Causal Fermion Systems and Octonions

We compare the structures and methods in the theory of causal fermion systemswith approaches to fundamental physics based on division algebras, inparticular the octonions. We find that octonions and, more generally, tensorproducts of division algebras come up naturally to describe the symmetries ofthe vacuum configuration of a causal fermion system. This is achieved byassociating the real and imaginary octonion basis elements with the neutrinoand charged sectors of the vacuum fermionic projector, respectively.Conversely, causal fermion systems provide octonionic theories with spacetimestructures and dynamical equations via the causal action principle. In thisway, octonionic theories and causal fermion systems complement each other.