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| '''Quantum inequalities''' are local constraints on the magnitude and extent of distributions of negative energy density in space-time. Initially conceived to clear up a long-standing problem in quantum field theory (namely, the potential for unconstrained negative energy density at a point), quantum inequalities have proven to have a diverse range of applications.
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| The form of the quantum inequalities is reminiscent of the [[uncertainty principle]]. | |
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| ==Energy conditions in classical field theory==
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| Einstein's theory of [[General Relativity]] amounts to a description of the relationship between the curvature of space-time, on the one hand, and the distribution of matter throughout space-time on the other. This precise details of this relationship are determined by the [[Einstein field equations|Einstein equations]]
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| <math>G_{\mu\nu}=\kappa T_{\mu\nu}</math>.
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| Here, the Einstein tensor <math>G_{\mu\nu}</math> describes the curvature of space-time, whilst the [[energy-momentum tensor]] <math>T_{\mu\nu}</math> describes the local distribution of matter. (<math>\kappa</math> is a constant.) The Einstein equations express ''local'' relationships between the quantities involved—specifically, this is a system of coupled non-linear second order partial differential equations.
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| A very simple observation can be made at this point: the zero-point of energy-momentum is not arbitrary. Adding a "constant" to the right-hand side of the Einstein equations will effect a change in the Einstein tensor, and thus also in the curvature properties of space-time.
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| All known classical matter fields obey certain "[[energy condition]]s". The most famous classical energy condition is the "weak energy condition"; this asserts that the local energy density, as measured by an observer moving along a time-like world line, is non-negative. The weak energy condition is essential for many of the most important and powerful results of classical relativity theory—in particular, the singularity theorems of Hawking ''et al.''
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| ==Energy conditions in quantum field theory==
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| The situation in [[quantum field theory]] is rather different: the expectation value of the energy density can be negative at any given point. In fact, things are even worse: by tuning the state of the quantum matter field, the expectation value of the local energy density can be made '''arbitrarily''' negative. | |
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| ==Applications==
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| Distributions of negative energy density comprise what is often referred to as [[exotic matter]], and allow for several intrgiuing possibilities: for example, the [[Alcubierre drive]] potentially allows for faster-than-light space travel.
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| Quantum inequalities constrain the magnitude and space-time extent of negative energy densities. In the case of the Alcubierre [[warp drive]] mentioned above, the quantum inequalities predict that the amount of exotic matter required to create and sustain the warp drive "bubble" far exceeds the total mass-energy of the universe.
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| ==People==
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| The earliest investigations into quantum inequalities were carried out by Larry Ford and Tom Roman; an early collaborator was Mitch Pfenning, one of Ford's students at Tufts University. Important work was also carried out by Eanna Flanagan.
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| More recently, Chris Fewster (of the [[University of York]], in the UK) has applied rigorous mathematics to produce a variety of quite general quantum inequalities. Collaborators have included Ford, Roman, Pfenning, Stefan Hollands and Rainer Verch.
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| ==Further reading==
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| ===Websites===
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| Quantum field theory on curved spacetime at the Erwin Schrödinger Institute
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| [http://www.phys.lsu.edu/mog/mog20/node16.html]
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| Quantum Energy Inequalities (University of York, UK)
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| [http://maths.york.ac.uk/www/PhysicsQIneq.htm] | |
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| ===Papers=== | |
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| Ford L H and Roman T A 1995 "Averaged Energy Conditions and Quantum Inequalities"
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| ''Phys. Rev.'' D '''51''' 4277-4286
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| [http://arxiv.org/abs/gr-qc/9410043]
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| Fewster C J 2000 "A general worldline quantum inequality"
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| ''Class. Quant. Grav.'' '''17''' 1897-1911
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| [http://arxiv.org/abs/gr-qc/9910060]
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| [[Category:Quantum field theory]]
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