Editing Zero-point energy (section)

=== Quantum field theory and beyond ===

In 1926, [[Pascual Jordan]]<ref>{{cite journal|last1=Born|first1=M.|last2=Heisenberg|first2=W.|last3=Jordan|first3=P.|title=Zur Quantenmechanik. II|language=de|trans-title=On quantum mechanics II|journal=Zeitschrift für Physik|date=1926|volume=35|issue=8|pages=557–615|doi=10.1007/BF01379806|bibcode=1926ZPhy...35..557B|s2cid=186237037}}</ref> published the first attempt to quantize the electromagnetic field. In a joint paper with [[Max Born]] and [[Werner Heisenberg]] he considered the field inside a cavity as a superposition of quantum harmonic oscillators. In his calculation he found that in addition to the "thermal energy" of the oscillators there also had to exist an infinite zero-point energy term. He was able to obtain the same fluctuation formula that Einstein had obtained in 1909.<ref>{{cite journal|last1=Einstein|first1=Albert|title=Zum gegenwärtigen Stand des Strahlungsproblems|journal=Physikalische Zeitschrift|date=1909|volume=10|pages=185–193|bibcode= 1909PhyZ...10..185E}}</ref> However, Jordan did not think that his infinite zero-point energy term was "real", writing to Einstein that "it is just a quantity of the calculation having no direct physical meaning".<ref>{{cite book|last1=Mehra|first1=J.|last2=Rechenberg|first2=H.|author-link1=Jagdish Mehra |author-link2=Helmut Rechenberg |title=The Historical Development of Quantum Theory |volume=6|date=2002|publisher=Springer|isbn=978-0-387-95262-8|page=57|oclc= 722601833}}</ref> Jordan found a way to get rid of the infinite term, publishing a joint work with Pauli in 1928,<ref>{{cite journal|last1=Jordan|first1=P.|last2=Pauli|first2=W.|title=Zur Quantenelektrodynamik ladungsfreier Felder|language=de|trans-title=On the quantum electrodynamics of charge-free fields|journal=Zeitschrift für Physik|date=1928|volume=47|issue=3|pages=151–173|doi= 10.1007/BF02055793|bibcode= 1928ZPhy...47..151J|s2cid=120536476}}</ref> performing what has been called "the first infinite subtraction, or renormalisation, in quantum field theory".<ref>{{cite book|last1=Schweber|first1=Silvan S.|title=QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga|url=https://archive.org/details/qedmenwhomadeitd0000schw|url-access=registration|date=1994|publisher=Princeton University Press|isbn=978-0-691-03327-3|pages=[https://archive.org/details/qedmenwhomadeitd0000schw/page/108 108]–112|oclc= 439849774}}</ref>

[[File:Paul Dirac, 1933.jpg|thumb|upright|Paul Dirac, 1933]]
Building on the work of Heisenberg and others, [[Paul Dirac]]'s theory of emission and absorption (1927){{sfnp|Dirac|1927}} was the first application of the quantum theory of radiation. Dirac's work was seen as crucially important to the emerging field of quantum mechanics; it dealt directly with the process in which "particles" are actually created: [[spontaneous emission]].<ref>{{cite journal|last1=Weinberg|first1=Steven|title=The Search for Unity: Notes for a History of Quantum Field Theory|journal=Daedalus|date=1977|volume=106|issue=4|pages=17–35|jstor=20024506}}</ref> Dirac described the quantization of the [[electromagnetic field]] as an ensemble of [[quantum harmonic oscillator|harmonic oscillator]]s with the introduction of the concept of [[creation and annihilation operators]] of particles. The theory showed that spontaneous emission depends upon the zero-point energy fluctuations of the electromagnetic field in order to get started.<ref name="Yokoyama">
{{cite book
 |last1=Yokoyama|first1=H.
 |last2=Ujihara|first2=K.
 |title=Spontaneous emission and laser oscillation in microcavities
 |publisher= CRC Press
 |location=Boca Raton
 |year=1995
 |isbn=978-0-8493-3786-4|oclc= 832589969
 |page=[https://books.google.com/books?id=J_0ZAwf6AQ0C&pg=PA6 6]
}}</ref>{{sfnp|Scully |Zubairy |1997|loc=[https://books.google.com/books?id=20ISsQCKKmQC&pg=PA22 §1.5.2 pp. 22–23]}} In a process in which a photon is annihilated (absorbed), the photon can be thought of as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. In the words of Dirac:{{sfnp|Dirac|1927}}

