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International Journal of Modern Physics B 20, 11/13 (2006) 1778 - 1791
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Finite dimensional quantizations of the (q,p) plane : new space and momentum inequalities
J.-P. Gazeau1, F.-X. Josse-Michaux1, Pascal Monceau2

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 non null smallest positive and largest eigenvalues, we infer the inequality $\delta_N(Q) \Delta_N(Q) = \sigma_N \overset{<}{\underset{N \to \infty}{\to}} 2 \pi$ (resp. $\delta_N(P) \Delta_N(P) = \sigma_N \overset{<}{\underset{N \to \infty}{\to}} 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 and connections with the finite Chern-Simons matrix model for the Quantum Hall effect are discussed.
1 :  APC - UMR 7164 - AstroParticule et Cosmologie
2 :  MSC - Matière et Systèmes Complexes
Physique/Physique Quantique
Quantization – coherent states – localisation – Chern-Simons
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