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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: 2023-08-11
Date submitted: 2023-01-07 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
  • Mathematical Physics
Approach: Theoretical


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 (+--)

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:10.1088/1126-6708/2003/04/040
[17] H.L. Carrion, M. Rojas, F. Toppan, Quaternionic and Octonionic Spinors. A Classification,
JHEP04. 2003, (2003), doi:10.1088/1126-6708/2003/04/040

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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Resubmission scipost_202211_00033v3 on 7 January 2023
Resubmission scipost_202211_00033v2 on 6 January 2023

Reports on this Submission

Anonymous Report 1 on 2023-1-9 (Invited Report)


The authors answered the question raised in the previous report and clarified the relation between quaternionic and hyperquaternionic representations of Clifford algebras.
The paper can be published as is.

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