Matrix representation of octonions and generalizations

Journal of Mathematical Physics

Volumn 40, Issue 8, 1999, Pages 4134-4150

  Daboul, Jamil ^a,b   Delbourgo, Robert ^a  

^a UNIVERSITY OF TASMANIA   (Australia)

^b BEN GURION UNIVERSITY OF THE NEGEV   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033241184     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532950     Document Type: Article

References (46)

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2. D. Finkelstein, J. M. Jauch, S. Shiminovitch, and D. Speiser, J. Math. Phys. 3, 207 (1962); 4, 136 (1963); 4, 788 (1963); G. G. Emch, Helv. Phys. Acta 36, 739 (1963); L. P. Horwitz and L. C. Biedenharn, ibid. 38, 385 (1965); Ann. Phys. (N.Y.) 157, 432 (1984); M. Gunaydin, C. Piron, and H. Ruegg, Commun. Math. Phys. 61, 69 (1978); C. G. Nash and G. C. Joshi, J. Math. Phys. 28, 2883 (1987); 28, 2886 (1987); S. de Leo and P. Rotelli, Prog. Theor. Phys. 92, 917 (1994).
3. D. Finkelstein, J. M. Jauch, S. Shiminovitch, and D. Speiser, J. Math. Phys. 3, 207 (1962); 4, 136 (1963); 4, 788 (1963); G. G. Emch, Helv. Phys. Acta 36, 739 (1963); L. P. Horwitz and L. C. Biedenharn, ibid. 38, 385 (1965); Ann. Phys. (N.Y.) 157, 432 (1984); M. Gunaydin, C. Piron, and H. Ruegg, Commun. Math. Phys. 61, 69 (1978); C. G. Nash and G. C. Joshi, J. Math. Phys. 28, 2883 (1987); 28, 2886 (1987); S. de Leo and P. Rotelli, Prog. Theor. Phys. 92, 917 (1994).
4. D. Finkelstein, J. M. Jauch, S. Shiminovitch, and D. Speiser, J. Math. Phys. 3, 207 (1962); 4, 136 (1963); 4, 788 (1963); G. G. Emch, Helv. Phys. Acta 36, 739 (1963); L. P. Horwitz and L. C. Biedenharn, ibid. 38, 385 (1965); Ann. Phys. (N.Y.) 157, 432 (1984); M. Gunaydin, C. Piron, and H. Ruegg, Commun. Math. Phys. 61, 69 (1978); C. G. Nash and G. C. Joshi, J. Math. Phys. 28, 2883 (1987); 28, 2886 (1987); S. de Leo and P. Rotelli, Prog. Theor. Phys. 92, 917 (1994).
5. D. Finkelstein, J. M. Jauch, S. Shiminovitch, and D. Speiser, J. Math. Phys. 3, 207 (1962); 4, 136 (1963); 4, 788 (1963); G. G. Emch, Helv. Phys. Acta 36, 739 (1963); L. P. Horwitz and L. C. Biedenharn, ibid. 38, 385 (1965); Ann. Phys. (N.Y.) 157, 432 (1984); M. Gunaydin, C. Piron, and H. Ruegg, Commun. Math. Phys. 61, 69 (1978); C. G. Nash and G. C. Joshi, J. Math. Phys. 28, 2883 (1987); 28, 2886 (1987); S. de Leo and P. Rotelli, Prog. Theor. Phys. 92, 917 (1994).
6. D. Finkelstein, J. M. Jauch, S. Shiminovitch, and D. Speiser, J. Math. Phys. 3, 207 (1962); 4, 136 (1963); 4, 788 (1963); G. G. Emch, Helv. Phys. Acta 36, 739 (1963); L. P. Horwitz and L. C. Biedenharn, ibid. 38, 385 (1965); Ann. Phys. (N.Y.) 157, 432 (1984); M. Gunaydin, C. Piron, and H. Ruegg, Commun. Math. Phys. 61, 69 (1978); C. G. Nash and G. C. Joshi, J. Math. Phys. 28, 2883 (1987); 28, 2886 (1987); S. de Leo and P. Rotelli, Prog. Theor. Phys. 92, 917 (1994).
