# The Theory of Causal Fermion Systems

## Online Course on Causal Fermion Systems

### Content

## Online Course on Causal Fermion Systems

This online course is based on the online lecture **“An Introduction to Causal Fermion Systems”** given at Universität Regensburg in the summer term 2021.

In order to get a first impression of what causal fermion systems are about, it might be a good idea to watch the **talks on causal fermion systems** given in **“Laws of Nature, Discussion series on Quantum Theory and Relativity”** in February 2022.

## Physical Preliminaries

**Minkowski space**: YouTube, PDF, GQE1- 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, GQE1- 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, GQE2**Dirac’s hole theory and the Dirac sea**: YouTube, PDF, GQE2

## Mathematical Preliminaries

**Basics on topology**: YouTube, PDF, GQE3- suggested literature:
*H. Schubert*, “Topology,” Oldbourne, 1967*K. Jänich*, “Topology,” Springer 1984

- suggested literature:
**Basics on abstract measure theory**: YouTube, PDF, GQE3- 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, GQE4- 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, GQE4- suggested literature:
*J. Rauch*, Sections 2.1, 2.2 and 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, PDF, GQE5- suggested literature:
*S. Lang*, “Introduction to Differentiable Manifolds,” 2nd edition, Springer, 2002*J.W. Milnor and J.D. Stasheff*, “Characteristic classes,” §1 and §2, Princeton University Press, 1974

- suggested literature:

## Basic Structures

**Basic definitions**: YouTube, website, GQE6 (introductory video)**Getting familiar with the basic definitions**:- Simple examples: [cfs16, Exercises 1.1 and 1.6]
- Why this form of the causal action? YouTube, PDF, GQE6
- Necessity of the constraints: YouTube, PDF, GQE6
- see also [cfs16, Exercises 1.2 and 1.4] or [intro, Section 12.4.]
- 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, website, GQE7 (introductory video)- Continuity of wave functions: YouTube, PDF, GQE7
- see also [cfs16, Section 1.1.4]

- For the connection to topological vector bundles see [topology14, Section 3.2].

- Continuity of wave functions: YouTube, PDF, GQE7
**The fermionic projector:**YouTube, website, GQE7 (introductory video)

## Correspondence to the Minkowski Vacuum

**Minkowski space as a causal fermion system**: YouTube, website, GQE8 (introductory video)**Introducing an ultraviolet regularization****Correspondence of****spacetime**: YouTube, PDF, GQE8- see also [cfs16, Section 1.2.3]
- For details on analytic properties of $F^\varepsilon(x)$ see [Op20].
- For the 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, GQE8- see also [cfs16, Section 1.2.4]

**Correspondence of the causal structure**: GQE8- see [cfs16, Section 1.2.5]

## The Euler-Lagrange Equations

**Causal variational principles**: YouTube, website, GQE9 (introductory video)- see also [intro, Sections 6.1-6.3]

**Short introduction and overview**: YouTube, website, GQE9 (introductory video)**The local trace is constant**: YouTube, PDF, GQE9**From the causal action principle to causal variational principles**: YouTube, PDF, GQE9- see [intro, Section 6.5] or for example [FKO21, Section 2.3]
- For the proof that $\F^\text{reg}$ is a manifold of operators see [FKi19] or [FL21].
- For a detailed treatment of the constraints see [lagrange12].

**Derivation of the Euler-Lagrange equations**: YouTube, PDF, GQE10

## Surface Layer Integrals

**Short introduction:**YouTube, website, GQE11 (introductory video)**Noether-like theorems**: YouTube, PDF, GQE11**General conservation laws**: YouTube, PDF, GQE11**The conserved one-form and the commutator inner product**: YouTube, PDF, GQE12**The symplectic form and the surface layer inner product**: YouTube, PDF, GQE12**The nonlinear surface layer integral**: YouTube, PDF, GQE12**Two-dimensional surface layer integrals**: YouTube, PDF, GQE12

## Existence Theory for Minimizing Measures

**Overview of existence theory**: YouTube, website, GQE13 (introductory video)**Measure-theoretic tools**: GQE13- The Banach-Alaoglu theorem: see for example [intro, Section 12.1]
- The Riesz representation theorem: see [intro, Section 12.2] or for more details Section 1.8 in
*L.C. Evans, R.F. Gariepy*, “Measure Theory and Fine Properties of Functions,” CRC Press, 1992 - The Radon-Nikodym theorem: see [intro, Section 12.5] or Section VII.31 in
*P.R. Halmos*, “Measure Theory,” Springer, 1974

**Existence of minimizers for causal variational principles in the compact setting**: YouTube, PDF, GQE13**Existence of minimizers for the causal action principle in the finite-dimensional setting**: YouTube, PDF, GQE14- see [intro, Sections 12.6 and 12.7] or [continuum08, Section 2]

**Existence of minimizers for causal variational principles in the non-compact setting**: YouTube, PDF, GQE14- see [FL20, Section 4]

## The Cauchy Problem for the Linearized Field Equations

## The Fermionic Projector in an External Field

**General concepts**: YouTube, PDF, GQE16**The mass oscillation properties**: YouTube, PDF, GQE16- see [intro, Section 15.2]
- See [infinite13] for globally hyperbolic spacetimes.

**Construction of the fermionic signature operator**: YouTube, PDF, GQE17- see [intro, Section 15.3]
- See again [infinite13] for globally hyperbolic spacetimes.

**Proof of the strong mass oscillation property in the Minkowski vacuum**: YouTube, PDF, GQE16- see [intro, Section 16.4]

**Overview of mass oscillation properties in various spactimes**: YouTube, PDF

## Basics on the Continuum Limit

**Short introduction**: YouTube, website, GQE17 (introductory video)**The causal perturbation expansion**: YouTube, PDF, GQE17**The light-cone expansion**: YouTube, PDF, GQE18**The formalism of the continuum limit**: YouTube, PDF, GQE19- see [cfs16, Sections 2.3 and 2.6]

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

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

## Topological Spinor Bundles

**Introduction and overview**: YouTube, PDF, GQE20**Topological fermion systems and simple examples**: see [topology14, Section 2]**Connection to topological vector bundles**: see [topology14, Section 3.2]**Clifford sections and topological spinor bundles**: see [topology14, Section 4]**The tangent cone measure**: see [topology14, Section 6.1]**Tangential Clifford sections**: see [topology14, Section 6.2]

## Geometric Structures and Connection to Lorentzian Spin Geometry

**Short introduction:**YouTube, website, GQE21 (introductory video)**Abstract construction of the spin connection**: see [lqg11, Section 3] and [intro, Section 11.2].**Causal fermion systems in globally hyperbolic spacetimes**: see [nrstg17, Section 1] and [intro, Section 11.3].**Correspondence to Lorentzian spin geometry**: see [lqg11, Section 5] and [intro, Section 11.3].

*The online course is still under construction.*

### Felix Finster

Lecturer of course