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.

Article web page: