Classical and quantum flux energy conditions for quantum vacuum states

The classical energy conditions are known to not be fundamental physics—they are typically violated by semiclassical quantum effects. Consequently, some effort has gone into finding possible semiclassical replacements for the classical energy conditions—the most well developed being the nonlocal time-integrated Ford-Roman quantum inequalities. In the current article we shall instead develop classical and quantum versions of a local “flux energy condition” (FEC and QFEC) based on the notion of constraining the possible fluxes measured by timelike observers. The naive classical FEC will be seen to be satisfied in some situations, and even for some quantum vacuum states, while its quantum analogue (the QFEC) is satisfied (for naturally defined quantum vacuum states) under a rather wide range of conditions. The situation for completely general (nonvacuum) quantum states is less clear.