# Zero-point energy

(Redirected from Zero point energy)
Zero-point radiation continually imparts random impulses on an electron, so that it never comes to a complete stop. Zero-point radiation gives the oscillator an average energy equal to the frequency of oscillation multiplied by one-half of Planck's constant

Zero-point energy (ZPE) or ground state energy is the lowest possible energy that a quantum mechanical system may have i.e. it is the energy of the system's ground state. Zero-point energy can have several different types of context e.g. it may be the energy associated with the ground state of an atom, a subatomic particle or even the quantum vacuum itself.

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

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

Given the equivalence of mass and energy expressed by Einstein's E = mc2, any point in space that contains energy can be thought of as having mass to create particles. Virtual particles spontaneously flash into existence at every point in space due to the energy of quantum fluctuations caused by the uncertainty principle. Modern physics has developed quantum field theory (QFT) to understand the fundamental interactions between matter and forces, it treats every single point of space as a quantum harmonic oscillator. According to QFT the universe is made up of matter fields whose quanta are fermions (e.g. electrons and quarks) and force fields, whose quanta are bosons (i.e. photons and gluons). All these fields have zero-point energy.[1] A vacuum can be viewed, not as empty space, but the combination of all zero-point fields. In QFT this combination of fields is called the vacuum state or sometimes the zero-point field (ZPF) and it's associated zero-point energy is called the vacuum energy. The QED vacuum is a part of the vacuum state which specifically deals with quantum electrodynamics (e.g. electromagnetic interactions between photons, electrons and the vacuum) and the QCD vacuum deals with quantum chromodynamics (e.g. color charge interactions between quarks, gluons and the vacuum). Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, and that all properties of matter are merely vacuum fluctuations arising from interactions of the zero-point field.[2]

The idea that "empty" space can have an intrinsic energy associated to it, and that there is no such thing as a "true vacuum" is seemingly unintuitive. For many practical calculations zero-point energy is dismissed in the mathematical model by fiat as a constant that may be canceled or as a term that has no physical effect. Such treatment causes problems however, as in Einstein's theory of general relativity the absolute energy value of space is not arbitrary and gives rise to the cosmological constant. Furthermore, many physical effects attributed to zero-point energy have been experimentally verified, such as the Casimir force, Lamb shift or magnetic moment of the electron. Physics currently lacks a full theoretical model for understanding zero-point energy, in particular the discrepancy between theorized and observed vacuum energy is a source of major contention.[3][4] Physicists John Wheeler and Richard Feynman calculated the zero-point radiation of the QED vacuum to be an order of magnitude greater than nuclear energy, with one teacup containing enough to boil all the world's oceans[5] while experimental evidence from both the expansion of the universe and the Casimir effect show any such force to be exceptionally weak. This discrepancy is known as the cosmological constant problem (or vacuum catastrophe) and is one of the greatest unsolved mysteries in physics.

Many physicists believe that understanding "the vacuum holds the key to a full understanding of nature" [6] and is critical in the search for the theory of everything. Active areas of research include the effects of virtual particles,[7] quantum entanglement,[8] the difference (if any) between inertial and gravitational mass,[9][10] variation in the speed of light,[11][12] a reason for the observed value of the cosmological constant[13] and the nature of dark energy.[14][15]

The concept of zero-point energy was developed by Max Planck in Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900.[16] The term zero-point energy is a translation from the German Nullpunktsenergie.[17]:275ff

## History

Zero-point energy evolved from the historical development of ideas about the vacuum. In the 17th century, 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.[18]

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

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

In 1900, Max Planck derived the average energy of a single energy radiator, e.g., a vibrating atomic unit, as a function of absolute temperature:[20]

${\displaystyle \epsilon ={\frac {h\nu }{e^{h\nu /kT}-1}}~,}$

where h is Planck's constant, ν is the frequency, k is Boltzmann's constant, and T is the absolute temperature. The zero-point energy makes no contribution to Planck's original law, as its existence to Planck was unknown in 1900.[21]

