Quantum harmonic analysis : an introduction

Quantum mechanics is arguably one of the most successful scientific theories ever and its applications to chemistry, optics, and information theory are innumerable. This book provides the reader with a rigorous treatment of the main mathematical tools from harmonic analysis which play an essential role in the modern formulation of quantum mechanics. This allows us at the same time to suggest some new ideas and methods, with a special focus on topics such as the Wigner phase space formalism and its applications to the theory of the density operator and its entanglement properties. This book can be used with profit by advanced undergraduate students in mathematics and physics, as well as by confirmed researchers. interesting and well-written manuscript by leading researchers Accessible approach, written in a user and student friendly way The Series Is Devoted To The Publication Of High-level Monographs And Specialized Graduate Texts Which Cover Classical And Modern Analysis, Partial Differential Equations With Natural Connections To Geometry And The Interplays Between These Fields And Their Applications To Mathematical Physics. Editor-in-chief Jie Xiao, Memorial University, Canada Editorial Board Der-chen Chang, Georgetown University, Usa Goong Chen, Texas A&m University, Usa Andrea Colesanti, University Of Florence, Italy Robert Mccann, University Of Toronto, Canada De-qi Zhang, National University Of Singapore, Singapore Kehe Zhu, University At Albany, Usa Please Send Any Book Proposals To Jie Xiao. Contents 7 Preface 13 Introduction 15 1 Preliminaries 19 2 Displacements and reflections 39 3 The cross-Wigner transform 51 4 Gaussians and hermite functions 67 5 The Weyl transform 79 6 The Cohen class 91 7 Born–Jordan quantization 101 8 Metaplectic operators 111 9 The property of symplectic covariance 121 10 The Feichtinger algebra 137 11 Hilbert–Schmidt operators 153 12 The trace class 163 13 The quantum Bochner theorem 175 14 The density operator 185 15 The uncertainty principle 199 16 Separability and entanglement 211 17 Separability of Gaussian states 223 Bibliography 233 Index 237