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March 25, 2015 By

In Praise of Quaternions

C. R. Math. Rep. Acad. Sci. Canada Vol. 35 (4) 2013, pp. 121–136

December 31, 2013

Joachim Lambek, Department of Mathematics and Statistics, McGill University, Montreal, PQ Canada; e-mail: lambek@math.mcgill.ca

Abstract/Résumé:

This is a survey of some of the applications of quaternions to physics in the 20th century. In the first half century, an elegant presentation of Maxwell’s equations and special relativity was achieved with the help of biquaternions, that is, quaternions with complex coefficients. However, a quaternionic derivation of Dirac’s celebrated equation of the electron depended on the observation that all $4\times 4$ real matrices can be generated by quaternions and their duals.

On examine quelques applications des quaternions à la physique du vingtième siècle. La première moitiè du siècle avait vu une présen-tation élégante des équations de Maxwell et de la relativité spéciale par les quaternions avec des coefficients complexes. Cependant, une dérivation de l’équation célèbre de Dirac dépendait sur l’observation que toutes les matrices $4\times 4$ réelles peuvent être generées par les représentations régulières des quaternions.

Keywords: Classification of elementary particles, Quaternionic treatment of Dirac equation, Three temporal dimensions
AMS Subject Classification: 16A40

[This journal is open access except for the current year and the preceding 5 years]

PDF(click to download): In Praise of Quaternions

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup cycles of ideals elliptic curves fixed point Fourier transform function fields. general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping noninterlacing numerical range orthogonal polynomials Predual space prime number property SP quadratic forms Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces Weak Markov set Whitney problems

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05C05 11A07 11A55 11B37 11B68 11D09 11D25 11D41 11E04 11F67 11G05 11R09 11R11 13B25 14J26 14M25 14P10 17B37 17B67 19K56 26A51 30C15 30H05 35B 37E10 37E20 37F20 37F25 39B72 42C05 43A07 43A62 46B20 46L05 46L35 46L40 46L55 46L80 47H10 53B25 53C55 54C60 60F10 60J75 83C05

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