Abstract
We discuss the construction of κ-Poincaré invariant actions for gauge theories on κ-Minkowski spaces. We consider various classes of untwisted and (bi)twisted differential calculi. Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deformed translations, combined with a twisted extension of the notion of connection, we prove an algebraic relation between the various twists and the classical dimension d of the κ-Minkowski space(-time) ensuring the gauge invariance of the candidate actions for gauge theories. We show that within a natural differential calculus based on a distinguished set of twisted derivations, d=5 is the unique value for the classical dimension at which the gauge action supports both the gauge invariance and the κ-Poincaré invariance. Within standard (untwisted) differential calculi, we show that the full gauge invariance cannot be achieved, although an invariance under a group of transformations constrained by the modular (Tomita) operator stemming from the κ-Poincaré invariance still holds.
Highlights
The characterization of the κ-Minkowski space has been carried out in [3] using the Hopf algebra bicrossproduct structure of the κ-Poincare quantum algebra Pκ [4, 5] whose coaction on κ-Minkowski is covariant, as being the dual of a subalgebra of Pκ often called the “algebra of deformed translations”
Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deformed translations, combined with a twisted extension of the notion of connection, we prove an algebraic relation between the various twists and the classical dimension d of the κ-Minkowski space(-time) ensuring the gauge invariance of the candidate actions for gauge theories
Within standard differential calculi, we show that the full gauge invariance cannot be achieved, an invariance under a group of transformations constrained by the modular (Tomita) operator stemming from the κ-Poincare invariance still holds
Summary
We first recall the main properties of the star product used to model the κ-deformation of the Minkowski space stemming from a Weyl-Wigner quantization This is summarized in the subsection 2.1 as well as in the appendix A together with the main properties of the algebra modeling the κ-Minkowski space. Insisting on the κ-Poincare invariance necessarily implies that these action functionals are KMS weight, implying the appearance of a KMS condition on the algebra of fields. This is (equivalently) reflected by the fact that the Lebesgue integral defines a twisted trace with respect to the star product, whose twist operator is linked to the Tomita operator generating the modular group of ∗-automorphisms of the KMS structure [41]
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