Zero-point energy

Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle.<ref name="FOOTNOTESciama1991137">Sciama (1991), p. 137.</ref> Therefore, even at absolute zero, atoms and molecules retain some vibrational motion. Apart from atoms and molecules, the empty space of the vacuum also has these properties. According to quantum field theory, the universe can be thought of not as isolated particles but continuous fluctuating fields: matter fields, whose quanta are fermions (i.e., leptons and quarks), and force fields, whose quanta are bosons (e.g., photons and gluons). All these fields have zero-point energy.<ref name="FOOTNOTEMilonni199435">Milonni (1994), p. 35.</ref> These fluctuating zero-point fields lead to a kind of reintroduction of an aether in physics<ref name="FOOTNOTESciama1991137">Sciama (1991), p. 137.</ref><ref name="FOOTNOTEDavies2011">Davies (2011).</ref> since some systems can detect the existence of this energy. However, this aether cannot be thought of as a physical medium if it is to be Lorentz invariant such that there is no contradiction with Einstein's theory of special relativity.<ref name="FOOTNOTESciama1991137">Sciama (1991), p. 137.</ref>

The notion of a zero-point energy is also important for cosmology, and physics currently lacks a full theoretical model for understanding zero-point energy in this context; in particular, the discrepancy between theorized and observed vacuum energy in the universe is a source of major contention.<ref>See Weinberg (1989) and Peebles & Ratra (2003) for review articles and Shiga (2005), Siegel (2016) for press comment</ref> Yet according to Einstein's theory of general relativity, any such energy would gravitate, and the experimental evidence from the expansion of the universe, dark energy and the Casimir effect shows any such energy to be exceptionally weak. One proposal that attempts to address this issue is to say that the fermion field has a negative zero-point energy, while the boson field has positive zero-point energy and thus these energies somehow cancel each other out.<ref name="FOOTNOTEWeinberg2015376">Weinberg (2015), p. 376.</ref><ref name="FOOTNOTESciama1991138">Sciama (1991), p. 138.</ref> This idea would be true if supersymmetry were an exact symmetry of nature; however, the LHC at CERN has so far found no evidence to support it. Moreover, it is known that if supersymmetry is valid at all, it is at most a broken symmetry, only true at very high energies, and no one has been able to show a theory where zero-point cancellations occur in the low-energy universe we observe today.<ref name="FOOTNOTESciama1991138">Sciama (1991), p. 138.</ref> This discrepancy is known as the cosmological constant problem and it is one of the greatest unsolved mysteries in physics. Many physicists believe that "the vacuum holds the key to a full understanding of nature".<ref name="FOOTNOTEDavies1985104">Davies (1985), p. 104.</ref>

Etymology and terminology

The term zero-point energy (ZPE) is a translation from the German Nullpunktsenergie.<ref name="FOOTNOTEEinstein1995270–285">Einstein (1995), pp. 270–285.</ref> Sometimes used interchangeably with it are the terms zero-point radiation and ground state energy. The term zero-point field (ZPF) can be used when referring to a specific vacuum field, for instance the QED vacuum which specifically deals with quantum electrodynamics (e.g., electromagnetic interactions between photons, electrons and the vacuum) or the QCD vacuum which deals with quantum chromodynamics (e.g., color charge interactions between quarks, gluons and the vacuum). A vacuum can be viewed not as empty space but as the combination of all zero-point fields. In quantum field theory this combination of fields is called the vacuum state, its associated zero-point energy is called the vacuum energy and the average energy value is called the vacuum expectation value (VEV) also called its condensate.

Overview

In classical mechanics all particles can be thought of as having some energy made up of their potential energy and kinetic energy. Temperature, for example, arises from the intensity of random particle motion caused by kinetic energy (known as Brownian motion). As temperature is reduced to absolute zero, it might be thought that all motion ceases and particles come completely to rest. In fact, however, kinetic energy is retained by particles even at the lowest possible temperature. The random motion corresponding to this zero-point energy never vanishes; it is a consequence of the uncertainty principle of quantum mechanics.

The uncertainty principle states that no object can ever have precise values of position and velocity simultaneously. The total energy of a quantum mechanical object (potential and kinetic) is described by its Hamiltonian which also describes the system as a harmonic oscillator, or wave function, that fluctuates between various energy states (see wave-particle duality). All quantum mechanical systems undergo fluctuations even in their ground state, a consequence of their wave-like nature. The uncertainty principle requires every quantum mechanical system to have a fluctuating zero-point energy greater than the minimum of its classical potential well. This results in motion even at absolute zero. For example, liquid helium does not freeze under atmospheric pressure regardless of temperature due to its zero-point energy.

Given the equivalence of mass and energy expressed by Albert Einstein's $E = mc 2$ , any point in space that contains energy can be thought of as having mass to create particles. Virtual particles spontaneously flash into existence at every point in space due to the energy of quantum fluctuations caused by the uncertainty principle. Modern physics has developed quantum field theory (QFT) to understand the fundamental interactions between matter and forces, it treats every single point of space as a quantum harmonic oscillator. According to QFT the universe is made up of matter fields, whose quanta are fermions (i.e. leptons and quarks), and force fields, whose quanta are bosons (e.g. photons and gluons). All these fields have zero-point energy.<ref name="FOOTNOTEMilonni199435">Milonni (1994), p. 35.</ref> Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, and that all properties of matter are merely vacuum fluctuations arising from interactions of the zero-point field.<ref name="FOOTNOTEBattersby2008">Battersby (2008).</ref>

The idea that "empty" space can have an intrinsic energy associated with it, and that there is no such thing as a "true vacuum" is seemingly unintuitive. It is often argued that the entire universe is completely bathed in the zero-point radiation, and as such it can add only some constant amount to calculations. Physical measurements will therefore reveal only deviations from this value.<ref name="FOOTNOTEItzyksonZuber1980111">Itzykson & Zuber (1980), p. 111.</ref> For many practical calculations zero-point energy is dismissed by fiat in the mathematical model as a term that has no physical effect. Such treatment causes problems however, as in Einstein's theory of general relativity the absolute energy value of space is not an arbitrary constant and gives rise to the cosmological constant. For decades most physicists assumed that there was some undiscovered fundamental principle that will remove the infinite zero-point energy and make it completely vanish. If the vacuum has no intrinsic, absolute value of energy it will not gravitate. It was believed that as the universe expands from the aftermath of the Big Bang, the energy contained in any unit of empty space will decrease as the total energy spreads out to fill the volume of the universe; galaxies and all matter in the universe should begin to decelerate. This possibility was ruled out in 1998 by the discovery that the expansion of the universe is not slowing down but is in fact accelerating, meaning empty space does indeed have some intrinsic energy. The discovery of dark energy is best explained by zero-point energy, though it still remains a mystery as to why the value appears to be so small compared to the huge value obtained through theory – the cosmological constant problem.<ref name="FOOTNOTEWeinberg2015376">Weinberg (2015), p. 376.</ref>

Many physical effects attributed to zero-point energy have been experimentally verified, such as spontaneous emission, Casimir force, Lamb shift, magnetic moment of the electron and Delbrück scattering.<ref name="FOOTNOTEMilonni1994111">Milonni (1994), p. 111.</ref><ref name="FOOTNOTEGreinerMüllerRafelski201212">Greiner, Müller & Rafelski (2012), p. 12.</ref> These effects are usually called "radiative corrections".<ref name="FOOTNOTEBordag et al.20094">Bordag et al. (2009), p. 4.</ref> In more complex nonlinear theories (e.g. QCD) zero-point energy can give rise to a variety of complex phenomena such as multiple stable states, symmetry breaking, chaos and emergence. Many physicists believe that "the vacuum holds the key to a full understanding of nature"<ref name="FOOTNOTEDavies1985104">Davies (1985), p. 104.</ref> and that studying it is critical in the search for the theory of everything. Active areas of research include the effects of virtual particles,<ref name="FOOTNOTECho2015">Cho (2015).</ref> quantum entanglement,<ref name="FOOTNOTEChoi2013">Choi (2013).</ref> the difference (if any) between inertial and gravitational mass,<ref name="Haisch et al. 1994">See Haisch, Rueda & Puthoff (1994) for proposal and Matthews (1994, 1995), Powell (1994) and Davies (1994) for comment.</ref> variation in the speed of light,<ref>See Urban et al. (2013), Leuchs & Sánchez-Soto (2013) and O'Carroll (2013) for comment.</ref> a reason for the observed value of the cosmological constant<ref name="FOOTNOTERughZinkernagel2002">Rugh & Zinkernagel (2002).</ref> and the nature of dark energy.<ref name="Dark Energy May Be Vacuum">"Dark Energy May Be Vacuum" (Press release). Niels Bohr Institute. 19 January 2007. Archived from the original on 31 May 2017.</ref><ref name="FOOTNOTEWall2014">Wall (2014).</ref>

History

Early aether theories

Zero-point energy evolved from historical ideas about the vacuum. To Aristotle the vacuum was τὸ κενόν, "the empty"; i.e., space independent of body. He believed this concept violated basic physical principles and asserted that the elements of fire, air, earth, and water were not made of atoms, but were continuous. To the atomists the concept of emptiness had absolute character: it was the distinction between existence and nonexistence.<ref name="FOOTNOTESaundersBrown19911">Saunders & Brown (1991), p. 1.</ref> Debate about the characteristics of the vacuum were largely confined to the realm of philosophy, it was not until much later on with the beginning of the renaissance, that Otto von Guericke invented the first vacuum pump and the first testable scientific ideas began to emerge. It was thought that a totally empty volume of space could be created by simply removing all gases. This was the first generally accepted concept of the vacuum.<ref name="FOOTNOTEConlon2011225">Conlon (2011), p. 225.</ref>

Late in the 19th century, however, it became apparent that the evacuated region still contained thermal radiation. The existence of the aether as a substitute for a true void was the most prevalent theory of the time. According to the successful electromagnetic aether theory based upon Maxwell's electrodynamics, this all-encompassing aether was endowed with energy and hence very different from nothingness. The fact that electromagnetic and gravitational phenomena were easily transmitted in empty space indicated that their associated aethers were part of the fabric of space itself. Maxwell himself noted that:

To those who maintained the existence of a plenum as a philosophical principle, nature's abhorrence of a vacuum was a sufficient reason for imagining an all-surrounding aether ... Aethers were invented for the planets to swim in, to constitute electric atmospheres and magnetic effluvia, to convey sensations from one part of our bodies to another, and so on, till a space had been filled three or four times with aethers.<ref name="FOOTNOTEKraghOverduin20147">Kragh & Overduin (2014), p. 7.</ref>

However, the results of the Michelson–Morley experiment in 1887 were the first strong evidence that the then-prevalent aether theories were seriously flawed, and initiated a line of research that eventually led to special relativity, which ruled out the idea of a stationary aether altogether. To scientists of the period, it seemed that a true vacuum in space might be created by cooling and thus eliminating all radiation or energy. From this idea evolved the second concept of achieving a real vacuum: cool a region of space down to absolute zero temperature after evacuation. Absolute zero was technically impossible to achieve in the 19th century, so the debate remained unsolved.

Second quantum theory

In 1900, Max Planck derived the average energy $ε$ of a single energy radiator, e.g., a vibrating atomic unit, as a function of absolute temperature:<ref name="FOOTNOTEPlanck1900">Planck (1900).</ref>

<math display="block"> \varepsilon = \frac{h\nu}{ e^{h\nu/(kT)}-1} \,,</math>

where $h$ is Planck's constant, $ν$ is the frequency, $k$ is the Boltzmann constant, and $T$ is the absolute temperature. The zero-point energy makes no contribution to Planck's original law, as its existence was unknown to Planck in 1900.<ref name="FOOTNOTELoudon20009">Loudon (2000), p. 9.</ref>

The concept of zero-point energy was developed by Max Planck in Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900.<ref name="FOOTNOTEKragh20127">Kragh (2012), p. 7.</ref>

In 1912, Max Planck published the first journal article to describe the discontinuous emission of radiation, based on the discrete quanta of energy.<ref name="FOOTNOTEPlanck1912a">Planck (1912a).</ref> In Planck's "second quantum theory" resonators absorbed energy continuously, but emitted energy in discrete energy quanta only when they reached the boundaries of finite cells in phase space, where their energies became integer multiples of $hν$ . This theory led Planck to his new radiation law, but in this version energy resonators possessed a zero-point energy, the smallest average energy a resonator could take on. Planck's radiation equation contained a residual energy factor, one $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}hν/2$ , as an additional term dependent on the frequency $ν$ , which was greater than zero (where $h$ is Planck's constant). It is therefore widely agreed that "Planck's equation marked the birth of the concept of zero-point energy."<ref name="FOOTNOTEMilonni199410">Milonni (1994), p. 10.</ref> In a series of papers from 1911 to 1913,<ref>See (Planck 1911, 1912a, 1912b, 1913) and Planck (1958) for reprints</ref> Planck found the average energy of an oscillator to be:<ref name="FOOTNOTEKragh20127">Kragh (2012), p. 7.</ref><ref name="FOOTNOTEKuhn1978235">Kuhn (1978), p. 235.</ref>

<math display="block">\varepsilon =\frac{h\nu} 2 + \frac{h\nu}{e^{h\nu/(kT)}-1} ~.</math>

Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistant Otto Stern.<ref>Einstein, Albert; Stern, Otto (1913). "Einige Argumente für die Annahme einer molekularen Agitation beim absoluten Nullpunkt" [Some arguments for the assumption of a molecular agitation at the absolute zero point]. Annalen der Physik (in Deutsch). 345 (3): 551–560. Bibcode:1913AnP...345..551E. doi:10.1002/andp.19133450309.</ref> In 1913 they published a paper that attempted to prove the existence of zero-point energy by calculating the specific heat of hydrogen gas and compared it with the experimental data. However, after assuming they had succeeded, they retracted support for the idea shortly after publication because they found Planck's second theory may not apply to their example. In a letter to Paul Ehrenfest of the same year Einstein declared zero-point energy "dead as a doornail".<ref name="FOOTNOTEEinstein1993563–565">Einstein (1993), pp. 563–565.</ref> Zero-point energy was also invoked by Peter Debye,<ref>Debye, Peter (1913). "Interferenz von Röntgenstrahlen und Wärmebewegung" [Interference of X-rays and thermal motion]. Annalen der Physik (in Deutsch). 348 (1): 49–92. Bibcode:1913AnP...348...49D. doi:10.1002/andp.19133480105.</ref> who noted that zero-point energy of the atoms of a crystal lattice would cause a reduction in the intensity of the diffracted radiation in X-ray diffraction even as the temperature approached absolute zero. In 1916 Walther Nernst proposed that empty space was filled with zero-point electromagnetic radiation.<ref>Nernst, Walther (1916). "Über einen Versuch, von quantentheoretischen Betrachtungen zur Annahme stetiger Energieänderungen zurückzukehren" [On an attempt to return from quantum-theoretical considerations to the assumption of constant energy changes]. Verhandlungen der Deutschen Physikalischen (in Deutsch). 18: 83–116.</ref> With the development of general relativity Einstein found the energy density of the vacuum to contribute towards a cosmological constant in order to obtain static solutions to his field equations; the idea that empty space, or the vacuum, could have some intrinsic energy associated with it had returned, with Einstein stating in 1920:

There is a weighty argument to be adduced in favour of the aether hypothesis. To deny the aether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view ... according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether. According to the general theory of relativity space without aether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this aether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.<ref>Einstein, Albert (1920). Äther und Relativitäts-Theorie [Aether and relativity theory] (in Deutsch). Berlin: Springer.</ref><ref>Einstein, Albert (1922). Jeffery, G. B.; Perrett, W. (eds.). Sidelights on Relativity: Ether and the Theory of Relativity. New York: Methuen & Co. pp. 1–24.</ref>

File:Heisenberg,Werner 1924 Göttingen - adjusted.jpeg

Heisenberg, 1924

Kurt Bennewitz [de] and Francis Simon (1923),<ref>Bennewitz, Kurt; Simon, Franz (1923). "Zur Frage der Nullpunktsenergie" [On the question of zero-point energy]. Zeitschrift für Physik (in Deutsch). 16 (1): 183–199. Bibcode:1923ZPhy...16..183B. doi:10.1007/BF01327389. S2CID 121049183.</ref> who worked at Walther Nernst's laboratory in Berlin, studied the melting process of chemicals at low temperatures. Their calculations of the melting points of hydrogen, argon and mercury led them to conclude that the results provided evidence for a zero-point energy. Moreover, they suggested correctly, as was later verified by Simon (1934),<ref>Simon, F. (1934). "Behaviour of Condensed Helium near Absolute Zero". Nature. 133 (3362): 529. Bibcode:1934Natur.133Q.529S. doi:10.1038/133529a0. S2CID 4130047.</ref><ref>Dugdale, J. S.; Simon, F. E. (1953). "Thermodynamic Properties and Melting of Solid Helium". Proceedings of the Royal Society. 218 (1134): 291. Bibcode:1953RSPSA.218..291D. doi:10.1098/rspa.1953.0105. S2CID 98061516.</ref> that this quantity was responsible for the difficulty in solidifying helium even at absolute zero. In 1924 Robert Mulliken<ref>Mulliken, Robert S. (1924). "The band spectrum of boron monoxide". Nature. 114 (2862): 349–350. Bibcode:1924Natur.114..349M. doi:10.1038/114349a0. S2CID 4121118.</ref> provided direct evidence for the zero-point energy of molecular vibrations by comparing the band spectrum of ¹⁰BO and ¹¹BO: the isotopic difference in the transition frequencies between the ground vibrational states of two different electronic levels would vanish if there were no zero-point energy, in contrast to the observed spectra. Then just a year later in 1925,<ref>Heisenberg, W. (1925). "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen" [On quantum-theoretical reinterpretation of kinematic and mechanical relationships]. In Blum, Walter; Rechenberg, Helmut; Dürr, Hans-Peter (eds.). Wissenschaftliche Originalarbeiten [Original Scientific Papers] (in Deutsch). Berlin, Heidelberg: Springer (published 1985). pp. 382–396. doi:10.1007/978-3-642-61659-4_26. ISBN 978-3-642-64900-4. OCLC 7331244990.</ref> with the development of matrix mechanics in Werner Heisenberg's famous article "Quantum theoretical re-interpretation of kinematic and mechanical relations" the zero-point energy was derived from quantum mechanics.<ref name="FOOTNOTEKragh2002162">Kragh (2002), p. 162.</ref>

In 1913 Niels Bohr had proposed what is now called the Bohr model of the atom,<ref>Bohr, Niels (1913). "On the Constitution of Atoms and Molecules, Part I" (PDF). Philosophical Magazine. 26 (151): 1–24. Bibcode:1913PMag...26....1B. doi:10.1080/14786441308634955.</ref><ref>Bohr, Niels (1913). "On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus" (PDF). Philosophical Magazine. 26 (153): 476–502. Bibcode:1913PMag...26..476B. doi:10.1080/14786441308634993.</ref><ref>Bohr, Niels (1913). "On the Constitution of Atoms and Molecules, Part III Systems containing several nuclei". Philosophical Magazine. 26 (155): 857–875. Bibcode:1913PMag...26..857B. doi:10.1080/14786441308635031.</ref> but despite this it remained a mystery as to why electrons do not fall into their nuclei. According to classical ideas, the fact that an accelerating charge loses energy by radiating implied that an electron should spiral into the nucleus and that atoms should not be stable. This problem of classical mechanics was nicely summarized by James Hopwood Jeans in 1915: "There would be a very real difficulty in supposing that the (force) law $1 / r 2$ held down to the zero values of $r$ . For the forces between two charges at zero distance would be infinite; we should have charges of opposite sign continually rushing together and, when once together, no force would tend to shrink into nothing or to diminish indefinitely in size."<ref>Jeans, James Hopwood (1915). The mathematical theory of electricity and magnetism (3rd ed.). Cambridge: Cambridge University Press. p. 168.</ref> The resolution to this puzzle came in 1926 with Schrödinger's famous equation.<ref>Schrödinger, Erwin (1926). "Quantisierung als Eigenwertproblem" [Quantization as an eigenvalue problem]. Annalen der Physik (in Deutsch). 79 (13): 361–376. Bibcode:1926AnP...385..437S. doi:10.1002/andp.19263851302.</ref> This equation explained the new, non-classical fact that an electron confined to be close to a nucleus would necessarily have a large kinetic energy so that the minimum total energy (kinetic plus potential) actually occurs at some positive separation rather than at zero separation; in other words, zero-point energy is essential for atomic stability.<ref>Lieb, E. H.; Seiringer, R. (2009). The Stability of Matter in Quantum Mechanics. Cambridge: Cambridge University Press. pp. 2–3. ISBN 978-0-521-19118-0. OCLC 638472161.</ref>

Quantum field theory and beyond

In 1926 Pascual Jordan<ref>Born, M.; Heisenberg, W.; Jordan, P. (1926). "Zur Quantenmechanik. II" [On quantum mechanics II]. Zeitschrift für Physik (in Deutsch). 35 (8): 557–615. Bibcode:1926ZPhy...35..557B. doi:10.1007/BF01379806. 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>Einstein, Albert (1909). "Zum gegenwärtigen Stand des Strahlungsproblems". Physikalische Zeitschrift. 10: 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>Mehra, J.; Rechenberg, H. (2002). The Historical Development of Quantum Theory. Vol. 6. Springer. p. 57. ISBN 978-0-387-95262-8. OCLC 722601833.</ref> Jordan found a way to get rid of the infinite term, publishing a joint work with Pauli in 1928,<ref>Jordan, P.; Pauli, W. (1928). "Zur Quantenelektrodynamik ladungsfreier Felder" [On the quantum electrodynamics of charge-free fields]. Zeitschrift für Physik (in Deutsch). 47 (3): 151–173. Bibcode:1928ZPhy...47..151J. doi:10.1007/BF02055793. S2CID 120536476.</ref> performing what has been called "the first infinite subtraction, or renormalisation, in quantum field theory".<ref>Schweber, Silvan S. (1994). QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga. Princeton University Press. pp. 108–112. ISBN 978-0-691-03327-3. OCLC 439849774.</ref>

Building on the work of Heisenberg and others, Paul Dirac's theory of emission and absorption (1927)<ref name="FOOTNOTEDirac1927">Dirac (1927).</ref> 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>Weinberg, Steven (1977). "The Search for Unity: Notes for a History of Quantum Field Theory". Daedalus. 106 (4): 17–35. JSTOR 20024506.</ref> Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators 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"> Yokoyama, H.; Ujihara, K. (1995). Spontaneous emission and laser oscillation in microcavities. Boca Raton: CRC Press. p. 6. ISBN 978-0-8493-3786-4. OCLC 832589969. </ref><ref name="FOOTNOTEScullyZubairy1997[httpsbooksgooglecombooksid20ISsQCKKmQCpgPA22 §1.5.2 pp. 22–23]">Scully & Zubairy (1997), §1.5.2 pp. 22–23.</ref> 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:<ref name="FOOTNOTEDirac1927">Dirac (1927).</ref>

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 ...

