A general worldline quantum inequality
Christopher J. Fewster
Abstract and summary · read the original at the source
In one page
Quantum field theory allows the energy density at a single point to sit below the ambient vacuum level, and by as much as you like — the same accounting that produces the Casimir force. What it does not allow is for that dip to stay deep for an arbitrary length of time. The limits are called quantum inequalities, and they set the budget any wormhole or warp geometry has to work inside. Christopher Fewster, at the University of York, gives the general version. Earlier bounds covered an observer sitting still in a static spacetime; his covers a real scalar field of any mass, on any globally hyperbolic spacetime of two or more dimensions, along any smooth timelike path, with any smooth sampling function and any pair of well-behaved quantum states. The bound is explicit, and the part that takes the work is proving that its right-hand side actually converges, which he does with the tools of microlocal analysis. For a stationary observer it collapses to one compact integral.
Why it matters hereChapter 4’s warp drives and wormholes are priced in how much energy may be held below the ambient vacuum level, how deeply and for how long, and this is the paper that states the rule in the general setting a real design occupies — a curved, arbitrary spacetime and an observer who is free to move. Chapter 2 gets the sharper picture underneath it: dips below the vacuum floor are limited in time rather than in space, which is why the vacuum’s bookkeeping is redistributive rather than creative.
What it claims
01For a real, minimally coupled Klein-Gordon field of any mass on any globally hyperbolic spacetime of dimension two or more, and for any two states whose two-point functions are globally Hadamard, the weighted average of the normal ordered energy density along an arbitrary smooth timelike worldline is bounded below by an explicit expression built from the reference state’s point-split energy density.Theorem 4.1, Eq. (4.1); Abstract
Published and peer-reviewed02The bound is non-trivial because its right-hand side is proved to converge for every smooth compactly supported sampling function — the step the author identifies as the most important of the proof — using microlocal analysis and the characterisation of Hadamard states by the microlocal spectral condition.Sect. 1, paragraph beginning ‘Sects. 2-4 will be concerned’; Sects. 2 to 4
Published and peer-reviewed03Because the energy density of a quantum field at a point is unbounded below as a function of the state, while worldline averages are bounded and purely spatial averages are not, dips below the ambient vacuum level are associated with localisation in time rather than in space.Sect. 1, Introduction, third paragraph
Published and peer-reviewed04The new result contains the earlier ones: it reduces to the static-spacetime bounds of Fewster and Eveson and of Fewster and Teo, so that the general derivation can be regarded as the correct setting for that earlier work.Sect. 1, Eq. (1.9); Sect. 6, Conclusion
Published and peer-reviewed05For a stationary observer in a stationary spacetime, with a stationary ground state as the reference, the bound takes a compact form: an integral of the squared Fourier transform of the sampling function against a polynomially bounded function whose growth is expected, on dimensional grounds, to go as the sampling frequency raised to the spacetime dimension.Sect. 1, Eq. (1.10); Sect. 5, Proposition 5.1
Published and peer-reviewed06The author states plainly that this is not expected to be the best possible worldline bound, that the bound presumably depends on the choice of orthonormal frame carried along the worldline, and that sharpening it — and extending the same method to spacetime averages of the stress-energy tensor — are the open next steps.Remark following Theorem 4.1; Sect. 6, Conclusion
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Abstract
Worldline quantum inequalities provide lower bounds on weighted averages of the renormalised energy density of a quantum field along the worldline of an observer. In the context of real, linear scalar field theory on an arbitrary globally hyperbolic spacetime, we establish a worldline quantum inequality on the normal ordered energy density, valid for arbitrary smooth timelike trajectories of the observer, arbitrary smooth compactly supported weight functions and arbitrary Hadamard quantum states. Normal ordering is performed relative to an arbitrary choice of Hadamard reference state. The inequality obtained generalises a previous result derived for static trajectories in a static spacetime. The underlying argument is straightforward and is made rigorous using the techniques of microlocal analysis. In particular, an important role is played by the characterisation of Hadamard states in terms of the microlocal spectral condition. We also give a compact form of our result for stationary trajectories in a stationary spacetime.
The way in
https://doi.org/10.1088/0264-9381/17/9/302Published as Classical and Quantum Gravity 17 (2000) 1897-1911 by Christopher J. Fewster of the Department of Mathematics, University of York. The version of record is free to read at IOP, but under the publisher’s own terms rather than a Creative Commons licence, and the author version on arXiv as gr-qc/9910060 — dated 18 October 1999 and revised 21 February 2000 — carries arXiv’s assumed licence for legacy submissions, checked on the arXiv abstract page on 2026-09-08. Neither is a Creative Commons licence, so this sheet carries the summary, the claims and the author’s own abstract and sends the reader to the source. The claims are located against the arXiv version’s sections, theorems and equations. Two companion sheets in this library carry the rest of this conversation: Ford, Helfer and Roman on why the purely spatial version of the bound does not exist in four dimensions, at /library/stm-755a293263, and Kontou and Olum on the bound in a spacetime with small curvature, at /library/stm-efbb5f52fe.
How to cite it
Christopher J. Fewster (2000) A general worldline quantum inequality. doi:10.1088/0264-9381/17/9/302
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