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Transactions of the American Mathematical Society

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Scattering for the Yang-Mills equations

Author: John C. Baez
Journal: Trans. Amer. Math. Soc. 315 (1989), 823-832
MSC: Primary 58G30; Secondary 35P25, 47F05, 58G25, 81E13
MathSciNet review: 949897
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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.

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Article copyright: © Copyright 1989 American Mathematical Society

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