Historical research guide · 0407 · 2000
A general worldline quantum inequality
Christopher J. Fewster
In one minute
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.
Connection
Chapter 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.
From the source
For 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
The 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
Because 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
The 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
For 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
The 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
The way in
https://doi.org/10.1088/0264-9381/17/9/302How to cite it
Christopher J. Fewster (2000) A general worldline quantum inequality doi:10.1088/0264-9381/17/9/302