On the functional formulation of quantum field theory

Abstract

This paper describes a functional formulation of Euclidean quantum field theory, based on a complete equivalence with classical statistical mechanics. One introduces an extra time variable and sets up a canonical scheme with new Lagrangian and Hamiltonian functions. The generating functional is then defined as the Gibbs average over the ensemble. This allows us, in particular, to control in a simple way the invariance properties of the integration measure. In several cases of physical interest it is seen that the invariance requirement leads to extra determinantal factors in the integration volume and, therefore, to a set of improved Feynman rules. In particular, enforcing dilatation invariance for the generating functional is shown to lead to a nonzero background,i.e. to spontaneous breaking of the symmetry. The application of the method to constrained systems is discussed in detail and in the case of Yang-Mills theories the Faddeev-Popov prescription for quantization is reproduced with remarkable simplicity. A discussion of the functional quantization of gravity is also offered.