General proof of the averaged null energy condition for a massless scalar field in twodimensional curved spacetime
Abstract
It is by now well known that the standard local (i.e., pointwise) energy conditions always can be violated in quantum field theory in curved (and flat) spacetime, even when these energy conditions hold for the corresponding classical field. Nevertheless, some global constraints on the stressenergy tensor may exist. Indeed recent work has shown that the averaged null energy condition (ANEC), which requires the positivity of energy suitably averaged along null geodesics, holds for a wide class of states of a minimally coupled scalar field on Minkowski spacetime, and also (in the massless case) on a wide class of states in curved twodimensional spacetimes satisfying certain asymptotic regularity properties. In this paper, we strengthen these results by proving that, for the massless scalar field in an arbitrary globally hyperbolic twodimensional spacetime, the ANEC holds for all Hadamard states along any complete, achronal null geodesic. In our analysis, the general, algebraic notion of state'' is used, so, in particular, it is not even assumed that our state belongs to any Fock representation. Our proof shows that the ANEC is a direct consequence of the general positivity condition which must hold for the twopoint function of any state. Our results also can bemore »
 Authors:

 Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637 (US)
 Publication Date:
 OSTI Identifier:
 5543299
 Resource Type:
 Journal Article
 Journal Name:
 Physical Review, D (Particles Fields); (USA)
 Additional Journal Information:
 Journal Volume: 44:2; Journal ID: ISSN 05562821
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; QUANTUM FIELD THEORY; SCALAR FIELDS; ENERGY DENSITY; GEODESICS; KLEINGORDON EQUATION; MASSLESS PARTICLES; MINKOWSKI SPACE; SPACETIME; TWODIMENSIONAL CALCULATIONS; DIFFERENTIAL EQUATIONS; ELEMENTARY PARTICLES; EQUATIONS; FIELD THEORIES; MATHEMATICAL SPACE; PARTIAL DIFFERENTIAL EQUATIONS; SPACE; WAVE EQUATIONS; 645400*  High Energy Physics Field Theory
Citation Formats
Wald, R, and Yurtsever, U. General proof of the averaged null energy condition for a massless scalar field in twodimensional curved spacetime. United States: N. p., 1991.
Web. doi:10.1103/PhysRevD.44.403.
Wald, R, & Yurtsever, U. General proof of the averaged null energy condition for a massless scalar field in twodimensional curved spacetime. United States. doi:10.1103/PhysRevD.44.403.
Wald, R, and Yurtsever, U. Mon .
"General proof of the averaged null energy condition for a massless scalar field in twodimensional curved spacetime". United States. doi:10.1103/PhysRevD.44.403.
@article{osti_5543299,
title = {General proof of the averaged null energy condition for a massless scalar field in twodimensional curved spacetime},
author = {Wald, R and Yurtsever, U},
abstractNote = {It is by now well known that the standard local (i.e., pointwise) energy conditions always can be violated in quantum field theory in curved (and flat) spacetime, even when these energy conditions hold for the corresponding classical field. Nevertheless, some global constraints on the stressenergy tensor may exist. Indeed recent work has shown that the averaged null energy condition (ANEC), which requires the positivity of energy suitably averaged along null geodesics, holds for a wide class of states of a minimally coupled scalar field on Minkowski spacetime, and also (in the massless case) on a wide class of states in curved twodimensional spacetimes satisfying certain asymptotic regularity properties. In this paper, we strengthen these results by proving that, for the massless scalar field in an arbitrary globally hyperbolic twodimensional spacetime, the ANEC holds for all Hadamard states along any complete, achronal null geodesic. In our analysis, the general, algebraic notion of state'' is used, so, in particular, it is not even assumed that our state belongs to any Fock representation. Our proof shows that the ANEC is a direct consequence of the general positivity condition which must hold for the twopoint function of any state. Our results also can be extended (with a restriction on states) to the massive scalar field in twodimensional Minkowski spacetime and (with an additional restriction on states) to the (massless or massive) minimally coupled scalar field on fourdimensional Minkowski spacetime. In the case of a (curved) fourdimensional spacetime with a bifurcate Killing horizon, our proof also extends to establish the ANEC for the null geodesic generators of the horizon (provided that there exists a stationary Hadamard state of the field). This latter result implies that the ANEC must hold for the massive KleinGordon field in de Sitter spacetime.},
doi = {10.1103/PhysRevD.44.403},
journal = {Physical Review, D (Particles Fields); (USA)},
issn = {05562821},
number = ,
volume = 44:2,
place = {United States},
year = {1991},
month = {7}
}