Zero-point energy

For related articles, seeQuantum vacuum (disambiguation).

Not to be confused withZero Point (photometry).

For other uses, seeZero point (disambiguation).

Zero-point energy(ZPE) is the lowest possibleenergythat aquantum mechanicalsystem may have. Unlike inclassical mechanics,quantum systems constantlyfluctuatein their lowest energy state as described by theHeisenberg uncertainty principle.Therefore, even atabsolute zero,atoms and molecules retain some vibrational motion. Apart fromatomsandmolecules,the empty space ofthe vacuumalso has these properties. According toquantum field theory,the universe can be thought of not as isolated particles but continuous fluctuatingfields:matterfields, whosequantaarefermions(i.e.,leptonsandquarks), andforce fields,whose quanta arebosons(e.g.,photonsandgluons). All these fields have zero-point energy.These fluctuating zero-point fields lead to a kind of reintroduction of anaetherin physicssince some systems can detect the existence of this energy.However, this aether cannot be thought of as a physical medium if it is to beLorentz invariantsuch that there is no contradiction withEinstein'stheory ofspecial relativity.

Liquid heliumretainskinetic energyand does not freeze regardless of temperature at standard atmospheric pressure due to zero-point energy. When cooled below itsLambda point,it exhibits properties ofsuperfluidity.

The notion of a zero-point energy is also important forcosmology,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.Yet according to Einstein's theory ofgeneral relativity,any such energy would gravitate, and the experimental evidence from theexpansion of the universe,dark energyand theCasimir effectshows any such energy to be exceptionally weak. One proposal that attempts to address this issue is to say that thefermion fieldhas a negative zero-point energy, while theboson fieldhas positive zero-point energy and thus these energies somehow cancel out each other.This idea would be true ifsupersymmetrywere an exactsymmetry of nature;however, theLarge Hadron CollideratCERNhas so far found no evidence to support it. Moreover, it is known that if supersymmetry is valid at all, it is at most abroken 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.This discrepancy is known as thecosmological constant problemand it is one of the greatestunsolved mysteries in physics.Many physicists believe that "the vacuum holds the key to a full understanding of nature".

Etymology and terminology

The termzero-point energy (ZPE)is a translationfrom the GermanNullpunktsenergie.Sometimes used interchangeably with it are the termszero-point radiationandground state energy.The termzero-point field(ZPF) can be used when referring to a specific vacuum field, for instance theQED vacuumwhich specifically deals withquantum electrodynamics(e.g., electromagnetic interactions between photons, electrons and the vacuum) or theQCD vacuumwhich deals withquantum chromodynamics(e.g.,color chargeinteractions between quarks, gluons and the vacuum). A vacuum can be viewed not as empty space but as the combination of all zero-point fields. Inquantum field theorythis combination of fields is called the vacuum state, its associated zero-point energy is called thevacuum energyand the average energy value is called thevacuum expectation value(VEV) also called itscondensate.

Overview

Kinetic energy vs temperature

Inclassical mechanicsallparticlescan be thought of as having someenergymade up of theirpotential energyandkinetic energy.Temperature,for example, arises from the intensity of random particle motion caused by kinetic energy (known asBrownian motion). As temperature is reduced toabsolute 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 theuncertainty principleofquantum mechanics.

Zero-point radiation continually imparts random impulses on anelectron,so that it never comes to a complete stop. Zero-point radiation gives theoscillatoran average energy equal to the frequency of oscillation multiplied by one-half of thePlanck constant.

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 itsHamiltonianwhich also describes the system as a harmonic oscillator, orwave function,that fluctuates between various energy states (seewave-particle duality). All quantum mechanical systems undergo fluctuations even in their ground state, a consequence of theirwave-like nature. The uncertainty principle requires every quantum mechanical system to have a fluctuating zero-point energy greater than the minimum of its classicalpotential well.This results in motion even at absolute zero. For example,liquid heliumdoes not freeze under atmospheric pressure regardless of temperature due to its zero-point energy.

Given the equivalence of mass and energy expressed byAlbert Einstein'sE=mc²,any point inspacethat contains energy can be thought of as having mass to create particles. 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 aquantum harmonic oscillator.According to QFT the universe is made up of matter fields, whose quanta arefermions(i.e.leptonsand quarks), and force fields, whose quanta arebosons(e.g.photonsandgluons). All these fields have zero-point energy.Recent experiments support the idea that particles themselves can be thought of as excited states of the underlyingquantum vacuum,and that all properties of matter are merely vacuum fluctuations arising from interactions of the zero-point field.

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.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 ofgeneral relativitythe absolute energy value of space is not an arbitrary constant and gives rise to thecosmological 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 theBig 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;galaxiesand 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 ofdark energyis 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 – thecosmological constant problem.

Many physical effects attributed to zero-point energy have been experimentally verified, such asspontaneous emission,Casimir force,Lamb shift,magnetic moment of the electronandDelbrück scattering.These effects are usually called "radiative corrections".In more complex nonlinear theories (e.g. QCD) zero-point energy can give rise to a variety of complex phenomena such asmultiple stable states,symmetry breaking,chaosandemergence.Active areas of research include the effects of virtual particles,quantum entanglement,the difference (if any) betweeninertial and gravitational mass,variation in thespeed of light,a reason for the observed value of thecosmological constantand the nature of dark energy.

History

Early aether theories

James Clerk Maxwell

Zero-point energy evolved from historical ideas about thevacuum.ToAristotlethe vacuum wasτὸ κενόν,"the empty"; i.e., space independent of body. He believed this concept violated basic physical principles and asserted thatthe elementsoffire,air,earth,andwaterwere not made of atoms, but were continuous. To theatomiststhe concept of emptiness had absolute character: it was the distinction between existence and nonexistence.Debate about the characteristics of the vacuum were largely confined to the realm ofphilosophy,it was not until much later on with the beginning ofthe renaissance,thatOtto von Guerickeinvented 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.

Late in the 19th century, however, it became apparent that the evacuated region still containedthermal radiation.The existence of theaetheras a substitute for a true void was the most prevalent theory of the time. According to the successfulelectromagneticaether theory based uponMaxwell'selectrodynamics,this all-encompassing aether was endowed with energy and hence very different from nothingness. The fact that electromagnetic and gravitational phenomena were transmitted in empty space was considered evidence that their associated aethers were part of the fabric of space itself. However Maxwell noted that for the most part these aethers weread hoc:

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.

Moreever, the results of theMichelson–Morley experimentin 1887 were the first strong evidence that the then-prevalent aether theories were seriously flawed, and initiated a line of research that eventually led tospecial 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

Planck in 1918, the year he received theNobel Prize in Physicsfor his work onquantum theory

In 1900,Max Planckderived the average energyεof a singleenergy radiator,e.g., a vibrating atomic unit, as a function of absolute temperature: $$\varepsilon ={\frac {h\nu }{e^{h\nu /(kT)}-1}}\,,$$ wherehis thePlanck constant,νis thefrequency,kis theBoltzmann constant,andTis the absolutetemperature.The zero-point energy makes no contribution to Planck's original law, as its existence was unknown to Planck in 1900.

The concept of zero-point energy was developed byMax Planckin Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900.

In 1912, Max Planck published the first journal article to describe the discontinuous emission of radiation, based on the discrete quanta of energy.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 ofhν.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, one⁠hν/2⁠,as an additional term dependent on the frequencyν,which was greater than zero (wherehis the Planck constant). It is therefore widely agreed that "Planck's equation marked the birth of the concept of zero-point energy."In a series of papers from 1911 to 1913,Planck found the average energy of an oscillator to be: $$\varepsilon ={\frac {h\nu }{2}}+{\frac {h\nu }{e^{h\nu /(kT)}-1}}~.$$

Einstein's official 1921 portrait after receiving the Nobel Prize in Physics

Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistantOtto Stern.In 1913 they published a paper that attempted to prove the existence of zero-point energy by calculating thespecific heatof 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 toPaul Ehrenfestof the same year Einstein declared zero-point energy "dead as a doornail".Zero-point energy was also invoked byPeter Debye,who noted that zero-point energy of the atoms of acrystal latticewould cause a reduction in the intensity of the diffracted radiation inX-ray diffractioneven as the temperature approached absolute zero. In 1916Walther Nernstproposed that empty space was filled with zero-pointelectromagnetic radiation.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.

