Reviews in Mathematical Physics

In this paper we represent the solutions of Maxwell's equations in terms of directional wavelet transform. Then, we make use of the polarization set, which refine the notion of wavefront set for vector-valued distributions as introduced by Dencker [2], in order to analyze the singularities of solutions of the Maxwell's equations.

The Bogoliubov–Dirac–Fock (BDF) model allows us to describe relativistic electrons interacting with the Dirac sea. It can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons are neglected.

This paper treats the case of an electron together with the Dirac sea in the absence of any external field. Such a system is described by its one-body density matrix, an infinite rank, self-adjoint operator.

The parameters of the model are the coupling constant α > 0 and the ultraviolet cut-off Λ > 0: we consider the subspace of squared integrable functions made of the functions whose Fourier transform vanishes outside the ball B(0, Λ). We prove the existence of minimizers of the BDF energy under the charge constraint of one electron and no external field provided that α, Λ^-1 and α log(Λ) are sufficiently small. The interpretation is the following: in this regime the electron creates a polarization in the Dirac vacuum which allows it to bind.

We then study the non-relativistic limit of such a system in which the speed of light tends to infinity (or equivalently α tends to zero) with αlog(Λ) fixed: after rescaling and translation the electronic solution tends to a Choquard–Pekar ground state.

We conclude our analysis of the linear response of charge transport in lattice systems of free fermions subjected to a random potential by deriving general mathematical properties of its conductivity at the macroscopic scale. The present paper belongs to a succession of studies on Ohm and Joule's laws from a thermodynamic viewpoint starting with [1-3]. We show, in particular, the existence and finiteness of the conductivity measure μ_Σ for macroscopic scales. Then we prove that, similar to the conductivity measure associated to Drude's model, μ_Σ converges in the weak*-topology to the trivial measure in the case of perfect insulators (strong disorder, complete localization), whereas in the limit of perfect conductors (absence of disorder) it converges to an atomic measure concentrated at frequency ν = 0. However, the AC-conductivity μ_Σ|_ℝ\{0} does not vanish in general: We show that μ_Σ(ℝ\{0}) > 0, at least for large temperatures and a certain regime of small disorder.