The Two-Component Majorana Equation-Novel Derivations and Known Symmetries

Author(s)
Eckart Marsch

ABSTRACT

We revisit the two-component Majorana equation and derive it in a new form by linearizing the relativistic dispersion relation of a massive particle, in a way similar to that used to derive the Dirac equation. We are using thereby the Pauli spin matrices, corresponding to an irreducible representation of the Lorentz group, and a lucid and transparent algebraic approach exploiting the newly introduced spin-flip operator. Thus we can readily build up the Majorana version of the Dirac equation in its chiral representation. The Lorentz-invariant complex conjugation operation involves the spin-flip operator, and its connection to chiral symmetry is discussed. The eigenfunctions of the Majorana equation are calculated in a concise way.

We revisit the two-component Majorana equation and derive it in a new form by linearizing the relativistic dispersion relation of a massive particle, in a way similar to that used to derive the Dirac equation. We are using thereby the Pauli spin matrices, corresponding to an irreducible representation of the Lorentz group, and a lucid and transparent algebraic approach exploiting the newly introduced spin-flip operator. Thus we can readily build up the Majorana version of the Dirac equation in its chiral representation. The Lorentz-invariant complex conjugation operation involves the spin-flip operator, and its connection to chiral symmetry is discussed. The eigenfunctions of the Majorana equation are calculated in a concise way.

Cite this paper

nullE. Marsch, "The Two-Component Majorana Equation-Novel Derivations and Known Symmetries,"*Journal of Modern Physics*, Vol. 2 No. 10, 2011, pp. 1109-1114. doi: 10.4236/jmp.2011.210137.

nullE. Marsch, "The Two-Component Majorana Equation-Novel Derivations and Known Symmetries,"

References

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[1] E. Majorana, “Teoria Simmetrica Dell’ Elettrone E Del Positrone,” Il Nuovo Cimento (1924-1942), Vol. 14, No. 4, 1937, pp. 171-184. doi:10.1007/BF02961314

[2] R. N. Mohapatra and P. B. Pal, “Massive Neutrinos in Physics and Astrophysics,” World Scientific, Singapore, 2004. doi:10.1142/9789812562203

[3] M. Fukugita and T. Yanagida, “Physics of Neutrinos and Applications to Astrophysics,” Springer, Berlin, 2003.

[4] M. Kaku, “Quantum Field Theory, A Modern Introduction,” Oxford University Press, New York, 1993.

[5] H. Weyl, “Elektron und Gravitation I,” Zeitschrift für Physik A Hadrons and Nuclei, Vol. 56, No. 5-6, 1929, pp. 330-352. doi:10.1007/BF01339504

[6] W. Pauli, “Zur Quantenmechanik des Magnetischen Elektrons,” Zeitschrift für Physik A Hadrons and Nuclei, Vol. 43, No. 9-10, 1927, pp. 601-623. doi:10.1007/BF01397326

[7] P. B. Pal, “Dirac, Majorana and Weyl Fermions,” arXiv:1006.1718v2 [hep-ph], 2010.

[8] P. M. A. Dirac, “The Quantum Theory of the Electron,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 117, No. 778, 1928, pp. 610-624.

[9] A. Das, “Lectures on Quantum Field Theory,” World Scientific, Singapore, 2008. doi:10.1142/9789812832870