A Real Version of the Dirac Equation and Its Coupling to the Electromagnetic Field
Author(s)
Eckart Marsch*
Affiliation(s)
Institute for Experimental and Applied Physics, Christian Albrechts University at Kiel, Kiel, Germany.
Institute for Experimental and Applied Physics, Christian Albrechts University at Kiel, Kiel, Germany.
ABSTRACT
A real version of the Dirac equation is derived and its coupling to the electromagnetic field considered. First the four-component real Majorana equation is briefly discussed. Then the complex Dirac equation including an electromagnetic field will be written as an eight-component real spinor equation by separating it into its real and imaginary parts. Through this decomposition, what becomes obvious is the way in which the electromagnetic field couples to charged fermions (electron and positron) when being described by real spinor fields. Thus, contrary to common expectation, charged fermions can also be described by a real Dirac equation if they are considered as a doublet related to the SO(2) symmetry group, which enables a matrix coupling to the electromagnetic field and corresponds to the usual U(1) gauge symmetry of the standard Dirac equation.
A real version of the Dirac equation is derived and its coupling to the electromagnetic field considered. First the four-component real Majorana equation is briefly discussed. Then the complex Dirac equation including an electromagnetic field will be written as an eight-component real spinor equation by separating it into its real and imaginary parts. Through this decomposition, what becomes obvious is the way in which the electromagnetic field couples to charged fermions (electron and positron) when being described by real spinor fields. Thus, contrary to common expectation, charged fermions can also be described by a real Dirac equation if they are considered as a doublet related to the SO(2) symmetry group, which enables a matrix coupling to the electromagnetic field and corresponds to the usual U(1) gauge symmetry of the standard Dirac equation.
Cite this paper
Marsch, E. (2015) A Real Version of the Dirac Equation and Its Coupling to the Electromagnetic Field. Journal of Modern Physics, 6, 1-11. doi: 10.4236/jmp.2015.61001.
Marsch, E. (2015) A Real Version of the Dirac Equation and Its Coupling to the Electromagnetic Field. Journal of Modern Physics, 6, 1-11. doi: 10.4236/jmp.2015.61001.
References
[1] Dirac, P.A.M. (1928) Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, A117, 610-624. [2] Lee, T.D. (1981) Particle Physics and Introduction to Field Theory. Harwood Academic Publishers, New York. [3] Kaku, M. (1993) Quantum Field Theory, A Modern Introduction. Oxford University Press, New York. [4] Weyl, H. (1929) Zeitschrift für Physik, 56, 330-352.
http://dx.doi.org/10.1007/BF01339504 [5] Hestenes, D. (1967) Journal of Mathematical Physics, 8, 798-808.
http://dx.doi.org/10.1063/1.1705279 [6] Ljolje, K. (2004) The Dirac Field in Real Domain. 1-15, arXiv quant-ph/0401068. [7] Majorana, E. (1937) Il Nuovo Cimento, 14, 171-184.
http://dx.doi.org/10.1007/BF02961314 [8] Case, K.M. (1957) Physical Review, 107, 307-316.
http://dx.doi.org/10.1103/PhysRev.107.307 [9] Mannheim, P.D. (1984) International Journal of Theoretical Physics, 23, 643-674.
http://dx.doi.org/10.1007/BF02214135 [10] Aste, A.A. (2010) Symmetry, 2, 1776-1809.
http://dx.doi.org/10.3390/sym2041776 [11] Marsch, E. (2012) International Scholarly Research Notices: Mathematical Physics, 2012, 1-17.
http://dx.doi.org/10.5402/2012/760239 [12] Marsch, E. (2012) Symmetry, 5, 271-286. [13] Pauli, W. (1927) Zeitschrift für Physik, 43, 601-623.
http://dx.doi.org/10.1007/BF01397326 [14] Klein, O. (1927) Zeitschrift für Physik, 41, 407-422.
http://dx.doi.org/10.1007/BF01400205 [15] Pauli, W. and Weisskopf, V. (1934) Helvetica Physica Acta, 7, 709-731. [16] Yang, C.N. and Mills, F. (1954) Physical Review, 96, 191-195.
http://dx.doi.org/10.1103/PhysRev.96.191 [17] Pal, P.B. (2011) American Journal of Physics, 79, 485-498.
http://dx.doi.org/10.1119/1.3549729
[1] Dirac, P.A.M. (1928) Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, A117, 610-624. [2] Lee, T.D. (1981) Particle Physics and Introduction to Field Theory. Harwood Academic Publishers, New York. [3] Kaku, M. (1993) Quantum Field Theory, A Modern Introduction. Oxford University Press, New York. [4] Weyl, H. (1929) Zeitschrift für Physik, 56, 330-352.
http://dx.doi.org/10.1007/BF01339504 [5] Hestenes, D. (1967) Journal of Mathematical Physics, 8, 798-808.
http://dx.doi.org/10.1063/1.1705279 [6] Ljolje, K. (2004) The Dirac Field in Real Domain. 1-15, arXiv quant-ph/0401068. [7] Majorana, E. (1937) Il Nuovo Cimento, 14, 171-184.
http://dx.doi.org/10.1007/BF02961314 [8] Case, K.M. (1957) Physical Review, 107, 307-316.
http://dx.doi.org/10.1103/PhysRev.107.307 [9] Mannheim, P.D. (1984) International Journal of Theoretical Physics, 23, 643-674.
http://dx.doi.org/10.1007/BF02214135 [10] Aste, A.A. (2010) Symmetry, 2, 1776-1809.
http://dx.doi.org/10.3390/sym2041776 [11] Marsch, E. (2012) International Scholarly Research Notices: Mathematical Physics, 2012, 1-17.
http://dx.doi.org/10.5402/2012/760239 [12] Marsch, E. (2012) Symmetry, 5, 271-286. [13] Pauli, W. (1927) Zeitschrift für Physik, 43, 601-623.
http://dx.doi.org/10.1007/BF01397326 [14] Klein, O. (1927) Zeitschrift für Physik, 41, 407-422.
http://dx.doi.org/10.1007/BF01400205 [15] Pauli, W. and Weisskopf, V. (1934) Helvetica Physica Acta, 7, 709-731. [16] Yang, C.N. and Mills, F. (1954) Physical Review, 96, 191-195.
http://dx.doi.org/10.1103/PhysRev.96.191 [17] Pal, P.B. (2011) American Journal of Physics, 79, 485-498.
http://dx.doi.org/10.1119/1.3549729