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In quantum field theory, the **quantum vacuum state** (also called the **quantum vacuum** or **vacuum state**) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. **Zero-point field** is sometimes used as a synonym for the vacuum state of an individual quantized field.

- Non-zero expectation value
- Energy
- Symmetry
- Non-linear permittivity
- Virtual particles
- Physical nature of the quantum vacuum
- Notations
- See also
- References and notes
- Further reading
- External links

According to present-day understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space".^{ [1] }^{ [2] } According to quantum mechanics, the vacuum state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into and out of the quantum field.^{ [3] }^{ [4] }^{ [5] }

The QED vacuum of quantum electrodynamics (or QED) was the first vacuum of quantum field theory to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s it was reformulated by Feynman, Tomonaga and Schwinger, who jointly received the Nobel prize for this work in 1965.^{ [6] } Today the electromagnetic interactions and the weak interactions are unified (at very high energies only) in the theory of the electroweak interaction.

The Standard Model is a generalization of the QED work to include all the known elementary particles and their interactions (except gravity). Quantum chromodynamics (or QCD) is the portion of the Standard Model that deals with strong interactions, and QCD vacuum is the vacuum of quantum chromodynamics. It is the object of study in the Large Hadron Collider and the Relativistic Heavy Ion Collider, and is related to the so-called vacuum structure of strong interactions.^{ [7] }

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 (VEV) 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 have non-vanishing vacuum expectation values called condensates. In the Standard Model, the non-zero vacuum expectation value of the Higgs field, arising from spontaneous symmetry breaking, is the mechanism by which the other fields in the theory acquire mass.

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. In the laboratory, it may be detected as the Casimir effect. In physical cosmology, the energy of the cosmological vacuum appears as the cosmological constant. In fact, the energy of a cubic centimeter of empty space has been calculated figuratively to be one trillionth of an erg (or 0.6 eV).^{ [8] } 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 be also proved directly without these axioms.^{ [9] } Poincaré invariance implies that only scalar combinations of field operators have non-vanishing VEV's. The VEV 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. See Higgs mechanism, standard model.

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 ε_{0} of vacuum permittivity.^{ [10] } These theoretical developments are described, for example, in Dittrich and Gies.^{ [5] } 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 permitivity is increased by a tiny amount with respect to ε_{0}. What's more, and what would be easier to observe (but still very difficult!), is that a strong electric field would modify the effective permeability of free space, becoming anisotropic with a value slightly below *μ*_{0} in the direction of the electric field and slightly exceeding *μ*_{0} in the perpendicular direction, thereby exhibiting birefringence for an electromagnetic wave travelling in a direction other than that of the electric field. The effect is similar to the Kerr effect but without matter being present.^{ [11] } This tiny nonlinearity can be interpreted in terms of virtual pair production ^{ [12] } The required electric field is predicted to be enormous, about V/m, known as the Schwinger limit; the equivalent Kerr constant has been estimated, being about 10^{20} times smaller than the Kerr constant of water. Explanations for dichroism from particle physics, outside quantum electrodynamics, also have been proposed.^{ [13] } Experimentally measuring such an effect is very difficult,^{ [14] } and has not yet been successful.

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.^{ [15] } The term "vacuum fluctuations" refers to the variance of the field strength in the minimal energy state,^{ [16] } and is described picturesquely as evidence of "virtual particles".^{ [17] } It is sometimes attempted to provide an intuitive picture of virtual particles, or variances, based upon the Heisenberg energy-time uncertainty principle:

(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 *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.^{ [18] } Although the phenomenon of virtual particles is accepted, this interpretation of the energy-time uncertainty relation is not universal.^{ [19] }^{ [20] } 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 *q* and momentum *p*, for example) do not satisfy a canonical commutation relation (such as [*q*, *p*] = i *ħ*).^{ [21] } 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.^{ [22] }^{ [23] } The very many approaches to the energy-time uncertainty principle are a long and continuing subject.^{ [23] }

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."^{ [1] } 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.

