Yang–Mills theory is a quantum field theory for nuclear binding devised byChen Ning Yang andRobert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is agauge theory based on aspecial unitary groupSU(n), or more generally anycompact Lie group. A Yang–Mills theory seeks to describe the behavior of elementary particles using thesenon-abelianLie groups and is at the core of the unification of theelectromagnetic force andweak forces (i.e.U(1) × SU(2)) as well asquantum chromodynamics, the theory of thestrong force (based onSU(3)). Thus it forms the basis of the understanding of theStandard Model of particle physics.

All known fundamental interactions can be described in terms of gauge theories, but working this out took decades.^[2]Hermann Weyl's pioneering work on this project started in 1915 when his colleagueEmmy 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.^[3]: 194 Weyl named the relevant symmetry inNoether's theorem the "gauge symmetry", by analogy to distance standardization inrailroad 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${\displaystyle \mathrm{U}(1)}$ produced a phase shift${\displaystyle e^{i\theta }}$ in electromagnetic fields that matched the conservation of electric charge.^[3]: 198 As the theory ofquantum electrodynamics developed in the 1930s and 1940s the${\displaystyle \mathrm{U}(1)}$ 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 interested in this possibility.

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 ofisospin, a quantum number that distinguishes a neutron from a proton, but he made no progress on a theory.^[3]: 200 Taking a break from Princeton in the summer of 1953, Yang met a collaborator who could help:Robert Mills. As Mills himself describes:

"During the academic year 1953–1954, Yang was a visitor toBrookhaven 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."^[4]

In the summer of 1953, Yang and Mills extended the concept of gauge theory forabelian groups, e.g.quantum electrodynamics, to non-abelian groups, selecting the groupSU(2) to provide an explanation for isospin conservation in collisions involving the strong interactions. Yang's presentation of the work at Princeton in February 1954 was challenged by Pauli, asking about the mass in the field developed with the gauge invariance idea.^[3]: 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'".^[2]: 13 Yang and Mills published in October 1954; near the end of the paper, they admit:

We next come to the question of the mass of the${\displaystyle b}$ quantum, to which we do not have a satisfactory answer.^[5]

This problem of unphysical massless excitation blocked further progress.^[3]

The idea was set aside until 1960, when the concept of particles acquiring mass throughsymmetry breaking in massless theories was put forward, initially byJeffrey Goldstone,Yoichiro Nambu, andGiovanni Jona-Lasinio. This prompted a significant restart of Yang–Mills theory studies that proved successful in the formulation of bothelectroweak unification andquantum chromodynamics (QCD). The electroweak interaction is described by the gauge groupSU(2) × U(1), while QCD is anSU(3) Yang–Mills theory. The massless gauge bosons of the electroweakSU(2) × U(1) mix afterspontaneous symmetry breaking to producethe three massive bosons of the weak interaction (W⁺
,W⁻
, andZ⁰
) as well as the still-masslessphoton field. The dynamics of the photon field and its interactions with matter are, in turn, governed by theU(1) gauge theory of quantum electrodynamics. TheStandard Model combines thestrong interaction with the unified electroweak interaction (unifying theweak andelectromagnetic interaction) through the symmetry groupSU(3) × SU(2) × U(1). In the current epoch the strong interaction is not unified with the electroweak interaction, but from the observedrunning of the coupling constants it is believed^{[citation needed]} they all converge to a single value at very high energies.

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 whyconfinement 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 theYang–Mills existence and mass gap problem is aMillennium Prize Problem.

