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

Abstract

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 $\delta \_N(Q)\Delta \_N(Q)=\sigma \_N\stackrel{<}{\underset{N\to \infty }{\to }}2\pi$ (resp. $\delta \_N(P)\Delta \_N(P)=\sigma \_N\stackrel{<}{\underset{N\to \infty }{\to }}2\pi$) 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.

References in corpus (2)

• Vector Coherent States from Plancherel's Theorem, Clifford Algebras and Matrix Domains
• Coherent state quantization of a particle in de Sitter space

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