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>Weisskopf, Viktor (1935). "Probleme der neueren Quantentheorie des Elektrons" [Problems of the new quantum theory of the electron]. Naturwissenschaften (in Deutsch). 23 (37): 631–637. Bibcode:1935NW.....23..631W. doi:10.1007/BF01492012. S2CID 6780937.</ref>

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 ... 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 Welton [de] (1948),<ref>Welton, Theodore Allen (1948). "Some observable effects of the quantum-mechanical fluctuations of the electromagnetic field". Physical Review. 74 (9): 1157. Bibcode:1948PhRv...74.1157W. doi:10.1103/PhysRev.74.1157.</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> Lamb, Willis; Retherford, Robert (1947). "Fine Structure of the Hydrogen Atom by a Microwave Method". Physical Review. 72 (3): 241–243. Bibcode:1947PhRv...72..241L. doi:10.1103/PhysRev.72.241.</ref> and measurement of the magnetic moment of the electron.<ref name=foley> Foley, H.; Kusch, P. (1948). "On the Intrinsic Moment of the Electron". Physical Review. 73 (3): 412. Bibcode:1948PhRv...73..412F. doi:10.1103/PhysRev.73.412.</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>Dresden, M. (1987). H. A. Kramers: Between Tradition and Revolution. New York: Springer. ISBN 978-1-461-29087-2. OCLC 1015092892.</ref> and also Victor Weisskopf (1936),<ref name="FOOTNOTEWeisskopf19366">Weisskopf (1936), p. 6.</ref> and first successfully applied to calculate a finite value for the Lamb shift by Hans Bethe (1947).<ref>Bethe, Hans Albrecht (1947). "The Electromagnetic Shift of Energy Levels". Physical Review. 72 (4): 339. Bibcode:1947PhRv...72..339B. doi:10.1103/PhysRev.72.339. S2CID 120434909.</ref> As per spontaneous emission, these effects can in part be understood with interactions with the zero-point field.<ref name="FOOTNOTEPower196435">Power (1964), p. 35.</ref><ref name="FOOTNOTEMilonni1994111">Milonni (1994), p. 111.</ref> 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>Pauli, Wolfgang (1946). "Exclusion principle and quantum mechanics" (PDF). nobelprize.org. Royal Swedish Academy of Sciences. Retrieved 20 October 2016.</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".

In 1948 Hendrik Casimir<ref>Casimir, Hendrik Brugt Gerhard; Polder, Dirk (1948). "The Influence of Retardation on the London–Van der Waals Forces". Physical Review. 73 (4): 360. Bibcode:1948PhRv...73..360C. doi:10.1103/PhysRev.73.360.</ref><ref>Casimir, Hendrik Brugt Gerhard (1948). "On the attraction between two perfectly conducting plates" (PDF). Proceedings of the Royal Netherlands Academy of Arts and Sciences. 51: 793–795. Retrieved 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 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>Eisenschitz, R. & London, F. (1930). "Über das Verhältnis der Van der Waalsschen Kräfte zu den homöopolaren Bindungskräften" [On the relationship of van der Waals forces to homeopolar binding forces]. Zeitschrift für Physik (in Deutsch). 60 (7–8): 491–527. Bibcode:1930ZPhy...60..491E. doi:10.1007/BF01341258. S2CID 125644826.</ref><ref>London, F. (1930). "Zur Theorie und Systematik der Molekularkräfte" [On the theory and systematics of molecular forces]. Zeitschrift für Physik (in Deutsch). 63 (3–4): 245. Bibcode:1930ZPhy...63..245L. doi:10.1007/BF01421741. 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>Lambrecht, Astrid (2002). "The Casimir effect: a force from nothing" (PDF). Physics World. Institute of Physics Publishing. 15 (9): 29–32. doi:10.1088/2058-7058/15/9/29. ISSN 0953-8585. Retrieved 24 October 2016.</ref> Soon afterwards after a conversation with 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 famous 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>Lifshitz, E. M. (1954). "The Theory of Molecular Attractive Forces between Solids". Journal of Experimental Theoretical Physics USSR. 29: 94–110.</ref><ref>Lifshitz, E. M. (1956). "The theory of molecular Attractive Forces between Solids". Soviet Physics. 2 (1): 73–83.</ref><ref>Derjaguin, B. V.; Abrikosova, I. I.; Lifshitz, E. M. (1956). "Direct measurement of molecular attraction between solids separated by a narrow gap". Quarterly Reviews, Chemical Society. 10 (3): 295–329. doi:10.1039/qr9561000295.</ref> in the case of plane parallel 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>Mahanty, J.; Ninham, B. W. (1976). Dispersion Forces. 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">Callen, Herbert; Welton, Theodore A. (1951). "Irreversibility and Generalized Noise". Physical Review. 83 (1): 34–40. Bibcode:1951PhRv...83...34C. doi:10.1103/PhysRev.83.34.</ref> proved the quantum fluctuation-dissipation theorem (FDT) which was originally formulated in classical form by Nyquist (1928)<ref name="ReferenceC">Nyquist, Harry (1928). "Thermal Agitation of Electric Charge in Conductors". Physical Review. 32 (1): 110–113. Bibcode:1928PhRv...32..110N. doi:10.1103/PhysRev.32.110.</ref> as an explanation for observed Johnson noise in electric circuits.<ref name="ReferenceD">Johnson, John Bertrand (1928). "Thermal Agitation of Electricity in Conductors". Physical Review. 32 (1): 97–109. Bibcode:1928PhRv...32...97J. doi:10.1103/PhysRev.32.97.</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.<ref name="FOOTNOTEMilonni199454">Milonni (1994), p. 54.</ref> FDT has been shown to be true experimentally under certain quantum, non-classical, conditions.<ref name="cloudfront.escholarship.org">Koch, Roger H.; Van Harlingen, D. J.; Clarke, John (1981). "Observation of Zero-Point Fluctuations in a Resistively Shunted Josephson Tunnel Junction" (PDF). Physical Review Letters. 47 (17): 1216–1219. Bibcode:1981PhRvL..47.1216K. doi:10.1103/PhysRevLett.47.1216. OSTI 1136482. S2CID 119728862.</ref><ref name=Allahverdyan-2000>Allahverdyan, A. E.; Nieuwenhuizen, Th. M. (2000). "Extraction of Work from a Single Thermal Bath in the Quantum Regime" (PDF). Physical Review Letters. 85 (9): 1799–1802. arXiv:cond-mat/0006404. Bibcode:2000PhRvL..85.1799A. doi:10.1103/PhysRevLett.85.1799. PMID 10970617. S2CID 32579381.</ref><ref name="FOOTNOTEScully et al.2003">Scully et al. (2003).</ref>

In 1963 the Jaynes–Cummings model<ref>Jaynes, E. T.; Cummings, F. W. (1963). "Comparison of quantum and semiclassical radiation theories with application to the beam maser". Proceedings of the IEEE. 51 (1): 89–109. doi:10.1109/PROC.1963.1664.</ref> was developed describing the system of a 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.<ref name="FOOTNOTEDrexhage1970">Drexhage (1970).</ref><ref name="FOOTNOTEDrexhage1974[page needed]">Drexhage (1974), p. ^{[page needed]}.</ref> 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>Hulet, Randall G.; Hilfer, Eric S.; Kleppner, Daniel (1985). "Inhibited Spontaneous Emission by a Rydberg Atom" (PDF). Physical Review Letters. 55 (20): 2137–2140. Bibcode:1985PhRvL..55.2137H. doi:10.1103/PhysRevLett.55.2137. hdl:1911/79433. PMID 10032058.</ref><ref>Yablonovitch, Eli (1987). "Inhibited Spontaneous Emission in Solid-State Physics and Electronics". Physical Review Letters. 58 (20): 2059–2062. Bibcode:1987PhRvL..58.2059Y. doi:10.1103/PhysRevLett.58.2059. PMID 10034639.</ref> or amplified. Amplification was first predicted by Purcell in 1946<ref>Purcell, E. M. (1946). "Proceedings of the American Physical Society". Physical Review. 69 (11–12): 674. Bibcode:1946PhRv...69Q.674.. doi:10.1103/PhysRev.69.674.</ref> (the Purcell effect) and has been experimentally verified.<ref name="FOOTNOTEGoy et al.1983">Goy et al. (1983).</ref> This phenomenon can be understood, partly, in terms of the action of the vacuum field on the atom.<ref name="FOOTNOTEMilonni1983">Milonni (1983).</ref>

The uncertainty principle

Zero-point energy is fundamentally related to the Heisenberg uncertainty principle.<ref name="Heisenberg_1927">Heisenberg, W. (1927). "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik" [On the evident content of quantum-theoretical kinematics and mechanics]. Zeitschrift für Physik (in Deutsch). 43 (3): 172–198. Bibcode:1927ZPhy...43..172H. doi:10.1007/BF01397280. S2CID 122763326.</ref> Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position and momentum, or a field's value and derivative at a point in space) cannot simultaneously be specified precisely by any given quantum state. In particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well, for then its position and momentum would both be completely determined to arbitrarily great precision. Therefore, the lowest-energy state (the ground state) of the system must have a distribution in position and momentum that satisfies the uncertainty principle, which implies its energy must be greater than the minimum of the potential well.

Near the bottom of a potential well, the Hamiltonian of a general system (the quantum-mechanical operator giving its energy) can be approximated as a quantum harmonic oscillator,

<math display="block">\hat{H} = V_0 + \tfrac{1}{2} k \left(\hat{x} - x_0\right)^2 + \frac{1}{2m} \hat{p}^2 \,,</math> where $V 0$ is the minimum of the classical potential well.

The uncertainty principle tells us that

<math display="block">\sqrt{\left\langle \left(\hat{x} - x_0\right)^2 \right\rangle} \sqrt{\left\langle \hat{p}^2 \right\rangle} \geq \frac{\hbar}{2} \,,</math>

making the expectation values of the kinetic and potential terms above satisfy

<math display="block">\left\langle \tfrac{1}{2} k \left(\hat{x} - x_0\right)^2 \right\rangle \left\langle \frac{1}{2m} \hat{p}^2 \right\rangle \geq \left(\frac{\hbar}{4}\right)^2 \frac{k}{m} \,.</math>

The expectation value of the energy must therefore be at least

<math display="block">\left\langle \hat{H} \right\rangle \geq V_0 + \frac{\hbar}{2} \sqrt{\frac{k}{m}} = V_0 + \frac{\hbar \omega}{2}</math>

where $ω = \sqrt k / m$ is the angular frequency at which the system oscillates.

A more thorough treatment, showing that the energy of the ground state actually saturates this bound and is exactly $E 0 = V 0 + ħω / 2$ , requires solving for the ground state of the system.

Atomic physics

The zero-point energy $E = ħω / 2$ causes the ground-state of a harmonic oscillator to advance its phase (color). This has measurable effects when several eigenstates are superimposed.

The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or a subatomic particle. In ordinary atomic physics, the zero-point energy is the energy associated with the ground state of the system. The professional physics literature tends to measure frequency, as denoted by $ν$ above, using angular frequency, denoted with $ω$ and defined by $ω = 2 πν$ . This leads to a convention of writing Planck's constant $h$ with a bar through its top ( $ħ$ ) to denote the quantity $h / 2π$ . In these terms, the most famous such example of zero-point energy is the above $E = ħω / 2$ associated with the ground state of the quantum harmonic oscillator. In quantum mechanical terms, the zero-point energy is the expectation value of the Hamiltonian of the system in the ground state.

If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator which acts non-trivially on a ground state and commutes with the Hamiltonian of the system.

According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature.

The wave function of the ground state of a particle in a one-dimensional well is a half-period sine wave which goes to zero at the two edges of the well. The energy of the particle is given by:

where $h$ is the Planck constant, $m$ is the mass of the particle, $n$ is the energy state ( $n = 1$ corresponds to the ground-state energy), and $L$ is the width of the well.

Quantum field theory

In quantum field theory (QFT), the fabric of "empty" space is visualized as consisting of fields, with the field at every point in space and time being a quantum harmonic oscillator, with neighboring oscillators interacting with each other. According to QFT the universe is made up of matter fields whose quanta are fermions (e.g. electrons and quarks), force fields whose quanta are bosons (i.e. photons and gluons) and a Higgs field whose quantum is the Higgs boson. The matter and force fields have zero-point energy.<ref name="FOOTNOTEMilonni199435">Milonni (1994), p. 35.</ref> A related term is zero-point field (ZPF), which is the lowest energy state of a particular field.<ref> Gribbin, J. R. (1998). Gribbin, M. (ed.). Q is for Quantum: An Encyclopedia of Particle Physics. Touchstone Books. Bibcode:1999qqep.book.....G. ISBN 978-0-684-86315-3. OCLC 869069919.</ref> The vacuum can be viewed not as empty space, but as the combination of all zero-point fields.

In QFT the zero-point energy of the vacuum state is called the vacuum energy and the average expectation value of the Hamiltonian is called the vacuum expectation value (also called condensate or simply VEV). The QED vacuum is a part of the vacuum state which specifically deals with quantum electrodynamics (e.g. electromagnetic interactions between photons, electrons and the vacuum) and the QCD vacuum deals with quantum chromodynamics (e.g. color charge interactions between quarks, gluons and the vacuum). Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, and that all properties of matter are merely vacuum fluctuations arising from interactions with the zero-point field.<ref name="FOOTNOTEBattersby2008">Battersby (2008).</ref>

Each point in space makes a contribution of $E = ħω / 2$ , resulting in a calculation of infinite zero-point energy in any finite volume; this is one reason renormalization is needed to make sense of quantum field theories. In cosmology, the vacuum energy is one possible explanation for the cosmological constant<ref name="FOOTNOTERughZinkernagel2002">Rugh & Zinkernagel (2002).</ref> and the source of dark energy.<ref name="Dark Energy May Be Vacuum"/><ref name="FOOTNOTEWall2014">Wall (2014).</ref>

Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large as Paul Dirac claimed it is, like a sea of energy. Other scientists specializing in General Relativity require the energy to be small enough for curvature of space to agree with observed astronomy. The Heisenberg uncertainty principle allows the energy to be as large as needed to promote quantum actions for a brief moment of time, even if the average energy is small enough to satisfy relativity and flat space. To cope with disagreements, the vacuum energy is described as a virtual energy potential of positive and negative energy.<ref name="FOOTNOTEPeskinSchroeder1995786–791">Peskin & Schroeder (1995), pp. 786–791.</ref>

In quantum perturbation theory, it is sometimes said that the contribution of one-loop and multi-loop Feynman diagrams to elementary particle propagators are the contribution of vacuum fluctuations, or the zero-point energy to the particle masses.

The quantum electrodynamic vacuum

The oldest and best known quantized force field is the electromagnetic field. Maxwell's equations have been superseded by quantum electrodynamics (QED). By considering the zero-point energy that arises from QED it is possible to gain a characteristic understanding of zero-point energy that arises not just through electromagnetic interactions but in all quantum field theories.

Redefining the zero of energy

In the quantum theory of the electromagnetic field, classical wave amplitudes $α$ and $α *$ are replaced by operators $a$ and $a †$ that satisfy:

<math display="block">\left[a,a^\dagger\right] = 1</math>

The classical quantity $| α | 2$ appearing in the classical expression for the energy of a field mode is replaced in quantum theory by the photon number operator $a † a$ . The fact that:

<math display="block">\left[a,a^\dagger a\right] \ne 1</math>

implies that quantum theory does not allow states of the radiation field for which the photon number and a field amplitude can be precisely defined, i.e., we cannot have simultaneous eigenstates for $a † a$ and $a$ . The reconciliation of wave and particle attributes of the field is accomplished via the association of a probability amplitude with a classical mode pattern. The calculation of field modes is entirely classical problem, while the quantum properties of the field are carried by the mode "amplitudes" $a †$ and $a$ associated with these classical modes.

The zero-point energy of the field arises formally from the non-commutativity of $a$ and $a †$ . This is true for any harmonic oscillator: the zero-point energy $ħω / 2$ appears when we write the Hamiltonian:

<math display="block">\begin{align} H_{cl} &= \frac{p^2}{2m} + \tfrac{1}{2} m \omega^2 {q}^2 \\ &= \tfrac{1}{2} \hbar \omega \left(a a^\dagger + a^\dagger a\right) \\ &=\hbar \omega \left(a^\dagger a +\tfrac{1}{2}\right) \end{align}</math>

It is often argued that the entire universe is completely bathed in the zero-point electromagnetic field, and as such it can add only some constant amount to expectation values. Physical measurements will therefore reveal only deviations from the vacuum state. Thus the zero-point energy can be dropped from the Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion. Thus we can choose to declare by fiat that the ground state has zero energy and a field Hamiltonian, for example, can be replaced by:<ref name="FOOTNOTEItzyksonZuber1980111">Itzykson & Zuber (1980), p. 111.</ref>

<math display="block">\begin{align} H_F - \left\langle 0|H_F|0\right\rangle &=\tfrac{1}{2} \hbar \omega \left(a a^\dagger + a^\dagger a\right)-\tfrac{1}{2}\hbar \omega \\ &= \hbar \omega \left(a^\dagger a + \tfrac{1}{2} \right)-\tfrac{1}{2}\hbar \omega \\ &= \hbar \omega a^\dagger a \end{align}</math>

without affecting any physical predictions of the theory. The new Hamiltonian is said to be normally ordered (or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted $: H F$ , i.e.:

<math display="block">:H_F : \equiv \hbar \omega \left(a a^\dagger + a^\dagger a\right) : \equiv \hbar \omega a^\dagger a</math>

In other words, within the normal ordering symbol we can commute $a$ and $a †$ . Since zero-point energy is intimately connected to the non-commutativity of $a$ and $a †$ , the normal ordering procedure eliminates any contribution from the zero-point field. This is especially reasonable in the case of the field Hamiltonian, since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition for the zero of energy. Moreover, this constant energy in the Hamiltonian obviously commutes with $a$ and $a †$ and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion.

