Nambu mechanics, $n$-ary operations and their quantization

Details Nambu mechanics, $n$-ary operations and their quantization Nambu mechanics, $n$-ary operations and their quantization

1997-03-10

arxiv.org/abs/q-alg/9703019v1

Mathematics

Quantum Algebra

23 pages, LaTeX2e with the LaTeX209 option. To be published in the proceedings of the Ascona meeting. Mathematical Physics Stu

Scientific paper

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 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).

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