Averaged energy conditions and quantum inequalities
Abstract
In this paper, connections are uncovered between the averaged weak (AWEC) and averaged null (ANEC) energy condtions, and quantum inequality restrictions (uncertaintyprincipletype inequalities) on negative energy. In two and fourdimensional Minkowski spacetime, we examine quantized, free massless, minimally coupled scalar fields. In a twodimensional spatially compactified Minkowski universe, we derive a covariant quantum inequalitytype bound on the difference of the expectation values of T_{μν}u^{μ}u^{ν} in an arbitrary quantum state and in the Casimir vacuum state. From this bound, it is shown that the difference of expectation values also obeys AWEC and ANECtype integral conditions. This is surprising, since it is well known that the expectation value of T_{μν}u^{μ}u^{ν} in the renormalized Casimir vacuum state alone satisfies neither quantum inequalities nor averaged energy conditions. Such ``difference inequalities,'' if suggestive of the general case, might represent limits on the degree of energy condition violation that is allowed over and above any violation due to negative energy densities in a background vacuum state. In our simple twodimensional model, they provide physically interesting examples of new constraints on negative energy which hold even when the usual AWEC, ANEC, and quantum inequality restrictions fail. In the limit when the size of the space is allowed to go to infinity, we derive quantum inequalities for timelike and null geodesics which, in appropriate limits, reduce to AWEC and ANEC in ordinary twodimensional Minskowski spacetime. Lastly, we also derive a covariant quantum inequality bound on the energy density seen by an arbitrary inertial observer in fourdimensional Minskowski spacetime. The bound implies that any inertial observer in flat spacetime cannot see an arbitrarily large negative energy density which lasts for an arbitrarily long period of time. From this latter bound, we derive AWEC and ANEC.
 Publication:

Physical Review D
 Pub Date:
 April 1995
 DOI:
 10.1103/PhysRevD.51.4277
 arXiv:
 arXiv:grqc/9410043
 Bibcode:
 1995PhRvD..51.4277F
 Keywords:

 04.62.+v;
 03.70.+k;
 11.10.Ef;
 Quantum field theory in curved spacetime;
 Theory of quantized fields;
 Lagrangian and Hamiltonian approach;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory
 EPrint:
 20pp, plain LATEX, TUTP9416