Advertisement

Advances in Applied Clifford Algebras

, Volume 22, Issue 1, pp 87–105 | Cite as

Glimpses of the Octonions and Quaternions History and Today’s Applications in Quantum Physics

  • Andrzej Krzysztof Kwaśniewski
Article

Abstract

Before we dive in this essay into the accessibility stream of nowadays indicatory applications of octonions and quaternions to computer and other sciences and to quantum physics (see for example [50-53], [41], [33]) and to Clifford algebras (see for example [17,16], 18) let us focus for a while on the crucially relevant events for today’s revival on interest to nonassociativities while the role of associative quaternions in eight periodicity constructive classification of associative Clifford algebras is now a text-book knowledge.

Our reflections keep wandering back to the Brahmagupta-Fibonacci Two-Square Identity and then via the Euler Four-Square Identity up to the Degen-Graves-Cayley Eight-Square Identity. These glimpses of history incline and invite us to re-tell the story on how about one month after quaternions have been carved on the Brougham bridge octonions were discovered by John Thomas Graves (1806-1870), jurist and mathematician - a friend of William Rowan Hamilton (1805-1865).

As for today we just mention en passant quaternionic and octonionic quantum mechanics, generalization of Cauchy-Riemann equations for octonions and Triality Principle and G 2 group in spinor language in a descriptive way in order not to daunt non-specialists. Relation to finite geometries is recalled and the links to the 7Stones of seven sphere, seven “imaginary” octonions’ units in out of the Plato’s Cave Reality applications are appointed. This way we are welcome back to primary ideas of Heisenberg, Wheeler and other distinguished founders of quantum mechanics and quantum gravity foundations.