However, things are not quite that simple. The zero-point energy cannot be eliminated by dropping its energy from the Hamiltonian: When we do this and solve the Heisenberg equation for a field operator, we must include the vacuum field, which is the homogeneous part of the solution for the field operator. In fact we can show that the vacuum field is essential for the preservation of the commutators and the formal consistency of QED. When we calculate the field energy we obtain not only a contribution from particles and forces that may be present but also a contribution from the vacuum field itself i.e. the zero-point field energy. In other words, the zero-point energy reappears even though we may have deleted it from the Hamiltonian.<ref name="FOOTNOTEMilonni199473–74">Milonni (1994), pp. 73–74.</ref>

The electromagnetic field in free space

From Maxwell's equations, the electromagnetic energy of a "free" field i.e. one with no sources, is described by:

<math display="block">\begin{align} H_F &= \frac{1}{8\pi}\int d^3r \left(\mathbf{E}^2 +\mathbf{B}^2\right) \\ &=\frac{k^2}{2\pi}|\alpha (t)|^2 \end{align}</math>

We introduce the "mode function" $A 0 (r)$ that satisfies the Helmholtz equation:

<math display="block"> \left( \nabla^2 + k^2 \right) \mathbf{A}_0(\mathbf{r}) = 0 </math>

where $k = ω / c$ and assume it is normalized such that:

<math display="block">\int d^3r \left|\mathbf{A}_0(\mathbf{r})\right|^2 = 1</math>

We wish to "quantize" the electromagnetic energy of free space for a multimode field. The field intensity of free space should be independent of position such that $| A 0 (r) | 2$ should be independent of $r$ for each mode of the field. The mode function satisfying these conditions is:

<math display="block"> \mathbf{A}_0(\mathbf{r}) = e_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}} </math>

where $k \cdot e k = 0$ in order to have the transversality condition $\nabla \cdot A (r, t)$ satisfied for the Coulomb gauge^{[dubious – discuss]} in which we are working.

To achieve the desired normalization we pretend space is divided into cubes of volume $V = L 3$ and impose on the field the periodic boundary condition:

<math display="block">\mathbf{A}(x+L,y+L,z+L,t)=\mathbf{A}(x,y,z,t)</math>

or equivalently

<math display="block"> \left(k_x,k_y,k_z\right)=\frac{2\pi}{L}\left(n_x,n_y,n_z\right)</math>

where $n$ can assume any integer value. This allows us to consider the field in any one of the imaginary cubes and to define the mode function:

<math display="block">\mathbf{A}_\mathbf{k}(\mathbf{r})= \frac{1}\sqrt{V} e_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}}</math>

which satisfies the Helmholtz equation, transversality, and the "box normalization":

<math display="block">\int_V d^3r \left|\mathbf{A}_\mathbf{k}(\mathbf{r})\right|^2 = 1</math>

where $e k$ is chosen to be a unit vector which specifies the polarization of the field mode. The condition $k \cdot e k = 0$ means that there are two independent choices of $e k$ , which we call $e k 1$ and $e k 2$ where $e k 1 \cdot e k 2 = 0$ and $e 2 k 1 = e 2 k 2 = 1$ . Thus we define the mode functions:

<math display="block">\mathbf{A}_{\mathbf{k}\lambda}(\mathbf{r})=\frac{1}\sqrt{V}e_{\mathbf{k}\lambda}e^{i\mathbf{k}\cdot\mathbf{r}} \, , \quad \lambda = \begin{cases} 1\\2 \end{cases}</math>

in terms of which the vector potential becomes^{[clarification needed]}:

<math display="block">\mathbf{A}_{\mathbf{k}\lambda}(\mathbf{r},t)=\sqrt{\frac{2\pi\hbar c^2}{\omega_k V}}\left[a_{\mathbf{k}\lambda}(0)e^{i\mathbf{k}\cdot\mathbf{r}}+a_{\mathbf{k}\lambda}^\dagger(0)e^{-i\mathbf{k}\cdot\mathbf{r}}\right]e_{\mathbf{k}\lambda}</math>

or:

<math display="block">\mathbf{A}_{\mathbf{k}\lambda}(\mathbf{r},t)=\sqrt{\frac{2\pi\hbar c^2}{\omega_k V}}\left[a_{\mathbf{k}\lambda}(0)e^{-i(\omega_k t-\mathbf{k}\cdot\mathbf{r})}+a_{\mathbf{k}\lambda}^\dagger(0)e^{i(\omega_k t-\mathbf{k}\cdot\mathbf{r})}\right] </math>

where $ω k = kc$ and $a k λ$ , $a † k λ$ are photon annihilation and creation operators for the mode with wave vector $k$ and polarization $λ$ . This gives the vector potential for a plane wave mode of the field. The condition for $(k x, k y, k z)$ shows that there are infinitely many such modes. The linearity of Maxwell's equations allows us to write:

<math display="block">\mathbf{A}(\mathbf{r}t)=\sum_{\mathbf{k}\lambda}\sqrt{\frac{2\pi\hbar c^2}{\omega_k V}}\left[a_{\mathbf{k}\lambda}(0)e^{i\mathbf{k}\cdot\mathbf{r}}+a_{\mathbf{k}\lambda}^\dagger(0)e^{-i\mathbf{k}\cdot\mathbf{r}}\right]e_{\mathbf{k}\lambda}</math>

for the total vector potential in free space. Using the fact that:

<math display="block">\int_V d^3r \mathbf{A}_{\mathbf{k}\lambda}(\mathbf{r})\cdot \mathbf{A}_{\mathbf{k}'\lambda'}^\ast(\mathbf{r})=\delta_{\mathbf{k},\mathbf{k}'}^3\delta_{\lambda,\lambda'}</math>

we find the field Hamiltonian is:

<math display="block">H_F=\sum_{\mathbf{k}\lambda}\hbar\omega_k\left(a_{\mathbf{k}\lambda}^\dagger a_{\mathbf{k}\lambda} + \tfrac{1}{2} \right) </math>

This is the Hamiltonian for an infinite number of uncoupled harmonic oscillators. Thus different modes of the field are independent and satisfy the commutation relations:

<math display="block">\begin{align} \left[a_{\mathbf{k}\lambda}(t),a_{\mathbf{k}'\lambda'}^\dagger(t)\right]&=\delta_{\mathbf{k},\mathbf{k}'}^3\delta_{\lambda,\lambda'} \\[10px] \left[a_{\mathbf{k}\lambda}(t),a_{\mathbf{k}'\lambda'}(t)\right]&=\left[a_{\mathbf{k}\lambda}^\dagger(t),a_{\mathbf{k}'\lambda'}^\dagger(t)\right]=0 \end{align}</math>

Clearly the least eigenvalue for $H F$ is:

<math display="block">\sum_{\mathbf{k}\lambda}\tfrac{1}{2}\hbar\omega_k</math>

This state describes the zero-point energy of the vacuum. It appears that this sum is divergent – in fact highly divergent, as putting in the density factor

shows. The summation becomes approximately the integral:

for high values of $v$ . It diverges proportional to $v 4$ for large $v$ .

There are two separate questions to consider. First, is the divergence a real one such that the zero-point energy really is infinite? If we consider the volume $V$ is contained by perfectly conducting walls, very high frequencies can only be contained by taking more and more perfect conduction. No actual method of containing the high frequencies is possible. Such modes will not be stationary in our box and thus not countable in the stationary energy content. So from this physical point of view the above sum should only extend to those frequencies which are countable; a cut-off energy is thus eminently reasonable. However, on the scale of a "universe" questions of general relativity must be included. Suppose even the boxes could be reproduced, fit together and closed nicely by curving spacetime. Then exact conditions for running waves may be possible. However the very high frequency quanta will still not be contained. As per John Wheeler's "geons"<ref>Wheeler, John Archibald (1955). "Geons". Physical Review. 97 (2): 511. Bibcode:1955PhRv...97..511W. doi:10.1103/PhysRev.97.511.</ref> these will leak out of the system. So again a cut-off is permissible, almost necessary. The question here becomes one of consistency since the very high energy quanta will act as a mass source and start curving the geometry.

This leads to the second question. Divergent or not, finite or infinite, is the zero-point energy of any physical significance? The ignoring of the whole zero-point energy is often encouraged for all practical calculations. The reason for this is that energies are not typically defined by an arbitrary data point, but rather changes in data points, so adding or subtracting a constant (even if infinite) should be allowed. However this is not the whole story, in reality energy is not so arbitrarily defined: in general relativity the seat of the curvature of spacetime is the energy content and there the absolute amount of energy has real physical meaning. There is no such thing as an arbitrary additive constant with density of field energy. Energy density curves space, and an increase in energy density produces an increase of curvature. Furthermore, the zero-point energy density has other physical consequences e.g. the Casimir effect, contribution to the Lamb shift, or anomalous magnetic moment of the electron, it is clear it is not just a mathematical constant or artifact that can be cancelled out.<ref name="FOOTNOTEPower196431–33">Power (1964), pp. 31–33.</ref>

Necessity of the vacuum field in QED

The vacuum state of the "free" electromagnetic field (that with no sources) is defined as the ground state in which $n k λ = 0$ for all modes $(k, λ)$ . The vacuum state, like all stationary states of the field, is an eigenstate of the Hamiltonian but not the electric and magnetic field operators. In the vacuum state, therefore, the electric and magnetic fields do not have definite values. We can imagine them to be fluctuating about their mean value of zero.

In a process in which a photon is annihilated (absorbed), we can think of the photon 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.<ref name="FOOTNOTEDirac1927">Dirac (1927).</ref> An atom, for instance, can be considered to be "dressed" by emission and reabsorption of "virtual photons" from the vacuum. The vacuum state energy described by $Σ k λ ħω k / 2$ is infinite. We can make the replacement:

<math display="block">\sum_{\mathbf{k}\lambda}\longrightarrow\sum_{\lambda}\left (\frac{1}{2\pi} \right )^3 \int d^3 k = \frac{V}{8\pi^3} \sum_\lambda \int d^3 k</math>

the zero-point energy density is:

<math display="block">\begin{align} \frac{1}{V}\sum_{\mathbf{k}\lambda}\tfrac{1}{2}\hbar\omega_k &=\frac{2}{8\pi^3}\int d^3 k \tfrac{1}{2}\hbar\omega_k \\ &= \frac{4\pi}{4\pi^3} \int dk\,k^2 \left(\tfrac{1}{2}\hbar\omega_k\right) \\ &=\frac{\hbar}{2\pi^2 c^3} \int d\omega\,\omega^3 \end{align}</math>

or in other words the spectral energy density of the vacuum field:

<math display="block">\rho_0(\omega)=\frac{\hbar\omega^3}{2\pi^2c^3}</math>

The zero-point energy density in the frequency range from $ω 1$ to $ω 2$ is therefore:

<math display="block">\int_{\omega_1}^{\omega_2} d\omega\rho_0(\omega) = \frac{\hbar}{8\pi^2c^3}\left(\omega_2^4-\omega_1^4\right)</math>

This can be large even in relatively narrow "low frequency" regions of the spectrum. In the optical region from 400 to 700 nm, for instance, the above equation yields around 220 erg/cm³.

We showed in the above section that the zero-point energy can be eliminated from the Hamiltonian by the normal ordering prescription. However, this elimination does not mean that the vacuum field has been rendered unimportant or without physical consequences. To illustrate this point we consider a linear dipole oscillator in the vacuum. The Hamiltonian for the oscillator plus the field with which it interacts is:

<math display="block">H=\frac{1}{2m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)^2 + \tfrac{1}{2}m\omega_0^2\mathbf{x}^2 + H_F</math>

This has the same form as the corresponding classical Hamiltonian and the Heisenberg equations of motion for the oscillator and the field are formally the same as their classical counterparts. For instance the Heisenberg equations for the coordinate $x$ and the canonical momentum $p = m ẋ + e A / c$ of the oscillator are:

<math display="block">\begin{align} \mathbf{\dot{x}}&=(i\hbar)^{-1}[\mathbf{x}.H] = \frac{1}{m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right) \\ \mathbf{\dot{p}}&=(i\hbar)^{-1}[\mathbf{p}.H] \begin{align}&=\tfrac{1}{2}\nabla\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)^2-m\omega_0^2\mathbf{\dot{x}} \\ &=-\frac{1}{m} \left[\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right) \cdot \nabla\right] \left[-\frac{e}{c}\mathbf{A}\right] - \frac{1}{m} \left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right) \times \nabla \times \left[-\frac{e}{c}\mathbf{A}\right] -m\omega_0^2 \mathbf{\dot{x}} \\ &= \frac{e}{c}(\mathbf{\dot{x}}\cdot\nabla)\mathbf{A} + \frac{e}{c}\mathbf{\dot{x}} \times \mathbf{B} -m\omega_0^2 \mathbf{\dot{x}} \end{align}\end{align}</math>

or:

<math display="block">\begin{align} m \mathbf{\ddot{x}} &= \mathbf{\dot{p}} - \frac{e}{c} \mathbf{\dot{A}} \\ &= -\frac{e}{c} \left[\mathbf{\dot{A}} - \left(\mathbf{\dot{x}} \cdot \nabla\right) \mathbf{A}\right] + \frac{e}{c} \mathbf{\dot{x}} \times \mathbf{B} - m\omega_0^2\mathbf{x} \\ &= e\mathbf{E} + \frac{e}{c} \mathbf{\dot{x}} \times \mathbf{B} - m\omega_0^2\mathbf{x} \end{align}</math>

since the rate of change of the vector potential in the frame of the moving charge is given by the convective derivative

<math display="block">\mathbf{\dot{A}}=\frac{\partial\mathbf{A}}{\partial t} + (\mathbf{\dot{x}} \cdot \nabla) \mathbf{A}^3 \,.</math>

For nonrelativistic motion we may neglect the magnetic force and replace the expression for $m ẍ$ by:

<math display="block">\begin{align} \mathbf{\ddot{x}}+\omega_0^2\mathbf{x} &\approx \frac{e}{m}\mathbf{E} \\ &\approx \sum_{\mathbf{k}\lambda} \sqrt{\frac{2\pi\hbar\omega_k}{V}} \left[a_{\mathbf{k}\lambda}(t) + a_{\mathbf{k}\lambda}^\dagger(t)\right] e_{\mathbf{k}\lambda} \end{align}</math>

Above we have made the electric dipole approximation in which the spatial dependence of the field is neglected. The Heisenberg equation for $a k λ$ is found similarly from the Hamiltonian to be:

<math display="block">\dot{a}_{\mathbf{k}\lambda} = i \omega_k a_{\mathbf{k}\lambda} + ie \sqrt\frac{2\pi}{\hbar \omega_k V} \mathbf{\dot{x}} \cdot e_{\mathbf{k}\lambda}</math>

In the electric dipole approximation.

In deriving these equations for $x$ , $p$ , and $a k λ$ we have used the fact that equal-time particle and field operators commute. This follows from the assumption that particle and field operators commute at some time (say, $t = 0$ ) when the matter-field interpretation is presumed to begin, together with the fact that a Heisenberg-picture operator $A (t)$ evolves in time as $A (t) = U † (t) A (0) U (t)$ , where $U (t)$ is the time evolution operator satisfying

<math display="block">i\hbar\dot{U} = HU \,,\quad U^\dagger(t) = U^{-1}(t) \,,\quad U(0) = 1 \,.</math>

Alternatively, we can argue that these operators must commute if we are to obtain the correct equations of motion from the Hamiltonian, just as the corresponding Poisson brackets in classical theory must vanish in order to generate the correct Hamilton equations. The formal solution of the field equation is:

<math display="block">a_{\mathbf{k}\lambda}(t)=a_{\mathbf{k}\lambda}(0)e^{-i\omega_{k}t}+ie \sqrt{\frac{2\pi}{\hbar \omega_k V}} \int^t_0dt'\,e_{\mathbf{k}\lambda}\cdot\mathbf{\dot{x}}(t')e^{i\omega_k\left(t'-t\right)}</math>

and therefore the equation for $ȧ k λ$ may be written:

<math display="block">\mathbf{\ddot{x}}+\omega^2_0\mathbf{x}=\frac{e}{m}\mathbf{E}_0(t)+\frac{e}{m}\mathbf{E}_{RR}(t)</math>

where:

<math display="block">\mathbf{E}_0(t)=i\sum_{\mathbf{k}\lambda} \sqrt{\frac{2\pi\hbar \omega_k}{V}}\left[a_{\mathbf{k}\lambda}(0)e^{-i\omega_kt}-a^\dagger_{\mathbf{k}\lambda}(0)e^{i\omega_kt}\right]e_{\mathbf{k}\lambda}</math>

and:

<math display="block">\mathbf{E}_{RR}(t)=-\frac{4\pi e}{V} \sum_{\mathbf{k}\lambda} \int^t_0dt'\left[e_{\mathbf{k}\lambda}\cdot\mathbf{\dot{x}}\left(t'\right)\right]\cos\omega_k\left(t'-t\right)</math>

It can be shown that in the radiation reaction field, if the mass $m$ is regarded as the "observed" mass then we can take:

<math display="block">\mathbf{E}_{RR}(t)=\frac{2e}{3c^3}\mathbf{\ddot{x}}</math>

The total field acting on the dipole has two parts, $E 0 (t)$ and $E RR (t)$ . $E 0 (t)$ is the free or zero-point field acting on the dipole. It is the homogeneous solution of the Maxwell equation for the field acting on the dipole, i.e., the solution, at the position of the dipole, of the wave equation

<math display="block">\left[\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right]\mathbf{E}=0</math>

satisfied by the field in the (source free) vacuum. For this reason $E 0 (t)$ is often referred to as the "vacuum field", although it is of course a Heisenberg-picture operator acting on whatever state of the field happens to be appropriate at $t = 0$ . $E RR (t)$ is the source field, the field generated by the dipole and acting on the dipole.

Using the above equation for $E RR (t)$ we obtain an equation for the Heisenberg-picture operator <math>\mathbf{x}(t)</math> that is formally the same as the classical equation for a linear dipole oscillator:

<math display="block"> \mathbf{\ddot{x}} + \omega^2_0\mathbf{x}-\tau \mathbf{\overset{...}{x}}=\frac{e}{m}\mathbf{E}_0(t) </math>

where $τ = 2 e 2 / 3 mc 3$ . in this instance we have considered a dipole in the vacuum, without any "external" field acting on it. the role of the external field in the above equation is played by the vacuum electric field acting on the dipole.

Classically, a dipole in the vacuum is not acted upon by any "external" field: if there are no sources other than the dipole itself, then the only field acting on the dipole is its own radiation reaction field. In quantum theory however there is always an "external" field, namely the source-free or vacuum field $E 0 (t)$ .

According to our earlier equation for $a k λ (t)$ the free field is the only field in existence at $t = 0$ as the time at which the interaction between the dipole and the field is "switched on". The state vector of the dipole-field system at $t = 0$ is therefore of the form

<math display="block">|\Psi\rangle=|\text{vac}\rangle|\psi_D\rangle \,,</math>

where $|vac⟩$ is the vacuum state of the field and $| ψ D ⟩$ is the initial state of the dipole oscillator. The expectation value of the free field is therefore at all times equal to zero:

<math display="block">\langle\mathbf{E}_0(t)\rangle=\langle\Psi|\mathbf{E}_0(t)|\Psi\rangle=0</math>

since $a k λ (0)|vac⟩ = 0$ . however, the energy density associated with the free field is infinite:

<math display="block">\begin{align} \frac{1}{4\pi} \left\langle \mathbf{E}^2_0(t) \right\rangle &= \frac{1}{4\pi} \sum_{\mathbf{k}\lambda} \sum_{\mathbf{k'}\lambda'} \sqrt{\frac{2\pi\hbar \omega_k}{V}} \sqrt{\frac{2\pi\hbar \omega_{k'}}{V}} \times \left\langle a_{\mathbf{k}\lambda}(0)a^\dagger_{\mathbf{k'}\lambda'}(0)\right\rangle \\ &= \frac{1}{4\pi}\sum_{\mathbf{k}\lambda}\left (\frac{2\pi\hbar \omega_k}{V} \right )\\ &= \int^\infin_0dw\,\rho_0(\omega) \end{align}</math>

The important point of this is that the zero-point field energy $H F$ does not affect the Heisenberg equation for $a k λ$ since it is a c-number or constant (i.e. an ordinary number rather than an operator) and commutes with $a k λ$ . We can therefore drop the zero-point field energy from the Hamiltonian, as is usually done. But the zero-point field re-emerges as the homogeneous solution for the field equation. A charged particle in the vacuum will therefore always see a zero-point field of infinite density. This is the origin of one of the infinities of quantum electrodynamics, and it cannot be eliminated by the trivial expedient dropping of the term $Σ k λ ħω k / 2$ in the field Hamiltonian.

