Editing 't Hooft–Polyakov monopole (section)

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{{short description|Yang–Mills–Higgs magnetic monopole}}
In [[theoretical physics]], the '''{{'}}t Hooft–Polyakov monopole''' is a [[topological soliton]] similar to the [[Dirac monopole]] but without the [[Dirac string]]. It arises in the case of a [[Yang–Mills theory]] with a [[gauge group]] <math>G</math>, coupled to a [[Higgs field]] which [[spontaneous symmetry breaking|spontaneously breaks]] it down to a smaller group <math>H</math> via the [[Higgs mechanism]]. It was first found independently by [[Gerardus 't Hooft|Gerard 't Hooft]] and [[Alexander Markovich Polyakov|Alexander Polyakov]].<ref>{{cite journal |last='t Hooft |first=G. |authorlink=Gerardus 't Hooft |year=1974 |title=Magnetic monopoles in unified gauge theories |journal=Nuclear Physics B |doi=10.1016/0550-3213(74)90486-6 |bibcode=1974NuPhB..79..276T |volume=79 |issue=2 |pages=276–284|hdl=1874/4686 |url=https://dspace.library.uu.nl/bitstream/1874/4686/2/14058.pdf }}</ref><ref>{{cite journal|last=Polyakov|first=A. M.|year=1974|title=Particle Spectrum in the Quantum Field Theory|url=http://www.jetpletters.ru/ps/1789/article_27297.pdf|journal=JETP Letters|volume=20|issue=6|pages=194–195|issn=0370-274X|authorlink=Alexander Markovich Polyakov}}</ref>

Unlike the Dirac monopole, the 't Hooft–Polyakov monopole is a smooth solution with a finite total [[energy]]. The solution is localized around <math>r=0</math>. Very far from the origin, the gauge group <math>G</math> is broken to <math>H</math>, and the 't Hooft–Polyakov monopole reduces to the Dirac monopole.

However, at the origin itself, the <math>G</math> [[gauge symmetry]] is unbroken and the solution is non-singular also near the origin. The Higgs field <math>H_i (i=1,2,3)</math>,
is proportional to <math>x_i f(|x|)</math>,
where the adjoint indices are identified with the three-dimensional spatial indices. The gauge field at infinity is such that the Higgs field's dependence on the angular directions is pure gauge. The precise configuration for the Higgs field and the gauge field near the origin is such that it satisfies the full [[Yang–Mills–Higgs equations]] of motion.