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 $\psi (x_1,x_2)$, 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.

22 pages

References in corpus (5)

• A relativistically interacting exactly solvable multi-time model for two mass-less Dirac particles in 1+1 dimensions
• New symmetry properties of pointlike scalar and Dirac particles
• Multi-Time Wave Functions
• Direct interaction along light cones at the quantum level
• Existence of relativistic dynamics for two directly interacting Dirac particles in 1+3 dimensions

Cited by in corpus (4)

• A new class of Volterra-type integral equations from relativistic quantum physics
• Consistency Proof for Multi-Time Schrodinger Equations with Particle Creation and Ultraviolet Cut-Off
• Probability conservation for multi-time integral equations
• Singular light cone interactions of scalar particles in 1+3 dimensions