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Quantization with maximally degenerate Poisson brackets: The harmonic oscillator!

Y. Nutku
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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.

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Y. Nutku. 2003-06-09. Quantization with maximally degenerate Poisson brackets: The harmonic oscillator!. https://doi.org/10.1088/0305-4470%2F36%2F27%2F308

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