Effective actions of local composite operators: The case of phi(4) theory, the itinerant electron model, and QED

The effective action Gamma[phi], defined from the generating functional W[J] through the Legendre transformation, plays the role of an action functional in the zero temperature field theory and of a generalized thermodynamical function(al) in equilibrium statistical physics. A compact graph rule for Gamma[phi] of a local composite operator is given in this paper. This long-standing problem of obtaining Gamma[phi] in this case is solved directly without introducing 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 summing-up rule. The latter is a technique for making the Legendre transformation in a graphical language. In the course of proof a special role is played by J((0))[phi], which is a function(a1) of the variable phi and is defined through the lowest inversion formula. Here J((0))[phi] has the meaning of the source J for the noninteracting system. Explicitly derived are the rules for the effective action of [phi(x)(2)] in the phi(4) theory, of the number density [n(r sigma)] in the itinerant electron model, and of the gauge-invariant operator [psi gamma(mu)psi] in QED.