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[[File:Quantum Fluctuations.gif|thumb|3D visualization of quantum fluctuations]]
{{For|related articles|Quantum vacuum (disambiguation)}}

In [[quantum physics]], a '''quantum fluctuation''' (or '''vacuum state fluctuation''' or '''vacuum fluctuation''') is the temporary change in the amount of energy in a point in [[space]],<ref>{{cite news|quote=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. |url=https://www.nytimes.com/1990/08/21/science/new-direction-in-physics-back-in-time.html?pagewanted=all|newspaper=[[The New York Times]]|title=New Direction in Physics: Back in Time | first=Malcolm W. | last=Browne | date=1990-08-21 | accessdate=2010-05-22}}</ref> as explained in [[Werner Heisenberg]]'s [[uncertainty principle]]. 

This allows the creation of particle-antiparticle pairs of virtual particles. The effects of these particles are measurable, for example, in the effective charge of the electron, different from its "naked" charge.

Quantum fluctuations may have been necessary in the origin of the structure of the [[universe]]: according to the model of expansive [[cosmic inflation|inflation]] the ones that existed when inflation began were amplified and formed the seed of all current observed structure. [[Vacuum energy]] may also be responsible for the current [[accelerating expansion of the universe]] ([[cosmological constant]]).

According to one formulation of the uncertainty principle, [[energy]] and [[time]] can be related by the relation<ref>{{citation|first1=Leonid|last1=Mandelshtam|authorlink1=Leonid Mandelshtam|first2=Igor|last2=Tamm|authorlink2=Igor Tamm|url=http://daarb.narod.ru/mandtamm/index-eng.html|title=The uncertainty relation between energy and time in nonrelativistic quantum mechanics|journal=Izv. Akad. Nauk SSSR (ser. Fiz.)|volume=9|pages=122–128|year=1945}}. English translation: J. Phys. (USSR) '''9''', 249–254 (1945).</ref>

:<math>\Delta E \Delta t \geq {\hbar \over 2 }</math>.
where [[Planck constant|''ħ''/2]] ≈ 5,27286×10<sup>-35</sup> Js

In the modern view, energy is always [[Conservation law (physics)|conserved]], but because the [[particle number operator]] does not [[commutative operation|commute]] with a field's [[Hamiltonian (quantum mechanics)|Hamiltonian]] or energy operator, the field's lowest-energy or [[ground state]], often called the [[vacuum state]], is not, as one might expect from that name, a state with no particles, but rather a [[quantum superposition]] of particle number [[eigenstate]]s with 0, 1, 2...etc. particles.

==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 <math>mc^2</math>). When the mass is very large like a [[Macroscopic scale|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]]{{how|date=June 2012}} of a quantum field (at least for a free field; for interacting fields, [[renormalization]] substantially complicates matters). For the [[Klein–Gordon equation|quantized Klein–Gordon field]] in the [[vacuum state]], we can calculate the probability density that we would observe a configuration <math>{\displaystyle\varphi_t(x)}</math> at a time <math>t</math> in terms of its Fourier transform <math>{\displaystyle\tilde\varphi_t(k)}</math> to be
:<math>\rho_0[\varphi_t] = \exp{\left[-\frac{1}{\hbar}
        \int\frac{d^3k}{(2\pi)^3}
            \tilde\varphi_t^*(k)\sqrt{|k|^2+m^2}\;\tilde \varphi_t(k)\right]}.
</math>
In contrast, for the [[Klein–Gordon equation|classical Klein–Gordon field]] at non-zero temperature, the [[Gibbs state|Gibbs probability density]] that we would observe a configuration <math>{\displaystyle\varphi_t(x)}</math> at a time <math>t</math> is
:<math>\rho_E[\varphi_t] = \exp{[-H[\varphi_t]/k_\mathrm{B}T]}=\exp{\left[-\frac{1}{k_\mathrm{B}T} \int\frac{d^3k}{(2\pi)^3}
            \tilde\varphi_t^*(k){\scriptstyle\frac{1}{2}}(|k|^2+m^2)\;\tilde \varphi_t(k)\right]}.
</math>
The amplitude of quantum fluctuations is controlled by [[planck constant|Planck's constant]] <math>\hbar</math>, just as the amplitude of thermal fluctuations is controlled by <math>k_\mathrm{B}T</math>, where <math>k_\mathrm{B}</math> is [[boltzmann constant|Boltzmann's constant]]. Note that the following three points are closely related:

# Planck's constant has units of [[Action (physics)|action]] (joule-seconds) instead of units of energy (joules),
# the quantum kernel is <math> \sqrt{|k|^2+m^2}</math> instead of <math> {\scriptstyle\frac{1}{2}}(|k|^2+m^2)</math> (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),{{citation needed|date=May 2015}}
# the quantum vacuum state is [[Lorentz invariance|Lorentz invariant]] (although not manifestly in the above), whereas the classical thermal state is not (the classical dynamics is Lorentz invariant, but the [[Gibbs state|Gibbs probability density]] is not a Lorentz invariant initial condition).

We can construct a [[Field (physics)#Continuous random fields|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 &mdash; 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==
{{Expand section|date=April 2017}}
{{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<ref>{{cite web|last1=Kennedy|first1=James (Jim) E.|title=Nature and Meaning of Information in Physics|url=http://science.jeksite.org/info1/pages/page4.htm|website=science.jeksite.org|accessdate=22 April 2017}}</ref>
* 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 counterintuitive quantum behavior.
:In the [[double slit experiment]] each particle makes an unpredictable choice between alternative possibilities, consistent with an interference pattern with the inherent fluctuations of the underlying quantum  field rendering the electron to do so.<ref>{{cite arxiv|author=[[Mani Lal Bhaumik]]|title=Comprehending Quantum Theory from Quantum Fields|eprint=1310.1251|date=4 October 2013|class=physics.gen-ph}}</ref>
: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.<ref>{{cite arxiv|author=[[Mani Lal Bhaumik]]|title=Reality of the wave function and quantum entanglement|eprint=1402.4764|date=15 December 2013|class=physics.gen-ph}}</ref>

== See also ==
* [[Casimir effect]]
* [[Cosmic microwave background]]
* [[Quantum annealing]]
* [[Quantum foam]]
* [[Virtual particle]]
* [[Virtual black hole]]
* [[Stochastic interpretation]]
* [[Zitterbewegung]]

==References==
{{reflist}}


[[Category:Quantum mechanics]]
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