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+{"text":"\\section{Introduction}\\label{intro}\n\nAssociation schemes~\\cite{BI,BCN} were first defined by Bose and Mesner~\\cite{BM} in the context of the design of experiments.\nPhilippe Delsarte used association schemes to unify the study \nof coding theory and design theory in his thesis~\\cite{PD}, where he derived his well-known linear programming bound which has since found many applications \nin combinatorics. There he identified two types of association schemes which were of particular interest: the so-called \n$P$-polynomial and $Q$-polynomial schemes. \nSchemes which are $P$-polynomial are precisely those arising from distance-regular graphs, and are well studied.\nIn particular, much effort has gone into the classification of distance-transitive graphs, the $P$-polynomial schemes \nwhich are the orbitals of a permutation group; and it is likely that all such examples are\nknown. Also well-studied are the schemes which are both $Q$-polynomial and\n$P$-polynomial. A well-known conjecture~\\cite[p.312]{BI} of Bannai and Ito is the following: for sufficiently\nlarge $d$, a primitive scheme is $P$-polynomial if and only if it is $Q$-polynomial.\n\nClassification efforts for $Q$-polynomial schemes are far less advanced than in the $P$-polynomial case; in particular it is likely that more\nexamples from permutation groups are yet to be found. The $Q$-polynomial property has no known combinatorial characterization, making their study more difficult. \nHowever, the list of known examples (see~\\cite{MMW,PW,VDMM}) indicates that these objects have interesting structure from the viewpoint\nof designs, lattices, coding theory and finite geometry.\n\nIn this paper, we give a new family of imprimitive $Q$-polynomial schemes with an\nunbounded number of classes. These schemes are formed by the orbitals of a group, giving a double cover of the scheme arising from the symplectic dual polar space graph. \nWe note that only one other family of imprimitive $Q$-polynomial schemes with an\nunbounded number of classes is known that is not $P$-polynomial, namely the bipartite doubles of the Hermitian dual polar space graphs, which are $Q$-bipartite and $Q$-antipodal.\nThe schemes in this paper are $Q$-bipartite, and have two $Q$-polynomial orderings.\nExcept when the field order $q$ is a square, the splitting field of these schemes is also\nirrational.  We note that this is the only known family of $Q$-polynomial schemes with\nunbounded number of classes and an irrational splitting field.\nIn the last section we give open parameters for hypothetical primitive $Q$-polynomial subschemes of this family. \n\nOur paper is organized as follows: Background material on Gaussian coefficients, two-graphs\nand double covers of graphs, are covered in Sections \\ref{gaussian}--\\ref{graphs}. In\nSection~\\ref{dual_polar} we recall the standard construction of the symplectic dual polar\ngraph $\\Gamma=\\Gamma(2n,q)$. There we also introduce the Maslov index, which we use in Section~\\ref{double_cover} to construct the double cover\n$\\widehat{\\Gamma}\\to\\Gamma$ when $q\\equiv1$ mod~4. In Section~\\ref{scheme} we construct a $(2n+1)$-class association scheme\n$\\mathcal{S}=\\mathcal{S}_{n,q}$ from $\\widehat{\\Gamma}$; and in Section~\\ref{Q-poly} we show that $\\mathcal{S}$ is\n$Q$-polynomial. The $P$-matrix of the scheme is constructed in Section~\\ref{P-matrix}.\nA particularly tantalizing open problem is the question whether $\\mathcal{S}$ is in general\nthe extended $Q$-bipartite double of a primitive $Q$-polynomial scheme; see Section~\\ref{subscheme}.\n\n\\section{Gaussian coefficients}\\label{gaussian}\n\nFor all integers $n,k$ we define the {\\em Gaussian coefficient\\\/}\n\\[\\gauss{n}k=\\gauss{n}k_q=\\begin{cases}\n\\frac{(q^n-1)(q^{n-1}-1)\\cdots(q^{n-k+1}-1)}{(q^k-1)(q^{k-1}-1)\\cdots(q-1)},\n&\\mbox{if\\ }k\\geqslant0;\\\\\n0,&\\mbox{if\\ }k<0.\n\\end{cases}\\]\nIn particular for $k=0$ the empty product gives $\\gauss{n}0=1$. In later sections, $q$ will be a fixed prime power; but here we may regard $q$ as an indeterminate, so that\nfor $n\\geqslant0$, after cancelling factors we find $\\gauss{n}k\\in{\\mathbb Z}[q]$; and specializing to\n$q=1$ gives the ordinary binomial coefficients $\\gauss{n}k_1=\\binom{n}k$. For general\n$n\\in{\\mathbb Z}$ we instead obtain a Laurent polynomial in $q$ with integer coefficients, i.e.\\ \n$\\gauss{n}k\\in{\\mathbb Z}[q,q^{-1}]$, as follows from conclusion (ii) of the following.\n\n\\begin{proposition}\\label{prop2.1}\nLet $n,k,\\ell\\in{\\mathbb Z}$. The Gaussian coefficients satisfy\n\\begin{itemize}\n\\item[(i)]$\\gauss{n}k=q^k\\gauss{n-1}k+\\gauss{n-1}{k-1}=\\gauss{n-1}k+q^{n-k}\\gauss{n-1}{k-1}$;\n\\item[(ii)]$\\gauss{-n}k=(-q^{-n})^k\\gauss{n+k-1}k$;\n\\item[(iii)]$\\gauss{n}k\\gauss{k}\\ell=\\gauss{n}\\ell\\gauss{n-\\ell}{k-\\ell}$;\n\\item[(iv)]$\\gauss{n}k=\\gauss{n}{n-k}$ whenever $0\\leqslant k\\leqslant n$.\\qed\n\\end{itemize}\n\\end{proposition}\nMost of the conclusions of Proposition~\\ref{prop2.1} are found in standard references such as~\\cite{A}. However, our definition of $\\gauss{n}k$ differs from the\nstandard definition found in most sources, which either leave $\\gauss{n}k$ undefined for $n<0$, or define it to be zero in that case.\nOur extension to all $n\\in{\\mathbb Z}$ means that the recurrence formulas~(i) hold for all integers $n,k$, unlike the `standard definition' which fails for $n=k=0$. Property~(i) plays a role in\nour later algebraic proofs using generating functions. In further defense of our definition,\nwe observe that it has become standard to extend the definition of binomial coefficients\n$\\binom{n}k$ so that $\\binom{-n}k=(-1)^k\\binom{n+k-1}k$ (see e.g.~\\cite[p.12]{A});\nand (ii) naturally generalizes this to Gaussian coefficients. We further note that (iii) holds for all $n,k\\in{\\mathbb Z}$\nwhether one takes the standard definition of $\\gauss{n}k$ or ours. The one advantage of\nthe standard definition is that it renders superfluous the extra restriction $0\\leqslant k\\leqslant n$\nin the symmetry condition~(iv). The interpretation of $\\gauss{n}k$ as the number of $k$-subspaces\nof an $n$-space over ${\\mathbb F}_q$ is valid for all $n\\geqslant0$.\n\nIn Section~\\ref{P-matrix} we will make use of the well-known generating polynomials\n\\[E_m(t)=\\prod_{i=0}^{m-1}(1+q^i t)=\\sum_{\\ell=0}^\\infty q^{\\binom{\\ell}2}\\gauss{m}\\ell t^\\ell\\quad\\mbox{for\\ }m=0,1,2,\\ldots;\\]\nnote that in the latter sum, the terms for $\\ell>m$ vanish, yielding $E_m(t)\\in{\\mathbb Z}[q,t]$ (or\nafter specializing to a fixed prime power $q$, we obtain $E_m(t)\\in{\\mathbb Z}[t]$).\nHere we see the usual binomial coefficient $\\binom{\\ell}2=\\frac12\\ell(\\ell-1)$. In Section~\\ref{P-matrix} we will make use of the following obvious relations:\n\n\\begin{proposition}\\label{prop2.2}\nFor all $m\\geqslant0$, the generating function $E_m(t)$ satisfies\n\\begin{itemize}\n\\item[(i)]$E_m(-q t)=\\frac{1- q^mt}{1-t} E_m(-t)$;\n\\item[(ii)]$E_m(q^2t)=\\frac{1+q^{m+1}t}{1+qt} E_m(qt)$; and\n\\item[(iii)]$E_m(r^3 t)=\\frac{1+rq^mt}{1+rt} E_m(rt)$ where $r=\\sqrt{q}$.\\qed\n\\end{itemize}\n\\end{proposition}\n\n\\section{Two-graphs and double covers of graphs}\n\\label{graphs}\n\nHere we describe the most basic connections between two-graphs and double covers of graphs; see \\cite{M,Se,BCN,Ta} for more details. Our notation is chosen to conform to that used in subsequent sections.\n\nLet $\\mathcal{V}$ be any set. Denote by $\\binom{\\mathcal{V}}k$ the collection of all $k$-subsets of $\\mathcal{V}$\n(i.e.\\ subsets of cardinality $k$). A {\\em two-graph on $\\mathcal{V}$\\\/} is a subset\n$\\Delta\\subseteq\\binom{\\mathcal{V}}3$ such that for\nevery 4-set $\\{x,y,z,w\\}\\in\\binom{\\mathcal{V}}4$, an even number, i.e.\\ 0, 2 or 4, of the triples\n$\\{x,y,z\\}$, $\\{x,y,w\\}$, $\\{x,z,w\\}$, $\\{y,z,w\\}$ is in $\\Delta$. If $\\Delta$ is a\ntwo-graph on $\\mathcal{V}$, then the complementary set of triples\n$\\overline\\Delta=\\bigl\\{\\{x,y,z\\}\\in\\binom\\V3:\\{x,y,z\\}\\notin\\Delta\\bigr\\}$ is also a\ntwo-graph, called the {\\em complementary two-graph on $\\mathcal{V}$\\\/}.\n\nA {\\em graph\\\/} on $\\mathcal{V}$ is a subset $\\Gamma\\subseteq\\binom\\V2$. Elements of\n$\\Gamma$ are called {\\em edges\\\/}.\nThe {\\em complete graph on $\\mathcal{V}$\\\/} is the graph $K_\\mathcal{V}$ with full edge set $\\binom{\\mathcal{V}}2$.\nIn general the complementary set of pairs\n$\\overline\\Gamma=\\bigl\\{\\{x,y\\}\\in\\binom\\V2:\\{x,y\\}\\notin\\Gamma\\bigr\\}$ is the\n{\\em complementary graph on $\\mathcal{V}$\\\/}.\n\nEvery graph on $\\mathcal{V}$ may be identified with a signing of the edges of the complete graph $K_\\mathcal{V}$, i.e.\\ a function $\\sigma:\\binom{\\mathcal{V}}2\\to\\{\\pm1\\}$. Under this correspondence, the graph corresponding to $\\sigma$ has as its edge set $\\sigma^{-1}(1)=\\bigl\\{\\{x,y\\}\\in\\binom{\\mathcal{V}}2:\\sigma(x,y)=1\\bigr\\}$. (Here we abbreviate\n$\\sigma(\\{x,y\\})=\\sigma(x,y)$.)\n\nGiven $\\Gamma$ and $\\sigma$ as above (which amounts to two graphs which may be\nentirely unrelated except for sharing the same vertex set $\\mathcal{V}$), we construct a new graph\n$\\widehat{\\Gamma}=\\widehat{\\Gamma}_\\sigma$ with vertex set $\\widehat{\\V}=\\mathcal{V}\\times\\{\\pm1\\}$ and adjacency relation defined by\n\\[(x,\\varepsilon)\\sim(y,\\varepsilon')\\;\\iff\\;x\\sim y\\mbox{\\ and\\ }\\varepsilon\\varepsilon'=\\sigma(x,y).\\]\n(Note that $(x,1)\\not\\sim(x,-1)$ since $\\Gamma$ has no loops.)\nThe map $(x,\\varepsilon)\\mapsto x$ is a {\\em double covering map\\\/} $\\theta:\\widehat{\\Gamma}\\to\\Gamma$, also called a {\\em double cover\\\/} or simply a {\\em cover\\\/}; and the {\\em fibers\\\/}\nof this map are the pairs $\\theta^{-1}(x)=\\{(x,1),(x,-1)\\}$ where $x\\in\\mathcal{V}$.\n(By definition, a {\\em covering map\\\/} of graphs is a graph homomorphism $\\theta:\\widehat{\\Gamma}\\to\\Gamma$ such that for any vertex $x\\in\\Gamma$, the preimage of the neighborhood graph $\\Gamma_x$ is isomorphic to a disjoint union of copies of $\\Gamma_x$; see e.g.~\\cite{GR}. `Double' refers to the condition that the covering map is\n2-to-1.) We also say that the vertices $(x,1)$ and $(x,-1)$ are {\\em antipodal\\\/}\nwith respect to the covering map. (Note that antipodal vertices must be at\ndistance${}\\geqslant2$; but we deviate from common custom by\n{\\em not requiring\\\/} pairs of antipodal vertices to be at maximal distance $\\diam\\widehat{\\Gamma}$.) We denote\nby $\\zeta$ the transposition interchanging antipodal vertices:\n$(x,1)\\stackrel\\zeta\\leftrightarrow(x,-1)$. Denote by\n$\\mathop{\\rm Aut}\\nolimits_\\zeta\\widehat{\\Gamma}\\leqslant\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}$\nthe subgroup consisting of all automorphisms of the graph $\\widehat{\\Gamma}$ which preserve the\nantipodality relation. In general, $\\mathop{\\rm Aut}\\nolimits_\\zeta\\widehat{\\Gamma}$ is the centralizer of $\\zeta$\nin the full automorphism group $\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}\\leqslant\\mathop{\\rm Sym}\\nolimits\\widehat{\\V}$; but in our case\nwe obtain equality $\\mathop{\\rm Aut}\\nolimits_\\zeta\\widehat{\\Gamma}=\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}$ (see\nLemma~\\ref{lemma5.4}).\nSimilarly, two covers $\\theta_i:\\widehat{\\Gamma}_i\\to\\Gamma$ of the\nsame graph $\\Gamma$ (for $i=1,2$) are {\\em equivalent\\\/} or {\\em isomorphic\\\/}\nif there is a graph isomorphism $\\rho:\\widehat{\\Gamma}_1\\to\\widehat{\\Gamma}_2$ which preserves\nantipodality, i.e.\\ $\\theta_1\\circ\\rho=\\theta_2$.\n\nGiven $\\sigma:\\binom{\\mathcal{V}}2\\to\\{\\pm1\\}$ as above, for every triple $\\{x,y,z\\}\\in\\binom{\\mathcal{V}}3$ we may define\n\\[\\sigma(x,y,z)=\\sigma(x,y)\\sigma(y,z)\\sigma(z,x)\\in\\{\\pm1\\}.\\]\nA triple $\\{x,y,z\\}\\in\\binom{\\mathcal{V}}3$ is called {\\em coherent\\\/} or {\\em non-coherent\\\/}\naccording as $\\sigma(x,y,z)=1$ or $-1$.\nThe set of all coherent triples forms a two-graph on $\\mathcal{V}$, denote by $\\Delta_\\sigma$; and\nthe set of non-coherent triples gives the complementary two-graph $\\overline{\\Delta}_\\sigma$.\n\nTwo sign functions $\\sigma_1,\\sigma_2:\\binom{\\mathcal{V}}2\\to\\{\\pm1\\}$ (or the corresponding graphs $\\sigma_1^{-1}(1)$, $\\sigma_2^{-1}(1)$ on $\\mathcal{V}$) are {\\em switching-equivalent\\\/}\nin the sense of Seidel~\\cite{Se} if there exists a map $f:\\mathcal{V}\\to\\{\\pm1\\}$ such that $\\sigma_2(x,y)=f(x)f(y)\\sigma_1(x,y)$ for all $\\{x,y\\}\\in\\binom{\\mathcal{V}}2$. We have\n$\\Delta_{\\sigma_1}=\\Delta_{\\sigma_2}$ iff $\\sigma_1$ and $\\sigma_2$ are\nswitching-equivalent. Assuming this holds,\nthen the corresponding covers $\\widehat{\\Gamma}_{\\sigma_1}$ and $\\widehat{\\Gamma}_{\\sigma_2}$\nare isomorphic via $(x,\\varepsilon)\\mapsto(x,f(x)\\varepsilon)$.\n\nIn the special case of the complete graph\n$\\Gamma=K_\\mathcal{V}$, the following three notions are equivalent (see \\cite[\\S1.5]{BCN}): two-graphs on $\\mathcal{V}$,\nswitching classes of graphs on $\\mathcal{V}$, and isomorphism classes of double covers of the complete graph $K_\\mathcal{V}$. For example given a double cover $\\widehat{K_\\mathcal{V}}\\to K_\\mathcal{V}$, the\ncorresponding two-graph is obtained as follows (see \\cite[p.488]{Se}): Each triple $\\{x,y,z\\}$ of distinct vertices in\n$\\mathcal{V}$ induces a triangle $K_{\\{x,y,z\\}}\\subseteq K_\\mathcal{V}$; and such a triple is coherent iff its preimage in $\\widehat{K_\\mathcal{V}}$ induces a pair of triangles, rather than a 6-cycle, in $\\widehat{K}_\\mathcal{V}$.\n\nAn {\\em automorphism of a two-graph $\\Delta$\\\/} is a permutation of the underlying\npoint set $\\mathcal{V}$ which preserves the set of coherent triples. We now relate $\\mathop{\\rm Aut}\\nolimits\\Delta$\nto the group $\\mathop{\\rm Aut}\\nolimits_\\zeta\\widehat{K}\\leqslant\\mathop{\\rm Aut}\\nolimits\\widehat{K}$ defined above for the associated double cover\n$\\widehat{K}\\to K$, where we abbreviate the complete graph $K_\\mathcal{V}=K$. The following is easy to verify (or see \\cite[\\S2]{Ta}, where this isomorphism is denoted $\\widehat{G}\/Z\\cong G$):\n\n\\begin{proposition}\\label{prop3.1}\nThe group $\\mathop{\\rm Aut}\\nolimits_\\zeta\\widehat{K}$ acts naturally on $\\Delta$, inducing the full automorphism group of $\\Delta$.\nThe kernel of this action is the central subgroup $\\langle\\zeta\\rangle$ of order~2; thus $(\\mathop{\\rm Aut}\\nolimits_\\zeta\\widehat{K})\/\\langle\\zeta\\rangle\\cong\\mathop{\\rm Aut}\\nolimits\\Delta$.\n\\end{proposition}\n\n\\section{Dual polar graphs of type $Sp(2n,q)$, $q$ odd}\n\\label{dual_polar}\n\nFix a finite field ${\\mathbb F}_q$ of odd prime power order $q$; an integer $n\\geqslant1$; a $2n$-dimensional vector space $V$ over ${\\mathbb F}_q$; and a symplectic (i.e.\\ nondegenerate alternating) bilinear form $B:V\\times V\\to{\\mathbb F}_q$. The {\\em symplectic group\\\/} $Sp(2n,q)$ consists of\nall {\\em(linear) isometries\\\/} of $B$, i.e.\n\\[Sp(2n,q)=\\{g\\in GL(V):B(x^g,y^g)=B(x,y)\\mbox{\\ for all\\ }x,y\\in V\\}.\\]\nThe group of all {\\em (linear) similarities\\\/} of $B$ is\n\\begin{align*}\nGSp(2n,q)&=\\{g\\in GL(V):\\mbox{for some nonzero\\ }\\mu\\in{\\mathbb F}_q\\mbox{\\ we have}\\\\ &\\qquad B(x^g,y^g)=\\mu B(x,y)\\mbox{\\ for all\\ }x,y\\in V\\};\n\\end{align*}\nsome other notations for this group are $GSp_n(q)$ in~\\cite{KL} or $CSp_n(q)$ in~\\cite[p.31]{BHR}.\nReplacing $GL(V)$ by $\\Gamma L(V)\\cong GL(V)\\rtimes\\mathop{\\rm Aut}\\nolimits{\\mathbb F}_q$, the group of all semilinear transformations of $V$, we obtain the group $\\Sigma Sp(2n,q)$ of\nall {\\em semi-isometries\\\/}, and\nthe group $\\Gamma Sp(2n,q)$ of all {\\em semi-similarities\\\/} of $B$, given by\n\\begin{align*}\n\\Sigma Sp(2n,q)&=\\{g\\in\\Gamma L(V):\\mbox{for some\\ }\\tau\\in\\mathop{\\rm Aut}\\nolimits{\\mathbb F}_q\\mbox{\\ we have}\\\\\n&\\qquad\\quad B(x^g,y^g)=B(x,y)^\\tau\\mbox{\\ for all\\ }x,y\\in V\\}\\\\\n&\\cong Sp(2n,q)\\rtimes\\mathop{\\rm Aut}\\nolimits{\\mathbb F}_q;\\\\\n\\Gamma Sp(2n,q)&=\\{g\\in\\Gamma L(V):\\mbox{for some nonzero\\ }\\mu\\in{\\mathbb F}_q\\mbox{\\ and\\ }\\tau\\in\\mathop{\\rm Aut}\\nolimits{\\mathbb F}_q\\\\\n&\\qquad\\quad\\mbox{we have\\ }B(x^g,y^g)=\\mu B(x,y)^\\tau\\mbox{\\ for all\\ }x,y\\in V\\}\\\\\n&\\cong GSp(2n,q)\\rtimes\\mathop{\\rm Aut}\\nolimits{\\mathbb F}_q.\n\\end{align*}\nThe projective versions of these groups are\n\\begin{align*}PSp(2n,q)&=Sp(2n,q)\/\\langle-I\\rangle,\\\\\nPGSp(2n,q)&=GSp(2n,q)\/Z,\\\\\nP\\Sigma Sp(2n,q)&=\\Sigma Sp(2n,q)\/\\langle-I\\rangle,\\\\\nP\\Gamma Sp(2n,q)&=\\Gamma Sp(2n,q)\/Z\n\\end{align*}\nwhere the central subgroup $Z$ of order $q-1$ consists of all scalar transformations $v\\mapsto\\lambda v$\nfor $0\\neq\\lambda\\in{\\mathbb F}_q$. We have\n\\[[P\\Gamma Sp(2n,q):P\\Sigma Sp(2n,q)]=[PGSp(2n,q):PSp(2n,q)]=2\\]\nwhere the nontrivial coset in both cases is represented by $h\\in GSp(2n,q)$\nsatisfying $B(u^h,v^h)=\\eta B(u,v)$ and $\\eta\\in{\\mathbb F}_q$ is a nonsquare.\n\nOur choice of notation for these groups, while not universal, is intended to conform\nreasonably with \\cite{Atlas,KL}. The group $P\\Gamma Sp(2n,q)$, for example, is denoted $PC\\Gamma Sp_n(q)$ in \\cite[p.31]{BHR}. It arises (see Theorem~\\ref{theorem4.1}) as the full automorphism\ngroup of the associated dual polar graph, which we now describe.\n\nDenote by $\\mathcal{V}$ be the collection of all maximal totally\nisotropic subspaces with respect to $B$, i.e.\\ \n\\[\\mathcal{V}=\\{X\\leqslant V:X^\\perp=X\\}\\]\nwhere by definition $X^\\perp=\\{v\\in V:B(x,v)=0\\mbox{\\ for all\\ }x\\in X\\}$. Members of\n$\\mathcal{V}$ are often called {\\em generators\\\/}, and every $X\\in\\mathcal{V}$ has dimension~$n$.\nDenote by $\\Gamma=\\Gamma(2n,q)$ the graph on $\\mathcal{V}$ where two vertices\n$X,Y\\in\\mathcal{V}$ are adjacent iff $X\\cap Y$ has codimension~1 in both $X$ and $Y$. More generally, the distance between $X$ and $Y$ in $\\Gamma$ is\n$d(X,Y)=k\\in\\{0,1,2,\\ldots,n\\}$ where the subspace $X\\cap Y$ has codimension $k$ in both $X$ and~$Y$. Let $\\Gamma_k$ denote the graph of the\ndistance-$k$ relation on $\\mathcal{V}$; i.e.\\ $\\Gamma_k$ has vertex set $\\mathcal{V}$ and two vertices $X,Y\\in\\mathcal{V}$ are adjacent in $\\Gamma_k$ iff $d(X,Y)=k$.\nThe graph $\\Gamma_1=\\Gamma$ is called the {\\em dual polar graph of type\\\/} Sp$(2n,q)$. It is {\\em distance\nregular\\\/}: given any two vertices $X,Y$ in $\\Gamma$ at distance $k\\in\\{0,1,2,\\ldots,n\\}$, the vertex $Y$ has $q^{\\binom{n-k}2}\\gauss{n}k$ neighbors $Z$ in $\\Gamma$, of which\n\\begin{align*}\na_k&=q^k-1\\mbox{\\ are at distance $k$ from\\ }X,\\\\\nb_k&=q^{k+1}\\gauss{n-k}1\\mbox{\\ are at distance $k+1$ from $X$, and}\\\\\nc_k&=\\gauss{k}1\\mbox{\\ are at distance $k-1$ from\\ }X;\n\\end{align*}\nsee \\cite[\\S9.4]{BCN}.\nThe edges of $\\Gamma_1,\\Gamma_2,\\ldots,\\Gamma_n$ partition the non-identical pairs\non $\\mathcal{V}$, viewed as the edges of the complete graph $K_\\mathcal{V}$; and together with the identity relation\n$\\Gamma_0=\\{(X,X):X\\in\\mathcal{V}\\}$ we obtain an $n$-class association scheme on $\\mathcal{V}$ (see Section~\\ref{scheme}). This scheme is $P$-polynomial since $\\Gamma$ is distance regular; see~\\cite{BCN}.\n\\begin{theorem}\\label{theorem4.1}\nFor $n\\geqslant2$, the full automorphism group of $\\Gamma=\\Gamma(2n,q)$ is the group $P\\Gamma Sp(2n,q)$ acting naturally on the projective space of $V$.\n\\end{theorem}\n\n\\begin{pf}\nSee \\cite[p.275]{BCN} (where this group is however denoted $P\\Sigma p(2n,q)$).\\qed\n\\end{pf}\nNote that when $n=1$, the dual polar graph $\\Gamma(2,q)$ is simply the complete graph\n$K_{q+1}$, whose full automorphism group is the symmetric group of degree $q+1$.\n\nFor use in Section~\\ref{double_cover} we record the following well-known fact. Although it follows easily from the axioms of polar geometry (or of near polygons), in the interest of self-containment we include a proof.\n\n\\begin{lemma}\\label{lemma4.2}\nThe `diamond' graph (as shown) is not an induced subgraph of the dual polar graph $\\Gamma$.\n\\end{lemma}\n\\[\\begin{tikzpicture}[scale=1.1]\n\\node at (-1,0) {$\\bullet$};\n\\node at (-1.25,0) {$X$};\n\\node at (0,.73) {$\\bullet$};\n\\node at (.22,.85) {$Y$};\n\\node at (0,-.73) {$\\bullet$};\n\\node at (.2,-.85) {$Z$};\n\\node at (1,0) {$\\bullet$};\n\\node at (1.3,0) {$W$};\n\\draw (-1,0) -- (0,.73) -- (1,0) -- (0,-.73) -- (-1,0);\n\\draw (0,.73) -- (0,-.73);\n\\end{tikzpicture}\\]\n\n\\begin{pf}If $X,Y,Z$ are mutually adjacent as shown, then $X\\cap Y$ and $X\\cap Z$ are distinct subspaces of codimension~1 in $X$, so $X=(X\\cap Y)+(X\\cap Z)$, whence $X\\subseteq Y+Z$. Thus $X=X^\\perp\\supseteq Y^\\perp\\cap Z^\\perp=Y\\cap Z$. Similarly,\n$W\\supseteq Y\\cap Z$. Now $X\\cap W$ contains a subspace of dimension $n{-}1$, contradicting $d(X,W)\\geqslant2$.\\qed\n\\end{pf}\n\nNow let $X$ be any $n$-dimensional vector space over ${\\mathbb F}_q$. An $n$-linear form $f:X^n\\to{\\mathbb F}_q$ (i.e.\\ linear in each argument whenever the other\n$n-1$ arguments are fixed) is {\\em alternating\\\/} if $f(x_1,x_2,\\ldots,x_n)=0$ whenever two $x_i$'s coincide; equivalently,\n$f(x_{1^\\tau},x_{2^\\tau},\\ldots,x_{n^\\tau})=-f(x_1,x_2,\\ldots,x_n)$ for every odd permutation $\\tau$ of the indices.\nThe space of all such alternating forms is one-dimensional, and is canonically identified with $(\\bigwedge^{\\mskip-2mu n}\\mskip-2mu X)^*$,\nthe dual space of $\\bigwedge^{\\mskip-2mu n}\\mskip-2mu X$. A {\\em determinant function on $X$\\\/} is any nonzero alternating form $X^n\\to{\\mathbb F}_q$. Since\n$\\dim(\\bigwedge^{\\mskip-2mu n}\\mskip-2mu X)^*=1$, a determinant function is determined up to nonzero scalar multiple.\n\nFix a choice of determinant function $\\delta_X$ for each $X\\in\\mathcal{V}$. Although these choices are not canonical, one may proceed by arbitrarily choosing a basis\n$\\psi_1,\\psi_2,\\ldots,\\psi_n$ for $X^*=\\mathop{\\rm Hom}\\nolimits(X,{\\mathbb F}_q)$; then we obtain a determinant function on $X$ by defining\n\\[\\delta_X(x_1,x_2,\\ldots,x_n)=\\det\\bigl(\\psi_i^*(x_j):1\\leqslant i,j\\leqslant n\\bigr).\\]\n\nWe need to define $\\sigma(X,Y)\\in\\{\\pm1\\}$ for any pair $X\\neq Y$ in $\\mathcal{V}$. Let $k\\in\\{1,2,\\ldots,n\\}$ be the codimension of $X\\cap Y$ in both\n$X$ and $Y$. Choose bases $x_1,x_2,\\ldots,x_n$ and $y_1,y_2,\\ldots,y_n$ for $X$ and $Y$ respectively, such that $x_i=y_i$ (for $k<i\\leqslant n$)\nis a common basis for $X\\cap Y$. (These bases depend on the choice of pair $(X,Y)$ and so are unrelated to any bases for $X$ and $Y$ used as a crutch\nfor constructing the corresponding determinant functions). Define\n\\[\\sigma(X,Y)=\\chi\\bigl(\\delta_X(x_1,x_2,\\ldots,x_n)\\delta_Y(y_1,y_2,\\ldots,y_n)\\det\\bigl[B(x_i,y_j):1\\leqslant i,j\\leqslant k\\bigr]\\bigr)\\]\nwhere $\\chi:{\\mathbb F}_q^{\\times}\\to\\{\\pm1\\}$ is the quadratic character:\n$\\chi(a)=1$ or $-1$ according as $a\\in{\\mathbb F}_q^{\\times}$ is a square or a nonsquare.\nThis definition is implicit in~\\cite{Th,KS}; and inspired by the literature, we refer to\n$\\sigma(X,Y)$ (or the ternary function $\\sigma(X,Y,Z)$ defined below) as the\n{\\em Maslov index\\\/}. Note that $B$ induces a nondegenerate bilinear form on the\n$2k$-space $(X+Y)\/(X\\cap Y)$, so that the $k\\times k$\nmatrix $\\bigl[B(x_i,y_j):1\\leqslant i,j\\leqslant k\\bigr]$ is nonsingular.\n\\bigskip\n\n\\begin{proposition}\\label{prop4.3}\nLet $X,Y\\in\\mathcal{V}$ at distance $d(X,Y)=k\\in\\{0,1,2,\\ldots,n\\}$.\n\\begin{itemize}\n\\item[(i)]The value of $\\sigma(X,Y)$ is independent of the choice of bases $x_i$ and $y_j$ as above.\n\\item[(ii)]Its dependence on the choice of determinant functions is expressed as follows:\nReplacing $\\delta_X$ by $c\\delta_X$ has the effect of multiplying $\\sigma(X,Y)$ by $\\chi(c)$.\n\\item[(iii)]$\\sigma(Y,X)=\\chi(-1)^k\\sigma(X,Y)=(-1)^{k(q-1)\/2}\\sigma(X,Y)$.\n\\item[(iv)]Let $g\\in\\Gamma Sp(2n,q)$, so that there exists a nonzero scalar $\\mu_g\\in{\\mathbb F}_q$\nand $\\tau_g\\in\\mathop{\\rm Aut}\\nolimits{\\mathbb F}_q$ satisfying $B(x^g,y^g)=\\mu_g B(x,y)^{\\tau_g}$ for all $x,y\\in Y$.\nThen there exist nonzero scalars $\\lambda_{g,U}\\in{\\mathbb F}_q$ for $U\\in\\mathcal{V}$, such that\n\\[\\sigma(X^g,Y^g)=\\chi\\bigl(\\mu_g^k\\lambda_{g,X}\\mskip1mu\\lambda_{g,Y}\\bigr)\\sigma(X,Y).\\]\n\\end{itemize}\n\\end{proposition}\n\n\\begin{pf}\nConsider a change of basis on $X\\cap Y$ specified by $x_i'=y_i'=\\sum_{k<j\\leqslant n}a_{ij}x_j$ where $A=\\bigl(a_{ij}:k<i,j\\leqslant n\\bigr)$\nis any invertible $(n-k)\\times(n-k)$ matrix. Then\n\\[\\delta_X(x_1,\\ldots,x_k,x_{k+1}',\\ldots,x_n')=(\\det A)\\delta_X(x_1,\\ldots,x_n)\\]\nand $\\delta_Y(y_1,\\ldots,y_n)$ is multiplied by the same factor, $\\det A$. The $(n-k)\\times (n-k)$ matrix $[B(x_i,y_j):k<i,j\\leqslant n]$ is unchanged, so the\nvalue of $\\sigma(X,Y)$ is multiplied by a net factor of $\\chi\\bigl((\\det A)^2\\bigr)=1$.\n\nNext consider replacing $x_1,\\ldots,x_k$ by $x_1',\\ldots,x_k'$ where\n\\[x_i'\\equiv\\sum_{1\\leqslant j\\leqslant k}a_{ij}x_j\\quad{\\rm mod\\ }(X\\cap Y)\\]\nfor $i=1,2,\\ldots,k$ where $A$ is an invertible $k\\times k$ matrix, and we leave the basis of $Y$ unchanged. Then\n\\[\\begin{array}{c}\n\\delta_X(x_1',\\ldots,x_k',x_{k+1},\\ldots,x_n)=(\\det A)\\delta_X(x_1,\\ldots,x_n);\\\\\n\\det\\bigl[B(x_i',y_j):1\\leqslant i,j\\leqslant k\\bigr]=(\\det A)\\det\\bigl[B(x_i,y_j):1\\leqslant i,j\\leqslant k\\bigr]\n\\end{array}\\]\nand the $\\delta_Y$ factor is unchanged; so once again, the value of $\\sigma(X,Y)$ is multiplied by $\\chi\\bigl((\\det A)^2\\bigr)=1$.\nThe same argument applies if $y_1,\\ldots,y_k$ are replaced by $y_1',\\ldots,y_k'$, and so~(i) follows. Conclusion~(ii) is clear.\n\nInterchanging $X$ and $Y$ has the effect of interchanging the $\\delta_X$ and $\\delta_Y$ factors, and replacing\n\\begin{multline*}\n\\bigl[B(x_i,y_j):1\\leqslant i,j\\leqslant k\\bigr]\\mapsto\\\\\n\\bigl[B(y_i,x_j):1\\leqslant i,j\\leqslant k\\bigr]=-\\bigl[B(x_i,y_j):1\\leqslant i,j\\leqslant k\\bigr].\n\\end{multline*}\nThe determinant of this matrix accrues a factor of $(-1)^k$, whence (iii) holds.\n\nLet $g\\in\\Gamma Sp(2n,q)$. There exists a nonzero $\\mu_g\\in{\\mathbb F}_q$ and\n$\\tau_g\\in\\mathop{\\rm Aut}\\nolimits{\\mathbb F}_q$ such that $(au+bv)^g=a^{\\tau_g}u^g+b^{\\tau_g}v^g$ and $B(u^g,v^g)=\\mu_g B(u,v)^{\\tau_g}$ for all\n$a,b\\in{\\mathbb F}_q$ and $u,v\\in V$. Now the map\n\\[X^n\\to{\\mathbb F}_q,\\quad(x_1,x_2,\\ldots,x_n)\\mapsto\\delta_{X^g}\\bigl(x_1^g,x_2^g,\\ldots,x_n^g\\bigr)^{\\tau_g^{-1}}\\]\nis a determinant function on $X$, so it is a scalar multiple of $\\delta_X(x_1,x_2,\\ldots,x_n)$. Hence there exists a nonzero scalar $\\lambda_X=\\lambda_{g,X}\\in{\\mathbb F}_q$ such that\n\\[\\delta_{X^g}\\bigl(x_1^g,x_2^g,\\ldots,x_n^g\\bigr)=\\lambda_{g,X}\\delta_X(x_1,x_2,\\ldots,x_n)^{\\tau_g}\\]\nfor all $x_1,x_2,\\ldots,x_n\\in X$.\n\nNow given $X,Y\\in\\mathcal{V}$ at distance $k$, fix bases $x_i,y_i$ as before; then\n\\begin{align*}\n\\sigma(X^g,Y^g)&=\\chi\\bigl(\\delta_{X^g}(x_1^g,x_2^g,\\ldots,x_n^g)\\delta_{Y^g}(y_1^g,y_2^g,\\ldots,y_n^g)\\\\\n&{\\mskip100mu}\\times\\det\\bigl[B(x_i^g,y_j^g):1\\leqslant i,j\\leqslant k\\bigr]\\bigr)\\\\\n&=\\chi\\bigl(\\lambda_{g,X}\\delta_X(x_1,x_2,\\ldots,x_n)^{\\tau_g}\\lambda_{g,Y}\\delta_Y(y_1,y_2,\\ldots,y_n)^{\\tau_g}\\\\\n&{\\mskip100mu}\\times\\det\\bigl[\\mu_g B(x_i,y_j)^{\\tau_g}:1\\leqslant i,j\\leqslant k\\bigr]\\bigr)\\\\\n&=\\chi\\bigl(\\mu_g^k\\lambda_{g,X}\\lambda_{g,Y}\\bigr)\\chi\\bigl(\\delta_X(x_1,x_2,\\ldots,x_n)^{\\tau_g}\\delta_Y(y_1,y_2,\\ldots,y_n)^{\\tau_g}\\\\\n&{\\mskip100mu}\\times\\det\\bigl[B(x_i,y_j):1\\leqslant i,j\\leqslant k\\bigr]^{\\tau_g}\\bigr)\\\\\n&=\\chi\\bigl(\\mu_g^k\\lambda_{g,X}\\mskip1mu\\lambda_{g,Y}\\bigr)\\sigma(X,Y)\n\\end{align*}\nsince $\\chi(a^\\tau)=\\chi(a)$. This proves (iv).\\qed\n\\end{pf}\n\nFor each triple $(X,Y,Z)$ with distinct $X,Y,Z\\in\\mathcal{V}$, define\n\\[\\sigma(X,Y,Z)=\\sigma(X,Y)\\sigma(Y,Z)\\sigma(Z,X)\\in\\{\\pm1\\}.\\]\nA triple $(X,Y,Z)$ of distinct elements of $\\mathcal{V}$ is {\\em coherent\\\/} or {\\em non-coherent\\\/}\naccording as $\\sigma(X,Y,Z)=1$ or $-1$.\n\n\\begin{theorem} \\label{theorem4.4}\nSuppose $q\\equiv1\\,$ mod~$4$. Then the set of coherent\ntriples forms a two-graph $\\Delta_\\sigma$ on $\\mathcal{V}$, invariant under $P\\Sigma Sp(2n,q)$.\n\\end{theorem}\n\n\\begin{pf}\nLet $X,Y,Z,W\\in\\mathcal{V}$ be distinct. Since $\\chi(-1)=1$, $(X,Y,Z)$ is coherent iff any permutation of its members yields a coherent triple; so\nthe set of coherent triples may be regarded as a collection of unordered triples $\\{X,Y,Z\\}$. Since\n\\begin{multline*}\n\\sigma(X,Y,Z)\\sigma(X,Y,W)\\sigma(X,Z,W)\\sigma(Y,Z,W)\\\\\n=\\sigma(X,Y)^2\\sigma(X,Z)^2\\cdots\\sigma(Z,W)^2=1,\n\\end{multline*}\nevenly many of the triples in $\\{X,Y,Z,W\\}$ are coherent. If $g\\in\\Gamma Sp(2n,q)$ with $B(x^g,y^g)=\\mu_g B(x,y)^{\\tau_g}$, then\n\\begin{align*}\n\\sigma(X^g,Y^g,Z^g)&=\\chi(\\mu_g^{d(X,Y)}\\lambda_{g,X}\\mskip1mu \\lambda_{g,Y})\\chi(\\mu_g^{d(Y,Z)}\\lambda_{g,Y}\\mskip1mu\\lambda_{g,Z})\\\\\n&\\phantom{{}={}}\\times\\chi(\\mu_g^{d(Z,X)}\\lambda_{g,Z}\\mskip1mu \\lambda_{g,X})\\sigma(X,Y,Z)\\\\\n\\phantom{\\sigma(X^g,Y^g,Z^g)}&=\\chi(\\mu_g)^{d(X,Y)+d(Y,Z)+d(Z,X)}\\sigma(X,Y,Z).\n\\end{align*}\nIn particular when $g\\in\\Sigma Sp(2n,q)$, $\\mu_g=1$ and $\\sigma(X^g,Y^g,Z^g)=\\sigma(X,Y,Z)$.\\qed\n\\end{pf}\n\nIf $q\\equiv3\\,$ mod~$4$, or $g\\in P\\Gamma Sp(2n,q)$ with $g\\notin P\\Sigma Sp(2n,q)$,\nthe situation is a little trickier: various subsets of the coherent triples form either two-graphs or skew two-graphs in the sense of \\cite{M},\ninvariant under $Sp(2n,q)$. We ignore this case here, and {\\em henceforth assume that\\\/}\n\\[q\\equiv1\\,\\hbox{\\ mod\\ }4.\\]\n\nWe next show that {\\em in a geodesic path, every triple of vertices is coherent.\\\/}\n\n\\begin{lemma} \\label{lemma4.5}\nSuppose $q\\equiv1\\,$ \\hbox{mod $4$}. Let $X,Y,Z\\in\\mathcal{V}$ such that $d(X,Y)=j$, $d(Y,Z)=k-j$ and $d(X,Z)=k$ where\n$1\\leqslant j<k\\leqslant n$. Then $\\sigma(X,Y,Z)=1$.\n\\end{lemma}\n\n\\begin{pf}\nChoose a hyperbolic basis $e_1,e_2,\\ldots,e_n,f_1,f_2,\\ldots,f_n$ for $V$, so that $B(e_i,e_j)=B(f_i,f_j)=0$ and $B(e_i,f_j)=\\delta_{ij}$.\nSince Sp$(2n,q)$ is transitive on triples of generators satisfying the given distance constraints, by Theorem~\\ref{theorem4.4} we may suppose that\n\\[\\begin{array}{c}\nX=\\langle e_1,e_2,\\ldots,e_n\\rangle,\\quad Y=\\langle f_1,f_2,\\ldots,f_j,e_{j+1},e_{j+2},\\ldots,e_n\\rangle,\\\\\nZ=\\langle f_1,f_2,\\ldots,f_k,e_{k+1},e_{k+2},\\ldots,e_n\\rangle.\n\\end{array}\\]\nWe choose the determinant function $\\delta_X$ on $X$ given by\n\\[\\delta_X(x_1,x_2,\\ldots,x_n)=\\det\\bigl[B(x_i,f_j)\\,:\\,1\\leqslant i,j\\leqslant n\\bigr]\\quad\\hbox{for\\ }x_1,x_2,\\ldots,x_n\\in X;\\]\nthis is nothing other than the determinant of the $n\\times n$ matrix whose columns are the coordinates of $x_1,\\ldots,x_n$ with respect to the\nbasis $e_1,e_2,\\ldots,e_n$. The determinant functions $\\delta_Y$, $\\delta_Z$ on $Y$ and on $Z$ are defined similarly, using the bases for $Y$ and on\n$Z$ listed above. The computation of $\\sigma(X,Z)$ is simplified by the fact that a basis for $X\\cap Z$ is $e_{k+1},e_{k+2},\\ldots,e_n$. We have\n\\[\\delta_X(e_1,e_2,\\ldots,e_n)=\\delta_Z(f_1,\\ldots,f_k,e_{k+1},\\ldots,e_n)=1\\]\nand $\\bigl[B(e_i,f_j)\\,:\\,1\\leqslant i,j\\leqslant k\\bigr]$ is a $k\\times k$ identity matrix, with determinant~1; thus $\\sigma(X,Z)=1$. Exactly the same reasoning\ngives $\\sigma(X,Y)=\\sigma(Y,Z)=1$, so $\\sigma(X,Y,Z)=1$.