Vilkovisky’s Unique Effective Action: An Introduction and Explicit Calculation

Abstract

The effective action is an important tool in modern quantum field theory. In particular it can be used to probe the non-perturbative vacuum structure of a theory. Examples of this are provided by the Coleman-Weinberg¹ mechanism in scalar QED, self-consistent dimensional reduction in Kaluza-Klein theory^2–12, and ß-function techniques for evaluating the field equations for the target space metric in non-linear σ-models^13–14. In non-perturbative applications, it is generally necessary to evaluate the effective action off-shell. It is well known, however, that the off-shell effective action as usually defined is both gauge dependent and reparametrization dependent. Reparametrization dependence means that under a field redefinition φ → φ’, the new effective action Γ’ [φ’] does not equal the original one Γ[φ] evaluated at the same phissical field φ. In terms more appropriate to the following discussion, one might say that the effective action is not a scalar function of the configuration space coordinates.

While I was walking on the stair I met a man who wasn’t there. He wasn’ there again today I wish that he would stay away. Author unknown