The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, $GAP(\rho_\beta)$, for a thermal density operator $\rho_\beta$ at inverse temperature $\beta$. More generally, $GAP(\rho)$ is a probability measure on the unit sphere in Hilbert space for any density operator $\rho$ (i.e., a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of $GAP(\rho)$ 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 $GAP(\rho)$ depends continuously on $\rho$ in the sense that convergence of $\rho$ in the trace norm implies weak convergence of $GAP(\rho)$.

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无限维希尔伯特空间中的热平衡分布

量子力学波函数上的热平衡分布是所谓的高斯调整投影 (GAP) 度量，$GAP(\rho_\beta)$，对于在逆温度 $\beta 下的热密度算子 $\rho_\beta$ $. 更一般地说，$GAP(\rho)$ 是希尔伯特空间中单位球面上对于任何密度算子$\rho$（即具有迹线1 的正算子）的概率测度。在这篇笔记中，我们收集了关于在无限维可分希尔伯特空间中 $GAP(\rho)$ 的严格定义的数学细节。它的存在性和唯一性遵循 Prohorov 关于在给定均值和协方差的 Hilbert 空间中高斯测度的存在性和唯一性的定理。我们还给出了另一种存在证明。最后，