In 1912, Max Planck published the first journal article[22] 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 of ${\displaystyle h\nu }$. This theory led Plank to his new radiation law, but in this version energy resonators possessed a zero-point energy, the smallest average energy a resonator could take on. Planck's radiation equation contained a residual energy factor, one ${\displaystyle 1/2h\nu }$, as an additional term dependent on the frequency ${\displaystyle \nu }$, which is was greater than zero (where ${\displaystyle h}$ = Planck's constant). It is therefore widely agreed that "Planck's equation marked the birth of the concept of zero-point energy."[23] In a series of papers from 1911-1913, Planck found that the average energy of an oscillator to be:[16]:sec 2[24]:235ff

${\displaystyle \epsilon ={\frac {h\nu }{2}}+{\frac {h\nu }{e^{h\nu /kT}-1}}~.}$

Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistant Otto Stern.[25] They attempted to prove the existence of zero-point energy by calculating the specific heat of hydrogen gas and compared it with the experimental data. However, after assuming they had succeeded and after publishing the findings, they retracted the support of the idea because they found Planck's second theory may not apply to their example.[17]:270ff Zero-point energy was also invoked by Debye,[26] who noted that zero-point energy of the atoms of a crystal lattice would cause a reduction in the intensity of the diffracted radiation in X-ray diffraction even as the temperature approached absolute zero. In 1916 Walther Nernst proposed that empty space was filled with zero-point electromagnetic radiation.[27] With the development of general relativity Einstein found the energy density of the vacuum to contribute towards to 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 to it had returned, with Einstein stating in 1920:

In 1924 Mulliken[30] provided direct evidence for the zero-point energy of molecular vibrations by comparing the band spectrum of B10O and B11O: 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,[31] with the development of matrix mechanics in Werner Heisenberg's famous article "Quantum theoretical re-interpretation of kinematic and mechanical relations" the existence of zero-point energy was shown to be "required by quantum mechanics, as a direct consequence of Heisenberg's uncertainty principle"[32]:162

In 1913 Bohr had proposed what is now called the Bohr model of the atom,[33][34][35] but despite this it remained a mystery as to why electrons do not fall into their nuclei. According to classical ideas, the fact that an accelerating charge loses energy by radiating implied that an electron should spiral into the nucleus and that atoms should not be stable. This problem of classical mechanics was nicely summarized by Jeans in 1915: "There would be a very real difficulty in supposing that the (force) law ${\displaystyle 1/r^{2}}$ held down to the zero values of . 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"[36] This resolution to this puzzle came in 1926 with Schrodinger's famous equation.[37] This equation explained the new, non-classical, fact that as an electron moves close to a nucleus its kinetic energy necessarily increases in such a way that the minimum total energy (kinetic plus potential) occurs at some positive separation rather than at zero separation; in other words, that zero-point energy is essential for atomic stability.[38]

Dirac's theory of emission and absorption (1927)[39] was the first application of the quantum theory of radiation and at the time was of crucial importance for the emerging field of quantum mechanics; it dealt directly with the process in which "particles" are actually created - spontaneous emission.[40] Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. The theory showed that spontaneous emission depends upon the zero-point energy fluctuations of the electromagnetic field in order to get started.[41][42] Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field. This view was popularized by Weisskopf (1935)[43] and later by Welton (1948),[44] who argued that spontaneous emission "can be thought of as forced emission taking place under the action of the fluctuating field." This new theory, called quantum electrodynamics (QED) predicted a fluctuating zero-point or "vacuum" field existing even in the absence of sources.

Throughout the 1940s improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom, now known as the Lamb shift,[45] and measurement of the magnetic moment of the electron.[46] Discrepancies between these experiments and Dirac's theory led to the idea of incorporating renormalisation into QED to deal with zero-point infinities. Renormalization was originally developed by Kramers[47] and also Weisskopf(1936),[48] and first successfully applied to calculate a finite value for the Lamb shift by Bethe (1947).[49] As per spontaneous emission, these effects can in part be understood with interactions with the zero-point field.[50][51] 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 fully eliminated. In Pauli's 1945 Nobel lecture[52] he made clear his opposition to the idea of zero-point energy stating "It is clear that this zero-point energy has no physical reality".