{{blockquote|The light-quantum has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely, the zero state, in which its momentum and therefore also its energy, are zero. When a light-quantum is absorbed it can be considered to jump into this zero state, and when one is emitted it can be considered to jump from the zero state to one in which it is physically in evidence, so that it appears to have been created. Since there is no limit to the number of light-quanta that may be created in this way, we must suppose that there are an infinite number of light quanta in the zero state&nbsp;...}}

Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field. This view was popularized by [[Victor Weisskopf]] who in 1935 wrote:<ref>{{cite journal|last1=Weisskopf|first1=Viktor|title=Probleme der neueren Quantentheorie des Elektrons|language=de|trans-title=Problems of the new quantum theory of the electron|journal=Naturwissenschaften|date=1935|volume=23|issue=37|pages=631–637|doi= 10.1007/BF01492012|bibcode= 1935NW.....23..631W|s2cid=6780937}}</ref>

{{blockquote|From quantum theory there follows the existence of so called zero-point oscillations; for example each oscillator in its lowest state is not completely at rest but always is moving about its equilibrium position. Therefore electromagnetic oscillations also can never cease completely. Thus the quantum nature of the electromagnetic field has as its consequence zero point oscillations of the field strength in the lowest energy state, in which there are no light quanta in space&nbsp;... The zero point oscillations act on an electron in the same way as ordinary electrical oscillations do. They can change the eigenstate of the electron, but only in a transition to a state with the lowest energy, since empty space can only take away energy, and not give it up. In this way spontaneous radiation arises as a consequence of the existence of these unique field strengths corresponding to zero point oscillations. Thus spontaneous radiation is induced radiation of light quanta produced by zero point oscillations of empty space}}

This view was also later supported by [[Theodore A. Welton|Theodore Welton]] (1948),<ref>{{cite journal|last1=Welton|first1=Theodore Allen|title=Some observable effects of the quantum-mechanical fluctuations of the electromagnetic field|journal=Physical Review|date=1948|volume=74|issue=9|page=1157|doi=10.1103/PhysRev.74.1157|bibcode=1948PhRv...74.1157W}}</ref> who argued that spontaneous emission "can be thought of as forced emission taking place under the action of the fluctuating field". This new theory, which Dirac coined [[quantum electrodynamics]] (QED), predicted a fluctuating zero-point or "vacuum" field existing even in the absence of sources.

