The continuum limit of causal fermion systems

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Fundamental Theories of Physics 186 Felix Finster The Continuum Limit of Causal Fermion Systems From Planck Scale Structures to Macroscopic Physics Fundamental Theories of Physics Volume 186 Series editors Henk van Beijeren, Utrecht, The Netherlands Philippe Blanchard, Bielefeld, Germany Paul Busch, York, United Kingdom Bob Coecke, Oxford, United Kingdom Dennis Dieks, Utrecht, The Netherlands Bianca Dittrich, Waterloo, Canada Detlef Dürr, München, Germany Ruth Durrer, Genève, Switzerland Roman Frigg, London, United Kingdom Christopher Fuchs, Boston, USA Giancarlo Ghirardi, Trieste, Italy Domenico J W Giulini, Bremen, Germany Gregg Jaeger, Boston, USA Claus Kiefer, Köln, Germany Nicolaas P Landsman, Nijmegen, The Netherlands Christian Maes, Leuven, Belgium Mio Murao, Tokyo, Japan Hermann Nicolai, Potsdam, Germany Vesselin Petkov, Montreal, Canada Laura Ruetsche, Ann Arbor, USA Mairi Sakellariadou, London, United Kingdom Alwyn van der Merwe, Denver, USA Rainer Verch, Leipzig, Germany Reinhard Werner, Hannover, Germany Christian Wüthrich, Geneva, Switzerland Lai-Sang Young, New York City, USA The international monograph series “Fundamental Theories of Physics” aims to stretch the boundaries of mainstream physics by clarifying and developing the theoretical and conceptual framework of physics and by applying it to a wide range of interdisciplinary scientific fields Original contributions in well-established fields such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome The series also provides a forum for non-conventional approaches to these fields Publications should present new and promising ideas, with prospects for their further development, and carefully show how they connect to conventional views of the topic Although the aim of this series is to go beyond established mainstream physics, a high profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard More information about this series at Felix Finster The Continuum Limit of Causal Fermion Systems From Planck Scale Structures to Macroscopic Physics 123 Felix Finster Fakultät für Mathematik Universität Regensburg Regensburg Germany ISSN 0168-1222 Fundamental Theories of Physics ISBN 978-3-319-42066-0 DOI 10.1007/978-3-319-42067-7 ISSN 2365-6425 (electronic) ISBN 978-3-319-42067-7 (eBook) Library of Congress Control Number: 2016945853 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Preface This book is devoted to explaining how the causal action principle gives rise to the interactions of the standard model plus gravity on the level of second-quantized fermionic fields coupled to classical bosonic fields It is the result of an endeavor which I was occupied with for many years Publishing the methods and results as a book gives me the opportunity to present the material in a coherent and comprehensible way The four chapters of this book evolved differently Chapters and are based on the notes of my lecture “The fermionic projector and causal variational principles” given at the University of Regensburg in the summer semester 2014 The intention of this lecture was to introduce the basic concepts Most of the material in these two chapters has been published previously, as is made clear in the text by references to the corresponding research articles We also included exercises in order to facilitate the self-study Chapters 3–5, however, are extended versions of three consecutive research papers written in the years 2007–2014 (arXiv:0908.1542 [math-ph], arXiv: 1211.3351 [math-ph], arXiv:1409.2568 [math-ph]) Thus the results of these chapters are new and have not been published elsewhere Similarly, the appendix is formed of the appendices of the above-mentioned papers and also contains results of original research The fact that Chaps 3–5 originated from separate research papers is still visible in their style In particular, each chapter has its own short introduction, where the notation is