Structure of Green's Functions in Quantum Field Theory

Abstract

The Green's functions are vacuum expectation values of time ordered products of field operators in the Heisenberg representation. They give us vital information about the nature of the interacting particles and quanta represented by these field operators. It is shown that the Fourier transforms of the Green's functions (for up to four operators) are expressible as parametric integrals involving invariant energy denominators and real, scalar weight functions which are termed the spectral functions. Relativity, causality, and some other fundamental assumptions of field theory are required to derive the result.The spectral functions have a simple physical interpretation, and completely specify the structure of the Green's functions. The equations of motion which hold between Green's functions of different order can be translated into the corresponding relations between the spectral functions. Renormalization can be carried out explicitly, bringing the equations for the spectral functions into a manifestly renormalized form. The causality condition also serves as a means of obtaining renormalized quantities without recourse to the usual subtraction procedure. The observed masses and coupling constants occur in the equations for the spectral functions as external parameters fixing the boundary condition. No unobservable bare masses and couplings, nor the (infinite) renormalization constants ever appear in the equations. These quantities, however, are shown to be expressible in terms of the spectral functions.The Green's functions involving more than four field operators are not considered in this paper.