ABSTRACT
A difficulty arises in defining the path integral representing the vacuumto-vacuum amplitude when a local symmetry, such as a gauge symmetry, is present in the classical action. The path integral is even more divergent than usual. The problem is as follows: Suppose we are quantizing a classical field theory with generic gauge dependent field At (x). The action, S[At ], is gauge invariant. The naive path integral, the path integral suggested by scalar field theory, is Z[J] = f DAIL exp(iS[A4, 4]). In this path integral we are integrating over all configurations of the field Ail , and each configuration in the sum is related to an infinite number of others by gauge transformations. However, the integrand is identical for two gauge equivalent configurations, hence the two configurations contribute the same information. This means that by integrating over all configurations, we are computing an infinite number of copies of some integral. Put in a slightly different fashion, Ai,(x) contains gauge dependent and gauge invariant components. When we integrate over the space of configurations, f DA.m , we are integrating with respect to both gauge invariant and gauge dependent parts. Yet, the action, which is gauge invariant, depends only on the gauge invariant part of Am . The integrand for the gauge dependent parts is unity, and so the integral over the gauge dependent parts diverges. This is the extra divergence we want to remove.
