Algebras of Hamieh and Abbas Used in the Dirac Equation

Abstract

Hamieh and Abbas [1] propose using a 3-dimensional real algebra in a solution of the Dirac equation. We show that this algebra, denoted by , belongs to a large class of quadratic Jordan algebras with subalgebras isomorphic to the complex numbers and that the spinor matrices associated with the solution of the Dirac equation generate a six-dimensional real noncommutative Jordan algebra.

Share and Cite:

G. Wene, "Algebras of Hamieh and Abbas Used in the Dirac Equation," Journal of Modern Physics, Vol. 3 No. 9, 2012, pp. 923-926. doi: 10.4236/jmp.2012.39120.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] S. Hamieh and H. Abbas, “Two Dimensional Representation of the Dirac Equation in Non-Associative Algebra,” Journal of Modern Physics, Vol. 3, No. 2, 2012, pp. 184-186. doi:10.4236/jmp.2012.32025
[2] P. Jordan, J. von Neumann and E. Wigner, “On an Algebraic Generalization of the Quantum Mechanical Formalism,” Annals of Mathematics, Vol. 35, No. 1, 1934, pp. 29-64. doi:10.2307/1968117
[3] A. A. Albert, “On Jordan Algebras of Linear Transformations,” Transactions of the American Mathematical Soci- ety, Vol. 59, No. 3, 1946, pp. 524-555. doi:10.1090/S0002-9947-1946-0016759-3
[4] J. L?hmas, E. Paal and L. Sorgsepp, “Nonassociative Al- gebras in Physics,” Hadronic Press, Palm Harbor, 1994.
[5] S. Okubo, “Introduction to Octonion and Other Non-Associative Algebras in Physics,” Cambridge University Press, New York, 1995. doi:10.1017/CBO9780511524479
[6] M. L. Tomber, “A Short History of Nonassociative Algebras,” Hadronic Journal, Vol. 1, 1979, pp. 1252-1387.
[7] R. D. Schafer, “An Introduction to Nonassociative Algebras,” Academic Press, New York, 1966.
[8] N. Jacobson, “Structure and Representations of Jordan Algebras,” American Mathematical Society, Providence, R.I., 1968.
[9] S. Okubo, “Pseudo-Quaternion and Pseudo-Octonion Algebras,” Hadronic Journal, Vol. 1, 1978, pp. 1250-1278.
[10] K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov and A. I. Shirshov, “Rings That Are Nearly Associative,” Academic Press, New York, 1982.
[11] H. Braun and M. Koecher, “Jordan-Algeben,” Springer-Verlag, Berlin, 1966. doi:10.1007/978-3-642-94947-0
[12] G. Domokos and S. K?vesi-Domokos, “The Algebra of Color,” Journal of Mathematical Physics, Vol. 19, No. 6, 1978, pp. 1477-1481. doi:10.1063/1.523815
[13] G. P. Wene, “An Example of a Flexible, Jordan-Admissible Algebra of Current Use in Physics,” Hadronic Journal, Vol. 1, 1978, pp. 944-954.
[14] G. P. Wene, “A Little Color in Abstract Algebra,” The American Mathematical Monthly, Vol. 89, 1982, pp. 417-419. doi:10.2307/2321659
[15] R. D. Schafer, “A Generalization of the Algebra of Color,” Journal of Algebra, Vol. 160, No. 1, 1993, pp. 93-129. doi:10.1006/jabr.1993.1180
[16] E. M. Corson, “Introduction to Tensors, Spinors, and Relativistic Wave-Equations,” Chelsea Publishing Company, New York, 1982.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.