Effective actions of local composite operators --- case of $φ^4$ theory, itinerant electron model, and QED

A compact graph rule for the effective action $Γ[ϕ]$ of a local composite operator is given in this paper. This long-standing problem of obtaining $Γ[ϕ]$ 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 $\langle φ(x)^2 \rangle$ in the $φ^4$ theory, of the number density $\langle n_{{\bf r}σ} \rangle$ in the itinerant electron model, and of the gauge invariant operator $\langle \barψγ^μψ\rangle$ in QED.

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