Symmetric Gauge Fields
Consider n scalar fields φ(x) = (φ 1(x),...,φ n (x)), interacting with K-valued gauge fields A u (x), where K is the Lie algebra of the gauge group K. As we have seen, the extremals of the energy functional are connected with quantum particles in the semiclassical approximation. How does one find such extremals? Using the results in the preceding chapter, one can take advantage of the invariance of the energy functional under gauge transformations and (usually) under spatial transformations such as rotations. More precisely, let L be the group generated by the group K ∞ of local gauge transformations together with the group O of spatial symmetries. If G ⊂ L is a subgroup, an extremal of the functional restricted to the space of G-invariant fields is also an extremal for the unrestricted functional. (We will see below that the non-degeneracy condition is met.)
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