How has the Theory of Entropicity (ToE) Derived the Schrödinger Equation of Quantum Mechanics?

The Theory of Entropicity derives the Schrödinger equation by starting from an entropic action principle on an information‑geometric manifold and then taking a reversible, small‑entropy‑production limit of the resulting dynamics.[1][2][3]

## Core route from ToE to Schrödinger

1. **Obidi Action and Master Entropic Equation**  

   - ToE introduces the Obidi Action for the entropic field $$S(x)$$, with a kinetic term, self‑interaction potential, and coupling to matter, and applies least action to obtain the Master Entropic Equation (MEE) governing entropy flow in spacetime.[2][3]

   - In appropriate regimes (weak entropic curvature, near‑equilibrium), this MEE reduces to a linear, wave‑like evolution equation for entropic configurations, which plays the same structural role as a generalized diffusion or wave equation.[1][3]

2. **Information geometry and complex structure**  

   - ToE embeds states on an information‑geometric manifold that simultaneously carries the Fisher–Rao metric (classical probabilities) and the Fubini–Study metric (quantum states), linked via the Amari–Čencov $$\alpha$$‑connection.[1][3]

   - In the special case where the entropic order parameter $$\alpha$$ selects the Fubini–Study limit, the statistical manifold acquires a Kähler (symplectic + metric + complex) structure, allowing entropic dynamics to be written as Hamiltonian flow on a complex Hilbert‑space‑like manifold.[1][3][4]

3. **Vuli–Ndlela Integral and quantum amplitudes**  

   - ToE replaces Feynman’s path integral by the Vuli–Ndlela Integral, which weights paths by an entropy functional instead of a purely phase‑based classical action, introducing intrinsic irreversibility and time asymmetry.[1][3]

   - In the reversible, near‑equilibrium limit where entropy production along allowed paths is minimized but not zero, the dominant contributions reduce to complex phase factors generated by an effective entropic Hamiltonian, recovering a unitary‑like propagation kernel analogous to the Feynman kernel for Schrödinger evolution.[1][3][4]

4. **From entropic Hamiltonian to Schrödinger form**  

   - On the Fubini–Study information manifold, the entropic Hamiltonian $$ \hat{H}_S $$ generates isometries of the quantum metric; imposing conservation of an entropic “energy” functional and compatibility with the symplectic structure leads to a first‑order time evolution law  

     $$\mathrm{i}\hbar_{\text{eff}}\,\partial_t \Psi = \hat{H}_S \Psi$$,  

     where $$\Psi$$ is now the complex amplitude associated with an entropic state.[1][3][4]

   - Identifying $$\hat{H}_S$$ with the usual kinetic+potential Hamiltonian in an appropriate limit, and interpreting $$|\Psi|^2$$ as the entropic probability density selected by the Fisher–Rao structure, this equation becomes the standard time‑dependent Schrödinger equation of quantum mechanics.[1][3][4]

## Role of entropy and limits

- **Entropy as generator of quantum behavior**: In ToE, quantum behavior (superposition, interference, and unitary‑like evolution) is not postulated but arises when the entropic field is constrained to evolve on a Kähler information manifold with minimal entropy production, corresponding to the reversible sector of the full entropic dynamics.[1][2][3]

- **Collapse as entropic phase transition**: Outside this reversible sector, when entropic thresholds are crossed, the same framework predicts entropy‑driven, irreversible “collapse” dynamics, so the Schrödinger equation appears as the limiting, low‑dissipation equation inside the broader entropic theory.[5][3][4]

Citations:

[1] The Theory of Entropicity (ToE) Derives and Explains Mass Increase ... https://client.prod.orp.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a

[2] Physics:Implications of the Obidi Action and the Theory of Entropicity (ToE) https://handwiki.org/wiki/Physics:Implications_of_the_Obidi_Action_and_the_Theory_of_Entropicity_(ToE)

[3] An Alternative Path toward Quantum Gravity and the Unification of ... http://www.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c

[4] Schrödinger equation - Wikipedia https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

[5] John Onimisi Obidi - Independent Researcher https://independent.academia.edu/JOHNOBIDI

[6] The Theory of Entropicity (ToE) Derives Einstein's Relativistic Speed ... https://www.cambridge.org/engage/coe/article-details/6908aca0113cc7cfffd949e3

[7] The Schrödinger equation and its Bohmian limit https://arxiv.org/pdf/1512.09084.pdf

[8] The Theory of Entropicity (ToE) https://entropicity.github.io/Theory-of-Entropicity-ToE/

[9] A Simple Explanation of the Unifying Mathematical Architecture of ... https://www.authorea.com/users/896400/articles/1348176-a-simple-explanation-of-the-unifying-mathematical-architecture-of-the-theory-of-entropicity-toe-crucial-elements-of-toe-as-a-field-theory

[10] Erwin Schrödinger and Negative Entropy – blog.rudnyi.ru http://blog.rudnyi.ru/2013/04/schrodinger-disorder-and-entropy.html