SciPost Submission Page
Hyperquaternions and Physics
by Patrick R. Girard, Romaric Pujol, Patrick Clarysse, Philippe Delachartre
Submission summary
| Authors (as registered SciPost users): | Patrick R. Girard |
| Submission information | |
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| Preprint Link: | scipost_202211_00033v3 (pdf) |
| Date accepted: | Aug. 11, 2023 |
| Date submitted: | Jan. 7, 2023, 5:21 p.m. |
| Submitted by: | Patrick R. Girard |
| Submitted to: | SciPost Physics Proceedings |
| Proceedings issue: | 34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| 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 HH 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, HHH and its subalgebra CH*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. »
Published as SciPost Phys. Proc. 14, 030 (2023)