Abstract
We consider a noncommutative field theory in three space-time dimensions, with space-time star commutators reproducing a solvable Lie algebra. The $\ensuremath{\star}$-product can be derived from a twist operator and it is shown to be invariant under twisted Poincar\'e transformations. In momentum space the noncommutativity manifests itself as a noncommutative $\ensuremath{\star}$-deformed sum for the momenta, which allows for an equivalent definition of the $\ensuremath{\star}$-product in terms of twisted convolution of plane waves. As an application, we analyze the $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ field theory at one loop and discuss its UV/IR behavior. We also analyze the kinematics of particle decay for two different situations: the first one corresponds to a splitting of space-time where only space is deformed, whereas the second one entails a nontrivial $\ensuremath{\star}$-multiplication for the time variable, while one of the three spatial coordinates stays commutative.
Highlights
The possibility that space-time is described by a noncommutative geometry is a fascinating idea that has deep physical motivations going back to Bronstein [1]
With θμν being a constant with the dimensions of length square. Field theories on these spaces have a peculiar behavior: some ultraviolet divergences are converted to infrared ones, a phenomenon known as UV/IR mixing [7]
The main result of the paper is the persistence of the UV/IR mixing at one loop for an interacting scalar field theory, λφ⋆4, which, not retaining Poincaresymmetry of its commutative analogue, is invariant under twisted Poincaretransformations
Summary
The possibility that space-time is described by a noncommutative geometry is a fascinating idea that has deep physical motivations going back to Bronstein [1] (for a more recent review see [2]). The most studied field theory is the one described by the Grönewold-Moyal product [3,4], which adapts to space-time the usual commutation rules of standard quantum mechanics [5], 1⁄2xμÃ;xν 1⁄4 iθμν; ð1:1Þ with θμν being a constant with the dimensions of length square Field theories on these spaces (for a review see [6]) have a peculiar behavior: some ultraviolet divergences are converted to infrared ones, a phenomenon known as UV/IR mixing [7]. It was shown in [10,11] that field theories based on the Grönewold-Moyal ⋆-product are invariant under the twisted θ-Poncaresymmetry Another example of a noncommutative (NC) space-time invariant under a quantum symmetry is the κ-Minkowski space-time. We conclude with a short summary and an appendix where the main calculations are performed
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