On momentum operators given by Killing vectors whose integral curves are geodesics

TL;DR

This paper investigates momentum operators derived from Killing vectors on curved manifolds, revealing geometric constraints, explicit representations on $S^3$, and the relationship between curvature, spectra, and position-momentum commutators.

Contribution

It characterizes manifolds admitting such momentum operators, provides explicit representations on $S^3$, and links geometric properties to quantum spectra and operator relations.

Findings

Manifolds with these momentum operators are either flat or of compact type with positive curvature in dimensions 1, 3, or 7.

Explicit momentum operator representations and Casimir elements are given for the 3-sphere $S^3$.

Maximum momentum resolution equals the de-Broglie wavelength, matching the manifold's diameter.

Abstract

We consider momentum operators on intrinsically curved manifolds. Given that the momentum operators are Killing vector fields whose integral curves are geodesics, it is shown that the corresponding manifold is either flat, or otherwise of compact type with positive constant sectional curvature and dimension equal to 1, 3 or 7. Explicit representations of momentum operators and the associated Casimir element will be discussed for the 3-sphere $S^3$. It will be verified that the structure constants of the underlying Lie algebra are proportional to $2\hbar/R$, where $R$ is the curvature radius of $S^3$. This results in a countable energy and momentum spectrum of freely moving particles in $S^3$. It is shown that the maximum resolution of the possible momenta is given by the de-Broglie wave length $\lambda_R=\pi R$, which is identical to the diameter of the manifold. The corresponding covariant position operators are defined in terms of geodesic normal coordinates and the associated commutator relations of position and momentum are established.