Non-Abelian local gauge invariance—Yang-Mills theories

4 Non-Abelian local gauge invariance—Yang-Mills theories 

We now turn to the much more subtle question of local non-Abelian gauge invariance. The first generalization of SU 2) to locally gauge invariant Lagrangians is due to Yang and Mills (1954) [a detailed account can be found in Taylor (1976)], but the treatment applies to any group with a finite number of generators [see Abers and Lee (1973)]. 

The aim is to introduce as many vector fields W x), gauge fields that are the analogue of the photon field A, as is necessary in order to construct a Lagrangian which is invariant under the local gauge transformations specified by Bj x). 

By analogy with electrodynamics we seek a covariant derivative such that 

If the group dimension is N we have to introduce one vector field W x) for each dimension, and we can then define 

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