]. This is needed because every such identifier
represents a Lucid expression to be evaluated by the engine, and as such
should also be part of the compiled GIPSY program. The corresponding class
diagram is in Figure 19. It includes the GIPSYProgram and all its associations
with GIPC, GEE, GEERGenerator, and the storage classes.
Figure 19: Class diagram describing GIPSYProgram.
To summarize, the GIPC-GEE interface is the GIPSYProgram representing
encapsulation of the five parts:
1. 1.
Linked AST(s)
2. 2.
Dictionary
3. 3.
A set of STs
4. 4.
A set of CPs
5. 5.
A set of ICs.
On the diagram in Figure 4 GIPSYProgram defines and corresponds to the GEER.
#### 4.1.2 JLucid
##### 4.1.2.1 Design
Figure 20: JLucid Design.
The class diagram describing JLucid is shown in Figure 20. The implementation
of JLucid parser-wise is heavily dependent on that of Indexical Lucid as the
largest chunk of the IPL work is the same. JLucid adds a preprocessor
JLucidPreprocessor class that is responsible for parsing initial source JLucid
program and extract Java and Lucid parts. The JLucidParser class is the one
that manipulates javacc-generated parsers amended to support embed() and
arrays. The sequence diagram describing the details of the compilation
sequence of JLucid is presented in Figure 21.
Figure 21: JLucid Compilation Sequence.
JLucid implements generation of Java sequential threads (STs) and their
communication procedures (CPs); thus, necessitating
JavaSequentialThreadGenerator and JavaCommunicationGenerator. For uniformity,
portability, and testing reasons, we also decided to send the source code
over, that can possibly be compiled on the remote machine. All this is done by
the GICF-integrated JavaCompiler, see Section 4.1.2.3.
##### 4.1.2.2 Grammar Generation
As it was shown in Chapter 3, the JLucid syntax extension to GIPL and
Indexical Lucid is minimal. The JavaCC grammars we use, are stored in the .jjt
files for the original two dialects. If we decide to have very similar grammar
files for JLucid to support JLucid extensions (arrays and embed()), then if
the original grammar has a bug, the fix will have to be propagated to all the
derived grammars, which will not scale from the maintenance point of view as
there will be similar small modifications from Objective Lucid and other
dialects. Thus, it was decided to only maintain the original grammars of GIPL
and Indexical Lucid and generate the ones for the dialects with the minimal
changes, so that each dialect only maintains the part that is relevant to its
syntactic extension.
For JLucid three bash shell scripts were created to process the original
JavaCC grammars of GIPL and Indexical Lucid and generate appropriate extended
versions for JLucid. These include jlucid.sh that generates JavaCC productions
for arrays and embed(), JGIPL.sh that alters the original GIPL.jjt grammar to
suit the needs of JLucid mostly in terms of class and package names and the
new productions. Similarly, the JIndexicalLucid.sh script exists for
processing of the IndexicalLucid.jjt file. The scripts are rather small and
presented in the Appendix D.
##### 4.1.2.3 Free Java Functions and Java Compiler
As defined in Chapter 3, by “free Java functions” we mean is that the
corresponding Java STs don’t have an enclosing Java class as far as JLucid
source code concerned. However, the enclosing class must exist when compiling
a Java program according to Java’s syntax and semantics. Thus, implementation-
wise we generate such a class internally that wraps all our sequential
threads, as e.g. in Section 4.1.1.8, and we compile that class. This job of
wrapping is delegated to the JavaCompiler, a member of the imperative
compilers framework (see Section 4.1.1.1). The JLucidCompiler as shown in
Figure 21 at some point invokes the JavaCompiler, and what the JavaCompiler
does internally is illustrated in Figure 22.
Figure 22: Java Compilation Sequence.
Being an imperative compiler, the JavaCompiler is obliged to produce the Java
STs and CPs among other things. The core of this process is the wrap() method
where the actual “wrapping” our pseudo-free Java functions into an internal
class occurs. The generated source code .java file is saved and is fed to the
external javac compiler as of this implementation. If there was no compilation
errors, a corresponding .class or series of .class files (for the case of
nested classes) is generated. The generated classes are reloaded back by the
JavaCompiler and their members that are of interest to us retrieved via the
Java Reflection Framework [Gre05], thus we obtain an array of references to
the ST methods and their parameters and assign them to our own data
structures. After this process completes, the corresponding FormatTag
describing the Java language and the compiler is created and all information
is embedded into the ImperativeNode, which represents a single and the only
node in the imperative AbstractSyntaxTree. Later on, this imperative node or
its pieces will replace a corresponding stub in the main intensional AST.
##### 4.1.2.4 Arrays
Implementation of arrays in JLucid coincides closely with the implementation
of objects in Objective Lucid in Section 4.1.3. As a part of the GIPSY Type
System (see Section 4.1.1.5), we employ the GIPSYArray (see Figure 8) type to
hold the array base type and its members and an overall value. As proposed
further, we treat arrays internally as objects (and objects as arrays), so
GIPSYArray is an extension of GIPSYObject that has a base type asserting the
data type of the all the elements in the arrays (as our arrays a homogenous
collection of elements). Thus, when a syntactic array token is parsed, a
corresponding instance of GIPSYArray is created to hold the type and value
information for later processing. The SemanticAnalyzer and the Executor are
made to understand the array type and apply similar type checking or execution
rules to a collection of values instead of a single value.
It might look like this approach will clash with the use of arrays in Java,
i.e., when a developer wishes to use Java arrays (or if a library already
implements some functionality via Java arrays). This should not be a problem
(though will require a more thorough investigation in the future work), when
we perform type matching by the base element type, as described in Section
4.1.1.4. The JavaCompiler is responsible for the appropriate conversion of the
native-to-GIPSY type conversions, by supplying a TypeMap such that it can also
be used by the GEE at run-time. Similar comments can be said of the native
array types that might exist in other imperative languages that we would be
hoping to support.
##### 4.1.2.5 Implementing embed()
To implement embed() we define a type GIPSYEmbed to fetch the file pointed by
the URL and hold it in there. In JLucid, a .java or .class file (later also a
.jar file) is loaded from either local or remote location pointed by the URL
as follows: if it is a .java file, it’s fetched and compiled similarly to the
generated class, but the name is static and known; with the .class file we
skip the compilation process, but extraction of the sequential threads is the
same; for the .jar its examined with the JarInputStream and JarEntry Java
classes to extract the class information.
##### 4.1.2.6 Abstract Syntax Tree and the Dictionary
When running the JLucid compiler in stand-alone mode, all the preprocessing
and re-assembling the intensional and imperative pieces into the combined main
AST happens in here, not in the GIPC, so the JLucid compiler returns a
complete linked AST with all imperative nodes linked in place and a proper
dictionary of identifiers, both intensional and imperative. JLucid compiler,
however, reused the Preprocessor and other parts of the new framework
internally instead of re-inventing the wheel.
The JLucidPreprocessor uses the general Preprocessor class to do the job of
chunkanizing the code segments and preparing initial imperative stubs. This
necessitated adding the #funcdecl segment in the JLucid programs that
previously did not have one in Chapter 3, to simplify preprocessing and
generation of the dictionary. The JLucidPreprocessor is set to reject any
other code segments than #JAVA, #JLUCID, or #funcdecl.
If the JLucidCompiler invoked from the GIPC as a part of general compilation
process (see Figure 15), the #JAVA segment will no longer be really processed
internally, and instead, GIPC will call JavaCompiler externally to the
JLucidCompiler, so essentially the JLucidCompiler will be responsible only for
the Lucid part (with arrays and embed()).
#### 4.1.3 Objective Lucid
This section addresses problems that arise when implementing Objective Lucid.
These include internal implementation to support the dot-notation, extension
to semantic analysis to be able to manipulate object data types (very likely
user-defined), and making it all work in the GICF and General Eduction Engine
(GEE) of the GIPSY by correctly forming the abstract syntax tree (AST) that
includes object data types.
##### 4.1.3.1 Design
Figure 23: Objective Lucid Design. Figure 24: Objective Lucid Compilation
Sequence.
The class diagram describing Objective Lucid is in Figure 23. Since the JLucid
compiler already does most of the legwork, Objective Lucid simply extends it
to add the dot-notation and some extra post-processing when unrolling the
objects. The corresponding compilation sequence is shown in Figure 24.
##### 4.1.3.2 Grammar Generation
Like with JLucid, the grammar files are generated for Objective Lucid using
bash shell scripts, ObjectiveGIPL.sh and ObjectiveIndexicalLucid.sh. These
scripts work with the grammars produced by the JLucid scripts (see Section
4.1.2.2) by simply extending them with the dot-notation production and fixing
up names of classes and packages. These scripts are presented in the Appendix
D.
##### 4.1.3.3 Object Instantiation
Normally, when a Lucid program refers to a Java object, it has to instantiate
it first by either calling a pseudo-free Java function that returns an object
instance or to call the constructor directly. This instantiation has to be
explicit at the beginning of the program to avoid Java’s NullPointerException
at run-time. Internally, the object instance is created using Java Reflection
[Gre05] by first loading and then initializing the needed class with
Class.forName("ClassXB").newInstance(). Referencing static members do not
require a class instance, and can be accessed using the class name, in this
case we just keep the Class.forName("ClassXB"). We also keep the needed
references to the object itself and its members in the GIPSYObject type of the
GIPSY Type System.
##### 4.1.3.4 The Dot-Notation
Implementing the dot-notation extension of JLucid is the easiest task of the
three. In fact, the E.id productions are just a syntactic sugar that can be
wrapped around already existing mechanisms of JLucid to include Java functions
as mentioned in Section 3.2.1.1. The compiler simply generates a set of
pseudo-free Java functions for every object member referenced from the
intensional program. These will be easy to place into the AST just the way
JLucid does it. In other words, this is achieved by automatic generation of
implicit accessor Java functions that had to be explicit in JLucid.
##### 4.1.3.5 Abstract Syntax Tree and the Dictionary
The GIPC (General Intensional Programming Compiler) generates abstract syntax
trees (AST) of all compiled GIPSY program parts, and constructs the GEER
(General Eduction Engine Resources), which is a data dictionary storing all
program identifiers, encapsulated with all ASTs generated at compile time.
Simply put, the GEER encapsulates all the meaning of a GIPSY program, and all
necessary resources to enable the GEE to execute the programs correctly. The
AST and the dictionary contain the generated accessor identifiers that are
processed by the JLucid mechanisms, as described previously. This is possible
because Java’s built-in class Class can provide us with all the meta-
information about its members through enumeration that we can place in the AST
and the dictionary. Little changes from the way JLucid processes that except
that the object members are put into the dictionary and acted upon as an array
of homogeneous types as described in the follow up section.
The ObjectiveLucidPreprocessor also makes use of the general Preprocessor, but
unlike JLucidPreprocessor, it also accepts the #typedecl segment as with
objects come user-defined types, so these have to be listed if used by the
Lucid part.
##### 4.1.3.6 Objects as Arrays and Arrays as Objects
Implementation-wise, we propose to treat arrays of JLucid as a special case of
objects and, the other way around, the objects be a generalization of arrays.
An array can be broken into its elements where every element is evaluated as
an expression under the same context. Thus, evaluating:
A[4] @ [d:4] where dimension d; A[#.d] = 42 * #.d fby.d (#.d - 1); end;
is equivalent to evaluating four Indexical Lucid expressions (possibly in
parallel). Under this point of view objects can be viewed as arrays where
every atomic member is evaluated as if it were an array element. Basically, we
denormalize an object into primitives and evaluate them. If an object
encapsulates other objects, then these are in turn denormalized and put into
the definition environment (dictionary). In other words, if you have an array
of four elements a[4], the elements are evaluated as four independent
expressions. Likewise, an object that has four data members, each of them is
evaluated as an expression under the same context.
Essentially, an array is a collection of atomic elements of the same type.
When evaluating say an array of four elements a[4] at some context [d:4], we
are, in fact, evaluating four ordinary Lucid expressions (possibly in
parallel) in the same context. Likewise, an object is a collection of atomic
elements of (possibly) different types. In case an object encapsulates another
object, that other object can in turn be split into atoms, and so on. All
atoms of an object evaluate as independent Lucid expressions, just like array
elements.
Thus, from Objective Lucid’s point of view, the following are equivalent:
(a) int a[4];
(b) class foo { int a1; int a2; int a3; int a4; }
So, internally, we represent (a) in the definition environment as:
a_4 // scope identifier a_4.a1 a_4.a2 a_4.a3 a_4.a4
Under the scope of array a_4 (a generated id) there are four members, and
a_4.a* comprise a denormalized identifier, also generated. And (b) will
become:
foo // scope identifier foo.a1 foo.a2 foo.a3 foo.a4
where foo.a* are generated variable identifiers in the definition environment.
Encapsulation will be handled in the following way:
class bar { int b1; int b2;
foo oFoo = new foo(); } bar bar.b1 bar.b2 bar.foo bar.foo.a1 bar.foo.a2
bar.foo.a3 bar.foo.a4
To paraphrase and explain in another example, if we have three separate Lucid
expressions:
// float a @ [d:2] where dimension d; a = 2.5 fby.d (a + 1); end; // integer b
@ [d:2] where dimension d; b = 1 fby.d (b + 1); end; // ASCII Char c @ [d:2]
where dimension d; c = ’a’ fby.d (c + 1); end;
Now if we group a, b, and c as a class:
class foo { float a = 2.5; int b = 1; char c = ’a’; public foo() {} }
So when we write:
f @ [d:2] where dimension d; f = foo() fby.d (f + 1); end;
we mean there start three subexpression evaluations:
f.a @ [d:2] where dimension d; f.a = foo().a fby.d (f.a + 1); end; f.b @ [d:2]
where dimension d; f.b = foo().b fby.d (f.b + 1); end; f.c @ [d:2] where
dimension d; f.c = foo().c fby.d (f.c + 1); end;
We say these are equivalent where the f in all expressions refers to the same
object’s instance (i.e. there are not three objects constructed, only one).
Similarly (nearly identically) we implement arrays:
a[3] @ [d:2] where dimension d; a = [1, 2, 3] fby.d (a + 1); end;
The above means:
array a { int a1 = 1; int a2 = 2; int a3 = 3; int length = 3; }
a1 @ [d:2]
where dimension d; a1 = 1 fby.d (a1 + 1); end; a2 @ [d:2]
where dimension d; a2 = 1 fby.d (a2 + 1); end; a3 @ [d:2]
where dimension d; a3 = 1 fby.d (a3 + 1); end;
The three subexpressions run in parallel, but refer back to the same array.
Should there be a need in one of the three subexpressions to use an array
value produced by another subexpression, they generate a demand for that
value.
### 4.2 External Design
The external design encompasses user interface design as well as external
software interfaces. In this work, a web interface to the GIPSY as well as
command-line interfaces are presented as a part of UI followed by the API of
the two external libraries used, JavaCC and MARF.
#### 4.2.1 User Interface
##### 4.2.1.1 WebEditor – A Web Front-End to the GIPSY
The user interface designed for the GIPSY in the scope of this thesis includes
a Servlet-driven web interface to the GIPSY daemon server running on our
development server for trying out GIPSY programs online. The web interface in
a form of a web page allows a connected user to enter, compile, run, and trace
GIPSY programs. Users are able to submit their own GIPSY programs (in any
supported Lucid dialect) or choose and modify from existing programs from the
GIPSY CVS repository (see [RG05a]) and then launch the computation. The GIPSY
servlet front-end generates demands through RIPE and returns back results
along with an execution trace to a web form. A screenshot of this interface is
illustrated in Figure 25.
Figure 25: GIPSY WebEditor Interface.
##### 4.2.1.2 GIPSY Command-Line Interface
Synopsis:
gipsy [ OPTIONS ]
gipsy --help | -h
This is an all-entry point for all of GIPSY that bundles all the modules. It
generally passes all the options to RIPE for further dispatching. When the
server part (see Section 7.10) is complete, this will be a GIPSY daemon
server. The command line interface includes the following options:
* •
\--help or -h displays application’s usage information.
* •
\--compile-only tells to compile a GIPSY program only and return the result of
the compilation (error or success messages) and the compiled program itself.
This will not invoke the GEE for execution after compilation. The option is
primarily for quick tests in development setups.
* •
\--debug tells to run in the debug/verbose mode.
It is possible to run the GIPSY by either invoking the GIPSY.class directly,
by running a corresponding gipsy.jar (see Appendix C.2) file, or using a
provided wrapper script gipsy. The latter is the simplest one to use as it
includes all the necessary options for the JVM and searches for the executable
.jar in several common places. A good idea is to put gipsy somewhere under
one’s PATH. (A similar approach applies to the other tools mentioned in the
follow up sections, such as ripe, gipc, gee, and regression. The tools exist
for both Unix and Windows in the form of shell scripts and batch files.)
Example uses of the GIPSY application include:
* •
gipsy or gipsy --help
* •
gipsy --compile-only
* •
gipsy --compile-only --debug
Where \--debug can be combined with any of these, otherwise the options are
exclusive.
##### 4.2.1.3 RIPE Command-Line Interface
Synopsis:
ripe [ OPTIONS ]
ripe --help | -h
The RIPE command-line interface right now acts mostly to activate various own
submodules (e.g. textual or DFG editors) or dispatch requests from users to
the other main modules, such as GIPC and GEE. The command-line interface
includes the following options:
* •
\--help or -h displays application’s usage information.
* •
\--gipc=‘$<$GIPC OPTIONS$>$’ tells RIPE to invoke GIPC with a set of GIPC
options (see Section 4.2.1.4).
* •
\--gee=‘$<$GEE OPTIONS$>$’ tells RIPE to invoke GEE with a set of GEE options
(see Section 4.2.1.5).
* •
\--regression=‘$<$REGRESSION OPTIONS$>$’ tells RIPE to invoke Regression
testing with a set of its options (see Section 4.2.1.6).
* •
\--dfg=‘$<$DFG EDITOR OPTIONS$>$’ tells RIPE to start the DFG editor with its
options. Currently, the DFGEditor Java class is a stub, and instead, the DFG
Editor of Yimin Ding [Din04] is started via a separate program, lefty. It is
planned the DFGEditor class would be a wrapper for the program in the future.
Therefore, all DFG editor options are ignored for now, but a provision is made
for the future.
* •
\--txt=‘$<$TEXTUAL EDITOR OPTIONS$>$’ tells RIPE to start the textual editor
with its options. Note, at the time of this writing TextualEditor is just a
stub, and as such does not have any options, but a provision is made when it
does.
* •
\--debug tells to run in the debug/verbose mode.
Example uses of the RIPE application include:
* •
ripe or ripe --help
* •
ripe --compile-only
* •
ripe --compile-only --debug
##### 4.2.1.4 GIPC Command-Line Interface
Synopsis:
gipc [ OPTIONS ] [ FILENAME1.ipl [ FILENAME2.ipl ] ... ]
gipc --help | -h
The command line interface for GIPC inherited some options from Lucid [Ren02]
and includes the following options:
* •
\--help or -h displays application’s usage information.
* •
[FILENAME1.ipl [FILENAME2.ipl] ...] tells GIPC to compile a GIPSY program as
indicated by the FILENAME. It is possible to have more the one input file for
compilation. If this is the case, the same number of instances of GIPC threads
will be initially spawned to compile those programs. Notice, however, this
does not mean all the files (in case of multiple .ipl files) comprise one
program and then linked together afterwards as in typical C or C++ compilers.
Instead, each .ipl file is treated as a stand-alone independent GIPSY program.
* •
\--stdin tells GIPC to interpret the standard input as a source GIPSY program.
This is the default if no FILENAME is supplied.
* •
\--gipl or -G (came from Lucid [Ren02] for backwards compatibility) tells GIPC
to interpret the source program unconditionally as a GIPL program (by default
no assumption is made and GIPC attempts to treat the incoming source code as a
general GIPSY program). It is primarily used to quickly test the GIPL compiler
only, without extra overhead or preprocessing. It is also used by the
Regression application for that same reason.
* •
\--indexical or -S (came from Lucid [Ren02]) tells GIPC to interpret the
source program unconditionally as an Indexical Lucid program.
* •
\--jlucid tells GIPC to interpret the source program unconditionally as a
JLucid program.
* •
\--objective tells GIPC to interpret the source program unconditionally as an
Objective Lucid program.
* •
\--translate or -T (came from Lucid [Ren02]) enables SIPL-to-GIPL translation.
This option is implied by default (as opposed to be optional in Lucid). It
tells the GIPC to interpret the input program unconditionally as a non-GIPL
program that requires operator and function translation. The option has no
effect with \--gipl as GIPL is the only intensional language that does not
require any further translation.
* •
\--disable-translate turns off automatic translation (in case the user knows
that an incoming non-GIPL program has nothing to translate, which is rarely
the case; otherwise, the GIPC will bail out with an error).
* •
\--warnings-as-errors tells to treat compilation warnings as errors and stop
compilation after displaying them.
* •
\--gee tells GIPC to run the compiled program immediately after compilation
(if successful) by feeding it directly to the GEE. The default is that the
compiled GIPSY program is saved into a file where the original name is
suffixed with the .gipsy extension.
* •
\--dfg tells GIPC to perform DFG code generation as a part of the compilation
process.
* •
\--debug to run in a debug/verbose mode.
Example uses of the GIPC application include:
* •
gipc or gipc --help or gipc -h
* •
gipc life.ipl
* •
gipc --disable-translate --gee --debug life.ipl
* •
gipc --gipl --debug gipl.ipl
* •
gipc --jlucid --stdin
##### 4.2.1.5 GEE Command-Line Interface
Synopsis:
gee [ OPTIONS ] [ FILENAME1.gipsy [ FILENAME2.gipsy ] ... ]
gee --help | -h
The command line interface includes the following options:
* •
\--help or -h displays application’s usage information.
* •
[FILENAME1.gipsy [FILENAME2.gipsy] ...] tells GEE to run a stored version of a
compiled GIPSY program as indicated by the FILENAME. It is possible to have
more than one input file for execution. If this is the case, the same number
of instances of GEE threads will be initially spawned to run those programs.
The programs will run concurrently, but there should not be any interference
or communication in their execution except they may share the output resource.
* •
\--stdin tells GEE to interpret the standard input as a compiled GIPSY
program. This is the default if no FILENAME is supplied.
* •
\--all tells GEE to start all implemented services/servers locally (threaded,
RMI, Jini, DCOM+, and CORBA).
* •
\--threaded tells GEE to start the threaded server only.
* •
\--rmi tells GEE to start the RMI service.
* •
\--jini tells GEE to start the Jini service.
* •
\--dcom tells GEE to start the DCOM+ service.
* •
\--corba tells GEE to start the CORBA service.
* •
\--debug tells GEE to run in the debug/verbose mode.
Example uses of the GEE application include:
* •
gee or gee --help or gee -h
* •
gee life.gipsy
* •
gee --disable-translate --threaded --debug life.gipsy
* •
gee --all --debug gipl.gipsy
* •
gipc --rmi --jini indexical.gipsy
##### 4.2.1.6 Regression Testing Application Command-Line Interface
Synopsis:
regression [ OPTIONS ]
regression --help | -h
The Regression application and its test suite are presented in detail in
Section 5.1. The application, based on options, invokes either GIPC or GEE or
both directly feeding a pre-selected list of test source programs. The command
line interface includes the following options:
* •
\--help or -h displays application’s usage information.
* •
\--sequential tells to run sequential tests (default).
* •
\--parallel tells to run parallel tests.
* •
\--gipl tells to test pure GIPL programs only.
* •
\--indexical tells to test pure GIPL and Indexical programs with the Indexical
Lucid compiler.
* •
\--gipsy tells to test general-style GIPSY programs with code segments.
* •
\--gee if specified, tells to run the GEE after compilation (default).
* •
\--all tells to do all of the above tests in one run (default).
* •
\--directory tells to pick source test files from a specified directory
instead of pre-set directories from the GIPSY source tree
* •
\--debug tells to run in the debug/verbose mode.
Example uses of the Regression application include:
* •
regression or regression --help or regression -h
* •
regression --gipl
* •
regression --parallel --indexical
* •
regression --all --debug
* •
regression --directory=/some/gipsy/misc/tests --all --debug
#### 4.2.2 External Software Interfaces
##### 4.2.2.1 JavaCC API
JavaCC-generated code contains a number of common classes and interfaces,
regardless of the language a parser is generated for. These have to do with
AST nodes, tokens, token types, character streams, and alike. The most often
used class out of this bundle is SimpleNode, which is a concrete node in the
AST. These classes have to be periodically refreshed by compiling the source
grammar when a newer version of javacc comes out.
Figure 26: JavaCC- and JJTree-generated Modules Used by Several GIPC Modules.
The below are JavaCC API/modules [VC05] used by the GIPSY and their
description. The corresponding class diagram is in Figure 26.
* •
Node is the common interface for all occurrences of SimpleNode to implement
(see below).
* •
The SimpleNode class represents a concrete node in every AST in the GIPC. Once
generated, this class is usually customized according to the needs of the
given parser/compiler. All concrete instances, however, implement the same
Node interface above. At the time of this writing, there are three SimpleNode
occurrences in the GIPSY source tree: the common one in gipsy.GIPC.intensional
for all the SIPLs and GIPL, as per original implementation presented in
[Ren02]. It is a basis for a GIPL AST aside from the related parsers known to
the SemanticAnalyzer and GEE’s Executor. This implementation is wrapped-around
by AbstractSyntaxTree that the rest of the modules know. Then, a customized
subclass of it is in gipsy.GIPC.DFG.DFGAnalyzer of Yimin Ding [Din04]. It was
made a subclass because a large portion of the code is identical. Finally, the
last one is in gipsy.GIPC.Preprocessing used by the Preprocessor. This
occurrence of SimpleNode was kept as-is due to the significant differences and
purpose with the former two.
* •
The ImperativeNode is another implementation of the Node interface created
manually for all the imperative language compilers. The ImperativeNode
represents an AST of a single node encapsulating STs, CPs, some meta
information that came from a given imperative compiler. The reason for this is
to maintain a global AST for a GIPSY program where all nodes implement the
same interface.
* •
SimpleCharStream is a common javacc utility that treats incoming source code
stream as a set of ASCII characters without extra UNICODE processing.
* •
ParseException is a common generated type of exception to indicate a parse
error. It was made manually to subclass GIPCException from the GIPSY
Exceptions Framework (see Section 4.2.3.2) for uniform exception handling.
* •
TokenMgrError a subclass of java.lang.Error primarily to signal lexical errors
in the incoming source code or token processing in general by a given parser
(e.g. by invoking a static parser twice).
##### 4.2.2.2 MARF Library API
Figure 27: MARF Utility Classes used by the GIPSY. Figure 28: Dictionary and
DictionaryItem API Figure 29: Dictionary Usage within the GIPSY
MARF (see Section 2.6.3) has a variety of useful utility and storage-related
modules that conveniently found their place in GIPSY. Most of these come from
the marf.util package as well as marf.Storage.222Later some natural language
processing (NLP) modules in marf.nlp of MARF might also get used in the GIPSY
as a part of another research project. The below are MARF API/modules used by
GIPSY and their description:
* •
marf.util.FreeVector is an extension of java.util.Vector that allows
theoretically vectors of infinite length, so it is possible to set or get an
element of the vector beyond its current physical bounds. Getting an element
beyond the boundaries returns null, as if the object at that index was never
set. Setting an element beyond bounds automatically grows the vector to that
element. In the GIPSY, marf.util.FreeVector is used as a base for our
Dictionary as shown in Figure 28. Figure 29 shows all the modules that are now
using Dictionary instead of java.util.Vector. The corresponding class diagram
of the MARF’s util API is shown in Figure 27.
* •
marf.util.OptionProcessor module is extensively used by the command-line user
interfaces (see Section 4.2.1) of GIPSY, GIPC, GEE, and Regression. A
convenient way of managing command-line options in a hash table and validating
them.
* •
marf.util.BaseThread class encapsulates some useful functionality used in
threaded versions of GEE and GIPC, which Java’s java.lang.Thread does not
provide:
* –
maintaining unique thread ID (TID) among multiple threads and reporting it
(for tracing, debugging, and RIPE). A note is added here that Java 1.5.* now
also provides a notion of a TID, but marf.util.BaseThread was written prior to
that and GIPSY remains Java 1.4-compliant still. Plus, MARF’s way of handling
this is more flexible.
* –
adapted human-readable trace information via toString()
* –
access to the Runnable target that was specified upon creation.
* –
integration with marf.util.ExpandedThreadGroup, see below.
* •
marf.util.ExpandedThreadGroup allows to start, stop, or other group operations
that Java’s java.lang.ThreadGroup doesn’t provide. ExpandedThreadGroup is, for
example, used in GIPC to create a group of compiler threads (in GIPSY every
compiler is a thread), one for each language chunk, that will run
concurrently. Additionally, a group of GEE, or rather, Executor threads would
run in the case of a forest of ASTs.
* •
marf.util.Arrays groups more array-related functionality together than the
java.util.Arrays class does, for example copying (homo- and heterogeneous
types) and converting to java.util.Vector, and provides some extras.
* •
marf.Storage.StorageManager provides basic implementation of the (possibly
compressed) object serialization, and in our case the GIPC and GEE are storage
manager with respect to a compiled GIPSY program.
* •
marf.util.Logger is primarily used by the Regression application to log
standard output before calling GIPC or GEE to a file, for future comparison
with an expected output.
* •
marf.util.Debug is used in many places for debugging convenience allowing to
issue debug messages only if the debug mode is globally on, which is also
maintained within the class.
##### 4.2.2.3 Servlets API
The Java Servlets technology from Sun [Mic05a] was used to implement the
WebEditor interface outlined earlier. While the actual API specification of
servlets is rather vast, the key used components used here are listed:
* •
The HttpServlet class is the base for all servlets, including WebEditor.
* •
The doGet() must be overridden to respond to the GET HTTP requests.
* •
The doPost() must be overridden to respond to the POST HTTP requests. In our
implementation, doPost() is a simply a wrapper around doGet(), so both GET and
POST requests are handled identically.
#### 4.2.3 Architectural Design and Unit Integration
Unit integration according to the initial design decisions of the GIPSY system
and setting up package hierarchy played an important role in the success of
this work. A proposed directory structure (see Appendix C.1) and a
corresponding breakdown of the Java packages (see Appendix C.1) hierarchy are
important to the success of GIPSY, especially for public use. The author of
this work inherited the previous GIPSY iteration without any structure or
packaging and proposed and restructured the system to what it is now.
##### 4.2.3.1 GIPSY
When integrating several components of a large system and redesigning some of
their API, the overall system design has to be considered. In Figure 30 is a
high-level view of the main GIPSY modules. These modules can be run as stand-
alone Java applications or start each other.
Figure 30: GIPSY Main Modules.
* •
The GIPSY class on the diagram represents a stand-alone server for a client-
server type of application, which is capable of spawning GIPC and GEE upon
client’s request. The prime goal of it is testing of intensional programs that
users can submit online and get the result in case they don’t have the
development environment set up from where they are working.
* •
The GIPC class when run as a stand-alone application invokes all the
intensional and imperative compilers required and produces a compiled version
of a submitted GIPSY program. Optionally, if requested, GIPC can pass the
compiled program on to GEE for execution. The GIPC along with GEE subsumes
what was previously known as Lucid and Facet defined by Chun Lei Ren in
[Ren02].
* •
The GEE when run as a stand-alone application, begins demand-driven execution
of a GIPSY program that was either compiled and stored or compiled and passed
from GIPC.
* •
The Regression class is the main driver for the Regression Testing Suite of
GIPSY, that also calls these modules for regression and unit testing.
##### 4.2.3.2 GIPSY Exceptions Framework
Figure 31: GIPSY Exceptions Framework.
The class diagram describing the GIPSY Exceptions Framework is in Figure 31.
The main exception type is GIPSYException that provides some machinery
encapsulating other exceptions. Every major module, like GIPC, GEE, or RIPE in
GIPSY defines its own sublcass of GIPSYException. By doing this, the
applications using the modules can differentiate the exception types and
handle them appropriately. The NotImplementedException is an easy way to use
to indicate some unimplemented but important stubs, if called. It is a
subclass of RuntimeException because it can happen virtually everywhere and
run-time exceptions do not need to be declared to be thrown or caught. The
GIPCException, GEEException, and RIPEException represent base exception
objects for the corresponding modules; the rest are primarily subclasses of
these.
##### 4.2.3.3 GEE Design
The general overview of GEE is in Figure 32. The several modules under the
gipsy.GEE package carry out a complex GIPSY program execution task. The GEE is
the facade and the main starting point for all of GEE. GEE may act as either
an application on its own or be invoked by the GIPC. For the stand-alone
execution a user has to supply a filename of a valid compiled GIPSYProgram.
This program is loaded and GEE starts the Executor thread to actually execute
it. Before Executor begins the GEE may optionally start the available demand
propagation services, such as local (just threads), RMI, Jini-based and the
like. The Executor while executing the program generates demands for the
identifiers listed in the program and then performs the final calculation
based on the results received. The Executor was formerly known as
XLucidInterpreter and the Java version of which was implemented by Bo Lu in
[Lu04] and reworked to handle sequential threads, arrays, objects, and other
than integer and float data types.
Figure 32: GEE Design.
###### Demand Dispatcher
In Figure 33 is a high-level overview of the DemandGenerator and related
classes. Most of the demand propagation in Jini and JavaSpaces is implemented
by Emil Vassev in [VP05]. The integration part included making sure the
IDemandList interface is consistently used by the DemandGenerator along with
the DemandDispatcherAgent to be compliant to the rest of the GEE. The
IDemandList interface was originally designed by Bo Lu in [Lu04] and
redesigned by the author of this thesis to be implemented by the RMI and
threaded versions of GEE and was formerly known as DemandList. Next, the
temporary class WorkTask was made to implement the ISequentialThread interface
according to the overall GIPSY design for sequential threads. This class is
marked as deprecated (and later on will be removed) as every sequential thread
class is generated by the SequentialThreadGenerator and is different from one
GIPSY program to another. Finally, the LUSException (service look up
exception) and DemandDispatcherException were made to be a part of the GIPSY
Exceptions Framework Section 4.2.3.2 by inheriting from the GEEException. For
further implementation details of the DemandDispatcher please refer to Emil’s
work [VP05].
Figure 33: The Demand Dispatcher Integrated and Implemented based on Jini.
###### Intensional Value Warehouse and Garbage Collection
Figure 34: Integration of the Intensional Value Warehouse and Garbage
Collection.
Intensional Value Warehouse and Garbage Collection were implemented by Lei Tao
in [Tao04]. After integration, his contributions became to look like as shown
in Figure 34. The IValueHouse and its extension IVWInterface are the ones used
by the Executor to communicate to a concrete implementation of a warehouse,
allowing adding/changing warehouse implementations easily without affecting
the Executor. All the exception handling is based on the GEEException.
##### 4.2.3.4 RIPE Design
Figure 35: RIPE Design.
The class diagram describing RIPE is in Figure 35. The RIPE class represents a
facade to the rest of the RIPE modules. It is semi-implemented, as many things
are not clear on this side of the project yet. The only part of RIPE that was
advanced well so far by Yimin Ding in [Din04] is the Data-Flow-Graph (DFG)
editor, which is not implemented in Java. The DFGEditor Java class is meant to
be main Java program acting like a bridge between Java and the LEFTY language,
but did not get implemented yet. The rest of the modules are planned stubs.
##### 4.2.3.5 Data Flow Graphs Integration
Figure 36: DFG Integration Design.
The integration of Yimin Ding’s [Din04] DFG-related work is presented in
Figure 36. The DFGAnalyzer was augmented to implement the ICompiler interface
as it follows the same structure as the rest of our compilers, which compiles
a Lucid code from DFG. The DFGException class, a subclass of GIPCException has
been created to indicate an error situation in the DFG processing.
DFGAnalyzer’s SimpleNode was updated to inherit from
GIPC.intesional.SimpleNode due to vast functionality overlap. The two analyzer
and generator modules have been placed under the GIPC.DFG.DFGAnalyzer and
GIPC.DFG.DFGGenerator packages.
### 4.3 Summary
This chapter presented most of the development effort went into integration,
design, and implementation of JLucid, Objective Lucid, and GICF. User
interfaces (both web and command line) has been outlined. Regression Test
Suite has been introduced. The follow up chapter presents a variety of testing
approaches went into the GIPSY to prove successful integration of the old and
implementation of new modules.
To summarize, Objective Lucid, as opposed to GLU [JD96, JDA97] and JLucid,
provides access to the object members and is real object-oriented hybrid
language. While JLucid may indirectly manipulate objects through pseudo-free
functions, the actual objects are still a “black box” to it.
The GICF and IPLCF gave an ability for an easier integration of intensional
and imperative languages in the GIPSY. The below are the steps one needs to
perform to add a new compiler to the GIPSY:
* •
create a package where the language compiler will reside (usually under
imperative/LANGUAGE or intensional/SIPL/LANGUAGE.
* •
add a compiler class that extends either one of IntensionalCompiler,
ImperativeCompiler, or implements one of their superinterfaces
* •
the code segment and fully qualified class name should be added to either
EImperativeLanguages or EIntensionalLanguages
* •
optionally implement a custom version of a preprocessor if it is a hybrid
language
* •
implement translation rules to GIPL if it is a SIPL if it is an intensional
language
* •
implement proper ST/CP generation for an imperative language according to that
language’s semantics and typing instructions
* •
implement type mapping table upon the need if it is an imperative language
The above might still sound complex, but it is much more easier and flexible
than before. Additionally, some of the steps can be abstracted and simplified,
but it is impossible to eliminate manual work altogether.
## Chapter 5 Testing
This chapter addresses the testing aspect of this thesis for the following two
main reasons: integration and refactoring of a variety of the GIPSY modules
including GICF and the development and operation of the two new Lucid dialects
developed in this work, namely JLucid and Objective Lucid. Notice, this
testing is far from comprehensive and does not include testing of the
execution performance of any of the programs and many compilation aspects are
still to be resolved as of this writing (and be resolved in the final
version). This is, however, a starting point of setting up the GIPSY testing
infrastructure for the projects to come to do mandatory systematic tests,
which are now a necessity given the size of the system, a centralized source
tree, and the number of subprojects developed simultaneously.
### 5.1 Regression Testing
#### 5.1.1 Introduction
The regression testing is a comprehensive set of tests for the implementation
and integration of the GIPSY modules. They test most of the operations and
capabilities of the GIPSY. The test cases primarily are various intensional
programs (hybrid or not) that exercise the main modules, such as GIPC and GEE
as well as their submodules with the major focus on GIPC.
#### 5.1.2 Regression Testing Suite
The regression tests can be run against already pre-compiled gipsy.jar, or by
using a temporary installation within the source tree using the Regression
application. Next, there are a “sequential” and “parallel” modes to run the
tests. In the sequential mode tests run in strict sequence, whereas in the
parallel mode multiple threads are started to run groups of tests in parallel.
##### 5.1.2.1 Unit Testing with JUnit
The core of the Regression application is based on the JUnit framework
introduced in Section 2.6.1.3. Regression represents a TestSuite, that
contains ParallelTestCase and SequentialTestCase, a subclasses of TestCase.
Both types of tests are customizable based on the options supplied to the
Regression application (see Section 4.2.1.6). JUnit helps to tell us what
errors happened and in which modules and the reason of the failures
dynamically at run-time.
##### 5.1.2.2 Unit Testing with diff
It becomes cumbersome to use JUnit for all possible cases, in a large system,
where often we are generally interested in the output behaviour changes only.
Here the Unix tool diff helps us. A collection of hand-checked outputs are
said to be “expected”, one ore more file for each test case. Then, when the
next time the test is run, a current directory is created with the current
outputs, and the current and expected output directories are compared with the
diff to show the differences in the output produced by the modules. This is
all achieved by the regression script.
##### 5.1.2.3 Tests
The actual test cases in the form of GIPL, Indexical Lucid, Objective Lucid,
JLucid, and GIPSY programs, are located under the corresponding src/tests/*
directories in the source tree in the form of *.ipl files. These comprise most
of the examples presented earlier in this work as well as developed in
[Paq99], [Ren02], [Wu02], and [Lu04]. The regression tests for the DFG
generation ([Din04]), Intensional Value Warehouse and Garbage Collector
[Tao04] and Demand Migration System (DMS) [VP05] are not present as of this
implementation.
### 5.2 Portability Testing
GIPSY has been tested and is known as expected (regression tests pass) to run
on Red Hat Linux 9, Fedora Core 2, Mac OS X, Solaris 9, Windows 98SE/2000/XP
systems under JDK 1.4 and 1.5. The corresponding hardware architectures were
Intel or Intel-compatible processors (Pentium II, III, and IV with 233 MHz to
1.4 GHz) and G3 and G4 processors from Apple and IBM. For the WebEditor
interface, Tomcat 5 on Mac OS X were tested, but it is believed to run on
other platforms the Jakarta Tomcat runs on.
### 5.3 Solving Problems
This section is targeting some common problems of synchronization in parallel
and distributed environment and how they are solved using the GIPSY system
relieving the programmer from the need of explicitly synchronize the objects.
They also illustrate the use of arrays and embedded Java, and Java objects.
These programs are among many other test cases from the Regression Tests
Suite.
#### 5.3.1 Prefix Sum
pseudocode (for thread ’j’)
’shared’ a ’future’ ’int’ ’array’ [1..logP, 1..P] := undefined;
’private’ sum ’int’ := j,
hop ’int’ := 1;
’do’ level = 1, logP --->
’if’ j <= P - hop ---> a[level, j] := sum ’fi’
’if’ j > hop ---> sum +:= a[level, j - hop] ’fi’
hop := 2 * hop
’od’
Figure 1: Pseudocode of a thread $j$ for the Prefix Sum Problem.
/*
* PREFIX SUM in GIPL-style JLucid program.
* Numbers are from 1 to 8.
* S[I] will contain prefix sum for number ’i’
*/
#JLUCID
// Array of prefix sums
S[8] @d 8
where
dimension d;
S[I] = if(#d = 0)
then 1
else (2 * S[I] - 1) @d (#d - 1)
fi;
// Index the array varies within.
I @i 8
where
dimension i;
I = if(#i = 0) 1 else (I - 1) @d (#i - 1);
end;
end;
Figure 2: The Prefix Sum Problem in JLucid in GIPL Style.
/*
* PREFIX SUM in Indexical Lucid-style JLucid
*/
#JLUCID
S[8] @d 8
where
dimension d;
S[I] = 1 fby.d (2 * S[I] - 1);
I @i 8
where
dimension i;
I = 1 fby.i (I - 1);
end;
end;
Figure 3: The Prefix Sum Problem in JLucid in Indexical Lucid Style.
The pseudocode of for a thread $j$ is in Figure 1 [Pro03a]. The Figure 2 shows
the program translated into Lucid. The Figure 3 shows the program translated
into Indexical Lucid for numbers from 1 to 8. Below is an equivalent
implementation of the problem (targeting only TLP) in Java; compare the
program’s line count and complexity to that of JLucid:
// Modified from Dr. Probst’s Cyclic.java
public class PrefixSum
{
public static final int P = 8; // number of workers
public static final int logP = 3; // number of rows in logP x P matrix
// For permutation of workers
private static int[] col = {3, 6, 5, 7, 4, 2, 1, 0};
// These two mimic a 2D array of future variables
public static int[][] a = new int [logP][P];
public static Semaphore[][] futures = new Semaphore[logP][P];
// The resulting sums are to be placed here.
public static int[] sums = new int[P];
public static void main(String[] argv)
{
Worker w[] = new Worker[P];
for(int j = 0; j < futures.length; j++ )
for(int k = 0; k < futures[j].length; k++)
futures[j][k] = new Semaphore(0);
for(int j = 0; j < P; j++)
{
w[col[j]] = new Worker(col[j] + 1);
w[col[j]].start();
}
for(int j = 0; j < P; j++)
{
try
{
w[j].join();
}
catch(InterruptedException e)
{
}
}
for(int j = 0; j < P; j++)
System.out.println ("Prefix Sum of " + (j + 1) + " = " + sums[j]);
System.out.println ("System terminates normally.");
}
}
class Semaphore
{
private int value;
Semaphore(int value1)
{
value = value1;
}
public synchronized void Wait()
{
try
{
while(value <= 0)
{
wait();
}
value--;
}
catch (InterruptedException e)
{
}
}
public synchronized void Signal()
{
++value;
notify();
}
}
class Worker extends Thread
{
private int j;
private int sum;
private int hop = 1;
public Worker(int col)
{
sum = j = col;
}
public void run()
{
System.out.println("Worker " + j + " begins execution.");
yield();
for(int level = 0; level < PrefixSum.logP; level++)
{
if(j <= PrefixSum.P - hop)
{
System.out.println
(
"Worker " + j +
" defines a[" + level + "," + (j-1) +"]."
);
PrefixSum.a[level][j - 1] = sum;
PrefixSum.futures[level][j - 1].Signal();
}
if(j > hop)
{
PrefixSum.futures[level][j - 1 - hop].Wait();
System.out.println
(
"Worker " + j +
" uses a[" + level + "," + (j - 1 - hop) + "]."
);
sum += PrefixSum.a[level][j - 1 - hop];
}
hop = 2 * hop;
}
PrefixSum.sums[j - 1] = sum;
System.out.println ("Worker " + j + " terminates.");
}
}
#### 5.3.2 Dining Philosophers
Below is a JLucid implementation of the Dining Philosophers problem [Dij65,
Dij71, Gin90]. We have arrays of 8 philosophers and 8 forks, each represented
as integers. A philosopher is either thinking (1) or eating (2); likewise for
forks, taken or not. A philosopher may eat when they have exactly two forks,
not less, if the forks are available. If none available, the philosopher waits
(implicit, guaranteed by the GEE). The special variable $I$ serves as an
intensional index for our arrays.
/**
* Dining Philosophers Problem
* in JLucid
*
* @author Serguei Mokhov, [email protected]
* @version $Revision: 1.10 $ $Date: 2005/03/02 02:57:31 $
*/
#funcdecl
int getIninitalRandomState();
boolean chew(int);
boolean brainstormIdea(int);
#JLUCID
/*
* Assume 8 philosophers and two states.
* States: 2 - eating, 1 - thinking
* Forks are either available or not; hence, 2 states as well.
*/
PHILOSOPHERS[8] @states 2
where
dimension states;
// Initialize all forks
FORKS[8] @availability 2
where
dimension availability;
FORKS[I] = getIninitalRandomState();
I @d 8
where
dimension d;
I = 1 fby.d (I - 1);
end;
end;
/*
* Run the actual algorithm.
* NOTE: in this implementation the computation
* never terminates (normally). It is an infinite loop.
*/
PHILOSOPHERS[I] =
if(#states == 1) then
eat(I) @states 2
eat(I) =
getForks(I) && chew(I);
getForks(I) = g(l, r)
where
l = FORK[I] @availability 1;
r = FORK[I] @availability 1;
end;
else
think(I) @states 1
think(I) =
putForks(I) && brainstormIdea(I);
putForks(I) = p(l, r)
where
l = FORK[I] @availability 2;
r = FORK[I] @availability 2;
end;
fi;
I @d 8
where
dimension d;
I = 1 fby.d (I - 1);
end;
end;
#JAVA
int getIninitalRandomState()
{
// Either 1 or 2
return new Random().nextInt(2) + 1;
}
boolean chew(int i)
{
try
{
System.out.println("Philo " + i + " is chewing smth tasty now.");
sleep(new Random().nextInt(i * 1200));
System.out.println("Philo " + i + " finished chewing.");
return true;
}
catch(InterruptedException e)
{
return false;
}
}
boolean brainstormIdea(int i)
{
try
{
System.out.println("Philo " + i + " is heavily thinking now.");
sleep(new Random().nextInt(i * 1200));
System.out.println("Philo " + i + " finished thinking.");
return true;
}
catch(InterruptedException e)
{
return false;
}
}
#### 5.3.3 Fast Fourier Transform
This is an example on how one would compute Fast Fourier Transform (FFT) in
the GIPSY for an array of double values. This is straightforward in Lucid
because it’s deterministic with plenty of parallelism. A JLucid program
implementing FFT is in Section 5.3.3.1. The algorithm is based on the Java
algorithm implemented in MARF [MCSN05, Pre93, Ber05], a code fragment of which
is in Section 5.3.3.2, originally written by Stephen Sinclair. The JLucid
version omits the imaginary part of the transform, but it would not be hard to
add it.
##### 5.3.3.1 Fast Fourier Transform in JLucid.
/*
* FFT implementation in JLucid.
* Serguei Mokhov
* $Id: fft.ipl,v 1.2 2005/08/13 01:37:23 mokhov Exp $
*/
#funcdecl
double sin(double);
double pi();
#JAVA
double sin(double pdValue)
{
return Math.sin(pdValue);
}
double pi()
{
return Math.PI;
}
#JLUCID
A
where
// A is an array of 9 FFT values with a
// normal FFT applied to the array below.
A = fft([1, 2, 3, 4, 6, 7, 8, 9], 9, 1);
fft(inputValues, length, sign) = fftValues
where
fftValues = apply(length, reverse(length, inputValues), sign);
apply(len, coeffs, direction) = coeffs @.s (N - 1)
where
dimension s;
N = 2 * len;
mmax = (2 fby.s istep) upon(mmax < N);
coeffs[J / 2] = coeffs[I / 2] - tempr;
coeffs[I / 2] = coeffs[I / 2] + tempr;
where
istep = mmax fby.s (istep) * 2;
M @.m mmax
where
dimension m;
M = (0 fby.m (M + 2)) upon (M < mmax);
tempr = wr * coeffs[J / 2] - wi * coeffs[J / 2];
J = I + mmax;
wr = 1.0 fby.m ((wtemp = wr) * wpr - wi * wpi + wr);
wi = 0.0 fby.m (wi * wpr + wtemp * wpi + wi);
where
dimension i;
I = (M fby.i (I + istep)) upon (I < N);
theta = (direction * 2 * pi()) / mmax;
wtemp = sin(0.5 * theta);
wpr = -2.0 * wtemp * wtemp;
wpi = sin(theta);
end;
end;
end;
end;
// Binary reversion
reverse(l, vals) = out @.i length
where
dimension i;
out[t] = vals[#.i] @ (#.i + 1) @.bit maxbits(length);
where
dimension bit;
t = 0 fby.bit ((t * 2) | (n & 1));
n = #i fby.bit (n / 2);
end;
end;
// Determine max number of bits
maxbits(len) = (mbits - 1) @.m 16
where
dimension m;
mbits = ( 0 fby.m (mbits + 1) ) upon (mbits < 16 && n != 0);
n = len fby.m (n / 2);
end;
end;
end;
// EOF
##### 5.3.3.2 Fast Fourier Transform code fragment in Java from MARF.
...
/**
* <p>FFT algorithm, translated from "Numerical Recipes in C++" that
* implements the Fast Fourier Transform, which performs a discrete Fourier transform
* in O(n*log(n)).</p>
*
* @param InputReal InputReal is real part of input array
* @param InputImag InputImag is imaginary part of input array
* @param OutputReal OutputReal is real part of output array
* @param OutputImag OutputImag is imaginary part of output array
* @param direction Direction is 1 for normal FFT, -1 for inverse FFT
* @throws MathException if the sizes or direction are wrong
*/
public static final void doFFT
(
final double[] InputReal,
double[] InputImag,
double[] OutputReal,
double[] OutputImag,
int direction
)
throws MathException
{
// Ensure input length is a power of two
int length = InputReal.length;
if((length < 1) | ((length & (length - 1)) != 0))
throw new MathException("Length of input (" + length + ") is not a power of 2.");
if((direction != 1) && (direction != -1))
throw new MathException("Bad direction specified. Should be 1 or -1.");
if(OutputReal.length < InputReal.length)
throw new MathException("Output length (" + OutputReal.length + ") < Input length (" + InputReal.length + ")");
// Determine max number of bits
int maxbits, n = length;
for(maxbits = 0; maxbits < 16; maxbits++)
{
if(n == 0) break;
n /= 2;
}
maxbits -= 1;
// Binary reversion & interlace result real/imaginary
int i, t, bit;
for(i = 0; i < length; i++)
{
t = 0;
n = i;
for(bit = 0; bit < maxbits; bit++)
{
t = (t * 2) | (n & 1);
n /= 2;
}
OutputReal[t] = InputReal[i];
OutputImag[t] = InputImag[i];
}
// put it all back together (Danielson-Lanczos butterfly)
int mmax = 2, istep, j, m; // counters
double theta, wtemp, wpr, wr, wpi, wi, tempr, tempi; // trigonometric recurrences
n = length * 2;
while(mmax < n)
{
istep = mmax * 2;
theta = (direction * 2 * Math.PI) / mmax;
wtemp = Math.sin(0.5 * theta);
wpr = -2.0 * wtemp * wtemp;
wpi = Math.sin(theta);
wr = 1.0;
wi = 0.0;
for(m = 0; m < mmax; m += 2)
{
for(i = m; i < n; i += istep)
{
j = i + mmax;
tempr = wr * OutputReal[j / 2] - wi * OutputImag[j / 2];
tempi = wr * OutputImag[j / 2] + wi * OutputReal[j / 2];
OutputReal[j / 2] = OutputReal[i / 2] - tempr;
OutputImag[j / 2] = OutputImag[i / 2] - tempi;
OutputReal[i / 2] += tempr;
OutputImag[i / 2] += tempi;
}
wr = (wtemp = wr) * wpr - wi * wpi + wr;
wi = wi * wpr + wtemp * wpi + wi;
}
mmax = istep;
}
}
...
#### 5.3.4 Moving Car
A less contrived example of an Objective Lucid program is presented in Figure
4. This is an example where a Car object changes with time. Eliminating $S$,
and ignoring the print call, we have have:
#typedecl
Car;
#JAVA
public class Car
{
public int x = 0;
public float speed;
public float speeddrop;
public float fuel;
public float fueldrainrate;
public Car()
{
// Assume initially car was already moving.
speed = 100.0; fuel = 40.5;
fueldrainrate = 0.018; speeddrop = 0.1;
}
// Move by a number of steps assuming constant speed
// and decelerate when ran out of fuel.
public Car move(int steps)
{
if(fuel > 0)
{
fuel -= fueldrainrate * speed * steps;
x += steps;
}
else if(speed > 0)
{
x += steps;
speed -= speeddrop * steps;
}
return this;
}
public void printCarState()
{
System.out.println
(
"Speed: " + speed + ", fuel: " + fuel +
", drain: " + fueldrainrate + ", x: " + x +
", speeddrop: " + speeddrop
);
}
}
#OBJECTIVELUCID
(C @.time 15).printCarState()
where
C = Car() fby.time S;
S = C.move(#time);
end;
Figure 4: Objective Lucid example of a Car object that changes in time.
C @.time 15 where C = Car() fby.time C.move(#.time)
Using the definition of fby gives:
C @.time 15 = (Car() fby.time C.move(#.time)) @.time 15 = if 15 <= 0 then
Car() else (C.move(#.time)) @.time (15 - 1) = C.move(14)
Our intention is that fby will give the sequence:
Car() Car.move(1) Car.move(2) ... Car.move(15)
This will work as follows. When one generates a demand for C.move(15) it’s not
satisfied until C.move(14) is until C.move(13) is … until C.move(1) is until
Car(), so it recurses back and finally the Car() object instance gets
constructed, and then the demands flow from 1 to 15 and the instance already
exists.
The car also does not accelerate indefinitely. It moves until it has enough
fuel, else it returns the car object with its members unmodified. The drop of
speed is also in place when fuel is depleted.
To further illustrate this idea let’s take the existing example of a simpler
problem of natural numbers presented in Figure 8 and convert it into Objective
Lucid as in Figure 6. First, we will present the eduction tree of the natural
numbers problem (see Figure 5, a corrected version of the one produced by
Paquet in [Paq99]) and then transmute it into the eduction tree of the
execution of the equivalent Objective Lucid propgram, as shown in Figure 7.
The program in Figure 6 exhibits the same properties as the Car example, so
the eduction tree will be similar but will take more space. The important
aspect here is to illustrate the difference between demands for STs and their
lazy execution (which is italisized, e.g. N.inc()); thus, the actual
invocation of a ST method happens at a later time after the demand is made so
we avoid not having called constructor prior execution of an instance method.
In the eduction trees the normal arrows correspond to demands made for
expressions and the bullet arrows correspond to the result of evaluation of
the demands, which are also bold and italic. In the Objective Lucid eduction
tree object instance is denoted as ClassName:MemberName:value and the {d:X}
presents the context of evaluation. The result of evaluation of the Objective
Lucid variant is said to be true because, as previously defined, void methods
are mapped to return true and the last expression bit that is evaluated here
is the print() method call of the instance of a Nat32 class, which returns
void.
Figure 5: Eduction Tree for the Natural Numbers Problem.
#typedecl
Nat42;
#JAVA
class Nat42
{
private int n;
public Nat42()
{
n = 42;
}
public Nat42 inc()
{
n++;
return this;
}
public void print()
{
System.out.println("n = " + n);
}
}
#OBJECTIVELUCID
(N @.d 2).print[d]()
where
dimension d;
N = Nat42[d]() fby.d N.inc[d]();
end
Figure 6: The Natural Numbers Problem in Objective Lucid. Figure 7: Eduction
Tree for the Natural Numbers Problem in Objective Lucid.
#### 5.3.5 Game of Life
The Game of Life [Gar70] would make a good benchmark for the GIPL. Life takes
place on a 2D grid and evolves in time, so it’s a 3D problem. The value of a
cell at time $T+1$ depends on the value of the cell and its 8 neighbours at
time $T$. Thus, there is a high branching factor and the IVW will get plenty
of exercise. Peter Grogono wrote a version in Haskell, which is functional and
lazy but is not concurrent and does not have an IVW. The author of this work
made a version in Indexical Lucid. In Figure 8 is the top-level function. The
Game of Life program is included in the test suite as a good elaborate test
case, but this work does not address any of the performance and efficiency
issues related to the execution and wareshousing, so no measurements have been
done two compare the efficiency of the program with and without the warehouse
nor with the Haskell program.
life = evolve T initial (conway life) where
initial = F(\i ->
if val Y i == 0 && 0 <= val X i && val X i < 5 then 1 else 0)
conway v = F(\i ->
let neighbours v =
ev v (n i) + ev v (ne i) + ev v (e i) + ev v (se i) +
ev v (s i) + ev v (sw i) + ev v (w i) + ev v (nw i) in
b2i(neighbours v == 3 || ev v i == 1 && neighbours v == 2))
evolve d s e = F(\i ->
if val d i == 0 then ev s i else ev e (prev d i))
b2i b = if b then 1 else 0
n i = F(...)
Figure 8: The Life in Haskell.
#INDEXICALLUCID
life = evolve(T, initial(T), conway(life, T))
where
dimension T;
evolve(d, u, v) = u fby.d v;
initial(d) =
if(Y == 0 && 0 <= X && X < 5) then 1 else 0
where
X = 0 fby.d X + 1;
Y = 0 fby.d Y + 1;
end;
conway(d, v) = b2i(neighbours == 3 || (v == 1 && neighbours == 2))
where
neighbours = n(d) + ne(d) + e(d) + se(d) + s(d) + sw(d) + w(d) + nw(d);
where
n(d) = v @.(d - 5);
ne(d) = v @.(d - 4);
e(d) = v @.(d + 1);
se(d) = v @.(d + 6);
s(d) = v @.(d + 5);
sw(d) = v @.(d + 4);
w(d) = v @.(d - 1);
nw(d) = v @.(d - 6);
end;
b2i(b) = if(b) then 1 else 0;
end;
end;
Figure 9: The Life in Indexical Lucid.
Explanations:
* •
$\mathit{evolve(d,u,v)}$ allows a value to evolve in the dimension $d$. The
first value of the stream is given by $u$ and subsequent values by $v$.
* •
$\mathit{initial(d)}$ defines the initial configuration (five ones in the row
0, zeroes everywhere else in the matrix 5-by-5).
* •
$\mathit{conway(d,v)}$ computes the successor of state $v$. The functions $n$,
$ne$, $e$, $se$, $s$, $sw$, $w$, and $nw$ are “navigators” that find values of
neighbours.
* •
$\mathit{b2i(d)}$ converts a Boolean to integer to decide the new value of an
entity.
### 5.4 Summary
There were many tests developed and exercised for the GIPSY. This section
attempted to show the reader the most representative ones and how the
Regression Tests Suite works in the GIPSY for the most modules of GIPC and GEE
and how JUnit is applied to make it possible and maintainable. Now, every new
module added to the GIPSY system will have to have a corresponding unit and/or
regression test (or several tests) exercising most of the features of this
module added.
## Chapter 6 Conclusion
To conclude, it is believed GIPSY is well off the ground and is steadily
getting ready for its first large public release to the research community. It
is becoming a lot more usable not only by a small circle of GIPSY developers,
but also by scientists and researchers from other research groups. Preliminary
testing (see Chapter 5) and results (Section 6.1) give confidence in the
success of an important step for the GIPSY in the are of flexible hybrid
intensional-imperative programming. To summarize, the newly introduced
features for the innovative intensional research platform GIPSY are a valuable
asset allowing us to release GIPSY to the masses and a new release will be
made at the SourceForge.net at http://sf.net/projects/sfgipsy circa the end of
December 2005 - January 2006.
### 6.1 Results
#### 6.1.1 Experiments
The experiments conducted on the GIPSY research platform were primarily
design, development, and testing of hybrid programming paradigms by fusing
together intensional and imperative languages. For test experiments please
refer to Chapter 5.
#### 6.1.2 Interpretation of Results
After extensive testing of the design and implementation of ideas presented in
Chapter 3 we can see an enhanced, more flexible GIPSY system taking off the
ground. Most of regression tests pass for the developed sample programs with
known errors and failures.
### 6.2 Discussions and Limitations
#### 6.2.1 Lack of Hybrid Intensional-Imperative Semantics Proofs
The semantics for the GIPSY Type System was not defined and the one of JLucid
and Objective Lucid was not formally proven to be correct.
#### 6.2.2 Genuine Imperative Compilers
The most serious limitation of the current implementation of the hybrid
paradigm is that there are no genuine imperative GIPSY compilers. The Java
wrapper compiler classes merely resort to the external tools from the library
of enumerated tools. This makes overall error checking and reporting
cumbersome. Additionally, this slows down the compilation process.
#### 6.2.3 Cross-Language Data Type Mapping
When implementing other imperative language compilers than Java, or a genuine
compiler for Java, a special mapping has to be explicitly established in the
form of TypeMap. We can avoid this for C/C++ with the JNI [Ste05], but not for
other popular languages.
#### 6.2.4 Dimension Index Overflow
While this limitation is not directly related to the main topics of this
thesis, it has to be mentioned. In the current implementation of the dimension
type in all Lucid variants is done as a simple Java integer, and as such, is
finite. Thus, incorrectly written Lucid programs or programs that may require
high dimension values may overflow the dimension index rendering execution of
the program incorrect. This limitation is not handled by the GEE nor
constrained in the operational semantics of Lucid.
#### 6.2.5 Hybrid-DFG Integration
This thesis does not address placement, rendering, and integration of the
hybrid AST nodes into DFGs.
#### 6.2.6 Dealing With Side Effects and Abrupt Termination
As of this implementation, GEE has very limited control over what’s happening
inside the STs in terms side effects, exceptions, non-termination, etc. in the
Java (or other imperative language) code causing it to exit prematurely or to
hang. Likewise, we cannot do warehousing of non-immutable STs due to the side
effects, i.e. when the same arguments are given to an ST may yield a different
result at different invocations. This is serious aspect, which is related to
the development of any future semantics of the hybrid programming languages
and deserves a separate publication.
#### 6.2.7 Imperative Function Overloading
It is an error to write the following:
#funcdecl int foo(int); int foo(float); ...
but it shouldn’t be. This is an error in the sense that only the last
declaration is retained due to the way function identifiers are handled, so no
function overloading at this moment is officially supported. The issue of
dealing with the semantics of a type system in which this is possible,
especially if we support multiple imperative PLs, where each may have
potentially its own type system or even paradigm is complex. However, this
feature is nice to have and some practical aspects can be implemented, which
will be a research topic on its own.
#### 6.2.8 Cross-Imperative Language Calls
Normally, an ST written say in #JAVA cannot call another ST in say #C. This
limitation is that only the intensional part can make calls to the imperative
functions. This eliminates the need to keep the type mappings between all
possible combinations of the imperative languages and semantics associated
with this.
However, depending on the language, procedures written in the same language
can possibly communicate by calling each other. E.g. in Java, defining free
members and passing state between free functions is possible as nothing is
done to prevent this.
#JAVA int i;
int foo() { return i + 1; }
int bar() { i++; return foo(); }
This is based on the knowledge about the internal implementation i.e. the “int
i;” bit will also be wrapped in the class, so it’d be legal to have it from
the Java’s point of view; however, is considered to be a kludge and non-
portable feature. To be on the safer side, the STs like that should be written
assuming no knowledge of internal state for communication is available.
#### 6.2.9 Security
JLucid, Objective Lucid, and GICF opened up doors for very flexible use of
external languages and resources as a part of intensional computation.
Unfortunately, there are security considerations to deal with when embedding a
vulnerable unsigned code from possibly untrusted remote location and then
propagate it to all the workers participating in computation can result
resulting either gaining some unwanted privileges to the attackers or DDoS.
## Chapter 7 Future Work
The future work to take on will focus in the following areas to either address
the limitations outlined in Section 6.2 or to introduce new features, not
necessarily all related to the topics of this thesis.
* •
Integration of the Demand Migration System (DMS) [VP05].
* •
Formal semantic verification from Indexical Lucid through Objective Lucid.
* •
Placement of hybrid nodes into DFGs.
* •
Security.
* •
Trial C compiler with JNI.
* •
Fully Explore Array Properties.
* •
Genuine imperative compilers in GICF.
* •
Introduction functional language compilers.
* •
Visualization and control of communication patterns and load balancing.
* •
Target Host Compilation.
* •
Java wrapper for the DFG Editor of Yimin Ding.
### 7.1 Formal Verification of Semantic Rules and the GIPSY Type System
One needs to formally conduct verification proofs of the semantic rules from
Indexical Lucid to Objective Lucid in PVS or Isabelle, so this project can be
undertaken in the near future and the work on it has already began.
Specifically, a relation to the semantic of objects and Java’s operational
semantics has to be made. Likewise, the semantics of the newly introduced
GIPSY type system has to be formally defined.
### 7.2 Dealing with Data Flow Graphs in Hybrid Programming
This thesis did not deal with the way on how to augment DFGAnalyzer and
DFGGenerator to support hybrid GIPSY programs. This can be addressed by adding
an unexpandable imperative DFG node to the graph. To make it more useful, i.e.
expandable and so it’s possible to generate the GIPSY code off it or reverse
it, would require having the genuine compilers as in Section 7.6 for
imperative languages, which is far from trivial.
### 7.3 Security
Security is a substantial concern in distributed computing. The great
flexibility provided by embedded Java in JLucid (and later in Objective Lucid)
can be misused and be a source of security breaches or DDoS attacks (e.g., due
to explicit oversynchronization using Java’s synchronization primitives
explicitly). Thus, the follow-up work in this direction would include
malicious code detection in embedding and distributing as well as explicit
synchronization points so that there are no deadlocks and DDoS potential is
reduced. This concern touches the compiler (GIPC), the Generator-Worker
architecture, the GIPSY Server, and the GIPSY Screen Saver components of the
GIPSY system.
### 7.4 Implementation of the C Compiler in GICF
An methodology of implementing a C compiler, and therefore, C CPs and STs has
been devised, but never implemented, so in the future a C compiler will be
implemented as a part of GICF with the JNI [Ste05].
### 7.5 Fully Explore Array Properties
The arrays in JLucid, Objective Lucid, and their generalization in GICF
requries further exploration and formalization and mapping of the GIPSY arrays
to their native equivalents.
### 7.6 Genuine Imperative and Functional Language Compilers
Future work in this area is to focus on writing our genuine compilers for the
mentioned imperative languages and extending support for more imperative and
functional languages (e.g. LISP, Scheme, or Haskell) and make it as much
automated as possible.
### 7.7 Visualization and Control of Communication Patterns and Load
Balancing
It is proposed to have a “3D editor” within RIPE’s DemandMonitor that will
render in 3D space the current communication patterns of a GIPSY program in
execution or replay it back and allow the user visually to redistribute
demands if they go off balance between workers. A kind of virtual 3D remote
control with a mini expert system, an input from which can be used to teach
the planning, caching, and load-balancing algorithms to perform efficiently
next time a similar GIPSY application is run.
### 7.8 Target Host Compilation
This has to do with enabling the GEE to deliver the ST source code around and
compile it on the target host instead of sending a pre-compiled version of the
STs. This is an experimental feature can be useful and dangerous and requires
a lot of research.
### 7.9 The GIPSY Screen Saver
This is a sample implementation of a worker, outlined in Section 3.3.3.4,
would represent an application for a PC as a way to contribute to a GIPSY
program execution. Three sample implementations of screen saver workers exist
one for Windows, one for Linux and one for MacOS X.
### 7.10 The GIPSY Server
A so-called “GIPSY server” will be implemented to be able to serve intensional
or otherwise requests primarily through the HTTP protocol, thus acting like a
mini-GIPSY intensional web server. It would accept request from remote clients
via HTTP or local clients via command line and be the starting point of
computation (an intensional computation resource) available to all those who
have no resources to set up GIPSY. This is not duplicate any of the DMS [VP05]
nor it is a part of RIPE, as it is primarily non-interactive and runs on the
background.
## Bibliography
* [AW76] Edward A. Ashcroft and William W. Wadge. Lucid - A Formal System for Writing and Proving Programs. volume 5. SIAM J. Comput. no. 3, 1976.
* [AW77] Edward A. Ashcroft and William W. Wadge. Erratum: Lucid - A Formal System for Writing and Proving Programs. volume 6(1):200. SIAM J. Comput., 1977.
* [BddzzP+05] Brian Berliner, david d ‘zoo’ zuhn, Jeff Polk, Larry Jones, Derek R. Price, Mark D. Baushke, and other authors. Concurrent Versions System (CVS). Free Software Foundation, Inc., 1989-2005. http://www.nongnu.org/cvs/.
* [Ber05] Stefan M. Bernsee. The DFT “à pied”: Mastering The Fourier Transform in One Day. DSPdimension.com, 1999-2005. http://www.dspdimension.com/data/html/dftapied.html.
* [BKU98] Gilbert Babin, Peter Kropf, and Herwig Unger. A Two-Level Communication Protocol for a Web Operating System (WOS). In Proceedings of IEEE Euromicro Workshop on Network Computing, Vasteras, Sweden, pages 934–944. IEEE, 1998.
* [BM96] Fancois Bourdoncle and Stephan Merz. On the integration of the functional programming, class-based object-oriented programming, and multi-methods. Technical report, Centre de Mathemathiques Appliquees, Ecole des Mines de Paris and Institut fur Informatik, Technische Universitat Munchen, October 1996.
* [Bor03] Borland. Borland JBuilder X. Borland Software Corporation, 1997-2003. http://www.borland.com/us/products/jbuilder/.
* [c+04] Eclipse contributors et al. Eclipse Platform. IBM, 2000-2004. http://www.eclipse.org/platform.
* [Car47] Rudolf Carnap. Meaning and Necessity: a Study in Semantics and Modal Logic. University of Chicago Press, Chicago, USA, 1947.
* [Con05] Ant Project Contributors. Apache Ant. The Apache Software Foundation, 2000-2005. http://ant.apache.org/.
* [CP05] Various Contributors and GNU Project. GNU Compiler Collection (GCC). Free Software Foundation, Inc., 1988-2005. http://gcc.gnu.org/onlinedocs/gcc/.
* [Dij65] E.W. Dijkstra. Cooperating Sequential Processes. Technical report, Technological University, Eidhoven, The Netherlands, 1965.
* [Dij71] E.W. Dijkstra. Hierarchical ordering of sequential processes. pages 115–138. Acta Inf., 1971.
* [Din04] Yi Min Ding. Bi-directional Translation Between Data-Flow Graphs and Lucid Programs in the GIPSY Environment. Master’s thesis, Department of Computer Science and Software Engineering, Concordia University, 2004.
* [Dod96] Chris Dodd. Intensional Programming I, chapter Rank analysis in the GLU compiler, pages 76-82. World Scientific, Singapore, 1996.
* [Fla97] D. Flanagan. Java in a Nutshell. O’Reily & Associates, Inc., second edition, 1997. ISBN 1-56592-262-X.
* [Fou05] Apache Foundation. Apache Jakarta Tomcat. apache.org, 1999-2005. http://jakarta.apache.org/tomcat/index.html.
* [Gar70] Martin Gardner. The Fantastic Combinations of John Horton Conway’s New Solitaire Game ‘Life’. pages 120–123. Scientific American, October 1970. http://ddi.cs.uni-potsdam.de/HyFISCH/Produzieren/lis_projekt/proj_gamel%ife/ConwayScientificAmerican.htm.
* [GB04] Erich Gamma and Kent Beck. JUnit. Object Mentor, Inc., 2001-2004. http://junit.org/.
* [Gin90] Armando R. Gingras. Dining Philosophers Revisited. In SIGCSE Bulletin, volume 22 No. 3, pages 21–28, September 1990\.
* [GP99] Jean-Raymond Gagné and John Plaice. Demand-Driven Real-Time Computing. World Scientific, September 1999.
* [Gre05] Dale Green. Java Reflection API. Sun Microsystems, Inc., 2001-2005. http://java.sun.com/docs/books/tutorial/reflect/index.html.
* [Gro01] Peter Grogono. A LaTeX2e Gallimaufry. Techniques, Tips, and Traps. Department of Computer Science and Software Engineering, Concordia University, March 2001. http://www.cse.concordia.ca/~grogono/documentation.html.
* [Gro02a] Peter Grogono. Semantics of Programming Languages, COMP745 Course Notes. Department of Computer Science and Software Engineering, Concordia University, 1996-2002.
* [Gro02b] Peter Grogono. GIPC Increments. Technical report, Department of Computer Science and Software Engineering, Concordia University, April 2002.
* [Gro04] Peter Grogono. Intensional programming in Onyx. Technical report, Department of Computer Science and Software Engineering, Concordia University, April 2004.
* [HJ02] Marieke Huisman and Bart Jacobs. Java Program Verification via a Hoare Logic with Abrupt Termination. Technical report, Computing Science Institute, University of Nijmegen, Nijmegen, The Netherlands, 2002. www.cs.ru.nl/B.Jacobs/PAPERS/FASE00.ps.Z.
* [JA03] Harry F. Jordan and Gita Alaghband. Fundamentals of Parallel Processing. Pearson Education, Inc., 2003. ISBN 0-13-901158-7.
* [JD96] Raganswamy Jagannathan and Chris Dodd. GLU programmer’s guide. Technical report, SRI International, Menlo Park, California, 1996.
* [JDA97] Raganswamy Jagannathan, Chris Dodd, and Iskender Agi. GLU: A high-level system for granular data-parallel programming. In Concurrency: Practice and Experience, volume 1, pages 63–83, 1997.
* [Kri59] Saul A. Kripke. A Completeness Theorem in Modal Logic. 1959\.
* [Kri63] Saul A. Kripke. Semantical Considerations on Modal Logic. 1963\.
* [LGP03] Bo Lu, Peter Grogono, and Joey Paquet. Distributed execution of multidimensional programming languages. In Proceedings 15th IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS 2003), volume 1, pages 284–289. International Association of Science and Technology for Development, November 2003\.
* [Lou97] Kenneth C. Louden. Compiler Construction: Principles and Practice. PWS Publishing Company, 1997. ISBN 0-564-93972-4.
* [Lu04] Bo Lu. Developing the Distributed Component of a Framework for Processing Intentional Programming Languages. PhD thesis, Department of Computer Science and Software Engineering, Concordia University, March 2004.
* [Mam05] Qusay H. Mamoud. Getting Started With JavaSpaces Technology: Beyond Conventional Distributed Programming Paradigms. Sun Microsystems, Inc., July 2005. http://java.sun.com/developer/technicalArticles/tools/JavaSpaces/.
* [MCSN05] Serguei Mokhov, Ian Clement, Stephen Sinclair, and Dimitrios Nicolacopoulos. Modular Audio Recognition Framework. Department of Computer Science and Software Engineering, Concordia University, 2002-2005. http://marf.sf.net.
* [Mic04] Sun Microsystems. NetBeans 4.0. Sun Microsystems, Inc., December 2004. http://www.netbeans.org.
* [Mic05a] Sun Microsystems. Java Servlet Technology. Sun Microsystems, Inc., 1994-2005. http://java.sun.com/products/servlets.
* [Mic05b] Sun Microsystems. The Java Website. Sun Microsystems, Inc., 1994-2005. `<http://java.sun.com>`.
* [Mic05c] Sun Microsystems. JavaServer Pages Technology. Sun Microsystems, Inc., 2001-2005. http://java.sun.com/products/jsp/.
* [Moe04] Anders Moeller. Program Verification with Hoare Logic. Technical report, University of Aarhus, 2004. http://www.brics.dk/~amoeller/talks/hoare.pdf.
* [Mok03a] Serguei Mokhov. GIPSY: CVS Service on Newton, A Crush Guide. Department of Computer Science and Software Engineering, Concordia University, June 2003.
* [Mok03b] Serguei Mokhov. Newton: Basic Security. Department of Computer Science and Software Engineering, Concordia University, June 2003.
* [Mok05a] Serguei Mokhov. Makefile Tutorial. Department of Computer Science and Software Engineering, Concordia University, 2003-2005.
* [Mok05b] Serguei Mokhov. Naming & Coding Conventions. Department of Computer Science and Software Engineering, Concordia University, 2003-2005.
* [MP05a] Serguei Mokhov and Joey Paquet. General Imperative Compiler Framework within the GIPSY. In Proceedings of PLC2005, Las Vegas, Nevada, USA, pages 36–42. CSREA Press, June 2005.
* [MP05b] Serguei Mokhov and Joey Paquet. Objective Lucid – First Step in Object-Oriented Intensional Programming in the GIPSY. In Proceedings of PLC2005, Las Vegas, Nevada, USA, pages 22–28. CSREA Press, June 2005.
* [MPG05] Serguei Mokhov, Joey Paquet, and Peter Grogono. Towards JLucid, Lucid with Embedded Java Functions in the GIPSY. In Proceedings of PLC2005, Las Vegas, Nevada, USA, pages 15–21. CSREA Press, June 2005.
* [MS01] Brian McNamara and Yannis Smaragdakis. Functional Programming in C++ using the FC++ Library. In SIGPLAN Notices, volume 36(4), pages 25–30, 2001.
* [Paq99] Joey Paquet. Scientific Intensional Programming. PhD thesis, Department of Computer Science, Laval University, Sainte-Foy, Canada, 1999.
* [PGW04] Joey Paquet, Peter Grogono, and Ai Hua Wu. Towards a Framework for the General Intensional Programming Compiler in the GIPSY. In 19th Annual ACM Conference on Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA 2004). Vancouver, Canada. ACM, October 2004.
* [PK04] Nikolaos Papaspyrou and Ioannis T. Kassios. GLU# embedded in C++: a marriage between multidimensional and object-oriented programming. In Softw., Pract. Exper., volume 34(7), pages 609–630, 2004.
* [Pre93] William H. Press. Numerical Recipes in C. Cambridge University Press. Cambridge, UK., second edition, 1993.
* [Pro03a] David K. Probst. Cyclic Reduction. Concordia University, Montreal, Quebec, Canada, February 2003.
* [Pro03b] David K. Probst. The Principle Of Least Bandwidth. Commentary from the HEC. Concordia University, Montreal, Quebec, Canada, February 2003.
* [Pro03c] David K. Probst. The Programmability Metric. Commentary from the HEC. Concordia University, Montreal, Quebec, Canada, February 2003.
* [PW05] Joey Paquet and Ai Hua Wu. GIPSY – A Platform for the Investigation on Intensional Programming Languages. In Proceedings of PLC2005, Las Vegas, Nevada, USA, pages 8–14. CSREA Press, June 2005.
* [Ren02] Chun Lei Ren. General Intensional Programming Compiler (GIPC) in the GIPSY. Master’s thesis, Department of Computer Science and Software Engineering, Concordia University, 2002.
* [RG05a] The GIPSY Research and Development Group. The GIPSY Project. Department of Computer Science and Software Engineering, Concordia University, 2002-2005. http://newton.cs.concordia.ca/~gipsy/.
* [RG05b] The GIPSY Research and Development Group. The GIPSYwiki: Online GIPSY collaboration platform. Department of Computer Science and Software Engineering, Concordia University, 2005. http://newton.cs.concordia.ca/~gipsy/gipsywiki.
* [Ron94] Panagiotis Rondogiannis. Higher-Order Functional Languages and Intensional Logic. PhD thesis, Department of Computer Science, University of Victoria, Victoria, Canada, 1994.
* [SM02] Yannis Smaragdakis and Brian McNamara. FC++: Functional tools for object-oriented tasks. In Softw., Pract. Exper., volume 32(10), pages 1015–1033, 2002\.
* [SMSP00] Richard Stallman, Roland McGrath, Paul Smith, and GNU Project. GNU Make. Free Software Foundation, Inc., 1997-2000. http://www.gnu.org/software/make/.
* [ST98] David B. Skillicorn and Domenico Talia. Models and Languages for Parallel Computation, volume 30, No. 2 of ACM Computing Surveys. ACM, Inc., June 1998. http://www.ece.cmu.edu/~ece767/papers/skillicorn.pdf.
* [Ste05] Beth Sterns. The Java Native Interface (JNI). Sun Microsystems, Inc., 2001-2005. http://java.sun.com/docs/books/tutorial/native1.1/index.html.
* [Tao04] Lei Tao. Warehouse and Garbage Collection in the GIPSY Environment. Master’s thesis, Department of Computer Science and Software Engineering, Concordia University, 2004.
* [VC05] Sreenivasa Viswanadha and Contributors. Java Compiler Compiler (JavaCC) - The Java Parser Generator. java.net, 2001-2005. https://javacc.dev.java.net/.
* [VP05] Emil Vassev and Joey Paquet. A Generic Framework for Migrating Demands in the GIPSY’s Demand-Driven Execution Engine. In Proceedings of PLC2005, Las Vegas, Nevada, USA, pages 29–35. CSREA Press, June 2005.
* [WAP05] Kaiyu Wan, Vasu Alagar, and Joey Paquet. Lucx: Lucid Enriched with Context. In Proceedings of PLC2005, Las Vegas, Nevada, USA, pages 48–14. CSREA Press, June 2005.
* [WP05] Ai Hua Wu and Joey Paquet. Object-Oriented Intensional Programming in the GIPSY: Preliminary Investigations. In Proceedings of PLC2005, Las Vegas, Nevada, USA, pages 43–47. CSREA Press, June 2005.
* [WPG03] Ai Hua Wu, Joey Paquet, and Peter Grogono. Design of a Compiler Framework in the GIPSY System. In Proceedings 15th IASTED International Conference on Parallel and Distributed Computing and Systems (PDCS 2003), volume 1, pages 320–328. International Association of Science and Technology for Development, November 2003\.
* [WSoafaotw05] Jimmy Wales, Larry Sanger, and other authors from all over the world. Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc., 2001-2005. http://wikipedia.org.
* [Wu02] Ai Hua Wu. Semantic Checking and Translation in the GIPSY. Master’s thesis, Department of Computer Science and Software Engineering, Concordia University, 2002.
## Appendix A Definitions and Abbreviations
### A.1 Abbreviations
* •
AST - Abstract Syntax Tree
* •
COM - Component Object Model
* •
CORBA - Common Object Requester Broker Architecture
* •
CLP - Cluster-Level Parallelism
* •
CP - Communication Procedure, Section 3.3.3.2
* •
CVS - Concurrent Versions System
* •
DCOM - Distributed COM
* •
DDoS - Distributed Denial of Service (attack).
* •
FFT - Fast Fourier Transform
* •
FTP - File Transfer Protocol
* •
DPR - Demand Propagation Resource, Section 2.5.4.1, [RG05a, PW05]
* •
GEE - General Eduction Engine
* •
GEER - GEE Resources, Section 4.1.1.14
* •
GIPC - General Intensional Program Compiler, Figure 10, [RG05a, PW05]
* •
GIPL - General Intensional Programming Language, [Paq99, RG05a, PW05]
* •
GIPSY - General Intensional Programming System, [RG05a, PW05]
* •
GLU - Granular Lucid, [JD96, JDA97, Paq99]
* •
HTTP - Hyper-Text Transfer Protocol
* •
IDP - Intensional Demand Propagator, Section 3.3.3.4, [RG05a, PW05]
* •
IDS - Intensional Data-dependency Structure
* •
IP - Intensional Programming
* •
IPL - Intensional Programming Language (e.g. GIPL, GLU, Lucid, Indexical
Lucid, JLucid, Tensor Lucid, Objective Lucid, Onyx [Gro04])
* •
IVW - Intensional Value Warehouse, Section 3.3.3.4, [RG05a, PW05]
* •
JDK - Java Developer’s Kit
* •
JNI - Java Native Interface
* •
JRE - Java Runtime Environment
* •
JSSE - Java Secure Socket Extension
* •
MARF - Modular Audio Recognition Framework [MCSN05]
* •
MPI - Message Passing Interface
* •
NCP - Native Communication Procedure
* •
NST - Native Sequential Thread
* •
NUMA - Non-Uniform Memory Access
* •
PVM - Parallel Virtual Memory System
* •
RFE - Ripe Function Executor, Section 3.3.3.4, [RG05a, PW05]
* •
RMI - Remote Method Invocation
* •
RPC - Remote Procedure Call
* •
SIPL - Specific IPL (e.g. Indexical Lucid, JLucid, Tensor Lucid, Objective
Lucid, Onyx)
* •
SLP - Stream-Level Parallelism
* •
ST - Sequential Thread, Section 3.3.3.1
* •
TLP - Thread-Level Parallelism
* •
TTS - Time To Solution
* •
UMA - Uniform Memory Access
* •
URI - Unified Resource Indentifier
* •
URL - Unified Resource Locatior
## Appendix B Sequential Thread and Communication Procedure Interfaces
In this section the actual definitions of the CP and ST interfaces, an example
of a generated wrapper class and a Worker are presented.
### B.1 Sequential Thread Interface
See Figure 1.
package gipsy.interfaces;
import java.io.Serializable;
import java.lang.reflect.Method;
/**
* <p>Sequential Thread represents a piece work to be done.
* Has to extend Serializable for RMI, CORBA, COM+, Jini to work.
* Runnable needed to run it in a separate thread.</p>
*
* $Id: ISequentialThread.java,v 1.13 2005/09/12 01:24:38 mokhov Exp $
*
* @version $Revision: 1.13 $
* @author Serguei Mokhov, [email protected]
* @since Inception
*/
public interface ISequentialThread
extends Runnable, Serializable
{
/**
* Work-piece to be done.
* @return WorkResult container
*/
public WorkResult work();
public WorkResult getWorkResult();
public void setMethod(Method poSTMethod);
}
// EOF
Figure 1: Sequential Thread Interface.
### B.2 Communication Procedure Interface
See Figure 2.
package gipsy.interfaces;
import gipsy.lang.GIPSYType;
import java.io.Serializable;
/**
* <p>CommunicationProcedure represents the means of delivery of sequential threads.</p>
* $Id: ICommunicationProcedure.java,v 1.11 2005/10/11 08:34:11 mokhov Exp $
* @version $Revision: 1.11 $
* @author Serguei Mokhov, [email protected]
* @since Inception
* @see gipsy.interfaces.SequentialThread
*/
public interface ICommunicationProcedure
extends Serializable
{
public GIPSYType getReturnType();
public GIPSYType getParamType(final int piParamNumber);
public GIPSYType[] getParamTypes();
public void setReturnType(GIPSYType poType);
public void setParamType(final int piParamNumber, GIPSYType poType);
public void setParamTypes(GIPSYType[] paoTypes);
public GIPSYType getParamType(String pstrLexeme);
public GIPSYType getParamType(String pstrLexeme, String pstrID);
public int getParamListSize();
/**
* Perform any initialization actions required.
* @return status object of the result of send operation.
* @throws CommunicationException in case of error
*/
public CommunicationStatus init()
throws CommunicationException;
/**
* Open a connection; whatever that means for a given protocol.
* @return status object of the result of send operation.
* @throws CommunicationException in case of error
*/
public CommunicationStatus open()
throws CommunicationException;
/**
* Close a connection; whatever that means for a given protocol.
* @return status object of the result of send operation.
* @throws CommunicationException in case of error
*/
public CommunicationStatus close()
throws CommunicationException;
/**
* Defines the means of sending data. Should be overridden by
* a concrete implementation, such as JINI, COM, CORBA, etc.
* @return status object of the result of send operation.
* @throws CommunicationException in case of error
*/
public CommunicationStatus send()
throws CommunicationException;
/**
* Defines the means of receiving data. Should be overridden by
* a concrete implementation, such as JINI, COM, CORBA, etc.
* @return status object of the result of receive operation.
* @throws CommunicationException in case of error
*/
public CommunicationStatus receive()
throws CommunicationException;
}
Figure 2: Communication Procedure Interface.
### B.3 Generated Sequential Thread Wrapper Class
This is a more complete version of the generated wrapper class for the code in
Figure 9.
import java.util.Hashtable;
import java.util.Vector;
public class <filename>_<machine_name>_<timestamp>
implements gipsy.interfaces.ISequentialThread
{
private OriginalSourceCodeInfo oOriginalSourceCodeInfo;
/**
* Inner class with original source code information
*/
public class OriginalSourceCodeInfo
{
/**
* For debugging / monitoring; generated statically
*/
private String strOriginalSource =
"int getN(int piDimension)" +
"{" +
" if(piDimension <= 0)" +
" return get42();" +
" else" +
" return getN(piDimension - 1) + 1;" +
"}" +
"" +
"int get42()" +
"{" +
" return 42;" +
"}";
/**
* Mapping to original source code position for error reporting
*/
private Hashtable oLineNumbers = new Hashtable();
/**
* Body is filled in by the preprocessor statically
*/
public OriginalSourceCodeInfo()
{
Vector int_getN_int_piDimension = new Vector();
// Start line and Length in lines
int_getN_int_piDimension.add(new Integer(3));
int_getN_int_piDimension.add(new Integer(7));
this.oLineNumbers.put
(
"int getN(int piDimension)",
int_getN_int_piDimension
);
Vector int_get42 = new Vector();
int_get42.add(new Integer(11));
int_get42.add(new Integer(4));
this.oLineNumbers.put
(
"int get42()",
int_get42
);
}
public Hashtable getLineNumbersHash()
{
return this.oLineNumbers;
}
public int getLineNumberForFunction(String pstrFunctionSignature)
{
}
public int getFunctionSourceLength(String pstrFunctionSignature)
{
}
public String toString()
{
}
}
/**
* Constructor
*/
public <filename>_<machine_name>_<timestamp>()
{
this.oOriginalSourceCodeInfo = new OriginalSourceCodeInfo();
}
public String toString()
{
return this.oOriginalSourceCodeInfo.toString();
}
/*
* Implementation of the SequentialThread interface
*/
// Body generated by the compiler
public void run()
{
Payload oPayload = new Payload();
oPayload.add("d", new Integer(42));
work(oPayload);
}
// Body generated by the compiler statically
public WorkResult work(Payload poPayload)
{
WorkResult oWorkresult = new WorkResult();
oWorkresult.add(getN(poPayload.getVaueOf("d")));
return oWorkResult;
}
/*
* ------------
* The below are generated off the source file nat2java.ipl
* ------------
*/
public static int getN(int piDimension)
{
if(piDimension <= 0)
return get42();
else
return getN(piDimension - 1) + 1;
}
public static int get42()
{
return 42;
}
}
### B.4 Sample Worker’s Implementation
package gipsy.wrappers;
//import gipsy.interfaces.SequentialThread;
import gipsy.interfaces.ICommunicationProcedure;
import gipsy.util.*;
import marf.util.BaseThread;
/**
* Worker Class Definition
*
* $Revision: 1.11 $ by $Author: mokhov $ on $Date: 2004/11/06 00:50:09 $
*
* @version $Revision: 1.11 $
* @author Serguei Mokhov
*/
public class Worker extends BaseThread
{
/**
* Aggregation of sequential threads.
*/
private Thread[] aoSequentialThreads = null;
/**
* Set of available communication procedures for different protocols.
*/
private ICommunicationProcedure[] aoCommuncationProcedures = null;
/**
* Default settings.
*/
public Worker()
{
super();
}
/**
* Generate a demand.
*/
public void demand()
{
}
/**
* Receive a result on a demand.
*/
public void receive()
{
}
/**
* Perform computation.
*/
public void work() throws GIPSYException
{
try
{
for(int i = 0; i < this.aoSequentialThreads.length; i++)
this.aoSequentialThreads[i].start();
}
catch(NullPointerException e)
{
throw new GIPSYException
(
"Worker TID=" + getTID() +
" did not have any sequential threads to work on."
);
}
}
/**
* Stops worker thread.
*/
public void stopWorker()
{
}
/**
* From Runnable interface, for TLP
*/
public void run()
{
try
{
work();
}
catch(GIPSYException e)
{
System.err.println(e);
}
}
}
// EOF
## Appendix C Architectural Module Layout
### C.1 GIPSY Java Packages and Directory Structure
Normally, a directory structure of a Java project corresponds to the package
naming; thus, the packages are named and declared after the directories. By
the means of Java packages, all the classes within the project and external
applications “know” how to identify and import the classes they intend to use.
A fully-qualified class name includes all the packages starting from the
“root” (the top-level directory of the hierarchy) all the way up to the class
itself, separated by a dot. The GIPSY Java packages breakdown as of this
writing corresponds to the Figure 1.
Figure 1: GIPSY Java Packages Hierarchy.
The logical breakdown was performed in accordance with the original conceptual
design primarily produced by Joey Paquet and further by Aihua Wu and Bo Lu,
has been the primary source of the hierarchy plus any exceptions and
extensions that various team members come up with or have been forced to
during implementation were taken into account.
The basic structure is as follows. The top root hierarchy is logically the
gipsy package. The major non-utility packages under it, which come from the
conceptual design, are GIPC, GEE, and RIPE. The major utility packages under
gipsy that are not present in the conceptual design are: interfaces for most
intermodule communication; wrappers for object wrapping; storage for the
serializable interface classes; util for most common exceptions and utility
modules (e.g. fast linked list [Din04]); and tests for the Unit and Regression
Testing Suites.
Under the GIPC package the major modules (to be discussed later in this
chapter) include Preprocessing for general GIPSY program preprocessing,
intensional and imperative language compilers and their necessary followers
(GenericTranslator for the former and CommunicationProcedureGenerator and
SequentialThreadGenerator for the latter). Then the DFG package for Lucid-to-
data-flow-graph and back generation.
The GEE’s main packages includes IDP for demand propagation and IVW for
caching and garbage collection.
Under RIPE we have interactive run-time editing and monitoring modules that
include textual editor, DFG editor, and the web-based editor.
### C.2 GIPSY Modules Packaging
GIPSY’s major and minor modules are packaged into a set of runnable .jar files
and distributed with wrapper scripts to be either used as ordinary command
line tools as a part of GIPSY Development Kit or the WebEditor web
application. Different .jar files include a subset of all GIPSY modules
depending on the need, e.g. GIPC includes GIPC-related classes plus GEE as we
allow to optionally invoke GEE after successful compilation. RIPE, except
itself, needs both GIPC and GEE, whereas GEE does not at all require presence
of any other module. Thus, the GIPSY binary distribution is broken down into
five major .jar files (notice, that these files do not include any external
libraries GIPSY references):
* •
gipsy.jar simply includes almost all of GIPSY.
* •
gipc.jar should be used/distributed as a part of so-called “GIPSY Development
Kit” if someone wants to be able to compile intensional programs and
optionally run them.
* •
gee.jar represents GIPSY’s non-interactive run-time environment, the GEE. This
can be distributed alone to the hosts that only wish to run pre-compiled GIPSY
programs and have no development environment set up.
* •
ripe.jar includes most of the interactive programming environment of the GIPSY
along with GIPC and GEE.
* •
Regression.jar includes the Regression Testing application plus all of GIPC
and GEE as the most exercised modules for testing as of this writing.
The Table 1 shows correspondence between the variety of modules and their
containment within a .jar file.
Table 1: Correspondence of the GIPSY .jar files and the modules. Module / Jar | gipsy.jar | ripe.jar | gipc.jar | gee.jar | Regression.jar
---|---|---|---|---|---
GIPSY | $\star$ | | | |
GIPC | $\star$ | $\star$ | $\star$ | | $\star$
RIPE | $\star$ | $\star$ | | |
GEE | $\star$ | $\star$ | $\star$ | $\star$ | $\star$
DFG/GIPC | $\star$ | $\star$ | $\star$ | | $\star$
DFGEditor | $\star$ | $\star$ | | |
Regression | | | | | $\star$
Interfaces | $\star$ | $\star$ | $\star$ | $\star$ | $\star$
WebEditor | | | | |
gipsy.lang | $\star$ | $\star$ | $\star$ | $\star$ | $\star$
gipsy.wrappers | $\star$ | $\star$ | $\star$ | $\star$ | $\star$
gipsy.util | $\star$ | $\star$ | $\star$ | $\star$ | $\star$
gipsy.storage | $\star$ | $\star$ | $\star$ | $\star$ | $\star$
## Appendix D Grammar Generation Scripts for JLucid and Objective Lucid
### D.1 jlucid.sh
#!/bin/bash
strDate=‘date‘
cat <<GRAMMAR_TAIL
/*
* Generated by jlucid.sh on $strDate
*/
/**
* @since $strDate
*/
void embed() #EMBED : {}
{
//<EMBED> <LPAREN> url() E() ( <COMMA> E() )* <RPAREN> <SEMICOLON>
<EMBED> <LPAREN> url() <COMMA> <STRING_LITERAL> ( <COMMA> E() )* <RPAREN> <SEMICOLON>
}
/**
* @since $strDate
*/
void array() #ARRAY : {}
{
<LBRACKET> E() ( <COMMA> E() )* <RBRACKET>
}
/**
* URL -> CHARACTER_LITERAL | STRING_LITERAL.
* @since $strDate
*/
void url() #URL :
{
Token oToken;
}
{
(
oToken = <CHARACTER_LITERAL>
| oToken = <STRING_LITERAL>
)
{
jjtThis.setImage(oToken.image);
}
}
// EOF
GRAMMAR_TAIL
# EOF
### D.2 JGIPL.sh
#!/bin/bash
cat ../../GIPL/GIPL.jjt | \
# Filter out unneeded stuff
grep -v ’// EOF’ | \
#grep -v ’import gipsy.GIPC.intensional.SimpleNode’ | \
# Fix package
sed ’s/intensional\.GIPL/intensional\.SIPL\.JLucid/g’ | \
# JLucid GIPL
sed ’s/GIPL/JGIPL/’ | \
sed ’s/\/\/{EXTEND-E}/\/\/{EXTEND-E}\n\t\t| embed()/’ | \
sed ’s/\/\/{EXTEND-FACTOR}/\/\/{EXTEND-FACTOR}\n\t| array()/’ | \
sed ’s/<WHERE: "where">/<WHERE: "where">\n\t| <EMBED: "embed">/g’ \
> JGIPL.jjt
./jlucid.sh >> JGIPL.jjt
# EOF
### D.3 JIndexicalLucid.sh
#!/bin/bash
cat ../../SIPL/IndexicalLucid/IndexicalLucid.jjt | \
# Filter out unneeded stuff
grep -v ’// EOF’ | \
#grep -v ’import gipsy.GIPC.intensional.SimpleNode’ | \
# Fix package
sed ’s/intensional\.SIPL\.IndexicalLucid/intensional\.SIPL\.JLucid/g’ | \
# JLucid Indexical
sed ’s/IndexicalLucid/JIndexicalLucid/’ | \
sed ’s/\/\/{EXTEND-E}/\/\/{EXTEND-E}\n\t\t| embed()/’ | \
sed ’s/\/\/{EXTEND-FACTOR}/\/\/{EXTEND-FACTOR}\n\t| array()/’ | \
sed ’s/<WHERE: "where">/<WHERE: "where">\n\t| <EMBED: "embed">/g’ \
> JIndexicalLucid.jjt
./jlucid.sh >> JIndexicalLucid.jjt
# EOF
### D.4 ObjectiveGIPL.sh
#!/bin/bash
cat JGIPL.jjt | \
# Filter out unneeded stuff
grep -v ’// EOF’ | \
# Fix package
sed ’s/intensional\.SIPL\.JLucid/intensional\.SIPL\.ObjectiveLucid/g’ | \
# ObjectiveLucid GIPL
sed ’s/JGIPL/ObjectiveGIPL/’ | \
sed ’s/\/\/{EXTEND-E1}/\/\/{EXTEND-E1}\n\t\t\t| ( <DOT> ID() ) #OBJREF E1()/’ \
> ObjectiveGIPL.jjt
# EOF
### D.5 ObjectiveIndexicalLucid.sh
#!/bin/bash
cat JIndexicalLucid.jjt | \
# Filter out unneeded stuff
grep -v ’// EOF’ | \
# Fix package
sed ’s/intensional\.SIPL\.JLucid/intensional\.SIPL\.ObjectiveLucid/g’ | \
# ObjectiveLucid Indexical
sed ’s/JIndexicalLucid/ObjectiveIndexicalLucid/’ | \
sed ’s/\/\/{EXTEND-E1}/\/\/{EXTEND-E1}\n\t\t\t| ( <DOT> ID() ) #OBJREF E1()/’ \
> ObjectiveIndexicalLucid.jjt
# EOF
## Index
* .NET Remoting §2.5.4
* API
* AbstractSyntaxTree §4.1.1.13, §4.1.1.9, §4.1.2.3, 2nd item
* addInvalidSegmentName() §4.1.1.4
* addValidSegmentName() §4.1.1.4
* bool §3.3.2
* boolean §3.3.2
* Car §5.3.4
* CCompiler §3.3.3.3, §3.3.3.3
* Class §2.6.1.1, §4.1.3.5
* Class.getConstructors() §2.6.1.1
* Class.newInstance() §2.6.1.1
* CodeSegment §4.1.1.4, §4.1.1.9
* CommunicationException §4.1.1.8
* CommunicationProcedureGenerator §C.1, §4.1.1.3, §4.1.1.5, §4.1.1.6, §4.1.1.8
* CommunicationStats §4.1.1.8
* Constructor §2.6.1.1
* DemandDispatcher §4.2.3.3
* DemandDispatcherAgent §4.2.3.3
* DemandDispatcherException §4.2.3.3
* DemandGenerator §4.2.3.3, §4.2.3.3
* DemandList §4.2.3.3
* DemandMonitor §7.7
* DFG §C.1
* DFGAnalyzer item 1, §4.1.1.9, §4.2.3.5, §4.2.3.5, §7.2
* DFGEditor 5th item, 5th item, §4.2.3.4
* DFGException §4.2.3.5
* DFGGenerator §2.6.2, §7.2
* Dictionary Figure 28, Figure 28, Figure 29, Figure 29, §4.1.1.9, 1st item, 1st item
* DictionaryItem Figure 28, Figure 28
* dimension §3.3.2, §3.3.2
* doGet() 2nd item, 3rd item
* doPost() 3rd item, 3rd item
* double §3.3.2, §3.3.2
* EImperativeLanguages §4.1.1.6, 3rd item
* EIntensionalLanguages §4.1.1.7, 3rd item
* embed() Figure 3, Figure 3, Figure 4, Figure 4, 3rd item, §3.1.1.2, §3.1.1.2, §3.1.1.2, §3.1.1.2, §3.1.1.2, §3.1.1.2, §3.1.1.2, §3.1.1.3, §3.1.2, 2nd item, 4th item, 3rd item, §4.1.1.4, §4.1.2.1, §4.1.2.2, §4.1.2.2, §4.1.2.5, §4.1.2.5, §4.1.2.6
* equals() §4.1.1.3
* Executor §4.1.1.11, §4.1.1.5, §4.1.2.4, 2nd item, 4th item, §4.2.3.3, §4.2.3.3, §4.2.3.3, §4.2.3.3, §4.2.3.3, §4.2.3.3
* ExpandedThreadGroup 4th item
* Facet 2nd item
* Float §4.1.1.5
* float §3.3.2, §3.3.2
* FormatTag Figure 5, Figure 5, §4.1.1.3, §4.1.1.9, §4.1.2.3
* Fun_Item §4.1.1.13
* GEE §C.1, §C.1, §2.5.4, §4.1.1.11, §4.1.1.14, 2nd item, 4th item, 6th item, 7th item, 1st item, 2nd item, 2nd item, 3rd item, 3rd item, 10th item, 2nd item, 3rd item, 4th item, 5th item, 6th item, 7th item, 8th item, 9th item, §4.2.1.3, §4.2.1.5, §4.2.1.6, §4.2.3.3, §4.2.3.3, §4.2.3.3, §4.2.3.3
* GEEException §4.2.3.2, §4.2.3.3, §4.2.3.3
* GEERGenerator §4.1.1.11, §4.1.1.12, §4.1.1.12, §4.1.1.14, §4.1.1.9
* generateCommunicationProcedures() §4.1.1.9
* generateSequentialThreads() §4.1.1.9
* GenericTranslator §C.1
* get42() §4.1.1.4
* getDeclaredMethods() §2.6.1.1
* getParameterTypes() §2.6.1.1
* getReturnType() §2.6.1.1
* GIPC §C.1, §C.1, 1st item, item 1, §4.1.1.10, §4.1.1.10, §4.1.1.11, §4.1.1.11, §4.1.1.11, §4.1.1.13, §4.1.1.14, §4.1.1.4, §4.1.1.4, §4.1.1.6, §4.1.1.7, §4.1.1.9, §4.1.1.9, §4.1.2.6, §4.1.2.6, §4.1.2.6, 2nd item, 4th item, 6th item, 7th item, 1st item, 2nd item, 2nd item, 2nd item, 3rd item, 2nd item, 11st item, 12nd item, 2nd item, 3rd item, 4th item, 4th item, 5th item, 6th item, 7th item, 8th item, 9th item, §4.2.1.3, §4.2.1.4, §4.2.1.4, §4.2.1.6, §4.2.3.3
* GIPC.DFG.DFGAnalyzer §4.2.3.5
* GIPC.DFG.DFGGenerator §4.2.3.5
* GIPC.intensional.GenericTranslator §4.1.1.7
* GIPC.intesional.SimpleNode §4.2.3.5
* GIPCException 5th item, §4.2.3.2, §4.2.3.5
* GIPLCompiler §4.1.1.9
* GIPSY 2nd item, 1st item, §4.2.1.2, §4.2.1.2
* gipsy §C.1, §C.1
* gipsy.GEE §4.2.3.3
* GIPSYArray §4.1.2.4, §4.1.2.4, §4.1.2.4
* GIPSYEmbed §4.1.1.5, §4.1.2.5
* GIPSYException §4.2.3.2, §4.2.3.2
* GIPSYFunction §4.1.1.5, §4.1.1.5
* GIPSYIdentifier §4.1.1.5
* GIPSYObject §4.1.2.4, §4.1.3.3
* GIPSYOperator §4.1.1.5
* GIPSYProgram Figure 19, Figure 19, §4.1.1.11, §4.1.1.11, §4.1.1.14, §4.1.1.14, §4.1.1.8, §4.2.3.3
* GIPSYType §4.1.1.13
* GIPSYVoid §4.1.1.5
* Hashtable §3.1.1.3
* HttpServlet 1st item
* ICommunicationProcedure §4.1.1.8, §4.1.1.8
* ICommunicationProceduresEnum §4.1.1.8
* ICompiler item 1, item 2, §4.1.1.9, §4.1.1.9, §4.2.3.5
* IDemandList §4.2.3.3, §4.2.3.3
* IdentifierContextCodeGenerator §4.1.1.9
* IDP §C.1
* IImperativeCompiler item 3, §4.1.1.6, §4.1.1.9, §4.1.1.9
* IIntensionalCompiler item 2, item 3, item 3, §4.1.1.7, §4.1.1.9, §4.1.1.9
* imperative §C.1
* ImperativeCompiler §4.1.1.2, §4.1.1.9, 2nd item
* ImperativeNode §4.1.1.11, §4.1.1.13, §4.1.1.7, §4.1.1.9, §4.1.2.3, 3rd item, 3rd item
* IndexicalLucidCompiler §4.1.1.9
* int §3.3.2, §3.3.2, §3.3.2
* Integer §3.2.1.1, §4.1.1.5
* intensional §C.1
* IntensionalCompiler item 2, 2nd item
* IntensionalCompiler.translate() §4.1.1.7
* interfaces §C.1
* IntesionalCompiler §4.1.1.9
* ISequentialThread §4.1.1.8, §4.2.3.3
* ISequentualThread §4.1.1.8
* Item_in_Dict §4.1.1.13
* IValueHouse §4.2.3.3
* IVW §C.1
* IVWInterface §4.2.3.3
* JarEntry §4.1.2.5
* JarInputStream §4.1.2.5
* JAVA §4.1.1.3
* java.lang §4.1.1.5
* java.lang.Error 6th item
* java.lang.Thread 3rd item
* java.lang.ThreadGroup 4th item
* java.reflect.* §2.6.1.1
* java.util.Arrays 5th item
* java.util.Vector §4.1.1.13, 1st item, 1st item, 5th item
* JavaCommunicationGenerator §4.1.2.1
* JavaCompiler 2nd item, §3.3.3.3, §4.1.1.12, §4.1.1.2, §4.1.1.2, §4.1.1.6, §4.1.1.9, §4.1.1.9, §4.1.2.1, §4.1.2.3, §4.1.2.3, §4.1.2.3, §4.1.2.3, §4.1.2.3, §4.1.2.4, §4.1.2.6
* JavaSequentialThreadGenerator §4.1.1.8, §4.1.2.1
* JGIPLParser §4.1.1.9
* JIndexicalLucidParser §4.1.1.9
* JLucidCompiler §4.1.1.9, §4.1.1.9, §4.1.2.3, §4.1.2.6, §4.1.2.6, §4.1.2.6
* JLucidParser §4.1.1.9, §4.1.2.1
* JLucidPreprocessor §4.1.1.2, §4.1.1.4, §4.1.2.1, §4.1.2.6, §4.1.2.6, §4.1.3.5
* Lucid 2nd item, 4th item, 5th item, 8th item, 8th item, §4.2.1.4
* LUSException §4.2.3.3
* main() item 1
* marf.nlp footnote 2
* marf.Storage §4.2.2.2
* marf.Storage.StorageManager 6th item
* marf.util §4.2.2.2
* marf.util.Arrays 5th item
* marf.util.BaseThread 1st item, 3rd item
* marf.util.Debug 8th item
* marf.util.ExpandedThreadGroup 4th item, 4th item
* marf.util.FreeVector 1st item, 1st item
* marf.util.Logger 7th item
* marf.util.OptionProcessor 2nd item
* Method §2.6.1.1, §2.6.1.1
* Nat32 §5.3.4
* native item 1, item 3, §3.3.3.3
* Node 1st item, 2nd item, 3rd item
* NotImplementedException §4.2.3.2
* NullCommunicationProcedure §4.1.1.8
* Object §4.1.1.3
* Object.notify() §2.5.4
* Object.notifyAll() §2.5.4
* Object.wait() §2.5.4
* ObjectiveLucidCompiler §4.1.1.9
* ObjectiveLucidPreprocessor §4.1.1.4, §4.1.3.5
* ParallelTestCase §5.1.2.1
* ParseException 5th item
* Preprocessing §C.1
* Preprocessor Figure 6, Figure 6, 2nd item, §4.1.1.10, §4.1.1.11, §4.1.1.11, §4.1.1.12, §4.1.1.2, §4.1.1.4, §4.1.1.4, §4.1.1.4, §4.1.1.4, §4.1.1.4, §4.1.1.4, §4.1.1.4, §4.1.1.5, §4.1.1.7, §4.1.1.9, §4.1.2.6, §4.1.2.6, §4.1.3.5, 2nd item
* PreprocessorParser §2.6.2
* Regression §2.6.1.3, 2nd item, 7th item, 4th item, 4th item, 4th item, §4.2.1.6, §4.2.1.6, §5.1.2, §5.1.2.1, §5.1.2.1, §5.1.2.1
* RIPE §C.1, §C.1, 2nd item, 3rd item, 4th item, 5th item, 6th item, §4.2.1.1, §4.2.1.2, §4.2.1.3, §4.2.1.3, §4.2.3.4
* RIPEException §4.2.3.2
* RMICommunicationProcedure §4.1.1.8
* run() §2.6.1.3, §2.6.1.3
* Runnable §4.1.1.8, §4.1.1.8, 3rd item
* runTest() §2.6.1.3
* RuntimeException §4.2.3.2
* Semantic §4.1.1.13
* SemanticAnalyzer §4.1.1.12, §4.1.1.13, §4.1.1.13, §4.1.1.5, §4.1.1.9, §4.1.2.4, 2nd item
* SequentialTestCase §5.1.2.1
* SequentialThreadGenerator §C.1, §4.1.1.3, §4.1.1.5, §4.1.1.6, §4.1.1.8, §4.2.3.3
* SequentialThreadSourceGenerator §4.1.1.3
* Serializable §4.1.1.8, §4.1.1.8
* setUp() §2.6.1.3
* SimpleCharStream 4th item
* SimpleNode §4.1.1.9, 1st item, 2nd item, 2nd item, 2nd item, §4.2.2.1, §4.2.3.5
* storage §C.1
* storage.Dictionary §4.1.1.13
* storage.DictionaryItem §4.1.1.13
* storage.FunctionItem §4.1.1.13
* String §3.3.2, §3.3.2, §3.3.2, §3.3.2
* string §3.3.2, §3.3.2, §3.3.2, §3.3.2
* synchronized §2.5.4
* System.loadLibrary() §3.3.3.3
* tearDown() §2.6.1.3
* Test §2.6.1.3
* TestCase §2.6.1.3, §5.1.2.1
* TestResult §2.6.1.3
* tests §C.1
* TestSuite §2.6.1.3, §5.1.2.1
* TextualEditor 6th item
* TokenMgrError 6th item
* toString() §4.1.1.3, §4.1.1.5, 2nd item
* translate() §4.1.1.9
* TranslationLexer §4.1.1.7
* TranslationParser §4.1.1.7
* Translator §4.1.1.7, §4.1.1.7, §4.1.1.7, §4.1.1.9
* true 2nd item, 2nd item, §3.3.2
* TypeMap §3.3.3.3, §4.1.2.4, §6.2.3
* util §C.1, 1st item
* void 2nd item, §3.3.2, §5.3.4, §5.3.4
* WebEditor §C.2, §2.6.5, 1st item, §4.2.2.3
* Worker Appendix B, §3.3.1, §4.1.1.8
* WorkResult §4.1.1.8
* WorkTask §4.2.3.3
* wrap() §4.1.2.3
* wrappers §C.1
* XLucidInterpreter §4.2.3.3
* Architecture
* Directory Structure §C.1
* GIPSY Java Packages §C.1
* GIPSY Modules Packaging §C.2
* Arrays
* JLucid §4.1.2.4
* AST 1st item, 1st item, §2.6.2, 3rd item, 4th item, 3rd item, 4th item, 6th item, 8th item, 4th item, §4.1.1.11, §4.1.1.11, §4.1.1.12, §4.1.1.12, §4.1.1.12, §4.1.1.14, §4.1.1.3, §4.1.1.3, §4.1.1.3, §4.1.2.3, §4.1.2.6, §4.1.3, §4.1.3.5, 2nd item, 3rd item, §6.2.5
* Background Chapter 2
* Build System §2.6.6
* Ant §2.6.6.4
* Eclipse §2.6.6.2
* JBuilder §2.6.6.3
* Makefiles §2.6.6.1
* NetBeans §2.6.6.5
* C §1.1, 4th item, 2nd item, §2.3, 1st item, 4th item, §2.5.3, item 1, item 4, §2.6.1.2, §2.6.2, §3.3.3.3, §3.3.3.3, §3.3.3.3, 1st item, §4.1.1.1, §4.1.1.1, §4.1.1.3, §4.1.1.6, §6.2.3
* C++ §1.1, 1st item, §2.3, §2.3.2, §2.3.4, 2nd item, 3rd item, 4th item, §3.3.3.3, §3.3.3.3, 1st item, 3rd item, §4.1.1.1, §4.1.1.3, §4.1.1.6, §6.2.3
* CLP 3rd item, §2.5.1
* Command-Line Interfaces
* GEE §4.2.1.5
* GIPC §4.2.1.4
* GIPSY §4.2.1.2
* Regression §4.2.1.6
* RIPE §4.2.1.3
* Communication Procedure §3.3.3, §3.3.3.2
* Interface §B.2
* Compilation Sequence
* Java Figure 22
* JLucid Figure 21
* Objective Lucid Figure 24
* Compiler Frameworks §2.4
* context §2.1
* CORBA 3rd item, 2nd item, §2.5.3, §2.5.4, §3.3.3.2, 4th item, 9th item
* CVS 6th item, §1.2, §2.6.4
* data types
* matching Lucid and Java §3.3.2
* DCOM+ 2nd item, §3.3.3.2, 4th item, 8th item
* Demand Dispatcher
* Integration §4.2.3.3
* Design
* Architectural Chapter 4
* Detailed Chapter 4
* External §4.2
* External Software Interfaces §4.2.2
* GEE §4.2.3.3
* GICF §4.1.1.6
* GIPC §4.1.1.9
* Internal §4.1
* JLucid §4.1.2.1
* Objective Lucid §4.1.3.1
* Semantic Analyzer §4.1.1.13
* User Interface §4.2.1
* DFG §2.6.1, §4.1.1.3, §5.1.2.3, §6.2.5
* Integration §4.2.3.5
* dimensions §2.1
* Dining Philosophers §5.3.2
* DMS §2.5.4, §5.1.2.3, §7.10
* DPR 11st item, §2.5.4.1, §2.5.4.1
* GIPSY Program §4.1.1.14
* eduction §2.3.3
* GLU §2.3.3
* embed() §3.1.1.2
* implementation of §4.1.2.5
* Examples
* Dining Philosophers §5.3.2
* FFT §5.3.3
* Game of Life §5.3.5
* Lucid §2.2.2.9
* Moving Car §5.3.4
* Natural Numbers Problem §2.2.2.9
* Prefix Sum §5.3.1
* The Hamming Problem §2.2.2.9
* Exceptions §4.2.3.2
* External Software Interfaces §4.2.2
* JavaCC API §4.2.2.1
* MARF Library API §4.2.2.2
* Servlets API §4.2.2.3
* Fast Fourier Transform §5.3.3
* FC++ 2nd item, §2.3.2, §2.3.2, 3rd item
* Fedora Core 2 §2.6.6.1, §5.2
* FFT 9th item, §5.3.3, §5.3.3.1, §5.3.3.2
* Files
* .c item 4, item 5, §3.3.3.3
* .class item 2, item 3, §3.1.1.2, §3.1.1.2, §3.1.1.2, §4.1.2.3, §4.1.2.3, §4.1.2.5, §4.1.2.5
* .h item 3, item 5, §3.3.3.3
* .ipl 2nd item, 2nd item
* .jar §C.2, §C.2, §C.2, §C.2, Table 1, Table 1, Chapter 4, §4.1.2.5, §4.1.2.5, §4.2.1.2
* .java §3.1.1.2, §3.1.1.2, §3.1.1.2, §4.1.2.3, §4.1.2.5, §4.1.2.5
* .jjt §4.1.2.2
* *.ipl §5.1.2.3
* build.xml §2.6.6.4, §2.6.6.4
* gee.jar 3rd item, Table 1
* gipc.jar 2nd item, Table 1
* GIPL.jjt §4.1.1.3, §4.1.2.2
* GIPSY.class §4.2.1.2
* gipsy.jar 1st item, Table 1, §4.2.1.2, §5.1.2
* GIPSY.jpx §2.6.6.3
* imperative/LANGUAGE 1st item
* IndexicalLucid.jjt §4.1.2.2
* IndexicalLucid.rul §4.1.1.7
* intensional/SIPL/LANGUAGE 1st item
* Java.jjt §4.1.1.3
* nbproject.xml §2.6.6.5
* PreprocessorParser.jjt §4.1.1.4
* README.dir §2.6.7
* Regression.jar 5th item, Table 1
* ripe.jar 4th item, Table 1
* src/tests/* §5.1.2.3
* Format Tag §4.1.1.3
* Fortran §1.1, 2nd item, 2nd item, 1st item, 4th item, §4.1.1.1, §4.1.1.1, §4.1.1.6
* Frameworks
* Compiler §2.4
* GICF §4.1.1.1, §4.1.1.3, §4.1.1.6, §4.1.2.3
* GIPC §4.1.1, §4.1.1.3, §4.1.1.9
* GIPSY Exceptions Figure 31, §4.2.3.2, §4.2.3.2
* GIPSY Type System §4.1.1.5
* IPLCF §4.1.1.7
* JUnit §5.1.2.1
* MARF §2.6.3
* RIPE §4.2.3.4
* Free Java Functions §4.1.2.3, §4.1.2.3
* JLucid §4.1.2.3
* FTP 10th item, §3.1.1.2
* GEE 12nd item, 3rd item, 4th item, 5th item, §C.2, Figure 12, 1st item, §2.5.1, §2.5.3, §2.5.4, §2.5.4, §2.5.4, §2.5.4, §2.5.4.1, §2.5.5, §2.6.1.3, §2.7, Figure 32, 4th item, §4.1.1.10, §4.1.1.12, §4.1.1.12, §4.1.1.14, §4.1.1.14, §4.1.1.2, §4.1.1.3, §4.1.1.3, §4.1.2.4, §4.1.3, §4.1.3.5, 2nd item, 2nd item, 3rd item, 11st item, 2nd item, 7th item, §4.2.3.2, §4.2.3.3, §4.2.3.3, §4.2.3.3, §5.1.1, §5.3.2, §5.4, §6.2.4, §6.2.6, §7.8
* Command-Line Interface §4.2.1.5
* Conceptual Design §2.5.4
* Design §4.2.3.3
* Integration §4.2.3.3
* Introduction §2.5.4
* GEER 13rd item, §2.5.4, §2.5.4.1, §2.5.4.1, §3.2.3, 8th item, §3.3.3.4, §4.1.1.14, §4.1.1.14, §4.1.1.3, §4.1.3.5
* GIPSY Program §4.1.1.14
* General Intensional Programming System §2.5
* GICF §1.1, 1st item, 2nd item, 4th item, §2.5.5, Chapter 3, §3.3, §3.3.1, §3.3.1, §3.3.3.3, 4th item, 5th item, 6th item, 7th item, 5th item, §3.4, Figure 9, §4.1, §4.1.1, §4.1.1.1, §4.1.1.1, §4.1.1.2, §4.1.1.3, §4.1.1.4, §4.1.1.6, §4.1.1.6, §4.1.1.7, §4.1.2.3, §4.1.3, §4.3, §4.3, Chapter 5, §6.2.9, 7th item, §7.4, §7.5
* Binary Compatibility §4.1.1.3
* Design §4.1.1.6
* Dictionary §4.1.1.3
* Format Tag §4.1.1.3
* GEER Generator as a Linker §4.1.1.12
* Generalization Issues §4.1.1.3
* Imperative Stubs §4.1.1.12
* Introduction §4.1.1.1
* Multiple Intensional Parts §4.1.1.12
* NCP Generator §4.1.1.12
* Sending Source Code Text §4.1.1.3
* Type Processor §4.1.1.12
* GIPC 14th item, 4th item, 5th item, §C.2, 2nd item, Figure 11, 1st item, 5th item, 1st item, §2.5.1, §2.5.3, §2.5.3, §2.5.3, §2.5.3, §2.5.4, §2.5.4.1, §2.5.5, §2.6.1.2, §2.6.1.3, §2.7, §3.1.1, §3.3.1, §3.3.3.1, §3.3.3.2, §3.3.3.4, §3.4, Figure 13, Figure 14, Figure 26, §4.1.1.14, §4.1.1.14, §4.1.1.2, §4.1.1.3, §4.1.1.4, §4.1.1.4, §4.1.1.9, §4.1.1.9, §4.1.3.5, 2nd item, 3rd item, 2nd item, §4.2.3.2, §5.1.1, §5.4, §7.3
* as a Meta Processor §4.1.1.10
* Command-Line Interface §4.2.1.4
* Initial Conceptual Design §2.5.3
* Introduction §2.5.3
* Linker §4.1.1.12
* Preprocessor §4.1.1.4
* Sequence Diagram §4.1.1.11
* GIPL 15th item, 22nd item, §1.1, Figure 4, Figure 7, Figure 7, §2.1, 1st item, item 4, 1st item, §2.2.2.6, §2.2.2.7, §2.2.2.8, §2.2.2.9, §2.5.3, §2.6.2, Figure 8, Figure 8, §3.1.1, §3.1.2, §3.2.1, §3.3.1, 1st item, Figure 11, §4.1.1.3, §4.1.1.9, §4.1.2.2, §4.1.2.2, 2nd item, 8th item, §5.1.2.3, §5.3.5
* Syntax §2.2.2.7
* GIPSY 16th item, Figure 1, 1st item, 3rd item, 4th item, §C.2, §1.1, §1.1, §1.1, 2nd item, §1.2, 1st item, §1.4, Figure 9, footnote 2, 1st item, 1st item, §2.3.3, §2.3.3, 5th item, §2.5, 6th item, §2.5.1, §2.5.1, §2.5.1, §2.5.1, §2.5.2, §2.5.3, §2.5.3, §2.5.4, §2.5.4, §2.6.1, §2.6.1.1, §2.6.1.2, §2.6.1.3, §2.6.3, §2.6.5, §2.6.6, §2.6.6.1, §2.6.6.2, §2.6.6.3, §2.6.6.4, §2.6.6.5, §2.7, §3.1.1, §3.2.1, §3.2.1.1, 5th item, §3.3.1, §3.3.1, §3.3.3.4, 1st item, 3rd item, 4th item, 5th item, 6th item, §3.4, Figure 25, Figure 27, Figure 29, Figure 3, Figure 30, Figure 4, footnote 2, 3rd item, 4th item, §4.1.1, §4.1.1.1, §4.1.1.1, §4.1.1.12, §4.1.1.2, §4.1.1.3, §4.1.1.4, §4.1.1.5, §4.1.1.6, §4.1.1.8, §4.1.3, 1st item, 1st item, 4th item, 4th item, §4.2, §4.2.1.1, §4.2.1.1, §4.2.1.2, §4.2.2.1, §4.2.2.2, §4.2.3, §4.2.3.1, §4.2.3.2, §4.2.3.3, §4.3, §4.3, Chapter 5, §5.1.1, §5.1.2.3, §5.2, §5.3, §5.3.3, §5.4, Chapter 6, §6.1.1, §6.1.2, §7.3
* Command-Line Interface §4.2.1.2
* Compilation process Figure 19
* GIPC Framework with Preprocessor Figure 4
* Goals §2.5.2
* Introduction §2.5.1
* Original GIPC Framework Figure 3
* Screen Saver §7.9
* Security §7.3
* Server §7.10
* Structure Figure 10
* Type System 5th item
* Types §4.1.1.5
* Web Front-End §4.2.1.1
* Web Portal §4.2.1.1
* WebEditor §4.2.1.1
* GIPSY Exceptions §4.2.3.2
* GIPSY Program §4.1.1.14
* Compiled §4.1.1.14
* GEER §4.1.1.14
* Intefacing GIPC and GEE §4.1.1.14
* Segments §4.1.1.4
* GIPSY Type System §4.1.1.5
* GLU 17th item, 22nd item, §1.1, 2nd item, 3rd item, item 2, 1st item, 3rd item, §2.3, §2.3.3, §2.3.3, §2.3.3, §2.3.4, 1st item, §2.5.4, §2.5.4.1, §2.6.1.2, §3.1.1, §3.3.3.1, §4.1.1.1, §4.3
* eduction §2.3.3
* GLU# 1st item, item 9, 4th item, §2.3.3, §2.3.4, §2.3.4, 2nd item, 3rd item
* GNU §2.6.6.1
* Grammar
* Generation, JLucid §4.1.2.2
* Generation, Objective Lucid §4.1.3.2
* Preprocessor Figure 7, Figure 7, §4.1.1.4
* Haskell §2.3.2, §2.5.4, Figure 8, Figure 8, §5.3.5, §7.6
* HTTP 18th item, §3.1.1.2
* hybrid
* JLucid 2nd item
* Hybrid Programming §2.3
* immutable §4.1.1.4
* Implementation Chapter 4
* Architectural Design §4.2.3
* Directory Structure §C.1
* GIPSY Java Packages §C.1
* GIPSY Modules Packaging §C.2
* JLucid §4.1.2
* Objective Lucid §4.1.3
* Unit Integration §4.2.3
* Indexical Lucid 22nd item, 37th item, §1.1, §1.1, Figure 1, Figure 6, Figure 6, §2.1, 1st item, 2nd item, item 2, 1st item, 3rd item, §2.2.2, §2.2.2, §2.2.2.6, §2.2.2.7, §2.2.2.8, §2.2.2.9, §2.5.1, §2.5.3, §2.6.2, Figure 1, Figure 1, Figure 2, Figure 2, Figure 3, Figure 3, 2nd item, 1st item, §3.1.1, §3.1.2, §3.2.1, 1st item, §4.1.1.3, §4.1.1.9, §4.1.2.1, §4.1.2.2, §4.1.2.2, Figure 9, Figure 9, §5.1.2.3, §5.3.5, 2nd item, §7.1
* asa §2.2.2.2
* fby §2.2.2.2, §2.2.2.2
* first §2.2.2.2
* next §2.2.2.2
* upon §2.2.2.2
* wvr §2.2.2.2
* Integration
* Demand Dispatcher §4.2.3.3
* DFG §4.2.3.5
* Garbage Collection §4.2.3.3
* GEE §4.2.3.3
* Intensional Value Warehouse §4.2.3.3
* Jini §4.2.3.3
* Semantic Analyzer §4.1.1.13
* Intensional
* Programming §2.1
* intensional
* logic §2.1
* operators §2.1
* Intensional Programming §2.1
* Interfaces
* Communication Procedure §B.2
* Sequential Thread §B.1
* Internal Design
* GICF §4.1.1.1
* GIPC §4.1.1
* IPLCF §4.1.1.7
* Introduction Chapter 1
* Contributions §1.2
* GICF §4.1.1.1
* GIPSY §2.5.1
* JLucid §3.1
* Scope of the Thesis §1.3
* Structure of the Thesis §1.4
* Thesis Statement §1.1
* IPLCF Figure 10, §4.1, §4.1.1, §4.1.1.7, §4.3
* Isabelle §7.1
* Java footnote 1, §1.1, §1.1, 2nd item, 4th item, 1st item, §2.3, 4th item, §2.5.3, §2.5.4, §2.5.4, §2.5.4, §2.6.1, §2.6.1.1, §2.6.1.2, §2.6.2, 1st item, §3.1.1, §3.1.1.1, §3.1.1.2, §3.1.1.3, §3.1.2, §3.2.1.1, §3.3.1, §3.3.1, §3.3.2, §3.3.2, §3.3.3.3, §3.3.3.3, 1st item, Figure 22, 1st item, 3rd item, 3rd item, §4.1.1.1, §4.1.1.6, §4.1.1.8, §4.1.2.3, §4.1.2.3, §4.1.2.4, §4.2.3.4, §5.3, §5.3.3.2, §6.2.3, §6.2.8, §6.2.8
* Reflection §2.5.4, §2.6.1.1
* Java Compiler
* JLucid §4.1.2.3
* Jini 2nd item, §2.5.3, §2.5.4, §2.5.4, §3.3.3.2, Figure 33, 4th item, 7th item, §4.2.3.3
* Integration §4.2.3.3
* JLucid 22nd item, 37th item, Appendix D, §1.1, §1.1, §1.1, 2nd item, 1st item, 4th item, 5th item, §1.4, item 6, §2.2.2.8, §2.6.2, Chapter 3, §3.1, 2nd item, 3rd item, §3.1.1, §3.1.1.3, §3.1.1.4, §3.1.2, §3.1.3, 3rd item, 6th item, §3.2.1, §3.2.1, §3.2.1.1, §3.2.1.1, §3.2.1.1, §3.2.3, §3.3.1, §3.3.1, 1st item, 2nd item, 1st item, 2nd item, 3rd item, §3.4, Figure 20, Figure 21, §4.1, §4.1.1.1, §4.1.1.14, §4.1.1.4, §4.1.1.9, §4.1.2, §4.1.2.1, §4.1.2.1, §4.1.2.2, §4.1.2.2, §4.1.2.4, §4.1.2.5, §4.1.3.2, §4.1.3.4, §4.1.3.5, §4.1.3.6, §4.3, §4.3, Figure 2, Figure 2, Figure 3, Figure 3, Chapter 5, §5.1.2.3, §5.3.1, §5.3.3.1, §6.2.1, §6.2.9, §7.3, §7.5
* Arrays §4.1.2.4
* AST §4.1.2.6
* Design §4.1.2.1
* Dictionary §4.1.2.6
* embed() §4.1.2.5
* Examples – FFT §5.3.3
* Free Java Functions §4.1.2.3
* Grammar Generation §4.1.2.2
* Implementation §4.1.2
* Introduction §3.1
* Java Compiler §4.1.2.3
* Non-Determinism §3.1.1.1
* Pseudo-Objectivism in §3.2.1.1
* Purpose §3.1.1
* Rationale §3.1.1
* Semantics §3.1.3
* SIPL §3.1.1.4
* Syntax §3.1.2
* JNI 25th item, §2.6.1.2, §3.3.3.3, §7.4
* JRE 26th item, §2.5.4
* JSSE 27th item, §2.5.4
* JUnit §2.6.1.3
* Layout
* Directory Structure §C.1
* GIPSY Java Packages §C.1
* GIPSY Modules Packaging §C.2
* Libraries
* MARF 28th item, §2.6.3, 1st item, §4.2, §5.3.3, §5.3.3.2
* Linux §2.6.6.1
* LISP 2nd item, §2.2.2.1, §2.5.4, §7.6
* logic
* Hoare §2.2.2.8
* intensional §2.1, §2.1
* non-intensional §2.1
* temporal §2.1
* Lucid 22nd item, 1st item, §2.1, §2.1, item 1, 1st item, 2nd item, 1st item, 1st item, 1st item, 1st item, 2nd item, 2nd item, §2.2, §2.2.1, §2.2.2.6, §2.2.2.6, §2.2.2.9, §2.2.3, §2.3, §2.3.3, §2.3.3, §2.3.4, 1st item, 2nd item, §2.5.1, §2.5.4, §2.7, footnote 1, §3.1.1, §3.1.1.1, §3.1.1.2, §3.1.1.2, §3.1.3, 2nd item, §3.2.1, §3.2.1.2, §3.2.2, §3.2.3, §3.2.3, 5th item, §3.3.2, §3.3.3.1, §3.3.3.1, §3.4, §4.1.1.5, §4.1.3.5, §5.3.1, §5.3.3, §6.2.4
* Abstract Syntax §2.2.2.6
* Arrays as Objects §4.1.3.6
* Basic Operators §2.2.2.2
* Examples §2.2.2.9
* Family §2.2.1
* GLU §3.3.3.1
* History §2.2.1
* Indexical §2.2.2, §2.5.1
* Introduction §2.2
* JLucid §3.1
* Non-Determinism §3.1.1.1
* Objective §3.2
* Objects as Arrays §4.1.3.6
* and # §2.2.2.4, §2.2.2.5
* Pipelined Dataflows item 1
* Semantics §2.2.2.8
* State of the Art §2.2.3
* Streams §2.2.2.1
* Tensor §2.5.1
* Lucx 1st item, item 7, §2.2.2.7
* Mac OS X §2.6.6.1, §5.2
* MARF
* FFT §5.3.3, §5.3.3.2
* Methodology Chapter 3
* ML §2.5.4
* ML≤ 1st item, §2.3.1, §2.3.1
* MPI 29th item, §2.5.4
* NetCDF §2.5.4
* Non-Determinism §3.1.1.1
* NUMA 32nd item
* Objective Lucid §3.2
* AST §4.1.3.5
* Design §4.1.3.1
* Dictionary §4.1.3.5
* Examples – Moving Car §5.3.4
* Grammar Generation §4.1.3.2
* Implementation §4.1.3
* Introduction §3.2.1
* Object Instantiation §4.1.3.3
* Semantic Rules Figure 18, Figure 18
* Semantics of §3.2.3
* Syntax §3.2.2
* The Dot-Notation §3.2.2, §4.1.3.4
* Onyx 22nd item, 37th item, 1st item, item 8, footnote 1
* Options
* -G 4th item
* -h 1st item, 1st item, 1st item, 1st item, 1st item
* -S 5th item
* –all 4th item, 8th item
* –compile-only 2nd item
* –corba 9th item
* –dcom 8th item
* –debug 3rd item, 7th item, 13rd item, 10th item, 10th item, §4.2.1.2
* –dfg 12nd item
* –dfg=‘$<$DFG EDITOR OPTIONS$>$’ 5th item
* –directory 9th item
* –disable-translate 9th item
* –gee 11st item, 7th item
* –gee=‘$<$GEE OPTIONS$>$’ 3rd item
* –gipc=‘$<$GIPC OPTIONS$>$’ 2nd item
* –gipl 4th item, 4th item
* –gipsy 6th item
* –help 1st item, 1st item, 1st item, 1st item, 1st item
* –indexical 5th item, 5th item
* –jini 7th item
* –jlucid 6th item
* –objective 7th item
* –parallel 3rd item
* –regression=‘$<$REGRESSION OPTIONS$>$’ 4th item
* –rmi 6th item
* –sequential 2nd item
* –stdin 3rd item, 3rd item
* –threaded 5th item
* –translate 8th item
* –txt=‘$<$TEXTUAL EDITOR OPTIONS$>$’ 6th item
* –warnings-as-errors 10th item
* [FILENAME1.gipsy [FILENAME2.gipsy] …] 2nd item
* [FILENAME1.ipl [FILENAME2.ipl] …] 2nd item
* Partial Lucid 1st item, item 3
* Perl §1.1, 2nd item, 3rd item, §4.1.1.1, §4.1.1.3, §4.1.1.6
* Prefix Sum §5.3.1
* Preprocessor §4.1.1.4
* GIPC §4.1.1.4
* Grammar §4.1.1.4
* Problems
* Dining Philosophers §5.3.2
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|
arxiv-papers
| 2009-07-15T16:24:05 |
2024-09-04T02:49:03.953892
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serguei A. Mokhov",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/0907.2640"
}
|
0907.2740
|
# Green functions and correlation functions of a solvable $S=1$ quantum Ising
spin model with dimerization
Zhi-Hua Yang1, Li-Ping Yang2, Hai-Na Wu3, Jianhui Dai1, and Tao Xiang4,2
1Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027,
China
2Institute of Theoretical Physics, Chinese Academy of Science, P.O. Box 2735,
Beijing 100080, China
3College of Science, Northeastern University, Shengyang 110006, China
4Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing
100080, China
###### Abstract
This is a supplementary material of our recent paperyangPRB , where a class of
exactly solvable $S=1$ quantum Ising spin models were studied based on the
hole decomposition scheme. Here we provide some details for the Green
functions, the spin-spin correlation functions, as well as the spin
susceptibility in the presence of dimerization.
Quantum Ising chains, Statistical lattice model; dimerization; quantum phase
transitions
## I Introduction
In Ref.yangPRB we have studied a class of the $S=1$ spin chains with the
nearest neighbor Ising coupling and both transverse and longitude single-ion
anisotropy by a combinational use of a hole decomposition scheme and a
recursive method. These models include the first example of the dimerized
$S=1$ quantum spin chain where all the eigen states can be solved exactly. In
this supplementary material we present some detailed derivations for the
physical quantities of the $S=1$ dimerized chain. All the notations are the
same as in Ref.yangPRB . In Sec. II, we discuss the Green functions of the
uniform or dimerized chains, respectively. In Sec. III, we study the
longitudinal spin-spin correlation function at zero- or finite-temperatures.
In Sec. V and VI we list some detailed formulae for the segmented M-matrices
and the partition functions.
## II Green functions
### II.1 Green functions of the uniform spin segments
The original $S=1$ quantum Ising model is mapped onto a large family of the
segmented $S=1/2$ transverse Ising models classified by the total number of
holesOitmaa ; YangPRL . These segmented $S=1/2$ models are then solved by
introducing the Bogoliubov fermionic quasi-particle operators
$\eta_{k}^{\dagger}$ and $\eta_{k}$ as defined in Eq. (14) in Ref. yangPRB .
Inversely, we have
$\begin{split}c_{j}^{\dagger}=\sum_{k}\frac{\Phi_{kj}+\Psi_{kj}}{2}\eta_{k}^{\dagger}+\frac{\Phi_{kj}^{*}-\Psi_{kj}^{*}}{2}\eta_{k},\\\
c_{j}=\sum_{k}\frac{\Phi_{kj}^{*}+\Psi_{kj}^{*}}{2}\eta_{k}+\frac{\Phi_{kj}-\Psi_{kj}}{2}\eta_{k}^{\dagger}.\end{split}$
The Green function, or the two-point correlation function, is defined by
$G_{jq}\equiv\langle F_{j}^{(-)}F_{q}^{(+)}\rangle,$ (1)
where $F_{j}^{(\pm)}\equiv c_{j}^{\dagger}\pm c_{j}$ .
For the uniform system, the wavefunctions $\Phi_{kj}$ and $\Psi_{kj}$ can be
taken as real, we have
$\begin{split}F_{j}^{(-)}=\sum_{k}\Psi_{kj}(\eta_{k}^{\dagger}-\eta_{k})~{},\\\
F_{j}^{(+)}=\sum_{k}\Phi_{kj}(\eta_{k}^{\dagger}-\eta_{k})~{}.\end{split}$ (2)
The Green function can be then expressed as
$\displaystyle
G_{jq}(\beta)=-\sum_{k}\Psi_{kj}\Phi_{kq}\tanh[\beta\Lambda(k)/2]~{}.$
Note that $\tanh[\beta\Lambda(k)/2]\rightarrow 1$ at the ground state
($\beta\rightarrow\infty$), so we have
$\displaystyle G_{jq}(\beta\rightarrow\infty)=-\sum_{k}\Psi_{kj}\Phi_{kq}~{}.$
We denote the wavefunctions for the chain with periodic boundary condition
(cyclic) and open boundary condition (free ends) by ($\Phi^{c}$, $\Psi^{c}$)
and ($\Phi^{f}$, $\Psi^{f}$), respectively. Then we have
$\displaystyle\begin{split}\Phi^{c}_{kj}&=\begin{cases}\sqrt{2/l}\sin
jk~{},~{}k>0,\\\ \sqrt{2/l}\cos jk~{},~{}k\leq 0,\end{cases}\\\
\Psi^{c}_{kj}&=-\frac{D}{\Lambda(k)}\left[(1+\lambda\cos
k)\Phi_{kj}^{c}+\lambda\sin k\Phi_{-kj}^{c}\right],\end{split}$ (3)
where $l$ is the length of the segment. The Green function is
$\displaystyle G^{c}_{r}=L_{r}+\lambda L_{r+1},$ (4)
where $r\equiv|j-q|$ and $L_{r}$ was defined in Refs. Lieb61 ; Pfeuty
$L_{r}=\frac{1}{\pi}\int_{0}^{\pi}dk\frac{1}{\sqrt{1+\lambda^{2}+2\lambda\cos
k}}\cos kr.$
Similarly,
$\displaystyle\begin{split}\Phi_{kj}^{f}&=A_{k}\sin(j-q+1)k,~{}\\\
\Psi_{kj}^{f}&=A_{k}\delta_{k}\sin jk,~{}\end{split}$ (5)
where
$\displaystyle A_{k}=\frac{1}{2}\left[2l+1-\frac{\sin{(2l+1)k}}{\sin
k}\right]^{-1/2}.$ (6)
Consequently, we have
$\displaystyle G^{f}_{jq}=-\sum_{k}A_{k}^{2}\delta_{k}\sin jk\sin(j-q+1)k.$
(7)
At the finite temperatures, we need to add the factor
$\tanh[\beta\Lambda(k)/2]$ to Eqs.(4) and (7).
### II.2 Green functions of the dimerized segments
In the presence of dimerization, the wavefunctions $\Phi_{kj}$ and $\Psi_{kj}$
are complex in general. So we now have,
$\begin{split}F_{j}^{(-)}=\sum_{k}\Psi_{kj}\eta_{k}^{\dagger}-\Psi_{kj}^{*}\eta_{k},\\\
F_{j}^{(+)}=\sum_{k}\Phi_{kj}\eta_{k}^{\dagger}+\Phi_{kj}^{*}\eta_{k}.\end{split}$
(8)
Then, the Green function is expressed by
$\begin{split}G_{jq}=&\sum_{k}(\Psi_{kj}\Phi_{kq}^{*}+\Psi_{kj}^{*}\Phi_{kq})\langle\eta_{k}^{\dagger}\eta_{k}\rangle-\sum_{k}\Psi_{kj}\Phi_{kq}^{*}.\end{split}$
(9)
Where,
$\langle\eta_{k}^{\dagger}\eta_{k}\rangle=[\exp{(\Lambda_{k}/(k_{B}T))}+1]^{-1}$,
satisfying Fermi-Dirac statistics. At the zero temperature, the Green function
can be written as
$\displaystyle G_{jq}=D_{j}Y[j,q]+2J_{j}Y[j+1,q],$ (10)
where
$\displaystyle Y[j,q]$ $\displaystyle=$
$\displaystyle-\sum_{k}\frac{e^{i(j-q)k}}{\Lambda(k)}[1+(-1)^{j+q}\gamma^{*}\gamma$
(11) $\displaystyle~{}~{}~{}~{}~{}~{}~{}+(-1)^{j}\gamma+(-1)^{q}\gamma^{*}].$
The dimerization parameter $\gamma$ is defined by
$\begin{split}\gamma=\frac{1-\tau}{1+\tau}\end{split}$ (12)
with $\tau$ being determined by Eqs. (19) in Ref. yangPRB .
Generally, $\tau$ has two solutions, corresponding to the upper/lower signs of
$\pm$ respectively in Eqs. (19) in Ref. yangPRB . In order to numerically
calculate the Green function, we need to express $Y[j,q]$-function in terms of
real variables. We introduce $p_{1,2}$, $q_{1,2}$ to express complex $\gamma$
as follows.
$\gamma_{1}=p_{1}+iq_{1},~{}~{}~{}\gamma_{2}=p_{2}+iq_{2},$ (13)
$p_{1,2}$ and $q_{1,2}$ are the real and imaginary parts of $\gamma_{1,2}$,
respectively,
$\displaystyle p_{1,2}$ $\displaystyle=$
$\displaystyle\frac{b_{1}^{2}+b_{2}^{2}+4b_{1}b_{2}\cos
2k-\left(\zeta_{1}\mp\zeta_{2}\right)^{2}}{\left[(b_{1}+b_{2})\cos
k-\zeta_{1}\pm\zeta_{2}\right]^{2}+(b_{2}-b_{1})^{2}\sin^{2}k},$
$\displaystyle q_{1,2}$ $\displaystyle=$
$\displaystyle\frac{-2(b_{2}-b_{1})\sin k\left[(b_{1}+b_{2})\cos
k+\zeta_{1}\mp\zeta_{2}\right]}{\left[(b_{1}+b_{2})\cos
k-\zeta_{1}\pm\zeta_{2}\right]^{2}+(b_{2}-b_{1})^{2}\sin^{2}k},$
where the subscript ${1}$ corresponds to the upper case, the subscript $2$
corresponds to the lower case. $\zeta_{1,2}$ are given by
$\displaystyle\zeta_{1}$ $\displaystyle=$ $\displaystyle{(a_{2}-a_{1})}/{2},$
$\displaystyle\zeta_{2}$ $\displaystyle=$
$\displaystyle\Gamma^{2}\sqrt{1-P+Q\cos 2k}~{}.$
where $a_{1}$, $a_{2}$, $P$, $Q$ and $\Gamma$ are defined in Ref.yangPRB .
For convenience, we divide $k$-region $[-\pi,\pi)$ into two subregions: ($I$)
for $[-\pi/2,\pi/2)$ and ($II$) for $[-\pi,-\pi/2)\cup[\pi/2,\pi)$,
respectively. Thus $G_{jq}$ can be expressed by
$G_{jq}=G_{jq}^{(I)}+G_{jq}^{(II)}.$ (14)
In Region ($I$), because of the symmetry between $k$ and $-k$, the Green
function can be reduced in $(0,\pi/2)$,
$\begin{split}G_{jq}^{(I)}=&-\sum_{(0,\pi/2)}\frac{2}{\Lambda_{-1}(k)}\\{D_{j}[1+(-1)^{j+q}(p_{1}^{2}+q_{1}^{2})\\\
&+(-1)^{j}p_{1}+(-1)^{q}p_{1}]\cos(j-q)k\\\
&+2J_{j}[1+(-1)^{j+q+1}(p_{1}^{2}+q_{1}^{2})\\\
&+(-1)^{j+1}p_{1}+(-1)^{q}p_{1}]\cos(j-q+1)k\\}~{}.\end{split}$ (15)
A similar Green function can be obtained for Region ($II$). The function
$Y[j,q]$ can be rewritten as
$\begin{split}Y[j,q]=&-\sum_{(0,\pi/2)}\frac{2}{\Lambda_{-1}(k)}[1+(-1)^{j+q}(p_{1}^{2}+q_{1}^{2})\\\
&+(-1)^{j}p_{1}+(-1)^{q}p_{1}]\cos(j-q)k\\\
&-\sum_{(\pi/2,\pi)}\frac{2}{\Lambda_{-2}(k)}[1+(-1)^{j+q}(p_{2}^{2}+q_{2}^{2})\\\
&+(-1)^{j}p_{2}+(-1)^{q}p_{2}]\cos(j-q)k.\end{split}$ (16)
So it is convenient to express the total Green function Eq. (10) in terms of
$Y[j,q]$. In the dimerization case, there are four such Green functions
associated with the four different parity combinations of the segments.
## III Correlation functions
### III.1 Zero temperature
In this subsection, we discuss the spin-spin correlations at zero temperature.
In Ref. yangPRB we show that the ground state has no hole if
$D_{z}>-\Delta_{h}(0)$, otherwise, it has holes once
$D_{z}\leq-\Delta_{h}(0)$. In the latter case, the holes break the original
chain into segments. We note that only the intra-segment spin-spin
correlations are non-zero.
For $D_{z}>-\Delta_{h}(0)$, the spin-spin correlation function of $S^{z}$ is
defined by $C^{z}_{mn}=\langle\Psi_{0}|S_{m}^{z}S_{n}^{z}|\Psi_{0}\rangle$,
where $|\Psi_{0}\rangle$ is the normalized ground state of the Hamiltonian. By
use of the Jordan-Wigner transformation, one has
$C_{mn}^{z}=\langle\Psi_{0}|F_{m}^{(-)}F_{m+1}^{(+)}F_{m+1}^{(-)}\cdots
F_{n-1}^{(-)}F_{n}^{(+)}|\Psi_{0}\rangle.$ (17)
It is straightforward to show that
$\langle\Psi_{0}|F_{j}^{(\pm)}F_{q}^{(\pm)}|\Psi_{0}\rangle=\pm\delta_{jq}$.
By further utilizing the Wick Theorem, we find that
$C_{mn}^{z}=\left|\begin{array}[]{cccc}G_{m,m+1}&G_{m,m+2}&\cdots&G_{m,n}\\\
G_{m+1,m+1}&G_{m+1,m+2}&\cdots&G_{m+1,n}\\\ \vdots&\vdots&\ddots&\vdots\\\
G_{n-1,m+1}&G_{n-1,m+2}&\cdots&G_{n-1,n}\end{array}\right|,$ (18)
for $n>m$, where,
$G_{jq}=\langle\Psi_{0}|F_{j}^{(-)}F_{q}^{(+)}|\Psi_{0}\rangle=-\langle\Psi_{0}|F_{j}^{(+)}F_{q}^{(-)}|\Psi_{0}\rangle$.
The general expression of $G_{jq}$ is derived in Sec. II.1 for the uniform
chain and in Sec. II.2 for the dimerized chain respectively. In general, one
has
$\displaystyle G_{jq}=D_{j}Y[j,q]+2J_{j}Y[j+1,q],$ (19)
where $Y[j,q]$ is given by Eq. (16). For a uniform system,
$Y[j,q]=Y[q,j]=\frac{1}{D}L_{j-q}$.
### III.2 Finite temperatures
At finite temperatures, the contribution from $p\neq 0$-sector should be taken
into account. A recursion formula similar to Eq. (36) in Ref. yangPRB can be
derived for the correlation function as following
$\displaystyle\sum_{m,n}^{L}C_{mn}^{z}(\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{Z(L)}\sum_{p=0}^{L}\sum_{l=0}^{L-p}\sum_{m,n}^{l}\alpha^{p}(p+1)\rho_{mn}^{z}$
(20) $\displaystyle z(l)Z^{(p-1)}(L-p-l).$
Where, $\rho_{mn}^{z}$ is the correlation function of individual segments. It
has a similar form with that in Eq. (18), but now $G_{jq}$ should be replaced
by $G_{jq}(\beta)$.
Figure 1: Temperature dependence of the spin-spin correlation function in a
uniform spin chain with $\lambda=1.5$.
In Fig. 1, we plotted the temperature dependence of the spin-spin correlation
function per site, $\sum_{m,n}^{L}C_{mn}^{z}(\beta)/L$. We find that when
$D_{z}\leq-\Delta_{h}(0)$, the correlation function approaches to zero in the
limit $T\rightarrow 0$. This indicates that the ground state is in the hole
condensation phase. On the other hand, when $D_{z}>-\Delta_{h}(0)$, the
correlation function approaches to a finite value (about 0.85 for the two
cases shown in the figure) in the zero temperature limit.
## IV Spin susceptibility
The spin susceptibility of the $S=1$ QIM can be also calculated using the
recursion formula introduced in the previous section. To do this, one needs to
first evaluate the partition functions of each $S=1/2$ Ising segments in the
applied magnetic field $\xi$, denoted by $z(l_{n},\xi)$. The partition
function of the original $S=1$ QIM is then given by
$Z(L,\xi)=\sum_{p=0}^{L}\sum_{\\{l_{n}\\}}\prod_{n=1}^{p+1}z(l_{n},\xi)\alpha^{p}$.
