A Bundle Isomorphism Relating Complex Velocity to Quantum Fisher Operators

We show that averaging matter dynamics over stochastic gravitational fluctuations gives rise to a complex velocity field η_μ = π_μ - i u_μ living as a section of the pullback bundle E = π₂^*\(T^*M\to \mathcal{C}\times M\). We prove that η_μ is isomorphic, via the Schrödinger representation, to the symmetric logarithmic derivative (SLD) operator L_μ on the Hilbert space mathcal{H}_x = L²\(mathcal{C}\), up to a trace-zero projection. This isomorphism widetilde{mathcal{T}}:Γ\(E / sim\to Γmathcal{L}\) is a bundle isomorphism preserving the flat U(1) connection proved in cite{meza2026topological} and the quantum Fisher metric. The quantum Fisher information metric g_μν^FS is expressed directly in terms of η_μ as g_μν^FS = - frac{4m²}{hbar²}Relangle \(η_μ - langle η_μrangleη_ν - langle η_νrangle\rangle_{\mathcal{P}}\). The holonomy of η_μ is quantized, leading to topological phases observable in atom interferometry.