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Foundations of Physics

, Volume 36, Issue 5, pp 715–744 | Cite as

On “Gauge Renormalization” in Classical Electrodynamics

  • Alexander L. Kholmetskii
Article

In this paper we pay attention to the inconsistency in the derivation of the symmetric electromagnetic energy–momentum tensor for a system of charged particles from its canonical form, when the homogeneous Maxwell’s equations are applied to the symmetrizing gauge transformation, while the non-homogeneous Maxwell’s equations are used to obtain the motional equation. Applying the appropriate non-homogeneous Maxwell’s equations to both operations, we obtained an additional symmetric term in the tensor, named as “compensating term”. Analyzing the structure of this “compensating term”, we suggested a method of “gauge renormalization”, which allows transforming the divergent terms of classical electrodynamics (infinite self-force, self-energy and self-momentum) to converging integrals. The motional equation obtained for a non-radiating charged particle does not contain its self-force, and the mass parameter includes the sum of mechanical and electromagnetic masses. The motional equation for a radiating particle also contains the sum of mechanical and electromagnetic masses, and does not yield any “runaway solutions”. It has been shown that the energy flux in a free electromagnetic field is guided by the Poynting vector, whereas the energy flux in a bound EM field is described by the generalized Umov’s vector, defined in the paper. The problem of electromagnetic momentum is also examined.

Keywords

classical electrodynamics energy–momentum tensor gauge transformation 

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References

  1. 1.
    Lorentz H.A, The Theory of Electrons, 2nd edn. (Dover, 1952).Google Scholar
  2. 2.
    Abraham M, (1903). “Die Prinzipien der Dynamik des Elektrons”. Ann. Phys. 10: 105Google Scholar
  3. 3.
    Dirac P.A.M., (1938). “Classical theory of radiating electrons”. Proc. Roy. Soc. (London) A 167: 148ADSCrossRefGoogle Scholar
  4. 4.
    Born M, Infeld L, (1934). “Foundations of the new field theory”. Proc. Roy. Soc. A 144, 145Google Scholar
  5. 5.
    Rohrlich F, (1965). Classical Charged Particles. Addison-Wesley, Reading MassMATHGoogle Scholar
  6. 6.
    Landau L.D, Lifshitz E.M, (1962). The Classical Theory of Fields, 2nd edn. Pergamon, New YorkMATHGoogle Scholar
  7. 7.
    Jackson J.D, (1975). Classical Electrodynamics. Wiley, New YorkMATHGoogle Scholar
  8. 8.
    Panofsky W.K.H, Phillips M, (1962). Classical Electricity and Magnetism, 2nd edn. Addison-Wesley, Reading MassMATHGoogle Scholar
  9. 9.
    Pauli W, (1958). Principles of quantum mechanics, Encyclopedia of Physics, Vol V/1. Springer, BerlinGoogle Scholar
  10. 10.
    Moniz E.J, Sharp D.H, (1977). “Radiation reaction in nonrelativistic quantum electrodynamics”. Phys. Rev. D 15: 2850CrossRefADSGoogle Scholar
  11. 11.
    Feynman R.P, Leighton R.B, Sands M, (1964). The Feynman Lectures in Physics, Vol 2. Addison-Wesley, Reading MassGoogle Scholar
  12. 12.
    Yang K.-H., (1976). “Gauge transformations and quantum mechanics II. Physical interpretation of classical gauge transformations”. Ann. Phys. 101, 97Google Scholar
  13. 13.
    Brown G.J.N, Crothers D.S.F, (1989). “Generalised gauge invariance of electromagnetism”. J. Phys. A: Math. Gen. 22: 2939CrossRefADSMathSciNetMATHGoogle Scholar
  14. 14.
    Møller C., (1972). The Theory of Relativity. Clarendon Press, OxfordGoogle Scholar
  15. 15.
    Chubykalo A, Espinoza A, Tzonchev R, (2004). “Experimental test of the compatibility of the definitions of the electromagnetic energy density and the Poynting vector”. Eur. Phys. J. D 31(1): 113CrossRefADSGoogle Scholar
  16. 16.
    Kholmetskii A.L, (2004). “Remarks on momentum and energy flux of a non-radiating electromagnetic field”. Annales de la Foundation Louis de Broglie 29, 549MathSciNetGoogle Scholar
  17. 17.
    Umov N.A, Izbrannye Sochineniya (Gostechizdat, Moscow, 1950) (in Russian).Google Scholar
  18. 18.
    Jackson J.D, (2002). “From Lorentz to Coulomb and other explicit gauge transformations”. Am. J. Phys. 70, 917CrossRefADSGoogle Scholar
  19. 19.
    Chubykalo A.E, Onoochin V.V, (2002). “On the theoretical possibility of the electromagnetic scalar potential wave spreading with an arbitrary velocity in vacuum”. Hadronic J. 25, 597MathSciNetMATHGoogle Scholar
  20. 20.
    Dmitriev V.P, (2004). “On vector potential of the Coulomb gauge”. Eur. J. Phys. 25, 23CrossRefMathSciNetGoogle Scholar
  21. 21.
    Aguirregabiria J.M, Hernández A., Rivas M, (1982). “A Lewis–Tolman-like paradox”. Eur. J. Phys. 3, 30CrossRefGoogle Scholar
  22. 22.
    Shockley W, James R.P, (1967). “Try simplest cases’ discovery of “hidden momentum” forces on magnetic currents”. Phys. Rev. Lett. 18, 876CrossRefADSGoogle Scholar
  23. 23.
    Aharonov Y, Pearle P, Vaidman L, (1988). “Comment on Proposed Aharonov–Casher effect: another example of an Aharonov–Bohm effect arising from a classical lag”’. Phys. Rev. A 37: 4052CrossRefADSGoogle Scholar
  24. 24.
    Graham M, Lahoz D.G, (1980). “Observation of static electromagnetic angular momentum in vacuo”. Nature 285, 154CrossRefADSGoogle Scholar
  25. 25.
    H. Poincaré, Rend. Circ. Mat. Palermo 21, 129 (1906). (Engl. trans. with modern notation in Schwartz H.M, “Poincaré’s Rendincoti paper on relativity” Am. J. Phys. 40, 862 (1972)).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Belarusian State UniversityMinskBelarus

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