Throughout the 1940s improvements in [[microwave]] technology made it possible to take more precise measurements of the shift of the levels of a [[hydrogen atom]], now known as the Lamb shift,<ref name=lamb>
{{cite journal
 | author-link1= Willis Lamb
 | author-link2=Robert Retherford
 | year=1947
 | title=Fine Structure of the Hydrogen Atom by a Microwave Method
 | journal=[[Physical Review]]
 | volume=72 | pages= 241–243
 | doi=10.1103/PhysRev.72.241
 | bibcode = 1947PhRv...72..241L
 | issue=3
 | last1= Lamb
 | first1= Willis
 | last2= Retherford
 | first2= Robert
 | doi-access=free
}}</ref> and measurement of the [[magnetic moment]] of the electron.<ref name=foley>
{{cite journal
 | author-link2=Polykarp Kusch
 | author-link1=Henry M. Foley
 | year=1948
 | title=On the Intrinsic Moment of the Electron
 | journal=[[Physical Review]]
 | volume=73 | pages=412
 | doi=10.1103/PhysRev.73.412
 | bibcode = 1948PhRv...73..412F
 | issue=3
 | last1= Foley
 | first1= H.
 | last2= Kusch
 | first2= P.
}}</ref> Discrepancies between these experiments and Dirac's theory led to the idea of incorporating [[renormalisation]] into QED to deal with zero-point infinities. Renormalization was originally developed by [[Hans Kramers]]<ref>{{cite book|last1=Dresden|first1=M.|title=H. A. Kramers: Between Tradition and Revolution|date=1987|publisher=Springer|location=New York|isbn=978-1-461-29087-2|oclc= 1015092892}}</ref> and also [[Victor Weisskopf]] (1936),{{sfnp|Weisskopf|1936|p=6}} and first successfully applied to calculate a finite value for the Lamb shift by [[Hans Bethe]] (1947).<ref>{{cite journal|last1=Bethe|first1=Hans Albrecht|title=The Electromagnetic Shift of Energy Levels|journal=Physical Review|date=1947|volume=72|issue=4|page=339|doi=10.1103/PhysRev.72.339|bibcode=1947PhRv...72..339B|s2cid=120434909 }}</ref> As per spontaneous emission, these effects can in part be understood with interactions with the zero-point field.{{sfnp|Power|1964|p=35}}{{sfnp|Milonni|1994|p=111}} But in light of renormalisation being able to remove some zero-point infinities from calculations, not all physicists were comfortable attributing zero-point energy any physical meaning, viewing it instead as a mathematical artifact that might one day be eliminated. In [[Wolfgang Pauli]]'s 1945 [[Nobel lecture]]<ref>{{cite web|last1=Pauli|first1=Wolfgang|title=Exclusion principle and quantum mechanics|url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1945/pauli-lecture.pdf|website=nobelprize.org|publisher=Royal Swedish Academy of Sciences|access-date=20 October 2016|date=1946}}</ref> he made clear his opposition to the idea of zero-point energy stating "It is clear that this zero-point energy has no physical reality".

[[File:Hendrik Casimir (1958).jpg|thumb|left|upright|Hendrik Casimir (1958)]]
In 1948 [[Hendrik Casimir]]<ref>{{cite journal|last1=Casimir|first1=Hendrik Brugt Gerhard|last2=Polder|first2=Dirk|title=The Influence of Retardation on the London–Van der Waals Forces|journal=Physical Review|date=1948|volume=73|issue=4|page=360|doi=10.1103/PhysRev.73.360|bibcode=1948PhRv...73..360C}}</ref><ref>{{cite journal|last1=Casimir|first1=Hendrik Brugt Gerhard|title=On the attraction between two perfectly conducting plates|journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences|date=1948|volume=51|pages=793–795|url=http://www.dwc.knaw.nl/DL/publications/PU00018547.pdf|access-date=19 October 2016}}</ref> showed that one consequence of the zero-point field is an attractive force between two uncharged, perfectly conducting parallel plates, the so-called Casimir effect. At the time, Casimir was studying the properties of [[colloid|colloidal solutions]]. These are viscous materials, such as paint and mayonnaise, that contain micron-sized particles in a liquid matrix. The properties of such solutions are determined by [[Van der Waals forces]] – short-range, attractive forces that exist between neutral atoms and molecules. One of Casimir's colleagues, Theo Overbeek, realized that the theory that was used at the time to explain Van der Waals forces, which had been developed by [[Fritz London]] in 1930,<ref>{{cite journal|first1=R. |last1=Eisenschitz |first2=F. |last2=London |name-list-style=amp |journal=Zeitschrift für Physik|volume=60|pages= 491–527 |year=1930|doi=10.1007/BF01341258|title=Über das Verhältnis der Van der Waalsschen Kräfte zu den homöopolaren Bindungskräften|language=de|trans-title=On the relationship of van der Waals forces to homeopolar binding forces|issue=7–8|bibcode=1930ZPhy...60..491E|s2cid=125644826 }}</ref><ref>{{cite journal|first=F. |last=London|journal= Zeitschrift für Physik |volume=63|page= 245 |year=1930|doi=10.1007/BF01421741|title=Zur Theorie und Systematik der Molekularkräfte|language=de|trans-title=On the theory and systematics of molecular forces|issue=3–4|bibcode=1930ZPhy...63..245L|s2cid=123122363}}</ref> did not properly explain the experimental measurements on colloids. Overbeek therefore asked Casimir to investigate the problem. Working with [[Dirk Polder]], Casimir discovered that the interaction between two neutral molecules could be correctly described only if the fact that light travels at a finite speed was taken into account.<ref>{{cite journal|last1=Lambrecht|first1=Astrid|title=The Casimir effect: a force from nothing|journal=Physics World|date=2002|volume=15|issue=9|pages=29–32|url=https://indico.cern.ch/event/247728/contributions/1569920/attachments/426300/591724/Casimir_Force_PhysWorld_2002.pdf|access-date=24 October 2016|publisher=Institute of Physics Publishing|doi=10.1088/2058-7058/15/9/29|issn=0953-8585}}</ref> Soon afterwards after a conversation with [[Niels Bohr|Bohr]] about zero-point energy, Casimir noticed that this result could be interpreted in terms of vacuum fluctuations. He then asked himself what would happen if there were two mirrors – rather than two molecules – facing each other in a vacuum. It was this work that led to his prediction of an attractive force between reflecting plates. The work by Casimir and Polder opened up the way to a unified theory of van der Waals and Casimir forces and a smooth continuum between the two phenomena. This was done by Lifshitz (1956)<ref>{{Cite journal|title = The Theory of Molecular Attractive Forces between Solids|last = Lifshitz|first = E. M.|date = 1954|journal = Journal of Experimental Theoretical Physics USSR|volume = 29|pages = 94–110}}</ref><ref>{{Cite journal|title = The theory of molecular Attractive Forces between Solids|last = Lifshitz|first = E. M.|date = 1956|journal = Soviet Physics|volume = 2|number = 1|pages = 73–83}}</ref><ref>{{Cite journal|title = Direct measurement of molecular attraction between solids separated by a narrow gap|last1 = Derjaguin|first1 = B. V.|date = 1956|journal = Quarterly Reviews, Chemical Society|doi = 10.1039/qr9561000295|last2 = Abrikosova|first2 = I. I.|last3 = Lifshitz|first3 = E. M.|volume = 10|issue = 3|pages = 295–329}}</ref> in the case of plane parallel [[Dielectric|dielectric plates]]. The generic name for both van der Waals and Casimir forces is dispersion forces, because both of them are caused by dispersions of the operator of the dipole moment.<ref>{{cite book|last1=Mahanty|first1=J.|last2=Ninham|first2=B. W.|title=Dispersion Forces|date=1976|publisher=Academic Press|isbn=978-0-124-65050-3|oclc= 925046024}}</ref> The role of relativistic forces becomes dominant at orders of a hundred nanometers.

