# Existence of relativistic dynamics for two directly interacting Dirac particles in 1+3 dimensions. (arXiv:1903.06020v1 [math-ph])

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

Schr\"odinger equation. Crucial use of a multi-time wave function

$\psi(x_1,x_2)$ 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.