Historical research guide · 0794 · 1996

On Quantum Inequalities

Adam D. Helfer

In one minute

Adam Helfer, a mathematician at the University of Missouri, takes the quantum inequalities Larry Ford introduced — the rule that a quantum field can sit below the ambient vacuum level only briefly, and the deeper the dip the briefer it has to be — and asks what an experimenter could actually do with one. His answer is that the instrument has to go on the books too. Any device that samples the field over a time t-nought needs a clock that resolves that interval, and a clock that fast carries at least as much energy as the dip it is trying to catch, so the sum of the two is never negative. Helfer calls this the operational weak energy condition, and by following a clock that is moving relative to the observer he pushes it to the operational dominant energy condition: every timelike part of the four-momentum density comes out positive once the apparatus is counted. Along the way he repairs a coherence assumption in the earlier derivations that an array of synchronised detectors would otherwise walk straight past.

Connection

Chapter 4’s wormhole and warp geometries and chapter 6’s vacuum-energy devices both turn on one number — how much below-ambient energy you can gather in one place, and for how long — and Helfer supplies the accounting rule that any such design has to be written against: count the instrument as well as the field.

From the source

Quantum field theory allows the smeared energy density of even a free Klein-Gordon field in flat spacetime to sit arbitrarily far below the ambient vacuum level, and the states that do so are dense in the Hilbert space, so below-vacuum energy densities are always available in principle.Introduction, Eq. 1 and the paragraph following it
Ford’s quantum inequality sets the exchange rate between depth and duration: the energy density sampled along an inertial worldline against a Lorentzian sampling function of characteristic width t-nought cannot fall below a numerical constant times Planck’s constant times the speed of light divided by the fourth power of that width, with the constant no larger than three over thirty-two pi squared.The Quantum Inequalities, Eqs. 3 and 4
The coherence assumption in the earlier derivations does not close the question on its own: an array of separately synchronised detectors, each individually obeying the bound, can be made numerous enough to trap an arbitrarily large below-ambient energy inside the same short interval, because nothing requires the detectors to sit near one another.The Quantum Inequalities, discussion following Eq. 5
Counting the instrument closes it. A device that weights the field by a sampling function of width t-nought needs clockwork that resolves times of that order, and by the Salecker-Wigner argument such a clock carries energy of at least Planck’s constant divided by that time — larger than the deficit it can catch, since the bound carries a factor of one over eight pi. Helfer states this as the operational weak energy condition: field plus apparatus is never negative.The Quantum Inequalities, Eq. 5 and the two paragraphs after it
The same reasoning applied to a clock moving relative to the observer gives the stronger operational dominant energy condition: for any future-pointing vector, the sum of the field’s momentum component and the clock’s own momentum component is positive, so operationally the four-momentum density stays future-pointing.The Dominant Energy Condition, Eqs. 6 to 12
Whether the operational conditions survive outside free fields in flat spacetime is the open question Helfer names: he expects the argument to extend to higher-spin linear fields, and to weakly coupled renormalizable theories scale by scale, while quantum chromodynamics should obey a Ford-type bound in the short-sampling limit where it is asymptotically free, and curved spacetime needs closer work because the subdominant short-distance terms there are more divergent than in Minkowski space.Generality of the Results, whole section

The way in

https://arxiv.org/abs/gr-qc/9612029

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

Licence checked on the source itself. The arXiv posting gr-qc/9612029, submitted 12 December 1996 from the Department of Mathematics, University of Missouri at Columbia, carries arXiv’s assumed licence for 1991 to 2003 submissions, which does not grant redistribution, and no Creative Commons statement appears in the eight-page text — so this sheet carries the summary, the claims and the author’s own abstract and sends the reader to the source, which is free to read on arXiv. The preprint has no journal reference on its arXiv record; Helfer developed the same operational argument in ‘Operational’ energy conditions, Classical and Quantum Gravity 15 (1998) 1169, preprint gr-qc/9709047. The claims below are read against the full preprint, whose sections are titled Introduction, The Quantum Inequalities, Generality of the Results, The Dominant Energy Condition and Conclusions. Three companion sheets in this library carry the rest of this conversation: Ford and Roman’s original bound at /library/stm-5e822858ce, their simpler plane-wave re-derivation at /library/stm-09a7d97555, and Fewster’s general worldline version at /library/stm-22ba5ed09b.

How to cite it

Adam D. Helfer (1996) On Quantum Inequalities arXiv:gr-qc/9612029

Where it sits in the curriculum

Wormholes, energy conditions and the negative-energy budget