Keywords

Clifford Algebras octonions discrete geometry quantum physics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Martin Huxley, Glimpses of History. An infinite Poem, http://www.cf.ac.uk/maths/numbertheory/history.html.
  2. 2.
    William Rowan Hamilton, On Quaternions: Letter to John Thomas Graves, 17th October, 1843. Published in the London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, vol. XXV (1844), pp 489-95. http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/QLetter/.
  3. 3.
    Arthur Cayley, On Jacobi’s elliptic functions, in reply to the Rev. Brice Bronwin; and on quaternions. Philosophical Magazine vol. 26 (1845) pp. 210-213. A. Cayley, Papers Collected Mathematical Papers (Cambridge 1889).Google Scholar
  4. 4.
    Leonard Eugene Dickson: On quaternions and their generalization and the history of the eight square theorem. Ann. of Math. 20, 155–171 (1919)MathSciNetCrossRefGoogle Scholar
  5. 5.
    John C. Baez, The Octonions. Bull. Amer. Math. Soc. 39 (2002), 145-205. Errata in Bull. Amer. Math. Soc. 42 (2005), 213.Google Scholar
  6. 6.
    John H. Conway and Derek A. Smith On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A K Peters Ltd.; 1 edition (2003).Google Scholar
  7. 7.
    Author: Seven Stones Multimedia. http://www.7stones.com/. XXI Century, The Planet Earth.
  8. 8.
    John C. Baez, This Week’s Finds In Mathematical Physics. (week104, week152, week192), http://math.ucr.edu/home/baez/twf_/ascii/weekXYZ.
  9. 9.
    Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Springer Verlag, 1994.Google Scholar
  10. 10.
    John C. Baez, This Week’s Finds In Mathematical Physics. (Week 59), http://math.ucr.edu/home/baez/week59.html.
  11. 11.
    Titus Piezas III, The Degen-Graves-Cayley Eight-Square Identity from The Ramanujan Pages, http://www.geocities.com/titus_piezas/DegenGraves1.htm , July 16, 2005.
  12. 12.
    Joe Malkevitch, Finite Geometries? American Mathematical Society Feature Column, June, 2006, York College (CUNY), http://www.ams.org/featurecolumn/archive/finitegeometries.html.
  13. 13.
    Kwaśniewski A.K.: Generalization of Cauchy-Riemann Equations. Reports on Math. Phys. 22, 133–148 (1985)ADSzbMATHCrossRefGoogle Scholar
  14. 14.
    A. K. Kwaśniewski, Cauchy-Riemann Equations for Cayley Numbers’ Functions. Bulletin de la Soc. des Sciences et de Lettres de Lodz (PAN) vol. XXVI (1998), 49-62, arXiv:0803.0035v1 [math.GM], (Sat, 1 Mar 2008).Google Scholar
  15. 15.
    John C. Baez, This Week’s Finds In Mathematical Physics (Week 61), http://math.ucr.edu/home/baez/week61.html.
  16. 16.
    Hasiewicz Z., Kwaśniewski A.K.: Triality Principle and G2 Group in Spinor Language. J. Math. Phys. 26, 6–11 (1985)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Hasiewicz Z., Kwaśniewski A.K., Morawiec P.: On Parallelizable Spheres, Division Algebras and Clifford Algebras. Reports on Math. Phys. 23, 161–168 (1986)ADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Nieto J.A., Alejo-Armenta L.N.: Hurwitz Theorem and Paralleziable Spheres from Tensor Analysis. International Journal of Modern Physics A 16(25), 4207–4222 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    van der Waerden B.L.: Hamilton’s Discovery of Quaternions. Mathematics Magazine 49(5), 227–234 (1976)zbMATHCrossRefGoogle Scholar
  20. 20.
    A. K. Kwaśniewski Ivan Bernoulli Series Universalissima. Ganita Bharati vol 28 No 1-2 (2006), pp. 93-101, http://front.math.ucdavis.edu/0601.5016.
  21. 21.
    H. Freudenthal Oktaven, Ausnahmegruppen, und Oktavengeometrie. Geometriae Dedicata, 19 1-73, (1985). (Informally published, Utrecht 1951).Google Scholar
  22. 22.
    Graves C. On the principles which regulate the interchange of symbols in certain symbolic equations. Proc. Royal Irish Academy vol. 6 (1853-1857), pp. 144-152 and Special Functions; Informational Bulletin, vol. 1 No 1 (1996), p. 2-4.Google Scholar
  23. 23.
    O. V. Viskov On One Result of George Boole. (In Russian) Integral Transforms and Special Functions-Bulletin. vol. 1 No 2 (1997), p. 2-7.Google Scholar
  24. 24.
    O. V. Viskov, The Lagrange formula and the Graves algebra. (See: epubs.siam.org/sam-bin/getfile/TVP/articles/97810.pdf).Google Scholar
  25. 25.
    A. K. Kwaśniewski, Extended finite operator calculus - an example of algebraization of analysis. Bulletin of the Allahabad Mathematical Society Vol 20, 2005 1-24. http://ii.uwb.edu.pl/akk/publ1.htm.
  26. 26.
    Xingmin Li and Lizhong Peng, The Cauchy integral formulas on the octonions. Bull. Belg. Math. Soc. Simon Stevin Volume 9 Number 1 (2002), 47-64.Google Scholar
  27. 27.
    Fred Y. Ye, from Zhejiang University Gauss-Bonnet Theorems in Quaternionic and Octonionic Structures. Sciencepaper Online Published-Date 2007-08-27.Google Scholar
  28. 28.
    Susumo Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge University Press, (1995).Google Scholar
  29. 29.
    Murray Bremner: Quantum Octonions. Communications in Algebra 27, 2809–2831 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lassig C.C., Joshi G.C.: An Octonionic Gauge Theory. Chaos Solitons Fractals 7, 769–775 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    Murat Gunaydin, Hermann Nicolai, Seven Dimensional Octonionic Yang-Mills Instanton and its Extension to an Heterotic String Soliton. Phys. Lett. B351 (1995), 169-172; Addendum-ibid. B376 (1996), 329.Google Scholar
