Causal Fermion Systems: An Introduction to Fundamental Structures, Methods and Applications

TL;DR

The book introduces causal fermion systems as a unified framework for fundamental physics, recovering quantum mechanics, general relativity, and quantum field theory in suitable limits via the causal action principle. It builds a robust mathematical apparatus to describe non-smooth and quantum geometries, including measure theory, operator theory, spinor geometry, and curved-spacetime Dirac operators, organized into four parts: background, core structures, analytical toolbox, and applications. A central aim is to provide a toolbox of rigorous methods for analyzing causal fermion systems and their dynamics, with extensive physical and mathematical preliminaries to support rigorous study. The text targets both mathematicians and physicists, offering detailed derivations, examples, and exercises to bridge disciplines and prepare readers for research in this evolving framework.

Abstract

This textbook introduces the basic concepts of the theory of causal fermion systems, a recent approach to the description of fundamental physics. The theory yields quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries and "quantum geometries." The dynamics is described by a novel variational principle, the causal action principle. The book includes a detailed summary of the mathematical and physical preliminaries. It explains the physical concepts behind the causal fermion system approach from the basics. Moreover, all the mathematical objects and structures are introduced step by step. The mathematical methods used for the analysis of causal fermion systems and the causal action principle are introduced in depth. Many examples and applications are worked out. The textbook is addressed to master and graduate students in mathematics or physics. Furthermore, it serves as a reference work for researchers working in the field.

Figures (29)

• Figure 1.1: The causal structure of Minkowski space.
• Figure 1.2: A spatially compact Dirac solution.
• Figure 1.3: The Dirac sea with particles and anti-particles.
• Figure 2.4: An open set $\Omega \subset E$.
• Figure 2.5: The Hausdorff property.

Theorems & Definitions (259)

• Lemma 1.3.1
• proof
• Definition 2.1.1
• Definition 2.1.2
• Lemma 2.1.3
• Definition 2.1.4
• Definition 2.2.1
• Definition 2.2.2
• Definition 2.2.3
• Theorem 2.2.4