7. D. Finkelstein, J. M. Jauch, S. Shiminovitch, and D. Speiser, J. Math. Phys. 3, 207 (1962); 4, 136 (1963); 4, 788 (1963); G. G. Emch, Helv. Phys. Acta 36, 739 (1963); L. P. Horwitz and L. C. Biedenharn, ibid. 38, 385 (1965); Ann. Phys. (N.Y.) 157, 432 (1984); M. Gunaydin, C. Piron, and H. Ruegg, Commun. Math. Phys. 61, 69 (1978); C. G. Nash and G. C. Joshi, J. Math. Phys. 28, 2883 (1987); 28, 2886 (1987); S. de Leo and P. Rotelli, Prog. Theor. Phys. 92, 917 (1994).
8. D. Finkelstein, J. M. Jauch, S. Shiminovitch, and D. Speiser, J. Math. Phys. 3, 207 (1962); 4, 136 (1963); 4, 788 (1963); G. G. Emch, Helv. Phys. Acta 36, 739 (1963); L. P. Horwitz and L. C. Biedenharn, ibid. 38, 385 (1965); Ann. Phys. (N.Y.) 157, 432 (1984); M. Gunaydin, C. Piron, and H. Ruegg, Commun. Math. Phys. 61, 69 (1978); C. G. Nash and G. C. Joshi, J. Math. Phys. 28, 2883 (1987); 28, 2886 (1987); S. de Leo and P. Rotelli, Prog. Theor. Phys. 92, 917 (1994).
9. D. Finkelstein, J. M. Jauch, S. Shiminovitch, and D. Speiser, J. Math. Phys. 3, 207 (1962); 4, 136 (1963); 4, 788 (1963); G. G. Emch, Helv. Phys. Acta 36, 739 (1963); L. P. Horwitz and L. C. Biedenharn, ibid. 38, 385 (1965); Ann. Phys. (N.Y.) 157, 432 (1984); M. Gunaydin, C. Piron, and H. Ruegg, Commun. Math. Phys. 61, 69 (1978); C. G. Nash and G. C. Joshi, J. Math. Phys. 28, 2883 (1987); 28, 2886 (1987); S. de Leo and P. Rotelli, Prog. Theor. Phys. 92, 917 (1994).
10. D. Finkelstein, J. M. Jauch, S. Shiminovitch, and D. Speiser, J. Math. Phys. 3, 207 (1962); 4, 136 (1963); 4, 788 (1963); G. G. Emch, Helv. Phys. Acta 36, 739 (1963); L. P. Horwitz and L. C. Biedenharn, ibid. 38, 385 (1965); Ann. Phys. (N.Y.) 157, 432 (1984); M. Gunaydin, C. Piron, and H. Ruegg, Commun. Math. Phys. 61, 69 (1978); C. G. Nash and G. C. Joshi, J. Math. Phys. 28, 2883 (1987); 28, 2886 (1987); S. de Leo and P. Rotelli, Prog. Theor. Phys. 92, 917 (1994).
11. F. Gursey, Proceedings of "Symmetries in Physics (1600-1980)" Sant Feliu de Guixols, 1992; S. de Leo, J. Math. Phys. 37, 2955 (1996). These articles contain reviews of this topic.
12. F. Gursey, Proceedings of "Symmetries in Physics (1600-1980)" Sant Feliu de Guixols, 1992; S. de Leo, J. Math. Phys. 37, 2955 (1996). These articles contain reviews of this topic.
13. F. Gursey, Quaternion Methods in Field Theory John Hopkins Workshop, QCD 161, J55, 1980; K. Morita, Prog. Theor. Phys. 67, 1860 (1982); S. L. Adler, Phys. Rev. Lett. 55, 783 (1985); Commun. Math. Phys. 104, 611 (1986); C. Nash and G. C. Joshi, J. Math. Phys. 28, 463 (1987); Int. J. Theor. Phys. 27, 409 (1988); S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, Oxford, 1995).
