Deformation quantization and Nambu Mechanics

Abstract

Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over R of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

p -branes from generalized Yang-Mills theory

“…. , X µ 4k−1 NPB which suggests a new formulation of both classical and quantum mechanics for this kind of object [23]. To identify the kind of physical object described by W GYM ∞ we need to recall the conformally invariant, 4k-dimensional, σ-model action introduced in [24], [14]:…”

On Deformations of n-Lie Algebras

“…The quantization problem of Nambu Mechanics was investigated by Dito, Flato, Sternheimer and Takhtajan [35,36], see also [16,17,18]. Let M be an m-dimensional C ∞ -manifold and A be the algebra of smooth real-valued functions on M.…”

“…A non-trivial abelian deformation of the algebra of polynomials on R m doesn't exist because of the vanishing of the second Harrison cohomology group. Nevertheless, it is possible to construct an abelian associative deformation of the usual pointwise product of the following form 27) where β maps a real polynomial on R 3 to the symmetric algebra constructed over the polynomials on R 3 (β : A → Symm(A )). T is an "evaluation map" which allows to go back to (deformed) polynomials (T : Symm(A ) −→ A ).…”

Nambu quantum mechanics: a nonlinear generalization of geometric quantum mechanics

“…Thus this formulation circumvents 8 some of the well known problems encountered in the quantization of the Nambu dynamics [27].…”