On Foundation of the Generalized Nambu Mechanics

TL;DR

This paper explores the foundational principles of Nambu mechanics, a generalization of Hamiltonian mechanics involving higher-order brackets, and discusses its mathematical structure, dynamics, and quantization approaches.

Contribution

It introduces the fundamental identity for Nambu brackets, explores lower-order structures, and presents explicit quantization representations for the case n=3.

Findings

Nambu brackets generalize Poisson brackets to higher orders.

Fundamental identity replaces Jacobi identity for consistency.

Explicit Nambu-Heisenberg relations for n=3 case.

Abstract

We outline the basic principles of canonical formalism for the Nambu mechanics---a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in 1973. It is based on the notion of Nambu bracket which generalizes the Poisson bracket to the multiple operation of higher order $n \geq 3$ on classical observables and is described by Hambu-Hamilton equations of motion given by $n-1$ Hamiltonians. We introduce the fundamental identity for the Nambu bracket which replaces Jacobi identity as a consistency condition for the dynamics. We show that Nambu structure of given order defines a family of subordinated structures of lower order, including the Poisson structure, satisfying certain matching conditions. We introduce analogs of action from and principle of the least action for the Nambu mechanics and show how dynamics of loops ($n-2$-dimensional objects) naturally appears in this formalism. We discuss several approaches to the quantization problem and present explicit representation of Nambu-Heisenberg commutation relation for $n=3$ case. We emphasize the role higher order algebraic operations and mathematical structures related with them play in passing from Hamilton's to Nambu's dynamical picture.