In terms of the segment magnetization
$m(l_{n},T)=-\frac{1}{\beta}\frac{\partial\ln z(l_{n},\xi)}{\partial\xi}$ and
the segment susceptibility $\chi(l_{n},T)=\frac{\partial
m(l_{n},T)}{\partial\xi}$, the total susceptibility $\chi(T)$ at zero-magnetic
field can be expressed as
$\displaystyle\chi(T)$ $\displaystyle=$
$\displaystyle\frac{1}{Z(L)}\sum_{p=0}^{L}\sum_{l=0}^{L-p}\alpha^{p}(p+1)$
(21) $\displaystyle\chi(l,T)z(l)Z^{(p-1)}(L-p-l).$
Thus the hole decomposition scheme provide an alternative approach to
calculate the susceptibility of the $S=1$ QIM. This approach is efficient
provided that the susceptibilities of the corresponding $S=1/2$ TIM’s with
varying chain length $L$ are available. We note that the susceptibility of the
$S=1/2$ TIM has already been studied by a number of groupsPfeuty70 ;
Ovchinnikov ; skew . So in principle these results could be used in the
numerical study of the susceptibility of the $S=1$ QIM.
## V Diagonalization of the M-matrix
For a periodic spin chain, the diagonalization of the M-matrix has been
discussed in Sec. IV A in Ref. yangPRB . Here we consider the diagonalization
of this $l\times l$ M-matrix for an open spin chain with the length $l$. The
aim here is to solve the following eigen equation
$\displaystyle M\Phi_{k}=\Lambda^{2}(k)\Phi_{k}$ (22)
in various cases, where $\Phi_{k}(j)$’s take the form of Eqs. (23) in Ref.
yangPRB .
We assume that the two ends of the open chain are located at the sites $r_{1}$
and $r_{2}$, respectively. $r_{1}$ and $r_{2}$ can be either odd or even, so
there are four kinds of $M$-matrices. In the following, we will present the
results for each cases.
### V.1 $(r_{1},r_{2})=(odd,\,even)$
In this case, the matrix $M$ is defined by
$M=\begin{pmatrix}a_{0}&b_{1}&0&\cdots&0&0\\\ b_{1}&a_{2}&b_{2}&\cdots&0&0\\\
0&b_{2}&a_{1}&\cdots&0&0\\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&0&0&\cdots&a_{1}&b_{1}\\\ 0&0&0&\cdots&b_{1}&a_{2}\end{pmatrix},$ (23)
where $a_{1,2},~{}b_{1,2}$ are defined in the main text and $a_{0}=D_{1}^{2}$.
The energy spectra can be solved following the approach introduced in Section
IV. The result is given by
$\displaystyle\Lambda^{2}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{e^{2ik}-t_{e}e^{-2ik}}[b_{1}\tau(e^{ik}-t_{o}e^{-ik})$
$\displaystyle+a_{2}(e^{2ik}-t_{e}e^{-2ik})+b_{2}\tau(e^{3ik}-t_{o}e^{-3ik})],$
The reflection parameters are
$\displaystyle t_{o}$ $\displaystyle=$ $\displaystyle e^{2i(l+1)k},~{}~{}$
(24) $\displaystyle t_{e}$ $\displaystyle=$
$\displaystyle\frac{t_{o}(b_{1}e^{ik}+b_{2}e^{-ik})}{(b_{1}e^{-ik}+b_{2}e^{ik})}.$
Then, the secular equation is given by
$\displaystyle\left[(a_{2}-a_{1})\pm W\right][b_{1}\sin(l+2)k+b_{2}\sin lk]$
(25) $\displaystyle=$
$\displaystyle\frac{2(a_{0}-a_{1})(b_{1}^{2}+b_{2}^{2}+2b_{1}b_{2}\cos 2k)\sin
lk}{b_{2}},$
where $W$ is defined as in Eq. (20) in Ref. yangPRB .
Other cases can be solved by the same way and the results are listed below.
### V.2 $(r_{1},r_{2})=(odd,\,odd)$
The reflection parameters $t_{o,e}$ are
$\displaystyle t_{e}$ $\displaystyle=$ $\displaystyle e^{2i(l+1)k},$ (26)
$\displaystyle t_{o}$ $\displaystyle=$
$\displaystyle\frac{t_{e}(b_{1}e^{-ik}+b_{2}e^{ik})}{(b_{1}e^{ik}+b_{2}e^{-ik})}.$
The secular equation is
$\displaystyle\left[(a_{1}-a_{2})\pm
W\right]\left[b_{1}\sin(l-1)k+b_{2}\sin(l+1)k\right]$ (27) $\displaystyle=$
$\displaystyle\frac{2b_{2}(b_{1}^{2}+b_{2}^{2}+2b_{1}b_{2}\cos
2k)\sin(l+1)k}{a_{0}-a_{1}}.$
### V.3 $(r_{1},r_{2})=(even,\,even)$
The reflection parameters $t_{o,e}$ are
$\displaystyle t_{e}$ $\displaystyle=$ $\displaystyle e^{2i(l+1)k},$ (28)
$\displaystyle t_{o}$ $\displaystyle=$
$\displaystyle\frac{t_{e}(b_{1}e^{ik}+b_{2}e^{-ik})}{(b_{1}e^{-ik}+b_{2}e^{ik})}.$
The secular equation is
$\displaystyle\left[(a_{2}-a_{1})\pm
W\right]\left[b_{1}\sin(l+1)k+b_{2}\sin(l-1)k\right]$ (29) $\displaystyle=$
$\displaystyle\frac{2b_{1}(b_{1}^{2}+b_{2}^{2}+2b_{1}b_{2}\cos
2k)\sin(l+1)k}{a_{3}-a_{2}}$
where, $a_{3}=D_{2}$.
### V.4 $(r_{1},r_{2})=(even,\,odd)$
The reflection parameters $t_{o,e}$ are
$\displaystyle t_{o}=e^{2i(l+1)k},$ (30) $\displaystyle
t_{e}=\frac{t_{o}(b_{1}e^{-ik}+b_{2}e^{ik})}{(b_{1}e^{ik}+b_{2}e^{-ik})}.$
The secular equation is
$\displaystyle\left[(a_{1}-a_{2})\pm W\right]$ (31) $\displaystyle=$
$\displaystyle\frac{2b_{1}[b_{1}\sin(lk)+b_{2}\sin(l+2)k]}{a_{3}-a_{2}}.$
## VI The partition functions of segments
The partition function of individual segment of length $l$ and parity
$(r_{1},r_{2})$ (defined in Sec. V) is given by
$z_{(r_{1},r_{2})}(l)=\prod_{\begin{subarray}{c}k_{1}\in(0,\pi/2),\\\
k_{2}\in(\pi/2,\pi)\end{subarray}}\cosh\left[\frac{\beta\Lambda_{1}(k_{1})}{2}\right]\cosh\left[\frac{\beta\Lambda_{2}(k_{2})}{2}\right],$
(32)
where, $k_{1,2}$ satisfy the corresponding secular equations.
## Acknowledgments
This work was supported in part by the National Natural Science Foundation of
China, the national program for basic research of China (the 973 program), the
PCSIRT (IRT-0754), and SRFDP (No.J20050335118) of Education Ministry of China.
## References
* (1) Z.H. Yang, L.P. Yang, H.N. Wu, J. Dai, and T. Xiang, Phys. Rev. B 79, 214427(2009).
* (2) J. Oitmaa and A.M.A. von Brasch, Phys. Rev. B 67, 172402 (2003).
* (3) Z.H. Yang, L.P. Yang, J. Dai, and T. Xiang, Phys. Rev. Lett. 100, 067203 (2008).
* (4) E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. N.Y. 16, 407 (1961).
* (5) P. Pfeuty, Ann. Phys. N.Y. 57, 79 (1970).
* (6) R.J. Elliott, P. Pfeuty and C. Wood, Phys. Rev. Lett. 25, 443 (1970).
* (7) A. A. Ovchinnikov, D. V. Dmitriev, V.Ya. Krivnov, and V. O. Cheranovskii, Phys. Rev. B 68, 214406 (2003) .
* (8) H. C. Fogedby, J. Phys. C. 11, 2801 (1978).
|
arxiv-papers
| 2009-07-16T03:58:33 |
2024-09-04T02:49:03.991170
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhi-Hua Yang, Li-Ping Yang, Hai-Na Wu, Jianhui Dai, Tao Xiang",
"submitter": "Zhihua Yang",
"url": "https://arxiv.org/abs/0907.2740"
}
|
0907.2772
|
# Decay $\eta_{b}\to J/\psi J/\psi$ in light cone formalism
V.V. Braguta [email protected] Institute for High Energy Physics, Protvino,
Russia Kartvelishvili, V [email protected] Lancaster
University, Lancaster, UK
###### Abstract
The decays of pseudoscalar bottomonium $\eta_{b}$ into a pair of vector
charmonia, $J/\psi J/\psi,J/\psi\psi^{\prime},\psi^{\prime}\psi^{\prime}$ are
considered in the light cone formalism. Relativistic and leading logarithmic
radiative corrections to the amplitudes of these processes are resummed. It is
shown that the small value for the branching ratio of the decay $\eta_{b}\to
J/\psi J/\psi$ obtained within the leading order nonrelativistic QCD is a
consequence of a fine-tuning between certain parameters, which is broken when
relativistic and leading logarithmic radiative corrections are taken into
account. As a result, the branching ratio obtained in this paper is enhanced
by an order of magnitude.
###### pacs:
12.38.-t, 12.38.Bx,
## I Introduction
Ever since the discovery of the $\Upsilon$ meson, there have been numerous
attempts of observing the lightest pseudoscalar bottomonium state, $\eta_{b}$.
However, only recently the first experimental evidence of the existence of
this meson was found by BaBar collaboration, in the radiative decay
$\Upsilon(3S)\to\eta_{b}+\gamma$ :2008vj . Its mass was found to be
$m_{\eta_{b}}=9388^{+3.1}_{-2.3}(stat)\pm 2.7(syst)$ MeV, but our knowledge of
its other properties remains rather poor.
In Braaten:2000cm it was proposed to look for the $\eta_{b}$ meson in the
decay $\eta_{b}\to J/\psi J/\psi$, but, despite its clean signature, this
process may be hard to observe due to its extremely small branching ratio:
contrary to other similar processes, such as the decays $\chi_{b}\to J/\psi
J/\psi$ Kartvelishvili:1984en , the rate of the decay $\eta_{b}\to J/\psi
J/\psi$ vanishes at the leading order of both relative velocity and
$1/M_{\eta_{b}}$ expansions. The calculations made within nonrelativisitic QCD
(NRQCD) Bodwin:1994jh yield $Br(\eta_{b}\to J/\psi J/\psi)\sim
10^{-8}-10^{-7}$ Jia:2006rx ; Gong:2008ue , however in Santorelli:2007xg it
was shown that the account of final-state interaction effects can enhance it
up to about $10^{-5}$.
A similar conclusion can be drawn from the comparison of the decays
$\eta_{b}\to J/\psi J/\psi,J/\psi\psi^{\prime},\psi^{\prime}\psi^{\prime}$ and
the processes of double charmonia production at B-factories. It is now clear
that these processes are greatly effected by radiative and relativistic
corrections Braaten:2002fi ; Liu:1 ; Liu:2 ; Zhang:2005ch ; Gong:2007db ;
Zhang:2008gp ; Bondar:2004sv ; Braguta:2005kr ; Berezhnoy:2007sp ;
Ebert:2008kj ; He:2007te ; Bodwin:2007ga ; Braguta:2008tg . With the mass of
$\eta_{b}$ being so close to the energy at which B-factories operate, it is
natural to expect that the same is true for the decays $\eta_{b}\to J/\psi
J/\psi,J/\psi\psi^{\prime},\psi^{\prime}\psi^{\prime}$, and hence the
consideration of these processes without accounting for radiative and
relativistic corrections is unreliable. This was also confirmed by the
calculation of radiative corrections within NRQCD, performed in Gong:2008ue .
In this paper, the processes $\eta_{b}\to J/\psi
J/\psi,J/\psi\psi^{\prime},\psi^{\prime}\psi^{\prime}$ are considered within
the light cone (LC) formalism Chernyak:1983ej . In this approach, the
amplitudes of these processes are expanded in
$(M_{c\bar{c}}/M_{b\bar{b}})^{2}\sim 0.1$, which is sufficiently small for the
applicability of the method Braguta:2009df .
In the LC formalism, the amplitude of a process under study is decomposed into
the perturbative part, dealing with the production of quarks and gluons at
small distances, and the large-distance part describing the hadronization of
the partons. For hard exclusive processes, the latter can be parameterized by
the process-independent distribution amplitudes (DA), which can be considered
as hadrons’ wave functions at lightlike separations between the partons inside
the hadron. It should be noted that DAs contain information about the
structure of mesons and effectively resum relativistic corrections to the
amplitude. Moreover, using renormalization group evolution of DAs, one can
take into account the leading logarithmic radiative corrections to the
amplitude.
This paper is organized as follows. In the next section DAs for charmonium are
defined, and various models for these DAs are discussed. In the third section,
the amplitude of the decay of $\eta_{b}$ into two vector mesons is derived.
Finally, in the last section the numerical results and their uncertainties are
presented and discussed.
## II Distribution amplitudes for charmonium
The amplitude of the process $\eta_{b}\to V_{1}V_{2}$, with $V_{1,2}$ standing
for either $J/\psi$ or $\psi^{\prime}$, can be parameterized with a single
formfactor $F$:
$\displaystyle
M=Fe_{\mu\nu\sigma\rho}p_{1}^{\mu}p_{2}^{\nu}\epsilon_{1}^{\sigma}\epsilon_{2}^{\rho},$
(1)
where $p_{1},p_{2}$ and $\epsilon_{1},\epsilon_{2}$ are the momenta and
polarization vectors of $V_{1}$ and $V_{2}$ respectively. Hence, the width of
the decay $\eta_{b}\to V_{1}V_{2}$ can be written in the form
$\displaystyle\Gamma[\eta_{b}\to V_{1}V_{2}]=|F|^{2}\frac{|{\bf
p}|^{3}}{4\pi},$ (2)
where ${\bf p}$ is the 3-momentum of a final meson in the $\eta_{b}$ rest
frame. If the final mesons are identical, $V_{1}=V_{2}$, the width $\Gamma$
should be divided by $2!$.
In the LC formalism, the amplitude of a hard exclusive process is expanded in
the inverse powers of the hard energy scale $E_{h}$, which for the decay
$\eta_{b}\to V_{1}V_{2}$ can be identified as $M_{\eta_{b}}$. The leading
order contribution in this expansion requires the two vector mesons to be
produced with polarizations $\lambda_{1}=\lambda_{2}=0$ Chernyak:1983ej , but
in this case the aplitude (1) vanishes. In order to obtain a non-zero result,
both vector mesons need to be transversely polarized, which in turn means that
the helicities of the quarks in both mesons must be flipped twice, and hence
leads to a suppression factor $\sim 1/(M_{\eta_{b}})^{2}$ Jia:2006rx .
Therefore, the decay $\eta_{b}\to V_{1}V_{2}$ is a next-to-next-to-leading
(NNLO) twist process, and in order for the calculations to be consistent one
needs DAs up to twist-4. In general, twist-4 DAs should contain terms
corresponding to higher Fock states in addition to the “valence” charm quark-
antiquark state, but we expect such higher states in charmonium to be
suppressed, and in the following we will neglect their contribution.
The DAs for a vector meson $V$ with momentum $p$ and polarization vector
$\epsilon$ can be defined as follows Ball:1998sk :
$\displaystyle\langle
V(p,\epsilon)|\bar{c}(x)\gamma_{\rho}[x,-x]c(-x)|0\rangle$ $\displaystyle=$
$\displaystyle f_{V}M_{V}\biggl{[}\frac{(\epsilon
x)}{(px)}p_{\rho}\int_{-1}^{1}d\xi
e^{i\xi(px)}\bigl{(}\varphi_{1}(\xi,\mu)+\frac{M_{V}^{2}x^{2}}{4}\varphi_{2}(\xi,\mu)\bigr{)}$
$\displaystyle+$ $\displaystyle\bigr{(}\epsilon_{\rho}-p_{\rho}\frac{(\epsilon
x)}{(px)}\bigl{)}\int_{-1}^{1}d\xi e^{i\xi(px)}\varphi_{3}(\xi,\mu)$
$\displaystyle-$ $\displaystyle\frac{1}{2}x_{\rho}\frac{(\epsilon
x)}{(px)^{2}}M_{V}^{2}\int_{-1}^{1}d\xi
e^{i\xi(px)}\varphi_{4}(\xi,\mu)\biggr{]},$ $\displaystyle\langle
V(p,\epsilon)|\bar{c}(x)\sigma_{\rho\lambda}[x,-x]c(-x)|0\rangle$
$\displaystyle=$ $\displaystyle
f_{T}(\mu)\biggl{[}\bigl{(}\epsilon_{\rho}p_{\lambda}-\epsilon_{\lambda}p_{\rho}\bigr{)}\int_{-1}^{1}d\xi
e^{i\xi(px)}\bigl{(}\chi_{1}(\xi,\mu)+\frac{M_{V}^{2}x^{2}}{4}\chi_{2}(\xi,\mu)\bigr{)}$
$\displaystyle+$
$\displaystyle\bigr{(}p_{\rho}x_{\lambda}-p_{\lambda}x_{\rho}\bigl{)}\frac{(\epsilon
x)}{(px)^{2}}M_{V}^{2}\int_{-1}^{1}d\xi e^{i\xi(px)}\chi_{3}(\xi,\mu)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\bigl{(}\epsilon_{\rho}x_{\lambda}-\epsilon_{\lambda}x_{\rho}\bigr{)}\frac{M_{V}^{2}}{(px)}\int_{-1}^{1}d\xi
e^{i\xi(px)}\chi_{4}(\xi,\mu)\biggr{]},$ $\displaystyle\langle
V(p,\epsilon)|\bar{c}(x)\gamma_{\rho}\gamma_{5}[x,-x]c(-x)|0\rangle$
$\displaystyle=$ $\displaystyle
f_{A}(\mu)e_{\rho\lambda\alpha\beta}\epsilon^{\lambda}p^{\alpha}x^{\beta}\int_{-1}^{1}d\xi
e^{i\xi(px)}\Phi_{1}(\xi,\mu),$ $\displaystyle\langle
V(p,\epsilon)|\bar{c}(x)[x,-x]c(-x)|0\rangle$ $\displaystyle=$ $\displaystyle-
if_{S}(\mu)(\epsilon x)\int_{-1}^{1}d\xi e^{i\xi(px)}\Phi_{2}(\xi,\mu).$
Here $[x,-x]$ is the gluon string which makes the matrix element gauge
invariant, $\xi$ is a dimensionless variable describing the relative motion of
the charmed quark and antiquark inside the meson, $\mu$ is the energy scale at
which the DAs are defined, while the constants $f_{V}$ and $f_{T}(\mu)$ are
defined by
$\displaystyle\langle V(p,\epsilon)|\bar{c}(0)\gamma_{\mu}c(0)|0\rangle$
$\displaystyle=$ $\displaystyle f_{V}M_{V}\epsilon_{\mu},$
$\displaystyle\langle V(p,\epsilon)|\bar{c}(0)\sigma_{\mu\nu}c(0)|0\rangle$
$\displaystyle=$ $\displaystyle
f_{T}(\mu)\bigl{(}\epsilon_{\mu}p_{\nu}-\epsilon_{\nu}p_{\mu}\bigr{)}.$ (4)
The constants $f_{A}(\mu),f_{S}(\mu)$ can be expressed through $f_{V},f_{T}$
as follows:
$\displaystyle f_{A}(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{(}f_{V}-f_{T}(\mu)\frac{2m_{c}(\mu)}{M_{V}}\biggr{)}M_{V},$
$\displaystyle f_{S}(\mu)$ $\displaystyle=$
$\displaystyle\biggl{(}f_{T}(\mu)-f_{V}\frac{2m_{c}(\mu)}{M_{V}}\biggr{)}M_{V}^{2},$
(5)
where $m_{c}(\mu)$ is the running mass of the $c$ quark.
Eqs. (II) contain 10 independent DAs, but only 4 of these are relevant for the
calculation of the $\eta_{b}\to V_{1}V_{2}$ decay rate:
$\varphi_{1}(\xi),\chi_{1}(\xi),\Phi_{1}(\xi)$ and $\Phi_{2}(\xi)$ (see
below). For the first two, $\varphi_{1}(\xi)$ and $\chi_{1}(\xi)$, we will use
models proposed in Braguta:2006wr ; Braguta:2007fh ; Braguta:2007tq ;
Braguta:2008qe . In Braguta:2008tg it was shown that, if the higher Fock
states are ignored, the functions $\Phi_{1}(\xi)$ and $\varphi_{3}(\xi)$ can
be unambiguously determined from the equations of motion. The same is true for
the functions $\Phi_{2}(\xi)$ and $\chi_{3}(\xi)$.
In the remainder of this section, a relation between
$\Phi_{2}(\xi),\chi_{3}(\xi)$ and $\varphi_{1}(\xi),\chi_{1}(\xi)$ will be
derived. The functions $\Phi_{2}(\xi)$ and $\chi_{3}(\xi)$ can be expanded
into a series of Gegenbauer polynomials Ball:1998sk :
$\displaystyle\chi_{3}(x,\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{[}1+\sum_{n=2,4..}c_{n}(\mu)C_{n}^{1/2}(2x-1)\biggr{]},$
$\displaystyle\Phi_{2}(x,\mu)$ $\displaystyle=$
$\displaystyle\frac{3}{4}(1-\xi^{2})\biggl{[}1+\sum_{n=2,4..}d_{n}(\mu)C_{n}^{3/2}(2x-1)\biggr{]}.$
(6)
The coefficients $c_{n}(\mu)$ and $d_{n}(\mu)$ are related to the moments of
the functions $\varphi_{1}(\xi),\chi_{1}(\xi)$ through the equations of motion
Ball:1998sk ,
$\displaystyle\frac{n+2}{2}\langle\xi^{n}\rangle_{\chi}$ $\displaystyle=$
$\displaystyle\langle\xi^{n}\rangle_{T}+\frac{n(n-1)}{2}(1-\delta(\mu))\langle\xi^{n-2}\rangle_{\Phi},$
$\displaystyle(n+1)(1-\delta(\mu))\langle\xi^{n}\rangle_{\Phi}$
$\displaystyle=$
$\displaystyle\langle\xi^{n}\rangle_{\chi}-\delta(\mu)\langle\xi^{n}\rangle_{L},$
(7)
where $\langle\xi^{n}\rangle_{L,T,\chi,\Phi}$ denote the moments of the DAs
$\phi_{1}(\xi),\chi_{1}(\xi),\chi_{3}(\xi),\Phi_{2}(\xi)$ respectively, while
$\delta(\mu)=2f_{V}/f_{T}(\mu)(m_{c}(\mu)/M_{V})$. By solving eqs. (7)
recursively, one can determine the functions $\Phi_{2}(\xi)$ and
$\chi_{3}(\xi)$. In Braguta:2006wr it was shown, that there is a fine-tuning
of the coefficients of the Gegenbauer expansion at the scale
$\mu\sim\overline{m}_{c}\equiv m_{c}(\mu=m_{c})$. Without this fine-tuning the
DAs of a nonrelativistic system would show an unphysical relativistic tail
already at the scale $\mu\sim\overline{m}_{c}$. In order to get rid of this
tail in the DAs $\Phi_{2}(\xi)$ and $\chi_{3}(\xi)$, fine-tuning is required
between the coefficients $c_{n},d_{n}$ and the parameter $\delta$, which is
related to the wave functions $\phi_{1}(\xi),\chi_{1}(\xi)$ Braguta:2008tg :
$\displaystyle\delta(\overline{m}_{c})=\frac{\int_{-1}^{1}\frac{d\xi}{1-\xi^{2}}\chi_{1}(\xi,\mu\sim\overline{m}_{c})}{\int_{-1}^{1}\frac{d\xi}{(1-\xi^{2})^{2}}\varphi_{1}(\xi,\mu\sim\overline{m}_{c})}.$
(8)
## III The amplitude of the process $\eta_{b}\to V_{1}V_{2}$
The diagrams that contribute to the amplitude of the process under study at
the leading order in the $\alpha_{s}$ expansion are shown in Fig. 1.
$\eta_{b}$$V_{1}$$V_{2}$$b$${\overline{b}}$$c$$c$
$\eta_{b}$$V_{1}$$V_{2}$$b$${\overline{b}}$$c$$c$
Figure 1: The diagrams contributing to the amplitude of the process
$\eta_{b}\to J/\psi J/\psi$ at the leading order in $\alpha_{s}$.
The procedure of calculating the amplitude is described in detail in
Chernyak:1983ej . This is a lengthy but straightforward exercise, yielding a
result which looks remarkably simple:
$\displaystyle F$ $\displaystyle=$ $\displaystyle\int
d\xi_{1}d\xi_{2}H(\xi_{1},\xi_{2},\mu)\biggl{(}f_{V1}f_{A2}(\mu)M_{V1}\varphi_{1}(\xi_{1},\mu)\Phi_{1}(\xi_{2},\mu)+f_{V2}f_{A1}(\mu)M_{V2}\varphi_{1}(\xi_{2},\mu)\Phi_{1}(\xi_{1},\mu)$
(9) $\displaystyle+$ $\displaystyle
f_{S1}(\mu)f_{T2}(\mu)\chi_{1}(\xi_{2},\mu)\Phi_{2}(\xi_{1},\mu)+f_{S2}(\mu)f_{T1}(\mu)\chi_{1}(\xi_{1},\mu)\Phi_{2}(\xi_{2},\mu)\biggr{)}.$
Here the function $H(\xi_{1},\xi_{2},\mu)$ represents the hard part of the
amplitude,
$\displaystyle
H(\xi_{1},\xi_{2},\mu)=\frac{1024\pi^{2}\alpha_{s}^{2}(\mu)}{27}f_{\eta_{b}}\frac{1}{M_{\eta_{b}}^{6}}\frac{1}{(1-\xi_{1}^{2})(1-\xi_{2}^{2})(1+\xi_{1}\xi_{2})},$
(10)
with the decay constant $f_{\eta_{b}}$ defined by
$\displaystyle\langle
0|\bar{b}(0)\gamma_{\rho}\gamma_{5}b(0)|\eta_{b}(p)\rangle$ $\displaystyle=$
$\displaystyle if_{\eta_{b}}p_{\rho}.$ (11)
At this point, some comments are in order.
1. 1.
In eq. (9) there is a clear separation of large- and small-distance
contributions. While $H(\xi_{1},\xi_{2},\mu)$ describes the hard part of the
amplitude, the large-distance part is parameterized by the combination of the
DAs, which effectively include resummation of the relativistic corrections to
the amplitude. A discussion of this point can be found in Braguta:2009df ;
Braguta:2008tg .
2. 2.
In eq. (9) the dependence of the hard part of the amplitude, the constants and
the DAs on the scale $\mu$ is explicitly shown. If the process in question
were a leading-twist process, one could perform an exact resummmation of all
leading-twist radiative corrections to the amplitude,
$\sim\alpha_{s}\log(M_{\eta_{b}}^{2}/M_{J/\psi}^{2})$, simply by taking
$\mu\sim M_{\eta_{b}}$ Chernyak:1983ej . Indeed, for a leading-twist process,
one would use the axial gauge, in which double-logarithmic and logarithmic
corrections only appear in the self-energy diagrams and re-scattering of final
particles. The double-logarithmic corrections are cancelled since final
particles are colorless objects, while the logarithmic corrections lead to the
renormalization of the DAs themselves. Although the decay $\eta_{b}\to
V_{1}V_{2}$ is a next-to-next-to-leading-twist process, all the arguments
given above still seem to be applicable. Note also that in eq. (9) there is no
divergence in the end-point region, $|\xi|\sim 1$, indicating that all
logarithms are collected. These arguments allow us to believe that eq. (9)
includes the exact resummation of leading logarithmic radiative corrections to
all loops.
3. 3.
Whenever NRQCD and LC approaches are used to describe the same process, one
should expect some kind of duality between the two results. For the process
$\eta_{b}\to VV$ this duality can be checked at the leading-order
approximation in relative velocity of the $c$-quark-antiquark pair inside
charmonia. In particular, by taking infinitely narrow DAs and the constants
$f_{T},f_{V}$ and masses $M_{V},2m_{c}$ at the next-to-leading order
approximation in relative velocity Braguta:2007ge ,
$\displaystyle\frac{f_{T}}{f_{V}}=1-\frac{\langle v^{2}\rangle}{3},$
$\displaystyle\frac{M_{V}}{2m_{c}}=1+\frac{\langle v^{2}\rangle}{2},$ (12)
and by neglecting all radiative corrections, one gets from eq. (9):
$\displaystyle
F=\frac{256\pi^{2}\alpha_{s}^{2}}{81}\frac{1}{m_{b}^{6}}f_{\eta_{b}}f_{V}^{2}m_{c}^{2}\langle
v^{2}\rangle,$ (13)
which coincides with the result obtained in Jia:2006rx . In these formulae,
$\langle v^{2}\rangle$ is the NRQCD matrix element, defined as
$\displaystyle\langle v^{2}\rangle=-\frac{1}{m_{c}^{2}}\frac{\langle
0|\chi^{+}(\vec{\sigma}\vec{\epsilon})({\overset{\leftrightarrow}{\bf
D}})^{2}\varphi|V(\epsilon)\rangle}{\langle
0|\chi^{+}(\vec{\sigma}\vec{\epsilon})\varphi|V(\epsilon)\rangle}.$ (14)
As noted in Braguta:2009df ; Braguta:2008tg , the duality between NRQCD and LC
allows us to estimate the size of power corrections. The idea is that if one
expands the NRQCD result in powers of $1/M_{\eta_{b}}$, than the first term
coincides with the LC prediction and the second term gives an estimate of
power corrections to the LC result. Thus, power corrections to the amplitude
of the $\eta_{b}\to VV$ decay can be estimated as $\sim
4v^{2}M_{V}^{2}/M_{\eta_{b}}^{2}$.
Now we have all the ingredients needed to calculate the rates of the decays
$\eta_{b}\to V_{1}V_{2}$.
## IV Numerical results and discussion
### IV.1 Input parameters
In order to obtain numerical results for the branching ratios of the decays
$\eta_{b}\to J/\psi J/\psi,J/\psi\psi^{\prime},\psi^{\prime}\psi^{\prime}$ the
following input parameters were used:
1. 1.
The strong coupling constant $\alpha_{s}(\mu)$ is taken at the one loop,
$\displaystyle\alpha_{s}(\mu)=\frac{4\pi}{\beta_{0}\log(\mu^{2}/\Lambda^{2})},$
(15)
with $\Lambda=0.2$ GeV, $\beta_{0}=25/3$.
2. 2.
The mass of the $c$-quark in $\overline{MS}$ scheme, ${\overline{m}}_{c}=1.2$
GeV.
3. 3.
The leptonic decay constants of the $J/\psi$ and $\psi^{\prime}$ mesons
$f_{V}^{J/\psi},f_{V}^{\psi^{\prime}}$ were determined directly from
experimental data, while the constants $f_{T}^{J/\psi}$ and
$f_{T}^{\psi^{\prime}}$ were calculated within NRQCD in Braguta:2007ge :
$\displaystyle(f_{V}^{J/\psi})^{2}$ $\displaystyle=$ $\displaystyle 0.173\pm
0.004~{}\mbox{GeV}^{2},\qquad~{}~{}~{}~{}~{}~{}(f_{V}^{\psi^{\prime}})^{2}=0.092\pm
0.002~{}\mbox{GeV}^{2},$ $\displaystyle(f_{T}^{J/\psi}(M_{J/\psi}))^{2}$
$\displaystyle=$ $\displaystyle 0.144\pm
0.016~{}\mbox{GeV}^{2},\quad(f_{T}^{\psi^{\prime}}(M_{J/\psi}))^{2}=0.068\pm
0.022~{}\mbox{GeV}^{2}.$ (16)
4. 4.
We assume that the total decay width of the $\eta_{b}$ meson
$\Gamma_{\mbox{tot}}(\eta_{b})$ can be approximated by its two-gluon decay
width $\Gamma(\eta_{b}\to gg)$ which, at the leading order in relative
velocity and $\alpha_{s}$, is equal to
$\displaystyle\Gamma_{\mbox{tot}}(\eta_{b})=\Gamma(\eta_{b}\to
gg)=\frac{8\pi}{9}\frac{\alpha_{s}^{2}}{M_{\eta_{b}}}f_{\eta_{b}}^{2}\;.$ (17)
5. 5.
The leading twist DAs needed for the calculations are taken from models
developed in Braguta:2006wr ; Braguta:2007fh ; Braguta:2007tq ; Braguta:2008qe
.
### IV.2 Estimation of uncertainties
The most important uncertainties come from the following sources:
1. 1.
Model-dependence of the DAs. These uncertainties can be estimated by varying
the parameters of these models (see Braguta:2006wr ; Braguta:2007fh ;
Braguta:2007tq ; Braguta:2008qe for more details). The calculations show that
for the processes $\eta_{b}\to J/\psi
J/\psi,J/\psi\psi^{\prime},\psi^{\prime}\psi^{\prime}$ these uncertainties are
no larger than $\sim 5\%,~{}13\%,~{}30\%$, respectively. In fact, these
uncertainties are expected to be rather low, due to the property that the
precision of any DA model improves with evolution Braguta:2006wr .
2. 2.
Radiative corrections. Within the approach used in this paper, the leading
logarithmic radiative corrections due to the evolution of the DAs and the
strong coupling constant were effectively resummed. Although we argued above
that this is also true for all leading logarithmic radiative corrections,
there is no strict proof of this statement. For this reason, we estimate the
uncertainty due to the radiative corrections as
$\sim\alpha_{s}(M_{\eta_{b}}/2)\log(M_{\eta_{b}}^{2}/(4M_{J/\psi}^{2}))\sim
50\%$.
3. 3.
Power corrections. As mentioned above, this source of uncertainty can be
estimated as $\sim 4\langle v^{2}\rangle M_{V}^{2}/M_{\eta_{b}}^{2}$, which is
the largest for the decay $\eta_{b}\to\psi^{\prime}\psi^{\prime}$, reaching
$\sim 4\langle
v^{2}\rangle_{\psi^{\prime}}M_{\psi^{\prime}}^{2}/M_{\eta_{b}}^{2}\sim 20\%$.
4. 4.
Relativistic corrections. This source of uncertainty appears because we
treated $\eta_{b}$ meson at the leading-order approximation in relative
velocity. It can be estimated as $\sim v_{\eta_{b}}^{2}\sim 10\%$.
5. 5.
The uncertainties in the values of constants (16). For the three processes
$\eta_{b}\to J/\psi J/\psi,J/\psi\psi^{\prime},\psi^{\prime}\psi^{\prime}$
these errors are estimated to be $\sim 16\%,27\%,49\%$, respectively.
6. 6.
Higher Fock states. It can be argued that at the scale $\mu$ relevant to
$\eta_{b}$ decay process, only a small fraction of quarkonium momentum is
carried by the quark-gluon sea, typically $\sim 5-10\%$ Kartvelishvili:1985ac
. Hence, we expect the effects of higher Fock states to be negligible,
compared to other uncertainties considered here.
The overall uncertainties of our calculations were obtained by adding the
above errors in quadrature.
### IV.3 Results and discussion
By substituting the expressions for DAs and the necessary constants into eqs.
(9) and (2), we get the following values for the three branching ratios:
$\displaystyle Br(\eta_{b}\to J/\psi J/\psi)$ $\displaystyle=$
$\displaystyle(6.2\pm 3.5)\times 10^{-7},$ $\displaystyle Br(\eta_{b}\to
J/\psi\psi^{\prime})$ $\displaystyle=$ $\displaystyle(10\pm 6)\times 10^{-7},$
(18) $\displaystyle Br(\eta_{b}\to\psi^{\prime}\psi^{\prime})$
$\displaystyle=$ $\displaystyle(3.7\pm 2.8)\times 10^{-7}.$
It is interesting to compare these results with previous calculations. In
particular, within the leading order NRQCD, one has Jia:2006rx :
$\displaystyle Br(\eta_{b}\to J/\psi J/\psi)=(2.4^{+4.2}_{-1.9})\times
10^{-8}.$ (19)
which is roughly 20 times smaller than our result shown above. The reason of
this suppression can be traced to the expression for the amplitude (9), where
all terms are in fact proportional to the constants $f_{A}$ and $f_{S}$,
which, in turn, are expressed through $f_{V}$ and $f_{T}$ (see eq. (5)). In
the absence of relativistic and radiative corrections, the fine-tuning between
$f_{V}$, $f_{T}$ and the masses, clearly visible in eqs. (12), guerantees that
$f_{A}$, $f_{S}$ and hence the formfactor $F$ are proportional to $\langle
v^{2}\rangle$, which is small for nonrelativistic systems. Taking relativistic
and leading logarithmic radiative corrections to the constants $f_{A}$ and
$f_{S}$ into account breaks the fine tuning, thus leading to a considerable
enhancement of the branching ratio. To illustrate the above argument
numerically, we take an infinitely narrow approximation for the DAs,
parameters with fine-tuning given by eqs. (12), and $\langle
v^{2}\rangle=0.25$, to obtain $Br(\eta_{b}\to J/\psi J/\psi)\simeq 2\times
10^{-8}$, in agreement the leading order NRQCD result Jia:2006rx . Next, we
take into account relativisitic and leading logarithmic radiative corrections
to the constants $f_{A}$ and $f_{S}$, but still use an infinitely narrow
approximation for the DAs. In this case fine-tuning is broken, and we get
$\sim 3\times 10^{-7}$, and order-of-magnitude increase compared to the NRQCD
value. By including renormalization group evolution and relativistic motion
into the DAs, we get a further increase of the branching ratio by a factor
$\sim 2$.
In Gong:2008ue the authors took into account one-loop radiative corrections
and obtained
$\displaystyle Br(\eta_{b}\to J/\psi J/\psi)=(2.1-18.6)\times 10^{-8}.$ (20)
Although this number seems to be compatible with ours shown in eq. (18), we do
not believe that the two results are in agreement with each other. In
particular, the analytical form of the formfactor $F$ obtained in Gong:2008ue
contains logarithmic terms:
$\displaystyle{\mbox{Re}}F\sim\frac{19}{32}\log^{2}{\frac{M_{\eta_{b}}^{2}}{M_{J/\psi}^{2}}}+...$
$\displaystyle{\mbox{Im}}F\sim\pi\frac{19}{16}\log{\frac{M_{\eta_{b}}^{2}}{M_{J/\psi}^{2}}}+...$
(21)
In the LC approach used in our calculation, all double logarithms cancel as
the final partciles are colourless objects Lepage:1980fj . Moreover, there are
only two reasons why a general QCD amplitude may contain large logarithms:
renormalization and collinear divergences Lepage:1980fj ; Smilga:1978bq .
Clearly, the imaginary part of $F$ is not renormalized at one loop, hence the
large logarithm in eq. (21) must be due to a collinear divergence. However, it
is known that collinear divergences can be factored out, and do not have an
imaginary part Smilga:1978bq . In light of these arguments, the result
obtained in Gong:2008ue looks strange.
The authors of Gong:2008ue believe that there is no need for renormalization
in their calculation of the radiative corrections, since the counterterms are
proportional to the leading order contribution, which vanishes at the leading
order in both $\alpha_{s}$ and $v_{c}$. We do not think that this statement is
correct, since the expansion is done in operators which are not
multiplicatively renormalizable. Therefore, the ultraviolet divergences may
arise at the leading order in $v_{c}$ due to the $v_{c}$-suppressed operators.
This effect violates NRQCD velocity scaling rules, and is discussed in detail
in Braguta:2008tg ; Braguta:2006wr .
Yet another estimate for the same branching ratio was obtained in
Santorelli:2007xg , where the final-state interaction effects due to a
different decay mechanism were taken into account, yielding
$\displaystyle Br(\eta_{b}\to J/\psi J/\psi)=(0.5\times 10^{-8}-1.2\times
10^{-5}).$ (22)
In conclusion, we have calculated the branching fractions of the decays
$\eta_{b}\to J/\psi J/\psi,J/\psi\psi^{\prime},\psi^{\prime}\psi^{\prime}$ in
the framework of the light cone formalism. The uncertainties of our
calculation have also been assessed. Our results, presented in eqs. (18), are
more than an order of magnitude larger than those obtained within NRQCD.
###### Acknowledgements.
The authors thank A.K. Likhoded and A.V. Luchinsky for useful discussion. This
work was partially supported by Russian Foundation of Basic Research under
grant 07-02-00417.
## References
* (1) B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 101, 071801 (2008) [Erratum-ibid. 102, 029901 (2009)] [arXiv:0807.1086 [hep-ex]].
* (2) E. Braaten, S. Fleming and A. K. Leibovich, Phys. Rev. D 63, 094006 (2001) [arXiv:hep-ph/0008091].
* (3) V. G. Kartvelishvili and A. K. Likhoded, Yad. Fiz. 40, 1273 (1984).
* (4) G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51, 1125 (1995) [Erratum-ibid. D 55, 5853 (1997)] [arXiv:hep-ph/9407339].
* (5) Y. Jia, Phys. Rev. D 78, 054003 (2008) [arXiv:hep-ph/0611130].
* (6) B. Gong, Y. Jia and J. X. Wang, Phys. Lett. B 670, 350 (2009) [arXiv:0808.1034 [hep-ph]].
* (7) P. Santorelli, Phys. Rev. D 77, 074012 (2008) [arXiv:hep-ph/0703232].
* (8) E. Braaten and J. Lee, Phys. Rev. D 67, 054007 (2003) [arXiv:hep-ph/0211085];
* (9) K. Y. Liu, Z. G. He and K. T. Chao, Phys. Lett. B 557, 45 (2003) [arXiv:hep-ph/0211181];
* (10) K. Y. Liu, Z. G. He and K. T. Chao, Phys. Rev. D 77, 014002 (2008) [arXiv:hep-ph/0408141].
* (11) Y. J. Zhang, Y. j. Gao and K. T. Chao, Phys. Rev. Lett. 96, 092001 (2006) [arXiv:hep-ph/0506076].
* (12) B. Gong and J. X. Wang, Phys. Rev. D 77, 054028 (2008) [arXiv:0712.4220 [hep-ph]].
* (13) Y. J. Zhang, Y. Q. Ma and K. T. Chao, Phys. Rev. D 78, 054006 (2008) [arXiv:0802.3655 [hep-ph]].
* (14) A. E. Bondar and V. L. Chernyak, Phys. Lett. B 612, 215 (2005) [arXiv:hep-ph/0412335].
* (15) V. V. Braguta, A. K. Likhoded and A. V. Luchinsky, Phys. Rev. D 72, 074019 (2005) [arXiv:hep-ph/0507275].
* (16) A. V. Berezhnoy, arXiv:hep-ph/0703143.
* (17) D. Ebert, R. N. Faustov, V. O. Galkin and A. P. Martynenko, arXiv:0803.2124 [hep-ph].
* (18) Z. G. He, Y. Fan and K. T. Chao, Phys. Rev. D 75, 074011 (2007) [arXiv:hep-ph/0702239].
* (19) G. T. Bodwin, J. Lee and C. Yu, Phys. Rev. D 77, 094018 (2008) [arXiv:0710.0995 [hep-ph]].
* (20) V. V. Braguta, arXiv:0811.2640 [hep-ph].
* (21) V. L. Chernyak and A. R. Zhitnitsky, Phys. Rept. 112, 173 (1984).
* (22) V. V. Braguta, A. K. Likhoded and A. V. Luchinsky, arXiv:0902.0459 [hep-ph].
* (23) P. Ball, V. M. Braun, Y. Koike and K. Tanaka, Nucl. Phys. B 529, 323 (1998) [arXiv:hep-ph/9802299].
* (24) V. V. Braguta, A. K. Likhoded and A. V. Luchinsky, Phys. Lett. B 646, 80 (2007) [arXiv:hep-ph/0611021].
* (25) V. V. Braguta, Phys. Rev. D 75, 094016 (2007) [arXiv:hep-ph/0701234].
* (26) V. V. Braguta, Phys. Rev. D 77, 034026 (2008) [arXiv:0709.3885[hep-ph]].
* (27) V. V. Braguta, A. K. Likhoded and A. V. Luchinsky, arXiv:0810.3607 [hep-ph].
* (28) V. V. Braguta, Phys. Rev. D 78, 054025 (2008) [arXiv:0712.1475 [hep-ph]].
* (29) V. G. Kartvelishvili and A. K. Likhoded, Sov. J. Nucl. Phys. 42 (1985) 823.
* (30) G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
* (31) A. V. Smilga and M. I. Vysotsky, Nucl. Phys. B 150, 173 (1979).
|
arxiv-papers
| 2009-07-16T08:06:36 |
2024-09-04T02:49:03.995997
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.V. Braguta, V.G. Kartvelishvili",
"submitter": "V Braguta",
"url": "https://arxiv.org/abs/0907.2772"
}
|
0907.2805
|
# FitSuite a general program for simultaneous fitting (and simulation) of
experimental data.
Szilárd Sajti [email protected] KFKI Research Institute for Particle and
Nuclear Physics, P.O. Box 49, H-1525 Budapest, Hungary László Deák KFKI
Research Institute for Particle and Nuclear Physics, P.O. Box 49, H-1525
Budapest, Hungary László Bottyán KFKI Research Institute for Particle and
Nuclear Physics, P.O. Box 49, H-1525 Budapest, Hungary
###### Abstract
In order to get accurate information about complex systems depending on a lot
of parameters, frequently different experimental methods and/or different
experimental conditions are used. The evaluation of these data sets is quite
often a problem. The correct approach is the simultaneous fitting, which is
rarely used, because only a very few programs are using it and even those
cover usually a narrow field of physics. FitSuite was written to tackle this
problem, by providing a general and extendable environment for simultaneous
fitting and simulation. Currently it is used for Mössbauer spectroscopy,
grazing incidence neutron and (non)resonant X-ray reflectometry, but in
principle other experimental methods can also be added.
data analysis; simultaneous fit; (X-ray; neutron; reflectometry; Mössbauer
spectroscopy)
###### pacs:
82.80.Ej, 83.83.Hf, 83.85.Ns
## I Introduction
Nowadays scientists examine more and more complex systems, which depend on a
lot of parameters. To get a correct, accurate and detailed picture about the
processes and phenomena in these systems we need more and more data. These can
be obtained by measurements performed on the same sample with different
experimental methods, which may be sensitive for different parameters, and/or
with the same method performed using slightly different experimental
conditions, such as temperature, pressure, magnetic field, etc. Such data
often depend partly on the same set of sample and experimental parameters,
therefore a simultaneous evaluation of all the data is prerequisite. However,
data evaluation programs are dominantly organized around a single experimental
method and a single theoretical approach used for simulation of this problem,
therefore a simultaneous access to the data for a common fitting algorithm is
not typical. Lacking suitable programs for simultaneous data evaluation,
experimentalist determine some of the parameters from one kind of measurement,
assume them error free and keep them constant when evaluating other
experiments, which is obviously an incorrect approach. Besides, for different
problems different programs are used, which makes it very difficult to tune
the parameters of such problems and their errors and correlations to each
other and to extend or modify the ‘codes’ used to simulate the results of
different experimental methods, which in science is a frequently arising
problem.
There are some programs which are able to perform simultaneous fitting, but
usually they are restricted to a ‘narrow’ field of physics, as e.g. IFEFFIT
IFEFFIT to X(-ray)A(bsorption)F(ine)S(tructure), RefFIT RefFIT used to
analyze optical spectra of solids, SimulReflec SIMULREFLEC for neutron and
X-ray reflectometry. They were written having in mind the specific problems,
and the requirements demanded by them. Even if they are written in a way, that
makes possible further extensions, generalizations; they are not apt to
include a problem from a quite different field, without writting essentially a
new program. Doing this we have to spend a lot of time with miscellaneous
problems having nothing to do with physics, in order to have just a feasible
user interface.
Over the past years, Hartmut Spiering has developed the general and versatile
data fitting environment EFFI (Environment For FItting) Spiering00 , which
aimed to solve these problems, and which has been very efficiently applied for
the evaluation of many sets of ‘conventional’ transmission and ‘synchrotron’
Mössbauer spectra the latter including grazing-incidence, i.e., synchrotron
Mössbauer reflectometry (SMR) NDL00 measurements, both time-differential and
time-integral. The main and yet essential disadvantage of this program was the
fact that its user interface was written using development tools available in
the early eighties, and therefore, in spite of its scientific merits, its
user-friendliness was considerably limited. Another problem is the lack of
documentation, which hinders effectively its development furher beyond a
certain degree.
Therefore it was targeted to write a new thoroughly documented program, called
FitSuite, with a graphical user interface, written in C++, retaining all the
good ideas, principles available in EFFI, but rethought in order to
generalize, extend them where it is possible.
In this article we are presenting the test version of FitSuite, freely
downloadable from the home page of the program EffiWebPage with the only
provision of properly acknowledging its usage in upcoming publications.
Presently, it is capable of simultaneously fitting several data sets of the
following kinds of experiments:
* •
Conventional Mössbauer absorption and emission spectroscopy
* •
X-ray reflectometry
* •
Nuclear resonant forward scattering of synchrotron radiation: time
differential mode
* •
Nuclear resonant forward scattering of synchrotron radiation: stroboscopic
mode
* •
Synchrotron Mössbauer reflectometry: time integral, time differential and
stroboscopic modes
* •
Specular polarized neutron reflectometry
* •
Off-specular polarized neutron reflectometry.
The addition of new kinds of experiments is possible.
## II Considerations, goals
Writting FitSuite we had several considerations, which the program should
satisfy. These will be summarized shortly in this section.
The program should provide a general abstract interface for simultaneous
fitting and/or simulation of different experimental methods, in order to be
able to add new type of problems with minimal effort, without changing the
program itself. There should be an interface for the rare users, who want to
add a new type of experimental method, giving just the functions, subroutines
needed for simulations, and some description of the parameters and the
concepts used in the modelled system (e.g.: sample, detector, source, layer).
As the addition of new methods should be possible without recompiling the
whole program, modularity is needed. For the goals of the program the object
oriented language C++ seemed to be the most appropriate. But as, there are a
lot of codes available in Fortran, and sorrily there are people, who do not
like to learn new program languages, it was an additional requirement to be
possible to write the functions (subroutines) not only in and/or C(++), but
also in Fortran.
There is another type of user (most of them), who just want to use the ‘codes’
provided by others in order to evaluate their experimental results or simulate
their problems. They need another interface, in order to be able to use the
program easily, with minimal effort. The interface should be a graphical user
interface (GUI), but the program ‘core’ should be separated from the GUI. This
is needed for several reasons, which are connected with further possible plans
about the extension of the features of the current program. Sometimes a
console interface can be more useful, than a GUI. If the user would like to
run the program on a cluster or a grid, there is only one GUI needed. The
change of the GUI will be easier, if it is separated from the core.
There was another requirement to use only packages, which make possible to
compile the program for different platforms (primarily Linux and Windows)
without much pain. Therefore we chose the Qt package from Trolltech for the
GUI development.
In the following we will try to summarize, what are the requirements to
describe an experimental method and its subject in an abstract way. This may
seem to be quite easy, as we usually are not aware the concepts we use without
hesitation and much thinking, describing or calculating problems related to a
physical system.
We will use a few concepts used in C++ and every object oriented language.
These will be concerned very slightly, therefore we hope that it will not
cause problems, even if the reader is not acquainted with them. If there is a
need of better understanding, we recommend any book related to these languages
(or just a fast search on the internet), and skip the parts in parenthesis
boldly.
## III Basic concepts
First we just sketch the main concepts used by FitSuite and their relations to
each other and we sunk into the details only thereafter. In FitSuite we have
always a simultaneous fit project (represented by the class
CLSimultanFitProject) which is consisted of fitting problems which the user
would like to fit simultaneously (represented by classes CLGenFitProblem and
CLFitProblem). A fitting problem is consisted of the experimental data
(represented by class CLExperimentalData) and of the computer model of the
experiment (represented by class CLModel). In the following we will see in
details, what a model is, what is it consisted of. We will get into the
details only to such a depth, which may be useful for a user, who would like
to add new problem types to the program.
### III.1 Model and its parts
A model of an experiment contains the ‘sketch’ of the experimental setup and
the system under study, as a physical system and the algorithms with which the
experiment can be simulated, its results can be calculated. Before this text
would start to get too complex and not too understandable, let see an ordinary
example by which we can explain what a model is in FitSuite more smoothly. Let
assume that someone has a model describing an experiment (or rather models of
experiments, if we want simultaneous fits) with a body, and try to answer the
questions: How should this model look like and what concepts it needs? First
we try to forget the experimental setup just concentrate on the subject of the
experiment, i.e. on the body. Clearly we cannot do this perfectly, as the
model of the body will depend very strongly on the experimental circumstances.
E.g. in a very simple throwing experiment in rare atmosphere the body can be
conceived just as a particle having mass. But we need a more detailed model in
dense atmosphere and to complicate it further we can allow the body to change
its shape. In these cases we have to know more about the structure, the
building blocks of the body which can influence its drag coefficient (air
resistance) and the parameters with which these structural elements can be
characterized. In FitSuite these parameters are called properties (represented
by class CLProperty) and the structural elements are called physical objects
or physical notions (represented by the class CLPhysObjNot the name is created
by putting together object and notion). The name physical notion is used
because in some cases the noun object is not appropriate (E.g. stating about
an object that it is consisted of a specific type of a material, it is
convenient to describe the material type with the same CLPhysObjNot class as
the physical objects, in spite of the fact that it cannot be called an object
and may not be a property as it may contain physical objects, as
characteristic atoms, molecule groups, in Mössbauer spectroscopy sites.
Naturally, we could define another class for notions, or properties which may
contain physical objects, but these ways would lead to a more complicated
program structure.) and I did not wanted to use the word concept. In the
following, in order to be short, we will write about physical object even if
it is a notion.
Thus we have now the subject of the experiment as a physical object, which is
built up from other physical objects and which are characterized by
properties. We can fix without making constraints on the generality, that the
hierarchy of the physical objects should have a simple tree structure (there
is one root object containing everything in the hierarchy, and there are no
loops in the corresponding graph). We want to describe not only the subject of
the experiment but the whole experimental setup, so we can have a similar
description of the experimental apparatus and the environment in which the
experiment is performed, but everything detailed only to an extent ensuring
that the problem can be simulated without having too much unnecessary
parameters. (It is not hard to see, that the requirement of having only the
necessary parameters would be too strict.) So we can fix that the main (or
root) object of the extended tree of physical objects should be always the
experimental scheme which contains the parts of the experimental setup and of
the system under study, as it can be seen in example shown in Fig. 1. (I am
not sure that this is the best word for this concept: experimental-setup,
-world, -universe, -system were also among candidates, but each may be mixed
with something else). This main object (CLPhysObjNot::MainObj) is contained by
the model (CLModel) and all the other physical objects are ‘children’
(CLPhysObjNot::ChildrenList) or ‘descendants’ (CLPhysObjNot::DescendantsList)
of this object.
Figure 1: An example of the tree structure belonging to an isotopic multilayer
system used to examine the self-diffusion of iron atoms in FePd alloys
DaniCikk .
Some of the physical objects may have their own models, which makes possible
to perform some simulations of the physical subsystem described by these
objects and the results of these simulations may be used by a model belonging
to an object containing these objects as children or descendants.
### III.2 Prototypes
If we want to give the possibility to the user to choose the subject (e.g. not
only two-winged but three-winged bodies also) of his or her experiment (and
the experimental setup also) flexible, but of course only within certain
limits, we have to (at least we went into this way to tackle this problem)
define prototypes of models, physical objects and properties (represented by
classes CLProtoModel, CLProtoPhysObjNot and CLProtoProperty respectively). The
relation of a model prototype object to the corresponding model objects (here
we use the word ‘objects’ in programming technique meaning) is similar to the
relation between the set of building block types plus the knowledge of the
connection possibilities and the ‘maquettes’ built up using these blocks and
rules (and not what is implied by the word prototype).
Now let see what informations these prototypes should contain, how they should
look like. First of all, we need something to identify, differentiate them. In
FitSuite we use for this purpose names and integer numbers. The first is for
humans, the second is generated internally and is used by the simulation
functions, in which the programmer can refer to them by variable names
generated from the above mentioned names (as the programmer is human).
Here arises the question, that when should we regard two prototypes different.
At the level of model prototypes there is no problem, they all have to have
different names at least the ones which can have a role in the same
simultaneous fit project. Therefore (but not only because of this) the model
prototypes are stored in repositories (represented by CLProtoModelRepository).
In a simultaneous fit project we can use only the model types of one
repository (at least in current version of FitSuite).
At the level of physical object prototypes we require uniqueness only within
the model prototype to which they belong. At the level of property prototypes
we require uniqueness only within the prototype of physical object to which it
belongs (e.g. the sample and the domains, or other miniature structures on it
also may have diameter, although their have different value and order of
magnitude). We have to know that to which object type a certain property
belongs, that is why we have the whole hierarchy of physical objects. In the
same way we have to know the parents (grand-…-grand-parents) of an object
(e.g. the body may have screws on its wings and fuselage as well). We have to
know that an object of a specific type which type of objects may and how many
may (or have to) contain or is there a limit at all. With these pieces of
information we can help the user when she builds up the model of her
experimental scheme. We may allow only the appropriate combination of building
blocks (or we can warn the user). Furthermore we may be able to select the
parameters, properties, objects, which we are interested in according to
complex type criteria (CLTypeSelectionCriterion, CLSelection) given by us.
The tree of the physical objects is an ordered structure by the parent-child
relations ‘vertically’ and is also ordered ‘horizontally’, as the list classes
(from standard C++ library), which are used for the storage of children of
physical objects can be conceived as an array with beginning and end into
(from) which we can insert (remove) elements at (from) arbitrary position.
Sometimes, this order is a requirement (e.g. thin layer systems) sometimes is
not, but even in the second case it is convenient. In the prototypes of
physical objects is also an order, the possible parent types and child types
are also ordered. therefore in the children list of a physical object the
sequences of different type of objects also have strict orders. (E.g. it is
not a requirement, but it is logical to have the order: source(s), sample(s),
detector(s) in a scattering, absorption, etc. experiment.) The properties also
have a strict order within a physical object, which is determined by the order
of their types in the corresponding physical object type. Furthermore, because
of the availability of these orders, we may have the physical objects of the
same type of a model in an ordered list
(CLProtoPhysObjNot::RepresentativesList). This may be convenient when we want
to perform some operation on the same type of objects and their properties
without going through the tree hierarchy.
### III.3 Group
In an ordered system as a layer structure, we may have periodic sequences. For
description of such sequences we use (as for thin layer structures is usual)
groups. In FitSuite a group is a physical object, which contains physical
objects of the same type that it belongs to, but it has no properties and its
repetition number (CLPhysObjNot::Nrep) is greater than 0. In the physical
object type we can specify whether that type may have group or not and how
deep these groups may be embedded into each other
(CLProtoPhysObjNot::GroupDepth).
### III.4 Property
Above, we just mentioned the concept of property, but did not examined it in
details or the requirements arising with it. The properties are (C++) objects
representing physical quantities and other numbers, which are needed in order
to simulate the problem properly, mainly arising because the calculations are
performed by a computer and because the experimental results are always
discrete data sets.
First let see what is needed in order to represent physical quantities. A
physical quantity has an algebraic structure. Even though all components of a
nonscalar physical quantity could be represented by independent scalars,
sometimes it may be convenient to know that these components belong together.
E.g. when the user lists out all the components of such a quantity it is good
to give only the name of the quantity and not all of the components. The
algebraic structure in computer representation should not be always identical
with the mathematical structure used in science. E.g. the components of a
symmetric tensor can be represented by a vector (it would be more appropriate
to call it an array) and not by a matrix. In current FitSuite the
E(lectric)F(ield)G(radient) tensor is represented as a 5 element vector. Three
of them are the Euler angles giving the orientation of the coordinate system
in which the EFG can be given by the remaining two parameters. In this case,
it would be more appropriate to speak about parameters determining the tensor,
than components, but we do not want to introduce a new concept just because of
this.
So the properties have (algebraic) structures, and are built up from scalar
components. Each component may have its unique name (within the property, and
if the user did not name it, the program will generate names using ordinal
numbers), its value, its minimum-, maximum value, order of magnitude. To a
physical quantity naturally belongs some unit. In case of a nonscalar quantity
the different components may be measured in different units
(CLProtoProperty::DefaultUnits). E.g. the magnetic induction vector given in
spherical coordinates has a radial component $B_{r}$ in Tesla (or Gauss) and
the angles $B_{\vartheta}$, $B_{\varphi}$ in degree (or radian). As it was
mentioned above the properties may represent numbers which are not physical
quantities. This means not in all cases that this type of property has no
physical significance at all. E.g. at the moment, symmetries of the sites are
represented by three integer numbers, $C_{nzn}$ is one of them, it determines
whether the axis $z$ is a symmetry axis or not, and if it is how many fold
this rotation symmetry is. Some numbers could be used as switches. E.g.:. But
the properties can also represent numbers which do not have real physical
significance. These typically just give an arbitrary size of an array, which
can say something about the (sampling frequency) resolution with which the
simulation or experiment was performed at most.
To each component belongs an integer number, which we call logical bit
collections (more specifically an enumeration type named EnLogicalCollection)
whose bits contain information specifying further the role of the
corresponding component in the model. E.g. whether they are constant,
independent variables, internal variable (handled internally during simulation
or fit and invisible for the user), free or fix; whether they were changed
since the last iteration step, etc.
To each property type belongs some help contained by a string
(CLProtoProperty::Notes) and an url reference (CLProtoProperty::UrlFragment)
to the place, where a more detailed help may be available.
The models, physical objects and properties have to contain some information
according to which their prototype can be determined. (This is solved in all
cases with the help of a pointer named Proto namely CLsubProperty::Proto,
CLPhysObjNot::Proto, CLModel::Proto pointing to the proper prototype namely an
object of class CLProtoProperty, CLProtoPhysObjNot and CLProtoModel,
respectively).
### III.5 Parameter name convention
The property uses always the name of its prototype (CLProtoProperty::Name), as
it is unique in the object (type), which is characterized by it. Therefore
with the object name and the property type name we can find it always. (E.g.
thickness is always thickness we just say that it is the thickness of the
body’s first left or back right wing, or its n-th screw on its right tail wing
upper part.) In case of physical objects, models the prototype name is not
enough, they have to have their own names. But of course we can use the
prototype names even in this case to generate an automatic name. Only those
physical object names are allowed, which are unique within a main model (i.e.
not a submodel). In a simultaneous fit project a property may be identified
unequivocally by the model name (main model and no submodel), the physical
object name and the property prototype name. As we will see later, from this
object tree structure we will generate a parameter list, used during the
(simultaneous) fit or simulation. Because of having these ‘constraints’ on the
names, each parameter belonging to a model can be and is identified using the
name convention: ModelName`=>`ObjectName`:>`PropertyName::ComponentName or in
case of scalars just ModelName`=>`ObjectName`:>`PropertyName. In case of
complex scalars we have automatic component names .re and .im. For complex
vectors (and other nonscalars), the component names get .re and .im as an
additional suffix. If we had no unique physical object name in a model, but
only its parent, then we should give the whole hierarchy of object names
(e.g.:
ModelName`=>`RootObjectName`->`Grand…GrandParentName`->`…`->`ParentName`-``>`ObjectName`:>`PropertyName::ComponentName),
which could be very long and quite impractical. Because of this parameter name
convention and some others coming later on, the names should not contain the
character sequences used as separators and suffixes: ‘`=>`’, ‘`:>`’, ‘::’,
‘.’, ‘,’, ‘`*>`’, ‘`>>`’, ‘.re’, ‘.im’. Use of whitespaces should be avoided
also, because it can cause bugs reading the simultaneous fit projects from
files.
### III.6 Beyond the tree structure
Sometimes we do not have ‘well defined’ physical objects, but rather a
statistical ensemble of them. In these cases we may need distributions and/or
correlation functions. In FitSuite presently, we have only correlation
functions of 2-order, and even those only for a very specific case. Later on
this should be rewritten if there is a requirement for it.
In order to be more understandable, let see the above mentioned problem. There
is a magnetic multilayer system. Some of the layers are consisted of magnetic
domains of $n$ type. The domains of different layers are antiferromagnetically
coupled to each other. For description of off-specular resonant X-ray
(Mössbauer) reflection DeakOffsp on such samples we have to know that: which
layer, what type of domains is consisted of; what is the fraction of the
$i$-th type of domain in the $m$-th layer. Besides this there are some
correlation functions between the domains in different layers, e.g.
$c_{ik,jl}(\dots)$ between the $i$-th layer‘s $k$-th type of domain and the
$j$-th layer‘s $l$-th type of domain. Each such correlation function has its
own parameters, which we should be able to fit.
It is obvious, that such a problem cannot be handled with the tree structure
shown before, as the correlation functions belong to two objects and it would
be a waste to add to each layer the same domain types. Therefore we created
classes to have symbolic objects also. This is also too specific currently and
perhaps unsatisfactorily tested. A symbolic physical object (CLSymbPhysObjNot)
is similar to the symbolic links in the Unix file systems, or the application
links in another well known operating system family, but there are a lot of
differences. The symbolic objects also have prototypes. (represented by the
class CLProtoSymbPhysObjNot. A CLProtoSymbPhysObjNot object has just a pointer
CLProtoSymbPhysObjNot::ObjectType to the physical object prototype
CLProtoPhysObjNot whose representatives may be symbolically linked as a child
to objects, whose type is restricted by a list
CLProtoSymbPhysObjNot::ParentTypes.) This way we can hinder the user creating
symbolic links which would be meaningless, or could result program faults,
i.e. this is a requirement of a ‘userproof’ program. Besides this we may have
additional constraints:
* •
Sometimes it may be useful to forbid to have some type of ‘brothers’
(CLProtoSymbPhysObjNot::ExcludedBrothers) and properties in the parent
(CLProtoSymbPhysObjNot::ExcludedProperties). In the first case we may not add
this type of symbolic object to an object containing already such a child
(listed among ExcludedBrothers), or if it contains already such a symbolic
child, we cannot add any child, whose type is listed among ExcludedBrothers.
In case of excluded properties, the properties are there, but we do not use
them (it is planned to hide them from the user in the future versions), for
this reason a logical bit (lcExcluded) are set to true for each component of
these properties, when we add such a symbolic child.
* •
We may specify some properties also, which may be different, for each symbolic
physical object, even if they are ‘links’ to the same object. These properties
we call overloaded properties, and they are given by a list
(CLProtoSymbPhysObjNot::NamesOfOverloadedProperties) containing their names.
In case of the off-specular example the fraction of domains in a layer,
changes from layer to layer.
The correlation function (CLCorrelationFunction) is a bit similar to a
physical object, it has properties and protototype
(CLProtoCorrelationFunction), which contains a reference (function pointer) to
the algorithm used for calculation of this function. (We have two special
cases, the fully correlated and the totally uncorrelated case, when the value
of the correlation function is identically 1 and 0, respectively. Therefore we
have an enumeration type (CLCorrelationFunction::Correlated), according to
which we can decide, that we have these two extreme cases, or we use a real
function belonging to the corresponding prototype.) Its name is the identical
with the name of its prototype (as e.g.: a Gauss or a Lorentz-function is
always the same, just its parameter values may be different). The parameters
belonging to a symbolic physical object and for a correlation function,
obviously should be different from what we have shown earlier. For the first
one we have
ModelName`=>`ObjectName`*>`SymbolicChildObjectName`:>`PropertyName::ComponentName
(e.g. ModelX`=>`nthLayer`*>`domainUp`:>`size), and for the second one
ModelName`=>`ObjectName_1`*>`SymbolicChildObjectName_1`,`ObjectName_2`*>`SymbolicChildObjectName_2`>>`FunctionName`:>`PropertyName::ComponentName
(e.g.
ModelX`=>`nthLayer`*>`domainUp,mthLayer`*>`domainDown`>>`Lorentz`:>`halfWidth).
### III.7 Plotting, independent variables for simulation
To have the results of a simulation or fit in an appropriate way we have to
plot the results, therefore we have to give some information for the computer,
what sort of plot we need: as the scaling (e.g. logarithmic or linear), the
labels of the axes, which arrays contain the results, which properties
determine the array sizes, etc. For this aim we use also a class (CLPlotType).
Each model prototype contains a list (CLProtoModel::PlotTypes) of them, from
which the user can choose, if (s)he would like to.
If we do not have data, but we would like to simulate, we have to tell to the
computer, for which independent variable values should be the simulation
performed. There may be several types of these also, depending on what is the
independent variable (e.g. in case of neutron reflectometry, the wavelength,
the scattering wavevector, the angle of incidence, etc.), which may be a
scalar or a vector independent variable. Besides this, even if we have data,
we should know what is there the independent variable. For this we have also a
class (CLSimulationPointsGenerator). This contains:
* •
the names of the properties and their components, which determine the range
and the distribution of the independent variables, i.e. their values;
* •
the names of related properties, which are used only for the generation of the
independent variables, and which should be hidden from the user, when (s)he
uses another type of simulation point generator, which does not depend on
them;
* •
the name of the property, in which we store the type identification number
belonging to the simulation point generator. This is needed, as in the
simulation functions (subroutines) we have to know, what should be calculated.
(Sometimes just the conversion of the independent variables could be enough,
but not always, this class is for that cases. The independent variable
conversion could be an additional step.)
### III.8 Transformation matrix technique, parameter list generation
The ‘optimization’ methods used for fitting require a parameter vector and not
an object tree structure with properties. Furthermore in case of simultaneous
fitting we usually have the results of experiments performed in a bit
different environment (external field, temperature, etc.) and/or different
type of experiments using the same ‘sample’. Therefore there is a lot of
common parameters. To eliminate this type of redundancy and as it is also
convenient for the user to use as few parameters as possible (as it is more
transparent for human and easier to fit in a parameter space with lower
dimension at least if we want to get correct results) transformation matrix
technique is used Kulcsar71 . For this we need also parameter vector (array).
Because of these considerations we have to generate the parameter vector and
the initial transformation matrix from the object tree structure. The model
parameters which still contain all the redundancy can be collected in an array
$\mathbf{p}=\left(\mathbf{p_{1}},\mathbf{p_{2}},\dots,\mathbf{p_{n}}\right),$
where $\mathbf{p_{i}}$ is the array containing all the parameters belonging to
the $i$-th model in the current simultaneous fit project. Let denote the array
of the fitting (or if you like simulation) parameters with $\mathbf{P}$ and
the transformation matrix with $\underline{\underline{\mathrm{T}}}.$ The
transformation matrix technique uses the expression
$\mathbf{p}=\underline{\underline{\mathrm{T}}}\mathbf{P},$ where
$\dim\mathbf{P}\leqslant\dim\mathbf{p}$. Above was mentioned that this
technique is used in order to eliminate the redundancy arising because of the
common parameters, but this is not the unique reason. We can take into account
some possible linear relations between the parameters also, which also is a
redundancy of course. Furthermore we could generalize this technique using
some additional nonlinear transformation operator. In that case we would have
$\mathbf{p}=\underline{\underline{\mathrm{T}}}\mathbf{P}+\mathbf{\mathrm{NL}}(\mathbf{P})$.
(The components of the nonlinear operator could be function pointers, given by
the user as an assembly like code, or using some mathematical parser package
as muParser and MTParser. The inhomogeneous transformation can be useful
sometimes also. Sorrily this is not available on the level of GUI either in
present program.)
Now arises the question, how to generate the initial
$\underline{\underline{\mathrm{T}}}$ matrix and the arrays $\mathbf{P}$ and
$\mathbf{p}$, which the user can change on the GUI according his ideas. It is
advisable to take into account that there are parameters which according to
expectations will not have interdependencies and therefore the
$\underline{\underline{\mathrm{T}}}$ matrix can be ‘block diagonalized’. It is
more transparent to handle submatrices with lower dimensions, than one
extended sparse matrix. Therefore we have to categorize the parameters
according to our expectation whether the subspace stretched by a subset of
them may have interdependencies or this is very unlikely. (If the user finds a
case, where our expectations are not met, (s)he is able to unite or split the
submatrices, thus our choice here is not a constraint.) The initial
submatrices generally are identity matrices, but not always. E.g. the
thickness of a multilayer sample will be the sum of the layer thicknesses; in
Mössbauer spectroscopy in a doublet site, the line positions and the measure
of the splitting and the isomer shift will not be independent, etc.
A model parameter type (e.g. layer thickness, magnitude of the external
magnetic field on the sample, etc.) can be specified by the physical object
types, by the property type and the property component. In a general case we
are able to differentiate not only by the object type, but by the branch of
objects starting from the root object in the given model type, we can take
into account this way the parents, grand…grandparents, the ‘pedigree’ of the
object. (E.g. in the throwing example, the length (material) of the screws on
the wings and on the fuselage can be quite different, and can be regarded
independent from each other.) They may belong to quite different subspace,
category. Each model has a partition (CLPartition ) class which contains a
category (CLCategory) list. Each parameter type belongs to only one category.
To each category belongs an initial transformation matrix, which is used
during the generation of the initial transformation submatrix belonging to the
category. The real and integer parameters are handled separately as we do not
want to guard the parameters against conversion (rounding) errors, and the
integer parameters are never fitted. Therefore the real and integer parameters
have separate partitions (CLProtoModel::Partition and
CLProtoModel::intPartition, respectively) and of course separate arrays and
transformation matrices.
### III.9 Arrays and algorithms
In order to simulate we have to provide the algorithms for the model also. To
each model belongs three function (in Fortran subroutine). One is used for the
simulation. In the simulation we use arrays, containing the spectra,
intermediate results, auxiliary arrays. These arrays, at least some of them
should be initialized with values different from 0, before the first
simulation of the fit iteration. For this we have another function. It is
clear, that we have to give the size of these arrays also. Therefore the array
initialization should be preceded by the array size initialization. The third
function is used for this. Later we will see how this functions should look
like, how we can write such one. We classify the arrays according to their
roles into five main groups:
* •
The input arrays are initialized before the first iteration step.
* •
The output arrays are (usually) set to 0 initially.
* •
The variable auxiliary arrays are used internally, usually are set to 0
initially.
* •
The constant auxiliary arrays are set only before the first iteration step,
and not changed thereafter.
* •
The constant integer auxiliary arrays are set also only before the first
iteration step, and not changed thereafter.
Thus we have parameter arrays and transformation matrices, and the simulation
and the two initialization functions, but we are still not done, as during the
simulation we may need the structure also. We have to provide this information
for the simulation functions someway. If we would use only C(++) language we
could use the CLPhysObjNot objects or something similar, e.g. generated
structures embedded into each other using (void* or just) pointers. But
sorrily we use also Fortran, where we cannot embed structures into each other.
(Even Fortran versions later than 77 - sorry Fortran believers - are childish,
a joke in this regard.) Therefore we use the ‘information array’ pinf
(CLModel::pinf) generated by the program. In the following we will not go into
the structure of this array, as it is quite complex, it can be found in the
program documentation, and to write the three type of function needed for
simulation and initialization it is enough to know the auxiliary functions,
some of which are shown in the appendix A.
### III.10 Following changes
During an iteration (fit) it is useful to calculate only if necessary. If an
auxiliary array was calculated in the former iteration step, and the
parameters and arrays on which it depends were not changed, there is no reason
to calculate it again, especially if it takes a lot of computation time. For
this reason we have functions to follow, the changes of the component of the
properties and arrays. The changes have three sources:
* -
change of the parameters, input arrays by the user, this happens always before
starting an iteration or simulation;
* -
change by the fitting method, this happens between the simulations, only the
free parameters and the arrays may be changed this way;
* -
change of internal variables, this is done by the simulation and
initialization functions, thus this is essentially the problem of the code
writer.
(Structural changes, as removing or adding a layer, would be another class of
changes, but that would lead us too far away, and the handling of this problem
is out of our plans in near future.) Therefore we have to know, whether was a
simulation (or iteration) run before the current calculation, had been there
an user interaction since then, are there free parameters, is the current
function call the first one during the fit. When the user changes a parameter
in the user interface, or changes an input array, the proper bit (lcChanged)
of the corresponding logical bit collection is set to true. Similarly in case
of free parameters another bit (lcFree) is set to true. The initialization and
simulation functions have logical arguments, determining whether the function
was already called, or is the first call during an iteration. Using these and
some auxiliary functions (not shown in this article), and proper coding, we
can decide when we can jump same code parts during simulation or fit, as there
was no change.
Besides the auxiliary functions needed to write simulation and initialization
functions, we also have to create the model types for the program. In future
for this task we will use another program with graphical user interface, in
order to decrease the number of possible errors, in this process. But now we
have to write a C++ program, as it can be seen in the appendix B this can be
done quite mechanically.
## IV The user interface of FitSuite
If you know already the principles used in FitSuite, which we have shown in
the former section, there is not much to know about the user interface.
Therefore we just skim over it shortly. Starting the program, the user can
start a new project or load a previously saved one. We can save our project
anytime. For building up the object tree structure of the models, we use an
interface similar to the treeviews used every day to handle our file system.
The main difference is, that here instead of directories we have physical
objects, and instead of files properties. Furthermore we are constrained by
the rules given by the model type. The data sets, parameters and
transformation matrices, have their own editors, using ‘spreadsheets’. For
plotting the package Qwt is used, gnuplot files are also generated.
For fitting several methods are available. Most of them is a slightly modified
version of the optimization functions available in Numerical Recipes NumRec .
During fitting, we optimize always the $\chi^{2}$ in current version. Later
this may be changed. Confidence limits, covariance matrix of the free fitting
parameters are calculated after fitting was finished.
Further details can be found in the User Manual and in the demos available at
the homepage of the program.
In the following we will show a few examples used for fitting. These are
experiments performed to determine the self-diffusion coefficient of iron
atoms in FePd alloys DaniCikk . For these experiments isotopic Pd(1 nm)
$[^{57}\text{Fe}_{47\%}\text{Pd}_{53\%}$(2 nm)
${}^{\text{nat}}\text{Fe}_{47\%}\text{Pd}_{53\%}$(3 nm)$]_{10}$ Pd(15 nm) Cr(3
nm) MgO(001) multilayers were grown, which thereafter were heated or
irradiated with $\text{He}^{+}$ ion beam. The effect of the latter treatment
can also be modelled as a diffusion of the iron atoms. X-ray reflectograms
(Fig. 2), and nuclear resonant reflection spectra in time integrated mode
(Fig. 3) were measured for samples treated with different ion fluxes. For the
evaluation of these spectra FitSuite was used with succes. For further details
of these experiments and their interpretation see DaniCikk .
Figure 2: X-ray reflectometry spectra of samples irradiated with different
$\text{He}^{+}$ doses and the fitted curves obtained with FitSuite. Figure 3:
Synchrotron Mössbauer reflectometry in time integrated spectra of samples
irradiated with different $\text{He}^{+}$ doses and the fitted curves obtained
with FitSuite.
As it is visible in Figs. 2-3, the theoretical results fit well to the
experimental data.
## V Summary
In this paper we presented a new general extendible program FitSuite for
simultaneous simulation and fitting of experimental data of measurements
performed on complex systems.
## VI Acknowledgement
This work was supported by the European Community under the Specific Targeted
Research Project Contract No. NMP4-CT-2003-001516 (DYNASYNC). FitSuite was
developed in frames of DYNASYNC and it is freely available from EffiWebPage
with the only provision of proper acknowledgement in future publications.
## Appendix A Auxiliary functions needed writting simulation and
initialization functions
Before the list we have to mention some additional facts, which we have to
know in order to use them:
## Appendix B Adding a new model type
In the previous appendix we saw the auxiliary functions needed to write
simulation and initialization functions. In this appendix we will see, how we
can add a new model type to the program. In future for this task we will use
another program with graphical user interface, in order to decrease the number
of possible errors in this process. But now we have to write a C++ program, as
we will see this can be done quite mechanically.
Here we will show only the most important steps, things, tricks, but not all
of them. In the following, short description parts, explanations will precede
the corresponding code fragments.
## References
* (1) H. Spiering, L. Deák, L. Bottyán, Hyperfine Interact. 125, 197, (2000)
* (2) D.L. Nagy, L. Bottyán, L. Deák, E. Szilágyi, H. Spiering, J. Dekoster, G. Langouche, Hyperfine Interact. 126, 353, (2000)
* (3) K. Kulcsár, D.L. Nagy, L. Pócs, in: Proc. Conf. on Mössbauer Spectrometry, Dresden (1971).
* (4) E.W. Müller, MOSFUN, Laboratory report, Anorganische Chemie und Analytische Chemie, Johannes Gutenberg-Universität , Mainz (1982).
* (5) http://www.fs.kfki.hu
* (6) M. Newville, J. Synchrotron Rad. 8, 322-324, (2001).
* (7) RefFIT written by A. Kuzmenko is available at http://optics.unige.ch/alexey/reffit.html
* (8) SimulReflec written by F. Ott and coworkers is available at http://www-llb.cea.fr/prism/programs/simulreflec/simulreflec.html
* (9) L. Deák, L. Bottyán, D.L. Nagy, H. Spiering, Yu.N. Khaidukov, Y. Yoda sent to journal, available at http://aps.arxiv.org/abs/0709.2763
* (10) W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery Numerical Recipes in C (Fortran), Cambridge University Press
* (11) D.G. Merkel, et al. to be published
|
arxiv-papers
| 2009-07-16T10:59:07 |
2024-09-04T02:49:04.002489
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sz. Sajti, L. De\\'ak, L. Botty\\'an",
"submitter": "Szil\\'ard Sajti",
"url": "https://arxiv.org/abs/0907.2805"
}
|
0907.2881
|
[labelstyle=]
# On epimorphisms and monomorphisms of Hopf algebras
Alexandru Chirvăsitu University of California, Berkeley, 970 Evans Hall #3480,
Berkeley, CA, 94720-3840 USA [email protected]
###### Abstract.
We provide examples of non-surjective epimorphisms $H\to K$ in the category of
Hopf algebras over a field, even with the additional requirement that $K$ have
bijective antipode, by showing that the universal map from a Hopf algebra to
its enveloping Hopf algebra with bijective antipode is an epimorphism in ${\rm
HopfAlg}$, although it is known that it need not be surjective. Dual results
are obtained for the problem of whether monomorphisms in the category of Hopf
algebras are necessarily injective. We also notice that these are
automatically examples of non-faithfully flat and respectively non-faithfully
coflat maps of Hopf algebras.
###### Key words and phrases:
Hopf algebra, epimorphism, monomorphism, faithfully flat, first Kaplansky
conjecture
###### 2000 Mathematics Subject Classification:
16W30, 18A20, 18A30, 18A40
## Introduction
In this paper, we are concerned primarily with the problem of whether
epimorphisms in the category ${\rm HopfAlg}$ of Hopf algebras over a field $k$
are surjective, and the dual question of whether monomorphisms are injective.
This makes sense in any concrete category; in [Re], for example, the
corresponding problem (on epimorphisms) is solved for some familiar
categories, such as groups, Lie algebras, $C^{*}$ and von Neumann algebras,
compact groups, locally compact groups, etc. To our knowledge, the problem has
not been treated in the literature in the context of Hopf algebras.
Aside from being interesting and natural in their own right, the two questions
do play a part in certain technical results on Hopf algebras. In [AD], for
example, a paper concerned with exact sequences of Hopf algebras, these
problems arise naturally several times. In the dual pair [AD, Lemmas 1.1.6,
1.1.10] it is shown that certain conditions on a morphism of Hopf algebras are
implied by injectivity, and imply that the morphism in question is a
monomorphism in ${\rm HopfAlg}$ (and similarly for surjectivity). Also, in a
remark after [AD, Prop. 1.2.3], the authors observe that in a diagram of the
form
$\begin{diagram}$
where the rows are what in that paper are called exact sequences of Hopf
algebras ([AD, Prop. 1.2.3]), $\theta$ is both a monomorphism and an
epimorphism of Hopf algebras. The authors then mention as unknown whether in
this case it follows that $\theta$ is an isomorphism, or, in general, whether
epimorphisms (monomorphisms) of Hopf algebras are surjective (injective). In
other words, this is a direct reference to our problem. It is, however, the
only such reference we could find in the literature.
A much more well-documented problem, on the other hand, is the one known as
Kaplansky’s first conjecture. Strictly speaking, the conjecture/problem has
undergone several transformations since its appearance in [Ka]. It initially
asked whether all Hopf algebras are (left and right) free modules over their
Hopf subalgebras. At the time, this was already known to be false: Oberst and
Schneider had constructed a counterexample in [OSch].
There are several positive results on the problem: it holds for instance if
the coradical of the large Hopf algebra is contained in the small one by a
result of Nichols (this also follows from [Ra2, Cor. 2.3]), or if the large
algebra is pointed ([Ra1]), or in the finite dimensional case by the now
famous Nichols-Zoeller theorem ([Mo, Theorem 3.1.5]).
In view of the general negative answer, it makes sense to weaken the
requirements: [Mo, Question 3.5.4] asks whether Hopf algebras are always (left
and right) faithfully flat over their Hopf subalgebras. Again, this holds in
various particular cases (commutative, or cocommutative, or even when the
large algebra has cocommutative coradical; we give some references below, in
Section 2, after Proposition 2.5).
In the commutative case, the problem of faithful flatness arose in the theory
of affine algebraic groups, for which we refer to [DG, Wa]. Indeed, faithful
flatness for commutative Hopf algebras ([Ta3, Th. 3.1]) is crucial in
Takeuchi’s purely algebraic proof in [Ta3] of the one-to-one correspondence
between normal closed subgroup schemes and quotient affine group schemes of an
affine group scheme. See [Ta3, Th. 5.2], and also [Wa, Chapters 13-16] for an
exposition of these results.
Despite all of these positive partial results, in general, Hopf algebras are
not faithfully flat over Hopf subalgebras ([Sc, Remark 2.6, Cor. 2.8]). At the
end of [Sc, $\S$2], Schauenburg asks what we refer to from now on as being the
current version of Kaplansky’s question (or problem):
Are Hopf algebras with bijective antipode (left and right) faithfully flat
over Hopf subalgebras with bijective antipode?
Our interest in the question of faithful (co)flatness for Hopf algebras stems
from the fact that there are strong connections between it and the problem of
whether epimorphisms are surjective. These are understood by first noticing
that epimorphisms of Hopf algebras can already be recognized at the level of
algebras (Proposition 2.4) through an adjunction, and then that a faithfully
flat epimorphism of algebras is an isomorphism (a well-known result, which we
prove however, for the sake of completeness, in Proposition 2.3).
It follows that whenever we have non-surjective epimorphisms, we automatically
have counterexamples to Kaplansky’s question. In particular, our
counterexamples to epi $\Rightarrow$ surjective in Section 2 and Section 3
recover those in [Sc] for Kaplansky’s problem, from this new point of view. On
the other hand, it follows that epimorphisms are surjective when the
conjecture holds (as mentioned above, for commutative or cocommutative, or
pointed Hopf algebras, for instance). In the commutative case, for example,
the fact that epi implies surjecivity can be translated into geometric
language as follows (see [Ta3, Th. 5.2, (i)]; we are using the same notations
as Takeuchi):
A morphism ${\rm Sp}(H)\to{\rm Sp}(K)$ of affine groups is a monomorphism if
and only if the corresponding Hopf algebra map $K\to H$ is surjective.
Indeed, the category of commutative Hopf algebras is the opposite of that of
affine groups, so a monomorphism in the latter is the same as an epimorphism
in the former.
The paper is organized as follows:
In Section 1 we introduce the notations and conventions to be used throughout.
We also very briefly recall two characterizations of monomorphisms of
coalgebras.
Section 2 is devoted to the questions asked above, in precisely that form.
They are quickly settled in the negative by the simple observation that the
antipode of a Hopf algebra $H$, regarded as a Hopf algebra map from $H$ to
$H^{op,cop}$ ($H$ with the opposite multiplication and coopposite
comultiplication) is both a monomorphism and an epimorphism in ${\rm
HopfAlg}$. We also need the facts, known for some time, that there are Hopf
algebras with non-surjective ([Ni]) or non-injective ([Ta2, Sc]) antipode.
In this same section, we highlight the interactions between the Kaplansky
conjecture and the problem of whether epimorphisms in ${\rm HopfAlg}$ (the
category of Hopf algebras) are surjective, as discussed above. We also look
briefly at the dual situation: the problem of whether surjective Hopf algebra
maps are faithfully coflat is linked to that of whether monomorphisms of Hopf
algebras are injective through Proposition 2.5 and Proposition 2.6.
Finally, as an interesting consequence of this discussion, we show in
Proposition 2.7 that the antipode of a Hopf algebra is surjective whenever its
image contains the coradical.
In Section 3 we modify our question by imposing stronger hypotheses (akin to
what is done in [Sc] for the Kaplansky problem): we ask whether an epimorphic
inclusion of Hopf algebras must be surjective if the larger Hopf algebra has
bijective antipode, as well as the dual question. Again, we prove that there
are counterexamples (Corollary 3.4). These are obtained through two
adjunctions between the categories of Hopf algebras and of Hopf algebras with
bijective antipode. One is the adjunction constructed in [Sc], where it is
shown that there is a free Hopf algebra with bijective antipode (denoted here
by $K^{*}(H)$) on every hopf algebra $H$. We prove that the universal map
$H\to K^{*}(H)$ is always an epimorphism of Hopf algebras, thus finding our
counterexamples whenever it is not surjective (and this does occur).
The other adjunction we use is the “dual” of the previous one: we prove that
there is a cofree Hopf algebra $K_{*}(H)$ with bijective antipode on every
Hopf algebra $H$, and that the universal map $K_{*}(H)\to H$ is always a
monomorphism of Hopf algebras. Again, this provides us with counterexamples to
mono $\Rightarrow$ injective whenever such a universal map is not injective.
Because we find the analogy interesting, we carry out a parallel discussion
for two adjunctions between the categories of bialgebras and Hopf algebras:
there exist both a free and a cofree Hopf algebra on a bialgebra $B$ (the
former follows from [Ta1] and is constructed explicitly in [Pa]; the existence
of the latter is proven in [Ag1], and we construct it here). We denote these
by $H^{*}(B)$ and $H_{*}(B)$ respectively. As before, we show that the unit of
the first adjunction provides us with epimorphisms $B\to H^{*}(B)$ of
bialgebras, and the counit of the other adjunction gives us monomorphisms
$H_{*}(B)\to B$ of bialgebras. See Theorem 3.2.
In Section 4 we finish with some problems for the reader.
First, there are the questions parallel to Kaplansky’s conjecture in its
current form and its dual: we would like to know whether epimorphisms
(monomorphisms) of Hopf algebras are surjective (injective) when all Hopf
algebras in question have bijective antipode.
Secondly, we ask for necessary and sufficient conditions on a bialgebra in
order that it be a quotient or a subbialgebra of a Hopf algebra, and also for
necessary and sufficient conditions on a Hopf algebra in order that it be a
quotient of one with bijective antipode. These are motivated by the result
(which is an immediate consequence of [Sc, Prop. 2.7]) that a Hopf algebra $H$
is a Hopf subalgebra of one with bijective antipode iff its antipode $S_{H}$
is injective.
## 1\. Preliminaries
Throughout this paper, $k$ will be an arbitrary field. Unless explicitly
specified otherwise, homomorphisms, tensor products, algebras, coalgebras, and
so on are over $k$. We work with several categories: ${\rm Alg}$, ${\rm
CoAlg}$, ${\rm BiAlg}$ and ${\rm HopfAlg}$ denote the categories of
$k$-algebras, coalgebras, bialgebras and Hopf algebras, respectively. ${\rm
SHopfAlg}$ stands for the category of Hopf algebras with a bijective antipode
(the $S$ in front is supposed to remind the reader of the usual notation $S$
for the antipode of a Hopf algebra). If $x,y$ are objects in a category
$\mathcal{C}$, we use the notation $\mathcal{C}(x,y)$ for the set of morphisms
from $x$ to $y$ in $\mathcal{C}$.
We use standard notations for opposite and coopposite structures: $A^{op}$ is
the opposite of the algebra $A$, and $C^{cop}$ is the coopposite of the
coalgebra $C$.
For an algebra $A,\ _{A}\mathcal{M}$ denotes the category of left $A$-modules,
and similarly, $\mathcal{M}_{A}$ is the category of right $A$-modules. For a
coalgebra $C,\ ^{C}\mathcal{M}$ and $\mathcal{M}^{C}$ are the categories of
left and, respectively, right $C$-comodules.
For basic notions of category theory such as limits, colimits, adjunctions,
comma categories and so on, we refer mainly to [MacL], but what we need can be
found in most sources. Another example is [Pa, Appendix]. We use the language
and notations in [MacL]. At some point we do make use of the notion of locally
presentable category, but only in passing. Everything we need on the subject
can be found for instance in [ARo, Chapter 1].
For the structure maps of our objects we reserve the usual notation:
$\eta,\Delta,\varepsilon,S,\bar{S}$ denote, respectively, the unit,
comultiplication, counit, antipode, and skew antipode of an appropriate object
(algebra, Hopf algebra, etc.). We sometimes use subscripts to indicate the
object in question: $S_{H}$ is the antipode of the Hopf algebra $H$, for
instance. For a coalgebra $C$ and an algebra $A$, we regard ${\rm Hom}(C,A)$
as an algebra in the usual way, under the convolution $*$; in Sweedler sigma
notation ([Mo, 1.4.2]; we have omitted the summation symbol), we have:
$(f*g)(c)=f(c_{(1)})g(c_{(2)}).$
Recall that when $H$ is a Hopf algebra with antipode $S,\ A$ is an algebra,
and $f\in{\rm Alg}(H,A)$, the composition $fS$ is the inverse of $f$ with
respect to the convolution operation $*$. Similarly, $Sf$ is the inverse of
$f\in{\rm CoAlg}(C,H)$ for a coalgebra $C$ ([Sw, Chapter IV, Lemma 4.0.3]).
We also require the notion of faithful coflatness over a coalgebra. The main
definitions and properties regarding (faithful) coflatness can be found in
[BW, Chapters 21]. Here, the notion replacing the tensor product is that of
cotensor product over a coalgebra, for which we refer to [Ta4, Appendix 2] or
[BW, Chapters 21,22].
We recall here a result on monomorphisms in ${\rm CoAlg}$. For a proof (of our
lemma and the converses to its two statements), the reader can consult for
example [NT], where quite a few characterizations of monomorphisms of
coalgebras can be found; for even more such characterizations see [Ag2, T.
2.1]. As is customary in the literature, we denote by $\square_{D}$ the
cotensor product over the coalgebra $D$.
###### Lemma 1.1.
Let $f:C\to D$ be a monomorphism in ${\rm CoAlg}$. Then the scalar
coresriction $\mathcal{M}^{C}\to\mathcal{M}^{D}$ is full, and the
comultiplication $\Delta_{C}$ is a bijection of $C$ onto
$C\square_{D}C\subseteq C\otimes C$.
## 2\. First version of the problem
The most general form of the problem we are concerned with in this paper
consists of the two analogous questions of whether epi(mono)morphisms in the
category ${\rm HopfAlg}$ are surjective (resp. injective). Notice that a map
of Hopf algebras $f:H\to K$ is an epimorphism iff the inclusion of the image
of $H$ in $K$ is epi. Similarly, when we investigate monomorphisms, we can
assume that they are surjective. We will sometimes do this without mentioning
it explicitly.
We shall see that the answers to the two questions are negative, using the
following simple observation:
###### Proposition 2.1.
The antipode $S$ of a Hopf algebra $H$ is both an epimorphism and a
monomorphism in ${\rm HopfAlg}$ from $H$ to $H^{op,cop}$.
###### Proof.
$S$ is an epimorphism iff for any Hopf algebra $K$, the map
${\rm HopfAlg}(H^{op,cop},K)\to{\rm HopfAlg}(H,K)$
induced by it and defined by $f\mapsto fS$ is injective. More generally, if
$A$ is an algebra and $f$ is an algebra map from $H^{op}$ to $A$, then $fS$ is
the inverse of $f$ in the monoid ${\rm Hom}(H^{cop},A)$ under convolution
(here, $H$ is viewed only as a coalgebra). It follows that $f$ is uniquely
determined by $fS$, which is what we needed.
The statement that $S$ is mono is proven similarly: we have to show that for
any Hopf algebra $K$, the map
${\rm HopfAlg}(K,H)\to{\rm HopfAlg}(K,H^{op,cop})$
given by $f\mapsto Sf$ is injective. Again, this holds more generally, if we
replace $K$ with a coalgebra $C$ and ${\rm HopfAlg}$ with ${\rm CoAlg}$,
simply by noticing that $Sf$ is the inverse of $f\in{\rm CoAlg}(C,H)$ in ${\rm
Hom}(C,H)$. ∎
The negative answers to our two questions now follow from the fact that there
exist Hopf algebras with pathological (non-surjective or non-injective)
antipode. A Hopf algebra with non-bijective antipode is already constructed in
[Ta1]. However, we need the more specific result ([Ni]) that Takeuchi’s
algebra has a non-surjective antipode. In fact, Nichols also shows in [Ni]
that the antipode is injective. The Hopf algebra in question is the free Hopf
algebra $H(M_{n}(k)^{*})$ (a construction introduced in [Ta1]) on the
coalgebra $M_{n}(k)^{*}$, the dual of the matrix algebra $M_{n}(k)$ for $n>1$.
We shall have more to say about such universal constructions in the next
section.
As for the injectivity of the antipode, Takeuchi proves ([Ta2, Theorem 9])
that either the same free Hopf algebra $H(M_{n}(k)^{*})$ has a non-injective
antipode (as mentioned above, we know this to be false from [Ni]), or some
quotient of $H(M_{2n}(k)^{*})$ does. Also, Schauenburg constructs in [Sc] a
Hopf algebra with a surjective, non-injective antipode. Given these
pathological examples and the previous proposition, we get
###### Corollary 2.2.
There exist (injective) non-surjective epimorphisms in ${\rm HopfAlg}$, as
well as (surjective) non-injective monomorphisms.
In the next section we will also see examples of non-surjective epimorphisms
$H\to K$ with $K$ having a bijective antipode, and of non-injective
monomorphisms $H\to K$ with $H$ having a bijective antipode. We do not know if
both algebras can be chosen to have bijective antipode in such
counterexamples.
As it turns out, the problem epi vs. surjective is linked to Kaplansky’s first
conjecture. The more modern version of this conjecture asked whether all Hopf
algebras are (left and right) faithfully flat over their Hopf subalgebras
([Mo, Question 3.5.4]). Schauenburg gave some counterexamples in [Sc], and
strengthened the hypotheses further: are Hopf algebras with bijective antipode
faithfully flat over Hopf subalgebras with bijective antipode? In order to see
the connection between the two problems, we need the following simple result
on faithful flatness:
###### Proposition 2.3.
Let $\iota:A\to B$ be a left faithfully flat extension of algebras. If $\iota$
is an epimorphism in ${\rm Alg}$, then it is an isomorphism.
###### Proof.
The fact that $\iota$ is epi implies that $b\otimes_{A}1=1\otimes_{A}b$ in
$B\otimes_{A}B$ for all $b\in B$ ([St, Chapter XI, Prop. 1.1]). It follows
immediately from this last condition that the map $\iota\otimes_{A}I_{B}:B\to
B\otimes_{A}B$ is an isomorphism of right $B$-modules (actually, it follows
that the map is surjective; the injectivity is clear from the fact that the
multiplication $B\otimes_{A}B\to B$ is a left inverse for
$\iota\otimes_{A}I_{B}$). By faithful flatness, $\iota$ must be an isomorphism
of right $A$-modules. ∎
The fact that the forgetful functor ${\rm HopfAlg}\to{\rm Alg}$ has a right
adjoint ([Ag1, Theorem 3.3]; the result is dual to Takeuchi’s construction of
a free Hopf algebra on a coalgebra in [Ta1]), together with the easy-to-prove
results that (a) left adjoints preserve epimorphisms and (b) faithful functors
reflect epimorphisms, imply
###### Proposition 2.4.
A morphism of Hopf algebras $f:H\to K$ is an epimorphism if and only if it is
an epimorphism in ${\rm Alg}$, when viewed as a map of algebras.
We also record the dual statement, which follows by the dual argument: by
[Ta1] the forgetful functor ${\rm HopfAlg}\to{\rm CoAlg}$ is a right adjoint,
and hence preserves monomorphisms.
###### Proposition 2.5.
A morphism of Hopf algebras $f:H\to K$ is a monomorphism if and only if it is
a monomorphism in ${\rm CoAlg}$, when viewed as a map of coalgebras.
Proposition 2.3 and Proposition 2.4 show that epimorphisms of Hopf algebras
are indeed surjective whenever Kaplansky’s conjecture holds, i.e. in those
stuations when we do have faithful flatness. Such situations are, for
instance, the case when (same notations as in the statement of Proposition
2.4) $K$ is commutative, or has cocommutative coradical, or is pointed ([Ta3,
Theorem 3.1] takes care of the cases when $K$ is either commutative or
cocommutative, but [Ta3, Theorem 3.2] easily implies the cocommutative
coradical and the pointed cases as well; later, Radford proved in [Ra1] that
pointed Hopf algebras are, in fact, free over their Hopf subalgebras).
The contrapositive is that counterexamples to epi $\Rightarrow$ surjective are
counterexamples to Kaplansky’s first conjecture. In particular, by Proposition
2.1, we recover Schauenburg’s example ([Sc, Remark 2.6]) $S(H)\subset H$ of a
non-faithfully flat inclusion of Hopf algebras whenever the antipode $S$ of
$H$ is not surjective.
The fact that epi implies surjectivity in the cocommutative case, for example,
can be used, together with some adjunctions, to prove the classical results
that epimorphisms are surjective in the categories of groups or Lie algebras.
See also [Re, Prop. 3,4] for an interesting method of proof, using split
extensions of groups and Lie algebras, respectively.
The discussion above on the connection between faithful flatness over Hopf
subalgebras and epimorphisms in ${\rm HopfAlg}$ can be dualized: one can ask
when a surjection of Hopf algebras is faithfully coflat (see Section 1), and
investigate the relation between this question and the problem of determining
if/when monomorphisms of Hopf algebras are injective. Faithful coflatness
appears in [AD], for example, along with faithful flatness, as an important
technical condition (see the dual pair of results [AD, Corollaries 1.2.5,
1.2.14]).
We now want to prove the dual of Proposition 2.3. Together with Proposition
2.5, it will establish the connection between faithful coflatness and the
injectivity of monomorphisms in ${\rm HopfAlg}$: if the surjective
monomorphism $H\to K$ happens to be faithfully coflat, then it is an
isomorphism. Again, the contrapositive is that whenever we have a non-
injective monomorphism in ${\rm HopfAlg}$ (which we may as well assume is
surjective), we have an example of non-faithfully coflat surjection of Hopf
algebras.
###### Proposition 2.6.
Let $f:C\to D$ be map of coalgebras, making $C$ left faithfully coflat over
$D$. If $f$ is a monomorphism in ${\rm CoAlg}$, then it is an isomorphism.
###### Proof.
Since $f$ is a monomorphism, we know from Lemma 1.1 that the canonical map
$C\to C\square_{D}C$ is bijective. The map
$f\square_{D}I_{C}:C\square_{D}C\to D\square_{D}C\cong C$
is a left inverse for $C\to C\square_{D}C$, so it must also be bijective.
Faithful coflatness implies that $-\square_{D}C$ reflects isomorphisms, so $f$
must be an isomorphism. ∎
Finally, we end this section with a consequence of Proposition 2.1 giving a
sufficient condition for the antipode of a Hopf algebra to be surjective. We
do not use this result elsewhere in the paper.
###### Proposition 2.7.
Let $H$ be a Hopf algebra with antipode $S$. If $S(H)$ contains the coradical
$H_{0}$ of $H$, then $S$ is surjective.
###### Proof.
Proposition 2.1 says that the inclusion $S(H)\to H$ is epi. On the other hand,
as $S(H)$ contains the coradical $H_{0}$, the inclusion is faithfully flat (in
fact, $H$ is even free over $S(H)$, by a result of Nichols; it is also an
immediate consequence of [Ra2, Cor. 2.3]). By Proposition 2.3, we are done:
the inclusion of $S(H)$ in $H$ must be surjective. ∎
## 3\. Adjunctions and bijective antipodes
We have seen in the previous section that one can find both non-surjective
epimorphisms and non-injective monomorphisms in the category ${\rm HopfAlg}$.
We now strengthen the hypotheses: for epimorphisms $H\to K$, we ask that $K$
have bijective antipode. Similarly, for monomorphisms $H\to K$, we ask that
$H$ have bijective antipode. Again, we find counterexamples in these
situations. I do not know what happens if both Hopf algebras are required to
have bijective antipodes.
The construction is as follows:
In [Sc], Schauenburg constructs the left adjoint, which we denote here by
$K^{*}$, of the inclusion $i:{\rm SHopfAlg}\to{\rm HopfAlg}$ (recall that
${\rm SHopfAlg}$ is the category of Hopf algebras with bijective antipode; we
will sometimes omit the inclusion functor), and proves ([Sc, Cor. 2.8]) that
the unit $H\to K^{*}(H)$ of the adjunction is a non-faithfully flat inclusion
of Hopf algebras whenever $H$ has injective non-bijective antipode (in fact,
he proves more, namely that the inclusion does not have a certain property
(P), weaker that faithful flatness). We show here that the unit $H\to
K^{*}(H)$ is always an epimorphism of Hopf algebras. We also prove that the
inclusion $i$ has a right adjoint $K_{*}$, and that the counit $K_{*}(H)\to H$
of the resulting adjunction is always a monomorphism of Hopf algebras. These
will be examples of non-surjective epimorphisms and non-injective
monomorphisms, with our extra requirements on the antipodes, when the antipode
of Hopf algebra $H$ is “pathological”.
There seems to be an interesting parallel between the pairs of categories
${\rm BiAlg},{\rm HopfAlg}$ on the one hand and ${\rm HopfAlg},{\rm SHopfAlg}$
on the other; in order to emphasize it, we also carry out the arguments
outlined above for the inclusion $j:{\rm HopfAlg}\to{\rm BiAlg}$. The
existence of the left adjoint to this inclusion is a classical result of
Takeuchi ([Ta1]; even though Takeuchi passes directly from coalgebras to Hopf
algebras, the intermediary adjoint from ${\rm HopfAlg}$ to ${\rm BiAlg}$ is
easily deduced, and the construction is given explicitly in [Pa, Theorem
2.6.3]), and the existence of a right adjoint is proven in [Ag1, Theorem 3.3].
We state here the existence result for these adjoints:
###### Theorem 3.1.
(a) The inclusion $j:{\rm HopfAlg}\to{\rm BiAlg}$ has both a left adjoint
$H^{*}$ and a right adjoint $H_{*}$.
(b) The inclusion $i:{\rm SHopfAlg}\to{\rm HopfAlg}$ has both a left adjoint
$K^{*}$ and a right adjoint $K_{*}$.
Before going into the proof (which will consist mainly of the constructions of
the right adjoints to the inclusions, since the left adjoints are constructed
explicitly in [Ta1, Pa] and [Sc] as indicated above), we state and prove the
main result of this section, and derive some consequences. We keep the
notations from the statement of Theorem 3.1.
###### Theorem 3.2.
(a) For every bialgebra $B$, the component $B\to H^{*}(B)$ of the unit of the
adjunction $(H^{*},j)$ is an epimorphism of bialgebras, and the component
$H_{*}(B)\to B$ of the counit of the adjunction $(j,H_{*})$ is a monomorphism
of bialgebras.
(b) For any Hopf algebra $H$, the unit $H\to K^{*}(H)$ of the adjunction
$(K^{*},i)$ is an epimorphism of Hopf algebras, and the counit $K_{*}(H)\to H$
of the adjunction $(i,K_{*})$ is a monomorphism of Hopf algebras.
For the proofs we require a category-theoretic lemma, which we state after
some notations.
Let $\mathcal{C},\mathcal{D}$ be two categories, and
$U:\mathcal{C}\to\mathcal{D}$ a functor with a left adjoint $F$ and a right
adjoint $G$. Denote by $\alpha:I_{\mathcal{D}}\to UF$ and $\beta:UG\to
I_{\mathcal{D}}$ the unit of the adjunction $(F,U)$ and the counit of the
adjunction $(U,G)$, respectively. We then have:
###### Lemma 3.3.
With the notations above, $\alpha_{d}:d\to UF(d)$ is an epimorphism for every
object $d\in\mathcal{D}$ iff $\beta_{d}:UG(d)\to d$ is a monomorphism for
every object $d\in\mathcal{D}$.
###### Proof.
For each pair of objects $d,d^{\prime}\in\mathcal{D}$, we have a commutative
diagram
$\begin{diagram}$
where the two vertical arrows are the bijections given by the two adjunctions,
and the two diagonal arrows are induced by $\alpha_{d}$ (the upper arrow) and
$\beta_{d}$ (the lower arrow).
The fact that $\alpha_{d}$ is an epimorphism for all $d$ is equivalent to the
upper diagonal arrow being an injection for all pairs $d,d^{\prime}$.
Similarly for $\beta_{d}$ and the lower diagonal arrow. But since the vertical
maps are bijections, the conditions that the upper and respectively lower
diagonal arrow be an injection for all pairs $d,d^{\prime}$ are equivalent. ∎
###### Proof of Theorem 3.2.
By applying Lemma 3.3 to the two situations depicted in (a) and (b) (with the
functor $U$ being the inclusion $j$ and $i$ respectively), we conclude that it
suffices to prove one of the two statements in each of (a) and (b). It is
enough, for instance, to show that the units of the two adjunctions
$(H^{*},j)$ and $(K^{*},i)$ are epimorphisms.
(a) We want to show that $\alpha:B\to H^{*}(B)$ is an epimorphism in ${\rm
BiAlg}$ (strictly speaking, it should be $jH^{*}(B)$). Let $S$ be the antipode
of $H^{*}(B)$. The subalgebra $H$ of $H^{*}(B)$ generated by
$S^{n}(\alpha(B)),\ n\geq 0$ is a Hopf subalgebra: it is an algebra by
definition, it is closed under $S$ again by definition, and it’s a
subcoalgebra because all the $S^{n}(\alpha(B))$ are. This means that $B\to H$
is a subobject of the initial object $B\to H^{*}(B)$ in the comma category
$B\downarrow{\rm HopfAlg}$ ([MacL, II$\S$6]), and hence that $H=H^{*}(B)$.
Now consider a morphism of bialgebras $f:H^{*}(B)\to B^{\prime}$. Then
$fS\alpha$ is the inverse of $f\alpha$ in ${\rm Hom}(B,B^{\prime})$ under
convolution, $fS^{2}\alpha$ is the inverse of $fS\alpha$ in ${\rm
Hom}(B^{cop},B^{\prime})$, and so on. Because, as we have just seen,
$H^{*}(B)$ is generated as an algebra by the iterations of $\alpha(B)$ under
$S$, $f$ is uniquely determined by $f\alpha$. This is precisely the condition
required in order that $\alpha$ be an epimorphism of bialgebras.
(b) The proof runs parallel to that from (a): instead of the antipode, we now
use the inverse $\bar{S}$ of the antipode $S$ of $K^{*}(H)$. Again, let $K$ be
the subalgebra of $K^{*}(H)$ generated by $\bar{S}^{n}(\alpha(H)),\ n\geq 0$.
Arguing as before, we conclude that $K=K^{*}(H)$, i.e. that $K^{*}(H)$ is
generated as an algebra by the images of $\alpha(H)$ through the iterations of
$\bar{S}$, and hence that a Hopf algebra map $f:K^{*}(H)\to H^{\prime}$ is
uniquely determined by $f\alpha:H\to H^{\prime}$. ∎
As a consequence, we have:
###### Corollary 3.4.
(a) If $B$ is a sub-bialgebra of a Hopf algebra such that $B$ itself is not
Hopf, then $B\to H^{*}(B)$ is an injective, non-surjective epimorphism of
bialgebras. Similarly, if the bialgebra $B$ is not Hopf but is a quotient of a
Hopf algebra, then $H_{*}(B)\to B$ is a surjective, non-injective monomorphism
of bialgebras.
(b) If $H$ does not have bijective antipode but is contained in a Hopf algebra
with bijective antipode, then $H\to K^{*}(H)$ is an injective, non-surjective
epimorphism of Hopf algebras. Similarly, if $H$ does not have bijective
antipode but is a quotient of a Hopf algebra with bijective antipode, then
$K_{*}(H)\to H$ is a non-injective, surjective monomorphism.
###### Proof.
(a) Since the inclusion of $B$ in a Hopf algebra factors through $B\to
H^{*}(B)$, the latter must be an injective map. The rest follows immediately
from Theorem 3.2. For the second statement the dual argument works.
(b) is entirely analogous to (a). ∎
Examples as in the previous corollary actually exist. Focusing on (b), the
Hopf algebra case, such examples can be found in [Sc]: any Hopf algebra $H$
with injective non-bijective antipode (such as the free Hopf algebra on the
coalgebra $M_{n}(k)^{*},\ n\geq 2$, according to [Ni]) injects properly into
$K^{*}(H)$, and also, an example is given of a Hopf algebra with non-bijective
antipode which is a quotient of a Hopf algebra with bijective antipode: it is
a quotient of the free Hopf algebra with bijective antipode on the coalgebra
$M_{4}(k)^{*}$. In conclusion, we have:
###### Corollary 3.5.
There is an epimorphic inclusion $H\to K$ of Hopf algebras with $K$ having a
bijective antipode. Similarly, there is a monomorphic surjection $H\to K$ of
Hopf algebras with $H$ having a bijective antipode.
Next, we give explicit constructions for the right adjoints to the inclusions
$j:{\rm HopfAlg}\to{\rm BiAlg}$ and $i:{\rm SHopfAlg}\to{\rm HopfAlg}$. In
particular, this solves [Ag1, Problem 2], which asks for a construction for
the right adjoint to $j$, shown there to exist by the Special Adjoint Functor
Theorem (the dual of [MacL, V$\S$8, Corollary]).
Throughout, we shall make free use of the fact that the following categories
are all complete and cocomplete: ${\rm Alg},{\rm CoAlg},{\rm BiAlg},{\rm
HopfAlg}$. In fact, they are locally presentable, and locally presentable
categories are cocomplete (by definition: [ARo, Def. 1.17]) and complete
([ARo, Remark 1.56]).
The local presentability is proven up to bialgebras in [Po1] in the more
general setting of monoids, comonoids and bimonoids in a symmetric monoidal
category with some extra assumptions, which are all satisfied by the category
of $k$-vector spaces (see Summary 4.3 in that paper); that ${\rm HopfAlg}$ is
locally presentable follows from [Po2, Prop. 4.3] and the fact that by [Ta1],
the forgetful functor ${\rm HopfAlg}\to{\rm CoAlg}$ has a left adjoint (this
is the argument used in the proof of [Ag1, Theorem 2.6]). Alternatively, one
could prove the local presentability of these categories directly, but we do
not go into these details here.
We start the construction of adjoints with the inclusion $j:{\rm
HopfAlg}\to{\rm BiAlg}$.
###### Proof of Theorem 3.1 (a).
As mentioned before, we only construct the right adjoints, since explicit
constructions for the left adjoints can be found in the literature, in the
sources cited above.
We simply dualize the construction from [Pa, Theorem 2.6.3]. As that proof is
very detailed, and most arguments here are simply dualizations of those, we
will only indicate how the construction goes, leaving out simple
verifications.
Let $B$ be a bialgebra, and let $P$ be the product (in the category ${\rm
BiAlg}$) of the bialgebras $B_{n},\ n\geq 0$, where $B_{n}=B$ if $n$ is even,
and $B_{n}=B^{op,cop}$ if $n$ is odd. Denote by $\pi_{n}$ the structure maps
$P\to B_{n}$ of the product of bialgebras, and let $\eta,\varepsilon$ be the
unit and counit of $P$ respectively. By the universality of the product, there
is a unique bialgebra map $S$ such that
$\begin{diagram}$
commutes for all $n\geq 0$.
Let $H_{*}(B)$ be the sum of all subcoalgebras $C\subseteq P$ on which $S$
behaves like an antipode. Specifically, the condition such a coalgebra $C$ is
supposed to satisfy is
$c_{(1)}S(c_{(2)})=\eta\varepsilon(c)=S(c_{(1)})c_{(2)},\ \forall c\in C.$ (1)
As the notation suggests, this is the object we are looking for. $H_{*}(B)$ is
by definition a subcoalgebra of $P$, and (1) holds with $H_{*}(B)$ instead of
$C$. In other words, $H_{*}(B)$ is the largest subcoalgebra of $P$ on which
$S$ acts as an antipode. It is an easy matter now to prove that $H_{*}(B)$ is
closed under multiplication and the action of $S$ (and it clearly contains the
unit $1_{P}$), so it is, in fact, a Hopf subalgebra of $P$ with antipode $S$.
We now want to prove that $\beta:H_{*}(B)\to B$, the composition of
$\pi_{0}:P\to B$ with the inclusion $H_{*}(B)\to P$, is universal from a Hopf
algebra to $B$. So let $f:H\to B$ be a bialgebra map from a Hopf algebra $H$
to $B$. The maps
$f_{n}=f\circ S_{H}^{n}:H\to B_{n},\ n\geq 0$
are bialgebra morphisms, and so define a bialgebra map $\tilde{f}:H\to P$ with
$\pi_{0}\circ\tilde{f}=f$. First, we want to show that $\tilde{f}$ intertwines
$S$ and $S_{H}$:
$\begin{diagram}$
In turn, this follows from the universality of the product $P$ if we show that
$\pi_{n}\circ\tilde{f}\circ S_{H}=\pi_{n}\circ S\circ\tilde{f},\ \forall n\geq
0$ (2)
as maps from $H$ to $B$. On the one hand, from the definition of $\tilde{f}$,
we get
$\pi_{n}\circ\tilde{f}\circ S_{H}=f_{n}\circ S_{H}=f\circ
S_{H}^{n+1}=\pi_{n+1}\circ\tilde{f},$ (3)
and on the other hand, from the definition of $S$, we have
$\pi_{n}\circ S\circ\tilde{f}=\pi_{n+1}\circ\tilde{f},$ (4)
because $\pi_{n}\circ S=\pi_{n+1}$ as maps from $P$ to $B$ (we identify the
underlying sets of all $B_{n}$). (3) and (4) now prove the desired equality
(2).
Because $\tilde{f}$ intertwines $S,S_{H}$ and $S_{H}$ is the antipode of $H$,
it follows that $S$ is the antipode of $\tilde{f}(H)$. The definition of
$H_{*}(B)$ now implies that the image of $\tilde{f}$ is contained in
$H_{*}(B)$, i.e. $\tilde{f}$ factors through $H_{*}(B)\subseteq P$. In other
words, we have just shown that any bialgebra map $f:H\to B$ factors as
$\begin{diagram}$
It remains to prove that in such a diagram, $\tilde{f}$ is unique. Again,
$\tilde{f}$ is determined by the sequence of maps $\pi_{n}\tilde{f}$ (also
regarding $\pi_{n}$ as a map from $H_{*}(B)\subseteq P$ to $B_{n}$). But
notice that, because $\tilde{f}$ commutes with the antipodes, we have
$\pi_{n}\circ\tilde{f}\circ S_{H}=\pi_{n}\circ
S\circ\tilde{f}=\pi_{n+1}\circ\tilde{f}.$
This means that $\pi_{n+1}\tilde{f}$ is the inverse of $\pi_{n}\tilde{f}$ in
${\rm Hom}(H,B_{n})$ under convolution, and hence that the sequence
$\pi_{n}\tilde{f}$ is uniquely determined by $\pi_{0}\tilde{f}=f$. This
finishes the proof. ∎
We now want to obtain the right adjoint to the inclusion $i:{\rm
SHopfAlg}\to{\rm HopfAlg}$ as a direct consequence of Theorem 3.1 (a) above.
For this, we need
###### Lemma 3.6.
Let $B$ be a bialgebra with a skew antipode $\bar{S}_{B}$. Then, the cofree
Hopf algebra $H_{*}(B)$ constructed above also has a skew antipode $\bar{S}$.
Consequently, the antipode $S$ of $H_{*}(B)$ is bijective.
###### Proof.
The last statement follows immediately from the first, as it is well-known
that a Hopf algebra has a skew antipode iff its antipode is bijective, in
which case the skew antipode is the inverse of the antipode ([Mo, Lemma
1.5.11]). We focus on showing that the antipode $S$ of $H_{*}(B)$ is
bijective.
We use the notations from the proof of Theorem 3.1 (a). Recall that there are
maps $\pi_{n}$ from $H_{*}(B)$ to $B_{n},\ n\geq 0$, where $B_{n}$ is $B$ for
even $n$ and $B^{op,cop}$ for odd $n$. $\pi_{0}$ is universal, and the maps
$\pi$ satisfy
$\pi_{n}S=\pi_{n+1},\ \forall n\geq 0.$ (5)
From the universality of $\pi_{0}:H_{*}(B)\to B$, we can find a unique Hopf
algebra map $\bar{S}$ making the following diagram of bialgebra morphisms
commutative:
$\begin{diagram}$
The aim is to show that $\bar{S}$ is a composition inverse to $S$. Complete
this diagram to the left with another square (commutative by (5) for $n=0$):
$\begin{diagram}$
Again by the universality of $\pi_{0}$, the composition $\bar{S}S$ is the
unique Hopf algebra map making the outer rectangle commutative. If we prove
that the identity on $H_{*}(B)$ also makes the outer rectangle commutative, we
will have shown that $\bar{S}$ is a left composition inverse for $S$. In other
words, we now want to show that
$\pi_{0}=\bar{S}_{B}\pi_{1}.$ (6)
Since $\bar{S}_{B}$ is an antipode for $B^{cop}$ and $\pi_{1}$ is in ${\rm
CoAlg}(H_{*}(B),B^{cop})$, the composition $\bar{S}_{B}\pi_{1}$ is the
convolution inverse of $\pi_{1}$ in ${\rm Hom}(H_{*}(B),B)$ (or ${\rm
Hom}(H_{*}(B),B^{cop})$, the algebra structure under convolution is the same).
On the other hand, (5) with $n=0$ shows that $\pi_{0}$ is also the convolution
inverse of $\pi_{1}$ in ${\rm Hom}(H_{*}(B),B)$. This implies the desired
equality (6).
We have just shown that $\bar{S}S=I_{H_{*}(B)}$. Deducing now that $S\bar{S}$
is also the identity is easy: $S=S\bar{S}S$ is the convolution inverse of both
$I_{H_{*}(B)}$ and of $S\bar{S}$ in ${\rm End}(H_{*}(B))$. ∎
We now have what we need to finish the proof of Theorem 3.1.
###### Proof of Theorem 3.1 (b).
Let $H$ be a Hopf algebra. The bialgebra $B=H^{op}$ has a skew antipode,
namely $S_{H}$. According to Lemma 3.6, the antipode of the cofree Hopf
algebra $H_{*}(B)$ on $B$ is bijective. The universal bialgebra map
$\beta:H_{*}(H^{op})\to H^{op}$ (7)
induces a bialgebra map denoted by the same symbol:
$\beta:(H_{*}(H^{op}))^{op}\to H.$
I claim that this is universal from a Hopf algebra with bijective antipode to
$H$. In other words, we have
$K_{*}(H)=(H_{*}(H^{op}))^{op},$
with the obvious universal map $\beta$ to $H$.
To see this, let $f:K\to H$ be a Hopf algebra map, with $K$ having bijective
antipode. $f$ is then also a bialgebra morphism from the Hopf algebra $K^{op}$
to $H^{op}$, and hence factors uniquely through $\beta$ by the universality of
(7). This gives a unique map $\tilde{f}$, say, from $K^{op}$ to
$H_{*}(H^{op})$. $\tilde{f}$ will then also be the unique Hopf algebra map
from $K$ to $(H_{*}(H^{op}))^{op}$ through which $f$ factors, and the proof is
finished. ∎
###### Remark 3.7.
Although we prefer the construction used above because it shows how Theorem
3.1 (b) follows directly from (a), there is more than one way of introducing
the right adjoint to $i:{\rm SHopfAlg}\to{\rm HopfAlg}$.
One idea, for instance, would be to dualize Schauenburg’s construction from
[Sc, Prop. 2.7]: $K_{*}(H)$ is the limit of the inverse system of Hopf
algebras $u_{n}:H_{n+1}\to H_{n},\ n\geq 0$, where all $H_{n}$ are $H$, and
all $f_{n}$ are equal to the square $S_{H}^{2}$ of the antipode $S_{H}$.
Alternatively, we could imitate the construction appearing in Theorem 3.1 (a),
by using a product of bialgebras $B_{n}$ indexed by the integers instead of
the natural numbers, with $B_{n}=B$ for $n$ even and $B_{n}=B^{op,cop}$ for
odd $n$ (just as before).
This observation works the other way around too: the left adjoint of the
inclusion $i:{\rm SHopfAlg}\to{\rm HopfAlg}$, denoted by $K^{*}$, can be
constructed in the same manner, using the left adjoint of $j:{\rm
HopfAlg}\to{\rm BiAlg}$ from Theorem 3.1 (a). Just as in the previous proof,
we have
$K^{*}(H)=(H^{*}(H^{op}))^{op}.$
## 4\. Some comments and problems
As remarked several times before, I do not know whether counterexamples as in
Corollary 3.5 still exist if we require that both Hopf algebras $H$ and $K$
have bijective antipode.
In the spirit of the connections we have noticed above between faithful
flatness/coflatness and the problem of category-theoretic conditions
(epimorphisms, monomorphisms) vs. set-theoretic conditions (surjectivity,
injectivity), we ask:
###### Question 1.
Is an epimorphism of Hopf algebras with bijective antipode necessarily
surjective?
And its dual:
###### Question 2.
Is a monomorphism of Hopf algebras with bijective antipode necessarily
injective?
These, we believe, should go hand in hand with the aforementioned Kaplansky
conjecture and its dual, regarding faithful coflatness.
We now turn our attention to the adjunctions which appear in Section 3. It
follows immediately from Theorem 3.1 (a) that a bialgebra $B$ has a largest
subbialgebra which is a quotient of a Hopf algebra (the image of $H_{*}(B)\to
B$), and dually, has a largest quotient bialgebra contained in a Hopf algebra
(the image of $B\to H^{*}(B)$). In an entirely analogous manner, Theorem 3.1
(b) implies that a Hopf algebra $H$ has a largest Hopf subalgebra which is a
quotient of one with bijective antipode (the image of $K_{*}(H)\to H$), and a
largest quotient Hopf algebra contained in one with bijective antipode (the
image of $H\to K^{*}(H)$). The natural problem arises of characterizing those
bialgebras (Hopf algebras) which are quotients or subbialgebras (resp.
quotients or Hopf subalgebras) of Hopf algebras (resp. Hopf algebras with
bijective antipode).
For one of the four adjunctions, at least, this question is settled: part of
[Sc, Prop. 2.7] says, in a slightly different formulation, that a Hopf algebra
is a Hopf subalgebra of one with bijective antipode iff it has injective
antipode. This is a consequence of Schauenburg’s construction of $K^{*}(H)$ as
the colimit of the inductive system $u_{n}:H_{n}\to H_{n+1},\ n\geq 0$, with
$H_{n}=H$ and $u_{n}=S_{H}^{2}$ for all $n\geq 0$ (see Remark 3.7). The result
just mentioned then follows from the fact that if in such a system all maps
are injections, the map sending $H_{0}$ to the colimit is also an injection.
As mentioned in Remark 3.7, we can dualize this construction. The dual
statement on inverse limits with surjective maps, however, no longer holds, in
general. At least not at the level of coalgebras (and the limit appearing
there is one of coalgebras, as the forgetful functor ${\rm HopfAlg}\to{\rm
CoAlg}$ is a right adjoint by [Ta1], so it preserves limits): one can easily
construct a sequence of surjections $C_{n+1}\to C_{n}$ where $C_{n}$ are
simple coalgebras and $\dim C_{n}\to\infty$, in which case the resulting limit
is none other than $0$.
Despite such examples, can we still find simple necessary and sufficient
conditions on a Hopf algebra in order that it be a quotient of a Hopf algebra
with bijective antipode?
###### Problem 1.
Characterize those Hopf algebras $H$ for which $K_{*}(H)\to H$ is surjective.
More specifically, we ask
###### Question 3.
Is it true that a Hopf algebra with surjective antipode is a quotient of one
with bijective antipode?
And what can be said about the other two adjunctions, between the categories
${\rm BiAlg}$ and ${\rm HopfAlg}$? We would like to find necessary and
sufficient conditions on a bialgebra, expressed intrinsically, in order that
it be a subbialgebra or a quotient bialgebra of a Hopf algebra.
###### Problem 2.
Characterize intrinsically those bialgebras $B$ for which (a) $B\to H^{*}(B)$
is injective, or (b) $H_{*}(B)\to B$ is surjective.
We take a moment here to point out that it is by no means true that all
bialgebras satisfy (a) (or (b)). In other words, $B\to H^{*}(B)$ is not always
injective, nor is $H_{*}(B)\to B$ always surjective. Some examples follow.
###### Example 4.1.
Let $M$ be a monoid, and $B=k[M]$ the monoid bialgebra. One sees easily that
the free Hopf algebra $H^{*}(B)$ on $B$ is precisely the group algebra of the
enveloping group $G(M)$ of $M$. If the canonical map $M\to G(M)$ happens to be
non-injective (and this happens whenever $M$ is not “cancellable”), $B\to
H^{*}(B)$ will be non-injective as well. This implies that $B$ is not a
subbialgebra of a Hopf algebra.
###### Example 4.2.
Let $H$ be a Hopf algebra with non-injective antipode. It is then clear that
$H\to K^{*}(H)$ cannot be an embedding. In view of Remark 3.7, $K^{*}(H)$ is
the opposite of $H^{*}(H^{op})$. Consequently, $B=H^{op}$ is not a
subbialgebra of a Hopf algebra.
###### Example 4.3.
The previous example can be dualized, using Remark 3.7 again: if $H$ is a Hopf
algebra with non-surjecive antipode, then $B=H^{op}$ is a bialgebra which is
not a quotient of a Hopf algebra.
## Acknowledgements
The author would like to thank Prof. Gigel Militaru for suggesting some of the
questions posed here, as well as colleagues Ana-Loredana Agore and Dragoş
Frăţilă for countless fruitful conversations on the topics treated in this
paper. Also, we thank the referee for valuable suggestions on the improvement
of this paper.
## References
* [AD] Andruskiewitsch, N. and Devoto, J. - Extensions of Hopf algebras, Algebra i Analiz 7 (1995), pp. 22 - 61
* [BW] Brzeziński, T. and Wisbauer, R. - Corings and comodules, Cambridge University Press (2003)
* [ARo] Adámek, J. and Rosický, J. - Locally presentable and accessible categories, Cambridge University Press (1994)
* [Ag1] Agore, A. L. - Categorical constructions for Hopf algebras, preprint, arXiv:0905.2613
* [Ag2] \- Monomorphisms of coalgebras, preprint, arXiv:0908.2959
* [DG] Demazure, M. and Gabriel, P. - Groupes algébriques 1, Masson (1970)
* [Ka] Kaplansky, I. - Bialgebras, Lecture notes, Univ. of Chicago, Chicago (1975)
* [MacL] Mac Lane, S. - Categories for the working mathematician, Springer-Verlag (1971)
* [Mo] Montgomery, S. - Hopf algebras and their actions on rings, vol. 82 of CBMS Regional Conference Series in Mathematics, AMS, Providence, Rhode Island (1993)
* [NT] Năstăsescu, C. and Torrecillas, B. - Torsion theories for coalgebras, J. Pure and Appl. Algebra 97 (1994), pp. 203 - 220
* [Ni] Nichols, W. D. - Quotients of Hopf algebras, Comm. Algebra 6 (1978), pp. 1789 - 1800
* [OSch] Oberst, U. and Schneider, H.-J. - Untergruppen formeller Gruppen von endlichem Index, J. Algebra 31 (1974), pp. 10 - 44
* [Pa] Pareigis, B. - Lectures on quantum groups and noncommutative geometry. Available on the author’s webpage
* [Po1] Porst, H. E. - On categories of monoids, comonoids, and bimonoids, Quaestiones Math. 31 (2008), pp. 127 - 139
* [Po2] \- Universal constructions for Hopf algebras, J. Pure Appl. Algebra 212 (2008), pp. 2547 - 2554
* [Ra1] Radford, D. E. - Pointed Hopf algebras are free over Hopf subalgebras, J. Algebra 45 (1977), pp. 266 - 273
* [Ra2] \- Operators on Hopf algebras, Amer. J. Math. 99 (1977), pp. 139 - 158
* [Re] Reid, G. A. - Epimorphisms and surjectivity, Inventiones Math. 9 (1970), pp. 295 - 307
* [Sc] Schauenburg, P. - Faithful flatness over Hopf subalgebras: Counterexamples, appeared in Interactions between ring theory and representations of algebras: proceedings of the conference held in Murcia, Spain, CRC Press (2000), pp. 331 - 344
* [St] Stenström, Bo - Rings of quotients, Springer-Verlag (1975)
* [Sw] Sweedler, M. E. - Hopf algebras, Benjamin New York (1969)
* [Ta1] Takeuchi, M. - Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), pp. 561 - 582
* [Ta2] \- There exists a Hopf algebra whose antipode is not injective, Sci. Papers Coll. Gen. Ed. Univ. Tokyo 21 (1971), pp. 127 - 130
* [Ta3] \- A correspondence between Hopf ideals and subHopf algebras, Manuscripta Math. ’bf 7 (1972), pp. 251 - 270
* [Ta4] \- Formal schemes over fields, Comm. Algebra 5 (1977), pp. 1483 - 1528
* [Wa] Waterhouse, William C. - Introduction to affine group schemes, Springer-Verlag (1979)
|
arxiv-papers
| 2009-07-16T16:34:18 |
2024-09-04T02:49:04.012653
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexandru Chirvasitu",
"submitter": "Alexandru Chirv{\\ba}situ L.",
"url": "https://arxiv.org/abs/0907.2881"
}
|
0907.2890
|
001–003
# Colour-Magnitude Diagrams of candidate age-gap filling LMC clusters
Eduardo Balbinot1 Basílio X. Santiago1 Leandro Kerber2 Beatriz Barbuy2 Bruno
M. S. Dias2 1 Departamento de Astronomia, Universidade Federal do Rio Grande
do Sul
Porto Alegre, RS, Brazil
email: (balbinot, santiago)@if.ufrgs.br
2 IAG, Universidade de Sao Paulo
Sao Paulo, SP, Brazil
(2008)
###### Abstract
The LMC has a rich star cluster system spanning a wide range of ages and
masses. One striking feature of the LMC cluster system is the existence of an
age gap between 3-10 Gyrs. Four LMC clusters whose integrated colours are
consistent with those of intermediate age simple stellar populations have been
imaged with the Optical Imager (SOI) at the Southern Telescope for
Astrophysical Research (SOAR). Their colour-magnitude diagrams (CMDs) reach V
$\sim$ 24\. Isochrone fits, based on Padova evolutionary models, were carried
out to these CMDs, after subtraction of field contamination. The preliminary
results are as follows: KMK88-38 has an age of about 1.3 Gyr, assuming typical
LMC metallicity and distance modulus, and a very low redenning. For OGLE-
LMC0531, the best eye fits to isochrones yield an age $\sim$ 1.6 Gyr and
E(B-V)=0.03. BSDL917 is younger, $\sim$ 150 Myrs, and subjected to larger
extinction (E(B-V)=0.08). The remaining cluster is currently under analysis.
Therefore, we conclude that these clusers are unlikely to fill in the LMC
cluster age gap, even when fitting uncertainties in the parameters are
considered.
###### keywords:
(galaxies:) Magellanic Clouds, galaxies: star clusters, (stars:) Hertzsprung-
Russell diagram
††volume: 256††journal: The Magellanic System: Stars, Gas, and
Galaxies††editors: Jacco Th. van Loon & Joana M. Oliveira, eds.
## 1 Introduction
The LMC has a rich star cluster system spanning a wide range of ages and
masses. One striking feature of the LMC cluster system is the existence of an
age gap between 3-10 Gyrs. Four LMC clusters whose integrated colours are
consistent with those of intermediate age simple stellar populations have been
imaged with the Optical Imager (SOI) at the Southern Telescope for
Astrophysical Research (SOAR).
## 2 Data
We have imaged 3 out of the 5 LMC clusters listed by Hunter et al (2003) as
belonging to the age gap. Two of them have been fully reduced.
The images were taken in 2007 with SOAR/SOI telescope in the B, V, and I
filters. A slow readout was used in order to minimise detector noise. A 2x2
pixel binning was adopted, yielding a spatial scale of 0.154 arcsec/pixel.
Seeing was always around 0.8 arcsec.
The exposures were flatfielded, bias subtracted, mosaiched and latter
combined. The photometry was carried out with standard point spread function
(PSF) fitting. The DAOPHOT package (Stetson 1994) was used to detect sources
(4 $\sigma$ above sky background) and measure magnitudes. The PSF was modelled
as a Moffat function with $\beta=1.5$. Photometric calibration was obtained
from the standard fields in the Northeast arm of the SMC (Sharpee et al.
2002). A typical SOAR/SOI image section is shown in Figure 1.
Figure 1: A 2.6’ x 3.6’ image section of the field around the cluster
KMK88-38, where two other known clusters are included. Their positions in the
image are indicated.
CMDs for the fields of the two age gap candidates already reduced are shown in
Figure 2. Their colour-magnitude diagrams (CMDs) reach $V\sim 23$. From left
to right, we show the full SOI field CMD, the CMD in the cluster region and
the field subtracted cluster CMD. Padova isochrones from Girardi et al. (2002)
are superposed to these latter. Field subtraction was performed statistically,
applying the method described in detail by Kerber et al. (2002).
Figure 2: V,(V-I) (top) and V,(B-V) (bottom) CMDs for the fields around
KMK88-38 (left) and LMC0531 (right) clusters. The panels from left to right
show: the entire SOI field, the region around the cluster, the result of field
subtraction. Padova isochrones are overlaid to the latter panels. Ages are
shown as $log(\tau_{max}(yrs))$, $log(\tau_{min})$, $\Delta log\tau$. The
adopted metallicity is also shown.
Besides the age gap candidates, the SOI images also covered other LMC clusters
listed in the catalogue by Bica et al. (2008). Figure 3 shows the field
subtracted CMDs for them, again with isochrones overlaid.
Figure 3: Field subtracted CMDs for the remaining clusters found in our
SOAR/SOI images. The symbols are the same as in the corresponding panels of
Figure 2.
Visual matching of the observed CMDs to the isochrone set has allowed us to
estimate age, metallicity, distance modulus and reddening for each cluster.
The resulting parameters are shown in Table 1.
## 3 Results and conclusions
The original targets, KMK88-38 and LMC0531, turn out to be the relatively old,
as expected, with ages $\sim 1-2$ Gyrs. However, they are still younger than
the lower age limit of the LMC gap. Interestingly, KMK88-39, LMC0214 and
LMC0523, which lie in the same SOI fields (5.5arcmin x 5.5 arcmin in size),
are much younger. LMC0523 final V,(B-V) CMD has 3 stars in the Red Clump
region. Even though they survived field subtraction, these stars have
relatively low probabilities of actually belonging to the cluster, and they
are, in fact, absent from the V,(V-I) CMD. For LMC0214 we have only B and V
images. The ages for this latter, as well as for KMK88-39 are upper limits, as
their upper main sequence is close to the saturation limit. Finally, NGC 1878
is a richer and denser cluster. Field subtraction was not as efficient in this
case, especially in V,(B-V). We attribute that to crowding effects, which make
photometric errors larger in the cluster region than in the field,
jeopardising the statistical field subtraction. Still, its V,(V-I) CMD
indicates that NGC 1878 is also younger than 0.5 Gyr.
$Name$ | $log(Age)$ | $Z$ | $E(B-V)$ | $(m-M)_{V}$
---|---|---|---|---
OGLE-LMC0214 | $8.4\pm 0.3$ | $0.013$ | $0.10$ | $18.50$
OGLE-LMC0523 | $7.8\pm 0.3$ | $0.013$ | $0.20$ | $18.40$
OGLE-LMC0531 | $9.2\pm 0.2$ | $0.014$ | $0.09$ | $18.50$
KMK88-38 | $9.2\pm 0.2$ | $0.006$ | $0.01$ | $18.65$
KMK88-39 | $8.5\pm 0.3$ | $0.011$ | $0.02$ | $18.50$
NGC1878 | $8.4\pm 0.3$ | $0.014$ | $0.17$ | $18.50$
Table 1: The parameters found for our sample.
We thus conclude that the sample analysed so far does not contain any cluster
located in the LMC age gap. We are currently reducing the field images of
another age gap candidate: LMC0169. We are also reducing lower exposure time
images of all clusters and perfecting the field subtraction algorithm; both
are important steps towards improving our age and metallicity constraints on
the clusters.
## References
* [Bica et al. (2008)] Bica E. et al., 2008, MNRAS, in press (arXiv:0806.3049)
* [Girardi et al.2002] Girardi L. et al., 2002, A&AS, 391, 195
* [Hunter et al.2003] Hunter D. et al., 2003, AJ, 126, 1836
* [Kerber et al.2002] Kerber, L.O. et al., 2002, A&A, 390, 121
* [Sharpee et al.2002] Sharpee, B. et al., 2002, AJ, 123, 3216
* [Stetson (1994)] Stetson, P.B., 1994, PASP, 106, 205
|
arxiv-papers
| 2009-07-16T17:16:00 |
2024-09-04T02:49:04.020072
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eduardo Balbinot, Basilio Santiago, Leandro Kerber, Batriz Barbuy and\n Bruno Dias",
"submitter": "Eduardo Balbinot",
"url": "https://arxiv.org/abs/0907.2890"
}
|
0907.2942
|
# What can break the Wandzura–Wilczek relation?
Alberto Accardia,b, Alessandro Bacchettaa,c, W. Melnitchouka, Marc Schlegela
aJefferson Lab, Newport News, VA 23606, USA
bHampton University, Hampton, VA 23668, USA
cUniversità degli Studi di Pavia, 27100 Pavia, Italy
###### Abstract
We analyze the breaking of the Wandzura–Wilczek relation for the $g_{2}$
structure function, emphasizing its connection with transverse momentum
dependent parton distribution functions. We find that the relation is broken
by two distinct twist-3 terms, and clarify how these can be separated in
measurements of double-spin asymmetries in semi-inclusive deep inelastic
scattering. Through a quantitative analysis of available $g_{2}$ data we also
show that the breaking of the Wandzura–Wilczek relation can be as large as
15–40% of the size of $g_{2}$.
††preprint: JLAB-THY-09-1033
## I Introduction
The spin structure of the nucleon remains one of the most challenging and
controversial problems in hadronic physics Bass (2005); Kuhn et al. (2009);
Burkardt et al. (2008). Experimentally it is now known, through many careful
measurements of the nucleon’s $g_{1}$ structure function, that quarks carry
only some 30% of the proton’s longitudinal spin, a feature which is now
qualitatively understood Myhrer and Thomas (2008). Moreover, polarized $pp$
scattering observables Morreale (2009) and open charm production in deep
inelastic scattering Alekseev et al. (2009) suggest that gluons carry an even
smaller fraction of the longitudinal spin. Presumably, the remainder arises
from quark and gluon orbital angular momentum.
Although less attention has been paid to it, there are a number of intriguing
questions associated with the transverse spin structure of the nucleon. An
example is the study of the $g_{2}$ structure function, which only in recent
years has been probed experimentally with high precision. Unlike all other
inclusive deep-inelastic scattering (DIS) observables, the $g_{2}$ structure
function is unique in directly revealing information on the long-range quark-
gluon correlations in the nucleon. In the language of the operator product
expansion (OPE) these are parametrized through matrix elements of higher twist
operators, which characterize the strength of nonperturbative multi-parton
interactions. (In the OPE “twist” is defined as the mass dimension minus the
spin of a local operator.) In other inclusive structure functions higher twist
contributions are suppressed by powers of the four-momentum transfer squared
$Q^{2}$, whereas in $g_{2}$ these appear at the same order as the leading
twist.
As discussed by Wandzura and Wilczek Wandzura and Wilczek (1977), the leading
twist contribution to the $g_{2}$ structure function, which is denoted by
$g_{2}^{\rm WW}$, can be expressed in terms of the leading twist (LT) part of
the $g_{1}$ structure function,
$g_{2}^{\rm WW}(x_{B})=-g_{1}^{\rm
LT}(x_{B})+\int_{x_{B}}^{1}\frac{dy}{y}g_{1}^{\rm LT}(y)\ ,$ (1)
where $x_{B}$ is the Bjorken scaling variable, and we suppress the explicit
dependence of the structure functions on $Q^{2}$. The Wandzura–Wilczek (WW)
relation asserts that the total $g_{2}$ structure function is given by the
leading twist approximation (1),
$g_{2}(x_{B})\stackrel{{\scriptstyle?}}{{=}}g_{2}^{\rm WW}(x_{B})\ ,$ (2)
which would be valid in the absence of higher twist contributions. In this
case the $g_{2}$ structure function would also satisfy the
Burkhardt–Cottingham (BC) sum rule Burkhardt and Cottingham (1970),
$\int_{0}^{1}dx_{B}\ g_{2}(x_{B})=0\ .$ (3)
Its violation would also signal the presence of twist-3 or higher
contributions. Unlike the WW relation, however, the validity of the BC sum
rule (which is yet to be conclusively demonstrated experimentally Anthony et
al. (2003); Amarian et al. (2004)) would not necessarily imply that higher
twist terms vanish Jaffe (1990); Jaffe and Ji (1991).
In this paper we explore the physics that can lead to the breaking of the WW
relation in QCD, preliminary results for which have appeared in Ref. Accardi
et al. (2009). In Sec. II we present a detailed theoretical analysis of quark-
quark and quark-gluon-quark correlation functions, and discuss the so-called
Lorentz invariance relations and equations of motion relations. From these we
show that the WW relation is valid if pure twist-3 and quark mass terms are
neglected, in agreement with OPE results. We find that there are two distinct
contributions with twist 3, denoted by $\widetilde{g}_{T}$ and
$\widehat{g}_{T}$, which correspond to two different “projections” of the
quark-gluon-quark correlator. An explicit demonstration of our findings is
made for the case of a point-like quark target, which shows that the twist-3
terms can in principle be as large as the twist-2 terms.
In Sec. III we discuss the phenomenology of the WW relation for both the
proton and neutron, and find that the available data from SLAC and Jefferson
Lab indicate a breaking of the WW relation at the level of 15–40% of the size
of $g_{2}$ within the 1-$\sigma$ confidence level. The two twist-3 terms can
be separated by measuring, in addition to $g_{2}$, the function $g_{1T}^{(1)}$
in semi-inclusive DIS, as we outline in Sec. IV. There we explain the
importance of measuring the two twist-3 functions $\widetilde{g}_{T}$ and
$\widehat{g}_{T}$ separately, and the insight which this can bring, for
example, to understanding the physics of quark-gluon-quark correlations
Burkardt (2008), or to determining the QCD evolution kernel for $g_{2}$ and
the large momentum tails of transverse momentum distributions (TMDs).
Finally, in Sec. V we briefly summarize our findings. Some technical details
for the analysis with a non-lightlike Wilson line and the model calculation of
parton correlation functions are presented in the appendices.
## II Theoretical analysis
In this section we set forth the framework for our analysis of the WW relation
by first defining quark-quark correlation functions and examining their most
general Lorentz and Dirac decomposition. This is followed by a discussion of
quark-gluon-quark correlators, and of the Lorentz invariance and equations of
motion relations from which a generalization of the WW relation is derived.
### II.1 Parton correlation functions
The quark-quark correlator for a quark of momentum $k$ in a nucleon with
momentum $P$ and spin $S$ is defined as
$\begin{split}&\Phi^{a}_{ij}(k,P,S;v)=\int\frac{d^{4}\xi}{(2\pi)^{4}}\,e^{ik\cdot\xi}\,\\\
&\quad\times\langle P,S\,|\,\overline{\psi}^{\,a}_{j}(0)\,{\cal
W}^{v}_{(0,\infty)}\,{\cal
W}^{v}_{(\infty,\xi)}\,\psi^{a}_{i}(\xi)\,|\,P,S\rangle\,,\end{split}$ (4)
where the quark fields $\psi^{a}_{i}$ are labeled by the flavor index $a$ and
Dirac index $i$. For ease of notation, the Dirac and flavor indices will be
suppressed in the following. The operator ${\cal W}^{v}_{(0,\infty)}$
represents a Wilson line (or gauge link) from the origin to infinity along the
direction specified by the vector $v$, and is necessary to ensure gauge
invariance of the correlator. The gauge links contain transverse pieces at
infinity Belitsky et al. (2003); Boer et al. (2003) and their precise form
depends on the process Collins (2002); Bomhof et al. (2006). In a covariant
gauge, the dependence of the correlator $\Phi$ on $v$ is evident from the
presence of the Wilson line in the direction conjugate to $v$. In light-cone
gauges the vector $v$ is orthogonal to the gauge field $A$, $v\cdot A=0$, and
the dependence on $v$ appears explicitly only in the gauge field propagators.
In tree-level analyses of semi-inclusive DIS (SIDIS) Mulders and Tangerman
(1996); Bacchetta et al. (2007) or the Drell-Yan process Tangerman and Mulders
(1995); Boer (1999); Arnold et al. (2009) $v$ is identified with the light-
cone vector $n_{-}$, where $n_{-}^{2}=0=n_{+}^{2}$ and $n_{-}\cdot n_{+}=1$,
with $n_{+}$ the corresponding orthogonal light-cone vector proportional to
$P$ (up to mass corrections). However, factorization theorems beyond tree-
level Collins and Soper (1981); Ji et al. (2005); Collins and Metz (2004);
Collins et al. (2008) demand a slightly non-lightlike vector $v$ in order to
regularize light-cone divergences. We leave a more detailed discussion of the
effect of the choice of $v$ to Appendix A and consider $v=n_{-}$ unless
otherwise specified.
The correlator $\Phi$ can be parametrized in terms of structures built from
the four vectors $P$, $S$, $k$ and $v$. Its full decomposition has been
studied in Ref. Goeke et al. (2005) (and further generalized in Ref. Meissner
et al. (2009)). It contains 12 scalar functions $A_{i}$ already known from
Refs. Mulders and Tangerman (1996); Tangerman and Mulders (1994), and 20
scalar functions $B_{i}$ which are multiplied by factors depending explicitly
on $v$, which were first introduced in Ref. Goeke et al. (2003) and called
parton correlation functions (PCFs) in Ref. Collins et al. (2008). For brevity
we consider only those terms of the full decomposition Goeke et al. (2005)
which are necessary for the present analysis,
$\begin{split}\Phi&(k,P,S;v)=MS/\,\gamma_{5}A_{6}+\frac{k\cdot
S}{M}P/\,\gamma_{5}A_{7}\\\ &+\frac{k\cdot
S}{M}k/\,\gamma_{5}A_{8}+M\frac{(S\cdot v)}{(P\cdot
v)}P/\,\gamma_{5}B_{11}+M\frac{(S\cdot v)}{(P\cdot v)}k/\,\gamma_{5}B_{12}\\\
&+M\frac{(k\cdot S)}{(P\cdot v)}v/\,\gamma_{5}B_{13}+M^{3}\frac{(S\cdot
v)}{(P\cdot v)^{2}}v/\,\gamma_{5}B_{14}+\cdots\ ,\end{split}$ (5)
where the nucleon mass $M$ is explicitly included to ensure that all PCFs have
the same mass dimension. (Any other hadronic scale, such as $\Lambda_{\rm
QCD}$, can be chosen, but we follow the choice used in the TMD literature
Mulders and Tangerman (1996).)
The PCFs $A_{i}$ and $B_{i}$ are in principle functions of the scalar products
$P\cdot k$, $k^{2}$, $P\cdot v$, $k\cdot v$ and $v^{2}$. However, because the
correlator $\Phi$ is invariant under the scale transformation $v\to\lambda v$,
where $\lambda$ is a constant, the PCFs can only depend on ratios of the
scalar products, $P\cdot k$, $k^{2}$ and $k\cdot v/P\cdot v$. We therefore
choose the PCFs to depend on the parton virtuality $\tau\equiv k^{2}$, on
$\sigma\equiv 2P\cdot k$, and on the parton momentum fraction $x=k\cdot
n_{-}/P\cdot n_{-}$. We emphasize that the explicit dependence on $x$ is
induced in general by the $v$ dependence of the correlator $\Phi$.
These considerations apply even when the correlator is integrated over the
parton transverse momentum, and in fact the $B_{i}$ terms give contributions
also to standard collinear parton distribution functions (PDFs), such as the
helicity distribution — see Eq. (22) below. However, when the correlator is
fully integrated over $d^{4}k$ the $B_{i}$ no longer contribute; indeed
$\displaystyle\int d^{4}k\,\Phi(k,P,S;v)=\langle
P,S\,|\,\overline{\psi}(0)\,\psi_{i}(0)\,|\,P,S\rangle\ ,$ (6)
and the dependence of the integral on $v$ disappears because ${\cal
W}^{v}_{(0,\infty)}{\cal W}^{v}_{(\infty,0)}=1$.
In TMD factorization the relevant objects are the integrals of $\Phi(k,P,S;v)$
over $k^{-}=k_{\mu}n_{+}^{\mu}$,
$\displaystyle\Phi(x,\bm{k}_{T})=\int
dk^{-}\,\Phi(k,P,S;v)=\int\frac{d\xi^{-}d^{2}\xi_{T}}{(2\pi)^{3}}\,e^{ik\cdot\xi}\,\langle
P,S\,|\,\overline{\psi}(0)\,{\cal W}^{v}_{(0,\infty)}\,{\cal
W}^{v}_{(\infty,\xi)}\,\psi(\xi)\,|\,P,S\rangle\Big{|}_{\xi^{+}=0}\,.$ (7)
It is also useful to define the $\bm{k}_{T}$-integrated correlators
$\displaystyle\begin{split}\Phi(x)&=\int
d^{2}\bm{k}_{T}\,\Phi(x,\bm{k}_{T})=\int\frac{d\xi^{-}}{2\pi}\,e^{ik\cdot\xi}\,\langle
P,S\,|\,\overline{\psi}(0)\,{\cal W}^{v}_{(0,\infty)}\,{\cal
W}^{v}_{(\infty,\xi)}\,\psi(\xi)\,|\,P,S\rangle\Big{|}_{\xi^{+}=\xi_{T}=0}\,\\\
&\stackrel{{\scriptstyle\text{LC}}}{{=}}\int\frac{d\xi^{-}}{2\pi}\,e^{ik\cdot\xi}\,\langle
P,S\,|\,\overline{\psi}(0)\,\psi(\xi)\,|\,P,S\rangle\Big{|}_{\xi^{+}=\xi_{T}=0}\,,\end{split}$
(8) $\displaystyle\begin{split}\Phi_{\partial}^{\alpha}(x)&=\int
d^{2}\bm{k}_{T}\,k_{T}^{\alpha}\Phi(x,\bm{k}_{T})=\int\frac{d\xi^{-}}{2\pi}\,e^{ik\cdot\xi}\,\langle
P,S\,|\,\overline{\psi}(0)\,{\cal
W}^{v}_{(0,\infty)}\,i\partial_{T}^{\alpha}\,{\cal
W}^{v}_{(\infty,\xi)}\,\psi(\xi)\,|\,P,S\rangle\Big{|}_{\xi^{+}=\xi_{T}=0}\,,\\\
&\stackrel{{\scriptstyle\text{LC}}}{{=}}\int\frac{d\xi^{-}}{2\pi}\,e^{ik\cdot\xi}\,\langle
P,S\,|\,\overline{\psi}(0)\,i\partial_{T}^{\alpha}\,\psi(\xi)\,|\,P,S\rangle\Big{|}_{\xi^{+}=\xi_{T}=0}\,.\end{split}$
(9)
where LC refers to the correlators in the light-cone gauge. The correlator
$\Phi_{\partial}^{\alpha}$ actually depends on the detailed form of the Wilson
line, and changes, for example, between the SIDIS and Drell–Yan processes.