In 1948 Casimir[53][54] showed that one consequence of the zero-point field is an attractive force between two uncharged, perfectly conducting parallel plates, the so-called Casimir effect.

In 1951 Callen and Welton[55] proved the quantum fluctuation-dissipation theorem (FDT) which was originally formulated in classical form by Nyquist (1928)[56] as an explanation for observed Johnson noise in electric circuits.[57] 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.[58] FDT has been shown to be true experimentally under certain quantum, non-classical, conditions.[59][60][61]

In 1963 the Jaynes-Cummings model[62] was developed describing the system of a two-level atom interacting with a quantized field mode (i.e the vacuum) within an optical cavity. It gave nonintuitive predictions i.e. 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.[63][64] These results were at first regarded with suspicion in some quarters: it was argued that no modification of a spontaneous emission rate would be possible, after all, how can the emission of a photon be affected by an atom's environment when the atom can only "see" its environment by emitting a photon in the first place? These experiments gave rise to cavity quantum electrodynamics (CQED), the study of effects of mirrors and cavities on radiative corrections. Spontaneous emission can be suppressed (or "inhibited")[65][66] or amplified. Amplification was first predicted by Purcell in 1946[67] (the Purcell effect) and has been experimentally verified.[68] This phenomena can be understood, partly, in terms of the action of the vacuum field on the atom.[69]

## The Uncertainty Principle

Main article: Uncertainty principle

Zero-point energy is fundamentally related to the Heisenberg uncertainty principle.[70] Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position and momentum, or a field's value and derivative at a point in space) cannot simultaneously be specified precisely by any given quantum state. In particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well: for, then, its position and momentum would both be completely determined to arbitrarily great precision. Therefore, instead, the lowest-energy state (the ground state) of the system must have a distribution in position and momentum that satisfies the uncertainty principle−−which implies its energy must be greater than the minimum of the potential well.

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

${\displaystyle {\hat {H}}=V_{0}+{\frac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}+{\frac {1}{2m}}{\hat {p}}^{2}~,}$

where V0 is the minimum of the classical potential well.

The uncertainty principle tells us that

${\displaystyle {\sqrt {\left\langle \left({\hat {x}}-x_{0}\right)^{2}\right\rangle }}{\sqrt {\left\langle {\hat {p}}^{2}\right\rangle }}\geq {\frac {\hbar }{2}}~,}$

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

${\displaystyle \left\langle {\frac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}\right\rangle \left\langle {\frac {1}{2m}}{\hat {p}}^{2}\right\rangle \geq \left({\frac {\hbar }{4}}\right)^{2}{\frac {k}{m}}~.}$

The expectation value of the energy must therefore be at least

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

where ${\displaystyle \omega ={\sqrt {k/m}}}$ is the angular frequency at which the system oscillates.

A more thorough treatment, showing that the energy of the ground state actually saturates this bound and is exactly E0=V0+ħω/2, requires solving for the ground state of the system.

## Atomic Physics

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

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

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

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

${\displaystyle {\frac {h^{2}n^{2}}{8mL^{2}}}}$

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

## Quantum Field Theory

In quantum field theory (QFT), the fabric of "empty" space is visualized as consisting of fields, with the field at every point in space and time being a quantum harmonic oscillator, with neighboring oscillators interacting with each other. According to QFT the universe is made up of matter fields whose quanta are fermions (e.g. electrons and quarks) and force fields, whose quanta are bosons (i.e. photons and gluons). All these fields have zero-point energy.[1] A related term is zero-point field (ZPF), which is the lowest energy state of a particular field.[71] The vacuum can be viewed, not as empty space, but the combination of all zero-point fields.

