Hyperquaternions: A New Tool for Physics

Patrick R. Girard (1) , Patrick Clarysse (1) , Romaric Pujol (2) , Robert Goutte (3) , Philippe Delachartre (4)

1 MOTIVATE - Imagerie et modélisation Vasculaires, Thoraciques et Cérébrales
2 Pôle de mathématiques
3 CREATIS - Centre de Recherche en Acquisition et Traitement de l'Image pour la Santé
4 Imagerie Ultrasonore

A hyperquaternion formulation of Clifford algebras in n dimensions is presented. The hyperquaternion algebra is defined as a tensor product of quaternion algebras 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 H⊗H≃m(4,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.

Mots clés: Clifford algebras; Hyperquaternions; Quaternions; Hyperconjugation

Domaines: Mathématiques [math]; Physique [physics]

Patrick R. Girard, Patrick Clarysse, Romaric Pujol, Robert Goutte, Philippe Delachartre. Hyperquaternions: A New Tool for Physics. Advances in Applied Clifford Algebras, 2018, 28 (3), ⟨10.1007/s00006-018-0881-8⟩. ⟨hal-01886860⟩