Effective actions of local composite operators --- case of ${\phi }^4$ theory, itinerant electron model, and QED

Abstract

A compact graph rule for the effective action $\Gamma [\varphi ]$ of a local composite operator is given in this paper. This long-standing problem of obtaining $\Gamma [\varphi ]$ in this case is solved directly without using the auxiliary field. The rule is first deduced with help of the inversion method, which is a technique for making the Legendre transformation perturbatively. It is then proved by using a topological relation and also by the sum-up rule. Explicitly derived are the rules for the effective action of $⟨\phi (x{)}^2⟩$ in the ${\phi }^4$ theory, of the number density $⟨n_{\mathbf{r}\sigma }⟩$ in the itinerant electron model, and of the gauge invariant operator $⟨\bar{\psi }{\gamma }^{\mu }\psi ⟩$ in QED.

([email protected]), 46p., LATEX, (figures are available on request)