Contact Geometry and Quantum Mechanics

TL;DR

This paper introduces a geometric, covariant framework for quantum mechanics that treats positions, momenta, and time uniformly as coordinates on a phase-spacetime, leading to a purely geometric and potentially topological formulation.

Contribution

It develops a covariant, geometric approach to quantum mechanics, generalizing Wigner functions and addressing time-dependent systems and observer ambiguities.

Findings

Quantum mechanics formulated as a flatness condition on phase-spacetime.

Provides a generalized derivation of Wigner functions.

Applicable to time-dependent and observer-ambiguous systems.

Abstract

We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental "phase-spacetime." We show that this covariant starting point makes quantization into a purely geometric flatness condition. This makes quantum mechanics purely geometric, and possibly even topological. Our approach is especially useful for time-dependent problems and systems subject to ambiguities in choices of clock or observer. As a byproduct, we give a derivation and generalization of the Wigner functions of standard quantum mechanics.