Hyperquaternions: A New Tool for Physics

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


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.


Clifford algebras Hyperquaternions Quaternions Hyperconjugation 



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).


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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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