Path Integrals in Field Theory

Abstract

Up to now we have investigated systems possessing one (q) or several (q _α α = 1,...,D) degrees of freedom and studied their quantization using path integrals. In this chapter we will pass over to (relativistic) field theories, starting with the case of a scalar neutral field ø(x). When discussing canonical quantization we already noticed that the field function ø(x, t) can be viewed as a generalized coordinate vector q _i (t) depending on the “continuous index” x in the place of the discrete index i. Thus a field is a system with an infinite number of degrees of freedom. Its dynamics is described by a Lagrange function which in local field theories can be written as an integral over the Lagrange density:

$$L = \int {{d^3}} xL(\phi (x),{\partial _\mu }\phi (x))$$

(12.1)

.