Abstract
A natural star product for 4-d $\kappa$-Minkowski space is used to investigate various classes of $\kappa$-Poincar\'e invariant scalar field theories with quartic interactions whose commutative limit coincides with the usual $\phi^4$ theory. $\kappa$-Poincar\'e invariance forces the integral involved in the actions to be a twisted trace, thus defining a KMS weight for the noncommutative (C*-)algebra modeling the $\kappa$-Minkowski space. The associated modular group and Tomita modular operator are characterized. In all the field theories, the twist generates different planar one-loop contributions to the 2-point function which are at most UV linearly diverging. Some of these theories are free of UV/IR mixing. In the others, UV/IR mixing shows up in non-planar contributions to the 2-point function as a polynomial singularity at exceptional zero external momenta while staying finite at non-zero external momenta. These results are discussed together with the possibility for the KMS weight relative to the quantum space algebra to trigger the appearance of KMS state on the algebra of observables.
Highlights
It is widely believed that the classical notion of spacetime is no longer adequate at the Planck scale to reconcile gravity with quantum mechanics
The κ-Minkowski spacetime appears in the physics literature to be one of the most studied noncommutative spaces with Lie algebra type noncommutativity and is sometimes regarded as a good candidate for a quantum space-time to be involved in a description of quantum gravity at least in some limit
A well-controlled star product for κ-Minkowski space is obtained from the representations of the convolution algebra of the affine group which here replaces the Heisenberg group underlying the popular quantization of a phase space
Summary
It is widely believed that the classical notion of spacetime is no longer adequate at the Planck scale to reconcile gravity with quantum mechanics. Note that the original κ-Minkowski space (2.1) (which we consider in this paper) does not fit in that description and breaks the classical relativity principle This leads us to the other approach widely studied in the literature, namely the extension of the usual notion of Lie algebra symmetries to the one of (deformed) Hopf algebra symmetries aiming to encode the new (canonical) symmetries for the quantum space-times. As far as we know, this product was amazingly not further exploited in the study of NCFT on κ-Minkowski space, despite its relatively simple expression and the associated tools of group harmonic analysis which make him well adapted to the study of quantum field theories
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