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A compact graph rule for the effective action $\Gamma[\phi]$ of a localcomposite operator is given in this paper. This long-standing problem ofobtaining $\Gamma[\phi]$ in this case is solved directly without using theauxiliary field. The rule is first deduced with help of the inversion method,which is a technique for making the Legendre transformation perturbatively. Itis then proved by using a topological relation and also by the sum-up rule.Explicitly derived are the rules for the effective action of $\langle\varphi(x)^2 \rangle$ in the $\varphi^4$ theory, of the number density $\langlen_{{\bf r}\sigma} \rangle$ in the itinerant electron model, and of the gaugeinvariant operator $\langle \bar{\psi}\gamma^\mu\psi \rangle$ in QED.Comment: (okumura@rk.phys.keio.ac.jp), 46p., LATEX, (figures are available on  request

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

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