Quantum fluctuation

In quantum physics, a quantum fluctuation (or vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space,[lower-alpha 1][2] as prescribed by Werner Heisenberg's uncertainty principle. They are tiny random fluctuations in the values of the fields which represent elementary particles, such as electric and magnetic fields which represent the electromagnetic force carried by photons, W and Z fields which carry the weak force, and gluon fields which carry the strong force.[3] Vacuum fluctuations appear as virtual particles, which are always created in particle-antiparticle pairs.[4] Since they are created spontaneously without a source of energy, vacuum fluctuations and virtual particles violate conservation of energy, however this is allowed because they annihilate each other within the time limit set by the uncertainty principle and so are not observable.[4][3] The uncertainty principle states the uncertainty in energy and time can be related by[5] ${\displaystyle \Delta E\,\Delta t\geq {\tfrac {1}{2}}\hbar ~}$, where 1/2ħ ≈ 5,27286×10⁻³⁵ Js. this means pairs of virtual particles with energy ${\displaystyle \Delta E}$ and lifetime shorter than ${\displaystyle \Delta t}$ are continually created and annihilated in empty space. Although the particles are not directly detectable, the cumulative effects of these particles are measurable. For example, without quantum fluctuations the "bare" mass and charge of elementary particles is infinite; from renormalization theory the shielding effect of the cloud of virtual particles is responsible for the finite mass and charge of elementary particles. Another consequence is the Casimir effect. One of the first observations which was evidence for vacuum fluctuations was the Lamb shift in hydrogen.

For related articles, see Quantum vacuum (disambiguation).

3D visualization of quantum fluctuations

Quantum fluctuations may have been necessary for the origin of the structure of the universe: According to the model of expansive inflation, the fluctuations that existed when inflation began were amplified and formed the seeds of all currently observed large-scale structure. Vacuum energy may also be responsible for the current accelerating expansion of the universe (cosmological constant).

Field fluctuations

A quantum fluctuation is the temporary appearance of energetic particles out of empty space, as allowed by the uncertainty principle. The uncertainty principle states that for a pair of conjugate variables such as position/momentum or energy/time, it is impossible to have a precisely determined value of each member of the pair at the same time. For example, a particle pair can pop out of the vacuum during a very short time interval.

An extension is applicable to the "uncertainty in time" and "uncertainty in energy" (including the rest mass energy mc². When the mass is very large like a macroscopic object, the uncertainties and thus the quantum effect become very small, and classical physics is applicable.

In quantum field theory, fields undergo quantum fluctuations. A reasonably clear distinction can be made between quantum fluctuations and thermal fluctuations of a quantum field (at least for a free field; for interacting fields, renormalization substantially complicates matters). An illustration of this distinction can be seen by considering quantum and classical Klein-Gordon fields: For the quantized Klein–Gordon field in the vacuum state, we can calculate the probability density that we would observe a configuration ${\displaystyle {\displaystyle \varphi _{t}(x)}}$ at a time t in terms of its Fourier transform ${\displaystyle {\displaystyle {\tilde {\varphi }}_{t}(k)}}$ to be

${\displaystyle \rho _{0}[\varphi _{t}]=\exp {\left[-{\frac {1}{\hbar }}\int {\frac {d^{3}k}{(2\pi )^{3}}}{\tilde {\varphi }}_{t}^{*}(k){\sqrt {|k|^{2}+m^{2}}}\;{\tilde {\varphi }}_{t}(k)\right]}~.}$

In contrast, for the classical Klein–Gordon field at non-zero temperature, the Gibbs probability density that we would observe a configuration ${\displaystyle {\displaystyle \varphi _{t}(x)}}$ at a time ${\displaystyle t}$ is

${\displaystyle \rho _{E}[\varphi _{t}]=\exp {[-H[\varphi _{t}]/k_{\mathrm {B} }T]}=\exp {\left[-{\frac {1}{k_{\mathrm {B} }T}}\int {\frac {d^{3}k}{(2\pi )^{3}}}{\tilde {\varphi }}_{t}^{*}(k){\scriptstyle {\frac {1}{2}}}(|k|^{2}+m^{2})\;{\tilde {\varphi }}_{t}(k)\right]}~.}$

These probability distributions illustrate that every possible configuration of the field is possible, with the amplitude of quantum fluctuations controlled by Planck's constant ${\displaystyle \hbar }$, just as the amplitude of thermal fluctuations is controlled by ${\displaystyle k_{\mathrm {B} }T}$, where k_B is Boltzmann's constant. Note that the following three points are closely related:

1. Planck's constant has units of action (joule-seconds) instead of units of energy (joules),
2. the quantum kernel is ${\displaystyle {\sqrt {|k|^{2}+m^{2}\,}}\,}$ instead of ${\displaystyle {\scriptstyle {\frac {1}{2}}}(|k|^{2}+m^{2})}$ (the quantum kernel is nonlocal from a classical heat kernel viewpoint, but it is local in the sense that it does not allow signals to be transmitted),
3. the quantum vacuum state is Lorentz invariant (although not manifestly in the above), whereas the classical thermal state is not (the classical dynamics is Lorentz invariant, but the Gibbs probability density is not a Lorentz invariant initial condition).

