Topological Quantization of Complex Velocity in Stochastic Spacetimes

Abstract

We establish a rigorous geometric framework for quantum fields on a stochastic gravitational background. Starting from a master partition function that averages over metric fluctuations, we define a matter amplitude $\mathcal{K}$, whose logarithmic derivative yields a complex velocity field $\eta_{\mu} = \pi_{\mu} - i u_{\mu}$. This object, originating in Nelson's stochastic mechanics, is a section of the pullback bundle $E = \pi_2^*(T^*M)$ over the product of configuration space $\mathcal{C}$ and spacetime $M$. We prove that $\eta_{\mu}$ defines a flat $U(1)$ connection with $\mathcal{K}$ as its horizontal section, and via a bundle isomorphism it maps to the symmetric logarithmic derivative of quantum estimation theory. The coupled dynamics collapse into $\mathcal{L}_{\eta}\eta = d(|\eta|^2)$. We resolve the tension between flatness and multi-valuedness: although the connection is flat, the potential can be multi-valued from topological terms or branch cuts. The total phase satisfies $\frac{m}{\hbar}\oint_\gamma \eta_{\mu} dx^{\mu} = 2\pi n + \Delta\phi_{\text{top}}$. We demonstrate this in a toy model: a scalar field on a conical spacetime with deficit angle $\alpha$, computing the matter amplitude in the Gaussian approximation, deriving the complex velocity, and calculating its holonomy. The resulting topological offset receives a quantized stochastic correction depending on the variance of metric fluctuations, providing an experimental signature for atom interferometry. This framework geometrizes quantum mechanics without hidden variables: stochasticity imprints spacetime fluctuations on matter, preserving the wave function's probabilistic nature while giving a geometric origin for the Born rule.