Topological Quantization of Complex Velocity in Stochastic Spacetimes
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arXiv
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| Format: | Preprint |
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2026
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| author | Meza-Domínguez, Jorge Matos, Tonatiuh |
| author_facet | Meza-Domínguez, Jorge Matos, Tonatiuh |
| contents | 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 $η_μ = π_μ - i u_μ$. This object, originating in Nelson's stochastic mechanics, is a section of the pullback bundle $E = π_2^*(T^*M)$ over the product of configuration space $\mathcal{C}$ and spacetime $M$. We prove that $η_μ$ 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 $\mathcal{L}_ηη= d(|η|^2)$. 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 $\frac{m}{\hbar}\oint_γη_μ dx^μ = 2πn + Δϕ_{\text{top}}$. We demonstrate this in a toy model: a scalar field on a conical spacetime with deficit angle $α$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25016 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological Quantization of Complex Velocity in Stochastic Spacetimes Meza-Domínguez, Jorge Matos, Tonatiuh General Relativity and Quantum Cosmology Mathematical Physics 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 $η_μ = π_μ - i u_μ$. This object, originating in Nelson's stochastic mechanics, is a section of the pullback bundle $E = π_2^*(T^*M)$ over the product of configuration space $\mathcal{C}$ and spacetime $M$. We prove that $η_μ$ 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 $\mathcal{L}_ηη= d(|η|^2)$. 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 $\frac{m}{\hbar}\oint_γη_μ dx^μ = 2πn + Δϕ_{\text{top}}$. We demonstrate this in a toy model: a scalar field on a conical spacetime with deficit angle $α$, 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. |
| title | Topological Quantization of Complex Velocity in Stochastic Spacetimes |
| topic | General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2603.25016 |