Historical research guide · 0441 · 1995
Averaged energy conditions and quantum inequalities
L. H. Ford · Thomas A. Roman
In one minute
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.
Connection
This 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.
From the source
For 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
Let 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
In 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’
The 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
Applied 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
Three 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
The way in
https://doi.org/10.1103/PhysRevD.51.4277How to cite it
L. H. Ford, Thomas A. Roman (1995) Averaged energy conditions and quantum inequalities doi:10.1103/PhysRevD.51.4277