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{{Short description|Random change in the energy inside a volume}}
{{For|related articles|Quantum vacuum (disambiguation)}}
{{use dmy dates|date=July 2020}}
[[File:Quantum Fluctuations.gif|thumb|upright=1|3D visualization of quantum fluctuations of the quantum chromodynamics [[QCD vacuum|(QCD) vacuum]]<ref>{{Cite web|title=Derek Leinweber|url=http://www.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/ImprovedOperators/index.html|access-date=2020-12-13|website=www.physics.adelaide.edu.au}}</ref>]]

In [[quantum physics]], a '''quantum fluctuation''' (also known as a '''vacuum state fluctuation''' or '''vacuum fluctuation''') is the temporary random change in the amount of energy in a point in [[space]],<ref name="Pahlavani">
{{cite book
 | last1  = Pahlavani
 | first1 = Mohammad Reza 
 | title  = Selected Topics in Applications of Quantum Mechanics
 | publisher = BoD
 | date   = 2015
 | pages  = 118
 | url    = https://books.google.com/books?id=MiyQDwAAQBAJ&q=%22virtual+particles%22+%22conservation+of+energy%22&pg=PA118
 | isbn   =  9789535121268
 }}</ref> as prescribed by [[Werner Heisenberg]]'s [[uncertainty principle]]. They are minute random fluctuations in the values of the fields which represent elementary particles, such as [[electric field|electric]] and [[magnetic field]]s which represent the [[electromagnetic force]] carried by [[photon]]s, [[W and Z boson|W and Z fields]] which carry the [[weak force]], and [[gluon]] fields which carry the [[strong force]].<ref name="Pagels">
{{cite book
 | last1  = Pagels
 | first1 = Heinz R. 
 | title  = The Cosmic Code: Quantum Physics as the Language of Nature
 | publisher = Courier Corp.
 | date   = 2012
 | pages  = 274–278
 | url    = https://books.google.com/books?id=6tLCAgAAQBAJ&q=%22vacuum+fluctuations%22+%22conservation+of+energy%22&pg=PA275
 | isbn   =  9780486287324
 }}</ref>

The [[uncertainty principle]] states the uncertainty in [[energy]] and [[time]] can be related by<ref>{{cite journal |first1=Leonid |last1=Mandelshtam |author-link1=Leonid Mandelshtam |first2=Igor |last2=Tamm |author-link2=Igor Tamm |year=1945 |title=Соотношение неопределённости энергия-время в нерелятивистской квантовой механике |trans-title=The uncertainty relation between energy and time in non-relativistic quantum mechanics |journal=Izv. Akad. Nauk SSSR (Ser. Fiz.) |volume=9 |pages=122–128 |url=http://daarb.narod.ru/mandtamm/index-eng.html |language=ru}} English translation: {{cite journal |year=1945 |title=The uncertainty relation between energy and time in non-relativistic quantum mechanics |journal=J. Phys. (USSR) |volume=9 |pages=249–254 |language=en}}</ref> <math>\Delta E \, \Delta t \geq \tfrac{1}{2}\hbar~</math>, where {{sfrac|1|2}}[[Planck constant|{{mvar|ħ}}]] ≈ {{val|5.27286|e=−35|u=J.s}}. This means that pairs of virtual particles with energy <math>\Delta E</math> and lifetime shorter than <math>\Delta t</math> 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|"bare" mass]] and charge of elementary particles would be 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 [[Quantum vacuum state|vacuum]] fluctuations was the [[Lamb shift]] in hydrogen. In July 2020, scientists reported that quantum vacuum fluctuations can influence the motion of macroscopic, human-scale objects by measuring correlations below the [[standard quantum limit]] between the position/momentum uncertainty of the mirrors of [[LIGO]] and the photon number/phase uncertainty of light that they reflect.<ref>{{cite news |title=Quantum fluctuations can jiggle objects on the human scale |url=https://phys.org/news/2020-07-quantum-fluctuations-jiggle-human-scale.html |access-date=15 August 2020 |work=phys.org |language=en}}</ref><ref>{{cite news |title=LIGO reveals quantum correlations at work in mirrors weighing tens of kilograms |url=https://physicsworld.com/a/ligo-reveals-quantum-correlations-at-work-in-mirrors-weighing-tens-of-kilograms/ |access-date=15 August 2020 |work=Physics World |date=1 July 2020}}</ref><ref>{{cite journal |last1=Yu |first1=Haocun |last2=McCuller |first2=L. |last3=Tse |first3=M. |last4=Kijbunchoo |first4=N. |last5=Barsotti |first5=L. |last6=Mavalvala |first6=N. |title=Quantum correlations between light and the kilogram-mass mirrors of LIGO |journal=Nature |date=July 2020 |volume=583 |issue=7814 |pages=43–47 |doi=10.1038/s41586-020-2420-8 |pmid=32612226 |url=https://www.nature.com/articles/s41586-020-2420-8 |language=en |issn=1476-4687|arxiv=2002.01519 |bibcode=2020Natur.583...43Y |s2cid=211031944 }}</ref>

== Field fluctuations ==
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 theory|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:<ref>{{cite arXiv |last=Morgan |first=Peter |title=A classical perspective on nonlocality in quantum field theory |year=2001 |language=en |eprint=quant-ph/0106141 <!--|bibcode=2001quant.ph..6141M--> }}</ref> For the [[Klein–Gordon equation|quantized Klein–Gordon field]] in the [[Quantum vacuum state|vacuum state]], we can calculate the probability density that we would observe a configuration <math>\varphi_t(x)</math> at a time {{mvar|t}} in terms of its [[Fourier transform]] <math>\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>\varphi_t(x)</math> at a time <math>t</math> is
: <math>\rho_E[\varphi_t] = \exp\big[-H[\varphi_t]/k_\text{B}T\big] = \exp{\left[-\frac{1}{k_\text{B}T} \int\frac{d^3k}{(2\pi)^3}
            \tilde\varphi_t^*(k) \frac{1}{2}\left(|k|^2 + m^2\right)\,\tilde\varphi_t(k)\right]}.</math>

These probability distributions illustrate that every possible configuration of the field is possible, with the amplitude of quantum fluctuations controlled by the [[Planck constant]] <math>\hbar</math>, just as the amplitude of thermal fluctuations is controlled by <math>k_\text{B}T</math>, where {{mvar|k}}{{sub|B}} is the [[Boltzmann constant]]. Note that the following three points are closely related:
# the Planck 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>\tfrac{1}{2} \big(|k|^2 + m^2\big)</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 probability density is not a Lorentz-invariant initial condition).

A [[Field (physics)#Continuous random fields|classical continuous random field]] can be constructed 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&nbsp;– in quantum-mechanical terms they always commute).

== See also ==
{{cols}}
* [[Cosmic microwave background]]
* [[False vacuum]]
* [[Hawking radiation]]
* [[Quantum annealing]]
* [[Quantum foam]]
* [[Stochastic interpretation]]
* [[Vacuum energy]]
* [[Vacuum polarization]]
* [[Virtual black hole]]
* [[Zitterbewegung]]
{{colend}}

== References ==
{{reflist|25em}}

{{Quantum field theories}}
{{Quantum mechanics topics}}

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