Topological Quantization of Complex Velocity in Stochastic Spacetimes

Abstract

Abstract We establish a formal 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 $\mathcal{K}[\Phi; x]$ with explicit parametric dependence on spacetime coordinates, whose logarithmic derivative yields a complex velocity field $\eta_{\mu} = \pi_{\mu} - i u_{\mu}$. This object is shown to be a section of the pullback bundle $E = \pi_2^*(T^*M)$ over the product of configuration space $\mathcal{C}$ and spacetime $M$, and defines a flat $\mathbb{C}^\times$ connection whose $U(1)$ part governs phase transport and whose $\mathbb{R}^+$ part governs scale transport. The coupled dynamics collapse into the complex geodesic equation $\mathcal{L}_{\eta}\eta = d(|\eta|^2)$. On non-simply connected spacetimes, the total phase satisfies $\frac{m}{\hbar}\oint_\gamma \eta_{\mu} dx^{\mu} = 2\pi n + \Delta\phi_{\mathrm{top}}$, demonstrated analytically for a scalar field with twisted boundary conditions on a conical spacetime. We provide a self-contained bundle isomorphism mapping $\eta_\mu$ to the symmetric logarithmic derivative of quantum estimation theory, and derive the associated Berry curvature. A prospective estimate for the MAGIS-100 interferometer yields $\gamma_B \sim 3.97 \times 10^{-9}$ rad for an astrophysical stochastic background (approximately five orders below projected sensitivity) and $\gamma_B \sim 3.97 \times 10^{-53}$ rad for Planck-scale fluctuations. This framework geometrizes quantum mechanics without hidden variables: stochasticity imprints spacetime fluctuations on matter, preserving the wave function's probabilistic nature while maintaining consistency with the Born rule.