Geometric View of One-Dimensional Quantum Mechanics

TL;DR

This work tests De Haro's Geometric View of Theories by encoding a spinless particle on a line and on a circle into a trivial Hilbert bundle over the classical phase space $M=T^*Q$, so that position and momentum representations arise as different global trivialisations related by the Fourier transform $\mathcal{F}$. The paper shows dualities are explicit coordinate changes on a single geometric object, with twists on the circle captured by flat connections or parameter-space holonomy. In the circle, Case A yields a fixed twist producing a shifted spectrum via $\varphi=\alpha/(2\pi)$, while Case B treats $\alpha$ as a base parameter and obtains a flat connection with nontrivial holonomy along the $\alpha$-circle, $\psi(\theta;\alpha) \to e^{i\theta}\psi(\theta;\alpha)$ after a loop. The results illustrate GV's program of organizing quantum representations and spectra via bundles, transitions, and connections, and suggest future work on quasi-dualities, higher-topology bases, and richer operator algebras in the geometric framework.

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.