  General linear electrodynamics allow for an arbitrary linear constitutive
relation between the field strength two-form and induction two-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 bi-refringent linear optical medium.
