Abstract
We present the relativistic quantum physics model hierarchy from Dirac-Maxwell to Vlasov/Euler-Poisson that models fast moving charges and their self-consistent electro-magnetic field. Our main interest is (asymptotic) analysis of these nonlinear time-dependent PDEs, with focus on the Pauli-Poisswell/Darwin system which is the consistent model at first/second order in 1/c (c = speed of light) that keeps both relativistic effects "spin" and "magnetism". Emphasis is on the (semi)classicial limit for vanishing Planck constant. We use both WKB methods and Wigner functions where we extend the 1993 results of P. L. Lions & Paul and Markowich & Mauser on the limit from Schrödinger-Poisson to Vlasov-Poisson, with similar subtilities of pure quantum states vs mixed states. In the hope of taming the mathematical complications stemming from the magnetic field, we are interested also in developing "quantum/semiclassical velocity averaging lemmata" building on the 1988 ideas of Golse, Perthame, Sentis and P.L.Lions. This talk aims to explain the models, the new results & ideas of proofs/techniques, as worked out in joint works mainly with Jakob Möller (X) and also Pierre Germain (ICL), Changhe Yang (Caltech), Francois Golse (X).