Heisenberg, 1924

Kurt BennewitzandFrancis Simon(1923),who worked atWalther Nernst's laboratory in Berlin, studied the melting process of chemicals at low temperatures. Their calculations of the melting points ofhydrogen,argonandmercuryled them to conclude that the results provided evidence for a zero-point energy. Moreover, they suggested correctly, as was later verified by Simon (1934),that this quantity was responsible for the difficulty in solidifying helium even at absolute zero. In 1924Robert Mullikenprovided direct evidence for the zero-point energy of molecular vibrations by comparing the band spectrum of¹⁰BO and¹¹BO: 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,with the development ofmatrix mechanicsinWerner Heisenberg's article "Quantum theoretical re-interpretation of kinematic and mechanical relations"the zero-point energy was derived from quantum mechanics.

In 1913Niels Bohrhad proposed what is now called theBohr modelof the atom,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 byJames Hopwood Jeansin 1915: "There would be a very real difficulty in supposing that the (force) law⁠1/r²⁠held down to the zero values ofr.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. "The resolution to this puzzle came in 1926 whenErwin Schrödingerintroduced theSchrödinger equation.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.

Quantum field theory and beyond

In 1926,Pascual Jordanpublished the first attempt to quantize the electromagnetic field. In a joint paper withMax BornandWerner Heisenberghe 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.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".Jordan found a way to get rid of the infinite term, publishing a joint work with Pauli in 1928,performing what has been called "the first infinite subtraction, or renormalisation, in quantum field theory".

Paul Dirac, 1933

Building on the work of Heisenberg and others,Paul Dirac's theory of emission and absorption (1927)was the first application of the quantum theory of radiation. Dirac's work was seen as crucially important to the emerging field of quantum mechanics; it dealt directly with the process in which "particles" are actually created:spontaneous emission.Dirac described the quantization of theelectromagnetic fieldas an ensemble ofharmonic oscillatorswith the introduction of the concept ofcreation and annihilation operatorsof particles. The theory showed that spontaneous emission depends upon the zero-point energy fluctuations of the electromagnetic field in order to get started.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:

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 byVictor Weisskopfwho in 1935 wrote:

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 byTheodore Welton(1948),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 coinedquantum electrodynamics(QED), predicted a fluctuating zero-point or "vacuum" field existing even in the absence of sources.

Throughout the 1940s improvements inmicrowavetechnology made it possible to take more precise measurements of the shift of the levels of ahydrogen atom,now known as the Lamb shift,and measurement of themagnetic momentof the electron.Discrepancies between these experiments and Dirac's theory led to the idea of incorporatingrenormalisationinto QED to deal with zero-point infinities. Renormalization was originally developed byHans Kramersand alsoVictor Weisskopf(1936),and first successfully applied to calculate a finite value for the Lamb shift byHans Bethe(1947).As per spontaneous emission, these effects can in part be understood with interactions with the zero-point field.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. InWolfgang Pauli's 1945Nobel lecturehe made clear his opposition to the idea of zero-point energy stating "It is clear that this zero-point energy has no physical reality".

Hendrik Casimir (1958)

In 1948Hendrik Casimirshowed 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 ofcolloidal 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 byVan 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 byFritz Londonin 1930,did not properly explain the experimental measurements on colloids. Overbeek therefore asked Casimir to investigate the problem. Working withDirk 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.Soon afterwards after a conversation withBohrabout zero-point energy, Casimir noticed that this result could be interpreted in terms of vacuum fluctuations. He then asked himself what would happen if there were two mirrors – rather than two molecules – facing each other in a vacuum. It was this work that led to his prediction of an attractive force between reflecting plates. The work by Casimir and Polder opened up the way to a unified theory of van der Waals and Casimir forces and a smooth continuum between the two phenomena. This was done by Lifshitz (1956)in the case of plane paralleldielectric 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.The role of relativistic forces becomes dominant at orders of a hundred nanometers.

In 1951Herbert Callenand Theodore Weltonproved the quantumfluctuation-dissipation theorem(FDT) which was originally formulated in classical form byNyquist(1928)as an explanation for observedJohnson noisein electric circuits.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.FDT has been shown to be true experimentally under certain quantum, non-classical, conditions.

In 1963 theJaynes–Cummings modelwas developed describing the system of atwo-level atominteracting 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.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 tocavity quantum electrodynamics(CQED), the study of effects of mirrors and cavities on radiative corrections. Spontaneous emission can be suppressed (or "inhibited" )or amplified. Amplification was first predicted by Purcell in 1946(thePurcell effect) and has been experimentally verified.This phenomenon can be understood, partly, in terms of the action of the vacuum field on the atom.

Uncertainty principle

Main article:Uncertainty principle

Zero-point energy is fundamentally related to the Heisenberg uncertainty principle.Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position andmomentum,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 apotential well,theHamiltonianof a general system (the quantum-mechanicaloperatorgiving its energy) can be approximated as aquantum harmonic oscillator, $${\hat {H}}=V_{0}+{\tfrac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}+{\frac {1}{2m}}{\hat {p}}^{2}\,,$$ whereV₀is the minimum of the classical potential well.

The uncertainty principle tells us that $${\sqrt {\left\langle \left({\hat {x}}-x_{0}\right)^{2}\right\rangle }}{\sqrt {\left\langle {\hat {p}}^{2}\right\rangle }}\geq {\frac {\hbar }{2}}\,,$$ making theexpectation valuesof the kinetic andpotentialterms above satisfy $$\left\langle {\tfrac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}\right\rangle \left\langle {\frac {1}{2m}}{\hat {p}}^{2}\right\rangle \geq \left({\frac {\hbar }{4}}\right)^{2}{\frac {k}{m}}\,.$$

The expectation value of the energy must therefore be at least

$$\left\langle {\hat {H}}\right\rangle \geq V_{0}+{\frac {\hbar }{2}}{\sqrt {\frac {k}{m}}}=V_{0}+{\frac {\hbar \ Omega }{2}}$$

whereω=√k/mis theangular frequencyat which the system oscillates.

A more thorough treatment, showing that the energy of the ground state actually saturates this bound and is exactlyE₀=V₀+⁠ħω/2⁠,requires solving for the ground state of the system.

Atomic physics

Main article:Ground state

The zero-point energyE=⁠ħω/2⁠causes 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 theground stateof the system. The professional physics literature tends to measure frequency, as denoted byνabove, usingangular frequency,denoted withωand defined byω= 2πν.This leads to a convention of writing the Planck constanthwith a bar through its top (ħ) to denote the quantity⁠h/2π⁠.In these terms, an example of zero-point energy is the aboveE=⁠ħω/2⁠associated with the ground state of the quantum harmonic oscillator. In quantum mechanical terms, the zero-point energy is theexpectation valueof the Hamiltonian of the system in the ground state.

If more than one ground state exists, they are said to bedegenerate.Many systems have degenerate ground states. Degeneracy occurs whenever there exists aunitary operatorwhich acts non-trivially on a ground state andcommuteswith the Hamiltonian of the system.

According to thethird law of thermodynamics,a system at absolute zero temperature exists in its ground state; thus, itsentropyis determined by the degeneracy of the ground state. Many systems, such as a perfectcrystal 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 exhibitnegative temperature.

Thewave functionof the ground state of aparticle in a one-dimensional wellis a half-periodsine wavewhich goes to zero at the two edges of the well. The energy of the particle is given by: $${\frac {h^{2}n^{2}}{8mL^{2}}}$$ wherehis thePlanck constant,mis the mass of the particle,nis the energy state (n= 1corresponds to the ground-state energy), andLis the width of the well.

Quantum field theory

Main articles:Vacuum expectation value,Vacuum energy,andVacuum state

In quantum field theory (QFT), the fabric of "empty" space is visualized as consisting offields,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 whosequantaare fermions (e.g.electronsand quarks), force fields whose quanta are bosons (i.e. photons and gluons) and a Higgs field whose quantum is theHiggs boson.The matter and force fields have zero-point energy.A related term iszero-point field(ZPF), which is the lowest energy state of a particular field.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 withquantum chromodynamics(e.g.color chargeinteractions between quarks, gluons and the vacuum). Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlyingquantum vacuum,and that all properties of matter are merely vacuum fluctuations arising from interactions with the zero-point field.

