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.

Examples and Limiting Cases

Minkowski Space as a Causal Fermion System

The Continuum Limit

Quantum Field Theory

General Relativity

Further Structures

Manifolds of Operators

Operator Algebras

Causal Cone Structures

Fock Space Structures

Positive Functionals

The Fermionic Signature Operator

Notions of Entropy