DEFORMATION QUANTIZATION AND NAMBU MECHANICS

Starting from deformation quantization (star-products), the quantizati on 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 approac h of Zariski quantization of fields (observables, functions, in this c ase 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 distr ibutive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identi ty of Nambu Mechanics also at the quantum level. Our construction is i n fact more general than the particular case considered here: it can b e utilized for quite general defining identities and for much more gen eral star-products.