Affiliation:
1. Belarusian State Pedagogical University
2. Octonion technology Ltd.
Abstract
In this paper, the q-generalization of the Higgs algebra is considered. The realization of this algebra is shown in an explicit form using a nonlinear transformation of the creation-annihilation operators of the q-harmonic oscillator. This transformation is the performance of two operations, namely, a “correction” using a function of the original Hamiltonian, and raising to the fourth power the creation and annihilation operators of a q-harmonic oscillator. The choice of the “correcting” function is justified by the standard form of commutation relations for the operators of the metaplectic realization Uq(SU(1,1)). Further possible directions of research are briefly discussed to summarize the results obtained. The first direction is quite obvious. It is the consideration of the problem when the dimension of the operator space increases or for any value N. The second direction can be associated with the analysis of the relationship between q-generalizations of the Higgs and Hahn algebras.
Publisher
Publishing House Belorusskaya Nauka
Subject
Computational Theory and Mathematics,General Physics and Astronomy,General Mathematics
Reference17 articles.
1. Higgs P.W. Dynamical symmetries in a spherical geometry. I. Journal of Physics A, 1979, vol. 12, no. 4, pp. 309–323. https://doi.org/10.1088/0305-4470/12/3/006
2. Kurochkin Yu. A., Otchik V. S. Analog of the Runge – Lenz vector and energy spectrum in the Kepler problem on a three-dimensional sphere. Doklady academii nauk BSSR [Doklady of the Academy of Sciences of BSSR], 1979, vol. 23, no. 11, pp. 987–990 (in Russian).
3. Bogush A. A., Kurochkin Yu. A., Otchik V. S. The quantum-mechanical Kepler problem in three-dimensional Lobačevskiĭ space. Doklady academii nauk BSSR [Doklady of the Academy of Sciences of BSSR], 1980, vol. 24, no. 1, pp. 19–22 (in Russian).
4. Chung W. S. Holstein-Primakoff realization of Higgs algebra and its q-extension. Modern Physics Letters A, 2014, vol. 29, no. 10, pp. 1450050–1450062. https://doi.org/10.1142/S0217732314500503
5. Frappat L., Gaboriaud J., Vinet L., Vinet S., Zhedanov A. S. The Higgs and Hahn algebras from a Howe duality perspective. Physics Letters A, 2019, vol. 383, no. 14, pp. 1531–1535. https://doi.org/10.1016/j.physleta.2019.02.024