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Vacuum Energy Mark D. Roberts, 117 Queen’s Road, Wimbledon, London SW19 8NS, Email:[email protected] http://cosmology.mth.uct.ac.za/ roberts ∼ February 1, 2008 Eprint: hep-th/0012062 Comments: A comprehensive review of Vacuum Energy, which is an extended version of a poster presented at L¨uderitz (2000). This is not a review of the cosmolog- ical constant per se, but rather vacuum energy in general, my approach to the cosmological constant is not standard. Lots of very small changes and several additions for the second and third versions: constructive feedback still welcome, but the next version will be sometime in coming due to my sporadiac internet access. First Version 153 pages, 368 references. Second Version 161 pages, 399 references. arXiv:hep-th/0012062v3 22 Jul 2001 Third Version 167 pages, 412 references. The 1999 PACS Physics and Astronomy Classification Scheme: http://publish.aps.org/eprint/gateway/pacslist 11.10.+x, 04.62.+v, 98.80.-k, 03.70.+k; The 2000 Mathematical Classification Scheme: http://www.ams.org/msc 81T20, 83E99, 81Q99, 83F05. 3 KEYPHRASES: Vacuum Energy, Inertial Mass, Principle of Equivalence. 1 Abstract There appears to be three, perhaps related, ways of approaching the nature of vacuum energy. The first is to say that it is just the lowest energy state of a given, usually quantum, system. The second is to equate vacuum energy with the Casimir energy. The third is to note that an energy difference from a complete vacuum might have some long range effect, typically this energy difference is interpreted as the cosmological constant. All three approaches are reviewed, with an emphasis on recent work. It is hoped that this review is comprehensive in scope. There is a discussion on whether there is a relation between vacuum energy and inertia. The solution suggested here to the nature of the vacuum is that Casimir energy can produce short range effects because of boundary conditions, but that at long range there is no over- all effect of vacuum energy, unless one considers lagrangians of higher order than Einstein’s as vacuum induced. No original calculations are presented in support of this position. Contents 1 Introduction. 4 1.1 Forward................................... 4 1.2 HistoricalBackground.. 4 1.2.1 The Zero-point Energy of a Harmonic Oscillator. .. 4 1.2.2 The Experimental Verification of Zero-point Energy. .... 6 1.3 Finding, Creating and Exploiting Vacua. ... 7 2 The Quantum Field Theory Vacuum. 9 2.1 TheVacuumStateofQFT’singeneral. 9 2.2 TheQEDVacuum............................. 14 2.3 TheToll-ScharnhorstEffect.. 15 2.4 TheYang-MillsVacuum.. 17 2.5 TheQCDVacuum............................. 20 2.6 TheSUSYVacuum. ........................... 23 2.7 TheSupergravityVacuum. 28 2.8 StringandBraneVacua.. 29 2.9 LatticeModels. .............................. 37 2.10 SymmetryBreaking. 39 2.11 Φ4 TheoryandtheRenormalizationGroup. 44 2.12 InstantonsandΘ-Vacua.. 46 2.13 Solitons, Integrable Models and Magnetic Vortices. ........ 48 2.14 TopologicalQuantumFieldTheory. 50 2.15SAZApproach. .............................. 50 2.16 SuperfluidsandCondensedMatter. 51 2 3 The Casimir and Related Effects. 53 3.1 Introduction. ............................... 53 3.2 HistoryoftheCasimirEffect. 54 3.3 Zero-Point Energy and Statistical Mechanics. ..... 57 3.4 CasimirCalculations.. 65 3.5 TheDynamicalCasimirEffect. 71 3.6 MechanicalAnalogsoftheCasimirEffect. .. 74 3.7 ApplicationsoftheCasimirEffect. 75 3.8 ExperimentalTesting of the Casimir Effect. ... 76 3.9 QuantumFieldTheoryonCurvedSpaces. 76 4 Vacuum Energy on Large Scales. 79 4.1 TheCosmologicalConstant.. 79 4.2 TheAnthropicPrinciple.. 91 4.3 Quintessence. ............................... 93 4.4 Inflation................................... 95 4.5 VacuumEnergyasCriticalDensity. 100 4.6 EquatingVacuumEnergywithDarkMatter. 103 4.7 Sonoluminescence. .. .. .. .. .. .. .. .. .. .. .. 106 4.8 QuantumCosmology. 107 5 Principle of Equivalence 109 6 Energy In General Relativity. 110 6.1 Does Vacuum Energy have Gravitational Effects? . 110 6.2 Various Approaches to Gravitational Energy. 111 7 Inertia. 114 7.1 InertiainGeneral. ............................ 114 7.2 SuperfluidAnalogy.. 118 8 Relativity of Motion in Vacuum. 118 9 The Vision Thing. 120 10 Acknowledgement. 120 11 Referencing Style. 121 3 1 Introduction. 