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arXiv1994

Spectral stochastic processes arising in quantum mechanical models with a non-L2 ground state

A functional integral representation is given for a large class of quantum mechanical models with a non--L2 ground state. As a prototype the particle in a periodic potential is discussed: a unique ground state is shown to exist as a state on the Weyl algebra, and a functional measure (spectral stochastic process) is constructed on trajectories taking values in the spectrum of the maximal abelian subalgebra of the Weyl algebra isomorphic to the algebra of almost periodic functions. The thermodynamical limit of the finite volume functional integrals for such models is discussed, and the superselection sectors associated to an observable subalgebra of the Weyl algebra are described in terms of boundary conditions and/or topological terms in the finite volume measures.

hep-th↗

arXiv1994

Orthogonal Decomposition of Some Affine Lie Algebras in Terms of their Heisenberg Subalgebras

In the present note we suggest an affinization of a theorem by Kostrikin et.al. about the decomposition of some complex simple Lie algebras ${\cal G}$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac-Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r, \, C_{2^m}, D_r,\, G_2,\, E_7,\, E_8$ can be decomposed into the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed.

hep-th↗

arXiv1996

Removal of the Energy Dependence from the Resolvent-like Energy-Dependent Interactions

The spectral problem $(A + V(z))ψ=zψ$ is considered with $A$, a self-adjoint Hamiltonian of sufficiently arbitrary nature. The perturbation $V(z)$ is assumed to depend on the energy $z$ as resolvent of another self-adjoint operator $A':$ $V(z)=-B(A'-z)^{-1}B^{*}$. It is supposed that operator $B$ has a finite Hilbert-Schmidt norm and spectra of operators $A$ and $A'$ are separated. The conditions are formulated when the perturbation $V(z)$ may be replaced with an energy-independent ``potential'' $W$ such that the Hamiltonian $H=A +W$ has the same spectrum (more exactly a part of spectrum) and the same eigenfunctions as the initial spectral problem. The orthogonality and expansion theorems are proved for eigenfunction systems of the Hamiltonian $ H=A + W $. Scattering theory is developed for $H$ in the case when operator $A$ has continuous spectrum. Applications of the results obtained to few-body problems are discussed.

nucl-th↗

arXiv1996

Representations for Three-Body T-Matrix on Unphysical Sheets

Explicit representations are formulated for the Faddeev components of three-body T-matrix continued analytically on unphysical sheets of the energy Riemann surface. According to the representations, the T-matrix on unphysical sheets is obviously expressed in terms of its components taken on the physical sheet only. The representations for T-matrix are used then to construct similar representations for analytical continuation of three-body scattering matrices and resolvent. Domains on unphysical sheets are described where the representations obtained can be applied.

nucl-th↗

arXiv1996

Representations for Three-Body T-Matrix on Unphysical Sheets: Proofs

A proof is given for the explicit representations which have been formulated in the author's previous work (nucl-th/9505028) for the Faddeev components of three-body T-matrix continued analytically on unphysical sheets of the energy Riemann surface. Also, the analogous representations for analytical continuation of the three-body scattering matrices and resolvent are proved. An algorithm to search for the three-body resonances on the base of the Faddeev differential equations is discussed.

nucl-th↗

arXiv1993

Modular Structure and Duality in Conformal Quantum Field Theory

Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and the vacuum vector concides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description is valid for the algebras associated with double cones. Moreover essential duality holds on the Minkowski space $M$, and Haag duality for double cones holds provided the net of local algebras is extended to a pre-cosheaf on the superworld $\tilde M$, i.e. the universal covering of the Dirac-Weyl compactification of $M$. As a consequence a PCT symmetry exists for any algebraic conformal field theory in even space-time dimension. Analogous results hold for a Poincaré covariant theory provided the modular groups corresponding to wedge algebras have the expected geometrical meaning and the split property is satisfied. In particular the Poincaré representation is unique in this case.