In 1951 [[Herbert Callen]] and Theodore Welton<ref name="ReferenceB">{{cite journal|last1=Callen|first1=Herbert|last2=Welton|first2=Theodore A.|title=Irreversibility and Generalized Noise|journal=Physical Review|date=1951|volume=83|issue=1|pages=34–40|doi=10.1103/PhysRev.83.34|bibcode=1951PhRv...83...34C}}</ref> proved the quantum [[fluctuation-dissipation theorem]] (FDT) which was originally formulated in classical form by [[Harry Nyquist|Nyquist]] (1928)<ref name="ReferenceC">{{cite journal|last1=Nyquist|first1=Harry|title=Thermal Agitation of Electric Charge in Conductors|journal=Physical Review|date=1928|volume=32|issue=1|pages=110–113|doi=10.1103/PhysRev.32.110|bibcode=1928PhRv...32..110N}}</ref> as an explanation for observed [[Johnson noise]] in electric circuits.<ref name="ReferenceD">{{cite journal|last1=Johnson|first1=John Bertrand|title=Thermal Agitation of Electricity in Conductors|journal=Physical Review|date=1928|volume=32|issue=1|pages=97–109|doi=10.1103/PhysRev.32.97|bibcode=1928PhRv...32...97J}}</ref> The fluctuation-dissipation theorem showed that when something dissipates energy, in an effectively irreversible way, a connected heat bath must also fluctuate. The fluctuations and the dissipation go hand in hand; it is impossible to have one without the other. The implication of FDT being that the vacuum could be treated as a heat bath coupled to a dissipative force and as such energy could, in part, be extracted from the vacuum for potentially useful work.{{sfnp|Milonni|1994|p=54}} FDT has been shown to be true experimentally under certain quantum, non-classical, conditions.<ref name="cloudfront.escholarship.org">{{cite journal|last1=Koch|first1=Roger H.|last2=Van Harlingen|first2=D. J.|last3=Clarke|first3=John|title=Observation of Zero-Point Fluctuations in a Resistively Shunted Josephson Tunnel Junction|journal=Physical Review Letters|date=1981|volume=47|issue=17|pages=1216–1219|doi=10.1103/PhysRevLett.47.1216|bibcode=1981PhRvL..47.1216K|osti=1136482|s2cid=119728862 |url=https://cloudfront.escholarship.org/dist/prd/content/qt7cb912p9/qt7cb912p9.pdf?t=maz7le}}</ref><ref name=Allahverdyan-2000>{{cite journal|last1=Allahverdyan|first1=A. E.|last2=Nieuwenhuizen|first2=Th. M.|title=Extraction of Work from a Single Thermal Bath in the Quantum Regime|journal=Physical Review Letters|date=2000|volume=85|issue=9|pages=1799–1802|doi=10.1103/PhysRevLett.85.1799|pmid=10970617|bibcode=2000PhRvL..85.1799A|arxiv=cond-mat/0006404|s2cid=32579381|url=https://pure.uva.nl/ws/files/3031844/12613_88222y.pdf}}</ref>{{sfnp|Scully et al.|2003}}

