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At least 253 records · Page 14Linked to original sources

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

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↗

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↗

arXiv1997

Magnetoresonances on a lasso graph

We consider a charged spinless quantum particle confined to a graph consisting of a loop to which a halfline lead is attached; this system is placed into a homogeneous magnetic field perpendicular to the loop plane. We derive the reflection amplitude and show that there is an infinite ladder of resonances; analyzing the resonance pole trajectories we show that half of them turn into true embedded eigenvalues provided the flux through the loop is an integer or halfinteger multiple of the flux unit $hc/e$. We also describe a general method to solve the scattering problem on graphs of which the present model is a simple particular case. Finally, we discuss ways in which a state localized initially at the loop decays.

quant-ph↗

arXiv1996

$q$-Gaussian processes: non-commutative and classical aspects

We examine, for $-1<q<1$, $q$-Gaussian processes, i.e. families of operators (non-commutative random variables) $X_t=a_t+a_t^*$ -- where the $a_t$ fulfill the $q$-commutation relations $a_sa_t^*-qa_t^*a_s=c(s,t)\cdot \id$ for some covariance function $c(\cdot,\cdot)$ -- equipped with the vacuum expectation state. We show that there is a $q$-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on $q$-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of $q$-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret \cite{FB}.

funct-an↗

arXiv1997

An Analogue of the Kac-Wakimoto Formula and Black Hole Conditional Entropy

A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac and Wakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the Connes-Stoermer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function the logarithm of an integer once the initial state or the final state is taken fixed.

gr-qc↗

arXiv1997

Pure E_0-semigroups and absorbing states

An E_0-semigroup acting on B(H) is called pure if the intersection of the ranges $α_t(B(H))$, $t>0$, is the algebra of scalars. We determine all pure E_0-semigroups which have a weakly continuous invariant state $ω$ and which are minimal in an appropriate sense. In such cases, for every normal state $ρ$ there is convergence to equilibrium in the trace norm $\lim_{t\to\infty}\| ρ\circα_t - ω\| = 0.$ A normal state $ω$ with this property is called an absorbing state. Such E_0-semigroups mest be cocycle perturbations of CAR/CCR flows, and we develop systematic methods for constructing those perturbations which have absorbing states with prescribed finite eigenvalue lists.

funct-an↗

arXiv1997

Elliptic algebra $A_{q,p}(\hat{sl_2})$ in the scaling limit

The scaling limit $A_{\hbar,η}(\hat{sl_2})$ of the elliptic algebra $A_{q,p}(\hat{sl_2})$ is investigated. The limiting algebra is defined in terms of a continuous family of generators being Fourier harmonics of Gauss coordinates of the $L$-operator. Ding-Frenkel isomorphism between $L$-operator's and current descriptions of the algebra $A_{\hbar,η}(\hat{sl_2})$ is established and is identified with the Riemann problem on a strip. The representations, coalgebraic structure and intertwining operators of the algebra are studied.

q-alg↗

arXiv1997

Asymptotics of a Class of Solutions to the Cylindrical Toda Equations

The small t asymptotics of a class of solutions to the 2D cylindrical Toda equations is computed. The solutions, q_k(t), have the representation q_k(t) = log det(I-lambda K_k) - log det(I-lambda K_{k-1}) where K_k are integral operators. This class includes the n-periodic cylindrical Toda equations. For n=2 our results reduce to the previously computed asymptotics of the 2D radial sinh-Gordon equation and for n=3 (and with an additional symmetry contraint) they reduce to earlier results for the radial Bullough-Dodd equation.

solv-int↗

arXiv1997

Extensions of Conformal Nets and Superselection Structures

Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Moebius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselsection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2,R), showing that they violate 3-regularity for n>2. When n>1, we obtain examples of non Moebius-covariant sectors of a 3-regular (non 4-regular) net.

hep-th↗

arXiv1997

Covariant Sectors with Infinite Dimension and Positivity of the Energy

We consider a Moebius covariant sector, possibly with infinite dimension, of a local conformal net of von Neumann algebras on the circle. If the sector has finite index, it has automatically positive energy. In the infinite index case, we show the spectrum of the energy always to contain the positive real line, but, as seen by an example, it may contain negative values. We then consider nets with Haag duality on the real line, or equivalently sectors with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely this is the case iff there exists an unbounded Moebius covariant left inverse. As a consequence the class of sectors with positive energy is stable under composition, conjugation and direct integral decomposition.

funct-an↗

arXiv1997

$O_\infty$ Realized on Bose Fock Space

We study the semigroup of Bogoliubov endomorphisms of the canonical commutation relations which give rise to representations of the Cuntz algebra $O_\infty$ on Fock space and describe the corresponding Cuntz algebra generators in detail.

math-ph↗

arXiv1997

Orbifold subfactors from Hecke algebras II --- Quantum doubles and braiding ---

A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M_infinity-M_infinity bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A_2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M_infinity-M_infinity bimodules are described by certain orbifolds (with ghosts) for SU(3)_3k. We actually compute several examples of the (dual) principal graphs of the asymptotic inclusions. As a corollary of the identification of Ocneanu's orbifolds with ours, we show that a non-degenerate braiding exists on the even vertices of D_2n, n>2.

funct-an↗

arXiv1993

Hidden algebra of the $N$-body Calogero problem

A certain generalization of the algebra $gl(N,{\bf R})$ of first-order differential operators acting on a space of inhomogeneous polynomials in ${\bf R}^{N-1}$ is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the $N$-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. Given representation implies that the Calogero Hamiltonian possesses infinitely-many, finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators.

hep-th↗

arXiv1995

Algebraization of difference eigenvalue equations related to $U_q(sl_2)$

A class of second order difference (discrete) operators with a partial algebraization of the spectrum is introduced. The eigenfuncions of the algebraized part of the spectrum are polinomials (discrete polinomials). Such difference operators can be constructed by means of $U_q(sl_2)$, the quantum deformation of the $sl_2$ algebra. The roots of polinomials determine the spectrum and obey the Bethe Ansatz equations. A particular case of difference equations for $q$-hypergeometric and Askey-Wilson polinomials is discussed. Applications to the problem of Bloch electrons in magnetic field are outlined. {abstract}

cond-mat↗