In 1953, in a private correspondence,Wolfgang Pauli formulated a six-dimensional theory ofEinstein's field equations ofgeneral relativity, extending thefive-dimensional theory ofTheodor Kaluza,Oskar Klein,Vladimir Fock, and others to a higher-dimensional internal space.^[6] However, there is no evidence that Pauli developed theLagrangian of agauge 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.^[6] Although Pauli did not publish his six-dimensional theory, he gave two seminar lectures about it in Zürich in November 1953.^[6]

In January 1954Ronald Shaw, a graduate student at theUniversity of Cambridge also developed a non-Abelian gauge theory for nuclear forces.^[7]However, the theory needed massless particles in order to maintaingauge invariance. Since no such massless particles were known at the time, Shaw and his supervisorAbdus Salam chose not to publish their work.^[7]Shortly after Yang and Mills published their paper in October 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.^[8][9]

Thedx¹⊗σ₃ coefficient of aBPST instanton on the(x¹,x²)-slice ofℝ⁴ whereσ₃ is the thirdPauli matrix (top left). Thedx²⊗σ₃ coefficient (top right). These coefficients determine the restriction of the BPST instantonA withg=2,ρ=1,z=0 to this slice. The corresponding field strength centered aroundz=0 (bottom left). A visual representation of the field strength of a BPST instanton with centerz on thecompactificationS⁴ ofℝ⁴ (bottom right). The BPST instanton is a classicalinstanton solution to theYang–Mills equations onℝ⁴.

Yang–Mills theories are special examples of gauge theories with a non-abelian symmetry group given by theLagrangian

${\displaystyle {\mathcal{L}}_{\mathrm{g}\mathrm{f}}=-\frac{1}{2}\mathrm{tr}(F^2)=-\frac{1}{4}{F}^{a\mu \nu }{F}_{\mu \nu }^a}$

with the generators${\displaystyle T^a}$ of theLie algebra, indexed bya, corresponding to theF-quantities (thecurvature or field-strength form) satisfying

${\displaystyle \mathrm{tr}\left(T^a{T}^b\right)=\frac{1}{2}{\delta }^{ab},\left[T^a,T^b\right]=i{f}^{abc}{T}^c.}$

Here, thef ^abc arestructure constants of the Lie algebra (totally antisymmetric if the generators of the Lie algebra are normalised such that${\displaystyle \mathrm{tr}(T^a{T}^b)}$ is proportional to${\displaystyle {\delta }^{ab}}$), thecovariant derivative is defined as

${\displaystyle D_{\mu }=I{\mathrm{\partial }}_{\mu }-ig{T}^a{A}_{\mu }^a,}$

I is theidentity matrix (matching the size of the generators),${\displaystyle A_{\mu }^a}$ is thevector potential, andg is thecoupling constant. In four dimensions, the coupling constantg is a pure number and for aSU(n) group one has${\displaystyle a,b,c=1\dots n^2-1.}$

The relation

${\displaystyle F_{\mu \nu }^a={\mathrm{\partial }}_{\mu }{A}_{\nu }^a-{\mathrm{\partial }}_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c}$

can be derived by thecommutator

${\displaystyle \left[D_{\mu },D_{\nu }\right]=-ig{T}^a{F}_{\mu \nu }^a.}$

The field has the property of being self-interacting and the equations of motion that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this theory only byperturbation theory with small nonlinearities.^{[citation needed]}

Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial fora indices (e.g.${\displaystyle f^{abc}=f_{abc}}$), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature,${\displaystyle {\eta }_{\mu \nu }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(+---).}$

From the given Lagrangian one can derive the equations of motion given by

${\displaystyle {\mathrm{\partial }}^{\mu }{F}_{\mu \nu }^a+g{f}^{abc}{A}^{\mu b}{F}_{\mu \nu }^c=0.}$

Putting${\displaystyle F_{\mu \nu }=T^a{F}_{\mu \nu }^a,}$ these can be rewritten as

${\displaystyle {\left(D^{\mu }{F}_{\mu \nu }\right)}^a=0.}$

ABianchi identity holds

${\displaystyle {\left(D_{\mu }{F}_{\nu \kappa }\right)}^a+{\left(D_{\kappa }{F}_{\mu \nu }\right)}^a+{\left(D_{\nu }{F}_{\kappa \mu }\right)}^a=0}$

which is equivalent to theJacobi identity

${\displaystyle \left[D_{\mu },\left[D_{\nu },D_{\kappa }\right]\right]+\left[D_{\kappa },\left[D_{\mu },D_{\nu }\right]\right]+\left[D_{\nu },\left[D_{\kappa },D_{\mu }\right]\right]=0}$

since${\displaystyle \left[D_{\mu },F_{\nu \kappa }^a\right]=D_{\mu }{F}_{\nu \kappa }^a.}$ Define thedual strength tensor${\displaystyle {\tilde{F}}^{\mu \nu }=\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{F}_{\rho \sigma },}$ then the Bianchi identity can be rewritten as

