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

Moshé Flato Université de Bourgogne, Dijon, France
Jacques C. H. Simon Université de Bourgogne, Dijon, France
Erik Taflin Université de Bourgogne, Dijon, France
Available Formats:
Electronic ISBN: 978-1-4704-0191-7
Product Code: MEMO/127/606.E
List Price: $70.00 MAA Member Price:$63.00
AMS Member Price: $42.00 Click above image for expanded view Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell-Dirac Equations Moshé Flato Université de Bourgogne, Dijon, France Jacques C. H. Simon Université de Bourgogne, Dijon, France Erik Taflin Université de Bourgogne, Dijon, France Available Formats:  Electronic ISBN: 978-1-4704-0191-7 Product Code: MEMO/127/606.E  List Price:$70.00 MAA Member Price: $63.00 AMS Member Price:$42.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1271997; 311 pp
MSC: 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 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.

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

• 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 $L^2-L^\infty$ estimates for the Dirac field
• 6. Construction of the modified wave operator and its inverse
• Appendix
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1271997; 311 pp
MSC: 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 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.

Readership

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 $L^2-L^\infty$ estimates for the Dirac field
• 6. Construction of the modified wave operator and its inverse
• Appendix
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.