Open Access
VOL. 39 | 2001 Electrons with self-field as solutions to nonlinear PDE
Hilary Booth

Editor(s) Andrew Hassell, Alexander Isaev, Adam Sikora

Proc. Centre Math. Appl., 2001: 1-14 (2001)


The Maxwell-Dirac equations give a model of an electron in an electromagnetic (e-m) field, in which neither the Dirac or the e-m fields are quantized. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equa- tion, resulting in a nonlinear system of partial differential equations (PDE's). In this way the self-field of the electron is included.

We review our results to date and give the four real consistency conditions (one of which is conservation of charge) which apply to the components of the wavefunction and its first derivatives. These must be met by any solutions to the Dirac equation. These conditions prove to be invaluable in the analysis of the nonlinear system, and generalizable to higher dimensional supersymmetric matter.

In earlier papers, we have shown analytically that in an isolated stationary system, the surrounding electon field must be equal and opposite to the central (external) field. The nonlinearity forces electric neutrality, at least in the static case. We illustrate these properties with a numerical family of orbits which occur in the (static) spherical and cylindrical ODE cases. These solutions are highly localized and die off exponentially with increasing distance from the central charge.


Published: 1 January 2001
First available in Project Euclid: 17 November 2014

zbMATH: 1122.81326
MathSciNet: MR1852690

Rights: Copyright © 2001, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.


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