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+{"text":"\\section{Introduction}\nThe quasi-one-dimensional (Q1D) charge order, \nwhich is well-known as the stripe order, has been \none of the significant issues in high-T$_{\\mathrm{c}}$ cuprates (HTC) for \nrecent years. The incommensurate peaks observed by the neutron \nscattering in La$_{\\mathrm{2-x}}$Ba$_{\\mathrm{x}}$CuO$_4$ \nare to be explained on the assumption that the static \nstripe order is present around 1\/8-filling.\\cite{Tranquada1995} The presence \nof the static stripe order in the ground state has been proved \ntheoretically by the density matrix renormalization group method on the basis \nof the Q1D {\\it t}-{\\it J} model.\\cite{White1998} \nOriginally, since the beginning of HTC \nstudies many experimentalists have pointed out the so-called 1\/8-problem in \nLa$_{\\mathrm{2-x}}$Ba$_{\\mathrm{x}}$CuO$_4$, i.e., the remarkable suppression \nof superconducting transition temperature at $x \\sim 1\/8$. \nMany theorists have \nthought that the strong on-site Coulomb interaction makes electrons \nalmost localized with the long-period spin and charge correlations \naround 1\/8-filling. They have studied these long-period correlations by the\n numerical or analytical methods on the basis of the \ntwo-dimensional (2D) Hubbard models and have clarified \nthat some long-period correlated \nstates can be the ground states around certain fillings.\n\\cite{Poilblanc1989,Zaanen1989,Kato1990,Salkola1996,Mizokawa1997,Machida1999,Ichioka1999,Kaneshita2001,Varlamov2002,Yanagisawa2002} These theoretically predicted long-period correlation \nstates are consistent with \nthe stripe state deduced from the neutron scattering experiments.\\cite{Tranquada1999} \n\nThe theoretical results on the basis of the 2D Hubbard models \nmean that the strong \non-site interaction is enough to stabilize the stripe state near 1\/8-filling \nas far as the ground state is considered. \nHowever, at finite temperature the strong \nfluctuations characteristic to the low dimensional strongly \ncorrelated electron systems exist. These strong fluctuations are composed of \nthe many elements, e.g., the anti-ferromagnetic (AF) fluctuation and \nthe charge ordering fluctuations with certain classes of super-lattice \nstructures. These strong fluctuations \naccompany with the instabilities of the spin and charge orderings \nand simultaneously tend to interrupt them. \nThe stripe with such strong fluctuations in metallic state \nis called a dynamical stripe.\\cite{Zaanen1996,Zaanen2000,Zaanen2003,Hasselmann2002} \nThe angle resolved photo-emission spectroscopy (ARPES) shows that \nthe stripes in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\\delta}$ has more dynamic \ncharacter than those in La$_{\\mathrm{2-x}}$Sr$_{\\mathrm{x}}$CuO$_4$. \nThe discrimination whether the stripe is static or dynamic is to be \njudged from the characteristic energy of the fluctuation mode. \nIf the fluctuation mode has low characteristic energy, \ni.e., the stripe fluctuates slowly, the mode \ncan easily couple with phonon and the static stripe long-range \norder will appear. On the other hand, \nif the fluctuation mode has high characteristic energy, \ni.e., the stripe fluctuates fast, the mode cannot couple with phonon and \nthe system will stay metallic together with the fluctuation. \nWhen we consider these fluctuations in the analysis of these models \nwithout the long-ranged Coulomb interaction, \nwe can expect that the metallic state should be recovered \nat higher temperature. In this state, however, \nthe electronic property is to be still affected by the stripe instability \nas far as its dynamical factor is investigated . \n\nIn this paper, we analyze the 2D three-band Hubbard model \non the basis of the unrestricted fluctuation exchange approximation (UFLEX). \nUFLEX can consider a strong correlation effect and a number of \nthe spatial inhomogeneities without any assumptions about the ground state. \nThese considered effects contain not only a static component \nbut a dynamic one. As a result, the electronic state \nreproduced on the basis of UFLEX is affected by \na number of spatially inhomogeneous fluctuations simultaneously. \nIn fact, when we calculate the one-particle spectral functions \nand the spin correlation functions for our fully self-consistent solutions, \nwe can find that the one-particle spectra of the electronic state \nreflect these exotic state accompanied with the anti-ferromagnetic \nor the further long-periodic instabilities. Especially, near $1\/8$-filling \nthe spin and charge correlation functions show the existence of the \nQ1D fluctuation at finite frequency. This fast Q1D fluctuation is originated \nfrom the striped spatial inhomogeneity. Thus, \nour results suggest that the spatial inhomogeneities should be concerned \nin the strongly correlated electron system when the electronic behavior \nat a finite frequency is discussed. \n\n\\section{2D three-band Hubbard model}\n\nOur 2D three-band Hubbard model Hamiltonian, $H$, is composed of {\\it d}-electrons at each Cu site and \n{\\it p}-electrons at O site. We consider only the on-site Coulomb repulsion $U$ between \n{\\it d}-electrons at each Cu site. Then, $H$ is divided \ninto the non-interacting part, $H_0$, and the interacting part, $H_1$, as \n\\begin{equation}\nH=H_0+H_1%\n-\\mu \\sum_{{\\mathbf k} \\sigma}\n(d_{{\\mathbf k} \\sigma}^{\\dagger}d_{{\\mathbf k} \\sigma}\n+p_{{\\mathbf k} \\sigma}^{x \\dagger}p_{{\\mathbf k} \\sigma}^x\n+p_{{\\mathbf k} \\sigma}^{y \\dagger}p_{{\\mathbf k} \\sigma}^y).\n\\end{equation} \nHere $d_{{\\mathbf k} \\sigma}(d_{{\\mathbf k} \\sigma}^{\\dagger})$ and \n$p_{{\\mathbf k} \\sigma}^{x(y)}(p_{{\\mathbf k} \\sigma}^{x(y) \\dagger})$ are \nthe annihilation (creation) operator for {\\it p}- and \n{\\it p}$^{x(y)}$-electron of momentum ${\\mathbf k}$ and spin $\\sigma$, \nrespectively. $\\mu$ is the chemical potential.\nThe non-interacting part $H_0$ is represented by \n\\begin{equation}\nH_0= {\\sum_{{\\mathbf k} \\sigma}}\n\\left(\nd_{{\\mathbf k} \\sigma}^\\dagger \\, \np_{{\\mathbf k} \\sigma}^{x \\dagger} \\, \np_{{\\mathbf k} \\sigma}^{y \\dagger}\n\\right) \\left(\n\\begin{array}{ccc} \\Delta_{dp} & \\zeta_{\\mathbf k}^x  & \\zeta_{\\mathbf k}^y \\\\ \n-\\zeta_{\\mathbf k}^x & 0 & \\zeta_{\\mathbf k}^p \\\\\n-\\zeta_{\\mathbf k}^y & \\zeta_{\\mathbf k}^p & 0 \\\\\n\\end{array} \\right)\\!\n\\left( \\begin{array}{c} d_{{\\mathbf k} \\sigma} \\\\ p_{{\\mathbf k} \\sigma}^x\n\\\\ p_{{\\mathbf k} \\sigma}^y \\\\\n\\end{array} \\right),\n\\end{equation}\nwhere $\\Delta_{dp}$ is the hybridization gap energy\nbetween {\\it d}- and {\\it p}-orbitals. \nWe take the lattice constant of the square\nlattice formed of Cu sites as the unit of length, and we can represent\n$\\zeta_{\\mathbf k}^{x(y)}=2{\\rm i}\\,t_{dp} \\sin \\frac{k_{x(y)}}{2}$ and\n$\\zeta_{\\mathbf k}^p=-4t_{pp} \\sin \\frac{k_x}{2} \\sin \\frac{k_y}{2}$,\nwhere $t_{dp}$ is the transfer energy between a {\\it d}-orbital and a\nneighboring {\\it p}$^{x(y)}$-orbital and $t_{pp}$ is that between a\n{\\it p}$^x$-orbital and a {\\it p}$^y$-orbital. In this\nstudy, we take $t_{dp}$ as the unit of energy. The residual part,\n$H_1$, is described as\n\\begin{equation}\nH_1= \\frac{U}{N} \\sum_{{\\mathbf k} {\\mathbf k}^\\prime} \\sum_{\\mathbf q}\n        d_{{\\mathbf k}+{\\mathbf q} \\uparrow}^{\\dagger} d_{{\\mathbf\n        k}^\\prime- {\\mathbf q} \\downarrow}^{\\dagger} d_{{\\mathbf k}^\\prime\n        \\downarrow} d_{{\\mathbf k} \\uparrow}.\n\\end{equation}\nwhere $U$ is the on-site Coulomb repulsion between {\\it d}-orbitals and\n$N$ is the number of ${\\mathbf k}$-space lattice points in the first\nBrillouin zone (FBZ). \n\n\\section{UFLEX}\n\nWe have developed UFLEX in order to analyze the spatially inhomogeneous \nsystem. In our UFLEX formulation we introduce 'cluster' momentum \nbesides 'usual' momentum. The notion of this cluster momentum is \nsuggested from the one in dynamical cluster approximation, \nin which the nonlocal correlations \nare considered beyond dynamical mean field theory.\n\\cite{Hettler1998,Maier2000,Imai2002}\n\nHowever, our cluster momentum is \nnot directly related with a coarse graining of the Brillouin zone. When \nthe most stable state of our system is spatially homogeneous, only a cluster \nmomentum, ${\\mathbf K}={\\mathbf 0}$, is enough to obtain \nthe {\\it true} solution for our problem. In that case, \nUFLEX is consistent to 'usual' FLEX.\\cite{Bickers1989} \nThe advantage of our introduction of cluster momenta is \nits flexibility in the correspondence to the problem. \nIf the problem have some important classes of the spatial inhomogeneities, \nwe will only need to provide the cluster momenta \ncorresponding to those classes. For example, when \nthe most important instability is the anti-ferromagnetic one, \ntwo cluster momenta, ${\\mathbf K}={\\mathbf 0}$ and ${\\mathbf K}\n=(\\frac{\\pi}{2},\\frac{\\pi}{2})$, are enough. Of course, the stripe \ninstabilities have their characteristic \ncluster momenta. Thus, when we anticipate to encounter \nthe stripe instabilities in our problem, we need to provide \nonly those cluster momenta.\n\nFirst, we define the unrestricted perturbed Green function, \n$G_{d\\,\\mathbf{K}}^{\\,\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n)$.\nHere we use the abbreviation of Fermion Matsubara frequencies, \n$\\epsilon_n=\\pi T(2n+1)$ with integer $n$, where $T$ is the temperature. $\\mathbf{K}$ indicates \nthe cluster momentum. The set, $\\{\\mathbf{K}\\}$, is chosen so that \nany sum, $\\mathbf{k}+\\mathbf{K}$, exists in the set, $\\{\\mathbf{k}\\}$. \n$G_{d\\,\\mathbf{K}}^{\\,\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n)$ is calculated by the Dyson equation :\n\\begin{equation}\n\\left[G_{d\\,\\mathbf{K}}^{\\,\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n)\n\\right]^{-1}=\\delta_{\\mathbf{K}} \\cdot \n\\left\\{G_d^{\\,\\sigma\\,(0)}\n(\\mathbf{k},\\mathrm{i}\\epsilon_n) \\right\\}^{-1}\n-\\Sigma_{\\mathbf{K}}^\\sigma(\\mathbf{k},\\mathrm{i}\\epsilon_n), \\\\\n\\label{eq:1}\n\\end{equation}\nwhere $\\delta_{\\mathbf{K}}$ means Kronecker's delta, \ni.e. $\\delta_{\\mathbf{K}=\\mathbf{0}}=1$ \nand $\\delta_{\\mathbf{K} \\neq \\mathbf{0}}=1$.\n$\\left[\\cdots\\right]^{-1}$ indicates \nthe inverse operation defined so as to satisfy the identity : \n\\begin{equation}\n\\delta_{\\mathbf{K}}= \\sum_{\\mathbf{K}^\\prime}G_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,\\sigma}\n(\\mathbf{k}+\\mathbf{K}^\\prime-\\mathbf{K},\\mathrm{i}\\epsilon_n)\n\\left[G_{d\\,\\mathbf{K}^\\prime}^{\\,\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n)\\right]^{-1}\n\\end{equation}\nfor all ${\\mathbf{k}}$ and $n$.  $G_d^{\\,\\sigma\\,(0)}(\\mathbf{k},\\mathrm{i}\\epsilon_n)$ \nin Eq.~(\\ref{eq:1}) is the unperturbed Green function derived by diagonalizing of $H-H_1$ as \n\\begin{align}\n& \\left\\{G_d^{\\,\\sigma\\,(0)}(\\mathbf{k},z-\\mu)\\right\\}^{-1}= \\nonumber \\\\\n& z-\\Delta_{dp}-\\frac{2z(2-\\cos k_x-\\cos k_y)+\n      8t_{pp}(1-\\cos k_x)(1-\\cos k_y)}\n{z^2-4t_{pp}^2(1-\\cos k_x)(1-\\cos k_y)}.\n\\end{align}\nIn order to estimate our unrestricted self-energy, $\\Sigma_{\\mathbf{K}}^\\sigma\n(\\mathbf{k},\\mathrm{i}\\epsilon_n)$, in Eq.~(\\ref{eq:1}), \nwe adopt the UFLEX as follows.\n\\begin{align}\n& \\Sigma_{\\mathbf{K}}^\\sigma\n(\\mathbf{k},\\mathrm{i}\\epsilon_n) = \\nonumber \\\\\n& \\frac{TU}{N}\\sum_{\\mathbf{q}\\,n^\\prime}\nG_{d\\,\\mathbf{K}}^{\\,-\\sigma}(\\mathbf{k}-\\mathbf{q},\n\\mathrm{i}\\epsilon_{n^\\prime})\\left.e^{ \\mathrm{i}\\epsilon_{n^\\prime}\\eta}\n\\right|_{\\eta \\rightarrow +0} \\nonumber \\\\\n& +\\frac{T}{N}\n\\sum_{\\mathbf{q}\\,\\mathbf{K}^\\prime\\,m}\nG_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,\\sigma}\n(\\mathbf{q}-\\mathbf{k}+\\mathbf{K}^\\prime-\\mathbf{K},\n\\mathrm{i}\\omega_m-\\mathrm{i}\\epsilon_n)\nV_{\\mathbf{K}^\\prime}^{(\\mathrm{pp})}\n(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\nonumber \\\\\n& +\\frac{T}{N}\n\\sum_{\\mathbf{q}\\,\\mathbf{K}^\\prime\\,m}\\left[\nG_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,-\\sigma}(\\mathbf{k}-\\mathbf{q},\n\\mathrm{i}\\epsilon_n-\\mathrm{i}\\omega_m)\nV_{\\mathbf{K}^\\prime}^{\\sigma\\,(\\mathrm{ph1})}(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\right. \\nonumber \\\\\n& \\hspace{5.5em}+\\left. G_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,\\sigma}(\\mathbf{k}-\\mathbf{q},\n\\mathrm{i}\\epsilon_n-\\mathrm{i}\\omega_m)\nV_{\\mathbf{K}^\\prime}^{\\sigma\\,(\\mathrm{ph2})}(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\right],\n\\label{eq:2}\n\\end{align}\n\\begin{align}\n\\label{eq:4}\n& V_{\\mathbf{K}}^{(\\mathrm{pp})}(\\mathbf{q},\\mathrm{i}\\omega_m) = \\nonumber \\\\ \n& U^2\\phi_{\\mathbf{K}}(\\mathbf{q},\\mathrm{i}\\omega_m) \\nonumber \\\\ \n& -U^2\\sum_{\\mathbf{K}^\\prime}\n\\phi_{\\mathbf{K}-\\mathbf{K}^\\prime}(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\left[\\,\n\\delta_{\\mathbf{K}^\\prime}\n+U\\phi_{\\mathbf{K}^\\prime}\n(\\mathbf{q}+\\mathbf{K}-\\mathbf{K}^\\prime,\\mathrm{i}\\omega_m)\\right]^{-1}, \\\\\n\\label{eq:5}\n& V_{\\mathbf{K}}^{\\sigma(\\mathrm{ph1})}(\\mathbf{q},\\mathrm{i}\\omega_m) = \\nonumber \\\\\n& -U^2\\chi_{\\mathbf{K}}^{\\sigma\\, -\\sigma}\n(\\mathbf{q},\\mathrm{i}\\omega_m) \\nonumber \\\\ \n&+U^2\\sum_{\\mathbf{K}^\\prime}\n\\chi_{\\mathbf{K}-\\mathbf{K}^\\prime}^{\\sigma\\, -\\sigma}\n(\\mathbf{q},\\mathrm{i}\\omega_m)\n\\left[\\delta_{\\mathbf{K}^\\prime}\n-U\\chi_{\\mathbf{K}^\\prime}^{\\sigma\\, -\\sigma}\n(\\mathbf{q}+\\mathbf{K}-\\mathbf{K}^\\prime,\\mathrm{i}\\omega_m)\n\\right]^{-1}, \n\\end{align}\nand\n\\begin{align}\n\\label{eq:6}\n& V_{\\mathbf{K}}^{\\sigma \\,(\\mathrm{ph2})}(\\mathbf{q},\\mathrm{i}\\omega_m) = \\nonumber \\\\\n& U^2\\sum_{\\mathbf{K}^\\prime}\n\\chi_{\\mathbf{K}-\\mathbf{K}^\\prime}^{-\\sigma -\\sigma}\n(\\mathbf{q},\\mathrm{i}\\omega_m) \\nonumber \\\\\n& \\times \\left[\\,\\delta_{\\mathbf{K}^\\prime}\\right.\n-\\!U^2\\!\\sum_{\\mathbf{K}^{\\prime \\prime}}\n\\chi_{\\mathbf{K}^\\prime-\\mathbf{K}^{\\prime \\prime}}^{\\sigma \\sigma}\n(\\mathbf{q}+\\mathbf{K}-\\mathbf{K}^\\prime,\\mathrm{i}\\omega_m) \\nonumber \\\\\n& \\hspace{7em}\n\\times \\left. \\chi_{\\mathbf{K}^{\\prime \\prime}}^{-\\sigma -\\sigma}\n(\\mathbf{q}+\\mathbf{K}-\\mathbf{K}^{\\prime \\prime},\\mathrm{i}\\omega_m)\n\\right]^{-1},\n\\end{align}\nwhere $\\omega_m=2m\\,\\pi T$ with integer $m$ are Boson Matsubara frequencies, \n\\begin{align}\n\\label{eq:3}\n& \\phi_{\\mathbf{K}}(\\mathbf{q},\\mathrm{i}\\omega_m)= \\nonumber \\\\ \n& \\frac{T}{N}\\,\\sum_{\\mathbf{k} \\mathbf{K}^\\prime\\, n}\nG_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\,\\sigma}(\\mathbf{q}-\\mathbf{k}, \n\\mathrm{i}\\omega_m-\\mathrm{i}\\epsilon_n)\n\\,G_{d\\,\\mathbf{K}^\\prime}^{-\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n),\n\\end{align}\nand\n\\begin{align}\n\\label{eq:7}\n& \\chi_{\\mathbf{K}}^{\\sigma \\sigma^\\prime}(\\mathbf{q},\\mathrm{i}\\omega_m)= \\nonumber \\\\ \n& -\\frac{T}{N}\\sum_{\\mathbf{k} \\mathbf{K}^\\prime\\,n}\nG_{d\\,\\mathbf{K}-\\mathbf{K}^\\prime}^{\\sigma}\n(\\mathbf{q}+\\mathbf{k},\\mathrm{i}\\omega_m+\\mathrm{i}\\epsilon_n)\n\\,G_{d\\,\\mathbf{K}^\\prime}^{\\sigma^\\prime}(\\mathbf{k}-\\mathbf{K}^\\prime\n,\\mathrm{i}\\epsilon_n) .\n\\end{align}\nIn order to obtain the solution satisfying the conserving law, \nwe need to solve \nEqs.~(\\ref{eq:1},\\ref{eq:2},\\ref{eq:4},\\ref{eq:5},\\ref{eq:6},\\ref{eq:3},\n\\ref{eq:7}) fully self-consistently. This self-consistent procedure is \nshown in Fig.~\\ref{figure:10} diagrammatically.\n\\begin{figure}\n\\includegraphics[width=6.1cm]{fig1.eps}\n\\caption{The self-consistent procedure in UFLEX. \n$i$ represents the iteration count, and $\\varepsilon$ is the small value.}\n\\label{figure:10}\n\\end{figure}\n\nIn numerical calculations we divide the FBZ into $16 \\times 16$ meshes. \nWe take $8\\times 8$ cluster momenta so that we can reproduce a \nhomogeneous state, inhomogeneous states \nwith 2-, 4-, or 8-lattice period charge density wave (CDW) \nor spin density wave (SDW) along \n$a(x)$-axis or $b(y)$-axis, or an AF state. We prepare $2^{11}=2048$ \nMatsubara frequencies for temperature $T=0.020 \\sim 180{\\mathrm{K}}$. \nThe other parameters :\n$t_{dp}=1.0 \\sim 0.80\\,{\\mathrm{eV}}$, \n$t_{pp}=0.60 \\sim 0.48\\,{\\mathrm{eV}}$, and \n$\\Delta_{dp}=0.70 \\sim 0.56\\,{\\mathrm{eV}}$, \n$U=4.0 \\sim 3.2\\,{\\mathrm{eV}}$, which are all common for our results.\n\n\\section{Fermi surface evolution with hole- or electron-doping}\n\\label{Fermi}\n\nWe have obtained the fully self-consistent solutions for \n$0.000 \\leq \\delta_{\\mathrm{h}} \\leq 0.057$, \n$0.112 \\leq \\delta_{\\mathrm{h}} \\leq 0.231$, \nand $0.000 \\leq \\delta_{\\mathrm{e}} \\leq 0.249$, \nwhere $\\delta_{\\mathrm{h}} \\equiv 1-n_{\\mathrm{d}}-n_{\\mathrm{p}}$ and   \n$\\delta_{\\mathrm{e}} \\equiv n_{\\mathrm{d}}+n_{\\mathrm{p}}-1$. \n$n_{\\mathrm{d}}$ and $n_{\\mathrm{p}}$ represent the number of \n$d$- and $p$-electrons, respectively. \nWe could not obtain any convergent solution for \n$0.057 < \\delta_{\\mathrm{h}} < 0.112$. According to the elaborated neutron scattering experiments on La$_{\\mathrm{2-x-y}}$Ba$_{\\mathrm{x}}$Sr$_{\\mathrm{y}}$CuO$_4$,\\cite{Fujita2002PRB,Fujita2002PRL,Tranquada2004} \nin this region the inhomogeneous states with longer \nthan 8-lattice period SDW along diagonal, which is called diagonal \nstripe state, could be realized. Our present \ncalculation would be short of provided cluster momenta to reproduce \nthe inhomogeneous states with the diagonal fluctuations. Hereafter, we \nrestrict our discussion to the states with the vertical fluctuations. \n\nIn this section we investigate the Fermi \nsurface evolution with doping in detail. In order to determine the Fermi \nsurface, we define the one-particle spectral \nweight as \n\\begin{equation}\nA^{\\sigma}(\\mathbf{k},E) \\equiv -\\frac{1}{\\pi}\\mathrm{Im}\n\\sum_{\\zeta=d,\\,p^x,p^y}\\left.\nG_{\\zeta\\,\\mathbf{0}}^\\sigma(\\mathbf{k},\\mathrm{i}\\epsilon_n)\n\\right|_{\\mathrm{i}\\epsilon_n \\rightarrow E},\n\\label{eq:8}\n\\end{equation}\nwhere\n\\begin{align}\n& G_{p^{x,y}\\,\\mathbf{0}}^\\sigma(\\mathbf{k},z-\\mu) \\nonumber \\\\\n&  = \\left\\{z[z-\\Delta_{dp}-\\Sigma_{\\mathbf{0}}^\\sigma(\\mathbf{k},z)]-2(1-\\cos k_{x,y})\\right\\} \\nonumber \\\\ \n& \\hspace{1em}\\times\\!\\left\\{[z^2-4t_{pp}^2(1-\\cos k_x)(1-\\cos k_y)][z-\\Delta_{dp}-\\Sigma_{\\mathbf{0}}^\\sigma(\\mathbf{k},z)]\\right. \\nonumber \\\\\n& \\hspace{2.5em} \n\\left.-2z(2-\\cos k_x-\\cos k_y)\\!-\\!8t_{pp}(1-\\cos k_x)(1-\\cos k_y)\\right\\}\n^{\\!-1}. \n\\label{eq:9}\n\\end{align}\nIn Eq.~(\\ref{eq:8}) we use Pad$\\acute{\\mathrm{e}}$ approximation \nfor the method of analytic continuation. In order to \ndetermine the Fermi surface on the basis of our calculated results, \nwe should calculate the one particle spectral weight at $E=0$, \n$A^{\\sigma}(\\mathbf{k},0)$, and find the $\\mathbf{k}$-points at which \n$A^{\\sigma}(\\mathbf{k},0)$ becomes large. \nBecause the thermal fluctuation with CDW, SDW, or anti-ferromagnetic \ninstabilities can generate the branched energy dispersion, we show \nour calculated results as follows. We choose the both $\\mathbf{k}$-points \non which the one-particle spectral weight at $E=0$ has \nthe largest and the second largest peaks for fixed $k_x$ \nas far as $0 < k_y \\leq k_x < \\pi$. And we repeat this operation for the other \nparts of FBZ. In the following figures, the points on which \nthe one-particle spectral weight at $E=0$ has\nthe largest and the second largest peaks are to be \nindicated with black and gray circles, respectively. \nOur results have been obtained at high temperature, and the \nthermal fluctuations make the Fermi level vague. \nHowever, if we have empty spaces surrounded by these points, \nwe can expect the branched energy dispersion near the Fermi level. The \nbranched energy dispersion is made by a spatial inhomogeneous instability. \nFurthermore, these points can suggest the Fermi surface connectivities at lower \ntemperature. \n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig2.eps}\n\\caption{Fermi surface evolutions near $1\/8$-filling. \nLeft sides : (a) for $\\delta_{\\mathrm{h}}=0.112$. \n(b) for $\\delta_{\\mathrm{h}}=0.127$. \nRight sides : The gray-scale images interpolated from raw data are attached \non account of the guide to the eye. The brighter spots have the larger spectral weights. (c) Illustration of nesting vectors. Dashed lines represent the planes connected with one another by the nesting vectors, $(\\pi,\\pi)$ or $(\\pi,-\\pi)$. Dotted lines represent the planes connected with one another by the nesting vectors, $(\\pi\/2,0)$ or $(0,\\pi\/2)$. \nThe illustration of nesting vectors \nis also applicable to the other figures representing \nFermi surface evolutions.}\n\\label{figure:1}\n\\end {figure}\nFirstly, we should mention our results for the hole-doped states \naround $1\/8$-filling, in which the dynamical stripe state might be realized. \nAs shown in Fig.~\\ref{figure:1}, \nnear $1\/8$-filling we have eight empty spots around the intersections \nof the planes indicated by dashed and dotted lines, which relate \nto the anti-ferromagnetic and the 4-lattice period \nCDW instabilities, respectively. \nIt means that the one-particle spectral weights are lost due \nto the fluctuations both of the anti-ferromagnetic instability and of \nthe 4-lattice period CDW instability. \nThese instabilities occur when some parts of the Fermi surface are \nconnected with one another by the nesting vectors. The parts of Fermi \nsurface connected with one another by the nesting vectors, $(\\pi,\\pi)$ \nor $(\\pi,-\\pi)$, are called 'hot spots'. On hot spots the quasi-particles \nplay important roles in the transport phenomena. \nTherefore, we expect some anomalous electronic behaviors near $1\/8$-filling, \nwhen the one-particle spectral weights are lost on hot spots. \nThis is the reason why so-called $1\/8$-problem occurs around $1\/8$-filling. \n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig3.eps}\n\\caption{Fermi surface evolutions for lightly-hole-doped states. (a) for $\\delta_{\\mathrm{h}}=0.039$. (b) for $\\delta_{\\mathrm{h}}=0.057$.}\n\\label{figure:2}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig4.eps}\n\\caption{Fermi surface evolutions for near half-filling. (a) for $\\delta_{\\mathrm{h}}=0.022$. (b) for half-filling. (b) for $\\delta_{\\mathrm{e}}=0.021$.}\n\\label{figure:4}\n\\end {figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig5.eps}\n\\caption{Fermi surface evolutions for lightly-electron-doped states. (a) for $\\delta_{\\mathrm{e}}=0.045$. (b) for $\\delta_{\\mathrm{e}}=0.066$. (b) for $\\delta_{\\mathrm{e}}=0.086$.}\n\\label{figure:5}\n\\end {figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig6.eps}\n\\caption{Fermi surface evolutions for heavily-electron-doped states. (a) for $\\delta_{\\mathrm{e}}=0.106$. (b) for $\\delta_{\\mathrm{e}}=0.144$. (b) for $\\delta_{\\mathrm{e}}=0.200$.}\n\\label{figure:6}\n\\end{figure}\nSecondly, we show our results for lightly-hole-doped states, \nin which the anti-ferromagnetic instability might exist. \nIn this case we have no more eight empty spots but four empty spots \ncentered on $(\\pm\\pi\/2,\\pm\\pi\/2)$ as shown in Fig.~\\ref{figure:2}. These \nempty spots are called 'Fermi arc's, which is believed to appear when \nthe strong anti-ferromagnetic fluctuation exists. In fact, these Fermi arcs \nhave been observed in La$_{1.97}$Sr$_{0.03}$CuO$_4$ \nby ARPES and interpreted as the evidence of the \nanti-ferromagnetic long-range order.\\cite{TYoshida2003} In our calculating \nresults, the spectral weights in these Fermi arcs are not completely zero, and \nthe anti-ferromagnetic long-range order does not exist. However, the existence \nof these Fermi arcs should be originated from \nthe strong anti-ferromagnetic fluctuation, evolving \ntoward the anti-ferromagnetic long-range order.\n \nWhen the doped holes become less, electrons should \nlocalize due to strong coulomb repulsion among them in real materials. \nHowever, in our three-band Hubbard model electrons cannot localize even \naround near half-filling and the Fermi surface are remained, as \nshown in Fig.~\\ref{figure:4}. If we could reproduce the Fermi surface \nevolution for the whole doping region, \nthe one electron spectral weight at the \nFermi level should be almost vanished near half-filling. \nThis result means that our formulation on \nthe basis of UFLEX is not efficient to describe the Mott-Hubbard \nmetal-insulator transition. Unfortunately this is the \nside-effect of UFLEX, in which we overestimate the thermal fluctuations \nand recover the itinerancy of electrons. It would be our future issue to \ndescribe the localized electrons but with strong fluctuations appropriately. \n\nOur results for electron-doped states will \nbe highlighted. The ARPES have already clarified the doping \ndependence of the Fermi surface in \nNd$_{\\rm 2-x}$Ce$_{\\rm x}$CuO$_{4 \\pm \\delta}$.\\cite{Armitage2002,Armitage2003,Shen2004} Except for completely vanished Fermi surface near half-filling, \ntheir observed evolution of Fermi surface with electron doping are consistent with our calculated result. As shown in Fig.~\\ref{figure:5}, \nin lightly-electron-doped region the eight pockets emerge \naround the cross-sectional point of Fermi surface with FBZ boundaries. \nThis reflects that the vertical charge fluctuation exists in the \nlightly-electron-doped region \nas well as near $1\/8$-hole-doped region. We can guess that \nthe vertical charge fluctuation gets to play an important role \nwhen the localized electron recovers its itinerancy \nas doped electron increases. \nWhen the doped-electron density density increases more, \neach two of eight pockets on the same \nFermi surface section come close to each other and unite into one \nas shown in Fig.~\\ref{figure:6}. Thus, \nin heavily-electron-doped region we have four pockets \naround the cross-sectional point of Fermi surface with $k_x=\\pm k_y$. \nThis means that the diagonal spin fluctuation, which is \nnot completely anti-ferromagnetic, \nis strengthened instead of the vertical fluctuation. \nThis diagonal spin fluctuation can cause pseudo-gap phenomena in electron-doped \nhigh-T$_{\\rm c}$ as observed at low temperature.\\cite{Onose2004} \nIt contrasts with the case of the hole-doped region, and the difference \nshould be caused by both a large $t_{pp}$. Such a $t_{pp}$ makes \nthe big difference of the energy dispersions between the hole-doped \nregion and the electron-doped region. If the electron were localized, \nthe difference of the energy dispersions should \nappear as the different charge distributions. \nIn our results the electron could not be localized, \nbut the charge fluctuations appear in the different way between the hole-doped \nregion and the electron-doped region instead. Hence, we can insist that \nthe inhomogeneity of the strongly correlated system should be considered. \n\n\\section{Spin and charge correlations}\n\nIn this section we show how the Fermi surface evolution \nwith doping is related to the development \nof the spin and charge correlations. The momentum dependences of \nspin and charge correlation functions \nreflect the spin and charge spatial inhomogeneities, respectively. \n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig7.eps}\n\\caption{The doping dependence of $S(\\mathbf{q})$ at its maximum momentum \n$\\mathbf{Q}_{\\mathrm{max}}$.}\n\\label{figure:7}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig8.eps}\n\\caption{The momentum dependences of $D(\\mathbf{q})$ at hole-doped regions.}\n\\label{figure:8}\n\\end {figure}\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig9.eps}\n\\caption{The momentum dependences of $D(\\mathbf{q})$ at electron-doped regions.}\n\\label{figure:9}\n\\end {figure}\nAt first we sum up our obtained results \nabout the static spin correlations as both the incommensurabilities and \nthe intensities of the peaks in the calculated elastic neutron scattering. \nThey are shown in Fig.~\\ref{figure:7}, where\n\\begin{equation}\nS(\\mathbf{q}) \\equiv\n\\sum_{\\mathbf{K}}\n\\chi_{-\\mathbf{K}}^{+-}\n(\\mathbf{q},0)\n\\left[\n\\delta_{\\mathbf{K}}\n-U\\chi_{\\mathbf{K}}^{+-}(\\mathbf{q}-\\mathbf{K},0)\n\\right]^{-1}.\n\\end{equation}\nIn the hole-doped region the incommensurabilities $\\delta_{\\rm inc}$, \ndefined by \n\\begin{equation}\n\\delta_{\\mathrm{inc}} \n\\equiv \\mathrm{Max}\\{[\\mathbf{Q}_{\\mathrm{max}}-\n(\\pi,\\pi)]_x,[\\mathbf{Q}_{\\mathrm{max}}-(\\pi,\\pi)]_y\\},\n\\end{equation}\nare almost proportional to $\\delta_{\\rm h}$. This has been already observed by  \nthe neutron scattering experiment on La$_{\\mathrm{2-x-y}}$Ba$_{\\mathrm{x}}$Sr$_{\\mathrm{y}}$CuO$_4$.\\cite{Yamada1998,Fujita2002PRB,Fujita2002PRL} The ratio of \n$\\delta_{\\rm inc}$ to $\\delta_{\\rm h}$ in our calculating results are almost \nhalf of the one in these experiments. However, we believe that this deviation \nis originated from the difference between the real hole number \n$\\delta_{\\rm h}$ introduced in CuO$_2$ plane, which is correspond to the \none in our results, and the instoicheiometric number $\\delta$ used in \nthe analysis in Ref.~\\citen{Yamada1998}.\n\nOn the other hand, in the electron-doped region \nwe cannot recognize such a relationship between $\\delta_{\\mathrm{inc}}$ and \n$\\delta_{\\mathrm{e}}$. It suggests that our spin correlation functions do not \nsimply reflect the Fermi surface nesting but rather the existence of \ninhomogeneities as discussed in Section~\\ref{Fermi}.  \n\nIn order to clear our insistence, we calculate the static \ncharge correlation functions at some regions. The momentum dependences \nof these correlation functions are shown in \nFigs.~\\ref{figure:8} and \\ref{figure:9}, where\n\\begin{equation}\nD(\\mathbf{q}) \\equiv \n\\frac{T}{N}\\,\\sum_{\\mathbf{k} \\mathbf{K}^\\prime\\, n}\nG_{d\\,\\mathbf{K}^\\prime}^{\\,\\sigma}(\\mathbf{q}-\\mathbf{k}, \n-\\mathrm{i}\\epsilon_n)\n\\,G_{d\\,-\\mathbf{K}^\\prime}^{-\\sigma}(\\mathbf{k},\\mathrm{i}\\epsilon_n).\n\\end{equation}\nFocusing on Fig.~\\ref{figure:8}(b) and Fig.~\\ref{figure:9}(b), \nwe can recognize that $D(\\mathbf{q})$ are slightly enhanced around the \nsymmetric lines, $q_x=0$ and $q_y=0$. These momentum dependences of \n$D(\\mathbf{q})$ suggest that the two collective modes \nwith the momenta $(\\pm 1,0)$ and $(0,\\pm 1)$ exist. These collective modes \ncan be translated as the instabilities toward \nvertical charge orderings along $x$-axis and $y$-axis, respectively. \nIn fact, when $\\delta_{\\mathrm{h}}=0.127$ and \n$\\delta_{\\mathrm{e}}=0.066$, the Fermi surfaces have eight pockets around \n$(\\pm\\pi,\\pm\\pi\/4)$ and $(\\pm\\pi\/4,\\pm\\pi)$ as shown in Fig.~\\ref{figure:1}(b) \nand Fig.~\\ref{figure:4}(b), respectively. As discussed in \\ref{Fermi}, \nthese anomalous Fermi surfaces are caused from the charge inhomogeneities, \nwhich appear as shown in Fig.~\\ref{figure:8}(b) and Fig.~\\ref{figure:9}(b). \n\n\\section{Conclusion}\n \nIn this paper, we have obtained fully self-consistent solutions \nfor the 2D three-band Hubbard model on the basis of UFLEX, in \nwhich the inhomogeneous distribution of d-electrons are allowed. \nBoth in hole-under-doped and electron-doped regions the one-particle \nspectral weights behave anomalously. Furthermore, \nthe spin and charge correlation functions \nreflect the existence of super-lattice structures around 1\/8-filling. \nOur numerical solutions show that the metallic state with spatially \ninhomogeneous spin or charge fast fluctuation can exist at high-temperature. \nThe electronic state in one of our microscopically derived \nsolutions corresponds to the dynamical stripe state mentioned in some \npioneering works.\\cite{Zaanen1996,Zaanen2000,Zaanen2003,Hasselmann2002} \nIn our 2D model, the electrons cannot have \nthe long-range order at finite temperature because of the strong \nfluctuation characteristic of low-dimensionality. However, \nin three-dimensional real materials, the spatially inhomogeneous fluctuation \nmay tend to have the long-range order and form the striped state, \nobserved in the neutron scattering experiments. This long-range order \nformation in three-dimensional real materials might not be reproduced \nby our simple model in this paper. The existences of the \nphonon or the long-range Coulomb potential might have an \nimportant role for these long-range order formation. \nThe theoretical researches concerning these complicated factors are \nto be expected as the future problem.\n\n\\section*{Acknowledgments}\nThe authors are grateful to Professor K. Yamaji, Professor Y. Aiura, Professor H. Eisaki, Dr. I. Nagai, Dr. M. Miyazaki, and Dr. S. Koike for their invaluable comments. The computation in this work has been done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The early development of the code for this computation has been achieved by using the IBM RS\/6000--SP at TACC and VT-Alpha servers at NeRI in AIST.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this talk we review recent developments on the interaction of vector mesons with baryons and nuclei. For this purpose we use effective field theory using \na combination of effective Lagrangians to account for hadron interactions,  implementing exactly unitarity in coupled channels. This approach has turned out to be a very efficient tool to \n face many problems in Hadron Physics. Using this  coupled channel unitary approach, usually referred to as chiral unitary approach, because the Lagrangians used account for chiral symmetry, the \ninteraction of the octet of\npseudoscalar mesons with the octet of stable baryons has been studied and \nleads to $J^P=1\/2^-$\nresonances which fit quite well the spectrum of the known low lying resonances\nwith these quantum numbers \n\\cite{Kaiser:1995cy,angels,ollerulf,carmenjuan,hyodo,ikeda}. \n New resonances are sometimes predicted, the most notable\nbeing the $\\Lambda(1405)$, where all the\nchiral approaches find two poles close by \n\\cite{Jido:2003cb,Borasoy:2005ie,Oller:2005ig,Oller:2006jw,Borasoy:2006sr,Hyodo:2008xr,Roca:2008kr}, rather than one, for which \nexperimental support is presented in \\cite{magas,sekihara}.  Another step forward in this\n direction has been the interpretation\nof low lying $J^P=1\/2^+$ states as molecular systems of two pseudoscalar mesons and one baryon\n\\cite{alberto,alberto2,kanchan,Jido:2008zz,KanadaEn'yo:2008wm}. \n\nMuch work has been done using pseudoscalar\nmesons as building blocks, but more recently, vectors instead of\npseudoscalars are also being considered. In the baryon sector the\ninteraction of the $\\rho \\Delta$ interaction has been recently addressed in\n\\cite{vijande}, where three degenerate $N^*$ states and three degenerate\n$\\Delta$ states around 1900 MeV, with $J^P=1\/2^-, 3\/2^-, 5\/2^-$, are found. The extrapolation to SU(3) with the interaction of the vectors of the nonet with\nthe baryons of the decuplet has been done in \\cite{sourav}. The\nunderlying theory for this study is the hidden gauge formalism\n\\cite{hidden1,hidden2,hidden4}, which deals with the interaction of vector mesons and\npseudoscalars, respecting chiral dynamics, providing the interaction of\npseudoscalars among themselves, with vector mesons, and vector mesons among\nthemselves. It also offers a perspective on the chiral Lagrangians as limiting\ncases at low energies of vector exchange diagrams occurring in the theory. \n\n\n In the meson sector, the interaction of $\\rho \\rho$ within this formalism has\nbeen addressed in \\cite{raquel}, where it has been shown to lead  to the\ndynamical generation of the $f_2(1270)$ and $f_0(1370)$  meson resonances. The extrapolation to SU(3) of the work \nof \\cite{raquel} has been done in \\cite{gengvec}, where many resonances are\nobtained, some of which can be associated to known meson states, while there are\npredictions for new ones.\n\n  In this talk we present the results of the interaction of the nonet\n   of vector mesons  with the\n   octet of baryons \\cite{angelsvec}, which have been obtained\n using the unitary approach in coupled\nchannels. The scattering amplitudes lead to poles in the\ncomplex plane which can be associated to some well known resonances. Under the\napproximation of neglecting the three momentum of the particles versus their\nmass, we obtain degenerate states of $J^P=1\/2^-,3\/2^-$ for the case of the\ninteraction of vectors with the octet of baryons. This degeneracy \nseems to be followed qualitatively by the experimental spectrum, although in\nsome cases the spin partners have not been identified. Improvements in the theory follow from the consideration of the decay of these states into a pseudoscalar meson and a baryon, and some results are presented here. \n\nMoreover we report on composite states of hidden charm emerging from the interaction of vector mesons and baryons with charm.  \nWe shall also report on some states coming from the three body system \n $\\Delta \\rho \\pi$ which can shed some light on the status of some $\\Delta$ states of $J^P=5\/2^+$ in the vicinity of 2000 MeV. Finally, we devote some attention to new developments around vector mesons in a nuclear medium, specifically the $K^*$ and $J\/\\psi$ in the medium. \n\n\n\n \n\n\\section{Formalism for $VV$ interaction}\n\nWe follow the formalism of the hidden gauge interaction for vector mesons of \n\\cite{hidden1,hidden2} (see also \\cite{hidekoroca} for a practical set of Feynman rules). \nThe Lagrangian involving the interaction of \nvector mesons amongst themselves is given by\n\\begin{equation}\n{\\cal L}_{III}=-\\frac{1}{4}\\langle V_{\\mu \\nu}V^{\\mu\\nu}\\rangle \\ ,\n\\label{lVV}\n\\end{equation}\nwhere the symbol $\\langle \\rangle$ stands for the trace in the SU(3) space \nand $V_{\\mu\\nu}$ is given by \n\\begin{equation}\nV_{\\mu\\nu}=\\partial_{\\mu} V_\\nu -\\partial_\\nu V_\\mu -ig[V_\\mu,V_\\nu]\\ ,\n\\label{Vmunu}\n\\end{equation}\nwith  $g$ given by $g=\\frac{M_V}{2f}$\nwhere $f=93\\,MeV$ is the pion decay constant. The magnitude $V_\\mu$ is the SU(3) \nmatrix of the vectors of the nonet of the $\\rho$\n\\begin{equation}\nV_\\mu=\\left(\n\\begin{array}{ccc}\n\\frac{\\rho^0}{\\sqrt{2}}+\\frac{\\omega}{\\sqrt{2}}&\\rho^+& K^{*+}\\\\\n\\rho^-& -\\frac{\\rho^0}{\\sqrt{2}}+\\frac{\\omega}{\\sqrt{2}}&K^{*0}\\\\\nK^{*-}& \\bar{K}^{*0}&\\phi\\\\\n\\end{array}\n\\right)_\\mu \\ .