The free field is in fact necessary for the formal consistency of the theory. In particular, it is necessary for the preservation of the commutation relations, which is required by the unitary of time evolution in quantum theory:

<math display="block">\begin{align} \left[z(t),p_z(t)\right]&=\left[U^\dagger(t)z(0)U(t),U^\dagger(t)p_z(0)U(t)\right]\\ &=U^\dagger(t)\left[z(0),p_z(0)\right]U(t)\\ &=i\hbar U^\dagger(t)U(t)\\ &=i\hbar \end{align}</math>

We can calculate $[z (t), p z (t)]$ from the formal solution of the operator equation of motion

<math display="block">\mathbf{\ddot{x}} + \omega^2_0\mathbf{x}-\tau \mathbf{\overset{...}{x}}=\frac{e}{m}\mathbf{E}_0(t)</math>

Using the fact that

<math display="block">\left[a_{\mathbf{k}\lambda}(0),a^\dagger_{\mathbf{k'}\lambda'}(0)\right]=\delta^3_\mathbf{kk'},\delta_{\lambda\lambda'}</math>

and that equal-time particle and field operators commute, we obtain:

<math display="block">\begin{align} [z(t),p_z(t)]&=\left[z(t),m\dot{z}(t)\right]+\left[z(t),\frac{e}{c}A_z(t)\right] \\ &=\left[z(t),m\dot{z}(t)\right] \\ &= \left (\frac{i\hbar e^2}{2\pi^2mc^3} \right ) \left (\frac{8\pi}{3} \right ) \int^\infin_0\frac{d\omega\,\omega^4}{\left(\omega^2-\omega^2_0\right)^2+\tau^2\omega^6} \end{align}</math>

For the dipole oscillator under consideration it can be assumed that the radiative damping rate is small compared with the natural oscillation frequency, i.e., $τω 0 ≪ 1$ . Then the integrand above is sharply peaked at $ω = ω 0$ and:

<math display="block">\begin{align} \left[z(t),p_z(t)\right]&\approx \frac{2i\hbar e^2}{3\pi mc^3}\omega^3_0 \int^\infin_{-\infin} \frac{dx}{x^2 + \tau^2\omega^6_0} \\ &= \left (\frac{2i\hbar e^2 \omega^3_0}{3\pi mc^3} \right )\left (\frac{\pi}{\tau\omega^3_0} \right ) \\ &=i\hbar \end{align}</math>

the necessity of the vacuum field can also be appreciated by making the small damping approximation in

<math display="block">\begin{align} &\mathbf{\ddot{x}} + \omega^2_0\mathbf{x}-\tau \mathbf{\overset{...}{x}}=\frac{e}{m}\mathbf{E}_0(t) \\ &\mathbf{\ddot{x}}\approx-\omega^2_0\mathbf{x}(t) && \mathbf{\overset{...}{x}}\approx-\omega^2_0\mathbf{\dot{x}} \end{align}</math>

and

<math display="block">\mathbf{\ddot{x}}+\tau\omega^2_0\mathbf{\dot{x}}+\omega^2_0\mathbf{x}\approx\frac{e}{m}\mathbf{E}_0(t)</math>

Without the free field $E 0 (t)$ in this equation the operator $x (t)$ would be exponentially dampened, and commutators like $[z (t), p z (t)]$ would approach zero for $t ≫ 1 / τω 20$ . With the vacuum field included, however, the commutator is $iħ$ at all times, as required by unitarity, and as we have just shown. A similar result is easily worked out for the case of a free particle instead of a dipole oscillator.<ref name="FOOTNOTEMilonni1981">Milonni (1981).</ref>

What we have here is an example of a "fluctuation-dissipation elation". Generally speaking if a system is coupled to a bath that can take energy from the system in an effectively irreversible way, then the bath must also cause fluctuations. The fluctuations and the dissipation go hand in hand we cannot have one without the other. In the current example the coupling of a dipole oscillator to the electromagnetic field has a dissipative component, in the form of the zero-point (vacuum) field; given the existence of radiation reaction, the vacuum field must also exist in order to preserve the canonical commutation rule and all it entails.

The spectral density of the vacuum field is fixed by the form of the radiation reaction field, or vice versa: because the radiation reaction field varies with the third derivative of $x$ , the spectral energy density of the vacuum field must be proportional to the third power of $ω$ in order for $[z (t), p z (t)]$ to hold. In the case of a dissipative force proportional to $ẋ$ , by contrast, the fluctuation force must be proportional to <math>\omega</math> in order to maintain the canonical commutation relation.<ref name="FOOTNOTEMilonni1981">Milonni (1981).</ref> This relation between the form of the dissipation and the spectral density of the fluctuation is the essence of the fluctuation-dissipation theorem.<ref name="ReferenceB"/>

The fact that the canonical commutation relation for a harmonic oscillator coupled to the vacuum field is preserved implies that the zero-point energy of the oscillator is preserved. it is easy to show that after a few damping times the zero-point motion of the oscillator is in fact sustained by the driving zero-point field.<ref name="ReferenceG">Senitzky, I. R. (1960). "Dissipation in Quantum Mechanics. The Harmonic Oscillator". Physical Review. 119 (2): 670. Bibcode:1960PhRv..119..670S. doi:10.1103/PhysRev.119.670.</ref>

The quantum chromodynamic vacuum

The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a non-perturbative vacuum state, characterized by a non-vanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the confined phase of quark matter. In technical terms, gluons are vector gauge bosons that mediate strong interactions of quarks in quantum chromodynamics (QCD). Gluons themselves carry the color charge of the strong interaction. This is unlike the photon, which mediates the electromagnetic interaction but lacks an electric charge. Gluons therefore participate in the strong interaction in addition to mediating it, making QCD significantly harder to analyze than QED (quantum electrodynamics) as it deals with nonlinear equations to characterize such interactions.

The Higgs field

The Standard Model hypothesises a field called the Higgs field (symbol: $ϕ$ ), which has the unusual property of a non-zero amplitude in its ground state (zero-point) energy after renormalization; i.e., a non-zero vacuum expectation value. It can have this effect because of its unusual "Mexican hat" shaped potential whose lowest "point" is not at its "centre". Below a certain extremely high energy level the existence of this non-zero vacuum expectation spontaneously breaks electroweak gauge symmetry which in turn gives rise to the Higgs mechanism and triggers the acquisition of mass by those particles interacting with the field. The Higgs mechanism occurs whenever a charged field has a vacuum expectation value. This effect occurs because scalar field components of the Higgs field are "absorbed" by the massive bosons as degrees of freedom, and couple to the fermions via Yukawa coupling, thereby producing the expected mass terms. The expectation value of $ϕ 0$ in the ground state (the vacuum expectation value or VEV) is then $⟨ ϕ 0 ⟩ = v / \sqrt 2$ , where $v = | μ | / \sqrt λ$ . The measured value of this parameter is approximately 246 GeV/c².<ref name="PDGreview2012">"Higgs bosons: theory and searches" (PDF). PDGLive. Particle Data Group. 12 July 2012. Retrieved 15 August 2012.</ref> It has units of mass, and is the only free parameter of the Standard Model that is not a dimensionless number.

The Higgs mechanism is a type of superconductivity which occurs in the vacuum. It occurs when all of space is filled with a sea of particles which are charged and thus the field has a nonzero vacuum expectation value. Interaction with the vacuum energy filling the space prevents certain forces from propagating over long distances (as it does in a superconducting medium; e.g., in the Ginzburg–Landau theory).

Experimental observations

Zero-point energy has many observed physical consequences.<ref name="FOOTNOTEMilonni1994111">Milonni (1994), p. 111.</ref> It is important to note that zero-point energy is not merely an artifact of mathematical formalism that can, for instance, be dropped from a Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion without latter consequence.<ref name="FOOTNOTEMilonni199442–43">Milonni (1994), pp. 42–43.</ref> Indeed, such treatment could create a problem at a deeper, as of yet undiscovered, theory.<ref name="FOOTNOTEPeskinSchroeder199522">Peskin & Schroeder (1995), p. 22.</ref> For instance, in general relativity the zero of energy (i.e. the energy density of the vacuum) contributes to a cosmological constant of the type introduced by Einstein in order to obtain static solutions to his field equations.<ref name="FOOTNOTEMilonni2009865">Milonni (2009), p. 865.</ref> The zero-point energy density of the vacuum, due to all quantum fields, is extremely large, even when we cut off the largest allowable frequencies based on plausible physical arguments. It implies a cosmological constant larger than the limits imposed by observation by about 120 orders of magnitude. This "cosmological constant problem" remains one of the greatest unsolved mysteries of physics.<ref name=scientificamerican0588-106>Abbott, Larry (1988). "The Mystery of the Cosmological Constant" (PDF). Scientific American. 258 (5): 106–113. Bibcode:1988SciAm.258e.106A. doi:10.1038/scientificamerican0588-106.</ref>

Casimir effect

A phenomenon that is commonly presented as evidence for the existence of zero-point energy in vacuum is the Casimir effect, proposed in 1948 by Dutch physicist Hendrik Casimir, who considered the quantized electromagnetic field between a pair of grounded, neutral metal plates. The vacuum energy contains contributions from all wavelengths, except those excluded by the spacing between plates. As the plates draw together, more wavelengths are excluded and the vacuum energy decreases. The decrease in energy means there must be a force doing work on the plates as they move.

Early experimental tests from the 1950s onwards gave positive results showing the force was real, but other external factors could not be ruled out as the primary cause, with the range of experimental error sometimes being nearly 100%.<ref>Derjaguin, B. V.; Abrikosova, I. I.; Lifshitz, E. M. (1956). "Direct measurement of molecular attraction between solids separated by a narrow gap". Quarterly Reviews, Chemical Society. 10 (3): 295–329. doi:10.1039/QR9561000295.</ref><ref>Sparnaay, M. J. (1958). "Measurements of attractive forces between flat plates". Physica. 24 (6–10): 751–764. Bibcode:1958Phy....24..751S. doi:10.1016/S0031-8914(58)80090-7.</ref><ref>Tabor, D.; Winterton, R. H. S. (1968). "Surface Forces: Direct Measurement of Normal and Retarded Van der Waals Forces". Nature. 219 (5159): 1120–1121. Bibcode:1968Natur.219.1120T. doi:10.1038/2191120a0. PMID 5675624. S2CID 4258508.</ref><ref>Hunklinger, S.; Geisselmann, H.; Arnold, W. (1972). "A Dynamic Method for Measuring the Van der Waals Forces between Macroscopic Bodies". Rev. Sci. Instrum. 43 (4): 584–587. Bibcode:1972RScI...43..584H. doi:10.1063/1.1685696.</ref><ref>Van Blokland, Peter H. G. M.; Overbeek, J. Theodoor G. (1978). "Van der Waals forces between objects covered with a chromium layer". J. Chem. Soc., Faraday Trans. 1. 74: 2637–2651. doi:10.1039/F19787402637.</ref> That changed in 1997 with Lamoreaux<ref>Lamoreaux, S. K. (1997). "Demonstration of the Casimir Force in the 0.6 to 6 μm Range" (PDF). Physical Review Letters. 78 (1): 5–8. Bibcode:1997PhRvL..78....5L. doi:10.1103/PhysRevLett.78.5.</ref> conclusively showing that the Casimir force was real. Results have been repeatedly replicated since then.<ref>Mohideen, Umar; Roy, Anushree (1998). "Precision Measurement of the Casimir Force from 0.1 to 0.9 μm". Physical Review Letters. 81 (21): 4549–4552. arXiv:physics/9805038. Bibcode:1998PhRvL..81.4549M. doi:10.1103/PhysRevLett.81.4549. S2CID 56132451.</ref><ref name="FOOTNOTEChan et al.2001">Chan et al. (2001).</ref><ref name="FOOTNOTEBressi et al.2002">Bressi et al. (2002).</ref><ref name="FOOTNOTEDecca et al.2003">Decca et al. (2003).</ref>

In 2009, Munday et al.<ref>Munday, J. N.; Capasso, Federico; Parsegian, V. Adrian (2009). "Measured long-range repulsive Casimir–Lifshitz forces" (PDF). Nature. 457 (7226): 170–173. Bibcode:2009Natur.457..170M. doi:10.1038/nature07610. PMC 4169270. PMID 19129843.</ref> published experimental proof that (as predicted in 1961<ref>Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, Lev P. (1961). "General Theory of Van der Waals' Forces". Soviet Physics Uspekhi. 4 (2): 154. Bibcode:1961SvPhU...4..153D. doi:10.1070/PU1961v004n02ABEH003330.</ref>) the Casimir force could also be repulsive as well as being attractive. Repulsive Casimir forces could allow quantum levitation of objects in a fluid and lead to a new class of switchable nanoscale devices with ultra-low static friction.<ref name="FOOTNOTECapasso et al.2007">Capasso et al. (2007).</ref>

An interesting hypothetical side effect of the Casimir effect is the Scharnhorst effect, a hypothetical phenomenon in which light signals travel slightly faster than $c$ between two closely spaced conducting plates.<ref name="Scharnhorst 1993">See Barton & Scharnhorst (1993) and Chown (1990)</ref>

Lamb shift

The quantum fluctuations of the electromagnetic field have important physical consequences. In addition to the Casimir effect, they also lead to a splitting between the two energy levels $2 S 1 / 2$ and $2 P 1 / 2$ (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which these states should have the same energy. Charged particles can interact with the fluctuations of the quantized vacuum field, leading to slight shifts in energy;<ref name="FOOTNOTEItzyksonZuber198080">Itzykson & Zuber (1980), p. 80.</ref> this effect is called the Lamb shift.<ref> Hawton, M. (1993). "Self-consistent frequencies of the electron–photon system". Physical Review A. 48 (3): 1824–1831. Bibcode:1993PhRvA..48.1824H. doi:10.1103/PhysRevA.48.1824. PMID 9909797.</ref> The shift of about 4.38×10⁻⁶ eV is roughly 10⁻⁷ of the difference between the energies of the 1s and 2s levels, and amounts to 1,058 MHz in frequency units. A small part of this shift (27 MHz ≈ 3%) arises not from fluctuations of the electromagnetic field, but from fluctuations of the electron–positron field. The creation of (virtual) electron–positron pairs has the effect of screening the Coulomb field and acts as a vacuum dielectric constant. This effect is much more important in muonic atoms.<ref name="FOOTNOTELe Bellac2006381">Le Bellac (2006), p. 381.</ref>

Fine-structure constant

Taking $ħ$ (Planck's constant divided by $2π$ ), $c$ (the speed of light), and $e 2 = q 2 e / 4π ε 0$ (the electromagnetic coupling constant i.e. a measure of the strength of the electromagnetic force (where $q e$ is the absolute value of the electronic charge and <math>\varepsilon_0</math> is the vacuum permittivity)) we can form a dimensionless quantity called the fine-structure constant:

<math display="block">\alpha = \frac{e^2}{\hbar c} = \frac{q_e^2}{4\pi\varepsilon_0\hbar c} \approx \frac{1}{137}</math>

The fine-structure constant is the coupling constant of quantum electrodynamics (QED) determining the strength of the interaction between electrons and photons. It turns out that the fine-structure constant is not really a constant at all owing to the zero-point energy fluctuations of the electron-positron field.<ref name="FOOTNOTELe Bellac200633">Le Bellac (2006), p. 33.</ref> The quantum fluctuations caused by zero-point energy have the effect of screening electric charges: owing to (virtual) electron-positron pair production, the charge of the particle measured far from the particle is far smaller than the charge measured when close to it.

The Heisenberg inequality where $ħ = h / 2π$ , and $Δ x$ , $Δ p$ are the standard deviations of position and momentum states that:

<math display="block">\Delta_x\Delta_p\ge\frac{1}{2}\hbar</math>

It means that a short distance implies large momentum and therefore high energy i.e. particles of high energy must be used to explore short distances. QED concludes that the fine-structure constant is an increasing function of energy. It has been shown that at energies of the order of the Z⁰ boson rest energy, $m z c 2 \approx$ 90 GeV, that:

<math display="block">\alpha\approx\frac{1}{129}</math>

rather than the low-energy $α \approx 1 / 137$ .<ref>Aitchison, Ian; Hey, Anthony (2012). Gauge Theories in Particle Physics: A Practical Introduction: Volume 1: From Relativistic Quantum Mechanics to QED (4th ed.). CRC Press. p. 343. ISBN 9781466512993.</ref><ref>Quigg, C (1998). Espriu, D; Pich, A (eds.). Advanced School on Electroweak Theory: Hadron Colliders, the Top Quark, and the Higgs Sector. World Scientific. p. 143. ISBN 9789814545143.</ref> The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for defining $α$ . All in all, $α$ depends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The energy dependence of $α$ has been observed for several years now in precision experiment in high-energy physics.

Vacuum birefringence

File:Eso1641a.ogv In the presence of strong electrostatic fields it is predicted that virtual particles become separated from the vacuum state and form real matter.^{[citation needed]} The fact that electromagnetic radiation can be transformed into matter and vice versa leads to fundamentally new features in quantum electrodynamics. One of the most important consequences is that, even in the vacuum, the Maxwell equations have to be exchanged by more complicated formulas. In general, it will be not possible to separate processes in the vacuum from the processes involving matter since electromagnetic fields can create matter if the field fluctuations are strong enough. This leads to highly complex nonlinear interaction - gravity will have an effect on the light at the same time the light has an effect on gravity. These effects were first predicted by Werner Heisenberg and Hans Heinrich Euler in 1936<ref name="FOOTNOTEHeisenbergEuler1936">Heisenberg & Euler (1936).</ref> and independently the same year by Victor Weisskopf who stated: "The physical properties of the vacuum originate in the "zero-point energy" of matter, which also depends on absent particles through the external field strengths and therefore contributes an additional term to the purely Maxwellian field energy".<ref name="FOOTNOTEWeisskopf19363">Weisskopf (1936), p. 3.</ref><ref name="FOOTNOTEGreinerMüllerRafelski2012278">Greiner, Müller & Rafelski (2012), p. 278.</ref> Thus strong magnetic fields vary the energy contained in the vacuum. The scale above which the electromagnetic field is expected to become nonlinear is known as the Schwinger limit. At this point the vacuum has all the properties of a birefringent medium, thus in principle a rotation of the polarization frame (the Faraday effect) can be observed in empty space.<ref name="FOOTNOTEGreinerMüllerRafelski2012291">Greiner, Müller & Rafelski (2012), p. 291.</ref><ref>See Dunne (2012) for a historical review of the subject.</ref>

Both Einstein's theory of special and general relativity state that light should pass freely through a vacuum without being altered, a principle known as Lorentz invariance. Yet, in theory, large nonlinear self-interaction of light due to quantum fluctuations should lead to this principle being measurably violated if the interactions are strong enough. Nearly all theories of quantum gravity predict that Lorentz invariance is not an exact symmetry of nature. It is predicted the speed at which light travels through the vacuum depends on its direction, polarization and the local strength of the magnetic field.<ref name="FOOTNOTEHeylShaviv20001">Heyl & Shaviv (2000), p. 1.</ref> There have been a number of inconclusive results which claim to show evidence of a Lorentz violation by finding a rotation of the polarization plane of light coming from distant galaxies.<ref>See Carroll & Field (1997) and Kostelecký and Mewes (2009, 2013) for an overview of this area.</ref> The first concrete evidence for vacuum birefringence was published in 2017 when a team of astronomers looked at the light coming from the star RX J1856.5-3754,<ref>See Mignani et al. (2017) for experiment and Cho (2016), Crane (2016) and Bennett (2016) for comment.</ref> the closest discovered neutron star to Earth.<ref name="FOOTNOTERees2012528">Rees (2012), p. 528.</ref>

Roberto Mignani at the National Institute for Astrophysics in Milan who led the team of astronomers has commented that "When Einstein came up with the theory of general relativity 100 years ago, he had no idea that it would be used for navigational systems. The consequences of this discovery probably will also have to be realised on a longer timescale."<ref name="FOOTNOTECrane2016">Crane (2016).</ref> The team found that visible light from the star had undergone linear polarisation^{[clarification needed]} of around 16%. If the birefringence had been caused by light passing through interstellar gas or plasma, the effect should have been no more than 1%. Definitive proof would require repeating the observation at other wavelengths and on other neutron stars. At X-ray wavelengths the polarization from the quantum fluctuations should be near 100%.<ref name="FOOTNOTECho2016">Cho (2016).</ref> Although no telescope currently exists that can make such measurements, there are several proposed X-ray telescopes that may soon be able to verify the result conclusively such as China's Hard X-ray Modulation Telescope (HXMT) and NASA's Imaging X-ray Polarimetry Explorer (IXPE).

