论文标题
来自Kähler结构
Bosonic and fermionic Gaussian states from Kähler structures
论文作者
论文摘要
我们表明，可以独特地以其线性复杂结构$ j $ J $唯一的特征来表明，玻色谐和费米子高斯状态（也称为“挤压相干状态”），这是经典相位空间上的线性图。这扩展了基于协方差矩阵的常规高斯方法，并提供了一个统一的框架来同时处理玻色子和费米子。可以用兼容的Kähler结构的三重$（g，ω，j）$识别纯高斯州，由正确定的度量$ g $，符号$ g $组成，符号$ g $ $ω$和线性复杂结构$ j $，带有$ j^2 = -1 \！\！\！1 $。混合高斯状态也可以用这种三倍来识别，但使用$ j^2 \ neq -1 \！\！1 $。我们应用这些方法来展示如何将涉及高斯状态的计算减少为这些对象的代数操作，从而导致许多已知和一些未知的身份。我们将这些方法应用于（a）纠缠和复杂性的研究，（b）稳定系统的动力学，（c）驱动系统的动力学。由此，我们汇总了数学结构和公式的全面列表，以并排比较骨气和费米子高斯状态。
We show that bosonic and fermionic Gaussian states (also known as "squeezed coherent states") can be uniquely characterized by their linear complex structure $J$ which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple $(G,Ω,J)$ of compatible Kähler structures, consisting of a positive definite metric $G$, a symplectic form $Ω$ and a linear complex structure $J$ with $J^2=-1\!\!1$. Mixed Gaussian states can also be identified with such a triple, but with $J^2\neq -1\!\!1$. We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. We apply these methods to the study of (A) entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems. From this, we compile a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.
