Covariant integral equations for Heisenberg operators

Abstract

Sets of rules are obtained for writing down directly the exact integral equations which are satisfied by certain functions of Heisenberg operators in quantum field theory. Three kinds of function are considered: the direct product, the chronological product and the M-product. The matrix elements of the M-product are equal to the Feynman amplitudes studied by Matthews & Salam (1) and the corresponding integral equation is called here the Matthews-Salam (M.-S.) equation. These authors have given a symbolic form of the M.-S. equation and a method of repeated differentiation and integration which can be used to obtain the explicit form of the integral equation in any particular example. In practice their method involves an immense amount of calculation even in quite simple examples. The rules obtained in the present paper make it possible to write down directly the M.-S. equation without any of the tedious calculations implied by the M.-S. method.

So long as the exact theory is used, the three sets of equations (for direct, chronological and M-products) are completely equivalent. When bound state theory is considered by an approximation based on a power series in the coupling constant different results are obtained. The approximation is inapplicable to the direct product equations, and leads to different approximate equations for the amplitudes obtained from the chronological and the M-products even when these amplitudes are identical. This paradox is explained and it is shown that the equation coming from the M-product corresponds to the Bethe-Salpeter equation.