Advertisement

Foundations of Physics

, Volume 41, Issue 1, pp 99–128 | Cite as

Two Mathematically Equivalent Versions of Maxwell’s Equations

Article

Abstract

This paper is a review of the canonical proper-time approach to relativistic mechanics and classical electrodynamics. The purpose is to provide a physically complete classical background for a new approach to relativistic quantum theory. Here, we first show that there are two versions of Maxwell’s equations. The new version fixes the clock of the field source for all inertial observers. However now, the (natural definition of the effective) speed of light is no longer an invariant for all observers, but depends on the motion of the source. This approach allows us to account for radiation reaction without the Lorentz-Dirac equation, self-energy (divergence), advanced potentials or any assumptions about the structure of the source. The theory provides a new invariance group which, in general, is a nonlinear and nonlocal representation of the Lorentz group. This approach also provides a natural (and unique) definition of simultaneity for all observers.

The corresponding particle theory is independent of particle number, noninvariant under time reversal (arrow of time), compatible with quantum mechanics and has a corresponding positive definite canonical Hamiltonian associated with the clock of the source.

We also provide a brief review of our work on the foundational aspects of the corresponding relativistic quantum theory. Here, we show that the standard square-root and Dirac equations are actually two distinct spin- \(\frac{1}{2}\) particle equations.