In QFT this combination of fields is called the vacuum state, it's associated zero-point energy is called the vacuum energy and the expectation value of the Hamiltonian is called the vacuum expectation value. The QED vacuum is a part of the vacuum state which specifically deals with quantum electrodynamics (e.g. electromagnetic interactions between photons, electrons and the vacuum) and the QCD vacuum deals with quantum chromodynamics (e.g. color charge interactions between quarks, gluons and the vacuum). Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, and that all properties of matter are merely vacuum fluctuations arising from interactions with the zero-point field.[72]

Each point in space makes a contribution of E=ħω/2, resulting in a calculation of infinite zero-point energy in any finite volume; this is one reason renormalization is needed to make sense of quantum field theories. In cosmology, the vacuum energy is one possible explanation for the cosmological constant[73] and the source of dark energy.[14][74]

Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large as Paul Dirac claimed it is, like a sea of energy. Other scientists specializing in General Relativity require the energy to be small enough for curvature of space to agree with observed astronomy. The Heisenberg uncertainty principle allows the energy to be as large as needed to promote quantum actions for a brief moment of time, even if the average energy is small enough to satisfy relativity and flat space. To cope with disagreements, the vacuum energy is described as a virtual energy potential of positive and negative energy.[75]

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

### The QED Vacuum

Main article: QED vacuum

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

#### Redefining the Zero of Energy

In the quantum theory of the electromagnetic field, classical wave amplitudes ${\displaystyle \alpha }$ and ${\displaystyle \alpha ^{*}}$are replaced by operators ${\displaystyle a}$ and ${\displaystyle a^{\dagger }}$ that satisfy:

${\displaystyle [a,a^{\dagger }]=1}$

The classical quantity ${\displaystyle |\alpha |^{2}}$ appearing in the classical expression for the energy of a field mode is replaced in quantum theory by the photon number operator ${\displaystyle a^{\dagger }a}$. The fact that:

${\displaystyle [a,a^{\dagger }a]\neq 1}$

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 for ${\displaystyle a^{\dagger }a}$ and ${\displaystyle a}$. The reconciliation of wave and particle attributes of the field is accomplished via the association of a probability amplitude with a classical mode pattern. The calculation of field modes is entirely classical problem, while the quantum properties of the field are carried by the mode "amplitudes" ${\displaystyle a^{\dagger }}$ and ${\displaystyle a}$ associated with these classical modes.

The zero-point energy of the field arises formally from the non-commutativity of ${\displaystyle a}$ and ${\displaystyle a^{\dagger }}$. This is true for any harmonic oscillator: the zero-point energy ${\displaystyle \hbar \omega /2}$ appears when we write the Hamiltonian:

{\displaystyle {\begin{aligned}H_{cl}&={\frac {{p}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}{p}^{2}\\&={\frac {1}{2}}\hbar \omega {\big (}aa^{\dagger }+a^{\dagger }a{\big )}\\&=\hbar \omega {\big (}a^{\dagger }a+{\frac {1}{2}}{\big )}\end{aligned}}}

It is often argued that the entire universe is completed 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:[76]

{\displaystyle {\begin{aligned}H_{F}-\langle 0|H_{F}|0\rangle &={\frac {1}{2}}\hbar \omega {\big (}aa^{\dagger }+a^{\dagger }a{\big )}-{\frac {1}{2}}\hbar \omega \\&={\frac {1}{2}}\hbar \omega {\big (}aa^{\dagger }+{\frac {1}{2}}{\big )}-{\frac {1}{2}}\hbar \omega \\&=\hbar \omega aa^{\dagger }\end{aligned}}}

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

${\displaystyle :H_{F}:\equiv \hbar \omega {\big (}aa^{\dagger }+a^{\dagger }a{\big )}:\equiv \hbar \omega aa^{\dagger }}$