fixed and some important formulas are stated Although this leads to some redundancy and a few repetitions, I decided to leave these introductions unchanged, because they might help the reader to revisit the prerequisites of each chapter We remark that, having the explicit analysis of the continuum limit in mind, the focus of this book is on the computational side This entails that more theoretical questions like the existence and uniqueness of solutions of Cauchy problems or the non-perturbative methods for constructing the fermionic projector are omitted To the reader interested in mathematical concepts from functional analysis and partial differential equations, we can recommend the book “An Introduction to the v vi Preface Fermionic Projector and Causal Fermion Systems” [FKT] The intention is that the book [FKT] explains the physical ideas in a non-technical way and introduces the mathematical background from a conceptual point of view It also includes the non-perturbative construction of the fermionic projector in the presence of an external potential and introduces spinors in curved space-time The present book, on the other hand, focuses on getting a rigorous connection between causal fermion systems and physical systems in Minkowski space Here we also introduce the mathematical tools and give all the technical and computational details needed for the analysis of the continuum limit With this different perspective, the two books should complement each other and when combined should give a mathematically and physically convincing introduction to causal fermion systems and to the analysis of the causal action principle in the continuum limit We point out that the connection to quantum field theory (in particular to second-quantized bosonic fields) is not covered in this book The reader interested in this direction is referred to [F17] and [F20] I would like to thank the participants of the spring school “Causal fermion systems” hold in Regensburg in March 2016 for their interest and feedback Moreover, I am grateful to David Cherney, Andreas Grotz, Christian Hainzl, Johannes Kleiner, Simone Murro, Joel Smoller and Alexander Strohmaier for helpful discussions and valuable comments on the manuscript Special thanks go to Johannes Kleiner for suggesting many of the exercises I would also like to thank the Max Planck Institute for Mathematics in the Sciences in Leipzig and the Center of Mathematical Sciences and Applications at Harvard University for hospitality while I was working on the manuscript I am grateful to the Deutsche Forschungsgemeinschaft (DFG) for financial support Regensburg, Germany May 2016 Felix Finster Contents Causal Fermion Systems—An Overview 1.1 The Abstract Framework 1.1.1 Basic Definitions 1.1.2 Space-Time and Causal Structure 1.1.3 The Kernel of the Fermionic Projector 1.1.4 Wave Functions and Spinors 1.1.5 The Fermionic Projector on the Krein Space 1.1.6 Geometric Structures 1.1.7 Topological Structures 1.2 Correspondence to Minkowski Space 1.2.1 Concepts Behind the Construction of Causal Fermion Systems 1.2.2 Introducing an Ultraviolet Regularization 1.2.3 Correspondence of Space-Time 1.2.4 Correspondence of Spinors and Physical Wave Functions 1.2.5 Correspondence of the Causal Structure 1.3 Underlying Physical Principles 1.4 The Dynamics of Causal Fermion Systems 1.4.1 The Euler-Lagrange Equations 1.4.2 Symmetries and Conserved Surface Layer Integrals 1.4.3 The Initial Value Problem and Time Evolution 1.5 Limiting Cases 1.5.1 The Quasi-free Dirac Field and Hadamard States 1.5.2 Effective Interaction via Classical Gauge Fields 1.5.3 Effective Interaction via Bosonic Quantum Fields Exercises 1 10 12 16 17 17 22 26 28 31 42 44 44 53 57 59 59 61 65 69 Computational Tools 2.1 The Fermionic Projector in an External Potential 2.1.1 The Fermionic Projector of the Vacuum 81 82 82 vii viii Contents 2.1.2 2.1.3 2.1.4 The External Field Problem Main Ingredients to the Construction The Perturbation Expansion of the Causal Green’s