\n\\label{Vmu}\n\\end{equation}\n\nThe interaction of ${\\cal L}_{III}$ gives rise to a contact term coming from \n$[V_\\mu,V_\\nu][V_\\mu,V_\\nu]$\n\\begin{equation}\n{\\cal L}^{(c)}_{III}=\\frac{g^2}{2}\\langle V_\\mu V_\\nu V^\\mu V^\\nu-V_\\nu V_\\mu\nV^\\mu V^\\nu\\rangle\\ ,\n\\label{lcont}\n\\end{equation}\n and on the other hand it gives rise to a three \nvector vertex from \n\\begin{equation}\n{\\cal L}^{(3V)}_{III}=ig\\langle (\\partial_\\mu V_\\nu -\\partial_\\nu V_\\mu) V^\\mu V^\\nu\\rangle\n\\label{l3V}=ig\\langle (V^\\mu\\partial_\\nu V_\\mu -\\partial_\\nu V_\\mu\nV^\\mu) V^\\nu\\rangle\n\\label{l3Vsimp}\\ ,\n\\end{equation}\n\nIn this latter case one finds an analogy with the coupling of vectors to\n pseudoscalars given in the same theory by \n \n\\begin{equation}\n{\\cal L}_{VPP}= -ig ~tr\\left([\nP,\\partial_{\\mu}P]V^{\\mu}\\right),\n\\label{lagrVpp}\n\\end{equation}\nwhere $P$ is the SU(3) matrix of the pseudoscalar fields. \n\nIn a similar way, we have the Lagrangian for the coupling of vector mesons to\nthe baryon octet given by\n\\cite{Klingl:1997kf,Palomar:2002hk} \n\n\n\\begin{equation}\n{\\cal L}_{BBV} =\n\\frac{g}{2}\\left(tr(\\bar{B}\\gamma_{\\mu}[V^{\\mu},B])+tr(\\bar{B}\\gamma_{\\mu}B)tr(V^{\\mu})\n\\right),\n\\label{lagr82}\n\\end{equation}\nwhere $B$ is now the SU(3) matrix of the baryon octet \\cite{Eck95,Be95}. Similarly,\none has also a lagrangian for the coupling of the vector mesons to the baryons\nof the decuplet, which can be found in \\cite{manohar}.\n\n\nWith these ingredients we can construct the Feynman diagrams that lead to the $PB\n\\to PB$ and $VB \\to VB$ interaction, by exchanging a vector meson between the\npseudoscalar or the vector meson and the baryon, as depicted in Fig.\\ref{f1} .\n\n\\begin{figure}[tb]\n\\epsfig{file=f1a.eps, width=7cm} \\epsfig{file=f1b.eps, width=7cm}\n\\caption{Diagrams obtained in the effective chiral Lagrangians for interaction\nof pseudoscalar [a] or vector [b] mesons with the octet or decuplet of baryons.}%\n\\label{f1}%\n\\end{figure}\n\n As shown in \\cite{angelsvec}, in the limit of small three momenta of the vector mesons, which we consider, the vertices of Eq. (\\ref{l3Vsimp}) and Eq. (\\ref{lagrVpp}) give rise to the same expression.  This makes the work technically easy allowing the use of many previous results. \n\n \n\n   A small amendment is in order in the case of vector mesons, which\n   is due to the mixing of $\\omega_8$ and the singlet of SU(3), $\\omega_1$, to give the\n   physical states of the $\\omega$ and the $\\phi$.\n    In this case, all one must do is to take the\n   matrix elements known for the $PB$ interaction and, wherever $P$ is the\n   $\\eta_8$, multiply the amplitude by the factor $1\/\\sqrt 3$ to get the\n   corresponding $\\omega $ contribution and by $-\\sqrt {2\/3}$ to get the\n   corresponding $\\phi$ contribution.  Upon the approximation consistent with\n   neglecting the three momentum versus the mass of the particles (in this\n   case the baryon), we can just take the $\\gamma^0$ component of \n   eq. (\\ref{lagr82})  and\n   then the transition potential corresponding to the diagram of Fig. 1(b ) is\n   given by\n   \n   \\begin{equation}\nV_{i j}= - C_{i j} \\, \\frac{1}{4 f^2} \\, (k^0 + k'^0)~ \\vec{\\epsilon}\\vec{\\epsilon\n} ',\n\\label{kernel}\n\\end{equation}\n where $k^0, k'^0$ are the energies of the incoming and outgoing vector mesons. \n   The same occurs in the case of the decuplet.  \n    \n    The $C_{ij}$ coefficients of Eq. (\\ref{kernel}) can be obtained directly from \n    \\cite{angels,bennhold,inoue}\n    with the simple rules given above for the $\\omega$ and the $\\phi$, and\n    substituting $\\pi$ by $\\rho$ and $K$ by $K^*$ in the matrix elements. The\n    coefficients are obtained both in the physical basis of states or in the\n    isospin basis. Here we will show results in isospin\n    basis. \n\n    The next step to construct the scattering matrix is done by solving the\n    coupled channels Bethe Salpeter equation in the on shell factorization approach of \n    \\cite{angels,ollerulf}\n   \\begin{equation}\nT = [1 - V \\, G]^{-1}\\, V,\n\\label{eq:Bethe}\n\\end{equation} \nwith $G$ the loop function of a vector meson and a baryon which we calculate in\ndimensional regularization using the formula of \\cite{ollerulf} and similar\nvalues for the subtraction constants. The $G$ function is convoluted with the \nspectral function for the vector mesons to take into account their width. \n\n  \n\n\n\n The iteration of diagrams implicit in the Bethe Salpeter equation in the case\n of the vector mesons propagates the $\\vec{\\epsilon}\\vec{\\epsilon }'$ term \n of the interaction, thus,\nthe factor $\\vec{\\epsilon}\\vec{\\epsilon }'$ appearing in the potential $V$\nfactorizes also in the $T$ matrix for the external vector mesons. This has as a consequence that the interaction is spin independent and we find degenerate states in $J^P=1\/2^-$ and $J^P=3\/2^-$.\n\n\\section{Results} \n\n The resonances obtained are summarized in Table  \\ref{tab:octet} .\n As one can see in Table \\ref{tab:octet} there are states which one can easily\nassociate to known resonances. There are ambiguities in other cases. One can also\nsee that in several cases the degeneracy in spin that the theory predicts is\nclearly visible in the experimental data, meaning that there are \nseveral states with about 50 MeV \nor less mass difference\nbetween them.  In some cases, the theory predicts quantum numbers for \nresonances which have no spin and parity associated. It would be interesting to\npursue the experimental research to test the theoretical predictions. \n\n\n\n\n\n\n\\begin{table}[ht]\n      \\renewcommand{\\arraystretch}{1.5}\n     \\setlength{\\tabcolsep}{0.2cm}\n\\begin{tabular}{c|c|cc|ccccc}\\hline\\hline\n$S,\\,I$&\\multicolumn{3}{c|}{Theory} & \\multicolumn{5}{c}{PDG data}\\\\\n\\hline\n    \\vspace*{-0.3cm}\n    & pole position    & \\multicolumn{2}{c|}{real axis} &  &  & &  &  \\\\\n    & {\\small (convolution)}    &\\multicolumn{2}{c|}{{\\small\n    (convolution)}} & \\\\\n    &   & mass & width &name & $J^P$ & status & mass & width \\\\\n    \\hline\n$0,1\/2$ & --- & 1696  & 92  & $N(1650)$ & $1\/2^-$ & $\\star\\star\\star\\star$ & 1645-1670\n& 145-185\\\\\n  &      &       &     & $N(1700)$ & $3\/2^-$ & $\\star\\star\\star$ &\n\t1650-1750 & 50-150\\\\\n       & $1977 + {\\rm i} 53$  & 1972  & 64  & $N(2080)$ & $3\/2^-$ & $\\star\\star$ & $\\approx 2080$\n& 180-450 \\\\\t\n   &     &       &     & $N(2090)$ & $1\/2^-$ & $\\star$ &\n $\\approx 2090$ & 100-400 \\\\\n \\hline\n$-1,0$ & $1784 + {\\rm i} 4$ & 1783  & 9  & $\\Lambda(1690)$ & $3\/2^-$ & $\\star\\star\\star\\star$ &\n1685-1695 & 50-70 \\\\\n  &       &       &    & $\\Lambda(1800)$ & $1\/2^-$ & $\\star\\star\\star$ &\n1720-1850 & 200-400 \\\\\n       & $1907 + {\\rm i} 70$ & 1900  & 54  & $\\Lambda(2000)$ & $?^?$ & $\\star$ & $\\approx 2000$\n& 73-240\\\\\n       & $2158 + {\\rm i} 13$ & 2158  & 23  &  &  &  & & \\\\\n       \\hline\n$-1,1$ & $ --- $ & 1830  & 42  & $\\Sigma(1750)$ & $1\/2^-$ & $\\star\\star\\star$ &\n1730-1800 & 60-160 \\\\\n  & $ --- $   & 1987  & 240  & $\\Sigma(1940)$ & $3\/2^-$ & $\\star\\star\\star$ & 1900-1950\n& 150-300\\\\\n   &     &       &   & $\\Sigma(2000)$ & $1\/2^-$ & $\\star$ &\n$\\approx 2000$ & 100-450 \\\\\\hline\n$-2,1\/2$ & $2039 + {\\rm i} 67$ & 2039  & 64  & $\\Xi(1950)$ & $?^?$ & $\\star\\star\\star$ &\n$1950\\pm15$ & $60\\pm 20$ \\\\\n         & $2083 + {\\rm i} 31 $ &  2077     & 29  &  $\\Xi(2120)$ & $?^?$ & $\\star$ &\n$\\approx 2120$ & 25  \\\\\n \\hline\\hline\n    \\end{tabular}\n\\caption{The properties of the 9 dynamically generated resonances stemming from the vector-baryon octet interaction and their possible PDG\ncounterparts.}\n\\label{tab:octet}\n\\end{table}\n\n\n\n\n\n\n  The predictions made here for resonances not observed should be a stimulus for\nfurther search of such states. In this\nsense it is worth noting the experimental program at Jefferson Lab \n\\cite{Price:2004xm} to investigate the $\\Xi$ resonances. We are\nconfident that the predictions  shown here stand on solid grounds and anticipate much\nprogress in the area of baryon spectroscopy and on the understanding of the\nnature of the baryonic resonances. \n\n\\section{Incorporating the pseudoscalar meson-baryon channels}\n\nImprovements in the states tabulated in Table \\ref{tab:octet} have been done by incorporating intermediate states of a pseudoscalar meson and a baryon \\cite{garzon}. This is done by including the diagrams of Fig.~(\\ref{box}). However, arguments of gauge invariance \\cite{Rapp:1997fs,Peters:1997va,Rapp:1999ej,Urban:1999im,Cabrera:2000dx,Cabrera:2002hc,kanchan1,kanchan2} demand that the meson pole term be accompanied by the corresponding Kroll Ruderman contact term, see Fig. \\ref{fig:vbpb}.\n\n\n\n\\begin{center}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.5]{intbox.eps} \n\n\\end{center}\n\\caption{ Diagram for the $VB \\rightarrow VB$ interaction incorporating the intermediate pseudoscalar-baryon states.}\n\\label{box}\n\\end{figure}\n\\end{center} \n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[scale=0.5]{vbpb.eps}\n\\end{center}\n\\caption{Diagrams of the $VB\\rightarrow PB$. (a) meson pole term, (b) Kroll Ruderman contact term.}\n\\label{fig:vbpb}\n\\end{figure}\n\n\nIn the intermediate B states of Fig. \\ref{box} we include baryons of the octet and the decuplet. The results of the calculations are a small shift and a broadening of the resonances obtained with the base of vector-baryon alone.The results are shown in Fig. \\ref{res1} and Tables \\ref{tab:pdg12} and \\ref{tab:pdg32}.  \n\nIn Fig.~\\ref{res1} we see two peaks for the state of $S=0$ and $I=1\/2$, one around 1700 MeV, in channels $\\rho N$ and $K^* \\Lambda$, and another peak near 1980 MeV, which appears in all the channels except for $\\rho N$. \nWe can see that the mixing of the PB channels affects differently the two spins, $J^P=1\/2^-$ and $3\/2^-$, as a consequence of the extra mechanisms contributing to the $J^P=1\/2^-$. The effect of the box diagram on the $J^P=3\/2^-$ sector is small, however the PB-VB mixing mechanism are more important in the $J^P=1\/2^-$ sector. Indeed, the Kroll Ruderman term only allows the $1\/2^-$ pseudoscalar baryon intermediate states in the box. The most important feature is a shift of the peak around 1700 MeV, which appears now around 1650 MeV. This breaking of the degeneracy is most welcome since this allows us to associate the $1\/2^-$ peak found at 1650 MeV with the $N^*(1650)(1\/2^-)$ while the peak for $3\/2^-$ at 1700 MeV can be naturally associated to the $N^*(1700)(3\/2^-)$. \n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[scale=1.5]{spin1.eps} \n\\end{center}\n\\caption{$|T|^2$ for the S=0, I=1\/2 states. Dashed lines correspond to tree level only and solid lines are calculated including the box diagram potential. Vertical dashed lines indicate the channel threshold.}\n\\label{res1}\n\\end{figure}\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{c|c|cc|ccccc}\n\\hline\\hline\n$S,\\,I$\t&\\multicolumn{3}{c|}{Theory} & \\multicolumn{5}{c}{PDG data}\\\\\n\\hline\n    \t& pole position\t& \\multicolumn{2}{c|}{real axis} &  &  & &  &  \\\\\n    \t& $M_R+i\\Gamma \/2$\t& mass & width &name & $J^P$ & status & mass & width \\\\\n\\hline\n$0,1\/2$ & $1690+i24^{*}\t\t\t$\t& 1658  & 98  \n\t\t& $N(1650)$ & $1\/2^-$ & $\\star\\star\\star\\star$ \t& 1645-1670\t& 145-185\\\\\n  \t\t\n       \t& $1979+i67\t\t\t$\t& 1973 & 85 \n   \t\t& $N(2090)$ & $1\/2^-$ & $\\star$ \t & $\\approx 2090$ & 100-400 \\\\\n\\hline\n$-1,0$\t& $1776+i39\t\t\t\t$\t& 1747 & 94  \n  \t\t& $\\Lambda(1800)$ & $1\/2^-$ & $\\star\\star\\star$ \t\t& 1720-1850 & 200-400 \\\\\n  \t\t\n        & $1906+i34^{*}\t\t\t$ \t& 1890 & 93  \n        & $\\Lambda(2000)$ & $?^?$ & $\\star$ \t\t\t\t\t& $\\approx 2000$ & 73-240\\\\\n\n        & $2163+i37\t\t\t\t$ \t& 2149 & 61  &  &  &  & & \\\\\n\\hline\n$-1,1$  & $ -\t\t\t$\t& 1829& 84  \n\t\t& $\\Sigma(1750)$ & $1\/2^-$ & $\\star\\star\\star$ & 1730-1800 & 60-160 \\\\\n   \t\t& $ -\t\t\t $ \t& 2116 & 200-240  \n   \t\t& $\\Sigma(2000)$ & $1\/2^-$ & $\\star$ \t\t\t& $\\approx 2000$ & 100-450 \\\\\n\\hline\n$-2,1\/2$& $2047+i19^{*}\t$\t& 2039 & 70  \n\t\t& $\\Xi(1950)$ & $?^?$ & $\\star\\star\\star$ \t& $1950\\pm15$ & $60\\pm 20$ \\\\\n\n        & $ -\t\t\t$\t& 2084 & 53   \n        & $\\Xi(2120)$ & $?^?$ & $\\star$ \t\t\t& $\\approx 2120$ & 25  \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{The properties of the nine dynamically generated resonances and their possible PDG\ncounterparts for $J^P=1\/2^-$. The numbers with asterisk in the imaginary part of the pole position are obtained without the convolution for the vector mass distribution of the $\\rho$ and $K^*$.}\n\\label{tab:pdg12}\n\\end{center}\n\\end{table}\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{c|c|cc|ccccc}\n\\hline\\hline\n$S,\\,I$\t&\\multicolumn{3}{c|}{Theory} & \\multicolumn{5}{c}{PDG data}\\\\\n\\hline\n    \t& pole position\t& \\multicolumn{2}{c|}{real axis} &  &  & &  &  \\\\\n    \t& $M_R+i\\Gamma \/2$\t& mass & width &name & $J^P$ & status & mass & width \\\\\n\\hline\n$0,1\/2$ & $1703+i4^{*}\t\t\t$\t& 1705  & 103  \n  \t\t& $N(1700)$ & $3\/2^-$ & $\\star\\star\\star$ \t\t& 1650-1750 & 50-150\\\\\n  \t\t\n       \t& $1979+i56\t\t\t$\t& 1975 & 72 \n       \t& $N(2080)$ & $3\/2^-$ & $\\star\\star$ & $\\approx 2080$ & 180-450 \\\\\t\n\\hline\n$-1,0$\t& $1786+i11\t\t\t\t$\t& 1785 & 19  \n\t\t& $\\Lambda(1690)$ & $3\/2^-$ & $\\star\\star\\star \\star$ \t& 1685-1695 & 50-70 \\\\\n  \t\t\n        & $1916+i13^{*}\t\t\t$ \t& 1914 & 59  \n        & $\\Lambda(2000)$ & $?^?$ & $\\star$ \t\t\t\t\t& $\\approx 2000$ & 73-240\\\\\n\n        & $2161+i17\t\t\t\t$ \t& 2158 & 29  &  &  &  & & \\\\\n\\hline\n$-1,1$  & $ -\t\t\t$\t& 1839& 58  \n  \t\t& $\\Sigma(1940)$ & $3\/2^-$ & $\\star\\star\\star$ & 1900-1950 & 150-300\\\\\n   \t\t& $ -\t\t\t $ \t& 2081 & 270  &  &  &  & & \\\\  \n\\hline\n$-2,1\/2$& $2044+i12^{*}\t$\t& 2040 & 53  \n\t\t& $\\Xi(1950)$ & $?^?$ & $\\star\\star\\star$ \t& $1950\\pm15$ & $60\\pm 20$ \\\\\n\n        & $2082+i5^{*} \t$\t& 2082 & 32   \n        & $\\Xi(2120)$ & $?^?$ & $\\star$ \t\t\t& $\\approx 2120$ & 25  \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{The properties of the nine dynamically generated resonances and their possible PDG\ncounterparts for $J^P=3\/2^-$. The numbers with asterisk in the imaginary part of the pole position are obtained without the convolution for the vector mass distribution of the $\\rho$ and $K^*$.}\n\\label{tab:pdg32}\n\\end{center}\n\\end{table}\n\n\n\n\\section{The $\\Delta \\rho \\pi$ system and $\\Delta$  $J^P=5\/2^+$ states around 2000 MeV}\nIn Refs.~\\cite{vijande,sourav} it was shown that the $\\Delta \\rho$ interaction\ngave rise to $N^*$ and $\\Delta$ states with degenerate spin-parity\n$1\/2^-,3\/2^-,5\/2^-$. In a recent work~\\cite{Xie:2011uw}, one extra\n$\\pi$ was introduced in the system, and via the Fixed Center\nApproximation to the Faddeev Equations, the new system was studied\nand new states were found.\n\n\\begin{center}%\n\\begin{table}[ptbh]\n\\caption{Assignment of $I=3\/2$ predicted states to $J^{P}=1\/2^{+}%\n,3\/2^{+},5\/2^{+}$ resonances. Estimated PDG masses for these\nresonances as well as their extracted values from references\n\\cite{Man92} and \\cite{Vra00} (in brackets) are shown for\ncomparison. N. C. stands for a non cataloged resonance in the PDG\nreview}\n\\begin{tabular}\n[c]{c|ccccc} \\hline\nPredicted & \\multicolumn{5}{c}{PDG data}\\\\\n&  &  &  &  & \\\\\nMass (MeV) & Name & $J^{P}$ & Estimated Mass (MeV) & Extracted Mass (MeV) & Status\\\\\n1770 & $\\Delta(1740)$ & $5\/2^{+}$ &  & $1752\\pm32$ & N.C.\\\\\n&  &  &  & $(1724\\pm61)$ & \\\\\n& $\\Delta(1600)$ & $3\/2^{+}$ & $1550-1700$ & $1706\\pm10$ & *** \\\\\n&  &  &  & $(1687\\pm44)$ & \\\\\n& $\\Delta(1750)$ & $1\/2^{+}$ & $\\approx1750$ & $1744\\pm36$ & * \\\\\n&  &  &  & $(1721\\pm61)$ & \\\\\\hline\n$1875$ & $\\Delta(1905)$ & $5\/2^{+}$ & $1865-1915$ & $1881\\pm18$ & ****\\\\\n&  &  &  & $(1873\\pm77)$ & \\\\\n& $\\Delta(1920)$ & $3\/2^{+}$ & $1900-1970$ & $2014\\pm16$ & ***\\\\\n&  &  &  & $(1889\\pm100)$ & \\\\\n& $\\Delta(1910)$ & $1\/2^{+}$ &\n$1870-1920$ & $1882\\pm10$ & ****\\\\\n&  &  &  & $(1995\\pm12)$ &\n\\\\\\hline\n\\end{tabular}\n\\label{threebody}\n\\end{table}\n\\end{center}\n\nWe show in Table \\ref{threebody} the two $\\Delta^*$ states obtained and\nthere is also a hint of another $\\Delta^*$ state around $2200$ MeV.\nExperimentally, only two resonances $\\Delta_{5\/2^+}(1905)(****)$ and\n$\\Delta_{5\/2^+}(2000)(**)$, are cataloged in the Particle Data Book\nReview~\\cite{pdg2010}. However, a careful look at\n$\\Delta_{5\/2^+}(2000)(**)$, shows that its nominal mass is in fact\nestimated from the mass $(1724\\pm61)$, $(1752\\pm32)$ and\n$(2200\\pm125)$ respectively, extracted from three independent\nanalyses of different character~\\cite{Man92,Vra00,Cut80}. Moreover a\nrecent new data analysis~\\cite{Suz10} has reported a\n$\\Delta_{5\/2^+}$ with a pole position at $1738$ MeV.\n\nOur results give quantitative theoretical support to the existence\nof two distinct $5\/2^+$ resonances, $\\Delta_{5\/2^+}(\\sim 1740)$ and\n$\\Delta_{5\/2^+}(\\sim 2200)$, apart from the one around 1905 MeV . We propose that these two resonances\nshould be cataloged instead of $\\Delta_{5\/2^+}(2000)$. This proposal\ngets further support from the possible assignment of the other\ncalculated baryon states in the $I=1\/2,3\/2$ and $J^P=1\/2^+,3\/2^+$\nsectors to known baryonic resonances. In particular the poorly\nestablished $\\Delta_{1\/2^+}(1750)(*)$ may be naturally interpreted\nas a $\\pi N_{1\/2^-}(1650)$ bound state.\n\n\\section{Hidden charm baryons from vector-baryon interaction}\nFollowing the idea of \\cite{angelsvec} in \\cite{wu} is was found that several baryon states emerged as hidden charm composite states of mesons and baryons with charm. In particular along the lines discussed here, we find a hidden charm baryon that couples to $J\/\\psi N$ and other channels, as shown in Tables \\ref{jpsicoupling},\\ref{jpsiwidth}.  This will play a role later on when we discuss the $J\/\\psi$ suppression in nuclei. To do the calculations in the charm sector an extension to SU(4) of the hidden gauge Lagrangians is made, but the symmetry is explicitly broken when considering the exchange of heavy vector mesons, where the appropriate reduction in the Feynman diagrams is taken into account. \n\n                                                                                   \\begin{table}[ht]\n      \\renewcommand{\\arraystretch}{1.1}\n     \\setlength{\\tabcolsep}{0.4cm}\n\\begin{center}\n\\begin{tabular}{cccccc}\\hline\n$(I, S)$&  $z_R$                   & \\multicolumn{4}{c}{$g_a$}\\\\\n\\hline\n$(1\/2, 0) $   &             & $\\bar{D}^{*} \\Sigma_{c}$ & $\\bar{D}^{*} \\Lambda^{+}_{c}$&  $J\/\\psi N$ \\\\\n          & $4415-9.5i$   & $2.83-0.19i     $          &$-0.07+0.05i   $            &  $-0.85+0.02i$ \\\\\n          &             &$ 2.83  $                 &$0.08  $                  &  $0.85$ \\\\\n\\hline\n$(0, -1) $ &                & $\\bar{D}^{*}_{s} \\Lambda^{+}_{c}$   & $\\bar{D}^{*} \\Xi_{c}$ & $\\bar{D}^{*} \\Xi'_{c}$ & $J\/\\psi \\Lambda$\\\\\n       &   $4368-2.8i  $  & $1.27-0.04i     $                     &$ 3.16-0.02i $          & $-0.10+0.13i  $          & $0.47+0.04i   $     \\\\\n       &                & $1.27 $                             & $3.16 $               & $0.16 $                & $0.47  $          \\\\\n       &   $4547-6.4i $   & $0.01+0.004i$                         & $0.05-0.02i$            & $2.61-0.13i $            & $-0.61-0.06i   $                \\\\\n       &                & $0.01   $                           & $0.05 $               & $2.61$                 & $0.61 $                  \\\\\n\\hline\\end{tabular} \\caption{Pole position ($z_R$) and coupling\nconstants ($g_a$) to various channels for the states from\n$PB\\rightarrow PB$ including the $J\/\\psi N$ and $J\/\\psi\\Lambda$\nchannels. }\n\\label{jpsicoupling}\n\\end{center}\n       \\renewcommand{\\arraystretch}{1.1}\n     \\setlength{\\tabcolsep}{0.4cm}\n\\begin{center}\n\\begin{tabular}{ccccc}\\hline\n$(I, S)$      &  $z_R$      & \\multicolumn{2}{c}{Real axis} & $\\Gamma_i$ \\\\\n          &    & $M$ & $\\Gamma$                  & \\\\\n\\hline\n$(1\/2, 0)$    &            &      &             &  $J\/\\psi N$\\\\\n          & $4415-9.5i$  & $4412$ & $47.3$        &  $19.2$       \\\\\n\\hline\n$(0, -1)$     &            &      &             &  $J\/\\psi\\Lambda$\\\\\n          & $4368-2.8i$  & $4368 $& $28.0 $       &  $5.4$        \\\\\n          & $4547-6.4i $ & $4544 $& $36.6 $       &  $13.8$        \\\\\n\\hline\\end{tabular} \\caption{Pole position ($z_R$), mass ($M$),\ntotal width ($\\Gamma$, including the contribution from the light\nmeson and baryon channel) and the decay widths for the $J\/\\psi N$\nand $J\/\\psi\\Lambda$ channels ($\\Gamma_i$). The unit are in MeV}\n\\label{jpsiwidth}\n\\end{center}\n\\end{table}  \n\n\\section{The properties of $K^*$ in nuclei}\nMuch work about the vector mesons in nuclei has been done in the last decade\n\\cite{rapp,hayano,mosel,na60,wood,nucl-th\/0610067,nanova}, mostly for the $\\rho,\\phi,\\omega$, which can be studied looking for dileptons. Maybe this technical detail was what prevented any attention being directed to the renormalization of the  $K^*$ in nuclei. However, recently this work has been addressed in \\cite{lauraraquel} with very interesting results. \n\n\n\n\nThe $K^{*-}$ width in vacuum is determined by the imaginary part of the free $\\bar K^*$ self-energy at rest,  ${\\rm Im} \\Pi^0_{\\bar K^*}$, due to the decay of the $\\bar{K}^*$ meson into $\\bar{K}\\pi$ pairs:  $\\Gamma_{K^{*-}}=-\\mathrm{Im}\\Pi_{\\bar{K}^*}^{0}\/m_{\\bar K^*}=42$ MeV \\cite{lauraraquel}. Note that this value is quite close to the experimental value $\\Gamma^{\\rm exp}_{K^{*-}}=50.8\\pm 0.9$ MeV.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5cm]{Fig2.eps}\n\\hfill\n\\caption{Self-energy diagrams from the decay of the $\\bar{K}^*$ meson in the medium.}\n\\label{fig:1}\n\\end{center}\n\\end{figure}\n\n\n\nThe $\\bar{K}^*$ self-energy  in matter, on one hand, results from its decay into ${\\bar K}\\pi$, $\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (a)}}$, including both the self-energy of the antikaon \\cite{Tolos:2006ny} and the pion \\cite{Oset:1989ey,Ramos:1994xy} (see first diagram of Fig.~\\ref{fig:1} and some specific contributions in diagrams $(a1)$ and $(a2)$ of Fig.~\\ref{fig:3}). Moreover, vertex corrections required by gauge invariance are also incorporated, which are associated to the last three diagrams in  Fig. \\ref{fig:1}.\n\n\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{new_diagrams.eps}\n\\caption{Contributions to the $\\bar K^*$ self-energy, depicting their different\ninelastic sources.}\n\\label{fig:3}\n\\end{center}\n\\end{figure}\n\n\n\n\nThe second contribution to the $\\bar K^*$ self-energy comes from its interaction\nwith the nucleons in the Fermi sea, as displayed in diagram (b) of \nFig.~\\ref{fig:3}. This accounts for the direct quasi-elastic process $\\bar K^* N \\to \\bar K^* N$ as well as other absorption channels $\\bar K^* N\\to \\rho Y, \\omega Y, \\phi Y, \\dots$ with $Y=\\Lambda,\\Sigma$. \nThis contribution is determined by\nintegrating the medium-modified $\\bar K^* N$ amplitudes, $T^{\\rho,I}_{\\bar\nK^*N}$, over the  Fermi sea of nucleons,\n\\begin{eqnarray}\n\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (b)}}(q^0,\\vec{q}\\,)&=&\\int \\frac{d^3p}{(2\\pi)^3} \\, n(\\vec{p}\\,)\\,\n\\left [~{T^\\rho}^{(I=0)}_{\\bar K^*N}(P^0,\\vec{P})+3 {T^\\rho}^{(I=1)}_{\\bar K^*N}(P^0,\\vec{P})\\right ] \\ ,\n \\label{eq:pid}\n\\end{eqnarray}\nwhere $P^0=q^0+E_N(\\vec{p}\\,)$ and $\\vec{P}=\\vec{q}+\\vec{p}$ are the\ntotal energy and momentum of the $\\bar K^*N$ pair in the nuclear\nmatter rest frame, and the values $(q^0,\\vec{q}\\,)$ stand for the\nenergy and momentum of the $\\bar K^*$ meson also in this frame. The\nself-energy $\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (b)}}$ has to be determined self-consistently\nsince it is obtained from the in-medium amplitude\n${T}^\\rho_{\\bar K^*N}$ which contains the $\\bar K^*N$ loop function\n${G}^\\rho_{\\bar K^*N}$, and this last quantity itself is a function of the complete self-energy\n$\\Pi_{\\bar K^*}^{\\rho}=\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (a)}}\n+\\Pi_{\\bar{K}^*}^{\\rho,{\\rm (b)}}$. \n\n\n\n\n\nWe note that the two contributions to the $\\bar K^*$ self-energy, coming from\nthe decay of\n$\\bar K \\pi$ pairs in the medium [Figs.~\\ref{fig:3}(a1) and \\ref{fig:3}(a2)] or\nfrom the  $\\bar K^* N$ interaction [Fig.~\\ref{fig:3}(b)] provide different\nsources\nof inelastic $\\bar K^* N$ scattering, which add incoherently in the $\\bar K^*$\nwidth.  \nAs seen in the upper two rows of Fig.~\\ref{fig:3}, the  $\\bar K^* N$\namplitudes mediated by intermediate $\\bar K N$ or $\\pi\nY$ states are not unitarized. Ideally, one would like to treat the vector\nmeson-baryon ($VB$) and pseudoscalar meson-baryon ($PB$) states on the same\nfooting. However, at the energies of interest, transitions of the type $\\bar K^*\nN \\to \\bar K N$ mediated by pion exchange may place this pion on its mass shell,\nforcing one to keep track of the proper analytical cuts contributing to the\nimaginary part of the amplitude and making the iterative process more\ncomplicated. A technical solution can be found by calculating the box diagrams\nof Figs.~\\ref{fig:3}(a1) and \\ref{fig:3}(a2), taking all the cuts into account\nproperly, and adding the resulting $\\bar K^* N \\to \\bar K^* N$ terms to the $VB\n\\to V^\\prime B^\\prime$ potential coming from vector-meson exchange, in a\nsimilar way as done for the study of the vector-vector interaction in\nRefs.~\\cite{raquel,gengvec}. As we saw in the former sections, the generated resonances barely change their position for spin 3\/2 and only by a moderate amount in some cases for spin 1\/2. Their widths are somewhat enhanced due to the opening of the newly\nallowed $PB$ decay channels \\cite{garzon}.\n\n\n\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{Fig5_colouronline.eps}\n\\hfill\n\\caption{  $\\bar K^*$ self-energy for\n $\\vec{q}=0 \\, {\\rm MeV\/c}$ and $\\rho_0$. }\n\\label{fig:auto-spec}\n\\end{center}\n\\end{figure}\n\n\n \nThe full $\\bar K^*$ self-energy as a function of the $\\bar K^*$ energy for zero\nmomentum at normal nuclear matter density is shown  in \nFig.~\\ref{fig:auto-spec}. We explicitly indicate the contribution to the\nself-energy coming from the self-consistent calculation of the $\\bar K^* N$\neffective interaction (dashed lines) and the self-energy from the $\\bar K^*\n\\rightarrow \\bar K \\pi$ decay mechanism (dot-dashed lines), as well as the\ncombined result from both sources (solid lines).\n\nAround $q^0= 800-900$ MeV we observe an enhancement of the width as well as some structures in the real part of the $\\bar K^*$ self-energy. The origin of these structures can be traced back to  the coupling of the $\\bar K^*$ to the in-medium $\\Lambda(1783) N^{-1}$ and  $\\Sigma(1830) N^{-1}$ excitations, which dominate the $\\bar K^*$ self-energy in this energy region. However, at lower energies where the $\\bar K^* N\\to V B$ channels \nare closed, or at large energies beyond the resonance-hole excitations,\nthe width of the $\\bar K^*$ is governed by the $\\bar K \\pi$ decay mechanism in dense matter. \n\n\nAs we can see, the $\\bar K^*$ feels a moderately attractive optical potential and acquires a width of $260$ MeV, which is about five times its width in vacuum. In the next section we devise a method to measure this large width experimentally. \n \n\n\n\\section{Transparency ratio for  $\\gamma A \\to K^+ K^{*-} A'$}\n\nThe width of the $\\bar K^*$ meson in nuclear matter can be analyzed experimentally by means of the nuclear transparency ratio. The aim is to compare the cross sections of the photoproduction reaction $\\gamma A \\to K^+ K^{*-} A'$ in different nuclei, and tracing the differences to the in medium $K^{*-}$ width.\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Fig8.eps}\n\\caption{Nuclear transparency ratio  for  $\\gamma A \\to K^+ K^{*-} A'$ for\ndifferent nuclei}\n\\label{fig:ratio}\n\\end{center}\n\\end{figure}\n\n\nThe normalized nuclear transparency ratio is defined as\n\\begin{equation}\nT_{A} = \\frac{\\tilde{T}_{A}}{\\tilde{T}_{^{12}C}} \\hspace{1cm} ,{\\rm with} \\ \\tilde{T}_{A} = \\frac{\\sigma_{\\gamma A \\to K^+ ~K^{*-}~ A'}}{A \\,\\sigma_{\\gamma N \\to K^+ ~K^{*-}~N}} \\ .\n\\end{equation}\nThe quantity $\\tilde{T}_A$ is the ratio of the nuclear $K^{*-}$-photoproduction cross section\ndivided by $A$ times the same quantity on a free nucleon. This describes the loss of flux of $K^{*-}$ mesons in the nucleus and is related to the absorptive part of the $K^{*-}$-nucleus optical potential and, therefore, to the $K^{*-}$ width in the nuclear medium.  In order to remove other nuclear effects not related to the absorption of the $K^{*-}$, we evaluate this ratio with respect to $^{12}$C, $T_A$.\n\nIn  Fig.~\\ref{fig:ratio} we show the transparency ratio for different nuclei and for two energies in the center of mass reference system, $\\sqrt{s}=3$ GeV and $3.5$ GeV, or, equivalently, two energies of the photon in the lab frame of $4.3$ GeV and $6$ GeV respectively. There is a very strong attenuation of the $\\bar{K}^*$ survival probability coming from the decay or absorption channels $\\bar{K}^*\\to \\bar{K}\\pi$ and $\\bar{K}^*N\\to \\bar K^* N, \\rho Y, \\omega Y, \\phi Y, \\dots$, with increasing nuclear-mass number $A$. The reason is the larger path that the $\\bar{K}^*$ has to follow before it leaves the nucleus, having then more chances of decaying or being absorbed.\n\n\\section{$J\/\\psi$ suppression}\n$J\/\\psi$ suppression in nuclei has been a hot topic \\cite{Vogt:1999cu}, among others for its possible interpretation as a signature of the formation of quark gluon plasma in heavy ion reactions \\cite{Matsui:1986dk}, but many other interpretations have been offered  \\cite{Vogt:2001ky,Kopeliovich:1991pu,Sibirtsev:2000aw}. In a recent paper \\cite{raquelxiao} a study has been done of different $J\/\\psi N$ reactions which lead to $J\/\\psi$ absorption in nuclei. The different reactions considered are \nthe transition of  $J\/\\psi N$ to $VN$ with $V$ being a light vector, $\\rho, \\omega,\\phi$, together with the inelastic channels, \n$J\/\\psi N \\to \\bar D \\Lambda_c$ and $J\/\\psi N \\to \\bar D \\Sigma_c$. \nAnalogously, we consider the mechanisms where the exchanged $D$ collides with a nucleon and gives $\\pi \\Lambda_c$ or $\\pi \\Sigma_c$. The cross section has a peak around $\\sqrt s=4415$ MeV, where the $J\/\\psi N$ couples to the resonance described in Tables \\ref{jpsicoupling} and \\ref{jpsiwidth}. With the inelastic cross section obtained we study the transparency ratio for electron induced $J\/\\psi$ production in nuclei at about 10 GeV and find that 30 - 35 \\% of the $J\/\\psi$ produced in heavy nuclei are absorbed inside the nucleus. This ratio is in line with depletions of $J\/\\psi$ in matter observed in other reactions. This offers a novelty in the interpretation of the $J\/\\psi$ suppression in terms of hadronic reactions, which has also been advocated before \\cite{Sibirtsev:2000aw}. Apart from novelties in the details of the calculations and the reaction channels considered, we find that the presence of the resonance that couples to $J\/\\psi N$ produces a peak in the inelastic $J\/\\psi N$ cross section and a dip in the transparency ratio. A measure of this magnitude could lead to an indirect observation of the existence of resonances that couple to $J\/\\psi N$. \n\nIn Fig. \\ref{crosec} we can see the total, elastic and inelastic cross sections in $J\/\\psi N$ to $J\/\\psi N$ through intermediate vector-baryon states. We can clearly see the peak around 4415 MeV produced by the hidden charm resonance dynamically generated form the interaction of $J\/\\psi N$ with its coupled $VB$ channels. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{crossection3.eps}\n\\caption{The total, elastic and inelastic cross sections in $J\/\\psi N$ to $J\/\\psi N$ through intermediate vector-baryon states.}\\label{crosec}\n\\end{center}\n\\end{figure}\n\nIn Fig. \\ref{figcro} we can see the cross section for $J\/\\psi N \\to \\bar{D} \\Lambda_c$  and $J\/\\psi N\\to \\bar{D} \\Sigma_c$. We can see that numerically the first cross section is sizeable, bigger than the one from the $VB$ channels. \n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{sigtot_sum1aa.eps}\n\\includegraphics[width=0.5\\textwidth]{sigtot_sum2aa.eps}\n\\caption{The cross section for $J\/\\psi N \\to \\bar{D} \\Lambda_c$ (up) and $J\/\\psi N\\to \\bar{D} \\Sigma_c$ (down).}\\label{figcro}\n\\end{center}\n\\end{figure}\n\nThe cross sections for $J\/\\psi N$ to $\\bar D \\pi \\Lambda_c$ or $\\bar D \\pi \\Sigma_c$ are small in size in the region of interest and are not plotted here. \n\nIn Fig. \\ref{sigin} we plot  the total $J\/\\psi N$ inelastic cross section, as the sum of all inelastic cross sections from the different sources discussed before.\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{sigtot_sum5.eps}\n\\caption{The total inelastic cross section of $J\/\\psi N$.}\\label{sigin}\n\\end{center}\n\\end{figure}\n\nFinally, in Fig. \\ref{transg} we show the transparency ratio of $^{208}Pb$ versus the one of $^{12}C$ as a function of the energy. We find sizeable reductions in the rate of $J\/\\psi$ production in electron induced reactions, and particularly a dip in the ratio that could be searched experimentally. \nIt should be noted that the calculation of the transparency ratio done so far does not consider the shadowing of the photons and assumes they can reach every point without being absorbed. However, for $\\gamma$ energies of around 10 GeV, as suggested here, the photon shadowing cannot be ignored. Talking it into account is easy since one can multiply the ratio $T_A'$ by the ratio of $N_{eff}$ for the nucleus of mass $A$ and $^{12}C$. This ratio for $^{208}Pb$ to $^{12}C$ at $E_\\gamma =$10 GeV is of the order $0.8$ but with uncertainties \\cite{Bianchi:1995vb}. We should then multiply $T_A'(^{208}Pb)$ in Fig. \\ref{transg} by this extra factor for a proper comparison with experiment.\nAlthough a good resolution should be implemented, the prospect of finding new states through this indirect method should serve as a motivation to perform such experiments.\n\n\\section{Conclusions}\n\nWe have made a survey of recent developments along the interaction of vector mesons with baryons and the properties of some vector mesons in a nuclear medium.  We showed that the interaction is strong enough to produce resonant states which can qualify as quasibound states of a vector meson and a baryon in coupled channels. This adds to the wealth of composite states already established from the interaction of pseudoscalar mesons with baryons. At the same time we offered the results of new pictures that mix the pseudoscalar-baryon states with the vector-baryon states and break the spin degeneracy that the original model had.  The method of vector-baryon interaction extended to the charm sector also produced some hidden charm states which couple to the $J\/\\psi N$  channel and had some repercussion in the $J\/\\psi$ suppression in nuclei. We also showed results for the spectacular renormalization of the $K^*$ in nuclei, where the width becomes as large as 250 MeV at normal nuclear matter density and we made suggestions of experiments that could test this large change.\n   \n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{totTA_g32.eps}\n\\caption{The transparency ratio of $J\/\\psi$ photoproduction  as a function of the energy in the CM of $J\/\\psi$ with nucleons of the nucleus. Solid line: represents  the effects due to $J\/\\psi$ absorption. Dashed line: includes photon shadowing \\cite{Bianchi:1995vb}.}\\label{transg}\n\\end{center}\n\\end{figure}\n\n\\section*{Acknowledgments}  \nThis work is partly supported by DGICYT contract number\nFIS2011-28853-C02-01, the Generalitat Valenciana in the program Prometeo, 2009\/090. L.T. acknowledges support from Ramon y Cajal Research Programme, and from FP7-PEOPLE-2011-CIG under contract PCIG09-GA-2011-291679. We acknowledge the support of the European Community-Research Infrastructure\nIntegrating Activity\nStudy of Strongly Interacting Matter (acronym HadronPhysics3, Grant Agreement\nn. 283286)\nunder the Seventh Framework Programme of EU.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\section{Introduction}\nJamming attacks pose a serious threat to the continuous operability of wireless communication systems \\cite{economist2021satellite, topgun}. \nEffective methods to mitigate such attacks are of paramount importance\nas wireless systems become increasingly critical to modern infrastructure~\\cite{popovski2014ultra, pirayesh2022jamming}.\nIn the uplink of massive multi-user multiple-input multiple-output (MU-MIMO) systems, effective jammer mitigation becomes possible by the asymmetry in the number of antennas between the basestation (BS), which has many antennas, and a mobile jamming device, which typically has one or few antennas.\nOne possibility, for instance, is to project the receive signals on the subspace \northogonal to the jammer's channel~\\cite{marti2021snips,yan2016jamming}. \nUnfortunately, such methods require accurate knowledge of the jammer's channel.\nIf a jammer transmits permanently and with a static signature (often called barrage jamming), the~BS~can estimate its channel, for instance during a dedicated period in which the user equipments (UEs) are not transmitting~\\cite{marti2021snips} or in which they transmit predefined symbols~\\cite{yan2016jamming}.\nIn contrast to barrage jamming, however, a smart jammer might jam the system only at specific time instants, such as when the UEs are transmitting data symbols,\nand thereby prevent the BS from estimating the jammer's channel using simple estimation algorithms.\n\n\n\\subsection{State of the Art}\n\nMulti-antenna wireless systems offer the unique potential to effectively mitigate jamming attacks.\nConsequently, a variety of multi-antenna methods have been proposed for the mitigation of jamming attacks in MIMO systems\n\\cite{pirayesh2022jamming, marti2021snips, shen14a, hoang2021suppression, yan2016jamming,zeng2017enabling, vinogradova16a, do18a, akhlaghpasand20a, akhlaghpasand20b, marti2021hybrid, wan2022robust, darsena2022anti}.\nCommon to all of them~is the assumption---in one way or other---that information about the jammer's transmit characteristics\n(e.g., the jammer's channel, or the covariance matrix between the UE transmit signals and the jammed receive signals)\ncan be estimated using some specific subset of the receive samples.\\footnote{The method of \\cite{vinogradova16a} is to some extent an exception as it estimates the~UEs' subspace and projects the receive signals thereon. This method, however, dist-inguishes the UEs' from the jammer's subspace based on the receive power, thereby presuming that the UEs and the jammer transmit with different power.}\n\\fref{fig:traditional}~illustrates the approach of such methods: \nThe data phase is preceded by an augmented \ntraining phase in which the jammer's transmit characteristics as well as the channel matrix are estimated.