In other words, within the normal ordering symbol we can commute ${\displaystyle a}$ and ${\displaystyle a^{\dagger }}$. Since zero-point energy is intimately connected to the non-commutativity of ${\displaystyle a}$ and ${\displaystyle a^{\dagger }}$, the normal ordering procedure eliminates any contribution from the zero-point field. This is especially reasonable in the case of the field Hamiltonian, since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition for the zero of energy. Moreover, this constant energy in the Hamiltonian obviously commutes with ${\displaystyle a}$ and ${\displaystyle a^{\dagger }}$ 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 consistent 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.[77]

#### The Electromagnetic Field in Free Space

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

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

We introduce the "mode function" ${\displaystyle \mathbf {A} _{0}(\mathbf {r} )}$ that satisfies the Helmholtz equation:

${\displaystyle (\nabla ^{2}+k^{2})\mathbf {A} _{0}(\mathbf {r} )=0}$

where ${\displaystyle k={\omega }/{c}}$ and assume it is normalized such that:

${\displaystyle \int d^{3}r|\mathbf {A} _{0}(\mathbf {r} )|^{2}=1}$

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 ${\displaystyle |\mathbf {A} _{0}(\mathbf {r} )|^{2}}$ should be independent of ${\displaystyle \mathbf {r} }$ for each mode of the field. The mode function satisfying these conditions is:

${\displaystyle \mathbf {A} _{0}(\mathbf {r} )=e_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {r} }}$

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

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

${\displaystyle \mathbf {A} (x+L,y+L,z+L,t)=\mathbf {A} (x,y,z,t)}$

or equivalently

${\displaystyle (k_{x},k_{y},k_{z})={\frac {2\pi }{L}}(n_{x},n_{y},n_{z})}$

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

${\displaystyle \mathbf {A} _{\mathbf {k} }(\mathbf {r} )=V^{-1/2}e_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {r} }}$

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

${\displaystyle \int \limits _{V}d^{3}r|\mathbf {A} _{\mathbf {k} }(\mathbf {r} )|^{2}=1}$

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

${\displaystyle \mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} )=V^{-1/2}e_{\mathbf {k} \lambda }e^{i\mathbf {k} \cdot \mathbf {r} }\ ,\ \lambda ={\begin{cases}1\\2\end{cases}}}$

in terms of which the vector potential becomes:

${\displaystyle \mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} t)=\left({\frac {2\pi \hbar c^{2}}{\omega _{k}V}}\right)^{\frac {1}{2}}[a_{\mathbf {k} \lambda }(0)e^{i\mathbf {k} \cdot \mathbf {r} }+a_{\mathbf {k} \lambda }^{\dagger }(0)e^{-i\mathbf {k} \cdot \mathbf {r} }]e_{\mathbf {k} \lambda }}$

or:

${\displaystyle \mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} t)=\left({\frac {2\pi \hbar c^{2}}{\omega _{k}V}}\right)^{\frac {1}{2}}[a_{\mathbf {k} \lambda }(0)e^{-i(\omega _{k}t-\mathbf {k} \cdot \mathbf {r} })+a_{\mathbf {k} \lambda }^{\dagger }(0)e^{i(\omega _{k}t-\mathbf {k} \cdot \mathbf {r} })}$

where ${\displaystyle \omega _{k}=kc}$ and ${\displaystyle a_{\mathbf {k} \lambda }}$, ${\displaystyle a_{\mathbf {k} \lambda }^{\dagger }}$ are photon annihilation and creation operators for the mode with wave vector ${\displaystyle k}$ and polarization ${\displaystyle \lambda }$. This gives the vector potential for a plane wave mode of the field. The condition for ${\displaystyle (k_{x},k_{y},k_{z})}$ shows that there are infinitely many such modes. The linearity of Maxwell's equations allows us to write:

${\displaystyle \mathbf {A} (\mathbf {r} t)=\sum _{\mathbf {k} \lambda }\left({\frac {2\pi \hbar c^{2}}{\omega _{k}V}}\right)^{\frac {1}{2}}[a_{\mathbf {k} \lambda }(0)e^{i\mathbf {k} \cdot \mathbf {r} }+a_{\mathbf {k} \lambda }^{\dagger }(0)e^{-i\mathbf {k} \cdot \mathbf {r} }]e_{\mathbf {k} \lambda }}$

