Existence of relativistic dynamics for two directly interacting Dirac particles in 1+3 dimensions

Abstract

Here we prove the existence and uniqueness of solutions of a class of integral equations describing two Dirac particles in 1+3 dimensions with direct interactions. This class of integral equations arises naturally as a relativistic generalization of the integral version of the two-particle Schrödinger equation. Crucial use of a multi-time wave function $ψ(x_1,x_2)$ with $x_1,x_2 \in \mathbb{R}^4$ is made. A central feature is the time delay of the interaction. Our main result is an existence and uniqueness theorem for a Minkowski half space, meaning that Minkowski spacetime is cut off before $t=0$. We furthermore show that the solutions are determined by Cauchy data at the initial time; however, no Cauchy problem is admissible at other times. A second result is to extend the first one to particular FLRW spacetimes with a Big Bang singularity, using the conformal invariance of the Dirac equation in the massless case. This shows that the cutoff at $t=0$ can arise naturally and be fully compatible with relativity. We thus obtain a class of interacting, manifestly covariant and rigorous models in 1+3 dimensions.