\nThis augmented training phase may (i) complement a traditional pilot phase with a dedicated period during which the UEs do not transmit in order to \nenable jammer estimation (e.g., \\cite{marti2021snips, shen14a, hoang2021suppression}) or (ii) consist of an extended pilot phase so that there exist pilot sequences that are unused by\nthe UEs and on whose span the receive signals can be projected to estimate the jammer subspace~\\mbox{(e.g., \\cite{do18a, akhlaghpasand20a, akhlaghpasand20b}).}\nThe estimated jammer characteristics are then used to perform jammer-mitigating data detection. \nSuch an approach succeeds in the case of barrage jammers, but is unreliable for estimating the propagation characteristics of smart jammers, see \\fref{sec:example}: \nA smart jammer can evade estimation and, thus, circumvent mitigation by not transmitting during the training phase, for instance because it is aware of the defense mechanism\nor simply because it jams in short bursts only. \nFor this reason, our proposed method does not estimate the jammer channel based on a dedicated training phase, \nbut instead utilizes the entire transmission period and unifies jammer estimation and mitigation, channel estimation and data detection; see \\fref{fig:maed}.\n\nMany studies have already shown how smart jammers can disrupt wireless communication systems by\ntargeting only specific parts\nof the wireless transmission process \\cite{miller2010subverting, miller2011vulnerabilities, clancy2011efficient, sodagari2012efficient, \nlichtman2013vulnerability, lichtman20185g, lichtman2016lte, girke2019towards,lapan2012jamming} \ninstead of using barrage jamming.\nJammers that target only the pilot phase have received considerable attention \n\\cite{miller2010subverting,miller2011vulnerabilities,clancy2011efficient,sodagari2012efficient,lichtman2013vulnerability}, \nas such attacks can be more energy-efficient than barrage jamming in disrupting communication systems that do not \ndefend themselves against jammers~\\cite{clancy2011efficient,sodagari2012efficient,lichtman2013vulnerability}.\nHowever, if a jammer is active during the pilot phase, then a BS that \\emph{does} defend itself against attacks\ncan estimate the jammer's channel by exploiting knowledge of the UE transmit symbols during the pilot phase, for instance with the aid of unused pilot sequences~\\cite{do18a, akhlaghpasand20a, akhlaghpasand20b}.\nTo disable such jammer-mitigating communication systems, a smart jammer might thus refrain from jamming the pilot phase and only target \nthe data phase, even if such jamming attacks have not received much attention so far \n\\cite{lichtman2016lte, girke2019towards}.\nOther threat models that have been analyzed include attacks on control \nchannels \\cite{lichtman2013vulnerability, lichtman2016lte, lichtman20185g}, the beam alignment procedure \\cite{darsena2022anti}, \nor the time synchronization phase~\\cite{lapan2012jamming}, but this paper will not consider such protocol or control channel attacking schemes. \n\n\\begin{figure}[tp]\n\\vspace{-1mm}\n\\centering\n~\n\\subfigure[Existing methods separate jammer estimation (JEST) and channel~estimation (CHEST) from the jammer-resilient\ndata detection (DET). They~are ineffective against jammers that jam the data phase but not the training~phase.]{\n\\includegraphics[width=0.95\\columnwidth]{figures\/sota.pdf}\n\\label{fig:traditional}\n}\n\\newline\n\\subfigure[Our method unifies jammer estimation and mitigation, channel estimation, and data detection \nto deal with jammers regardless of their activity~pattern.]{\n\\includegraphics[width=0.95\\columnwidth]{figures\/maed.pdf}\n\\label{fig:maed}\n}\n\\caption{\nThe approach to jammer mitigation taken by existing methods (a) compared to the proposed method (b).\nIn the figure, $\\bmy_1,\\dots,\\bmy_K$ are the receive signals, \nand $\\hat\\Hj, \\hat\\bH$, and $\\hat\\bS_D$ are the estimates of the jammer channel, the UE channel matrix, and the UE transmit symbols, respectively.\n}\n\\vspace{-2mm}\n\\label{fig:maed_vs_trad}\n\\end{figure}\n\n\n\\subsection{Contributions}\nTo mitigate smart jammers, we propose a novel approach that does not depend on the jammer being active during \n\\textit{any} specific period.\nLeveraging the fact that a jammer cannot change its subspace instantaneously,\nwe utilize a problem formulation which unifies jammer estimation and mitigation, \nchannel estimation, and data detection, instead of dealing with these tasks independently (cf.~\\fref{fig:maed}).\nWe support the soundness of the proposed optimization problem by proving that its global minimum is unique and recovers the transmitted data symbols, given that certain sensible conditions are satisfied. \nBy building on techniques for joint channel estimation and data detection\n\\cite{vikalo2006efficient, xu2008exact, kofidis2017joint, castaneda2018vlsi, yilmaz2019channel, he2020model, song2021soft},\nwe then develop two efficient iterative algorithms for approximately solving the optimization \nproblem. The first algorithm is called MAED (short for MitigAtion, Estimation, and Detection) and \nsolves the problem approximately using forward-backward splitting (FBS) \\cite{goldstein16a}. \nThe second algorithm is called SO-MAED (short for Soft-Output MAED) and extends MAED with a more informative \nprior on the data symbols to produce soft symbol estimates. SO-MAED also relies on deep unfolding to optimize its parameters {\\cite{song2021soft, hershey2014deep, balatsoukas2019deep, goutay2020deep, monga2021algorithm}.\nWe use simulations with different propagation models to demonstrate that MAED and SO-MAED effectively mitigate a wide variety of \nna\\\"ive and smart  jamming attacks without requiring any knowledge about the attack type.\n\n\\subsection{Notation}\nMatrices and column vectors are represented by boldface uppercase and lowercase letters, respectively.\nFor a matrix~$\\bA$, the transpose is $\\tp{\\bA}$, the conjugate transpose is $\\herm{\\bA}$, \nthe Moore-Penrose pseudoinverse is $\\pinv{\\bA}$,\nthe entry in the $\\ell$th row and $k$th column is $[\\bA]_{\\ell,k}$, \nthe $k$th column is $\\bma_k$,\nthe submatrix consisting of the columns from $n$ through $m$ is $\\bA_{[n:m]}$,\nand the Frobenius norm is $\\| \\bA \\|_F$.\nThe $N\\!\\times\\!N$ identity matrix is $\\bI_N$.\nFor a vector~$\\bma$, the $\\ell_2$-norm is $\\|\\bma\\|_2$, the real part is $\\Re\\{\\bma\\}$, the imaginary part is $\\Im\\{\\bma\\}$, \nand the span is $\\textit{span}(\\bma)$.\nThe $k$th standard unit vector is denoted $\\bme_k$, where the dimension is implicit. \nExpectation with respect to a random vector~$\\bmx$ is denoted by \\Ex{\\bmx}{\\cdot}.\nWe define $i^2=-1$. \nThe complex $n$-hypersphere of radius $r$ is denoted by $\\mathbb{S}_r^n$,\nand~$[n:m]$ are the integers from $n$ through~$m$.\n\n\n\n\\section{System Setup}\\label{sec:setup}\nWe consider the uplink of a massive MU-MIMO system in which $U$ single-antenna UEs transmit data to \na $B$ antenna BS in the presence of a single-antenna jammer. \nThe channels are assumed to be frequency flat and block-fading with coherence time $K=T+D$.\nThe first $T$ time slots are used to \ntransmit pilot symbols; the remaining $D$ time slots are used to transmit data symbols.\nThe UE transmit matrix is $\\bS = [\\bS_T,\\bS_D]$, where $\\bS_T\\in \\opC^{U\\times T}$\nand $\\bS_D\\in\\setS^{U\\times D}$ contain the pilots~and the transmit symbols, respectively. The set $\\setS$ is the transmit constellation, which is normalized to have unit average symbol energy.\nWe assume that the jammer does not prevent the UEs and the BS from establishing synchronization,\nwhich allows us to use the following discrete-time input-output relation:\n\\begin{align}\n\t\\bY = \\bH\\bS + \\Hj\\tp{\\bsj} + \\bN. \\label{eq:io}\n\\end{align}\nHere, $\\bY\\in\\opC^{B\\times K}$ is the BS receive matrix that contains the \\mbox{$B$-dimensional} receive vectors over all $K$ time slots, \n\\mbox{$\\bH\\!\\in\\!\\opC^{B\\times U}$} models the MIMO uplink channel,\n$\\Hj\\in\\opC^B$ models the channel between the jammer and the BS, $\\bsj=\\tp{[\\tp{\\bsj_T},\\tp{\\bsj_D}]}\\in\\opC^K$  contains the jammer transmit symbols over all $K$ time slots, \nand $\\bN\\in\\opC^{B\\times K}$ models thermal noise consisting of independently and identically distributed (i.i.d.) circularly-symmetric complex Gaussian entries with variance~$N_0$.\nUnless stated otherwise, we assume that the jammer's transmit symbols $\\bsj$ are independent of $\\bS$. \nNo other assumptions about the distribution of $\\bsj$ are made;\nin particular, we do not assume that these entries are~i.i.d.\n\nIn what follows, we use plain symbols for the true channels and transmit signals, variables with a tilde for optimization variables, and quantities with a hat for (approximate) solutions to optimization problems, e.g., $\\hat\\bS_D$ is the the estimate of the UE transmit symbol matrix~$\\bS_D$ as determined by solving an optimization problem with respect to $\\tilde\\bS_D$.\n\n\\section{Motivating Example} \\label{sec:example}\nWe start by considering the motivating example of \\fref{fig:example},\nwhich shows uncoded bit error-rates (BERs) of different receivers for an i.i.d. Rayleigh \nfading MU-MIMO system with \\mbox{$B=128$} BS antennas and $U=32$ UEs that transmit 16-QAM symbols under~a jamming attack.\nIn \\fref{fig:example:success} the system is attacked by a barrage jammer that\ntransmits i.i.d. Gaussian symbols and whose \nreceive power exceeds that of the average UE by 30\\,dB. The ``LMMSE'' curve \nshows the performance of a non-mitigating receiver that estimates the UE channel matrix \nbased on orthogonal pilots with a least squares (LS) estimator followed by a linear minimum mean square error (LMMSE) detector. \nThe \\mbox{``JL-LMMSE''} curve shows the performance of an identical receiver but for \na jammerless system without a jamming attack. \nThe ``geniePOS'' receiver is furnished with ground-truth knowledge of the jammer channel~$\\Hj$. This baseline method  \nnulls the jammer by orthogonally projecting the receive signals on the orthogonal \ncomplement of $\\textit{span}(\\Hj)$ using the matrix $\\bP_\\Hj = \\bI_B - \\Hj\\pinv{\\Hj}$, \nwhere $\\pinv{\\Hj}=\\herm{\\Hj}\/\\|\\Hj\\|_2^2$,~as\n\\begin{align}\n\t\\bP_\\Hj\\bY \n\t&= \\bP_\\Hj\\,\\bH\\bS + \\bP_\\Hj\\,\\Hj\\tp{\\bsj} + \\bP_{\\Hj}\\,\\bN \\label{eq:pos} \\\\\n\t&= \\bP_\\Hj\\,\\bH\\bS + \\bP_\\Hj\\,\\bN,\n\\end{align}\nsince $\\bP_\\Hj\\,\\Hj=\\mathbf{0}$.\nThe result is an effective jammerless system with receive signal $\\bY_{\\bP} = \\bP_\\Hj \\bY$, \neffective channel matrix \\mbox{$\\bH_{\\bP} = \\bP_\\Hj\\bH$}, and (colored) noise \n$\\bN_\\bP = \\bP_\\Hj \\bN \\sim \\setC\\setN(\\mathbf{0},\\No\\bP_\\Hj)$.\nFinally, geniePOS performs LS channel estimation and subsequent LMMSE data detection in this \nprojected system  \\cite{marti2021snips}. \nThe ``POS'' receiver works analogously to geniePOS, except that it is not furnished with ground-truth \nknowledge of the jammer channel---instead, this method estimates the jammer subspace $\\Hj\/\\|\\Hj\\|_2$\nbased on ten receive samples in which the UEs do not transmit and only the jammer is active. \nIf the matrix received in that period is denoted by $\\bY_\\text{J}$, then the jammer subspace is estimated\nas the left-singular vector of the largest singular value of $\\bY_\\text{J}$.\n\\fref{fig:example:success} shows that geniePOS effectively mitigates the jammer, achieving a performance\nvirtually identical to that of the jammer-free \\mbox{JL-LMMSE} receiver. Indeed, geniePOS nulls the jammer \nperfectly, so that the only performance loss comes from the loss of one degree-of-freedom\nin the receive signal. POS is not as effective,~since it nulls the jammer only imperfectly due\nto its noisy estimate of the jammer subspace. However, this method still mitigates the jammer with\na loss of less than 2\\,dB in SNR (at $0.1\\%$ BER) compared to the jammer-free JL-LMMSE receiver.\n\\begin{remark}\nWe point out that reserving time slots for jammer estimation in which the UEs can not transmit\ndirectly reduces the achievable data~rates.\t\n\\end{remark}\n\nContrastingly, in \\fref{fig:example:fail} the attacking (smart) jammer is aware of the POS receiver's mitigation scheme\nand suspends transmission during the time slots that are used to estimate its subspace. \nThe POS receiver's subspace estimate is thus based entirely on noise and is completely independent\nof the jammer's true channel~$\\Hj$. Consequently, the mitigation mechanism fails spectacularly, yielding\na bit error-rate identical to the non-mitigating LMMSE receiver.\n\\begin{figure}[tp]\n\\centering\n\\!\\!\n\\subfigure[mitigation of a barrage jammer]{\n\\includegraphics[height=3.85cm]{figures\/128x32_16QAM_I1_D128_barrage_gaussian_rho30_100Trials_success}\n\\label{fig:example:success}\n}\\!\\!\\!\n\\subfigure[failed mitigation of a smart jammer]{\n\\includegraphics[height=3.85cm]{figures\/128x32_16QAM_I1_D128_barrage_gaussian_rho30_100Trials_fail}\n\\label{fig:example:fail}\n}\\!\\!\n\\caption{Example that illustrates how methods that estimate the jammer's channel based on \na subset of samples fail when facing a smart jammer.}\n\\label{fig:example}\n\\end{figure}\n\n\n\\section{Joint Jammer Estimation and Mitigation, Channel Estimation, and Data Detection}\nThe foregoing example has demonstrated the danger of estimating the jammer's subspace\n(or other characteristics of~the jammer, such as its spatial covariance) \nbased on a certain subset of receive samples when facing a smart jammer.\nWe therefore propose a method that does not depend on the jammer being active during any specific period.\nThis independence is achieved by considering the receive signal over an entire coherence interval \nat once and exploiting the fact that the jammer subspace stays fixed within that period, \nregardless of the jammer's activity pattern or transmit sequence. \nSpecifically, we first propose a novel optimization problem that combines a tripartite goal of \n(i) mitigating the jammer's interference by locating its subspace $\\textit{span}(\\Hj)$\nand projecting the receive matrix $\\bY$ onto the orthogonal complement of~that subspace, \n(ii) estimating the channel matrix~$\\bH$, and (iii) recovering the data matrix $\\bS_D$.\nWe then establish the soundness of the proposed optimization problem by proving that, \nunder certain sensible conditions, and assuming negligible thermal noise, \nits minimum is unique and corresponds to the desired solution; in particular, \nthe problem recovers the data matrix~$\\bS_D$.   \nFinally, we develop efficient iterative algorithms that approximately solve the proposed optimization problem.\n\n\n\\subsection{The Optimization Problem}\n\nWe start our derivation by considering the maximum-likelihood problem for joint channel estimation and data detection in the absence of\njamming, which is \\cite{vikalo2006efficient}\n\\begin{align}\n\t \\big\\{\\hat\\bH, \\hat\\bS_D\\big\\}\n\t&= \\argmin_{\\substack{\\hspace{1.3mm}\\tilde\\bH\\in\\opC^{B\\times U}\\\\ \\tilde\\bS_D\\in\\setS^{U\\times D}}}\\!\n\t\\big\\|\\bY - \\tilde\\bH \\tilde\\bS \\big\\|^2_F, \\label{eq:ml_jed}\n\\end{align}\nwhere we define $\\tilde\\bS \\triangleq [\\bS_T,\\tilde\\bS_D]$ for brevity and leave the dependence on\n$\\tilde\\bS_D$ implicit. This objective already integrates the goals of estimating the channel matrix\nand detecting the data symbols: If the noise $\\bN$ is small enough to be negligible, \nthe problem is minimized by the true channel and data matrices,\n\\begin{align}\n\t\\|\\bY - \\bH \\bS \\|^2_F \\approx 0,\n\\end{align}\nwhere the pilot matrix $\\bS_T$ ensures uniqueness.\\footnote{If the noise $\\bN$ \nis not strictly equal to zero, then the channel estimate $\\hat\\bH$ for which \\eqref{eq:ml_jed} is minimized\ndoes not coincide \\emph{exactly} \nwith the true channel matrix~$\\bH$. But thanks to the discrete search space, the minimizing data \nestimate $\\hat\\bS_D$ still coincides exactly with the true data matrix $\\bS_D$ if $\\bN$ is small enough.}\nHowever, in case of a jamming attack, the jammer will cause a residual\n\\begin{align}\n\t\\|\\bY - \\bH\\bS\\|^2_F &= \\|\\Hj\\tp{\\bsj} + \\bN\\|^2_F \\approx \\|\\Hj\\tp{\\bsj}\\|^2_F \\gg 0 \\label{eq:jed_residual}\n\\end{align}\nwhen plugging the true channel and data matrices into \\fref{eq:ml_jed}, and there is no reason to assume that \nthere exists no tuple $\\{\\tilde\\bH,\\tilde\\bS_D\\}$ with $\\tilde\\bS_D\\neq\\bS_D$ such that \n$\\|\\bY - \\tilde\\bH \\tilde\\bS\\|^2_F < \\|\\bY - \\bH\\bS\\|^2_F$.\nNote, however, that the residual $\\Hj\\tp{\\bsj}$ in \\eqref{eq:jed_residual} is a rank-one matrix whose\ncolumns are all in $\\textit{span}(\\Hj)$, regardless of the jamming signal~$\\bsj$.\nConsider therefore what happens when we take the~matrix\\footnote{The dependence\nof $\\tilde\\bP$ on $\\tilde\\bmp$ is left implicit here and throughout the paper.}\n\\begin{align}\n\\tilde\\bP\\triangleq\\bI-\\tilde\\bmp\\herm{\\tilde\\bmp},~\\tilde\\bmp\\in \\mathbb{S}_1^B,\n\\end{align} \nwhich projects a signal \nonto the orthogonal complement of some arbitrary one-dimensional subspace $\\textit{span}(\\tilde\\bmp)$,\nand then apply that projection to the objective of \\eqref{eq:ml_jed} as follows: \n\\begin{align}\n\t\\|\\tilde\\bP(\\bY - \\tilde\\bH\\tilde\\bS)\\|^2_F. \\label{eq:ml_p_jed}\n\\end{align}\nIf we now plug the true channel and data matrices into \\fref{eq:ml_p_jed} (still assuming negligibility of the noise $\\bN$), then we obtain\n\\begin{align}\n\t\\|\\tilde\\bP(\\bY - \\bH\\bS)\\|^2_F \n\t&= \\|\\tilde\\bP\\Hj\\tp{\\bsj} + \\tilde\\bP\\bN\\|^2_F \\\\\n\t&\\approx  \\|\\tilde\\bP\\Hj\\tp{\\bsj}\\|^2_F \\geq 0, \n\\end{align}\nwith equality if and only if $\\tilde\\bmp$ is collinear with $\\Hj$. \nIn other words, the unit vector $\\tilde\\bmp$ which in combination with the true channel and data matrices minimizes \\eqref{eq:ml_p_jed}\nis collinear with the jammer's channel, in which case $\\tilde\\bP$ is the POS matrix $\\bP_\\Hj$ from~\\eqref{eq:pos}.\n\nThus, if the noise $\\bN$ is negligible, and if\n(i)~$\\tilde\\bP$ is the projection onto the orthogonal complement of $\\textit{span}(\\Hj)$,\n(ii)~$\\tilde\\bH$ is the true channel matrix,\nand (iii) $\\tilde\\bS$ contains the true data matrix, \nthen the tuple $\\{\\tilde\\bmp,\\tilde\\bH,\\tilde\\bS\\}$ minimizes \\eqref{eq:ml_p_jed}.\nThese~are, of course, exactly the goals which we want to attain.\n %\nWe thus formulate our joint jammer estimation and mitigation, channel estimation, and data detection problem as follows:\n\\begin{align}\n\t \\big\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\big\\}\n\t&= \\argmin_{\\substack{\\tilde\\bmp\\in \\mathbb{S}_1^B\\hspace{1.4mm}\\\\ \\hspace{1.3mm}\\tilde\\bH_\\bP\\in\\opC^{B\\times U}\\\\ \\tilde\\bS_D\\in\\setS^{U\\times D}}}\\!\n\t\\big\\|\\tilde\\bP\\bY - \\tilde\\bH_\\bP \\tilde\\bS \\big\\|^2_F.\\!\n\t\\label{eq:obj1}\n\\end{align}\nNote that, compared to \\eqref{eq:ml_p_jed}, we have absorbed the projection matrix $\\tilde\\bP$ directly into \nthe unknown channel matrix $\\tilde\\bH_\\bP$, which replaces the product $\\tilde\\bP\\tilde\\bH$ in \\eqref{eq:ml_p_jed}. \nThis approach avoids the issue that the columns~of~$\\tilde\\bH$ would be indeterminate with respect to the length of their components in the direction of $\\tilde\\bmp\\approx\\Hj$, \nso that there would be no distinction between channel estimates $\\tilde\\bH + \\Hj\\tp{\\tilde{\\bsj}}$\nwith different  jamming sequences~$\\tilde{\\bsj}$.\n\n\\subsection{Theory}\\label{sec:theory}\nWe have derived the optimization problem \\eqref{eq:obj1} based on intuitive but non-rigorous arguments.\nThus, we will now support the soundness of \\eqref{eq:obj1} by proving that, under certain sensible conditions, \nand assuming that the noise is negligible, its solution is unique \nand guaranteed to recover the true data matrix.\n\nWe make the following assumptions: \nThe channel matrix $\\bH$ has full column rank $U$, \n the jammer channel $\\Hj$ is not included in the columnspace of $\\bH$, \nand the pilot matrix $\\bS_T$ has full row rank $U$.\nIn addition, we define a concept which may seem cryptic at first, but which will be clarified later.\n\n\\begin{defi} \\label{def:eclipse}\nWe say that the jammer is \\emph{eclipsed} in a given coherence interval if there exists a matrix $\\tilde{\\bS}_D\\in\\setS^{U\\times D}$\nsuch that \\mbox{$\\text{rank}(\\bS_D - \\tilde{\\bS}_D)\\leq 1$} and $\\tp{\\bsj_D} = \\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D$.\n\\end{defi}\nWe can now state our result; the proof is in \\fref{app:proof1}.\n\\begin{thm} \\label{thm:maed}\nIn the absence of noise, $\\bN=\\mathbf{0}$, and if the jammer is not eclipsed, then the problem in \\eqref{eq:obj1} has the unique solution $\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\}=\\{\\bmp, \\bP\\bH, \\bS_D\\}$. \n(In fact, $\\hat{\\bmp}$ is unique only up to an immaterial complex rotation, $\\hat{\\bmp}=\\alpha\\bmp, |\\alpha|=1$.)\n\\end{thm}\n\n\nIn other words, as long as the jammer is not ecplised, the problem in \\eqref{eq:obj1} is uniquely minimized by the true jammer subspace, projected channel matrix, and data matrix. \nWe now shed light on the notion of eclipsedness, and we answer in the affirmative the important question of whether\none can expect the jammer to typically be un-eclipsed.\nIn essence, the jammer is eclipsed if its jamming signal $\\bmw$ is such that multiple possible ``explanations''\nof the receive signal $\\bY$ exist which are consistent with the pilot matrix $\\bS_T$ \nand under some of which the jammer is not recognized as the jammer; cf. the discussion of \\eqref{eq:eclipsing_equation}\nin \\fref{app:proof1}.\nThis is best explained by considering the two emblematic cases of an eclipsed jammer:\n\\subsubsection{An inactive jammer (or no jammer)} \nClearly, if $\\bsj=\\mathbf{0}$, then $\\tp{\\bsj_D}=\\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D$ for all $\\tilde{\\bS}_D$, \nincluding \\mbox{$\\tilde{\\bS}_D=\\bS_D$}.~In~this case, there is a mismatch between the jammerless actual wireless transmission and the jammed model in \\eqref{eq:io}. \nSince there is no jammer subspace to identify, the choice of the projection~$\\tilde\\bP$ is undetermined, so that \n\\fref{thm:maed} no longer~applies.   \n\n\\subsubsection{The jammer transmits a valid pilot sequence}\nIf the jammer transmits the $k$th UE's pilot sequence in the training phase and constellation symbols\nin the data phase, then there are no formal grounds for the receiver to distinguish between the jammer and the $k$th UE. \nIt can readily be shown that, besides the desired solution \n$\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\}=\\{\\bmp, \\bP\\bH, \\bS_D\\}$, there exists then another solution to~\\eqref{eq:obj1}\nwhich identifies the $k$th UE as the jammer, nulls that UE by setting $\\hat\\bmp = \\bmh_k\/\\|\\bmh_k\\|$, \nand instead identifies the jammer as the $k$th UE by estimating\n\\begin{align}\n\t\\hat\\bH_\\bP &= \\hat\\bP [ \\bmh_1, \\dots, \\bmh_{k-1}, \\Hj, \\bmh_{k+1}, \\dots, \\bmh_U ], \\\\\n\t\\hat\\bS_D &= \\tp{[ \\tp{\\bms_{D,1}}, \\dots, \\tp{\\bms_{D,k-1}}, \\tp{\\bmw_D}, \\tp{\\bms_{D,k+1}}, \\dots, \\tp{\\bms_{D,U}} ]},\n\\end{align}\nwhere $\\bms_{D,u}$ is the $u$th row of $\\bS_D$.\n\nIn addition to these two emblematic cases, eclipsing can also happen in more accidental cases where the symbol\nerror matrix $\\bS_D - \\tilde\\bS_D$ has rank one for some $\\tilde\\bS_D$ such that by some (un)fortunate\ncoincidence $\\tp{\\bsj_D} = \\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D$ holds true. \n\nHowever, we will now show that if the jammer does not know the pilot sequences, e.g., because\nthey are drawn at random by the BS and secretly communicated to the UEs, then an active jammer (where $\\bsj\\neq \\mathbf{0}$)\nis typically not eclipsed. To show this, we consider a case in which the pilot matrix $\\bS_T$ is square; the proof is relegated to \\fref{app:proof2}.\n\n\\begin{thm} \\label{thm:maed2}\n\tIf the pilot matrix $\\bS_T$ is drawn uniformly over the set of $U\\times U$ unitary matrices and if $\\bsj\\neq\\mathbf{0}$ is\n\tindependent of $\\bS_T$,\tthen the probability that the jammer is eclipsed is zero. \n\\end{thm}\n\n\n\\begin{remark} \\label{rem:rare}\n\tIt is by no means necessary to use random pilots to avoid eclipsing. \n\tAnother sufficient condition for the jammer to be eclipsed only with zero probability  \n\tis if $\\tp{\\bmw_D}$ is independent of $\\tp{\\bsj_T}\\pinv{\\bS_T}$ and if\n\tat least one of the marginals of $\\tp{\\bmw_D}$ or of $\\tp{\\bsj_T}\\pinv{\\bS_T}$ \n\thas no mass points. \n\tIn essence, unless the jammer choses its input sequence as some (partially randomized) \n\tfunction of the pilot matrix $\\bS_T$, eclipsing is the rare exception, not the norm. \n\tIn this regard, see also the simulation results in \\fref{sec:results}.\n\\end{remark}\n\n\\begin{remark} \nThe fact that error-free communication in the presence of jamming can be assured if \nthe BS and UEs~share~a common secret that enables them to use a randomized~communication scheme, but \nnot otherwise, is reminiscent of~information-theoretic results which prove a similar \ndichotomy on a more fundamental level. See \\cite[Sec. V]{lapidoth1998reliable} and references therein.\n\\end{remark}\n\n\n\\section{Forward-Backward Splitting with a Box Prior}\n\nWe now provide the first of two algorithms for approximately solving the\njoint jammer estimation and mitigation, channel estimation, and data detection problem in \\eqref{eq:obj1}. \nNote first of all that the objective is quadratic in $\\tilde\\bH_\\bP$, \nso we can derive the optimal value of $\\tilde\\bH_\\bP$ as a function of $\\tilde\\bP$ and $\\tilde\\bS$ as\n\\begin{align}\n\t\\hat\\bH_\\bP = \\tilde\\bP\\bY\\pinv{\\tilde\\bS}, \n\\end{align}\nwhere $\\pinv{\\tilde\\bS}=\\herm{\\tilde\\bS}\\inv{(\\tilde\\bS\\herm{\\tilde\\bS})}$. \nSubstituting $\\hat\\bH_\\bP$ back into \\eqref{eq:obj1} yields \nan optimization problem which only depends on $\\tilde\\bmp$ and $\\tilde\\bS_D$:\n\\begin{align}\n\t\\big\\{\\hat\\bmp, \\hat\\bS_D\\big\\} = \n\t\\argmin_{\\substack{\\tilde\\bmp\\in \\mathbb{S}_1^B\\hspace{1.4mm}\\\\ \\tilde\\bS_D\\in\\setS^{D\\times U}}}\n\t\\big\\|\\tilde\\bP\\bY(\\bI_K - \\pinv{\\tilde\\bS}\\tilde\\bS)\\big\\|^2_F. \\label{eq:obj3}\n\\end{align}\nSolving \\eqref{eq:obj3} remains difficult due to its combinatorial nature, so we resort to solving it approximately. \nFirst, we relax the constraint set $\\setS$ to its convex hull $\\setC\\triangleq\\textit{conv}(\\setS)$ as in \\cite{castaneda2018vlsi}.\nThis can be viewed as replacing the probability mass function over the constellation\n$\\setS$, which represents the true symbol prior, with a box prior that is uniform \nover $\\setC$ and zero elsewhere \\cite{jeon2021mismatched}. \nWe then approximately solve this~relaxed problem formulation in an iterative fashion by alternating between \na forward-backward splitting descent step in $\\tilde\\bS$ and a minimization step in $\\tilde\\bP$.\n\n\\subsection{Forward-Backward Splitting Step in $\\tilde\\bS$} \\label{sec:fbs}\nForward-backward splitting (FBS) \\cite{goldstein16a}, also called proximal gradient descent,\nis an iterative method for solving convex optimization problems of the form\n\\begin{align}\n\t\\argmin_{\\tilde\\bms}\\, f(\\tilde\\bms) + g(\\tilde\\bms), \\label{eq:fbs1}\n\\end{align}\nwhere $f$ is convex and differentiable, and $g$ is convex but not necessarily\ndifferentiable, smooth, or bounded. Starting from an initialization vector $\\tilde\\bms^{(0)}$, \nFBS solves the problem in~\\eqref{eq:fbs1} iteratively by computing \n\\begin{align}\n\t\\tilde\\bms^{(t+1)} = \\proxg\\big(\\tilde\\bms^{(t)} - \\tau^{(t)}\\nabla f(\\tilde\\bms^{(t)}); \\tau^{(t)}\\big). \\label{eq:fbs2}\n\\end{align}\nHere, $\\tau^{(t)}$ is the stepsize at iteration $t$, $\\nabla f(\\tilde\\bms)$ is the gradient~of $f(\\tilde\\bms)$,  \nand $\\proxg$ is the proximal operator of $g$, defined as \\cite{parikh13a}\n\\begin{align}\n\t\\proxg(\\bmx, \\tau) = \\argmin_{\\tilde\\bmx} \\tau g(\\tilde\\bmx) + \\frac12 \\|\\bmx - \\tilde\\bmx\\|_2^2.\n\\end{align}\nFor a suitable sequence of stepsizes $\\{\\tau^{(t)}\\}$, FBS solves convex optimization problems exactly.\nFBS can also be used to approximately solve non-convex\nproblems, although there are typically no guarantees for optimality or even convergence~\\cite{goldstein16a}.\nFor the optimization problem in \\fref{eq:obj3}, we define $f$ and $g$ as \n\\begin{align}\n\tf(\\tilde\\bS) &= \\big\\|\\tilde\\bP\\bY(\\bI_K - \\pinv{\\tilde\\bS}\\tilde\\bS)\\big\\|^2_F\n\\end{align}\nand\n\\begin{align}\n\tg(\\tilde\\bS) &= \\begin{cases}\n\t\t0 &\\text{if }\\,\\tilde\\bS_{[1:T]}=\\bS_T \\text{ and } \\tilde\\bS_{[T+1:K]}\\in\\setC^{U\\times D}\n\t\t\\!\\!\\!\\\\\n\t\t\\infty &\\text{else}.\n\t\\end{cases}\n\\end{align}\nThe gradient of $f$ in $\\tilde\\bS$ is given by \n\\begin{align}\n\t\\nabla f(\\tilde\\bS) = -\\herm{(\\bY\\pinv{\\tilde\\bS})}\\tilde\\bP\\bY(\\bI_K - \\pinv{\\tilde\\bS}\\tilde\\bS), \\label{eq:gradient}\n\\end{align}\nand the proximal operator for $g$ is simply the orthogonal projection onto $\\setC$, which  \nacts entrywise on $\\tilde\\bS$~as\n\\begin{align}\n\t[\\proxg(\\tilde\\bS)]_{u,k} = \\begin{cases}\n\t\t[\\bS_T]_{u,k} &\\text{ if } k\\in[1:T] \\\\\n\t\t\\text{proj}_\\setC([\\tilde\\bS_{u,k}]) &\\text{ else,}\n\t\\end{cases} \\label{eq:proxg} \n\\end{align}\nwhere the function $\\text{proj}_\\setC$ is given as \n\\begin{align}\n\t\\text{proj}_\\setC(x) =\\, & \\min\\{\\max\\{\\Re(x),-\\lambda\\},\\lambda\\} \\nonumber\\\\\n\t&+ i\\min\\{\\max\\{\\Im(x),-\\lambda\\},\\lambda\\},\n\\end{align}\nwith $\\lambda=\\sqrt{\\sfrac{1}{2}}$ for a QPSK constellation \nand $\\lambda=\\sqrt{\\sfrac{9}{10}}$ for a 16-QAM constellation, see \\fref{fig:constellations}. \nTo select the per-iteration stepsizes~$\\{\\tau^{(t)}\\}$, we use the Barzilai-Borwein method \n\\cite{barzilai1988two}. \n\n\\subsection{Minimization Step in $\\tilde\\bP$}\nAfter each FBS step in $\\tilde\\bS$, we minimize \\eqref{eq:obj3}\nwith respect to the vector~$\\tilde\\bmp$. Defining the residual matrix\n$\\tilde\\bE\\triangleq \\bY(\\bI_K - \\pinv{\\tilde{\\bS}}\\tilde{\\bS})$\nand performing standard algebraic manipulations yields\n\\begin{align}\n\t\\hat\\bmp &= \\argmin_{\\tilde\\bmp\\in \\mathbb{S}_1^B} \\big\\|\\tilde\\bP \\tilde\\bE \\big\\|^2_F\t\\\\\n\t&= \\argmax_{\\tilde\\bmp\\in \\mathbb{S}_1^B} \\, \\herm{\\tilde\\bmp} \\tilde\\bE \\herm{\\tilde\\bE} \\tilde\\bmp. \\label{eq:rayleigh}\n\\end{align}\nIt follows that the vector $\\hat\\bmp$ minimizing \\eqref{eq:obj3} for a fixed~$\\tilde\\bS$ is the unit vector that maximizes the \nRayleigh quotient of $\\tilde\\bE \\herm{\\tilde\\bE}$. \nThe~solution is the unit-length eigenvector \n$\\bmv_1(\\tilde\\bE \\herm{\\tilde\\bE})$ associated with the largest eigenvalue of $\\tilde\\bE \\herm{\\tilde\\bE}$~\\cite[Thm.\\,4.2.2]{horn2013matrix},\n\\begin{align}\n\t\\hat\\bmp=\\bmv_1(\\tilde\\bE \\herm{\\tilde\\bE}).\n\\end{align}\nCalculating this eigenvector in every iteration of our algorithm would be computationally expensive, \nso we approximate it using a single power iteration \\cite[Sec.\\,8.2.1]{GV96}, i.e., \nwe estimate \n\\begin{align}\n\t\\hat\\bmp^{(t+1)} = \\frac{\\tilde\\bE^{(t+1)} \\herm{(\\tilde\\bE^{(t+1)})}\\hat\\bmp^{(t)}}{\\|\\tilde\\bE^{(t+1)} \\herm{(\\tilde\\bE^{(t+1)})}\\hat\\bmp^{(t)}\\|_2},\n\\end{align}\nwhere the power method is initialized with the subspace estimate $\\hat\\bmp^{(t)}$ from the previous algorithm iteration.\n\n\\subsection{Preprocessing}\nIf the algorithm starts directly with a gradient descent step in the direction of \\eqref{eq:gradient}, \none runs the risk of advancing significantly into the wrong direction---especially if the jammer is extremely strong,\nsince a strong jammer will also lead to a large gradient amplitude. Empirically, we observe that such a large initial digression \ncan be problematic (if, e.g., the jammer is $\\geq\\!50$\\,dB stronger than the average UE).\nIt might therefore be tempting to start the algorithm directly with a projection step: \nIf one initializes $\\tilde\\bS^{(0)}=\\mathbf{0}_{U\\times D}$, then $\\tilde\\bE^{(0)}=\\bY$, so that the algorithm starts by \nnulling the dimension of $\\bY$ which contains the most energy. In the presence of a strong jammer, this is a sensible strategy\nsince this dimension then corresponds to the jammer subspace. However, if the received jamming energy is \nsmall compared to the energy received from the UEs (e.g., because the jammer does not transmit at all during a given coherence interval), \nthen such a projection would inadvertently null the strongest user. \nTo thread the needle between these two cases---largely removing a strong jammer before the first gradient step, \nbut not removing any legitimate UEs when a strong jammer is absent---we propose to start with a \\emph{regularized} projection step:\nThe algorithm starts by a projection onto the orthogonal complement of the eigenvector of the largest eigenvalue\nof \n\\begin{align}\n\t\\bY\\herm{\\bY} + \\mathbf{\\Gamma}, \\label{eq:regularizer}\n\\end{align}\nwhere $\\mathbf{\\Gamma}\\in\\opC^{B\\times B}$ is a constant regularization matrix. The basic idea is that this regularization matrix is\nstill overshadowed by very strong jammers, so that these are largely nulled within the preprocessing, \nwhile, in the presence of only a weak jammer (or no jammer), the regularization matrix has a sufficiently diverting impact \non the eigenvectors to prevent the nulling of a legitimate UE. \nThere are countless ways of choosing such a regularization matrix. (Note, however, that $\\mathbf{\\Gamma}$ should \nnot be a multiple of the identity matrix $\\bI_B$, which does not affect the eigenvectors of \\eqref{eq:regularizer}.) For simplicity, we set $\\mathbf{\\Gamma}$ to the all-zero matrix, except for \nthe top left entry, which is set to $0.1BUK$. \n\n\\subsection{Algorithm Complexity} \nWe now have all the ingredients for MAED, which is summarized in \\fref{alg:maed}. Its only input is the receive matrix~$\\bY$, \nas it does not even require an estimate of the thermal noise variance $\\No$.\nMAED is initialized with $\\tilde\\bS^{(0)}=[\\bS_T, \\mathbf{0}_{U\\times D}]$ and $\\tau^{(0)}=\\tau=0.1$, \nand runs for a fixed number of $t_{\\max}$~iterations.\n\nThe complexity of MAED is dominated by the eigenvector calculation in the preprocessing step\n(which could be reduced by approximating it using the power method) as well \nas the gradient computation in line 5 of \\fref{alg:maed},  \nwhich has a complexity of $O(3BUK+2U^2K+U^3)$.\nThe overall complexity of MAED is therefore \\mbox{$O(t_{\\max}(3BUK+2U^2K+U^3))$.}\nNote, however, that MAED detects $D$ data vectors at once.\nThus, the computational complexity per detected symbol is only \\mbox{$O(t_{\\max}(3BK+2KU+U^2)\/D)$.}\n\n\n\\section{Soft-Output Estimates with Deep Unfolding}\nMAED, wich corresponds to the algorithm proposed in~\\cite{marti2022smart} (newly adding the preprocessing step), \nalready attains the goal of mitigating smart jammers, see \\fref{sec:results}. \nHowever, its detection performance can be suboptimal, especially when higher-order transmit constellations \nsuch as 16-QAM are used. \nThe culprit is the box prior of MAED, which does not fully exploit the \ndiscrete nature of the transmit constellation. In particular, the box prior is uninformative about \nthe  constellation symbols in the interior.\nTo improve detection performance, especially in the cases where MAED performs suboptimally, \nwe now provide a second algorithm for approximately solving the problem \nin \\eqref{eq:obj1}. This second algorithm builds on MAED but replaces the proximal operator in \\eqref{eq:proxg}, which enforces\nMAED's box prior, by an approximate posterior mean estimator (PME) based on the discrete symbol prior as in \\cite{song2021soft}.  \nSince the PME also enables meaningful soft-output estimates of the bits that underlie the transmitted data symbols, we refer to \nthis second algorithm as soft-output MAED (SO-MAED).\n\n\\begin{algorithm}[tp]\n\\caption{MAED}\n\\label{alg:maed}\n\\begin{algorithmic}[1]\n\\setstretch{1.0}\n\\State {\\bfseries input:} $\\bY$\n\\State \\text{initialize} $\\tilde\\bS^{(0)} \\!=\\! \\left[\\bS_T, \\mathbf{0}_{U\\!\\times\\! D} \\right]\\!, \n\\tilde\\bmp^{(0)} \\!=\\! \\bmv_1(\\bY \\herm{\\bY} \\!+ \\mathbf{\\Gamma} ), \\tau^{(0)} \\!= \\tau$\n\\State $\\tilde\\bP^{(0)} = \\bI_B - \\tilde\\bmp^{(0)}\\tilde\\bmp^{(0)}{}^\\text{H}$\n\\For{$t=0$ {\\bfseries to} $t_{\\max}-1$}\n\t\\State $\\nabla f(\\tilde\\bS^{(t)}) = -\\herm{\\big(\\bY\\tilde\\bS^{(t)}{}^\\dagger\\big)} \\tilde\\bP^{(t)}\\bY(\\bI_K - \\tilde\\bS^{(t)}{}^\\dagger \\tilde\\bS^{(t)})$\n\t\\State $\\tilde\\bS^{(t+1)} = \\proxg\\big(\\tilde\\bS^{(t)} - \\tau^{(t)}\\nabla f(\\tilde\\bS^{(t)})\\big)$ \n\t\\State $\\tilde\\bE^{(t+1)} = \\bY(\\bI_K - \\tilde\\bS^{(t+1)}{}^\\dagger \\tilde\\bS^{(t+1)})$\n\t\\State $\\tilde\\bmp^{(t+1)} = \\tilde\\bE^{(t+1)} \\tilde{\\bE}^{(t+1)}{}^\\text{H}\\, \\tilde\\bmp^{(t)}\/\\|\\tilde\\bE^{(t+1)} \\tilde{\\bE}^{(t+1)}{}^\\text{H}\\, \\tilde\\bmp^{(t)}\\|_2$\n\t\\State $\\tilde\\bP^{(t+1)} = \\bI_B - \\tilde\\bmp^{(t+1)}\\tilde\\bmp^{(t+1)}{}^\\text{H}$\n\t\\State compute $\\tau^{(t+1)}$ by following \\cite[Sec.\\,4.1]{goldstein16a}\n\t\\EndFor\n\t\\State \\textbf{output:} $\\tilde\\bS^{(t_{\\max})}_{[T+1:K]}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Approximate Posterior-Mean Estimation}\n\n\nTo replace the proximal operator following the gradient descent step in \\eqref{eq:fbs2}\nwith a more appropriate data symbol estimator which takes into account the discrete constellation~$\\setS$,\nwe model the per-iteration outputs of the gradient descent step\n\\begin{align}\n\t\\tilde\\bX^{(t)} = \\tilde\\bS^{(t)} - \\tau^{(t)}\\nabla f(\\tilde\\bS^{(t)})\n\\end{align} \nas \n\\begin{align}\n\t\\tilde\\bX^{(t)} &= \\bS + \\bZ^{(t)} = \\big[ \\bS_T, \\bS_D \\big] + \\big[ \\bZ_T^{(t)}, \\bZ_D^{(t)} \\big], \\label{eq:symbol_noise}\n\\end{align}\ni.e., as the true transmit symbol matrix $\\bS$ corrupted by an additive error~$\\bZ^{(t)}$.\nIf the distribution of $\\bZ^{(t)}$ were known, \none could compute the posterior mean $\\mathbb{E}[\\bS\\,|\\,\\tilde\\bX^{(t)}]$\nand use it as a constellation-aware replacement of the proximal step \\eqref{eq:proxg}, \n\\begin{align}\n\t\\tilde\\bS^{(t+1)} = \\mathbb{E}\\big[\\bS\\,|\\,\\tilde\\bX^{(t)}\\big].\n\\end{align}\nUnfortunately, the distribution of $\\bZ^{(t)}$ is unknown in practice, but the~submatrix $\\bS_T$ of $\\bS$ (and hence its mean)\t\nis known at the receiver. To estimate the mean of the submatrix $\\bS_D$, we assume that\nthe entries of $\\bZ_D^{(t)}$ are distributed independently~of $\\bS$ as i.i.d.\\ circularly-symmetric \ncomplex Gaussians with variance~$\\nu^{(t)}$, \n\\begin{align}\n\t\\big[\\bZ^{(t)}\\big]_{u,k}\\sim\\setC\\setN(0,\\nu^{(t)}).