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

${\displaystyle \int \limits _{V}d^{3}r\mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} )\cdot \mathbf {A} _{\mathbf {k} '\lambda '}^{\ast }(\mathbf {r} )=\delta _{\mathbf {k} ,\mathbf {k} '}^{3}\delta _{\lambda ,\lambda '}}$

we find the field hamiltonian is:

${\displaystyle H_{F}=\sum _{\mathbf {k} \lambda }\left(\hbar \omega _{k}(a_{\mathbf {k} \lambda }^{\dagger }a_{\mathbf {k} \lambda }+{\frac {1}{2}}\right)}$

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:

${\displaystyle [a_{\mathbf {k} \lambda }(t),a_{\mathbf {k} '\lambda '}^{\dagger }(t)]=\delta _{\mathbf {k} ,\mathbf {k} '}^{3}\delta _{\lambda ,\lambda '}}$
${\displaystyle [a_{\mathbf {k} \lambda }(t),a_{\mathbf {k} '\lambda '}(t)]=[a_{\mathbf {k} \lambda }^{\dagger }(t),a_{\mathbf {k} '\lambda '}^{\dagger }(t)]=0}$

Clearly the least eigenvalue for ${\displaystyle H_{F}}$ is:

${\displaystyle \sum _{\mathbf {k} \lambda }{\frac {1}{2}}\hbar \omega _{k}}$

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

${\displaystyle {\frac {8\pi v^{2}dv}{c^{3}}}V}$

shows. The summation becomes approximately the integral:

${\displaystyle {\frac {1}{2}}\cdot {\frac {8\pi hV}{c^{3}}}\int v^{3}dv}$

for high values of ${\displaystyle v}$. It diverges like ${\displaystyle v^{4}}$ for large ${\displaystyle v}$.

There are two separate questions to consider. First, is the divergence a real one such that the zero-point energy really is infinite? If we consider the volume ${\displaystyle V}$ is contained by perfectly conducting walls, very high frequencies can only be contained by taking more and more perfect conduction. No actual method of containing the high frequencies is possible. Such modes will not be stationary in our box and thus not countable in the stationary energy content. So from this physical point of view the above sum should only extend to those frequencies which are countable; a cut-off energy is thus eminently reasonable. However, on the scale of a "universe" questions of general relativity must be included. Suppose even the boxes could be reproduced, fitted together and closed nicely by curving the space-time. 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"[78] 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 i.e. 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 to 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 space-time 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.[79]

### The QCD Vacuum

Main article: QCD vacuum

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

## Experimental observations

Zero-point energy has many observed physical consequences.[51] It is important to note that zero-point energy is not merely an artefact 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.[80] Indeed, such treatment could create a problem at a deeper, as of yet undiscovered, theory.[81] 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.[82] 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.[83]

### Casimir Effect

Main article: Casimir effect

A phenomenon that is commonly presented as evidence for the existence of zero-point energy in vacuum is the Casimir effect, proposed in 1948 by Dutch physicist Hendrik B. G. Casimir (Philips Research), 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 results from the 1950s onwards gave positive results but external factors could not be ruled out, with the range of experimental error sometimes being nearly 100%.[84][85][86][87][88] That changed in 1997 with Lamoreaux[89] conculsively showing such the Casimir force was real. Results have been repeatedly replicated since then.[90][91][92][93]

However, there is still some debate on whether vacuum energy is necessary to explain the Casimir effect. Robert Jaffe of MIT argues that the Casimir force should not be considered evidence for vacuum energy, since it can be derived in QED without reference to vacuum energy by considering charge-current interactions (the radiation-reaction picture).[94]

### Fine Structure Constant

Taking ħ (Planck's constant divided by 2π), ${\displaystyle c}$ (the speed of light), and ${\displaystyle e^{2}={q_{e}}^{2}/{4\pi \varepsilon _{0}}}$ (the electromagnetic coupling constant i.e. a measure of the strength of the electromagnetic force (where ${\displaystyle q_{e}}$ is the absolute value of the electronic charge and ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity)) we can form a dimensionless quantity called the fine-structure constant:

${\displaystyle \alpha ={\frac {e^{2}}{\hbar c}}={\frac {{q_{e}}^{2}}{4\pi \varepsilon _{0}\hbar c}}\cong {\frac {1}{137}}}$

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.[95] 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 charged measured when close to it.

