On the relation between classical and quantum dynamics --- a mathematician's perspective:
Abstract:
Classical mechanics and quantum mechanics give very different descriptions of the laws of evolution of a physical system which at
high energies should be quite similar. I will discuss one aspect of this in a very simple system: one spineless particle constraint to lie on a closed and bounded manifold with no external force. In this case the quantum unique ergodicity conjecture states that if the classical dynamics is uniformly hyperbolic ("chaotic" in a strong sense) the steady states of the Schroedinger evolution (which are just the eigenfunctions of the Laplace-Beltrami operator) become equidistributed in the high energy limit. I will present some of the rigorous results in this direction, and focus on the case of arithmetic manifolds which have a subtle but very rich symmetry which can be used to aid the analysis.