A7 Yang Mills fields

It is important to observe that the Yang-Mills equations, that form the basis of our current understanding of both the strong and the weak forces of particle interactions, are also conformally invariant, so long as we can ignore the introduction of mass which may be taken to be through the subsequent agency of the Higgs field. The Yang-Mills field strengths can be described by a tensor quantity (a 'bundle curvature')

Fab&f = — [email protected], where the (abstract) indices 0, r, ... refer to the internal symmetry group (U(2), SU(3), or whatever) of relevance to the particle symmetries. We can represent this bundle curvature in terms of the spinor quantity <AB0r (P&R 5.5.36) by

where, for a unitary internal group, the complex conjugate of a lower internal index becomes an upper internal index, and vice versa. The field equations mirror those for the Maxwell equations, where we supply the additional internal indices as indicated above. Accordingly the conformal invariance of Maxwell theory also applies to the Yang-Mills equations, since the internal indices 0, r, ... are unaffected by the conformal rescaling.

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