Mean-field derivation of the Klein–Gordon–Maxwell equations

Develop a rigorous mean-field derivation of the massive Klein–Gordon–Maxwell equations starting from an N-body microscopic model with appropriate relativistic interactions and gauge coupling (e.g., a Pauli–Fierz–type or related quantum field framework). Identify the scaling regime and assumptions under which the N-body dynamics converges, in the mean-field limit, to the Klein–Gordon–Maxwell system.

Background

The Schrödinger–Poisson system has well-established mean-field derivations from N-body Schrödinger dynamics. By contrast, the authors emphasize that such a derivation is missing for mKGM, despite its interpretation as a U(1) Yang–Mills–Higgs model for a spinless charged scalar field.

A rigorous derivation would require specifying the microscopic model, interaction, and scaling such that the macroscopic limit yields mKGM, bridging relativistic quantum particle-field models with the classical gauge-coupled Klein–Gordon field.