Speculated involvement in other phenomena

Dark energy

Unsolved problem in physics:

{{{2}}}

(more unsolved problems in physics)

In the late 1990s it was discovered that very distant supernovae were dimmer than expected suggesting that the universe's expansion was accelerating rather than slowing down.<ref name="FOOTNOTERiess et al.1998">Riess et al. (1998).</ref><ref name="FOOTNOTEPerlmutter et al.1998">Perlmutter et al. (1998).</ref> This revived discussion that Einstein's cosmological constant, long disregarded by physicists as being equal to zero, was in fact some small positive value. This would indicate empty space exerted some form of negative pressure or energy.

There is no natural candidate for what might cause what has been called dark energy but the current best guess is that it is the zero-point energy of the vacuum.<ref>Clark, Stuart (2016). "The Universe is Flat as a Pancake". New Scientist. Vol. 232, no. 3097. p. 35.</ref> One difficulty with this assumption is that the zero-point energy of the vacuum is absurdly large compared to the observed cosmological constant. This issue, called the cosmological constant problem, is one of the greatest unsolved mysteries in physics.

The European Space Agency's Euclid telescope, launched on 1 July 2023, will map galaxies up to 10 billion light years away.<ref>Miller, Katrina (1 July 2023). "The Dark Universe Is Waiting. What Will the Euclid Telescope Reveal?". The New York Times. Retrieved 23 August 2023.</ref> By seeing how dark energy influences their arrangement and shape, the mission will allow scientists to see if the strength of dark energy has changed. If dark energy is found to vary throughout time it would indicate it is due to quintessence, where observed acceleration is due to the energy of a scalar field, rather than the cosmological constant. No evidence of quintessence is yet available, but it has not been ruled out either. It generally predicts a slightly slower acceleration of the expansion of the universe than the cosmological constant. Some scientists think that the best evidence for quintessence would come from violations of Einstein's equivalence principle and variation of the fundamental constants in space or time.<ref name="Carroll1998">Carroll, Sean M. (1998). "Quintessence and the Rest of the World: Suppressing Long-Range Interactions" (PDF). Physical Review Letters. 81 (15): 3067–3070. arXiv:astro-ph/9806099. Bibcode:1998PhRvL..81.3067C. doi:10.1103/PhysRevLett.81.3067. ISSN 0031-9007. S2CID 14539052.</ref> Scalar fields are predicted by the Standard Model of particle physics and string theory, but an analogous problem to the cosmological constant problem (or the problem of constructing models of cosmological inflation) occurs: renormalization theory predicts that scalar fields should acquire large masses again due to zero-point energy.

Cosmic inflation

Unsolved problem in physics:

Why does the observable universe have more matter than antimatter?

(more unsolved problems in physics)

Cosmic inflation is phase of accelerated cosmic expansion just after the Big Bang. It explains the origin of the large-scale structure of the cosmos. It is believed quantum vacuum fluctuations caused by zero-point energy arising in the microscopic inflationary period, later became magnified to a cosmic size, becoming the gravitational seeds for galaxies and structure in the Universe (see galaxy formation and evolution and structure formation).<ref>Tyson, Neil deGrasse and Donald Goldsmith (2004), Origins: Fourteen Billion Years of Cosmic Evolution, W. W. Norton & Co., pp. 84–85.</ref> Many physicists also believe that inflation explains why the Universe appears to be the same in all directions (isotropic), why the cosmic microwave background radiation is distributed evenly, why the Universe is flat, and why no magnetic monopoles have been observed.

The mechanism for inflation is unclear, it is similar in effect to dark energy but is a far more energetic and short lived process. As with dark energy the best explanation is some form of vacuum energy arising from quantum fluctuations. It may be that inflation caused baryogenesis, the hypothetical physical processes that produced an asymmetry (imbalance) between baryons and antibaryons produced in the very early universe, but this is far from certain.

Cosmology

Paul S. Wesson examined the cosmological implications of assuming that zero-point energy is real.<ref>Wesson, Paul S. "Cosmological constraints on the zero-point electromagnetic field." Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 378, Sept. 10, 1991, p. 466-470. Research supported by NSERC. 378 (1991): 466-470.</ref> Among numerous difficulties, general relativity requires that such energy not gravitate, so it cannot be similar to electromagnetic radiation.

Alternative theories

There has been a long debate<ref>Enz, Charles P. (1974). Enz, C. P.; Mehra, J. (eds.). Physical Reality and Mathematical Description Is the Zero-Point Energy Real?. Dordrecht: D. Reidel Publishing Company. pp. 124–132. doi:10.1007/978-94-010-2274-3. ISBN 978-94-010-2274-3. S2CID 118779716.</ref> over the question of whether zero-point fluctuations of quantized vacuum fields are "real" i.e. do they have physical effects that cannot be interpreted by an equally valid alternative theory? Schwinger, in particular, attempted to formulate QED without reference to zero-point fluctuations via his "source theory".<ref>See Schwinger (1998a, 1998b, 1998c)</ref> From such an approach it is possible to derive the Casimir Effect without reference to a fluctuating field. Such a derivation was first given by Schwinger (1975)<ref>Schwinger, Julian (1975). "Casimir effect in source theory". Letters in Mathematical Physics. 1 (1): 43–47. Bibcode:1975LMaPh...1...43S. doi:10.1007/BF00405585. S2CID 126297065.</ref> for a scalar field, and then generalized to the electromagnetic case by Schwinger, DeRaad, and Milton (1978).<ref>Schwinger, Julian; DeRaad, Lester L.; Milton, Kimball 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.</ref> in which they state "the vacuum is regarded as truly a state with all physical properties equal to zero". More recently Jaffe (2005)<ref>Jaffe, R. L. (2005). "Casimir effect and the quantum vacuum". Physical Review D. 72 (2): 021301. arXiv:hep-th/0503158. Bibcode:2005PhRvD..72b1301J. doi:10.1103/PhysRevD.72.021301. S2CID 13171179.</ref> has highlighted a similar approach in deriving the Casimir effect stating "the concept of zero-point fluctuations is a heuristic and calculational aid in the description of the Casimir effect, but not a necessity in QED."

Nevertheless, as Jaffe himself notes in his paper, "no one has shown that source theory or another S-matrix based approach can provide a complete description of QED to all orders." Furthermore, Milonni has shown the necessity of the vacuum field for the formal consistency of QED.<ref name="FOOTNOTEMilonni199448">Milonni (1994), p. 48.</ref> In QCD, color confinement has led physicists to abandon the source theory or S-matrix based approach for the strong interactions. The Higgs mechanism, Hawking Radiation and the Unruh effect are also theorized to be dependent on zero-point vacuum fluctuations, the field contribution being an inseparable parts of these theories. Jaffe continues "Even if one could argue away zero-point contributions to the quantum vacuum energy, the problem of spontaneous symmetry breaking remains: condensates [ground state vacua] that carry energy appear at many energy scales in the Standard Model. So there is good reason to be skeptical of attempts to avoid the standard formulation of quantum field theory and the zero-point energies it brings with it." It is difficult to judge the physical reality of infinite zero-point energies that are inherent in field theories, but modern physics does not know any better way to construct gauge-invariant, renormalizable theories than with zero-point energy and they would seem to be a necessity for any attempt at a unified theory.<ref name="FOOTNOTEGreinerMüllerRafelski201220">Greiner, Müller & Rafelski (2012), p. 20.</ref>

Chaotic and emergent phenomena

The mathematical models used in classical electromagnetism, quantum electrodynamics (QED) and the Standard Model all view the electromagnetic vacuum as a linear system with no overall observable consequence. For example, in the case of the Casimir effect, Lamb shift, and so on these phenomena can be explained by alternative mechanisms other than action of the vacuum by arbitrary changes to the normal ordering of field operators. See the alternative theories section. This is a consequence of viewing electromagnetism as a U(1) gauge theory, which topologically does not allow the complex interaction of a field with and on itself.<ref>Barrett, Terence W. (2008). Topological Foundations of Electromagnetism. Singapore: World Scientific. p. 2. ISBN 9789812779977.</ref> In higher symmetry groups and in reality, the vacuum is not a calm, randomly fluctuating, largely immaterial and passive substance, but at times can be viewed as a turbulent virtual plasma that can have complex vortices (i.e. solitons vis-à-vis particles), entangled states and a rich nonlinear structure.<ref name="FOOTNOTEGreinerMüllerRafelski201223">Greiner, Müller & Rafelski (2012), p. 23.</ref> There are many observed nonlinear physical electromagnetic phenomena such as Aharonov–Bohm (AB)<ref> Ehrenberg, W; Siday, RE (1949). "The Refractive Index in Electron Optics and the Principles of Dynamics". Proceedings of the Physical Society. Series B. 62 (1): 8–21. Bibcode:1949PPSB...62....8E. CiteSeerX 10.1.1.205.6343. doi:10.1088/0370-1301/62/1/303.</ref><ref name="Significance of electromagnetic potentials in quantum theory"> Aharonov, Y; Bohm, D (1959). "Significance of electromagnetic potentials in quantum theory". Physical Review. 115 (3): 485–491. arXiv:1911.10555. Bibcode:1959PhRv..115..485A. doi:10.1103/PhysRev.115.485. S2CID 121421318.</ref> and Altshuler–Aronov–Spivak (AAS) effects,<ref>Altshuler, B. L.; Aronov, A. G.; Spivak, B. Z. (1981). "The Aaronov-Bohm effect in disordered conductors" (PDF). Pisma Zh. Eksp. Teor. Fiz. 33: 101. Bibcode:1981JETPL..33...94A. Archived from the original (PDF) on 4 November 2016. Retrieved 3 November 2016.</ref> Berry,<ref>Berry, M. V. (1984). "Quantal Phase Factors Accompanying Adiabatic Changes". Proc. R. Soc. A392 (1802): 45–57. Bibcode:1984RSPSA.392...45B. doi:10.1098/rspa.1984.0023. S2CID 46623507.</ref> Aharonov–Anandan,<ref>Aharonov, Y.; Anandan, J. (1987). "Phase change during a cyclic quantum evolution". Physical Review Letters. 58 (16): 1593–1596. Bibcode:1987PhRvL..58.1593A. doi:10.1103/PhysRevLett.58.1593. PMID 10034484.</ref> Pancharatnam<ref>Pancharatnam, S. (1956). "Generalized theory of interference, and its applications". Proceedings of the Indian Academy of Sciences. 44 (5): 247–262. doi:10.1007/BF03046050. S2CID 118184376.</ref> and Chiao–Wu<ref>Chiao, Raymond Y.; Wu, Yong-Shi (1986). "Manifestations of Berry's Topological Phase for the Photon". Physical Review Letters. 57 (8): 933–936. Bibcode:1986PhRvL..57..933C. doi:10.1103/PhysRevLett.57.933. PMID 10034203.</ref> phase rotation effects, Josephson effect,<ref>B. D. Josephson (1962). "Possible new effects in superconductive tunnelling". Phys. Lett. 1 (7): 251–253. Bibcode:1962PhL.....1..251J. doi:10.1016/0031-9163(62)91369-0. </ref><ref name=Joe>B. D. Josephson (1974). "The discovery of tunnelling supercurrents". Rev. Mod. Phys. 46 (2): 251–254. Bibcode:1974RvMP...46..251J. doi:10.1103/RevModPhys.46.251. </ref> Quantum Hall effect,<ref name=vonKlitzing:1980> K. v. Klitzing; G. Dorda; M. Pepper (1980). "New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance". Physical Review Letters. 45 (6): 494–497. Bibcode:1980PhRvL..45..494K. doi:10.1103/PhysRevLett.45.494.</ref> the De Haas–Van Alphen effect,<ref>De Haas, W. J.; Van Alphen, P. M. (1930). "The dependance of the susceptibility of diamagnetic metals upon the field". Proc. Netherlands R. Acad. Sci. 33: 1106.</ref> the Sagnac effect and many other physically observable phenomena which would indicate that the electromagnetic potential field has real physical meaning rather than being a mathematical artifact<ref name="FOOTNOTEPenrose2004453–454">Penrose (2004), pp. 453–454.</ref> and therefore an all encompassing theory would not confine electromagnetism as a local force as is currently done, but as a SU(2) gauge theory or higher geometry. Higher symmetries allow for nonlinear, aperiodic behaviour which manifest as a variety of complex non-equilibrium phenomena that do not arise in the linearised U(1) theory, such as multiple stable states, symmetry breaking, chaos and emergence.<ref>Feng, J. H.; Kneubühl, F. K. (1995). Barrett, Terence William; Grimes, Dale M. (eds.). Solitons and Chaos in Periodic Nonlinear Optical Media and Lasers: Advanced Electromagnetism: Foundations, Theory and Applications. Singapore: World Scientific. p. 438. ISBN 978-981-02-2095-2.</ref>

What are called Maxwell's equations today, are in fact a simplified version of the original equations reformulated by Heaviside, FitzGerald, Lodge and Hertz. The original equations used Hamilton's more expressive quaternion notation,<ref>Hunt, Bruce J. (2005). The Maxwellians. Cornell: Cornell University Press. p. 17. ISBN 9780801482342.</ref> a kind of Clifford algebra, which fully subsumes the standard Maxwell vectorial equations largely used today.<ref>Josephs, H.J. (1959). "The Heaviside papers found at Paignton in 1957". Proceedings of the IEE - Part C: Monographs. 106 (9): 70. doi:10.1049/pi-c.1959.0012.</ref> In the late 1880s there was a debate over the relative merits of vector analysis and quaternions. According to Heaviside the electromagnetic potential field was purely metaphysical, an arbitrary mathematical fiction, that needed to be "murdered".<ref>Hunt, Bruce J. (2005). The Maxwellians. Cornell: Cornell University Press. pp. 165–166. ISBN 9780801482342.</ref> It was concluded that there was no need for the greater physical insights provided by the quaternions if the theory was purely local in nature. Local vector analysis has become the dominant way of using Maxwell's equations ever since. However, this strictly vectorial approach has led to a restrictive topological understanding in some areas of electromagnetism, for example, a full understanding of the energy transfer dynamics in Tesla's oscillator-shuttle-circuit can only be achieved in quaternionic algebra or higher SU(2) symmetries.<ref>Barrett, T. W. (1991). "Tesla's Nonlinear Oscillator-Shuttle-Circuit (OSC) Theory" (PDF). Annales de la Fondation Louis de Broglie. 16 (1): 23–41. ISSN 0182-4295. Archived from the original (PDF) on 13 September 2016. Retrieved 3 November 2016.</ref> It has often been argued that quaternions are not compatible with special relativity,<ref name="FOOTNOTEPenrose2004201">Penrose (2004), p. 201.</ref> but multiple papers have shown ways of incorporating relativity.<ref>Rocher, E. Y. (1972). "Noumenon: Elementary entity of a new mechanics". J. Math. Phys. 13 (12): 1919. Bibcode:1972JMP....13.1919R. doi:10.1063/1.1665933.</ref><ref>Imaeda, K. (1976). "A new formulation of classical electrodynamics". Il Nuovo Cimento B. 32 (1): 138–162. Bibcode:1976NCimB..32..138I. doi:10.1007/BF02726749. S2CID 123315936.</ref><ref>Kauffmann, T.; Sun, Wen IyJ (1993). "Quaternion mechanics and electromagnetism". Annales de la Fondation Louis de Broglie. 18 (2): 213–219.</ref><ref>Lambek, Joachim. "QUATERNIONS AND THREE TEMPORAL DIMENSIONS" (PDF).</ref>

A good example of nonlinear electromagnetics is in high energy dense plasmas, where vortical phenomena occur which seemingly violate the second law of thermodynamics by increasing the energy gradient within the electromagnetic field and violate Maxwell's laws by creating ion currents which capture and concentrate their own and surrounding magnetic fields. In particular Lorentz force law, which elaborates Maxwell's equations is violated by these force free vortices.<ref name="FOOTNOTEBostick et al.1966">Bostick et al. (1966).</ref><ref>Ferraro, V .; Plumpton, C. (1961). An Introduction to Magneto-Fluid Mechanics. Oxford: Oxford University Press.</ref><ref>White, Carol (1977). Energy potential: Toward a new electro-magnetic field theory. New York: Campaigner Pub. ISBN 978-0918388049.</ref> These apparent violations are due to the fact that the traditional conservation laws in classical and quantum electrodynamics (QED) only display linear U(1) symmetry (in particular, by the extended Noether theorem,<ref>Noether, E. (1918). "Invariante Variationsprobleme". Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse. 1918: 235–257.</ref> conservation laws such as the laws of thermodynamics need not always apply to dissipative systems,<ref name="FOOTNOTEScott2006[httpsbooksgooglecombooksidKC7gZmIEAiwCpgPA163 163]">Scott (2006), p. 163.</ref><ref>Pismen, L. M. (2006). Patterns and Interfaces in Dissipative Dynamics. Springer. p. 3. ISBN 9783540304319.</ref> which are expressed in gauges of higher symmetry). The second law of thermodynamics states that in a closed linear system entropy flow can only be positive (or exactly zero at the end of a cycle). However, negative entropy (i.e. increased order, structure or self-organisation) can spontaneously appear in an open nonlinear thermodynamic system that is far from equilibrium, so long as this emergent order accelerates the overall flow of entropy in the total system. The 1977 Nobel Prize in Chemistry was awarded to thermodynamicist Ilya Prigogine<ref>The Nobel Foundation (1977). "The Nobel Prize in Chemistry 1977". nobelprize.org. Royal Swedish Academy of Sciences. Retrieved 3 November 2016.</ref> for his theory of dissipative systems that described this notion. Prigogine described the principle as "order through fluctuations"<ref>Nicolis, G.; Prigogine, I. (1977). Self-organization in Nonequilibrium Systems: From Dissipative Structures to Order Through Fluctuations. Wiley-Blackwell. ISBN 978-0471024019.</ref> or "order out of chaos".<ref>Prigogine, Ilya; Stengers, Isabelle (1984). Order out of Chaos. Flamingo. ISBN 978-0-00-654115-8.</ref> It has been argued by some that all emergent order in the universe from galaxies, solar systems, planets, weather, complex chemistry, evolutionary biology to even consciousness, technology and civilizations are themselves examples of thermodynamic dissipative systems; nature having naturally selected these structures to accelerate entropy flow within the universe to an ever-increasing degree.<ref>Gleick, James (1987). Chaos: Making a New Science (1998 ed.). Vintage. p. 308. ISBN 9780749386061.</ref> For example, it has been estimated that human body is 10,000 times more effective at dissipating energy per unit of mass than the sun.<ref>Chaisson, Eric J. (2002). Cosmic Evolution: The Rise of Complexity in Nature. Harvard University Press. p. 139. ISBN 978-0674009875.</ref>

One may query what this has to do with zero-point energy. Given the complex and adaptive behaviour that arises from nonlinear systems considerable attention in recent years has gone into studying a new class of phase transitions which occur at absolute zero temperature. These are quantum phase transitions which are driven by EM field fluctuations as a consequence of zero-point energy.<ref>Kais, Sabre (2011). Popelier, Paul (ed.). Finite Size Scaling for Criticality of the Schrödinger Equation: Solving the Schrödinger Equation: Has Everything Been Tried?. Singapore: Imperial College Press. pp. 91–92. ISBN 978-1-84816-724-7.</ref> A good example of a spontaneous phase transition that are attributed to zero-point fluctuations can be found in superconductors. Superconductivity is one of the best known empirically quantified macroscopic electromagnetic phenomena whose basis is recognised to be quantum mechanical in origin. The behaviour of the electric and magnetic fields under superconductivity is governed by the London equations. However, it has been questioned in a series of journal articles whether the quantum mechanically canonised London equations can be given a purely classical derivation.<ref>"Classical Physics Makes a Comeback". The Times. London. 14 January 1982.</ref> Bostick,<ref>Bostick, W. (1985). "On the Controversy over Whether Classical Systems Like Plasmas Can Behave Like Superconductors (Which Have Heretofore Been Supposed to Be Strictly Quantum Mechanically Dominated)" (PDF). International Journal of Fusion Energy. 3 (2): 47–51. Archived (PDF) from the original on 3 April 2016. Retrieved 22 May 2020.</ref><ref>Bostick, W. (1985). "The Morphology of the Electron" (PDF). International Journal of Fusion Energy. 3 (1): 9–52. Archived (PDF) from the original on 3 April 2016. Retrieved 22 May 2020.</ref> for instance, has claimed to show that the London equations do indeed have a classical origin that applies to superconductors and to some collisionless plasmas as well. In particular it has been asserted that the Beltrami vortices in the plasma focus display the same paired flux-tube morphology as Type II superconductors.<ref>Bostick, W. (1985). "Recent Experimental Results of The Plasma-Focus Group at Darmstadt, West Germany: A Review and Critique" (PDF). International Journal of Fusion Energy. 3 (1): 68. Archived (PDF) from the original on 3 April 2016. Retrieved 22 May 2020.</ref><ref>Edwards, W. Farrell (1981). "Classical Derivation of the London Equations". Physical Review Letters. 47 (26): 1863–1866. Bibcode:1981PhRvL..47.1863E. doi:10.1103/PhysRevLett.47.1863.</ref> Others have also pointed out this connection, Fröhlich<ref>Fröhlich, H (1966). "Macroscopic wave functions in superconductors". Proceedings of the Physical Society. 87 (1): 330–332. Bibcode:1966PPS....87..330F. doi:10.1088/0370-1328/87/1/137.</ref> has shown that the hydrodynamic equations of compressible fluids, together with the London equations, lead to a macroscopic parameter (<math>\mu</math> = electric charge density / mass density), without involving either quantum phase factors or Planck's constant. In essence, it has been asserted that Beltrami plasma vortex structures are able to at least simulate the morphology of Type I and Type II superconductors. This occurs because the "organised" dissipative energy of the vortex configuration comprising the ions and electrons far exceeds the "disorganised" dissipative random thermal energy. The transition from disorganised fluctuations to organised helical structures is a phase transition involving a change in the condensate's energy (i.e. the ground state or zero-point energy) but without any associated rise in temperature.<ref name="FOOTNOTEReed1995[httpsbooksgooglecombooksidOdnsCgAAQBAJpgPA226 226]">Reed (1995), p. 226.</ref> This is an example of zero-point energy having multiple stable states (see Quantum phase transition, Quantum critical point, Topological degeneracy, Topological order<ref>Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang (2010). "Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order". Physical Review B. 82 (15): 155138. arXiv:1004.3835. Bibcode:2010PhRvB..82o5138C. doi:10.1103/PhysRevB.82.155138. S2CID 14593420.</ref>) and where the overall system structure is independent of a reductionist or deterministic view, that "classical" macroscopic order can also causally affect quantum phenomena. Furthermore, the pair production of Beltrami vortices has been compared to the morphology of pair production of virtual particles in the vacuum.

