Deformation Quantization of Superintegrable Systems and Nambu Mechanics

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.

LateX2e, 18 pages

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• Classical and Quantal Ternary Algebras
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• A toy model of open membrane field theory in constant 3-form flux
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• Extra Dimensions and Nonlinear Equations
• D1-brane in Constant R-R 3-form Flux and Nambu Dynamics in String Theory
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• Matrix Quantization of Classical Nambu Brackets and Super $p$-Branes
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• Study of the "Non-Abelian" Current Algebra of a Non-linear $\sigma$-Model
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• Classical dynamics in deformed spaces