Each point in space makes a contribution ofE=⁠ħω/2⁠,resulting in a calculation of infinite zero-point energy in any finite volume; this is one reasonrenormalizationis needed to make sense of quantum field theories. Incosmology,the vacuum energy is one possible explanation for the cosmological constantand the source of dark energy.

Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large asPaul Diracclaimed it is, like asea of energy.Other scientists specializing inGeneral Relativityrequire the energy to be small enough forcurvature of spaceto agree with observedastronomy.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 avirtual energypotentialof positive and negative energy.

In quantumperturbation theory,it is sometimes said that the contribution ofone-loopand multi-loopFeynman diagramstoelementary particlepropagatorsare the contribution ofvacuum fluctuations,or the zero-point energy to the particlemasses.

Quantum electrodynamic vacuum

Main article:QED vacuum

The oldest and best known quantized force field is theelectromagnetic field.Maxwell's equationshave 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 allquantum field theories.

Redefining the zero of energy

In the quantum theory of the electromagnetic field, classical wave amplitudesαandα*are replaced by operatorsaanda^†that satisfy: $$\left[a,a^{\dagger }\right]=1$$

The classical quantity|α|²appearing in the classical expression for the energy of a field mode is replaced in quantum theory by the photon number operatora^†a.The fact that: $$\left[a,a^{\dagger }a\right]\neq 1$$ 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 fora^†aanda.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^†andaassociated with these classical modes.

The zero-point energy of the field arises formally from the non-commutativity ofaanda^†.This is true for any harmonic oscillator: the zero-point energy⁠ħω/2⁠appears when we write the Hamiltonian: $${\begin{aligned}H_{cl}&={\frac {p^{2}}{2m}}+{\tfrac {1}{2}}m\ Omega ^{2}{q}^{2}\\&={\tfrac {1}{2}}\hbar \ Omega \left(aa^{\dagger }+a^{\dagger }a\right)\\&=\hbar \ Omega \left(a^{\dagger }a+{\tfrac {1}{2}}\right)\end{aligned}}$$

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: $${\begin{aligned}H_{F}-\left\langle 0|H_{F}|0\right\rangle &={\tfrac {1}{2}}\hbar \ Omega \left(aa^{\dagger }+a^{\dagger }a\right)-{\tfrac {1}{2}}\hbar \ Omega \\&=\hbar \ Omega \left(a^{\dagger }a+{\tfrac {1}{2}}\right)-{\tfrac {1}{2}}\hbar \ Omega \\&=\hbar \ Omega a^{\dagger }a\end{aligned}}$$ without affecting any physical predictions of the theory. The new Hamiltonian is said to benormally ordered(or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted:H_F,i.e.: $${\displaystyle:H_{F}:\equiv \hbar \ Omega \left(aa^{\dagger }+a^{\dagger }a\right):\equiv \hbar \ Omega a^{\dagger }a}$$

In other words, within the normal ordering symbol we can commuteaanda^†.Since zero-point energy is intimately connected to the non-commutativity ofaanda^†,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 withaanda^†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.

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: $${\begin{aligned}H_{F}&={\frac {1}{8\pi }}\int d^{3}r\left(\mathbf {E} ^{2}+\mathbf {B} ^{2}\right)\\&={\frac {k^{2}}{2\pi }}|\ Alpha (t)|^{2}\end{aligned}}$$

We introduce the "mode function"A₀(r)that satisfies theHelmholtz equation: $$\left(\nabla ^{2}+k^{2}\right)\mathbf {A} _{0}(\mathbf {r} )=0$$ wherek=⁠ω/c⁠and assume it is normalized such that: $$\int d^{3}r\left|\mathbf {A} _{0}(\mathbf {r} )\right|^{2}=1$$

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|A₀(r)|²should be independent ofrfor each mode of the field. The mode function satisfying these conditions is: $$\mathbf {A} _{0}(\mathbf {r} )=e_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {r} }$$ wherek·e_k= 0in order to have the transversality condition∇·A(r,t)satisfied for the Coulomb gaugein which we are working.

To achieve the desired normalization we pretend space is divided into cubes of volumeV=L³and impose on the field the periodic boundary condition: $$\mathbf {A} (x+L,y+L,z+L,t)=\mathbf {A} (x,y,z,t)$$ or equivalently $$\left(k_{x},k_{y},k_{z}\right)={\frac {2\pi }{L}}\left(n_{x},n_{y},n_{z}\right)$$ wherencan assume any integer value. This allows us to consider the field in any one of the imaginary cubes and to define the mode function: $$\mathbf {A} _{\mathbf {k} }(\mathbf {r} )={\frac {1}{\sqrt {V}}}e_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {r} }$$ which satisfies the Helmholtz equation, transversality, and the "box normalization": $$\int _{V}d^{3}r\left|\mathbf {A} _{\mathbf {k} }(\mathbf {r} )\right|^{2}=1$$ wheree_kis chosen to be a unit vector which specifies the polarization of the field mode. The conditionk·e_k= 0means that there are two independent choices ofe_k,which we calle_k1ande_k2wheree_k1·e_k2= 0ande²
_k1=e²
_k2= 1.Thus we define the mode functions: $$\mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} )={\frac {1}{\sqrt {V}}}e_{\mathbf {k} \lambda }e^{i\mathbf {k} \cdot \mathbf {r} }\,,\quad \lambda ={\begin{cases}1\\2\end{cases}}$$ in terms of which the vector potential becomes: $$\mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r},t)={\sqrt {\frac {2\pi \hbar c^{2}}{\ Omega _{k}V}}}\left[a_{\mathbf {k} \lambda }(0)e^{i\mathbf {k} \cdot \mathbf {r} }+a_{\mathbf {k} \lambda }^{\dagger }(0)e^{-i\mathbf {k} \cdot \mathbf {r} }\right]e_{\mathbf {k} \lambda }$$ or: $$\mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r},t)={\sqrt {\frac {2\pi \hbar c^{2}}{\ Omega _{k}V}}}\left[a_{\mathbf {k} \lambda }(0)e^{-i(\ Omega _{k}t-\mathbf {k} \cdot \mathbf {r} )}+a_{\mathbf {k} \lambda }^{\dagger }(0)e^{i(\ Omega _{k}t-\mathbf {k} \cdot \mathbf {r} )}\right]$$ whereω_k=kcanda_kλ,a^†
_kλare photon annihilation and creation operators for the mode with wave vectorkand polarizationλ.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: $$\mathbf {A} (\mathbf {r} t)=\sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar c^{2}}{\ Omega _{k}V}}}\left[a_{\mathbf {k} \lambda }(0)e^{i\mathbf {k} \cdot \mathbf {r} }+a_{\mathbf {k} \lambda }^{\dagger }(0)e^{-i\mathbf {k} \cdot \mathbf {r} }\right]e_{\mathbf {k} \lambda }$$ for the total vector potential in free space. Using the fact that: $$\int _{V}d^{3}r\mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} )\cdot \mathbf {A} _{\mathbf {k} '\lambda '}^{\ast }(\mathbf {r} )=\delta _{\mathbf {k},\mathbf {k} '}^{3}\delta _{\lambda,\lambda '}$$ we find the field Hamiltonian is: $$H_{F}=\sum _{\mathbf {k} \lambda }\hbar \ Omega _{k}\left(a_{\mathbf {k} \lambda }^{\dagger }a_{\mathbf {k} \lambda }+{\tfrac {1}{2}}\right)$$

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: $${\begin{aligned}\left[a_{\mathbf {k} \lambda }(t),a_{\mathbf {k} '\lambda '}^{\dagger }(t)\right]&=\delta _{\mathbf {k},\mathbf {k} '}^{3}\delta _{\lambda,\lambda '}\\[10px]\left[a_{\mathbf {k} \lambda }(t),a_{\mathbf {k} '\lambda '}(t)\right]&=\left[a_{\mathbf {k} \lambda }^{\dagger }(t),a_{\mathbf {k} '\lambda '}^{\dagger }(t)\right]=0\end{aligned}}$$

Clearly the least eigenvalue forH_Fis: $$\sum _{\mathbf {k} \lambda }{\tfrac {1}{2}}\hbar \ Omega _{k}$$

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 $${\frac {8\pi v^{2}dv}{c^{3}}}V$$ shows. The summation becomes approximately the integral: $${\frac {4\pi hV}{c^{3}}}\int v^{3}\,dv$$ for high values ofv.It diverges proportional tov⁴for largev.

