Editing Yang–Mills theory (section)

== History and qualitative description ==
=== Gauge theory in electrodynamics ===
All known fundamental interactions can be described in terms of gauge theories, but working this out took decades.<ref name=oraifearthagh>{{Cite journal |last1=O’Raifeartaigh |first1=Lochlainn |last2=Straumann |first2=Norbert |date=2000-01-01 |title=Gauge theory: Historical origins and some modern developments |url=https://link.aps.org/doi/10.1103/RevModPhys.72.1 |journal=Reviews of Modern Physics |language=en |volume=72 |issue=1 |pages=1–23 |doi=10.1103/RevModPhys.72.1 |issn=0034-6861|url-access=subscription }}</ref> [[Hermann Weyl]]'s pioneering work on this project started in 1915 when his colleague [[Emmy Noether]] proved that every conserved physical quantity has a matching symmetry, and culminated in 1928 when he published his book applying the geometrical theory of symmetry ([[group theory]]) to quantum mechanics.<ref name=Baggott40/>{{rp|194}} Weyl named the relevant symmetry in [[Noether's theorem]] the "gauge symmetry", by analogy to distance standardization in [[Track gauge|railroad gauges]].

[[Erwin Schrödinger]] in 1922, three years before working on his equation, connected Weyl's group concept to electron charge. Schrödinger showed that the group <math>U(1)</math> produced a phase shift <math>e^{i\theta}</math> in electromagnetic fields that matched the conservation of electric charge.<ref name=Baggott40/>{{rp|198}} As the theory of [[quantum electrodynamics]] developed in the 1930's and 1940's the <math>U(1)</math> group transformations played a central role. Many physicists thought there must be an analog for the dynamics of nucleons. 
[[Chen Ning Yang]] in particular was obsessed with this possibility.

=== 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]].

=== Parallel work on non-Abelian gauge theories ===
In 1953, in a private correspondence, [[Wolfgang Pauli]] formulated a six-dimensional theory of [[Einstein's field equations]] of [[general relativity]], extending the five-dimensional theory of [[Kaluza–Klein theory|Kaluza, Klein]], [[Vladimir Fock|Fock]], and others to a higher-dimensional internal space.<ref name = Straumann>{{cite arXiv |last=Straumann |first=N. |year=2000 |title=On Pauli's invention of non-abelian Kaluza-Klein Theory in 1953 |eprint=gr-qc/0012054}}</ref> However, there is no evidence that Pauli developed the [[Lagrangian (field theory)|Lagrangian]] of a [[gauge field]] or the quantization of it. Because Pauli found that his theory "leads to some rather unphysical shadow particles", he refrained from publishing his results formally.<ref name=Straumann/> Although Pauli did not publish his six-dimensional theory, he gave two seminar lectures about it in Zürich in November 1953.<ref name = Straumann />

In January&nbsp;1954 [[Ronald Shaw (physicist)|Ronald Shaw]], a graduate student at the [[University of Cambridge]] also developed a non-Abelian gauge theory for nuclear forces.<ref name=shaw>
{{cite journal
 |last1=Atiyah |first1=M. |authorlink1=Michael Atiyah
 |date=2017
 |title=Ronald Shaw 1929–2016 by Michael Atiyah (1954)
 |journal=Trinity College Annual Record
 |volume=2017 |issue= |pages=137–146
 |type=memorial
 |url=https://issuu.com/trinityalumni/docs/trinity_ar_2017_web
}}
</ref>
However, the theory needed massless particles in order to maintain [[gauge invariance]].  Since no such massless particles were known at the time, Shaw and his supervisor [[Abdus Salam]] chose not to publish their work.<ref name=shaw/>
Shortly after Yang and Mills published their paper in October&nbsp;1954, Salam encouraged Shaw to publish his work to mark his contribution. Shaw declined, and instead it only forms a chapter of his PhD thesis published in 1956.<ref>
{{cite thesis
 |last=Shaw |first=Ronald
 |date=September 1956
 |title=The problem of particle types and other contributions to the theory of elementary particles
 |degree=Ph.D.
 |at=ch.&nbsp;3, pp.&nbsp;34–46
 |publisher=[[University of Cambridge]]
}}
</ref><ref>
{{cite book
 |last=Fraser |first=Gordon
 |date=2008
 |title=Cosmic Anger: Abdus Salam – the first Muslim Nobel scientist
 |place=Oxford, UK
 |publisher=Oxford University Press
 |isbn=978-0199208463
 |page=117
}}
</ref>