However, for our discussion this will not be relevant and we can consider the
average between the correlator for SIDIS and Drell–Yan Boer et al. (2003).
For any correlator, we can introduce the Dirac projections
$\Phi^{[\Gamma]}\equiv\frac{1}{2}\text{Tr}[\Gamma\Phi]\ ,$ (10)
where $\Gamma$ is a matrix in Dirac space. The transverse momentum dependent
parton distribution functions then appear as terms of the general
decomposition of the projections $\Phi^{[\Gamma]}(x,\bm{k}_{T})$, the full
list of which can be found in Refs. Goeke et al. (2005); Bacchetta et al.
(2007). Usually a TMD is defined to have “twist” equal to $n$ if in the
expansion of the correlator it appears at order $(M/P^{+})^{n-2}$, where
$P^{+}=P_{\mu}n_{-}^{\mu}$. In physical observables, TMDs of twist $n$ appear
with a suppression factor $(M/Q)^{n-2}$ compared to twist-2 TMDs. We finally
note that at present TMD factorization for SIDIS has been proven for twist-2
TMDs only Ji et al. (2005), and problems are known to occur at twist 3,
indicating that the formalism may not yet be complete Gamberg et al. (2006);
Bacchetta et al. (2008).
For the following discussion we shall need the definitions of certain TMDs
(note that here and in the following $\alpha$ is restricted to be a transverse
index) Bacchetta et al. (2007)
$\displaystyle\Phi^{[\gamma^{+}\gamma_{5}]}(x,\bm{k}_{T})$
$\displaystyle=S_{L}\,g_{1L}(x,\bm{k}_{T}^{2})+\frac{\bm{k}_{T}\cdot\bm{S}_{T}}{M}\,g_{1T}(x,\bm{k}_{T}^{2})\
,$ (11)
$\displaystyle\begin{split}\Phi^{[\gamma^{\alpha}\gamma_{5}]}(x,\bm{k}_{T})&=\frac{M}{P^{+}}\bigg{[}S_{T}^{\alpha}\,g_{T}(x,\bm{k}_{T}^{2})+S_{L}\,\frac{k_{T}^{\alpha}}{M}\,g_{L}^{\perp}(x,\bm{k}_{T}^{2})\\\
&\quad\quad-\frac{k_{T}^{\alpha}\,k_{T}^{\rho}+\frac{1}{2}\,\bm{k}_{T}^{2}\,g^{\alpha\rho}_{T}}{M^{2}}\,S_{T\rho}\,g_{T}^{\perp}(x,\bm{k}_{T}^{2})\\\
&\quad\quad-\frac{\epsilon_{T}^{\alpha\rho}k_{T\rho}}{M}\,g^{\perp}(x,\bm{k}_{T}^{2})\bigg{]},\end{split}$
(12)
$\displaystyle\begin{split}\Phi^{[i\sigma^{\alpha+}\gamma_{5}]}(x,\bm{k}_{T})&=S_{T}^{\alpha}\,h_{1}(x,\bm{k}_{T}^{2})+S_{L}\,\frac{p_{T}^{\alpha}}{M}\,h_{1L}^{\perp}(x,\bm{k}_{T}^{2})\\\
&\quad-\frac{p_{T}^{\alpha}p_{T}^{\rho}-\frac{1}{2}\,p_{T}^{2}\,g^{\alpha\rho}_{T}}{M^{2}}\,S_{T\rho}\,h_{1T}^{\perp}(x,\bm{k}_{T}^{2})\\\
&\quad-\frac{\epsilon_{T}^{\alpha\rho}p_{T\rho}}{M}\,h_{1}^{\perp}(x,\bm{k}_{T}^{2})\,,\end{split}$
(13)
where $S_{L}=S^{+}M/P^{+}$, and the transverse tensors $g^{\alpha\rho}_{T}$
and $\epsilon_{T}^{\alpha\rho}$ are defined as
$\displaystyle g^{\alpha\rho}_{T}$
$\displaystyle=g^{\alpha\rho}-n_{+}^{\alpha}n_{-}^{\rho}-n_{-}^{\alpha}n_{+}^{\rho}\,,$
(14) $\displaystyle\epsilon_{T}^{\alpha\rho}$
$\displaystyle=\epsilon^{\alpha\rho\beta\sigma}(n_{+})_{\beta}(n_{-})_{\sigma}.$
(15)
For the $\bm{k}_{T}$-integrated distributions, we correspondingly have
$\displaystyle\Phi^{[\gamma^{+}\gamma_{5}]}(x)$
$\displaystyle=S_{L}\,g_{1L}(x)\,,$ (16)
$\displaystyle\Phi^{[i\sigma^{\alpha+}\gamma_{5}]}(x)$
$\displaystyle=S_{T}^{\alpha}\,h_{1}(x)\,,$ (17)
$\displaystyle\Phi^{\alpha[\gamma^{+}\gamma_{5}]}_{\partial}(x)$
$\displaystyle=S_{T}^{\alpha}Mg_{1T}^{(1)}(x)\,,$ (18)
$\displaystyle\Phi^{[\gamma^{\alpha}\gamma_{5}]}(x)$
$\displaystyle=\frac{M}{P^{+}}S_{T}^{\alpha}\,g_{T}(x)\,,$ (19)
where for any TMD $f=f(x,\bm{k}_{T}^{2})$ we define
$\displaystyle f^{(1)}(x,\bm{k}_{T}^{2})$
$\displaystyle=\frac{\bm{k}_{T}^{2}}{2M^{2}}f(x,\bm{k}_{T}^{2})\ ,$ (20)
$\displaystyle f^{(1)}(x)$ $\displaystyle=\int
d^{2}\bm{k}_{T}\,f^{(1)}(x,\bm{k}_{T}^{2})\ .$ (21)
To avoid confusion with the structure function $g_{1}$, here we use the
notation $g_{1L}$ also for the helicity-dependent PDF, contrary to what is
used in some of the TMD literature Bacchetta et al. (2007).
The connection between the TMDs and the $A_{i}$ and $B_{i}$ amplitudes has
been worked out in detail in the Appendix of Ref. Metz et al. (2008) for
$v=n_{-}$. In Appendix A we extend these results to a non-lightlike vector
$v$. We shall not repeat here the calculations but only quote the results
relevant for our discussion, namely
$\displaystyle\begin{split}g_{1L}(x,{\bm{k}}_{T}^{2})&=\int d\sigma
d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\\\
&\quad\times\Bigl{(}-A_{6}-B_{11}-xB_{12})\\\
&\quad\quad-\frac{\sigma-2xM^{2}}{2M^{2}}(A_{7}+xA_{8})\Bigr{)}\,,\end{split}$
(22) $\displaystyle\begin{split}g_{1T}(x,{\bm{k}}_{T}^{2})&=\int d\sigma
d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\\\
&\quad\times\Bigl{(}A_{7}+xA_{8}\Bigr{)}\,,\end{split}$ (23)
$\displaystyle\begin{split}g_{T}(x,{\bm{k}}_{T}^{2})&=\int d\sigma
d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\\\
&\quad\times\Bigl{(}-A_{6}-\frac{\tau-x\sigma+x^{2}M^{2}}{2M^{2}}A_{8}\Bigr{)}\,.\end{split}$
(24)
As anticipated, we see that $B_{i}$ terms appear also in the function
$g_{1L}$, which survives if the correlator is integrated over $\bm{k}_{T}$.
### II.2 Lorentz invariance relations
From the preceding discussion, using the techniques discussed for example in
Ref. Tangerman and Mulders (1994), it is possible to derive the so-called
Lorentz invariance relation (LIR)
$\displaystyle g_{T}(x)$
$\displaystyle=g_{1L}(x)+\frac{d}{dx}\,g_{1T}^{(1)}(x)+\widehat{g}_{T}(x)\,,$
(25)
where the function $\widehat{g}_{T}$ is given by
$\begin{split}\widehat{g}_{T}(x)&=\int d^{2}{\bm{k}}_{T}d\sigma
d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\\\
&\quad\times\Big{[}B_{11}+xB_{12}-\frac{{\bm{k}}_{T}^{2}}{2M^{2}}\Big{(}\frac{\partial
A_{7}}{\partial x}+x\frac{\partial A_{8}}{\partial x}\Big{)}\Big{]}\\\
&\quad+\pi\int d\sigma
d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\,{\bm{k}}_{T}^{2}\\\
&\quad\quad\quad\times\left.\frac{\sigma-2xM^{2}}{2M^{2}}\Bigl{(}A_{7}+xA_{8}\Bigr{)}\right|_{{\bm{k}}_{T}^{2}\to
0}^{{\bm{k}}_{T}^{2}\to\infty}\ .\end{split}$ (26)
The proper operator definition for $\widehat{g}_{T}$ can be traced back to
Ref. Bukhvostov et al. (1983) (see also Belitsky (1997); Kundu and Metz
(2002)), and requires the introduction of the twist-3 quark-gluon-quark
correlator
$\begin{split}&i\Phi_{F}^{\alpha}(x,x^{\prime})=\int\frac{d\xi^{-}d\eta^{-}}{(2\pi)^{2}}\,e^{ik\cdot\xi}\,e^{i(k^{\prime}-k)\cdot\eta}\,\delta_{T}^{\alpha\rho}\\\
&\quad\times\langle P|\overline{\psi}(0)\,{\cal
W}^{v}_{(0,\eta)}\,ig\,F^{+\alpha}(\eta)\,{\cal
W}^{v}_{(\eta,\xi)}\,\psi(\xi)|P\rangle\Big{|}_{\begin{subarray}{c}\xi^{+}=\xi_{T}=0\\\
\eta^{+}=\eta_{T}=0\end{subarray}}\\\
&\stackrel{{\scriptstyle\text{LC}}}{{=}}\int\frac{d\xi^{-}d\eta^{-}}{(2\pi)^{2}}\,e^{ik\cdot\xi}\,e^{i(k^{\prime}-k)\cdot\eta}\\\
&\quad\times\langle
P|\overline{\psi}(0)\,ig\,\partial^{+}_{\eta}A_{T}^{\alpha}(\eta)\,\psi(\xi)|P\rangle\Big{|}_{\begin{subarray}{c}\xi^{+}=\xi_{T}=0\\\
\eta^{+}=\eta_{T}=0\end{subarray}}\ ,\end{split}$ (27)
where $F^{+\alpha}$ is the gluon field strength tensor, $k^{\prime}$ is the
gluon momentum, and $x^{\prime}=k^{\prime}\cdot n_{-}/P\cdot n_{-}$. Note that
this correlator has been discussed in slightly different forms in Refs. Boer
et al. (1998); Kanazawa and Koike (2000); Eguchi et al. (2007); Boer et al.
(2003), for example. It can be expanded in terms of four scalar functions
$G_{F}$, $\widetilde{G}_{F}$, $H_{F}$ and $E_{F}$ according to Boer et al.
(1998); Kanazawa and Koike (2000)
$\begin{split}i\Phi_{F}^{\alpha}(x,x^{\prime})&=\frac{M}{4}\biggl{[}G_{F}(x,x^{\prime})i\epsilon_{T}^{\alpha\rho}S_{T\rho}+\widetilde{G}_{F}(x,x^{\prime})S_{T}^{\alpha}\gamma_{5}\\\
&\quad+H_{F}(x,x^{\prime})\,S_{L}\gamma_{5}\gamma_{T}^{\alpha}+E_{F}(x,x^{\prime})\,\gamma_{T}^{\alpha}\biggr{]}n/\,_{+}\
.\end{split}$ (28)
Hermiticity and parity invariance impose that these functions are real and
either odd or even under the interchange of $x$ and $x^{\prime}$ Kanazawa and
Koike (2000),
$\displaystyle G_{F}(x,x^{\prime})$ $\displaystyle=G_{F}(x^{\prime},x)\,,$
$\displaystyle\widetilde{G}_{F}(x,x^{\prime})$
$\displaystyle=-\widetilde{G}_{F}(x^{\prime},x)\,,$ (29) $\displaystyle
E_{F}(x,x^{\prime})$ $\displaystyle=E_{F}(x^{\prime},x)\,,$ $\displaystyle
H_{F}(x,x^{\prime})$ $\displaystyle=-H_{F}(x^{\prime},x)\,.$ (30)
We can then express the function $\widehat{g}_{T}$ as
$\begin{split}MS_{T}^{\alpha}\,\widehat{g}_{T}(x)&=-\int
dx^{\prime}\,\frac{i\Phi_{F}^{\alpha[\gamma^{+}\gamma_{5}]}(x^{\prime},x)}{(x-x^{\prime})^{2}}\\\
&=MS_{T}^{\alpha}\ {\cal P}\int
dx^{\prime}\,\frac{\widetilde{G}_{F}(x,x^{\prime})/(x-x^{\prime})}{x-x^{\prime}},\end{split}$
(31)
where ${\cal P}$ denotes the principal value integral. (The need for the
principal value was apparently overlooked in Refs. Belitsky (1997); Kundu and
Metz (2002).) The imaginary part arising from the pole at $x=x^{\prime}$
cannot give a contribution to the LIR in Eq. (25), but rather contributes to a
LIR involving the functions $f_{T}$ and $f_{1T}^{\perp(1)}$, which we do not
discuss here. We note that $\widehat{g}_{T}$ is a “pure twist-3” function,
being part of the twist-3 correlator of Eq. (27). Since the integrand in Eq.
(31) is antisymmetric in $x\leftrightarrow x^{\prime}$, one obtains the
nontrivial property
$\int_{0}^{1}dx\,\widehat{g}_{T}(x)=0\ .$ (32)
In some analyses Tangerman and Mulders (1994); Boer and Mulders (1998)
$\widehat{g}_{T}$ was believed to vanish because (i) the $B_{i}$ parton
correlation functions were not taken into account, (ii) the partial
derivatives in Eq. (26) were neglected since an explicit $x$-dependence of the
PCFs is generated only through the additional $v$-dependence, (iii) the
boundary terms like the last terms in (26) were neglected. However, none of
these assumptions is justified, as we show explicitly in a quark-target
perturbative calculation in Appendix B. We can further draw some model-
independent conclusions about the boundary terms by comparing them with the
expression for $g_{1T}$ in Eq. (23). Positivity bounds imply that
$|\bm{k}_{T}^{2}g_{1T}|\leq M|\bm{k}_{T}|f_{1}$ Bacchetta et al. (2000), which
is sufficient to guarantee that the $\bm{k}_{T}^{2}=0$ boundary term indeed
vanishes. However, since $g_{1T}$ behaves as $1/\bm{k}_{T}^{4}$ at large
$\bm{k}_{T}$ Bacchetta et al. (2008), the boundary term at
$\bm{k}_{T}^{2}=\infty$ cannot be neglected.
If $\widehat{g}_{T}$ is nonetheless neglected, it is possible to express the
twist-3 function $g_{T}$ in terms of the twist-2 functions $g_{1L}$ and
$g_{1T}$ Mulders and Tangerman (1996); Tangerman and Mulders (1994). Relations
of this kind have been often mistakenly called Lorentz invariance relations
Mulders and Tangerman (1996); Tangerman and Mulders (1994); Henneman et al.
(2002), but should not be confused with the correct Lorentz invariance
relations such as in Eq. (25).
In the literature, model calculations have been used to argue that the pure
twist-3 terms are not necessarily small Jaffe and Ji (1991); Harindranath and
Zhang (1997). For example, $\widehat{g}_{T}$ can be computed perturbatively in
the quark-target model of Refs. Harindranath and Zhang (1997); Kundu and Metz
(2002). Using Eqs. (38), (40) and (42) of Ref. Kundu and Metz (2002) one finds
$\displaystyle g_{T}(x)-g_{1L}(x)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}C_{F}\ln{\frac{Q^{2}}{\mu^{2}}}\bigl{[}2x-\delta(1-x)\bigr{]}\,,$
(33) $\displaystyle g_{1T}^{(1)}(x)$
$\displaystyle=-\frac{\alpha_{s}}{2\pi}C_{F}\ln{\frac{Q^{2}}{\mu^{2}}}\,x(1-x)\,,$
(34)
where $C_{F}=4/3$, $\mu$ is an infrared cutoff, and from Eq. (25) one has
$\displaystyle\widehat{g}_{T}(x)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}C_{F}\ln{\frac{Q^{2}}{\mu^{2}}}\,\bigl{[}1-\delta(1-x)\bigr{]}\,.$
(35)
From this calculation one can see that $\widehat{g}_{T}$ is comparable to the
size of the other twist-2 functions. Moreover, its lowest moment vanishes, so
that the nontrivial requirement of Eq. (32) is fulfilled. In Appendix C we
confirm the above result (for $x<1$ only) starting directly from the
definition in Eq. (31).
### II.3 Equations of motion relations
The equations of motion (EOM) for quarks, $D/\,\psi=m\psi$, imply further
relations between twist-2 and pure twist-3 functions (namely, between $qq$ and
$qgq$ matrix elements). They are referred to as “equations of motion
relations”, and for the case of interest here are expressed as
$\displaystyle g_{1T}^{(1)}(x)$
$\displaystyle=xg_{T}(x)-x\widetilde{g}_{T}(x)-\frac{m}{M}\,h_{1}(x)\ ,$ (36)
where
$\begin{split}xM&S_{T}^{\sigma}\,\widetilde{g}_{T}(x)={\cal P}\int
dx^{\prime}\,\frac{i\Phi_{F\rho}^{[\gamma^{+}\gamma_{T}^{\sigma}\gamma_{T}^{\rho}\gamma_{5}]}(x^{\prime},x)}{x-x^{\prime}}\\\
&=MS_{T}^{\sigma}\left({\cal P}\int
dx^{\prime}\,\frac{G_{F}(x,x^{\prime})}{2(x^{\prime}-x)}+\int
dx^{\prime}\,\frac{\widetilde{G}_{F}(x,x^{\prime})}{2(x^{\prime}-x)}\right)\
.\end{split}$ (37)
The full list of EOM relations can be found in Ref. Bacchetta et al. (2007).
Using Eq. (36) to eliminate $g_{1T}^{(1)}(x)$ in Eq. (25), one finds the
differential equation
$\displaystyle
x\frac{d}{dx}\left(g_{T}-\widetilde{g}_{T}-\frac{m}{M}\frac{h_{1}}{x}\right)+g_{1L}-\widetilde{g}_{T}-\frac{m}{M}\frac{h_{1}}{x}+\widehat{g}_{T}=0\
.$ (38)
Assuming that the relevant functions are integrable by $\int_{x}^{1}(dy/y)$
and solving for $g_{T}$ one finds
$\displaystyle\begin{split}g_{T}(x)&=\int_{x}^{1}\frac{dy}{y}\Bigl{(}g_{1L}(y)+\widehat{g}_{T}(y)\Bigr{)}\\\
&\quad+\widetilde{g}_{T}^{\star}(x)+\frac{m}{M}(h_{1}/x)^{\star}(x)\
,\end{split}$ (39)
where we have introduced the shorthand notation
$\displaystyle f^{\star}(x)\equiv
f(x)-\int_{x}^{1}\frac{dy}{y}f(y)=-\int_{x}^{1}\frac{dy}{y}\frac{d}{dy}\left[yf(y)\right]\
.$ (40)
Note that if the integrals over $x$ and $y$ can be exchanged, the function $f$
satisfies
$\displaystyle\int_{0}^{1}dx\,f^{\star}(x)=0\ .$ (41)
In general, however, this is not necessarily true, as stressed in Refs. Jaffe
(1990); Jaffe and Ji (1991).
In DIS on a quark-target, $\widetilde{g}_{T}$ can be computed using Eqs. (38)
and (43) of Ref. Kundu and Metz (2002), giving
$\begin{split}xg_{T}(x)-\frac{m}{M}h_{1}(x)&=\frac{\alpha_{s}}{2\pi}C_{F}\ln{\frac{Q^{2}}{\mu^{2}}}\\\
&\quad\times\biggl{[}-x(1-x)+\frac{\delta(1-x)}{2}\biggr{]}\,,\end{split}$
(42)
and using Eq. (36) we obtain
$\displaystyle\widetilde{g}_{T}(x)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}C_{F}\ln{\frac{Q^{2}}{\mu^{2}}}\,\frac{\delta(1-x)}{2}\,.$
(43)
Again we see that the twist-3 function $\widetilde{g}_{T}$ has a size
comparable to that of the other twist-2 functions.
### II.4 Breaking of the Wandzura–Wilczek relation
The hadronic tensor relevant for spin-dependent DIS structure functions is
given by the standard Lorentz decomposition
$\displaystyle\begin{split}&W^{\mu\nu}(P,q)=\frac{1}{P\cdot
q}\varepsilon^{\mu\nu\rho\sigma}q_{\rho}\\\
&\quad\times\Big{[}S_{\sigma}g_{1}(x_{B},Q^{2})+\Big{(}S_{\sigma}-\frac{S\cdot
q}{P\cdot q}\,p_{\sigma}\Big{)}g_{2}(x_{B},Q^{2})\Big{]}\ ,\end{split}$ (44)
where $q$ is the momentum of the exchanged photon and $x_{B}=Q^{2}/(2P\cdot
q)$ is the Bjorken variable. In general the structure functions $g_{1}$ and
$g_{2}$ in Eq. (44) are functions of the physical (external) variable $x_{B}$
and are given by convolutions of the hard $\gamma^{*}$–parton scattering
coefficient functions and the relevant PDFs. At leading order in $\alpha_{s}$,
and including terms up to twist 3, they can be expressed in terms of the
distributions $g_{1L}^{a}$ and $g_{T}^{a}$ (where we now explicitly include
the flavor index $a$) introduced above as Bacchetta et al. (2007)
$\displaystyle g_{1}(x)$
$\displaystyle=\frac{1}{2}\,\sum_{a}e_{a}^{2}\;g_{1L}^{a}(x)\ \,,$ (45)
$\displaystyle g_{1}(x)+g_{2}(x)$
$\displaystyle=\frac{1}{2}\,\sum_{a}e_{a}^{2}\;g_{T}^{a}(x)\,,$ (46)
where for simplicity we have suppressed the $Q^{2}$ dependence. This then
enables the difference between the full $g_{2}$ structure function and the WW
approximation (1) to be written as
$\begin{split}g_{2}&(x)-g_{2}^{\rm WW}(x)\\\
&=\frac{1}{2}\,\sum_{a}e_{a}^{2}\biggl{(}\widetilde{g}_{T}^{a\star}(x)+\frac{m}{M}(h_{1}^{a}/x)^{\star}(x)+\int_{x}^{1}\frac{dy}{y}\widehat{g}_{T}^{a}(y)\Biggr{)}\
,\end{split}$ (47)
which represents the breaking of the WW relation. Note that the right-hand-
side of Eq. (47) contains a quark mass term and two pure twist-3 terms. This
is the main result of our analysis.
From Eq. (41) the $x$ integral of the pure twist-3 function containing
$\widetilde{g}^{a}_{T}$ and the mass term vanish. Using Eq. (32), and assuming
that $\widehat{g}^{a}_{T}$ is regular enough to exchange the $x$ and $y$
integrals, we see that the $\widehat{g}^{a}_{T}$ term also vanishes. This
implies that the above expression for $g_{2}$ satisfies the
Burkhardt–Cottingham sum rule, Eq. (3), which is not in general guaranteed in
the OPE Jaffe (1990); Jaffe and Ji (1991).
To obtain the WW relation one must neglect quark mass terms compared to the
hadron mass (which can be reasonably done for light quarks), and either
neglect both of the pure twist-3 terms, or assume that they cancel each other.
The explicit quark-target perturbative calculations show that such a
cancellation does not take place in general, and that the size of the WW
breaking term can be comparable to the size of $g_{2}^{\rm WW}$,
$\displaystyle\begin{split}&g_{2}^{\rm
WW}(x)=1-\delta(1-x)-\frac{\alpha_{s}}{2\pi}C_{F}\ln{\frac{Q^{2}}{\mu^{2}}}\\\
&\times\biggl{[}-\log\frac{(1-x)^{2}}{x}+\frac{3}{2}\,\delta(1-x)+\frac{2x^{2}}{(1-x)_{+}}+\frac{1}{2}\biggr{]},\end{split}$
(48) $\displaystyle\begin{split}g_{2}(x)-&g_{2}^{\rm
WW}(x)=\delta(1-x)-1+\frac{\alpha_{s}}{2\pi}C_{F}\ln{\frac{Q^{2}}{\mu^{2}}}\\\
&\times\biggl{[}-\log\frac{(1-x)^{2}}{x}+\frac{1}{2}\,\delta(1-x)+\frac{2}{(1-x)_{+}}-\frac{3}{2}\biggr{]}.\end{split}$
(49)
To obtain the above expressions we again made use of the results in Ref. Kundu
and Metz (2002). Note that both $g_{2}^{\rm WW}$ and the total $g_{2}$
structure function in the quark-target model respect the BC sum rule.
## III Constraints from data
It is often stated in the literature (see e.g. Ref. Avakian et al. (2008))
that the WW relation holds experimentally to a good accuracy. While there are
certainly indications that this may indeed be so Anthony et al. (2003);
Amarian et al. (2004), it is important to quantify the degree to which this
relation holds and place limits on the size of its violation. This is the
focus of this section.
We define the experimental WW breaking term $\Delta_{\rm ex}(x_{B})$ as the
difference between the experimental data and $g_{2}^{\rm WW}$,
$\displaystyle\Delta_{\rm
ex}(x_{B},Q^{2})=g_{2}^{\rm{ex}}(x_{B},Q^{2})-g_{2}^{\rm{WW}}(x_{B},Q^{2})\ ,$
(50)
with the Wandzura–Wilczek term computed using the LSS2006 (set 1) fit of the
$g_{1}$ structure function Leader et al. (2007). The fit was performed
including a phenomenological higher-twist term and target mass corrections in
order to extract the pure twist-2 contribution, $g_{1}^{\rm LT}$. Using
parametrizations of $g_{1}$ which do not account for the $1/Q$ power
corrections de Florian et al. (2008); Hirai et al. (2006) would risk
inadvertantly including spurious higher twist contributions when computing the
WW approximation. We will demonstrate the impact of this difference by
comparing our $g_{2}^{\rm WW}$ with $(g_{2}^{\rm WW})^{\prime}$ computed using
the total $g_{1}$ instead of $g_{1}^{\rm LT}$ in Eq. (1).
For proton targets we consider data from the SLAC E142 Abe et al. (1998) and
E155x Anthony et al. (2003) experiments, while for the neutron only the high-
precision data sets from the SLAC E155x Anthony et al. (2003), and Jefferson
Lab E99-117 Zheng et al. (2004) and E01-012 Kramer et al. (2005) experiments,
obtained using 2H or 3He targets, are included. We checked explicitly that
including the lower-precision data sets from Refs. Abe et al. (1997, 1998);
Anthony et al. (1996) does not alter the fit results, except for artificially
lowering the $\chi^{2}$ values due to the much larger errors compared to the
higher-precision data sets. In total, there are 52 data points for the proton
and 18 points for the neutron, which are used separately to fit the WW
breaking term $\Delta$. Systematic errors, when quoted, were added in
quadrature. For the shape of $\Delta$ we choose the form
$\Delta(x_{B},\alpha,\beta)=\alpha(1-x_{B})^{\beta}\bigl{(}(\beta+2)x_{B}-1\bigr{)}\,,$
(51)
which vanishes at $x_{B}=1$, has no divergences at $x_{B}=0$, fulfills the BC
sum rule, and only has a single node. We do not consider its $Q^{2}$ QCD
evolution. The evolution of $g_{2}$ has been studied numerically in Ref.
Stratmann (1993) in the limit of a large number of colors. Most of the data
considered lie in the range 1 GeV${}^{2}\leq Q^{2}\leq$ 10 GeV2 where the
effect of QCD evolution is rather mild, as indicated also by the results of
the E01-012 experiment Kramer et al. (2005).
The goodness of the fit is estimated using the $\chi^{2}$ function
$\chi^{2}=\sum_{i=1}^{N}\frac{\bigl{[}\Delta(x_{Bi})-\Delta_{\rm
ex}(x_{Bi})\bigr{]}^{2}}{\sigma_{\rm ex}^{2}(x_{Bi})}\,.$ (52)
To quantify the size of the breaking term $\Delta$ compared to $g_{2}^{\rm
WW}$ we define, for any interval $[x_{B}^{\rm{min}},x_{B}^{\rm{max}}]$, the
ratio of their quadratic integrals
$\displaystyle
r^{2}=\frac{\int_{y^{\rm{min}}}^{y^{\rm{max}}}dy\,x_{B}^{2}\Delta^{2}(x_{B})}{\int_{y^{\rm{min}}}^{y^{\rm{max}}}dy\,x_{B}^{2}g_{2}^{2}(x_{B})}\
,$ (53)
with $y=\log(x_{B})$. The value of $r$ is a good indicator of the relative
magnitude of $\Delta$ and $g_{2}$, which change sign as a function of $x_{B}$.
In practice we compute $r$ at the average kinematics of the E155 experiment
Anthony et al. (2003). For the proton, we consider three intervals: the entire
measured $x_{B}$ range, [0.02,1]; the low-$x_{B}$ region, [0.02,0.15]; and the
high-$x_{B}$ region, [0.15,1]. For the neutron, due to the limited statistical
significance of the low-$x_{B}$ data, we limit ourselves to quoting the value
of $r$ for the large-$x_{B}$ region, [0.15,1].
| proton | $\chi^{2}$/d.o.f. | $r_{\text{tot}}$ | $r_{\text{low}}$ | $r_{\text{hi}}$
---|---|---|---|---|---
(I) | $\Delta$ | = 0 | 1.22 | | |
(II) | $\Delta$ | = $\alpha(1-x_{B})^{\beta}\bigl{(}(\beta+2)x_{B}-1\bigr{)}$ | | | |
| $\alpha$ | = $0.13\pm 0.05$ | | | |
| $\beta$ | = $4.4\pm 1.0$ | 1.05 | 15–32% | 18–36% | 14–31%
| neutron | | | |
(I) | $\Delta$ | = 0 | 1.66 | | |
(II) | $\Delta$ | = $\alpha(1-x_{B})^{\beta}\bigl{(}(\beta+2)x_{B}-1\bigr{)}$ | | | |
| $\alpha$ | = $0.64\pm 0.92$ | | | |
| $\beta$ | = $24\pm 10$ | 1.11 | | | 18–40%
Table 1: Results of the 1-parameter fits of the WW breaking term $\Delta$ for
different choices of its functional form. The value $r$ of the relative size
of the breaking term is computed for three regions of $x_{B}$: the entire
measured $x_{B}$ range, [0.02,1]; the low-$x_{B}$ region, [0.02,0.15]; and the
high-$x_{B}$ region, [0.15,1]. See text for further details.
Figure 1: Top panels: Experimental proton and neutron $g_{2}$ structure
functions compared to $g_{2}^{\rm WW}$. The crosses represent $g_{2}^{\rm WW}$
computed at the experimental kinematics, while the solid lines are $g_{2}^{\rm
WW}$ computed at the average $Q^{2}$ of the E155x experiment. Data points for
the proton target Abe et al. (1998); Anthony et al. (2003) have been slightly
shifted in $x_{B}$ for clarity. For the neutron only the high-precision data
from Anthony et al. (2003); Zheng et al. (2004); Kramer et al. (2005) are
included. Bottom panels: The WW-breaking term $\Delta$ fitted to $\Delta_{\rm
ex}$ computed using the LSS2006 $g_{!}^{\rm LT}$ (hashed region). The dashed
line represents $g_{2}^{\rm WW}-(g_{2}^{\rm WW})^{\prime}$, the spurious HT
contribution to $\Delta$ that would be obtained using the total $g_{1}$ to
compute $\Delta_{\rm ex}$.
The results of the fits are presented in Table 1 and Figure 1. The proton fit
displays a positive WW breaking at large-$x_{B}$ and a negative breaking at
small-$x_{B}$. The size of the breaking term is typically 15–35% of the size
of $g_{2}$ (see the $r$ values in Table 1). The neutron fit is completely
dominated by the high-precision E01-012 data, which are concentrated on a very
limited $x_{B}$ range; it clearly indicates a 18–40% breaking of the WW
relation at high $x_{B}$, but cannot be used to conclude much at lower $x_{B}$
values. A striking feature of the proton WW-breaking term in Fig. 1 is that it
is comparable in size and opposite in sign to $g_{2}^{\rm WW}-(g_{2}^{\rm
WW})^{\prime}$. It is essential, therefore, to use fits of $g_{1}$ that
subtract higher twist terms, which would otherwise largely cancel the proton
WW-breaking term and obscure the violation of the WW relation. In the case of
the neutron one would generally obtain an enhancement of the WW-breaking term,
although the experimental uncertainties there are considerably larger.
In summary, we have found that the experimental data are consistent with a
substantial breaking of the WW relation (2). Previous analyses have verified
the WW relation only qualitatively, and using parametrizations which do not
subtract higher twist terms in $g_{1}$. The present analysis clearly
demonstrates that this can give the misleading impression that the WW relation
holds to much better accuracy than it does in more complete analyses where the
higher twist corrections have been consistently taken into account. More data
are certainly needed to pin down the breaking of the WW relation to higher
precision. New data are expected soon from the HERMES Collaboration and from
the d2n (E06-014) and SANE (E07-003) experiments at Jefferson Lab JLab
experiment E07-002 , S. Choi, M. Jones, Z.-E. Meziani and O. Rondon
(spokespersons)() (SANE); JLab experiment E06-014 , S. Choi, X. Jiang, Z.-E.
Meziani and B. Sawatzky (spokespersons)() (d2n).
## IV Toward a deeper understanding of quark-gluon-quark correlations
In the past, since the LIR-breaking $\widehat{g}_{T}$ term was not considered
in Eq. (47) and the quark-mass term with $h_{1}$ was neglected, the breaking
of the WW relation was considered to be a direct measurement of the pure
twist-3 term $\widetilde{g}_{T}$. The presumed experimental validity of the WW
relation was therefore taken as evidence that $\widetilde{g}_{T}$ is small.
This observation was then generalized to assume that all pure twist-3 terms
are small. In contrast, the present analysis shows that, precisely due to the
presence of $\widehat{g}_{T}$, the measurement of the breaking of the WW
relation does not provide information on a single pure twist-3 matrix element.
Even if in future the WW relation were to be found to be satisfied to greater
accuracy than the present data suggest, one could only conclude that the sum
of the terms in (47) is small,
$\displaystyle\sum_{a}e_{a}^{2}\biggl{(}-\widetilde{g}_{T}^{a}(x)+\int_{x}^{1}\frac{dy}{y}\Bigl{(}\widehat{g}_{T}^{a}(y)+\widetilde{g}_{T}^{a}(y)\Bigr{)}\biggr{)}\approx
0\ .$ (54)
This can occur either because $\widehat{g}_{T}^{a}$ and
$\widetilde{g}_{T}^{a}$ are both small, or because they (accidentally) cancel
each other. No information can be obtained on the size of the twist-3 quark-
gluon-quark term $\widetilde{g}_{T}$ from the experimental data on $g_{2}$
alone. Note that these results were essentially already obtained in Ref. Metz
et al. (2008). In that work, however, the authors considered the WW breaking
to be small and assumed that $\widetilde{g}_{T}^{a}$ was small (which we argue
is not necessarily the case), concluding that $\widehat{g}_{T}^{a}$ is also
small.
Of course it is desirable to test our conclusions empirically. A reliable way
to investigate $\widetilde{g}_{T}$ experimentally is through measurement of
the function $g_{1T}^{(1)}$. This function is accessible in semi-inclusive
deep inelastic scattering with transversely polarized targets and
longitudinally polarized lepton beams (see, e.g., the second line of Tab. IV
in Ref. Boer and Mulders (1998)). Preliminary data related to this function
have been presented by the COMPASS Collaboration Parsamyan (2008) and more are
expected from the HERMES Collaboration and from the E06-010 experiment at
Jefferson Lab JLab experiment E06-010/E06-011, J.-P. Chen, E. Cisbani, H. Gao,
X. Jiang, J.-C. Peng, spokespersons . Using the EOM relation (36) and assuming
$m=0$, one obtains
$\displaystyle x\widetilde{g}_{T}(x)$
$\displaystyle=xg_{T}(x)-g_{1T}^{(1)}(x)\ .$ (55)
In combination with the measurement of the WW breaking, this can be used to
determine the size of twist-3 function $\widehat{g}_{T}$. (Alternatively, one
can use the LIR (25).)
The importance of separately studying $\widetilde{g}_{T}$ and
$\widehat{g}_{T}$ resides in the fact that these are projections of different
combinations of the twist-3 functions $G_{F}(x,x^{\prime})$ and
$\widetilde{G}_{F}(x,x^{\prime})$. As with all other terms in the
decomposition of the quark-gluon-quark correlator in Eq. (28), these functions
are involved in the evolution equation of twist-3 collinear PDFs Balitsky and
Braun (1989); Belitsky and Mueller (1997), in the evolution of the transverse
moments of the TMDs Kang and Qiu (2009); Vogelsang and Yuan (2009), in the
calculation of processes at high transverse momentum Eguchi et al. (2007), and
in the calculation of the high transverse momentum tails of TMDs Ji et al.
(2006); Koike et al. (2008). Ultimately, through a global study of all of
these observables, one could simultaneously obtain better knowledge of twist-3
collinear functions and twist-2 TMDs, and at the same time test the validity
of the formalism. Gathering as much information as one can on the quark-gluon-
quark correlator is essential to reach this goal. The separation of the
functions $\widetilde{g}_{T}$ and $\widehat{g}_{T}$ is an important first step
in this direction.
## V Conclusions
In this analysis we have shown that the Wandzura–Wilczek relation for the
$g_{2}$ structure function is violated by a quark mass term, and two distinct
pure twist-3 contributions, containing the parton distribution functions
$\widehat{g}_{T}$ and $\widetilde{g}_{T}$. As evident from their definitions
in Eqs. (31) and (37) respectively, these correspond to two different
projections of the general quark-gluon-quark correlator in Eq. (27). Their
measurement can give unique and complementary information on twist-3 physics.
The two twist-3 functions have some interesting connections with the formalism
of transverse momentum distributions. One of them is involved in the equation-
of-motion relation expressed in Eq. (36), while the other is involved in the
Lorentz invariance relation in Eq. (25). Both relations contain the same
moment of the transverse momentum distribution $g_{1T}$. From the theoretical
point of view, this is another intriguing example of the interplay between
transverse momentum distributions and (collinear) twist-3 distributions. From
the phenomenological point of view, this means that a measurement of the
function $g_{1T}$ in semi-inclusive DIS in principle allows one to separately
measure $\widehat{g}_{T}$ and $\widetilde{g}_{T}$.
Although the Wandzura–Wilczek relation is often used to simplify the treatment
of twist-3 and TMD physics, we stress that there are no compelling theoretical
or phenomenological grounds supporting its validity. In fact, using the
experimental information currently available, we were able to provide a
quantitative assessment of the violation of the Wandzura–Wilczek relation.
Assuming a simple functional form for the WW-breaking term, we found that it
can be as large as 15–40% at the 1-$\sigma$ confidence level.
As new data become available, it should be possible to better pin down the
violation of the Wandzura–Wilczek relation and measure the transverse momentum
distribution $g_{1T}$ in semi-inclusive DIS. This will offer us a deeper look
into the physics of quark-gluon-quark correlations and its connection to
transverse momentum distributions.
###### Acknowledgements.
We are grateful to M. Burkardt and A. Metz for helpful discussions. This work
was supported by the DOE contract No. DE-AC05-06OR23177, under which Jefferson
Science Associates, LLC operates Jefferson Lab, and NSF award No. 0653508.
## Appendix A TMDs with a non-lightlike Wilson line direction
Factorization theorems beyond tree-level Collins et al. (1988); Ji et al.
(2005, 2004); Collins and Metz (2004); Collins et al. (2008) demand a slightly
non-lightlike vector $v$ in order to regularize the lightcone (or rapidity)
divergences Collins (2003, 2008). In Ref. Ji et al. (2005) the Wilson line
vector is chosen to be timelike and a parameter $\zeta^{2}=4(P\cdot
v)^{2}/v^{2}$ is used as a regulator, with the requirement that $\zeta^{2}\gg
M^{2},\bm{k}_{T}^{2}$. In other articles in the literature $v$ has been chosen
to be spacelike Collins and Soper (1981).
In addition to $k\cdot P$, $k^{2}$, $P\cdot v$ and $k\cdot v$, the PCFs
$A_{i}$ and $B_{i}$ can now in principle depend also on $v^{2}$. We can derive
the following relation between the invariants
$\frac{k\cdot v}{P\cdot v}=ax+\frac{2\sigma}{\zeta^{2}(1+a)}\,,$ (56)
with $a=\sqrt{1-4M^{2}/\zeta^{2}}$. Neglecting terms of order
$M^{2}/\zeta^{2}$ and $\sigma/\zeta^{2}$, the above expression reduces to $x$.
We therefore conclude that the PCFs depend on $\sigma,\tau,x$ and additionally
on $\zeta^{2}$. To be precise, the definition of parton correlation functions
in Collins et al. (2008) involves an additional soft factor which is not
included in the correlator $\Phi$. The inclusion of the soft factor leads to
an additional dependence on a gluon rapidity parameter. However, we leave this
soft factor aside since it plays no role in our subsequent discussion.
The expressions for the TMDs in Eqs. (22), (23) and (24) then become
$\begin{split}g_{1L}(x,&{\bm{k}}_{T}^{2},\zeta^{2})=\int d\sigma
d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\\\
&\quad\times\Bigl{[}-A_{6}-a\Bigl{(}B_{11}+xB_{12}+\frac{4M^{2}}{\zeta^{2}(1+a)}B_{14}\Bigr{)}\\\
&\quad\quad-\frac{\sigma-2xM^{2}}{2M^{2}}\Bigl{(}A_{7}+xA_{8}+\frac{4M^{2}}{\zeta^{2}(1+a)}B_{13}\Bigr{)}\Bigr{]},\end{split}$
(57) $\displaystyle\begin{split}g_{1T}(x,{\bm{k}}_{T}^{2},\zeta^{2})&=\int
d\sigma d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\\\
&\quad\times\Bigl{[}A_{7}+xA_{8}+\frac{4M^{2}}{\zeta^{2}(1+a)}B_{13}\Bigr{]},\end{split}$
(58) $\displaystyle\begin{split}g_{T}(x,{\bm{k}}_{T}^{2},\zeta^{2})&=\int
d\sigma d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\\\
&\quad\times\Bigl{[}-A_{6}-\frac{\tau-x\sigma+x^{2}M^{2}}{2M^{2}}A_{8}\Bigr{]},\end{split}$
(59)
The full expression for $\widehat{g}_{T}$ which generalizes Eq. (26) then
becomes
$\begin{split}\widehat{g}&{}_{T}(x)=\int d^{2}{\bm{k}}_{T}\,d\sigma
d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\\\
&\times\Big{[}B_{11}+xB_{12}+\frac{4M^{2}}{\zeta^{2}(1+a)}B_{14}\\\
&\quad-\frac{{\bm{k}}_{T}^{2}}{2M^{2}}\Big{(}\frac{\partial A_{7}}{\partial
x}+x\frac{\partial A_{8}}{\partial
x}+\frac{4M^{2}}{\zeta^{2}(1+a)}\frac{\partial B_{13}}{\partial
x}\Big{)}\Big{]}\\\ &+\pi\int d\sigma
d\tau\,\delta(\tau-x\sigma+x^{2}M^{2}+{\bm{k}}_{T}^{2})\,{\bm{k}}_{T}^{2}\\\
&\quad\times\frac{\sigma-2xM^{2}}{2M^{2}}\Bigl{(}A_{7}+xA_{8}+\frac{4M^{2}}{\zeta^{2}(1+a)}B_{13})\Bigr{)}\Big{|}_{{\bm{k}}_{T}^{2}\rightarrow
0}^{{\bm{k}}_{T}^{2}\rightarrow\infty}.\end{split}$ (60)
## Appendix B Parton correlation functions for a quark target
In this Appendix we compute the parton correlation functions relevant for our
discussion of the WW relation for the case of a point-like quark target. The
calculations are performed in the first non-trivial order in perturbative QCD
(ı.e., at order $\alpha_{s}$) Harindranath and Zhang (1997); Kundu and Metz
(2002). To this end we insert a complete set of intermediate states into Eq.
(LABEL:e:corr1). To order $\alpha_{s}$, only the vacuum state and a one-gluon
state are relevant. The involved Feynman diagrams are shown in Fig. 2 (real
gluon contributions) and Fig. 3 (virtual gluon contributions).
Figure 2: Diagrams in the quark-target calculation involving only real gluons.
The Hermitean conjugate diagrams, which are not shown, are also taken into
account in the calculation.
Figure 3: As in Fig. 2 but for diagrams involving virtual gluons.
The correlator may be written as
$\begin{split}\Phi_{ij}(k,P,S;v)&=\delta^{(4)}(P-k)\Phi_{ij}^{\rm{vir}}(m^{2},\lambda^{2},\zeta^{2},\mu_{R}^{2})\\\
&\quad+\Phi_{ij}^{\rm{real}}(k,P,S;v)\,,\end{split}$ (61)
where $\Phi^{\rm{vir}}$ denotes the contributions from the vacuum intermediate
state. Its kinematics is totally determined by the four-dimensional delta-
function $\delta^{(4)}(P-k)$ and depends only on the quark mass $m$, with a
small gluon mass $\lambda$ serving here as an infrared regulator, and the
parameter $\zeta^{2}=4(P\cdot v)^{2}/v^{2}$ which regulates lightcone
divergences. By applying a renormalization procedure we can subtract ultra-
violet divergences in $\Phi^{\rm vir}$, which introduces a dependence on the
renormalization point $\mu_{R}^{2}$. The virtual corrections can be written as
$\begin{split}\Phi_{ij}^{\rm{vir}}(k,P,S;v)&=\delta^{(4)}(P-k)\langle
P,S,d|\,\bar{\psi}_{j}(0)\,{\cal W}^{v}_{(0,\infty)}\,|0\rangle\\\
&\quad\times\langle 0|\,{\cal
W}^{v}_{(\infty,0)}\,\psi_{i}(0)\,|P,S,d\rangle\,,\end{split}$ (62)
where the incoming on-shell quark is described by the state $|P,S,d\rangle$,
with $d$ a color index of the quark in the fundamental SU(3) representation.
For the sake of brevity we will omit the explicit dependence on and summation
over the color indices in the following. Since we work in Feynman gauge,
possible contributions from gauge links at lightcone infinity are irrelevant
Belitsky et al. (2003).
The second contribution in Eq. (61) is generated by one gluon in the
intermediate state. To order $\alpha_{s}$ it is given by
$\begin{split}\Phi_{ij}^{\rm{real}}&(k,P,S;v)=\frac{1}{(2\pi)3}\sum_{\sigma,\beta}\delta^{+}((P-k)^{2}-\lambda^{2})\\\
&\quad\times\overline{M}^{\sigma,\beta}_{j}(k,P,S;v)\,M^{\sigma,\beta}_{i}(k,P,S;v)\,,\end{split}$
(63)
with $\overline{M}\equiv M^{\dagger}\gamma^{0}$,
$\delta^{+}(a^{2})\equiv\delta(a^{2})\Theta(a^{0})$, $\sigma$ denotes the
polarization of the gluon in the intermediate state, and $\beta$ is its color
index in the adjoint representation of SU(3). The matrix element $M$ is then
represented by
$\begin{split}M^{\sigma,\beta}_{i}&(k,P,S;v)=\langle
P-k,\sigma,\beta|\psi_{i}(0)|P,S,d\rangle\\\
&+ig\int_{0}^{\infty}d\lambda\,\langle P-k,\sigma,\beta|v\cdot A(\lambda
v)\,\psi_{i}(0)|P,S,d\rangle\,,\end{split}$ (64)
where $|P-k,\sigma,\beta\rangle$ denotes the intermediate gluon state with a
color index $\beta$. The leading perturbative contribution in $\alpha_{s}$ to
the matrix element $M$ gives
$\begin{split}M_{i}^{\sigma,\beta}&(k,P,S;v)=-gt^{\beta}\biggl{(}\frac{(k/\,+m)\varepsilon/\,_{\sigma}^{*}(P-k)}{[k^{2}-m^{2}+i\epsilon]}\\\
&\quad+\frac{v\cdot\varepsilon^{*}_{\sigma}(P-k)}{[v\cdot(P-k)+i\epsilon]}\biggr{)}_{il}u_{l}(P,S)\,,\end{split}$
(65)
where $\varepsilon(P-k)$ denotes the gluon polarization vector and $u$ is the
quark spinor. The color flow is given by the color matrix $t^{\beta}$ in the
fundamental representation. Inserting (65) into (63) then yields
$\begin{split}\Phi^{\rm{real}}_{ij}&(k,P,S;v)=-\frac{\alpha_{s}}{(2\pi)^{2}}C_{F}\delta^{+}((P-k)^{2}-\lambda^{2})\\\
&\times\Bigg{[}\frac{(k/\,+m)\gamma_{\mu}(P/\,+m)(1+\gamma_{5}S/\,)\gamma^{\mu}(k/\,+m)}{[k^{2}-m^{2}+i\epsilon][k^{2}-m^{2}-i\epsilon]}\\\
&\quad+\frac{(P/\,+m)(1+\gamma_{5}S/\,)v/\,(k/\,+m)}{[k^{2}-m^{2}-i\epsilon][v\cdot(P-k)+i\epsilon]}\\\
&\quad+\frac{(k/\,+m)v/\,(P/\,+m)(1+\gamma_{5}S/\,)}{[k^{2}-m^{2}+i\epsilon][v\cdot(P-k)-i\epsilon]}\\\
&\quad+\frac{v^{2}(P/\,+m)(1+\gamma_{5}S/\,)}{[v\cdot(P-k)+i\epsilon][v\cdot(P-k)-i\epsilon]}\Bigg{]}_{ij}\,.\end{split}$
(66)
The various parton correlation functions in Eq. (5) can be extracted from Eq.
(66) by decomposing the numerators in terms of the basis matrices $1$,
$\gamma_{5}$, $\gamma^{\mu}$, $\gamma^{\mu}\gamma_{5}$ and $\sigma^{\mu\nu}$.
In this way we obtain expressions for parton correlation functions at leading
order in $\alpha_{s}$ for a quark target. In the following we list only the
PCFs $A_{6-8}$ and $B_{11-14}$ which are relevant for the discussion of the
Wandzura–Wilczek relation, cf. Eqs. (22)–(24). Setting
$a=\sqrt{\smash[b]{1-4m^{2}/\zeta^{2}}}$, we find (to order $\alpha_{s}$)
$\begin{split}&A^{\rm{real}}_{6}(\tau,\sigma,x,\zeta^{2})=\frac{C_{F}\alpha_{s}}{2\pi^{2}}\delta^{+}(\tau-\sigma+m^{2}-\lambda^{2})\\\
&\quad\times\Biggl{[}\frac{\tau+m^{2}}{\bigl{(}\tau-m^{2}\bigr{)}^{2}}+\frac{(1+a)(1+ax)+2\sigma/\zeta^{2}}{\bigl{[}\tau-m^{2}\bigr{]}\bigl{[}(1+a)(1-ax)-2\sigma/\zeta^{2}\bigr{]}}\\\
&\quad+\frac{2(1+a)^{2}}{\bigl{[}(1-ax)^{2}(1+a)^{2}\zeta^{2}-4\sigma(1-ax)(1+a)+4\sigma^{2}/\zeta^{2}\bigr{]}}\Biggr{]},\end{split}$
(67)
$\displaystyle A^{\rm{real}}_{7}$ $\displaystyle(\tau,\sigma,x,\zeta^{2})=0,$
(68)
$\displaystyle\begin{split}A^{\rm{real}}_{8}&(\tau,\sigma,x,\zeta^{2})=\frac{C_{F}\alpha_{s}}{2\pi^{2}}\delta^{+}(\tau-\sigma+m^{2}-\lambda^{2})\\\
&\times\Biggl{[}\frac{-2m^{2}}{\bigl{(}\tau-m^{2}\bigr{)}^{2}}\Biggr{]},\end{split}$
(69)
$\displaystyle\begin{split}B^{\rm{real}}_{11}&(\tau,\sigma,x,\zeta^{2})=\frac{C_{F}\alpha_{s}}{2\pi^{2}}\delta^{+}(\tau-\sigma+m^{2}-\lambda^{2})\\\
&\times\Biggl{[}\frac{-(1+a)}{\bigl{[}\tau-m^{2}\bigr{]}\bigl{[}(1+a)(1-ax)-2\sigma/\zeta^{2}\bigr{]}}\Biggr{]},\end{split}$
(70)
$\displaystyle\begin{split}B^{\rm{real}}_{12}&(\tau,\sigma,x,\zeta^{2})=\frac{C_{F}\alpha_{s}}{2\pi^{2}}\delta^{+}(\tau-\sigma+m^{2}-\lambda^{2})\\\
&\times\Biggl{[}\frac{(1+a)}{\bigl{[}\tau-m^{2}\bigr{]}\bigl{[}(1+a)(1-ax)-2\sigma/\zeta^{2}\bigr{]}}\Biggr{]},\end{split}$
(71)
$\displaystyle\begin{split}B^{\rm{real}}_{13}&(\tau,\sigma,x,\zeta^{2})=\frac{C_{F}\alpha_{s}}{2\pi^{2}}\delta^{+}(\tau-\sigma+m^{2}-\lambda^{2})\\\
&\times\Biggl{[}\frac{-(1+a)}{\bigl{[}\tau-m^{2}\bigr{]}\bigl{[}(1+a)(1-ax)-2\sigma/\zeta^{2}\bigr{]}}\Biggr{]},\end{split}$
(72) $\displaystyle B^{\rm{real}}_{14}$
$\displaystyle(\tau,\sigma,x,\zeta^{2})=0\,.$ (73)
These results demonstrate that all terms in Eq. (60) contribute to generate a
nonzero $\widehat{g}_{T}$ since (i) the $B_{i}$ terms are nonzero, (ii) the
PCFs can depend explicitly on $x$, and (iii) the boundary term at
$\bm{k}_{T}^{2}=\infty$ cannot be neglected.
## Appendix C Quark target TMDs and PDFs at $x<1$
We are now in a position to calculate the TMDs for a quark target defined in
Eqs. (57)–(59), their $\bm{k}_{T}$-integrals appearing in the LIR of Eq. (25),
and the function $\widehat{g}_{T}$ as defined in Eq. (60). Similar
calculations have been performed in Harindranath and Zhang (1997); Kundu and
Metz (2002); Ji et al. (2005); Schlegel and Metz (2004); Schlegel et al.
(2004). Without entering into details, we note that the light-cone divergences
occurring for $\zeta\to\infty$ can be moved to $x=1$, introducing the well-
known “plus” distribution Ji et al. (2005); Bacchetta et al. (2008). If we
restrict ourselves to the region $x<1$, the results are free of light-cone
divergences and do not depend on $\zeta$. In this region we can use either
Eqs. (57)–(59) or (22)–(24). The resulting functions are then given by
$\displaystyle\begin{split}&g_{1L}(x<1,{\bm{k}}^{2}_{T})=\frac{2C_{F}\alpha_{s}}{(2\pi)^{2}}\frac{1}{{\bm{k}}^{2}_{T}+x\lambda^{2}+(1-x)^{2}m^{2}}\\\
&\times\biggl{[}1-x-\frac{2(1-x)(1-x(1-x))m^{2}}{{\bm{k}}^{2}_{T}+x\lambda^{2}+(1-x)^{2}m^{2}}+\frac{2x}{(1-x)_{+}}\biggr{]},\end{split}$
(74) $\displaystyle
g_{1T}(x<1,{\bm{k}}^{2}_{T})=-\frac{2C_{F}\alpha_{s}}{(2\pi)^{2}}\frac{2x(1-x)m^{2}}{({\bm{k}}^{2}_{T}+x\lambda^{2}+(1-x)^{2}m^{2})^{2}},$
(75)
$\displaystyle\begin{split}&g_{T}(x<1,{\bm{k}}^{2}_{T})=\frac{2C_{F}\alpha_{s}}{(2\pi)^{2}}\frac{1}{{\bm{k}}^{2}_{T}+x\lambda^{2}+(1-x)^{2}m^{2}}\\\
&\quad\times\biggl{[}x-\frac{(1-x)^{2}(1+x)m^{2}}{{\bm{k}}^{2}_{T}+x\lambda^{2}+(1-x)^{2}m^{2}}+\frac{1+x}{(1-x)_{+}}\biggr{]}.\end{split}$
(76)
When working with non-lightlike Wilson lines, it is not clear how to obtain
the collinear parton distribution functions upon integration over the
transverse momentum Ji et al. (2005). However, at the one-loop level these
subtleties are relevant only at $x=1$. Since we restrict ourselves to the
region $x<1$, we can safely compute collinear PDFs through
$\bm{k}_{T}$-integration. For simplicity we choose an upper boundary $Q$ for
the $\bm{k}_{T}$-integration, and shift quark mass effects into the finite
part by introducing an arbitrary infrared cutoff $\mu$ in order to obtain
agreement with the results of Refs. Harindranath and Zhang (1997); Kundu and
Metz (2002). The divergent parts of the parton distributions, i.e., the terms
including the upper cutoff $Q$, are given by
$\displaystyle g_{1L}(x<1)$
$\displaystyle=\frac{\alpha_{s}C_{F}}{2\pi}\,\frac{1+x^{2}}{(1-x)_{+}}\ln\frac{Q^{2}}{\mu^{2}},$
(77) $\displaystyle g_{T}(x<1)$
$\displaystyle=\frac{\alpha_{s}C_{F}}{2\pi}\,\frac{1+2x-x^{2}}{(1-x)_{+}}\ln\frac{Q^{2}}{\mu^{2}},$
(78) $\displaystyle g_{1T}^{(1)}(x<1)$
$\displaystyle=-\frac{\alpha_{s}C_{F}}{2\pi}x(1-x)\ln\frac{Q^{2}}{\mu^{2}}.$
(79)
These results have appeared earlier in Refs. Harindranath and Zhang (1997);
Kundu and Metz (2002); Ji et al. (2005); Schlegel and Metz (2004); Schlegel et
al. (2004), but have been derived here for the first time starting from the
PCFs.
For $\widehat{g}_{T}$ at $x<1$, using either Eq. (60) or Eq. (26) we obtain
$\widehat{g}_{T}(x<1)=\frac{\alpha_{s}C_{F}}{2\pi}\,\ln\frac{Q^{2}}{\mu^{2}}\,,$
(80)
confirming the result in Eq. (35), which was not obtained directly but rather
using the LIR relation Eq. (25).
## References
* Bass (2005) S. D. Bass, Rev. Mod. Phys. 77, 1257 (2005).
* Kuhn et al. (2009) S. E. Kuhn, J. P. Chen, and E. Leader, Prog. Part. Nucl. Phys. 63, 1 (2009).
* Burkardt et al. (2008) M. Burkardt, A. Miller, and W. D. Nowak (2008), arXiv:0812.2208 [hep-ph].
* Myhrer and Thomas (2008) F. Myhrer and A. W. Thomas, Phys. Lett. B663, 302 (2008).
* Morreale (2009) A. Morreale (PHENIX) (2009), arXiv:0905.2632 [nucl-ex].
* Alekseev et al. (2009) Alekseev et al. (COMPASS) (2009), arXiv:0904.3209 [hep-ex].
* Wandzura and Wilczek (1977) S. Wandzura and F. Wilczek, Phys. Lett. B72, 195 (1977).
* Burkhardt and Cottingham (1970) H. Burkhardt and W. N. Cottingham, Annals Phys. 56, 453 (1970).
* Anthony et al. (2003) P. L. Anthony et al. (E155), Phys. Lett. B553, 18 (2003).
* Amarian et al. (2004) M. Amarian et al. (Jefferson Lab E94-010), Phys. Rev. Lett. 92, 022301 (2004).
* Jaffe and Ji (1991) R. L. Jaffe and X. Ji, Phys. Rev. D43, 724 (1991).
* Jaffe (1990) R. L. Jaffe, Comments Nucl. Part. Phys. 19, 239 (1990).
* Accardi et al. (2009) A. Accardi, A. Bacchetta, and M. Schlegel (2009), arXiv:0905.3118 [hep-ph].
* Burkardt (2008) M. Burkardt (2008), arXiv:0810.3589 [hep-ph].
* Boer et al. (2003) D. Boer, P. J. Mulders, and F. Pijlman, Nucl. Phys. B667, 201 (2003).
* Belitsky et al. (2003) A. V. Belitsky, X. Ji, and F. Yuan, Nucl. Phys. B656, 165 (2003).
* Collins (2002) J. C. Collins, Phys. Lett. B536, 43 (2002).
* Bomhof et al. (2006) C. J. Bomhof, P. J. Mulders, and F. Pijlman, Eur. Phys. J. C47, 147 (2006).
* Mulders and Tangerman (1996) P. J. Mulders and R. D. Tangerman, Nucl. Phys. B461, 197 (1996), erratum-ibid. B484 (1997) 538.
* Bacchetta et al. (2007) A. Bacchetta, M. Diehl, K. Goeke, A. Metz, P. J. Mulders, and M. Schlegel, JHEP 02, 093 (2007).
* Tangerman and Mulders (1995) R. D. Tangerman and P. J. Mulders, Phys. Rev. D51, 3357 (1995).
* Boer (1999) D. Boer, Phys. Rev. D60, 014012 (1999).
* Arnold et al. (2009) S. Arnold, A. Metz, and M. Schlegel, Phys. Rev. D79, 034005 (2009).
* Ji et al. (2005) X. Ji, J.-P. Ma, and F. Yuan, Phys. Rev. D71, 034005 (2005).
* Collins and Soper (1981) J. C. Collins and D. E. Soper, Nucl. Phys. B193, 381 (1981).
* Collins and Metz (2004) J. C. Collins and A. Metz, Phys. Rev. Lett. 93, 252001 (2004).
* Collins et al. (2008) J. C. Collins, T. C. Rogers, and A. M. Stasto, Phys. Rev. D77, 085009 (2008).
* Goeke et al. (2005) K. Goeke, A. Metz, and M. Schlegel, Phys. Lett. B618, 90 (2005).
* Meissner et al. (2009) S. Meissner, A. Metz, and M. Schlegel (2009), arXiv:0906.5323 [hep-ph].
* Tangerman and Mulders (1994) R. D. Tangerman and P. J. Mulders (1994), arXiv:hep-ph/9408305.
* Goeke et al. (2003) K. Goeke, A. Metz, P. V. Pobylitsa, and M. V. Polyakov, Phys. Lett. B567, 27 (2003).
* Gamberg et al. (2006) L. P. Gamberg, D. S. Hwang, A. Metz, and M. Schlegel, Phys. Lett. B639, 508 (2006).
* Bacchetta et al. (2008) A. Bacchetta, D. Boer, M. Diehl, and P. J. Mulders, JHEP 08, 023 (2008).
* Metz et al. (2008) A. Metz, P. Schweitzer, and T. Teckentrup (2008), arXiv:0810.5212 [hep-ph].
* Bukhvostov et al. (1983) A. P. Bukhvostov, E. A. Kuraev, and L. N. Lipatov, JETP Lett. 37, 482 (1983).
* Belitsky (1997) A. V. Belitsky (1997), arXiv:hep-ph/9703432.
* Kundu and Metz (2002) R. Kundu and A. Metz, Phys. Rev. D65, 014009 (2002).
* Eguchi et al. (2007) H. Eguchi, Y. Koike, and K. Tanaka, Nucl. Phys. B763, 198 (2007).
* Boer et al. (1998) D. Boer, P. J. Mulders, and O. V. Teryaev, Phys. Rev. D57, 3057 (1998).
* Kanazawa and Koike (2000) Y. Kanazawa and Y. Koike, Phys. Lett. B478, 121 (2000).
* Boer and Mulders (1998) D. Boer and P. J. Mulders, Phys. Rev. D57, 5780 (1998).
* Bacchetta et al. (2000) A. Bacchetta, M. Boglione, A. Henneman, and P. J. Mulders, Phys. Rev. Lett. 85, 712 (2000).
* Henneman et al. (2002) A. A. Henneman, D. Boer, and P. J. Mulders, Nucl. Phys. B620, 331 (2002).
* Harindranath and Zhang (1997) A. Harindranath and W.-M. Zhang, Phys. Lett. B408, 347 (1997).
* Avakian et al. (2008) H. Avakian et al., Phys. Rev. D77, 014023 (2008).
* Leader et al. (2007) E. Leader, A. V. Sidorov, and D. B. Stamenov, Phys. Rev. D75, 074027 (2007).
* de Florian et al. (2008) D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang, Phys. Rev. Lett. 101, 072001 (2008).
* Hirai et al. (2006) M. Hirai, S. Kumano, and N. Saito, Phys. Rev. D74, 014015 (2006).
* Abe et al. (1998) K. Abe et al. (E143), Phys. Rev. D58, 112003 (1998).
* Zheng et al. (2004) X. Zheng et al. (Jefferson Lab Hall A), Phys. Rev. C70, 065207 (2004).
* Kramer et al. (2005) K. Kramer et al., Phys. Rev. Lett. 95, 142002 (2005).
* Abe et al. (1997) K. Abe et al. (E154), Phys. Lett. B404, 377 (1997).
* Anthony et al. (1996) P. L. Anthony et al. (E142), Phys. Rev. D54, 6620 (1996).
* Stratmann (1993) M. Stratmann, Z. Phys. C60, 763 (1993).
* JLab experiment E07-002 , S. Choi, M. Jones, Z.-E. Meziani and O. Rondon (spokespersons)() (SANE) JLab experiment E07-002 (SANE), S. Choi, M. Jones, Z.-E. Meziani and O. Rondon (spokespersons).
* JLab experiment E06-014 , S. Choi, X. Jiang, Z.-E. Meziani and B. Sawatzky (spokespersons)() (d2n) JLab experiment E06-014 (d2n), S. Choi, X. Jiang, Z.-E. Meziani and B. Sawatzky (spokespersons).
* Parsamyan (2008) B. Parsamyan (COMPASS), Eur. Phys. J. ST 162, 89 (2008).
* (58) JLab experiment E06-010/E06-011, J.-P. Chen, E. Cisbani, H. Gao, X. Jiang, J.-C. Peng, spokespersons.
* Balitsky and Braun (1989) I. I. Balitsky and V. M. Braun, Nucl. Phys. B311, 541 (1989).
* Belitsky and Mueller (1997) A. V. Belitsky and D. Mueller, Nucl. Phys. B503, 279 (1997).
* Kang and Qiu (2009) Z.-B. Kang and J.-W. Qiu, Phys. Rev. D79, 016003 (2009).
* Vogelsang and Yuan (2009) W. Vogelsang and F. Yuan (2009), arXiv:0904.0410 [hep-ph].
* Ji et al. (2006) X. Ji, J.-W. Qiu, W. Vogelsang, and F. Yuan, Phys. Rev. Lett. 97, 082002 (2006).
* Koike et al. (2008) Y. Koike, W. Vogelsang, and F. Yuan, Phys. Lett. B659, 878 (2008).
* Collins et al. (1988) J. C. Collins, D. E. Soper, and G. Sterman, Adv. Ser. Direct. High Energy Phys. 5, 1 (1988).
* Ji et al. (2004) X. Ji, J.-P. Ma, and F. Yuan, Phys. Lett. B597, 299 (2004).
* Collins (2003) J. C. Collins, Acta Phys. Polon. B34, 3103 (2003).
* Collins (2008) J. Collins, PoS LC2008, 028 (2008).
* Schlegel and Metz (2004) M. Schlegel and A. Metz (2004), arXiv:hep-ph/0406289.
* Schlegel et al. (2004) M. Schlegel, K. Goeke, A. Metz, and M. V. Polyakov, Phys. Part. Nucl. 35, S44 (2004).
|
arxiv-papers
| 2009-07-17T19:08:58 |
2024-09-04T02:49:04.024813
|
{
"license": "Public Domain",
"authors": "Alberto Accardi, Alessandro Bacchetta, W. Melnitchouk, Marc Schlegel",
"submitter": "Alessandro Bacchetta",
"url": "https://arxiv.org/abs/0907.2942"
}
|
0907.2974
|
language=Java, frame=single, basicstyle=, captionpos=b,
showstringspaces=false, showspaces=false, extendedchars=true,
linewidth=1breaklines=true, float=phtb
# Service-Oriented Architectures and Web Services:
Course Tutorial Notes
Serguei A. Mokhov
Computer Science and Software Engineering
Faculty of Engineering and Computer Science
Concordia University
[email protected]
###### Abstract
This document presents a number of quick-step instructions to get started on
writing mini-service-oriented web services-based applications using NetBeans
6.5.x, Tomcat 6, GlassFish 2.1, and Java 1.6 primarily in Fedora 9 Linux with
user quota restrictions. While the tutorial notes are oriented towards the
students taking the SOEN691A course on service-oriented architectures (SOA) at
Computer Science and Software Engineering (CSE) Department, Faculty of
Engineering and Computer Science (ENCS), other may find some of it useful as
well outside of CSE or Concordia. The notes are compiled mostly based on the
students’ needs and feedback.
###### Contents
1. 1 Introduction
1. 1.1 Linux
1. 1.1.1 Accounts
2. 1.1.2 Java 1.6
3. 1.1.3 NetBeans 6.5.1
2. 2 Configuring NetBeans and GlassFish for BPEL
3. 3 Step-by-Step Environment Setup
4. 4 Step-by-Step Simple Application and Web Service Creation and Testing
5. 5 BPEL Composite Applications
6. 6 Conclusion
1. 6.1 See Also
###### List of Figures
1. 1 Terminal Window
2. 2 Setting up Java 1.6 as a Default in the Terminal
3. 3 Setting up HOME to the Group Directory
4. 4 NetBeans 6.5.1 Start-up Screen
5. 5 NetBeans: Services $\rightarrow$ Server $\rightarrow$ GlassFish V2
6. 6 Right-click GlassFish V2 $\rightarrow$ Properties
7. 7 GlassFish Admin Console Login Screen
8. 8 Downloading Additional Libraries in a Terminal with wget
9. 9 List of Components and Shared Libraries Installed in GlassFish
10. 10 “Java EE” $\rightarrow$ “Enterprise Application”
11. 11 NetBeans Programming Projects Location
12. 12 A1’s Example Server and Client Settings
13. 13 A1 Project Tree
14. 14 New Login Web Service
15. 15 A1 Project Tree after Login Web Service Creation
16. 16 Adding a Web Method login()
17. 17 Implementing a Simple Web login() Method for Quick Unit Testing
18. 18 Unit-testing Page for the Login WS
19. 19 login() Web Method Invocation Trace
20. 20 Creating a New Web Services Client in the Client Application Package from a Project
21. 21 Selecting the Service to Create a Client For from the Project
22. 22 Creating a New Web Services Client Nearly Done. Notice the URL
## 1 Introduction
### 1.1 Linux
We are using Fedora 9 Linux during the labs. For your own work you can use any
platform of your choice, e.g. Windows or MacOS X on your laptops. You will
have to do the installation and configuration of NetBeans, Java, Tomcat and so
on there.
On ENCS Windows the software was not made readily available (in particular
more recent NetBeans with the ALL option, and Tomcat 6 [Apa09]).
#### 1.1.1 Accounts
Under UNIX, disk space (for a sample account acs691a1) would be accessible
under e.g. /groups/a/ac_as691_a1. Under Windows, that path would be `\\filer-
groups\groups\a\ac_soen691a_1` (“S:” drive). There is a 1GB storage space
there and your in-school work related to the assignments and courses can be
put there, as the generated data files can be large at times.
#### 1.1.2 Java 1.6
Java 1.6 is not a default Java in ENCS. You need to make it default. In order
to use this version all you need to prepend:
/encs/pkg/jdk-6/root/bin
in your path. To do so there are simple instructions:
People using tcsh:
[serguei@lucid ~] % setenv PATH /encs/pkg/jdk-6/root/bin:$PATH
[serguei@lucid ~] % rehash
[serguei@lucid ~] % java -version
java version "1.6.0_14"
Java(TM) SE Runtime Environment (build 1.6.0_14-b08)
Java HotSpot(TM) Client VM (build 14.0-b16, mixed mode, sharing)
[serguei@lucid ~] %
People using bash:
bash-2.05b$ export PATH=/encs/pkg/jdk-6/root/bin:$PATH
bash-2.05b$ rehash
bash: rehash: command not found
bash-2.05b$ java -version
java version "1.6.0_14"
Java(TM) SE Runtime Environment (build 1.6.0_14-b08)
Java HotSpot(TM) Client VM (build 14.0-b16, mixed mode, sharing)
bash-2.05b$
You can avoid typing the above commands to set the PATH each time you open a
terminal under Linux by recording it in ~/.cshrc. If you do not have this file
in your home directory you can create one with the following content (e.g.
using vim [MC07]):
set path=( /encs/pkg/jdk-6/root/bin $path )
or copy an example from [Mok09] and update the path to include the above
directory first. Thus, next time when you login and open terminal, Java 1.6
will always be your default. The same applies if you click on the NetBeans
shortcut in the menu.
#### 1.1.3 NetBeans 6.5.1
NetBeans [Sun09b] is accessible as a simple command netbeans or from the
“Applications” $\rightarrow$ “Programming” $\rightarrow$ “NetBeans” menu with
a corresponding icon.
## 2 Configuring NetBeans and GlassFish for BPEL
The ALL option typically installs GlassFish 2.1 [Sun09a] as well as Tomcat 6
bundled by default with NetBeans, as well as some of the components. This
includes some of the BPEL [Wik09] components as well. To complete all the
needed extensions for BPEL for GlassFish you’d need to download WSDL
extensions and Saxon shared libraries and deploy them within your running
GlassFish instance. Download libraries for BPEL SE [Ope09], specifically:
wsdlextlib.jar and saxonlib.jar. That’s all you need for your setup in the
lab. For your home computer you may need to download and install the actual
BPEL service engine component from the same web page [Ope09], called
bpelserviceengine.jar.
## 3 Step-by-Step Environment Setup
1. 1.
Login to Linux. If you never did before likely your default Window manager is
GNOME.
2. 2.
Open up the terminal: “Applications” $\rightarrow$ “System Tools”
$\rightarrow$ “Terminal”. The window similar to Figure 1 should pop-up.
Figure 1: Terminal Window
3. 3.
Configure your Java 1.6 to be the default as outlined in Section 1.1.2, and an
example is shown in Figure 2.
Figure 2: Setting up Java 1.6 as a Default in the Terminal
4. 4.
In the same terminal window, change your HOME environment variable to that of
your 1GB group directory. This will allow most portions of NetBeans to write
the temporary and configuration files there by default instead of your main
Unix home directory. I use a temporary directory of mine
/tmp/groups/s/sm_s691a_1, as an example – and you should be using the
directory assigned to you with your group 1GB quota. An example to do so is
very similar as to set up PATH, except it is a single entry. It is exemplified
in Figure 3. Unlike PATH, it is not recommended to put these commands to
change your HOME into .cshrc.
Figure 3: Setting up HOME to the Group Directory
5. 5.
Create the following directories in your new HOME (your 1GB group directory):
mkdir .netbeans .netbeans-derby .netbeans-registration
ls -al
These directories will hold all the configuration and deployment files
pertaining to NetBeans, the Derby security controller, and the personal domain
for GlassFish operation. The overall content may easily reach 80MB in total
disk usage for all these directories.
6. 6.
Disk usage, quota, and big files:
quota
du -h
bigfiles
7. 7.
In your real home directory, remove any previous NetBeans et co. setup files
you may have generated from the previous runs:
.asadminpass
.asadmintruststore
.netbeans*
.personalDomain*
(assuming no important data for you are saved there):
\rm -rf .netbeans* .personalDomain* .asadmin*
8. 8.
In your real home directory create symbolic links (“shortcuts”) to the same
NetBeans directories, so in case it all still goes to the group directory
without impending your main quota:
[serguei@alfredo ~] % pwd
/nfs/home/s/serguei
[serguei@alfredo ~] % ln -s /tmp/groups/s/sm_s691a_1/.netbeans* .
[serguei@alfredo ~] % ls -ld .netbeans*
lrwxrwxrwx 1 serguei serguei 34 2009-07-11 08:05 .netbeans -> /tmp/groups/s/sm_s691a_1/.netbeans
lrwxrwxrwx 1 serguei serguei 40 2009-07-11 08:05 .netbeans-derby -> /tmp/groups/s/sm_s691a_1/.netbeans-derby
lrwxrwxrwx 1 serguei serguei 47 2009-07-11 08:05 .netbeans-registration -> /tmp/groups/s/sm_s691a_1/.netbeans-registration
9. 9.
Again, in the same terminal window launch NetBeans, by executing command
netbeans &, and after some time it should fully start up without of any
errors. You will be prompted to allow Sun to collect your usage information
and register; it is recommended to answer “No” to both. And then you will see
a left-hand-side (LHS) menu, the main editor page with the default browsed
info, and the top menu of the NetBeans, as shown in Figure 4. This is NetBeans
6.5.1, the latest released by the project is 6.7, and it will look slightly
different in some places, but overall it is more-or-less the same.
Figure 4: NetBeans 6.5.1 Start-up Screen
10. 10.
Navigate to the Services tab and expand the Server tree in the LHS menu. You
should be able to see a GlassFish V2 entry there, as shown in Figure 5.
Figure 5: NetBeans: Services $\rightarrow$ Server $\rightarrow$ GlassFish V2
11. 11.
Right-click on “GlassFish V2” and then “Properties”, as in Figure 6. Observe
the “Domains folder” and “Domain Name”. If the folder points within your
normal home directory, you have to change it as follows:
Figure 6: Right-click GlassFish V2 $\rightarrow$ Properties
1. (a)
Close the properties window.
2. (b)
Right-click on “GlassFish V2” and then “Remove”. Confirm with “Yes” the
removal.
3. (c)
Right-click on “Servers” and then “Add Server…”.
4. (d)
Select “GlassFish V2” and then “Next”, and “Next”.
5. (e)
Then for the “Domain Folder Location” Browse or paste your group directory,
e.g. /tmp/groups/s/sm_s691a_1/.domain in my case, notice where .domain is an
arbitrary name of a directory under your group directory that is not existing
yet, give it any name you like, and then press “Next”.
6. (f)
Pick a user name and a password for the admin console (web-based) of
GlassFish. The NetBeans default (of the GlassFish we removed) is ‘admin’ and
‘adminadmin’. It is strongly suggested however you do NOT follow the default,
and pick something else. Do NOT make it equal to your ENCS account either.
7. (g)
“Next” and “Finish”. Keep the ports at their defaults. Notice it may take time
to restart the new GlassFish instance and recreate your personal domain you
indicated in the group folder.
12. 12.
Right-click on GlassFish again and select “Start”. It may also take some time
to actually start GlassFish; watch the bottom-right corner as well as the
output window for the startup messages and status. There should be no errors.
Apache Derby network service should have started.
13. 13.
Once started, right-click on GlassFish again, and select “View Admin Console”.
You should see the GlassFish login window pop-up in the Firefox web browser,
looking as shown in Figure 7.
Figure 7: GlassFish Admin Console Login Screen
14. 14.
To log in, use the username and password you created earlier in Step 11f.
15. 15.
In your group home terminal, download additional libraries from [Ope09]. You
will only need 2 (wsdlextlib.jar and saxonlib.jar) out of typical 3, because
the version installed in ENCS already includes the 3rd
(bpelserviceengine.jar). You will likely need the 3rd file however, for your
laptop or home desktop in Windows. You can either download them directly from
the browser, or using the wget command, as shown in Figure 8.
Figure 8: Downloading Additional Libraries in a Terminal with wget
16. 16.
In your GlassFish console web page, under “Common Tasks” $\rightarrow$ “JBI”
$\rightarrow$ “Shared Libraries” you need to install the two libraries we
downloaded (3 for your Windows laptop or home desktop) by clicking “Install”
and following the steps by browsing to the directory where you downloaded the
files and installing them. Then, once installed sun-saxon-library and sun-
wsdl-ext-library should be listed under the “Shared Libraries”.
17. 17.
Make sure under “Components” you have sun-bpel-engine. Linux boxes in the labs
should have it installed with the NetBeans, at home it’s the 3rd file –
bpelserviceengine.jar, that may need to be installed using the similar
procedure as in the previous step. Roughly, how your “Components” and “Shared
Libraries” should look like is in Figure 9.
Figure 9: List of Components and Shared Libraries Installed in GlassFish
On this the environment setup should be complete. You will technically not
need to repeat except if you remove all the files from your group directory.
## 4 Step-by-Step Simple Application and Web Service Creation and Testing
1. 1.
Go to the “Projects” tab in NetBeans.
2. 2.
Then “File” $\rightarrow$ “New Project”.
3. 3.
Choose “Java EE” $\rightarrow$ “Enterprise Application”, as shown in Figure
10, and then “Next”.
Figure 10: “Java EE” $\rightarrow$ “Enterprise Application”
4. 4.
Give the project properties, like Project Name to be “A1”, project location
somewhere in your group directory, e.g. as for me shown in Figure 11, and then
“Next”.
Figure 11: NetBeans Programming Projects Location
5. 5.
In the next tab, you can optionally enable “Application Client Module” for an
example, and keep the rest at their defaults, e.g. as shown in Figure 12.
Notice, I altered the client package Main class to be in soen691a.a1.Main. It
is not strictly required in here as you can test your web services using web
service unit testing tools built-into the IDE.
Figure 12: A1’s Example Server and Client Settings
6. 6.
Click “Finish” to create your first project with the above settings. You
should see something that looks like as shown in Figure 13, after some of the
tree elements expanded.
Figure 13: A1 Project Tree
7. 7.
Under A1-war, create a package, called soen691a by right-clicking under “A1”
$\rightarrow$ “Source Packages” $\rightarrow$ “New” $\rightarrow$ “Java
Package” $\rightarrow$ “Package Name”: soen691a. Then “Finish”.
8. 8.
Create a “Web Service” under that package, by right-click on the newly created
package $\rightarrow$ “New” $\rightarrow$ “Web Service” $\rightarrow$ “Web
Service Name” $\rightarrow$ Login, as shown in Figure 14.
Figure 14: New Login Web Service
9. 9.
The LHS project tree if expanded would look like shown in Figure 14.
Figure 15: A1 Project Tree after Login Web Service Creation
10. 10.
Right-click on Login WS, and select “Add Operation…” and create a web method
login(), as shown in Figure 16.
Figure 16: Adding a Web Method login()
11. 11.
After the web method login() appears as a stub inside the Login class with
return false; by default. For quick unit testing of the new method, implement
it with some test user name and password as shown in Figure 17, which will
later be replaced to be read from the XML file.
Figure 17: Implementing a Simple Web login() Method for Quick Unit Testing
12. 12.
Perform a simple unit test for the web method. Your GlassFish must be running
and you have to “start” your project by deploying – just press the green angle
“play” button. You should see a “Hello World” page appearing in your browser.
13. 13.
Then, under “A1-war” $\rightarrow$ “Web Services” $\rightarrow$ “Login” right-
click on Login and select “Test Web Service”. It should pop-up another browser
window (or tab) titled something like “LoginService Web Service Tester” with a
pre-made form to test inputs to your web method(s), as shown in Figure 18.
Figure 18: Unit-testing Page for the Login WS
14. 14.
Fill-in the correct test values that we defined earlier for login and press
the “login” button. Observe the exchanged SOAP XML messages and the true value
returned as a result, as shown in Figure 19.
Then try any wrong combination of the username and password and see that it
returns false. This completes basic verification of your web service – that is
can be successfully deployed and ran, and its method(s) unit-tested on the
page.
Figure 19: login() Web Method Invocation Trace
15. 15.
Java-based client callee of a web service has to be defined e.g. as a WS
client, as shown in earlier screenshots as “A1-app-client”, which has a
Main.main() method. In that method you simply invoke the desired service by
calling its web method after a number of instantiations. It may look like you
are calling a local method of a local class, but, in fact, on the background
there is a SOAP message exchange, marshaling/demarshaling of data types, etc.
and actually connection to a web service, posting a request, receiving and
parsing HTTP response, etc. all done by the middleware.
Steps:
1. (a)
Right-click “A1-app-client” $\rightarrow$ “New” $\rightarrow$ “Web Service
Client”. A dialog shown in Figure 20 should appear. Click “Browse”.
Figure 20: Creating a New Web Services Client in the Client Application
Package from a Project
2. (b)
Select your web service to generate a reference client for, as e.g. shown in
Figure 21 and click “OK”.
Figure 21: Selecting the Service to Create a Client For from the Project
3. (c)
Having selected the service to generate the WS client code for, you should see
the URL, as shown in Figure 22 “Finish”, re-deploy (green “Play” button).
Figure 22: Creating a New Web Services Client Nearly Done. Notice the URL
4. (d)
Then, in Main, import the generated code classes to invoke the service, as
shown in Listing 1.
Listing 1: Invoking a Web Service from a Plain Java Class
⬇
package soen691a.a1;
import soen691a.Login;
import soen691a.LoginService;
/**
* @author serguei
*/
public class Main {
/**
* @param args the command line arguments
*/
public static void main(String[] args) {
LoginService service = new LoginService();
Login login = service.getLoginPort();
//…
// Must be false
boolean success = login.login(”wrongusername”, ”wrongpasword”);
// Must be false
success = login.login(”wrongusername”, ”pa$$+3$T”);
// Must be false
success = login.login(”userTest”, ”wrongpasword”);
// Must be true
success = login.login(”userTest”, ”pa$$+3$T”);
//…
}
}
See also an example from DMARF [Mok06].
16. 16.
Relative path for loading XML can be found using
System.getProperty(‘‘user.dir’’) to find out your current working directory of
the application, which is actually relative to the config/ subdirectory in
your personal domain folder, so it would be based on your deployment, but
roughly:
System.getProperty("user.dir") + "../generated/....../users.xml"
where “......” is the path leading to where your users.xml and others actually
are. You can configure Ant’s build.xml (actually build-impl.xml and other
related files for deployment to copy your XML data files into config/
automatically.
17. 17.
Loading and querying XML with SAX is exemplified in TestNN with MARF [CMt09,
The09], specifically at these CVS URLs:
http://marf.cvs.sf.net/viewvc/marf/apps/TestNN/
http://marf.cvs.sf.net/viewvc/marf/marf/src/marf/Classification/NeuralNetwork/
Do not validate your XML unless you specified a DTD schema (not necessary
here), just make sure your tags are matching, properly nested, and closed.
## 5 BPEL Composite Applications
GlassFish is needed for BPEL (while the previous could be done with Tomcat 6).
E.g. tutorial from NetBeans:
http://www.netbeans.org/kb/61/soa/loanprocessing.html
Similarly, there are good application samples available in the betbeans to
start the process of a BPEL composite application: “New” $\rightarrow$
“Samples” $\rightarrow$ “SOA”; specifically “Travel Resevation Service” and
“BPEL BluePrint 1”.
## 6 Conclusion
Please direct any problems and errors with these notes or any other
constructive feedback to [email protected].
### 6.1 See Also
* •
GlassFish website [Sun09a].
* •
Unix commands [Mok05].
* •
ENCS help: http://www.encs.concordia.ca/helpdesk/.
* •
An example of the XML parsing application, TestNN with MARF [CMt09, The09]
using the built-in SAX parser.
## References
* [Apa09] Apache Foundation. Apache Jakarta Tomcat. [online], 1999–2009. http://jakarta.apache.org/tomcat/index.html.
* [CMt09] Ian Clement, Serguei A. Mokhov, and the MARF Research & Development Group. TestNN – Testing Artificial Neural Network in MARF. Published electronically within the MARF project, http://marf.sf.net, 2002–2009. Last viewed April 2008.
* [MC07] Bram Moolenaar and Contributors. Vim the editor – Vi Improved. [online], 2007. http://www.vim.org/.
* [Mok05] Serguei A. Mokhov. UNIX commands, revision 1.4. [online], 2003 – 2005. http://users.encs.concordia.ca/~mokhov/comp444/tutorials/unix-commands.%pdf.
* [Mok06] Serguei A. Mokhov. On design and implementation of distributed modular audio recognition framework: Requirements and specification design document. [online], August 2006. Project report, http://arxiv.org/abs/0905.2459, last viewed May 2009\.
* [Mok09] Serguei A. Mokhov. A .cshrc example, 2000–2009. http://users.encs.concordia.ca/~mokhov/comp346/.cshrc.
* [Ope09] OpenESB Contributors. BPEL service engine. [online], 2009. https://open-esb.dev.java.net/BPELSE.html.
* [Sun09a] Sun Microsystems. Sun GlassFish: Open web application platform. [online], 1994–2009. http://www.sun.com/glassfish.
* [Sun09b] Sun Microsystems. NetBeans 6.5.1. [online], July 2004–2009. http://www.netbeans.org.
* [The09] The MARF Research and Development Group. The Modular Audio Recognition Framework and its Applications. SourceForge.net, 2002–2009. http://marf.sf.net, last viewed December 2008.
* [Wik09] Wikipedia. Business Process Execution Language (BPEL) — Wikipedia, the free encyclopedia. [Online; accessed 14-July-2009], 2009. http://en.wikipedia.org/w/index.php?title=Business_Process_Execution_La%nguage&oldid=302021294.
## Index
* API
* A1-war item 7
* false item 14
* HOME Figure 3, Figure 3, item 4, item 4, item 5
* Login Figure 14, Figure 14, Figure 18, Figure 18, item 10, item 11, item 13, item 8
* login() Figure 16, Figure 16, Figure 17, Figure 17, Figure 19, Figure 19, item 10, item 11
* Main item 15d, item 5
* Main.main() item 15
* PATH §1.1.2, item 4, item 4
* path §1.1.2
* soen691a item 7, item 7
* soen691a.a1.Main item 5
* System.getProperty(“user.dir”) item 16
* TestNN item 17, 4th item
* true item 14
* Files
* .asadminpass item 7
* .asadmintruststore item 7
* .cshrc item 4
* .domain item 11e
* .netbeans* item 7
* .personalDomain* item 7
* /.cshrc §1.1.2
* /encs/pkg/jdk-6/root/bin §1.1.2
* /tmp/groups/s/sm_s691a_1 item 4
* /tmp/groups/s/sm_s691a_1/.domain item 11e
* bpelserviceengine.jar §2, item 15, item 17
* build-impl.xml item 16
* build.xml item 16
* config/ item 16, item 16
* saxonlib.jar §2, item 15
* users.xml item 16
* wsdlextlib.jar §2, item 15
* Frameworks
* Distributed MARF item 15
* MARF item 17, 4th item
* Java Service-Oriented Architectures and Web Services: Course Tutorial Notes, §1.1.2
* Libraries
* Distributed MARF item 15
* MARF item 17, 4th item
* MARF item 17, 4th item
* Distributed item 15
* Tools
* bash §1.1.2
* netbeans §1.1.3
* netbeans & item 9
* vim §1.1.2
* wget Figure 8, Figure 8, item 15
|
arxiv-papers
| 2009-07-17T03:36:56 |
2024-09-04T02:49:04.032955
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Serguei A. Mokhov, Shahriar Rostami, Hammad Ali, Min Chen and Yuhong\n Yan",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/0907.2974"
}
|
0907.3055
|
# Electromagnons and instabilities in magnetoelectric materials with non-
collinear spin orders
M. A. van der Vegte, C. P. van der Vegte, and M. Mostovoy Zernike Institute
for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG
Groningen, The Netherlands
###### Abstract
We show that strong electromagnon peaks can be found in absorption spectra of
non-collinear magnets exhibiting a linear magnetoelectric effect. The
frequencies of these peaks coincide with the frequencies of antiferromagnetic
resonances and the ratio of the spectral weights of the electromagnon and
antiferromagnetic resonance is related to the ratio of the static
magnetoelectric constant and magnetic susceptibility. Using a Kagomé lattice
antiferromagnet as an example, we show that frustration of spin ordering gives
rise to magnetoelastic instabilities at strong spin-lattice coupling, which
transform a non-collinear magnetoelectric spin state into a collinear
multiferroic state with a spontaneous electric polarization and magnetization.
The Kagomé lattice antiferromagnet also shows a ferroelectric incommensurate-
spiral phase, where polarization is induced by the exchange striction
mechanism.
###### pacs:
75.80.+q, 75.30.Ds, 78.20.-e, 75.10.Hk, 75.30.Et, 75.25.+z
###### pacs:
75.80.+q,71.45.Gm,76.50.+g,75.10.Hk
## I Introduction
The recent renewal of interest in multiferroic materials led to discovery of
many novel compounds where electric polarization is induced by ordered
magnetic states with broken inversion symmetry.CheongNatMat2007 ;
KimuraARMR2007 ; RameshNatMat2007 The electric polarization in multiferroics
is very susceptible to changes in spin ordering produced by an applied
magnetic field, which gives rise to dramatic effects such as the magnetically-
induced polarization flops and colossal magnetocapacitance.KimuraNature2003 ;
HurNature2004 ; GotoPRL2004 Magnetoelectric interactions also couple spin
waves to polar phonon modes and make possible to excite magnons by an
oscillating electric field of light, which gives rise to the so-called
electromagnon peaks in photoabsorption.SmolenskiiSPU1982
Electromagnons were recently observed in two groups of multiferroic
orthorombic manganites, $R$MnO3 ($R$ = Gd,Tb,Dy,Eu1-xYx) and $R$Mn2O5 ($R$ =
Y,Tb). PimenovNaturePhys2006 ; PimenovPRB2006 ; ValdesPRB2007 ; SushkovPRL2007
Ferroelectricity in $R$MnO3 appears in a non-collinear antiferromagnetic state
with the cycloidal spiral ordering and the magnetoelectric coupling originates
from the so-called inverse Dzyaloshinskii-Moriya mechanism. KatsuraPRL2005 ;
KenzelmannPRL2005 ; SergienkoPRB2006 ; MostovoyPRL2006 ; MalashevichPRL2008
In Ref. [KatsuraPRL2007, ] it was noted that the same mechanism can couple
magnons to photons and that an oscillating electric field of light can excite
rotations of the spiral plane. However, the selection rule for the
electromagnon polarization resulting from this coupling does not agree with
recent experimental dataValdesPRB2007 ; KidaPRB2008 ; SushkovJPCM2008 ;
TakahashiPRL2008 and, moreover, the inverse Dzyaloshinskii-Moriya mechanism
of relativistic nature is too weak to explain the strength of the
electromagnon peaks in $R$MnO3.
These peaks seem to originate from the exchange striction, i.e. ionic shifts
induced by changes in the Heisenberg exchange energy when spins order or
oscillate.ValdesPRL2009 This mechanism explains the experimentally observed
polarization of electromagnons. Since the Heisenberg exchange interaction is
stronger than the Dzyaloshinskii-Moriya interaction, it can induce larger
electric dipoles. In Ref. [ValdesPRL2009, ] it was shown that the magnitude of
the spectral weight of the giant electromagnon peak in the spiral state of
rare earth manganites is in good agreement with the large spontaneous
polarization in the E-type antiferromagnetic state,LorenzPRB2006 which has
not been reliably measured yet but is expected to exceed the polarization in
the spiral state by 1-2 orders of magnitude.SergienkoPRL2006 ; PicozziPRL99
From the fact that the mechanism that couples magnons to light in rare earth
manganites is different from the coupling that induces the static polarization
in these materials we can conclude that electromagnons can also be observed in
non-multiferroic magnets. In this paper we focus on electromagnons in
materials exhibiting a linear magnetoelectric effect, i.e. when an applied
magnetic field, $\mathbf{H}$, induces an electric polarization, $\mathbf{P}$,
proportional to the field, while an applied electric field, $\mathbf{E}$,
induces a magnetization, $\mathbf{M}$. This unusual coupling takes place in
antiferromagnets where both time reversal and inversion symmetries are
spontaneously broken.Landaubook1984 ; FiebigJAPD2005
It is natural to expect that when an electric field applied to a
magnetoelectric material oscillates, the induced magnetization will oscillate
too. Such a dynamical magnetoelectric response, however, requires presence of
excitations that are coupled both to electric and magnetic fields. They appear
when magnons, which can be excited by an oscillating magnetic field
(antiferromagnetic resonances), mix with polar phonons, which are coupled to
an electric field. Thus in materials showing a linear magnetoelectric effect,
for each electromagnon peak there is an antiferromagnetic resonance with the
same frequency.
This reasoning does not apply to all magnetoelectrics and the dc
magnetoelectric effect is not necessarily related to hybrid spin-lattice
excitations. As will be discussed below, in materials with collinear spin
orders electromagnons either do not exist or have a relatively low spectral
weight. In this paper we argue that electromagnons should be present in non-
collinear antiferromagnets showing strong static magnetoelectric response. As
a simple example, we consider a Kagomé lattice antiferromagnet with the 120∘
spin ordering, shown in Fig. 1. Such an ordering has a nonzero magnetic
monopole moment, which allows for a linear magnetoelectric effect with the
magnetoelectric tensor $\alpha_{ij}=\alpha\delta_{ij}$ for electric and
magnetic fields applied in the plane of the Kagomé lattice.SpaldinJPCM2008 A
relatively strong magnetoelectric response was recently predicted for Kagomé
magnets with the KITPite crystal structure, in which magnetic ions are located
inside oxygen bipyramids.DelaneyPRL2009 In this structure the oxygen ions
mediating the superexchange in basal planes are located outside the up-
triangles forming the Kagomé lattice and inside the down-triangles or vice
versa (see Fig. 1), in which case magnetoelectric responses of all triangles
add giving rise to a large magnetoelectric constant.
Figure 1: (Color online) The Kagomé magnet with the KITPite crystal structure,
in which the ligand ions (open circles) mediating the superexchange between
spins are positioned in a way that gives rise to a strong linear
magnetoelectric response in the $120^{\circ}$ spin state. Here, $J_{1}$ and
$J_{2}$ denote, respectively, the nearest-neighbor and next-nearest-neighbor
exchange constants, the solid arrows denote spins, while the empty arrows
denote the shifts of the ligand ions.
This paper is organized as follows. In Sec. II we analyze the symmetry of
magnon modes and the magnetoelectric coupling in the Kagomé lattice magnet
with the KITPite structure and show that the dc magnetoelectric effect in this
system is related to presence of electromagnon modes. The common origin of the
dc and ac magnetoelectric responses implies existence of relations between
static and dynamic properties of magnetoelectric materials, derived in Sec.
III. In Sec. IV we discuss softening of (electro)magnons and the resulting
divergence of the coupled magnetoelectric response. In Sec. V we discuss the
transition from a magnetoelectric to a multiferroic state at a strong spin-
lattice coupling and plot the phase diagram of our model system. In section VI
we discuss the importance of non-collinearity of spins for dynamic
magnetoelectric response and possible electromagnons in known magnetoelectric
materials. In Sec. VII we conclude.
## II Symmetry considerations
The coupling of magnetic excitations in the Kagomé magnet to electric field,
resulting from the Heisenberg exchange striction or any other non-relativistic
interaction, can be found using the method outlined in Ref. [ValdesPRL2009, ].
To simplify notation, we consider a single up-triangle, which has the same
point symmetry as the whole Kagomé lattice with the $120^{\circ}$ spin
ordering shown in Fig. 1. The form of the magnetoelectric coupling is
constrained by the $3_{z}$ and $m_{x}$ symmetry operations:BulaevskiiPRB2008
$\displaystyle H_{\rm me}$ $\displaystyle=$
$\displaystyle-\gamma\left\\{\frac{E_{x}}{\sqrt{2}}\left[\left(\mathbf{S}_{2}\cdot\mathbf{S}_{3}\right)-\left(\mathbf{S}_{1}\cdot\mathbf{S}_{3}\right)\right]\right.$
(1) $\displaystyle+$
$\displaystyle\left.\frac{E_{y}}{\sqrt{6}}\left[\left(\mathbf{S}_{1}\cdot\mathbf{S}_{3}\right)+\left(\mathbf{S}_{2}\cdot\mathbf{S}_{3}\right)-2\left(\mathbf{S}_{1}\cdot\mathbf{S}_{2}\right)\right]\right\\}.$
We then replace $\mathbf{S}_{i}$ by $\langle
S\rangle\mathbf{n}_{i}+\delta\mathbf{S}_{i}$, where the unit vectors
$\left(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}\right)=\left(-\frac{\sqrt{3}}{2}{\hat{x}}-\frac{1}{2}{\hat{y}},\frac{\sqrt{3}}{2}{\hat{x}}-\frac{1}{2}{\hat{y}},{\hat{y}}\right)$
describe the $120^{\circ}$ spin ordering in the $xy$ plane and
$\delta\mathbf{S}_{i}\perp\mathbf{n}_{i}$ is the oscillating part, which is
the superposition of the orthogonal magnon modes in the triangle (the zero
wave vector magnons for the Kagomé lattice):
$\delta\mathbf{S}_{i}=\sum_{\alpha}\left(q_{\alpha}\mbox{\boldmath$\psi$}_{\alpha
i}+\langle S\rangle p_{\alpha}\mbox{\boldmath$\varphi$}_{\alpha i}\right),$
(2)
where $\alpha=0,x,y$ labels the magnon,
$\left\\{\begin{array}[]{rcl}\mbox{\boldmath$\varphi$}_{0i}&=&{\hat{z}}\frac{1}{\sqrt{3}}\left(1,1,1\right),\\\
\mbox{\boldmath$\varphi$}_{xi}&=&{\hat{z}}\frac{1}{\sqrt{6}}\left(1,1,-2\right),\\\
\mbox{\boldmath$\varphi$}_{yi}&=&{\hat{z}}\frac{1}{\sqrt{2}}\left(-1,1,0\right),\end{array}\right.$
(3)
are the out-of-plane components of the magnons and
$\mbox{\boldmath$\psi$}_{\alpha i}=\mbox{\boldmath$\varphi$}_{\alpha
i}\times\mathbf{n}_{i}$ are the in-plane components (see Fig. 2),
$\left\\{\begin{array}[]{rcl}\mbox{\boldmath$\psi$}_{0}&=&\frac{1}{\sqrt{3}}\left(\frac{1}{2}{\hat{x}}-\frac{\sqrt{3}}{2}{\hat{y}},\frac{1}{2}{\hat{x}}+\frac{\sqrt{3}}{2}{\hat{y}},-{\hat{x}}\right),\\\
\mbox{\boldmath$\psi$}_{x}&=&\frac{1}{\sqrt{6}}\left(\frac{1}{2}{\hat{x}}-\frac{\sqrt{3}}{2}{\hat{y}},\frac{1}{2}{\hat{x}}+\frac{\sqrt{3}}{2}{\hat{y}},2{\hat{x}}\right),\\\
\mbox{\boldmath$\psi$}_{y}&=&\frac{1}{\sqrt{2}}\left(-\frac{1}{2}{\hat{x}}+\frac{\sqrt{3}}{2}{\hat{y}},\frac{1}{2}{\hat{x}}+\frac{\sqrt{3}}{2}{\hat{y}},0\right).\end{array}\right.$
(4)
The single-magnon excitation by an electric field is described by the terms
linear in $\delta\mathbf{S}$, while the terms quadratic in $\delta\mathbf{S}$
give rise to the photoexcitation of a two-magnon continuum. Since spins order
in plane, the polarization oscillations are induced by the in-plane
oscillations of spins and the coupling of electric field to magnons, obtained
from Eq.(1), has the form:
$H_{\rm me}=-g_{E}\left(q_{x}E_{x}+q_{y}E_{y}\right),$ (5)
where $g_{E}=\frac{3}{2}\gamma\langle S\rangle$. This magnetoelectric coupling
is only nonzero in the magnetically ordered state with broken time reversal
symmetry, which is why the coupling constant is proportional to $\langle
S\rangle$.
Figure 2: (Color online) The magnon modes in a triangle with the $120^{\circ}$
spin ordering. The thin arrows indicate the directions of average spins, while
the thick arrows show the in-plane components of the magnon modes.
We note that Eq.(5) can also be obtained by a conventional symmetry analysis
of magnetic modes. The vector $\mbox{\boldmath$\psi$}_{0}$ and, hence, the
corresponding amplitude, $q_{0}$, forms a one-dimensional representation
$\Gamma_{1}$, while
$\left(\mbox{\boldmath$\psi$}_{x},\mbox{\boldmath$\psi$}_{y}\right)$ and
$\left(\\!\begin{array}[]{c}q_{x}\\\ q_{y}\end{array}\\!\right)$ form a two-
dimensional representation $\Gamma_{3}$ (see Table 1). The direct product,
$\Gamma_{2}\times\Gamma_{3}$, where $\Gamma_{2}$ is the symmetry of the spin
ordering, transforms as the doublet of the in-plane components of the electric
fields, $\Gamma_{4}$, which leads to Eq.(5). This general symmetry analysis
is, however, insensitive to the microscopic mechanism of the magnetoelectric
coupling, whereas the derivation staring from Eq.(1) only applies to non-
relativistic mechanisms.
| | $3_{z}$ | $m_{x}$ | $T$
---|---|---|---|---
$\Gamma_{1}$ | $q_{0}$ | +1 | +1 | $-1$
$\Gamma_{2}$ | $\left\langle\mathbf{S}\right\rangle$ | +1 | $-1$ | $-1$
$\Gamma_{3}$ | $\left(\\!\begin{array}[]{c}q_{x}\\\ q_{y}\end{array}\\!\right),\left(\\!\begin{array}[]{c}H_{x}\\\ H_{y}\end{array}\\!\right)$ | $\left(\\!\begin{array}[]{cc}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\\ +\frac{\sqrt{3}}{2}&-\frac{1}{2}\end{array}\\!\\!\right)$ | $\left(\\!\begin{array}[]{cc}+1&0\\\ 0&-1\end{array}\\!\\!\right)$ | $\left(\\!\\!\begin{array}[]{cc}-1&0\\\ 0&-1\end{array}\\!\\!\right)$
$\Gamma_{4}$ | $\left(\\!\begin{array}[]{c}E_{x}\\\ E_{y}\end{array}\\!\right)$ | $\left(\\!\begin{array}[]{cc}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\\ +\frac{\sqrt{3}}{2}&-\frac{1}{2}\end{array}\\!\right)$ | $\left(\\!\\!\begin{array}[]{cc}-1&0\\\ 0&+1\end{array}\\!\\!\right)$ | $\left(\\!\\!\begin{array}[]{cc}+1&0\\\ 0&+1\end{array}\\!\\!\right)$
Table 1: The transformation properties of several irreducible representations
of the space group of the Kagomé lattice and time reversal operation $T$.
Table 1 shows that the coupling to the in-plane components of magnetic field
has the form,
$H_{\rm m}=-g_{H}\left(q_{x}H_{x}+q_{y}H_{y}\right),$ (6)
while the Hamiltonian describing magnon modes with zero wave vector in the
Kagomé layer is,
$H(p,q)=\frac{p_{0}^{2}}{2m_{0}}+\frac{1}{2m}\left(p_{x}^{2}+p_{y}^{2}\right)+\frac{\kappa_{0}q_{0}^{2}}{2}+\frac{\kappa}{2}\left(q_{x}^{2}+q_{y}^{2}\right).$
(7)
If we consider, for example, the microscopic spin Hamiltonian describing the
antiferromagnetic nearest-neighbor and next-nearest-neighbor Heisenberg
exchange interactions with the exchange constants, respectively, $J_{1}$ and
$J_{2}$ and the easy plane magnetic anisotropy $\Delta$,
$H=J_{1}\sum_{\langle
ij\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+J_{2}\sum_{\langle\langle
ij\rangle\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+\frac{\Delta}{2}\sum_{i}\left(S_{i}^{z}\right)^{2},$
(8)
for which the $120^{\circ}$ spin ordering shown in Fig. 1 is a classical
ground state,HarrisPRB1992 ; ElhajalPRB2002 we get
$\begin{array}[]{ll}m_{0}^{-1}=\left[6(J+J^{\prime})+\Delta\right]\langle
S\rangle^{2},&m^{-1}=\Delta\langle S\rangle^{2},\\\
\kappa_{0}=0,&\kappa=3\left(J_{1}+J_{2}\right).\end{array}$ (9)
The linearized equations of motion for spins in applied electric and magnetic
fields are obtained from Eqs.(5),(6), and (7), if we impose the commutation
relations for the amplitudes of the in-plane and out-of-plane parts of
${\delta\mathbf{S}}$:
$\left[q_{\alpha},p_{\beta}\right]=i\delta_{\alpha,\beta}.$ (10)
From these equations we find the frequencies of the three magnon modes with
zero wave vector: $\omega_{0}^{2}=\kappa_{0}m_{0}^{-1}=0$ and
$\omega_{x}^{2}=\omega_{y}^{2}=\kappa
m^{-1}=3\left(J_{1}+J_{2}\right)\Delta\langle S\rangle^{2}$.
Minimizing the spin energy with respect to $q_{x}$ and $q_{y}$ in the static
limit, we obtain an effective magnetoelectric coupling,
$H_{\rm me}=-\alpha\left(H_{x}E_{x}+H_{y}E_{y}\right).$ (11)
where the magnetoelectric coefficient $\alpha=\frac{g_{E}g_{H}}{2\kappa}$.
Furthermore, the $q_{x}$-mode can be excited by both electric and magnetic
field oscillating with the frequency $\omega_{x}$ in the direction parallel to
the $x$ axis, while the $q_{y}$-mode can be excited by both $E_{y}$ and
$H_{y}$, which shows that the static linear magnetoelectric effect in this
non-collinear magnet is related to the presence of electromagnon and
antiferromagnetic resonance peaks with equal frequencies in the optical
absorption spectrum.
## III Relations between static and dynamic magnetoelectric response
The common origin of the static and dynamic magnetoelectric response of non-
collinear magnets leads to quantitative relations between dc susceptibilities
and spectral weights of peaks in the optical absorption spectrum. These
relations simplify when the coupling of magnons to electric field is mediated
by a single optical phonon. The description of magnons in terms of conjugated
coordinates and momenta is very convenient for derivation of these relations,
since the coupled magnon and optical phonon are in this approach just a pair
of coupled oscillators:
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2m}\left(p_{x}^{2}+p_{y}^{2}\right)+\frac{\kappa}{2}\left(q_{x}^{2}+q_{y}^{2}\right)$
(12)
$\displaystyle+\frac{1}{2M}\left(P_{x}^{2}+P_{y}^{2}\right)+\frac{K}{2}\left(Q_{x}^{2}+Q_{y}^{2}\right)$
$\displaystyle-\lambda\left(q_{x}Q_{x}+q_{y}Q_{y}\right)-f\left(Q_{x}E_{x}+Q_{y}E_{y}\right)$
$\displaystyle-g_{H}\left(q_{x}H_{x}+q_{y}H_{y}\right),$
where $(Q_{x},P_{x})$ and $(Q_{y},P_{y})$ are the coordinates and momenta of
the optical phonons coupled to magnons, which also form a two-dimensional
representation.
The magnetoelectric response of such a system is easy to calculate. The result
can be expressed in terms of observable quantities, such as the ‘dressed’
magnon and phonon frequencies, $\omega_{\rm mag}$ and $\omega_{\rm ph}$, and
the spectral weights of the magnon and phonon peaks excited by an electric and
magnetic field. We denote the spectral weight of the electromagnon peak by
$S_{\rm mag}^{E}=8\int\\!\\!d\omega\omega{\chi}^{\prime\prime}_{\rm
e}(\omega),$ (13)
where ${\chi}^{\prime\prime}_{\rm e}(\omega)$ is the imaginary part of the
dielectric ac susceptibility, while the spectral weight of the
antiferromagnetic resonance is,
$S_{\rm mag}^{H}=8\int\\!\\!d\omega\omega{\chi}^{\prime\prime}_{\rm
m}(\omega),$ (14)
where ${\chi}^{\prime\prime}_{\rm m}(\omega)$ is the imaginary part of the
magnetic ac susceptibility. The integration in Eqs.(13) and Eq.(14) goes over
an interval of frequencies around $\omega_{\rm mag}$. The two spectral weights
for the phonon $S_{\rm ph}^{E}$ and $S_{\rm ph}^{H}$ are defined in a similar
way. We assume that the magnon and phonon peaks are sufficiently narrow and
can be separated from each other. The four spectral weights satisfy a
relation,
$S_{\rm mag}^{E}S_{\rm mag}^{H}=S_{\rm ph}^{E}S_{\rm ph}^{H},$ (15)
following from the fact that an electric field only interacts with the ‘bare’
phonon, while a magnetic field is only coupled to the ‘bare’ magnon.
The relations between the dc and ac magnetoelectric responses of the coupled
spin-lattice system have the form,
$\left\\{\begin{array}[]{rcl}\Delta{\epsilon}&=&\frac{S_{\rm
mag}^{E}}{\omega_{\rm mag}^{2}}+\frac{S_{\rm ph}^{E}}{\omega_{\rm ph}^{2}},\\\
\\\ \Delta{\mu}&=&\frac{S_{\rm mag}^{H}}{\omega_{\rm mag}^{2}}+\frac{S_{\rm
ph}^{H}}{\omega_{\rm ph}^{2}},\\\ \\\ 4\pi|\alpha|&=&\sqrt{S_{\rm
mag}^{E}S_{\rm mag}^{H}}\left|\frac{1}{\omega_{\rm
mag}^{2}}-\frac{1}{\omega_{\rm ph}^{2}}\right|,\end{array}\right.$ (16)
where $\Delta{\epsilon}$($\Delta{\mu}$) is the increase of the real part of
the dielectric constant (magnetic permeability) at zero frequency resulting
from the magnon and phonon peaks (we use the Gaussian units). The first two
equations are the Kramers-Kronig relations for the real and imaginary parts of
dielectric and magnetic susceptibilities, while the last relation follows from
equations of motion. Combining Eqs.(15) and (16) we can express the ratio of
the spectral weights of the electromagnon and the antiferromagnetic resonance
through the ratio of the static magnetoelectric constant $\alpha$ and magnetic
susceptibility ${\chi}_{\rm m}={\chi}^{\prime}_{\rm m}(\omega=0)$:
$\frac{S_{\rm mag}^{E}}{S_{\rm mag}^{H}}=\left(\frac{\alpha}{{\chi}_{\rm
m}}\right)^{2}\left(\frac{1+\frac{\omega_{\rm mag}^{2}}{\omega_{\rm
ph}^{2}}\frac{S_{\rm mag}^{E}}{S_{\rm ph}^{E}}}{1-\frac{\omega_{\rm
mag}^{2}}{\omega_{\rm ph}^{2}}}\right)^{2}$ (17)
For $\omega_{\rm mag}^{2}\ll\omega_{\rm ph}^{2}$, the ratio of the spectral
weights is just the square of the ratio of the dc magnetoelectric constant and
magnetic susceptibility.
Due to the spin-lattice coupling, the ‘dressed’ magnon frequency, $\omega_{\rm
mag}$, is lower than its ‘bare’ value, $\sqrt{\frac{\kappa}{m}}$ (assuming
that the ‘bare’ magnon frequency is smaller than the ‘bare’ phonon frequency).
As the spin-lattice coupling increases, $\omega_{\rm mag}$ vanishes at a
critical value of the coupling. According to Eq.(16), this results in the
simultaneous divergency of $\epsilon$, $\mu$ and $\alpha$, indicating an
instability towards a multiferroic phase, which is both ferroelectric and
ferromagnetic. Another manifestation of this instability is the fact that as
the spin-lattice coupling approaches the critical value, the magnetoelectric
constant $\alpha$ tends to its upper bound equal the geometric mean of the
dielectric and magnetic susceptibilities, $\sqrt{\chi_{\rm e}\chi_{\rm m}}$,
imposed by the requirement of stability with respect to applied electric and
magnetic fields.BrownPR1968
## IV Magnon softening
To study the transition from the magnetoelectric state of the Kagomé magnet to
the multiferroic state in more detail, we consider a simple microscopic model,
in which positions of magnetic ions are fixed, while ligand ions are allowed
to move. The spin-lattice coupling originates from the dependence of the
exchange constants on displacements of ligand ions mediating the superexchange
in the direction perpendicular to the straight line connecting two neighboring
spins. We denote the positions of the three ligand ions outside up-triangles
by $\mathbf{u}_{1}$, $\mathbf{u}_{2}$, and $\mathbf{u}_{3}$, while the
position of a ligand ion inside a down-triangle is denoted by $\mathbf{v}$.
Then, for example, the exchange constant for the spins $\mathbf{S}_{1}$ and
$\mathbf{S}_{2}$ is $J_{1}+J_{1}^{\prime}\left(u_{3}\right)_{y},$ while for
the spins $\mathbf{S}_{4}$ and $\mathbf{S}_{5}$ it is
$J_{1}+J_{1}^{\prime}v_{y}$ (see Fig. 1). Furthermore, we assume that phonons
are dispersionless and the lattice energy for a pair of the up- and down-
triangles is,
$U_{lat}=\frac{K}{2}\left(\sum_{i=1}^{3}\mathbf{u}_{i}^{2}+\mathbf{v}^{2}\right),$
(18)
where $K$ is the spring constant.
For the $120^{\circ}$ structure shown in Fig. 3(a), the magnetoelectric
constant $\alpha$ in Eq.(11) is given by
$\alpha=\frac{1}{\left(1-g\right)}\frac{3Q\langle\mu\rangle
J_{1}^{\prime}}{2(J_{1}+J_{2})Kv},$ (19)
where $Q=-2e$ is the charge of the oxygen ion,
$\langle\mu\rangle=2\mu_{B}\langle S\rangle$ is the average magnetic moment on
each site, $v$ is the unit cell volume, and
$g=\frac{15}{8}\frac{\left(J_{1}^{\prime}\langle
S\rangle\right)^{2}}{(J_{1}+J_{2})K}$ (20)
is the dimensionless spin-lattice coupling constant.
An estimate for the magnetoelectric constant, $\alpha\sim 10^{-3}$, for the
model parameters appropriate for the KITPite structure ($S=2$, $J_{1}\sim
3$meV, $K\sim 6\mbox{eV}\cdot\mbox{\AA}^{-2}$,
$\frac{J^{\prime}_{1}}{J_{1}}\sim 3.5\mbox{\AA}^{-1}$, and
$v=177\mbox{\AA}^{3}$) agrees well with the result of ab initio
calculations.DelaneyPRL2009 Furthermore, $\chi_{\rm m}\sim 2\cdot 10^{-4}$,
so that $\left(\frac{\alpha}{{\chi}_{\rm m}}\right)^{2}\sim 25$. Thus, the
electromagnon peak in KITPite should be much stronger than the
antiferromagnetic resonance peak, which is also the case for rare earth
manganites with a spiral ordering.ValdesPRL2009
At $g=1$, the magnetoelectric constant diverges and so do the dielectric and
magnetic susceptibilities:
$\chi_{e},\chi_{m}\propto\frac{1}{1-g}.$ (21)
Since in our model there are two polar phonons coupled to a magnon with a
given polarization (one in the up-triangle and another in the down-triangle),
the magnetoelectric constant comes close to its upper bound but does not reach
it at $g=1$:
$\left[\frac{\alpha}{\sqrt{\chi_{e}\chi_{m}}}\right]_{g=1}\approx 0.985.$ (22)
Surprisingly, the softening of the $q_{0}$ magnon mode, which is not coupled
to polar lattice distortions, occurs at a lower value $g_{0}<1$. Since
$\omega_{0}=0$, the softening in this case means vanishing velocity of the
$q_{0}$-mode. The velocity vanishes, because away from the $\Gamma$-point in
the magnetic Brillouin zone magnons with different symmetry become mixed and
the $q_{0}$-mode is coupled to the electromagnon modes. As the spin-lattice
coupling grows and the electromagnon frequency decreases, the lowest-frequency
magnon branch is pushed down, which ultimately reduces the velocity of the
$q_{0}$-mode to zero.
## V Magnetoelastic instabilities
Figure 3: (Color online) The minimal-energy spin configurations of the Kagomé
magnet for three different values of the spin-lattice coupling: (a) the
120∘-state with zero wave vector, (b) the incommensurate ferroelectric state,
and (c) the collinear multiferroic state. The short solid arrows show the spin
directions, while the short empty arrows denote the shifts of the ligand ions
in these states. The long solid and empty arrows show the direction of,
respectively, the spontaneous magnetization and polarization.
Although KITPite, for which $g\sim 0.05$, is far away from the instabilities
discussed in the previous section, it is interesting to study behavior of
magnetoelectric materials when the spin-lattice coupling becomes strong, in
particular, in view of the dramatic magnetoelectric effects recently observed
in multiferroics. As the magnetoelectric constant becomes large close to the
transition between magnetoelectric and multiferroic states, it is important to
understand possible scenarios of such a transition.
In this section we present the analytical and numerical study of the phase
diagram of the KITPite layer for strong spin-lattice couplings. In particular,
we show that none of the continuous transitions involving magnon softening,
discussed in Sec. IV, actually takes place, as the strong spin-lattice
coupling makes the frustrated Kagomé magnet unstable towards a first-order
magnetoelastic transition that relieves the frustration. This frustration-
driven instability is similar to the one found in spinels, where a collinear
ordering of spins appears together with a lattice deformation.LeePRL2000 ;
TchernyshyovPRL2002 ; TchernyshyovPRB2002 ; PencPRL2004 We show that the
transformation of a non-collinear magnetoelectric state into a collinear
multiferroic state can involve two transitions and an intermediate phase,
which is ferroelectric but not ferromagnetic.
To understand the origin of magnetoelastic instabilities at strong spin-
lattice coupling, we first consider a single up-triangle and integrate out the
lattice degrees of freedom, $\mathbf{u}_{i}$ ($i=1,2,3$). Then the total
energy of the triangle takes the form of an effective spin Hamiltonian with
quadratic and bi-quadratic interactions:
$E_{\triangle}=\sum_{\langle
i,j\rangle}\left[J_{1}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\frac{\left(J^{\prime}_{1}\right)^{2}}{2K}(\mathbf{S}_{i}\cdot\mathbf{S}_{j})^{2}\right].$
(23)
The bi-quadratic interactions favor collinear spins and for
${\tilde{g}}=\frac{15}{8}\frac{\left(J_{1}^{\prime}S\right)^{2}}{J_{1}K}=\frac{\left(J_{1}+J_{2}\right)}{J_{1}}g>\frac{5}{6}$,
the lowest-energy spin configuration is a collinear state of the
$\uparrow\uparrow\downarrow$ type (the spins lie in the lattice plane).
Similarly, an effective spin Hamiltonian for a down-triangle, where the
exchange along all bonds is mediated by a single ligand ion located inside the
triangle, has the form,
$\displaystyle E_{\nabla}$ $\displaystyle=$ $\displaystyle\sum_{\langle
i,j\rangle}\left[J_{1}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\frac{\left(J^{\prime}_{1}\right)^{2}}{2K}(\mathbf{S}_{i}\cdot\mathbf{S}_{j})^{2}\right]$
(24) $\displaystyle+\frac{\left(J^{\prime}_{1}\right)^{2}}{2K}\sum_{i\neq
j\neq
k}\left(\mathbf{S}_{i}\cdot\mathbf{S}_{j}\right)\left(\mathbf{S}_{j}\cdot\mathbf{S}_{k}\right).$
In this case the critical coupling is lower: ${\tilde{g}}=\frac{15}{32}$. Due
to the inequivalence of the up- and down-triangles in the KITPite structure
the transition from the $120^{\circ}$-state shown in Fig. 3a to a fully
collinear state shown in Fig. 3c goes in two steps via an intermediate state,
where only the spins in the down-triangles are collinear while the spins in
the up-triangles are still non-collinear (see Fig. 3b).
Figure 4: (Color online) Plotted is the zero-temperature phase diagram of the
Kagomé layer for large values of the spin-lattice coupling
${\tilde{g}}=\frac{15}{8}\frac{\left(J_{1}^{\prime}\langle
S\rangle\right)^{2}}{J_{1}K}=\frac{\left(J_{1}+J_{2}\right)}{J_{1}}g$. For
relatively small $\frac{J_{2}}{J_{1}}$ the magnetoelectric (ME) phase with the
120∘ spin ordering [see Fig. 3(a)] undergoes a first-order transition into the
incommensurate (IC) spin state, which is ferroelectric (FE) [see Fig. 3(b)],
as the coupling constant ${\tilde{g}}$ increases. Further increase of
${\tilde{g}}$ results in the transition into the collinear multiferroic (MF)
phase [see Fig. 3(c)], with the parallel spontaneous polarization and
magnetization, $\mathbf{P}\parallel\mathbf{M}$. For larger
$\frac{J_{2}}{J_{1}}$, the ME state undergoes a direct transition into the
fully collinear MF state. Also plotted are the dot-dash line, where the
$q_{0}$-mode in the ME state softens and the dotted line, at which the
magnetoelectric response of the ME state diverges (both phenomena do not occur
because of the intervening first-order transitions to the FE and MF states).
The transition to the intermediate state does not fully lift the frustration:
the number of combinations of the down-triangles with collinear spins and up-
triangles with the $120^{\circ}$-angle between spins grows exponentially with
the system size. In our model this degeneracy is removed by the next-nearest-
neighbor interactions between spins, which select the state shown in Fig. 3b.
The next-nearest-neighbor interactions in the vertical direction induce a
small spin canting, as a result of which spins in down-triangles are not
strictly collinear and the angles between spins in up-triangles deviate from
$120^{\circ}$ by the angle $\varphi\propto\frac{J_{2}}{J_{1}}$. Furthermore,
the nearest-neighbor interactions in the remaining two directions are
frustrated in the commensurate spin state with the wave vector
$\mathbf{Q}=2k\left(1,0\right)+k\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$,
where $k=\frac{\pi}{3a}$ ($a$ being the lattice constant). This frustration is
lifted, when the spin ordering becomes incommensurate with the lattice:
$k=\frac{\pi}{3a}+\delta$, where $\delta\propto\frac{J_{2}}{J_{1}}$.
The incommensurate state has zero net magnetization, but its electric
polarization is nonzero. For the state shown in Fig. 3(b) the polarization
vector is parallel to the $y$ axis. This polarization originates not from the
inverse Dzyaloshinskii-Moriya mechanism, which makes incommensurate spiral
states in e.g. $R$MnO3 ferroelectric, but from the fact that all bonds
connecting parallel spins are parallel to each other, which forces the ligand
ions in all down-triangles to shift in the same direction, since
$\left\\{\begin{array}[]{rcl}v_{x}&=&\eta\frac{1}{\sqrt{2}}\left(\mathbf{S}_{3}\cdot\mathbf{S}_{5}-\mathbf{S}_{3}\cdot\mathbf{S}_{4}\right),\\\
\\\
v_{y}&=&\eta\frac{1}{\sqrt{6}}\left(2\left(\mathbf{S}_{4}\cdot\mathbf{S}_{5}\right)-\mathbf{S}_{3}\cdot\mathbf{S}_{4}-\mathbf{S}_{3}\cdot\mathbf{S}_{5}\right),\end{array}\right.$
(25)
where $\eta=\sqrt{\frac{3}{2}}\frac{J^{\prime}_{1}}{K}$ and the labeling of
spins is the same as in Fig. 1. For the state shown in Fig. 3(b), where bonds
connecting (nearly) parallel spins are oriented along the $x$ axis, $v_{x}=0$,
while $v_{y}$ and hence the polarization $P_{y}$ is nonzero.
We note that the coupling of exchange interactions to strains, which gives
rise to magnetoelastic transitions in frustrated spinels, LeePRL2000 ;
TchernyshyovPRL2002 ; TchernyshyovPRB2002 also favors the ferroelectric
state. The absence of inversion symmetry in the KITPite layer allows for the
piezoelectric coupling,
$2u_{xy}E_{x}+(u_{xx}-u_{yy})E_{y},$ (26)
where $u_{ij}$ is the strain tensor, so that the ligand displacement,
$\mathbf{v}$, the electric polarization, $\mathbf{P}$, and the strains are
coupled to each other.
The zero-temperature phase diagram of the Kagomé layer with ${\tilde{g}}$ and
$\frac{J_{2}}{J_{1}}$ along the horizontal and vertical axes is shown in Fig.
4. For small $\frac{J_{2}}{J_{1}}$, the incommensurate ferroelectric (IC FE)
state, discussed above, intervenes between the magnetoelectric (ME) state with
the 120∘ spin ordering [see Fig. 3(a)] and the fully collinear multiferroic
(MF) state shown in Fig. 3(c), in which the spontaneous polarization and
magnetization are parallel to each other, $\mathbf{P}\parallel\mathbf{M}$. As
the ratio $\frac{J_{2}}{J_{1}}$ grows, the interval of the coupling constant
${\tilde{g}}$ where the intermediate state is stabilized shrinks and above the
tricritical point the ME state undergoes a direct transition into the
collinear MF state along the critical line
$\frac{J_{2}}{J_{1}}=\frac{5}{3}{\tilde{g}}-1$. Also plotted are the dot-dash
line, at which the $q_{0}$-mode would soften and the dotted line, at which
$\alpha$, $\chi_{\rm e}$, and $\chi_{\rm m}$ would diverge, if the ME state
would survive at strong spin-lattice couplings.
## VI Discussion
We showed that the static magnetoelectric response of non-collinear
antiferromagnets can be related to hybrid magnon-phonon modes coupled to both
electric and magnetic fields. Such magnetoelectric materials are analogs of
displacive ferroelectrics the dielectric response of which is governed by
optical phonon modes. If spins in an ordered state are collinear, the exchange
striction cannot couple an electric field to a single magnon, as the expansion
of scalar products of parallel or antiparallel spins begins with terms of
second order in $\delta\mathbf{S}$, which give rise to photoexcitation of a
two-magnon continuum (the so-called “charged magnons”DamascelliPRL1998 ).
Electromagnons in collinear magnets can still originate from mechanisms
involving relativistic effects, such as the exchange striction induced by the
antisymmetric Dzyaloshinskii-Moriya interaction, which is proportional to the
vector product of two spins. In $3d$ transition metal compounds such couplings
are weak compared to the exchange striction driven by the Heisenberg exchange,
so that the spectral weight of electromagnons in collinear magnets should be
relatively low.
Magnetoelectric materials with collinear spin orders may rather be analogs of
‘order-disorder’ ferroelectrics with the static magnetoelectric response
originating from thermal spin fluctuations. Cr2O3 seems to be an example of
such a material: its magnetoelectric coefficient passes through a maximum
below Néel temperature and then strongly decreases when temperature goes to
zero and spin fluctuations become suppressed.RadoPR1962 ; YatomPR1969
We note that the rotationally invariant coupling Eq.(1) may also originate
from purely electronic mechanisms, such as the polarization of electronic
orbitals induced by a magnetic ordering. SergienkoPRL2006 ; PicozziPRL99 ;
BulaevskiiPRB2008 ; SpaldinJPCM2008 ; Furukawa2009 Ab initio calculations
suggest that in rare earth manganites the electronic mechanisms of
magnetoelectric coupling are as important as the exchange
striction.PicozziPRL99 On the other hand, the increase of the spectral weight
of the electromagnon peaks in $R$MnO3 below the spiral ordering temperature
occurs largely at the expense of the strength of the optical phonon peak at
$\sim 100$cm-1, suggesting the dominant role of the spin-lattice coupling.
SushkovJPCM2008 ; ValdesPRL2009 If electronic mechanisms dominate and an
electromagnon gets its spectral weight from frequencies much higher than those
of optical phonons, Eq.(16) should to be modified in an obvious way, while
Eq.(17), where $\frac{\omega_{\rm mag}}{\omega_{\rm ph}}$ should be replaced
by $0$, is still valid.
We note that the non-collinearity of spins by itself does not guarantee strong
magnetoelectric effect and electromagnon peaks – the crystal structure is
equally important. Thus, in the layered Kagomé antiferromagnet, the iron
jarosite KFe3(OH)6(SO4)2,GroholNatMat2005 which has the spin ordering shown
in Fig. 1, the ligand ions are located outside of both up- and down-triangles,
which cancels the magnetoelectric effect due to the Heisenberg exchange
striction. The cancellation also occurs in triangular magnets with the
$120^{\circ}$ spin ordering, as they contain three different spin triangles,
such that spins in one triangle are rotated by $\pm 120^{\circ}$ with respect
to spins in two other trianglesDelaneyPRL2009 (more generally, the linear
magnetoelectric effect can only be induced by a spin ordering with zero wave
vector). We note, however, that the lattice trimerization in hexagonal
manganitesAkenNatMat2004 makes the three types of spin triangles inequivalent
and destroys the cancellation. This can be also seen from the symmetry
properties of the A1,2 and B1,2 phases of hexagonal manganitesFiebigJAP2003
allowing for the magnetoelectric term $E_{x}H_{y}-E_{y}H_{x}$ in the A1-phase,
which has a toroidal moment, and the term $E_{x}H_{x}+E_{y}H_{y}$ in the
A2-phase, which has a magnetic monopole moment. Whether electromagnons in
these phases can be observed, depends on the magnitude of the trimerization
and remains to be explored.LeeNature2008
## VII Conclusions
In conclusion, we showed that magnets with non-collinear spin orders resulting
in a linear magnetoelectric effect may also show electromagnon peaks in
optical absorption spectrum. While electromagnons should be present in many
non-collinear magnets, the specific feature of magnetoelectric materials is
that some magnon modes can be excited by both electric and magnetic fields,
i.e. electromagnons are also antiferromagnetic resonances. We derived a simple
relation Eq.(17) between the ratio of the spectral weights of the
electromagnon and antiferromagnetic resonance peaks and the ratio of the
static magnetoelectric constant and magnetic susceptibility, which can be used
to estimate the strength of electromagnon peaks on the basis of dc
measurements.
To make our consideration more specific, we considered a Kagomé lattice magnet
with the KITPite structure, where the ligand ions are positioned in a way that
gives rise to a relatively strong linear magnetoelectric effect.DelaneyPRL2009
Using the symmetry analysis we identified the magnon modes that are coupled to
both electric and magnetic fields and give rise to the linear magnetoelectric
effect. We showed that the softening of these modes at a strong spin-lattice
coupling results in the divergence of the magnetoelectric constant as well as
of magnetic and dielectric susceptibilities, signaling the instability of the
magnetoelectric state towards a multiferroic state with spontaneously
generated $\mathbf{P}$ and $\mathbf{M}$. However, the detailed study of the
phase diagram of this model revealed that the electromagnon softening does not
actually take place, since the first-order transition to the collinear
multiferroic state occurs at a lower value of the spin-lattice coupling. In
some region of model parameters a ferroelectric incommensurate-spiral phase
intervenes between the magnetoelectric and multiferroic phases. While in known
spiral magnets, ferroelectricity is likely induced by the inverse
Dzyaloshinskii-Moriya mechanism, in our model it results from the stronger
exchange striction mechanism due to the collinearity of spins in half of the
triangles. These magnetoelastic instabilities are typical for frustrated
magnets, where non-collinear spin orders usually occur. We also discussed the
possibility to observe electromagnons in known magnetoelectric materials. We
hope that our study will stimulate experimental work in this direction.
###### Acknowledgements.
This work was supported by the Zernike Institute for Advanced Materials and by
the Stichting voor Fundamenteel Onderzoek der Materie (FOM).
## References
* (1) A review: S.-W. Cheong and M. Mostovoy, Nature Mater. 6, 13 (2007).
* (2) A review: T. Kimura, Annu. Rev. Mater. Res. 37, 387 (2007).
* (3) A review: R. Ramesh and N. Spaldin, Nature Mater. 6, 21 (2007).
* (4) T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature (London) 426, 55 (2003).
* (5) N. Hur, S. Park, P.A. Sharma, J.S. Ahn, S. Guha, and S-W. Cheong, Nature (London) 429, 392 (2004).
* (6) T. Goto, T. Kimura, G. Lawes, A.P. Ramirez, and Y. Tokura, Phys. Rev. Lett. 92, 257201 (2004).
* (7) G. Smolenskii and I. Chupis, Sov. Phys. Usp. 25, 475 (1982).
* (8) A. Pimenov, A.A. Mukhin, V.Yu. Ivanov, V.D. Travkin, A.M. Balbashov and A. Loidl, Nature Physics 2, 97 (2006).
* (9) A. Pimenov, T. Rudolf, F. Mayr, A. Loidl, A.A. Mukhin, and A.M. Balbashov, Phys. Rev. B 74, 100403(R) (2006).
* (10) R. Valdés Aguilar, A.B. Sushkov, C.L. Zhang, Y.J. Choi, S.-W. Cheong, and H.D. Drew, Phys. Rev. B 76, 060404(R) (2007).
* (11) A.B. Sushkov, R.V. Aguilar,S. Park, S.-W. Cheong, and H.D. Drew, Phys. Rev. Lett. 98, 027202 (2007).
* (12) H. Katsura, N. Nagaosa, and A.V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005).
* (13) M. Kenzelmann, A.B. Harris, S. Jonas, C. Broholm, J. Schefer, S.B. Kim, C.L. Zhang, S.-W. Cheong, O.P. Vajk, and J.W. Lynn , Phys. Rev. Lett. 95, 087206 (2005).
* (14) I.A. Sergienko and E. Dagotto, Phys. Rev. B 73, 094434 (2006).
* (15) M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006).
* (16) A. Malashevich and D. Vanderbilt, Phys. Rev. Lett. 101, 037210 (2008).
* (17) H. Katsura, A. V. Balatsky, and N. Nagaosa, Phys. Rev. Lett. 98, 027203 (2007).
* (18) N. Kida, Y. Ikebe, Y. Takahashi, J.P. He, Y. Kaneko, Y. Yamasaki, R. Shimano, T. Arima, N. Nagaosa, and Y. Tokura, Phys. Rev. B 78, 104414 (2008).
* (19) A.B. Sushkov, M. Mostovoy, R. Valdés Aguilar, S.-W. Cheong and H. D. Drew, J. Phys.: Condens. Matter 20,434210 (2008).
* (20) Y. Takahashi, N. Kida, Y. Yamasaki, J. Fujioka, T. Arima, R. Shimano, S. Miyahara, M. Mochizuki, N. Furukawa, and Y. Tokura, Phys. Rev. Lett. 101, 187201 (2008).
* (21) R. Valdés Aguilar, M. Mostovoy, A.B. Sushkov, C.L. Zhang, Y.J. Choi, S-W. Cheong, and H.D. Drew, Phys. Rev. Lett. 102, 047203 (2009).
* (22) B. Lorenz, Y.-Q. Wang, and C.-W. Chu, Phys. Rev. B 76, 104405 (2007).
* (23) I.A. Sergienko, C. Sen, and E. Dagotto, Phys. Rev. Lett. 97, 227204 (2006).
* (24) S. Picozzi, K. Yamauchi, B. Sanyal, I.A. Sergienko, and E. Dagotto, Phys. Rev. Lett. 99, 227201 (2007).
* (25) L.D. Landau and E.M. Lifshitz, Electrodynamics of Contiuous Media (Pergamon, Oxford, 1984).
* (26) For a recent review see, M. Fiebig, J. Appl. Phys. D 38, R123 (2005).
* (27) N.A. Spaldin, M. Fiebig, and M. Mostovoy, J. Phys.: Condens. Matter 20, 434203 (2008).
* (28) K.T. Delaney, M. Mostovoy, and N.A. Spaldin, Phys. Rev. Lett. 102, 157203 (2009).
* (29) L.N. Bulaevskii, C.D. Batista, M.V. Mostovoy, and D.I. Khomskii, Phys. Rev. B 78, 024402 (2008).
* (30) A. B. Harris, C. Kallin and A. J. Berlinsky, Phys. Rev. B 45, 2899 (1992).
* (31) M. Elhajal, B. Canals and C. Lacroix, Phys. Rev. B 66,014422 (2002).
* (32) W.F. Brown, Jr., R.M. Hornreich, and S. Shtrikman, Phys. Rev. 168, 574 (1968).
* (33) S.-H. Lee, C. Broholm, T. H. Kim, W. Ratcliff, and S.-W. Cheong, Phys. Rev. Lett. 84, 3718 (2000).
* (34) O. Tchernyshyov, R. Moessner, and S.L. Sondhi, Phys. Rev. Lett. 88, 067203 (2002).
* (35) O. Tchernyshyov, R. Moessner, and S. L. Sondhi, Phys. Rev. B 66, 064403 (2002).
* (36) K. Penc, N. Shannon, and H. Shiba, Phys. Rev. Lett. 93, 197203 (2004).
* (37) A. Damascelli, D. van der Marel, M. Grüninger, C. Presura, T.T.M. Palstra, J. Jegoudez and A. Revcolevschi, Phys. Rev. Lett 81, 918 (1998).
* (38) G. Rado, Phys. Rev. 128, 2546 (1962).
* (39) H. Yatom and R. Englman, Phys. Rev. 188, 793 (1969).
* (40) S. Miyahara and N. Furukawa, preprint ArXive:0811.4082.
* (41) D. Grohol, K. Matan, J.-H. Cho, S.-H. Lee, J.W. Lynn, D.G. Nocera, and Y.S. Lee, Nature Mater. 4, 323 (2005).
* (42) See e.g. B.B. van Aken, T.T.M. Palstra, A. Filippetti, and N.A. Spaldin, Nature Mater. 3, 164 (2004).
* (43) The phases of hexagonal manganites are described e.g. in M. Fiebig, Th. Lottermoser, and R. V. Pisarev, J. Appl. Phys. 93, 8194 (2003).
* (44) A large trimerization below Néel temperature was reported in S. Lee, A. Pirogov, M. Kang, K.-H. Jang, M. Yonemura, T. Kamiyama, S.-W. Cheong, F. Gozzo, N. Shin, H. Kimura, Y. Noda, and J.-G. Park, Nature 451, 805 (2207).
|
arxiv-papers
| 2009-07-17T12:10:31 |
2024-09-04T02:49:04.040731
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. A. van der Vegte, C. P. van der Vegte, and M. Mostovoy",
"submitter": "M.A. van der Vegte",
"url": "https://arxiv.org/abs/0907.3055"
}
|
0907.3285
|
# $CP$–Violation in $K,$ $B$ and $B_{s}$ decays
Fayyazuddin
National Centre for Physics and Physics Department Quaid-i-Azam University,
Islamabad
([email protected])
###### Abstract
In this review we give an overview of $CP$-violation for $K^{0}(\bar{K}^{0}),$
$B_{q}^{0}(\bar{B}_{q}^{0}),$ $q=d,s$ systems. Direct $CP-$violation and
mixing induced $CP$-violation are discussed.
## 1 Introduction
Symmetries have played an important role in particle physics. In quantum
mechanics a symmetry is associated with a group of transformations under which
a Lagrangian remains invariant. Symmetries limit the possible terms in a
Lagrangian and are associated with conservation laws. Here we will be
concerned with the role of discrete symmetries: Space Reflection (Parity) $P$:
$\vec{x}\rightarrow-\vec{x}$, Time Reversal $T$: $t\rightarrow-t$ and Charge
Conjugation $C$: $particle\rightarrow antiparticle$.
Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD) respect all
these symmetries. Also, all Lorentz invariant local quantum field theories are
$CPT$ invariant. However, in weak interactions $C$ and $P$ are maximally
violated separately but as we will see below, $CP$ is conserved.
First indication of parity violation was revealed in the decay of a particle
with spin parity $J^{P}=0^{-},$ called $K$-meson into two modes
$K^{0}\rightarrow\pi^{+}\pi^{-}$ (parity violating), and
$K^{0}\rightarrow\pi^{+}\pi^{-}$ $\pi^{0}$(parity conserving).
Lee and Yang in 1956, suggested that there is no experimental evidence for
parity conservation in weak interaction. They suggested number of experiments
to test the validity of space reflection invariance in weak decays. One way to
test this is to measure the helicity of outgoing muon in the decay:
$\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$
The helicity of muon comes out to be negative, showing that parity
conservation does not hold in this decay. In the rest frame of the pion, since
$\mu^{+}$ comes out with negative helicity, the neutrino must also come out
with negative helicity because of the spin conservation. Thus confirming the
fact that neutrino is left handed.
$\pi^{+}\rightarrow\mu^{+}(-)+\nu_{\mu}$
Under charge conjugation,
$\pi^{+}\overset{C}{\rightarrow}\pi^{-}\qquad\mu^{+}\overset{C}{\rightarrow}\mu^{-}\qquad\nu_{\mu}\overset{C}{\rightarrow}\bar{\nu}_{\mu}$
Helicity $\mathcal{H}=\frac{\vec{\sigma}\cdot\vec{p}}{\left|\vec{p}\right|}$
under $C$ and $P$ transforms as,
$\mathcal{H}\overset{C}{\rightarrow}\mathcal{H},\qquad\mathcal{H}\overset{P}{\rightarrow}-\mathcal{H}$
Invariance under $C$ gives,
$\Gamma_{\pi^{+}\rightarrow\mu^{+}(-)\nu_{\mu}}=\Gamma_{\pi^{-}\rightarrow\mu^{-}(-)\bar{\nu}_{\mu}}$
Experimentally,
$\Gamma_{\pi^{+}\rightarrow\mu^{+}(-)\nu_{\mu}}>>\Gamma_{\pi^{-}\rightarrow\mu^{-}(-)\bar{\nu}_{\mu}}$
showing that $C$ is also violated in weak interactions. However, under $CP$,
$\Gamma_{\pi^{+}\rightarrow\mu^{+}(-)\nu_{\mu}}\overset{CP}{\rightarrow}\text{
\ }\Gamma_{\pi^{-}\rightarrow\mu^{-}(+)\bar{\nu}_{\mu}}$
which is seen experimentally. Thus, $CP$ conservation holds in weak
interaction.
In the Standard Model, the fermions for each generation in their left handed
chirality state belong to the representation,
$\displaystyle\left(\begin{array}[]{c}u_{i}\\\ d_{i}\end{array}\right)$
$\displaystyle:$ $\displaystyle q(3,2,1/3)$ $\displaystyle\bar{u}_{i}$
$\displaystyle:$ $\displaystyle(\bar{3},1,-4/3)$ $\displaystyle\bar{d}_{i}$
$\displaystyle:$ $\displaystyle(\bar{3},1,2/3)$
$\displaystyle\left(\begin{array}[]{c}\nu_{e^{-}}\\\
e_{i}^{-}\end{array}\right)$ $\displaystyle:$ $\displaystyle l(1,2,-1/2)$
$\displaystyle e_{i}^{+}$ $\displaystyle:$ $\displaystyle(1,1,1)$
of the electroweak unification group $SU_{C}(3)\times SU_{L}(2)\times
U_{Y}(1)$. Hence, the weak interaction Lagrangian for the charged current in
the Standard Model is given by,
$\mathcal{L}_{W}=\bar{\psi}_{i}\gamma^{\mu}(1-\gamma^{5})\psi_{j}W_{\mu}^{+}+h.c.$
where $\psi_{i}$ is any of the left-handed doublet ($i$ is the generation
index). We note that the weak eigenstates $d^{\prime},s^{\prime}$ and
$b^{\prime}$ are not equal to the mass eigenstates $d,s$ and $b$. They are
related to each other by a unitarity transformation,
$\left(\begin{array}[]{c}d^{\prime}\\\ s^{\prime}\\\
b^{\prime}\end{array}\right)=V\left(\begin{array}[]{c}d\\\ s\\\
b\end{array}\right)$ (3)
where $V$ is called the $CKM$ matrix.
$V=\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\ V_{cd}&V_{cs}&V_{cb}\\\
V_{td}&V_{ts}&V_{tb}\end{array}\right)$
$\simeq\left(\begin{array}[]{ccc}1-\frac{1}{2}\lambda^{2}&\lambda&A\lambda^{3}\left(\rho-i\eta\right)\\\
-\lambda&1-\frac{1}{2}\lambda^{2}&A\lambda^{2}\\\
A\lambda^{3}\left(1-\rho-i\eta\right)&-A\lambda^{2}&1\end{array}\right)+O\left(\lambda^{4}\right),\,\left.\lambda=0.22\right.$
(4)
The unitarity of $V$, $VV^{\dagger}=1$ gives,$\left[\text{Fig.1}\right]$
$V_{ud}^{\ast}V_{ub}+V_{cb}^{\ast}V_{cd}+V_{td}^{\ast}V_{tb}=0$ (5)
The second line in equation (4) expresses $V$ in terms of Wolfenstien
parametrization. Thus,
$\displaystyle V_{cb}$ $\displaystyle=$ $\displaystyle A\lambda^{2}$
$\displaystyle V_{ub}$ $\displaystyle=$
$\displaystyle\left|V_{ub}\right|e^{-i\gamma}$ $\displaystyle V_{td}$
$\displaystyle=$ $\displaystyle\left|V_{td}\right|e^{-i\beta}$
where,
$\tan{\gamma}=\frac{\eta}{\rho}=\frac{\bar{\eta}}{\bar{\rho}},\quad\tan{\beta}=\frac{\bar{\eta}}{1-\bar{\rho}},\quad\bar{\rho}=\rho(1-\frac{\lambda^{2}}{2}),\quad\bar{\eta}=\eta(1-\frac{\lambda^{2}}{2}).$
In order to show that $\mathcal{L}_{W}$ is $CP$-invariant, we first note that
under $C$, $P$ and $T$ operations the Dirac spinor $\Psi$ transforms as
follows:
$\displaystyle P\Psi\left(t,\vec{x}\right)P^{-1}$ $\displaystyle=$
$\displaystyle\gamma^{0}\Psi\left(t,-\vec{x}\right)$ $\displaystyle
C\Psi\left(t,\vec{x}\right)C^{-1}$ $\displaystyle=$
$\displaystyle-i\gamma^{2}\gamma^{0}\bar{\Psi}^{T}\left(t,\vec{x}\right)$ (6)
$\displaystyle T\Psi\left(t,\vec{x}\right)T^{-1}$ $\displaystyle=$
$\displaystyle\gamma^{1}\gamma^{3}\Psi\left(-t,\vec{x}\right)$
The effect of transformations $C$, $P$ and $CP$ on various quantities that
appear in a gauge theory Lagrangian are given below:
$\begin{array}[]{ccccc}\text{Transformation}&\text{Scalar}&\text{Pseudoscalar}&\text{
Vector}&\text{Axial vector}\\\
&\bar{\Psi}_{i}\Psi_{j}&i\bar{\Psi}_{i}\gamma_{5}\Psi_{j}&\bar{\Psi}_{i}\gamma^{\mu}\Psi_{j}&\bar{\Psi}_{i}\gamma^{\mu}\gamma^{5}\Psi_{j}\\\
P&\bar{\Psi}_{i}\Psi_{j}&-i\bar{\Psi}_{i}\gamma_{5}\Psi_{j}&\eta\left(\mu\right)\bar{\Psi}_{i}\gamma^{\mu}\Psi_{j}&-\eta\left(\mu\right)\bar{\Psi}_{i}\gamma^{\mu}\gamma^{5}\Psi_{j}\\\
C&\bar{\Psi}_{j}\Psi_{i}&i\bar{\Psi}_{j}\gamma_{5}\Psi_{i}&-\bar{\Psi}_{j}\gamma^{\mu}\Psi_{i}&\bar{\Psi}_{j}\gamma^{\mu}\gamma^{5}\Psi_{i}\\\
CP&\bar{\Psi}_{j}\Psi_{i}&-i\bar{\Psi}_{j}\gamma_{5}\Psi_{i}&-\eta\left(\mu\right)\bar{\Psi}_{j}\gamma^{\mu}\Psi_{i}&-\eta\left(\mu\right)\bar{\Psi}_{j}\gamma^{\mu}\gamma^{5}\Psi_{i}\end{array}$
The vector bosons associated with the electroweak unification group
$SU_{L}\left(2\right)\times U\left(1\right)$ transform under $CP$ as:
$\displaystyle
W_{\mu}^{\pm}\left(\vec{x},t\right)\overset{CP}{\rightarrow}-\eta\left(\mu\right)W_{\mu}^{\mp}\left(-\vec{x},t\right)$
$\displaystyle
Z_{\mu}\left(\vec{x},t\right)\overset{CP}{\rightarrow}-\eta\left(\mu\right)Z_{\mu}\left(-\vec{x},t\right)$
(7) $\displaystyle
A_{\mu}\left(\vec{x},t\right)\overset{CP}{\rightarrow}-\eta\left(\mu\right)A_{\mu}\left(-\vec{x},t\right)$
where,
$\eta\left(\mu\right)=\begin{cases}+1,&\text{if $\mu$=0}\\\ -1,&\text{if
$\mu$=1,2,3}\end{cases}$
The Lagrangian transforms as:
$\displaystyle\mathcal{L}_{W}$ $\displaystyle=$
$\displaystyle\bar{\psi}_{i}\gamma^{\mu}(1-\gamma^{5})\psi_{j}W_{\mu}^{+}+h.c.$
$\displaystyle\overset{CP}{\rightarrow}$
$\displaystyle-\eta(\mu)\bar{\psi}_{j}\gamma^{\mu}(1-\gamma^{5})\psi_{i}(-\eta(\mu))W_{\mu}^{-}+h.c.$
Thus, the weak interaction Lagrangian in the Standard Model violates $C$ and
$P$ but is $CP$-invariant.
It is instructive to discuss the restrictions imposed by $CPT$ invariance.
