Non-monokinetic semiclassical limit of massive Klein–Gordon–Maxwell to relativistic Vlasov–Maxwell

Establish the semiclassical limit of the (3+1)-dimensional massive Klein–Gordon–Maxwell equations to the relativistic Vlasov–Maxwell system for general (non-monokinetic) initial data, such as mixed states represented by density matrices, and develop an appropriate relativistic Wigner-measure framework for Klein–Gordon–type fields. Prove convergence of observables in suitable norms and address the passage of the electromagnetic nonlinearity to the limit.

Background

The paper proves the monokinetic semiclassical limit from mKGM to relativistic Euler–Maxwell (and interprets it as a monokinetic relativistic Vlasov–Maxwell limit). However, extending this to general (mixed state) data—analogous to SP→VP results that rely on density matrices and Wigner transforms—is not available for Klein–Gordon fields.

The authors note that mixed-state frameworks used in the non-relativistic case (SP→VP) are not typically defined for Klein–Gordon equations, and that no relativistic adaptation of the key semiclassical methods exists. This leaves a gap in understanding the full non-monokinetic semiclassical limit for mKGM.