Abstract
A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with quantum corrections. The self-consistency of the operator algebra is proved for the product of three operators. Some properties of the non-associative quantum mechanics are considered. It is proposed that some generalization of the non-associative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interaction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
1. J. C. Baez, “The Octonions,” math.ra/0105155.
2. P. Jordan, J. von Neumann, and E. Wigner, “On an algebraic generalization of the quantum mechanical formalism,” Ann. Math. 35, 29–64 (1934).
3. T. Kugo and P.-K. Townsend, “Supersymmetry and the division algebras,” Nucl. Phys. B 221, 357–380 (1983).
4. M. Gogberashvili, “Octonionic geometry,” hep-th/0409173.
5. V. Dzhunushaliev, “Non-perturbative operator quantization of strongly interacting fields,” Found. Phys. Lett. 16, 57 (2003).
6. B. A. Bernevig, J. p. Hu, N. Toumbas and S. C. Zhang, “The eight-dimensional quantum Hall effect and the octonions,” Phys. Rev. Lett. 91, 236803 (2003).
7. A. I. Nesterov and L. V. Sabinin, “Nonassociative geometry: Towards discrete structure of spacetime,” Phys. Rev. D 62, 081501 (2000).
8. B. Grossman, “A three cocycle in quantum mechanics,” Phys. Lett. B 152, 93, (1985).
9. R. Jackiw, “3 - cocycle in mathematics and physics,” Phys. Rev. Lett. 54, 159, (1985).
10. A. I. Nesterov, “Three-cocycles, nonassociative gauge transformations and Dirac's monopole,” Phys. Lett. A 328, 110 (2004).
11. M. Günaydin and F. Gürsey “Quark structure and octonions,” J. Math. Phys. 14, 1651 (1973); “ Quark statistics and octonions,” Phys. Rev. D 9, 3387 (1974).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dzhunushaliev, V. A Non-Associative Quantum Mechanics. Found Phys Lett 19, 157–167 (2006). https://doi.org/10.1007/s10702-006-0373-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10702-006-0373-2