Self-consistent approximations and the Legendre transform

Abstract

It is well-known in quantum field theory and statistical physics that the effective action (i.e., the Legendre transform of the free energy) is the generating function for one particle irreducible diagrams. We present an explicit diagrammatic proof of this result which permits a clear statement concerning its validity in the context of self-consistent approximation schemes. In particular, we find a necessary and sufficient condition on the approximate set of Green's functions for the Legendre transform theorem to hold. This condition is exactly what one expects on physical grounds. Our proof also lends itself to an examination of more extended diagrammatic schemes, such as those involving composite operators in renormalization theory.