skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: General proof of the averaged null energy condition for a massless scalar field in two-dimensional 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 stress-energy 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 two-dimensional 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 two-dimensional 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 two-point function of any state. Our results also can bemore » extended (with a restriction on states) to the massive scalar field in two-dimensional Minkowski spacetime and (with an additional restriction on states) to the (massless or massive) minimally coupled scalar field on four-dimensional Minkowski spacetime. In the case of a (curved) four-dimensional 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 Klein-Gordon field in de Sitter spacetime.« less

Authors:
;  [1]
  1. 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 0556-2821
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; QUANTUM FIELD THEORY; SCALAR FIELDS; ENERGY DENSITY; GEODESICS; KLEIN-GORDON EQUATION; MASSLESS PARTICLES; MINKOWSKI SPACE; SPACE-TIME; TWO-DIMENSIONAL 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 two-dimensional 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 two-dimensional 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 two-dimensional 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 two-dimensional 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 stress-energy 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 two-dimensional 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 two-dimensional 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 two-point function of any state. Our results also can be extended (with a restriction on states) to the massive scalar field in two-dimensional Minkowski spacetime and (with an additional restriction on states) to the (massless or massive) minimally coupled scalar field on four-dimensional Minkowski spacetime. In the case of a (curved) four-dimensional 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 Klein-Gordon field in de Sitter spacetime.},
doi = {10.1103/PhysRevD.44.403},
journal = {Physical Review, D (Particles Fields); (USA)},
issn = {0556-2821},
number = ,
volume = 44:2,
place = {United States},
year = {1991},
month = {7}
}