Giuseppe Dito, Moshe Flato, Daniel Sternheimer, Leon Takhtajan
Deformation Quantization and Nambu Mechanics
(72K, LaTeX)
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 what we call the
Zariski quantization of fields (observables, functions, in this case
polynomials). This quantization is based on the factorization over ${\Bbb R}$
of polynomials in several real variables. We quantize the 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.