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

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 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. Published under license by AIP Publishing.