# 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.

→ overview of the fundamental mathematical structures

→ basic definitions

$
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\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}
$

### Felix Finster

Author