The gauge fields in Yang Mills theory are

[Lapidus]

The gauge fields in Yang Mills theory are matrices:

A$_{\mu}$ = A$^{a}_{\mu}$ T$^{a}$


But A$^{a}_{\mu}$ are vector fields, i.e. a=1,..,n four-vectors. Should not there be a U(1) gauge symmetry for each of them in addition to the non-abelian gauge symmetry?

In Lagrangian for the strong force, does not each of these four vectors correspond to a gluon? Gluons or weak bosons are spin-1 particles, so they most be described by four vectors. How do they follow from matrices??

And how can a vector field/ a four-vector be non-abelian??

help, please!

 

[Finbar]

In 4-dimensions each 4-vector corresponds to a spin-one particle(not 4 spin one particles).
In SU(N) there are N^2-1 generators so a,b goes from 1 to N^2 -1(not N). In SU(3) that makes 3^2 -1 = 8 gluons $A^a_\mu$

 

[Bill_K]

Lapidus, as you say, the gauge fields A^a_μ in Yang-Mills theory are a set of four-vectors, a=1,..,n. The T_a are not fields and not matrices, they are the group generators. They will be represented by matrices if you consider their action on particles making up a particular group representation. A^a_μ and T_a occur together in the covariant derivative, D_μ = ∂_μ + ig A^a_μT_a.

For example for QCD there are 8 gauge fields and 8 generators, a=1,..,8. Quarks belong to a 3-d representation labeled by color, i=1,2,3, and in the term of the Lagrangian where D_μ acts on them, the T^a will be represented by eight 3x3 matrices. Elsewhere in the Lagrangian, D_μ acts on the eight gauge fields themselves, and in that term the T^a will be represented by eight 8x8 matrices.

 

[Simon_Tyler]

Each generator of the group does lead to a U(1) gauge symmetry, but since the generators have non-trivial commutation relations, these U(1)s are all linked together to form part of a larger group.

Think of the case of rotations in 3-space. There are three basis elements that together generate SO(3). But each generator alone makes rotations in the plane, which is SO(2)~U(1).

 

[Lapidus]

thank so much you, guys! Got it

 

[tom.stoer]

Theoretically there could very well be an U(1) generator as well.

In the case of QCD there could be a U(3) = U(1) * SU(3) symmetry which would result a 9th generator represented by the 3*3 identity matrix. But this U(1) symmetry would result in a new color-force similar to an el.-mag. like long range force. b/c we do not observe this long range force in nature this extra U(1) factor has to be excluded.