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Advances in Mathematics Scientific Journal

ON SOME ASPECTS OF COHERENT STATES QUANTIZATION WITH RELATED EXAMPLES

Isiaka Aremua, Komi Sodoga, P.K. Tchakpélé

Published Nov 21, 2021• 1 citation
Publisher

Abstract

This work addresses the general procedure of quantization also known as the Berezin-Klauder-Toeplitz quantization, or as coherent state (CS) (anti-Wick) quantization. The method is first illustrated by the motion of a particle on the circle. Then, we take as second example, a set of generalized photon-added coherent states related to associated hypergeometric functions. The nonclassical behaviour of this set of coherent states is also investigated.

DOI
10.37418/amsj.10.11.2

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This work addresses the general procedure of quantization also known as the Berezin-Klauder-Toeplitz quantization, or as coherent state (CS) (anti-Wick) quantization. The method is first illustrated by the motion of a particle on the circle. Then, we take as second example, a set of generalized photon-added coherent states related to associated hypergeometric functions. The nonclassical behaviour of this set of coherent states is also investigated.
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 investigate the consistency of coherent state (or Berezin-Klauder-Toeplitz, or anti-Wick) quantization in regard to physical observations in the non- relativistic (or Galilean) regime. We compare this procedure with the canonical quantization (on both mathematical and physical levels) and examine whether they are or not equivalent in their predictions: is it possible to dif- ferentiate them on a strictly physical level? As far as only usual dynamical observables (position, momentum, energy, ...) are concerned, the quantization through coherent states is proved to be a perfectly valid alternative. We successfully put to the test the validity of CS quantization in the case of data obtained from vibrational spectroscopy (data that allowed to validate canonical quantization in the early period of Quantum Mechanics).
We discuss the construction of coherent states (CS) for systems with continuous spectra. First, we propose to adopt the Malkin–Manko approach, developed for systems with discrete spectra, to the case under consideration. Following this approach, we consider two examples, a free particle and a particle in a linear potential. Second, we generalize the approach of action-angle CS to systems with continuous spectra. In the first approach we start with a well-defined quantum formulation (canonical quantization) of a physical system and the construction of CS follows from such a quantization. In the second approach, the quantization procedure is inherent to the CS construction itself.
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
We present a quantization scheme of an arbitrary measure space based on overcomplete families of states and generalizing the Klauder and the Berezin-Toeplitz approaches. This scheme could reveal itself as an efficient tool for quantizing physical systems for which more traditional methods like geometric quantization are uneasy to implement. The procedure is illustrated by (mostly two-dimensional) elementary examples in which the measure space is a NNN-element set and the unit interval. Spaces of states for the NNN-element set and the unit interval are the 2-dimensional euclidean R2\R^2R2 and hermitian $\C^2$ planes.