Aspects of perturbative quantum field theory on non-commutative spaces

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.

The Gribov problem in noncommutative gauge theory

“…The problem shares some similarities with the UV/IR problem of noncommutative gauge theory [27]. Indeed, a propagator of the form of the Gribov-Zwanziger-dell'Antonio propagator has already been proposed by hand in the NC field theory framework, emerging from the necessity of curing the IR/UV phenomenon in scalar translation invariant models on the Moyal plane [22] and it has later been argued (see [27] for an up to date review) that the same modification could be applied to NC gauge models, which are known to present the same kind of problem. Thus, the Gribov-Zwanziger restriction would solve, at the same time, the problem of the zero-modes of the noncommutative Faddeev-Popov operator and the UV/IR mixing, clarifying the common origin of both problems.…”

“…Indeed it has been shown [4] that noncommutative QED similarly to commutative non-Abelian gauge theories, exhibits Gribov copies. For a review of noncommutative gauge theories see [27] and refs. therein.…”

Introduction to noncommutative field and gauge theory

“…As an example, we will review the case of noncommutative QED on R 3 𝜆 in Section 10. The existence of the so-called Gribov copies in noncommutative Moyal QED has been shown in [97][98][99][100]. In standard gauge theory, Gribov ambiguity is a feature of non-abelian gauge theories, consisting in the fact that they exhibit different field configurations that obey the same gaugefixing condition, yet they are related by a gauge transformation, meaning that they are on the same gauge orbit.…”

Photon polarization tensor on deformed spacetime: A four-photon-tadpole contribution

“…After two decades of development, many open questions still remain in the field of the perturbative quantization of the noncommutative (NC) field theories on Moyal space [2]. One of most renowned issues is the quadratic infrared (IR) divergence in the one-loop 1-PI two point function of the gauge fields [3,4,5].…”

“…Actually its existence in a very large and important subcategory of the deformed gauge theories on Moyal space, namely those defined via Seiberg-Witten (SW) map [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21], was confirmed not long ago [22,23]. Unlike the quadratic UV/IR mixing in the scalar field theory on Moyal space [24], this quadratic IR divergence does not have a quadratic UV divergent counterpart, and appears to be difficult to control [26,27,28,29,2].…”