Absolute Quantum Energy Inequalities in Curved Spacetime

Quantum Energy Inequalities (QEIs) are results which limit the extent to which the smeared renormalized energy density of the quantum field can be negative, when averaged along a timelike curve or over a more general timelike submanifold in spacetime. On globally hyperbolic spacetimes the minimally-coupled massive quantum Klein–Gordon field is known to obey a ‘difference’ QEI that depends on a reference state chosen arbitrarily from the class of Hadamard states. In many spacetimes of interest this bound cannot be evaluated explicitly. In this paper we obtain the first ‘absolute’ QEI for the minimally-coupled massive quantum Klein–Gordon field on four dimensional globally hyperbolic spacetimes; that is, a bound which depends only on the local geometry. The argument is an adaptation of that used to prove the difference QEI and utilizes the Sobolev wave-front set to give a complete characterization of the singularities of the Hadamard series. Moreover, the bound is explicit and can be formulated covariantly under additional (general) conditions. We also generalise our results to incorporate adiabatic states. Dedicated to Klaus Fredenhagen on the occasion of his 60th birthday. Submitted: February 27, 2007. Accepted: November 22, 2007. Calvin J. Smith: Address from 1 January 2007.