dorsal/arxiv
View SchemaNambu mechanics, $n$-ary operations and their quantization
| Authors | Moshe Flato, Giuseppe Dito, Daniel Sternheimer |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9703019 |
| URL | https://arxiv.org/abs/q-alg/9703019 |
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 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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"abstract": "We start with an overview of the \"generalized Hamiltonian dynamics\"\nintroduced in 1973 by Y. Nambu, its motivations, mathematical background and\nsubsequent developments -- all of it on the classical level. This includes the\nnotion (not present in Nambu\u0027s work) of a generalization of the Jacobi identity\ncalled Fundamental Identity. We then briefly describe the difficulties\nencountered in the quantization of such $n$-ary structures, explain their\nreason and present the recently obtained solution combining deformation\nquantization with a \"second quantization\" type of approach on ${\\Bbb R}^n$. The\nsolution is called \"Zariski quantization\" because it is based on the\nfactorization of (real) polynomials into irreducibles. Since we want to\nquantize composition laws of the determinant (Jacobian) type and need a Leibniz\nrule, we need to take care also of derivatives and this requires going one step\nfurther (Taylor developments of polynomials over polynomials). We also discuss\na (closer to the root, \"first quantized\") approach in various circumstances,\nespecially in the case of covariant star products (exemplified by the case of\nsu(2)). Finally we address the question of equivalence and triviality of such\ndeformation quantizations of a new type (the deformations of algebras are more\ngeneral than those considered by Gerstenhaber).",
"arxiv_id": "q-alg/9703019",
"authors": [
"Moshe Flato",
"Giuseppe Dito",
"Daniel Sternheimer"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Nambu mechanics, $n$-ary operations and their quantization",
"url": "https://arxiv.org/abs/q-alg/9703019"
},
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