The Heisenberg inequality where ħ = h/2π, and ${\displaystyle \Delta _{x}}$, ${\displaystyle \Delta _{p}}$ are the standard deviations of position and momentum states that:

${\displaystyle \Delta _{x}\Delta _{p}\geq {\frac {1}{2}}\hbar }$

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 the Z0 boson rest energy, ${\displaystyle m_{z}c^{2}\cong }$ 90GeV, that:

${\displaystyle \alpha \cong {\frac {1}{129}}}$

rather than the low energy ${\displaystyle \alpha \cong 1/137}$.[96][97] The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for defining ${\displaystyle \alpha }$. All in all, ${\displaystyle \alpha }$ depends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The energy dependence of ${\displaystyle \alpha }$ has been observed for several years now in precision experiment in high-energy physics.

### Lamb Shift

Main article: lamb shift

The quantum fluctuations of the electromagnetic field have important physical consequences. In addition to the Casimir effect, they also lead to a splitting between the two energy levels 2S1/2 and 2P1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which these states should have the same energy. Charged particles can interact with the fluctuations of the quantized vacuum field, leading to slight shifts in energy,[98] this effect is called the Lamb shift.[99] The shift of about 4.38 × 10−6eV is roughly 10−7 of the difference between the energies of the 1s and 2s levels, and amounts to 1058 MHz in frequency units. A small part of this shift (27Mhz ≈ 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.[100]

### Dark Energy & the Cosmological Constant

Main article: dark energy
 Unsolved problem in physics: why does the large zero-point energy of the vacuum not cause a large cosmological constant? What cancels it out? Why the zero-point energy density not decrease with the expansion of the universe?[101][102][103] (more unsolved problems in physics)

In cosmology, the zero-point energy offers an intriguing possibility for explaining the speculative positive values of the proposed cosmological constant. [104] In brief, if the energy is "really there", then it should exert a gravitational force.[105] In general relativity, mass and energy are equivalent; both produce a gravitational field. One obvious difficulty with this association is that the zero-point energy of the vacuum is absurdly large. Naively, it is infinite, because it includes the energy of waves with arbitrarily short wavelengths. But since only differences in energy are physically measurable, the infinity can be removed by renormalization. In all practical calculations, this is how the infinity is handled.

## Potential Applications

Throughout space there is energy. Is this energy static or kinetic? If static our hopes are in vain; if kinetic – and we know it is, for certain – then it is a mere question of time when men will succeed in attaching their machinery to the very wheel work of Nature. Many generations may pass, but in time our machinery will be driven by a power obtainable at any point in the Universe.

Nikola Tesla (1889)[106]

Nikola Tesla was the first to propose that the vacuum energy, or æther, might be harnessed for useful work;[106] ever since then many people have claimed to exploit zero-point energy with a large amount of pseudoscientific literature causing ridicule around the subject.[107][108]

Despite controversy, harnessing zero-point energy is an ongoing area of worldwide research, particularly in China, Germany, Russia and Brazil.[107] The Casimir force between two plates, which is caused by zero-point energy, was first predicted in 1948 by Dutch physicist Hendrik Casimir.[109] Steve K. Lamoreaux initially measured the tiny force in 1997.[110] It had long been assumed that the Casimir force had little practical use; it was assumed 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 force in nature.[107]

