Fermions and octonions
type: Item
instance of: scholarly article
title: Fermions and octonions
zbMATH Open document ID: 0632.17009
DOI: 10.1007/BF01218483
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: Let \(\psi_{\omega}(z)=z^{\omega /2} : e^{i\omega \cdot Q(z)} : c_{\omega}\) be the vertex operator, where \[ Q^ j(z)=q^ j-ip^ j \log z+i\sum_{n\neq 0}z^{-n}\alpha^ j_ n/n. \] This vertex operator provides a complex Fermi field when \(\omega^ 2=1\). By the standard product identity of vertex operators, one has \[ \psi_{\omega_ 1}(z_ 1)\psi_{\omega_ 2}(z_ 2)=\epsilon (\omega_ 1,\omega_ 2)(1-z_ 2/z_ 1)^{\omega_ 1\cdot \omega_ 2}[\psi_{\omega_ 1+\omega_ 2}(z_ 2)+O(z_ 1-z_ 2)] \] where \(\epsilon\) denotes a cocycle. The six fermion fields \(\psi_{\pm \omega_ 1}\), \(\psi_{\pm \omega_ 2}\) and \(\psi_{\pm (\omega_ 1+\omega_ 2)}\) form a closed set under the above operator product expansion. The main purpose of this paper is to analyse the algebraic structure of these fermion fields: a relation between the operator product and the product law of the division algebras formed by the complex numbers, quaternions and octonions.
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17A35
Mathematics Subject Classification ID: 81T30
zbMATH DE Number: 4027714
zbMATH Keywords: vertex operator
zbMATH Keywords: fermion fields
zbMATH Keywords: division algebras
author: Peter Goddard
author: Werner Nahm
author: David I. Olive
author: Henri Ruegg
author: Adam Schwimmer
reviewed by: Hiromichi Yamada
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01218483
OpenAlex ID: W2006431728
description: scientific article; zbMATH DE number 4027714