Quantum vacuum state

{{main|Vacuum expectation value}}

File:Vacuum fluctuations revealed through spontaneous parametric down-conversion.ogv (in the red ring) amplified by spontaneous parametric down-conversion. ]]

If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator, or more accurately, the ground state of a measurement problem. In this case, the vacuum expectation value of any field operator vanishes. For quantum field theories in which perturbation theory breaks down at low energies (for example, Quantum chromodynamics or the BCS theory of superconductivity), field operators may obtain non-vanishing vacuum expectation values by spontaneous symmetry breaking. In the Standard Model, the Higgs field acquires a non-zero expectation value when the electroweak symmetry is broken, and this explains part of the masses of other particles.

{{main|Vacuum energy}}

The vacuum state is associated with a zero-point energy, and this zero-point energy (equivalent to the lowest possible energy state) has measurable effects. It may be detected as the Casimir effect in the laboratory. In physical cosmology, the energy of the cosmological vacuum appears as the cosmological constant. The energy of a cubic centimeter of empty space has been calculated figuratively to be one trillionth of an erg (or 0.6 eV).Sean Carroll, Sr Research Associate – Physics, California Institute of Technology, June 22, 2006 C-SPAN broadcast of Cosmology at Yearly Kos Science Panel, Part 1. An outstanding requirement imposed on a potential Theory of Everything is that the energy of the quantum vacuum state must explain the physically observed cosmological constant.

For a relativistic field theory, the vacuum is Poincaré invariant, which follows from

Wightman axioms but can also be proved directly without these axioms.{{cite journal|last=Bednorz|first=Adam|title=Relativistic invariance of the vacuum|journal=The European Physical Journal C|date=November 2013|volume=73|issue=12|pages=2654|doi=10.1140/epjc/s10052-013-2654-9|arxiv = 1209.0209 |bibcode = 2013EPJC...73.2654B |s2cid=39308527}} Poincaré invariance implies that only scalar combinations of field operators have non-vanishing vacuum expectation values. The vacuum may break some of the internal symmetries of the Lagrangian of the field theory. In this case, the vacuum has less symmetry than the theory allows, and one says that spontaneous symmetry breaking has occurred.

{{main|Schwinger limit}}

Quantum corrections to Maxwell's equations are expected to result in a tiny nonlinear electric polarization term in the vacuum, resulting in a field-dependent electrical permittivity ε deviating from the nominal value ε₀ of vacuum permittivity.{{cite arXiv |eprint=hep-th/0610088 |first=David |last=Delphenich |title=Nonlinear Electrodynamics and QED |date=2006}} These theoretical developments are described, for example, in Dittrich and Gies. The theory of quantum electrodynamics predicts that the QED vacuum should exhibit a slight nonlinearity so that in the presence of a very strong electric field, the permittivity is increased by a tiny amount with respect to ε₀. Subject to ongoing experimental efforts{{cite journal |last1=Battesti |first1=Rémy |last2=Beard |first2=Jerome |last3=Böser |first3=Sebastian |last4=Bruyant |first4=Nicolas |last5=Budker |first5=Dmitry |last6=Crooker |first6=Scott A. |last7=Daw |first7=Edward J. |last8=Flambaum |first8=Victor V. |last9=Inada |first9=Toshiaki |last10=Irastorza |first10=Igor G. |last11=Karbstein |first11=Felix |last12=Kim |first12=Dong Lak |last13=Kozlov |first13=Mikhail G. |last14=Melhem |first14=Ziad |last15=Phipps |first15=Arran |last16=Pugnat |first16=Pierre |last17=Rikken |first17=Geert |last18=Rizzo |first18=Carlo |last19=Schott |first19=Matthias |last20=Semertzidis |first20=Yannis K. |last21=ten Kate |first21=Herman H.J. |last22=Zavattini |first22=Guido |display-authors=1 |title=High magnetic fields for fundamental physics |journal=Physics Reports |date=November 2018 |volume=765-766 |pages=1–39 |doi=10.1016/j.physrep.2018.07.005|arxiv=1803.07547 |bibcode=2018PhR...765....1B |s2cid=4931745 }} is the possibility that a strong electric field would modify the effective permeability of free space, becoming anisotropic with a value slightly below μ₀ in the direction of the electric field and slightly exceeding μ₀ in the perpendicular direction. The quantum vacuum exposed to an electric field exhibits birefringence for an electromagnetic wave traveling in a direction other than the electric field. The effect is similar to the Kerr effect but without matter being present.Mourou, G. A.; T. Tajima, and S. V. Bulanov, [http://link.aps.org/doi/10.1103/RevModPhys.78.309 Optics in the relativistic regime; § XI Nonlinear QED], Reviews of Modern Physics vol. 78 (no. 2), pp. 309–371, (2006) [https://web.archive.org/web/20030928093337/http://acc-physics.kek.jp/sokensympo/frontier_accelerator/FACW1-PROC/FACW1-23.1.pdf pdf file]. This tiny nonlinearity can be interpreted in terms of virtual pair production{{cite journal | url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.135.B1279 | doi=10.1103/PhysRev.135.B1279 | title=Birefringence of the Vacuum | date=1964 | last1=Klein | first1=James J. | last2=Nigam | first2=B. P. | journal=Physical Review | volume=135 | issue=5B | pages=B1279–B1280 | bibcode=1964PhRv..135.1279K | url-access=subscription }} A characteristic electric field strength for which the nonlinearities become sizable is predicted to be enormous, about $1.32\; \backslash times\; 10^\{18\}$V/m, known as the Schwinger limit; the equivalent Kerr constant has been estimated, being about 10²⁰ times smaller than the Kerr constant of water. Explanations for dichroism from particle physics, outside quantum electrodynamics, also have been proposed.{{cite journal |last1=Gies |first1=Holger |last2=Jaeckel |first2=Joerg |last3=Ringwald |first3=Andreas |date=2006 |title=Polarized Light Propagating in a Magnetic Field as a Probe of Millicharged Fermions |journal=Physical Review Letters |volume=97 |issue=14 |page=140402 |arxiv=hep-ph/0607118 |bibcode=2006PhRvL..97n0402G |doi=10.1103/PhysRevLett.97.140402 |pmid=17155223 |s2cid=43654455}} Experimentally measuring such an effect is challenging,{{cite arXiv |eprint=0704.0748 |class=hep-th |last1=Davis |first2=Joseph |last2=Harris |title=Experimental Challenges Involved in Searches for Axion-Like Particles and Nonlinear Quantum Electrodynamic Effects by Sensitive Optical Techniques |date=2007 |author3=Gammon |author4=Smolyaninov |author5=Kyuman Cho}} and has not yet been successful.

