Finite dimensional quantizations of the particle motion: toward new space and momentum inequalities?
Résumé
We present a N-dimensional quantization à 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 largest and non null smallest positive eigenvalues, we infer the inequality \delta_N(Q) \Delta_N(Q) \approx < 2 \pi (resp. \delta_N(P) \Delta_N(P)\approx < 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 are discussed.