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

TL;DR

This paper constructs a gauge-invariant isomorphism linking complex velocity fields from stochastic gravity to quantum estimation operators, revealing intrinsic geometric structures and topological phases in spacetime.

Contribution

It establishes a rigorous, gauge-invariant bundle isomorphism connecting matter dynamics with quantum Fisher information, highlighting topological effects in quantum gravity models.

Findings

The isomorphism is independent of the choice of Gaussian measure.

The Fisher metric simplifies in terms of Madelung--Bohm velocities.

Non-contractible spacetime loops produce observable topological phases.

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.