Foundations of Physics

, Volume 49, Issue 4, pp 365–389 | Cite as

Electromagnetism as Quantum Physics

  • Charles T. SebensEmail author


One can interpret the Dirac equation either as giving the dynamics for a classical field or a quantum wave function. Here I examine whether Maxwell’s equations, which are standardly interpreted as giving the dynamics for the classical electromagnetic field, can alternatively be interpreted as giving the dynamics for the photon’s quantum wave function. I explain why this quantum interpretation would only be viable if the electromagnetic field were sufficiently weak, then motivate a particular approach to introducing a wave function for the photon (following Good in Phys Rev 105(6):1914–1919, 1957). This wave function ultimately turns out to be unsatisfactory because the probabilities derived from it do not always transform properly under Lorentz transformations. The fact that such a quantum interpretation of Maxwell’s equations is unsatisfactory suggests that the electromagnetic field is more fundamental than the photon.


Photon wave function Electromagnetism Weber vector Bohmian mechanics Field Particle Quantum field theory 



Thank you to Steve Carlip, Sheldon Goldstein, Chris Hitchcock, Michael Kiessling, Matthias Lienert, Ward Struyve, A. Shadi Tahvildar-Zadeh, Roderich Tumulka, David Wallace, and an anonymous referee for helpful feedback and discussion. This project was supported in part by funding from the President’s Research Fellowships in the Humanities, University of California (for research conducted while at the University of California, San Diego).


