# The Theory of Causal Fermion Systems

## Online Course on Causal Fermion Systems

### Content

## Online Course on Causal Fermion Systems

Welcome to the **online lecture “An Introduction to Causal Fermion Systems”** given at Universität Regensburg in the summer term 2021.

You see the material of the current week in red.

The videos which still need to be added are indicated in grey.

The **question hour** takes place every *Wednesday at 10:15* (first on April 14; always German time).

The **exercise class** takes place every *Thursday at 12:15*. On April 15 there will be a Präsenzübung (see GQE0). The first exercise sheet will be uploaded on April 15. The solutions must be submitted within seven days, at latest on Thursday at 12:00. Please send your solutions per Email to Marco Oppio.

If you are a student enrolled in Regensburg, then please register to the course on Grips. On the Grips page, you will find the Meeting-IDs and passwords for the question hour and the exercises class.

**You can also participate if you are not a student** (or a student not enrolled in Regensburg). In this case, please let us know via e*mail*. We will send you the Meeting-IDs and passwords for the question hour and the exercises class.

Here are the **guiding questions** and the **exercises** uploaded so far:

Here are the notes of the question hours: 14.4.

## Physical Preliminaries

**Minkowski space**: YouTube, PDF, GQE0- suggested literature:
*W. Rindler*, “Introduction to Special Relativity,” Clarendon Press, 1982*G.N. Naber*, “The Geometry of Minkowski Spacetime,” Dover Publications, 1992

- suggested literature:
**The Dirac equation**: YouTube, PDF, GQE0- suggested literature:
*J.D. Bjorken and S.D. Drell*, “Relativistic Quantum Mechanics,” McGraw-Hill, 1964*B. Thaller*, “The Dirac Equation,” Springer, 1992*M.E. Peskin and D.V. Schroeder*, “An Introduction to Quantum Field Theory,” Westview Press, 1995

- suggested literature:
**Dirac spinors and wave functions:**YouTube, PDF**Dirac’s hole theory and the Dirac sea:**YouTube, PDF

## Mathematical Preliminaries

**Basics on topology:**YouTube, PDF- suggested literature:
*H. Schubert*, “Topology,” Oldbourne, 1967*K. Jänich*, “Topology,” Springer 1984

- suggested literature:
**Basics on abstract measure theory:**YouTube, PDF- suggested literature:
*W. Rudin*, “Real and Complex Analysis,” McGraw Hill, 1966*P.R. Halmos*, “Measure Theory,” Springer, 1974

- suggested literature:
**Hilbert spaces and linear operators:**YouTube, PDF- suggested literature:
*W. Rudin*, “Real and Complex Analysis,” McGraw Hill, third ed. 1987*M. Reed and B. Simon*, “Methods of Modern Mathematical Physics. I, Functional analysis,” Academic Press, 1980*P. Lax*, “Functional Analysis,” Wiley-Interscience, 2002

- suggested literature:
**Distributions and Fourier transform:**YouTube, PDF- suggested literature:
*J. Rauch*, “A crash course in distribution theory,” Appendix A in “Partial Differential Equations,” 2nd edition, Springer, 1997*F.G. Friedlander and M. Joshi*, “Introduction to the Theory of Distributions,” Cambridge University Press, 1998

- suggested literature:
**Manifolds and vector bundles:**YouTube- suggested literature:
*S. Lang*, “Introduction to Differentiable Manifolds,” 2nd edition, Springer, 2002

- suggested literature:

## Basic Structures

**Basic definitions:**YouTube (introductory video)**Getting familiar with the definitions**:- Simple examples: [cfs16, Exercises 1.1 and 1.6]
- Why this form of the causal action? YouTube, PDF
- Necessity of the constraints: YouTube, PDF
- see also [cfs16, Exercises 1.2 and 1.4] or [intro, Section 10.1.]
- For other lower bounds of the causal Lagrangian involving the local trace: see [discrete05, Proposition 4.3]
- For local Hölder continuity of the causal Lagrangian: see [FL21, Section 5.1]

**Spin spaces and physical wave functions:**YouTube (introductory video)**The fermionic projector:**YouTube (introductory video)

## Correspondence to the Minkowski Vacuum

**Minkowski space as a causal fermion system:**YouTube (introductory video)**Introducing an ultraviolet regularization****Correspondence of****spacetime**: YouTube, PDF- see also [cfs16, Section 1.2.3]
- For details on analytic properties of $F^\varepsilon(x)$ see [Op20].
- For interpretation of local correlation operators and comparison to ETH approach see [FFOP20].
- For related operator algebras see [FO20].

**Correspondence of spinors and physical wave****functions**: YouTube, PDF- see also [cfs16, Section 1.2.4]

**Correspondence of causal structure**: see [cfs16, Section 1.2.5]

## The Euler-Lagrange Equations

**Causal variational principles:**YouTube (introductory video)**Short introduction and overview:**YouTube (introductory video)**The local trace is constant**: YouTube, PDF- see also [cfs16, Proposition 1.4.1]
- For a detailed treatment of constraints see [lagrange12].

**From the causal action principle to causal variational principles**: YouTube, PDF- see for example [FKO21, Section 2.3]
- For the proof that $\F^\text{reg}$ is a manifold of operators see [FKi19] or [FL21].

**Derivation of the Euler-Lagrange equations**: YouTube, PDF

## The Linearized Field Equations

## Existence Theory for Minimizing Measures

**Overview of existence theory:**YouTube (introductory video)**Measure-theoretic tools:****Existence of minimizers in the compact setting**- see [intro, Section 10.5]

**Existence of minimizers in the non-compact setting**- see [FL20]

**Existence of minimizers in the finite-dimensional setting**- see [intro, Sections 10.6 and 10.7]

## The Fermionic Projector in an External Field

- Some functional analytic constructions from [intro, Chapters 14 and 15]; see also [infinite13].

## Geometric Structures and Connection to Lorentzian Spin Geometry

**Short introduction:**YouTube (introductory video)**Abstract construction of the spin connection**: see [lqg11, Section 3] and [intro, Section 9.2].**Causal fermion systems in globally hyperbolic spacetimes**: see [nrstg17, Section 1] and [intro, Section 9.1].**Correspondence to Lorentzian spin geometry**: see [lqg11, Section 5] and [intro, Section 9.3].

## Topological Spinor Bundles

- Some examples and selected topics from [topology14]

## Basics on the Continuum Limit

**Short introduction**: YouTube (introductory video)**The causal perturbation expansion**- see [cfs16, Section 2.1]

**The light-cone expansion**- see [cfs16, Section 2.2]

**The formalism of the continuum limit**- see [cfs16, Sections 2.3 and 2.6]

**Overview of results of the analysis of the continuum limit**: YouTube- see [cfs16, Sections 3.1, 4.1 and 5.1]

**Derivation of the Einstein equations**: YouTube (introductory video)- see [cfs16, Sections 4.5 and 4.9]

*The online course is still under construction.*

### Felix Finster

Lecturer of course