Quantum jumps and attractors of the Maxwell-Schr\"odinger equations

Abstract

Our goal is the discussion of the problem of mathematical interpretation of basic postulates (or `principles') of Quantum Mechanics: transitions to quantum stationary orbits, the wave-particle duality, and the probabilistic interpretation, in the context of semiclassical self-consistent Maxwell--Schr\"odinger equations. We discuss possible relations of these postulates to the theory of attractors of Hamiltonian nonlinear PDEs and to a new general mathematical conjecture on global attractors of G-invariant nonlinear Hamiltonian partial differential equations with a Lie symmetry group G. This conjecture is inspired by our results on global attractors of nonlinear Hamiltonian PDEs obtained since 1990 for a list of model equations with three basic symmetry groups: the trivial group, the group of translations, and the unitary group U(1). We present sketchy these results.