$CPT$ invariance implies,
${}_{\text{out}}\left\langle f\left|\mathcal{L}\right|X\right\rangle$
$\displaystyle=$ ${}_{\text{out}}\left\langle
f\left|\left(CPT\right)^{-1}\mathcal{L}CPT\right|X\right\rangle$ (8)
$\displaystyle=$
$\displaystyle\eta_{T}^{x\ast}\eta_{T}^{f}\,\,{}_{\text{in}}\left\langle\tilde{f}\left|\left(CP\right)^{\dagger}\mathcal{L}^{\dagger}\left(CP\right)^{-1\dagger}\right|X\right\rangle^{\ast}$
$\displaystyle=$ $\displaystyle\eta_{T}^{x\ast}\eta_{T}^{f}\left\langle
X\left|\left(CP\right)^{-1}\mathcal{L}\left(CP\right)\right|f\right\rangle_{\text{in}}$
$\displaystyle=$
$\displaystyle-\eta_{T}^{x\ast}\eta_{T}^{f}\eta_{CP}^{f}\left\langle\bar{X}\left|\mathcal{L}S_{f}\right|\bar{f}\right\rangle_{\text{out}}$
$\displaystyle=$
$\displaystyle\eta_{f}\,\,{}_{\text{out}}\left\langle\bar{f}\left|S_{f}^{\dagger}\mathcal{L}^{\dagger}\right|\bar{X}\right\rangle^{\ast}$
$\displaystyle=$
$\displaystyle\eta_{f}\,\exp(2i\delta_{f})_{\text{out}}\left\langle\bar{f}\left|\mathcal{L}\right|\bar{X}\right\rangle^{\ast}$
Hence, we get:
${}_{\text{out}}\left\langle\bar{f}\left|\mathcal{L}\right|\bar{X}\right\rangle$
$\displaystyle=$
$\displaystyle\eta_{f}\,\exp(2i\delta_{f})_{\text{out}}\left\langle
f\left|\mathcal{L}\right|X\right\rangle^{\ast}$
$\displaystyle\bar{A}_{\bar{f}}$ $\displaystyle=$
$\displaystyle\eta_{f}\exp(2i\delta_{f})A_{f}^{\ast}$ (9)
In deriving the above result, we have put $\tilde{f}=f$ where $\tilde{f}$
means that momenta and spin are reversed. Since we are in the rest frame of
$X$, $T$ will reverse only magnetic quantum number and we can drop
$\tilde{f}$. Further we have used,
$CP\left|X\right\rangle=-\left|\bar{X}\right\rangle$ (10)
$CP\left|f\right\rangle=\eta_{f}^{CP}\left|\bar{f}\right\rangle$ (11)
$\left|f\right\rangle_{\text{in}}=S_{f}\left|f\right\rangle_{\text{out}}=\exp(2i\delta_{f})\left|f\right\rangle_{\text{in}}$
(12)
where $\delta_{f}$ is the strong interaction phase shift. If $CP$-invariance
holds, then,
${}_{\text{out}}\left\langle f\left|\mathcal{L}\right|X\right\rangle=_{\text{
out}}\left\langle\bar{f}\left|\mathcal{L}\right|\bar{X}\right\rangle\newline
\Rightarrow\bar{A}_{\bar{f}}=A_{f}.$
Thus, the necessary condition for $CP$-violation is that the decay amplitude
$A$ should be complex. In view of our discussion above, under $CP$ an operator
$O\left(\vec{x},t\right)$ is replaced by,
$O\left(\vec{x},t\right)\rightarrow O^{\dagger}\left(-\vec{x},t\right)$ (13)
The effective Lagrangian has the structure
($\mathcal{L}^{\dagger}=\mathcal{L}$),
$\mathcal{L}=aO+a^{\ast}O^{\dagger}$ (14)
Hence, $CP$-violation requires $a^{\ast}\neq a$. We now discuss the
implication of $CPT$ constraint with respect to $CP$ violation of weak decays.
The weak amplitude is complex; it contains the final state strong phase
$\delta_{f}$ and in addition it may also contain a weak phase $\phi$. Taking
out both these phases,
$A_{f}=\exp(i\phi)F_{f}=\exp(i\phi)\exp(i\delta_{f})\left|F_{f}\right|$
$CPT$ (Eq. (9)) gives,
$\bar{A}_{\bar{f}}=\exp(2i\delta_{f})\exp(-i\phi)\exp(-i\delta_{f})\left|F_{f}\right|=\exp(-i\phi)F_{f}$
We conclude that the weak interaction Lagrangian in the Standard Model is $CP$
invariant and since $CP$ violation has been observed in hadronic sector (only
in $B,B_{s}$ and $K$ decays) and not in leptonic sector, it is a consequence
of mismatch between weak and mass eigenstates (i.e. the phases in $CKM$
matrix) and/or the mismatch between $CP$-eigenstates,
$\left|X_{1,2}^{0}\right\rangle=\frac{1}{\sqrt{2}}\left[\left|X^{0}\right\rangle\mp\left|\bar{X}^{0}\right\rangle\right];\text{
}CP\left|X_{1,2}^{0}\right\rangle=\pm\left|X_{1,2}^{0}\right\rangle$ (15)
and the mass eigenstates i.e. $CP$-violation in the mass matrix. $CP$\-
violation due to mass mixing and in the decay amplitude has been
experimentally observed in $K^{0}$ and $B_{d}^{0}$. For $B_{s}$ decays, the
$CP$-violation in the mass matrix is not expected in the Standard Model. In
fact time dependent $CP$-violation asymmetry gives a clear way to observe
direct $CP$-violation in $B$ and $B_{s}$ decays.
If $CP$ is conserved,
$\displaystyle\left\langle X_{2}\left|H\right|X_{1}\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
X_{2}\left|\left(CP\right)^{-1}H\left(CP\right)\right|X_{1}\right\rangle$
$\displaystyle=$ $\displaystyle-\left\langle
X_{2}\left|H\right|X_{1}\right\rangle$
then,
$\left\langle X_{2}\left|H\right|X_{1}\right\rangle=0.$
Thus $\left|X_{1}\right\rangle$ and $\left|X_{2}\right\rangle$ are also mass
eigenstates. They form a complete set (in units $\hbar=c=1$),
$\displaystyle\left|\psi\left(t\right)\right\rangle$ $\displaystyle=$
$\displaystyle
a\left(t\right)\left|X_{1}\right\rangle+b\left(t\right)\left|X_{2}\right\rangle$
$\displaystyle i\frac{d\left|\psi\left(t\right)\right\rangle}{dt}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}m_{1}-\frac{i}{2}\Gamma_{1}&0\\\
0&m_{2}-\frac{i}{2}\Gamma_{2}\end{array}\right)\left|\psi\left(t\right)\right\rangle.$
(18)
The solution is,
$\displaystyle a\left(t\right)$ $\displaystyle=$ $\displaystyle
a\left(0\right)\exp\left(-im_{1}t-\frac{1}{2}\Gamma_{1}t\right)$
$\displaystyle b\left(t\right)$ $\displaystyle=$ $\displaystyle
b\left(0\right)\exp\left(-im_{2}t-\frac{1}{2}\Gamma_{2}t\right)$
Suppose we start with the state $\left|X^{0}\right\rangle$, i.e.,
$\left|\psi\left(0\right)\right\rangle=\left|X^{0}\right\rangle$
Then we get,
$\displaystyle\left|\psi\left(t\right)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\exp\left(-im_{1}t-\frac{1}{2}\Gamma_{1}t\right)\left|X_{1}\right\rangle\right.$
(19)
$\displaystyle+\left.\exp\left(-im_{2}t-\frac{1}{2}\Gamma_{2}t\right)\left|X_{2}\right\rangle\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left\\{\left[\exp\left(-im_{1}t-\frac{1}{2}\Gamma_{1}t\right)\right.\right.$
$\displaystyle\left.+\exp\left(-im_{2}t-\frac{1}{2}\Gamma_{2}t\right)\right]\left|X^{0}\right\rangle$
$\displaystyle-\left[\exp\left(-im_{1}t-\frac{1}{2}\Gamma_{1}t\right)\right.$
$\displaystyle\left.\left.-\exp\left(-im_{2}t-\frac{1}{2}\Gamma_{2}t\right)\right]\left|\bar{X}^{0}\right\rangle\right\\}$
However, in $\left|X^{0}\right\rangle-\left|\bar{X}^{0}\right\rangle$ basis,
$\displaystyle|\psi(t)\rangle$ $\displaystyle=$ $\displaystyle
a(t)|X^{0}\rangle+\bar{a}(t)|\bar{X}^{0}\rangle$
$\displaystyle\frac{i}{dt}|\psi(t)\rangle$ $\displaystyle=$ $\displaystyle
M|\psi(t)\rangle$
the mass matrix $M$ is not diagonal and is given by,
$M=m-\frac{i}{2}\Gamma=\left(\begin{array}[]{cc}m_{11}-\frac{i}{2}\Gamma_{11}&m_{12}-\frac{i}{2}\Gamma_{12}\\\
m_{21}-\frac{i}{2}\Gamma_{21}&m_{22}-\frac{i}{2}\Gamma_{22}\end{array}\right)$
(20)
Hermiticity of matrices $m_{\alpha\alpha^{\prime}}$ and
$\Gamma_{\alpha\alpha^{\prime}}$ gives ($\alpha=\alpha^{\prime}=1,2$),
$\displaystyle\left(m\right)_{\alpha\alpha^{\prime}}$ $\displaystyle=$
$\displaystyle\left(m^{\dagger}\right)_{\alpha\alpha^{\prime}}=\left(m^{\ast}\right)_{\alpha^{\prime}\alpha},\qquad\Gamma_{\alpha\alpha^{\prime}}=\Gamma_{\alpha^{\prime}\alpha}^{\ast}$
$\displaystyle m_{21}$ $\displaystyle=$ $\displaystyle
m_{12\,}^{\ast}\qquad\Gamma_{21}=\Gamma_{12}^{\ast}$ (21)
$CPT$ invariance gives,
$\left\langle
X^{0}\left|M\right|X^{0}\right\rangle=\left\langle\bar{X}^{0}\left|M\right|\bar{X}^{0}\right\rangle$
$m_{11}=m_{22},\qquad\Gamma_{11}=\Gamma_{22}$
$\left\langle\bar{X}^{0}\left|M\right|X^{0}\right\rangle=\left\langle\bar{X}^{0}\left|M\right|X^{0}\right\rangle\text{:
identity}$ (22)
Diagonalization of mass matrix $M$ in eq. (20) gives,
$\displaystyle m_{11}-\frac{i}{2}\Gamma_{11}-pq$ $\displaystyle=$
$\displaystyle m_{1}-\frac{i}{2}\Gamma_{1}$ $\displaystyle
m_{11}-\frac{i}{2}\Gamma_{11}+pq$ $\displaystyle=$ $\displaystyle
m_{2}-\frac{i}{2}\Gamma_{2}$ (23)
where,
$p^{2}=m_{12}-\frac{i}{2}\Gamma_{12},\qquad
q^{2}=m_{12}^{\ast}-\frac{i}{2}\Gamma_{12}^{\ast}$ (24)
The eigenstates are given by,
$|X_{1,2}\rangle=\frac{1}{\sqrt{\left|p\right|^{2}+\left|q\right|^{2}}}\left[p|X^{0}\rangle\mp
q|\bar{X}^{0}\rangle\right]$
## 2 $K^{0}-\bar{K}^{0}$ Complex and $CP$–Violation in $K$-Decay
Consider the process,
$K^{0}\rightarrow\pi^{+}\pi^{-}\rightarrow\bar{K}^{0},\qquad\left|\Delta
Y\right|=2$
Thus, weak interaction can mix $K^{0}$ and $\bar{K}^{0}$,
$\left\langle K^{0}\left|H\right|\bar{K}^{0}\right\rangle\neq 0.$
Off diagonal matrix elements are not zero. Thus, $K^{0}$ and $\bar{K}^{0}$
cannot be mass eigenstates.
Select the phase:
$CP\left|K^{0}\right\rangle=-\left|\bar{K}^{0}\right\rangle.$
Define,
$\displaystyle\left|K_{1}^{0}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\left|K^{0}\right\rangle-\left|\bar{K}^{0}\right\rangle\right]$
$\displaystyle\left|K_{2}^{0}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\left|K^{0}\right\rangle+\left|\bar{K}^{0}\right\rangle\right]$
Choose:
$CP\left|K_{1}^{0}\right\rangle=+\left|K_{1}^{0}\right\rangle\qquad
CP\left|K_{2}^{0}\right\rangle=-\left|K_{2}^{0}\right\rangle$
where $K_{1}^{0}$ and $K_{2}^{0}$ are eigenstates of $CP$ with eigenvalues
$+1$ and $-1$.
Assuming $CP$ conservation,
$\left\langle\bar{K}^{0}\left|M\right|K^{0}\right\rangle=\left\langle
K^{0}\left|M\right|\bar{K}^{0}\right\rangle$ (25)
$m_{21}=m_{12}\qquad\Gamma_{21}=\Gamma_{12}$
where $m_{12}$ and $\Gamma_{12}$ are real. Thus,
$\displaystyle pq$ $\displaystyle=$ $\displaystyle
m_{12}-\frac{i}{2}\Gamma_{12}$ $\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle m_{11}-m_{12},\qquad\Gamma_{1}=\Gamma_{11}-\Gamma_{12}$
$\displaystyle m_{2}$ $\displaystyle=$ $\displaystyle
m_{11}+m_{12},\qquad\Gamma_{2}=\Gamma_{11}+\Gamma_{12}$ $\displaystyle\Delta
m$ $\displaystyle=$ $\displaystyle m_{2}-m_{1}=2m_{12},$ (26)
$\displaystyle\,\Delta\Gamma$ $\displaystyle=$
$\displaystyle\Gamma_{2}-\Gamma_{1}=2\Gamma_{12}$ (27)
Since,
$CP\left(\pi^{+}\,\pi^{-}\right)=\left(-1\right)^{l}\left(-1\right)^{l}=1$
therefore, it is clear that,
$K_{1}^{0}\longrightarrow\pi^{+}\,\pi^{-}$
is allowed by $CP$ conservation.
However, experimentally it was found that long lived $K_{2}^{0}$ also decay to
$\pi^{+}\,\pi^{-}$ but with very small probability. Small $CP$ non
conservation can be taken into account by defining,
$\displaystyle\left|K_{S}\right\rangle$ $\displaystyle=$
$\displaystyle\left|K_{1}^{0}\right\rangle+\varepsilon\left|K_{2}^{0}\right\rangle$
$\displaystyle\left|K_{L}\right\rangle$ $\displaystyle=$
$\displaystyle\left|K_{2}^{0}\right\rangle+\varepsilon\left|K_{1}^{0}\right\rangle$
(28)
where $\varepsilon$ is a small number. Thus $CP$ non conservation manifests
itself by the ratio:
$\displaystyle\eta_{+-}$ $\displaystyle=$
$\displaystyle\frac{A\left(K_{L}\rightarrow\pi^{+}\,\pi^{-}\right)}{A\left(K_{S}\rightarrow\pi^{+}\,\pi^{-}\right)}=\varepsilon$
(29) $\displaystyle\left|\eta_{+-}\right|$ $\displaystyle\simeq$
$\displaystyle\left(2.286\pm 0.017\right)\times 10^{-3}$
Now $CP$ non conservation implies,
$m_{12}\neq m_{12}^{\ast},\qquad\Gamma_{12}\neq\Gamma_{12}^{\ast}.$ (30)
Since $CP$ violation is a small effect, therefore,
$\text{Im}m_{12}\ll\text{Re}m_{12}\qquad\text{Im}\Gamma_{12}\ll\text{Re}\Gamma_{12}.$
(31)
Further, if $CP$\- violation arises from mass matrix, then,
$\Gamma_{12}=\Gamma_{12}^{\ast}.$ (32)
Thus, $CP$–violation can result by a small term $i\text{Im}m_{12}$ in the mass
matrix given in Eq. (18),
$M=\left(\begin{array}[]{cc}m_{1}-\frac{i}{2}\Gamma_{1}&i\text{Im}m_{12}\\\
-i\text{Im}m_{12}&m_{2}-\frac{i}{2}\Gamma_{2}\end{array}\right).$ (33)
Diagonalization gives,
$\varepsilon=\frac{i\text{Im}m_{12}}{\left(m_{2}-m_{1}\right)-i\left(\Gamma_{2}-\Gamma_{1}\right)/2}.$
(34)
Then from Eq. (27) up to first order, we get,
$\displaystyle\Delta m$ $\displaystyle=$ $\displaystyle m_{2}-m_{1}\rightarrow
m_{K_{L}}-m_{K_{S}}$ $\displaystyle=$ $\displaystyle 2\text{Re}m_{12}$
$\displaystyle\Delta\Gamma$ $\displaystyle=$
$\displaystyle\Gamma_{2}-\Gamma_{1}=\Gamma_{L}-\Gamma_{S}=2\Gamma_{12}$ (35)
Eq. (19) is unchanged, replace,
$m_{1}\rightarrow m_{S},\qquad m_{2}\rightarrow m_{L}$
$\Gamma_{1}\rightarrow\Gamma_{S},\qquad\Gamma_{2}\rightarrow\Gamma_{L}$
Now,
$\displaystyle\Delta m$ $\displaystyle=$ $\displaystyle m_{L}-m_{S}$
$\displaystyle\Delta\Gamma$ $\displaystyle=$
$\displaystyle\Gamma_{L}-\Gamma_{S}$ $\displaystyle\Gamma_{S}$
$\displaystyle=$ $\displaystyle\frac{\hbar}{\tau_{S}}=7.367\times
10^{-12}\text{ MeV},\,\,\,$ $\displaystyle\left.\tau_{S}=\left(0.8935\pm
0.0008\right)\times 10^{-10}\text{ s}\right.$ $\displaystyle\Gamma_{L}$
$\displaystyle=$ $\displaystyle\frac{\hbar}{\tau_{L}}=1.273\times
10^{-14}\text{ MeV},\,\,$ $\displaystyle\left.\,\tau_{L}=\left(5.17\pm
0.04\right)\times 10^{-8}\text{ s}\right.$ $\displaystyle\Delta\Gamma$
$\displaystyle\simeq$ $\displaystyle-\Gamma_{S}$ $\displaystyle m_{L}$
$\displaystyle=$ $\displaystyle m+\frac{1}{2}\Delta m$ $\displaystyle m_{S}$
$\displaystyle=$ $\displaystyle m-\frac{1}{2}\Delta m$ (36)
Hence from Eq. (19),
$\left|\psi\left(t\right)\right\rangle\approx
e^{\frac{-i}{2}mt}\left\\{\begin{array}[]{c}\left[e^{\frac{-1}{2}\Gamma_{S}t}e^{\frac{i}{2}\Delta
mt}+e^{-\frac{i}{2}\Delta mt}\right]\left|K^{0}\right\rangle\\\
-\left[e^{\frac{-1}{2}\Gamma_{S}t}e^{\frac{i}{2}\Delta
mt}-e^{-\frac{i}{2}\Delta
mt}\right]\left|\bar{K}^{0}\right\rangle\end{array}\right\\}$ (37)
Therefore, probability of finding $\bar{K}^{0}$ at time $t$ (recall that we
started with $K^{0}$),
$\displaystyle P\left(K^{0}\rightarrow\bar{K}^{0},t\right)$ $\displaystyle=$
$\displaystyle\left|\left\langle\bar{K}^{0}\left|{}\right.\psi\left(t\right)\right\rangle\right|^{2}$
(38) $\displaystyle=$
$\displaystyle\frac{1}{4}\left(1+e^{-\Gamma_{S}t}-2e^{-\frac{1}{2}\Gamma_{S}t}\cos\left(\Delta
m\right)t\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(1+e^{-t/\tau_{S}}-2e^{-\frac{1}{2}t/\tau_{S}}\cos\left(\Delta
m\right)t\right)$
If kaons were stable $(\tau_{S}\rightarrow\infty)$, then,
$P\left(K^{0}\rightarrow\bar{K}^{0},t\right)=\frac{1}{2}\left[1-\cos\left(\Delta
m\right)t\right]$ (39)
which shows that a state produced as pure $Y=1$ state at $t=0$ continuously
oscillates between $Y=1$ and $Y=-1$ state with frequency $\omega=\frac{\Delta
m}{\hbar}$ and period of oscillation,
$\tau=\frac{2\pi}{\left(\Delta m/\hbar\right)}.$ (40)
Kaons, however, decay and their oscillations are damped.
By measuring the period of oscillation, $\Delta m$ can be determined.
$\Delta m=m_{L}-m_{S}=\left(3.489\pm 0.008\right)\times 10^{-12}\text{ MeV.}$
(41)
Such a small number is measured as a consequence of superposition principle in
quantum mechanics,
$\displaystyle\pi^{-}p$ $\displaystyle\rightarrow$ $\displaystyle
K^{0}\Lambda^{0}$
$\displaystyle\left.{}^{|}\\!\\!\\!\longrightarrow\bar{K}^{0}p\rightarrow\pi^{+}\Lambda^{0}\right.$
$\pi^{+}$ can only be produced by $\bar{K}^{0}$ in the final state. This would
give a clear indication of oscillation.
Coming back to $CP$-violation,
$\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle\frac{i\text{Im}m_{12}}{\Delta
m-i\Delta\Gamma/2}\qquad\varepsilon=\left|\varepsilon\right|e^{i\phi_{\varepsilon}}$
(42) $\displaystyle\tan\phi_{\varepsilon}$ $\displaystyle=$
$\displaystyle-2\Delta m/\Delta\Gamma=\Delta m/\Gamma_{S}-\Gamma_{L}$ (43)
$\displaystyle\approx$ $\displaystyle\frac{2\times
0.474\Gamma_{S}}{0.998\Gamma_{S}}$ $\displaystyle\Rightarrow$
$\displaystyle\phi_{\varepsilon}=43.59\pm 0.05^{0}$
$\displaystyle\left|\epsilon\right|$ $\displaystyle=$ $\displaystyle(2.229\pm
0.012)\times 10^{-3}$ (44)
So far we have considered $CP$-violation due to mixing in the mass matrix. It
is important to detect the $CP$-violation in the decay amplitude if any. This
is done by looking for a difference between $CP$-violation for the final
$\pi^{0}\pi^{0}$ state and that for $\pi^{+}\pi^{-}$. Now due to Bose
statistics, the two pions can be either in $I=0$ or $I=2$ states. Using
Clebsch-Gordon (CG) coefficients,
$\displaystyle A\left(K^{0}\rightarrow\pi^{+}\pi^{-}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left[\sqrt{2}A_{0}e^{i\delta_{0}}+A_{2}e^{i\delta_{2}}\right]$
$\displaystyle A\left(K^{0}\rightarrow\pi^{0}\pi^{0}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left[A_{0}e^{i\delta_{0}}-\sqrt{2}A_{2}e^{i\delta_{2}}\right]$
(45)
Now $CPT$-invariance gives,
$\displaystyle A\left(\bar{K}^{0}\rightarrow\pi^{+}\pi^{-}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left[\sqrt{2}A_{0}^{\ast}e^{i\delta_{0}}+A_{2}^{\ast}e^{i\delta_{2}}\right]$
$\displaystyle A\left(\bar{K}^{0}\rightarrow\pi^{0}\pi^{0}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left[A_{0}^{\ast}e^{i\delta_{0}}-\sqrt{2}A_{2}^{\ast}e^{i\delta_{2}}\right]$
(46)
The dominant decay amplitude is $A_{0}$ due to $\Delta I=1/2$ rule,
$\left|A_{2}/A_{0}\right|\simeq 1/22$. Using the Wu–Yang phase convention, we
can take $A_{0}$ to be real. Neglecting terms of order
$\varepsilon\text{Re}\frac{A_{2}}{A_{0}}$ and
$\varepsilon\text{Im}\frac{A_{2}}{A_{0}}$, we get,
$\displaystyle\eta_{+-}$ $\displaystyle\equiv$
$\displaystyle\left|\eta_{+-}\right|e^{i\phi_{+-}}\simeq\varepsilon+\varepsilon^{\prime}$
$\displaystyle\eta_{00}$ $\displaystyle\equiv$
$\displaystyle\left|\eta_{00}\right|e^{i\phi_{00}}\simeq\varepsilon-2\varepsilon^{\prime}$
(47)
where,
$\varepsilon^{\prime}=\frac{i}{\sqrt{2}}e^{i\left(\delta_{2}-\delta_{0}\right)}\text{Im}\frac{A_{2}}{A_{0}}$
(48)
Clearly $\varepsilon^{\prime}$ measures the $CP$-violation in the decay
amplitude, since $CP$-invariance implies $A_{2}$ to be real.
After $35$ years of experiments at Fermilab and CERN, results have converged
on a definitive non-zero result for $\varepsilon^{\prime}$,
$\displaystyle R$ $\displaystyle=$
$\displaystyle\left|\frac{\eta_{00}}{\eta_{+-}}\right|^{2}=\left|\frac{\varepsilon-2\varepsilon^{\prime}}{\varepsilon+\varepsilon^{\prime}}\right|^{2},\qquad\varepsilon^{\prime}\ll\varepsilon$
$\displaystyle\simeq$
$\displaystyle\left|1-\frac{3\varepsilon^{\prime}}{\varepsilon}\right|^{2}\simeq
1-6\text{Re}\left(\varepsilon^{\prime}/\varepsilon\right)$
$\displaystyle\text{Re}\left(\varepsilon^{\prime}/\varepsilon\right)$
$\displaystyle=$ $\displaystyle\frac{1-R}{6}$ (49) $\displaystyle=$
$\displaystyle\left(1.65\pm 0.26\right)\times 10^{-3}.$ (50)
This is an evidence that although $\varepsilon^{\prime}$ is a very small, but
$CP$-violation does occur in the decay amplitude. Further we note from Eq.
(48),
$\phi_{\varepsilon^{\prime}}=\delta_{2}-\delta_{0}+\frac{\pi}{2}\approx
42.3\pm 1.5^{0}$
where numerical value is based on an analysis of $\pi\pi$ scattering.
We now discuss the CP-asymmetry in leptonic decays of kaon.
$\displaystyle\frac{\Delta S}{\Delta Q}$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle K^{+}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{0}+l^{+}+\nu_{l}$ $\displaystyle K^{0}$
$\displaystyle\rightarrow$ $\displaystyle\pi^{-}+l^{+}+\nu_{l}=f$
$\displaystyle\overline{K}^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{+}+l^{-}+\overline{\nu}_{l}=f^{*}\text{ CPT}$
$\displaystyle\frac{\Delta S}{\Delta Q}$ $\displaystyle=$ $\displaystyle-1$
$\displaystyle K^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{+}+l^{-}+\overline{\nu}_{l}=g^{*}$
$\displaystyle\overline{K}^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{-}+l^{+}+\nu_{l}=g\text{ CPT}$ $\displaystyle A(K_{L}^{0}$
$\displaystyle\rightarrow$
$\displaystyle\pi^{-}+l^{+}+\nu_{l})=\frac{1}{\sqrt{2}}[(1+\epsilon)f+(1-\epsilon)g]$
$\displaystyle A(K_{L}^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{+}+l^{-}+\overline{\nu}_{l})=\frac{1}{\sqrt{2}}[(1+\epsilon)g^{*}+(1-\epsilon)f*]$
The CP-asymmetry parameter $\delta_{l}:$
$\displaystyle\delta_{l}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(K_{L}^{0}\rightarrow\pi^{-}l^{+}\nu_{l})-\Gamma(K_{L}^{0}\rightarrow\pi^{+}l^{-}\overline{\nu}_{l})}{\Gamma(K_{L}^{0}\rightarrow\pi^{-}l^{+}\nu_{l})+\Gamma(K_{L}^{0}\rightarrow\pi^{+}l^{-}\overline{\nu}_{l})}$
$\displaystyle=$
$\displaystyle\frac{2\text{Re}\epsilon[\left|f\right|^{2}-\left|g\right|^{2}]}{\left|f\right|^{2}+\left|g\right|^{2}+(fg^{*}+f^{*}g)+O(\epsilon^{2})}$
In the standard model $\frac{\Delta S}{\Delta Q}=-1$ transitions are not
allowed, thus $g=0$. Hence
$\delta_{l}\approx 2\text{Re}\epsilon=(3.32\pm 0.06)10^{-3}[\text{Expt.
value}]$
From Eq. (44), we get
$2\text{Re}\epsilon=2\left|\epsilon\right|\cos\phi_{\epsilon}$
which gives on using expermintal values for $\left|\epsilon\right|$ and
$\phi_{\epsilon}$
$2\text{Re}\epsilon=(3.23\pm 0.02\times 10^{-3})$
in agreement with the expermimental value for $\delta_{l}$
Finally we discuss CP-asymmetries for $K\rightarrow 3\pi$ decays. The decays
$\displaystyle K^{+}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{+}\pi^{0}\pi^{0}\text{, }\pi^{+}\pi^{+}\pi^{-}$
$\displaystyle K^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{+}\pi^{-}\pi^{0}\text{, }\pi^{0}\pi^{0}\pi^{0}$
are partiy conserving decays i.e. the parity of the final state is $-1$. Now
the C-partiy of $\pi^{0}$ and ($\pi^{+}\pi^{-})_{l^{\prime}}$ are given by
$C(\pi^{0})=1,\text{ }C(\pi^{+}\pi^{-})=(-1)^{l^{\prime}}$
and G-parity of pion is $-1.$ Thus
$\displaystyle CP|\pi^{0}\pi^{0}\pi^{0}$ $\displaystyle>$
$\displaystyle=-|\pi^{0}\pi^{0}\pi^{0}>$ $\displaystyle
CP|\pi^{+}\pi^{-}\pi^{0}$ $\displaystyle>$
$\displaystyle=(-1)^{l^{\prime}+1}|\pi^{+}\pi^{-}\pi^{0}>$
Hence CP-conservation implies
$\displaystyle K_{2}^{0}$ $\displaystyle\rightarrow$ $\displaystyle\text{
}\pi^{0}\pi^{0}\pi^{0}\text{ allowed.}$ $\displaystyle K_{1}^{0}$
$\displaystyle\rightarrow$ $\displaystyle\text{ }\pi^{0}\pi^{0}\pi^{0}\text{
is forbidden.}$ $\displaystyle K_{1}^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{+}\pi^{-}\pi^{0}\text{ allowed if }l_{1}\text{ is odd.}$
$\displaystyle K_{2}^{0}$ $\displaystyle\rightarrow$
$\displaystyle\pi^{+}\pi^{-}\pi^{0}\text{ allowed if }l_{1}\text{ is even.}$
Now G-partiy of three pions $\pi^{+}\pi^{-}\pi^{0}:$
$\displaystyle G$ $\displaystyle=$ $\displaystyle
C(-1)^{I}=(-1)^{l^{\prime}+I}=-1$ $\displaystyle\text{Hence }l^{\prime}$
$\displaystyle=$ $\displaystyle\text{even},\text{ }I(\text{odd});\text{
}I=1,3$ $\displaystyle l^{\prime}$ $\displaystyle=$
$\displaystyle\text{odd},\text{ }I(\text{even});\text{ }I=0,2$
Only $l^{\prime}=0$ decays are favored as the decays for $l^{\prime}>0$ are
highly suppressed due to centrifugal barrier. Hence
$K_{1}^{0}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ is highly suppressed. Thus we have
to take into account $I=1,3$ amplitudes viz $a_{1}$ and $a_{3}$. $I=3$
contribution is expected to be suppressed as it requires $\Delta
I=\frac{5}{2}$ transition.
Hence CP-asymmetries of $K^{0}\rightarrow 3\pi$ decays are given by
$\displaystyle\eta_{000}$ $\displaystyle=$
$\displaystyle\frac{A(K_{s}\rightarrow\text{
}\pi^{0}\pi^{0}\pi^{0})}{A(K_{L}\rightarrow\text{
}\pi^{0}\pi^{0}\pi^{0})}=\frac{[i\text{Im}a_{1}+\epsilon\text{Re}a_{1}]}{\text{Re}a_{1}+i\epsilon\text{Im}a_{1}}$
$\displaystyle\approx$
$\displaystyle\epsilon+i\frac{\text{Im}a_{1}}{\text{Re}a_{1}}$
$\displaystyle\eta_{+-0}$ $\displaystyle=$
$\displaystyle\frac{A(K_{s}\rightarrow\pi^{+}\pi^{-}\pi^{0})}{A(K_{L}\rightarrow\pi^{+}\pi^{-}\pi^{0})}\approx\epsilon+i\frac{\text{Im}a_{1}}{\text{Re}a_{1}}=\eta_{000}$
## 3 $B^{0}-\bar{B}^{0}$ Complex
For $B_{q}^{0}$ (q=d or s) we show below that both $m_{12}$ and $\Gamma_{12}$
have the same phase. Thus,
$\displaystyle m_{12}$ $\displaystyle=$
$\displaystyle\left|m_{12}\right|e^{-2i\phi_{M}}$ $\displaystyle\Gamma_{12}$
$\displaystyle=$ $\displaystyle\left|\Gamma_{12}\right|e^{-2i\phi_{M}}$ (51)
$\displaystyle\left|\Gamma_{12}\right|$ $\displaystyle\ll$
$\displaystyle\left|m_{12}\right|$ $\displaystyle p^{2}$ $\displaystyle=$
$\displaystyle
e^{-2i\phi_{M}}\left[\left|m_{12}\right|-i\left|\Gamma_{12}\right|\right]\simeq\left|m_{12}\right|e^{-2i\phi_{M}}$
$\displaystyle q^{2}$ $\displaystyle=$ $\displaystyle
e^{+2i\phi_{M}}\left[\left|m_{12}\right|-i\left|\Gamma_{12}\right|\right]\simeq\left|m_{12}\right|e^{2i\phi_{M}}$
(52) $\displaystyle 2pq$ $\displaystyle=$ $\displaystyle
2\left|m_{12}\right|=(m_{2}-m_{1})-\frac{i}{2}\left(\Gamma_{2}-\Gamma_{1}\right)$
$\displaystyle\Rightarrow\Delta m_{B}$ $\displaystyle=$
$\displaystyle\frac{\left(m_{2}-m_{1}\right)}{2}=\left|m_{12}\right|$ (53)
$\displaystyle\Delta\Gamma$ $\displaystyle=$
$\displaystyle\Gamma_{2}-\Gamma_{1}=0$
The above equations follow from the fact that,
$m_{12}-i\Gamma_{12}=\langle\bar{B}_{q}^{0}\left|H_{eff}^{\Delta
B=2}\right|B_{q}^{0}\rangle$
$H_{eff}^{\Delta B=2}$ induces particle-antiparticle transition. For $\Delta
m_{12},$ $H_{eff}^{\Delta B=2}$ arises from the box diagram as shown in Fig.
2, where the dominant contribution comes out from the $t-$quark. Thus,
$m_{12}\varpropto(V_{tb})^{2}(V_{tq}^{\ast})^{2}m_{t}^{2}$
Now,
$\Gamma_{12}\varpropto\sum_{f}\langle\bar{B}^{0}\left|H_{W}\right|f\rangle\langle
f\left|H_{W}\right|B^{0}\rangle$
where the sum is over all the final states which contribute to both
$B_{q}^{0}$ and $\bar{B}_{q}^{0}$ decays. Thus,
$\Gamma_{12}\varpropto\left(V_{cb}V_{cq}^{\ast}+V_{ub}V_{uq}^{\ast}\right)^{2}m_{b}^{2}\propto(V_{tb})^{2}(V_{tq}^{\ast})^{2}m_{b}^{2}$
Hence we have the result that,
$\frac{\left|\Gamma_{12}\right|}{\left|m_{12}\right|}\sim\frac{m_{b}^{2}}{m_{t}^{2}}$
Now $B_{d}^{0}\rightarrow\bar{B}_{d}^{0}$ transition:
$\left(V_{tb}\right)^{2}\left(V_{td}^{\ast}\right)^{2}=A^{2}\lambda^{6}\left[\left(1+\rho\right)^{2}+\eta^{2}\right]e^{2i\beta}$
Hence,
$m_{12}=\left|m_{12}\right|e^{2i\beta},\qquad\Gamma_{12}=\left|\Gamma_{12}\right|e^{2i\beta},\qquad\phi_{M}=-\beta$
On the other hand, $B_{s}^{0}\rightarrow\bar{B}_{s}^{0}$ transition:
$\left(V_{tb}\right)^{2}\left(V_{ts}^{\ast}\right)^{2}=\left|V_{ts}\right|^{2}\approx
A^{2}\lambda^{4}$ (54)
$m_{12}=\left|m_{12}\right|,\qquad\Gamma_{12}=\left|\Gamma_{12}\right|$ (55)
$\phi_{M}=0$ (56)
Also we have,
$\frac{\Delta m_{B_{s}}}{\Delta
m_{B_{d}}}=\frac{\left|m_{12}\right|_{s}}{\left|m_{12}\right|_{d}}=\frac{1}{\lambda^{2}\left[\left(1+\rho\right)^{2}+\eta^{2}\right]}\xi$
where $\xi$ is $SU(3)$ breaking parameter.
Hence the mass eigenstates $B_{L}^{0}$ and $B_{H}^{0}$ can be written as:
$\displaystyle\left|B_{L}^{0}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\left|B^{0}\right\rangle-e^{2i\phi_{M}}\left|\bar{B}^{0}\right\rangle\right]\quad
CP=+1,\phi_{M}\rightarrow 0$ (57) $\displaystyle\left|B_{H}^{0}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\left|\bar{B}^{0}\right\rangle+e^{2i\phi_{M}}\left|B^{0}\right\rangle\right]\quad
CP=-1,\phi_{M}\rightarrow 0$ (58)
In this case, $CP$ violation occurs due to phase factor $e^{2i\phi_{M}}$ in
the mass matrix.
Now one gets (from Eq. (19)), using Eqs.(53), (57) and (58),
$\displaystyle\left|B^{0}\left(t\right)\right\rangle$ $\displaystyle=$
$\displaystyle e^{-imt}e^{-\frac{1}{2}\Gamma t}\left\\{\cos\left(\frac{\Delta
m}{2}t\right)\left|B^{0}\right\rangle\right.$ (59)
$\displaystyle\left.-ie^{+2i\phi_{M}}\sin\left(\frac{\Delta
m}{2}t\right)\left|\bar{B}^{0}\right\rangle\right\\}$
Similarly we get,
$\displaystyle\left|\bar{B}^{0}\left(t\right)\right\rangle$ $\displaystyle=$
$\displaystyle-e^{-imt}e^{-\frac{1}{2}\Gamma t}\left\\{\cos\left(\frac{\Delta
m}{2}t\right)\left|\bar{B}^{0}\right\rangle\right.$ (60)
$\displaystyle\left.-ie^{-2i\phi_{M}}\sin\left(\frac{\Delta
m}{2}t\right)\left|B^{0}\right\rangle\right\\}$
Suppose we start with $B^{0}$ viz $|B^{0}\left(0\right)\rangle=|B^{0}\rangle,$
the probabilities of finding $\bar{B}^{0}$ and $B^{0}$ at time $t$ is given
by,
$\displaystyle P\left(B^{0}\rightarrow\bar{B}^{0},t\right)$ $\displaystyle=$
$\displaystyle\left|\langle\bar{B}^{0}|B^{0}\left(t\right)\rangle\right|^{2}$
$\displaystyle=$ $\displaystyle\frac{1}{2}e^{-\Gamma t}\left(1-\cos(\Delta
m\right)t)$ $\displaystyle P\left(B^{0}\rightarrow B^{0},t\right)$
$\displaystyle=$ $\displaystyle\left|\langle
B^{0}|B^{0}\left(t\right)\rangle\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-\Gamma t}\left(1+\cos(\Delta m\right)t)$
These are equations of a damped harmonic oscillator, the angular frequency of
which is,
$\omega=\frac{\Delta m}{\hslash}$
We define the mixing parameter,
$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{T}\left|\langle\bar{B}^{0}|B^{0}\left(t\right)\rangle\right|^{2}dt}{\int_{0}^{T}\left|\langle
B^{0}|B^{0}\left(t\right)\rangle\right|^{2}dt}=\frac{\chi}{1-\chi}$
$\displaystyle\xrightarrow{T\rightarrow\infty}$
$\displaystyle\frac{\left(\Delta m/\Gamma\right)^{2}}{2+\left(\Delta
m/\Gamma\right)^{2}}=\frac{x^{2}}{2+x^{2}}$
Experimentally, for $B_{d}^{0}$ and $B_{s}^{0}$,
$\displaystyle\Delta m_{B_{d}^{0}}$ $\displaystyle=$ $\displaystyle(0.507\pm
0.005)\times 10^{-12}\hbar s^{-1}=(3.337\pm 0.033)\times 10^{-10}\text{MeV}$
$\displaystyle\Delta m_{B_{s}^{0}}$ $\displaystyle=$ $\displaystyle(17.77\pm
0.10\pm 0.007)\times 10^{-12}\hbar s^{-1}=(1.17\pm 0.01)\times
10^{-10}\text{MeV}$ $\displaystyle x_{d}$ $\displaystyle=$
$\displaystyle\left(\frac{\Delta
m_{B_{d}^{0}}}{\Gamma_{B_{d}^{0}}}\right)=0.77\pm 0.008$ $\displaystyle x_{s}$
$\displaystyle=$ $\displaystyle\left(\frac{\Delta
m_{B_{s}^{0}}}{\Gamma_{B_{s}^{0}}}\right)=26.05\pm 0.25$
From Eq. (59) and (60), the decay amplitudes for,
$\displaystyle B^{0}\left(t\right)$ $\displaystyle\rightarrow$ $\displaystyle
f\,\,\,\,\,\,\,\,\,\,\,\,\,A_{f}\left(t\right)=\left\langle
f\left|H_{w}\right|B^{0}\left(t\right)\right\rangle$
$\displaystyle\bar{B}^{0}\left(t\right)$ $\displaystyle\rightarrow$
$\displaystyle\bar{f}\,\,\,\,\,\,\,\,\,\,\,\,\bar{A}_{\bar{f}}\left(t\right)=\left\langle\bar{f}\left|H_{w}\right|\bar{B}^{0}\left(t\right)\right\rangle$
(61)
are given by,
$\displaystyle\,\,\,A_{f}\left(t\right)$ $\displaystyle=$ $\displaystyle
e^{-imt}e^{-\frac{1}{2}\Gamma t}\left\\{\cos\left(\frac{\Delta
m}{2}t\right)A_{f}\right.$ (62)
$\displaystyle\,\,\,\,\,\,\,\,\left.-ie^{+2i\phi_{M}}\sin\left(\frac{\Delta
m}{2}t\right)\bar{A}_{\bar{f}}\right\\}$
$\displaystyle\bar{A}_{\bar{f}}\left(t\right)$ $\displaystyle=$ $\displaystyle
e^{-imt}e^{-\frac{1}{2}\Gamma t}\left\\{\cos\left(\frac{\Delta
m}{2}t\right)\bar{A}_{\bar{f}}\right.$ (63)
$\displaystyle\left.-ie^{-2i\phi_{M}}\sin\left(\frac{\Delta
m}{2}t\right)A_{\bar{f}}\right\\}.$
From Eqs.(62) and (63), we get for the decay rates,
$\displaystyle\Gamma_{f}(t)$ $\displaystyle=$ $\displaystyle e^{-\Gamma
t}\left[\begin{array}[]{c}\frac{1}{2}\left(\left|A_{f}\right|^{2}+\left|\bar{A}_{f}\right|^{2}\right)+\frac{1}{2}\left(\left|A_{f}\right|^{2}-\left|\bar{A}_{f}\right|^{2}\right)\cos\Delta
mt\\\
-\frac{i}{2}\left(2i\text{Im}e^{2i\phi_{M}}A_{f}^{\ast}\bar{A}_{f}\right)\sin\Delta
mt\end{array}\right]$ (66) $\displaystyle\bar{\Gamma}_{\bar{f}}(t)$
$\displaystyle=$ $\displaystyle e^{-\Gamma
t}\left[\begin{array}[]{c}\frac{1}{2}\left(\left|A_{\bar{f}}\right|^{2}+\left|\bar{A}_{\bar{f}}\right|^{2}\right)-\frac{1}{2}\left(\left|A_{\bar{f}}\right|^{2}-\left|\bar{A}_{\bar{f}}\right|^{2}\right)\cos\Delta
mt\\\
+\frac{i}{2}\left(2i\text{Im}e^{2i\phi_{M}}A_{\bar{f}}^{\ast}\bar{A}_{\bar{f}}\right)\sin\Delta
mt\end{array}\right]$ (69)
for $\Gamma_{\bar{f}}$ and $\bar{\Gamma}_{f}$ change $f\rightarrow\bar{f}$ and
$\bar{f}\rightarrow f$ in $\Gamma_{f}$ and $\bar{\Gamma}_{\bar{f}}$
respectively.
As a simple application of the above equations, consider the semi-leptonic
decays of $B^{0}$,
$\displaystyle B^{0}$ $\displaystyle\rightarrow$ $\displaystyle l^{+}\nu
X^{-}:f\text{ \ for example }X^{-}=D^{-}$ $\displaystyle\bar{B}^{0}$
$\displaystyle\rightarrow$ $\displaystyle l^{-}\bar{\nu}X^{+}:\bar{f}\text{ \
for example }X^{+}=D^{+}$
In the standard model, $\bar{B}^{0}$ decay into $l^{+}\nu X^{-}$ and $B^{0}$
decay into $l^{-}\bar{\nu}X^{+}$ is forbidden. Thus,
$\displaystyle\bar{A}_{f}$ $\displaystyle=$ $\displaystyle 0,\qquad
A_{\bar{f}}=0$ $\displaystyle\Gamma_{f}(t)$ $\displaystyle=$ $\displaystyle
e^{-\Gamma t}\frac{1}{2}\left|A_{f}\right|^{2}\left(1+\cos\Delta mt\right)$
$\displaystyle\Gamma_{\bar{f}}(t)$ $\displaystyle=$ $\displaystyle e^{-\Gamma
t}\frac{1}{2}\left|\bar{A}_{\bar{f}}\right|^{2}\left(1-\cos\Delta
mt\right),\qquad\because\left|\bar{A}_{\bar{f}}\right|=\left|A_{f}\right|$
Hence,
$\delta=\frac{\int_{0}^{\infty}\Gamma_{\bar{f}}(t)dt}{\int_{0}^{\infty}\Gamma_{f}(t)dt}=\frac{x_{d}^{2}}{2+x_{d}^{2}}=r_{d}$
Non zero value of $\delta$ would indicate mixing. If, however,
$\bar{A}_{f}\neq 0$ and $A_{\bar{f}}\neq 0$ due to some exotic mechanism, then
$\delta\neq 0$ even without mixing. Thus
$\displaystyle\frac{\Gamma\left(\mu^{-}X^{+}\right)}{\Gamma\left(\mu^{+}X^{-}\right)+\Gamma\left(\mu^{-}X^{+}\right)}$
$\displaystyle=$ $\displaystyle\frac{r_{d}}{1+r_{d}}=\chi_{d}$
$\displaystyle=$ $\displaystyle 0.172\pm 0.010\text{ (Expt value)}$
which gives,
$x_{d}=0.723\pm 0.032$
## 4 $CP$-Violation in $B$-Decays
Case-I:
$|\bar{f}\rangle=CP|f\rangle=|f\rangle$
For this case we get, from Eqs. (62) and (63),
$\displaystyle\mathcal{A}_{f}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{f}\left(t\right)-\bar{\Gamma}_{f}\left(t\right)}{\Gamma_{f}\left(t\right)+\bar{\Gamma}_{f}\left(t\right)}=\cos\left(\Delta
mt\right)\left(\left|A_{f}\right|^{2}-\left|\bar{A}_{f}\right|^{2}\right)$
(70) $\displaystyle-i\sin\left(\Delta
mt\right)\left(e^{2i\phi_{M}}A_{f}^{\ast}\bar{A}_{f}-e^{-2i\phi_{M}}A_{f}\bar{A}_{f}^{\ast}\right)/\left(\left|A_{f}\right|^{2}+\left|\bar{A}_{f}\right|^{2}\right)$
$\displaystyle=$ $\displaystyle\cos\left(\Delta
mt\right)C_{\pi\pi}+\sin\left(\Delta mt\right)S_{\pi\pi}$ (71)
where,
$C_{\pi\pi}=\frac{1-\left|\bar{A}_{f}\right|^{2}/\left|A_{f}\right|^{2}}{1+\left|\bar{A}_{f}\right|^{2}/\left|A_{f}\right|^{2}}=\frac{1-\left|\lambda\right|^{2}}{1+\left|\lambda\right|^{2}}\qquad\lambda=\frac{\bar{A}_{f}}{A_{f}}$
This is the direct $CP$ violation and,
$S_{\pi\pi}=\frac{2\text{Im}\left(e^{2i\phi_{M}}\lambda\right)}{1+\left|\lambda\right|^{2}}$
is the mixing induced $CP$-violation.
If the decay proceeds through a single diagram (for example tree graph),
$\bar{A}_{f}/A_{f}$ is given by (see Eqs. (15) and (16)),
$\lambda=\frac{\bar{A}_{f}}{A_{f}}=\frac{e^{i\left(\phi+\delta_{f}\right)}}{e^{i\left(-\phi+\delta_{f}\right)}}=e^{2i\phi}$
where $\phi$ is the weak phase in the decay amplitude. Hence from Eq. (70), we
obtain,
$\mathcal{A}_{f}(t)=\sin\left(\Delta
mt\right)\sin\left(2\phi_{M}+2\phi\right)$ (72)
In particular for the decay,
$B^{0}\rightarrow J/\psi\,K_{s},\,\,\,\,\phi=0$
we obtain,
$\mathcal{A}_{\psi
K_{s}}\left(t\right)=\sin\left(2\phi_{M}\right)\sin\left(\Delta
mt\right)=-\sin 2\beta\sin(\Delta mt)$ (73)
and,
$\displaystyle\mathcal{A}_{\psi K_{s}}$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{\infty}\left[\Gamma_{f}\left(t\right)-\bar{\Gamma}_{f}\left(t\right)\right]dt}{\int_{0}^{\infty}\left[\Gamma_{f}\left(t\right)+\bar{\Gamma}_{f}\left(t\right)\right]dt}$
$\displaystyle\mathcal{A}_{\psi K_{s}}$ $\displaystyle=$
$\displaystyle-\sin\left(2\beta\right)\,\,\frac{\left(\Delta
m/\Gamma\right)}{1+\left(\Delta m/\Gamma\right)^{2}}$ (74) Experiment
$\displaystyle:$ $\displaystyle\left(\frac{\Delta
m}{\Gamma}\right)_{B_{d}^{0}}=0.776\pm 0.008$ (75)
$\mathcal{A}_{\psi K_{s}}$ has been experimentally measured. It gives,
$\sin 2\beta=0.678\pm 0.025$
Corresponding to the decay $B^{0}\rightarrow J/\psi\,K_{s}$, we have the decay
$B_{s}^{0}\rightarrow J/\psi\,\phi.$ Thus for this decay
$\displaystyle\mathcal{A}_{J/\psi\phi}^{(t)}$ $\displaystyle=$
$\displaystyle-\sin 2\beta_{s}\sin(\Delta m_{B_{s}}t)$
$\displaystyle\mathcal{A}_{J/\psi\phi}$ $\displaystyle=$ $\displaystyle-\sin
2\beta_{s}\frac{(\Delta m_{B_{s}}/\Gamma_{s})}{1+(\Delta
m_{B_{s}}/\Gamma_{s})^{2}}$
In the standard model, $\beta_{s}=0,$ $\mathcal{A}_{J/\psi\phi}=0.$
This is an example of $CP$-violation in the mass matrix. We now discuss the
direct $CP$-violation.
Direct $CP$-violation in $B$ decays involves the weak phase in the decay
amplitude. The reason for this being that necessary condition for direct $CP$
-violation is that decay amplitude should be complex as discussed in section
1\. But this is not sufficient because in the limit of no final state
interactions, the direct $CP$-violation in $B\rightarrow f$,
$\bar{B}\rightarrow\bar{f}$ decay vanishes. To illustrate this point, we
discuss the decays $\bar{B}^{0}\rightarrow\pi^{+}\pi^{-}$. The main
contribution to this decay is from tree graph (see Fig. 3); But this decay can
also proceed via the penguin diagram (see Fig. 4).
The contribution of penguin diagram can be written as
$P=V_{ub}V_{ud}^{\ast}f\left(u\right)+V_{cb}V_{cd}^{\ast}f\left(c\right)+V_{tb}V_{td}^{\ast}f\left(t\right)$
(76)
where $f\left(u\right)$, $f\left(c\right)$ and $f\left(d\right)$ denote the
contributions of $u$, $c$ and $t$ quarks in the loop. Now using the unitarity
equation (5), we can rewrite Eq. (76) as,
$\displaystyle P_{c}$ $\displaystyle=$ $\displaystyle
V_{ub}V_{ud}^{\ast}\left(f\left(u\right)-f\left(t\right)\right)+V_{cb}V_{cd}^{\ast}\left(f\left(c\right)-f\left(t\right)\right)$
(77) $\displaystyle\text{or }P_{t}$ $\displaystyle=$ $\displaystyle
V_{ub}V_{ud}^{\ast}\left(f\left(u\right)-f\left(c\right)\right)+V_{tb}V_{td}^{\ast}\left(f\left(t\right)-f\left(c\right)\right)$
Due to loop integration $P$ is suppressed relative to $T$ but still its
contribution is not negligible. The first part of Eq. (77) has the same CKM
matrix elements as for the tree graph, so we can absorb it in the tree graph.
Hence we can write (with $f=\pi^{+}\pi^{-}$),
$\bar{A}_{f}=A\left(\bar{B}^{0}\rightarrow\pi^{+}\pi^{-}\right)=\left|T\right|e^{i\left(-\gamma+\delta_{T}\right)}+\left|P\right|e^{i\left(\phi+\delta_{P}\right)}$
(78)
where $\delta_{T}$ and $\delta_{P}$ are strong interaction phases which have
been taken out so that $T$ and $P$ are real. $\phi$ is the weak phase in
Penguin graph. $CPT$ invariance gives,
$A_{f}\equiv{}A\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)=\left|T\right|e^{-i\left(-\gamma-\delta_{T}\right)}+\left|P\right|e^{-i\left(\phi-\delta_{P}\right)}.$
(79)
Hence direct $CP$–violation asymmetry is given by,
$\displaystyle A_{CP}$ $\displaystyle=$
$\displaystyle\frac{-\Gamma\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)+\Gamma\left(\bar{B}^{0}\rightarrow\pi^{+}\pi^{-}\right)}{\Gamma\left(B^{0}\rightarrow\pi^{+}\pi^{-}\right)+\Gamma\left(\bar{B}^{0}\rightarrow\pi^{+}\pi^{-}\right)}$
$\displaystyle=$
$\displaystyle-\frac{1-\left|\lambda\right|^{2}}{1+\left|\lambda\right|^{2}}$
$\displaystyle=$
$\displaystyle\frac{-2r\sin\delta\sin\left(\phi+\gamma\right)}{1+2r\cos\delta\cos\left(\gamma+\phi\right)+r^{2}}$
where ($F_{\text{CKM}}$ = CKM factor),
$\delta=\delta_{P}-\delta_{T}\qquad r\rightarrow
F_{\text{CKM}}\frac{\left|P\right|}{\left|T\right|}$
For the time dependent $CP$-asymmetry for $B^{0}\rightarrow\pi^{+}\pi^{-}$
decay we obtain from Eqs. (70) and (78),
$\mathcal{A}(t)=C_{\pi\pi}(\cos\Delta mt)+S_{\pi\pi}(\sin\Delta mt),$ (80a)
where the direct CP–violation parameter $C_{\pi\pi}$ and the mixing induced
parameter $S_{\pi\pi}$ are given by,
$S_{\pi\pi}=\frac{2\text{Im}[e^{2i\phi_{M}}\lambda]}{1+|\lambda|^{2}}=-\frac{\sin\left(2\beta+2\gamma\right)+2r\cos\delta\sin\left(2\beta+\gamma-\phi\right)+r^{2}\sin(2\beta-2\phi)}{1+2r\cos\delta\cos(\gamma+\phi)+r^{2}}$
(80b)
$C_{\pi\pi}=-A_{CP}$ (80c)
As discussed above, we have two choices in selecting the Penguin contribution.
For the first choice,
$\phi=\pi,\quad
F_{\text{CKM}}=\frac{\left|V_{cb}\right|\left|V_{cd}\right|}{\left|V_{ub}\right|\left|V_{ud}\right|}=\frac{1}{\left(\bar{\rho}^{2}+\bar{\eta}^{2}\right)^{1/2}}$
$r=\frac{1}{R_{b}}\frac{\left|P_{C}\right|}{\left|T\right|}$
For this case we have,
$\displaystyle C_{\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{2r\sin\delta\sin\gamma}{1+2r\cos\delta\cos\gamma+r^{2}}$
$\displaystyle S_{\pi\pi}$ $\displaystyle=$
$\displaystyle-\frac{\sin(2\beta+2\gamma)+2r\cos\delta\sin(2\beta+\gamma)+r^{2}\sin
2\beta}{1+2r\cos\delta\cos\gamma+r^{2}}$ $\displaystyle=$
$\displaystyle\frac{\sin(2\alpha)+2r\cos\delta\sin(\beta-\alpha)-r^{2}\sin
2\beta}{1+2r\cos\delta\cos\left(\alpha+\beta\right)+r^{2}}$
For the second choice,
$\phi=\beta,\quad
F_{\text{CKM}}=\frac{\left|V_{tb}\right|\left|V_{td}\right|}{\left|V_{ub}\right|\left|V_{ud}\right|}\approx\frac{\sqrt{\left(1-\bar{\rho}\right)^{2}+\bar{\eta}^{2}}}{\sqrt{\bar{\rho}^{2}+\bar{\eta^{2}}}}$
$r=\frac{R_{t}}{R_{b}}\frac{\left|P_{t}\right|}{\left|T\right|}$
So that in this case we get,
$\displaystyle C_{\pi\pi}$ $\displaystyle=$ $\displaystyle-
A_{CP}=\frac{2r\sin\delta\sin\alpha}{1-2r\cos\delta\cos\alpha+r^{2}}$
$\displaystyle S_{\pi\pi}$ $\displaystyle=$ $\displaystyle\frac{\sin
2\alpha-2r\cos\delta\sin\alpha}{1-2r\cos\delta\cos\alpha+r^{2}}$
For $B^{+}\rightarrow\pi^{+}\pi^{-}$, $B^{0}\rightarrow\pi^{0}\pi^{0}$, the
decay amplitudes are given by,
$\displaystyle A_{00}$ $\displaystyle=$ $\displaystyle
A(B^{0}\rightarrow\pi^{0}\pi^{0})=\frac{1}{\sqrt{2}}Te^{i\delta_{T}}e^{i\gamma}\left[-r_{c}e^{i\delta_{CT}}+re^{-i(\phi+\gamma-\delta+\delta_{CT})}\right]$
$\displaystyle A_{+0}$ $\displaystyle=$ $\displaystyle
A(B^{+}\rightarrow\pi^{+}\pi^{0})=\frac{1}{\sqrt{2}}Te^{i\delta_{T}}e^{i\gamma}\left[1+r_{C}e^{i\delta_{CT}}\right]$
$\displaystyle r_{C}$ $\displaystyle=$
$\displaystyle\frac{C}{T},\qquad\delta_{CT}=\delta_{C}-\delta_{T}$
Hence for $B^{0}\rightarrow\pi^{0}\pi^{0}$, the $CP$–asymmetries are given by
$\displaystyle C_{\pi^{0}\pi^{0}}$ $\displaystyle=$ $\displaystyle-
A_{00}^{CP}=\frac{-2r/r_{C}\sin(\delta-\delta_{CT})\sin(\gamma+\phi)}{1+r/r_{C}^{2}+2r/r_{C}\cos(\delta-\delta_{CT})\cos(\gamma+\phi)}$
$\displaystyle S_{\pi^{0}\pi^{0}}$ $\displaystyle=$
$\displaystyle-\frac{\sin(2\beta+2\gamma)-2r/r_{C}\cos(\delta-\delta_{CT})\sin(2\beta+\gamma-\phi)+r^{2}/r_{C}^{2}\sin(2\beta-2\phi)}{1++r^{2}/r_{C}^{2}2r/r_{C}\cos(\delta-\delta_{CT})\cos(\gamma+\phi)}$
For the case $\phi=\beta$, we get,
$\displaystyle C_{\pi^{0}\pi^{0}}$ $\displaystyle=$ $\displaystyle-
A_{00}^{CP}=\frac{-2r/r_{C}\sin(\delta-\delta_{CT})\sin\alpha}{1+r^{2}/r_{C}^{2}-2r/r_{C}\cos(\delta-\delta_{CT})\cos\alpha}$
$\displaystyle S_{\pi^{0}\pi^{0}}$ $\displaystyle=$ $\displaystyle-\frac{\sin
2\alpha-2r/r_{C}\cos(\delta-\delta_{CT})\sin\alpha}{1+r^{2}/r_{C}^{2}-2r/r_{C}\cos(\delta-\delta_{CT})\cos\alpha}$
We end this section by considering the decays
$\bar{B}^{0}\rightarrow\phi K_{s},\omega K_{s}\text{ and }\rho K_{s}$
These decays satisfy the relations
$\displaystyle\left[\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\langle\rho^{0}\bar{K}^{0}\left|H_{W}\left(s)\right)\right|\bar{B}^{0}\rangle-\langle\omega\bar{K}^{0}\left|H_{W}\left(s)\right)\right|\bar{B}^{0}\rangle\\\
-\langle\phi\bar{K}^{0}\left|H_{W}\left(s)\right)\right|\bar{B}^{0}\rangle\end{array}\right)\right]$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\frac{1}{2}\left(C-P+P_{EW}\right)-\frac{1}{2}\left(C+P+\frac{1}{3}P_{EW}\right)\\\
-(-P+\frac{1}{3}P_{EW})\end{array}\right]=0$
where $C,P$ and $P_{EW}$ are color suppressed, penguin and electroweak penguin
amplitudes for these decay.
From the above equation, we obtain,
$\displaystyle\frac{\langle\Gamma\rangle_{\omega
K}+\langle\Gamma\rangle_{\rho^{0}K}}{\langle\Gamma\rangle_{\phi K}}$
$\displaystyle\approx$ $\displaystyle 1$
$\displaystyle\frac{S(\rho^{0}K_{s})+S(\omega K_{s})}{2}$
$\displaystyle\approx$ $\displaystyle S(\phi K_{s})=-\sin 2\beta$
$\displaystyle C(\rho^{0}K_{s})$ $\displaystyle\approx$
$\displaystyle-C(\omega K_{s})$
where we have neglected the terms of the order $r^{2}$. The parameter $r$ is
defined below,
$\displaystyle\langle\rho^{0}\bar{K}^{0}\left|H_{W}(s)\right|\bar{B}^{0}\rangle$
$\displaystyle=$
$\displaystyle-\left|V_{cb}\right|\left|V_{cd}\right|\left|P\right|e^{i\delta_{P}}\left[1-re^{i(\delta_{C}-\delta_{P})}e^{-i\gamma}\right]$
$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{\left|C\right|}{\left|P\right|}\lambda^{2}R_{b}$
$\displaystyle\frac{\left|C\right|}{\left|P\right|}$ $\displaystyle\sim$
$\displaystyle\frac{a_{2}}{a_{4}}$
Assuming factorization for the electroweak penguin, we get from the above
equation an interesting sum rule,
$f_{\rho}F_{1}^{B-K}(m_{\rho}^{2})-\frac{1}{3}f_{\omega}F_{1}^{B-K}(m_{\omega}^{2})-\frac{2}{3}f_{\phi}F_{1}^{B-K}(m_{\phi}^{2})=0$
Assuming $F_{1}(m_{\rho}^{2})=F_{1}(m_{\omega}^{2})=F_{1}(m_{\phi}^{2})\approx
F_{1}(1$ GeV${}^{2})$, we get from the relation,
$f_{\rho}-\frac{1}{3}f_{\omega}-\frac{2}{3}f_{\phi}=0$
which is reminiscent of current algebra and spectral function sum rules of
1960’s.
The above sum rule is very well satisfied by the experimental values,
$f_{\rho}=\left(209\pm 1\right)\text{MeV},\qquad f_{\omega}=\left(187\pm
3\right)\text{MeV},\qquad f_{\phi}=\left(221\pm 3\right)\text{ MeV.}$
It is convenient to write, from Eqs.(66) and (69), the decay rates in the
following form,
$\displaystyle\left[\Gamma_{f}(t)-\bar{\Gamma}_{\bar{f}}(t)\right]+\left[\Gamma_{\bar{f}}-\bar{\Gamma}_{f}(t)\right]$
$\displaystyle=$ $\displaystyle e^{-\Gamma t}\left\\{\cos\Delta
mt\left[\left(\left|A_{f}\right|^{2}-\left|\bar{A}_{\bar{f}}\right|^{2}\right)+\left(\left|A_{\bar{f}}\right|^{2}-\left|\bar{A}_{f}\right|^{2}\right)\right]\right.$
$\displaystyle\left.+2\sin\Delta
mt\left[\text{Im}\left(e^{2i\phi_{M}}A_{f}^{\ast}\bar{A}_{f}\right)+\text{Im}\left(e^{2i\phi_{M}}A_{\bar{f}}^{\ast}\bar{A}_{\bar{f}}\right)\right]\right\\}$
$\displaystyle\left[\Gamma_{f}(t)+\bar{\Gamma}_{\bar{f}}(t)\right]-\left[\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{f}(t)\right]$
$\displaystyle=$ $\displaystyle e^{-\Gamma t}\left\\{\cos\Delta
mt\left[\left(\left|A_{f}\right|^{2}+\left|\bar{A}_{\bar{f}}\right|^{2}\right)-\left(\left|A_{\bar{f}}\right|^{2}+\left|\bar{A}_{f}\right|^{2}\right)\right]\right.$
$\displaystyle\left.+2\sin\Delta
mt\left[\text{Im}\left(e^{2i\phi_{M}}A_{f}^{\ast}\bar{A}_{f}\right)-\text{Im}\left(e^{2i\phi_{M}}A_{\bar{f}}^{\ast}\bar{A}_{\bar{f}}\right)\right]\right\\}$
We now use the above equations to obtain some interesting results for the $CP$
asymmetries for B-decays.
Case-II:
We first consider the case in which single weak amplitudes $A_{f}$ and
$A_{\bar{f}}^{{}^{\prime}}$ with different weak phases describe the decays:
$\displaystyle A_{f}$ $\displaystyle=$ $\displaystyle\langle
f\left|\mathcal{L_{W}}\right|B^{0}\rangle=e^{i\phi}F_{f}$ $\displaystyle
A_{\bar{f}}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\langle\bar{f}\left|\mathcal{L_{W}^{{}^{\prime}}}\right|B^{0}\rangle=e^{i\phi^{{}^{\prime}}}F_{\bar{f}}^{{}^{\prime}}$
$CPT$ gives,
$\displaystyle\bar{A}_{\bar{f}}$ $\displaystyle=$
$\displaystyle\langle\bar{f}\left|\mathcal{L_{W}}\right|\bar{B}^{0}\rangle=e^{2i\delta_{f}}A_{f}^{\ast}$
$\displaystyle\bar{A}_{f}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\langle
f\left|\mathcal{L_{W}}^{{}^{\prime}}\right|\bar{B}^{0}\rangle=e^{2i\delta^{{}^{\prime}}_{\bar{f}}}A_{\bar{f}}^{\ast^{\prime}}$
Note $\delta_{f}$ and $\delta_{\bar{f}}^{\prime}$ are strong phases; $\phi$
and $\phi^{\prime}$ are weak phases. The states $|f>$ and $|\overline{f}>$ are
C-conjugate of each other such as states $D^{(*)-}\pi^{+}(D^{(*)+}\pi^{-}),$
$D_{s}^{(*)-}K^{+}(D_{s}^{(*)+}K^{-}),$ $D^{-}\rho^{+}(D^{+}\rho^{-})$
Hence, we get from Eqs.(LABEL:e1), (LABEL:e2), (LABEL:e3) and (LABEL:e4),
$\displaystyle\mathcal{A}\left(t\right)$ $\displaystyle\equiv$
$\displaystyle\frac{[\Gamma_{f}(t)-\bar{\Gamma}_{\bar{f}}(t)]+[\Gamma_{\bar{f}}(t)-\bar{\Gamma}_{f}(t)]}{[\Gamma_{f}(t)+\bar{\Gamma}_{\bar{f}}(t)]+[\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{f}]}$
(87) $\displaystyle=$
$\displaystyle\frac{2\bigl{|}F_{f}\bigr{|}\bigl{|}\overset{{}^{\prime}}{F}_{\bar{f}}\bigr{|}}{\bigl{|}F_{f}\bigr{|}^{2}+\bigl{|}\overset{{}^{\prime}}{F}_{\bar{f}}\bigr{|}^{2}}\sin\Delta
mt\sin\bigl{(}2\phi_{M}-\phi-\phi^{{}^{\prime}}\bigr{)}\cos\bigl{(}\delta_{f}-\overset{{}^{\prime}}{\delta}_{\bar{f}}\bigr{)}$
$\displaystyle\mathcal{F}\left(t\right)$ $\displaystyle\equiv$
$\displaystyle\frac{\left[\Gamma_{f}(t)+\bar{\Gamma}_{\bar{f}}\right]-\left[\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{f}\right]}{\left[\Gamma_{f}(t)+\bar{\Gamma}_{\bar{f}}\right]+\left[\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{{f}}\right]}$
(88) $\displaystyle=$
$\displaystyle\frac{\bigl{|}F_{f}\bigr{|}^{2}-\bigl{|}\overset{{}^{\prime}}{F}_{\bar{f}}\bigr{|}^{2}}{\bigl{|}F_{f}\bigr{|}^{2}+\bigl{|}\overset{{}^{\prime}}{F}_{\bar{f}}\bigr{|}^{2}}\cos\Delta
mt$ $\displaystyle-$
$\displaystyle\frac{2\bigl{|}F_{f}\bigr{|}\bigl{|}\overset{{}^{\prime}}{F}_{\bar{f}}\bigr{|}}{\bigl{|}F_{f}\bigr{|}^{2}+\bigl{|}\overset{{}^{\prime}}{F}_{\bar{f}}\bigr{|}^{2}}\sin\Delta
mt\cos\left(2\phi_{M}-\phi-\phi^{{}^{\prime}}\right)\sin\bigl{(}\delta_{f}-\overset{{}^{\prime}}{\delta}_{\bar{f}}\bigr{)}$
We now apply the above formula to $B\rightarrow\pi D$ and $B_{s}\rightarrow
KD_{s}$ decays. For these decays,
$\phi=0,\qquad\phi^{\prime}=\gamma$ $\phi_{M}=\begin{cases}-\beta,&\text{for
$B^{0}$}\\\ -\beta_{s},&\text{for $B_{s}^{0}$}\end{cases}$ $\displaystyle
A_{f}$ $\displaystyle=$ $\displaystyle\langle
D^{-}\pi^{+}\left|\mathcal{L_{W}}\right|B^{0}\rangle=F_{f}$ $\displaystyle
A_{\bar{f}}^{\prime}$ $\displaystyle=$ $\displaystyle\langle
D^{+}\pi^{-}\left|\mathcal{L_{W}}^{\prime}\right|B^{0}\rangle=e^{i\gamma}F_{\bar{f}}^{{}^{\prime}}$
$\displaystyle A_{f_{s}}$ $\displaystyle=$ $\displaystyle\langle
K^{+}D_{s}^{-}\left|\mathcal{L_{W}}\right|B_{s}^{0}\rangle=F_{f_{s}}$
$\displaystyle A_{\bar{f}_{s}}^{\prime}$ $\displaystyle=$
$\displaystyle\langle
K^{-}D_{s}^{+}\left|\mathcal{L_{W}}^{\prime}\right|B_{s}^{0}\rangle=e^{i\gamma}F_{\bar{f}_{s}}^{{}^{\prime}}$
Note that the effective Lagrangians for decays $(q=d,s)$ are given by,
$\displaystyle\mathcal{L_{W}}=V_{cb}V_{uq}^{\ast}\left[\bar{q}\gamma^{\mu}\left(1-\gamma_{5}\right)u\right]\left[\bar{c}\gamma_{\mu}\left(1-\gamma_{5}\right)b\right]$
(89a)
$\displaystyle\mathcal{L_{W}}^{\prime}=V_{ub}V_{cq}^{\ast}\left[\bar{q}\gamma^{\mu}\left(1-\gamma_{5}\right)c\right]\left[\bar{u}\gamma_{\mu}\left(1-\gamma_{5}\right)b\right]$
(89b)
respectively. In the Wolfenstein parametrization of $CKM$ matrix,
$\frac{\left|V_{ub}\right|\left|V_{cq}\right|}{\left|V_{cb}\right|\left|V_{uq}\right|}=\lambda^{2}\sqrt{\bar{\rho}^{2}+\bar{\eta}^{2}},\qquad$
(90)
Define,
$r=\lambda^{2}R_{b}\frac{\bigl{|}\overset{{}^{\prime}}{F}_{\bar{f}}\bigr{|}}{\bigl{|}F_{f}\bigr{|}}andr_{s}=R_{b}\frac{\bigl{|}\overset{{}^{\prime}}{F}_{\bar{f}_{s}}\bigr{|}}{\bigl{|}F_{f_{s}}\bigr{|}}$
Thus, we get from Eqs. $\eqref{e6}$ and $\eqref{e7}$ for $B^{0}$ decays,
(replacing
$\frac{\bigl{|}\overset{{}^{\prime}}{F}_{\bar{f}}\bigr{|}}{\bigl{|}F_{f}\bigr{|}}$
by $r$),
$\displaystyle\mathcal{A}\left(t\right)$ $\displaystyle=$
$\displaystyle-\frac{2r}{1+r^{2}}\sin\Delta
m_{B}t\sin\left(2\beta+\gamma\right)\cos\left(\delta_{f}-\overset{{}^{\prime}}{\delta}_{\bar{f}}\right)$
$\displaystyle\mathcal{F}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{1-r^{2}}{1+r^{2}}\cos\Delta
m_{B}t-\frac{2r}{1+r^{2}}\sin\Delta
m_{B}t\cos\left(2\beta+\gamma\right)\sin\left(\delta_{f}-\overset{{}^{\prime}}{\delta}_{\bar{f}}\right)$
(91)
For the decays,
$\displaystyle\bar{B}_{s}^{0}\left(B_{s}^{0}\right)$
$\displaystyle\rightarrow$ $\displaystyle
K^{-}D_{s}^{+}\left(K^{+}D_{s}^{-}\right)$
$\displaystyle\bar{B}_{s}^{0}\left(B_{s}^{0}\right)$
$\displaystyle\rightarrow$ $\displaystyle
K^{+}D_{s}^{-}\left(K^{-}D_{s}^{+}\right)$
we get,
$\displaystyle\mathcal{A}_{s}\left(t\right)$ $\displaystyle=$
$\displaystyle-\frac{2r_{s}}{1+r_{s}^{2}}\sin(\Delta
m_{B_{s}}t)\sin\left(2\beta_{s}+\gamma\right)\cos\left(\delta_{f_{s}}-\overset{{}^{\prime}}{\delta}_{\bar{f}_{s}}\right)$
$\displaystyle\mathcal{F}_{s}(t)$ $\displaystyle=$
$\displaystyle\frac{1-r_{s}^{2}}{1+r_{s}^{2}}\cos\Delta
m_{B_{s}}t-\frac{2r_{s}}{1+r_{s}^{2}}\sin\Delta
m_{B_{s}}t\cos\left(2\beta_{s}+\gamma\right)\sin\left(\delta_{f_{s}}-\overset{{}^{\prime}}{\delta}_{\bar{f}_{s}}\right)$
(92)
We note that for time integrated $CP$-asymmetry,
$\displaystyle\mathcal{A}_{s}$ $\displaystyle\equiv$
$\displaystyle\frac{\int_{0}^{\infty}\left[\Gamma_{fs}\left(t\right)-\bar{\Gamma}_{fs}\left(t\right)\right]dt}{\int_{0}^{\infty}\left[\Gamma_{fs}\left(t\right)+\bar{\Gamma}_{fs}\left(t\right)\right]dt}$
(93) $\displaystyle=$
$\displaystyle-\frac{2r}{1+r^{2}}\sin\left(2\beta_{s}+\gamma\right)\frac{\Delta
m_{B_{s}}/\Gamma_{s}}{1+\left(\Delta
m_{B_{s}}/\Gamma_{s}\right)^{2}}\cos(\delta_{f_{s}}-\overset{{}^{\prime}}{\delta}_{\bar{f}_{s}})$
The $CP$–asymmetry $\mathcal{A}_{s}\left(t\right)$ or $\mathcal{A}_{s}$
involves two experimentally unknown parameters
$\sin\left(2\beta_{s}-\gamma\right)$ and $\Delta m_{B_{s}}$. Both these
parameters are of importance in order to test the unitarity of $CKM$ matrix
viz whether $CKM$ matrix is a sole source of $CP$–violation in the processes
in which $CP$–violation has been observed.
Within the case II, we discuss $B$ decays into baryons and antibaryons.
So far we have discussed the $CP$-violation in kaon and
$B_{q}^{0}-\bar{B}_{q}^{0}$ systems. There is thus a need to study
$CP$-violation outside these systems.
The decays of $B(\bar{B})$ mesons to baryon-antibaryon pair $N_{1}$
$\bar{N}_{2}$ $(\bar{N}_{1}$ $N_{2})$ and subsequent decays of
$N_{2},\bar{N}_{2}$ or $(N_{1},\bar{N}_{1})$ to a lighter hyperon
(antihyperon) plus a meson provide a means to study $CP$-odd observables as
for example in the process,
$e^{-}e^{+}\rightarrow B,\bar{B}\rightarrow N_{1}\bar{N}_{2}\rightarrow
N_{1}\bar{N}_{2}^{\prime}\bar{\pi},\qquad\bar{N}_{1}N_{2}\rightarrow\bar{N}_{1}N_{2}^{\prime}\pi$
The decay $B\rightarrow N_{1}\bar{N}_{2}(f)$ is described by the matrix
element,
$M_{f}=F_{q}e^{+i\phi}\left[\bar{u}(\mathbf{p}_{1})(A_{f}+\gamma_{5}B_{f})v(\mathbf{p}_{2})\right]$
(94)
where as $B\rightarrow\overline{N}_{1}N_{2}(\overline{f})$ is described by the
matrix elements
$\overset{}{M^{\prime}}_{f}=\overset{}{F^{\prime}}_{q}e^{+i\phi^{{}^{\prime}}}\left[\bar{u}(\mathbf{p}_{2})(\overset{}{A^{\prime}}_{\overline{f}}+\gamma_{5}\overset{}{B^{\prime}}_{\overline{f}})v(\mathbf{p}_{1})\right]$
where $F_{q}$ is a constant containing CKM factor, $\phi$ is the weak phase.
The amplitude $A_{f}$ and $B_{f}$ are in general complex in the sense that
they incorporate the final state phases $\delta_{p}^{f}$ and $\delta_{s}^{f}$
and they may also contain weak phases $\phi_{s}$ and $\phi_{p}$ Note that
$A_{f}$ is the parity violating amplitude ($p$-wave) whereas $B_{f}$ is parity
conserving amplitude ($s$-wave). The $CPT$ invariance gives the matrix
elements for the decay $\bar{B}\rightarrow\bar{N}_{1}N_{2}(\bar{f}):$
$\bar{M}_{\bar{f}}=F_{q}e^{-i\phi}\left[\bar{u}(\mathbf{p}_{2})(-A_{f}^{\ast}e^{2i\delta_{p}^{f}}+\gamma_{5}B_{f}^{\ast}e^{2i\delta_{s}^{f}})v(\mathbf{p}_{1})\right]$
(95)
if the decays are described by a single matrix element $M_{f}$. If
$\phi_{s}=0=\phi_{p}$ then $CPT$ and $CP$ invariance give the same predictions
viz
$\bar{\Gamma}_{\bar{f}}=\Gamma_{f},\qquad\bar{\alpha}_{\bar{f}}=-\alpha_{f},\qquad\bar{\beta}_{\bar{f}}=-\beta_{f},\qquad\bar{\gamma}_{\bar{f}}=\gamma_{f}$
(96)
The decay width for the mode $B\rightarrow N_{1}\bar{N}_{2}(f)$ is given by,
$\displaystyle\Gamma_{f}$ $\displaystyle=$
$\displaystyle\frac{m_{1}m_{2}}{2\pi
m_{B}^{2}}\left|\mathbf{p}\right|\left|M_{f}\right|^{2}$ (97) $\displaystyle=$
$\displaystyle\frac{F_{q}^{2}}{2\pi
m_{B}^{2}}\left|\mathbf{p}\right|\left[(p_{1}\cdot
p_{2}-m_{1}m_{2})\left|A_{f}\right|^{2}+(p_{1}\cdot
p_{2}+m_{1}m_{2})\left|B_{f}\right|^{2}\right]$
In order to take into account the polarization of $N_{1}$ and $\bar{N}_{2},$
we give the general expression for $\left|M_{f}\right|^{2}$,
$\displaystyle\left|M_{f}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{F_{q}^{2}}{16m_{1}m_{2}}Tr\left[\begin{array}[]{c}(\not{p}_{1}+m_{1})(1+\gamma_{5}\gamma\cdot
s_{1})(A_{f}+\gamma_{5}B_{f})(\not{p}_{2}-m_{2})\\\
\times(1+\gamma_{5}\gamma\cdot
s_{2})(A_{f}^{\ast}-\gamma_{5}B_{f}^{\ast})\end{array}\right]$ (100)
where $s_{1}^{\mu},s_{2}^{\mu}$ are polarization vectors of $N_{1}$ and
$\bar{N}_{2}$ respectively $(p_{1}.s_{1}=0,\quad p_{2}.s_{2}=0,\quad
s_{1}^{2}=-1=s_{2}^{2})$.
In the rest frame of $B$, we get,
$\displaystyle\left|M_{f}\right|^{2}$ $\displaystyle=$ $\displaystyle
F_{q}^{2}\frac{2E_{1}E_{2}}{4m_{1}m_{2}}\left[\left|a_{s}^{f}\right|^{2}+\left|a_{p}^{f}\right|^{2}\right]$
(104)
$\displaystyle\left\\{\begin{array}[]{c}1+\alpha_{f}\left(\frac{m_{1}}{E_{1}}\mathbf{n}\cdot\mathbf{s}_{1}-\frac{m_{2}}{E_{2}}\mathbf{n}\cdot\mathbf{s}_{2}\right)\\\
+\beta_{f}\mathbf{n}\cdot(\mathbf{s}_{1}\times\mathbf{s}_{2})+\gamma_{f}\left[(\mathbf{n}\cdot\mathbf{s}_{1})(\mathbf{n}\cdot\mathbf{s}_{2})-\mathbf{s}_{1}\cdot\mathbf{s}_{2}\right]\\\
-\frac{m_{1}m_{2}}{E_{1}E_{2}}(\mathbf{n}\cdot\mathbf{s}_{1})(\mathbf{n}\cdot\mathbf{s}_{2})\end{array}\right\\}$
where,
$\displaystyle a_{s}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{p_{1}\cdot
p_{2}+m_{1}m_{2}}{2E_{1}E_{2}}}B,\quad a_{p}=-\sqrt{\frac{p_{1}\cdot
p_{2}-m_{1}m_{2}}{2E_{1}E_{2}}}A$ (105) $\displaystyle\alpha_{f}$
$\displaystyle=$
$\displaystyle\frac{2S_{f}P_{f}\cos(\delta_{s}^{f}-\delta_{p}^{f})}{S_{f}^{2}+P_{f}^{2}},\quad\beta_{f}=\frac{2S_{f}P_{f}\sin(\delta_{s}^{f}-\delta_{p}^{f})}{S_{f}^{2}+P_{f}^{2}}$
$\displaystyle\gamma_{f}$ $\displaystyle=$
$\displaystyle\frac{S_{f}^{2}-P_{f}^{2}}{S_{f}^{2}+P_{f}^{2}},\quad
a_{s}=S_{f}e^{i\delta_{s}^{f}},\quad a_{p}^{f}=P_{f}e^{i\delta_{p}^{f}}$ (106)
In the rest frame of $B$, due to spin conservation,
$\left(\lambda_{1}\equiv\frac{E_{1}}{m_{1}}\mathbf{n}\cdot\mathbf{s}_{1}\right)=\left(\lambda_{2}\equiv\frac{-E_{2}}{m_{2}}\mathbf{n}\cdot\mathbf{s}_{2}\right)=\pm
1$ (107)
Thus, invariants multiplying $\beta_{f}$ and $\gamma_{f}$ vanish. Hence we
have,
$\displaystyle\left|M_{f}\right|^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{2E_{1}E_{2}}{m_{1}m_{2}}\right)F_{q}^{2}(S_{f}^{2}+P_{f}^{2})\left[(1+\lambda_{1}\lambda_{2})+\alpha_{f}(\lambda_{1}+\lambda_{2})\right]$
(108) $\displaystyle\Gamma_{f}$ $\displaystyle=$
$\displaystyle\Gamma_{f}^{++}+\Gamma_{f}^{--}=\frac{2E_{1}E_{2}}{2\pi
m_{B}^{2}}\left|\vec{p}\right|F_{q}^{2}\left[S_{f}^{2}+P_{f}^{2}\right]=\bar{\Gamma}_{\bar{f}}$
(109) $\displaystyle\Delta\Gamma_{f}$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{f}^{++}-\Gamma_{f}^{--}}{\Gamma_{f}^{++}+\Gamma_{f}^{--}}=\alpha_{f},\qquad\Delta\bar{\Gamma}_{\bar{f}}=\bar{\alpha}_{\bar{f}}=-\alpha_{f}$
(110)
Eqs. (109) and (110) follow from $CP$ invariance. It will be of interest to
test these equations.
Now $B_{q}^{0},$ $\bar{B}_{q}^{0}$ annihilate into baryon-antibaryon pair
$N_{1}\bar{N}_{2}$ through $W$-exchange as depicted in Figs (5a) and (5b).
$B^{-}\rightarrow N_{1}\bar{N}_{2}$ through annihilation diagram is shown in
Fig (6). It is clear from Fig (5a) and (5b), that we have the same final state
configuration for $B_{q}^{0},$ $\bar{B}_{q}^{0}\rightarrow N_{1}\bar{N}_{2}.$
Thus, one would expect,
$\displaystyle S_{\bar{f}}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
S_{f},\qquad P_{\bar{f}}^{{}^{\prime}}=P_{f}$
$\displaystyle\overset{{}^{\prime}}{\delta}_{s}^{\bar{f}}$ $\displaystyle=$
$\displaystyle\delta_{s}^{f},\qquad\overset{{}^{\prime}}{\delta}_{p}^{\bar{f}}=\delta_{p}^{f}$
(111)
Hence we have,
$\displaystyle\Gamma_{\bar{f}}^{\prime}$ $\displaystyle=$
$\displaystyle\bar{\Gamma}_{f}^{\prime}=r^{2}\Gamma_{f};\ \ \ \
r^{2}=\frac{\left|F_{q}^{\prime}\right|^{2}}{\left|F_{q}\right|^{2}}$ (112)
$\displaystyle\bar{\alpha}_{f}^{\prime}$ $\displaystyle=$
$\displaystyle-\alpha_{\bar{f}}^{\prime}=\alpha_{f}=-\bar{\alpha}_{\bar{f}}$
(113)
Above predictions can be tested in future experiments on baryon decay modes of
$B$-mesons. In particular $\bar{\alpha}_{f}^{\prime}=\alpha_{f}$ would give
direct confirmation of Eqs. (111).
For the time dependent baryon decay modes of $B_{q}^{0}-\bar{B}_{q}^{0}$, we
have: $\left(\phi=\gamma,\ \phi^{\prime}=0\right)$
$\displaystyle\mathcal{A(}t)$ $\displaystyle=$
$\displaystyle\mathcal{A}^{++}(t)+\mathcal{A}^{--}(t)=\frac{2r\sin\Delta
mt\sin(2\phi_{M}-\gamma)}{1+r^{2}}$ (114) $\displaystyle\Delta\mathcal{A}(t)$
$\displaystyle=$ $\displaystyle\mathcal{A}^{++}(t)-\mathcal{A}^{--}(t)=0$
(115) $\displaystyle\mathcal{F}(t)$ $\displaystyle=$
$\displaystyle\mathcal{F}^{++}(t)+\mathcal{F}^{--}(t)=\frac{1-r^{2}}{1+r^{2}}\cos\Delta
mt$ (116) $\displaystyle\Delta\mathcal{F}(t)$ $\displaystyle=$
$\displaystyle\mathcal{F}^{++}(t)-\mathcal{F}^{--}(t)=\frac{1-r^{2}}{2(1+r^{2})}(\alpha_{f}+\bar{\alpha}_{\bar{f}})\cos\Delta
mt$ (117) $\displaystyle-\frac{4r\sin\Delta
mt\sin(2\phi_{M}-\gamma)S_{f}P_{f}}{(1+r^{2})(S_{f}^{2}+P_{f}^{2})}$
where we have used Eqs. (113). For $B_{d}^{0},$
$r=-\lambda^{2}\sqrt{\bar{\rho}^{2}+\bar{\eta}^{2}}\approx-(0.02\pm 0.006)$
[4], $\phi_{M}=-\beta;$ for $B_{s}^{0},$
$r=-\sqrt{\bar{\rho}^{2}+\bar{\eta}^{2}}\approx-(0.40\pm 0.13)$,
$\phi_{M}=-\beta_{s}$.
Eq.(114) gives a means to determine the weak phase $2\beta+\gamma$ or $\gamma$
in the baryon decay modes of $B_{d}^{0}$ and $B_{s}^{0}$ respectively. Non-
zero $\cos\Delta mt$ term in $\Delta\mathcal{F}(t)$ would give clear
indication of $CP$ violation especially for baryon decay modes of $B_{d}^{0},$
for which $r^{2}\leq 1,$ so that $\frac{1-r^{2}}{1+r^{2}}\approx 1$. It may be
noted that the time-dependent asymmetries arises because there are two
independent amplitudes for the decays $B_{q}^{0}\rightarrow
N_{1}\overline{N}_{2},$ $\overline{N}_{1}N_{2}:M_{f}\overset{}{,\
M^{\prime}}_{\overline{f}}.$
The baryon decay modes of $B$-mesons not only provide a means to test
prediction of $CP$ asymmetry viz $\alpha_{f}+\bar{\alpha}_{\bar{f}}=0$ for
charmed baryons (discussed above) but also to test the $CP$-asymmetry in
hyperon (antihyperon) decays viz absence of $CP$-odd observables
$\Delta\Gamma,\Delta\alpha,\Delta\beta$ discussed in [8]. Consider for example
the decays,
$B_{q}^{0}\rightarrow p\bar{\Lambda}_{c}^{-}\rightarrow
p\bar{p}K^{0}(p\bar{\Lambda}\pi^{-}\rightarrow p\overline{p}\pi^{+}\pi^{-}),$
$\bar{B}_{q}^{0}\rightarrow\overline{p}\Lambda_{c}^{+}\rightarrow\bar{p}p\bar{K}^{0}(\overline{p}\Lambda\pi^{+}\rightarrow\bar{p}p\overline{b}\pi^{-}\pi^{+})$
By analyzing the final state $\bar{p}p\bar{K}^{0},p\bar{p}K^{0},$ one may test
$\alpha_{f}=-\bar{\alpha}_{\bar{f}}$ for the charmed hyperon. We note that for
$\Lambda_{c}^{+},$ $c\tau=59.9\mu$m, whereas $c\tau=7.8$cm for
$\Lambda-$hyperon, so that the decays of $\Lambda_{c}^{+}$ and $\Lambda$ would
not interfere with each other. By analysing the final state
$\bar{p}p\pi^{-}\pi^{+}$ and $p\bar{p}\pi^{+}\pi^{-},$ one may check
$CP$–violation for hyperon decays. One may also note that for
$(B_{d}^{0},\bar{B}_{d}^{0})$ complex, the competing channels viz
$B_{d}^{0}\rightarrow\bar{p}\Lambda_{c}^{+},$ $\bar{B}_{d}^{0}\rightarrow
p\bar{\Lambda}_{c}^{-}$ are doubly Cabibbo suppressed by
$r^{2}=\lambda^{2}\left(\bar{\rho}^{2}+\bar{\eta}^{2}\right)$ unlike
$(B_{s}^{0}-\bar{B}_{s}^{0})$ complex where the competing channels are
suppressed by a factor of $\left(\bar{\rho}^{2}+\bar{\eta}^{2}\right)$. Hence
$B_{d}^{0}($ $\bar{B}_{d}^{0})$ decays are more suitable for this type of
analysis. Other decays of interest are,
$\displaystyle B^{-}$ $\displaystyle\rightarrow$
$\displaystyle\Lambda\bar{\Lambda}_{c}^{-}\rightarrow\Lambda\bar{\Lambda}\pi^{-}\rightarrow
p\pi^{-}\bar{p}\pi^{+}\pi^{-}$ $\displaystyle B^{+}$
$\displaystyle\rightarrow$
$\displaystyle\bar{\Lambda}\Lambda_{c}^{+}\rightarrow\bar{\Lambda}\Lambda\pi^{+}\rightarrow\bar{p}\pi^{+}p\pi^{-}\pi^{+}$
$\displaystyle B_{c}^{-}$ $\displaystyle\rightarrow$
$\displaystyle\bar{p}\Lambda\rightarrow\bar{p}p\pi^{-}$ $\displaystyle
B_{c}^{+}$ $\displaystyle\rightarrow$ $\displaystyle p\bar{\Lambda}\rightarrow
p\bar{p}\pi^{+}$
The non-leptonic hyperon (antihyperon) decays $N\rightarrow
N^{\prime}\pi(\bar{N}\rightarrow\bar{N}^{\prime}\bar{\pi})$ are related to
each other by $CPT$,
$\displaystyle a_{l}(I)$ $\displaystyle=$ $\displaystyle\left\langle
f_{lI}^{out}\left|H_{W}\right|N\right\rangle=\eta_{f}e^{2i\delta_{l}(I)}\left\langle\bar{f}_{lI}^{out}\left|H_{W}\right|\bar{N}\right\rangle$
$\displaystyle=$
$\displaystyle\eta_{f}e^{2i\delta_{l}(I)}\bar{a}_{l}^{\ast}(I)$
Hence,
$\bar{a}_{l}(I)=\eta_{f}e^{2i\delta_{l}(I)}\bar{a}_{l}^{\ast}(I)=(-1)^{l+1}e^{i\delta_{l}(I)}e^{-i\phi}\left|a_{l}\right|$
where we have selected the phase $\eta_{f}=(-1)^{l+1}$. Here $I$ is the
isospin of the final state and $\phi$ is the weak phase. Thus necessary
condition for non-zero $CP$ odd observables is that the weak phase for each
partial wave amplitude should be different. For instance for the decays
$B^{0}(\bar{B}^{0})\rightarrow p\bar{\Lambda}_{c}^{-}(\bar{p}\Lambda_{c}^{+})$
we have,
$\displaystyle\delta\Gamma$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\delta\alpha_{f}$ $\displaystyle=$
$\displaystyle-\tan\left(\delta_{s}-\delta_{p}\right)\tan\left(\phi_{s}-\phi_{p}\right)$
$\displaystyle\approx$
$\displaystyle-\tan\left(\delta_{s}-\delta_{p}\right)\sin\left(\phi_{s}-\phi_{p}\right)$
Case III:
Here $A_{f}\neq A_{\bar{f}}$.
$\displaystyle A_{f}$ $\displaystyle=$ $\displaystyle\langle
f\left|\mathcal{L_{W}}\right|B^{0}\rangle=\left[e^{i\phi_{1}}F_{1f}+e^{i\phi_{2}}F_{2f}\right]$
$\displaystyle A_{\bar{f}}$ $\displaystyle=$
$\displaystyle\langle\bar{f}\left|\mathcal{L_{W}}\right|B^{0}\rangle=\left[e^{i\phi_{1}}F_{1\bar{f}}+e^{i\phi_{2}}F_{2\bar{f}}\right]$
Examples:
$B^{0}\rightarrow\rho^{-}\pi^{+}(f):\text{ }A_{f}\qquad
B^{0}\rightarrow\rho^{+}\pi^{-}(\bar{f}):A_{\bar{f}}$ $B_{s}^{0}\rightarrow
K^{\ast-}K^{+}\qquad B_{s}^{0}\rightarrow K^{\ast+}K^{-}$
$CPT$ gives,
$\bar{A}_{\bar{f},f}=\sum_{i}[e^{-i\phi_{i}}e^{2i\delta^{i}_{f,\bar{f}}}F^{\ast}_{if\bar{f}}]$
Subtracting and adding Eqs. (LABEL:e2) and (LABEL:e1), we get,
We now discuss the decays listed in case (ii) where $A_{f}\neq A_{\bar{f}}$.
Subtracting and adding Eqs. $(\ref{e2})$ and $(\ref{e1})$, we get,
$\displaystyle\frac{\Gamma_{f}(t)-\bar{\Gamma}_{f}(t)}{\Gamma_{f}(t)+\bar{\Gamma}_{f}(t)}=$
$\displaystyle C_{f}\cos\Delta mt+S_{f}\sin\Delta mt$ $\displaystyle=$
$\displaystyle(C-\Delta C)\cos\Delta mt+(S-\Delta S)\sin\Delta mt$ (118)
$\displaystyle\frac{\Gamma_{\bar{f}}(t)-\bar{\Gamma}_{\bar{f}}(t)}{\Gamma_{\bar{f}}(t)+\bar{\Gamma}_{\bar{f}}(t)}=$
$\displaystyle C_{\bar{f}}\cos\Delta mt+S_{\bar{f}}\sin\Delta mt$
$\displaystyle=$ $\displaystyle(C+\Delta C)\cos\Delta mt+(S+\Delta
S)\sin\Delta mt$ (119)
where
$\displaystyle C_{\bar{f},f}$ $\displaystyle=(C\pm\Delta C)$
$\displaystyle=\frac{\bigl{|}A_{\bar{f},f}\bigr{|}^{2}-\bigl{|}\bar{A}_{\bar{f},f}\bigr{|}^{2}}{\bigl{|}A_{\bar{f},f}\bigr{|}^{2}+\bigl{|}\bar{A}_{\bar{f},f}\bigr{|}^{2}}$
$\displaystyle=\frac{\Gamma_{\bar{f},f}-\bar{\Gamma}_{\bar{f},f}}{\Gamma_{\bar{f},f}+\bar{\Gamma}_{\bar{f},f}}$
$\displaystyle=\frac{R_{\bar{f},f}(1-A_{CP}^{\bar{f},f})-R_{\bar{f},f}(1+A_{CP}^{\bar{f},f})}{\Gamma(1\pm
A_{CP})}$ (120) $\displaystyle S_{\bar{f},f}$ $\displaystyle=(S\pm\Delta S)$
(121)
$\displaystyle=\frac{2\text{Im}[e^{2i\phi_{M}}A^{\ast}_{\bar{f},f}\bar{A}_{\bar{f},f}]}{\Gamma_{\bar{f},f}+\bar{\Gamma}_{\bar{f},f}}$
(122) $\displaystyle A_{CP}^{\bar{f}}$
$\displaystyle=\frac{\bar{\Gamma}_{f}-\Gamma_{\bar{f}}}{\Gamma_{\bar{f}}+\bar{\Gamma}_{f}}$
$\displaystyle A_{CP}^{f}$
$\displaystyle=\frac{\bar{\Gamma}_{\bar{f}}-\Gamma_{f}}{\Gamma_{f}+\bar{\Gamma}_{\bar{f}}}$
(123) $\displaystyle A_{CP}$
$\displaystyle=\frac{(\Gamma_{\bar{f}}+\bar{\Gamma}_{\bar{f}})-(\bar{\Gamma_{f}}+\Gamma_{f})}{(\Gamma_{\bar{f}}-\bar{\Gamma}_{\bar{f}})-(\bar{\Gamma_{f}}+\Gamma_{f})}$
(124)
$\displaystyle=\frac{R_{f}A^{f}_{CP}-R_{\bar{f}}A^{\bar{f}}_{CP}}{\Gamma}$
(125)
where
$\displaystyle R_{f}$
$\displaystyle=\frac{1}{2}(\Gamma_{f}+\bar{\Gamma}_{\bar{f}}),\qquad
R_{\bar{f}}=\frac{1}{2}(\Gamma_{\bar{f}}+\bar{\Gamma}_{f})$
$\displaystyle\Gamma$ $\displaystyle=R_{f}+R_{\bar{f}}$ (126)
The following relations are also useful which can be easily derived from above
equations
$\displaystyle\frac{R_{\bar{f},f}}{R_{f}+R_{\bar{f}}}$
$\displaystyle=\frac{1}{2}[(1\pm\Delta C)\pm A_{CP}C]$ (127)
$\displaystyle\frac{R_{\bar{f}}-R_{f}}{R_{f}+R_{\bar{f}}}$
$\displaystyle=[\Delta C+A_{CP}C]$ (128)
$\displaystyle\frac{R_{\bar{f}}A_{CP}^{\bar{f}}+R_{f}A_{CP}^{f}}{R_{f}+R_{\bar{f}}}$
$\displaystyle=[C+A_{CP}\Delta C]$ (129)
For these decays, the decay amplitudes can be written in terms of tree
amplitude $e^{i\phi_{T}}T_{f}$ and the penguin amplitude $e^{i\phi_{P}}P_{f}$:
$\displaystyle A_{f}$
$\displaystyle=e^{i\phi_{T}}e^{i\delta_{f}^{T}}\bigl{|}T_{f}\bigr{|}[1+r_{f}e^{i(\phi_{P}-\phi_{T})}e^{i\delta_{f}}]$
$\displaystyle A_{\bar{f}}$
$\displaystyle=e^{i\phi_{T}}e^{i\delta_{\bar{f}}^{T}}\bigl{|}T_{\bar{f}}\bigr{|}[1+r_{\bar{f}}e^{i(\phi_{P}-\phi_{T})}e^{i\delta_{\bar{f}}}]$
(130)
where
$r_{f,\bar{f}}=\frac{\bigl{|}P_{f,\bar{f}}\bigr{|}}{\bigl{|}T_{f,\bar{f}}\bigr{|}},\quad\delta_{f,\bar{f}}=\delta^{P}_{f,\bar{f}}-\delta^{T}_{f,\bar{f}}$.
$\displaystyle\bar{A}_{\bar{f}}$
$\displaystyle=e^{-i\phi_{T}}e^{i\delta_{f}^{T}}\bigl{|}T_{f}\bigr{|}[1+r_{f}e^{-i(\phi_{P}-\phi_{T})}e^{i\delta_{f}}]$
$\displaystyle\bar{A}_{f}$
$\displaystyle=e^{-i\phi_{T}}e^{i\delta_{\bar{f}}^{T}}\bigl{|}T_{\bar{f}}\bigr{|}[1+r_{\bar{f}}e^{-i(\phi_{P}-\phi_{T})}e^{i\delta_{\bar{f}}}]$
(131)
$\text{For}B^{0}\rightarrow\rho^{-}\pi^{+}:A_{f};\qquad
B^{0}\rightarrow\rho^{+}\pi^{-}:A_{\bar{f}};\quad\phi_{T}=\gamma,\phi_{P}=-\beta$
(132)
$\text{For}B^{0}\rightarrow D^{\ast-}D^{+}:A^{D}_{f};\qquad B^{0}\rightarrow
D^{\ast+}D^{-}:A^{D}_{\bar{f}};\quad\phi_{T}=0,\phi_{P}=-\beta$ (133)
Hence for $B^{0}\rightarrow\rho^{-}\pi^{+},B^{0}\rightarrow\rho^{+}\pi^{-}$,
we have
$\displaystyle A_{f}$
$\displaystyle=\bigl{|}T_{f}\bigr{|}e^{-i\gamma}e^{i\delta^{T}_{f}}[1-r_{f}e^{i(\alpha+\delta_{f})}]$
$\displaystyle A_{\bar{f}}$
$\displaystyle=\bigl{|}T_{\bar{f}}\bigr{|}e^{-i\gamma}e^{i\delta^{T}_{\bar{f}}}[1-r_{\bar{f}}e^{i(\alpha+\delta_{\bar{f}})}]$
(134) $\displaystyle\text{where}\qquad r_{f,\bar{f}}$
$\displaystyle=\frac{|V_{tb}||V_{td}|}{|V_{ub}||V_{ud}|}\frac{\bigl{|}P_{f,\bar{f}}\bigr{|}}{\bigl{|}T_{f,\bar{f}}\bigr{|}}=\frac{R_{t}}{R_{b}}\frac{\bigl{|}P_{f,\bar{f}}\bigr{|}}{\bigl{|}T_{f,\bar{f}}\bigr{|}}$
(135)
and for $\text{B}^{0}\rightarrow D^{*-}D^{+}$, $\text{B}^{0}\rightarrow
D^{*+}D^{-}$, we have
$\displaystyle A_{f}^{D}$
$\displaystyle=\bigl{|}T_{f}^{D}\bigr{|}e^{i\delta_{f}^{TD}}[1-r_{f}^{D}e^{i(-\beta+\delta_{f}^{D})}]$
$\displaystyle A_{\bar{f}}^{D}$
$\displaystyle=\bigl{|}T_{\bar{f}}^{D}\bigr{|}e^{i\delta_{\bar{f}}^{TD}}[1-r_{\bar{f}}^{D}e^{i(-\beta+\delta_{\bar{f}}^{D})}]$
(136) $\displaystyle\text{where}\qquad r_{f,\bar{f}}$
$\displaystyle=R_{t}\frac{\bigl{|}P_{f,\bar{f}}^{D}\bigr{|}}{\bigl{|}T_{f,\bar{f}}^{D}\bigr{|}}$
## 5 Final State Strong Phases
As we have seen the CP asymmetries in the hadronic decays of B and K mesons
involve strong final state phases. Thus strong interactions in these decays
play a crucial role. The short distance strong interactions effects at the
quark level are taken care of by perturbative QCD in terms of Wilson
coefficients. The CKM matrix which connects the weak eigenstates will mass
eigenstates is another aspect of strong interactions at quark level. In the
case of semi leptonic decays, the long distance strong interaction effects
manifest themselves in the form factors of final states after hadronization.
Likewise the strong interaction final state phases are long distance effects.
These phase shifts essentially arise in terms of S-matrix which changes an
’in’ state into an ’out’ state viz.
$|f\rangle_{out}=S|f\rangle_{in}=e^{2i\delta_{f}}|f\rangle_{in}$ (137)
In fact, the CPT invariance of weak interaction Lagrangian gives for the weak
decay $B(\bar{B})\rightarrow f(\bar{f})$
$\bar{A}_{\bar{f}}\equiv_{out}\langle\bar{f}|\mathcal{L}_{w}|\bar{B}\rangle=\eta_{f}e^{2i\delta_{f}}A_{f}{\ast}$
(138)
It is difficult to reliably estimate the final state strong phase shifts. It
involves the hadronic dynamics. However, using isospin, C-invariance of
S-matrix and unitarity of S-matrix, we can relate these phases. In this
regard, the decays $B^{0}\rightarrow f,\bar{f}$ described by two independent
single amplitudes $A_{f}$ and $A_{\bar{f}}^{\prime}$ discussed in section 4
case (ii) and the decays described by the weak amplitudes $A_{f}\neq
A_{\bar{f}}$, described in section case (iii) are of interest
The invariance of S-matrix viz. $S_{\bar{f}}=S_{f}$ would imply
$\delta_{f}=\delta_{\bar{f}}^{\prime},\qquad\delta_{1f}=\delta_{1\bar{f}},\qquad\delta_{2f}=\delta_{2\bar{f}}$
In the above decays, b is converted into $b\rightarrow c(u)+\bar{u}+d$. In
particular, for the tree graph, the configuration is such that $\bar{u}$ and d
essentially go together into color singlet states will the third quark c(u)
recoiling; there is a significant probability that system will hadronize as a
two body final state. Thus at least for the tree amplitude $\delta_{f}^{T}$
should be equal to $\delta_{\bar{f}}^{T}$. To proceed further, we use the
unitarity of S-matrix to relate the final state strong phases. The time
reversal invariance gives
$F_{f}=_{out}\langle f|\mathcal{L}_{W}|B\rangle=_{in}\langle
f|\mathcal{L}_{W}|B\rangle^{*}$ (139)
where $\mathcal{L}_{W}$ is the weak interaction Lagrangian without the CKM
factor such as $V_{ud}^{*}V_{ub}$. From Eq. $\eqref{4.68}$, we have
$\displaystyle F_{f}^{*}=$ ${}_{out}\langle
f|S^{\dagger}\mathcal{L}_{W}|B\rangle$ $\displaystyle=$
$\displaystyle\sum_{n}S_{nf}^{*}F_{n}$ (140)
It is understood that the unitarity equation which follows from time reversal
invaraince holds for each amplitude with the same weak phase. Above equation
can be written in two equivalent forms:
1. 1.
Exclusive version of Unitarity
Writing
$S_{nf}=\delta_{nf}+iM_{nf}$ (141)
we get from Eq. $\eqref{4.69}$,
$ImF_{f}=\sum_{n}M_{nf}^{*}F_{n}$ (142)
where $M_{nf}$ is the scattering amplitude for $f\rightarrow n$. In this
version, the sum is over all allowed exclusive channels. This version is more
suitable in a situation where a single exclusive channel is dominant one. To
get the final result, one uses the dispersion relation.
2. 2.
Inclusive version of Unitarity
This version is more suitable for our analysis. For this case, we write Eq.
$\eqref{4.69}$ in the form
$F_{f}^{*}-S_{ff}^{*}F_{f}=\sum_{n\neq f}S_{nf}^{*}F_{n}$ (143)
Parametrizing S-matrix as $S_{ff}\equiv S=\eta e^{2i\Delta}$, we get after
taking the absolute square of both sides of Eq. $\eqref{4.72}$
$|F_{f}|^{2}[(q+\eta^{2})-2\eta\cos
2(\delta_{f}-\Delta)]=\sum_{n,n^{\prime}\neq
f}F_{n}S_{nf}^{\ast}F^{\ast}_{n^{\prime}}S_{n^{\prime}f}$ (144)
The above equation is an exact equation. In the random phase approximation, we
can put
$\displaystyle\sum_{n^{\prime},n\neq
f}F_{n}S_{nf}^{\ast}F_{n^{\prime}}S_{n^{\prime}f}=$ $\displaystyle\sum_{n\neq
f}|F_{n}|^{2}|S_{nf}|^{2}$ $\displaystyle=$
$\displaystyle\bar{|F_{n}|^{2}}(1-\eta^{2})$ (145)
We note that in a single channel description:
$(Flux)_{in}-(Flux)_{out}=1-|\eta
e^{2i\Delta}|^{2}=1-\eta^{2}=\text{Absorption}$
The absorption takes care of all the inelastic channels.
Similarly for the amplitude $F_{\bar{f}}$, we have
$F_{\bar{f}}^{\ast}-S^{\ast}_{\bar{f}\bar{f}}F_{\bar{f}}=\sum_{\bar{n}\neq\bar{f}}S^{\ast}_{\bar{n}\bar{f}}F_{\bar{n}}$
(146)
The C-invariance of S-matrix gives:
$\displaystyle S_{fn}=$ $\displaystyle\langle f|S|n\rangle=\langle
f|C^{-1}CSC^{-1}C|n\rangle$ $\displaystyle=$
$\displaystyle\langle\bar{f}|S|\bar{n}\rangle=S_{\bar{f}\bar{n}}$ (147)
Thus in particular C-invariance of S-matrix gives
$S_{\bar{f}\bar{f}}=S_{ff}=\eta e^{2i\Delta}$ (148)
Hence from Eq. $\eqref{4.73}$, using Eqs. ($\ref{4.74}-\ref{12}$), we get
$\frac{1}{1-\eta^{2}}[(1+\eta^{2})-2\eta\cos
2(\delta_{f,\bar{f}}-\Delta)]=\rho^{2},\bar{\rho}^{2}$ (149)
where
$\rho^{2}=\frac{\overline{\bigl{|}F_{n}\bigr{|}}^{2}}{\bigl{|}F_{f}\bigr{|}^{2}},\qquad\bar{\rho}^{2}=\frac{\overline{\bigl{|}F_{\bar{n}}\bigr{|}}^{2}}{\bigl{|}F_{\bar{f}}\bigr{|}^{2}}$
(150)
It is convenient to write Eq. $\eqref{4.78}$ in the form
$\displaystyle\sin^{2}(\delta_{f,\bar{f}}-\Delta)$
$\displaystyle=\frac{1-\eta^{2}}{4\eta}\left[\rho^{2},\bar{\rho}^{2}-\frac{1-\eta}{1+\eta}\right]$
(151) $\displaystyle 0$
$\displaystyle\leq(\delta_{f,\bar{f}}-\Delta)\leq\theta$ (152)
$\displaystyle-\theta$ $\displaystyle\leq(\delta_{f,\bar{f}}-\Delta)\leq 0$
(153)
where $\theta=\sin^{-1}\sqrt{\frac{1-\eta}{2}}$.
The strong interaction parameters $\Delta\ $and$\ \eta\ $can be determond by
strong interaction dynamics. Using $SU\left(2\right)$, C-invarience of strong
interactions and Regge pole phenomonology, the scattering aplitude
$M\left(s,t\right)$ for two particle final state can be calculated.(For
details see ref. [12]). The s-wave scattering amplitude $f$ for the decay
modes $\pi^{+}D^{-}\left(\pi^{-}D^{+}\right),\ K^{+}\pi^{-},\ \pi^{+}\pi^{-}$
which are s-wave decay modes of $B^{0}$ is given by
$f\left(s\right)=\frac{1}{16\pi s}\int_{-s}^{0}M\left(s^{\prime}t\right)dt$
where
$t\approx-\frac{1}{2}s\left(1-\cos\theta\right)$
Using the relation $S=\eta e^{2i\Delta}=1+2if$, the phase shift $\Delta,$ the
parameter $\eta$ and the phase angle $\theta$ can be determind. One gets
$\left(s=m_{B}^{2}\right)$
$\displaystyle\pi^{+}D^{-}\left(\pi^{-}D^{+}\right)$ $\displaystyle:$
$\displaystyle\ \Delta\approx-7^{o},\ \eta\approx 0.62,\ \rho_{\min}\approx
0.23,\ \theta\approx 26^{o}$ $\displaystyle K^{+}\pi^{-}\ or\ K^{0}\pi^{+}$
$\displaystyle:$ $\displaystyle\ \Delta\approx-9^{o},\ \eta\approx 0.52,\
\rho_{\min}\approx 0.31,\ \theta\approx 29^{o}$ $\displaystyle\pi^{+}\pi^{-}$
$\displaystyle:$ $\displaystyle\ \Delta\approx-21^{o},\ \eta\approx 0.48,\
\rho_{\min}\approx 0.35,\ \theta\approx 31^{o}$
Hence we get the following bounds
$\displaystyle\pi^{+}D^{-}\left(\pi^{-}D^{+}\right)$ $\displaystyle:$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ 0\leq\delta_{f,\ \bar{f}}-\Delta\leq
26^{o}$ $\displaystyle K^{+}\pi^{-}\ or\ K^{0}\pi^{+}$ $\displaystyle:$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ 0\leq\delta_{f}-\Delta\leq 29^{o}$
$\displaystyle\pi^{+}\pi^{-}$ $\displaystyle:$ $\displaystyle\ \ \ \ \ \ \ \ \
\ \ \ 0\leq\delta_{f}-\Delta\leq 31^{o}$
For the tree amplitude, factorization implies $\delta_{f}^{T}=0.$ We can
therefore take the point of view, the effective final state phase shift is
given by $\delta_{f}-\Delta.\ $We take the lower bounds for the tree amplitude
so that final state effective phase shift $\delta_{f}^{T}=0.$ For the penguin
we assume that the effective value of the final state phase shift
$\delta_{f}^{P}$ is near the upper bound. Thus for
$\pi^{+}D^{-}\left(\pi^{-}D^{+}\right),$
$\delta_{f}^{T}=\delta_{\bar{f}}^{\prime T}\approx 0$; for $K^{+}\pi^{-},$ the
phase shift $\delta_{+-}=\delta_{+-}^{P}\sim 29^{o}$ where as for
$\pi^{+}\pi^{-},$ the phase shift $\delta_{+-}=\delta_{+-}^{P}\sim 31^{o}.$
These phase shifts are relavent for the Direct CP-asymmetries for
$B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}\rightarrow\pi^{+}\pi^{-}$ decays.
The decay $B^{0}\rightarrow K^{+}\pi^{-}$ is described by two amplitudes (For
details see ref.[13])
$\displaystyle A\left(B^{0}\rightarrow K^{+}\pi^{-}\right)$ $\displaystyle=$
$\displaystyle-\left[P+e^{i\gamma_{T}}\right]=\left|P\right|\left[1-re^{i\left(\gamma+\delta_{+-}\right)}\right]$
$\displaystyle P$ $\displaystyle=$
$\displaystyle-\left|P\right|e^{-i\delta_{p}},\ \ T=\left|T\right|$
$\displaystyle\delta_{+-}$ $\displaystyle=$ $\displaystyle\delta_{P},\text{ \
}r=\frac{\left|T\right|}{\left|P\right|}$ $\displaystyle
A_{CP}\left(B^{0}\rightarrow\pi^{-}K^{+}\right)$ $\displaystyle=$
$\displaystyle\frac{-2r\sin\gamma\sin\delta_{+-}}{R}$ $\displaystyle R$
$\displaystyle=$ $\displaystyle 1-2r\cos\gamma\cos\delta_{+-}+r^{2}$
Neglecting the terms of order $r^{2},$
$\tan\gamma\tan\delta_{+-}=\frac{-A_{CP}\left(B^{0}\rightarrow\pi^{-}K^{+}\right)}{1-R}$
From the experimental values of $A_{CP}=\left(-0.097\pm 0.012\right)$ and
$R=0.899\pm 0.048,$ with $\delta_{+-}\approx 29^{o},$ we get
$\gamma=\left(60\pm 3\right)^{o}.$However for $\delta_{+-}\approx 20^{o},$
(which corresponds to $\delta_{f}-\Delta\approx 20^{o};$ the value one gets
for $\rho^{2}=0.65$), we get $\gamma=\left(69\pm 3\right)^{o}.$
The phase shift $\delta_{+-}\approx\left(20\sim 29\right)^{o}$ for the
$K^{+}\pi^{-}$ is compatible with the experimental value of the direct CP-
asymmetry for $B^{0}\rightarrow K^{+}\pi^{-}$ decay mode. For
$\pi^{+}\pi^{-},$ $\delta_{+-}\sim 31^{o}$ is compatible with the value
$\left(33\pm 7\begin{array}[]{c}+8\\\ -10\end{array}\right)^{o}$ obtained by
the authers of ref. [13]. In any case, our analysis shows that the upper limit
for final state stronge phase $\delta_{f}$ is around $30^{o}$. Finally, we
note that the actual value of the effective final state phase shift
$\left(\delta_{f}-\Delta\right)$ depends on one free parameter $\rho;$ the
factorization implies $\delta_{f}^{T}=0$ $i.e.\
\left(\delta_{f}-\Delta\right)=0$ for the tree amplitude; for the penguin
amplitude, $\delta_{f}^{P}$ depends on $\rho;$ in any case it can not be
greator than the upper bound.
## 6 CP Asymmetries and Strong Phases
Case II:
Now, we discuss the experimental tests to verify the equality (implied by
C-invariance of S-marix) of phase shifts $\delta_{f}$ and $\delta_{\bar{f}}$
for the decays $B\rightarrow\pi D,\pi D^{*},\rho D$ and $B_{s}\rightarrow
KD_{s},KD_{s}^{*},K^{*}D_{s}$.
From Eqs.$\eqref{4.35b}$, we note that CP-asymetries:
$\displaystyle-\frac{S_{-}+S_{+}}{2}$ $\displaystyle=$
$\displaystyle\frac{2r_{D}}{1+r_{D}^{2}}\sin(2\beta+\gamma)\cos(\delta_{f}-\delta_{\overline{f}}^{\prime})$
$\displaystyle-\frac{S_{+}-S_{-}}{2}$ $\displaystyle=$
$\displaystyle\frac{2r_{D}}{1+r_{D}^{2}}\cos(2\beta+\gamma)\sin(\delta_{f}-\delta_{\overline{f}}^{\prime})$
involve dthe weak phase $2\beta+\gamma$ and strong phase
$\delta_{f}-\delta_{\overline{f}}^{\prime}.$ These asymmetries are of interst
because for
$\delta_{f}=\delta_{\overline{f}}^{\prime},\frac{S_{+}-S_{-}}{2}=0$
and
$-\frac{S_{-}+S_{+}}{2}=\frac{2r_{D}}{1+r_{D}^{2}}\sin(2\beta+\gamma)$
Hence we can verify the equality of phases $\delta_{f}$ and
$\delta_{\overline{f}}^{\prime}$ and determine the weak phase $2\beta+\gamma.$
For $B_{s}^{0},$ replace $r_{D}\rightarrow r_{s}$,
$\delta_{f}\rightarrow\delta_{f_{s}}$,
$\delta_{\overline{f}}^{\prime}=\delta_{\overline{f}_{s}}^{\prime}$ and
$\beta$ by $\beta_{s}.$ In standared model $\beta_{s}=0.$
The experimental results for the B decays are as follows discussed in section
4
$\begin{array}[]{cccc}&D^{-}\pi^{+}&D^{*-}\pi^{+}&D^{-}\rho^{+}\\\
\frac{S_{-}+S_{+}}{2}&-0.046\pm 0.023&-0.037\pm 0.012&-0.024\pm 0.031\pm
0.009\\\ \frac{S_{-}-S_{+}}{2}&-0.022\pm 0.021&-0.006\pm 0.016&-0.098\pm
0.055\pm 0.018\end{array}$
To determine the parameter $r_{D}$ or $r_{s}$, we assume factorization for the
tree amplitude. Factorization gives for the decays $\bar{B}^{0}\rightarrow
D^{+}\pi^{-},D^{*+}\pi^{-},D^{+}\rho^{-},D^{+}a_{1}^{-}$:
$\displaystyle|\bar{F}_{\bar{f}}|=|\bar{T}_{\bar{f}}|$
$\displaystyle=G[f_{\pi}(m_{B}^{2}-m_{D}^{2})f_{0}^{B-D}(m_{\pi}^{2}),2f_{\pi}m_{B}|\vec{p}|A_{0}^{B-D^{*}}(m_{\pi}^{2}),$
$\displaystyle
2f_{\rho}m_{B}|\vec{p}|f_{+}^{B-D}(m_{\rho}^{2}),2f_{a_{1}}m_{B}|\vec{p}|f_{+}^{B-D}(a_{1}^{2})]$
(154)
$\displaystyle|\bar{F}_{\bar{f}}^{{}^{\prime}}|=|\bar{T}_{\bar{f}}^{{}^{\prime}}|$
$\displaystyle=G^{{}^{\prime}}[f_{D}(m_{B}^{2}-m_{\pi}^{2})f_{0}^{B-\pi}(m_{D}^{2}),2f_{D^{*}}m_{B}|\vec{p}|f_{+}^{B-\pi}(m_{D^{*}}^{2}),$
$\displaystyle
2f_{D}m_{B}|\vec{p}|A_{0}^{B-\rho}(m_{D}^{2}),2f_{D}m_{B}|\vec{p}|A_{0}^{B-a_{1}}(m_{B}^{2})]$
(155) $\displaystyle G$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}|V_{ud}||V_{cb}|a_{1},\quad
G^{{}^{\prime}}=\frac{G_{F}}{\sqrt{2}}|V_{cd}||V_{ub}|$ (156)
$\displaystyle\Gamma(\bar{B}^{0}\rightarrow D^{+}\pi^{-})$
$\displaystyle=|V_{cb}|^{2}|f_{0}^{B-D}(m_{\pi}^{2})|^{2}(2.281\times
10^{-9})MeV$ $\displaystyle\Gamma(\bar{B}^{0}\rightarrow D^{*+}\pi^{-})$
$\displaystyle=|V_{cb}|^{2}|A_{0}^{B-D^{*}}(m_{\pi}^{2})|^{2}(2.129\times
10^{-9})MeV$ $\displaystyle\Gamma(\bar{B}^{0}\rightarrow D^{+}\rho^{-})$
$\displaystyle=|V_{cb}|^{2}|f_{+}^{B-D}(m_{\rho}^{2})|^{2}(5.276\times
10^{-9})MeV$ $\displaystyle\Gamma(\bar{B}^{0}\rightarrow D^{+}a_{1}^{-})$
$\displaystyle=|V_{cb}|^{2}|f_{+}^{B-D}(m_{a_{1}}^{2})|^{2}(5.414\times
10^{-9})MeV$ (157)
Decay | Decay Width $(10^{-9}$ MeV $\times|V_{cb}|^{2}$) | Form Factor | Form Factors $h(w^{(\ast)})$
---|---|---|---
$\bar{B}^{0}\rightarrow D^{+}\pi^{-}$ | $(2.281)|f_{0}^{B-D}(m_{\pi}^{2})|^{2}$ | $0.58\pm 0.05$ | $0.51\pm 0.03$
$\bar{B}^{0}\rightarrow D^{\ast+}\pi^{-}$ | $(2.129)|A_{0}^{B-D\ast}(m_{\pi}^{2})|^{2}$ | $0.61\pm 0.04$ | $0.54\pm 0.03$
$\bar{B}^{0}\rightarrow D^{+}\rho^{+}$ | $(5.276)|f_{+}^{B-D}(m_{\rho}^{2})|^{2}$ | $0.65\pm 0.11$ | $0.57\pm 0.10$
$\bar{B}^{0}\rightarrow D^{+}a_{1}$ | $(5.414)|f_{+}^{B-D}(m_{a_{1}}^{2})|^{2}$ | $0.57\pm 0.31$ | $0.50\pm 0.27$
Table 1: Form Factors
The decay widths for the above channels are given in the table 1
where we have used
$a_{1}^{2}|V_{ud}|^{2}\approx 1,\quad f_{\pi}=131MeV,\quad
f_{\rho}=209MeV,\quad f_{a_{1}}=229MeV$
Using the experimental branching ratios and
$|V_{cb}|=(38.3\pm 1.3)\times 10^{-3}$ (158)
we obtain the corresponding form factors given in Table 1.
$\displaystyle|f_{0}^{B-D}(m_{\pi}^{2})|$ $\displaystyle=0.58\pm 0.05$
$\displaystyle|A_{0}^{B-D^{\ast}}(m_{\pi}^{2})|$ $\displaystyle=0.61\pm 0.04$
$\displaystyle|f_{+}^{B-D}(m_{\rho}^{2})|$ $\displaystyle=0.65\pm 0.11$
$\displaystyle|f_{+}^{B-D}(m_{a_{1}}^{2})|$ $\displaystyle=0.57\pm 0.31$ (159)
In terms of Isgur Wise variables:
$\omega=v\cdot v^{{}^{\prime}},\quad v^{2}=v^{{}^{\prime}2}=1,\quad
t=q^{2}=m_{B}^{2}+m_{D^{*}}^{2}-2m_{B}m_{D^{*}}\omega$ (160)
the form factors can be put in the following form
$\displaystyle f_{+}^{B-D}(t)$
$\displaystyle=\frac{m_{B}+m_{D}}{2\sqrt{m_{B}m_{D}}}h_{+}(\omega),\quad
f_{0}^{B-D}(t)=\frac{\sqrt{m_{B}m_{D}}}{m_{B}+m_{D}}(1+\omega)h_{0}(\omega)$
$\displaystyle A_{2}^{B-D^{\ast}}(t)$
$\displaystyle=\frac{m_{B}+m_{D^{\ast}}}{2\sqrt{m_{B}m_{D^{\ast}}}}(1+\omega)h_{A_{2}}(\omega),\quad
A_{0}^{B-D^{\ast}}(t)=\frac{m_{B}+m_{D^{\ast}}}{2\sqrt{m_{B}m_{D^{\ast}}}}h_{A_{0}}(\omega)$
$\displaystyle A_{1}^{B-D^{\ast}}(t)$
$\displaystyle=\frac{\sqrt{m_{B}m_{D^{\ast}}}}{m_{B}+m_{D^{\ast}}}(1+\omega)h_{A_{1}}(\omega)$
(161)
Heavy Quark Effective Theory (HQET) gives:
$h_{+}(\omega)=h_{0}(\omega)=h_{A_{0}}(\omega)=h_{A_{1}}(\omega)=h_{A_{2}}(\omega)=\zeta(\omega)$
where $\zeta(\omega)$ is Isgur-Wise form factor, with normalization
$\zeta(1)=1$. For
$\displaystyle t$ $\displaystyle=m_{\pi}^{2},m_{\rho}^{2},m_{a_{1}}^{2}$
$\displaystyle\omega^{\ast}$ $\displaystyle=1.589(1.504),1.559,1.508$
we get the form factors h’s given in Table 1.
In reference , the value quoted for $h_{A_{1}}(\omega_{max}^{*})$ is
$|h_{A_{1}}(\omega_{max}^{*})|=0.52\pm 0.03$ (162)
Since $\omega_{max}^{\ast}=1.504$, the value for $|h_{A_{0}}(max)|$ obtained
in Table 1 is in remarkable agreement with the value given in Eq.
$\eqref{c10}$that assumption for $B^{0}\rightarrow\pi D^{(\ast)}$ decays is
experimentally on solid footing and is in agreement with HQET.
From Eqs. $\eqref{c1}$ and $\eqref{c2}$, we obtain
$\displaystyle r_{D}$
$\displaystyle=\lambda^{2}R_{b}\frac{|\bar{T}_{f}^{{}^{\prime}}|}{|\bar{T}_{\bar{f}}|}$
$\displaystyle=\lambda^{2}R_{b}\left[\frac{f_{D}(m_{B}^{2}-m_{\pi}^{2})f_{0}^{B-\pi}(m_{D}^{2})}{f_{\pi}(m_{B}^{2}-m_{D}^{2})f_{0}^{B-D}(m_{\pi}^{2})},\quad\frac{f_{D^{\ast}}f_{+}^{B-\pi}(m_{D^{\ast}}^{2})}{f_{\pi}A_{0}^{B-D}(m_{\pi}^{2})},\quad\frac{f_{D}A_{0}^{B-\rho}(m_{D}^{2})}{f_{\rho}f_{+}^{B-D}(m_{\rho^{2}})}\right]$
(163)
where
$\frac{|V_{ub}||V_{cd}|}{|V_{cb}||V_{ud}|}=\lambda^{2}R_{b}\approx(0.227)^{2}(0.40)\approx
0.021$ (164)
To determine $r_{D}$, we need information for the form factors
$f_{0}^{B-\pi}(m_{D}^{2}),f_{+}^{B-\pi}(m_{D}^{2}),A_{0}^{B-\rho}(m_{D}^{2})$.
For these form factors, we use the following values:
$\displaystyle A_{0}^{B-\rho}(0)$ $\displaystyle=0.30\pm
0.03,A_{0}^{B-\rho}(m_{D}^{2})=0.38\pm 0.04$ $\displaystyle f_{+}^{B-\pi}(0)$
$\displaystyle=f_{0}^{B-\pi}(0)=0.26\pm 0.04,\quad
f_{+}^{B-\pi}(m_{D^{\ast}}^{2})=0.32\pm 0.05,\quad
f_{0}^{B-D}(m_{D}^{2})=0.28\pm 0.04$
Along with the remaining form factors given in Table, we obtain
$r_{D}=[0.018\pm 0.002,\quad 0.017\pm 0.003,\quad 0.012\pm 0.002]$ (165)
The above value for $r_{D}^{\ast}$ gives
$-\left(\frac{S_{+}+S_{-}}{2}\right)_{D^{\ast}\pi}=2(0.017\pm
0.003)\sin(2\beta+\gamma)$ (166)
The experimental value of the CP asymmetry for $B^{0}\rightarrow D^{*}\pi$
decay has the least error. Hence we obtain the following bounds
$\displaystyle\sin(2\beta+\gamma)$ $\displaystyle>0.69$ (167) $\displaystyle
44^{\circ}$ $\displaystyle\leq(2\beta+\gamma)\leq 90^{\circ}$ (168)
$\displaystyle\text{or}\quad 90^{\circ}$ $\displaystyle\leq(2\beta+\gamma)\leq
136^{\circ}$ (169)
Selecting the second solution, and using $\beta\approx 43^{\circ}$, we get
$\gamma=(70\pm 23)^{\circ}$ (170)
To end this section, we discuss the decays $\bar{B}_{s}^{0}\rightarrow
D_{s}^{+}K^{-},D_{s}^{*+}K^{-}$ for which no experimental data is available.
However, using facorization, we get
$\displaystyle\Gamma(\bar{B}_{s}^{0}\rightarrow D_{s}^{+}K^{-})$
$\displaystyle=(1.75\times
10^{-10})|V_{cb}f_{0}^{B_{s}-D_{s}}(m_{K}^{2})|^{2}MeV$ (171)
$\displaystyle\Gamma(\bar{B}_{s}^{0}\rightarrow D_{s}^{*+}K^{-})$
$\displaystyle=(1.57\times
10^{-10})|V_{cb}A_{0}^{B_{s}-D_{s}^{*}}(m_{K}^{2})|^{2}MeV$ (172)
SU(3) gives
$\displaystyle|V_{cb}f_{0}^{B_{s}-D_{s}}(m_{K}^{2})|^{2}$
$\displaystyle\approx|V_{cb}||f_{0}^{B-D}(m_{\pi}^{2})|^{2}=(0.50\pm
0.04)\times 10^{-3}$
$\displaystyle|V_{cb}A_{0}^{B_{s}-D_{s}^{\ast}}(m_{K}^{2})|^{2}$
$\displaystyle\approx|V_{cb}||A_{0}^{B-D}(m_{\pi}^{2})|^{2}=(0.56\pm
0.04)\times 10^{-3}$ (173)
From the above equations, we get the following branching ratios
$\frac{\Gamma(\bar{B_{s}}^{0}\rightarrow
D_{s}^{(\ast)+}K^{-})}{\Gamma_{\bar{B}_{s}^{0}}}=(1.94\pm 0.07)\times
10^{-4}[(1.96\pm 0.07)\times 10^{-4}]$ (174)
For $\bar{B}_{s}^{0}\rightarrow D_{s}^{\ast+}K^{-}$
$r_{s}=R_{b}\left[\frac{f_{D_{s}^{\ast}}f_{+}^{B_{s}-K}(m_{D_{s}^{\ast}}^{2})}{f_{K}A_{0}^{B_{s}-D_{s}^{\ast}}(m_{K}^{2})}\right]$
(175)
Hence we get
$\displaystyle-(\frac{S_{+}+S_{-}}{2})_{D_{s}^{\ast}K}$
$\displaystyle=(0.41\pm 0.08)\sin(2\beta_{s}+\gamma)$ $\displaystyle=(0.41\pm
0.08)\sin\gamma$ (176)
where we have used
$\displaystyle R_{b}$
$\displaystyle=0.40,\quad\frac{f_{D_{s}}}{f_{K}}=\frac{f_{D_{s}^{*}}}{f_{K}}=1.75\pm
0.06,\quad f_{+}^{B_{s}-K}(m_{D_{s}^{*}}^{2})=0.34\pm 0.06$ $\displaystyle
A_{0}^{B_{s}-D_{s}^{*}}(m_{K}^{2})$
$\displaystyle=A_{0}^{B_{s}-D_{s}^{*}}(0)=\frac{m_{B_{s}}+m_{D_{s}^{*}}}{2\sqrt{m_{B_{s}m_{D_{s}^{*}}}}}\left[h_{0}(\omega_{s}^{*}=1.453)=0.52\pm.03\right]$
$\displaystyle=0.58\pm 0.03$ (177)
Case III
We now confine ourselves to
$B^{0}(\bar{B}^{0})\rightarrow\rho^{-}\pi^{+},\rho^{+}\pi^{-}(\rho^{+}\pi^{-},\rho^{-},\pi^{+})$
decays only [13,14]. The experimental results for these decays are [6] as
$\displaystyle\Gamma$ $\displaystyle=R_{f}+R_{\bar{f}}=(22.8\pm 2.5)\times
10^{-6}$ (178) $\displaystyle A_{CP}^{f}$ $\displaystyle=-0.16\pm 0.23,\quad
A_{CP}^{\bar{f}}=0.08\pm 0.12$ (179) $\displaystyle C$ $\displaystyle=0.01\pm
0.14,\quad\Delta C=0.37\pm 0.08$ (180) $\displaystyle S$
$\displaystyle=0.01\pm 0.09,\quad\Delta S=-0.05\pm 0.10$ (181)
With the above values, it is hard to draw any reliable conclusion. Neglecting
the term $A_{CP}C$ in Eqs. $\eqref{ccc9}$ and $\eqref{ccc10}$, we get
$\displaystyle R_{\bar{f},f}$ $\displaystyle=\frac{1}{2}\Gamma(1\pm\Delta C)$
(182) $\displaystyle R_{\bar{f}}-R_{f}$ $\displaystyle=\Delta C$ (183)
Using the above value for $\Delta C$, we obtain
$\displaystyle R_{\bar{f}}$ $\displaystyle=(15.6\pm 1.7)\times 10^{-6}$
$\displaystyle R_{f}$ $\displaystyle=(7.2\pm 0.8)\times 10^{-6}$ (184)
We analyze these decays by assuming factorization for the tree
graphs$\left[\text{19}\right]$. This assumption gives
$\displaystyle T_{\bar{f}}$ $\displaystyle=\bar{T}_{f}\sim
2m_{B}f_{\rho}|\vec{p}|f_{+}(m_{\rho}^{2})$ (185) $\displaystyle T_{f}$
$\displaystyle=\bar{T}_{\bar{f}}\sim 2m_{B}f_{\pi}|\vec{p}|A_{0}(m_{\pi}^{2})$
(186)
Using $f_{+}(m_{\rho}^{2})\approx 0.26\pm 0.04$ and $A_{0}(m_{\pi}^{2})\approx
A_{0}(0)=0.29\pm 0.03$ and $|V_{ub}|=(3.5\pm 0.6)\times 10^{-3}$, we get the
following values for the tree amplitude contribution to the branching ratios
$\displaystyle\Gamma_{\bar{f}}^{\text{tree}}$ $\displaystyle=(15.6\pm
1.1)\times 10^{-6}\equiv|T_{\bar{f}}|^{2}$ (187)
$\displaystyle\Gamma_{f}^{\text{tree}}$ $\displaystyle=(7.6\pm 1.4)\times
10^{-6}\equiv|T_{f}|^{2}$ (188) $\displaystyle t$
$\displaystyle=\frac{T_{f}}{T_{\bar{f}}}=\frac{f_{\pi}A_{0}(m_{\pi}^{2})}{f_{\rho}f_{+}(m_{\rho}^{2})}=0.70\pm
0.12$ (189)
Now
$\displaystyle B_{\bar{f}}$
$\displaystyle=\frac{R_{\bar{f}}}{|T_{\bar{f}}|^{2}}=1-2r_{\bar{f}}\cos\alpha\cos\delta_{\bar{f}}+r_{\bar{f}}^{2}$
(190) $\displaystyle B_{f}$
$\displaystyle=\frac{R_{f}}{|T_{f}|^{2}}=1-2r_{f}\cos\alpha\cos\delta_{f}+r_{f}^{2}$
(191)
Hence from Eqs. $\eqref{ccc25}$ and $\eqref{ccc29}$, we get
$\displaystyle B_{\bar{f}}$ $\displaystyle=1.00\pm 0.12$ $\displaystyle B_{f}$
$\displaystyle=0.95\pm 0.11$ (192)
In order to take into account the contribution of penguin diagram, we
introduce the angles $\alpha_{eff}^{f,\bar{f}}$ , defined as follows
$\displaystyle e^{i\beta}A_{f,\bar{f}}$
$\displaystyle=|A_{f,\bar{f}}|e^{-i\alpha_{eff}^{f,\bar{f}}}$ $\displaystyle
e^{-i\beta}\bar{A}_{f,\bar{f}}$
$\displaystyle=|\bar{A}_{f,\bar{f}}|e^{i\alpha_{eff}^{f,\bar{f}}}$ (193)
With this definition, we separate out tree and penguin contributions:
$\displaystyle e^{i\beta}A_{f,\bar{f}}-e^{-i\beta}\bar{A}_{f,\bar{f}}$
$\displaystyle=|A_{f,\bar{f}}|e^{-i\alpha^{f,\bar{f}}}-|\bar{A}_{f,\bar{f}}|e^{i\alpha^{f,\bar{f}}}$
$\displaystyle=2iT_{f,\bar{f}}\sin\alpha$ (194) $\displaystyle
e^{i(\alpha+\beta)}A_{f,\bar{f}}-e^{-i(\alpha+\beta)}\bar{A}_{f,\bar{f}}$
$\displaystyle=|A_{f,\bar{f}}|e^{-i(\alpha_{eff}^{f,\bar{f}}-\alpha)}$
$\displaystyle=(2iT_{f,\bar{f}}\sin\alpha)r_{f,\bar{f}}e^{i\delta_{f,\bar{f}}}$
$\displaystyle=2iP_{f,\bar{f}}\sin\alpha$ (195)
From Eq. $\eqref{ccc35}$, we get
$\displaystyle 2\frac{|T_{f,\bar{f}}|^{2}}{R_{f,\bar{f}}}\sin^{2}\alpha$
$\displaystyle\equiv\frac{2\sin^{2}\alpha}{B_{f,\bar{f}}}=1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}$ (196) $\displaystyle\sin 2\delta_{f,\bar{f}}^{T}$
$\displaystyle=-A_{CP}^{f,\bar{f}}\frac{\sin
2\alpha_{eff}^{f,\bar{f}}}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (197) $\displaystyle\cos 2\delta_{f,\bar{f}}^{T}$
$\displaystyle=\frac{\sqrt{1-A_{CP}^{f,\bar{f}2}}-\cos
2\alpha_{eff}^{f,\bar{f}}}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (198)
From Eqs. $\eqref{ccc35}$ and $\eqref{ccc36}$, we get
$\displaystyle r_{f,\bar{f}}^{2}$
$\displaystyle=\frac{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos(2\alpha_{eff}^{f,\bar{f}}-2\alpha)}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (200) $\displaystyle
r_{f,\bar{f}}\cos\delta_{f,\bar{f}}$
$\displaystyle=\frac{\cos\alpha-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos(2\alpha_{eff}^{f,\bar{f}}-\alpha)}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (201) $\displaystyle
r_{f,\bar{f}}\sin\delta_{f,\bar{f}}$
$\displaystyle=\frac{-A_{CP}^{f,\bar{f}}/\sin\alpha}{1-\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha_{eff}^{f,\bar{f}}}$ (202)
Now factorization implies [23]
$\delta_{f}^{T}=0=\delta_{\bar{f}}^{T}$ (203)
Thus in the limit $\delta_{f}^{T}\rightarrow 0$, we get for Eq.
$\eqref{ccc38b}$
$\displaystyle\cos 2\alpha_{eff}^{f,\bar{f}}$
$\displaystyle=-1,\qquad\alpha_{eff}^{f,\bar{f}}=90^{\circ}$ (204)
$\displaystyle r_{f,\bar{f}}\cos\delta_{f,\bar{f}}$ $\displaystyle=\cos\alpha$
(205) $\displaystyle r_{f,\bar{f}}\sin\delta_{f,\bar{f}}$
$\displaystyle=\frac{-A_{CP}^{f,\bar{f}}/\sin\alpha}{1+\sqrt{1-A_{CP}^{f,\bar{f}2}}}$
(206) $\displaystyle r_{f,\bar{f}}^{2}$
$\displaystyle=\frac{1+\sqrt{1-A_{CP}^{f,\bar{f}2}}\cos
2\alpha}{1+\sqrt{1-A_{CP}^{f,\bar{f}2}}}$ (207)
$\displaystyle\approx\cos^{2}\alpha+\frac{1}{4}A_{CP}^{f,\bar{f}2}\sin^{2}\alpha$
(208)
The solution of Eq. $\eqref{ccc44}$ is graphically shown in Fig. 7 for
$\alpha$ in the range $80^{\circ}\leq\alpha<103^{\circ}$ for
$r_{f,\bar{f}}=0.10,015,0.20,0.25,0.30$. From the figure, the final state
phases $\delta_{f,\bar{f}}$ for various values of $r_{f,\bar{f}}$ can be read
for each value of $\alpha$ in the above range. Few examples are given in Table
2
$\alpha$ | $r_{f}$ | $\delta_{f}$ | $A_{CP}^{f}\approx-2r_{f}\sin\delta_{f}\sin\alpha$
---|---|---|---
$80^{\circ}$ | 0.20 | $29^{\circ}$ | -0.19
| 0.25 | $46^{\circ}$ | -0.36
$82^{\circ}$ | 0.15 | $22^{\circ}$ | -0.11
| 0.20 | $46^{\circ}$ | -0.28
$85^{\circ}$ | 0.10 | $29^{\circ}$ | -0.10
| 0.15 | $54^{\circ}$ | -0.24
$86^{\circ}$ | 0.10 | $46^{\circ}$ | -0.14
| 0.15 | $62^{\circ}$ | -0.26
$88^{\circ}$ | 0.10 | $70^{\circ}$ | -0.19
Table 2:
For $\alpha>90^{\circ}$, change $\alpha\rightarrow\pi-\alpha$,
$\delta_{f}\rightarrow\pi-\delta_{f}$. For example, for $\alpha=103^{\circ}$
$\displaystyle r_{f}$ $\displaystyle=0.25,\quad\delta_{f}=154^{\circ},\quad
A_{CP}^{f}\approx-0.22$ $\displaystyle r_{f}$
$\displaystyle=0.30,\quad\delta_{f}=138^{\circ},\quad A_{CP}^{f}\approx-0.40$
These examples have been selected keeping in view that final state phases
$\delta_{f,\bar{f}}$ are not too large. For $A^{f,\bar{f}}_{CP}$, we have used
Eq. $\eqref{ccc45}$ neglecting the second order term. An attractive option is
$A_{CP}^{f}=A_{CP}^{\bar{f}}$ for each value of $\alpha$; although
$A_{CP}^{f}\neq A_{CP}^{\bar{f}}$ is also a possibility.
$A^{f}_{CP}=A_{CP}^{\bar{f}}$ implies
$r_{f}=r_{\bar{f}},\delta_{f}=\delta_{\bar{f}}$.
Neglecting terms of order $r_{f,\bar{f}}^{2}$, we have
$\displaystyle
A_{CP}\approx\frac{2\sin\alpha(r_{\bar{f}}\sin\delta_{\bar{f}}-t^{2}r_{f}\sin\delta_{f})}{1+t^{2}}=-\frac{A_{CP}^{\bar{f}}-t^{2}A_{CP}^{f}}{1+t^{2}}$
(209) $\displaystyle
C\approx-\frac{2t^{2}}{(1+t)^{2}}(A_{CP}^{\bar{f}}+A_{CP}^{f})$ (210)
$\displaystyle\Delta
C\approx\frac{1-t^{2}}{1+t^{2}}-\frac{4t^{2}\cos\alpha}{(1+t^{2})^{2}}(r_{\bar{f}}\cos\delta_{\bar{f}}-r_{f}\cos\delta_{f})$
(211)
Now the second term in Eq. $\eqref{cccc3}$ vanishes and using the value of $t$
given in Eq. $\eqref{ccc30}$, we get
$\Delta C\approx 0.34\pm 0.06$ (212)
Assuming $A_{CP}^{\bar{f}}=A_{CP}^{f}$, we obtain
$\displaystyle A_{CP}$
$\displaystyle=-\frac{1-t^{2}}{1+t^{2}}A_{CP}^{\bar{f}}$
$\displaystyle=(0.34\pm 0.06)(-A_{CP}^{\bar{f}})$ (213) $\displaystyle C$
$\displaystyle\approx-\frac{4t^{2}}{(1+t^{2})^{2}}A_{CP}^{\bar{f}}\approx-(0.88\pm
0.14)A_{CP}^{\bar{f}}$ (214)
Finally the CP asymmetries in the limit $\delta_{f,\bar{f}}^{T}\rightarrow 0$
$\displaystyle S_{\bar{f}}=S+\Delta S$
$\displaystyle=\frac{2\text{Im}[e^{2i\phi_{M}}A_{\bar{f}^{*}}\bar{A}_{\bar{f}}]}{\Gamma(1+A_{CP})}$
$\displaystyle=\sqrt{1-C_{\bar{f}}^{2}}\sin(2\alpha_{eff}^{\bar{f}}+\delta)$
$\displaystyle=-\sqrt{1-C_{\bar{f}}^{2}}\cos\delta$ (215) $\displaystyle
S_{f}=S-\Delta S$
$\displaystyle=\frac{2\text{Im}[e^{2i\phi_{M}}A_{f}^{*}\bar{A}_{f}]}{\Gamma(1-A_{CP})}$
$\displaystyle=\sqrt{1-C_{f}^{2}}\sin(2\alpha_{eff}^{\bar{f}}-\delta)$
$\displaystyle=\sqrt{1-C_{f}^{2}}\cos\delta$ (216)
The phase $\delta$ is defined as
$\bar{A}_{\bar{f}}=\frac{|\bar{A}_{\bar{f}|}}{|\bar{A}_{f}|}\bar{A}_{f}e^{i\delta}$
(217)
Hence we have
$\frac{S+\Delta S}{S-\Delta
S}=-\frac{\sqrt{1-C_{\bar{f}}^{2}}}{\sqrt{1-C_{f}^{2}}}$
## 7 Conclusion
In weak interaction, both P and C are violated but CP is conserved by the weak
interaction Lagrangian. Hence for $X^{0}-\bar{X}^{0}$ complex
$(X^{0}=K^{0},B^{0},B_{s}^{0})$; the mass matrix is not diagonal in
$|X^{0}\rangle$ and $|\bar{X}^{0}\rangle$ basis. However, assuming $CP$
conservation, the $CP$ eigenstates $|X_{1}^{0}\rangle$ and $|X_{2}^{0}\rangle$
can be mass eigenstates and hence mass matrix is diagonal in this basis. The
two sets of states are related to each other by superposition principle of
quantum mechanics. This gives rise to quantum mechanical interference so that
even if we start with a state $|X^{0}\rangle$, the time evolution of this
state can generate the state $|X^{0}\rangle$. This is a source of mixing
induced $CP$ violation. However, both in $K^{0}-\bar{K}^{0}$ and
$B^{0}-\bar{B}^{0}$ complex, the mass eigenstates $|K_{S}^{0}\rangle$,
$|K_{L}^{0}\rangle$ and $|B_{L}^{0}\rangle$, $|B_{H}^{0}\rangle$ are not $CP$
eigenstates. In the case of $K^{0}-\bar{K}^{0}$ complex, there is a small
admixture of wrong $CP$ state characterized by a small parameter $\epsilon$,
which gives rise to the $CP$ violating decay
$K_{L}^{0}\rightarrow\pi^{+}\pi^{-}$. This was the first $CP$ violating decay
observed experimentally. For $B^{0}-\bar{B}^{0}$ complex, the mismatch between
mass eigenstates and $CP$ eigenstates $|B_{1}^{0}\rangle$ and
$|B_{2}^{0}\rangle$ is given by the phase factor $e^{2i\phi_{M}}$ where the
phase factor is $\phi_{M}=-\beta$ in the standard model viz. one of the phases
in the CKM matrix. For $B_{s}^{0}-\bar{B}_{s}^{0}$, there is no mismatch
between $CP$ eigenstates $|B_{1s}^{0}\rangle$ and $|B_{2s}^{0}\rangle$ and the
mass eigenstates. There is no extra phase available in CKM matrix, with three
generations of quarks to accomodate more than two independent phases $\beta$
and $\gamma$; the unitarity of CKM matrix requires $\alpha+\beta+\gamma=\pi$.
The quantum mechanical interference gives rise to non zero mass differences
$\Delta m_{K}$, $\Delta m_{B}$ and $\Delta m_{B_{s}}$ between mass
eigenstates. The mixing induced $CP$ violation involves these mass
differences.
The $CPT$ invariance plays an important role in $CP$ violation in weak decays.
$CPT$ invariance gives
$\bar{A}_{\bar{f}}=\eta_{f}e^{2i\delta_{f}}A_{f}^{\ast},\quad
A_{f}=e^{i\delta_{f}}e^{i\phi}|A_{f}|$
where $A_{f}$ and $\bar{A}_{\bar{f}}$ are the amplitudes for the decays
$X\rightarrow f$ and $\bar{X}\rightarrow\bar{f}$, the states $|f\rangle$ and
$|\bar{f}\rangle$ being $CP$ conjugate of each other. For direct $CP$
violation, at least two amplitudes with different weak phase are required:
$A_{f}=A_{1f}+A_{2f}$
$CPT$ gives:
$\displaystyle\bar{A}_{\bar{f}}$
$\displaystyle=e^{2i\delta_{1f}}A_{1f}^{\ast}+e^{2i\delta_{2f}}A_{2f}^{\ast}$
$\displaystyle A_{if}$ $\displaystyle=e^{i\delta_{if}}e^{i\phi_{i}}|A_{if}|$
where $(\delta_{1f},\delta_{2f})$, $(\phi_{1},\phi_{2})$ are strong final
state phases and the weak phases respectively. Thus the direct $CP$ violation
is given by
$A_{CP}=\frac{\bar{\Gamma}(\bar{X}\rightarrow\bar{f})-\Gamma(X\rightarrow
f)}{\bar{\Gamma}(\bar{X}\rightarrow\bar{f})+\Gamma(X\rightarrow f)}$
where $\delta_{f}=\delta_{2f}-\delta_{1f}$, $\phi=\phi_{2}-\phi_{1}$. Hence
the necessary condition for non-zero direct $CP$ violation is $\delta_{f}\neq
0$ and $\phi\neq 0$.
In section 2, the $CP$ violation due to mismatch between $CP$ eigenstates
$|K_{1}^{0}\rangle$, $|K_{2}^{0}\rangle$ and mass eigenstates
$|K_{S}^{0}\rangle$ and $|K_{L}^{0}\rangle$ in terms of the parameter
$\epsilon$ and direct $CP$ violation due to different weak phases bewteen the
cecay amplitudes $A_{0}$ and $A_{2}$ are discussed.
Section 4:
Case I
The $CP$ violation for $B^{0}\rightarrow f$ decay where
$|\bar{f}\rangle=CP|f\rangle=|f\rangle$ are discussed. In particular for the
decay $B^{0}\rightarrow J/\psi K_{S}^{0}$ described by a single amplitude
$A_{f}$, the $CP$ asymmetry is given by
$A_{J/\psi K_{S}}=-\sin 2\beta\frac{(\Delta m_{B}/\Gamma)}{1+(\Delta
m_{B}/\Gamma)}$
It is a good illustration of $CP$ violation due to mismatch between mass and
$CP$ eigenstates, involving the mixing parameter $\Delta m_{B}$. From the
experimental values of $A_{J/\psi K_{S}}$ and $(\Delta m/\Gamma)_{B^{0}}$, the
weak phase $2\beta$ is found to be $(43\pm 3)^{\circ}$. Corresponding to
$B^{0}\rightarrow J/\psi K^{0}_{S}$, we have $B^{0}_{S}\rightarrow J/\psi\phi$
and for this decay
$A_{J/\psi\phi}=-\sin 2\beta_{s}\frac{(\Delta
m_{B_{S}^{0}}/\Gamma_{S})}{1+(\Delta m_{B_{S}^{0}}/\Gamma_{S})^{2}}$
Any finite value of $A_{J/\psi\phi}$ would imply $\beta_{s}\neq 0$ in
contradiction with the standard model.
In this section for the case (i), both direct and mixing induced $CP$
violation viz. $A_{CP}$, $C_{f}$ and $S_{f}$ for
$B^{0}\rightarrow\pi^{+}\pi^{-}$ described by two amplitudes $T$ and $P_{t}$
given by tree and penguin diagrams is discussed. We find
$C_{\pi\pi}=-A_{CP}(\pi\pi)$ and $S_{\pi\pi}$ is essentially given by
$S_{\pi\pi}\approx(\sin 2\alpha+2r\cos\delta\sin\alpha\cos 2\alpha),\ \ \ \ \
\ \ \ \ \ \ r=\frac{R_{t}}{R_{b}}\frac{|P_{t}|}{|T|}$
$S_{\pi\pi}\neq 0$ even when final state phase $\delta=0$.