^{ [24] }(See also.^{ [25] }^{ [26] }^{ [27] })

Photon-photon interaction can occur only through interaction with the vacuum state of some other field, for example through the Dirac electron-positron vacuum field; this is associated with the concept of vacuum polarization.^{ [28] } According to Milonni (1994): "... all quantum fields have zero-point energies and vacuum fluctuations."^{ [29] } 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."^{ [30] }^{ [31] } 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."

^{ [32] }

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, *α*, goes to zero."^{ [33] }

The vacuum state is written as or . The vacuum expectation value (see also Expectation value) of any field should be written as .

- 1 2 Astrid Lambrecht (2002). Hartmut Figger; Dieter Meschede; Claus Zimmermann (eds.).
*Observing mechanical dissipation in the quantum vacuum: an experimental challenge; in*. Berlin/New York: Springer. p. 197. ISBN 978-3-540-42418-5.**Laser physics at the limits** - ↑ Christopher Ray (1991).
*Time, space and philosophy*. London/New York: Routledge. Chapter 10, p. 205. ISBN 978-0-415-03221-6. - ↑ AIP Physics News Update,1996
- ↑ Physical Review Focus Dec. 1998
- 1 2 Walter Dittrich & Gies H (2000).
*Probing the quantum vacuum: perturbative effective action approach*. Berlin: Springer. ISBN 978-3-540-67428-3. - ↑ For an historical discussion, see for example Ari Ben-Menaḥem, ed. (2009). "Quantum electrodynamics (QED)".
*Historical Encyclopedia of Natural and Mathematical Sciences*.**1**(5th ed.). Springer. pp. 4892*ff*. ISBN 978-3-540-68831-0. For the Nobel prize details and the Nobel lectures by these authors see "The Nobel Prize in Physics 1965". Nobelprize.org. Retrieved 2012-02-06. - ↑ Jean Letessier; Johann Rafelski (2002).
*Hadrons and Quark-Gluon Plasma*. Cambridge University Press. p. 37*ff*. ISBN 978-0-521-38536-7. - ↑ 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
- ↑ Bednorz, Adam (November 2013). "Relativistic invariance of the vacuum".
*The European Physical Journal C*.**73**(12): 2654. arXiv: 1209.0209 . Bibcode:2013EPJC...73.2654B. doi:10.1140/epjc/s10052-013-2654-9. S2CID 39308527. - ↑ David Delphenich (2006). "Nonlinear Electrodynamics and QED". arXiv: hep-th/0610088 .
- ↑ Mourou, G. A., T. Tajima, and S. V. Bulanov,
*Optics in the relativistic regime*; § XI*Nonlinear QED*,*Reviews of Modern Physics*vol.**78**(no. 2), 309-371 (2006) pdf file. - ↑ Klein, James J. and B. P. Nigam,
*Birefringence of the vacuum*,*Physical Review*vol.**135**, p. B1279-B1280 (1964). - ↑ Holger Gies; Joerg Jaeckel; Andreas Ringwald (2006). "Polarized Light Propagating in a Magnetic Field as a Probe of Millicharged Fermions".
*Physical Review Letters*.**97**(14): 140402. arXiv: hep-ph/0607118 . Bibcode:2006PhRvL..97n0402G. doi:10.1103/PhysRevLett.97.140402. PMID 17155223. S2CID 43654455. - ↑ Davis; Joseph Harris; Gammon; Smolyaninov; Kyuman Cho (2007). "Experimental Challenges Involved in Searches for Axion-Like Particles and Nonlinear Quantum Electrodynamic Effects by Sensitive Optical Techniques". arXiv: 0704.0748 [hep-th].
- ↑ Myron Wyn Evans; Stanisław Kielich (1994).
*Modern nonlinear optics, Volume 85, Part 3*. John Wiley & Sons. p. 462. ISBN 978-0-471-57548-1.For all field states that have classical analog the field quadrature variances are also greater than or equal to this commutator.

- ↑ David Nikolaevich Klyshko (1988).
*Photons and nonlinear optics*. Taylor & Francis. p. 126. ISBN 978-2-88124-669-2. - ↑ Milton K. Munitz (1990).
*Cosmic Understanding: Philosophy and Science of the Universe*. Princeton University Press. p. 132. ISBN 978-0-691-02059-4.The spontaneous, temporary emergence of particles from vacuum is called a "vacuum fluctuation".

