dorsal/arxiv
View SchemaFinite dimensional quantizations of the (q,p) plane : new space and momentum inequalities
| Authors | Jean-Pierre Gazeau, François-Xavier Josse-Michaux, Pascal Monceau |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0411210 |
| URL | https://arxiv.org/abs/quant-ph/0411210 |
| DOI | 10.1142/S0217979206034285 |
| Journal | International Journal of Modern Physics B 20, 11/13 (2006) 1778 - 1791 |
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 $\delta\_N(Q) \Delta\_N(Q) = \sigma\_N \overset{<}{\underset{N \to \infty}{\to}} 2 \pi$ (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.
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"abstract": "We present a N-dimensional quantization a la Berezin-Klauder or frame\nquantization of the complex plane based on overcomplete families of states\n(coherent states) generated by the N first harmonic oscillator eigenstates. The\nspectra of position and momentum operators are finite and eigenvalues are\nequal, up to a factor, to the zeros of Hermite polynomials. From numerical and\ntheoretical studies of the large $N$ behavior of the product $\\lambda\\_m(N)\n\\lambda\\_M(N)$ of non null smallest positive and largest eigenvalues, we infer\nthe inequality $\\delta\\_N(Q) \\Delta\\_N(Q) = \\sigma\\_N \\overset{\u003c}{\\underset{N\n\\to \\infty}{\\to}} 2 \\pi$ (resp. $\\delta\\_N(P) \\Delta\\_N(P) = \\sigma\\_N\n\\overset{\u003c}{\\underset{N \\to \\infty}{\\to}} 2 \\pi $) involving, in suitable\nunits, the minimal ($\\delta\\_N(Q)$) and maximal ($\\Delta\\_N(Q)$) sizes of\nregions of space (resp. momentum) which are accessible to exploration within\nthis finite-dimensional quantum framework. Interesting issues on the\nmeasurement process and connections with the finite Chern-Simons matrix model\nfor the Quantum Hall effect are discussed.",
"arxiv_id": "quant-ph/0411210",
"authors": [
"Jean-Pierre Gazeau",
"Fran\u00e7ois-Xavier Josse-Michaux",
"Pascal Monceau"
],
"categories": [
"quant-ph"
],
"doi": "10.1142/S0217979206034285",
"journal_ref": "International Journal of Modern Physics B 20, 11/13 (2006) 1778 -\n 1791",
"title": "Finite dimensional quantizations of the (q,p) plane : new space and momentum inequalities",
"url": "https://arxiv.org/abs/quant-ph/0411210"
},
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