Yang–Mills theory

The Yang-Mills theory ( according to Chen Ning Yang and Robert L. Mills ) is a non-Abelian gauge theory, which is used to describe the strong and weak interactions. In contrast, the quantum electrodynamics is an example of an Abelian gauge theory.

The Yang-Mills theory in 1954 was introduced by Yang and Mills and formulated independently around the same time in the dissertation of Ronald Shaw at Abdus Salam.

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

Yang-Mills field energy

The Yang-Mills theory is based on the Yang-Mills action for the gauge bosons:

The size is called Yang-Mills field strength; is the dual to Yang-Mills field strength. The positive quantity g is the constant of interaction in physics.

Applying now the principle of least action to the represented by Eichbosonenfelder to, we obtain the corresponding Euler -Lagrange equations, the Yang-Mills equations:

Where the term ~ g is the Yang-Mills charges contains.

Here the mathematical language of differential forms was used, which allows a compact notation. The Yang-Mills field strength is defined by the second Maurer - Cartan structure equation or a relationship (more precisely, its local representation ) of a principal fiber bundle ( in physics gauge potential or Eichbosonfeld called ) with the curvature of the same ( in the physics field strength called field strength tensor ) into connection:

As above is a Lie algebra -valued 1-form on the principal bundle and a Lie algebra -valued 2- form on this principal bundle. Further illustrates the derivation and outer, the outer product of the differential shapes, which does not disappear here between, since the Lie algebra components of generally not interchange. For this reason, the field shape is not " closed " as opposed to abelian gauge theories such as electrodynamics.

In component form, and with the coupling constant g, is considered as in quantum chromodynamics

And the Yang-Mills equations in this notation ( if, as usual, on the right side still inserts a source term)

The aforementioned duality operator * with respect to the indices μ and ν to be formed, for example, with ( ---) the signature of the Minkowski space. With respect to the indices a have to proceed according to the group under consideration. The same also applies for the track (Tr is the English term, an abbreviation for "trace "). Upper and lower indices, and the order of double indexes are interchanged by the * operation. The Yang-Mills functional can therefore also be written in the explicit form.

In physics, usually considered a compact, semi- simple Lie group, for example, or whose hermitian generators satisfy the following Kommutationsrelation:

The hot (real) structure constants of the Lie group. Any element of is represented by the following equation:

Dirac particles in the Yang-Mills theory

The wave function ( Dirac ) field of a charged ( with Yang-Mills charges ) particle transformed as:

(applies only to particles that transform according to the fundamental representation of the gauge group). The Lagrangian for the Dirac field, from the follow on the Euler - Lagrange equations, the equations of motion of the charged fermion particle thus described, looks due to the coupling of the Dirac field ψ and Yang-Mills field A ( " gauge field " ) as follows:

The above mentioned coupling constant. This Lagrangian describes the coupling of the field to the matter fields. The term is called the covariant derivative or minimal coupling. The variables are the four components of the vector additionally Lie algebra 1 -valent form ( i.e., the indices a have been omitted for simplicity, usually allowed to clear the symbol ^, which here for the sake of clarity of the covariant derivative does not happen ). Of course also comes in consideration of Dirac particles in the overall effect should still the above-mentioned field component, which is indicated here by the points and do not depend explicitly on ψ.

If the Yang-Mills theory is used to describe the strong interaction ( in the form of a gauge theory, the already mentioned quantum chromodynamics ), then describes the so-called gluon field, and indeed represent ( the eight Gluonenarten has eight generators, commonly used is the so-called Gell- Mann matrices ). Some important Yang-Mills theories with charged fermion matter fields have the property of so-called asymptotic freedom at high energies or short distances, depending on the gauge group and the number of Fermionentypen.

Open Issues

A major advance in the enforcement of the Yang-Mills theories in physics was the proof of renormalizability by Gerardus ' t Hooft in the early 1970s. The renormalization is true even if the gauge bosons are massive as in the electroweak interaction. The masses are acquired according to the standard model by the so-called Higgs mechanism.

In mathematics, the Yang-Mills theory is current research area and served, for example, Simon Donaldson for the classification of differentiable structures on 4- manifolds. The theory was taken up by the Clay Mathematics Institute's list of the Millennium problems. In particular, it is at this price problem to prove it, that the lowest excitations of a pure Yang - Mills theory ( ie without matter fields) must have a finite mass and excitation energy (that is, there is a so-called Mass - Gap - in the solid state physics, one would say: an energy gap - the vacuum state ). A related open problem is further proof of the presumed confinement property of Yang-Mills fields interacting with Fermionenfeldern.

In physics, the study of Yang-Mills theories done now no longer perturbative analytical methods, but through lattice calculations ( lattice gauge theories ) or so-called functional methods such as Dyson -Schwinger equations.