How does the Spectral Obidi Action (SOA) Incorporate Fubini-Study metric in the Theory of Entropicity (ToE)?

The Spectral Obidi Action (SOA) incorporates the Fubini-Study metric by weighting it with the entropy field $$ S $$, transforming the standard quantum projective geometry into an **entropy-deformed quantum manifold** that governs state transitions and modular flow.[1][3]

Fubini-Study in Quantum Context

The Fubini-Study (FS) metric $$ g_\text{FS} $$ naturally arises on complex projective space $$ \mathbb{CP}^n $$ (rays in Hilbert space), measuring infinitesimal distances between pure quantum states:  

$$ ds^2_\text{FS} = g_{i\bar{j}} dz^i d\bar{z}^j = \frac{ \langle \delta \psi | \delta \psi \rangle \langle \psi | \psi \rangle - |\langle \psi | \delta \psi \rangle|^2 }{ \langle \psi | \psi \rangle^2 }. $$  

It encodes quantum distinguishability and overlaps, central to path integrals and Born probabilities.[2][3]

Entropy Weighting Mechanism

In ToE's Spectral Action  

$$ \mathcal{A}_\text{Spectral} = \text{Tr} \left[ \rho \log \left( \frac{\rho}{\rho_0 e^{S/k_B}} \right) \right], $$  

the modular operator $$ \Delta = \rho \otimes \rho_0^{-1} e^{S/k_B} $$ (from Tomita-Takesaki theory) induces an **entropy-weighted FS metric**:  

$$ g^{(S)}_{i\bar{j}} = e^{S/k_B} g_\text{FS}^{i\bar{j}}. $$  

This exponential boost reflects irreversible entropy production, curving quantum state space like gravity curves spacetime.[1][3]

Role in Spectral Dynamics

- **Spectral flow**: Extremizing $$ \mathcal{A}_\text{Spectral} $$ drives states along deformed FS geodesics, where $$ \nabla S $$ adds an "entropic force" to quantum transitions, recovering ETL (no-rush) delays and Unruh-like temperatures.[1]

- **Unification bridge**: FS + Fisher-Rao (classical info metric) project onto the Amari-Čencov α-connections of the full entropic manifold, with α parameterizing duality between local/spectral views.[3]

- **Physical output**: Entanglement entropy gradients via weighted FS yield emergent metric curvature and particle masses as excitations.[9]

This makes quantum irreversibility geometric, subsuming standard QM as the $$ S \to 0 $$ limit.[3]

Citations:

[1] John Onimisi Obidi https://www.authorea.com/doi/pdf/10.22541/au.176340906.62496480

[2] Fubini–Study metric - Wikipedia https://en.wikipedia.org/wiki/Fubini%E2%80%93Study_metric

[3] John Onimisi Obidi 1 1Affiliation not available October 17, 2025 https://d197for5662m48.cloudfront.net/documents/publicationstatus/284761/preprint_pdf/a59997ba8ff6f388fae888a3e35f0908.pdf

[4] Contents https://arxiv.org/html/2505.11330v3

[5] Geometric Quaternionic Quantum Mechanics https://math.mit.edu/documents/rsi/2020Xu.pdf

[6] The Fubini–Study metric on an ‘odd’ Grassmannian is rigid https://arxiv.org/html/2403.18757v1

[7] Fubini-Study metric in nLab https://ncatlab.org/nlab/show/Fubini-Study+metric

[8] [리만의 복소해석 2기 서브-스터디] CP^1에서의 Fubini-Study metric 계산 https://www.youtube.com/watch?v=typQZJ0jqZU

[9] A New Theory Says Gravity May Come From Entropy— ... https://www.popularmechanics.com/science/a64069299/gravity-entropy-unified-theory/