PIER M
 
Progress In Electromagnetics Research M
ISSN: 1937-8726
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 16 > pp. 197-211

SPINOR AND HERTZIAN DIFFERENTIAL FORMS IN ELECTROMAGNETISM

By P. Hillion

Full Article PDF (166 KB)

Abstract:
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.

Citation:
P. Hillion, "Spinor and Hertzian Differential Forms in Electromagnetism," Progress In Electromagnetics Research M, Vol. 16, 197-211, 2011.
doi:10.2528/PIERM10111501

References:
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.
doi:10.1109/PROC.1981.12048

10. Hehl, , F. W. and Y. Obhukov, Foundations of Classical Electrodynamics, Birkhauser, Basel, 2003.
doi:10.1007/978-1-4612-0051-2

11. Hehl, F. W., "Maxwell's equations in Minkowski's world," Annalen. der Physik, Vol. 17, 691-704, 2008.
doi:10.1002/andp.200810320

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.
doi:10.1002/0471723096

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.
doi:10.2529/PIERS071019000855

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.
doi:10.1103/PhysRev.37.1380

22. Hillion, P. and S. Quinnez, "Proca and electromagnetic fields," International Journal of Theortetical Physics, Vol. 25, 727-733, 1986.
doi:10.1007/BF00668718

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.
doi:10.1088/0264-9381/4/6/020

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.
doi:10.1002/(SICI)1099-1204(199601)9:1/2<81::AID-JNM229>3.0.CO;2-J

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.
doi:10.1088/0150-536X/16/1/001

31. Penrose, R., "Twitor algebra," J. Math. Phys., Vol. 8, 345-367, 1967.
doi:10.1063/1.1705200

32. Witten, E., "Perturbative gauge theory as a string theory in twistor space," Comm. Math. Phys., Vol. 252, 189-258, 2004.
doi:10.1007/s00220-004-1187-3

33. Oliveira, C. C. and de Amaral C. Marcio, "Spinor formalism in gravitation," II Nuovo Cimento., Vol. 47, No. 1, 9-18, 1967.
doi:10.1007/BF02771370

34. Berkovitz, N., "Explaining the pure spinor formalism for the superstring," Journal of the High Energy Physics, Vol. 2008, 2008.
doi:10.1088/1126-6708/2008/01/009

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.
doi:10.1088/0031-8949/75/4/003

39. Hillion, P., "Hertz potentials in uniaxially anisotropic media," Journal of Phsyics A: Mathematical Theory, Vol. 41, 365401, 2008.
doi:10.1088/1751-8113/41/36/365401

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.
doi:10.1119/1.10955

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.
doi:10.1088/0305-4470/21/9/036

46. Przezriecki, S. S. and R. A. Hurd, "A note on scalar Hertz potentials for gyrotropic media," Applied Physics, Vol. 20, 313-317, 1979.
doi:10.1007/BF00895002

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.


© Copyright 2010 EMW Publishing. All Rights Reserved