Generalized Abelian Deformations: Application to Nambu Mechanics

TL;DR

This paper explores Abelian generalized deformations of polynomial products, providing an explicit example related to Nambu Mechanics, and introduces notions of triviality to analyze their properties in a quantum-mechanical setting.

Contribution

It constructs a specific non-trivial Abelian deformation example for su/2, linking it to a potential quantum description of Nambu Mechanics.

Findings

Zariski product is never trivial in strong or weak sense

Constructed example is strongly non-trivial

Provides a quantum-mechanical context for Nambu Mechanics

Abstract

We study Abelian generalized deformations of the usual product of polynomials introduced in hep-th/9602016. We construct an explicit example for the case of $su/2$ which provides a tentative of a quantum-mechanical description of Nambu Mechanics on $R^3$. By introducing the notions of strong and weak triviality of generalized deformations, we show that the Zariski product is never trivial in either sense, while the example constructed here in a quantum-mechanical context is only strongly non-trivial.