\\qed\n\\end{pf}\n\nIn the case of triples $X,Y,Z$ not lying on geodesic paths, however, $\\sigma$ (or its two-graph $\\Delta_\\sigma$) yields interesting nontrivial information. In particular, the restriction of $\\Delta_\\sigma$\nto partial spreads (sets of vertices of $\\Gamma$ mutually at distance $n$) was investigated in~\\cite[\\S6]{M}.\nHere we consider triangles in $\\Gamma$:\n\n\\begin{lemma}\\label{lemma4.6}\nSuppose $q\\equiv1\\,$ \\hbox{mod $4$}. Let $X,Y\\in\\mathcal{V}$ such that $d(X,Y)=1$, i.e.\\ $X$ and $Y$ are adjacent in $\\Gamma$.\nThere are $a_1=q{-}1$ common neighbors $Z$ of $X$ and $Y$ in $\\Gamma$; and exactly half of the resulting triples $(X,Y,Z)$ are coherent.\n\\end{lemma}\n\n\\begin{pf}\nChoose a hyperbolic basis $e_i,f_i$ as in the proof of Lemma~\\ref{lemma4.5}. Again without loss of generality,\n\\[X=\\langle e_1,e_2,\\ldots,e_n\\rangle,\\quad Y=\\langle f_1,e_2,\\ldots,e_n\\rangle,\\quad Z=\\langle e_1{+}\\alpha f_1,e_2,\\ldots,e_n\\rangle\\]\nwhere $0\\neq\\alpha\\in{\\mathbb F}_q$. The $q{-}1$ choices of $\\alpha$ give exactly the $q{-}1$ common neighbors of $X$ and $Y$ in $\\Gamma$.\nThese bases of $X,Y,Z$ give rise to natural choices of determinant functions $\\delta_X$,\n$\\delta_Y$, $\\delta_Z$ as described in the proof of Lemma~\\ref{lemma4.5}. When computing\n$\\sigma(X,Y),\\sigma(Y,Z),\\sigma(Z,X)$, we use $e_2,e_3,\\ldots,e_n$ as the basis of $X\\cap Y=X\\cap Z=Y\\cap Z$. Now\n\\[\\delta_X(e_1,e_2,\\ldots,e_n)=\\delta_Z(e_1{+}\\alpha f_1,e_2,\\ldots,e_n)=1\\]\nand $B(e_1,e_1{+}\\alpha f_1)=\\alpha$, so $\\sigma(X,Z)=\\chi(\\alpha)$. Similarly, $\\sigma(X,Y)=\\sigma(Y,Z)=1$ and\n\\[\\sigma(X,Y,Z)=\\chi(\\alpha).\\]\nSince exactly half the nonzero elements of ${\\mathbb F}_q$ are squares, the result follows.\\qed\n\\end{pf}\n\n\\begin{theorem}\\label{theorem4.7}\nSuppose $q\\equiv1\\,$ \\hbox{mod $4$}. Let $X,Y\\in\\mathcal{V}$ such that $d(X,Y)=k$. Then $Y$ has exactly $a_k=q^k{-}1$ neighbors $Z\\in\\mathcal{V}$\nat distance $k$ from $X$ in $\\Gamma$; and exactly half of the resulting triples $(X,Y,Z)$ are coherent.\n\\end{theorem}\n\n\\begin{pf}\nThe result holds for $k=1$ by Lemma~\\ref{lemma4.6}, so we may assume $k\\geqslant2$.\nGiven $X,Y\\in\\mathcal{V}$ with $d(X,Y)=k$, there are $\\gauss{k}1$ choices of hyperplane $H<Y$ containing $X\\cap Y$. Each such $H$ yields an Sp$(2,q)$-space\n$H^\\perp\/H$, which contains $q{+}1$ subspaces of the form $Z\/H$ with $Z\\in\\mathcal{V}$. One such $Z$ has distance $k{-}1$ from $X$, this being the subspace\n$W=(Y{\\cap}Z)+X\\cap(Y{+}Z)=(Y{+}Z)\\cap(X{+}(Y{\\cap}Z))$. If we exclude $W$ and $Y$ itself, this leaves exactly $q{-}1$ choices of $Z$ having the required\ndistances from $X$ and $Y$; and this gives $(q{-}1)\\gauss{k}1=q^k{-}1=a_k$ choices of $Z$, the full number. But for how many such $Z$ is the resulting\ntriple $(X,Y,Z)$ coherent? In each case $\\sigma(X,W,Y)=\\sigma(X,W,Z)=1$ by Lemma~\\ref{lemma4.5}; therefore $\\sigma(X,Y,Z)=\\sigma(W,Y,Z)$.\nBut by Lemma~\\ref{lemma4.6}, given\n$W,Y$ at distance~1, exactly half of the $q{-}1$ choices of $Z$ yield coherent triples $(W,Y,Z)$. Therefore among the $a_k=(q{-}1)\\gauss{k}1$ triples\n$(X,Y,Z)$ with fixed $X$ and $Y$, exactly $\\frac{q{-}1}{2}\\gauss{k}1=(q^k{-}1)\/2$ such triples are coherent.\\qed\n\\end{pf}\n\n\\section{The Double Cover $\\widehat{\\Gamma}\\to\\Gamma$}\n\\label{double_cover}\n\nThe resulting double cover $\\widehat{\\Gamma}=\\widehat{\\Gamma}(2n,q)\\to\\Gamma(2n,q)$ has vertex set $\\widehat{\\V}=\\mathcal{V}\\times\\{\\pm1\\}$ and adjacency relation\n\\[(X,\\varepsilon)\\sim(Y,\\varepsilon')\\qquad\\iff\\qquad d(X,Y)=1\\hbox{\\ and\\ }\\varepsilon\\varepsilon'=\\sigma(X,Y).\\]\nThe covering map is given by $(X,\\varepsilon)\\mapsto X$. \n\n\\begin{theorem} \\label{theorem5.1}\nEvery geodesic path\n\\[X_0\\sim X_1\\sim\\cdots\\sim X_k\\]\nin $\\Gamma$ (meaning that $d(X_i,X_j)=|j-i|$) lifts to exactly two paths\n\\[(X_0,\\varepsilon_0)\\sim(X_1,\\varepsilon_1)\\sim\\cdots\\sim(X_k,\\varepsilon_k)\\]\nin $\\widehat{\\Gamma}$, in which $\\varepsilon_k=\\varepsilon_0\\sigma(X_0,X_k)$ for each $k\\geqslant1$; thus any one of the $\\varepsilon_i$ determines all the others\nalong this path.\n\\end{theorem}\n\n\\begin{pf}\nWe have $\\varepsilon_1=\\varepsilon_0\\sigma(X_0,X_1)$ by definition of adjacency in $\\widehat{\\Gamma}$. Assuming that\n$\\varepsilon_i=\\varepsilon_0\\sigma(X_0,X_i)$ for some $i\\in\\{1,2,\\ldots,k{-}1\\}$,\n\\[\\varepsilon_{i+1}=\\varepsilon_i\\sigma(X_i,X_{i+1})=\\varepsilon_0\\sigma(X_0,X_i)\\sigma(X_i,X_{i+1})=\\varepsilon_0\\sigma(X_0,X_{i+1})\\]\nsince $\\sigma(X_0,X_i,X_{i+1})=1$ by Lemma~\\ref{lemma4.5}.\\qed\n\\end{pf}\n\n\\noindent However, not every geodesic path in $\\widehat{\\Gamma}$ is obtained by lifting a geodesic path in~$\\Gamma$. For example if $(X,Y,Z)$ is an incoherent\ntriangle in $\\Gamma$, say with $\\sigma(X,Y)=\\varepsilon$, $\\sigma(Y,Z)=\\varepsilon'$ and $\\sigma(X,Z)=-\\varepsilon\\varepsilon'$, then\n\\[(X,1)\\sim(Y,\\varepsilon)\\sim(Z,\\varepsilon\\varepsilon')\\sim(X,-1)\\]\nis a geodesic path of length~3 in $\\widehat{\\Gamma}$, obtained by lifting a closed path of length~3 (not a geodesic path) in~$\\Gamma$.\n\n\\begin{lemma} \\label{lemma5.2}\nLet $X_0\\sim X_1\\sim\\cdots\\sim X_k$ be a geodesic path of length $k\\geqslant1$ in $\\Gamma$, so that $d(X_i,X_j)=|j-i|$, and let\n$\\varepsilon,\\varepsilon'\\in\\{\\pm1\\}$. Then $(X_0,\\varepsilon)$ and $(X_k,\\varepsilon')$ have distance $k$ or $k{+}1$ in $\\widehat{\\Gamma}$, according\nas $\\sigma(X_0,X_k)=\\varepsilon\\varepsilon'$ or $-\\varepsilon\\varepsilon'$. In particular, the diameter of $\\widehat{\\Gamma}$ is $\\max\\{n{+}1,3\\}$.\n\\end{lemma}\n\n\\begin{pf}\nIf $\\sigma(X_0,X_k)=\\varepsilon\\varepsilon'$, then we have a path\n\\[(X_0,\\varepsilon_0)\\sim(X_1,\\varepsilon_1)\\sim\\cdots\\sim(X_k,\\varepsilon_k)\\]\nin $\\widehat{\\Gamma}$ where $\\varepsilon_i=\\varepsilon_0\\sigma(X_0,X_i)$ for $i=1,2,\\ldots,k$; in particular if $\\varepsilon_0=\\varepsilon$ then\n$\\varepsilon_k=\\varepsilon'$. Clearly this path in $\\widehat{\\Gamma}$ is shortest possible.\n\nNow suppose $\\sigma(X_0,X_k)=-\\varepsilon\\varepsilon'$. We first obtain a path\n\\[(X_0,\\varepsilon)\\sim(X_1,\\varepsilon_1)\\sim\\cdots\\sim(X_{k-1},\\varepsilon_{k-1})\\]\nin $\\widehat{\\Gamma}$ where $\\varepsilon_i=\\varepsilon_0\\sigma(X_0,X_i)$ for $i=1,2,\\ldots,k{-}1$. Let $Y\\in\\mathcal{V}$ be adjacent to both $X_{k-1}$ and $X_k$ in\n$\\Gamma$, such that $\\sigma(X_{k-1},Y,X_k)=-1$. (By Lemma~\\ref{lemma4.6}, there are $\\frac{q-1}{2}\\geqslant1$ choices of such $Y\\in\\mathcal{V}$.) Appending\n\\[(X_{k-1},\\varepsilon_{k-1})\\sim(Y,\\varepsilon'')\\sim(X_k,\\varepsilon'),\\]\nwhere $\\varepsilon''=\\varepsilon_{k-1}\\sigma(X_{k-1},Y)=\\varepsilon'\\sigma(Y,X_k)$, we obtain a path of length $k{+}1$ from $(X_0,\\varepsilon)$ to\n$(X_k,\\varepsilon')$ in $\\widehat{\\Gamma}$; once again this path is shortest possible.\\qed\n\\end{pf}\n\nThe fibers of the covering map $\\widehat{\\Gamma}\\to\\Gamma$ are the {\\it antipodal\\\/} pairs\n$\\{(X,1),(X,-1)\\}$ for $X\\in\\mathcal{V}$.\n\n\\begin{lemma}\\label{lemma5.3}\nLet $(X,\\varepsilon)$ and $(W,\\varepsilon')$ be any two vertices of $\\widehat{\\Gamma}$. Then\n$(X,\\varepsilon)$ and $(W,\\varepsilon')$ are antipodal iff they are at distance~3 in $\\widehat{\\Gamma}$ and are joined by exactly $\\frac12 q(q^n{-}1)$ paths of length~3 in $\\widehat{\\Gamma}$.\n\\end{lemma}\n\n\\begin{pf}Consider a typical antipodal pair $\\{(X,1),(X,-1)\\}$ where $X\\in\\mathcal{V}$. There exist $b_0=q\\gauss{n}1$ vertices $Y\\in\\mathcal{V}$ adjacent to $X$ in $\\Gamma$; and each such vertex $Y$ has $a_1=q{-}1$ neighbors $Z$ in common with $X$.\nBy Lemma~\\ref{lemma4.6}, exactly half of these choices of the vertex $Z$ yield coherent triples $(X,Y,Z)$. In particular, $X$ lies in exactly $b_0{\\cdot}\\frac12 a_1=\\frac12 q\\bigl(q^n-1)$ incoherent triples $(X,Y,Z)$, giving\nthe same number of paths $(X,1)\\sim(Y,\\varepsilon)\\sim(Z,\\varepsilon')\\sim(X,-1)$\nof length~3 in $\\widehat{\\Gamma}$. There is no path of length${}<3$ from $(X,1)$ to $(X,-1)$ in\n$\\widehat{\\Gamma}$, otherwise the covering map would give a closed\npath of length${}<3$ from $X$ to $X$ in $\\Gamma$. This shows that any two antipodal\nvertices $(X,1),(X,-1)$ are at distance~3 in $\\widehat{\\Gamma}$; and in each case there are exactly $\\frac12 q(q^n{-}1)$ geodesic paths from $(X,1)$ to $(X,-1)$.\n\nConversely, let $(X,\\varepsilon)$ and $(W,\\varepsilon')$ be any two vertices at distance~3 in\n$\\widehat{\\Gamma}$. By Lemma~\\ref{lemma5.2}, $d(X,W)\\in\\{0,2,3\\}$\nin $\\Gamma$. Consider first the case that $d(X,W)=3$; then by Theorem~\\ref{theorem5.1}, every geodesic path from $(X,\\varepsilon)$ to\n$(W,\\varepsilon')$ in $\\widehat{\\Gamma}$ arises from a unique geodesic path $X\\sim Y\\sim Z\\sim W$ in $\\Gamma$. There are exactly $c_3 c_2 c_1=(q^2{+}q{+}1)(q{+}1)$ such\ngeodesic paths from $X$ to $W$; and this number clearly cannot equal $\\frac12 q(q^n-1)$.\n\nNext suppose $d(X,W)=2$ in $\\Gamma$. For every geodesic path\n\\[(X,\\varepsilon)\\sim(Y,\\varepsilon'')\\sim(Z,\\varepsilon''')\\sim(W,\\varepsilon')\\]\nin $\\widehat{\\Gamma}$, we have $X\\sim Y\\sim Z\\sim W$ in $\\Gamma$. Further, the condition $d(X,W)=2$ requires either $X\\sim Z$ or $Y\\sim W$ (but {\\it not both,\\\/} by Lemma~\\ref{lemma4.2}). We first count\ngeodesic paths satisfying $X\\sim Z$, noting that the vertex $W$ has $c_2=q{+}1$ neighbors $Z$ in common with $X$; and in each case $\\sigma(X,Z,W)=1$ by Lemma~\\ref{lemma4.5}.\nMoreover $Z$ has $a_1=q{-}1$ neighbors $Y$ in common with $X$ (all of which satisfy $\\sigma(Y,Z,W)=1$, again by Lemma~\\ref{lemma4.5}). By the two-graph\ncondition, we have $\\sigma(X,Y,Z)=-1$ iff $\\sigma(X,Y,W)=-1$. By Lemma~\\ref{lemma4.6}, for each $Z$ there are exactly $\\frac12\\bigl(q{-}1\\bigr)$\nchoices of $Y$ satisfying the latter condition; and each such pair $(Y,Z)$ yields a unique geodesic path\n$(X,\\varepsilon)\\sim(Y,\\varepsilon'')\\sim(Z,\\varepsilon''')\\sim(W,\\varepsilon')$. We obtain\n$(q{+}1){\\cdot}\\frac12\\bigl(q{-}1\\bigr)=\\frac12\\bigl(q^2{-}1\\bigr)$ geodesic paths in this case. There are another $\\frac12\\bigl(q^2{-}1\\bigr)$ geodesic paths from $(X,\\varepsilon)$ to $(W,\\varepsilon')$ satisfying $Y\\sim W$, for a total of $q^2{-}1$ geodesic paths. Once again, this number cannot equal $\\frac12 q(q^n{-}1)$.\\qed\n\\end{pf}\n\n\\begin{lemma}\\label{lemma5.4}\n$\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}$ acts naturally on $\\Gamma$, with kernel $\\langle\\zeta\\rangle$,\ninducing a {\\em proper\\\/} subgroup $\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}\/\\langle\\zeta\\rangle<\\mathop{\\rm Aut}\\nolimits\\Gamma$.\n\\end{lemma}\n\n\\begin{pf}\nBy Lemma~\\ref{lemma5.3}, $\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}$ permutes fibres of the covering\nmap $\\widehat{\\Gamma}\\to\\Gamma$, and so $\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}$ acts naturally on $\\Gamma$.\nIt remains to be shown that the induced subgroup $\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}\/\\langle\\zeta\\rangle\\leqslant\\mathop{\\rm Aut}\\nolimits\\Gamma$ is proper.\n\nChoose a hyperbolic basis $e_1,e_2,\\ldots,e_n,f_1,f_2,\\ldots,f_n$ for $V$, so that\n$B(e_i,e_j)=B(f_i,f_j)=0$ and $B(e_i,f_j)=\\delta_{ij}$, and let $\\eta\\in{\\mathbb F}_q$ be a nonsquare.\nConsider the subspaces $X,Y,Z,Z'\\in\\mathcal{V}$ defined by $X=\\langle e_1,e_2,\\ldots,e_n\\rangle$,\n$Y=\\langle f_1,e_2,\\ldots,e_n\\rangle$, $Z=\\langle e_1{+}f_1,e_2,\\ldots,e_n\\rangle$ and\n$Z'=\\langle e_1{+}\\eta f_1,e_2,\\ldots,e_n\\rangle$. By straightforward computation,\n$\\sigma(X,Y,Z)=1$ and $\\sigma(X,Y,Z')=-1$. Now consider $g\\in GL(V)$ mapping our\noriginal ordered basis to the new ordered basis $e_1,\\eta e_2,\\ldots,\\eta e_n,\\eta f_1,f_2,\\ldots,f_n$ so that $B(u^g,v^g)=\\eta B(u,v)$ for all $u,v\\in V$; thus $g\\in GSp(2n,q)$\ninduces an automorphism of the dual polar graph $\\Gamma=\\Gamma(2n,q)$.\nHowever, $g$ maps the coherent triple $\\{X,Y,Z\\}$ to the non-coherent triple $\\{X,Y,Z'\\}$\nand so does not preserve $\\Delta_\\sigma$. If $g$ were induced by an automorphism of\n$\\widehat{\\Gamma}$, this automorphism would map $\\{X,Y,Z\\}\\times\\{\\pm1\\}$ to\n$\\{X,Y,Z'\\}\\times\\{\\pm1\\}$. However, the induced subgraphs of $\\widehat{\\Gamma}$ on these\ntwo 6-sets of vertices are not isomorphic (a pair of triangles and a 6-cycle, respectively;\nsee Section~\\ref{graphs}).\\qed\n\\end{pf}\n\nThe natural action of $\\Sigma Sp(2n,q)$ on $\\mathcal{V}$ lifts to an action on $\\widehat{\\V}$ as follows:\nLet $g\\in\\Sigma Sp(2n,q)$ with associated field automorphism $\\tau_g$ in the earlier notation of this section.\nGiven $(U,\\varepsilon)\\in\\widehat{\\V}$, the map\n\\[U^n\\to{\\mathbb F}_q,\\quad (u_1,u_2,\\ldots,u_n)\\mapsto\\delta_{U^g}(u_1^g,u_2^g,\\ldots,u_n^g)^{\\tau_g^{-1}}\\]\nis a determinant function; so there exists a nonzero constant $\\lambda_{g,U}\\in{\\mathbb F}_q$ such that\n\\[\\delta_{U^g}(u_1^g,u_2^g,\\ldots,u_n^g)=\\lambda_{g,U}\\delta_U(u_1,u_2,\\ldots,u_n)^{\\tau_g}.\\]\nDefine $(U,\\varepsilon)^g=(U^g,\\chi(\\lambda_{g,U})\\varepsilon)$. One easily checks that\nthis defines an action of $\\Sigma Sp(2n,q)$\non $\\widehat{\\V}$. The central element $-I\\in Sp(2n,q)$ fixes every $U\\in\\mathcal{V}$ and since\n\\[\\delta_U(-u_1,-u_2,\\ldots,-u_n)=(-1)^n\\delta_U(u_1,u_2,\\ldots,u_n)\\]\nwhere $\\chi(-1)^n=1$,\n$-I$ acts trivially on $\\widehat{\\V}$; thus $\\Sigma Sp(2n,q)$ induces a permutation group\n$P\\Sigma Sp(2n,q)$ on~$\\widehat{\\V}$. The transposition $\\zeta$ which exchanges antipodal\nvertices via $(U,1)\\stackrel\\zeta\\leftrightarrow(U,-1)$ is not induced by any element of\n$P\\Sigma Sp(2n,q)$ since $Z(P\\Sigma Sp(2n,q))=1$, so we obtain a permutation group\n$\\langle\\zeta\\rangle\\times P\\Sigma Sp(2n,q)$ acting faithfully on $\\widehat{\\V}$. We show that this\npermutation group preserves the graph $\\widehat{\\Gamma}$, and is in fact its full automorphism group:\n\n\\begin{theorem}\\label{theorem5.5}\n$\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}\\cong2\\times P\\Sigma Sp(2n,q)$ where this group acts as defined above.\nThe full automorphism group of the two-graph associated to $\\sigma$ is $\\mathop{\\rm Aut}\\nolimits\\Delta_\\sigma\\cong P\\Sigma Sp(2n,q)$.\n\\end{theorem}\n\n\\begin{pf}\nSuppose $(X,\\varepsilon)\\sim(Y,\\varepsilon')$ in $\\widehat{\\Gamma}$, so that\n$\\sigma(X,Y)=\\varepsilon\\varepsilon'$; and let $g\\in\\Sigma Sp(2n,q)$ with\n$\\tau_g\\in\\mathop{\\rm Aut}\\nolimits{\\mathbb F}_q$ as above. Then by Proposition~\\ref{prop4.3}(iv) we have\n\\[\\sigma(X^g,Y^g)=\\chi(\\lambda_{g,X}\\lambda_{g,Y})\\sigma(X,Y)\n=\\bigl(\\chi(\\lambda_{g,X})\\varepsilon\\bigr)\\bigl(\\chi(\\lambda_{g,Y})\\varepsilon'\\bigr)\\]\nso that by definition, $(X,\\varepsilon)^g\\sim(Y,\\varepsilon')^g$. Thus $P\\Sigma Sp(2n,q)$,\nacting on $\\widehat{\\Gamma}$ as defined above, preserves the graph $\\widehat{\\Gamma}$.\nIt is clear that the central factor $\\langle\\zeta\\rangle$ also preserves $\\widehat{\\Gamma}$,\nso that $\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}$ has a subgroup isomorphic to\n$\\langle\\zeta\\rangle\\times P\\Sigma Sp(2n,q)$. Moreover by Proposition~\\ref{prop3.1},\n$\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}\/\\langle\\zeta\\rangle\\cong\\mathop{\\rm Aut}\\nolimits\\Delta_\\sigma$. (We use the fact that by\nLemma~\\ref{lemma5.4}, $\\mathop{\\rm Aut}\\nolimits_\\zeta\\widehat{\\Gamma}=\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}$ in the notation\nof Proposition~\\ref{prop3.1}.)\n\nSuppose now that $n\\geqslant2$, so that $\\mathop{\\rm Aut}\\nolimits\\Gamma\\cong P\\Gamma Sp(2n,q)$ by\nTheorem~\\ref{theorem4.1}. By Lemma~\\ref{lemma5.4}, $\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}$\nacts on $\\Gamma$, inducing a group of automorphisms satisfying\n\\[P\\Sigma Sp(2n,q)\\leqslant\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}\/\\langle\\zeta\\rangle<P\\Gamma Sp(2n,q).\\]\nThis forces $\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}\\cong\\langle\\zeta\\rangle\\times P\\Sigma Sp(2n,q)$ and\n$\\mathop{\\rm Aut}\\nolimits\\Delta_\\sigma\\cong P\\Sigma Sp(2n,q)$.\n\nFinally suppose $n=1$, so that $\\Delta_\\sigma$ is the\nTaylor-Paley two-graph on $q+1$ vertices, with full automorphism group\n$\\mathop{\\rm Aut}\\nolimits\\Delta_\\sigma\\cong P\\Sigma Sp(2,q)=P\\Sigma L(2,q)$ by \\cite[Theorem 2]{Ta};\nsee also~\\cite[\\S4]{M}. As above, $\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}$ has a subgroup\nisomorphic to $\\langle\\zeta\\rangle\\times P\\Sigma Sp(2,q)$, and\n$\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}\/\\langle\\zeta\\rangle\\cong\\mathop{\\rm Aut}\\nolimits\\Delta_\\sigma\\cong P\\Sigma Sp(2,q)$,\nso we must have equality:\n$\\mathop{\\rm Aut}\\nolimits\\widehat{\\Gamma}\\cong2\\times P\\Sigma Sp(2,q)=2\\times P\\Sigma L(2,q)$.\\qed\n\\end{pf}\n\n\\section{The Association Scheme}\n\\label{scheme}\n\nFrom the double cover $\\widehat{\\Gamma}\\to\\Gamma$, we now construct association schemes.\nAs we will see in Section~\\ref{Q-poly}, this gives\na new family of $Q$-polynomial association schemes. We begin with the\nrelevant definitions, following \\cite[Chapter~2]{BCN}.\n\nLet $\\Omega$ be a finite set. A {\\em (symmetric) $d$-class association scheme on $\\Omega$\\\/} is a pair $(\\Omega,\\mathcal{R})$ such that\n\\begin{enumerate}\n\\item $\\mathcal{R}=\\{R_0,\\ldots,R_d\\}$ is a partition of $\\Omega\\times\\Omega$;\n\\item $R_0$ is the identity relation on $\\Omega$;\n\\item $R_i=R_i^{\\top}$ for $0\\leqslant i\\leqslant d$; and\n\\item there are constants $p_{ij}^k$ such that for any pair $(x,y)\\in R_k$, the number of $z \\in\\Omega$ such that $(x,z)\\in R_i$ and $(z,y)\\in R_j$ equals $p_{ij}^k$.\n\\end{enumerate}\nFor the rest of this paper, all association schemes are symmetric (i.e.\\ the third property above holds).\nEach relation $R_i$ has adjacency matrix $A_i$ defined by\n\\[(A_i)_{x,y}=\\begin{cases}\n1,&\\mbox{if\\ }(x,y)\\in R_i;\\\\\n0,&\\mbox{otherwise.}\n\\end{cases}\\]\nThe axioms above imply that $p_{ji}^k=p_{ij}^k$ and the matrices $A_0,\\ldots,A_d$ form an algebra of symmetric matrices satisfying\n$A_iA_j=\\sum_k p_{ij}^k A_k$. This matrix algebra is also closed under Schur (entrywise) multiplication, which we will denote by \n`$\\circ$'. This algebra is referred to as the {\\em Bose-Mesner algebra\\\/} ${\\mathfrak{A}}$ of the association scheme.  \n\nSince ${\\mathfrak{A}}$ is a commutative algebra consisting of symmetric matrices, its elements are simultaneously diagonalizable, and ${\\mathfrak{A}}$ has a second basis consisting of primitive idempotents $E_0,\\ldots,E_d$.  \nWe define the parameters $Q_{ij}$ by $E_j =\\frac{1}{|\\Omega|}\\sum_i Q_{ij}A_i$.  Similarly we define the parameters $P_{ij}$ by the relation \n$A_j=\\sum_i P_{ij} E_i$.  The matrix $P$ of parameters $P_{ij}$ is often referred to as the character table of the scheme.   \nThe matrix $Q$ of parameters $Q_{ij}$ satisfies $Q=|\\Omega|P^{-1}$. \n\nWe say an association scheme is {\\em$Q$-polynomial\\\/} if, after suitably reindexing its idempotents, \nthe idempotent $E_j$ is a degree $j$ polynomial in $E_1$ (where multiplication is done entrywise). This is equivalent to the condition\nthat the $j$th column of the $Q$-matrix is a degree $j$ polynomial of the column 1 of the $Q$-matrix (note that we start indexing the columns at 0).\n\nPermutation groups give many examples of association schemes.  Let $G$ be a transitive\npermutation group acting on a finite set $\\Omega$, and suppose the orbits of $G$ on\n$\\Omega\\times\\Omega$ happen to be symmetric relations; such a group is called {\\em generously transitive\\\/}).\nIt is not hard to check that the orbits of $G$ on $\\Omega\\times\\Omega$ form an association scheme. We will refer to these schemes as {\\em Schurian schemes\\\/}.\n\\bigskip\n\nWe will now construct a $(2n{+}1)$-class Schurian association scheme $\\mathcal{S}=\\mathcal{S}_{n,q}$ with vertex set $\\widehat{\\V}=\\mathcal{V}\\times\\{\\pm1\\}$ of cardinality\n$|\\widehat{\\V}|=2|\\mathcal{V}|=2q^{n^2}\\!\\prod_{i=1}^n(q^{2i}-1)$ using the Maslov index $\\sigma$\nand the double cover $\\widehat{\\Gamma}\\to\\Gamma$ (defined as in Section~\\ref{double_cover}).\nFor $k=0,1,2,\\ldots,n$, the $k$th and $(2n{+}1{-}k)$th relations are given by\n\\begin{align*}\nR_k&=\\{((X,\\varepsilon),(Y,\\varepsilon'))\\in\\widehat{\\V}\\times\\widehat{\\V}\\;:\\;d(X,Y)=k,\\;\\varepsilon\\varepsilon'=\\sigma(X,Y)\\};\\\\\nR_{2n+1-k}&=\\{((X,\\varepsilon),(Y,\\varepsilon'))\\in\\widehat{\\V}\\times\\widehat{\\V}\\;:\\;d(X,Y)=k,\\;\\varepsilon\\varepsilon'=-\\sigma(X,Y)\\}.\n\\end{align*}\nThese are symmetric relations which clearly partition $\\widehat{\\V}\\times\\widehat{\\V}$. In particular,\n$R_1$ is the adjacency relation of our graph $\\widehat{\\Gamma}$ of Section~\\ref{double_cover}; and the identity and antipodality relations are\n\\begin{align*}\nR_0&=\\{((X,\\varepsilon),(X,\\varepsilon)):X\\in\\mathcal{V},\\;\\varepsilon=\\pm1\\};\\\\\nR_{2n+1}&=\\{((X,\\varepsilon),(X,-\\varepsilon)):X\\in\\mathcal{V},\\;\\varepsilon=\\pm1\\}.\n\\end{align*}\nWe will write\n\\[(X,\\varepsilon)\\irel{i}(Y,\\varepsilon')\\iff((X,\\varepsilon),(Y,\\varepsilon'))\\in R_i\\,.\\]\nIn the following, the parameters $a_i,b_i,c_i$ are those of the dual polar graph $\\Gamma$\nas given in Section~\\ref{dual_polar}.\n\n\\begin{lemma}\\label{lemma6.1}\nLet $(X,\\varepsilon)\\irel{k}(Y,\\varepsilon')$ where $k\\in\\{0,1,2,\\ldots,2n{+}1\\}$. The number of $(Z,\\varepsilon'')\\in\\widehat{\\V}$\nsuch that $(X,\\varepsilon)\\irel{i}(Z,\\varepsilon'')\\irel{1}(Y,\\varepsilon')$ is\n\\[p_{i,1}^k=\\begin{cases}\nc_k=\\gauss{k}1,&\\mbox{if }\\,i=k{-}1\\leqslant n;\\\\\n\\noalign{\\smallskip}\n\\frac12 a_k=\\frac{1}{2}\\bigl(q^k-1),&\\mbox{if }\\,i=k;\\\\\n\\noalign{\\smallskip}\nb_k=q^{k+1}\\gauss{n-k}1,&\\mbox{if }\\,i=k{+}1\\leqslant n{+}2;\\\\\n\\noalign{\\smallskip}\nb_{2n+1-k}=q^{2n+2-k}\\gauss{k-n-1}1,&\\mbox{if }i=k{-}1\\geqslant n{-}1;\\\\\n\\noalign{\\smallskip}\n\\frac12 a_{2n+1-k}=\\frac{1}{2}\\bigl(q^{2n+1-k}-1),&\\mbox{if }\\,i=2n{+}1{-}k;\\\\\n\\noalign{\\smallskip}\nc_{2n+1-k}=\\gauss{2n+1-k}1,&\\mbox{if }\\,i=k{+}1\\geqslant n{+}1;\\\\\n\\noalign{\\smallskip}\n0,&\\hbox{otherwise.}\n\\end{cases}\\]\n\\end{lemma}\n\n\\begin{pf}(i) First suppose $d(X,Y)=k\\leqslant n$, so $\\varepsilon\\varepsilon'=\\sigma(X,Y)$. Then $(Z,\\varepsilon'')\\in\\widehat{\\V}$ satisfies\n$(X,\\varepsilon)\\irel{i}(Z,\\varepsilon'')\\irel{1}(Y,\\varepsilon')$ iff\n\\[\\begin{cases}\n\\mbox{\\qquad case (i.a)}\\\\\n\\hline\ni=d(X,Z)\\leqslant n\\\\\nd(Z,Y)=1\\\\\n\\varepsilon''=\\varepsilon\\sigma(X,Z)=\\varepsilon'\\sigma(Y,Z)\n\\end{cases}\\mbox{\\quad\\em or\\quad}\n\\begin{cases}\n\\mbox{\\qquad case (i.b)}\\\\\n\\hline\ni=2n{+}1{-}d(X,Z)\\geqslant n{+}1\\\\\nd(Z,Y)=1\\\\\n\\varepsilon''=-\\varepsilon\\sigma(X,Z)=\\varepsilon'\\sigma(Y,Z)\n\\end{cases}\\]\nMoreover, each such $(Z,\\varepsilon'')$ satisfies $d(X,Z)\\in\\{k{-}1,k,k{+}1\\}$ by the triangle inequality.\n\nThere are exactly $c_k=\\gauss{k}1$ choices of $Z\\in\\mathcal{V}$ satisfying $d(X,Z)=k{-}1$ and $d(Z,Y)=1$. Each such $Z$ yields a coherent\ntriple $(X,Y,Z)$ by Lemma~\\ref{lemma4.5}, so $\\varepsilon\\varepsilon'=\\sigma(X,Y)=\\sigma(X,Z)\\sigma(Y,Z)$. This yields $c_k$ pairs $(Z,\\varepsilon'')$,\nall of which satisfy~(i.a).\n\nThere are exactly $b_k=q^{k+1}\\gauss{n-k}1$ choices of $Z\\in\\mathcal{V}$ satisfying $d(X,Z)=k{+}1$ and $d(Z,Y)=1$. Each such $Z$ yields a coherent\ntriple $(X,Y,Z)$ by Lemma~\\ref{lemma4.5}, once again with $\\varepsilon\\varepsilon'=\\sigma(X,Y)=\\sigma(X,Z)\\sigma(Y,Z)$. This yields $b_k$ pairs $(Z,\\varepsilon'')$,\nall of which satisfy~(i.a).\n\nThere are exactly $a_k=q^k-1$ choices of $Z\\in\\mathcal{V}$ satisfying $d(X,Z)=k$ and $d(Z,Y)=1$. By Theorem~\\ref{theorem4.7}, exactly $a_k\/2$ of these $Z$ yield coherent\ntriples $(X,Y,Z)$, in which case $\\varepsilon\\varepsilon'=\\sigma(X,Y)=\\sigma(X,Z)\\sigma(Y,Z)$; this yields $a_k\/2$ pairs $(Z,\\varepsilon'')$\nsatisfying~(i.a). The remaining $a_k\/2$ of these $Z$ yield incoherent\ntriples $(X,Y,Z)$, with $\\varepsilon\\varepsilon'=\\sigma(X,Y)=-\\sigma(X,Z)\\sigma(Y,Z)$; and the resulting pairs $(Z,\\varepsilon'')$ satisfy~(i.b).\\qed\n\\end{pf}\n\nIn Section~\\ref{double_cover} we lifted the action of $P\\Sigma Sp(2n,q)$ on $\\mathcal{V}$, to a\ntransitive permutation action of $\\langle\\zeta\\rangle\\times P\\Sigma Sp(2n,q)$ on $\\widehat{\\V}$\n(below Lemma~\\ref{lemma5.4}). Theorem~\\ref{theorem5.5} shows\nthat this group preserves $R_1$ (the adjacency relation of the graph $\\widehat{\\Gamma}$).\nWe next show that this group preserves {\\em each\\\/} of the relations $R_i$, and\nso gives the full automorphism group of the scheme.\n\n\\begin{lemma}\\label{lemma6.2}\nThe diagonal action of $2\\times PSp(2n,q)$ on $\\widehat{\\V}\\times\\widehat{\\V}$ preserves each\nof the relations $R_i$. The same conclusion holds for the subgroup $2\\times P\\Sigma Sp(2n,q)$.\n\\end{lemma}\n\n\\begin{pf}\nClearly the central factor $(U,\\varepsilon)\\stackrel\\zeta\\leftrightarrow(U,-\\varepsilon)$ preserves each $R_i$.\nNow let $g\\in Sp(2n,q)$, and suppose\n$X,Y\\in\\mathcal{V}$ such that $d(X,Y)=k\\in\\{0,1,2,\\ldots,n\\}$. Also let $\\varepsilon,\\varepsilon'\\in\\{\\pm1\\}$, so that $((X,\\varepsilon),(Y,\\varepsilon'))\\in R_k$ or $R_{2n+1-k}$\naccording as $\\varepsilon\\varepsilon'\\sigma(X,Y)=1$ or $-1$.\nSince $g$ preserves distances in $\\Gamma$, $d(X^g,Y^g)=k$. Let $x_1,x_2,\\ldots,x_n$ and\n$y_1,y_2,\\ldots,y_n$ be bases for $X$ and $Y$ respectively, such that a basis for\n$X\\cap Y$ is formed by $x_{k+1}{=}y_{k+1}$, $x_{k+2}{=}y_{k+2}$, \\dots,\n$x_n{=}y_n$. Then $(X,\\varepsilon)^g=(X^g,\\chi(\\lambda_{g,X})\\varepsilon)$ and\n$(Y,\\varepsilon')^g=(Y^g,\\chi(\\lambda_{g,Y})\\varepsilon')$ where\n\\begin{align*}\n&\\chi(\\lambda_{g,X})\\varepsilon\\chi(\\lambda_{g,Y})\\varepsilon'\\sigma(X^g,Y^g)\\\\\n&\\qquad=\\varepsilon\\varepsilon'\\chi\\bigl(\\lambda_{g,X}\\delta_{X^g}(x_1^g,\\ldots,x_n^g)\\lambda_{g,Y}\\delta_{Y^g}(y_1^g,\\ldots,y_n^g)\\\\\n&\\qquad\\phantom{{}={}}\\times\\det\\bigl[B(x_i^g,y_j^g):1\\leqslant i,j\\leqslant k\\bigr]\\bigr)\\\\\n&\\qquad=\\varepsilon\\varepsilon'\\chi\\bigl(\\lambda_{g,X}^2\\lambda_{g,Y}^2\\delta_X(x_1,\\ldots,x_n)\\delta_Y(y_1,\\ldots,y_n)\\det\\bigl[B(x_i,y_j)\\,:\\,1\\leqslant i,j\\leqslant k\\bigr]\\bigr)\\\\\n&\\qquad=\\varepsilon\\varepsilon'\\sigma(X,Y).\n\\end{align*}\nIf this value is $1$, then both $(X,\\varepsilon)\\irel{k}(Y,\\varepsilon')$ and\n$(X,\\varepsilon)^g\\irel{k}(Y,\\varepsilon')^g$; but if the latter value is $-1$, then\n$(X,\\varepsilon)\\irel{\\scriptscriptstyle{2n{+}1{-}k}}(Y,\\varepsilon')$ and\n$(X,\\varepsilon)^g\\irel{\\scriptscriptstyle{2n{+}1{-}k}}(Y,\\varepsilon')^g$.\n\nThus $2\\times PSp(2n,q)$ preserves the relations $R_i$ as claimed. A similar argument\nholds for $2\\times P\\Sigma Sp(2n,q)$.\\qed\n\\end{pf}\n\nIt is easy to see that $\\langle\\zeta\\rangle\\times P\\Sigma Sp(2n,q)$ acts transitively on\neach $R_i$, and similarly for $\\langle\\zeta\\rangle\\times PSp(2n,q)$. This yields\n\n\\begin{theorem}\\label{theorem6.3}\nThe diagonal action of the group $2\\times PSp(2n,q)$ on $\\widehat{\\V}\\times\\widehat{\\V}$ has orbits $R_0$, $R_1$, \\dots, $R_{2n+1}$; so these form the relations of a $(2n+1)$-class Schurian association scheme. The same conclusion holds for $2\\times P\\Sigma Sp(2n,q)$,\nwhich is therefore the full automorphism group of the association scheme $\\mathcal{S}$.\\qed\n\\end{theorem}\n\n\\section{The $Q$-polynomial property}\n\\label{Q-poly}\n\nIn this section we will use some parameters of the scheme to prove that the association scheme $\\mathcal{S}$ is $Q$-polynomial.\nWe will benefit from the action of the $A_i$'s by left-multiplication on the Bose-Mesner algebra,\nresulting in matrices $L_i$ defined by $(L_i)_{kj}=p_{ij}^k$.\nIn particular, the parameter $p_{1j}^k$ of the scheme from Lemma~\\ref{lemma6.1}, is the $(k,j)$-entry of the matrix\n\\[L_1=\\begin{pmatrix}  \n0&b_0&0&0&0&\\cdots &0&0&0&0 \\\\\n1&\\frac{a_1}{2}&b_1&0&0&\\cdots &0&0&\\frac{a_1}{2}&0 \\\\\n0&c_2&\\frac{a_2}{2}&\\ddots&0&\\cdots &0&\\iddots&0&0 \\\\\n0&0&\\ddots&\\ddots&b_{d-1}&0&\\frac{a_{d-1}}{2}&0&0&0 \\\\\n0&0&0&c_d&\\frac{a_d}{2}&\\frac{a_d}{2} &0&0&0&0 \\\\\n0&0&0&0&\\frac{a_d}{2}&\\frac{a_d}{2} &c_d&0&0&0 \\\\\n0&0&0&\\frac{a_{d-1}}{2}&0&b_{d-1}&\\ddots&\\ddots&0&0 \\\\\n0&0&\\iddots&0&0&0 &\\ddots&\\frac{a_2}{2}&c_2&0 \\\\\n0&\\frac{a_1}{2}&0&0&0&0 &0&b_1&\\frac{a_1}{2}&1 \\\\\n0&0&0&0&0&0 &0&0&b_0&0\n\\end{pmatrix}.\\]\n\nAs it turns out, this matrix has distinct eigenvalues, which in turn will give us a great deal of information about the scheme.\nIn particular by \\cite[Proposition 2.2.2]{BCN}, the columns of $Q$ are right eigenvectors of $L_1$.\nWe will use the following generalization of \\cite[Theorem 8.1.1]{BCN} to prove that $\\mathcal{S}$ is $Q$-polynomial.\n\n\\begin{theorem}\\label{theorem7.1}\nSuppose  $A_i$ is a matrix in a $d$-class association scheme $(\\Omega,\\mathcal{R})$ with $d+1$ distinct eigenvalues.\nThen $(\\Omega,\\mathcal{R})$ is $Q$-polynomial if and only if there is a sequence of {\\em distinct\\\/} complex numbers $\\sigma_0,\\sigma_1,\\ldots,\\sigma_d$ and polynomials \n$s_0(x),s_1(x),\\ldots,s_d(x)$ of degree $0,1,\\ldots,d$, respectively, with\n\\[\\sum_j p_{ij}^k\\sigma_j^\\ell=s_\\ell(\\sigma_k)\\]\nfor $0 \\leqslant\\ell\\leqslant d$. Furthermore, the leading coefficients of the polynomials $s_0(x)$, $s_1(x)$, \\dots, $s_d(x)$ are precisely the eigenvalues of $A_i$ in a $Q$-polynomial ordering.\n\\end{theorem}\n\n\\begin{pf}\nWithout loss of generality we assume $A_1$ has this property.  Let $L_1$ be the corresponding intersection matrix.  Let \n$S,T$ be the $d+1$ by $d+1$ matrices with $S_{jk}=\\sigma_j^k$ and $T_{jk}$ equal to the coefficient of $x^j$ in the polynomial $s_k(x)$. Then the above statement is equivalent to \n$L_1 S=ST$.  Then $L_i$ is similar to $T$ and since $T$ is upper triangular, the diagonal entries of $T$ are precisely the eigenvalues of $L_1$. \nSince $T$ is an upper triangular matrix with distinct diagonal entries, an easy induction shows that it can be diagonalized by an upper triangular matrix.  Namely, there is an invertible matrix $U$ and a diagonal matrix $D$ with $D_{jj}=T_{jj}$ such that $U^{-1}TU=D$.\nThen $L_1(SU)=(SU)D$. This implies that the columns of $SU$ are eigenvectors of $L_1$, hence there is a diagonal matrix $D'$ such that $SUD'=Q$.\nSince the $j$th column of $SUD'$ is a degree $j$ polynomial of the first column of $T$, which is a linear combination of columns 0 and 1 of $SUD'$, it is clear that the $j$th column of $SUD'$ is a degree $j$ polynomial of the \nfirst column of $SUD'$.  This implies that the columns of $Q$ are in a given $Q$-polynomial ordering, which in turn implies that the ordering of the eigenvalues in $T$ is a $Q$-polynomial ordering.