A Gauge-Invariant Bundle Isomorphism Between Complex Velocity Fields and Symmetric Logarithmic Derivatives

We establish a rigorous bundle isomorphism between the complex velocity field ${\eta }_{\mu }={\pi }_{\mu }-i{u}_{\mu }$, obtained by averaging matter dynamics over stochastic gravitational fluctuations, and the symmetric logarithmic derivative (SLD) operator $L_{\mu }$ of quantum estimation theory. The isomorphism $\tilde{\mathcal{T}}:\mathrm{\Gamma }(E\mathrm{/}\sim )\to \mathrm{\Gamma }(\mathcal{L})$ maps gauge-equivalence classes of sections of the pullback bundle $E={\pi }_2^{\ast }(T^{\ast }M)$ over $\mathcal{C}\times M$ to SLD operators on the Hilbert space ${\mathcal{H}}_0=L^2(\mathcal{C},{\nu }_0)$, where $\mathcal{C}$ is the infinite-dimensional Fréchet manifold of matter fields and ${\nu }_0$ is a fixed Gaussian measure. We prove that $\tilde{\mathcal{T}}$ and the associated quantum Fisher metric are independent of the choice of ${\nu }_0$, rendering the construction intrinsic to the physical probability density. The Fisher metric acquires a simple form in terms of the Madelung–Bohm velocities: $g_{\mu \nu }^{\mathrm{F}\mathrm{S}}=\frac{4m^2}{{\mathrm{\hslash }}^2}[\mathrm{Cov}({\pi }_{\mu },{\pi }_{\nu })+\mathrm{Cov}(u_{\mu },u_{\nu }){]}_{\mathcal{P}}$. As a consequence, the flat $U(1)$ connection defined by ${\eta }_{\mu }$ yields a quantized holonomy for non-contractible spacetime loops, predicting topological phases that may be observable in atom interferometry.