Deformation Quantization and Nambu Mechanics

TL;DR

This paper introduces a novel Zariski quantization method for Nambu Mechanics, utilizing polynomial factorization to achieve a consistent deformation quantization that preserves fundamental identities.

Contribution

It presents a new approach to quantize Nambu Mechanics through Zariski quantization of fields, ensuring algebraic consistency and extending to general star-products.

Findings

Successfully quantized the algebra of fields with a deformation preserving Nambu's fundamental identity.

Demonstrated the method's applicability to general identities and star-products.

Provided a framework for consistent deformation quantization in infinite-dimensional field algebras.

Abstract

Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ${\Bbb R}$ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.