Universal Journal of Physics and Application Vol. 2(1-2), pp. 1 - 38
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QUANTIZATION IS JUST A CERTAIN REGARD TO...

AUTHOR(S) INFORMATION: Gazeau J.-P.

ABSTRACT

In signal analysis, the Hilbertian structure associated to a measure set is the mathematical framework for analyzing signals which “live” precisely on the set. A frame or coherent states quantization consists in selecting a Hilbert subspace which is reproducing. The selection can be motivated either by a statistical reading of experimental data or by the need of focusing on certain aspects of signals. This frame quantization scheme could reveal itself as an efficient tool for quantizing physical systems for which the implementing of more traditional methods is unmanageable. The procedure is first illustrated by the example of infinite- and finite-dimensional quantizations of the particle motion on the line. Interesting new inequalities concerning observables emerge from the finite-dimensional quantization, in particular in the context of the quantum Hall effect. We next apply the procedure to the still problematic quantization of the particle motion on the circle. Related to the latter problem is the quantization of dynamics of a test particle in the two-dimensional de Sitter space, the group of symmetry of which is SO₀(1, 2). Our quantization procedure then yields the realization of the corresponding principal series representation of SO₀(1, 2). We also present an application of the method to a toy model for quantum geometry, namely the Ashtekar-Fairhurst-Willis polymer particle representation.