Quantization with maximally degenerate Poisson brackets: The harmonic oscillator!
Abstract
Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.
Keywords
Cite
@article{arxiv.quant-ph/0306059,
title = {Quantization with maximally degenerate Poisson brackets: The harmonic oscillator!},
author = {Y. Nutku},
journal= {arXiv preprint arXiv:quant-ph/0306059},
year = {2008}
}