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Topological Quantization of Complex Velocity in Stochastic Spacetimes

Jorge Meza-Domíguez, Tonatiuh Matos

Abstract

The hydrodynamic formulation of quantum mechanics features two velocity fields: a geodesic (classical) velocity $π_μ$ and a stochastic (quantum) velocity $u_μ$. We show that averaging over a stochastic gravitational wave background unifies these into a single complex velocity $η_μ=π_μ-iu_μ$, derived from the logarithmic derivative of a matter amplitude $\mathcal{K}$. This object lives as a section of the pullback bundle $π_{2}^{*}(T^{*}M)$ over configuration space and defines a flat $U(1)$ connection, satisfying $D_μ\mathcal{K}=0$. Crucially, $η_μ$ acts as a fundamental information-geometric carrier, where $u_μ$ maps the variance of metric fluctuations $\langle h_{μν}h_{αβ}\rangle$ to the Fisher metric and von Neumann entropy. The resulting geometric structure collapses into an elegant complex geodesic equation $η^ν\nabla_νη_μ=\nabla_μ(\frac{1}{2}η^νη_ν)$, while non-trivial spacetime topology imposes a holonomy quantization condition. This topological phase suggests observable signatures in atom interferometry and cosmological correlations, providing an experimental window into the stochastic nature of spacetime at the Planck scale.

Topological Quantization of Complex Velocity in Stochastic Spacetimes

Abstract

The hydrodynamic formulation of quantum mechanics features two velocity fields: a geodesic (classical) velocity and a stochastic (quantum) velocity . We show that averaging over a stochastic gravitational wave background unifies these into a single complex velocity , derived from the logarithmic derivative of a matter amplitude . This object lives as a section of the pullback bundle over configuration space and defines a flat connection, satisfying . Crucially, acts as a fundamental information-geometric carrier, where maps the variance of metric fluctuations to the Fisher metric and von Neumann entropy. The resulting geometric structure collapses into an elegant complex geodesic equation , while non-trivial spacetime topology imposes a holonomy quantization condition. This topological phase suggests observable signatures in atom interferometry and cosmological correlations, providing an experimental window into the stochastic nature of spacetime at the Planck scale.

Paper Structure

This paper contains 7 sections, 1 theorem, 18 equations.

Table of Contents

  1. Introduction
  2. Complex Velocity from Stochastic Gravity
  3. Geometric Structure: Pullback Bundle and $U(1)$ Connection
  4. Holonomy Quantization and Topological Phases
  5. Unified Geometrical Equation
  6. Observable Signatures and Implications
  7. Discussion and Conclusions

Key Result

Proposition 1

$\eta_\mu$ is a section of $E$: $\eta \in \Gamma(\pi_2^*(T^*M) \to \mathcal{C}\times M)$. Equivalently, $\eta: \mathcal{C} \to \Gamma(T^*M)$. Smoothness is understood in the sense of Fréchet manifolds, ensuring well-defined variational calculus on $\mathcal{C}$.

Theorems & Definitions (2)

  • Definition 1
  • Proposition 1