Dynamics of Causal Fermion Systems Field Equations and Correction Terms for a New Unified Physical Theory Dynamik kausaler Fermionensysteme Feldgleichungen und Korrekturterme f¨ureine neue vereinheitlichte Theorie Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat) der Fakult¨at f¨ur Mathematik der Universit¨at Regensburg vorgelegt von Johannes Kleiner aus Kempten (Allg¨au) im Jahr 2017 Promotionsgesuch eingereicht am 19.07.2017. Die Arbeit wurde angeleitet von Prof. Dr. Felix Finster. Pr¨ufungsausschuss: Vorsitzender: Prof. Dr. Denis-Charles Cisinski 1. Gutachter: Prof. Dr. Felix Finster 2. Gutachter: Prof. Dr. Peter Pickl (LMU M¨unchen) weiterer Pr¨ufer: Prof. Dr. Helmut Abels I dedicate this thesis to humanity in the hope that it will continue its bright path to establish a world which is just and good to all. Abstract The theory of causal fermion systems is a new physical theory which aims to describe a fundamental level of physical reality. Its mathematical core is the causal action principle. In this thesis, we develop a formalism which connects the causal action principle to a suitable notion of fields on space-time. We derive field equations from the causal action principle and find that the dynamics induced by the field equations conserve a symplectic form which gives rise to an Hamiltonian time evolution if the causal fermion system admits a notion of `time'. In this way, we establish the dynamics of causal fermion systems. Remarkably, the causal action principle implies that there are correction terms to the field equations, which we subsequently derive and study. In particular, we prove that there is a stochastic and a non-linear correction term and investigate how they relate to the Hamiltonian time evolution. Furthermore, we give theorems which generalize the connection between symmetries and conservation laws in Noether's theorems to the theory of causal fermion systems. The appearance of the particular correction terms is reminiscent of dynamical collapse models in quantum theory. Kurzzusammenfassung Die Theorie der kausalen Fermionensysteme ist eine k¨urzlich entwickelte physika- lische Theorie, welche die fundamentalen Theorien in der Physik vereinheitlicht und somit einen neuen Vorschlag f¨urdie Beschreibung der grundlegenden Strukturen der physikalischen Realit¨atdarstellt. Die maßgebliche mathematische Struktur in dieser Theorie bildet das sogenannte kausale Wirkungsprinzip. Gegenstand dieser Arbeit ist es, einen Formalismus zu entwickeln, welcher aufbauend auf diesem abstrakten Prinzip eine Beschreibung liefert, die Parallelen zu den ¨ublichen Theorien in der Physik hat: Die Dynamik von kausalen Fermionensystemen. Ergebnis dieser Forschungen ist der sogenannte Jet-Formalismus von kausalen Fermio- nensystemen, welcher als verallgemeinerte physikalische Felder 1-Jets zu Grunde legt. Basierend auf den Euler-Lagrange Gleichungen des kausalen Wirkungsprinzips leiten wir Feldgleichungen ab und studieren deren L¨osungen. Es zeigt sich, dass eine sym- plektische Form konstruiert werden kann, welche unter der von den Feldgleichungen induzierten Dynamik erhalten ist. Dies f¨uhrtzur Definition einer Hamiltonschen Zeitentwicklung f¨urkausale Fermionensysteme. Bemerkenswerterweise f¨uhrtdas kausale Wirkungsprinzip zu Korrekturtermen f¨ur die Feldgleichungen, die wir im weiteren Verlauf der Arbeit untersuchen. Unsere The- oreme etablieren einen stochastischen und einen nicht-linearen Korrekturterm, sowie deren Beziehung zu markoskopischen Gleichungen und zur eingangs erw¨ahnten sym- plektischen Form. Das Auftreten dieser speziellen Korrekturterme deutet auf Paralle- len zu dynamischen Kollaps-Theorien in der nicht-relativistischen Quantenmechanik hin. Die Arbeit enth¨altaußerdem Theoreme, welche den durch die Noetherschen Theoreme gegebenen Zusammenhang zwischen Symmetrien und Erhaltungsgr¨oßen auf die Theorie der kausalen Fermionensysteme ¨ubertragen. Acknowledgements First of all, I want to thank the one person whom I could talk to about the details of my research, my supervisor Felix Finster. I am grateful for the inspiration, conversations, ways of thinking and knowledge which you gave me, and deeply appreciate the freedom you provided for me to study other fields and visit other institutions. Next, I want to thank my personal friends and my family who enriched my life so considerably during my studies and during my PhD. For providing for all my needs and comforts, both material and mental, I want to thank Lucia, Teresa and Meinrad. For being the most supportive partner, I want to thank Barbara. For spurring my mind with the most inspiring conversations about the foundations of reality, I want to thank Robin and Clemens. For the most enlightening late hour coffee house physics conversations and the most fascinating dance moves, I want to thank Jan-Hendrik. For being the most reliable and honest friends over many years, and for being hidden safe harbours in my life, I