FINITE DIMENSIONAL QUANTIZATIONS OF THE (q,p) PLANE: NEW SPACE AND MOMENTUM (OR QUADRATURES) INEQUALITIES

Abstract

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 number eigenstates. The spectra of position and momentum operators (or of quadrature operators in quantum optics) 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 λm(N) λM(N) of non null smallest positive and largest eigenvalues, we infer the inequality [Formula: see text] (resp. [Formula: see text]) involving, in suitable units, the minimal (δN(Q)) and maximal (ΔN(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.