\\qed\n\\end{pf}\n\nThis leads to our main result:\n\n\\begin{theorem}\\label{theorem7.2}\nThe scheme $\\mathcal{S}$ is $Q$-polynomial. Furthermore, it has two $Q$-polynomial orderings.\n\\end{theorem}\n\n\\begin{pf}\nLet $r=\\sqrt{q}$ and $d=2n+1$. We define the sequence of polynomials\n\\[s_\\ell(x)=\\begin{cases}\nr^\\ell\\gauss{n-\\ell+1}{1}x^\\ell+\\frac{1}{r^{\\ell-2}}\\gauss{\\ell-1}{1}x^{\\ell-2},&\\mbox{for $\\ell$ odd};\\\\\n\\noalign{\\smallskip}\nr^\\ell\\left(\\gauss{n-\\ell+1}{1}-\\frac{1}{r^\\ell}\\right)x^\\ell+\\frac{1}{r^{\\ell-2}}\\left( \\gauss{\\ell-1}{1}+r^{\\ell-2} \\right)x^{\\ell-2},&\\mbox{for $\\ell$ even}\n\\end{cases}\\]\nand constants\n\\[\\sigma_j=\\begin{cases}\n\\frac{1}{r^j},&\\mbox{for\\ }0\\leqslant j\\leqslant n;\\\\\n-\\frac{1}{r^{2n+1-j}},&\\mbox{for\\ }n+1\\leqslant j\\leqslant2n+1.\n\\end{cases}\\]\nThe polynomials $s_0(x),\\ldots,s_{2n+1}(x)$ realize $\\sigma_0,\\ldots,\\sigma_{2n+1}$\nas a $Q$-sequence for $\\mathcal{S}$, as we proceed to show by direct computation.\nFor $k\\leqslant n$ we have $\\sum_j p_{1j}^k \\sigma_j^\\ell=c_k \\sigma_{k-1}^\\ell+\\frac{a_k}{2} \\sigma_{k}^\\ell+b_k \\sigma_{k+1}^\\ell+\\frac{a_k}{2} \\sigma_{2n+1-k}^\\ell $.\nFor odd $\\ell$ this reduces to\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align*}\nc_k \\sigma_{k-1}^\\ell+b_k \\sigma_{k+1}^\\ell\n&=\\tfrac{1}{r^{(k-1)\\ell}}\\gauss{k}{1}+q\\left(\\gauss{k}{1}-\\gauss{n}{1}-\\gauss{k}{1} \\right)\\tfrac{1}{r^{(k+1)\\ell}}\\\\\n&=\\tfrac{1}{r^{(k+1)\\ell}}\\left(\\gauss{k}{1}q^\\ell+q\\bigl(\\gauss{n}{1}-\\gauss{k}{1}\\bigr)\\right)\\\\\n&=\\tfrac{1}{r^{(k+1)\\ell}}\\left(\\gauss{n+1}{1}-\\gauss{\\ell}{1}+\\gauss{k+\\ell}{1}- \\gauss{k+1}{1}\\right)\\\\\n&=\\tfrac{1}{r^{(k+1)\\ell}}\\left( \\gauss{n-\\ell+1}{1}r^{2\\ell}+\\gauss{\\ell-1}{1}r^{2(k+1)}\\right)\\\\\n&=r^\\ell\\gauss{n-\\ell+1}{1}\\tfrac{1}{r^{k\\ell}}+\\tfrac{1}{r^{\\ell-2}}\\gauss{\\ell-1}{1}\\tfrac{1}{r^{k(\\ell-2)}}\\\\\n&=s_\\ell(\\sigma_k),\n\\end{align*}\nwhereas for even $\\ell$ we have\n\\begin{align*}\n\\textstyle{\\sum_j p_{1j}^k \\sigma_j^\\ell}\n&=c_k\\sigma_{k-1}^\\ell+a_k \\sigma_{k}^\\ell+b_k \\sigma_{k+1}^\\ell\\\\\n&=\\gauss{k}{1}\\tfrac{1}{r^{(k-1)\\ell}}+(q-1)\\gauss{k}{1}\\tfrac{1}{r^{k\\ell}}+q\\left(\\gauss{k}{1}-\\gauss{n}{1}-\\gauss{k}{1}\\right)\\tfrac{1}{r^{(k+1)\\ell}}\\\\\n&=\\tfrac{1}{r^{(k+1)\\ell}}\\left(\\gauss{k}{1}q^{\\ell}+q\\bigl(\\gauss{n}{1}-\\gauss{k}{1}\\bigr)+r^{2k+\\ell}-r^\\ell\\right)\\\\\n&=\\tfrac{1}{r^{(k+1)\\ell}}\\left(\\gauss{n+1}{1}-\\gauss{\\ell}{1}+\\gauss{k+\\ell}{1}- \\gauss{k+1}{1}+r^{2k+\\ell}-r^\\ell\\right)\\\\\n&=\\tfrac{1}{r^{(k+1)\\ell}}\\left(\\gauss{n-\\ell+1}{1}r^{2\\ell}-r^\\ell+\\gauss{\\ell-1}{1}r^{2(k+1)}+r^{2k+\\ell}\\right)\\\\\n&=r^\\ell\\left(\\gauss{n-\\ell+1}{1}-\\tfrac{1}{r^\\ell}\\right)\\tfrac{1}{r^{k\\ell}}+\\tfrac{1}{r^{\\ell-2}}\\left(\\gauss{\\ell-1}{1}+r^{\\ell-2}\\right)\\tfrac{1}{r^{k(\\ell-2)}}\\\\\n&=s_\\ell(\\sigma_k).\n\\end{align*}\n\\endgroup\n\nNow we deal with $k\\geqslant n+1$, noting that\n\\[\\sum_j p_{1j}^k \\sigma_j^\\ell=b_{2n+1-k} \\sigma_{k-1}^\\ell+\\tfrac{a_{2n+1-k}}{2} \\sigma_{k}^\\ell+c_{2n+1-k} \\sigma_{k+1}^\\ell+\\tfrac{a_{2n+1-k}}{2} \\sigma_{2n+1-k}^\\ell.\\]\nFor odd $\\ell$ this reduces to\n\\begin{align*}\nb_{2n+1-k} \\sigma_{k-1}^\\ell+c_{2n+1-k} \\sigma_{k+1}^\\ell&=- b_{2n+1-k} \\sigma_{2n+2-k}^\\ell - c_{2n+1-k} \\sigma_{2n-k}^\\ell\\\\\n&=-s_\\ell( \\sigma_{2n+1-k})=s_\\ell(-\\sigma_{2n+1-k})=s_\\ell(\\sigma_{k}),\n\\end{align*}\nwhile for even $\\ell$ we obtain\n\\begin{align*}\nb_{2n+1-k}\\sigma_{k-1}^\\ell+{}&a_{2n+1-k}\\sigma_{k}^\\ell+c_{2n+1-k} \\sigma_{k+1}^\\ell\\\\\n&=b_{2n+1-k}\\sigma_{2n+2-k}^\\ell+a_{2n+1-k}\\sigma_{2n+1-k}^\\ell+c_{2n+1-k}\\sigma_{2n-k}^\\ell\\\\\n&=s_\\ell( \\sigma_{2n+1-k})=s_\\ell(-\\sigma_{2n+1-k})=s_\\ell(\\sigma_{k}).\n\\end{align*}\n\nFor nonsquare $q$ the splitting field of $\\mathcal{S}$ is irrational, implying that it is a quadratic extension of the rationals, namely \n$\\mathbb{Q}(r)$. The Galois group acts faithfully on the idempotents of the scheme, yielding a second $Q$-polynomial ordering.  \nThis second $Q$-polynomial ordering can also be obtained by replacing $r\\mapsto-r$ in both the $\\sigma_j$ and the polynomials $s_\\ell(x)$, showing that this second ordering exists for square $q$ as well. \\qed\n\\end{pf}\n\nWe note that by a result of Suzuki \\cite{SUZ}, $Q$-polynomial schemes can have at most two $Q$-polynomial orderings.\n\n\\section{The $P$-matrix}\n\\label{P-matrix}\n\nWe now compute the $P$-matrix of the scheme $\\mathcal{S}$, expressing it in terms of the auxiliary matrices $\\widetilde{P}$ and $\\widehat{P}$ whose entries are defined by\n\\begin{align*}\n\\widetilde{P}_{ij}&=\\sum\\limits_{l=0}^j (-1)^\\ell r^{j-2\\ell+(j-\\ell)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell};\\\\\n\\widehat{P}_{ij}&=\\sum\\limits_{\\ell=0}^j (-1)^\\ell r^{(j-\\ell)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}.\n\\end{align*}\nBy \\cite[Proposition 2.2.2]{BCN}, the $P$-matrix is determined by the left-normalized left eigenvectors of $L_1$. We first show that the rows of $\\widetilde{P}$ and $\\widehat{P}$ are left\neigenvectors of the matrices defined by\n\\[M\\llap{$\\widetilde{\\phantom{D}}$}=\\begin{pmatrix}  \n0&b_0&&&&\\\\\n1&a_1&b_1&&&\\\\\n&c_2&a_2&\\smash\\ddots&&\\\\\n\\noalign{\\vskip-2pt}\n&&\\ddots&\\ddots&\\ddots&\\\\\n\\noalign{\\vskip-3pt}\n&&&\\ddots&a_{d-1}&b_{d-1}\\\\\n&&&&c_{d}&a_d\n\\end{pmatrix},\\quad\nM\\llap{$\\widehat{\\phantom{D}}$}=\\begin{pmatrix}  \n0&b_0&&&&\\\\\n1&0&b_1&&&\\\\\n\\noalign{\\vskip-4pt}\n&c_2&0&\\ddots&&\\\\\n\\noalign{\\vskip-2pt}\n&&\\ddots&\\ddots&\\ddots&\\\\\n\\noalign{\\vskip-5pt}\n&&&\\ddots&0&b_{d-1}\\\\\n&&&&c_{d}&0\n\\end{pmatrix}\\]\nrespectively. We will show that the corresponding diagonal forms are\n\\[\\widetilde{D}=\\diag(\\widetilde{P}_{i0},\\widetilde{P}_{i1},\\ldots,\\widetilde{P}_{in}),\\quad\\widehat{D}=\\diag(\\widehat{P}_{i0},\\widehat{P}_{i1},\\ldots,\\widehat{P}_{in}).\\]\nThe ordering we give to the eigenvectors of $M\\llap{$\\widetilde{\\phantom{D}}$}$ and $M\\llap{$\\widehat{\\phantom{D}}$}$ may seem arbitrary, but will be important later.\n\n\\begin{theorem}\\label{theorem8.1}\n$\\widetilde{P}M\\llap{$\\widetilde{\\phantom{D}}$}=\\widetilde{D}\\widetilde{P}$ and $\\widehat{P}M\\llap{$\\widehat{\\phantom{D}}$}=\\widehat{D}\\widehat{P}$.\n\\end{theorem}\n\n\\begin{pf}\nFix $i$ and let $v_i=(\\widetilde{P}_{i0},\\widetilde{P}_{i1},\\ldots,\\widetilde{P}_{in})$.\nWe must show that $v_iM\\llap{$\\widetilde{\\phantom{D}}$}=\\widetilde{P}_{i1}v_i$. In particular, we need to show the following recurrence holds for all $j$:\n\\begin{align*}\nb_{j-1}\\sum\\limits_{\\ell=0}^{j-1}(-1)^\\ell&r^{j-1-2\\ell +(j-1-\\ell)^2+\\ell^2}\n\\gauss{i}{\\ell}\\gauss{n-i}{j-1-\\ell} + a_j \\sum\\limits_{\\ell=0}^{j}(-1)^\\ell r^{j-2\\ell +(j-\\ell)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}\\\\\n&\\qquad+c_{j+1}\\sum\\limits_{\\ell=0}^{j+1}(-1)^\\ell r^{j+1-2\\ell+(j+1-\\ell)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j+1-\\ell}\\\\\n&=\\widetilde{P}_{i1} \\sum\\limits_{\\ell=0}^j(-1)^\\ell r^{j-2\\ell+(j-\\ell)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}.\n\\end{align*}\nMultiplying both sides by $q-1$ and substituting for $b_{j-1}, a_j$ and $c_{j+1}$, we find this is equivalent to showing that the quantity $z_j$, defined as follows, vanishes for all $j$:\n\\begin{align*}\nz_j={}&(q^{n+1}-q^j)\\sum\\limits_{\\ell=0}^{j-1}(-1)^\\ell r^{j-2\\ell-1+(j-\\ell-1)^2+\\ell^2} \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell-1}\\\\\n&+(q-1)(q^j-1)\\sum\\limits_{\\ell=0}^{j}(-1)^\\ell r^{j-2\\ell+(j-\\ell)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}\\\\\n&+(q^{j+1}-1) \\sum\\limits_{\\ell=0}^{j+1}(-1)^\\ell r^{ j-2\\ell+1+(j-\\ell+1)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell+1}\\\\\n&-\\bigl(q^i(q^{n-2i+1}-1)-q+1\\bigr)\\sum\\limits_{\\ell=0}^j (-1)^\\ell r^{ j-2\\ell+(j-\\ell)^2+\\ell^2} \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}.\n\\end{align*}\nThe second and last sums combine, simplifying to\n\\begin{align*}\nz_j={}&(q^{n+1}-q^j) \\sum\\limits_{\\ell=0}^{j-1} (-1)^\\ell r^{ j-2\\ell-1 +(j-\\ell-1)^2+\\ell^2} \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell-1}\\\\\n&+\\bigl((q-1)q^{j}+q^i - q^{n-i+1}\\bigr) \\sum\\limits_{\\ell=0}^{j} (-1)^\\ell r^{ j-2\\ell +(j-\\ell)^2+\\ell^2} \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}\\\\\n&+(q^{j+1}-1) \\sum\\limits_{\\ell=0}^{j+1} (-1)^\\ell r^{ j-2\\ell+1 + (j-\\ell+1)^2+\\ell^2} \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell+1}.\n\\end{align*}\nNow it suffices to show that the generating function $Z(t)=\\sum_{j=0}^\\infty z_j t^j$ vanishes. We first express $Z(t)$ in terms of the\npolynomials $E_m(t)$ defined in Section~\\ref{gaussian}.\nUsing Proposition~\\ref{prop2.2}(iii),\nwe are able to rewrite our generating function as $Z(t)=\\Sigma_1+\\Sigma_2+\\cdots+\\Sigma_6$ where\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align*}\n\\Sigma_1&=q^{n+1}\\sum\\limits_{j=0}^{\\infty}\\sum\\limits_{\\ell=0}^{j-1}(-1)^\\ell r^{ j-2\\ell-1 +(j-\\ell-1)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell-1}t^j\\\\\n&=q^{n+1}tE_i(-t)E_{n-i}(qt);\\\\\n\\Sigma_2&=-\\sum\\limits_{j=0}^{\\infty}q^j\\sum\\limits_{\\ell=0}^{j-1}(-1)^\\ell r^{ j-2\\ell-1 +(j-\\ell-1)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell-1}t^j\\\\\n&=-qtE_i(-qt)E_{n-i}(q^2t);\\\\\n\\Sigma_3&=(q-1)\\sum\\limits_{j=0}^{\\infty}q^j\\sum\\limits_{\\ell=0}^{j+1}(-1)^\\ell r^{ j-2\\ell +(j-\\ell)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}t^j\\\\\n&=(q-1)E_i(-qt)E_{n-i}(q^2t);\\\\\n\\Sigma_4&=(q^i - q^{n-i+1})\\sum\\limits_{j=0}^{\\infty}\\sum\\limits_{\\ell=0}^{j+1} (-1)^\\ell r^{ j-2\\ell+(j-\\ell)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}t^j\\\\\n&=(q^i-q^{n-i+1})E_i(-t)E_{n-i}(qt);\\\\\n\\Sigma_5&=\\sum\\limits_{j=0}^{\\infty}q^{j+1}\\sum\\limits_{\\ell=0}^j(-1)^\\ell r^{ j-2\\ell+1 +(j-\\ell+1)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell+1}t^j\\\\\n&=\\tfrac{1}{t}E_i(-qt)E_{n-i}(q^2t);\\\\\n\\Sigma_6&=-\\sum\\limits_{j=0}^{\\infty}\\sum\\limits_{\\ell=0}^j (-1)^\\ell r^{ j-2\\ell+1 +(j-\\ell+1)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell+1}t^j\\\\\n&=-\\tfrac{1}{t}E_i(-t)E_{n-i}(qt).\n\\end{align*}\n\\endgroup\nUsing Proposition~\\ref{prop2.2}(i,ii), we find \n\\begin{align*}\nZ(t)&{}=\\Sigma_1+\\Sigma_2+ \\Sigma_3+\\Sigma_4+\\Sigma_5+\\Sigma_6\\\\\n&{}=\\Bigl(q^{n+1}t+q^i-q^{n-i+1}-\\tfrac{1}{t}\\\\\n&\\qquad{}+\\tfrac{(1-q^it)}{(1-t)}\\tfrac{(1+q^{n-i+1}t)}{(1+qt)}\\bigl(-qt+q-1+\\tfrac{1}{t}\\bigr)\\Bigr) E_i(-t) E_{n-i}(qt)=0\n\\end{align*}\nas required.\n\n\nThe strategy for showing $\\widehat{P}M\\llap{$\\widehat{\\phantom{D}}$}=\\widehat{D}\\widehat{P}$ is very similar but the details are sufficiently different that we provide the details here.\nFix $i$ and let $v_i=(\\widehat{P}_{i0},\\widehat{P}_{i1},\\ldots,\\widehat{P}_{in})$.\nWe must show that $v_iM\\llap{$\\widehat{\\phantom{D}}$}=\\widehat{P}_{i1}v_i$. In particular, we need to show the following recurrence holds for all $j$:\n\\[b_{j-1}\\sum\\limits_{\\ell=0}^{j-1} (-1)^\\ell r^{ (j-1-\\ell)^2+\\ell^2} \\gauss{i}{\\ell}\\gauss{n-i}{j-1-\\ell} + c_{j+1}\\sum\\limits_{\\ell=0}^{j+1} (-1)^\\ell r^{ (j+1-\\ell)^2+\\ell^2}  \\gauss{i}{\\ell}\\gauss{n-i}{j+1-\\ell}\\]\n\\[=\\widehat{P}_{i1} \\sum\\limits_{\\ell=0}^j (-1)^\\ell r^{ (j-\\ell)^2+\\ell^2} \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}.\\]\nMultiplying both sides by $q-1$ and substituting for $b_{j-1}, c_{j+1}$, we find this is equivalent to showing that the following is zero for all $j$: \n\\begin{align*}\nz_j={}&(q^{n+1}-q^j) \\sum\\limits_{\\ell=0}^{j-1} (-1)^\\ell r^{(j-\\ell-1)^2+\\ell^2} \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell-1}\\\\\n&+(q^{j+1}-1) \\sum\\limits_{\\ell=0}^{j+1} (-1)^\\ell r^{(j-\\ell+1)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell+1}\\\\\n&-q^i(q^{2n-i}-1) \\sum\\limits_{\\ell=0}^j (-1)^\\ell r^{(j-\\ell)^2+\\ell^2}\\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}.\n\\end{align*}\nAgain, it suffices to show that the generating function $Z(t)=\\sum_{j=0}^\\infty z_j t^j$ vanishes. As before, we first\nrewrite our generating function as $Z(t)=\\Sigma_1+\\Sigma_2+\\cdots+\\Sigma_6$ where\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align*}\n\\Sigma_1&=q^{n+1} \\sum\\limits_{\\ell=0}^{j-1} (-1)^\\ell r^{ (j-\\ell-1)^2+\\ell^2} t^\\ell \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell-1} t^m=q^{n+1}tE_i(-rt) E_{n-i}(rt);\\\\\n\\Sigma_2&=-q^{j} \\sum\\limits_{\\ell=0}^{j-1} (-1)^\\ell r^{ (j-\\ell-1)^2+\\ell^2}t^\\ell \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell-1}t^m=-qtE_i(-r^3t) E_{n-i}(r^3t);\\\\\n\\Sigma_3&=q^{j+1} \\sum\\limits_{\\ell=0}^{j+1} (-1)^\\ell r^{ (j-\\ell+1)^2+\\ell^2}t^\\ell  \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell+1}t^m=\\tfrac{1}{t}E_i(-r^3t) E_{n-i}(r^3t);\\\\\n\\Sigma_4&=-\\sum\\limits_{\\ell=0}^{j+1} (-1)^\\ell r^{ (j-\\ell+1)^2+\\ell^2}t^\\ell  \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell+1}t^m=-\\tfrac{1}{t}E_i(-rt) E_{n-i}(rt);\\\\\n\\Sigma_5&=-rq^{2n} \\sum\\limits_{\\ell=0}^j (-1)^\\ell r^{ (j-\\ell)^2+\\ell^2}t^\\ell \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}t^m=-r q^{n-i}E_i(-rt) E_{n-i}(rt);\\\\\n\\Sigma_6&=rq^i \\sum\\limits_{\\ell=0}^j (-1)^\\ell r^{ (j-\\ell)^2+\\ell^2}t^\\ell \\gauss{i}{\\ell}\\gauss{n-i}{j-\\ell}t^m=r q^{i}E_i(-rt) E_{n-i}(rt)\n\\end{align*}\n\\endgroup\nin terms of the polynomials $E_m(t)$ defined\nin Section~\\ref{gaussian}. Using Proposition~\\ref{prop2.2}(iii), we find \n\\begin{align*}\nZ(t)&{}=\\Sigma_1+\\Sigma_2+\\Sigma_3+\\Sigma_4+\\Sigma_5+\\Sigma_6\\\\\n&=\\Bigl(q^{n+1}t-\\tfrac{1}{t}-rq^{n-i}+rq^i\\\\\n&\\qquad+\\bigl(\\tfrac{1}{t}-qt\\bigr)\\tfrac{(1-rq^it)(1+rq^{n-i}t)}{(1-rt)(1+rt)}\\Bigr)E_i(-rt) E_{n-i}(rt)=0\n\\end{align*}\nas required.\\qed\n\\end{pf} \n\n\\bigskip\n\n\\begin{corollary}\\label{corollary8.2}\nThe $P$-matrix of the $Q$-polynomial scheme $\\mathcal{S}$ is given by\n\\[\\begin{cases}\nP_{i,j}=P_{i,2n+1-j}=\\widetilde{P}_{\\frac{i}2,j}&\\mbox{for $i$ even, $0\\leqslant j\\leqslant n$;}\\\\\n\\noalign{\\smallskip}\nP_{i,j}=-P_{i,2n+1-j}=\\widehat{P}_{\\lfloor\\frac{i}2\\rfloor,j}&\\mbox{for $i$ odd, $0\\leqslant j\\leqslant n$.}\n\\end{cases}\\]\n\\end{corollary}\n\n\\begin{pf}\nIf $(v_0,\\ldots,v_n)$ is a left eigenvector of $M\\llap{$\\widetilde{\\phantom{D}}$}$ or $M\\llap{$\\widehat{\\phantom{D}}$}$, it is easily seen that\neither $(v_0,\\ldots,v_n, v_n,\\dots,v_0)$ or  \n$(v_0,\\ldots, v_n,-v_n,\\ldots,-v_0)$ is a left eigenvector of $L_1$, respectively.  The fact that this ordering of the eigenvalues of $L_1$ is a $Q$-polynomial ordering follows from Theorem~\\ref{theorem7.2}.\\qed\n\\end{pf}\n\n\\section{A hypothetical subscheme}\n\\label{subscheme}\n\nWe ask whether $\\mathcal{S}$ is the extended $Q$-bipartite double (in the sense of \\cite{MMW}) of a primitive $Q$-polynomial scheme.  \nWe investigated these parameters up to $n=20$ and found they satisfied the Krein conditions, had integral eigenvalue multiplicities and nonnegative integral $p^k_{ij}$, and satisfy the handshaking lemma for all square $q$.\nThis appears to give an infinite family of feasible parameters for primitive $Q$-polynomial schemes with an unbounded number of classes. \nDetailed parameters and a proof of feasibility will be given in a forthcoming paper of Eiichi Bannai and Jianmin Ma.\n\nWe give the smallest case below for which existence is unknown:\n\\bigskip\n\n\\[P=\\begin{pmatrix} \n 1 &  \\frac{r^4 + r^3 + r^2 + r}{2} &  \\frac{r^4 - r^3 + r^2 - r}{2} &  \\frac{r^6 + r^4}{2} &  \\frac{r^6 - r^4}{2} \\\\\n  1 &  \\frac{r^3 + r^2 + r - 1}{2} &  \\frac{-r^3 + r^2 - r - 1}{2} &  \\frac{r^4 - r^2}{2} &  \\frac{-r^4 - r^2}{2} \\\\\n  1 &  \\frac{r^2 - 1}{2} &  \\frac{r^2 - 1}{2} &  -r^2 &  0 \\\\\n  1 &  \\frac{-r^2 - 1}{2} &  \\frac{-r^2 - 1}{2} &  0 &  r^2 \\\\\n  1 &  \\frac{-r^3 - r^2 - r - 1}{2} &  \\frac{r^3 - r^2 + r - 1}{2} &  \\frac{r^4 + r^2}{2} &  \\frac{-r^4 + r^2}{2}\n \\end{pmatrix}\\]\n\n\\[Q= \\begin{pmatrix} \n 1 &  \\frac{r^4 - 1}{2} &  \\frac{r^6 + r^4 + r^2 + 1}{2} &  \\frac{r^6 - r^4 + r^2 - 1}{2} &  \\frac{r^4 + 1}{2} \\\\\n  1 &  \\frac{r^4 - 2r + 1}{2r} &  \\frac{r^5 - r^4 + r - 1}{2r} &  \\frac{-r^5 + r^4 - r + 1}{2r} &  \\frac{-r^4 - 1}{2r} \\\\\n  1 &  \\frac{-r^4 - 2r - 1}{2r} &  \\frac{r^5 + r^4 + r + 1}{2r} &  \\frac{-r^5 - r^4 - r - 1}{2r} &  \\frac{r^4 + 1}{2r} \\\\\n  1 &  \\frac{r^4 - 2r^2 + 1}{2r^2} &  \\frac{-r^4 - 1}{r^2} &  0 &  \\frac{r^4 + 1}{2r^2} \\\\\n  1 &  \\frac{-r^4 - 2r^2 - 1}{2r^2} &  0 &  \\frac{r^4 + 1}{r^2} &  \\frac{-r^4 - 1}{2r^2}\n\\end{pmatrix}\\]\n\n\\[L_0= \\begin{pmatrix} \n 1 &  0 &  0 &  0 &  0 \\\\\n  0 &  1 &  0 &  0 &  0 \\\\\n  0 &  0 &  1 &  0 &  0 \\\\\n  0 &  0 &  0 &  1 &  0 \\\\\n  0 &  0 &  0 &  0 &  1\n\\end{pmatrix}\\]\n\n\\[L_1= \\begin{pmatrix} \n 0 &  \\frac{r^4 + r^3 + r^2 + r}{2} &  0 &  0 &  0 \\\\\n  1 &  \\frac{r^2 + 2r - 3}{4} &  \\frac{r^2 - 1}{4} &  \\frac{r^4 + r^3}{2} &  0 \\\\\n  0 &  \\frac{r^2 + 2r + 1}{4} &  \\frac{r^2 - 1}{4} &  0 &  \\frac{r^4 + r^3}{2} \\\\\n  0 &  \\frac{r^2 + 2r + 1}{2} &  0 &  \\frac{r^4 + r^3 + r^2 - r - 2}{4} &  \\frac{r^4 + r^3 - r^2 - r}{4} \\\\\n  0 &  0 &  \\frac{r^2 + 1}{2} &  \\frac{r^4 + r^3 + r^2 + r}{4} &  \\frac{r^4 + r^3 - r^2 + r - 2}{4}\n\\end{pmatrix}\\]\n\n\\[L_2= \\begin{pmatrix} \n 0 &  0 &  \\frac{r^4 - r^3 + r^2 - r}{2} &  0 &  0 \\\\\n  0 &  \\frac{r^2 - 1}{4} &  \\frac{r^2 - 2r + 1}{4} &  0 &  \\frac{r^4 - r^3}{2} \\\\\n  1 &  \\frac{r^2 - 1}{4} &  \\frac{r^2 - 2r - 3}{4} &  \\frac{r^4 - r^3}{2} &  0 \\\\\n  0 &  0 &  \\frac{r^2 - 2r + 1}{2} &  \\frac{r^4 - r^3 + r^2 + r - 2}{4} &  \\frac{r^4 - r^3 - r^2 + r}{4} \\\\\n  0 &  \\frac{r^2 + 1}{2} &  0 &  \\frac{r^4 - r^3 + r^2 - r}{4} &  \\frac{r^4 - r^3 - r^2 - r - 2}{4}\n\\end{pmatrix}\\]\n\n\\[L_3= \\begin{pmatrix} \n 0 &  0 &  0 &  \\frac{r^6 + r^4}{2} &  0 \\\\\n  0 &  \\frac{r^4 + r^3}{2} &  0 &  \\frac{r^6 + r^4 - 2r^3}{4} &  \\frac{r^6 - r^4}{4} \\\\\n  0 &  0 &  \\frac{r^4 - r^3}{2} &  \\frac{r^6 + r^4 + 2r^3}{4} &  \\frac{r^6 - r^4}{4} \\\\\n  1 &  \\frac{r^4 + r^3 + r^2 - r - 2}{4} &  \\frac{r^4 - r^3 + r^2 + r - 2}{4} &  \\frac{r^6 + 2r^4 - 3r^2}{4} &  \\frac{r^6 - 2r^4 + r^2}{4} \\\\\n  0 &  \\frac{r^4 + r^3 + r^2 + r}{4} &  \\frac{r^4 - r^3 + r^2 - r}{4} &  \\frac{r^6 - r^2}{4} &  \\frac{r^6 - r^2}{4}\n\\end{pmatrix}\\]\n\n\\[L_4= \\begin{pmatrix} \n 0 &  0 &  0 &  0 &  \\frac{r^6 - r^4}{2} \\\\\n  0 &  0 &  \\frac{r^4 - r^3}{2} &  \\frac{r^6 - r^4}{4} &  \\frac{r^6 - 3r^4 + 2r^3}{4} \\\\\n  0 &  \\frac{r^4 + r^3}{2} &  0 &  \\frac{r^6 - r^4}{4} &  \\frac{r^6 - 3r^4 - 2r^3}{4} \\\\\n  0 &  \\frac{r^4 + r^3 - r^2 - r}{4} &  \\frac{r^4 - r^3 - r^2 + r}{4} &  \\frac{r^6 - 2r^4 + r^2}{4} &  \\frac{r^6 - 2r^4 + r^2}{4} \\\\\n  1 &  \\frac{r^4 + r^3 - r^2 + r - 2}{4} &  \\frac{r^4 - r^3 - r^2 - r - 2}{4} &  \\frac{r^6 - r^2}{4} &  \\frac{r^6 - 4r^4 + 3r^2}{4}\n\\end{pmatrix}\\]\n\n\\[L_0^*=\\begin{pmatrix}\n1&0&0&0&0\\\\\n0&1&0&0&0\\\\\n0&0&1&0&0\\\\\n0&0&0&1&0\\\\\n0&0&0&0&1\n\\end{pmatrix}\\]\n\n\\[L_1^*=\\begin{pmatrix}\n0&\\frac{r^4 - 1}{2}&0&0&0\\\\\n1&\\frac{r^6 - 5r^4 - 3r^2 - 1}{2(r^4 + r^2)} &\\frac{r^8 +2r^4 + 1}{2(r^4 + r^2)} &0&0\\\\\n0&\\frac{r^6 - r^4 + r^2 - 1}{2(r^4 + r^2)} &\\frac{r^8 - 4r^2 + 3}{4(r^4 + r^2)} &\\frac{r^6 - r^4 + r^2 - 1}{4r^2}&0\\\\\n0&0&\\frac{r^6 + r^4 + r^2 + 1}{4r^2}&\\frac{r^6 - 3r^4 - 3r^2 - 3}{4r^2}&\\frac{r^4 + 1}{2r^2}\\\\\n0&0&0&\\frac{r^6 - r^4 + r^2 - 1}{2r^2}&\\frac{r^4 - 2r^2 + 1}{2r^2}\n\\end{pmatrix}\\]\n\n\\[L_2^*=\\begin{pmatrix}\n0&0&\\frac{r^6 + r^4 + r^2 + 1}{2}&0&0\\\\\n0&\\frac{r^8 + 2r^4 + 1}{2(r^4 + r^2)} &\\frac{r^{10} + r^8 + 2r^6 - 2r^4 + r^2 - 3}{4(r^4 + r^2)} &\\frac{r^8 + 2r^4 + 1}{4r^2}&0\\\\\n1&\\frac{r^8 - 4r^2 + 3}{4(r^4 + r^2)} &\\frac{r^{10} + 3r^8 + 2r^6 - 2r^4 + r^2 - 5}{4(r^4 + r^2)} &\\frac{r^8 - 2r^6 + 2r^4 - 2r^2 + 1}{4r^2}&\n\\frac{r^6 + r^4 + r^2 + 1}{4r^2}\\\\\n0&\\frac{r^6 + r^4 + r^2 + 1}{4r^2}&\\frac{r^8 - 1}{4r^2}&\\frac{r^8 + 2r^4 + 1}{4r^2}&\\frac{r^6 - r^4 + r^2 - 1}{4r^2}\\\\\n0&0&\\frac{r^8 + 2r^6 + 2r^4 + 2r^2 + 1}{4r^2}&\\frac{r^8 - 2r^6 + 2r^4 - 2r^2 + 1}{4r^2}&\\frac{r^6 - r^4 + r^2 - 1}{2r^2}\n\\end{pmatrix}\\]\n\n\\[L_3^*=\\begin{pmatrix}\n0&0&0&\\frac{r^6 - r^4 + r^2 - 1}{2}&0\\\\\n0&0&\\frac{r^8 + 2r^4 + 1}{4r^2}&\\frac{r^{10} - 3r^8 - 2r^6 - 6r^4 - 3r^2 - 3}{4(r^4 + r^2)} &\\frac{r^8 + 2r^4 + 1}{2(r^4 + r^2)} \\\\\n0&\\frac{r^6 - r^4 + r^2 - 1}{4r^2}&\\frac{r^8 - 2r^6 + 2r^4 - 2r^2 + 1}{4r^2}&\\frac{r^{10} - r^8 + 2r^6 - 2r^4 + r^2 - 1}{4(r^4 + r^2)} &\\frac{r^8 - 2r^6 + 2r^4 - 2r^2 + 1}{4(r^4 + r^2)} \\\\\n1&\\frac{r^6 - 3r^4 - 3r^2 - 3}{4r^2}&\\frac{r^8 + 2r^4 + 1}{4r^2}&\\frac{r^8 - 4r^6 + 4r^4 - 4r^2 + 3}{4r^2}&\\frac{r^6 - r^4 + r^2 - 1}{4r^2}\\\\\n0&\\frac{r^6 - r^4 + r^2 - 1}{2r^2}&\\frac{r^8 - 2r^6 + 2r^4 - 2r^2 + 1}{4r^2}&\\frac{r^8 - 2r^6 + 2r^4 - 2r^2 + 1}{4r^2}&0\n\\end{pmatrix}\\]\n\n\\[L_4^*=\\begin{pmatrix}\n0&0&0&0&\\frac{r^4+1}{2}\\\\\n0&0&0&\\frac{r^8 + 2r^4 + 1}{2(r^4 + r^2)} &\\frac{r^6 - r^4 + r^2 - 1}{2(r^4 + r^2)} \\\\\n0&0&\\frac{r^6 + r^4 + r^2 + 1}{4r^2}&\\frac{r^8 - 2r^6 + 2r^4 - 2r^2 + 1}{4(r^4 + r^2)} &\\frac{r^6 - r^4 + r^2 - 1}{2(r^4 + r^2)} \\\\\n0&\\frac{r^4 + 1}{2r^2}&\\frac{r^6 - r^4 + r^2 - 1}{4r^2}&\\frac{r^6 - r^4 + r^2 - 1}{4r^2}&0\\\\\n1&\\frac{r^4 - 2r^2 + 1}{2r^2}&\\frac{r^6 - r^4 + r^2 - 1}{2r^2}&0&0\n\\end{pmatrix}\\]\n\n\\section*{Acknowledgements}\n\nThis research was supported in part by NSF grant DMS-1400281. The software package\nMAGMA~\\cite{Magma} was instrumental in finding the initial examples \nof these schemes, as well as in computations related to verifying our proofs.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDue to asymptotic freedom, the large distance behavior of\n\\mbox{SU$(N)$}~gauge theories must be treated non-perturbatively.\nA convenient method is to put the model in a finite spatial box\nof length $L$ and calculate the low-lying energy eigenstates as\nfunction of $L$. Obviously the wave functionals of\nsuch states are concentrated at small $L$\nin the vacua of {\\em classical}\nconfiguration space. At larger $L$ they can spread out over low energy\nbarriers between these vacua. This spreading causes the breakdown\nof conventional perturbation theory.\n\nThis picture has been used~\\cite{LuKoBa,BaDa} for reduction of the\nfield theory to a finite dimensional system. The remaining degrees\nof freedom are expected to correspond to the set of vacua (called\nthe vacuum valley) and suitable paths over the barriers between\nvacua. One of the requirements on such paths is that they cross\na barrier at its lowest point. This point, a sphaleron~\\cite{Ta,KlMa},\nis defined through a mini-max procedure. One first finds the maximal\nenergy on a path connecting two vacua. Then one minimizes this maximum\nover the space of all such paths.\nIn local terms a sphaleron is\na saddle point of the energy functional with exactly one\nunstable mode.\n\nWell-known paths between two vacua are instantons~\\cite{BePo,BePoScTy},\ninterpreting Euclidean time as a path parameter. In the\nWKB-approximation the tunneling amplitude is dominated by paths near\ninstantons.\nTherefore it seems natural to assume that amongst the instanton paths\nthere is one which crosses a barrier at its lowest point, i.e. it\ngoes through a sphaleron. Let us assume that the set of instantons\n(called the instanton moduli-space) has a scale parameter $\\rho$.\nSince all instantons have the same Euclidean action, a dimensional\nargument easily shows that the maximal energy along the tunneling\npath goes as $1\/\\rho$. Hence it is likely that instantons with\nmaximal width, as set by $L$, go through a sphaleron.\n\nThe latter assumption was made in~\\cite{BaDa} for the following\nreasons.\nFirst, in a direct approach it is often easier to find\ninstantons than sphalerons. Then, if the assumption holds,\none can find all sphalerons via instantons. Second,\nfor small $L$ (high barriers) the spreading of a\nlow energy wave function is determined by tunneling. For large\n$L$ (low barriers) it is determined by classically allowed motion\nthrough a sphaleron. If the\nassumption does not hold, there is some range of $L$~values in\nwhich the most important paths between vacua change dramatically.\n\nThe conjecture that a sphaleron lies on an instanton path is\n{\\em in general} not true. For example, consider the two-dimensional\npotential $V(q_1,q_2)=(q_1^2-1)^2\\mbox{$((q_2^2-1)^2(2+q_2^2)+1)$}\n+(1+b^2\/q_2^2)^{-1}$ for small values of the parameter $b$.\nIt has vacua at $(\\pm 1, 0)$  with zero energy\nand (to leading order in $b$) sphalerons\nat $(0,\\pm 1)$ with energy 2. However, for small enough $b$\nthe instanton does not go through a sphaleron. Instead,\nit goes straight through the saddle point at $(0,0)$, which has\none unstable mode and energy 3. It is even possible that an\ninstanton goes through several saddle points with one unstable\nmode (we think of a one-dimensional example), in which only the\none with {\\em highest} energy would be a candidate sphaleron.\n\nThe above clearly shows that in field theory the conjecture\nneeds to be checked. For \\mbox{SU$(2)$}~gauge theory on a space-time\n$\\mbox{${\\rm T}^3$}\\times\\mbox{$[-\\half T, \\half T]$}$ we~\\cite{PeGoSnBa}\nhave developed a numerical procedure to find the widest\ninstantons (for $T\\rightarrow\\infty$). Subsequent numerical\ninvestigation~\\cite{GaBa} has shown that the widest instanton\ngoes through a saddle point with one unstable mode, like for\n\\mbox{SU$(2)$}\\ on a space-time $S^3\\times\\mbox{I$\\!$R}$~\\cite{BaDa}. It is\nlikely these saddle points are sphalerons. For the two-dimensional\n\\mbox{O$(3)\\,\\sigma$-model}, which is studied in the present paper, this can be\nproven rigorously.\n\nIt is well-known that this $\\sigma$-model shares with the\nfour-dimensional gauge theories features like renormalizability and\nasymptotic freedom. Another\nsimilarity is still more important to us now. Both models,\nwhen put in a spatial cube with periodic boundary conditions,\nhave vacuum valleys of nonzero dimension. This\ncan increase the number of\ninstantons, as they\nmust have endpoints in the vacuum valley. In this context\nwe also mention the\nabsence of instantons on compact\nspace-times \\mbox{${\\rm T}^3\\times {\\rm T}^1$}\\ and \\mbox{${\\rm T}^1\\times {\\rm T}^1$},\nrespectively~\\cite{BrBa,RiRo,MiSp}. This does not rule out\nthe existence of instantons on an\ninfinite time interval. It merely suggests the vacuum valley\ncannot be reached in a finite amount of time.\n\nThe outline of this paper is as follows: in\nsection~\\ref{sec:genform} we will set up a convenient formalism,\nwhich will be used in\nsection~\\ref{sec:stat} to derive all static solutions of the\nEuler-Lagrange equations. In particular we\nfind the vacuum valley and the\nsphaleron solutions. Section~\\ref{sec:inst} is devoted\nto constructing all instanton solutions and interpreting their\nmoduli-space. A scale parameter will emerge that relates the\ninstanton field at $t=+\\infty$ to that at $t=-\\infty$.\nWe conclude in section~\\ref{sec:concl} by\nputting our results in\nperspective with respect to the four-dimensional \\mbox{SU$(2)$}\\ case.\n\n\n\\section{The \\mbox{O$(3)\\,\\sigma$-model}\\ in general coordinates}\n\\label{sec:genform}\n\nThe action for the \\mbox{O$(3)\\,\\sigma$-model}\\ on a cylinder reads\n\\begin{equation}\nS[\\mbox{$\\vec{n}$}]=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\mbox{$\\int\\!\\!d^2x\\,\\,$} |\\partial_{\\mu}\\mbox{$\\vec{n}(x)$}|^2,\\ \\ \\ \\mbox{$\\vec{n}(x)$}\\in\\mbox{I$\\!$R}^3,\\ \\ \\\n|\\mbox{$\\vec{n}(x)$}|^2=1,\n\\label{eq:nact}\n\\end{equation}\nwhere the integration runs over space-time\n$\\{(x_2,x_1)\\in\\mbox{${\\rm T}^1\\times\\re$}\\}$. We use\noverall scale invariance to fix the length of the spatial 1-torus\n${\\rm T}^1$ (the circle) to be $2\\pi$. So $\\mbox{$\\vec{n}$}(x+2\\pi\\hat{e}_2)=\\mbox{$\\vec{n}$}(x)$.\nThe metric on space-time is Euclidean, and we use the summation\nconvention over repeated indices throughout.\n\nBy definition, $\\mbox{$\\vec{n}(x)$}\\in S^2$. If we use coordinates $v^i$ ($i=1,2$)\nand a metric $g_{ij}$ on $S^2$, \\refeq{nact} can be\nrewritten as\n\\begin{equation}\nS[v]=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\mbox{$\\int\\!\\!d^2x\\,\\,$} g_{ij}(v(x))\\partial_{\\mu} v^i(x)\\partial_{\\mu} v^j(x).\n\\label{eq:vact}\n\\end{equation}\nIn this paper we will use both\nspherical coordinates $(\\vartheta,\\varphi)$ and\nstereographic projection $(u_1,u_2)$ which will be paired as\n$u\\equiv u_1+i u_2$, with complex conjugate $\\bar{u}=u_1-i u_2$\n(c.f.~\\cite{BePo,RiRo}):\n\\begin{equation}\n\\mbox{$\\vec{n}$}=\\left(\\!\\!\\begin{array}{c}\\sin\\vartheta\\cos\\varphi \\\\\n\\sin\\vartheta\\sin\\varphi \\\\ \\cos\\vartheta\\end{array}\\!\\!\\right)=\n\\frac{1}{1+|u|^2}\\left(\\!\\!\\begin{array}{c}u+\\mbox{$\\bar{u}$} \\\\ (u-\\mbox{$\\bar{u}$})\/i\n\\\\ |u|^2-1\\end{array}\\!\\!\\right)\n\\label{eq:npar}\n\\end{equation}\n(hence $u=\\cot\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\vartheta\\,e^{i\\varphi}$).\n\nIn section~\\ref{sec:stat} we will need the Euler-Lagrange\nequations and the Hessian of \\refeq{vact}. A\nstraightforward computation~\\cite{AlFrMu} gives (for vanishing\n$\\delta v(x_1\\rightarrow\\pm\\infty,x_2)$):\n\\begin{eqnarray}\nS[v+\\delta v]&=&S[v]+S^{(1)}[v,\\xi]+S^{(2)}[v,\\xi]+\\mbox{$\\cal O$}(\\xi^3),\n\\nonumber\\\\\nS^{(1)}[v,\\xi]&=&-\\mbox{$\\int\\!\\!d^2x\\,\\,$} \\left(D_{\\mu}\\partial_{\\mu} v\\right)_i\\xi^i,\n\\label{eq:pertact}\\\\\nS^{(2)}[v,\\xi]&=&\\mbox{$\\int\\!\\!d^2x\\,\\,$} \\xi^i\\mbox{$\\cal H$}_{ij}[v]\\xi^j,\\ \\ \\\n\\mbox{$\\cal H$}_{ij}[v]=-\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\left(D_{\\mu}D_{\\mu}\\right)_{ij}+\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$} R_{kijl}\\partial_{\\mu}\nv^k\\partial_{\\mu} v^l.\n\\nonumber\n\\end{eqnarray}\nHere $\\xi=\\delta v+\\mbox{$\\cal O$}(\\delta v^2)$ is defined in such a way that it\ntransforms covariantly; see~\\cite{AlFrMu} for details. Since the\naction is a scalar, this guarantees that \\mbox{$\\cal H$}\\ is a tensor, as can be\nverified from the explicit form. Furthermore, $D_{\\mu}$ is\nthe covariant derivative and $R$ is the Riemann tensor, both at the\npoint $v$:\n\\begin{eqnarray}\n\\left(D_{\\mu}\\lambda\\right)_i&=&\\partial_{\\mu}\\lambda_i-{\\Gamma^j}_{ik}\\partial_{\\mu}\nv^k\\lambda_j,\\ \\ \\ \\mbox{(for any vector $\\lambda$)} \\nonumber\\\\\n\\Gamma_{kij}&=&\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\left(\\hat{\\partial}_j g_{ik}+\\hat{\\partial}_i\ng_{jk}-\\hat{\\partial}_k g_{ij}\\right),\\ \\ \\\n(\\hat{\\partial}_i\\equiv\\frac{\\partial}{\\partial v^i})\n\\\\\n{R^l}_{ijk}&=&\\hat{\\partial}_j{\\Gamma^l}_{ik}-\\hat{\\partial}_k\n{\\Gamma^l}_{ij}\n+{\\Gamma^m}_{ik}{\\Gamma^l}_{mj}-{\\Gamma^m}_{ij}{\\Gamma^l}_{mk}.\n\\nonumber\n\\end{eqnarray}\nThe Euclidean action, \\refeq{vact}, is precisely half the\ntotal volume of \\mbox{$v(\\mbox{${\\rm T}^1\\times\\re$})\\subset S^2$}.\nThis action also naturally\noccurs in string theory (see e.g.~\\cite{GrScWi}). Therefore,\nextremizing the action (which amounts to putting $D_{\\mu}\\partial_{\\mu} v=0$,\n\\refeq{pertact}) gives a geodesic surface (in affine parametrization)\non the space $S^2$.