Scattering for the Yang-Mills equations

Abstract:

We construct wave and scattering operators for the Yang-Mills equations on Minkowski space, ${{\mathbf {M}}_0} \cong {{\mathbf {R}}^4}$. Sufficiently regular solutions of the Yang-Mills equations on ${{\mathbf {M}}_0}$ are known to extend uniquely to solutions of the corresponding equations on the universal cover of its conformal compactification, $\tilde {\mathbf {M}} \cong {\mathbf {R}} \times {S^3}$. Moreover, the boundary of ${{\mathbf {M}}_0}$ as embedded in $\tilde {\mathbf {M}}$ is the union of "lightcones at future and past infinity", ${C_ \pm }$. We construct wave operators ${W_ \pm }$ as continuous maps from a space ${\mathbf {X}}$ of time-zero Cauchy data for the Yang-Mills equations to Hilbert spaces ${\mathbf {H}}({C_ \pm })$ of Goursat data on ${C_ \pm }$. The scattering operator is then a homeomorphism $S:{\mathbf {X}} \to {\mathbf {X}}$ such that $U{W_ + } = {W_ - }S$, where $U:{\mathbf {H}}({C_ + }) \to {\mathbf {H}}({C_ - })$ is the linear isomorphism arising from a conformal transformation of $\tilde {\mathbf {M}}$ mapping ${C_ - }$ onto ${C_ + }$. The maps ${W_ \pm }$ and $S$ are shown to be smooth in a certain sense.