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Hyperquaternions: A New Tool for Physics

  • Patrick R. GirardEmail author
  • Patrick Clarysse
  • Romaric Pujol
  • Robert Goutte
  • Philippe Delachartre
Article
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Part of the following topical collections:
  1. Proceedings ICCA 11, Ghent, 2017

Abstract

A hyperquaternion formulation of Clifford algebras in n dimensions is presented. The hyperquaternion algebra is defined as a tensor product of quaternion algebras \(\mathbb {H}\) (or a subalgebra thereof). An advantage of this formulation is that the hyperquaternion product is defined independently of the choice of the generators. The paper gives an explicit expression of the generators and develops a generalized multivector calculus. Due to the isomorphism \(\mathbb {H}\otimes \mathbb {H}\simeq m(4, \mathbb {R})\), hyperquaternions yield all real, complex and quaternion square matrices. A hyperconjugation is introduced which generalizes the concepts of transposition, adjunction and transpose quaternion conjugate. As applications, simple expressions of the unitary and unitary symplectic groups are obtained. Finally, the hyperquaternions are compared, in the context of physical applications, to another algebraic structure based on octonions which has been proposed recently.

Keywords

Clifford algebras Hyperquaternions Quaternions Hyperconjugation 

Notes

Acknowledgements

This work was performed in the framework of the LABEX PRIMES (ANR-11-LABX-0063) and LABEX CELYA (ANR-10-LABX-0060) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

References

  1. 1.
    Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  2. 2.
    Angles, P.: The structure of Clifford algebra. Adv. Appl. Clifford Algebras 19, 585–610 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Binz, E., de Gosson, M.A., Hiley, B.J.: Clifford Algebras in symplectic geometry and quantum mechanics. Found. Phys. 43, 424–439 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brackx, F., de Schepper, H., Eelbode, D., Lavika, R., Soucek, V.: Fundaments of quaternionic Clifford analysis I: quaternionic structure. Adv. Appl. Clifford Algebras 24, 955–980 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Casanova, G.: L’algèbre vectorielle. PUF, Paris (1976)Google Scholar
  6. 6.
    Crumeyrolle, A.: Orthogonal and Symplectic Clifford Algebras: Spinor Structures. Kluwer Academic Publishers, Dordrecht (1990)CrossRefzbMATHGoogle Scholar
  7. 7.
    Deheuvels, R.: Formes quadratiques et groupes classiques. Presses Universitaires de France, Paris (1981)zbMATHGoogle Scholar
  8. 8.
    Furey, C.: Unified theory of ideals. Phys. Rev. D86, 025024 (2012)ADSGoogle Scholar
  9. 9.
    Garling, D.J.H.: Clifford Algebras: An Introduction. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Girard, P.R.: The quaternion group and modern physics. Eur. J. Phys. 5, 25–32 (1984)CrossRefGoogle Scholar
  11. 11.
    Girard, P.R.: Einstein’s equations and Clifford algebra. Adv. Appl. Clifford Algebras 9(2), 225–230 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Girard, P.R.: Quaternions. Clifford algebras and relativistic physics. Birkhäuser, Basel (2007)zbMATHGoogle Scholar
  13. 13.
    Girard, P.R., Clarysse, P., Pujol, R.A., Wang, L., Delachartre, P.: Differential geometry revisited by Biquaternion Clifford algebra. In: Boissonnat, J.-D., et al. (eds.) Curves and Surfaces. Springer, Berlin (2015)Google Scholar
  14. 14.
    Greiner, M.: Mécanique Quantique: Symétries. Springer, Berlin (1999)Google Scholar
  15. 15.
    Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York (1997)zbMATHGoogle Scholar
  16. 16.
    Lagally, M.: Vorlesungen über Vektorrechnung, 5th edn. Akademische Verlagsgesellschaft, Leipzig (1956)zbMATHGoogle Scholar
  17. 17.
    Moore, C.L.E.: Hyperquaternions. J. Math. Phys. 1, 63–77 (1922)CrossRefzbMATHGoogle Scholar
  18. 18.
    Shirokov, D.S.: Quaternion typification of Clifford algebra elements. Adv. Appl. Clifford Algebras 22(2), 243–256 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shirokov, D.S.: Symplectic, orthogonal and linear groups in Clifford algebra. Adv. Appl. Clifford Algebras 25, 707–718 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Patrick R. Girard
    • 1
    Email author
  • Patrick Clarysse
    • 1
  • Romaric Pujol
    • 2
  • Robert Goutte
    • 1
  • Philippe Delachartre
    • 1
  1. 1.Université de Lyon, CREATIS; CNRS UMR 5220; INSERM U1206; INSA-Lyon, Université Lyon 1, FranceVilleurbanneFrance
  2. 2.Pôle de Mathématiques, INSA-LyonVilleurbanneFrance

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