# Quantum Energy Inequalities in Pre-Metric Electrodynamics

September 06, 2017

Pre-metric electrodynamics is a covariant framework for electromagnetism with
a general constitutive law. Its lightcone 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 pre-metric electrodynamics theories with
linear constitutive laws 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
lightcones, 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 lightcones.

Keywords:

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