[Paper Review] Generalized Abelian Deformations: Application to Nambu Mechanics

Giuseppe Dito, Moshé Flato|arXiv (Cornell University)|Sep 15, 1996

Advanced Topics in Algebra6 references3 citations

TL;DR

This paper introduces generalized Abelian deformations of polynomial products, constructing a quantum-mechanical model for Nambu mechanics on R³ using su(2) algebra. It demonstrates that the deformation is strongly non-trivial, providing a non-trivial quantum realization of Nambu brackets via a specific product structure that avoids triviality in both strong and weak senses.

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.

Motivation & Objective

• To develop a generalized Abelian deformation framework for polynomial products beyond standard star products.
• To apply this deformation formalism to the quantization of Nambu mechanics, a generalization of Hamiltonian mechanics with three-fold brackets.
• To construct a concrete quantum model on R³ using the su(2) algebra as a realization of Nambu structure.
• To analyze the triviality of deformations using strong and weak notions of triviality in deformation theory.
• To show that the constructed deformation is non-trivial in the strong sense, distinguishing it from trivial or weakly trivial deformations.

Proposed method

• Generalizing Abelian deformations by extending the product structure on polynomial algebras using cohomological techniques.
• Introducing a new product, the Zariski product, as a candidate for deformation in the context of Nambu structures.
• Constructing a specific deformation using the su(2) Lie algebra to model quantum Nambu mechanics on R³.
• Defining strong and weak triviality of deformations via cohomological obstructions and equivalence classes.
• Using the RIMS preprint no. 1103 and arXiv:hep-th/9609114 as the formal basis for the construction.
• Applying techniques from quantum algebra and deformation quantization to ensure consistency with Nambu bracket axioms.

Experimental results

Research questions

• RQ1 Can generalized Abelian deformations provide a consistent quantum framework for Nambu mechanics?
• RQ2 Is the Zariski product, as a deformation candidate, trivial in the strong or weak sense?
• RQ3 Does a non-trivial quantum realization of Nambu mechanics exist on R³ using su(2) algebra?
• RQ4 How do strong and weak triviality conditions constrain the possible deformations of polynomial algebras?
• RQ5 Can a deformation be constructed that is strongly non-trivial while still respecting the Nambu bracket structure?

Key findings

• The Zariski product is shown to be non-trivial in both the strong and weak senses, indicating its robustness as a deformation structure.
• The constructed deformation using su(2) is strongly non-trivial, meaning it cannot be reduced to a trivial product via equivalence transformations.
• The deformation provides a concrete quantum-mechanical model for Nambu mechanics on R³, realized through a non-trivial product on polynomials.
• The method successfully avoids weak triviality, ensuring the deformation is not cohomologically trivial.
• The result establishes a link between generalized Abelian deformations and higher-order Poisson structures via Nambu brackets.
• The construction is consistent with the axioms of Nambu mechanics and provides a new path toward deformation quantization of higher-order brackets.