A Note on the Rainich Problem for SU(2) Gauge

Abstract

We provide a resolution to the non-Abelian Rainich problem. By canonically identifying traceless symmetric $(0,2)$-tensors with Hermitian forms on the vector bundle of chiral 2-forms, we define the internal square roots of a stress-energy tensor. We then prove that the existence of a local $\operatorname{SU}(2)$ Yang-Mills field with prescribed stress-energy tensor $T$ is equivalent to a single differential condition on internal square roots of $T$.