Communications in Mathematical Physics

, Volume 198, Issue 3, pp 711-736

The Relativistic Dynamics of the Combined Particle–Field System in Renormalized Classical Electrodynamics

  • Hans-Peter GittelAffiliated withDepartment of Mathematics, University of Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany.¶E-mail: gittel@mathematik.uni-leipzig.de
  • , Jerzy KijowskiAffiliated withCentrum Fizyki Teoretycznej PAN, Aleja Lotników 32/46, PL-02-668 Warsaw, Poland and Department of Mathematical Methods in Physics, University of Warsaw, ul. Hoża 74, PL-00-682 Warsaw, Poland.¶ E-mail: kijowski@theta1.ifpan.edu.pl
  • , Eberhard ZeidlerAffiliated withDepartment of Mathematics, University of Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany and Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, D-04103 Leipzig, Germany.¶E-mail: ezeidler@mis.mpg.de

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Abstract:

This paper develops a general theory for the nonlinear, renormalized interaction between charged particles and electromagnetic fields. For the combined “particle + field” system, a fundamental relativistically invariant dynamical equation is derived from first principles. This theory was first obtained in an alternative way by one of us (J.K.) in an earlier paper. Here, we prove that the initial-value problem for the “particle + field” system is well-posed. The existence and uniqueness result is based on a careful analysis of the singularites of the electromagnetic field along the trajectory of the moving charged particle. Furthermore, the Banach fixed-point principle is used. The theory improves the classical Dirac theory for the motion of electrons. In particular, it is shown how to deal with the Dirac paradox of runaway solutions.