STM-D-0839Paper2003Published and peer-reviewed

Null energy conditions in quantum field theory

Christopher J. Fewster · Thomas A. Roman

Abstract and summary · read the original at the source

In one page

General relativity keeps its predictive power by assuming matter is well behaved — in particular that the energy measured along a light ray never runs negative. Quantum fields do not obey that rule, so for two decades physicists looked for the weaker rule that survives: a quantum inequality, which allows the energy to dip below the ambient vacuum level but says exactly how deep and for how long. Christopher Fewster and Thomas Roman ask whether such a bound exists along a light ray in ordinary four-dimensional spacetime, and answer it by construction. They build a family of quantum states — the vacuum plus a two-particle piece whose momenta crowd into a narrowing cone around the ray — that drives the averaged null energy as far down as you please, while the states themselves converge on the vacuum. In two dimensions the bound does exist; four dimensions are different. What does survive is a bound along timelike paths, which means a deep dip on one light ray has to be paid for on the rays beside it.

Why it matters hereChapter 4 lives or dies on the rulebook for below-ambient-vacuum energy, because that rulebook is what decides how much of a warp or wormhole geometry the quantum vacuum will actually support — and this is the paper that says where the rulebook has no page. Along a single light ray in four dimensions there is no bound at all; the constraint moves instead to the transverse direction, where a timelike-averaged inequality still bites. Chapter 2 gets the mechanism, because the states doing the work are nothing exotic — an ordinary superposition of the vacuum with a two-particle excitation, which is the structured vacuum behaving as the theory allows. The warp-drive spacetime whose energy conditions this bears on is treated on the site at /library/stm-d33809e031.

What it claims

01For the quantised massless minimally coupled real scalar field in four-dimensional Minkowski space, weighted averages of the null-contracted stress-energy tensor along a null geodesic are unbounded from below over the class of Hadamard states — so there is no quantum null energy inequality along a null worldline in four dimensions.Theorem II.1, Sect. II A
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02The dimension is what matters. An affine-parameter-invariant bound of exactly this kind does exist in two-dimensional flat spacetime, and a scaling argument in the introduction shows why the extension to higher dimensions was expected to be problematic: the left side of such a bound scales as the square of the rescaling factor while dimensional analysis puts the right side at the factor raised to the spacetime dimension.Sect. I, the two-dimensional bound and the scaling argument that follows it
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03The counterexample states are superpositions of the vacuum with multimode two-particle states whose three-momenta lie in a cone about the chosen null vector; the construction pushes the momenta arbitrarily high while shrinking the cone to zero, and the striking feature is that the sampled null energy runs to minus infinity while the sequence of states converges on the vacuum itself.Sect. II A, the vacuum-plus-two-particle vector; Sect. IV Conclusion
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04Those same states still satisfy the averaged null energy condition: integrated over the whole geodesic rather than sampled with a weight, the renormalised null-contracted stress-energy is non-negative, so the nonexistence of a null-worldline quantum inequality does not amount to unbounded violation of the averaged condition.Sect. II; Sect. IV Conclusion
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05In any globally hyperbolic spacetime, and for massive as well as massless fields, the null-contracted stress-energy averaged along a timelike worldline does obey a quantum inequality; evaluated in four-dimensional Minkowski space with a Gaussian sampling function of width tau, the massless bound falls off as the inverse fourth power of tau, taking the same form as the corresponding weak-energy bound.Theorem III.1 and the Minkowski evaluation, Sect. III
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06Two consequences the authors state as open questions: large below-ambient values along one null geodesic must be compensated by positive values on neighbouring geodesics, since the transverse extent is limited by the timelike bound; and no Penrose-type singularity theorem can be proved from a null-worldline quantum inequality, leaving transversely smeared inequalities of the Flanagan–Wald kind as the route under investigation.Sect. IV Conclusion, closing paragraphs
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Abstract

For the quantised, massless, minimally coupled real scalar field in four-dimensional Minkowski space, we show (by an explicit construction) that weighted averages of the null-contracted stress-energy tensor along null geodesics are unbounded from below on the class of Hadamard states. Thus there are no quantum inequalities along null geodesics in four-dimensional Minkowski spacetime. This is in contrast to the case for two-dimensional flat spacetime, where such inequalities do exist. We discuss in detail the properties of the quantum states used in our analysis, and also show that the renormalized expectation value of the stress energy tensor evaluated in these states satisfies the averaged null energy condition (as expected), despite the nonexistence of a null-averaged quantum inequality. However, we also show that in any globally hyperbolic spacetime the null-contracted stress energy averaged over a timelike worldline does satisfy a quantum inequality bound (for both massive and massless fields). We comment briefly on the implications of our results for singularity theorems.

Christopher J. Fewster, Department of Mathematics, University of York; Thomas A. Roman, Department of Physics and Earth Sciences, Central Connecticut State University. Physical Review D 67, 044003 (2003).

(Abstract only. The complete paper is free to read at https://arxiv.org/abs/gr-qc/0209036 and via https://doi.org/10.1103/PhysRevD.67.044003 — see the rights note for why the full text is not reproduced here.)

The way in

https://doi.org/10.1103/PhysRevD.67.044003Published as Physical Review D 67, 044003 (2003); dated 10 September 2002 and revised 25 November 2002, issued as Erwin Schrödinger Institute preprint ESI 1205 (2002). LICENCE. Checked directly rather than taken from an aggregator label: the journal version carries the APS default licence, and the green open-access copy, arXiv gr-qc/0209036v2, is filed under arXiv’s assumed licence for 1991 to 2003 submissions, which grants arXiv distribution rights only and no Creative Commons re-use. So this page carries the summary, the claims and the authors’ own abstract, and sends the reader to the full text at the source. The summary and claims were written from the complete published text.

How to cite it

Christopher J. Fewster, Thomas A. Roman (2003) Null energy conditions in quantum field theory. doi:10.1103/PhysRevD.67.044003

Where it sits in the curriculum

The metric, warp drives and wormholes What the vacuum is