Gauge Theories on Deformed Spaces

Abstract

Abstract. The aim of this review is to present an overview over available models and approaches to non-commutative gauge theory. Our main focus thereby is on gauge models formulated on flat Groenewold-Moyal spaces and renormalizability, but we will also review other deformations and try to point out common features. This review will by no means be complete and cover all approaches, it rather reflects a highly biased selection.

On the Physics of the Minimal Length: The Question of Gauge Invariance

“…Note that our discussion generalizes the notion of gauge invariance found in the context of non-commutative field theory (NCFT) [97][98][99][100][101][102][103] . Given the geometric formulation presented in the appendix, it should be possible to deform the usual discussion from NCFT based on the Moyal product by realizing gauge transformations in terms of the isometries of the curved momentum space, associated with either a spherical or hyperbolic geometry.…”

“…The question of gauge invariance in the presence of a minimal length has been addressed in the context of non-commutative field theories (NCFT) [97][98][99][100][101][102][103] . In that case the minimal length can be related, in a particular realization, to an effective magnetic length.…”

Field Theory with Coordinate Dependent Noncommutativity

“…Over the last twenty years a great amount of work has been devoted to the study of structural aspects and phenomenological applications of field theories on the simplest quantized space, namely the Groenewold-Moyal (or θ-deformed) space [61,85], e.g., see [4,9,21,62,91,103,104,108,110,111] and references therein for a review. On this space the theories are formulated in terms of ordinary functions by means of a deformed associative product, the so-called Groenewold Here, the noncommutativity parameters θ µν = −θ νµ are real constants and expression (1.1) implies that the space-time coordinates x µ fulfill a Heisenberg-type algebra,…”

“…One refers to this case as the canonical deformation. Quite generally, the interest in this and more general deformed spaces was triggered by their link with quantum gravity, quantum geometry, string theory and D-branes, matrix models, the quantum Hall effect as well as other physical systems (see, e.g., [4,9,21,41,62,67,91,103,104,108,110,111] and references therein). Moreover, it was realized that quantum field theory is in some ways better behaved on noncommutative space-time than in ordinary space-time, e.g., see [91] for an assessment.…”

A new approach to non-commutative U ⋆ (N) gauge fields

“…For a review on the topic see refs. [5][6][7]. Only some years ago, Grosse and Wulkenhaar were able to resolve the UV/IR mixing problem in the case of a scalar field theory by adding an oscillator-like term to the (Euclidean) action [8,9], thereby rendering it renormalizable to all orders of perturbation theory [10][11][12].…”