Deformation Quantization of Superintegrable Systems and Nambu Mechanics

TL;DR

This paper explores phase space quantization of superintegrable systems using deformation quantization and Nambu brackets, demonstrating new applications to nonlinear sigma models and validating Nambu's quantization approach.

Contribution

It introduces a phase space quantization method for superintegrable systems that incorporates Nambu brackets, with new applications to nonlinear sigma models and validation of Nambu's quantization.

Findings

Successful quantization of nonlinear sigma models on de Sitter N-spheres and Chiral Models.

Introduction of classical Nambu brackets and their quantization.

Validation of Nambu's original quantization proposal through comparison with Moyal quantization.

Abstract

Phase Space is the framework best suited for quantizing superintegrable systems, naturally preserving the symmetry algebras of the respective hamiltonian invariants. The power and simplicity of the method is fully illustrated through new applications to nonlinear sigma models, specifically for de Sitter N-spheres and Chiral Models, where the symmetric quantum hamiltonians amount to compact and elegant expressions. Additional power and elegance is provided by the use of Nambu Brackets to incorporate the extra invariants of superintegrable models. Some new classical results are given for these brackets, and their quantization is successfully compared to that of Moyal, validating Nambu's original proposal.