置换不变代数、Fock空间实现和Calogero模型
Permutation Invariant Algebras, a Fock Space Realization and the Calogero Model
摘要：
本文利用三种等价的方法研究了置换不变振子代数及其Fock空间表示，即（i）在Fock空间中建立和湮灭算符的正常序展开，（ii）湮灭算符对单项式态的作用，（iii）Fock空间中内积的Gram矩阵.分别讨论了具有厄米数算子的置换不变代数和具有非厄米数算子的置换不变代数。将一般分析的结果应用于M体Calogero模型下的S_M扩展Heisenberg代数。特别注意的是专门用于分析的Calogero模型的革兰氏矩阵。我们讨论了它们的结构、本征值和本征态。得到了本征值为正的一般条件，即满足这个条件时，Fock空间中所有态的范数都为正。我们发现了一个普遍的临界点，在这个临界点上，物理自由度发生了减少。我们构造对偶算子，从而得到自由玻色振子的普通海森堡代数。本文从Fock空间的观点出发，简要讨论了Calogero振子到自由玻色振子以及自由玻色振子到Calogero振子映射的存在性。
英文摘要：
We study permutation invariant oscillator algebras and their Fock space representations using three equivalent techniques, i.e. (i) a normally ordered expansion in creation and annihilation operators, (ii) the action of annihilation operators on monomial states in Fock space and (iii) Gram matrices of inner products in Fock space. We separately discuss permutation invariant algebras which possess hermitean number operators and permutation invariant algebras which possess non-hermitean number operators. The results of a general analysis are applied to the S_M extended Heisenberg algebra, underlying the M-body Calogero model. Particular attention is devoted to the analysis of Gram matrices for the Calogero model. We discuss their structure, eigenvalues and eigenstates. We obtain a general condition for positivity of eigenvalues, meaning that all norms of states in Fock space are positive if this condition is satisfied. We find a universal critical point at which the reduction of the physical degrees of freedom occurs. We construct dual operators, leading to the ordinary Heisenberg algebra of free Bose oscillators. From the Fock-space point of view, we briefly discuss the existence of mapping from the Calogero oscillators to the free Bose oscillators and vice versa.
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