${\displaystyle D_{\mu }{\tilde{F}}^{\mu \nu }=0.}$

A source${\displaystyle J_{\mu }^a}$ enters into the equations of motion as

${\displaystyle {\mathrm{\partial }}^{\mu }{F}_{\mu \nu }^a+g{f}^{abc}{A}^{b\mu }{F}_{\mu \nu }^c=-J_{\nu }^a.}$

Note that the currents must properly change under gauge group transformations.

We give here some comments about the physical dimensions of the coupling. InD dimensions, the field scales as${\displaystyle \left[A\right]=\left[L^{\left(\frac{2-D}{2}\right)}\right]}$ and so the coupling must scale as${\displaystyle \left[g^2\right]=\left[L^{\left(D-4\right)}\right].}$ This implies that Yang–Mills theory is notrenormalizable for dimensions greater than four. Furthermore, forD = 4 , the coupling is dimensionless and both the field and the square of the coupling have the same dimensions of the field and the coupling of a massless quarticscalar field theory. So, these theories share thescale invariance at the classical level.

A method of quantizing the Yang–Mills theory is by functional methods, i.e.path integrals. One introduces a generating functional forn-point functions as

${\displaystyle Z[j]=\int [\mathrm{d}A]\exp \left[-\frac{i}{2}\int {\mathrm{d}}^{4}x\mathrm{tr}\left(F^{\mu \nu }{F}_{\mu \nu }\right)+i\int {\mathrm{d}}^{4}x{j}_{\mu }^a(x)A^{a\mu }(x)\right],}$

but this integral has no meaning as it is because the potential vector can be arbitrarily chosen due to thegauge freedom. This problem was already known for quantum electrodynamics but here becomes more severe due to non-abelian properties of the gauge group. A way out has been given byLudvig Faddeev andVictor Popov with the introduction of aghost field (seeFaddeev–Popov ghost) that has the property of being unphysical since, although it agrees withFermi–Dirac statistics, it is a complex scalar field, which violates thespin–statistics theorem. So, we can write the generating functional as

being

${\displaystyle S_F=-\frac{1}{2}\int {\mathrm{d}}^{4}x\mathrm{tr}\left(F^{\mu \nu }{F}_{\mu \nu }\right)}$

for the field,

${\displaystyle S_{gf}=-\frac{1}{2\xi }\int {\mathrm{d}}^{4}x(\mathrm{\partial }\cdot A{)}^2}$

for the gauge fixing and

${\displaystyle S_g=-\int {\mathrm{d}}^{4}x\left({\overline{c}}^a{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}^{\mu }{c}^a+g{\overline{c}}^a{f}^{abc}{\mathrm{\partial }}_{\mu }{A}^{b\mu }{c}^c\right)}$

for the ghost. This is the expression commonly used to derive Feynman's rules (seeFeynman diagram). Here we havec^a for the ghost field whileξ fixes the gauge's choice for the quantization. Feynman's rules obtained from this functional are the following^[10]

Diagram	Momentum-space Feynman rule
Gluon propagator	${\displaystyle D_{\mu \nu }^{ab}(k)=\frac{-i{\delta }^{ab}}{k^2+iϵ}\left(g_{\mu \nu }-(1-\xi )\frac{k_{\mu }{k}_{\nu }}{k^2}\right)}$
3-gluon vertex	${\displaystyle V_{\mu \nu \rho }^{abc}(p,q,r)=g{f}^{abc}[g_{\mu \nu }(p-q{)}_{\rho }+g_{\nu \rho }(q-r{)}_{\mu }+g_{\rho \mu }(r-p{)}_{\nu }],p+q+r=0}$
4-gluon vertex
Ghost propagator	${\displaystyle D_{\mathrm{g}\mathrm{h}}^{ab}(k)=\frac{i{\delta }^{ab}}{k^2+iϵ}}$
Ghost--ghost--gluon (ccg) vertex	${\displaystyle V_{\mu }^{abc}({\overline{c}}^a(p),c^b(q),A_{\mu }^c(r))=-g{f}^{abc}{p}_{\mu },p+q+r=0}$

