UV/IR mixing as a twisted Poincaré anomaly

Abstract

We analyze symmetries of the 1-loop effective action of φ 4 noncommutative field theory. It is shown, that despite the twisted Poincaré invariance of the classical noncommutative action, its 1-loop quantum counterpart lacks this invariance. Though Noether analysis of the model is somewhat obscure, it is still possible to interpret this symmetry breaking as a quantum anomaly due to inappropriate choice of the quantization method. * apinzul@unb.br 1 IntroductionBased on a quite general argument [1], it is believed that one or another form of a noncommutative (NC) space-time should emerge as a quasi-classical approximation to quantum gravity in a sense that quantum fields on such a space-time will probe the quantum gravity scale through noncommutativity. This observation, as well as the appearance of NC field theory in super strings [2], sparked considerable interest in noncommutative physics. Most of the efforts have been devoted to the analysis of theories on a NC space-time with constant noncommutativity, a so-called Moyal space. (Literature on the subject is vast, for a review see, e.g. [3]) This is the simplest possible type of noncommutativity, defined by the following commutation relation between the coordinatesHere θ µν gives the scale of quantum gravity, usually it is thought to be related to the Planck scale. It is clear that if θ µν is some non-dynamical constant, which is the case in string theory, then (1) breaks Lorentz invariance.The study of Lorentz-breaking theories has a long history by now [4]. Up to the first order in θ µν , NC quantum field theory with constant noncommutativity is just a special case of the general situation (e.g., see discussion in [5]). The quantization of NC field theories (NCFT) had been performed within this framework, i.e quantization of a Lorentz violating theory. Perturbative analysis led to some noncommutative corrections but also, what is more important, to a quite universal property of NCFT -UV/IR mixing [6]. (Though, see some recent proposals on models without UV/IR mixing [7][8][9].) This property, which is due to the non-local nature of noncommutativity, put an end to hopes that noncommutativity (at least in the simple form of a constant θ µν ) would serve as a UV regulator, which was the major motivation for one of the first works on a NC space-time [10].After works [11,12], another possibility to look at NCFT had become possible: instead of thinking in terms of broken Lorentz invariance, one could now think about NCFT as a theory with preserved, but twisted Lorentz symmetry.1 On the classical level, this is just a point of view that does not add anything new to the picture. But the transition to quantization changes the situation drastically: one has to maintain this new twisted symmetry on quantum level. Essentially, there exist two approaches to quantization of NCFT with twisted Lorentz symmetry:• The quantization should be done according to the usual rules. In other words, the usual Feynman rules, derived from the path integral with the standard quant...

Renormalization of the Commutative Scalar Theory with Harmonic Term to All Orders

“…The resulting theory is translationinvariant, but does not possess the properties exposed above for the Grosse-Wulkenhaar model (see also [26,27,28,29]). Some new features can also be found in other noncommutative scalar theories [30,31]. A noncommutative gauge theory involving a harmonic term has been constructed in [32,33], which is therefore strongly related to the Grosse-Wulkenhaar model.…”

Scattering in a three-dimensional fuzzy space

“…In the case of 3+1-dimensional quantum field theories, it was recently shown that the Lorentz symmetry is restored upon twisting [6]. Despite this insight several outstanding and controversial issues still plague the twisted implementation of the Lorentz symmetry [7]. The first difficulty one encounters is to carry out the standard Noether analysis and to identify conserved charges for the twisted Lorentz symmetry.…”

“…This has rather drastic consequences such as the absence of UV/IR mixing. Indeed, in [7] it is argued that UV/IR mixing may be related to a quantum anomaly of the Lorentz group, which is very closely related to the choice of functional integral measure. At this point there seems to be no consensus between these different points of view.…”

“…At this point there seems to be no consensus between these different points of view. See [7] and references therein for further discussions.…”

Renormalization of noncommutative quantum field theories

“…Because of (5) the quantum fields written on GM plane, unlike ordinary quantum fields, follow an unusual statistics called twisted statistics. Noncommutative field theories without twisted commutation relations do not preserve the classical twisted Poincaré invariance at quantum level and suffer from UV/IR mixing [8]. The twisted statistics is a novel feature of fields on GM plane.…”