Vol. 16

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues

Spinor and Hertzian Differential Forms in Electromagnetism

By Pierre Hillion
Progress In Electromagnetics Research M, Vol. 16, 197-211, 2011


The purpose of this paper is to extend to spinor electromagnetism the differential forms, based on the Cartan exterior derivative and originally developed for tensor fields, in a very compact way. To this end, differential electromagnetic forms are first compared to conventional tensors. Then, using the local isomorphism between the O (3,C) and SL (2,C) groups supplying the well known connection between complex vectors and traceless second rank spinors, they are generalized to spinor electromagnetism and to Proca fields. These differential forms are finally expressed in terms of Hertz potentials.


Pierre Hillion, "Spinor and Hertzian Differential Forms in Electromagnetism," Progress In Electromagnetics Research M, Vol. 16, 197-211, 2011.


    1. Cartan, E., Lecons sur les Invariants Integraux, Hermann, Paris, 1958.

    2. Jackson, J. D., Classical Electrodynamics, Wiley, New York, 1976.

    3. Jones, D. S., Acoustic and Electromagnetic Waves, Clarendon, Oxford, 1986.

    4. Moller, C., The Theory of Relativity, Clarendon, Oxford, 1952.

    5. Eddington , A. S., The Mathematical Theory of Relativity, University Press, Cambridge, 1951.

    6. De Rham, "Differential Manifolds," Springer, 1984.

    7. Misner, C. W. , K. S. Thorne, and J. A. Wheeler, Gravitation, , W. H. Freeman, San Francisco, 1973.

    8. Meetz, L and W. L. Engl, Electromagnetic Felder, Springer, Berlin, 1980.

    9. Deschamps, G. A., "Electromagnetism and differential forms," IEEE Proceedings, Vol. 69, 676-696, 1981.

    10. Hehl, , F. W. and Y. Obhukov, Foundations of Classical Electrodynamics, Birkhauser, Basel, 2003.

    11. Hehl, F. W., "Maxwell's equations in Minkowski's world," Annalen. der Physik, Vol. 17, 691-704, 2008.

    12. Warnick, K. F. and P. Russer, "Two, three and four dimensional electromagnetism using differential forms," Turkish Journal of Electrical. Engineering, Vol. 14, 151-172, 2006.

    13. Lindell, I. V., Differential Forms in Electromagnetism, Wiley IEEE, Hoboken, 2004.

    14. Bossavit , A., "Differential forms and the computation of fields and forces in electromagnetism," European Journal of Mechanics B, Fluids, Vol. 10, 474-488, 1991.

    15. Stern, A. , Y. Tong, M. Desbrun, and J. E. Marsden, "Variational integrators for Maxwell's equations with sources," PIERS Online, Vol. 4, No. 7, 711-715, 2008.

    16. Russer, P., "Geometrical concepts in teaching electromagnetics," Course, Nottingham available on Google; See also: P. Russer,Electromagnetic Circuit and Antenna Design for Communications Engineering, Artech House, Boston, 2006.

    17. Post, E. J., Formal Structure of Electromagnetism, North Holland, Amsterdam, 1962.

    18. Cartan, E., "Lecons sur la theorie des Spineurs," Hermann, Paris, 1938.

    19. Corson, E. M., Introduction to Spinors, Tensors and Relativistic Wave Equations, Blackie & Sons, London, 1954.

    20. Penrose, R. and W. Rindler, Spinors and Space-time, University Press, Cambridge, 1968.

    21. Laporte, O. and G. E. Uhlenbeck, "Application of spinor analysis to the Maxwell and Dirac equations," Physical Review, Vol. 37, 1380-1387, 1931.

    22. Hillion, P. and S. Quinnez, "Proca and electromagnetic fields," International Journal of Theortetical Physics, Vol. 25, 727-733, 1986.

    23. Mustafa, E. and J. M. Cohen, "Hertz and Debye potentials and electromagnetic fields in general relativity," Classical and Quantum Gravity, Vol. 4, 1623-1631, 1987.

