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Hyperquaternions and Physics
by Patrick R. Girard, Romaric Pujol, Patrick Clarysse, Philippe Delachartre
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Patrick Girard 
Submission information  

Preprint Link:  scipost_202211_00033v3 (pdf) 
Date accepted:  20230811 
Date submitted:  20230107 17:21 
Submitted by:  Girard, Patrick 
Submitted to:  SciPost Physics Proceedings 
Proceedings issue:  34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The paper develops, within a new representation of Clifford algebras in terms of tensor products of quaternions called hyperquaternions, several applications. The first application is a quaternion 2D representation in contradistinction to the frequently used 3D one. The second one is a new representation of the conformal group in (1+2) space (signature +) within the Dirac algebra C5(2,3)=C*H*H subalgebra of H*H*H (* tensor product). A numerical example and a canonical decomposition into simple planes are given. The third application is a classification of all hyperquaternion algebras into four types, providing the general formulas of the signatures and relating them to the symmetry groups of physics.
List of changes
1) Introduction (+) changed into (+)
2)
Section 5 addition of
« Concerning the matrix representation of hyperquaternion algebras, which is beyond the scope of this paper, the above
isomorphisms show that H*H can be represented either by a reducible real matrix R(16) (real
16×16 matrix) or by an irreducible R(4) matrix (H being represented by an irreducible
R(4) matrix). Similarly, H*H*H and its subalgebra C*H*H can be represented either by a reducible
matrix R(64) or by an irreducible matrix R(16) . A classification of real irreducible
representations of quaternionic Clifford algebras can be found in [16,17]. »
Addition of two references
[16] S. Okubo, Real representations of finite Clifford algebras. I. Classification, J. Math. Phys.
32, 1657 (1991), doi:10.1088/11266708/2003/04/040
[17] H.L. Carrion, M. Rojas, F. Toppan, Quaternionic and Octonionic Spinors. A Classification,
JHEP04. 2003, (2003), doi:10.1088/11266708/2003/04/040
3) Acknowledgements : addition of
« The authors acknowledge and thank an anonymous referee for comments clarifying the
matrix quaternionic Clifford algebra representation issue. »
Published as SciPost Phys. Proc. 14, 030 (2023)