Gauge theories on \(\kappa\)-Minkowski spaces: twist and modular operators. (English) Zbl 1437.83041
Summary: We discuss the construction of \(\kappa\)-Poincaré invariant actions for gauge theories on \(\kappa\)-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 \(\kappa\)-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 \(\kappa\)-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 \(\kappa\)-Poincaré invariance still holds.
MSC:
| 83C45 | Quantization of the gravitational field |
| 83C65 | Methods of noncommutative geometry in general relativity |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
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