On foundation of the generalized Nambu mechanics

Abstract

We outline 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 bracketa "binary" operation on classical observables on the phase space, to the "multiple" operation of higher order n ≥ 3. Nambu dynamics is described by the phase flow given by Nambu-Hamilton equations of motion-a system of ODE's which involves n − 1 "Hamiltonians". We introduce the fundamental identity for the Nambu bracket-a generalization of the Jacobi identity, as a consistency condition for the dynamics. We show that Nambu bracket structure defines an hierarchy of infinite families of "subordinated" structures of lower order, including Poisson bracket structure, which satisfy certain matching conditions. The notion of Nambu bracket enables to define Nambu-Poisson manifolds-phase spaces for the Nambu mechanics, which turn out to be more 'rigid" than Poisson manifolds-phase spaces for the Hamiltonian mechanics. We introduce the analog of the action form and the action principle for the Nambu mechanics. In its formulation dynamics of loops (n − 2-dimensional chains for the general n-ary case) naturally appears. We discuss several approaches to the quantization of 1 Nambu mechanics, based on the deformation theory, path integral formulation and on Nambu-Heisenberg "commutation" relations. In the latter formalism we present explicit representation of Nambu-Heisenberg relation in the n = 3 case.We emphasize the role ternary and higher order algebraic operations, and mathematical structures related to them, play in passing from Hamilton's to Nambu's dynamical picture.

Hopf structure in nambu-lien-algebras

“…Such a generalized Jacobi identity has been analyzed by several authors [3,6,7,10], in different contexts, and it was called fundamental identity in Ref. [7].…”

“…During the past two decades Nambu proposal has been a matter for many investigations [2] - [12] and the permanent interest for this issue is related with the recognition of the feasible physical richness and the mathematical beauty of ternary and higher algebraic systems. Recently such an algebraic structure has been analyzed and reformulated by Tachtajan [7] - [9] in an invariant geometrical form. He proposed the notion of Nambu-Lie "gebra", which is a generalization of Lie algebras for ternary (in general n-ary) case.…”

Deformation quantization and Nambu Mechanics

“…This turns out to be a nonstraightforward task [1], [20] and the usual approaches to quantization failed to give an appropriate solution. See Sect.…”

“…Let us first review some basic notions on Nambu-Poisson manifolds (the reader is referred to [20] for further details). Let M be a m-dimensional C ∞ -manifold.…”

“…Such a situation no longer prevails in Nambu Mechanics. The FI imposes strong constraints on Nambu-Poisson structures and the linear superposition of two Nambu-Poisson structures does not define in general a Nambu-Poisson structure (see [20]). In that sense, it seems hopeless to have some notion of point-particles in Nambu Mechanics and this fact suggests that quantization here will have more to do with a field-like approach than with a quantummechanical one, and we shall have to quantize the observables (functions) rather than the dynamical variables themselves.…”