Journal ArticleDOI
Quantization in complex symmetric spaces
TLDR
In this paper, the authors constructed the quantization of a classical mechanics whose phase space is a classical complex symmetric space, and established the important qualitative differences between the quantisation of such mechanics and the quantification of ordinary mechanics with plane phase spaces.Abstract:
By means of the method described in the author's paper Quantization (Math. USSR Izv. 8 (1974), 1109-1165), we construct the quantization of a classical mechanics whose phase space is a classical complex symmetric space. We establish the important qualitative differences between the quantization of such mechanics and the quantization of ordinary mechanics with plane phase spaces: for all the spaces considered, except for the sphere, Planck's constant is bounded above. Moreover, in the compact case Planck's constant takes on only discrete values.Bibliography: 17 items.read more
Citations
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Journal ArticleDOI
Toeplitz quantization of Kähler manifolds ang gl(N), N→∞ limits
TL;DR: For general compact Kahler manifolds, it was shown in this paper that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit.
Journal ArticleDOI
Quantum systems with finite Hilbert space
TL;DR: In this paper, a factorization of finite quantum systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied, and the general formalism is applied to the case of angular momentum.
Journal ArticleDOI
Quantization methods: a guide for physicists and analysts
S. Twareque Ali,Miroslav Engliš +1 more
TL;DR: An overview of some of the better known quantization techniques for systems with finite numbers of degrees-of-freedom can be found in this paper, including canonical quantization and the related Dirac scheme, introduced in the early days of quantum mechanics.
Journal ArticleDOI
Mixing Quantum and Classical Mechanics
TL;DR: In this paper, a quantum-classical mixing is studied by a group-theoretical approach, and a quantumclassical equation of motion is derived, which preserves the Lie algebra structure of quantum and classical mechanics.
Book ChapterDOI
Between classical and quantum
TL;DR: In this article, the authors discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit h -> 0 of small Planck's constant (in a finite system), and through decoherence and consistent histories.
References
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Book
Differential Geometry and Symmetric Spaces
TL;DR: In this article, the classification of symmetric spaces has been studied in the context of Lie groups and Lie algebras, and a list of notational conventions has been proposed.
BookDOI
The classical groups : their invariants and representations
TL;DR: Weyl as discussed by the authors discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations using basic concepts from algebra, and examines the various properties of the groups.
Journal ArticleDOI
General Concept of Quantization
F. A. Berezin,F. A. Berezin +1 more
TL;DR: In this article, the general definition of quantization is proposed and two classical systems are considered: the phase space is a Lobachevskii plane and the two-dimensional sphere.
Journal ArticleDOI
The Classical Limit of Quantum Spin Systems
TL;DR: In this article, the authors derived a classical integral representation for the partition function, Z Q, of a quantum spin system and obtained upper and lower bounds to the quantum free energy in terms of two classical free energies (or ground state energies).