  1. 1.
    Akhiezer, A.I., Berestetskii, V.B.: Quantum Electrodynamics. Interscience, New York. Translated from the second Russian edition by G.M, Volkoff (1965)Google Scholar
  2. 2.
    Albert, D.: Time and Chance. Harvard University Press, Cambridge (2000)zbMATHGoogle Scholar
  3. 3.
    Archibald, W.J.: Field equations from particle equations. Can. J. Phys. 33(9), 565–572 (1955)ADSMathSciNetzbMATHGoogle Scholar
  4. 4.
    Baker, D.J.: Against field interpretations of quantum field theory. Br. J. Philos. Sci. 60(3), 585–609 (2009)ADSMathSciNetzbMATHGoogle Scholar
  5. 5.
    Baker, D.: The philosophy of quantum field theory. Oxford Handbooks Online, Oxford (2016)Google Scholar
  6. 6.
    Barrett, J.A.: Typical worlds. Stud. Hist. Philos. Mod. Phys. 58, 31–40 (2017)zbMATHGoogle Scholar
  7. 7.
    Bialynicki-Birula, I.: On the wave function of the photon. Acta Phys. Pol. A 86(1–2), 97–116 (1994)MathSciNetGoogle Scholar
  8. 8.
    Bialynicki-Birula, I.: The photon wave function. In: Eberly, J.H., Mandel, L., Wolf, E. (eds.) Coherence and Quantum Optics VII: Proceedings of the Seventh Rochester Conference on Coherence and Quantum Optics, held at the University of Rochester, June 7–10, 1995 pp. 313–322. Springer (1996)Google Scholar
  9. 9.
    Bialynicki-Birula, I., Bialynicka-Birula, Z.: The role of the Riemann-Silberstein vector in classical and quantum theories of electromagnetism. J. Phys. A 46(5), 053001 (2013)ADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Bjorken, J.D., Drell, S.D.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1964)zbMATHGoogle Scholar
  11. 11.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables II. Phys. Rev. 85, 180–193 (1952)ADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    Bohm, D.: Comments on an article of Takabayasi concerning the formulation of quantum mechanics with classical pictures. Prog. Theor. Phys. 9(3), 273–287 (1953)ADSMathSciNetzbMATHGoogle Scholar
  13. 13.
    Bohm, D., Hiley, B.J.: The Undivided Universe: An Ontological Interpretation of Quantum Theory. Routledge, Abingdon (1993)Google Scholar
  14. 14.
    Callender, C.: Is time ‘handed’ in a quantum world? Proc. Aristot. Soc. 100(1), 247–269 (2000)Google Scholar
  15. 15.
    Carroll, S.M., Sebens, C.T.: Many worlds, the Born rule, and self-locating uncertainty. In: Struppa, D., Tollaksen, J. (eds.) Quantum Theory: A Two-Time Success Story. Springer, Milano (2014)Google Scholar
  16. 16.
    Chandrasekar, N.: Quantum mechanics of photons. Adv. Stud. Theoret. Phys. 6(8), 391–397 (2012)zbMATHGoogle Scholar
  17. 17.
    Cugnon, J.: The photon wave function. Open J. Microphys. 1, 41–52 (2011)ADSGoogle Scholar
  18. 18.
    Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, Oxford (1958)zbMATHGoogle Scholar
  19. 19.
    Dresden, M.: HA Kramers: Between Tradition and Revolution. Springer, New York (1987)Google Scholar
  20. 20.
    Duncan, A.: The Conceptual Framework of Quantum Field Theory. Oxford University Press, Oxford (2012)zbMATHGoogle Scholar
  21. 21.
    Dürr, D., Goldstein, S., Tumulka, R., Zanghì, N.: Bohmian mechanics and quantum field theory. Phys. Rev. Lett. 93, 090402 (2004)MathSciNetADSGoogle Scholar
  22. 22.
    Einstein, A.: Zur elektrodynamik bewegter körper (On the electrodynamics of moving bodies). Annal. Phys. 17, 891–921 (1905)ADSzbMATHGoogle Scholar
  23. 23.
    Esposito, S.: Photon wave mechanics: a de Broglie-Bohm approach. Found. Phys. Lett. 12(6), 533–545 (1999)MathSciNetGoogle Scholar
  24. 24.
    Flack, R, Hiley, B.J.: Weak values of momentum of the electromagnetic field: average momentum flow lines, not photon trajectories (2016). arXiv:1611.06510
  25. 25.
    Fleming, G., Butterfield, J.: Strange positions. In: Butterfield, J., Pagonis, C. (eds.) From Physics to Philosophy, pp. 108–165. Cambridge University Press, Cambridge (1999)Google Scholar
  26. 26.
    Fraser, D.: The fate of ‘particles’ in quantum field theories with interactions. Stud. Hist. Philos. Mod. Phys. 39(4), 841–859 (2008)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Frenkel, J.: Wave Mechanics: Advanced General Theory. Oxford University Press, Oxford (1934)zbMATHGoogle Scholar
  28. 28.
    Good Jr., R.H.: Particle aspect of the electromagnetic field equations. Phys. Rev. 105(6), 1914–1919 (1957)ADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Good Jr., R.H.: Photon. In: Besancon, R.M. (ed.) The Encyclopedia of Physics, pp. 921–925. Van Nostrand Reinhold, New York (1985)Google Scholar
  30. 30.
    Good Jr., R.H., Nelson, T.J.: Classical Theory of Electric and Magnetic Fields. Academic Press, Cambridge (1971)Google Scholar
  31. 31.
    Greiner, W., Reinhardt, J.: Field Quantization. Springer, New York (1996)zbMATHGoogle Scholar
  32. 32.
    Griffiths, D.J.: Introduction to Electrodynamics, 3rd edn. Prentice Hall, Upper Saddle River (1999)Google Scholar
  33. 33.
    Halvorson, H., Clifton, R.: No place for particles in relativistic quantum theories? Philos. Sci. 69(1), 1–28 (2002)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Hatfield, B.: Quantum Theory of Point Particles and Strings. Frontiers in Physics, vol. 75. Addison-Wesley, Boston (1992)Google Scholar
  35. 35.
    Hestenes, D.: Space-Time Algebra. Gordon and Breach, London (1966)zbMATHGoogle Scholar
  36. 36.
    Holland, P.R.: The de Broglie-Bohm theory of motion and quantum field theory. Phys. Rep. 224(3), 95–150 (1993)ADSMathSciNetGoogle Scholar
  37. 37.
    Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1993)Google Scholar
  38. 38.
    Holland, P.R.: Hydrodynamic construction of the electromagnetic field. Proc. R. Soc. Lond. A 461, 3659–3679 (2005)ADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    Itzykson, C., Zuber, J.-B.: Quantum Field Theory. McGraw-Hill, New York (1980)zbMATHGoogle Scholar
