Nowadays functional integrals are used in various branches of theoretical physics, and may be regarded as an ‘integral calculus’ of modern physics. Solutions of differential or functional equations arising in diffusion theory, quantum mechanics, quantum field theory and quantum statistical mechanics can be written in the form of functional integrals.

Functional integral methods are widely applied in quantum field theory, especially in gauge fields. There exist numerous interesting applications of functional integrals to the study of infrared and ultraviolet asymptotic behaviour of Green's functions in quantum field theory and also to the theory of extended objects (vortex-like excitations, solitons, instantons).

In statistical physics functional methods are very useful in problems dealing with collective modes (long-wave phonons and quantum vortices in superfluids and superconductors, plasma oscillations in systems of charged particles, collective modes in 3He-type systems and so on).

This book is devoted to some applications of functional integrals for describing collective excitations in statistical physics. The main idea is to go in the functional integral from the initial variables to some new fields corresponding to ‘collective’ degrees of freedom. The choice of specific examples is to a large extent determined by the scientific interests of the author.

We dwell on modifications of the functional integral scheme developed for the description of collective modes, such as longwave phonons and quantum vortices in superfluids, superconductors and 3He, plasma oscillations in systems of charged particles.