Self-consistent form of the random phase approximation

A method of approximate calculation of statistical sum is suggested which is based on the representation of the latter in the form of the path integral and improving the random phase approximation. Essential in the improving is the using of the Finberg method for the evaluation of the Green functions in external field. In contrast to the usual random phase approximation the scheme presented is self-consistent. The method is illustrated by the example of the transition into superconducting state. The equation formulating the self-consistency condition is investigated. It is shown that values of the critical indices coincide with those for the ideal Bose gas and the spherical model.