The transformations of non-abelian gauge fields under translations

Abstract

I consider infinitesimal translations $x'^α=x^α+δx^α$ and demand that Noether's approach gives a symmetric energy-momentum tensor as it is required for gravitational sources. This argument determines the transformations of non-abelian gauge fields under infinitesimal translations to differ from the usually assumed invariance by the gauge transformation, $A'^a_γ (x') - A^a_γ(x) = \partial_γ [ δx_β A^{a β}(x)] + C^a_{bc} δx_β A^{c β}(x) A^{b}_γ(x)$ where the $C^a_{bc}$ are the structure constants of the gauge group.