Irreducibility of dynamics and representation of KMS states in terms of Lévy processes

Abstract

Quantum systems described by the Schrödinger operators H = N j =1(p j ) + W (x 1 , . . . , x N ), p j = −ı∇ j , x j ∈ R ν with being continuous functions such that the pseudodifferential operators (p j ) generate Lévy processes, are considered. It is proven that the linear span of the operators α t 1 (F 1 ) · · · α t n (F n ) is dense in the algebra of all observables in the σ -strong and hence in the σ -weak and strong topologies. Here α t (F ) = exp(ıtH)F exp(−ıtH) are time automorphisms and the F 's are taken from families of multiplication operators obeying conditions described in the paper. This result implies that a linear functional continuous in either of these topologies is fully determined by its values on such products. In the case of KMS states this yields a representation of such states in terms of path integrals.1. Introduction. Gibbs (equilibrium) states of infinite-particle quantum systems described by unbounded Schrödinger operators cannot be constructed directly as KuboMartin-Schwinger (KMS) states. The only way here is to use path integrals to represent local Gibbs states (describing finite subsystems) in terms of probability measures. Then the Gibbs states of the whole system are constructed as probability measures with the help of the DLR technique known in classical statistical mechanics, see [5]. For models like quantum crystals, the corresponding approach is well elaborated, see e.g., [1]. In the present article we make first steps in developing a similar technique for a wider class of quantum systems by proving the basic statement for representing local Gibbs states in terms of probability measures.We consider a system of interacting quantum particles with the kinetic energy of a particle being a pseudo-differential operator (p), p = −ı∇. Such operators appear if one 'quantizes' the relativistic expression [p 2 c 2 + m 2 c 4 ] 1/2 − mc 2 and were used in particular in studying problems of stability of matter, see [12]-[14] and the references therein. At the same time, such Schrödinger operators generate stochastic processes, see [4], [8], which

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