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Semiclassical Dynamics and Magnetic Weyl Calculus

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Semiclassical Dynamics and Magnetic Weyl Calculus TECHNISCHE UNIVERSITÄT MÜNCHEN Zentrum Mathematik Semiclassical Dynamics and Magnetic Weyl Calculus Maximilian Stefan Lein Vollständiger Abdruck der von der Fakultät für Mathematik der Technischen Univer- sität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzende: Univ.-Prof. Dr. Simone Warzel Prüfer der Dissertation: 1. Univ.-Prof. Dr. Herbert Spohn 2. Prof. Dr. Radu Purice, IMAR, Bukarest, Rumänien 3. Univ.-Prof. Dr. Stefan Teufel, Eberhard-Karls-Universität Tübingen (schriftliches Gutachten) Die Dissertation wurde am 18.10.2010 bei der Technischen Universität München ein- gereicht und durch die Fakultät für Mathematik am 19.01.2011 angenommen. Acknowledgements I would like to express my gratitude towards a few people who have directly or indi- rectly contributed to this thesis. First and foremost, I would like to thank Herbert Spohn for his support and faith in my abilities. He has encouraged and supported frequent visits to conferences and other countries. But the thing I am by far most grateful for is his instrumental role in the inception of my lecture ‘Quantization and Semiclassics’ in winter 2009/2010. This was a very, very exhilarating experience for me that I will not forget. Secondly, I would like to thank my coworkers Fabian Belmonte, Giuseppe De Nit- tis, Martin Fürst, Marius Măntoiu, Serge Richard David Sattlegger, and Marcello Seri. Without their contributions, this thesis would be at most a third of what it is now. I thank Giuseppe for his friendship and hospitality not just during my visit at SISSA. Marius’ invitation to Santiago enabled me to get to know a new continent. I continue to draw a lot of pleasure from discussions and interaction with you and other col- leagues in the field. There are other people who deserve special mention: I owe Radu Purice a debt of gratitude for many stimulating discussions and the possibility of organizing ROGER 2009 in Sibiu as well as his invitation to Bukarest in 2007. I appreciate he has agreed to be one of the referees to this thesis. Similarly, I would like to mention Stefan Teufel whose lecture on Quantum Dynamics in 2002 gave me the last push I needed towards the mathematical aspects of physics. Furthermore, the graduate seminar he gave to- gether with Herbert Spohn and Detlef Dürr was the single most enjoyable class I have sat in during my studies. He has also kindly agreed to be the third referee to this thesis. I benefitted from the work environment at M5 which all its members helped create. One person deserves mention here and that is Wilma Ghamam who is the good soul of the department. She always knows where things are, who to ask and where people are. Her unpretentious attitude is really unique. I also recognize the kind support of DAAD for a 4-month short term scholarship that allowed me to visit Prof. Littlejohn at UC Berkeley and the financial support of DFG and Fondecyt under the Grant 1090008 which have made my 4-week visit to Santiago possible. In that context, I also thank Georgi Raikov and Rafael de Tiedra Aldecoa for preparing my visit to Chile. iii The other, non-scientific half of my life is just as important: my family and friends have kept my life interesting and fun. The memories from the last four years I share with Julius Bahr, Ann Etienne, Stephanie Farnleitner, Sabrina Oberacher, my brother Fabian and my sister Alexandra are invaluable. iv Abstracts Kurzzusammenfassung Weyl Quantisierung und semiklassische Techniken können benutzt werden, um Lei- tungseigenschaften von kristallinen Festkörpern zu verstehen, die externen, langsam variierenden elektromagnetischen Feldern ausgesetzt werden. Der Fall, in dem das Magnetfeld schwach, aber konstant ist, wird von bisherigen mathematischen Ergeb- nissen nicht abgedeckt. Genau das ist das Regime des Quanten-Hall-Effekts und es gilt zu verstehen, wieso die transversale Leitfähigkeit quantisiert ist. Möchte man für die- sen Fall semiklassische Bewegungsgleichungen rigoros herleiten, muss man den kon- ventionellen Weyl-Kalkül durch einen magnetischen ersetzen, der einen semiklassi- schen Parameter enthält. Mathematisch gesehen hat man es mit magnetischen Pseudodifferentialoperatoren zu tun, die auch für sich gesehen von Interesse sind. Daher widmen wir diesen zwei weitere Kapitel, die sich mit deren Eigenschaften befassen. Abstract Weyl quantization and related semiclassical techniques can be used to study conduc- tion properties of crystalline solids subjected to slowly-varying, external electromag- netic fields. The case where the external magnetic field is constant, is not covered by existing theory as proofs involving usual Weyl calculus break down. This is the regime of the so-called quantum Hall effect where quantization of transverse conduc- tance is observed. To rigorously derive semiclassical equations of motion, one needs to systematically develop a magnetic Weyl calculus which contains a semiclassical pa- rameter. Mathematically, the operators involved in the analysis are magnetic pseudodiffer- ential operators, a topic which by itself is of interest for the mathematics and math- ematical physics community alike. Hence, we will devote two additional chapters to further understanding of properties of those operators. v Contents 1 Introduction 1 1.1 Physical aspects: quantization of magnetic systems ........... 