Case II
We consider the cdecays described by two independent decay amplitudes $A_{f}$
and $A_{\bar{f}}^{{}^{\prime}}$ with different weak phases $(O$ and $\gamma)$
where the final states $|f\rangle$ and $|\bar{f}\rangle$ are $C$ and $CP$
conjugate of each other such as the states $D^{(\ast)-}\pi^{+}$
$(D^{(\ast)+}\pi^{-})$, $D_{s}^{(\ast)-}K^{+}$ $(D_{s}^{(\ast)+}K^{-})$,
$D^{-}\rho^{+}$ $(D^{+}\rho^{-})$.
It is argued in section 5, that $C$ and $CP$ invariance of hadronic
interactions imply $\delta_{f}=\delta_{\bar{f}}^{{}^{\prime}}$.
As discussed in section 6, the equality of phases
$\delta_{f}=\delta_{\bar{f}}^{{}^{\prime}}$ implies that time-dependent $CP$
asymmetries:
$\displaystyle-\left(\frac{S_{+}+S_{-}}{2}\right)$
$\displaystyle=\frac{2r_{D^{(\ast)}}}{1+2r^{2}_{D^{(\ast)}}}\sin(2\beta+\gamma)$
$\displaystyle\frac{S_{+}-S_{-}}{2}$ $\displaystyle=0$
It is further shown that from the experimental value of
$\frac{S_{+}+S_{-}}{2}$ for $B^{0}\rightarrow D^{\ast-}\pi^{+}$
$\displaystyle\sin(2\beta+\gamma)$ $\displaystyle>0.69$ $\displaystyle
44^{\circ}\leq 2\beta+\gamma\leq 90^{\circ}\quad$ $\displaystyle or\quad
90^{\circ}\leq 2\beta+\gamma\leq 136^{\circ}$
Selecting the second solution and using $2\beta\approx 43^{\circ}$, we get
$\gamma=(70\pm 23)^{\circ}$
Using $SU(3)$, for the form factors for $B_{s}^{0}\rightarrow D^{\ast-}K^{+}$,
we predict
$-\left(\frac{S_{+}+S_{-}}{2}\right)=(0.41\pm 0.08)\sin(2\beta_{s}+\gamma)$
In the standard model $\beta_{s}=0$.
Case III
For the case (III) for which $A_{f}\neq A_{\bar{f}}$ such as
$B^{0}\rightarrow\rho^{+}\pi^{-}:A_{\bar{f}}$ and
$B^{0}\rightarrow\rho^{-}\pi^{+}:A_{f}$ where $A_{f,\bar{f}}$ are given by
tree amplitude $e^{i\gamma}T_{f,\bar{f}}$ and penguin amplitude
$e^{-i\beta}P_{f,\bar{f}}$ are discussed.
In section 6 case (iii), the factorization for the tree graph implies
$\delta_{f}^{T}\approx\delta_{\bar{f}}^{T}\approx 0$. In the limit
$\delta_{f,\bar{f}}^{T}\rightarrow 0$, it is shown that
$\displaystyle r_{f,\bar{f}}\cos\delta_{f,\bar{f}}$ $\displaystyle=\cos\alpha$
$\displaystyle r_{f,\bar{f}}^{2}$
$\displaystyle\approx\cos^{2}\alpha+A_{CP}^{f,\bar{f}2}\sin^{2}\alpha$
Finally, in the limit $\delta_{f,\bar{f}}^{T}\rightarrow 0$, we get
$\frac{S_{\bar{f}}}{S_{f}}=\frac{S+\Delta S}{S-\Delta
S}=-\sqrt{\frac{1-C_{\bar{f}}^{2}}{1-C_{f}^{2}}}$
To conclude:
1. 1.
No evidence that space-time symmeries are violated by fundamental laws of
nature. The Translational and Rotational symmetries imply that space is
homogeneous and isotropic.
Translational Symmetry $\displaystyle\Rightarrow\text{Energy Momentum
Conservation}$ Rotational Symmetry $\displaystyle\Rightarrow\text{Angular
Momentum Conservation}$
If we examine the light emitted by a distant object billions of light years
away, we find that atoms have been following the same laws as they are here
and now. (Translational Symmetry)
2. 2.
Discrete Symmetries are not universal; both C and P are violated in the weak
interaction but repsected by electromagnetic and strong interactions. There is
no evidence for violation of time reversal invariance by any of the
fundamental laws of nature.
3. 3.
Basic weak interaction Lagrangian is CP conserving. CP violation in weak
interactions is a consequence of mismatch between mass eigenstates and CP
eigenstates and or mismatch between weak and mass eigenstates at quark level.
There is no evidence of CP violation in Lepton sector. There is no evidence
that CP invariance is violated by any of the fundamentals laws of nature as
implied by CPT invariance and T-invariance.
4. 4.
CP violation in weak decays is an example where basic laws are CP invariant
but states at quark level contain CP violating phases.
5. 5.
The fundamental interaction governing atoms and molecules is the
electromagnetic interaction which does not violate bilateral symmetry (left-
right symmetry). In nature we find organic molecules in asymmetric form, i.e.
left handed or right handed. This is another example where the basic laws
governing these molecules are bilateric symmetric but states are not.
(Asymmetric intial conditions?)
6. 6.
Baryon Asymmetry of the Universe: Baryogenesis: No evidence for existence of
antibaryons in the universe. $\eta=n_{B}/n_{\gamma}\sim 3\times 10^{-10}$. The
universe started with a complete matter antimatter symmetry in big bang
picture. In subsequent evolution of the universe, a net baryon number is
generated. This is possible provided the following conditions of Sakharov are
satisfied
1. (a)
There exists a baryon number violating interaction.
2. (b)
There exist C and CP violation to induce the asymmetry between particle and
antiparticle processes.
3. (c)
Departure from thermal equilibrium of X-particles which mediate the baryon
number violating interactions.
7. 7.
There seems to be no connection between CP violation required by baryogenesis
and CP violation observed in weak decays.
Selective List of References.
## References
* [1] For a review, see for example CP violation edited by C. Jarlskog, World Scientific (1989).
* [2] Fayyazuddin and Riazuddin. A Modern Introduction to Particle Physics Second Ed. 2000, World Scientific Singapore.
* [3] H. Quinn. B Physics and CP violation. hep-ph/0111177 v1.
* [4] R. D. Peccei. Thoughts on CP violation. hep-ph/0209245.
* [5] L. Wolfenstein. CP violation: The past as prologue. hep-ph/0210025. Section 4 Case(II)
* [6] C. Amsler, et.al. Particle Data Group, Phys. Lett B667,1 (2008).
* [7] Fayyazuddin. Phys. Rev. D 70 114018 (2004).
* [8] Fayyazuddin. Phys. Rev. D 77 014007 (2008). arXiv: hep-ph/0709.3364. Section 5
* [9] J. F. Donohue et.al. Phys. Rev. Lett. 77, 2187 (1996).
* [10] M. Suzuki and L. Wofenstein, Phys. Rev. D 60, 074019 (1999).
* [11] Fayyazuddin, JHEP 09, 055 (2002).
* [12] Fayyazuddin, arXiv: hep-ph/0909.2085
* [13] M.Gronau and J.L. Rosner, hep-ph/0807.3080 v3 Section 6 Case(II) Form Factors
* [14] S. Balk, J. G. Korner, G. Thompson, F. Hussain J. Phys. C 59, 283-293 (1993).
* [15] N. Isgur and M. B. Wise Phys. Lett B 232, 113 (1989) Phys. Lett. B 237, 527 (1990).
* [16] S. Faller et.al. hep-ph/0809.0222 v1.
* [17] P. Ball, R. Zweicky and W. I. Fine. hep-ph/0412079 v1.
* [18] G. Duplancic et.al. hep-ph/0801.1796 v2. Section 6 Case(II) and (III) Factorization
* [19] J. D. Bjorken, Topics in B-physics, Nucl. Phys. 11 (proc.suppl.) 325 (1989); M. Beneke, G. Buchalla, M. Neubart and C. T. Sachrajda, Phys. Rev. Lett, 83, 1914 (1999), Nucl. Phys. B591 313 (2000); C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. Lett. 87, 201806 (2001) hep-ph/0107002. Case (III)
* [20] V. Page and D. London, Phys. Rev. D 70, 017501 (2004).
* [21] M. Gronau and J. Zupan: hep-ph/0407002, 2004 Refernces to earlier literature can be found in this reference.
* [22] Y. Grossman and H. R. Quinn. Phys. Rev. D 58 017504 (1998); J. Charles. Phys Rev. D 59 054007 (1999); M. Gronau et. al. Phys. Lett B 514 315 (2001).
* [23] M. Beneke and M. Neuebert, Nucl. Phys. B675, 338 (2003).
Figure Captions:
1. Figure 1
The Unitarity triangle
2. Figure 2
The Box Diagram
3. Figure 3
The Tree Diagram
4. Figure 4
The Penguin Diagram
5. Figure 5
(a) $W$-exchange diagram for $B_{q}^{0}\rightarrow
N_{1}\bar{N}_{2}\left(M_{f}\right);$
(b) $W$-exchange diagram for $B_{q}^{0}\rightarrow
N_{1}\bar{N}_{2}\left(M_{f}^{\prime}\right)\ $
6. Figure 6
Annihilation diagram for $B^{-}\rightarrow N_{1}\bar{N}_{2}$
7. Figure 7
Plot of equation $r_{f}\cos\delta_{\left(f\right)}=\cos\alpha$ for different
values of $r.$ For $80^{o}\leq\alpha\leq 103^{o}.\ $Where solid curve, dashed
curve, dashed doted curve, dashed bouble doted and double dashed doted curve
are corresponding to $r=0.1,\ r=0.15,\ r=0.2,\ r=0.25$ and $r=0.3$
respectively.
|
arxiv-papers
| 2009-07-19T12:28:50 |
2024-09-04T02:49:04.051364
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fayyazuddin",
"submitter": "Aqeel Ahmed",
"url": "https://arxiv.org/abs/0907.3285"
}
|
0907.3626
|
# $J/\psi$ production at mid and forward rapidity at RHIC
Zhen Qu1 Yunpeng Liu1 Nu Xu2 Pengfei Zhuang1 1Physics Department, Tsinghua
University, Beijing 100084, China
2Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA
###### Abstract
We calculate the rapidity dependence of $J/\psi$ nuclear modification factor
and averaged transverse momentum square in heavy ion collisions at RHIC in a
3-dimensional transport approach with regeneration mechanism.
###### keywords:
$J/\psi$ production, regeneration, heavy ion collisions, quark-gluon plasma
###### PACS:
25.75.-q , 12.38.Mh , 24.85.+p
††journal: Nuclear Physics A
$J/\psi$ suppression [1] is widely accepted as a probe of quark-gluon plasma
(QGP) formed in relativistic heavy ion collisions and was first observed at
SPS [2] more than ten years ago. At RHIC and LHC energy, a significant number
of charm quarks are generated in central heavy ion collisions, and the
recombination of these uncorrelated charm quarks offers another source for
$J/\psi$ production [3]. There are different ways to describe the $J/\psi$
regeneration. In the statistical model [4], all the $J/\psi$s are produced at
hadronization of the system through thermal distributions and charm
conservation. In some other models, $J/\psi$s come from both the continuous
regeneration inside the hot medium and primordial production through initial
nucleon-nucleon collisions [3, 5]. The regeneration is used to describe [6]
the $J/\psi$ nuclear modification factor $R_{AA}$ and averaged transverse
momentum square $\langle p_{t}^{2}\rangle$. From the experimental data [7] at
RHIC, the $R_{AA}$ is almost the same as that at SPS, which seems difficult to
explain in models with only primordial production mechanism.
The rapidity dependence of $J/\psi$ production was also measured at RHIC [7,
8] and discussed in models [9, 10, 11]. The surprising finding of the
experimental result is that the apparent suppression at forward rapidity is
stronger than that at midrapidity, i.e. $R_{AA}^{mid}>R_{AA}^{forward}$. This
is again difficult to explain in models with only initial production
mechanism, since the suppression at midrapidity should be stronger than that
at forward rapidity. Not only $R_{AA}$ but also $\langle p_{t}^{2}\rangle$
depends on the rapidity [7]. In semi-central and central Au+Au collisions the
value of $\langle p_{t}^{2}\rangle$ at midrapidity is lower than that at
forward rapidity, i.e. $\langle p_{T}^{2}\rangle^{mid}<\langle
p_{t}^{2}\rangle^{forward}$. In this paper, we will discuss the rapidity
dependence of $R_{AA}$ and $\langle p_{t}^{2}\rangle$ in a 3-dimensional
transport model with both initial production and continuous regeneration
mechanisms.
At RHIC $J/\psi$s are detected at midrapidity $|y|<0.35$ and forward rapidity
$1.2<y<2.2$, both are located in the plateau of the rapidity distribution of
light hadrons [12]. Therefore, the space-time evolution of the QGP can be
approximately described by the transverse hydrodynamic equations at
midrapidity [13], and the $J/\psi$ motion is controlled by a 3-dimensional
transport equation in an explicitly boost invariant form
$\left[\cosh(y_{\Psi}-\eta)\ \partial/\partial\tau+\sinh(y_{\Psi}-\eta)/\tau\
\partial/\partial\eta+{\bf
v}_{t}^{\Psi}\cdot\nabla_{t}\right]f_{\Psi}=-\alpha_{\Psi}f_{\Psi}+\beta_{\Psi},$
(1)
where $f_{\Psi}=f_{\Psi}({\bf p}_{t},y,{\bf x}_{t},\eta,\tau|{\bf b})$ is the
charmonium distribution function in phase space at fixed impact parameter
${\bf b}$, and we have used transverse energy
$E_{t}=\sqrt{E_{\Psi}^{2}-p_{z}^{2}}$, rapidity
$y_{\Psi}=1/2\ln[(E_{\psi}+p_{z})/(E_{\Psi}-p_{z})]$, proper time
$\tau=\sqrt{t^{2}-z^{2}}$ and space-time rapidity
$\eta=1/2\ln{[(t+z)}/{(t-z)]}$ to replace the charmonium energy
$E_{\Psi}=\sqrt{{\bf p}^{2}+m_{\Psi}^{2}}$, longitudinal momentum $p_{t}$,
time $t$ and longitudinal coordinate $z$. The term with transverse velocity
${\bf v}_{t}^{\Psi}={\bf p}_{t}/E_{t}$ reflects the leakage effect in
charmonium motion. To take into account the decay of the charmonium excitation
states into $J/\psi$, the symbol $\Psi$ here stands for $J/\psi,\chi_{c}$ and
$\psi^{\prime}$ and the ratio of their contributions in the initial condition
is taken as 6:3:1. The suppression and regeneration in the QGP are described
by the loss and gain terms on the right hand side of the transport equation.
Considering the gluon dissociation process, $\alpha$ can be explicitly written
as [14]
$\alpha_{\Psi}({\bf p},{\bf x},t|{\bf b})=\int d^{3}{\bf
q}/\left((2\pi)^{3}4E_{t}E_{g}\right)W_{g\Psi}^{c\bar{c}}(s)f_{g}({\bf
q},T,u)\Theta(T-T_{c})/\Theta(T_{d}^{\Psi}-T),$ (2)
where $W_{g\Psi}^{c\bar{c}}$ is the transition probability [15] as a function
of the colliding energy $\sqrt{s}$ of the dissociation process, $E_{g}$ and
$f_{g}$ are the gluon energy and gluon thermal distribution, and $T_{c}$ and
$T_{d}^{\Psi}$ are the critical temperature of the deconfinement phase
transition and dissociation temperature of $\Psi$, taken as $T_{c}=165$ MeV,
$T_{d}^{J/\psi}/T_{c}=1.9$ and
$T_{d}^{\chi_{c}}/T_{c}=T_{d}^{\psi^{\prime}}/T_{c}=1$. The two step functions
$\Theta$ in $\alpha$ indicate that the suppression is finite in the QGP phase
at temperature $T<T_{d}^{\Psi}$ and becomes infinite at $T>T_{d}^{\Psi}$. We
have here neglected the suppression process in hadron phase [13, 6].
The gain term $\beta_{\Psi}$ can be obtained from the loss term $\alpha$ by
considering detailed balance [3]. We assume local thermalization of charm
quarks in the QGP and take the charm quark distribution as
$f_{c}({\bf k},{\bf x},t)={\rho_{c}({\bf x},t)}f_{q}({\bf k})$ (3)
with $\rho_{c}$ being the density of charm quarks in coordinate space,
$\rho_{c}({\bf x},t)=T_{A}({\bf x}_{t})T_{B}({\bf x}_{t}-{\bf
b})\cosh\eta/\tau\ d\sigma_{NN}^{c\bar{c}}/d\eta$ (4)
and $f_{q}$ the normalized Fermi distribution in momentum space, where $T_{A}$
and $T_{B}$ are the thickness functions for the two colliding nuclei
determined by nuclear geometry. Since the large uncertainty of charm quark
production cross section in pp collisions for both experimental and
theoretical studies, we assume the rapidity dependence of charm production as
a Gauss distribution
$d\sigma_{pp}^{c\bar{c}}/d\eta=d\sigma_{pp}^{c\bar{c}}/d\eta\big{|}_{\eta=0}e^{-\eta^{2}/2\eta_{0}^{2}}$
with $d\sigma_{pp}^{c\bar{c}}/d\eta\big{|}_{\eta=0}=120\ \mu$b which agrees
with the experimental data [16] and
$(d\sigma_{pp}^{c\bar{c}}/d\eta\big{|}_{\eta=1.7})/(d\sigma_{pp}^{c\bar{c}}/d\eta\big{|}_{\eta=0})=1/3$
to determine the parameter $\eta_{0}$ which is in between the smallest and
largest theoretical estimation [16].
The contribution from the primordial charmonium production is reflected in the
initial condition of the transport equation at the starting time $\tau_{0}$.
By fitting the experimental data [17] for pp collisions at RHIC, the initial
charmonium momentum distribution is extracted as
$f_{pp}({\bf p}_{t},y)=5g(y)/\left(4\pi\langle
p_{t}^{2}\rangle(y)\right)\left[1+p_{t}^{2}/\left(4\langle
p_{t}^{2}\rangle(y)\right)\right]^{-6},$ (5)
where the rapidity distribution $g(y)$ is a double Gauss function [17], and
the rapidity dependence of the averaged transverse momentum square is taken as
$\langle p_{t}^{2}\rangle(y)=\langle p_{t}^{2}\rangle(0)(1-y^{2}/y_{max}^{2})$
with the parameters $\langle p_{t}^{2}\rangle(0)=$ 4.1 (GeV/c)2 and
$y_{max}=\textrm{arccosh}(\sqrt{s}/{2m_{J/\psi}})$. Note that, to include the
Cronin effect in the initial state of heavy ion collisions [18], we add an
extra term to $\langle p_{t}^{2}\rangle$ which comes from the gluon multi-
scattering with nucleons [6, 11].
The charmonium production, including initial production and regeneration, is
related to the QGP evolution through the local temperature $T$ and fluid
velocity $u_{\mu}$ appearing in the thermal gluon and charm quark
distributions, they are determined by the ideal hydrodynamics [13].
Figure 1: The nuclear modification factor $R_{AA}$ (left panel) and averaged
transverse momentum square $\langle p_{t}^{2}\rangle$ (right panel) at mid and
forward rapidity as functions of number of participants $N_{p}$. The
theoretical calculations with only initial production (dot-dashed lines), only
regeneration (dashed lines) and both (solid lines) are compared with the
experimental data [7, 8].
With the known distribution $f_{J/\psi}({\bf p}_{t},y,{\bf
x}_{t},\eta,\tau|{\bf b})$, one can calculate the $J/\psi$ yield and momentum
spectra. The nuclear modification factor $R_{AA}$ and averaged transverse
momentum square $\langle p_{t}^{2}\rangle$ at mid and forward rapidity are
shown in Fig.1 as functions of centrality. Since $R_{AA}$ is normalized to the
pp collisions, the assumption of the same medium at mid and forward rapidity
leads to similar $R_{AA}$ in the two rapidity regions, when we consider only
initial production, as shown in the left panel. The regeneration at forward
rapidity is, however, much less than that at mid rapidity. As a result of the
competition, the total $R_{AA}$ at forward rapidity is less than that at mid
rapidity, consistent with the experimental observation.
While the population is dominated by low momentum $J/\psi$s, the averaged
transverse momentum carries more information on high momentum $J/\psi$s and
can tell us more about the dynamics of charmonium production and suppression.
The initially produced $J/\psi$s are from the hard nucleon-nucleon process at
the very beginning of the collision and their $p_{t}$ spectrum is harder. From
the gluon multi-scattering with nucleons before the two gluons fuse into a
$J/\psi$, there is a $p_{t}$ broadening for the initially produced $J/\psi$s.
Considering further the leakage effect which enables the high momentum
$J/\psi$s escape from the anomalous suppression in the hot medium, the
initially produced $\langle p_{t}^{2}\rangle$ increases smoothly with
centrality and becomes saturated at large $N_{p}$. Since the regenerated
$J/\psi$s are from the thermalized charm quarks inside the QGP, their averaged
momentum is small and almost independent of the centrality. Both the initially
produced and regenerated $\langle p_{t}^{2}\rangle$ is not sensitive to the
rapidity region. While the difference between the initially produced and
regenerated $R_{AA}$ decreases with increasing $N_{p}$, the difference between
the values of $\langle p_{t}^{2}\rangle$ from the two rapidity regions
increases smoothly with centrality! The total $\langle p_{t}^{2}\rangle$
depends strongly on the fraction of the regeneration. At mid rapidity, the
regeneration and initial production are equally important in central
collisions, see the left panel of Fig.1. The large contribution from the
regeneration leads to a remarkable decrease of the value of $\langle
p_{t}^{2}\rangle$ at mid rapidity. At forward rapidity, the regeneration
contribution is, however, very small even for central collisions, see the left
panel again. In this case, the total $\langle p_{t}^{2}\rangle$ is dominated
by the initial production in the whole $N_{p}$ region.
In summary, we calculated the $J/\psi$ nuclear modification factor and
averaged transverse momentum square at mid and forward rapidity in a three
dimensional transport approach. The experimentally observed rapidity
dependence of $R_{AA}$ and $\langle p_{t}^{2}\rangle$ in Au+Au collisions at
$\sqrt{s}$=200 GeV can well be explained by our model calculation where the
continuous regeneration of $J/\psi$ from thermalized charm quarks in QGP is an
important ingredient. We predict that at higher colliding energies, for
example at LHC, the regeneration will become the dominant ingredient.
## Acknowledgments
We are grateful to Dr. Xianglei Zhu for the help in numerical calculations.
The work is supported by the NSFC grant No. 10735040, the National Research
Program Grants 2006CB921404 and 2007CB815000. and the U.S. Department of
Energy under Contract No. DE-AC03-76SF00098.
## References
* [1] T. Matsui and H. Satz, Phys. Lett. B178, 416 (1986).
* [2] M. Gonin et al., [NA50 Collaboration], Nucl. Phys. A610, 404c (1996).
* [3] R. L. Thews, M. Schroedter, and J. Rafelski, Phys. Rev. C63, 054905 (2001); J. Phys. G27, 715 (2001).
* [4] P. Braun-Munzinger and J. Stachel, Phys. Lett. B490, 196 (2000); Nucl. Phys. A690, 119 (2001).
* [5] L. Grandchamp and R. Rapp, Phys. Lett. B523, 60 (2001); Nucl. Phys. A709, 415 (2002).
* [6] L. Yan, P. Zhuang, and N. Xu, Phys. Rev. Lett. 97, 232301 (2006).
* [7] A. Adare et al., [PHENIX Collaboration], Phys. Rev. Lett. 98, 232301 (2007).
* [8] J. Lajoie, [PHENIX Collaboration], J. Phys. G34, S191 (2007).
* [9] D. Kharzeev, E. Levin, M. Nardi, and K. Tuchin, ArXiv:0809.2933.
* [10] A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, Phys. Lett. B652, 259 (2007).
* [11] X. Zhao and R. Rapp, ArXiv:0810.4566.
* [12] I.G. Bearden et al., [BRAHMS Collaboration], Phys. Rev. Lett. 88, 202301 (2002).
* [13] X. Zhu, P. Zhuang and N. Xu, Phys. lett. B607, 107(2005).
* [14] Y. Liu, Z. Qu, N. Xu, and P. Zhuang, Phys. Lett. B678, 72 (2009).
* [15] M.E. Peskin, Nucl. Phys. B156, 365(1979); G. Bhanot, M.E. Peskin, Nucl. Phys. B156, 391(1979).
* [16] Y. Zhang, J. Phys. G35, 104022(2008).
* [17] A. Adare et al., [PHENIX Collaboration], Phys. Rev. Lett. 98, 232002(2007).
* [18] S. Gavin and M. Gyulassy, Phys. Lett. B214, 24 1(1988); J. Hüfner, Y. Kurihara and H.J. Pirner, Phys. Lett. B215, 218(1988).
|
arxiv-papers
| 2009-07-21T16:34:06 |
2024-09-04T02:49:04.072123
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhen Qu, Yunpeng Liu, Nu Xu, Pengfei Zhuang",
"submitter": "Zhen Qu",
"url": "https://arxiv.org/abs/0907.3626"
}
|
0907.3710
|
# Throughput metrics and packet delay in TCP/IP networks
[Work in progress]
Andrei M. Sukhov
Timur Sultanov
Mikhail V. Strizhov
Samara State Aerospace University Moskovskoe sh., 34 Samara, 443086, Russia
[email protected] Samara State Aerospace University, Togliatti branch
Voskresenskaya st., 1 Togliatti, 445000, Russia [email protected] Samara
State Aerospace University Moskovskoe sh., 34 Samara, 443086, Russia
[email protected] Alexey P. Platonov
Russian Institute for Public Networks Kurchatova sq. 1 Moscow, 123182,
Russia [email protected]
(20 April 2009)
###### Abstract
In the paper the method for estimation of throughput metrics like available
bandwidth and end-t-end capacity is supposed. This method is based on
measurement of network delay $D_{i}$ for packets of different sizes $W_{i}$.
The simple expression for available bandwidth
$B_{av}=(W_{2}-W_{1})/(D_{2}-D_{1})$ is substantiated. The number of
experiments on matching of the results received new and traditional methods is
spent. The received results testify to possibility of application of new
model.
###### category:
C.2.3 Computer-communication networks Network Operations
###### keywords:
network monitoring
###### category:
C.4 Performance of systems Measurement techniques
###### keywords:
new model for available bandwidth, end-to-end capacity, delay for packets of
different sizes, RIPE Test Box
††titlenote: corresponding author
## 1 Introduction
Measurement of throughput metrics like available bandwidth and capacity gives
a great chance to predict the end-to-end performance of applications, for
dynamic path selection and traffic engineering, and select among numbers of
differentiated classes of service. The throughput metric is an important
metric for several applications, such as grid, video and voice streaming,
overlay routing, p2p file transfers, server selection, and interdomain path
monitoring.
Various real-time applications in Internet, first of all, transmission audio
and video information become more and more popular, however for their
qualitative transmission high-speed networks are required. The major factors
defining quality of the service are: quality of the equipment (the codec and a
video server) and an available bandwidth of the channel. Providers and their
customers should provide a demanded available bandwidth for voice and video
applications to guarantee presence of demanded services in a global network.
In this paper we define a network path as the sequence of links that forward
packets from the path sender to the receiver. There are various definitions
for the throughput metrics, but we will adhere to the approaches accepted in a
series of papers by Dovrolis et al [5, 10, 18].
Two bandwidth metrics that are commonly associated with a path are the
capacity $C$ and the available bandwidth $B_{av}$. The capacity $C$ is the
maximum IP-layer throughput that the path can provide to a flow, when there is
no competing traffic load (cross traffic). The available bandwidth $B_{av}$,
on the other hand, is the maximum IP-layer throughput that the path can
provide to a flow, given the path’s current cross traffic load. The link with
the minimum transmission rate determines the capacity of the path, while the
link with the minimum unused capacity limits the available bandwidth.
Measuring available bandwidth is not only for knowing the network status, but
also to provide information to network applications on how to control their
outgoing traffic and fairly share the network bandwidth.
Another related throughput metric is the Bulk-Transfer-Capacity (BTC). The BTC
of a path in a certain time period is the throughput of a bulk TCP transfer,
when the transfer is only limited by the network resources and not by
limitations at the end-systems. The intuitive definition of BTC is the
expected long-term average data rate (bits per second) of a single ideal TCP
implementation over the path in question.
In order to measure different capacity metrics, the installation of special
utilities [12] is required at both ends of path. This is uncomfortable process
especially for usual Internet users who try to install modern network
applications like videoconference service.
For today also there are the various systems, allowing defining an available
bandwidth, but they have the disadvantages, therefore search of new solutions
is claimed. Among them, such as iperf, netperf, pathrate, pathload and abget,
and also a number of little-known programs ncs, netest, pipechar. We will
consider the cores from the above described products.
Each of the products set forth above has disadvantages. Utilities Iperf,
netperf, pathrate have one feature which is their essential disadvantage. To
estimate capacity of a network it is required to instal client and server
parts of the program. The utility abget demands HTTP a server on the remote
server and the privilege of the superuser, and as to programs ncs, netest,
pipechar so they are not adapted for operation with network screens that in
modern conditions does their a little used.
At the same time these programs use algorithms of an estimation of available
bandwidth, grounded on transmission the considerable quantity of packages on a
data link that reduces capacity of a network suffices and demands considerable
time.
In order to construct a perfect picture of a global network (monitoring and
bottlenecks troubleshooting) and develop the standards describing new
appendices, the modern measuring infrastructure should be installed. In Russia
different measurement projects are realized in the area of networking, for
example, PingER [16] in Institute of Theoretical and Experimental Physics
(ITEP), but full access to the collected data is limited for researchers.
Unfortunately, current measuring area do not reflect structure of the Russian
segment of a global network.
At present time powerful measurement system like RIPE Test Box is expanded
[7]. Unfortunately, this system doesn’t measure the available bandwidth, but
it collects the numerical values characterized the network heals like delay,
jitter, routing path, etc. This data allows us to investigate the basic
interdependencies of available bandwidth from basic network parameters. Our
aim is to estimate the available bandwidth from the delay value, received from
one point of path.
In our work we try to present the uniform model, allowing measuring all known
throughput metrics. Our method is based on testing of a network by packages of
the different size. Earlier such technique called Variable Packet Size (VPS)
was applied in work [6]. The VPS technique can estimate the capacity of a hop
$i$ based on the relation between the Round-Trip Time (RTT) up to hop $i$ and
the probing packet size $W$.
## 2 Model
The well-known expression for throughput metric describing a ratio between a
network delay and the packet size is a version of the Little’s Law [13].
$B=W/D$ (1)
Here $W$ is the size of transmitted packet and $D$ is the networking packet
delay. This formula is ideally for calculation of available bandwidth between
two network points that are connected immediately (in other words for
distantion in one hop). In general case the delay value is caused by such
constant network factors as propagation delay, transmission delay, per-packet
router processing time, etc [18].
In 1999 Downey [6] for the first time has detected linear dependence of the
minimum possible round trip time on the size of transferred packets. In 2004
precise experiments by Choi et al [2] proved that the minimum fixed delay
component for a packet of size $W$ is a linear (or precisely, an affine)
function of its size,
$D^{fixed}(W)=W\sum_{i=1}^{h}1/C_{i}+\sum_{i=1}^{h}\delta_{i}$ (2)
where $C_{i}$ is each link of capacity of $h$ hops and $\delta_{i}$ is
propagation delay. To validate this assumption, they check the minimum delay
of packets of the same size for three path, and plot the minimum delay against
the packet size.
Let $D(W)$ represents the point-to-point delay of a packet. Here we refer to
it as the minimum path transit time for the given packet size $W$, denoted by
$D^{fixed}(W)=\min D(W)$. With the fixed delay component $D^{fixed}(W)$
identified, we can now substract it from the point-to-point delay of each
packet to study the variable delay component $d^{var}$. The variable delay
component of the packet, $d^{var}$, is given by
$D(W)=D^{fixed}(W)+d^{var}$ (3)
Figure 1: Packet Size vs Delay
On the Fig. 1 the graphic shows the linear dependence between average network
delay $D_{av}(W)=\mathbb{E}[D(W)]$ and packet size $W$ like it is constructed
in paper [2]. Slope angle concerning $Y$ axe could be considered as available
bandwidth $B_{av}$ in contrast to bottleneck capacity $C$ (maximum throughput)
for computed minimal delay $D^{fixed}(W)$:
$D^{fixed}(W)=D_{min}+W/C,$ (4)
where
$D_{min}=\lim_{W\rightarrow 0}D^{fixed}(W)$ (5)
Prolongation of line $D(W)$ from Fig. 1 to $Y$ axe gives the intercept value
$a=\sum_{i=1}^{h}\delta_{i}$. Then the Equation (1) for the throughput metric
which path consists of two or more hops should be modernized to the following
view:
$B_{av}=W/(D_{av}-a)$ (6)
The value $a$ is related to the distance between the sites (i.e. propagation
delay) and per-packet router processing time at each hop along the path
between the sites [3, 4]. This value represents as the minimum delay $D_{min}$
for which the very small package can be transmitted on a network from one
point in another. In the general case $a(n,l)$ could be considered as the
linear function depended on $n$ and $l$,
$a=f(n,l)\approx\alpha n+\beta l$ (7)
where $n$ is the number of hops (routers) that is measured by the traceroute
utility and $l=\sum_{n}l_{n}$ is the sum of single length of routing path.
The Equation (6) gives us the simple way for estimation of throughput metrics
including active bandwidth $B_{av}$ and capacity $C$. Our method supposes the
variation of packet size on the same path for measurement of the throughput
metrics. If the testing process between two fixed points is organized by
packets with different sizes $W_{1}$ and $W_{2}$ then the delay times $D_{i}$
get two different values. Experiments should show the identical value for
available bandwidth $B_{av}$ independently from packet size $W_{i}$. The
system from two equations with different values of variables
$D_{i}=\mathbb{E}[D(W_{i})]$ and $W_{i}$ is easy solved to find $B_{av}$ and
$a$:
$B_{av}=\frac{W_{2}-W_{1}}{D_{2}-D_{1}}$ (8)
It should be noted that similar result was first time received for bandwidth-
dominated path in classical paper of Jacobson [9] dedicated congestion and
avoidance control.
Fig. 2 illustrates a schematic representation of transfer of packages of the
different sizes on the slowest link in the path (the bottleneck). The vertical
dimension is bandwidth, the horizontal dimension is time.
Another result for capacity $C$ will turn out, if instead of the average value
$D_{av}(W)$ in an analogue of the equation (6)
$C=\frac{W}{D^{fixed}(W)-D_{min}}$ (9)
the minimum fixed delay component $D^{fixed}(W)$ is used
$C=\frac{W_{2}-W_{1}}{D^{fixed}(W_{2})-D^{fixed}(W_{1})}$ (10)
Figure 2: Available Bandwidth Illustration
It is necessary to notice that experimental definition of any throughput
metrics demands carrying out of several measurements for a network delay.
After these measurements are spent for packages of the different sizes, it is
necessary to choose from them the minimum and average values. The minimum
value will be used for calculation of available bandwidth $B_{av}$, and
average value for capacity $C$. Even in work of Downey [6] it was noticed that
are many data points near the minimum and we can find the minimum delay
$D_{min}$ with a small number of probes at each packet size. It should be
noted that the method presented in given work allows measuring the available
bandwidth and capacity of the outgoing channel.
The minimal delay of datagram transmission $D_{min}$ may be calculated as
$D_{min}=\frac{W_{2}D_{1}-W_{1}D_{2}}{W_{2}-W_{1}}$ (11)
This value as well as the methods of its measurement has a important
significance in applied tasks of control theory [19]. The second significant
question of networking control theory is the distribution type for variable
delay component $d^{var}$ which should be studied. To know the expression for
this parameter we may easy calculate the duration of buffer for streaming
aplication on receiving side.
## 3 Precise Experiments
A number of measurements in a global network have been spent for
acknowledgement of our method. In this work the very first results which are
already processed are presented only.
For practical realization of our method the sizes $W_{1}$ and $W_{2}$ should
different in several times, it is reasonable to choose 64 and 1064 bytes for
Linux based systems, 32 and 1032 bytes for Windows correspondingly. The basic
problem of experimental testing is the precise of delay measurements that is
necessary for accurate result. The exact metering demands micro second
precision for delay measurements; we are reaching such accuracy with help of
RIPE Test Box mechanism [17]. In order to prepare the experiments three Test
Boxes have been installed in Moscow, Samara and Rostov on Don during 2006-2008
years in framework of RFBR grant 06-07-89074. Each RIPE Test Box represents a
server under management of an FreeBSD operating system with the GPS receiver
connected to it.
Characteristic times of investigated processes (a packet delay, jitter) have
the order from 10 $ms$ to 1 $sec$, therefore is quite enough accuracy of
system hours of a RIPE Test Box for their reliable measurement. At the first
stage experiment between tt01.ripe.net (RIPE NCC at AMS-IX, Amsterdam) and
tt143.ripe.net (Samara, SSAU) have been made which included
* •
Precision measurement of packet delay in the size 100 and 1100 bytes with
accuracy 2-12 $\mu s$
* •
Measurement of available bandwidth by means of utility iperf [12]
* •
Measurement of bandwidth by a method of downloading of a file on FTP
Thus, at us it will be generated alternatively measured three sizes of
throughput metrics for the subsequent comparative analysis.
It is necessary to notice that the utility iperf is started with an option -u
and measures speed of a stream between two points that precisely enough
corresponds to available bandwidth. Speed of downloading on ftp measures a
Bulk-Transfer-Capacity (BTC) and gives strongly underestimated value.
Unfortunately, at the given stage we could not spend more exact measurements,
but further we assume to find partners for installation of exact utilities.
The design of the RIPE TTM system meets all requirements shown by our method,
namely it allows to change the size of a testing package and to find network
delay with a split-hair accuracy.
By default, testing is conducted by packages in the size of 100 byte, but
there is a page corresponding to point of the menu Configuration of local Test
Box. On which it is possible to add testing packages to RIPE Box up to 1500
byte in size with demanded frequency.
In our case it is reasonable to add testing 1100 (1024) byte packages with
frequency of 60 times in a minute. It is necessary to notice that the results
of tests will be available on next day.
Testing results are available in telnet to RIPE Test Box on port 9142. It is
important to come and write down simultaneously the data on both ends of the
investigated channel, in the case presented here it is tt01.ripe.net and
tt143.ripe.net. Obtained data will contain required delay of packages of the
different sizes. Also, we need to distinguish packages.
Therefore at first it is reversible to sending Box and we will find lines, see
Table 1.
SNDP | 9 | 1240234684 | -h | tt01.ripe.net | -p | 6000 | -n | 1024 | -s | 1039148464
---|---|---|---|---|---|---|---|---|---|---
SNDP | 9 | 1240234685 | -h | tt164.ripe.net | -p | 6000 | -n | 100 | -s | 1039148548
SNDP | 9 | 1240234685 | -h | tt01.ripe.net | -p | 6000 | -n | 100 | -s | 1039148557
Table 1: The data of sending box
Last value in string is sequence number of the packet. It is necessary to us
to find this number on the receiving side at the channel. The string example
on the receiving side is lower resulted, see Table 2.
RCDP 12 2 89.186.245.200 60322 193.0.0.228 6000 | 1240234684.785799 | 0.044084 0X2107 0X2107 1039148464 0.000002 0.000008
---|---|---
RCDP 12 2 89.186.245.200 53571 193.0.0.228 6000 | 1240234685.788367 | 0.043591 0X2107 0X2107 1039148557 0.000002 0.000008
Table 2: The data of receivig box
For set number of a package it is easy to find network delay, in our case it
makes 0.044084 seconds. The following package 1039148557 has the size of 100
bytes and its delay makes 0.043591 seconds. Thus, the difference will make
0.000493 second.
Our model assumes operations with minimal and average values; therefore we
should note average values, not less than five pairs for the delay, going
consistently. In our case, average difference
$\mathbb{E}[D(1024)-\mathbb{E}[D(100)$ is 0.000571 seconds. (tt143 -> tt01).
Then the required bandwidth of the link (tt143 -> tt01) can be calculated as
$B_{av}(tt143\rightarrow tt01)=\frac{924\times 8}{0.000571}=12.9[Mbps]$ (12)
The minimal and average values of the return link (tt01 -> tt143) are
$\mathbb{E}[D(1024)]-\mathbb{E}[D(100)]=0.000511$ second and
$D^{fixed}(1024)-D^{fixed}(100)=0.000492$ second/ Then available bandwidth and
capacity can be calculated as
$\displaystyle C(tt01\rightarrow tt143)=\frac{924\times
8}{0.000492}=15.0[Mbps]$ (13) $\displaystyle B_{av}(tt01\rightarrow
tt143)=\frac{924\times 8}{0.000511}=14.7[Mbps]$ (14)
The main problem of the offered method consists in understanding, what value
is measured. Actually, it can be bulk transport capacity or available
bandwidth. Alternative measurements of the given values are necessary for
specification.
It is ideal to compare the width received by our method to the values measured
by alternative methods, first of all by means of the utility iperf.
Unfortunately, such tests are not spent yet, we allocate only in the speed of
FTP downloading. It makes 3.04 - 3.20 Mbps in a direction from tt143.ripe.net
to tt01.ripe.net and 3.2-3.3 Mbps in the opposite direction. That is
additional researches for which carrying out partners are required are
necessary.
It should be noted that Table II from paper [2] gives us these values;
calculated slope is inverse value to end-to-end capacity. The corresponding
capacities for data set 1, 2, 3 (path 1 and 2) are 285 Mbps, 128 Mbps, 222
Mbps and 205 Mbps.
## 4 AvBand Utility
Routinely the special utilities could be used for delay measurements; we tried
to test traditional ping, the new UDPping and other utility. In result of test
the simplest utility ping was found to be a best choice for delay
measurements.
Utility AvBand (Available Bandwidth) has been developed, realizing the above
described method, using in the basis algorithm ping. This algorithm has been
developed by Mike Muus in 1983 in the USA for operating system BSD [14]. Its
advantage consists that it is possible to work with any router or the host
which responds to packages of inquiries ICMP Echo. The given version of the
utility is developed for platform Windows and uses library ICMP Windows API.
In the near future we plan working out of the utility for Unix systems, first
of all for family Linux.
The given utility defines available bandwidth of outgoing channel between host
from which measurement and a remote server interesting us is spent. For this
purpose the program measures RTT (Round Trip Time) that is the time between
sending of inquiry and answer reception. Thus at first packages in 32 bytes
(standard Windows size) are generated and their RTT is defined, and the
following step forms packages of the size in 1032 bytes and is measured their
RTT. On Fig. 3 the screenshot of the program is presented.
Figure 3: The AvBand Screenshot
In the field “Host” it is entered a host name, available bandwidth to which we
are going to measure. In the field “Retries” the quantity of the echo-
inquiries which will be sent on a remote host is underlined. After that enough
to press button“Start” and the utility will send the set quantity of packages
of the size of 32 bytes, further the same quantity of packages in the size of
1032 bytes. The collected values of the received delays on each of groups of
packages are averaged, and then by means of our model the available bandwidth
of the channel pays off and is displayed. It is necessary to notice that the
available bandwidth of the outgoing channel is measured.
Hosts | Available bandwidth
---|---
testing | remote | ping or | FTP | Iperf
server | host | AvBand | |
SSAU | IOC RAS | 20-20.6 | 17.6-27.4 |
| | Mbps | Mbps |
SSAU | server2.hosting.reg.ru | 1150 | 1140 |
| | Kbps | Kbps |
OSU | SSAU | 2500 | | 2450
| | Kbps | | Kbps
AIST | SSAU | 536 | 600 | 659
| | Kbps | Kbps | Kbps
Infolada | SSAU | 346 | 374 |
| | Kbps | Kbps |
VolgaTelecom | SSAU | 274 | | 283
| | Kbps | | Kbps
Table 3: Experimental results
For check of utility AvBand a series of experiences with use of following
measuring mechanisms also has been spent:
* •
Utility AvBand
* •
Standard ping
* •
Iperf
* •
FTP
Measurements with Samara State Aerospace University (SSAU), Institute of
Organic Chemistry of the Russian Academy of Sciences (IOC RAS), control centre
RIPE in Amsterdam (RIPE), Ohio State University (OSU), and also a number of
local experiments with use of networks of various Internet Service Providers
of the Samara region (Infolada, AIST, VolgaTelecom, etc.) have been currently
spent. All data on experiments is resulted in the table more low.
As a case in point ADSL connection in Samara region could be chosen for
illustration of our approach. The delay measurements give $D_{1}=18$ $ms$,
$D_{2}=42$ $ms$, that corresponds to 350 Kbps of available bandwidth. During
FTP session the delay grows to 300 ms and 425 ms that corresponds
approximately to 60 Kbps of available bandwidth. This is very rough
computation, but it could be made quickly and independently.
## 5 Conclusion
Now measurements are not completed yet, is planned to type the data from not
less than 50 points scattered on territory of a planet. From these
measurements not less than 10 should be fulfilled with application of RIPE
Test Boxes. Thus, summing up to the done operation, it is possible to draw the
main output: the theoretical model of calculation of an available bandwidth
proves to be true.
Further it is planned to continue researches to establish type of distribution
for a network delay. At definition of type of distribution it is supposed to
use analogy to molecular physics, namely about distribution of molecules in
the speeds Maxswell. Probably, in our case required distribution should be
presented in the form of product of normally (Gaussian) distribution and the
inverse function defined by the Equation 6. The knowledge of density of
distribution in TCP/IP networks will help to find a new class the decision in
the networked control systems.
In summary we would like to express special gratitude of Prasad Calyam and
Gregg Trueb from Ohio State University for the invaluable help at carrying out
of measurements. Also it would be desirable to thank all collective of
technical service RIPE ncc and especially Ruben van Staveren and Roman
Kalyakin for constant assistance in comprehension of subtleties of a measuring
infrastructure.
## References
* [1] Ben Fredj, S., Bonald, T., Proutiere, A., Regnie, G., Roberts, J.: Statistical Bandwidth Sharing: A Study of Congestion at Flow Level. In: ACM SIGCOMM (2001)
* [2] Choi, B.-Y., Moon, S., Zhang, Z.-L., Papagiannaki, K. and Diot, C.: Analysis of Point-To-Point Packet Delay In an Operational Network. In: Infocom 2004, Hong Kong, pp. 1797-1807 (2004)
* [3] Cottrell, L., Matthews, W. and Logg C.: Tutorial on Internet Monitoring $\&$ PingER at SLAC. http://www.slac.stanford.edu/comp/net/wan-mon/tutorial.html
* [4] Crovella, M.E. and Carter, R.L.: Dynamic Server Selection in the Internet. In: Proc. of the Third IEEE Workshop on the Architecture and Implementation of High Performance Communication Subsystems (1995)
* [5] Dovrolis C., Ramanathan P., and Moore D., Packet-Dispersion Techniques and a Capacity-Estimation Methodology, IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 12, NO. 6, DECEMBER 2004, p. 963-977
* [6] Downey A.B., Using Pathchar to estimate internet link characteristics, in Proc. ACM SICCOMM, Sept. 1999, pp. 222 223.
* [7] Georgatos, F., Gruber, F., Karrenberg, D., Santcroos, M., Susanj, A., Uijterwaal, H. and Wilhelm R., Providing active measurements as a regular service for ISP’s. In: PAM2001
* [8] Guojun, J.: Available Bandwidth Measurement and Sampling, http://www.caida.org/workshops/isma/0312/abstracts/guojun.pdf
* [9] Jacobson, V. Congestion avoidance and control. In Proceedings of SIGCOMM 88 (Stanford, CA, Aug. 1988), ACM
* [10] Jain, M., Dovrolis, K.: End-to-end Estimation of the Available Bandwidth Variation Range. In: SIGMETRICS’05, Banff, Alberta, Canada (2005)
* [11] H.323 Beacon Tool, http://www.osc.edu/networking/itecohio.net/beacon/
* [12] Iperf, dast.nlanr.net/Projects/Iperf/
* [13] Kleinrock, L. Queueing Systems, vol. II. John Wiley & Sons, 1976.
* [14] Mike Muus, Ping documentation, http://ftp.arl.mil/ mike/ping.html
* [15] Padhye, J., Firoiu, V., Towsley, D., Kurose, J.: Modeling TCP Throughput: A Simple Model and its Empirical Validation. In: Proc. SIGCOMM Symp. Communications Architectures and Protocols, pp. 304-314 (1998)
* [16] PingER, http://www-iepm.slac.stanford.edu/pinger/
* [17] Ripe Test Box, http://ripe.net/projects/ttm/
* [18] Prasad R.S., Dovrolis C., and B. A. Mah B.A., The effect of layer-2 storeand-forward devices on per-hop capacity estimation, in Proc. IEEE INFOCOM, Mar. 2003, pp. 2090 2100.
* [19] Zhang, W., Branicky, M.S., Phillips S.M.: Stability of Networked Control Systems. In: IEEE Control System Magazine, 21(1), pp. 84-99 (2001)
|
arxiv-papers
| 2009-07-21T17:35:44 |
2024-09-04T02:49:04.079862
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.V. Sukhov, T.G. Sultanov, M.V. Strizhov, A.P. Platonov",
"submitter": "Andrei Sukhov M",
"url": "https://arxiv.org/abs/0907.3710"
}
|
0907.3766
|
# Semiclassical Approach to Survival Probability at Quantum Phase Transitions
Wen-ge Wang∗, Pinquan Qin, Lewei He, and Ping Wang Department of Modern
Physics, University of Science and Technology of China, Hefei, 230026, China
###### Abstract
We study the decay of survival probability at quantum phase transitions (QPT)
with infinitely-degenerate ground levels at critical points. For relatively
long times, the semiclassical theory predicts power law decay of the survival
probability in systems with $d=1$ and exponential decay in systems with
sufficiently large $d$, where $d$ is the degrees of freedom of the classical
counterpart of the system. The predictions are checked numerically in four
models.
###### pacs:
05.45.Mt, 05.70.Jk, 73.43.Nq, 64.60.Ht
## I Introduction
A quantum phase transition (QPT) is characterized by non-analyticity of the
ground level of the system at the critical point in the large size limit. At a
QPT, certain fundamental properties of the ground state (GS) change
drastically under small variation of a controlling parameter, e.g., strength
of a magnetic field. Most of the works in QPT have focused on properties of
equilibrium states (including GS at zero temperature) Sach99 . While, the non-
analyticity influences in fact both equilibrium and non-equilibrium
properties. Indeed, when the time scale of interest is smaller than the
relaxation time, which diverges at the critical point, usually the system is
not in an equilibrium state and unitary dynamics should be considered (Fig.1).
Due to significant progress in cold atom experiments, time dependent
simulation of models undergoing QPT is becoming realizable Sadler06 ; Lewen07
, hence, investigation in the unitary dynamics at QPT is of interest both
theoretically and experimentally. For example, resorting to theoretical
technique such as a quantum version of the Kibble-Zurek theory Kibble ;
Zurek85 , it has been shown that slow change of the controlling parameter
passing the critical point may induce some intriguing effects dyn-qpt .
In this paper, we study a different dynamics at QPT, which is induced by a
sudden small change in the controlling parameter,
$\lambda\to\lambda^{\prime}$, in the vicinity of a critical point
$\lambda_{c}$. A measure of the effect of this dynamics is the survival
probability (SP) of an initial state prepared in the GS $|0_{\lambda}\rangle$
of $H(\lambda)$,
$M(t)=|\langle
0_{\lambda}|e^{-iH(\lambda^{\prime})t/\hbar}|0_{\lambda}\rangle|^{2}.$ (1)
The SP, sometimes called autocorrelation function, is a quantity accessible
experimentally experiment-SP . Recent it was found that relatively significant
and fast decay of the SP may indicate the position of QPT Quan06 ; LE-qpt ;
Rossini07 , which has been demonstrated experimentally Peng08 . Short time
decay of the SP has been studied in these works. Of further interest, while
still unknown, is the law for relatively-long-time decay of the SP at QPT and
whether it may be useful in revealing characteristic properties of QPT foot-
osc ; foot-deco .
Figure 1: A schematic plot, where $\tau$ is the time scale of interest and
$\tau_{r}$ is the relaxation time, $\lambda$ is a controlling parameter with
critical value $\lambda_{c}$ of a QPT. Below the solid curves,
$\tau<\tau_{r}$, the system is usually not in an equilibrium state and its
unitary dynamics should be considered.
To find an answer to the above question, here we focus on those QPT, at the
critical points of which the ground levels have infinite degeneracy in the
large size limit. This is a type of QPT met in many cases (see models
discussed below and those in Ref. Sach99 ). At such a QPT, the non-analyticity
may be a consequence of avoided crossings of infinite levels, not a few
levels.
We find that the semiclassical theory may be used in the study of the SP decay
when $\lambda^{\prime}$ is sufficiently close to $\lambda_{c}$. The theory
predicts a power law decay of the SP in some systems and an exponential decay
in some other systems. Numerical results obtained in four models confirm these
predictions.
## II Semiclassical approach
We first discuss a condition for the applicability of the semiclassical theory
in the study of the SP of GS. We use notations:
$\epsilon=\lambda^{\prime}-\lambda,\delta=\lambda^{\prime}-\lambda_{c},\Delta\lambda=\lambda-\lambda_{c}$
(Fig.1), and $\eta=\epsilon/\Delta\lambda$. We use $|\alpha_{\lambda}\rangle$
with $\alpha=0,1,\ldots$ to denote eigenstates of $H(\lambda)$ with
eigenenergies $E_{\alpha}(\lambda)$ in increasing energy order. When the
ground level of $H(\lambda_{c})$ is infinitely degenerate and those of
$H(\lambda^{\prime})$ are non-degenerate (or have finite degeneracy),
infinitely many low-lying levels of $H(\lambda^{\prime})$ must join its ground
level in the limit $\lambda^{\prime}\to\lambda_{c}$, i.e.,
$\lim_{\lambda^{\prime}\to\lambda_{c}}E_{\alpha}(\lambda^{\prime})=E_{0}(\lambda_{c}),\hskip
28.45274pt\text{for many}\ \alpha.$ (2)
This has two consequences: (i) $H(\lambda^{\prime})$ of $\lambda^{\prime}$
sufficiently close to $\lambda_{c}$ must have a high density of states near
its ground level. (ii) For a fixed $\lambda$ near $\lambda_{c}$, when
$\lambda^{\prime}$ is sufficiently close to $\lambda_{c}$,
$H(\lambda^{\prime})$ may have many levels below $\overline{E}$, where
$\overline{E}=\langle 0_{\lambda}|H(\lambda^{\prime})|0_{\lambda}\rangle$,
i.e., the initial state $|0_{\lambda}\rangle$ may have a relatively high mean
energy in the system $H(\lambda^{\prime})$. This is in agreement with a
property revealed in recent study of the fidelity of GS near critical points,
which has close relationship to the SP, namely, for a fixed small $\epsilon$,
the overlap $|\langle 0_{\lambda}|0_{\lambda^{\prime}}\rangle|$ decreases
significantly when $\lambda^{\prime}$ approaches $\lambda_{c}$ GS-fid .
Moreover, suppose the system has a classical counterpart in the low energy
region. Here, a classical counterpart means a classical system, the
quantization of which gives a system mathematically equivalent to the original
quantum system; its components are not required to be directly related to
components of the original system. The property (2) implies that in the
process $\lambda^{\prime}\to\lambda_{c}$ longer and longer trajectories in the
classical system may be of relevance. For a fixed initial state
$|0_{\lambda}\rangle$, one may assume that the initial value of the Lagrangian
$L$ does not change notably in this process. Then, trajectories of relevance
may have large action $S=\int_{0}^{t}Ldt^{\prime}$ for $\lambda^{\prime}$
sufficiently close to $\lambda_{c}$.
The above discussed properties for $\lambda^{\prime}$ sufficiently close to
$\lambda_{c}$, namely, high density of states, relative highness of
$\overline{E}$, and large action of some relevant classical trajectories,
imply that a semiclassical approach may be valid. To be specific, for any
given $\lambda$ near $\lambda_{c}$, it is reasonable to expect that the
semiclassical theory may be used in the study of the SP when
$\lambda^{\prime}$ is sufficiently close to $\lambda_{c}$.
According to the semiclassical theory, qualitative difference in classical
trajectories may have quantum manifestation. Specifically, in the case of
$d=1$ where $d$ is the degree(s) of freedom of the classical counterpart in
the configuration space, the classical motion may show periodicity within a
time scale of interest; on the other hand, for a large $d$, even in a regular
system, classical trajectories may show no signature of periodicity within
times of practical interest. This difference suggests that the SP decay in the
former case may be slower than in the latter case, which we discuss below.
We consider small $\epsilon$, such that
$H(\lambda^{\prime})=H(\lambda)+\epsilon V$, with
$V\simeq\frac{dH(\lambda)}{d\lambda}$. The SP of the GS of $H(\lambda)$ is a
special case of the so-called quantum Loschmidt echo or (Peres) fidelity
Peres84 , $M_{L}(t)=|m(t)|^{2}$, where
$m(t)=\langle\Psi_{0}|{\rm exp}(iH(\lambda^{\prime})t/\hbar){\rm
exp}(-iH(\lambda)t/\hbar)|\Psi_{0}\rangle.$ (3)
In studying the SP, one may employ a semiclassical approach that has been
found successful in the study of Loschmidt echo JP01 ; PZ02 ; CT02 ; VH03 ;
wwg-LEc ; WL05 ; wwg-LEr ; JAB03 . For an initial Gaussian wave packet, narrow
in the coordinate space with width $\sigma$ and centered at ($\widetilde{\bf
r}_{0},\widetilde{\bf p}_{0}$) in the phase space, using the semiclassical Van
Vleck-Gutzwiller propagator, it has been shown that JP01 ; VH03
$\displaystyle m_{\rm sc}(t)\simeq\left(\pi w^{2}\right)^{-d/2}\int d{{\bf
p}_{0}}\exp{\left[\frac{i}{\hbar}\Delta S-\frac{({\bf p}_{0}-\widetilde{\bf
p}_{0})^{2}}{w^{2}}\right]}$ (4)
for small $\epsilon$, which works in both regular and chaotic cases VH03 ;
wwg-LEr . Here, $\Delta S$ is the action difference between two nearby
trajectories in the two systems starting at $({\bf p}_{0},\widetilde{\bf
r}_{0})$ and approximately can be evaluated along one trajectory, $\Delta
S\simeq\epsilon\int_{0}^{t}dt^{\prime}V[{\bf r}(t^{\prime}),{\bf
p}(t^{\prime})]$ JP01 . The quantity $w$ is $\hbar/\sigma$ for sufficiently
small $\sigma$ and depends on both $\sigma$ and the local instability of the
classical trajectory when $\sigma$ is not very small WL05 .
We first discuss the SP in the case of $d=1$ with a regular dynamics. We
assume that the GS can be (approximately) written as a Gaussian wave packet in
certain coordinate of the classical counterpart. This is possible, e.g., in
the models discussed below. In this case, as shown in Ref. wwg-LEr , for
$t>T$, due to the periodicity of the classical motion, the main contribution
of $\Delta S$ to the SP is given by its average part $\epsilon Ut$, where
$U=\frac{1}{T}\int_{0}^{T}V(t)dt$ and $T$ is the period of the classical
motion in $H(\lambda)$. Upto the first order expansion of $U$ in $p_{0}$, Eq.
(4) predicts a Gaussian decay of the SP PZ02 ; wwg-LEr . For relatively long
times, higher order terms of $U$ induces power law decay of the SP wwg-LEr ;
note-power . For example, to the second order term,
$M_{1}(t)\simeq{c_{0}}{(1+\xi^{2}t^{2})^{-1/2}}e^{-\Gamma
t^{2}/(1+\xi^{2}t^{2})},$ (5)
where $c_{0}\sim 1$, $\Gamma=(\frac{\epsilon
w}{\hbar}\frac{\partial\widetilde{U}}{\partial p_{0}})^{2}/2$,
$\xi=|\frac{\epsilon w^{2}}{2\hbar}\frac{\partial^{2}\widetilde{U}}{\partial
p_{0}^{2}}|$, with tilde indicating evaluation at $\widetilde{p}_{0}$ wwg-LEr
. It is seen that $M_{1}$ has a Gaussian decay $e^{-\Gamma t^{2}}$ for initial
times and has a $1/{\xi t}$ decay for long times.
Next, we consider the case of a regular classical counterpart with large $d$.
In this case, the underlying classical motion is typically quasi-periodic with
many different frequencies, as a result, $T$ is usually much longer than time
scales of practical interest. For times $t\ll T$, classical trajectories may
look random in the torus, due to the difference in the frequencies. To
calculate the SP in this case, one may write it in terms of the distribution
$P(\Delta S)$ of $\Delta S$ (with the Gaussian weight taken into account),
$M_{\rm sc}(t)\simeq\left|\int d\Delta Se^{i\Delta S/\hbar}P(\Delta
S)\right|^{2}$. When the trajectories can be effectively regarded as random
walks for times $t\ll T$ due to the many frequencies, $P(\Delta S)$ is close
to a Gaussian distribution, independent of the initial state. In this case,
the SP can be calculated in the same way as in a chaotic system CT02 , which
has an exponential decay determined by the variance of $\Delta S$,
$\displaystyle M_{2}(t)\simeq e^{-K_{s}\epsilon^{2}t/\hbar^{2}},$ (6)
where
$K_{s}\simeq\frac{1}{t}\left\langle\left[\int Vdt\right]^{2}-\left\langle\int
Vdt\right\rangle^{2}\right\rangle,$ (7)
with $\int Vdt=\int_{0}^{t}dt^{\prime}V[{\bf r}(t^{\prime}),{\bf
p}(t^{\prime})]$ note-expon .
To summarize, for small $\epsilon$ and sufficiently small $\delta$, and for
relatively long times, the SP may have a power law decay when $d=1$, and has
the exponential decay $M_{2}(t)$ when $d$ is sufficiently large. We remark
that, for $\lambda$ far from $\lambda_{c}$, the SP is always close to 1 for
small $\epsilon$.
Figure 2: (Color online) Decay of the SP (dashed curves) in the normal phase
of Dicke model. Parameters: $\omega=\omega_{0}=1$, $\epsilon=10^{-5}$, and
$\delta=-10^{-m}$ with $m=6,7,8,9,10,11$ from top to bottom. The solid curve
is a fitting curve of the form in Eq. (5), having an initial Gaussian decay
$e^{-\Gamma t^{2}}$ followed by a $1/\xi t$ decay. The $1/t$ decay becomes
clear with increasing $m$, i.e., with $\lambda^{\prime}$ approaching
$\lambda_{c}$. Upper right inset: $(\ln M)/\epsilon t^{2}$ versus $\eta$ for
different pairs of $(\epsilon,t)$ with short $t$, in agreement with the
prediction $\Gamma\sim|\eta\epsilon|$. Lower left inset: $\ln M$ versus
$\ln(\epsilon^{1/2}t)$ for $\epsilon\in(10^{-6},10^{-5})$ and $\ln
t\in(8.6,9.5)$ in the $1/t$ decay region. $\delta=-10^{-10}$, thus,
$|\eta|\simeq 1$. The results are in agreement with the prediction
$\xi\sim|\eta\epsilon|^{1/2}$.
## III Numerical simulations
The first model we study is the single-mode Dicke model Dicke54 , describing
the interaction between a single bosonic mode and a collection of $N$ two-
level atoms. In terms of collective operators ${\bf J}$ for the $N$ atoms, the
Dicke Hamiltonian is written as (hereafter we take $\hbar=1$) EB03 ,
$H=\omega_{0}J_{z}+\omega
a^{{\dagger}}a+({\lambda}/{\sqrt{N}})(a^{{\dagger}}+a)(J_{+}+J_{-}).$ (8)
In the limit $N\to\infty$, the system undergoes a QPT at
$\lambda_{c}=\frac{1}{2}\sqrt{\omega\omega_{0}}$, with a normal phase for
$\lambda<\lambda_{c}$ and a super-radiant phase for $\lambda>\lambda_{c}$. The
Hamiltonian can be diagonalized in this limit,
$H(\lambda)=\sum_{k=1,2}e_{k\lambda}c_{k\lambda}^{{\dagger}}c_{k\lambda}+g,$
(9)
where $c_{k\lambda}^{{\dagger}}$ and $c_{k\lambda}$ are bosonic creation and
annihilation operators, $e_{k\lambda}$ are single quasi-particle energies, and
$g$ is a c-number function EB03 . To be specific, in the normal phase,
$e_{k\lambda}^{2}=\frac{1}{2}\left\\{\omega^{2}+\omega_{0}^{2}+(-1)^{k}\sqrt{(\omega_{0}^{2}-\omega^{2})^{2}+16\lambda^{2}\omega\omega_{0}}\right\\}.$
(10)
It is seen that $e_{1\lambda_{c}}=0$, hence, the ground level of
$H(\lambda_{c})$ is infinitely degenerate. Since
$e_{2\lambda_{c}}=\sqrt{\omega^{2}+\omega_{0}^{2}}$ is finite, at the QPT one
may consider the effective Hamiltonian $H_{\rm
eff}(\lambda)=e_{1\lambda}c_{1\lambda}^{{\dagger}}c_{1\lambda}$ with $d=1$.
Direct calculation shows $e_{1\lambda}\simeq A|\Delta\lambda|^{1/2}$, with
$A=\frac{2(\omega\omega_{0})^{3/4}}{\sqrt{\omega^{2}+\omega_{0}^{2}}}$, and
$V=-\frac{A^{2}}{2e_{1\lambda}}\left(c_{1\lambda}^{{\dagger}}c_{1\lambda}+2(c_{1\lambda}^{{\dagger}})^{2}+2c_{1\lambda}^{2}\right)\sim|\Delta\lambda|^{-1/2}.$
(11)
The semiclassical result Eq. (5) predicts that the SP has a Gaussian decay
followed by a power law decay, with scaling properties
$\Gamma\sim\frac{\epsilon^{2}}{|\Delta\lambda|^{-1}}=|\eta\epsilon|,\hskip
28.45274pt\xi\sim|\eta\epsilon|^{1/2}.$ (12)
These predictions have been confirmed in our numerical simulations (Fig. 2).
Numerically, the SP was calculated by making use of relations between
$(c_{k\lambda}^{{\dagger}},c_{k\lambda})$ and
$(c_{k\lambda^{\prime}}^{{\dagger}},c_{k\lambda^{\prime}})$, which can be
directly derived from formulas given in Ref. EB03 . Our numerical results
support the prediction that the semiclassical theory may work for sufficiently
small $|\delta|$. Similar results have also been found in the super-radiant
phase.
The second model we have studied is the LMG model lipkin , with the
Hamiltonian $H=-\frac{1}{N}(S_{x}^{2}+{\gamma}S_{y}^{2})-\lambda S_{z}$, which
has a critical point at $\lambda_{c}=1$Dusuel . The model has a classical
counterpart with $d=1$. Direct computation shows a $1/t$ decay of the SP for
relatively long times in the neighborhood of $\lambda_{c}$ WZW09 .
As a third model, we study a 1-dimensional Ising chain in a transverse field,
$H(\lambda)=-\sum_{i=1}^{N}\sigma_{i}^{z}\sigma_{i+1}^{z}+\lambda\sigma_{i}^{x}.$
(13)
The Hamiltonian can be diagonalized by using Jordan-Wigner and Bogoliubov
transformations, giving $H(\lambda)=\sum_{k}e_{k}(b_{k}^{{\dagger}}b_{k}-1/2)$
Sach99 . Here, $b_{k}^{{\dagger}}$ and $b_{k}$ are creation and annihilation
operators for fermions and $e_{k}$ are single quasi-particle energies,
$e_{k}=2\sqrt{1+\lambda^{2}-2\lambda\cos(ka)}$ (14)
with lattice spacing $a$, where $k=\frac{2\pi m}{aN}$ with
$m=-M,-M+1,\ldots,M$ and $N=2M+1$. Note that $(ka)$ in Eq. (14) is in fact
independent of the lattice spacing $a$, with $ka=2\pi m/N$.
To understand the degeneracy property of the ground level in the large $N$
limit, let us consider those $m$ satisfying $|m|<N^{\beta}$ for large $N$,
where $\beta\in(0,1)$ is an arbitrary number independent of $N$. In the limit
$N\to\infty$, one has $ka\to 0$ for these $m$. As a result, Eq. (14) gives
$e_{k}=2|\Delta\lambda|$ with $\lambda_{c}=1$, in particular, at the critical
point $\lambda=\lambda_{c}$, $e_{k}=0$ for these modes $m$. The number of
these modes $m$ is infinitely large in the limit $N\to\infty$, hence, the
ground level is infinitely degenerate.
Figure 3: (Color online) Decay of the SP (circles) in a 1-dimensional Ising
chain in a transverse field, with $N=2\times 10^{8}$, $\epsilon=8\times
10^{-6}$, and $\delta=-4\times 10^{-6}$. It has the expected exponential decay
(solid straight line). Lower left inset: Dependence of $\ln(-\ln M)$ on
$\ln|\epsilon|$ for a fixed time $t$. The straight line has a slope 2, as
predicted in Eq. (6). Upper right inset: The SP increases slowly with
increasing $|\delta|$ for fixed $\epsilon$ and $t$, in agreement with the
prediction for $K_{s}$ given in the text.
In a sufficiently low energy region and for $\lambda\simeq\lambda_{c}$, a
classical counterpart of the system can be introduced as follows. For
$\lambda=\lambda_{c}$, $e_{k}\simeq 4\pi|m|/N$ for sufficiently large $N$ and
small $|m|$. In the low energy region, due to this linear dependence of
$e_{k}$ on $m$, using the method of bosonization (see Ref. Sach99 ), one can
express fermionic states $b_{k_{1}}^{{\dagger}}\ldots
b_{k_{n}}^{{\dagger}}|{\rm vacuum}\rangle$ in terms of (many) bosonic modes.
Each bosonic mode has a classical counterpart with one degree of freedom,
hence, $H(\lambda_{c})$ has a classical counterpart in the low energy region
with a large value of $d$ ($d\to\infty$ in the large $N$ limit). This implies
that $H(\lambda)$ with $\lambda\simeq\lambda_{c}$ also has a classical
counterpart with large $d$, as a result, typically the SP should have an
exponential decay $M_{2}(t)$ in Eq. (6).
Direct derivation shows that the perturbation in this model is
$V=\frac{\lambda-\cos
ka}{e_{k}/4}(b_{k}^{{\dagger}}b_{k}-\frac{1}{2})+\frac{\sin
ka}{e_{k}/2}i(b_{k}b_{-k}-b^{{\dagger}}_{k}b^{{\dagger}}_{-k}).$ (15)
Further analysis shows that $V$ has no singularity at the critical point,
e.g.,
$\displaystyle\frac{\sin ka}{e_{k}}\sim\left\\{\begin{array}[]{l}\sin
ka/|\Delta\lambda|,\ \hskip 42.67912pt\text{for}\
|ka|\lesssim|\Delta\lambda|\\\ \sin ka/\sqrt{1-\cos ka},\hskip
14.22636pt\text{for}\ |ka|>|\Delta\lambda|\end{array}\right..\ $ (18)
Therefore, $K_{s}$ in Eq. (6) has no singularity in the vicinity of
$\lambda_{c}$. For large and fixed $N$ and for $|\Delta\lambda|\gg 1/N$, since
the coupling strength of $V$ in the eigenbasis of $H(\lambda)$ increases with
decreasing $|\Delta\lambda|$, $K_{s}$ should increase slowly with decreasing
$|\Delta\lambda|$.
Numerical computation of the SP can be done by using the following expression
given in Ref.Quan06 ,
$M(t)=\prod_{k>0}F_{k},$ (19)
where
$\displaystyle
F_{k}=1-\sin^{2}(\theta_{\lambda}-\theta_{\lambda^{\prime}})\sin^{2}(e_{k}t),$
(20)
$\displaystyle\theta_{\lambda}=\arctan\frac{-\sin(ka)}{\cos(ka)-\lambda},$
(21)
and $e_{k}$ are evaluated at $\lambda^{\prime}$ . Our numerical computations
confirm not only the prediction of an exponential decay of the SP at the
criticality, but also some details in the exponent of $M_{2}(t)$ discussed
above (see Fig.3), namely, the $\epsilon^{2}$ dependence and the properties of
$K_{s}$.
As a fourth model, we have studied the XY model Sach99 , with the Hamiltonian
$H=-\sum_{i}\frac{1+\gamma}{2}\sigma_{i}^{x}\sigma_{i+1}^{x}+\frac{1-\gamma}{2}\sigma_{i}^{y}\sigma_{i+1}^{y}+\frac{\lambda}{2}\sigma_{i}^{z},$
(22)
which has critical points $\lambda_{c}=\pm 1$. As in the Ising chain, in the
low energy region around $\lambda_{c}$, the XY model has a classical
counterpart with large $d$. The SP in this model can be calculated in a way
similar to that in the Ising chain discussed above, and our numerical
simulations also confirmed the semiclassically predicted exponential decay of
the SP.
## IV Conclusions and discussions
We have shown that the semiclassical theory may be used in the study of the
decay of SP (survival probability) of GS (ground states) in the vicinity of
those QPT with infinitely degenerate ground levels at the critical points. Two
qualitatively different decaying behaviors of the SP have been found for
relatively long times: power law decay in systems with $d=1$ and exponential
decay in systems with sufficiently large $d$, where $d$ is the degrees of
freedom of the classical counterpart of the quantum system.
The above results suggest that the SP decay may be useful in the
classification of QPT, an important topic far from being completely solved, in
particular, in the non-equilibrium regime. Here, we have found two classes:
one class with power law decay and another class with exponential decay. It
needs further investigation whether other types of SP decay may appear at QPT,
e.g., relatively-long-time Gaussian decay or a decay between power-law and
exponential.
W.-G.W. is grateful to P.Braun, F.Haake, R.Schützhold, J.Gong, T.Prosen,
G.Benenti, and G.Casati for helpful discussions. This work is partly supported
by the Natural Science Foundation of China under Grant Nos. 10775123 and
10975123 and the National Fundamental Research Programme of China Grant
No.2007CB925200.
## References
* (1)
* (2) [] *Email address: [email protected]
* (3) S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999).
* (4) L.E. Sadler et al., Nature 443, 312 (2006).
* (5) M. Lewenstein, et al., Adv. Phys. 56, 243 (2007).
* (6) T.W.B. Kibble, J.Phys.A 9, 1387 (1976); Phys. Rep. 67, 183 (1980).
* (7) W.H. Zurek, Nature 317, 505 (1985); Phys. Rep. 276, 177 (1996).
* (8) J. Dziarmaga, A. Smerzi1, W. H. Zurek, and A. R. Bishop, Phys. Rev. Lett. 88, 167001 (2002); U. Dorner, P. Fedichev, D. Jaksch, M. Lewenstein, and P. Zoller, ibid. 91, 073601 (2003); W.H. Zurek, U. Dorner, and P. Zoller, ibid. 95, 105701 (2005); B. Damski and W.H. Zurek, ibid. 99, 130402 (2007).
* (9) P.Felker and A.Zewail, Adv.Chem.Phys.70, 265 (1988).
* (10) H.T. Quan, Z. Song, X.F. Liu, P. Zanardi, and C.P. Sun, Phys. Rev. Lett. 96, 140604 (2006).
* (11) Z.G.Yuan, P.Zhang, and S.S.Li, Phys.Rev.A 75, 012102 (2007); Y.C.Li and S.S.Li, ibid. 76, 032117 (2007).
* (12) D. Rossini, T. Calarco, V. Giovannetti, S. Montangero, and R. Fazio, Phys. Rev. A 75, 032333 (2007).
* (13) J. Zhang, X. Peng, N. Rajendran, and D. Suter, Phys. Rev. Lett. 100, 100501 (2008); J. Zhang, et al., Phys.Rev.A 79, 012305 (2009).
* (14) In systems with finite $N$, the SP may show a periodic behavior for sufficiently long times Quan06 . Here, we study the decay law of the SP for relatively long times in the thermodynamic limit.
* (15) At transition between localized and delocalized regimes, the SP may have power law decay as shown in S.A.Schofield, P.G.Wolynes, and R.E.Wyatt, Phys. Rev. Lett. 74, 3720 (1995); A.Ossipov, M.Weiss, T.Kottos, and T.Geisel, Phys. Rev. B 64, 224210 (2001).
* (16) P. Zanardi and N. Paunković, Phys. Rev. E 74, 031123 (2006); L.C. Campos Venuti and P. Zanardi, Phys. Rev. Lett. 99, 095701 (2007); W.-L. You, Y.-W. Li, and S.-J. Gu, Phys. Rev. E 76, 022101 (2007); S.-J. Gu, H. M. Kwok, W. Q. Ning, and H. Q. Lin, Phys. Rev. B 77, 245109 (2008).
* (17) A. Peres, Phys. Rev. A 30, 1610 (1984).
* (18) R.A. Jalabert and H.M. Pastawski, Phys.Rev.Lett.86 2490 (2001); G.Benenti and G.Casati, Phys.Rev.E 65, 066205 (2002); T.Gorin et al., Phys.Rep.435, 33 (2006); Ph.Jacquod and C.Petitjean, Adv.Phys.58, 67 (2009).
* (19) N. R. Cerruti and S. Tomsovic, Phys. Rev. Lett. 88, 054103 (2002); J. Phys. A 36, 3451 (2003);
* (20) J. Vaníček and E.J. Heller, Phys. Rev. E 68, 056208 (2003).
* (21) W.-G.Wang, G.Casati, and B.Li, Phys. Rev. E 69, 025201(R) (2004); W.-G.Wang, G. Casati, B. Li, and T. Prosen, ibid. 71, 037202 (2005).
* (22) W.-G.Wang and B. Li, Phys. Rev. E 71, 066203 (2005);
* (23) T. Prosen and M. Žnidarič, J. Phys. A 35, 1455 (2002).
* (24) Ph. Jacquod, I. Adagideli, and C.W.J. Beenakker, Europhys. Lett. 61, 729 (2003).
* (25) W.-G. Wang, G. Casati, and B. Li, Phys. Rev. E 75, 016201 (2007).
* (26) The Gaussian shape of the initial state is irrelevant in the derivation of the power law decay of $M_{L}(t)$. Moreover, as shown in PZ02 , $M_{L}(t)$ has a long-time $t^{-d}$ decay for initial random states. These imply that the power law feature might be insensitive to the initial condition.
* (27) Dividing $\overline{M}_{L}(t)$ in regular systems into diagonal and off-diagonal parts, Ref. JAB03 shows that the off-diagonal part may have an exponential decay under certain condition.
* (28) R.H. Dicke, Phys. Rev. 93, 99 (1954).
* (29) C. Emary and T. Brandes, Phys. Rev. E 67, 066203 (2003).
* (30) H.J. Lipkin, N. Meshkov,and A.J. Glick, Nucl. Phys. 62, 188 (1965).
* (31) S. Dusuel and J. Vidal, Phys. Rev. B 71, 224420 (2005).
* (32) P. Wang, Q. Zheng, and W.-G. Wang, unpublished.
|
arxiv-papers
| 2009-07-22T02:37:50 |
2024-09-04T02:49:04.086797
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wen-ge Wang, Pinquan Qin, Lewei He, and Ping Wang",
"submitter": "Wen-Ge Wang",
"url": "https://arxiv.org/abs/0907.3766"
}
|
0907.3832
|
KEK-TH-1322 IPMU-09-0083 arXiv:0907.3832
Topological String on OSP(1$|$2)/U(1)
Gaston Giribeta***E-mail: [email protected], Yasuaki Hikidab†††E-mail:
[email protected] and Tadashi Takayanagic‡‡‡E-mail:
[email protected]
aPhysics Department, University of Buenos Aires, and Conicet,
Ciudad Universitaria, Pab. I, 1428, Buenos Aires, Argentina
bKEK Theory Group, Tsukuba, Ibaraki 305-0801, Japan
cInstitute for the Physics and Mathematics of the Universe (IPMU),
University of Tokyo, Kashiwa, Chiba 277-8582, Japan
We propose an equivalence between topological string on OSP(1$|$2)/U(1) and
$\hat{c}\leq 1$ superstring with ${\cal N}=1$ world-sheet supersymmetry. We
examine this by employing a free field representation of OSP(1$|$2) WZNW model
and find an agreement on the spectrum. We also analyze this proposal at the
level of scattering amplitudes by applying a map between correlation functions
of OSP(1$|$2) WZNW model and those of ${\cal N}=1$ Liouville theory.
###### Contents
1. 1 Introduction
2. 2 ${\cal N}=2$ Coset OSP(1$|$2)/U(1) and 2D Superstring
1. 2.1 OSP(1$|$2) Current Algebra
2. 2.2 ${\cal N}=2$ Supersymmetric Coset Model
3. 2.3 Topological Twisting
4. 2.4 Chiral Primaries
3. 3 OSP(1$|$2)/U(1) Coset from ${\cal N}=1$ Super Liouville
1. 3.1 OSP(1$|$2) WZNW Model
2. 3.2 OSP(1$|$2)–Super Liouville Correspondence
3. 3.3 Amplitudes of OSP(1$|$2)/U(1) Coset Model
4. 4 Correspondence to $\hat{c}\leq 1$ Superstring Theory
1. 4.1 $\hat{c}\leq 1$ Superstring Theory
2. 4.2 Amplitudes of Topological Model
3. 4.3 Comparison of Correlation Functions
5. 5 Conclusion and Discussions
6. A Free Field Correlation Functions
## 1 Introduction
Superstrings on AdS spaces have been widely studied by virtue of their
applications to the holographic duality, i.e. the AdS/CFT correspondence [1];
and it has become clear that the structure of supergroup $\sigma$-models is of
great importance to investigate superstrings in these spaces. For instance,
the supergroup PSU(2,2$|$4) turns out to be important to construct superstring
theory on $AdS_{5}\times S^{5}$ [2]. Besides, superstring theory on
$AdS_{3}\times S^{3}$ can be described in terms of the PSL(1,1$|$2) WZNW model
[3]. However, in spite of its importance, quantizing supergroup
$\sigma$-models is a quite difficult problem, and hence solving superstring
theory on AdS spaces exactly still remains as an unsolved question.
Fortunately, there is a simpler type of duality for which string world-sheet
theory is still described by a supergroup WZNW model. It has been established
in [4, 5] that two-dimensional superstring (type 0 string) can be
holographically described by a simple Hermitian matrix model. At present, this
is the only dynamical model of string theory which is non-perturbatively well-
defined and is exactly solvable even at finite temperature. The two-
dimensional type 0 string theory is originally defined by string world-sheet
theory with the $\hat{c}=1$ matter coupled to ${\cal N}=1$ super Liouville
theory. Boosting the linear dilaton with Liouville potential kept the same,
this theory can be extended to $\hat{c}\leq 1$ type 0 string theory as it has
been done for bosonic string in [6, 7]. Note that dual matrix model can be
constructed even for $\hat{c}<1$ case, as shown in [6].
In this paper, we argue that these $\hat{c}\leq 1$ superstring theories can be
described by utilizing the supergroup OSP(1$|$2).111Current superalgebra of
OSP type also appears in an attempt [8] to generalize heterotic string so as
to be dual to Type I string theory with a OSP gauge symmetry. Precisely
speaking, we propose that the $\hat{c}\leq 1$ superstring is equivalent to
topological string on ${\cal N}=2$ superconformal coset
OSP(1$|$2)/U(1).222Topological strings on cosets based on sugerpgroups have
been studied for the analysis of Maldacena conjecture via world-sheet theory
in [9, 10, 11, 12, 13]. This relation can be thought of as a supersymmetric
version of the known relation between $c\leq 1$ bosonic string theory and
topological string on SL(2)/U(1) [14, 7]. This extension might be guessed from
the quantum Hamiltonian reduction since OSP(1$|$2) WZNW model is reduced to
${\cal N}=1$ super Liouville theory [15], just like SL(2) WZNW model is
reduced to bosonic Liouville theory [16].
This paper is organized as follows. In the next section, we explicitly
construct the ${\cal N}=2$ superconformal coset OSP(1$|$2)/U(1) as a natural
extension of Kazama-Suzuki model for bosonic cosets [17, 18].333Generic
construction of Kazama-Suzuki model for cosets of supergoups was given in [19,
20] very recently. We analyze it in the free field theory and show that the
$\hat{c}\leq 1$ string world-sheet appears after the topological twisting. In
particular, we show that free fields in the coset model become the matter
contents of $\hat{c}\leq 1$ superstring, and the chiral primaries of the coset
model are identified with the physical operators of $\hat{c}\leq 1$
superstring. In section 3, we review and extend the map between the
correlation functions in OSP(1$|$2) WZNW model and those in the ${\cal N}=1$
Liouville theory. This relation was originally obtained in [21] as an
generalization of $H_{3}^{+}$-Liouville relation [22, 23]. In section 4, after
briefly reviewing $\hat{c}\leq 1$ superstrings, we apply this map to study the
scattering S-matrices. We explicitly show that the correlation functions of
physical operators in the topological model are mapped to those of physical
operators in the $\hat{c}\leq 1$ superstring. In section 5, we summarize the
conclusion. In the appendix, we discuss correlation functions of OSP(1$|2$)
WZNW model in the free field representation.
## 2 ${\cal N}=2$ Coset OSP(1$|$2)/U(1) and 2D Superstring
In this section we construct and analyze ${\cal N}=2$ supersymmetric coset
(Kazama-Suzuki model [17, 18]) based on OSP(1$|$2)/U(1). After its topological
twisting, we show explicitly from the free field theory analysis that the
world-sheet theory of $\hat{c}\leq 1$ superstring indeed appears. We also
discuss chiral primary states which are the physical states in the
topologically twisted theory.
### 2.1 OSP(1$|$2) Current Algebra
The current algebra of OSP(1$|$2) includes SL(2) bosonic subalgebra, which is
generated by $J^{3}(z)$ and $J^{\pm}(0)$ with their OPEs444For a while we
concentrate on the holomorphic part.
$\displaystyle J^{+}(z)J^{-}(0)\sim\frac{k}{z^{2}}-\frac{2J^{3}(0)}{z},\qquad
J^{3}(z)J^{\pm}(0)\sim\pm\frac{J^{\pm}(0)}{z},\qquad
J^{3}(z)J^{3}(0)\sim-\frac{k}{2z^{2}}.$ (2.1)
In addition to these bosonic generators, there are fermionic ones with
$\displaystyle J^{3}(z)j^{\pm}(0)\sim\pm\frac{j^{\pm}(0)}{2z},\qquad
J^{\pm}(z)j^{\mp}(0)\sim\mp\frac{j^{\pm}(0)}{z},$ (2.2) $\displaystyle
j^{+}(z)j^{-}(0)\sim\frac{2k}{z^{2}}-\frac{2J^{3}(0)}{z},\qquad
j^{\pm}(z)j^{\pm}(0)\sim-\frac{2J^{\pm}(0)}{z}.$
The energy momentum tensor is given by Sugawara construction and the central
charge is $c=2k/(2k-3)$. These are the definition of OSP(1$|$2) current
algebra with level $k$.
In a free field representation [24, 15, 25, 26] the above currents may be
expressed as
$\displaystyle J^{-}=\beta,\qquad
J^{+}=\beta\gamma^{2}-\frac{1}{b}\gamma\partial\phi+\gamma\theta
p+k\partial\gamma-(k-1)\theta\partial\theta,$ (2.3) $\displaystyle
J^{3}=\beta\gamma-\frac{1}{2b}\partial\phi+\frac{1}{2}\theta p,\qquad
j^{-}=p-\beta\theta,\qquad j^{+}=\gamma
p-\beta\gamma\theta+\frac{1}{b}\theta\partial\phi-(2k-1)\partial\theta,$
where the OPEs of these free fields are
$\phi(z)\phi(0)\sim-\ln z,\qquad\beta(z)\gamma(0)\sim\frac{1}{z},\qquad
p(z)\theta(0)\sim\frac{1}{z}~{}.$ (2.4)
The field $\phi$ has the background charge $Q_{\phi}=b$ and the central charge
is $c=1+3Q_{\phi}^{2}$. Here the parameter $b$ is related to the level $k$ as
$1/b^{2}=2k-3$. The bosonic fields $(\beta,\gamma)$ have conformal weights
$(1,0)$ and the central charge of this system is $c=2$. On the other hand,
$(p,\theta)$ are fermions with conformal weights $(1,0)$ and central charge
$c=-2$. In the following analysis, it is useful to bosonize the fermionic
fields $(p,\theta)$ as
$\theta=e^{iY},\qquad p=e^{-iY}.$ (2.5)
For instance, the $J^{3}$ current takes the form
$J^{3}=\beta\partial\gamma-\frac{1}{2b}\partial\phi+\frac{i}{2}\partial Y.$
(2.6)
The energy momentum tensor is given by
$\displaystyle
T=\beta\partial\gamma-\frac{1}{2}\partial\phi\partial\phi+\frac{b}{2}\partial^{2}\phi-p\partial\theta=\beta\partial\gamma-\frac{1}{2}\partial\phi\partial\phi+\frac{b}{2}\partial^{2}\phi-\frac{1}{2}\partial
Y\partial Y+\frac{i}{2}\partial^{2}Y$ (2.7)
in terms of these free fields.
One of the merits to utilize the free field representation is that vertex
operators can be expressed in a simple form. Namely, the vertex operators of
OSP(1$|$2) model can be written in terms of free fields as
$\displaystyle\Phi^{s}_{j,m}\sim e^{isY}\gamma^{j-s/2+m}e^{-2bj\phi},$ (2.8)
whose conformal weight with respect to (2.7) is
$\Delta=-2b^{2}j(j+\tfrac{1}{2})+\tfrac{1}{2}s(s-1).$ (2.9)
Here we set $s=0,1/2,1$. For $s=0,1$ we can express the vertex operator even
in terms of $\theta$, but not for $s=1/2$. The operator with $s=1/2$
corresponds to a twist operator in R-sector, whose role was argued for
GL(1$|$1) WZNW model in [27], see also [28]. In order to compute correlation
functions, the overall normalization of vertex operators should be fixed. More
precise definition will be given in section 3. Correlation functions of
vertices (2.8) are discussed in the appendix, where the Coulomb gas
prescription is given.
Notice that the expression (2.9) is invariant under the Weyl transformation
$j\to-j-1/2$ and under $s\to 1-s$. This allows us to consider a second
contribution to (2.8) which goes like $\sim e^{2b(j+1/2)\phi}$ and it would
dominate the large $\phi$ regime for $j>-1/4$. In addition, there exist
conjugate representations which are similar to those that exist in the free
field realization of SU(2) model [29, 30, 25, 26]. For instance, one finds the
operator
$\displaystyle\hat{\Phi}_{j,j+\frac{1}{2}}^{0}\sim\beta^{k-2-2j}e^{b(2k-3-2j)\phi}$
(2.10)
which represents a Kac-Moody primary of conformal dimension (2.9), with
$m=j+1/2$ and $s=0$. It can be also thought of as a conjugate representation
for the state with $m=j$, $s=1$.
In order to define the OSP(1$|$2)/U(1) coset theory, we utilize the
representation introduced in [31] and [32] to realize the $SL(2)/U(1)$ coset
theory. This amount to introduce a boson $X^{3}(z)$ with
$X^{3}(z)X^{3}(0)\sim-\ln z$, as well as a $(b,c)$ ghost system, which are
used to mode out the $U(1)$ factor. Then the vertex operators of the coset
theory are given by
$\displaystyle\Phi^{s}_{j,m}=\Psi^{s}_{j,m}e^{-i\sqrt{\frac{2}{k}}mX^{3}}.$
(2.11)
As we will discuss below, the OSP(1$|$2) current algebra admits the symmetry
under the spectral flow action as in the case of SL(2) WZNW model [33]. For
OSP(1$|$2) model, spectrally flowed states are defined in this form as
$\displaystyle\Phi^{s,w}_{j,m}=\Psi^{s}_{j,m}e^{-i\sqrt{\frac{2}{k}}(m+\frac{k}{2}w)X^{3}}$
(2.12)
with the index of spectral flow $w$.
### 2.2 ${\cal N}=2$ Supersymmetric Coset Model
In this subsection we would like to construct ${\cal N}=2$ supersymmetric
model based on the coset OSP(1$|$2)/U(1). For the purpose we first generalize
the OSP(1$|$2) current algebra into ${\cal N}=1$ supersymmetric version,
therefore we need superpartners of currents $(J^{3},J^{\pm},j^{\pm})$. While
we introduce fermions $(\psi^{3},\psi^{\pm})$ with spin $1/2$ for the bosonic
currents $(J^{3},J^{\pm})$, we include bosons $\varphi^{\pm}$ with spin $1/2$
for the fermionic currents $j^{\pm}$. We assume the OPEs of these fields as
$\displaystyle\psi^{+}(z)\psi^{-}(0)\sim\frac{1}{z},\qquad\psi^{3}(z)\psi^{3}(0)\sim\frac{1}{z},\qquad\varphi^{+}(z)\varphi^{-}(0)\sim-\frac{1}{z}.$
(2.13)
Notice that the extra bosons $\varphi^{\pm}$ satisfy wrong spin-statistic
relation. With these new fields we can define ${\cal N}=1$ supercurrents as
$\displaystyle\hat{J}^{\pm}=J^{\pm}+\sqrt{2}\psi^{\pm}\psi^{3}+\frac{1}{2}\varphi^{\pm}\varphi^{\pm},\qquad\hat{j}^{\pm}=j^{\pm}\pm
i\sqrt{2}\psi^{\pm}\varphi^{\mp}\pm i\psi^{3}\varphi^{\pm},$ (2.14)
along with
$\displaystyle\hat{J}^{3}=J^{3}+\psi^{+}\psi^{-}+\frac{1}{2}\varphi^{+}\varphi^{-}.$
(2.15)
We construct the coset model by using the last current as well as removing one
of the fermions $\psi^{3}$ to preserve ${\cal N}=1$ world-sheet supersymmetry.
We can show that the coset model actually has enhanced ${\cal N}=2$
supersymmetry as Kazama-Suzuki models for bosonic cosets [17, 18]. We find
that the generators of ${\cal N}=2$ superconformal symmetry are
$\displaystyle
J_{R}=-\frac{1}{2k-3}\left(2J^{3}+(2k-1)\psi^{+}\psi^{-}+(2k-2)\varphi^{+}\varphi^{-}\right),$
(2.16) $\displaystyle
G^{\pm}=\frac{1}{\sqrt{2k-3}}\left(2J^{\pm}\psi^{\mp}\pm\sqrt{2}j^{\pm}\varphi^{\mp}+(\varphi^{\mp})^{2}\psi^{\pm}\right),$
$\displaystyle
T=\frac{1}{2k-3}\Bigl{[}J^{+}J^{-}+J^{-}J^{+}+\frac{1}{2}(j^{-}j^{+}-j^{+}j^{-})+4J^{3}\psi^{+}\psi^{-}+2J^{3}\varphi^{+}\varphi^{-}+2\psi^{+}\psi^{-}\varphi^{+}\varphi^{-}$
$\displaystyle\qquad-\frac{2k+1}{2}(\psi^{+}\partial\psi^{-}+\psi^{-}\partial\psi^{+})-(k-1)(\varphi^{+}\partial\varphi^{-}-\varphi^{-}\partial\varphi^{+})+\frac{1}{2}(\varphi^{+})^{2}(\varphi^{-})^{2}\Bigr{]}.$
In fact, we can compute the OPEs of generators as
$\displaystyle
T(z)T(0)\sim\frac{c/2}{z^{4}}+\frac{2T(0)}{z^{2}}+\frac{\partial
T(0)}{z}~{},\qquad
T(z)G^{\pm}(0)\sim\frac{\frac{3}{2}G^{\pm}(0)}{z^{2}}+\frac{\partial
G^{\pm}(0)}{z},$ (2.17) $\displaystyle
T(z)J_{R}(0)\sim\frac{J_{R}(0)}{z^{2}}+\frac{\partial J_{R}(0)}{z},\qquad
J_{R}(z)G^{\pm}(0)\sim\pm\frac{G^{\pm}(0)}{z},\qquad
J_{R}(z)J_{R}(0)\sim\frac{c/3}{z^{2}},$ $\displaystyle
G^{\pm}(z)G^{\mp}(0)\sim\frac{\frac{2}{3}c}{z^{3}}\pm\frac{2J_{R}(0)}{z^{2}}+\frac{2T(0)}{z}\pm\frac{\partial
J_{R}(0)}{z},\qquad G^{\pm}(z)G^{\pm}(0)\sim 0.$
In this way we have explicitly shown that these generators satisfy the ${\cal
N}=2$ superconformal algebra with central charge $\hat{c}=c/3=1/(2k-3)$.
### 2.3 Topological Twisting
To realize the U(1) quotient more explicitly, we combine a free scalar field
$X$ with the ${\cal N}=1$ OSP current algebra and finally take a quotient by
the complexified U(1) (i.e. the complex plane ${\mathbb{C}}$). This quotient
can be done by taking BRST invariant state about the BRST operator
$\displaystyle Q_{B}=\int dzC(z)J_{g}(z),$ (2.18)
where we introduced fermionic ghosts $(B,C)$ with the conformal weights
$(1,0)$. Here the BRST current $J_{g}$ is defined by
$\displaystyle J_{g}=\hat{J}^{3}-\frac{i}{2b}\partial X,$ (2.19)
and it is easy to see that $J_{g}(z)J_{g}(0)\sim 0$ which guarantees
$Q_{B}^{2}=0$.
Notice that in this formalism we can always set $J_{g}(z)$ to zero since it is
gauged. Using this fact we can use the following expression of the R-current
of ${\cal N}=2$ superconformal algebra as
$\displaystyle
J^{\prime}_{R}=J_{R}-\frac{4k-8}{2k-3}J_{g}=-2J^{3}-3\psi^{+}\psi^{-}-2\varphi^{+}\varphi^{-}+i\frac{2k-4}{\sqrt{2k-3}}\partial
X.$ (2.20)
In the anti-holomorphic part, we use the same expression with bars.555 This
choice means that we gauge the vector $U(1)$ current instead of the axial
$U(1)$ current. The former may produce a trumpet like geometry and the latter
a black hole like geometry [32]. We choose the vector gauge just for the
simplicity of expression. We will find that this form of R-current is useful
to construct the topological model as in the bosonic case [14, 7].
Now, we perform topological twists [34, 35] by using the expression (2.20) of
R-current. Namely, we redefine the energy momentum tensor by
$T^{top}=T+\frac{1}{2}\partial J^{\prime}_{R}$ and
$\bar{T}^{top}=\bar{T}+\frac{1}{2}\bar{\partial}\bar{J}^{\prime}_{R}$.
Employing the free field representation (2.3), we then find the following maps
of fields. First of all, the background charge of the field $\phi$ is shifted
from $Q_{\phi}=b$ to $Q_{\phi}=b+1/b$. After the twist, the field $\phi$
corresponds to the Liouville field. Recall that the central charge is written
as $c=1+3Q^{2}$ in terms of background charge $Q$. Next, the field $X$ would
have background charge $Q_{X}=i(1/b-b)$ after the twist, and this field
becomes the bosonic part of the $\hat{c}\leq 1$ matter. The conformal weights
of fermions $(\theta,p)$ are shifted from $(0,1)$ to $(1/2,1/2)$ and they
become superpartners of the above bosonic fields. The other fields
$\psi^{\pm}$ and $\varphi^{\pm}$ are mapped to the superghosts $(b,c)$ and
$(\beta^{\prime},\gamma^{\prime})$ of type 0 superstring theory.666Here we use
the notation $(\beta^{\prime},\gamma^{\prime})$ to represent superghosts of
superstring in order to distinguish them from the ones in (2.3). In table 1
the changes of conformal weights are summarized. In the end we expect that
$(\beta,\gamma)$ would be canceled out with $(B,C)$ as in the bosonic string
case [14]. In this way we obtain the same field contents as the world-sheet
theory of the type 0 $\hat{c}\leq 1$ string including ghosts. More detailed
explanation of the type 0 string will be given in subsection 4.1.
| Before twisting | After twisting
---|---|---
| Central charge | Conformal weights | Central charge | Conformal weights
$(\theta,p)$ | $-2$ | $(0,1)$ | $1$ | $(1/2,1/2)$
$(\psi^{+},\psi^{-})$ | $1$ | $(1/2,1/2)$ | $-26$ | $(2,-1)$
$(\varphi^{+},\varphi^{-})$ | $-1$ | $(1/2,1/2)$ | $11$ | $(3/2,-1/2)$
$(\beta,\gamma)$ | $2$ | $(1,0)$ | $2$ | $(0,1)$
$(B,C)$ | $-2$ | $(1,0)$ | $-2$ | $(1,0)$
Table 1: Changes of central charges and conformal weights after topological
twisting.
### 2.4 Chiral Primaries
In the previous subsection we have shown that free fields in the ${\cal N}=2$
coset are mapped to the matter contents of the $\hat{c}\leq 1$ superstring
theory after the topological twist. In fact we can identify physical operators
of the topological model with those of the $\hat{c}\leq 1$ superstring, which
is the subject of this subsection. In order to define the coset model we have
introduced two spin $1/2$ fermions $\psi^{\pm}$ and two spin $1/2$ bosons
$\varphi^{\pm}$. With the bosonization formula, they can be written as
$\displaystyle\psi^{+}=e^{iH},\qquad\psi^{-}=e^{-iH},\qquad\varphi^{+}=e^{-\chi}\partial\xi,\qquad\varphi^{-}=e^{\chi}\eta,$
(2.21)
where $H,\chi$ are free bosons without background charges and free fermions
$\xi,\eta$ are with $\Delta_{\xi}=0,\Delta_{\eta}=1$. The non-trivial OPEs are
given as
$\displaystyle H(z)H(0)\sim-\ln z,\qquad\chi(z)\chi(0)\sim-\ln
z,\qquad\eta(z)\xi(0)\sim\frac{1}{z}.$ (2.22)
These bosonized expressions of fermions are useful to define vertex operators.
Since the operators of the coset model must be invariant under the BRST charge
(2.18), they should take the form
$\displaystyle e^{irH+u\chi}\Phi_{j,m}^{s}e^{2ib(m+r-\frac{u}{2})X},$ (2.23)
whose conformal weight is
$\displaystyle\Delta=-2b^{2}j(j+\tfrac{1}{2})+\frac{s(s-1)}{2}+\frac{r^{2}}{2}-\frac{u^{2}}{2}+2b^{2}\left(m+r-\frac{u}{2}\right)^{2}.$
(2.24)
Here we have used $\Phi_{j,m}^{s}$ as the vertex operator of OSP(1$|$2) WZNW
model as defined in (2.8).
Physical operators of the topological model can be constructed from chiral
primaries of the ${\cal N}=2$ coset model. Here we review how to perform the
topological twist to the chiral primaries by following [14, 7]. First we find
chiral primary states of the coset in NS-sector,777Notice that there are three
different spin structures that appear in this paper. One is for the OSP(1$|$2)
current algebra, which is defined such that an an integer $s$ in (2.8) means
the NS-sector, while a half integer $s$ implies R-sector. The second spin
structure is the ordinary one for the $N=2$ superconformal field theory. The
third one is for the $\hat{c}\leq 1$ superstring. In this section the notion
of NS,R is with respect to the second spin structure. which satisfy
$\displaystyle G^{+}_{r-1/2}|{\rm NS}\rangle=G^{-}_{r+1/2}|{\rm NS}\rangle=0$
(2.25)
for $r=0,1,\cdots$. Among the vertex operators of the form (2.23), there are
chiral primary operators ${\cal O}^{NS,s}_{j}$ corresponding to the above
chiral primary states. These chiral primaries can be mapped to R-ground states
by spectral flow operation. Redefining the U(1)R current (2.16) as
$\displaystyle
J_{R}=-2b^{2}\hat{J}^{3}-\psi^{+}\psi^{-}-\varphi^{+}\varphi^{-}=-ib\partial
X_{R},$ (2.26)
with $X_{R}(z)X_{R}(0)\sim\ln z$, the R-ground states are obtained by ${\cal
O}^{R,s}_{j}=e^{\frac{i}{2}bX_{R}}{\cal O}^{NS,s}_{j}$. Finally, the elements
of cohomology for the topological theory are obtained by the topological twist
as ${\cal O}^{s=1}_{j}=e^{-\frac{i}{2}\sqrt{k}bX^{\prime}_{R}}{\cal
O}^{R,s}_{j}$. Here we define $X^{\prime}_{R}$ as
$\displaystyle
J_{R}^{\prime}=-2J^{3}-3\psi^{+}\psi^{-}-2\varphi^{+}\varphi^{-}+2ib(k-2)\partial
X=:-i\sqrt{k}b\partial X_{R}^{\prime}$ (2.27)
from the expression of R-current (2.20).
Among the chiral primaries of the ${\cal N}=2$ coset model, we focus on the
following two types of operators in NS-sector;
$\displaystyle{\cal
O}^{NS,\frac{1}{2}}_{j}=e^{-\frac{1}{2}\chi}\Phi^{\frac{1}{2}}_{j,j-\frac{1}{4}}e^{2ibjX},\qquad{\cal
O}^{NS,1}_{j}=\Phi^{1}_{j,j}e^{2ibjX},$ (2.28)
which satisfy $\Delta=q_{R}/2=-jb^{2}-1/4$ and $\Delta=q_{R}/2=-jb^{2}$,
respectively. The R-sector ground states are constructed as
$\displaystyle{\cal
O}^{R,\frac{1}{2}}_{j}=e^{\frac{i}{2}H}\Phi^{\frac{1}{2}}_{j,j-\frac{1}{4}}e^{2ib(j+\frac{1}{4})X},\qquad{\cal
O}^{R,1}_{j}=e^{\frac{i}{2}H+\frac{1}{2}\chi}\Phi^{1}_{j,j}e^{2ib(j+\frac{1}{4})X}.$
(2.29)
After the topological twist we finally obtain
$\displaystyle{\cal
O}^{\frac{1}{2}}_{j}=e^{-iH-\chi}\Phi^{\frac{1}{2},w=1}_{j,j-\frac{1}{4}}e^{2ib(j+\frac{1}{4b^{2}})X},\qquad{\cal
O}^{1}_{j}=e^{-iH-\frac{1}{2}\chi}\Phi^{1,w=1}_{j,j}e^{2ib(j+\frac{1}{4b^{2}})X}.$
(2.30)
Notice that vertex operators are spectrally flowed in the sense of OSP(1$|$2)
WZNW model as in (2.12) during the procedure of topological twist. Under the
spectral flow action we may identify
$\Phi_{j,j+\frac{s-1}{2}}^{s,w=1}=\Phi_{-j-\frac{k}{2}+\frac{1}{4},j+\frac{s-1}{2}+\frac{k}{2}}^{s-\frac{1}{2}}$.
Combining with the free field representation of vertex operators (2.8), we
find
$\displaystyle{\cal O}^{\frac{1}{2}}_{j}\sim
ce^{-\chi}e^{2ib(j+\frac{1}{4b^{2}})X+2b(j+\frac{k}{2}-\frac{1}{4})\phi},\qquad{\cal
O}^{1}_{j}=ce^{-\frac{1}{2}\chi}e^{\frac{i}{2}Y+2ib(j+\frac{1}{4b^{2}})X+2b(j+\frac{k}{2}-\frac{1}{4})\phi}.$
(2.31)
In the above, we renamed $c=\exp(-iH)$ as suggested by the previous
discussion. Moreover the $\beta$-ghost in the superstring should be written as
$\beta^{\prime}=\partial\xi\exp(-\chi)$. Therefore, we can say that these
operators have one $c$-ghost and picture $-1$.888The operators corresponding
to those in the other picture may be obtained by the action of operator
similar to the picture changing operator.
Notice that the above two operators (2.31) indeed coincide with the tachyon
and RR field vertex operators in the two dimensional type 0 superstring,
respectively (see subsection 4.1). Actually they complete the list of physical
operators since there are no massive stringy modes in two dimensional
superstring. This fact may be seen by taking the light-cone gauge. In this
way, we have learned that the physical states (chiral primary states) in the
topological string on OSP(1$|$2)/U(1) are mapped into the physical states in
the two dimensional type 0 string. We will study the relation between these
two theories in more detail below.
## 3 OSP(1$|$2)/U(1) Coset from ${\cal N}=1$ Super Liouville
In references [22, 23] it was shown that arbitrary correlation functions of
primary fields in SL(2) WZNW model can be written in terms of correlation
functions of Liouville field theory. This property may be useful to show the
equivalence between the scattering amplitudes in $c\leq 1$ bosonic string and
the topological string on $SL(2)/U(1)$. The agreement for three-point
functions between them has been shown in [7], and this is generalized by [36]
to arbitrary tree level amplitudes by utilizing the generalized relation of
[37]. Recently it was shown in [21] that correlation functions of OSP(1$|$2)
WZNW model can be written in terms of those of ${\cal N}=1$ super Liouville
field theory. Later we would like to show the equivalence between ${\cal N}=2$
coset model of OSP(1$|$2)/U(1) and the $\hat{c}\leq 1$ superstring in the
level of amplitudes. For the purpose we generalize the relation such as to
include RR-sectors of fermions and spectrally flowed sectors of OSP(1$|$2)
model. In this section we derive the generalized relation in the path integral
formulation following [23, 21].
### 3.1 OSP(1$|$2) WZNW Model
Let us start from the action of OSP(1$|$2) WZNW model. In terms of free fields
the action may be written as999 A derivation of this action can be found in
[21]. This action hare is a bit different from the one in [21], but it is easy
to see the equivalence between the two expressions.
$\displaystyle S^{\text{WZNW}}(g)=\frac{1}{2\pi}\int
d^{2}z\Bigl{[}\frac{1}{2}\partial\phi\bar{\partial}\phi+\frac{b}{8}\sqrt{g}{\cal
R}\phi+\beta\bar{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}+p\bar{\partial}\theta+\bar{p}\partial\bar{\theta}$
(3.1)
$\displaystyle-\frac{1}{k}\beta\bar{\beta}e^{2b\phi}-\frac{1}{2k}(p+\beta\theta)(\bar{p}+\bar{\beta}\bar{\theta})e^{b\phi}\Bigr{]},$
where $\phi,\gamma,\bar{\gamma},\theta,\bar{\theta}$ are related to the
parameters of elements $g\in\,$ OSP(1$|$2) and
$\beta,\bar{\beta},\theta,\bar{\theta}$ are conjugate variables. The
generators of current algebra symmetry are written as in (2.3) in these
variables. Here we use the form of vertex operator as
$\displaystyle
V_{j}^{s,\bar{s}}(\mu|z)=\mu^{j+\frac{1}{2}+\frac{s}{2}}\bar{\mu}^{j+\frac{1}{2}+\frac{\bar{s}}{2}}e^{isY+i\bar{s}\bar{Y}}e^{\mu\gamma-\bar{\mu}\bar{\gamma}}e^{2b(j+\frac{1}{2})\phi}~{}.$
(3.2)
For the NSNS-sector with $s,\bar{s}=0,1$ these vertex operators are the same
as in [21]. The conformal weights are given as $\Delta=-2b^{2}j(j+1/2)$. The
vertex operators in the RR-sector are given by spin fields with
$s=\bar{s}=1/2$, and the conformal weights are $\Delta=-2b^{2}j(j+1/2)+1/8$.
The above expression in so-called $\mu$-basis is useful for our purpose, and
it can be mapped to the $m$-basis expression given in (2.8) by101010In order
to compare with the previous notation, we may need to perform a flip
$j\to-j-1/2$. Moreover, it might be natural to multiply the factor
$N^{s.\bar{s}}_{j,m,\bar{m}}=\frac{\Gamma(-j+1/2-s/2+m)}{\Gamma(j+1/2+\bar{s}/2-\bar{m})}$
as, e.g., in [22]; see also [38]. Here we remove it since it may diverge in
our case.
$\displaystyle\Phi^{s,\bar{s}}_{j,m,\bar{m}}=\int\frac{d^{2}\mu}{|\mu|^{2}}\mu^{-m}\bar{\mu}^{-\bar{m}}V^{s,\bar{s}}_{j}(\mu|z).$
(3.3)
In some sense, the $\mu$-basis expression can be thought of generating
function of the $m$-basis expression.
Since the operators of topological model in (2.30) are written in terms of
OSP(1$|$2) vertex operators with spectral flow index $w=1$, it is important to
understand the symmetry under the spectral flow. The spectral flow action
$\rho^{w}$ can be defined as
$\displaystyle\rho^{w}(J_{n}^{3})=J^{3}_{n}-\frac{k}{2}w\delta_{n,0},\qquad\rho^{w}(J_{n}^{\pm})=J^{\pm}_{n\pm
w},\qquad\rho^{w}(j_{r}^{\pm})=j^{\pm}_{r\pm\frac{w}{2}},$ (3.4)
where the mode expansions are $J^{A}(z)=\sum_{n}J^{A}_{n}z^{-n-1}$ with
$A=\pm,3$ and $j^{\pm}(z)=\sum_{r}j^{\pm}_{r}z^{-r-1}$. We can easily see that
the new currents satisfy the same (anti-)commutation relations as before,
which implies that the spectral flow is the symmetry of the current algebra
OSP(1$|$2). The vacuum state is defined such as
$\displaystyle\rho^{w}(J^{A}_{n})|w\rangle=0,\qquad\rho^{w}(j^{\pm}_{r})|w\rangle=0$
(3.5)
for $n,r\geq 0$. In terms of free fields, the vacuum state
$|w\rangle=|w\rangle_{(\beta,\gamma)}\otimes|w\rangle_{\phi}\otimes|w\rangle_{Y}$
is characterized as
$\displaystyle\beta_{n-w}|w\rangle_{(\beta,\gamma)}=0,\qquad\gamma_{n+w}|w\rangle_{(\beta,\gamma)}=0$
(3.6)
for $n\geq 0$, and moreover
$\displaystyle|w\rangle_{\phi}=e^{\frac{w}{2b}\phi}|0\rangle_{\phi},\qquad|w\rangle_{Y}=e^{-\frac{iw}{2}Y}|0\rangle_{Y}.$
(3.7)
In the following we assume $w\geq 0$ and denote $v^{w}(0)$ as the operator
corresponding to the state $|w\rangle$.
As discussed in [37, 39], generic $N$-point functions of operators with
spectral flow can be reduced to $N$-point functions of (3.2) with the
inversion of $v^{w}(\xi)$. We choose the position of insertion $v^{w}(\xi)$ as
$\xi=0$ since it does not affect the following discussion. In the path
integral formulation they are given as
$\displaystyle\left\langle\prod_{\nu=1}^{N}V^{s_{\nu},\bar{s}_{\nu}}_{j_{\nu}}(\mu_{\nu}|z_{\nu})v^{w}(0)\right\rangle=\int_{(w)}{\cal
D}\phi{\cal D}^{2}\beta{\cal D}^{2}\gamma{\cal D}^{2}\theta{\cal
D}^{2}pe^{-S^{\text{WZNW}}(g)}\times$ (3.8)
$\displaystyle\times\prod_{\nu=1}^{N}V^{s_{\nu},\bar{s}_{\nu}}(\mu_{\nu}|z_{\nu})e^{w(\phi(0)/2b-iY(0)/2)}.$
The effects of insertion $v^{w}(0)$ appears in the right hand side in two
ways. One is the extra insertion of vertex operator
$e^{w(\phi(0)/2b-iY(0)/2)}$, and the other is the restriction to the
integration domain of $\beta,\bar{\beta}$ such that $\beta,\bar{\beta}$ have a
zero of order $w$ at $\xi=0$. For more detail see [39].
### 3.2 OSP(1$|$2)–Super Liouville Correspondence
Now that we have the OSP(1$|$2) WZNW model, we can derive the relation between
the correlation functions of OSP(1$|$2) WZNW model and those of ${\cal N}=1$
super Liouville field theory by following the analysis of [21]. For this
purpose we first integrate $\beta,\gamma$ as in [21]. Integrations over
$\gamma$ and $\bar{\gamma}$ lead to delta functionals for $\beta$ and
$\bar{\beta}$, which replace the fields $\beta,\bar{\beta}$ by
$\displaystyle\beta(x)=\sum_{\nu=1}^{N}\frac{\mu_{\nu}}{x-z_{\nu}}=u\frac{x^{w}\prod_{i=1}^{N-2-w}(x-y_{i})}{\prod_{\nu=1}^{N}(x-z_{\nu})}=:u{\cal
B}(x).$ (3.9)
The insertion of $v^{w}(0)$ forces $\beta(x)$ to have a zero of order $w$ at
$x=0$ and this requirement gives constraints
$\displaystyle\sum_{\nu=1}^{N}\mu_{\nu}z_{\nu}^{-n}=0$ (3.10)
for $n=0,1,\cdots,w$. Since a 1-form with $N$ poles on a sphere has $N-2$
zero’s, $\beta$ can be represented as in the right hand side by the positions
of $N-2-w$ more zero’s $y_{i}$. In other words, the parameters $y_{i}$ are
defined by the equation (3.9), and the new parameters are essential to relate
the model to super Liouville theory as seen below. Moreover, we can see that
the number of spectral flow $w$ is restricted as $w\leq N-2$.
After the integration over $\beta,\gamma$, the action becomes something
similar to super Liouville theory, but the coefficients include functions
${\cal B}(z),\bar{\cal B}(\bar{z})$. Following [21, 39] these can be removed
by the redefinition of fields as
$\displaystyle\phi^{\prime}:=\phi+\frac{1}{2b}\ln|u{\cal B}|^{2},\qquad
Y^{\prime}:=Y-\frac{i}{2}\ln|u{\cal B}|^{2}.$ (3.11)
Moreover, after some manipulations we find the relation
$\displaystyle\left\langle\prod_{\nu=1}^{N}V^{s_{\nu},\bar{s}_{\nu}}_{j_{\nu}}(\mu_{\nu}|z_{\nu})v^{w}(0)\right\rangle$
$\displaystyle\qquad=\prod_{n=0}^{w}\delta^{2}(\sum_{\nu}\mu_{\nu}z^{-n})|u|^{2-\frac{w}{2b^{2}}+\frac{w}{2}}|\Theta^{w}_{N}|^{2}\left\langle\prod_{\nu=1}^{N}V^{s_{\nu}-\frac{1}{2},\bar{s}_{\nu}-\frac{1}{2}}_{\alpha_{\nu}}(z_{\nu})\prod_{j=1}^{N-2-w}V^{\frac{1}{2},\frac{1}{2}}_{-\frac{1}{2b}}(y_{j})\right\rangle$
(3.12)
with $\alpha_{\nu}=2b(j_{\nu}+1/2)+1/2b$. The right hand side is computed by
the sum of ${\cal N}=1$ super Liouville theory
$(\phi^{\prime},\psi,\bar{\psi})$ and massless fermions
$(\psi_{X},\bar{\psi}_{X})$
$\displaystyle S[\phi^{\prime},\psi,\psi_{X}]$ $\displaystyle=\
\frac{1}{4\pi}\int
d^{2}z\,\Bigl{[}\,\partial\phi^{\prime}\bar{\partial}\phi^{\prime}+\frac{Q_{\phi^{\prime}}}{4}\sqrt{g}R\phi^{\prime}+\frac{2}{k}e^{2b\phi^{\prime}}\,+$
$\displaystyle\hskip
56.9055pt+\psi\bar{\partial}\psi+\bar{\psi}\partial\bar{\psi}+\psi_{X}\bar{\partial}\psi_{X}+\bar{\psi}_{X}\partial\bar{\psi}_{X}-\frac{2}{k}\psi\bar{\psi}e^{b\phi^{\prime}}\,\Bigr{]}$
(3.13)
with $Q_{\phi^{\prime}}=b+1/b$. The fermions are defined by
$\displaystyle\psi\pm i\psi_{X}=\sqrt{2}e^{\pm
iY^{\prime}},\qquad\bar{\psi}\pm i\bar{\psi}_{X}=\sqrt{2}e^{\pm
i\bar{Y}^{\prime}},$ (3.14)
and the vertex operators are
$\displaystyle
V^{s,\bar{s}}_{\alpha}(z)=e^{isY+is\bar{Y}}e^{\alpha\phi^{\prime}}$ (3.15)
with conformal weights $\Delta=\alpha(Q_{\phi^{\prime}}-\alpha)/2+s^{2}/2$.
The twist factor is
$\displaystyle\Theta^{w}_{N}=\prod_{\mu<\nu}^{N}(z_{\mu}-z_{\nu})^{\frac{1}{4b^{2}}-\frac{1}{4}}\prod_{i<j}^{N-2-w}(y_{i}-y_{j})^{\frac{1}{4b^{2}}-\frac{1}{4}}\prod_{\nu=1}^{N}\prod_{i=1}^{N-2-w}(z_{\nu}-y_{j})^{-\frac{1}{4b^{2}}+\frac{1}{4}}.$
(3.16)
Here we should notice that the operators in the NSNS(RR)-sector are mapped to
those in the RR(NSNS)-sector. Moreover, if the winding number is violated
maximally as $w=N-2$, then there is no extra insertion of operator at
$z=y_{i}$.
### 3.3 Amplitudes of OSP(1$|$2)/U(1) Coset Model
Utilizing the fundamental relation (3.12), we can rewrite correlation
functions of OSP(1$|$2)/U(1) coset in terms of ${\cal N}-1$ super Liouville
theory with a supersymmetric pair of free boson and fermion in a manner
similar to the bosonic case [39]. Here the vertex operators of coset model are
defined as in (2.11)111111Only in this subsection we construct the coset model
by gauging the axial U(1) symmetry in order to use the trick of [39].
$\displaystyle\Psi^{s,\bar{s}}_{j,m,\bar{m}}(z,\bar{z})=V^{X^{3}}_{m,\bar{m}}(z,\bar{z})\Phi^{s,\bar{s}}_{j,m,\bar{m}}(z,\bar{z})$
(3.17)
with
$\displaystyle
V^{X^{3}}_{m,\bar{m}}(z,\bar{z})=e^{i\sqrt{\frac{2}{k}}(-mX^{3}+\bar{m}\bar{X}^{3})}.$
(3.18)
Moreover, the vertex operators of OSP(1$|$2) model with spectral flow index
$w$ are related to vertex operators of OSP(1$|$2)/U(1) model as (see (2.12))
$\displaystyle\Psi^{s,\bar{s}}_{j,m,\bar{m}}(z,\bar{z})=V^{X^{3}}_{m+\frac{kw}{2},\bar{m}+\frac{kw}{2}}(z,\bar{z})\Phi^{s,\bar{s},w}_{j,m,\bar{m}}(z,\bar{z}).$
(3.19)
Therefore we can also obtain a formula for correlation functions of OSP(1$|$2)
model with non-trivial spectral flow actions. In particular, we will be
interested in a specific $N$-point function in OSP(1$|$2) WZNW model as
$\displaystyle{\cal
M}=\left\langle\Phi^{s_{1},\bar{s}_{1}}_{j_{1},m_{1},\bar{m}_{1}}(z_{1})\Phi^{s_{2},\bar{s}_{2}}_{j_{2},m_{2},\bar{m}_{2}}(z_{2})\prod_{\nu=3}^{N}\Phi^{s_{\nu},\bar{s}_{\nu},w_{\nu}=1}_{j_{\nu},m_{\nu},\bar{m}_{\nu}}(z_{\nu})\right\rangle.$
(3.20)
Since the amplitude has $N-2$ number of winding violation, it should be
written in terms of $N$-point function of ${\cal N}=1$ super Liouville theory.
Let us first study $N$-point function of OSP(1$|$2)/U(1) coset model. As
before we introduce a new field $\hat{X}^{3}$ by
$\displaystyle\hat{X}^{3}_{L}:=X^{3}_{L}-i\sqrt{\frac{k}{2}}\ln(u{\cal B}),$
(3.21)
and the right mover defined by its complex conjugate. By closely following
[23] and utilizing the formula (3.12), we finally obtain
$\displaystyle\left\langle\prod_{\nu=1}^{N}\Psi^{s_{\nu},\bar{s}_{\nu}}_{j_{\nu},m_{\nu},\bar{m}_{\nu}}\right\rangle=\int\frac{\prod_{i=1}^{N-2-w}d^{2}y_{i}}{(N-2-w)!}\times$
(3.22)
$\displaystyle\qquad\qquad\times\left\langle\prod_{\nu=1}^{N}V^{\hat{X}^{3}}_{m_{\nu}+\frac{k}{2},\bar{m}_{\nu}+\frac{k}{2}}(z_{\nu})V^{s_{\nu}-\frac{1}{2},\bar{s}_{\nu}-\frac{1}{2}}_{\alpha_{\nu}}(z_{\nu})\prod_{i=1}^{N-2-w}V^{\hat{X}^{3}}_{-\frac{k}{2},-\frac{k}{2}}(y_{i})V^{\frac{1}{2},\frac{1}{2}}_{-\frac{1}{2b}}(y_{i})\right\rangle.$
The label $w$ is related to the winding number violation as
$\sum_{\nu}m_{\nu}=\sum_{\nu}\bar{m}_{\nu}=-\frac{kw}{2}$. The right hand side
should be computed by the theory with the action
$S[\phi^{\prime},\psi,\psi_{X}]$ for ${\cal N}=1$ super Liouville theory and a
free fermion $(\psi_{X},\bar{\psi}_{X})$ and a free boson $\hat{X}^{3}$ with
background charge $Q=-i\sqrt{k}$ for its dual field.
With the formula (3.22) and (3.19) we can write down generic correlation
functions in the OSP(1$|$2) model with spectral flow action considered in
terms of super Liouville theory. Here we only compute the amplitude (3.20)
since it is the case used in the later analysis. With the formula (3.19) we
can relate the amplitude (3.20) to a $N$-point function of the coset as
$\displaystyle\left\langle\prod_{\nu=1}^{N}\Psi^{s_{\nu},\bar{s}_{\nu}}_{j_{\nu},m_{\nu},\bar{m}_{\nu}}\right\rangle$
$\displaystyle={\cal M}\times\left\langle
V^{X^{3}}_{m_{1},\bar{m}_{1}}(z_{1})V^{X^{3}}_{m_{2},\bar{m}_{2}}(z_{2})\prod_{\nu=3}^{N}V^{X^{3}}_{m_{\nu}+\frac{k}{2},\bar{m}_{\nu}+\frac{k}{2}}(z_{\nu})\right\rangle.$
(3.23)
Then, by combining with the formula (3.22), we find
$\displaystyle{\cal
M}=|\Theta_{s}(z_{\nu})|^{2}\left\langle\prod_{\nu=1}^{N}V^{s_{\nu}-\frac{1}{2},\bar{s}_{\nu}-\frac{1}{2}}_{\alpha_{\nu}}(z_{\nu})\right\rangle,$
(3.24)
where the right hand side is computed by the ${\cal N}=1$ super Liouville
theory and a free fermion with the action (3.13). The coefficient is given by
$\displaystyle\Theta_{s}(z_{\nu})=(z_{1}-z_{2})^{\frac{k}{2}+m_{1}+m_{2}}\prod_{\nu=3}^{N}[(z_{1}-z_{\nu})(z_{2}-z_{\nu})]^{\frac{k}{2}+m_{\nu}}~{},$
(3.25)
and bared expression for $\bar{\Theta}_{s}(\bar{z}_{\nu})$. This formula will
be important to relate amplitudes of the topological model and the
$\hat{c}\leq 1$ superstring theory.
## 4 Correspondence to $\hat{c}\leq 1$ Superstring Theory
In section 2, we have studied the topological model based on the ${\cal N}=2$
supersymmetric coset OSP(1$|$2)/U(1). In particular, we have observed that
free fields and chiral primaries of the coset model are mapped to matter
contents and physical operators in the $\hat{c}\leq 1$ superstring theory. In
this section, we establish the relation in more detail. After briefly
reviewing the $\hat{c}\leq 1$ superstring theory and the method to compute
amplitudes in topological models, we compare the amplitudes of both theories.
### 4.1 $\hat{c}\leq 1$ Superstring Theory
In section 2 we have already encountered the $\hat{c}\leq 1$ superstring
theory during constructing the topological model of the coset OSP(1$|$2)/U(1).
In this subsection we define the $\hat{c}\leq 1$ superstring theory in a more
precise way.121212Notice that we are setting $\alpha^{\prime}=2$ in this
paper. The matter part consists of a linear dilaton $X$ with background charge
$Q_{X}=i(1/b-b)$ and a free fermion $\psi_{X}$. The action of these fields is
given by
$\displaystyle S_{X}=\frac{1}{4\pi}\int d^{2}z\left[\partial
X\bar{\partial}X+\frac{Q_{X}}{4}\sqrt{g}{\cal
R}X+\psi_{X}\bar{\partial}\psi_{X}+\bar{\psi}_{X}\partial\bar{\psi}_{X}\right].$
(4.1)
The theory also includes the ${\cal N}=1$ super Liouville theory, whose action
is
$\displaystyle S=\frac{1}{4\pi}\int
d^{2}z\left[\partial\phi\bar{\partial}\phi+\frac{Q_{\phi}}{4}\sqrt{g}{\cal
R}\phi+\psi\bar{\partial}\psi+\bar{\psi}\partial\bar{\psi}+\mu_{L}\psi\bar{\psi}e^{b\phi}\right]$
(4.2)
with $Q_{\phi}=b+1/b$. The total central charge is now
$c=3/2+3Q_{X}^{2}+3/2+3Q_{\phi}^{2}=15$, and hence we can construct a critical
superstring theory by coupling the world-sheet superghosts $(b,c)$ and
$(\beta^{\prime},\gamma^{\prime})$.
Primary operators of this theory may take the form $\exp(\alpha X+\beta\phi)$,
which has the conformal weight
$\Delta=\alpha(Q_{X}-\alpha)/2+\beta(Q_{\phi}-\beta)/2$. Following the
standard method to construct BRST invariant operators, we can find out
physical operators in the $\hat{c}\leq 1$ superstring theory. The tachyon
vertex operator is given as
$\displaystyle c\bar{c}{\cal
T}^{(-1)}_{p}=c\bar{c}e^{-(\chi+\bar{\chi})}e^{ik_{X}(X+\bar{X})+k_{\phi}^{\pm}\phi},$
(4.3)
where the momenta run over $ik_{X}=Q_{X}/2+ip$ and
$k_{\phi}^{\pm}=Q_{\phi}/2\pm p$ with $p\in\mathbb{R}$. We bosonize the
superghost $\beta^{\prime},\gamma^{\prime}$ like in (2.21), which yields the
new fields $\chi$. In other words, the above expression is in $(-1,-1)$
picture; and in $(0,0)$ picture it is written as
$\displaystyle c\bar{c}{\cal
T}^{(0)}_{p}=c\bar{c}(ik_{X}\psi_{X}+k_{\phi}^{\pm}\psi)(ik_{X}\bar{\psi}_{X}+k_{\phi}^{\pm}\bar{\psi})e^{ik_{X}(X+\bar{X})+k_{\phi}^{\pm}\phi}.$
(4.4)
There are other physical operators in the RR-sector. The Ramond vertex
operator in $(-1/2,-1/2)$ picture is written as
$\displaystyle c\bar{c}{\cal
R}^{(-1/2)}_{p}=c\bar{c}e^{-\frac{1}{2}(\chi+\bar{\chi})}e^{\pm\frac{i}{2}(Y+\bar{Y})+ik_{X}(X_{L}+X_{R})+k^{\pm}_{\phi}\phi}.$
(4.5)
Indeed, these vertex operators (4.3) and (4.5) are the same as those obtained
from the topological string on OSP(1$|$2)/U(1) as observed in (2.31).
If $X$ direction is compactified with radius $R$, then the momentum takes
discrete values $p=n/R$ with $n\in{\mathbb{Z}}$. For winding modes we should
replace $X_{R}\to-X_{R}$, $\bar{H}\to-\bar{H}$ and $p=wR/2$ with
$w\in{\mathbb{Z}}$. After the proper GSO projection, type 0B theory has the
tachyon modes and the momentum modes in the RR-sector, On the other hand, type
0A theory has the tachyon modes and the winding modes in the RR-sector. See,
e.g. [4, 5] for more detail.
### 4.2 Amplitudes of Topological Model
Before dealing with the specific case of OSP(1$|$2)/U(1), we give generic
arguments on amplitudes in topological models. Let us consider a topological
field theory obtained by the topological twist $T^{top}(z)=T(z)+1/2\partial
J_{R}(z)$ of a ${\cal N}=2$ super conformal field theory [34, 35]. We consider
the B model, namely twist the same way for the anti-holomorphic part as
$\bar{T}^{top}(\bar{z})=\bar{T}(\bar{z})+1/2\bar{\partial}\bar{J}_{R}(\bar{z})$.
Then the physical spectrum can be computed from the cohomology of BRST
operator $Q=\oint G^{+}(z)dz$. Let us write a basis of physical operators (in
NS sector) as $\phi_{i}$, then we can obtain other types of physical operators
as
$\displaystyle\oint dzG^{-}_{-\frac{1}{2}}\cdot\phi_{i},\qquad\oint
d\bar{z}\bar{G}^{-}_{-\frac{1}{2}}\cdot\phi_{i},\qquad\int
d^{2}zG^{-}_{-\frac{1}{2}}\bar{G}^{-}_{-\frac{1}{2}}\cdot\phi_{i}.$ (4.6)
Following the arguments on [40, 35, 41], we compute amplitudes of the form
$\displaystyle{\cal
F}=\left\langle\phi_{i_{1}}(z_{1})\phi_{i_{2}}(z_{2})\phi_{i_{3}}(z_{3})\prod_{\nu=4}^{N}\left[\int
d^{2}z_{\nu}\tilde{\phi}_{i_{\nu}}(z_{\nu})\right]\right\rangle,$ (4.7)
which would give us non-trivial information of the topological model. Here we
have defined
$\tilde{\phi}_{i}=G^{-}_{-\frac{1}{2}}\bar{G}^{-}_{-\frac{1}{2}}\phi_{i}$.
An important fact is that the above amplitudes of topological model can be
computed in the original untwisted model [35]. After the topological twist the
$U(1)_{R}$ current becomes anomalous, and hence we should insert $U(1)_{R}$
fields into correlators of original model to reproduce the topological
amplitudes. Here we insert the operator
$\mu(z,\bar{z})=e^{\frac{i}{2}\sqrt{\frac{c}{3}}(X_{R}(z))+\bar{X}_{R}(\bar{z}))}$
at two points $z=z_{1},z_{2}$, where the free boson $X_{R}$ is related to the
R-current as $J_{R}(z)=-i\sqrt{\frac{c}{3}}\partial X_{R}(z)$. This operator
maps the physical operators $\phi_{i}$ of the topological model to operators
$\phi_{i}^{R}$ in the R-ground states of the original model. In this way, we
can write the topological amplitude (4.7) as
$\displaystyle{\cal F}$
$\displaystyle=|z_{1}-z_{2}|^{q_{1}+q_{2}}(|z_{1}-z_{3}||z_{2}-z_{3}|)^{q_{3}}\times$
(4.8)
$\displaystyle\times\left\langle\phi^{R}_{i_{1}}(z_{1})\phi^{R}_{i_{2}}(z_{2})\phi_{i_{3}}(z_{3})\prod_{\nu=4}^{N}\left[\int
d^{2}z_{\nu}(|z_{1}-z_{\nu}||z_{2}-z_{\nu}|)^{q_{\nu}-1}\tilde{\phi}_{i_{\nu}}(z_{\nu})\right]\right\rangle,$
where the right hand side is computed in the original model before the
topological twisting. Here we have denoted $q_{\nu}$ as the $U(1)_{R}$ charge
of $\phi_{i_{\nu}}$.
### 4.3 Comparison of Correlation Functions
After the preparation we can now compare correlation functions of topological
model on OSP(1$|$2)/U(1) and of the $\hat{c}\leq 1$ superstring. We start from
the amplitude of the topological model, and then show the equivalence by using
the formula (3.24) obtained above. Here we only consider the amplitudes of
operator of the first type in (2.30), which corresponds to the tachyon
operator in the $\hat{c}\leq 1$ superstring. The case with the second type in
(2.30) can be analyzed in a similar way.
Since the conservation of U(1) current $J=\partial\chi$ is violated by the
amount of $-2$, non vanishing amplitudes may be given as
$\displaystyle{\cal F}=\left\langle{\cal O}^{\frac{1}{2}}_{j_{1}}(z_{1}){\cal
O}^{\frac{1}{2}}_{j_{2}}(z_{2}){\cal
O}^{-\frac{1}{2}}_{j_{3}}(z_{3})\prod_{\nu=4}^{N}\left[\int
d^{2}z_{\nu}\tilde{\cal
O}^{-\frac{1}{2}}_{j_{\nu}}(z_{\nu})\right]\right\rangle.$ (4.9)
This violation corresponds to the fact that the sum of picture must be $-2$ in
superstring theory. The operator ${\cal O}^{\frac{1}{2}}_{j}$ is defined in
(2.30) as
$\displaystyle{\cal
O}^{\frac{1}{2}}_{j}=e^{-i(H+\bar{H})-(\chi+\bar{\chi})}\Phi^{\frac{1}{2},\frac{1}{2},w=1}_{j,j-\frac{1}{4},j-\frac{1}{4}}e^{2ib(j+\frac{1}{4b^{2}})(X+\bar{X})}.$
(4.10)
Following the argument in the previous subsection, this operator would be
mapped to the R-ground state operator of the original model
$\displaystyle{\cal
O}^{R,\frac{1}{2}}_{j}=e^{\frac{i}{2}(H+\bar{H})}\Phi^{\frac{1}{2},\frac{1}{2}}_{j,j-\frac{1}{4},j-\frac{1}{4}}e^{2ib(j+\frac{1}{4})(X+\bar{X})}.$
(4.11)
Another operator ${\cal O}^{-\frac{1}{2}}_{j}$ is given by a linear
combination of
$\displaystyle
e^{-i(H+\bar{H})}\Phi^{s,\bar{s},w=1}_{j,j+\frac{1}{4},j+\frac{1}{4}}e^{2ib(j+\frac{1}{4b^{2}})(X+\bar{X})}$
(4.12)
with $s,\bar{s}=-1/2,3/2$. This operator should correspond to picture $(0,0)$
tachyon, and can be constructed by following the analysis in section 2. The
other operator $\tilde{\cal O}^{-\frac{1}{2}}_{j}$ is then generated by the
action of $G^{-}_{-1/2}\bar{G}^{-}_{-1/2}$ as mentioned before and written as
a linear combination of
$\displaystyle\Phi^{s,\bar{s},w=1}_{j,j-\frac{3}{4},j-\frac{3}{4}}e^{2ib(j+\frac{1}{4b^{2}})(X+\bar{X})}$
(4.13)
with $s,\bar{s}=-1/2,3/2$.
As argued in the previous subsection, we first map the amplitude of
topological model (4.9) to that of original model before twisting as in (4.8).
Then we can use the formula (3.24) to relate it to the amplitude of ${\cal
N}=1$ super Liouville theory. Following the analysis of [36, 7] we can then
show that
$\displaystyle{\cal F}=\left\langle c\bar{c}{\cal
T}^{(-1)}_{j_{1}}(z_{1})c\bar{c}{\cal T}^{(-1)}_{j_{2}}(z_{2})c\bar{c}{\cal
T}^{(0)}_{j_{3}}(z_{3})\prod_{\nu=4}^{N}\left[\int d^{2}z_{\nu}{\cal
T}^{(0)}_{j_{\nu}}(z_{\nu})\right]\right\rangle$ (4.14)
up to some coefficients. Here, the operators ${\cal T}^{(p)}$ are tachyon
operators in the $p$-th picture and they are given by
$\displaystyle{\cal
T}^{(-1)}_{p}=e^{-(\chi+\bar{\chi})}e^{ik_{X}(X+\bar{X})+k_{\phi}^{+}\phi}~{},$
(4.15) $\displaystyle{\cal
T}^{(0)}_{p}=(ik_{X}\psi_{X}+k_{\phi}^{+}\psi)(ik_{X}\bar{\psi}_{X}+k_{\phi}^{+}\bar{\psi})e^{ik_{X}(X+\bar{X})+k_{\phi}^{+}\phi}$
(4.16)
with $k_{X}=(1/b-b)/2+2b(j+1/4)$ and $k_{\phi}^{+}=(1/b+b)/2+2b(j+1/4)$. In
this way we have shown that the amplitude of topological string on
OSP(1$|$2)/U(1) can be identified with that of the $\hat{c}\leq 1$
superstring.
## 5 Conclusion and Discussions
In this paper we have proposed an equivalence between the topological string
theory based on the coset OSP(1$|$2)/U(1) and the $\widehat{c}\leq 1$
superstring theory. The latter is constructed by coupling a $\hat{c}\leq 1$
matter to the $\mathcal{N}=1$ super Liouville theory. This can be regarded as
a supersymmetric version of the equivalence between the topological string on
$SL(2)/U(1)$ and the $c\leq 1$ bosonic string, which was originally discovered
by Mukhi and Vafa [14] for the case $c=1$ and was later generalized to the
$c<1$ case in [7]. First we showed in the free field description that the
field contents and the physical operators of the world-sheet theories of both
string theories match. Moreover, we investigated the proposed equivalence at
the level of scattering amplitudes by applying the map [21] between
correlation functions in the OSP(1$|$2) WZNW model and in super Liouville
field theory.
This map is a supersymmetric version of the one found by Ribault and Teschner
to relate correlation functions in the SL(2) WZNW model and those in Liouville
theory [22, 23]. In the last years, the result has been used with great
success to investigate different dualities between non-rational conformal
field theories. In particular, it has led to the proof of Fateev-
Zamolodchikov-Zamolodchikov conjecture in [39]. The duality between different
non-rational two-dimensional conformal field theories has a long story, and
now a considerable list of examples is available: quantum Hamiltonian
reduction [16], Mukhi-Vafa duality [14, 7], and Fateev-Zamolodchikov-
Zamolodchikov duality [42] (and its supersymmetric version [43]) are probably
the most renowned examples. These examples were, in fact, very useful to study
string theory. For instance, it was the Fateev-Zamolodchikov-Zamolodchikov
duality what really permitted to construct a dual matrix model for strings in
the the 2D black hole background [44]. It is our hope that the new equivalence
between conformal theories we studied in this paper will be relevant to
understand new aspects of superstring dualities as well.
There are a number of issues which should be understood in the future. First
we would like to understand better the spin structure and the picture changing
operation of the topological string theory on the supercoset. It is also
important to prove the complete equivalence of physical states between these
two theories. An exhaustive analysis of the cohomology of the theory is needed
to this end. Finally, it would be nice if we could understand a geometrical
interpretation of the supercoset OSP(1$|$2)/U(1) in terms of a certain (maybe
super) Calabi-Yau manifold, as SL(2)/U(1) coset model is related to the
conifold.
### Acknowledgement
We are very grateful to K. Hori for a useful discussion. GG thanks the members
of the IPMU for their hospitality during his stay. YH would like to thank T.
Creutzig, H. Irie and P. B. Rønne for useful discussions. The work of GG has
been partially supported by ANPCyT grant PICT-2007-00849, by UBACyT grants
X861 and X432, and by JSPS-CONICET cooperation programme. The work of YH is
supported by JSPS Research Fellowship. TT is supported by World Premier
International Research Center Initiative (WPI Initiative), MEXT, Japan. The
work of TT is supported in part by JSPS Grant-in-Aid for Scientific Research
No.20740132 and by JSPS Grant-in-Aid for Creative Scientific Research No.
19GS0219.
## Appendix A Free Field Correlation Functions
Although the computation of $N$-point functions in the OSP(1$|$2) WZNW model
also involves fermion contributions and the insertion of picture changing
operators, the building blocks to construct such observables are the
correlation functions of vertices (2.8). Let us discuss these correlation
functions in the free field representation proposed here. Consider the vertex
operators
$\Phi_{j,m}^{s}(z)=N_{j,m}^{s}\
\gamma_{(z)}^{-j-1/2+m-s/2}e^{2(j+1/2)b\phi(z)}e^{isY(z)}\times h.c.$ (A.1)
where $h.c.$ stands for the anti-holomorphic portion of the
operator,131313More precisely, the $h.c.$ refers to the ”bared contribution”,
and not necessarily to the anti-holomorphic part. Actually, labels $s,m$ and
$\bar{s},\bar{m}$ are not necessarily related by complex conjugation. and
$N_{j,m}^{s}$ refers to the normalization.
Correlation functions are defined as follows;
$\left\langle\prod\nolimits_{\nu=1}^{N}\Phi_{j_{n},m_{\nu}}^{s_{\nu}}(z_{\nu})\right\rangle=\int\mathcal{D}\phi\mathcal{D}^{2}\beta\mathcal{D}^{2}\gamma\mathcal{D}^{2}\theta\mathcal{D}^{2}p\
e^{-S^{\text{WZNW}}(g)}\prod\nolimits_{\nu=1}^{N}\Phi_{j_{n},m_{\nu}}^{s_{\nu}}(z_{\nu})$
where $S^{\text{WZNW}}(g)$ refers to the action of the WZNW model (3.1). It is
convenient to consider again the bosonization (2.5); that is, defining
$\theta=e^{iY}$, $p=e^{-iY}$. The existence of non-trivial background charges
associated to the fields $\phi$ and $Y$ requires special treatment of
correlators. As usual in the Coulomb gas representation, this charge
compensation is achieved by inserting additional operators that contribute to
screen the charges at infinity. Screening operators are exact marginal
deformations of the affine theory. In this theory four operators of this kind
are available; namely
$\displaystyle\mathcal{S}_{++}(z,\bar{z})$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2k}S_{+}(z)\bar{S}_{+}(\bar{z}),\qquad\mathcal{S}_{+-}(z,\bar{z})=\frac{\lambda}{2k}S_{+}(z)\bar{S}_{-}(\bar{z}),$
(A.2) $\displaystyle\mathcal{S}_{-+}(z,\bar{z})$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2k}S_{-}(z)\bar{S}_{+}(\bar{z}),\qquad\mathcal{S}_{--}(z,\bar{z})=\frac{\lambda}{2k}S_{-}(z)\bar{S}_{-}(\bar{z}),$
(A.3)
with
$S_{+}(z)=e^{-iY(z)}e^{b\phi(z)},\qquad
S_{-}(z)=\beta_{(z)}e^{+iY(z)}e^{b\phi(z)},$ (A.4)
and where $\lambda$ is a constant (see below).
The $N$-point correlation functions are thus defined by inserting different
amount of screening operators (A.2)-(A.3) in the correlators, in addition to
the $N$ vertex operators. Non-vanishing correlation functions are given by
$\displaystyle n_{+}-n_{-}$ $\displaystyle=$
$\displaystyle\sum\nolimits_{\nu=1}^{N}s_{\nu}-1,\qquad\bar{n}_{+}-\bar{n}_{-}=\sum\nolimits_{\nu=1}^{N}\bar{s}_{\nu}-1$
(A.5) $\displaystyle n_{+}+n_{-}$ $\displaystyle=$
$\displaystyle\bar{n}_{+}+\bar{n}_{-}=-2\sum\nolimits_{\nu=1}^{N}j_{\nu}+1-N$
(A.6)
together with
$\sum\nolimits_{\nu=1}^{N}m_{\nu}=\sum\nolimits_{\nu=1}^{N}\bar{m}_{\nu}=0,$
(A.7)
where $n_{\pm}$ (and $\bar{n}_{\pm}$) are the amount of operators of the type
$S_{\pm}(z)$ (resp. $\bar{S}_{\pm}(\bar{z})$) in the correlators. Equations
(A.5)-(A.7) determine the amount of screening operators in terms of the
quantum number of the vertices involved in the correlators.
To illustrate the Coulomb gas prescription, let us consider the sector
$s_{\nu}=\bar{s}_{\nu}$, which yields $n_{\pm}=\bar{n}_{\pm}$. In this case,
correlation functions are given by contributions of the form
$\frac{(\lambda/2k)^{n_{+}+n_{-}}}{n_{+}!n_{-}!}\int\prod\nolimits_{r=1}^{n_{+}}d^{2}w_{r}\prod\nolimits_{l=1}^{n_{-}}d^{2}y_{l}\left\langle\prod\nolimits_{\nu=1}^{N}\Phi_{j_{n},m_{\nu}}^{s_{\nu}}(z_{\nu})\prod\nolimits_{r=1}^{n_{+}}\mathcal{S}_{++}(w_{r})\prod\nolimits_{l=1}^{n_{-}}\mathcal{S}_{--}(y_{l})\right\rangle_{\text{free}}$
(A.8)
where the subscript ”$\mathrm{free"}$ refers to the fact that this correlator
is defined in the free field theory. The amount of screening insertions
$n_{\pm}$ in (A.8) is given by (A.5) and (A.6). Correlators similar to (A.8)
but with a different amount of screening insertions
$\prod\nolimits_{r=1}^{n_{+}-n}\mathcal{S}_{++}(w_{r})$
$\prod\nolimits_{l=1}^{n_{-}-n}\mathcal{S}_{--}(y_{l})$
$\prod\nolimits_{t=1}^{n}\mathcal{S}_{-+}(\hat{w}_{t})$
$\prod\nolimits_{s=1}^{n}\mathcal{S}_{+-}(\hat{y}_{s})$ also contribute. All
the contributions are gathered with an appropriate prescription to integrate
the screening operators in the world-sheet.
The inclusion of screening operators (A.2)-(A.3) in (A.8) can be also thought
of as coming from the interaction terms in the action $S^{\text{WZNW}}(g)$. In
this picture, conditions (A.5)-(A.7) emerge from the integration over the
zero-modes of the fields. The scale $\lambda$ is easily introduced by shifting
the zero mode of $\phi$ as $\phi(z)\rightarrow\phi(z)+b^{-1}\log(\lambda)$.
The parameter $\lambda$ allows to keep track of the KPZ scaling of correlation
functions [45].
Correlation functions (A.8) can be computed by using free field propagators
(2.4),
$\displaystyle\left\langle\prod\nolimits_{\nu=1}^{N}\Phi_{j_{n},m_{\nu}}^{s_{\nu}}(z_{\nu})\right\rangle$
$\displaystyle=$
$\displaystyle\prod\nolimits_{\nu=1}^{N}N_{j_{\nu},m_{\nu}}^{s_{\nu}}\frac{(\lambda/2k)^{n_{+}+n_{-}}}{n_{+}!n_{-}!}\int\prod\nolimits_{r=1}^{n_{+}}d^{2}w_{r}\prod\nolimits_{l=1}^{n_{-}}d^{2}y_{l}$
(A.9)
$\displaystyle\times\left\langle\prod\nolimits_{\nu=1}^{N}e^{is_{\nu}Y(z_{\nu})}\prod\nolimits_{r=1}^{n_{+}}e^{-iY(w_{r})}\prod\nolimits_{l=1}^{n_{-}}e^{iY(y_{l})}\right\rangle_{\text{free}}\times$
$\displaystyle\times\left\langle\prod\nolimits_{\nu=1}^{N}e^{b(2j_{\nu}+1)\phi(z_{\nu})}\prod\nolimits_{r=1}^{n_{+}}e^{b\phi(w_{r})}\prod\nolimits_{l=1}^{n_{-}}e^{b\phi(y_{l})}\right\rangle_{\text{free}}\times$
$\displaystyle\times\left\langle\prod\nolimits_{\nu=1}^{N}\gamma_{(z_{\nu})}^{m_{\nu}-j_{\nu}-(s_{\nu}+1)/2}\prod\nolimits_{l=1}^{n_{-}}\beta_{(y_{l})}\right\rangle_{\text{free}}\times
h.c.$
where, again, $N_{j,m}^{s}$ is the normalization of the vertex. The standard
normalization
$N_{j,m}^{s}=\frac{\Gamma(-j+1/2-s/2+m)}{\Gamma(j+1/2+\bar{s}/2+\bar{m})}$
yields
$\left\langle\Phi_{j,m}^{s}(z_{1})\Phi_{-j-1/2,-m}^{1-s}(z_{2})\right\rangle=|z_{1}-z_{2}|^{-4\Delta}.$
By expanding this expression, after Wick contracting all the contributions, it
takes the form
$\left\langle\prod\nolimits_{\nu=1}^{N}\Phi_{j_{n},m_{\nu}}^{s_{\nu}}(z_{\nu})\right\rangle=\frac{1}{n_{+}!n_{-}!}\left(\frac{\lambda}{2k}\right)^{-2(j_{1}+...j_{N})+1-N)}\prod\nolimits_{\nu=1}^{N}N_{j_{\nu},m_{\nu}}^{s_{\nu}}\times$
$\times\prod\nolimits_{\mu<\nu}^{N}(z_{\mu}-z_{\nu})^{s_{\mu}s_{\nu}-b^{2}(2j_{\mu}+1)(2j_{\nu}+1)}\int\prod\nolimits_{r=1}^{n_{+}}d^{2}w_{r}\prod\nolimits_{l=1}^{n_{-}}d^{2}y_{l}\times$
$\times\prod\nolimits_{l=1}^{n_{-}}\prod\nolimits_{r=1}^{n_{+}}(w_{r}-y_{l})^{-1-b^{2}}\prod\nolimits_{l<l^{\prime}}^{n_{-}}(y_{l}-y_{l^{\prime}})^{1-b^{2}}\prod\nolimits_{r<r^{\prime}}^{n_{+}}(w_{r}-w_{r^{\prime}})^{1-b^{2}}\times$
$\times\prod\nolimits_{\nu=1}^{N}\prod\nolimits_{r=1}^{n_{+}}(z_{\nu}-w_{r})^{-s_{\nu}-b^{2}(2j_{\nu}+1)}\prod\nolimits_{\nu=1}^{N}\prod\nolimits_{l=1}^{n_{-}}(z_{\nu}-y_{l})^{s_{\nu}-b^{2}(2j_{\nu}+1)}\times$
$\times\left\langle\prod\nolimits_{\nu=1}^{N}\gamma_{(z_{\nu})}^{m_{\nu}-j_{\nu}-(s_{\nu}+1)/2}\prod\nolimits_{l=1}^{n_{-}}\beta_{(y_{l})}\right\rangle_{\text{free}}\times
h.c.$ (A.10)
In addition, we may resort to projective invariance to fix three points at
$z_{1}=0$, $z_{2}=1$, and $z_{N}=\infty$.
