×

zbMATH — the first resource for mathematics

Rotations in the space of Split octonions. (English) Zbl 1201.81066
Summary: The geometrical application of split octonions is considered. The new representation of products of the basis units of split octonionic having David’s star shape (instead of the Fano triangle) is presented. It is shown that active and passive transformations of coordinates in octonionic “eight-space” are not equivalent. The group of passive transformations that leave invariant the pseudonorm of split octonions is SO\((4,4)\), while active rotations are done by the direct product of \(O(3,4)\)-boosts and real noncompact form of the exceptional group \(G_{2}\). In classical limit, these transformations reduce to the standard Lorentz group.
MSC:
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
83A05 Special relativity
11R52 Quaternion and other division algebras: arithmetic, zeta functions
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
17B25 Exceptional (super)algebras
17A35 Nonassociative division algebras
PDF BibTeX XML Cite
Full Text: DOI EuDML arXiv
References:
[1] R. Schafer, Introduction to Non-Associative Algebras, Dover, New York, NY, USA, 1995. · Zbl 0875.62584
[2] T. A. Springer and F. D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics, Springer, Berlin, Germany, 2000. · Zbl 1087.17001
[3] J. C. Baez, “The octonions,” Bulletin of the American Mathematical Society, vol. 39, no. 2, pp. 145-205, 2002. · Zbl 1026.17001
[4] D. Finkelstein, Quantum Relativity: A Synthesis of the Ideas of Einstein and Heisenberg, Springer, Berlin, Germany, 1996. · Zbl 0869.00010
[5] G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, John Wiley & Sons, New York, NY, USA, 1972. · Zbl 0235.46085
[6] F. Gürsey and C. Tze, On the Role of Division, Jordan and Related Algebras in Particle Physics, World Scientific, Singapore, 1996. · Zbl 0956.81502
[7] J. Lõhmus, E. Paal, and L. Sorgsepp, Nonassociative Algebras in Physics, Hadronic Press, Palm Harbor, Fla, USA, 1994. · Zbl 0840.17001
[8] M. Günaydin and F. Gürsey, “Quark statistics and octonions,” Journal of Mathematical Physics, vol. 14, no. 11, pp. 1651-1667, 1973. · Zbl 0338.17004
[9] M. Günaydin and F. Gürsey, “Quark statistics and octonions,” Physical Review D, vol. 9, no. 12, pp. 3387-3391, 1974.
[10] S. L. Adler, “Quaternionic chromodynamics as a theory of composite quarks and leptons,” Physical Review D, vol. 21, no. 10, pp. 2903-2915, 1980.
[11] K. Morita, “Octonions, quarks and QCD,” Progress of Theoretical Physics, vol. 65, no. 2, pp. 787-790, 1981.
[12] T. Kugo and P. Townsend, “Supersymmetry and the division algebras,” Nuclear Physics B, vol. 221, no. 2, pp. 357-380, 1983.
[13] A. Sudbery, “Division algebras, (pseudo)orthogonal groups and spinors,” Journal of Physics A, vol. 17, no. 5, pp. 939-955, 1984. · Zbl 0544.22010
[14] G. Dixon, “Derivation of the standard model,” Il Nuovo Cimento B, vol. 105, no. 3, pp. 349-364, 1990.
[15] S. De Leo, “Quaternions for GUTs,” International Journal of Theoretical Physics, vol. 35, no. 9, pp. 1821-1837, 1996. · Zbl 0873.15009
[16] I. R. Porteous, Clifford Algebras and the Classical Groups, Cambridge University Press, Cambridge, UK, 1995. · Zbl 0855.15019
[17] S. Okubo, Introduction to Octonions and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge, UK, 1995. · Zbl 0841.17001
[18] P. Lounesto, Clifford Algebras and Spinors, Cambridge University Press, Cambridge, UK, 2001. · Zbl 0973.15022
[19] H. L. Carrion, M. Rojas, and F. Toppan, “Quaternionic and octonionic spinors. A classification,” Journal of High Energy Physics, vol. 7, no. 4, pp. 901-928, 2003. · Zbl 1076.81522
[20] D. Finkelstein, J. M. Jauch, S. Schiminovich, and D. Speiser, “Foundations of quaternion quantum mechanics,” Journal of Mathematical Physics, vol. 3, no. 2, pp. 207-220, 1962.
[21] L. P. Horwitz and L. C. Biedenharn, “Quaternion quantum mechanics: second quantization and gauge fields,” Annals of Physics, vol. 157, no. 2, pp. 432-488, 1984. · Zbl 0558.46039
[22] M. Günaydin, C. Piron, and H. Ruegg, “Moufang plane and octonionic quantum mechanics,” Communications in Mathematical Physics, vol. 61, no. 1, pp. 69-85, 1978. · Zbl 0409.22018
[23] S. De Leo and P. Rotelli, “Translations between quaternion and complex quantum mechanics,” Progress of Theoretical Physics, vol. 92, no. 5, pp. 917-926, 1994.
