Quantum Harmonic Analysis: An Introduction: 4 (Advances in Analysis and Geometry, 4)

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 Preface Introduction 1 Preliminaries 2 Displacements and reflections 3 The cross-Wigner transform 4 Gaussians and hermite functions 5 The Weyl transform 6 The Cohen class 7 Born–Jordan quantization 8 Metaplectic operators 9 The property of symplectic covariance 10 The Feichtinger algebra 11 Hilbert–Schmidt operators 12 The trace class 13 The quantum Bochner theorem 14 The density operator 15 The uncertainty principle 16 Separability and entanglement 17 Separability of Gaussian states Bibliography Index