Zero-point energy

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Zero-point energy (ZPE) is the lowest possible

The term zero-point energy (ZPE) is a translation from the German Nullpunktsenergie.^[8] Sometimes used interchangeably with it are the terms zero-point radiation and ground state energy. The term zero-point field (ZPF) can be used when referring to a specific vacuum field, for instance the QED vacuum which specifically deals with quantum electrodynamics (e.g., electromagnetic interactions between photons, electrons and the vacuum) or the QCD vacuum which deals with quantum chromodynamics (e.g., color charge interactions between quarks, gluons and the vacuum). A vacuum can be viewed not as empty space but as the combination of all zero-point fields. In quantum field theory this combination of fields is called the vacuum state, its associated zero-point energy is called the vacuum energy and the average energy value is called the vacuum expectation value (VEV) also called its condensate.

Overview

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

The uncertainty principle states that no object can ever have precise values of position and velocity simultaneously. The total energy of a quantum mechanical object (potential and kinetic) is described by its

Zero-point energy evolved from historical ideas about the

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.[23]

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

Second quantum theory

In 1900, Max Planck derived the average energy ε of a single energy radiator, e.g., a vibrating atomic unit, as a function of absolute temperature:^[24]

$${\displaystyle \varepsilon ={\frac {h\nu }{e^{h\nu /(kT)}-1}}\,,}$$

where h is

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.^[26]

In 1912, Max Planck published the first journal article to describe the discontinuous emission of radiation, based on the discrete quanta of energy.^[27] In Planck's "second quantum theory" resonators absorbed energy continuously, but emitted energy in discrete energy quanta only when they reached the boundaries of finite cells in phase space, where their energies became integer multiples of hν. This theory led Planck to his new radiation law, but in this version energy resonators possessed a zero-point energy, the smallest average energy a resonator could take on. Planck's radiation equation contained a residual energy factor, one hν/2, as an additional term dependent on the frequency ν, which was greater than zero (where h is Planck's constant). It is therefore widely agreed that "Planck's equation marked the birth of the concept of zero-point energy."^[28] In a series of papers from 1911 to 1913,^[29] Planck found the average energy of an oscillator to be:^[26][30]

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

Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistant

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.[35]^[36]

In 1926 Pascual Jordan^[49] published the first attempt to quantize the electromagnetic field. In a joint paper with Max Born and Werner Heisenberg he considered the field inside a cavity as a superposition of quantum harmonic oscillators. In his calculation he found that in addition to the "thermal energy" of the oscillators there also had to exist an infinite zero-point energy term. He was able to obtain the same fluctuation formula that Einstein had obtained in 1909.^[50] 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".^[51] Jordan found a way to get rid of the infinite term, publishing a joint work with Pauli in 1928,^[52] performing what has been called "the first infinite subtraction, or renormalisation, in quantum field theory".^[53]

Building on the work of Heisenberg and others, Paul Dirac's theory of emission and absorption (1927)^[54] 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.^[55] 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.^[56][57] 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:^[54]

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

Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field. This view was popularized by Victor Weisskopf who in 1935 wrote:^[58]

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

This view was also later supported by

In 1948

Zero-point energy is fundamentally related to the Heisenberg uncertainty principle.^[91] Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position and momentum, or a field's value and derivative at a point in space) cannot simultaneously be specified precisely by any given quantum state. In particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well, for then its position and momentum would both be completely determined to arbitrarily great precision. Therefore, the lowest-energy state (the ground state) of the system must have a distribution in position and momentum that satisfies the uncertainty principle, which implies its energy must be greater than the minimum of the potential well.

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

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

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

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

The expectation value of the energy must therefore be at least

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

where ω = √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

The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or a subatomic particle. In ordinary atomic physics, the zero-point energy is the energy associated with the

If more than one ground state exists, they are said to be

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.

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In quantum field theory (QFT), the fabric of "empty" space is visualized as consisting of fields, with the field at every point in space and time being a quantum harmonic oscillator, with neighboring oscillators interacting with each other. According to QFT the universe is made up of matter fields whose quanta are fermions (e.g. electrons and quarks), force fields whose quanta are bosons (i.e. photons and gluons) and a Higgs field whose quantum is the Higgs boson. The matter and force fields have zero-point energy.^[2] A related term is zero-point field (ZPF), which is the lowest energy state of a particular field.^[92] The vacuum can be viewed not as empty space, but as the combination of all zero-point fields.

In QFT the zero-point energy of the vacuum state is called the vacuum energy and the average expectation value of the Hamiltonian is called the vacuum expectation value (also called condensate or simply VEV). The QED vacuum is a part of the vacuum state which specifically deals with quantum electrodynamics (e.g. electromagnetic interactions between photons, electrons and the vacuum) and the QCD vacuum deals with

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^[18] and the source of dark energy.^[19][20]

Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large as

The oldest and best known quantized force field is the

Redefining the zero of energy

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

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

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

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

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

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

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

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:^[10]

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

without affecting any physical predictions of the theory. The new Hamiltonian is said to be normally 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 commute a and a^†. Since zero-point energy is intimately connected to the non-commutativity of a and a^†, the normal ordering procedure eliminates any contribution from the zero-point field. This is especially reasonable in the case of the field Hamiltonian, since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition for the zero of energy. Moreover, this constant energy in the Hamiltonian obviously commutes with a and a^† and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion.

