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

Preprint Link:  scipost_202211_00033v1 (pdf) 
Date submitted:  20221118 16:15 
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.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 20221220 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202211_00033v1, delivered 20221220, doi: 10.21468/SciPost.Report.6353
Strengths
6/10
Weaknesses
6/10
Report
The paper is a clear assessment of the properties of hyperquaternions
with applications to 2D physics and conformal algebras.
The papers could be published, but before publications
some points should be addressed by the authors.
1  In the Introduction, line 9, there is an obvious mistake: the signature of
the (1+2) space is +—, instead of +.
2  The hyperquaternionic algebras C_5(2,3) and C_6(2,4) are nonminimal representations of the corresponding Clifford algebras. This point needs
to be clarified.
In:
 S. Okubo, Real representations of finite Clifford algebras,1. Classification,
J. Math. Phys. 32 (1991) 1657.,
the quaternionic Clifford algebras associated to the different spacetime signatures were presented.
In:
 H.L. Carrion et al.,Quaternionic and octonionic spinors. A classification, JHEP04(2003)040; arXiv:hepth/0302113
a table was presented (Table 4 in JHEP, Table 7 in the arXiv version)
with the dimension of the quaternionic Clifford algebras.
In real counting (as matrices with real entries) both C_5(2,3) and C_6(2,4) are given by
16x16 matrices.
The hyperquaternionic construction of C_5(2,3) and C_6(2,4) is nonminimal.
C_5(2,3) is presented as tensor products of CxHxH which, in real counting, gives
matrices with 32x32 real entries.
C_6(2,4) is presented as tensor products HxHxH which, in real counting,
gives matrices with 64x64 real entries.
The hyperquaternionic representations of C_5(2,3) and C_6(2,4) are
reducible, while the quaternionic representations of C_5(2,3) and C_6(2,4)
are known to be irreducible.
It is unclear which advantages would be produced by a reducible hyperquaternionic
representation with respect to the reducible quaternionic representation.
Requested changes
Correction of signature at line 9 of the Introduction.
Explanation of the reducibility of the hyperquaternionic reps of the Clifford algebras
C_5(2,3) and C_6(2,4).