2005/07/14 by Matthew Cargo, Cargo, Matthew, Alfonso Gracia-Saz +4
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.math-ph/0507032
34 pages, 2 figures
arxiv created 2005/07/14 · openalex publication_date 2005/07/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A normal form transformation is carried out on the operators of a complete set of commuting observables in a multidimensional, integrable quantum system, mapping them by unitary conjugation into functions of the harmonic oscillators in the various degrees of freedom. The transformation works at the level of the Weyl symbols of the operators, which are manipulated as formal power series in hbar by use of the Moyal star product. It is assumed that the Weyl symbol of one of the operators (the Hamiltonian) has a generic, stable fixed point in phase space. The normal form transformation takes place in a neighborhood of this fixed point. Once the normal form has been achieved, the Einstein-Brillouin-Keller or torus quantization rule follows easily, including higher order corrections in hbar. Crucial parts of the normal form transformation are not obvious generalizations of the one-dimensional case, nor is final quantization rule. The result raises some issues of differential geometry not found in the one-dimensional case.