The transformations of non-abelian gauge fields under translations

Abstract

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

References to work by Jackiw and others are added