View source for Yang–Mills theory

{{Use American English|date = March 2019}}
{{Short description|Physical theory unifying the electromagnetic, weak and strong interactions}}
{{unsolved|physics|Yang–Mills theory in the non-[[Perturbation theory (quantum mechanics)|perturbative]] regime:
The equations of Yang–Mills remain unsolved at [[energy scale]]s relevant for describing [[Atomic nucleus|atomic nuclei]]. How does Yang–Mills theory give rise to the physics of [[Atomic nucleus|nuclei]] and [[hadron|nuclear constituents]]?}}
{{Quantum field theory}}
'''Yang–Mills theory''' is a [[gauge theory]] based on a [[special unitary group]] SU(''N''), or more generally any [[Compact Lie algebra|compact]], [[reductive Lie algebra]]. Yang–Mills theory seeks to describe the behavior of elementary particles using these [[non-abelian group|non-abelian]] Lie groups and is at the core of the unification of the [[electromagnetic force]] and [[weak force]]s (i.e. U(1) × SU(2)) as well as [[quantum chromodynamics]], the theory of the [[strong force]] (based on SU(3)). Thus it forms the basis of our understanding of the [[Standard Model]] of particle physics.

== History and theoretical description ==
In a private correspondence, [[Wolfgang Pauli]] formulated in 1953 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 talks about it in Zürich.<ref>See Abraham Pais' account of this period as well as L. Susskind's "Superstrings, Physics World on the first non-abelian gauge theory" where Susskind wrote that Yang–Mills was "rediscovered" only because Pauli had chosen not to publish.</ref> Recent research shows that an extended Kaluza–Klein theory is in general not equivalent to Yang–Mills theory, as the former contains additional terms.<ref name=Reifler>{{cite arXiv |last=Reifler|first=N |year=2007 |title=Conditions for exact equivalence of Kaluza-Klein and Yang–Mills theories |eprint=0707.3790|class=gr-qc }}</ref>

In early 1954, [[Chen Ning Yang]] and [[Robert Mills (physicist)|Robert Mills]]<ref name=YM>{{cite journal |authorlink1=Chen-Ning Yang |first1=C. N. |last1=Yang |authorlink2=Robert Mills (physicist) |first2=R. |last2=Mills |title=Conservation of Isotopic Spin and Isotopic Gauge Invariance |journal=[[Physical Review]] |volume=96 |issue=1 |pages=191–195 |year=1954 |doi=10.1103/PhysRev.96.191|bibcode = 1954PhRv...96..191Y }}</ref> extended the concept of gauge theory for [[abelian group]]s, e.g. [[quantum electrodynamics]], to nonabelian groups to provide an explanation for strong interactions. The idea by Yang–Mills was criticized by Pauli,<ref>[http://universe-review.ca/R15-21-YangPauli.htm An Anecdote by C. N. Yang]</ref> as the [[quantum|quanta]] of the Yang–Mills field must be massless in order to maintain [[gauge invariance]]. 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 SU(2) × U(1), while QCD is an [[SU(3)]] Yang–Mills theory. The massless gauge bosons of the electroweak SU(2) × U(1) mix after [[spontaneous symmetry breaking]] to produce the 3 massive weak bosons ({{SubatomicParticle|W boson+}}, {{SubatomicParticle|W boson-}}, and {{SubatomicParticle|Z boson}}) 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 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 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]].

== Mathematical overview ==
Yang–Mills theories are special examples of gauge theories with a non-abelian symmetry group given by the [[Lagrangian (field theory)|Lagrangian]]

:<math>\mathcal{L}_\mathrm{gf} = -\frac{1}{2}\operatorname{Tr}(F^2)=- \frac{1}{4}F^{a\mu \nu} F_{\mu \nu}^a </math>

with the generators of the [[Lie algebra]], indexed by {{mvar|a}}, corresponding to the ''F''-quantities (the [[curvature]] or field-strength form) satisfying

:<math>\operatorname{Tr}(T^aT^b)=\frac{1}{2}\delta^{ab},\quad [T^a,T^b]=if^{abc}T^c.</math>