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 volumeVis 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"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.

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 whichn_kλ= 0for 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.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 byΣ_kλ⁠ħω_k/2⁠is infinite. We can make the replacement: $$\sum _{\mathbf {k} \lambda }\longrightarrow \sum _{\lambda }\left({\frac {1}{2\pi }}\right)^{3}\int d^{3}k={\frac {V}{8\pi ^{3}}}\sum _{\lambda }\int d^{3}k$$ the zero-point energy density is: $${\begin{aligned}{\frac {1}{V}}\sum _{\mathbf {k} \lambda }{\tfrac {1}{2}}\hbar \ Omega _{k}&={\frac {2}{8\pi ^{3}}}\int d^{3}k{\tfrac {1}{2}}\hbar \ Omega _{k}\\&={\frac {4\pi }{4\pi ^{3}}}\int dk\,k^{2}\left({\tfrac {1}{2}}\hbar \ Omega _{k}\right)\\&={\frac {\hbar }{2\pi ^{2}c^{3}}}\int d\ Omega \,\ Omega ^{3}\end{aligned}}$$ or in other words the spectral energy density of the vacuum field: $$\rho _{0}(\ Omega )={\frac {\hbar \ Omega ^{3}}{2\pi ^{2}c^{3}}}$$

The zero-point energy density in the frequency range fromω₁toω₂is therefore: $$\int _{\ Omega _{1}}^{\ Omega _{2}}d\ Omega \rho _{0}(\ Omega )={\frac {\hbar }{8\pi ^{2}c^{3}}}\left(\ Omega _{2}^{4}-\ Omega _{1}^{4}\right)$$

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 220erg/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: $$H={\frac {1}{2m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)^{2}+{\tfrac {1}{2}}m\ Omega _{0}^{2}\mathbf {x} ^{2}+H_{F}$$

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 coordinatexand the canonical momentump=mẋ+⁠eA/c⁠of the oscillator are: $${\begin{aligned}\mathbf {\dot {x}} &=(i\hbar )^{-1}[\mathbf {x}.H]={\frac {1}{m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\\\mathbf {\dot {p}} &=(i\hbar )^{-1}[\mathbf {p}.H]{\begin{aligned}&={\tfrac {1}{2}}\nabla \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)^{2}-m\ Omega _{0}^{2}\mathbf {\dot {x}} \\&=-{\frac {1}{m}}\left[\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\cdot \nabla \right]\left[-{\frac {e}{c}}\mathbf {A} \right]-{\frac {1}{m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\times \nabla \times \left[-{\frac {e}{c}}\mathbf {A} \right]-m\ Omega _{0}^{2}\mathbf {\dot {x}} \\&={\frac {e}{c}}(\mathbf {\dot {x}} \cdot \nabla )\mathbf {A} +{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\ Omega _{0}^{2}\mathbf {\dot {x}} \end{aligned}}\end{aligned}}$$ or: $${\begin{aligned}m\mathbf {\ddot {x}} &=\mathbf {\dot {p}} -{\frac {e}{c}}\mathbf {\dot {A}} \\&=-{\frac {e}{c}}\left[\mathbf {\dot {A}} -\left(\mathbf {\dot {x}} \cdot \nabla \right)\mathbf {A} \right]+{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\ Omega _{0}^{2}\mathbf {x} \\&=e\mathbf {E} +{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\ Omega _{0}^{2}\mathbf {x} \end{aligned}}$$ since the rate of change of the vector potential in the frame of the moving charge is given by the convective derivative $$\mathbf {\dot {A}} ={\frac {\partial \mathbf {A} }{\partial t}}+(\mathbf {\dot {x}} \cdot \nabla )\mathbf {A} ^{3}\,.$$

For nonrelativistic motion we may neglect the magnetic force and replace the expression formẍby: $${\begin{aligned}\mathbf {\ddot {x}} +\ Omega _{0}^{2}\mathbf {x} &\approx {\frac {e}{m}}\mathbf {E} \\&\approx \sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar \ Omega _{k}}{V}}}\left[a_{\mathbf {k} \lambda }(t)+a_{\mathbf {k} \lambda }^{\dagger }(t)\right]e_{\mathbf {k} \lambda }\end{aligned}}$$

Above we have made the electric dipole approximation in which the spatial dependence of the field is neglected. The Heisenberg equation fora_kλis found similarly from the Hamiltonian to be: $${\dot {a}}_{\mathbf {k} \lambda }=i\ Omega _{k}a_{\mathbf {k} \lambda }+ie{\sqrt {\frac {2\pi }{\hbar \ Omega _{k}V}}}\mathbf {\dot {x}} \cdot e_{\mathbf {k} \lambda }$$ in the electric dipole approximation.

In deriving these equations forx,p,anda_kλwe 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 operatorA(t)evolves in time asA(t) =U^†(t)A(0)U(t),whereU(t)is the time evolution operator satisfying $$i\hbar {\dot {U}}=HU\,,\quad U^{\dagger }(t)=U^{-1}(t)\,,\quad U(0)=1\,.$$

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: $$a_{\mathbf {k} \lambda }(t)=a_{\mathbf {k} \lambda }(0)e^{-i\ Omega _{k}t}+ie{\sqrt {\frac {2\pi }{\hbar \ Omega _{k}V}}}\int _{0}^{t}dt'\,e_{\mathbf {k} \lambda }\cdot \mathbf {\dot {x}} (t')e^{i\ Omega _{k}\left(t'-t\right)}$$ and therefore the equation forȧ_kλmay be written: $$\mathbf {\ddot {x}} +\ Omega _{0}^{2}\mathbf {x} ={\frac {e}{m}}\mathbf {E} _{0}(t)+{\frac {e}{m}}\mathbf {E} _{RR}(t)$$ where $$\mathbf {E} _{0}(t)=i\sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar \ Omega _{k}}{V}}}\left[a_{\mathbf {k} \lambda }(0)e^{-i\ Omega _{k}t}-a_{\mathbf {k} \lambda }^{\dagger }(0)e^{i\ Omega _{k}t}\right]e_{\mathbf {k} \lambda }$$ and $$\mathbf {E} _{RR}(t)=-{\frac {4\pi e}{V}}\sum _{\mathbf {k} \lambda }\int _{0}^{t}dt'\left[e_{\mathbf {k} \lambda }\cdot \mathbf {\dot {x}} \left(t'\right)\right]\cos \ Omega _{k}\left(t'-t\right)$$

It can be shown that in theradiation reactionfield, if the massmis regarded as the "observed" mass then we can take $$\mathbf {E} _{RR}(t)={\frac {2e}{3c^{3}}}\mathbf {\ddot {x}}$$

The total field acting on the dipole has two parts,E₀(t)andE_RR(t).E₀(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 $$\left[\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right]\mathbf {E} =0$$ satisfied by the field in the (source free) vacuum. For this reasonE₀(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 att= 0.E_RR(t)is the source field, the field generated by the dipole and acting on the dipole.

Using the above equation forE_RR(t)we obtain an equation for the Heisenberg-picture operator$\mathbf {x} (t)$that is formally the same as the classical equation for a linear dipole oscillator: $$\mathbf {\ddot {x}} +\ Omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)$$ whereτ=⁠2e²/3mc³⁠.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 fieldE₀(t).

