A non-associative quantum mechanics. (English) Zbl 1099.81004

Summary: A nonassociative quantum mechanics is proposed in which the product of three and more operators can be nonassociative. 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 nonassociative quantum mechanics are considered. It is proposed that some generalization of the nonassociative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interaction.


81P05 General and philosophical questions in quantum theory
81R15 Operator algebra methods applied to problems in quantum theory
47N50 Applications of operator theory in the physical sciences
Full Text: DOI arXiv


[1] 1. J. C. Baez, ”The Octonions,” math.ra/0105155. · Zbl 1026.17001
[2] 2. P. Jordan, J. von Neumann, and E. Wigner, ”On an algebraic generalization of the quantum mechanical formalism,” Ann. Math. 35, 29–64 (1934). · JFM 60.0902.02
[3] 3. T. Kugo and P.-K. Townsend, ”Supersymmetry and the division algebras,” Nucl. Phys. B 221, 357–380 (1983).
[4] 4. M. Gogberashvili, ”Octonionic geometry,” hep-th/0409173. · Zbl 1110.51300
[5] 5. V. Dzhunushaliev, ”Non-perturbative operator quantization of strongly interacting fields,” Found. Phys. Lett. 16, 57 (2003).
[6] 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] 7. A. I. Nesterov and L. V. Sabinin, ”Nonassociative geometry: Towards discrete structure of spacetime,” Phys. Rev. D 62, 081501 (2000). · Zbl 1036.53010
[8] 8. B. Grossman, ”A three cocycle in quantum mechanics,” Phys. Lett. B 152, 93, (1985). · Zbl 1177.81097
[9] 9. R. Jackiw, ”3 - cocycle in mathematics and physics,” Phys. Rev. Lett. 54, 159, (1985).
[10] 10. A. I. Nesterov, ”Three-cocycles, nonassociative gauge transformations and Dirac’s monopole,” Phys. Lett. A 328, 110 (2004). · Zbl 1134.81393
[11] 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). · Zbl 0338.17004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.