The correlator of the ($\beta$,$\gamma$) ghost fields in (A.10) yields a
rather complicated expression in general. Nevertheless, it simplifies
substantially in some particular cases. For instance, in the case of two
insertions it reads [46, 47]
$\left\langle\gamma_{(z_{1}=0)}^{m_{1}-j_{1}-(s_{1}+1)/2}\gamma_{(z_{2}=1)}^{m_{2}-j_{2}-(s_{2}+1)/2}\prod\nolimits_{l=1}^{n_{-}}\beta_{(y_{l})}\right\rangle_{\text{free}}\times
h.c.=\prod\nolimits_{l=1}^{n_{-}}|y_{l}|^{-2}|1-y_{l}|^{-2}\times$
$\times(-1)^{n_{-}}\frac{\Gamma(1/2-j_{1}-s_{1}/2+m_{1})}{\Gamma(1/2+j_{1}+s_{1}/2-m_{1})}\frac{\Gamma(1/2-j_{2}-s_{2}/2+m_{2})}{\Gamma(1/2+j_{2}+s_{2}/2-m_{2})}.$
World-sheet integral (A.10) can in principle be computed by using generalized
Selberg integral formulas of the type worked out in [48, 49, 50]. To do this
one has to give a precise prescription for contour integration. We will not
address the details of such prescription in this appendix.
Integral representation (A.8) gathers the residues associated to the $N$
-point correlation functions of the OSP(1$|$2) WZNW model, and after analytic
continuation in $n_{\pm}$ and $j_{i}$ the exact form of the correlation
functions are obtained. The exact expressions for two- and three-point
correlation functions in the OSP(1$|$2) WZNW model were found in [21].
Representation (A.8) gives important information about the correlators. For
instance, the KPZ scaling properties can be read from this expression.
Correlators (A.10) scale as $\sim\left(\lambda/2k\right)^{n_{+}+n_{-}}$,
where, according to (A.6), $n_{+}+n_{-}=1-2(j_{1}+j_{2}+...j_{N})-N$. In
particular, for the two-point function, where $N=2$ and $j_{1}=j_{2}=j$, we
obtain $\sim\left(\lambda/2k\right)^{-4j-1}$. So, let us compare this with the
scaling properties of the exact exact solution of the OPS(1$|$2) WZNW model
found in [21]. First, let us notice that in comparing the conventions of [21]
with ours here we have to redefine $j$ as follows $j\rightarrow j+1/2$. Thus,
the KPZ scaling is $\sim\left(\lambda/2k\right)^{-4j-3}$ which precisely
agrees with the result in [21]. Actually, it is not hard to see that if one
introduces the scale $\lambda$ in the formulas of [21] then eq. (4.7) therein
scales like
$\sim\left(\frac{2kb^{2}}{i\lambda\gamma(\frac{b^{2}+1}{2})}\right)^{4j+3}.$
(A.11)
Analogously, for the three-point functions one finds
$\sim\left(\lambda/2k\right)^{-2(j_{1}+j_{2}+j_{3})-5}$ which also coincides
with the scaling of eq. (4.21) of [21].
One can also see that in the coincidence limit $\lim_{z_{1}\rightarrow
z_{2}}\Phi_{j_{1}m\,_{1}}^{s_{1}}(z_{1})\Phi_{j_{2}m\,_{2}}^{s_{2}}(z_{2})$,
where two of the vertices hit each other, the pole condition that appears at
$z_{1}=z_{2}$ can be interpreted as a mass-shell condition
$L_{0}-1=-2b^{2}j(j+1/2)+s(s-1)/2+l=0$ of a level-$l$ excited intermediate
state carrying momenta $j=j_{1}+j_{2}-1+(N+n_{+}^{\prime}+n_{-}^{\prime})/2$
and $s=s_{1}+s_{2}+n_{+}^{\prime}-n_{-}^{\prime}$, where $n_{\pm}^{\prime}\leq
n_{\pm}$ are the amount of screening operators of the type $S_{\pm}$ whose
inserting points also tend to $z_{2}$ together with $z_{1}$. In this limit,
the $N$-point function factorizes in the product of one three-point function
times one $N-2$-point function.
## References
* [1] J. M. Maldacena, “The large $N$ limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200];
* [2] R. R. Metsaev and A. A. Tseytlin, “Type IIB superstring action in $AdS_{5}\times S^{5}$ background,” Nucl. Phys. B 533, 109 (1998) [arXiv:hep-th/9805028].
* [3] N. Berkovits, C. Vafa and E. Witten, “Conformal field theory of AdS background with Ramond-Ramond flux,” JHEP 9903, 018 (1999) [arXiv:hep-th/9902098].
* [4] T. Takayanagi and N. Toumbas, “A matrix model dual of type 0B string theory in two dimensions,” JHEP 0307 (2003) 064 [arXiv:hep-th/0307083].
* [5] M. R. Douglas, I. R. Klebanov, D. Kutasov, J. M. Maldacena, E. J. Martinec and N. Seiberg, “A new hat for the $c=1$ matrix model,” arXiv:hep-th/0307195.
* [6] T. Takayanagi, “Matrix model and time-like linear dilaton matter,” JHEP 0412 (2004) 071 [arXiv:hep-th/0411019].
* [7] T. Takayanagi, “$c<1$ string from two dimensional black holes,” JHEP 0507, 050 (2005) [arXiv:hep-th/0503237].
* [8] T. Okuda and T. Takayanagi, “Ghost D-branes,” JHEP 0603 (2006) 062 [arXiv:hep-th/0601024].
* [9] N. Berkovits, “A new limit of the $AdS_{5}\times S^{5}$ sigma model,” JHEP 0708, 011 (2007) [arXiv:hep-th/0703282].
* [10] N. Berkovits and C. Vafa, “Towards a worldsheet derivation of the Maldacena conjecture,” JHEP 0803, 031 (2008) [AIP Conf. Proc. 1031, 21 (2008)] [arXiv:0711.1799 [hep-th]].
* [11] G. Bonelli and H. Safaai, “On gauge/string correspondence and mirror symmetry,” JHEP 0806, 050 (2008) [arXiv:0804.2629 [hep-th]].
* [12] N. Berkovits, “Perturbative super-Yang-Mills from the topological $AdS_{5}\times S^{5}$ sigma model,” JHEP 0809, 088 (2008) [arXiv:0806.1960 [hep-th]].
* [13] G. Bonelli, P. A. Grassi and H. Safaai, “Exploring pure spinor string theory on $AdS_{4}\times\mathbb{CP}^{3}$,” JHEP 0810, 085 (2008) [arXiv:0808.1051 [hep-th]].
* [14] S. Mukhi and C. Vafa, “Two-dimensional black hole as a topological coset model of $c=1$ string theory,” Nucl. Phys. B 407, 667 (1993) [arXiv:hep-th/9301083].
* [15] M. Bershadsky and H. Ooguri, “Hidden OSP($N|$2) symmetries in superconformal field theories,” Phys. Lett. B 229, 374 (1989).
* [16] M. Bershadsky and H. Ooguri, “Hidden SL$(n)$ symmetry in conformal field theories,” Comm. Math. Phys. 126 (1989) 49.
* [17] Y. Kazama and H. Suzuki, “New ${\cal N}=2$ superconformal field theories and superstring compactification,” Nucl. Phys. B 321, 232 (1989).
* [18] Y. Kazama and H. Suzuki, “Characterization of ${\cal N}=2$ superconformal models generated by coset space method,” Phys. Lett. B 216, 112 (1989).
* [19] T. Creutzig, P. B. Ronne and V. Schomerus, “${\cal N}=2$ superconformal symmetry in super coset models,” arXiv:0907.3902 [hep-th].
* [20] T. Creutzig, “Branes in supergroups,” DESY-THESIS-2009-018.
* [21] Y. Hikida and V. Schomerus, “Structure constants of the OSP(1$|$2) WZNW model,” JHEP 0712, 100 (2007) [arXiv:0711.0338 [hep-th]].
* [22] S. Ribault and J. Teschner, “$H_{3}^{+}$ WZNW correlators from Liouville theory,” JHEP 0506, 014 (2005) [arXiv:hep-th/0502048].
* [23] Y. Hikida and V. Schomerus, “$H^{+}_{3}$ WZNW model from Liouville field theory,” JHEP 0710, 064 (2007) [arXiv:0706.1030 [hep-th]].
* [24] M. Wakimoto, “Fock representations of the affine lie algebra $A_{1}(1)$,” Commun. Math. Phys. 104 (1986) 605.
* [25] I. P. Ennes, A. V. Ramallo and J. M. Sanchez de Santos, “On the free field realization of the osp(1$|$2) current algebra,” Phys. Lett. B 389, 485 (1996) [arXiv:hep-th/9606180].
* [26] I. P. Ennes, A. V. Ramallo and J. M. Sanchez de Santos, “Structure constants for the osp(1$|$2) current algebra,” Nucl. Phys. B 491, 574 (1997) [arXiv:hep-th/9610224].
* [27] A. LeClair, “The gl(1$|$1) super-current algebra: the role of twist and logarithmic fields,” arXiv:0710.2906 [hep-th].
* [28] T. Creutzig and P. B. Ronne, “The GL(1$|$1)-symplectic fermion correspondence,” Nucl. Phys. B 815, 95 (2009) [arXiv:0812.2835 [hep-th]].
* [29] V. S. Dotsenko, “The free field representation of the SU(2) conformal field theory,” Nucl. Phys. B 338 (1990) 747.
* [30] V. S. Dotsenko, “Solving the SU(2) conformal field theory with the Wakimoto free field representation,” Nucl. Phys. B 358, 547 (1991).
* [31] M. Bershadsky and D. Kutasov, “Comment on gauged WZW theory,” Phys. Lett. B 266, 345 (1991).
* [32] R. Dijkgraaf, H. L. Verlinde and E. P. Verlinde, “String propagation in a black hole geometry,” Nucl. Phys. B 371, 269 (1992).
* [33] J. M. Maldacena and H. Ooguri, “Strings in $AdS_{3}$ and SL(2,${\mathbb{R}}$) WZW model. I,” J. Math. Phys. 42, 2929 (2001) [arXiv:hep-th/0001053].
* [34] T. Eguchi and S. K. Yang, “${\cal N}=2$ superconformal models as topological field theories,” Mod. Phys. Lett. A 5 (1990) 1693.
* [35] E. Witten, “Mirror manifolds and topological field theory,” arXiv:hep-th/9112056.
* [36] S. Nakamura and V. Niarchos, “Notes on the S-matrix of bosonic and topological non-critical strings,” JHEP 0510, 025 (2005) [arXiv:hep-th/0507252].
* [37] S. Ribault, “Knizhnik-Zamolodchikov equations and spectral flow in $AdS_{3}$ string theory,” JHEP 0509, 045 (2005) [arXiv:hep-th/0507114].
* [38] G. Giribet, “The string theory on $AdS_{3}$ as a marginal deformation of a linear dilaton background,” Nucl. Phys. B 737, 209 (2006) [arXiv:hep-th/0511252].
* [39] Y. Hikida and V. Schomerus, “The FZZ-duality conjecture - A proof,” arXiv:0805.3931 [hep-th].
* [40] R. Dijkgraaf, H. L. Verlinde and E. P. Verlinde, “Topological strings in $d<1$,” Nucl. Phys. B 352, 59 (1991).
* [41] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, “Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes,” Commun. Math. Phys. 165 (1994) 311 [arXiv:hep-th/9309140].
* [42] V. Fateev, A.B. Zamolodchikov and Al. Zamolodchikov, unpublished note.
* [43] K. Hori and A. Kapustin, “Duality of the fermionic 2d black hole and ${\cal N}=2$ Liouville theory as mirror symmetry,” JHEP 0108 (2001) 045 [arXiv:hep-th/0104202].
* [44] V. Kazakov, I. K. Kostov and D. Kutasov, “A matrix model for the two-dimensional black hole,” Nucl. Phys. B 622, 141 (2002) [arXiv:hep-th/0101011].
* [45] V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov, “Fractal structure of 2d-quantum gravity,” Mod. Phys. Lett. A 3, 819 (1988).
* [46] K. Becker and M. Becker, “Interactions in the SL(2,$\mathbb{R}$) / U(1) black hole background,” Nucl. Phys. B 418, 206 (1994) [arXiv:hep-th/9310046].
* [47] G. Giribet and C. A. Nunez, “Correlators in $AdS_{3}$ string theory,” JHEP 0106, 010 (2001) [arXiv:hep-th/0105200].
* [48] V. S. Dotsenko and V. A. Fateev, “Four point correlation functions and the operator algebra in the two-dimensional conformal invariant theories with the central charge $c<1$,” Nucl. Phys. B 251, 691 (1985).
* [49] T. Fukuda and K. Hosomichi, “Three-point functions in sine-Liouville theory,” JHEP 0109, 003 (2001) [arXiv:hep-th/0105217].
* [50] T. Fukuda and K. Hosomichi, “Super Liouville theory with boundary,” Nucl. Phys. B 635, 215 (2002) [arXiv:hep-th/0202032].
|
arxiv-papers
| 2009-07-22T13:08:55 |
2024-09-04T02:49:04.093316
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gaston Giribet, Yasuaki Hikida and Tadashi Takayanagi",
"submitter": "Yasuaki Hikida",
"url": "https://arxiv.org/abs/0907.3832"
}
|
0907.3845
|
# Discrete coherent and squeezed states of many-qudit systems
Andrei B. Klimov Departamento de Física, Universidad de Guadalajara, 44420
Guadalajara, Jalisco, Mexico Carlos Muñoz Departamento de Física,
Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico Luis L.
Sánchez-Soto Departamento de Óptica, Facultad de Física, Universidad
Complutense, 28040 Madrid, Spain
###### Abstract
We consider the phase space for a system of $n$ identical qudits (each one of
dimension $d$, with $d$ a primer number) as a grid of $d^{n}\times d^{n}$
points and use the finite field $\mathrm{GF}(d^{n})$ to label the
corresponding axes. The associated displacement operators permit to define
$s$-parametrized quasidistribution functions in this grid, with properties
analogous to their continuous counterparts. These displacements allow also for
the construction of finite coherent states, once a fiducial state is fixed. We
take this reference as one eigenstate of the discrete Fourier transform and
study the factorization properties of the resulting coherent states. We extend
these ideas to include discrete squeezed states, and show their intriguing
relation with entangled states between different qudits.
###### pacs:
03.65.Ta,03.65.Fd,42.50.Dv
## I Introduction
The concept of phase-space representation of quantum mechanics, introduced in
the pioneering works of Weyl Weyl (1928), Wigner Wigner (1932), and Moyal
Moyal (1949), is a very useful and enlightening approach that sheds light on
the correspondence between quantum and classical worlds.
Numerous applications of the phase-space methods to physical problems have
been extensively discussed in the last decades Lee (1995); Schroek (1996);
Schleich (2001); Zachos et al. (2005). However, much of this subject is
usually illustrated in terms of continuous variables, most often position and
momentum.
For discrete systems, or qudits in the modern parlance of quantum information,
things are less straightforward. Since the dynamical symmetry group for a
qudit is SU($d$), one may be tempted to interpret its associated phase space
as a generalized Bloch sphere Kimura (2003); Schirmer et al. (2004), which is
supported by the rigorous construction of Kostant Kostant (1970) and Kirilov
Kirillov (1976.) in terms of coadjoint orbits. Even if this picture is quite
popular, especially when applied to qubits, one can rightly argue that there
is a lot of information redundancy there and that the phase space should be a
grid of points, as one could expect for a truly discrete system.
Indeed, apart from some noteworthy exceptions Hannay and Berry (1980);
Leonhardt (1995); Miquel et al. (2002), nowadays there is a wide consensus in
picturing the phase space for a qudit as a $d\times d$ grid. This can be
traced back to the elegant approach proposed by Schwinger Schwinger (1960a, b,
c), who clearly recognized that the expansion of arbitrary operators in terms
of certain operator basis was the crucial mathematical concept in setting such
a grid. These ideas have been rediscovered and developed further by several
authors Buot (1973); Cohendet et al. (1988); Kasperkovitz and Peev (1994);
Opatrný et al. (1995); Rivas and de Almeida (1999); Mukunda et al. (2004);
Chaturvedi et al. (2006), although the contributions of Wootters Wootters
(1986); Wootters and Fields (1989); Wootters (2006); Wootters and Sussman
(2007) and Galetti and coworkers Galetti and de Toledo Piza (1988, 1992, 1995)
are worth stressing.
To equip this grid with properties analogous to the geometry of an ordinary
plane, it turns out essential Klimov et al. (2005, 2007, 2009) to label the
axes in terms of the elements of a finite field $\mathrm{GF}(d)$ with $d$
elements. It is well known that such a field exist only when the dimension is
a prime or a power of a prime Lidl and Niederreiter (1986). This, of course,
gives a special role to qudits in prime dimensions, but also is ideally suited
to deal with systems of $n$ of these qudits.
Once the natural arena is properly established, the next question is how to
represent states (and operators) on phase space. This is done through
quasidistribution functions, which allow for the calculation of quantum
averages in a way that exactly parallels classical mechanics. There are,
however, important differences with respect to a classical description: they
come from the noncommuting nature of conjugate quantities (like position and
momentum), which precludes their simultaneous precise measurement and,
therefore, imposes a fundamental limit on the accuracy with which we can
determine a point in phase space. As a distinctive consequence of this, there
is no unique rule by which we can associate a classical phase-space variable
to a quantum operator. Therefore, depending on the operator ordering, various
quasidistributions can be defined. For continuous variables, the best known
are the Glauber-Sudarshan $P$ function Glauber (1963); Sudarshan (1963), the
Husimi $Q$ function Husimi (1940), and the Wigner $W$ function Hillery et al.
(1984), corresponding to normal, antinormal, and symmetric order,
respectively, in the associated characteristic functions. In fact, all of them
are special cases of the $s$-parametrized quasidistributions introduced by
Cahill and Glauber Cahill and Glauber (1969).
The problem of generalizing these quasidistributions (mainly the Wigner
function) to finite systems has also a long history. Much of the previous
literature has focused on spin variables, trying to represent spin states by
continuous functions of angle variables. This idea was initiated by
Stratonovich Stratonovich (1956), Berezin Berezin (1975), and Agarwal Agarwal
(1981). The resulting Wigner function, naturally related to the SU(2)
dynamical group, has been further studied by a number of authors Scully
(1983); Cohen and Scully (1986); Varilly and Gracia-Bondía (1989); Heiss and
Weigert (2000), has been applied to some problems in quantum optics Dowling et
al. (1994); Benedict and Czirják (1999) and extended to more general groups
Brif and Mann (1998).
However, these Wigner functions are not defined in a discrete phase space. A
detailed review of possible solutions can be found in Ref. Björk et al.
(2008). Perhaps, the most popular one is due to Wootters and coworkers
Wootters (1987); Gibbons et al. (2004); Wootters (2004), which imposes a
structure by assigning a quantum state to each line in phase space. Any good
assignment of quantum states to lines is called a “quantum net” and can be
used to define a discrete Wigner function. In this paper, we show how to
introduce a set of $s$-parametrized functions, in close correspondence with
the continuous case. We emphasize that, although some interesting work has
been done in this direction by using a mod $d$ invariance Ruzzi et al. (2005);
Marchiolli et al. (2005), our approach works quite well for many-qudit
systems.
Another essential ingredient in any phase-space description is the notion of
coherent states Klauder and Skagerstam (1999). This is firmly established for
continuous variables and can easily extended for other dynamical symmetry
groups Perelomov (1986). However, for discrete systems we have again a big gap
waiting to be filled. The reason for this can be traced back to the fact that,
as brightly pointed out in Ref. Ruzzi (2006), in the continuum we have one,
and only one, harmonic oscillator, while in the discrete there are a lot of
candidates for that role, each one surely with its virtues, but surely no
undisputed champion.
The strategy we adopt to deal with this problem is to look for eigenstates of
the discrete Fourier transform Galetti and Marchiolli (1996). For continuous
variables, they have a very distinguishable behavior that is at the basis of
the remarkable properties of coherent states. We explore this approach,
getting a strikingly simple family of discrete coherent states (even for many
qudits) fulfilling all the requirements.
To put the icing on the cake, we also extend the notion of squeezed states for
these systems Marchiolli et al. (2007). For a single qudit, the resulting
states have nice and expected properties. However, when they really appear as
highly interesting is for multipartite systems, since they present an
intriguing relation with entanglement.
In short, the program developed in this paper can be seen as a handy toolbox
for the phase-space analysis of many-qudit systems, which should be of
interest to a large interdisciplinary community working in these topics.
## II Phase space for continuous variables
In this Section we briefly recall the relevant structures needed to set up a
phase-space description of Cartesian quantum mechanics. This will facilitate
comparison with the discrete case later on. For simplicity, we choose one
degree of freedom only, so the associated phase space is the plane
$\mathbb{R}^{2}$.
The relevant observables are the Hermitian coordinate and momentum operators
$\hat{q}$ and $\hat{p}$, with canonical commutation relation (with $\hbar=1$
throughout)
$[\hat{q},\hat{p}]=i\,\hat{\openone}\,,$ (1)
so that they are the generators of the Heisenberg-Weyl algebra. Ubiquitous and
profound, this algebra has become the hallmark of noncommutativity in quantum
theory. To avoid technical problems with the unbounded operators $\hat{q}$ and
$\hat{p}$, it is convenient to work with their unitary counterparts Putnam
(1967)
$\hat{U}(q)=\exp(-iq\,\hat{p})\,,\qquad\hat{V}(p)=\exp(ip\,\hat{q})\,,$ (2)
which generate translations in position and momentum, respectively. The
commutation relations are then expressed in the Weyl form
$\hat{V}(p)\hat{U}(q)=e^{iqp}\,\hat{U}(q)\hat{V}(p)\,.$ (3)
Their infinitesimal form immediately gives (1), but (3) is more useful in many
instances.
In terms of $\hat{U}$ and $\hat{V}$ a displacement operator can be introduced
as
$\hat{D}(q,p)=e^{-iqp/2}\,\hat{U}(p)\hat{V}(q)\,,$ (4)
which usually is presented in the entangled form
$\hat{D}(q,p)=\exp[i(p\hat{q}-q\hat{p})]$. However, this cannot be done in
more general situations.
The $\hat{D}(q,p)$ form a complete orthonormal set (in the trace sense) in the
space of operators acting on $\mathcal{H}$ (the Hilbert space of square
integrable functions on $\mathbb{R}$). The unitarity imposes that
$\hat{D}^{\dagger}(q,p)=\hat{D}(-q,-p)$, and $\hat{D}(0,0)=\hat{\openone}$. In
addition, they obey the simple composition law
$\hat{D}(\hat{q}_{1},\hat{p}_{1})\hat{D}(\hat{q}_{2},\hat{p}_{2})=e^{i(p_{1}q_{2}-q_{1}p_{2})/2}\,\hat{D}(\hat{q}_{1}+\hat{q}_{2},\hat{q}_{2}+\hat{p}_{2})\,.$
(5)
The displacement operators constitute a basic element for the notion of
coherent states. Indeed, if we choose a fixed normalized reference state
$|\psi_{0}\rangle$, we can define these coherent states as Perelomov (1986)
$|q,p\rangle=\hat{D}(q,p)\,|\psi_{0}\rangle\,,$ (6)
so they are parametrized by phase-space points. These states have a number of
remarkable properties, inherited from those of $\hat{D}(q,p)$. In particular,
$\hat{D}(q,p)$ transforms any coherent state in another coherent state:
$\hat{D}(\hat{q}_{1},\hat{p}_{1})\,|q_{2},p_{2}\rangle=e^{i(p_{1}q_{2}-q_{1}p_{2})/2}\,|q_{1}+q_{2},q_{2}+p_{2}\rangle\,.$
(7)
We need to determine the fiducial vector $|\psi_{0}\rangle$. The standard
choice is to take it as the vacuum $|0\rangle$. This has quite a number of
relevant properties, but the one we want to stress for what follows is that
$|0\rangle$ is an eigenstate of the Fourier transform (as they are all the
Fock states) Peres (1993), and so is the Gaussian
$\psi_{0}(q)=\langle q|0\rangle=\frac{1}{\pi^{1/4}}\,\exp(-q^{2}/2)\,,$ (8)
in appropriate units. In addition, this wavefunction represents a minimum
uncertainty state, namely
$(\Delta q)^{2}\,(\Delta p)^{2}=\frac{1}{4}\,,$ (9)
where $(\Delta q)^{2}$ and $(\Delta p)^{2}$ are the corresponding variances.
Our next task is to map the density matrix $\hat{\varrho}$ into a function
defined on $\mathbb{R}^{2}$. There exists a whole class of these
quasidistribution functions, related to different orderings of $\hat{q}$ and
$\hat{p}$. The corresponding mappings are generated by an $s$-ordered kernel
$W_{\hat{\varrho}}^{(s)}(q,p)=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}\,\hat{w}^{(s)}(q,p)]\,,$
(10)
where $\hat{w}^{(s)}$ is the double Fourier transform of the displacement
operator with a weight fixed by the operator ordering
$\displaystyle\hat{w}^{(s)}(q,p)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\exp[-i(pq^{\prime}-qp^{\prime})]\,\hat{D}(q^{\prime},p^{\prime})$
(11) $\displaystyle\times$
$\displaystyle\langle\psi_{0}|\hat{D}(q^{\prime},p^{\prime})|\psi_{0}\rangle^{-s}\,dq^{\prime}dp^{\prime}\,,$
and $s\in[-1,1]$. The mapping is invertible, so that
$\hat{\varrho}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\hat{w}^{(-s)}(q,p)\,W^{(s)}(q,p)\,dqdp\,.$
(12)
The Hermitian kernels $\hat{w}^{(s)}(q,p)$ are also a complete trace-
orthonormal set and they transform properly under displacements
$\hat{w}^{(s)}(q,p)=\hat{D}(q,p)\,\hat{w}^{(s)}(0,0)\,\hat{D}^{\dagger}(q,p)\,.$
(13)
The symmetric ordering ($s=0$) corresponds to the Wigner function $W(q,p)$ and
the associated kernel $\hat{w}^{(0)}(0,0)$ is just $2\hat{\mathcal{P}}$, where
$\hat{\mathcal{P}}$ is the parity operator. For the antinormal ordering
($s=-1$), which corresponds to the Husimi $Q$ function,
$\hat{w}^{(0)}(0,0)=|0\rangle\langle 0|/\pi$.
The quasidistribution functions (10) fulfill all the basic properties required
for any good probabilistic description. First, due to the Hermiticity of
$\hat{w}^{(s)}(q,p)$, they are real for Hermitian operators. Second, on
integrating $W^{(s)}(q,p)$ over one variable, the probability distribution of
the conjugate variable is reproduced. And finally, $W^{(s)}(q,p)$ is
covariant, which means that for the displaced state
$\hat{\varrho}^{\prime}=\hat{D}(q_{0},p_{0})\,\hat{\varrho}\,\hat{D}^{\dagger}(q_{0},p_{0})$,
one has
$W_{\hat{\varrho}^{\prime}}^{(s)}(q,p)=W_{\hat{\varrho}}^{(s)}(q-q_{0},p-p_{0})\,,$
(14)
so that these functions follow displacements rigidly without changing their
form, reflecting the fact that physics should not depend on a certain choice
of the origin.
## III Single qudit
### III.1 Discrete phase space
We consider a system living in a Hilbert space $\mathcal{H}_{d}$, whose
dimension $d$ is assumed from now on to be a prime number. We choose a
computational basis $|\ell\rangle$ in $\mathcal{H}_{d}$ ($\ell=0,\ldots,d-1$)
which we arbitrarily interpret as the “position” basis, with periodic boundary
conditions $|\ell+d\rangle=|\ell\rangle$. The conjugate “momentum” basis can
be introduced by means of the discrete Fourier transform Vourdas (2004), that
is
$|\tilde{\ell}\rangle=\hat{\mathcal{F}}\,|\ell\rangle\,,$ (15)
where
$\hat{\mathcal{F}}=\frac{1}{\sqrt{d}}\sum_{\ell,\ell^{\prime}=0}^{d-1}\omega(\ell\,\ell^{\prime})\,|\ell\rangle\langle\ell^{\prime}|\,,$
(16)
and we use the notation
$\omega(\ell)=\omega^{\ell}=\exp(i2\pi\ell/d)\,,$ (17)
$\omega=\exp(i2\pi/d)$ being a $d$th root of the unity. Whenever we do not
specify the ranges in a sum, we understand the index running all its natural
domain.
Once we have position and momentum basis, the phase space turns out to be a
periodic $d\times d$ grid of points; i.e., the torus
$\mathbb{Z}_{d}\times\mathbb{Z}_{d}$, where $\mathbb{Z}_{d}$ is the field of
the integer numbers modulo $d$.
Mimicking the approach in Sec. II, we introduce the operators $\hat{U}$ and
$\hat{V}$, which generate finite translations in position and momentum,
respectively. In fact, $\hat{U}$ generates cyclic shifts in the position
basis, while $\hat{V}$ is diagonal
$\hat{U}^{n}|\ell\rangle=|\ell+n\rangle\,,\qquad\hat{V}^{m}|\ell\rangle=\omega(m\ell)\,|\ell\rangle\,,$
(18)
where addition and multiplication must be understood modulo $d$. Conversely,
$\hat{U}$ is diagonal in the momentum basis and $\hat{V}$ acts as a shift,
which is reflected also by the fact that
$\hat{V}=\mathcal{F}\,\hat{U}\,\mathcal{F}^{\dagger}\,,$ (19)
much in the spirit of the standard way of looking at complementary variables
in the infinite-dimensional Hilbert space: the position and momentum
eigenstates are Fourier transform one of the other. Note that the operators
$\hat{U}$ and $\hat{V}$ are generalizations of the Pauli matrices $\sigma_{x}$
and $\sigma_{z}$, so many authors use the notation $\hat{X}$ and $\hat{Z}$ for
them.
One can directly check the identity
$\hat{V}^{m}\hat{U}^{n}=\omega(mn)\,\hat{U}^{n}\hat{V}^{m}\,,$ (20)
which is the finite-dimensional version of the Weyl form of the commutation
relations and show that they obey a generalized Clifford algebra Chuang and
Nielsen (2000).
One may be tempted to define discrete position and momentum operators. A
possible way to achieve this is to write Vourdas (2005)
$\hat{U}=\exp(-i2\pi\hat{P}/d)\,,\qquad\hat{V}=\exp(i2\pi\hat{Q}/d)\,,$ (21)
with
$\hat{Q}=\sum_{\ell}\ell\,|\ell\rangle\langle\ell|\,,\qquad\hat{P}=\sum_{\tilde{\ell}}\tilde{\ell}\,|\tilde{\ell}\rangle\langle\tilde{\ell}|\,.$
(22)
However, for finite quantum systems the Heisenberg-Weyl group is discrete,
there is no Lie algebra (that is, there are no infinitesimal displacements)
and the role of position and momentum is limited. For this reason our
formalism is mainly based on the operators $\hat{U}$ and $\hat{V}$.
Next we introduce the displacement operators
$\hat{D}(m,n)=\phi(m,n)\,\hat{U}^{n}\hat{V}^{m}\,,$ (23)
where $\phi(m,n)$ is a phase required to avoid plugging extra factors when
acting with $\hat{D}$. The conditions of unitarity and periodicity restrict
the possible values of $\phi$. One relevant choice (for $d>2$) that have been
analyzed in the literature is Björk et al. (2008)
$\phi(m,n)=\omega(2^{-1}\,mn)\,,$ (24)
where $2^{-1}$ is the multiplicative inverse of $2$ in $\mathbb{Z}_{d}$. For
qubits, $\phi(m,n)$ may be taken as $\phi(m,n)=\pm i^{mn}$.
Without entering in technical details, this choice guarantees all the good
properties, in particular the analogous to Eq. (5):
$\displaystyle\hat{D}(m_{1},n_{1})\hat{D}(m_{2},n_{2})=\omega[2^{-1}(m_{1}n_{2}-m_{2}n_{1})]$
$\displaystyle\times\hat{D}(m_{1}+m_{2},n_{1}+n_{2})\,,$ (25)
and the following relation
$\frac{1}{d}\sum_{m,n}\hat{D}(m,n)=\hat{\mathcal{P}}\,,$ (26)
where $\hat{\mathcal{P}}$ is the parity operator
$\hat{\mathcal{P}}|\ell\rangle=|-\ell\rangle$ modulo $d$. Physically, this is
the basis for translational covariance and this also means that $\hat{D}(m,n)$
translates the standard basis states cyclically $m$ places in one direction
and $n$ places in the orthogonal one, as one would expect from a displacement
operator.
### III.2 Coherent states
Once a proper displacement operator has been settled, the coherent states for
a single qudit can be defined as
$|m,n\rangle=\hat{D}(m,n)\,|\psi_{0}\rangle\,,$ (27)
where $|\psi_{0}\rangle$ is again a reference state. These states are also
labeled by points of the discrete phase space, as it should be.
A possible choice Saraceno (1990); Paz et al. (2004) is to use for
$|\psi_{0}\rangle$ the ground state of the Harper Hamiltonian Harper (1955)
$\hat{H}=2-\frac{\hat{U}+\hat{U}^{\dagger}}{2}-\frac{\hat{V}+\hat{V}^{\dagger}}{2}\,,$
(28)
which is considered as the discrete counterpart of the harmonic oscillator
with the proper periodicity conditions. While such a replacement is
interesting, it is by no means unique.
We prefer to take a different route, pioneered by Galetti and coworkers
Galetti and Marchiolli (1996). We use again as a guide the analogy with the
continuous case and look for eigenstates $|f\rangle$ of the discrete Fourier
transform, which play the role of Fock states for our problem and are
determined by
$\langle\ell|\hat{\mathcal{F}}|f\rangle=i^{\ell}\,\langle\ell|f\rangle\,.$
(29)
Obviously, the fact that $\hat{\mathcal{F}}^{4}=\hat{\openone}$ implies that
it has four eigenvalues: 1, $-1$, $i$, and $-i$. The solutions of this
equation were fully studied by Mehta Mehta (1987) (see also Ruzzi Ruzzi
(2006)). Taking $|\psi_{0}\rangle$ as the “ground” state (i.e., $\ell=0$) one
gets
$|\psi_{0}\rangle=\frac{1}{\sqrt{C}}\sum_{k\in\mathbb{Z}}\sum_{\ell}\omega(k\ell)\,e^{-\frac{\pi}{d}k^{2}}\,|\ell\rangle\,,$
(30)
and the normalization constant $C$ is given by
$C=\sum_{k\in\mathbb{Z}}e^{-\frac{2\pi}{d}k^{2}}=\vartheta_{3}\left(0\bigl{|}e^{-\frac{2\pi}{d}}\right)\,,$
(31)
$\vartheta_{3}$ being the third Jacobi function Mumford (1983). Note in
passing that this fiducial state can be alternatively represented as
$|\psi_{0}\rangle=\frac{1}{\sqrt{C}}\sum_{\ell}\vartheta_{3}\left(\frac{\pi\ell}{d}\bigl{|}e^{-\frac{\pi}{d}}\right)|\ell\rangle\,.$
(32)
The appearance of the Jacobi function in the present context can be directly
understood by realizing that this function is a periodic eigenstate of the
discrete Fourier operator with eigenvalue $+1$ and period $d$. In addition, it
plays the role of the Gaussian for periodic variables, which makes this
approach even more appealing Řeháček et al. (2008).
We also observe that $|\psi_{0}\rangle$ satisfies a “parity” condition: if we
write it as $|\psi_{0}\rangle=\sum_{\ell}c_{\ell}\,|\ell\rangle$, then
$c_{\ell}=c_{-\ell}$. This guarantees that the average values of $\hat{U}$ and
$\hat{V}$ in $|\psi_{0}\rangle$ are the same:
$\langle\psi_{0}|\hat{U}|\psi_{0}\rangle=\langle\psi_{0}|\hat{V}|\psi_{0}\rangle$.
The coherent states (27) have properties fully analogous to the standard ones
for continuous variables, as one can check with little effort.
The Harper Hamiltonian commutes with the Fourier operator
$[\hat{\mathcal{F}},\hat{H}]=0$. In fact, the state (30) is an approximate
eigenstate of (28) in the high-dimensional limit
$\hat{H}|\psi_{0}\rangle\simeq\left(\frac{\pi}{d}-\frac{\pi^{2}}{2d^{2}}+\frac{\pi^{3}}{6d^{3}}\right)|\psi_{0}\rangle\,,\qquad
d\gg 1\,,$ (33)
which provides another argument for its use as a reference.
Finally, according to the recent results in Refs. Forbes et al. (2003) and
Massar and Spindel (2008), the following uncertainty relation holds
$(\Delta U)^{2}\,(\Delta V)^{2}\geq\frac{\pi^{2}}{d^{2}}\,,$ (34)
where $(\Delta U)^{2}=1-|\langle\psi|\hat{U}|\psi\rangle|^{2}$ [and an
analogous expression for $(\Delta V)^{2}$] denotes the circular dispersion,
which is the natural generalization of variance for unitary operators. One can
check that $|\psi_{0}\rangle$ saturates this inequality, confirming that it is
also a minimum uncertainty state.
### III.3 Quasidistribution functions
The displacement operators lead us to introduce a Hermitian $s$-ordered kernel
$\displaystyle\hat{w}^{(s)}(m,n)$ $\displaystyle=$
$\displaystyle\frac{1}{d}\sum_{k,l}\omega(nk-ml)\,\hat{D}(m,n)$ (35)
$\displaystyle\times$
$\displaystyle\langle\psi_{0}|\hat{D}(m,n)|\psi_{0}\rangle^{-s}\,,$
which, as $\hat{w}^{(s)}(q,p)$ in Eq. (11), appears as a double Fourier
transform of $\hat{D}$ with a weight determined by the operator ordering.
However, here the parameter $s$ takes only discrete values ($s=-1,0,1$).
These kernels are normalized and covariant under transformations of the
generalized Pauli group
$\hat{D}(m,n)\,\hat{w}^{(s)}(k,l)\,\hat{D}^{\dagger}(m,n)=\hat{w}^{(s)}(k+m,l+n)\,.$
(36)
They can be then conveniently represented as
$\hat{w}^{(s)}(m,n)=\hat{D}(m,n)\,\hat{w}^{(s)}(0,0)\,\hat{D}^{\dagger}(m,n)\,,$
(37)
where, according to Eq. (26), $\hat{w}^{(s)}(0,0)$ coincides with the parity
for $s=0$ ($d\neq 2$), as in the continuous case.
The $s$-ordered quasidistribution functions $W^{(s)}_{\hat{\varrho}}$ are
generated through the mapping
$W^{(s)}_{\hat{\varrho}}(m,n)=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}\,\hat{w}^{(s)}(m,n)]\,,$
(38)
which is invertible, so that
$\hat{\varrho}=\frac{1}{d}\sum_{m,n}\hat{w}^{-(s)}(m,n)\,W^{(s)}(m,n)\,.$ (39)
These functions fulfill all the basic properties required for the
probabilistic description we are looking for. Let us apply them to the
reference state $|\psi_{0}\rangle$ (notice that any other coherent state is
just a displaced copy of this one). The corresponding Wigner function ($s=0$)
can be obtained after some algebra. We omit the details and merely quote the
final result:
$\displaystyle W_{|\psi_{0}\rangle}(m,n)$ $\displaystyle=$
$\displaystyle\frac{d}{C}\sum_{k}\sum_{p,q\in\mathbb{Z}}\omega[(2k-1-2m)n]$
(40) $\displaystyle\times$ $\displaystyle\exp[-k+2m+qd-(d-1)/2]^{2}$
$\displaystyle\times$ $\displaystyle\exp[-(\pi/d)(k+pd-d/2)^{2}]\,,$
which, in the limit $d\gg 1$, can be approximated by the compact expression
$\displaystyle W^{(0)}_{|\psi_{0}\rangle}(m,n)$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{2}}{d^{3/2}}\sum_{k,l}(-1)^{kl}\,\omega(mk-nl)$ (41)
$\displaystyle\times$ $\displaystyle\exp[-\pi(k^{2}+l^{2})/(2d)]\,.$
Figure 1: $Q$ function (42) for the reference state $|\psi_{0}\rangle$, which
plays the role of vacuum for continuous states.
The $Q$ function ($s=-1$) for the same state $|\psi_{0}\rangle$ reduces to
$Q_{|\psi_{0}\rangle}=|\langle\psi_{0}|\hat{D}(m,n)|\psi_{0}\rangle|^{2}\,,$
(42)
which exhibits the additional interesting symmetry
$Q_{|\psi_{0}\rangle}(m,n)=Q_{|\psi_{0}\rangle}(-n,m)\,.$ (43)
In Fig. 1 we have plotted this $Q$ function for a 31-dimensional system. The
aspect of the figure confirms the issues one expects from a fairly localized
Gaussian state.
## IV Many qudits
### IV.1 Discrete phase space
Next, we consider a system of $n$ identical qudits, living in the Hilbert
space $\mathcal{H}_{d^{n}}$. Instead of natural numbers, it is convenient to
use elements of the finite field $\mathrm{GF}(d^{n})$ to label states: in this
way we can almost directly translate all the properties studied before for a
single qudit and we can endow the phase-space with many of the geometrical
properties of the ordinary plane Gibbons et al. (2004). In the Appendix we
briefly summarize the basic notions of finite fields needed to proceed.
We denote by $|\lambda\rangle$ [from here on, Greek letters will label
elements in the field $\mathrm{GF}(d^{n})$] an orthonormal basis in the
Hilbert space of the system. Operationally, the elements of the basis can be
labeled by powers of a primitive element using, for instance, the polynomial
or the normal basis.
The generators of the Pauli group act now as
$\hat{U}_{\nu}|\lambda\rangle=|\lambda+\nu\rangle\,,\qquad\hat{V}_{\mu}|\lambda\rangle=\chi(\mu\lambda)\,|\lambda\rangle\,,$
(44)
where $\chi(\lambda)$ is an additive character (defined in the Appendix) and
the Weyl form of the commutation relations reads as
$\hat{V}_{\mu}\hat{U}_{\nu}=\chi(\mu\nu)\,\hat{U}_{\nu}\hat{V}_{\mu}\,.$ (45)
The finite Fourier transform Vourdas (2007)
$\hat{\mathcal{F}}=\frac{1}{\sqrt{d^{n}}}\sum_{\lambda,\lambda^{\prime}}\chi(\lambda\,\lambda^{\prime})\,|\lambda\rangle\langle\lambda^{\prime}|$
(46)
allows us to introduce the conjugate basis
$|\hat{\lambda}\rangle=\hat{\mathcal{F}}|\lambda\rangle$ and also we have
$\hat{V}_{\mu}=\hat{\mathcal{F}}\,\hat{U}_{\mu}\,\hat{\mathcal{F}}^{\dagger}\,.$
(47)
In this way, the concepts delineated in the previous section can be
immediately generalized. For example, the displacement operators are
$\hat{D}(\mu,\nu)=\phi(\mu,\nu)\,\hat{U}_{\nu}\hat{V}_{\mu}\,,$ (48)
where the phase $\phi(\mu,\nu)$ must satisfy the conditions
$\phi(\mu,\nu)\,\phi^{\ast}(\mu,\nu)=1\,,\qquad\phi(\mu,\nu)\,\phi(-\mu,-\nu)=\chi(-\mu\nu)\,,$
(49)
to guarantee the unitarity and orthogonality of $\hat{D}$. We also impose
$\phi(\mu,0)=1$ and $\phi(0,\nu)=1$, which means that the displacements along
the “position” axis $\mu$ and the “momentum” axis $\nu$ are not associated
with any phase.
For fields of odd characteristics one possible form of this phase is
$\phi(\mu,\nu)=\chi(-2^{-1}\,\mu\nu)\,,$ (50)
and we have then the same composition law as in Eq. (III.1), namely
$\displaystyle\hat{D}(\mu_{1},\nu_{1})\hat{D}(\mu_{2},\nu_{2})$
$\displaystyle=$ $\displaystyle\chi[2^{-1}(\mu_{1}\nu_{2}-\mu_{2}\nu_{1})]$
(51) $\displaystyle\times$
$\displaystyle\hat{D}(\mu_{1}+\mu_{2},\nu_{1}+\nu_{2})\,.$
### IV.2 Coherent states
Given our previous discussion, it seems reasonable to extend the coherent
states (27) in the form
$|\mu,\nu\rangle=\hat{D}(\mu,\nu)\,|\Psi_{0}\rangle\,,$ (52)
where $|\Psi_{0}\rangle$ is a reference state to be determined. In the
continuous case, the extension of coherent states (6) to many degrees of
freedom is straightforward: they are simply obtained by taking the direct
product of single-mode coherent states. To reinterpret (52) in the same
spirit, one needs first to map the abstract Hilbert space
$\mathcal{H}_{d^{n}}$, where the $n$-qudit system lives, into $n$ single-qudit
Hilbert spaces $\mathcal{H}_{d}\otimes\cdots\otimes\mathcal{H}_{d}$. This is
achieved by expanding any field element in a convenient basis
$\\{\theta_{j}\\}$ ($j=1,\ldots,n$), so that
$\lambda=\sum_{j}\ell_{j}\,\theta_{j}\,,$ (53)
where $\ell_{j}\in\mathbb{Z}_{d}$. Then, we can represent the states as
$|\lambda\rangle=|\ell_{1}\rangle\otimes\cdots\otimes|\ell_{n}\rangle=|\ell_{1},\ldots,\ell_{n}\rangle\,,$
(54)
and the coefficients $\ell_{j}$ play the role of quantum numbers for each
qudit.
The use of the selfdual basis is especially advantageous, since only then the
basic operators (and the Fourier operator) factorize in terms of single-qudit
analogues
$\hat{U}_{\nu}=\hat{U}^{n_{1}}\otimes\cdots\otimes\hat{U}^{n_{n}}\,,\qquad\hat{V}_{\mu}=\hat{V}^{m_{1}}\otimes\cdots\otimes\hat{V}^{m_{n}}\,,$
(55)
and the displacement operators factorize accordingly
$\hat{D}(\mu,\nu)=\hat{D}(m_{1},n_{1})\otimes\cdots\otimes\hat{D}(m_{n},n_{n})\,,$
(56)
where $m_{j},n_{j}\in\mathbb{Z}_{d}$ are the coefficients of the expansion of
$\mu$ and $\nu$ in the basis, respectively. In consequence, the eigenstates of
the Fourier transform are direct product of single-qudit eigenstates and we
can write for the reference state
$|\Psi_{0}\rangle=\bigotimes_{j=1}^{n}|\psi_{0j}\rangle\,,$ (57)
where $|\psi_{0j}\rangle$ are of the form (30) for each qudit (with $d>2$).
For qubits, we have Muñoz et al. (2009)
$|\Psi_{0}\rangle=\bigotimes_{j=1}^{n}\frac{(|0\rangle+\xi|1\rangle)_{j}}{(1+\xi^{2})^{1/2}}\,,$
(58)
with $\xi=\sqrt{2}-1$.
Unfortunately, the selfdual basis can be constructed only if either $d$ is
even or both $n$ and $d$ are odd. This means that for such a simple system as
two qutrits, this privileged basis does not exist. Nevertheless, one can
always find an almost selfdual basis and one can proceed much along the same
lines with minor modifications (the interested reader can consult the
comprehensive review Björk et al. (2008) for a full account of these methods).
It is interesting to stress that for $n$ qubits, the reference state (58) can
be elegantly written in terms of the field elements $\mathrm{GF}(2^{n})$ as
follows
$|\Psi_{0}\rangle=\frac{1}{(1+\xi^{2})^{n/2}}\sum_{\alpha\in\mathrm{GF}(2^{n})}\xi^{h(\alpha)}\,|\alpha\rangle\,,$
(59)
where the function $h(\alpha)$ counts the number of nonzero coefficients
$a_{j}$ in the expansion of $\alpha$ in the basis.
The operator transforming from an arbitrary basis $\\{\theta^{\prime}_{j}\\}$
into the selfdual one $\\{\theta_{j}\\}$ is given by
$\hat{\mathcal{T}}=\sum_{\mu\in GF(2^{n})}|m_{1},\ldots,m_{n}\rangle\langle
m_{1}^{\prime},\ldots,m_{n}^{\prime}|\,,$ (60)
where
$\mu=\sum_{j}m_{j}^{\prime}\theta_{j}^{\prime}=\sum_{j}m_{j}\theta_{j}\,.$
(61)
The operator $\hat{\mathcal{T}}$ is always a permutation and plays the role of
an entangling (nonlocal) operator.
Let us examine the simple yet illustrative example of a two-qubit coherent
state. According to Eq. (59), we have
$|\Psi_{0}\rangle=\frac{1}{1+\xi^{2}}(|0\rangle+\xi|\sigma\rangle+\xi|\sigma^{2}\rangle+\xi^{2}|\sigma^{3}\rangle)\,,$
(62)
where $\sigma$ is a primitive element. The selfdual basis is
$\\{\sigma,\sigma^{2}\\}$, and we have the representation
$\displaystyle|0\rangle=|00\rangle=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\
1\end{array}\right)\,,\qquad|\sigma\rangle=|10\rangle=\left(\begin{array}[]{c}0\\\
0\\\ 1\\\ 0\end{array}\right)\,,$ (71) (72)
$\displaystyle|\sigma^{2}\rangle=|01\rangle=\left(\begin{array}[]{c}0\\\ 1\\\
0\\\
0\end{array}\right)\,,\qquad|\sigma^{3}\rangle=|11\rangle=\left(\begin{array}[]{c}1\\\
0\\\ 0\\\ 0\end{array}\right)\,.$ (81)
In consequence,
$\displaystyle|\Psi_{0}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{1+\xi^{2}}\left(\begin{array}[]{c}\xi^{2}\\\ \xi\\\
\xi\\\ 1\end{array}\right)$ (86) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\xi^{2}}}\left(\begin{array}[]{c}\xi\\\
1\end{array}\right)\otimes\frac{1}{\sqrt{1+\xi^{2}}}\left(\begin{array}[]{c}\xi\\\
1\end{array}\right)\,.$ (91)
In a non selfdual basis, such as $\\{\sigma,\sigma^{3}\\}$, we have
$|0\rangle=|00\rangle\,,\quad|\sigma\rangle=|10\rangle\,,\quad|\sigma^{3}\rangle=|01\rangle\,,\quad|\sigma^{2}\rangle=|11\rangle\,,$
(92)
and
$|\Psi_{0}\rangle=\frac{1}{1+\xi^{2}}\left(\begin{array}[]{c}\xi\\\ \xi^{2}\\\
\xi\\\ 1\end{array}\right)\,.$ (93)
The transition operator (60) turns out to be
$\hat{\mathcal{T}}=\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right)\,,$ (94)
which is nothing but a matrix representation of the CNOT operator.
### IV.3 Quasidistribution functions
The displacement operators (50) immediately suggest to introduce an
$s$-ordered kernel
$\displaystyle\hat{w}^{(s)}(\mu,\nu)$ $\displaystyle=$
$\displaystyle\frac{1}{d^{n}}\sum_{\lambda,\kappa}\chi(\mu\lambda-\nu\kappa)\,\hat{D}(\mu,\nu)$
(95) $\displaystyle\times$
$\displaystyle\langle\Psi_{0}|\hat{D}(\mu,\nu)|\Psi_{0}\rangle^{-s}\,,$
which, in view of Eq. (46), can also be interpreted as a double Fourier
transform of $\hat{D}(\mu,\nu)$. We can next introduce $s$-ordered
quasidistribution functions through
$W^{(s)}_{\hat{\varrho}}(\mu,\nu)=\mathop{\mathrm{Tr}}\nolimits[\hat{\varrho}\,\hat{w}^{(s)}(\mu,\nu)]\,,$
(96)
and the inversion relation reads as
$\hat{\varrho}=\frac{1}{d^{n}}\sum_{\mu,\nu}\hat{w}^{(-s)}(\mu,\nu)\,W^{(s)}_{\hat{\varrho}}(\mu,\nu)\,.$
(97)
Due to the factorization of the character in the selfdual basis, the kernels
$\hat{w}^{(s)}(\mu,\nu)$ factorize in this basis
$\hat{w}^{(s)}(\mu,\nu)=\prod_{j}\hat{w}^{(s)}(m_{j},n_{j})\,,$ (98)
and, consequently, also do the corresponding quasidistributions
$W_{\hat{\varrho}}^{(s)}(\mu,\nu)=\prod_{j}W_{\hat{\varrho}_{j}}^{(s)}(m_{j},n_{j})\,.$
(99)
For the particular case of the Wigner function, one can check that
$\sum_{\mu,\nu}W_{\hat{\varrho}}(\mu,\nu)\,\delta_{\nu,\alpha\mu+\beta}=\sum_{\mu,\nu}W_{\hat{\varrho}}(\mu,\nu)\,\delta_{\nu,-\alpha^{-1}\mu-\beta}\,,$
(100)
that is, the sum over a line of slope $\alpha$ is the same as over a line of
slope $-\alpha^{-1}$. The sum over the axes $\mu$ and $\nu$ are thus equal
$\sum_{\mu,\nu}W_{\hat{\varrho}}(\mu,\nu)\,\delta_{\nu,0}=\sum_{\mu,\nu}W_{\hat{\varrho}}(\mu,\nu)\,\delta_{\mu,0}\,.$
(101)
Note also, that the $Q$ function reduces to
$Q_{\hat{\varrho}}(\mu,\nu)=\langle\mu,\nu|\hat{\varrho}|\mu,\nu\rangle\,.$
(102)
In Fig. 2 we have plotted this $Q$ function for the reference state
$|\Psi_{0}\rangle$ in a system of three qutrits. The selfdual basis here is
$\\{\sigma,\sigma^{3},\sigma^{9}\\}$ and the primitive element is a solution
of the irreducible polynomial $x^{3}+2x^{2}+1=0$.
Figure 2: $Q$ function for the reference state $|\Psi_{0}\rangle$, for a
system of three qutrits. The order in the axes is as follows: $\sigma^{13}$,
$\sigma^{17}$, $\sigma^{14}$, $\sigma$, $\sigma^{2}$, $\sigma^{21}$,
$\sigma^{23}$, $\sigma^{7}$, $\sigma^{15}$, $\sigma^{4}$, $\sigma^{16}$,
$\sigma^{6}$, $\sigma^{8}$, 0, $\sigma^{9}$, $\sigma^{12}$, $\sigma^{25}$,
$\sigma^{24}$, $\sigma^{5}$, $\sigma^{3}$, $\sigma^{19}$, $\sigma^{11}$,
$\sigma^{22}$, $\sigma^{10}$, $\sigma^{20}$, $\sigma^{18}$, $\sigma^{26}$,
with $\sigma$ the primitive element.
## V Squeezed states
Squeezed states constitute a simple nontrivial enlargement of the notion of
coherent states. In continuous variables, a squeezed state is a minimum
uncertainty state that my have less fluctuations in one quadrature ($\hat{q}$
or $\hat{p}$) than a coherent state. They are generated from the vacuum by
using the unitary squeeze operator
$\hat{S}(\mathfrak{s})=\exp[-i\mathfrak{s}\,(\hat{q}\hat{p}+\hat{p}\hat{q})]\,,$
(103)
with a subsequent displacement to an arbitrary point in the complex plane
$|q,p;\mathfrak{s}\rangle=\hat{D}(q,p)\hat{S}(\mathfrak{s})\,|\psi_{0}\rangle\,.$
(104)
It is easy to check that
$\hat{S}(\mathfrak{s})\,\hat{q}\,\hat{S}^{\dagger}(\mathfrak{s})=\hat{q}\,e^{\mathfrak{s}}\,,\qquad\hat{S}(\mathfrak{s})\,\hat{p}\,\hat{S}^{\dagger}(\mathfrak{s})=\hat{p}\,e^{-\mathfrak{s}}\,,$
(105)
so that, the operator $\hat{S}(\mathfrak{s})$ attenuates one quadrature and
amplifies the canonical one by the same factor determined by the squeeze
factor $\mathfrak{s}$, which, for simplicity, we have taken as real. As a
simple consequence of (105) one can verify the transformations for
$\hat{U}(q)$ and $\hat{V}(p)$:
$\hat{S}(\mathfrak{s})\,\hat{U}(q)\,\hat{S}^{\dagger}(\mathfrak{s})=U^{\mathfrak{s}}(q)\,,\quad\hat{S}(\mathfrak{s})\,\hat{V}(p)\,\hat{S}^{\dagger}(\mathfrak{s})=\hat{V}^{-\mathfrak{s}}(p)\,.$
(106)
For a single qudit, squeezed states have been recently considered in detail in
Ref. Marchiolli et al. (2007), using an extended Cahill-Glauber formalism.
Here, we prefer to follow an alternative approach and define a squeeze
operator as
$\hat{S}_{s}=\sum_{\ell}|\ell\rangle\langle s\ell|\,,\qquad
s\in\mathbb{Z}_{d}\,.$ (107)
At first sight, this can appear as a rather abstract choice. However, notice
that
$\hat{S}_{s}^{\dagger}\,\hat{U}^{n}\,\hat{S}^{\dagger}_{s}=\hat{U}^{n\,s}\,,\qquad\hat{S}_{s}\,\hat{V}^{m}\,\hat{S}_{s}=\hat{V}^{m\,s^{-1}}\,,$
(108)
which is a direct translation of the action (106) to this discrete case. This
also means that in the squeezed “vacuum”
$|\psi_{0};s\rangle=\hat{S}_{s}|\psi_{0}\rangle\,,$ (109)
the average values of some powers of the displacement operators are the same
$\langle\psi_{0};s|\hat{U}|\psi_{0};s\rangle=\langle\psi_{0};s|\hat{V}^{s^{2}}|\psi_{0};s\rangle\,.$
(110)
Perhaps, the clearest way to visualize this squeezing is to use a
quasidistribution, such as, e.g., the Wigner function. If
$\hat{\varrho}_{s}=\hat{S}_{s}\,\hat{\varrho}\,\hat{S}_{s}^{\dagger}$ denotes
the density operator of a squeezed state, we have
$W_{\hat{\varrho}_{s}}(m,n)=W_{\hat{\varrho}}(sm,s^{-1}n)\,,$ (111)
whose geometrical interpretation is obvious and is the phase-space counterpart
of the property (105). For reasons that will become evident soon, we refer to
this as “geometrical squeezing”. We also note the following symmetry property
of the Wigner function
$\sum_{m,n}W_{\hat{\varrho}_{s}}(m,n)\,\delta_{n,0}=\sum_{m,n}W_{\hat{\varrho}_{s^{-1}}}(m,n)\,\delta_{m,0}\,.$
(112)
For many qudits, our developed intuition suggests a direct translation of
(107) in terms of the field elements in $\mathrm{GF}(d^{n})$, namely
$\hat{S}_{\varsigma}=\sum_{\lambda}|\lambda\rangle\langle\varsigma\lambda|\,,\qquad\varsigma\in\mathrm{GF}(d^{n})\,,$
(113)
in terms of which we can write relations similar to Eqs.(108)-(111). In fact,
one can define a squeezed “vacumm” as in Eq. (109), i.e.,
$|\Psi_{0};\varsigma\rangle=\hat{S}_{\varsigma}|\Psi_{0}\rangle$. In Fig. 3 we
plot the Wigner function for this squeezed state in a system of three qutrits
with $\varsigma=\sigma^{7}$.
Figure 3: Wigner function for a squeezed “vacuum” state
$|\Psi_{0},\varsigma\rangle$, for a system of three qutrits, with the same
order in the axes as in Fig. 2.
Nevertheless, now the squeezing acquires a new physical perspective: the
squeeze operator (113) cannot be, in general, factorized into a product of
single qudit squeezing operators. This means that by applying
$\hat{S}_{\varsigma}$ to a factorized state we generate correlations between
qudits; i.e., we create entangled states. The most striking example is of
course the $n$ qubit case, since there is no single qubit squeezing.
To understand these correlations consider a general factorized state
$|\Psi\rangle=\sum_{\lambda}\\!C_{\lambda}|\lambda\rangle=\sum_{c_{\ell_{1}},\ldots,c_{\ell_{n}}}\\!\\!\\!c_{\ell_{1}}\ldots
c_{\ell_{n}}\,|\ell_{1},\ldots,\ell_{n}\rangle\,,$ (114)
and apply (113). The resulting state turns out to be
$\hat{S}_{\varsigma}|\Psi\rangle=\sum_{\ell_{1},\cdots,\ell_{n}}C_{m_{1}\theta_{1}+\ldots+m_{n}\theta_{n}}\,|\ell_{1},\ldots,\ell_{n}\rangle\,,$
(115)
where
$\displaystyle\displaystyle m_{i}=\sum_{j,k=0}^{d-1}f_{ijk}\ell_{j}h_{k}\,,$
(116) $\displaystyle
f_{ijk}=\mathop{\mathrm{tr}}\nolimits(\theta_{i}\theta_{j}\theta_{k})\,,\quad
h_{k}=\mathop{\mathrm{tr}}\nolimits(\varsigma^{-1}\theta_{k})\,,$
$\mathop{\mathrm{tr}}\nolimits$ (written in lower case) is the trace operation
in the field (see the Appendix) and $\\{\theta_{j}\\}$ is the basis.
As a example let us consider a three-qubit system. Now the selfdual basis is
$\\{\theta_{1}=\sigma^{3},\theta_{2}=\sigma^{5},\theta_{3}=\sigma^{6}\\}$,
where $\sigma$ is a primitive element, solution of the irreducible polynomial
$x^{3}+x+1=0$. The result of applying $\hat{S}_{\sigma^{k}}$ to the state
(114) can be expressed in terms of
$\displaystyle\hat{S}_{\sigma}|\Psi\rangle$ $\displaystyle=$
$\displaystyle\sum_{\lambda\in\mathrm{GF}(2^{3})}\\!\\!C_{\sigma^{6}\lambda}|\lambda\rangle=\sum_{p,q,r\in\mathbb{Z}_{2}}c_{p+q}c_{p+r}c_{q}|p,q,r\rangle\,,$
$\displaystyle\hat{S}_{\sigma^{3}}|\Psi\rangle$ $\displaystyle=$
$\displaystyle\sum_{\lambda\in\mathrm{GF}(2^{3})}\\!\\!C_{\sigma^{4}\lambda}|\lambda\rangle=\sum_{p,q,r\in\mathbb{Z}_{2}}c_{p+q+r}c_{p+r}c_{r}|p,q,r\rangle\,.$
In fact, the transformations $\\{\hat{S}_{\sigma^{5}},\hat{S}_{\sigma^{6}}\\}$
generate the same entanglement (except for permutations) as
$\hat{S}_{\sigma^{3}}$, while
$\\{\hat{S}_{\theta^{2}},\hat{S}_{\theta^{4}}\\}$ generate the same
entanglement (again except for permutations) as $\hat{S}_{\sigma}$.
## VI Concluding remarks
In summary, we have provided a handy toolbox for dealing with many-qudit
systems in phase space. The mathematical basis of our approach is the use
algebraic field extensions that produce results in composite dimensions in a
manner very close to the continuous case.
Another major advantage of our theory relies on the use of the finite Fourier
transform and its eigenstates for the definition of coherent states. We
believe that this makes a clear connection with the standard coherent states
for continuous variables and constitutes an elegant solution to this problem.
The factorization properties of the resulting coherent states in different
bases is also an interesting question.
We have also established a set of important results that have allowed us to
obtain discrete analogs of squeezed states. While for a single qudit, these
squeezed states have the properties one would expect from our continuous-
variable experience, for many qudits an amazing relation with entanglement
appears.
We think that the techniques presented here are more than a mere academic
curiosity, for they are immediately applicable to a variety of experiments
involving qudit systems.
## Appendix A Finite fields
In this appendix we briefly recall the minimum background needed in this
paper. The reader interested in more mathematical details is referred, e.g.,
to the excellent monograph by Lidl and Niederreiter Lidl and Niederreiter
(1986).
A commutative ring is a nonempty set $R$ furnished with two binary operations,
called addition and multiplication, such that it is an Abelian group with
respect the addition, and the multiplication is associative. Perhaps, the
motivating example is the ring of integers $\mathbb{Z}$, with the standard sum
and multiplication. On the other hand, the simplest example of a finite ring
is the set $\mathbb{Z}_{n}$ of integers modulo $n$, which has exactly $n$
elements.
A field $F$ is a commutative ring with division, that is, such that 0 does not
equal 1 and all elements of $F$ except 0 have a multiplicative inverse (note
that 0 and 1 here stand for the identity elements for the addition and
multiplication, respectively, which may differ from the familiar real numbers
0 and 1). Elements of a field form Abelian groups with respect to addition and
multiplication (in this latter case, the zero element is excluded).
The characteristic of a finite field is the smallest integer $d$ such that
$d\,1=\underbrace{1+1+\ldots+1}_{\mbox{\scriptsize$d$ times}}=0$ (118)
and it is always a prime number. Any finite field contains a prime subfield
$\mathbb{Z}_{d}$ and has $d^{n}$ elements, where $n$ is a natural number.
Moreover, the finite field containing $d^{n}$ elements is unique and is called
the Galois field $\mathrm{GF}(d^{n})$.
Let us denote as $\mathbb{Z}_{d}[x]$ the ring of polynomials with coefficients
in $\mathbb{Z}_{d}$. Let $P(x)$ be an irreducible polynomial of degree $n$
(i.e., one that cannot be factorized over $\mathbb{Z}_{d}$). Then, the
quotient space $\mathbb{Z}_{d}[X]/P(x)$ provides an adequate representation of
$\mathrm{GF}(d^{n})$. Its elements can be written as polynomials that are
defined modulo the irreducible polynomial $P(x)$. The multiplicative group of
$\mathrm{GF}(d^{n})$ is cyclic and its generator is called a primitive element
of the field.
As a simple example of a nonprime field, we consider the polynomial
$x^{2}+x+1=0$, which is irreducible in $\mathbb{Z}_{2}$. If $\sigma$ is a root
of this polynomial, the elements
$\\{0,1,\sigma,\sigma^{2}=\sigma+1=\sigma^{-1}\\}$ form the finite field
$\mathrm{GF}(2^{2})$ and $\sigma$ is a primitive element.
A basic map is the trace
$\mathop{\mathrm{tr}}\nolimits(\lambda)=\lambda+\lambda^{2}+\ldots+\lambda^{d^{n-1}}\,.$
(119)
It is always in the prime field $\mathbb{Z}_{d}$ and satisfies
$\mathop{\mathrm{tr}}\nolimits(\lambda+\lambda^{\prime})=\mathop{\mathrm{tr}}\nolimits(\lambda)+\mathop{\mathrm{tr}}\nolimits(\lambda^{\prime})\,.$
(120)
In terms of it we define the additive characters as
$\chi(\lambda)=\exp\left[\frac{2\pi
i}{p}\mathop{\mathrm{tr}}\nolimits(\lambda)\right]\,,$ (121)
which posses two important properties:
$\chi(\lambda+\lambda^{\prime})=\chi(\lambda)\chi(\lambda^{\prime}),\qquad\sum_{\lambda^{\prime}\in\mathrm{GF}(d^{n})}\chi(\lambda\lambda^{\prime})=d^{n}\delta_{0,\lambda}\,.$
(122)
Any finite field $\mathrm{GF}(d^{n})$ can be also considered as an
$n$-dimensional linear vector space. Given a basis $\\{\theta_{j}\\}$,
($j=1,\ldots,n$) in this vector space, any field element can be represented as
$\lambda=\sum_{j=1}^{n}\ell_{j}\,\theta_{j},$ (123)
with $\ell_{j}\in\mathbb{Z}_{d}$. In this way, we map each element of
$\mathrm{GF}(d^{n})$ onto an ordered set of natural numbers
$\lambda\Leftrightarrow(\ell_{1},\ldots,\ell_{n})$.
Two bases $\\{\theta_{1},\ldots,\theta_{n}\\}$ and
$\\{\theta_{1}^{\prime},\ldots,\theta_{n}^{\prime}\\}$ are dual when
$\mathop{\mathrm{tr}}\nolimits(\theta_{k}\theta_{l}^{\prime})=\delta_{k,l}.$
(124)
A basis that is dual to itself is called selfdual.
There are several natural bases in $\mathrm{GF}(d^{n})$. One is the polynomial
basis, defined as
$\\{1,\sigma,\sigma^{2},\ldots,\sigma^{n-1}\\},$ (125)
where $\sigma$ is a primitive element. An alternative is the normal basis,
constituted of
$\\{\sigma,\sigma^{d},\ldots,\sigma^{d^{n-1}}\\}.$ (126)
The choice of the appropriate basis depends on the specific problem at hand.
For example, in $\mathrm{GF}(2^{2})$ the elements $\\{\sigma,\sigma^{2}\\}$
are both roots of the irreducible polynomial. The polynomial basis is
$\\{1,\sigma\\}$ and its dual is $\\{\sigma^{2},1\\}$, while the normal basis
$\\{\sigma,\sigma^{2}\\}$ is selfdual.
The selfdual basis exists if and only if either $d$ is even or both $n$ and
$d$ are odd. However for every prime power $d^{n}$, there exists an almost
selfdual basis of $\mathrm{GF}(d^{n})$, which satisfies the properties:
$\mathop{\mathrm{tr}}\nolimits(\theta_{i}\theta_{j})=0$ when $i\neq j$ and
$\mathop{\mathrm{tr}}\nolimits(\theta_{i}^{2})=1$, with one possible
exception. For instance, in the case of two qutrits $\mathrm{GF}(3^{2})$, a
selfdual basis does not exist and two elements $\\{\sigma^{2},\sigma^{4}\\}$,
$\sigma$ being a root of the irreducible polynomial $x^{2}+x+2=0$, form an
almost selfdual basis
$\mathop{\mathrm{tr}}\nolimits(\sigma^{2}\sigma^{2})=1\,,\quad\mathop{\mathrm{tr}}\nolimits(\sigma^{4}\sigma^{4})=2\,,\quad\mathop{\mathrm{tr}}\nolimits(\sigma^{2}\sigma^{4})=0\,.$
(127)
## References
* Weyl (1928) H. Weyl, _Gruppentheorie und Quantemechanik_ (Hirzel-Verlag, Leipzig, 1928).
* Wigner (1932) E. P. Wigner, Phys. Rev. 40, 749 (1932).
* Moyal (1949) J. E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949).
* Lee (1995) H.-W. Lee, Phys. Rep. 259, 147 (1995).
* Schroek (1996) F. E. Schroek, _Quantum Mechanics on Phase Space_ (Kluwer, Dordrecht, 1996).
* Schleich (2001) W. P. Schleich, _Quantum Optics in Phase Space_ (Wiley-VCH, Berlin, 2001).
* Zachos et al. (2005) C. K. Zachos, D. B. Fairlie, and T. L. Curtright, eds., _Quantum mechanics in phase space_ (World Scientific, Singapore, 2005).
* Kimura (2003) G. Kimura, Phys. Lett. A 314, 339 (2003).
* Schirmer et al. (2004) S. G. Schirmer, T. Zhang, and J. V. Leahy, J. Phys. A 37, 1389 (2004).
* Kostant (1970) B. Kostant, Lect. Notes Math. 170, 87 (1970).
* Kirillov (1976.) A. A. Kirillov, _Elements of the Theory of Representations_ (Springer-Verlag, Berlin, 1976.).
* Hannay and Berry (1980) J. H. Hannay and M. V. Berry, Physica D 1, 267 (1980).
* Leonhardt (1995) U. Leonhardt, Phys. Rev. Lett. 74, 4101 (1995).
* Miquel et al. (2002) C. Miquel, J. P. Paz, M. Saraceno, E. Knill, R. Laflamme, and C. Negrevergne, Nature (London) 418, 59 (2002).
* Schwinger (1960a) J. Schwinger, Proc. Natl. Acad. Sci. USA 46, 570 (1960a).
* Schwinger (1960b) J. Schwinger, Prod. Natl. Acad. Sci. USA 46, 883 (1960b).
* Schwinger (1960c) J. Schwinger, Proc. Natl. Acad. Sci. USA 46, 1401 (1960c).
* Buot (1973) F. A. Buot, Phys. Rev. B 10, 3700 (1973).
* Cohendet et al. (1988) O. Cohendet, P. Combe, M. Sirugue, and M. Sirugue-Collin, J. Phys. A 21, 2875 (1988).
* Kasperkovitz and Peev (1994) P. Kasperkovitz and M. Peev, Ann. Phys. 230, 21 (1994).
* Opatrný et al. (1995) T. Opatrný, V. Bužek, J. Bajer, and G. Drobný, Phys. Rev. A 52, 2419 (1995).
* Rivas and de Almeida (1999) A. M. F. Rivas and A. M. O. de Almeida, Ann. Phys. 276, 223 (1999).
* Mukunda et al. (2004) N. Mukunda, S. Chaturvedi, and R. Simon, J. Math. Phys. 45, 114 (2004).
* Chaturvedi et al. (2006) S. Chaturvedi, E. Ercolessi, G. Marmo, G. Morandi, N. Mukunda, and R. Simon, J. Phys. A 39, 1405 (2006).
* Wootters (1986) W. K. Wootters, Found. Phys. 16, 391 (1986).
* Wootters and Fields (1989) W. K. Wootters and B. D. Fields, Ann. Phys. 191, 363 (1989).
* Wootters (2006) W. K. Wootters, Found. Phys. 36, 112 (2006).
* Wootters and Sussman (2007) W. K. Wootters and D. M. Sussman, arXiv:0704.1277 (2007).
* Galetti and de Toledo Piza (1988) D. Galetti and A. F. R. de Toledo Piza, Physica A 149, 267 (1988).
* Galetti and de Toledo Piza (1992) D. Galetti and A. F. R. de Toledo Piza, Physica A 186, 513 (1992).
* Galetti and de Toledo Piza (1995) D. Galetti and A. de Toledo Piza, Physica A 214, 207 (1995).
* Klimov et al. (2005) A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, J. Phys. A 38, 2747 (2005).
* Klimov et al. (2007) A. B. Klimov, J. L. Romero, G. Björk, and L. L. Sánchez-Soto, J. Phys. A 40, 3987 (2007).
* Klimov et al. (2009) A. B. Klimov, J. L. Romero, G. Björk, and L. L. Sánchez-Soto, Ann. Phys. 324, 53 (2009).
* Lidl and Niederreiter (1986) R. Lidl and H. Niederreiter, _Introduction to Finite Fields and their Applications_ (Cambridge University Press, Cambridge, 1986).
* Glauber (1963) R. J. Glauber, Phys. Rev. 131, 2766 (1963).
* Sudarshan (1963) E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
* Husimi (1940) K. Husimi, Proc. Phys. Math. Soc. Japan 22, 264 (1940).
* Hillery et al. (1984) M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
* Cahill and Glauber (1969) K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969).
* Stratonovich (1956) R. L. Stratonovich, JETP 31, 1012 (1956).
* Berezin (1975) F. A. Berezin, Commun. Math. Phys. 40, 153 (1975).
* Agarwal (1981) G. S. Agarwal, Phys. Rev. A 24, 2889 (1981).
* Scully (1983) M. O. Scully, Phys. Rev. D 28, 2477 (1983).
* Cohen and Scully (1986) L. Cohen and M. O. Scully, Found. Phys. 16, 295 (1986).
* Varilly and Gracia-Bondía (1989) J. C. Varilly and J. M. Gracia-Bondía, Ann. Phys. 190, 107 (1989).
* Heiss and Weigert (2000) S. Heiss and S. Weigert, Phys. Rev. A 63, 012105 (2000).
* Dowling et al. (1994) J. P. Dowling, G. S. Agarwal, and W. P. Schleich, Phys. Rev. A 49, 4101 (1994).
* Benedict and Czirják (1999) M. G. Benedict and A. Czirják, Phys. Rev. A 60, 4034 (1999).
* Brif and Mann (1998) C. Brif and A. Mann, J. Phys. A 31, L9 (1998).
* Björk et al. (2008) G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, Prog. Opt. 51, 469 (2008).
* Wootters (1987) W. K. Wootters, Ann. Phys. 176, 1 (1987).
* Gibbons et al. (2004) K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A 70, 062101 (2004).
* Wootters (2004) W. K. Wootters, IBM J. Res. Dev. 48, 99 (2004).
* Ruzzi et al. (2005) M. Ruzzi, M. A. Marchiolli, and D. Galetti, J. Phys. A pp. 6239–6251 (2005).
* Marchiolli et al. (2005) M. A. Marchiolli, M. Ruzzi, and D. B. F. Galetti, Phys. Rev. A 72, 042308 (2005).
* Klauder and Skagerstam (1999) J. R. Klauder and B. S. Skagerstam, _Coherent States: Applications in Physics and Mathematical Physics_ (World Scientific Publishing, Singapore, 1999).
* Perelomov (1986) A. Perelomov, _Generalized Coherent States and their Applications_ (Springer, Berlin, 1986).
* Ruzzi (2006) M. Ruzzi, J. Math. Phys 47, 063507 (2006).
* Galetti and Marchiolli (1996) D. Galetti and M. A. Marchiolli, Ann. Phys. 249, 454 (1996).
* Marchiolli et al. (2007) M. A. Marchiolli, M. Ruzzi, and D. Galetti, Phys. Rev. A 76, 032102 (2007).
* Putnam (1967) C. R. Putnam, _Commutation Properties of Hilbert Space Operators and Related Topics_ (Springer, Berlin, 1967).
* Peres (1993) A. Peres, _Quantum Theory: Concepts and Methods_ (Kluwer, Dordrecht, 1993).
* Vourdas (2004) A. Vourdas, Rep. Prog. Phys. 67, 267 (2004).
* Chuang and Nielsen (2000) I. Chuang and M. Nielsen, _Quantum Computation and Quantum Information_ (Cambridge University Press, Cambridge, 2000).
* Vourdas (2005) A. Vourdas, J. Phys. A 38, 8453 (2005).
* Saraceno (1990) M. Saraceno, Ann. Phys. 199, 37 (1990).
* Paz et al. (2004) J. P. Paz, A. J. Roncaglia, and M. Saraceno, Phys. Rev. A 69, 032312 (2004).
* Harper (1955) P. G. Harper, Proc. Phys. Soc. A 68, 874 (1955).
* Mehta (1987) M. L. Mehta, J. Math. Phys. 28, 781 (1987).
* Mumford (1983) D. Mumford, _Tata Lectures on Theta I_ (Birkhauser, Boston, 1983).
* Řeháček et al. (2008) J. Řeháček, Z. Bouchal, R. Čelechovský, Z. Hradil, and L. L. Sánchez-Soto, Phys. Rev. A 77, 032110 (2008).
* Forbes et al. (2003) G. W. Forbes, M. A. Alonso, and A. E. Siegman, J. Phys. A 36, 7027 (2003).
* Massar and Spindel (2008) S. Massar and P. Spindel, Phys. Rev. Lett. 100, 190401 (2008).
* Vourdas (2007) A. Vourdas, J. Phys. A 40, R285 (2007).
* Muñoz et al. (2009) C. Muñoz, A. B. Klimov, L. L. Sánchez-Soto, and G. Björk, Int. J. Quantum Inf. 7, 17 (2009).
|
arxiv-papers
| 2009-07-22T13:58:13 |
2024-09-04T02:49:04.102472
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. B. Klimov, C. Munoz and L. L. Sanchez-Soto",
"submitter": "Luis L. Sanchez. Soto",
"url": "https://arxiv.org/abs/0907.3845"
}
|
0907.3906
|
# Does stability of relativistic dissipative fluid dynamics imply causality?
Shi Pua,c Tomoi Koideb Dirk H. Rischkea,b aInstitut für Theoretische Physik,
Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am
Main, Germany bFrankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1,
D-60438 Frankfurt am Main, Germany cDepartment of Modern Physics, University
of Science and Technology of China, Hefei 230026, P.R. China
###### Abstract
We investigate the causality and stability of relativistic dissipative fluid
dynamics in the absence of conserved charges. We perform a linear stability
analysis in the rest frame of the fluid and find that the equations of
relativistic dissipative fluid dynamics are always stable. We then perform a
linear stability analysis in a Lorentz-boosted frame. Provided that the ratio
of the relaxation time for the shear stress tensor, $\tau_{\pi}$, to the sound
attenuation length, $\Gamma_{s}=4\eta/3(\varepsilon+P)$, fulfills a certain
asymptotic causality condition, the equations of motion give rise to stable
solutions. Although the group velocity associated with perturbations may
exceed the velocity of light in a certain finite range of wavenumbers, we
demonstrate that this does not violate causality, as long as the asymptotic
causality condition is fulfilled. Finally, we compute the characteristic
velocities and show that they remain below the velocity of light if the ratio
$\tau_{\pi}/\Gamma_{s}$ fulfills the asymptotic causality condition.
## I Introduction
Data from the Relativistic Heavy-Ion Collider (RHIC) on the collective flow of
matter in nucleus-nucleus collisions have delivered a surprising result: the
elliptic flow coefficient $v_{2}$ is sufficiently large Adams:2003zg ;
Adams:2003am ; Sorensen:2003kp ; Adler:2002pu to be compatible with
calculations performed in the framework of ideal fluid dynamics review . This
has given rise to the notion that “RHIC physicists serve up the perfect
liquid” press ; Gyulassy:2004zy ; Shuryak:2004cy .
Of course, no real liquid can have zero viscosity: for all weakly coupled
theories, i.e., theories with well-defined quasi-particles, in the dilute
limit there is a lower bound which one can derive from the uncertainty
principle Danielewicz:1984ww : the ratio of shear viscosity to entropy density
$\eta/s\gtrsim 1/12$. For certain strongly coupled theories without
quasiparticles, there is also a lower bound which can be obtained from the
AdS/CFT conjecture Kovtun:2004de , $\eta/s\geq 1/(4\pi)$, i.e., surprisingly
close to the bound for dilute, weakly coupled systems.
In order to see whether the shear viscosity of the hot and dense matter
created in nuclear collisions at RHIC is close to the lower bound, one has to
perform calculations in the framework of relativistic dissipative fluid
dynamics. This program has only been recently initiated, but has already led
to an enormous activity in the literature muronga ; roma ; luzum1 ; luzum2 ;
Song:2007ux ; Song:2007fn ; chau ; du ; pasi ; pratt ; bha ; mol ; Betz:2008me
; gue ; dkkm1 ; dkkm2 ; dkkm3 ; dkkm4 ; dkkm5 ; knk .
Fluid dynamics is an effective theory for the long-wavelength, small-frequency
modes of a given theory. In order to see this, let us introduce three length
scales: (a) a microscopic length scale, $\ell_{\rm micro}$. In all theories,
at sufficiently large temperatures this length scale can be defined as the
thermal wavelength $\lambda_{\rm th}\sim 1/T$. In weakly coupled theories with
well-defined quasi-particles, this can be interpreted as the interparticle
distance. (b) A mesoscopic length scale, $\ell_{\rm meso}$. In weakly coupled
theories and in the dilute limit, this can be identified with the mean-free
path of particles between collisions. In strongly coupled theories, such a
scale is not known and should be identified with $\ell_{\rm micro}$. (c) A
macroscopic length scale, $\ell_{\rm macro}$. This is the scale over which the
conserved densities (e.g. the charge density, $n$, or the energy density,
$\varepsilon$) of the theory vary. Thus, $\ell_{\rm
macro}^{-1}\sim|\partial\varepsilon|/\varepsilon$, i.e., $\ell_{\rm
macro}^{-1}$ is proportional to the gradients of the conserved quantities.
We now define the quantity $K\equiv\ell_{\rm meso}/\ell_{\rm macro}$. For
dilute systems, this quantity is identical to the so-called Knudsen number. If
$K$ is sufficiently small, fluid dynamics as an effective theory can be
derived in a controlled way as a power series in terms $K$. Since
$K\sim\ell_{\rm macro}^{-1}$, this series expansion is equivalent to a
gradient expansion.
To zeroth order in $K$, one obtains the equations of ideal fluid dynamics. To
first order in $K$, one obtains the Navier-Stokes (NS) equations. So-called
second-order theories contain terms of second order in $K$. Examples for the
latter are the Burnett equations samojeden , the Israel-Stewart equations for
relativistic dissipative fluid dynamics is , the memory function theory dkkm1
; dkkm4 , extended thermodynamics jou ; dkkm4 , and others else . The main
difference between first and second-order theories is the velocity of signal
propagation. The relativistic NS equations allow for infinite signal
propagation speeds and are therefore acausal. On the other hand, all second-
order theories are considered to be causal in the sense that all signal
velocities are smaller than the speed of light, provided that the parameters
of the theory are suitably chosen.
The stability and causality of fluid-dynamical theories are usually studied
around a hydrostatic state (i.e., for vanishing macroscopic flow velocity)
which is in thermodynamical equilibrium. However, if a theory is stable around
a hydrostatic state, it does not necessarily imply that it is stable in a
state of nonzero flow velocity. Following this idea, the stability and
causality of first and second-order fluid dynamics for a state with nonzero
background flow velocity (mathematically realized by a Lorentz boost) were
studied for the case of nonzero bulk viscosity, but for vanishing shear stress
and heat flow in Ref. dkkm3 . There it was found that causality and stability
are intimately related: for all parameters considered, the theory becomes
unstable if and only if there is a mode which propagates faster than the speed
of light.
In this paper, we extend this analysis to the case of nonvanishing shear
viscosity in second-order theories of relativistic dissipative fluid dynamics.
A similar analysis for a hydrostatic background has already been done by
Hiscock, Lindblom, and Olson his ; his2 , but they discussed exclusively the
low- and high-wavenumber limits his2 . As we shall show in this paper, their
analysis missed a divergence of the group velocity of a shear mode at
intermediate wavenumbers. This anomalous behavior is generic, i.e., it cannot
be removed by tuning the parameters of the theory, e.g., the relaxation time
for the shear stress tensor, $\tau_{\pi}$, and the shear viscosity, $\eta$.
However, if the ratio $\tau_{\pi}/\Gamma_{s}$, where
$\Gamma_{s}=2(D-2)\eta/[(D-1)(\varepsilon+P)]$ is the sound attenuation length
in $D$ space-time dimensions, is chosen such that the large-momentum limit of
the group velocity associated with the perturbation remains below the velocity
of light (the so-called asymptotic causality condition), one can ensure that
the divergence is restricted to a finite range of momenta. It will be
demonstrated that in this case, the causality of the theory is not
compromised. On the other hand, second-order fluid dynamics is always stable
in the rest frame of the fluid, even if we use a parameter set which violates
the asymptotic causality condition.
We also study the causality and stability for a state with nonzero background
flow velocity, i.e., in a Lorentz-boosted frame. We find that the divergence
of the group velocity is removed. However, depending on the boost velocity the
group velocity of either the shear or the sound mode may still exceed the
speed of light in a certain range of wavenumbers. Nevertheless, provided that
the ratio $\tau_{\pi}/\Gamma_{s}$ fulfills the asymptotic causality condition,
we can show that the equations are stable. In contrast to the analysis in the
rest frame, however, they become unstable if the asymptotic causality
condition is violated. We shall demonstrate that if the asymptotic causality
condition is fulfilled, the causality of the theory as a whole is not
compromised. In this sense, causality and stability are intimately related.
So far, the discussion was limited to the fluid-dynamical equations in the
linear approximation. Therefore, we expect the results to be valid for all
versions of second-order theories presently discussed in the literature, since
they differ only by nonlinear terms. We also compute the characteristic
velocities for the so-called simplified IS equations Song:2007ux without
linearizing these equations. Our analysis strongly indicates that the
characteristic velocities remain below the velocity of light if the ratio
$\tau_{\pi}/\Gamma_{s}$ is chosen such that the asymptotic causality condition
is fulfilled.
The asymptotic causality condition implies that, for a given
$\Gamma_{s}\sim\eta$, $\tau_{\pi}$ must not be arbitrarily small. This
explains why relativistic NS theory is acausal, because there
$\tau_{\pi}\rightarrow 0$, while $\eta$ is non-zero. It also implies that
second-order theories are not per se causal; they can violate causality (and
become unstable) if a too small value for $\tau_{\pi}$ is chosen. The
statement that second-order theories automatically cure the shortcomings of NS
theory is therefore not true.
This paper is organized as follows. In Sec. II, we discuss the causality and
stability of the linearized second-order fluid-dynamical equations in the
local rest frame. We also extend this analysis to nonzero bulk viscosity and
show that the divergence of the group velocity still exists in this case. In
Sec. III, this discussion is generalized to a Lorentz-boosted frame. We
discuss Lorentz boosts both in and orthogonal to the direction of propagation
of the perturbation. It will be demonstrated that superluminal group
velocities will not compromise the causality of the theory as long as the
asymptotic causality condition is fulfilled. In Sec. IV, we compute the
characteristic velocities in the nonlinear case. A summary of our results
concludes this work in Sec. V. An Appendix contains details of our
calculations in Sec. IV. The metric tensor is $g^{\mu\nu}={\rm
diag}(+,-,-,-)$; our units are $\hbar=c=k_{B}=1$.
## II Stability in the rest frame
As mentioned in the Introduction, there are several approaches to formulate a
second-order theory of relativistic dissipative fluids is ; dkkm1 ; dkkm3 ;
dkkm4 ; jou ; else . These approaches differ only by nonlinear (second-order)
terms. However, since we shall apply a linear stability analysis in the
following, these differences vanish and all approaches lead to the same set of
linearized fluid-dynamical equations. In this work, we do not consider any
conserved charges and thus are left with energy-momentum conservation,
$\partial_{\mu}T^{\mu\nu}=0\;,$ (1)
where
$T^{\mu\nu}=\varepsilon\,u^{\mu}u^{\nu}-(P+\Pi)\Delta^{\mu\nu}+\pi^{\mu\nu}$
(2)
is the energy-momentum tensor. Here, $\varepsilon$ and $P$ are the energy
density and the pressure, while $u^{\mu}$, $\Pi$, and $\pi^{\mu\nu}$ are the
fluid velocity, the bulk viscous pressure, and the shear stress tensor,
respectively. We also introduced the projection operator
$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}\;,$ (3)
which projects onto the $(D-1)$-dimensional subspace orthogonal to the fluid
velocity. We compute in the Landau frame LL , where there is no energy flow in
the local rest frame.
In second-order theories of relativistic dissipative fluid dynamics, the bulk
viscous pressure and the shear stress tensor are determined from evolution
equations. In $D$ space-time dimensions ($D\geq 3$), these equations are given
by
$\displaystyle\tau_{\Pi}\,\frac{d}{d\tau}\Pi+\Pi$ $\displaystyle=$
$\displaystyle-\zeta\,\partial_{\mu}u^{\mu}\;,$ (4a)
$\displaystyle\tau_{\pi}\,P^{\mu\nu\alpha\beta}\,\frac{d}{d\tau}\pi_{\alpha\beta}+\pi^{\mu\nu}$
$\displaystyle=$ $\displaystyle
2\eta\,P^{\mu\nu\alpha\beta}\,\partial_{\alpha}u_{\beta}\;;$ (4b)
possible other second-order terms Betz:2008me can be neglected for the
purpose of a linear stability analysis. In Eqs. (4), the comoving derivative
is denoted by $u^{\mu}\partial_{\mu}\equiv d/d\tau$. The relaxation times for
the bulk viscous pressure and the shear stress tensor are denoted by
$\tau_{\Pi}$ and $\tau_{\pi}$, respectively. The coefficients $\zeta,\,\eta$
are the bulk and shear viscosities, respectively. We also introduced the
symmetric rank-four projection operator
$P^{\mu\nu\alpha\beta}=\frac{1}{2}\left(\Delta^{\mu\alpha}\Delta^{\nu\beta}+\Delta^{\nu\alpha}\Delta^{\mu\beta}\right)-\frac{1}{D-1}\,\Delta^{\mu\nu}\Delta^{\alpha\beta}\;.$
(5)
The shear stress tensor is traceless $\pi^{\mu}{}_{\mu}=0$ and orthogonal to
the fluid velocity $u_{\mu}\pi^{\mu\nu}=0$.
The stability and causality of a relativistic dissipative fluid with bulk
viscous pressure only have been investigated in Ref. dkkm3 . Thus, for the
sake of simplicity, we shall first ignore the effects from bulk viscous
pressure and discuss the properties of the fluid-dynamical equations of motion
including only shear viscosity. The interplay between shear and bulk viscosity
will be discussed afterwards.
### II.1 Shear viscosity only
For convenience, we introduce the following parameterization:
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle as\;,$ (6a)
$\displaystyle\tau_{\pi}$ $\displaystyle=$
$\displaystyle\frac{\eta}{\varepsilon+P}\,b=\frac{ab}{T}\;,$ (6b)
where $s$ and $T$ are the entropy density and the temperature, respectively.
From the second equation we obtain $\tau_{\pi}(\varepsilon+P)/\eta=b$. The
parametrization (6) is motivated by the leading-order results for the causal
shear viscosity coefficient and the relaxation time obtained in Ref. knk
where the relation $\tau_{\pi}=\eta/P$ was found. For a massless ideal gas
equation of state, $\varepsilon=(D-1)P$, this result is reproduced by choosing
$b=D$.
In this section, we discuss the stability of second-order relativistic fluid
dynamics in the local rest frame. Following Ref. his ; dkkm3 , let us
introduce a perturbation $\sim e^{i\omega t-ikx}$ around the hydrostatic
equilibrium state,
$\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle\varepsilon_{0}+\delta\varepsilon\,e^{i\omega t-ikx}\;,$ (7a)
$\displaystyle\pi^{\mu\nu}$ $\displaystyle=$
$\displaystyle\pi^{\mu\nu}_{0}+\delta\pi^{\mu\nu}\,e^{i\omega t-ikx}\;,$ (7b)
$\displaystyle u^{\mu}$ $\displaystyle=$ $\displaystyle u^{\mu}_{0}+\delta
u^{\mu}\,e^{i\omega t-ikx}\;,$ (7c)
where $\varepsilon_{0}={\rm const.}$, $\pi^{\mu\nu}_{0}=0$, and
$u^{\mu}_{0}=(1,0,0,\ldots)$, respectively. In the linear approximation, the
velocity perturbation has no zeroth component,
$\delta u^{\mu}=(0,\delta u^{1},\delta u^{2},\ldots,\delta u^{D-1})\;,$ (8)
because $u^{\mu}u_{\mu}=1$. Moreover, in the local rest frame,
$\delta\pi^{0\nu}\equiv 0$ on account of the orthogonality condition
$u_{\mu}\pi^{\mu\nu}=0$. Since $\pi^{\mu\nu}$ is traceless,
$\delta\pi^{(D-1)(D-1)}$ is not an independent variable. Taking all of this
into account, the linearized fluid-dynamical equations can be written as
$AX=0\;,$ (9)
where
$\displaystyle X$ $\displaystyle=$ $\displaystyle(\delta\varepsilon,\delta
u^{1},\delta\pi^{11},\delta u^{2},\delta\pi^{12},\ldots,\delta
u^{D-1},\delta\pi^{1(D-1)},$
$\displaystyle\;\;\delta\pi^{22},\delta\pi^{33},\ldots,\delta\pi^{(D-2)(D-2)},\delta\pi^{23},\delta\pi^{24},\ldots,\delta\pi^{2(D-1)},\delta\pi^{34},\ldots,\delta\pi^{(D-2)(D-1)})^{T}\;.$
The matrix $A$ is expressed as
$A=\left(\begin{array}[]{cccc}T&0&0&0\\\ 0&B&0&0\\\ G&0&C&0\\\
0&0&0&E\end{array}\right)\;,$ (10)
with
$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}i\omega&f_{1}&0\\\
-ikc_{s}^{2}&f_{2}&-ik\\\ 0&\Gamma&f\end{array}\right)\;,$ (11d)
$\displaystyle B$ $\displaystyle=$ $\displaystyle{\rm
diag}(B_{0},\ldots,B_{0})_{(D-2)\times(D-2)}\;,\;\;B_{0}=\left(\begin{array}[]{cc}f_{2}&-ik\\\
\Gamma_{1}&f\end{array}\right)\;,$ (11g) $\displaystyle G$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\Gamma_{2}&0\\\ &\ldots&\\\
0&\Gamma_{2}&0\end{array}\right)_{(D-3)\times 3}\;,$ (11k) $\displaystyle C$
$\displaystyle=$ $\displaystyle{\rm diag}(f,\ldots,f)_{(D-3)\times(D-3)}\;,$
(11l) $\displaystyle E$ $\displaystyle=$ $\displaystyle{\rm
diag}(f,\ldots,f)_{\frac{1}{2}(D-2)(D-3)\times\frac{1}{2}(D-2)(D-3)}\;,$ (11m)
where $c_{s}=\sqrt{\partial P/\partial\varepsilon}$ is the velocity of sound.
Here, we introduced the abbreviations
$\displaystyle f$ $\displaystyle=$ $\displaystyle
i\omega\,\tau_{\pi}+1\;,\qquad\;f_{1}=-ik\,(\varepsilon+P)\;,$ $\displaystyle
f_{2}$ $\displaystyle=$ $\displaystyle
i\omega\,(\varepsilon+P)\;,\qquad\Gamma=-ik\,\frac{2(D-2)}{D-1}\,\eta\;,$
$\displaystyle\Gamma_{1}$ $\displaystyle=$ $\displaystyle-
ik\,\eta\;,\qquad\qquad\Gamma_{2}=ik\,\frac{2}{D-1}\,\eta\;.$
For nontrivial solutions of Eq. (9), the determinant of the matrix $A$ should
vanish. This leads to the following conditions for the dispersion relations
$\omega(k)$:
$\displaystyle f$ $\displaystyle=$ $\displaystyle 0\;,$ (12a)
$\displaystyle\det B=\left(\det B_{0}\right)^{D-2}$ $\displaystyle=$
$\displaystyle 0\;,$ (12b) $\displaystyle\det
T=\det\left(\begin{array}[]{ccc}i\omega&f_{1}&0\\\ -ik\,c_{s}^{2}&f_{2}&-ik\\\
0&\Gamma&f\end{array}\right)$ $\displaystyle=$ $\displaystyle 0\;.$ (12f)
Equation (12a) gives a purely imaginary frequency
$\omega=\frac{i}{\tau_{\pi}}\;,$ (13)
which corresponds to a nonpropagating mode. The degeneracy of this mode is
$(D-3)[1+(D-2)/2]$.
Equation (12b) leads to a complex frequency,
$\omega=\frac{1}{2\tau_{\pi}}\left(i\pm\sqrt{\frac{4\,\eta\,\tau_{\pi}}{\varepsilon+P}\,k^{2}-1}\right)\;,$
(14)
corresponding to two propagating modes, if $k$ is larger than the critical
wavenumber
$k_{c}=\sqrt{\frac{\varepsilon+P}{4\,\eta\,\tau_{\pi}}}\equiv\frac{\sqrt{b}}{2\,\tau_{\pi}}\;.$
(15)
Following Ref. baier , we shall call these modes shear modes. There are in
total $2(D-2)$ shear modes.
Equation (12f) gives the same dispersion relation as Eq. (16) of Ref. dkkm3 ,
after replacing $2(D-2)\eta/(D-1)$ with $\zeta_{0}$. Introducing the sound
attenuation length in $D$ space-time dimensions
$\Gamma_{s}\equiv\frac{2(D-2)}{D-1}\,\frac{\eta}{\varepsilon+P}\equiv\frac{2(D-2)}{D-1}\,\frac{\tau_{\pi}}{b}\;,$
(16)
the analytic solution in the limit of small wavenumber $k$ is
$\omega=\left\\{\begin{array}[]{l}\displaystyle\frac{i}{\tau_{\pi}}\;,\\\
\displaystyle\pm\,k\,c_{s}+i\,\frac{\Gamma_{s}}{2}\,k^{2}\;,\end{array}\right.$
(17)
while for large wavenumber we obtain
$\omega=\left\\{\begin{array}[]{l}\displaystyle\frac{i}{\tau_{\pi}}\,\left[1+\frac{\Gamma_{s}}{\tau_{\pi}c_{s}^{2}}\right]^{-1}\;,\\\\[8.5359pt]
\displaystyle\pm\,k\,c_{s}\sqrt{1+\frac{\Gamma_{s}}{\tau_{\pi}c_{s}^{2}}}+\frac{i}{2\tau_{\pi}}\,\left[1+\frac{\tau_{\pi}c_{s}^{2}}{\Gamma_{s}}\right]^{-1}\;.\end{array}\right.$
(18)
This corresponds to another nonpropagating mode and two propagating modes
which we call sound modes in accordance with Ref. baier . All imaginary parts
are positive and therefore the nonpropagating, as well as the shear and sound
modes are stable around the hydrostatic equilibrium state. This fact is
already known from the study of Hiscock and Lindblom his .
In order to discuss the issue of causality, we follow Ref. his ; dkkm3 and
study the group velocity defined as
$v_{g}=\frac{\partial{\rm Re}\,\omega}{\partial k}\;.$ (19)
For the two nonpropagating modes, ${\rm Re}\,\omega=0$. Consequently, in order
to discuss causality, we have to consider the behavior of the imaginary part
dkkm3 . Let us digress for the moment and consider the diffusion equation with
diffusion constant $D_{0}$. There is a nonpropagating mode with dispersion
relation $\omega=iD_{0}k^{2}$. Moreover, it is known that the diffusion
equation is acausal. Therefore, we conjecture that a $k^{2}$ dependence of any
nonpropagating mode can be considered a sign of acausality. In our case, the
nonpropagating modes are either independent of $k$, or have a weak $k$
dependence (cf. Fig. 1). According to our conjecture, we conclude that the
nonpropagating modes do not violate causality.
Figure 1: The real parts (left panel) and the imaginary parts (right panel) of
the dispersion relations for the sound modes (full lines) and the
nonpropagating mode (dashed line) obtained from Eq. (12f). The parameters are
$a=\frac{1}{4\pi}\,,\;b=6\,,\;c_{s}^{2}=\frac{1}{3}$ for the 3+1-dimensional
case, $D=4$. Figure 2: The group velocity (22) for
$a=1/(4\pi)\,,\;D=4\,,\;c_{s}^{2}=\frac{1}{3}$, and $b=6$ (full line), $b=2$
(dashed line), as well as $b=1.5$ (dotted line).
The dispersion relations resulting from Eq. (12f) are shown in Fig. 1, and the
corresponding group velocity resulting from Eq. (19) in Fig.2. The group
velocity has a maximum for a finite value of $k/T$ and approaches its
asymptotic value ($k\rightarrow\infty$) from above. For small values of $b$,
it may thus happen that the group velocity becomes superluminal. Nevertheless,
in Sec. III.3 we shall show that only the asymptotic value determines whether
the theory as a whole is causal or not. The asymptotic value of the group
velocity is
$v_{g,{\rm sound}}^{\rm as}=\lim_{k\rightarrow\infty}\frac{\partial
Re\,\omega}{\partial
k}=c_{s}\,\sqrt{1+\frac{\Gamma_{s}}{\tau_{\pi}c_{s}^{2}}}\;.$ (20)
Consequently, for the asymptotic group velocity of sound waves to be less than
the speed of light, $\tau_{\pi}$ and $\Gamma_{s}$ should satisfy the
following, so-called asymptotic causality condition:
$\frac{\Gamma_{s}}{\tau_{\pi}}\leq
1-c_{s}^{2}\;\;\Longleftrightarrow\;\;\frac{1}{b}\equiv\frac{\eta}{\tau_{\pi}(\varepsilon+P)}\leq\frac{D-1}{2(D-2)}(1-c_{s}^{2})\;.$
(21)
This is similar to the causality condition for the group velocity in the case
of bulk viscosity, Eq. (21) of Ref. dkkm3 . For conformal fluids, where
$c_{s}^{2}=1/(D-1)$, the condition (21) simplifies to
$\Gamma_{s}\leq(D-2)\tau_{\pi}/(D-1)$ or, equivalently, $b\geq 2$. For
example, for the values of $\eta$ and $\tau_{\pi}$ deduced from the AdS/CFT
correspondence baier ; Heller:2007qt ; push , $\eta=s/(4\pi)$,
$\tau_{\pi}=(2-\ln 2)/(2\pi T)$, the condition (21) is always satisfied
because $b=2(2-\ln 2)\simeq 2.614>2$.
Figure 3: The real parts (left panel) and the imaginary parts (right panel) of
the dispersion relations for the shear modes obtained from Eq. (12b). The
parameters are $a=\frac{1}{4\pi}\,,\;b=6\,,\;c_{s}^{2}=\frac{1}{3}$ for the
3+1-dimensional case, $D=4$. Figure 4: The group velocity (22) for
$D=4\,,\;b=6\,,\;c_{s}^{2}=\frac{1}{3}$, and $a=1/(4\pi)$ (full line), $a=1/4$
(dashed line), as well as $a=1$ (dotted line).
The dispersion relations for the shear modes resulting from Eq. (12b) change
their behavior from nonpropagating to propagating at the critical wavenumber
(15), as shown in Fig. 3. It should be noted that a similar behavior is
observed in the case of bulk viscosity, cf. Fig. 1 in Ref. dkkm3 . For
wavenumbers larger than $k_{c}$, the (modulus of the) group velocity of the
propagating mode is
$v_{g}=v_{g,{\rm shear}}^{\rm as}\,\frac{k/k_{c}}{\sqrt{(k/k_{c})^{2}-1}}\;,$
(22)
where
$v_{g,{\rm shear}}^{\rm
as}\equiv\frac{1}{\sqrt{2\tau_{\pi}k_{c}}}\equiv\sqrt{\frac{\eta}{\tau_{\pi}(\varepsilon+P)}}\equiv\frac{1}{\sqrt{b}}$
(23)
is the asymptotic value of $v_{g}$ in the large-wavenumber limit. If the
asymptotic causality condition (21) is satisfied, $v_{g,{\rm shear}}^{\rm
as}\leq\sqrt{(D-1)(1-c_{s}^{2})/2(D-2)}$. This is smaller than 1 for any value
of $c_{s}$ and $D\geq 3$. However, near the critical wavenumber $k_{c}$ the
group velocity diverges, as shown in Fig. 4. From the definitions of $k_{c}$,
Eq. (15), and the parameters $a,b$, Eqs. (6), we observe that
$k_{c}/T=(2a\sqrt{b})^{-1}$. The $1/a$-scaling of $k_{c}/T$ for fixed $b$ can
be nicely observed in Fig. 4.
In Sec. III.3 we shall show that the apparent violation of causality of the
group velocity does not cause the theory as a whole to become acausal. The
important issue is whether the asymptotic causality condition is fulfilled. If
yes, the theory is causal.
Figure 5: The real parts (left panel) and the imaginary parts (right panel) of
the dispersion relations for the sound modes obtained from Eq. (12f). The
parameters are $a=\frac{1}{4\pi}\,,\;b=1\,,\;c_{s}^{2}=\frac{1}{3}$ for the
3+1-dimensional case, $D=4$.
Figure 6: The real parts (left panel) and the imaginary parts (right panel) of
the dispersion relations for the shear modes obtained from Eq. (12b). The
parameters are $a=\frac{1}{4\pi}\,,\;b=1\,,\;c_{s}^{2}=\frac{1}{3}$ for the
3+1-dimensional case, $D=4$.
We remark that, in the local rest frame, the stability of the system of fluid-
dynamical equations is not affected if we choose a parameter set which
violates the asymptotic causality condition (21), for instance a conformal
fluid in $D=4$ dimensions and $b=1$. This is demonstrated for the sound modes
in Fig. 5, and for the shear modes in Fig. 6.
### II.2 Competition of bulk and shear
The question we would like to answer in this section is whether the problem of
the divergent group velocity can be removed by adding bulk viscosity to the
discussion. For the sake of simplicity, we consider only the 2+1-dimensional
case, i.e., $D=3$. Similarly to Eqs. (6), we introduce the parametrization
$\zeta=a_{1}s\;,\qquad\tau_{\Pi}=\frac{\zeta}{\varepsilon+P}\,b_{1}\;.$ (24)
As before, the equations of motion (4) have to be linearized, yielding Eq.
(9), where now
$X=(\delta\varepsilon,\delta u^{x},\delta\pi^{xx},\delta
u^{y},\delta\pi^{xy},\delta\Pi)^{T}\;,$ (25)
and
$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}i\omega&-ik\,(\varepsilon+P)&0&0&0&0\\\
-ik\,c_{s}^{2}&i\omega\,(\varepsilon+P)&-ik&0&0&-ik\\\
0&-ik\,\eta&i\omega\,\tau_{\pi}+1&0&0&0\\\
0&0&0&i\omega\,(\varepsilon+P)&-ik&0\\\
0&0&0&-ik\,\eta&i\omega\,\tau_{\pi}+1&0\\\
0&-ik\,\zeta&0&0&0&i\omega\,\tau_{\Pi}+1\end{array}\right)\;.$ (32)
Then, the dispersion relations are given by solving the following equations:
$\displaystyle k^{2}\eta+i\omega\,(1+i\omega\,\tau_{\pi})(\varepsilon+P)$
$\displaystyle=$ $\displaystyle 0\;,$ (33a) $\displaystyle i\omega
k^{2}\,(1+i\omega\,\tau_{\Pi})\,\eta+(1+i\omega\,\tau_{\pi})\left[i\omega
k^{2}\,\zeta+(1+i\omega\,\tau_{\Pi})(\varepsilon+P)(c_{s}^{2}k^{2}-\omega^{2})\right]$
$\displaystyle=$ $\displaystyle 0\;.$ (33b)
The dispersion relations resulting from sound and bulk viscous modes, Eq.
(33b), are
$\omega=\left\\{\begin{array}[]{l}\displaystyle\frac{T}{2aa_{1}(b+b_{1}+bb_{1}c_{s}^{2})}\left\\{\frac{}{}ia(1+bc_{s}^{2})+ia_{1}(1+b_{1}c_{s}^{2})\right.\\\
\qquad\left.\pm\left[4aa_{1}c_{s}^{2}(b+b_{1}+bb_{1}c_{s}^{2})-(a+a_{1}+abc_{s}^{2}+a_{1}b_{1}c_{s}^{2})^{2}\right]^{1/2}\right\\}\;,\\\
\displaystyle\pm
k\sqrt{\frac{1}{b}+\frac{1}{b_{1}}+c_{s}^{2}}+\frac{i\,T}{2(b+b_{1}+bb_{1}c_{s}^{2})}\left(\frac{b}{a_{1}b_{1}}+\frac{b_{1}}{ab}\right)\;,\end{array}\right.$
(34)
for large $k$, and
$\omega=\left\\{\begin{array}[]{l}\displaystyle\frac{i}{\tau_{\pi}}\;,\\\\[8.5359pt]
\displaystyle\frac{i}{\tau_{\Pi}}\;,\\\ \pm c_{s}^{2}k\;,\end{array}\right.$
(35)
for small $k$.
Thus the asymptotic causality condition reads
$\frac{1}{b_{1}}+\frac{1}{b}\equiv\frac{\zeta}{\tau_{\Pi}(\varepsilon+P)}+\frac{\eta}{\tau_{\pi}(\varepsilon+P)}\leq
1-c^{2}_{s}\;.$ (36)
On the other hand, the equation for the shear modes, Eq. (33a), is the same as
Eq. (12b) and hence the corresponding group velocity again shows a divergence.
Thus, the inclusion of bulk viscosity does not solve the problem of the
divergent group velocity.
## III Stability in Lorentz-boosted frame
The discussion of causality and stability in the case of nonzero bulk
viscosity in a Lorentz-boosted frame in Ref. dkkm3 has shown that causality
and stability are intimately related. Relativistic dissipative fluid dynamics
becomes unstable if the group velocity exceeds the speed of light. If this is
still true in the case of nonzero shear viscosity, the divergence of the group
velocity found in the rest frame may induce an instability in a moving frame.
In order to investigate this question, we consider the stability of the
hydrostatic state observed from a Lorentz-boosted frame, following Ref. dkkm3
. In this section, we restrict our investigations to the case $D=4$.
We consider a frame moving with a velocity $\vec{V}$ with respect to the
hydrostatic state. Then, the total fluid velocity $u^{\prime\;\mu}$ is given
by
$u^{\prime\;\mu}=\left(\begin{array}[]{cc}\gamma_{V}&V\gamma_{V}\vec{n}^{T}\\\
V\gamma_{V}\vec{n}&\gamma_{V}P_{\parallel}+Q_{\perp}\end{array}\right)u^{\mu},$
(37)
where $\gamma_{V}=1/\sqrt{1-V^{2}}$, $P_{\parallel}=\vec{n}\vec{n}^{T}$, and
$Q_{\perp}=1-P_{\parallel}$, with $\vec{n}=\vec{V}/|\vec{V}|$. We consider the
two cases where the direction of the Lorentz boost is parallel and where it is
perpendicular to the direction of propagation of the perturbation; the latter
we take to be the $x$ direction.
### III.1 Boost along the $x$ direction
The perturbation of the fluid velocity is given by
$u^{\prime\;\mu}=u^{\prime\;\mu}_{0}+\delta u^{\prime\;\mu}\;e^{i\omega
t-ikx}\;,$ (38)
where
$\displaystyle u^{\prime\;\mu}_{0}$ $\displaystyle=$
$\displaystyle\gamma_{V}(1,V,0,0)\;,$ (39a) $\displaystyle\delta
u^{\prime\;\mu}$ $\displaystyle=$ $\displaystyle(V\gamma_{V}\delta
u^{x},\gamma_{V}\delta u^{x},\delta u^{y},\delta u^{z})\;,$ (39b)
where $\delta u^{\mu}$ is the velocity perturbation in the local rest frame.
The linearized fluid-dynamical equations are again given by Eq. (9), with
$X=(\delta\varepsilon,\delta u^{x},\delta\pi^{xx},\delta
u^{y},\delta\pi^{xy},\delta
u^{z},\delta\pi^{xz},\delta\pi^{yy},\delta\pi^{yz})^{T}\;,$ (40)
and
$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}T_{1}&0&0&0\\\ 0&B_{1}&0&0\\\
G_{1}&0&C_{1}&0\\\ 0&0&0&E_{1}\end{array}\right)\;.$ (45)
The submatrices are given by
$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\gamma_{V}^{2}\left(\begin{array}[]{ccc}i\omega(1+V^{2}c_{s}^{2})-ikV(1+c_{s}^{2})&\;\;i[2\omega
V-k(1+V^{2})](\varepsilon+P)&\;\;i\gamma_{V}^{-2}V(\omega V-k)\\\ i\omega
V(1+c_{s}^{2})-ik(V^{2}+c_{s}^{2})&\;\;i\left[\omega(1+V^{2})-2kV\right](\varepsilon+P)&\;\;i\gamma_{V}^{-2}(\omega
V-k)\\\ 0&\frac{4}{3}i\eta\gamma_{V}(\omega
V-k)&\;\;\gamma_{V}^{-2}F\end{array}\right)\;,$ (46d) $\displaystyle B_{1}$
$\displaystyle=$ $\displaystyle{\rm diag}(B_{01},B_{01})\;,\qquad
B_{01}=\left(\begin{array}[]{cc}i\gamma_{V}(\omega-
kV)(\varepsilon+P)&\;\;i(\omega V-k)\\\ i\eta\gamma_{V}^{2}(\omega
V-k)&\;\;F\end{array}\right)\;,$ (46h) $\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle\left(\frac{}{}0\qquad-\frac{2}{3}i\eta\gamma_{V}(\omega
V-k)\qquad 0\right)\;,$ (46i) $\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle E_{1}=F\;.$ (46j) Here we abbreviated $F=i\gamma_{V}(\omega-
kV)\tau_{\pi}+1\;.$ (46k)
Obviously,
$\displaystyle{\rm det}A={\rm det}T_{1}\times{\rm det}B_{1}\times F^{2}\;.$
(47)
From $F^{2}=0$, we only obtain two trivial propagating modes
$\omega=\frac{i}{\gamma_{V}\tau_{\pi}}+kV\;.$ (48)
The group velocity is $v_{g}=V$, which implies that these modes correspond to
the nonpropagating modes in the LRF.
From ${\rm det}B_{1}=0$, we obtain
$[iT+ab\gamma_{V}(kV-\omega)](kV-\omega)+a\gamma_{V}(kV-\omega)^{2}T=0\;,$
(49)
corresponding to the shear modes. There are in total four modes satisfying
this relation. The solutions are given by
$\omega_{\pm}=\frac{1}{2a(b-V^{2})\gamma_{V}}\left[i\,T-2a(1-b)kV\gamma_{V}\pm\sqrt{-T^{2}+4iakTV\gamma_{V}^{-1}+4a^{2}bk^{2}\gamma_{V}^{-2}}\right]\;.$
(50)
On the other hand, the sound modes result from
$\displaystyle
c_{s}^{2}(\varepsilon+P)\left[\frac{}{}1-i\gamma_{V}\tau_{\pi}(kV-\omega)\right]\left\\{\frac{}{}k^{2}\left[\frac{}{}V^{2}+(V-1)^{2}V\gamma_{V}^{2}+1\right]\right.$
(51) $\displaystyle+$
$\displaystyle\left.2kV\omega\left[\frac{}{}(V-1)V\gamma_{V}^{2}-1\right]+V^{2}\omega^{2}-c_{s}^{-2}(\omega-
kV)^{2}\frac{}{}\right\\}$ $\displaystyle+$
$\displaystyle\frac{4}{3}i\gamma_{V}\eta(k-V\omega)^{2}\left\\{\frac{}{}kV\left[\frac{}{}c_{s}^{2}\gamma_{V}^{2}V(1-V)-1\right]+\omega\right\\}\qquad=0\;.$
In Fig. 7, the dependence of the group velocity on the wavenumber is shown for
various values of the boost velocity $V$. The left panel shows the behavior of
one of the shear modes and the right panel one of the sound modes. The
parameter set used here is $a=\frac{1}{4\pi},\,b=6,\,c_{s}^{2}=\frac{1}{3}$,
which satisfies the asymptotic causality condition. We observe that the
divergence of the group velocity of the shear mode in the rest frame is
tempered by the Lorentz boost to result in a peak of finite height. However,
the group velocity may still exceed the speed of light in a certain range of
wavenumbers. As we increase the boost velocity, the peak height diminishes,
until the group velocity remains below the speed of light for all wavenumbers.
However, further increasing the boost velocity leads to an acausal group
velocity in the sound mode.
Figure 7: The group velocity calculated for one of the shear modes (left
panel) and one of the sound modes (right panel). We set
$a=1/(4\pi),b=6,c_{s}^{2}=1/3$. The solid line is for a boost velocity
$V=0.05$, the dashed line for $V=0.4$ and the dotted line for $V=0.99$,
respectively.
Although the group velocity of the shear or the sound mode may exceed the
speed of light, as long as the asymptotic causality condition is fulfilled,
the theory is still stable. This is demonstrated in the left panel of Fig. 8,
where the imaginary parts of the modes are shown for the parameter set
$a=\frac{1}{4\pi},\,b=6,\,c_{s}^{2}=\frac{1}{3}$. We observe that all
imaginary parts are positive, indicating the stability of the theory.
In contrast to the rest frame, where the theory is stable even for parameters
which violate the asymptotic causality condition (21), this is no longer the
case in a Lorentz-boosted frame. In the right panel of Fig. 8, the imaginary
parts of the modes are calculated with the parameter set
$a=\frac{1}{4\pi},\,b=1,\,c_{s}^{2}=\frac{1}{3}$. Now one observes the
appearance of negative imaginary parts, indicating that the theory becomes
unstable.
Figure 8: The imaginary parts of the dispersion relations for a boost in $x$
direction with velocity $V=0.9$. The left panel shows the results for the
parameter set $a=\frac{1}{4\pi},\,b=6,\,c_{s}^{2}=\frac{1}{3}$, which fulfills
the asymptotic causality condition, while the right panel is for
$a=\frac{1}{4\pi},\,b=1,\,c_{s}^{2}=\frac{1}{3}$, which violates this
condition. The dashed lines are for the shear modes, while the solid lines are
for the sound modes.
### III.2 Boost along the $y$ direction
Now we consider a Lorentz boost along the $y$ direction. The perturbation of
the fluid velocity is given by
$u^{\prime\;\mu}=u^{\prime\;\mu}_{0}+\delta u^{\prime\;\mu}\;e^{i\omega
t-ikx}\;,$ (52)
where
$\displaystyle u^{\prime\;\mu}_{0}$ $\displaystyle=$
$\displaystyle\gamma_{V}(1,0,V,0)\;,$ (53a) $\displaystyle\delta
u^{\prime\;\mu}$ $\displaystyle=$ $\displaystyle(V\gamma_{V}\delta
u^{y},\delta u^{x},\gamma_{V}\delta u^{y},\delta u^{z})\;.$ (53b)
Similarly to the preceding discussion, the linearized fluid-dynamical
equations take the form (9), where the matrix $A$ is
$A=\left(\begin{array}[]{cccc}T_{2}&H_{1}&H_{2}&0\\\
H_{3}&B_{2}&H_{4}&H_{5}\\\ G_{2}&H_{6}&C_{2}&0\\\
0&H_{7}&0&E_{2}\end{array}\right)\;,$ (54)
with
$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}i\omega\gamma_{V}^{2}(1+c_{s}^{2}V^{2})&-ik\gamma_{V}(\varepsilon+P)&0\\\
-ikc_{s}^{2}&i\omega\gamma_{V}(\varepsilon+P)&-ik\\\
0&-\frac{4}{3}ik\eta&F_{1}\end{array}\right)\;,$ (55d) $\displaystyle H_{1}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}2i\omega
V(\varepsilon+P)\gamma_{V}^{2}&-ikV&0&0\\\ 0&i\omega V&0&0\\\
-\frac{2}{3}i\omega V\eta\gamma_{V}&0&0&0\end{array}\right)\;.$ (55h)
$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}i\omega
V^{2}&0&0\end{array}\right)^{T}\;,$ (55j) $\displaystyle H_{3}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}i\omega
V\gamma_{V}^{2}(1+c_{s}^{2})&-ikV\gamma_{V}(\varepsilon+P)&0\\\ 0&i\omega
V\gamma_{V}^{2}\eta&0\\\ 0&0&0\\\ 0&0&0\end{array}\right)\;,$ (55o)
$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}i\omega\gamma_{V}^{2}(1+V^{2})(\varepsilon+P)&-ik&0&0\\\
-ik\gamma_{V}\eta&F_{1}&0&0\\\ 0&0&i\omega\gamma_{V}(\varepsilon+P)&-ik\\\
0&0&-ik\eta&F_{1}\end{array}\right)\;,$ (55t) $\displaystyle H_{4}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}i\omega
V&0&0&0\end{array}\right)^{T}\;,\qquad\qquad
H_{5}=\left(\begin{array}[]{cccc}0&0&i\omega V&0\end{array}\right)^{T}\;,$
(55w) $\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\frac{2}{3}ik\gamma_{V}^{2}\eta&0\end{array}\right)\;,\qquad\qquad
H_{6}=\left(\begin{array}[]{cccc}\frac{4}{3}i\omega
V\gamma_{V}^{3}\eta&0&0&0\end{array}\right)\;,$ (55z) $\displaystyle H_{7}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}0&0&i\omega
V\gamma_{V}^{2}\eta&0\end{array}\right)\;,\qquad\;\;\;C_{2}=E_{2}=F_{1}\;.$
(55ab)
Here we abbreviated
$F_{1}=i\omega\gamma_{V}\tau_{\pi}+1\;.$
The condition ${\rm det}A=0$ leads again to the following nine modes: three
nonpropagating modes, four shear modes and two sound modes.
The nonpropagating mode has almost the same form as that in the LRF,
$\omega=\frac{i}{\gamma_{V}\tau_{\pi}}\;.$ (56)
The shear modes are given by the solution of the following equation
$\displaystyle
k^{2}\eta+\gamma_{V}\omega\left[V^{2}\gamma_{V}\eta\omega+(\varepsilon+P)(i-\gamma_{V}\tau_{\pi}\omega)\right]=0\;,$
(57)
and the solutions are given by
$\omega_{\pm}=\frac{1}{2a(b-V^{2})\gamma_{V}}\left[i\,T\pm\sqrt{-T^{2}+4a^{2}bk^{2}-4a^{2}k^{2}V^{2}}\right]\;.$
(58)
We find that the critical wavenumber is now given by
$\tilde{k}_{c}=T/(2a\sqrt{b-V^{2}})$, below which the shear modes become
nonpropagating modes.
On the other hand, the sound modes and another nonpropagating mode result from
$\displaystyle
3c_{s}^{2}(\varepsilon+P)(-i+\gamma_{V}\tau_{\pi}\omega)(k^{2}+V^{2}\gamma_{V}^{2}\omega^{2})$
(59) $\displaystyle+$
$\displaystyle\gamma_{V}\omega\left\\{4k^{2}\eta+\gamma_{V}\omega\left[3i(\varepsilon+P)+4V^{2}\gamma_{V}\eta\omega-3(\varepsilon+P)\gamma_{V}\tau_{\pi}\omega\right]\right\\}=0\;.$
The real and imaginary parts of this dispersion relation are calculated with a
parameter set satisfying the asymptotic causality condition. The results are
shown in Fig. 9. One observes that the real parts are symmetric around
$\omega=0$. This symmetry is due to the fact that the direction of the Lorentz
boost is orthogonal to the direction of the perturbation. The critical wave
number $\tilde{k_{c}}$ where the shear mode changes from nonpropagating to
propagating mode can be clearly seen. The imaginary parts are seen to be
positive. We confirmed that the imaginary parts become negative if we use a
parameter set which violates the asymptotic causality condition.
Figure 9: The real and imaginary parts for the dispersion relations of the
shear modes (dashed lines) and sound modes (solid lines), for a Lorentz boost
in $y$ direction. We use
$a=\frac{1}{4\pi},\,b=6,\,c_{s}^{2}=\frac{1}{3},\,V=0.9$ in the
3+1-dimensional case.
### III.3 Causality of wave propagation
In the preceding discussion we have seen that the theory is stable if the
asymptotic causality condition is fulfilled. The reverse is in general not
true, as the discussion in the local rest frame has shown, since a stable
theory may also violate the asymptotic causality condition. However, the
discussion in the Lorentz-boosted frame has revealed that the stability of a
theory is contingent upon whether the asymptotic causality condition is
fulfilled.
In this section, we shall show that the causality of the theory as a whole is
guaranteed if the asymptotic stability condition is fulfilled. The group
velocity may become superluminal, or even diverge, as long as this apparent
violation of causality is restricted to a finite range of momenta. The
argument leading to this conclusion is analogous to that of Sommerfeld and
Brillouin in classical electrodynamics jackson ; bri . For instance, in the
case of anomalous dispersion the group velocity may become superluminal, but
the causality of the theory as a whole is not affected.
The change in a fluid-dynamical variable induced by a general perturbation is
given by
$\delta X(x,t)=\sum_{j}\int d\omega\,\widetilde{\delta
X}_{j}(\omega)\,e^{i\omega t-ik_{j}(\omega)x}\;,$ (60)
where $\delta X(x,t)$ stands for $\delta\varepsilon$, $\delta u^{\mu}$, and
$\delta\pi^{\mu\nu}$. The index $j$ denotes the different modes, i.e., the
shear modes, the sound modes etc. The function $k_{j}(\omega)$ is the inverted
dispersion relation $\omega_{j}(k)$ of the respective mode. The Fourier
components are given by
$\sum_{j}\widetilde{\delta
X}_{j}(\omega)=\frac{1}{2\pi}\int^{\infty}_{-\infty}dt\,\delta
X(0,t)\,e^{-i\omega t}\;.$ (61)
We assume that the incident wave has a well-defined front that reaches $x=0$
not before $t=0$. Thus $\delta X(0,t)=0$ for $t<0$. This condition on $\delta
X(0,t)$ ensures that $\sum_{j}\widetilde{\delta X}_{j}(\omega)$ is analytic in
the lower half of the complex $\omega$ plane jackson . On the other hand, in
Sec. II.1 we have found that the group velocity of the shear modes diverges
for certain values of $k$. These divergences correspond to singularities in
the complex $\omega$ plane. However, if the asymptotic causality condition is
fulfilled, the imaginary part of the dispersion relation is always positive,
i.e., the singularities only appear in the upper half of the complex $\omega$
plane. In this case, the system is also stable. On the other hand, if the
asymptotic causality condition is violated, the singularities may appear also
in the lower half-plane, i.e., for negative imaginary part of the dispersion
relation, and the system is unstable.
We shall now demonstrate that the divergences in the group velocity do not
violate causality as long as the asymptotic causality condition is satisfied,
i.e., as long as the asymptotic group velocity remains subluminal. To this
end, we compute Eq. (60) by contour integration in the complex $\omega$ plane.
To close the contour, we have to know the asymptotic behavior of the
dispersion relations. In our calculation, we found that the real part of the
dispersion relation at large $k$ is proportional to $k$ [see Eq. (18)], with a
coefficient which is the large-$k$ limit of the group velocity, i.e.,
$v_{gj}^{\rm as}$,
$\lim_{k\rightarrow\infty}{\rm Re}~{}\omega_{j}(k)=v_{gj}^{\rm as}\,k\;.$ (62)
Then, in the large-$k$ limit, the exponential becomes
$\exp[i\omega
t-ik_{j}(\omega)x]\rightarrow\exp\left[-i\,\frac{\omega}{v_{gj}^{\rm
as}}\,(x-v_{gj}^{\rm as}\,t)\right]\;.$ (63)
In the case $x>v_{gj}^{\rm as}\,t$, we have to close the integral contour in
the lower half plane. If the asymptotic causality condition is fulfilled,
there are no singularities in the lower half plane, and Eq. (60) vanishes. On
the other hand, the contour should be closed in the upper half plane if $x\leq
v_{gj}^{\rm as}\,t$. Then, because of the singularities, Eq. (60) may have a
nonzero value. However, as long as we choose a parameter set for which the
asymptotic group velocity $v_{gj}^{\rm as}$ is smaller than the speed of
light, i.e., for which the asymptotic causality condition is fulfilled, the
signal propagation does not violate causality, since the locations $x$ where
the disturbance has travelled lie within the cone given by $v_{gj}^{\rm as}$
which, in turn, lies within the lightcone, q.e.d.
To conclude this section, we have shown that the asymptotic causality
condition not only implies stability in a general (Lorentz-boosted) frame, but
also causality of the theory as a whole.
## IV Characteristic velocities
So far, we have analyzed the causality and stability of relativistic
dissipative fluid dynamics with shear viscosity using a linear stability
analysis. However, there is another possibility to analyze causality, namely
by studying the characteristic velocities. For the sake of simplicity, we
consider the 2+1-dimensional case with shear viscosity only. The fluid-
dynamical equations can be written in the following form:
$\left(A_{ab}^{t}\partial_{t}+A_{ab}^{x}\partial_{x}+A_{ab}^{y}\partial_{y}\right)Y_{b}=B_{a}\;,$
(64)
where $Y_{b}^{T}=(\varepsilon,u^{x},u^{y},\pi^{xx},\pi^{xy})$ and
$B_{a}^{T}=(0,\;0,\;0,\;\pi^{xx},\;\pi^{xy})$. The expressions for the
components of $A$ are given in the Appendix. Then, as discussed in Ref. his ,
the characteristic velocities are defined as the roots of the following
equations,
$\displaystyle\det(v_{x}A^{t}-A^{x})$ $\displaystyle=$ $\displaystyle 0\;,$
(65a) $\displaystyle\det(v_{y}A^{t}-A^{y})$ $\displaystyle=$ $\displaystyle
0\;.$ (65b)
For the case of bulk viscosity, see Ref. dkkm3 .
For the sake of simplicity, we consider $u^{\mu}=(1,\;0,\;0)$ and
$\pi^{xx}=\pi^{xy}=0$. Then, the characteristic velocities are given by
$v_{x}=v_{y}=\left\\{\begin{array}[]{l}0\;,\\\
\displaystyle\pm\sqrt{\frac{1}{b}}\;,\\\
\displaystyle\pm\sqrt{\frac{1}{b}+c_{s}^{2}}\;.\end{array}\right.$ (66)
Interestingly, the second velocity is identical to the asymptotic group
velocity (23) for the shear modes and the third velocity is the same as the
asymptotic group velocity (20) for the sound modes (since $D=3$). As a matter
of fact, if the asymptotic causality condition (21) is satisfied, the velocity
(66) is smaller than the speed of light.
In Fig. 10, we show the $b$ dependence of one of the five characteristic
velocities. We set $u^{\mu}=(\sqrt{5}/2,\;1/2,\;0),\;\pi^{xx}=\pi^{xy}=0$, and
$c_{s}^{2}=1/2$. The velocity exhibits a divergence at small values of $b$,
and thus exceeds the speed of light. This divergence occurs also for at least
one other characteristic velocity. As far as we have checked numerically, in
order to satisfy causality, one should use a value of $b$ which is larger than
about 2. This condition is consistent with the asymptotic causality condition
(21).
Figure 10: One of the five characteristic velocities determined from the roots
of Eqs. (65). The left panel is for $v_{x}$ and the right panel is for
$v_{y}$. We set $u^{\mu}=(\sqrt{5}/2,\;1/2,\;0),\;\pi^{xx}=\pi^{xy}=0$, and
$c_{s}^{2}=1/2$.
## V Concluding remarks
In this work, we have discussed the stability and causality of relativistic
dissipative fluid dynamics, based on a linear stability analysis around a
hydrostatic state. Following the usual argument, we calculated the group
velocity from the dispersion relation of the perturbation. We found that the
group velocity diverges at a critical wavenumber $k_{c}$. The appearance of
the divergence is independent of the dimensionality of space-time and can be
removed neither by tuning the parameters of the theory nor by adding bulk
viscosity to the discussion.
Nevertheless, in the rest frame of the background this acausal group velocity
does not cause the fluid to become unstable. Moreover, investigating causality
and stability in a Lorentz-boosted frame, we found that the fluid-dynamical
equations of motion are stable, if we choose parameters which satisfy a so-
called asymptotic causality condition. They become unstable if this condition
is violated. In this sense, the problems of acausality and instability are
still correlated even in the case of shear viscosity, as was already found for
the case of bulk viscosity dkkm3 .
We have then demonstrated that the causality of the theory as a whole is
guaranteed if the asymptotic causality condition is fulfilled. Therefore, a
superluminal group velocity in a finite range of momenta can cause the theory
neither to become acausal nor unstable. Finally, we studied the characteristic
velocities and found a violation of causality for small values of
$\tau_{\pi}(\varepsilon+P)/\eta$, but not for values which satisfy the
asymptotic causality condition.
The asymptotic causality condition requires that the ratio
$\tau_{\pi}/\Gamma_{s}$ is sufficiently large, i.e., that the time scale
$\tau_{\pi}$ over which the shear viscous pressure relaxes towards its NS
value is not too small compared to the sound attenuation length
$\Gamma_{s}\sim\eta/(\varepsilon+P)\equiv\eta/(Ts)$. This is an important
finding for practitioners of fluid dynamics, who frequently consider
$\tau_{\pi}$ and the shear viscosity-to-entropy density ratio $\eta/s$ to be
independent from each other. We have demonstrated that this is not the case if
one wants the theory to remain causal. Therefore, second-order theories of
relativistic dissipative fluid dynamics are not automatically causal by
construction. Our findings also illuminate why NS theory violates causality
from a different perspective, because there $\tau_{\pi}\rightarrow 0$ while
$\eta$ remains non-zero.
## Acknowledgement
Shi Pu thanks Zhe Xu and Qun Wang for helpful discussions. We acknowledge G.
Moore and the referee for valuable comments concerning causality and the
divergence of the group velocity which have resulted in the discussion
presented in Sec. III.3. This work was (financially) supported by the
Helmholtz International Center for FAIR within the framework of the LOEWE
program (Landesoffensive zur Entwicklung Wissenschaftlich-Ökonomischer
Exzellenz) launched by the State of Hesse.
## Appendix A Matrix elements in Eq. (64)
The fluid-dynamical equations can be expressed in the form (64). Let us
parameterize the velocity of the fluid as
$u^{\mu}=(\cosh\theta,\sinh\theta\cos\phi,\sinh\theta\sin\phi)$. The matrix
elements of $A_{ab}^{x}$ are
$\displaystyle A^{x}_{11}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh\theta\cosh\theta\cos\phi\;,$
$\displaystyle A^{x}_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\text{sech}^{3}\theta\left\\{\frac{}{}2\sinh^{2}\theta\left[(2w+\pi^{xx})\sin^{2}\phi+3w\cos^{2}\phi-\pi^{xy}\sin\phi\cos\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{4}\theta(\cos(2\phi)+3)+w+\pi^{xx}\frac{}{}\right\\}\;,$
$\displaystyle A^{x}_{13}$ $\displaystyle=$
$\displaystyle\text{sech}^{3}\theta\left\\{\frac{}{}\sinh^{2}\theta\cos\phi\left[(w-\pi^{xx})\sin\phi+\pi^{xy}\cos\phi\right]+w\sinh^{4}\theta\sin\phi\cos\phi+\pi^{xy}\right\\}\;,$
$\displaystyle A^{x}_{14}$ $\displaystyle=$
$\displaystyle\tanh\theta\cos\phi\;,$ $\displaystyle A^{x}_{15}$
$\displaystyle=$ $\displaystyle\tanh\theta\sin\phi\;,$ $\displaystyle
A^{x}_{21}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh^{2}\theta\cos^{2}\phi+c_{s}^{2}\;,$
$\displaystyle A^{x}_{22}$ $\displaystyle=$ $\displaystyle
2w\sinh\theta\cos\phi\;,$ $\displaystyle A^{x}_{24}$ $\displaystyle=$
$\displaystyle A^{x}_{35}=1\;,$ $\displaystyle A^{x}_{31}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh^{2}\theta\sin\phi\cos\phi\;,$
$\displaystyle A^{x}_{32}$ $\displaystyle=$ $\displaystyle
w\sinh\theta\sin\phi\;,$ $\displaystyle A^{x}_{33}$ $\displaystyle=$
$\displaystyle w\sinh\theta\cos\phi\;,$ $\displaystyle A^{x}_{42}$
$\displaystyle=$
$\displaystyle\text{sech}^{2}\theta\left\\{\frac{}{}\sinh^{4}\theta\cos^{2}\phi\left[\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}+\tau_{\pi}\pi^{xy}\sin(2\phi)\right]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{2}\theta\left[2(\eta-\tau_{\pi}\pi^{xx})\cos^{2}\phi+\eta\sin^{2}\phi\right]+\eta\frac{}{}\right\\}\;,$
$\displaystyle A^{x}_{43}$ $\displaystyle=$
$\displaystyle-2\tau_{\pi}\tanh^{2}\theta\cos^{2}\phi\left[\sinh^{2}\theta\cos\phi(\pi^{xy}\cos\phi-\pi^{xx}\sin\phi)+\pi^{xy}\right]\;,$
$\displaystyle A^{x}_{44}$ $\displaystyle=$ $\displaystyle
A^{x}_{55}=\tau_{\pi}\sinh\theta\cos\phi\;,$ $\displaystyle A^{x}_{52}$
$\displaystyle=$
$\displaystyle\frac{\tanh^{2}\theta\cos\phi}{2(\sinh^{2}\theta\cos^{2}\phi+1)}\left\\{\frac{}{}-2\sinh^{2}\theta\left(\pi^{xx}\sin^{3}\phi+2\pi^{xx}\sin\phi\cos^{2}\phi+\pi^{xy}\cos^{3}\phi\right)\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{4}\theta\sin^{2}(2\phi)(\pi^{xy}\cos\phi-2\pi^{xx}\sin\phi)-2\pi^{xx}\sin\phi-2\pi^{xy}\cos\phi\frac{}{}\right\\}\;,$
$\displaystyle A^{x}_{53}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\text{sech}^{2}\theta\left\\{\frac{}{}2\sinh^{4}\theta\cos^{2}\phi\left[\eta-\tau_{\pi}\pi^{xx}\cos(2\phi)+\tau_{\pi}\pi^{xx}-\tau_{\pi}\pi^{xy}\sin(2\phi)\right]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{2}\theta\left[(\eta+\tau_{\pi}\pi^{xx})\cos(2\phi)+3\eta+\tau_{\pi}\pi^{xx}-\tau_{\pi}\pi^{xy}\sin(2\phi)\right]+2\eta\frac{}{}\right\\}\;.$
The matrix elements of $A_{ab}^{t}$ are given by
$\displaystyle A^{t}_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(c_{s}^{2}+1\right)\cosh(2\theta)-c_{s}^{2}+1\right]\;,$
$\displaystyle A^{t}_{12}$ $\displaystyle=$
$\displaystyle\frac{2\sinh\theta}{\left(\sinh^{2}\theta\cos^{2}\phi+1\right)^{2}}\left\\{\frac{}{}\sinh^{2}\theta\cos\phi\left(2w\cos^{2}\phi+\pi^{xx}\sin^{2}\phi-\pi^{xy}\sin\phi\cos\phi\right)\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{4}\theta\cos^{5}\phi+(w+\pi^{xx})\cos\phi+\pi^{xy}\sin\phi\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{13}$ $\displaystyle=$ $\displaystyle
2\sinh\theta\left(w\sin\phi+\frac{\pi^{xy}\cos\phi-\pi^{xx}\sin\phi}{\sinh^{2}\theta\cos^{2}\phi+1}\right)\;,$
$\displaystyle A^{t}_{14}$ $\displaystyle=$
$\displaystyle\frac{\cos(2\phi)}{\text{csch}^{2}\theta+\cos^{2}\phi}\;,$
$\displaystyle A^{t}_{15}$ $\displaystyle=$
$\displaystyle\frac{\sin(2\phi)}{\text{csch}^{2}\theta+\cos^{2}\phi}\;,$
$\displaystyle A^{t}_{21}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh\theta\cosh\theta\cos\phi\;,$
$\displaystyle A^{t}_{31}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh\theta\cosh\theta\sin\phi\;,$
$\displaystyle A^{t}_{22}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{2}\left\\{\frac{}{}2\sinh^{2}\theta\left[(2w+\pi^{xx})\sin^{2}\phi+3w\cos^{2}\phi-\pi^{xy}\sin\phi\cos\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{4}\theta\left[\cos(2\phi)+3\right]+2w+2\pi^{xx}\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{23}$ $\displaystyle=$
$\displaystyle\text{sech}^{3}\theta\left\\{\sinh^{2}\theta\cos\phi\left[w\sinh^{2}\theta\sin\phi+(w-\pi^{xx})\sin\phi+\pi^{xy}\cos\phi\right]+\pi^{xy}\right\\}\;,$
$\displaystyle A^{t}_{24}$ $\displaystyle=$
$\displaystyle\tanh\theta\cos\phi\;,$ $\displaystyle A^{t}_{25}$
$\displaystyle=$ $\displaystyle\tanh\theta\sin\phi\;,$ $\displaystyle
A^{t}_{32}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{\left(\sinh^{2}\theta\cos^{2}\phi+1\right)^{2}}\left\\{\frac{}{}\sinh^{2}\theta\left[(w+3\pi^{xx})\sin\phi\cos\phi+3\pi^{xy}\sin^{2}\phi+2\pi^{xy}\cos^{2}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{4}\theta\left[3(w+\pi^{xx})\sin\phi\cos^{3}\phi+(w+5\pi^{xx})\sin^{3}\phi\cos\phi+2\pi^{xy}\sin^{4}\phi+\pi^{xy}\cos^{4}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{16}\sinh^{6}\theta\left[10\sin(2\phi)+\sin(4\phi)\right]\left[(w-\pi^{xx})\cos(2\phi)+w+\pi^{xx}-\pi^{xy}\sin(2\phi)\right]\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{8}\theta\sin\phi\cos^{5}\phi+\pi^{xy}\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{33}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{8\left(\sinh^{2}\theta\cos^{2}\phi+1\right)}\left\\{\frac{}{}\sinh^{4}\theta\left[4(w+2\pi^{xx})\cos(2\phi)+(\pi^{xx}-w)\cos(4\phi)+21w\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.9\pi^{xx}+10\pi^{xy}\sin(2\phi)+\pi^{xy}\sin(4\phi)\right]+4\sinh^{2}\theta\left[6w+2\pi^{xx}\cos(2\phi)-4\pi^{xx}\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.3\pi^{xy}\sin(2\phi)\right]-4w\sinh^{6}\theta\cos^{2}\phi\left[\cos(2\phi)-3\right]+8w-8\pi^{xx}\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{34}$ $\displaystyle=$
$\displaystyle-\frac{\tanh\theta\sin\phi\left(\sinh^{2}\theta\sin^{2}\phi+1\right)}{\sinh^{2}\theta\cos^{2}\phi+1}\;,$
$\displaystyle A^{t}_{35}$ $\displaystyle=$
$\displaystyle\frac{\tanh\theta\cos\phi}{2\sinh^{2}\theta\cos^{2}\phi+2}\left\\{\frac{}{}2-\sinh^{2}\theta[\cos(2\phi)-3]\right\\}\;,$
$\displaystyle A^{t}_{42}$ $\displaystyle=$
$\displaystyle\tanh\theta\cos\phi\left\\{\frac{}{}\sinh^{2}\theta\left\\{2\sin\phi\left[(\eta-\tau_{\pi}\pi^{xx})\sin\phi+\tau_{\pi}\pi^{xy}\cos\phi\right]+\eta\cos^{2}\phi\right\\}+\eta-2\tau_{\pi}\pi^{xx}\right\\}\;,$
$\displaystyle A^{t}_{43}$ $\displaystyle=$
$\displaystyle-\tanh\theta\left\\{\frac{}{}\sinh^{2}\theta\cos^{2}\phi\left[(\eta-2\tau_{\pi}\pi^{xx})\sin\phi+2\tau_{\pi}\pi^{xy}\cos\phi\right]+\eta\sin\phi+2\tau_{\pi}\pi^{xy}\cos\phi\right\\}\;,$
$\displaystyle A^{t}_{44}$ $\displaystyle=$ $\displaystyle
A^{t}_{55}=\tau_{\pi}\cosh\theta\;,$ $\displaystyle A^{t}_{52}$
$\displaystyle=$
$\displaystyle\frac{\tanh\theta}{4\sinh^{2}\theta\cos^{2}\phi+4}\left\\{\frac{}{}-2\sinh^{2}\theta\left\\{\sin\phi\left[-2\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)+3\tau_{\pi}\pi^{xx}\right]+2\tau_{\pi}\pi^{xy}\cos^{3}\phi\right\\}\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{4}\theta\sin^{2}(2\phi)\left[(\eta-2\tau_{\pi}\pi^{xx})\sin\phi+2\tau_{\pi}\pi^{xy}\cos\phi\right]+4(\eta-\tau_{\pi}\pi^{xx})\sin\phi-4\tau_{\pi}\pi^{xy}\cos\phi\frac{}{}\right\\}\;,$
$\displaystyle A^{t}_{53}$ $\displaystyle=$
$\displaystyle\tanh\theta\left\\{\frac{}{}\sinh^{2}\theta\left[\eta\cos^{3}\phi+\tau_{\pi}\pi^{xx}\sin\phi\sin(2\phi)-2\tau_{\pi}\pi^{xy}\sin\phi\cos^{2}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.(\eta+\tau_{\pi}\pi^{xx})\cos\phi-\tau_{\pi}\pi^{xy}\sin\phi\frac{}{}\right\\}\;.$
The matrix elements of $A_{ab}^{y}$ are
$\displaystyle A^{y}_{11}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh\theta\cosh\theta\sin\phi\;,$
$\displaystyle A^{y}_{21}$ $\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh^{2}\theta\sin\phi\cos\phi\;,$
$\displaystyle A^{y}_{12}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{\left(\sinh^{2}\theta\cos^{2}\phi+1\right)^{2}}\left\\{\frac{}{}\sinh^{2}\theta\left[(w+3\pi^{xx})\sin\phi\cos\phi+3\pi^{xy}\sin^{2}\phi+2\pi^{xy}\cos^{2}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{4}\theta\left[3(w+\pi^{xx})\sin\phi\cos^{3}\phi+(w+5\pi^{xx})\sin^{3}\phi\cos\phi+2\pi^{xy}\sin^{4}\phi+\pi^{xy}\cos^{4}\phi\right]\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{16}\sinh^{6}\theta[10\sin(2\phi)+\sin(4\phi)][(w-\pi^{xx})\cos(2\phi)+w+\pi^{xx}-\pi^{xy}\sin(2\phi)]\right.$
$\displaystyle+$
$\displaystyle\left.w\sinh^{8}\theta\sin\phi\cos^{5}\phi+\pi^{xy}\frac{}{}\right\\}\;,$
$\displaystyle A^{y}_{13}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{3}\theta}{8\left(\sinh^{2}\theta\cos^{2}\phi+1\right)}\left\\{\frac{}{}\sinh^{4}\theta[4(w+2\pi^{xx})\cos(2\phi)+(\pi^{xx}-w)\cos(4\phi)+21w\right.$
$\displaystyle-$
$\displaystyle\left.9\pi^{xx}+10\pi^{xy}\sin(2\phi)+\pi^{xy}\sin(4\phi)]+4\sinh^{2}\theta[6w+2\pi^{xx}\cos(2\phi)-4\pi^{xx}\right.$
$\displaystyle+$
$\displaystyle\left.3\pi^{xy}\sin(2\phi)]-4w\sinh^{6}\theta\cos^{2}\phi[\cos(2\phi)-3]+8w-8\pi^{xx}\frac{}{}\right\\}\;,$
$\displaystyle A^{y}_{14}$ $\displaystyle=$
$\displaystyle-\frac{\tanh\theta\sin\phi\left(\sinh^{2}\theta\sin^{2}\phi+1\right)}{\sinh^{2}\theta\cos^{2}\phi+1}\;,$
$\displaystyle A^{y}_{15}$ $\displaystyle=$
$\displaystyle\frac{\tanh\theta\cos\phi}{2\sinh^{2}\theta\cos^{2}\phi+2}\left\\{\frac{}{}2-\sinh^{2}\theta\left[\cos(2\phi)-3\right]\right\\}\;,$
$\displaystyle A^{y}_{22}$ $\displaystyle=$ $\displaystyle
w\sinh\theta\sin\phi\;,$ $\displaystyle A^{y}_{23}$ $\displaystyle=$
$\displaystyle w\sinh\theta\cos\phi\;,$ $\displaystyle A^{y}_{25}$
$\displaystyle=$ $\displaystyle 1\;,$ $\displaystyle A^{y}_{31}$
$\displaystyle=$
$\displaystyle\left(c_{s}^{2}+1\right)\sinh^{2}\theta\sin^{2}\phi+c_{s}^{2}\;,$
$\displaystyle A^{y}_{32}$ $\displaystyle=$
$\displaystyle\frac{2\sinh\theta\left[\sinh^{2}\theta\sin\phi\cos\phi(\pi^{xx}\sin\phi-\pi^{xy}\cos\phi)+\pi^{xx}\cos\phi+\pi^{xy}\sin\phi\right]}{\left(\sinh^{2}\theta\cos^{2}\phi+1\right)^{2}}\;,$
$\displaystyle A^{y}_{33}$ $\displaystyle=$ $\displaystyle
2\sinh\theta\left(w\sin\phi+\frac{\pi^{xy}\cos\phi-\pi^{xx}\sin\phi}{\sinh^{2}\theta\cos^{2}\phi+1}\right)\;,$
$\displaystyle A^{y}_{34}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}\theta\sin^{2}\phi+1}{\sinh^{2}\theta\cos^{2}\phi+1}\;,$
$\displaystyle A^{y}_{35}$ $\displaystyle=$
$\displaystyle\frac{\sin(2\phi)}{\text{csch}^{2}\theta+\cos^{2}\phi}\;,$
$\displaystyle A^{y}_{42}$ $\displaystyle=$
$\displaystyle\tanh^{2}\theta\sin\phi\cos\phi\left\\{\frac{}{}\sinh^{2}\theta[2\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}+\tau_{\pi}\pi^{xy}\sin(2\phi)]+2\eta-2\tau_{\pi}\pi^{xx}\right\\}\;,$
$\displaystyle A^{y}_{43}$ $\displaystyle=$
$\displaystyle-\frac{\text{sech}^{2}\theta}{2}\left\\{\frac{}{}2\sinh^{4}\theta\cos^{2}\phi[\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}+\tau_{\pi}\pi^{xy}\sin(2\phi)]\right.$
$\displaystyle+$
$\displaystyle\left.\sinh^{2}\theta\left\\{\eta[\cos(2\phi)+3]+2\tau_{\pi}\pi^{xy}\sin(2\phi)\right\\}+2\eta\frac{}{}\right\\}\;,$
$\displaystyle A^{y}_{44}$ $\displaystyle=$ $\displaystyle
A^{y}_{55}=\tau_{\pi}\sinh\theta\sin\phi\;,$ $\displaystyle A^{y}_{52}$
$\displaystyle=$
$\displaystyle\frac{\tanh^{2}\theta}{8(\sinh^{2}\theta\cos^{2}\phi+1)}\left\\{\frac{}{}\sinh^{2}\theta\left[(\tau_{\pi}\pi^{xx}-\eta)\cos(4\phi)+9\eta+4\tau_{\pi}\pi^{xx}\cos(2\phi)-5\tau_{\pi}\pi^{xx}\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.8\tau_{\pi}\pi^{xy}\sin\phi\cos^{3}\phi\right]+2\sinh^{4}\theta\sin^{2}(2\phi)[\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}+\tau_{\pi}\pi^{xy}\sin(2\phi)]\right.$
$\displaystyle+$
$\displaystyle\left.4[4\eta+\tau_{\pi}\pi^{xx}\cos(2\phi)-\tau_{\pi}\pi^{xx}-\tau_{\pi}\pi^{xy}\sin(2\phi)]+8\eta\text{csch}^{2}\theta\frac{}{}\right\\}\;,$
$\displaystyle A^{y}_{53}$ $\displaystyle=$
$\displaystyle\tau_{\pi}\tanh^{2}\theta\sin\phi\left[\sinh^{2}\theta\sin(2\phi)(\pi^{xx}\sin\phi-\pi^{xy}\cos\phi)+\pi^{xx}\cos\phi-\pi^{xy}\sin\phi\right]\;,$
where we defined $w=\varepsilon+P$. All other elements vanish.
## References
* (1) J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 92, 062301 (2004) [arXiv:nucl-ex/0310029].
* (2) J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 92, 052302 (2004) [arXiv:nucl-ex/0306007].
* (3) P. R. Sorensen, arXiv:nucl-ex/0309003.
* (4) C. Adler et al. [STAR Collaboration], Phys. Rev. C 66, 034904 (2002) [arXiv:nucl-ex/0206001].
* (5) See for example, P. Huovinen and P.V. Ruuskanen, Ann. Rev. Nucl. Part. Sci. 56, 163 (2006); Y. Hama, T. Kodama and O. Socolowski, Braz. J. Phys. 35, 24 (2005), Jean-Yves Ollitrault, Euro. J. Phys. 29, 275 (2008) and references therein.
* (6) _RHIC Scientists Serve Up ”Perfect” Liquid_ ,
http://www.bnl.gov/bnlweb/pubaf/pr/PR_display.asp?prID=05-38
* (7) M. Gyulassy and L. McLerran, Nucl. Phys. A 750, 30 (2005) [arXiv:nucl-th/0405013].
* (8) E. V. Shuryak, Nucl. Phys. A 750, 64 (2005) [arXiv:hep-ph/0405066].
* (9) P. Danielewicz and M. Gyulassy, Phys. Rev. D 31, 53 (1985).
* (10) P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005) [arXiv:hep-th/0405231].
* (11) L. D. Landau, E. M. Lifshitz: _Fluid Mechanics_ , (Pergamon Press, New York, 1959), Sections 133-136
* (12) A. Muronga, Phys. Rev. Lett. 88, 062302 (2002) [Erratum ibid. 89, 159901 (2002)].