[24] V. Dzhunushaliev, “A non-associative quantum mechanics,” Foundations of Physics Letters, vol. 19, no. 2, pp. 157-167, 2006. · Zbl 1099.81004
[25] F. Gürsey, “Symmetries in physics (1600-1980),” in Proceedings of the 1st International Meeting on the History of Scientific Ideas, Seminario d’Història de las Cièncias, Barcelona, Spain, 1987.
[26] S. De Leo, “Quaternions and special relativity,” Journal of Mathematical Physics, vol. 37, no. 6, pp. 2955-2968, 1996. · Zbl 0870.53061
[27] K. Morita, “Quaternionic Weinberg-Salam theory,” Progress of Theoretical Physics, vol. 67, no. 6, pp. 1860-1876, 1982.
[28] C. Nash and G. C. Joshi, “Spontaneous symmetry breaking and the Higgs mechanism for quaternion fields,” Journal of Mathematical Physics, vol. 28, no. 2, pp. 463-467, 1987.
[29] S. L. Adler, Quaternion Quantum Mechanics and Quantum Field, Oxford University Press, New York, NY, USA, 1995. · Zbl 0885.00019
[30] D. F. Kurdgelaidze, “Fundamentals of nonassociative classical field theory,” Soviet Physics Journal, vol. 29, no. 11, pp. 883-887, 1986.
[31] A. J. Davies and G. C. Joshi, “A bimodular representation of ten-dimensional fermions,” Journal of Mathematical Physics, vol. 27, no. 12, pp. 3036-3039, 1986. · Zbl 0614.17001
[32] S. De Leo and P. Rotelli, “The quaternionic Dirac Lagrangian,” Modern Physics Letters A, vol. 11, no. 5, pp. 357-366, 1996. · Zbl 1022.81518
[33] B. A. Bernevig, J. Hu, N. Toumbas, and S.-C. Zhang, “Eight-dimensional quantum hall effect and “octonions”,” Physical Review Letters, vol. 91, no. 23, Article ID 236803, 2003.
[34] F. D. Smith Jr., “Spin(8) gauge field theory,” International Journal of Theoretical Physics, vol. 25, no. 4, pp. 355-403, 1986. · Zbl 0603.58047
[35] M. Pav\vsi\vc, “A novel view on the physical origin of E8,” Journal of Physics A, vol. 41, no. 33, Article ID 332001, 2008. · Zbl 1144.81026
[36] C. Castro, “On the noncommutative and nonassociative geometry of octonionic space time, modified dispersion relations and grand unification,” Journal of Mathematical Physics, vol. 48, no. 7, Article ID 073517, 2007. · Zbl 1144.81321
[37] D. B. Fairlie and C. A. Manogue, “Lorentz invariance and the composite string,” Physical Review D, vol. 34, no. 6, pp. 1832-1834, 1986.
[38] K.-W. Chung and A. Sudbery, “Octonions and the Lorentz and conformal groups of ten-dimensional space-time,” Physics Letters B, vol. 198, no. 2, pp. 161-164, 1987.
[39] J. Lukierski and F. Toppan, “Generalized space-time supersymmetries, division algebras and octonionic M-theory,” Physics Letters B, vol. 539, no. 3-4, pp. 266-276, 2002. · Zbl 0996.17002
[40] L. Boya, GROUP 24: Physical and Mathematical Aspects of Symmetries, CRC Press, Paris, France, 2002.
[41] M. Gogberashvili, “Octonionic electrodynamics,” Journal of Physics A, vol. 39, no. 22, pp. 7099-7104, 2006. · Zbl 1122.78001
[42] M. Gogberashvili, “Octonionic version of Dirac equations,” International Journal of Modern Physics A, vol. 21, no. 17, pp. 3513-3523, 2006. · Zbl 1095.81022
[43] M. Gogberashvili, “Octonionic geometry,” Advances in Applied Clifford Algebras, vol. 15, no. 1, pp. 55-66, 2005. · Zbl 1110.51300
[44] G. Amelino-Camelia, “Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale,” International Journal of Modern Physics D, vol. 11, no. 1, pp. 35-59, 2002. · Zbl 1062.83500
[45] J. Magueijo and L. Smolin, “Lorentz invariance with an invariant energy scale,” Physical Review Letters, vol. 88, no. 19, Article ID 190403, 2002.
[46] C. A. Manogue and J. Schray, “Finite Lorentz transformations, automorphisms, and division algebras,” Journal of Mathematical Physics, vol. 34, no. 8, pp. 3746-3767, 1993. · Zbl 0797.53075
[47] J. Beckers, V. Hussin, and P. Winternitz, “Nonlinear equations with superposition formulas and the exceptional group G2. I. Complex and real forms of g2 and their maximal subalgebras,” Journal of Mathematical Physics, vol. 27, no. 9, pp. 2217-2227, 1986. · Zbl 0614.22008
[48] J. Beckers, V. Hussin, and P. Winternitz, “Nonlinear equations with superposition formulas and the exceptional group G2. II. Classification of the equations,” Journal of Mathematical Physics, vol. 28, no. 3, pp. 520-529, 1987. · Zbl 0614.22009
[49] M. Zorn, “The automorphisms of Cayley’s non-associative algebra,” Proceedings of the National Academy of Sciences of the United States of America, vol. 21, no. 6, pp. 355-358, 1935. · Zbl 0011.38903
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.