The quantum interest conjecture
L. H. Ford · Thomas A. Roman
Abstract and summary · read the original at the source
In one page
Lawrence Ford and Thomas Roman turn a hunch from an earlier paper into a theorem. Quantum field theory lets a field sit below the ambient vacuum level for a while, but it keeps the books: their quantum inequalities say the deeper the dip, the briefer it must be. Here they show the ledger carries one more line. Separate a pulse below the vacuum level from the compensating pulse above it, and the second pulse must be strictly larger than the first — with a surcharge that grows as the gap between them widens. They call it quantum interest: an energy loan is always repaid, and repaid with interest set by the size and the duration of the loan. They prove it for massless scalar fields in two and four dimensional flat spacetime, show that a moving mirror pays that interest through the Doppler shift of its own second pulse, and derive a hard ceiling on how far apart the two pulses can be placed at all.
Why it matters hereChapter 4’s wormhole and warp geometries and chapter 6’s vacuum-energy devices both run on the same ledger — how far below the ambient vacuum level a field can be pushed, for how long, and what it costs to put back. This is the paper that adds the interest term to that ledger and proves it rather than assuming it, which is why every serious later warp design either satisfies it or says explicitly how it means to. It sits directly downstream of the two Ford and Roman quantum-inequality papers already in this library, stm-5e822858ce and stm-09a7d97555.
What it claims
01Quantum field theory permits energy densities below the ambient vacuum level, violating the classical pointwise energy conditions, and two such states have been produced in the laboratory — the Casimir effect and squeezed states of light — although the energy densities themselves have not been directly measured.Section 1, first paragraph
Settled physics02The quantum inequalities constrain how far below the vacuum level a sampled energy density can go: for a massless minimally coupled scalar field in four dimensional Minkowski spacetime, folded against a Lorentzian sampling function of width t-nought, the sampled density is bounded below by minus three over thirty-two pi squared t-nought to the fourth power, and later work by Flanagan and by Fewster and Eveson generalized this to arbitrary sampling functions, including ones with compact support — the freedom this paper exploits.Section 1, Eqs. 1, 3 and 4
Published and peer-reviewed03Using a compactly supported sampling function, the quantum inequalities force a maximum time separation between the two pulses, so the larger the pulse below the vacuum level, the sooner its compensating pulse must arrive: in two dimensions the separation is at most about zero point one three one divided by the pulse magnitude, and in four dimensions at most about zero point three three eight times the cube root of the collecting area divided by the pulse magnitude.Section 3, Eqs. 25 to 29
Published and peer-reviewed04A mirror that starts accelerating emits a delta-function pulse below the vacuum level and, when its acceleration is halted, a compensating pulse above it; working the two-dimensional case exactly gives an overcompensation fraction that is a monotonically increasing function of the product of pulse magnitude and separation, approaching twenty-four pi times that product in the slow limit. The physical cause is plain in this example — the second pulse is Doppler shifted because the mirror is already moving when it emits it.Section 2, Eqs. 14 to 16 and Figure 1
Published and peer-reviewed05The conjecture is proved rather than assumed: in four dimensional flat spacetime, setting the overcompensation to exactly zero makes the quantum inequality demand that the product of pulse magnitude and the cube of the separation go to zero, which leaves no non-trivial pulses at all — so the overcompensation must be strictly greater than zero. In the small-interest limit it is bounded below by four twenty-sevenths of the square of that product divided by the inequality constant times pi times the area, which means the interest grows as the sixth power of the pulse separation at fixed pulse magnitude.Section 4.1, Eqs. 33 to 39
Published and peer-reviewed06Two questions are left open and named: the optimal value of the maximum pulse separation is not known, nor is which sampling function maximizes the interest for a fixed separation. Moving mirrors pay far more interest than the proved bounds require — an overcompensation fraction of about four at half the mirror’s own maximum separation — so either that state is unusually generous or the general bounds are not yet tight.Section 5, Conclusions
What to watch
Read it · abstract
Abstract
Although quantum field theory allows local negative energy densities and fluxes, it also places severe restrictions upon the magnitude and extent of the negative energy. The restrictions take the form of quantum inequalities. These inequalities imply that a pulse of negative energy must not only be followed by a compensating pulse of positive energy, but that the temporal separation between the pulses is inversely proportional to their amplitude. In an earlier paper we conjectured that there is a further constraint upon a negative and positive energy delta-function pulse pair. This conjecture (the quantum interest conjecture) states that a positive energy pulse must overcompensate the negative energy pulse by an amount which is a monotonically increasing function of the pulse separation. In the present paper we prove the conjecture for massless quantized scalar fields in two and four-dimensional flat spacetime, and show that it is implied by the quantum inequalities.
The way in
https://doi.org/10.1103/PhysRevD.60.104018Published as Physical Review D 60, 104018 (1999) under the APS default licence. The preprint is on arXiv as gr-qc/9901074, filed 26 January 1999 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. Ford writes from the Institute of Cosmology at Tufts, Roman from Central Connecticut State University. Supported by NSF grant Phy-9800965 and a CCSU/AAUP faculty research grant. The two quantum-inequality papers this one builds on are on this site as stm-5e822858ce and stm-09a7d97555.
How to cite it
L. H. Ford, Thomas A. Roman (1999) The quantum interest conjecture. doi:10.1103/PhysRevD.60.104018
Where it sits in the curriculum
The evidence ladderWhat the vacuum isThe metric, warp drives and wormholesEnergy from the vacuum