* (13) P. Romatschke and U. Romatschke, Phys. Rev. Lett. 99, 172301 (2007).
* (14) M. Luzum and P. Romatschke, Phys. Rev. Lett. 103, 262302 (2009) [arXiv:0901.4588 [nucl-th]].
* (15) M. Luzum and P. Romatschke, Phys. Rev. C 78, 034915 (2008) [Erratum-ibid. C 79, 039903 (2009)] [arXiv:0804.4015 [nucl-th]].
* (16) H. Song and U. W. Heinz, Phys. Rev. C 77, 064901 (2008) [arXiv:0712.3715 [nucl-th]].
* (17) H. Song and U. W. Heinz, Phys. Lett. B 658, 279 (2008) [arXiv:0709.0742 [nucl-th]].
* (18) A. Chaudhuri, J. Phys. G35, 104015 (2008).
* (19) K. Dusling and D. Teaney, Phys. Rev. C 77, 034905 (2008).
* (20) D. Molnar and P. Huovinen, J. Phys. G35, 104125 (2008).
* (21) S. Pratt, Phys. Rev. C77, 024910 (2008).
* (22) R. S. Bhalerao and S. Gupta, Phys. Rev. C77 014902 (2008).
* (23) I. Bouras, E. Molnar, H. Niemi, Z. Xu, A. El, O. Fochler, C. Greiner and D.H. Rischke, Phys. Rev. Lett. 103, 032301 (2009) [arXiv:0902.1927 [hep-ph]].
* (24) B. Betz, D. Henkel and D. H. Rischke, Prog. Part. Nucl. Phys. 62, 556 (2009) [arXiv:0812.1440 [nucl-th]]; J. Phys. G 36, 064029 (2009).
* (25) M. Martinez and M. Strickland, Phys. Rev. C 79, 044903 (2009).
* (26) T. Koide, G. S. Denicol, Ph. Mota and T. Kodama, Phys. Rev. C75, 034909 (2007).
* (27) G.S.Denicol, T. Kodama, T. Koide and Ph. Mota, Phys. Rev. C78, 034901 (2008).
* (28) G.S. Denicol, T. Kodama, T. Koide and Ph. Mota, J. Phys. G 35, 115102 (2008).
* (29) G. S. Denicol, T. Kodama, T. Koide and Ph. Mota, J. Phys. G36, 035103 (2009).
* (30) G. S. Denicol, T. Kodama, T. Koide and Ph. Mota, Phys. Rev. C 80, 064901 (2009) [arXiv:0903.3595 [hep-ph]].
* (31) T. Koide, E. Nakano and T. Kodama, Phys. Rev. Lett. 103, 052301 (2009).
* (32) L.L. Samojeden and G.M. Kremer, Physica A 307, 354 (2002).
* (33) W. Israel and J. M. Stewart, Ann. Phys. (N.Y.) 118, 341 (1979).
* (34) D. Jou, J. Casas-Vázquez, and G. Lebon, Rep. Prog. Phys. 51, 1105 (1988); 62, 1035 (1999).
* (35) M. Grmela and H. C. Öttinger, Phys. Rev. E 56, 6620 (1997); B. Carter, Proc. R. Soc. A433, 45 (1991).
* (36) R. Baier, P. Romatschke, D. T. Son, A. O. Starinets and M. A. Stephanov, JHEP 0804, 100 (2008).
* (37) Z. Xu and C. Greiner, Phys. Rev. Lett. 100, 172301 (2008) [arXiv:0710.5719 [nucl-th]].
* (38) W. A Hiscock and L. Lindblom, Ann. Phys. (N.Y.) 151, 466 (1983); Phys. Rev. D31, 725 (1985); ibid D35, 3723 (1987); Phys. Lett. A131, 509 (1988); W. A. Hiscock and T. S. Olson, Phys. Lett. A141, 125 (1989).
* (39) W. A Hiscock and L. Lindblom, Phys. Rev. D31, 725 (1985); ibid D35, 3723 (1987).
* (40) M. P. Heller and R. A. Janik, Phys. Rev. D 76, 025027 (2007).
* (41) S. Pu and Q. Wang, arXiv:0810.5271 [hep-ph].
* (42) J. D. Jackson: Classical Electrodynamics, (John Wiley $\&$ Sons, Inc., 1999).
* (43) L. Brillouin: Wave Propagation and Group Velocity, (Academic Press, London, 1960).
|
arxiv-papers
| 2009-07-22T19:17:31 |
2024-09-04T02:49:04.111640
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shi Pu, Tomoi Koide, Dirk H. Rischke",
"submitter": "Shi Pu",
"url": "https://arxiv.org/abs/0907.3906"
}
|
0907.3990
|
# A Deformation of Commutative Polynomial Algebras in Even Numbers of
Variables
Wenhua Zhao
###### Abstract.
We introduce and study a deformation of commutative polynomial algebras in
even numbers of variables. We also discuss some connections and applications
of this deformation to the generalized Laguerre orthogonal polynomials and the
interchanges of right and left total symbols of differential operators of
polynomial algebras. Furthermore, a more conceptual re-formulation for the
image conjecture [Z3] is also given in terms of the deformed algebras.
Consequently, the well-known Jacobian conjecture [Ke] is reduced to an open
problem on this deformation of polynomial algebras.
###### Key words and phrases:
The generalized Laguerre polynomials, total symbols of differential operators,
the image conjecture, the Jacobian conjecture.
###### 2000 Mathematics Subject Classification:
33C45, 32W99, 14R15
## 1\. Introduction
Let $\xi=(\xi_{1},\xi_{2},...,\xi_{n})$ and $z=(z_{1},z_{2},...,z_{n})$ be
$2n$ commutative free variables. Throughout this paper, we denote by
$\mathbb{C}[\xi,z]$, $\mathbb{C}[z]$ and $\mathbb{C}[\xi]$ the vector spaces
(without any algebra structures) over $\mathbb{C}$ of polynomials in
$({\xi,z})$, in $z$ and in $\xi$, respectively. The corresponding polynomial
algebras will be denoted respectively by ${\mathcal{A}}[\xi,z]$,
${\mathcal{A}}[z]$ and ${\mathcal{A}}[\xi]$.
For any $1\leq i\leq n$, we set $\partial_{i}\\!:=\partial_{z_{i}}$ and
$\delta_{i}=\partial_{\xi_{i}}$. Denote by
$\partial=(\partial_{1},\partial_{2},...,\partial_{n})$ and
$\delta=(\delta_{1},\delta_{2},...,\delta_{n})$. We also occasionally use
$\partial_{z}$ and $\partial_{\xi}$ to denote $\partial$ and $\delta$,
respectively.
Set
$\Omega\\!:=\sum_{i=1}^{n}(\partial_{i}\otimes\delta_{i}+\delta_{i}\otimes\partial_{i})$.
For any $t\in\mathbb{C}$, we define a new product $\ast_{t}$ for the vector
space $\mathbb{C}[\xi,z]$ by setting, for any $f,g\in\mathbb{C}[\xi,z]$,
(1.1) $\displaystyle f\ast_{t}g\\!:=\mu\left(e^{-t\,\Omega}(f\otimes
g)\right),$
where $\mu:\mathbb{C}[\xi,z]\otimes\mathbb{C}[\xi,z]\to\mathbb{C}[\xi,z]$
denotes the product map of the polynomial algebra ${\mathcal{A}}[{\xi,z}]$.
Denote by ${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$ the new algebra
$(\mathbb{C}[\xi,z],\ast_{t})$. For the case $t=1$, we also introduce the
following short notation:
(1.2) $\displaystyle\ast\\!:$ $\displaystyle=\ast_{t=1}.$ (1.3)
$\displaystyle{\mathcal{B}}[{\xi,z}]\\!:$
$\displaystyle={\mathcal{B}}_{t=1}[{\xi,z}].$
Note that, when $t=0$, the algebra ${\mathcal{B}}_{t=0}[{\xi,z}]$ coincides
with the usual polynomial algebra ${\mathcal{A}}[{\xi,z}]$.
In this paper, we first show that ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$ gives a deformation of the polynomial algebra
${\mathcal{A}}[{\xi,z}]$. Actually, it is a trivial deformation in the sense
of deformation theory. To be more precise, set
(1.4) $\displaystyle\Lambda\\!:$
$\displaystyle=\sum_{i=1}^{n}\delta_{i}\partial_{i}.$ (1.5)
$\displaystyle\Phi_{t}\\!:$ $\displaystyle=e^{t\Lambda}=\sum_{m\geq
0}\frac{t^{m}\Lambda^{m}}{m!}.$ (1.6) $\displaystyle\Phi\\!:$
$\displaystyle=\Phi_{t=1}.$
Note that, $\Phi_{t}$ for any $t\in\mathbb{C}$ is a well-defined bijective
linear map from $\mathbb{C}[\xi,z]$ to $\mathbb{C}[\xi,z]$, whose inverse map
is given by $\Phi_{-t}=e^{-t\Lambda}$. This is because the differential
operator $\Lambda$ of $\mathbb{C}[\xi,z]$ is locally nilpotent, i.e. for any
$f(\xi,z)\in\mathbb{C}[\xi,z]$, $\Lambda^{m}f(\xi,z)=0$ when $m\gg 0$.
With the notation fixed above, we will show that, for any $t\in\mathbb{C}$,
$\Phi_{t}:{\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[{\xi,z}]$ actually is an
isomorphism of $\mathbb{C}$-algebras (See Proposition 2.1 and Corollary 2.2).
Note that, from the point view of deformation theory, the deformation
${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$ is not interesting at all.
But, surprisingly, as we will show in this paper, the algebra
${\mathcal{B}}_{t}[{\xi,z}]$ and the isomorphism $\Phi_{t}$ are actually
closely related with the generalized Laguerre polynomials (See [Sz], [PS] and
[AAR]) and the interchanges of right and left total symbols of differential
operators of polynomial algebras.
Furthermore, as we will show in Section 4, the algebras
${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$ and the isomorphism $\Phi_{t}$
via their connections with the image conjecture proposed in [Z3] are also
related with the Jacobian conjecture which was first proposed by O. H. Keller
[Ke] in $1939$ (See also [BCW] and [E]). Actually, the Jacobian conjecture can
be viewed as a conjecture which, in some sense, just claims that the algebra
${\mathcal{B}}_{t}[{\xi,z}]$ $(t\neq 0)$ should not differ or change too much
from the polynomial algebra
${\mathcal{A}}[{\xi,z}]={\mathcal{B}}_{t=0}[{\xi,z}]$. Therefore, from this
point of view, the triviality of the deformation ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$ (in the sense of deformation theory) can be viewed as a
fact in favor of the Jacobian conjecture. For another interesting application
of the isomorphism $\Phi$ to the Jacobian conjecture, see [Z5].
Considering the length of the paper, below we give a more detailed description
for the contents and the arrangement of the paper.
In Subsection 2.1, we prove some simple properties of the deformation
${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$ and the isomorphism
$\Phi_{t}:{\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[{\xi,z}]$, which will be
needed for the rest of this paper. In particular, in this subsection the
triviality of the deformation ${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$
in the sense of deformation theory is proved in Proposition 2.1. and Corollary
2.2.
In Subsection 2.2, we show that, for different $t\in\mathbb{C}$, the
$\ell$-adic topologies induced by ${\mathcal{B}}_{t}[{\xi,z}]$ on the common
base vector space $\mathbb{C}[\xi,z]$ are different. But they are all
homeomorphic to the $\ell$-adic topology induced by the polynomial algebra
${\mathcal{A}}[{\xi,z}]$ under the isomorphism
$\Phi_{t}:{\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[{\xi,z}]$ (viewed as an
automorphism of $\mathbb{C}[\xi,z]$). See Proposition 2.9 and also Corollary
2.10 for the precise statements.
In Subsection 2.3, we study the induced isomorphism $(\Phi_{t})_{*}$
$(t\in\mathbb{C})$ of $\Phi_{t}$ from the differential operator algebra, or
the Weyl algebra ${\mathcal{D}}_{t}[{\xi,z}]$ of ${\mathcal{B}}_{t}[{\xi,z}]$
to the Weyl algebra ${\mathcal{D}}[{\xi,z}]$ of ${\mathcal{A}}[{\xi,z}]$. The
main results of this subsection are Propositions 2.11 and 2.13. Proposition
2.11 says that the derivations $\partial_{z_{i}}$ and $\partial_{\xi_{i}}$
$(1\leq i\leq n)$ of ${\mathcal{A}}[{\xi,z}]$ are also derivations of
${\mathcal{B}}_{t}[{\xi,z}]$ for all $t\in\mathbb{C}$ and are fixed by the
isomorphism $(\Phi_{t})_{*}$. Proposition 2.13 gives explicitly the images
under $(\Phi_{t})_{*}$ of the multiplication operators with respect to the
product $\ast_{t}$ of ${\mathcal{B}}_{t}[{\xi,z}]$.
In Section 3, by using some results derived in Section 2, we show in Theorem
3.1 that $\Phi=\Phi_{t=1}$ (resp. $\Phi_{t=-1}$) as an automorphism of
$\mathbb{C}[\xi,z]$ actually coincides with the linear map which changes left
(resp. right) total symbols of differential operators of ${\mathcal{A}}[z]$ to
their right (resp. left) total symbols. Consequently, the products
$\ast_{t=\pm 1}$ appear naturally when one derives left or right total symbols
of certain differential operators of ${\mathcal{A}}[z]$ (See Corollary 3.2).
The results derived in this subsection also play some important roles in [Z5]
in which among some other results a more straightforward proof for the
equivalence of the Jacobian conjecture and the vanishing conjecture (See [Z1]
and [Z2]) will be given.
In Subsection 4.1, we study the Taylor series expansion of polynomials in
$\mathbb{C}[{\xi,z}]$ with respect to the new product $\ast_{t}$ and use it to
give a more conceptual proof for the expansion of polynomials given in Eq.
(4.6). This expansion was first proved in [Z3] and played a crucial role there
in the proof of the implication of the Jacobian conjecture from the image
conjecture (See Conjecture 4.3).
In Subsection 4.2, we first recall the notion of the so-called Mathieu
subspaces of commutative algebras (See Definition 4.2), which was first
introduced in [Z4], and also the image conjecture (See Conjecture 4.3) for the
differential operators $\xi_{i}-t\partial_{i}$ $(1\leq i\leq n)$ in terms of
the notion of Mathieu subspaces. We then give a re-formulation of Conjecture
4.3 in terms of the algebra ${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$
(See Conjecture 4.5) and show in Theorem 4.6 that these two conjectures are
equivalent to each other. Since it has been shown in [Z3] that Conjecture 4.3
implies Jacobian conjecture, hence so does Conjecture 4.5.
Consequently, via its connections with Conjecture 4.5, the Jacobian conjecture
is reduced to an open problem on the deformation ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$ of the polynomial algebra ${\mathcal{A}}[{\xi,z}]$. The
open problem asks if the ideal $\xi\mathbb{C}[{\xi,z}]$ of
${\mathcal{A}}[{\xi,z}]$ generated by $\xi$ will remain to be a Mathieu
subspace in the algebra ${\mathcal{B}}_{t}[{\xi,z}]$ for any $t\neq 0$. Note
that any ideal is automatically a Mathieu subspace, but not conversely.
Therefore, the triviality (in the sense of deformation theory) of the
deformation ${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$ proved in
Proposition 2.1 can be viewed as a fact in favor of the Jacobian conjecture.
Section 5 is mainly on a connection of the algebra ${\mathcal{B}}[{\xi,z}]$,
especially, its product $\ast$ with the multi-variable generalized Laguerre
polynomials, and also some of the applications of this connection to both
${\mathcal{B}}[{\xi,z}]$ and the generalized Laguerre polynomials.
In Subsection 5.1, we very briefly recall the definition of the (generalized)
Laguerre polynomials $L_{\alpha}^{[\bf k]}(z)$ $({\bf
k},\alpha\in{\mathbb{N}}^{n})$ (See Eqs. (5.1)–(5.3)) and also the orthogonal
property (See Theorem 5.1) of these polynomials.
In Subsection 5.2, we show in Theorem 5.2 that, for any ${\bf
k},\alpha\in{\mathbb{N}}^{n}$, we have
(1.7) $\displaystyle L_{\alpha}^{[{\bf k}]}(\xi
z)=\frac{(-1)^{|\alpha|}}{\alpha!}\,\xi^{-\bf k}(\xi^{\alpha+{\bf k}}\ast
z^{\alpha})=\frac{(-1)^{|\alpha|}}{\alpha!}\,z^{-{\bf k}}(\xi^{\alpha}\ast
z^{\alpha+{\bf k}}).$
Consequently, the generalized Laguerre polynomials $L_{\alpha}^{[\bf k]}(z)$
$({\bf k},\alpha\in{\mathbb{N}}^{n})$ can be obtained by evaluating the
polynomials $\xi^{-\bf k}(\xi^{\alpha+{\bf k}}\ast z^{\alpha})$ or $z^{-{\bf
k}}(\xi^{\alpha}\ast z^{\alpha+{\bf k}})$ at $\xi=(1,1,...,1)$. Note that the
evaluation map at $\xi=(1,1,...,1)$ is not an algebra homomorphism from
${\mathcal{B}}[{\xi,z}]$ to ${\mathcal{A}}[{\xi,z}]$. Otherwise, the
generalized Laguerre polynomials would be trivialized.
In the first part of Subsection 5.3, we use certain results of the generalized
Laguerre polynomials and the connection in Eq. (1.7) above to derive more
properties on the polynomials $\xi^{\alpha}\ast z^{\alpha}$ which, by
Proposition 2.6, $(c)$, are actually the monomials of $\xi$ and $z$ in the new
algebra ${\mathcal{B}}[{\xi,z}]$.
For example, by using the connection in Eq. (1.7) and I. Schur’s
irreducibility theorem [Sc1] of the Laguerre polynomials in one variable, we
immediately have that, when $n=1$, the monomials $\xi^{m}\ast z^{m}$ $(m\geq
2)$ of ${\mathcal{B}}[{\xi,z}]$ are actually irreducible over ${\mathbb{Q}}$
(See Theorem 5.9). Furthermore, by using I. Schur’s irreducibility theorem
[Sc2] and M. Filaseta and T.-Y. Lam’s irreducibility theorem [FL] on the
generalized Laguerre polynomials, we have that, all but finitely many of the
polynomials $\xi^{-k}(\xi^{m+k}\ast z^{m})$ and $z^{-k}(\xi^{m}\ast z^{m+k})$
$(m,k\in{\mathbb{N}})$ are irreducible over ${\mathbb{Q}}$ (See Theorem 5.10).
In the second part of Subsection 5.3, we use the connection given in Eq. (1.7)
and certain results of ${\mathcal{B}}[{\xi,z}]$ derived in Section 2 to give
new proofs, first, for some recurrent formulas of the generalized Laguerre
polynomials (See Proposition 5.11) and, second, for the fact that the
generalized Laguerre polynomials satisfy the so-called associate Laguerre
differential equation (See Theorem 5.12). At the end of this subsection, we
draw the reader’s attention to a conjecture, Conjecture 5.13, on the
generalized Laguerre polynomials, which is still open even for the classical
Laguerre polynomials in one variable.
Acknowledgment The author would like to thank the anonymous referees for
pointing out many typos, minor errors of the previous version of this paper,
and also for suggesting the new proof of Lemma 5.8 without the condition that
the base filed $K$ has infinitely many elements.
## 2\. The Deformation ${\mathcal{B}}_{t}[{\xi,z}]$ of the Polynomial Algebra
${\mathcal{A}}[{\xi,z}]$
In this section, we first derive in Subsection 2.1 some properties and
identities for the algebra ${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$. In
Subsection 2.2, we show that, for different $t\in\mathbb{C}$, the $\ell$-adic
topologies induced by the algebras ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$ on the common base vector space $\mathbb{C}[\xi,z]$ are
different. But they are all homeomorphic under the isomorphism
$\Phi_{t}:{\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[{\xi,z}]$ to the
$\ell$-adic topology on $\mathbb{C}[\xi,z]$ induced by the polynomial algebra
${\mathcal{A}}[{\xi,z}]$ (See Proposition 2.9 and also Corollary 2.10).
In Subsection 2.3, we study the isomorphism $(\Phi_{t})_{*}$ induced by
$\Phi_{t}$ from the Weyl algebra of ${\mathcal{B}}_{t}[{\xi,z}]$ to the Weyl
algebra of ${\mathcal{A}}[{\xi,z}]$. The main results in this subsection are
Propositions 2.11 and 2.13.
### 2.1. Some Properties of the Algebras ${\mathcal{B}}_{t}[{\xi,z}]$
First, one remark on notation and convention is that, we will freely use
throughout this paper some commonly used multi-index notations and
conventions. For instance, for $n$-tuples $\alpha=(k_{1},k_{2},...,k_{n})$ and
$\beta=(m_{1},m_{2},...,m_{n})$ of non-negative integers, we have
$\displaystyle|\alpha|$ $\displaystyle=\sum_{i=1}^{n}k_{i}.$
$\displaystyle\alpha!$ $\displaystyle=k_{1}!k_{2}!\cdots k_{n}!.$
$\displaystyle\binom{\alpha}{\beta}$
$\displaystyle=\begin{cases}\frac{\alpha!}{\beta!(\alpha-\beta)!}&\mbox{if
$k_{i}\geq m_{i}$ for all $1\leq i\leq n$};\\\
0,&\mbox{otherwise.}\end{cases}$
The notation and convention fixed in the previous section will also be used
throughout this paper.
The first main result of this section is the following proposition.
###### Proposition 2.1.
For any $t\in\mathbb{C}$ and $f,g\in\mathbb{C}[\xi,z]$, we have
(2.1) $\displaystyle\Phi_{t}(f\ast_{t}g)=\Phi_{t}(f)\Phi_{t}(g).$
Proof: We first set
(2.2) $\displaystyle
f\ast_{t}^{\prime}g\\!:=\Phi_{t}^{-1}\big{(}\Phi_{t}(f)\Phi_{t}(g)\big{)}=\Phi_{-t}\big{(}\Phi_{t}(f)\Phi_{t}(g)\big{)}.$
for any $f,g\in\mathbb{C}[{\xi,z}]$.
We view $t$ as a formal parameter which commutes with $\xi$ and $z$. Then, by
Eqs. (1.1), (2.2) and the fact that the differential operators $\Lambda$ and
$\Omega$ are locally nilpotent on $\mathbb{C}[\xi,z]$ and
$\mathbb{C}[\xi,z]\otimes\mathbb{C}[\xi,z]$, respectively, we see that
$f\ast_{t}g$ and $f\ast_{t}^{\prime}g$ are polynomials in $t$ with
coefficients in $\mathbb{C}[{\xi,z}]$. Furthermore, by setting $t=0$ in Eqs.
(1.1) and (2.2), we see that the constant terms (with respect to $t$) of
$f\ast_{t}g$ and $f\ast_{t}^{\prime}g$ are both $fg\in\mathbb{C}[{\xi,z}]$. In
other words, we have
(2.3) $\displaystyle
f\ast_{t}g\left.\right|_{t=0}=f\ast_{t}^{\prime}g\left.\right|_{t=0}=fg.$
From Eq. (1.1), we have,
(2.4) $\displaystyle\frac{\partial}{\partial t}(f\ast_{t}g)$
$\displaystyle=-\mu\left(\,e^{-t\Omega}(\Omega(f\otimes g))\,\right)$
$\displaystyle=-\sum_{i=1}^{n}\mu\left(e^{-t\Omega}\left((\delta_{i}f)\otimes(\partial_{i}g)+(\partial_{i}f)\otimes(\delta_{i}g)\right)\right)$
$\displaystyle=-\sum_{i=1}^{n}\left((\delta_{i}f)\ast_{t}(\partial_{i}g)+(\partial_{i}f)\ast_{t}(\delta_{i}g)\right).$
On the other hand, from Eq. (2.2), we have,
$\displaystyle\frac{\partial}{\partial t}(f$
$\displaystyle\ast_{t}^{\prime}g)=\frac{\partial}{\partial
t}\left(e^{-t\Lambda}((e^{t\Lambda}f)\,(e^{t\Lambda}g))\right)$
$\displaystyle=e^{-t\Lambda}\left(-\Lambda((e^{t\Lambda}f)\,(e^{t\Lambda}g))+(e^{t\Lambda}\Lambda
f)\,(e^{t\Lambda}g)+(e^{t\Lambda}f)\,(e^{t\Lambda}\Lambda g)\right).$
Note that, for any $u,v\in\mathbb{C}[\xi,z]$, it is easy to check that we have
the following identity:
$\displaystyle\Lambda(uv)=(\Lambda u)v+u(\Lambda
v)+\sum_{i=1}^{n}\left(\,(\delta_{i}u)(\partial_{i}v)+(\partial_{i}u)(\delta_{i}v)\,\right).$
By the last two equations above and also Eq. (2.2), we have
(2.5) $\displaystyle\frac{\partial}{\partial t}(f\ast_{t}^{\prime}g)$
$\displaystyle=-\sum_{i=1}^{n}e^{-t\Lambda}\left(\,((e^{t\Lambda}\delta_{i}f)\,(e^{t\Lambda}\partial_{i}g))+(e^{t\Lambda}\partial_{i}f)\,(e^{t\Lambda}\delta_{i}g)\,\right)$
$\displaystyle=-\sum_{i=1}^{n}\left(\,(\delta_{i}f)\ast_{t}^{\prime}(\partial_{i}g)+(\partial_{i}f)\ast_{t}^{\prime}(\delta_{i}g)\,\right).$
Next, we use the induction on $(\deg f+\deg g)$ to show Eq. (2.1). First, when
$\deg f+\deg g=0$, i.e. both $f$ and $g$ have degree zero, it is easy to see
from Eqs. (1.1) and (2.2) that $f\ast_{t}g=f\ast_{t}^{\prime}g=fg$ in this
case.
In general, by Eqs. (2.4), (2.5) and also the induction assumption, we have
(2.6) $\displaystyle\frac{\partial}{\partial
t}(f\ast_{t}g)=\frac{\partial}{\partial t}(f\ast_{t}^{\prime}g).$
Since $f\ast_{t}g$ and $f\ast_{t}^{\prime}g$ are polynomials in $t$ with
coefficients in $\mathbb{C}[{\xi,z}]$ and both satisfy Eqs. (2.3) and (2.6),
it is easy to see that they must be equal to each other. Hence, Eq. (2.1)
holds. $\Box$
###### Corollary 2.2.
For any $t\in\mathbb{C}$,
$\Phi_{t}:{\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[{\xi,z}]$ is an
isomorphism of algebras. Therefore, in the sense of deformation theory, the
deformation ${\mathcal{B}}_{t}[{\xi,z}]$ is a trivial deformation of the
commutative polynomial algebra ${\mathcal{A}}[{\xi,z}]$.
Next we derive some properties of the algebras ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$, which will be needed for the rest of this paper.
###### Lemma 2.3.
For any $f,g\in\mathbb{C}[{\xi,z}]$, we have
(2.7) $\displaystyle
f\ast_{t}g=\sum_{\alpha,\beta\in{\mathbb{N}}^{n}}\frac{(-t)^{|\alpha|+|\beta|}}{\alpha!\beta!}\,\,(\delta^{\beta}\partial^{\alpha}f)(\partial^{\beta}\delta^{\alpha}g).$
Proof: Note first that, for any $1\leq i,j\leq n$,
$\partial_{i}\otimes\delta_{i}$ and $\delta_{j}\otimes\partial_{j}$ commute
with each other. So we have
(2.8) $\displaystyle e^{-t\,\Omega}$
$\displaystyle=e^{-t\sum_{i=1}^{n}\delta_{i}\otimes\partial_{i}}\,e^{-t\sum_{i=1}^{n}\partial_{i}\otimes\delta_{i}},$
(2.9) $\displaystyle e^{-t\sum_{i=1}^{n}\partial_{i}\otimes\delta_{i}}$
$\displaystyle=\prod_{i=1}^{n}e^{-t(\partial_{i}\otimes\delta_{i})}=\prod_{i=1}^{n}\sum_{k_{i}\geq
0}\frac{(-t)^{k_{i}}}{k_{i}!}\,(\partial_{i}^{k_{i}}\otimes\delta_{i}^{k_{i}})$
$\displaystyle=\sum_{\alpha\in{\mathbb{N}}^{n}}\frac{(-t)^{|\alpha|}}{\alpha!}\,\,(\partial^{\alpha}\otimes\delta^{\alpha}).$
Similarly,
(2.10) $\displaystyle
e^{-t\sum_{i=1}^{n}\delta_{i}\otimes\partial_{i}}=\sum_{\beta\in{\mathbb{N}}^{n}}\frac{(-t)^{|\beta|}}{\beta!}\,\,(\delta^{\beta}\otimes\partial^{\beta}).$
Then it is easy to see that Eq. (2.7) follows directly from Eq. (1.1) and the
last three equations above. $\Box$
###### Proposition 2.4.
$(a)$ For any $\lambda_{i}(\xi)\in\mathbb{C}[\xi]$, $p_{i}(z)\in\mathbb{C}[z]$
$(i=1,2)$, we have
(2.11) $\displaystyle\lambda_{1}(\xi)\ast_{t}\lambda_{2}(\xi)$
$\displaystyle=\lambda_{1}(\xi)\lambda_{2}(\xi).$ (2.12) $\displaystyle
p_{1}(z)\ast_{t}p_{2}(z)$ $\displaystyle=p_{1}(z)p_{2}(z).$ (2.13)
$\displaystyle\lambda(\xi)\ast_{t}p(z)=\sum_{\alpha\in{\mathbb{N}}^{n}}$
$\displaystyle\frac{(-1)^{|\alpha|}t^{|\alpha|}}{\alpha!}(\delta^{\alpha}\lambda(\xi))(\partial^{\alpha}p(z)).$
$(b)$ For any $\lambda(\xi)\in\mathbb{C}[\xi]$, $p(z)\in\mathbb{C}[z]$ and
$g({\xi,z})\in\mathbb{C}[{\xi,z}]$, we have
(2.14) $\displaystyle\lambda(\xi)\ast_{t}g({\xi,z})$
$\displaystyle=\lambda(\xi-t\partial)g({\xi,z}).$ (2.15) $\displaystyle
p(z)\ast_{t}g({\xi,z})$ $\displaystyle=p(z-t\delta)g({\xi,z}).$
Note that the components $\xi_{i}-t\partial_{i}$ $(1\leq i\leq n)$ of the
$n$-tuple $\xi-t\partial$ in Eq. (2.14) commute with one another. So the
substitution $\lambda(\xi-t\partial)$ of $\xi-t\partial$ into the polynomial
$\lambda(\xi)$ is well-defined. Similarly, the substitution $p(z-t\delta)$ in
Eq. (2.15) is also well-defined.
Proof: Eqs. (2.11)–(2.13) follow directly from Eq. (2.7).
To show Eq. (2.14), first, by Eq. (2.7), we have
(2.16)
$\displaystyle\lambda(\xi)\ast_{t}g({\xi,z})=\sum_{\alpha\in{\mathbb{N}}^{n}}$
$\displaystyle\frac{(-1)^{|\alpha|}t^{|\alpha|}}{\alpha!}(\delta^{\alpha}\lambda(\xi))(\partial^{\alpha}g({\xi,z})).$
Second, note that the multiplication operators by $\xi_{i}$ $(1\leq i\leq n)$
and the derivations $\partial_{j}$ $(1\leq j\leq n)$ commute. By using the
Taylor series expansion of $\lambda(\xi-t\partial)$ at $\xi$, we have
(2.17) $\displaystyle\lambda(\xi-t\partial)g({\xi,z})$
$\displaystyle=\left(\sum_{\alpha\in{\mathbb{N}}^{n}}\frac{(-1)^{|\alpha|}t^{|\alpha|}}{\alpha!}(\delta^{\alpha}\lambda)(\xi)\partial^{\alpha}\right)g({\xi,z})$
$\displaystyle=\sum_{\alpha\in{\mathbb{N}}^{n}}\frac{(-1)^{|\alpha|}t^{|\alpha|}}{\alpha!}(\delta^{\alpha}\lambda(\xi))(\partial^{\alpha}g({\xi,z})).$
Hence, Eq. (2.14) follows from the last two equations. Eq. (2.15) can be
proved similarly. $\Box$
###### Lemma 2.5.
For any $t\in\mathbb{C}$, $\lambda(\xi)\in\mathbb{C}[\xi]$ and
$p(z)\in\mathbb{C}[z]$, we have
(2.18) $\displaystyle\Phi_{t}(\lambda(\xi))$ $\displaystyle=\lambda(\xi).$
(2.19) $\displaystyle\Phi_{t}(p(z))$ $\displaystyle=p(z).$ (2.20)
$\displaystyle\Phi_{t}(\lambda(\xi)p(z))$
$\displaystyle=\lambda(\xi)\ast_{-t}p(z).$
Proof: Since $\Lambda(\lambda(\xi))=\Lambda(p(z))=0$, $\Phi_{t}=e^{t\Lambda}$
fixes $\lambda(\xi)$ and $p(z)$. Hence we have Eqs. (2.18) and (2.19).
To show Eq. (2.20), by Eqs. (2.1), (2.18) and (2.19), we have
$\displaystyle\lambda(\xi)\ast_{t}p(z)$
$\displaystyle=\Phi_{-t}\left(\,\Phi_{t}(\lambda(\xi))\Phi_{t}(p(z))\,\right)$
$\displaystyle=\Phi_{-t}(\lambda(\xi)p(z)).$
Replacing $t$ be $-t$ in the equation above, we get Eq. (2.20). $\Box$
###### Proposition 2.6.
For any $t\in\mathbb{C}$, the following statements hold.
$(a)$ The subspaces $\mathbb{C}[\xi]$ and $\mathbb{C}[z]$ of
${\mathcal{B}}_{t}[{\xi,z}]$ are closed under the product $\ast_{t}$ and
hence, are actually subalgebras of ${\mathcal{B}}_{t}[{\xi,z}]$.
$(b)$ As associative algebras, $(\mathbb{C}[\xi],\ast_{t})$ and
$(\mathbb{C}[z],\ast_{t})$ are identical as the usual polynomial algebras
${\mathcal{A}}[\xi]$ and ${\mathcal{A}}[z]$ in $\xi$ and $z$, respectively.
$(c)$ ${\mathcal{B}}_{t}[{\xi,z}]$ is a commutative free algebra generated
freely by $\xi_{i}$ and $z_{i}$ $(1\leq i\leq n)$. The set of the monomials
generated by $\xi_{i}$ and $z_{i}$ $(1\leq i\leq n)$ in
${\mathcal{B}}_{t}[{\xi,z}]$ is given by
$\\{\xi^{\alpha}\ast_{t}z^{\beta}\,|\,\alpha,\beta\in{\mathbb{N}}^{n}\\}$.
Proof: Note that $(a)$ and $(b)$ follow immediately from Eqs. (2.11) and
(2.12).
To show $(c)$, first, by Eqs. (2.18) and (2.19), we know that the algebra
isomorphism
$\Phi_{-t}=\Phi_{t}^{-1}:{\mathcal{A}}[{\xi,z}]\to{\mathcal{B}}_{t}[{\xi,z}]$
as a linear map from $\mathbb{C}[\xi,z]$ to $\mathbb{C}[\xi,z]$ fixes
$\xi_{i}$ and $z_{i}$ $(1\leq i\leq n)$. Hence, ${\mathcal{B}}_{t}[{\xi,z}]$
is a commutative free algebra generated freely by $\xi_{i}$ and $z_{i}$
$(1\leq i\leq n)$.
The second part of $(c)$ follows from Eqs. (2.11), (2.12) and the fact that
the product $\ast_{t}$ is associative and commutative. $\Box$
The next two lemmas will be needed in Subsection 5.3.
###### Lemma 2.7.
For any $t\in\mathbb{C}$ and $\alpha,\beta\in{\mathbb{N}}$,
(2.21)
$\displaystyle(z\partial-\xi\delta)(\xi^{\alpha}\ast_{t}z^{\beta})=(|\beta|-|\alpha|)(\xi^{\alpha}\ast_{t}z^{\beta}),$
where
$z\partial-\xi\delta\\!:=\sum_{i=1}^{n}(z_{i}\partial_{i}-\xi_{i}\delta_{i})$.
Proof: First, by Euler’s lemma, we have
(2.22)
$\displaystyle(z\partial-\xi\delta)(\xi^{\alpha}z^{\beta})=(|\beta|-|\alpha|)(\xi^{\alpha}z^{\beta}).$
Second, note that $z\partial-\xi\delta$ commutes with $\Lambda$, hence also
with $\Phi_{t}$ for any $t\in\mathbb{C}$. Apply $\Phi_{-t}$ to Eq. (2.22), we
get
$\displaystyle(z\partial-\xi\delta)\Phi_{-t}(\xi^{\alpha}z^{\beta})=(|\beta|-|\alpha|)\Phi_{-t}(\xi^{\alpha}z^{\beta}).$
Then, by Eq. (2.20) with $t$ replaced by $-t$, Eq. (2.21) follows from the
equation above. $\Box$
###### Lemma 2.8.
For any $\lambda_{i}(\xi)\in\mathbb{C}[\xi]$ and $p_{i}(z)\in\mathbb{C}[z]$
$(i=1,2)$, we have
(2.23)
$\displaystyle(\lambda_{1}(\xi)p_{1}(z))\ast_{t}(\lambda_{2}(\xi)p_{2}(z))=(\lambda_{1}(\xi)\ast_{t}p_{2}(z))\,(\lambda_{2}(\xi)\ast_{t}p_{1}(z)).$
Proof: First, by Eq. (2.7), we have
$\displaystyle\quad(\lambda_{1}(\xi)p_{1}(z))\ast_{t}(\lambda_{2}(\xi)p_{2}(z))$
$\displaystyle=\sum_{\alpha,\beta\in{\mathbb{N}}^{n}}\frac{(-t)^{|\alpha|+|\beta|}}{\alpha!\beta!}\left((\delta^{\alpha}\lambda_{1}(\xi))(\partial^{\beta}p_{1}(z))\right)\left((\delta^{\beta}\lambda_{2}(\xi))(\partial^{\alpha}p_{2}(z))\right)$
Taking sum over $\alpha\in{\mathbb{N}}^{n}$ and applying Eq. (2.13):
$\displaystyle=(\lambda_{1}(\xi)\ast_{t}p_{2}(z))\sum_{\beta\in{\mathbb{N}}^{n}}\frac{(-t)^{|\beta|}}{\beta!}(\partial^{\beta}p_{1}(z))(\delta^{\beta}\lambda_{2}(\xi))$
Taking sum over $\beta\in{\mathbb{N}}^{n}$ and applying Eq. (2.13):
$\displaystyle=(\lambda_{1}(\xi)\ast_{t}p_{2}(z))(\lambda_{2}(\xi)\ast_{t}p_{1}(z)).$
Hence we get Eq. (2.23). $\Box$
### 2.2. The $\ell$-adic Topologies Induced by ${\mathcal{B}}_{t}[{\xi,z}]$
on $\mathbb{C}[\xi,z]$
We have seen that the algebras ${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$
share the same base vector space $\mathbb{C}[\xi,z]$ and, by Proposition 2.6,
$(c)$, they are all commutative free algebras generated freely by $\xi$ and
$z$. Therefore, we may talk about the $\ell$-adic topologies on
$\mathbb{C}[\xi,z]$ induced by the algebras ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$, which are defined as follows.
For any $t\in\mathbb{C}[{\xi,z}]$ and $m\geq 0$, set $U_{t,m}$ to be the
subspace of $\mathbb{C}[\xi,z]$ spanned by the monomials
$\xi^{\alpha}\ast_{t}z^{\beta}$ of ${\mathcal{B}}_{t}[{\xi,z}]$ with
$\alpha,\beta\in{\mathbb{N}}^{n}$ and $|\alpha+\beta|\geq m$. The $\ell$-adic
topology induced from the algebra ${\mathcal{B}}_{t}[{\xi,z}]$ is the topology
whose open subsets are the subsets generated by $U_{t,m}$ $(m\in{\mathbb{N}})$
and their translations by elements of ${\mathcal{B}}_{t}[{\xi,z}]$. We denote
by ${\mathcal{T}}_{t}$ this topology on $\mathbb{C}[\xi,z]$.
The main result of this subsection is the following proposition.
###### Proposition 2.9.
$(a)$ For any $s\neq t\in\mathbb{C}$, we have
${\mathcal{T}}_{s}\neq{\mathcal{T}}_{t}$.
$(b)$ For any $t\in\mathbb{C}$, the algebra isomorphism
$\Phi_{t}:({\mathcal{B}}_{t}[{\xi,z}],{\mathcal{T}}_{t})\to({\mathcal{A}}[{\xi,z}],{\mathcal{T}}_{0})$
is also a homeomorphism of topological spaces. Consequently,
$({\mathcal{B}}_{t}[{\xi,z}],{\mathcal{T}}_{t})$ $(t\in\mathbb{C})$ as
topological spaces are all homeomorphic.
Proof: $(a)$ Let $\\{\alpha_{m}\in{\mathbb{N}}^{n}\,|\,m\geq 1\\}$ be any
sequence of elements of ${\mathbb{N}}^{n}$ such that $|\alpha_{m}|=m$ for any
$m\geq 1$.
Set $u_{m}\\!:=\xi^{\alpha_{m}}\ast_{t}z^{\alpha_{m}}$ for any $m\geq 1$.
Then, by the definition of ${\mathcal{T}}_{t}$, we see that the sequence
$\\{u_{m}\\}$ converges to $0\in\mathbb{C}[\xi,z]$ with respect to the
topology ${\mathcal{T}}_{t}$.
But, on the other hand, set $r:=s-t\neq 0$. Then, by Eq. (2.14), we have
$\displaystyle u_{m}$
$\displaystyle=\xi^{\alpha_{m}}\ast_{t}z^{\alpha_{m}}=(\xi-t\partial)^{\alpha_{m}}z^{\alpha_{m}}=((\xi-s\partial)+r\partial)^{\alpha_{m}}z^{\alpha_{m}}$
$\displaystyle=\sum_{\begin{subarray}{c}\beta,\gamma\in{\mathbb{N}}^{n}\\\
\beta+\gamma=\alpha_{m}\end{subarray}}\binom{\alpha_{m}}{\gamma}(\xi-s\partial)^{\gamma}(\partial^{\beta}z^{\alpha_{m}})$
$\displaystyle=\sum_{\begin{subarray}{c}\beta,\gamma\in{\mathbb{N}}^{n}\\\
\beta+\gamma=\alpha_{m}\end{subarray}}\binom{\alpha_{m}}{\gamma}\xi^{\gamma}\ast_{s}(\partial^{\beta}z^{\alpha_{m}})\equiv\alpha_{m}!\mod(U_{s,0}).$
From the equation above, we see that the sequence $\\{u_{m}\\}$ does not
converge to $0\in\mathbb{C}[\xi,z]$ with respect to the topology
${\mathcal{T}}_{s}$. Hence ${\mathcal{T}}_{s}\neq{\mathcal{T}}_{t}$.
$(b)$ Note that ${\mathcal{B}}_{t=0}[{\xi,z}]$ is the usual polynomial algebra
${\mathcal{A}}[{\xi,z}]$ and
$\Phi_{t}:{\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[{\xi,z}]$ is an algebra
isomorphism. Furthermore, from Eqs. (2.1), (2.18) and (2.19), we have
(2.24)
$\displaystyle\Phi_{t}(\xi^{\alpha}\ast_{t}z^{\beta})=\xi^{\alpha}z^{\beta}$
for any $\alpha,\beta\in{\mathbb{N}}^{n}$.
Therefore, for any $m\geq 0$, we have, $\Phi_{t}(U_{t,m})=U_{0,m}$ and
$\Phi_{t}^{-1}(U_{0,m})=U_{t,m}$. Hence, we have $(b)$. $\Box$
Actually, the proof above also shows that Proposition 2.9 also holds for the
following topologies on $\mathbb{C}[\xi,z]$ induced by the free algebras
${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$.
For any $t\in\mathbb{C}[{\xi,z}]$ and $m\geq 0$, set
(2.25) $\displaystyle
U_{t,m}^{\prime}\\!:=\sum_{\begin{subarray}{c}\alpha\in{\mathbb{N}}^{n};\,\\\
|\alpha|\geq m\end{subarray}}\xi^{\alpha}\ast_{t}\mathbb{C}[z].$
Denote by ${\mathcal{T}}^{\prime}_{t}$ the topology on $\mathbb{C}[\xi,z]$
generated by $U_{t,m}^{\prime}$ and their translations (as open subsets).
Then, by a similar argument as in the proof of Proposition 2.9, it is easy to
see that the following corollary also holds.
###### Corollary 2.10.
$(a)$ For any $s\neq t\in\mathbb{C}$, we have
${\mathcal{T}}_{s}{}^{\prime}\neq{\mathcal{T}}_{t}{}^{\prime}$.
$(b)$ For any $t\in\mathbb{C}$, the algebra isomorphism
$\Phi_{t}:({\mathcal{B}}_{t}[{\xi,z}],{\mathcal{T}}_{t}{}^{\prime})\to({\mathcal{A}}[{\xi,z}],{\mathcal{T}}_{0}{}^{\prime})$
is also a homeomorphism of topological spaces.
Note that, due to the symmetric roles played by $\xi$ and $z$, the corollary
above also holds if $\xi$ in Eq. (2.25) is replaced by $z$.
### 2.3. The Induced Isomorphism $(\Phi_{t})_{\ast}$ on Differential Operator
Algebras
For any $t\in\mathbb{C}$, denote by ${\mathcal{D}}_{t}[{\xi,z}]$ the
differential operator algebra or the Weyl algebra of
${\mathcal{B}}_{t}[{\xi,z}]$, i.e. the associative algebra generated by the
$\mathbb{C}$-derivations and the multiplication operators of the algebra
${\mathcal{B}}_{t}[{\xi,z}]$. Since
$\Phi_{t}:{\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[{\xi,z}]$ is an algebra
isomorphism (See Corollary 2.2), it induces an algebra isomorphism, denoted by
$(\Phi_{t})_{*}:{\mathcal{D}}_{t}[{\xi,z}]\to{\mathcal{D}}[{\xi,z}]$, from the
Weyl algebra ${\mathcal{D}}_{t}[{\xi,z}]$ of ${\mathcal{B}}_{t}[{\xi,z}]$ to
the Weyl algebra ${\mathcal{D}}[{\xi,z}]$ of ${\mathcal{A}}[{\xi,z}]$.
Recall that the induced map $(\Phi_{t})_{*}$ is defined by setting
(2.26)
$\displaystyle(\Phi_{t})_{*}(\psi)=\Phi_{t}\circ\psi\circ\Phi_{t}^{-1}=\Phi_{t}\circ\psi\circ\Phi_{-t}$
for any $\psi\in{\mathcal{D}}_{t}[{\xi,z}]$.
The main result of this subsection are the following two propositions, even
though their proofs are very simple.
###### Proposition 2.11.
For any $t\in\mathbb{C}$, the following statements hold.
$(a)$ $\partial_{i}$ and $\delta_{i}$ $(1\leq i\leq n)$ are also derivations
of ${\mathcal{B}}_{t}[{\xi,z}]$.
$(b)$ For any $1\leq i\leq n$, we have
(2.27) $\displaystyle(\Phi_{t})_{*}(\partial_{i})$
$\displaystyle=\partial_{i},$ (2.28) $\displaystyle(\Phi_{t})_{*}(\delta_{i})$
$\displaystyle=\delta_{i},$
Proof: Note first that $\partial_{i}$ and $\delta_{i}$ $(1\leq i\leq n)$
commute with $\Lambda$, hence also with $\Phi_{t}$ for any $t\in\mathbb{C}$.
Then, Eqs. (2.27) and (2.28) follows immediately from this fact and the
definition of $(\Phi_{t})_{*}$ given in Eq. (2.26).
$(a)$ follows from the general fact that the induced map of any algebra
isomorphism maps derivations to derivations. It can also be checked directly
as follows.
For any $f,g\in{\mathcal{B}}_{t}[{\xi,z}]$, by Eq. (2.1) and the fact that
$\partial_{i}$ $(1\leq i\leq n)$ commute with $\Phi_{t}$ $(t\in\mathbb{C})$,
we have
$\displaystyle\partial_{i}(f\ast_{t}g)$
$\displaystyle=\partial_{i}\Big{(}\Phi_{-t}\big{(}\Phi_{t}(f)\Phi_{t}(g)\big{)}\Big{)}=\Phi_{-t}\Big{(}\partial_{i}\big{(}\Phi_{t}(f)\Phi_{t}(g)\big{)}\Big{)}$
$\displaystyle=\Phi_{-t}\big{(}(\partial_{i}\Phi_{t}(f))\Phi_{t}(g)\big{)}+\Phi_{-t}\big{(}\Phi_{t}(f)(\partial_{i}\Phi_{t}(g))\big{)}$
$\displaystyle=\Phi_{-t}\big{(}\Phi_{t}(\partial_{i}f)\Phi_{t}(g)\big{)}+\Phi_{-t}\big{(}\Phi_{t}(f)\Phi_{t}(\partial_{i}g)\big{)}$
$\displaystyle=(\partial_{i}f)\ast_{t}g+f\ast_{t}(\partial_{i}g).$
Similarly, we can show that $\delta_{i}$ $(1\leq i\leq n)$ are also
derivations of ${\mathcal{B}}_{t}[{\xi,z}]$. $\Box$
###### Corollary 2.12.
For any $\alpha,\beta,\gamma\in{\mathbb{N}}^{n}$, we have
(2.29) $\displaystyle\partial^{\gamma}(\xi^{\alpha}\ast_{t}z^{\beta})$
$\displaystyle=\gamma!\,\binom{\beta}{\gamma}\,(\xi^{\alpha}\ast_{t}z^{\beta-\gamma}),$
(2.30) $\displaystyle\delta^{\gamma}(\xi^{\alpha}\ast_{t}z^{\beta})$
$\displaystyle=\gamma!\,\binom{\alpha}{\gamma}\,(\xi^{\alpha-\gamma}\ast_{t}z^{\beta}).$
Proof: Note that, by Eqs. (2.11) and (2.12), we know that, for any
$\alpha,\beta\in{\mathbb{N}}^{n}$, $\xi^{\alpha}\ast_{t}z^{\beta}$ will remain
the same if we replace the (usual) product of ${\mathcal{A}}[{\xi,z}]$ in the
factors $\xi^{\alpha}$ and $z^{\beta}$ by the product $\ast_{t}$ of
${\mathcal{B}}_{t}[{\xi,z}]$. By Proposition 2.11, $(a)$, we know that
$\partial_{i}$ and $\delta_{i}$ $(1\leq i\leq n)$ are also the derivations of
${\mathcal{B}}_{t}[{\xi,z}]$. From these two facts, it is easy to see that
both equations in the corollary hold. $\Box$
###### Proposition 2.13.
For any $t\in\mathbb{C}$ and $f(\xi,z)\in\mathbb{C}[{\xi,z}]$,
$(\Phi_{t})_{*}$ maps the multiplication operator of
${\mathcal{B}}_{t}[{\xi,z}]$ by $f(\xi,z)$ $($with respect to the product
$\ast_{t}$$)$ to the multiplication operator of ${\mathcal{A}}[{\xi,z}]$ by
$\Phi_{t}(f(\xi,z))$ $($with respect to the product of
${\mathcal{A}}[{\xi,z}]$$)$.
Proof: We denote by $\psi_{f}$ the multiplication operator of
${\mathcal{B}}_{t}[{\xi,z}]$ by $f(\xi,z)$ $($with respect to the product
$\ast_{t}$$)$. Then for any $u({\xi,z})\in\mathbb{C}[{\xi,z}]$, by Eqs. (2.26)
and (2.1) we have
$\displaystyle(\Phi_{t})_{*}(\psi_{f})u({\xi,z})$
$\displaystyle=(\Phi_{t}\circ\psi_{f}\circ\Phi_{t}^{-1})u({\xi,z})=\Phi_{t}\big{(}f(\xi,z)\ast_{t}\Phi_{t}^{-1}(u({\xi,z}))\big{)}$
$\displaystyle=\Phi_{t}(f(\xi,z))\,\Phi_{t}\big{(}\Phi_{t}^{-1}(u(\xi,z))\big{)}=\Phi_{t}(f(\xi,z))\,u({\xi,z}).$
Hence, the proposition follows. $\Box$
By the proposition above and Eqs. (2.18) and (2.19), we also have the
following corollary.
###### Corollary 2.14.
For any $t\in\mathbb{C}$, $\lambda(\xi)\in\mathbb{C}[\xi]$ and
$p(z)\in\mathbb{C}[z]$, $(\Phi_{t})_{*}$ maps the multiplication operators of
${\mathcal{B}}_{t}[{\xi,z}]$ by $\lambda(\xi)$ and $p(z)$ $($with respect to
the product $\ast_{t}$$)$ to the multiplication operators of
${\mathcal{A}}[{\xi,z}]$ by $\lambda(\xi)$ and $p(z)$ $($with respect to the
product of ${\mathcal{A}}[{\xi,z}]$$)$, respectively.
Note that, as pointed out before, the algebras ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$ share the same base vectors space $\mathbb{C}[\xi,z]$.
Therefore, their Weyl algebras ${\mathcal{D}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$
are all subalgebras of the algebra of linear endomorphisms of
$\mathbb{C}[\xi,z]$. The following corollary says that all these subalgebras
turn out to be same, i.e. they do not depend on the parameter
$t\in\mathbb{C}$.
###### Corollary 2.15.
For any $t\in\mathbb{C}$, as subalgebras of the algebra of linear
endomorphisms of $\mathbb{C}[\xi,z]$,
${\mathcal{D}}_{t}[{\xi,z}]={\mathcal{D}}[{\xi,z}]$.
Proof: By Proposition 2.6, $(c)$, we know that ${\mathcal{B}}_{t}[{\xi,z}]$ is
a commuative free algebra generated freely by $\xi$ and $z$. By Proposition
2.11, $(a)$, we know that $\partial_{i}$ and $\delta_{i}$ $(1\leq i\leq n)$
are also derivations of ${\mathcal{B}}_{t}[{\xi,z}]$. Therefore, the Weyl
algebra ${\mathcal{D}}_{t}[{\xi,z}]$ as an associative algebra over
$\mathbb{C}$ is generated by the derivations $\partial_{i}$, $\delta_{i}$
$(1\leq i\leq n)$ and the multiplication operators (with respect to the
product $\ast_{t}$) by $\xi_{i},z_{i}\in{\mathcal{B}}_{t}[{\xi,z}]$ $(1\leq
i\leq n)$.
By Eqs. (2.14) and (2.15), we see that the multiplication operators by
$\xi_{i},z_{i}\in{\mathcal{B}}_{t}[{\xi,z}]$ $(1\leq i\leq n)$ are same as the
operators $\xi_{i}-t\partial_{i}$ and $z_{i}-t\delta_{i}$ which lie in
${\mathcal{D}}[{\xi,z}]$. Hence we have
${\mathcal{D}}_{t}[{\xi,z}]\subseteq{\mathcal{D}}[{\xi,z}]$.
To show ${\mathcal{D}}[{\xi,z}]\subseteq{\mathcal{D}}_{t}[{\xi,z}]$, by
Proposition 2.11, $(a)$, it will be enough to show that the multiplication
operators (with respect to the product of ${\mathcal{A}}[{\xi,z}]$) by
$\xi_{i},z_{i}\in{\mathcal{A}}[{\xi,z}]$ $(1\leq i\leq n)$ also belong to
${\mathcal{D}}_{t}[{\xi,z}]$.
But, for any $f({\xi,z})\in\mathbb{C}[{\xi,z}]$, by Eqs. (2.14) and (2.15), we
have
$\displaystyle\xi_{i}f({\xi,z})$
$\displaystyle=(\xi_{i}-t\partial_{i})f({\xi,z})+t\partial_{i}f({\xi,z})=\xi_{i}\ast_{t}f({\xi,z})+t\partial_{i}f({\xi,z}),$
$\displaystyle z_{i}f({\xi,z})$
$\displaystyle=(z_{i}-t\delta_{i})f({\xi,z})+t\delta_{i}f({\xi,z})=z_{i}\ast_{t}f({\xi,z})+t\delta_{i}f({\xi,z}).$
From the equations above, we see that the multiplication operators (with
respect to the product of ${\mathcal{A}}[{\xi,z}]$) by
$\xi_{i},z_{i}\in{\mathcal{A}}[{\xi,z}]$ $(1\leq i\leq n)$ do belong to
${\mathcal{D}}_{t}[{\xi,z}]$. $\Box$
## 3\. Connections with Interchanges of Right and Left Total Symbols of
Differential Operators
In this section, we show in Theorem 3.1 that the isomorphisms $\Phi_{t}$ with
$t=\pm 1$ coincide with the interchanges between total left and right symbols
of differential operators of the polynomial algebra ${\mathcal{A}}[z]$.
First, let us fix the following notation and convention for the differential
operators of ${\mathcal{A}}[z]$.
We denote by ${\mathcal{D}}[z]$ the differential operator algebra or the Weyl
algebra of ${\mathcal{A}}[z]$. For any differential operator
$\phi\in{\mathcal{D}}[z]$ and polynomial $u(z)\in{\mathcal{A}}[z]$, the
notation $\phi\,u(z)$ usually denotes the composition of $\phi$ and the
multiplication operator by $u(z)$. So $\phi\,u(z)$ is still a differential
operator of ${\mathcal{A}}[z]$. The polynomial obtained by applying $\phi$ to
$u(z)$ will be denoted by $\phi(u(z))$.
Next, let us recall the right and left total symbols of differential operators
of the polynomial algebra ${\mathcal{A}}[z]$.
For any $\phi\in{\mathcal{D}}[z]$, it is well-known (e.g. see Proposition 2.2
(pp. 4) in [B] or Theorem 3.1 (pp. 58) in [C]) that $\phi$ can be written
uniquely as the following two finite sums:
(3.1)
$\displaystyle\phi=\sum_{\alpha\in{\mathbb{N}}^{n}}a_{\alpha}(z)\partial^{\alpha}=\sum_{\beta\in{\mathbb{N}}^{n}}\partial^{\beta}b_{\beta}(z)$
where $a_{\alpha}(z),b_{\beta}(z)\in\mathbb{C}[z]$ but denote the
multiplication operators by $a_{\alpha}(z)$ and $b_{\beta}(z)$, respectively.
For the differential operator $\phi\in{\mathcal{D}}[z]$ in Eq. (3.1), the
right and left total symbols are defined to be the polynomials
$\sum_{\alpha\in{\mathbb{N}}^{n}}a_{\alpha}(z)\xi^{\alpha}\in\mathbb{C}[{\xi,z}]$
and
$\sum_{\beta\in{\mathbb{N}}^{n}}b_{\beta}(z)\xi^{\beta}\in\mathbb{C}[{\xi,z}]$,
respectively. We denote by
${\mathcal{R}}:{\mathcal{D}}[z]\to\mathbb{C}[\xi,z]$ (resp.
${\mathcal{L}}:{\mathcal{D}}[z]\to\mathbb{C}[\xi,z]$) the linear map which
maps any $\phi\in{\mathcal{D}}[z]$ to its right total symbol (resp. left total
symbol).
Note that, by the uniqueness of the expressions in Eq. (3.1), both
${\mathcal{R}}$ and ${\mathcal{L}}$ are isomorphisms of vector spaces over
$\mathbb{C}$. The interchange of the left (resp. right) total symbol of
differential operators to their right (resp. left) total symbols is given by
the isomorphism ${\mathcal{R}}\circ{\mathcal{L}}^{-1}$ (resp.
${\mathcal{L}}\circ{\mathcal{R}}^{-1}$) from $\mathbb{C}[\xi,z]$ to
$\mathbb{C}[\xi,z]$.
The main result of this section is the following theorem.
###### Theorem 3.1.
As linear maps from $\mathbb{C}[\xi,z]$ to $\mathbb{C}[\xi,z]$, we have
(3.2) $\displaystyle\Phi$
$\displaystyle={\mathcal{R}}\circ{\mathcal{L}}^{-1}.$ (3.3)
$\displaystyle\Phi_{t=-1}$
$\displaystyle={\mathcal{L}}\circ{\mathcal{R}}^{-1}.$
Proof: Note first that, Eq. (3.3) follows from Eq. (3.2) and the fact that
$\Phi_{t=-1}=\Phi_{t=1}^{-1}=\Phi^{-1}$.
To show Eq. (3.2), since both $\Phi$ and
${\mathcal{R}}\circ{\mathcal{L}}^{-1}$ are linear maps, it is enough to show
that, for any $\alpha,\beta\in{\mathbb{N}}^{n}$, we have
(3.4)
$\displaystyle\Phi(\xi^{\alpha}z^{\beta})=({\mathcal{R}}\circ{\mathcal{L}}^{-1})(\xi^{\alpha}z^{\beta}).$
Since
(3.5)
$\displaystyle({\mathcal{R}}\circ{\mathcal{L}}^{-1})(\xi^{\alpha}z^{\beta})={\mathcal{R}}(\partial^{\alpha}z^{\beta}),$
so we have to find the right total symbol of the differential operator
$\partial^{\alpha}z^{\beta}\in{\mathcal{D}}[z]$.
Note that, for any dummy $u(z)\in\mathbb{C}[z]$, by the Leibniz rule, we have
(3.6) $\displaystyle\partial^{\alpha}(z^{\beta}u(z))$
$\displaystyle=\sum_{\gamma\in{\mathbb{N}}^{n}}\binom{\alpha}{\gamma}(\partial^{\gamma}z^{\beta})(\partial^{\alpha-\gamma}u(z))$
$\displaystyle=\left(\sum_{\gamma\in{\mathbb{N}}^{n}}\binom{\alpha}{\gamma}(\partial^{\gamma}z^{\beta})\partial^{\alpha-\gamma}\right)u(z).$
Therefore, the right total symbol of the differential operator
$\partial^{\alpha}z^{\beta}\in{\mathcal{D}}[z]$ is given by
$\displaystyle{\mathcal{R}}(\partial^{\alpha}z^{\beta})$
$\displaystyle=\sum_{\gamma\in{\mathbb{N}}^{n}}\binom{\alpha}{\gamma}(\partial^{\gamma}z^{\beta})\xi^{\alpha-\gamma}=\sum_{\gamma\in{\mathbb{N}}^{n}}\binom{\alpha}{\gamma}\xi^{\alpha-\gamma}(\partial^{\gamma}z^{\beta})$
$\displaystyle=\sum_{\gamma\in{\mathbb{N}}^{n}}\frac{1}{\gamma!}(\delta^{\gamma}\xi^{\alpha})(\partial^{\gamma}z^{\beta})$
Combining the equation above with Eqs. (2.13) and (2.20) with $t=-1$, we have
(3.7)
$\displaystyle{\mathcal{R}}(\partial^{\alpha}z^{\beta})=\xi^{\alpha}\ast_{t=-1}z^{\beta}=\Phi_{t=1}(\xi^{\alpha}z^{\beta})=\Phi(\xi^{\alpha}z^{\beta}).$
Hence, we have proved Eq. (3.4) and also the theorem. $\Box$
###### Corollary 3.2.
For any $\lambda(\xi)\in\mathbb{C}[\xi]$ and $p(z)\in\mathbb{C}[z]$, we have
(3.8) $\displaystyle{\mathcal{R}}(\lambda(\partial)p(z))$
$\displaystyle=\lambda(\xi)\ast_{t=-1}p(z).$ (3.9)
$\displaystyle{\mathcal{L}}(p(z)\lambda(\partial))$
$\displaystyle=\lambda(\xi)\ast p(z).$
Proof: By Eqs. (3.2) and (2.20) with $t=1$, we have
$\displaystyle{\mathcal{R}}(\lambda(\partial)p(z))$
$\displaystyle={\mathcal{R}}({\mathcal{L}}^{-1}(\lambda(\xi)p(z)))=({\mathcal{R}}\circ{\mathcal{L}}^{-1})(\lambda(\xi)p(z))$
$\displaystyle=\Phi_{t=1}(\lambda(\xi)p(z))=\lambda(\xi)\ast_{t=-1}p(z).$
So we have Eq. (3.8). Eq. (3.9) can be proved similarly by using Eqs. (3.3)
and (2.20) with $t=-1$. $\Box$
Finally, we end this section with the following one-variable example.
###### Example 3.3.
Let $n=1$ and $\phi=z^{2}\partial^{3}$. Then,
$\displaystyle{\mathcal{R}}(\phi)$
$\displaystyle={\mathcal{R}}(z^{2}\partial^{3})=\xi^{3}z^{2}.$
$\displaystyle{\mathcal{L}}(\phi)$
$\displaystyle={\mathcal{L}}(z^{2}\partial^{3})=\xi^{3}\ast
z^{2}=(z-\delta)^{2}\xi^{3}$
$\displaystyle=(z^{2}-2z\delta+\delta^{2})\xi^{3}=\xi^{3}z^{2}-6\xi^{2}z+6\xi.$
Therefore, we have
$\displaystyle\phi=z^{2}\partial^{3}=\partial^{3}z^{2}-6\partial^{2}z+6\partial.$
## 4\. A Re-formulation of the Image Conjecture on Commuting Differential
Operators of Order One with Constant Leading Coefficients
In this section, we show that the algebra ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$ is closely related with a theorem (See Theorem 4.1) first
proved in [Z3] and also with the so-called image conjecture (See Conjecture
4.3) proposed in [Z3] on the differential operators $\xi-t\partial$
$(t\in\mathbb{C})$.
In Subsection 4.1, we use certain Taylor series expansion of elements of
${\mathcal{B}}_{t}[{\xi,z}]$ to give a new and more conceptual proof for
Theorem 4.1. In Subsection 4.2, we first give a new formulation (See
Conjecture 4.5) for Conjecture 4.3 in terms of the algebra
${\mathcal{B}}_{t}[{\xi,z}]$ and the notion of Mathieu subspaces (see
Definition 4.2) introduced in [Z4], and then show in Theorem 4.6 that the new
formulation is indeed equivalent to Conjecture 4.3.
### 4.1. The Taylor Series with Respect to the Product $\ast_{t}$
First, let us recall the following elementary fact on polynomials in $\xi$ and
$z$.
For any $f({\xi,z})\in{\mathcal{A}}[{\xi,z}]$, we may view $f({\xi,z})$ as a
polynomial in $\xi$ with coefficients in ${\mathcal{A}}[z]$. Then it has the
following Taylor series expansion
(4.1) $\displaystyle
f({\xi,z})=\sum_{\alpha\in{\mathbb{N}}^{n}}\frac{1}{\alpha!}\,\,\xi^{\alpha}c_{\alpha}(z)$
for some $c_{\alpha}(z)\in{\mathcal{A}}[z]$.
Let $ev_{{}_{0}}:{\mathcal{A}}[{\xi,z}]\to{\mathcal{A}}[z]$ be the evaluation
map of ${\mathcal{A}}[{\xi,z}]$ at $\xi=0$, i.e. for any
$u({\xi,z})\in{\mathcal{A}}[{\xi,z}]$, $ev_{{}_{0}}(u)\\!:=u(0,z)$. Then, the
$c_{\alpha}(z)$ $(\alpha\in{\mathbb{N}}^{n})$ in Eq. (4.1) are given by
(4.2) $\displaystyle c_{\alpha}(z)=ev_{{}_{0}}(\delta^{\alpha}f).$
Note that another characterization of the evaluation map $ev_{{}_{0}}$ is that
$ev_{{}_{0}}$ is the (unique) algebra homomorphism from
${\mathcal{A}}[{\xi,z}]$ to ${\mathcal{A}}[z]$ with $ev_{{}_{0}}(\xi_{i})=0$
and $ev_{{}_{0}}(z_{i})=z_{i}$ for any $1\leq i\leq n$.
Now, come back to our algebras ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$. By Proposition 2.6, $(c)$, we know that
${\mathcal{B}}_{t}[{\xi,z}]$ is also a commutative free algebra generated
freely by $\xi$ and $z$ with the same base vector space $\mathbb{C}[\xi,z]$.
Hence, we should expect similar expansions as in Eq. (4.1) for polynomials
$f({\xi,z})\in\mathbb{C}[\xi,z]$ with respect to the product $\ast_{t}$.
But, in order to formulate the expected expansions precisely, we need first to
introduce the analogue of the evaluation map $ev_{{}_{0}}$ for the algebra
${\mathcal{B}}_{t}[{\xi,z}]$.
Note that, by Proposition 2.6, $(b)$, the subalgebra of
${\mathcal{B}}_{t}[{\xi,z}]$ generated by $z$ is also
${\mathcal{A}}[z]\subset\mathbb{C}[{\xi,z}]$. Parallel to the second
characterization of the evaluation map $ev_{{}_{0}}$ mentioned above, we let
${\mathcal{E}}_{t}$ be the unique algebra homomorphism from
${\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[z]$ such that
${\mathcal{E}}_{t}(\xi_{i})=0$ and ${\mathcal{E}}_{t}(z_{i})=z_{i}$ for any
$1\leq i\leq n$.
Note also that, by Eqs. (2.18) and (2.19), the algebra isomorphism
$\Phi_{t}:{\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[{\xi,z}]$ maps $\xi_{i}$
(resp. $z_{i}$) to $\xi_{i}$ (resp. $z_{i}$) for any $1\leq i\leq n$. Hence
the composition
$ev_{{}_{0}}\circ\Phi_{t}:{\mathcal{B}}_{t}[{\xi,z}]\to{\mathcal{A}}[z]$ has
the same characterizing property of ${\mathcal{E}}_{t}$. Therefore, we have
(4.3) $\displaystyle{\mathcal{E}}_{t}=ev_{{}_{0}}\circ\Phi_{t}.$
Furthermore, we can also derive a more explicit formula for
${\mathcal{E}}_{t}$ as follows.
For any $\alpha\in{\mathbb{N}}^{n}$ and $p(z)\in\mathbb{C}[z]$, consider
(4.4) $\displaystyle{\mathcal{E}}_{t}(\xi^{\alpha}p(z))$
$\displaystyle=ev_{{}_{0}}(\Phi_{t}(\xi^{\alpha}p(z)))$ Applying Eq. (2.20)
and then Eq. (2.14) with $t$ replaced by $-t$:
$\displaystyle=ev_{{}_{0}}(\xi^{\alpha}\ast_{-t}p(z)))$
$\displaystyle=ev_{{}_{0}}((\xi+t\partial)^{\alpha}(p(z)))=t^{|\alpha|}\partial^{\alpha}(p(z)).$
From the formula above, we see that, for any $g(z,\xi)\in\mathbb{C}[z,\xi]$,
${\mathcal{E}}_{t}(g(z,\xi))\in\mathbb{C}[z]$ can be obtained by, first,
writing each monomial of $g(z,\xi)$ as $\xi^{\beta}z^{\gamma}$
$(\beta,\gamma\in{\mathbb{N}}^{n})$, i.e. putting the free variables
$\xi_{i}$’s to the most left in each monomial of $g(z,\xi)$, and then
replacing the part $\xi^{\beta}$ by the differential operator
$t^{|\beta|}\partial^{\beta}$ and applying it to the other part $z^{\gamma}$
of the monomial. For examples, we have
$\displaystyle{\mathcal{E}}_{t}(1)$ $\displaystyle=1;$
$\displaystyle{\mathcal{E}}_{t}(z^{\alpha})$
$\displaystyle=(t\partial)^{0}(z^{\alpha})=z^{\alpha}\qquad\quad\,\,\qquad\qquad\mbox{for
any }\alpha\in{\mathbb{N}}^{n};$
$\displaystyle{\mathcal{E}}_{t}(\xi^{\alpha})$
$\displaystyle=t^{|\alpha|}\partial^{\alpha}(1)=0\qquad\qquad\qquad\quad\quad\mbox{for
any }0\neq\alpha\in{\mathbb{N}}^{n};$
$\displaystyle{\mathcal{E}}_{t}(z_{1}^{m}\xi_{1}^{2})$
$\displaystyle=t^{2}\partial_{1}^{2}(z_{1}^{m})=m(m-1)t^{2}z_{1}^{m-2}\qquad\,\mbox{for
any }m\geq 2.$
Now we are ready to formulate and prove the expected expansion of polynomials
with respect to the new product $\ast_{t}$, which is parallel to the Taylor
expansion in Eq. (4.1).
###### Theorem 4.1.
For any $t\in\mathbb{C}$ and $f(\xi,z)\in\mathbb{C}[\xi,z]$, we have
(4.5) $\displaystyle f(\xi,z)$
$\displaystyle=\sum_{\alpha\in{\mathbb{N}}^{n}}\frac{1}{\alpha!}\,\,\xi^{\alpha}\ast_{t}a_{\alpha}(z),$
(4.6) $\displaystyle f(\xi,z)$
$\displaystyle=\sum_{\alpha\in{\mathbb{N}}^{n}}\frac{1}{\alpha!}\,(\xi-t\partial_{z})^{\alpha}a_{\alpha}(z),$
where, for any $\alpha\in{\mathbb{N}}^{n}$,
(4.7) $\displaystyle a_{\alpha}(z)={\mathcal{E}}_{t}(\delta^{\alpha}f).$
Furthermore, the expansions of the forms in Eqs. $(\ref{D_t-Taylor-e1})$ and
$(\ref{D_t-Taylor-e2})$ for $f({\xi,z})$ are unique.
Proof: Note first that, by Eq. (2.14) in Proposition 2.4, Eq. (4.5) and Eq.
(4.6) are actually equivalent. So we will focus only on Eq. (4.5).
The uniqueness of the expansion in Eq. (4.5) follows directly from Proposition
2.6, $(a)$-$(c)$.
To show that Eq. (4.5) with $a_{\alpha}(z)$ $(\alpha\in{\mathbb{N}}^{n})$
given in Eq. (4.7) does hold, we first write the Taylor series expansion of
$\Phi_{t}(f({\xi,z}))$ as in Eq. (4.1):
(4.8)
$\displaystyle\Phi_{t}(f({\xi,z}))=\sum_{\alpha\in{\mathbb{N}}^{n}}\frac{1}{\alpha!}\,\,\xi^{\alpha}a_{\alpha}(z)$
where $a_{\alpha}(z)\in\mathbb{C}[z]$ $(\alpha\in{\mathbb{N}}^{n})$ are given
by
(4.9) $\displaystyle a_{\alpha}(z)=ev_{{}_{0}}(\delta^{\alpha}\Phi_{t}(f)).$
Applying $\Phi_{-t}$ to Eq.(4.8) and, by Eq. (2.20) with $t$ replaced by $-t$,
we get Eq. (4.5).
Next, note that $\delta^{\alpha}$ $(\alpha\in{\mathbb{N}}^{n})$ commute with
$\Lambda$, hence they also commute with $\Phi_{t}=e^{t\Lambda}$. Then, by Eqs.
(4.9) and (4.3), we have
$\displaystyle
a_{\alpha}(z)=ev_{{}_{0}}(\Phi_{t}(\delta^{\alpha}f))=(ev_{{}_{0}}\circ\Phi_{t})(\delta^{\alpha}f)={\mathcal{E}}_{t}(\delta^{\alpha}f).$
Therefore, Eq. (4.7) also holds. $\Box$
Several remarks on Theorem 4.1 and the proof above are as follows.
First, Theorem 4.1 with $t=1$ was first proved in [Z3]. The proof in [Z3] is
more straightforward. It does not use the algebra ${\mathcal{B}}_{t}[{\xi,z}]$
and the product $\ast_{t}$. But the proof given here is more conceptual. For
example, the expansion in Eq. $(\ref{D_t-Taylor-e2})$ becomes much more
natural after we show here that it is just the usual Taylor series expansion
of polynomials as in Eq. $(\ref{Taylor-e1})$ but in the new context of the
algebra ${\mathcal{B}}_{t}[{\xi,z}]$.
Second, Eq. (4.7) can also be derived directly from Eq. (4.6) as in [Z3].
Namely, apply $\delta^{\alpha}$ to Eq. (4.6) and then replace $\xi$ by
$t\partial$ in both sides of the resulting equation.
Third, not all formal power series $f({\xi,z})\in{\mathcal{A}}[[{\xi,z}]]$ can
be expanded in the form of Eq. $(\ref{D_t-Taylor-e1})$ or $(\ref{D_t-
Taylor-e2})$. For example, let $n=1$ and $f({\xi,z})=e^{\xi z}$ and assume
that $(\ref{D_t-Taylor-e2})$ holds for $f({\xi,z})$. Then, by the argument in
the previous paragraph, we see that $a_{m}(z)$ $(m\geq 0)$ must be given by
Eq. (4.7). But, for the series $\delta^{m}f({\xi,z})=z^{m}\sum_{k\geq
0}\frac{(\xi z)^{k}}{k!}$, ${\mathcal{E}}_{t}$ is not well-defined, which is a
contradiction.
Another way to look at the fact above is as follows. Even though
${\mathcal{B}}_{t}[{\xi,z}]$ $(t\neq 0)$ and ${\mathcal{A}}[{\xi,z}]$ share
the same base vector space $\mathbb{C}[\xi,z]$, by Proposition 2.9, we know
that they induce different $\ell$-adic topologies on $\mathbb{C}[\xi,z]$.
Therefore, their completions with respect to the different $\ell$-adic
topologies will be different. In other words, the formal power series algebras
with respect to the product $\ast_{t}$ $(t\neq 0)$ and the usual formal power
series algebra ${\mathcal{A}}[[{\xi,z}]]$ do not share the same base vector
space anymore.
For the example $f({\xi,z})=e^{\xi z}$ above, we have
$f({\xi,z})\in{\mathcal{A}}[[{\xi,z}]]$. But, by the argument in the proof of
Proposition 2.9 with $\alpha_{m}$ $(m\geq 1)$ replaced by $m$, it is easy to
see that, for any $t\neq 0$, $f({\xi,z})=e^{\xi z}$ does not lie in the
completion of ${\mathcal{B}}_{t}[{\xi,z}]$ with respect to the $\ell$-adic
topology on $\mathbb{C}[{\xi,z}]$ induced by ${\mathcal{B}}_{t}[{\xi,z}]$.
Therefore, $f({\xi,z})=e^{\xi z}$ can not be written as a formal power series
with respect to the product $\ast_{t}$ as in Eq. (4.5).
### 4.2. Re-Formulation of the Image Conjecture in Terms of the Algebra
${\mathcal{B}}_{t}[{\xi,z}]$
First let us recall the following notion introduced recently in [Z4].
###### Definition 4.2.
Let $R$ be any commutative ring and ${\mathcal{A}}$ a commutative $R$-algebra.
We say that an $R$-subspace ${\mathcal{M}}$ of ${\mathcal{A}}$ is a Mathieu
subspace of ${\mathcal{A}}$ if the following property holds: if
$a\in{\mathcal{A}}$ satisfies $a^{m}\in{\mathcal{M}}$ for all $m\geq 1$, then,
for any $b\in{\mathcal{A}}$, we have $ba^{m}\in{\mathcal{M}}$ for all $m\gg
0$, i.e. there exists $N\geq 1$ $($depending on $a$ and $b$$)$ such that
$ba^{m}\in{\mathcal{M}}$ for all $m\geq N$.
From the definition above, it is easy to see that any ideal of ${\mathcal{A}}$
is automatically a Mathieu subspace of ${\mathcal{A}}$, but not conversely
(See [Z4] for some examples of Mathieu subspaces which are not ideals).
Therefore, the notion of Mathieu subspaces can be viewed as a generalization
of the notion of ideals.
Next, for any $t\in\mathbb{C}$, set
(4.10)
$\displaystyle\rm{Im\,}(\xi-t\partial)\\!:=\sum_{i=1}^{n}(\xi_{i}-t\partial_{i})\mathbb{C}[{\xi,z}].$
We call $\rm{Im\,}(\xi-t\partial)$ the image of the commuting differential
operators $(\xi_{i}-t\partial_{i})$ $(1\leq i\leq n)$.
With the notion and notation fixed above, the image conjecture proposed in
[Z4] for the commuting differential operators $(\xi-t\partial)$ can be re-
stated as follows.
###### Conjecture 4.3.
For any $t\in\mathbb{C}$, $\rm{Im\,}(\xi-t\partial)$ is a Mathieu subspace of
the polynomial algebra ${\mathcal{A}}[{\xi,z}]$.
One of the motivations of the conjecture above is the following theorem proved
in [Z3].
###### Theorem 4.4.
Conjecture 4.3 implies the Jacobian conjecture.
Actually, it has been shown in [Z3] that the Jacobian conjecture is equivalent
to some very special cases of Conjecture 4.3. For more detail, see [Z3].
The main result of this subsection is to show that the conjecture above can
actually be re-formulated as follows.
###### Conjecture 4.5.
Set $\xi\mathbb{C}[{\xi,z}]\\!:=\sum_{i=1}^{m}\xi_{i}\mathbb{C}[\xi,z]$. Then,
for any $t\in\mathbb{C}$, $\xi\mathbb{C}[{\xi,z}]$ as a subspace of
${\mathcal{B}}_{t}[{\xi,z}]$ is a Mathieu subspace of
${\mathcal{B}}_{t}[{\xi,z}]$.
###### Theorem 4.6.
Conjecture 4.5 is equivalent to Conjecture 4.3.
Proof: First, denote by $\xi\ast_{t}\mathbb{C}[{\xi,z}]$ the ideal of
${\mathcal{B}}_{t}[{\xi,z}]$ generated by $\xi_{i}$ $(1\leq i\leq n)$. View
$\xi\ast_{t}\mathbb{C}[{\xi,z}]$ as a subspace of ${\mathcal{A}}[{\xi,z}]$ and
apply Eqs. (4.10) and (2.14), we have
(4.11)
$\displaystyle\rm{Im\,}(\xi-t\partial)=\sum_{i=1}^{n}\xi_{i}\ast_{t}\mathbb{C}[{\xi,z}]=\xi\ast_{t}\mathbb{C}[{\xi,z}].$
Second, by Eqs. (2.1) and (2.18), we have
$\displaystyle\Phi_{t}(\xi\ast_{t}\mathbb{C}[{\xi,z}])=\Phi_{t}(\xi)\Phi_{t}(\mathbb{C}[{\xi,z}])=\xi\mathbb{C}[{\xi,z}]$
Hence, we also have
(4.12)
$\displaystyle\xi\ast_{t}\mathbb{C}[{\xi,z}]=\Phi_{t}^{-1}(\xi\mathbb{C}[{\xi,z}])=\Phi_{-t}(\xi\mathbb{C}[{\xi,z}]).$
Combine Eqs. (4.11) and (4.12), we get
(4.13)
$\displaystyle\Phi_{-t}(\xi\mathbb{C}[{\xi,z}])=\rm{Im\,}(\xi-t\partial).$
Third, by Proposition $4.9$ in [Z4], we know that pre-images of Mathieu
subspaces under algebra homomorphisms are still Mathieu subspaces, from which
it is easy to check that Mathieu subspaces are preserved by algebra
isomorphisms. By using this fact (on the algebra isomorphism
$\Phi_{-t}:{\mathcal{B}}_{-t}[{\xi,z}]\to{\mathcal{A}}[{\xi,z}]$) and also Eq.
(4.13), we see that, $\xi\mathbb{C}[{\xi,z}]$ is a Mathieu subspace of
${\mathcal{B}}_{-t}[{\xi,z}]$ iff $\rm{Im\,}(\xi-t\partial)$ is a Mathieu
subspace of ${\mathcal{A}}[{\xi,z}]$.
Replacing $t$ by $-t$ in the equivalence above, we have that,
$\xi\mathbb{C}[{\xi,z}]$ is a Mathieu subspace of ${\mathcal{B}}_{t}[{\xi,z}]$
for any $t\in\mathbb{C}$ iff $\rm{Im\,}(\xi+t\partial)$ is a Mathieu subspace
of ${\mathcal{A}}[{\xi,z}]$ for any $t\in\mathbb{C}$ iff
$\rm{Im\,}(\xi-t\partial)$ is a Mathieu subspace of ${\mathcal{A}}[{\xi,z}]$
for any $t\in\mathbb{C}$. Hence, we have proved the theorem. $\Box$
From Theorems 4.4 and 4.6, we immediately have the following corollary.
###### Corollary 4.7.
Conjecture 4.5 implies the Jacobian conjecture.
###### Remark 4.8.
Note that, when $t=0$, Conjecture 4.5 is trivial since $\xi\mathbb{C}[\xi,z]$
is an ideal of the algebra
${\mathcal{B}}_{t=0}[{\xi,z}]={\mathcal{A}}[{\xi,z}]$. In general, Conjecture
4.5 in some sense just claims that the algebras ${\mathcal{B}}_{t}[{\xi,z}]$
$(t\in\mathbb{C})$ do not differ or change too much from
${\mathcal{A}}[{\xi,z}]$ so that the vector subspace $\xi\mathbb{C}[\xi,z]$
still remains as a Mathieu subspace of ${\mathcal{B}}_{t}[{\xi,z}]$.
From this point of view, the triviality of the deformation
${\mathcal{B}}_{t}[{\xi,z}]$ $(t\in\mathbb{C})$ of the polynomial algebra
${\mathcal{A}}[{\xi,z}]$ given in Corollary 2.2 may be viewed as a fact in
favor of Conjecture 4.5, hence also to the Jacobian conjecture via the
implication in Corollary 4.7.
###### Remark 4.9.
Conjecture 4.5 and also the Jacobian conjecture can be viewed as problems
caused by the following fact. Namely, due to the change of the algebra
structure from ${\mathcal{A}}[{\xi,z}]$ to ${\mathcal{B}}_{t}[{\xi,z}]$, the
evaluation map at $\xi=0$, which is an algebra homomorphism from
${\mathcal{A}}[{\xi,z}]$ to ${\mathcal{A}}[z]$, is not an algebra homomorphism
from ${\mathcal{B}}_{t}[{\xi,z}]$ to ${\mathcal{A}}[z]$ if $t\neq 0$.
Therefore, its kernel $\xi\mathbb{C}[{\xi,z}]$ does not remain to be an ideal
of ${\mathcal{B}}_{t}[{\xi,z}]$ anymore.
But, on the other hand, as we will see later in Subsection 5.2 $($See Theorem
5.2 and Remark 5.4$)$, the same fact for the evaluation map at $\xi=1$, i.e.
$\xi_{i}=1$ $(1\leq i\leq n)$, in some sense also causes something truely
remarkable, namely, the generalized Laguerre polynomials.
## 5\. Connections with the Generalized Laguerre Polynomials
In this section, we study some connections and interactions of the monomials
of the algebra ${\mathcal{B}}[{\xi,z}]$ in $\xi$ and $z$ with the generalized
Laguerre polynomials in one or more variables.
In Subsection 5.1, we briefly recall the definition and the orthogonal
property of the generalized Laguerre polynomials. In Subsection 5.2, we show
that the generalized Laguerre polynomials can be obtained from certain
monomials of the algebra ${\mathcal{B}}[{\xi,z}]$ in $\xi$ and $z$ (See
Theorem 5.2 and Corollary 5.3).
In Subsection 5.3, we study some applications of the connection given in
Theorem 5.2. We first use certain properties of the generalized Laguerre
polynomials to derive some results on some monomials of
${\mathcal{B}}[{\xi,z}]$ in $\xi$ and $z$. We then use some results derived in
Section 2 on the algebra ${\mathcal{B}}[{\xi,z}]$ to give new proofs for some
important properties of the generalized Laguerre polynomials (see Proposition
5.11 and Theorem 5.12).
### 5.1. The Generalized Laguerre Orthogonal Polynomials
First, let us recall the generalized Laguerre orthogonal polynomials in one
variable.
For any $k\in{\mathbb{R}}$ and $m\in{\mathbb{N}}$, the generalized Laguerre
polynomial $L_{m}^{[k]}(z)$ in one variable is given by
(5.1) $\displaystyle
L_{m}^{[k]}(z)=\sum_{j=0}^{m}\binom{m+k}{m-j}\,\frac{(-z)^{j}}{j!}.$
Here we are only interested in the case that $k\in{\mathbb{N}}$. For any fixed
$k\in{\mathbb{N}}$, the generating function of the generalized Laguerre
polynomials $L_{m}^{[k]}(z)$ $(m\geq 0)$ is given by
(5.2)
$\displaystyle\frac{\exp(-\frac{zu}{1-u})}{(1-u)^{k+1}}=\sum_{m=0}^{+\infty}L_{m}^{[k]}(z)\,u^{m},$
where $u$ above denotes a formal variable which commutes with $z$.
The multi-variable generalized Laguerre polynomials are defined as follows.
Let ${\bf k}=(k_{1},k_{2},...,k_{n})\in{\mathbb{N}}^{n}$ and
${\alpha}=(a_{1},a_{2},...,a_{n})\in{\mathbb{N}}^{n}$. The generalized
Laguerre polynomials in $n$-variable $z=(z_{1},z_{2},...,z_{n})$ is defined by
(5.3) $\displaystyle L_{\alpha}^{[{\bf
k}]}(z)\\!:=L_{a_{1}}^{[k_{1}]}(z_{1})L_{a_{2}}^{[k_{2}]}(z_{2})\cdots
L_{a_{n}}^{[k_{n}]}(z_{n}).$
The polynomials $L_{\alpha}(z)\\!:=L_{\alpha}^{[0]}(z)$
$(\alpha\in{\mathbb{N}}^{n})$ are the so-called the (classical) Laguerre
polynomials. They were named after Edmond. N. Laguerre [L]. The generalized
Laguerre polynomials were introduced much later by G. Pólya and G. Szegö [PS]
in $1976$.
One of the most important properties of the generalized Laguerre polynomials
is the following theorem.
###### Theorem 5.1.
For any ${\bf k},\alpha,\beta\in{\mathbb{N}}^{n}$, we have
(5.4) $\displaystyle\int_{({\mathbb{R}}_{>0})^{n}}L_{\alpha}^{[{\bf
k}]}(z)L_{\beta}^{[{\bf
k}]}(z)\,w(z)\,dz=\delta_{\alpha,\beta}\,\frac{(\alpha+{\bf k})!}{\alpha!},$
where $\delta_{\alpha,\beta}$ is the Kronecker delta function and $w(z)$ given
by
(5.5) $\displaystyle w(z)\\!:=z^{\bf k}e^{-\sum_{i=1}^{n}z_{i}}.$
The function $w(z)$ above is called the weight function of the generalized
Laguerre polynomials $L_{\alpha}^{[\bf k]}(z)$ $(\alpha\in{\mathbb{N}}^{n})$.
Consequently, with any fixed ${\bf k}$, the generalized Laguerre polynomials
$L_{\alpha}^{[{\bf k}]}(z)$ $(\alpha\in{\mathbb{N}}^{n})$ form an orthogonal
basis of $\mathbb{C}[z]$ with respect to the Hermitian form defined by
(5.6)
$\displaystyle(f,g)=\int_{({\mathbb{R}}_{>0})^{n}}f(z)\bar{g}(z)\,w(z)\,dz,$
where $\bar{g}(z)$ denotes the complex conjugation of the polynomial
$g(z)\in\mathbb{C}[z]$.
There are many other interesting and important properties of the generalized
Laguerre polynomials. We refer the reader to [Sz], [PS], [AAR] and [DX] for
very thorough study on this family of orthogonal polynomials. See also the
Wolfram Research web sources [W1] and [W2] for over one hundred formulas and
identities on the (generalized) Laguerre polynomials.
### 5.2. The Generalized Laguerre Polynomials in Terms of the Product $\ast$
The main result of this subsection is the following theorem.
###### Theorem 5.2.
For any ${\bf k},\alpha\in{\mathbb{N}}^{n}$, we have
(5.7) $\displaystyle L_{\alpha}^{[{\bf k}]}(\xi z)$
$\displaystyle=\frac{(-1)^{|\alpha|}}{\alpha!}\,\xi^{-{\bf
k}}(\xi^{\alpha+{\bf k}}\ast z^{\alpha}).$ (5.8) $\displaystyle
L_{\alpha}^{[{\bf k}]}(\xi z)$
$\displaystyle=\frac{(-1)^{|\alpha|}}{\alpha!}\,z^{-{\bf k}}(\xi^{\alpha}\ast
z^{\alpha+{\bf k}}),$
where $\xi z\\!:=(\xi_{1}z_{1},\xi_{2}z_{2},...,\xi_{n}z_{n})$.
In particular, for the Laguerre polynomials, we have
(5.9) $\displaystyle L_{\alpha}(\xi
z)=\frac{(-1)^{|\alpha|}}{\alpha!}\,\,\xi^{\alpha}\ast z^{\alpha}.$
Proof: We first prove Eq. (5.9). Note first that, as pointed out in Subsection
$2.1$ [Z4], the Laguerre polynomials $L_{m}(z)$ $(m\in{\mathbb{N}})$ in one
variable can be obtained as
(5.10) $\displaystyle L_{m}(z)=\frac{1}{m!}(\partial-1)^{m}(z^{m}).$
Changing the variable $z\to\xi z$ in the equation above, we get
$\displaystyle L_{m}(\xi z)$
$\displaystyle=\frac{1}{m!}(\xi^{-1}\partial-1)^{m}(\xi^{m}z^{m})=\frac{1}{m!}\xi^{-m}(\partial-\xi)^{m}(\xi^{m}z^{m})$
$\displaystyle=\frac{1}{m!}(\partial-\xi)^{m}(z^{m})=\frac{(-1)^{m}}{m!}(\xi-\partial)^{m}(z^{m}).$
By Eq. (5.3) with ${\bf k}=0$ and the equation above, we see that the multi-
variable Laguerre polynomials $L_{\alpha}(z)$ $(\alpha\in{\mathbb{N}}^{n})$
can be given by
(5.11) $\displaystyle L_{\alpha}(\xi
z)=\frac{(-1)^{|\alpha|}}{\alpha!}(\xi-\partial)^{\alpha}(z^{\alpha}).$
Then, apply Eq. (2.14) with $\lambda(\xi)=\xi^{\alpha}$ and $t=1$, we get Eq.
(5.9).
To show Eq. (5.7), recall that we have the following well-known identity for
the one-variable generalized Laguerre polynomials, which can be easily derived
from the generating functions of the generalized Laguerre polynomials in Eq.
(5.2):
(5.12) $\displaystyle L_{m}^{[k]}(z)=(-1)^{k}\partial^{k}L_{m+k}(z).$
Now, by Eq. (5.3) and the equation above, we see that the multi-variable
generalized Laguerre polynomials can be given by
(5.13) $\displaystyle L_{\alpha}^{[{\bf k}]}(z)=(-1)^{|\bf k|}\partial^{\bf
k}L_{\alpha+{\bf k}}(z).$
Changing the variable $z\to\xi z$ in the equation above, we get
(5.14) $\displaystyle L_{\alpha}^{[{\bf k}]}(\xi z)$ $\displaystyle=(-1)^{|\bf
k|}(\partial^{\bf k}L_{\alpha+{\bf k}})(\xi z)$ $\displaystyle=(-1)^{|\bf
k|}\xi^{-{\bf k}}\partial^{\bf k}(L_{\alpha+{\bf k}}(\xi z))$ Applying Eq.
(5.9) and then Eq.(2.29): $\displaystyle=\frac{(-1)^{|\alpha|}}{(\alpha+{\bf
k})!}\xi^{-{\bf k}}\partial^{\bf k}(\xi^{\alpha+{\bf k}}\ast z^{\alpha+{\bf
k}}).$ $\displaystyle=\frac{(-1)^{|\alpha|}}{\alpha!}\xi^{-{\bf
k}}(\xi^{\alpha+{\bf k}}\ast z^{\alpha}).$
Hence, we get Eq. (5.7). Switching $\xi$ and $z$ in Eq. (5.7) and using the
commutativity of the product $\ast$, we get Eq. (5.8). $\Box$
###### Corollary 5.3.
For any ${\bf k},\alpha\in{\mathbb{N}}^{n}$, we have
$\displaystyle L_{\alpha}(z)$
$\displaystyle=\frac{(-1)^{|\alpha|}}{\alpha!}\,\,\left.(\xi^{\alpha}\ast
z^{\alpha})\right|_{\xi=1};$ $\displaystyle L_{\alpha}^{[{\bf k}]}(z)$
$\displaystyle=\frac{(-1)^{|\alpha|}}{\alpha!}\,\left.(\xi^{\alpha+{\bf
k}}\ast z^{\alpha})\right|_{\xi=1};$ $\displaystyle L_{\alpha}^{[{\bf k}]}(z)$
$\displaystyle=\frac{(-1)^{|\alpha|}}{\alpha!}\,\left.z^{-{\bf
k}}(\xi^{\alpha}\ast z^{\alpha+{\bf k}})\right|_{\xi=1},$
where $|_{{}_{\xi=1}}$ denotes the evaluation map from $\mathbb{C}[{\xi,z}]$
to $\mathbb{C}[z]$ by setting $\xi_{i}=1$ for any $1\leq i\leq n$.
###### Remark 5.4.
Note that, the evaluation map $|_{{}_{\xi=1}}$ viewed as a linear map from
${\mathcal{A}}[{\xi,z}]$ to ${\mathcal{A}}[z]$ is a homomorphism of algebras.
But, as a linear map from the algebra ${\mathcal{B}}[{\xi,z}]$ to the
polynomial algebra ${\mathcal{A}}[z]$, it is not a homomorphism of algebras
anymore. In particular, we have
$\displaystyle\left.(\xi^{\alpha}\ast z^{\alpha})\right|_{\xi=1}\neq 1\ast
z^{\alpha}=z^{\alpha}.$
Otherwise the generalized Laguerre polynomials would be trivialized.
Therefore, in some sense, the fact that the evaluation map
$|_{{}_{\xi=1}}:{\mathcal{B}}[{\xi,z}]\to{\mathcal{A}}[z]$ fails to be an
algebra homomorphism causes the non-trivial, actually truly remarkable,
generalized Laguerre polynomials. But, on the other hand, as we have discussed
in Subsection 4.2 $($See Remark 4.9$)$, the same fact for the evaluation map
at $\xi=0$ also causes some extremely difficult open problems such as
Conjecture 4.5 and the Jacobian conjecture.
Another immediate consequence of Theorem 5.2 is the following corollary.
###### Corollary 5.5.
For any $\alpha,\beta\in{\mathbb{N}}^{n}$, we have
(5.15) $\displaystyle\xi^{\beta}(\xi^{\alpha}\ast
z^{\alpha+\beta})=z^{\beta}(\xi^{\alpha+\beta}\ast z^{\alpha}).$
Note that the corollary follows immediately from Eqs. (5.7) and (5.8) with
${\bf k}=\beta$. But here we also give a more straightforward proof.
Proof: Consider
$\displaystyle\xi^{\beta}(\xi^{\alpha}\ast z^{\alpha+\beta})$
$\displaystyle=(\xi-\partial+\partial)^{\beta}(\xi^{\alpha}\ast
z^{\alpha+\beta})$
$\displaystyle=\sum_{\gamma\in{\mathbb{N}}^{n}}\binom{\beta}{\gamma}(\xi-\partial)^{\beta-\gamma}\partial^{\gamma}(\xi^{\alpha}\ast
z^{\alpha+\beta})$ Applying Eq. (2.29) and then Eq.(2.14):
$\displaystyle=\sum_{\gamma\in{\mathbb{N}}^{n}}\binom{\beta}{\gamma}\frac{(\alpha+\beta)!}{(\alpha+\beta-\gamma)!}\,(\xi-\partial)^{\beta-\gamma}(\xi^{\alpha}\ast
z^{\alpha+\beta-\gamma})$
$\displaystyle=\sum_{\gamma\in{\mathbb{N}}^{n}}\binom{\beta}{\gamma}\frac{(\alpha+\beta)!}{(\alpha+\beta-\gamma)!}\,\xi^{\beta-\gamma}\ast(\xi^{\alpha}\ast
z^{\alpha+\beta-\gamma})$
$\displaystyle=\sum_{\gamma\in{\mathbb{N}}^{n}}^{\beta}\binom{\beta}{\gamma}\frac{(\alpha+\beta)!}{(\alpha+\beta-\gamma)!}\,(\xi^{\alpha+\beta-\gamma}\ast
z^{\alpha+\beta-\gamma}).$
By switching $\xi\leftrightarrow z$ in the argument above and using the
commutativity of the product $\ast$, it is easy to see that we also have
$\displaystyle z^{\beta}(\xi^{\alpha+\beta}\ast
z^{\alpha})=\sum_{\gamma\in{\mathbb{N}}^{n}}^{\beta}\binom{\beta}{\gamma}\frac{(\alpha+\beta)!}{(\alpha+\beta-\gamma)!}\,(\xi^{\alpha+\beta-\gamma}\ast
z^{\alpha+\beta-\gamma}).$
Hence Eq. (5.15) follows. $\Box$
### 5.3. Some Applications of Theorem 5.2
First, let us derive some identities for the exponential series
$\exp_{*}(\cdot)=e_{\ast}^{\\{\cdot\\}}$ of the algebra
${\mathcal{B}}[{\xi,z}]$, i.e. the usual exponential series but with the
product replaced by $\ast$.
###### Proposition 5.6.
Let $u=(u_{1},u_{2},...,u_{n})$ be $n$ free commutative variables. Set
$\xi\ast z\\!:=(\xi_{1}\ast z_{1},\,\xi_{2}\ast z_{2},...,\,\xi_{n}\ast
z_{n})$ and $(\xi\ast z)u\\!:=\sum_{i=1}^{n}(\xi_{i}\ast z_{i})u_{i}$. Then,
for any ${\bf k}=(k_{1},k_{2},...,k_{n})\in{\mathbb{N}}^{n}$, we have
(5.16) $\displaystyle\xi^{-\bf k}\left(\xi^{\bf k}\ast e_{*}^{-(\xi\ast
z)u}\right)$
$\displaystyle=\prod_{i=1}^{n}\,\,\frac{\exp(-\,\frac{(\xi_{i}z_{i})\,u_{i}}{1-u_{i}})}{(1-u_{i})^{k_{i}+1}}.$
(5.17) $\displaystyle z^{-\bf k}\left(z^{\bf k}\ast e_{*}^{-(\xi\ast
z)u}\right)$
$\displaystyle=\prod_{i=1}^{n}\,\,\frac{\exp(-\,\frac{(\xi_{i}z_{i})\,u_{i}}{1-u_{i}})}{(1-u_{i})^{k_{i}+1}}.$
In particular, when ${\bf k}=0$, we have the following expression of the
exponential $\exp_{*}(-(\xi\ast z)u)$:
(5.18) $\displaystyle\exp_{*}(-(\xi\ast z)u)$
$\displaystyle=\prod_{i=1}^{n}\,\,\frac{\exp(-\,\frac{(\xi_{i}z_{i})\,u_{i}}{1-u_{i}})}{(1-u_{i})}.$
Proof: We give a proof for Eq. (5.16). The proof of Eq. (5.17) is similar.
First, by the commutativity and associativity of the product $\ast$ and also
by Proposition 2.6, $(b)$, it is easy to see that, for any
$\alpha,\beta\in{\mathbb{N}}^{n}$, we have
(5.19) $\displaystyle(\xi^{\alpha}\ast z^{\alpha})\ast(\xi^{\beta}\ast
z^{\beta})$ $\displaystyle=\xi^{\alpha+\beta}\ast z^{\alpha+\beta}.$ (5.20)
$\displaystyle(\xi\ast z)^{\ast\alpha}$ $\displaystyle=(\xi^{\alpha}\ast
z^{\alpha}),$
where $(\xi\ast z)^{\ast\alpha}$ denotes the “$\alpha^{\rm th}$” power of
$(\xi\ast z)$ with respect to the new product $\ast$.
By the last two equations above and Eq. (5.7), we have
(5.21) $\displaystyle\xi^{-{\bf k}}\left(\xi^{\bf k}\ast\exp_{*}^{-(\xi\ast
z)u}\right)$
$\displaystyle=\sum_{\alpha\in{\mathbb{N}}^{n}}\frac{(-1)^{|\alpha|}}{\alpha!}\,\xi^{-\bf
k}(\xi^{\alpha+{\bf k}}\ast z^{\alpha})u^{\alpha}$
$\displaystyle=\sum_{\alpha\in{\mathbb{N}}^{n}}L_{\alpha}^{[\bf k]}(\xi
z)u^{\alpha}.$
On the other hand, by Eqs. (5.2) and (5.3), we see that the generating
function of the multi-variable generalized Laguerre polynomials
$L_{\alpha}^{[\bf k]}(z)$ $(\alpha\in{\mathbb{N}}^{n})$ is given by
(5.22)
$\displaystyle\prod_{i=1}^{n}\,\,\frac{\exp(-\,\frac{z_{i}\,u_{i}}{1-u_{i}})}{(1-u_{i})^{k_{i}+1}}=\sum_{\alpha\in{\mathbb{N}}^{n}}L_{\alpha}^{[\bf
k]}(z)u^{\alpha}.$
Replacing $z$ by $\xi z$ in the equation above, we get
(5.23)
$\displaystyle\prod_{i=1}^{n}\,\,\frac{\exp(-\,\frac{(\xi_{i}z_{i})\,u_{i}}{1-u_{i}})}{(1-u_{i})^{k_{i}+1}}=\sum_{\alpha\in{\mathbb{N}}^{n}}L_{\alpha}^{[\bf
k]}(\xi z)u^{\alpha}.$
Combining Eqs. (5.21) and (5.23), we get Eq. (5.16). $\Box$
Next we use the connection given in Theorem 5.2 to derive more properties on
the monomials in $\xi$ and $z$ with respect to the product $\ast$ from certain
results on the generalized Laguerre polynomials.
For convenience, for any $\alpha\in{\mathbb{N}}^{n}$, we set
(5.24) $\displaystyle L_{\alpha}(z;\xi)\\!:=\xi^{\alpha}\ast z^{\alpha}.$
Note that, by Eqs. (5.1) and (5.9), the polynomials $L_{\alpha}(z;\xi)$
$(\alpha\in{\mathbb{N}}^{n})$ are polynomials with coefficients in
${\mathbb{Q}}$. In particular, for any fixed $\xi\in({\mathbb{R}}_{>0})^{n}$,
by Eqs. (5.1) and (5.9), it is easy to see that the polynomials
$L_{\alpha}(z;\xi)$ $(\alpha\in{\mathbb{N}}^{n})$ are polynomials in $z$ with
real coefficients and form a linear basis of $\mathbb{C}[z]$.
The next proposition says that this basis is also orthogonal with respect to
the following weight function:
(5.25) $\displaystyle
w_{\xi}(z)\\!:=e^{-\langle\xi,z\rangle}\prod_{i=1}^{n}\xi_{i},$
###### Proposition 5.7.
For any $\alpha,\beta\in{\mathbb{N}}^{n}$, we have
(5.26)
$\displaystyle\int_{({\mathbb{R}}_{>0})^{n}}L_{\alpha}(z;\xi)L_{\beta}(z;\xi)\,w_{\xi}(z)\,dz=(\alpha!)^{2}\delta_{\alpha,\beta}.$
Proof: Note that, under the change of variables $z_{i}\to\xi_{i}z_{i}$ $(1\leq
i\leq n)$, by Eqs. (5.9) and (5.24) the Laguerre polynomials $L_{\alpha}(z)$
will be changed to
(5.27) $\displaystyle L_{\alpha}(z)\to L_{\alpha}(\xi
z)=\frac{(-1)^{|\alpha|}}{\alpha!}L_{\alpha}(z;\xi).$
By Eq. (5.25) and also Eq. (5.5) with ${\bf k}=0$, the weight function $w(z)$
of the Laguerre polynomials is changed to
(5.28) $\displaystyle w(z)\to w_{\xi}(z)\prod_{i=1}^{n}\xi_{i}^{-1}.$
Now, apply the same changing of the variables to the integral in Eq. (5.4)
with ${\bf k}=0$, by the last two equations above, we get
(5.29) $\displaystyle\delta_{\alpha,\beta}$
$\displaystyle=\frac{(-1)^{|\alpha+\beta|}}{\alpha!\beta!}\int_{({\mathbb{R}}_{>0})^{n}}L_{\alpha}(z;\xi)L_{\beta}(z;\xi)\,w_{\xi}(z)\,dz$
Hence Eq. (5.26) follows. $\Box$
Denote by ${\mathcal{A}}_{\mathbb{Q}}[{\xi,z}]$ the polynomial algebra in
$\xi$ and $z$ over ${\mathbb{Q}}$. Next we assume $n=1$ and consider the
irreducibility of the polynomial $L_{\alpha}(z;\xi)$
$(\alpha\in{\mathbb{N}}^{n})$ as elements of
${\mathcal{A}}_{\mathbb{Q}}[{\xi,z}]$. But, first, we need to prove the
following lemma.
###### Lemma 5.8.
Let $\xi$ and $z$ be two commutative free variables and $K$ any field. Then,
for any $f(z)\in K[z]$ with $\deg f\geq 2$, $f(z)$ is irreducible over $K$ iff
$f(\xi z)\in K[\xi,z]$ $($as a polynomial in two variables$)$ is irreducible
over $K$.
Proof: The $(\Leftarrow)$ part of the lemma is trivial. We use the
contradiction method to show the $(\Rightarrow)$ part of the lemma.
Assume that $f(\xi z)$ is reducible in $K[{\xi,z}]$. Write
(5.30) $\displaystyle f(\xi z)=g({\xi,z})h({\xi,z})$
for some $g({\xi,z}),h({\xi,z})\in K[{\xi,z}]$ with $\deg g,\deg h\geq 1$.
Setting $\xi=1$ in the equation above, we also have
(5.31) $\displaystyle f(z)=g(1,z)h(1,z).$
Let $\bar{K}$ be the algebraic closure of $K$. Write
$f(z)=b\prod_{i=1}^{d}(z-a_{i})$ for some $b\in K\backslash\\{0\\}$ and
$a_{i}\in\bar{K}$ $(1\leq i\leq d)$. Then we have
(5.32) $\displaystyle f(\xi z)=b\prod_{i=1}^{d}(\xi z-a_{i}).$
Since $f(z)$ is irreducible over $K$ and $\deg f\geq 2$ by the assumption, we
have $a_{i}\neq 0$ $(1\leq i\leq d)$. Hence, for each $i$, $\xi z-a_{i}$ is
irreducible in $\bar{K}[{\xi,z}]$. Then by Eqs. (5.30) and (5.32), we have
(5.33) $\displaystyle g(\xi,z)=c\prod_{k=1}^{m}(\xi z-a_{i_{k}})$
for some $c\in\bar{K}\backslash\\{0\\}$, $1\leq m<d$ and $1\leq
i_{1}<i_{2}<\cdots<i_{m}\leq d$.
However, the equation above implies $g(1,z)=c\prod_{k=1}^{m}(z-a_{i_{k}})$.
Since $g(\xi,z)\in K[{\xi,z}]$, we also have $g(1,z)\in K[z]$. Then by Eq.
(5.31), $g(1,z)$ is a divisor of $f(z)$ in $K[z]$ with $1\leq\deg
g(1,z)=m<d=\deg f(z)$, which contradicts to the assumption that $f(z)$ is
irreducible in $K[z]$. $\Box$
###### Theorem 5.9.
Let $\xi$ and $z$ be two commutative free variables. For any $m\geq 2$,
$L_{m}(z;\xi)=\xi^{m}\ast z^{m}$ is irreducible in
${\mathcal{A}}_{\mathbb{Q}}[{\xi,z}]$.
Proof: By a theorem proved by I. Schur [Sc1], we know that, for any $m\geq 1$,
the Laguerre polynomials $L_{m}(z)$ in one variable is irreducible over
${\mathbb{Q}}$. Hence, by Eq. (5.9) and Lemma 5.8, the theorem holds. $\Box$
Note that I. Schur also proved in [Sc2] that the generalized Laguerre
polynomials $L_{m}^{[1]}(z)$ $(m\geq 0)$ in one variable are also irreducible
over ${\mathbb{Q}}$. Furthermore, M. Filaseta and T.-Y. Lam proved in [FL]
that, for any non-negative $k\in{\mathbb{Q}}$, all but finitely many of the
generalized Laguerre polynomials $L_{m}^{[k]}(z)$ $(m\geq 0)$ in one variable
are irreducible over ${\mathbb{Q}}$. Hence, by a similar argument as for
Theorem 5.9, we also have the following theorem.
###### Theorem 5.10.
Let $\xi$ and $z$ be two commutative free variables. Then, for any
$k\in{\mathbb{N}}$, all but only finitely many of the polynomials
$z^{-k}(\xi^{m}\ast z^{m+k})$ and $\xi^{-k}(\xi^{m+k}\ast z^{m})$
$(m\in{\mathbb{N}})$ are irreducible over ${\mathbb{Q}}$.
Next, we re-prove some important properties of the generalized Laguerre
polynomials by using their expressions given in Theorem 5.2. For simplicity,
we here only consider the one-variable case. Similar results for the multi-
variable generalized Laguerre polynomials can be simply derived from the one-
variable case via Eq. (5.3).
First, let us look at the following recurrent formulas of the Laguerre
polynomials in one variable.
###### Proposition 5.11.
For any $m\geq 1$, we have
(5.34) $\displaystyle(m+1)L_{m+1}(z)$
$\displaystyle=(2m+1-z)L_{m}(z)-mL_{m-1}(z),$ (5.35) $\displaystyle
zL^{\prime}_{m}(z)$ $\displaystyle=m(L_{m}(z)-L_{m-1}(z))$
Proof: Note first that, for any $m\geq 1$, by Eqs. (2.14) and (2.15), we have
$\displaystyle\xi\ast z^{m}$ $\displaystyle=(\xi-\partial)z^{m}=\xi
z^{m}-mz^{m-1},$ $\displaystyle z\ast\xi^{m}$
$\displaystyle=(z-\delta)\xi^{m}=z\xi^{m}-m\xi^{m-1}.$
Hence, we also have
$\displaystyle\xi\ast z$ $\displaystyle=\xi z-1,$ $\displaystyle\xi z^{m}$
$\displaystyle=\xi\ast z^{m}+mz^{m-1},$ $\displaystyle z\xi^{m}$
$\displaystyle=z\ast\xi^{m}+m\xi^{m-1}.$
By the last three equations above and also Eq. (2.23), we have
$\displaystyle(\xi z-1)(\xi^{m}\ast z^{m})$ $\displaystyle=(\xi\ast
z)(\xi^{m}\ast z^{m})=(z\xi^{m})\ast(\xi z^{m})$
$\displaystyle=(z\ast\xi^{m}+m\xi^{m-1})\ast(\xi\ast z^{m}+mz^{m-1})$
$\displaystyle=\xi^{m+1}\ast z^{m+1}+2m\,\xi^{m}\ast z^{m}+m^{2}\xi^{m-1}\ast
z^{m-1}.$
Multiply $(-1)^{m}/m!$ to the equation above and then apply Eq. (5.9), we get
$\displaystyle(\xi z-1)L_{m}(\xi z)=-(m+1)L_{m+1}(\xi z)+2mL_{m}(\xi
z)-mL_{m-1}(\xi z).$
Replace $\xi z$ by $z$ in the equation above, we get
$\displaystyle(z-1)L_{m}(z)=-(m+1)L_{m+1}(z)+2mL_{m}(z)-mL_{m-1}(z),$
Hence Eq. (5.34) follows.
To show Eq. (5.35), by Eqs. (2.15) and (2.30), we have,
$\displaystyle\xi^{m}\ast z^{m}$ $\displaystyle=z\ast(\xi^{m}\ast
z^{m-1})=(z-\delta)(\xi^{m}\ast z^{m-1})$ $\displaystyle=z(\xi^{m}\ast
z^{m-1})-m(\xi^{m-1}\ast z^{m-1})$
$\displaystyle=\frac{1}{m}z\partial(\xi^{m}\ast z^{m})-m(\xi^{m-1}\ast
z^{m-1}).$
Multiply $(-1)^{m}/m!$ to the equation above and then apply Eq. (5.9), we get
$\displaystyle L_{m}(\xi z)=\frac{1}{m}z\partial(L_{m}(\xi z))+L_{m-1}(\xi
z)=\frac{1}{m}\xi zL^{\prime}_{m}(\xi z)+L_{m-1}(\xi z).$
Replace $\xi z$ by $z$ in the equation above, we get
$\displaystyle L_{m}(z)=\frac{1}{m}zL^{\prime}_{m}(z)+L_{m-1}(z).$
Hence Eq. (5.35) follows. $\Box$
Next, we give a new proof for the following important property of the
generalized Laguerre polynomials in one variable.
###### Theorem 5.12.
For any $k,m\in{\mathbb{N}}$, $L_{m}^{[k]}(z)$ solves the following so-called
associated Laguerre differential equation:
(5.36) $\displaystyle zf^{\prime\prime}(z)+(k+1-z)f^{\prime}(z)+mf(z)=0.$
Proof: First, by Eq. (5.1), we have $L_{0}^{[k]}(z)=1$. It is easy to see that
the theorem holds for this case.
Assume $m\geq 1$. Then, by Eq. (2.15), we have
$\displaystyle\xi(\xi^{m+k}\ast z^{m})$ $\displaystyle=\xi(z\ast(\xi^{m+k}\ast
z^{m-1}))$ $\displaystyle=\xi(z-\delta)(\xi^{m+k}\ast z^{m-1})$
$\displaystyle=\xi z(\xi^{m+k}\ast z^{m-1})-\xi\delta(\xi^{m+k}\ast z^{m-1}).$
Add $z\partial(\xi^{m+k}\ast z^{m-1})$ to the equation above and apply Eq.
(2.21), we have
$\displaystyle\xi(\xi^{m+k}\ast z^{m})+z\partial(\xi^{m+k}\ast z^{m-1})$
$\displaystyle=\xi z(\xi^{m+k}\ast
z^{m-1})-(\xi\delta-z\partial)(\xi^{m+k}\ast z^{m-1})$ $\displaystyle=(\xi
z-k-1)(\xi^{m+k}\ast z^{m-1}).$
By Eq. (2.29), we may re-write the equation above as
$\displaystyle\xi(\xi^{m+k}\ast z^{m})+\frac{1}{m}z\partial^{2}(\xi^{m+k}\ast
z^{m})=\frac{1}{m}(\xi z-k-1)\partial(\xi^{m+k}\ast z^{m}).$
Multiply $\frac{(-1)^{m}\xi^{-k-1}}{(m-1)!}$ to both sides of the equation
above and then apply Eq. (5.7), we have
$\displaystyle mL_{m}^{[k]}(\xi z)+z\xi^{-1}\partial^{2}(L_{m}^{[k]}(\xi
z))=(\xi z-k-1)\xi^{-1}\partial(L_{m}^{[k]}(\xi z)).$
By the Chain rule, the equation above is same as
$\displaystyle mL_{m}^{[k]}(\xi z)+z\xi(\partial^{2}L_{m}^{[k]})(\xi z)=(\xi
z-k-1)(\partial L_{m}^{[k]})(\xi z).$
Replace $\xi z$ by $z$, or $z$ by $\xi^{-1}z$ in the equation above, we get
$\displaystyle mL_{m}^{[k]}(z)+z\partial^{2}L_{m}^{[k]}(z)=(z-k-1)\partial
L_{m}^{[k]}(z).$
Hence we have proved the theorem. $\Box$
Finally, let us point out the following conjecture on the generalized Laguerre
polynomials, which is a special case of Conjecture $3.5$ in [Z4] for all the
classical orthogonal polynomials.
###### Conjecture 5.13.
For any ${\bf k}\in{\mathbb{N}}^{n}$, the subspace ${\mathcal{M}}$ of the
polynomial algebra ${\mathcal{A}}[z]$ spanned by the generalized Laguerre
polynomials $L^{[\bf k]}_{\alpha}(z)$ $(0\neq\alpha\in{\mathbb{N}}^{n})$ is a
Mathieu subspace of ${\mathcal{A}}[z]$.
Despite the vast amount of known results on the generalized Laguerre
polynomials in the literature, the conjecture above is even still open for the
classical Laguerre polynomials, (i.e. the case with ${\bf k}=0$) in one
variable.
## References
* [AAR] G. F. Andrews, R. Askey and R. Roy, Special functions. Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. [MR1688958].
* [BCW] H. Bass, E. Connell, D. Wright, The Jacobian conjecture, reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc. 7, (1982), 287–330. [MR 83k:14028].
* [B] J.-E. Björk, Rings of differential operators. North-Holland Publishing Co., Amsterdam-New York, 1979. [MR0549189].
* [C] S. C. Coutinho, A primer of algebraic $D$-modules. London Mathematical Society Student Texts, 33. Cambridge University Press, Cambridge, 1995. [MR1356713].
* [DX] C. Dunkl and Y. Xu, Orthogonal polynomials of several variables. Encyclopedia of Mathematics and its Applications, 81. Cambridge University Press, Cambridge, 2001. [MR1827871].
* [E] A. van den Essen, _P_ olynomial automorphisms and the Jacobian conjecture. Progress in Mathematics, 190. Birkhäuser Verlag, Basel, 2000. [MR1790619].
* [FL] M. Filaseta, and T.-Y. Lam, On the irreducibility of the generalized Laguerre polynomials. Acta Arith. 105 (2002), no. 2, 177–182. [MR1932764].
* [Ke] O. H. Keller, Ganze Gremona-Transformationen, Monats. Math. Physik 47 (1939), no. 1, 299-306. [MR1550818].
* [L] E. Laguerre, Sur $l^{\prime}$intégrale $\int_{0}^{\infty}\frac{e^{-x}dx}{x}$. Bull. Soc. math. France 7 (1879) 72 C81. Reprinted in Oeuvres, Vol. 1. New York: Chelsea, 428–437, 1971.
* [PS] G. Pólya and G. Szegö, Problems and theorems in analysis. Vol. II. Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Springer Study Edition, Springer, New York, 1976. [MR0465631].
* [Sc1] I. Schur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, I. Sitzungsber. Preuss. Akad. Wiss. Berlin Phys.-Math. Kl., 14 (1929), 125–136.
* [Sc2] I. Schur, Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome. Journal für die reine und angewandte Mathematik 165 (1931), 52–58.
* [Sz] G. Szegö, Orthogonal Polynomials. 4th edition. American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975. [MR0372517].
* [W1] Wolfram Research, http://functions.wolfram.com/Polynomials/LaguerreL/.
* [W2] Wolfram Research, http://functions.wolfram.com/Polynomials/LaguerreL3/.
* [Z1] W. Zhao, Hessian Nilpotent Polynomials and the Jacobian Conjecture, Trans. Amer. Math. Soc. 359 (2007), no. 1, 249–274 (electronic). [MR2247890]. See also math.CV/0409534.
* [Z2] W. Zhao, A Vanishing Conjecture on Differential Operators with Constant Coefficients, Acta Mathematica Vietnamica, vol 32 (2007), no. 3, 259–285. [MR2368014]. See also arXiv:0704.1691v2 [math.CV].
* [Z3] W. Zhao, Images of commuting Differential Operators of Order One with Constant Leading Coefficients. arXiv:0902.0210 [math.CV]. Submitted.
* [Z4] W. Zhao, Generalizations of the Image Conjecture and the Mathieu Conjecture. To appear in J. Pure Appl. Algebra. DOI:10.1016/j.jpaa.2009.10.007. See also arXiv:0902.0212 [math.CV].
* [Z5] W. Zhao, New Proofs for the Abhyankar-Gujar Inversion Formula and the Equivalence of the Jacobian Conjecture and the Vanishing Conjecture. Submitted. See also arXiv:0907.3991 [math.AG].
Department of Mathematics, Illinois State University, Normal, IL 61790-4520.
E-mail: [email protected].
|
arxiv-papers
| 2009-07-23T06:55:31 |
2024-09-04T02:49:04.121966
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenhua Zhao",
"submitter": "Wenhua Zhao",
"url": "https://arxiv.org/abs/0907.3990"
}
|
0907.4056
|
# Series Evaluation of a Quartic Integral
###### Abstract.
We present a new single sum series evaluation of Moll’s quartic integral and
present two new generalizations.
Moa Apagodu
Department of Mathematics, Virginia Commonwealth University,
Richmond, VA 23284, USA
In a beautiful personal story [6] Victor Moll describes his encounter with
certain quartic integral and derives its evaluation and goes on to study
analytic and number theoretic properties (_log-concavity_ , _p-adic
valuations_ , _location of the zeros_ , etc.) of a polynomial associated with
the evaluation of the integral [1,2,4,5,6,7]. In this article we use the
Almkvist-Zeilberger algorithm ([3,8,9,10]) to derive a new series evaluation
of this integral. In addition, we give two new generalizations of the
identity. In [1], T. Amdeberhan and V. Moll presented a survey of old and new
proofs of the evaluation and the formula:
Theorem 1 [T. Amdeberhan and V. Moll, [1]]:
$\int_{0}^{\infty}\frac{dx}{(x^{4}+2x^{2}a+1)^{m+1}}=\frac{\pi}{2}\frac{{2m\choose
m}}{4^{m}(2(a+1))^{m+1/2}}{}_{2}F_{1}\left({{-m,m+1}\atop{-m+1/2}}\,;\,(a+1)/2\right)\,\,.$
where
${}_{2}F_{1}\left({{a,b}\atop{c}}\,;\,x\right)=\sum_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}}{(c)_{k}(1)_{k}}x^{k}\,\,$
and $(z)_{k}=z(z+1)(z+2)\ldots(z+k-1)$.
The polynomial associated with the evaluation of the integral that is the
subject of study in [1,2,3,4] is
$P_{m}(a)=\frac{{2m\choose
m}}{4^{m}}{}_{2}F_{1}\left({{-m,m+1}\atop{-m+1/2}}\,;\,(a+1)/2\right)\,.$
Next we state the main results of this article:
Theorem 2:
$\displaystyle\int_{0}^{\infty}\frac{dx}{(x^{4}+2ax^{2}+1)^{m+1}}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{l=0}^{\infty}(-1)^{l}\frac{2^{l}(\frac{l}{2}-\frac{3}{4})!(m+\frac{l}{2}-\frac{1}{4})!}{l!m!}a^{l}\,\,.$
Proof:
We use the Almkvist-Zeilberger algorithm ([3,8,9,10]), and the reader is
assumed to be familiar with this method. In particular, we used Zeilberger’s
Maple package EKHAD8 (procedure AZc) that computes differential operators and
certificates for single variable hyper-exponential functions accompanying [3],
available from
http://www.math.rutgers.edu/~zeilberg/tokhniot/EKHAD .
We cleverly construct the (certificate) function
$R(x,a)=-\frac{x(4m+3+4ax^{2}m+2ax^{2}-x^{4})}{(x^{4}+2ax^{2}+1)}$
with the motives
$-4m-3-4a(2m+3)D_{a}(F(x,a))-4(a^{2}-1)D_{a}^{2}F(x,a)=D_{x}(R(x,a)F(x,a))\,\,,$
where $F(x,a)$ is the integrand and $D_{a}$ is differentiation operator with
respect to the variable $a$. If we integrate both sides with respect to $x$ on
the limits of integration and observe that the right-hand side vanishes, we
get a differential operator
$-4m-3-4a(2m+3)D_{a}-4(a^{2}-1)D_{a}^{2}\,\,,$
that annihilates the left side of the theorem. Using the standard technique
(or use Paul Zimmermann and Bruno Salvy’s $gfun$ from Maple library if you
wish) of translating a differential equation satisfied by a power series into
a recurrence relation for its coefficients $a_{l}(m)$, we get
$(-4l^{2}+(-8m-8)l-4m-3)a_{l}(m)+(4l^{2}+12l+8)a_{l}(m)(l+2)=0\,\,,$
a homogeneous recurrence relation satisfied by the discrete coefficient
function ${a_{l}(m)}$. Finally, the theorem follows by solving the recurrence
relation with the initial conditions calculated directly from the integral:
$a_{0}(m)=I(0,m)$ and $a_{1}(m)=I^{\prime}(0,m)$, where $I(a,m)$ is the the
integral on the left. Q.E.D.
Comparing the right-hand side of our theorem with that of (_theorem 1_), we
get
$None$
$P_{m}(a)=\frac{2^{m+3/2}(a+1)^{m+1/2}}{4\pi}\sum_{l=0}^{\infty}(-1)^{l}\frac{2^{l}(\frac{l}{2}-\frac{3}{4})!(m+\frac{l}{2}-\frac{1}{4})!}{l!m!}a^{l}\,\,$
Using Newton’s Binomial theorem,
$(1+a)^{m+1/2}=\sum_{k=0}^{\infty}{m+1/2\choose k}a^{k}\,\,,$
and multiplication of series, the coefficient of $a^{n}$, $d_{n}(m)$, in the
polynomial $P_{m}(a)$ is
$\displaystyle d_{n}(m)$ $\displaystyle=$
$\displaystyle\frac{2^{m+3/2}}{4\pi}\sum_{k+l=n}{m+\frac{1}{2}\choose
k}(-1)^{l}\frac{2^{l}(\frac{l}{2}-\frac{3}{4})!(m+\frac{l}{2}-\frac{1}{4})!}{l!m!}\,\,$
$\displaystyle=$
$\displaystyle\frac{2^{m+3/2}}{4\pi}\sum_{l=0}^{n}{m+\frac{1}{2}\choose
n-l}(-1)^{l}\frac{2^{l}(\frac{l}{2}-\frac{3}{4})!(m+\frac{l}{2}-\frac{1}{4})!}{l!m!}\,\,.$
Next, we give the first of two generalizations in which $2$ in the integral of
_theorem 2_ is replaced by any integer $n$ for which the integral exists.
Theorem 3:
$\displaystyle\int_{0}^{\infty}\frac{dx}{(x^{2n}+nax^{n}+1)^{m+1}}$
$\displaystyle=$
$\displaystyle\frac{1}{2n}\sum_{l=0}^{\infty}(-1)^{l}\frac{n^{l}(\frac{l}{2}-\frac{2n-1}{2n})!(\frac{l}{2}+m-\frac{1}{2n})!}{l!m!}a^{l}\,\,.$
any integer $n$ for which the integral exists.
Proof:
First, we make the change of variables $z=x^{n}$ and the question reduces to
evaluating
$\int_{0}^{\infty}\frac{dz}{n(z^{2}+2az+1)^{m+1}z^{1-1/n}}\,\,.$
Then, EKHAD gives a differential operator
$-2n-2nm+1-(2m+3)n^{2}aD_{a}-n^{2}(a^{2}-1)D_{a}^{2}\,\,.$
with certificate function
$R(z,a)=-\frac{nz(2n+2nm-1+nzma+naz-az-z^{2})}{z^{2}+az+1}\,\,.$
That is,
$(-2n-2nm+1-(2m+3)n^{2}aD_{a}-n^{2}(a^{2}-1)D_{a}^{2})F(x,a)=D_{x}(R(x,a)F(x,a))\,\,.$
where $F(x,a)$ is the integrand and $D_{a}$ is differentiation operator with
respect to the variable $a$. Now integrate both sides and convert the
resulting differential operator for the series into a recurrence relation for
the coefficients and solve. Q.E.D.
The second generalization where $n$ is replaced by any parameter $\alpha$ for
which the integral exists whose proof follows from _theorem 3_ by writing
$\alpha a$ as $n\left(\frac{a\alpha}{n}\right)$.
Theorem 4:
$\displaystyle\int_{0}^{\infty}\frac{dx}{(x^{2n}+\alpha ax^{n}+1)^{m+1}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{l=0}^{\infty}(-1)^{l}n^{l-1}(\frac{\alpha}{n})^{l}\frac{(\frac{l}{2}-\frac{2n-1}{2n})!(\frac{l}{2}+m-\frac{1}{2n})!}{l!m!}a^{l}\,\,.$
for any integer $n>0$ and indeterminate $\alpha$ for which the integral
exists.
Problem: Find analogous polynomial _Polypart_ as in theorem 1 associated with
the evaluation of the generalization in _theorem 3_ if it exists, i.e. find
$closedForm(n,m,a)$ such that
$\displaystyle P_{m}^{n}(a)$ $\displaystyle:=$ $\displaystyle
closedForm(n,m,a)\times\frac{1}{2n}\sum_{l=0}^{\infty}(-1)^{l}\frac{n^{l}(\frac{l}{2}-\frac{2n-1}{2n})!(\frac{l}{2}+m-\frac{1}{2n})!}{l!m!}a^{l}\,\,,$
is a polynomial in $a$.
For the special case $n=2$ (_Polypart_),
$closedForm(2,m,a)=\frac{\pi}{2}\frac{1}{(2(a+1))^{m+1/2}}$.
References
[1] T. Amdeberhan and V. Moll, A formula for a quartic integral: a survey of
old proofs and some new ones, Ramanujan J 18: 91 102 (2009).
[2] T. Amdeberhan, D. Manna, and V. Moll, The 2-Adic Valuation of a Sequence
Arising from a Rational Integral, Journal of Combinatorial Theory, Series A,
115, 2008, 1474-1486.
[3] M. Apagodu and D. Zeilberger, Multi-Variable Zeilberger and Almkvist-
Zeilberger Algorithms and the Sharpening of Wilf-Zeilberger Theory, Adv. Appl.
Math. 37(2006)(Special Regev issue), 139-152.
[4] G. Boros, Victor H. Moll, and Sarah RileyAn, An Elementary evaluation of a
quartic integral, Scientia, Series A: Math. Sciences 11, 2005, 1-12.
[5] M. Kauers and P. Paule, A Computer Proof of Moll’s Log-Concavity
Conjuctur, Proceedings of AMS, 135(12):3847–3856(2007).
[6] V. Moll, The Evaluation of Integrals: a Personal Story, Notices Amer.
Math. Soc. 49, March 2002, 311-317.
[7] V. Moll and D. Manna, REMARKABLE SEQUENCE OF INTEGERS, to appear in
Expositiones Mathematicae.
[8] M. Petkov sek, H.S. Wilf, D. Zeilberger, ”A=B”, A.K. Peters Ltd., 1996.
[9] H. Wilf, D. Zeilberger, ”Rational Functions Certify Combinatorial
Identities”, J. Amer. Math. Soc. 3 147-158 (1990).
[10] D. Zeilberger, ”The method of creative telescoping”, J. Symbolic
Computation, 11 195-204 (1991).
|
arxiv-papers
| 2009-07-23T13:09:16 |
2024-09-04T02:49:04.131993
|
{
"license": "Public Domain",
"authors": "Moa Apagodu",
"submitter": "Moa Apagodu Dr.",
"url": "https://arxiv.org/abs/0907.4056"
}
|
0907.4217
|
Parabolic nef currents
on hyperkähler manifolds
Misha Verbitsky111The work is partially supported by the grant RFBR for
support of scientific schools NSh-3036.2008.2 and RFBR grant 09-01-00242-a
Abstract
Let $M$ be a compact, holomorphically symplectic Kähler manifold, and $\eta$ a
(1,1)-current which is nef (a limit of Kähler forms). Assume that the
cohomology class of $\eta$ is parabolic, that is, its top power vanishes. We
prove that all Lelong sets of $\eta$ are coisotropic. When $M$ is generic,
this is used to show that all Lelong numbers of $\eta$ vanish. We prove that
any hyperkähler manifold with $\operatorname{Pic}(M)={\mathbb{Z}}$ has non-
trivial coisotropic subvarieties, if a generator of $\operatorname{Pic}(M)$ is
parabolic.
###### Contents
1. 1 Introduction
1. 1.1 Hyperkähler manifolds
2. 1.2 The Bogomolov-Beauville-Fujiki form
3. 1.3 The hyperkähler SYZ conjecture
4. 1.4 Lelong numbers and hyperkähler geometry
2. 2 Hyperkähler geometry: preliminary results
1. 2.1 The structure of a Kähler cone
2. 2.2 Subvarieties in generic hyperkähler manifolds
3. 2.3 Cohomology of hyperkähler manifolds
3. 3 Cohomology classes dominated by a nef class
1. 3.1 Positive forms and positive currents
2. 3.2 Regularization for nef currents
3. 3.3 Cohomology classes dominated by a nef current
4. 3.4 $\eta$-coisotropic subvarieties and cohomology classes
## 1 Introduction
### 1.1 Hyperkähler manifolds
Definition 1.1: A hyperkähler manifold is a compact, Kähler, holomorphically
symplectic manifold.
Definition 1.2: A hyperkähler manifold $M$ is called simple if $H^{1}(M)=0$,
$H^{2,0}(M)={\mathbb{C}}$.
Theorem 1.3: (Bogomolov’s Decomposition Theorem, [Bo1], [Bes]). Any
hyperkähler manifold admits a finite covering, which is a product of a torus
and several simple hyperkähler manifolds.
Remark 1.4: Further on, all hyperkähler manifolds are silently assumed to be
simple.
A note on terminology. Speaking of hyperkähler manifolds, people usually mean
one of two different notions. One either speaks of holomorphically symplectic
Kähler manifold, or of a manifold with a hyperkähler structure, that is, a
triple of complex structures satisfying quaternionic relations and parallel
with respect to the Levi-Civita connection. The equivalence (in compact case)
between these two notions is provided by the Yau’s solution of Calabi-Yau
conjecture ([Bes]). Throughout this paper, we use the complex algebraic
geometry point of view, where “hyperkähler” is synonymous with “Kähler
holomorphically symplectic”, in lieu of the differential-geometric approach.
To avoid the terminological confusion, we tried not mention quaternionic
structures (except Subsection 2.2, where it was impossible to avoid). The
reader may check [Bes] for an introduction to hyperkähler geometry from the
differential-geometric point of view.
Notice also that we included compactness in our definition of a hyperkähler
manifold. In the differential-geometric setting, one does not usually assume
that the manifold is compact.
### 1.2 The Bogomolov-Beauville-Fujiki form
Theorem 1.5: ([F]) Let $\eta\in H^{2}(M)$, and $\dim M=2n$, where $M$ is
hyperkähler. Then $\int_{M}\eta^{2n}=q(\eta,\eta)^{n}$, for some integer
quadratic form $q$ on $H^{2}(M)$.
Definition 1.6: This form is called Bogomolov-Beauville-Fujiki form. It is
defined by this relation uniquely, up to a sign. The sign is determined from
the following formula (Bogomolov, Beauville; [Bea], [Hu1], 23.5)
$\displaystyle\lambda q(\eta,\eta)$
$\displaystyle=(n/2)\int_{X}\eta\wedge\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n-1}-$
$\displaystyle-(1-n)\frac{\left(\int_{X}\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}\right)\left(\int_{X}\eta\wedge\Omega^{n}\wedge\overline{\Omega}^{n-1}\right)}{\int_{M}\Omega^{n}\wedge\overline{\Omega}^{n}}$
where $\Omega$ is the holomorphic symplectic form, and $\lambda$ a positive
constant.
Remark 1.7: The form $q$ has signature $(b_{2}-3,3)$. It is negative definite
on primitive forms, and positive definite on the space
$\langle\Omega,\overline{\Omega},\omega\rangle$ where $\omega$ is a Kähler
form, as seen from the following formula
$\mu q(\eta_{1},\eta_{2})=\\\
\int_{X}\omega^{2n-2}\wedge\eta_{1}\wedge\eta_{2}-\frac{2n-2}{(2n-1)^{2}}\frac{\int_{X}\omega^{2n-1}\wedge\eta_{1}\cdot\int_{X}\omega^{2n-1}\wedge\eta_{2}}{\int_{M}\omega^{2n}},\
\ \mu>0$ (1.1)
(see e. g. [V4], Theorem 6.1, or [Hu1], Corollary 23.9).
Definition 1.8: Let $[\eta]\in H^{1,1}(M)$ be a real (1,1)-class on a
hyperkähler manifold $M$. We say that $[\eta]$ is parabolic if
$q([\eta],[\eta])=0$. A line bundle $L$ is called parabolic if $c_{1}(L)$ is
parabolic.
The present paper is a study of algebro-geometric properties of parabolic
bundles and cohomology classes, in hope to find criteria for effectivity.
### 1.3 The hyperkähler SYZ conjecture
Theorem 1.9: (D. Matsushita, see [Ma1]). Let $\pi:\;M{\>\longrightarrow\>}X$
be a surjective holomorphic map from a hyperkähler manifold $M$ to $X$, with
$0<\dim X<\dim M$. Then $\dim X=1/2\dim M$, and the fibers of $\pi$ are
holomorphic Lagrangian (this means that the symplectic form vanishes on the
fibers).111Here, as elsewhere, we silently assume that the hyperkähler
manifold $M$ is simple.
Definition 1.10: Such a map is called a holomorphic Lagrangian fibration.
Remark 1.11: The base of $\pi$ is conjectured to be rational. J.-M. Hwang
([Hw]) proved that $X\cong{\mathbb{C}}P^{n}$, if it is smooth. D. Matsushita
([Ma2]) proved that it has the same rational cohomology as
${\mathbb{C}}P^{n}$.
Remark 1.12: The base of $\pi$ has a natural flat connection on the smooth
locus of $\pi$. The combinatorics of this connection can be used to determine
the topology of $M$ ([KZ], [G]),
Remark 1.13: Matsushita’s theorem is implied by the following formula of
Fujiki. Let $M$ be a hyperkähler manifold, $\dim_{\mathbb{C}}M=2n$, and
$\eta_{1},...,\eta_{2n}\in H^{2}(M)$ cohomology classes. Then
$\eta_{1}\wedge\eta_{2}\wedge...=\frac{1}{2n!}\sum_{\sigma}q(\eta_{\sigma_{1}}\eta_{\sigma_{2}})q(\eta_{\sigma_{3}}\eta_{\sigma_{3}})q(\eta_{\sigma_{2n-1}}\eta_{\sigma_{2n}})$
(1.2)
with the sum taken over all permutations. An algebraic argument (see e.g. 2.3)
allows to deduce from this formula that for any non-zero $\eta\in H^{2}(M)$,
one would have $\eta^{n}\neq 0$, and $\eta^{n+1}=0$, if $q(\eta,\eta)=0$, and
$\eta^{2n}\neq 0$ otherwise. Applying this to the pullback $\pi^{*}\omega_{X}$
of the Kähler class from $X$, we immediately obtain that
$\dim_{\mathbb{C}}X=n$ or $\dim_{\mathbb{C}}X=2n$. Indeed,
$\omega_{X}^{\dim_{\mathbb{C}}X}\neq 0$ and
$\omega_{X}^{\dim_{\mathbb{C}}X+1}=0$.
Definition 1.14: Let $(M,\omega)$ be a Calabi-Yau manifold, $\Omega$ the
holomorphic volume form, and $Z\subset M$ a real analytic subvariety,
Lagrangian with respect to $\omega$. If
$\Omega{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}$ is proportional to the
Riemannian volume form, $Z$ is called special Lagrangian (SpLag).
The special Lagrangian varieties were defined in [HL] by Harvey and Lawson,
who proved that they minimize the Riemannian volume in their cohomology class.
This implies, in particular, that their moduli are finite-dimensional. In
[McL], McLean studied deformations of non-singular special Lagrangian
subvarieties and showed that they are unobstructed.
In [SYZ], Strominger-Yau-Zaslow tried to explain the mirror symmetry
phenomenon using the special Lagrangian fibrations. They conjectured that any
Calabi-Yau manifold admits a Lagrangian fibration with special Lagrangian
fibers. Taking its dual fibration, one obtains “the mirror dual” Calabi-Yau
manifold.
Remark 1.15: It is easy to see that a holomorphic Lagrangian subvariety of a
hyperkähler manifold $(M,I)$ is special Lagrangian on $(M,J)$, where $(I,J,K)$
is a quaternionic structure associated with the hyperkähler structure on $M$
(Subsection 2.2). Therefore, existence of holomorphic Lagrangian fibrations
implies existence of special Lagrangian fibrations postulated by Strominger-
Yau-Zaslow.
Definition 1.16: A line bundle is called semiample if $L^{N}$ is generated by
its holomorphic sections, which have no common zeros.
Remark 1.17: From semiampleness it obviously follows that $L$ is nef. Indeed,
let $\pi:\;M{\>\longrightarrow\>}{\mathbb{P}}H^{0}(L^{N})^{*}$ the the
standard map. Since sections of $L$ have no common zeros, $\pi$ is
holomorphic. Then $L\cong\pi^{*}{\cal O}(1)$, and the curvature of $L$ is a
pullback of the Kähler form on ${\mathbb{C}}P^{n}$. However, the converse is
false: a nef bundle is not necessarily semiample (see e.g. [DPS1, Example
1.7]).
Remark 1.18: Let $\pi:\;M{\>\longrightarrow\>}X$ be a holomorphic Lagrangian
fibration, and $\omega_{X}$ a Kähler class on $X$. Then
$\eta:=\pi^{*}\omega_{X}$ is semiample and parabolic. The converse is also
true, by Matsushita’s theorem: if $L$ is semiample and parabolic, $L$ induces
a Lagrangian fibration. This is the only known source of non-trivial special
Lagrangian fibrations.
Conjecture 1.19: (Hyperkähler SYZ conjecture) Let $L$ be a parabolic nef line
bundle on a hyperkähler manifold. Then $L$ is semiample.
Remark 1.20: This conjecture was stated by many people (Tyurin, Bogomolov,
Hassett-Tschinkel, Huybrechts, Sawon); please see [Saw] for an interesting and
historically important discussion, and [V5] for details and reference.
Remark 1.21: The SYZ conjecture can be seen as a hyperkaehler version of
“abundance conjecture” (see e.g. [DPS2], 2.7.2).
### 1.4 Lelong numbers and hyperkähler geometry
In [V5], it was shown that any parabolic line bundle $L$ with a smooth metric
of semipositive curvature is ${\mathbb{Q}}$-effective (this means that
$c_{1}(L)$ is represented by a rational divisor). Further results in this
direction require detailed study of singularities of positive currents on
hyperkähler manifolds. The present paper is an attempt to understand these
singularities.
Let $[\eta]$ be a nef cohomology class. Using weak compactness of positive
currents, it is possible to show that $[\eta]$ is represented by a positive,
closed $(1,1)$-current $\eta$ (3.1). Locally, $\eta$ can be considered as a
curvature of a singular metric on a line bundle.
Using a local $dd^{c}$-lemma, we may assume that $\eta=dd^{c}\varphi$, for
some function $\varphi$, which is plurisubharmonic, because $\eta$ is
positive. Then $\eta$ is a curvature of a trivial bundle with a singular
metric $h{\>\longrightarrow\>}e^{-2\varphi}|h|^{2}$.
A multiplier ideal sheaf ${\cal I}(\eta)$ of a current $\eta$ is an ideal of
all holomorphic functions $h$ on $M$ for which $e^{-2\varphi}|h|^{2}$ is
locally integrable. Nadel has shown that a multiplier ideal sheaf of a
positive current is always coherent.
The notion of a multiplier ideal has many applications in algebraic geometry,
due to the Nadel’s vanishing theorem.
Theorem 1.22: (Nadel’s Vanishing Theorem; see [N], [D2]). Let $(M,\omega)$ be
a Kähler manifold, $\eta$ a closed, positive (1,1)-current,
$\eta>\varepsilon\omega$, and $L$ a holomorphic line bundle with
$c_{1}(L)=[\eta]$. Consider a singular metric on $L$ associated with $\eta$,
and let ${\cal I}(L)$ be the sheaf of $L^{2}$-integrable sections. Then
$H^{i}({\cal I}(L)\otimes K_{M})=0$ for all $i>0$.
The Lelong number $\nu_{x}(\Theta)$ of a $(p,p)$-current $\Theta$ at $x\in M$,
as defined in [D5], is a mass of a measure $\Theta\wedge\mu_{x}^{n-p}$ carried
at $x$, where $\mu_{x}=dd^{c}(\log\operatorname{\text{\it dist}}_{x}^{2})$,
and $\operatorname{\text{\it dist}}_{x}^{2}$ is a square of a distance from
$x$. The current $\mu_{x}$ can be approximated by smooth, closed, positive
currents using a regularized maximum function (see Subsection 3.2), and this
allows one to define the product $\Theta\wedge\mu_{x}^{n-p}$ as a limit of
closed, positive currents with bounded mass, well defined because of a weak
compactness principle.
For a positive number $c>0$, the Lelong set $F_{c}$ of a (1,1)-current $\eta$
is a set of all points $x\in M$ with $\nu_{x}(\eta)\geqslant c$. By Siu’s
theorem ([Si]), a Lelong set of a positive, closed current is complex
analytic.
The following theorem was proven in [V5], using an advanced version of Nadel’s
vanishing, due to [DPS2].
Theorem 1.23: ([V5, Theorem 4.1]) Let $L$ be a parabolic nef bundle on a
hyperkähler manifold, and $\eta$ a positive closed current, representing
$c_{1}(L)$. Assume that all Lelong numbers of $\eta$ vanish. Then $L$ is
${\mathbb{Q}}$-effective.
In the present paper, we show that all Lelong sets of a parabolic nef current
on a hyperkähler manifold are coisotropic with respect to its holomorphic
symplectic form (3.4).
Comparing 3.4 and 1.4, we obtain the following. Let $L$ be a parabolic nef
bundle on a hyperkähler manifold $M$. Then either $L$ is
${\mathbb{Q}}$-effective, or $M$ has non-trivial coisotropic subvarieties. A
similar result was proven in by Campana-Oguiso-Peternell, who have shown that
such a manifold always contains a subvariety of dimension $\geqslant 2$ ([COP,
Theorem 6.2]).
For a generic hyperkähler manifold, all complex subvarieties are
holomorphically symplectic ([V1], [V2]). Therefore, such a manifold does not
have any coisotropic subvarieties (2.2). This implies that all Lelong numbers
of a parabolic nef current on a generic hyperkähler manifold vanish (3.4).
## 2 Hyperkähler geometry: preliminary results
### 2.1 The structure of a Kähler cone
Definition 2.1: A class $\eta\in H^{1,1}(M)$ is called pseudoeffective if it
can be represented by a positive current, and nef if it lies in a closure of a
Kähler cone.
The following useful theorem, due to S. Boucksom, is known as the divisorial
Zariski decomposition theorem.
Theorem 2.2: ([Bou]) Let $M$ be a hyperkähler manifold. Then every
pseudoeffective class can be decomposed as a sum
$\eta=\nu+\sum_{i}a_{i}E_{i},$
where $\nu$ is nef, $a_{i}$ positive numbers, and $E_{i}$ exceptional divisors
satisfying $q(E_{i},E_{i})<0$.
Remark 2.3: Let $M_{1},M_{2}$ be holomorphic symplectic manifolds,
bimeromorphically equivalent. Then $H^{2}(M_{1})$ is naturally isomorphic to
$H^{2}(M_{2})$, and this isomorphism is compatible with Bogomolov-Beauville-
Fujiki form. Indeed, the manifolds $M_{i}$ have trivial canonical bundle,
hence a bimeromorphic equivalence is non-singular in codimension 1.
Definition 2.4: A modified nef cone (also “birational nef cone” and “movable
nef cone”) is a closure of a union of all nef cones for all bimeromorphic
models of a holomorphically symplectic manifold $M$.
Theorem 2.5: ([Bou], [Hu2]). On a hyperkähler manifold, the modified nef cone
is dual to the pseudoeffective cone under the Bogomolov-Beauville-Fujiki
pairing.
Corollary 2.6: Let $M$ be a simple hyperkähler manifold such that all integer
$(1,1)$-classes satisfy $q(\nu,\nu)\geqslant 0$. Then its Kähler cone is one
of two components $K_{+}$ of a set $K:=\\{\nu\in H^{1,1}(M,{\mathbb{R}})\ \ |\
\ q(\nu,\nu)>0\\}$.
Proof: The pseudoeffective cone $K_{ps}$ of $M$ is equal to the nef cone
$K_{n}$ by the divisorial Zariski decomposition. A square of a Kähler form is
positive, hence $K_{n}=K_{ps}$ is contained in one of components of $K$,
denoted by $K_{+}$. This gives inclusions
$K_{ps}=K_{n}\subset K_{mn}\subset K_{+}$ (2.1)
Since $K_{+}$ is self-dual, dualising (2.1) gives
$K_{+}\subset K_{ps}\subset K_{mn}=K_{n}^{*}$ (2.2)
However, all elements of $K_{mn}$ satisfy $q(\eta,\eta)\geqslant 0$, hence
$K_{mn}\subset K_{+}$. Then (2.2) gives
$K_{+}\subset K_{ps}\subset K_{mn}=K_{n}^{*}\subset K_{+},$
and all these cones are equal.
Remark 2.7: From the Hodge index theorem, it follows immediately that the
condition
$\forall\eta\in\operatorname{Pic}(M)\ \ q(\eta,\eta)\geqslant 0$
implies that $\operatorname{Pic}(M)$ has rank 1.
Remark 2.8: From 2.1 it follows that on a hyperkähler manifold with
$\operatorname{Pic}(M)={\mathbb{Z}}$, for any rational class $\eta\in
H^{1,1}(M)$ with $q(\eta,\eta)\geqslant 0$ either $\eta$ or $-\eta$ is nef.
### 2.2 Subvarieties in generic hyperkähler manifolds
This is a brief introduction to the theory of subvarieties in generic
hyperkähler manifolds. For more details and missing reference, please see [V2]
and [V3].
Recall now that any Kähler manifold with trivial canonical class admits a
unique Ricci-flat Kähler metric in a given Kähler class ([Y]). Using Bochner’s
vanishing, it is possible to show that any holomorphic form on a compact
Ricci-flat manifold is parallel with respect to the Levi-Civita connection.
If the manifold $M$ is holomorphically symplectic, a Ricci-flat metric
together with the holomorphic symplectic form can be used to construct a
triple of complex structures $(I,J,K)$ satisfying quaternionic relations
$I\circ J=-J\circ I=K$, and parallel with respect to the Levi-Civita
connection. In differential geometry and physics, hyperkähler manifolds are
usually defined in terms of this quaternionic structure ([Bes]).
Consider an operator $L=aI+bJ+cK$, with $a,b,c\in{\mathbb{R}}$ satisfying
$a^{2}+b^{2}+c^{2}=1$. Since $I,J,K$ are parallel with respect to the Levi-
Civita connection, $L$ is also parallel. Using the quaternionic relations, we
obtain $L^{2}=-1$. Since $L$ is parallel, it is an integrable complex
structure. Such a complex structure is called induced by the quaternionic
action. The set of induced complex structures is parametrized by the
2-dimensional sphere $S^{2}$. It is easy to check that this gives a
holomorphic family of complex structures on $M$ over ${\mathbb{C}}P^{1}$. The
total space of this family is called the twistor space of $M$. Denote the base
of the twistor family by $C$, $C\cong{\mathbb{C}}P^{1}$.
The group $SU(2)$ of unitary quaternions acts on $TM$. We extend this action
to the bundle $\Lambda^{*}M$ of differential forms by multiplicativity. This
action is parallel, hence it commutes with the Laplacian. This gives a natural
$SU(2)$-action on $H^{*}(M)$, analogous to the Hodge decomposition in Kähler
geometry.
Given a class $v\in H^{2p}(M)$ which is not $SU(2)$-invariant, let
$S_{v}\subset C$ be the set of all induced complex structures $L\in C$ for
which $v\in H^{p,p}(M)$. For an $SU(2)$-class, we set $S_{v}=\emptyset$. Since
the Hodge decomposition on $(M,L)$ is induced by the $SU(2)$-action, $S_{v}$
can be expressed through the action of $SU(2)$. Then it is easy to check that
$S_{v}$ is finite, for all $v$.
