A Gauge-Invariant Bundle Isomorphism Between Complex Velocity Fields and Symmetric Logarithmic Derivatives
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913068939739136 |
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| author | Meza-Domínguez, Jorge |
| author_facet | Meza-Domínguez, Jorge |
| contents | We establish a rigorous bundle isomorphism between the complex velocity field $η_μ = π_μ - i u_μ$, obtained by averaging matter dynamics over stochastic gravitational fluctuations, and the symmetric logarithmic derivative (SLD) operator $L_μ$ of quantum estimation theory. The isomorphism $\widetilde{\mathcal{T}}: Γ(E/{\sim}) \to Γ(\mathcal{L})$ maps gauge-equivalence classes of sections of the pullback bundle $E = π_2^*(T^*M)$ over $\mathcal{C} \times M$ to SLD operators on the Hilbert space $\mathcal{H}_0 = L^2(\mathcal{C}, ν_0)$, where $\mathcal{C}$ is the infinite-dimensional Fréchet manifold of matter fields and $ν_0$ is a fixed Gaussian measure. We prove that $\widetilde{\mathcal{T}}$ and the associated quantum Fisher metric are independent of the choice of $ν_0$, rendering the construction intrinsic to the physical probability density. The Fisher metric acquires a simple form in terms of the Madelung--Bohm velocities: $g_{μν}^{\mathrm{FS}} = \frac{4m^2}{\hbar^2} \bigl[\operatorname{Cov}(π_μ,π_ν) + \operatorname{Cov}(u_μ,u_ν)\bigr]_{\mathcal{P}}$. As a consequence, the flat $U(1)$ connection defined by $η_μ$ yields a quantized holonomy for non-contractible spacetime loops, predicting topological phases that may be observable in atom interferometry. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12187 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Gauge-Invariant Bundle Isomorphism Between Complex Velocity Fields and Symmetric Logarithmic Derivatives Meza-Domínguez, Jorge Quantum Physics Information Theory General Relativity and Quantum Cosmology Mathematical Physics Quantum Algebra We establish a rigorous bundle isomorphism between the complex velocity field $η_μ = π_μ - i u_μ$, obtained by averaging matter dynamics over stochastic gravitational fluctuations, and the symmetric logarithmic derivative (SLD) operator $L_μ$ of quantum estimation theory. The isomorphism $\widetilde{\mathcal{T}}: Γ(E/{\sim}) \to Γ(\mathcal{L})$ maps gauge-equivalence classes of sections of the pullback bundle $E = π_2^*(T^*M)$ over $\mathcal{C} \times M$ to SLD operators on the Hilbert space $\mathcal{H}_0 = L^2(\mathcal{C}, ν_0)$, where $\mathcal{C}$ is the infinite-dimensional Fréchet manifold of matter fields and $ν_0$ is a fixed Gaussian measure. We prove that $\widetilde{\mathcal{T}}$ and the associated quantum Fisher metric are independent of the choice of $ν_0$, rendering the construction intrinsic to the physical probability density. The Fisher metric acquires a simple form in terms of the Madelung--Bohm velocities: $g_{μν}^{\mathrm{FS}} = \frac{4m^2}{\hbar^2} \bigl[\operatorname{Cov}(π_μ,π_ν) + \operatorname{Cov}(u_μ,u_ν)\bigr]_{\mathcal{P}}$. As a consequence, the flat $U(1)$ connection defined by $η_μ$ yields a quantized holonomy for non-contractible spacetime loops, predicting topological phases that may be observable in atom interferometry. |
| title | A Gauge-Invariant Bundle Isomorphism Between Complex Velocity Fields and Symmetric Logarithmic Derivatives |
| topic | Quantum Physics Information Theory General Relativity and Quantum Cosmology Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2604.12187 |