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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.
Published as SciPost Phys. Proc. 14, 030 (2023)
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. »
Submission & Refereeing History
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