How does a non-abelian gauge symmetry lead to
asymptotic freedom for quarks?
In a non-Abelian gauge theory, the gauge fields also carry the charge. (I'm sure you've been told before that gluons carry color). The self-interaction of the gauge fields is non-trivial, but in the case of QCD they cause a decrease in the coupling constant with increasing energy. In fact, the actual behavior depends on the number of colors and flavors - more colors favor decreasing coupling and more flavors favor increasing coupling. With 3 colors and 6 flavors, the result is decreasing.
In what respect is the colour field non-commutative?
The gauge group is SU(3) which is non-abelian. The gauge group of QED is simply the unit circle U(1) which is abelian (the phases add in the unit cirlce).
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