论文标题
通用量子统计的统一框架：规范分区功能，最大职业数量和波函数的排列阶段
Unified framework for generalized quantum statistics: canonical partition function, maximum occupation number, and permutation phase of wave function
论文作者
论文摘要
除了Bose和Fermi统计数据外，仍然存在各种广义量子统计。接近广义量子统计的两种方法：（1）在量子力学中，将波函数的排列对称性和（2）在统计力学中概括，概括了量子统计的最大职业数量。但是，这两种方法之间的联系是晦涩的。在本文中，我们建议一个统一的框架来描述各种广义量子统计。我们首先提供了遵守各种广义量子统计的理想$ n $粒子气体的规范分区功能的一般公式。然后，我们通过构造一种方法来获得置换阶段和从规范分区函数中获得最大职业数量来揭示波函数的排列阶段与最大职业数量之间的联系。在我们的方案中，波函数的置换阶段被推广到矩阵阶段，而不是数字。通常认为，不同种类的统计数据以最大数字区分。我们表明，最大职业人数不足以区分不同种类的广义量子统计。作为示例，我们讨论了统一框架中的一系列广义量子统计，从而提供相应的规范分区功能，最大职业数量和波函数的排列阶段。尤其是，我们提出了三种新型的广义量子统计数据，这些量子统计似乎是难题中缺失的作品。该方案的数学基础是不变矩阵的数学理论，schur-weyl二元性，对称函数以及置换组和单一组的表示理论。本文的结果在统计力学和此类数学理论之间建立了桥梁。
Beyond Bose and Fermi statistics, there still exist various kinds of generalized quantum statistics. Two ways to approach generalized quantum statistics: (1) in quantum mechanics, generalize the permutation symmetry of the wave function and (2) in statistical mechanics, generalize the maximum occupation number of quantum statistics. The connection between these two approaches, however, is obscure. In this paper, we suggest a unified framework to describe various kinds of generalized quantum statistics. We first provide a general formula of canonical partition functions of ideal $N$-particle gases obeying various kinds of generalized quantum statistics. Then we reveal the connection between the permutation phase of the wave function and the maximum occupation number, through constructing a method to obtain the permutation phase and the maximum occupation number from the canonical partition function. In our scheme, the permutation phase of wave functions is generalized to a matrix phase, rather than a number. It is commonly accepted that different kinds of statistics are distinguished by the maximum number. We show that the maximum occupation number is not sufficient to distinguish different kinds of generalized quantum statistics. As examples, we discuss a series of generalized quantum statistics in the unified framework, giving the corresponding canonical partition functions, maximum occupation numbers, and the permutation phase of wave functions. Especially, we propose three new kinds of generalized quantum statistics which seem to be the missing pieces in the puzzle. The mathematical basis of the scheme are the mathematical theory of the invariant matrix, the Schur-Weyl duality, the symmetric function, and the representation theory of the permutation group and the unitary group. The result in this paper builds a bridge between the statistical mechanics and such mathematical theories.