The idea that the vacuum energy can have multiple stable energy states is a leading hypothesis for the cause of cosmic inflation. In fact, it has been argued that these early vacuum fluctuations led to the expansion of the universe and in turn have guaranteed the non-equilibrium conditions necessary to drive order from chaos, as without such expansion the universe would have reached thermal equilibrium and no complexity could have existed. With the continued accelerated expansion of the universe, the cosmos generates an energy gradient that increases the "free energy" (i.e. the available, usable or potential energy for useful work) which the universe is able to use to create ever more complex forms of order.<ref>Chaisson, Eric J. (2005). "2 Non-equilibrium Thermodynamics in an Energy-Rich Universe". Non-equilibrium Thermodynamics in an Energy-Rich Universe. Understanding Complex Systems. pp. 21–31. doi:10.1007/11672906_2. ISBN 978-3-540-22495-2.</ref><ref>Chaisson, Eric J. (2002). Cosmic Evolution: The Rise of Complexity in Nature. Harvard University Press. p. 216. ISBN 978-0674009875.</ref> The only reason Earth's environment does not decay into an equilibrium state is that it receives a daily dose of sunshine and that, in turn, is due to the sun "polluting" interstellar space with entropy. The sun's fusion power is only possible due to the gravitational disequilibrium of matter that arose from cosmic expansion. In this essence, the vacuum energy can be viewed as the key cause of the structure throughout the universe. That humanity might alter the morphology of the vacuum energy to create an energy gradient for useful work is the subject of much controversy.

Purported applications

Physicists overwhelmingly reject any possibility that the zero-point energy field can be exploited to obtain useful energy (work) or uncompensated momentum; such efforts are seen as tantamount to perpetual motion machines.^{[citation needed]}

Nevertheless, the allure of free energy has motivated such research, usually falling in the category of fringe science. As long ago as 1889 (before quantum theory or discovery of the zero point energy) Nikola Tesla proposed that useful energy could be obtained from free space, or what was assumed at that time to be an all-pervasive aether.<ref>Peterson, I (1997). "Peeking inside an electron's screen". Science News. 151: 89. Retrieved 24 October 2016.</ref> Others have since claimed to exploit zero-point or vacuum energy with a large amount of pseudoscientific literature causing ridicule around the subject.<ref name=army>Amber M. Aiken. "Zero-Point Energy: Can We Get Something From Nothing?" (PDF). U.S. Army National Ground Intelligence Center. Forays into "free energy" inventions and perpetual-motion machines using ZPE are considered by the broader scientific community to be pseudoscience.</ref><ref name=saf>"Zero-point energy, on season 8 , episode 2". Scientific American Frontiers. Chedd-Angier Production Company. 1997–1998. PBS. Archived from the original on 1 January 2006.</ref> Despite rejection by the scientific community, harnessing zero-point energy remains an interest of research, particularly in the US where it has attracted the attention of major aerospace/defence contractors and the U.S. Department of Defense as well as in China, Germany, Russia and Brazil.<ref name=army/><ref name="FOOTNOTEScott2004">Scott (2004).</ref>

Casimir batteries and engines

A common assumption is that the Casimir force is of little practical use; the argument is made that the only way to actually gain energy from the two plates is to allow them to come together (getting them apart again would then require more energy), and therefore it is a one-use-only tiny force in nature.<ref name=army/> In 1984 Robert Forward published work showing how a "vacuum-fluctuation battery" could be constructed; the battery can be recharged by making the electrical forces slightly stronger than the Casimir force to reexpand the plates.<ref>Forward, Robert L. (1985). "Extracting electrical energy from the vacuum by cohesion of charged foliated conductors". Physical Review B. 30 (4): 1700. Bibcode:1984PhRvB..30.1700F. doi:10.1103/PhysRevB.30.1700.</ref>

In 1999, Pinto, a former scientist at NASA's Jet Propulsion Laboratory at Caltech in Pasadena, published in Physical Review his thought experiment (Gedankenexperiment) for a "Casimir engine". The paper showed that continuous positive net exchange of energy from the Casimir effect was possible, even stating in the abstract "In the event of no other alternative explanations, one should conclude that major technological advances in the area of endless, by-product free-energy production could be achieved."<ref name="FOOTNOTEPinto1999">Pinto (1999).</ref>

Garret Moddel at University of Colorado has highlighted that he believes such devices hinge on the assumption that the Casimir force is a nonconservative force, he argues that there is sufficient evidence (e.g. analysis by Scandurra (2001)<ref>Scandurra, M. (2001). "Thermodynamic properties of the quantum vacuum". arXiv:hep-th/0104127.</ref>) to say that the Casimir effect is a conservative force and therefore even though such an engine can exploit the Casimir force for useful work it cannot produce more output energy than has been input into the system.<ref>Moddel, Garret; Dmitriyevaa, Olga (2009). "Extraction of Zero-Point Energy from the Vacuum: Assessment of Stochastic Electrodynamics-Based Approach as Compared to Other Methods". Atoms. 7 (2). 51. arXiv:0910.5893. doi:10.3390/atoms7020051. S2CID 17095906.</ref>

In 2008, DARPA solicited research proposals in the area of Casimir Effect Enhancement (CEE). The goal of the program is to develop new methods to control and manipulate attractive and repulsive forces at surfaces based on engineering of the Casimir force.<ref>"Research in a Vacuum: DARPA Tries to Tap Elusive Casimir Effect for Breakthrough Technology". www.scientificamerican.com. Scientific American. 2008. Retrieved 22 February 2024.</ref>

A 2008 patent by Haisch and Moddel<ref>U.S. patent 7,379,286</ref> details a device that is able to extract power from zero-point fluctuations using a gas that circulates through a Casimir cavity. A published test of this concept by Moddel<ref>Dmitriyevaa, Olga; Moddel, Garret (2012). "Test of zero-point energy emission from gases flowing through Casimir cavities" (PDF). Physics Procedia. 38: 8–17. Bibcode:2012PhPro..38....8D. doi:10.1016/j.phpro.2012.08.007. Archived from the original (PDF) on 7 May 2021. Retrieved 1 November 2016.</ref> was performed in 2012 and seemed to give excess energy that could not be attributed to another source. However it has not been conclusively shown to be from zero-point energy and the theory requires further investigation.<ref>Henriques, Carlos (2014). Study of atomic energy shifts induced by Casimir cavities (Thesis for: MS). Advisors: Fernandes, Luis & Amaro, F. doi:10.13140/RG.2.1.4297.1608.</ref>

Single heat baths

In 1951 Callen and Welton<ref name="ReferenceB"/> proved the quantum fluctuation-dissipation theorem (FDT) which was originally formulated in classical form by Nyquist (1928)<ref name="ReferenceC"/> as an explanation for observed Johnson noise<ref name="ReferenceD"/> in electric circuits. 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.<ref name="FOOTNOTEMilonni199454">Milonni (1994), p. 54.</ref> Such a theory has met with resistance: Macdonald (1962)<ref>MacDonald, D.K.C. (1962). "On Brownian Movement and irreversibility". Physica. 28 (4): 409–416. Bibcode:1962Phy....28..409M. doi:10.1016/0031-8914(62)90019-8.</ref> and Harris (1971)<ref>Harris, I. A. (1971). "Zero-point fluctuations and thermal-noise standards". Electron. Lett. 7 (7): 148–149. Bibcode:1971ElL.....7..148H. doi:10.1049/el:19710095.</ref> claimed that extracting power from the zero-point energy to be impossible, so FDT could not be true. Grau and Kleen (1982)<ref>Grau, G.; Kleen, W. (1982). "Comments on zero-point energy, quantum noise and spontaneous-emission noise". Solid-State Electronics. 25 (8): 749–751. Bibcode:1982SSEle..25..749G. doi:10.1016/0038-1101(82)90204-0.</ref> and Kleen (1986),<ref>Kleen, W. (1985). "Thermal noise and zero-point-energy". Noise in Physical Systems and 1/F Noise 1985. pp. 331–332. doi:10.1016/B978-0-444-86992-0.50072-2. ISBN 9780444869920.</ref> argued that the Johnson noise of a resistor connected to an antenna must satisfy Planck's thermal radiation formula, thus the noise must be zero at zero temperature and FDT must be invalid. Kiss (1988)<ref>Kiss, L. B. (1988). "To the problem of zero-point energy and thermal noise". Solid State Communications. 67 (7): 749–751. Bibcode:1988SSCom..67..749K. doi:10.1016/0038-1098(88)91020-4.</ref> pointed out that the existence of the zero-point term may indicate that there is a renormalization problem—i.e., a mathematical artifact—producing an unphysical term that is not actually present in measurements (in analogy with renormalization problems of ground states in quantum electrodynamics). Later, Abbott et al. (1996) arrived at a different but unclear conclusion that "zero-point energy is infinite thus it should be renormalized but not the 'zero-point fluctuations'".<ref name="FOOTNOTEAbbott et al.1996">Abbott et al. (1996).</ref> Despite such criticism, FDT has been shown to be true experimentally under certain quantum, non-classical conditions. Zero-point fluctuations can, and do, contribute towards systems which dissipate energy.<ref name="cloudfront.escholarship.org"/> A paper by Armen Allahverdyan and Theo Nieuwenhuizen in 2000 showed the feasibility of extracting zero-point energy for useful work from a single bath, without contradicting the laws of thermodynamics, by exploiting certain quantum mechanical properties.<ref name=Allahverdyan-2000/>

There have been a growing number of papers showing that in some instances the classical laws of thermodynamics, such as limits on the Carnot efficiency, can be violated by exploiting negative entropy of quantum fluctuations.<ref name="FOOTNOTEScully et al.2003">Scully et al. (2003).</ref><ref name="FOOTNOTEScully2001">Scully (2001).</ref><ref>Galve, Fernando; Lutz, Eric (2009). "Nonequilibrium thermodynamic analysis of squeezing". Physical Review A. 79 (5): 055804. Bibcode:2009PhRvA..79e5804G. doi:10.1103/PhysRevA.79.055804.</ref><ref>Dillenschneider, R.; Lutz, E. (2009). "Energetics of quantum correlations". EPL. 88 (5): 50003. arXiv:0803.4067. Bibcode:2009EL.....8850003D. doi:10.1209/0295-5075/88/50003. S2CID 119262651.</ref><ref>Huang, X. L.; Wang, Tao; Yi, X. X. (2012). "Effects of reservoir squeezing on quantum systems and work extraction". Physical Review E. 86 (5): 051105. Bibcode:2012PhRvE..86e1105H. doi:10.1103/PhysRevE.86.051105. PMID 23214736.</ref><ref>Boukobza, E.; Ritsch, H. (2013). "Breaking the Carnot limit without violating the second law: A thermodynamic analysis of off-resonant quantum light generation". Physical Review A. 87 (6): 063845. Bibcode:2013PhRvA..87f3845B. doi:10.1103/PhysRevA.87.063845.</ref><ref name="FOOTNOTERoßnagel et al.2014">Roßnagel et al. (2014).</ref><ref name="FOOTNOTECorrea et al.2014">Correa et al. (2014).</ref><ref>Abah, Obinna; Lutz, Eric (2014). "Efficiency of heat engines coupled to nonequilibrium reservoirs". EPL. 106 (2): 20001. arXiv:1303.6558. Bibcode:2014EL....10620001A. doi:10.1209/0295-5075/106/20001. S2CID 118468331.</ref><ref>Gardas, Bartłomiej; Deffner, Sebastian; Saxena, Avadh (2016). "Non-hermitian quantum thermodynamics". Scientific Reports. 6: 23408. arXiv:1511.06256. Bibcode:2016NatSR...623408G. doi:10.1038/srep23408. PMC 4802220. PMID 27003686.</ref>

Despite efforts to reconcile quantum mechanics and thermodynamics over the years, their compatibility is still an open fundamental problem. The full extent that quantum properties can alter classical thermodynamic bounds is unknown<ref>Gemmer, Jochen; Michel, M.; Mahler, Günter (2009). Quantum Thermodynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems. Springer. doi:10.1007/978-3-540-70510-9. ISBN 978-3-540-70510-9.</ref>

Space travel and gravitational shielding

The use of zero-point energy for space travel is speculative and does not form part of the mainstream scientific consensus. A complete quantum theory of gravitation (that would deal with the role of quantum phenomena like zero-point energy) does not yet exist. Speculative papers explaining a relationship between zero-point energy and gravitational shielding effects have been proposed,<ref name="Haisch et al. 1994"/><ref>Noever, David; Bremner, Christopher (1999). "Large-scale Sakharov condition". AIAA 35th Joint Propulsion Conference and Exhibit. doi:10.2514/6.1999-2146.</ref><ref>Haisch, B.; Rueda, A.; Dobyns, Y. (2001). "Inertial mass and the quantum vacuum fields" (PDF). Annalen der Physik. 10 (5): 393–414. arXiv:gr-qc/0009036. Bibcode:2001AnP...513..393H. doi:10.1002/1521-3889(200105)10:5<393::AID-ANDP393>3.0.CO;2-Z. S2CID 15382105.</ref><ref>Podkletnov, Evgeny; Modanese, Giovanni (2001). "Impulse Gravity Generator Based on Charged YBa₂Cu₃O_7−y Superconductor with Composite Crystal Structure". arXiv:physics/0108005.</ref> but the interaction (if any) is not yet fully understood. According to the general theory of relativity, rotating matter can generate a new force of nature, known as the gravitomagnetic interaction, whose intensity is proportional to the rate of spin.<ref>Matthews, Robert (21 September 1996). "Antigravity machine weighed down by controversy". New Scientist. Retrieved 26 October 2016.</ref> In certain conditions the gravitomagnetic field can be repulsive. In neutrons stars for example it can produce a gravitational analogue of the Meissner effect, but the force produced in such an example is theorized to be exceedingly weak.<ref>Lano, R. P. (1996). "Gravitational Meissner Effect". arXiv:hep-th/9603077.</ref>

In 1963 Robert Forward, a physicist and aerospace engineer at Hughes Research Laboratories, published a paper showing how within the framework of general relativity "anti-gravitational" effects might be achieved.<ref>Forward, R. L. (1963). "Guidelines to Antigravity" (PDF). American Journal of Physics. 31 (3): 166–170. Bibcode:1963AmJPh..31..166F. doi:10.1119/1.1969340.</ref> Since all atoms have spin, gravitational permeability may be able to differ from material to material. A strong toroidal gravitational field that acts against the force of gravity could be generated by materials that have nonlinear properties that enhance time-varying gravitational fields. Such an effect would be analogous to the nonlinear electromagnetic permeability of iron making it an effective core (i.e. the doughnut of iron) in a transformer, whose properties are dependent on magnetic permeability.<ref>Forward, R. L. (1961). "General Relativity for the Experimentalist". Proceedings of the IRE. 49 (5): 892–904. Bibcode:1961PIRE...49..892F. doi:10.1109/JRPROC.1961.287932. S2CID 51650940.</ref><ref>Swain, John (2010). "Gravitatomagnetic Analogs of Electric Transformers". arXiv:1006.5754 [gr-qc].</ref><ref>"Physicist Predicts Gravitational Analogue Of Electrical Transformers". MIT Technology Review. 6 July 2010. Retrieved 28 October 2016.</ref> In 1966 Dewitt<ref>DeWitt, Bryce S. (1966). "Superconductors and Gravitational Drag". Physical Review Letters. 16 (24): 1092–1093. Bibcode:1966PhRvL..16.1092D. doi:10.1103/PhysRevLett.16.1092.</ref> was first to identify the significance of gravitational effects in superconductors. Dewitt demonstrated that a magnetic-type gravitational field must result in the presence of fluxoid quantization. In 1983, Dewitt's work was substantially expanded by Ross.<ref>Ross, D. K. (1983). "The London equations for superconductors in a gravitational field". Journal of Physics A. 16 (6): 1331–1335. Bibcode:1983JPhA...16.1331R. doi:10.1088/0305-4470/16/6/026.</ref>

From 1971 to 1974 Henry William Wallace, a scientist at GE Aerospace was issued with three patents.<ref>U.S. patent 3,626,606</ref><ref>U.S. patent 3,626,605</ref><ref>U.S. patent 3,823,570</ref> Wallace used Dewitt's theory to develop an experimental apparatus for generating and detecting a secondary gravitational field, which he named the kinemassic field (now better known as the gravitomagnetic field). In his three patents, Wallace describes three different methods used for detection of the gravitomagnetic field – change in the motion of a body on a pivot, detection of a transverse voltage in a semiconductor crystal, and a change in the specific heat of a crystal material having spin-aligned nuclei. There are no publicly available independent tests verifying Wallace's devices. Such an effect if any would be small.<ref>Barker, B. M.; O'Connell, R. F. (1979). "The gravitational interaction: Spin, rotation, and quantum effects-a review". General Relativity and Gravitation. 11 (2): 149–175. Bibcode:1979GReGr..11..149B. doi:10.1007/BF00756587. S2CID 121728055.</ref><ref>O'Connell, R. F. (1970). "The gravitational field of the electron". Physics Letters A. 32 (6): 402–403. Bibcode:1970PhLA...32..402O. doi:10.1016/0375-9601(70)90022-8.</ref><ref>O'Connell, R. F.; Rasband, S. N. (1971). "Lense-Thirring Type Gravitational Forces Between Disks and Cylinders". Nature. 232 (35): 193–195. Bibcode:1971NPhS..232..193O. doi:10.1038/physci232193a0.</ref><ref>Peres, Asher (1978). "Test of equivalence principle for particles with spin". Physical Review D. 18 (8): 2739–2740. Bibcode:1978PhRvD..18.2739P. doi:10.1103/PhysRevD.18.2739.</ref><ref>Obukhov, Yuri N. (2001). "Spin, gravity, and inertia". Physical Review Letters. 86 (2): 192–195. arXiv:gr-qc/0012102. Bibcode:2001PhRvL..86..192O. doi:10.1103/PhysRevLett.86.192. PMID 11177789. S2CID 35509153.</ref><ref>Ritter, R. C.; Winkler, L. I.; Gillies, G. T. (1993). "Search for anomalous spin-dependent forces with a polarized-mass torsion pendulum". Physical Review Letters. 70 (6): 701–704. Bibcode:1993PhRvL..70..701R. doi:10.1103/PhysRevLett.70.701. PMID 10054182.</ref> Referring to Wallace's patents, a New Scientist article in 1980 stated "Although the Wallace patents were initially ignored as cranky, observers believe that his invention is now under serious but secret investigation by the military authorities in the USA. The military may now regret that the patents have already been granted and so are available for anyone to read."<ref>"Antigravity Not So Crazy After All". Patents Review. New Scientist. Vol. 85, no. 1194. 14 February 1980. p. 485.</ref> A further reference to Wallace's patents occur in an electric propulsion study prepared for the Astronautics Laboratory at Edwards Air Force Base which states: "The patents are written in a very believable style which include part numbers, sources for some components, and diagrams of data. Attempts were made to contact Wallace using patent addresses and other sources but he was not located nor is there a trace of what became of his work. The concept can be somewhat justified on general relativistic grounds since rotating frames of time varying fields are expected to emit gravitational waves."<ref>Cravens, D. L. (1990). "Electric Propulsion Study: Final Report" (PDF). Contract F04611-88-C-0014, Astronautics Laboratory (AFSC), Air Force Space Technology Center, Space Systems Division, Air Force Systems Command, Edwards AFB, CA. Archived from the original (PDF) on 12 August 2011. Retrieved 26 October 2016.</ref>

