Abstract
In this paper I will first derive, based on energy estimations and geometric invariance, the asymptotic behavior of solutions of linear spin field equations in Minkowski space. It generalizes the result in [3] where it was proved for the spin-1 and spin-2 cases. The techniques are then applied to Yang-Mills equations, the result improves the previous one in [1] by allowing the initial data to have charge, dipole and quadrupole moments. The Lie derivative operator for spinors and some properties will be also discussed; they can be used to simplify some algebraic calculations of [4].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Choquet-Bruhat, Y., Christodoulou, D.: Existence of global solutions of the Yang-Mills, Higgs and spinor field equations in 3+1 dimensions, Ann. Sci. Ecole Norm. Sup.14, 481–506 (1981)
Christodoulou, D.: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math.39 (2), 267–282 (1986)
Christodoulou, D., Klainerman, S.: Asymptotic behavior of linear field equations in Minkowski space. Commun. Pure Appl. Math. 1990
Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space (in press)
Eardley, D. M., Moncrief, V.: The global existence of Yang-Mills fields in 4-dimensional Minkowski space, Part 1: Local existence and smoothness properties. Commun. Math. Phys.83, 171–191 (1982)
Eardley, D. M., Moncrief, V.: The global existence of Yang-Mills fields in 4-dimensional Minkowski space, Part 2: Completion of Proof. Commun. Math. Phys.83, 193–212 (1982)
Ginbre, J. Velo, G.: The Cauchy Problem for coupled Yang-Mills and Scalar fields in Temporal Gauge. Commun. Math. Phys.82, 1–28 (1982); on pairs of null directions. J. Math. Phys.14 (7). 874–881 (1973)
Klainerman, S.: Uniform decay estimates and the Lorentz invariance of the classical wave equation. Commun. Pure Appl. Math.38 (3), 321–332 (1985)
Klainerman, S.: The null condition and global existence to nonlinear wave equations. Lectures in Appl. Math.23 (Part I), Providence, RI: Am. Math. Soc. 1986
Newman, E. T., Penrose, R.: An Approach to gravitational radiation by a method of spin coefficients. J. Math. Phys.3, 896–902 (1962)
Penrose, R.: Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. R. Soc. LondonA284, 159–203 (1965)
Penrose, R., Rindler, W.: Spinors and Space-time, Vol. 1, Two-spinors calculus and relativistic fields. Cambridge: Cambridge University Press 1986
Shu, W.: Spin Field Equations and Yang-Mills Equation, Ph.D. thesis, Princeton University
Author information
Authors and Affiliations
Additional information
Communicated by T. Spencer
This research is partially supported by a grant from NSF under DMS-8610730
Rights and permissions
About this article
Cite this article
Shu, WT. Asymptotic properties of the solutions of linear and nonlinear spin field equations in Minkowski space. Commun.Math. Phys. 140, 449–480 (1991). https://doi.org/10.1007/BF02099131
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02099131