# Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell-Dirac Equations

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*Moshé Flato; Jacques C. H. Simon; Erik Taflin*

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.

#### Table of Contents

# Table of Contents

## Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell-Dirac Equations

- Contents vii8 free
- 1. Introduction 112 free
- 2. The nonlinear representation T and spaces of differentiate vectors 2031 free
- 3. The asymptotic nonlinear representation 5162
- 4. Construction of the approximate solution 7889
- 5. Energy estimates and L[sup(2)] L[sup(∞)] estimates for the Dirac field 120131
- 6. Construction of the modified wave operator and its inverse 203214
- Appendix 294305
- References 309320