The connection of two-particle relativistic quantum mechanics with the Bethe–Salpeter equation

Abstract

The formal equivalence between the wave equations of two-particle relativistic quantum mechanics, based on the manifestly covariant Hamiltonian formalism with constraints, and the Bethe–Salpeter equation are shown. This is achieved by algebraically transforming the latter so as to separate it into two independent equations that match the equations of Hamiltonian relativistic quantum mechanics. The first equation determines the relative time evolution of the system, while the second one yields a three-dimensional eigenvalue equation. A connection is thus established between the Bethe–Salpeter wave function and its kernel on the one hand and the quantum mechanical wave function and interaction potential on the other. For the sector of solutions of the Bethe–Salpeter equation having nonrelativistic limits, this relationship can be evaluated in perturbation theory. A generalized form of the instantaneous approximation that simplifies the various expressions involved in the above relations is also devised. It also permits the evaluation of the normalization condition of the quantum mechanical wave function as a three-dimensional integral.