E-Book Content
The Principle of the Fermionic Projector
arXiv:hep-th/0001048v5 8 Oct 2009
Chapters 0-4 Felix Finster
Abstract The “principle of the fermionic projector” provides a new mathematical framework for the formulation of physical theories and is a promising approach for physics beyond the standard model. The book begins with a brief review of relativity, relativistic quantum mechanics and classical gauge theories, with the emphasis on the basic physical concepts and the mathematical foundations. The external field problem and Klein’s paradox are discussed and then resolved by introducing the so-called fermionic projector, a global object in space-time which generalizes the notion of the Dirac sea. The mathematical core of the book is to give a precise definition of the fermionic projector and to employ methods of hyperbolic differential equations for its detailed analysis. The fermionic projector makes it possible to formulate a new type of variational principles in space-time. The mathematical tools for the analysis of the corresponding Euler-Lagrange equations are developed. A particular variational principle is proposed which gives rise to an effective interaction showing many similarities to the interactions of the standard model. The main chapters of the book are easily accessible for beginning graduate students in mathematics or physics. Several appendices provide supplementary material which will be useful to the experienced researcher.
Contents Preface
vii
Preface to the Online Edition
ix
Chapter 0. The Principle of the Fermionic Projector – A New Mathematical Model of Space-Time
1
Chapter 1. Preliminaries 1.1. Relativity 1.2. Relativistic Quantum Mechanics 1.3. Fock Space Quantization of the Free Dirac Field 1.4. Classical Gauge Theories 1.5. Dirac Spinors in Curved Space-Time
7 7 10 16 19 21
Chapter 2. The Fermionic Projector in the Continuum 2.1. The External Field Problem 2.2. The Causal Perturbation Expansion 2.3. Definition of the Fermionic Projector 2.4. Interpretation and Consequences 2.5. The Light-Cone Expansion 2.6. Normalization of the Fermionic States
29 29 32 41 45 49 62
Chapter 3. The Principle of the Fermionic Projector 3.1. Connection between Local Gauge Freedom and the Measurability of Position and Time 3.2. Projection on Fermionic States 3.3. Discretization of Space-Time 3.4. The Principle of the Fermionic Projector 3.5. A Variational Principle 3.6. Discussion
71 71 74 76 78 78 85
Chapter 4. The Continuum Limit 4.1. The Method of Variable Regularization 4.2. The Regularized Product P (x, y) P (y, x) in the Vacuum 4.3. The Regularized Vacuum on the Light Cone, Scalar Component 4.4. The Regularized Vacuum on the Light Cone, Vector Component 4.5. The General Formalism
89 90 94 96 104 111
Chapter 5. The Euler-Lagrange Equations in the Vacuum 5.1. The Fermion Configuration of the Standard Model 5.2. The General Two-Point Action 5.3. The Spectral Decomposition of P (x, y) P (y, x)
123 123 124 126
v
vi
CONTENTS
5.4. 5.5. 5.6.
Strong Spectral Analysis of the Euler-Lagrange Equations Motivation of the Lagrangian, the Mass Degeneracy Assumption Stability of the Vacuum
132 134 138
Chapter 6. The Dynamical Gauge Group 6.1. The Euler-Lagrange Equations to Highest Degree on the Light Cone 6.2. The Gauge Terms in the Euler-Lagrange Equations
149 150 156
Chapter 7. Spontaneous Block Formation 7.1. The Partial Trace and the Dynamical Mass Matrices 7.2. Analysis of Degeneracies 7.3. The Dynamical Mass Matrices in the Quark and Neutrino Blocks
167 168 171 185
Chapter 8. The Effective Gauge Group 8.1. The Chiral Transformation in the Quark Blocks 8.2. The Chiral Transformation in the Lept