Gianfausto Dell'Antonio Lectures on the Mathematics of Quantum Mechanics Volume II: Selected Topics February 17, 2016 Mathematical Department, Universita' Sapienza (Rome) Mathematics Area, ISAS (Trieste) 2 A Caterina, Fiammetta, Simonetta Il ne faut pas toujours tellement epuiser un sujet q'on ne laisse rien a fair au lecteur. Il ne s'agit pas de fair lire, mais de faire penser Charles de Secondat, Baron de Montesquieu Contents 1 Lecture 1. Wigner functions. Coherent states. Gabor transform. Semiclassical correlation functions ........................ 11 1.1 Coherent states . 15 1.2 Husimi distribution . 17 1.3 Semiclassical limit using Wigner functions . 21 1.4 Gabor transform . 24 1.5 Semiclassical limit of joint distribution function . 25 1.6 Semiclassical limit using coherent states . 26 1.7 Convergence of quantum solutions to classical solutions . 29 1.8 References for Lecture 1 . 34 2 Lecture 2 Pseudifferential operators . Berezin, Kohn-Nirenberg, Born-Jordan quantizations ................................. 35 2.1 Weyl symbols . 36 2.2 Pseudodifferential operators . 36 2.3 Calderon - Vaillantcourt theorem . 39 2.4 Classes of Pseudodifferential operators. Regularity properties. 44 2.5 Product of Operator versus products of symbols . 46 2.6 Correspondence between commutators and Poisson brackets; time evolution . 49 2.7 Berezin quantization . 51 2.8 Toeplitz operators . 53 2.9 Kohn-Nirenberg Quantization . 54 2.10 Shubin Quantization . 55 2.11 Born-Jordarn quantization . 56 2.12 References for Lecture 2 . 57 4 Contents 3 Lecture 3 Compact and Schatten class operators. Compactness criteria. Bouquet of Inequalities ........................... 59 3.1 Schatten Classes . 64 3.2 General traces . 65 3.3 General Lp spaces . 66 3.4 Carleman operators . 69 3.5 Criteria for compactness . 70 3.6 Appendix to Lecture 3: Inequalities . 76 3.6.1 Lebesgue decomposition theorem . 77 3.6.2 Further inequalities . 78 3.6.3 Interpolation inequalities . 81 3.6.4 Young inequalities . 85 3.6.5 Sobolev-type inequalities . 87 3.7 References for Lecture 3 . 90 4 Lecture 4 Periodic potentials. Wigner-Seitz cell and Brillouen zone. Bloch and Wannier functions .............................. 91 4.1 Fermi surface, Fermi energy . 92 4.2 Periodic potentials. Wigner-Satz cell. Brillouin zone. The Theory of Bloch-Floquet-Zak . 95 4.3 Decompositions . 96 4.4 One particle in a periodic potential . 99 4.5 the Mathieu equation . 103 4.6 The case d ≥ 2. Fibration in momentum space . 104 4.7 Direct integral decomposition . 107 4.8 Wannier functions . 111 4.9 Chern class . 114 4.10 References for Chapter 4 . 117 5 Lecture 5 Connection with the properties of a crystal. Born- Oppenheimer approximation. Edge states and role of topology ...................................................119 5.1 Crystal in a magnetic field . 121 5.2 Slowly varying electric field . 122 5.3 Heisenberg representation . 126 5.4 Pseudifferential point of view . 126 5.5 Topology induced by a magnetic field . 129 5.6 Algebraic-geometric formulation . 131 5.7 Determination of a topological index . 132 5.8 Gauge transformation, relative index and Quantum pumps . 136 5.9 References for Lecture 5 . 138 Contents 5 6 Lecture 6 Lie-Trotter Formula, Wiener Process, Feynmann-Kac formula ....................................................141 6.1 The Feynmann-Kac formula . 146 6.2 Stationary Action; the Fujiwara's approach . 147 6.3 Generalizations of Fresnel integral . 149 6.4 Relation with stochastic processes . 149 6.5 Random variables. Independence . 151 6.6 Stochastic processes, Markov processes . 152 6.7 Construction of Markov processes . 153 6.8 Measurability . 155 6.9 Wiener measure . 158 6.10 The Feynman-Kac formula I: bounded continuous potentials . 159 6.11 The Feynman-Kac formula II: more general potentials . 160 6.12 References for Lecture 6 . 162 7 Lecture 7 Elements of probabiity theory. Construction of Brownian motion. Diffusions .........................................163 7.1 Inequalities . 164 7.2 Independent random variables. 166 7.3 Criteria of convergence . 167 7.4 Laws of large numbers; Kolmogorov theorems . 169 7.5 Central limit theorem . 170 7.6 Construction of probability spaces . 172 7.7 Construction of Brownian motion (Wiener measure) . 174 7.8 Brownian motion as limit of random walks. 175 7.9 Relative compactness . 177 7.10 Modification of Wiener paths. Martingales. 178 7.11 Ito integral . 181 7.12 References for Lecture 7 . 184 8 Lecture 8 Ornstein-Uhlenbeck process. Markov structure . Semigroup property. Paths over function spaces . 185 8.1 Mehler kernel . 185 8.2 Ornstein-Uhlenbeck measure . 187 8.3 Markov processes on function spaces . 189 8.4 Processes with (continuous) paths on space of distributions. The free-field process . 192 8.5 Osterwalder path spaces . 194 8.6 Strong Markov property . 196 8.7 Positive semigroup structure . 197 8.8 Markov Fields . Euclidian invariance. Local Markov property . 201 8.9 Quantum Field . 202 6 Contents 8.10 Euclidian Free Field . 205 8.11 Connection with a local field in Minkowski space . 206 8.12 Modifications of the O.U. process. Modification of euclidian fields . 207 8.13 Refences for Lecture 8 . 209 9 Lecture 9 Modular Operator. Tomita-Takesaki theory Non- commutative integration ...................................211 9.1 The trace. Regular measure (gage) spaces . 212 9.2 Brief review of the K-M-S. condition . 214 9.3 The Tomita-Takesaki theory . 216 9.4 Modular structure, Modular operator, Modular group . 220 9.5 Intertwining properties . 223 9.6 Modular condition. Non-commutative Radon-Nikodym derivative . 227 9.7 Positive cones . 233 9.8 References for Lecture 9 . 234 10 Lecture 10 Scattering theory. Time-dependent formalism. Wave Operators .................................................235 10.1 Scattering Theory . ..