Bimetric QED

We study, as a model of Lorentz symmetry breaking, the quantization and renormalization of an extension of QED in a flat spacetime where the photons and electrons propagate differently and do not share the same lightcone. We will refer to this model as bimetric QED (BIMQED). As a preliminary we discuss the formulation of electrodynamics in a premetric formalism showing nevertheless that there is, on the basis of a simple criteron, a preferred metric. Arising from this choice of metric is a Weyl-like tensor (WLT). The Petrov classification of the WLT gives rise to a corresponding classification of Lorentz symmetry breaking. We do not impose any constraint on the strength of the symmetry breaking and are able to obtain explicit dispersion relations for photon propagation in each of the Petrov classes. The associated birefringence appears in some cases as two distinct polarization dependent lightcones and in other cases as a more complicated structure that cannot be disentangled in a simple way. We show how in BIMQED the renormalization procedure can, in addition to its effect on standard parameters such as charge and mass, force the renormalization of the metrics and the WLT. Two particularly tractable cases are studied in detail for which we can obtain renormalization group flows for the parameters of the model together with an analysis of fixed point structure. Of course these results are consistent with previous studies but we are not constrained to treat Lorentz symmetry breaking as necessarily weak. As we found in a previous study of a scalar field theory model an acceptable causal structure for the model imposes constraints on relationship between the various lightcones in BIMQED.

Figure 1

Contributions to the photon propagator to $O(e_0^2)$.

Figure 2

Contributions to the electron propagator to $O(e_0^2)$.

Figure 3

Contributions to the electron-photon vertex to $O(e_0^3)$.

Figure 4

Renormalization group trajectories Petrov class O: rotationally invariant case.