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Existence of relativistic dynamics for two directly interacting Dirac particles in \(1+3\) dimensions. (English) Zbl 1483.81078

Summary: 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 \(\psi( x_1, x_2)\) with \(x_1, x_2\in \mathbb{R}^4\) 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 the 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.

MSC:

81Q40 Bethe-Salpeter and other integral equations arising in quantum theory
45E99 Singular integral equations
45P05 Integral operators
81V25 Other elementary particle theory in quantum theory
81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
51B20 Minkowski geometries in nonlinear incidence geometry
83F05 Relativistic cosmology
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