\n\nFinally, it is important to introduce the winding number which\nmeasures the number of times \\mbox{${\\rm T}^1\\times\\re$}\\ is wrapped around $S^2$ by\n$v$ (and therefore is invariant under continuous deformations\nof $v$):\n\\begin{equation}\nQ[v]=-\\raisebox{.15ex}{\\scriptsize$\\frac{1}{8\\pi}$}\\mbox{$\\int\\!\\!d^2x\\,\\,$} \\varepsilon_{\\mu\\nu}\\mbox{$\\vec{n}$}\\cdot\\left(\\partial_{\\mu}\\mbox{$\\vec{n}$}\n\\times\\partial_{\\nu}\\mbox{$\\vec{n}$}\\right)=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{8\\pi}$}\\mbox{$\\int\\!\\!d^2x\\,\\,$}\\varepsilon_{\\mu\\nu}\\partial_{\\mu}\\mbox{$\\vec{n}$}\\cdot\\left(\n\\mbox{$\\vec{n}$}\\times\\partial_{\\nu}\\mbox{$\\vec{n}$}\\right)=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{8\\pi}$}\\mbox{$\\int\\!\\!d^2x\\,\\,$}\\varepsilon_{\\mu\\nu}g_{ij}\n\\partial_{\\mu} v^i\\lambda^j_{\\nu},\n\\label{eq:topch}\n\\end{equation}\nwhere $\\lambda^j_{\\nu}$ is the vector in the tangent space of $S^2$\n(at the point $v$) corresponding to $\\mbox{$\\vec{n}$}\\times\\partial_{\\nu}\\mbox{$\\vec{n}$}$; up to orientation\n$\\lambda_{\\nu}$ is defined by $g_{ij}\\lambda^i_{\\nu}\\lambda^j_{\\nu}=\ng_{ij}\\partial_{\\nu} v^i\\partial_{\\nu} v^j,\\;g_{ij}\\lambda^i_{\\nu}\\partial_{\\nu} v^j=0$ (no summations\nover $\\nu$). From this follows the well-known formula~\\cite{BePo}\n\\begin{equation}\nS=\\tilde{S}_{\\pm}\\mp4\\pi Q,\\ \\ \\ \\tilde{S}_{\\pm}[v]=\n\\raisebox{.15ex}{\\scriptsize$\\frac{1}{4}$}\\mbox{$\\int\\!\\!d^2x\\,\\,$} g_{ij}\\left(\\partial_{\\mu} v^i\\pm\\varepsilon_{\\mu\\nu}\n\\lambda^i_{\\nu}\\right)\n\\left(\\partial_{\\mu} v^j\\pm\\varepsilon_{\\mu\\rho}\\lambda^j_{\\rho}\\right).\n\\label{eq:niceact}\n\\end{equation}\nWe define an instanton to be a minimal action configuration in the\nsector $Q=+1$. Therefore it has action $4\\pi$ and satisfies the instanton\nequation\n\\begin{equation}\n\\partial_{\\mu} v^i=\\varepsilon_{\\mu\\nu}\\lambda^i_{\\nu}.\n\\label{eq:insteq}\n\\end{equation}\n\n\\section{Sphalerons, vacua and other static solutions}\n\\label{sec:stat}\n\nIn this section we study the potential, or energy functional,\nwhich is the action (\\ref{eq:vact}) restricted to space, i.e.\\\nwithout time-dependence and without integration over time:\n\\begin{equation}\nV[v]=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\int_{{\\rm T}^1}\\!\\!dx_2\\,\\, g_{ij}(v)\\partial_2 v^i\n\\partial_2 v^j.\\ \\ \\ (v=v(x_2))\n\\label{eq:pot}\n\\end{equation}\nThis is the geodesic action for a curve $v(x_2)$ on $S^2$.\nTherefore all static solutions of\nthe Euler-Lagrange equations, being extrema of \\refeq{pot}, are big\ncircles on $S^2$ (affinely parametrized by $x_2$). Remembering\nthe requirement $v(x_2+2\\pi)=v(x_2)$ we conclude that the most\ngeneral static solution reads (after an \\mbox{SO$(3)$}~rotation)\n\\begin{equation}\n\\mbox{$\\vec{n}$}_k(x_2)=\\left(\\begin{array}{c}\\cos(kx_2) \\\\\n\\sin(kx_2) \\\\ 0\\end{array}\\right),\\ \\ \\ k\\in{\\rm Z \\!\\! Z}.\n\\label{eq:statsol}\n\\end{equation}\nOne easily computes the energy\n\\begin{equation}\nV[\\mbox{$\\vec{n}$}_k]=\\pi k^2.\n\\label{eq:staten}\n\\end{equation}\nIn particular $\\mbox{$\\vec{n}$}_0$ is a classical vacuum. Due to \\mbox{SO$(3)$}~symmetry\nthe vacuum valley is isomorphic to $S^2$.\n\nNow we turn to the sphalerons. For this we have to determine the Hessian\nof the energy functional, \\refeq{pot}, at the field\n$\\mbox{$\\vec{n}$}_k$, \\refeq{statsol}. This is easy in spherical coordinates where\n\\refeq{statsol} reads\n$\\vartheta_k(x_2)=\\frac{\\pi}{2},\\;\\varphi_k(x_2)=kx_2$ and the metric is\ngiven by $(g_{ij})={\\rm diag}(1,\\sin^2\\vartheta)$.\nSubstituting these formulas in the static version of \\refeq{pertact}\nwe obtain\n\\begin{equation}\n\\mbox{$\\cal H$}[\\mbox{$\\vec{n}$}_k]=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\left(\\begin{array}{cc}-\\partial_2^2-k^2 & 0 \\\\ 0 &\n-\\partial_2^2\\end{array}\\right).\n\\end{equation}\nOne immediately sees that\n$k=\\pm1$ is the only solution with exactly one unstable mode\n($\\delta\\vartheta(x_2)=1$, $\\delta\\varphi(x_2)=0$).\nSo\\footnote{At this point we do not prove that these saddle points are\ntrue sphalerons in the mini-max sense. This proof\ncan be constructed easily with the results of the next section.}\n the sphaleron\nsolutions are given by $\\mbox{$\\vec{n}$}_1$\nand \\mbox{SO$(3)$}~transformations thereof. In particular $k\\rightarrow-k$ can\nbe undone by a rotation over $\\pi$ around any vector\n$\\mbox{$\\vec{n}$}_k(x_2)$ ($x_2$ fixed). Thus, sphaleron moduli-space\nis isomorphic to $\\mbox{SO$(3)$}$.\n\nOne can verify that\n\\mbox{SO$(3)$}~rotations are responsible for the 3 zero-modes of the\nHessian. Note that the sphaleron is invariant\nunder an $x_2$-translation in combination with a specific\n${\\rm SO}(2)\\subset\\mbox{SO$(3)$}$~rotation (in the case of \\refeq{statsol} around\nthe 3-axis). Therefore spatial translations do not give new sphaleron\nsolutions. Also one can check that the discrete symmetries\n$x_2\\rightarrow-x_2$ and $\\mbox{$\\vec{n}$}\\rightarrow-\\mbox{$\\vec{n}$}$ leave the sphaleron\nmoduli-space invariant.\n\n\\section{Instanton solutions}\n\\label{sec:inst}\n\\subsection{Construction of the solutions}\n\nIn stereographic projection~\\cite{BePo,RiRo} the instanton equation\n(\\ref{eq:insteq}) is particularly simple:\n\\begin{equation}\n\\mbox{$\\partial_{\\bar{z}}$} u=0.\n\\label{eq:analyt}\n\\end{equation}\nHere we have introduced complex coordinates on space-time, $z=x_1+ix_2$,\n$\\bar{z}=x_1-ix_2$, with derivatives $\\mbox{$\\partial_z$}=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}(\\partial_1-i\\partial_2)$,\n$\\mbox{$\\partial_{\\bar{z}}$}=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}(\\partial_1+i\\partial_2)$. The construction of instantons\nreduces to finding\nall analytic functions on \\mbox{${\\rm T}^1\\times\\re$}\\ with topological charge $Q=1$.\n\nSubstitution of \\refeq{npar} in \\refeq{topch} gives, for any $u$\nsatisfying \\refeq{analyt},\n\\begin{equation}\nQ[u]=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{\\pi}$}\\intxD{\\mbox{\\scriptsize${\\rm T}^1\\times\\re$}\\,}\\frac{|\\mbox{$\\partial_z$} u|^2}{(1+|u|^2)^2}=\n-\\raisebox{.15ex}{\\scriptsize$\\frac{1}{\\pi}$}\\intxD{\\mbox{\\scriptsize${\\rm T}^1\\times\\re$}\\,}\\mbox{$\\partial_{\\bar{z}}$}\\left(\\frac{1}{1+|u|^2}\\frac{\\mbox{$\\partial_z$} u}{u}\\right).\n\\label{eq:utopch}\n\\end{equation}\nHandling carefully the poles of $\\frac{1}{1+|u|^2}\\frac{\\mbox{$\\partial_z$} u}{u}$,\ni.e.\\ the zeros of $u$, one finds\n\\begin{equation}\nQ[u]=-\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2\\pi}$}\\left(\\int_{x_1=\\infty} - \\int_{x_1=-\\infty}\\right)dx_2\\,\\,\n\\frac{1}{1+|u|^2}\\frac{\\mbox{$\\partial_z$} u}{u}+\\sum_i n_i,\n\\label{eq:nicetopch}\n\\end{equation}\nwhere the $x_2$~integrations run from $0$ to $2\\pi$ and $i$ runs over\nthe zeros of $u$, of degree $n_i\\in\\mbox{I$\\!$N}$. In order to simplify this\nformula, we observe that\nboth the kinetic and the potential term in the Lagrangian\nare semi-positive definite. Hence any finite-action configuration, in\nparticular an instanton, must\napproach a point in the vacuum valley for $x_1\\rightarrow\\pm\\infty$:\n$\\lim_{x_1\\rightarrow\\pm\\infty}u(z)=u_{\\pm}$. In the derivation below\nwe will assume that\n$0<|u_{\\pm}|<\\infty$, as can always be achieved by an\n\\mbox{SO$(3)$}~rotation\\footnote{This is equivalent to changing coordinates\non $S^2$ by shifting the pole used in stereographic\nprojection.}. Under this\nassumption \\refeq{nicetopch} reduces to $Q[u]=\\sum_i n_i$.\n\nNow we are fully prepared to determine all instanton solutions. The\nstrategy is first to find a class of solutions and then to prove no\nsolutions exist outside this class. In order to construct a\nsolution observe that\n\\begin{enumerate}\n\\item Since $e^{z+2\\pi i}=e^z$, any $u=h(e^z)$ ($h$ single-valued and\nanalytic) is a function on \\mbox{${\\rm T}^1\\times\\re$}\\ satisfying the instanton equation.\n\\item Under the above assumption any instanton can have only one zero\n$z_1$ of degree $n_1=1$.\n\\end{enumerate}\nA class of functions satisfying all requirements is\n\\begin{equation}\nu^{\\rm inst}_{a,b,c,d}(z)=-\\frac{c+d e^z}{a+b e^z},\n\\label{eq:inst}\n\\end{equation}\nwith certain restrictions on the complex coefficients $a,b,c,d$; note\nthat $u_{+}=-d\/b$, $u_{-}=-c\/a$ and $z_1=\\ln(-c\/d)$ (which is unique on\n\\mbox{${\\rm T}^1\\times\\re$}). So in order to satisfy the assumption $0<|u_{\\pm}|<\\infty$, we\nmust require $a,b,c,d\\not=0$. Also we must demand $a\/b\\not=c\/d$ as\notherwise the zero of $u^{\\rm inst}_{a,b,c,d}$ cancels against its\npole and we have a trivial solution with $Q=0$.\n\nTo prove that all instanton solutions are of the form (\\ref{eq:inst}) is\neasy: suppose $\\tilde{u}$ is an instanton. We can still assume\n$0<|\\tilde{u}_{\\pm}|<\\infty$, so $\\tilde{u}$ can have only\none zero of degree 1, say at $z=\\tilde{z}_1$. Now define a function\n$f$ by\n\\begin{equation}\n\\tilde{u}(z)=u^{\\rm inst}_{a,b,c,d}(z)f(z).\n\\nonumber\n\\end{equation}\nBy choosing $-c\/a=\\tilde{u}_{-}$, $-d\/b=\\tilde{u}_{+}$,\n$c\/d=-e^{\\tilde{z}_1}$ and imposing the\ninstanton requirements $\\mbox{$\\partial_{\\bar{z}}$} \\tilde{u}=0$, $Q[\\tilde{u}]=1$ and\n$\\tilde{u}(z+2\\pi i)=\\tilde{u}(z)$, that are already met by $u$, we\nsee that $f$ has to satisfy\n\\begin{equation}\n\\left\\{ \\begin{array}{l}\\mbox{$\\partial_{\\bar{z}}$} f=0 \\\\ \\lim_{{\\rm Re}(z)\\rightarrow\\pm\n\\infty}f(z)=1 \\\\ f(z+2\\pi i)=f(z) \\\\ f(z)\\not=0.\\end{array}\\right.\n\\label{eq:fisone}\n\\end{equation}\nThus, $1\/f$ is analytic and bounded on \\mbox{$l\\!\\!\\!$C}. By Liouville's theorem this\nimplies, using the second condition in\n\\refeq{fisone}, that $f(z)=1$. This completes the proof.\n\nFinally we drop the assumption $0<|u_{\\pm}|<\\infty$.\nTaking into account the boundary terms in\n\\refeq{nicetopch}, one finds that the only restriction on $(a,b,c,d)$ is\nthat $u^{\\rm inst}_{a,b,c,d}(z)$ is not constant, corresponding to\n\\begin{equation}\nad-bc\\not=0.\n\\label{eq:coefreq}\n\\end{equation}\nSo the class of instantons, \\refeq{inst}, is precisely the set of\nconformal mappings of $e^z$.\n\nWe end this part by noting that by the same line of argument each\nmulti-instanton with topological charge $Q$ can be written as\n$\\prod_{n=1}^Q u^{\\rm inst}_{a_n,b_n,c_n,d_n}$. For\n(multi-)anti-instantons one substitutes\n$z\\rightarrow\\bar{z}$.\n\n\\subsection{Physical interpretation of the moduli-space}\n\nIt is clear that the moduli-space has six real dimensions: $(a,b,c,d)\\in\n\\mbox{$l\\!\\!\\!$C}^4$ with the projective character $u^{\\rm\ninst}_{ga,gb,gc,gd}=u^{\\rm inst}_{a,b,c,d}$ for any $g\\in\\mbox{$l\\!\\!\\!$C}\\backslash\n\\{0\\}$ (while the set of coefficients not satisfying \\refeq{coefreq} has\nzero measure). A physical parametrization of this moduli-space is\n$(-c\/a,-d\/b,\\ln(-c\/d))$, corresponding to the starting point $u_{-}$ in\nthe\nvacuum valley $S^2$, the end point $u_{+}$ and a space-time translation\nparameter, which one can show to be in $1-1$~relation with the\ninstanton position (cf. \\refeq{instsol} below). The\ndisadvantage of this parametrization is that it leaves unclear what kind\nof manifold the moduli-space is; the parametrization is singular for\n$u_{-}=u_{+}$, since the requirement of \\refeq{coefreq} is not met in\nthis case. Note that this means that even for $T\\rightarrow\\infty$\nperiodic boundary conditions do not admit instanton solutions. We\nwill see below that $u_{-}\\rightarrow u_{+}$ corresponds to\n$\\rho\\rightarrow0$, where $\\rho$ is the instanton scale parameter.\n\nFor a better description of moduli-space it is necessary first to\nconsider the transformation of instantons under space-time translations\nand \\mbox{SO$(3)$}~rotations. We will see that most\ninstantons are not invariant under these symmetries which therefore give\nrise to five of the six dimensions of moduli-space. The interesting\nsixth parameter, not related to a symmetry of the action, will play the\nrole of a scale parameter, related to the geodesic distance between the\npoints $u_{\\pm}$ in the vacuum valley $S^2$.\n\n{}From \\refeq{inst} it is trivial to see that under a translation\n$z\\rightarrow z+z_0$, $(a,b,c,d)\\rightarrow(a,b e^{z_0},c,d e^{z_0})$.\nNote that \\refeq{coefreq} is therefore invariant under this shift, as\nit should be. Any continuous symmetry of the action must be present\nin the instanton moduli-space.\n\nThe effect of an \\mbox{SO$(3)$}~rotation is more difficult to derive, because\nwhile it acts linearly on \\mbox{$\\vec{n}$}, \\mbox{$\\vec{n}$}\\ and $u$ are related non-linearly\nby \\refeq{npar}. Nevertheless, one can show\nthat $(a,b,c,d)$ again transform linearly. After some effort one sees\nthat the rotation\n\\begin{equation}\n\\mbox{$\\vec{n}(x)$}\\rightarrow R\\mbox{$\\vec{n}(x)$},\\ \\ \\ R=e^{\\alpha_a L^a},\\ \\ \\ {L^a}_{ij}=\n-\\varepsilon_{aij},\\ \\ \\ (\\alpha_a\\in\\mbox{I$\\!$R})\n\\end{equation}\ninduces\n\\begin{equation}\n\\mbox{\\scriptsize $\\left(\\!\\!\\begin{array}{c}a\\\\ b\\\\ c\\\\ d\\end{array}\n\\!\\!\\right)$}\\rightarrow\n\\tilde{R}\\mbox{\\scriptsize $\\left(\\!\\!\\begin{array}{c}a\\\\ b\\\\ c\\\\ d\n\\end{array}\\!\\!\\right)$},\\ \\ \\\n\\tilde{R}=e^{\\alpha_a \\tilde{L}^a},\\ \\ \\ \\tilde{L}^a=-\\frac{i}{2}\\sigma^a\n\\otimes{\\bf 1},\\ \\ \\ {\\rm (i.e.\\ } \\tilde{L}^1= -\\frac{i}{2}\n\\mbox{\\scriptsize $\\left(\\!\\!\n\\begin{array}{cccc}0&0&1&0\\\\ 0&0&0&1\\\\ 1&0&0&0\\\\\n0&1&0&0\\end{array}\\!\\!\\right)$} {\\rm\\ etc.)}\n\\label{eq:rot}\n\\end{equation}\nwhere $\\sigma^a$ are the Pauli-matrices. Since $\\pm(a,b,c,d)$ are\nidentified, this is a representation of \\mbox{SO$(3)$}. Notice that only\n$(a,c)$ and $(b,d)$ mix. Hence both $|a|^2+|c|^2$ and\n$|b|^2+|d|^2$ are rotationally invariant, as is \\refeq{coefreq}.\n\nUsing the projective character of moduli-space, and an \\mbox{SO$(3)$}~rotation\n$\\tilde{R}$, it is always possible to bring\n$u^{\\rm inst}_{a,b,c,d}$ ($ad\\not= bc$) to the form\n$u^{\\rm inst}_{-1,0,\\tilde{c},\\tilde{d}}$ with $\\tilde{c}\\in\\mbox{I$\\!$R}$,\n$\\tilde{c}\\geq0$, $\\tilde{d}\\in\\mbox{$l\\!\\!\\!$C}\\backslash\\{0\\}$. If $\\tilde{c}>0$,\nthis fixes $\\tilde{R}$ completely. If\n$\\tilde{c}=0$, then $\\tilde{R}$ is only unique up to a factor\n$e^{\\alpha_3\\tilde{L}^3}$. This can be fixed by requiring\n$\\tilde{d}=|\\tilde{d}|$. Therefore, we can parametrize\n$u^{\\rm inst}_{a,b,c,d}$ uniquely by $\\tilde{R}$, $|\\tilde{d}|$ and\n$\\tilde{c}e^{i\\phi}$ ($\\tilde{c}\\geq0$). Here $\\phi={\\rm\nArg}(\\tilde{d})$ ($\\phi\\in[0,2\\pi)$) if $\\tilde{c}>0$ and $\\phi$ is\nundetermined if $\\tilde{c}=0$. We conclude that $(\\tilde{c},\\phi)$ can\nbe viewed as polar coordinates on $\\mbox{I$\\!$R}^2$. Furthermore, $|\\tilde{d}|\n>0$ due to \\refeq{coefreq}, so $\\ln|\\tilde{d}|\\in\\mbox{I$\\!$R}$. Thus the instanton\nmoduli space is isomorphic to $\\mbox{SO$(3)$}\\times\\mbox{I$\\!$R}^2\\times\\mbox{I$\\!$R}$. The discrete\nsymmetry transformations $\\mbox{$\\vec{n}$}\\rightarrow-\\mbox{$\\vec{n}$}$\n(in stereographic coordinates $u\\rightarrow-1\/\\bar{u}$),\n$x_2\\rightarrow-x_2$ or $x_1\\rightarrow-x_1$ make an instanton solution\nanti-analytic, and therefore are transformations from instanton\nmoduli-space into anti-instanton moduli-space. This should be compared\nto the sphaleron moduli space which is invariant under such discrete\ntransformations.\n\nNote that $\\tilde{d} e^z=e^{z+\\ln|\\tilde{d}|+i\\phi}$. So after an\n\\mbox{SO$(3)$}~rotation and a translation in space-time \\mbox{${\\rm T}^1\\times\\re$}, any instanton\nsolution can be brought to the form\n\\begin{equation}\nu^{\\rm inst}_c(z)\\equiv c+e^{z+\\ln(\\sqrt{1+c^2})+i\\pi}.\\ \\ \\ (c\\geq0)\n\\label{eq:instsol}\n\\end{equation}\nThe factor $e^{\\ln(\\sqrt{1+c^2})+i\\pi}=-\\sqrt{1+c^2}$ centers the\ninstanton around $z=0$ (see below). From the above equation it\nfollows that $\\lim_{x_1\\rightarrow\\infty}|u^{\\rm\ninst}_c(z)|=\\infty$ and $\\lim_{x_1\\rightarrow-\\infty}u^{\\rm\ninst}_c(z)=c$, hence\n(using \\refeq{npar}) this instanton `tunnels' from\n$(\\sin\\vartheta_{-},0, \\cos\\vartheta_{-})$ to $(0,0,1)$, with\n$\\cot\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\vartheta_{-}=c$. Note that\n$\\theta_{-}$ is the geodesic distance between these vacua.\n\nLet us determine the instanton size $\\rho$ as function of\n$c=2{\\rm arccot}\\vartheta_{-}$. Substituting \\refeq{instsol} into the\nLagrangian density, which for an instanton in stereographic coordinates\nis $4\\pi$ times the integrand in \\refeq{utopch} (see\neqs.(\\ref{eq:niceact},\\ref{eq:insteq})), one obtains\n\\begin{equation}\n\\mbox{$\\cal L$}[u^{\\rm inst}_c](x_1,x_2)=4\\frac{|\\mbox{$\\partial_z$} u^{\\rm inst}_c|^2}{(1+|u^{\\rm\ninst}_c|^2)^2}=\\frac{1}{(\\sqrt{1+c^2}\\cosh x_1-c\\cos x_2)^2}.\n\\label{eq:instlagr}\n\\end{equation}\nThis function is plotted in fig.\\ref{fig:lagr} for different values of\n$c$. From the formula it is clear that $u^{\\rm inst}_c$ is\ncentered at $z=0$. Now consider the potential along the instanton path,\n\\begin{equation}\nV[u^{\\rm inst}_c](x_1)=\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\int_0^{2\\pi}\\!\\!\ndx_2\\,\\,\\mbox{$\\cal L$}[u^{\\rm inst}_c](x_1,x_2).\n\\end{equation}\nThe factor $\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}$ comes from the fact that the kinetic energy,\n$\\raisebox{.15ex}{\\scriptsize$\\frac{1}{2}$}\\int_{{\\rm T}^1}\\!dx_2\\, g_{ij}(v)\\partial_1v^i\\partial_1v^j$,\nis equal to the potential energy, \\refeq{pot} (this `self-duality'\nfollows from \\refeq{insteq}). We see that the potential is maximal at\n$x_1=0$ where it satisfies\n\\begin{equation}\nV^{\\max}_c\\equiv V[u^{\\rm inst}_c](0)=\\pi\\sqrt{1+c^2}.\n\\label{eq:insten}\n\\end{equation}\nSince all instantons have equal action, it is natural to define the\ninstanton size\n\\begin{equation}\n\\rho(c)\\equiv\\frac{\\pi}{V^{\\max}_c}=\\frac{1}{\\sqrt{1+c^2}}.\n\\label{eq:instrho}\n\\end{equation}\nNote however that for small $c$ there are two different scales; the\nshape of $\\mbox{$\\cal L$}[u^{\\rm inst}_c](x_1,x_2)$ is anisotropic (see\nfig.\\ref{fig:lagr}). Only for $c\\gg1$ and $x_1^2+x_2^2\\ll1$ the\nboundary effects disappear and $\\mbox{$\\cal L$}[u^{\\rm inst}_c](x_1,x_2)\\approx\n\\frac{4c^2}{(1+c^2(x_1^2+x_2^2))^2}$ becomes rotationally invariant.\n\nThe relationship between instantons and sphalerons is now also clear.\n{}From eqs.(\\ref{eq:insten},\\ref{eq:staten}) we see that only\n$u^{\\rm inst}_{c=0}$ can go through a sphaleron at the time of maximal\n$V[u^{\\rm inst}_c]$ (i.e.\\ $x_1=0$). Indeed this does happen, since\n\\refeq{instsol} gives $u^{\\rm inst}_{c=0}(x_1=0,x_2)=-e^{ix_2}$,\nwhich up to a rotation is just the sphaleron solution $\\mbox{$\\vec{n}$}_1$,\n\\refeq{statsol},\nin stereographic coordinates. As the instanton is a self-dual solution\nof the equations of motion, it has to follow streamlines of the energy\nfunctional. This explains why\n$\\partial_1 u^{\\rm inst}_{c=0}|_{x_1=0}=-\\exp(ix_2)$\ncorresponds to the unstable mode of the sphaleron solution.\n\nFinally note that\n$u^{\\rm inst}_{c=0}$, unlike $u^{\\rm inst}_{c>0}$, is a point in\ninstanton moduli-space that is symmetric under a joint\n${\\rm SO}(2)$~rotation $e^{\\alpha_{3}\\tilde{L}^3}$ and a spatial\ntranslation $x_2\\rightarrow x_2-\\alpha_3$. The sphaleron has of\ncourse the same symmetry, as mentioned in section~\\ref{sec:stat}.\nAlso a natural correspondence between sphaleron moduli-space \\mbox{SO$(3)$}\\\nand the subspace of widest instantons emerges. From the paragraph\nabove \\refeq{instsol} it follows that the latter is isomorphic to\n$\\mbox{SO$(3)$}\\times\\mbox{I$\\!$R}$, $\\mbox{I$\\!$R}$ corresponding to time translations.\n\\begin{figure}[t]\n\\epsfxsize=\\textwidth\n\\epsffile{fig.ps}\n\\caption{The Lagrangian densities of three instantons,\n\\protect\\refeq{instsol},\nwith from left to right $c=0,\\:0.25,\\:0.5$.}\n\\label{fig:lagr}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nWe have proven that the \\mbox{O$(3)\\,\\sigma$-model}\\ on a space-time \\mbox{${\\rm T}^1\\times\\re$}\\ admits instantons.\nThe moduli-space is 6-dimensional ($\\mbox{SO$(3)$}\\times\\mbox{I$\\!$R}^2\\times\\mbox{I$\\!$R}$):\n3 parameters for \\mbox{SO$(3)$}, 2 for scaling and spatial translations, 1 for\ntime translations. It is possible to `tunnel' between any different\npoints $\\mbox{$\\vec{n}$}_{\\pm}$ in the vacuum valley ($S^2$),\nbut this gives no independent parameters. Three parameters describing\n$\\mbox{$\\vec{n}$}_{\\pm}$ can be removed by an \\mbox{SO$(3)$}~rotation, while\nthe fourth (the geodesic distance between the points) depends\nuniquely on the scale parameter $\\rho$. Instantons with maximum scale, as\nset by the extent of spatial ${\\rm T}^1$, satisfy $\\mbox{$\\vec{n}$}_{+}=-\\mbox{$\\vec{n}$}_{-}$. These,\nand only these, instantons go through sphalerons. None of the exotic\npossibilities sketched in the introduction take place in this model.\nOn the other hand,\n$\\rho\\rightarrow0$ corresponds to $\\mbox{$\\vec{n}$}_{+}\\rightarrow\\mbox{$\\vec{n}$}_{-}$. Exact\nequality cannot be reached. We think this peculiar size-dependence\nis important for improving on instanton gas calculations as\nin~\\cite{BeLu}. By doing a proper convolution with the vacuum wave\nfunction at $t\\rightarrow\\pm\\infty$ it might be possible to remove the\nwell-known UV divergence for $\\rho\\rightarrow0$, which was also\nencountered in recent numerical studies~\\cite{MiSp}.\n\nOur results do not admit a straightforward generalization to\n\\mbox{SU$(2)$}~gauge theory on a space-time \\mbox{${\\rm T}^3\\times\\re$}. In that model\nthe vacuum valley is isomorphic to \\mbox{${\\rm T}^3$}~\\cite{BaKo},\nwhich can be parametrized\nby three Polyakov lines $P_i$. Instantons again must have endpoints,\n$P_i^{\\pm}$, in the vacuum valley. For the special case\n$P_i^{+}=-P_i^{-}$ it has already been known for some time\nthat an 8-dimensional moduli-space exists. This anti-periodic\nsituation corresponds to time-like twist~\\cite{Ho}, which has been\nanalyzed~\\cite{BrMaTo} on any space-time $\\mbox{$[-\\half T, \\half T]$}\\times\\mbox{${\\rm T}^3$}$. For\n$T\\rightarrow\\infty$ it is very likely~\\cite{PeGoSnBa} that the\nmoduli-space includes a scale parameter. This is not the case\nfor the \\mbox{O$(3)\\,\\sigma$-model}\\ on a space-time \\mbox{${\\rm T}^1\\times\\re$}. We have just proven that\nanti-periodic boundary conditions, $\\mbox{$\\vec{n}$}_{+}=-\\mbox{$\\vec{n}$}_{-}$, fix the\ninstanton size\\footnote{We also note that no instantons exist\nwith anti-periodic boundary conditions in finite time $T$,\n\\mbox{$\\mbox{$\\vec{n}$}(x_1+T,x_2)=-\\mbox{$\\vec{n}$}(x_1,x_2)$}\n(while the winding number $Q$ is still a well-defined integer object).\nThe reason is simple: in stereographic coordinates anti-pbc read\n$u(z+T)=-1\/\\bar{u}(z)$, which is incompatible with the instanton\nequation $\\mbox{$\\partial_{\\bar{z}}$} u=0$.}. It would be nice to understand the cause of\nsuch different behavior\nbetween two models that are so similar in other respects. This might\nbe a starting point for finding new instanton parameters in \\mbox{SU$(2)$}~gauge\ntheory on \\mbox{${\\rm T}^3\\times\\re$}, by relaxing the condition $P_i^{+}=-P_i^{-}$.\n\n\\section{Acknowledgments}\n\nI am grateful to Pierre van Baal for fruitful discussions and for\ncarefully reading this manuscript. Also I thank Pieter Rijken for\nexpanding my knowledge of analytic functions, and Margarita\nGarc\\'{\\i}a P\\'{e}rez for discussions on various sphaleron-related\ntopics.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:1}\nThe combined MAXIMA-1 \\cite{MAXIMA-1} ,  BOOMERANG \\cite{BOOMERANG} , \nDASI \\cite{DASI} ,  COBE\/DMR Cosmic Microwave Background (CMB)\nobservations \\cite{COBE} ,  the recent WMAP data \\cite{SPERGEL} and\nSDSS\n \\cite{SDSS} imply that the\nUniverse is flat \\cite{flat01} and that most of the matter in\nthe Universe is dark, i.e. exotic.These results have been confirmed and improved\nby the recent WMAP data \\cite{WMAP06}. The deduced cosmological expansion is\nconsistent with  the luminosity distance as a function of redshift of distant supernovae \n\\cite{supernova1,supernova2,supernova3}.\nAccording to the scenario favored by the observations \nthere are various contributions to the energy content of our Universe.\nThe most accessible energy component is baryonic matter, which accounts for\n$\\sim 5\\%$ of the total energy density. A component that has\nnot been directly observed is cold dark matter (CDM)): a pressureless fluid that is\nresponsible for the growth of cosmological perturbations through\ngravitational instability. Its contribution to the total energy density is\nestimated at \n$\\sim 25\\%$. The dark matter is expected to become more abundant \nin extensive halos, that\nstretch up to 100--200 kpc from the center of galaxies.\nThe component with the biggest contribution to the energy density has\nan equation of state similar to that of a cosmological constant and is characterized as dark energy. \nThe ratio $w=p\/\\rho$ is negative and close to $-1$. \nThis component is responsible for $\\sim 70\\%$ of the total energy density \nand induces\nthe observed acceleration of the Universe \\cite{supernova1}$^-$\\cite{supernova3} . \nThe total energy density of our Universe \nis believed to take the critical value consistent with spatial flatness.\nAdditional indirect information about the existence of dark matter\ncomes from  the rotational curves \\cite{Jung} . The rotational velocity of an object increases so long\nis surrounded by matter. Once outside matter the velocity of rotation drops as the square root  of the \ndistance. Such observations are not possible in our own galaxy. The observations of other galaxies, \nsimilar to our own, indicate that the rotational velocities of objects outside the luminous matter\ndo not drop. So there must be a halo of dark matter out there.\n\nSince the non exotic component cannot exceed $40\\%$ of the CDM\n~\\cite {Benne} , there is room for exotic WIMP's (Weakly\nInteracting Massive Particles).\\\\\n  In fact the DAMA experiment ~\\cite {BERNA2} has claimed the observation of one signal in direct\ndetection of a WIMP, which with better statistics has subsequently\nbeen interpreted as a modulation signal \\cite{BERNA1} . These  data,\nhowever, if they are due to the coherent process, are not\nconsistent with other recent experiments, see e.g. EDELWEISS and\nCDMS \\cite{EDELWEISS} . It could still be interpreted as due to the\nspin cross section, but with a new interpretation of the extracted\nnucleon cross section. \n\nSince the WIMP  is expected to be very massive, $m_{\\chi} \\geq 30 GeV$, and\nextremely non relativistic with average kinetic energy $T \\leq 100 KeV$,\nit can be directly detected mainly via the recoiling\nof a nucleus  in the WIMP-nucleus elastic scattering.\n\nThe above developments are in line with particle physics considerations.\n\\begin{enumerate} \n\\item Dark matter in supersymmetric theories\\\\\nThe lightest supersymmetric particle (LSP) or neutralino\n is the most natural WIMP candidate. In the most favored scenarios the\nLSP can be simply described as a Majorana fermion, a linear \ncombination of the neutral components of the gauginos and Higgsinos\n\\cite{Jung}$^-$\\cite{Hab-Ka} . \n\n In order to compute the event rate one needs\n an effective Lagrangian at the elementary particle \n(quark) level obtained in the framework of supersymmetry \n~\\cite{Jung,ref2,Hab-Ka} .\n One starts with   \nrepresentative input in the restricted SUSY parameter space as described in\nthe literature, e.g. Ellis {\\it et al} \\cite{EOSS04} , Bottino {\\it et al} , \nKane {\\it et al} , Castano {\\it et al} and Arnowitt {\\it et al} \\cite{ref2} as well as elsewhere\n\\cite{GOODWIT}$^-$\\cite{UK01} .\nWe will not, however, elaborate on how one gets the needed parameters from\nsupersymmetry. \n\nEven though the SUSY WIMPs have been well studied, for tor the reader's convenience we will give a description  in sec. \\ref{sec:diagrams} of the basic \nSUSY ingredients needed to calculate LSP-nucleus scattering\n cross section.\n\n\\item  Kaluza-Klein (K-K) WIMPs. \\\\\nThese arise in  extensions of the standard model with compact extra dimensions. In\nsuch models  a tower of massive particles appear as Kaluza-Klein\nexcitations.\nIn this scheme the ordinary particles are associated with the zero\nmodes and are assigned K-K parity $+1$.\nIn models with Universal Extra Dimensions one can have\ncosmologically stable particles in the excited modes because of a\ndiscreet symmetry yielding  K-K parity $-1$ (see previous work \\cite{ST02a,ST02b,CFM02} as well as\nthe recent review by Servant\n\\cite{SERVANT}).\n \\\\The kinematics involved is similar to\nthat of the neutralino, leading to cross sections which are\nproportional $\\mu^2_r$, $\\mu_r$ being the WIMP-nucleus reduced\nmass. Furthermore the\n nuclear physics input\n  is independent of the WIMP mass, since for heavy WIMP $mu_r\\simeq Am_p$.\n   There are appear two differences compared to the\n  neutralino, though,  both related to its larger mass.\n  \\\\i) First the density\n  (number of particles per unit volume) of a WIMP\n falls inversely proportional to its mass. Thus,\n  if the WIMP's considered are much heavier than the nuclear\n targets, the corresponding event rate takes the form:\n  \\beq\n R(m_{WIMP})=R(A)\\frac{A \\mbox{ GeV\n}}{m_{WIMP}}\n \\label{eq:rate}\n \\eeq\n where $R(A)$ are the rates extracted from experiment up to WIMP\n masses of the order of the mass of the target.\n\\\\ii) Second the average WIMP energy is now  higher. In fact one\nfinds that $\\langle T_{WIMP}\\rangle =\\frac{3}{4}m_{WIMP}\n\\upsilon^2_0\\simeq 40 \\left ({m_{WIMP}}\/({100 \\mbox{\nGeV)}}\\right )$keV ($\\upsilon_0\\simeq 2.2\\times 10^5$km\/s).\n  Thus for a K-K WIMP with mass $1$ TeV, the average\nWIMP energy is $0.4$ MeV. Hence, due to the high velocity tail of\nthe velocity distribution,  one expects {\\bf an energy transfer to\nthe nucleus  in the MeV region. Thus many nuclear\n targets can now be excited by the WIMP-nucleus interaction and the de-excitation photons\n can be detected.}\n\n \n \\end{enumerate}\nIn addition to the particle model one needs the following  ingredients:\n\\begin{itemize}\n\\item A procedure in going from the quark to the nucleon level, i.e. a quark \nmodel for the nucleon. The results depend crucially on the content of the\nnucleon in quarks other than u and d. This is particularly true for the scalar\ncouplings as well as the isoscalar axial coupling ~\\cite{Dree}$^-$\\cite{Chen} . Such topics will be discussed in sec.\n\\ref{sec:nuc}.\n\\item computation of  the relevant nuclear matrix elements~\\cite{Ress}$^-$\\cite{SUHONEN03}\nusing as reliable as possible many body nuclear wave functions.\nBy putting as accurate nuclear physics input as possible, \none will be able to constrain the SUSY parameters as much as possible.\nThe situation is a bit simpler in the case of the scalar coupling, in which\ncase one only needs the nuclear form factor.\n\\item  Convolution with the LSP velocity Distribution.\nTo this end we will consider here  Maxwell-Boltzmann \\cite {Jung} (MB) velocity distributions, with an upper velocity cut off put in by hand. The characteristic velocity of the M-B distribution can be increased by\na factor $n$ ($\\upsilon_0\\rightarrow n \\upsilon_0,~n\\ge1 $)by considering the interaction of dark matter and dark energy \\cite{TETRVER06}.