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

Abstract

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 $\widetilde{\mathcal{T}}: \Gamma(E/{\sim}) \to \Gamma(\mathcal{L})$ maps gauge-equivalence classes of sections of the pullback bundle $E = \pi_2^*(T^*M)$ 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 Fr\'echet manifold of matter fields and $\nu_0$ is a fixed Gaussian measure. We prove that $\widetilde{\mathcal{T}}$ 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: $g_{\mu\nu}^{\mathrm{FS}} = \frac{4m^2}{\hbar^2} \bigl[\operatorname{Cov}(\pi_\mu,\pi_\nu) + \operatorname{Cov}(u_\mu,u_\nu)\bigr]_{\mathcal{P}}$. 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.