\n\\end{align}\nThe variances $\\{\\nu^{(t)}\\}_{t=0}^{t_{\\max}-1}$ are treated as algorithm parameters\nand will be optimized using deep unfolding as detailed below. \n\nBased on this idealized model, we utilize a three-step procedure as in \\cite{song2021soft} for computing \nsymbol estimates. First, we use~\\fref{eq:symbol_noise} to compute log-likelihood ratios (LLRs)\nfor every transmitted bit. We then convert these LLRs to the probabilities of the respective bits being $1$. \nThis step also provides the aforementioned soft-output estimates. \nFinally, we convert the bit probabilities back to symbol estimates by calculating the symbol mean.  \n\\mbox{The LLRs can be computed following \\cite{collings2004low, jeon2021mismatched} as}\n\\begin{align}\n\tL_{i,u,k}^{(t)} = \\frac{\\ell\\big(\\tilde X_{u,k}^{(t)}\\big)\\!}{\\nu^{(t)}}, \n\t~ i\\!\\in\\![1\\!:\\!\\log_2\\!|\\setS|],~ u\\!\\in\\![1\\!:\\!U],~ k\\!\\in\\![T\\!+\\!1\\!:\\!K],\n\t\t \\label{eq:llrs}\n\\end{align}\nwhere $\\ell(\\cdot)$ is specified in Table~\\ref{tab:llr} (cf.~\\fref{fig:constellations}).\nThe LLR values are exact for QPSK and use the max-log \napproximation for 16-QAM~\\cite{jeon2021mismatched}.\nThe LLRs can then be converted to probabilities~via\n\\begin{align}\n\tp_{i,u,k}^{(t)} = \\frac12\\left( 1 + \\tanh{\\left(\\frac{L_{i,u,k}^{(t)}}{2}\\right)} \\right).\n\t\\label{eq:bit_probabs}\n\\end{align}\nFinally, the probabilities of \\eqref{eq:bit_probabs} can be used to compute symbol estimates according to Table~\\ref{tab:symbol_estimates}.\n\n\nTo summarize, SO-MAED replaces MAED's proximal operator in \\eqref{eq:proxg}\nwith the symbol estimator that consists of \\eqref{eq:llrs}, \\eqref{eq:bit_probabs}, and Table~\\ref{tab:symbol_estimates}.\nWe refer to this symbol estimation as posterior-mean approximation (PMA) and denote it as \n\\begin{align}\n\t\\tilde\\bS^{(t+1)} = \\pma(\\tilde\\bX^{(t)},\\nu^{(t)}),\t\t\n\\end{align}\nwhere the subscript $\\setS$ makes explicit the dependence of the PMA on the symbol constellation.\nSince the PMA involves only scalar computations, its complexity is negligible compared to the\nmatrix-vector and matrix-matrix operations of SO-MAED. The complexity order of SO-MAED is \ntherefore identical to that of MAED, namely \\mbox{$O(t_{\\max}(3BUK+2U^2K+U^3))$.}\n\n\n\\subsection{Deep Unfolding of SO-MAED}\nThe procedure outlined in the previous subsection requires the variances $\\{\\nu^{(t)}\\}_{t=0}^{t_{\\max}-1}$ \nof the per-iteration estimation errors~$\\bZ^{(t)}$, which are generally unknown. We treat these variances \nas parameters of SO-MAED and optimize them using deep unfolding \\cite{song2021soft, hershey2014deep, balatsoukas2019deep, goutay2020deep, monga2021algorithm}.\nDeep unfolding is an emerging paradigm under which iterative algorithms are unfolded into artificial neural networks\nwith one layer per iteration, so that the algorithm parameters can be\nregarded as trainable weights of that network. These weights are then learned from training data with  \nstandard deep learning tools~\\cite{Baydin2018automatic, tensorflow2015whitepaper}. \n\n\nTo improve stability of learning, we use the error precisions $\\{\\rho^{(t)}\\}_{t=0}^{t_{\\max}-1}$ \ninstead of the variances $\\{\\nu^{(t)}\\}_{t=0}^{t_{\\max}-1}$\nas parameters of the unfolded network, with $\\rho^{(t)}=1\/\\nu^{(t)}$.\nIn addition, we also regard the gradient step sizes $\\{\\tau^{(t)}\\}_{t=0}^{t_{\\max}-1}$\nas trainable weights (instead of computing them according to the Barzilai-Borwein method). \nFurthermore, we add a momentum term with per-iteration weights $\\{\\lambda^{(t)}\\}_{t=0}^{t_{\\max}-1}$ to our gradient descent procedure.\nFinally, inspired by the Bussgang decomposition \\cite{bussgang52a, minkoff85a}, we add per-iteration scale factors \n$\\{\\alpha^{(t)}\\}_{t=0}^{t_{\\max}-1}$ to the output of \\eqref{eq:symbol_noise}, \nwith the goal of accommodating uncorrelatedness (if not independence) between $\\bZ_D^{(t)}$ and $\\bS_D$ in \\eqref{eq:symbol_noise}.\nThe final algorithm is summarized in \\fref{alg:so_maed}.\n\n\nWe implement the unfolded algorithm in TensorFlow \\cite{tensorflow2015whitepaper} and train it \nusing as cost function the empirical binary cross-entropy (BCE) between the transmitted bits \nand the estimated bit probabilities \\eqref{eq:bit_probabs} from the last iteration as the output of our network.\nThe loss function is given as\n\\begin{align}\n\t\\sum_{\\text{sample}\\in\\setD} \\beta_{\\text{sample}} \\!\\left( \\sum_{i=1}^{\\log_2|\\setS|} \\sum_{u=1}^U \\sum_{k=T+1}^K \\text{BCE}(b_{i,u,k}, p_{i,u,k}^{t_{\\max}}) \\right)\\!\\!,\\!\n\\end{align}\nwhere \n\\begin{align}\n\t\\text{BCE}(b,p(b)) = b \\log_2(p(b)) + (1-b)\\log_2(1-p(b)),\n\\end{align}\nand where $\\beta_{\\text{sample}}$ are the weights given to the different samples in the training set $\\setD$.\nWe only learn a single set of weights per system dimensions $\\{U,B,K\\}$, which is used for all \nsignal-to-noise ratios (SNRs) and, most importantly, \nall jamming attacks (since a receiver does not typically know in advance\nwhich type of a jamming attack it is facing). For this reason, we train using samples from different SNRs and \ndifferent jamming attacks. \nWe also have to avoid overfitting to a specific type of jamming attack.\nIf our evaluation in \\fref{sec:results} would feature only the exact same types of jammers \nthat were used for training, this would raise questions about the ability of SO-MAED to generalize to \njamming attacks which differ from those explicitly included in the training set.\nHowever, the principles underlying the SO-MAED algorithm are essentially invariant with respect to the type of a \njamming attack. For this reason, we only train on a single type of jammers, namely pilot jammers, cf. \\fref{sec:setup} (which we have empirically recognized to be the most difficult to mitigate), while evaluating the trained algorithm on many other jammer types besides pilot jammers, cf. \\fref{sec:results}.\n The attacks used for training also comprise different jammer receive strengths,\nnamely $\\{-\\infty\\,\\text{dB}, 0\\,\\text{dB}, 10\\,\\text{dB}, 20\\,\\text{dB}, 40\\,\\text{dB}, 80\\,\\text{dB}\\}$ \ncompared to the average UE. \n\\begin{figure}[tp]\n\\centering\n\n\\subfigure[QPSK\\hspace{1cm}]{\n\\includegraphics[height=3.4cm]{figures\/constellation_qpsk}\n\\label{fig:constellations:qpsk}\n}\n\\subfigure[16-QAM\\hspace{1cm}]{\n\\includegraphics[height=3.4cm]{figures\/constellation_16qam}\n\\label{fig:constellations:16qam}\n}\n\\caption{Transmit constellations $\\setS$ (including the used Gray mapping) and their convex hulls $\\setC$. \n$\\lambda=\\sqrt{\\sfrac{1}{2}}$ for QPSK and $\\lambda=\\sqrt{\\sfrac{9}{10}}$ for 16-QAM.\n}\n\\label{fig:constellations}\n\\end{figure}\n\\begin{table}[tp!]\n\\centering\n\\caption{LLR computation according to \\cite[Tbl. 1]{collings2004low}, \\cite{jeon2021mismatched}}\n\\vspace{-5mm}\n\\setstretch{1.1}\n\\begin{tabular}[t]{@{}lcl@{}}\n\\toprule\n&\\!\\!Bit $i$\\!\\!& $\\ell(x)$ \\\\\n\\midrule\n\\multirow{2}{*}{\\!QPSK\\!}& $1$ & $4\\lambda\\Re\\{x\\}$  \\\\ \n& $2$ & $4\\lambda\\Im\\{x\\}$ \\vspace{1mm} \\\\\n\\multirow{4}{*}{\\!16-QAM\\!}& $1$ & $\\frac{2\\lambda}{3} \\left( 4\\Re\\{x\\} + \\left|\\Re\\{x\\} \\!-\\! \\frac{2\\lambda}{3} \\right| - \\left|\\Re\\{x\\} \\!+\\! \\frac{2\\lambda}{3} \\right| \\right)\\!$ \\\\\n& $2$ & $\\frac{4\\lambda}{3} \\left( \\frac{2\\lambda}{3} - \\left|\\Re\\{x\\} \\right| \\right) $ \\\\\n& $3$ & $\\frac{2\\lambda}{3} \\left( 4\\Im\\{x\\} + \\left|\\Im\\{x\\} \\!-\\! \\frac{2\\lambda}{3} \\right| - \\left|\\Im\\{x\\} \\!+\\!\\frac{2\\lambda}{3} \\right| \\right)\\! $ \\\\\n& $4$ & $\\frac{4\\lambda}{3} \\left( \\frac{2\\lambda}{3} - \\left|\\Im\\{x\\} \\right| \\right) $ \\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{5mm}\n\\label{tab:llr}\n\\centering\n\\caption{Mapping the probabilities in \\eqref{eq:bit_probabs} to symbol estimates\n\\cite{jeon2021mismatched, tomasoni2006low}}\n\\vspace{-5mm}\n\\setstretch{1.1}\n\\begin{tabular}[t]{@{}lcc@{}}\n\\toprule\n& $\\Re\\{\\hat{s}\\}$\\!& $\\Im\\{\\hat{s}\\}$ \\\\\n\\midrule\nQPSK & $\\lambda(2p_{1}-1)$ & $\\lambda(2p_{2}-1)$  \\\\\n16-QAM& $\\lambda(2p_1-1)(3-2p_2)$ &  $\\lambda(2p_3-1)(3-2p_4)$ \\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{-2.5mm}\n\\label{tab:symbol_estimates} \n\\end{table\nAlso with regard to the jammer receive strength, the evaluation in \\fref{sec:results} will consider jammers with different strengths\nthan have been used for training.\nThe sample weights $\\beta_{\\text{sample}}$ are used to prevent certain training samples (e.g., those at low SNR with strong jammers)\nfrom dominating the learning process by drowning out the loss contribution from training samples with inherently lower BCE. \nFor this, we fix a ``baseline performance'' and select the weight of a training sample as the inverse of this sample's BCE loss \naccording to the baseline. The baseline is set by an untrained version of SO-MAED with reasonably initialized weights\n(its performance in general already exceeds that of~MAED).\nFor training, we use the Adam optimizer \\cite{kingma2014adam} from Keras with default values. The batch size \nstarts at one sample, but is increased first to five and then to ten samples whenever the training loss does not improve for two consecutive\nepochs.\n \n \n\\begin{algorithm}[tp]\n\\caption{SO-MAED}\n\\label{alg:so_maed}\n\\begin{algorithmic}[1]\n\\setstretch{1.0}\n\\State {\\bfseries input:} $\\bY, \\{\\tau^{(t)}, \\alpha^{(t)}, \\lambda^{(t)}, \\rho^{(t)} \\}_{t=0}^{t_{\\max} -1}$\n\\State \\text{init} $\\tilde\\bS^{(0)} \\!=\\! \\left[\\bS_T, \\mathbf{0}_{U\\!\\times\\! D} \\right], \n\\tilde\\bmp^{(0)} \\!=\\! \\bmv_1(\\bY \\herm{\\bY} \\!+ \\mathbf{\\Gamma} ), \\mathbf{\\Delta}^{(-1)} = \\mathbf{0}$\n\\State $\\tilde\\bP^{(0)} = \\bI_B - \\tilde\\bmp^{(0)}\\tilde\\bmp^{(0)}{}^\\text{H}$\n\\For{$t=0$ {\\bfseries to} $t_{\\max}-1$}\n\t\\State $\\nabla f(\\tilde\\bS^{(t)}) = -\\herm{\\big(\\bY\\tilde\\bS^{(t)}{}^\\dagger\\big)} \\tilde\\bP^{(t)}\\bY(\\bI_K - \\tilde\\bS^{(t)}{}^\\dagger \\tilde\\bS^{(t)})$\n\t\\State $\\mathbf{\\Delta}^{(t)} = - \\tau^{(t)} \\nabla f(\\tilde\\bS^{(t)}) + \\lambda^{(t)} \\mathbf{\\Delta}^{(t-1)}$\n\t\\State $\\tilde\\bX^{(t)} = \\tilde\\bS^{(t)} + \\mathbf{\\Delta}^{(t)} $\n\t\\State $\\tilde\\bS^{(t+1)} = \\pma\\big( \\alpha^{(t)}\\tilde\\bX^{(t)}, 1\/\\rho^{(t)} \\big)$ \n\t\\State $\\tilde\\bE^{(t+1)} = \\bY(\\bI_K - \\tilde\\bS^{(t+1)}{}^\\dagger \\tilde\\bS^{(t+1)})$\n\t\\State $\\tilde\\bmp^{(t+1)} = \\tilde\\bE^{(t+1)} \\tilde{\\bE}^{(t+1)}{}^\\text{H}\\, \\tilde\\bmp^{(t)}\/\\|\\tilde\\bE^{(t+1)} \\tilde{\\bE}^{(t+1)}{}^\\text{H}\\, \\tilde\\bmp^{(t)}\\|_2$\n\t\\State $\\tilde\\bP^{(t+1)} = \\bI_B - \\tilde\\bmp^{(t+1)}\\tilde\\bmp^{(t+1)}{}^\\text{H}$\n\t\\EndFor\n\t\\State \\textbf{output:} $\\tilde\\bS^{(t_{\\max})}_{[T+1:K]}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\\section{Simulation Results} \\label{sec:results}\n\\subsection{Simulation Setup} \\label{sec:setup}\nWe simulate a massive MU-MIMO system with $B=128$~BS antennas, \n$U=32$ single-antenna UEs, and one single-antenna jammer. \nThe UEs transmit for $K=160$ time slots, where the first\n$T=32$ slots are used for orthogonal pilots $\\bS_T$ in the form of a\n $32\\times32$ Hadamard matrix with unit symbol energy.\nThe remaining $D=128$ slots are used to transmit QPSK or 16-QAM payload data.\nUnless noted otherwise, the channels are modelled as i.i.d. Rayleigh fading. \nWe define the average receive signal-to-noise ratio (SNR) as \n\\begin{align}\n\\textit{SNR} \\define \\frac{\\Ex{\\bS}{\\|\\bH\\bS\\|_F^2}}{\\Ex{\\bN}{\\|\\bN\\|_F^2}}.\n\\end{align}\nWe consider four different jammer types:  \n(J1) barrage jammers that transmit i.i.d.\\ jamming symbols during the entire coherence interval,\n(J2) pilot jammers that transmit i.i.d.\\ jamming symbols during the pilot phase but do not jam the data phase, \n(J3) data jammers that transmit i.i.d.\\ jamming symbols during the data phase but do not jam the pilot phase, \nand (J4) sparse jammers that transmit i.i.d.\\ jamming symbols during some fraction $\\alpha$ of randomly selected bursts of unit length (i.e., one time slot).\nThe jamming symbols are either circularly symmetric complex Gaussian or drawn uniformly from the transmit constellation \n(i.e., QPSK or 16-QAM).\nThey are also independent of the UE transmit symbols $\\bS$, unless stated otherwise.\nWe quantify the strength of the jammer's interference relative to the strength of the average UE, either as the ratio \nbetween total receive~\\textit{energy}\n\\begin{align}\n\\rE \\define \\frac{\\Ex{\\bsj}{\\|\\Hj\\bsj\\|_2^2}}{\\frac1U\\Ex{\\bS}{\\|\\bH\\bS\\|_2^2}},\n\\end{align}\nor as the ratio between receive \\textit{power during those phases that the jammer is jamming}\n\\begin{align}\n\t\\rP \\triangleq \\frac{\\rE}{\\gamma},\n\\end{align}\nwhere $\\gamma$ is the jammer's duty cycle and equals $1,\\frac{T}{K},\\frac{D}{K}$, or~$\\alpha$\nfor barrage, pilot, data, or sparse jammers, respectively. This allows us to either precisely \ncontrol the jammer energy (for jammers which are assumed to be essentially energy-limited)\nor the transmit intensity (for jammers which may want to pass themselves off as a legitimate UE, for instance).\n\n\n\\subsection{Performance Baselines} \\label{sec:baseline}\nWe set the number of iterations for MAED and \\mbox{SO-MAED} to $t_{\\max}=20$ and \nemphasize again that we use only two different sets of weights for \\mbox{SO-MAED}: one for QPSK transmission and one\nfor 16-QAM transmission. Neither \\mbox{SO-MAED} nor MAED is adapted to the different jammer scenarios.\nWe compare our algorithms to the following baseline methods: \nThe first baseline is the ``LMMSE'' method from \\fref{sec:example}, which does not mitigate the jammer and \nseparately performs least-squares (LS) channel estimation and LMMSE data detection.\nThe second baseline is the ``geniePOS'' method from \\fref{sec:example}, which projects the receive\nsignals onto the orthogonal complement of the  true  jammer subspace \nand then separately performs LS channel estimation and LMMSE data detection in this projected space.\nThe last baseline, \\mbox{``JL-SIMO,''} serves as a lower bound for attainable error-rate performance. \nThis method operates in a jammerless but otherwise equivalent system and implements (with perfect channel knowledge) \nthe single-input multiple-output (SIMO) bound\ncorresponding to the idealized case in which no inter-user interference is present. \n\n\n\\begin{figure*}[tp]\n\\!\\!\\!\\!\\!\n\\subfigure[strong barrage jammer (J1)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/rayleigh_qpsk\/128x32_QPSK_I1_D128_barrage_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:qpsk:strong:static}\n}\\!\\!\n\\subfigure[strong pilot jammer (J2)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/rayleigh_qpsk\/128x32_QPSK_I1_D128_pilot_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:qpsk:strong:pilot}\n}\\!\\!\n\\subfigure[strong data jammer (J3)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/rayleigh_qpsk\/128x32_QPSK_I1_D128_data_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:qpsk:strong:data}\n}\\!\\!\n\\subfigure[strong sparse jammer (J4)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/rayleigh_qpsk\/128x32_QPSK_I1_D128_sparse_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:qpsk:strong:burst}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) for \\emph{QPSK} transmission in the presence of a \\emph{strong} ($\\rE=30$\\,dB) jammer which\ntransmits Gaussian symbols (a) during the entire coherence interval,\n(b) during the pilot phase only, (c) during the data phase only, or (d) in random unit-symbol bursts \nwith a duty cycle of $\\alpha=20\\%$. \n}\n\\label{fig:qpsk_strong_jammers}\n\\end{figure*}\n\\begin{figure*}[tp]\n\\vspace{-1mm}\n\\!\\!\\!\\!\\!\n\\subfigure[strong barrage jammer (J1)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/30dB_16qam\/128x32_16QAM_I1_D128_barrage_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:strong:static}\n}\\!\\!\n\\subfigure[strong pilot jammer (J2)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/30dB_16qam\/128x32_16QAM_I1_D128_pilot_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:strong:pilot}\n}\\!\\!\n\\subfigure[strong data jammer (J3)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/30dB_16qam\/128x32_16QAM_I1_D128_data_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:strong:data}\n}\\!\\!\n\\subfigure[strong sparse jammer (J4)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/30dB_16qam\/128x32_16QAM_I1_D128_sparse_gaussian_rho30_NJE1_T20_1000Trials}\n\\label{fig:strong:burst}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) for \\emph{16-QAM} transmission in the presence of a \\emph{strong} ($\\rE\\!=\\!30$\\,dB) jammer which transmits Gaussian symbols~(a)~during the entire coherence interval,\n(b) during the pilot phase only, (c) during the data phase only, or (d) in random unit-symbol bursts \nwith a duty cycle of $\\alpha=20\\%$. \n}\n\\label{fig:strong_jammers}\n\\end{figure*}\n\\begin{figure*}[h!]\n\\vspace{-1mm}\n\\!\\!\\!\\!\\!\n\\subfigure[weak barrage jammer (J1)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/0dB_16qam\/128x32_16QAM_I1_D128_barrage_constellation_rho0_NJE0_T20_1000Trials}\n\\label{fig:strong:static}\n}\\!\\!\n\\subfigure[weak pilot jammer (J2)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/0dB_16qam\/128x32_16QAM_I1_D128_pilot_constellation_rho0_NJE0_T20_1000Trials}\n\\label{fig:strong:pilot}\n}\\!\\!\n\\subfigure[weak data jammer (J3)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/0dB_16qam\/128x32_16QAM_I1_D128_data_constellation_rho0_NJE0_T20_1000Trials}\n\\label{fig:strong:data}\n}\\!\\!\n\\subfigure[weak sparse jammer (J4)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/0dB_16qam\/128x32_16QAM_I1_D128_sparse_constellation_rho0_NJE0_T20_1000Trials}\n\\label{fig:strong:burst}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) for \\emph{16-QAM} transmission in the presence of a \\emph{weak} ($\\rP=0$\\,dB) jammer\nwhich transmits 16-QAM symbols (a) during the entire coherence interval,\n(b) during the pilot phase only, (c) during the data phase only, or (d) in random unit-symbol bursts \nwith a duty cycle of $\\alpha=20\\%$. \n}\n\\label{fig:weak_jammers}\n\\end{figure*}\n\n\\begin{figure*}[tp]\n\\!\\!\\!\\!\\!\n\\subfigure[barrage jammers (J1)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/many_barrage_jammers\/128x32_16QAM_I1_D128_barrage_gaussian_NJE0_T20_1000Trials}\n\\label{fig:many:static}\n}\\!\\!\n\\subfigure[pilot jammers (J2)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/many_pilot_jammers\/128x32_16QAM_I1_D128_pilot_gaussian_NJE0_T20_1000Trials}\n\\label{fig:many:pilot}\n}\\!\\!\n\\subfigure[data jammers (J3)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/many_data_jammers\/128x32_16QAM_I1_D128_data_gaussian_NJE0_T20_1000Trials}\n\\label{fig:many:data}\n}\\!\\!\n\\subfigure[sparse jammers (J4)]{\n\\includegraphics[height=4cm]{figures\/firm_final_weights\/many_sparse_jammers\/128x32_16QAM_I1_D128_sparse_gaussian_NJE0_T20_1000Trials}\n\\label{fig:many:burst}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) performance curves of SO-MAED in the presence of jammers with different receive powers\ncompared to the average UE,\n$\\rP\\in\\{-20\\,\\text{dB}, -10\\,\\text{dB}, 0\\,\\text{dB}, 10\\,\\text{dB}, 20\\,\\text{dB}, 40\\,\\text{dB}, 80\\,\\text{dB}\\}$. \nThe subfigures correspond to the different jammer types (J1)\\,-\\,(J4) and show one curve \nper~jammer power (plotted with 25\\% opacity to depict the degree of overlap between curves). \nCurves that level off into an error floor are labeled with their jammer power, e.g.,\nin \\fref{fig:many:static}, the barrage jammer with receive power $\\rP=-20$\\,dB has an error floor while all other barrage jammers\nhave virtually identical BER curves.\n}\n\\label{fig:many_jammers}\n\\end{figure*}\n\n\\subsection{Mitigation of Strong Gaussian Jammers}\nWe first investigate the ability of MAED and SO-MAED to mitigate strong jamming attacks. \nFor this, we simulate Gaussian  jammers with \\mbox{$\\rE=30$\\,dB} of all four types introduced in Section~\\ref{sec:setup} \nand evaluate the performance of our algorithms compared to the baselines of Section~\\ref{sec:baseline} for QPSK transmission (\\fref{fig:qpsk_strong_jammers}) \nas well as for 16-QAM transmission (\\fref{fig:strong_jammers}).\nWe note at this point that the performances of geniePOS and JL-SIMO are independent of the considered jammer type: geniePOS uses the genie-provided\njammer channel to null the jammer perfectly, regardless of its transmit sequence, and JL-SIMO operates on a jammer-free system.\nUnsurprisingly, the jammer-oblivious LMMSE baseline performs significantly worse than the jammer-robust geniePOS baseline under all attack scenarios,\nwith the data jamming attack turning out to be the most harmful and the pilot jamming attack the least harmful.\nBoth MAED and SO-MAED succeed in mitigating all four jamming attacks with highest effectiveness, even outperforming the genie-assisted geniePOS method by a\nconsiderable margin.\\footnote{The potential for MAED and SO-MAED to outperform geniePOS is a consequence of the superiority of joint channel estimation and data detection over \nseparating channel estimation from data detection.}\nTheir efficacy is further reflected in the fact that SO-MAED and MAED approach the performance of the jammerless and MU interference-free\nJL-SIMO lower bound to within less than $2$\\,dB and $3$\\,dB at $0.1$\\%~BER, respectively, in all considered~scenarios.\n\nThe behavior is largely similar when 16-QAM instead of QPSK is used as transmit constellation~(\\fref{fig:strong_jammers}). \nHowever, due to the decreased informativeness of the box prior for such higher-order constellations, MAED performs now closer \nto geniePOS, while SO-MAED still performs within $2$\\,dB (at $0.1$\\% BER) of the JL-SIMO lower bound.\nThe increased performance gap between them notwithstanding, both MAED and SO-MAED are able to  effectively mitigate all four attack types.\n\n\n\n\n\\subsection{Mitigation of Weak Constellation Jammers} \\label{sec:results:weak}\nWe now turn to the analysis of more restrained jamming attacks in which the jammer transmits constellation symbols \nwith relative power $\\rP=0$\\,dB during its on-phase (to pass itself off as just another UE, for instance \\cite{vinogradova16a}).\nSimulation results for 16-QAM transmission under all four types of jamming attacks are shown in~\\fref{fig:weak_jammers}. \nBecause of the weaker jamming attacks, the jammer-oblivious LMMSE baseline now performs closer to the jammer-resistant geniePOS baseline \nthan it does in \\fref{fig:strong_jammers}.\nMAED again mitigates all attack types rather successfully, outperforming geniePOS in the low-SNR regime but slightly leveling off at high SNR.\nInterestingly, MAED shows worse performance under these weak jamming attacks than under the strong jamming attacks of \\fref{fig:strong_jammers}.\nThe reason is the following: MAED searches for the jamming subspace by looking for the dominant dimension \nof the iterative residual error $\\tilde\\bE^{(t)}$, see~\\fref{eq:rayleigh}. If the received jamming energy is small compared to the \nreceived signal energy, then it becomes hard to distinguish the residual errors caused by the jamming signal from those caused by \nerrors in estimating the channel and data matrices $\\tilde\\bH_\\bP^{(t)}$ and $\\tilde\\bS_D^{(t)}$.\nNote in contrast that, due to its superior signal prior, the equivalent performance loss of SO-MAED is only so small as to be \nvirtually unnoticeable. Thus, SO-MAED outperforms MAED by a large margin and still approaches the JL-SIMO bound by less than 2dB at a BER of 0.1\\%.\\footnote{We \nmention that MAED does not suffer such a performance loss under weak jamming attacks when the transmit constellation is QPSK, \nsince in that case the box signal prior of MAED is sufficiently accurate, see also \\cite{marti2022smart}.}\n\n\n\\subsection{How Versatile is SO-MAED Really?}\nIn the remainder of our evaluation, we focus mostly on \\mbox{SO-MAED}, since it is clearly the better of the two proposed algorithms. \nTo show that our approach indeed succeeds in mitigating arbitrary jamming attacks without need for fine-tuning of the algorithm or its parameters, \n\\fref{fig:many_jammers} depicts performance results for a series of jamming attacks spanning a dynamic range from $\\rP=-20$\\,dB to \n$\\rP=80$\\,dB. Specifically, \\fref{fig:many_jammers} shows results for all four jammer types, where every \nsubfigure plots BER curves for jamming attacks with \n$\\rP\\in\\{-20\\,\\text{dB}, -10\\,\\text{dB}, 0\\,\\text{dB}, 10\\,\\text{dB}, 20\\,\\text{dB}, 40\\,\\text{dB}, 80\\,\\text{dB}\\}$.\nThe purpose of these plots is to illustrate that, apart from jamming attacks where the jammer is significantly weaker than \nthe average UE\\footnote{A jammer that is much weaker than the average UE resembles\na non-transmitting, and thus eclipsed, jammer; see Sections \\ref{sec:theory} and \\ref{sec:results:eclipsed}.}, \nthe curves are virtually indistinguishable, meaning that the performance of SO-MAED is virtually independent of the specific\ntype of jamming attack that it is facing.\n\n\n\\subsection{Eclipsed Jammers} \\label{sec:results:eclipsed}\n\n\n\\begin{figure}[tp]\n\\centering\n\\!\\!\\!\\!\\!\n\\subfigure[no jammer]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/eclipsed\/128x32_16QAM_I1_D128_all-zero_eclipsed_gaussian_rho0_NJE0_T20_1000Trials}\n\\label{fig:eclipsed:no_jam}\n}\\!\\!\\!\n\\subfigure[weak UE-impersonating jammer]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/eclipsed\/128x32_16QAM_I1_D128_pilot_eclipsed_gaussian_rho0_NJE0_T20_1000Trials}\n\\label{fig:eclipsed:weak_impers}\n}\\!\\! \\\\\n\\!\\!\\!\\!\\!\n\\subfigure[strong UE-impersonating jammer]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/eclipsed\/128x32_16QAM_I1_D128_pilot_eclipsed_gaussian_rho30_NJE0_T20_1000Trials}\n\\label{fig:eclipsed:strong_impers}\n}\\!\\!\\!\n\\subfigure[data-dependent jammer]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/eclipsed\/128x32_16QAM_I1_D128_row-difference_eclipsed_gaussian_rho30_NJE0_T20_1000Trials}\n\\label{fig:eclipsed:row_eclipsed}\n}\\!\\!\n\\caption{Uncoded bit error-rate of SO-MAED for different types of eclipsed jammers: (a) no jammer, \n(b) $\\rP=0$\\,dB jammer impersonating the $j$th UE by transmitting its pilot sequence \n($\\text{UE}_j$ denotes the BER of the impersonated UE, and $\\overline{\\text{UE}}_j$ the BER among all other UEs),\n(c) $\\rP=30$\\,dB jammer impersonating the $j$th UE by transmitting its pilot sequence, \nand (d) jammer causes eclipsing by transmitting a jamming sequence that depends on the UE transmit matrix~$\\bS$.\nDashed lines represent the BER of the impersonated UE, transparent lines represent the BER among \nthe UEs that are not impersonated by the jammer.\n}\n\\label{fig:eclipsed}\n\\vspace{-2mm}\n\\end{figure}\n\nUp to this point, the jamming signal $\\bmw$ has always been generated independently from the UE transmit matrix $\\bS$.\nThe strong performance results of both MAED and SO-MAED have supported the claim in Remark~\\ref{rem:rare} that,\nin this case, eclipsing is the (rare) exception, not the norm.  \nWe now turn to an empirical analysis of how SO-MAED behaves when eclipsing \\emph{does}\noccur (\\fref{fig:eclipsed}). To this end, we consider scenarios in which the jammer is eclipsed \nbecause there is no jamming activity (\\fref{fig:eclipsed:no_jam}), \nbecause the jammer transmits a UE's pilot sequence (\\fref{fig:eclipsed:weak_impers}, \\fref{fig:eclipsed:strong_impers}),\nor because the jamming sequence~$\\bmw$ depends on the transmit matrix $\\bS$ (which in reality would be unknown to the jammer) \nin a way that causes eclipsing (\\fref{fig:eclipsed:row_eclipsed}). \n\n\nIn the case of no jammer (\\fref{fig:eclipsed:no_jam}), or no jamming activity within a coherence interval,\nwe see that SO-MAED still reliably detects the transmit data. However, it now suffers from an error floor\n(albeit significantly below $0.1$\\% BER). The reason for this error floor is that, in the absence of \njamming energy to guide the choice of the nulled direction $\\tilde\\bmp$, there is the temptation to instead ``cover up''\ndetection errors (similar to the phenomenon discussed in \\fref{sec:results:weak}).\nHowever, the low level of the error floor shows that this potential pitfall does not cause a systematic breakdown \nof SO-MAED. \nWe emphasize also that SO-MAED does not simply null the strongest UE. Such (degenerate) behavior would only occur\nif one UE were \\emph{far} stronger than the others. With any reasonable power control scheme, UE nulling is not an \nissue.\\footnote{This is exemplified by our experiments with i.i.d. Rayleigh fading chanels, which also exhibit minor imbalances in receive power between different UEs. Cf. also our results in \\fref{sec:beyond}, \nwhere we use $\\pm1.5$\\,dB power control.}\nIn the case of a jammer that impersonates the $j$th UE by transmitting its pilot sequence in the training phase \nand constellation symbols in the data phase, with the same power ($\\rP=0$\\,dB) as the average~UE, \nSO-MAED indeed suffers a performance breakdown (\\fref{fig:eclipsed:weak_impers}). However, \ncloser analysis shows that this error floor is caused solely by errors in detecting the symbols of the \nimpersonated UE. \nThis is not surprising: The jammer is statistically indistinguishable from the $j$th UE,\nso that is impossible to reliably separate the UE transmit symbols from the fake jammer transmit symbols. \nIn this regard, we refer again to the information-theoretic discussion of \\mbox{\\cite[Sec. V]{lapidoth1998reliable}}.\nSuch impersonation attacks could be forestalled by using encrypted pilots~\\cite{basciftci2015securing}.\nIf the jammer transmits the $j$th UE's pilot sequence and constellation symbols, but with much more\npower ($\\rP=30$\\,dB), then the iterative detection procedure of \\mbox{SO-MAED} will separate the jammer subspace from the \n$j$th UE's subspace (\\fref{fig:eclipsed:strong_impers}), since, being so much stronger than any UE, \nthe jammer subspace will dominate the residual matrix $\\tilde\\bE^{(t)}$ in~\\fref{eq:rayleigh}.\nFinally, \\fref{fig:eclipsed:row_eclipsed} shows results for a case where the jammer knows $\\bS$\nand choses~$\\tilde\\bS_D$ to differ from $\\bS_D$ in a single row (with valid constellation symbols in the differing row), \nso that $\\textit{rank}(\\bS_D-\\tilde\\bS_D)=1$. It first draws $\\bmw_T\\sim\\setC\\setN(\\mathbf{0},\\bI_D)$ and then\nsets $\\tp{\\bsj_D} = \\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D$ to cause eclipsing. \nThe jammer strength is $\\rP=30$\\,dB. \nThe results show an error floor at roughly 0.2\\% BER caused by the presence of an alternative spurious solution.\nHowever, the results in Figs.~\\ref{fig:qpsk_strong_jammers}\\,--\\,\\ref{fig:many_jammers} show \nthat, when the jammer has to select~$\\bmw$ without knowing $\\bS_D$, such accidental eclipsing is \nextremely rare.\n\n\n\n\\subsection{Beyond i.i.d. Rayleigh Fading} \\label{sec:beyond}\nSo far, our experiments were based on i.i.d. Rayleigh fading channels, but our method does not depend\nin any way on this particular channel model.  \nTo demonstrate that MAED and SO-MAED are also applicable in scenarios that deviate strongly from the i.i.d. Rayleigh model,\nwe now evaluate our algorithms on realistic mmWave channels generated \nwith the commercial  Wireless InSite ray-tracer \\cite{Remcom}. The simulated scenario is depicted in \\fref{fig:remcom_scenario}.\nWe simulate a mmWave massive MU-MIMO system with a carrier frequency of $60$\\,GHz and a bandwidth \nof $100$\\,MHz. The BS is placed at a height of $10$\\,m and consists of a horizontal uniform linear array with $B=128$ omnidirectional antennas spaced at half \na wavelength. The omnidirectional single-antenna UEs and the jammer are located at a height of $1.65$\\,m and placed in a $150^\\circ$ sector \nspanning $180$\\,m$\\times$$90$\\,m in front of the BS; see \\fref{fig:remcom_scenario}. The~UEs~and the jammer are drawn at random from a grid with $5$\\,m pitch while ensuring\nthat the minimum angular separation between any two UEs, as well as between the jammer and any UE, is $2.5^\\circ$.\nWe assume $\\pm1.5$\\,dB power control, so that the ratio between the maximum and minimum per-UE receive power is~2.\n\\begin{figure}[tp]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{figures\/sector_marked_cropped}\n\\caption{Simulated scenario. The location of the BS his highlighted by the white circle while the red squares depict all possible UE locations.}\n\\label{fig:remcom_scenario}\n\\vspace{-2mm}\n\\end{figure}\n\\begin{figure}[tp]\n\\centering\n\\!\\!\\!\\!\\!\n\\subfigure[barrage jammer (J1)]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/remcom\/128x32_QPSK_I1_D128_barrage_gaussian_rho30_NJE1_T30_1000Trials}\n\\label{fig:remcom:barrage}\n}\\!\\!\\!\n\\subfigure[pilot jammer (J2)]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/remcom\/128x32_QPSK_I1_D128_pilot_gaussian_rho30_NJE1_T30_1000Trials}\n\\label{fig:remcom:pilot}\n}\\!\\! \\\\\n\\!\\!\\!\\!\\!\n\\subfigure[data jammer (J3)]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/remcom\/128x32_QPSK_I1_D128_data_gaussian_rho30_NJE1_T30_1000Trials}\n\\label{fig:remcom:data}\n}\\!\\!\\!\n\\subfigure[sparse jammer (J4)]{\n\\includegraphics[height=3.95cm]{figures\/firm_final_weights\/remcom\/128x32_QPSK_I1_D128_sparse_gaussian_rho30_NJE1_T30_1000Trials}\n\\label{fig:remcom:sparse}\n}\\!\\!\n\\caption{Uncoded bit error-rate (BER) for \\emph{QPSK} transmission over realistic mmWave channels in the presence of a \\emph{strong} ($\\rE=30$\\,dB) jammer.}\n\\label{fig:remcom_results}\n\\vspace{-2mm}\n\\end{figure}\nThe high correlation exhibited by these mmWave channels slows convergence of MAED and SO-MAED, so we increase their number of \niterations to $t_{\\max}=30$. We also retrain the parameters from SO-MAED on mmWave channels (while making a clear split between the training\nset and the~evaluation~set).\n\nThe results for QPSK transmission in the presence of $\\rE=30$\\,dB are shown in \\fref{fig:remcom_results}. \nThe performance hierarchy is identical as in the equivalent Rayleigh-fading setup of \\fref{fig:qpsk_strong_jammers}:\ngeniePOS is clearly outperformed by MAED, which is in turn outperformed by SO-MAED. \nHowever, the more challenging nature of mmWave channels amplifies performance differences: \nDue to its artificial immunity from the high inter-user interference of mmWave channels, JL-SIMO is now in a class of its own. \nHowever, MAED and SO-MAED gain almost $4$\\,dB and $6$\\,dB in SNR on geniePOS at $0.1\\%$ BER, respectively, \nregardless of the jammer type. This shows that MAED and SO-MAED are also well suited \nfor scenarios that deviate significantly from the i.i.d. Rayleigh model.\n\\vspace{-1mm}\n\n\n\n\\section{Conclusions}\n\nWe have proposed an approach for the mitigation of smart jamming attacks on the massive\nMU-MIMO uplink and supported its basic soundness with theoretical results. \nIn contrast to existing mitigation methods, our approach does not rely on jamming activity \nduring any particular time instant.\nInstead, our approach utilizes a newly proposed problem formulation which exploits the fact \nthat the jammer's subspace remains constant within a coherence interval.\nWe have developed two efficient iterative algorithms, MAED and SO-MAED, which approximately solve the \nproposed optimization problem. \nOur simulation results have shown that MAED and SO-MAED are able to effectively mitigate a wide range of jamming attacks.\nIn particular, they succeed in mitigating attack types like data jamming and sparse jamming,\nfor which---to the best of our knowledge---no mitigation methods have existed so far.\n\nFuture work could focus on jammer mitigation with iterative detection and decoding, \nfor which the soft-output estimates of SO-MAED are well suited. Other directions for \nfuture work are the extension to mutli-antenna jammers as well as to jammer mitigation\nin wideband systems.\n\n\\appendices\n\n\\vspace{-1mm}\n\n\\section{Proof of \\fref{thm:maed}} \\label{app:proof1}\nObviously, if $\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\}=\\{\\bmp, \\bP\\bH, \\bS_D\\}$, then\n\\begin{align}\n\t\\hat\\bP\\bY - \\hat\\bH_\\bP \\hat\\bS \n\t&= \\bP\\bY - \\bP\\bH \\bS \\\\\n\t&= \\bP(\\bH\\bS + \\Hj\\tp{\\bsj}) - \\bP\\bH \\bS \\\\\n\t&= \\bP \\Hj\\tp{\\bsj} = \\mathbf{0}, \n\\end{align}\nand so the objective value of \\eqref{eq:obj1} is zero. Since the objective function in \\eqref{eq:obj1} is nonnegative, \nit follows that $\\{\\hat\\bmp, \\hat\\bH_\\bP, \\hat\\bS_D\\}=\\{\\bmp, \\bP\\bH, \\bS_D\\}$ is a solution to \\eqref{eq:obj1}. \nIt remains to prove uniqueness. For this, we rewrite the objective in \\eqref{eq:obj1} as\n\\begin{align}\n\t\\big\\|\\tilde\\bP\\bY \\!- \\tilde\\bH_\\bP \\tilde\\bS \\big\\|^2_F \n\t= \\big\\|\\tilde\\bP\\bY_T \\!- \\tilde\\bH_\\bP \\bS_T \\big\\|^2_F + \\big\\|\\tilde\\bP\\bY_D \\!- \\tilde\\bH_\\bP \\tilde\\bS_D \\big\\|^2_F\n\t\\label{eq:decomposition}\n\\end{align}\nThe objective can only be zero if both terms on the right-hand-side (RHS) of \\eqref{eq:decomposition} are zero.\nThe first term is zero iff\n\\begin{align}\n\t& \\tilde\\bP\\bY_T - \\tilde\\bH_\\bP \\bS_T = \\mathbf{0},\n\\end{align}\nwhich implies\n\\begin{align}\n\t\\tilde\\bH_\\bP = \\tilde\\bP\\bY_T \\pinv{\\bS_T}, \\label{eq:optimal_h}\n\\end{align}\nsince $\\bS_T$ has full row rank.\nPlugging this back into the second term on the RHS of \\eqref{eq:decomposition} gives\n\\begin{align}\n\t& \\tilde\\bP\\bY_D - \\tilde\\bH_\\bP \\tilde\\bS_D \\\\\n\t&= \\tilde\\bP(\\bY_D - \\bY_T \\pinv{\\bS_T} \\tilde\\bS_D) \\\\\n\t&= \\tilde\\bP \\left(\\bH\\bS_D + \\Hj\\tp{\\bsj_D} - (\\bH\\bS_T + \\Hj\\tp{\\bsj_T}) \\pinv{\\bS_T} \\tilde\\bS_D \\right) \\\\\n\t&= \\tilde\\bP \\left(\\bH [\\bS_D- \\tilde\\bS_D] + \\Hj[\\tp{\\bsj_D}  - \\tp{\\bsj_T}\\pinv{\\bS_T} \\tilde\\bS_D] \\right). \\label{eq:term_mat}\n\\end{align}\nThe second term on the RHS of \\eqref{eq:decomposition} (and, hence, the objective) is zero if and only if\nthe matrix in \\eqref{eq:term_mat} is the zero matrix.\nThe projector $\\tilde{\\bP}$ can null a matrix of (at most) rank one. \nIt follows that the objective function in \\eqref{eq:decomposition} can be zero only if \n\\begin{align}\n\t\\bH [\\bS_D- \\tilde\\bS_D] + \\Hj[\\tp{\\bsj_D}  - \\tp{\\bsj_T}\\pinv{\\bS_T} \\tilde\\bS_D] \\label{eq:eclipsing_equation}\n\\end{align}\nis a matrix of (at most) rank one. Since $\\bH$ has full column rank and $\\Hj$ is not included in the columnspace of $\\bH$, \nthis requires that either $\\text{rank}(\\bS_D- \\tilde\\bS_D)=1$ and $\\tp{\\bsj_D}  - \\tp{\\bsj_T}\\pinv{\\bS_T} \\tilde\\bS_D=\\mathbf{0}$, \nor that $\\bS_D - \\tilde\\bS_D=\\mathbf{0}$. But the first case is precluded since it implies that the jammer is eclipsed,\nin contrast to our assumption. \n\nIn the second case, we have $\\tilde\\bS_D = \\bS_D$, so the estimated data matrix coincides with the true data matrix.