97-112 Moshe Flato, Giuseppe Dito and Daniel Sternheimer

Nambu mechanics, $n$-ary operations and their quantization (72K, LaTeX) Mar 10, 97

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Abstract. We start with an overview of the ``generalized Hamiltonian dynamics" introduced in 1973 by Y. Nambu, its motivations, mathematical background and subsequent developments -- all of it on the classical level. This includes the notion (not present in Nambu's work) of a generalization of the Jacobi identity called Fundamental Identity. We then briefly describe the difficulties encountered in the quantization of such $n$-ary structures, explain their reason and present the recently obtained solution combining deformation quantization with a ``second quantization" type of approach on ${\Bbb R}^n$. The solution is called ``Zariski quantization" because it is based on the factorization of (real) polynomials into irreducibles. Since we want to quantize composition laws of the determinant (Jacobian) type and need a Leibniz rule, we need to take care also of derivatives and this requires going one step further (Taylor developments of polynomials over polynomials). We also discuss a (closer to the root, ``first quantized") approach in various circumstances, especially in the case of covariant star products (exemplified by the case of ${\frak {su}}(2)$). Finally we address the question of equivalence and triviality of such deformation quantizations of a new type (the deformations of algebras are more general than those considered by Gerstenhaber). (Comments: 23 pages, LaTeX2e with the LaTeX209 option. To be published in the proceedings of the Ascona meeting. Mathematical Physics Studies, volume 20, Kluwer.)

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