Memoirs of the American Mathematical Society 1997; 311 pp; softcover Volume: 127 ISBN10: 0821806831 ISBN13: 9780821806838 List Price: US$66 Individual Members: US$39.60 Institutional Members: US$52.80 Order Code: MEMO/127/606
 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 MaxwellDirac 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 MaxwellDirac 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 smalldata Cauchy problem for the MaxwellDirac 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. Readership 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 \(L^2L^\infty\) estimates for the Dirac field
 Construction of the modified wave operator and its inverse
 Appendix
