Quantization of general linear electrodynamics

General linear electrodynamics allow for an arbitrary linear constitutive relation between the field strength 2-form and induction 2-form density if crucial hyperbolicity and energy conditions are satisfied, which render the theory predictive and physically interpretable. Taking into account the higher-order polynomial dispersion relation and associated causal structure of general linear electrodynamics, we carefully develop its Hamiltonian formulation from first principles. Canonical quantization of the resulting constrained system then results in a quantum vacuum which is sensitive to the constitutive tensor of the classical theory. As an application we calculate the Casimir effect in a birefringent linear optical medium.

Figure 1

Causal structure of a typical bihyperbolic and energy-distinguishing polynomial dispersion relation. Dotted surfaces indicate sets of covectors and vectors that are null with respect to the polynomial $P$ and its dual $P^{\#}$, respectively. The cone $C$ of hyperbolic covectors and the cone $C^{\#}$ of observers both arise as hyperbolicity cones. Purely spatial directions, as seen by an observer with worldline tangent $T$, are those vectors annihilated by the preimage of $T$ under the Legendre map $L$.