Fibre bundle formulation of time-dependent mechanics

TL;DR

This paper develops a geometric framework for classical and quantum time-dependent mechanics using fibre bundles, connections, and leafwise geometric quantization, providing a unified approach to reference frames and transformations.

Contribution

It introduces a fibre bundle formulation for both classical and quantum mechanics with a novel geometric quantization scheme for time-dependent systems.

Findings

Classical mechanics formulated as a field theory on fibre bundles.

Quantum mechanics described using Banach and Hilbert bundles.

Leafwise geometric quantization scheme proposed for quantization.

Abstract

We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over a time axis. Connections on these bundles describe reference frames. Quantum time-dependent mechanics is phrased in geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is leafwise geometric quantization.