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arXiv:quant-ph/0303090 · CRM proceedings & lecture notes

Examples of Berezin-Toeplitz quantization: Finite sets and unit interval

Jean‐Pierre Gazeau, Tarik Garidi, Eric Huguet, M. Lachièze-Rey, J. Renaud

Published Mar 2, 2004• 10 citations
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Abstract

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 $N$-element set and the unit interval. Spaces of states for the $N$-element set and the unit interval are the 2-dimensional euclidean $\R^2$ and hermitian $\C^2$ planes.

DOI
10.1090/crmp/034/07

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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 \C2\C^2\C2 planes.
We present a N-dimensional quantization a la Berezin-Klauder or frame quantization of the complex plane based on overcomplete families of states (coherent states) generated by the N first harmonic oscillator eigenstates. The spectra of position and momentum operators are finite and eigenvalues are equal, up to a factor, to the zeros of Hermite polynomials. From numerical and theoretical studies of the large N behavior of the product \\lambda_M(N) \\lambda_m(N) of largest and non null smallest positive eigenvalues, we infer the inequality \\delta_N(Q) \\Delta_N(Q) \\approx < 2 \\pi (resp. \\delta_N(P) \\Delta_N(P)\\approx < 2 \\pi) involving, in suitable units, the minimal (\\delta_N(Q)) and maximal (\\Delta_N(Q)) sizes of regions of space (resp. momentum) which are accessible to exploration within this finite-dimensional quantum framework. Interesting issues on the measurement process are discussed.
The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.
We present a N-dimensional quantization à la Berezin-Klauder or frame quantization of the complex plane based on overcomplete families of states (coherent states) generated by the N first number eigenstates. The spectra of position and momentum operators (or of quadrature operators in quantum optics) are finite and eigenvalues are equal, up to a factor, to the zeros of Hermite polynomials. From numerical and theoretical studies of the large N behavior of the product λm(N) λM(N) of non null smallest positive and largest eigenvalues, we infer the inequality [Formula: see text] (resp. [Formula: see text]) involving, in suitable units, the minimal (δN(Q)) and maximal (ΔN(Q)) sizes of regions of space (resp. momentum) which are accessible to exploration within this finite-dimensional quantum framework. Interesting issues on the measurement process and connections with the finite Chern-Simons matrix model for the Quantum Hall effect are discussed.
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.