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

DOI:
https://doi.org/10.1090/S0002-9947-1989-0949897-7

MathSciNet review:
949897

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Abstract: We construct wave and scattering operators for the Yang-Mills equations on Minkowski space, . Sufficiently regular solutions of the Yang-Mills equations on are known to extend uniquely to solutions of the corresponding equations on the universal cover of its conformal compactification, . Moreover, the boundary of as embedded in is the union of "lightcones at future and past infinity", . We construct wave operators as continuous maps from a space of time-zero Cauchy data for the Yang-Mills equations to Hilbert spaces of Goursat data on . The scattering operator is then a homeomorphism such that , where is the linear isomorphism arising from a conformal transformation of mapping onto . The maps and are shown to be smooth in a certain sense.

**[1]**S. M. Paneitz and I. E. Segal,*Analysis in space-time bundles*. I, J. Funct. Anal.**47**(1982), 78-142. MR**663834 (83k:22042)****[2]**Y. Choquet-Bruhat and D. Christodoulou,*Existence of global solutions the Yang-Mills, Higgs, and spinor field equations in**dimensions*, Ann. Ecole. Norm. Sup. (4)**14**(1981), 481-506. MR**654209 (84c:81041)****[3]**D. Eardley and V. Moncrief,*The global existence of Yang-Mills-Higgs fields on*-*dimensional Minkowski space*, Comm. Math. Phys.**83**(1982), 121-191, 193-212. MR**649159 (83e:35106b)****[4]**Y. Choquet-Bruhat, S. M. Paneitz and I. E. Segal,*The Yang-Mills equations on the universal cosmos*, J. Funct. Anal.**53**(1983), 112-150. MR**722506 (85i:58111)****[5]**I. E. Segal,*Reduction of scattering to an invariant finite displacement in an ambient space-time*, Proc. Nat. Acad. Sci. U.S.A.**81**(1984), 7266-7268. MR**768161 (86b:81090)****[6]**J. Baez, I. E. Segal and Z. Zhou,*The global Goursat problem and scattering for nonlinear wave equations*(to appear).**[7]**J. Baez,*Scattering and the geometry of the solution manifold of*, J. Funct. Anal. (to appear).**[8]**D. Bleecker,*Gauge theory and variational principles*, Addison-Wesley, Reading, Mass. 1981. MR**643361 (83h:53049)****[9]**V. Moncrief,*Reduction of the Yang-Mills equations*, Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Math., vol. 836, Springer, New York 1980. MR**607702 (83a:58041)****[10]**I. E. Segal,*Non-linear semi-groups*, Ann. of Math.**78**(1964), 339-364. MR**0152908 (27:2879)****[11]**-,*Differential operators in the manifold of solutions of a non-linear differential equation*, J. Math. Pures Appl.**44**(1965), 71-132. MR**0192369 (33:594)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0949897-7

Article copyright:
© Copyright 1989
American Mathematical Society