# General covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere

## TL;DR

This paper develops a covariant geometric momentum framework for particles on curved surfaces, interprets the spin connection as a gauge potential, and analyzes a Dirac fermion on a sphere, revealing algebraic structures and absence of geometric potential.

## Contribution

It introduces a general covariant geometric momentum formalism and applies it to Dirac fermions on a sphere, uncovering algebraic properties and the lack of curvature-induced potential.

## Key findings

- General covariant geometric momentum framework established.
- Generalized total angular momentum forms an su(2) algebra.
- No curvature-induced geometric potential for the Dirac fermion.

## Abstract

For a particle that is constrained on an ($N-1$)-dimensional ($N\geq2$) curved surface, the Cartesian components of its momentum in $N$-dimensional flat space is believed to offer a proper form of momentum for the particle on the surface, which is called the geometric momentum as it depends on the mean curvature. Once the momentum is made general covariance, the spin connection part can be interpreted as a gauge potential. The present study consists in two parts, the first is a discussion of the general framework for the general covariant geometric momentum. The second is devoted to a study of a Dirac fermion on a two-dimensional sphere and we show that there is the generalized total angular momentum whose three cartesian components form the $su(2)$ algebra, obtained before by consideration of dynamics of the particle, and we demonstrate that there is no curvature-induced geometric potential for the fermion.