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Thermal Equilibrium Distribution in Infinite-Dimensional Hilbert Spaces

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 $GAP(ρ)$.