{{main|Virtual particle}}

The presence of virtual particles can be rigorously based upon the non-commutation of the quantized electromagnetic fields. Non-commutation means that although the average values of the fields vanish in a quantum vacuum, their variances do not.{{cite book |author=Evans |first1=Myron Wyn |author-link1=Myron Wyn Evans |url=https://books.google.com/books?id=25LX8F2ybCsC&pg=PA462 |title=Modern nonlinear optics, Volume 85, Part 3 |last2=Kielich |first2=Stanislaw |date=1994 |publisher=John Wiley & Sons |isbn=978-0-471-57548-1 |page=462 |quote=For all field states that have classical analog the field quadrature variances are also greater than or equal to this commutator.}} The term "vacuum fluctuations" refers to the variance of the field strength in the minimal energy state,{{cite book |author=Klyshko |first=David Nikolaevich |url=https://books.google.com/books?id=IPfwdhR4TaYC&pg=PA126 |title=Photons and nonlinear optics |date=1988 |publisher=Taylor & Francis |isbn=978-2-88124-669-2 |page=126}} and is described picturesquely as evidence of "virtual particles".

{{cite book |author=Munitz |first=Milton K. |url=https://books.google.com/books?id=HkOg14hXqi8C&pg=PA132. |title=Cosmic Understanding: Philosophy and Science of the Universe |date=1990 |publisher=Princeton University Press |isbn=978-0-691-02059-4 |page=132 |quote=The spontaneous, temporary emergence of particles from vacuum is called a "vacuum fluctuation".}}

It is sometimes attempted to provide an intuitive picture of virtual particles, or variances, based upon the Heisenberg energy-time uncertainty principle:

$$\backslash Delta\; E\; \backslash Delta\; t\; \backslash ge\; \backslash frac\{\backslash hbar\}\{2\}\; \backslash ,\; ,$$

(with ΔE and Δt being the energy and time variations respectively; ΔE is the accuracy in the measurement of energy and Δt is the time taken in the measurement, and {{math|ħ}} is the Reduced Planck constant) arguing along the lines that the short lifetime of virtual particles allows the "borrowing" of large energies from the vacuum and thus permits particle generation for short times. Although the phenomenon of virtual particles is accepted, this interpretation of the energy-time uncertainty relation is not universal. One issue is the use of an uncertainty relation limiting measurement accuracy as though a time uncertainty Δt determines a "budget" for borrowing energy ΔE. Another issue is the meaning of "time" in this relation because energy and time (unlike position {{math|q}} and momentum {{math|p}}, for example) do not satisfy a canonical commutation relation (such as {{math|[q, p] {{=}} i ħ}}). Various schemes have been advanced to construct an observable that has some kind of time interpretation, and yet does satisfy a canonical commutation relation with energy. Many approaches to the energy-time uncertainty principle are a long and continuing subject.

