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Octonions, Exceptional Jordan Algebra and The Role of The Group \(F_4\) in Particle Physics

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Abstract

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan–von Neumann–Wigner classification of finite euclidean Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra \({\mathcal {J}}\). The automorphism group \(F_4\) of \({\mathcal {J}}\) and its maximal Borel-de Siebenthal subgroups \(\frac{SU(3)\times SU(3)}{{{\mathbb {Z}}_3}}\) and Spin(9) are studied in detail and applied to the classification of fundamental fermions and gauge bosons. Their intersection in \(F_4\) is demonstrated to coincide with the gauge group \(S(U(2)\times U(3))\) of the Standard Model of particle physics.

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Acknowledgements

I.T. thanks Michel Dubois-Violette for enlightening discussions and acknowledges the hospitality of the IHES (Bures-sur-Yvette) and the NCCR SwissMAP (Geneva) where part of this work was done. S. Drenska’s work has been supported by the Bulgarian National Science Fund, DFNI E02/6.

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  1. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, 1784, Sofia, Bulgaria

    Ivan Todorov & Svetla Drenska

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  1. Ivan Todorov
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Correspondence to Ivan Todorov.

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Extended version of lectures presented by I.T. at the Institute for Nuclear Research and Nuclear Energy during the spring of 2017 (see, [48]) and in May 2018.

This article is part of the Topical Collection on Homage to Prof. W.A. Rodrigues Jr edited by Jayme Vaz.

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Todorov, I., Drenska, S. Octonions, Exceptional Jordan Algebra and The Role of The Group \(F_4\) in Particle Physics. Adv. Appl. Clifford Algebras 28, 82 (2018). https://doi.org/10.1007/s00006-018-0899-y

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