\n\nOther distributions are possible, such as\n  non symmetric ones,  like those \nof Drukier \\cite {Druk} and Green \\cite{GREEN02} , or non isothermal ones, e.g. those arising from late in-fall of   \ndark matter into our galaxy, like  Sikivie's caustic rings \\cite {SIKIVIE} . In any event in a proper treatment the velocity distribution ought to be consistent with the dark matter density as, e.g., in the context of the Eddington theory \\cite{OWVER} .\n\\end{itemize}\n\n\n\n Since the expected rates are extremely low or even undetectable\nwith present techniques, one would like to exploit the characteristic\nsignatures provided by the reaction. Such are: \n\\begin{enumerate}\n\\item The modulation\n effect, i.e the dependence of the event rate on the velocity of\nthe Earth \n\\item The directional event rate, which depends on the\n velocity of the sun around the galaxy as well as the the velocity\nof the Earth. \nhas recently begun to appear feasible by the planned experiments\n\\cite {UKDMC,DRIFT} .\n\\item Detection of signals other than nuclear recoils, such as\n\\begin{itemize}\n\\item Detection of $\\gamma$ rays following nuclear de-excitation, whenever possible \\cite{eji93,VQS04} .\nThis seems to become feasible for heavy WIMPs especially in connection with modified M-B distributions due to \nthe coupling of dark matter and dark energy ($\\langle T_{WIMP} \\rangle \\simeq n^2  40 \\left ({m_{WIMP}}\/({100 \\mbox{\nGeV})}\\right ),~n\\ge 1$keV)\n\\item Detection of ionization electrons produced directly in the LSP-nucleus collisions \\cite{VE05,MVE05} . \n\\item Observations of hard X-rays produced\\cite{EMV05} , when the inner shell electron holes produced as above are filled.\n\\end{itemize}\n\\end{enumerate}\n\n In all calculations we will, of course, include an appropriate \nnuclear form factor and take into account\nthe influence on the rates of the detector energy cut off.\nWe will present our results a function of the\nLSP mass, $m_{\\chi}$, in a way\nwhich can be easily understood by the experimentalists.\n\n\n\\section{The Feynman Diagrams Entering the Direct Detection of WIMPS.}\n\\label{sec:diagrams}\n\\subsection{The Feynman Diagrams involving the neutralino}\n\\label{sec:FEYLSP}\n The neutralino is perhaps the most viable WIMP candidate and has been extensively \nstudied (see, e.g., our recent review \\cite{JDV06}). Here we will give a very brief \nsummary of the most important aspects entering the direct neutralino searches.\n In currently favorable supergravity models the LSP is a linear\ncombination~\\cite{Jung} of the neutral four fermions \n${\\tilde B}, {\\tilde W}_3, {\\tilde H}_1$ and ${\\tilde H}_2$ \nwhich are the supersymmetric partners of the gauge bosons $B_\\mu$ and\n$W^3_\\mu$ and the Higgs scalars\n$H_1$ and $H_2$. Admixtures of s-neutrinos are expected to be negligible.\nThe relevant Feynman diagrams involve Z-exchange, s-quark exchange and Higgs \nexchange.\n\\subsubsection{The Z-exchange contribution.}\n\\label{sec:Z-exc}\nThe relevant Feynman diagram is shown in Fig. \\ref{LSPZH}. It does not lead to coherence, since $\\bar{\\Psi}\\gamma_{\\lambda}\\Psi=0$ for a Majorana fermion like the neutralino (the Majorana fermions do not\nhave electromagnetic properties). The coupling $\\bar{\\Psi}\\gamma_{\\lambda}\\gamma_5\\Psi$ yields negligible contribution for a non relativistic particle in the case of the spin independent cross section \\cite{JDV96}. It may be important in the case of the spin contribution, which arises\nfrom the axial current).\n\\begin{figure}\n\\psfig{file=zvec.ps,width=2.0in}\n\\psfig{file=higgs.ps,width=2.0in}\n\\caption{The LSP-quark interaction mediated by Z and Higgs exchange.}\n\\label{LSPZH}\n\\end{figure}\n\\subsubsection{The $s$-quark Mediated Interaction }\n\\label{sec:sq-exc}\nThe other interesting possibility arises from the other two components of\n$\\chi_1$, namely ${\\tilde B}$ and ${\\tilde W}_3$. Their corresponding\ncouplings to $s$-quarks (see Fig. \\ref{LSPSQVS} ) can be read from the appendix C4 of Ref.~\\cite{Hab-Ka} and our earlier review \\cite{JDV06}.\n\\begin{figure}\n\\begin{center}\n\\psfig{file=sqvec.ps,width=2.0in}\n\\psfig{file=sqsca.ps,width=2.0in}\n\\end{center}\n\\caption{The LSP-quark interaction mediated by s-quark exchange. Normally it yields V-A interaction \nwhich does not lead to coherence at the nuclear level. If, however, the isodoublet\n s-quark is admixed with isosinglet one to yield a scalar interaction at the quark level.}\n\\label{LSPSQVS}\n\\end{figure}\nNormally this contribution yields vector like contribution, i.e it does not lead to coherence. If,\nhowever, there exists mixing between the s-quarks with  isospin $1\/2$ ($\\tilde{q}_L$) and the isospin 0 ($\\tilde{q}_R$), the s-quark exchange \nmay lead to a scalar interaction at the quark level and hence to coherence over all nucleons\nat the nuclear level \\cite{JDV06}.\n\\subsubsection{The Intermediate Higgs Contribution}\n\\label{sec:Higgs-exc}\nThe most important contribution to coherent scattering can be achieved via the intermediate Higgs\nparticles which survive as physical particles. In supersymmetry there exist two such physical\nHiggs particles, one light $h$ with a mass $m_h\\leq$120 GeV and one heavy $H$ with mass $m_H$, which\n is much larger. The relevant interaction can arise out of the\nHiggs-Higgsino-gaugino interaction \\cite{JDV06} leading to a Feynman diagram shown in Fig.\n\\ref{LSPZH}. \n\nIn the case of the scalar interaction the resulting amplitude is proportional to the quark mass.\n \\subsection{The Feynman Diagrams involving the K-K WIMPs}\n\\label{sec:FEYKK}\n\n \\subsubsection{The Kaluza-Klein Boson as a dark matter candidate}\n \\label{KK}\n We will assume that the lightest exotic particle, which can serve as a dark matter candidate, is a gauge boson $B^{1}$\n having the same quantum numbers and couplings with the standard model gauge boson $B$, except that it has K-K parity\n $-1$. Thus its couplings must involve another negative K-K parity particle. In this work we will assume that such  a particle can be one of\n  the K-K quarks, partners of the ordinary quarks, but much heavier  \\cite{ST02a,ST02b,CFM02} .\n \\begin{itemize}\n\\item Intermediate K-K quarks.\\\\\nthis case the relevant\nFeynman diagrams are\n shown in fig. \\ref{fig:kkq}.\n \\begin{figure}\n\\psfig{file=kkqa.ps,width=2.2in}\n\\psfig{file=kkqb.ps,width=2.2in}\n \\caption{K-K quarks mediating the interaction of K-K gauge boson $B^{1}$ with quarks at tree\nlevel.}\n \\label{fig:kkq}\n  \\end{figure}\n\\\\The amplitude at the nucleon level can be written as:\n \\beq {\\cal M}_{coh}= \\Lambda(\\mbox{\\boldmath\n$\\epsilon^{*'}$}.\\mbox{\\boldmath $\\epsilon$})N\\left [\n\\frac{11+12\\tau_3}{54} \\frac{m_p\nm_W}{(m_{B^{(1)}})^2} f_1(\\Delta )+\\frac{1+\\tau_3}{3} \\frac{m_W}{\nm_{B^{(1)}}} f_2(\\Delta ) \\right ] N \\eeq\n$$\\Lambda=i 4 \\sqrt{2} G_F m_W \\tan^2{\\theta_W\n},f_1(\\Delta )=\\frac{1+\\Delta +\\Delta ^2 \/2}{\\Delta ^2(1+\\Delta\n\/2)^2},$$\n$$ f_2 (\\Delta )=\\frac{1+\\Delta }{\\Delta (1+\\Delta \/2)}~,\n~\\Delta =\\frac{m_{q^{(1)}}}{m_{B^{(1)}}}-1$$ We see that the\namplitude is very sensitive to the parameter $\\Delta $ (\"resonance\neffect\").\n\nIn going from the quark to the nucleon level the best procedure is\nto replace the quark energy by the constituent quark mass $\\simeq\n1\/3m_p$, as opposed to adopting  \\cite{ST02a,ST02b,CFM02} a procedure related to\nthe current mass encountered in the neutralino case \\cite{JDV06}.\n\n\n\n\nIn the case of the spin contribution we find at the nucleon level\nthat:\n\\barr {\\cal M}_{spin}&=& -i 4 \\sqrt{2} G_F m_W \\tan^2{\\theta_W\n}\\frac{1}{3} \\frac{m_p m_W}{(m_{B^{(1)}})^2} f_1(\\Delta )\ni(\\mbox{\\boldmath $\\epsilon^{*'}$}\\times \\mbox{\\boldmath\n$\\epsilon$}).\n\\nonumber\\\\\n&&\\left [ N\\mbox{\\boldmath $\\sigma$} (g_0+g_1 \\tau_3) N\n\\right ] \\earr\n$$g_0=\\frac{17}{18}\\Delta u+\\frac{5}{18} \\Delta d+\\frac{5}{18} \\Delta s~,\n~g_1=\\frac{17}{18}\\Delta u-\\frac{5}{18} \\Delta d$$\nfor the isoscalar and isovector quantities \\cite{JDV06}. The\nquantities $\\Delta_q$ are given by \\cite{JDV06}\n$$\\Delta u=0.78\\pm 0.02~,~\\Delta d=-0.48\\pm 0.02~,~\\Delta s=-0.15\\pm 0.02$$\nWe thus find $g_0=0.26~,~g_1=0.41\\Rightarrow\na_p=0.67~,~a_n=-0.15$.\n\\item Intermediate Higgs Scalars.\\\\\nThe corresponding\nFeynman diagram is shown in Fig. \\ref{fig:kkhz}\n   \\begin{figure}[!ht]\n\\psfig{file=kkh.eps,width=2.2in}\n\\psfig{file=kknu.eps,width=2.2in}\n \\caption{The Higgs H mediating interaction of K-K gauge boson $B^{1}$ with quarks at tree\nlevel (on the left). The Z-boson mediating the interaction of K-K\nneutrino $\\nu^{(1)}$ with quarks at tree level (on the right).}\n \\label{fig:kkhz}\n  \\end{figure}\nThe relevant amplitude is given by:\n  \\beq\n {\\cal M}_N(h)= -i~4 \\sqrt{2}G_F m^2_W \\tan^2{\\theta_W}~\\left [\\frac{1}{4}\\frac{m_p}{m^2_h}\n \\left (-\\mbox{\\boldmath\n$\\epsilon^{*'}$}.\\mbox{\\boldmath $\\epsilon$} \\right ) \\prec\nN|N\\succ  \\sum_q f_q\\right ]\n  \\eeq\n   In going from the quark to the nucleon level we follow a procedure analogous to that\n of the of the neutralino,\n  i.e.\n  $\\prec  N |m_q q \\bar{q}|N  \\succ  \\Rightarrow f_q m_p$\n\\end{itemize}\n  \\subsubsection{K-K neutrinos as dark matter candidates}\n  The other possibility is the dark matter candidate to be a heavy K-K neutrino.\n   We will distinguish the following cases:\n\\begin{itemize}\n\\item Process mediated by Z-exchange.\\\\\n\n The amplitude associated with the diagram of Fig. \\ref{fig:kkhz} becomes:\n\n\n\n  \\beq\n\n{\\cal M}_{\\nu^{(1)}}=-\\frac{1}{2 \\sqrt{2}}G_FJ^{\\lambda}(\\nu^{(1)}) J_{\\lambda}(NNZ)\n \\eeq\n with $J_{\\lambda}(NNZ)$ the standard nucleon neutral current and\n$$J_{\\lambda}(\\nu^{(1)})= \\bar{\\nu}^{(1)}\\gamma _{\\lambda\n}\\gamma_5\\nu^{(1)}~,~J_{\\lambda}(\\nu^{(1)})= \\bar{\\nu}^{(1)}\\gamma\n_{\\lambda }(1-\\gamma_5)\\nu^{(1)}$$\n for Majorana and Dirac neutrinos respectively.\n\\item Process mediated by right handed currents  via Z'-boson exchange.\\\\\n The process is similar to that exhibited by Fig. \\ref{fig:kkhz}, except that instead of Z we encounter Z', which is much\nheavier.  Assuming that the couplings of the $Z'$ are similar\nto those of $Z$, the above results apply except that now the\namplitudes are retarded by the multiplicative factor\n$\\kappa=m^2_{Z}\/m^2_{Z'}$\n\\item Process mediated by Higgs exchange.\\\\\n In this case in Fig \\ref{fig:kkhz} the Z is replaced by the Higgs particle.\n\n Proceeding as above we find that the amplitude at the nucleon level is:\n  \\beq\n {\\cal M}_{\\nu^{(1)}}(h)=\n-2 \\sqrt{2}G_F \\frac{m_p m_{\\nu^{(1)}}}{m_h^2}\n\\bar{\\nu}^{(1)}~\\nu^{(1)} \\prec N|N \\succ \\sum_q f_q\n \\eeq\n  In the evaluation of the parameters\n$f_q$ one encounters both theoretical and experimental errors.\n\\end{itemize}\n\\section{Other non SUSY Models}\n\\label{Zmodel}\n We should mention that there exist extensions of the standard model not motivated by symmetry. Such are:\n\\begin{itemize}\n\\item Models which introduce extra higgs particles and impose a discrete symmetry which leads to a \"parity\" \na la R-parity or K-K parity \\cite{MA06}.\n\\item Extensions of the standard model, which do not require a parity, but introduce high weak isospin\nmultiplets \\cite{CFA06} with Y=0. So the WIMP-nucleus interaction via Z-exchange at tree level is absent and the dominant contribution to the WIMP-nucleus scattering occurs at the one loop level.\n\\item Another interesting extension of the standard model is in the direction of tecnicolor \\cite{GKS06}. In this case the WIMP is\nthe neutral LTP (lightest neutral technibaryon).\n This is scalar particle, which couples to the quarks via derivative coupling\nthrough Z-exchange.\n\\end{itemize}\n\\section{Going from the Quark to the Nucleon Level} \n\\label{sec:nuc}\n In going from the quark to the nucleon level one has to be a bit more \ncareful in handling the quarks other than $u$ and $d$. This is especially true in the\ncase of the scalar interaction, since in this case the coupling \nof the WIMP to the quarks\nis proportional to their mass~\\cite{JDV06} .\nThus one has to consider in the nucleon not only\nsea quarks ($u {\\bar u}, d {\\bar d}$ and $s {\\bar s}$) but the heavier\nquarks as well due to QCD effects ~\\cite{Dree00} . \nThis way one obtains the scalar Higgs-nucleon\ncoupling by using  effective parameters $f_q$ defined as follows:\n\\beq\n\\Big<N| m_q \\bar{q}q|N \\Big> = f_q m_N\n\\label{fofq}\n\\eeq\nwhere $m_N$ is the nucleon mass. \n The parameters $f_q,~q=u,d,s$ can be obtained by chiral\nsymmetry breaking \nterms in relation to phase shift and dispersion analysis (for a recent review see\n\\cite{JDV06}). We like to emphasize here that since the current masses of the u and d \nquarks are small, the heavier quarks tend to dominate even though the probability of\nfinding them in the nucleus is quite small. In fact the s quark contribution may become dominant,\ne.g. allowed by the above analysis is the choice:\n$$f_d=0.046,f_u=0.025,f_s=0.400,f_c=0.050,f_b=0.055,f_t=0.095$$\n\n  The isoscalar and the isovector axial current in the case of K-K theories has already been discussed\n  above. In the case of the neutralino these\ncouplings at the nucleon level, $ f^0_A$, $f^1_A$, are obtained from the corresponding ones given by the SUSY\n models at the quark level, $ f^0_A(q)$, $f^1_A(q)$, via renormalization\ncoefficients $g^0_A$, $g_A^1$, i.e.\n$ f^0_A=g_A^0 f^0_A(q),f^1_A=g_A^1 f^1_A(q).$\nThe  renormalization coefficients are given terms of $\\Delta q$ defined above \\cite{JELLIS},\nvia the relations\n$$g_A^0=\\Delta u+\\Delta d+\\Delta s=0.77-0.49-0.15=0.13~,~g_A^1=\\Delta u-\\Delta d=1.26$$\nWe see that, barring very unusual circumstances at the quark level, the isoscalar contribution is\nnegligible. It is for this reason that one might prefer to work in the isospin basis.\n\\section{The allowed SUSY Parameter Space}\n\\label{sec:parameter}\n It is clear from the above discussion that the LSP-nucleon cross section depends, among other things,\non the parameters of supersymmetry.\n One starts with a set of parameters at the GUT scale and predicts the low energy\nobservables via the renormalization group equations (RGE). Conversely starting from the low energy phenomenology\none can constrain the input parameters at the GUT scale. \n The parameter space is the most crucial. In SUSY models derived from minimal SUGRA\nthe allowed parameter space is characterized at the GUT scale  by five \nparameters:\n\\begin{itemize}\n\\item two universal mass parameters, one for the scalars, $m_0$, and one for the\nfermions, $m_{1\/2}$.\n\\item $tan\\beta $, i.e the ratio of the Higgs expectation values, $\\left < H_2 \\right>\/\\left < H_1 \\right>$.\n\\item The trilinear coupling  $ A_0 $ (or  $ m^{pole}_t $)  and \n\\item The sign of $\\mu $ in the Higgs self-coupling  $\\mu H_1H_2$.\n\\end{itemize}\nThe experimental constraints \\cite{JDV06} restrict the values of the above\nparameters yielding the {\\bf allowed SUSY parameter space}.\n\\section{Event rates}\n\\label{sec:rates}\nThe differential non directional  rate can be written as\n\\begin{equation}\ndR_{undir} = \\frac{\\rho (0)}{m_{\\chi}} \\frac{m}{A m_N}\n d\\sigma (u,\\upsilon) | \\mbox{\\boldmath $\\upsilon$}|\n\\label{2.18}\n\\end{equation}\nwhere A is the nuclear mass number, \n$\\rho (0) \\approx 0.3 GeV\/cm^3$ is the WIMP density in our vicinity,\n m is the detector mass, \n $m_{\\chi}$ is the WIMP mass and $d\\sigma(u,\\upsilon )$ is the differential cross section.\n \n The directional differential rate, i.e. that obtained, if nuclei recoiling in the direction $\\hat{e}$ are \nobserved, is given by \\cite{JDVSPIN04,JDV06} :\n\\beq\ndR_{dir} = \\frac{\\rho (0)}{m_{\\chi}} \\frac{m}{A m_N}\n|\\upsilon| \\hat{\\upsilon}.\\hat{e} ~\\Theta(\\hat{\\upsilon}.\\hat{e})\n ~\\frac{1}{2 \\pi}~\nd\\sigma (u,\\upsilon\\\n\\nonumber \\delta(\\frac{\\sqrt{u}}{\\mu_r \\upsilon\n\\sqrt{2}}-\\hat{\\upsilon}.\\hat{e})\n \\label{2.20}\n\\eeq\nwhere $\\Theta(x)$ is the Heaviside function.\n\nThe differential cross section is given by:\n\\beq\nd\\sigma (u,\\upsilon)== \\frac{du}{2 (\\mu _r b\\upsilon )^2}\n [(\\bar{\\Sigma} _{S}F(u)^2\n                       +\\bar{\\Sigma} _{spin} F_{11}(u)]\n\\label{2.9}\n\\end{equation}\nwhere $ u$ the energy transfer $Q$ in dimensionless units given by\n\\begin{equation}\n u=\\frac{Q}{Q_0}~~,~~Q_{0}=[m_pAb]^{-2}=40A^{-4\/3}~MeV\n\\label{defineu}\n\\end{equation}\n with  $b$ is the nuclear (harmonic oscillator) size parameter. $F(u)$ is the\nnuclear form factor and $F_{11}(u)$ is the spin response function associated with\nthe isovector channel.\n\nThe scalar contribution is given by:\n\\begin{equation}\n\\bar{\\Sigma} _S  =  (\\frac{\\mu_r}{\\mu_r(p)})^2\n                           \\sigma^{S}_{p,\\chi^0} A^2\n \\left [\\frac{1+\\frac{f^1_S}{f^0_S}\\frac{2Z-A}{A}}{1+\\frac{f^1_S}{f^0_S}}\\right]^2\n\\approx  \\sigma^{S}_{N,\\chi^0} (\\frac{\\mu_r}{\\mu_r (p)})^2 A^2\n\\label{2.10}\n\\end{equation}\n(since the heavy quarks dominate the isovector contribution is\nnegligible). $\\sigma^S_{N,\\chi^0}$ is the LSP-nucleon scalar cross section.\n\nThe spin contribution is given by:\n\\begin{equation}\n\\bar{\\Sigma} _{spin}  =  (\\frac{\\mu_r}{\\mu_r(p)})^2\n                           \\sigma^{spin}_{p,\\chi^0}~\\zeta_{spin},\n\\zeta_{spin}= \\frac{1}{3(1+\\frac{f^0_A}{f^1_A})^2}S(u)\n\\label{2.10a}\n\\end{equation}\n\\begin{equation}\nS(u)\\approx S(0)=[(\\frac{f^0_A}{f^1_A} \\Omega_0(0))^2\n  +  2\\frac{f^0_A}{ f^1_A} \\Omega_0(0) \\Omega_1(0)+  \\Omega_1(0))^2  \\, ]\n\\label{s(u)}\n \\end{equation}\n The couplings $f^1_A$ ($f^0_A$) and the nuclear matrix elements $\\Omega_1(0)$ ($\\Omega_0(0)$) associated\n with the isovector (isoscalar) components are normalized so that, in the case\n of the proton at $u=0$, they yield $\\zeta_{spin}=1$.\n\n With these definitions in the proton neutron representation we get:\n\n\n\n\n \\beq\n \\zeta_{spin}= \\frac{1}{3}S^{'}(0)~,~S^{'}(0)=\\left[(\\frac{a_n}{a_p}\\Omega_n(0))^2+2 \\frac{a_n}{a_p}\\Omega_n(0) \\Omega_p(0)+\\Omega^2_p(0)\\right]\n \\label{Spn}\n \\eeq\n where $\\Omega_p(0)$ and $\\Omega_n(0)$ are the proton and neutron components of the static spin nuclear matrix elements. In extracting limits on the nucleon cross sections from the data we will find it convenient to\n write:\n \\begin{equation}\n                          \\sigma^{spin}_{p,\\chi^0}~\\zeta_{spin} =\\frac{\\Omega^2_p(0)}{3}|\\sqrt{\\sigma_p}+\\frac{\\Omega_n}{\\Omega_p} \\sqrt{\\sigma_n}\n e^{i \\delta}|^2\n\\label{2.10ab}\n\\end{equation}\n In Eq. (\\ref{2.10ab}) $\\delta$ the relative\nphase between the two amplitudes $a_p$ and $a_n$, which in most models is 0 or $\\pi$, i.e. one expects them to be relatively real.\n The static spin matrix elements are obtained in the context of a given nuclear model. Some such matrix elements of interest to the planned experiments  can be found in \\cite{JDV06}.\n \nThe spin ME are defined as follows:\n\\beq\n\\Omega_p(0)=\\sqrt{\\frac{J+1}{J}}\\prec J~J| \\sigma_z(p)|J~J\\succ ~~,~~\n\\Omega_n(0)=\\sqrt{\\frac{J+1}{J}}\\prec J~J| \\sigma_z(n)|J~J\\succ\n\\label{Omegapn}\n\\eeq\nwhere $J$ is the total angular momentum of the nucleus and $\\sigma_z=2 S_z$. The spin operator is defined by $S_z(p)=\\sum_{i=1}^{Z} S_z(i)$, i.e. a sum over all protons in the nucleus,  and\n$S_z(n)=\\sum_{i=1}^{N}S_z(i)$, i.e. a sum over all neutrons. Furthermore $\\Omega_0(0)=\\Omega_p(0)+\\Omega_n(0)~~,~~\\Omega_1(0)=\\Omega_p(0)-\\Omega_n(0)$\n\\section{The WIMP velocity distribution}\nTo obtain the total rates one must fold with WIMP velocity distribution and integrate  the\nabove expressions  over the\nenergy transfer from $Q_{min}$ determined by the detector energy cutoff to $Q_{max}$\ndetermined by the maximum LSP velocity (escape velocity, put in by hand in the\nMaxwellian distribution), i.e. $\\upsilon_{esc}=2.84~\\upsilon_0$ with  $\\upsilon_0$\nthe velocity of the sun around the center of the galaxy($229~Km\/s$).\n\nFor a given velocity distribution f(\\mbox{\\boldmath $\\upsilon$}$^{\\prime}$),\n with respect to the center of the galaxy,\none can find the velocity distribution in the Lab\nf(\\mbox{\\boldmath $\\upsilon$},\\mbox{\\boldmath $\\upsilon$}$_E$)\nby writing \n\\mbox{\\boldmath $\\upsilon$}$^{'}$=\n          \\mbox{\\boldmath $\\upsilon$}$ \\, + \\,$ \\mbox{\\boldmath $\\upsilon$}$_E \\, ,$\n\\mbox{\\boldmath $\\upsilon$}$_E$=\\mbox{\\boldmath $\\upsilon$}$_0$+\n \\mbox{\\boldmath $\\upsilon$}$_1$, with\n\\mbox{\\boldmath $\\upsilon$} $_1 \\,$ the  Earth's velocity\n around the sun.\n  \nIt is convenient to choose a coordinate system so  that\n $\\hat{x}$  is radially out in the plane of the galaxy,\n $\\hat{z}$ in the sun's direction of motion and \n $\\hat{y}=\\hat{z}\\times\\hat{x}$.\n\nSince the axis of the ecliptic  \nlies very close to the $x,y$ plane ($\\omega=186.3^0$) only  the angle\n $\\gamma=29.8^0$\nbecomes relevant.\nThus the velocity of the earth around the\nsun is given by \n\\begin{equation}\n\\mbox{\\boldmath $\\upsilon$}_E  = \\mbox{\\boldmath $\\upsilon$}_0 \\hat{z} +\n                                  \\mbox{\\boldmath $\\upsilon$}_1  \n(\\, sin{\\alpha} \\, {\\bf \\hat x}\n-cos {\\alpha} \\, cos{\\gamma} \\, {\\bf \\hat y}\n+ cos {\\alpha} \\, sin{\\gamma} \\, {\\bf \\hat z} \\,)\n\\label{3.6}  \n\\end{equation}\nwhere $\\alpha$ is  phase of the earth's orbital motion.\n\nThe WIMP velocity distribution f(\\mbox{\\boldmath $\\upsilon$}$^{\\prime}$) is not known. Many velocity distributions\nhave been used. The most common one is the M-B distribution with characteristic velocity $\\upsilon_0$ \nwith an upper bound $\\upsilon_{esc}=2.84 \\upsilon_0$. \n\\beq\nf(\\mbox{\\boldmath $\\upsilon$}^{\\prime})=\\frac{1}{(\\sqrt{\\pi}\\mbox{\\boldmath $\\upsilon_0$})^3}\ne^{-(\\mbox{\\boldmath $\\upsilon$}^{\\prime}\/\\mbox{\\boldmath $\\upsilon_0$})^2}\n\\label{fv}\n\\eeq\nModifications of this velocity distribution\nhave also been considered such as: i) Axially symmetric M-B distribution  \\cite {Druk,Verg00}. and ii) \nmodifications of the characteristic parameters of the M-B distribution by considering a coupling\nbetween dark matter and dark energy \\cite{TETRVER06} \n($\\upsilon_0 \\rightarrow n \\upsilon_0,\\upsilon_{esc}\\rightarrow n \\upsilon_{esc}$). Other possibilities are adiabatic velocity distribution following the Eddington approach \n\\cite{EDDIN}$^-$\\cite{VEROW06} ,\ncaustic rings \\cite{SIKIVI1}$^-$\\cite{Gelmini} and  Sagittarius dark matter \\cite{GREEN02} .\n  \nFor a given energy transfer the velocity $\\upsilon$ is constrained to be\n\\beq\n\\upsilon\\geq \\upsilon_{min}~,~\\upsilon_{min}= \\sqrt{\\frac{ Q A m_p}{2}}\\frac{1}{\\mu_r}.\n\\eeq\n\\section{The Direct detection rate}\nThe event rate for the coherent WIMP-nucleus elastic scattering is given by \\cite{Verg01,JDV03,JDVSPIN04,JDV06}:\n\\beq\nR= \\frac{\\rho (0)}{m_{\\chi^0}} \\frac{m}{m_p}~\n              \\sqrt{\\langle v^2 \\rangle } \\left [f_{coh}(A,\\mu_r(A)) \\sigma_{p,\\chi^0}^{S}+f_{spin}(A,\\mu_r(A))\\sigma _{p,\\chi^0}^{spin}~\\zeta_{spin} \\right]\n\\label{fullrate}\n\\eeq\nwith\n\\beq\nf_{coh}(A, \\mu_r(A))=\\frac{100\\mbox{GeV}}{m_{\\chi^0}}\\left[ \\frac{\\mu_r(A)}{\\mu_r(p)} \\right]^2 A~t_{coh}\\left(1+h_{coh}cos\\alpha \\right)\n\\eeq\n\\beq\nf_{spin}(A, \\mu_r(A))=\\left[ \\frac{\\mu_r(A)}{\\mu_r(p)} \\right]^2 \\frac{t_{spin}(A)}{A}t_{spin}\\left(1+h_{spin}cos\\alpha \\right)\n\\eeq\nwith $\\sigma_{p,\\chi^0}^{S}$ and $\\sigma _{p,\\chi^0}^{spin}$ the scalar and spin proton cross sections\n$~\\zeta_{spin}$ the nuclear spin ME. In the above expressions $h$ is the modulation amplitude.\n\n The number of events in time $t$ due to the scalar interaction, which leads to coherence, is:\n\\beq\n R\\simeq 1.60~10^{-3}\n\\frac{t}{1 \\mbox{y}} \\frac{\\rho(0)}{ {\\mbox0.3GeVcm^{-3}}}\n\\frac{m}{\\mbox{1Kg}}\\frac{ \\sqrt{\\langle\nv^2 \\rangle }}{280 {\\mbox kms^{-1}}}\\frac{\\sigma_{p,\\chi^0}^{S}}{10^{-6} \\mbox{ pb}} f_{coh}(A, \\mu_r(A))\n\\label{scalareventrate}\n\\eeq\nIn the above expression\n $m$ is the target mass, $A$ is the number of nucleons\nin the nucleus and $\\langle v^2 \\rangle$ is the average value of the square of the WIMP velocity.\n\nIn the case of the spin interaction we write:\n\\beq\n R\\simeq 16\n\\frac{t}{1 \\mbox{y}} \\frac{\\rho(0)}{ {\\mbox0.3GeVcm^{-3}}}\n\\frac{m}{\\mbox{1Kg}}\\frac{ \\sqrt{\\langle\nv^2 \\rangle }}{280 {\\mbox kms^{-1}}}\\frac{\\sigma_{p,\\chi^0}^{S}}{10^{-2} \\mbox{ pb}} f_{spin}(A, \\mu_r(A))\n\\label{spineventrate}\n\\eeq\nNote the different scale for the proton spin cross section.\nThe parameters $f_{coh}(A,\\mu_r(A))$, $f_{spin}(A,\\mu_r(A))$, which give the relative merit\n for the coherent and the spin contributions in the case of a nuclear\ntarget compared to those of the proton,  have already been  tabulated \\cite{JDV06}\n for energy cutoff $Q_{min}=0,~10$ keV. It is clear that for large A the coherent process is\nexpected to dominate unless for some reason the scalar proton cross section is very suppressed.\n\nIn the case of directional experiments the event rate is given\n by Eqs (\\ref{scalareventrate}) and (\\ref{spineventrate}) except that now:\n \\beq\nf_{coh}(A, \\mu_r(A))=\\frac{100\\mbox{GeV}}{m_{\\chi^0}}\\left[ \\frac{\\mu_r(A)}{\\mu_r(p)} \\right]^2 A\\frac{\\kappa}{2 \\pi}t_{coh}\\left(1+h_m(coh)cos{(\\alpha+\\alpha_m \\pi)} \\right)\n\\eeq\n\\beq\nf_{spin}(A, \\mu_r(A))=\\frac{100\\mbox{GeV}}{m_{\\chi^0}}\\left[ \\frac{\\mu_r(A)}{\\mu_r(p)} \\right]^2 \\frac{\\kappa}{2 \\pi}\\frac{t_{spin}}{A}\\left(1+h_m(spin)cos{(\\alpha+\\alpha_m \\pi)} \\right)\n\\eeq\nIn the above expressions $h_m$ is the modulation amplitude and $\\alpha _m$ the shift in the phase of the modulation (in units of $\\pi$) relative to the phase of the Earth. $\\kappa\/(2 \\pi)$, $\\kappa\\leq 1$, is the suppression factor entering due to the restriction of\nthe phase space.  $\\kappa$, $h_m$ and $\\alpha_m$ depend on the direction of observation. It is precisely this\ndependence as well as the large values of $h_m$, which can be exploited to reject background \\cite{JDV06}, that makes the directional experiments quite attractive in spite of the suppression factor relative to the standard\nexperiments.\n\\section{Bounds on the scalar proton cross section}\nUsing the above formalism one can obtain the quantities of interest $t$ and $h$ both for the standard as\nwell as the directional experiments. Due to lack of space we are not going to present the obtained results\nhere. The interested reader can find some of these results elsewhere \\cite{JDVSPIN04,JDV06} . Here we are \nsimply going to show how\none can employ such results to extract the nucleon cross section from the data.\nDue to space considerations we are not going to discuss the limits extracted from the data on the spin cross\nsections, since in this case one has to deal with two amplitudes (one for the proton and one for the neutron). We will only\nextract some limits imposed on the\nscalar nucleon cross section (the proton and neutron cross section are essentially the same).\nIn what follows we will employ for all targets \\cite{BCFS02}$^-$\\cite{PAVAN01} the limit of CDMS II for the Ge target \\cite{CDMSII04} ,\n i.e.  $<2.3$ events for \nan exposure of $52.5$ Kg-d with a threshold of $10$ keV. This event rate is similar to that for other systems \\cite{SGF05}. The thus obtained limits are exhibited in Fig. \\ref{b127.73}. For larger WIMP masses one can\nextrapolate these curves, assuming an increase as $\\sqrt{m_{\\chi}}$.\n\\begin{figure}\n\\rotatebox{90}{\\hspace{0.0cm} $\\sigma_p\\rightarrow 10^{-5}$pb}\n\\psfig{file=bcoh127.eps,width=2.0in}\n\\rotatebox{90}{\\hspace{0.0cm} $\\sigma_p\\rightarrow 10^{-5}$pb}\n\\psfig{file=bcoh73.eps,width=2.0in}\n\\hspace{-2.0cm} $m_{\\chi}\\rightarrow$ GeV\n\\caption{ The limits on the scalar proton cross section for A$=127$ on the left and A$=73$ on the right as functions of $m_{\\chi}$. The continuous (dashed) curves correspond to $Q_{min}=0~(10)$ keV respectively. Note that the advantage of the larger nuclear mass number of the A$=127$ system is counterbalanced by the favorable form factor dependence of the A$=73$ system.}\n \\label{b127.73}\n\\end{figure}\n\\section{Transitions to excited states}\nThe above formalism can easily be extended to cover transitions to excited states. Only the kinematics and \nthe nuclear physics is different. In other words one now needs:\n\\begin{itemize}\n\\item The inelastic scalar form factor.\\\\\n The transition amplitude is non zero due to the momentum transfer involved. The relevant multipolarities\n are determined by the spin and parity of the final state.\n \\item Spin induced transitions.\\\\\n In this case one can even have a Gamow-Teller like transition, if the final state is judiciously chosen.\n \\end{itemize}\n \n In the case of $^{127}I$ the static spin matrix element involving the first excited state around \n50 keV is twice as large compared to that of the\n ground state \\cite{VQS04}. The spin response function was assumed to be the same with that of the ground\n state. The results obtained \\cite{VQS04} are shown in Fig. \\ref{ratio}.\n \\begin{figure}\n\\begin{center}\n\\rotatebox{90}{\\hspace{1.0cm} {\\tiny BRR}$\\rightarrow$}\n\\includegraphics[height=.17\\textheight]{ratio0.eps}\n\n \\rotatebox{90}{\\hspace{1.0cm} {\\tiny BRR}$\\rightarrow$}\n\\includegraphics[height=.17\\textheight]{ratioQ.eps}\\\\\n \\hspace{0.0cm}$m_{LSP}\\rightarrow$ ($GeV$)\n \\caption{ The ratio of the rate to\nthe excited state divided by that of the ground state as a\nfunction of the LSP mass (in GeV) for $^{127}I$.\n We found that\nthe static spin matrix element of the transition from the ground\nto the excited state is a factor of 1.9 larger than that\ninvolving the ground state and assumed that the spin response functions\n$F_{11}(u)$ are the same. \nOn the left we show the results for $Q_{min}=0$ and on\nthe right for $Q_{min}=10~KeV$. \\label{ratio} }.\n\\end{center}\n\\end{figure}\nThese results are very encouraging, since, as we have mentioned, for heavier WIMPS like those involved\nin K-K theories, the branching ratios are expected to be much larger. Thus one may consider such\ntransitions, since the detection of de-excitation $\\gamma$ rays is much easier than the detection of\nrecoiling nuclei.\n\\section{Other non recoil experiments}\nAs we have already mentioned the nucleon recoil experiments are very hard. It is therefore necessary to consider\nother possibilities. One such possibility is to detect the electrons produced during the WIMP-nucleus \ncollisions \\cite{VE05,MVE05} employing detectors with low energy threshold with a high Z target. Better yet\none may attempt to detect the very hard X-rays generated when the inner shell electron holes are \nfilled \\cite{EMV05}. The relative X-ray to nucleon recoil probabilities $[Z \\sigma _K \/\\sigma _r]_i$, \nfor $i=L (m_{\\chi}\\leq \\mbox{100GeV}),~M(\\mbox{100 GeV}\\leq m_{\\chi}\\leq  \\mbox{200 GeV})$ and $H (m_{\\chi}\\simeq \\mbox{200 GeV})$ are shown in table \\ref{table:X-rays}. For even heavier WIMPs, like\nthose expected in K-K theories, the relative probability is expected to be even larger.\n\n\\begin{table}\n\\begin{center}\n\\caption{K X-ray cross sections relative to the nuclear recoil,\nrates and energies in WIMPs nuclear interactions with $^{131}$Xe.