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 \(η_μ\): \(g_{μν}^{\mathrm{FS}} = -\frac{4m^2}{\hbar^2} \operatorname{Re}\langle(η_μ - \langleη_μ\rangle) (η_ν - \langleη_ν\rangle)\rangle_{\mathcal{P}}\). As a consequence, the flat \(U(1)\) connection defined by \(η_μ\) yields a quantized holonomy for non-contractible spacetime loops, predicting topological phases observable in atom interferometry.

翻译：我们建立了复速度场 \(η_μ = π_μ - i u_μ\)（通过平均随机引力涨落下物质动力学获得）与量子估计理论中的对称对数导数（SLD）算子 \(L_μ\) 之间的严格丛同构。同构 \(\widetilde{\mathcal{T}}: Γ(E/{\sim}) \to Γ(\mathcal{L})\) 将拉回丛 \(E = π_2^*(T^*M)\) 在 \(\mathcal{C} \times M\) 上的截面的规范等价类映射到希尔伯特空间 \(\mathcal{H}_0 = L^2(\mathcal{C}, ν_0)\) 上的 SLD 算子，其中 \(\mathcal{C}\) 是物质场的无限维 Fréchet 流形，\(ν_0\) 是固定的高斯测度。我们证明 \(\widetilde{\mathcal{T}}\) 及相关的量子 Fisher 度量与 \(ν_0\) 的选择无关，从而使得该构造内禀于物理概率密度。Fisher 度量可以用 \(η_μ\) 表示为简洁形式：\(g_{μν}^{\mathrm{FS}} = -\frac{4m^2}{\hbar^2} \operatorname{Re}\langle(η_μ - \langleη_μ\rangle) (η_ν - \langleη_ν\rangle)\rangle_{\mathcal{P}}\)。由此，由 \(η_μ\) 定义的平坦 \(U(1)\) 联络在非单连通的时空环路上产生量子化 holonomy，预言了可在原子干涉测量中观测的拓扑相位。