EFFECTIVE ACTIONS OF LOCAL COMPOSITE OPERATORS: THE CASE OF φ4 THEORY, THE ITINERANT ELECTRON MODEL, AND QED

Abstract

The effective action Γ[ɸ], 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 Γ[ɸ] of a local composite operator is given in this paper. This long-standing problem of obtaining Γ[ɸ] 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)[ɸ], which is a function(al) of the variable ɸ and is defined through the lowest inversion formula. Here J(0)[ɸ] has the meaning of the source J for the noninteracting system. Explicitly derived are the rules for the effective action of <φ(x)2> in the φ4 theory, of the number density <nrσ> in the itinerant electron model, and of the gauge-invariant operator [Formula: see text] in QED.