According to our earlier equation fora_kλ(t)the free field is the only field in existence att= 0as the time at which the interaction between the dipole and the field is "switched on". The state vector of the dipole-field system att= 0is therefore of the form $$|\Psi \rangle =|{\text{vac}}\rangle |\psi _{D}\rangle \,,$$ 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: $$\langle \mathbf {E} _{0}(t)\rangle =\langle \Psi |\mathbf {E} _{0}(t)|\Psi \rangle =0$$ sincea_kλ(0)|vac⟩ = 0.however, the energy density associated with the free field is infinite: $${\begin{aligned}{\frac {1}{4\pi }}\left\langle \mathbf {E} _{0}^{2}(t)\right\rangle &={\frac {1}{4\pi }}\sum _{\mathbf {k} \lambda }\sum _{\mathbf {k'} \lambda '}{\sqrt {\frac {2\pi \hbar \ Omega _{k}}{V}}}{\sqrt {\frac {2\pi \hbar \ Omega _{k'}}{V}}}\times \left\langle a_{\mathbf {k} \lambda }(0)a_{\mathbf {k'} \lambda '}^{\dagger }(0)\right\rangle \\&={\frac {1}{4\pi }}\sum _{\mathbf {k} \lambda }\left({\frac {2\pi \hbar \ Omega _{k}}{V}}\right)\\&=\int _{0}^{\infty }dw\,\rho _{0}(\ Omega )\end{aligned}}$$

The important point of this is that the zero-point field energyH_Fdoes not affect the Heisenberg equation fora_kλsince it is a c-number or constant (i.e. an ordinary number rather than an operator) and commutes witha_kλ.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 termΣ_kλ⁠ħω_k/2⁠in 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: $${\begin{aligned}\left[z(t),p_{z}(t)\right]&=\left[U^{\dagger }(t)z(0)U(t),U^{\dagger }(t)p_{z}(0)U(t)\right]\\&=U^{\dagger }(t)\left[z(0),p_{z}(0)\right]U(t)\\&=i\hbar U^{\dagger }(t)U(t)\\&=i\hbar \end{aligned}}$$

We can calculate[z(t),p_z(t)]from the formal solution of the operator equation of motion $$\mathbf {\ddot {x}} +\ Omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)$$

Using the fact that $$\left[a_{\mathbf {k} \lambda }(0),a_{\mathbf {k'} \lambda '}^{\dagger }(0)\right]=\delta _{\mathbf {kk'} }^{3},\delta _{\lambda \lambda '}$$ and that equal-time particle and field operators commute, we obtain: $${\begin{aligned}[z(t),p_{z}(t)]&=\left[z(t),m{\dot {z}}(t)\right]+\left[z(t),{\frac {e}{c}}A_{z}(t)\right]\\&=\left[z(t),m{\dot {z}}(t)\right]\\&=\left({\frac {i\hbar e^{2}}{2\pi ^{2}mc^{3}}}\right)\left({\frac {8\pi }{3}}\right)\int _{0}^{\infty }{\frac {d\ Omega \,\ Omega ^{4}}{\left(\ Omega ^{2}-\ Omega _{0}^{2}\right)^{2}+\tau ^{2}\ Omega ^{6}}}\end{aligned}}$$

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.,τω₀≪ 1.Then the integrand above is sharply peaked atω=ω₀and: $${\begin{aligned}\left[z(t),p_{z}(t)\right]&\approx {\frac {2i\hbar e^{2}}{3\pi mc^{3}}}\ Omega _{0}^{3}\int _{-\infty }^{\infty }{\frac {dx}{x^{2}+\tau ^{2}\ Omega _{0}^{6}}}\\&=\left({\frac {2i\hbar e^{2}\ Omega _{0}^{3}}{3\pi mc^{3}}}\right)\left({\frac {\pi }{\tau \ Omega _{0}^{3}}}\right)\\&=i\hbar \end{aligned}}$$ the necessity of the vacuum field can also be appreciated by making the small damping approximation in $${\begin{aligned}&\mathbf {\ddot {x}} +\ Omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)\\&\mathbf {\ddot {x}} \approx -\ Omega _{0}^{2}\mathbf {x} (t)&&\mathbf {\overset {...}{x}} \approx -\ Omega _{0}^{2}\mathbf {\dot {x}} \end{aligned}}$$ and $$\mathbf {\ddot {x}} +\tau \ Omega _{0}^{2}\mathbf {\dot {x}} +\ Omega _{0}^{2}\mathbf {x} \approx {\frac {e}{m}}\mathbf {E} _{0}(t)$$

Without the free fieldE₀(t)in this equation the operatorx(t)would be exponentially dampened, and commutators like[z(t),p_z(t)]would approach zero fort≫⁠1/τω²
₀⁠.With the vacuum field included, however, the commutator isiħat 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.

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 ofx,the spectral energy density of the vacuum field must be proportional to the third power ofωin order for[z(t),p_z(t)]to hold. In the case of a dissipative force proportional toẋ,by contrast, the fluctuation force must be proportional to$\ Omega$in order to maintain the canonical commutation relation.This relation between the form of the dissipation and the spectral density of the fluctuation is the essence of the fluctuation-dissipation theorem.

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.

Quantum chromodynamic vacuum

Main article:QCD vacuum

The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of anon-perturbativevacuum state, characterized by a non-vanishingcondensatessuch as thegluon condensateand thequark condensatein the complete theory which includes quarks. The presence of these condensates characterizes the confined phase ofquark matter.In technical terms, gluons arevectorgauge bosonsthat mediatestrong interactionsof quarks in quantum chromodynamics (QCD). Gluons themselves carry the color charge of the strong interaction. This is unlike the photon, which mediates theelectromagnetic interactionbut 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 withnonlinear equationsto characterize such interactions.

Higgs field

Main article:Higgs mechanism

The potential for the Higgs field, plotted as function ofϕ⁰andϕ³.It has aMexican-hatorchampagne-bottle profileat the ground.

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 expectationspontaneously breakselectroweakgauge symmetrywhich 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 ofϕ⁰in the ground state (the vacuum expectation value or VEV) is then⟨ϕ⁰⟩ =⁠v/√2⁠,wherev=⁠|μ|/√λ⁠.The measured value of this parameter is approximately246 GeV/c².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 ofsuperconductivitywhich 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 theGinzburg–Landau theory).

Experimental observations

Zero-point energy has many observed physical consequences.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.Indeed, such treatment could create a problem at a deeper, as of yet undiscovered, theory.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.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.

Casimir effect

Main article:Casimir effect

Casimir forces on parallel plates

A phenomenon that is commonly presented as evidence for the existence of zero-point energy in vacuum is the Casimir effect, proposed in 1948 byDutchphysicistHendrik 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%.That changed in 1997 with Lamoreauxconclusively showing that the Casimir force was real. Results have been repeatedly replicated since then.

In 2009, Munday et al.published experimental proof that (as predicted in 1961) 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.

An interesting hypothetical side effect of the Casimir effect is theScharnhorst effect,a hypothetical phenomenon in which light signals travel slightlyfaster thancbetween two closely spaced conducting plates.

Lamb shift

Main article:Lamb shift

Fine structureof energy levels in hydrogen – relativistic corrections to theBohr model

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 twoenergy levels²S_⁠1/2⁠and²P_⁠1/2⁠(interm symbolnotation) of thehydrogen atomwhich was not predicted by theDirac 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;this effect is called the Lamb shift.The shift of about4.38×10⁻⁶eVis roughly10⁻⁷of 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.

Fine-structure constant

Main article:Fine-structure constant

Takingħ(thePlanck constantdivided by2π),c(thespeed of light), ande²=⁠q²
_e/4πε₀⁠(the electromagneticcoupling constanti.e. a measure of the strength of theelectromagnetic force(whereq_eis the absolute value of theelectronic chargeand$\varepsilon _{0}$is thevacuum permittivity)) we can form a dimensionless quantity called thefine-structure constant: $$\ Alpha ={\frac {e^{2}}{\hbar c}}={\frac {q_{e}^{2}}{4\pi \varepsilon _{0}\hbar c}}\approx {\frac {1}{137}}$$

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.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 whereħ=⁠h/2π⁠,andΔ_x,Δ_pare the standard deviations of position and momentum states that: $$\Delta _{x}\Delta _{p}\geq {\frac {1}{2}}\hbar$$

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 theZ⁰bosonrest energy,m_zc²≈90 GeV, that: $$\ Alpha \approx {\frac {1}{129}}$$ rather than the low-energyα≈⁠1/137⁠.The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for definingα.All in all,αdepends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The energy dependence ofαhas been observed for several years now in precision experiment in high-energy physics.

Vacuum birefringence

Main articles:Lorentz-violating electrodynamicsandEuler–Heisenberg Lagrangian

Light coming from the surface of a strongly magneticneutron star(left) becomes linearly polarised as it travels through the vacuum.

In the presence of strong electrostatic fields it is predicted that virtual particles become separated from the vacuum state and form real matter.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 andHans Heinrich Eulerin 1936and 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".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 theSchwinger limit.At this point the vacuum has all the properties of abirefringent medium,thus in principle a rotation of the polarization frame (theFaraday effect) can be observed in empty space.