\nIn that case,~\\eqref{eq:term_mat} is\n\\begin{align}\n\t\\tilde\\bP \\Hj[\\tp{\\bsj_D}  - \\tp{\\bsj_T}\\pinv{\\bS_T} \\tilde\\bS_D], \n\\end{align}\nwhich (again by the assumption that the jammer is not ecplised) is zero if and only if $\\tilde\\bmp$ is collinear with $\\bmj$, \nmeaning that $\\tilde\\bmp = \\alpha \\bmp, |\\alpha|=1$. This means that also the estimated jammer subspace coincides with the\ntrue jammer subspace.\nFinally, plugging this value of $\\tilde\\bmp$ back into \\eqref{eq:optimal_h} yields\n\\begin{align}\n\t\\tilde\\bH_\\bP &= \\tilde\\bP\\bY_T \\pinv{\\bS_T}  \\\\\n\t&= \\tilde\\bP (\\bH\\bS_T + \\Hj\\tp{\\bsj_T}) \\pinv{\\bS_T} \\\\\n\t&= \\tilde\\bP\\bH\\bS_T \\pinv{\\bS_T} = \\bH_\\bP, \n\\end{align}\nshowing that also the estimated channel matrix coincides with the projection of the true channel matrix. \nWe have thereby shown that $\\big\\|\\tilde\\bP\\bY - \\tilde\\bH_\\bP \\tilde\\bS \\big\\|^2_F$ is zero if and only if\n$\\tilde\\bS_D=\\bS_D$, $\\tilde\\bmp = \\alpha\\bmp, |\\alpha|=1$, and $\\tilde\\bH_\\bP = \\bH_\\bP$.\n\\hfill $\\blacksquare$ \n\n\n\n\n\n\\section{Proof of \\fref{thm:maed2}} \\label{app:proof2}\n$\\bS_T$ is unitary, so the jammer eclipses if\n\\begin{align}\n\t\\tp{\\bsj_D} &= \\tp{\\bsj_T}\\pinv{\\bS_T}\\tilde{\\bS}_D \\\\\n\t&= \\tp{\\bsj_T}\\bS_T^H\\tilde{\\bS}_D = (\\underbrace{\\bS_T\\bsj_T^\\ast}_{\\triangleq \\bmx})^H \\tilde{\\bS}_D\n\\end{align}\nfor some $\\tilde{\\bS}_D\\in\\setS^{U\\times D}$ such that $\\text{rank}(\\bS_D - \\tilde{\\bS}_D)\\leq 1$.\nSince~$\\bS_T$ is Haar distributed\\footnote{The uniform distribution over unitary matrices is called Haar distribution.}, \nthe vector $\\bmx$ is distributed uniformly over the\ncomplex $U$-dimensional sphere of radius $\\|\\bmw\\|$ \\cite[p. 16]{meckes2019random}.\nIn particular, $\\bmx$ is independent of $\\bmw_D$.\nWe can now show that \n\\begin{align}\n\t\\tp{\\bmw_D} \\neq \\bmx^H\\tilde{\\bS}_D \n\\end{align}\nholds with probability one by showing that already for the first entry, we have\n\\begin{align}\n\tw_{D,1} \\neq \\bmx^H \\tilde\\bms_{D,1} \\label{eq:first_entry_criterion}\n\\end{align}\nwith probability one, where $ \\tilde\\bms_{D,1}$ is the leftmost column of~$\\tilde{\\bS}_D$:\nWe can interpret $\\bmx^H \\tilde\\bms_{D,1}=\\|\\tilde\\bms_{D,1}\\| \\langle \\tilde\\bms_{D,1}\/\\|\\tilde\\bms_{D,1}\\|, \\bmx  \\rangle$ \nas the top left entry of a matrix product $\\|\\tilde\\bms_{D,1}\\|\\bX\\bZ$, \nwhere $\\bX$ is a Haar-distributed matrix whose first row is $\\bmx^H$, and where $\\bZ$ is an unitary matrix whose first \ncolumn is $\\tilde\\bms_{D,1}\/\\|\\tilde\\bms_{D,1}\\|$. \nIt then follows from \\cite[p. 7]{meckes2019random} that $\\bX\\bZ$ is Haar distributed.\nThus, the distribution of its top left entry and hence also of $\\bmx^H\\tilde\\bms_{D,1}$ has no mass points. \nSince $w_{D,1}$ is independent of $\\bmx$, \\eqref{eq:first_entry_criterion} must therefore hold with probability one. \n\\hfill $\\blacksquare$\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\t\\label{S1}\n\\paragraph{The topos approach.} The topos approach to quantum theory was initiated by Isham in \\cite{Isham97} and Butterfield and Isham in \\cite{IB98,IB99,IB00,IB02}. It was developed and broadened into an approach to the formulation of physical theories in general by Isham and the author in \\cite{DI(1),DI(2),DI(3),DI(4)}. The long article \\cite{DI08} gives a more-or-less exhaustive\\footnote{and probably exhausting...} and coherent overview of the approach. More recent developments are the description of arbitrary states by probability measures \\cite{Doe08} and further developments \\cite{Doe09} concerning the new form of quantum logic that constitutes a central part of the topos approach. For background, motivation and the main ideas see also Isham's article in this volume \\cite{Ish10}.\n\nMost of the work so far has been done on standard non-relativistic quantum theory. A quantum system is described by its \\emph{algebra of physical quantities}. Often, this can be assumed to be $\\BH$, the algebra of all bounded operators on the (separable) Hilbert space $\\Hi$ of the system. More generally, one can use a suitable operator algebra. For conceptual and pragmatical reasons, we assume that the algebra of physical quantities is a \\emph{von Neumann algebra $\\N$} (see e.g. \\cite{KR83a,Tak79}). For our purposes, this poses no additional technical difficulty. Physically, it allows the description of quantum systems with symmetry and\/or superselection rules. The reader unfamiliar with von Neumann algebras can always assume that $\\N=\\BH$.\n\nQuantum theory usually is identified with the Hilbert space formalism together with some interpretation. This works fine in a vast number of applications. Moreover, the Hilbert space formalism is very rigid. One cannot just change some part of the structure since this typically brings down the whole edifice. Yet, there are serious conceptual problems with the usual instrumentalist interpretations of quantum theory which become even more severe when one tries to apply quantum theory to gravity and cosmology. For a discussion of some of these conceptual problems, see Isham's article in this volume.\n\nThe topos approach provides not merely another interpretation of the Hilbert space formalism, but a mathematical reformulation of quantum theory, based upon structural and conceptual considerations. The resulting formalism is \\emph{not} a Hilbert space formalism. The fact that such a reformulation is possible at all is somewhat surprising. (Needless to say, many open questions remain.)\n\n\\paragraph{Daseinisation.} In this article, we mainly focus on how the new topos formalism relates to the standard Hilbert space formalism. The main ingredient is the process that we coined \\emph{daseinisation}. It will be shown how daseinisation relates familiar structures in quantum theory to structures within the topos associated with a quantum system. Hence, daseinisation gives the `translation' from the ordinary Hilbert space formalism to the topos formalism.\n\nIn our presentation, we will keep the use of topos theory to an absolute minimum and do not assume any familiarity with category theory beyond the very basics. Some notions and results from functional analysis, e.g. the spectral theorem, will be used. As an illustration, we will show how all constructions look like concretely for the algebra $\\N=\\mc B (\\bbC^3)$ and the spin-$z$ operator, $\\hat S_z\\in\\mc B (\\bbC^3)$, of a spin-$1$ particle. \n\nIn fact, there are two processes called daseinisation. The first is \\emph{daseinisation of projections}, which maps projection operators to certain subobjects of the state object. (The state object and its subobjects will be defined below.) This provides the bridge from ordinary Birkhoff-von Neumann quantum logic to a new form of quantum logic that is based upon the internal logic of a topos associated with the quantum system. The second is \\emph{daseinisation of self-adjoint operators}, which maps each self-adjoint operator to an arrow in the topos. Mathematically, the two forms of daseinisation are related, but conceptually they are quite different.\n\nThe topos approach emphasises the r\\^ole of \\emph{classical perspectives} onto a quantum system. A classical perspective or \\emph{context} is nothing but a set of commuting physical quantities, or more precisely the abelian von Neumann algebra generated by such a set. One of the main ideas is that \\emph{all} classical perspectives should be taken into account simultaneously. But given e.g. a projection operator, which represents a proposition about the value of a physical quantity in standard quantum theory, one is immediately faced with the problem that the projection operator is contained in some contexts, but not in all. The idea is to approximate the projection in all those contexts that do not contain it. Likewise, self-adjoint operators, which represent physical quantities, must be approximated suitably to all contexts. Daseinisation is nothing but a process of systematic approximation to all classical contexts.\n\nThis article is organised as follows: in section \\ref{S2}, we discuss propositions and their representation in classical physics and standard quantum theory. Section \\ref{S3} presents some basic structures in the topos approach to quantum theory. In section \\ref{S4}, the representation of propositions in the topos approach is discussed -- this is via daseinisation of projections. Section \\ref{S5}, which is the longest and most technical, shows how physical quantities are represented in the topos approach; here we present daseinisation of self-adjoint operators is discussed. Section \\ref{S6} concludes.\n\n\\section{\\mbox{Propositions and their representation in}\\\\ classical physics and standard quantum\\\\ theory}\t\\label{S2}\n\\paragraph{Propositions.} Let $S$ be some physical system, and let $A$ denote a physical quantity pertaining to the system $S$ (e.g. position, momentum, energy, angular momentum, spin...). We are concerned with propositions of the form ``the physical quantity $A$ has a value in the set $\\De$ of real numbers'', for which we use the shorthand notation ``$\\Ain{\\De}$''. In all applications, $\\De$ will be a Borel subset of the real numbers $\\bbR$.\n\nArguably, physics is fundamentally about what we can say about the truth values of propositions of the form ``$\\Ain{\\De}$'' (where $A$ varies over the physical quantities of the system and $\\De$ varies over the Borel subsets of $\\bbR$) when the system is in a given state. Of course, the state may change in time and hence the truth values also will change in time.\\footnote{One could also think of propositions about \\emph{histories} of a physical system $S$ such that propositions refer to multiple times, and the assignment of truth values to such propositions (in a given, evolving state). For some very interesting recent results on a topos formulation of histories see Flori's recent article \\cite{Flo08}.}\n\nSpeaking about propositions like ``$\\Ain{\\De}$'' requires a certain amount of conceptualising. We accept that it is sensible to talk about physical systems as suitably separated entities, that each such physical system is characterised by its physical quantities, and that the range of values of a physical quantity $A$ is a subset of the real numbers. If we want to assign truth values to propositions, then we need the concept of a state of a physical system $S$, and the truth values of propositions will depend on the state. If we regard these concepts as natural and basic (and potentially as prerequisites for doing physics at all), then a proposition ``$\\Ain{\\De}$'' \\emph{refers to the world} in the most direct conceivable sense.\\footnote{Of course, as explained by Isham in his article \\cite{Ish10}, it is one of the central motivations for the whole topos programme to find a mathematical framework for physical theories that is not depending fundamentally on the real numbers. In particular, the premise that physical quantities have real values is doubtful in this light. How this seeming dilemma is solved in the topos approach to quantum theory will become clear later in section \\ref{Sec_RepOfPhysQuants}.} We will deviate from common practice, though, by insisting that also in quantum theory, a proposition is about `how things are', and is not to be understood as a counterfactual statement, i.e., it is not merely about what we would obtain as a measurement result if we were to perform a measurement.\n\n\\paragraph{Classical physics, state spaces and realism.} In classical physics, a state of the system $S$ is represented by an element of a set, namely by a point of the state space $\\mathcal{S}$ of the system.\\footnote{We avoid the usual synonym `phase space', which seems to be a historical misnomer. In any case, there are no phases in a phase space.} Each physical quantity $A$ is represented by a real-valued function $f_A:\\mathcal{S}\\rightarrow\\bbR$ on the state space. A proposition ``$\\Ain{\\De}$'' is represented by a certain subset of the state space, namely the set $f_A^{-1}(\\De)$. We assume that $f_A$ is (at least) measurable. Since $\\De$ is a Borel set, the subset $f_A^{-1}(\\De)$ of state space representing ``$\\Ain{\\De}$'' is a Borel set, too.\n\nIn any state, each physical quantity $A$ has a value, which is simply given by evaluation of the function $f_A$ representing $A$ at the point $s\\in\\mc{S}$ of state space representing the state, i.e., the value is $f_A(s)$. Every proposition ``$\\Ain{\\De}$'' has a Boolean truth value in each given state $s\\in\\mc{S}$:\n\\begin{equation}\n\t\t\tv(\\SAin{\\De};s)=\\left\\{\n\t\t\t\\begin{tabular}\n\t\t\t\t\t\t[c]{ll\n\t\t\t\t\t\t$true$ & if $s\\in f_ A^{-1}(\\De)$\\\\\n\t\t\t\t\t\t$false$ & if $s\\notin f_A^{-1}(\\De)$.\n\t\t\t\\end{tabular}\n\t\t\t\\ \\right.\n\\end{equation}\nLet ``$\\Ain{\\De}$'' and ``$B\\varepsilon\\Gamma$'' be two different propositions, represented by the Borel subsets $f_A^{-1}(\\De)$ and $f_B^{-1}(\\Gamma)$ of $\\mc{S}$, respectively. The union $f_A^{-1}(\\De)\\cup f_B^{-1}(\\Gamma)$ of the two Borel subsets is another Borel subset, and this subset represents the proposition ``$\\Ain{\\De}$ or $B\\varepsilon\\Gamma$'' (disjunction). Similarly, the intersection $f_A^{-1}(\\De)\\cap f_B^{-1}(\\Gamma)$ is a Borel subset of $\\mc{S}$, and this subset represents the proposition ``$\\Ain{\\De}$ and $B\\varepsilon\\Gamma$'' (conjunction). Both conjunction and disjunction can be extended to arbitrary countable families of Borel sets representing propositions. The set-theoretic operations of taking unions and intersections distribute over each other. Moreover, the negation of a proposition ``$\\Ain{\\De}$'' is represented by the complement $\\mc{S}\\backslash f_A^{-1}(\\De)$ of the Borel set $f_A^{-1}(\\De)$. Clearly, the empty subset $\\emptyset$ of $\\mc{S}$ represents the trivially false proposition, while the maximal Borel subset, $\\mc{S}$ itself, represents the trivially true proposition. The set $\\mc{B(S)}$ of Borel subsets of the state space $\\mc{S}$ of a classical system thus is a Boolean $\\sigma$-algebra, i.e., a $\\sigma$-complete distributive lattice with complement. Stone's theorem shows that every Boolean algebra is isomorphic to the algebra of subsets of a suitable space, so Boolean logic is closely tied to the use of sets.\n\nThe fact that in a given state $s$ each physical quantity has a value and each proposition has a truth value makes classical physics a \\emph{realist} theory.\\footnote{Here, we could enter into an interesting, but potentially never-ending discussion on physical reality, ontology and epistemology etc. We avoid that and posit that for the purpose of this paper, a realist theory is one that, in any given state, allows to assign truth values to all propositions of the form ``$\\Ain{\\De}$''. Moreover, we require that there is a suitable logical structure, in particular a deductive system, in which we can argue about (representatives of) propositions.}\n\n\\paragraph{Representation of propositions in standard quantum theory.} In contrast to that, in quantum physics it is not possible to assign real values to all physical quantities at once. This is the content of the Kochen-Specker theorem. In the standard Hilbert space formulation of quantum theory, each physical quantity $A$ is represented by a self-adjoint operator $\\A$ on some Hilbert space $\\Hi$. The range of possible real values that $A$ can take is given by the spectrum $\\spec{\\A}$ of the operator $\\A$. Of course, there is a notion of states in quantum theory: in the simplest version, they are given by unit vectors in the Hilbert space $\\Hi$. There is a particular mapping taking self-adjoint operators and unit vectors to real numbers, namely the evaluation\n\\begin{equation}\n\t\t\t(\\A,\\ket\\psi)\\longmapsto\\bra\\psi\\A\\ket\\psi.\n\\end{equation}\nIn general (unless $\\ket\\psi$ is an eigenstate of $\\A$), this real value is \\emph{not} the value of the physical quantity $A$ in the state described by $\\psi$. It rather is the expectation value, which is a statistical and instrumentalist notion. The physical interpretation of the mathematical formalism of quantum theory fundamentally depends on measurements and observers.\n\nAccording to the spectral theorem, propositions like ``$\\Ain{\\De}$'' are represented by projection operators $\\P=\\hat E[\\Ain{\\De}]$. Each projection $\\P$ corresponds to a unique closed subspace $U_\\P$ of the Hilbert space $\\Hi$ of the system and vice versa. The intersection of two closed subspaces is a closed subspace, which can be taken as the definition of a conjunction. The closure of the subspace generated by two closed subspaces is a candidate for the disjunction, and the function sending a projection $\\P$ to $\\hat 1-\\P$ is an orthocomplement. Birkhoff and von Neumann suggested in their seminal paper \\cite{BvN36} to interpret these mathematical operations as providing a logic for quantum systems.\n\nAt first sight, this looks similar to the classical case: the Hilbert space $\\Hi$ now takes the r\\^ole of a state space, while its closed subspaces represent propositions. But there is an immediate, severe problem: if the Hilbert space $\\Hi$ is at least two-dimensional, then the lattice $\\mc{L(H)}$ of closed subspaces is \\emph{non-distributive} (and so is the isomorphic lattice of projections). This makes it very hard to find a proper semantics of quantum logic.\\footnote{We cannot discuss the merits and shortcomings of quantum logic here. Suffice it to say that there are more conceptual and interpretational problems, and a large number of developments and abstractions from standard quantum logic, i.e., the lattice of closed subspaces of Hilbert space. An excellent review is \\cite{DCG02}.}\n\n\\section{Basic structures in the topos approach to quantum theory}\t\\label{S3}\n\\subsection{Contexts}\nAs argued by Isham in this volume, the topos formulation of quantum theory is based upon the idea that one takes the collection of \\emph{all} classical perspectives on a quantum system. No single classical perspective can deliver a complete picture of the quantum system, but the collection of all of them may well do. As we will see, it is also of great importance how the classical perspectives relate to each other.\n\nThese ideas are formalised in the following way: we consider a quantum system as being described by a \\emph{von Neumann algebra} $\\N$ (see e.g. \\cite{KR83a}). Such an algebra is always given as a subalgebra of $\\BH$, the algebra of all bounded operators on a suitable Hilbert space $\\Hi$. We can always assume that the identity in $\\N$ is the identity operator $\\hat 1$ on $\\Hi$. Since von Neumann algebras are much more general than just the algebras of the form $\\BH$, and since all our constructions work for arbitrary von Neumann algebras, we present the results in this general form. For a very good introduction to the use of operator algebras in quantum theory see \\cite{Emch84}. Von Neumann algebras can be used to describe quantum systems with symmetry and\/or superselection rules. Throughout, we will use the very simple example of $\\N=\\mc{B}(\\bbC^3)$ to illustrate the constructions.\n\nWe assume that the self-adjoint operators in the von Neumann algebra $\\N$ associated with our quantum system represent the \\emph{physical quantities} of the system. Since a quantum system has non-commuting physical quantities, the algebra $\\N$ is \\emph{non-abelian}. The $\\bbR$-vector space of self-adjoint operators in $\\N$ is denoted as $\\N_\\sa$.\n\nA classical perspective is given by a collection of commuting physical quantities. Such a collection determines an \\emph{abelian subalgebra}, typically denoted as $V$, of the non-abelian von Neumann algebra $\\N$. An abelian subalgebra is often called a \\emph{context}, and we will use the notions classical perspective, context and abelian subalgebra synonymously. We will only consider abelian subalgebras that\n\\begin{itemize}\n\t\\item[(a)] are von Neumann algebras, i.e., they are closed in the weak topology. The technical advantage is that the spectral theorem holds in both $\\N$ and its abelian von Neumann subalgebras, and that the lattices of projections in $\\N$ and its abelian von Neumann subalgebras are complete;\n\t\\item[(b)] contain the identity operator $\\hat 1$.\n\\end{itemize}\nGiven a von Neumann algebra $\\N$, let $\\VN$ be the set of all its abelian von Neumann subalgebras which contain $\\hat 1$. By convention, the trivial abelian subalgebra $V_0=\\bbC\\hat 1$ is not contained in $\\VN$. If some subalgebra $V'\\in\\VN$ is contained in a larger subalgebra $V\\in\\VN$, then we denote the inclusion as $i_{V'V}:V'\\rightarrow V$. Clearly, $\\VN$ is a partially ordered set under inclusion, and as such is a simple kind of \\emph{category} (see e.g. \\cite{McL98}). The objects in this category are the abelian von Neumann subalgebras, and the arrows are the inclusions. Clearly, there is at most one arrow between any two objects. We call $\\VN$ the \\emph{category of contexts} of the quantum system described by the von Neumann algebra $\\N$.\n\nConsidering the abelian parts of a non-abelian structure may seem trivial, yet in fact it is not, since the context category $\\VN$ keeps track of the relations between contexts: whenever two abelian subalgebras $V,\\tilde V$ have a non-trivial intersection, then there are inclusion arrows\n\\begin{equation}\n\t\t\tV\\longleftarrow V\\cap\\tilde V\\longrightarrow\\tilde V\n\\end{equation}\nin $\\VN$. Every self-adjoint operator $\\A\\in V\\cap\\tilde V$ can be written as $g(\\hat B)$ for some $\\hat B\\in V_{\\sa}$, where $g:\\bbR\\rightarrow\\bbR$ is a Borel function, and, at the same time, as another $h(\\hat C)$ for some $\\hat C\\in\\tilde V_\\sa$ and some other Borel function $h$.\\footnote{This follows from the fact that each abelian von Neumann algebra $V$ is generated by a single self-adjoint operator, see e.g. Prop. 1.21 in \\cite{Tak79}. A Borel function $g:\\bbR\\rightarrow\\bbC$ takes a self-adjoint operator $\\A$ in a von Neumann algebra to another operator $g(\\A)$ in the same algebra. If $g$ is real-valued, then $g(\\A)$ is self-adjoint. For details, see \\cite{KR83a,Tak79}.} The point is that while $\\A$ commutes with $\\hat B$ and $\\A$ commutes with $\\hat C$, the operators $\\hat B$ and $\\hat C$ do not necessarily commute. In that way, the context category $\\VN$ encodes a lot of information about the algebraic structure of $\\N$, not just between commuting operators, and about the relations between contexts.\n\nIf $V'\\subset V$, then the context $V'$ contains less self-adjoint operators and less projections than the context $V$, so one can describe less physics from the perspective of $V'$ than from the perspective of $V$. The step from $V$ to $V'$ hence involves a suitable kind of \\emph{coarse-graining}. We will see later how daseinisation of projections resp. self-adjoint operators implements this informal idea of coarse-graining.\n\n\\begin{example}\t\t\t\\label{Ex1}\nLet $\\Hi=\\bbC^3$, and let $\\N=\\mc{B}(\\bbC^3)$, the algebra of all bounded linear operators on $\\bbC^3$. $\\mc{B}(\\bbC^3)$ is the algebra $M_3(\\bbC)$ of $3\\times 3$-matrices with complex entries, acting as linear transformations on $\\bbC^3$. Let $(\\psi_1,\\psi_2,\\psi_3)$ be an orthonormal basis of $\\bbC^3$, and let $(\\P_1,\\P_2,\\P_3)$ be the three projections onto the one-dimensional subspaces $\\bbC\\psi_1,\\bbC\\psi_2$ and $\\bbC\\psi_3$, respectively. Clearly, the projections $\\P_1,\\P_2,\\P_3$ are pairwise orthogonal, i.e., $\\P_i\\P_j=\\delta_{ij}\\P_i$.\n\nThere is an abelian subalgebra $V$ of $\\mc{B}(\\bbC^3)$ generated by the three projections. One can use von Neumann's double commutant construction and define $V=\\{\\P_1,\\P_2,\\P_3\\}''$ (see \\cite{KR83a}). More concretely, $V=\\rm{lin}_{\\bbC}(\\P_1,\\P_2,\\P_3)$. Even more explicitly, one can pick the matrix representation of the projections $\\P_i$ such that\n\\begin{equation}\n\t\t\t\\P_1=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t1 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right),\\ \\ \\\n\t\t\t\\P_2=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 1 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right),\\ \\ \\\n\t\t\t\\P_3=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 1\n\t\t\t\\end{array}\\right).\t\t\t\n\\end{equation}\nThe abelian algebra $V$ generated by these projections then consists of all diagonal $3\\times 3$-matrices (with complex entries on the diagonal). Clearly, there is no larger abelian subalgebra of $\\mc{B}(\\bbC^3)$ that contains $V$, so $V$ is maximal abelian.\n\nEvery orthonormal basis $(\\tilde\\psi_1,\\tilde\\psi_2,\\tilde\\psi_3)$ determines a maximal abelian subalgebra $\\tilde V$ of $\\mc{B}(\\bbC^3)$. There is some redundancy, though: if two bases differ only by a permutation and\/or phase factors of the basis vectors, then they generate the same maximal abelian subalgebra. Evidently, there are uncountably many maximal abelian subalgebras of $\\mc{B}(\\bbC^3)$.\n\nThere also are non-maximal abelian subalgebras. Consider for example the algebra generated by $\\P_1$ and $\\P_2+\\P_3$. This algebra is denoted as $V_{\\P_1}$ and is given as $V_{\\P_1}=\\rm{lin}_\\bbC(\\P_1,\\P_2+\\P_3)=\\bbC\\P_1+\\bbC(\\hat 1-\\P_1)$. Here, we use that $\\P_2+\\P_3=\\hat 1-\\P_1$. There are uncountably many non-maximal subalgebras of $\\mc{B}(\\bbC^3)$.\n\nThe trivial projections $\\hat 0$ and $\\hat 1$ are contained in every abelian subalgebra. An algebra of the form $V_{\\P_1}$ contains two non-trivial projections: $\\P_1$ and $\\hat 1-\\P_1$. There is the trivial abelian subalgebra $V_0=\\bbC\\hat 1$ consisting of multiples of the identity operator only. By convention, we will not consider the trivial algebra (it is not included in the partially ordered set $\\VN$).\n\nThe set $\\VN=\\mc{V}(\\mc{B}(\\bbC^3))=:\\mc{V}(\\bbC^3)$ of abelian subalgebras of $\\mc{B}(\\bbC^3)$ hence can be divided into two subsets: the maximal abelian subalgebras, corresponding to orthonormal bases as described, and the ones of the form $V_{\\P_1}$ that are generated by a single projection.\\footnote{In other (larger) algebras than $\\mc{B}(\\bbC^3)$, the situation is more complicated, of course.} Clearly, given two different maximal abelian subalgebras $V$ and $\\tilde V$, neither one contains the other. Similarly, if we have two non-maximal algebras $V_{\\P_1}$ and $V_{\\P_2}$, neither one will contain the other, with one exception: if $\\P_2=\\hat 1-\\P_1$, then $V_{\\P_1}=V_{\\P_2}$. This comes from the more general fact that if a unital operator algebra contains an operator $\\A$, then it also contains the operator $\\hat 1-\\A$.\n\nOf course, non-maximal subalgebras can be contained in maximal ones. Let $V$ be the maximal abelian subalgebra generated by three pairwise orthogonal projections $\\P_1,\\P_2,\\P_3$. Then $V$ contains three non-maximal abelian subalgebras, namely $V_{\\P_1},V_{\\P_2}$ and $V_{\\P_3}$. Importantly, each non-maximal abelian subalgebra is contained in many different maximal ones: consider two projections $\\Q_2,\\Q_3$ onto one-dimensional subspaces in $\\bbC^3$ such that $\\P_1,\\Q_2,\\Q_3$ are pairwise orthogonal. We assume that $\\Q_2,\\Q_3$ are such that the maximal abelian subalgebra $\\tilde V$ generated by $\\P_1,\\Q_2,\\Q_3$ is different from the algebra $V$ generated by the three pairwise orthogonal projections $\\P_1,\\P_2,\\P_3$. Then both $V$ and $\\tilde V$ contain $V_{\\P_1}$ as a subalgebra. This argument makes clear that $V_{\\P_1}$ is contained in every maximal abelian subalgebra that contains the projection $\\P_1$.\n\nWe take inclusion of smaller into larger abelian subalgebras (here: of non-maximal into maximal ones) as a partial order on the set $\\mc{V}(\\bbC^3)$.\n\nSince $\\bbC^3$ is finite-dimensional, the algebra $\\mc{B}(\\bbC^3)$ is both a von Neumann and a $C^*$-algebra. The same holds for the abelian subalgebras. We do not have to worry about weak closedness. On infinite-dimensional Hilbert spaces, these questions become important. In general, an abelian von Neumann subalgebra $V$ is generated by the collection $\\PV$ of its projections (which all commute with each other, of course) via the double commutant construction.\n\\end{example}\n\n\\subsection{Gel'fand spectra and the spectral presheaf}\n\\paragraph{Introduction.} We will now make use of the fact that each abelian ($C^*$-) algebra $V$ of operators can be seen as an algebra of continuous functions on a suitable topological space, the \\emph{Gel'fand spectrum} of $V$. At least locally, for each context, we thus obtain a mathematical formulation of quantum theory that is similar to classical physics, with a state space and physical quantities as real-valued functions on this space. The `local state spaces' are then combined into one global structure, the \\emph{spectral presheaf}, which serves as a state space analogue for quantum theory.\n\n\\paragraph{Gel'fand spectra.} Let $V\\in\\VN$ be an abelian subalgebra of $\\N$, and let $\\Si_V$ denote its Gel'fand spectrum. $\\Si_V$ consists of the multiplicative states on $V$, i.e., the positive linear functionals $\\ld:V\\longrightarrow\\bbC$ of norm $1$ such that\n\\begin{equation}\n\t\t\t\\forall\\A,\\hat B\\in V:\\ld(\\A\\hat B)=\\ld(\\A)\\ld(\\hat B).\n\\end{equation}\nThe elements $\\ld$ of $\\Si_V$ also are pure states of $V$ and algebra homomorphisms from $V$ to $\\bbC$. (It is useful to have these different aspects in mind.) The set $\\Si_V$ is a compact Hausdorff space in the weak$^*$ topology \\cite{KR83a,Tak79}.\n\nLet $\\ld\\in\\Si_V$ be given. It can be shown that for any self-adjoint operator $\\A\\in V_\\sa$, it holds that\n\\begin{equation}\n\t\t\t\\ld(\\A)\\in\\spec{\\A},\n\\end{equation}\nand every element $s\\in\\spec{\\A}$ is given as $s=\\tilde\\ld(\\A)$ for some $\\tilde\\ld\\in\\Si_V$. Moreover, if $g:\\bbR\\rightarrow\\bbR$ is a Borel function, then\n\\begin{equation}\t\t\t\\label{FUNC}\n\t\t\t\\ld(g(\\A)=g(\\ld(\\A)).\n\\end{equation}\nThis implies that an element $\\ld$ of the Gel'fand spectrum $\\Si_V$ can be seen as a \\emph{valuation} on $V_\\sa$, i.e., a function sending each self-adjoint operator in $V$ to some element of its spectrum in such a way that equation \\eq{FUNC}, the FUNC principle, holds. Within each context $V$, all physical quantities in $V$ (and only those!) can be assigned values at once, and the assignment of values commutes with taking (Borel) functions of the operators. Every element $\\ld$ of the Gel'fand spectrum gives a different valuation on $V_\\sa$.\n\nThe well-known \\emph{Gel'fand representation theorem} shows that the abelian von Neumann algebra $V$ is isometrically $*$-isomorphic (i.e., isomorphic as a $C^*$-algebra) to the algebra of continuous, complex-valued functions on its Gel'fand spectrum $\\Si_V$.\\footnote{The Gel'fand representation theorem holds for abelian $C^*$-algebras. Since every von Neumann algebra is a $C^*$-algebra, the theorem applies to our situation.} The isomorphism is given by\n\\begin{eqnarray}\n\t\t\tV &\\longrightarrow& C(\\Si_V)\\\\\n\t\t\t\\A &\\longmapsto& \\overline{A},\n\\end{eqnarray}\nwhere $\\overline{A}(\\ld):=\\ld(\\A)$ for all $\\ld\\in\\Si_V$. The function $\\overline{A}$ is called the \\emph{Gel'fand transform} of the operator $\\A$. If $\\A$ is a self-adjoint operator, then $\\overline{A}$ is real-valued, and $\\overline{A}(\\Si_V)=\\spec{\\A}$.\n\nThe fact that the self-adjoint operators in an abelian (sub)algebra $V$ can be written as real-valued functions on the compact Hausdorff space $\\Si_V$ means that the Gel'fand spectrum $\\Si_V$ plays a r\\^ole exactly like the state space of a classical system. Of course, since $V$ is abelian, not all self-adjoint operators $\\A\\in\\N_{\\sa}$ representing physical quantities of the quantum system are contained in $V$. We will interpret the Gel'fand spectrum $\\Si_V$ as a \\emph{local state space} at $V$. Local here means `at the abelian part $V$ of the global non-abelian algebra $\\N$'.\n\nFor a projection $\\P\\in\\PV$, we have\n\\begin{equation}\n\t\t\t\\ld(\\P)=\\ld(\\P^{2})=\\ld(\\P)\\ld(\\P),\n\\end{equation}\nso $\\ld(\\P)\\in\\{0,1\\}$ for all projections. This implies that the Gel'fand transform $\\overline{P}$ of $\\P$ is the characteristic function of some subset of $\\Si_V$.\\footnote{The fact that a characteristic function can be continuous shows that the Gel'fand topology on $\\Si_V$ is pretty wild: the Gel'fand spectrum of an abelian von Neumann algebra is \\emph{extremely disconnected}, that is, the closure of each open set is open.} Let $A$ be a physical quantity that is represented by some self-adjoint operator in $V$, and let ``$\\Ain{\\De}$'' be a proposition about the value of $A$. Then we know from the spectral theorem that there is a projection $\\hat E[\\Ain{\\De}]$ in $V$ that represents the proposition. We saw that $\\ld(\\hat E[\\Ain{\\De}])\\in\\{0,1\\}$, and by interpreting $1$ as `true' and $0$ as `false', we see that a valuation $\\ld\\in\\Si_V$ assigns a Boolean truth-value to each proposition ``$\\Ain{\\De}$''.\n\nThe precise relation between projections in $V$ and subsets of the Gel'fand spectrum of $V$ is as follows: let $\\P\\in\\PV$, and define\n\\begin{equation}\n\t\t\t\tS_\\P:=\\{\\ld\\in\\Si_V\\mid\\ld(\\P)=1\\}.\n\\end{equation}\nOne can show that $S_\\P\\subseteq\\Si_V$ is \\emph{clopen}, i.e., closed and open in $\\Si_V$. Conversely, every clopen subset $S$ of $\\Si_V$ determines a unique projection $\\P_S\\in\\PV$, given as the inverse Gel'fand transform of the characteristic function of $S$. There is a lattice isomorphism\n\\begin{eqnarray}\t\t\t\\label{alpha}\n\t\t\t\\alpha:\\PV &\\longrightarrow& \\mc{C}l(\\Si_V)\\\\\t\t\t\\nonumber\n\t\t\t\\P &\\longmapsto& S_\\P\n\\end{eqnarray}\nbetween the lattice of projections in $V$ and the lattice of clopen subsets of $\\Si_V$. Thus, starting from a proposition ``$\\Ain{\\De}$'', we obtain a clopen subset $S_{\\hat E[\\Ain{\\De}]}$ of $\\Si_V$. This subset consists of all valuations, i.e., pure states $\\ld$ of $V$ such that $\\ld(\\hat E[\\Ain\\De])=1$, which means that the proposition ``$\\Ain\\De$'' is true in the state\/under the valuation $\\ld$. This strengthens the interpretation of $\\Si_V$ as a local state space.\n\nWe saw that each context $V$ provides us with a local state space, and one of the main ideas in the work by Isham and Butterfield \\cite{IB98,IB99,IB00,IB02} was to form a single global object from all these local state spaces.\n\nIn order to keep track of the inclusion relations between the abelian subalgebras, we must relate the Gel'fand spectra of larger and smaller subalgebras in a natural way. Let $V,V'\\in\\VN$ such that $V'\\subseteq V$. Then there is a mapping\n\\begin{eqnarray}\t\t\t\\label{Restriction}\n\t\t\tr:\\Si_V &\\longrightarrow& \\Si_{V'}\\\\\t\t\t \\nonumber\n\t\t\t\\ld &\\longmapsto& \\ld|_{V'},\n\\end{eqnarray}\nsending each element $\\ld$ of the Gel'fand spectrum of the larger algebra $V$ to its restriction $\\ld|_{V'}$ to the smaller algebra $V'$. It is well-known that the mapping $r:\\Si_V\\rightarrow\\Si_{V'}$ is continuous with respect to the Gel'fand topologies and surjective. Physically, this means that every valuation $\\ld'$ on the smaller algebra is given as the restriction of a valuation on the larger algebra.\n\n\\paragraph{The spectral presheaf.} We can now define the central object in the topos approach to quantum theory: the \\emph{spectral presheaf} $\\Sig$. A \\emph{presheaf} is a contravariant, $\\Set$-valued functor, see e.g. \\cite{McL98}.\\footnote{We will only make minimal use of category theory in this article. In particular, the following definition should be understandable in itself, without further knowledge of functors etc.} Our base category, the domain of this functor, is the context category $\\VN$. To each object in $\\VN$, i.e., to each context $V$, we assign its Gel'fand spectrum $\\Sig_V:=\\Si_V$, and to each arrow in $\\VN$, i.e., each inclusion $i_{V'V}$, we assign a function from the set $\\Sig_V$ to the set $\\Sig_{V'}$. This function is $\\Sig(i_{V'V}):=r$ (see equation \\eq{Restriction}). It implements the concept of coarse-graining on the level of the local state spaces (Gel'fand spectra).\n\nThe spectral presheaf $\\Sig$ associated with the von Neumann algebra $\\N$ of a quantum system is the analogue of the state space of a classical system. The spectral presheaf is not a set, and hence not a space in the usual sense. It rather is a particular, $\\Set$-valued functor, built from the Gel'fand spectra of the abelian subalgebras of $\\N$, and the canonical restriction functions between them.\n\nWe come back to our example, $\\N=\\mc{B}(\\bbC^3)$, and describe its spectral presheaf:\n\n\\begin{example}\t\t\t\\label{Ex_SigOfB(C3)}\nLet $\\N=\\mc{B}(\\bbC^3)$ as in Ex. \\ref{Ex1}, and let $V\\in\\mc{V}(\\bbC^3)$ be a maximal abelian subalgebra. As discussed above, $V$ is of the form $V=\\{\\P_1,\\P_2,\\P_3\\}''$ for three pairwise orthogonal rank-$1$ projections $\\P_1,\\P_2,\\P_3$. The Gel'fand spectrum $\\Si_V$ of $V$ has three elements (and is equipped with the discrete topology, of course). The spectral elements are given as\n\\begin{equation}\n\t\t\t\\ld_i(\\P_j)=\\delta_{ij}\\ \\ \\ \\ (i=1,2,3).\n\\end{equation}\nLet $\\A\\in V$ be an arbitrary operator. Then $\\A=\\sum_{i=1}^3 a_i\\P_i$ for some (unique) complex coefficients $a_i$. We have $\\ld_i(\\A)=a_i$. If $\\A$ is self-adjoint, then the $a_i$ are real and are the eigenvalues of $\\A$.\n\nThe non-maximal abelian subalgebra $V_{\\P_1}$ has two elements in its Gel'fand spectrum. Let us call them $\\ld'_1$ and $\\ld'_2$, where $\\ld'_1(\\P)=1$ and $\\ld'_1(\\hat 1-\\P)=0$, while $\\ld'_2(\\P)=0$ and $\\ld'_2(\\hat 1-\\P)=1$. The restriction mapping $r:\\Si_V\\rightarrow\\Si_{V'}$ sends $\\ld_1$ to $\\ld'_1$. Both $\\ld_2$ and $\\ld_3$ are mapped to $\\ld'_2$.\n\nAnalogous relations hold for the other non-maximal abelian subalgebras $V_{\\P_2}$ and $V_{\\P_3}$ of $V$. Since the spectral presheaf is given by assigning to each abelian subalgebra its Gel'fand spectrum, and to each inclusion the corresponding restriction function between the Gel'fand spectra, we have a complete, explicit description of the spectral presheaf $\\Sig$ of the von Neumann algebra $\\mc{B}(\\bbC^3)$.\n\nThe generalisation to higher dimensions is straightforward. In particular, if $\\rm{dim}(\\Hi)=n$ and $V$ is a maximal abelian subalgebra of $\\mc{B}(\\bbC^n)$, then then Gel'fand spectrum consists of $n$ elements. Topologically, the Gel'fand spectrum $\\Si_V$ is a discrete space. (For infinite-dimensional Hilbert spaces, where more complicated von Neumann algebras than $\\mc{B}(\\bbC^n)$ exist, this is not true in general.)