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

## TL;DR

This paper explores the mathematical foundations of quantum mechanics through the lens of Maxwell-Schrödinger equations, proposing a new conjecture on global attractors of Hamiltonian PDEs related to quantum principles.

## Contribution

It introduces a novel conjecture linking quantum postulates to attractors of Hamiltonian PDEs, supported by results on global attractors for models with symmetry groups.

## Key findings

- Results on global attractors for nonlinear Hamiltonian PDEs since 1990.
- Proposed mathematical conjecture on attractors of G-invariant Hamiltonian PDEs.
- Discussion of quantum principles in the context of attractors and symmetry groups.

## 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.