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2.3 The role of other symmetries

There are many other symmetries that affect how Lorentz violation might manifest itself below the Planck scale. The standard model in Minkowski space is invariant under four main symmetries, three continuous and one discrete. There are two continuous spacetime symmetries, Lorentz symmetry and translation symmetry, as well as gauge and CPT symmetry. Supersymmetry can also have profound effects on how Lorentz violation can occur. Finally, including gravity means that we must take into account diffeomorphism invariance. The fate of these other symmetries in conjunction with Lorentz violation can often have significant observational ramifications.

2.3.1 CPT invariance

Lorentz symmetry is intimately tied up with CPT symmetry in that the assumption of Lorentz invariance is required for the CPT theorem [162]. Lorentz violation therefore allows for (but does not require) CPT violation, even if the other properties of standard quantum field theory are assumed. Conversely, however, CPT violation implies Lorentz violation for local field theories [134]. Furthermore, many observational results are sensitive to CPT violation but not directly to Lorentz violation. Examples of such experiments are kaon decay (see Section 5.5) and g-ray birefringence (see Section 6.3), both of which indirectly provide stringent bounds on Lorentz violation that incorporates CPT violation. Hence CPT tests are very important tools for constraining Lorentz violation. In effective field theory CPT invariance can explicitly be imposed to forbid a number of strongly constrained operators. For more discussion on this point see Section 4.3.

2.3.2 Supersymmetry

SUSY, while related to Lorentz symmetry, can still be an exact symmetry even in the presence of Lorentz violation. Imposing exact SUSY provides another custodial symmetry that can forbid certain operators in Lorentz violating field theories. If, for example, exact SUSY is imposed in the MSSM (minimal supersymmetric standard model), then the only Lorentz violating operators that can appear have mass dimension five or above [137Jump To The Next Citation Point]. Of course, we do not have exact SUSY in nature. The size of low dimension Lorentz violating operators in a theory with Planck scale Lorentz violation and low energy broken SUSY has recently been analyzed in [65Jump To The Next Citation Point]. For more discussion on this point see Section 4.3.

2.3.3 Poincaré invariance

In many astrophysics approaches to Lorentz violation, conservation of energy-momentum is used along with Lorentz violating dispersion relations to give rise to new particle reactions. Absence of these reactions then yields constraints. Energy/momentum conservation between initial and final particle states requires translation invariance of the underlying spacetime and the Lorentz violating physics. Therefore we can apply the usual conservation laws only if the translation subgroup of the Poincaré group is left unmodified. If Lorentz violation happens in conjunction with a modification of the rest of the Poincaré group, then it can happen that modified conservation laws must be applied to threshold reactions. This is the situation in DSR: All reactions that are forbidden by conservation in ordinary Lorentz invariant physics are also forbidden in DSR [146Jump To The Next Citation Point], even though particle dispersion relations in DSR would naively allow new reactions. The conservation equations change in such a way as to compensate for the modified dispersion relations (see Section 3.4). Due to this unusual (and useful) feature, DSR evades many of the constraints on effective field theory formulations of Lorentz violation.

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