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Yang–Mills theory
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== Open problems == Yang–Mills theories met with general acceptance in the physics community after [[Gerard 't Hooft]], in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisor [[Martinus Veltman]].<ref> {{cite journal | last1 = 't Hooft | first1 = G. |author1-link = Gerard 't Hooft | last2 = Veltman | first2 = M. |author2-link = Martinus Veltman | doi = 10.1016/0550-3213(72)90279-9 | title = Regularization and renormalization of gauge fields | journal = Nuclear Physics B | volume = 44 | issue = 1 | pages = 189–213 | year = 1972 | bibcode = 1972NuPhB..44..189T | hdl = 1874/4845 | url = https://repositorio.unal.edu.co/handle/unal/81144 | hdl-access = free }} </ref> Renormalizability is obtained even if the gauge bosons described by this theory are massive, as in the electroweak theory, provided the mass is only an "acquired" one, generated by the [[Higgs mechanism]]. The mathematics of the Yang–Mills theory is a very active field of research, yielding e.g. invariants of differentiable structures on four-dimensional manifolds via work of [[Simon Donaldson]]. Furthermore, the field of Yang–Mills theories was included in the [[Clay Mathematics Institute]]'s list of "[[Millennium Prize Problems]]". Here [[Yang–Mills existence and mass gap|the prize-problem]] consists, especially, in a proof of the conjecture that the lowest excitations of a pure Yang–Mills theory (i.e. without matter fields) have a finite mass-gap with regard to the vacuum state. Another open problem, connected with this conjecture, is a proof of the [[Color confinement|confinement]] property in the presence of additional fermions. In physics the survey of Yang–Mills theories does not usually start from perturbation analysis or analytical methods, but more recently from systematic application of numerical methods to [[lattice gauge theory|lattice gauge theories]].
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