According to Astrid Lambrecht (2002): "When one empties out a space of all matter and lowers the temperature to absolute zero, one produces in a Gedankenexperiment [thought experiment] the quantum vacuum state." According to Fowler & Guggenheim (1939/1965), the third law of thermodynamics may be precisely enunciated as follows:

It is impossible by any procedure, no matter how idealized, to reduce any assembly to the absolute zero in a finite number of operations.Fowler, Ralph; Guggenheim, Edward A. (1965). Statistical Thermodynamics. A Version of Statistical Mechanics for Students of Physics and Chemistry, reprinted with corrections, Cambridge University Press, London, England, p. 224. (See also.Partington, J. R. (1949). An Advanced Treatise on Physical Chemistry, volume 1, Fundamental Principles. The Properties of Gases, Longmans, Green and Company, London, England, p. 220.Wilks, J. (1971). The Third Law of Thermodynamics, Chapter 6 in Thermodynamics, volume 1, ed. W. Jost, of H. Eyring, D. Henderson, W. Jost, Physical Chemistry. An Advanced Treatise, Academic Press, New York, p. 477.Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics, New York, {{ISBN|0-88318-797-3}}, p. 342.)

Photon-photon interaction can occur only through interaction with the vacuum state of some other field, such as the Dirac electron-positron vacuum field; this is associated with the concept of vacuum polarization.Jauch, J. M.; Rohrlich, F. (1955/1980). The Theory of Photons and Electrons. The Relativistic Quantum Field Theory of Charged Particles with Spin One-half, second expanded edition, Springer-Verlag, New York, {{ISBN|0-387-07295-0}}, pp. 287–288. According to Milonni (1994): "... all quantum fields have zero-point energies and vacuum fluctuations."Milonni, P. W. (1994). The Quantum Vacuum. An Introduction to Quantum Electrodynamics, Academic Press, Incorporated, Boston, Massachusetts, {{ISBN|0-12-498080-5}}, p. xv. This means that there is a component of the quantum vacuum respectively for each component field (considered in the conceptual absence of the other fields), such as the electromagnetic field, the Dirac electron-positron field, and so on. According to Milonni (1994), some of the effects attributed to the vacuum electromagnetic field can have several physical interpretations, some more conventional than others. The Casimir attraction between uncharged conductive plates is often proposed as an example of an effect of the vacuum electromagnetic field. Schwinger, DeRaad, and Milton (1978) are cited by Milonni (1994) as validly, though unconventionally, explaining the Casimir effect with a model in which "the vacuum is regarded as truly a state with all physical properties equal to zero."Milonni, P. W. (1994). The Quantum Vacuum. An Introduction to Quantum Electrodynamics, Academic Press, Incorporated, Boston, Massachusetts, {{ISBN|0-12-498080-5}}, p. 239.{{cite journal |last1=Schwinger |first1=J. |last2=DeRaad |first2=L. L. |last3=Milton |first3=K. A. |year=1978 |title=Casimir effect in dielectrics |journal=Annals of Physics |volume=115 |issue=1 |pages=1–23 |bibcode=1978AnPhy.115....1S |doi=10.1016/0003-4916(78)90172-0}} In this model, the observed phenomena are explained as the effects of the electron motions on the electromagnetic field, called the source field effect. Milonni writes:

The basic idea here will be that the Casimir force may be derived from the source fields alone even in completely conventional QED, ... Milonni provides detailed argument that the measurable physical effects usually attributed to the vacuum electromagnetic field cannot be explained by that field alone, but require in addition a contribution from the self-energy of the electrons, or their radiation reaction. He writes: "The radiation reaction and the vacuum fields are two aspects of the same thing when it comes to physical interpretations of various QED processes including the Lamb shift, van der Waals forces, and Casimir effects."Milonni, P. W. (1994). The Quantum Vacuum. An Introduction to Quantum Electrodynamics, Academic Press, Incorporated, Boston, Massachusetts, {{ISBN|0-12-498080-5}}, p. 418.

