arXiv quant-ph/06070742006-07quant-phmath-phmath.COmath.MP

玻色子算子函数的非交叉正规序

Noncrossing normal ordering for functions of boson operators

Toufik Mansour, Matthias Schork, Simone Severini

备注：: 16 pages, 10 figures
DOI：: 10.1007/s10773-007-9508-x

摘要：

玻色子算符的函数的正常有序形式在许多情况下是重要的，特别是关于量子场论和量子光学。从Katriel的开创性工作开始[Lett. Nuovo Cimento，10（13）：565- 567，1974]，在过去的几年里，正常有序形式已经被证明具有丰富的组合结构，主要是凭借与划分理论的联系。在本文中，我们试图丰富这一联系。通过考虑非交叉分拆的线性表示，我们定义了非交叉正规序的概念。由于非交叉划分有许多意想不到的联系（例如，与自由概率的联系），非交叉正规序似乎是一个有趣的概念。我们明确地给出了函数（a^{r}（a^<$）^{s}）^{n}）和（a^{r}+（a^<$）^{s}）^{n}的非交叉正规序形式，以及各种特殊情况。我们能够建立这些功能，k元树和集的格路径之间的非交叉收缩的第一次双射。

英文摘要：

Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento, 10(13):565--567, 1974], in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (a^{r}(a^†)^{s})^{n}) and (a^{r}+(a^†)^{s})^{n}, plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.

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摘要：

英文摘要：

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