Research Article | Open Access
Vladimir Dzhunushaliev, "Toy Models of a Nonassociative Quantum Mechanics", Advances in High Energy Physics, vol. 2007, Article ID 012387, 10 pages, 2007. https://doi.org/10.1155/2007/12387
Toy Models of a Nonassociative Quantum Mechanics
Toy models of a nonassociative quantum mechanics are presented. The Heisenberg equation of motion is modified using a nonassociative commutator. Possible physical applications of a nonassociative quantum mechanics are considered. The idea is discussed that a nonassociative algebra could be the operator language for the nonperturbative quantum theory. In such approach the nonperturbative quantum theory has observables and unobservables quantities.
- P. Jordan, J. von Neumann, and E. Wigner, “On an algebraic generalization of the quantum mechanical formalism,” Annals of Mathematics, vol. 35, no. 1, pp. 29–64, 1934.
- S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Montroll Memorial Lecture Series in Mathematical Physics, Cambridge University Press, Cambridge, UK, 1995.
- J. C. Baez, “The octonions,” Bulletin of the American Mathematical Society, vol. 39, no. 2, pp. 145–205, 2002.
- B. Grossman, “A 3-cocycle in quantum mechanics,” Physics Letters B, vol. 152, no. 1-2, pp. 93–97, 1985.
- R. Jackiw, “Three-cocycle in mathematics and physics,” Physical Review Letters, vol. 54, no. 3, pp. 159–162, 1985.
- M. Gogberashvili, “Octonionic version of Dirac equations,” International Journal of Modern Physics A, vol. 21, no. 17, pp. 3513–3523, 2006.
- M. Gogberashvili, “Octonionic electrodynamics,” Journal of Physics A, vol. 39, no. 22, pp. 7099–7104, 2006.
- V. Dzhunushaliev, “Observables and unobservables in a non-associative quantum theory,” preprint, 2007, http://arxiv.org/abs/quant-ph/0702263.
- Y. Nambu, “Generalized hamiltonian dynamics,” Physical Review D, vol. 7, no. 8, pp. 2405–2412, 1973.
- S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, Oxford, UK, 2001.
- B. Thaller, Advanced Visual Quantum Mechanics, Springer Science & Business Media, New York, NY, USA, 2005.
- S. L. Glashow, “Partial-symmetries of weak interactions,” Nuclear Physics, vol. 22, no. 4, pp. 579–588, 1961.
- S. Weinberg, “A model of leptons,” Physical Review Letters, vol. 19, no. 21, pp. 1264–1266, 1967.
- A. Salam, “Relativistic groups and analiticity,” in Proceedings of the 8th Nobel Symposium on Elementary Particle Theory, N. Svartholm, Ed., p. 367, Almquist & Wiksell, Stockholm, Sweden, May 1968.
- K. Carmody, “Circular and hyperbolic quaternions, octonions, and sedenions,” Applied Mathematics and Computation, vol. 28, no. 1, pp. 47–72, 1988.
- K. Carmody, “Circular and hyperbolic quaternions, octonions, and sedenions—further results,” Applied Mathematics and Computation, vol. 84, no. 1, pp. 27–47, 1997.
Copyright © 2007 Vladimir Dzhunushaliev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.