Wide field view of the neutron starRX J1856.5-3754

Both Einstein's theory of special and general relativity state that light should pass freely through a vacuum without being altered, a principle known asLorentz 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 ofquantum gravitypredict 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.There have been a number of inconclusive results which claim to show evidence of aLorentz violationby finding a rotation of the polarization plane of light coming from distant galaxies.The first concrete evidence for vacuum birefringence was published in 2017 when a team ofastronomerslooked at the light coming from the starRX J1856.5-3754,the closest discoveredneutron startoEarth.

Roberto Mignani at theNational Institute for AstrophysicsinMilanwho led the team ofastronomershas 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."The team found that visible light from the star had undergone linear polarisationof around 16%. If the birefringence had been caused by light passing throughinterstellar gasor 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. AtX-raywavelengths the polarization from the quantum fluctuations should be near 100%.Although notelescopecurrently 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'sHard X-ray Modulation Telescope(HXMT) and NASA's Imaging X-ray Polarimetry Explorer (IXPE).

Speculated involvement in other phenomena

Dark energy

Main article:Dark energy

Unsolved problem in physics:

Why does the large zero-point energy of thevacuumnot cause a large cosmological constant? What cancels it out?

(more unsolved problems in physics)

In the late 1990s it was discovered that very distantsupernovaewere dimmer than expected suggesting that the universe's expansion was accelerating rather than slowing down.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 ofnegative 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, but this guess is known to be off by 120orders of magnitude.

TheEuropean Space Agency'sEuclid telescope,launched on 1 July 2023, will map galaxies up to 10 billion light years away.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 toquintessence,where observed acceleration is due to the energy of ascalar 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'sequivalence principleandvariation of the fundamental constantsin space or time.Scalar fields are predicted by theStandard Modelof particle physicsandstring theory,but an analogous problem to the cosmological constant problem (or the problem of constructing models ofcosmological inflation) occurs:renormalizationtheory predicts that scalar fields should acquire large masses again due to zero-point energy.

Cosmic inflation

Main article:Inflation (cosmology)

Unsolved problem in physics:

Why does the observable universe have more matter than antimatter?

(more unsolved problems in physics)

Cosmic inflationis phase of acceleratedcosmic expansionjust after the Big Bang. It explains the origin of thelarge-scale structure of the cosmos.It is believedquantum vacuum fluctuationscaused 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 (seegalaxy formation and evolutionandstructure formation).Many physicists also believe that inflation explains why the Universe appears to be the same in all directions (isotropic), why thecosmic microwave backgroundradiation is distributed evenly, why the Universe is flat, and why nomagnetic monopoleshave 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 causedbaryogenesis,the hypothetical physical processes that produced anasymmetry(imbalance) betweenbaryonsand antibaryons produced in thevery early universe,but this is far from certain.

Cosmology

Paul S. Wessonexamined the cosmological implications of assuming that zero-point energy is real.Among numerous difficulties,general relativityrequires that such energy not gravitate, so it cannot be similar to electromagnetic radiation.

Alternative theories

There has been a long debateover 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".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)for a scalar field, and then generalized to the electromagnetic case by Schwinger, DeRaad, and Milton (1978).in which they state "the vacuum is regarded as truly a state with all physical properties equal to zero".Jaffe(2005)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."

Milonnihas shown the necessity of the vacuum field for the formal consistency of QED.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 aunified theory. Nevertheless, as pointed out by Jaffe, "no known phenomenon, including the Casimir effect, demonstrates that zero point energies are “real” "

Chaotic and emergent phenomena

See also:Chaos theory,Emergence,andSelf-organization

The mathematical models used inclassical electromagnetism,quantum electrodynamics (QED) and theStandard Modelall view theelectromagnetic vacuumas 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 thealternative theoriessection. 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.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 virtualplasmathat can have complex vortices (i.e.solitonsvis-à-vis particles),entangled statesand a rich nonlinear structure.There are many observed nonlinear physical electromagnetic phenomena such asAharonov–Bohm(AB)and Altshuler–Aronov–Spivak (AAS) effects,Berry,Aharonov–Anandan,Pancharatnamand Chiao–Wuphase rotation effects,Josephson effect,Quantum Hall effect,theDe Haas–Van Alphen effect,theSagnac effectand many other physically observable phenomena which would indicate that the electromagnetic potential field has real physical meaning rather than being a mathematical artifactand 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 andemergence.

What are called Maxwell's equations today, are in fact a simplified version of the original equations reformulated byHeaviside,FitzGerald,LodgeandHertz.The original equations usedHamilton's more expressivequaternionnotation,a kind ofClifford algebra,which fully subsumes the standard Maxwell vectorial equations largely used today.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".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 inTesla'soscillator-shuttle-circuit can only be achieved in quaternionic algebra or higher SU(2) symmetries.It has often been argued that quaternions are not compatible with special relativity,but multiple papers have shown ways of incorporating relativity.

A good example of nonlinear electromagnetics is in high energy dense plasmas, wherevortical phenomenaoccur which seemingly violate thesecond law of thermodynamicsby increasing the energy gradient within the electromagnetic field and violateMaxwell's lawsby creating ion currents which capture and concentrate their own and surrounding magnetic fields. In particularLorentz force law,which elaborates Maxwell's equations is violated by these force free vortices.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 extendedNoether theorem,conservation lawssuch as thelaws of thermodynamicsneed not always apply todissipative systems,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 1977Nobel Prize in Chemistrywas awarded to thermodynamicistIlya Prigoginefor his theory of dissipative systems that described this notion. Prigogine described the principle as "order through fluctuations"or "order out of chaos".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.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.

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 ofphase transitionswhich occur at absolute zero temperature. These are quantum phase transitions which are driven by EM field fluctuations as a consequence of zero-point energy.A good example of a spontaneous phase transition that are attributed to zero-point fluctuations can be found insuperconductors.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 theLondon 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.Bostick,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 theBeltrami vorticesin the plasma focus display the same pairedflux-tubemorphology asType II superconductors.Others have also pointed out this connection, Fröhlichhas shown that the hydrodynamic equations of compressible fluids, together with the London equations, lead to a macroscopic parameter ($\mu$= electric charge density / mass density), without involving eitherquantum phase factorsor the Planck constant. In essence, it has been asserted that Beltrami plasma vortex structures are able to at least simulate the morphology ofType IandType 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) butwithout any associated rise in temperature.This is an example of zero-point energy having multiple stable states (seeQuantum phase transition,Quantum critical point,Topological degeneracy,Topological order) 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.

Artist's conception of theexpansion of the universe.On the left, the dramatic expansion occurs in theinflationary epoch.

The idea that the vacuum energy can have multiple stable energy states is a leading hypothesis for the cause ofcosmic 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.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 toperpetual motion machines.

Nevertheless, the allure of free energy has motivated such research, usually falling in the category offringe science.As long ago as 1889 (before quantum theory or discovery of the zero point energy)Nikola Teslaproposed that useful energy could be obtained from free space, or what was assumed at that time to be an all-pervasiveaether.Others have since claimed to exploit zero-point or vacuum energy with a large amount ofpseudoscientificliterature causing ridicule around the subject.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 theU.S. Department of Defenseas well as in China, Germany, Russia and Brazil.

Casimir batteries and engines

A common assumption is that theCasimir forceis 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.In 1984Robert Forwardpublished 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.

In 1999, Pinto, a former scientist atNASA'sJet Propulsion Laboratory at Caltechin Pasadena, published inPhysical Reviewhisthought 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."

Garret Moddel atUniversity of Coloradohas highlighted that he believes such devices hinge on the assumption that the Casimir force is anonconservative force,he argues that there is sufficient evidence (e.g. analysis by Scandurra (2001)) 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.

In 2008,DARPAsolicited 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.

A 2008 patent by Haisch and Moddeldetails 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 Moddelwas 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.

Single heat baths

In 1951Callenand Weltonproved the quantumfluctuation-dissipation theorem(FDT) which was originally formulated in classical form byNyquist(1928)as an explanation for observedJohnson noisein 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.Such a theory has met with resistance: Macdonald (1962)and Harris (1971)claimed that extracting power from the zero-point energy to be impossible, so FDT could not be true. Grau and Kleen (1982)and Kleen (1986),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)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'".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.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 thelaws of thermodynamics,by exploiting certain quantum mechanical properties.

