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.
Highlights
This article discusses some work on the space-time or integral over trajectories method in quantum mechanics, considering in particular its application to problems in statistical mechanics and its relation to the theory of a system of interacting bosons
We evaluate the average in the right-hand side of the last relation using the T -product definition (4), which leads to the formula
Let us apply formulas (13) and (14) in order to establish a relation between the Gibbs equilibrium averages of Bose operators and the functional integral
Summary
This article discusses some work on the space-time or integral over trajectories method in quantum mechanics, considering in particular its application to problems in statistical mechanics and its relation to the theory of a system of interacting bosons. The Daniell theory is based on a family H(X) of elementary functions h(x) on a set X with an elementary integral I(h) defined for them Under certain conditions, this family can be extended to a broader family L to which the integral I is extended so that L becomes a Banach space with the norm φ = I |φ|. The idea of expressing physical observables as continual integrals was developed in quantum field theory for representing Green functions. Two methods for such a representation appeared almost simultaneously. Bogolyubov returned to this construction within the framework of statistical mechanics when investigating a polaron model [17] It was shown in [18] that the measure arising in the Bogolyubov approach is the Gaussian measure in an appropriate space of continuous functions.
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