In 1999 however, Fabrizio Pinto, a former scientist at NASA's Jet Propulsion Laboratory at Caltech in Pasadena, published in Physical Review his 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." [111] Despite this and several similar peer reviewed papers, there is not a consensus as to whether such devices will actually work in practice. Garret Moddel at University of Colorado has highlighted that he believes such a device hinges on the assumption that the Casimir force is a nonconservative force, he argues that there is sufficient evidence to say that it 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 then has been input into the system.[112]

In 1951 Callen and Welton[55] proved the quantum fluctuation-dissipation theorem (FDT) which was originally formulated in classical form by Nyquist (1928)[56] as an explanation for observed Johnson noise[113] in electric circuits. Fluctuation-dissipation theorem showed that when something dissipates energy, in an effectively irreversible way, a connected heat bath must also fluctuate. The fluctuations and the dissipation go hand in hand; it is impossible to have one without the other. The implication of FDT being that the vacuum could be treated as a heat bath coupled to a dissipative force and as such energy could, in part, be extracted from the vacuum for potentially useful work.[58] Such a theory has met with resistance: Macdonald (1962)[114] and Harris (1971)[115] claimed that extracting power from the zero-point energy to be impossible, so FDT could not be true. Grau and Kleen (1982)[116] and Kleen (1986),[117] argued that the Johnson noise of a resistor connected to an antenna must satisfy Planck's thermal radiation formulaa, thus the noise must be zero at zero temperature and FDT must be invalid. Kiss (1988)[118] 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)[119] 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.[120][121][122]

There have been several promising breakthroughs in the field of thermodynamics; a paper by Armen Allahverdyan and Theo Nieuwenhuizen in 2000[123] and then by Marlan Scully et al. in 2003 published in Science[124] showed the feasibility of extracting zero-point energy for useful work from a single bath, without contradicting the laws of thermodynamics, by exploiting certain quantum mechanical properties.

In 2014 NASA's Eagleworks Laboratories[125] announced that they had successfully validated the use of a Quantum Vacuum Plasma Thruster which makes use of the Casimir effect for propulsion.[126][127]

## In popular culture

"Zero-point energy" has been invoked in science fiction movies and video games, often as an explanation for "impossible" technology that provides free energy or otherwise contradicts known laws of physics.

Science skeptic and writer Martin Gardner has called claims of such zero-point-energy-based systems "as hopeless as past efforts to build perpetual motion machines".[128] A perpetual motion machine is a device that can operate indefinitely, with optional output of excess energy, without any source of fuel. Such a device would violate the laws of thermodynamics. Despite the science, numerous articles and books have been published addressing and discussing the potential of tapping zero-point-energy from the quantum vacuum or elsewhere. Examples of such are the work of the following authors: Claus Wilhelm Turtur,[129] Jeane Manning, Joel Garbon,[130] John Bedini,[131] Tom Bearden,[132][133][134] Thomas Valone,[135][136][137] Moray B King,[138][139][140] Christopher Toussaint, Bill Jenkins,[141] Nick Cook[142] and William James.[143]

The 2004 video game Half-Life 2 features a weapon called the "zero-point energy field manipulator" also known as the "Gravity gun".

In Disney/Pixar's animated film The Incredibles, the main villain Syndrome refers to his weapons as using zero-point energy.[144][145] The fan fiction community devoted to the character is named "Zero Point" because of this.[146]

In the Stargate series of television shows, Zero Point Modules are advanced power sources built by the Ancients to power their cities and outposts. Weighing only a few kilograms,[147] a single ZPM has been quoted as able to power the entire city of Atlantis for thousands of years. ZPMs supposedly extract vacuum energy from a small artificially-created region of subspace,[148] based on the concept of zero-point energy.[149] ZPMs are depicted as more powerful and efficient than fictional Naquadah generators or any conventional energy source on present day Earth.[148]

The motion picture adaptation, Atlas Shrugged, Part 1, proposed an entirely different sort of revolutionary motor. In the film, Henry Rearden and Dagny Taggart mention the Casimir Effect, which classically is an attraction between two metal conducting plates brought close together in a quantum vacuum.[150]

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