\n$[Z \\sigma _K \/\\sigma _r]_L, [Z \\sigma _K \/\\sigma _r]_M$ and $[Z\n\\sigma _K \/\\sigma _r]_H$ are the ratios for light (30 GeV), medium\n(100 GeV) and heavy (300 GeV) WIMPs.} \\label{t:2} \\vspace{0.5cm}\n\\label{table:X-rays}\n\\begin{tabular}{|ccccc|}\n\\hline K X-ray & $E_K(K_{ij})$ keV    & $[\\frac{Z\n\\sigma _K(K_{ij})}{\\sigma _r}]_{L}$ &  $[\\frac{Z \\sigma\n_K(K_{ij})}{\\sigma _r} ]_{M} $ & $[\\frac{Z \\sigma\n_K(K_{ij})}{\\sigma _r}]_{H} $ \\\\ \\hline\n K$_{\\alpha 2}$  & 29.5  & 0.0086 & 0.0560 & 0.0645 \\\\\n K$_{\\alpha 1}$ & 29.8  & 0.0160 & 0.1036 & 0.1196 \\\\\n K$_{\\beta 1}$  & 33.6   & 0.0047 & 0.0303 & 0.0350 \\\\\n K$_{\\beta 2}$  & 34.4   & 0.0010 & 0.0067 & 0.0077 \\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe K$_{\\alpha}$ and K$_{\\beta}$ lines can be\nseparated experimentally by using good energy-resolution\ndetectors, but the sum of all K lines can be measured in modest\nenergy-resolution experiments.\n\\section{Conclusions}\nWe examined the various signatures expected in the direct detection of WIMPs via their\ninteraction with nuclei. We specially considered WIMPs predicted in supersymmetric models (LSP or neutralino) as well as  theories with extra dimensions. We presented the formalism for the modulation \namplitude for non directional as well as directional experiments. We discussed the role played by\nnuclear physics on the extraction of the nucleon cross sections from the data. We also considered\nnon recoil experiments, such as measuring the $\\gamma$ rays following the de-excitation of the nucleus\nand\/or the hard X-rays after the de-excitation of the inner shell electron holes produced during the WIMP \nnucleus interaction. These are favored by very heavy MIMPs in the TeV region and velocity distributions\nexpected in models allowing interaction of dark matter and dark energy.\n\n{\\bf Acknowledgments}:\nThis work was supported in part by the European Union contract MRTN-CT-2004-503369.  Special thanks to Professor Raduta for support and hospitality during  the Predeal Summer School.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGravitational lensing of photons by black holes has received great attention in the last few years, mainly due to the increasing evidence of the presence of supermassive black holes at the center of galaxies. Theoretical studies of black hole lenses, both numerically and analytically, were made with Schwarzschild \\cite{virbha1, bozza1, schw}, Reissner--Nordstr\\\"{o}m \\cite{eiroto}, general spherically symmetric \\cite{bozza2, bozman1} and rotating \\cite{bozza3, rotating} geometries, and also for black holes coming from alternative theories \\cite{alternative} or braneworld cosmologies \\cite{braneworld}. Even naked singularities were considered as lenses \\cite{nakedsing}. Photons (or null mass particles) passing close enough to the photon sphere of the lens will have large deflection angles, and they can even make one or more turns around the deflector before reaching the observer. By this mechanism, two infinite sets of strong deflection images, one at each side of the lens, are produced. The presence of images with large deflection angles is not a new fact, since they were obtained already in 1959 for the Scharzschild spacetime \\cite{darwin}. The analytical study of these images is more simple if one adopts the strong deflection limit, which consists in a logarithmic approximation of the deflection angle, first obtained for the Schwarzschild metric \\cite{darwin}, revisited by several authors \\cite{old, bozza1},  extended to Reissner--Nordstr\\\"{o}m geometry \\cite{eiroto}, to general spherically symmetric spacetimes \\cite{bozza2} and to Kerr metric \\cite{bozza3}.  For some lensing configurations two weak deflection images are also obtained, which are analyzed by making a first order Taylor expansion of the deflection angle (weak deflection limit), as it is usually done for more standard astrophysical objects, such as stars and galaxies (see, e.g., \\cite{schneider}). Intermediate cases can be treated analytically by perturbative \\cite{perturba} or variational methods \\cite{amore}. A special configuration, where no weak deflection images are present, is when the source is in front of the lens instead of behind it, which is called retrolensing \\cite{retrolens}. Recently, the strong deflection limit was extended to include sources very close to the black hole \\cite{bozza4}.\\\\\n\nLensing of neutrinos have been previously studied by other authors. In Ref. \\cite {escribano}, gravitational lensing of neutrinos by stars and galaxies was analyzed, and in Ref. \\cite{mena}, the lensing effects of supernova neutrinos by the Galactic center black hole was considered, in the weak deflection limit. However, perhaps the most interesting cosmological sources of neutrinos from the point of view of lensing are transients associated with gamma-ray bursts (GRBs). It is expected that proton-photon interactions during the GRB will result into copious photopion production and hence neutrinos would be generated from the decay of charged pions and muons (e.g. \\cite{neutrinos, meszaros, dermer, murase}). Since GRBs occur frequently, say once per day, and can be detected at gamma-rays by SWIFT satellite and then the follow up of the afterglows usually allows the identification of the host galaxy and the corresponding redshift (see, e.g., \\cite{meszaros2}), they are outstanding candidates for lensing produced by massive black holes in the center of interposed galaxies. \\\\\n\nWe notice, however, that in the collapsar scenario for long GRBs \\cite{woosley1, woosley2} the jet not always is expected to be able to make its way through the star, so no observable gamma-ray emission would result in such cases \\cite{meszaros}. Nonetheless, the neutrino emission might be important. If the event is lensed, the neutrino signal should repeat and hence be identified, despite the absence of electromagnetic counterparts. \\\\\n\nIn this letter we investigate gravitational lensing of neutrinos by Schwarzschild black holes. We pay special attention to neutrino transients lensed by supermassive black holes located at the center of galaxies. In Section \\ref{pm} we present the expressions that give the positions and magnifications of the weak and strong deflection images, and in Section \\ref{td} we calculate the time delays between the arrival signals. Then, in Section \\ref{lnb}, we calculate the specific time delays produced by some interposed supermassive black holes for neutrino transient at a distance of $\\sim 10^{28}$ cm. Finally, in Section \\ref{conc}, a brief summary and the conclusions are presented.\n\n\\section{Positions and magnifications of the images}\\label{pm} \n\nNeutrinos have zero or negligible mass, so we assume that they follow null geodesics as photons do. We consider a point source of neutrinos, with angular diameter distance $D_{\\rm{os}}$ to the observer, behind a Schwarzschild black hole lens, placed at an angular diameter distance $D_{\\rm{ol}}$. The angular diameter distance between the lens and the source is dubbed $D_{\\rm{ls}}$. The optical axis is defined by the line that joins the observer with the deflector. The distances are very large compared to the Schwarzschild radius of the black hole and the angles are measured from the observer. We restrict our analysis to high alignment, which is more interesting from an astrophysical point of view, since the images are more prominent. Then the angular position of the source $\\beta $, taken positive here, is small. For this configuration, we have two weak deflection images and two infinite sets of strong deflection (also called relativistic \\cite{virbha1}) images. Neutrinos with closest approach distance $r_{0}$ much larger than the photon sphere radius $r_{\\rm{ps}}=3MG\/c^2$, which corresponds to the unstable circular orbit around the black hole\\footnote{For a complete study of the photon sphere in a spherically symmetric geometry see \\cite{atkinson}; and for a general definition of the photon surface in an arbitrary spacetime see \\cite{claudel}.}, will have a small deflection angle $\\alpha$, which can be approximated to first order in $1\/r_{0}$ by $\\alpha= 4GM\/(c^{2}r_{0})$ (weak deflection limit). Within this approximation, the lens equation has the form \\cite{schneider}\n\\begin{equation}\n\\beta =\\theta-\\frac{\\theta_{\\rm{E}}^{2}}{\\theta},\n\\label{pm1} \n\\end{equation}\nwhere $\\theta $ is the angular position of the image and $\\theta_{\\rm{E}}$ is the angular Einstein radius, given by\n\\begin{equation}\n\\theta_{\\rm{E}}=\\sqrt{\\frac{2R_{\\rm{s}}D_{\\rm{ls}}}{D_{\\rm{ol}}D_{\\rm{os}}}},\n\\label{pm2} \n\\end{equation} \nwith $R_{\\rm{s}}=2MG\/c^{2}$ the Schwarzschild radius of the lens. The lens equation has two solutions:\n\\begin{equation}\n\\theta_{\\rm{p},s}=\\frac{1}{2}\\left( \\beta\\pm\\sqrt{\\beta^{2}+4\\theta_{\\rm{E}}^{2}}\\right),\n\\label{pm3} \n\\end{equation} \nthat give the positions of the primary (upper sign) and the secondary (lower sign) images. The primary image lies inside the Einstein radius and the secondary image outside. When $\\beta =0$, instead of two images, an Einstein ring with radius $\\theta_{\\rm{E}}$ is obtained. Another important aspect is the magnification of the images, defined as the ratio between the observed and intrinsic fluxes of the source. As a consequence of the Liouville theorem in curved spacetimes \\cite{misner}, gravitational lensing preserves surface brightness for neutrinos and photons, so the magnifications of the images are given by the ratio of the solid angles subtended by the images and the source, which result in \\cite{schneider}:\n\\begin{equation}\n\\mu _{\\rm{p},s}=\\frac{1}{4}\\left( \\frac{\\beta}{\\sqrt{\\beta^{2}+4\\theta_{\\rm{E}}^{2}}}+\n\\frac{\\sqrt{\\beta^{2}+4\\theta_{\\rm{E}}^{2}}}{\\beta}\\pm 2 \\right),\n\\label{pm4} \n\\end{equation}\nwhere the plus sign corresponds to the primary image and the minus sign to the secondary one. If the position of the source $\\beta$ is close to zero, the magnifications of both images are large. If $\\beta =0$ the approximation of point source breaks down and the magnifications become infinite. It is not difficult to see that $\\mu _{\\rm{p}}>1$ for all $\\beta $, and $\\mu _{\\rm{s}}>1$ only if $\\beta \/\\theta_{\\rm{E}}< \\sqrt{(3\\sqrt{2}-4)\/2}\\approx 0.35$. When $\\beta \/\\theta_{\\rm{E}}$ is large we have that $\\mu _{\\rm{p}}\\approx 1$ and $\\mu _{\\rm{s}}\\approx 0$.\\\\\n\nBesides the weak deflection images, two infinite sets of relativistic images are formed by neutrinos that make one or more loops, in both directions of winding, around the black hole lens. For high alignment, the deflection angle corresponding to the relativistic images is close to an even number of $\\pi$, $\\alpha=\\pm (2n\\pi + \\Delta\\alpha_{n})$ with $0<\\Delta\\alpha_{n}\\ll 1$, the upper sign corresponding to one set of images and the lower one to the other set. The other angles involved are small, then the lens equation \\cite{virbha1}\\footnote{Eq. (\\ref{pm5a}) is valid for asymptotically flat spacetimes, with the source and the observer in the flat region; for more general lens equations see \\cite{lensequa}.} \n\\begin{equation}\n\\tan \\beta =\\tan \\theta -\\frac{D_{\\mathrm{ls}}}{D_{\\mathrm{os}}}\\left( \\tan\n\\theta + \\tan(\\alpha -\\theta )\\right),\n\\label{pm5a}\n\\end{equation}\ntakes the form \\cite{bozza1, bozza2}\n\\begin{equation}\n\\beta=\\theta \\mp \\frac{D_{\\rm{ls}}}{D_{\\rm{os}}}\\Delta\\alpha_{n}.\n\\label{pm5b}\n\\end{equation}\nIn the strong deflection limit, i.e. for trajectories passing close to the photon sphere of the black hole, the deflection angle can be approximated by a logarithmic function of the impact parameter $b$, defined as the perpendicular distance from the deflector to the asymptotic path at infinite. For the Schwarzschild geometry, it can be shown that \\cite{darwin, bozza1, bozza2}\n\\begin{equation}\n\\alpha =\\pm \\left[ -c_{1}\\ln \\left( \\frac{b}{b_{\\rm{ps}}}-1\\right)+ c_{2}\\right] +O(b-b_{\\rm{ps}}),\n\\label{pm6} \n\\end{equation} \nwith $c_{1}=1$, $c_{2}=\\ln [216(7-4\\sqrt{3})]-\\pi$ and $b_{\\rm{ps}}=\\sqrt{3}r_{\\rm{ps}} =3\\sqrt{3}R_{\\rm{s}}\/2$ the critical impact parameter. Neutrinos with impact parameter smaller than the critical value will spiral inside the photon sphere into the black hole, not reaching the observer, and those with $b$ larger than $b_{\\rm{ps}}$ will make one or more outward turns outside the photon sphere, finally getting to the observer. As in the case of photons, using that $b=\\sin D_{\\rm{ol}} \\theta \\approx \\theta D_{\\rm{ol}}$, inverting Eq. (\\ref{pm6}) and Taylor expanding it around $\\alpha =2n\\pi$ to obtain $\\Delta\\alpha_{n}$, then replacing the result in the lens equation (\\ref{pm5b}) and finally inverting it, the positions of the relativistic images can be approximated (keeping only the lower order terms) by \\cite{bozza1, bozza2}:\n\\begin{equation}\n\\theta_{n}=\\pm \\theta_{n}^{\\rm{E}}+\\frac{D_{\\rm{os}}b_{\\rm{ps}}}{D_{\\rm{ls}}D_{\\rm{ol}}c_{1}}e_{n}\\beta,\n\\label{pm7} \n\\end{equation} \nwhere\n\\begin{equation}\n e_{n}=e^{(c_{2}-2n\\pi)\/c_{1}},\n\\nonumber\n\\end{equation} \nand\n\\begin{equation}\n\\theta_{n}^{\\rm{E}}=\\frac{b_{\\rm{ps}}}{D_{\\rm{ol}}}\\left( 1- \\frac{D_{\\rm{os}}b_{\\rm{ps}}}{D_{\\rm{ls}}D_{\\rm{ol}}c_{1}}e_{n}\\right)(1+e_{n}),\n\\label{pm8} \n\\end{equation} \nis the $n$-th relativistic Einstein ring radius. For perfect alignment an infinite sequence of Einstein rings is obtained instead of point images. With the same considerations given above for the weak deflection images, the magnification of the $n$-th image has the same expression that was found previously for photons \\cite{bozza1, bozza2}:\n\\begin{equation}\n\\mu_{n}=\\frac{1}{\\beta}\\frac{b_{\\rm{ps}}^2D_{\\rm{os}}}{D_{\\rm{ol}}^2 D_{\\rm{ls}} c_{1}}(1+e_{n})e_{n},\n\\label{pm9} \n\\end{equation} \nfor both sets of relativistic images. The first image ($n=1$) is the strongest one and the others have magnifications that decrease exponentially with $n$. For a given source angle $\\beta$, the relativistic images are very faint compared with the weak deflection ones\\footnote{For example, if $\\beta \/\\theta_{\\rm{E}}\\ll 1$  we have that $\\mu_{1}\/\\mu_{\\rm{p}}\\propto (R_{\\rm{S}}\/D_{\\rm{ol}})^{3\/2}$, which is usually a very small number.}.\n\n\\section{Time delays}\\label{td} \n\nNeutrinos that form distinct images take different paths, resulting in time delays between the images. Considering again that neutrinos follow null geodesics as photons do, the time delay between the primary and the secondary images is given by \\cite{schneider}:\n\\begin{equation}\n\\Delta t_{\\rm{p,s}}=\\frac{2R_{\\rm{s}}}{c}(1+z_{\\rm{d}})\\left( \\frac{\\theta_{\\rm{s}}^{2}-\\theta_{\\rm{p}}^{2}}{2|\\theta_{\\rm{p}}\\theta_{\\rm{s}}|} +\\ln \\left|\\frac{\\theta_{\\rm{s}}}{\\theta_{\\rm{p}}}\\right|\\right),\n\\label{td1}\n\\end{equation}\nwhere $z_{\\rm{d}}$ is the redshift of the deflector. The last equation can be written in the form\n\\begin{equation}\n\\Delta t_{\\rm{p,s}}=\\frac{2R_{\\rm{s}}}{c}(1+z_{\\rm{d}})\\left( \\frac{-\\beta \\sqrt{\\beta^{2}+4\\theta_{\\rm{E}}^{2}}}{2\\theta_{\\rm{E}}^{2}}+\\ln \\left| \\frac{\\beta -\\sqrt{\\beta^{2}+4\\theta_{\\rm{E}}^{2}}}{\\beta +\\sqrt{\\beta^{2}+4\\theta_{\\rm{E}}^{2}}}\\right|\\right).\n\\label{td2}\n\\end{equation}\nWhen $\\beta =0$ there is no time delay. Large time delays can be obtained if $\\beta \/\\theta_{\\rm{E}}\\gg 1$, but in this case the magnification of the primary image is close to one and the secondary image is very faint. The optimal situation for a variable source is when $\\beta \/\\theta_{\\rm{E}}$ is small enough to have large magnifications of both images, but not too close to zero, so the time delay can be longer than the typical time scale of the transient source.\\\\\n\nIn the case of relativistic images, the time delay between the images formed at the same side of the lens is given by \\cite{bozman1}\\footnote{The expressions from Ref. \\cite{bozman1} have been rewritten here using physical units, adding the cosmological factor $1+z_{d}$ and expanding them to first order in the source position angle (measured from the observer instead of from the source).}:\n\\begin{equation}\n\\Delta t^{\\rm{s}}_{n,m}=\\frac{b_{\\rm{ps}}}{c}(1+z_{\\rm{d}})\\left[ 2\\pi (n-m)+2\\sqrt{2}(w_{m}-w_{n})\\pm \\dfrac{\\sqrt{2}D_{\\rm{os}}(w_{m}-w_{n})}{c_{1}D_{\\rm{ls}}}\\beta \\right] ,\n\\label{td3}\n\\end{equation}\nwhere\n\\begin{equation}\nw_{k}=e^{(c_{2}-2k\\pi)\/(2c_{1})},\n\\nonumber\n\\end{equation}\nand the upper\/lower sign corresponds if both images are on the same\/opposite side of the source. For the images at the opposite side of the lens we have \\cite{bozman1}:\n\\begin{equation}\n\\Delta t^{\\rm{o}}_{n,m}=\\frac{b_{\\rm{ps}}}{c}(1+z_{\\rm{d}})\\left[ 2\\pi (n-m)+2\\sqrt{2}(w_{m}-w_{n})+\\left(  \\dfrac{\\sqrt{2}D_{\\rm{os}}(w_{m}+w_{n})}{c_{1}D_{\\rm{ls}}}-\\frac{2D_{\\rm{os}}}{D_{\\rm{ls}}}\\right) \\beta \\right],\n\\label{td4}\n\\end{equation}\nwhere the image with winding number $n$ is on the same side of the source and the other one on the opposite side. The first term in Eqs. (\\ref{td3}) and (\\ref{td4}) is by large the most important one \\cite{bozman1}. The time delays between the relativistic images are longer than the time delay between the primary and the secondary images.\n\n\\section{Lensing of neutrino transients}\\label{lnb} \n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nGalaxy & Black hole mass ($M_{\\odot}$) & Distance (Mpc) & $\\theta_{\\rm{E}}$ (arcsec) & $\\Delta t_{\\rm{s,p}}$ (s)\\\\\n\\hline\nMilky Way & $2.8 \\times 10^{6}$ & $0.0085$ & $1.6$  & $11$ \\\\\n\\hline\nNGC0224   & $ 3.0\\times 10^{7}$ & $0.7$ & $0.6$  & $1.2\\times 10^{2}$\\\\\n\\hline\nNGC3115   & $ 2.0\\times 10^{9}$ & $8.4$ & $1.4$  & $7.9\\times 10^{3}$\\\\\n\\hline\nNGC3377   & $ 1.8\\times 10^{8}$ & $9.9$ & $0.4$  & $7.1\\times 10^{2}$\\\\\n\\hline\nNGC4486B   & $ 5.7\\times 10^{8}$ & $15.3$ & $0.5$  & $2.2\\times 10^{3}$\\\\\n\\hline\nNGC4486   & $ 3.3\\times 10^{9}$ & $15.3$ & $1.3$ & $1.3\\times 10^{4}$\\\\\n\\hline\nNGC4261   & $ 4.5\\times 10^{8}$ & $27.4$ & $0.4$ & $1.8\\times 10^{3}$\\\\\n\\hline\nNGC7052   & $ 3.3\\times 10^{8}$ & $58.7$ & $0.2$ & $1.3\\times 10^{3}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Time delays between the weak deflection images of neutrino burst sources at a distance of $10^{28}$ cm. The lenses are supermassive black holes at the center of the galaxies indicated. The Schwarzschild geometry was adopted to model the black holes. The source angular position is $\\beta =0.1 \\, \\theta_{\\rm{E}}$, with $\\theta_{\\rm{E}}$ the angular Einstein radius. In this case, the angular positions of the primary and the secondary images are $\\theta_{\\rm{p}}=1.05 \\, \\theta_{\\rm{E}}$ and $\\theta_{\\rm{s}}=-0.95 \\, \\theta_{\\rm{E}}$, while their respective magnifications are $\\mu_{\\rm{p}}=5.5$ and $\\mu_{\\rm{s}}=4.5$.}\n\\label{table1}\n\\end{table}\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nGalaxy & Black hole mass ($M_{\\odot}$) & Distance (Mpc) & $\\theta_{\\rm{E}}$ (arcsec) & $\\Delta t_{\\rm{s,p}}$ (s)\\\\\n\\hline\nMilky Way & $2.8 \\times 10^{6}$ & $0.0085$ & 1.6 & $55$\\\\\n\\hline\nNGC0224   & $ 3.0\\times 10^{7}$ & $0.7$ &  0.6 & $6.0\\times 10^{2}$\\\\\n\\hline\nNGC3115   & $ 2.0\\times 10^{9}$ & $8.4$ & 1.4  & $4.0\\times 10^{4}$\\\\\n\\hline\nNGC3377   & $ 1.8\\times 10^{8}$ & $9.9$ & 0.4 & $3.6\\times 10^{3}$\\\\\n\\hline\nNGC4486B  & $ 5.7\\times 10^{8}$ & $15.3$ & 0.5  & $1.1\\times 10^{4}$\\\\\n\\hline\nNGC4486   & $ 3.3\\times 10^{9}$ & $15.3$ & 1.3  & $6.6\\times 10^{4}$\\\\\n\\hline\nNGC4261   & $ 4.5\\times 10^{8}$ & $27.4$ & 0.4  & $8.9\\times 10^{3}$\\\\\n\\hline\nNGC7052   & $ 3.3\\times 10^{8}$ & $58.7$ & $0.2$ & $6.6\\times 10^{3}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Time delays between the weak deflection images of neutrino burst sources at a distance of $10^{28}$ cm. The lenses are supermassive black holes at the center of the galaxies indicated. The Schwarzschild geometry was adopted to model the black holes. The source angular position is $\\beta =0.5 \\, \\theta_{\\rm{E}}$, with $\\theta_{\\rm{E}}$ the angular Einstein radius. In this case, the angular positions of the primary and the secondary images are $\\theta_{\\rm{p}}=1.28 \\, \\theta_{\\rm{E}}$ and $\\theta_{\\rm{s}}=-0.78 \\, \\theta_{\\rm{E}}$, while their respective magnifications are $\\mu_{\\rm{p}}=1.6$ and $\\mu_{\\rm{s}}=0.6$.}\n\\label{table2}\n\\end{table}\n\nThe angular resolution of the primary and secondary images is beyond the capability of current and near future neutrino detectors, which is of the order of one tenth of a degree, but the temporal resolution of the images of individual transient events, which have typical durations in the range of $\\sim$10 s to 100 s for long GRBs \\cite{meszaros2}, is possible. As an example of this, we consider neutrino transients acting as possible sources situated at distances of the order of $10^{28}$ cm, with supermassive black holes at the center of interposed galaxies as lenses. Some results of our calculations for specific cases of lenses in the local universe ($z_{\\rm{d}}\\sim 0$) are shown in Tables \\ref{table1} and \\ref{table2}, with the masses and distances taken from Ref. \\cite{richstone}. We see that the separation between the primary and secondary images of neutrino transients is of the order of a second of arc, so they cannot be resolved. For suitable values of the parameters involved, the weak deflection images can be both magnified several times, with time delays of $10^2 - 10^4$ s, larger than the intrinsic time of variation of the sources. \\\\\n\nIf one fixes $\\beta \/\\theta_{\\rm{E}}$ to obtain from Eq. (\\ref{pm4}) the desired values of magnification of the images, it is clear from Eq. (\\ref{td2}) that the time delay increases linearly with the redshift of the lens. Then, with a typical source with redshift $z_{\\rm{s}}\\sim 1$, the values of time delays between the primary and the secondary images for lenses closer to the neutrino sources can be up to twice of those obtained in Tables \\ref{table1} and \\ref{table2}. But far lenses require better alignment to have large magnifications because $\\theta _{\\rm{E}}$ decreases with the distance to the lens.\\\\\n\nThroughout this letter we have assumed that the neutrinos move like photons in null geodesics, so the time delays do not depend on the energy of the neutrinos. The time lag between neutrinos with an energy $E_{\\nu}$ and a small mass $m_{\\nu}$ and photons travelling the same distance $d$ can be approximated by $\\Delta t\\approx (1\/2)(d\/c)(m_{\\nu}c^2\/E_{\\nu})^2$, which using $E_{\\nu} \\gtrsim 1$ TeV (for neutrinos associated with GRBs), $m_{\\nu}c^2 \\lesssim 1$ eV, $d \\sim 10^{28}$ cm,  gives $\\Delta t < 10^{-6}$ s. This time lag corresponds to a total travelling time of about $3 \\times 10^{17}$ s. Then, if neutrinos have mass, the time delays given in Tables \\ref{table1} and \\ref{table2}  should be modified in the same proportion, i.e. in less than one part in $10^{23}$. There is also observational evidence related to the supernova SN1987A which shows that the time delay due to the presence of the galaxy for photons and neutrinos with different energies is the same within a $0.5\\%$ or better accuracy \\cite{supernova}. All these justify our assumption, and the results obtained are excellent approximations if the neutrinos have sub-eV mass.\\\\\n\nConcerning the relativistic images, if we choose $\\beta \/\\theta_{\\rm{E}}=0.1$ as it was done in Table \\ref{table1}, we have from Eq. (\\ref{pm9}) that the magnification of the brightest image is $\\mu _{1}=1.1\\times 10^{-17}$ for the Galactic black hole and $\\mu _{1}=1.8\\times 10^{-22}$ for NGC4486. Similar values are obtained for the other black holes considered in Tables \\ref{table1} and \\ref{table2}. The other image magnifications decrease exponentially with $n$. To obtain magnifications about one or larger, a closer alignment is necessary, i.e. $\\beta \\sim b_{\\rm{ps}}\/d_{\\rm{ol}}$ instead of  $\\beta \\sim \\theta _{\\rm{E}}$. Then, while the primary and secondary images are amplified several times for $\\beta \/\\theta_{\\rm{E}}=0.1$, the strong deflection ones are highly demagnified. Using Eq. (\\ref{pm7}), it can be seen that the angular separation between the strong deflection images is of the order of micro arc seconds or less. The time delays between the relativistic images, given by Eqs. (\\ref{td3}) and (\\ref{td4}), can be large, but they are too faint to be detected. So, in what follows, we restrict ourselves to the weak deflection images.\\\\\n\nThe probability of supermassive black holes located at the center of galaxies in the line of sight to GRBs is not negligible because of the high-redshift of most GRBs. Moreover, optical spectroscopic observations can detect absorbing lines of the interposed galaxy in the afterglow, hence allowing a direct determination of the different distances involved in the scenario. In the case of choked GRBs, where only neutrinos are produced, the time delays and the relative magnifications of the signals could used for the unequivocal identifications of {\\sl dark} neutrino transients. A neutrino transient associated with a GRBs might have an fluence of several times $10^{-4}$ erg cm$^{-2}$ \\cite{dermer}. With a mild amplification as obtained for the parameters adopted in Tables \\ref{table1} and \\ref{table2}, this might imply the detection of a few neutrinos by a km$^{3}$-detector. Even if the signal-to-noise ratio is not at a high confidence level, the repetition of the signal on a time scale from minutes to hours from the same location in the sky would render the identification of the neutrino transient source unequivocal. If the detection of the GRB afterglow allows a clear determination of the redshifts involved, then Eqs. (\\ref{pm4}) and (\\ref{td2}) can be used to obtain an independent estimated of the central black hole mass in the interposed galaxy. \\\\\n\nThe analysis of current databases indicates that the space-time clustering of GRBs is only marginal, at the level of $5\\%$ or less \\cite{romero}, but as we have mentioned in the Introduction, choked collapsars can result in transient neutrino sources without electromagnetic counterparts, so the total number of neutrino transients that is affected by lensing effects could be significantly larger from what is inferred from GRB population studies. The detection of a single event could be of paramount importance for our understanding of physical processes governing the GRBs.\n\n\n\\section{Final remarks}\\label{conc} \n\nIn this letter we have shown that the primary and secondary images of neutrino transient sources lensed by supermassive black holes cannot be angularly resolved but they could be temporally resolved by next generation instruments. The relativistic images, instead, are too faint and packed to be detected. Thus, we have found that neutrino transients produced by long GRBs can act as sources for gravitational lensing when supermassive black holes are present in foreground galaxies. This sources would have a unique signature, that will allow an easy detection above the background despite a possible low signal-to-noise ratio: repetition. The neutrino fluence can be magnified, but more importantly, the arriving signal will repeat, leading to an unequivocal identification. We conclude that neutrino gravitational lensing can help to establish GRBs as sources of relativistic protons and neutrinos, as proposed by several authors \\cite{meszaros,dermer}.\n\n\\section*{Acknowledgments}\n\nE.F.E. is supported by UBA and CONICET. G.E.R. is supported by grants PIP 5375 (CONICET) and PICT 03-13291 BID 1728\/OC-AR (ANPCyT) and by the Ministerio de Educaci\\'on y Ciencia (Spain) under grant AYA 2007-68034-C03-01, FEDER funds.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nQuantum mechanics traditionally places the observer `outside' of the system being studied, and refers only to ensembles of data. \nThe theory does not refer to the primary data obtained in individual trials.  So standard quantum mechanics is incomplete for two\nreasons -- it excludes the primary observer \\emph{and} the primary data.  For a theory that claims to be a fundamental law of\nnature, it excludes too much of nature.\n\nStandard theory also makes a distinction between microscopic and macroscopic systems, and this is ambiguous at best.  In addition,\nif one tries to use a quantum mechanical superposition of states to define an ontology in an individual trial, and if a secondary\nobserver like Schr\\\"{o}dinger's cat is included in the system, then the result will not only be paradoxical, it will be wrong.  The\ncat is not in a dual state of consciousness at any time during that famous experiment.\n\nStandard quantum mechanics posits four auxiliary rules that lead to these unacceptable  results.  These rules (called the\n\\emph{sRules}) are: (\\textbf{1}) Quantum measurement occurs only when a quantum mechanical microscopic (or possibly mesoscopic)\nsystem engages a macroscopic measuring device.  (\\textbf{2}) Primary data is collected during individual trials of this kind in\nwhich single eigenvalues are chosen by a stochastic process.  (\\textbf{3}) This choice is followed by a state reduction (i.e., the\ncollapse) in which a stochastically chosen eigenvalue is the sole survivor.  And (\\textbf{4}), the probability of choosing a given\neigenvalue is equal to its square modulus (i.e., the Born rule).  These auxiliary rules are not contained in the Schr\\\"{o}dinger\nequation.  They are supplementary instructions that tell us how to use the Schr\\\"{o}dinger equation\\footnote{These are Copenhagen\nsRules.  There may be any number of variations.  Their essential feature is the Born Rule that establishes the \\emph{only} link\nbetween  observation and the symbols of the theory.}.\n\nOther auxiliary rules are possible -- rules that avoid the difficulties described above.  There are at least two such rule-sets\ncalled the \\emph{nRules} and the \\emph{oRules} that correctly relate Schr\\\"{o}dinger's equation to observation.  Both are\nontologically based, for they place the primary observer `inside' the system -- or at least they allow for that possibility.  With\neither of these rule-sets (as in classical physics) the observer of an external system can extend the system to include himself so\nhe can become a continuous part of the wider system.  That is not possible with the auxiliary rules of standard quantum mechanics,\nwhere the observer is only allowed to ``peek\" at the system from time to time. \n\nIn addition, the nRules provide an ontological description of individual trials, so they include the primary data obtained in each\ntrial.   They give us a running description of each trial.  They are sequentially deterministic except for the time interval between\nstochastic choices, inasmuch as they cannot predict when the next stochastic choice will occur. The nRules are more deterministic\nthan the sRules, but they are less deterministic than classical physics.  The oRules are the least deterministic.  Both\nthe nRules and the oRules present an ontology for individual trials that removes the ambiguity that is now associated with\nSchr\\\"{o}dinger's cat.\n\nThe proposed rule-sets do not make a fundamental distinction between microscopic and macroscopic things.  