Singular light cone interactions of scalar particles in 1+3 dimensions

Abstract

Here we consider an integral equation describing a fixed number of scalar particles which interact not through boson exchange but directly along light cones, similarly as in bound state equations such as the Bethe-Salpeter equation. The equation involves a multi-time wave function $ψ(x_1,...,x_N)$ with $x_i=(t_i,\mathbf{x}_i) \in \mathbb{R}^4$ as a crucial concept. Assuming a cutoff in time, we prove that it has a unique solution for all data at the initial time. The cutoff is justified by considering the integral equation for a particular curved spacetime with a Big Bang singularity where an initial time occurs naturally without violating any spacetime symmetries. The main feature of our work is that we treat the highly singular case that interactions occur exactly at zero Minkowski distance, reflected by a delta distribution along the light cone. We also extend the existence and uniqueness result to an arbitrary number $N \geq 2$ of particles. Overall, we provide a rigorous example for a certain type of interacting relativistic quantum dynamics in 1+3 spacetime dimensions.