Local algebras for causal fermion systems in Minkowski space

Abstract

A notion of local algebras is introduced in the theory of causal fermion systems. Their properties are studied in the example of the regularized Dirac sea vacuum in Minkowski space. The commutation relations are worked out, and the differences to the canonical commutation relations are discussed. It is shown that the spacetime point operators associated to a Cauchy surface satisfy a time slice axiom. It is proven that the algebra generated by operators in an open set is irreducible as a consequence of Hegerfeldt's theorem. The light cone structure is recovered by analyzing expectation values of the operators in the algebra in the limit when the regularization is removed. It is shown that every spacetime point operator commutes with the algebras localized away from its null cone, up to small corrections involving the regularization length.