3 1.2 Mathematical aspects: magnetic ΨDOs .................. 7 1.3 Structure and main results ........................ 9 2 Magnetic Weyl Calculus 13 2.1 Standard Weyl calculus .......................... 13 2.1.1 Comparison of classical and quantum mechanical frameworks 13 2.1.1.1 Hamiltonian framework of classical mechanics ... 14 2.1.1.2 Quantum mechanics .................. 15 2.1.1.3 Comparison of the two frameworks .......... 15 2.1.2 The Weyl system .......................... 17 2.1.3 Weyl quantization ......................... 18 2.1.4 The Wigner transform ...................... 19 2.1.5 The Weyl product ......................... 25 2.1.6 Quantization of Hörmander symbols ............... 26 2.2 Magnetic Weyl calculus .......................... 27 2.2.1 Standard ansatz: minimal coupling ............... 28 2.2.2 Covariant quantization formula ................. 29 2.2.3 The Magnetic Wigner transform ................. 33 2.2.4 The magnetic Weyl product ................... 34 2.3 Extension to larger classes of functions ................. 37 2.3.1 Extension via duality ....................... 37 2.3.2 The magnetic Moyal algebra ................... 41 2.3.3 Important subclasses ....................... 43 2.4 Important results ............................. 44 2.4.1 L2-continuity and selfadjointness ................ 44 2.4.2 Commutator criteria ....................... 46 2.4.3 Inversion and holomorphic functional calculus ......... 51 3 Asymptotic Expansions and Semiclassical Limit 55 3.1 Scalings ................................... 55 vii Contents 3.2 Magnetic Weyl quantization ....................... 58 3.3 Semiclassical symbols and precision ................... 59 3.4 The Magnetic Wigner transform ..................... 61 3.5 Asymptotic expansions of the product .................. 62 3.6 Semiclassical limit ............................. 73 3.7 Relation between magnetic and ordinary Weyl calculus ........ 81 4 Magnetic Space-adiabatic Perturbation Theory 85 4.1 The model ................................. 86 4.2 Rewriting the problem ........................... 88 4.2.1 The Bloch-Floquet-Zak transform ................ 89 4.2.2 Equivariant magnetic Weyl calculus ............... 91 4.3 The magnetic Bloch electron as a space-adiabatic problem ....... 93 4.3.1 Slow variation: the adiabatic point of view ........... 93 4.3.2 Effective quantum dynamics to any order ............ 95 4.4 Adiabatic decoupling ........................... 96 4.4.1 Effective quantum dynamics ................... 96 4.4.2 Effective dynamics for a single band ............... 101 4.4.3 Semiclassical equations of motion ................ 102 4.4.3.1 Connection of the effective dynamics with the origi- nal dynamics ...................... 103 4.4.3.2 An Egorov-type theorem ................ 107 4.5 Physical relevance for the quantum Hall effect ............. 109 5 An Algebraic Point of View 111 5.1 Twisted crossed products ......................... 112 5.1.1 Gelfand theory ........................... 112 5.1.2 Crossed Products ......................... 114 5.1.3 Twisted crossed products ..................... 116 5.1.4 Special case of X -algebras .................... 119 5.2 Generalized Weyl calculus ......................... 126 5.3 The concept of affiliation ......................... 130 5.3.1 Observables affiliated to C∗-algebras .............. 130 5.3.2 Spectrum of affiliated observables ................ 133 5.3.3 Tensor products and families of observables .......... 135 6 Pseudodifferential theory revisited 141 6.1 Magnetic composition of anisotropic symbols .............. 143 6.1.1 Anisotropic symbol spaces .................... 143 6.1.2 Magnetic composition ....................... 147 viii Contents 6.2 Relevant C∗-algebras ........................... 150 6.3 Inversion and affiliation .......................... 152 6.4 Spectral properties ............................. 156 6.4.1 Families of ideals ......................... 157 6.4.2 Decomposition of SA into quasi orbits .............. 158 6.4.3 The essential spectrum of anisotropic magnetic operators ... 163 7 Outlook 167 7.1 Magnetic Weyl calculus for other gauge groups ............. 167
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