Momentum Gauge Fields and Non-Commutative Space-Time

TL;DR

This paper introduces a novel gauge theory based on momentum space representation of the position operator, exploring its implications for non-commutative space-time and quantum phenomena like Landau levels.

Contribution

It proposes a new gauge principle in momentum space and demonstrates its ability to produce non-commutative space-time and related physical effects.

Findings

Non-commutative space-time can depend on momentum.

Space-time can be commutative at low momentum and non-commutative at high momentum.

Momentum gauge fields can reproduce Landau level phenomena.

Abstract

In this work we present a gauge principle that starts with the momentum space representation of the position operator (${\hat x}_i = i \hbar \frac{\partial}{\partial p_i}$) rather than starting with the position space representation of the momentum operator (${\hat p}_i = -i \hbar \frac{\partial}{\partial x_i}$). We discuss some simple examples with this new type of gauge theory: (i) analog solutions from ordinary gauge theory in this momentum gauge theory, (ii) Landau levels using momentum gauge fields, (iii) the emergence of non-commutative space-times from the momentum gauge fields. We find that the non-commutative space-time parameter can be momentum dependent, and one can construct a model where space-time is commutative at low momentum but becomes non-commutative at high momentum.