# The Theory of Causal Fermion Systems

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

## 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
• For how to get from discrete spacetime to causal fermion systems: see [pfp06, Preface to second online edition]
• 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)
• Continuity of  wave functions: YouTube, PDF
• The fermionic projector: YouTube (introductory video)
• Connection to wave evaluation operator:  [cfs16, Section 1.1.4]
• Krein structure: [cfs16, Section 1.1.5]

## Correspondence to the Minkowski Vacuum

• Minkowski space as a causal fermion system: YouTube (introductory video)
• Introducing an ultraviolet regularization
• see [cfs16, Sections 1.2.1 and 1.2.2] or [intro, Section 4.3]
• Correspondence of spacetime: YouTube, PDF
• 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
• 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
• For a detailed treatment of constraints see [lagrange12].
• From the causal action principle to causal variational principles: YouTube, PDF
• Derivation of the Euler-Lagrange equations: YouTube, PDF
• see also [intro, Sections 6.1 and 6.2] and [jet16, Section 2.2].
• For details on jet spaces see [FK18, Section 2.2] and [FL21, Section 5].

## The Linearized Field Equations

• Short introduction: YouTube (introductory video)
• Derivation of the linearized field equations: YouTube, PDF
• see [intro, Section 6.3]
• For a detailed treatment in non-smooth setting see [jet16].
• Commutator jets and inner solutions:  YouTube
• see [intro, Sections 7.5 and Section 7.6]

## Surface Layer Integrals

• Short introduction: YouTube (introductory video)
• The nonlinear surface layer integral:  YouTube
• Two-dimensional surface layer integrals:  YouTube

## Existence Theory for Minimizing Measures

• Overview of existence theory: YouTube (introductory video)
• Measure-theoretic tools:
• The Banach-Alaoglu theorem: see [intro, Section 10.2]
• The Riesz representation theorem: see [intro, Section 10.3]
• The Radon-Nikodym theorem: see [intro, Section 10.4]
• Existence of minimizers in the compact setting
• see [intro, Section 10.5]
• Existence of minimizers in the non-compact setting
• 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].

## The Dynamical Wave Equation

• Short introduction: YouTube (introductory video)
• Some material from [FKO21]

## Topological Spinor Bundles

• Some examples and selected topics from [topology14]

## Basics on the Continuum Limit

• Short introduction: YouTube (introductory video)
• The causal perturbation expansion
• The light-cone expansion
• 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 equationsYouTube (introductory video)
• see [cfs16, Sections 4.5 and 4.9]

## Connection to Quantum Field Theory

• Short introduction: YouTube (introductory video)
• Some material from [FK18] and [FK21]

The online course is still under construction.

### Felix Finster

Lecturer of course