  We study, as a model of Lorentz symmetry breaking, the quantisation and
renormalisation 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 pre-metric 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 polarisation dependent
lightcones and in other cases as a a more complicated structure that cannot be
disentangled in a simple way.
  We show how in BIMQED the renormalisation procedure can, in addition to its
effect on standard parameters such as charge and mass, force the
renormalisation of the metrics and the WLT. Two particularly tractable cases
are studied in detail for which we can obtain renormalisation 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.