funct-an↗

arXiv1994

Non-Local Matrix Generalizations of W-Algebras

There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinary linear differential operators of order $m$, $L = -d^m + U_1 d^{m-1} + U_2 d^{m-2} + \ldots + U_m$. In this paper, I consider in detail the case where the $U_k$ are $n\times n$-matrix-valued functions, with particular emphasis on the (more interesting) second Gelfand-Dikii bracket. Of particular interest is the reduction to the symplectic submanifold $U_1=0$. This reduction gives rise to matrix generalizations of (the classical version of) the {\it non-linear} $W_m$-algebras, called $V_{m,n}$-algebras. The non-commutativity of the matrices leads to {\it non-local} terms in these $V_{m,n}$-algebras. I show that these algebras contain a conformal Virasoro subalgebra and that combinations $W_k$ of the $U_k$ can be formed that are $n\times n$-matrices of conformally primary fields of spin $k$, in analogy with the scalar case $n=1$. In general however, the $V_{m,n}$-algebras have a much richer structure than the $W_m$-algebras as can be seen on the examples of the {\it non-linear} and {\it non-local} Poisson brackets of any two matrix elements of $U_2$ or $W_3$ which I work out explicitly for all $m$ and $n$. A matrix Miura transformation is derived, mapping these complicated second Gelfand-Dikii brackets of the $U_k$ to a set of much simpler Poisson brackets, providing the analogue of the free-field realization of the $W_m$-algebras.

hep-th↗

arXiv1994

Transparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy Dispersion Relations for the Fast Decaying Potentials

For the two-dimensional Schrödinger equation $$ [- Δ+v(x)]ψ=Eψ,\ x\in \R^2,\ E=E_{fixed}>0 \ \ \ \ \ (*)$$ at a fixed positive energy with a fast decaying at infinity potential $v(x)$ dispersion relations on the scattering data are given.Under "small norm" assumption using these dispersion relations we give (without a complete proof of sufficiency) a characterization of scattering data for the potentials from the Schwartz class $S=C_{\infty}^{(\infty)} (\hbox{\bf R}^2).$ For the potentials with zero scattering amplitude at a fixed energy $\scriptstyle E_{fixed}$ (transparent potentials) we give a complete proof of this characterization. As a consequence we construct a family (parameterized by a function of one variable) of two-dimensional spherically-symmetric real potentials from the Schwartz class $S$ transparent at a given energy. For the two-dimensional case (without assumption that the potential is small) we show that there are no nonzero real exponentially decreasing at infinity, potentials transparent at a fixed energy. For any dimension greater or equal 1 we prove that there are no nonzero real potentials with zero forward scattering amplitude at an energy interval. We show that KdV-type equations in dimension 2+1 related with the scattering problem $(*)$ (the Novikov-Veselov equations) do not preserve, in general, these dispersion relations starting from the second one. As a corollary these equations do not preserve, in general , the decay rate faster then $|x|^{-3}$ for initial data from the Schwartz class.

solv-int↗

arXiv1994

Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$

For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras $gl(N)$, $N\to\infty$.

hep-th↗

arXiv1993

Coherent States of the q--Canonical Commutation Relations

For the $q$-deformed canonical commutation relations $a(f)a^\dagger(g) = (1-q)\,\langle f,g\rangle{\bf1}+q\,a^\dagger(g)a(f)$ for $f,g$ in some Hilbert space ${\cal H}$ we consider representations generated from a vector $Ω$ satisfying $a(f)Ω=\langle f,ϕ\rangleΩ$, where $ϕ\in{\cal H}$. We show that such a representation exists if and only if $\Vertϕ\Vert\leq1$. Moreover, for $\Vertϕ\Vert<1$ these representations are unitarily equivalent to the Fock representation (obtained for $ϕ=0$). On the other hand representations obtained for different unit vectors $ϕ$ are disjoint. We show that the universal C*-algebra for the relations has a largest proper, closed, two-sided ideal. The quotient by this ideal is a natural $q$-analogue of the Cuntz algebra (obtained for $q=0$). We discuss the Conjecture that, for $d<\infty$, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting cases $q=\pm1$ we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.

funct-an↗

arXiv1994

Schwinger Terms and Cohomology of Pseudodifferential Operators

We study the cohomology of the Schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators in 3+1 dimensions of this type, the Schwinger term is equivalent to the ``twisted'' Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we also show how the ordinary Radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component.

