Editing Quantum fluctuation (section)

== 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).