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In mathematical physics, the correspondence between quantum and classical mechanics is a central topic, which this book explores in more detail in the particular context of spin systems, that is, SU(2)-symmetric mechanical systems. A detailed presentation of quantum spin-j systems, with emphasis on the SO(3)-invariant decomposition of their operator algebras, is first followed by an introduction to the Poisson algebra of the classical spin system and then by a similarly detailed examination of its SO(3)-invariant decomposition. The book next proceeds with a detailed and systematic study of general quantum-classical symbol correspondences for spin-j systems and their induced twisted products of functions on the 2-sphere. This original systematic presentation culminates with the study of twisted products in the asymptotic limit of high spin numbers. In the context of spin systems it shows how classical mechanics may or may not emerge as an asymptotic limit of quantum mechanics. The book will be a valuable guide for researchers in this field and its self-contained approach also makes it a helpful resource for graduate students in mathematics and physics.
List of contents
Preface.- 1 Introduction.- 2 Preliminaries.- 3 Quantum Spin Systems and Their Operator Algebras.- 4 The Poisson Algebra of the Classical Spin System.- 5 Intermission.- 6 Symbol Correspondences for a Spin-j System.- 7 Multiplications of Symbols on the 2-Sphere.- 8 Beginning Asymptotic Analysis of Twisted Products.- 9 Conclusion.- Appendix.- Bibliography.- Index.
Summary
In mathematical physics, the correspondence between quantum and classical mechanics is a central topic, which this book explores in more detail in the particular context of spin systems, that is, SU(2)-symmetric mechanical systems. A detailed presentation of quantum spin-j systems, with emphasis on the SO(3)-invariant decomposition of their operator algebras, is first followed by an introduction to the Poisson algebra of the classical spin system and then by a similarly detailed examination of its SO(3)-invariant decomposition. The book next proceeds with a detailed and systematic study of general quantum-classical symbol correspondences for spin-j systems and their induced twisted products of functions on the 2-sphere. This original systematic presentation culminates with the study of twisted products in the asymptotic limit of high spin numbers. In the context of spin systems it shows how classical mechanics may or may not emerge as an asymptotic limit of quantum mechanics. The book will be a valuable guide for researchers in this field and its self-contained approach also makes it a helpful resource for graduate students in mathematics and physics.
Additional text
From the book reviews:
“This book constitutes an interesting and highly useful monograph devoted to symbol correspondences, that will help the reader to better understand the existing relation between classical and quantum mechanics. For the particular case of physicists, this work will clarify the mathematical context and formalism that is not usually presented with such an amount of detail in other books on the subject. … This book is highly recommended to the specialist as well as to the non-specialist interested on spin systems.” (Rutwig Campoamor-Stursberg, zbMATH, Vol. 1305, 2015)
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From the book reviews:
"This book constitutes an interesting and highly useful monograph devoted to symbol correspondences, that will help the reader to better understand the existing relation between classical and quantum mechanics. For the particular case of physicists, this work will clarify the mathematical context and formalism that is not usually presented with such an amount of detail in other books on the subject. ... This book is highly recommended to the specialist as well as to the non-specialist interested on spin systems." (Rutwig Campoamor-Stursberg, zbMATH, Vol. 1305, 2015)