Here, the {{math|''f''<sup>''abc''</sup>}} are [[structure constant]]s of the Lie algebra, the [[covariant derivative]] is defined as

:<math>D_\mu=I\partial_\mu-igT^aA^a_\mu, </math>

{{mvar|I}} is the [[identity matrix]] (matching the size of the generators), <math>A^a_\mu</math> is the [[Four-vector|vector]] potential, and ''g'' is the [[coupling constant]]. In four dimensions, the coupling constant ''g'' is a pure number and for a SU(''N'') group one has <math>a,b,c=1\ldots N^2-1.</math>

The relation

:<math>F_{\mu \nu}^a = \partial_\mu A_\nu^a-\partial_\nu A_\mu^a+gf^{abc}A_\mu^bA_\nu^c </math>

can be derived by the [[commutator]]

:<math>[D_\mu, D_\nu] = -igT^aF_{\mu\nu}^a.</math>

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 by [[perturbation theory]] with small nonlinearities.

Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for ''a'' indices (e.g. <math>f^{abc}=f_{abc}</math>), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature, <math>\eta_{\mu \nu }={\rm diag}(+---)</math>.

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

:<math>\partial^\mu F_{\mu\nu}^a+gf^{abc}A^{\mu b}F_{\mu\nu}^c=0.</math>

Putting <math>F_{\mu\nu}=T^aF^a_{\mu\nu}</math>, these can be rewritten as

:<math>(D^\mu F_{\mu\nu})^a=0.</math>

A [[Bianchi identity]] holds

:<math>(D_\mu F_{\nu \kappa})^a+(D_\kappa F_{\mu \nu})^a+(D_\nu F_{\kappa \mu})^a=0</math>

which is equivalent to the [[Jacobi identity]]

:<math>[D_{\mu}, [D_{\nu},D_{\kappa}]]+[D_{\kappa},[D_{\mu},D_{\nu}]]+[D_{\nu},[D_{\kappa},D_{\mu}]]=0</math>

since <math>[D_{\mu},F^a_{\nu\kappa}]=D_{\mu}F^a_{\nu\kappa}</math>. Define the [[Hodge star operator|dual]] strength tensor
<math>\tilde{F}^{\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}</math>, then the Bianchi identity can be rewritten as

:<math>D_{\mu}\tilde{F}^{\mu\nu}=0.</math>

A source <math>J_\mu^a</math> enters into the equations of motion as

:<math>\partial^\mu F_{\mu\nu}^a+gf^{abc}A^{b\mu}F_{\mu\nu}^c=-J_\nu^a.</math>

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

We give here some comments about the physical dimensions of the coupling. In ''D'' dimensions, the field scales as <math>[A]=[L^\frac{2-D}{2}]</math>{{Citation needed|date=November 2017}} and so the coupling must scale as <math>[g^2]=[L^{D-4}]</math>. This implies that Yang–Mills theory is not [[renormalization|renormalizable]] for dimensions greater than four. Furthermore, for ''D'' = 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 quartic [[scalar field theory]]. So, these theories share the [[scale invariance]] at the classical level.

== Quantization ==
A method of quantizing the Yang–Mills theory is by functional methods, i.e. [[Path integral formulation|path integrals]]. One introduces a generating functional for ''n''-point functions as

:<math>Z[j]=\int [dA]\exp\left[- \frac{i}{2} \int d^4x\operatorname{Tr}(F^{\mu \nu} F_{\mu \nu})+i\int d^4x \, j^a_\mu(x)A^{a\mu}(x)\right] ,</math>

but this integral has no meaning as it is because the potential vector can be arbitrarily chosen due to the [[gauge 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 by [[Ludvig Faddeev]] and [[Victor Popov]] with the introduction of a '''ghost field''' (see [[Faddeev–Popov ghost]]) that has the property of being unphysical since, although it agrees with [[Fermi–Dirac statistics]], it is a complex scalar field, which violates the [[spin–statistics theorem]]. So, we can write the generating functional as

