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The structure of the solution set for the Yang-Mills equations

Published online by Cambridge University Press:  24 October 2008

Judith M. Arms
Judith M. Arms
Affiliation:
University of Washington, Seattle
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Abstract

The solution set for the (sourceless) Yang-Mills equations on a spacetime with compact Cauchy surface is a smooth manifold (i.e. the equations are linearization stable) except at solutions that are symmetric. At such symmetric solutions, the structure is described by a homogeneous quadratic form. The degeneracy space for this form is tangent to a manifold of symmetric solutions. Symmetry breaking occurs for perturbations in the nondegenerate directions of the quadratic form. The terms ‘symmetry’ and ‘stability’ in the present work are compared to these terms as used elsewhere in the literature.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 2 , September 1981 , pp. 361 - 372
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

Abraham, R. and Marsden, J. (1978). Foundations of mechanics, 2nd ed., Addison-Wesley.Google Scholar
Arms, J. (1979). Linearization stability of gravitational and gauge fields. J. Math. Phys. 20, 443–453.CrossRefGoogle Scholar
Arms, J., Fischer, A. and Marsden, J. (1975). Une approche symplectique pour des théorèmes de décomposition en géométrie ou relativité générale. C.R. Acad. Sci. Paris, 281, 517–520.Google Scholar
Arms, J., Marsden, J. and Moncrief, V. (1982). The structure of the space of solutions of Einstein's equations. II. Many killing fields. (In preparation.)Google Scholar
Arms, J., Marsden, J. and Moncrief, V. (1981). Bifurcations of momentum mappings. Comm. Math. Phys. 78, 455–478.CrossRefGoogle Scholar
Atiyah, M. R., Hitchin, N. J. and Singer, I. M. (1978). Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London A 362, 425–461.Google Scholar
Bergmann, P. and Flaherty, E. (1978). Symmetries in gauge theories. J. Math. Phys. 19, 212–214.CrossRefGoogle Scholar
Branson, T. (1979). The Yang-Mills equations: quasi-invariance, special solutions, and Banach manifold geometry. Ph.D. thesis, MIT.Google Scholar
Brill, D. & Deser, S. (1973). Instability of closed spaces in General Relativity. Comm. Math. Phys. 32, 291–304.CrossRefGoogle Scholar
Choquet-Bruhat, Y. and Deser, S. (1972). Stabilité initiale de l'espace temps de Minkowski. C.R. Acad. Sci. Paris 275, 1019–1027.Google Scholar
Fischer, A. and Marsden, J. (1973). Linearization stability of the Einstein equations. Bull. Amer. Math. Soc. 79, 995–1001.CrossRefGoogle Scholar
Fischer, A. and Marsden, J. (1975). Linearization stability of non-linear partial differential equations. Proc. Symp. Pure Math. AMS 27 (part 2), 219–263.CrossRefGoogle Scholar
Fischer, A., Marsden, J. and Moncrief, V. (1979). The structure of the space of solutions of Einstein's equations; One killing field. (Preprint.)Google Scholar
Harnad, J., Shnider, S. and Vinet, L. (1979). The Yang-Mills system in compactified Minkowski space; Invariance conditions and SU(2) invariant solutions. J. Math. Phys. 20, 931–942.CrossRefGoogle Scholar
Hörmander, L. (1966). Pseudo-differential operators and non-elliptic boundary problems. Ann. of Math. 83, 129–209.CrossRefGoogle Scholar
Jackiw, R. and Rossi, P. (1980). Stability and bifurcation in Yang-Mills theory. Phys. Rev., Sect. D 21, 426–445.Google Scholar
Kuranishi, M. (1965). New proof for the existence of locally complete families of complex structures. Proc. Conf. on Complex Analysis, ed. Aeppli, A. et al. (Springer).Google Scholar
Moncrief, V. (1975), (1976). Spacetime symmetries and linearization stability of the Einstein equations: I: J. Math. Phys. 16, 493–498; II: 17, 1893–1902.CrossRefGoogle Scholar
Moncrief, V. (1977). Gauge symmetries of Yang-Mills fields. Ann. Phys. 108, 387–400.CrossRefGoogle Scholar
O'mtrchadha, N. and York, J. (1974). Initial value problem of general relativity. II. Stability of solutions of the initial value problem. Phys. Rev., Sect. D 10, 437–446.Google Scholar