CPT breaking in noncommutative Magueijo-Smolin model

TL;DR

This work analyzes CPT-related issues in a noncommutative spacetime realization of doubly special relativity proposed by Magueijo and Smolin, focusing on the Granik basis linked to the κ-Poincaré algebra. It shows that the Casimir $C=\frac{-p_0^2+p_i^2}{(1-p_0/\kappa)^2}$ is not invariant under $p_0\to -p_0$, which leads to a breaking of standard charge conjugation and potentially CPT symmetry in the first-quantized and quantum-field-theoretic formulations. The Hopf-algebra structure reveals that the antipode-based definitions of discrete symmetries do not generally restore invariance in the Granik basis, though an alternative Kowalski–Glikman–Nowak realization can formalize CPT invariance, at the cost of awkward conjugation rules. The results underscore the tension between noncommutative geometry realizations and discrete symmetries, with implications for Planck-scale phenomenology and the interpretation of antiparticles in DSR models.

Abstract

We review an instance of noncommutative geometry based on a specific realization of the model of doubly special relativity proposed by Magueijo and Smolin (MS) on noncommutative spacetime. In particular, we discuss the Hopf algebra associated to it, which has not been considered in the literature till now. We show that the momentum sector of this model can be viewed as a particular basis of the \kp model. An interesting property is that the MS Hamiltonian is not invariant under the reversal of the sign of the energy, and in particular it is not invariant under the standard definition of charge conjugation. Therefore, if following Dirac one identifies the negative energy states with antiparticles, their mass differs from that of particles. We examine the possible consequences of this fact in the context of first quantization and discuss its interpretation from the point of view of quantum field theory, taking into account possible alternative definitions of charge conjugation proposed in the noncommutative framework.