Dynamical variables in the Bethe-Salpeter formalism

Abstract

It is shown that a knowledge of the behaviour of the propagators around their singularities enables one to determine not only the masses of bound states, but also the matrix element of any dynamical variable between two bound states. One is thus enabled to find such a matrix element, to any order in the coupling constant, by the integration of certain expressions over the corresponding Bethe-Salpeter wave-functions. As a consequence, it is possible to find normalization and orthogonality properties of these wave-functions, which in turn lead to the condition which must be imposed on their singularities a t the origin. More light is thus shed on Goldstein’s difficulty concerning the existence of a continuous infinity of bound states. The formalism is extended to scattering states in which some of the particles may be composite—in particular, an expression for the S -matrix is obtained