GEOMETRY OF THE OCTONIONS, THE

by Oregon State University USA Tevian Dray, Corinne A Manogue

There are precisely two further generalizations of the real and complex numbers, namely, the quaternions and the octonions. The quaternions naturally describe rotations in three dimensions. In fact, all (continuous) symmetry groups are based on one of these four number systems. This book provides an elementary introduction to the properties of the octonions, with emphasis on their geometric structure. Elementary applications covered include the rotation groups and their spacetime generalization, the Lorentz group, as well as the eigenvalue problem for Hermitian matrices. In addition, more sophisticated applications include the exceptional Lie groups, octonionic projective spaces, and applications to particle physics including the remarkable fact that classical supersymmetry only exists in particular spacetime dimensions.

Genres: Mathematics

228 pages, Hardcover

First published March 1, 2013

Reviews

[Astragen · Rating 4 out of 5]

If you're into stuff like this, you can read the full review.


Clifford Algebras: "The Geometry of the Octonions" by Tevian Dray, Corinne Manogue



All of quantum field theory (with the exception of gravity, but people are working on that) can be derived from the following statement: "Assume maximal self-consistent unambiguous algebraic structure and neglect trivial results". The only such set of dimensions which can have non-trivial maximal structure are 3 positive 1 negative (left-hand rule physics) and 3 negative 1 positive (right-hand rule physics). From the maximal algebraic structure (Clifford algebras) we can then derive all of the underlying laws which govern electromagnetism, strong force, weak force, and the Higgs field. We can additionally derive the existence and relative charges/masses/etc. of all particles in the Standard Model by identifying ideal groups in the maths' group theory (except for why neutrinos have mass, but people are working on that too).

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