This point of view is also stated by Jaffe (2005): "The Casimir force can be calculated without reference to vacuum fluctuations, and like all other observable effects in QED, it vanishes as the fine structure constant, {{math|α}}, goes to zero."Jaffe, R. L. (2005). Casimir effect and the quantum vacuum, Physical Review D, 72: 021301(R), http://1–5.cua.mit.edu/8.422_s07/jaffe2005_casimir.pdf{{Dead link|date=July 2018 |bot=InternetArchiveBot |fix-attempted=yes }}.

{{div col|colwidth=22em}}

• Pair production
• Vacuum energy
• Lamb shift
• False vacuum decay
• Squeezed coherent state
• Quantum fluctuation
• Scharnhorst effect
• Van der Waals force
• Casimir effect

{{div col end}}

{{Reflist|refs=

A vaguer description is provided by {{cite book |title=Quarks, leptons and the big bang |author=Jonathan Allday |url=https://books.google.com/books?id=kgsBbv3-9xwC&pg=PA224 |pages=224 ff |quote=The interaction will last for a certain duration Δt. This implies that the amplitude for the total energy involved in the interaction is spread over a range of energies ΔE. |isbn=978-0-7503-0806-9 |edition=2nd |publisher=CRC Press |date=2002}}

For a review, see {{cite book |title=Time in Quantum Mechanics |volume=734 |editor=J.G. Muga |editor2=R. Sala Mayato |editor3=Í.L. Egusquiza |pages=73–105 |chapter=Chapter 3: The Time–Energy Uncertainty Relation |author=Paul Busch |author-link=Paul Busch (physicist) |isbn=978-3-540-73472-7 |date=2008 |edition=2nd |publisher=Springer|arxiv=quant-ph/0105049 |bibcode=2002tqm..conf...69B |doi=10.1007/978-3-540-73473-4_3 |series=Lecture Notes in Physics |s2cid=14119708 }}

{{cite book |title=Operational quantum physics |url=https://archive.org/details/operationalquant00busc |url-access=limited |author=Paul Busch |author-link1=Paul Busch (physicist) |author2=Marian Grabowski |author3=Pekka J. Lahti |chapter=§III.4: Energy and time |pages=[https://archive.org/details/operationalquant00busc/page/n86 77]ff |isbn=978-3-540-59358-4 |date=1995 |publisher=Springer}}

Quantities satisfying a canonical commutation rule are noncompatible observables, meaning they can both be measured simultaneously, only with limited precision. See {{cite book |title=Encyclopedic dictionary of mathematics |author=Kiyosi Itô |chapter-url=https://books.google.com/books?id=azS2ktxrz3EC&pg=PA1303 |pages=1303 |chapter=§ 351 (XX.23) C: Canonical commutation relations |isbn=978-0-262-59020-4 |date=1993 |edition=2nd |publisher=MIT Press}}

For an example, see {{cite book |title=The accidental universe |author=P. C. W. Davies |url=https://archive.org/details/accidentaluniver0000davi |url-access=registration |pages=[https://archive.org/details/accidentaluniver0000davi/page/106 106] |isbn=978-0-521-28692-3 |date=1982 |publisher=Cambridge University Press}}

This "borrowing" idea has led to proposals for using the zero-point energy of vacuum as an infinite reservoir and various "camps" about this interpretation. See, for example, {{cite book |title=Quest for zero point energy: engineering principles for 'free energy' inventions |author=Moray B. King |url=https://books.google.com/books?id=0RmkmrFxHM0C&pg=PA124 |pages=124 ff |isbn=978-0-932813-94-7 |date=2001 |publisher=Adventures Unlimited Press}}

}}

• Free pdf copy of [http://www.physics.arizona.edu/~rafelski/Books/StructVacuumE.pdf The Structured Vacuum – thinking about nothing] by Johann Rafelski and Berndt Muller (1985) {{ISBN|3-87144-889-3}}.
• M. E. Peskin and D. V. Schroeder, An introduction to Quantum Field Theory.
• H. Genz, Nothingness: The Science of Empty Space.
• {{cite arXiv |eprint=astro-ph/0107316|last1= Puthoff|first1= H. E.|title= Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight|last2= Little|first2= S. R.|last3= Ibison|first3= M.|year= 2001}}
• E. W. Davis, V. L. Teofilo, B. Haisch, H. E. Puthoff, L. J. Nickisch, A. Rueda and D. C. Cole (2006), "[http://www.calphysics.org/articles/Davis_STAIF06.pdf Review of Experimental Concepts for Studying the Quantum Vacuum Field]".

• [https://web.archive.org/web/19980224172207/http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970724a.html Energy into Matter]

{{Quantum mechanics topics}}

Category:Quantum field theory

Category:Vacuum

Category:Quantum states

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