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Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell-Dirac Equations
Moshé Flato, Jacques C. H. Simon, and Erik Taflin, Université de Bourgogne, Dijon, France

Memoirs of the American Mathematical Society
1997; 311 pp; softcover
Volume: 127
ISBN-10: 0-8218-0683-1
ISBN-13: 978-0-8218-0683-8
List Price: US$70
Individual Members: US$42
Institutional Members: US$56
Order Code: MEMO/127/606
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The purpose of this work is to present and give full proofs of new original research results concerning integration of and scattering for the classical Maxwell-Dirac equations. These equations govern first quantized electrodynamics and are the starting point for a rigorous formulation of quantum electrodynamics. The presentation is given within the formalism of nonlinear group and Lie algebra representations, i.e. the powerful new approach to nonlinear evolution equations covariant under a group action.

The authors prove that the nonlinear Lie algebra representation given by the manifestly covariant Maxwell-Dirac equations is integrable to a global nonlinear representation of the Poincaré group on a differentiable manifold of small initial conditions. This solves, in particular, the small-data Cauchy problem for the Maxwell-Dirac equations globally in time. The existence of modified wave operators and asymptotic completeness is proved. The asymptotic representations (at infinite time) turn out to be nonlinear. A cohomological interpretation of the results in the spirit of nonlinear representation theory and its connection to the infrared tail of the electron are developed.


Graduate students, research mathematicians, and mathematical physicists interested in new methods for nonlinear partial differential equations and applications to quantum field theories.

Table of Contents

  • Introduction
  • The nonlinear representation \(T\) and spaces of differentiable vectors
  • The asymptotic nonlinear representation
  • Construction of the approximate solution
  • Energy estimates and \(L^2-L^\infty\) estimates for the Dirac field
  • Construction of the modified wave operator and its inverse
  • Appendix
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