In 1986 the U.S. Air Force's then Rocket Propulsion Laboratory (RPL) at Edwards Air Force Base solicited "Non Conventional Propulsion Concepts" under a small business research and innovation program. One of the six areas of interest was "Esoteric energy sources for propulsion, including the quantum dynamic energy of vacuum space..." In the same year BAE Systems launched "Project Greenglow" to provide a "focus for research into novel propulsion systems and the means to power them".<ref name="FOOTNOTEScott2004">Scott (2004).</ref><ref>Allen, J. E. (2005). "Aeronautics-1903; aerospace-2003; ? ? 2103". Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. 219 (3): 235–260. doi:10.1243/095441005X30252. S2CID 110771631.</ref>

In 1988 Kip Thorne et al.<ref>Thorne, Kip; Michael Morris; Ulvi Yurtsever (1988). "Wormholes, Time Machines, and the Weak Energy Condition" (PDF). Physical Review Letters. 61 (13): 1446–1449. Bibcode:1988PhRvL..61.1446M. doi:10.1103/PhysRevLett.61.1446. PMID 10038800.</ref> published work showing how traversable wormholes can exist in spacetime only if they are threaded by quantum fields generated by some form of exotic matter that has negative energy. In 1993 Scharnhorst and Barton<ref name="Scharnhorst 1993"/> showed that the speed of a photon will be increased if it travels between two Casimir plates, an example of negative energy. In the most general sense, the exotic matter needed to create wormholes would share the repulsive properties of the inflationary energy, dark energy or zero-point radiation of the vacuum.<ref>Wheeler, J Craig (2007). Cosmic Catastrophes (2nd ed.). New York: Cambridge University Press. p. 228. ISBN 978-0521857147.</ref> Building on the work of Thorne, in 1994 Miguel Alcubierre<ref>Alcubierre, Miguel (1994). "The warp drive: hyper-fast travel within general relativity". Classical and Quantum Gravity. 11 (5): L73–L77. arXiv:gr-qc/0009013. Bibcode:1994CQGra..11L..73A. doi:10.1088/0264-9381/11/5/001. S2CID 4797900.</ref> proposed a method for changing the geometry of space by creating a wave that would cause the fabric of space ahead of a spacecraft to contract and the space behind it to expand (see Alcubierre drive). The ship would then ride this wave inside a region of flat space, known as a warp bubble and would not move within this bubble but instead be carried along as the region itself moves due to the actions of the drive.

In 1992 Evgeny Podkletnov<ref>Podkletnov, E.; Nieminen, R. (1992). "A possibility of gravitational force shielding by bulk YBa₂Cu₃O_7−x superconductor". Physica C: Superconductivity. 203 (3–4): 441–444. Bibcode:1992PhyC..203..441P. doi:10.1016/0921-4534(92)90055-H.</ref> published a heavily debated<ref>Rounds, Frederic N. (1998). "Anomalous Weight Behavior in YBa₂Cu₃O₇ Compounds at Low Temperature". Proc. NASA Breakthrough Propulsion Phys. Workshop. 279: physics/9705043. arXiv:physics/9705043. Bibcode:1997physics...5043R.</ref><ref name="FOOTNOTEWoods et al.2001">Woods et al. (2001).</ref><ref>Tajmar, M.; Plesescu, F.; Marhold, K. & de Matos, C.J. (2006). "Experimental Detection of the Gravitomagnetic London Moment". arXiv:gr-qc/0603033v1.</ref><ref>Robertson, Glen A. (1999). "On the Mechanism for a Gravity Effect using Type II Superconductors" (PDF). NASA Technical Reports Server. Retrieved 26 October 2016.</ref> journal article claiming a specific type of rotating superconductor could shield gravitational force. Independently of this, from 1991 to 1993 Ning Li and Douglas Torr published a number of articles<ref>Li, N.; Torr, D. G. (1991). "Effects of a gravitomagnetic field on pure superconductors". Physical Review D. 43 (2): 457–459. Bibcode:1991PhRvD..43..457L. doi:10.1103/PhysRevD.43.457. PMID 10013404.</ref><ref>Li, Ning; Torr, D. G. (1992). "Gravitational effects on the magnetic attenuation of superconductors". Physical Review B. 46 (9): 5489–5495. Bibcode:1992PhRvB..46.5489L. doi:10.1103/PhysRevB.46.5489. PMID 10004334.</ref><ref>Torr, Douglas G.; Li, Ning (1993). "Gravitoelectric-electric coupling via superconductivity". Foundations of Physics Letters. 6 (4): 371–383. Bibcode:1993FoPhL...6..371T. doi:10.1007/BF00665654. S2CID 122075917.</ref> about gravitational effects in superconductors. One finding they derived is the source of gravitomagnetic flux in a type II superconductor material is due to spin alignment of the lattice ions. Quoting from their third paper: "It is shown that the coherent alignment of lattice ion spins will generate a detectable gravitomagnetic field, and in the presence of a time-dependent applied magnetic vector potential field, a detectable gravitoelectric field." The claimed size of the generated force has been disputed by some<ref name="FOOTNOTEKowitt1994">Kowitt (1994).</ref><ref>Harris, Edward G. (1999). "Comments on "Gravitoelectric-Electric Coupling via Superconductivity" by Douglas G. Torr and Ning Li". Foundations of Physics Letters. 12 (2): 201–208. doi:10.1023/A:1021621425670. S2CID 115204136.</ref> but defended by others.<ref name="FOOTNOTEWoods2005">Woods (2005).</ref><ref>Tajmar, Martin; de Matos, Clovis (2006). "Gravitomagnetic Fields in Rotating Superconductors to Solve Tate's Cooper Pair Mass Anomaly" (PDF). AIP Conf. Proc. 813: 1415–1420. arXiv:gr-qc/0607086. Bibcode:2006AIPC..813.1415T. doi:10.1063/1.2169327. S2CID 24997124.</ref> In 1997 Li published a paper attempting to replicate Podkletnov's results and showed the effect was very small, if it existed at all.<ref>Li, N.; Noever, D.; Robertson, T.; Koczor, R.; Brantley, W. (August 1997). "Static Test for a Gravitational Force Coupled to Type II YBCO Superconductors". Physica C. 281 (2–3): 260–267. Bibcode:1997PhyC..281..260L. doi:10.1016/S0921-4534(97)01462-7.</ref> Li is reported to have left the University of Alabama in 1999 to found the company AC Gravity LLC.<ref name="FOOTNOTELucentini2000">Lucentini (2000).</ref> AC Gravity was awarded a U.S. DOD grant for $448,970 in 2001 to continue anti-gravity research. The grant period ended in 2002 but no results from this research were ever made public.<ref>"Annual Report on Cooperative Agreements and Other Transactions Entered into During FY2001 Under 10 USC 2371". DOD. p. 66. Archived from the original on 1 August 2021. Retrieved 6 March 2014.</ref>

In 2002 Phantom Works, Boeing's advanced research and development facility in Seattle, approached Evgeny Podkletnov directly. Phantom Works was blocked by Russian technology transfer controls. At this time Lieutenant General George Muellner, the outgoing head of the Boeing Phantom Works, confirmed that attempts by Boeing to work with Podkletnov had been blocked by Moscow, also commenting that "The physical principles – and Podkletnov's device is not the only one – appear to be valid... There is basic science there. They're not breaking the laws of physics. The issue is whether the science can be engineered into something workable"<ref name="FOOTNOTECook2002">Cook (2002).</ref>

Froning and Roach (2002)<ref>Froning, H.; Roach, R. (2002). "Preliminary Simulations of Vehicle Interactions with the Quantum Vacuum by Fluid Dynamic Approximations". AIAA 38th Joint Propulsion Conference & Exhibit. p. 52236. doi:10.2514/6.2002-3925. ISBN 978-1-62410-115-1.</ref> put forward a paper that builds on the work of Puthoff, Haisch and Alcubierre. They used fluid dynamic simulations to model the interaction of a vehicle (like that proposed by Alcubierre) with the zero-point field. Vacuum field perturbations are simulated by fluid field perturbations and the aerodynamic resistance of viscous drag exerted on the interior of the vehicle is compared to the Lorentz force exerted by the zero-point field (a Casimir-like force is exerted on the exterior by unbalanced zero-point radiation pressures). They find that the optimized negative energy required for an Alcubierre drive is where it is a saucer-shaped vehicle with toroidal electromagnetic fields. The EM fields distort the vacuum field perturbations surrounding the craft sufficiently to affect the permeability and permittivity of space.

In 2009 Giorgio Fontana and Bernd Binder presented a new method to potentially extract the Zero-point energy of the electromagnetic field and nuclear forces in the form of gravitational waves.<ref>Fontana, Giorgio; Binder, Bernd (16 March 2009). "Electromagnetic to Gravitational wave Conversion via Nuclear Holonomy". AIP Conference Proceedings. 1103 (1): 524–531. Bibcode:2009AIPC.1103..524F. doi:10.1063/1.3115561. ISSN 0094-243X.</ref> In the spheron model of the nucleus,<ref>Pauling, Linus (October 1965). "The Close-Packed-Spheron Model of Atomic Nuclei and ITS Relation to the Shell Model". Proceedings of the National Academy of Sciences. 54 (4): 989–994. Bibcode:1965PNAS...54..989P. doi:10.1073/pnas.54.4.989. ISSN 0027-8424. PMC 219778. PMID 16578621.</ref> proposed by the two times Nobel laureate Linus Pauling, dineutrons are among the components of this structure. Similarly to a dumbbell put in a suitable rotational state, but with nuclear mass density, dineutrons are nearly ideal sources of gravitational waves at X-ray and gamma-ray frequencies. The dynamical interplay, mediated by nuclear forces, between the electrically neutral dineutrons and the electrically charged core nucleus is the fundamental mechanism by which nuclear vibrations can be converted to a rotational state of dineutrons with emission of gravitational waves. Gravity and gravitational waves are well described by General Relativity, that is not a quantum theory, this implies that there is no Zero-point energy for gravity in this theory, therefore dineutrons will emit gravitational waves like any other known source of gravitational waves. In Fontana and Binder paper, nuclear species with dynamical instabilites, related to the Zero-point energy of the electromagnetic field and nuclear forces, and possessing dineutrons, will emit gravitational waves. In experimental physics this approach is still unexplored.

In 2014 NASA's Eagleworks Laboratories announced that they had successfully validated the use of a Quantum Vacuum Plasma Thruster which makes use of the Casimir effect for propulsion.<ref name="FOOTNOTEWhite, March, Williams et al.2011">White, March, Williams et al. (2011).</ref><ref>Maxey, Kyle (11 December 2012). "Propulsion on an Interstellar Scale – the Quantum Vacuum Plasma Thruster". engineering.com. Retrieved 24 October 2016.</ref><ref>Hambling, David (31 July 2014). "Nasa validates 'impossible' space drive". Wired UK. Retrieved 24 October 2016.</ref> In 2016 a scientific paper by the team of NASA scientists passed peer review for the first time.<ref name="FOOTNOTEWhite, March, Lawrence et al.2016">White, March, Lawrence et al. (2016).</ref> The paper suggests that the zero-point field acts as pilot-wave and that the thrust may be due to particles pushing off the quantum vacuum. While peer review doesn't guarantee that a finding or observation is valid, it does indicate that independent scientists looked over the experimental setup, results, and interpretation and that they could not find any obvious errors in the methodology and that they found the results reasonable. In the paper, the authors identify and discuss nine potential sources of experimental errors, including rogue air currents, leaky electromagnetic radiation, and magnetic interactions. Not all of them could be completely ruled out, and further peer-reviewed experimentation is needed in order to rule these potential errors out.<ref>Drake, Nadia; Greshko, Michael (21 November 2016). "NASA Team Claims 'Impossible' Space Engine Works—Get the Facts". National Geographic. Archived from the original on 22 November 2016. Retrieved 22 November 2016.</ref>

References

Notes

Articles in the press

Battersby, S. (20 November 2008). "It's Confirmed: Matter is Merely Vacuum Fluctuations". New Scientist. Archived from the original on 27 May 2017.
Bennett, J. (30 November 2016). "Scientists Catch "Virtual Particles" Hopping In and Out of Existence". Popular Mechanics. Archived from the original on 29 May 2017.
Cho, A. (1 October 2015). "Physicists Observe Weird Quantum Fluctuations Of Empty Space—Maybe". Science. doi:10.1126/science.aad4655. Archived from the original on 27 May 2017.
Cho, A. (30 November 2016). "Astronomers Spot Signs of Weird Quantum Distortion in Space". Science. doi:10.1126/science.aal0437. Archived from the original on 29 May 2017.
Choi, C. Q. (12 February 2013). "Something from Nothing? A Vacuum Can Yield Flashes of Light". Scientific American. Archived from the original on 30 May 2017.
Chown, M. (7 April 1990). "Science: Can Photons Travel 'Faster Than Light'?". New Scientist. Archived from the original on 30 May 2017.
Cook, N. (29 July 2002). "Anti-Gravity Propulsion Comes "Out of the Closet"". Jane's Defence Weekly. London. Archived from the original on 2 August 2002.
Crane, L. (30 November 2016). "Quantum Particles Seen Distorting Light From a Neutron Star". New Scientist. Archived from the original on 29 May 2017.
Davies, P. C. W. (22 September 1994). "Inertia Theory: Magic Roundabout. Paul Davies on the Meaning of Mach's Principle". The Guardian. Archived from the original on 17 January 1999.
Davies, P. C. W. (19 November 2011). "Out of the Ether: The Changing Face of the Vacuum". New Scientist. Vol. 212, no. 2839. pp. 50–52. Bibcode:2011NewSc.212Q..50D. doi:10.1016/S0262-4079(11)62858-3.
Lucentini, J. (28 September 2000). "Weighty Implications: NASA Funds Controversial Gravity Shield". SPACE.com. Archived from the original on 9 November 2000.
Matthews, R. (25 February 1995). "Nothing Like a Vacuum". New Scientist. Vol. 145, no. 1966. pp. 30–33. Archived from the original on 13 April 2016. Via Calphysics Institute.
O'Carroll, E. (25 March 2013). "Scientists Examine Nothing, Find Something". Christian Science Monitor. Archived from the original on 31 March 2013.
Pilkington, M. (17 July 2003). "Zero Point Energy". The Guardian. Archived from the original on 7 February 2017.
Powell, C. S. (1994). "Unbearable Lightness: A New Theory May Explain Why Objects Tend to Stay Put". Scientific American. 270 (5): 30–31. doi:10.1038/scientificamerican0594-27.
Scott, W. B. (March 2004). "To the Stars" (PDF). Aviation Week & Space Technology. pp. 50–53. Archived from the original (PDF) on 26 February 2017. Retrieved 25 October 2016.
Siegel, E. (22 September 2016). "What Is The Physics Of Nothing?". Forbes. Archived from the original on 27 May 2017.
Shiga, D. (28 September 2005). "Vacuum Energy: Something For Nothing?". New Scientist. Archived from the original on 27 May 2017.
Wall, M. (27 March 2014). "Does Dark Energy Spring From the 'Quantum Vacuum?'". SPACE.com. Archived from the original on 29 March 2014.

