Causal Fermion Systems and Octonions
Felix Finster, Niels G. Gresnigt, J. M. Isidro, Antonino Marciano, Claudio F. Paganini, Tejinder P. Singh
TL;DR
This work examines deep links between octonionic, division-algebraic approaches and causal fermion systems (CFS) to fundamental physics. It shows that octonions naturally describe the symmetries of the Minkowski vacuum in a CFS, while the causal action principle provides the spacetime and dynamical content that octonionic models lack. Conversely, octonionic constructions inform the internal (gauge and generation) structure of CFS vacua and suggest algebraic routes to the Standard Model within a dynamical spacetime framework. The study highlights mutual benefits: octonionic theories gain a dynamical context from CFS, and CFS gains a more structured vacuum and symmetry origin from octonionic algebra, with future work focusing on exceptional Lie groups and chiral symmetry breaking to further unify the approaches.
Abstract
We compare the structures and methods in the theory of causal fermion systems with approaches to fundamental physics based on division algebras, in particular the octonions. We find that octonions and, more generally, tensor products of division algebras come up naturally to describe the symmetries of the vacuum configuration of a causal fermion system. This is achieved by associating the real and imaginary octonion basis elements with the neutrino and charged sectors of the vacuum fermionic projector, respectively. Conversely, causal fermion systems provide octonionic theories with spacetime structures and dynamical equations via the causal action principle. In this way, octonionic theories and causal fermion systems complement each other..