The Photon Wave Function

Joseph Cugnon

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DOI: 10.4236/ojm.2011.13008                

Abstract

The properties of a wave equation for a six-component wave function of a photon are re-analyzed. It is shown that the wave equation presents all the properties required by quantum mechanics, except for the ones that are linked with the definition of the position operator. The situation is contrasted with the three-component formulation based on the Riemann-Silberstein wave function. The inconsistency of the latter with the principles of quantum mechanics is shown to arise from the usual interpretation of the wave function. Finally, the Lorentz invariance of the six-component wave equation is demonstrated explicitly for Lorentz boosts and space inversion.

Keywords

Photon, Wave Function, Quantum Mechanics, Lorentz Invariance

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	V. V. Dvoeglazov, “2(2S + 1)-Component Model and Its Connection with Other Field Theories,” XVII Symposium on Nuclear Physics, Mexico, 4-7 January 1994.
[2]	I. Bialynicki-Birula, “On the Wave Function of the Pho- ton,” Acta Physica Polonica A, Vol. 86, 1994, pp. 97- 116.
[3]	I. Bialynicki-Birula, “Photon Wave Function,” In: E. Wolf Ed., Progress in Optics XXXVI, Elsevier, Amster- dam, 1996.
[4]	I. Bialynicki-Birula, “Exponential Localization of Pho- tons,” Physical Review Letters, Vol. 80, No. 24, 1998, pp. 5247-5250. doi:10.1103/PhysRevLett.80.5247
[5]	J. E. Sipe, “Photon Wave Functions,” Physical Review A, Vol. 52, No. 3, 1995, pp. 1875-1883. doi:10.1103/PhysRevLett.80.5247
[6]	D. H. Kobe, “A Relativistic Schr?dinger-Like Equation for a Photon and Its Second Quantization,” Foundations of Physics, Vol. 29, No. 8, 1999, pp. 1203-1231. doi:10.1023/A:1018855630724
[7]	M. Hawton, “Lorentz-Invariant Photon Number Density,” Physical Review A, Vol. 78, No. 1, 2008, Article ID 012111. doi:10.1103/PhysRevA.78.012111
[8]	H. Feshbach and F. Villars, “Elementary Relativistic Wave Mechanics of Spin 0 and Spin 1/2 Particles,” Re- views of Modern Physics, Vol. 30, No. 1, 1958, pp. 24- 45. doi:10.1103/RevModPhys.30.24
[9]	Z.-Y. Wang, C.-D. Xiong and O. Keller, “The First- Quantized Theory of Photons,” Chinese Physics Letters, Vol. 24, No. 2, 2007, pp. 418-420. doi:10.1088/0256-307X/24/2/032
[10]	Z.-Y. Wang, C.-D. Xiong and O. Keller, “Photon Wave Mechanics,” Chinese Physics Letters, Vol. 24, No. 2, 2007, arXiv:quant-ph/0511181v5.
[11]	S. Esposito, E. Recami, A. Van der Merwe and R. Battiston, “Ettore Majorana: Unpublished Research Notes on Theoretical Physics,” Springer, Berlin, 2009. doi:10.1007/978-1-4020-9114-8
[12]	M. G. Raymer and B. J. Smith, “The Maxwell Wave Function of the Photon,” SPIE Conference Optics and Photonics, San Diego, August 2005.
[13]	H. Weber, “Notes on Differential Equations by Riemann,” In: Riemann and G. F. Berhard, Eds., Die Partiellen Differential-Gleichungen der Mathematischen Physik Nach Riemann’s Vorlesungen, Friedrich Vieweg und Sohn, Bruns- wick, 1901, p. 348.
[14]	L. Silberstein, “Elektromagnetische Grundgleichungen in Bivectorieller Behandlung,” Annalen der Physik, Vol. 22, 1907, p. 579. doi:10.1002/andp.19073270313
[15]	J. J. Sakurai, “Advanced Quantum Mechanics,” Addison- Wesley Publication Co., London, 1967.
[16]	J. D. Jackson, “Classical Electrodynamics,” John Wiley, New York, 1975, pp. 552-554.