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Universal Journal of Physics and Application

Bogolyubov Gaussian Measure in Quantum Statistical Mechanics

Дмитрий Петрович Санкович

Published Mar 1, 2019• 0 citations
Publisher

Abstract

The first steps in the application of methods for integrating functions defined on abstract sets were taken by Wiener. Most widely, the ideas of functional integration were developed in Feynman's works. The Feynman continual integral is well known to a wide community of physicists. Along with this, there is another approach to the construction of a functional integral in quantum physics. This approach was proposed by Bogolyubov. Bogolyubov's methods are relevant in quantum statistical physics, and have natural ties with probability theory. We review some mathematical results of integration with respect to a special Gaussian measure that arises in the statistical theory for quantum systems. It is shown that the Gibbs equilibrium averages of the chronological products of Bose operators can be represented as functional integrals with respect to this measure (the Bogolyubov measure). Some properties of this measure are studied. We rewrite partition function of many particle Bose systems in terms of Bogolyubov functional integral.

DOI
10.13189/ujpa.2019.130201

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The first steps in the application of methods for integrating functions defined on abstract sets were taken by Wiener. Most widely, the ideas of functional integration were developed in Feynman's works. The Feynman continual integral is well known to a wide community of physicists. Along with this, there is another approach to the construction of a functional integral in quantum physics. This approach was proposed by Bogolyubov. Bogolyubov's methods are relevant in quantum statistical physics, and have natural ties with probability theory. We review some mathematical results of integration with respect to a special Gaussian measure that arises in the statistical theory for quantum systems. It is shown that the Gibbs equilibrium averages of the chronological products of Bose operators can be represented as functional integrals with respect to this measure (the Bogolyubov measure). Some properties of this measure are studied. We rewrite partition function of many particle Bose systems in terms of Bogolyubov functional integral.
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.
This translation of the survey article by I. M. Gel'fand and A. M. Yaglom on the theory and applications of integration in functional spaces in problems of quantum physics was prepared because it was felt that such a review would be of interest and of use to mathematical physicists working in several different fields. The article begins with a discussion of Wiener measure, after which the extension is made to the complex measure introduced by Feynman in his formulation of quantum mechanics, and examples are given of the use of these methods in quantum mechanics, quantum field theory, and quantum statistical physics. A comprehensive bibliography of works devoted to the theory and applications of functional integration methods is included.
In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.
Quantum statistical mechanics has developed primarily through two approaches, pioneered by Gibbs and Feynman, respectively. In Gibbs' method one calculates partition functions from phase-space integrations or sums over stationary states. Alternatively, in Feynman's approach, the focus is on the path-integral formulation. The Hubbard-Stratonovich transformation leads to a functional-integral formulation for calculating partition functions. We outline here the functional integral approach to quantum statistical mechanics, including generalizations and improvements to Hubbard's formulation. We show how the dimensionality of the integrals is reduced exactly, how the problem of assuming an unknown canonical transformation is avoided, how the reality of the partition function in the complex representation is guaranteed, and how the extremum conditions are simplified. This formulation can be applied to general systems, including superconductors.
This analysis of the survey paper by I. M. and A. M. Gel'fand. M. Yaglom was prepared on integration theory and implementations in the practical space in the problems of quantum physics since he thought that such an analysis would be interesting and beneficial for mathematical physicists employed in various fields.It starts with a description of the Wiener test, after which the complex test proposed by Feynman is broadened to provide descriptions of the application of these measures in quantum mechanics, quantum fields and quantum statistical physics. The article offers explanations of these measures.