Mapping an Instanton to Spacetime

A mapping from the Lie algebra of the complexified Lorentz group to the $\mathfrak{su}(2)\times\mathfrak{su}(2) \sim\mathfrak{sp}(1)\times\mathfrak{sp}(1)$ part of the algebra the coset space $Sp(2)/[Sp(1)\times Sp(1)]$ is presented. The coset space is shown to be home to the instanton, the curvature form that optimizes the Yang-Mills functional. Arguments are presented to support the generalization to $Sp(n)/Sp(1)^n$ to yield a self-consistent many-body theory for $n$ particles interacting with one another via fields that reside in the coset space.