Fermions and octonions
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.
scientific article; zbMATH DE number 4027714