无限维Hilbert空间中的热平衡分布
Thermal Equilibrium Distribution in Infinite-Dimensional Hilbert Spaces
摘要：
量子机械波函数上的热平衡分布是所谓的高斯调整投影（GAP）测量，$GAP(ρ_β)$，用于在逆温度$β$下的热密度算子$ρ_β$。更一般地，$GAP(ρ)$是希尔伯特空间中的单位球面上的任何密度算子$ρ$的概率测度（即，迹为1的正算子）。本文在无限维可分Hilbert空间中给出了$GAP(ρ)$的严格定义.它的存在性和唯一性遵循Prohorov定理的存在性和唯一性的高斯措施在希尔伯特空间与给定的均值和协方差。我们也给出了一个替代存在性证明。最后，证明了$GAP(ρ)$连续依赖于$ρ$，即$ρ$在迹范数下的收敛蕴涵着$GAP(ρ)$的弱收敛.
英文摘要：
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 $GAP(ρ)$.
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