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13 Coherent State or Frame Quantization
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131 Introduction
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Physics is part of the natural sciences and its prime object is what we call nature , or rather, in a more restrictive sense time , space , matter , energy , and interaction , which appear at a certain moment of the process in the form of signi cant data, such as position, speed, and frequencies So the question arises how to process those data, and this raises the question of a selected point of view or frame Faced with a set of raw collected data encoded into a certain mathematical form and provided by a measure, that is, a function which attributes a weight of importance to subsets of data, we give in addition more or less importance to different aspects of those data by choosing, opportunistically, the most appropriate frame of analysis We include in this general scheme the quantum processing, that is, the way of considering objects from a quantum point of view, exactly like we quantize the classical phase space in quantum mechanics To a certain extent, quantization pertains to a larger discipline than just restricting ourselves to speci c domains of physics such as mechanics or eld theory The aim of this chapter is to provide a generalization of the Berezin Klauder Toeplitz quantization illustrated in 12, precisely the quantization of a measure space X once we are given a family of coherent states or frame constructed along the lines given in 5 We also develop the notion of lower and upper symbols resulting from such a quantization scheme, and nally discuss the probabilistic content of the construction A quite elementary example, the quantization of the circle with 2 ~ 2 matrices, is presented as an immediate illustration of the formalism
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132 Some Ideas on Quantization
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Many reviews exist on the quantization problem and the variety of approaches for solving it; see, for instance, the recent ones by Ali and Englis [155] and by Landsman [174] For simpli cation, let us say that a quantization is a procedure that
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Coherent States in Quantum Physics Jean-Pierre Gazeau Copyright 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40709-5
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13 Coherent State or Frame Quantization
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associates with an algebra Acl of classical observables an algebra Aq of quantum observables The algebra Acl is usually realized as a commutative Poisson algebra 13) of derivable functions on a symplectic (or phase) space X The algebra Aq is, however, noncommutative in general and the quantization procedure must provide a correspondence Acl Aq : f A f Various procedures of quantization exist, and minimally require the following conditions, which loosely parallel those listed in Section 123: With the constant function 1 is associated the unity of Aq , The commutation relations of Aq reproduce the Poisson relations of Acl Moreover, they offer a realization of the Heisenberg algebra Aq is realized as an algebra of operators acting in some Hilbert space Most physical quantum theories may be obtained as the result of a canonical quantization procedure However, the prescriptions for the latter appear quite arbitrary Moreover, it is dif cult, if not impossible, to implement it covariantly It is thus dif cult to generalize this procedure to many systems Geometrical quantization [175] exploits fully the symplectic structure of the phase space, but generally requires more structure, such as a symplectic potential, for example, the Legendre form on the cotangent bundle of a con guration space In this regard, the deformation quantization appears more general in the sense that it is based on the symplectic structure only and it preserves symmetries (symplectomorphisms) [176, 177] The coherent state quantization that is presented here is by far more universal since it does not even require a symplectic or Poisson structure The only structure that a space X must possess is a measure This procedure can be considered from different viewpoints: It is mostly genuine in the sense that it veri es all the requirements above, including those relative to the Poisson structure when the latter is present It may also be viewed as a fuzzy cation of X: an algebra Acl of functions on X is replaced by an algebra Aq of operators, which may be considered as the coordinates of a fuzzy version of X, even though the original X is not equipped with a manifold structure The term fuzzy manifold, where points have noncommutative coordinates, was proposed by Madore [178] in his presentation of noncommutative geometry of simple manifolds such as the sphere As a matter of fact, ordinary quantum mechanics may also be viewed as a noncommutative version of the geometry of the phase space, where position and momentum operators do not commute [179] In this regard, the quantization of a set of data makes a fuzzy or noncommutative geometry emerge
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13) A Poisson algebra A is an associative algebra together with a Lie bracket X , Y {X , Y } that also satis es the Leibniz law: for any X A the action D X (Y ) = {X , Y } on A is a derivation {X , Y Z } = {X , Y }Z + Y {X , Z }
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As a matter of fact, the space of real-valued smooth functions over a symplectic manifold, when equipped with the Poisson bracket, forms a Poisson algebra
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