ABSTRACT
The extension of the formalism based on path integrals to the description of gauge theories, such as QED, involves a non trivial problem, arising from fact that the field tensor and the Lagrangian density of the theory, and by consequence any observable quantity described by a functional https://www.w3.org/1998/Math/MathML" display="inline"> O [ A μ ] , are left unchanged by gauge transformations of the fields. This invariance implies that there is an infinite set of field configurations for which the integrand of a path integral involving https://www.w3.org/1998/Math/MathML" display="inline"> O [ A μ ] remains the same. Therefore, the results of the integration turns out to be infinite. Conceptually, this difficulty can be circumvented by a change of variables which, given a field configuration https://www.w3.org/1998/Math/MathML" display="inline"> A μ , allows to distinguish the contribution of all configurations—referred to as gauge trajectories—that can be obtained from https://www.w3.org/1998/Math/MathML" display="inline"> A μ through a gauge transformation. The integral is still divergent, bur if the contribution of the subspace of gauge trajectories can be factorised it can be ignored, because the calculation of the Green's function always involve ratios of path integrals. The main issue associated with this procedure is the appearance of the Jacobian determinant of the transformation. In the case of QED, however, the Jacobian turns out to be independent of the fields, and appears as an irrelevant multiplicative factor in the results of the path integrals. The application of this gauge fixing method is illustrated considering the calculation of the generating functional of QED and the photon propagator in the Lorenz gauge. The generalisation to the case of non-abelian theories, due to Faddeev and Popov, is discussed in Chapter 14.
