Symbol correspondences for spin systems

Abstract

The present monograph explores the correspondence between quantum and classical mechanics in the particular context of spin systems, that is, SU(2)-symmetric mechanical systems. Here, a detailed presentation of quantum spin-j systems, with emphasis on the SO(3)-invariant decomposition of their operator algebras, is followed by an introduction to the Poisson algebra of the classical spin system and a similarly detailed presentation of its SO(3)-invariant decomposition. Subsequently, this monograph 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.

Research Monograph, 171 pages (book format, preliminary version)

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Cited by in corpus (9)

• A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
• Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension
• On the family of Wigner functions for N-level quantum system
• Eulerian and Lagrangian stability in Zeitlin's model of hydrodynamics
• Classical and quantum KMS states on spin lattice systems
• On the infinite dimension limit of invariant measures and solutions of Zeitlin's 2D Euler equations
• Correspondence rules for Wigner functions over SU(3)/U(2)
• Semi-Classical Discretization and Long-Time Evolution of Variable Spin Systems
• Asymptotic localization of symbol correspondences for spin systems and sequential quantizations of $S^2$