Geometric View of One-Dimensional Quantum Mechanics

TL;DR

This paper applies De Haro's Geometric View to simple quantum systems, illustrating how quantum representations and dualities can be understood geometrically through phase space and fiber bundle structures.

Contribution

It demonstrates a geometric framework for understanding quantum mechanics of a particle on a line and circle, including boundary conditions and dualities, using fiber bundles and connections.

Findings

Position and momentum representations are different trivializations of a Hilbert bundle.

Fourier transform acts as a fiberwise unitary transition function.

Twisted boundary conditions can be incorporated as boundary conditions or base coordinates.

Abstract

We apply De Haro's Geometric View of Theories to one of the simplest quantum systems: a spinless particle on a line and on a circle. The classical phase space M = T*Q is taken as the base of a trivial Hilbert bundle E ~ M x H, and the familiar position and momentum representations are realised as different global trivialisations of this bundle. The Fourier transform appears as a fibrewise unitary transition function, so that the standard position-momentum duality is made precise as a change of coordinates on a single geometric object. For the circle, we also discuss twisted boundary conditions and show how a twist parameter can be incorporated either as a fixed boundary condition or as a base coordinate, in which case it gives rise to a flat U(H)-connection with nontrivial holonomy. These examples provide a concrete illustration of how the Geometric View organises quantum-mechanical representations and dualities in geometric terms.