Each has four rules that\napply equally to all parts of nature.  As a result, both microscopic and macroscopic systems may be said to experience quantum\njumps; and in the nRule case, microscopic states can sometimes undergo state reduction.  Examples are given in this and related\npapers.  Neither one of the proposed rule-sets directly includes the Born Rule that connects probability with square modulus; for in\nboth cases, probability is introduced only through \\emph{probability current}.\n\nThis paper is concerned only with the nRules.  Their adequacy is demonstrated in a number of cases, and their properties (described\nabove) are made apparent.\n\n\n\n\n\n\n\n\n\\section*{Ontology and Epistemology}\n\nThe method of this paper differs from that of traditional quantum mechanics in that it sees the observer in an ontological rather\nthan an epistemological context.  Traditional or standard quantum theory (i.e., Copenhagen) places the observer outside of the\nsystem where operators and\/or operations are used to obtain information about the system.  This is the epistemological model shown\nin \\mbox{Fig.\\ 1}.  \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.8]{aXnRulesFig1.eps}\n\\center{Figure 1: Epistemological Model (Copenhagen)}\n\\end{figure}\n\nThe large OP in Fig.\\ 1 might be a mathematical `operator' or a corresponding physical `operation'.  The observer makes a\nmeasurement by choosing a formal operator that is associated with a chosen laboratory operation.  As a result, the observer is\nforever outside of the observed system -- making operational choices.  The observer is forced to act apart from the system as one who\nposes theoretical and experimental questions to the system, and he can only get answers  through `instantaneous'\ncontacts with the system at a given time.  This model is both useful and epistemologically sound.  \n\n\tHowever, the special rules developed in this paper apply to the system by itself, independent of the possibility that an observer\nmay be inside, and disregarding everything on the outside.  This is the ontological model shown in \\mbox{Fig.\\ 2}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.8]{aXnRulesFig2.eps}\n\\center{Figure 2: Ontological Model (requires special rules)}\n\\end{figure}\n\nA measurement occurring inside this system is not represented by a formal operator.  Rather, it is represented by a measuring\ndevice that is itself part of the system.  If the sub-system being measured is $S$ and a detector is $D$, then a measurement\ninteraction is given by $\\Phi = SD$.  If an observer joins the system in order to look at the detector, then the system becomes\n$\\Phi = SDB$, where $B$ is the brain state of the observer.  Contact between the observer and the observed is continuous in this\ncase.\n\nThe ontological model is able to place the observer inside the universe of things and give a full account of his conscious\nexperience there.  It is a departure from traditional quantum mechanics and has three defining characteristics:  (1) It includes\nobservations given by $\\Phi = SDB$ as described above, (2) it allows all conscious observations to be continuous, and (3) it rejects\nthe long-standing Born interpretation of quantum mechanics by introducing probability (only) through the notion of\n\\emph{probability current}. \n\nQuantum mechanical measurement is  said to refer to ensembles of observations but not to individual observations.  In this\npaper I propose a set of four \\emph{nRules (1-4)} that  apply to individual measurements in the ontological model.  I claim that\nthey are a consistent and complete set of rules that can give an ontological description of any individual measurement or\ninteraction in quantum mechanics.  These rules are not themselves a formal theory of measurement.  I make no attempt to understand\n\\emph{why} they work, but strive only to insure that they do work. Presumably, a formal theory can one day be found to explain these\nrules in the same way that atomic theory now explains the empirically discovered rules of atomic spectra, or in the way that current\ntheories of measurement aspire to merge with standard quantum mechanics, or make the neurological connection with conscious\nobservation.\n\n\n\n\n\n\n\n\\section*{The oRules}\n Other papers \\cite{RM1, RM2, RM3} propose another set of rules called the oRules (1-4).  These are similar to the nRules\nexcept that the basis states of reduction are confined to observer brain states, reflecting the views of Wigner and von Neumann. \nLike the nRules, they introduce probability through the notion of `probability current' rather than through square modulus, and they\naddress the state reduction of conscious individuals in an ontological context, thereby giving us an alternative quantum friendly\nontology.  In Ref.\\ 3 they are called simply the \\emph{rules (1-4)}.  Like the nRules, the oRules are not a formal theory of\nmeasurement  for they require a wider theoretical framework to be understood.  I do not finally choose one of the rule-sets or\npropose an explanatory theory.  I am only concerned with how state reduction might occur in each case.  For the above stated\nreasons, both of these rule-sets are more acceptable than the sRules; and so far as I am aware, there is no observation that can\ndistinguish between the two.\n\n\n\n\n\n\n\\section*{The Interaction: Particle and Detector}\n\n\t\tBefore introducing the nRules, we will apply  Schr\\\"{o}dinger's equation to a `microscopic' particle interacting with a\n`macroscopic' detector in order to see what difficulties arise.  These two objects are assumed to be initially independent and\ngiven by the equation \n \n\\begin{equation}\n\\Phi(t)=exp(-iHt)\\psi_i\\otimes d_i\n\\end{equation}\nwhere $\\psi_i$ is the initial particle state and $d_i$ is the initial detector state.  The particle is then allowed to pass over the\ndetector, where the two interact with a cross section that may or may not result in a capture.  After the interaction begins at a\ntime $t_0$, the state is an entanglement in which the particle variables and the detector variables are not separable.  \n\nThe first component of the resulting system is the detector $d_0$ in its ground state prior to capture, and the second, third, and\nfourth components are the detector in various states of capture   given by $d_w$, $d_{m}$, and $d_d$.  \n\\begin{equation}\n\\Phi(t \\ge t_0) =\\psi(t)d_0 + d_w(t) \\rightarrow d_{m}(t) \\rightarrow d_d(t)\n\\end{equation}\nwhere $d_w(t)$ represents the entire detector immediately after a capture when only the window side of the detector is affected, and\n$d_d(t)$ represents the entire detector when the result of a capture has worked its way through to the display side of the\ndetector.   The middle state $d_{m}(t)$ represents the entire detector during stages in between, when the effects of the capture\nhave found their way into the interior of the detector, but not as far as the display. \n\nThe state $\\psi(t)$ is a free particle as a function of time, including all the incoming and scattered components.  It does no harm\nand it is convenient to let $\\psi(t)$ carry the total time dependence of the first component, and to let $d_0$ be normalized\nthroughout The first component in Eq.\\ 2 is a superposition of all possible scattered waves of $\\psi(t)$ in\nproduct with all possible recoil states of the ground state detector, so $d_0$ is a spread of detector states including all the\nrecoil possibilities together with their correlated environments.  Subsequent components are also superpositions of this kind. \nThey include all of the recoil components of the detector that have captured the particle. \n  \nThere is a clear discontinuity or ``quantum jump\" between the two components $d_0$ and $d_w$ at the detector's window interface. \nThis discontinuity is represented by a ``plus\" sign and can only be bridged by a stochastic jump.  The remaining evolution from\n$d_w(t)$ to $d_d(t)$ is connected by ``arrows\"  and is continuous and classical.  These three detector states develop in time and\nmay be represented by the single component \n\\begin{displaymath}\nd_1(t) = d_w(t) \\rightarrow d_{m}(t)\\rightarrow d_d(t)\n\\end{displaymath}\n so\n\\begin{equation}\n\\Phi(t \\ge t_0)=\\psi(t)d_0 + d_1(t) \n\\end{equation}\nThe capture state $d_1(t)$ in Eq.\\ 3 is equal to zero at $t_0$ and increases with time\\footnote{Each component in Eq.\\ 3 has an\nattached environmental term $E_0$ and $E_1$ that is not shown.  These are orthogonal to one another, insuring local decoherence. \nBut even though Eq.\\ 3 may be decoherent locally, we  assume that the macroscopic states $d_0$ and $d_1$ are fully coherent when\n$E_0$ and $E_1$ are included.  So Eq.\\ 3 and others like it in this paper are understood to be coherent when universally\nconsidered.  We  call them ``superpositions\", reflecting their global rather than their local\nproperties.}$^,$\\footnote{Superpositions of macroscopic states have been found at low temperatures \\cite{JRF}.  The components of\nthese states are locally coherent for a measurable period of time.}$^,$\\footnote{Equation  3 can be written with coefficients\n$c_0(t)$ and $c_1(t)$ giving $\\Phi(t \\ge t_0) = c_0(t)\\psi(t)d_0 + c_1(t)d_1$, where the states $\\psi(t)$, $d_0$, and\n$d_1$ are normalized throughout.  We let $c_0(t)\\psi(t)$ in this expression be equal to $\\psi(t)$ in Eq.\\ 3, and let $c_1(t)d_1$ be\nequal to $d_1(t)$ in Eq.\\ 3.}$^,$\\footnote{It is important to realize that\nthe interaction Hamiltonian can only connect $\\psi(t)d_0$ with the window state $d_w(t)$, which is the \\emph{launch state} of the\nactivated detector.  It cannot pour probability current directly into a state that is more dynamically advanced, like $d_m(t)$ or\n$d_d(t)$.  Therefore, at every moment during the current flow, a new state $d_w(t)$ is launched, since the state that was launched\nimmediately before that time has moved on (dynamically) to become $d_m(t)$. This means that the component $d_1(t)$ in Eq.\\ 3 is  a\nsuperposition of a continuum of all the detector states that have been launched at all previous times during the interaction.  For\nthe time being we will ignore this complication and come back to it (in Footnote 6) after we have examined this case from the point\nof view of the nRules.}. \n\n\n\n\n\n\n\n\n\\section*{Add an Observer}\nAssume that an observer is looking at the detector in Eq. 1 from the beginning.  \n\\begin{displaymath}\n\\Phi(t)=exp(-iHt)\\psi_i\\otimes D_iB_i\n\\end{displaymath}\nwhere $B_i$ is the observer's initial brain state that is entangled with the detector $D_i$.  This brain is understood to\ninclude \\emph{only} higher order brain parts -- that is, the physiology of the brain that is directly associated with consciousness\nafter all image processing is complete.  All lower order physiology leading to $B_i$ is assumed to be part of the detector.  The\ndetector is now represented by a capital $D$, indicating that it includes the bare detector  \\emph{plus} the low-level\nphysiology of the observer.\n\nFollowing the interaction between the particle and the detector, we  have\n\\begin{eqnarray}\n\\Phi(t \\ge t_0) &=& \\psi(t)D_0B_0 + D_w(t)B_0 \\rightarrow D_{m}(t)B_0 \\rightarrow D_d(t)B_1 \\hspace{.5cm}  \\\\\n\\mbox{or} \\hspace{.5cm} \\Phi(t \\ge t_0) &=& \\psi(t)D_0B_0 + D_1(t)B_1 \\nonumber\n\\end{eqnarray}\nwhere $B_0$ is the observer's brain when the detector is observed to be in its ground state $D_0$, and $B_1$ is the brain when the\ndetector is observed (only at the display end) to be in its capture state $D_1$.   As before, a discontinuous quantum jump is\nrepresented by a plus sign, and the continuous evolution of a single component is represented by an arrow.  \n\nIf the interaction is long lived compared to the time it takes for the signal to travel through the detector in Eq.\\ 4 (as in the\ncase of a long lived radioactive decay), then the superposition in that equation might exist for a long time before a capture.  \nThis means that there can be two active brain states of this observer in superposition, where one sees the detector in its ground\nstate and the other simultaneously sees the detector in its capture state.  Equation 4 therefore invites a paradoxical\ninterpretation like that associated with Schr\\\"{o}dinger's cat.  This ambiguity cannot be allowed.  It is not acceptable on\nempirical grounds.  An observer who watches a detector in these circumstances will \\emph{not} experience dual states of\nconsciousness.  \n\nThe nRules of this paper must not only provide for a stochastic trigger that gives rise to a state reduction, and describe that\nreduction, they must also insure than an empirical ambiguity of this kind will not occur.  \n\n\n\n\n\n\n\\section*{The nRules}\n\tThe first rule establishes the existence of a stochastic trigger.  This is a property of the system that has nothing to do with\nthe kind of interaction taking place or its representation.  Apart from making a choice, the trigger by itself has no effect on\nanything.  It initiates a state reduction only when it is combined with nRules 2 and 3.\n\n\\vspace{.4cm}\n\n\\noindent\n\\textbf{nRule (1)}: \\emph{For any subsystem of n complete components in a system having a total square modulus equal to s, the\nprobability per unit time of a stochastic choice of one of those components at time t is given by $(\\Sigma_nJ_n)\/s$, where\nthe net probability current $J_n$ going into the $n^{th}$ component at that time is positive.}\n\n\\noindent\n[\\textbf{note}: A \\emph{complete component} is a solution of Schr\\\"{o}dinger's equation that includes all of the (symmetrized)\nobjects in the universe.  It is made up of \\emph{complete states} of those objects  including all their state variables.  If\n$\\psi(x_1, x_2)$ is a two particle system with inseparable variables $x_1$ and $x_2$, then $\\psi$ is considered to be a single \nobject.  All such  objects are included in a complete component.  A component that is a sum of\nless than  the full range of a variable (such as a partial Fourier expansion) is not complete. This is why representation does not\nmatter to the stochastic trigger.]\n\n\\noindent\n[\\textbf{note}: Functions are not normalized in this treatment.  Instead, probability currents are normalized at each moment of\ntime by dividing $J$ by the value of $s$ at each moment of time. \n\n\\vspace{.4cm}\n\nThe second rule specifies the conditions under which \\emph{ready states} appear in solutions of Schr\\\"{o}dinger's equation.\nThese are understood to be the basis states of a state reduction. Ready states are always ``underlined\" in this treatment.\n\n\\vspace{.4cm}\n\n\\noindent\n\\textbf{nRule (2)}: \\emph{If a noncyclic interaction produces complete components that are discontinuous with the initial \ncomponent, then all of the new states that appear in these components will be ready states.}\n\n\\noindent\n[\\textbf{note}: A cyclic interaction between two component is one that produces continuous oscillations.  A \\emph{noncyclic\ninteraction} goes in one direction only.\n\n\\noindent\n[\\textbf{note}: Continuous means continuous in all variables. Although solutions to Schr\\\"{o}dinger's equation change\ncontinuously in time, they can be \\emph{discontinuous} in other variables -- e.g., the separation between the $n^{th}$ and the $(n +\n1)^{th}$ orbit of an atom with no orbits in between.  A discontinuity  can also exist between macroscopic states that are locally\ndecoherent.   For instance, the displaced detector states $d_0$ (ground state) and $d_w$ (window capture state) appearing in\n\\mbox{Eq.\\ 2} are discontinuous with respect to detector variables.  There is no state in between.  Like atomic orbits, these\ndetector states are a `quantum jump' apart.]\n\n\\noindent\n[\\textbf{note}: The \\emph{initial component} is the first complete component that appears in a given solution of Schr\\\"{o}dinger's\nequation.  A  solution is defined by a specific set of boundary conditions.  So Eqs.\\ 1 and 3 are both included in the single\nsolution that contains the discontinuity between $d_0$ and $d_1$, where Eq.\\ 1 (together with its complete environment) is the\ninitial state.  However, boundary conditions change with the collapse of the wave function.  The   component that\nsurvives a collapse will be complete, and will be the initial component of the new solution.]\n\n\\noindent\n[\\textbf{note}: If a noncyclic interaction does not produce complete components that are discontinuous with the\ninitial component, then the Hamiltonian will develop the state in the usual way, independent of these rules.]\n\n\\vspace{.4cm}\n\nThe collapse of a wave function and the change of a ready state to a \\emph{realized} state is provided for by nRule (3).  If a\ncomplete state is not `ready' it will be called `realized'.  We therefore introduce dual state categories where ready\nstates are the basis states of a collapse.  They are on stand-by, ready to be stochastically chosen and converted by nRule (3) to\nrealized states.  In this paper, realized states are \\emph{not} underlined.    \n\n\\vspace{.4cm}\n\n\\noindent\n\\textbf{nRule (3)}: \\emph{If a component is stochastically chosen during an interaction, then all of the ready states that result\nfrom that interaction (using nRule 2)  will become realized, and all other components in the superposition will be immediately\nreduced to zero.}\n\n\\noindent\n[\\textbf{note}: The claim of an immediate (i.e., discontinuous) reduction is the simplest possible way to describe the collapse\nof the state function.  A collapse is brought about by an instantaneous change in the boundary conditions of the\nSchr\\\"{o}dinger equation, rather than by the introduction of a new `continuous' mechanism of some kind.]\n\n\n\\noindent\n[\\textbf{note}:  This collapse does not preserve normalization.  That does not alter probability of subsequent\nreductions because of the way probability per unit time is defined in nRule (1); that is, current $J$ is divided by the total square\nmodulus.  Again, currents are normalized in this treatment -- not functions.]\n\n\n\\noindent\n[\\textbf{note}: If the stochastic trigger selects a component that does not contain ready states, then there will be no nRule (3) \nstate reduction.]\n\n\\vspace{.4cm}\n\nOnly positive current going into a \\emph{ready component} (i.e., a component containing ready states) is physically meaningful\nbecause it represents positive probability.  A negative current (coming out of a ready component) is not physically meaningful and\nis not allowed by nRule (4).  Without this restriction, probability current might flow in-and-out of one ready component and into\nanother.  The same probability current  would then be `used' and `reused'.  Given  the above rules, this would \ndistort the total probability of a process.  If the nRules are to work, the total integrated positive current (divided by $s$) must\nbe no greater than 1.0.  To insure this we say     \n\n\\vspace{.4cm}\n\n\n\\noindent\n\\textbf{nRule (4)}: \\emph{A ready component cannot transmit probability current.}\n\n\\vspace{.4cm}\nAlthough it can receive current that increases its square modulus, a ready state is dynamically terminal.  It cannot develop beyond\nitself.  If a ready state $\\underline{S}_1(t)$ evolves from a state $S_0(t)$, then Schr\\\"{o}dinger's equation $H[S_0(t) +\n\\underline{S}_1(t)] = i\\hbar\\frac{d}{dt}[S_0(t) + \\underline{S}_1(t)]$ will change $\\underline{S}_1(t)$ in the usual way. \nHowever, nRule (4) will prevent the creation of a second order component.  The state $\\underline{S}_1(t)$ is time dependent because\nits square modulus increases \\emph{and} because it reflects the dynamical changes  coming from $S_0(t)$ at every moment of time, \nbut  it does not advance dynamically on its own.  It is the launch state of a new solution of Schr\\\"{o}dinger's equation if and when\nit is stochastically chosen, and it will contain all the initial conditions of that solution.  Those initial conditions are not\napplied until the moment of choice. \n\nWhile no theoretical reason can be given to explain Rule(4), its important consequences are demonstrated throughout the rest of\nthis paper.\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Particle\/Detector Revisited}\nWhen the nRules are applied to the particle\/detector interaction, Eq.\\ 2 becomes \n\\begin{displaymath}\n\\Phi(t \\ge t_0) = \\psi(t)d_0 + \\underline{d}_w(t)\n\\end{displaymath}\nwhere the quantum jump (+ sign)  is discontinuous and noncyclic, where \\mbox{nRule (2)} requires that\n$\\underline{d}_w(t)$ is a ready state, and where the other components of the detector ($d_m$ and $d_d$) are zero because nRule (4)\nwill not allow $\\underline{d}_w(t)$ to pass current to them.  At $t_0$, the first component is maximum and the second is zero, but\n$\\underline{d}_w(t)$ increases in time because of the probability current flow from $\\psi(t)d_0$.  These components are\ncomplete because the environment of each (not shown) is assumed to be present.\n\n\tThe ready component in this equation is time dependent because of its increase in square modulus \\emph{and} because it duplicates\nthe moment-to-moment changes that occur within the detector (such as molecular changes) \\emph{although} it will not advance\ndynamically to $d_m$.  So long a $\\underline{d}_w(t)$ is a ready state, it will not pass current along to its successors.  As\nindicated above, it is a function that contains the boundary conditions of the next solution of Schr\\\"{o}dinger's equation -- that\nis, the `collapsed' solution that is realized when $\\underline{d}_w(t)$ is stochastically chosen.  So the effect of nRule (4) is to\nput the boundary conditions of the `next' solution on hold until there is a stochastic hit.  The nRules  then launch the new\nsolution with boundary conditions that are inherited at that moment from the old solution.  \n\n\t \tIf there is a stochastic hit at time $t_{sc}$, then a continuous classical evolution will give\n\\begin{displaymath}\n\\Phi(t\\ge t_{sc} > t_0)= d_w(t) \\rightarrow d_m(t) \\rightarrow d_d(t) \n\\end{displaymath}\nso the capture is eventually registered at the display end of the detector.\n\n\t If there is \\emph{no} stochastic hit on $\\underline{d}_w(t)$ it will become a \\emph{phantom} component.  A component is a phantom\nwhen there is no longer probability current flowing into it (in this case because the interaction is complete), and when there can\nbe no current flowing out of it because it is a ready component that complies with nRule (4).  A phantom component can be dropped\nout of the equation without consequence.  Doing so only changes the definition of the system -- it changes the total square modulus\n$s$ that normalizes the current in nRule (1).  This is the same kind of redefinition that occurs in standard practice when one\nchooses to renormalize a system at some new starting time.  Keeping a phantom is like keeping the initial system.  Because of nRule\n(3), kept phantoms are reduced to zero whenever another component is stochastically chosen.  \n\n\tThe nRules place this particle\/detector system in an ontological setting, for they give an \\emph{insider's} answer to the\nprobability question.  The nRules ask:  What is the probability that $\\underline{d}_w(t)$ will be stochastically chosen during the\nnext time interval $dt$; and then, what is the probability that it will be stochastically chosen during the time interval $dt$\nafter that, etc?  The nRules are always concerned with what happens next.  Probabilities that are associated with the second order\nstates like $d_m(t)$ and $d_d(t)$ are ruled out by nRule (4) until the fate of $\\underline{d}_w(t)$ is determined.  This is the\nmost important consequence of nRule (4) -- it does not allow a stochastic leap over the \\emph{next} (ready) state of the system. \nThe insider is concerned with the temporal ordering of states, and the nRules address that concern.\n\nIn contrast, the sRules ask:  What is the probability of finding $d_0$, $d_w$, $d_m$, or $d_d$ at some finite time $T$ after\n$t_0$?  This is an outsider's question.  It is the question asked by one who can only observer the system at distinct and finitely\nseparated times like $t_0$ and $T$.\n\n\n\n\n\n\n\n\n\n\n\\section*{Particle\/Detector\/Observer Revisited}\nTo see how the nRules carry out a particle capture when an observer is a witness, we apply them to the first row of Eq.\\ 4.  As\nbefore, this only affects the first two components\n\\begin{equation}\n\\Phi(t \\ge t_0)=\\psi(t)D_0B_0 + \\underline{D}_w(t)B_0 \n\\end{equation}\nbecause nRule (4) will not allow $\\underline{D}_w(t)$ to pass probability current to the other components.  Component\n$\\underline{D}_w(t)$ is `ready' because it is a discontinuous and noncyclic quantum jump away from $\\psi(t)D_0B_0$ and it is complete\nbecause it includes the (not shown) entangled environment.  Again, the time dependence of $\\underline{D}_w(t)$ does not\nrepresent a dynamical evolution beyond the changes given to it by the first component.  It does not evolve dynamically\non its own.  And again, the sub-0 on $B_0$ indicates an awareness of the ground state $D_0$ of the detector.  Since the second brain\nstate in Eq.\\ 5 is the same as the first, there is only one brain state $B_0$ in this superposition.  A cat-like ambiguity is\nthereby avoided.  Equation 5 now \\emph{replaces} Eq.\\ 4.   \n\n\tEquation 5 is the state of the system before there is a stochastic hit that produces a state reduction.  If there is a capture,\nthen there will be a stochastic hit on the second component of Eq.\\ 5 at a time $t_{sc}$. This will reduce the first component to\nzero according to nRule (3), and convert the ready state in the second component to a realized state.\n\\begin{displaymath}\n\\Phi(t = t_{sc} > t_0) = D_w(t)B_0\n\\end{displaymath}\nThe observer is still conscious of the detector's ground state in this equation because the capture has only affected the window end\nof the detector.  But after $t_{sc}$, a continuous evolution will produce\n\\begin{equation}\n\\Phi(t \\ge t_{sc} > t_0) = D_w(t)B_0 \\rightarrow D_{m}(t)B_0 \\rightarrow D_d(t)B_1\n\\end{equation}\nSince this equation represents a \\emph{single component} that evolves in time as shown, there is no time at which both $B_0$ and\n$B_1$ appear simultaneously.  There is therefore no cat-like ambiguity in this equation.\n\n\\vspace{.4cm}\n\n\tStandard quantum mechanics  gives us Eq.\\ 4 by the same logic that it gives us Schr\\\"{o}dinger's cat and\nEverett's many worlds.  \\mbox{Equation 4} (top or bottom row) is a single equation that simultaneously presents two different\nconscious brain states, resulting in an unacceptable ambiguity.  But with the nRules, each Schr\\\"{o}dinger solution is\nseparately grounded by its own stochastically selected boundary, allowing the rules to correctly and unambiguously predict the\ncontinuous experience of the observer in two stages.  This is accomplished by replacing `one' equation in \\mbox{Eq.\\ 4} with `two'\nequations in Eqs.\\ 5 and 6.   Equation 5 describes the state of the system \\emph{before} capture, and Eq.\\ 6 describes the state of\nthe system \\emph{after} capture.  Before and after are two \\emph{different} solutions to Schr\\\"{o}dinger's equation, specified by\ndifferent boundary conditions.  Remember we said that the stochastic trigger selects the new boundary that applies to the reduced\nstate.  So it is the stochastic event that separates the two solutions \\mbox{-- defining} before and after.  \n\n\t\tIf there is no stochastic hit on the second component in Eq.\\ 5 it will become a phantom component.  The new\nsystem is then just the first component of that equation.  This corresponds to the observer continuing to see the ground state\ndetector $D_0$, as he should in this \ncase\\footnote{Had the reasoning of Footnote 5 been applied to Eq.\\ 4, the component $D_1(t)B_1$\nwould be a superposition of the continuum of launch possibilities.  It would include a superposition of all the brain states $B_0$\nthat existed before the signal could have traveled through the detector to reach the brain, plus all the brain states $B_1$ that\nwere reached after that time.  This would have produced a massive cat-like paradox prior to a stochastic hit or state reduction of\nany kind.  However, the nRules also avoid this difficulty because they produce only \\emph{one} launch component\n$D_1(t_{sc})B_1$.}.\n\n\n\n\n\n\n\n\n\\section*{A Terminal Observation}\n\tAn observer who is inside a system must be able to confirm the validity of the Born rule that is normally applied from the\noutside.  To show this, suppose our observer is not aware of the detector during the interaction with the particle, but\nhe looks at the detector after  a time $t_f$ when the primary interaction is\ncomplete. Assume initial normalization equal to 1.0. During that interaction we have \n\\begin{equation}\n\\Phi(t_f > t \\ge t_0) = [\\psi(t)d_0 + \\underline{d}_w(t)]\\otimes X\n\\end{equation}\nwhere $X$ is the unknown state of the observer prior to the physiological \ninteraction. \n\nAssume there has \\emph{not} been a capture.  Then after the interaction is complete and before the observer looks at the detector\nwe have\n\\begin{displaymath}\n\\Phi(t \\ge t_f > t_0) = [\\psi(t)d_0 + \\underline{d}_w(t_f)]\\otimes X\n\\end{displaymath}\nwhere there is no longer a probability current flow inside the bracket. The second component in the bracket is therefore a\nphantom.  There is no current flowing into it, and none can flow out of it because of nRule (4).  So the equation is\nessentially   \n\\begin{displaymath}\n\\Phi(t \\ge t_f > t_0) = \\psi(t)d_0\\otimes X\n\\end{displaymath}\nWhen the observer finally observes the detector at $t_{ob}$ he will get\n\\begin{displaymath}\n\\Phi(t \\ge t_{ob} > t_f > t_0) = \\psi(t)d_0\\otimes X \\rightarrow \\psi(t)D_0B_0 \n\\end{displaymath}\nwhere the physiological process (represented by the arrow) carries $\\otimes X$ into $B_0$ and $d_0$ into $D_0$ by a continuous\nclassical progression leading from independence to entanglement.   This corresponds to the observer coming on board to witness\nthe detector in its ground state as he should when is no capture.  The probability of this happening (according to the sRules) is\nequal to the square modulus of $\\psi(t)d_0\\otimes X$ in Eq.\\ 7.\n\nIf the particle \\emph{is} captured during the primary interaction, there will be a stochastic hit on the second component inside\nthe bracket of Eq.\\ 7 at some time $t_{sc} < t_f$.  This results in a capture given by\n\\begin{displaymath}\n\\Phi(t_f > t = t_{sc} > t_0) = d_w(t)\\otimes X\n\\end{displaymath}\nafter which \n\\begin{displaymath}\n\\Phi(t_f > t \\ge  t_{sc} > t_0) = [d_w(t) \\rightarrow d_m(t) \\rightarrow d_d(t)]\\otimes X \n\\end{displaymath}\n\\begin{displaymath}\n\\Phi(t \\ge t_f >   t_{sc} > t_0) =  d_1(t)\\otimes X\n\\end{displaymath}\nas a result of the classical progression inside the detector.  When the observer does become aware of the detector at $t_{ob} > t_f$\nwe finally get\n\\begin{displaymath}\n\\Phi(t \\ge t_{ob} > t_f > t_{sc} > t_0) = d_1\\otimes X \\rightarrow D_1B_1\n\\end{displaymath}\nSo the observer comes on board to witness the detector in its capture state with a probability (according to the sRules) equal to\nthe square modulus of\n$\\underline{d}_w(t_f)\\otimes X$ in Eq.