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.

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

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 completequantum 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,but the interaction (if any) is not yet fully understood. According to thegeneral 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.In certain conditions the gravitomagnetic field can be repulsive. In neutron stars for example, it can produce a gravitational analogue of theMeissner effect,but the force produced in such an example is theorized to be exceedingly weak.

In 1963Robert Forward,a physicist and aerospace engineer atHughes Research Laboratories,published a paper showing how within the framework of general relativity "anti-gravitational" effects might be achieved.Since all atoms havespin,gravitational permeability may be able to differ from material to material. A strongtoroidalgravitational field that acts against the force of gravity could be generated by materials that havenonlinear propertiesthat 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.In 1966Dewittwas first to identify the significance of gravitational effects in superconductors. Dewitt demonstrated that a magnetic-type gravitational field must result in the presence offluxoid quantization.In 1983, Dewitt's work was substantially expanded by Ross.

From 1971 to 1974 Henry William Wallace, a scientist atGE Aerospacewas issued with three patents.Wallace usedDewitt'stheory 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.Referring to Wallace's patents, aNew Scientistarticle 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."A further reference to Wallace's patents occur in an electric propulsion study prepared for theAstronautics LaboratoryatEdwards Air Force Basewhich 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."

In 1986 theU.S. Air Force's then Rocket Propulsion Laboratory (RPL) atEdwards Air Force Basesolicited "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 yearBAE Systemslaunched "Project Greenglow" to provide a "focus for research into novel propulsion systems and the means to power them".

In 1988Kip Thorneet al.published work showing how traversablewormholescan exist in spacetime only if they are threaded by quantum fields generated by some form ofexotic matterthat hasnegative energy.In 1993 Scharnhorst and Bartonshowed thatthe speed of a photon will be increasedif 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 theinflationary energy,dark energy or zero-point radiation of the vacuum.Building on the work of Thorne, in 1994Miguel Alcubierreproposed 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 (seeAlcubierre drive). The ship would then ride this wave inside a region of flat space, known as awarp bubbleand would not move within this bubble but instead be carried along as the region itself moves due to the actions of the drive.

In 1992Evgeny Podkletnovpublished a heavily debatedjournal article claiming a specific type of rotating superconductor could shield gravitational force. Independently of this, from 1991 to 1993Ning Liand Douglas Torr published a number of articlesabout gravitational effects in superconductors. One finding they derived is the source ofgravitomagnetic fluxin atype II superconductormaterial is due tospin alignmentof 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 somebut defended by others.In 1997 Li published a paper attempting to replicate Podkletnov's results and showed the effect was very small, if it existed at all.Li is reported to have left the University of Alabama in 1999 to found the companyAC Gravity LLC.AC Gravity was awarded aU.S. Department of Defensegrant for $448,970 in 2001 to continue anti-gravity research. The grant period ended in 2002 but no results from this research were made public.

In 2002Phantom Works,Boeing's advanced research and development facility inSeattle,approachedEvgeny Podkletnovdirectly. 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 Russian government, lso 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"

Froning and Roach (2002)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 withtoroidalelectromagnetic 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 ofgravitational waves.In the spheron model of the nucleus,proposed by the two times Nobel laureateLinus Pauling,dineutronsare among the components of this structure. Similarly to adumbbellput in asuitable 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 2014NASA'sEagleworksLaboratories announced that they had successfully validated the use of aQuantum Vacuum Plasma Thrusterwhich makes use of the Casimir effect for propulsion.In 2016 a scientific paper by the team of NASA scientists passed peer review for the first time.The paper suggests that the zero-point field acts aspilot-waveand 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.

Zero-point energy in fiction

The concept of Zero-point energy used as an energy source has been an element used inscience fictionand related media.

See also

• Casimir effect
• Ground state
• Lamb shift
• QED vacuum
• QCD vacuum
• Quantum fluctuation
• Quantum foam
• Scalar field
• Time crystal
• Topological order
• Unruh effect
• Vacuum energy
• Vacuum expectation value
• Vacuum state
• Virtual particle

