Chiral Dirac Equation Derived From Quaternionic Maxwell’s Systems

Hector Torres-Silva

University School of Electrical and Electronic Engineering, Arica, Chile..

DOI: 10.4236/jemaa.2013.53017                

Abstract

In the present article we propose a simple equality involving the Dirac operator and the Maxwell operators under chiral approach. This equality establishes a direct connection between solutions of the two systems and moreover, we show that it is valid when the natural relation between the frequency of the electromagnetic wave and the energy of the Dirac particle is fulfilled if the electric field E is parallel to the magnetic field H. Our analysis is based on the quaternionic form of the Dirac equation and on the quaternionic form of the Maxwell equations. In both cases these reformulations are completely equivalent to the traditional form of the Dirac and Maxwell systems. This theory is a new quantum mechanics (QM) interpretation. The below research proves that the QM represents the electrodynamics of the curvilinear closed chiral waves. It is entirely according to the modern interpretation and explains the particularities and the results of the quantum field theory. Also this work may help to clarify the controversial relation between Maxwell and Dirac equations while presenting an original way to derive the Dirac equation from the chiral electrodynamics, leading, perhaps, to novel conception in interactions between matter and electromagnetic fields. This approach may give a reinterpretation of Majorana equation, neutrino mass, violation of Heinsenberg’s measurement-disturbation relationship and mass generation in systems like graphene devices.

Keywords

Quaternion; Dirac Equation; Maxwell System

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	W. I. Fushchyld, “On the Connection between Solutions of Dirac and Maxwell Equations,” Scientific Works, Vol. 4, No. 1, 2002, pp. 320-336.
[2]	A. A. Campolattaro, “Generalized Maxwell Equations and Quantum Mechanics,” International Journal of Theoretical Physics, Vol. 29, No. 2, 1990, pp. 141-155. doi:10.1007/BF00671324
[3]	V. V. Dvoeglazov, “Generalized Maxwell and Weyl Equations for Massless Particles,” Revista Mexicana de Física, Vol. 49S1, No. 6, 2003, pp. 99-103.
[4]	R. H. Good, “Particle Aspect of the Electromagnetic Field Equations,” Physical Review, Vol. 105, No. 6, 1957, pp. 1914-1919. doi:10.1103/PhysRev.105.1914
[5]	H. E. Moses, “Solution of Maxwell’s Equations in Spinor Notation,” Physical Review, Vol. 113, No. 6, 1959, pp. 1670-1679. doi:10.1103/PhysRev.113.1670
[6]	K. Imaeda, “A New Formulation of Classical Electrodynamics,” Nuovo Cimento, Vol. 32B, No. 1, 1976, pp. 138- 162.
[7]	H. Campos, V. Kravchenko and L. Méndez, “Complete Families of Solutions of the Dirac Equation: An Application of Bicomplex Pseudoanalytic Function Theory and Transmutation Operators,” Advances in Applied Clifford Algebras, Vol. 22, No. 3, 2011, pp. 557-594.
[8]	V. V. Kravchenko, “On the Relation between the Maxwell System and the Dirac Equation,” WSEAS Transaction on Systems, Vol. 1, No. 2, 2002, pp. 115-118.
[9]	V. V. Kravchenko and M. P. Ramirez, “On Bers Generating Functions for First Order Systems of Mathematical Physics,” Advances in Applied Clifford Algebras, Vol. 21, No. 3, 2011, pp. 547-559. doi:10.1007/s00006-010-0261-5
[10]	I. Yu. Krivsky, V. M. Simulik, “Unitary connection in Maxwell-Dirac isomorphism and the Clifford algebra”, Advances in Applied Clifford Algebras, v. 6, No. 2, 1996, pp. 249-259.
[11]	A. Lakhtakia, “Beltrami Fields in Chiral Media,” World Scientific Series in Contemporary Chemical Physics, Vol. 2, No. 1, 1994.
[12]	J. Vaz Jr. and W. Rodrigues Jr., “Equivalence of Dirac and Maxwell Equations and Quantum Mechanics,” International Journal of Theoretical Physics, Vol. 32, No. 6, 1993, pp. 945-959. doi:10.1007/BF01215301
[13]	A. Gsponer, “On the ‘Equivalence’ of the Maxwell and Dirac Equations,” International Journal of Theoretical Physics, Vol. 41, No. 4, 2002, pp. 689-964. doi:10.1023/A:1015232427515
[14]	V. V. Kravchenko and H. Oviedo, “On the Quaternionic Reformulation of Maxwell’s Equations for Chiral Media and Its Applications, Zeischrift für Analysis und ihre Anwendungen,” Journal for Analysis and Its Applications,” Vol. 22, No. 3, 2003, pp. 569-589.
[15]	B. Schneider and E. Karapinar, “A Note on Biquaternionic MIT Bag Model,” International Journal of Contemporary Mathematical Sciences, Vol. 1, No. 10, 2006, pp. 449-461.
[16]	T. D. Lee and C. N. Yang, “Question of Parity Conservation in Weak Interactions,” Physical Review, Vol. 104, No. 1, 1956, pp. 254-258. doi:10.1103/PhysRev.104.254
[17]	S. Adler, “Axial-Vector Vertex in Spinor Electrodynamics,” Physical Review, Vol. 177, No. 5, 1969, pp. 2426- 2438. doi:10.1103/PhysRev.177.2426
[18]	V. Ginzburg and L. Landau, “On the Theory of Superconductivity,” Zhurnal Eksperimentalnoi i Teoreticheskoi Fisiki. Vol. 20, No. 1, 1950, p. 1064.
[19]	A. Zee, “Broken-Symmetric Theory of Gravity,” Physical Review Letter, Vol. 42, No. 7, 1979, pp. 417-421. doi:10.1103/PhysRevLett.42.417
[20]	H. Torres-Silva and D. Torres, “Chiral Current in a Graphene Battery,” Journal of Electromagnetic Analysis and Applications, Vol. 4, No. 10, 2012, pp. 426-431.
[21]	H. Torres-Silva, “Chiral Transverse Electromagnetic Standing Waves with E II H in the Dirac Equation and the Spectra of the Hydrogen Atom,” In: A. Akdagli, Ed., Behavior of Electromagnetic Waves in Different Media and Structures, Chapter 15, Book Intech, Rijeka, 2011, pp. 301-324.
[22]	H. Torres-Silva, “Chiral Waves in Graphene Medium and Optical Simulation with Metamaterial,” In: A. Kishk, Ed., Solutions and Applications of Scattering, Propagation, Radiation and Emission of Electromagnetic Waves, Chapter 2, Book Intech, Rijeka, 2012, pp. 25-55.
[23]	E. Chubykalo, et al., “Self Dual Electromagnetic Fields,” American Journal of Physics, Vol. 78, No. 8, 2010, pp 858-861.
[24]	L. Rozena, et al., “Violation of Heisenbergs’s Measurement-Disturbance Relationship by Weak Measurements,” Physical Review Letters, Vol. 109, 2012, Article ID: 100 404.
[25]	J. Erhart, et al., “Experimental Demonstration of a Universally Valid Error-Disturbance Uncertainty Relation in Spin Measurements,” Nature Physics, Vol. 8, 2012, pp 185-189. doi:10.1038/nphys2398
[26]	A. Studenikin, “Neutrino Magnetic Moment,” Nuclear Physics B, Vol. 188, No. 1, 2009, pp. 220-222.
[27]	H. Torres-Silva and D. Torres Cabezas, “Chiral Seismic Attenuation with Acoustic Metamaterials,” Journal of Electromagnetics Analysis and Applications, 2013, in Press.