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 $\hat{I}{\cdot }_{\hat{I}}¼=\ddot{I}{€}_{\hat{I}}¼-i{u}_{\hat{I}}¼$, obtained by averaging matter dynamics over stochastic gravitational fluctuations, and the symmetric logarithmic derivative (SLD) operator $L_{\hat{I}}¼$ of quantum estimation theory. The isomorphism $\tilde{\mathcal{T}}:\hat{I}“(E/\sim )\to \hat{I}“(\mathcal{L})$ maps gauge-equivalence classes of sections of the pullback bundle $E=\ddot{I}{€}_2^{\ast }(T^{\ast }M)$ over $\mathcal{C}\times M$ to SLD operators on the Hilbert space ${\mathcal{H}}_0=L^2(\mathcal{C},\hat{I}{½}_0)$, where $\mathcal{C}$ is the infinite-dimensional Fréchet manifold of matter fields and $\hat{I}{½}_0$ is a fixed Gaussian measure. We prove that $\tilde{\mathcal{T}}$ and the associated quantum Fisher metric are independent of the choice of $\hat{I}{½}_0$, rendering the construction intrinsic to the physical probability density. The Fisher metric acquires a simple form in terms of the Madelung--Bohm velocities: $g_{\hat{I}¼\hat{I}½}^{\mathrm{FS}}=\frac{4m^2}{{\hslash }^2}[\mathrm{Cov}(\ddot{I}{€}_{\hat{I}}¼,\ddot{I}{€}_{\hat{I}}½)+\mathrm{Cov}(u_{\hat{I}}¼,u_{\hat{I}}½){]}_{\mathcal{P}}$. As a consequence, the flat $U(1)$ connection defined by $\hat{I}{\cdot }_{\hat{I}}¼$ yields a quantized holonomy for non-contractible spacetime loops, predicting topological phases that may be observable in atom interferometry.