The transformations of non-abelian gauge fields under translations

Abstract

I consider infinitesimal translations $x'^{\alpha}=x^{\alpha}+\delta x^{\alpha}$ 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, $A'^a_{\gamma} (x') - A^a_{\gamma}(x) = \partial_{\gamma} [ \delta x_{\beta} A^{a \beta}(x)] + C^a_{bc} \delta x_{\beta} A^{c \beta}(x) A^{b}_{\gamma}(x)$ where the $C^a_{bc}$ are the structure constants of the gauge group.