hep-th↗

arXiv1995

The Conformal Spin and Statistics Theorem

We prove the equality between the statistics phase and the conformal univalence for a superselection sector with finite index in Conformal Quantum Field Theory on $S^1$. A relevant point is the description of the PCT symmetry and the construction of the global conjugate charge.

hep-th↗

arXiv1995

An Algebraic Spin and Statistics Theorem

A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional space-time. Our main assumption is the requirement that the modular groups of the von Neumann algebras of local observables associated with wedge regions act geometrically as pure Lorentz transformations. Such a property, satisfied by the local algebras generated by Wightman fields because of the Bisognano-Wichmann theorem, is regarded as a natural primitive assumption.

funct-an↗

arXiv1995

On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schrödinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. We prove the inequality \sum_i|E_i(H)|^γ\leq L_{γ,1}\int_{\Bbb R} V^{γ+1/2}(x)dx, (1) for the "limit" case $γ=1/2.$ This will imply improved estimates for the best constants $L_{γ,1}$ in (1), as $1/2<γ<3/2.

quant-ph↗

arXiv1994

Rigorous mean field model for CPA: Anderson model with free random variables

A model of a randomly disordered system with site-diagonal random energy fluctuations is introduced. It is an extension of Wegner's $n$-orbital model to arbitrary eigenvalue distribution in the electronic level space. The new feature is that the random energy values are not assumed to be independent at different sites but free. Freeness of random variables is an analogue of the concept of independence for non-commuting random operators. A possible realization is the ensemble of at different lattice-sites randomly rotated matrices. The one- and two-particle Green functions of the proposed hamiltonian are calculated exactly. The eigenstates are extended and the conductivity is nonvanishing everywhere inside the band. The long-range behaviour and the zero-frequency limit of the two-particle Green function are universal with respect to the eigenvalue distribution in the electronic level space. The solutions solve the CPA-equation for the one- and two-particle Green function of the corresponding Anderson model. Thus our (multi-site) model is a rigorous mean field model for the (single-site) CPA. We show how the Llyod model is included in our model and treat various kinds of noises.

cond-mat↗

arXiv1997

Fourier Duality as a Quantization Principle

The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes essential use of Fourier duality. The extension of this property to more general phase spaces requires the use of Kac algebras, which provide the necessary background for the implementation of Fourier duality on general locally compact groups. Kac algebras -- and the duality they incorporate -- are consequently examined as candidates for a general quantization framework extending the usual formalism. Using as a test case the simplest non-trivial phase space, the half-plane, it is shown how the structures present in the complete-plane case must be modified. Traces, for example, must be replaced by their noncommutative generalizations - weights - and the correspondence embodied in the Weyl-Wigner formalism is no more complete. Provided the underlying algebraic structure is suitably adapted to each case, Fourier duality is shown to be indeed a very powerful guide to the quantization of general physical systems.

funct-an↗

arXiv1997

Projective Fourier Duality and Weyl Quantization

The Weyl-Wigner correspondence prescription, which makes large use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. An Abelian and a symmetric projective Kac algebras are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyl formula arises naturally as an irreducible component of the duality mapping between these projective algebras.

funct-an↗

arXiv1995

Lefschetz Numbers and Geometry of Operators in W*-modules

The main goal of the present paper is to generalize the results of~\cite{TroLNM,TroBoch} in the following way: To be able to define $K_0(A)ø\C$-valued Lefschetz numbers of the first type of an endomorphism $V$ on a C*-elliptic complex one usually assumes that $V=T_g$ for some representation $T_g$ of a compact group $G$ on the C*-elliptic complex. We try to refuse this restriction in the present paper. The price to pay for this is twofold: (i) $ $ We have to define Lefschetz numbers valued in some larger group as $K_0(A)ø\C$. (ii) We have to deal with W*-algebras instead of general unital C*-algebras. To obtain these results we have got a number of by-product facts on the theory of Hilbert W*- and C*-modules and on bounded module operators on them which are of independent interest.

funct-an↗