In addition to the acoustic (phonon) excitations described above, specific excitations called ‘quantum vortices’ may exist in superfluid Bose systems.

Attempts to incorporate the vortex excitations have already been made in the original version of Landau's theory (Landau, 1941). The existence of a periodical lattice of quantum vortices in a superconductor located in a magnetic field was conjectured by Abrikosov (1952, 1957). The idea of quantum vortices occurring in rotating helium below the λ-point was introduced independently by Onsager (1949) and Feynman (1955).

In this section we present the description of quantum vortices in the functional integral formalism. We will also discuss the role of quantum vortices in the phase transition for the superfluid to the normal state.

Our starting point is the functional integration scheme over fast and slow variables. We will apply this formalism mainly to two-dimensional Bose systems and will try to take into account vortex-like configurations of the Bose field. We thus come to the conclusion that the system of vortices is equivalent to the system of charged particles interacting via a new field which play the role of electromagnetic field. At small temperatures there are only pairs with opposite signs, bounded by the long-range logarithm potential. It turns out that the phase transition in the two-dimensional Bose system is nothing but the dissociation of bounded pairs. A similar approach to the three-dimensional systems leads to the conclusion that phase transition is accompanied here by creation of long vortex filaments.

In this section we will develop an approach to the microscopic theory of periodic structures in the framework of the functional integration method. This approach was suggested by Kapitonov & Popov (1981) and was developed by Andrianov, Kapitonov & Popov (1982, 1983).

Our starting point will be a system of electrons and ions with the Coulomb interaction. The properties of crystals are determined by the collective excitations (phonons). Clearly, a microscopic theory must describe phonons and their interactions starting from the system of electrons and ions. The functional integral method allows us to realize this aim. The main idea is to go from the initial action of electrons and ions to the effective action functional in terms of the electric potential field φ(x, τ). This field has an immediate physical meaning and provides the collective variable we need.

We can find the static field φ0(x), corresponding to the crystalline structure from the stationary condition for the effective action functional Seff[φ]. If φ0(x) is known we can consider small fluctuations in the vicinity of the stationary point of Seff. In order to do this, we have to expand Seff in this neighbourhood and to separate the quadratic form of φ(x, τ) − φ0(x). It is this quadratic form that defines the spectrum of collective excitations. Forms of the third and higher degrees describe the interaction of these excitations.

The model described below is immediately applicable to the description of metallic hydrogen.

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.