However, things are not quite that simple. The zero-point energy cannot be eliminated by dropping its energy from the Hamiltonian: When we do this and solve the Heisenberg equation for a field operator, we must include the vacuum field, which is the homogeneous part of the solution for the field operator. In fact we can show that the vacuum field is essential for the preservation of the commutators and the formal consistency of QED. When we calculate the field energy we obtain not only a contribution from particles and forces that may be present but also a contribution from the vacuum field itself i.e. the zero-point field energy. In other words, the zero-point energy reappears even though we may have deleted it from the Hamiltonian.^[94]

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:

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

We introduce the "mode function" A₀(r) that satisfies the Helmholtz equation:

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

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

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

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)|² should be independent of 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 k · e_k = 0 in order to have the transversality condition ∇ · A(r,t) satisfied for the Coulomb gauge^{[dubious – discuss]} in which we are working.

To achieve the desired normalization we pretend space is divided into cubes of volume V = L³ 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

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

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

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

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

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

where e_k is chosen to be a unit vector which specifies the polarization of the field mode. The condition k · e_k = 0 means that there are two independent choices of e_k, which we call e_k1 and e_k2 where e_k1 · e_k2 = 0 and e²
_k1 = e²
_k2 = 1. Thus we define the mode functions:

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

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

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

or:

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

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

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

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

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

we find the field Hamiltonian is:

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

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:

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

Clearly the least eigenvalue for H_F is:

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

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

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

shows. The summation becomes approximately the integral:

$${\displaystyle {\frac {4\pi hV}{c^{3}}}\int v^{3}\,dv}$$

for high values of v. It diverges proportional to v⁴ for large v.

There are two separate questions to consider. First, is the divergence a real one such that the zero-point energy really is infinite? If we consider the volume V is contained by perfectly conducting walls, very high frequencies can only be contained by taking more and more perfect conduction. No actual method of containing the high frequencies is possible. Such modes will not be stationary in our box and thus not countable in the stationary energy content. So from this physical point of view the above sum should only extend to those frequencies which are countable; a cut-off energy is thus eminently reasonable. However, on the scale of a "universe" questions of general relativity must be included. Suppose even the boxes could be reproduced, fit together and closed nicely by curving spacetime. Then exact conditions for running waves may be possible. However the very high frequency quanta will still not be contained. As per John Wheeler's "geons"^[95] 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.^[96]

Necessity of the vacuum field in QED

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

In a process in which a photon is annihilated (absorbed), we can think of the photon as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state.^[54] An atom, for instance, can be considered to be "dressed" by emission and reabsorption of "virtual photons" from the vacuum. The vacuum state energy described by Σ_kλ ħω_k/2 is infinite. We can make the replacement:

$${\displaystyle \sum _{\mathbf {k} \lambda }\longrightarrow \sum _{\lambda }\left({\frac {1}{2\pi }}\right)^{3}\int d^{3}k={\frac {V}{8\pi ^{3}}}\sum _{\lambda }\int d^{3}k}$$

the zero-point energy density is:

$${\displaystyle {\begin{aligned}{\frac {1}{V}}\sum _{\mathbf {k} \lambda }{\tfrac {1}{2}}\hbar \omega _{k}&={\frac {2}{8\pi ^{3}}}\int d^{3}k{\tfrac {1}{2}}\hbar \omega _{k}\\&={\frac {4\pi }{4\pi ^{3}}}\int dk\,k^{2}\left({\tfrac {1}{2}}\hbar \omega _{k}\right)\\&={\frac {\hbar }{2\pi ^{2}c^{3}}}\int d\omega \,\omega ^{3}\end{aligned}}}$$

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

$${\displaystyle \rho _{0}(\omega )={\frac {\hbar \omega ^{3}}{2\pi ^{2}c^{3}}}}$$

The zero-point energy density in the frequency range from ω₁ to ω₂ is therefore:

$${\displaystyle \int _{\omega _{1}}^{\omega _{2}}d\omega \rho _{0}(\omega )={\frac {\hbar }{8\pi ^{2}c^{3}}}\left(\omega _{2}^{4}-\omega _{1}^{4}\right)}$$

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 220 erg/cm³.

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

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

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 coordinate x and the canonical momentum p = mẋ +eA/c of the oscillator are:

$${\displaystyle {\begin{aligned}\mathbf {\dot {x}} &=(i\hbar )^{-1}[\mathbf {x} .H]={\frac {1}{m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\\\mathbf {\dot {p}} &=(i\hbar )^{-1}[\mathbf {p} .H]{\begin{aligned}&={\tfrac {1}{2}}\nabla \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)^{2}-m\omega _{0}^{2}\mathbf {\dot {x}} \\&=-{\frac {1}{m}}\left[\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\cdot \nabla \right]\left[-{\frac {e}{c}}\mathbf {A} \right]-{\frac {1}{m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\times \nabla \times \left[-{\frac {e}{c}}\mathbf {A} \right]-m\omega _{0}^{2}\mathbf {\dot {x}} \\&={\frac {e}{c}}(\mathbf {\dot {x}} \cdot \nabla )\mathbf {A} +{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\omega _{0}^{2}\mathbf {\dot {x}} \end{aligned}}\end{aligned}}}$$

or:

$${\displaystyle {\begin{aligned}m\mathbf {\ddot {x}} &=\mathbf {\dot {p}} -{\frac {e}{c}}\mathbf {\dot {A}} \\&=-{\frac {e}{c}}\left[\mathbf {\dot {A}} -\left(\mathbf {\dot {x}} \cdot \nabla \right)\mathbf {A} \right]+{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\omega _{0}^{2}\mathbf {x} \\&=e\mathbf {E} +{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\omega _{0}^{2}\mathbf {x} \end{aligned}}}$$

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

$${\displaystyle \mathbf {\dot {A}} ={\frac {\partial \mathbf {A} }{\partial t}}+(\mathbf {\dot {x}} \cdot \nabla )\mathbf {A} ^{3}\,.}$$