Keywords

Special relativity Proper time Radiation reaction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Einstein, A.: Ann. Phys. (Leipzig) 17, 891 (1905) ADSMATHGoogle Scholar
  2. 2.
    Bridgman, P.W.: A Sophisticate’s Primer of Relativity. Wesleyan University Press, Middletown (1962) Google Scholar
  3. 3.
    Purcell, E.M.: Electricity and Magnetism, vol. II. McGraw-Hill, New York (1984) Google Scholar
  4. 4.
    Miller, A.I.: Frontiers of Physics 1900–1911. Birkhäuser, Boston (1986) Google Scholar
  5. 5.
    Einstein, A.: Ann. Phys. (Leipzig) 17, 919 (1905) Google Scholar
  6. 6.
    Planck, M.: Phys. Z. 10, 825 (1909) Google Scholar
  7. 7.
    Lorentz, L.V.: Philos. Mag. 34, 287 (1867) Google Scholar
  8. 8.
    Hamdan, N., Hariri, A.K., López-Bonilla, J.: Hadron. J. 30, 513 (2007) MATHGoogle Scholar
  9. 9.
    Ritz, W.: Ann. Chim. Phys. 13, 145 (1908) MATHGoogle Scholar
  10. 10.
    Einstein, A.: Phys. Z. 10, 821 (1909) Google Scholar
  11. 11.
    Schilpp, P.A. (ed.): Autobiographical Notes. Albert Einstein: Philosopher-Scientist, vol. 1. Open Court Press, Peru (1969) Google Scholar
  12. 12.
    Brown, H.R.: Eur. J. Phys. 26, S85 (2005) MATHCrossRefGoogle Scholar
  13. 13.
    Wheeler, J.A., Feynman, R.P.: Rev. Mod. Phys. 21, 425 (1949) MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Dirac, P.A.M.: Proc. R. Soc. Lond. A 167, 148 (1938) CrossRefADSGoogle Scholar
  15. 15.
    Damour, T.: Ann. Phys. (Leipzig) 17(8), 619–630 (2008) MATHMathSciNetADSGoogle Scholar
  16. 16.
    Lorentz, H.A., Einstein, A., Minkowski, H., Weyl, H.: The Principle of Relativity. Dover, New York (1952). W. Perret and G.B. Jeffery (translators, with additional notes by A. Sommerfeld) MATHGoogle Scholar
  17. 17.
    Walters, S.: In: Gonner, H., Renn, J., Ritter, J. (eds.) The Expanding Worlds of General Relativity. Einstein Studies, vol. 7, pp. 45–86. Birkhäuser, Boston (1999) CrossRefGoogle Scholar
  18. 18.
    Gill, T.L., Zachary, W.W., Lindesay, J.: Found. Phys. 31, 1299 (2001) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Dresden, M.: In: Brown, L.M. (ed.) Renormalization: From Lorentz to Landau (and Beyond). Springer, New York (1993) Google Scholar
  20. 20.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. II. Addison-Wesley, New York (1974) Google Scholar
  21. 21.
    Pryce, M.H.L.: Proc. R. Soc. Lond. A 195, 400 (1948) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Currie, D.G., Jordan, T.F., Sudarshan, E.C.G.: Rev. Mod. Phys. 35, 350 (1963) CrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Gill, T.L., Zachary, W.W., Lindesay, J.: Int. J. Theor. Phys. 37, 2573 (1998) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Horwitz, L.P., Piron, C.: Helv. Phys. Acta 46, 316 (1981) Google Scholar
  25. 25.
    Arnol’d, V.I.: Mathematical Methods in Classical Mechanics. Springer, New York (1993) Google Scholar
  26. 26.
    Penzias, A.A., Wilson, R.W.: Astrophys. J. 142, 419 (1965) CrossRefADSGoogle Scholar
  27. 27.
    Peebles, P.J.E.: Principles of Physical Cosmology. Princeton University Press, London (1993) Google Scholar
  28. 28.
    Cohen, A.G., Glashow, S.L.: arXiv:hep-ph/0605036v1
  29. 29.
    Santilli, R.M.: Elements of Hadronic Mechanics, I: Mathematical Foundations. Ukraine Academy of Sciences, Kiev (1993) Google Scholar
  30. 30.
    Feynman, R.P.: Phys. Rev. 74, 939 (1948) MATHCrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Stueckelberg, E.C.G.: Helv. Phys. Acta 15, 23 (1942) MATHMathSciNetGoogle Scholar
  32. 32.
    Paul, H.: Photonen: Experimente und ihre Deutung. Vieweg, Braunschweig (1993) Google Scholar
  33. 33.
    Buenker, R.J.: J. Chem. Phys. 22, 124 (2003) Google Scholar
  34. 34.
    Bargmann, V., Wigner, E.P.: Proc. Natl. Acad. Sci. 34, 211 (1948) MATHCrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Schrödinger, E., Bass, L.: Proc. R. Soc. Lond. A 232, 1 (1938) Google Scholar
  36. 36.
    Goldhaber, A., Nieto, M.: Rev. Mod. Phys. 43, 277 (1971) CrossRefADSGoogle Scholar
  37. 37.
    Jackiw, R.: Comments Mod. Phys. A 1, 1 (1999) Google Scholar
  38. 38.
    Feynman, R.P.: Quantum Electrodynamics. Benjamin, New York (1964) Google Scholar
  39. 39.
    Akhiezer, A.I., Berestetskii, V.D.: Quantum Electrodynamics. Wiley-Interscience, New York (1965) Google Scholar
  40. 40.
    Pound, R.V., Snider, J.L.: Phys. Rev. 140, B788 (1965) CrossRefADSGoogle Scholar
  41. 41.
    Curtis, H.D.: Publ. Lick Obs. 13 (1918) Google Scholar
  42. 42.
    Pearson, T.J., Zensus, J.A.: In: Zensus, J.A., Pearson, T.J. (eds.) Superluminal Radio Sources. Cambridge University Press, London (1987) Google Scholar
  43. 43.
    Zensus, J.A.: Ann. Rev. Astron. Astrophys. 35, 607 (1997) CrossRefADSGoogle Scholar
  44. 44.
    Mirabel, I.F., Rodríguez, L.F.: Ann. Rev. Astron. Astrophys. 37, 409 (1999) CrossRefADSGoogle Scholar
  45. 45.
    Rees, M.J.: Nature 211, 468 (1966) CrossRefADSGoogle Scholar
  46. 46.
    De Rújula, A.: arXiv:hep-ph/0412094v1
  47. 47.
    Greisen, K.: Phys. Rev. Lett. 16, 748 (1966) CrossRefADSGoogle Scholar
  48. 48.
    Zatsepin, G.T., Kuzmin, V.A.: Sov. Phys. JETP. Lett. 4, 78 (1966) (Engl. Transl.) ADSGoogle Scholar
  49. 49.
    Takeda, M., et al.: Phys. Rev. Lett. 81, 1163 (1998) CrossRefADSGoogle Scholar
  50. 50.
    Takeda, M., et al.: Astrophys. J. 522, 225 (1999) CrossRefADSGoogle Scholar
  51. 51.
    Jui, C.H. (HiRes Collaboration): In: Proc. 27th International Cosmic Ray Conference 2001, Hamburg Google Scholar
  52. 52.
    Dar, A.: arXiv:0906.0973v1 (2009)
  53. 53.
    Payne-Gaposchkin, C., Haramundanis, K.: Introduction to Astronomy, 2nd edn. Prentice-Hall, Englewood Cliffs (1970) Google Scholar
  54. 54.
    Gill, T.L., Zachary, W.W.: J. Math. Phys. 43, 69–93 (2002) MATHCrossRefMathSciNetADSGoogle Scholar
  55. 55.
    Feynman, R.P.: Phys. Rev. 81, 108–128 (1951) CrossRefMathSciNetADSGoogle Scholar
  56. 56.
    Dyson, F.J.: Phys. Rev. 75, 1736–1755 (1949) MATHCrossRefMathSciNetADSGoogle Scholar
  57. 57.
    Gill, T.L., Zachary, W.W.: J. Phys. A, Math. Gen. 38, 2479–2496 (2005); Corrigendum: Ibid. 39, 1537–1538 (2006) MATHCrossRefMathSciNetADSGoogle Scholar
  58. 58.
    Gill, T.L., Zachary, W.W., Alfred, M.: J. Phys. A, Math. Gen. 38, 6955–6976 (2005) MATHCrossRefMathSciNetADSGoogle Scholar
  59. 59.
    Afshar, S.S., Flores, E., McDonald, K.F., Knoesel, E.: Found. Phys. 37, 295–305 (2007) MATHCrossRefMathSciNetADSGoogle Scholar
  60. 60.
    Drozdov, I.V., Stahlhofen, A.A.: arXiv:0803.2596v1 (2008)

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Electrical EngineeringHoward UniversityWashingtonUSA

Personalised recommendations