Yang Mills Theory

The Yang-Mills theory (after the physicists Chen Ning Yang and Robert L. Mills ) is a non - Abelian gauge theory that is used to describe the strong and weak interaction . It was introduced in 1954 by Yang and Mills and independently of it at the same time in the dissertation of Ronald Shaw with the physicist Abdus Salam and in Japan by Ryoyu Utiyama .

This article mainly describes the mathematical aspects of the interdisciplinary phenomenon. The physical aspects are mainly discussed in one of the most important examples of Yang Mills theories, quantum chromodynamics .

The theory is generally non-Abelian, i.e. not commutative . However, it also contains quantum electrodynamics as an Abelian gauge theory as a special case .

Yang-Mills effect and field equations

The Yang-Mills theory is based on the Yang-Mills effect for the gauge bosons : ${\displaystyle \mathbf {S} _{\mathrm {YM} }}$

${\displaystyle \mathbf {S} _{\mathrm {YM} }={\frac {1}{4}}\int \operatorname {Tr} \left(*F\wedge F\right)}$

• The size is called the Yang Mills field strength ${\displaystyle F}$
• ${\displaystyle *F}$is the too dual Yang-Mills field strength. The duality operator * is to be formed with the signature of the Minkowski space with regard to the indices  μ and  ν (see below) , e.g. B. with (+ −−−). Regarding the indices  a one must proceed according to the considered group. The same applies to the Tr track (abbreviation for  trace ). Upper and lower indices and the order of double indices are swapped by the * operation. The Yang-Mills functional can also be written in the explicit form: ${\displaystyle F}$ ${\displaystyle \mathbb {M} ^{4}}$ 

${\displaystyle \mathbf {S} _{\mathrm {YM} }={\frac {1}{4}}\int \,\mathrm {d} ^{4}x\,F_{a}^{\nu \mu }\cdot F_{\mu \nu }^{a}}$

If one now applies the principle of the smallest effect to the gauge boson fields in , one obtains the Yang-Mills equations as the associated Euler-Lagrange equations : ${\displaystyle \mathbf {S} _{\mathrm {YM} }}$

${\displaystyle {\mathcal {D}}F:=\mathrm {d} F+g\,A\wedge F\equiv 0}$

where the term contains  the Yang-Mills charges . In physics, the positive quantity  means the interaction constant . ${\displaystyle \sim g}$ ${\displaystyle g}$

Here the mathematical language of differential forms was used, which allows compact notation. Likewise, the effect in terms of design is not limited to four dimensions and can be used in this representation, for. B. for a Yang-Mills theory in a -dimensional Minkowski space with metric signature . Yang-Mills theories in higher dimensions and their supersymmetric extensions are e.g. B. relevant for AdS / CFT correspondence . ${\displaystyle d+1}$ ${\displaystyle 1-d}$

The Yang-Mills field strength is defined by the second Maurer-Cartan structural equation, which describes the differential geometric relationship (more precisely its local representation ) of a main fiber bundle ( called calibration potential or calibration boson field in physics ) with its curvature ( field strength or field strength in physics ). Called field strength tensor): ${\displaystyle A}$ ${\displaystyle F}$

${\displaystyle F:=\mathrm {d} A+gA\wedge A}$

As above is

• ${\displaystyle A}$a Lie algebra valued 1-form over the main fiber bundle
• ${\displaystyle F}$ a Lie-algebra-valued 2-form over this main fiber bundle
• ${\displaystyle \mathrm {d} A}$the outer derivative
• ${\displaystyle A\wedge A}$the outer product of differential forms, which here does not vanish between the , since the Lie algebra components of do not generally interchange. ${\displaystyle A}$ ${\displaystyle A}$

For this reason, the field shape is not "closed" in contrast to Abelian gauge theories such as electrodynamics . ${\displaystyle F}$ ${\displaystyle (\mathrm {d} F=0),}$

In component notation, as in quantum chromodynamics, the following applies:

${\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf_{bc}^{a}A_{\mu }^{b}A_{\nu }^{c}}$

and the Yang-Mills equations are written in this way (if, as usual, a source term is added on the right-hand side):

