From Klein-Gordon-Wave to Schr\"odinger-Wave: a Normal Form Approach

Abstract

We consider a Klein-Gordon-Wave system, describing the evolution of a massive field and a massless one interacting through a Yukawa-like coupling, and we explicitly derive its Hamiltonian normal form to first and second order. To the first-order approximation, the normal form results in a Schr\"odinger-Wave system, which reduces to the Schr\"odinger-Poisson one in the singular limit of vanishing perturbative parameter. The second-order approximation provides the successive corrections to the Schr\"odinger-Wave system, and is presented in order to show that higher-order approximations to all orders can be obtained by iterating our constructive procedure. The normal form technique adopted here formally extends the standard Birkhoff normal form procedure for harmonic oscillators to include a set of free particles in the unperturbed problem. The mathematical result obtained here might explain, for example, the "cooling" process of ultra-light dark matter, the approximate validity of the Schr\"odinger-Poisson system describing its dynamics and the long term conservation of the total dark matter mass.