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On classical electrodynamics of point particles and mass renormalization: Some preliminary results

Abstract

Apparently, no rigorous results exist for the dynamics of a classical point particle interacting with the electromagnetic field, as described by the standard Maxwell-Lorentz equations. Some results are given here for the corresponding linearized system (dipole approximation) in the presence of a mechanical linear restoring force. We consider a regularization of the system (Pauli-Fierz model), and explicitly solve the Cauchy problem in terms of normal modes. Then we study the limit of the particle's motion as the regularization is removed. We prove that the particle's motion corresponding to smooth initial data for the field has a well-defined limit if mass is renormalized, while the motion is trivial (i.e. the particle does not move at all) if mass is not renormalized. Moreover, the limit particle's motion corresponding to an interesting class of initial data satifies exactly the Abraham-Lorentz-Dirac equation. Finally, for generic initial data the limit motion is runaway.

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References

  1. Nelson, E.: Quantum Fluctuations, Princeton U.P., Princeton, 1985.

    Google Scholar 

  2. Jackson, J. D.: Classical Electrodynamics, Wiley, New York, 1975.

    Google Scholar 

  3. Schweber, S.: An Introduction to Relativistic Quantum Field Theory, Row Peterson, New York, 1961.

    Google Scholar 

  4. Lorentz, H. A.: The Theory of Electrons, Dover, New York, 1952 (first edition 1909).

    Google Scholar 

  5. Dirac, P. A. M.: Classical theory of radiating electrons, Proc. Roy. Soc. A 167 (1938), 148–168.

    Google Scholar 

  6. Bohm, D. and Weinstein, M.: The self oscillations of a charged particle, Phys. Rev. 74 (1948), 1789–1798.

    Google Scholar 

  7. Feynman, R. P., Leighton, R. B. and Sands, M.: The Feynman Lectures in Physics, Mainly Electromagnetics and Matter, Vol. II, Addison-Wesley, Redwood City, 1963.

    Google Scholar 

  8. Yaghjian, A. D.: Relativistic Dynamics of a Charged Sphere, Lecture Notes in Phys. 11, Springer-Verlag, Berlin, Heidelberg, 1992.

    Google Scholar 

  9. Arai, A.: Rigorous theory for spectra and radiation for a model in quantum electrodynamics, J. Math. Phys. 24 (1983), 1986.

    Google Scholar 

  10. Bambusi, D. and Noja, D.: Rigorous results on classical electrodynamics of point particles in the dipole approximation and proof of the Abraham-Lorentz-Dirac equation, Quaderno No. 24/94 Dipartimento di Matematica ‘F. Enriques’, University of Milan, 1994.

  11. Dautray, R. and Lions, J. L.: Analyse mathematique et calcul numérique pour les sciences et les techniques, Masson, São Paulo, 1985.

    Google Scholar 

  12. Abraham, M.: Theorie der Elektrizität, vol. II, Teubner, Leipzig, Berlin, 1908.

    Google Scholar 

  13. Kramers, H. A.: Nonrelativistic Quantum Electrodynamics and the Correspondence Principle, Rapports et discussion du huitieme Conseil de Physique Solvay. Published by R. Stoops, Brussels, 1950.

    Google Scholar 

  14. Bambusi, D. and Galgani, L.: Some Rigorous Results on the Pauli-Fierz Model of Classical Electrodynamics, Ann. Inst. H. Poincaré, Physique théorique 58 (1993), 155–171.

    Google Scholar 

  15. Lamb, H.: Proc. London Math. Soc. 2 (1900), 88.

    Google Scholar 

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Bambusi, D., Noja, D. On classical electrodynamics of point particles and mass renormalization: Some preliminary results. Lett Math Phys 37, 449–460 (1996). https://doi.org/10.1007/BF00312675

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  • DOI: https://doi.org/10.1007/BF00312675

Mathematics Subject Classifications (1991)

  • 81T15
  • 83C50

Key words

  • classical electrodynamics
  • point particles
  • mass renormalization
  • dipole approximation