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

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 $ψ(x_1,x_2)$, where $x_1$ and $x_2$ are spacetime points. Specifically, we consider a natural class of covariant equations governing the time evolution of $ψ$ 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.