For nonrelativistic motion we may neglect the magnetic force and replace the expression for mẍ by:

$${\displaystyle {\begin{aligned}\mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} &\approx {\frac {e}{m}}\mathbf {E} \\&\approx \sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar \omega _{k}}{V}}}\left[a_{\mathbf {k} \lambda }(t)+a_{\mathbf {k} \lambda }^{\dagger }(t)\right]e_{\mathbf {k} \lambda }\end{aligned}}}$$

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

$${\displaystyle {\dot {a}}_{\mathbf {k} \lambda }=i\omega _{k}a_{\mathbf {k} \lambda }+ie{\sqrt {\frac {2\pi }{\hbar \omega _{k}V}}}\mathbf {\dot {x}} \cdot e_{\mathbf {k} \lambda }}$$

In the electric dipole approximation.

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

$${\displaystyle i\hbar {\dot {U}}=HU\,,\quad U^{\dagger }(t)=U^{-1}(t)\,,\quad U(0)=1\,.}$$

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:

$${\displaystyle a_{\mathbf {k} \lambda }(t)=a_{\mathbf {k} \lambda }(0)e^{-i\omega _{k}t}+ie{\sqrt {\frac {2\pi }{\hbar \omega _{k}V}}}\int _{0}^{t}dt'\,e_{\mathbf {k} \lambda }\cdot \mathbf {\dot {x}} (t')e^{i\omega _{k}\left(t'-t\right)}}$$

and therefore the equation for ȧ_kλ may be written:

$${\displaystyle \mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} ={\frac {e}{m}}\mathbf {E} _{0}(t)+{\frac {e}{m}}\mathbf {E} _{RR}(t)}$$

where:

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

and:

$${\displaystyle \mathbf {E} _{RR}(t)=-{\frac {4\pi e}{V}}\sum _{\mathbf {k} \lambda }\int _{0}^{t}dt'\left[e_{\mathbf {k} \lambda }\cdot \mathbf {\dot {x}} \left(t'\right)\right]\cos \omega _{k}\left(t'-t\right)}$$

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

$${\displaystyle \mathbf {E} _{RR}(t)={\frac {2e}{3c^{3}}}\mathbf {\ddot {x}} }$$

The total field acting on the dipole has two parts, E₀(t) and E_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

$${\displaystyle \left[\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right]\mathbf {E} =0}$$

satisfied by the field in the (source free) vacuum. For this reason E₀(t) is often referred to as the "vacuum field", although it is of course a Heisenberg-picture operator acting on whatever state of the field happens to be appropriate at t = 0. E_RR(t) is the source field, the field generated by the dipole and acting on the dipole.

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

$${\displaystyle \mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)}$$

where τ = 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 field E₀(t).

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

$${\displaystyle |\Psi \rangle =|{\text{vac}}\rangle |\psi _{D}\rangle \,,}$$

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:

$${\displaystyle \langle \mathbf {E} _{0}(t)\rangle =\langle \Psi |\mathbf {E} _{0}(t)|\Psi \rangle =0}$$

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

$${\displaystyle {\begin{aligned}{\frac {1}{4\pi }}\left\langle \mathbf {E} _{0}^{2}(t)\right\rangle &={\frac {1}{4\pi }}\sum _{\mathbf {k} \lambda }\sum _{\mathbf {k'} \lambda '}{\sqrt {\frac {2\pi \hbar \omega _{k}}{V}}}{\sqrt {\frac {2\pi \hbar \omega _{k'}}{V}}}\times \left\langle a_{\mathbf {k} \lambda }(0)a_{\mathbf {k'} \lambda '}^{\dagger }(0)\right\rangle \\&={\frac {1}{4\pi }}\sum _{\mathbf {k} \lambda }\left({\frac {2\pi \hbar \omega _{k}}{V}}\right)\\&=\int _{0}^{\infty }dw\,\rho _{0}(\omega )\end{aligned}}}$$

The important point of this is that the zero-point field energy H_F does not affect the Heisenberg equation for a_kλ since it is a c-number or constant (i.e. an ordinary number rather than an operator) and commutes with a_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 term Σ_kλ ħω_k/2 in the field Hamiltonian.

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

$${\displaystyle {\begin{aligned}\left[z(t),p_{z}(t)\right]&=\left[U^{\dagger }(t)z(0)U(t),U^{\dagger }(t)p_{z}(0)U(t)\right]\\&=U^{\dagger }(t)\left[z(0),p_{z}(0)\right]U(t)\\&=i\hbar U^{\dagger }(t)U(t)\\&=i\hbar \end{aligned}}}$$

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

$${\displaystyle \mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)}$$

Using the fact that

$${\displaystyle \left[a_{\mathbf {k} \lambda }(0),a_{\mathbf {k'} \lambda '}^{\dagger }(0)\right]=\delta _{\mathbf {kk'} }^{3},\delta _{\lambda \lambda '}}$$

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

$${\displaystyle {\begin{aligned}[z(t),p_{z}(t)]&=\left[z(t),m{\dot {z}}(t)\right]+\left[z(t),{\frac {e}{c}}A_{z}(t)\right]\\&=\left[z(t),m{\dot {z}}(t)\right]\\&=\left({\frac {i\hbar e^{2}}{2\pi ^{2}mc^{3}}}\right)\left({\frac {8\pi }{3}}\right)\int _{0}^{\infty }{\frac {d\omega \,\omega ^{4}}{\left(\omega ^{2}-\omega _{0}^{2}\right)^{2}+\tau ^{2}\omega ^{6}}}\end{aligned}}}$$

