Editing Yang–Mills theory (section)

=== Yang and Mills find the nuclear force gauge theory ===
Yang's core idea was to look for a conserved quantity in nuclear physics comparable to electric charge and use it to develop a corresponding gauge theory comparable to electrodynamics. He settled on conservation of [[isospin]], a quantum number that distinguishes a neutron from a proton, but he made no progress on a theory.<ref name=Baggott40/>{{rp|200}} Taking a break from Princeton in the summer of 1953, Yang met a collaborator who could help: [[Robert Mills (physicist)|Robert Mills]]. As Mills himself describes:<blockquote>"During the academic year 1953–1954, Yang was a visitor to [[Brookhaven National Laboratory]] ... I was at Brookhaven also ... and was assigned to the same office as Yang. Yang, who has demonstrated on a number of occasions his generosity to physicists beginning their careers, told me about his idea of generalizing gauge invariance and we discussed it at some length ... I was able to contribute something to the discussions, especially with regard to the quantization procedures, and to a small degree in working out the formalism; however, the key ideas were Yang's."<ref>
{{cite book
 |last1=Gray |first1=Jeremy
 |last2=Wilson |first2=Robin
 |date=2012-12-06
 |title=Mathematical Conversations: Selections from the ''Mathematical Intelligencer''
 |publisher=Springer Science & Business Media
 |isbn=9781461301950
 |page=63
 |url=https://books.google.com/books?id=e0TTBwAAQBAJ&q=during+the+academic+year+1953-1954+yang+was+a+visitor+to+the+brookhaven+national+laboratory...+I+was+at+brookhaven+also...+and+was+assigned&pg=PA63
 |via=Google Books
}}
</ref>
</blockquote>

In the summer 1953, Yang and Mills extended the concept of gauge theory for [[abelian group]]s, e.g. [[quantum electrodynamics]], to non-abelian groups, selecting the group {{math| [[SU(2)]] }} to provide an explanation for isospin conservation in collisions involving the strong interactions. Yang's presentation of the work at Princeton in February&nbsp;1954 was challenged by Pauli, asking about the mass in the field developed with the gauge invariance idea.<ref name=Baggott40>{{Cite book |last=Baggott |first=J.E. |year=2013 |title=The Quantum Story: A history in 40&nbsp;moments |edition=Impression&nbsp;3 |place=Oxford, UK |publisher=Oxford University Press |isbn=978-0-19-956684-6 }}</ref>{{rp|202}} Pauli knew that this might be an issue as he had worked on applying gauge invariance but chose not to publish it, viewing the massless excitations of the theory to be "unphysical 'shadow particles'".<ref name=oraifearthagh/>{{rp|13}} Yang and Mills published in October&nbsp;1954; near the end of the paper, they admit:
{{blockquote| We next come to the question of the mass of the <math>b</math> quantum, to which we do not have a satisfactory answer.<ref name=YM>
{{cite journal
 |first1=C.N. |last1=Yang   |author-link1=Chen-Ning Yang
 |first2=R.   |last2=Mills  |author-link2=Robert Mills (physicist)
 |year=1954
 |title=Conservation of isotopic spin and isotopic gauge invariance
 |journal=[[Physical Review]]
 |volume=96 |issue=1 |pages=191–195
 |doi=10.1103/PhysRev.96.191 |doi-access=free
 |bibcode = 1954PhRv...96..191Y
}}
</ref>}}
This problem of unphysical massless excitation blocked further progress.<ref name=Baggott40/>

The idea was set aside until 1960, when the concept of particles acquiring mass through [[symmetry breaking]] in massless theories was put forward, initially by [[Jeffrey Goldstone]], [[Yoichiro Nambu]], and [[Giovanni Jona-Lasinio]]. This prompted a significant restart of Yang–Mills theory studies that proved successful in the formulation of both [[electroweak interaction|electroweak unification]] and [[quantum chromodynamics]] (QCD). The electroweak interaction is described by the gauge group {{math|SU(2) × U(1)}}, while QCD is an {{math| [[SU(3)]] }} Yang–Mills theory. The massless gauge bosons of the electroweak {{math|SU(2) × U(1)}} mix after [[spontaneous symmetry breaking]] to produce [[W and Z bosons|the three massive bosons]] of the weak interaction ({{math|{{SubatomicParticle|W boson+}}}}, {{math|{{SubatomicParticle|W boson-}}}}, and {{math|{{SubatomicParticle|Z boson0}}}}) as well as the still-massless [[photon]] field. The dynamics of the photon field and its interactions with matter are, in turn, governed by the {{math|U(1)}} gauge theory of quantum electrodynamics. The [[Standard Model]] combines the [[strong interaction]] with the unified electroweak interaction (unifying the [[weak interaction|weak]] and [[electromagnetic interaction]]) through the symmetry group {{math|SU(3) × SU(2) × U(1)}}. In the current epoch the strong interaction is not unified with the electroweak interaction, but from the observed [[Running coupling|running of the coupling]] constants it is believed{{citation needed|reason=By whom?|date=January 2016}} they all converge to a single value at very high energies.

[[Phenomenology (particle physics)|Phenomenology]] at lower energies in quantum chromodynamics is not completely understood due to the difficulties of managing such a theory with a strong coupling. This may be the reason why [[color confinement|confinement]] has not been theoretically proven, though it is a consistent experimental observation.  This shows why QCD confinement at low energy is a mathematical problem of great relevance, and why the [[Yang–Mills existence and mass gap]] problem is a [[Millennium Prize Problems|Millennium Prize Problem]].