Application of the Feynman Integral for Approximate Calculations

Abstract

In this chapter we discuss a semi-classical approximation in quantum mechanics and quantum field theory. We restrict ourselves to a formal expansion in $$\hbar $$ resulting in a direct way from the path integral. We formulate the semi-classical approximation as a stationary phase method of the calculation of integrals. We perform such an integral for harmonic and anharmonic oscillators. In application to QFT we show that the result of the stationary phase method can be considered as a resummation of the Feynman diagrams with one and more loops (depending on the power of the small parameter $$\hbar $$ ). In Euclidean field theory the corresponding method of computations can be associated with the Laplace method of an approximate calculation of integrals. The result of calculations by the Laplace method can be expressed as an effective action. The effective action in the first order of perturbation can be represented by a determinant of a differential operator. In this chapter the determinants are studied first in a perturbation expansion and next in an exact expression by means of the heat kernel using the Feynman-Kac path integral representation of the heat kernel.