Direct semi-classical/non-relativistic limit of massive Klein–Gordon–Maxwell to Vlasov–Poisson

Establish the combined semi-classical and non-relativistic limit of the (3+1)-dimensional massive Klein–Gordon–Maxwell equations and prove convergence, in an appropriate topology for observables (density, current, electromagnetic field), to the Vlasov–Poisson system. Precisely, show that under a simultaneous regime where the Planck parameter tends to zero and the speed-of-light scaling yields the non-relativistic limit, solutions of the massive Klein–Gordon–Maxwell equations converge to solutions of Vlasov–Poisson without passing through the intermediate Schrödinger–Poisson system.

Background

The paper reviews known limits between quantum and kinetic/fluid models. It is classical that massive Klein–Gordon–Maxwell (mKGM) converges to Schrödinger–Poisson (SP) in the non-relativistic limit, and that SP converges to Vlasov–Poisson (VP) in the semi-classical limit. The authors highlight the absence of a direct result that bypasses SP and establishes a combined semi-classical/non-relativistic limit from mKGM straight to VP.

They reference analogous double-limit results for related systems (e.g., Dirac–Maxwell to SP) and note this direct limit remains unproved for mKGM. A rigorous formulation would require specifying the joint scaling, functional framework, and convergence of key observables.