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Iranian Journal of Physics Research

Berezin’s quantization approach for determining the ladder operators associated to a movement particle on the circle and sphere

Rezaei, S, Ardeshir Rabeie

Published May 1, 2021• 0 citations
Publisher

Abstract

The quantum formulation of a physical system is essentially based on the associated creation and annihilation operators. In this article, we introduce these ladder operators for a movement particle on the circle and 2-dimensional sphere by Berezin’s quantization .This approach is derived from the resolution of the unity condition in coherent states. In other words, the coherent states provide a straight forward quantization scheme from a classical state to corresponding quantum state. In this article, we study the coherent states of these systems from heat kernel function point of view.

DOI
10.47176/ijpr.21.1.51054

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The quantum formulation of a physical system is essentially based on the associated creation and annihilation operators. In this article, we introduce these ladder operators for a movement particle on the circle and 2-dimensional sphere by Berezin’s quantization .This approach is derived from the resolution of the unity condition in coherent states. In other words, the coherent states provide a straight forward quantization scheme from a classical state to corresponding quantum state. In this article, we study the coherent states of these systems from heat kernel function point of view.
Berezin-Klauder-Toeplitz (“anti-Wick”) or “coherent state” quantization of the complex plane, viewed as the phase space of a particle moving on the line, is derived from the resolution of the unity provided by the standard (or gaussian) coherent states. The construction of these states and their attractive properties are essentially based on the energy spectrum of the harmonic oscillator, that is on natural numbers. We follow in this work the same path by considering sequences of non-negative numbers and their associated “non-linear” coherent states. We illustrate our approach with the 2-d motion of a charged particle in a uniform magnetic field. By solving the involved Stieltjes moment problem we construct a family of coherent states for this model. We then proceed with the corresponding coherent state quantization and we show that this procedure takes into account the circle topology of the classical motion.
We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S 1 including the Aharonov–Bohm-type quantum description. Coherent states are constructed by Perelomov's method as group-related coherent states generated by Weyl operators on the quantum phase space . Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction, we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was . The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, some of their further properties are also studied demonstrating similarities as well as substantial differences. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
In this paper we try to formulate the Berezin quantization on projective Hilbert space P(H) and use its geometric structure. It will be shown that the star product in Berezin quantiza- tion is equivalent to the Poisson bracket on P(H) and the Berezin method to construction a correspondence between a given classical theory and a given quantum theory is used to define a classical limit for geometric quantum mechanics. the covariant symbols. These symbols are expectation values of the observables in terms of coherent states: the holomorphic functions on classical phase space M that is assumed to be aKmanifold. Berezin (2) showed that the covariant symbols form a ∗-algebra which in limit � → 0l eads to the Poisson algebra between the corresponding classical observables: The functions on the phase space M. In this paper we will see that the Berezin ∗-algebra is in fact a Poisson algebra which is induced by the Fubini-Study 2-form on space of coherent states. This space is defined as follows: coherent states span a dense subspace H of Hilbert space H. P( H), which is denoted by M, is aKmanifold with induced symplectic structure from P(H). Therefore the covariant symbols can be considered as functions on M. It is shown (13) that there exists an embedding mapping between the classical phase space M and M, by which to any point z ∈ M is associated ap ointZ ∈M. With this construction to all of the quantum observables are associated their covariant symbols, which form a Poisson algebra on M and since the corresponding classical observables form a Poisson algebra on M , the Berezin quantization is a systematic procedure to relate these two Poisson algebras. Also the relation of Berezin quantization and geometric for- mulation of quantum mechanics will be evident as follows. The geometric quantum mechanics is a formulation of quantum mechanics in projective Hilbert spaces. With our construction one sees that the Berezin quantization is an equivalent formulation and in addition gives a prescription as classical limit for geometric quantum mechanics (9).
Abstract. We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere S d. The coherent states are labeled by points in the associated phase space T ∗ (S d). These coherent states are not of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal–Bargmann representation for the system, the associated unitary Segal–Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of B. Hall and M. Stenzel, we give here a substantially different description based on ideas of T. Thiemann and of K. Kowalski and J. Rembieliński. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to carry out the calculations in a self-contained and explicit way.
We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere Sd. The coherent states are labeled by points in the associated phase space T*(Sd). These coherent states are not of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal–Bargmann representation for the system, the associated unitary Segal–Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of Hall and Stenzel, we give here a substantially different description based on ideas of Thiemann and of Kowalski and Rembieliński. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to carry out the calculations in a self-contained and explicit way.