The union $R:=\bigcup_{v\in H^{*}(M,{\mathbb{Z}})}S_{v}$ is countable.
Clearly, for any induced complex structure $L\notin R$,
$v\in H^{p,p}(M)\cap H^{2p}(M,{\mathbb{Z}})\Rightarrow\text{$v$ is
$SU(2)$-invariant}.$
Definition 2.9: An induced complex structure $L$ is called generic if $L\notin
R$.
As shown in [V1], a closed complex subvariety $X\subset M$ with fundamental
class $[X]\in H^{2p}(M)$ $SU(2)$-invariant is necessarily holomorphically
symplectic outside of its singularities.
Theorem 2.10: ([V1]) Let $(M,I,J,K)$ be a hyperkähler manifold equipped with
a quaternionic structure, and $L$ a generic induced complex structure. Then
all complex subvarietis $X\subset(M,L)$ are holomorphically symplectic outside
of singularities.
Remark 2.11: In [V3] it was also shown that a normalization of $X$ is smooth
and holomorphically symplectic.
Definition 2.12: A hyperkähler manifold $(M,I)$ is generic if $I$ is generic
for some quaternionic structure constructed as above.
Remark 2.13: Let $M$ be a generic hyperkähler manifold. Then all complex
subvarieties of $M$ are holomorphically symplectic, by 2.2. In particular, $M$
has no divisors.
### 2.3 Cohomology of hyperkähler manifolds
In the sequel, some basic results about cohomology of hyperkähler manifolds
will be used. The following theorem was proving in [V4], using representation
theory.
Theorem 2.14: ([V4]) Let $M$ be a simple hyperkähler manifold, and
$H^{*}_{r}(M)$ the part of cohomology generated by $H^{2}(M)$. Then
$H^{*}_{r}(M)$ is isomorphic to the symmetric algebra (up to the middle
degree). Moreover, the Poincare pairing on $H^{*}_{r}(M)$ is non-degenerate.
This brings the following corollary.
Corollary 2.15: Let $\eta_{1},...\eta_{n+1}\in H^{2}(M)$ be cohomology
classes on a simple hyperkähler manifold, $\dim_{\mathbb{C}}M=2n$. Suppose
that $q(\eta_{i},\eta_{j})=0$ for all $i,j$. Then
$\eta_{1}\wedge\eta_{2}\wedge...\wedge\eta_{n+1}=0$.
Proof: Let $H:=\eta_{1}\wedge\eta_{2}\wedge...\wedge\eta_{n+1}$. From the
Fujiki’s formula (1.2) it follows directly that
$H\wedge\rho_{1}\wedge...\wedge\rho_{n-1}=0,$
for any cohomology classes $\rho_{1},...,\rho_{n-1}\in H^{2}(M)$. Therefore,
for any $v\in H^{2n-2}_{r}(M)$, $H\wedge v=0$. Since the Poincare form is non-
degenerate on $H^{2n-2}_{r}(M)$ (2.3), this implies that $H=0$.
## 3 Cohomology classes dominated by a nef class
### 3.1 Positive forms and positive currents
In this Subsection, we recall standard notions of positivity for $(p,p)$-forms
and currents. A reader may consult [D5] for more details.
Recall that a real $(p,p)$-form $\eta$ on a complex manifold is called weakly
positive if for any complex subspace $V\subset TM$, $\dim_{\mathbb{C}}V=p$,
the restriction $\rho{\left|{}_{{\phantom{|}\\!\\!}_{V}}\right.}$ is a non-
negative volume form. Equivalently, this means that
$(\sqrt{-1}\>)^{p}\rho(x_{1},\overline{x}_{1},x_{2},\overline{x}_{2},...,x_{p},\overline{x}_{p})\geqslant
0,$
for any vectors $x_{1},...x_{p}\in T_{x}^{1,0}M$. A form is called strongly
positive if it can be expressed as a sum
$\eta=(-\sqrt{-1}\>)^{p}\sum_{i_{1},...i_{p}}\alpha_{i_{1},...i_{p}}\xi_{i_{1}}\wedge\overline{\xi}_{i_{1}}\wedge...\wedge\xi_{i_{p}}\wedge\overline{\xi}_{i_{p}},\
\ $
running over some set of $p$-tuples
$\xi_{i_{1}},\xi_{i_{2}},...,\xi_{i_{p}}\in\Lambda^{1,0}(M)$, with
$\alpha_{i_{1},...,i_{p}}$ real and non-negative functions on $M$.
The strongly positive and the weakly positive forms form closed, convex cones
in the space $\Lambda^{p,p}(M,{\mathbb{R}})$ of real $(p,p)$-forms. These two
cones are dual with respect to the Poincare pairing
$\Lambda^{p,p}(M,{\mathbb{R}})\times\Lambda^{n-p,n-p}(M,{\mathbb{R}}){\>\longrightarrow\>}\Lambda^{n,n}(M,{\mathbb{R}})$
For (1,1)-forms and $(n-1,n-1)$-forms, the strong positivity is equivalent to
weak positivity.
Remark 3.1: A strongly positive form is a linear combination of products
$\alpha(\sqrt{-1}\>)^{p}z_{i_{1}}\wedge\overline{z}_{i_{1}}\wedge
z_{i_{2}}\wedge\overline{z}_{i_{2}}\wedge z_{i_{k}}\wedge\overline{z}_{i_{k}}$
where $\alpha$ is a smooth, positive function, and
$z_{1},...,z_{n}\in\Lambda^{1,0}(M)$ is a basis in $(0,1)$ forms. In the
sequel, we shall abbreviate such a form as $\alpha(z\wedge\overline{z})_{I}$,
where $I=(i_{1},...,i_{k})$ is a multiindex.
A current is a form taking values in distributions. The space of
$(p,q)$-currents on $M$ is denoted by $D^{p,q}(M)$. A strongly positive
current111In the present paper, we shall often omit “strongly”, because we are
only interested in strong positivity. is a linear combination
$\sum_{I}\alpha_{I}(z\wedge\overline{z})_{I}$
where $\alpha_{I}$ are positive, measurable functions, and the sum is taken
over all multi-indices $I$. An integration current of a closed complex
subvariety is a strongly positive current.
Notice that “strongly positive” should not be confused with “strictly
positive” (the latter means that a class belongs to the inner part of a
positive cone). For instance, 0 is a strongly positive current.
Positivity of a current $\nu$ is often expressed as $\nu\geqslant 0$. If
$\nu_{1}-\nu_{2}$ is positive, one often writes $\nu_{1}\geqslant\nu_{2}$.
It is easy to define the de Rham differential on currents, and check that its
cohomology coinside with the de Rham cohomology of a manifold.
A mass of a positive $(p,p)$-current $\rho$ on a compact $n$-dimensional
Kähler manifold $(M,\omega)$ is a number $\int_{M}\rho\wedge\omega^{n-p}$.
This number is non-negative, and never vanishes, unless $\rho=0$.
Claim 3.2: (“weak compactness of positive currents”) Let $\\{\eta_{i}\\}$ be
a sequence of positive $(p,p)$-currents with bounded mass. Then
$\\{\eta_{i}\\}$ has a subsequence converging to a positive current, in weak
topology.
The de Rham differential is by definition continuous in the topology of
currents, and the projection from closed currents to the de Rham cohomology
also continuous. Then, weak compactness implies the following useful result.
Corollary 3.3: Let $\eta_{i}\in H^{p,p}(M)$ be a sequence of cohomology
classes represented by closed, positive currents, and $\eta$ its limit. Then
$\eta$ also can be represented by a closed, positive current.
Definition 3.4: A nef current is a positive, closed current, obtained as a
weak limit of strongly positive, closed forms.
Definition 3.5: Let $\eta$, $\eta^{\prime}$ be nef currents. Choose sequences
$\\{\eta_{i}\\}$, $\\{\eta^{\prime}_{i}\\}$ of closed, strongly positive forms
converging to $\eta$, $\eta^{\prime}$. Then
$\\{\eta_{i}\wedge\eta^{\prime}_{i}\\}$ is a bounded sequence of closed,
strongly positive forms. From weak compactness it follows that
$\\{\eta_{i}\wedge\eta^{\prime}_{i}\\}$ has a limit. We define a product
$\eta\wedge\eta^{\prime}$ of nef currents as a form which can be obtained as a
limit of $\\{\eta_{i}\wedge\eta^{\prime}_{i}\\}$, for some choice of sequences
$\\{\eta_{i}\\}$, $\\{\eta^{\prime}_{i}\\}$. The limit
$\\{\eta_{i}\wedge\eta^{\prime}_{i}\\}$ is non-unique (see the example below).
However, it is a closed, positive current, which represents the product of the
corresponding cohomology classes.
Example 3.6: Let $M={\mathbb{C}}P^{2}$. Given a hyperplane $H$, we choose a
sequence of positive, closed (1,1)-forms $\eta_{i}(H)$ converging to a current
of integration $[H]$ of $H$. Suppose that the absolute value of $\eta_{i}(H)$
is bounded everywhere by $C_{i}$, and the mass of $[H]-\eta_{i}(H)$ is bounded
by $\varepsilon_{i}$. Let $\alpha$ be a positive (1,1)-current. Then the mass
of $([H]-\eta_{i}(H))\wedge\alpha$ is bounded by
$\varepsilon_{i}\sup|\alpha|$:
$\int_{{\mathbb{C}}P^{2}}\bigg{|}([H]-\eta_{i}(H))\wedge\alpha\bigg{|}\leqslant\varepsilon_{i}\sup|\alpha|$
(3.1)
Let now $H,H^{\prime}$ be two distinct hyperplanes, and $\eta_{i}(H)$,
$\eta_{i}(H^{\prime})$ the sequences of positive, closed forms approximating
$H,H^{\prime}$ as above. Then (3.1) implies that
$\int_{{\mathbb{C}}P^{2}}\bigg{|}([H]-\eta_{i}(H))\wedge\eta_{j}(H^{\prime})\bigg{|}\leqslant\varepsilon_{i}C_{j}.$
(3.2)
Choosing a sequence $i_{k},j_{k}$ in such a way that
$\lim\limits_{k\rightarrow\infty}\varepsilon_{i_{k}}C_{j_{k}}=0$, and applying
(3.2), we obtain that the sequence
$\eta_{i_{k}}(H)\wedge\eta_{j_{k}}(H^{\prime})$ has the same limit as
$\lim[H]\wedge\eta_{j}(H^{\prime})=[p]$, where $p=H\cap H^{\prime}$ is a point
where $H$ and $H^{\prime}$ intersect. Given a sequence $H_{l}$ of planes
converging to $H$, with $H_{l}\cap H=p$, and applying the same argument, we
obtain a sequence $\eta_{i_{k}}(H)\wedge\eta_{j_{k}}(H_{k})$ converging to
$[p]$. However, $\eta_{j_{k}}(H_{k})$, for appropriate choice of an
approximating sequence, clearly converges to $H$. This gives a sequence of
closed, positive forms $\eta_{i},\eta^{\prime}_{i}$ converging to $[H]$, and
the product $\eta_{i}\wedge\eta^{\prime}_{i}$ converges to the current of
integration $[p]$, associated with an arbitrary point $p\in H$.
### 3.2 Regularization for nef currents
In [D1], the notion of a regularized maximum of two functions was defined.
Choose $\varepsilon>0$, and let ${\rm
max}_{\varepsilon}:\;{\mathbb{R}}^{2}{\>\longrightarrow\>}{\mathbb{R}}$ be a
smooth, convex function which is monotonous in both arguments and satisfies
${\rm max}_{\varepsilon}(x,y)={\rm max}(x,y)$ whenever $|x-y|>\varepsilon$.
Then ${\rm max}_{\varepsilon}$ is called a regularized maximum. It is easy to
show ([D1]) that a regularized maximum of two strictly plurisubharmonic
functions is again strictly plurisubharmonic. Moreover, for any smooth form
$A$ and $L^{1}$-functions $x,y$ which satisfy $A+dd^{c}x\geqslant 0$ and
$A+dd^{c}y\geqslant 0$, one would have $A+dd^{c}{\rm
max}_{\varepsilon}(x,y)\geqslant 0$.
Recall that an almost plurisubharmonic function is a generalized function $f$
which satisfies $dd^{c}f+A\geqslant 0$ for some smooth (1,1)-form $A$.
Clearly, almost plurisubharmonic functions are locally integrable.
The Demailly’s Regularization Theorem ([D3], Theorem 1.1, [D5], 21.3) implies
that any positive, closed (1,1)-current $T$ on a Kähler manifold $(M,\omega)$
can be weakly approximated by a sequence $T_{k}$ of closed, real
$(1,1)$-currents in the same cohomology class satisfying the following
assumptions
(i)
$T_{k}+\delta_{k}\omega\geqslant 0$, where $\\{\delta_{k}\\}$ is a sequence of
real numbers converging to 0.
(ii)
$T_{k}$ are smooth outside of a complex analytic subset $Z_{k}\subset M$, with
$Z_{1}\subset Z_{2}\subset...$
(iii)
Let $T_{0}$ be a smooth form cohomologous to $T$. Then
$T_{k}=T_{0}+dd^{c}\psi_{k}$, where $\psi_{k}$ is a non-increasing sequence of
almost plurisubharmonic functions converging to an almost plurisubharmonic
$\psi$, which satisfies $dd^{c}\psi+T_{0}=T$.
(iv)
Locally around $Z_{k}$, the functions $\psi_{k}$ have logarithmic poles,
namely
$\psi_{k}=\lambda_{k}\log\sum|g_{k,l}|^{2}+\tau_{k},$
where $g_{k,l}$ are holomorphic functions vanishing on $Z_{k}$, and $\tau_{k}$
is smooth.
(v)
The Lelong numbers $\nu(T_{k},x)$ of $T_{k}$ are non-decreasing in $k$ for any
$x\in M$ and converge to $\nu(T,x)$.
Claim 3.7: Let $T=\eta$ be a nef $(1,1)$-current. Then the corresponding
approximation currents $T_{k}+\delta_{k}\omega$ of the Demailly’s
regularization procedure can be also chosen nef.
Proof: Let $T_{0}$ be a smooth, closed form cohomologous to $\eta$. Then
$\eta=T_{0}+dd^{c}\psi$, where $\psi=\lim_{\downarrow}\psi_{k}$. Let $\nu_{i}$
be a sequence of smooth functions such that the form
$T_{0}+dd^{c}\nu_{i}+\varepsilon_{i}\omega$ is positive, closed, and weakly
converges to $\eta=T_{0}+dd^{c}\psi$, for $\varepsilon_{i}$ a sequence of real
numbers converging to 0. Such $\\{\nu_{i}\\}$ exists, because $\eta$ is nef.
Indeed, there exists a sequence of smooth, positive forms $\eta_{i}$
converging to $\eta$, with the cohomology class
$[\eta_{i}]=[\eta]+[\alpha_{i}]$, where $[\alpha_{i}]\in H^{1,1}(M)$
converging to 0. Choose smooth, closed representatives $\alpha_{i}$ with
$\lim_{i}(\sup|\alpha_{i}|)=0$, and set $\varepsilon_{i}=\sup|\alpha_{i}|$.
Then $\varepsilon_{i}\omega_{i}+\alpha_{i}$ is positive. Choose now $\nu_{i}$
in such a way that $\eta_{i}=dd^{c}\nu_{i}+\alpha_{i}+T_{0}$. Then
$d^{c}\nu_{i}+T_{0}+\varepsilon_{i}\omega>dd^{c}\nu_{i}+\alpha_{i}+T_{0}$,
hence positive.
Adding constant terms if necessary, we may assume that $\lim\nu_{i}=\psi$. Fix
$k\in{\mathbb{Z}}^{>0}$. The function $\mu_{i}(k):={\rm
max}_{\varepsilon}(\nu_{i},\psi_{k})$ is smooth, because $\psi_{k}$ is smooth
outside of its poles. The limit $\lim\limits_{i\rightarrow\infty}\mu_{i}(k)$
is equals to ${\rm max}_{\varepsilon}(\psi,\psi_{k})=\psi_{k}$ (the last
equation holds because $\psi\leqslant\psi_{k}$). Therefore, $\mu_{i}(k)$
converges to $\psi_{k}$. On the other hand,
$T_{0}+dd^{c}\nu_{i}+\varepsilon_{i}\omega$ is positive, and
$T_{0}+dd^{c}\psi_{k}+\delta_{k}\omega$ is positive by approximation property.
From the properties of a regularized maximum it follows that
$T_{0}+(\delta_{k}+\varepsilon_{k})\omega+dd^{c}\mu_{i}(k)$ is also positive.
We proved that the current
$T_{k}+(\delta_{k}+\varepsilon_{k})\omega=T_{0}+dd^{c}\psi_{k}+(\delta_{k}+\varepsilon_{k})\omega$
is nef.
### 3.3 Cohomology classes dominated by a nef current
Definition 3.8: Let $M$ be a compact Kähler manifold, $\eta$ a nef current,
obtained as a limit of positive, closed forms $\eta_{i}$, and $\eta^{p}$ a
limit of $\eta_{i}^{p}$, which exists by weak compactness. A current $\nu$ is
called dominated by $\eta$ if $C\eta^{p}+\nu$ and $C\eta^{p}-\nu$ are strongly
positive, for a sufficiently big $C>0$.
For an example of a current dominated by a nef current $\eta$, we look at the
Lelong sets of $\eta$. From Demailly’s regularization and the Siu’s
decomposition theorem ([Si], [D5]), the following result can be easily
deduced.
Theorem 3.9: Let $\eta$ be a nef current, and $Z$ a $p$-dimensional
irreducible component of its Lelong set $F_{c}$. Denote by $[Z]$ its
integration current. Then $[Z]$ is dominated by $\eta$, and moreover,
$\eta^{p}-c^{p}[Z]$ is positive.
Proof: By Siu’s theorem ([Si], [D5, 2.10]), it follows immediately that the
Lelong sets of $\eta^{p}$ have dimension $\geqslant p$. By Siu’s decomposition
formula ([D5, 2.18]), $\eta^{p}$ can be written as
$\eta^{p}=\sum_{i}c_{i}[Z_{i}]+R,$
where $R$ is a positive, closed current, $Z_{i}$ are all $p$-dimensional
components of the Lelong set of $\Theta$, and $c_{i}=\nu_{x}(\Theta)$ for a
generic point $x\in Z_{i}$. Therefore, to prove 3.3, it suffices to show that
$\nu_{x}(\eta^{p})\geqslant\nu_{x}(\eta)^{p}$ (3.3)
at a generic point of $Z$, where $Z$ is an irreducible $p$-dimensional
component of the Lelong set of $\eta$.
Using the regularization theorem, 3.2 and semicontinuity of Lelong numbers, we
find that it suffices to prove inequality (3.3) for the nef currents with
logarithmic singularities approximating $\eta$. Therefore, we may assume that
the singularities of $\eta$ are logarithmic. Since the equality (3.3) is
local, we can also assume that $\eta=dd^{c}\varphi$, for some plurisubharmonic
function $\varphi$ with logarithmic singularities.
Locally around a generic point of $Z$, we have $F_{c}(\eta)=Z$. Therefore
$\eta$ must have a logarithmic pole of order $\geqslant c$ at $Z$. Splitting
the poles onto a part corresponding to $Z$ and the rest, we can write $\eta$
as $\eta=dd^{c}\varphi$, where $\varphi=u+v+A$, $u=c\log\sum|g_{i}|^{2}$,
$v=c^{\prime}\log\sum|f_{i}|^{2}$, with $\\{f_{i}\\}$ a finite set of
holomorphic functions, $g_{i}$ generators of the ideal of $Z$, $c^{\prime}<c$,
and $A$ smooth. Consider a sequence $\nu_{i}$ of smooth plurisubharmonic
functions converging to $\varphi$ (such a sequence exists, because $\eta$ is
nef). Then $\mu_{i}:=u+{\rm max}_{\varepsilon}(v,\nu_{i}-C_{i})+A$ converges
to $\varphi$, for an appropriate choice of a sequence $C_{i}\gg 0$. Moreover,
the limit $\lim(dd^{c}\mu_{i})^{p}$ is by construction equal to
$\lim(dd^{c}\nu_{i})^{p}$, hence we may assume that
$\lim(dd^{c}\mu_{i})^{p}=\eta^{p}$. Clearly,
$\nu_{x}((dd^{c}\mu_{i})^{p})=c^{p}$. Then
$\nu_{x}(\eta^{p})=\nu_{x}((dd^{c}\varphi)^{p})\geqslant\nu_{x}((dd^{c}\mu_{i})^{p})=\nu_{x}((dd^{c}u)^{p})=c^{p}.$
by semicontinuity. We proved (3.3).
### 3.4 $\eta$-coisotropic subvarieties and cohomology classes
Definition 3.10: Let $M$ be a hyperkähler manifold, $[\eta]\in H^{1,1}(M)$ a
parabolic nef class on $M$, and $\eta$ a nef current representing $[\eta]$. We
say that a subvariety $Z\subset M$ is $[\eta]$-coisotropic if $\eta$ dominates
the current of integration $[Z]$.
Definition 3.11: Let $(M,\Omega)$ be a a holomorphically symplectic manifold,
$\dim_{\mathbb{C}}Z=2n$, and $Z\subset M$ a complex subvariety of codimension
$p\leqslant n$. Then $Z$ is called coisotropic if the restriction
$\Omega^{n-p+1}{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}$ vanishes on all
smooth points of $Z$.
Remark 3.12: This is equivalent to $\Omega$ having rank $\leqslant n-p$ on
$TZ$ in the smooth points of $Z$, which is the minimal possible rank for a
$2n-p$-dimensional subspace in a $2n$-dimensional symplectic space.
Proposition 3.13: Let $M$ be a hyperkähler manifold, $[\eta]\in H^{1,1}(M)$ a
parabolic nef class on $M$, and $Z\subset M$ an $[\eta]$-coisotropic
subvariety of complex codimension $p$. Then
(i)
$p\leqslant n$,
(ii)
$Z$ is coisotropic with respect to a holomorphic symplectic form on $M$, and
(iii)
$[\eta]^{n-p+1}{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}=0$.
Proof: Since $[\eta]$ is nef, we may chose a representative nef current
$\eta$, which is a limit of positive, closed forms $\\{\eta_{i}\\}$. Choose
this sequence in such a way that $\eta_{i}^{k}$ converges for all $k>0$, and
denote the respective limits by $\eta^{k}$.
The current $\eta^{n+1}$ is by definition positive, and cohomologous to 0,
because $[\eta]^{n+1}=0$ (2.3). The domination of $Z$ by $\eta$ means that
$\eta^{p}-c[Z]$ is strongly positive, for some $c>0$. Since $\eta^{n+1}=0$,
$\eta^{p}-c[Z]\geqslant 0$ implies that
$0=\eta^{n+1}=\eta^{p}\wedge\eta^{n-p+1}\geqslant[Z]\wedge\eta^{n-p+1}$ (3.4)
Choosing a subsequence in $\eta_{i}$ if necessary, we may assume that the
restriction $\eta_{i}^{n-p+1}{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}$
converges to a positive current. Then (3.4) gives that
$\eta_{i}^{n-p+1}{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}=[Z]\wedge\eta^{n-p+1}$
vanishes everywhere. This proves 3.4 (i) and (iii).
Let $\Omega$ be a holomorphic symplectic form on $M$. It is easy to check that
$\Omega^{i}\wedge\overline{\Omega}^{i}$ is weakly positive. A product of a
strongly positive current and a weakly positive form is weakly positive, hence
the product $\eta^{p}\wedge\Omega^{n-p+1}\wedge\overline{\Omega}^{n-p+1}$ is
positive. However, this product is cohomologous to 0, as follows from 2.3, and
therefore
$\eta^{p}\wedge\Omega^{n-p+1}\wedge\overline{\Omega}^{n-p+1}=0$
Using the same argument as above, we obtain
$0=\eta^{p}\wedge\Omega^{n-p+1}\wedge\overline{\Omega}^{n-p+1}\geqslant[Z]\wedge\Omega^{n-p+1}\wedge\overline{\Omega}^{n-p+1},$
(3.5)
hence $\Omega^{n-p+1}\wedge\overline{\Omega}^{n-p+1}$ vanishes on $Z$. Using
3.4, we obtain that this is equivalent to $Z$ being coisotropic. We proved 3.4
(ii).
As follows from 2.2, on a generic hyperkähler manifold $M$, all complex
subvarieties are holomorphically symplectic. Then $M$ does not have non-
trivial coisotropic subvarieties. This gives
Corollary 3.14: Let $M$ be a generic hyperkähler manifold, and $[\eta]\in
H^{1,1}(M)$ a parabolic nef class, represented by a positive current $\eta$.
Then all Lelong numbers of $\eta$ vanish.
Comparing 3.4, 3.3, and 1.4, we obtain the following.
Corollary 3.15: Let $L$ be a parabolic line bundle on a hyperkähler manifold,
equipped with a singular metric with positive curvature current $\eta$, which
is nef, and $Z$ a component of its Lelong set. Then $Z$ is $\eta$-coisotropic.
In particular, $\dim Z\geqslant\frac{1}{2}\dim M$, and $Z$ is coisotropic with
respect to the standard holomorphic symplectic structure on $M$. Moreover,
either $c_{1}(L)$ is represented by a rational divisor, or the Lelong sets of
$L$ are non-empty.
Comparing this with 2.1, we obtain
Corollary 3.16: Let $M$ be a hyperkähler manifold with
$\operatorname{Pic}(M)={\mathbb{Z}}$, and $L$ a line bundle generating
$\operatorname{Pic}(M)$. Assume that $q(L,L)=0$. Then $c_{1}(M)$ can be
represented by a divisor, or $M$ has non-trivial coisotropic subvarieties.
Remark 3.17: Since all divisors are coisotropic, the first alternative in the
Corollary above in fact implies the second one.
Acknowledgements: I am grateful to S. Boucksom, J.-P. Demailly, D. Kaledin, A.
Kuznetsov and M. Paun for many valuable discussions. Many thanks to Tony
Pantev for a useful e-mail exchange. An early version of this paper was used
as a source of a mini-series of lectures at a conference “Holomorphically
symplectic varieties and moduli spaces”, in Lille, June 2-6, 2009. I am
grateful to the organizers for this opportunity and to the participants for
their insight and many useful comments.
## References
* [Bea] Beauville, A. Varietes Kähleriennes dont la première classe de Chern est nulle. J. Diff. Geom. 18, pp. 755-782 (1983).
* [Bes] Besse, A., Einstein Manifolds, Springer-Verlag, New York (1987)
* [Bo1] Bogomolov, F. A., On the decomposition of Kähler manifolds with trivial canonical class, Math. USSR-Sb. 22 (1974), 580-583.
* [Bo2] F. Bogomolov, Hamiltonian Kähler manifolds, Sov. Math. Dokl. 19 (1978), 1462–1465.
* [Bou] Boucksom, S., Higher dimensional Zariski decompositions, Ann. Sci. Ecole Norm. Sup. (4) 37 (2004), no. 1, 45–76, arXiv:math/0204336
* [COP] Frederic Campana, Keiji Oguiso, Thomas Peternell, Non-algebraic hyperkähler manifolds, arXiv:0804.1682, 18 pages.
* [D1] Demailly, J.-P., Estimations $L^{2}$ pour l’opérateur $\overline{\partial}$ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Ecole Norm. Sup. 4e Sér. 15 (1982) 457–511.
* [D2] Demailly, J.-P., Une généralisation du théoréme d’annulation de Kawamata-Viehweg, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), 123–126.
* [D3] Demailly, Jean-Pierre, Regularization of closed positive currents and Intersection Theory, J. Alg. Geom. 1 (1992) 361-409
* [D4] Demailly, Jean-Pierre, Regularization of closed positive currents of type (1,1) by the flow of a Chern connection, Actes du Colloque en l’honneur de P. Dolbeault (Juin 1992), Aspects of Mathematics, Vol. E 26, Vieweg, (1994) 105-126
* [D5] Demailly, Jean-Pierre, Analytic methods in algebraic geometry, Lecture Notes, École d’éé de Math matiques de Grenoble ”Géométrie des variétés projectives complexes : programme du modèle minimal” (June-July 2007)
* [DP] Demailly, J.-P., Paun, M., Numerical characterization of the Kähler cone of a compact Kähler manifold, math.AG/0105176 also in Annals of Mathematics, 159 (2004), 1247-1274.
* [DPS1] Jean-Pierre Demailly, Thomas Peternell, Michael Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geometry 3 (1994) 295-345
* [DPS2] Jean-Pierre Demailly, Thomas Peternell, Michael Schneider, Pseudo-effective line bundles on compact Kähhler manifolds, International Journal of Math. 6 (2001), pp. 689-741.
* [F] Fujiki, A. On the de Rham Cohomology Group of a Compact Kähler Symplectic Manifold, Adv. Stud. Pure Math. 10 (1987), 105-165.
* [G] Mark Gross, The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations, arXiv:0802.3407, 44 pages.
* [HL] R. Harvey, B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47-157.
* [Hu1] Huybrechts, Daniel, Compact hyperkähler manifolds, Calabi-Yau manifolds and related geometries, Universitext, Springer-Verlag, Berlin, 2003, Lectures from the Summer School held in Nordfjordeid, June 2001, pp. 161-225.
* [Hu2] Huybrechts, Daniel, The Kähhler cone of a compact hyperkähler manifold, Math. Ann. 326 (2003), no. 3, 499–513, arXiv:math/9909109.
* [Hw] Jun-Muk Hwang, Base manifolds for fibrations of projective irreducible symplectic manifolds, arXiv:0711.3224, Inventiones mathematicae, vol. 174, issue 3, pp. 625-644.
* [KV] Verbitsky M., Hyperholomorphic sheaves and new examples of hyperkähler manifolds, alg-geom 9712012, a chapter in a book “Hyperkähler manifolds” (joint with D. Kaledin), Int’l Press, Boston, 2001.
* [KZ] Maxim Kontsevich, Yan Soibelman, Homological mirror symmetry and torus fibrations, arXiv:math/0011041, Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publishing, River Edge, NJ, 2001, pp. 203-263.
* [McL] McLean, R.C. Deformations of calibrated submanifolds, Comm. Anal. Geom. 6, (1998), 705-747.
* [Ma1] D. Matsushita, On fibre space structures of a projective irreducible symplectic manifold, alg-geom/9709033, math.AG/9903045, also in Topology 38 (1999), No. 1, 79-83. Addendum, Topology 40 (2001), No. 2, 431-432.
* [Ma2] Matsushita, D., Higher direct images of Lagrangian fibrations, Amer. J. Math. 127 (2005), arXiv:math/0010283.
* [N] A. M. Nadel. Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Proc. Nat. Acad. Sci. U.S.A., 86 (1989), 7299–7300 and Annals of Math., 132 (1990), 549–596.
* [Saw] Sawon, J. Abelian fibred holomorphic symplectic manifolds, Turkish Jour. Math. 27 (2003), no. 1, 197-230, math.AG/0404362.
* [Si] Y.T. Siu., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., 27 (1974), 53-156.
* [SYZ] A. Strominger, S.-T. Yau, and E. Zaslow, Mirror Symmetry is T -duality, Nucl. Phys. B479, (1996) 243-259.
* [V1] Verbitsky M., Hyperkähler and holomorphic symplectic geometry I, Journ. of Alg. Geom., 5 no. 3 (1996) pp. 401-415, also in alg-geom/9307009
* [V2] Verbitsky M., Hyperkähler embeddings and holomorphic symplectic geometry II, GAFA 5 no. 1 (1995), 92-104, also in alg-geom/9403006
* [V3] Verbitsky, M., Hypercomplex Varieties, alg-geom/9703016, Comm. Anal. Geom. 7 (1999), no. 2, 355–396.
* [V4] Verbitsky, M., Cohomology of compact hyperkähler manifolds. alg-geom electronic preprint 9501001, 89 pages, LaTeX.
* [V5] Verbitsky, M., Hyperkahler SYZ conjecture and semipositive line bundles, arXiv:0811.0639, 21 pages, to appear in GAFA.
* [Y] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I. Comm. on Pure and Appl. Math. 31, 339-411 (1978).
Misha Verbitsky
Institute of Theoretical and Experimental Physics
B. Cheremushkinskaya, 25, Moscow, 117259, Russia
[email protected]
|
arxiv-papers
| 2009-07-24T05:20:30 |
2024-09-04T02:49:04.140253
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/0907.4217"
}
|
0907.4364
|
# Dynamic Deformation of Uniform Elastic Two-Layer Objects
Miao Song
###### Abstract
This thesis presents a two-layer uniform facet elastic object for real-time
simulation based on physics modeling method. It describes the elastic object
procedural modeling algorithm with particle system from the simplest one-
dimensional object, to more complex two-dimensional and three-dimensional
objects.
The double-layered elastic object consists of inner and outer elastic mass
spring surfaces and compressible internal pressure. The density of the inner
layer can be set different from the density of the outer layer; the motion of
the inner layer can be opposite to the motion of the outer layer. These
special features, which cannot be achieved by a single layered object, result
in improved imitation of a soft body, such as tissue’s liquidity non-uniform
deformation. The construction of the double-layered elastic object is closer
to the real tissue’s physical structure.
The inertial behavior of the elastic object is well illustrated in
environments with gravity and collisions with walls, ceiling, and floor. The
collision detection is defined by elastic collision penalty method and the
motion of the object is guided by the Ordinary Differential Equation
computation.
Users can interact with the modeled objects, deform them, and observe the
response to their action in real time.
## Acknowledgments
This thesis is made possible by these important people in my life:
I would like to thank Dr. Peter Grogono, my supervisor, for his sure guidance,
careful and knowledgeable support, and his patience. I also want to thank
Prof. Jason Lewis for his constant encouragement. Special thanks to Ms.
Catherine LeBel, my immediate superior, and the CSLP (Center for the Study of
Learning and Performance) software development team, who I work with, for
their able support. Many thanks to Serguei Mokhov for his sturdy belief in the
completion of this thesis and for introducing me to many application software
tools. I would also like to thank my cherished little treasure, my daughter
Deschanel, for her faith in me. Finally, I must thank my parents, brother, and
all of my loved ones for their care and support throughout my studies.
###### Contents
1. Acknowledgments
2. 1 Introduction
1. 1.1 Definitions
1. 1.1.1 Deformable Object
2. 1.1.2 Elastic Object
2. 1.2 Animation Techniques
3. 1.3 Elastic Animation
4. 1.4 Applications
5. 1.5 Thesis Goal
6. 1.6 Organization
3. 2 Related Work and Background Material
1. 2.1 Existing Elastic Object Models
2. 2.2 Summary of The Existing Models
4. 3 Procedural Modeling Methodology
1. 3.1 Graphics Objects Modeling Methods
2. 3.2 Procedural Methods
3. 3.3 1D
1. 3.3.1 Geometric Data Type
2. 3.3.2 Modeling Algorithm
4. 3.4 2D
1. 3.4.1 Geometric Data Type
2. 3.4.2 Modeling Algorithm
5. 3.5 3D
1. 3.5.1 Non-Uniform Sphere
1. 3.5.1.1 Geometric Data Type
2. 3.5.1.2 Modeling Algorithm
2. 3.5.2 Uniform Sphere
1. 3.5.2.1 Geometric Data Type
2. 3.5.2.2 Modeling Algorithm
3. 3.5.3 Comparison of Non-uniform and Uniform Methods
5. 4 Physical Based Modeling Methodology
1. 4.1 Gravity Force
2. 4.2 Spring Hooke’s Force
3. 4.3 Spring Damping Force
4. 4.4 Drag Force
5. 4.5 Air Pressure Force
1. 4.5.1 Volume
2. 4.5.2 Normals
1. 4.5.2.1 2D Normals
2. 4.5.2.2 3D Normals
6. 4.6 Collision Force
1. 4.6.1 Collision Detection
2. 4.6.2 Collision Response
7. 4.7 Force Accumulation Algorithm
6. 5 Numerical Integration Methodology
1. 5.1 Differential Equations
1. 5.1.1 Explicit Euler Integrator
2. 5.1.2 Midpoint Integrator
3. 5.1.3 Runge Kutta Fourth Order Integrator
2. 5.2 Newton’s Laws
1. 5.2.1 Newton’s Laws in Euler Integrator
2. 5.2.2 Newton’s Laws in Midpoint Integrator
3. 5.2.3 Newton’s Laws in the Runge Kutta Fourth Order Integrator
3. 5.3 Comparison of Three Integrators
1. 5.3.1 Efficiency
2. 5.3.2 Accuracy
3. 5.3.3 Stability
7. 6 Design and Implementation
1. 6.1 Elastic Object Simulation System Design
1. 6.1.1 Domain Analysis-Based Modeling
2. 6.2 Elastic Object Simulation System Implementation
1. 6.2.1 Design and Implementation of Data Types
2. 6.2.2 Design and Implementation of Components: Model
3. 6.2.3 Design and Implementation of Components: Controller
4. 6.2.4 Simulation Loop Sequence
8. 7 Experimental Results
1. 7.1 Animation Sequence
1. 7.1.1 1D
2. 7.1.2 2D
3. 7.1.3 3D
2. 7.2 Simulation Parameters
1. 7.2.1 Summary of the Adjustable Parameters
2. 7.2.2 Stability vs. Time Step
3. 7.2.3 Efficiency and Accuracy
3. 7.3 Computational Errors
1. 7.3.1 Collision Detection
2. 7.3.2 Subdivision Method
9. 8 Conclusion and Future Work
1. 8.1 Contribution
2. 8.2 Conclusion
3. 8.3 Future Work
###### List of Figures
1. 1 Soft Body Deformation
2. 2 Human Tissue Layers
3. 1 Particle System
4. 2 Mass-spring Model
5. 3 Fluid-Based Soft-Object Model
6. 4 Pressure Soft Body Model
7. 1 One Dimensional Elastic Object
8. 2 Data Structure for One-dimensional Object Representation
9. 3 Two-dimensional Elastic Object with Single Layer
10. 4 Four Types of Springs on a Two-dimensional Object
11. 5 Two-dimensional Elastic Object Facets
12. 6 Data Structure for Two-dimensional Object Representation 1
13. 7 Data Structure for Two-dimensional Object Representation 2
14. 8 Data Structure for Two-dimensional Object Representation 3
15. 9 Polar Cartesian Coordinates Non-uniform Sphere Generation
16. 10 Quadrilaterals and Triangles on Non-uniform Sphere
17. 11 Data Structure for Three-dimensional Non-uniform Object Representation
18. 12 Uniform Sphere Generation
19. 13 Subdivision of A Triangle By Bisecting Sides
20. 14 Data Structure for Three-dimensional Uniform Object Representation without Subdivision
21. 15 Data Structure for Three-dimensional Uniform Object Representation with the Number of Subdivision n=1
22. 1 Double-layered Two-dimensional Elastic Object Filled With Air
23. 2 Particle Inelastic Collision and Impact
24. 1 Elastic Object at Different Time States
25. 2 Euler Integrator
26. 3 Midpoint Integrator
27. 4 Runge Kutta 4th Order Integrator
28. 1 Model-View-Controller
29. 2 Class Diagram
30. 3 Face-Spring-Particle Class Diagram
31. 4 Special 3D Uniform Modeling Face Constructor
32. 5 Model Object Class Diagram
33. 6 $Idle()$ Model Updates
34. 7 General $Update()$ Function
35. 8 Integrator Framework Class Diagram
36. 9 General $integrate()$ and $AccumulateForces()$ Functions
37. 10 Simulation Loop Sequence Diagram
38. 1 Animation Sequence of One Dimensional Elastic Object
39. 2 Animation Sequence of Two Dimensional Elastic Object
40. 3 Animation Sequence of Three Dimensional Elastic Object
41. 4 Elastic Object at Timestep = 0.003
42. 5 Elastic Object at Timestep = 0.03
43. 6 Elastic Object at Timestep = 0.3
44. 7 Second Subdivision Iteration
45. 1 Uniform Shape Modeling
46. 2 Non-Uniform Density
47. 3 Liquid Motion and Inertia
## Chapter 1 Introduction
In our real physical world there exist not only rigid bodies but also soft
bodies, such as human and animal’s soft parts and tissue, and other non-living
soft objects, such as cloth, gel, liquid, and gas.
Soft body simulation, which is also known as deformable object simulation, is
a vast research topic and has a long history in computer graphics. It has been
used increasingly nowadays to improve the quality and efficiency in the new
generation of computer graphics for character animation, computer games, and
surgical training. So far, various elastically deformable models have been
developed and used for this purpose.
In this chapter, we will introduce the concepts about deformable as well as
elastic objects. Moreover, we will explain how important this research is and
its present applications.
### 1.1 Definitions
Soft Body DeformationElastic DeformationPlastic DeformationFracture
DeformationSmall Elastic DeformationLarge Elastic DeformationTissue
AnimationFluid Animation
Figure 1: Soft Body Deformation
#### 1.1.1 Deformable Object
In engineering mechanics, “deformable object” refers to an object whose shape
can be changed due to an applied force, such as tensile (pulling), compressive
(pushing), bending, or tearing forces. The deformation can be categorized as
the following, depending on the types of material and the forces applied:
* •
Elastic deformation (small deformation) is reversible. The object shape is
temporarily deformed when tension is applied and it returns to its original
shape when force is removed. An object made of rubber has a large elastic
deformation range; silk cloth material has a moderate elastic deformation
range; crystal has almost no elastic deformation range.
* •
Plastic deformation (moderate deformation) is not reversible. The object shape
is deformed when tension is applied and its shape is partially returned to its
original form when the force is removed. Objects such as silver and gum, which
can be stretched at their original length, cannot completely restore their
original shapes after deformation.
* •
Fracture deformation (large deformation) is not reversible, but is different
from the plastic deformation. The object is permanently deformed when it is
irreversibly bent, torn, or broken apart after the material has reached the
end of the elastic deformation ranges. All materials will experience fracture
deformation when sufficient force is applied.
#### 1.1.2 Elastic Object
Elastic objects belong to a subset of soft body deformable objects. They are
dynamic objects that change shape significantly and keep constant volume in
response to collision. They can be bent, stretched, and squeezed. Moreover,
they restore their previous shape after deformation. Elastic objects can be
divided into two domains:
* •
Large elastic deformation, such as fluid deformation, which focuses on flows
through space. It tracks velocity and material properties at fixed points in
space.
* •
Small elastic deformation, such as tissue deformation, which uses particle
systems to identify chunks of matter and track their position, acceleration,
and velocity.
Within this wide research range of soft body simulation, this work has focused
on small elastic deformable object simulation, such as tissue animation. Even
though there has been many valuable contribution related to this field, there
are still many difficulties in accomplishing to realistic and efficient
deformable simulation.
### 1.2 Animation Techniques
This section introduces some basic concepts related to the elastic simulation,
such as the subject animation method. Animation relies on persistence of
vision and refers to a series motion illusions resulting from the display of
static images in rapid-shown succession. In traditional animation, squash and
stretch are exaggerated for elastic objects. In order to be efficient when
working with many of single frame images (or simply frames), inbetweening and
cel animation [TJ84] have been introduced by Disney for manual traditional
animation.
The rate of the animation refers to how many frames are displayed within a
given amount of time. If the rate is too low, which is lower than the brain
visual retention, the animation becomes jerky because the brain retains the
empty frame from the previous image to the next image.
A frame rate, which is the time between two updates of the display, describes
the update frequency. In computer games, frames are often discussed in terms
of frames per second (fps). The lower bound for smooth animation is between 22
to 30 frames per second.
For many years’ research, computer-animation has been developed dramatically
to replace the amount of manual traditional animation. The techniques of key-
framing, morphing, and motion capture [HO99] have been widely used.
* •
Key-frame animation: is based on manual animation. It is a sequence of images
of the same object with its local transformations, e.g. values for
translation, rotation and scale, computed by interpolating between keyframes.
* •
Morphing: is a method usually used to estimate and generate a sequence of
frames between one object to the other object with same number of vertices.
Morphing is a good animation technique when using skeletal animation would be
too complex, e.g. facial animation.
* •
Motion capture: is the method that attaches sensors on actors bodies and
records the data for their movements and apply these data to a computer
generated characters.
### 1.3 Elastic Animation
There are two different methods about elastic animation modeling, which
depends on the predefined simulation or simulation in real time.
###### Kinematic modeling
predefines the positions and velocities of objects. It does not concern what
causes movement and how things get where they are in the first place and only
deals with the actual movement. For example, given that a ball’s initial speed
is 10 kilometers per hour on a perfect smooth plane, we can use kinematic
method to calculate how far it travels in two hours.
###### Dynamic modeling
also termed as physically based modeling, is the study of masses and forces
that cause the kinematic quantities, such as acceleration, velocity, and
position, to change as time progresses. For example, when we know the ball’s
initial speed, we need to know how far it travels after an external force
dynamically applied to it.
For elastic object movement, the dynamic methods calculate how the soft body
behaves after external force applied dynamically. The animator does not need
to specify the exact path of an object compared to using the kinematic
modeling method. The system predefines the initial condition of the elastic
object, such as position and gravity force. The animation of the object
movement is updated each time step based on the acceleration derived from
Newton’s Laws of motion. The dynamic simulation method is more advanced,
easier to achieve the realistic motion than kinematic method. Therefore, we
will only represent dynamic simulation of elastic object in this thesis.
### 1.4 Applications
Elastic modeling has been developed and used in several fields, including
geometric modeling, computer vision, computational mathematics, physics
engines, bio-mechanics, engineering, character animation, and many other
fields [GM97].
###### Character Animation
There is much advanced animation modeling software, which has advanced
features for modeling, texturing, and lighting. However, for modeling the
simulation of elastic objects, 3D artists have to do it manually, frame-by-
frame because most of the current 3D software does not provide soft object
simulation functionality. Artists have to use not only their drawing skills
and intuition, but also posses some knowledge of physics to make the objects
behave as if they are in the real world or close.
The techniques of the non-physically based modeling of the elastic object
include modeling the group of control points and changing their property
parameters manually frame-by-frame. The virtual objects will not convince
audiences because no natural laws of physics are applied. Moreover, key-frame
animation is an inefficient way to model elastic objects without functionality
provided by software. Hence, most of 3D film animators have to ignore the
movement details of soft objects.
The latest version of the most advanced animation tool, Maya, provides the
Soft Mod Deformer tool, which allows smooth sculpting of a group of objects
[Wag07]. However, users need to have knowledge about how to use this complex
software in order to access this advanced functionality. Moreover, users can
only animate elastic object with Kinematic modeling method by setting values
through the software interface rather than interact with the object in real
time.
###### Computer Games
Compared to the fancy and lifelike character animation widely used in 3D
films, computer games are more concerned about computation efficiency because
users interact with the software in real time. As one might notice, the
majority of computer games do not portray the characters in detail, even with
the mesh and texture modeling. It is not likely that elastic simulation will
be widely introduced to computer games because existing elastic models usually
require expensive calculation and are inconvenient to use in real time
simulation.
###### Surgical Training
Surgeons benefit from the rapid development of computer graphics and hardware
techniques. The computer generated visual virtual environment imitates the
reality of medical operations and organ construction to fulfill the training
purpose. This new application improves surgical outcomes and decreases the
research costs. However, the reality and accuracy of the software always
require high-end knowledge of physics, mathematic and heavy computation. It
makes it difficult for users to interact with virtual objects in real time.
### 1.5 Thesis Goal
The elastic object for dynamic simulation, which has been widely used, is the
one layer elastic surface with different content within. The soft objects can
be squashed and stretched according to external and internal forces applied on
them. Its computation depends on geometric modeling methods and physical
equations. However, this method is inefficient to imitate the behaviors of
real human’s tissue because human’s or animal’s soft body does not consist of
only one layer with either liquid or air inside. Figure 2 from a biological
research group demonstrates the complexity of human tissue [McE05]. A tissue
is composed of epidermis, dermis, fat, fascia, and muscle layer.
Figure 2: Human Tissue Layers
* •
The epidermis is skin’s outermost layer. It is responsible for the skin
coloring because it contains the skin’s pigment.
* •
The dermis, which is the layer that lies below the epidermis, consists
entirely of living cells. It provides the skin elasticity because this layer
is composed of bundles of fibers and small blood vessels.
* •
The fat, the fascia, and the muscle layer are hypodermis layer of skin. It is
an inner layer of and cushion for the skin. This subcutaneous tissue layer
varies throughout the body region and hormonal influence. The fat and muscle
increase the tension of the skin and protect the bones.
Soft tissues are multi-composite layers; therefore, one layer elastic object
is not accurate to model the kind of soft body exemplified by human tissue.
Moreover, it is difficult to represent the object’s inertia caused by the
internal material realistically and its liquidity motion based on the various
material densities.
In this thesis, we investigate a more accurate two-layer elastic object. The
outer layer of the elastic object represents the epidermis and the dermis
layer of a real tissue. The inner layer of this object corresponds to the
hypodermis layers of skin. This two-layer computer generated elastic object
provides more accurate modeling method based on the main feature of human
tissue. Its deformation is based on the realistic physical consistency of
tissues and the laws of established physics. The objective of this new model
is to be convincing and to have distinct realism to the animated scene by
applying proper physics. The program should be easy in implementation,
convenient to re-use, and gives best elastic body behavior at the minimum cost
rather than model the absolute complex object with the exact accurate physical
equations. Users should be able to interact with the soft body in real-time
and the collision detection and response must be handled correctly.
### 1.6 Organization
This chapter starts with the introduction of elastic objects, their
applications, some basic concepts related to physical based deformable
simulation, and the thesis goal. Chapter 2 surveys and analyzes the existing
elastic simulation system and its problems. Chapter 3 describes the modeling
methodology of elastic objects in one-dimension, two-dimension, and three-
dimension. Physically-based modeling and simulation map a natural phenomena to
a computer simulation program. There are two basic processes in this mapping:
mathematical modeling and numerical solution [Lin06]. Chapter 4 introduces
mathematical modeling, which describes natural phenomena by mathematical
equations. Chapter 5 presents the dynamics numerical equation of motion by
using ODE (ordinary differential equation) associated with our elastic
simulation system, and discusses the complexity and improvement of the
different numerical time integrator of Euler, Midpoint, and Runge Kutta 4th
order. Chapter 6 presents the detailed design and implementation of the
simulation system. Chapter 7 shows our experimental results with the animation
sequences of the elastic object simulation and discusses the effects of
changing the simulation parameters. Chapter 8 gives the conclusion of our
current system, summarizes our contributions based on the existing elastic
simulation models, and analyzes the possible development and related work in
the future.
## Chapter 2 Related Work and Background Material
Research about modeling deformable objects in computer graphics field has been
going on for over 40 years and a wide variety techniques have been developed.
In this chapter, we will review the existing geometric approaches for modeling
elastic objects. These models are all based on physical laws. From the early
elastic model, such as particle model, mass-spring model, finite element
model, to recent development such as fluid based model, and pressure model, we
briefly introduce their physically-based modeling methods and compare these
approaches with their advantages and disadvantages.
### 2.1 Existing Elastic Object Models
###### Particle Model
has been used by Reeves [Ree83] and to model the natural phenomena such as
fire, water, liquid as shown in Figure 1. Particles move under the force field
and constraint without interacting with each other.
Figure 1: Particle System
The advantage of this particle model is that the method is easy to implement.
It is the earliest model to imitate the natural phenomena.
The disadvantage is that all the particles are independent and there are no
forces connecting the particles. Therefore, for some phenomena, such as
springs and mass, the method cannot achieve.
###### Deformable Surface
was introduced first time by Terzopoulos et al. [TPBF87], using finite
difference for the integration of energy-based Lagrange equations based on
Hooke’s law.
It was very successful in creating and animating surfaces. However, this
method requires not only the discretization of the surfaces into spline
patches, but also the specification of local connectivity for spring mass
systems. These involve a significant amount of manual preprocessing before the
surface model can be used.
###### Linear Mass Spring System
has been widely used for modeling elastic objects as shown in Figure 2. It is
actually derived from the particle model; however, it simplifies the modeling
of the inter-particle connection by using flexible springs. Three dimensional
systems contain a finite set of masses connected by springs, which are assumed
to obey Hooke’s Law.
Figure 2: Mass-spring Model
This method was first introduced by Terzopoulos [TJF89] to describe melting
effects. Particles, which are connected by springs, have an associated
temperature as one of their properties. The stiffness of the spring is
dependent on the temperature of the linked two particles. Increased
temperature decreases the spring stiffness. When the temperature reaches the
melting point, the stiffness becomes zero.
The advantages of mass-spring model are that it is easy to construct and
display the simulation dynamically.
The disadvantages are that such system restricts to only the objects with
small elastic deformation with approximation of the true physics, not for the
objects that require large elastic deformation, such as fluid. This method
also has difficulties to simulate the separation and fusion of a constant
volume object. Moreover, the spring stiffness is problematic. If the spring is
too weak, for the closed shape model with only simple springs to model the
surface will be very easy to collapse. If we try to avoid the collapse, we
need to model with spring stiffer, and then we will have difficulty to choose
the elasticity because the object is nearly rigid. Another disadvantage is
that the mass spring system has less stability and requires the numerical
integrator to take small time steps [DW92] than FEM model.
###### Finite Element Method
known as FEM Model [GM97], is the most accurate physical model compared to
other models. It treats deformable object as a continuum, which means the
solid bodies with mass and energy distributed all over the object. This
continuum model is derived from equations of continuum mechanics. The whole
model can be considered as the equilibrium of a general object subjected to
external forces. The deformation of the elastic object is a function of these
forces and the material property. The object will stop deformation and reach
the equilibrium state when the potential energy is minimized. The applied
forces must be converted to the associated force vectors and the mass and
stiffness are computed by numerically integrating over the object at each time
step, so the re-evaluation of the object deformation is necessary and requires
heavy pre-processing time [GM97].
The advantage of FEM model is that it gives more realistic deformation result
than mass-spring system because the physics are more accurate.
The disadvantage is that the system lacks efficiency. Because the energy
equation will be used, the FEM is only efficient for the small deformation of
the elastic object, such as application to the plastic material, which has a
small deformation range. Alternatively, the object has less control elements
needed to be computed, as in cloth deformation. If we need to simulate the
human soft body parts or facial animation, the deformation rate is very high.
It will be very difficult and sometimes impossible to carry out the
integration procedure over the entire body. Therefore, it has been limited to
apply in real-time system because of the heavy computational effort (usually
it is done off-line). Moreover, the implementation is complicated.
###### Fluid Based Model
[DL02] consists of two components: an elastic surface and a compressible fluid
as shown in Figure 3. The surface is represented as a mass spring system. The
fluid is modeled using finite difference approximations to the Navier-Stokes
equations of fluid flow. Figure 3 describes how this model simulates the fluid
flows down a sink simulated. The inner layer is modeled by a particle system,
which is similar to real water molecules. Using the numerical methods, the
motion of each particle can be computed. In this example, the motion of the
each particle is at the center of the basin, and points down to the sink.
Figure 3: Fluid-Based Soft-Object Model
The fluid based model uses physically based modeling and it produces realistic
fluid animation. It illustrates the behavior of fluid in environments with
gravity and collisions with planes.
The disadvantage of this model is the heavy computation for both elastic
surface and density inside fluid. It also provides a solution to the constant
volume problem.
###### Pressure Model
was introduced by M. Matyka [Mat03, Mat04b, Mat04b]. It simulates an elastic
deformable object with a internal pressure based on the ideal gas law as shown
in Figure 4. The object volume is calculated approximately by bounding box,
shaped as sphere, cube, or ellipsoid. Another method to determine the object
volume is based on Gauss’s Theorem.
Figure 4: Pressure Soft Body Model
Advantage of this model is that it gives very convincing effects about elastic
property in real time simulation. The object behaves like a balloon filled
only with air.
However, it can not imitate more interesting effects, such as human tissue. It
can not achieve the effect of semi-liquid deformable object because the air
pressure density is uniform inside of the object, which is different from
liquid with non-uniform density. It is not accurate for describing the inertia
of the semi-liquid object.
### 2.2 Summary of The Existing Models
Previous work on deformable object animation uses physically-based methods
with local and global deformations applied directly to the geometric models.
Based on the survey of the existing elastic models, we conclude their usage as
the two types:
* •
Interactive models are used when speed and low latency are most important and
physical accuracy is secondary. Typical examples include mass-spring models
and spline surfaces used as deformable models. These can satisfy the character
animation with exaggerated unrealistic deformation.
* •
Accurate models are chosen when physical accuracy is important in order to
accomplish the surgical training purpose which requires the accurate tissue
modeling. The continuum simulation model, for instance, the most accurate
model, FEM, is not ideal for simulation requiring real time interaction and
the object undergoing large deformation.
In short, elastic object simulation is a dilemma of demanding accuracy and
interactivity.
## Chapter 3 Procedural Modeling Methodology
### 3.1 Graphics Objects Modeling Methods
###### Polygonal Methods
create geometric objects that can be described by their convex planar
polygonal surfaces. These methods are easy to describe the shapes of
mathematical objects rendered on graphics system. However, they are difficult
to describe physical objects, such as cloud and fire, and their complex
behaviors combined with physical laws [Ang03].
###### Procedural Methods
build natural phenomena, 3D models and textures in an algorithmic manner and
generate polygons only during the rendering process. The details of the object
modeling can be controlled upon vary requests. Meanwhile, these methods are
easy to combine computer graphics with physical laws [Ang03, Wik07].
### 3.2 Procedural Methods
We use procedural modeling methods in our elastic object simulation system.
One of the possible approaches to procedural modeling, a particle system, is
designed to model elastic objects as described in this section. This particle
system is also capable of describing the complex behaviors of elastic objects
based on physical laws, such as solving differential equations of thousands of
particles on those elastic objects. Another approach is language-based
procedural method [Ang03], which generates complex objects with simple
programs.
In order to model an elastic object, we need to study the following basic data
structures, which are varied in one-dimensional, two-dimensional, and three-
dimensional modeling methods.
###### Particles
are objects that have mass, position, velocity, and forces applied on them,
but have no spatial extent. Moreover, the differential position and velocity
change are two properties for these computation of each particle.
###### Springs
are massless with natural length not equal to zero. They are the linkage of
particles. There must be at least one spring connects with two particles
paired by modeling algorithm.
###### Faces
are the data type that consists of springs as the edges and particles as the
vertices.
### 3.3 1D
The techniques used in an one-dimensional object are presented here, which are
applied subsequently to models in two and three dimensions. An one-dimensional
object with one end fixed as shown in Figure 1(a). The other end is interacted
by users with mouse as in Figure 1(b). The interacted force is restricted to
one dimension.
(a) The Initial Spring System is at Rest
(b) The Compressed Force is Applied
(c) The Stretched Force is Applied
Figure 1: One Dimensional Elastic Object
#### 3.3.1 Geometric Data Type
###### Particle
There are two mass particles $P_{0}$ and $P_{1}$ on a single spring $SP_{1}$.
###### Spring
In one-dimensional object, only one type of the spring, structural spring, is
introduced. Structural spring, is used to model the object shape, connected by
the two mass particles in this case. In Figure 1(a), the spring is at the
initial state of equilibrium. The natural length of the spring is $l$ and the
force $f$ is zero. In Figure 1(b), the spring is compressed by an external
mouse force. The current spring length $l^{\prime}<l$ and the spring force
restores the elastic object to its equilibrium position $f>0$. When the spring
is stretched out as shown in Figure 1(c), the current spring length
$l^{\prime}>l$ and the force of the spring $f<0$.
#### 3.3.2 Modeling Algorithm
* •
Step1: Create two particles $P_{0}$ and $P_{1}$ with positions $(x_{0},y_{0})$
and $(x_{1},y_{1})$ shown in Figure 2.
* •
Step2: Create a spring $S_{1}$ with these two particles as two ends $Sp_{1}$
and $Sp_{2}$.
Figure 2: Data Structure for One-dimensional Object Representation
### 3.4 2D
In this section, we extend the one-dimensional elastic object to two
dimensions. We create two separated elastic circles, inner circle and outer
circle. Both of them consist of the same modeling structure as one-dimensional
mass particles and springs. Then, the two concentric circles, inner and outer,
are connected by various springs to become one two-layered elastic object.
However, the distinct features in two-dimensional object have more types of
springs presented and the air pressure inside the two-layer close shape is
calculated. The spring surface prevents infinite expansion of the air;
meanwhile the internal pressure avoids the surface collapsing.
#### 3.4.1 Geometric Data Type
Two-dimensional object is made of three types of primitives, mass particles,
springs, and indexed triangular faces.
###### Particles
are defined based on their coordinates related to $x$ and $y$ axes. Consider a
unit circle with twelve particles as an example shown in Figure 3. The spatial
position for each particle $P_{i}$ is $(x_{i},y_{i})$ can be defined by the
two equations:
Figure 3: Two-dimensional Elastic Object with Single Layer
$x(\theta)=\cos(\theta+\Delta\theta)$
$y(\theta)=\sin(\theta+\Delta\theta)$
where
$0^{0}\leq\theta\leq 360^{0}$
$\Delta\theta$ is a small angle stepping along $\theta$
###### Springs
In additional to the structural spring, which also exists in one-dimensional
object, there are two other types of springs, radius spring and shear spring.
(a) Structural Springs
(b) Radius Springs
(c) Shear Left Springs
(d) Shear Right Springs
Figure 4: Four Types of Springs on a Two-dimensional Object
* $\diamond$
Structural springs: give the basic structure of inner circle and outer circle
to prevent neighboring particles from getting too close to one another as
shown in Figure 4(a). Linkage of each structural spring i is to connect with
two particles${P}_{i}$ and ${P}_{i+1}$ or ${P}_{i-1}$ and ${P}_{i}$.
* $\diamond$
Radius springs: are the springs connected from particles on inner circle to
the particle on the outer circle as part of the circle radius in order to
prevent the bending of the surface as shown in Figure 4(b). Linkage of each
radius spring $i$ is to connect with particle ${P}^{inner}_{i}$ and the
particle ${P}^{outer}_{i}$.
* $\diamond$
Shear springs: are springs connected from particles on inner circle to their
diagonal neighbors on outer circle in order to avoid the surface fold over .
Linkage of each left shear spring i is to connect with particle
${P}^{outer}_{i}$ diagonally and the particle ${P}^{inner}_{i-1}$; connect
with particle ${P}^{outer}_{i+1}$ diagonally and the particle
${P}^{inner}_{i}$ and so on as shown in Figure 4(c). Linkage of each right
shear spring i is to connect with particle ${P}^{outer}_{i-1}$ diagonally and
the particle ${P}^{inner}_{i}$; connect with particle ${P}^{outer}_{i}$
diagonally and the particle ${P}^{inner}_{i+1}$ and so on as shown in Figure
4(d).
###### Faces
are the data structure for the only purpose of drawing and displaying a filled
object to a two-dimensional object. The triangular facets can be drawn
separately and visualized as a filled circle as shown in Figure 5.
Figure 5: Two-dimensional Elastic Object Facets
#### 3.4.2 Modeling Algorithm
* •
Step 1: Define the number of particles as $n=12$ in our example. Then, the
step size is $\Delta\theta=\frac{360^{0}}{12}=30$ degrees.
* •
Step 2: Define the group of particles’ position on inner circle and the ones
on outer circle as shown in Figure 6. The first particle $P_{0}$ is at
$(\cos\theta,\sin\theta)$ where $\theta=0^{0}$, the second particle is at
$(\cos\theta,\sin\theta)$ where $\theta=\theta+\Delta\theta=30^{0}$… By
multiplying the inner and outer coordinates with a different radius value, for
example, $Radius_{inner}=1.5$, and $Radius_{outer}=2$ to create two concentric
circles.
* •
Step 3: Add the structural springs $S_{0}$, $S_{1}$, …$S_{11}$ to the inner
circle according to the spring index of inner particles as shown in Figure 6.
The same method is applied to outer structural springs on outer circle. The
last spring, $S_{11}$ in our example, is composed of two particles $P_{11}$
and $P_{0}$ as two ends in order to make a close shape.
Figure 6: Data Structure for Two-dimensional Object Representation 1
* •
Step 4: Loop through the number of structural springs $n=12$ to add the same
number of radius springs according to the linkage of the inner and outer
particles as shown in Figure 7.
Figure 7: Data Structure for Two-dimensional Object Representation 2
* •
Step 5: Loop through the number of structural springs to add the same number
of shear left springs and shear right springs according to the linkage of the
inner and outer particles as shown in Figure 8.
Figure 8: Data Structure for Two-dimensional Object Representation 3
### 3.5 3D
In this section, a more complicated three-dimensional elastic object is
extended from the two-dimensional object. In the two-dimensional model, the
structural springs’ index is the most important key data structure to link up
all the particles and reference about the index of the particles. This spring
linkage method will still work for the model based on the non-uniform sphere
geometric modeling method. However, in the other geometric modeling method,
the uniform sphere modeling, the faces’ index is the key data structure of the
linkage to other data structure, such as particles and springs. The reason is
because in the later geometric modeling method, each facet on the object is
used for subdivision of other facets in each iteration. Compared with a two-
dimensional object, the three-dimensional object consists the same types of
primitives, such as particles, springs, and faces, but extended to $z$ axis.
#### 3.5.1 Non-Uniform Sphere
##### 3.5.1.1 Geometric Data Type
One of the simplest non-uniform modeling methods to generate an approximate
facet sphere uses Polar to Cartesian Coordinates method. Consider $\theta$ the
angle on $xy$-plane (around $z$-axis), known as the Azimuthal Coordinate. The
angle $\phi$ is from $z$-axis, known as the Polar Coordinate. If we fix
$\theta$ and draw curves as we change $\phi$, we get circles of constant
longitude; if we fix $\phi$ and vary $\theta$, we obtain circles of constant
latitude [Ang03].
Figure 9: Polar Cartesian Coordinates Non-uniform Sphere Generation
###### Particles
The spherical coordinates for a particle $i$ can be defined by the three
equations:
$x(\theta,\phi)=\cos\theta\sin\phi$
$y(\theta,\phi)=\cos\theta\cos\phi$
$z(\theta,\phi)=\sin\phi$
where
$0^{0}<=\theta<=360^{0}$
$-90^{0}<=\phi<=90^{0}$
By stepping $\theta$ and $\phi$ in small angles $\Delta\theta$ and
$\Delta\phi$ between their bounds as the number of slices and stacks, the
particles are:
$P_{0}(x_{0},y_{0},z_{0})$= $(\sin\theta\cos\phi,\cos\theta\cos\phi,\sin\phi)$
$P_{1}(x_{1},y_{1},z_{1})$=
$(\sin\theta\cos(\phi+\Delta\phi),\cos\theta\cos(\phi+\Delta\phi),\sin(\phi+\Delta\phi))$
$P_{2}(x_{2},y_{2},z_{2})$=
$(\sin(\theta+\Delta\theta)\cos\phi,\cos(\theta+\Delta\theta)\cos\phi,\sin\phi)$
$P_{3}(x_{3},y_{3},z_{3})$=
$(\sin(\theta+\Delta\theta)\cos(\phi+\Delta\phi),\cos(\theta+\Delta\theta)\cos(\phi+\Delta\phi),\sin(\phi+\Delta\phi))$
$\cdots$
However, at the North and South Pole areas, we can only use triangles to
present because all lines of latitude are converged.
The particle at the North Pole area can be presented as:
$P(x,y,z)$=
$(\sin(\theta+\Delta\theta)\cos{90}^{0},\cos(\theta+\Delta\theta)\cos{90}^{0},\sin{90}^{0})$
The particle at the South Pole area can be presented as:
$P(x,y,z)$=
$(\sin(\theta+\Delta\theta)\cos{90}^{0},\cos(\theta+\Delta\theta)\cos{90}^{0},-\sin{90}^{0})$
###### Springs
There are also three types of springs in three-dimensional objects as we
described in two-dimensional objects, such as structural, radius, and shear
springs.
Figure 10: Quadrilaterals and Triangles on Non-uniform Sphere
* •
Structural spring is still the basic data structure to form the shapes of
inner and outer spheres. Four particles define four springs as the proper
order. Taking the first four particles $P_{0}$, $P_{1}$, $P_{2}$, and $P_{3}$
as an example, the first four springs’s are $S_{0}=P_{0}P_{2}$,
$S_{1}=P_{2}P_{3}$, $S_{2}=P_{3}P_{1}$, $S_{3}=P_{1}P_{0}$ shown in Figure 10.
The structural springs on two poles are also defined by the particles on poles
as proper order.
* •
Radius and shear springs, which connect inner and outer layers, follow the
same methods as in two-dimensional object.
###### Faces
Any quadrilateral-facet on the body of sphere can be represented by four
springs: $S_{i}$, $S_{i+1}$, $S_{i+2}$, and $S_{i+3}$. Any triangular-facet on
the poles can be represented by three springs: $S_{j}$, $S_{j+1}$, and
$S_{j+2}$.
##### 3.5.1.2 Modeling Algorithm
* •
Step 1: Define the number of slices and stacks of a sphere, $n_{slice}=10$ and
$n_{stack}=10$ in our example. Then, the step size is
$\Delta\theta=\frac{360^{0}}{10}=36^{0}$ and
$\Delta\phi=\frac{180^{0}}{10}=18^{0}$.
* •
Step 2: Define the group of particles’ position on inner circle and the ones
on outer circle shown in Figure 11. By multiplying the inner and outer
coordinates with a different radius value, for example, $Radius_{inner}=1.5$,
and $Radius_{outer}=2$ to create two concentric spheres.
Figure 11: Data Structure for Three-dimensional Non-uniform Object
Representation
* •
Step 3: Add the structural springs $S_{0}$, $S_{1}$, … to the inner circle
according to the spring index of inner particles as shown in Figure 11. The
same method is applied to outer structural springs on outer circle.
* •
Step 4: Loop through the number of structural springs to add the same number
of radius springs and shear springs according to the linkage of the inner and
outer particles as described in two-dimensional object modeling method, shown
in Figure 7 and Figure 8.
#### 3.5.2 Uniform Sphere
An important drawback of the non-uniform sphere model is that the faces vary
in both shape (some are triangles and some are quadrilaterals) and size.
##### 3.5.2.1 Geometric Data Type
Surface refinement is a simple way for uniform modeling. It is started with a
kernel polyhedron, which is a regular polyhedron with faces that are
equilateral triangles. We have used an octahedron with bisecting each face at
the same time recursively. This method is a powerful technique for generating
approximations to curves and surfaces of a sphere to any desired level of
accuracy.
(a) The Initial Octahedron Shape
(b) The Unit Facet Sphere Object With One Iteration
(c) The Unit Facet Sphere Object With Second Iteration
Figure 12: Uniform Sphere Generation
###### Particles
The algorithm starts with a regular octahedron shown in Figure 12(a). The
shape is composed of eight equilateral triangles, determined by six vertices,
$P_{0}(0,0,1)$, $P_{1}(0,0,-1)$, $P_{2}(-1,-1,0)$, $P_{3}(1,-1,0)$,
$P_{4}(1,1,0)$, and $P_{5}(-1,1,0)$. The vertices of the kernel polyhedron are
known to lie on the surface of a unit sphere of radius $r=1$. We fix the two
vertices $P_{0}$ and $P_{1}$ on $z$ axis and normalize the other five vertices
by multiplying $\frac{1}{\sqrt{2}}$ in order to make them lie on the unit
sphere, centered at the origin. The six vertices after normalization are
$P_{0}(0,0,1)$, $P_{1}(0,0,-1)$, $P_{2}(-0.7,-0.7,0)$, $P_{3}(0.7,-0.7,0)$,
$P_{4}(0.7,0.7,0)$, and $P_{5}(-0.7,0.7,0)$.
###### Faces
We talk about faces before talking about springs because the face is the key
data structure for recursive subdivision and its index is referenced by spring
index. The first eight triangular faces defined by the six particles are
$f_{0}=P_{0}P_{3}P_{4}$, $f_{1}=P_{0}P_{4}P_{5}$, $f_{2}=P_{0}P_{5}P_{2}$,
$f_{3}=P_{0}P_{2}P_{3}$, $f_{4}=P_{1}P_{4}P_{3}$, $f_{5}=P_{1}P_{5}P_{4}$,
$f_{6}=P_{1}P_{2}P_{5}$, $f_{7}=P_{1}P_{3}P_{2}$.
###### Springs
Each face is composed of three springs. Therefore, the first twelve springs on
the octahedron are $S_{0}=P_{0}P_{3}$, $S_{1}=P_{3}P_{4}$, $S_{2}=P_{4}P_{0}$,
$S_{3}=P_{4}P_{5}$, $S_{4}=P_{5}P_{0}$, $S_{5}=P_{5}P_{2}$,
$S_{6}=P_{2}P_{0}$, $S_{7}=P_{2}P_{3}$, $S_{8}=P_{1}P_{4}$,
$S_{9}=P_{1}P_{3}$, $S_{10}=P_{1}P_{5}$, and $S_{11}=P_{2}P_{1}$.
###### Subdivision
We can subdivide a single triangular face of the kernel polyhedron by
projecting the midpoints $pa$, $pb$, $pc$ of its three edges onto the surface
of the sphere as shown in Figure 13.
Figure 13: Subdivision of A Triangle By Bisecting Sides
This face is split into four faces by bisecting each edge. The four new
triangles are still in the same plane as the original triangle. We move the
new vertices $pa$, $pb$, $pc$ to the unit sphere by normalizing each new
vertices. The number of particles increases by a factor of 2. The number of
springs increases by a factor of 3. The number of facets increases by a factor
of 4. We subdivide another 7 triangles with the same method. After subdividing
the octahedron once, the number of particles are 12, the number of triangular
faces is 32, and the number of springs is 36. We can repeat the subdivision
process $n$ times to generate successively closer approximations to the
sphere.
##### 3.5.2.2 Modeling Algorithm
* •
Step 1: Define a collection of particles to create a closed equilateral
triangles shape of the elastic object. Define an octahedron object as the
initial object with 6 particles, 8 triangular faces, and 12 structural springs
as shown in Figure 14.
Figure 14: Data Structure for Three-dimensional Uniform Object Representation
without Subdivision
Figure 15: Data Structure for Three-dimensional Uniform Object Representation
with the Number of Subdivision n=1
* •
Step 2: Connect the particles with the structural springs according to the
edge order of the octahedron to make an inner layer of the three-dimensional
object. Each particle is separated equidistantly from its neighbors.
* •
Step 3: Check if there is need of subdivision to approach a more spherical
object.
* •
Step 4: In the first subdivision, the object becomes 12 particles, 32
triangles, and 36 structural springs. The Figure 15 shows how new data can be
inserted into a collection of particles, faces, and springs after
subdivision.Use the first triangle as a concrete example. In the initial
octahedron shape, find the midpoint of each edge on $F_{0}$. Normalize the
coordinates of these three new particles to make them lie on the sphere. Push
these three new particles to the particle container. The first face of the
initial octahedron has three pointers that point to particles $P_{0}$,
$P_{3}$, and $P_{4}$ as shown in Figure 14. After subdividing this triangle,
it becomes four smaller triangles connected by the bisectors $pa$, $pb$, and
$pc$. The three new triangles are pushed onto container $Faces$. The middle
triangle replaces the original big triangle because the original triangle does
not exist anymore. The pointers on each face point to the correspondent
particles as indicated in Figure 15. New structural springs are added to
Spring container correspondent to new faces only if there is no such spring
has existed yet. The subdivision of the remaining faces follows the same
method. More faces are approaching the object to a unit facet sphere.
* •
Step 5: Repeat Step 1 to 5 to create an outer layer with larger radius value
than the inner layer.
* •
Step 6: Loop through the number of structural springs to add the same number
of radius springs according to the linkage of the inner and outer particles.
* •
Step 7: Loop through the number of structural springs to add the same number
of shear left springs and shear right springs according to the linkage of the
inner and outer particles.
#### 3.5.3 Comparison of Non-uniform and Uniform Methods
The advantages of both methods are that they can be used to describe complex
behaviors combined with physical laws, such as elasticity. Additionally, the
level of detail (LoD) of the object can be adjusted depending on the proximity
of the object on the display to the human’s eye.
The disadvantage of the non-uniform sphere modeling method is that the facets
of the sphere do not have approximately equal size. The facets become smaller
at the poles and bigger at the “equator”. Therefore, the springs are shorter
at the poles and longer at the equator. The normal of each spring varies from
equator and the ones on the poles. Consequently, this non-uniform modeling
method increases errors in force computations for each particle.
The disadvantage of the uniform facet sphere generation algorithm is that it
can not generate surfaces of arbitrary resolution. It can be shown that at all
levels of recursion, particles at the kernel points are connected to four
springs if the kernel object is an octahedron (as shown in Figure 12(b)). In
other cases, all the particles at the kernel points are connected to five
springs if the kernel object is an icosahedron (20 faces); all the particles
at the kernel points are connected to three springs if the kernel object is a
tetrahedron. All particles at recursively generated points are connected to
six springs. This will result the irregular surface stiffness and might cause
the non-spherical shape because the same pressure will displace the regions of
a surface about the kernel points further than the rest of the surface.
To solve this problem, the sum of the spring forces accumulated at a particle
can be normalized by multiplying a factor of $\frac{6}{n_{springs}}$, where
$n_{springs}$ is the number of springs connected to this particle. For
example, if $particle_{a}$ is a kernel point, which is connected to four
springs and the sum of the spring forces is $f_{a}$; and if $particle_{b}$,
which is the point generated from subdivision, connects to six springs and the
sum of the six spring forces is $f_{b}$. $f_{a}$ is multiplied by a factor of
$\frac{6}{4}$ and $f_{b}$ is multiplied by a factor of $\frac{6}{6}$.
Our simulation system ignores the described drawback resulting from the
uniform sphere modeling method. We find a set of air pressure and spring
stiffness parameter values at which the simulation is stable by trial and
error. Thus, the difference of the forces for every particle either connected
to four springs or six springs is not addressed in this work.
## Chapter 4 Physical Based Modeling Methodology
A one-dimensional object model includes gravity force ${\bf F}^{g}$, user
applied force ${\bf F}^{a}$, and collision force ${\bf F}^{c}$ as external
forces; linear structural spring force ${\bf F}^{h}$ and spring damping force
${\bf F}^{d}$ as internal forces.
${\bf F}={\bf F}^{g}+{\bf F}^{h}+{\bf F}^{d}+{\bf F}^{a}+{\bf F}^{c}$ (1)
A two-dimensional object model is considered as a closed shape with air
pressure inside. Then, the air pressure ${\bf F}^{p}$ is a new internal force
exist in two-dimensional object in addition to the common forces in one-
dimensional object. Accumulation of forces on a three-dimensional object is
similar to forces applied on the two-dimensional one. The only difference is
that all forces on three-dimensional objects are extended to axis z.
${\bf F}={\bf F}^{g}+{\bf F}^{h}+{\bf F}^{d}+{\bf F}^{a}+{\bf F}^{p}+{\bf
F}^{c}$ (2)
### 4.1 Gravity Force
Gravity force is a constant force at which the earth attracts objects based on
their masses. In most cases, the particle system does not include gravitation,
but, in our system, particle gravities represent object’s density. Users can
set particle gravities to a non-zero value. g is a constant scalar of the
gravitational field.
${\bf F}^{g}=m\textit{g}$ (3)
### 4.2 Spring Hooke’s Force
Spring force is a linear force exerted by a compressed or stretched spring
upon two connected particles. The particles which compress or stretch a spring
are always acted upon by this spring force which restores them to their
equilibrium positions. It is calculated as following according to Hooke’s law,
which describes the opposing force exerted by a spring.
${\bf F}_{12}^{h}=-\left(||{\bf r}_{2}-{\bf r}_{1}||-r_{l}\right)\,{k}_{s}$
(4)
where
${\bf r}_{1}$ is the first particle position,
${\bf r}_{2}$ is the second particle position,
${r}_{l}$ is default length of the resting spring between the two particles,
${k}_{s}$ is the stiffness of the spring,
when $||{\bf r}_{2}-{\bf r}_{1}||-r_{l}=0$, the spring is resting,
when $||{\bf r}_{2}-{\bf r}_{1}||-r_{l}>0$, the spring is extending,
when $||{\bf r}_{2}-{\bf r}_{1}||-r_{l}<0$, the spring is contracting.
We have discussed the type of structural spring in one-dimensional object. In
two and three dimensional object model, the same method applies on the other
three types of spring, such as radius springs, shear left springs, and shear
right springs with different spring stiffness and spring damping factor. So,
the total Hooke’s spring force is:
$\displaystyle{\bf F}_{total}^{h}={\bf F}_{structure}^{h}+{\bf
F}_{radius}^{h}+{\bf F}_{shearleft}^{h}+{\bf F}_{shearright}^{h}$ (5)
### 4.3 Spring Damping Force
Spring damping force is also called viscous damping. It is opposite force of
the Hook spring force in order to simulate the natural damping and resist the
motion. It is also opposite to the velocity of the moving mass particle and is
proportional to the velocity because the spring is not completely elastic and
it absorbs some of the energy and tends to decrease the velocity of the mass
particle attached to it. It is needed to simulate the natural damping due to
the forces of friction. More importantly, it is useful to enhance numerical
stability and is required for the model to be physically correct [BA97].
${\bf F}_{12}^{d}=\left({\bf v}_{2}-{\bf v}_{1}\right)\cdot\left(\frac{{\bf
r}_{2}-{\bf r}_{1}}{||{\bf r}_{2}-{\bf r}_{1}||}\right)\,{k}_{d}$ (6)
where
$\left(\frac{{\bf r}_{2}-{\bf r}_{1}}{||{\bf r}_{2}-{\bf r}_{1}||}\right)$ is
the direction of the spring,
${\bf v}_{1}$ and ${\bf v}_{2}$ is the velocity of the two masses,
${k}_{d}$ is spring damping coefficient.
When the two endpoints moving away from each other, the force imparted from
the damper will act against that motion; when the two endpoints moving toward
each other, the damper will act against the squeeze motion. The damper will
always acts against the motion. The total spring damping force is:
$\displaystyle{\bf F}_{total}^{d}={\bf F}_{structure}^{d}+{\bf
F}_{radius}^{d}+{\bf F}_{shearleft}^{d}+{\bf F}_{shearright}^{d}$ (7)
### 4.4 Drag Force
Drag force is the force when users interact with the elastic object through
mouse. At the moment users click the mouse, the simulation system finds the
nearest particle $i$ to the current position of the mouse. If users drag this
particle $i$, the drag force contributes to force of this nearest particle.
The forces applied on rest particles are effected by the new user applied
force, which is passed through by springs.
We consider one end of the string connects to the mouse position and the other
end of the string connects to the nearest particle on the object. This string
is elastic, so it has all the spring’s properties, such as hook spring force
and damping force. The drag force can be presented as following:
${\bf F}^{a}=-\left(||{\bf r}_{m}-{\bf
r}_{i}||-r_{lm}\right)\,{k}_{sm}+\left({\bf v}_{m}-{\bf
v}_{i}\right)\cdot\left(\frac{{\bf r}_{m}-{\bf r}_{i}}{||{\bf r}_{m}-{\bf
r}_{i}||}\right)\,{k}_{dm}$ (8)
where
${\bf r}_{m}$ is the mouse position,
${\bf r}_{i}$ is the particle position nearest to mouse,
${\bf r}_{lm}$ is default length of the resting mouse spring,
${k}_{sm}$ is the stiffness of the mouse spring,
${\bf v}_{m}$ is the velocity of the mouse represented as a mass
${\bf v}_{i}$ is the mass for the nearest particle,
${k}_{dm}$ is spring damping coefficient for the mouse spring.
${\bf F}^{a}$ is a momentary force for interacting with the elastic simulation
system. This force is accumulated to the current forces already applied on
this nearest particle.
In a one-dimensional object simulation system, the nearest particle to mouse
is either $P_{0}$ or $P_{1}$. In a two-dimensional and three-dimensional
object system, the drag force is only applied on the outer layer of the double
layered object when user interacts the object with mouse.
### 4.5 Air Pressure Force
In order to describe an elastic object more accurately, especially soft body
of human beings and animals, the calculation only about the elastic force on
the object’s surface is not enough. We add the flow pressure force inside of
the elastic object to make the object wobbly looking when it is deformed.
The pressure force will be calculated for every spring, then update each
particle’s direction. The pressure vector is always acting in a direction of
normal vectors to the surface, so the shape will not deform completely. If
pressure is simulated without also simulating the mass-spring system, the
object will explode.
(a) The External Gravity Force Is Applied Producing A Pressure Wave
(b) The Object Restores Its Shape With Internal Air Pressure Force
Figure 1: Double-layered Two-dimensional Elastic Object Filled With Air
In Figure 1(a), the object is deformed from bottom because of the gravity
force. If there is no internal air pressure, the object will collapse unless
the springs are hard enough to avoid the failure. With the very hard springs,
it is difficult to simulate the reality of the elasticity. The real elastic
object restores its shape as described in Figure 1(b). The simplified version
of the Ideal Gas Law [Mat03], also known as Clausius Clapeyron Equation, is
used to describe such effect:
$PV=NRT$ (9)
where
$P$ is the pressure value,
$V$ is the volume of the object,
$N$ is number of mols,
$R$ is the gas constant,
$T$ is the gas temperature. Therefore, the pressure force is:
${\bf F}^{p}=P\,{\bf n}$ (10)
where
${\bf F}^{p}$ is the pressure force vector,
${\bf n}$ is the normal vector to the springs on the object.
$P=\frac{NRT}{V}$ (11)
#### 4.5.1 Volume
In order to find an estimate pressure inside of the object which will be
applied to particles later, we need to calculate the volume of the object. The
approximation of the volume is calculated with Gauss’ Theorem [Ker07]:
$V=\int\int\int_{v}f(x,y,z)\,dx\,dy\,dz\Longleftrightarrow
V=\int\int\int_{v}f(x,y,z)\,dV$ (12)
where triple integrals of a function $f(x,y,z)$ define a volume integral of an
elastic sphere. Moreover, triple integrals can be transformed into surface
double integrals over the boundary surface of a region if the three-
dimensional object is closed shape by divergence theorem [Ker07]:
$V=\int\int\int_{V}\Delta{\bf F}\,dV\Longleftrightarrow S=\int\int_{S}{\bf
F}\,dS$ (13)
where
${\bf F}$ is a vector field,
$V$ is the object volume,
$S$ is the object surface.
Double integrals over a plane region may be transformed into line integrals by
Green’s Theorem in the Plane:
$\int\int_{S}\Delta{\bf F}\,dx\,dy\Longleftrightarrow\int_{L}{\bf F}\,dL$ (14)
where $L$ is the object edge and $dL$ is the length of the edge.
Therefore, a triple integrals function $f(x,y,z)$ shown in Eq.12, which
defines a volume integral of an elastic sphere, can be transformed to line
integrals as shown in Eq.15.
$V\approx\int_{L}{\bf F}\,dL$ (15)
We assume on the line, the vector field ${\bf F}=(x,0)$, the simplified
integration of body volume is [Mat03, Ker07]:
$\int_{L}{\bf F}\,dL=\displaystyle\sum_{i=0}^{i=NUMS-1}\frac{1}{2}({\bf
x_{1}}-{\bf x_{2}})\,{\bf n_{x}}\,dL$ (16)
where
$V$ is the volume of the object,
$({\bf x_{1}}-{\bf x_{2}})$ is the absolute difference of the line (represents
spring here) of the start and end particles at axis x,
${\bf n_{x}}$ is the normal vector to this line (spring) at axis x,
$dL$ is the line’s (spring’s) length.
#### 4.5.2 Normals
Normals are unit vectors perpendicular to specified data structure, such as
particle (vertex psuedo-normals), spring (line), and face (polygonal facets)
on the object.
* •
Particle normal, or vertex psuedo-normals, does not exist for vertices;
however, it can be considered as the average of the normals of the subtended
neighbor particles. To calculate the particle normal is to sum up the normals
for each face adjoining this particle, and then to normalize the sum.
* •
Spring (line) normal in two dimension is based on the two particles
$P_{1},P_{2}$ connected on the spring. It is perpendicular to the spring
itself.
* •
Face (plane) normal in three dimension is determined by right-hand rule, which
is perpendicular to its surface based on the any pair of springs on the
surface. The normal for a triangle surface composed with three particles
$P_{1},P_{2},P_{3}$ is computed as the vector cross product of the springs
$P_{2}-P_{1}$ and $P_{2}-P_{3}$.
The usage of normal calculation method in our elastic object simulation system
is for analysis of the direction of the pressure force inside of the object.
Only spring normal is calculated here because all the internal and external
forces will apply on each spring, and the spring will define the particles’
force which connect onto it.
##### 4.5.2.1 2D Normals
For the single spring $Spring_{12}$, the Cartesian coordinates for particle
$P_{1}$ is $(x_{1},y_{1})$; the Cartesian coordinates for particle $P_{2}$ is
$(x_{2},y_{2})$. The normal to this spring is the spring rotate $90^{0}$ at
axis z according to the space position. So, we can get the components of the
normal in x axis and y axis as following
$\left[\begin{array}[]{c}{x^{\prime}}\\\
{y^{\prime}}\end{array}\right]=\left[\begin{array}[]{cc}\cos{90^{0}}&-\sin{90^{0}}\\\
\sin{90^{0}}&\cos{90^{0}}\end{array}\right]\left[\begin{array}[]{c}{x_{2}-x_{1}}\\\
{y_{2}-y_{1}}\end{array}\right]$
$\left[\begin{array}[]{c}{x^{\prime}}\\\
{y^{\prime}}\end{array}\right]=\left[\begin{array}[]{c}-\left({y_{2}-y_{1}}\right)\\\
{x_{2}-x_{1}}\end{array}\right]$
##### 4.5.2.2 3D Normals
The calculation of the 3D normals of springs is important because it will
define the direction of the internal air pressure force, either compress the
elastic object or expand its volume. In theory, in three-dimensional
simulation, the normal of a spring in the space position is represented as an
average of the normals of faces connected to it. However, in our elastic
object simulation system, we use a simplified estimated normal based on the
normal algorithm of the two-dimensional calculation discussed above. Instead
of rotating a line $90^{0}$ at z axis to get its normal vector in two-
dimension, the estimated algorithm is rotating a line $90^{0}$ at z axis, y
axis, and x axis to get its normal in three-dimension.
$\left[\begin{array}[]{c}{x^{\prime}}\\\ {y^{\prime}}\\\ {z^{\prime}}\\\
{1}\end{array}\right]=\left[\begin{array}[]{cccc}\cos{90^{0}}&-\sin{90^{0}}&0&0\\\
\sin{90^{0}}&\cos{90^{0}}&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right]\left[\begin{array}[]{cccc}\cos{90^{0}}&0&-\sin{90^{0}}&0\\\
0&1&0&0\\\ \sin{90^{0}}&0&\cos{90^{0}}&0\\\ 0&0&0&1\end{array}\right]$
$\left[\begin{array}[]{cccc}1&0&0&0\\\ 0&\cos{90^{0}}&\sin{90^{0}}&0\\\
0&-\sin{90^{0}}&\cos{90^{0}}&0\\\
0&0&0&1\end{array}\right]\left[\begin{array}[]{c}{x_{2}-x_{1}}\\\
{y_{2}-y_{1}}\\\ {z_{2}-z_{1}}\\\ {1}\end{array}\right]$
Therefore,
$\left[\begin{array}[]{c}{x^{\prime}}\\\ {y^{\prime}}\\\ {z^{\prime}}\\\
{1}\end{array}\right]=\left[\begin{array}[]{c}{z_{2}-z_{1}}\\\
{y_{2}-y_{1}}\\\ -\left({x_{2}-x_{1}}\right)\\\ {1}\end{array}\right]$
We use a vector $(1,0,0)$ as an example to prove this algorithm, the normal
vector for this vector is:
$\left[\begin{array}[]{c}{x^{\prime}}\\\ {y^{\prime}}\\\ {z^{\prime}}\\\
{1}\end{array}\right]=\left[\begin{array}[]{c}{0-0}\\\ {0-0}\\\
-\left({1-0}\right)\\\ {1}\end{array}\right]=\left[\begin{array}[]{c}{0}\\\
{0}\\\ {-1}\\\ {1}\end{array}\right]$
This result is reasonably correct and believable despite of the fact that if
vector $(1,0,0)$ lies in the $xz$-plane or lies in the $xy$-plane.
However, it shows that this estimation algorithm has the limitation for some
cases, for example vector $(0,1,0)$, the normal vector is:
$\left[\begin{array}[]{c}{x^{\prime}}\\\ {y^{\prime}}\\\ {z^{\prime}}\\\
{1}\end{array}\right]=\left[\begin{array}[]{c}{0-0}\\\ {1-0}\\\
-\left({0-0}\right)\\\ {1}\end{array}\right]=\left[\begin{array}[]{c}{0}\\\
{1}\\\ {0}\\\ {1}\end{array}\right]$
This result shows the normal vector is the vector itself, which is obviously
wrong. However, with this estimated algorithm, the simulation result appears
enough realistic; moreover, it requires less computational effort111Our
estimation only takes 3 additions vs. 12 multiplications and 6 additions for
two cross products and three more additions and divisions for the averaging..
### 4.6 Collision Force
If an object continues traveling under a force without colliding with other
objects, it will be very difficult to describe objects’ motion and elastic
response in reality. Collision force is the force to make object bounce away
from the fixed interacting plane when elastic object collision happens.There
are two steps to describe the collision effects: detection and reaction.
Detect the elastic object if particles hit anything; adjust their position by
computing the impulse.
#### 4.6.1 Collision Detection
Collision Detection is a geometric problem of determining if a moving object
intersected with other objects at some point between an initial and final
configuration. In our elastic object simulation system, we are concerned with
the problem of determining if any of $n$ particles collide with any of $m$
solid planes.
###### Perfect Elastic Collision
Figure 2: Particle Inelastic Collision and Impact
We take one particle collides with a plane shown in Figure 2 as an example. We
can detect this collision by inserting the particle position into the plane
equation:
$P(x,y,z)=ax+by+cz+d$ (17)
If $P(x,y,z)>0$, the particle is within the plane. If $P(x,y,z)=0$, the
particle collides with the plane. If $P(x,y,z)<0$, the particle penetrates the
plane. At each time step, looping through all the particles on the object,
each particle is checked if it is outside of the interacting plane.
When the particle $i$ collides with the plane, if there is a perfect elastic
collision as in Figure 2, the particle does not lose its energy, so its speed
does not change. However, its direction after the collision is in the
direction of a perfect reflection.
${\bf F}^{c}=2((P-P^{\prime})\cdot\,{\bf n})\,{\bf n}-(P-P^{\prime})$ (18)
where
${\bf F}^{c}$ is the the direction of a perfect reflection
${\bf n}$ is the normal at the point of collision $P^{\prime}$ and the
previous position of particle $P$
$P-P^{\prime}$ is the vector from the particle to the surface.
###### Damped Elastic Collision
If there is a damped elastic collision, the particle cannot penetrate the
surface, and it cannot bounce from the surface because of the force being
applied to it, then we need to apply the damped elastic collision method. The
particle loses some of its energy when it collides with another object. The
coefficient of restitution of a particle is the friction of the normal
velocity retained after the collision. Therefore, the angle of reflection is
computed as for the inelastic collision, and the normal component of the
velocity is reduced by the coefficient of restitution.
#### 4.6.2 Collision Response
Collision Response is a physics problem of determining the forces of the
collision. In elastic collision, elastic object should bounce away from the
colliding plane and some energy is lost in the collision response as described
in the penalty method.
${\bf F}^{c}=-e\,{\bf F}^{c}$ (19)
where $e$ is elasticity of the collision and $0.0\leq e\leq 1.0$. At $e=0$,
the particle does not bounce at all; $e=1$, the particle bounces with no
friction.
In an one-dimensional object, the boundaries are the walls and floors. In a
two-dimensional and three-dimensional object, the particles on the outer layer
still follow the same method and same pre-defined boundary as the one-
dimensional object. However, for the particles on the inner layer, the
boundary is constrained to the outer layer instead of the wall and floor.
### 4.7 Force Accumulation Algorithm
The following algorithm describes how different forces are accumulated and
applied to an elastic object. For a one-dimensional object, some steps will be
skipped, for example, there are no other types of spring computations except
structural springs because other types of springs only apply on two-layer 2D
or 3D objects. Moreover, there are no pressure force accumulation and volume
computation because these steps are only available for closed shape objects.
* •
Step 1: Loop through the number of particles to assign particles with mass
value $m$ and compute gravity force ${\bf F}^{g}$. Gravity force, which is
independently on each particle, either depends on a constant force, or one or
more of particle position, particle velocity, and time [Wit97]. If the object
is one-dimensional, the mass of each particle can be different. If the object
is two or three dimensional, the mass of the particles on inner or outer layer
can also be set differently.
* •
Step 2: Loop through the number of the structural springs to accumulate the
structural spring force.
* •
Step 3: Loop through the number of the radius springs to accumulate the radius
spring force.
* •
Step 4: Loop through the number of the shear springs to accumulate the shear
spring force.
* •
Step 5: Initialize density as gas, liquid, or rubber inside of the body and
introduce some simple physics to describe it. In the current system, only air
pressure material is considered and only pressure equation will be used for
this extra force computation.
* •
Step 6: Calculate volume of the inner layer and outer layer of the elastic
object.
* •
Step 7: Calculate the normals of springs on each triangular face to define the
pressure force direction.
* •
Step 8: Calculate the force from the internal air pressure by multiplying the
force value by normal vector of the spring.
* •
Step 9: Accumulate pressure force to each particle.
* •
Step 10: If users apply the drag force, compute the user applied force and
accumulate this force to the dragged particle.
* •
Step 11: Integrate the object’s momentum motion by calculating the derived
velocity and its new position for each particle. This step will be explained
in next chapter.
* •
Step 12: Resolve collision detection and response and define the updated
position.
## Chapter 5 Numerical Integration Methodology
Assume, after the elastic object simulation system creates an elastic object
based on the methodology described in Chapter 3 with its initial force state
in Figure 1(a) as described in Chapter 4, the system starts the simulation.
The simulation system is updated a finite number of times. The object is at
the state in Figure 1(b) after 50 discrete time steps.
(a) Elastic Object at the Initial Step
(b) Elastic Object at the Step 50
Figure 1: Elastic Object at Different Time States
In each update, the accumulated impact forces on the object tell it how to
change the velocity for next step and result in a re-computation of the
forces. The dynamic force applied on this object may be the collision force
when the object reaches the boundary; or, the mouse dragging force when user
interacts with the object. Overall, the shape deformation, a mapping of the
positions of every particle in the original object to those in the deformed
body of this elastic object, is also computed in real time. Therefore, it is
important to study differential equations, which govern dynamics and geometric
representation of objects [Lin06] and tell us how the velocity and
displacement of the particles are integrated dynamically from the knowledge of
force applied onto them.
### 5.1 Differential Equations
Differential equations describe a relation between a function and one or more
of its derivatives. The order of the equation is the order of the highest
derivative it contains. The elastic object simulation system is associated
with initial value problems because it always seeks the particles’ velocity
and position at next time step $t+h$ from their initial state at time $t$. We
will concentrate on ODE (ordinary differential equation), where all
derivatives are with respect to single independent variable, often
representing time, such as position and velocity, during the derivate of the
state at discrete time steps [Ang03].
${y}^{\prime}={\bf A}(y,t)$ (1)
where
${\bf A}$ is a function of $y$ and $t$,
$y$ is a vector, which is the state of the system,
${y}^{\prime}$ is a vector, which is $y$’s time derivative.
Suppose that we integrate the Eq.1 over a short time $h$
$y(t+h)-y(t)=\int_{t}^{t+h}{\bf A}(y,t)\,dt$ (2)
where
$h$ is the small stepsize of time,
$y(t)$ is the initial state at the start point $t$,
$y(t+h)$ is the value we need to find over time thereafter.
Thus
$y(t+h)\approx y(t)+h{\bf A}(y(t),t)$ (3)
#### 5.1.1 Explicit Euler Integrator
The simplest ODE integration method is Explicit Euler Integration method or
Forward Euler method. It evaluates the forces at time $t$, compute derivatives
${\bf A}$ at the state of $t$ by multiplying the interval $h$, and add it to
the current state $t$. Consider a Taylor series expansion as in Eq.4:
$y({t}+{h})=y({t})+{h}{y}^{\prime}({t})+\frac{{{h}^{2}}}{2!}{y}^{\prime\prime}({t})+\frac{{{h}^{3}}}{3!}{y}^{\prime\prime\prime}({t})+\cdots+\frac{{{h}^{n}}}{n!}\left(\frac{\partial^{n}y}{\partial{t}^{n}}\right)+\cdots$
(4)
Euler method retains only first derivative:
$y({t}+{h})=y({t})+{h}{y}^{\prime}({t})+O(h^{2})$ (5)
Figure 2: Euler Integrator
We split the series into elements, which we will later use in a re-usable
manner throughout integrator framework, where
$k_{0}$ which represents the first term in Eq.5, is the initial state
$k_{0}=y({t})$ (6)
$k_{1}$ which represents the second term in Eq.5, is the function to find the
simplest estimation, the Euler slope of the interval.
$k_{1}={y}^{\prime}({t})={\bf A}(y(t),t)$ (7)
Thus
$y({t}+{h})=k_{0}+{{h}}k_{1}$ (8)
We can apply this method iteratively to compute further values at state
$t+2h$, $t+3h$,…. [BD03] This method is easy to implement; however, it is a
low accuracy prototype ODE. In Figure 2, we can see Euler method only
calculates the derivative, also called slope, at the beginning of the interval
and adds it to the value at the initial state; therefore, it is asymmetric and
not stable. .
###### Pseudocode for Euler Method
`Line 1: define A(y(t), t)`
`Line 2: initial values y0 and t0`
`Line 3: stepsize h and number of steps n`
`Line 4: for i from 1 to n do`
`Line 5: k1 = A(y(t), t)`
`Line 6: y = y + hk1`
`Line 7: t = t + h`
#### 5.1.2 Midpoint Integrator
Compared to the Euler method, the one-sided estimate algorithm, midpoint
integrator is a symmetric estimate method with a higher per-step accuracy. It
computes the derivative at the center of the interval first, then computes the
end of the interval.
The midpoint integrator, just like others, is based on the Taylor’s series. It
retains only first three derivative term:
$y({t}+{h})=y({t})+{h}{y}^{\prime}({t})+\frac{{{h}^{2}}}{2!}{y}^{\prime}({t})+O({h}^{3})$
(9)
Figure 3: Midpoint Integrator
We split the series into elements again for explanation of the method, where
$k_{0}$, which represents the first term in Eq.9, is the initial state at time
$t$.
$k_{0}=y({t})$ (10)
$k_{1}$ which represents the second term in Eq.9, is the function to find the
the simplest Euler slope of the interval at time $t$.
$k_{1}={y}^{\prime}({t})={\bf A}(y(t),t)$ (11)
$k_{2}$ is the function to find the the simplest Euler slope of the interval
at time $t+h$.
$k_{2}={y}^{\prime}({t+h})={\bf A}(y(t+h),t+h)$ (12)
Since the unknown $(y+h)$ appears on the right side of Eq.13, in ${\bf
A}(y(t+h),t+h)$ as one of the arguments of function ${\bf A}$, we can use the
value obtained using the Euler method in Eq.5.
${\bf A}(y(t+h),t+h)\approx{\bf A}(y(t)+h{\bf}(y(t),t),t+h)={\bf
A}(y(t)+hk1,t+h)$ (13)
The midpoint integration technique obtains a more accurate estimate of the
slope than Euler’s technique. The following equation computes the integrand at
the middle of the interval of $t$ and $t+h$ shown in Figure 3. Thus,
$y({t}+{h})=k_{0}+{{h}}\frac{k_{1}+k_{2}}{2}$ (14)
Compared to Euler Method, Midpoint Method, also called the Runge-Kutta method
of order 2, goes from $t$ to $t+h$, we must evaluate function ${\bf A}$ twice.
By using Taylor’s theorem to evaluate the per-step error, we would find that
it is now $O(h^{3})$. Therefore, this method is more stable than Euler Method
with same step size.
###### Pseudocode for Midpoint Method
`Line 1: define A(y(t), t)`
`Line 2: initial values y0 and t0`
`Line 3: stepsize h and number of steps n`
`Line 4: for i from 1 to n do`
`Line 5: k1 = A(y(t), t)`
`Line 6: k2 = A(y(t+h), t+h)= A(y+hk1, t+h)`
`Line 7: y = y + h/2(k1+k2)`
`Line 8: t = t + h`
#### 5.1.3 Runge Kutta Fourth Order Integrator
Runge Kutta Fourth Order integrator evaluates the derivative four times. It is
the most accurate integrator that we describe compared to Euler and Midpoint.
Figure 4: Runge Kutta 4th Order Integrator
The Runge Kutta Fourth integrator, is also based on the Taylor’s series. It
retains only first five derivative term with a local truncation error
$O(h^{5})$:
$y({t}+{h})=y({t})+{h}{y}{{}^{\prime}}({t})+\frac{{{h}^{2}}}{2!}{y}{{}^{\prime\prime}}({t})+\frac{{{h}^{3}}}{3!}{y}{{}^{\prime\prime\prime}}({t})+\frac{{{h}^{4}}}{4!}{y}{{}^{\prime\prime\prime\prime}}({t})+O({h}^{5})$
(15)
$k_{0}=y({t})$ (16)
${k}_{1}={y}^{\prime}({t})={\bf A}(y(t),t)$ (17)
${k}_{2}={\bf A}(y(t)+{h}\frac{{k}_{1}}{2},t+\frac{h}{2})$ (18)
${k}_{3}={\bf A}(y(t)+{h}\frac{{k}_{2}}{2},t+\frac{h}{2})$ (19)
${k}_{4}={\bf A}(y(t)+{h}{k}_{3},t+h)$ (20)
$y({t}+{h})={k}_{0}+\frac{1}{6}{h}({k}_{1}+2\,{k}_{2}+2\,{k}_{3}+{k}_{4})$
(21)
where
$k_{0}$ is the initial state
$k_{1}$ is the slope at the left end of interval,
$k_{2}$ is the slope at the middle point using the Euler formula to go from
$t$ to $t+\frac{h}{2}$,
$k_{3}$ is the second approximation to the slope at the midpoint,
$k_{4}$ is the slope at $t+h$ using the Euler formula and the slope $k_{3}$ to
go from $t$ to $t+h$.
###### Pseudocode for Runge Kutta Fourth Order Method
`Line 1: define A(y(t), t)`
`Line 2: initial values y0 and t0`
`Line 3: stepsize h and number of steps n`
`Line 4: for i from 1 to n do`
`Line 5: k1 = A(y(t), t)`
`Line 6: k2 = A(y+h/2(k1), t+h/2)`
`Line 7: k3 = A(y+h/2(k2), t+h/2)`
`Line 8: k4 = A(y+hk3, t+h)`
`Line 9: y = y + h/6(k1+2*k2+2*k3+k4)`
`Line 10: t = t + h`
### 5.2 Newton’s Laws
After the force accumulation on the object, it is important to find the
acceleration a in order to define the motion of objects in their next time
step. The physical law that governs the motion of objects is the Newton’s
Second law. It states that the force ${\bf F}$ is proportional to the time
rate of change of its linear momentum. Momentum is the product of mass $m$ and
velocity v.
${\bf F}\approx{m}\frac{\Delta{\bf v}}{\Delta t}$ (22)
###### Velocity
v is the integral of acceleration a with respect to the time $t$. Therefore,
integrating the acceleration gives us the new velocity v.
${\bf v}=\int{\bf a}{dt}$ (23)
###### Position
r is the integral of velocity v with respect to the time $t$. Therefore,
integrating the velocity gives us the new position r.
${\bf r}=\int{\bf v}{dt}$ (24)
Let’s take one particle on the object as an example and understand how the
different integrators work.
#### 5.2.1 Newton’s Laws in Euler Integrator
Based on the Euler Integrator method shown in Eq.5, the new velocity and
position of a particle can be integrated follows.
###### Velocity
can be represented as the following equation:
${\bf v}(t+h)\approx{\bf v}(t)+{h}{\bf v}^{\prime}({t})$ (25)
$v_{k0}$ represents the first term in Eq.25, which is the initial velocity at
time $t$
${v_{k0}}={\bf v}(t)$ (26)
$v_{k1}$ represents the second term in Eq.25, which is the function to compute
the derivative velocity in the period $h$
$v_{k1}={h}{\bf v}^{\prime}({t})={{\bf a}(t)}h$ (27)
###### Position
can be represented as the following equation
${\bf r}(t+h)\approx{\bf r}(t)+{h}{\bf r}^{\prime}({t})$ (28)
$r_{k0}$ is the initial position at time $t$
${r_{k0}}={\bf r}(t)$ (29)
$r_{k1}$ is the function to find the travel position in the period $h$
$r_{k1}={h}{\bf r}^{\prime}({t})={{\bf v}(t)}h$ (30)
#### 5.2.2 Newton’s Laws in Midpoint Integrator
We apply the midpoint algorithm theory on the Newton’s law in order to achieve
higher accuracy in the the relationship between the velocity and the position
according the Eq.9.
###### Velocity
can be represented as the following equation
${\bf v}({t}+{h})\approx{\bf v}({t})+{h}{\bf
v}^{\prime}({t})+\frac{{{h}^{2}}}{2!}{{\bf v}}^{\prime\prime}({t})$ (31)
$v_{k0}$ is the initial velocity at state $t$
${v_{k0}}={\bf v}(t)$ (32)
$v_{k1}$ is the function to compute the derivative velocity in the period $h$
$v_{k1}={\bf v}^{\prime}({t})={{\bf a}(t)}h$ (33)
$v_{k2}$ is the function to compute the derivative velocity in the period
$t+h$
$v_{k2}={\bf v}^{\prime}({t+h})={\bf v}(t)+{{\bf a}(t)}h$ (34)
Therefore, the new velocity of a particle is
${\bf v}(t+h)=v_{k0}+\frac{v_{k1}+v_{k2}}{2}$ (35)
###### Position
can be represented as the following equation
${\bf r}({t}+{h})\approx{\bf r}({t})+{h}{\bf
r}^{\prime}({t})+\frac{{{h}^{2}}}{2!}{\bf r}^{\prime\prime}({t})$ (36)
$r_{k0}$ is the initial position at state $t$
${r_{k0}}={\bf r}(t)$ (37)
$r_{k1}$ is the function to find the travel position in the period $h$
$r_{k1}={\bf r}^{\prime}({t})={{\bf v}(t)}h$ (38)
$r_{k2}$ is the function to find the travel position in the period $t+h$
$r_{k2}={\bf r}^{\prime}({t+h})={\bf r}(t)+{{\bf v}(t)}h$ (39)
Therefore, the new position of a particle is
${\bf r}(t+h)=r_{k0}+\frac{r_{k1}+r_{k2}}{2}$ (40)
#### 5.2.3 Newton’s Laws in the Runge Kutta Fourth Order Integrator
Based on the Runge Kutta Fourth Order method we have shown in Eq.9, the new
velocity and position of a particle can be integrated as following.
###### Velocity
can be represented as the following equation
${\bf v}({t}+{h})\approx{\bf v}({t})+{h}{\bf
v}{{}^{\prime}}({t})+\frac{{{h}^{2}}}{2!}{\bf
v}{{}^{\prime\prime}}({t})+\frac{{{h}^{3}}}{3!}{\bf
v}{{}^{\prime\prime\prime}}({t})+\frac{{{h}^{4}}}{4!}{\bf
v}{{}^{\prime\prime\prime\prime}}({t})$ (41)
$v_{k0}$ is the initial velocity at time $t$
${v_{k0}}={\bf v}(t)$ (42)
$v_{k1}$ is the function to compute the derivative velocity in the period $h$
$v_{k1}={{\bf a}(t)}h$ (43)
$v_{k2}$ is the function to compute the derivative velocity of the Euler
integration in the period $h/2$ based on the previous step
$v_{k2}=v_{k0}+\frac{v_{k1}}{2}$ (44)
$v_{k3}$ is the function to compute the derivative velocity of the second
approximation based on the $v_{k2}$ in the period $h/2$
$v_{k3}=v_{k0}+\frac{v_{k2}}{2}$ (45)
$v_{k4}$ is the function to compute the final resulting velocity change of
$v_{k3}$ from $v_{k0}$
$v_{k4}=v_{k0}+v_{k3}$ (46)
Therefore, the new velocity of the particle is
${\bf
v}({t}+{h})=v_{k0}+\frac{1}{6}{h}({v_{k1}}+2\,{v_{k2}}+2\,{v_{k3}}+{v_{k4}})$
(47)
If we integrate the velocity vector over time, it gives us how the position
vector changed over this time.
###### Position
can be represented as the following equation
${\bf r}({t}+{h})\approx{\bf r}({t})+{h}{\bf
r}{{}^{\prime}}({t})+\frac{{{h}^{2}}}{2!}{\bf
r}{{}^{\prime\prime}}({t})+\frac{{{h}^{3}}}{3!}{\bf
r}{{}^{\prime\prime\prime}}({t})+\frac{{{h}^{4}}}{4!}{\bf
r}{{}^{\prime\prime\prime\prime}}({t})$ (48)
$r_{k0}$ is the initial position at time $t$
${r_{k0}}={\bf r}(t)$ (49)
$r_{k1}$ is the function to find the travel position in the period $h$
$r_{k1}={{\bf v}(t)}h$ (50)
$r_{k2}$ is the function to find the travel position of the Euler integration
in the period $h/2$ based on the previous step
$r_{k2}=r_{k0}+\frac{r_{k1}}{2}={\bf r}(t)+\frac{{{\bf v}(t)}h}{2}$ (51)
$r_{k3}$ is the function to find the travel position of the second
approximation based on the $r_{k2}$ in the period $h/2$
$r_{k3}=r_{k0}+\frac{r_{k2}}{2}$ (52)
$r_{k4}$ is the function to find the travel position change of $r_{k3}$ from
$r_{k0}$
$r_{k4}=r_{k0}+{r_{k3}}$ (53)
Therefore, the new position of the particle is
${\bf
r}({t}+{h})=r_{k0}+\frac{1}{6}{h}({r_{k1}}+2\,{r_{k2}}+2\,{r_{k3}}+{r_{k4}})$
(54)
### 5.3 Comparison of Three Integrators
#### 5.3.1 Efficiency
For a given step size, Euler is more efficient because it requires only one
derivative evaluation per step. Mid Point requires about twice as much
computation than the Euler integrator because Mid Point uses two steps to
calculate velocity and position at the next time. Runge Kutta Fourth Order
requires about four times as much computation as Euler integrator because it
use four steps to calculate the velocity and position at the next time step
[BD03]. For some configuration, if speed is the priority, Euler integration is
convenient to use, but at the expense of accuracy and stability.
#### 5.3.2 Accuracy
Smaller time steps means more stability and accuracy. But also means more
computation. If a given step size is $h$, error of Euler method is
$O({h}^{2})$ as a first-order method, error of midpoint is $O({h}^{3})$, and
error of RK 4 is $O({h}^{5})$ [BD03].
* •
The Euler method is based on keep the first two terms of the Taylor series
expansion
$y({t}+{h})=y({t})+{h}y^{\prime}({t})+O(h^{2})$ (55)
* •
An improved method which involves the second derivative is Midpoint method as
following
$y({t}+{h})=y({t})+{h}y^{\prime}({t})+\frac{{{h}^{2}}}{2!}y^{\prime\prime}({t})+O({h}^{3})$
(56)
* •
An improved method which involves the four derivative is Runge Kutta method as
following
$y({t}+{h})=y({t})+{h}{y}{{}^{\prime}}({t})+\frac{{{h}^{2}}}{2!}{y}{{}^{\prime\prime}}({t})+\frac{{{h}^{3}}}{3!}{y}{{}^{\prime\prime\prime}}({t})+\frac{{{h}^{4}}}{4!}{y}{{}^{\prime\prime\prime\prime}}({t})+O({h}^{5})$
(57)
#### 5.3.3 Stability
With smaller step time value, such as 10 ms, the system integrated by any of
the three methods is stable. However, if we give the system a higher step time
value, such as 50 or 100 ms, with same mass, damping coefficient, gravity
acceleration, the elastic object under Euler system will explode after a short
period because its numerical instability causes the mass to oscillate out of
control; midpoint and Runge Kutta Fourth Order integrator are more stable
[BD03].
## Chapter 6 Design and Implementation
In this chapter, we will present the detailed design of the two-layer elastic
object physical based simulation system and its implementation.
### 6.1 Elastic Object Simulation System Design
In this section, an overview of the framework and the algorithm for the
elastic simulation system is given.
#### 6.1.1 Domain Analysis-Based Modeling
Figure 1: Model-View-Controller
This elastic object simulation system has been designed and implemented
according to the well known architectural pattern, Model-View-
Controller[Wik07]. This pattern is ideal for real time simulation because it
simplifies the dynamic tasks handling by separating data (Model) from user
interface (View). Thus, the user’s interaction with the software does not
impact the data handling; the data can be reorganized without changing the
user interface. The communication between the Model and the View is done
through another component: Controller. In our current simulation system, the
application has been split into these three separated components:
* •
Model is an application of object modeling. It stores the geometric modeling
methods of the elastic objects and the data of the objects themselves, such as
one-dimensional, two-dimensional, and three-dimensional elastic objects and
their associated data structure, such as vector, particles, springs, and
faces.
* •
View is the screen presentation to render the Model and a user interface for
dynamical simulation. The view in my system is the GLUT window which displays
the elastic object and allows the user to use mouse and keyboard to interact
with the elastic object.
* •
Controller handles the processes and responds from the user interaction and
invokes the changes to the model. When the user interacts with the elastic
object through the GLUT window by dragging it with mouse, the controller
handles the new dragging force from the user interface, integrates the new
force to find out the change of the acceleration and velocity, and where the
object should move to in next display update. This is done through the series
of registered GLUT callback functions that process the input from the user.
### 6.2 Elastic Object Simulation System Implementation
The system is implemented using OpenGL and the C++ programming language with
object oriented programming paradigm. Figure 2 describes structure of the
software based on the classes.
Figure 2: Class Diagram
* •
The three data structures, such as particle, spring, and face compose an
elastic object.
* •
The elastic object types can be varied by the dimensionality: one-, two-, or
three-dimensional.
* •
The types of integrators are also varied by their complexities, such as Euler,
Midpoint, and Runge-Kutta.
* •
An “Object” instance contains an instance of an “Integrator”. The relationship
between them is aggregation rather than a common composition because when the
elastic object is destroyed, the integrator object is not necessary destroyed.
The “Object” has an aggregation of the “Integrator” by containing only a
reference or pointer to the “Integrator”.
* •
The classes “Object”, “ViewSpace”, and “Integrator” are associated to each
other based the Model-View-Controller model.
Let’s have a close view at each model and the related classes with their
parameters and member functions.
#### 6.2.1 Design and Implementation of Data Types
Figure 3: Face-Spring-Particle Class Diagram
The basic data structure is the object vector, which defines the the scalar
value with direction. For the second basic data structure, particle, whose
properties, such as position, velocity are made up of the object vector. The
next higher data structure is spring, which is defined by two particle
objects. Face, which is the highest data structure in this simulation system,
is composed of three connected springs.
###### Particle
In Figure 3, the particle class shows that each particle has mass $mass$,
position ${\bf r}$, velocity ${\bf v}$, derivative of position ${\bf dr}$,
derivative of velocity ${\bf dv}$, and force vector ${\bf f}$. Particle
constructor sets up its properties with default values.
###### Spring
As shown in Figure 3, the spring class, there are different types of springs
to construct the object, such as structural, radius, shear-left, and shear-
right springs, declared in the enum type $spring\\_type$ and the default
spring type is structural. $*sp1$ is the head of the spring and points to a
particle; $*sp2$ is the tail of the spring and points to a particle. $restLen$
is the spring length when it is in the resting state. $ks$ is Hooke’s spring
constant and $kd$ is the spring damping factor. The spring normal vector will
be calculated and needed in pressure force calculation.
###### Face
In Figure 3, the face class shows that a face contains $*fp1$, $*fp2$, and
$*fp3$ point to the first, the second, and the third particles as three of its
vertices. It also contains $*fs1$, $*fs2$, and $*fs3$ point to the first,
second, and third spring as three of its edges. There are two face
constructors. The first one stores the information of three vertices that
point to three particles. It represents faces on two-dimensional objects. The
faces will only be needed at the display process.
Figure 4 represents another face constructor along with its algorithm
implementation. It accepts three vertices on each face that point to the three
particles, and constructs a spring and stores the spring information into the
spring vector. This constructor is called by three-dimensional uniform
modeling method. The index of face is the key data structure for subdivision
method in subroutine. The constructor initializes the three springs based on
the three particles. First spring contains particle $p1$ and $p2$; the second
spring contains particle $p2$ and $p3$; the third spring contains particle
$p3$ and $p1$. A special care is taken not to duplicate existing springs
(which would result in incorrect behaviour of the model); therefore, we only
allow the new and non-existing springs to be saved in the spring vector. If
the first spring already exists with particles $p1$ and $p2$, the new spring
$fs1$ will point to the existing spring. Same method is applied on the second
spring $fs2$ and third spring $fs3$. Otherwise, the new spring will be pushed
and saved into the spring vector. Please refer to the actual code for the
complete implementation.
Face(Particle *Ap1, Particle *Ap2, Particle *Ap3, vector<Spring*> &springs)
: fp1(Ap1), fp2(Ap2), fp3(Ap3) {
fs1 = new Spring(Ap1, Ap2); fs2 = new Spring(Ap2, Ap3); fs3 = new Spring(Ap3, Ap1);
bool a = false, b = false, c = false;
for(int o = 0; o < springs.size(); o++) {
if(springs[o]->sp1 == Ap1Ψ&& springs[o]->sp2 == Ap2) {
delete fs1; fs1 = springs[o]; a = true;
}
if(springs[o]->sp1 == Ap2Ψ&& springs[o]->sp2 == Ap3) {
delete fs2; fs2 = springs[o]; b = true;
}
if(springs[o]->sp1 == Ap3Ψ&& springs[o]->sp2 == Ap1) {
delete fs3; fs3 = springs[o]; c = true;
}
}
if(!a) springs.push_back(fs1);
if(!b) springs.push_back(fs2);
if(!c) springs.push_back(fs3);
}
Figure 4: Special 3D Uniform Modeling Face Constructor
#### 6.2.2 Design and Implementation of Components: Model
Figure 5: Model Object Class Diagram
The class “Object” is the base class for elastic object of any supported
dimensionality. It contains the most common data structure and properties of
an elastic object. The geometric complexity is increased according to the
dimensions. The “Object1D” inherits from the parent class “Object”, “Object2D”
inherits from “Object1D”, and “Object3D” inherits from “Object2D”. This type
of inheritance hierarchy is in place because when each dimensionality is
added, the new object type depends on some of the previous implementation and
the new things that come with each additional dimension. For example, 1D
object has a notion of structural springs varying in a single dimension; 2D
takes the notion of structural springs and augments it with radius and shear
springs as well as the notion of pressure inside an enclosed object; 3D
extends 2D by adding the notion of face subdivision and volume making object
more dynamic in terms of run-time number of vertices (to make it more or less
smooth depending on the trade off between quality and performance). All
objects share the same $Update()$/$Draw()$ mechanism, which is used by the
OpenGL state machine to update all the vertices of an object in the Model and
reflect the changes in the View by drawing the deformations in real-time.
###### Object
As shown in Figure 5, the object class, an elastic object contains a particle
object, a spring object, a face object, and an integrator object. The data
structure varies from inner to outer layers, for example, the pointers to the
particles on the inner layer and on the outer layer of the object are saved in
different data vectors. $SetObject()$ constructs the geometric shape of the
elastic object, which, in turn, constructs the particles $SetParticles()$ and
connects the particles by the structural springs via the
$Add\\_Structural\\_Spring()$ call. The enum type $dimensionality$ has one of
the values $(DIM1D,DIM2D,DIM3D)$ to determine the object’s dimensionality
type: 1D, 2D, or 3D; the enum type $integrator\\_type$ determines which type
of integrator the simulation system uses, Euler, Midpoint, or Runge Kutta
Fourth Order integrator. Such design allows extension to add new integrators
and select existing integrators at run-time. The variable $closest_{i}$ is the
closest point on the outer layer to mouse position and $FindClosestPoint()$ is
the function to find such a particle (used in dragging force application when
dragging the object across the simulation window). The function $Update()$
modifies the simulated object’s state (either each time point when idle or
application of the drag force by the user), and determines the object’s
overall forces, velocity, position in the next time step. $Draw()$ visualizes
the object after each update.
void Idle() {
object1D.Update(DT, mousedown != 0, xMouse, yMouse);
object2D.Update(DT, mousedown != 0, xMouse, yMouse);
object3D.Update(DT, mousedown != 0, xMouse, yMouse);
glutPostRedisplay();
}
Figure 6: $Idle()$ Model Updates
void Object::Update(float deltaT, bool drag, float xDrag, float yDrag) {
if(integrator == NULL) {
switch(integratorType) {
case EULER:
integrator = new EulerIntegrator(*this);
break;
case MIDPOINT:
integrator = new MidpointIntegrator(*this);
break;
case RK4:
integrator = new RungeKutta4Integrator(*this);
break;
default:
assert(false);
return;
}
integrator->setDimension(dim);
}
integrator->integrate(deltaT, drag, xDrag, yDrag);
}
Figure 7: General $Update()$ Function
In the main simulation, the $Idle()$ function shown in Figure 6, elastic
objects update at every time step $DT$ to tell the the system how the objects
behave and the change for their velocity and position. There are four
parameters for $Update()$ as shown in Figure 7, the time step $deltaT$, if
there exists user interaction $drag=0$ by default, the mouse position on $x$
and $y$ axises (for dragging upon mouse release) is at 0 by default. The
general algorithm of the $Update()$ presented, illustrates that the most of
the actual modifications are based on the dynamically selected integrator and
the dimensionality of the simulation object being integrated. If in the
feature a new integrator is added, this function has to be updated to account
for it in the framework.
###### 1D Object
In Figure 5, the “Object1D” class shows that an one-dimensional object
contains two particles and one spring. The type of particles is
$outer\\_points$ and spring type is structural $outer\\_springs$.
###### 2D Object
In Figure 5, the “Object2D” class shows that an two-dimensional object
contains inner and outer layers. The type of particles is $inner\\_points$ and
$outer\\_points$. The spring type is structural $inner\\_springs$ and
$outer\\_springs$; moreover, there are another three new types of springs,
$radius\\_springs$, $shear\\_springs\\_left$, and $shear\\_springs\\_right$.
The function $Add\\_Structural\\_Spring()$ models the shape of the inner
circle by connecting $inner\\_springs$ and the outer circle by connecting the
$outer\\_springs$ separately. $Add\\_Radius\\_Spring()$ adds the radius
springs with the inner point $i$ and outer point $i$. $Add\\_Shear\\_Spring()$
adds the left shear springs with inner point $i$ and outer point $i+1$ and the
right shear springs with inner point $i+1$ and outer point $i$. The variable
$pressure$, which is an additional inner force compared to “Object1D”, is at
each spring along its normal.
###### 3D Object
In Figure 5, the “Object3D” class shows that a three-dimensional object uses
similar method as a two-dimensional object by extending the variables into the
$z$ axis. However, there are two methods introduced to create a three-
dimensional object, such as $nonunitsphere()$ and $SetObject()$, which uses
iteration to define an uniform sphere. The base shape for subdivision a sphere
is defined in $Octahedron()$ and $Iteration()$ computes the coordinates of the
newly generated particles and springs based on the level of detail, the
variable $Iterations$.
#### 6.2.3 Design and Implementation of Components: Controller
Figure 8: Integrator Framework Class Diagram
The types of integrators are varied by their complexities, such as Euler,
Midpoint, and Runge-Kutta. The common attributes and methods are defined in
the parent class “Integrator”, as shown in Figure 8. The subclasses
“EulerIntegrator”, “MidpointIntegrator”, and “RungeKuttaIntegrator” inherit
the super classes based on the complexity. The Euler integrator is a basic
building block for other integrators which provides the first step of
computation of $k_{1}$ in $k1()$. Midpoint integrator uses Euler’s $k1()$
implementation and provides the 2nd step, $k_{2}$ implemented in $k2()$.
Finally, the RK4 integrator adds the last two refinement steps $k_{3}$
(function $k3()$) and $k_{4}$ (function $k4()$) in addition to what Euler and
midpoint have provided. Thus, RK4 implementation depends on the midpoint
which, in turn, depends on the Euler integrator with different parameters.
void Integrator::integrate(float deltaT, bool drag, float xDrag, float yDrag) {
dragExists = drag; mDragX = xDrag; mDragY = yDrag;
AccumulateForces();
Derivatives(deltaT, 1.0);
}
...
void Integrator::AccumulateForces() {
ExternalForces();
SpringForces();
switch(dim) {
case DIM1D:
break;
case DIM2D:
case DIM3D:
PressureForces();
break;
}
}
Figure 9: General $integrate()$ and $AccumulateForces()$ Functions
In Figure 9 there is a general $integrate()$ function (which is called from
$Object::Update()$) and a general $AccumulateForces()$ function, both of which
play a vital role in the integrator framework in this thesis. They illustrate
the general algorithm of integration applied to the Model’s data: first, the
effect of all the forces is accumulated (which includes external forces, such
as gravity and drag, as well as forces induced by springs and pressure); then,
the integrator-specific derivation is performed to each particle of an object.
In the general “Integrator” the $Derivatives()$ function is pure virtual as is
left to be overridden by the “EulerIntegrator”, “MidpointIntegrator”, and
“RungeKutta4Integrator” concrete implementations. It is important to note that
the reverse forces are also accounted at the collision detection at the end of
each $Derivatives()$ implementation. Another note worth mentioning is that the
pressure forces are not applicable in the 1D case as there is no enclosed
object, which can hold pressure in this cases. $ExternalForces()$ checks for
the existence of the mouse drag force (from the user) as well as gravity and
sums them up. $SpringForces()$ accumulates contributions for all spring types
(a subject to dimensionality as well, e.g. 1D case does not have radius or
shear springs, only one structural spring).
#### 6.2.4 Simulation Loop Sequence
The sequence diagram in Figure 10 describes the control-flow of the simulation
sequence and logic of the elastic object simulation system. The following
sequence of steps describes all of the possible states of the elastic object
as events occur in greater detail. There we track the different states how the
physical simulation loop works, such as display of the objects, accumulation
of forces, integration of forces, and so on. In other words, this is the main
algorithm of the entire simulation system.
Figure 10: Simulation Loop Sequence Diagram
* •
Step 1: “ViewSpace” initializes the virtual world and provides the user an
interactive environment. It provides the interface to allow user to drag the
object, or choose the parameters. For example, user can choose the object
type, one-dimensional, two-dimensional, or three-dimensional. User can choose
the integrator type, Euler, Midpoint, or Runge Kutta 4. User can set up the
springs’ stiffness, damping variable, and the pressure.
* •
Step 2: $SetObject()$ function creates an elastic object based on the
interface variable set from Step 1.
* •
Step 3: $SetParticles()$ function sets up the particles’ position and their
other initial properties, such as mass and velocity.
* •
Step 4: $AddSprings()$ function connects particles with springs according to
their index.
* •
Step 5: $AddFaces()$ connects the springs with faces based on proper index.
This step will be ignored if the object is one-dimensional.
* •
Step 6: $SetIntegratorType()$ function tells the Controller which integrator
users select through the interface.
* •
Step 7: $Update()$ updates the integrator’s time step.
* •
Step 8: $Integrate()$ contains two functions, $AccumulateForces()$ and
$Derivatives()$. It is based on all the object geometric information modeled
and all the forces information accumulated, to integrate over the time step to
get new object position and orientation.
* •
Step 9: $AccumulateForces()$ state is to sum up the forces accumulated on each
particle.
* •
Step 10: $GravityForce()$ is to accumulate gravity force based on the
particles’ masses.
* •
Step 11: $MouseForce()$ is the external force from the interface when user
interacts with the object. It will be added or subtracted from the particles
depends on the force’s direction.
* •
Step 12: $SpringForce()$ is to accumulate internal force of the particles
connected by springs.
* •
Step 13: $PressureForce()$ is to accumulate the internal pressure acted on the
particles. For one-dimensional object, this state is omitted.
* •
Step 14: $Derivatives()$ does the real derivative computation of acceleration
and velocity in order to get new velocity and position of elastic objects
based on the integrator type defined by users.
* •
Step 15: $CollisionForce()$ is to check if the object is out of boundaries
after the integration state. If the new position is outside of the boundary,
then it will be corrected and reset on the edge of the boundary. Moreover, the
new collision force will be added to the object.
* •
Step 16: $Draw()$ displays the object with new position, velocity, and
deformed shape.
## Chapter 7 Experimental Results
In this chapter, the one-dimensional, two-dimensional, and three-dimensional
objects are illustrated at different animation sequences, with different
simulation parameters, and by simulation with different numerical integration
methods.
### 7.1 Animation Sequence
The screenshots in this section present the animation sequence of the one-
dimensional, two-dimensional, and three-dimensional objects when they are at
the initial state, colliding with floor, bouncing back from the floor,
responding to user’s external dragging, and at the resting state.
#### 7.1.1 1D
This simulation shows two masses connected with one spring. The one-
dimensional object moves in a three-dimensional environment, which consists of
ceiling, walls, and floor. Users can drag the mass with the mouse to change
the object’s position and direction. Figure 1(a) presents the initial state of
the object; Figure 1(b) shows the object collides with the floor when it drops
with gravity force; Figure 1(c) displays the collision response of the object
based on the penalty method; Figure 1(d) shows the moment when users drag the
object; Figure 1(e) shows how the object reacts on the external impact, such
as mouse dragging force or bouncing force with walls; Figure 1(f) displays the
object resting on the floor after a while when there is no interaction from
the user.
(a) The initial state
(b) Collide with floor
(c) Bounce back from the floor
(d) Drag the object
(e) Response to compact
(f) The resting state
Figure 1: Animation Sequence of One Dimensional Elastic Object
#### 7.1.2 2D
The simulation as shown in Figure 2(a) through Figure 2(f) is how a two-
dimensional object moves in a three-dimensional environment. This two-layer
object consists of 10 particles and 10 structural springs on both inner and
outer circles. Moreover, it contains 10 radius springs, 10 shear left springs,
and 10 shear right springs between the inner and outer layers. If a two-
dimensional object with only one layer, or the object has no pressure force
within, the spring’s stiffness has to be a larger value than without, then the
object will not collapse. However, as shown in Figure 2(b), if the spring
stiffness is small enough, the object does not collapse, neither overlap with
the layers because of the stability of the two-layer structure.
(a) The initial state
(b) Collide with floor
(c) Bounce back from the floor
(d) Drag the object
(e) Response to compact
(f) The resting state
Figure 2: Animation Sequence of Two Dimensional Elastic Object
#### 7.1.3 3D
The simulation as shown in Figure 3(a) through Figure 3(f) is how a three-
dimensional uniform facet object moves in a three-dimensional environment.
This two-layer object, which is generated by subdividing an octahedron once,
consists of 12 particles, 36 structural springs, and 32 faces, on both inner
and outer spheres. Moreover, the object also contains 36 radius springs, 36
shear left springs, and 36 shear right springs between the inner and outer
layers. Just like in two dimensions, the two-layer structure gives the three-
dimensional sphere more stability.
(a) The initial state
(b) Collide with floor
(c) Bounce back from the floor
(d) Drag the object
(e) Response to compact
(f) The resting state
Figure 3: Animation Sequence of Three Dimensional Elastic Object
### 7.2 Simulation Parameters
The parameters in the simulation such as mass, spring stiffness, and friction
(damping) can be changed. One can drag the object mass with a mouse to change
its position. Effects of different simulation parameters are discussed.
#### 7.2.1 Summary of the Adjustable Parameters
The parameters that influence the behavior of the simulated environment are
summarized below, with their default values. Most initial and default values
were based on the 2D case from [Mat03]; otherwise, the values are empirical
and are partially dependent on the hardware the simulation is executing on.
* •
KS = 800.0f where KS is structural spring stiffness constant. The larger this
value is, the less elastic the object is and it is more resistant to the inner
pressure and deformation. The lesser this value is the more object is
deformable and a subject to break up if the inner pressure force is high.
* •
KD = 15.0f where KD is structural spring damping constant, opposite to the
spring retraction force. It denotes how fast the object is to resist its
motion.
* •
RKS = 700.0f where RKS is radius and shear spring stiffness constant, similar
to KS, but for radius and shear springs as opposed to the structural springs.
* •
RKD = 50.0f where RKD is radius and shear spring damping constant, similar to
KD, but for radius and shear springs.
* •
MKS = 150.0f where MKS is the spring stiffness constant of the spring
connected with the mouse and the approximate nearest particle on the object.
This constitutes the elasticity of the “drag” spring connected to the mouse:
the lesser the value is, the more elastic it is, and the harder it is to drag
the object as a result.
* •
MKD = 25.0f where MKD is the damping constant of the spring connect with the
mouse and the approximate nearest point on the object.
* •
PRESSURE = 20.0f where PRESSURE is gas constant used in the ideal gas equation
mentioned earlier to determine the pressure force inside the enclosed object.
If this constant is too high, and the combined spring stiffness for all the
spring types is low enough, the object can “blow up”.
* •
MASS = 1.0f where MASS is the mass for each particle. The object can be made
heavier or lighter if this value is larger or smaller respectively, in order
to experiment with the gravity effects. Naturally, the heavier objects will be
more difficult to drag upwards in the simulation environment. Conversely, the
smaller-mass object can be dragged around with less effort given the rest of
the parameters remain constant.
#### 7.2.2 Stability vs. Time Step
First, the figures in this section (Figure 4(a), Figure 4(b), and Figure 4(c))
show the stability of the three integrators. We consider the integration time
step parameter in these scenarios only, assuming all the other parameters
(discussed later) are not change for the described simulations. As shown in
those figures, when the time step is small, such as $DT=0.003$111This is an
empirical value; dependent on the performance of the hardware., three of the
integrators behave well and the object does not “blow up”. However, when one
increases the time step by a factor of 10 to $DT=0.03$, the midpoint (see
Figure 5(b)) and RK4 (see Figure 5(c)) integrators are still stable and the
object integrated with Euler integrator “blows up” as in Figure 5(a).
Furthermore, when the time step is increased 10-fold more to $DT=0.3$, only
the object integrated with RK4 (see Figure 6(c)) is stable and another two
objects integrated with Euler (Figure 6(a)) and Midpoint (Figure 6(b)) methods
“blow up”.
(a) The object integrated with Euler Method
(b) The object integrated with Midpoint Method
(c) The object integrated with RK4
Figure 4: Elastic Object at Timestep = 0.003
(a) The object integrated with Euler Method
(b) The object integrated with Midpoint Method
(c) The object integrated with RK4
Figure 5: Elastic Object at Timestep = 0.03
(a) The object integrated with Euler Method
(b) The object integrated with Midpoint Method
(c) The object integrated with RK4 Method
Figure 6: Elastic Object at Timestep = 0.3
#### 7.2.3 Efficiency and Accuracy
The more computational effort is required, the less efficient algorithm is.
Likewise, the more accurate algorithm is, the more computation effort it
requires, the less efficient it is. Thus, in our simulation system the most
efficient and least accurate integration method is Euler’s, followed by
Midpoint (about twice as more accurate and slower), followed by RK4 (four
times slower than Euler’s and the most accurate of the three). This can be
illustrated in Figure 4(a), Figure 4(b), and Figure 4(c) running concurrently
with the same time step of $0.003$, where one can see the simulation with
Euler’s method reaches the floor fastest and RK4 slowest. Of course, the
efficiency of the simulation and the accuracy of the shape and movement
depends on the amount of particles (and as a result, all kinds of springs) in
the object.
### 7.3 Computational Errors
This section briefly summarizes the error accumulated in the application of
the described algorithms and their effects.
#### 7.3.1 Collision Detection
We have applied the Penalty Method in our simulation system. This simple but
inaccurate algorithm causes the object to “stick” on the collision surface
when dragging the object at the same time and it may become difficult to drag
the object away for a period of time.
#### 7.3.2 Subdivision Method
The spherical shape is not perfect round because the number of springs
associated to each particle is not uniform. If one wants more quality
subdivision has to be done in more than one subdivision operation, but the
simulation may rapidly become very slow as the number of particles grow
requiring a much greater computational effort, which is suitable only for the
high-end hardware if one wishes to do it in real-time. In Figure 7 is an
example of the two iterations of the subdivision.
Figure 7: Second Subdivision Iteration
## Chapter 8 Conclusion and Future Work
This chapter describes our contribution based on the existing elastic model
and analyzes the possible development and related work in the future.
### 8.1 Contribution
The new model, two-layer elastic object with uniform-surfaces is a simple,
efficient approach to imitate the liquid effects of elastic object, such as
human’s tissue and soft body. Since the modeling and structure of the tissue
kind elastic object is closer to real tissue than an one layer object, the
level of realism has been increased. The images in this chapter are
screenshots from the elastic simulation system we have developed. The modeling
method and the density setting provides significant improvements on the
conflicts of accuracy and interactivity on previous models. The realism of the
results, such as liquid motion and inertia effects are also enhanced.
###### Procedural Modeling
We have applied the procedural modeling method with particle system to model
elastic objects. From simple one-dimensional to most complicated three-
dimensional object, we introduced the modeling method for different
dimensional objects and related physics knowledge gradually. In the elastic
object simulation system, each particle has its local coordinate which is easy
to be computed at every time step. Moreover, this modeling method can
efficiently control the level of detail as required by graphics artists and
computer hardware available. As shown in Figure 1(a) and Figure 1(b), this
modeling method also most approximately approaches the ideal equal faces;
therefore, the edges(springs) on the faces and the forces on each particle are
approximately to be equal at initial state in order to minimize the
computation error caused by the object geometry.
(a) Two-dimensional Object
(b) Three-dimensional Object
Figure 1: Uniform Shape Modeling
###### Density
As shown in Figure 2(a) and Figure 2(b), the density is defined only for each
particle on the elastic surface and the internal density is represented by air
pressure physics equation. The weights of particles on inner and outer layer
can be set differently. For example, a balloon half filled with liquid, the
bottom is heavier than the top part because the density is at the bottom is
liquid and top part is air. The weights on inner layer can be set much heavier
than outer layer. This special feature gives us flexibilities to imitate
different material effects with such simple model.
(a) Two-dimensional Object
(b) Three-dimensional Object
Figure 2: Non-Uniform Density
###### Inertia
Inertia effect is a unique effect in two layer-elastic simulation system,
which can not be achieved with one-layer object. Figure 3(a) and Figure 3(b)
show the inertial movement of a two-dimensional and three-dimensional elastic
object. In Figure 3(a), the inner layer and the outer layer have the opposite
internal force drive them along axis x. Since the two layers are connected by
springs, the inner particles and outer particles have an extra force applied
on them, interactive force between inner and outer particles. And their
movement, position, and acceleration will be computed according to the
contribution of this extra interactive force. This interactive force does not
exist in a single layer object. Figure 3(b) displays the moment when the
elastic object drops down onto the ground. The outer and inner particles will
fall with the object based on their gravity and springs force. Here, the
inertia for inner particle and outer particle are dependent not only on the
force from their own motion, the force from the neighbors on the same layer,
but also from the interaction on the other layer. This simulation system is
more accurate to describe the inertia property happened in the liquid object.
(a) Two-dimensional Object
(b) Three-dimensional Object
Figure 3: Liquid Motion and Inertia
###### Stability
The two layered system is stable. Even without the internal pressure force,
the shape will not collapse because the two layers are connected by different
types of springs. The simulation system works well even with the very
inaccurate Euler integrator at large time step, which will result shape
collapse or blow up on a one-layer object with the same set of values. We have
also implemented the higher level integrators, such as Midpoint and Runga
Kutta 4.
###### Re-usability
The design of this simulation system is based on well-known software design
pattern. It decomposes the novel concepts into concrete small components. The
functions and classes are easy to be plugged and adapted into other program.
This elastic simulation model simplifies the physical modeling method with a
group of masses and springs. Also, the simulation is computed in real time
based on the numerical integration of the physical laws of dynamics.
### 8.2 Conclusion
We have developed a one-dimensional elastic object, a two-layer two-
dimensional elastic object, and extend it into three-dimensions. These models
are all physically based, making use of results from gravity and pressure
forces and are implemented with three types of integrations: Euler, Midpoint,
and Runge Kutta Fourth Order. The procedural uniform surface generation
algorithm provides a convenient mechanism for collision detection. It can
generate convincing behaviors when the objects collide with rigid floors or
walls because all the particles are checked in every update cycle. Moreover,
the rendering is fast because graphics software and hardware renders
triangular facets very efficiently.
### 8.3 Future Work
###### Character Animation
The functionality development of elastic simulation modeling for 3D software
design and implementation has emerged as a new challenge in computer graphics.
One of the existing software with the elastic modeling functionality is Maya,
which provides shape deformation, especially facial animation, for a group of
objects. It is more convenient than traditional frame animation. However, the
elastic object movement is not attached to skeleton animation. Furthermore,
this elastic simulation is not in real time.
A possible future work that can be done based on the elastic simulation is to
define a skeleton system and to map the mesh body onto it. The different parts
of the body can be defined as the different freedom of deformable based on the
elasticity. For example, the mesh is less elastic on the arms, legs; the mesh
is more elastic on the areas that consist fats, like breast, belly. The weight
of the elastic property of the muscles can be mapped and dynamically set
according to the skeleton. The system can be integrated into advanced
animation software as a Plug-in.
###### Collision Detection
between soft objects is a complex phenomenon, which has not been widely
developed in physics. In our current system, we are using the penalty methods
[MJ88], which do not generate the contact surface between the interacting
objects. This method uses the amount of inter-penetration for computing a
force which pushes the objects apart instead. Even though the result is fair
enough based on estimation, in reality, the contact surfaces should be
generated rather than local inter-penetrations. Especially, if we want to use
computer animation to imitate organ surgery and help surgeon practice as if
interact with real objects, the penalty method is no longer appropriate. There
must be a more accurate algorithm to define the collision between rigid body
and soft body, or soft body to soft body. Our software should be able to
describe other soft body deformation, such as fractures.
## Bibliography
* [Ang03] Edward Angel. Interactive Computer Graphics: A top-Down Approach Using OpenGL. Addison-Wesley, 2003.
* [BA97] D. Baraff and A.Witkin. Physically based modeling: Principles and practice. SIGGRAPH97 ACM, Course 34, August 1997.
* [BD03] William E. Boyce and Richard C. Diprima. Elementary Differential Equations and Boundary Value Problems. John Wiley and Sons, Inc, 2003.
* [Bou92] Paul Bourke. Sphere Generation. http://local.wasp.uwa.edu.au/~pbourke/modelling_rendering/sphere_cylind%er, 1992.
* [Bou97] Paul Bourke. Solving Systems Defined by Differential Equations. http://local.wasp.uwa.edu.au/~pbourke/modelling_rendering/solver/, 1997\.
* [Bou98] Paul Bourke. Particle System Example. http://local.wasp.uwa.edu.au/~pbourke/modelling_rendering/particle/inde%x.html, 1998.
* [Bri98] M. James Bridge. The way balls really bounce. volume 33, pages 236–241. Physics Education, November 1998.
* [BW98] David Baraff and Andrew Witkin. Large steps in cloth simulation. pages 43–54. SIGGRAPH 98 Conference Proceedings, July 1998.
* [CGCP02] Steve Capell, Seth Green, Brain Curless, and Zoran Popvic. Collisions and deformations: A multi resolution framework for dynamic deformations. pages 41–48. Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation, July 2002.
* [DL02] Nixon D. and R. Lobb. A fluid-based soft-object model. volume 22, Iss. 4, pages 68–75. Comp. Graph. and App. IEEE, July 2002\.
* [DMB00] Desbrun, M.P.Cani, and A. Barr. Adaptive simulation of soft bodies in real-time. pages 133–144. Computer Animation, May 2000.
* [DW92] D.Baraff and A. Witkin. Dynamic simulation of non-penetrating flexible bodies. volume 26, No. 2, pages 303–308. Computer Graphics, 1992.
* [FW98] Jessica Fitch and J. Reed Walker. Deformable ball simulation. http://vorlon.case.edu/~jrw24/CG/, 1998.
* [GM97] F. Gibson and Brain Mirtich. A survey of deformable models in computer graphics. Mitsubishi Electric Research Laboratories TR97 -19, November 1997.
* [Gro01] Peter Grogono. A Latex2e Gallimaufry Techniques, Tips, and Traps. Department of Computer Science, Concordia University, 2001.
* [Gro02a] Peter Grogono. Getting Started with OpenGL. Department of Computer Science, Concordia University, 2002.
* [Gro02b] Peter Grogono. Lecture Notes of Advanced Computer Graphics. Department of Computer Science, Concordia University, 2002.
* [HB02] Donald Hearn and M.Pauline Baker. Computer Graphics. Prentice Hall, 2002.
* [Hec96a] Chris Hecker. Behind the screen: Physics, part 2: Angular effects. pages 14–22. Game Developer Magazine, December 1996.
* [Hec96b] Chris Hecker. Physics, the next frontier. pages 12–20. Game Developer Magazine, October 1996.
* [Hec97a] Chris Hecker. Physics, part 3: Collision response. pages 11–18. Game Developer Magazine, March 1997.
* [Hec97b] Chris Hecker. The third dimension. pages 17–18. Game Developer Magazine, June 1997.
* [HO99] J.K Hodgins and J.F. O’Brien. Computer animation. volume 3, pages 686–690. The Wiley Encyclopedia of Electrical and Electronics Engineering, John G. Webster Ed., 1999.
* [JS03] J.Teran and S.Blemker. Cloth and deformable bodies: Finite volume methods for the simulation of skeletal muscle. pages 68–74. Euro.Symp. on Comp.Anim. (SIGGRAPH Proc), 2003.
* [KCH00] Y.M. Kang, J.H. Choi, and H.G.Cho. Fast and stable animation of cloth with an approximated implicit method. pages 247–262. Computer Graphics International, 2000.
* [Ker07] Erwin Keryszig. Advanced Engineering Mathematics. John Wiley and Sons, Inc., 2007.
* [KO03] Kazuya G. Kobayashi and Katsutoshi Ootsubo. Ffd: free-form deformation by using triangular mesh. pages 226–234. Proceedings of the eighth ACM symposium on Solid modeling and applications, 2003.
* [Lan99a] Jeff Lander. Devil in the blue faceted dress: Real-time cloth animation. pages 17–21. Game Developer Magazine, May 1999.
* [Lan99b] Jeff Lander. Lone game developer battles physics simulator. pages 15–17. Game Developer Magazine, April 1999.
* [LH02] Matti Larsson and Christian Holmboe. VR Dynamics - A primer. http://www.cs.umu.se/kurser/TDBD07/VT02/uppsproj/resultat/uppsats/Matti%&Christian/VR_Dynamics_023.doc, 2002.
* [Lin06] Ming C. Lin. Lecture for Class: Physically-Based Modeling, Simulation and Animation. Department of Computer Science College of Arts and Sciences, The University of North Carolina at Chapel Hill, 2006.
* [Mat03] Maciej Matyja. Pressure Model of Soft Body Simulation. SIGRAD2003 conference in UMEA (Sweden) paper, November 2003.
* [Mat04a] Maciej Matyja. How to Implement a Pressure Soft Body Model. http://panoramix.ift.uni.wroc.pl/~maq/soft2d/index.php, 2004.
* [Mat04b] Maciej Matyja. Inverse Dynamic Displacement Constraints in Real-Time Cloth and Soft-Body Models in Graphics Programming Methods. Charles River Media, Inc., 2004.
* [McE05] Kevin J. McElwee. Keratin - Introduction to skin and hair biology. http://www.instylehair.com/modules.php?name=News&file=article&sid=7, 2005\.
* [MDMJ02] Matthias Muller, Julie Dorsey, Leonard McMillan, and Robert Jagnow. Collisions and deformations: Stable real-time deformations. pages 49–54. Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation, July 2002.
* [MJ88] M.Moore and J.Wilhelms. Collision detection and response for computer animation. volume 22, No. 4, pages 289–298. Computer Graphics, August 1988.
* [Oli01] Gustavo Oliveira. Exploring Spring Models. www.gamasutra.com/features/20011005/oliveira_01.htm, 2001.
* [Pla88] Barr Platt. Constraint methods for flexible models. volume 22, No. 4, pages 279–288. SIGGRAPH88, August 1988.
* [Pro95] Xavier Provot. Deformation constrains in a spring-mass model to describe rigid cloth behavior. pages 147–154. Graphics Interface, May 1995.
* [Ree83] William T Reeves. Particle systems - a technique for modeling a class of fuzzy objects. volume 17, pages 359–376. Computer Graphics, 1983.
* [TF96] Thomas and Finney. Calculus with Analytic Geometry. Addison-Wesley, 1996.
* [TF98] Demetri Terzopoulos and Kurt Fleischer. A survey of deformable models in computer graphics, volume 7. The Visual Computer, November 1998.
* [TJ84] Frank Thomas and Ollie Johnston. Disney Animation: The Illusion of Life. Abbrville Press, 1984.
* [TJF89] Demetri Terzopoulos, J.Platt, and K. Fleischer. From gloop to glop: heating and melting deformable objects. pages 219–226. Proceedings of Graphics Interface, 1989.
* [TM92] Demetri Terzopoulos and Dimitri Metaxas. Dynamic deformation of solid primitives with constraints. volume 26, No. 2, pages 269–278. ACM SIGGRAPH Computer Graphics, July 1992.
* [TPBF87] D. Terzopoulos, J. Platt, A.H. Barr, and K.W. Fleischer. Elastically deformable models. 21(4):205–214, 1987.
* [TW88] Demetri Terzopoulos and Andrew Witkin. Physically-based models with rigid and deformable components. In Proc. Graphics Interface, pages 146–154, June 1988.
* [TW90] Demetri Terzopoulos and K. Waters. Physically-based facial modeling, analysis, and animation. volume 1, pages 73–80. Journal of Visualization and Computer Animation, 1990.
* [Wag07] Sean Wagstaff. Maya Complete 6: Industry-Strength Visual-Effects Application Upgrades Tools, Adds Photoshop Support. http://www.macworld.com/2004/12/reviews/mayacomplete6/index.php, 2007.
* [WB93] A. Witkin and D. Baraff. An introduction to physically based modeling. SIGGRAPH Course Notes 32, 1993.
* [Wik07] Wikipedia. Procedural Modeling. http://en.wikipedia.org/wiki/, 2007.
* [Wit97] Andrew Witkin. Particle System Dynamics. SIGGRAPH97 Course Notes http://www.cs.cmu.edu/afs/cs/user/baraff/www/sigcourse/notesc.pdf, 1997\.
* [WM86] G. Wyvill and C. McPheeters. Data Structures for Soft Objects, volume 22, No. 3. The Visual Computer, April 1986.
|
arxiv-papers
| 2009-07-24T19:13:02 |
2024-09-04T02:49:04.151908
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Miao Song",
"submitter": "Miao Song",
"url": "https://arxiv.org/abs/0907.4364"
}
|
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