Bibliography

Abbott, D.; Davis, B. R.; Phillips, N. J.; Eshraghian, K. (1996). "Simple derivation of the thermal noise formula using window-limited Fourier transforms and other conundrums". IEEE Transactions on Education. 39 (1): 1–13. Bibcode:1996ITEdu..39....1A. CiteSeerX 10.1.1.129.5792. doi:10.1109/13.485226.
Barton, G.; Scharnhorst, K. (1993). "QED Between Parallel Mirrors: Light Signals Faster Than c, or Amplified by the Vacuum". Journal of Physics A: Mathematical and General. 26 (8): 2037–2046. Bibcode:1993JPhA...26.2037B. doi:10.1088/0305-4470/26/8/024. ISSN 0305-4470.
Battersby, Stephen (2016). "Dark energy: Staring into darkness". Nature. 537 (7622): 201–204. Bibcode:2016Natur.537S.201B. doi:10.1038/537S201a. PMID 27681049. S2CID 4398296.
Bordag, M; Klimchitskaya, G. L.; Mohideen, U.; Mostepanenko, V. M. (2009). Advances in the Casimir Effect. Oxford: Oxford University Press. ISBN 978-0-19-923874-3. LCCN 2009279136. OCLC 319209483.
Bostick, W. H.; Prior, W.; Grunberger, L.; Emmert, G. (1966). "Pair Production of Plasma Vortices". Physics of Fluids. 9 (10): 2078. Bibcode:1966PhFl....9.2078B. doi:10.1063/1.1761572.
Bressi, G.; Carugno, G.; Onofrio, R.; Ruoso, G. (2002). "Measurement of the Casimir Force between Parallel Metallic Surfaces". Physical Review Letters. 88 (4): 041804. arXiv:quant-ph/0203002. Bibcode:2002PhRvL..88d1804B. doi:10.1103/PhysRevLett.88.041804. PMID 11801108. S2CID 43354557.
Capasso, F.; Munday, J. N.; Iannuzzi, D.; Chan, H. B. (2007). "Casimir Forces and Quantum Electrodynamical Torques: Physics and Nanomechanics" (PDF). IEEE Journal of Selected Topics in Quantum Electronics. 13 (2): 400–414. Bibcode:2007IJSTQ..13..400C. doi:10.1109/JSTQE.2007.893082. S2CID 32996610.
Carroll, S. M.; Field, G. B. (1997). "Is There Evidence for Cosmic Anisotropy in the Polarization of Distant Radio Sources?" (PDF). Physical Review Letters. 79 (13): 2394–2397. arXiv:astro-ph/9704263. Bibcode:1997PhRvL..79.2394C. doi:10.1103/PhysRevLett.79.2394. ISSN 0031-9007. S2CID 13943605.
Chan, H. B.; Aksyuk, V. A.; Kleiman, R. N.; Bishop, D. J.; Capasso, F. (2001). "Quantum Mechanical Actuation of Microelectromechanical Systems by the Casimir Force" (PDF). Science. 291 (5510): 1941–1944. Bibcode:2001Sci...291.1941C. doi:10.1126/science.1057984. PMID 11239149. S2CID 17072357.
Conlon, T. E. (2011). Thinking About Nothing : Otto Von Guericke and The Magdeburg Experiments on the Vacuum. San Francisco: Saint Austin Press. ISBN 978-1-4478-3916-3. OCLC 840927124.
Correa, L. A.; Palao, J. P.; Alonso, D.; Adesso, G. (2014). "Quantum-enhanced absorption refrigerators". Scientific Reports. 4 (3949): 3949. arXiv:1308.4174. Bibcode:2014NatSR...4E3949C. doi:10.1038/srep03949. PMC 3912482. PMID 24492860.
Davies, P. C. W. (1985). Superforce: The Search for a Grand Unified Theory of Nature. New York: Simon and Schuster. ISBN 978-0-671-47685-4. LCCN 84005473. OCLC 12397205.
Decca, R. S.; López, D.; Fischbach, E.; Krause, D. E. (2003). "Measurement of the Casimir Force between Dissimilar Metals". Physical Review Letters. 91 (5): 050402. arXiv:quant-ph/0306136. Bibcode:2003PhRvL..91e0402D. doi:10.1103/PhysRevLett.91.050402. PMID 12906584. S2CID 20243276.
Dirac, Paul A. M. (1927). "The Quantum Theory of the Emission and Absorption of Radiation" (PDF). Proc. R. Soc. A. 114 (767): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039. Archived from the original (PDF) on 25 October 2016. Retrieved 17 October 2016.
Drexhage, K. H. (1970). "Monomolecular Layers and Light". Scientific American. 222 (3): 108–119. Bibcode:1970SciAm.222c.108D. doi:10.1038/scientificamerican0370-108.
Drexhage, K. H. (1974). "IV Interaction of Light with Monomolecular Dye Layers". In Wolf, E. (ed.). Progress in Optics. Vol. 12. pp. 163–232. doi:10.1016/S0079-6638(08)70266-X. ISBN 978-0-444-10571-4.
Dunne, G. V. (2012). "The Heisenberg-Euler Effective Action: 75 years on". International Journal of Modern Physics A. 27 (15): 1260004. arXiv:1202.1557. Bibcode:2012IJMPA..2760004D. doi:10.1142/S0217751X12600044. ISSN 0217-751X. S2CID 119258601.
Einstein, A. (1995). Klein, Martin J.; Kox, A. J.; Renn, Jürgen; Schulmann, Robert (eds.). The Collected Papers of Albert Einstein Vol. 4 The Swiss Years: Writings, 1912–1914. Princeton: Princeton University Press. ISBN 978-0-691-03705-9. OCLC 929349643.
Einstein, A. (1993). Klein, Martin J.; Kox, A. J.; Schulmann, Robert (eds.). The Collected Papers of Albert Einstein Vol. 5 The Swiss Years: Correspondence, 1902–1914. Princeton: Princeton University Press. ISBN 978-0-691-03322-8. OCLC 921496342.
Goy, P.; Raimond, J. M.; Gross, M.; Haroche, S. (1983). "Observation of Cavity-Enhanced Single-Atom Spontaneous Emission". Physical Review Letters. 50 (24): 1903–1906. Bibcode:1983PhRvL..50.1903G. doi:10.1103/PhysRevLett.50.1903.
Greiner, W.; Müller, B.; Rafelski, J. (2012). Quantum Electrodynamics of Strong Fields: With an Introduction into Modern Relativistic Quantum Mechanics. Springer. doi:10.1007/978-3-642-82272-8. ISBN 978-0-387-13404-8. LCCN 84026824. OCLC 317097176.
Haisch, B.; Rueda, A.; Puthoff, H. E. (1994). "Inertia as a Zero-Point-Field Lorentz Force" (PDF). Physical Review A. 49 (2): 678–694. Bibcode:1994PhRvA..49..678H. doi:10.1103/PhysRevA.49.678. PMID 9910287. Archived from the original (PDF) on 31 July 2018. Retrieved 17 October 2016.
Heisenberg, W.; Euler, H. (1936). "Folgerungen aus der Diracschen Theorie des Positrons". Zeitschrift für Physik. 98 (11–12): 714–732. arXiv:physics/0605038. Bibcode:1936ZPhy...98..714H. doi:10.1007/BF01343663. ISSN 1434-6001. S2CID 120354480.
Heyl, J. S.; Shaviv, N. J. (2000). "Polarization evolution in strong magnetic fields". Monthly Notices of the Royal Astronomical Society. 311 (3): 555–564. arXiv:astro-ph/9909339. Bibcode:2000MNRAS.311..555H. doi:10.1046/j.1365-8711.2000.03076.x. ISSN 0035-8711. S2CID 11717019.
Itzykson, C.; Zuber, J.-B. (1980). Quantum Field Theory (2005 ed.). Mineola, New York: Dover Publications. ISBN 978-0486445687. LCCN 2005053026. OCLC 61200849.
Kostelecký, V. Alan; Mewes, M. (2009). "Electrodynamics with Lorentz-violating operators of arbitrary dimension". Physical Review D. 80 (1): 015020. arXiv:0905.0031. Bibcode:2009PhRvD..80a5020K. doi:10.1103/PhysRevD.80.015020. ISSN 1550-7998. S2CID 119241509.
Kostelecký, V. Alan; Mewes, M. (2013). "Constraints on Relativity Violations from Gamma-Ray Bursts". Physical Review Letters. 110 (20): 201601. arXiv:1301.5367. Bibcode:2013PhRvL.110t1601K. doi:10.1103/PhysRevLett.110.201601. ISSN 0031-9007. PMID 25167393. S2CID 8579347.
Kowitt, Mark (1994). "Gravitomagnetism and magnetic permeability in superconductors". Physical Review B. 49 (1): 704–708. Bibcode:1994PhRvB..49..704K. doi:10.1103/PhysRevB.49.704. PMID 10009347.
Kragh, H. (2002). Quantum Generations: A History of Physics in the Twentieth Century. Princeton University Press. ISBN 978-0-691-09552-3. OCLC 248763258.
Kragh, H. (2012). "Preludes to Dark Energy: Zero-Point Energy and Vacuum Speculations". Archive for History of Exact Sciences. 66 (3): 199–240. arXiv:1111.4623. doi:10.1007/s00407-011-0092-3. ISSN 0003-9519. S2CID 118593162.
Kragh, H. S.; Overduin, J. M. (2014). The Weight of the Vacuum : A Scientific History of Dark Energy. New York: Springer. ISBN 978-3-642-55089-8. LCCN 2014938218. OCLC 884863929.
Kuhn, T. (1978). Black-Body Theory and the Quantum Discontinuity, 1894-1912. New York: Oxford University Press. ISBN 978-0-19-502383-1. LCCN 77019022. OCLC 803538583.
Le Bellac, M. (2006). Quantum Physics (2012 ed.). Cambridge University Press. ISBN 978-1-107-60276-2. OCLC 957316740.
Leuchs, G.; Sánchez-Soto, L. L. (2013). "A Sum Rule For Charged Elementary Particles". The European Physical Journal D. 67 (3): 57. arXiv:1301.3923. Bibcode:2013EPJD...67...57L. doi:10.1140/epjd/e2013-30577-8. ISSN 1434-6060. S2CID 118643015.
Loudon, R. (2000). The Quantum Theory of Light (3rd ed.). Oxford: Oxford University Press. ISBN 978-0198501770. LCCN 2001265846. OCLC 44602993.
Matthews, R. (1994). "Inertia: Does Empty Space Put Up the Resistance?". Science. 263 (5147): 612–613. Bibcode:1994Sci...263..612M. doi:10.1126/science.263.5147.612. ISSN 0036-8075. JSTOR 2883066. PMID 17747645 – via Calphysics Institute.
Mignani, R. P.; Testa, V.; González Caniulef, D.; Taverna, R.; Turolla, R.; Zane, S.; Wu, K. (2017). "Evidence for vacuum birefringence from the first optical-polarimetry measurement of the isolated neutron star RX J1856.5−3754" (PDF). Monthly Notices of the Royal Astronomical Society. 465 (1): 492–500. arXiv:1610.08323. Bibcode:2017MNRAS.465..492M. doi:10.1093/mnras/stw2798. ISSN 0035-8711. S2CID 38736536.
Milonni, P. W. (1981). "Radiation reaction and the nonrelativistic theory of the electron". Physics Letters A. 82 (5): 225–226. Bibcode:1981PhLA...82..225M. doi:10.1016/0375-9601(81)90191-2.
Milonni, P. W. (1983). "Does the Electromagnetic Mass of an Electron Depend on Where It Is?". International Journal of Theoretical Physics. 22 (4): 323–328. Bibcode:1983IJTP...22..323M. doi:10.1007/BF02082897. S2CID 119991060.
Milonni, P. W. (1994). The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Boston: Academic Press. ISBN 978-0124980808. LCCN 93029780. OCLC 422797902.
Milonni, P. W. (2009). "Zero-Point Energy". In Greenberger, D.; Hentschel, K.; Weinert, F. (eds.). Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy. Berlin, Heidelberg: Springer. pp. 864–866. arXiv:0811.2516. doi:10.1007/978-3-540-70626-7. ISBN 9783540706229. LCCN 2008942038. OCLC 297803628.
Peebles, P. J. E.; Ratra, Bharat (2003). "The Cosmological Constant and Dark Energy". Reviews of Modern Physics. 75 (2): 559–606. arXiv:astro-ph/0207347. Bibcode:2003RvMP...75..559P. doi:10.1103/RevModPhys.75.559. ISSN 0034-6861. S2CID 118961123.
Penrose, Roger (2004). The Road to Reality (8th ed.). New York: Alfred A. Knopf. ISBN 978-0-679-45443-4. OCLC 474890537.
Perlmutter, S.; Aldering; Goldhaber; Knop; Nugent; Castro; Deustua; Fabbro; Goobar; et al. (1999). "Measurements of Ω and Λ from 42 High-Redshift Supernovae". Astrophysical Journal. 517 (2): 565–86. arXiv:astro-ph/9812133. Bibcode:1999ApJ...517..565P. doi:10.1086/307221. S2CID 118910636.
Peskin, M. E.; Schroeder, D. V. (1995). An Introduction To Quantum Field Theory. Addison-Wesley. ISBN 978-0-201-50397-5. OCLC 635667163.
Pinto, F. (1999). "Engine cycle of an optically controlled vacuum energy transducer". Physical Review B. 60 (21): 14740. Bibcode:1999PhRvB..6014740P. doi:10.1103/PhysRevB.60.14740.
Planck, M. (1900). "Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum". Verhandlungen der Deutschen Physikalischen Gesellschaft. 2: 237–245.
Planck, M. (1911). "Eine neue Strahlungshypothese". Verhandlungen der Deutschen Physikalischen Gesellschaft. 13: 138–148.
Planck, M. (1912a). "Über die Begründung das Gesetzes des schwarzen Strahlung". Annalen der Physik. 37 (4): 642–656. Bibcode:1912AnP...342..642P. doi:10.1002/andp.19123420403.
Planck, M. (1912b). "La loi du rayonnement noir et l'hypothèse des quantités élémentaires d'action". In Langevin, P.; Solvay, E.; de Broglie, M. (eds.). La Théorie du Rayonnement et les Quanta. Paris: Gauthier-Villars. pp. 93–114. LCCN unk84021539. OCLC 5894537227.
Planck, M. (1913). Vorlesungen über die Theorie der Wärmestrahlung. Leipzig: J. A. Barth. OCLC 924400975.
Planck, M. (1958). Physikalische Abhandlungen und Vorträge. Vol. 2. Braunschweig: Vieweg & Sohn. LCCN 59047616. OCLC 603096370.
Power, E. A. (1964). Introductory Quantum Electrodynamics. London: Longmans. LCCN 65020006. OCLC 490279969.
Reed, D. (1995). "Foundational Electrodynamics and Beltrami Vector Fields". In Barrett, Terence William; Grimes, Dale M. (eds.). Advanced Electromagnetism: Foundations, Theory and Applications. Singapore: World Scientific. pp. 217–249. ISBN 978-981-02-2095-2.
Rees, Martin, ed. (2012). Universe. New York: DK Pub. ISBN 978-0-7566-9841-6. LCCN 2011277855. OCLC 851193468.
Riek, C.; Seletskiy, D. V.; Moskalenko, A. S.; Schmidt, J. F.; Krauspe, P.; Eckart, S.; Eggert, S.; Burkard, G.; Leitenstorfer, A. (2015). "Direct Sampling of Electric-Field Vacuum Fluctuations" (PDF). Science. 350 (6259): 420–423. Bibcode:2015Sci...350..420R. doi:10.1126/science.aac9788. ISSN 0036-8075. PMID 26429882. S2CID 40368170.
Riess, A. G.; Filippenko; Challis; Clocchiatti; Diercks; Garnavich; Gilliland; Hogan; Jha; et al. (1998). "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant". Astronomical Journal. 116 (3): 1009–38. arXiv:astro-ph/9805201. Bibcode:1998AJ....116.1009R. doi:10.1086/300499. S2CID 15640044.
Roßnagel, J.; Abah, O.; Schmidt-Kaler, F.; Singer, K.; Lutz, E. (2014). "Nanoscale Heat Engine Beyond the Carnot Limit". Physical Review Letters. 112 (3): 030602. arXiv:1308.5935. Bibcode:2014PhRvL.112c0602R. doi:10.1103/PhysRevLett.112.030602. PMID 24484127. S2CID 1826585.
Rugh, S. E.; Zinkernagel, H. (2002). "The Quantum Vacuum and the Cosmological Constant Problem". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 33 (4): 663–705. arXiv:hep-th/0012253. Bibcode:2002SHPMP..33..663R. doi:10.1016/S1355-2198(02)00033-3. ISSN 1355-2198. S2CID 9007190.
Saunders, Simon; Brown, Harvey R., eds. (1991). The Philosophy of Vacuum. Oxford: Oxford University Press. ISBN 978-0198244493. LCCN 90048906. OCLC 774073198.
Schwinger, J. (1998a). Particles, Sources, and Fields: Volume I. Reading, Massachusetts: Advanced Book Program, Perseus Books. ISBN 978-0-7382-0053-8. LCCN 98087896. OCLC 40544377.
Schwinger, J. (1998b). Particles, Sources, and Fields: Volume II. Reading, Massachusetts: Advanced Book Program, Perseus Books. ISBN 978-0-7382-0054-5. LCCN 98087896. OCLC 40544377.
Schwinger, J. (1998c). Particles, Sources, and Fields: Volume III. Reading, Massachusetts: Advanced Book Program, Perseus Books. ISBN 978-0-7382-0055-2. LCCN 98087896. OCLC 40544377.
Sciama, D. W. (1991). "The Physical Significance of the Vacuum State of a Quantum Field". In Saunders, Simon; Brown, Harvey R. (eds.). The Philosophy of Vacuum. Oxford: Oxford University Press. ISBN 978-0198244493. LCCN 90048906. OCLC 774073198.
Scott, Alwyn (2006). Encyclopedia of Nonlinear Science. Routledge. ISBN 978-1-57958-385-9. OCLC 937249213.
Scully, M. O.; Zubairy, M. S. (1997). Quantum optics. Cambridge UK: Cambridge University Press. ISBN 978-0-521-43595-6. OCLC 444869786.
Scully, M. O. (2001). "Extracting Work from a Single Thermal Bath via Quantum Negentropy". Physical Review Letters. 87 (22). 220601. Bibcode:2001PhRvL..87v0601S. doi:10.1103/PhysRevLett.87.220601. PMID 11736390.
Scully, M. O.; Zubairy, M. S.; Agarwal, G. S.; Walther, H. (2003). "Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence". Science. 299 (5608): 862–863. Bibcode:2003Sci...299..862S. doi:10.1126/science.1078955. PMID 12511655. S2CID 120884236.
Urban, M.; Couchot, F.; Sarazin, X.; Djannati-Atai, A. (2013). "The Quantum Vacuum as the Origin of the Speed of Light". The European Physical Journal D. 67 (3): 58. arXiv:1302.6165. Bibcode:2013EPJD...67...58U. doi:10.1140/epjd/e2013-30578-7. ISSN 1434-6060. S2CID 15753833.
Weinberg, S. (1989). "The Cosmological Constant Problem" (PDF). Reviews of Modern Physics. 61 (1): 1–23. Bibcode:1989RvMP...61....1W. doi:10.1103/RevModPhys.61.1. hdl:2152/61094. ISSN 0034-6861. S2CID 122259372.
Weinberg, S. (2015). Lectures on Quantum Mechanics (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-107-11166-0. LCCN 2015021123. OCLC 910664598.
Weisskopf, V. (1936). "Über die Elektrodynamik des Vakuums auf Grund des Quantentheorie des Elektrons" [On the Elecrodynamics of the Vacuum on the Basis of the Quantum Theory of the Electron] (PDF). Kongelige Danske Videnskabernes Selskab, Mathematisk-fysiske Meddelelse. 24 (6): 3–39.
White, H.; March, P.; Lawrence, J.; Vera, J.; Sylvester, A.; Brady, D.; Bailey, P. (2016). "Measurement of Impulsive Thrust from a Closed Radio-Frequency Cavity in Vacuum". Journal of Propulsion and Power. 33 (4): 830–841. doi:10.2514/1.B36120. hdl:2060/20170000277. S2CID 126303009.
White, H.; March, P.; Williams, N.; O'Neill, W. (5 December 2011). Eagleworks Laboratories: Advanced Propulsion Physics Research (PDF). JANNAF Joint Propulsion Meeting; 5–9 December 2011; Huntsville, AL. Retrieved 24 October 2016.
Wilson, C. M.; Johansson, G.; Pourkabirian, A.; Simoen, M.; Johansson, J. R.; Duty, T.; Nori, F.; Delsing, P. (2011). "Observation of the Dynamical Casimir Effect in a Superconducting Circuit". Nature. 479 (7373): 376–379. arXiv:1105.4714. Bibcode:2011Natur.479..376W. doi:10.1038/nature10561. ISSN 0028-0836. PMID 22094697. S2CID 219735.
Woods, R. C. (2005). "Manipulation of gravitational waves for communications applications using superconductors". Physica C: Superconductivity. 433 (1–2): 101–107. Bibcode:2005PhyC..433..101W. doi:10.1016/j.physc.2005.10.003.
Woods, R. C.; Cooke, S. G.; Helme, J.; Caldwell, C. H. (2001). "Gravity modification by high-temperature superconductors". AIAA 37th Joint Propulsion Conference and Exhibit. doi:10.2514/6.2001-3363.

External links

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v t e Major branches of physics
Divisions	Pure Applied Engineering
Approaches	Experimental Theoretical Computational
Classical	Classical mechanics Newtonian Analytical Celestial Continuum Acoustics Classical electromagnetism Classical optics Ray Wave Thermodynamics Statistical Non-equilibrium
Modern	Relativistic mechanics Special General Nuclear physics Quantum mechanics Particle physics Atomic, molecular, and optical physics Atomic Molecular Modern optics Condensed matter physics
Interdisciplinary	Astrophysics Atmospheric physics Biophysics Chemical physics Geophysics Materials science Mathematical physics Medical physics Ocean physics Quantum information science
Related	History of physics Nobel Prize in Physics Philosophy of physics Physics education Timeline of physics discoveries

v t e Quantum mechanics
Background	Introduction History Timeline Classical mechanics Old quantum theory Glossary
Fundamentals	Born rule Bra–ket notation Complementarity Density matrix Energy level Ground state Excited state Degenerate levels Zero-point energy Entanglement Hamiltonian Interference Decoherence Measurement Nonlocality Quantum state Superposition Tunnelling Scattering theory Symmetry in quantum mechanics Uncertainty Wave function Collapse Wave–particle duality
Formulations	Formulations Heisenberg Interaction Matrix mechanics Schrödinger Path integral formulation Phase space
Equations	Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations	Bayesian Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Local Superdeterminism Many-worlds Objective collapse Quantum logic Relational Transactional Von Neumann–Wigner
Experiments	Bell test Davisson–Germer Delayed-choice quantum eraser Double-slit Franck–Hertz Mach–Zehnder interferometer Elitzur–Vaidman Popper Quantum eraser Stern–Gerlach Wheeler's delayed choice
Science	Quantum biology Quantum chemistry Quantum chaos Quantum cosmology Quantum differential calculus Quantum dynamics Quantum geometry Quantum measurement problem Quantum mind Quantum stochastic calculus Quantum spacetime
Technology	Quantum algorithms Quantum amplifier Quantum bus Quantum cellular automata Quantum finite automata Quantum channel Quantum circuit Quantum complexity theory Quantum computing Timeline Quantum cryptography Quantum electronics Quantum error correction Quantum imaging Quantum image processing Quantum information Quantum key distribution Quantum logic Quantum logic gates Quantum machine Quantum machine learning Quantum metamaterial Quantum metrology Quantum network Quantum neural network Quantum optics Quantum programming Quantum sensing Quantum simulator Quantum teleportation
Extensions	Quantum fluctuation Casimir effect Quantum statistical mechanics Quantum field theory History Quantum gravity Relativistic quantum mechanics
Related	Schrödinger's cat in popular culture Wigner's friend EPR paradox Quantum mysticism
Category

v t e Quantum electrodynamics
Formalism	Euler–Heisenberg Lagrangian Feynman diagram Gupta–Bleuler formalism Path integral formulation
Particles	Dual photon Electron Faddeev–Popov ghost Photon Positron Positronium Virtual particles
Concepts	Anomalous magnetic dipole moment Furry's theorem Klein–Nishina formula Landau pole QED vacuum Self-energy Schwinger limit Uehling potential Vacuum polarization Vertex function Ward–Takahashi identity
Processes	Bhabha scattering Breit–Wheeler process Bremsstrahlung Compton scattering Delbrück scattering Lamb shift Møller scattering Schwinger effect Photon-photon scattering
See also: Template:Quantum mechanics topics

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