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The gauge fields in Yang Mills theory are 
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#1
Sep211, 02:30 PM

P: 283

The gauge fields in Yang Mills theory are matrices:
A[itex]_{\mu}[/itex] = A[itex]^{a}_{\mu}[/itex] T[itex]^{a}[/itex] But A[itex]^{a}_{\mu}[/itex] are vector fields, i.e. a=1,..,n fourvectors. Should not there be a U(1) gauge symmetry for each of them in addition to the nonabelian gauge symmetry? In Lagrangian for the strong force, does not each of these four vectors correspond to a gluon? Gluons or weak bosons are spin1 particles, so they most be described by four vectors. How do they follow from matrices?? And how can a vector field/ a fourvector be nonabelian?? help, please! 


#2
Sep211, 06:32 PM

P: 343

In 4dimensions each 4vector corresponds to a spinone particle(not 4 spin one particles).
In SU(N) there are N^21 generators so a,b goes from 1 to N^2 1(not N). In SU(3) that makes 3^2 1 = 8 gluons [itex] A^a_\mu [/itex] 


#3
Sep311, 06:29 PM

Sci Advisor
Thanks
P: 4,160

Lapidus, as you say, the gauge fields A^{a}_{μ} in YangMills theory are a set of fourvectors, 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}_{μ}T_{a}.
For example for QCD there are 8 gauge fields and 8 generators, a=1,..,8. Quarks belong to a 3d 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. 


#4
Sep311, 06:40 PM

P: 313

The gauge fields in Yang Mills theory are
Each generator of the group does lead to a U(1) gauge symmetry, but since the generators have nontrivial commutation relations, these U(1)s are all linked together to form part of a larger group.
Think of the case of rotations in 3space. 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). 


#5
Sep311, 07:02 PM

P: 283

thank so much you, guys! Got it



#6
Sep411, 03:51 AM

Sci Advisor
P: 5,451

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 colorforce 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. 


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