Interacting relativistic quantum dynamics of two particles on spacetimes with a Big Bang singularity

TL;DR

This paper develops a relativistic quantum theory for two particles interacting along light cones in curved spacetimes with a Big Bang, proving existence and uniqueness of solutions under certain conditions.

Contribution

It introduces a covariant integral equation framework for relativistic quantum interactions without potentials, extending results to cosmological spacetimes with singularities.

Findings

Unique solutions exist for retarded interactions in Big Bang spacetimes.

Solutions are parametrized differently for time-symmetric interactions with Big Bang and Big Crunch.

The approach extends Minkowski space results to curved cosmological spacetimes.

Abstract

Relativistic quantum theories are usually thought of as being quantum field theories, but this is not the only possibility. Here we consider relativistic quantum theories with a fixed number of particles that interact neither through potentials nor through exchange of bosons. Instead, the interaction can occur directly along light cones, in a way similar to the Wheeler-Feynman formulation of classical electrodynamics. For two particles, the wave function is here of the form $\psi(x_1,x_2)$, where $x_1$ and $x_2$ are spacetime points. Specifically, we consider a natural class of covariant equations governing the time evolution of $\psi$ involving integration over light cones, or even more general spacetime regions. It is not obvious, however, whether these equations possess a unique solution for every initial datum. We prove for Friedmann-Lemaitre-Robertson-Walker spacetimes (certain cosmological curved spacetimes with a Big Bang singularity, i.e., with a beginning in time) that in the case of purely retarded interactions there does, in fact, exist a unique solution for every datum on the initial hypersurface. The proof is based on carrying over similar results for a Minkowski half-space (i.e., the future of a spacelike hyperplane) to curved spacetime. Furthermore, we show that also in the case of time-symmetric interactions and for spacetimes with both a Big Bang and a Big Crunch solutions do exist. However, initial data are then not appropriate anymore; the solution space gets parametrized in a different way.