Aspects of perturbative quantum field theory on non-commutative spaces

TL;DR

This paper reviews quantum field theories on non-commutative Moyal space, focusing on renormalization challenges, the UV/IR mixing problem, and recent progress in constructing renormalizable gauge models.

Contribution

It provides an overview of the development of renormalization techniques for non-commutative quantum field theories and discusses new models addressing UV/IR mixing.

Findings

UV/IR mixing was solved for certain models

Progress in constructing renormalizable gauge theories on non-commutative space

Outline of a generalized renormalization scheme for non-local models

Abstract

In this contribution to the proceedings of the Corfu Summer Institute 2015, I give an overview over quantum field theories on non-commutative Moyal space and renormalization. In particular, I review the new features and challenges one faces when constructing various scalar, fermionic and gauge field theories on Moyal space, and especially how the UV/IR mixing problem was solved for certain models. Finally, I outline more recent progress in constructing a renormalizable gauge field model on non-commutative space, and how one might attempt to prove renormalizability of such a model using a generalized renormalization scheme adapted to the non-commutative (and hence non-local) setting.