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12 Coherent State Quantization: The Klauder Berezin Approach We explain in 12 the way in which standard coherent states allow a natural quantization of a large class of functions and distributions, including tempered distributions, on the complex plane viewed as the phase space of the particle motion on the line We show how they offer a classical-like representation of the evolution of quantum observables They also help to set Heisenberg inequalities concerning the phase operator and the number operator for the oscillator Fock states By restricting the formalism to the nite dimension, we present new quantum inequalities concerning the respective spectra of position and momentum matrices that result from such a coherent state quantization scheme for the motion on the line 13 Coherent State or Frame Quantization In 13 we extend the procedure of standard coherent state quantization to any measure space labeling a total family of vectors solving the identity in some Hilbert space We thus advocate the idea that, to a certain extent, quantization pertains to a larger discipline than just being restricted to speci c domains of physics such as mechanics or eld theory We also develop the notion of lower and upper symbols resulting from such a quantization scheme, and we discuss the probabilistic content of the construction 14 Elementary Examples of Coherent State Quantization The examples which are presented in 14 are, although elementary, rather unusual In particular, we start with measure sets that are not necessarily phase spaces Such sets are far from having any physical meaning in the common sense Finite Set We rst consider a two-dimensional quantization of a N-element set that leads, for N v 4, to a Pauli algebra of observables Unit Interval We study two-dimensional (and higher-dimensional) quantizations of the unit segment



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$ ./test18 -a -b test1 -cd test2 test3 test4 Found the -a option Found the -b option, with parameter value test1 Found the -c option Parameter #1: test2 Parameter #2: test3 Parameter #3: test4 $

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Unit Circle We apply the same quantization procedure to the unit circle in the plane As an interesting byproduct of this fuzzy circle , we give an expression for the phase or angle operator, and we discuss its relevance in comparison with various phase operators proposed by many authors 15 Motions on Simple Geometries Two examples of coherent state quantization of classical motions taking place in simple geometries are presented in 15 Motion on the Circle Quantization of the motion of a particle on the circle (like the quantization of polar coordinates in a plane) is an old question with so far no really satisfactory answers Many questions concerning this subject have been addressed, more speci cally devoted to the problem of angular localization and related Heisenberg inequalities We apply our scheme of coherent state quantization to this particular problem Motion on the Hyperboloid Viewed as a 1 + 1 de Sitter Space-Time To a certain extent, the motion of a massive particle on a 1 + 1 de Sitter background, which means a one-sheeted hyperboloid embedded in a 2+1 Minkowski space, has characteristics similar to those of the phase space for the motion on the circle Hence, the same type of coherent state is used to perform the quantization Motion in an Interval We revisit the quantum motion in an in nite square well with our coherent state approach by exploiting the fact that the quantization problem is similar, to a certain extent, to the quantization of the motion on the circle S 1 However, the boundary conditions are different, and this leads us to introduce vector coherent states to carry out the quantization Motion on a Discrete Set of Points We end this series of examples by the consideration of a problem inspired by modern quantum geometry, where geometrical entities are treated as quantum observables, as they have to be in order for them to be promoted to the status of objects and not to be simply considered as a substantial arena in which physical objects live 16 Motion on the Torus 16 is devoted to the coherent states associated with the discrete Weyl Heisenberg group and to their utilization for the quantization of the chaotic motion on the torus 17 Fuzzy Geometries: Sphere and Hyperboloid In 17, we end this series of examples of coherent state quantization with the application of the procedure to familiar geometries, yielding a noncommutative or fuzzy structure for these objects.

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