\n\\end{example}\n\n\\section{Representation of propositions in the topos approach -- Daseinisation of projections}\t\\label{S4}\nWe now consider the representation of propositions like ``$\\Ain{\\De}$'' in our topos scheme. We already saw that the spectral presheaf $\\Sig$ is an analogue of the state space $\\mc{S}$ of a classical system. In analogy to classical physics, where propositions are represented by (Borel) subsets of state space, in the topos approach propositions will be represented by suitable subobjects of the spectral presheaf.\n\n\\paragraph{Coarse-graining of propositions.} Consider a proposition ``$\\Ain{\\De}$'', where $A$ is some physical quantity of the quantum system under consideration. We assume that there is a self-adjoint operator $\\A$ in the von Neumann algebra $\\N$ of the system that represents $A$. From the spectral theorem, we know that there is a projection operator $\\P:=\\hat E[\\Ain{\\De}]$ that represents the proposition ``$\\Ain{\\De}$''. As is well-known, the mapping from propositions to projections is many-to-one. One can form equivalence classes of propositions to obtain a bijection. In a slight abuse of language, we will usually refer to \\emph{the} proposition represented by a projection. Since $\\N$ is a von Neumann algebra, the spectral theorem holds. In particular, for all propositions ``$\\Ain\\De$'', we have $\\P=\\hat E[\\Ain\\De]\\in\\PN$. The projections representing propositions are all contained in the von Neumann algebra. If we had chosen a more general $C^*$-algebra instead, this would not have been the case.\n\nWe stipulate that in the topos formulation of quantum theory all classical perspectives\/contexts must be taken into account at the same time, so we have to adapt the proposition ``$\\Ain{\\De}$'' resp. its representing projection $\\P$ to all contexts. The idea is very simple: in every context $V$, we pick the strongest proposition implied by ``$\\Ain{\\De}$'' that can be made from the perspective of this context. On the level of projections, this amounts to taking the \\emph{smallest} projection in any context $V$ that is larger than or equal to $\\P$.\\footnote{Here, we use the interpretation that $\\P<\\Q$ for two projections $\\P,\\Q\\in\\PN$ means that the proposition represented by $\\P$ implies the proposition represented by $\\Q$. This is customary in quantum logic, and we use it to motivate our construction, though the partial order on projections will \\emph{not} be the implication relation for the form of quantum logic resulting from our scheme.} We write, for all $V\\in\\VN$,\n\\begin{equation}\t\t\t\\label{Eq_dastooVP}\n\t\t\t\\dastoo{V}{P}:=\\bigwedge\\{\\Q\\in\\PV\\mid\\Q\\geq\\P\\}\n\\end{equation}\nfor this approximation of $\\P$ to $V$. If $\\P\\in\\PV$, then clearly the approximation will give $\\P$ itself, i.e. $\\dastoo{V}{P}=\\P$. If $\\P\\notin\\PV$, then $\\dastoo{V}{P}>\\P$. For many contexts $V$, it holds that $\\dastoo{V}{P}=\\hat 1$.\n\nSince the projection $\\dastoo{V}{P}$ lies in $V$, it corresponds to a proposition about some physical quantity described by a self-adjoint operator \\emph{in} $V$. If $\\dastoo{V}{P}$ happens to be a spectral projection of the operator $\\A$, then the proposition corresponding to $\\dastoo{V}{P}$ is of the form ``$\\Ain\\Ga$'' for some Borel set $\\Ga$ of real numbers that is larger than the set $\\De$ in the original proposition ``$\\Ain\\De$''.\\footnote{We remark that in order to have $\\dastoo{V}{P}\\in\\PV$, it is sufficient, but \\emph{not} necessary that $\\A\\in V_\\sa$.} The fact that $\\Ga\\supset\\De$ shows that the mapping\n\\begin{eqnarray}\n\t\t\t\\delta^o_V:\\PN &\\longrightarrow& \\PV\\\\\t\t\t\\nonumber\n\t\t\t\\P &\\longmapsto& \\dastoo{V}{P}\n\\end{eqnarray}\nimplements the idea of coarse-graining on the level of projections (resp. the corresponding propositions).\n\nIf $\\dastoo{V}{P}$ is not a spectral projection of $\\A$, it still holds that it corresponds to a proposition ``$B\\varepsilon\\Ga$'' about the value of some physical quantity $B$ represented by a self-adjoint operator $\\hat B$ in $V$, and since $\\P<\\dastoo{V}{P}$, this proposition can still be seen as a coarse-graining of the original proposition ``$\\Ain\\De$'' corresponding to $\\P$ (even though $A\\neq B$).\n\nEvery projection is contained in some abelian subalgebra, so there is always some $V$ such that $\\dastoo{V}{P}=\\P$. From the perspective of this context, no coarse-graining takes place.\n\nWe call the original proposition ``$\\Ain\\De$'' that we want to represent the \\emph{global} proposition, while a proposition ``$B\\varepsilon\\Ga$'' corresponding to the projection $\\dastoo{V}{P}$ is called a \\emph{local} proposition. `Local' here again means `at the abelian part $V$' and has no spatio-temporal connotation.\n\nWe can collect all the local approximations into a mapping\n\\begin{equation}\n\t\t\t\\P\\mapsto(\\dastoo{V}{P})_{V\\in\\VN}\n\\end{equation}\nwhose component at $V$ is given by \\eq{Eq_dastooVP}. From every global proposition we thus obtain one coarse-grained local proposition for each context $V$. In the next step, we will consider the whole collection of coarse-grained local propositions, relate it to a suitable subobject of the spectral presheaf, and regard this object as the representative of the global proposition.\n\n\\paragraph{Daseinisation of projections.} Let $V$ be a context. Since $V$ is an abelian von Neumann algebra, the lattice $\\PV$ of projections in $V$ is a distributive lattice. Moreover, $\\PV$ is complete and orthocomplemented.  We saw in \\eq{alpha} that there is a lattice isomorphism between $\\PV$ and $\\mc{C}l(\\Sig_V)$, the lattice of clopen subsets of $\\Sig_V$.\n\nWe already have constructed a family $(\\dastoo{V}{P})_{V\\in\\VN}$ of projections from a single projection $\\P\\in\\PN$ representing a global proposition. For each context $V$, we now consider the clopen subset $\\alpha(\\dastoo{V}{P})=S_{\\dastoo{V}{P}}\\subseteq\\Si_V$. We thus obtain a family $(S_{\\dastoo{V}{P}})_{V\\in\\VN}$ of clopen subsets, one for each context. One can show that whenever $V'\\subset V$, then\n\\begin{equation}\n\t\t\tS_{\\dastoo{V}{P}}|_{V'}=\\{\\ld|_{V'} \\mid \\ld\\in S_{\\dastoo{V}{P}}\\}=S_{\\dastoo{V'}{P}},\n\\end{equation}\nthat is, the clopen subsets in the family $(S_{\\dastoo{V}{P}})_{V\\in\\VN}$ `fit together' under the restriction mappings of the spectral presheaf $\\Sig$ (see Thm. 3.1 in \\cite{DI08}). This means that the family $(S_{\\dastoo{V}{P}})_{V\\in\\VN}$ of clopen subsets, together with the restriction mappings between them, forms a \\emph{subobject} of the spectral presheaf. This particular subobject will be denoted as $\\ps{\\das{P}}$ and is called the \\emph{daseinisation of $\\P$}.\n\nA subobject of the spectral presheaf (or any other presheaf) is the analogue of a subset of a space. Concretely, for each $V\\in\\VN$, the set $S_{\\dastoo{V}{P}}$ is a subset of the Gel'fand spectrum of $V$. Moreover, the subsets for different $V$ are not arbitrary, but fit together under the restriction mappings of the spectral presheaf.\n\nThe collection of all subobjects of the spectral presheaf is a complete \\emph{Heyting algebra} and is denoted as $\\Sub{\\Sig}$. It is the analogue of the power set of the state space of a classical system.\n\nLet $\\ps S$ be a subobject of the spectral presheaf $\\Sig$ such that the component $S_V\\subseteq\\Sig_V$ is a clopen subset for all $V\\in\\VN$. We call such a subobject a \\emph{clopen} subobject. All subobjects obtained from daseinisation of projections are clopen. One can show that the clopen subobjects form a complete Heyting algebra $\\Subcl{\\Sig}$ (see Thm. 2.5 in \\cite{DI08}). This complete Heyting algebra can be seen as the analogue of the $\\sigma$-complete Boolean algebra of Borel subsets of the state space of a classical system. In our constructions, the Heyting algebra $\\Subcl{\\Sig}$ will be more important than the bigger Heyting algebra $\\Sub{\\Sig}$, just as in classical physics, where the Boolean algebra of measurable subsets of state space is technically more important than the algebra of all subsets.\n\nStarting from a global proposition ``$\\Ain\\De$'' with corresponding projection $\\P$, we have constructed the clopen subobject $\\ps{\\das{P}}$ of the spectral presheaf. This subobject, the analogue of the measurable subset $f_A^{-1}(\\De)$ of the state space of a classical system, is the representative of the global proposition ``$\\Ain\\De$''. The following mapping is called \\emph{daseinisation of projections}:\n\\begin{eqnarray}\n\t\t\\ps\\delta:\\PN &\\longrightarrow& \\Subcl{\\Sig}\\\\\t\t\t\\nonumber\n\t\t\\P &\\longmapsto& \\ps{\\das{P}}.\n\\end{eqnarray}\nIt has the following properties:\n\\begin{itemize}\n\t\\item[(1)] If $\\P<\\Q$, then $\\ps{\\delta(\\P)}<\\ps{\\delta(\\Q)}$, i.e., daseinisation is order-preserving;\n\t\\item[(2)] the mapping $\\ps{\\delta}:\\PN\\rightarrow\\Subcl{\\Sig}$ is injective, that is, two (inequivalent) propositions correspond to two different subobjects;\n\t\\item[(3)] $\\ps{\\delta(\\hat 0)}=\\ps 0$, the empty subobject, and $\\ps{\\delta(\\hat 1)}=\\Sig$. The trivially false proposition is represented by the empty subobject, the trivially true proposition is represented by the whole of $\\Sig$.\n\t\\item[(4)] for all $\\P,\\Q\\in\\PN$, it holds that $\\ps{\\delta(\\P\\vee\\Q)}=\\ps{\\delta(\\P)}\\vee\\ps{\\delta(\\Q)}$, that is, daseinisation preserves the disjunction (Or) of propositions;\n\t\\item[(5)] for all $\\P,\\Q\\in\\PN$, it holds that $\\ps{\\delta(\\P\\wedge\\Q)}\\leq\\ps{\\delta(\\P)}\\wedge\\ps{\\delta(\\Q)}$, that is, daseinisation does not preserve the conjunction (And) of propositions;\n\t\\item[(6)] in general, $\\ps{\\delta(\\P)}\\wedge\\ps{\\delta(\\Q)}$ is not of the form $\\ps{\\delta(\\hat R)}$ for a projection $\\hat R\\in\\PN$, and daseinisation is not surjective.\n\\end{itemize}\nThe domain of the mapping $\\ps\\delta$, the lattice $\\PN$ of projections in the von Neumann algebra $\\N$ of physical quantities, is the lattice that Birkhoff and von Neumann suggested as the algebraic representative of propositional quantum logic (\\cite{BvN36}; in fact, they considered the case $\\N=\\BH$.) Thus daseinisation of projections can be seen as a `translation' mapping between ordinary, Birkhoff-von Neumann quantum logic, which is based upon the non-distributive lattice of projections $\\PN$, and the topos form of propositional quantum logic, which is based upon the distributive lattice $\\Subcl{\\Sig}$. The latter more precisely is a Heyting algebra.\n\nThe quantum logic aspects of the topos formalism and the physical and conceptual interpretation of the relations above are discussed in some detail in \\cite{Doe09}, building upon \\cite{IB98,DI(2),DI08}. The resulting new form of quantum logic has many attractive features and avoids a number of the well-known conceptual problems of standard quantum logic \\cite{DCG02}.\n\n\\begin{example}\nLet $\\Hi=\\bbC^3$, and let $S_z$ be the physical quantity `spin in $z$-direction'. We consider the proposition ``$S_z\\varepsilon(-0.1,0.1)$'', i.e., ``the spin in $z$-direction has a value between $-0.1$ and $0.1$''. (Since the eigenvalues are $-\\frac{1}{\\sqrt 2},0$ and $\\frac{1}{\\sqrt 2}$, this amounts to saying that the spin in $z$-direction is $0$.) The self-adjoint operator representing $S_z$ is\n\\begin{equation}\n\t\t\t\\hat S_z=\\frac{1}{\\sqrt{2}}\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t1 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & -1\n\t\t\t\\end{array}\\right),\n\\end{equation}\nand the eigenvector corresponding to the eigenvalue $0$ is $\\ket\\psi=(0,1,0)$. The projection $\\P:=\\hat E[S_z\\varepsilon(-0.1,0.1)]=\\ket\\psi\\bra\\psi$ onto this eigenvector hence is\n\\begin{equation}\n\t\t\t\\P=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 1 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right).\n\\end{equation}\nThere is exactly one context $V\\in\\mc{V}(\\bbC^3)$ that contains the operator $\\hat S_z$, namely the (maximal) context $V_{\\hat S_z}=\\{\\P_1,\\P_2,\\P_3\\}''$ generated by the projections \n\\begin{equation}\n\t\t\t\\P_1=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t1 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right),\\ \\ \\\n\t\t\t\\P_2=\\P=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 1 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\n\t\t\t\\end{array}\\right),\\ \\ \\\n\t\t\t\\P_3=\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 1\n\t\t\t\\end{array}\\right).\t\t\t\n\\end{equation}\nIn the following, we consider different kinds of contexts $V$ and the approximations $\\dasout{\\P}{V}$ of $\\P=\\P_2$ to these contexts.\n\n\\textbf{(1) $V_{\\hat S_z}$ and its subcontexts:} Of course, $\\dasout{\\P}{V_{\\hat S_z}}=\\P$. Let $V_{\\P_1}=\\{\\P_1,\\P+\\P_3\\}''$, then $\\dasout{\\P}{V_{\\P_1}}=\\P+\\P_3$. Analogously, $\\dasout{\\P}{V_{\\P_3}}=\\P+\\P_3$, but $\\dasout{\\P}{V_{\\P}}=\\P$, since $V_\\P$ contains the projection $\\P$.\n\n\\textbf{(2) Maximal contexts which share the projection $\\P$ with $V_{\\hat S_z}$, and their subcontexts:} Let $\\Q_1,\\Q_3$ be two orthogonal rank-$1$ projections that are both orthogonal to $\\P$ (e.g. projections obtained from $\\P_1,\\P_3$ by a rotation about the axis determined by $\\P$). Then $V=\\{\\Q_1,\\P,\\Q_3\\}''$ is a maximal algebra that contains $\\P$, so $\\dasout{\\P}{V}=\\P$. Clearly, there are uncountably many contexts $V$ of this form.\n\nFor projections $\\Q_1,\\P,\\Q_3$ as described above, we have $\\dasout{\\P}{V_{\\Q_1}}=\\P+\\Q_3$, $\\dasout{\\P}{V_{\\Q_3}}=\\Q_1+\\P$ (and $\\dasout{\\P}{V_\\P}=\\P$, as mentioned before).\n\n\\textbf{(3) Contexts which contain a rank-$2$ projection $\\Q$ such that $\\Q>\\P$:} We first note that there are uncountably many projections $\\Q$ of this form.\n\\begin{enumerate}\n\t\\item [(a)] The trivial cases of contexts containing $\\Q>\\P$ are (1) maximal contexts containing $\\P$; and (2) the context $V_{\\Q}=V_{\\hat 1-\\Q}$.\n\t\\item [(b)] There are also maximal contexts which do not contain $\\P$, but do contain $\\Q$: let $\\Q_1,\\Q_2$ be two orthogonal rank-$1$ projections such that $\\Q_1+\\Q_2=\\Q$, where $\\Q_1,\\Q_2\\neq\\P$. Let $\\Q_3:=\\hat 1-(\\Q_1+\\Q_2)$. For any given $\\Q$, there are uncountably many contexts of the form $V=\\{\\Q_1,\\Q_2,\\Q_3\\}$ (since one can rotate around the axis determined by $\\Q_3$).\n\\end{enumerate}\nFor all these contexts, $\\dasout{\\P}{V}=\\Q$, of course.\n\n\\textbf{(4) Other contexts:} All other contexts neither contain $\\P$ nor a projection $\\Q>\\P$ that is not the identity $\\hat 1$. This means that the smallest projection larger than $\\P$ contained in such a context is the identity, so $\\dasout{\\P}{V}=1$.\n\nWe have constructed $\\dasout{\\P}{V}$ for all contexts. Recalling that for each context $V$, there is a lattice isomorphism $\\alpha$ from the projections to the clopen subsets of the Gel'fand spectrum $\\Si_V$ (see \\eq{alpha}), we can easily write down the clopen subsets $S_{\\dasout{\\P}{V}}\\;(V\\in\\mc V (\\bbC^3))$ corresponding to the projections $\\dasout{\\P}{V}\\;(V\\in\\mc V (\\bbC^3))$. The daseinisation $\\ps{\\das{P}}$ of the projection $\\P$ is nothing but the collection $(S_{\\dasout{\\P}{V}})_{V\\in\\mc V (\\bbC^3)}$ (together with the canonical restriction mappings described in \\eq{Restriction} and Example \\ref{Ex_SigOfB(C3)}).\n\\end{example}\n\n\n\\section{Representation of physical quantities -- Daseinisation of self-adjoint operators}\t\\label{S5}\t\\label{Sec_RepOfPhysQuants}\n\\subsection{Physical quantities as arrows}\t\t\t\\label{SubS_PhysQuantitiesAsArrows}\nIn classical physics, a physical quantity $A$ is represented by a function $f_A$ from the state space $\\mc{S}$ of the system to the real numbers. Both the state space and the real numbers are sets (with extra structure), so they are objects in the topos $\\Set$ of sets and functions.\\footnote{We will not give the technical definition of a topos here. The idea is that a topos is a category that is structurally similar to $\\Set$, the category of small sets and functions. Of course, $\\Set$ itself is a topos.} The function $f_A$ representing a physical quantity is an arrow in the topos $\\Set$.\n\nThe topos reformulation of quantum physics uses structures in the topos $\\SetVNop$, the topos of presheaves over the context category $\\VN$. The objects in the topos, called presheaves, are the analogues of sets, and the arrows between them, called natural transformations, are the analogues of functions. We will now represent a physical quantity $A$ by a suitable arrow, denoted $\\dasB{\\A}$, from the spectral presheaf $\\Sig$ to some other presheaf related to the real numbers. Indeed, there is a real-number object $\\ps{\\bbR}$ in $\\SetVNop$ that is very much like the familiar real numbers: for every $V$, the component $\\ps{\\bbR}_V$ simply is the ordinary real numbers, and all restriction functions are just the identity. But the Kochen-Specker theorem tells us that it is impossible to assign (sharp) real values to all physical quantities, which suggests that the presheaf $\\ps{\\bbR}$ is \\emph{not} the right codomain for our arrows $\\dasB{\\A}$ representing physical quantities.\n\nInstead, we will be using a presheaf, denoted as $\\Rlr$, that takes into account that (a) one cannot assign sharp values to all physical quantities in any given state in quantum theory; and (b) we have coarse-graining. We first explain how coarse-graining will show up: if we want to assign a `value' to a physical quantity $A$, then we again have to find some global expression, involving all contexts $V\\in\\VN$. Let $\\A\\in\\N_\\sa$ be the self-adjoint operator representing $A$. For those contexts $V$ that contain $\\A$, there is no problem, but if $\\A\\notin V$, we will have to approximate $\\A$ by a self-adjoint operator \\emph{in} $V$. Actually, we will use two approximations: one self-adjoint operator in $V$ approximating $\\A$ from below (in a suitable order, specified below), and one operator approximating $\\A$ from above. Daseinisation of self-adjoint operators is nothing but the technical device achieving this approximation.\n\nWe will show that the presheaf $\\Rlr$ basically consists of real intervals, to be interpreted as `unsharp values'. Upon coarse-graining, the intervals can only get bigger. Assume we have given a physical quantity $A$, some state of the system and two contexts $V,V'$ such that $V'\\subset V$. We use the state to assign some interval $[a,b]$ as an `unsharp value' to $A$ at $V$. Then, at $V'$, we will have a bigger interval $[c,d]\\supseteq[a,b]$, being an even more unsharp value of $A$. It it important to note, though, that every self-adjoint operator is contained in some context $V$. If the state is an eigenstate of $\\A$ with eigenvalue $a$, then at $V$, we will assign the `interval' $[a,a]$ to $A$. In this sense, the eigenvector-eigenvalue link is preserved, regardless of the unsharpness in values introduced by coarse-graining. As always in the topos scheme, it is of central importance that the interpretationally relevant structures -- here, the `values' of physical quantities -- are global in nature. One has to consider all contexts at once, and not just argue locally at some context $V$, which will necessarily only give partial information.\n\n\\paragraph{Approximation in the spectral order.} Let $\\A\\in\\N_\\sa$ be a self-adjoint operator, and let $V\\in\\VN$ be a context. We assume that $\\A\\notin V_\\sa$, so we have to approximate $\\A$ in $V$. The idea is very simple: we take a pair of self-adjoint operators, consisting of the largest operator in $V_\\sa$ smaller than $\\A$ and the smallest operator in $V_\\sa$ larger than $\\A$. The only non-trivial point is the question which order on self-adjoint operators to use.\n\nThe most commonly used order is the \\emph{linear order}: $\\A\\leq\\hat B$ if and only if $\\bra\\psi\\A\\ket\\psi\\leq\\bra\\psi\\hat B\\ket\\psi$ for all vector states $\\bra\\psi\\_\\ket\\psi:\\N\\rightarrow\\bbC$. Yet, we will approximate in another order on the self-adjoint operators, the so-called \\emph{spectral order} \\cite{Ols71,deG04}. This has the advantage that the spectra of the approximated operators are subsets of the spectrum of the original operator, which is not the case for the linear order in general.\n\nThe spectral order is defined in the following way: let $\\A,\\hat B\\in\\N_\\sa$ be two self-adjoint operators in a von Neumann algebra $\\N$, and let $(\\hat E^\\A_r)_{r\\in\\bbR},(\\hat E^{\\hat B}_r)_{r\\in\\bbR}$ be their spectral families (see e.g. \\cite{KR83a}). Then\n\\begin{equation}\t\t\t\\label{Def_SpecOrder}\n\t\t\t\\A\\leq_s\\hat B\\text{\\ \\ \\ if and only if\\ \\ \\ }\\forall r\\in\\bbR:\\hat E^\\A_r\\geq\\hat E^{\\hat B}_r.\n\\end{equation}\nEquipped with the spectral order, $\\N_\\sa$ becomes a boundedly complete lattice. The spectral order is coarser than the linear order, i.e., $\\A\\leq_s\\hat B$ implies $\\A\\leq\\hat B$. The two orders coincide on projections and for commuting operators.\n\nConsider the set $\\{\\hat B\\in V_\\sa \\mid \\hat B\\leq_s\\A\\}$, i.e., those self-adjoint operators in $V$ that are spectrally smaller than $\\A$. Since $V$ is a von Neumann algebra, its self-adjoint operators form a boundedly complete lattice under the spectral order, so the above set has a well-defined maximum with respect to the spectral order. It is denoted as\n\\begin{equation}\t\t\\label{dastoiVA}\n\t\t\t\\dastoi{V}{A}:=\\bigvee\\{\\hat B\\in V_\\sa \\mid \\hat B\\leq_s\\A\\}.\n\\end{equation}\nSimilarly, we define\n\\begin{equation}\t\t\\label{dastooVA}\n\t\t\t\\dastoo{V}{A}:=\\bigwedge\\{\\hat B\\in V_\\sa \\mid \\hat B\\geq_s\\A\\},\n\\end{equation}\nwhich is the smallest self-adjoint operator in $V$ that is spectrally larger than $\\A$.\n\nUsing the definition \\eq{Def_SpecOrder} of the spectral order, it is easy to describe the spectral families of the operators $\\dastoi{V}{A}$ and $\\dastoo{V}{A}$:\n\\begin{equation}\t\t\t\\label{Eq_SpecFamOutDas}\n\t\t\t \\forall r\\in\\bbR:\\hat E^{\\dastoo{V}{A}}_r=\\dasinn{\\hat E^\\A_r}{V}\n\\end{equation}\nand\n\\begin{equation}\n\t\t\t\\forall r\\in\\bbR:\\hat E^{\\dastoi{V}{A}}_r=\\dasout{\\hat E^\\A_r}{V}.\n\\end{equation}\nIt turns out (see \\cite{deG07,DI(2)}) that the spectral family defined by the former definition is right-continuous. The latter expression must be amended slightly if we want a right-continuous spectral family: we simply enforce right continuity by setting\n\\begin{equation}\t\t\t\\label{Eq_SpecFamInnDas}\n\t\t\t\\forall r\\in\\bbR:\\hat E^{\\dastoi{V}{A}}_r=\\bigwedge_{s>r}\\dasout{\\hat E^\\A_s}{V}.\n\\end{equation}\nThese equations show the close mathematical link between the approximations in the spectral order of projections and self-adjoint operators.\n\nThe self-adjoint operators $\\dastoi{V}{A},\\dastoo{V}{A}$ are the best approximations to $\\A$ in $V_\\sa$ with respect to the spectral order. $\\dastoi{V}{A}$ approximates $\\A$ from below, $\\dastoo{V}{A}$ from above. Since $V$ is an abelian $C^*$-algebra, we can consider the Gel'fand transforms of these operators to obtain a pair\n\\begin{equation}\n\t\t\t(\\fu{\\dastoi{V}{A}},\\fu{\\dastoo{V}{A}})\n\\end{equation}\nof continuous functions from the Gel'fand spectrum $\\Si_V$ of $V$ to the real numbers. One can show that \\cite{deG05b,DI(3)}\n\\begin{equation}\n\t\t\t\\spec{\\dastoi{V}{A}}\\subseteq\\spec{\\A},\\ \\ \\ \\ \\spec{\\dastoo{V}{A}}\\subseteq\\spec{\\A},\n\\end{equation}\nso the Gel'fand transforms actually map into the spectrum $\\spec{\\A}$ of the original operator $\\A$. Let $\\ld\\in\\Si_V$ be a pure state of the algebra $V$.\\footnote{We remark that in general, this cannot be identified with a state of the non-abelian algebra $\\N$, so we still have a local argument here.} Then we can evaluate $(\\fu{\\dastoi{V}{A}}(\\ld),\\fu{\\dastoo{V}{A}}(\\ld))$ to obtain a pair of real numbers in the spectrum of $\\A$. The first number is smaller than the second, since the first operator $\\dastoi{V}{A}$ approximates $\\A$ from below in the spectral order (and $\\dastoi{V}{A}\\leq_s\\A$ implies $\\dastoi{V}{A}\\leq\\A$), while the second operator $\\dastoo{V}{A}$ approximates $\\A$ from above. The pair $(\\fu{\\dastoi{V}{A}}(\\ld),\\fu{\\dastoo{V}{A}}(\\ld))$ of real numbers is identified with the interval $[\\fu{\\dastoi{V}{A}}(\\ld),\\fu{\\dastoo{V}{A}}(\\ld)]$ and interpreted as the component at $V$ of the unsharp value of $\\A$ in the (local) state $\\ld$.\n\n\\paragraph{The presheaf of `values'.} We now see in which way intervals show up as `values'. If we want to define a presheaf encoding this, we encounter a certain difficulty. It is no problem to assign to each context $V$ the collection $\\mathbb{IR}_V$ of real intervals, but if $V'\\subset V$, then we need a restriction mapping from $\\mathbb{IR}_V$ to $\\mathbb{IR}_{V'}$. Since both sets are the same, the naive guess is to take an interval in $\\mathbb{IR}_V$ and to map it to the same interval in $\\mathbb{IR}_{V'}$. But we already know that the `values' of our physical quantities come from approximations of a self-adjoint operator $\\A$ to all the contexts. If $V'\\subset V$, then in general we have $\\dastoi{V'}{A}<_s\\dastoi{V'}{A}$,\\footnote{This, of course, is nothing but coarse-graining on the level of self-adjoint operators.} which implies $\\fu{\\dastoi{V'}{A}}(\\ld|_{V'})\\leq\\fu{\\dastoi{V}{A}}(\\ld)$ for $\\ld\\in\\Si_V$. Similarly, $\\fu{\\dastoo{V'}{A}}(\\ld|_{V'})\\geq\\fu{\\dastoo{V}{A}}(\\ld)$, so the intervals that we obtain get bigger when we go from larger contexts to smaller ones, due to coarse-graining. Our restriction mapping from the intervals at $V$ to the intervals at $V'$ must take this into account.\n\nThe intervals at different contexts clearly also depend on the state $\\ld\\in\\Si_V$ that one picks. Moreover, we want a presheaf that is not tied to the operator $\\A$, but can provide `values' for all physical quantities and their corresponding self-adjoint operators.\n\nThe idea is to define a presheaf such that at each context $V$, we have all intervals (including those of the form $[a,a]$), plus all possible restrictions to the same or larger intervals at smaller contexts $V'\\subset V$. While this sounds daunting, there is a very simple way to encode this: let $\\downarrow\\!\\!V:=\\{V'\\in\\VN \\mid V'\\subseteq V\\}$. This is a partially ordered set. We now consider a function $\\mu:\\downarrow\\!\\!V\\rightarrow\\bbR$ that \\emph{preserves the order}, that is, $V'\\subset V$ implies $\\mu(V')\\leq\\mu(V)$. Analogously, let $\\nu:\\downarrow\\!\\!V\\rightarrow\\bbR$ denote an \\emph{order-reversing} function, that is, $V'\\subset V$ implies $\\mu(V')\\geq\\mu(V)$. Additionally, we assume that for all $V'\\subseteq V$, it holds that $\\mu(V')\\leq\\nu(V')$. The pair $(\\mu,\\nu)$ thus gives one interval $(\\mu(V'),\\nu(V'))$ for each context $V'\\subseteq V$, and `going down the line' from $V$ to smaller subalgebras $V',V'',...$, these intervals can only get bigger (or stay the same). Of course, each pair $(\\mu,\\nu)$ gives a specific sequence of intervals (for each given $\\ld$). In order to have all possible intervals and sequences built from them, we simply consider the collection of all pairs of order-preserving and -reversing functions.\n\nThis is formalised in the following way: let $\\Rlr$ be the presheaf given\n\\begin{itemize}\n\t\\item [(a)] on objects: for all $V\\in\\VN$, let\n\t\\begin{equation}\n\t\t\t\t\\Rlr_V:=\\{(\\mu,\\nu) \\mid \\mu,\\nu:\\downarrow\\!\\!V\\rightarrow\\bbR,\\mu\\text{ order-preserving, }\\nu\\text{ order-reversing},\\ \\mu\\leq\\nu\\};\n\t\\end{equation}\n\t\\item [(b)] on arrows: for all inclusions $i_{V'V}:V'\\rightarrow V$, let\n\t\\begin{eqnarray}\n\t\t\t\t\\Rlr(i_{V'V}):\\Rlr_V &\\longrightarrow& \\Rlr_{V'}\\\\\t\t\t\\nonumber\n\t\t\t\t(\\mu,\\nu) &\\longmapsto& (\\mu|_{V'},\\nu|_{V'}).\n\t\\end{eqnarray}\n\\end{itemize}\nThe presheaf $\\Rlr$ is where physical quantities take their `values' in the topos version of quantum theory. It is the analogue of the set of real numbers, where physical quantities in classical physics take their values.\n\n\\paragraph{Daseinisation of a self-adjoint operator.} We now start to assemble the approximations to the self-adjoint operator $\\A$ into an arrow from the spectral presheaf $\\Sig$ to the presheaf $\\Rlr$ of values. Such an arrow is a so-called natural transformation (see e.g. \\cite{McL98}), since these are the arrows in the topos $\\SetVNop$. For each context $V\\in\\VN$, we define a function, which we will denote as $\\dasB{\\A}_V$, from the Gel'fand spectrum $\\Sig_V$ of $V$ to the collection $\\Rlr_V$ of pairs of order-preserving and -reversing functions from $\\downarrow\\!\\!V$ to $\\bbR$. In a second step, we will see that these functions fit together in the appropriate sense to form a natural transformation.\n\nThe function $\\dasB{\\A}_V$ is constructed as follows: let $\\ld\\in\\Sig_V$. In each $V'\\in\\downarrow\\!\\!V$, we have one approximation $\\delta^i(\\A)_{V'}$ to $\\A$. If $V''\\subset V'$, then $\\delta^i(\\A)_{V''}\\leq_s\\delta^i(\\A)_{V'}$, which implies $\\delta^i(\\A)_{V''}\\leq\\delta^i(\\A)_{V'}$. One can evaluate $\\ld$ at $\\delta^i(\\A)_{V'}$ for each $V'$, giving an order-preserving function\n\\begin{align}\t\t\t\\label{mu_ld}\n\t\t\t\t\\mu_\\ld:\\downarrow\\!\\!V &\\longrightarrow \\bbR\\\\\t\t\t \\nonumber\n\t\t\t\tV' &\\longmapsto \\ld|_{V'}(\\delta^i(\\A)_{V'})=\\ld(\\delta^i(\\A)_{V'}).\n\\end{align}\nSimilarly, using the approximations $\\delta^o(\\A)_{V'}$ to $\\A$ from above, we obtain an order-reversing function\n\\begin{align}\t\t\t\\label{nu_ld}\n\t\t\t\t\\nu_\\ld:\\downarrow\\!\\!V &\\longrightarrow \\bbR\\\\\t\t\t\\nonumber\n\t\t\t\tV' &\\longmapsto \\ld|_{V'}(\\delta^o(\\A)_{V'})=\\ld(\\delta^o(\\A)_{V'}).\n\\end{align}\nThis allows us to define the desired function:\n\\begin{align}\t\t\t\\label{dasB(A)_V}\n\t\t\t\\dasB{\\A}_V:\\Sig_V &\\longrightarrow \\Rlr_V\\\\\t\t\t\\nonumber\n\t\t\t\\ld &\\longmapsto (\\mu_\\ld,\\nu_\\ld).\n\\end{align}\n\nObviously, we obtain one function $\\dasB{\\A}_V$ for each context $V\\in\\VN$. The condition for these functions to form a natural transformation is the following: whenever $V'\\subset V$, one must have $\\dasB{\\A}_V(\\ld)|_{V'}=\\dasB{\\A}_{V'}(\\ld|_{V'})$. In fact, this equality follows trivially from the definition. We thus have arrived at an arrow\n\\begin{equation}\n\t\t\t\\dasB{\\A}:\\Sig\\longrightarrow\\Rlr\n\\end{equation}\nrepresenting a physical quantity $A$. The arrow $\\dasB{\\A}$ is called the \\emph{daseinisation of $\\A$}. This arrow is the analogue of the function $f_A:\\mc{S}\\rightarrow\\bbR$ from the state space to the real numbers representing the physical quantity $A$ in classical physics.\n\n\\paragraph{Pure states and assignment of `values' to physical quantities.} Up to now, we have always argued with elements $\\ld\\in\\Sig_V$. This is a local argument, and we clearly need a global representation of states. Let $\\psi$ be a unit vector in Hilbert space, identified with the pure state $\\bra\\psi\\_\\ket\\psi:\\N\\rightarrow\\bbC$ as usual. In the topos approach, such a pure state is represented by the subobject $\\ps{\\delta(\\P_\\psi)}$ of the spectral presheaf $\\Sig$, i.e., the daseinisation of the projection $\\P_\\psi$ onto the line $\\bbC\\psi$ in Hilbert space. Details can be found in \\cite{DI08}.\n\nThe subobject $\\ps\\wpsi:=\\ps{\\delta(\\P_\\psi)}$ is called the \\emph{pseudo-state} associated to $\\psi$. It is the analogue of a point $s\\in\\mc{S}$ of the state space of a classical system. Importantly, the pseudo-state is not a global element of the presheaf $\\Sig$. Such a global element would be the closest analogue of a point in a set, but as Isham and Butterfield observed \\cite{IB98}, the spectral presheaf $\\Sig$ has no global elements at all -- a fact that is equivalent to the Kochen-Specker theorem. Nonetheless, subobjects of $\\Sig$ of the form $\\ps\\wpsi=\\ps{\\delta(\\P_\\psi)}$ are minimal subobjects in an appropriate sense and as such are as close to points as possible (see \\cite{DI08,Doe08}).\n\nThe `value' of a physical quantity $A$, represented by an arrow $\\dasB{\\A}$, in a state described by $\\ps\\wpsi$ then is\n\\begin{equation}\n            \\dasB{\\A}(\\ps\\wpsi).\n\\end{equation}\nThis of course is the analogue of the expression $f_A(s)$ in classical physics, where $f_A$ is the real-valued function on the state space $\\mc{S}$ of the system representing the physical quantity $A$, and $s\\in\\mc{S}$ is the state of the system. \n\nThe expression $\\dasB{\\A}(\\ps\\wpsi)$ turns out to describe a considerably more complicated object than the classical expression $f_A(s)$. One can easily show that $\\dasB{\\A}(\\ps\\wpsi)$ is a subobject of the presheaf $\\Rlr$ of `values', see section 8.5 in \\cite{DI08}. Concretely, it is given at each context $V\\in\\VN$ by\n\\begin{equation}\n            (\\dasB{\\A}(\\ps\\wpsi))_V=\\dasB{\\A}_V(\\ps\\wpsi_V)=\\{\\dasB{\\A}_V(\\ld) \\mid \\ld\\in\\ps\\wpsi_V\\}.\n\\end{equation}\nAs we saw above, each $\\dasB{\\A}_V(\\ld)$ is a pair of functions $(\\mu_\\ld,\\nu_\\ld)$ from the set $\\downarrow\\!\\!V$ to the reals, and more specifically to the spectrum $\\spec{\\A}$ of the operator $\\A$. The function $\\mu_\\ld$ is order-preserving, so it takes smaller and smaller values as one goes down from $V$ to smaller subalgebras $V',V'',...$, while $\\nu_\\ld$ is order-reversing and takes larger and larger values. Together, they determine one real interval $[\\mu_\\ld(V'),\\nu_\\ld(V')]$ for every context $V'\\in\\downarrow\\!\\!V$. Every $\\ld\\in\\ps\\wpsi_V$ determines one such sequence of nested intervals. The `value' of the physical quantity $A$ in the given state is the collection of all these sequences of nested intervals for all the different contexts $V\\in\\VN$.\n\nWe conjecture that the usual expectation value $\\bra\\psi\\A\\ket\\psi$ of the physical quantity $A$, which of course is a single real number, can be calculated from the `value' $\\dasB{\\A}(\\ps\\wpsi)$. It is easy to see that each real interval showing up in (the local expressions of) $\\dasB{\\A}(\\ps\\wpsi)$ contains the expectation value $\\bra\\psi\\A\\ket\\psi$.\n\nIn \\cite{Doe08}, it was shown that arbitrary quantum states (not just pure ones) can be represented by probability measures on the spectral presheaf. It was also shown how the expectation values of physical quantities can be calculated using these measures.\n\nInstead of closing this section by showing how the relevant structures look like concretely for our standard example $\\N=\\mc{B}(\\bbC^3)$, we will introduce an efficient way of calculating the approximations $\\dastoi{V}{A}$ and $\\dastoo{V}{A}$ first, which works uniformly for all contexts $V\\in\\VN$. This will simplify the calculation of the arrow $\\dasB{\\A}$ significantly.\n\n\\subsection{Antonymous and observable functions}\nOur aim is to define two functions $g_\\A,f_\\A$ from which we can calculate the approximations $\\dastoi{V}{A}$ resp. $\\dastoo{V}{A}$ to $\\A$ for all contexts $V\\in\\VN$. More precisely, we will describe the Gel'fand transforms of the operators $\\dastoi{V}{A}$ and $\\dastoo{V}{A}$ (for all $V$). These are functions from the Gel'fand spectrum $\\Si_V$ to the reals. The main idea, due to de Groote, is to use the fact that each $\\ld\\in\\Si_V$ corresponds to a maximal \\emph{filter} in the projection lattice $\\PV$ of $V$. Hence, we will consider the Gel'fand transforms as functions on filters, and by `extending' each filter in $\\PV$ to a filter in $\\PN$ (non-maximal in general), we will arrive at functions on filters in $\\PN$, the projection lattice of the non-abelian von Neumann algebra $\\N$. First, we collect a few basic definitions and facts about filters:\n\n\\paragraph{Filters.} Let $\\mathbb{L}$ be a lattice with zero element $0$. A subset $F$ of elements of $\\mathbb{L}$ is a \\emph{(proper) filter} (or \\emph{(proper) dual ideal}) if (i) $0\\notin F$, (ii) $a,b\\in F$ implies $a\\wedge b\\in F$ and (iii) $a\\in F$ and $b\\geq a$ imply $b\\in F$. When we speak of filters, we always mean proper filters. We write $\\mc{F}(\\mathbb{L})$ for the set of filters in a lattice $\\mathbb{L}$. If $\\N$ is a von Neumann algebra, then we write $\\mc{F}(\\N)$ for $\\mc{F}(\\PN)$.\n\nA subset $B$ of elements of $\\mathbb{L}$ is a \\emph{filter base} if (i) $0\\neq B$ and (ii) for all $a,b\\in B$, there exists a $c\\in B$ such that $c\\leq a\\wedge b$.\n\nMaximal filters and maximal filter bases in a lattice $\\mathbb{L}$ with $0$ exist by Zorn's lemma, and each maximal filter is a maximal filter base and vice versa. The maximal filters in the projection lattice of a von Neumann algebra $\\N$ are denoted by $\\mc{Q}(\\N)$. Following de Groote \\cite{deG05c}, one can equip this set with a topology and consider its relations to well-known constructions (in particular, Gel'fand spectra of abelian von  Neumann algebras). Here, we will not consider the topological aspects.\n\nIf $V$ is an abelian subalgebra of a von Neumann algebra $\\N$, then the projections in $V$ form a distributive lattice $\\PV$, that is, for all $\\P,\\Q,\\hat{R}\\in\\PV$ we have\n\\begin{eqnarray}\n\t\t\t\\P\\wedge(\\Q\\vee\\hat{R})  &=& (\\P\\wedge\\Q)\\vee(\\P\\wedge\\hat{R}),\\\\\n\t\t\t\\P\\vee(\\Q\\wedge\\hat{R})  &=& (\\P\\vee\\hat{Q})\\wedge(\\P\\vee\\hat{R}).\n\\end{eqnarray}\nThe projection lattice of a von Neumann algebra is distributive if and only if the algebra is abelian.\n\nA maximal filter in a complemented, distributive lattice $\\mathbb{L}$ is called an \\emph{ultrafilter}, hence $\\mc{Q}(\\mathbb{L})$ is the space of ultrafilters in $\\mathbb{L}$. We will not use the notion `ultrafilter' for maximal filters in \\emph{non}-distributive lattices like the projection lattices of non-abelian von Neumann algebras.\n\nThe characterising property of an ultrafilter $U$ in a distributive, complemented lattice $\\mathbb{L}$ is that for each element $a\\in\\mathbb{L}$, either $a\\in U$ or $a^{c}\\in U$, where $a^{c}$ denotes the complement of $a$. This can easily be seen: a complemented lattice has a maximal element $1$, and we have $a\\vee a^{c}=1$ by definition. Let us assume that $U$ is an ultrafilter, but $a\\notin U$ and $a^{c}\\notin U$. In particular, $a\\notin U$ means that there is some $b\\in U$ such that $b\\wedge a=0$. Using distributivity of the lattice $\\mathbb{L}$, we get\n\\begin{equation}\n\t\t\tb=b\\wedge(a\\vee a^{c})=(b\\wedge a)\\vee(b\\wedge a^{c})=b\\wedge a^{c},\n\\end{equation}\nso $b\\leq a^{c}$. Since $b\\in U$ and $U$ is a filter, this implies $a^{c}\\in U$, contradicting our assumption. If $\\mathbb{L}$ is the projection lattice $\\PV$ of an abelian von Neumann algebra $V$, then the maximal element is the identity operator $\\hat{1}$ and the complement of a projection is given as $\\P^{c}:=\\hat{1}-\\P$. Each ultrafilter $q\\in\\mc{Q}(V)$ hence contains either $\\P$ or $\\hat{1}-\\P$ for all $\\P\\in\\PV$.\n\n\\paragraph{Elements of the Gel'fand spectrum and filters.