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 at ω = ω₀ and:

$${\displaystyle {\begin{aligned}\left[z(t),p_{z}(t)\right]&\approx {\frac {2i\hbar e^{2}}{3\pi mc^{3}}}\omega _{0}^{3}\int _{-\infty }^{\infty }{\frac {dx}{x^{2}+\tau ^{2}\omega _{0}^{6}}}\\&=\left({\frac {2i\hbar e^{2}\omega _{0}^{3}}{3\pi mc^{3}}}\right)\left({\frac {\pi }{\tau \omega _{0}^{3}}}\right)\\&=i\hbar \end{aligned}}}$$

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

$${\displaystyle {\begin{aligned}&\mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)\\&\mathbf {\ddot {x}} \approx -\omega _{0}^{2}\mathbf {x} (t)&&\mathbf {\overset {...}{x}} \approx -\omega _{0}^{2}\mathbf {\dot {x}} \end{aligned}}}$$

and

$${\displaystyle \mathbf {\ddot {x}} +\tau \omega _{0}^{2}\mathbf {\dot {x}} +\omega _{0}^{2}\mathbf {x} \approx {\frac {e}{m}}\mathbf {E} _{0}(t)}$$

Without the free field E₀(t) in this equation the operator x(t) would be exponentially dampened, and commutators like [z(t),p_z(t)] would approach zero for t ≫ 1/τω²
₀. With the vacuum field included, however, the commutator is iħ at all times, as required by unitarity, and as we have just shown. A similar result is easily worked out for the case of a free particle instead of a dipole oscillator.^[97]

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

The spectral density of the vacuum field is fixed by the form of the radiation reaction field, or vice versa: because the radiation reaction field varies with the third derivative of x, the spectral energy density of the vacuum field must be proportional to the third power of ω in order for [z(t),p_z(t)] to hold. In the case of a dissipative force proportional to ẋ, by contrast, the fluctuation force must be proportional to ${\displaystyle \omega }$ in order to maintain the canonical commutation relation.^[97] This relation between the form of the dissipation and the spectral density of the fluctuation is the essence of the fluctuation-dissipation theorem.^[76]

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.^[98]

The quantum chromodynamic vacuum

The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a

to characterize such interactions.

The Higgs field

Main article: Higgs mechanism

The Standard Model hypothesises a field called the Higgs field (symbol: ϕ), which has the unusual property of a non-zero amplitude in its ground state (zero-point) energy after renormalization; i.e., a non-zero vacuum expectation value. It can have this effect because of its unusual "Mexican hat" shaped potential whose lowest "point" is not at its "centre". Below a certain extremely high energy level the existence of this non-zero vacuum expectation

gauge symmetry which in turn gives rise to the Higgs mechanism and triggers the acquisition of mass by those particles interacting with the field. The Higgs mechanism occurs whenever a charged field has a vacuum expectation value. This effect occurs because scalar field components of the Higgs field are "absorbed" by the massive bosons as degrees of freedom, and couple to the fermions via Yukawa coupling, thereby producing the expected mass terms. The expectation value of ϕ⁰ in the ground state (the vacuum expectation value or VEV) is then ⟨ϕ⁰⟩ = v/√2, where v = |μ|/√λ. The measured value of this parameter is approximately 246 GeV/c².^[99]

It has units of mass, and is the only free parameter of the Standard Model that is not a dimensionless number.

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

Experimental observations

Zero-point energy has many observed physical consequences.^[11] 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.^[100] Indeed, such treatment could create a problem at a deeper, as of yet undiscovered, theory.^[101] 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.^[102] 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.^[103]

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 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%.^{[104][105][106][107][108]} That changed in 1997 with Lamoreaux^[109] conclusively showing that the Casimir force was real. Results have been repeatedly replicated since then.^{[110][111][112][113]}

In 2009, Munday et al.^[114] published experimental proof that (as predicted in 1961^[115]) 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.^[116]

An interesting hypothetical side effect of the Casimir effect is the Scharnhorst effect, a hypothetical phenomenon in which light signals travel slightly faster than c between two closely spaced conducting plates.^[117]

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 ²S_1/2 and ²P_1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which these states should have the same energy. Charged particles can interact with the fluctuations of the quantized vacuum field, leading to slight shifts in energy;^[118] this effect is called the Lamb shift.^[119] The shift of about 4.38×10⁻⁶ eV is roughly 10⁻⁷ of the difference between the energies of the 1s and 2s levels, and amounts to 1,058 MHz in frequency units. A small part of this shift (27 MHz ≈ 3%) arises not from fluctuations of the electromagnetic field, but from fluctuations of the electron–positron field. The creation of (virtual) electron–positron pairs has the effect of screening the Coulomb field and acts as a vacuum dielectric constant. This effect is much more important in muonic atoms.^[120]

Fine-structure constant

Main article: Fine-structure constant

Taking ħ (

electronic charge

and ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity)) we can form a dimensionless quantity called the fine-structure constant:

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

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.^[121] The quantum fluctuations caused by zero-point energy have the effect of screening electric charges: owing to (virtual) electron-positron pair production, the charge of the particle measured far from the particle is far smaller than the charge measured when close to it.

The Heisenberg inequality where ħ = h/2π, and Δ_x, Δ_p are the standard deviations of position and momentum states that:

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

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, m_zc² ≈ 90 GeV, that:

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

rather than the low-energy α ≈ 1/137.^[122][123] The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for defining α. All in all, α depends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The energy dependence of α has been observed for several years now in precision experiment in high-energy physics.