These rules for Feynman's diagrams can be obtained when the generating functional given above is rewritten as

with

${\displaystyle Z_0[j,\overline{\epsilon },\epsilon ]=\exp \left(-\int {\mathrm{d}}^{4}x{\mathrm{d}}^{4}y{\overline{\epsilon }}^a(x)C^{ab}(x-y){\epsilon }^b(y)\right)\exp \left(\frac{1}{2}\int {\mathrm{d}}^{4}x{\mathrm{d}}^{4}y{j}_{\mu }^a(x)D^{ab\mu \nu }(x-y)j_{\nu }^b(y)\right)}$

being the generating functional of the free theory. Expanding ing and computing thefunctional derivatives, we are able to obtain all then-point functions with perturbation theory. UsingLSZ reduction formula we get from then-point functions the corresponding process amplitudes,cross sections anddecay rates. The theory is renormalizable and corrections are finite at any order of perturbation theory.

For quantum electrodynamics the ghost field decouples because the gauge group is abelian. This can be seen from the coupling between the gauge field and the ghost field that is${\displaystyle {\overline{c}}^a{f}^{abc}{\mathrm{\partial }}_{\mu }{A}^{b\mu }{c}^c.}$ For the abelian case, all the structure constants${\displaystyle f^{abc}}$ are zero and so there is no coupling. In the non-abelian case, the ghost field appears as a useful way to rewrite the quantum field theory without physical consequences on the observables of the theory such as cross sections or decay rates.

One of the most important results obtained for Yang–Mills theory isasymptotic freedom. This result can be obtained by assuming that thecoupling constantg is small (so small nonlinearities), as for high energies, and applyingperturbation theory. The relevance of this result is due to the fact that a Yang–Mills theory that describes strong interaction and asymptotic freedom permits proper treatment of experimental results coming fromdeep inelastic scattering.

To obtain the behavior of the Yang–Mills theory at high energies, and so to prove asymptotic freedom, one applies perturbation theory assuming a small coupling. This is verifieda posteriori in theultraviolet limit. In the opposite limit, the infrared limit, the situation is the opposite, as the coupling is too large for perturbation theory to be reliable. Most of the difficulties that research meets is just managing the theory at low energies. That is the interesting case, being inherent to the description of hadronic matter and, more generally, to all the observed bound states of gluons and quarks and their confinement (seehadrons). The most used method to study the theory in this limit is to try to solve it on computers (seelattice gauge theory). In this case, large computational resources are needed to be sure the correct limit of infinite volume (smaller lattice spacing) is obtained. This is the limit the results must be compared with. Smaller spacing and larger coupling are not independent of each other, and larger computational resources are needed for each. As of today, the situation appears somewhat satisfactory for the hadronic spectrum and the computation of the gluon and ghost propagators, but theglueball andhybrids spectra are yet a questioned matter in view of the experimental observation of such exotic states. Indeed, theσ resonance^[11][12]is not seen in any of such lattice computations and contrasting interpretations have been put forward. This is a hotly debated issue.

Yang–Mills theories met with general acceptance in the physics community afterGerard 't Hooft, in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisorMartinus Veltman.^[13]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 theHiggs 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 ofSimon Donaldson. Furthermore, the field of Yang–Mills theories was included in theClay Mathematics Institute's list of "Millennium Prize Problems". Herethe 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 theconfinement 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 tolattice gauge theories.

• "Yang-Mills field",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• "Yang–Mills theory".DispersiveWiki. Archived fromthe original on 2021-06-03. Retrieved2018-08-30.
• "The Millennium Prize Problems". TheClay Mathematics Institute. Archived fromthe original on 2009-01-16. Retrieved2008-11-24.

• Gauge theories
• Symmetry

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