近日，华中科技大学郭武中课题组研究了量子场论中密度矩阵的谱投影。2025年4月2日出版的《高能物理杂志》发表了这项成果。

课题组研究了量子场论中密度矩阵的谱投影。通过适当的正则化，密度矩阵的光谱投影有望得到良好的定义。这些投影仪可以使用Riesz投影公式获得，该公式允许研究组计算特征值的密度和投影状态中局部算子的期望值。结果发现，应力能量张量的期望值中存在普遍的发散项，其中系数普遍取决于特征值的密度和描述特征值对边界位置依赖性的函数。

利用投影态，研究组可以在量子场论中构建一系列新态，并讨论它们的一般性质，重点讨论全息方面。他们观察到量子涨落在半经典极限内被抑制。研究组还证明，之前使用引力路径积分构建的固定面积态可以通过适当叠加适当数量的投影态来构建。此外，他们将谱投影应用于非厄米算子，如转移矩阵，以获得它们的特征值和密度。最后，研究组强调了光谱投影的潜在应用，包括构建新的密度和转移矩阵，以及理解几何态的叠加。

附：英文原文

Title: Spectral projections for density matrices in quantum field theories

Author: Guo, Wu-zhong

Issue&Volume: 2025-04-02

Abstract: In this paper, we investigate the spectral projection of density matrices in quantum field theory. With appropriate regularization, the spectral projectors of density matrices are expected to be well-defined. These projectors can be obtained using the Riesz projection formula, which allows us to compute both the density of eigenvalues and the expectation values of local operators in the projected states. We find that there are universal divergent terms in the expectation value of the stress energy tensor, where the coefficients depend universally on the density of eigenvalues and a function that describes the dependence of eigenvalues on boundary location. Using projection states, we can construct a series of new states in quantum field theories and discuss their general properties, focusing on the holographic aspects. We observe that quantum fluctuations are suppressed in the semiclassical limit. We also demonstrate that the fixed area state, previously constructed using gravitational path integrals, can be constructed by suitably superposition of appromiate amount of projection states. Additionally, we apply spectral projection to non-Hermitian operators, such as transition matrices, to obtain their eigenvalues and densities. Finally, we highlight potential applications of spectral projections, including the construction of new density and transition matrices and the understanding of superpositions of geometric states.

DOI: 10.1007/JHEP04(2025)033

Source: https://link.springer.com/article/10.1007/JHEP04(2025)033