The Linear Dynamics of Wave Functions in Causal Fermion Systems

TL;DR

This paper develops a mathematical framework for the dynamics of wave functions in causal fermion systems, deriving a wave equation, proving existence and uniqueness of solutions, and analyzing Green's operators with an application to Minkowski space.

Contribution

It introduces a new dynamical wave equation for causal fermion systems and proves the well-posedness of its initial value problem.

Findings

Solutions form a Hilbert space with a conserved scalar product

Existence and uniqueness of global solutions are established

Green's operators are constructed and analyzed

Abstract

The dynamics of spinorial wave functions in a causal fermion system is studied. A so-called dynamical wave equation is derived. Its solutions form a Hilbert space, whose scalar product is represented by a conserved surface layer integral. We prove under general assumptions that the initial value problem for the dynamical wave equation admits a unique global solution. Causal Green's operators are constructed and analyzed. Our findings are illustrated in the example of the regularized Minkowski vacuum.