  32. 32.
    Tongsong Jiang and Li Chen, Algebraic algorithms for least squares problem in quaternionic quantum theory. Computer Physics Communications Volume 176 Issue 7, 1 April 2007, 481-485.Google Scholar
  33. 33.
    Stephen L. Adler, Quaternionic Quantum Mechanics and Quantum Fields. (1995), Oxford University Press, US.Google Scholar
  34. 34.
    Jacobs K. C., Quantum Cosmologies in the Quaternion Formulation of General Relativity Theory. Bulletin of the American Astronomical Society Vol. 3 (1971), p.480.Google Scholar
  35. 35.
    J. Tits, Le plan projectif des octaves et les groupes de Lie exceptionels. Bull. Acad. Roy. Belg. Sci. 39 (1953), 300–329.Google Scholar
  36. 36.
    Yaglom I.M.: Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century. Birkhäuser, Boston (1988)zbMATHGoogle Scholar
  37. 37.
    Hardy G.H., Wright E.M.: An Introduction to the Theory of Numbers. Clarendon Press, Oxford (1959)Google Scholar
  38. 38.
    A. K. Kwaniewski, Could We Manage without Micro-UFOs? Physics Essays; (An International Journal Dedicated to Fundamental Questions in Physics) Ottawa; Vol. 9 no 2 (June 1996), 274-281, http://ii.uwb.edu.pl/akk/publ1.htm.
  39. 39.
    P Kainen, An octonion model for physics. (2000) Online Article posted to physics number-theory mathematics fun algebra by jrw on 2007-05-26 20:16:11 see: http://www.citeulike.org/tag/number-theory.
  40. 40.
    Albuquerque H., Majid S., Quasialgebra structure of the octonions. J. of Algebra 220 no. 1 (1999), 188-224.Google Scholar
  41. 41.
    T. Kugo and P. K. Townsend, Supersymmetry and the division algebras. Nucl. Phys. B221 (1983), 357-380.Google Scholar
  42. 42.
    A. Dimakis and F. Muller-Hoissen, Nonassociativity and Integrable Hierarchies. Phys. Rev. Lett. 91 (2003), 236803.Google Scholar
  43. 43.
    Merab Gogberashvili, Octonionic Version of Dirac Equations. Int. J. Mod. Phys. A21 (2006) 3513-3524, http://front.math.ucdavis.edu/0505.3101.
  44. 44.
    Merab Gogberashvili, Octonionic electrodynamics. J. Phys. A39 (2006), 7099- 7104, http://arxiv.org/abs/hep-th/0512258.
  45. 45.
    Vladimir D. Dzhunushaliev, Hamilton’s equations in a non-associative quantum theory. http://arxiv.org/abs/hep-th/0602047, Sun, 5 Feb 2006 .
  46. 46.
    Vladimir D. Dzhunushaliev, Colorless operators in a non-associative quantum theory. Physics Letters A vol. 355 no. 4-5 (2006), pp. 298-302.Google Scholar
  47. 47.
    Vladimir D. Dzhunushaliev, A Non-Associative Quantum Mechanics. Foundations of Physics Letters vol. 19 no. 2 (2006), pp. 157-167.Google Scholar
  48. 48.
    Vladimir D. Dzhunushaliev Toy Models of a Nonassociative Quantum Mechanics. Advances in High Energy Physics, Volume 2007 (2007), Article ID 12387, 10 pages, doi: 10.1155/2007/12387, http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/12387.
  49. 49.
    Vladimir D. Dzhunushaliev Non-associativity, supersymmetry and hidden variables. http://arxiv.org/abs/0712.1647 , 31 January 2008.
  50. 50.
    Stefano De Leo., Khaled Abdel-Khalek: Octonionic Quantum Mechanics and Complex Geometry. Prog. Theor. Phys. 96, 823–832 (1996)ADSCrossRefGoogle Scholar
  51. 51.
    Horwitz L.P., Biedenharn L.C.: Quaternion quantum mechanics: second quantisation and gauge fields. Ann. of Phys. 157, 432 (1984)MathSciNetADSzbMATHCrossRefGoogle Scholar
  52. 52.
    Rembieliski J.: Tensor product of the octonionic Hilbert spaces and colour confinement. J. of Phys. A11, 2323 (1978)ADSGoogle Scholar
  53. 53.
    Pascual Jordan, von John Neumann, Eugene Wigner.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934)CrossRefGoogle Scholar
  54. 54.
    Tongsong Jiang.: Cramer ruler for quaternionic linear equations in quaternionic quantum theory. Rep. Math. Phys. 57(3), 463 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Tongsong Jiang, Y. Liu, M. Wei, Quaternion generalized singular value decomposition and applications. Applied Math. (J. Chin. Univ.) 2006, 21 (1): 113.Google Scholar
  56. 56.
    Tongsong Jiang, Algebraic methods for diagonalization of a quaternion matrix in quaternion quantum theory. J. Math. Phys. (2005), 46 (5).Google Scholar
  57. 57.
    Tongsong Jiang and M. Wei, On a Solution of the Quaternion Matrix Equation and Its Application. Acta Math. Sinica (2005), 21, (3).Google Scholar
  58. 58.
    Tongsong Jiang: An algorithm for quaternionic linear equations in quaternionic quantum theory. J. Math. Phys. 45(11), 4218 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Tongsong Jiang: An algorithm for eigenvalue and eigenvector of quaternion matrices in quaternion mechanics. J. Math. Phys. 45(8), 3334 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Tongsong Jiang., Wei M.: On solutions of the matrix equations XAXB = C and XA . . . B = C. Lin. Alg. Appl. 367, 225 (2003)zbMATHCrossRefGoogle Scholar
  61. 61.
    Tongsong Jiang, Wei M.: Equality constrained least squares problem over quaternion field. Appl. Math. Lett. 16, 883 (2003)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Member of the Institute of Combinatorics and its ApplicationsWinnipeg, ManitobaCanada
  2. 2.BiałystokPoland

Personalised recommendations