Global gauge invariance—the non-Abelian case

The generalization to non-Abelian transformations is straightforward in the global case but is fairly complex in the local case. 

As already mentioned the simplest non-Abelian invariance is isospin where the fields are assumed to come in multiplets 

Naturally the Lj, representing the Tj, also satisfy this relation. 

The formalism generalizes immediately to higher global non-Abelian gauge symmetries. Let Tj j = be the generators 

The are called the structure constants of the group and are antisymmetric under interchange of any pair of indices. Given that the fields transform according to some representation of G, the generators Tj will be represented by matrices Lj satisfying (2.3.25). The gauge transformations, specified by N parameters 0 = On) are 

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