We can construct a classical continuous random field that has the same probability density as the quantum vacuum state, so that the principal difference from quantum field theory is the measurement theory (measurement in quantum theory is different from measurement for a classical continuous random field, in that classical measurements are always mutually compatible – in quantum mechanical terms they always commute). Quantum effects that are consequences only of quantum fluctuations, not of subtleties of measurement incompatibility, can alternatively be models of classical continuous random fields.

Interpretations

See also: Interpretations of quantum mechanics

The success of quantum fluctuation theories have given way to metaphysical interpretations on the nature of reality and their potential role in the origin and structure of the universe:

• The fluctuations are a manifestation of the innate uncertainty on the quantum level[6]
• Fluctuations of the fields in each element of our universe's spacetime could be coherent throughout the universe by mesoscopic quantum entanglement.
• A fundamental particle arising out of its quantum field is always inescapably subject to this reality and is thus describable by an associated wave function.

The wave function of a quantum particle represents the reality of the innate quantum fluctuations at the core of the universe and bestows the particle its counter-intuitive quantum behavior.

In the double slit experiment each particle makes an unpredictable choice between alternative possibilities. Cumulatively, those choices are consistent with an interference pattern with the inherent fluctuations of the underlying quantum field.[7] Such an underlying immutable quantum field by which quantum fluctuations are correlated in a universal scale may explain the non-locality of quantum entanglement as a natural process.[8]

As has been recently demonstrated, charged extended particles can experience self-oscillatory dynamics as a result of classical electrodynamic self-interactions.[9] This trembling motion has a frequency that is closely related to the zitterbewegung frequency appearing in Dirac's equation. The mechanism producing these fluctuations arises because some parts of an accelerated charged composite particle emit perturbing electromagnetic fields that can affect other parts of the particle, producing self-forces.

Using the Liénard-Wiechert potential as solutions to Maxwell's equations with sources, it can be shown that these forces can be described in terms of differential equations with state-dependent delay, which display limit cycle behavior. Therefore, the principle of inertia, as appearing in Newton's first law, would only hold in an average sense, since uniform motion is unstable. Consequently, pilot waves would be necessary attached to any electromagnetically interacting body.

See also

• Casimir effect
• Cosmic microwave background
• Quantum annealing
• Quantum foam
• Virtual particle
• Virtual black hole
• Stochastic interpretation
• Zitterbewegung

Footnotes

1. According to quantum theory, the vacuum contains neither matter nor energy, but it does contain fluctuations, transitions between something and nothing in which potential existence can be transformed into real existence by the addition of energy. (Energy and matter are equivalent since all matter ultimately consists of packets of energy.) Thus, the vacuum's totally empty space is actually a seething turmoil of creation and annihilation, of which to the ordinary world appears calm because the scale of fluctuations in the vacuum is tiny and the fluctuations tend to cancel each other out. Even though they appear calm, they are in a state of restlessness, looking for compatible matter or fluctuations. — M.W. Browne (1990)[1]

References

1. Browne, Malcolm W. (21 August 1990). "New direction in physics: Back in time". The New York Times. Retrieved 22 May 2010.
2. Pahlavani, Mohammad Reza (2015). Selected Topics in Applications of Quantum Mechanics. BoD. p. 118. ISBN 9789535121268.
3. Pagels, Heinz R. (2012). The Cosmic Code: Quantum Physics as the Language of Nature. Courier Corp. pp. 274–278. ISBN 9780486287324.
4. Kane, Gordon (9 October 2006). "Are virtual particles really constantly popping in and out of existence? Or are they merely a mathematical bookkeeping device for quantum mechanics?". Sciences FAQ. Scientific American website, Springer Nature America. Retrieved 5 August 2020.
5. Mandelshtam, Leonid; Tamm, Igor (1945). "Соотношение неопределённости энергия-время в нерелятивистской квантовой механике" [The uncertainty relation between energy and time in non-relativistic quantum mechanics]. Izv. Akad. Nauk SSSR (Ser. Fiz.) (in Russian). 9: 122–128. English translation: "The uncertainty relation between energy and time in non-relativistic quantum mechanics". J. Phys. (USSR). 9: 249–254. 1945.
6. Kennedy, James (Jim) E. "Nature and meaning of information in physics". science.jeksite.org. Retrieved 22 April 2017.
7. Bhaumik, Mani Lal (4 October 2013). "Comprehending quantum theory from quantum fields". arXiv:1310.1251 [physics.gen-ph].
8. Bhaumik, Mani Lal (15 December 2013). "Reality of the wave function and quantum entanglement". arXiv:1402.4764 [physics.gen-ph].
9. López, Álvaro G. (30 January 2020). "On an electrodynamic origin of quantum fluctuations". arXiv:2001.07392 [quant-ph].