Quantum mechanics and classical trajectories

TL;DR

This paper explores a geometric approach to quantum mechanics using Fedosov's deformation quantization, providing a new perspective on the classical limit and Hamiltonian dynamics.

Contribution

It introduces a novel representation of wave functions via flat vector bundles over phase space, linking classical and quantum dynamics through geometric conditions.

Findings

Hamilton's equation emerges as a condition on phase space curves.

The approach generalizes the Schrödinger representation.

Provides a geometric framework for quantum-classical correspondence.

Abstract

The classical limit $\hbar$->0 of quantum mechanics is known to be delicate, in particular there seems to be no simple derivation of the classical Hamilton equation, starting from the Schr\"odinger equation. In this paper I elaborate on an idea of M. Reuter to represent wave functions by parallel sections of a flat vector bundle over phase space, using the connection of Fedosov's construction of deformation quantization. This generalizes the ordinary Schr\"odinger representation, and allows naturally for a description of quantum states in terms of a curve plus a wave function. Hamilton's equation arises in this context as a condition on the curve, ensuring the dynamics to split into a classical and a quantum part.