Functions 2.1.5 Computation of Operator Products 2.1.6 The Causal Perturbation Expansion 2.1.7 Introducing Particles and Anti-Particles 2.2 The Light-Cone Expansion 2.2.1 Basic Definition 2.2.2 Inductive Light-Cone Expansion of the Green’s Functions 2.2.3 Structural Results for Chiral Potentials 2.2.4 Reduction to the Phase-Free Contribution 2.2.5 The Residual Argument 2.2.6 The Non-causal Low Energy Contribution 2.2.7 The Non-causal High Energy Contribution 2.2.8 The Unregularized Fermionic Projector in Position Space 2.3 Description of Linearized Gravity 2.4 The Formalism of the Continuum Limit 2.4.1 Example: The ie-Regularization 2.4.2 Example: Linear Combinations of ie-Regularizations 2.4.3 Further Regularization Effects 2.4.4 The Formalism of the Continuum Limit 2.4.5 Outline of the Derivation 2.5 Computation of the Local Trace 2.6 Spectral Analysis of the Closed Chain 2.6.1 Spectral Decomposition of the Regularized Vacuum 2.6.2 The Double Null Spinor Frame 2.6.3 Perturbing the Spectral Decomposition 2.6.4 General Properties of the Spectral Decomposition 2.6.5 Spectral Analysis of the Euler-Lagrange Equations Exercises An Action Principle for an Interacting Fermion System and Its Analysis in the Continuum Limit 3.1 Introduction 3.2 An Action Principle for Fermion Systems in Minkowski Space 3.3 Assuming a Vacuum Minimizer 3.4 Introducing an Interaction 3.4.1 A Dirac Equation for the Fermionic Projector 3.4.2 The Interacting Dirac Sea 83 84 89 92 97 103 105 105 106 114 122 134 141 143 147 148 151 151 159 162 163 167 176 179 180 184 187 189 191 193 209 209 212 215 219 220 221 Contents 3.5 3.6 3.7 3.8 3.9 ix 3.4.3 Introducing Particles and Anti-Particles 3.4.4 The Light-Cone Expansion and Resummation 3.4.5 Clarifying Remarks 3.4.6 Relation to Other Approaches The Continuum Limit 3.5.1 Weak Evaluation on the Light Cone 3.5.2 The Euler-Lagrange Equations in the Continuum Limit The Euler-Lagrange Equations to Degree Five 3.6.1 The Vacuum 3.6.2 Chiral Gauge Potentials The Euler-Lagrange Equations to Degree Four 3.7.1 The Axial Current Terms and the Mass Terms 3.7.2 The Dirac Current Terms 3.7.3 The Logarithmic Poles on the Light Cone 3.7.4 A Pseudoscalar Differential Potential 3.7.5 A Vector Differential Potential 3.7.6 Recovering the Differential Potentials by a Local Axial Transformation 3.7.7 General Local Transformations 3.7.8 The Shear Contributions by the Local Axial Transformation 3.7.9 Homogeneous Transformations in the High-Frequency Limit 3.7.10 The Microlocal Chiral Transformation 3.7.11 The Shear Contributions by the Microlocal Chiral Transformation The Field Equations 3.8.1 The Smooth Contributions to the Fermionic Projector at the Origin 3.8.2 Violation of Causality and the Vacuum Polarization 3.8.3 Higher Order Non-Causal Corrections to the Field Equations 3.8.4 The Standard Quantum Corrections to the Field Equations 3.8.5 The Absence of the Higgs Boson 3.8.6 The Coupling Constant and the Bosonic Mass in Examples The Euler-Lagrange Equations to Degree Three and Lower 3.9.1 Scalar and Pseudoscalar Currents 3.9.2 Bilinear Currents and Potentials 3.9.3 Further Potentials and Fields 3.9.4 The Non-Dynamical Character of the EL Equations to Lower Degree 222 222 225 226 229 229 234 238 239 243 246 247 249 250 252 254 256 259 260 263 271 275 278 278 283 290 292 298 300 302 302 303 304 306 ... naturally when analyzing the mathematical structure of the causal action principle The kernel of the fermionic operator as defined by (1.1.13) is also referred to as the kernel of the fermionic projector,... complement of I Then the non-trivial eigenvalues of the operator product x y are given as the zeros of the characteristic polynomial of the restriction x y| I : I → I The coefficients of this... convincing introduction to causal fermion systems and to the analysis of the causal action principle in the continuum limit We point out that the connection to quantum field theory (in particular to
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