How do the quark charges relate to the gluon charges?
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The popular idea about the strong charges is that quark have R,G,B and gluons have a "pair" of one of these and its anticharge (like red-antigreen).
A bit deeper, the concept is that the strong field has SU(3), which is the group of the 1-determinant 3x3 complex matrices. It has 8 independent infinitesimal generators, making 8 independent gluons.
Fine.
What is with the quark charges? How do they apply into this picture? How are they constructed from SU(3)?
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