${\displaystyle \partial ^{\mu }F_{\mu \nu }^{a}+gf_{bc}^{a}A^{\mu b}F_{\mu \nu }^{c}\equiv J_{\nu }^{a}}$

In physics, one usually considers a compact, semi-simple Lie group , such as or , whose Hermitian generators fulfill the following commutation relation : ${\displaystyle G}$ ${\displaystyle SU(N)}$ ${\displaystyle SO(N)}$

${\displaystyle \left[T_{a},T_{b}\right]=if_{ab}^{c}\,T_{c}}$

The are called (real) structural constants of the Lie group. ${\displaystyle f_{ab}^{c}}$

Any element of is represented by the following equation: ${\displaystyle U}$ ${\displaystyle G}$

${\displaystyle U=e^{ig\theta ^{a}\,T_{a}}}$

Dirac particles in the Yang-Mills theory

The wave function (Dirac field) of a particle charged (with Yang-Mills charges) transforms as follows: ${\displaystyle \psi }$ ${\displaystyle U\in G}$

${\displaystyle \psi \to U\,\psi }$ or.

${\displaystyle {\bar {\psi }}\to {\bar {\psi }}\,U^{\dagger }}$

However, this only applies to particles that transform according to the fundamental representation of the gauge group .

The Lagrange function for the Dirac field, from which the equations of motion of the charged fermion described by the Euler-Lagrange equations follow, looks like this:

${\displaystyle {\mathcal {L}}(\psi ,A):={\bar {\psi }}\,\left[\mathrm {i} \,\gamma ^{\mu }\left(\partial _{\mu }-ig\,{\hat {A}}_{\mu }\right)+m\right]\psi +\dots \,}$

This Lagrange function describes the coupling of the Yang-Mills field ("calibration field") to the matter or Dirac fields : ${\displaystyle A}$ ${\displaystyle \psi }$

• ${\displaystyle g}$ is the coupling constant given above,
• ${\displaystyle \gamma }$a Dirac matrix
• The term is called a covariant derivative or minimal coupling . ${\displaystyle \partial _{\mu }-ig\,{\hat {A}}_{\mu }=:\nabla _{\mu }}$
• The variables form the four-vector components of the additional Lie algebra-valued 1-form  (i.e. the indices  a are omitted for the sake of simplicity; usually the symbol ^ is also omitted, which for the sake of clarity does not happen here with the covariant derivation ) . ${\displaystyle {\hat {A}}_{\mu }}$ ${\displaystyle A}$
• If Dirac particles are taken into account, the above-mentioned field component is added to the overall effect, which is indicated here by points and does not explicitly  depend on. ${\displaystyle \psi }$

If the Yang-Mills theory is used to describe the strong interaction (in the form of a gauge theory, the aforementioned quantum chromodynamics ), then describes the gluon field. The above represent the eight types of gluons (it has 8 generators, the Gell-Mann matrices are usually used to represent them ). ${\displaystyle SU(3)}$ ${\displaystyle A}$ ${\displaystyle T_{a}}$ ${\displaystyle SU(3)}$

Some important Yang-Mills theories with charged fermion matter fields have the property of asymptotic freedom at high energies or short distances, which depends on the gauge group and the number of fermion types.

Open issues

A major step forward in the implementation of the Yang Mills theories in physics was the proof of their renormalizability by Gerardus' t Hooft in the early 1970s. The renormalizability also applies if the calibration bosons are massive as in the electroweak interaction . The masses are acquired through the Higgs mechanism according to the standard model .

In mathematics, the Yang-Mills theory is a current research area and has served e.g. B. Simon Donaldson for the classification of differentiable structures on 4- manifolds . Yang Mills Theory was added to the list of Millennium Problems by the Clay Mathematics Institute . In particular, this price problem is about proving that the lowest excitations of a pure Yang-Mills theory (i.e. without matter fields) must have a finite (i.e. here, non-vanishing) mass or excitation energy (i.e. there is a measure -Gap - in solid-state physics one would say: an energy gap - to the vacuum state ). A related further open problem is the detection of the presumed confinement property of Yang-Mills fields in interaction with fermion fields.

In physics, the investigation of Yang-Mills theories is no longer done using perturbative analytical methods, but rather using lattice calculations ( lattice range theories ) or functional methods such as B. Dyson-Schwinger equations .