References

Articles in the press

• Battersby, S. (20 November 2008)."It's Confirmed: Matter is Merely Vacuum Fluctuations".New Scientist.Archivedfrom the original on 27 May 2017.
• Bennett, J. (30 November 2016)."Scientists Catch" Virtual Particles "Hopping In and Out of Existence".Popular Mechanics.Archivedfrom 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.Archivedfrom 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.Archivedfrom the original on 29 May 2017.
• Choi, C. Q. (12 February 2013)."Something from Nothing? A Vacuum Can Yield Flashes of Light".Scientific American.Archivedfrom the original on 30 May 2017.
• Chown, M. (7 April 1990)."Science: Can Photons Travel 'Faster Than Light'?".New Scientist.Archivedfrom 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 fromthe originalon 2 August 2002.
• Crane, L. (30 November 2016)."Quantum Particles Seen Distorting Light From a Neutron Star".New Scientist.Archivedfrom 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 fromthe originalon 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.Archived fromthe originalon 9 November 2000.
• Matthews, R. (25 February 1995)."Nothing Like a Vacuum".New Scientist.Vol. 145, no. 1966. pp. 30–33.Archivedfrom the original on 13 April 2016.Via Calphysics Institute.
• O'Carroll, E. (25 March 2013)."Scientists Examine Nothing, Find Something".Christian Science Monitor.Archivedfrom the original on 31 March 2013.
• Pilkington, M. (17 July 2003)."Zero Point Energy".The Guardian.Archivedfrom 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 fromthe original(PDF)on 26 February 2017.Retrieved25 October2016.
• Siegel, E.(22 September 2016)."What Is The Physics Of Nothing?".Forbes.Archivedfrom the original on 27 May 2017.
• Shiga, D. (28 September 2005)."Vacuum Energy: Something For Nothing?".New Scientist.Archivedfrom the original on 27 May 2017.
• Wall, M. (27 March 2014)."Does Dark Energy Spring From the 'Quantum Vacuum?'".SPACE.Archivedfrom 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.CiteSeerX10.1.1.129.5792.doi:10.1109/13.485226.
• Barton, G.; Scharnhorst, K. (1993). "QED Between Parallel Mirrors: Light Signals Faster Thanc,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.ISSN0305-4470.
• Battersby, Stephen (2016)."Dark energy: Staring into darkness".Nature.537(7622): 201–204.Bibcode:2016Natur.537S.201B.doi:10.1038/537S201a.PMID27681049.S2CID4398296.
• Bordag, M; Klimchitskaya, G. L.; Mohideen, U.; Mostepanenko, V. M. (2009).Advances in the Casimir Effect.Oxford: Oxford University Press.ISBN978-0-19-923874-3.LCCN2009279136.OCLC319209483.
• 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.PMID11801108.S2CID43354557.
• 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.S2CID32996610.
• 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.ISSN0031-9007.S2CID13943605.
• 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.PMID11239149.S2CID17072357.
• Conlon, T. E. (2011).Thinking About Nothing: Otto Von Guericke and The Magdeburg Experiments on the Vacuum.San Francisco: Saint Austin Press.ISBN978-1-4478-3916-3.OCLC840927124.
• 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.PMC3912482.PMID24492860.
• Davies, P. C. W.(1985).Superforce: The Search for a Grand Unified Theory of Nature.New York: Simon and Schuster.ISBN978-0-671-47685-4.LCCN84005473.OCLC12397205.
• 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.PMID12906584.S2CID20243276.
• 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 fromthe original(PDF)on 25 October 2016.Retrieved17 October2016.
• 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.ISBN978-0-444-10571-4.
• 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.ISSN0217-751X.S2CID119258601.
• 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.ISBN978-0-691-03705-9.OCLC929349643.
• 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.ISBN978-0-691-03322-8.OCLC921496342.
• 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.ISBN978-0-387-13404-8.LCCN84026824.OCLC317097176.
• 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.PMID9910287.Archived fromthe original(PDF)on 31 July 2018.Retrieved17 October2016.
• 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.ISSN1434-6001.S2CID120354480.
• 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.ISSN0035-8711.S2CID11717019.
• Itzykson, C.; Zuber, J.-B. (1980).Quantum Field Theory(2005 ed.). Mineola, New York: Dover Publications.ISBN978-0486445687.LCCN2005053026.OCLC61200849.
• 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.ISSN1550-7998.S2CID119241509.
• 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.ISSN0031-9007.PMID25167393.S2CID8579347.
• 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.PMID10009347.
• Kragh, H. (2002).Quantum Generations: A History of Physics in the Twentieth Century.Princeton University Press.ISBN978-0-691-09552-3.OCLC248763258.
• 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.ISSN0003-9519.S2CID118593162.
• Kragh, H. S.; Overduin, J. M. (2014).The Weight of the Vacuum: A Scientific History of Dark Energy.New York: Springer.ISBN978-3-642-55089-8.LCCN2014938218.OCLC884863929.
• Kuhn, T.(1978).Black-Body Theory and the Quantum Discontinuity, 1894-1912.New York: Oxford University Press.ISBN978-0-19-502383-1.LCCN77019022.OCLC803538583.
• Le Bellac, M. (2006).Quantum Physics(2012 ed.). Cambridge University Press.ISBN978-1-107-60276-2.OCLC957316740.
• 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.ISSN1434-6060.S2CID118643015.
• Loudon, R.(2000).The Quantum Theory of Light(3rd ed.). Oxford: Oxford University Press.ISBN978-0198501770.LCCN2001265846.OCLC44602993.
• 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.ISSN0036-8075.JSTOR2883066.PMID17747645– 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.ISSN0035-8711.S2CID38736536.
• 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.S2CID119991060.
• Milonni, P. W.(1994).The Quantum Vacuum: An Introduction to Quantum Electrodynamics.Boston: Academic Press.ISBN978-0124980808.LCCN93029780.OCLC422797902.
• 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.ISBN9783540706229.LCCN2008942038.OCLC297803628.
• 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.ISSN0034-6861.S2CID118961123.
• Penrose, Roger (2004).The Road to Reality(8th ed.). New York: Alfred A. Knopf.ISBN978-0-679-45443-4.OCLC474890537.
• 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.S2CID118910636.
• Peskin, M. E.; Schroeder, D. V. (1995).An Introduction To Quantum Field Theory.Addison-Wesley.ISBN978-0-201-50397-5.OCLC635667163.
• 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". InLangevin, P.;Solvay, E.;de Broglie, M.(eds.).La Théorie du Rayonnement et les Quanta.Paris: Gauthier-Villars. pp.93–114.LCCNunk84021539.OCLC5894537227.
• Planck, M.(1913).Vorlesungen über die Theorie der Wärmestrahlung.Leipzig: J. A. Barth.OCLC924400975.
• Planck, M.(1958).Physikalische Abhandlungen und Vorträge. Vol. 2.Braunschweig: Vieweg & Sohn.LCCN59047616.OCLC603096370.
• Power, E. A.(1964).Introductory Quantum Electrodynamics.London: Longmans.LCCN65020006.OCLC490279969.
• 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.ISBN978-981-02-2095-2.
• Rees, Martin,ed. (2012).Universe.New York: DK Pub.ISBN978-0-7566-9841-6.LCCN2011277855.OCLC851193468.
• 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.ISSN0036-8075.PMID26429882.S2CID40368170.
• 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.S2CID15640044.
• 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.PMID24484127.S2CID1826585.
• 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.ISSN1355-2198.S2CID9007190.
• Saunders, Simon;Brown, Harvey R.,eds. (1991).The Philosophy of Vacuum.Oxford: Oxford University Press.ISBN978-0198244493.LCCN90048906.OCLC774073198.
• Schwinger, J.(1998a).Particles, Sources, and Fields: Volume I.Reading, Massachusetts: Advanced Book Program, Perseus Books.ISBN978-0-7382-0053-8.LCCN98087896.OCLC40544377.
• Schwinger, J.(1998b).Particles, Sources, and Fields: Volume II.Reading, Massachusetts: Advanced Book Program, Perseus Books.ISBN978-0-7382-0054-5.LCCN98087896.OCLC40544377.
• Schwinger, J.(1998c).Particles, Sources, and Fields: Volume III.Reading, Massachusetts: Advanced Book Program, Perseus Books.ISBN978-0-7382-0055-2.LCCN98087896.OCLC40544377.
• Sciama, D. W.(1991). "The Physical Significance of the Vacuum State of a Quantum Field". InSaunders, Simon;Brown, Harvey R.(eds.).The Philosophy of Vacuum.Oxford: Oxford University Press.ISBN978-0198244493.LCCN90048906.OCLC774073198.
• Scott, Alwyn (2006).Encyclopedia of Nonlinear Science.Routledge.ISBN978-1-57958-385-9.OCLC937249213.
• Scully, M. O.; Zubairy, M. S. (1997).Quantum optics.Cambridge UK: Cambridge University Press.ISBN978-0-521-43595-6.OCLC444869786.
• 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.PMID11736390.
• 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.PMID12511655.S2CID120884236.
• 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.ISSN1434-6060.S2CID15753833.
• 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.ISSN0034-6861.S2CID122259372.
• Weinberg, S.(2015).Lectures on Quantum Mechanics(2nd ed.). Cambridge: Cambridge University Press.ISBN978-1-107-11166-0.LCCN2015021123.OCLC910664598.
• 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.S2CID126303009.
• 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.Retrieved24 October2016.
• 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.ISSN0028-0836.PMID22094697.S2CID219735.
• 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.

Further reading

Press articles

• Brumfiel, G. (3 June 2011)."Moving Mirrors Make Light From Nothing".Nature.doi:10.1038/news.2011.346.
• Brooks, M. (16 November 2011)."Light Pulled Out of Empty Space".New Scientist.Archivedfrom the original on 30 May 2017.
• Cartlidge, E. (17 November 2011)."How to Turn Darkness into Light".Physics World.Institute of Physics.Archivedfrom the original on 30 May 2017.
• Marcus, A. (12 October 2009)."Research in a Vacuum: DARPA Tries to Tap Elusive Casimir Effect for Breakthrough Technology".Scientific American.Archivedfrom the original on 2 March 2015.
• Matthews, R.; Sample, I. (1 September 1996)."'Anti-Gravity' Device Gives Science a Lift ".The Sunday Telegraph.Archived fromthe originalon 6 March 2003.
• Sciama, D. W.(2 February 1978)."The Ether Transmogrified".New Scientist.Vol. 77, no. 1088. pp. 298–300 – via Google Books.
• Yirka, B. (2 October 2015)."Research Team Claims to Have Directly Sampled Electric-Field Vacuum Fluctuations".Phys.org.Archivedfrom the original on 27 May 2017.

Journal articles

• Boyer, T. H. (1970). "Quantum Zero-Point Energy and Long-Range Forces".Annals of Physics.56(2): 474–503.Bibcode:1970AnPhy..56..474B.doi:10.1016/0003-4916(70)90027-8.ISSN0003-4916.
• Bulsara, A. R.; Gammaitoni, L. (1996)."Tuning in to Noise"(PDF).Physics Today.49(3): 39–45.Bibcode:1996PhT....49c..39B.doi:10.1063/1.881491.ISSN0031-9228.Archived fromthe original(PDF)on 12 July 2018.Retrieved31 May2017.
• Lahteenmaki, P.; Paraoanu, G. S.; Hassel, J.; Hakonen, P. J. (2013)."Dynamical Casimir Effect in a Josephson Metamaterial".Proceedings of the National Academy of Sciences.110(11): 4234–4238.arXiv:1111.5608.Bibcode:2013PNAS..110.4234L.doi:10.1073/pnas.1212705110.ISSN0027-8424.S2CID10972781.

Books

• Beiser, A. (2003).Concepts of Modern Physics(6th ed.). Boston: McGraw-Hill.ISBN978-0072448481.LCCN2001044743.OCLC48965418.
• Heitler, W.(1984).The Quantum Theory of Radiation(1954 reprint 3rd ed.). New York: Dover Publications.ISBN978-0486645582.LCCN83005201.OCLC924845769.
• Rafelski, J.;Muller, B. (1985).Structured Vacuum: Thinking About Nothing(PDF).H. Deutsch: Thun.ISBN978-3871448898.LCCN86175968.OCLC946050522.

External links

• Nima Arkani-Hamedonthe issue of vacuum energy and dark energy.
• Steven Weinbergonthe cosmological constant problem.