On Foundation of the Generalized Nambu Mechanics
Leon Takhtajan

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 on classical observables and is described by Hambu-Hamilton equations of motion given by 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 (-dimensional objects) naturally appears in this…
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