STM-D-0441Paper1995Published and peer-reviewed

Averaged energy conditions and quantum inequalities

L. H. Ford · Thomas A. Roman

Abstract and summary · read the original at the source

In one page

This is the paper where Lawrence Ford and Thomas Roman, at the Tufts Institute of Cosmology, first pin a number on how far a quantum field can be pushed below the ambient vacuum level. Quantum field theory happily allows it — the Casimir effect and squeezed light are the two cases with experimental support — but Ford and Roman show the books balance over time. Sample the energy density an inertial observer measures against a bell-shaped window of width t-nought and the average cannot fall below a quantity that scales as one over t-nought to the fourth power. Push deeper and you get less time. Stretch the window to infinity and the classical averaged energy conditions drop out as a limiting case, so they are consequences rather than separate assumptions. In a two-dimensional model universe they find something further: the difference between an arbitrary state and the Casimir vacuum obeys its own bound, even where the Casimir vacuum alone obeys none.

Why it matters hereThis is the origin of the Ford-Roman quantum inequality — the specification that chapter 4’s warp and wormhole geometries are designed to satisfy rather than dodge, and the reason later designs are built around positive-energy shells. Chapter 2 gets the sharper picture: the vacuum floor can be dipped below, and the price of depth is duration. The simpler 1997 re-derivation, extended to the electromagnetic and massive fields, is on this site as stm-09a7d97555.

What it claims

01For a quantized free, minimally coupled, massless scalar field in four-dimensional Minkowski spacetime, the energy density along an inertial observer’s worldline, sampled against a Lorentzian window of proper-time width t-nought, is bounded below by minus three divided by thirty-two pi squared times t-nought to the fourth power — and by Lorentz invariance the bound holds in every inertial frame, so it can be written covariantly.Section 4, Eqs. 64 and 65
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02Let the sampling width go to infinity, so the window covers the whole worldline, and the quantum inequality returns the averaged weak energy condition along every timelike geodesic; taking the null limit of that result returns the averaged null energy condition in four-dimensional Minkowski spacetime. The averaged conditions are consequences of the quantum inequality, not independent postulates.Section 4, Eqs. 66 and 67
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03In a two-dimensional spatially compactified Minkowski universe the authors derive a covariant bound not on the energy density itself but on its difference between an arbitrary quantum state and the Casimir vacuum, and show that difference also obeys averaged-energy-condition-type integral conditions. This is the surprise of the paper: the Casimir vacuum expectation value alone satisfies neither the quantum inequalities nor the averaged conditions, yet the difference from it does.Abstract; Section 2, ‘A Difference Inequality in 2D’
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04The authors read that difference inequality physically. Of the two ways to deepen the effect in their model — shrinking the space, which changes the Casimir background, or changing the quantum state of the field — changing the state is not very effective at magnifying the depth beyond the background; shrinking the space makes the density more negative but over a correspondingly smaller region. Whether the two-dimensional model is suggestive of the general case is left open, along with whether similar difference inequalities exist in four-dimensional curved spacetimes.Section 5, second paragraph
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05Applied to warp drive, the bound — proved for free massless fields in flat spacetime — suggests that starting in flat space and collecting enough below-vacuum energy by the procedure of the reference cited would be extremely difficult, because compensating positive energy arrives before enough is gathered to curve space significantly. The authors name the route past their own result in the same paragraph: begin instead with other kinds of fields, not necessarily free or massless, in curved spacetime.Section 5, third paragraph
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06Three questions are left explicitly open: whether difference inequalities can be constructed in four-dimensional curved spacetimes where the averaged conditions fail, which would limit how far they can be violated; whether a scale-invariant bound exists in the regions of an evaporating black hole spacetime where the averaged null condition is violated, perhaps keyed to the hole’s mass; and what to make of Kuo and Ford’s result that below-vacuum energy densities in flat spacetime undergo large fluctuations, which makes naive use of semiclassical gravity suspect.Section 5, final three paragraphs
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Read it · abstract

Abstract

In this paper, connections are uncovered between the averaged weak (AWEC) and averaged null (ANEC) energy conditions, and quantum inequality restrictions (uncertainty principle-type inequalities) on negative energy. In two- and four-dimensional Minkowski spacetime, we examine quantized, free massless, minimally-coupled scalar fields. In a two-dimensional spatially compactified Minkowski universe, we derive a covariant quantum inequality-type bound on the difference of the expectation values of Tµν uµ uν 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 ANEC-type integral conditions. This is surprising, since it is well-known that the expectation value of Tµν uµ uν 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 two-dimensional 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 two-dimensional Minkowski spacetime. Lastly, we also derive a covariant quantum inequality bound on the energy density seen by an arbitrary inertial observer in four-dimensional Minkowski 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.

The way in

https://doi.org/10.1103/PhysRevD.51.4277Published as Physical Review D 51, 4277 (1995) under the APS default licence. The preprint is on arXiv as gr-qc/9410043, filed in October 1994 under arXiv’s assumed licence for submissions of that era, which does not grant redistribution, and no Creative Commons statement appears in the text — so this sheet carries the summary, the claims and the authors’ own abstract and sends the reader to the source. Preprint number TUTP-94-16, Tufts Institute of Cosmology. The simpler plane-wave re-derivation Ford and Roman published two years later is on this site as stm-09a7d97555.

How to cite it

L. H. Ford, Thomas A. Roman (1995) Averaged energy conditions and quantum inequalities. doi:10.1103/PhysRevD.51.4277

Where it sits in the curriculum

The metric, warp drives and wormholes What the vacuum is The evidence ladder Energy from the vacuum