14. F. Gursey, Quaternion Methods in Field Theory John Hopkins Workshop, QCD 161, J55, 1980; K. Morita, Prog. Theor. Phys. 67, 1860 (1982); S. L. Adler, Phys. Rev. Lett. 55, 783 (1985); Commun. Math. Phys. 104, 611 (1986); C. Nash and G. C. Joshi, J. Math. Phys. 28, 463 (1987); Int. J. Theor. Phys. 27, 409 (1988); S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, Oxford, 1995).
15. F. Gursey, Quaternion Methods in Field Theory John Hopkins Workshop, QCD 161, J55, 1980; K. Morita, Prog. Theor. Phys. 67, 1860 (1982); S. L. Adler, Phys. Rev. Lett. 55, 783 (1985); Commun. Math. Phys. 104, 611 (1986); C. Nash and G. C. Joshi, J. Math. Phys. 28, 463 (1987); Int. J. Theor. Phys. 27, 409 (1988); S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, Oxford, 1995).
16. F. Gursey, Quaternion Methods in Field Theory John Hopkins Workshop, QCD 161, J55, 1980; K. Morita, Prog. Theor. Phys. 67, 1860 (1982); S. L. Adler, Phys. Rev. Lett. 55, 783 (1985); Commun. Math. Phys. 104, 611 (1986); C. Nash and G. C. Joshi, J. Math. Phys. 28, 463 (1987); Int. J. Theor. Phys. 27, 409 (1988); S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, Oxford, 1995).
17. F. Gursey, Quaternion Methods in Field Theory John Hopkins Workshop, QCD 161, J55, 1980; K. Morita, Prog. Theor. Phys. 67, 1860 (1982); S. L. Adler, Phys. Rev. Lett. 55, 783 (1985); Commun. Math. Phys. 104, 611 (1986); C. Nash and G. C. Joshi, J. Math. Phys. 28, 463 (1987); Int. J. Theor. Phys. 27, 409 (1988); S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, Oxford, 1995).
18. F. Gursey, Quaternion Methods in Field Theory John Hopkins Workshop, QCD 161, J55, 1980; K. Morita, Prog. Theor. Phys. 67, 1860 (1982); S. L. Adler, Phys. Rev. Lett. 55, 783 (1985); Commun. Math. Phys. 104, 611 (1986); C. Nash and G. C. Joshi, J. Math. Phys. 28, 463 (1987); Int. J. Theor. Phys. 27, 409 (1988); S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, Oxford, 1995).
19. F. Gursey, Quaternion Methods in Field Theory John Hopkins Workshop, QCD 161, J55, 1980; K. Morita, Prog. Theor. Phys. 67, 1860 (1982); S. L. Adler, Phys. Rev. Lett. 55, 783 (1985); Commun. Math. Phys. 104, 611 (1986); C. Nash and G. C. Joshi, J. Math. Phys. 28, 463 (1987); Int. J. Theor. Phys. 27, 409 (1988); S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, Oxford, 1995).
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41. 2) = 0, ∀ x, y ∈ T. See N. Jacobson, Structure and Representations of Jordan Algebras, (American Mathematical Society, 1968). We shall find another class of flexible algebras below.
42. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer-Verlag New York, 1972).
43. K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, and A. I. Shirshov, Rings That Are Nearly Associative (Academic, New York, 1982).
44. j) = diag(-2,-2,2).
45. -, where α± = (α±β)/√2. We have refrained from doing so in the text.
46. Thus, if we reverse the sign in front of the σ in (17), we only need to change the sign of the commutators in (18) to fix the multiplication rule.