On the Many-Particle Structure of Green's Functions in Quantum Field Theory

Abstract

The structure of the expectation values of retarded multiple commutators (r functions) is analyzed in terms of the number of particles in the decomposition of absorptive parts. As to the one-particle structure, it is found that an r function is a sum of a finite number of terms, each of them except one (that one being called one-particle irreducible) being in momentum space a product of one-particle irreducible factors, linked by one-particle propagation functions. As to the two-particle structure, it is found that a one-particle irreducible function is the solution of an inhomogeneous Bethe-Salpeter equation, whose kernel and inhomogeneous term both are two-particle irreducible functions. This structure, which could be generalized to higher particle numbers, closely resembles perturbation theory but is here derived from locality and the asymptotic condition alone, by converting the nonlinear system of integral equations for r functions stepwise into one in which neither one- or two-particle reducible functions, nor one- or two-particle intermediate states appear. The implication of such structure analysis for an interpretation of perturbation theory, improvements of present methods to derive analytic properties of scattering amplitudes, and a formalism with unstable particles are discussed, and the strength of singularities of various functions investigated.