Gauge Fixing - Maximum Abelian Gauge

In any non-Abelian gauge theory, any maximum Abelian gauge is an incomplete gauge which fixes the gauge freedom outside of the maximum Abelian subgroup. Examples are

• For SU(2) gauge theory in D dimensions, the maximum Abelian subgroup is a U(1) subgroup. If this is chosen to be the one generated by the Pauli matrix σ₃, then the maximum Abelian gauge is that which maximizes the function

where

• For SU(3) gauge theory in D dimensions, the maximum Abelian subgroup is a U(1)×U(1) subgroup. If this is chosen to be the one generated by the Gell-Mann matrices λ₃ and λ₈, then the maximum Abelian gauge is that which maximizes the function

where