1.1 Forward. In physics there are a great variety of views of what vacuum energy is. It is understood differently in many formalisms and domains of study, and ideas from one area are sometimes used in another; for example a microscopic origin for the cosmological constant is often used to justify its inclusion in cosmology. A narrow view is that vacuum energy is synonymous with the cosmological constant; why this is the case is explained in 2.1 2. A wide § ¶ view is that vacuum energy is just the lowest energy of any system under consideration. Finkelstein [125] (1991) takes an extreme view of what the vacuum is: The structure of the vacuum is the central problem of physics today: the fusion of the theories of gravity and the quantum is a subproblem. In general relativity gravitational energy is hard to calculate, and often am- biguous, especially in non-asymptotically flat spacetimes. Now how vacuum energy fits into gravitation is not straightforward as the gravitational field might have its own vacuum energy, this is discussed in 6. Einstein [111] § (1924) discusses the effect of “ether”, perhaps the ether can be thought of as vacuum energy. Vacuum energy, like any other energy, can be thought of as having an equivalent mass via E = mc2. Moving masses have inertia, and this is one way of looking at how inertia is related to vacuum energy, this is discussed in 7. The principle of equivalence, 5, can be formulated in a § § way which says that the laws of physics are the same in all inertial frames. An inertial frame is thus a primitive concept, and one way of thinking of it is of it occurring because of some property of the vacuum. 1.2 Historical Background. The presentation here of the history of zero-point energy largely follows Sciama [310] (1991), the is also a historical discussion in Lima and Maia [391] (1995). 1.2.1 The Zero-point Energy of a Harmonic Oscillator. Zero-point energy was introduced into physics by Max Planck in 1911. In a renewed attempt to understand the interaction between matter and radi- 4 ation and its relationship to the black body spectrum Planck put forward the hypothesis that the absorption of radiation occurs continuously while its emission is discrete and in energy units of hν, On this hypothesis the average energyε ¯ of a harmonic oscillator at temperature T would be given by 1 ε¯ = hν + B(ν, T ) (1.1) 2 where hν B(ν, T ) . (1.2) ≡ ehν/kT 1 − 1 Thus the oscillator would have a finite zero-point energy 2 hν even at the absolute zero of temperature. When it comes to quantum field theory, there can be thought of as being a quantum oscillation at each point in spacetime, this leads to the zero-point energy giving an overall infinite energy contribu- tion, so that there is a problem in how to adjust this contribution in order to allow quantum field theory to be well defined. Planck abandoned his discrete emission of energy, in units of hν, hy- pothesis in 1914 when Fokker showed that an assembly of rotating dipoles interacting classically with electromagnetic radiation would posses statistical properties (such as specific heat) in conflict with observation. Planck then became convinced that no classical discussion could lead in a satisfactory manner to a derivation of his distribution for black body radiation, in other words the distribution was fundamental and h could not be derived from previously know properties. Nevertheless according to Sciama [310] (1991) the idea of zero-point motion for a quantum harmonic oscillator continued to intrigue physicists, and the possibility was much discussed before its ex- istence was definitely shown in 1925 to be required by quantum mechanics, as a direct consequence of Heisenberg’s uncertainty principle. The Einstein-Hopf (1910) derivation of the black body radiation spec- trum, requires an expression for zero-point motion. That derivation involved a study of the interchange of energy and momentum between a harmonic oscillator and the radiation field. It led to the Rayleigh-Jeans distribution rather than to Planck’s because of the classical assumptions which were originally used for the harmonic oscillators and the radiation field. In the Einstein-Hopf calculation the mean square momentum of an oscillator was found to be proportional to the mean energy of the oscillator and to the mean energy density of the radiation field. Einstein and Stern now included the zero-point energy of the oscillator in its mean energy, in the hope of deriving the Planck rather than the Rayleigh-Jeans distribution, but found 5 1 that they could do so only if they took hν rather than 2 hν for the zero-point energy. According to Sciama [310] (1991) the reason for this difficulty is their neglect of the zero-point energy of the radiation field itself. It is not a consistent procedure to link together two physical systems only one of which possesses zero-point fluctuations in a steady state. These fluctuations would simply drive zero-point fluctuations in the other system or be damped out, depending on the relative number of degrees of freedom in the two systems. Nerst (1916), first proposed that ’empty’ space was everywhere filled with zero-point electromagnetic radiation. According to Sciama [310] (1991) the main considerations of Einstein and Stern are wrong; when one takes the classical limit kT >> hν for the Planck distribution, one finds 1 hν B(ν, T ) kT hν + hν O , (1.3) 2 kT → − × whereas 1 hν hν + B(ν, T ) kT + hν O . (1.4) 2 → × kT Thus the correct classical limit is obtained only if the zero-point energy is included. Einstein and Stern gave two independent arguments in favour of re- taining the zero-point energy, the first involved rotational specific heats of molecular gases and the second from an attempted, derivation of the Planck distribution itself.
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