Vacuum birefringence

Main articles: Lorentz-violating electrodynamics and Euler–Heisenberg Lagrangian

In the presence of strong electrostatic fields it is predicted that virtual particles become separated from the vacuum state and form real matter.^{[citation needed]} The fact that electromagnetic radiation can be transformed into matter and vice versa leads to fundamentally new features in quantum electrodynamics. One of the most important consequences is that, even in the vacuum, the Maxwell equations have to be exchanged by more complicated formulas. In general, it will be not possible to separate processes in the vacuum from the processes involving matter since electromagnetic fields can create matter if the field fluctuations are strong enough. This leads to highly complex nonlinear interaction - gravity will have an effect on the light at the same time the light has an effect on gravity. These effects were first predicted by Werner Heisenberg and Hans Heinrich Euler in 1936^[124] and independently the same year by Victor Weisskopf who stated: "The physical properties of the vacuum originate in the "zero-point energy" of matter, which also depends on absent particles through the external field strengths and therefore contributes an additional term to the purely Maxwellian field energy".^[125][126] Thus strong magnetic fields vary the energy contained in the vacuum. The scale above which the electromagnetic field is expected to become nonlinear is known as the Schwinger limit. At this point the vacuum has all the properties of a birefringent medium, thus in principle a rotation of the polarization frame (the Faraday effect) can be observed in empty space.^[127][128]

RX 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 as

RX J1856.5-3754,^[131] the closest discovered neutron star to Earth.^[132]

Roberto Mignani at the

interstellar gas or plasma, the effect should have been no more than 1%. Definitive proof would require repeating the observation at other wavelengths and on other neutron stars. At X-ray wavelengths the polarization from the quantum fluctuations should be near 100%.^[134] Although no telescope currently exists that can make such measurements, there are several proposed X-ray telescopes that may soon be able to verify the result conclusively such as China's Hard X-ray Modulation Telescope

(HXMT) and NASA's Imaging X-ray Polarimetry Explorer (IXPE).

Speculated involvement in other phenomena

Dark energy

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?^{[18][103][135]}

(more unsolved problems in physics)

In the late 1990s it was discovered that very distant supernovae were dimmer than expected suggesting that the universe's expansion was accelerating rather than slowing down.^[136][137] This revived discussion that Einstein's cosmological constant, long disregarded by physicists as being equal to zero, was in fact some small positive value. This would indicate empty space exerted some form of negative pressure or energy.

There is no natural candidate for what might cause what has been called dark energy but the current best guess is that it is the zero-point energy of the vacuum.^[138] One difficulty with this assumption is that the zero-point energy of the vacuum is absurdly large compared to the observed cosmological constant. This issue, called the cosmological constant problem, is one of the greatest unsolved mysteries in physics.

The

cosmological inflation) occurs: renormalization

theory 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)

isotropic), why the cosmic microwave background radiation is distributed evenly, why the Universe is flat, and why no magnetic monopoles

have been observed.

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

Cosmology

Paul S. Wesson examined the cosmological implications of assuming that zero-point energy is real.^[142] Among numerous difficulties, general relativity requires that such energy not gravitate, so it cannot be similar to electromagnetic radiation.

Alternative theories

There has been a long debate^[143] over the question of whether zero-point fluctuations of quantized vacuum fields are "real" i.e. do they have physical effects that cannot be interpreted by an equally valid alternative theory? Schwinger, in particular, attempted to formulate QED without reference to zero-point fluctuations via his "source theory".^[144] 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)^[145] for a scalar field, and then generalized to the electromagnetic case by Schwinger, DeRaad, and Milton (1978).^[146] in which they state "the vacuum is regarded as truly a state with all physical properties equal to zero". More recently Jaffe (2005)^[147] has highlighted a similar approach in deriving the Casimir effect stating "the concept of zero-point fluctuations is a heuristic and calculational aid in the description of the Casimir effect, but not a necessity in QED."

Nevertheless, as Jaffe himself notes in his paper, "no one has shown that source theory or another S-matrix based approach can provide a complete description of QED to all orders." Furthermore,

Hawking Radiation and the Unruh effect are also theorized to be dependent on zero-point vacuum fluctuations, the field contribution being an inseparable parts of these theories. Jaffe continues "Even if one could argue away zero-point contributions to the quantum vacuum energy, the problem of spontaneous symmetry breaking remains: condensates [ground state vacua] that carry energy appear at many energy scales in the Standard Model. So there is good reason to be skeptical of attempts to avoid the standard formulation of quantum field theory and the zero-point energies it brings with it." It is difficult to judge the physical reality of infinite zero-point energies that are inherent in field theories, but modern physics does not know any better way to construct gauge-invariant, renormalizable theories than with zero-point energy and they would seem to be a necessity for any attempt at a unified theory.^[149]

Chaotic and emergent phenomena

Beyond the Standard Model
Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons
Standard Model
Evidence Hierarchy problem Dark matter Dark energy Quintessence Phantom energy Dark radiation Dark photon Cosmological constant problem Strong CP problem Neutrino oscillation
Theories Brans–Dicke theory Cosmic censorship hypothesis Fifth force F-theory Theory of everything Unified field theory Grand Unified Theory Technicolor Kaluza–Klein theory 6D (2,0) superconformal field theory Noncommutative quantum field theory Quantum cosmology Brane cosmology String theory Superstring theory M-theory Mathematical universe hypothesis Mirror matter Randall–Sundrum model N = 4 supersymmetric Yang–Mills theory Twistor string theory Dark fluid Doubly special relativity de Sitter invariant special relativity Causal fermion systems Black hole thermodynamics Unparticle physics Graviphoton Graviscalar Graviton Gravitino Massive gravity Gauge gravitation theory Gauge theory gravity CPT symmetry
Supersymmetry MSSM NMSSM Superstring theory M-theory Supergravity Supersymmetry breaking Extra dimensions Large extra dimensions
Quantum gravity False vacuum String theory Spin foam Quantum foam Quantum geometry Loop quantum gravity Quantum cosmology Loop quantum cosmology Causal dynamical triangulation Causal fermion systems Causal sets Canonical quantum gravity Semiclassical gravity Superfluid vacuum theory
Experiments ANNIE Gran Sasso INO LHC SNO Super-K Tevatron NOvA
v t e

