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-Lemaître-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.

A new class of Volterra-type integral equations from relativistic quantum physics

“…Furthermore, we expect additional singularities of the Green's functions to appear on spacetimes with a Big Bang, as a consequence of the latter. We address this circle of questions in a subsequent paper [17].…”

“…In fact, this is also a possible cosmological scenario: the Big Crunch. It would be of interest to develop existence results for this case (see [17]).…”

Consistency Proof for Multi-time Schrödinger Equations with Particle Creation and Ultraviolet Cut-Off

“…The commutator condition ( 9) is closely related to the integrability condition of the Tomonaga-Schwinger equation, which in turn is closely related to the causality axiom of the Wightman axioms [44, p. 65]. Since interaction potentials violate (9) [35,38], interaction needs to be implemented in a different way, such as through particle creation [25,39,41], zero-range interaction [17,20,22,24], interaction along light cones [23,26,31,32], or other ways [9][10][11]36]. In this paper, we focus on particle creation.…”

Existence of relativistic dynamics for two directly interacting Dirac particles in 1+3 dimensions

“…An interesting class of such dynamics has recently been suggested in [26] and has been subsequently analyzed rigorously in [29,30]: multi-time integral equations. But why study integral equations instead of PDEs?…”

“…While such a cutoff destroys Lorentz invariance, there could be physical justification for a beginning in time which is compatible with relativity. Such a justification has been provided in [30]. There, the integral equation was extended to curved spacetimes and analyzed in more detail for certain spacetimes which feature a Big Bang singularity, Friedman-Lemaître-Robertson-Walker (FLRW) spacetimes.…”