:<math>\begin{align}
Z[j,\bar\varepsilon,\varepsilon] & = \int [dA] [d\bar c] [dc] \exp\left\{iS_F[\partial A,A]+iS_{gf}[\partial A]+iS_g[\partial c,\partial\bar c,c,\bar c,A]\right\} \\
&\exp\left\{i\int d^4x j^a_\mu(x)A^{a\mu}(x)+i\int d^4x[\bar c^a(x)\varepsilon^a(x)+\bar\varepsilon^a(x) c^a(x)]\right\}
\end{align}</math>

being

:<math>S_F=- \frac{1}{2} \int  \operatorname{d}\!^4 x \operatorname{Tr}(F^{\mu \nu} F_{\mu \nu})</math>

for the field,

:<math>S_{gf}=-\frac{1}{2\xi} \int  \operatorname{d}\!^4 x (\partial\cdot A)^2</math>

for the gauge fixing and

:<math>S_g=-\int  \operatorname{d}\!^4 x  (\bar c^a\partial_\mu\partial^\mu c^a+g\bar c^a f^{abc}\partial_\mu A^{b\mu}c^c)</math>

for the ghost. This is the expression commonly used to derive Feynman's rules (see [[Feynman diagram]]). Here we have ''c<sup>a</sup>'' for the ghost field while ξ fixes the gauge's choice for the quantization. Feynman's rules obtained from this functional are the following

<center>[[File:FeynRulesEN.jpg|488px]]</center>

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

:<math>\begin{align}
Z[j,\bar\varepsilon,\varepsilon] &= \exp\left(-ig\int d^4x \, \frac{\delta}{i\delta\bar\varepsilon^a(x)} f^{abc}\partial_\mu\frac{i\delta}{\delta j^b_\mu(x)} \frac{i\delta}{\delta\varepsilon^c(x)}\right)\\
& \qquad \times \exp\left(-ig\int d^4xf^{abc}\partial_\mu\frac{i\delta}{\delta j^a_\nu(x)}\frac{i\delta}{\delta j^b_\mu(x)}\frac{i\delta}{\delta j^{c\nu}(x)}\right)\\
& \qquad \qquad \times \exp\left(-i\frac{g^2}{4}\int d^4xf^{abc}f^{ars}\frac{i\delta}{\delta j^b_\mu(x)} \frac{i\delta}{\delta j^c_\nu(x)} \frac{i\delta}{\delta j^{r\mu}(x)} \frac{i\delta}{\delta j^{s\nu}(x)}\right) \\
& \qquad \qquad \qquad \times Z_0[j,\bar\varepsilon,\varepsilon]
\end{align}</math>

with

:<math>Z_0[j,\bar\varepsilon,\varepsilon]=\exp\left(-\int d^4xd^4y\bar\varepsilon^a(x)C^{ab}(x-y)\varepsilon^b(y)\right)\exp\left(\tfrac{1}{2}\int d^4xd^4yj^a_\mu(x)D^{ab\mu\nu}(x-y)j^b_\nu(y)\right)</math>

