Deformation Quantization, Superintegrability, and Nambu Mechanics

TL;DR

This paper explores phase space quantization of superintegrable systems, demonstrating the effectiveness of Nambu brackets and comparing different quantization methods, with applications to nonlinear sigma-models and validation of Nambu's approach.

Contribution

It introduces a novel application of phase space quantization to superintegrable systems using Nambu brackets, validating Nambu's original proposal over other methods.

Findings

Successful quantization of Nambu brackets consistent with Nambu's original idea

Application to nonlinear sigma-models with elegant quantum Hamiltonians

Validation of Nambu brackets against Moyal quantization

Abstract

Phase Space is the framework best suited for quantizing superintegrable systems--systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally. We illustrate the power and simplicity of the method through new applications to nonlinear sigma-models, specifically for Chiral Models and de Sitter N-spheres, where the symmetric quantum hamiltonians amount to compact and elegant expressions, in accord with the Groenewold-van Hove theorem. Additional power and elegance is provided by the use of Nambu Brackets (linked to Dirac Brackets) involving the extra invariants of superintegrable models. The quantization of Nambu Brackets is then successfully compared to that of Moyal, validating Nambu's original proposal, while invalidating other proposals.