Hindawi Publishing Corporation Advances in High Energy Physics Volume 2007, Article ID 12387, 10 pages doi:10.1155/2007/12387 Research Article Toy Models of a Nonassociative Quantum Mechanics Vladimir Dzhunushaliev Department of Physics and Microelectronics Engineering, Kyrgyz-Russian Slavic University, 44 Kievskaya Street, Bishkek 720021, Kyrgyzstan Correspondence should be addressed to Vladimir Dzhunushaliev, dzhun@krsu.edu.kg Received 19 July 2007; Revised 8 September 2007; Accepted 18 September 2007 Recommended by George Siopsis 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 nonas- sociative 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 non- perturbative quantum theory has observables and unobservables quantities. Copyright q 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. 1. Introduction In 1, the attempt was made to obtain a possible generalization of quantum mechanics on any numbers including nonassociative numbers: octonions. In 2, the author applies nonas- sociative algebras to physics. This book covers topics ranging from algebras of observables in quantum mechanics, to angular momentum and octonions, division algebras, triple-linear products, and Yang-Baxter equations. The nonassociative gauge theoretic reformulation of Ein- stein’s general relativity theory is also discussed. In 3, one can find the review of mathemat- ical definitions and physical applications for the octonions. The modern applications of the nonassociativity in physics are as follows: in 4, 5 it is shown that the requirement that finite translations be associative leads to Dirac’s monopole quantization condition; in 6, 7 Dirac’s operator and Maxwell’s equations are derived in the algebra of split-octonions. In this paper, we attempt to give toy models of a nonassociative quantum mechanics us- ing finite dimensional nonassociative algebras—octonions or sedenions. In the previous paper 8, we have shown that in a nonassociative quantum theory the observables can be presented only by elements of an associative subalgebra of a nonassociative algebra of nonperturbative