We establish a rigorous bundle isomorphism between the complex velocity field $η_μ = π_μ - i u_μ$, obtained by averaging matter dynamics over stochastic gravitational fluctuations, and the symmetric logarithmic derivative (SLD) operator $L_μ$ of quantum estimation theory. The isomorphism $\widetilde{\mathcal{T}}: Γ(E/{\sim}) \to Γ(\mathcal{L})$ maps gauge-equivalence classes of sections of the pullback bundle $E = π_2^*(T^*M)$ over $\mathcal{C} \times M$ to SLD operators on the Hilbert space $\mathcal{H}_0 = L^2(\mathcal{C}, ν_0)$, where $\mathcal{C}$ is the infinite-dimensional Fréchet manifold of matter fields and $ν_0$ is a fixed Gaussian measure. We prove that $\widetilde{\mathcal{T}}$ and the associated quantum Fisher metric are independent of the choice of $ν_0$, rendering the construction intrinsic to the physical probability density. The Fisher metric acquires a simple form in terms of the Madelung--Bohm velocities: $g_{μν}^{\mathrm{FS}} = \frac{4m^2}{\hbar^2} \bigl[\operatorname{Cov}(π_μ,π_ν) + \operatorname{Cov}(u_μ,u_ν)\bigr]_{\mathcal{P}}$. As a consequence, the flat $U(1)$ connection defined by $η_μ$ yields a quantized holonomy for non-contractible spacetime loops, predicting topological phases that may be observable in atom interferometry.

翻译：摘要：我们建立了复速度场 $\eta_\mu = \pi_\mu - i u_\mu$（通过平均物质动力学在随机引力涨落上的作用得到）与量子估计理论中的对称对数导数算子 $L_\mu$ 之间的严格丛同构。同构 $\widetilde{\mathcal{T}}: \Gamma(E/{\sim}) \to \Gamma(\mathcal{L})$ 将拉回丛 $E = \pi_2^*(T^*M)$（定义在 $\mathcal{C} \times M$ 上）的截面的规范等价类映射到希尔伯特空间 $\mathcal{H}_0 = L^2(\mathcal{C}, \nu_0)$ 上的对称对数导数算子，其中 $\mathcal{C}$ 是无限维 Fréchet 流形（物质场空间），$\nu_0$ 是固定的高斯测度。我们证明 $\widetilde{\mathcal{T}}$ 及关联的量子 Fisher 度量与 $\nu_0$ 的选择无关，从而使得构造内蕴于物理概率密度。Fisher 度量采用马德隆-玻姆速度的简洁形式：$g_{\mu\nu}^{\mathrm{FS}} = \frac{4m^2}{\hbar^2} \bigl[\operatorname{Cov}(\pi_\mu,\pi_\nu) + \operatorname{Cov}(u_\mu,u_\nu)\bigr]_{\mathcal{P}}$。由此，由 $\eta_\mu$ 定义的平坦 $U(1)$ 联络对非可缩时空回路产生量子化全纯性，预言了可能在原子干涉测量中观测到的拓扑相位。