\\ 7.  The nRules therefore confirm the Born rule, in this case as a theorem.\n\n\n\n\n\n\n\\section*{An Intermediate Case}\n\t\tIn Eq.\\ 5 the observer is assumed to interact with the detector from the beginning.  Suppose that the incoming particle results\nfrom a long half-life decay, and that the observer's physiological involvement only \\emph{begins}  in the middle of the\nprimary interaction.  Before that time we will have\n\\begin{displaymath}\n\\Phi(t \\ge t_0) = [\\psi(t)d_0 + \\underline{d}_w(t)]\\otimes X\n\\end{displaymath}\nwhere again $X$ is the unknown brain state of the observer prior to the physiological interaction.  Primary probability current here\nflows between the detector components inside the bracket.  Let the physiological interaction begin at a time $t_{look}$ after\n$\\underline{d}_w(t)$ has gained some amplitude, and suppose it lasts for a period of time equal to $\\pi$.   Then\n\\begin{eqnarray}\n&\\Phi(t=t_{look} > t_0) = \\psi(t)d_0\\otimes X + \\underline{d}_w(t) \\otimes X&  \\\\\n&\\hspace{3.2cm}\\downarrow\\hspace{1.6cm}\\downarrow&\\nonumber\\\\\n&\\Phi(t=t_{look} + \\pi > t_{look} > t_0) = \\psi(t)D_0B_0 + \\underline{D}_w(t) B_0& \\nonumber\n\\end{eqnarray}\nwhere the arrows mean that the evolution from the first to the second row is classical and continuous. That evolution carries\n$\\otimes X$ into $B_0$ and $d_0$ into $D_0$ by a process that leads from independence to entanglement.  So while the second\ncomponent in each row gains square modulus because of the primary interaction (plus signs), the observer is simultaneously coming on\nboard by a continuous process (arrows).  \n\nThe second component at each moment of time during the physiological interaction in Eq.\\ 8 is the launch state of the new solution\nof the Schr\\\"{o}dinger equation -- if there is a stochastic hit at that moment.  This component establishes the boundary conditions\nof any  newly launched solution.   \n\nAfter the time $t_{ob} =  t_{look} + \\pi$ we can write Eq.\\ 8 as \n\\begin{equation}\n\\Phi(t \\ge t_{ob} > t_0) = \\psi(t)D_0B_0 + \\underline{D}_w(t)B_0\n\\end{equation}\nThis equation is identical with Eq.\\ 5; so from this moment on, it is as though the observer has been on board from the beginning.  \n\n  \tIf there is a subsequent capture at a time $t_{sc}$, this will become like Eq.\\ 6.\n\\begin{equation}\n\\Phi(t \\ge t_{sc} > t_{ob} > t_0) = D_w(t)B_0 \\rightarrow  D_m(t)B_0 \\rightarrow D_d(t)B_0\n\\end{equation}\n\n\tIf a stochastic hit occurs between $t_{look}$ and $t_{ob}$, then the ready component at that moment (the second component in Eq.\\\n8) will be chosen and made a realized state.  It will then proceed classically and continuously to $D_dB_1$ as in Eq.\\ 10. \n\n\n\n\n\n\n\n\n\n\\section*{A Second Observer}\nIf a second observer is standing by while the first observer interacts with the detector during the primary interaction, the state\nof the system will be\n\\begin{displaymath}\n\\Phi(t \\ge  t_0) = [\\psi(t)D_0B_0 + \\underline{D}_w(t)B_0]\\otimes X\n\\end{displaymath}\nwhere $X$ is an unknown state of the second observer prior to his interacting with the system.  The detector $D$ here includes the\nlow-level physiology of the first observer.  A further expansion of the detector will include the second observer's low-level\nphysiology when he comes on board.  The detector will then split into two parallel paths, one connecting to the first observer and\nthe other  connecting to the second observer.    When a product of brain states appears in the form $BB$ or $B\\otimes X$, the\nfirst term will refer to the first observer and the second to the second observer.  \n\t\nThe result of the second observer looking at the detector will be the same as that found for the first observer in the previous\nsection, except now the first observer will be present in each case.  In particular, the equations similar to \\mbox{Eqs.\\ 8, 9, and\n10} are now \n\\begin{eqnarray}\n&\\Phi(t=t_{look} + t) = \\psi(t)d_0\\otimes X + \\underline{d}_w(t) \\otimes X \\rightarrow& \\nonumber\\\\\n &\\Phi(t = t_{look} + \\pi > t_{look} > t_0) = [\\psi(t)D_0B_0B_0 + \\underline{D}_w(t)B_0B_0]&  \\nonumber\\\\\n&\\Phi(t \\ge t_{ob}  > t_0) = \\psi(t)D_0B_0B_0 + \\underline{D}_w(t)B_0B_0&  \\nonumber \\\\\n&\\Phi(t \\ge t_{sc} > t_{ob}  > t_0) = D_w(t)B_0B_0 \\rightarrow D_m(t)B_0B_0 \\rightarrow D_d(t)B_1B_1&  \\nonumber \n\\end{eqnarray}\nThese will all yield the same result for the new observer as they did for the old observer.  In no case will the nRules produce a\nresult like $B_1B_2$ or $B_2B_1$.\n\n\\vspace{.4cm}\n\nUp to this point we have seen how the nRules go about including observers inside a system in an ontological model.  These rules\ndescribe when and how the observer becomes conscious of measuring instruments, and replicate common empirical experience in these\nsituations. The nRules are also successfully applied in another paper \\cite{RM5}  where two versions of the Schr\\\"{o}dinger cat\nexperiment are examined.  In the first version a conscious cat is made unconscious by a stochastically initiated process; and in the\nsecond version an unconscious cat is made conscious by a stochastically initiated process. \n\nIn the following sections we turn attention to another problem -- the requirement that macroscopic states must appear in\ntheir normal sequence.   This sequencing chore represents a major application of nRule (4) that is best illustrated in the case of a\nmacroscopic counter.\n\n\n\n\n\n\n\n\n\\section*{A Counter}\n\nIf a beta counter that is exposed to a radioactive source is turned on at time $t_0$, its state function will be given by\n\\begin{displaymath}\n\\Phi(t \\ge   t_0) = C_0(t) + \\underline{C}_1(t)\\\n\\end{displaymath}\nwhere $C_0$ is a counter that reads zero counts, $C_1$ reads one count, and $C_2$ (not shown) reads two counts, etc.  The second\ncomponent $\\underline{C}_1(t)$ is zero at $t_0$ and increases in time.  The underline indicates that it is a ready state as\nrequired by nRule (2).  $C_2$ and higher states do not appear in this equation because \\mbox{nRule (4)} forbids current to leave\n$\\underline{C}_1(t)$.  Ignore the time required for the capture effects to go from the window to the display end of the counter.  \n\nThe $0^{th}$ and the $1^{st}$ components are the only ones that are initially active, where the current flow is $J_{01}$\nfrom the $0^{th}$ to the $1^{st}$ component.  The resulting distribution at some time $t$ before $t_{sc}$ is shown in Fig.\\ 3,\nwhere $t_{sc}$ is the time of a stochastic hit on the second component.  \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.7]{aXnRulesFig3.eps}\n\\center{Figure 3: Square moduli before the first stochastic hit}\n\\end{figure}\n\nThis means that the $1^{st}$ component \\emph{will} be chosen because all of the current from the (say,\nnormalized) $0^{th}$ component will pore into the $1^{st}$ component making $\\int J_{01}dt = 1.0$.  Following the stochastic hit\non the $1^{st}$ component, there will be a collapse to that component because of \\mbox{nRule (3)}.  The first two dial readings\nwill therefore be sequential, going from 0 to 1 without skipping a step such as going directly from 0 to 2.  It is nRule (4) that\nenforces the no-skip behavior of a counter; for without it, any component in the superposition might be chosen as a result of\nprobability current flowing into it.    It is empirically mandated that these states should always follow in sequence without\nskipping a step. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=1.2]{aXnRulesFig4.eps}\n\\center{Figure 4: Square moduli before second hit}\n\\end{figure}\n\nWith the stochastic choice of the $1^{st}$ component at $t_{sc}$, the process will begin again as shown in \nFig.\\ 4.  This  leads with certainty to a stochastic choice of the $2^{nd}$ component.  That certainty is accomplished by the\nwording of nRule (1) that requires current normalization at each moment of time; that is, the  current  $J_{12}$ is divided by the\ntotal square modulus at each moment.  The total integral $\\int J_{12}dt$  is less than 1.0 in  Fig.\\ 4 because of the reduction that\noccurred in \\mbox{Fig.\\ 3};  but it is restored to 1.0 when divided by the (new) total square modulus.  It is therefore certain that\nthe $2^{nd}$ component will be chosen.\n\nAnd finally, with the choice of the $2^{nd}$ component, the process will resume again with current $J_{23}\/s$ going from the\n$2^{nd}$ to the $3^{rd}$ component.  This leads with certainty to a stochastic choice of the $3^{rd}$ component.  \n\n If an observer watches the counter from the beginning he will be able to see it go sequentially from $C_0$ to $C_1$, to $C_2$,\netc., for he is himself a continuous part of the system.  He does not have to `peek' intermittently like an\nepistemological observer who is not part of the system as has been said.  Although the empirical experience of the ontological\n(nRule) observer is different from the epistemological (sRule) observer, there is no contradiction between the two.  That's because\nthe nRules and the sRules answer two different questions about probability as previously noted.  The nRules ask: What is the\nprobability that the system will jump to the \\emph{next} counter state in the next differentially small increment of time; and the\nsRules ask: What it the probability that an outside observer will find the system in any one of its possible counter states if he\nlooks after some finite time $T$?  \n\n\n\n\n\n\n\n\n\n\\section*{The Parallel Case} \nNow imagine parallel states in which a quantum process may go either clockwise or counterclockwise as shown in Fig. 5.  Each\ncomponent includes a macroscopic piece of laboratory apparatus $A$, where the Hamiltonian provides for a discontinuous clockwise\ninteraction going from the $0^{th}$ to the $r^{th}$ state, and another one going from there to the final state $f$; as well as a\ncomparable counterclockwise interaction from the $0^{th}$ to the $l^{th}$ state and from there to the final \\mbox{state $f$}.  The\nHamiltonian does not provide a direct route from the $0^{th}$ to the final state, so the system will choose stochastically between a\nclockwise and a counterclockwise route.  Ready states $\\underline{A}_l$ and $\\underline{A}_r$ are the \\emph{eigenstates} of that\nchoice, and contain the boundary conditions for each separate path.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.8]{aXnRulesFig5.eps}\n\\center{Figure 5: Possible parallel decay routes}\n\\end{figure}\n\nWith nRule (4) in place, probability current cannot initially flow from either of the intermediate states to the final state, for\nthat would require current flow from a ready state.  The dashed lines in Fig.\\ 5 indicate the forbidden\ntransitions.  But once the state $\\underline{A}_l$ (or $\\underline{A}_r$) has been stochastically chosen, it will become a\nrealized state $A_l$ (or $A_r$) and a subsequent transition to $\\underline{A}_f$ can occur that realizes $A_f$.  \n\nThe effect of nRule (4) is therefore to force this macroscopic system into a classical sequence that goes either clockwise or\ncounterclockwise.  Without it, the system might make a  second order transition (through one of the intermediate states) to the final\nstate, without the intermediate state being realized.  The observer would then see the initial state followed by the final state,\nwithout knowing (in principle) which pathway was followed.  This is a familiar property of continuous microscopic evolution.  In\nHeisenberg's famous example formalized by Feynman, a microscopic particle observed at point $a$ and later at point $b$ will travel\nover a quantum mechanical superposition of all possible paths in between.  Without nRule (4), macroscopic objects facing\ndiscontinuous and noncyclic parallel choices would do the same thing.  But that should not occur.  The fourth nRule forces this\nparallel system into one or the other classical path, so it is not a quantum mechanical superposition of both paths.  \n\n\n\n\n\n\n\n\n\n\\section*{A Continuous Variable}\nIn the above examples nRule (4) guarantees that sequential discontinuous steps in a superposition are not passed over. \nIf the variable itself is classical and continuous, then continuous observation is possible without the\nnecessity of stochastic jumps.  In that case we do not need \\mbox{nRule (4)} or any of the \\mbox{nRules (1-4)}, for they do not\nprevent or in any way qualify the motion.  \n\nHowever, a classical variable may require a quantum mechanical jump-start.  For instance, the\nmechanical device that is used to  seal the fate of Schr\\\"{o}dinger's cat (e.g., a falling hammer) begins its motion with a\nstochastic hit.  That is, the decision to begin the motion (or not) is left to a $\\beta$-decay.  In this case \\mbox{nRule (4)}\nforces the motion to begin at the beginning, insuring that no value of the classical variable is passed over;  so the hammer\nwill always fall from its \\emph{initial} angle with the horizontal.   Without nRule (4), the hammer might begin its fall at some\nother angle because probability current will flow into angles other than the initial one.  \n\n\n\n\n\n\n\n\\section*{Microscopic Systems}\nThe discussion so far has been limited to experiments or procedures whose outcome is empirically known.  Our claim has been that the\nnRules are chosen to work without regard to a theory as to `why' they work.  Therefore, our attention has always gone to\nmacroscopic situations in which the results are directly available to conscious experience.  However, if the nRules are correct we\nwould also want know how they apply to microscopic systems, even though the predicted results in these cases are more\nspeculative.  In this section we will consider the implications of the nRules in three microscopic cases.  The important question to\nask in each case is: Under what circumstances will these rules result in a state reduction of a microscopic system?\n\n\n\n\\begin{displaymath}\n\\mbox{\\textbf{Case 1 -- spin states}}\n\\end{displaymath}\n\nNo state reduction will result from changing the representation.    In particular, replacing the\nspin state $+z$ with the sum of states $+x$ and $-x$ will not result in either one of the $x$-states becoming a ready state.  \n\n\tThis will be true for a spin $+z$ particle even if the common environment of $+x$ and $-x$ includes a magnetic field that is\ncontinuously applied in (say) the $x$-direction.  So long as the magnetic field  is the same for both $+x$ and $-x$, the result will\nbe the same (i.e., neither one will become a ready state).  \n\nMore generally, let the environments $E^p(t)$ of $+x$ and $E^n(t)$ of $-x$ change continuously over time such that\n\\begin{equation}\n\\Phi(\\tau > t \\ge t_0) = \\frac{1}{\\sqrt{2}}(+z)E^p(t) + \\frac{1}{\\sqrt{2}}(-z)E^n(t)\n\\end{equation}\nwhere $E^p(0) = E^n(0) = E$, and $E^p(\\tau) = E_+,  E^n(\\tau) = E_-$.  States $E_+$ and $E_-$ are the final environments of $+x$\nand $-x$ at a time $\\tau$ that will be either the time at which the process is complete, or  complete to some\ndesired extent.  It may be that the environments $E_+$ and $E_-$ are \\emph{similar} to $E$ and to each other (i.e., same temperature\nand pressure, same particles, same radiation field, etc.), but time will change $E_+$ and $E_-$ so they can no longer be\n\\emph{identical} with $E$ or with each other.  Of course, it might also be that $E_+$ and $E_-$ are not even similar \\mbox{to E}.\n\nTaking the first and last term in Eq.\\ 11 we write\n\\begin{eqnarray}\n\\Phi(\\tau > t \\ge t_0) &=& \\frac{1}{\\sqrt{2}}[(+z) + (-z)]E \\rightarrow \\frac{1}{\\sqrt{2}}[(+z)E_+ + (-z)E_-] \\\\\n&&   \\hspace{1cm}  t =t_ 0  \\hspace{2.8cm} t = \\tau  \\nonumber\n\\end{eqnarray}\nwhere the arrow indicates a continuous transition. Therefore, there will be \\emph{no state reduction} in this case.  It may be\nnoncyclic, but this process does not lead to a collapse of the wave to either $+x$ or $-x$.   \n\n\tFor example, if the magnetic field is non-homogeneous there will be a physical separation of the $+x$ and $-x$ states.  One will\nmove into a stronger magnetic field and the other will move into a weaker magnetic field.  Assuming that this field is continuously\n(i.e., classically) applied, there will be no ready state and no state reduction.  Ready states will appear only when the $+x$ and\n$-x$ components are picked up by different detectors at different locations, resulting in a \\emph{detector related} discontinuity. \nAccording to nRule (3), a state reduction can only occur if and when that happens.\n\n\tThere are many other `quantum' processes that are continuous and therefore not subject to stochastic reduction, such\nas scattering, interference, diffraction, and tunneling.  Vacuum fluctuations are cyclic. \n\n\\begin{displaymath}\n\\mbox{\\textbf{Case 2 -- Free neutron decay}}\n\\end{displaymath}\n\n\nA free neutron decay is written $\\Phi(t) = n(t) + \\underline{ep\\overline{\\nu}}(t)$, where the second component is zero at $t = 0$\nand increases in time as probability current flows into it.  This component contains three entangled particles  making a whole \nobject, where all three  are `ready' states as indicated by the underline \\mbox{(see nRule 2)}. Each component  is multiplied by a\nterm representing the environment (not shown).  Each  is complete for this reason and also because the variables of each particle\ntake on all of the values that are allowed by the boundary conditions.  Following \\mbox{nRule (3)} there will be a\nstochastic hit on $\\underline{ep\\overline{\\nu}(t)}$ at some time $t_{sc}$, reducing the system to the realized correlated states\n$ep\\overline{\\nu}(t_{sc})$.  \n\nSpecific values of, say, the electron's momentum are not stochastically chosen by this reduction because all possible values of\nmomentum are included in $ep\\overline{\\nu}(t_{sc})$ and its subsequent evolution.  For the electron's momentum to be determined in\na specific direction away from the decay site, a detector in that direction must be activated.  That will require another stochastic\nhit on the component that includes that detector.   \n\nThis case provides a good example of how  $\\underline{ep\\overline{\\nu}}(t)$ is a function time beyond its increase in square\nmodulus.  Assume that the neutron moves across the laboratory in a wave packet of finite width.  At each moment the ready component\n$\\underline{ep\\overline{\\nu}}(t)$ will ride with the packet, having the same size, shape, and group velocity.  It is the launch\ncomponent that contains the boundary conditions of the next solution of Schr\\\"{o}dinger's equation -- the solution that appears when\n$\\underline{ep\\overline{\\nu}}(t)$ is stochastically chosen at $t_{sc}$.  Before this `collapse', $\\underline{ep\\overline{\\nu}}(t)$\nis time dependent because it increases in square modulus \\emph{and} because it follows the motion of the neutron; however, nRule (4)\ninsures that it will not evolve dynamically beyond itself until it becomes a realized component at the time of  stochastic choice. \nThe neutron\n$n(t)$ will then disappear and the separate particles $e(t)p(t)\\overline{\\nu}(t)$ will spread out on their own, still\ncorrelated in conserved quantities.   \n\n\\begin{displaymath}\n\\mbox{\\textbf{Case 3 -- Atomic absorption and emission}}\n\\end{displaymath}\n\nIf an atom is raised to an excited state by a passing photons, the absorption part of this interaction is cyclic because the atom\nmight fall back to its ground state by a stimulated emission.  However, the excited state atom might also emit `another' photon by\nsimultaneous emission and that is noncyclic.  That could lead to a stochastic hit and collapse of the wave to the newly formed decay\nstate.  The  pulse of $N$ photons is represented by $\\gamma_N(t)$ in the first component of Eq.\\ 13, and the spontaneously emitted\nphoton is $\\underline{\\gamma}(t)$ in the third component. \n\n\\begin{equation} \n\\Phi(t \\ge t_0) = \\gamma_N(t)A_0(t) \\leftrightarrow \\gamma_{N-1}(t)A_1(t)+ \\gamma_{N-1}(t)\\underline{\\gamma}(t)\\underline{A}_0(t)\n\\end{equation}\nThe continuous (cyclic) oscillation is represented by the reversible arrow $\\leftrightarrow$ between the first and second\ncomponents.\n\nIf the incoming photons have a small cross section with the atom, the second component in Eq.\\ 13 will not fully discharge into the\nthird component.  In that case the latter will become a phantom that can be subsequently ignored.  There will be no collapse.  The\nsecond component will then dampen out and the first component, modified by the encounter, will be the only survivor.   \n\nOtherwise, there will be a stochastic hit on the third component at some time $t_{sc}$, resulting in \n\\begin{displaymath}\n\\Phi(t \\ge t_{sc} < t_0) = \\gamma_{N-1}(t)\\gamma(t)A_0(t)\n\\end{displaymath}  \nThe newly emitted photon $\\gamma(t)$ will evolve until it has a final pulse width $\\Delta T_f$ that is\nassociated with its final spread of energy $\\Delta E$.  There is no necessary relationship between $\\Delta T_f$ and the half-life\n$T_{1\/2}$ of the decay from the second to the third component in Eq.\\ 13.  There might be any number of influences affecting\n$T_{1\/2}$, including the number of photons in the stimulating pulse $\\gamma_N(t)$.  If there are no incoming photons, and if the\nexcited state $A_1(t)$ is realized by a quantum jump from a higher energy level at some stochastically chosen moment $t_{scc}$,\nthen \n\\begin{equation} \n\\Phi(t \\ge t_{scc}) = A_1(t)+ \\underline{\\gamma}(t)\\underline{A}_0(t)\n\\end{equation}\nwhere the second component is zero at $t_{scc}$ and increases in time.  The state $A_1(t)$  decays \\emph{only} into the\nsecond component in this equation; whereas in \\mbox{Eq.\\ 13} it shares time with the initial state\n$\\gamma_N(t)A_0(t)$.  In Eq. 13 the  average current flow into $\\underline{\\gamma}(t)\\underline{A}_0(t)$ is therefore decreased,\nso the half-life $T_{1\/2}$ of the spontaneous decay is increased.\n\nIn contrast, $\\Delta T_f$ is determined only by the time dependent solutions of the excited state $A_1(t)$.  So $\\Delta T_f$ is\nnot causally connected to $T_{1\/2}$.  The value of $\\Delta T_f$ is transmitted from $A_1(t)$ to the launch component\n$\\underline{\\gamma} (t)\\underline{A}_0(t)$ at each moment of time, setting the boundary conditions of that launch site at that\nmoment.  Therefore, it will not matter to the properties of the emitted photon (in particular $\\Delta T_f$) if a stochastic hit\non $\\underline{\\gamma}(t)\\underline{A}_0(t)$ in Eq.\\ 13 or Eq.\\ 14 aborts the decay before it is complete.\n\n\n\n\n\n\n\\section*{Decoherence}\nSuppose that two states $A$ and $B$ that are initially in coherent Rabi oscillation and are exposed to a phase disrupting\nenvironment.  This may be expressed by the equation\n\\begin{equation}\n\\Phi(\\tau > t \\ge t_0) = (A \\leftrightarrow B)E(t) \\rightarrow [AE_A(t) + BE_B(\\tau)] \n\\end{equation}\nwhere $A$ is at a higher energy level than $B$.  The right pointing arrow indicates a continuous process.  When the environments\n$E_A$ and $E_B$ are sufficiently different to be orthogonal, then states $A$ and $B$ in Eq.\\ 15 will become totally decoherent. \nStatistically, this leaves a local mixture of $A$ and $B$ with no current flow between them.  The subsequent \ndecay of $AE_A$ is generally much slower than decoherence, so decoherence will be essentially complete before there is a\nstochastic interruption.  Evidence for this is found in low temperature experiments \\cite{DV, YY} where Rabi oscillations undergo\ndecoherent decay without any sign of interruption due to state reduction.     \n\n\\vspace{.4cm}\n\nThe ammonia molecule is generally found in a partially decoherent state.  In a rarified atmosphere the molecule will most likely be\nin its symmetric coherent form $(A\\leftrightarrow B)$, where $A$ has the nitrogen atom on one side of the\nhydrogen plane, and $B$ has the nitrogen atom placed symmetrically on the other side.  This is the lowest energy state available to\nthe molecule.  In this case the states $A$ and $B$ (Eq.\\ 15) taken by themselves are equally energetic at a higher energy level.  \n  \nCollisions with other molecules in the environment will tend to destroy the coherence between $A$ and $B$, causing the ammonia\nmolecule to become decoherent to some extent.  This decoherence can be reversed by decreasing the pressure.  Since an ammonia\nmolecule wants to fall into its lowest energy level, it will tend to return to the symmetric  state when outside pressure\nis reduced.  In general, equilibrium can be found between a given environment and some degree of decoherence.    \n\n\tThe ammonia molecule  cannot assume the symmetric form $(A \\leftrightarrow B)$ if the environmental collisions are too frequent\n-- i.e., if the pressure exceeds about 0.5 atm.\\ at room temperature \\cite{JZ}.   At low pressures the molecule is a stable coherent\nsystem, and at high pressures it is a stable decoherent system.  It seems to change from a microscopic object to a macroscopic\nobject as a function of its environment.  This further supports the idea that the micro\/macroscopic distinction is not\nfundamental.  \n\n\\vspace{.4cm}\n\nTo accommodate the main example of this paper, which is a detector that may or may not capture a\nparticle, a more general form is adopted.  We now use time dependent coefficients $m(t)$ and $n(t)$, where $m(t_0) = 1, n(t_0) = 0$,\nand where $m(t)$ decreases in time keeping\n$m(t)^2 + n(t)^2 = 1$.  These coefficients describe the progress of the primary interaction, giving  \n\\begin{eqnarray}\n\\Phi(\\tau > t \\ge t_0) &=& [m(t)A + n(t)\\underline{B}]E \\rightarrow [m(t)AE_A(t) + n(t)\\underline{B}E_B(t)] \\nonumber\\\\\n&&   \\hspace{.95cm}  t = t_0  \\hspace{3.3cm} t = \\tau \\nonumber\n\\end{eqnarray} \nwhere the transitions \\emph{inside} the brackets are now discontinuous and noncyclc, making $\\underline{B}$ a ready state as\nrequired by nRule (2).  This can be written\n\\begin{eqnarray}\n\\Phi(t = t_0) &=& AE \\hspace{.5cm}\\mbox{(intial  component)}\\hspace{1.8cm} m(t_0) = 1 \\\\\n\\Phi(\\tau \\ge t \\ge t_0) &=&  AE \\rightarrow [m(t)AE^m(t) + n(t)\\underline{B}E^n(t)]\\hspace{.4cm}  m(t) \\rightarrow 0 \n\\nonumber\n\\end{eqnarray} \nSince state $\\underline{B}$ in this equation is a `ready' state and  the  primary current flows from $A$ to\n$\\underline{B}$ inside the  square bracket, $\\underline{B}$ is a candidate for state reduction according to nRule (3).  \n\n\tEquation 16 applied to our example in Eq.\\ 3 with $m(t)A = \\psi(t)d_0$ and $n(t)\\underline{B} = \\underline{d}_w(t)$ says that \ninitially coherent states $\\psi(t)d_0$ and $\\underline{d}_w$ rapidly become decoherent.  Because of the macroscopic nature of the\ndetector, decoherence at time $\\tau$ may be assumed to be complete, and the decay time from $t_0$ to $\\tau$ extremely short lived. \nThe  time for any newly created pair of macroscopic objects to approach full decoherence is so brief that it is not measurable in\npractice.  Still, we see that decoherence does not happen immediately when a new macroscopic state is  created.\n\n\n\n\n\n\n\n\\section*{Grounding the Schr\\\"{o}dinger Solutions}\n\tStandard quantum mechanics is not completely grounded because it does not recognize all of the boundary conditions (beyond \nthe initial conditions) that are stochastically chosen during the lifetime of the system.  With either one of the proposed\nrule-sets, every stochastic hit sets a new boundary (i.e., the chosen eigenvalue) for a new solution of Schr\\\"{o}dinger's equation. \nOn the other hand, traditional quantum theory accumulates all the possible solutions as though they were all simultaneously valid;\nand as a result, this model encourages bizarre speculations such as the many-world interpretation of Everett or the cat paradox of\nSchr\\\"{o}dinger.  With the proposed rule-sets, these empirical distortions disappear.  It is because standard quantum mechanics\nfails to respond to the system's ongoing state reductions that these fanciful excursions seem plausible.  \n\n\n\n\n\n\n\n\\section*{Limitation of the Born Rule}\nUsing the Born rule in standard theory, the observer can only record an observation at a given instant of time, and he must do so\nconsistently over an ensemble of observations.  He cannot himself become part of the system for any finite period of\ntime.  When discussing the Zeno effect it is said that continuous observation can be simulated by rapidly increasing the number of\ninstantaneous observations; but of course, that is not really  continuous.    \n\n\t On the other hand, the observer in an ontological model can \\emph{only} be continuously involved with the observed system.  Once\nhe is on board and fully conscious of a system, the observer can certainly try to remove himself ``immediately\".  However, that\neffort is not likely to result in a truly instantaneous conscious observation.    So the epistemological observer claims to make\ninstantaneous observations but cannot make continuous ones; and the ontological observer makes continuous observations but cannot\n(in practice) make instantaneous ones.  Evidently the Born rule would require the ontological observer to do something that cannot\nbe realistically done.  Epistemologically we can ignore this difficulty, but a consistent ontology should not match a continuous\nphysical process with continuous observation by using a discontinuous rule of correspondence.  Therefore, no ontological model\nshould  make fundamental use of the Born interpretation that places unrealistic demands on an observer.  \n\n\n\n\n\n\n\n\\section*{Status of the Rules}\nNo attempt has been made to relate conscious brain states to particular neurological configurations.  The nRules are an empirically\ndiscovered set of macro relationships that exist on another level than microphysiology, and there is no need to connect these two\ndomains.  These rules preside over physiological detail in the same way that thermodynamics presides over molecular detail.  It is\ndesirable to eventually connect these domains as thermodynamics is now connected to molecular motion; and hopefully, this is what a\ncovering theory will do.  But for the present we are left to investigate the rules by themselves without the benefit of a wider\ntheoretical understanding of state reduction or of conscious systems.  There are two rule-sets of this kind giving us two different\nquantum friendly ontologies -- the nRules of this paper and the oRules of Refs. 1-3.  \n  \n\\hspace{.4cm}\n\nThe question is, which if either of these two rule-sets is correct (or most correct)?  Without the availability of a wider\ntheoretical structure or a discriminating observation, there is no way to tell.  Reduction theories that are currently being\nconsidered may accommodate a conscious observer, but none are fully accepted.  So the search goes on for an extension of quantum\nmechanics that is sufficiently comprehensive to cover the collapse associated with an individual measurement.  I expect that\nany such theory will support one of the ontological rule-sets, so these rules might server as a guide for the construction of a\n\\mbox{wider theory}.  \n\n\n\n\\pagebreak\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}