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16 Quantizations of the Motion on the Torus
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of the quantized versions of observables for both types of quantizations Theorem 163 Suppose that for all smooth observable f on the torus, that is, f C (T2 ), for all k N, there exists C > 0 so that for all N M ( ) k Op
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Here is an area-preserving map on the torus such that N [0, 2 a [0, 2 and the unitary representative M ( ) satis es (1652) b Write M ( ) N = N N j j j for the eigenvalues and eigenfunctions of M ( ) Then there exists a set of indices E (N ) [1, N ] satisfying lim
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that, for all functions f C (T ) and for all maps J : N N J (N ) E (N ), we have the ergodicity property of the lower symbols: lim N (N ) Op J
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17 Fuzzy Geometries: Sphere and Hyperboloid
171 Introduction
This is an extension to the sphere S 2 and to the one-sheeted hyperboloid in R3 , the 1 + 1 de Sitter space-time, of the quantization of the unit circle as was described in 14 We show explicitly how the coherent state quantization of these manifolds leads to fuzzy or noncommutative geometries The quantization of the sphere rests upon the unitary irreducible representations D j of SO(3) or SU (2), j N/2 The quantization of the hyperboloid is carried out with unitary irreducible 1 representations of SO 0 (1, 2) or SU (1, 1) in the principal series, namely, U 2 +i , R The limit at in nite values of the representation parameters j and , restores commutativity for the geometries considered It should be stressed that we proceed to a noncommutative reading of a given geometry and not of a given dynamical system This means that we do not consider in our approach any time parameter and related evolution
172 Quantizations of the 2-Sphere
In this section, we proceed to the fuzzy quantization [212] of the sphere S 2 by using the coherent states built in 6 from orthonormal families of spin spherical harmonics Y jm j m j We recall that for a given such that 2 Z and j such that 2| | 2 j N there corresponds the continuous family of coherent states (69) living in a (2 j + 1)-dimensional Hermitian space For a given j, we thus get 2 j + 1 realizations, corresponding to the possible values of These realizations yield a family of 2 j + 1 nonequivalent quantizations of the sphere We show in particular that the case = 0 is singular in the sense that it maps the Cartesian coordinates of the 2-sphere to null operators We then establish the link between this coherent state quantization approach to the 2-sphere and the original Madore construction [178] of the fuzzy sphere and we examine the question of equivalence between the two procedures Note that a construction of the fuzzy sphere based on Gilmore Perelomov Radcliffe coherCoherent States in Quantum Physics Jean-Pierre Gazeau Copyright 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40709-5
17 Fuzzy Geometries: Sphere and Hyperboloid
ent states (in the case = j) was also carried out by Grosse and Pre najder [213] They proceeded with a covariant symbol calculus la Berezin with its corresponding -product However, their approach is different from the coherent state quantization illustrated here Appendix C contains a set of formulas concerning the group SU (2), its unitary representations, and the spin spherical harmonics, specially needed for a complete description of our coherent state approach to the 2-sphere