want to thank Martin, Sven and Clemens. For getting entangled in spiralling scientific and philosophic debates, I want to thank the many friends which I have made during the Rethinking workshops, and in particular my co-organizers Robin, Fede, Jan-Hendrik and Franz. For putting up with my ever critical attitude, and for much enlightenment, I want to thank the members of my reading circle in Regensburg. For co-organizing and being a source of countless ideas, I want to thank Hermann Josef, Sven and Barbara. For being the most fun company and for keeping my bounds with home, I want to thank my Allg¨aufriends, in particular Thommy, as well as Soldo and Elfriede. Finally, my thanks go to the many friends I cannot mention here, in particular from my studies in Regensburg, Heidelberg and Freiburg. In endless conversations, you have given me orientation in life and goals to strive for. Thanks so much for that. They go to my former supervisors, who have invested so much of their time and effort into furthering my education, to my former and current colleagues, who I have shared with many jolly and amusing moments and to the many wonderful people who I have met in other scientific institutions, in particular in Oxford and Marseille. All of you have given me confidence and trust in science, and have shown me the warm-hearted part of this joint endeavour of ours. Concerning institutions, most of all I want to thank the Studienstiftung des Deutschen Volkes for providing trust in my capabilities and visions by financing my PhD and many journeys. Next, I want to thank the Department of Mathematics at the University of Regensburg for providing my daily needs and a very lively mathematical atmosphere, and in particular the DFG Graduate School GRK 1692 for financing journeys and for enabling one of the most dearest activities of my PhD, the organization of the Rethinking Foundations of Physics Workshops, by providing financial means and infrastructure. Thank you so much for that! Finally, I want to thank the Center of Mathematical Sciences and Applications of Harvard University, the Department of Computer Science of Oxford University and the Centre de Physique Th´eoriqueof Aix-Marseille Universit´e for hospitality and support. Contents 1 Introduction 1 1.1 Noether-Like Theorems . .3 1.2 Hamiltonian Formulation and Linearized Field Equations . .5 1.3 Stochastic and Non-Linear Correction Terms . .9 1.4 Connection to Foundations of Quantum Theory . 12 2 Causal Fermion Systems as a Candidate for a Unified Physical Theory 15 2.1 The Theory . 15 2.2 Example: Dirac Wave Functions in Minkowski Space . 17 2.3 Inherent Structures . 20 2.4 The Minkowski Vacuum . 23 2.5 Description of More General Space-Times . 25 2.6 The Continuum Limit . 26 2.7 Entanglement and Nonlocality . 29 2.8 Clarifying Remarks . 30 3 Noether-Like Theorems 35 3.1 Preliminaries . 36 3.1.1 The Classical Noether Theorem . 36 3.1.2 Causal Variational Principles in the Compact Setting . 38 3.1.3 The Concept of Surface Layer Integrals . 39 3.2 Noether-Like Theorems in the Compact Setting . 41 3.2.1 Symmetries of the Lagrangian . 41 3.2.2 Symmetries of the Universal Measure . 43 3.2.3 Generalized Integrated Symmetries . 44 3.3 The Setting of Causal Fermion Systems . 46 3.3.1 Basic Definitions and the Euler-Lagrange Equations . 46 3.3.1.1 The finite-dimensional setting . 47 3.3.1.2 The infinite-dimensional setting . 49 3.3.2 Noether-Like Theorems . 50 3.4 Example: Current Conservation . 53 3.4.1 A General Conservation Law . 53 3.4.2 Correspondence to Dirac Current Conservation . 54 i Contents 3.4.3 Clarifying Remarks . 68 3.5 Example: Conservation of Energy-Momentum . 70 3.5.1 Generalized Killing Symmetries and Conservation Laws . 71 3.5.2 Correspondence to the Dirac Energy-Momentum Tensor . 73 3.6 Example: Symmetries of the Universal Measure . 77 3.7 Conservation Laws and Microscopic Mixing . 78 4 Hamiltonian Formulation and Linearized Field Equations 81 4.1 Causal Variational Principles and Causal Fermion Systems . 81 4.1.1 Causal Variational Principles in the Non-Compact Setting . 81 4.1.2 The Euler-Lagrange Equations . 83 4.1.3 Local Minimizers and Second Variations . 84 4.1.4 The Setting of Causal Fermion Systems . 88 4.2 The Symplectic Form in the Smooth Setting . 89 4.2.1 The Weak Euler-Lagrange Equations . 90 4.2.2 The Nonlinear Solution Space . 90 4.2.3 The Symplectic Form and Hamiltonian Time Evolution . 93 4.3 The Lower Semi-Continuous Setting . 95 4.3.1 The Weak Euler-Lagrange Equations . 96 4.3.2 Families of Solutions and Linearized Solutions . 97 4.3.3 The Symplectic Form and Hamiltonian Time Evolution . 102 1;1 1 4.4 Example: A Lattice System in R × S ................... 105 4.4.1 The Lagrangian . 105 4.4.2 A Local Minimizer . 106 4.4.3 The Jet Spaces . 107 4.4.4 The Symplectic Form . 109 Appendix A: The Fr´echet Manifold Structure of B ................ 111 5 Stochastic and Non-Linear Correction Terms 115 5.1 Preliminaries .