Electronic ISBN:  9781470401917 
Product Code:  MEMO/127/606.E 
List Price:  $70.00 
MAA Member Price:  $63.00 
AMS Member Price:  $42.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 127; 1997; 311 ppMSC: Primary 35; Secondary 81; 22;
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.ReadershipGraduate students, research mathematicians, and mathematical physicists interested in new methods for nonlinear partial differential equations and applications to quantum field theories.

Table of Contents

Chapters

1. Introduction

2. The nonlinear representation $T$ and spaces of differentiable vectors

3. The asymptotic nonlinear representation

4. Construction of the approximate solution

5. Energy estimates and $L^2L^\infty $ estimates for the Dirac field

6. Construction of the modified wave operator and its inverse

Appendix


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

Chapters

1. Introduction

2. The nonlinear representation $T$ and spaces of differentiable vectors

3. The asymptotic nonlinear representation

4. Construction of the approximate solution

5. Energy estimates and $L^2L^\infty $ estimates for the Dirac field

6. Construction of the modified wave operator and its inverse

Appendix