 97112 Moshe Flato, Giuseppe Dito and Daniel Sternheimer
 Nambu mechanics, $n$ary operations and their quantization
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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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