Book review

Path integrals were introduced into theoretical physics by Feynman, and now they are recognized as a powerful tool for constructing perturbative expansions and semiclassical approximations in quantum mechanics and, especially, in quantum field theory. Actually, the integral over the system trajectories, or more generally, the functional integral representation for generating functional of Wightman functions (correlators), is the most elegant way to get Feynman graphs for transition probability amplitudes. The functional method has got a special significance after Faddeev and Popov (the author of the present book) proposed, in 1967, a ghost-field representation for the quantized gauge field theories. That discovery opened the way to progress in modern particle physics, which is based essentially upon non-Abelian gauge fields.

It was also found that the functional integral representation can be used successfully in statistical physics of many-body systems, in particular, for construction of temperature correlators (the Matzubara functions). The approach was based upon the fact that the equilibrium Gibbs state is described by means of the density operator which is given by exponential of the Hamiltonian, like the time evolution operator, where time has to be substituted for (Ņi/T), where T stands for the temperature. The power of the method was brilliantly demonstrated in Feynman's lectures on statistical physics ("Statistical Mechanics" by R. P. Feynman, W.A. Benjamin, Inc., Reading, 1972).

The book by Popov contains a collection of elegant solutions for a number of problems in statistical physics, obtained by means of the functional integral method. The approach is uniform: starting from the integral representation, one has to change the variables, selecting an essential direction in the functional space (a collective coordinate) and evaluating the other integrals approximately. The equivalent approach in the operator formalism has been known as the method of collective excitations. The following problems have been considered: i) weakly interacting Bose systems, including superfluidity and vortices, ii) weakly interacting Fermi systems and the superconductivity, including plasma dynamics and the theory of fluid 3He, iii) spectra of many-electron systems in crystals and atoms, including the theory of quantum crystals.

There is a dozen of books on theoretical physics treating various aspects of the functional integral method. "Techniques and Applications of Path Integrals" by L.S. Schulman (J. Wiley, New York, 1981) is probably the most fundamental work published in that field in the 80-ties. With all that, the monograph by Popov is an important contribution because of its practical character and the abundance of concrete applications of the method of functional integration. It is a significant addition to Feynman's "Statistical Mechanics", as it presents results obtained recently. The book is especially valuable, since a good portion of it is based upon original results by the author and his co-workers, some of which have been never published in English before.

It was a good idea to print the relatively cheap paperback edition, to the benefit of physicists, students in particular, working in the theory of condensed matter.