Editing Instanton (section)

{{short description|Solitons in Euclidean spacetime}}
{{multiple image|perrow = 2|total_width=300
| image1 = -y-(x^2+y^2+1) plot; BPST instanton.png
| image2 = X-(x^2+y^2+1) plot; BPST instanton.png
| image3 = Curvature of BPST Instanton.png
| image4 = BPST on sphere.png
| footer = The ''dx<sup>1</sup>⊗σ<sub>3</sub>'' coefficient of a [[BPST instanton]] on the ''(x<sup>1</sup>,x<sup>2</sup>)''-slice of '''R'''<sup>4</sup> where ''σ<sub>3</sub>'' is the third [[Pauli matrix]] (top left). The ''dx<sup>2</sup>⊗σ<sub>3</sub>'' coefficient (top right). These coefficients determine the restriction of the BPST instanton ''A'' with ''g=2,ρ=1,z=0'' to this slice. The corresponding field strength centered around ''z=0'' (bottom left). A visual representation of the field strength of a BPST instanton with center ''z'' on the [[compactification (mathematics)|compactification]] ''S<sup>4</sup>'' of '''R'''<sup>4</sup> (bottom right). The BPST instanton is a classical instanton solution to the [[Yang–Mills equations]] on '''R'''<sup>4</sup>.
}}
An '''instanton''' (or '''pseudoparticle'''<ref>Instantons in Gauge Theories. Edited by Mikhail A. Shifman. World Scientific, 1994.</ref><ref>Interactions Between Charged Particles in a Magnetic Field. By Hrachya Nersisyan, Christian Toepffer, Günter Zwicknagel. Springer, Apr 19, 2007. Pg 23</ref><ref>Large-Order Behaviour of Perturbation Theory. Edited by J.C. Le Guillou, J. Zinn-Justin. Elsevier, Dec 2, 2012. Pg. 170.</ref>) is a notion appearing in theoretical and [[mathematical physics]]. An instanton is a classical solution to [[equations of motion]] with a finite, [[Vacuum state|non-zero action]], either in [[quantum mechanics]] or in [[quantum field theory]]. More precisely, it is a solution to the equations of motion of the [[classical field theory]] on a [[Euclidean space|Euclidean]] [[spacetime]].<ref name=":0">{{Cite journal |last=Vaĭnshteĭn |first=A. I. |last2=Zakharov |first2=Valentin I. |last3=Novikov |first3=Viktor A. |last4=Shifman |first4=Mikhail A. |date=1982-04-30 |title=ABC of instantons |url=https://iopscience.iop.org/article/10.1070/PU1982v025n04ABEH004533/meta |journal=Soviet Physics Uspekhi |language=en |volume=25 |issue=4 |pages=195 |doi=10.1070/PU1982v025n04ABEH004533 |issn=0038-5670|url-access=subscription }}</ref>

In such quantum theories, solutions to the equations of motion may be thought of as [[critical point (mathematics)|critical points]] of the [[Action (physics)|action]]. The critical points of the action may be [[maxima and minima|local maxima]] of the action, [[maxima and minima|local minima]], or [[saddle point]]s.  Instantons are important in [[quantum field theory]] because:

* they appear in the [[functional integration|path integral]] as the leading quantum corrections to the classical behavior of a system, and
* they can be used to study the tunneling behavior in various systems such as a [[Yang–Mills theory]].

Relevant to [[Dynamics (mechanics)|dynamics]], families of instantons permit that instantons, i.e. different critical points of the equation of motion, be related to one another. In physics instantons are particularly important because the condensation of instantons (and noise-induced anti-instantons) is believed to be the explanation of the [[Supersymmetric theory of stochastic dynamics#Classification of stochastic dynamics|noise-induced chaotic phase]] known as [[self-organized criticality]].