Quantum energy inequalities in premetric electrodynamics

Premetric electrodynamics is a covariant framework for electromagnetism with a general constitutive relation. Its light-cone structure can be more complicated than that of Maxwell theory as is shown by the phenomenon of birefringence. We study the energy density of quantized premetric electrodynamics theories with linear constitutive relations admitting a single hyperbolicity double cone and show that averages of the energy density along the worldlines of suitable observers obey a quantum energy inequality (QEI) in states that satisfy a microlocal spectrum condition. The worldlines must meet two conditions: (a) the classical weak energy condition must hold along them, and (b) their velocity vectors have positive contractions with all positive frequency null covectors (we call such trajectories "subluminal"). After stating our general results, we explicitly quantize the electromagnetic potential in a translationally invariant uniaxial birefringent crystal. Since the propagation of light in such a crystal is governed by two nested light cones, the theory shows features absent in ordinary (quantized) Maxwell electrodynamics. We then compute a QEI bound for worldlines of inertial subluminal observers, which generalizes known results from the Maxwell theory. Finally, it is shown that the QEIs fail along trajectories that have velocity vectors which are timelike with respect to only one of the light cones.