Stochastically positive structures on Weyl algebras. The case of quasi-free states

Abstract

We consider quasi-free stochastically positive ground and thermal states on Weyl algebras in Euclidean time formulation. In particular, we obtain a new derivation of a general form of thermal quasi-free state and give conditions when such state is stochastically positive i.e. when it defines periodic stochastic process with respect to Euclidean time, so called thermal process. Then we show that thermal process completely determines modular structure canonically associated with quasifree thermal state on Weyl algebra. We discuss a variety of examples connected with free quantum field theories on globally hyperbolic stationary space-times and models of quantum statistical mechanics.

Some Models of Nonideal Bose Gas With Bose-Einstein Condensate

“…Additionally those commutative analysis methods open the doors for applications of the methods of classical statistical mechanics for studying the perturbations of the free thermal structures by the so called thermo-eld like perturbations. Such programme was initiated in 9, 10] and developed in certain directions 11,12]. The present contribution continues the analysis of 9, 10] for the so called gentle perturbations of the free Bose Matter.…”

“…In a series of papers 9,10,11,12] new mathematical technologies for studying the nonrelativistic Bose Matter at thermal equilibrium and in nonzero temperature were invented. They are based on the observation that the modular structure of the free Bose Matter has the so called stochastic positivity property 11].…”

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“…It is important to point out here that the canonical procedure of reconstructing the whole noncommutative modular structure of the free Bose matter from the process ( ) cr τ τ ξ ξ is well known [8,10,11]. It is the non-ergodicity of the process ξ that reflects the presence of Bose-Einstein condensate (see, e.g., [3,5] in the critical case: the spontaneous breaking of U(l)-gauge invariance).…”