Semiclassical derivation of the Vlasov–Maxwell equation

Establish the rigorous semiclassical limit from the non-relativistic Maxwell–Schrödinger equation to the non-relativistic Vlasov–Maxwell equation as Planck’s constant tends to zero, without restricting to the monokinetic Euler–Maxwell case or introducing a regularization.

Background

The paper studies the Maxwell–Schrödinger system as a quantum model coupled to a self-consistent electromagnetic field and derives the pressureless Euler–Maxwell system in the semiclassical limit for sufficiently small monokinetic initial data. The broader expected kinetic limit is the non-relativistic Vlasov–Maxwell equation, in which the distribution function depends on both position and velocity.

The author explicitly identifies the full derivation as an outstanding open problem. The result established in the paper addresses only the monokinetic regime and is presented as a first step toward resolving the kinetic problem.

References

The rigorous derivation leading from Maxwell Schrodinger eq to Vlasov Maxwell in the limit as $\hbar \rightarrow 0$ is well recognized as an outstanding open problem.

— From the Maxwell-Schrödinger equation to the Euler-Maxwell equation in the semiclassical limit  (2610.11199 - Porat, 8 Oct 2026) in Section 1, “Background”