Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian

Abstract

We consider the spinless Pauli-Fierz Hamiltonian which describes a quantum system of non-relativistic identical particles coupled to the quantized electromagnetic field. We study the time evolution in a mean-field limit where the number $N$ of charged particles gets large while the coupling to the radiation field is rescaled by $1/\sqrt{N}$. At time zero we assume that almost all charged particles are in the same one-body state (a Bose-Einstein condensate) and we assume also the photons to be close to a coherent state. We show that at later times and in the limit $N\to \infty$ the charged particles as well as the photons exhibit condensation, with the time evolution approximately described by the Maxwell-Schrödinger system, which models the coupling of a non-relativistic particle to the classical electromagnetic field. Our result is obtained by an extension of the "method of counting", introduced by Pickl, to condensates of charged particles in interaction with their radiation field.

improved presentation, extension of Theorem II.2 to a larger class of initial states

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• Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron
• Derivation of the Landau-Pekar equations in a many-body mean-field limit
• Mean-field limits of particles in interaction with quantized radiation fields
• Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model
• Semi-classical limit of the massive Klein-Gordon-Maxwell system toward the relativistic Euler-Maxwell system via an adapted modulated energy method