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On the Quaternion Representation for Octonion Generalization of Lorentz Boosts

Date: May 23,2022 |Hits: 218 Download PDF How to cite this paper

M. Kharinov

St. Petersburg Federal Research Center of the Russian Academy of Sciences (SPC RAS), 39, 14th Line V.O., St. Petersburg, 199178, Russia.

*Corresponding author: M. Kharinov


The paper proposes an approach to the generalization of Lorentz transformations for the real Euclidean spacetime of double dimensions. The approach is based on symmetry considerations. It provides: a) Generalized additive decomposition of a linear operator into self-adjoint (symmetric) and skewsymmetric parts; b) development of the apparatus of non-associative octonions due to the double generalization of the cross vector product for three arguments and eight-dimensional space; c) quaternion representation of Lorentz transformations as a linear combination of spatial rotation and one more orthogonal transformation; d) analytical solution of the eigenvector/eigenvalue problem for the composition of Lorentz boosts, in order to extend the quaternion record of Lorentz boost composition to the octonionic case. In general, our studies are consistent with those of Tevian Dray and Сorinne A. Manogue, but are limited to using only ordinary quaternions and octonions.


[1] Hamilton, W. R. (1853). Lectures on quaternions. Dublin: Hodges and Smith, p. 890.

[2] Kantor, I. L., Solodovnikov, A. S. (1989). Hypercomplex numbers: an elementary introduction to algebras. Springer, p. 169. 

[3] Berry, T. and Visser, M. (2020). Relativistic combination of non-collinear 3-velocities using quaternions. Universe, 6(12), 237. 

[4] Berry, T. and Visser, M. (2021). Lorentz boosts and Wigner rotations: self-adjoint complexified quaternions. Physics, 3(2), 352-366. 

[5] Sweetser, D. B. (2010). Lorentz Boosts Using Quaternions. https://www.youtube.com/watch?v=DrVm1JTM8X4, from 3:24 till 3:32. 

[6] Madelung, E. (1960). Mathematical Аpparatus of Physics. State publishing house of physical and mathematical literature, Moscow, p. 618.

[7] Manogue, C. A. and Schray, J. (1993). Finite Lorentz transformations, automorphisms, and division algebras. Jour-nal of Mathematical Physics, 34(8), 3746-3767.

[8] Korn, G. A. and Korn, T. M. (1973) Mathematical handbook for engineers and scientists. Moscow, p. 832. 

[9] Dray, T., Manogue, C. A., and Okubo, S. (2001). Orthonormal eigenbases over the octonions. arXiv preprint math/0106021.

[10] Kharinov, M. V. (2018). Product of Three Octonions. Adv. Appl. Clifford Algebras, Springer Nature, 29(1), p. 16. 

[11] Silagadze, Z. K. (2002). Multi-dimensional vector product. Journal of Physics A: Mathematical and General, Insti-tute of Physics Publishing, UK, 35 (23), 4949-4953. 

[12] Okubo, S. (1993). Triple products and Yang–Baxter equation. I. Octonionic and quaternionic triple systems. Journal of mathematical physics, 34(7), 3273-3291.

[13] Dray, T. and Manogue C. A. (1998). The octonionic eigenvalue problem. Adv. Appl. Clifford Algebra, 8(2), 341-364.

[14] Salamon, D. A. and Walpuski, T. (2010). Notes on the octonions. ArXiv preprint, arXiv: 1005.2820, p. 95.

[15] Kharinov, M. V. (2020). The Quartet of Eigenvectors for Quaternionic Lorentz Transformation. Adv. Appl. Clifford Algebras, Springer Nature, 30(25), p. 20.

[16] Casanova, G. (1976). L’algebre vectorielle. Presses Universitaires de France-PUF, p. 118.

[17] Møller, C. (1955). The Theory of Relativity. Oxford University Press, 1955.

[18] Ungar, A. A. (2013). Hyperbolic geometry. Fifteenth International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, 259-282, doi: 10.7546/giq-15-2014-259-282.

[19] Barrett, J. F. (2015). Minkovski Space-Time and Hyperbolic Geometry, MASSEE International Congress on Mathematics MICOM-2015, https://www.researchgate.net/publication/287988654_Minkowski_space-time_and_ hyperbolic_geometry_Original_2015_version.

How to cite this paper

On the Quaternion Representation for Octonion Generalization of Lorentz Boosts

How to cite this paper: M. Kharinov. (2022) On the Quaternion Representation for Octonion Generalization of Lorentz Boosts. Journal of Applied Mathematics and Computation6(2), 198-205.

DOI: http://dx.doi.org/10.26855/jamc.2022.06.004

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