being the generating functional of the free theory. Expanding in ''g'' and computing the [[functional derivative]]s, we are able to obtain all the ''n''-point functions with perturbation theory. Using [[LSZ reduction formula]] we get from the ''n''-point functions the corresponding process amplitudes, [[Cross section (physics)|cross sections]] and [[decay rate]]s. 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 <math>\bar c^a f^{abc}\partial_\mu A^{b\mu}c^c</math>. For the abelian case, all the structure constants <math>f^{abc}</math> 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 is [[asymptotic freedom]]. This result can be obtained by assuming that the [[coupling constant]] ''g'' is small (so small nonlinearities), as for high energies, and applying [[perturbation 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 from [[deep 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 verified [[a posteriori]] in the [[Ultraviolet divergence|ultraviolet 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 (see [[hadrons]]). The most used method to study the theory in this limit is to try to solve it on computers (see [[lattice 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 the [[glueball]] and [[exotic meson|hybrids]] spectra are yet a questioned matter in view of the experimental observation of such exotic states. Indeed, the σ resonance<ref name=ccl>{{cite journal |first=I. |last=Caprini |first2=G. |last2=Colangelo |first3=H. |last3=Leutwyler |year=2006 |title=Mass and width of the lowest resonance in QCD |journal=[[Physical Review Letters]] |volume=96 |issue=13 |page=132001 |doi=10.1103/PhysRevLett.96.132001 |bibcode=2006PhRvL..96m2001C |arxiv = hep-ph/0512364 |pmid=16711979}}</ref><ref name=ygp>{{cite journal |first=F. J. |last=Yndurain |first2=R. |last2=Garcia-Martin |first3=J. R. |last3=Pelaez |title=Experimental status of the ππ isoscalar S wave at low energy: ''f''<sub>0</sub>(600) pole and scattering length |journal=[[Physical Review D]] |volume=76 |issue=7 |page=074034 |year=2007 |doi=10.1103/PhysRevD.76.074034|arxiv = hep-ph/0701025 |bibcode = 2007PhRvD..76g4034G }}</ref> is not seen in any of such lattice computations and contrasting interpretations have been put forward. This is a hotly debated issue.

== 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. | last2 = Veltman | first2 = M. | 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 | pmid =  | pmc = |bibcode = 1972NuPhB..44..189T }}</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 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 Fermion particles.

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]].

== See also ==
{{div col|colwidth=30em}}

* [[Aharonov–Bohm effect]]
* [[Coulomb gauge]]
* [[Gauge covariant derivative]]
* [[Kaluza–Klein theory]]
* [[Lattice gauge theory]]
* [[Lorenz gauge]]
* [[N = 4 supersymmetric Yang–Mills theory|''N'' = 4 supersymmetric Yang–Mills theory]]
* [[Propagator]]
* [[Quantum gauge theory]]
* [[Standard model (basic details)|Field theoretical formulation of the standard model]]
* [[Symmetry in physics]]
* [[Weyl gauge]]
* [[Yang–Mills existence and mass gap]]
* [[Yang–Mills–Higgs equations]]

{{div col end}}

== References ==
{{Reflist}}

== Further reading ==
; Books
* {{cite book |last = Frampton |first=P. |author-link  =Paul Frampton |title = Gauge Field Theories |edition=3rd |publisher=[[Wiley-VCH]] |year=2008 |isbn = 978-3-527-40835-1 }}
* {{cite book |last1 = Cheng |first1 = T.-P. |last2 = Li |first2 = L.-F.  |title = Gauge Theory of Elementary Particle Physics |publisher = [[Oxford University Press]] |year=1983 |isbn = 0-19-851961-3 }}
* {{cite book |editor-last = 't Hooft |editor-first = Gerardus |editor-link = Gerard 't Hooft |title = 50 Years of Yang–Mills theory |location = Singapore |publisher = [[World Scientific]] |year = 2005 |isbn = 981-238-934-2 }}

; Articles
* {{cite arXiv |year=1999 |eprint=math-ph/9902027 |title = Preparation for Gauge Theory |last = Svetlichny |first = George }}
* {{cite web |last=Gross |first=D. |author-link = David Gross |year=1992 |url = https://scholar.google.se/scholar_url?url=http://www.colin-baxter.com/academic/bib/downloads/955.pdf&hl=sv&sa=X&scisig=AAGBfm0MBLXQuVbYC78zgUqhhplqMGAWfw&nossl=1&oi=scholarr&ei=-DRJVafYG8KUsgHOuIDYBQ&ved=0CB8QgAMoADAA |title = Gauge theory – Past, Present and Future |access-date = 2015-05-05 }}

== External links ==
{{wikiquote}}
* {{springer|title=Yang-Mills field|id=p/y099030}}
* [https://dispersivewiki.org/DispersiveWiki/index.php?title=Yang-Mills_equations Yang–Mills theory on DispersiveWiki]
* [http://www.claymath.org/ The Clay Mathematics Institute]
* [http://www.claymath.org/prizeproblems The Millennium Prize Problems]

{{Quantum field theories}}

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