Momentum gauge fields from curved momentum space through Kaluza-Klein reduction

TL;DR

This paper explores how curved momentum space and momentum-dependent gauge fields are interconnected, deriving these effects via Kaluza-Klein reduction, revealing modifications to quantum mechanics and noncommutative geometries.

Contribution

It introduces a novel derivation of momentum-dependent gauge fields from higher-dimensional curved momentum space using Kaluza-Klein reduction.

Findings

Gauge fields lead to Moyal-type noncommutativity.

Curved momentum space results in Snyder-type noncommutative geometry.

Modified Heisenberg algebra due to gauge and curvature effects.

Abstract

In this work we investigate the relation between curved momentum space and momentum-dependent gauge fields. While the former is a classic idea that has been shown to be tied to minimal-length models, the latter constitutes a relatively recent development in quantum gravity phenomenology. In particular, the gauge principle in momentum space amounts to a modification of the position operator of the form $\hat{X}^\mu\rightarrow\hat{X}^\mu-g A^\mu (\hat{P})$ akin to a gauge-covariant derivative in momentum space according to the minimal coupling prescription. Here, we derive both effects from a Kaluza-Klein reduction of a higher-dimensional geometry exhibiting curvature in momentum space. The interplay of the emerging gauge fields as well as the remaining curved momentum space lead to modifications of the Heisenberg algebra. While the gauge fields imply Moyal-type noncommutativity dependent on the analogue field strength tensor, the dimensionally reduced curved momentum space geometry translates to a Snyder-type noncommutative geometry.