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Finite dimensional quantizations of the (q,p) plane : new space and momentum inequalities

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 $λ\_m(N) λ\_M(N)$ of non null smallest positive and largest eigenvalues, we infer the inequality $δ\_N(Q) Δ\_N(Q) = σ\_N \overset{<}{\underset{N \to \infty}{\to}} 2 π$ (resp. $δ\_N(P) Δ\_N(P) = σ\_N \overset{<}{\underset{N \to \infty}{\to}} 2 π$) involving, in suitable units, the minimal ($δ\_N(Q)$) and maximal ($Δ\_N(Q)$) 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.