论文标题
无限维克斯空间中的热平衡分布
Thermal Equilibrium Distribution in Infinite-Dimensional Hilbert Spaces
论文作者
论文摘要
对量子力学波函数的热平衡分布是一种所谓的高斯调整后的投影（GAP）度量，$ GAP（ρ_β）$，对于热密度运算符$ρ_β$在相反的温度下$β$。更一般而言，$ GAP（ρ）$是任何密度操作员$ρ$（即带有跟踪1的正面操作员）的希尔伯特空间单位球的概率度量。在本说明中，我们收集了有关无限二维可分离的希尔伯特空间的严格定义（ρ）$的严格定义的数学详细信息。它的存在和唯一性来自Prohorov的定理，关于具有给定平均值和协方差的希尔伯特空间中高斯措施的存在和独特性。我们还提供了另一种证明。最后，我们给出证明$ GAP（ρ）$在$ρ$上不断取决于$ρ$在跟踪规范中的收敛意味着$ GAP（ρ）$的弱收敛性。
The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, $GAP(ρ_β)$, for a thermal density operator $ρ_β$ at inverse temperature $β$. More generally, $GAP(ρ)$ is a probability measure on the unit sphere in Hilbert space for any density operator $ρ$ (i.e., a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of $GAP(ρ)$ in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prohorov's theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Finally, we give a proof that $GAP(ρ)$ depends continuously on $ρ$ in the sense that convergence of $ρ$ in the trace norm implies weak convergence of $GAP(ρ)$.
