Deformation quantization and Nambu Mechanics
Deformation quantization and Nambu Mechanics
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变形量化和南部力学
DOI：
10.1007/bf02509794
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发表时间：
1996
影响因子：
2.4
通讯作者：
L. Takhtajan
中科院分区：
文献类型：
--
作者：
G. Dito;M. Flato;D. Sternheimer;L. Takhtajan
从形变量子化(星积)出发，研究了Nambu力学的量子化问题。在考虑了一些不可能的情况并将其与场量子化进行类比之后，提出了场的Zariski量子化的新方法(在这种情况下是多项式)，解决了量子化问题。这种量化是基于多个实变量的多项式在ℝ上的因式分解。我们通过定义由多项式生成的无限维域代数的变形来量子化这个代数，它是交换的、结合的和分配的。然后，这一过程适用于导数(Nambu括号所需的)，这确保了Nambu力学的基本恒等式也在量子水平上的有效性。我们的构造实际上比这里考虑的特殊情况更一般：它可以用于非常一般的定义恒等式和更一般的星积。
Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ℝ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.