Geometrical gauge conditions in Yang–Mills theory: Some nonexistence results

Abstract

We investigate the possibility of defining an ’’orthogonal’’ gauge for non-Abelian Yang–Mills theory. Such a gauge would generalize, in a geometrical way, the orthogonality features of the Coulomb gauge in electrodynamics. We show however that such a gauge does not exist (even locally) in the non-Abelian case. Specifically we prove that the tangent spaces defined at every point by orthogonality to the gauge group orbits admit no integral submanifolds. We also study the question of existence of global gauge conditions in phase space. We show that such global gauges, should they exist, would induce globally defined gauges in configuration space contradicting Singer’s result. We thus conclude (modulo certain technical points which might render Singer’s argument inapplicable to our function spaces) that global gauge conditions in phase space do not exist.