Ground states and spectral properties in quantum field theories Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium vorgelegt dem Rat der Fakultät für Mathematik und Informatik der Friedrich–Schiller–Universität Jena von M.Sc. Markus Lange geboren am 10.02.1987 in Münster 1. Gutachter: Prof. Dr. David Hasler (Friedrich-Schiller-University Jena, Ernst-Abbe- Platz 2, 07743 Jena, Germany) 2. Gutachter: Prof. Dr. Marcel Griesemer (University Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany) 3. Gutachter: Prof. Dr. Jacob Schach Møller (Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark) Tag der öffentlichen Verteidigung: 29. Juni 2018 Abstract In this thesis we consider non-relativistic quantum electrodynamics in dipole approximation and study low-energy phenomenons of quantum mechanical systems. We investigate the analytic dependence of the lowest-energy eigenvalue and eigenvector on spectral parameters of the system. In particular we study situations where the ground-state eigenvalue is assumed to be degenerate. In the first situation the eigenspace of a degenerate ground-state eigenvalue is assumed to split up in a specific way in second order formal perturbation theory. We show, using a mild infrared assumption, that the emerging unique ground state and the corresponding ground-state eigenvalue are analytic functions of the coupling constant in a cone with apex at the origin. Secondly we analyse the situation that the degeneracy is protected by a set of symmetries for the considered quantum mechanical system. We prove, in accordance with known results for the non-degenerate situation, that the ground-state eigenvalue and eigenvectors depend analytically on the coupling constant. In order to show these results we extend operator-theoretic renormalization to such degenerate situations. To complement the analyticity results we additionally show that an asymptotic expansion of the ground state and the ground-state eigenvalue exists up to arbitrary order. The infrared assumption needed for the asymptotic expansion is weaker than the usual assumptions required for other methods such as operator theoretic renormalization to be applicable. Zusammenfassung in deutscher Sprache In der vorliegenden Arbeit betrachten wir Modelle der nicht-relativistischen Quantenelektrodynamik in Dipol-Approximation und studieren Phänomene, die bei kleinen Energien in diesen quanten-mechanischen Systemen auftreten. Wir untersuchen die analytische Abhängigkeit des kleinsten Energie-Eigenwertes und der zugehörigen Eigenvektoren im Bezug auf spektrale Parameter des Systems. Insbesondere un- tersuchen wir Situationen, in denen der Grundzustandseigenwert entartet ist. In der ersten Situation wird angenommen, dass sich der Eigenraum des Grundzustandseigenwertes in zweiter Ordnung in for- maler Störungstheorie in bestimmter Weise aufspaltet. Wir zeigen unter Annahme einer schwachen Infrarot-Bedingung, dass der durch die Aufspaltung entstandene nicht-entartete Grundzustand und die zugehörige Grundzustandsenergie analytische Funktionen der Kopplungskonstante in einem Kegel mit Spitze im Ursprung sind. Im zweiten Fall untersuchen wir die Situation, dass die Entartung durch eine Menge von Symmetrien im Systems erzeugt wird. Unter bestimmten Voraussetzungen zeigen wir, dass der Grundzustand und die Grundzustandsenergie analytische Funktionen der Kopplungskonstante sind. Dieses Ergebnis stimmt mit entsprechenden Ergebnissen für nicht-entartete Situationen überein. Um diese Resultate zu zeigen erweitern wir die Methode ’operator-theoretic renormalization’ auf diese entarteten Situationen. Ergänzend zu den obrigen Analytizitätsergebnisse zeigen wir, dass eine asympto- tische Entwicklung, zu beliebiger Ordnung, des Grundzustandes und der Grundzustandsenergie existiert. Die dafür benötigte Infrarot-Bedingung ist schwächer als die üblichen Bedingungen die gebraucht werden damit andere Methoden, wie zum Beispiel ’operator-theoretic renormalization’, anwendbar sind. Acknowledgements I genuinely thank my adviser David Hasler for his guidance and support during my doctorate. We had a lot of fruitful and enlightening discussions and he always had an open door for my questions and very good advises. Moreover I thank Gerhard Bräunlich for interesting discussions and a fruitful collaboration that resulted in a joint work with him and David Hasler. I am thankful to Professor Israel Michael Sigal for his hospitality in September 2016 at the University of Toronto. I learned a lot about Born-Oppenheimer approximation during that time. Additionally I want to thank him for connecting me with other researchers in the community on multiple occasions. I am very grateful to Siegfried Beckus, Johanna Nabrotzki and Andreas Vollmer for reading parts of