See also: Chaos theory, Emergence, and Self-organization

The mathematical models used in

entangled states and a rich nonlinear structure.^[151] There are many observed nonlinear physical electromagnetic phenomena such as Aharonov–Bohm (AB)^[152][153] and Altshuler–Aronov–Spivak (AAS) effects,^[154] Berry,^[155] Aharonov–Anandan,^[156] Pancharatnam^[157] and Chiao–Wu^[158] phase rotation effects, Josephson effect,^[159][160] Quantum Hall effect,^[161] the De Haas–Van Alphen effect,^[162] the Sagnac effect and many other physically observable phenomena which would indicate that the electromagnetic potential field has real physical meaning rather than being a mathematical artifact^[163] and therefore an all encompassing theory would not confine electromagnetism as a local force as is currently done, but as a SU(2) gauge theory or higher geometry. Higher symmetries allow for nonlinear, aperiodic behaviour which manifest as a variety of complex non-equilibrium phenomena that do not arise in the linearised U(1) theory, such as multiple stable states, symmetry breaking, chaos and emergence.^[164]

What are called Maxwell's equations today, are in fact a simplified version of the original equations reformulated by Heaviside, FitzGerald, Lodge and Hertz. The original equations used Hamilton's more expressive quaternion notation,^[165] a kind of Clifford algebra, which fully subsumes the standard Maxwell vectorial equations largely used today.^[166] 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".^[167] It was concluded that there was no need for the greater physical insights provided by the quaternions if the theory was purely local in nature. Local vector analysis has become the dominant way of using Maxwell's equations ever since. However, this strictly vectorial approach has led to a restrictive topological understanding in some areas of electromagnetism, for example, a full understanding of the energy transfer dynamics in Tesla's oscillator-shuttle-circuit can only be achieved in quaternionic algebra or higher SU(2) symmetries.^[168] It has often been argued that quaternions are not compatible with special relativity,^[169] but multiple papers have shown ways of incorporating relativity.^{[170][171][172][173]}

A good example of nonlinear electromagnetics is in high energy dense plasmas, where

dissipative systems,^[178][179] which are expressed in gauges of higher symmetry). The second law of thermodynamics states that in a closed linear system entropy flow can only be positive (or exactly zero at the end of a cycle). However, negative entropy (i.e. increased order, structure or self-organisation) can spontaneously appear in an open nonlinear thermodynamic system that is far from equilibrium, so long as this emergent order accelerates the overall flow of entropy in the total system. The 1977 Nobel Prize in Chemistry was awarded to thermodynamicist Ilya Prigogine^[180] for his theory of dissipative systems that described this notion. Prigogine described the principle as "order through fluctuations"^[181] or "order out of chaos".^[182] 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.^[183] 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.^[184]

One may query what this has to do with zero-point energy. Given the complex and adaptive behaviour that arises from nonlinear systems considerable attention in recent years has gone into studying a new class of

Type II superconductors.^[189][190] Others have also pointed out this connection, Fröhlich^[191]

has shown that the hydrodynamic equations of compressible fluids, together with the London equations, lead to a macroscopic parameter (${\displaystyle \mu }$ = electric charge density / mass density), without involving either quantum phase factors or Planck's constant. In essence, it has been asserted that Beltrami plasma vortex structures are able to at least simulate the morphology of Type I and Type II superconductors. This occurs because the "organised" dissipative energy of the vortex configuration comprising the ions and electrons far exceeds the "disorganised" dissipative random thermal energy. The transition from disorganised fluctuations to organised helical structures is a phase transition involving a change in the condensate's energy (i.e. the ground state or zero-point energy) but without any associated rise in temperature.^[192] This is an example of zero-point energy having multiple stable states (see Quantum phase transition, Quantum critical point, Topological degeneracy, Topological order^[193]) and where the overall system structure is independent of a reductionist or deterministic view, that "classical" macroscopic order can also causally affect quantum phenomena. Furthermore, the pair production of Beltrami vortices has been compared to the morphology of pair production of virtual particles in the vacuum.

The idea that the vacuum energy can have multiple stable energy states is a leading hypothesis for the cause of

cosmic inflation. In fact, it has been argued that these early vacuum fluctuations led to the expansion of the universe and in turn have guaranteed the non-equilibrium conditions necessary to drive order from chaos, as without such expansion the universe would have reached thermal equilibrium and no complexity could have existed. With the continued accelerated expansion of the universe, the cosmos generates an energy gradient that increases the "free energy" (i.e. the available, usable or potential energy for useful work) which the universe is able to use to create ever more complex forms of order.^[194][195]

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 (

perpetual motion machines.^{[citation needed}

]

Nevertheless, the allure of free energy has motivated such research, usually falling in the category of

U.S. Department of Defense as well as in China, Germany, Russia and Brazil.^[197][199]

