Abstract
We present a construction of $\kappa$-deformed complex scalar field theory with the objective of shedding light on the way discrete symmetries and CPT invariance are affected by the deformation. Our starting point is the observation that, in order to have an appropriate action of Lorentz symmetries on antiparticle states, these should be described by four-momenta living on the complement of the portion of de Sitter group manifold to which $\kappa$-deformed particle four-momenta belong. Once the equations of motions are properly worked out from the deformed action we obtain that particle and antiparticle are characterized by different mass-shell constraints leading to a subtle form of departure from CPT invariance. The remaining part of our work is dedicated to a detailed description of the action of deformed Poincar\'e and discrete symmetries on the complex field.
Highlights
It is commonly expected that the usual description of space-time as a smooth manifold is no longer reliable as we approach the Planck scale when quantum effects of the geometry can no longer be neglected
We laid down the basic ingredients for the construction of a complex field theory on κ-Minkowski space covariant under the action of deformed relativistic symmetries described by the κ-Poincarealgebra
The guiding principle which we followed in the definition of the field and its action was the requirement of an appropriate transformation of the former under the action of discrete symmetries
Summary
It is commonly expected that the usual description of space-time as a smooth manifold is no longer reliable as we approach the Planck scale when quantum effects of the geometry can no longer be neglected. Most of the proposed observational frameworks having sufficient sensitivity to capture effects of quantum gravity origin [9,10] were based on purely kinematical models, like, for example, the well-known case of measuring the time of flight of gamma-ray-burst photons of different energies [11,12] It has, been argued that κ deformations may have a subtle, and, in principle, measurable, effect on elementary particles, linked to the deformation of CPT symmetry [13]. This change of definition is a consequence of the assumed nice behavior of the field with respect to the discrete CPT transformations and leads to one of the major results of this paper, that the mass-shell relations of particles and antiparticles differ from each other, as a manifold the mass shell in both cases is the same hyperboloid in momentum space, as anticipated in Refs. We mention papers that influenced us [16,17,18,19,20,21,22,23,24,25,26,27] in working on this project, but we stress that the crucial aspects of the present construction, like the doubling of momentum space and insistence on the proper action of discrete symmetries, are new
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