Moshe Flato, Giuseppe Dito and Daniel Sternheimer
Nambu mechanics, $n$-ary operations and their quantization
(72K, LaTeX)
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.)