- ↑ For an example, see P. C. W. Davies (1982).
*The accidental universe*. Cambridge University Press. pp. 106. ISBN 978-0-521-28692-3. - ↑ A vaguer description is provided by Jonathan Allday (2002).
*Quarks, leptons and the big bang*(2nd ed.). CRC Press. pp. 224*ff*. ISBN 978-0-7503-0806-9.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*. - ↑ This "borrowing" idea has led to proposals for using the zero-point energy of vacuum as an infinite reservoir and a variety of "camps" about this interpretation. See, for example, Moray B. King (2001).
*Quest for zero point energy: engineering principles for 'free energy' inventions*. Adventures Unlimited Press. pp. 124*ff*. ISBN 978-0-932813-94-7. - ↑ Quantities satisfying a canonical commutation rule are said to be noncompatible observables, by which is meant that they can both be measured simultaneously only with limited precision. See Kiyosi Itô (1993). "§ 351 (XX.23) C: Canonical commutation relations".
*Encyclopedic dictionary of mathematics*(2nd ed.). MIT Press. p. 1303. ISBN 978-0-262-59020-4. - ↑ Paul Busch; Marian Grabowski; Pekka J. Lahti (1995). "§III.4: Energy and time".
*Operational quantum physics*. Springer. pp. 77*ff*. ISBN 978-3-540-59358-4. - 1 2 For a review, see Paul Busch (2008). "Chapter 3: The Time–Energy Uncertainty Relation". In J.G. Muga; R. Sala Mayato; Í.L. Egusquiza (eds.).
*Time in Quantum Mechanics*. Lecture Notes in Physics.**734**(2nd ed.). Springer. pp. 73–105. arXiv: quant-ph/0105049 . Bibcode:2002tqm..conf...69B. doi:10.1007/978-3-540-73473-4_3. ISBN 978-3-540-73472-7. S2CID 14119708. - ↑ Fowler, R., Guggenheim, E.A. (1965).
*Statistical Thermodynamics. A Version of Statistical Mechanics for Students of Physics and Chemistry*, reprinted with corrections, Cambridge University Press, London, page 224. - ↑ Partington, J.R. (1949).
*An Advanced Treatise on Physical Chemistry*, volume 1,*Fundamental Principles. The Properties of Gases*, Longmans, Green and Co., London, page 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, page 477. - ↑ Bailyn, M. (1994).
*A Survey of Thermodynamics*, American Institute of Physics, New York, ISBN 0-88318-797-3, page 342. - ↑ 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, pages 287–288. - ↑ Milonni, P.W. (1994).
*The Quantum Vacuum. An Introduction to Quantum Electrodynamics*, Academic Press, Inc., Boston, ISBN 0-12-498080-5, page xv. - ↑ Milonni, P.W. (1994).
*The Quantum Vacuum. An Introduction to Quantum Electrodynamics*, Academic Press, Inc., Boston, ISBN 0-12-498080-5, page 239. - ↑ Schwinger, J.; DeRaad, L.L.; Milton, K.A. (1978). "Casimir effect in dielectrics".
*Annals of Physics*.**115**(1): 1–23. Bibcode:1978AnPhy.115....1S. doi:10.1016/0003-4916(78)90172-0. - ↑ Milonni, P.W. (1994).
*The Quantum Vacuum. An Introduction to Quantum Electrodynamics*, Academic Press, Inc., Boston, ISBN 0-12-498080-5, page 418. - ↑ Jaffe, R.L. (2005). Casimir effect and the quantum vacuum,
*Phys. Rev. D***72**: 021301(R), http://1–5.cua.mit.edu/8.422_s07/jaffe2005_casimir.pdf%5B%5D

- Free pdf copy of 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* - Puthoff, H. E.; Little, S. R.; Ibison, M. (2001). "Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight". arXiv: astro-ph/0107316 .
- E. W. Davis, V. L. Teofilo, B. Haisch, H. E. Puthoff, L. J. Nickisch, A. Rueda and D. C. Cole(2006)"Review of Experimental Concepts for Studying the Quantum Vacuum Field"

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