The Theory of Causal Fermion Systems

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
• 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
• Dirac spinors and wave functions: YouTube, PDF, GQE2
• Dirac’s hole theory and the Dirac sea: YouTube, PDF, GQE2

Mathematical Preliminaries

• Basics on topologyYouTube, PDF, GQE3
• suggested literature:
• H. Schubert, “Topology,” Oldbourne, 1967
• K. Jänich, “Topology,” Springer 1984
• Basics on abstract measure theoryYouTube, PDF, GQE3
• suggested literature:
• W. Rudin, “Real and Complex Analysis,” McGraw Hill, 1966
• P.R. Halmos, “Measure Theory,” Springer, 1974
• Hilbert spaces and linear operatorsYouTube, 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
• Distributions and Fourier transformYouTube, 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
• Manifolds and vector bundlesYouTube, 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

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
• For how to get from discrete spacetime to causal fermion systems: see [pfp06, Preface to second online edition]
• 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
• For the connection to topological vector bundles see [topology14, Section 3.2].
• The fermionic projector: YouTube, website, GQE7 (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, website, GQE8 (introductory video)
• Introducing an ultraviolet regularization
• see [cfs16, Sections 1.2.1 and 1.2.2] or [intro, Section 5.4]
• Correspondence of spacetime: YouTube, PDF, GQE8
• 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
• Correspondence of the causal structure: GQE8
• see [cfs16, Section 1.2.5]

The Linearized Field Equations

• Short introduction: YouTube, website, GQE10 (introductory video)
• Derivation of the linearized field equations: YouTube, PDF, GQE10
• see [intro, Section 8.1]
• For a detailed treatment in the non-smooth setting see [jet16].
• Commutator jets and inner solutions: YouTube, PDF, GQE11
• see [intro, Sections 8.2 and Section 8.3] and [FKO21, Section 3.1]
• For details on decay and regularity conditions on inner solutions see [FK18, Section 3].

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
• see [intro, Section 12.3]
• For lower semi-continuous Lagrangians see [FL20, Section 3.2]
• 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 Fermionic Projector in an External Field

• General concepts: YouTube, PDF, GQE16
• see [cfs16, Section 2.1.1] and [intro, Section 15.1]
• 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
• For external potentials in Minkowski space see [intro, Section 17] or [FMRö15, Section 4].
• For spacetimes with symmetries see [FR17].
• For various examples of curved spacetimes see [FMRö16, FR16, DFMRa19]
• For the mass decomposition in a black hole spacetime see [FR18].

Basics on the Continuum Limit

• Short introduction: YouTube, website, GQE17 (introductory video)
• The causal perturbation expansion: YouTube, PDF, GQE17
• see [cfs16, Section 2.1]
• For more details on the functional calculus in spacetime see [norm14].
• The light-cone expansion: YouTube, PDF, GQE18
• see [cfs16, Section 2.2]
• More details can be found in [light98].
• For a non-perturbative proof of the Hadamard property see [FMRö15, Section 5].
• For the general context of the Hadamard condition and quasi-free states see for example the introduction to [FMRö15] and the references therein.
• 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 equationsYouTube, 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 Dynamical Wave Equation

• Short introduction: YouTube, website, GQE22 (introductory video)
• The extended Hilbert space: see [FKO21, Section 4]
• Derivation of the dynamical wave equation: see [FKO21, Section 5]
• The Cauchy problem and finite propagation speed: see [FKO21, Section 6.2]

Connection to Quantum Field Theory

• Short introduction: YouTube, website, GQE23 (introductory video)
• Linear dynamics on bosonic fock spaces: see [FK18, Section 7]
• The partition function: see [FK21, Section 3]
• Construction of a quantum state: see [FK21, Sections 4.1-4.4]

The online course is still under construction.

Felix Finster

Lecturer of course