# The Theory of Causal Fermion Systems

## Learning

## Introductory Videos

These videos describe the content of corresponding websites. They are intended to give a nontechnical introduction and an impressionistic overview.

- Basic definitions: YouTube
- Generalizations and special cases: YouTube
- Existence theory: YouTube
- Spin spaces and physical wave functions: YouTube
- The fermionic projector: YouTube
- Geometric structures: YouTube
- Surface layer integrals: YouTube
- The Euler-Lagrange equations: YouTube
- The linearized field equations: YouTube
- The dynamical wave equation: YouTube
- Construction of linear fields and waves: YouTube
- Minkowski space as a causal fermion system: YouTube
- The continuum limit: YouTube
- Connection to quantum field theory: YouTube
- Connection to general relativity: YouTube

The online course originates from online lectures given at Universität Regensburg in the summer term 2021. It includes introductory videos, more specialized lectures, exercises and many links to books and articles.

- Physical preliminaries
- Mathematical preliminaries
- Basic structures
- Correspondence to the Minkowski vacuum
- The Euler-Lagrange equations
- The linearized field equations
- Surface layer integrals
- Existence theory for minimizing measures
- The Cauchy problem for the linearized field equations
- The fermionic projector in an external field
- Basics on the continuum limit
- Topological spinor bundles
- Geometric structures and connection to Lorentzian spin geometry
- The dynamical wave equation
- Connection to quantum field theory

## Literature on Causal Fermion Systems

## Introductory and survey articles

- F. Finster,
notes of invited talk given at the 16th Marcel Grossmann Meeting, July 2021**Causal fermion systems: Classical gravity and beyond,**

This article gives a review on the status of the theory of causal fermion systems with regard to the gravitational interaction. - F. Finster, M. Jokel,
in “Progress and Visions in Quantum Theory in View of Gravity,” Birkhäuser (2020) 63-92**Causal fermion systems: An elementary introduction to physical ideas and mathematical concepts,**

This article gives an elementary introduction to the theory of causal fermion systems, with a focus on the underlying physical ideas and the conceptual and mathematical foundations. - F. Finster,
**Causal fermion systems: Discrete space-times, causation and finite propagation speed**, J. Phys.: Conf. Ser.**1275**(2019) 012009

These are notes of a talk for physicists. The focus is on discrete space-time structures. The linearized field equations and surface layer integrals are explained in this simpler discrete setting. - F. Finster,
**Causal fermion systems: A primer for Lorentzian geometers**, J. Phys.: Conf. Ser.**968**(2018) 012004

This article is addressed to Lorentzian geometers or readers familiar with general relativity. It is outlined how a causal fermion system can be constructed in curve space-time, and how it enodes the geometric structures. - F. Finster, J. Kleiner,
**Causal fermion systems as a candidate for a unified physical theory**, J. Phys.: Conf. Ser.**626**(2015) 012020

This article gives a non-technical introduction. It gives an outline of the continuum limit, and discusses the connections to quantum field theory and the foundations of quantum theory. - F. Finster, A. Grotz, D. Schiefeneder,
**Causal fermion systems: A quantum space-time emerging from an action principle**, in “Quantum Field Theory and Gravity,” Birkhäuser (2012) 157-182

This is the article where causal fermion systems were first introduced. - F. Finster,
**A formulation of quantum field theory realizing a sea of interacting Dirac particles**, Lett. Math. Phys.**97**(2011) 165-183

This article outlines the connection to quantum field theory and the Dirac sea picture. - F. Finster,
**From discrete space-time to Minkowski space: Basic mechanisms, methods and perspectives**, in “Quantum Field Theory,” Birkhäuser (2009) 235-259 - F. Finster,
**Fermion systems in discrete space-time**, J. Phys.: Conf. Ser.**67**(2007) 012048

The causal action is introduced for discrete space-times. - F. Finster,
**The principle of the fermionic projector: An approach for quantum gravity?**, in “Quantum Gravity,” Birkhäuser (2006) 263-281

In this early paper, the connection to quantum gravity is discussed.

## Books

- F. Finster, N. Kamran,
**Spinors on Singular Spaces and the Topology of Causal Fermion Systems**, Mem. Amer. Math. Soc.**259**(2019), no. 1251, v+83 pp

This thin book introduces causal fermion systems from the perspective of topology and differential geometry. The guiding theme is to encode the topology and geometry in terms of linear operators on a Hilbert space. This is explained and illustrated in many simple examples. The causal action principle, however, is not covered. - F. Finster,
**The Continuum Limit of Causal Fermion Systems**, Fundamental Theories of Physics**186**, Springer, 2016

This is the only textbook on and at present the best reference to causal fermion systems. Chapter 1 is a general introduction. Chapter 2 provides the computational tools needed for analyzing the continuum limit. In Chapters 3-5 the continuum limit is worked out for Dirac systems of increasing complexity, giving all the interactions of the standard model plus classical gravity. - F. Finster,
**The Principle of the Fermionic Projector**, AMS/IP Studies in Advanced Mathematics Series**35**(2006)The “principle of the fermionic projector” was a major preliminary step towards developing causal fermion systems. It states the general mathematical structure of the causal action principle in the setting where wave functions are varied in a discrete space-time. Moreover, the causal action is stated for the first time, but ony as an example and without all the constraints. In the more recent foreword, the developments leading to causal fermion systems are outlined.

