Gauge theory, gravitation, and symmetry

Abstract

The coupled Einstein-Yang-Mills (or Einstein-Maxwell in the case of an Abelian internal symmetry) theory is shown to be the unique gauge theory of ${\mathrm{T}}_{4} \ensuremath{\bigotimes}G$, where ${\mathrm{T}}_{4}$ is the 4-dimensional translation group and $G$ is an internal-symmetry group. In the case of ${\mathrm{P}}_{4} \ensuremath{\bigotimes}G$ where ${\mathrm{P}}_{4}$ is the 4-dimensional Poincar\'e group one obtains Einstein-Cartan theory coupled with the internal gauge fields. As in the Abelian case internal Yang-Mills fields do not create any extra torsion, owing to the gauge invariance of the internal symmetry. The arguments are given in terms of the conventional gauge formalism, without using the bundle language.