Hyperquaternions and Physics: Historical Outlook
Résumé
Hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof), it follows that they constitute Clifford algebras due to a theorem by Clifford (1845−1879). Examples of hyperquaternions are quaternions H(e1=i, e2=j), biquaternions H⊗C(e1=iI, e2=jI, e3=kI), tetraquaternions H⊗H(e0=j, e1=kI, e2=kJ, e3=kK), and so on H⊗H⊗C,H⊗H⊗H.... Quaternions were discovered in 1843 by Hamilton (1805−1865) after a long vain search of triplets and satisfy the multiplication rule i2=j2=k2=ijk=−1. Containing R and C as particular cases, quaternions were immediately perceived as a major discovery and were to lead to the 3D vector calculus still in use today. Hamilton also introduced complex quaternions which he named biquaternions. Independently of Clifford, Lipschitz (1832−1903) discovered in 1880, as an extension of quaternions, what is called today the even subalgebra as well as the formula of ndimensional rotations in euclidean spaces y=axa−1(a∈C+n). Moore(1876−1931), working on a canonical decomposition of rotations was to call the elements of Lipschitz’s algebra hyperquaternions which justifies the terminology adopted above. Hyperquaternions constitute a new mathematical formalism which seems to be particularly well adapted to physics. Since H⊗H'M4(R), [H⊗H]⊗C'M4(C),[H⊗H]⊗H'M4(H) it follows that hyperquaternions yield all real matrices aswell as the complex and quaternionic ones. Introducing a hyperconjugation, one obtains a simple expression of major symmetry groups of physics. Prof. Dr Girard who has a Ph.D. in History of Science (UW-Madison) retraces historically the uses (and frequent misuses) of hyperquaternions up to the present day in physics.References:[1] Girard, Patrick R. et al., Differential Geometry Revisited by Biquaternion Clifford Algebra. In J.-D. Boissonat et al (Eds.): Curves and Surfaces (Springer, 2015).[2] Girard, Patrick R., Quaternions, Clifford Algebras and Relativistic Physics (Birkhauser, Basel, 2007).[3] Clifford, W. K., Applications of Grassmann’s extensive algebra, Amer. J. Math., 1 (1878), pp. 350-358.[4] Lipschitz R., Principes d’un calcul algébrique qui contient comme espèces particulières le calcul des quantités imaginaires et des quaternions, C. R. Acad. Sci. Paris, 91 (1880), pp. 619-621, 660-664.[5] Moore, C. J. E., Hyperquaternions, Journal of Mathematical Physics, 1 (1922), pp. 63-77.[6] Hankins, Thomas L., Sir William Rowan Hamilton (The Johns Hopkins University Press, Baltimore, 1980).