Bogolyubov Measure in Quantum Equilibrium Statistical Mechanics

TL;DR

This paper explores the application of functional integration methods in quantum Bose-systems, representing Gibbs averages as path integrals over a special non-Wiener Gaussian measure called the Bogolyubov measure, and investigates its properties and related equations.

Contribution

It introduces the Bogolyubov measure in the context of quantum statistical mechanics, analyzes its properties, and develops approximation formulas and connections to PDEs.

Findings

Representation of Gibbs averages as path integrals over the Bogolyubov measure.

Proof of nondifferentiability and quadratic variation of Bogolyubov trajectories.

Identification of independent increments and relation to parabolic PDEs.

Abstract

Application of the functional integration methods in equilibrium statistical mechanics of quantum Bose-systems is considered. We show that Gibbs equilibrium averages of Bose-operators can be represented as path integrals over a special Gauss measure defined in the corresponding space of continuous functions. This measure arises in the Bogolyubov T-product approach and is non-Wiener. We consider problems related to integration with respect to the Bogolyubov measure in the space of continuous functions and calculate some functional integrals with respect to this measure. Approximate formulas that are exact for functional polynomials of a given degree and also some formulas that are exact for integrable functionals belonging to a broader class are constructed. We establish the nondifferentiability of the Bogolyubov trajectories in the corresponding function space and prove a theorem on the quadratic variation of trajectories and study the properties implied by this theorem for the scale transformations. We construct some examples of semigroups related to the Bogolyubov measure. Independent increments are found for this measure. We consider the relation between the Bogolyubov measure and parabolic partial differential equations. An inequality for some traces is proved.