Terminology comparison

In mathematics and physics there are very different terminologies, which are systematically compared here: In mathematics, for example, the differential geometric context is generated , while in physics one speaks of the vector potential of the field, which among other things generates its particles (e.g. . the calibration particles of elementary particle physics ). In mathematics, or denotes the curvature , in physics, on the other hand, the field tensor . In both terminologies, the term denotes an antisymmetric part (Lie commutator) of the curvature or of the field strength tensor. In this context, physicists speak of structural constants of the tensor. ${\displaystyle A}$ ${\displaystyle \Omega }$ ${\displaystyle F}$ ${\displaystyle \,A\wedge A}$

Yang Mills Theory and Gravitation

Utiyama recognized that the structures of general relativity also fit the form of Yang Mills theories. He then tried to understand the general theory of relativity as the Yang-Mills theory of the Lorentz group . This is special insofar as the underlying geometry is allowed to be calibrated, while other Yang-Mills theories such as quantum chromodynamics assume a Minkowski-like geometry (i.e. the special theory of relativity ). Utiyama discovered that the coupling of the new field actually has the form of the covariant derivative in the Riemannian space , but only if he ignored anti-symmetrical parts of the relationship and assumed the symmetry of the metric ad hoc. This theory differs from the theories described above in that the field strength tensor , which here is the Riemann curvature tensor , only appears in the first order in the Lagrangian .

Kibble later realized that it was more practical to start from the Poincaré group (called "complete Lorentz group" by Kibble). In this case, two field equations are obtained, since the Poincaré group breaks down into a Lorentz part and a translation part. This leads to the Einstein-Cartan (-Sciama-Kibble) theory of gravity. In this, the ad-hoc assumptions of Utiyama are no longer necessary: ​​The context may have antisymmetric components ( called torsion ) and instead of the metric one gets tetrad fields, which do not necessarily have to be symmetric.

In the discourse on generalizations of the general relativity theory, based on these considerations, approaches emerge to demand a Lagrangian function for gravity, which is quadratic in the field strength tensor.

These Yang-Mills theories of gravity do not automatically mean that the quantization of gravity is possible with them. Since the underlying geometry is calibrated here, theorems regarding renormalizability no longer apply without further ado.

literature

• Gerardus' t Hooft (editor, physicist): 50 years of Yang-Mills theory. World Scientific, Singapore 2005, ISBN 981-256-007-6
• Keith J. Devlin (mathematician): The Millennium problems - the seven greatest unsolved mathematical puzzles of our time. Granta Books, London 2005, pp. 63-97, ISBN 1-86207-735-5
• Michael F. Atiyah (mathematician): Geometry of Yang-Mills fields. Scuola Normale Superiore, 1979
• Mikio Nakahara (physicist): Geometry, Topology and Physics. Second edition. Graduate Student Series in Physics. Institute of Physics Publishing, Bristol and Philadelphia, 2003, pp. 374-418, ISBN 0-7503-0606-8

Individual evidence

1. ↑ Yang, Mills Conservation of isotopic spin and isotopic gauge invariance , Physical Review, Volume 96, 1954, pp. 191-195, abstract
2. ↑ ^{a b} Utiyama Invariant Theoretical Interpretation of Interaction , Physical Review, vol 101, 1956, pp 1597-1607 Abstract
3. ↑ Other discoverers were Wolfgang Pauli , but only unpublished in letters to Abraham Pais (1953). Yang and Mills were also the only ones to strike a link with the strong interaction. A Kaluza-Klein theory with SU (2) Eichgruppe was already presented by Oscar Klein in 1938 at a conference in Kazimierz, Poland, and also applied it to the strong interaction, which was largely ignored. See Lochlainn O'Raifeartaigh, The Dawning of Gauge Theory, Princeton UP 1997, pp. 8f
4. ↑ Kibble Lorentz Invariance and the Gravitational Field , Journal of Mathematical Physics, Volume 2, 1961, pp. 212-221
5. ↑ E.g. Hehl, Nitsch, von der Heyde Gravitation and the Poincaré Gauge Field Theory with Quadratic Lagrangian , General Relativity and Gravitation - One Hundred Years after the Birth of Albert Einstein, Volume 1, 1980, pp. 329-355