The Photon Wave Function

  • Iwo Bialynicki-Birula


Quantization of the electromagnetic field is traditionally introduced at the level of second quantization: the classical field variables are replaced by field operators. I believe that the reasons why a first-quantized theory of photons has never been fully developed are mainly historical. Had Dirac discovered his relativistic wave equation [1] prior to his quantization of the electromagnetic field [2], he would have noticed and most probably further explored a great similarity between the wave equation for the electron (or even better for the neutrino) and the Maxwell equations. As it happened, this similarity was noticed later (for the first time apparently by Majorana [3]) and played no role in the development of the quantum theory of electromagnetism because the quantized electromagnetic field has been introduced from the very beginning and accounted for all quantum properties electromagnetic radiation. Subsequently quantum electrodynamics has become so successful in explaining with utmost accuracy all experiments within its range of applicability that there was no need to search for an alternative formulation that would employ the concept of the photon wave function. Considering our trust in quantum electrodynamics and our familiarity with its formal apparatus one may even ask if there is any justification at all for, what it essentially amounts to, a reconstruction of the notion of the photon function from QED, only to face a not so familiar object whose properties are yet to be uncovered.


Wave Function Wave Equation Maxwell Equation Total Angular Momentum Quantum Electrodynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. A. M. Dirac, Proc. Roy. Soc.(London)A117, 610 (1928); A118, 351 (1928).zbMATHGoogle Scholar
  2. [2]
    P. A. M. Dirac, Proc. Roy. Soc.(London)A114, 243 (1927).Google Scholar
  3. [3]
    E. Majorana (unpublished notes), quoted after R. Miguani, E. Bec. mi, and M. Baldo, Lett.Nuovo Cime ito 11, 568 (1974).Google Scholar
  4. [4]
    G. Wentzel, Quantum Theory of Fields, lnterscience, New York 1949.Google Scholar
  5. [5]
    R. London, The Quantum Theory of Light, Clarendon, Oxford 1973.Google Scholar
  6. [6]
    W. H. Louisell, Quantum Theory of Radiation, Wiley, New York 1990.zbMATHGoogle Scholar
  7. [7]
    C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms, Wiley, New York 1989.Google Scholar
  8. [8]
    I. Bialynicki-Birula and Z. Bialynicka-Birula, Quantum Electrodynamics, Pergamon, Oxford 1976.Google Scholar
  9. [9]
    P. A. M. Dirac, The Principles of Quantum Theory, Clarendon Press, Oxford 1958.Google Scholar
  10. [10]
    G. Baym, Lectures on Quantum Mechanics, Benjamin, Reading 1969.zbMATHGoogle Scholar
  11. [11]
    II. J. Lipkin, Quantum Mechanics, North-Holland, Amsterdam 1973.Google Scholar
  12. [12]
    C. Cohen-’Fannoudji, B. Din, and F. Laboë, Quantum Mechanics, Vol.!, Wiley, New York 1977.Google Scholar
  13. [13]
    H. A. Nramers, Quantentheorie des Elektrons und der.5’trahlungin Hand-und.1ahrbuch der Chemischen Physik, Eucken-Wolf, Leipzig, 1938 (English translation Quantum Mechanics, North-Volland, Amsterdam 1957 ).Google Scholar
  14. [14]
    D. Bohm, Quantum Theory, Constable, London 1954.Google Scholar
  15. [15]
    E. A. Power, Introductory Quantum Electrodynamics, Longmans, London 1964.zbMATHGoogle Scholar
  16. [16]
    L. Silberstein, Ann. d. Phys.22,579 (1907); 24, 783 (1907). After publishing his first paper Silberstein discovered that the complex form of Maxwell equations has been already known to Riemann (Die partiellen Differential-Gleichungen der mathematische Physik, Lectures by B. Riemann edited by H. Weber vol. 2, Vieweg, Braunschweig 1901 ).Google Scholar
  17. [17]
    H. Bateman, The Mathematical Analysis of Electrical and Optical Wave Motion on the Basis of Maxwell’s Equations, Cambridge, 1915 (reprinted by Dover, New York 1955 ).Google Scholar
  18. [18]
    I. Bialynicki-Birula, Acta Phys. Polon. A 86, 97 (1994).Google Scholar
  19. [19]
    L. D. Landau and R. Peierls, Z. Phys. 62, 188 (1930).CrossRefGoogle Scholar
  20. [20]
    R. J. Cook, Phys. Rev. A25, 2164 (1982); 26, 2754 (1982).Google Scholar
  21. [21]
    W. Pauli, Prinzipien der Quantentheorie, Handbuch der Physik, Vol.24, Springer, Berlin, 1933, (English translation: General Principles of Quantum Mechanics, Springer, Berlin, 1980 ).Google Scholar
  22. [22]
    H. Weyl, Z. Phys. 56, 330 (1929).CrossRefzbMATHGoogle Scholar
  23. [23]
    R. Penrose and W. Rindler, Spinors and Space-Time,Cambridge University Press, Cambridge 1984, Vol.I, Ch. 5.Google Scholar
  24. [24]
    C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York 1980, p. 50.Google Scholar
  25. [25]
    A. Messiah, Quantum Mechanics, North-Holland, Amsterdam 1962.zbMATHGoogle Scholar
  26. [26]
    L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Addison-Wesley, Reading 1981.zbMATHGoogle Scholar
  27. [27]
    A. Ashtekar, private communication.Google Scholar
  28. [28]
    A. Ashtekar and A.Magnon, Proc. Roy. Soc. A346, 375 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Ashtekar and A.Magnon, Gen. Rel. Gray. 12, 205 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    V. Bargmann and E. P. Wigner, Proc. Nat. Acad. Sci. USA 34, 211 (1948).MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    L. Gross, J. Math. Phys. 5, 687 (1964).CrossRefzbMATHGoogle Scholar
  32. [32]
    G. Kaiser, A Friendly Guide to WaveletsBirkhäuser, Boston 1994. Google Scholar
  33. [33]
    J. D. Jackson, Classical Electrodynamics, Wiley, New York 1975.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Iwo Bialynicki-Birula
    • 1
  1. 1.Center for Theoretical Physics, Polish Academy of Sciences Lotników 32/46, 02-668 Warsaw, Poland and Abteilung für QuantenphysikUniversität UlmUlmGermany

Personalised recommendations