Abstract
The relativistic first-order wave equations for massive particles with spin 0,1,1/2 are formulated in terms of a factorization of the Klein–Fock equation by means of the algebra of octonions. An analogous method applied to Hamiltonian of the quantum isotropic oscillator leads to the natural generalization of the model. The class of supersymmetric oscillators with dimension N≤7 associated with te algebras of the Cayley–Dickson series is introduced.
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References
Penney, R.: Octonions and Dirac equation, Amer. J. Phys. 36 (1968), 871-873.
Bogush, A. A. and Kurochkin, Yu. A.: A Formulation of the Relativistic Wave Equations for Particles with Spin 0, 1/2 and 1 in Terms of Octonion Algebra, Proc. Internat. Sem., Zvenigorod, 1979; vol. 2, Nauka, Moscow, 1980 (in Russian).
Kalashnikov, S. K., Konstein, S. E., and Fradkin, E. S.: Octonions and exceptional groups in field theory, Soviet J. Nuclear Phys. 729 (1979), 1660-1669 (in Russian).
Kurdgelaidze, D. F.: The foundations of non-associative classical field theory, Acta Phys. Hungar. 75 (1985), 79-98.
Kurdgelaidze, D. F.: Introduction in the Non-Associative Classical Field Theory, Tbilisi, 1987, (in Russian).
Sorgsepp, L. and Lôhmus, J.: Dirac equation in the regular bimodule representation of octonions, Trans. Inst. Phys. Estonian Acad. Sci. 62 (1987), 159-173.
Kantor, I. L. and Solodovnikov, A. S.: Hypercomplex Numbers, Nauka, Moscow, 1973 (in Russian).
Berezin, A. V., Kurochkin, Yu. A., and Tolkachev, E. A.: Quaternions in Relativistic Physics, Nauka i Technika, Minsk, 1989 (in Russian).
Gürsey, F.: Relation of charge independence and baryon conservation to Pauli's transformation, Nuovo Cimento 7 (1958), 411-415.
Ermolaev, E. A.: Quaternion factorization of the Klein-Fock operator, Vesti Acad. Sci. BSSR. Ser. Phys.-Math. 1 (1979), 75-78 (in Russian).
Fedorov, F. I.: On reduction of wave equations for spin 0 and 1 to the Hamiltonian form, Soviet J. JETP 31 (1956), 140-141 (in Russian).
Arnold, V. I.: Mathematical Methods in Classical Mechanics, Nauka, Moscow, 1974 (in Russian).
Valuev, B. N.: Application of the Clifford algebra to the Ising-Onsager problem, Tech. Rep. P17-11020 (1977), Joint Inst. for Nuclear Res., Dubna (in Russian).
Gendenstein, L. E. and Krive, I. V.: Supersymmetry in quantum mechanics, Uspekhi Fiz. Nauk 146 (1985), 553-590.
Lôhmus, J. and Sorgsepp, L.: Ternary algebra of sedenions, Trans. Inst. Phys. Estonian Acad. Sci. 66 (1990), 170-179.
Tomil'chik, L. M.: On the mass spectrum of nucleon izobars, in Proc. VIIIth Soviet Conf. on Elementary Particles, Uzhgorod, 1971, Kiev, 1971, pp. 31-33.
Tomil'chik, L. M.: Supersymmetric two-dimensional Dirac oscillator, Rapid Comm. Theoret. Phys. 660(3), Inst. Phys. Acad. Sci. Belarus, Minsk, 1992.
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Bogush, A.A., Kurochkin, Y.A. Cayley–Dickson Procedure, Relativistic Wave Equations and Supersymmetric Oscillators. Acta Applicandae Mathematicae 50, 121–129 (1998). https://doi.org/10.1023/A:1005875403156
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DOI: https://doi.org/10.1023/A:1005875403156