Geometric Characterization of Anisotropic Correlations via Mutual Information Tomography
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912754178195456 |
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| author | Leighton-Trudel, Beau |
| author_facet | Leighton-Trudel, Beau |
| contents | Characterizing anisotropic correlations in quantum and statistical systems requires a coordinate-invariant framework. We introduce a geometric map based on the local informational line element, calibrated by the Euclidean benchmark scale $C_{\mathrm{vac}}$: $ds^{2} = C_{\mathrm{vac}}/I(x,x+ε)$. We prove that this map yields a smooth Riemannian structure $g_{ij}$ if and only if the short-distance mutual information (MI) follows the anisotropic inverse-quadratic law (local exponent $X_{\text{loc}}=2$). A key insight is that anisotropy is necessary to activate tensor geometry; isotropic MI forces conformal flatness $g_{ij} \propto δ_{ij}$, suppressing shear degrees of freedom. We employ a parameterization-invariant unimodular split $g_{ij} = V^{2/D}γ_{ij}$, which rigorously separates local density fluctuations (volume $V$) from directional anisotropy (shape/shear $γ_{ij}$). We introduce ``MI Tomography,'' an operational protocol to reconstruct these geometric components from finite directional measurements. The protocol is validated using the equal-time ground state of an anisotropic 2D quantum harmonic lattice (massless relativistic scalar) on a torus, where the reconstructed shape tensor $γ_{ij}$ quantitatively recovers the physical coupling anisotropy. We work strictly in the local, fixed-coarse-graining $X_{\text{loc}}=2$ branch; the line element is used solely to extract the local kinematic structure (the local metric tensor), deferring global distance claims. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Characterization of Anisotropic Correlations via Mutual Information Tomography Leighton-Trudel, Beau Statistical Mechanics Characterizing anisotropic correlations in quantum and statistical systems requires a coordinate-invariant framework. We introduce a geometric map based on the local informational line element, calibrated by the Euclidean benchmark scale $C_{\mathrm{vac}}$: $ds^{2} = C_{\mathrm{vac}}/I(x,x+ε)$. We prove that this map yields a smooth Riemannian structure $g_{ij}$ if and only if the short-distance mutual information (MI) follows the anisotropic inverse-quadratic law (local exponent $X_{\text{loc}}=2$). A key insight is that anisotropy is necessary to activate tensor geometry; isotropic MI forces conformal flatness $g_{ij} \propto δ_{ij}$, suppressing shear degrees of freedom. We employ a parameterization-invariant unimodular split $g_{ij} = V^{2/D}γ_{ij}$, which rigorously separates local density fluctuations (volume $V$) from directional anisotropy (shape/shear $γ_{ij}$). We introduce ``MI Tomography,'' an operational protocol to reconstruct these geometric components from finite directional measurements. The protocol is validated using the equal-time ground state of an anisotropic 2D quantum harmonic lattice (massless relativistic scalar) on a torus, where the reconstructed shape tensor $γ_{ij}$ quantitatively recovers the physical coupling anisotropy. We work strictly in the local, fixed-coarse-graining $X_{\text{loc}}=2$ branch; the line element is used solely to extract the local kinematic structure (the local metric tensor), deferring global distance claims. |
| title | Geometric Characterization of Anisotropic Correlations via Mutual Information Tomography |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2512.07659 |