# The Theory of Causal Fermion Systems

## Introduction for Mathematicians

### Introduction for mathematicians

It is a basic concept in the theory of Causal Fermion Systems that all the space-time structures (geometry, topology, fields, wave functions, etc.) are encoded in linear operators on a Hilbert space. More precisely, every space-time point corresponds to one operator. The relations between different space-time points are encoded in the corresponding operator products. The physical equations are formulated via a nonlinear variational principle, the causal action principle.

Causal fermion systems use methods of and give close connections to the following areas of mathematics:

• functional analysis
• measure theory
• calculus of variations
• Lorentzian geometry
• Hyperbolic partial differential equations
• Fourier analysis and microlocal analysis
• Methods of quantum field theory: Fock spaces, Feynman diagrams, renormalization
• Methods of non-smooth and infinite-dimensional analysis.
$\renewcommand{\H}{\mathscr{H}} \newcommand{\Lin}{\mathrm{L}} \newcommand{\F}{{\mathscr{F}}} \newcommand{\Sact}{{\mathcal{S}}} \newcommand{\T}{{\mathcal{T}}} \renewcommand{\L}{{\mathcal{L}}} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\R}{\mathbb{R}} \newcommand{\1}{\mathbb{1}} \DeclareMathOperator{\tr}{tr}$