Classical mechanics with lapse

TL;DR

This paper develops a covariant approach to classical and quantum mechanics by treating time as a Riemannian manifold with a lapse function, with implications for quantum cosmology.

Contribution

It introduces a covariant quantization method using a lapse function for time, extending classical mechanics on manifolds to a quantum framework.

Findings

Covariant Schrödinger operator with lapse function

Implications for quantum cosmology discussed

Framework unifies classical and quantum mechanics on manifolds

Abstract

Mechanics is developed over a differentiable manifold as space of possible positions. Time is considered to fill a one--dimensional Riemannian manifold, so having the metric as lapse. Then the system is quantized with covariant instead of partial derivatives in the Schr\"odinger operator. Consequences for quantum cosmology are shortly discussed.