the thesis in advance. Their useful remarks and suggestions were very helpful in the formation process of this thesis. In addition, I want to thank my colleagues Alexandra, Andreas, Benjamin, Daniel, Jannis, Marcel, Melchior, Nina, Oliver, Siegfried and Therese for the nice and pleasant working atmosphere and the many relaxing breaks we had together. I thank the Research Training Group (1523/2) “Quantum and Gravitational Fields” which is located at the Friedrich-Schiller-University Jena, Germany, for funding my doctoral position for three years and for their financial support on most of my business trips during my nearly four years inJena. Last but not least I want to thank my family and friends for their mental and emotional support. In particular I am deeply thankful to my parents Mechthild and Dieter Lange for their faith in me and for laying a very reliable foundation for my ongoing life journey. In addition I am thankful to Johanna Nabrotzki for being there and diverting my thoughts at the right moments. Moreover I want to thank my dearest friends Simon Frerking, Thomas Goldbeck, Florian Peschka and Hendrik Sommer for all the joyful times we had and will have together. Contents Abstract iii Acknowledgements iv 1 Introduction 1 2 The Spin-Boson model 4 2.1 The atomic space . .4 2.2 The Fock space . .6 2.2.1 Definition of the Fock space . .6 2.2.2 The bosonic Fock space . .9 2.3 Properties of the Spin-Boson Hamiltonian . 11 2.3.1 Examples of generalized Spin-Boson Models . 13 3 Operator-theoretic renormalization 15 3.1 Perturbation theory . 15 3.2 The operator-theoretic renormalization group method . 17 3.2.1 The Feshbach map . 17 3.2.2 The operator-theoretic renormalization group method . 19 3.3 Operator-valued integral kernels . 20 3.3.1 Banach spaces of integral kernels . 21 3.3.2 Field operators, Pull-Through Formula and Wick’s theorem . 23 4 Degenerate perturbation theory 27 4.1 Second order split-up for the Spin-Boson model . 27 4.1.1 Definition of the model and statement of result . 28 4.1.2 Initial Feshbach step . 30 4.1.3 Banach space estimate for the first step . 32 4.1.4 Second Feshbach step . 37 4.1.5 Banach Space estimate for the second step . 41 4.1.6 Proof of main result for the split-up Spin-Boson model . 47 4.2 Symmetries for the Spin-Boson Hamiltonian . 50 4.2.1 Preliminaries . 50 4.2.2 Description of the model and statement of result . 53 4.2.3 Construction of an effective Hamiltonian . 56 4.2.4 The renormalization transformation . 60 4.2.5 Iterating the renormalization transformation Rρ ................... 61 4.2.6 Proof of main result for the symmetry-protected Spin-Boson model . 69 4.3 The hydrogen atom in dipole approximation . 71 5 Asymptotic expansions 75 5.1 Model and statement of results . 75 5.2 Asymptotic perturbation theory . 79 5.2.1 Expansion method . 79 5.2.2 Resolvent method . 81 5.3 Existence of an asymptotic expansion . 84 1 Introduction The general theory of quantum mechanics, which includes the quantum theory of fields is a fundamental theory of nature. This theory gives us a way to describe what happens at small scales and energy levels of atoms and subatomic particles. For example the process of absorption and emission of photons by electrons that are bounded to an atomic nucleus can be described using tools from quantum mechanics. Physicists already worked a lot on important and interesting questions involving quantum mechanics. Moreover a part of the physics community has shifted its research focus to questions concerning relativistic field theories and string theory. From a mathematical point of view even the seemingly simple question: “Does the lowest-energy eigenvalue of a system of non-relativistic matter that interacts with a quantized field of massless particles depend analytically on the spectral parameters of the overall system?”is conceptually very complicated and there are still open problems concerning this question. Likewise similar questions regarding excited energy eigenvalues of such interacting matter-radiation systems do only have partial answers. In this thesis we extend pre-existing answers concerning the above mentioned question to degenerate situations. In particular, we generalize the operator-theoretic renormalization group method so that we can handle Hamiltonians with degenerate eigenvalues. Such renormalization group methods play an essential role in the spectral analysis of quantum mechanical systems. About 100 years ago Erwin Schrödinger, Werner Heisenberg, Max Born and others developed an early version of quantum theory that is based on work of Max Planck on black body radiation [113] and Albert Einstein on the photoelectric effect [44]. Later Paul Dirac, David Hilbert, John von Neumann and Hermann Weyl developed a mathematically