论文标题
大N矩阵量子力学和分区代数中的置换对称性
Permutation symmetry in large N Matrix Quantum Mechanics and Partition Algebras
论文作者
论文摘要
我们描述了置换对称性对状态空间的含义以及一般尺寸$ n $矩阵的量子机械系统的动态。我们解决了一般的11-参数置换不变量子矩阵谐波振荡器哈密顿量，并计算规范分区函数。可以使用分区代数图来描述希尔伯特空间的置换不变扇区，用于将其描述，形成分区塔代数$ p_k（n）$的基础。整数$ k $被解释为量子力学中矩阵振荡器多项式的程度。描述了相互作用的汉密尔顿人的家族，这些家族是通过代表理论基础对置换不变子空间进行对角线化的，我们为$ n \ ge 2k $构建。其中包括低能国家是排列不变的汉密尔顿人，并可能引起与分区代数尺寸相关的较大基态脱落。基于对称性的量子机制在文献中讨论的许多身体伤疤可以在具有置换对称性的这些矩阵系统中实现。将矩阵索引值映射到晶格位点，可以在修改后的Bose-Hubbard模型的背景下实现机制。与对称组的Clebsch-Gordan多重性（Kronecker系数）相似，类似于AD/CFT研究的极端相关因子遵守选择规则。
We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size $N$. We solve the general 11- parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras $P_k(N)$. The integer $k$ is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalised by a representation theoretic basis for the permutation invariant subspace which we construct for $ N \ge 2k $. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realised in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realisation of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in AdS/CFT are shown to obey selection rules based on Clebsch-Gordan multiplicities (Kronecker coefficients) of symmetric groups.
