Abstract
We present a class of interacting nonlocal quantum field theories, in which the CPT invariance is violated while the Lorentz invariance is present. This result rules out a previous claim in the literature that the CPT violation implies the violation of Lorentz invariance. Furthermore, there exists the reciprocal of this theorem, namely that the violation of Lorentz invariance does not lead to the CPT violation, provided that the residual symmetry of Lorentz invariance admits the proper representation theory for the particles. The latter occurs in the case of quantum field theories on a noncommutative space–time, which in place of the broken Lorentz symmetry possesses the twisted Poincaré invariance. With such a CPT-violating interaction and the addition of a C-violating (e.g., electroweak) interaction, the quantum corrections due to the combined interactions could lead to different properties for the particle and antiparticle, including their masses.
Highlights
Lorentz symmetry and the CP T invariance are two of the most fundamental symmetries of Nature, whose violation has not yet been observed
The worst is that in such a model, where Lorentz invariance is violated by the free fields, there is no concept of spin to start with altogether
The quantum treatment of such theories as well should be performed through the use of Yang-Feldman equation [24] with the fields, denoted by ΦH (x), in the Heisenberg picture. With such a CP T -violating interaction as in (3.1)-(3.5) or (4.8), and the addition of a Cviolating interaction, the quantum corrections due to the combined interactions could lead to different properties for the particle and antiparticle, including their masses
Summary
Lorentz symmetry and the CP T invariance are two of the most fundamental symmetries of Nature, whose violation has not yet been observed. Later on the theorem was proven by Jost [3] (see [4,5,6]) within the axiomatic formulation of quantum field theory without reference to any specific form of interaction. The issue was taken up in relation with the Lorentz symmetry by Greenberg [17], the conclusion of Greenberg’s analysis being that CP T violation implies violation of Lorentz invariance. This result was given as a ”theorem”, the dispute on the validity of which is the subject of this Letter. Is the violation complete or is any subgroup of Lorentz symmetry left, which should have the needed spin-representations to which the particles are assigned? Does the corresponding theory which violates both CP T and Lorentz invariance contain fields with a plausible description in terms of equations of motion?
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