    24. Olmsted, J. M. H., "Advanced Calculus," Appleton-Century Crofts, 1961.

    25. Dautray, R. and J. L. Lions, Analyse Mathematique et Calcul Numerique Pour les Sciences et Les Techniques, Masson, Paris, 1985.

    26. Bossavit, A., Computational Electromagnetism, Academic Press, San Diego, 1997.

    27. Ren, Z. and A. Razeh, "Computation of the 3D electromagnetic field using differential forms based elements and dual formalism," International Journal of Numerical Modelling, Vol. 9, 81-96, 1996.

    28. Ren, Z. and A. Bossavit, "A new approach to eddy current problems and numerical evidence of its validity," International Journal of Applied Electromagnetics in Materials, Vol. 3, 39-46, 1992.

    29. Hillion, P., "The Wilsons' experiment," Apeiron, Vol. 6, 1-8, 1999.

    30. Hillion, P. and S. Quinnez, "Diffraction patterns of circular and rectangular apertures in the spinor formalism of electromagnetism," Journal of Optics, Vol. 16, 5-19, 1985.

    31. Penrose, R., "Twitor algebra," J. Math. Phys., Vol. 8, 345-367, 1967.

    32. Witten, E., "Perturbative gauge theory as a string theory in twistor space," Comm. Math. Phys., Vol. 252, 189-258, 2004.

    33. Oliveira, C. C. and de Amaral C. Marcio, "Spinor formalism in gravitation," II Nuovo Cimento., Vol. 47, No. 1, 9-18, 1967.

    34. Berkovitz, N., "Explaining the pure spinor formalism for the superstring," Journal of the High Energy Physics, Vol. 2008, 2008.

    35. Mafra, C. R., "Superstring amplitude in the pure spinor ormalism," Nuclear Physics B, Vol. 171, 292-294, 2007.

    36. Stratton, J. A., Electromagnetic Theory, Mac Graw Hill, New York, 1941.

    37. Felsen, L. B. and N. Marcuwitz, Radiation and Scattering of Waves, Wiley, Hoboken, 2003.

    38. Hillion, P., "Hertz potentials in Boys-Post isotropic chiral media ," Physica. Scripta, Vol. 75, 404-406, 2007.

    39. Hillion, P., "Hertz potentials in uniaxially anisotropic media," Journal of Phsyics A: Mathematical Theory, Vol. 41, 365401, 2008.

    40. Mc Crea , W. H., "Hertzian electromagnetic potentials," Proceedings Royal Society A, Vol. 290, 447-457, London, 1957.

    41. Essex, E. A., "Hertz vectror potentials of electromagnetic theory," American Journal of Physics, Vol. 45, 1099-1101, 1977.

    42. Cough, W., "An alternative approach to Hertz vectors," Progress In Electromagnetic Research, Vol. 12, 205-217, 1996.

    43. Wu, A. C. T., "Debye scalar potentials for electromagnetic fields," Physical Review, Vol. 34, 3109-3114, 1986.

    44. Lindell, I. V., "Potential representation of electromagnetic fields in decomposable anisotropic media," Journal of Physics D, Vol. 33, 3169-3172, 2001.

    45. Weiglhofer, W. S., "Isotropic chiral media and scalar Hertz potentials," Journal of Physics A: Mathematics, General, Vol. 21, 2249-2251, 1988.

    46. Przezriecki, S. S. and R. A. Hurd, "A note on scalar Hertz potentials for gyrotropic media," Applied Physics, Vol. 20, 313-317, 1979.

    47. Hillion, P., "Self-dual electromagnetism in isotropic media," Nuovo. Cimento., Vol. 121B, 11-25, 2006.

    48. Synge, J. L., Relativity: The Special Theory, North-Holland, Amsterdam, 1958.

    49. Christianto, V., F. L. Smarandache, F. Lichtenberg, and , "A note on extended Proca equation," Progress in Physics, Vol. 1, 40-44, 2009.