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

Abstract

Here we prove the existence and uniqueness of solutions of a class of integral equations describing two Dirac particles in 1+3 dimensions with direct interactions. This class of integral equations arises naturally as a relativistic generalization of the integral version of the two-particle Schrödinger equation. Crucial use of a multi-time wave function ψpx 1 , x 2 q with x 1 , x 2 P R 4 is made. A central feature is the time delay of the interaction. Our main result is an existence and uniqueness theorem for a Minkowski half space, meaning that Minkowski spacetime is cut off before t " 0. We furthermore show that the solutions are determined by Cauchy data at the initial time; however, no Cauchy problem is admissible at other times. A second result is to extend the first one to particular FLRW spacetimes with a Big Bang singularity, using the conformal invariance of the Dirac equation in the massless case. This shows that the cutoff at t " 0 can arise naturally and be fully compatible with relativity. We thus obtain a class of interacting, manifestly covariant and rigorous models in 1+3 dimensions.

Multi-time dynamics of the Dirac-Fock-Podolsky model of QED

“…with K = 4 N denoting the dimension of spinor space of the N Dirac electrons. In view of (10) and (11), we use the notation for a.e. (x 1 , ..., x N ) :…”

“…In recent years, there has been a renewed interest in constructing mathematically rigoros multi-time models, see [5] for an overview. Some of the current efforts to understand Dirac's multi-time models focus on the well-posedness of the corresponding initial value problems [6,7,8,9,10], other works also ask the question how the multi-time formalism could be exploited to avoid the infamous ultraviolet divergence of relativistic QFT and how a varying number of particles by means of creation and annihilation processes can be addressed [11,12,13]. Beside being candidate models for fundamental formulations of relativistic wave mechanics, a better mathematical understanding of such multi-time evolutions may also be beneficial regarding more technical discussions, such as the control of scattering estimates on vacuum expectation values of products of interacting field operators; see e.g.…”