In 1963 the [[Jaynes–Cummings model]]<ref>{{cite journal|last1=Jaynes|first1=E. T.|last2=Cummings|first2=F. W.|title=Comparison of quantum and semiclassical radiation theories with application to the beam maser|journal=Proceedings of the IEEE|date=1963|volume=51|issue=1|doi=10.1109/PROC.1963.1664|pages=89–109}}</ref> was developed describing the system of a [[Two-level system|two-level atom]] interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave nonintuitive predictions such as that an atom's spontaneous emission could be driven by field of effectively constant frequency ([[Rabi frequency]]). In the 1970s experiments were being performed to test aspects of quantum optics and showed that the rate of spontaneous emission of an atom could be controlled using reflecting surfaces.{{sfnp|Drexhage|1970}}{{sfnp|Drexhage|1974|p={{page needed|date=May 2020}}}} These results were at first regarded with suspicion in some quarters: it was argued that no modification of a spontaneous emission rate would be possible, after all, how can the emission of a photon be affected by an atom's environment when the atom can only "see" its environment by emitting a photon in the first place? These experiments gave rise to [[cavity quantum electrodynamics]] (CQED), the study of effects of mirrors and cavities on radiative corrections. Spontaneous emission can be suppressed (or "inhibited")<ref>{{cite journal|last1=Hulet|first1=Randall G.|last2=Hilfer|first2=Eric S.|last3=Kleppner|first3=Daniel|title=Inhibited Spontaneous Emission by a Rydberg Atom|journal=Physical Review Letters|date=1985|volume=55|issue=20|pages=2137–2140|doi=10.1103/PhysRevLett.55.2137|pmid=10032058|bibcode=1985PhRvL..55.2137H|url=https://scholarship.rice.edu/bitstream/1911/79433/1/PhysRevLett.55.2137.pdf|hdl=1911/79433}}</ref><ref>{{cite journal|last1=Yablonovitch|first1=Eli|title=Inhibited Spontaneous Emission in Solid-State Physics and Electronics|journal=Physical Review Letters|date=1987|volume=58|issue=20|pages=2059–2062|doi=10.1103/PhysRevLett.58.2059|bibcode=1987PhRvL..58.2059Y|pmid=10034639|doi-access=free}}</ref> or amplified. Amplification was first predicted by Purcell in 1946<ref>{{cite journal|last1=Purcell|first1=E. M.|title=Proceedings of the American Physical Society|journal=Physical Review|date=1946|volume=69|issue=11–12|page=674|doi=10.1103/PhysRev.69.674|bibcode=1946PhRv...69Q.674.}}</ref> (the [[Purcell effect]]) and has been experimentally verified.{{sfnp|Goy et al.|1983}} This phenomenon can be understood, partly, in terms of the action of the vacuum field on the atom.{{sfnp|Milonni|1983}}