Spatially averaged quantum inequalities do not exist in four-dimensional spacetime
L. H. Ford · Adam D. Helfer · Thomas A. Roman
Abstract and summary · read the original at the source
In one page
Quantum field theory lets the energy density at a point sit below the ambient vacuum level — the same accounting that gives the Casimir force. If nothing limited how deep that dip could go, traversable wormholes and warp bubbles would be easy, so physicists went looking for the limits. The ones they found are the quantum inequalities, and they constrain what a single observer measures along their own worldline over time. Larry Ford and Thomas Roman at the Tufts Institute of Cosmology, with Adam Helfer at the University of Missouri, ask whether there is a matching limit on space alone. They build an explicit counterexample: a normalizable state of a massless scalar field made from the vacuum superposed with pairs of particles whose momenta nearly cancel while their energies cannot. Averaged over a region of space at one instant, its energy density can be made as negative as you like, over a region as large as you like. The worldline inequality still holds, and the state’s total energy is still positive.
Why it matters hereChapter 4’s exotic-matter budget is usually quoted as if the quantum inequalities capped how much below-ambient energy density you may gather in one place; this paper shows that in four dimensions no such spatial cap exists, and hands the field a sharper question — what the spacetime-averaged bounds actually are. It is also chapter 2 in miniature: the vacuum is a structured medium whose energy bookkeeping is redistributive, not creative.
What it claims
01The authors construct a class of quantum states for a massless, minimally coupled free scalar field — the vacuum superposed with multi-mode two-particle states, cut off at momentum Lambda and normalizable in the limit where that cutoff is removed — which exhibit local energy densities below the ambient vacuum level.Abstract; Section II A, Eqs. 6 to 14
Published and peer-reviewed02In those states the spatially averaged energy density S of F, the integral of a normalized spatial sampling function against the renormalized energy density at one instant, is unbounded below for any sampling function of finite width — so no spatially averaged quantum inequality exists over bounded regions in four-dimensional Minkowski spacetime.Section III, first paragraph, Eqs. 42 and 43
Published and peer-reviewed03The mechanism is that the pairs of particles have nearly, but not exactly, opposite momenta: the three-momenta of a pair can cancel while their energies cannot, which lets the spatial scale of the energy density be far larger than its temporal scale. Two-dimensional spacetime is the special case where the pair’s momenta must be exactly parallel or anti-parallel, which is why spatial bounds exist there and not here.Section III, second paragraph; Figure 1; Eqs. 21 and 24
Published and peer-reviewed04The same states still satisfy the worldline quantum inequality, and the total energy of any state remains non-negative. Integrating over all space removes the cross term between the vacuum and the two-particle component, leaving only the non-negative term — the compensating positive energy is real, it simply sits arbitrarily far from the negative region.Section I, Eqs. 2 and 3; Section III, paragraphs 1 and 4
Published and peer-reviewed05The unbounded negativity is not an artefact of a sharp-edged sampling region: in these examples the energy density is uniformly negative throughout the sampled region, which shows that the earlier edge-effect account of Garfinkle’s box result is one explanation but not the only one.Section III, paragraphs 5 and 6
Published and peer-reviewed06Spacetime averaged quantum inequalities do exist, so their bounds must diverge as the temporal averaging scale goes to zero; the authors conjecture that this divergence matches the cutoff divergence of the spatial average, and name the two live programmes — finding the spacetime averaged inequalities, and extracting more from the worldline ones — as both under investigation.Section III, final three paragraphs, Eqs. 44 and 45
What to watch
Read it · abstract
Abstract
We construct a particular class of quantum states for a massless, minimally coupled free scalar field which are of the form of a superposition of the vacuum and multi-mode two-particle states. These states can exhibit local negative energy densities. Furthermore, they can produce an arbitrarily large amount of negative energy in a given region of space at a fixed time. This class of states thus provides an explicit counterexample to the existence of a spatially averaged quantum inequality in four-dimensional spacetime.
The way in
https://doi.org/10.1103/PhysRevD.66.124012Published as Physical Review D 66, 124012 (2002) by L. H. Ford and Thomas A. Roman of the Tufts Institute of Cosmology with Adam D. Helfer of the University of Missouri, and supported by NSF grants Phy-9800965 and Phy-9988464. The author version is free to read on arXiv as gr-qc/0208045, posted 14 August 2002 and revised 24 October 2002, but that posting carries arXiv’s assumed licence for legacy submissions and the version of record carries the APS default licence — neither is a Creative Commons licence. So this page holds the summary, the claims and the authors’ own abstract, and sends the reader to the source. The claims are located against the arXiv version’s sections, equations and figure.
How to cite it
L. H. Ford, Adam D. Helfer, Thomas A. Roman (2002) Spatially averaged quantum inequalities do not exist in four-dimensional spacetime. doi:10.1103/PhysRevD.66.124012
Where it sits in the curriculum