Introduction to noncommutative field and gauge theory

TL;DR

This paper provides an introductory overview of noncommutative field and gauge theories, highlighting key concepts, specific models like Moyal spacetime, and symmetry considerations, aimed at readers new to the subject.

Contribution

It offers a comprehensive introduction to noncommutative field and gauge theories, emphasizing derivations-based calculus and specific models such as Moyal spacetime and $ ext{R}^3_ ext{lambda}$.

Findings

Review of quantum mechanics as a noncommutative prototype

Analysis of scalar and gauge theories on noncommutative spaces

Discussion of noncommutative spacetime symmetries and twist approach

Abstract

These are lecture notes for an introductory course on noncommutative field and gauge theory. We begin by reviewing quantum mechanics as the prototypical noncommutative theory, as well as the geometrical language of standard gauge theory. Then, we review a specific approach to noncommutative field and gauge theory, which relies on the introduction of a derivations-based differential calculus. We focus on the cases of constant and linear noncommutativity, e.g., the Moyal spacetime and the so-called $\mathbb{R}^3_\lambda$, respectively. In particular, we review the $g\varphi^4$ scalar field theory and the $U(1)$ gauge theory on such noncommutative spaces. Finally, we discuss noncommutative spacetime symmetries from both the observer and particle point of view. In this context, the twist approach is reviewed and the $\lambda$-Minkowski $g\varphi^4$ model is discussed.