Causality in Plasma Electrodynamics

Causality in electrodynamics is reviewed in regard to the interrelations among the causal requirement, the analyticity of the dielectric permittivity, and the Kramers-Kronig relations. We show that the collisionless damping (Landau damping) of a plasma wave can be formally derived from the causal requirement imposed on the susceptibility of a Vlasov-Poisson plasma. Here, the causal requirement is that the susceptibility χ(t) be nil for t < 0, which means the future electric field has nothing to do with the response effected in the medium at the present time. We show that this single requirement provides the analyticity of χ(ω) in the upper half-ω plane, the Kramers- Kronig relations, and Landau damping. Cerenkov emission which is the inverse process of Landau damping is also discussed in the light of causality. We present an easy way to calculate the electric fluctuation in a magnetized plasma by regarding a plasma as an assembly of non-interacting cold beams. We investigate the case of a separable distribution function g(x, t, v) = g(v)f(x − v0t), which corresponds to a special type of Benstein-Greene-Kruskal wave and a slight generalization of Van Kampen's distribution function. We review Van Kampen's theory of a Vlasov-Poisson system, which corresponds to the Sturm-Liouville theory of differential equations, and show that the Vlasov- Poisson equations in this separable case are trivially solved in terms of the separation constant. It is philosophically interesting that the collisionless damping of a plasma wave can be attributed to causality, which defines the direction of time and is operative in general electrodynamics.