} We already saw that for all projections $\\P\\in\\PV$ and all elements $\\ld$ of the Gel'fand spectrum $\\Si_V$ of $V$, it holds that $\\ld(\\P)\\in\\{0,1\\}$. Given $\\ld\\in\\Si_V$, it is easy to construct a maximal filter $F_\\ld$ in the projection lattice $\\PV$ of $V$: let\n\\begin{equation}\n\t\t\tF_\\ld:=\\{\\P\\in\\PV\\mid\\ld(\\P)=1\\}.\n\\end{equation}\nIt is clear that $\\P\\in F_\\ld$ implies $\\Q\\in F_\\ld$ for all projections $\\Q\\geq\\P$. Let $\\P_{1},\\P_{2}\\in F_{\\ld}$. From the fact that $\\ld$ is a multiplicative state of $V$, we obtain $\\ld(\\P_{1}\\wedge\\P_{2})=\\ld(\\P_{1}\\P_{2})=\\ld(\\P_{1})\\ld(\\P_{2})=1$, so $\\P_{1}\\wedge\\P_{2}\\in F_{\\ld}$. Finally, we have $1=\\ld(\\hat 1)=\\ld(\\P+\\hat 1-\\P)=\\ld(\\P)+\\ld(\\hat 1-\\P)$ for all $\\P\\in\\PV$, so either $\\P\\in F_\\ld$ or $\\hat 1-\\P\\in F_\\ld$ for all $\\P$, which shows that $F_{\\ld}$ is a maximal filter in $\\PV$, indeed. It is straightforward to see that the mapping\n\\begin{align}\t\t\t\\label{Eq_MaxFilterFromld}\n\t\t\t\\beta:\\Si_V &\\longrightarrow \\mathcal{Q}(V)\\\\\t\t\t\\nonumber\n\t\t\t\\ld &\\longmapsto F_\\ld\n\\end{align}\nfrom the Gel'fand spectrum of $V$ to the set $\\mathcal{Q}(V)$ of maximal filters in $\\mathcal{P}(V)$ is injective. This is the first step in the construction of a homeomorphism between these two spaces. De Groote has shown the existence of such a homeomorphism in Thm. 3.2 of \\cite{deG05c}.\n\nGiven a filter $F$ in the projection lattice $\\PV$ of some context $V$, one can `extend' it to a filter in the projection lattice $\\PN$ of the full non-abelian algebra $\\N$ of physical quantities. One simply defines the \\textit{$\\N$-cone over $F$} as\n\\begin{equation}\n\t\t\t\\mc C_{\\N}(F):=\\uparrow\\!\\!F=\\{\\Q\\in\\PN \\mid \\exists\\P\\in F:\\P\\leq\\Q\\}.\n\\end{equation}\nThis is the smallest filter in $\\PN$ containing $F$.\n\nWe need the following technical, but elementary lemma:\n\\begin{lemma}\t\t\t\\label{L_deltaT^-1(D)=Cone(D)}\nLet $\\N$ be a von Neumann algebra, and let $\\mc{T}$ be a von Neumann subalgebra such that $\\hat 1_{\\mc{T}}=\\hat 1_\\N$. Let \n\\begin{align}\n\t\t\t\\delta_{\\mc{T}}^{i}:\\PN &\\longrightarrow \\mc{P(T)}\\\\\t\t\t\\nonumber\n\t\t\t\\P &\\longmapsto \\dastoi{\\mc{T}}{P}:=\\bigvee\\{\\Q\\in\\mc{P(T)} \\mid \\Q\\leq\\P\\}.\n\\end{align}\t\t\t\nThen, for all filters $F\\in\\mc{F}(\\mc{T})$,\n\\begin{equation}\n\t\t\t(\\delta_{\\mc{T}}^{i})^{-1}(F)=\\mc{C}_{\\N}(F).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nIf $\\Q\\in F\\subset\\mc{P(T)}$, then $(\\delta_{\\mc{T}}^{i})^{-1}(\\Q)=\\{\\P\\in\\PN\\mid\\delta^{i}(\\hat{P})_{\\mc{T}}=\\Q\\}$. Let $\\P\\in\\PN$ be such that there is a $\\Q\\in F$ with $\\Q\\leq\\P$, i.e., $\\P\\in\\mc{C}_{\\N}(F)$. Then $\\delta^{i}(\\hat{P})_{\\mc{T}}\\geq\\Q$, which implies $\\delta^{i}(\\hat{P})_{\\mc{T}}\\in F$, since $F$ is a filter in $\\mc{P(T)}$. This shows that $\\mc{C}_{\\N}(F)\\subseteq(\\delta_{\\mc{T}}^{i})^{-1}(F)$. Now let $\\P\\in\\PN$ be such that there is no $\\Q\\in F$ with $\\Q\\leq\\P$. Since $\\delta^{i}(\\P)_{\\mc{T}}\\leq\\P$, there also is no $\\hat{Q}\\in F$ with $\\Q\\leq\\delta^{i}(\\P)_{\\mc{T}}$, so $\\P\\notin(\\delta_{\\mc{T}}^{i})^{-1}(F)$. This shows that $(\\delta_{\\mc{T}}^{i})^{-1}(F)\\subseteq\\mc{C}_{\\N}(F)$.\n\\end{proof}\n\n\\paragraph{Definition of antonymous and observable functions.}\nLet $\\N$ be a von Neumann algebra, and let $\\A$ be a self-adjoint operator in $\\A$. As before, we denote the spectral family of $\\A$ as $(\\hat E^\\A_r)_{r\\in\\bbR}=\\hat E^\\A$.\n\\begin{definition}\nThe \\emph{antonymous function of $\\A$} is defined as\n\\begin{align}\n\t\t\tg_\\A:\\mc F (\\N) &\\longrightarrow \\spec{\\A}\\\\\n\t\t\tF &\\longmapsto \\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^\\A_r\\in F\\}.\n\\end{align}\nThe \\emph{observable function of $\\A$} is\n\\begin{align}\n\t\t\tf_\\A:\\mc F (\\N) &\\longrightarrow \\spec{\\A}\\\\\n\t\t\tF &\\longmapsto \\on{inf}\\{r\\in\\bbR \\mid \\hat E^\\A_r\\in F\\}.\n\\end{align}\n\\end{definition}\nIt is straightforward to see that the range of these functions indeed is the spectrum $\\spec{\\A}$ of $\\A$.\n\nWe observe that the mapping described in \\eq{dastoiVA} can be generalised: when approximating a self-adjoint operator $\\A$ with respect to the spectral order in a von Neumann subalgebra of $\\N$, then one can use an arbitrary, not necessarily abelian subalgebra $\\mc T$ of the non-abelian algebra $\\N$. The only condition is that the unit elements in $\\N$ and $\\mc T$ coincide. We define\n\\begin{align}\n\t\t\t\\delta^i_{\\mc T}:\\N_\\sa &\\longrightarrow \\mc T_\\sa\\\\\t\t\\nonumber\n\t\t\t\\A &\\longmapsto \\dastoi{\\mc T}{A}=\\bigvee\\{\\hat B\\in\\mc T_\\sa \\mid \\hat B\\leq_s\\A\\}.\n\\end{align}\nAnalogously, generalising \\eq{dastooVA}\n\\begin{align}\n\t\t\t\\delta^o_{\\mc T}:\\N_\\sa &\\longrightarrow \\mc T_\\sa\\\\\t\t\\nonumber\n\t\t\t\\A &\\longmapsto \\dastoo{\\mc T}{A}=\\bigwedge\\{\\hat B\\in\\mc T_\\sa \\mid \\hat B\\geq_s\\A\\}.\n\\end{align}\nThe following proposition is central:\n\\begin{proposition}\t\t\t\\label{P_g_AEncodesAllg_delta^i(A)}\nLet $\\A\\in\\N_{sa}$. For all von Neumann subalgebras $\\mc{T}\\subseteq\\N$ such that $\\hat 1_{\\mc{T}}=\\hat 1_\\N$ and all filters $F\\in\\mc{F(T)}$, we have\n\\begin{equation}\n\t\t\tg_{\\dastoi{\\mc{T}}{A}}(F)=g_{\\A}(\\mc{C}_{\\N}(F)).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nWe hav\n\\begin{eqnarray*}\n\t\t\tg_{\\delta^{i}(\\A)_{\\mc{T}}}(F) &=&  \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\hat{E}_{r}^{\\dastoi{\\mc{T}}{A}}\\in F\\}\\\\\n\t\t\t&\\stackrel{\\eq{Eq_SpecFamInnDas}}{=}&  \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\bigwedge_{\\mu>r}\\delta^{o}(\\hat{E}_{\\mu}^{\\hat A})_{\\mc{T}}\\in F\\}\\\\\n\t\t\t&=&  \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\delta^{o}(\\hat{E}_{r}^{\\hat A})_{\\mc{T}}\\in F\\}\\\\\n\t\t\t&=&  \\sup\\{r\\in\\bbR \\mid \\delta^{i}(\\hat{1}-\\hat{E}_{r}^{\\hat A})_{\\mc{T}}\\in F\\}\\\\\n\t\t\t&=&  \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\hat{E}_{r}^{\\hat A}\\in(\\delta_{\\mc{T}}^{i})^{-1}(F)\\}\\\\\n\t\t\t&\\stackrel{\\text{Lemma \\ref{L_deltaT^-1(D)=Cone(D)}}}{=}&  \\sup\\{r\\in\\bbR \\mid \\hat{1}-\\hat{E}_{r}^{\\hat A}\\in\\mc{C}_{\\N}(F)\\}\\\\\n\t\t\t&=&  g_{\\A}(\\mc{C}_{\\N}(F))\n\\end{eqnarray*}\n\\end{proof}\n\nThis shows that the antonymous function $g_{\\A}:\\mc{F}(\\N)\\rightarrow\\spec{\\A}$ of $\\A$ encodes in a simple way \\emph{all} the antonymous functions $g_{\\delta^{i}(\\A)_{\\mc{T}}}:\\mc{F(T)}\\rightarrow\\spec{\\delta^{i}(\\A)}_{\\mc{T}}$ corresponding to the approximations $\\delta^{i}(\\A)_{\\mc{T}}$ (from below in the spectral order) to $\\A$ to von Neumann subalgebras $\\mc{T}$ of $\\N$. If, in particular, $\\mc{T}$\nis an abelian subalgebra, then the set $\\mc{Q(T)}$ of \\emph{maximal} filters in $\\mc{D(T)}$ can be identified with the Gel'fand spectrum of $\\mc{T}$ (see \\cite{deG05b}), and $g_{\\delta^{i}(\\A)_{\\mc{T}}}|_{\\mc{Q(T)}}$ can be identified with the Gel'fand transform of\n$\\delta^{i}(\\A)_{\\mc{T}}$. Thus we get\n\n\\begin{corollary}\nThe antonymous function $g_{\\A}:\\mc{F}(\\N)\\rightarrow\\spec{\\A}$ encodes all the Gel'fand transforms of the inner daseinised operators of the form $\\dastoi{V}{A}$, where $V\\in\\VN$ is an abelian von Neumann subalgebra of $\\N$ (such that $\\hat 1_V=\\hat 1_\\N$). Concretely, if $\\mc{Q}(V)$ is the space of maximal filters in $\\PV$ and $\\Si_V$ is the Gel'fand spectrum of $V$, then we have an identification $\\beta:\\Sig_V\\rightarrow\\mc{Q}(V)$, $\\ld\\mapsto F_\\ld$ (see \\eq{Eq_MaxFilterFromld}). Let $\\fu{\\dastoi{V}{A}}:\\Si_V\\rightarrow\\spec{\\dastoi{V}{A}}$ be the Gel'fand transform of $\\dastoi{V}{A}$. We can identify $\\fu{\\dastoi{V}{A}}$ with $g_{\\dastoi{V}{A}}|_{\\mc{Q}(V)}$, and hence, from Prop. \\ref{P_g_AEncodesAllg_delta^i(A)},\n\\begin{equation}\t\t\t\\label{EqInnDasFromAnton}\n\t\t\t\\fu{\\dastoi{V}{A}}(\\ld)=g_{\\dastoi{V}{A}}(F_\\ld)=g_{\\A}(\\mc{C}_{\\N}(F_\\ld)).\n\\end{equation}\n\\end{corollary}\n\nNot surprisingly, there is a similar result for observable functions and outer daseinisation. This result was first proved in a similar form by de Groote in \\cite{deG07}.\n\n\\begin{proposition}\t\t\t\\label{P_f_AEncodesAllf_delta^o(A)}\nLet $\\A\\in\\N_{sa}$. For all von Neumann subalgebras $\\mc{T}\\subseteq\\N$ such that $\\hat 1_{\\mc{T}}=\\hat 1_\\N$ and all\nfilters $F\\in\\mc{F(T)}$, we have\n\\begin{equation}\n\t\t\tf_{\\dastoo{\\mc{T}}{A}}(F)=f_{\\A}(\\mc{C}_{\\N}(F)).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nWe have\n\\begin{eqnarray}\n\t\t\tf_{\\dastoo{\\mc{T}}{A}}(F) &=& \\inf\\{r\\in\\bbR \\mid \\hat{E}_{r}^{\\dastoo{\\mc{T}}{A}}\\in F\\}\\\\\n\t\t\t&\\stackrel{\\eq{Eq_SpecFamOutDas}}{=}& \\inf\\{r\\in\\bbR \\mid \\delta^{i}(\\hat{E}_{r}^{\\hat A})_{\\mc{T}}\\in F\\}\\\\\n\t\t\t&=& \\inf\\{r\\in\\bbR \\mid \\hat{E}_{r}^{\\hat A}\\in(\\delta_{\\mc{T}}^{i})^{-1}(F)\\}\\\\\n\t\t\t&\\stackrel{\\text{Lemma \\ref{L_deltaT^-1(D)=Cone(D)}}}{=}& \\inf\\{r\\in\\bbR \\mid \\hat{E}_{r}^{\\hat A}\\in\\mc{C}_{\\N}(F)\\}\\\\\n\t\t\t&=& f_{\\A}(\\mc{C}_{\\N}(F)).\n\\end{eqnarray}\n\\end{proof}\n\nThis shows that the observable function $f_\\A:\\mc{F}(\\N)\\rightarrow\\spec{\\A}$ of $\\A$ encodes in a simple way \\emph{all} the observable functions $f_{\\dastoo{\\mc{T}}{A}}:\\mc{F}(\\N)\\rightarrow\\spec{\\dastoo{\\mc{T}}{A}}$ corresponding to approximations $\\dastoo{\\mc{T}}{A}$ (from above in the spectral order) to $\\A$ to von Neumann subalgebras $\\mc{T}$ of $\\N$. If, in particular, $\\mc{T}=V$ is an abelian subalgebra, then we get\n\\begin{corollary}\nThe observable function $f_{\\A}:\\mc{F}(\\N)\\rightarrow\\spec{\\A}$ encodes all the Gel'fand transforms of the outer daseinised operators of the form $\\dastoo{V}{A}$, where $V\\in\\VN$ is an abelian von Neumann subalgebra of $\\N$ (such that $\\hat 1_V=\\hat 1_\\N$). Using the identification $\\beta:\\Sig_V\\rightarrow\\mc{Q}(V)$, $\\ld\\mapsto F_{\\ld}$ between the Gel'fand and the space of maximal filters (see \\eq{Eq_MaxFilterFromld}), we can identify the Gel'fand transform $\\fu{\\dastoo{V}{A}:\\Sig_V\\rightarrow\n\\spec{\\dastoo{V}{A}}}$ of $\\dastoo{V}{A}$ with $f_{\\dastoo{V}{A}}|_{\\mc{Q(T)}}$, and hence, from Prop. \\ref{P_f_AEncodesAllf_delta^o(A)},\n\\begin{equation}\n\t\t\t\\fu{\\dastoo{V}{A}}(\\ld)=f_{\\dastoo{V}{A}}(F_{\\ld})=f_\\A(\\mc{C}_{\\N}(F_{\\ld})).\n\\end{equation}\n\\end{corollary}\n\n\\paragraph{Vector states from elements of Gel'fand spectra.} As emphasised in section \\ref{SubS_PhysQuantitiesAsArrows}, an element of $\\ld$ of the Gel'fand spectrum $\\Sig_V$ of some context $V$ is only a local state: it is a pure state of the abelian subalgebra $V$, but it is not a state of the full non-abelian algebra $\\N$ of physical quantities. We will now show how certain local states $\\ld$ can be extended to global states, i.e., states of $\\N$.\n\nLet $V$ be a context that contains at least one rank-$1$ projection.\\footnote{Depending on whether the von Neumann algebra $\\N$ contains rank-$1$ projections, there may or may not be such contexts. For the case $\\N=\\BH$, there of course are many rank-$1$ projections and hence contexts $V$ of the required form.} Let $\\P$ be such a projection, and let $\\psi\\in\\Hi$ be a unit vector such that $\\P$ is the projection onto the ray $\\bbC\\psi$. ($\\psi$ is fixed up to a phase.) We write $\\P=\\P_\\psi$. Then\n\\begin{equation}\t\t\t\\label{F_P_psi}\n\t\t\tF_{\\P_\\psi}:=\\{\\Q\\in\\PV \\mid \\Q\\geq\\P_\\psi\\}\n\\end{equation}\nclearly is an ultrafilter in $\\PV$. It thus corresponds to some element $\\ld_\\psi$ of the Gel'fand spectrum $\\Sig_V$ of $V$. Being an element of the Gel'fand spectrum, $\\ld_\\psi$ is a pure state of $V$. The cone\n\\begin{equation}\n\t\t\t\\mc C_{\\N}(F_{\\P_\\psi})=\\{\\Q\\in\\PN \\mid \\Q\\geq\\P_\\psi\\}\n\\end{equation}\nis a maximal filter in $\\PN$. It contains all projections in $\\N$ that represent propositions ``$\\Ain\\De$'' that are (totally) true in the vector state $\\psi$ on $\\N$.\\footnote{Here, we again use the usual identification between unit vectors and vector states, $\\psi\\mapsto\\bra\\psi\\_\\ket\\psi$.} This vector state is uniquely determined by the cone $\\mc C_{\\N}(F_{\\P_\\psi})$. The `local state' $\\ld_\\psi$ on the context $V\\subset\\N$ can thus be extended to a `global' vector state. On the level of filters, this extension corresponds to the cone construction.\n\nFor a finite-dimensional Hilbert space $\\Hi$, one necessarily has $\\N=\\BH$. Let $V$ be an arbitrary maximal context, and let $\\ld\\in\\Sig_V$. Then $\\ld$ is of the form $\\ld=\\ld_{\\P_\\psi}$ for some unit vector $\\psi\\in\\Hi$ and corresponding rank-$1$ projection $\\P_\\psi$. Hence, every such $\\ld_{\\P_\\psi}$ can be extended to a vector state $\\psi$ on the whole of $\\BH$.\n\n\\paragraph{The eigenstate-eigenvalue link.} We will now show that the arrow $\\dasB{\\A}$ constructed from a self-adjoint operator $\\A\\in\\N_{\\sa}$ preserves the eigenstate-eigenvalue link in a suitable sense. We employ the relation between ultrafilters in $\\PV$ of the form $F_{\\P_\\psi}$ (see equation \\eq{F_P_psi}) and maximal filters in $\\PN$ established in the previous paragraph, but `read it backwards'.\n\nLet $\\A$ be some self-adjoint operator, and let $\\psi$ be an eigenstate of $\\A$ with eigenvalue $a$. Let $V$ be a context that contains $\\A$, and let $\\P_\\psi$ be the projection determined by $\\psi$, i.e., the projection onto the one-dimensional subspace (ray) $\\bbC\\psi$ of Hilbert space. Then $\\P_\\psi\\in V$ from the spectral theorem. Consider the maximal filter\n\\begin{equation}\n\t\t\tF:=\\{\\Q\\in\\PN \\mid \\Q\\geq\\P_\\psi\\}\n\\end{equation}\nin $\\PN$ determined by $\\P_\\psi$. Clearly, $F\\cap\\PV=\\{\\Q\\in\\PV \\mid \\Q\\geq\\P_\\psi\\}$ is an ultrafilter in $\\PV$, namely the ultrafilter $F_{\\P_\\psi}$ from equation \\eq{F_P_psi}, and $F=\\mc C_{\\N}(F_{\\P_\\psi})$. Let $\\ld_\\psi$ be the element of $\\Sig_V$ (i.e., local state of $V$) determined by $F_{\\P_\\psi}$. The state $\\ld_\\psi$ is nothing but the restriction of the vector state $\\bra\\psi\\_\\ket\\psi$ (on $\\N$) to the context $V$, and hence \n\\begin{equation}\n\t\t\t\\bra\\psi\\A\\ket\\psi=\\ld_\\psi(\\A)=a.\n\\end{equation}\nSince $\\A\\in V_{\\sa}$, we have $\\dastoi{V}{A}=\\dastoo{V}{A}=\\A$. This implies that \n\\begin{align}\t\t\t\\nonumber\n\t\t\t\\dasB{\\A}_V(\\ld_\\psi)(V) &= [g_{\\dastoi{V}{A}}(\\ld_\\psi),f_{\\dastoo{V}{A}}(\\ld_\\psi)]\\\\\t\t\t\\nonumber\n\t\t\t&= [g_\\A(\\ld_\\psi),f_\\A(\\ld_\\psi)]\\\\\n\t\t\t&= [\\fu{A}(\\ld_\\psi),\\fu{A}(\\ld_\\psi)]\\\\\t\t\\nonumber\n\t\t\t&= [\\ld_\\psi(\\A),\\ld_\\psi(\\A)]\\\\\t\t\\nonumber\n\t\t\t&= [a,a],\n\\end{align}\ni.e., the arrow $\\dasB{\\A}$ delivers the (interval only containing the) eigenvalue $a$ for an eigenstate $\\psi$ (resp. the corresponding $\\ld_\\psi$) at a context $V$ which actually contains $\\A$. In this sense, the eigenstate-eigenvalue link is preserved, and locally at $V$ the value of $\\A$ actually becomes a single real number as expected.\n\n\\begin{example}\t\t\t\\label{Ex_ValueOfS_z}\nWe return to our example, the spin-$1$ system. It is described by the algebra $\\mc{B}(\\bbC^3)$ of bounded operators on the three-dimensional Hilbert space $\\bbC^3$. In particular, the spin-$z$ operator is given by\n\\begin{equation}\n\t\t\t\\hat S_z=\\frac{1}{\\sqrt{2}}\n\t\t\t\\left(\\begin{array}\n\t\t\t\t\t\t[c]{ccc\n\t\t\t\t\t\t1 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & 0\\\\\n\t\t\t\t\t\t0 & 0 & -1\n\t\t\t\\end{array}\\right).\n\\end{equation}\nThis is the matrix expression for $\\hat S_z$ with respect to the basis $e_1,e_2,e_3$ of eigenvectors of $\\hat S_z$. Let $\\P_1,\\P_2,\\P_3$ be the corresponding rank-$1$ projections onto the eigenspaces. The algebra $V_{\\hat S_z}:=\\{\\P_1,\\P_2,P_3\\}''$ generated by these projections is a maximal abelian subalgebra of $\\mc{B}(\\bbC^3)$, and it is the only maximal abelian subalgebra containing $\\hat S_z$.\n\nLet us now consider how the approximations $\\dastoi{V}{S_z}$ and $\\dastoo{V}{S_z}$ of $\\hat S_z$ to other contexts $V$ look like. In order to do so, we first determine the antonymous and the observable function of $\\hat S_z$. The spectral family of $\\hat S_z$ is given by\n\\begin{equation}\t\t\t\\label{SpecFamOfSz}\n\t\t\t\\hat E^{\\hat S_z}_\\ld=\\left\\{\n\t\t\t\\begin{tabular}\n\t\t\t\t\t\t[c]{ll\n\t\t\t\t\t\t$\\hat 0$ & if $\\ld<-\\frac{1}{\\sqrt 2}$\\\\\n\t\t\t\t\t\t$\\P_3$ & if $-\\frac{1}{\\sqrt 2}\\leq\\ld<0$\\\\\n\t\t\t\t\t\t$\\P_3+\\P_2$ & if $0\\leq\\ld<\\frac{1}{\\sqrt 2}$\\\\\n\t\t\t\t\t\t$\\hat 1$ & if $\\ld\\geq \\frac{1}{\\sqrt 2}$.\n\t\t\t\\end{tabular}\n\t\t\t\\ \\right.\n\\end{equation}\nThe antonymous function $g_{\\hat S_z}$ of $\\hat S_z$ is\n\\begin{align}\n\t\t\tg_{\\hat S_z}:\\mc{F}(\\mc{B}(\\bbC^3)) &\\longrightarrow \\spec{\\hat S_z}\\\\    \\nonumber\n\t\t\tF &\\longmapsto \\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^{\\hat S_z}_r\\in F\\},\n\\end{align}\nand the observable function is\n\\begin{align}\n\t\t\tf_{\\hat S_z}:\\mc{F}(\\mc{B}(\\bbC^3)) &\\longrightarrow \\spec{\\hat S_z}\\\\    \\nonumber\n\t\t\tF &\\longmapsto \\on{inf}\\{r\\in\\bbR \\mid \\hat E^{\\hat S_z}_r\\in F\\}.\n\\end{align}\nLet $\\Q_1,\\Q_2,\\Q_3$ be three pairwise orthogonal rank-$1$ projections, and let $V=\\{\\Q_1,\\Q_2,\\Q_3\\}''\\in\\mc V (\\mc B (\\bbC^3))$ be the maximal context determined by them. Then the Gel'fand spectrum $\\Sig_V$ has three elements; let us denote them $\\ld_1,\\ld_2$ and $\\ld_3$ (where $\\ld_i(\\Q_j)=\\delta_{ij}$). Clearly, $\\Q_i$ is the smallest projection in $V$ such that $\\ld_i(\\Q_i)=1\\;(i=1,2,3)$. Each $\\ld_i$ defines a maximal filter in the projection lattice $\\PV$ of $V$:\n\\begin{equation}\n\t\t\tF_{\\ld_i}=\\{\\Q\\in\\PV \\mid \\Q\\geq\\Q_i\\}.\n\\end{equation}\nThe Gel'fand transform $\\fu{\\dastoi{V}{S_z}}$ of $\\dastoi{V}{S_z}$ is a function from the Gel'fand spectrum $\\Sig_V$ of $V$ to the spectrum of $\\hat S_z$. Equation \\eq{EqInnDasFromAnton} shows how to calculate this function from the antonymous function $g_{\\hat S_z}$:\n\\begin{equation}\n\t\t\t\t\\fu{\\dastoi{V}{S_z}}(\\ld_i)=g_{\\hat S_z}(\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i})),\n\\end{equation}\nwhere $\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i})$ is the cone over the filter $F_{\\ld_i}$, i.e.,\n\\begin{align}\t\t\t\\nonumber\n\t\t\t\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i}) \n\t\t\t&= \\{\\hat R\\in\\mc P (\\mc B(\\bbC^3)) \\mid \\exists \\Q\\in F_{\\ld_i}:\\hat R\\geq\\Q\\}\\\\\n\t\t\t&= \\{\\hat R\\in\\mc P (\\mc B(\\bbC^3)) \\mid \\hat R\\geq\\Q_i\\}.\n\\end{align}\nNow it is easy to actually calculate $\\fu{\\dastoi{V}{S_z}}$: for all $\\ld_i\\in\\Sig_V$,\n\\begin{align}\t\t\t\\label{deltai(Sz)FromAnton}\t\\nonumber\n\t\t\t\\fu{\\dastoi{V}{S_z}}(\\ld_i)\n\t\t\t&= \\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^{\\hat S_z}_r\\in\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i})\\}\\\\\n\t\t\t&= \\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^{\\hat S_z}_r\\geq\\Q_i\\}.\n\\end{align}\nAnalogously, we obtain $\\fu{\\dastoo{V}{S_z}}$: for all $\\ld_i$,\n\\begin{align}\t\t\t\\label{deltao(Sz)FromAnton}\t\\nonumber\n\t\t\t\\fu{\\dastoo{V}{S_z}}(\\ld_i)\n\t\t\t&= \\on{inf}\\{r\\in\\bbR \\mid \\hat E^{\\hat S_z}_r\\in\\mc C_{\\mc B (\\bbC^3)}(F_{\\ld_i})\\}\\\\\n\t\t\t&= \\on{inf}\\{r\\in\\bbR \\mid \\hat E^{\\hat S_z}_r\\geq\\Q_i\\}.\n\\end{align}\nUsing the expression \\eq{SpecFamOfSz} for the spectral family of $\\hat S_z$, we can see directly that the values $\\fu{\\dastoi{V}{S_z}}(\\ld_i),\\fu{\\dastoo{V}{S_z}}(\\ld_i)$ lie in the spectrum of $\\hat S_z$ as expected. A little less obviously, $\\fu{\\dastoi{V}{S_z}}(\\ld_i)\\leq\\fu{\\dastoo{V}{S_z}}(\\ld_i)$, so we can think of the pair of values as an interval $[\\fu{\\dastoi{V}{S_z}}(\\ld_i),\\fu{\\dastoo{V}{S_z}}(\\ld_i)]$.\n\nWe had assumed that $V=\\{\\Q_1,\\Q_2,\\Q_2\\}''$ is a \\emph{maximal} abelian subalgebra, but this is no actual restriction. A non-maximal subalgebra $V'$ has a Gel'fand spectrum $\\Sig_{V'}$ consisting of two elements. All arguments work analogously. The important point is that for each element $\\ld$ of the Gel'fand spectrum, there is a unique projection $\\Q$ in $V'$ corresponding to $\\ld$, given as the smallest projection in $V'$ such that $\\ld(\\Q)=1$.\n\nThis means that we can now write down explicitly the natural transformation $\\dasB{S_z}:\\Sig\\rightarrow\\Rlr$, the arrow in the presheaf topos representing the physical quantity `spin in $z$-direction', initially given by the self-adjoint operator $\\hat S_z$. For each context $V\\in\\mc V (\\mc B (\\bbC^3))$, we have a function\n\\begin{align}\t\t\t\\label{dasB(S_z)_V}\n\t\t\t\\dasB{S_z}_V:\\Sig_V &\\longrightarrow \\Rlr_V\\\\\t\t\t\\nonumber\n\t\t\t\\ld &\\longmapsto (\\mu_{\\ld},\\nu_{\\ld}),\n\\end{align}\ncompare \\eq{dasB(A)_V}. According to \\eq{mu_ld}, $\\mu_\\ld:\\downarrow\\!\\!V\\rightarrow\\bbR$ is given as\n\\begin{align}\n\t\t\t\t\\mu_\\ld:\\downarrow\\!\\!V &\\longrightarrow \\bbR\\\\\t\t\t \\nonumber\n\t\t\t\tV' &\\longmapsto \\ld(\\delta^i(\\hat S_z)_{V'})=\\fu{\\delta^i(\\hat S_z)_{V'}}(\\ld).\n\\end{align}\nLet $\\Q_{V'}\\in\\mc P (V')$ be the projection in $V'$ corresponding to $\\ld$, that is, the smallest projection in $V'$ such that $\\ld(\\Q_{V'})=1$. Note that this projection $\\Q_{V'}$ depends on $V'$. For each $V'\\in\\downarrow\\!\\!V$, the element $\\ld$ (originally an element of the Gel'fand spectrum $\\Sig_V$) is considered as an element of $\\Sig_{V'}$, given by the restriction $\\ld|_{V'}$ of the original $\\ld$ to the smaller algebra $V'$. Then \\eq{deltai(Sz)FromAnton} implies, for all $V'\\in\\downarrow\\!\\!V$,\n\\begin{equation}\n\t\t\t\t\\mu_\\ld(V')=\\on{sup}\\{r\\in\\bbR \\mid \\hat 1-\\hat E^{\\hat S_z}\\geq\\Q_{V'}\\}.\n\\end{equation}\nFrom \\eq{nu_ld}, we obtain\n\\begin{align}\n\t\t\t\t\\nu_\\ld:\\downarrow\\!\\!V &\\longrightarrow \\bbR\\\\\t\t\t\\nonumber\n\t\t\t\tV' &\\longmapsto \\ld(\\delta^o(\\hat S_z)_{V'})=\\fu{\\delta^o(\\hat S_z)_{V'}}(\\ld),\n\\end{align}\nand with \\eq{deltao(Sz)FromAnton}, we get for all $V'\\in\\downarrow\\!\\!V$,\n\\begin{equation}\n\t\t\t\t\\nu_\\ld(V')=\\on{inf}\\{r\\in\\bbR \\mid \\hat E^{\\hat S_z}\\geq\\Q_{V'}\\}.\n\\end{equation}\n\nWe finally want to calculate the `value' $\\dasB{\\hat S_z}(\\ps\\wpsi)$ of the physical quantity `spin in $z$-direction' in the (pseudo-)state $\\ps\\wpsi$. As described in section \\ref{SubS_PhysQuantitiesAsArrows}, $\\psi$ is a unit vector in the Hilbert space (resp. a vector state) and $\\ps\\wpsi=\\ps{\\delta(\\P_\\psi)}$ is the corresponding pseudo-state, a subobject of $\\Sig$. The `value' $\\dasB{\\hat S_z}(\\ps\\wpsi)$ is given at $V\\in\\mc V (\\mc B (\\bbC^3))$ as\n\\begin{equation}\n            (\\dasB{\\hat S_z}(\\ps\\wpsi))_V=\\dasB{\\hat S_z}_V(\\ps\\wpsi_V)\n            =\\{\\dasB{\\hat S_z}_V(\\ld) \\mid \\ld\\in\\ps\\wpsi_V\\}.\n\\end{equation}\nHere, $\\ps\\wpsi_V=\\alpha(\\delta^o(\\P_\\psi)_V)=S_{\\delta^o(\\P_\\psi)_V}$ is a (clopen) subset of $\\Sig_V$ (compare \\eq{alpha}). This means that for each context $V$, the state determines a collection $\\ps\\wpsi_V$ of elements of the Gel'fand spectrum $\\Sig_V$ of $V$. We then evaluate the component $\\dasB{\\hat S_z}_V$ of the arrow\/natural transformation representing spin-$z$, given by \\eq{dasB(S_z)_V} (and subsequent equations), on all the $\\ld\\in\\ps\\wpsi_V$ to obtain the component at $V$ of the `value' $\\dasB{\\hat S_z}(\\ps\\wpsi)$. Each $\\ld\\in\\ps\\wpsi$ gives a sequence of intervals; one interval for each $V'\\in\\downarrow\\!\\!V$ such that if $V''\\subset V'$, the interval at $V''$ contains the interval at $V'$.\n\\end{example}\n\nThe example can easily be generalised to other operators and higher dimensions. Other finite-dimensional Hilbert spaces present no further conceptual difficulty at all. Of course, infinite-dimensional Hilbert spaces bring a host of new technical challenges, but the main tools used in the calculation, the antonymous and observable functions ($g_\\A$ resp. $f_\\A$), are still available. Since they encode the approximations in the spectral order of an operator $\\A$ to \\emph{all} contexts $V\\in\\VN$, the natural transformation $\\dasB{\\A}$ corresponding to a self-adjoint operator $\\A$ can be written down efficiently, without the need to actually calculate the approximations to all contexts separately.\n\n\\section{Conclusion}\t\\label{S6}\nWe have shown how daseinisation relates central aspects of the standard Hilbert space formalism of quantum theory to the topos formalism. Daseinisation of projections gives subobjects of the spectral presheaf $\\Sig$. These subobjects form a Heyting algebra, and every pure state allows to assign truth-values to all propositions. The resulting new form of quantum logic is contextual, multi-valued and intuitionistic. For more details on the logical aspects, see \\cite{DI08,Doe07b,Doe09}.\n\nThe daseinisation of self-adjoint operators gives arrows from the state object $\\Sig$ to the presheaf $\\Rlr$ of `values'. Of course, the `value' $\\dasB{\\A}(\\ps\\wpsi)$ of a physical quantity in a pseudo-state $\\ps\\wpsi$ is considerably more complicated than the value of a physical quantity in classical physics, which is just a real number. The main point, though, is that in the topos approach all physical quantities \\emph{do} have (generalised) values in any given state -- something that clearly is not the case in ordinary quantum mechanics. Moreover, we have shown in Example \\ref{Ex_ValueOfS_z} how to calculate the `value' $\\dasB{\\hat S_z}(\\ps\\wpsi)$.\n\nRecently, Heunen, Landsman and Spitters suggested a closely related scheme using topoi in quantum theory (see \\cite{HLS09} as well as their contribution to this volume and references therein). All the basic ingredients are the same: a quantum system is described by an algebra of physical quantities, in their case a Rickart $C^*$-algebra, associated with this algebra is a spectral object whose subobjects represent propositions, and physical quantities are represented by arrows in a topos associated with the quantum system. The choice of topos is very similar to ours: as the base category, one considers all abelian subalgebras (i.e., contexts) of the algebra of physical quantities and orders them partially under inclusion. In our scheme, we choose the topos to be contravariant, $\\Set$-valued functors (called \\emph{presheaves}) over the context category, while Heunen et al. choose \\emph{co}variant functors. The use of covariant functors allows the construction of the spectral object as the topos-internal Gel'fand spectrum of a topos-internal abelian $C^*$-algebra canonically defined from the external non-abelian algebra of physical quantities. \n\nDespite the similarities, there are some important conceptual and interpretational differences between the original contravariant approach and the covariant approach. These differences and their physical consequences will be discussed in a forthcoming article \\cite{Doe09b}.\n\nSumming up, the topos approach provides a reformulation of quantum theory in a way that seemed impossible up to now due to powerful no-go theorems like the Kochen-Specker theorem. In spite of the Kochen-Specker theorem, there is a suitable notion of state `space' for a quantum system in analogy to classical physics: the spectral presheaf. \n\n\\paragraph{Acknowledgements.} I would like to thank Chris Isham for the great collaboration and his constant support. I am very grateful to Hans Halvorson for organising the very enjoyable \\textsl{Deep Beauty} meeting.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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length}}}}\n\n\\nc{\\Cliff}{{\\mathsf{Cliff}}}\n\\nc{\\Fl}{{\\mathsf{Fl}}}\n\\nc{\\Fib}{{\\mathsf{Fib}}}\n\\nc{\\Coh}{{\\mathsf{Coh}}}\n\\nc{\\FCoh}{{\\mathsf{FCoh}}}\n\n\\nc{\\reg}{{\\text{\\normalshape reg}}}\n\n\\nc{\\cplus}{{\\mathbf{C}_+}}\n\\nc{\\cminus}{{\\mathbf{C}_-}}\n\\nc{\\cthree}{{\\mathbf{C}_*}}\n\\nc{\\Qbar}{{\\bar{Q}}}\n\n\\nc{\\bh}{{\\bar{h}}}\n\\nc{\\bOmega}{{\\overline{\\Omega}}}\n\\nc\\tGr{\\widetilde{\\Gr}}\n\n\\nc{\\seq}[1]{\\stackrel{#1}{\\sim}}\n\\nc\\ogu{\\overline{G\/U}}\n\\nc\\chlam{\\check{\\lam}}\n\n\\nc\\St{\\operatorname{St}}\n\n\\renewcommand\\gg{\\Gr_G}\n\\nc\\uS{\\underline{S}}\n\\nc\\QM{\\mathcal{QM}}\n\\nc{\\chmu}{\\check{\\mu}}\n\n\\begin{document}\n\\title{Weyl modules and $q$-Whittaker functions}\n\\dedication{To the memory of Andrei Zelevinsky who taught us the beauty of\nsymmetric functions}\n\\author{Alexander Braverman\\inst{1} \\and Michael Finkelberg\\inst{2}}\n\\institute{Department of Mathematics, Brown University,\n151 Thayer St., Providence RI 02912, USA, \\email{braval@math.brown.edu} \\and\nIMU, IITP and National Research University\nHigher School of Economics\nDepartment of Mathematics, 20 Myasnitskaya st, Moscow 101000 Russia,\n\\email{fnklberg@gmail.com}}\n\\date{Received: \/ Revised version: }\n\\maketitle\n\\begin{abstract}\nLet $G$ be a semi-simple simply connected group over $\\CC$. Following \\cite{GLOI} we use the $q$-Toda integrable system obtained by quantum group version\nof the\nKostant-Whittaker reduction (cf. \\cite{Et} and \\cite{Sev}) to define the notion of $q$-Whittaker functions $\\Psi_{\\chlam}(q,z)$.\nThis is a family of invariant polynomials on the maximal torus $T\\subset G$ (here $z\\in T$) depending\non a dominant weight $\\check\\lam$ of $G$ whose coefficients are rational functions in a variable $q\\in \\CC^*$.\nFor a conjecturally the same (but a priori different) definition of the $q$-Toda system these functions were studied\nby B.~Ion in \\cite{Ion} and by I.~Cherednik in \\cite{C} (we shall denote the $q$-Whittaker functions from \\cite{C}\nby $\\Psi'_{\\chlam}(q,z)$). For $G=SL(N)$ these functions were extensively studied in \\cite{GLOI}-\\cite{GLOIII}.\n\nWe show that when $G$ is simply laced, the function $\\hatPsi_{\\check\\lam}(q,z)=\\Psi_{\\chlam}(q,z)\\cdot{\\prod\\limits_{i\\in I}\\prod\\limits_{r=1}^{\\langle\\alpha_i,\\check\\lambda\\rangle}(1-q^r)}$\n(here $I$ denotes the set of vertices of the Dynkin diagram\nof $G$) is equal to the character of a certain finite-dimensional $G[[\\st]]\\rtimes\\CC^*$-module $D(\\check\\lam)$\n(the Demazure module). When $G$ is not simply laced a twisted version of the above statement holds. This result is known for $\\Psi_{\\chlam}$ replaced by $\\Psi'_{\\chlam}$\n(cf. \\cite{San} and \\cite{Ion}); however our proofs are algebro-geometric (and rely on our previous work \\cite{BF11}) and thus they are completely\ndifferent from \\cite{San} and\n\\cite{Ion} (in particular, we give an apparently new algebro-geometric interpretation of the modules $D(\\chlam)$).\n\\subclass{19E08}\n\\end{abstract}\n\n\\sec{int}{Introduction}\n\\ssec{}{The $q$-Whittaker functions}Let $G$ be a semi-simple, simply connected group over $\\CC$ with Lie algebra\n$\\grg$; we choose a pair of opposite Borel subgroups $B, B_-$ of $G$ with unipotent radicals\n$U, U_-$; the intersection $B\\cap B_-$ is a maximal torus $T$ of $G$. It will be convenient for us to denote\nthe weight lattice of $T$ by $\\check \\Lam$ and the coweight lattice by $\\Lam$. In this paper\nwe study certain invariant polynomials $\\Psi_{\\chlam}(q,z)$ on $T$ (the invariance is with respect\nto the Weyl group $W$ of $G$). Here $z\\in T$, $q\\in \\CC^*$ and $\\check\\lam:T\\to \\CC^*$ is a dominant weight of $G$. The function\n $\\Psi_{\\chlam}(q,z)$ is a polynomial function of $z$ with coefficients which are rational functions of $q$ (in fact, later\n were are going to work with a certain modification $\\hatPsi_{\\chlam}(q,z)$ of $\\Psi_{\\chlam}(q,z)$ which will be polynomial in\n$q$).\n\nThe definition of $\\Psi_{\\chlam}(q,z)$ is as follows. Let $\\check{G}$ denote the Langlands dual\ngroup of $G$ with its maximal torus $\\check{T}$. In \\cite{Et} and \\cite{Sev} the authors define (by adapting the so called\nKostant-Whittaker reduction to the case of quantum groups) a homomorphism\n$\\calM:\\CC[T]^W\\to \\End_{\\CC(q)}\\CC(q)[\\check T]$ called the quantum difference Toda integrable system associated with\n$\\check G$.\nFor each $f\\in \\CC[T]^W$ the operator $\\calM_f:=\\calM(f)$ is indeed a difference operator: it is a $\\CC(q)$-linear combination of\nshift operators $\\bfT_{\\check\\beta}$ where $\\check\\beta\\in\\check\\Lam$ and\n$$\n\\bfT_{\\check\\beta}(F(x))=F(q^{\\check\\beta}x).\n$$\n\n\\noindent\n{\\bf Remark.} In principle the constructions of \\cite{Et} and \\cite{Sev} depend on a choice of orientation of the Dynkin diagram of\n$\\check{G}$; however one can deduce from the main result of \\cite{ferm} that the resulting homomorphism is independent of this choice.\n\n\\medskip\n\\noindent\nIn particular, the above operators can be restricted to operators acting in the space of functions on the lattice\n$\\check\\Lam$ by means\nof the embedding $\\check\\Lam\\hookrightarrow \\check T$ sending every $\\chlam$ to $q^{\\chlam}$. For any $f\\in \\CC[T]^W$\nwe shall\ndenote the corresponding operator by $\\calM_f^{\\on{lat}}$. The following conjecture should probably be not very difficult; however, at the moment we don't know how to prove it:\n\\conj{whit-exist-co}\n\\begin{enumerate}\n\\item\nThere exists a unique collection of $\\CC(q)$-valued polynomials $\\Psi_{\\chlam}(q,z)$ on $T$ satisfying the following properties:\n\na) $\\Psi_{\\chlam}(q,z)=0$ if $\\chlam$ is not dominant.\n\nb) $\\Psi_{0}(q,z)=1$.\n\nc) Let us consider all the functions $\\Psi_{\\chlam}(q,z)$ as one function $\\Psi(q,z):\\check \\Lam\\to \\CC(q)$ depending\non $z\\in  T$. Then for every $f\\in \\CC[T]^W$ we have\n$$\n\\calM^{\\on{lat}}_f(\\Psi(q,z))=f(z)\\Psi(q,z).\n$$\n\\item The polynomials $\\Psi_{\\chlam}(q,z)$ are $W$-invariant.\n\\end{enumerate}\n\\end{conjec}\nOf course, the second statement follows from the ``uniqueness\" part of the first.\n\nSome remarks about the literature are necessary here. First of all, \\refco{whit-exist-co} is easy for $G=SL(N)$. In this case, the functions\n$\\Psi_{\\chlam}(q,z)$ are extensively studied in \\cite{GLOI}-\\cite{GLOIII}. Second, for general $G$ there exists another definition of the\n$q$-Toda system using double affine Hecke algebras, studied for example in \\cite{C}. Since it is not clear to us\nhow to prove that the definition of $q$-Toda from \\cite{C} and the definition of \\cite{Et} and \\cite{Sev} are the same,\nwe shall denote the operators from \\cite{C} by $\\calM_f'$. It is easy to see\nthat $\\calM_f=\\calM_f'$ for $G=SL(N)$.\\footnote{In fact, as we are going to explain later, the results of this\npaper together with the results of \\cite{Ion} imply that $\\calM_f=\\calM_f'$\nfor any $G$, but we would like to have a more direct proof of this fact.}\nSimilarly we shall denote by $(\\calM^{\\on{lat}}_f)'$ their ``lattice'' version. Then it is shown in \\cite{C} that the existence part of\n\\refco{whit-exist-co} holds for any $G$ if the operators $\\calM^{\\on{lat}}_f$ are replaced by $(\\calM^{\\on{lat}}_f)'$.\nWe shall denote the corresponding polynomials by $\\Psi_{\\chlam}'(q,z)$.\n\n\nThe main result of this paper  will imply the following:\n\\th{whit-exist}\n\\begin{enumerate}\n\\item\nThere exists a collection of $W$-invariant polynomials $\\Psi_{\\chlam}(q,z)$ on $T$ with coefficients in $\\CC(q)$ satisfying a), b) and c) above.\n\\item\nLet $\\hatPsi_{\\check\\lam}(q,z)=\\Psi_{\\chlam}(q,z)\\cdot{\\prod\\limits_{i\\in I}\\prod\\limits_{r=1}^{\\langle\\alpha_i,\\check\\lambda\\rangle}(1-q^r)}$. Then $\\hatPsi_{\\chlam}(q,z)$ is a polynomial\n    function on $\\AA^1\\x  T$.\n\\end{enumerate}\n\\eth\nWe are going to construct the above polynomials explicitly by algebro-geometric\nmeans. Thus we prove the existence part of~\\refco{whit-exist-co}.\n\nWe shall usually refer to the polynomials $\\Psi_{\\chlam}$ and $\\hatPsi_{\\chlam}$ as $q$-Whittaker functions (following\n\\cite{GLOI}-\\cite{GLOIII}).\nIt is not difficult to see that\n$$\n\\lim\\limits_{q\\to 0}\\Psi_{\\chlam}=\\lim\\limits_{q\\to 0}\\hatPsi_{\\chlam}=\\chi(L(\\chlam))\n$$\nwhere $\\chi(L(\\lam))$ stands for the character of the irreducible representation $L(\\chlam)$ of $G$ with highest\nweight $\\chlam$.\n\nThe main purpose of this paper is to give several (algebro-geometric and representation-theoretic) interpretations of\nthe functions $\\Psi_{\\chlam}$ and $\\hatPsi_{\\chlam}$; as a byproduct we shall show that $\\hatPsi_{\\chlam}(q,z)$ is {\\em positive},\ni.e. it is a linear combination of the functions $\\chi(L(\\chmu))$ with coefficients in $\\ZZ_{\\geq 0}[q]$ (this also implies that\n$\\Psi_{\\chlam}$ is a linear combination of the $\\chi(L(\\chmu))$'s with coefficients in $\\ZZ_{\\geq 0}[[q]]$). All of our results are known\nfor the polynomials $\\Psi_{\\chlam}'$  (and thus, in particular, we can show that $\\Psi_{\\chlam}=\\Psi_{\\chlam}'$) due to\n\\cite{C} and~\\cite{San},~\\cite{Ion} but our proofs are totally different from {\\em loc. cit.}\n\\ssec{}{Weyl modules}\nRecall the notion of Weyl ${\\mathfrak g}[\\st]$-module $\\CW(\\check\\lambda)$\nfor dominant $\\check\\lambda\\in\\Lambda^\\vee_+$, see e.g.~\\cite{CFK}.\nIt is the maximal $G$-integrable ${\\mathfrak g}[\\st]$-quotient module of\n$\\on{Ind}_{\\fu[\\st]\\oplus\\ft}^{{\\mathfrak g}[\\st]}{\\mathbb C}_{\\check\\lambda}$\nwhere $\\fu\\subset {\\mathfrak g}$ is the nilpotent radical of a Borel subalgebra, containing $\\grt$. There is also a natural\nnotion of {\\em dual Weyl module} $\\calW(\\chlam)^{\\vee}$ (one has to replace the induction by coinduction and ``quotient module\"\nby ``submodule\"). Both $\\calW(\\chlam)$ and $\\calW(\\chlam)^{\\vee}$ are endowed with a natural action of\n$\\CC^*$ by ``loop rotation\". When restricted to $G\\x \\CC^*$ the module\n$\\calW(\\chlam)$ becomes a direct sum of finite-dimensional representations and the character $\\chi(\\calW(\\chlam))$ makes sense;\nmoreover it is a linear combination of $\\chi(L(\\chmu))$'s with coefficients in $\\ZZ_{\\geq 0}[[q]]$. Also we have\n$\\chi(\\calW(\\chlam))=\\chi(\\calW(\\chlam)^{\\vee})$.\n\nLet $\\AA^{\\chlam}$ denote the space of all formal linear combinations $\\sum\\gam_i x_i$ where\n$x_i\\in \\AA^1$ and $\\gam_i$ are dominant weights of $G$ such that $\\sum \\gam_i=\\chlam$.\nThe character of $\\CC[\\AA^{\\chlam}]$ with respect to the natural action of\n$\\CC^*$ is equal to ${\\prod\\limits_{i\\in I}\\prod\\limits_{r=1}^{\\langle\\alpha_i,\\check\\lambda\\rangle}(1-q^r)}$. According to~\\cite{CFK} there exists\nan action of $\\CC[\\AA^{\\chlam}]$ on $\\calW(\\chlam)$ such that\n\n1) This action commutes with $G[\\st]\\rtimes \\CC^*$;\n\n2) $\\calW(\\chlam)$ is finitely generated and free\nover $\\CC[\\AA^{\\chlam}]$.\n\nLet $D(\\chlam)$ be the fiber of $\\calW(\\chlam)$ at $\\chlam\\cdot 0\\in \\AA^{\\chlam}$. This module is called\na Demazure module (for reasons explained in~\\cite{CL} and~\\cite{FL}).\nThis is a finite-dimensional $G[\\st]\\rtimes \\CC^*$-module\n(in fact, it is easy to see that the action of $G[\\st]$ on $D(\\chlam)$ extends to an action of $G[[\\st]]$).