## Lecture Notes

- F. Finster, S. Kindermann, J.-H. Treude,
**An Introductory Course on Causal Fermion Systems**

This book draft is based on notes of lectures on causal variational principles given at the University of Regensburg in the spring terms 2017 and 2021. The intention is to address both physicists and mathematicians interested in the subject. Part 3 can be regarded as a mathematical toolbox which provides useful methods and techniques.

## Videos on Causal Fermion Systems

## International Spring School "Causal fermion systems," Regensburg, February 2018

See here for more details on the spring school.

**Preliminary Lecture.**Mathematical and physical preliminaries (lecturer: A. Platzer)**Lecture 1.**Introduction to the theory and inherent structures (F. Finster)**Lecture 2.**Correspondence to Minkowski space (F. Finster)**Lecture 3.**The field equations in the jet formulation (J. Kleiner)**Lecture 4.**Conserved surface layer integrals (J. Kleiner)**Lecture 5.**Fermionic and bosonic jets in Minkowski space and surface layer integrals (F. Finster)**Lecture 6.**Outline of the continuum limit (J. Kleiner)**Lecture 7.**Correspondence to classical field equations (F. Finster)**Lecture 8.**Existence theory for causal variational principles (J. Kleiner)**Lecture 9.**Geometric structures (F. Finster)**Lecture 1o.**Classical and quantum space-times (F. Finster)**Lecture 11.**Correspondence to quantum field theory (F. Finster)

## International Spring School "Causal fermion systems," Regensburg, February 2016

See here for more details on the spring school.

**Preliminary Lecture 1.**Physical preliminaries (lecturer: J. Kleiner)**Preliminary Lecture 2.**Mathematical preliminaries (J.-H. Treude)**Lecture 1.**Causal fermion systems: the abstract framework (F. Finster)**Lecture 2.**Correspondence to Minkowski space I (F. Finster)**Lecture 3.**Correspondence to Minkowski space I and overview of the analysis in the continuum limit (F. Finster)**Lecture 4.**Causal perturbation expansion and light cone expansion (F. Finster)**Lecture 5.**The continuum limit I (F. Finster)**Lecture 6.**The Euler-Lagrange equations (F. Finster)**Lecture 7.**The continuum limit II (F. Finster)**Lecture 8.**Connection to the standard model (F. Finster)**Lecture 9.**An introduction to spinors in curved space-time (F. Finster)**Lecture 1o.**Connections to the Einstein equations and geometric structures of a causal fermion system (F. Finster)**Discussion.**Exerpts from one discussion session (moderated by J. Kleiner)

## Public Talks and Discussions

**Laws of Nature, Discussion series on Quantum Theory and Relativity**, February 2022: YouTube, Slides**Diskussionsreihe “Was ist wirklich?”**, Kausale Fermionsysteme: Eine neue vereinheitlichte Theorie entsteht, Universität Regensburg, February 2017 (in German)

## Live Recordings of Talks at Conferences, Workshops and Seminars

Here are a few selected videos of talks given at conferences, workshops or seminars:

**Workshop “Mathematical and Conceptual Aspects of Quantum Theory,”**Banff International Research Station, Oaxaca, México, F. Finster, An introduction to causal fermion systems and the causal action principle, June 2022 (Slides)- F. Finster, Quantum states of causal fermion systems and collapse,
**Mathematical Physics Colloquium Regensburg-München**(online via Zoom), February 2021 (Slides) - F. Finster, Causal fermion systems and the causal action principle,
**Mathematical Physics Colloquium**at Universität Tübingen (online via Zoom), July 2020 (Slides) - F. Finster, A positive mass theorem for static causal fermion systems,
**Oberseminar “Geometric Analysis”**at Universität Tübingen (online via Zoom), July 2020 (Slides) **Workshop “Information theoretic foundations for physics,”**F. Finster, Causal fermion systems from an information theoretic perspective, Perimeter Institute, Waterloo, Canada, May 2015**Conference “Quantum Mathematical Physics – A Bridge between Mathematics and Physics,”**F. Finster, Causal fermion systems as an approach to quantum theory, Regensburg, October 2014