The theory of causal fermion systems represents a novel approach to fundamental physics and is a promising candidate for a unified physical theory. This book offers a comprehensive overview of the theory, structured in four parts: the first lays the necessary mathematical and physical foundations; the second offers an introduction to the theory and the causal action principle; the third describes the mathematical tools for analyzing causal fermion systems; and the fourth gives an outlook on the key physical applications. With relevance across mathematical and theoretical physics, the book is aimed at graduate students and researchers interested in novel approaches to the structure of spacetime and alternative perspectives to the more established quantum field theories. It can be used for advanced courses in the subject or as a reference for research and self-guided study. Exercises are included at the end of each chapter to build and develop key concepts.
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A comprehensive resource offering an in-depth introduction to the theory of causal fermion systems, a novel approach to fundamental physics.
How to use this book; Part I. Physical and Mathematical Background:
1.
Physical preliminaries;
2. Mathematical preliminaries;
3. Elements of
operator theory;
4. Spinors in curved spacetime; Part II. Causal Fermion
Systems: Fundamental Structures:
5. A brief introduction to causal fermion
systems;
6. Causal variational principles;
7. The Euler-Lagrange equations;
8. The linearized field equations;
9. Surface layer integrals and
conservation laws;
10. Positive functionals;
11. Topological and geometric
structures; Part III. Mathematical Methods and Analytic Constructions:
12.
Measure-theoretic methods;
13. Methods of hyperbolic partial differential
equations;
14. Energy methods for the linearized field equations;
15.
Functional analytic methods in spacetime;
16. Fourier methods;
17. Methods of
scattering theory;
18. Methods of perturbation theory;
19. Methods of
microlocal analysis; Part IV. Applications and Outlook:
20. A few explicit
examples of causal variational principles;
21. Basics on the continuum limit;
22. Connection to quantum field theory; Appendix. The spin coefficients;
References; Notation index in order of appearance; Notation index
thematic order; Subject index.
Felix Finster is Full Professor of Mathematics at the University of Regensburg. Following postdoctoral research at Harvard University, he was a member of the Max Planck Institute for Mathematics in the Sciences in Leipzig for four years. He has been at the University of Regensburg since 2002 and works on general relativity and quantum field theory. Sebastian Kindermann is currently a teacher at the Comenius Gymnasium in Deggendorf, Germany. He studied physics and mathematics at the University of Regensburg, completing his master's degree in 2020. Jan-Hendrik Treude wrote his PhD thesis at the University of Regensburg, graduating in 2015. Currently, he is Fachbereichsreferent (manager of department) of the Department of Mathematics and Statistics at the University of Konstanz, Germany.
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