Casimir batteries and engines

A common assumption is that the Casimir force is of little practical use; the argument is made that the only way to actually gain energy from the two plates is to allow them to come together (getting them apart again would then require more energy), and therefore it is a one-use-only tiny force in nature.^[197] In 1984 Robert Forward published work showing how a "vacuum-fluctuation battery" could be constructed; the battery can be recharged by making the electrical forces slightly stronger than the Casimir force to reexpand the plates.^[200]

In 1999, Pinto, a former scientist at NASA's Jet Propulsion Laboratory at Caltech in Pasadena, published in Physical Review his thought experiment (Gedankenexperiment) for a "Casimir engine". The paper showed that continuous positive net exchange of energy from the Casimir effect was possible, even stating in the abstract "In the event of no other alternative explanations, one should conclude that major technological advances in the area of endless, by-product free-energy production could be achieved."^[201]

Garret Moddel at University of Colorado has highlighted that he believes such devices hinge on the assumption that the Casimir force is a nonconservative force, he argues that there is sufficient evidence (e.g. analysis by Scandurra (2001)^[202]) 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.^[203]

In 2008, DARPA solicited research proposals in the area of Casimir Effect Enhancement (CEE). The goal of the program is to develop new methods to control and manipulate attractive and repulsive forces at surfaces based on engineering of the Casimir force.^[204]

A 2008 patent by Haisch and Moddel^[205] details a device that is able to extract power from zero-point fluctuations using a gas that circulates through a Casimir cavity. A published test of this concept by Moddel^[206] was performed in 2012 and seemed to give excess energy that could not be attributed to another source. However it has not been conclusively shown to be from zero-point energy and the theory requires further investigation.^[207]

Single heat baths

In 1951

Johnson noise^[78] 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.^[79] Such a theory has met with resistance: Macdonald (1962)^[208] and Harris (1971)^[209] claimed that extracting power from the zero-point energy to be impossible, so FDT could not be true. Grau and Kleen (1982)^[210] and Kleen (1986),^[211] 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)^[212] 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'".^[213] 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.^[80] A paper by Armen Allahverdyan and Theo Nieuwenhuizen in 2000 showed the feasibility of extracting zero-point energy for useful work from a single bath, without contradicting the laws of thermodynamics, by exploiting certain quantum mechanical properties.^[81]

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.[82]^{[214][215][216][217][218][219][220][221][222]}

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^[223]

Space travel and gravitational shielding

The use of zero-point energy for space travel is speculative and does not form part of the mainstream scientific consensus. A complete

general theory of relativity, rotating matter can generate a new force of nature, known as the gravitomagnetic interaction, whose intensity is proportional to the rate of spin.^[227] In certain conditions the gravitomagnetic field can be repulsive. In neutrons stars for example it can produce a gravitational analogue of the Meissner effect, but the force produced in such an example is theorized to be exceedingly weak.^[228]

In 1963

Hughes Research Laboratories, published a paper showing how within the framework of general relativity "anti-gravitational" effects might be achieved.^[229] Since all atoms have spin, gravitational permeability may be able to differ from material to material. A strong toroidal gravitational field that acts against the force of gravity could be generated by materials that have nonlinear properties that enhance time-varying gravitational fields. Such an effect would be analogous to the nonlinear electromagnetic permeability of iron making it an effective core (i.e. the doughnut of iron) in a transformer, whose properties are dependent on magnetic permeability.^{[230][231][232]} In 1966 Dewitt^[233] was first to identify the significance of gravitational effects in superconductors. Dewitt demonstrated that a magnetic-type gravitational field must result in the presence of fluxoid quantization. In 1983, Dewitt's work was substantially expanded by Ross.^[234]

From 1971 to 1974 Henry William Wallace, a scientist at GE Aerospace was issued with three patents.^{[235][236][237]} Wallace used Dewitt's theory to develop an experimental apparatus for generating and detecting a secondary gravitational field, which he named the kinemassic field (now better known as the gravitomagnetic field). In his three patents, Wallace describes three different methods used for detection of the gravitomagnetic field – change in the motion of a body on a pivot, detection of a transverse voltage in a semiconductor crystal, and a change in the specific heat of a crystal material having spin-aligned nuclei. There are no publicly available independent tests verifying Wallace's devices. Such an effect if any would be small.^{[238][239][240][241][242][243]} Referring to Wallace's patents, a New Scientist article in 1980 stated "Although the Wallace patents were initially ignored as cranky, observers believe that his invention is now under serious but secret investigation by the military authorities in the USA. The military may now regret that the patents have already been granted and so are available for anyone to read."^[244] A further reference to Wallace's patents occur in an electric propulsion study prepared for the Astronautics Laboratory at Edwards Air Force Base which states: "The patents are written in a very believable style which include part numbers, sources for some components, and diagrams of data. Attempts were made to contact Wallace using patent addresses and other sources but he was not located nor is there a trace of what became of his work. The concept can be somewhat justified on general relativistic grounds since rotating frames of time varying fields are expected to emit gravitational waves."^[245]

In 1986 the

U.S. Air Force's then Rocket Propulsion Laboratory (RPL) at Edwards Air Force Base solicited "Non Conventional Propulsion Concepts" under a small business research and innovation program. One of the six areas of interest was "Esoteric energy sources for propulsion, including the quantum dynamic energy of vacuum space..." In the same year BAE Systems launched "Project Greenglow" to provide a "focus for research into novel propulsion systems and the means to power them".^[199][246]

In 1988

wormholes can exist in spacetime only if they are threaded by quantum fields generated by some form of exotic matter that has negative energy. In 1993 Scharnhorst and Barton^[117] showed that the speed of a photon will be increased if it travels between two Casimir plates, an example of negative energy. In the most general sense, the exotic matter needed to create wormholes would share the repulsive properties of the inflationary energy, dark energy or zero-point radiation of the vacuum.^[248] Building on the work of Thorne, in 1994 Miguel Alcubierre^[249] proposed a method for changing the geometry of space by creating a wave that would cause the fabric of space ahead of a spacecraft to contract and the space behind it to expand (see Alcubierre drive

). The ship would then ride this wave inside a region of flat space, known as a warp bubble and would not move within this bubble but instead be carried along as the region itself moves due to the actions of the drive.

