Construction of Currents in Causal Fermion Systems

Abstract: This paper presents a novel and systematic formalism for deriving classical field equations within the framework ofcausal fermion systems, explicitly accounting for higher-order corrections such as quantum effects and those arising from spacetime discreteness. Our method, which also generalizes to non-abelian gauge fields and gravitation, gives a systematic procedure for evaluating the linearized field equations of causal fermion systems. By probing these equations with specific wave functions and employing Taylor expansions, we reformulate them as a family of tensorial equations of increasing rank. We show that, for rank one, this approach recovers the established classical dynamics corresponding to Maxwell's equations. In addition, the approach gives rise to higher-rank tensorial equations, where the second-rank equations are expected to encode the Einstein equations, and higher-rank tensors potentially reveal new physics and systematic corrections.