  40. 40.
    Keller, O.: On the theory of spatial localization of photons. Phys. Rep. 411(1), 1–232 (2005)ADSGoogle Scholar
  41. 41.
    Kiessling, M.K.-H., Tahvildar-Zadeh, S.A.: On the quantum-mechanics of a single photon (2017). arXiv:1801.00268
  42. 42.
    Kobe, D.H.: A relativistic Schrödinger-like equation for a photon and its second quantization. Found. Phys. 29(8), 1203–1231 (1999)MathSciNetGoogle Scholar
  43. 43.
    Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, Volume 2: The Classical Theory of Fields, 3rd edn. Addison-Wesley Publishing Company, Boston (1971)Google Scholar
  44. 44.
    Landau, L., Peierls, R.: Quantenelektrodynamik im konfigurationsraum. Z. Phys. 62(3), 188–200 (1930)ADSzbMATHGoogle Scholar
  45. 45.
    Lindell, I.V.: Methods for Electromagnetic Field Analysis. Oxford University Press, Oxford (1992)Google Scholar
  46. 46.
    Malament, D.B.: In defense of dogma: why there cannot be a relativistic quantum mechanics of (localizable) particles. In: Clifton, R. (ed.) Perspectives on Quantum Reality, pp. 1–10 . Springer, New York (1996)Google Scholar
  47. 47.
    Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge (1995)Google Scholar
  48. 48.
    Messiah, A.: Quantum Mechanics: Volume II. North-Holland Publishing Company, Amsterdam (1962)zbMATHGoogle Scholar
  49. 49.
    Mignani, E., Recami, E., Baldo, M.: About a Dirac-like equation for the photon according to Ettore Majorana. Lett. Nuovo Cimento (1971–1985) 11(12), 568–572 (1974)Google Scholar
  50. 50.
    Moses, H.E.: Solution of Maxwell’s equations in terms of a spinor notation: the direct and inverse problem. Phys. Rev. 113, 1670–1679 (1959)Google Scholar
  51. 51.
    Ney, A., Albert, D.Z.: The Wave Function: Essays on the Metaphysics of Quantum Mechanics. Oxford University Press, Oxford (2013)zbMATHGoogle Scholar
  52. 52.
    Norsen, T.: Foundations of Quantum Mechanics. Springer, New York (2017)zbMATHGoogle Scholar
  53. 53.
    Pais, A.: ‘Subtle is the Lord...’: The Science and Life of Albert Einstein. Oxford University Press, Oxford (1982)Google Scholar
  54. 54.
    Pauli, W.: General Principles of Quantum Mechanics. Springer, New York (1980)Google Scholar
  55. 55.
    Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Westview Press, Boulder (1995)Google Scholar
  56. 56.
    Pollack, G.L., Stump, D.R.: Electromagnetism. Addison-Wesley, Boston (2002)Google Scholar
  57. 57.
    Raymer, MG., Smith, B.J.: The Maxwell wave function of the photon. In: The Nature of Light: What is a Photon? vol. 5866, pp. 293–298. International Society for Optics and Photonics (2005)Google Scholar
  58. 58.
    Riesz, M.: Clifford numbers and spinors. In: Bolinder, E.F., Lounesto, P. (eds). Lectures delivered October 1957–January 1958, Kluwer (1993)Google Scholar
  59. 59.
    Rumer, G.: Zur wellentheorie des lichtquants. Z. Phys. 65(3), 244–252 (1930)ADSzbMATHGoogle Scholar
  60. 60.
    Ryder, L.H.: Quantum Field Theory. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  61. 61.
    Schweber, S.S.: Introduction to Relativistic Quantum Field Theory. Harper & Row, New York (1961)zbMATHGoogle Scholar
  62. 62.
    Sebens, C.T.: Forces on fields. Studies in history and philosophy of modern physics 63, 1–11 (2018)ADSMathSciNetzbMATHGoogle Scholar
  63. 63.
    Sebens, C.T.: How electrons spin (2018). arXiv:1806.01121
  64. 64.
    Sebens, C.T., Carroll, S.M.: Self-locating uncertainty and the origin of probability in Everettian quantum mechanics. Br. J. Philos. Sci. 69(1), 25–74 (2018)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Silberstein, L.: Elektromagnetische grundgleichungen in bivektorieller behandlung. Ann. Phys. 327(3), 579–586 (1907)zbMATHGoogle Scholar
  66. 66.
    Silberstein, L.: Nachtrag zur abhandlung über ‘elektromagnetische grundgleichungen in bivektorieller behandlung’. Ann. Phys. 329(14), 783–784 (1907)Google Scholar
  67. 67.
    Struyve, W.: Pilot-wave approaches to quantum field theory. In: Journal of Physics: Conference Series, vol. 306, pp. 012047. IOP Publishing, Bristol (2011)Google Scholar
  68. 68.
    Thaller, B.: The Dirac Equation. Springer, New York (1992)zbMATHGoogle Scholar
  69. 69.
    Valentini, A.: On the pilot-wave theory of classical, quantum and subquantum physics. Ph.D. thesis, ISAS, Trieste (1992)Google Scholar
  70. 70.
    Wallace, D.: The Emergent Multiverse: Quantum theory according to the Everett interpretation. Oxford University Press, Oxford (2012)zbMATHGoogle Scholar
  71. 71.
    Wallace, D.: The quantum theory of fields. In: Knox, E., Wilson, A. (eds.) Handbook of Philosophy of Physics (forthcoming)Google Scholar
  72. 72.
    Weber, H., Riemann, B.: Die Partiellen Differential-Gleichungen Der Mathematischen Physik, Nach Riemann’s Vorlesungen Bearbeitet von Heinrich Weber. Braunschweig (1901)Google Scholar
  73. 73.
    Wesley, J.P.: A resolution of the classical wave-particle problem. Found. Phys. 14, 155–170 (1984)ADSGoogle Scholar
  74. 74.
    Wigner, E.P.: Thirty years of knowing Einstein. In: Woolf, H. (ed.) Some Strangeness in the Proportion, pp. 461–468. Addison-Wesley, Boston (1980)Google Scholar
  75. 75.
    Wigner, E.P.: Interpretation of quantum mechanics. In: Wheeler, J.A., Zurek, W.H. (eds.) Quantum Theory and Measurement, pp. 260–314. Princeton University Press, Princeton (1983)Google Scholar
  76. 76.
    Zel’dovich, Y.B.: Number of quanta as an invariant of the classical electromagnetic field. Sov. Phys. Dokl. 10(8), 771–772 (1966)ADSGoogle Scholar

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Authors and Affiliations

  1. 1.Division of the Humanities and Social SciencesCalifornia Institute of TechnologyPasadenaUSA

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