The Dirac-Klein-Gordon system in the strong coupling limit

TL;DR

This paper analyzes the strong coupling limit of the Dirac-Klein-Gordon system, demonstrating convergence to a non-linear Dirac equation and its implications for relativistic nuclear models.

Contribution

It proves the convergence of solutions in the strong coupling limit and extends the results to a many-body Dirac-Fock framework.

Findings

Solutions converge to a cubic non-linear Dirac equation

Retarded interactions are approximated by local self-interactions

Results are relevant to relativistic mean-field theory of nuclei

Abstract

We study the Dirac equation coupled to scalar and vector Klein-Gordon fields in the limit of strong coupling and large masses of the fields. We prove convergence of the solutions to those of a cubic non-linear Dirac equation, given that the initial spinors coincide. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. We generalize this result to a many-body Dirac-Fock equation on the space of Hilbert-Schmidt operators.