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.
[CI89] From quantum electrodynamics to mean-field theory. I. The Bogoliubov–Dirac–Fock formalism, J. Phys. B: At. Mol. Opt. Phys., Volume 22 (1989) no. 23, pp. 3791-3814 | DOI
[Dir28a] The quantum theory of the electron, Proc. R. Soc. Lond., Ser. A, Volume 117 (1928) no. 778, pp. 610-624 | Zbl
[Dir28b] The quantum theory of the electron. Part II, Proc. R. Soc. Lond., Ser. A, Volume 118 (1928) no. 779, pp. 351-361 | Zbl
[Rin96] Relativistic mean field theory in finite nuclei, Prog. Part. Nucl. Phys., Volume 37 (1996), pp. 193-263 | DOI
[Sim05] Trace ideals and their applications, Mathematical Surveys and Monographs, American Mathematical Society, 2005 no. 120 | Zbl
[Sol70] Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, Volume 1 (1970) no. 10, pp. 2766-2769 | DOI
[Tha91] The Dirac equation, Texts and Monographs in Physics, Springer, 1991 | Zbl