\nWe are going to prove the following\n\n\\th{demazure}Assume that $G$ is simply laced. Then\n\\begin{enumerate}\n\\item\n\\eq{dem-1}\n\\chi(\\calW(\\chlam))=\\Psi_{\\chlam}(q,z)\n\\end{equation}\n\\item\n\\eq{dem-2}\n\\chi(D(\\chlam))=\\hatPsi_{\\chlam}(q,z).\n\\end{equation}\n\\end{enumerate}\nIn particular, $\\hatPsi_{\\chlam}(q,z)$ is positive in the sense discussed above.\n\\eth\nWhen $G$ is not simply laced, the above result is still true, if one replaces $G[[\\st]]$ by some twisted (in the sense of Kac-Moody groups) version\nof it; we shall not give the details here (cf. \\refss{non-simp} for a discussion of the non-simply laced case).\n\n\\reft{demazure}(2) is proved in \\cite{Ion} for $\\hatPsi_{\\chlam}'$ instead of $\\hatPsi_{\\chlam}$.\n\\footnote{It is important to emphasize that the definition of Demazure modules used in this paper (as fibers of Weyl modules)\nis not obviously equivalent to the standard definition used in \\cite{Ion}; however, the equivalence of the two definitions\nis proved in~\\cite{CL} in type $A$, and in~\\cite{FL} in general.}\nThus \\reft{demazure} together with \\cite{Ion} imply the following\n\\cor{twotoda}\nAssume that $G$ is simply laced. Then we have $\\hatPsi_{\\chlam}'=\\hatPsi_{\\chlam}$. Hence\nfor any $f\\in \\CC[T]^W$ we have $\\calM_f=\\calM'_f$.\n\\end{corol}\nAs was mentioned earlier we would like to have a more direct proof of this result\n(independent of the results of \\cite{Ion} and this paper).\nWe would also like to emphasize that our proof of \\reft{demazure} is geometric (in fact it follows easily from the main\nresult of \\cite{BF11}) and thus it is quite different from the proof in \\cite{Ion}. Also, \\refc{twotoda} is\nwrong if $G$ is not simply laced, cf.~\\refss{non-simp}.\n\\ssec{}{Geometric interpretation and spaces of (quasi-)maps}\nTo prove \\reft{demazure} it is clearly enough to prove \\refe{dem-1}.\nThis will be done by interpreting both the LHS and the RHS in terms of algebraic geometry.\n\nLet us first do it for the LHS. The quotient $G[[\\st]]\/T\\cdot U_-[[\\st]]$\ncan naturally be regarded as a scheme\nover $\\CC$. Any weight $\\chlam$ defines a $G[[\\st]]\\rtimes \\CC^*$-equivariant line bundle on this scheme in the standard way.\nWe shall prove\n\\th{weyl}\nThere is a natural isomorphism $\\Gam(G[[\\st]]\/T\\cdot U_-[[\\st]],\\calO(\\chlam))\\simeq \\calW(\\chlam)^{\\vee}$. Similarly,\n$\\Gam(G[[\\st]]\/B_-[[\\st]],\\calO(\\chlam))\\simeq D(\\chlam)^{\\vee}$.\n\\eth\n{\\bf Remark.} \\reft{weyl} is not difficult; it can be thought of as an analog of Borel-Weil-Bott theorem for $G[[\\st]]$. Let us\nalso stress, that while the dual Weyl module $\\calW(\\chlam)^{\\vee}$ has a natural action of $G[[\\st]]$, the Weyl module\n$\\calW(\\chlam)$ itself only has an action of $G[\\st]$.\n\n\\medskip\n\\noindent\nOn the other hand, there is a well known connection between the quotient $G[[\\st]]\/T\\cdot U_-[[\\st]]$ and the space of based maps\n$\\PP^1\\to G\/B$. Moreover, in \\cite{BF11} we have given a construction of the universal eigen-function of the operators\n$\\calM_f$ via the geometry of the above spaces of maps. Using this construction, we can obtain \\refe{dem-1} from \\reft{weyl}\nby a (simple) sequence of formal manipulations. Technically, in order to perform this we shall need to consider a\ncompactification of the space of maps by the corresponding space of quasi-maps.\n\n\\ssec{non-simp}{The case of non-simply laced $G$}\nFormally, the above results do not hold when $G$ is not simply laced. However, it is\neasy to adjust all the results to the non-simply laced case following~Section~7 of~\\cite{BF11}; in particular, in the non-simply laced case the\nfunctions $\\Psi_{\\chlam}$ and $\\hatPsi_{\\chlam}$ should be interpreted as\nthe characters of global (resp. local) Weyl modules for the distinguished\nmaximal parahoric subalgebra in a certain twisted affine algebra corresponding\nto ${\\mathfrak g}$ (cf.~Section~7 of~\\cite{BF11} for more detail). The relevant theory\nof Weyl modules and their relation to Demazure\nmodules in the twisted case is developed in \\cite{FoKu}.\nOn the other hand, the character of {\\em nontwisted} local Weyl modules are\nidentified with $\\hatPsi'_{\\chlam}$ in~\\cite{Len}.\n\\ssec{}{Plan of the paper} This paper is organized as follows.\n In \\refs{2} we discuss certain line\nbundles on the space of (quasi-)maps and relate those to sections of a line bundle on $G[[\\st]]\/T\\cdot U_-[[\\st]]$.\n\\refs{3} is devoted to the proof of certain cohomology vanishing on the space of quasi-maps.\nIn \\refs{4} we give an interpretation of $\\Psi_{\\chlam}$ via quasi-maps. Finally in \\refs{5} we give a proof of\n\\reft{demazure}.\n\n\n\n\n\\sec{2}{Quasimaps' scheme}\n\nWe follow the notations of~\\cite{BF11}, unless specified otherwise.\n\n\\ssec{ind}{Ind-scheme $\\fQ$}\nGiven $\\beta\\geq\\alpha\\in\\Lambda_+$ (the cone of positive integral combinations\nof the simple coroots) we consider the closed embedding\n$\\varphi_{\\alpha,\\beta}:\\ \\QM_{\\mathfrak g}^\\alpha\\hookrightarrow\\QM_{\\mathfrak g}^\\beta$ adding the\ndefect $(\\beta-\\alpha)\\cdot0$ at the point $0\\in\\bC$. We denote by\n$\\fQ$ the direct limit of this system.\n\nRecall that $V_{\\check\\omega_i},\\ i\\in I$, are the fundamental ${\\mathfrak g}$-modules,\nand $\\QM^\\alpha_{\\mathfrak g}$ is equipped with a closed embedding\n$\\psi_\\alpha:\\ \\QM^\\alpha_{\\mathfrak g}\\hookrightarrow\n\\prod_{i\\in I}{\\mathbb P}\\Gamma(\\bC,V_{\\check\\omega_i}\\otimes\\CO(\\langle\\alpha,\n\\check\\omega_i\\rangle))$.\nGiven a ${\\mathfrak g}$-weight $\\check\\lambda=\n\\sum_{i\\in I}d_i\\check\\omega_i\\in\\Lambda^\\vee$ we define a line bundle\n$\\CO(\\check\\lambda)^\\alpha$ on $\\QM^\\alpha_{\\mathfrak g}$ as\n$\\psi_\\alpha^*\\bigotimes_{i\\in I}\\CO(d_i)$.\nNote that if $\\check\\lambda$ is dominant, i.e. $d_i\\geq0\\ \\forall i$,\nthen $\\CO(\\check\\lambda)^\\alpha$ is the inverse image of $\\CO(1)$ on\n${\\mathbb P}\\Gamma(\\bC,V_{\\check\\lambda}\\otimes\\CO(\\langle\\alpha,\n\\check\\lambda\\rangle))$ under the natural morphism\n$\\QM^\\alpha_{\\mathfrak g}\\to{\\mathbb P}\\Gamma(\\bC,V_{\\check\\lambda}\\otimes\\CO(\\langle\\alpha,\n\\check\\lambda\\rangle))$.\nClearly,\n$\\varphi_{\\alpha,\\beta}^*\\CO(\\check\\lambda)^\\beta\\simeq\\CO(\\check\\lambda)^\\alpha$.\nThe resulting line bundle on the ind-scheme $\\fQ$ is denoted\n$\\CO(\\check\\lambda)$.\n\n\\ssec{pro}{Infinite type scheme $\\bQ$}\nWe denote ${\\mathbb C}[[t^{-1}]]$ by $R$, and ${\\mathbb C}((t^{-1}))$ by $F$. Recall that\n$R_n=R\/(t^{-n})$. We denote the projection $R\\twoheadrightarrow R_n$ by $p_n$.\nThe ${\\mathbb C}$-points of the infinite type scheme $\\overline{G\/U_-}(R)$\nare the collections of\nvectors $v_{\\check\\lambda}\\in V_{\\check\\lambda}\\otimes R,\\ \\check\\lambda\\in\n\\Lambda^\\vee_+$ (dominant ${\\mathfrak g}$-weights), satisfying the Pl\\\"ucker equations.\nWe denote by $\\widehat\\bQ\\subset\\overline{G\/U_-}(R)$ the open subscheme\nformed by all the maps $\\on{Spec}R\\to\\overline{G\/U_-}$ whose restriction\nto the generic point of $\\on{Spec}R$ lands into\n$G\/U_-\\subset\\overline{G\/U_-}(R)$.\nIt is equipped with a free action\nof the Cartan torus $T:\\ h(v_{\\check\\lambda})=\\check\\lambda(h)v_{\\check\\lambda}$.\nThe quotient scheme $\\bQ=\\widehat\\bQ\/T$ is a closed subscheme in\n$\\prod_{i\\in I}{\\mathbb P}(V_{\\check\\omega_i}\\otimes R)$. Any weight $\\check\\lambda\\in\n\\Lambda^\\vee$ gives rise to a line bundle $\\CO(\\check\\lambda)$ on $\\bQ$.\n\n\\ssec{indpro}{The embedding $\\fQ\\hookrightarrow\\bQ$}\nWe fix a coordinate $t$ on $\\bC$ such that $t(0)=0,\\ t(\\infty)=\\infty$.\nFor $\\alpha\\in\\Lambda_+$ we define a $T$-torsor\n$\\widehat\\QM{}^\\alpha_{\\mathfrak g}\\stackrel{p}{\\to}\\QM^\\alpha_{\\mathfrak g}$ as follows.\nThe ${\\mathbb C}$-points of $\\widehat\\QM{}^\\alpha_{\\mathfrak g}$ are the collections\n$(v_{\\check\\lambda}\\in\\CL_{\\check\\lambda}\\subset V_{\\check\\lambda}\\otimes\\CO_\\bC),\\\n\\check\\lambda\\in\\Lambda^\\vee_+$, such that\n\n(a)\n$(\\CL_{\\check\\lambda}\\subset\nV_{\\check\\lambda}\\otimes\\CO_\\bC)_{\\check\\lambda\\in\\Lambda^\\vee_+}\\in\\QM^\\alpha_{\\mathfrak g}$;\n(b) $v_{\\check\\lambda}\\in\\Gamma(\\bC-0,\\CL_{\\check\\lambda})$ are the\nnonvanishing sections satisfying the Pl\\\"ucker equations.\n\nThe projection $p$ forgets the sections $v_{\\check\\lambda}$.\nThe action of $T$ on $\\widehat\\QM{}^\\alpha_{\\mathfrak g}$ is defined as follows:\n$h(v_{\\check\\lambda}\\in\\CL_{\\check\\lambda})=\n(\\check\\lambda(h)v_{\\check\\lambda}\\in\\CL_{\\check\\lambda})$.\n\nTaking a formal expansion of $v_{\\check\\lambda}$ at $\\infty\\in\\bC$ we obtain\na closed embedding $s_\\alpha:\\ \\widehat\\QM{}^\\alpha_{\\mathfrak g}\\hookrightarrow\n\\widehat\\bQ$. Clearly, $s_\\alpha$ is $T$-equivariant, and gives rise to the\nsame named closed embedding $s_\\alpha:\\ \\QM^\\alpha_{\\mathfrak g}\\hookrightarrow\\bQ$.\nEvidently, for $\\beta\\geq\\alpha$ we have\n$s_\\alpha=s_\\beta\\circ\\varphi_{\\alpha,\\beta}$. Hence we obtain the closed\nembedding $s:\\ \\fQ\\hookrightarrow\\bQ$. The restriction of the line bundle\n$\\CO(\\check\\lambda)$ on $\\bQ$ to $\\fQ$ coincides with the line bundle\n$\\CO(\\check\\lambda)$ on $\\fQ$.\n\n\\ssec{open}{Open subschemes $\\fQ_\\infty\\subset\\fQ$ and $\\bQ_\\infty\\subset\\bQ$}\nWe define an open subscheme\n$\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}\\subset\\QM^\\alpha_{\\mathfrak g}$ formed by all the\nquasimaps without defect at $\\infty\\in\\bC$. Clearly,\n$\\varphi_{\\alpha,\\beta}(\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g})\\subset\n\\overset{\\circ}\\QM{}^\\beta_{\\mathfrak g}$. The direct limit of this system is denoted by\n$\\fQ_\\infty$; it is an open sub ind-scheme of $\\fQ$.\n\nNote that $s(\\fQ_\\infty)\\subset G(R)\/T\\cdot U_-(R)\\subset\\bQ$. We are going to denote the\nopen subscheme $G(R)\/T\\cdot U_-(R)\\subset\\bQ$ by $\\bQ_\\infty$. For $n\\geq1$,\nwe have a natural projection $p_n:\\ \\bQ_\\infty\\to G\/U_-(R_n)\/T=:\\bQ_n$.\n\n\\begin{lemma}\n\\label{codim2}\nThe restriction\n$\\Gamma(\\fQ,\\CO(\\check\\lambda))\\to\\Gamma(\\fQ_\\infty,\\CO(\\check\\lambda))$\nis an isomorphism for any $\\check\\lambda\\in\\Lambda^\\vee$.\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to prove that the restriction\n$\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))\\to\n\\Gamma(\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))$ is an isomorphism\nfor any $\\alpha\\in\\Lambda_+$. Since the complement of\n$\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}$ in $\\QM^\\alpha_{\\mathfrak g}$ has codimension two, it\nsuffices to know that $\\QM^\\alpha_{\\mathfrak g}$ is normal. However, locally in the\n\\'etale topology, $\\QM^\\alpha_{\\mathfrak g}$ is isomorphic to the product of the\nZastava space $Z^\\alpha_{\\mathfrak g}$ and the flag variety $\\CB_{\\mathfrak g}$. Finally, the\nnormality of $Z^\\alpha_{\\mathfrak g}$ is proved in~\\cite[Corollary~2.10]{BF11}.\n\\end{proof}\n\nThe following conjecture is not needed in this paper, but it might be useful for future purposes.\n\\conj{codim two}\nThe restriction $\\Gamma(\\bQ,\\CO(\\check\\lambda))\\to\n\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))$ is an isomorphism\nfor any $\\check\\lambda\\in\\Lambda^\\vee$.\n\\end{conjec}\n\nLet us make a few remarks about \\refco{codim two}.\nAs in the proof of~Lemma~\\ref{codim2}, it suffices to know that the scheme\n$\\bQ$ is normal. According to~\\cite{D},~\\cite{GK}, the formal completion of\n$\\bQ$ at a closed point $x\\in\\bQ$ is isomorphic to the product of the formal\ncompletion of a certain $\\QM^\\alpha_{\\mathfrak g}$ at a closed point\n$\\phi\\in\\QM^\\alpha_{\\mathfrak g}$, and countably many copies of the formal disc.\nSo the normality of the formal neighborhood of every closed point follows from the normality of $\\QM^\\alpha_{\\mathfrak g}$.\nUnfortunately, since $\\bQ$ is not noetherian it does not imply the normality of $\\bQ$ itself.\n\n\n\\medskip\n\\noindent\nThe group $\\BG_m$ acts on $\\fQ$ and $\\bQ$ by loop rotations, and the line\nbundles $\\CO(\\check\\lambda)$ are $\\BG_m$-equivariant. Hence $\\BG_m$ acts\non the global sections of these line bundles. We will denote by\n$\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda))\\subset\\Gamma(\\fQ,\\CO(\\check\\lambda))$\nthe subspace of $\\BG_m$-finite sections.\n\n\n\\th{isoq}\nThe restriction $\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))\\to\n\\widetilde\\Gamma(\\fQ_\\infty,\\CO(\\check\\lambda))=\n\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda))$ is an isomorphism\nfor any $\\check\\lambda\\in\\Lambda^\\vee$.\n\\eth\n\n\\begin{proof}\nThe closed embedding $\\varphi_{\\alpha,\\beta}:\\ \\QM^\\alpha_{\\mathfrak g}\\hookrightarrow\n\\QM^\\beta_{\\mathfrak g}$ lifts in an evident way to the same named closed embedding of\n$T$-torsors $\\widehat\\QM{}^\\alpha_{\\mathfrak g}\\hookrightarrow\\widehat\\QM{}^\\beta_{\\mathfrak g}$.\nWe denote the limit of this system by $\\widehat\\fQ$, a $T$-torsor over $\\fQ$.\nThe construction of~\\refss{indpro} defines a $T$-equivariant closed embedding\n$s:\\ \\widehat\\fQ\\hookrightarrow\\widehat\\bQ_\\infty:=G\/U_-(R)$. We have to prove\nthat the restriction ${\\mathbb C}[\\widehat\\bQ_\\infty]\\to\n\\widetilde{\\mathbb C}[\\widehat\\fQ_\\infty]=\\widetilde{\\mathbb C}[\\widehat\\fQ]$ is an isomorphism.\nHere $\\widetilde{\\mathbb C}[\\widehat\\fQ_\\infty]$ (resp. $\\widetilde{\\mathbb C}[\\widehat\\fQ]$)\nstands for the ring of $\\BG_m$-finite functions on $\\widehat\\fQ_\\infty$\n(resp. $\\widehat\\fQ$).\n\nTo this end we mimick the argument of~\\cite[Section~2]{BF11}.\nWe choose a {\\em regular} dominant $\\mu\\in\\Lambda^+$, and consider the\ncorresponding $T$-fixed point $t^\\mu\\in\\Gr_G$. Its stabilizer $\\on{St}_\\mu$\nin $G[t^{-1}]$ has the unipotent radical $\\on{RadSt}_\\mu$, and the quotient\n$\\on{St}_\\mu\/\\on{RadSt}_\\mu$ is canonically isomorphic to $T$.\nThe quotient $G[t^{-1}]\/\\on{St}_\\mu$ is the $G[t^{-1}]$-orbit\n$\\sW_{G,\\mu}\\subset\\Gr_G$ of $t^\\mu$ (see~\\cite[Section~2.4]{BF11}),\nand the quotient $G[t^{-1}]\/\\on{RadSt}_\\mu$ is a $T$-torsor\n$\\widehat\\sW_{G,\\mu}$.\n\n{\\em NB:} The group denoted $\\on{St}_\\mu$ in~\\cite[Section~2.6]{BF11} is\nthe intersection of our present $\\on{St}_\\mu$ with the first congruence\nsubgroup $G_1\\subset G[t^{-1}]$.\n\nIn modular terms, $\\sW_{G,\\mu}$ parametrizes the $G$-bundles on $\\bC$\nof isomorphism type $W\\mu$ equipped with a trivialization on $\\bC-0$\n(see~\\cite[Proof of~Theorem~2.8]{BF11}). Such a bundle ${\\mathcal F}_G$\npossesses a canonical Harder-Narasimhan flag $HN({\\mathcal F}_G)$. Note that this flag\nis complete, i.e. it is a reduction to the Borel, since $\\mu$ is regular.\nIn particular, the fiber ${\\mathcal F}_{G,\\infty}$ of ${\\mathcal F}_G$ at $\\infty\\in\\bC$ is\nequipped with a\ncanonical reduction to the Borel. Now $\\widehat\\sW_{G,\\mu}$ parametrizes\nthe data as above along with a further reduction of ${\\mathcal F}_{G,\\infty}$ to the\nunipotent radical of the Borel.\n\nIn complete similarity with~\\cite[Lemma~2.7]{BF11} we have\n\n\\begin{lemma}\n\\label{2.7}\n(1) Fix $n\\geq1$, and let $\\mu\\in\\Lambda^+_{\\on{reg}}$ satisfy the following\ncondition: $\\langle\\mu,\\check\\alpha\\rangle\\geq n$ for every positive root\n$\\check\\alpha$ of ${\\mathfrak g}$. Then the image of $\\on{RadSt}_\\mu$ in\n$G[t^{-1}]\/G_n=G(R_n)$ is equal to $U_-(R_n)$. In particular, we have a\nnatural map $\\pi_{\\mu,n}:\\ \\widehat\\sW_{G,\\mu}\\to G(R_n)\/U_-(R_n)$.\n\n(2) Under the assumption of (1), for every $k<n$, the map\n$\\pi_{\\mu,n}^*:\\ {\\mathbb C}[G(R_n)\/U_-(R_n)]\\to{\\mathbb C}[\\widehat\\sW_{G,\\mu}]$ induces\nan isomorphism on functions of homogeneity degree $k$ with respect to $\\BG_m$.\n\\qed\n\\end{lemma}\n\nWe denote the intersection of $\\sW_{G,\\mu}\\subset\\Gr_G$ with\n$\\overline\\Gr{}^\\lambda_G$ by $\\overline\\sW{}^\\lambda_{G,\\mu}$. We denote the\npreimage of $\\overline\\sW{}^\\lambda_{G,\\mu}\\subset\\sW_{G,\\mu}$ in\n$\\widehat\\sW_{G,\\mu}$ by $\\widehat\\sW{}^\\lambda_{G,\\mu}$.\nIn complete similarity with~\\cite[Theorem~2.8]{BF11} we have\n\n\\begin{lemma}\n\\label{2.8}\n(1) Let $\\lambda\\geq\\mu\\in\\Lambda^+_{\\on{reg}}$, and let $\\alpha=\\lambda-\\mu$.\nThen there exists a natural birational $T\\times\\BG_m$-equivariant morphism\n$s^\\lambda_\\mu:\\ \\widehat\\sW{}^\\lambda_{G,\\mu}\\to\\widehat\\QM{}^\\alpha_{\\mathfrak g}$\nsuch that for any $n$ satisfying the condition in~Lemma~\\ref{2.7}(1), the\nfollowing diagram is commutative:\n\\eq{CD}\n\\begin{CD}\n\\widehat\\sW{}^\\lambda_{G,\\mu} @>s^{\\lam}_{\\mu}>> \\widehat\\QM{}^\\alpha_{\\mathfrak g}\\\\\n@V\\pi_{\\mu,n}VV @VVp_n\\circ s_\\alpha V\\\\\n G(R_n)\/U_-(R_n)@>\\on{id}>> G(R_n)\/U_-(R_n)\n \\end{CD}\n\\end{equation}\n($s_\\alpha$ was constructed in~\\refss{indpro}).\n\n(2) The map $(s^\\lambda_\\mu)^*:\\ {\\mathbb C}[\\widehat\\QM{}^\\alpha_{\\mathfrak g}]\\to\n{\\mathbb C}[\\widehat\\sW{}^\\lambda_{G,\\mu}]$ induces an isomorphism on functions of\ndegree $<n$ for any $n$ satisfying the condition in~Lemma~\\ref{2.7}(1). \\qed\n\\end{lemma}\n\nNow~\\reft{isoq} immediately follows from~Lemma~\\ref{2.7} and~Lemma~\\ref{2.8}.\n\\end{proof}\n\nNote that if one assumes \\refco{codim two} then it follows that\nthe restriction $\\Gamma(\\bQ,\\CO(\\check\\lambda))\\to\n\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda))$ is an isomorphism\nfor any $\\check\\lambda\\in\\Lambda^\\vee$.\n(this follows immediately from~\\reft{isoq},~Lemma~\\ref{codim2}, and~\\refco{codim two}).\n\n\n\n\n\n\\sec{3}{Cohomology vanishing}\n\nFrom now on we assume that $G$ is simply laced.\n\n\\ssec{noname}{}\nThe group $\\BG_m$ acts on $\\fQ$ and $\\bQ$ by loop rotations, and the line\nbundles $\\CO(\\check\\lambda)$ are $\\BG_m$-equivariant. Hence $\\BG_m$ acts\non the cohomology $H^n(\\fQ,\\CO(\\check\\lambda)):=\\displaystyle{\\lim_\\leftarrow\nH^n(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))}$\nof these line bundles. We will denote by\n$\\widetilde{H}{}^n(\\fQ,\\CO(\\check\\lambda))\\subset H^n(\\fQ,\\CO(\\check\\lambda))$\nthe subspace of $\\BG_m$-finite classes.\n\nRecall that $\\alpha\\mapsto\\alpha^*$ stands for the natural (linear) isomorphism\nbetween the coroot lattice of ${\\mathfrak g}$ and its root lattice, taking the simple\ncoroots to the corresponding simple roots. Now $\\Lambda_+$ contains a cofinal\nsubsystem $\\Lambda_+^{\\check\\lambda}$ formed by $\\alpha$ such that\n$\\alpha^*+\\check\\lambda$ is dominant.\n\n\\th{vanish}\n(1) For $n>0$ and $\\alpha\\in\\Lambda_+^{\\check\\lambda}$ we have\n$H^n(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))=0$.\n\n(2) For $n>0$ and $\\check\\lambda\\in\\Lambda^\\vee$ we have\n$\\widetilde{H}{}^n(\\fQ,\\CO(\\check\\lambda))=0$.\n\n(3) For $\\check\\lambda\\not\\in\\Lambda^\\vee_+$ we have\n$\\widetilde{H}{}^0(\\fQ,\\CO(\\check\\lambda))=0$.\n\\eth\n\n\\begin{proof}\n(3) is clear, and (2) follows from (1). We prove (1).\n\nAccording to~\\cite[Proposition~5.1]{BF11}, $Z^\\alpha_{\\mathfrak g}$ is a Gorenstein\nvariety with rational singularities. Since $\\QM^\\alpha_{\\mathfrak g}$ is, locally in\n\\'etale topology, isomorphic to $Z^\\alpha_{\\mathfrak g}\\times\\CB_{\\mathfrak g}$, we conclude that\n$\\QM^\\alpha_{\\mathfrak g}$ is a Gorenstein variety with rational singularities as well.\n(It is here that we use the assumption that $G$ is simply laced.)\nLet us denote the dualizing sheaf of $\\QM^\\alpha_{\\mathfrak g}$ by $\\omega^\\alpha$.\n\n\\begin{lemma}\n\\label{givlee}\n$\\omega^\\alpha\\simeq\\CO(-\\alpha^*-2\\check\\rho)$.\n\\end{lemma}\n\n\\begin{proof}\nIn case $G=\\SL(N)$, the lemma is proved in~\\cite[Theorem~3]{GL}.\nFor arbitrary simply laced $G$ we first prove that $\\omega^\\alpha\\simeq\n\\CO(\\check\\lambda)$ for some $\\check\\lambda$. It is enough to check this\non the open subscheme $\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}$ since the complement\nis of codimension two. We have the morphism of evaluation at $\\infty\\in\\bC:\\\n\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}\\stackrel{ev_\\infty}{\\longrightarrow}\\CB_{\\mathfrak g}$.\nIt is a $G$-equivariant fibration with fibers isomorphic to $Z^\\alpha_{\\mathfrak g}$.\nSince the big cell $U\\cdot e_-\\subset\\CB_{\\mathfrak g}$ is a free orbit of $U$, we have\n$ev_\\infty^{-1}(U\\cdot e_-)\\simeq Z^\\alpha_{\\mathfrak g}\\times U$. The canonical class\nof $Z^\\alpha_{\\mathfrak g}$ is trivial (see~\\cite[Proof of~Proposition~5.1]{BF11}), hence\nthe canonical class of $ev_\\infty^{-1}(U\\cdot e_-)$ is trivial as well.\nThus $\\omega^\\alpha$ has a nowhere vanishing section $\\sigma$ on\n$ev_\\infty^{-1}(U\\cdot e_-)$. Hence the class of $\\omega^\\alpha$ on\n$\\overset{\\circ}\\QM{}^\\alpha_{\\mathfrak g}$ is a linear combination of the pullbacks\nunder $ev_\\infty$ of the Schubert divisors on $\\CB_{\\mathfrak g}$. The pullback of an\nirreducible Schubert divisor being $\\CO(\\check\\omega_i)$ we conclude that\nthere exists $\\check\\lambda$ such that $\\omega^\\alpha\\simeq\\CO(\\check\\lambda)$.\n\nIt remains to check $\\check\\lambda=-\\alpha^*-2\\check\\rho$. We will do this on\nanother open subscheme $\\overset{\\bullet}\\QM{}^\\alpha_{\\mathfrak g}\\subset\\QM^\\alpha_{\\mathfrak g}$ with the\ncomplement of codimension two. Namely, $\\overset{\\bullet}\\QM{}^\\alpha_{\\mathfrak g}$\nis the moduli space of quasimaps with defect at most a simple coroot (or no\ndefect at all). Note that $\\overset{\\bullet}\\QM{}^\\alpha_{\\mathfrak g}$ is smooth,\nand the Kontsevich resolution is an isomorphism over it. Let us fix a\nquasimap without defect $\\phi\\in\\QM^{\\alpha-\\alpha_i}_{\\mathfrak g}$, and consider a curve\n$C^\\phi_i\\subset\\overset{\\bullet}\\QM{}^\\alpha_{\\mathfrak g}$ formed by all the\nquasimaps $\\phi(\\alpha_i\\cdot c),\\ c\\in\\bC$ (twisting $\\phi$ by an arbitrary point of $\\bC$).\nIt is easy to see that $\\deg(\\CO(\\check\\omega_j)|_{C^\\phi_i})=\\delta_{ij}=\n\\langle\\alpha_i,\\check\\omega_j\\rangle$. Hence it remains to check that\n$\\deg(\\omega^\\alpha|_{C^\\phi_i})=-\\langle\\alpha_i,\\alpha^*+2\\check\\rho\\rangle$.\nThis is done in~\\cite[Proposition~4.4]{FK}. Although {\\em loc. cit.} is\nformulated for $G=\\SL(N)$, its proof goes through word for word for arbitrary\nsimple $G$.\n\nThe lemma is proved. \\end{proof}\n\nWe are ready to finish the proof of the theorem. For\n$\\alpha\\in\\Lambda^{\\check\\lambda}_+$ the line bundle\n$\\CL=\\CO(\\check\\lambda)\\otimes(\\omega^\\alpha)^*$ on $\\QM^\\alpha_{\\mathfrak g}$ is\nvery ample. We have to prove that $H^n(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))=\nH^n(\\QM^\\alpha_{\\mathfrak g},\\CL\\otimes\\omega^\\alpha)=0$ for $n>0$. According\nto~\\cite[Proposition~5.1]{BF11}, $\\QM^\\alpha_{\\mathfrak g}$ has rational singularities.\nLet $\\pi:\\ X\\to\\QM^\\alpha_{\\mathfrak g}$ be a resolution of singularities. Then for the\ncanonical line bundle $\\omega_X$ of $X$ we have $R\\pi_*\\omega_X=\\omega^\\alpha$.\nHence $H^n(\\QM^\\alpha_{\\mathfrak g},\\CL\\otimes\\omega^\\alpha)=\nH^n(X,\\pi^*\\CL\\otimes\\omega_X)=0$ (for $n>0$) by Kawamata-Viehweg vanishing\nsince $\\pi^*\\CL$ is nef and big.\n\nThis completes the proof of the theorem.\n\\end{proof}\n\n\n\\sec{4}{$q$-Whittaker functions}\n\n\\ssec{charac}{The character of $R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))$}\nRecall~\\cite[Introduction]{BF11} that $\\fJ_\\alpha(q,z)$\nis the character of $T\\times\\BG_m$-module ${\\mathbb C}[Z^\\alpha_{\\mathfrak g}]$, a rational\nfunction on $T\\times\\BG_m$. Let $x_i$ stand for the character of the dual\ntorus $\\check T$ corresponding to the simple coroot $\\alpha_i$. For\n$\\alpha\\in\\Lambda_+$ the corresponding character of $\\check T$ is denoted by\n$x^\\alpha$. We consider the formal generating functions\n$J_{\\mathfrak g}(q,z,x)=\\sum_{\\alpha\\in\\Lambda_+}x^\\alpha\\fJ_\\alpha$, and\n$\\fJ_{\\mathfrak g}(q,z,x)=\\prod_{i\\in I}x_i^{\\log(\\check\\omega_i)\/\\log q}J_{\\mathfrak g}(q,z,x)$,\ncf.~\\cite[Equation~(18)]{BF0}.\n\nAccording to~\\cite[Corollary~1.6]{BF11}, the function $\\fJ_{\\mathfrak g}(q,z,x)$\nis an eigen-function of the quantum difference Toda integrable system\nassociated with ${\\mathfrak g}$. For example, if $G=\\SL(N)$, the function\n$\\fJ_{\\mathfrak g}(q,z,x)$ is an eigenfunction of the operator\n$\\fG=T_1+T_2(1-x_1)+\\ldots+T_N(1-x_{N-1})$, cf.~\\cite[Equation~(16)]{BF0},\nwhere $T_k(F(q,z,x_1,\\ldots,x_{N-1}))=F(q,z,x_1,\\ldots,x_{k-2},q^{-1}x_{k-1},\nqx_k,x_{k+1},\\ldots,x_{N-1})$.\n\nNote that if we plug $x=q^{\\check\\lambda}$ into $J_{\\mathfrak g}(q^{-1},z,x)$ or into\n$\\fJ_{\\mathfrak g}(q^{-1},z,x)$, then for $\\check\\lambda\\in\\Lambda^\\vee_+$ these formal\nseries converge, and we have $\\fJ_{\\mathfrak g}(q^{-1},z,q^{\\check\\lambda}):=\\prod_{i\\in I}\n(q^{\\langle\\alpha_i,\\check\\lambda\\rangle})^{\\log(\\check\\omega_i)\/\\log q}\nJ_{\\mathfrak g}(q^{-1},z,q^{\\check\\lambda})=\nz^{\\check\\lambda}J_{\\mathfrak g}(q^{-1},z,q^{\\check\\lambda})$ (a formal Taylor series in\n$q$ with coefficients in Laurent polynomials in $z$).\n\nThe following lemma is a reformulation of~\\cite[Proposition~2]{GL}:\n\n\\begin{lemma}\n\\label{gile}\nThe class of $R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))$ in\n$K_{T\\times\\BG_m}(pt)$ equals\n$$\\sum_{\\substack{\\gamma+\\beta=\\alpha\\\\ w\\in W}}\nz^{w\\check\\lambda}q^{\\langle\\gamma,\\check\\lambda\\rangle}\n\\fJ_\\gamma(q^{-1},wz)\\fJ_\\beta(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $\\pi:\\ \\overline{M}_{0,0}({\\mathbb P}^1\\times\\CB_{\\mathfrak g},(1,\\alpha))\\to\\QM^\\alpha_{\\mathfrak g}$\n(resp. $\\varpi:\\ M^\\alpha_{\\mathfrak g}\\to Z^\\alpha_{\\mathfrak g}$) be the Kontsevich resolution,\nsee e.g.~\\cite[Appendix]{FFKM} (resp.~\\cite[Proof of~Proposition~5.1]{BF11}).\nSince the singularities of $\\QM^\\alpha_{\\mathfrak g}$ (resp. $Z^\\alpha_{\\mathfrak g}$) are rational,\nwe have $R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))=\nR\\Gamma(\\overline{M}_{0,0}({\\mathbb P}^1\\times\\CB_{\\mathfrak g},(1,\\alpha)),\n\\pi^*\\CO(\\check\\lambda))$ (resp. ${\\mathbb C}[Z^\\alpha_{\\mathfrak g}]={\\mathbb C}[M^\\alpha_{\\mathfrak g}]$).\nHence we have to express the character of\n$R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda))$ via the characters of\n${\\mathbb C}[M^\\beta_{\\mathfrak g}]$. This is done in~\\cite[Proof of~Proposition~2]{GL} via\nthe Atiyah-Bott-Lefschetz localization to $T\\times\\BG_m$-fixed points of\n$\\overline{M}_{0,0}({\\mathbb P}^1\\times\\CB_{\\mathfrak g},(1,\\alpha))$. As usually, we have to\nadd that {\\em loc. cit.} deals with $G=\\SL(N)$, however, the proof\ngoes through word for word for arbitrary semisimple $G$.\n\\end{proof}\n\n\\ssec{lim charac}{The character of $\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda))$}\nBy the proof of~\\reft{isoq} and~Lemma~\\ref{2.8}(2), the character\n$\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))$ is the limit of the characters\n$\\chi (R^0\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda)))$\nas $\\alpha\\to\\infty$. By~\\reft{vanish}(1), as $\\alpha\\to\\infty$, the limit\nof the characters $\\chi (R^{>0}\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda)))$\nvanishes. Thus, the character $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))$\nis the limit of the characters $\\chi (R\\Gamma(\\QM^\\alpha_{\\mathfrak g},\\CO(\\check\\lambda)))$\nas $\\alpha\\to\\infty$. We define\n$\\fJ_\\infty(q,z):=\\displaystyle{\\lim_{\\alpha\\to\\infty}\\fJ_\\alpha(q,z)}$\n(it is easy to see that the latter limit exists).\n\n\\prop{brav}\n$$\n\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))=\n\\sum_{w\\in W}\\fJ_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})\\fJ_\\infty(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}.\n$$\n\\end{propos}\n\n\\begin{proof}\nAs $\\alpha$ goes to $\\infty$, the formula of~Lemma~\\ref{gile} goes to\n$$\\sum_{\\substack{\\gamma\\in\\Lambda_+\\\\ w\\in W}}\nz^{w\\check\\lambda}q^{\\langle\\gamma,\\check\\lambda\\rangle}\n\\fJ_\\gamma(q^{-1},wz)\\fJ_\\infty(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}=$$\n$$\\sum_{w\\in W}\nz^{w\\check\\lambda}J_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})\\fJ_\\infty(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}=$$\n$$\\sum_{w\\in W}\\fJ_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})\\fJ_\\infty(q,wz)\n\\prod_{\\check\\alpha\\in\\check{R}{}^+}(1-wz^{\\check\\alpha})^{-1}.$$\n\\end{proof}\n\n\\cor{gera}\nLet $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))=\\Psi_{\\chlam}(q,z)$. Then the functions\n$\\Psi_{\\chlam}(q,z)$ satisfy all the conditions of \\refco{whit-exist-co}.\n\\end{corol}\n\n\\begin{proof}\nPart 2 of \\refco{whit-exist-co} is obvious by construction. Also \\refco{whit-exist-co}(1b) is obvious.\nAccording to~\\reft{vanish}(2), $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))=0$\nif $\\check\\lambda\\not\\in\\Lambda^\\vee_+$, which proves \\refco{whit-exist-co}(1a).\n\nLet us prove \\refco{whit-exist-co}(1c). The function\n$\\fJ_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})$ on the lattice $\\Lambda^\\vee$ is an\neigenfunction of the quantum difference Toda restricted to the lattice.\nAccording to~\\refp{brav}, $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))$ is\na linear combination of the functions $\\fJ_{\\mathfrak g}(q^{-1},wz,q^{\\check\\lambda})$\nwith coefficients independent of $\\check\\lambda$. Hence $\\Psi_{\\chlam}(q,z)$\nis an eigenfunction of the quantum difference Toda as well.\n\\end{proof}\n\n\n\n\n\n\\sec{5}{Weyl modules}\n\n\\ssec{nameless}{} Recall that $R={\\mathbb C}[[t^{-1}]]$. We introduce a new variable\n$\\st=t^{-1}$, so that $R={\\mathbb C}[[\\st]]$. We set $\\widetilde R:={\\mathbb C}[\\st]\\subset R$.\nThe proalgebraic group $G(R)$ acts naturally on the profinite dimensional\nvector space $\\Gamma(\\fQ,\\CO(\\check\\lambda))$. The continuous dual\n$\\Gamma(\\fQ,\\CO(\\check\\lambda))^\\vee$ coincides with the graded dual\n$\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda))^\\vee$, and is equipped with a\nnatural action of $G(\\widetilde R):\\ g\\cdot v^*(v):=v^*(\\tau g\\cdot v)$.\nHere $g\\mapsto\\tau g$ is the Chevalley antiinvolution of $G$ identical on $T$.\nThe derivative of these actions gives rise to the actions of ${\\mathfrak g}(R)$\nand ${\\mathfrak g}(\\widetilde R)$. According to~\\reft{isoq}, the\n${\\mathfrak g}(\\widetilde R)$-module $\\Gamma(\\fQ,\\CO(\\check\\lambda))^\\vee$ coincides\nwith the graded dual $\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))^\\vee$.\n\nWe denote the preimage of the big cell $U\\cdot e_-\\subset\\CB_{\\mathfrak g}$ in\n$G\/U_-\\to\\CB_{\\mathfrak g}$ by $C\\subset G\/U_-$. We denote the open subscheme\n$C(R)\/T\\subset G(R)\/T\\cdot U_-(R)=\\bQ_\\infty$ by $\\overset{\\circ}{\\bQ}$.\nWe have the restriction morphism of ${\\mathfrak g}(R)$-modules\n$\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))\\hookrightarrow\n\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))$. Now $C(R)$ is a free\norbit of $B(R)\\subset G(R)$, and\n$\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))=\n\\on{CoInd}_{\\fu(R)\\oplus\\ft}^{{\\mathfrak g}(R)}{\\mathbb C}_{\\check\\lambda}$. The graded dual\n$\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))^\\vee=\n\\on{Ind}_{\\fu(\\widetilde R)\\oplus\\ft}^{{\\mathfrak g}(\\widetilde R)}{\\mathbb C}_{\\check\\lambda}$.\n\n\\begin{lemma}\n\\label{?}\n$\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))\\subset\n\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))$ is the maximal\n$G$-integrable ${\\mathfrak g}(R)$-submodule. Equivalently,\n$\\Gamma(\\bQ_\\infty,\\CO(\\check\\lambda))^\\vee$ is the maximal $G$-integrable\n${\\mathfrak g}(\\widetilde R)$-quotient module of\n$\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))^\\vee$.\n\\end{lemma}\n\n\\begin{proof}\nNote that $\\bQ_\\infty$ is the $G$-saturation of $\\overset{\\circ}{\\bQ}$.\nLet $v\\in\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))$ lie in a\nfinite-dimensional ${\\mathfrak g}$-submodule\n$V\\subset\\Gamma(\\overset{\\circ}{\\bQ},\\CO(\\check\\lambda))$. The action of\n${\\mathfrak g}$ on $V$ integrates to the action of $G$. Let us view $v$ as a\n$\\check\\lambda$-covariant function on $C(R)$. We have to check that\n$v$ is the restriction of a $\\check\\lambda$-covariant function $\\hat v$\non $G\/U_-(R)$ to $C(R)$. Given a point $y\\in G\/U_-(R)$ we can find $g\\in G$\nsuch that $g(y)\\in C(R)$. Then we define $\\hat{v}(y):=u(gy)$ where we view\n$u:=gv\\in V$ as a $\\check\\lambda$-covariant function on $C(R)$. Clearly, this\nis well defined, i.e. independent of a choice of $g$.\n\\end{proof}\n\nRecall the notion of Weyl ${\\mathfrak g}(\\widetilde R)$-module $\\CW(\\check\\lambda)$\nfor dominant $\\check\\lambda\\in\\Lambda^\\vee_+$, see e.g.~\\cite{CFK}.\nIt is the maximal $G$-integrable ${\\mathfrak g}(\\widetilde R)$-quotient module of\n$\\on{Ind}_{\\fu(\\widetilde R)\\oplus\\ft}^{{\\mathfrak g}(\\widetilde R)}{\\mathbb C}_{\\check\\lambda}$\n({\\em loc. cit.}). Thus~Lemma~\\ref{?} implies the first part of \\reft{demazure}.\n\nOn the other hand, taking into account \\reft{isoq} we also get\n\\prop{!}\nFor $\\check\\lambda\\in\\Lambda^\\vee_+$, we have a natural isomorphism of\n${\\mathfrak g}(\\widetilde R)$-modules $\\Gamma(\\fQ,\\CO(\\check\\lambda))^\\vee\\simeq\n\\CW(\\check\\lambda)$. \\qed\n\\end{propos}\n\nCombining this with \\refc{gera} we get the following\n\\cor{weyl-whit}\n$\\chi(\\calW(\\chlam))=\\Psi_{\\chlam}(q,z)$.\n\\end{corol}\nThis is actually the statement of \\reft{demazure}(1).\nTo prove \\reft{demazure}(2) let us\nrecall that the Demazure module $D(\\check\\lambda)$ is a certain\n${\\mathfrak g}(\\widetilde R)$-submodule of an irreducible integrable level one\nrepresentation of\n${\\mathfrak g}_{\\on{aff}}$, see e.g.~\\cite[2.2]{FL}. In addition, according\nto~\\cite{CFK},~\\cite{FL} there exists\nan action of $\\CC[\\AA^{\\chlam}]$ on $\\calW(\\chlam)$ such that\n\n1) This action commutes with $G(R)\\rtimes \\CC^*$.\n\n2) $\\calW(\\chlam)$ is finitely generated and free\nover $\\CC[\\AA^{\\chlam}]$.\n\n3) The fiber of $\\calW(\\chlam)$ at $\\chlam\\cdot 0$ is isomorphic to $D(\\chlam)$.\n\n\\medskip\n\\noindent\nThus we get the following corollary, which is actually the statement of \\reft{demazure}(2)\n(as was mentioned in the introduction it was proved in~\\cite{GLOIII} for $G=\\SL(N)$):\n\\cor{dem}\nThe product $\\chi(\\widetilde\\Gamma(\\fQ,\\CO(\\check\\lambda)))\\cdot\\displaystyle\n{\\prod_{i\\in I}\\prod_{r=1}^{\\langle\\alpha_i,\\check\\lambda\\rangle}(1-q^r)}=\\hatPsi_{\\chlam}(q,z)$\nis equal to\nthe character of the (finite dimensional) Demazure module $D(\\check\\lambda)$. In particular, it is a finite\nlinear combination of $\\chi(L(\\chmu))$'s with coefficients in $\\ZZ_{\\geq 0}[q]$.\n\\end{corol}\n\n\\ssec{}{Geometric interpretation of the $\\CC[\\AA^{\\chlam}]$-action}\nWe conclude the paper by giving an interpretation of the $\\CC[\\AA^{\\chlam}]$-action on $\\calW(\\chlam)$ in terms of\n\\reft{demazure}(1). This will enable us to prove the second assertion of \\reft{weyl}.\nIt would be nice to prove that this action is free directly by geometric means (without referring to\n\\cite{FL}).\n\nLet $T(R)_1$ denote the first congruence subgroup in $T(R)$ (i.e. the kernel of the natural map\n$T(R)\\to T$). Let $\\grt(R)_1$ denote its (abelian) Lie algebra (i.e. the kernel of the natural map\n$\\grt(R)\\to \\grt$). We denote by $\\grt(\\tilR)_1\\subset \\grt(R)_1$ the corresponding subspace (consisting of all\nmappings $\\AA^1\\to \\grt$ which are equal to $0$ at $0$). Then for every $\\chlam\\in\\Lam^{\\vee}_+$ there exists a natural epimorphism\n$\\pi_{\\chlam}:U(\\grt(R)_1)=\\Sym\\grt(R)_1)\\to \\CC[\\AA^{\\chlam}]$ defined by the following formula:\n$$\n\\pi_{\\chlam} (h\\st^n)(\\sum_i \\gam_i x_i)=\\sum_i \\langle h,\\gam_i)x_i^n.\n$$\nHere $h\\in \\grt$ and $\\sum_i \\gam_i x_i\\in \\AA^{\\chlam}$.\n\nClearly, the group $T(R)_1$ acts (on the right) on the scheme $\\bQ_{\\infty}=G(R)\/T\\cdot U_-(R)$. Hence we get a natural action\nof $\\Sym(\\grt(R)_1)$ on $\\Gam(\\bQ_{\\infty}, \\calO(\\chlam))$ for every $\\chlam\\in \\Lam^{\\vee}$.\nThe following result is easy to prove; we leave\nthe details to the reader:\n\\prop{config}\n\\begin{enumerate}\n\\item The above action\nof $\\Sym(\\grt(\\tilR)_1)$ on $\\Gam(\\bQ_{\\infty}, \\calO(\\chlam))$ factors through $\\pi_{\\chlam}$.\n\\item\nThe resulting action\nof $\\CC[\\AA^{\\chlam}]$ on $\\Gam(\\bQ_{\\infty}, \\calO(\\chlam))^{\\vee}=\\calW(\\chlam)$\ncoincides with the action considered in~\\cite{CFK} and \\cite{FL}.\n\\end{enumerate}\n\\end{propos}\nFrom \\refp{config} we immediately get the following\n\\cor{}\nWe have\n$\\Gam(G(R)\/B_-(R),\\calO(\\chlam))\\simeq D(\\chlam)^{\\vee}$ (this is the second assertion of \\reft{weyl}).\n\\end{corol}\n\\begin{proof}\nIt follows from  \\refp{config} and from the fact that $D(\\chlam)$ is the fiber of $\\calW(\\chlam)$ over\n$\\chlam\\cdot 0\\in \\CC[\\AA^{\\chlam}]$ that $D(\\chlam)^{\\vee}$ is isomorphic to the invariants of $\\grt(\\tilR)$ on $\\calW(\\chlam)^{\\vee}$.\nSince $\\grt(\\tilR)_1$ is dense in $\\grt(R)_1$, it follows that\n$$\n(\\calW(\\chlam)^{\\vee})^{\\grt(\\tilR)_1}=(\\calW(\\chlam)^{\\vee})^{\\grt(R)_1}.\n$$\nFrom \\refp{!} we get\n$$\n(\\calW(\\chlam)^{\\vee})^{\\grt(R)_1}=\\Gam(G(R)\/T\\cdot U_-(R),\\calO(\\chlam))^{\\grt(R)_1}=\\Gam(G(R)\/B_-(R),\\calO(\\chlam)).\n$$\n\n\n\\end{proof}\n\n\\begin{acknowledgement}\nThis paper emerged as a result of numerous conversations held between the first author and\nA.~Borodin in IHES in June 2011. In particular, A.~Borodin has introduced the first author to the notion of $q$-Whittaker\nfunctions and brought to his attention the papers \\cite{GLOI}-\\cite{GLOIII}. The first author wishes\nalso to thank IHES staff for exceptionally pleasant working conditions and hospitality.\nWe are grateful to B.~Feigin and S.~Loktev for\ntheir numerous patient explanations about Weyl modules. The observation that\n$\\Gamma(G[[\\st]]\/T\\cdot U_-[[\\st]],\\CO(\\check\\lambda))$ is a dual Weyl module is essentially due to them.\nWe would like to thank K.~Schwede and other mathoverflowers for their\nexplanations about Kodaira vanishing. The first author is also deeply indebted to Patrick\nClark for his patient explanations about Macdonald polynomials and to P.~Etingof and I.~Cherednik for explanations about the\n$q$-Toda system.\nM.F. was partially supported by the RFBR grants 12-01-00944, 12-01-33101,\n13-01-12401\/13,\nthe National Research University Higher School of Economics' Academic Fund\naward No.12-09-0062 and\nthe AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023.\nThis study comprises research findings from the ``Representation Theory\nin Geometry and in Mathematical Physics\" carried out within The\nNational Research University Higher School of Economics' Academic Fund Program\nin 2012, grant No 12-05-0014.\n\\end{acknowledgement}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}