In 1992

type II superconductor material is due to spin alignment of the lattice ions. Quoting from their third paper: "It is shown that the coherent alignment of lattice ion spins will generate a detectable gravitomagnetic field, and in the presence of a time-dependent applied magnetic vector potential field, a detectable gravitoelectric field." The claimed size of the generated force has been disputed by some^[258][259] but defended by others.^[260][261] In 1997 Li published a paper attempting to replicate Podkletnov's results and showed the effect was very small, if it existed at all.^[262] Li is reported to have left the University of Alabama in 1999 to found the company AC Gravity LLC.^[263] AC Gravity was awarded a U.S. DOD grant for $448,970 in 2001 to continue anti-gravity research. The grant period ended in 2002 but no results from this research were ever made public.^[264]

In 2002

Evgeny Podkletnov directly. Phantom Works was blocked by Russian technology transfer controls. At this time Lieutenant General George Muellner, the outgoing head of the Boeing Phantom Works, confirmed that attempts by Boeing to work with Podkletnov had been blocked by Moscow, also commenting that "The physical principles – and Podkletnov's device is not the only one – appear to be valid... There is basic science there. They're not breaking the laws of physics. The issue is whether the science can be engineered into something workable"^[265]

Froning and Roach (2002)[266] put forward a paper that builds on the work of Puthoff, Haisch and Alcubierre. They used fluid dynamic simulations to model the interaction of a vehicle (like that proposed by Alcubierre) with the zero-point field. Vacuum field perturbations are simulated by fluid field perturbations and the aerodynamic resistance of viscous drag exerted on the interior of the vehicle is compared to the Lorentz force exerted by the zero-point field (a Casimir-like force is exerted on the exterior by unbalanced zero-point radiation pressures). They find that the optimized negative energy required for an Alcubierre drive is where it is a saucer-shaped vehicle with toroidal electromagnetic fields. The EM fields distort the vacuum field perturbations surrounding the craft sufficiently to affect the permeability and permittivity of space.

In 2009 Giorgio Fontana and Bernd Binder presented a new method to potentially extract the Zero-point energy of the electromagnetic field and nuclear forces in the form of gravitational waves.^[267] In the spheron model of the nucleus,^[268] proposed by the two times Nobel laureate Linus Pauling, dineutrons are among the components of this structure. Similarly to a dumbbell put in a suitable rotational state, but with nuclear mass density, dineutrons are nearly ideal sources of gravitational waves at X-ray and gamma-ray frequencies. The dynamical interplay, mediated by nuclear forces, between the electrically neutral dineutrons and the electrically charged core nucleus is the fundamental mechanism by which nuclear vibrations can be converted to a rotational state of dineutrons with emission of gravitational waves. Gravity and gravitational waves are well described by General Relativity, that is not a quantum theory, this implies that there is no Zero-point energy for gravity in this theory, therefore dineutrons will emit gravitational waves like any other known source of gravitational waves. In Fontana and Binder paper, nuclear species with dynamical instabilites, related to the Zero-point energy of the electromagnetic field and nuclear forces, and possessing dineutrons, will emit gravitational waves. In experimental physics this approach is still unexplored.

In 2014

Quantum Vacuum Plasma Thruster which makes use of the Casimir effect for propulsion.^{[269][270][271]} In 2016 a scientific paper by the team of NASA scientists passed peer review for the first time.^[272] The paper suggests that the zero-point field acts as pilot-wave and that the thrust may be due to particles pushing off the quantum vacuum. While peer review doesn't guarantee that a finding or observation is valid, it does indicate that independent scientists looked over the experimental setup, results, and interpretation and that they could not find any obvious errors in the methodology and that they found the results reasonable. In the paper, the authors identify and discuss nine potential sources of experimental errors, including rogue air currents, leaky electromagnetic radiation, and magnetic interactions. Not all of them could be completely ruled out, and further peer-reviewed experimentation is needed in order to rule these potential errors out.^[273]

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

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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. Archived from the original on 30 May 2017.
• Cartlidge, E. (17 November 2011). "How to Turn Darkness into Light". Physics World. Institute of Physics. Archived from 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. Archived from the original on 2 March 2015.
• Matthews, R.; Sample, I. (1 September 1996). "'Anti-Gravity' Device Gives Science a Lift". The Sunday Telegraph. Archived from the original on 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. Archived from 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.
  ISSN 0003-4916
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• Bulsara, A. R.; Gammaitoni, L. (1996). "Tuning in to Noise" (PDF). Physics Today. 49 (3): 39–45.
  ISSN 0031-9228. Archived from the original
  (PDF) on 12 July 2018. Retrieved 31 May 2017.
• 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.
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Books

• Beiser, A. (2003). Concepts of Modern Physics (6th ed.). Boston: McGraw-Hill.
  OCLC 48965418
  .
• OCLC 924845769
  .
• OCLC 946050522
  .

External links

Wikimedia Commons has media related to Zero-point energy.

Wikiquote has quotations related to Zero-point energy.

Look up zero-point energy in Wiktionary, the free dictionary.

• Nima Arkani-Hamed on the issue of vacuum energy and dark energy.
• Steven Weinberg on the cosmological constant problem.