The Theory of Causal Fermion Systems
Overview of Mathematical Aspects
Basic Definitions
The general definitions of a causal fermion system and the causal structure are given.
Generalizations and Special Cases
An overview of different generalizations and specializations is given.
Existence Theory
The existence theory for minimizers of the causal action is outlined.
Inherent Structures
A causal fermion system involves structures which are inherent in the sense that they do not give additional input but merely give information already encoded in the causal fermion system a useful name. Here is an overview of the most important inherent structures:
Spin Spaces and Physical Wave Functions
These structures bear similarity with a topological vector bundle, with the physical wave functions as sections thereof.
The Fermionic Projector
The kernel of the fermionic projector induces relations between space-time points. It is also the kernel of an integral operator.
Geometric Structures
The spin connection and curvature are introduced.
Surface Layer Integrals
Analytic Structures
The Euler-Lagrange Equations
Critical points of the causal action principle satisfy the Euler-Lagrange equations.
The Linearized Field Equations
Linearizing families of solutions of the Euler-Lagrange equations gives rise to the linearized field equations.
The Dynamical Wave Equation
Considering the behavior of the physical wave functions under linear perturbations of the causal fermion system gives rise to the dynamical wave equations.
Existence Theory for Linear Fields and Waves
The existence theory for solutions of the linearized field equations and the dynamical wave equation are outlined. The methods are based on energy estimates.