The Bogolyubov Functional Integral

Problems of integration with respect to a special Gaussian measure (the Bogolyubov measure) that arises in the statistical equilibrium theory for quantum systems are considered. It is shown that the Gibbs equilibrium means of Bose operators can be represented as functional integrals with respect to this measure. Certain functional integrals with respect to the Bogolyubov measure are calculated. Approximate formulas are constructed that are exact for functional polynomials of a given degree, as well as formulas that are exact for integrable functionals of a wider class. The nondifferentiability of Bogolyubov trajectories in the corresponding function space is established. A theorem on the quadratic variation of trajectories is proved. The properties of scale transformations that follow from this theorem are studied. Examples of semigroups associated with the Bogolyubov measure are constructed. Independent increments for this measure are found. A relation between the Bogolyubov measure and parabolic partial differential equations is considered. An inequality for traces is proved, and an upper estimate is obtained for the Gibbs equilibrium mean of the square of the coordinate operator in the case of a one-dimensional nonlinear oscillator with a positive symmetric interaction.