JT引力中的系综平均源于矩阵量子力学中的纠缠
Ensemble averaging in JT gravity from entanglement in Matrix Quantum Mechanics
AI总结：
本文将任意谱曲线的矩阵积分推广至矩阵量子力学，构建了特征值的流体力学有效理论，证明JT引力对偶于MQM的单时间点约缩，并计算了特征值二分的Rényi熵及子集约化密度矩阵。
摘要：
我们考虑将具有任意谱曲线 $ρ_0(E)$ 的矩阵积分推广为矩阵量子力学（MQM）的0+1D理论。利用大-$N$ 下1D量子系统的最新技术，我们为特征值构建了流体力学有效理论。结果是在弯曲背景上的简单2D自由玻色子BCFT，描述了特征值围绕 $ρ_0(E)$ 的量子涨落，此时 $ρ_0(E)$ 成为特征值密度算符 $\hatρ(E)$ 量子期望值的大-$N$ 极限。对随机矩阵系综的平均变成了量子期望值。等时密度关联重现了随机矩阵理论的结果（包括非微扰修正）。这表明可将JT引力解释为MQM的 $\textit{单时间点}$ 约缩的对偶。作为应用，我们计算了与特征值二分相关的Rényi熵。我们匹配了Hartnoll和Mazenc关于对偶于二维弦理论的 $c=1$ 矩阵模型的先前结果，并将其推广至任意 $ρ_0(E)$。该流体力学理论为二维弦理论中时空的涌现提供了清晰图景。熵自然是有限的，并展现出大量短程纠缠，与微正则熵成正比。我们还计算了 $n<N$ 个特征值子集的约化密度矩阵。
英文摘要：
We consider the generalization of a matrix integral with arbitrary spectral curve $ρ_0(E)$ to a 0+1D theory of matrix quantum mechanics (MQM). Using recent techniques for 1D quantum systems at large-$N$, we formulate a hydrodynamical effective theory for the eigenvalues. The result is a simple 2D free boson BCFT on a curved background, describing the quantum fluctuations of the eigenvalues around $ρ_0(E)$, which is now the large-$N$ limit of the quantum expectation value of the eigenvalue density operator $\hatρ(E)$. The average over the ensemble of random matrices becomes a quantum expectation value. Equal-time density correlations reproduce the results (including non-perturbative corrections) of random matrix theory. This suggests an interpretation of JT gravity as dual to a $\textit{one-time-point}$ reduction of MQM. As an application, we compute the Rényi entropy associated to a bipartition of the eigenvalues. We match a previous result by Hartnoll and Mazenc for the $c=1$ matrix model dual to two-dimensional string theory and extend it to arbitrary $ρ_0(E)$. The hydrodynamical theory provides a clear picture of the emergence of spacetime in two dimensional string theory. The entropy is naturally finite and displays a large amount of short range entanglement, proportional to the microcanonical entropy. We also compute the reduced density matrix for a subset of $n<N$ eigenvalues.
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