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

How Does the Theory of Entropicity (ToE) Explain the Aharonov-Bohm (AB) Effect of Quantum Field Theory (QFT)?

The Theory of Entropicity (ToE), proposed by researcher John Onimisi Obidi in 2025, explains the Aharonov-Bohm (AB) Effect by reinterpreting electromagnetic potentials as manifestations of a fundamental, dynamic entropic field. In this framework, the phase shift observed in the AB effect is not caused by "action at a distance" or a mathematical vector potential, but by the physical interaction of a particle with the informational gradients of the entropic field. 

Core Explanatory Mechanisms in ToE

According to the Theory of Entropicity, the AB effect is explained through the following principles:

• Entropy as a Fundamental Field: ToE posits that entropy is not just a measure of disorder but a primary, physical field from which space, time, and forces emerge. The magnetic vector potential 
  

  $$
   is viewed as a specific configuration of this entropic field that carries "organizational information" about the environment.
• The No-Rush Theorem: This principle asserts that no physical interaction can occur instantaneously. In the AB effect, the phase shift is seen as the finite time the entropic field takes to redistribute and synchronize the state of the electron as it passes around a solenoid.
• Entropic Force and Gradients: The theory replaces classical forces with "entropic gradients". Even if the magnetic field 
  

  $$
   is zero outside a solenoid, the entropic field still possesses a gradient (the vector potential 
  

  $$
  ) that dictates the "optimal path" and phase of the electron.
• Vuli–Ndlela Integral: This is a reformulation of Feynman's path integral that weights paths based on their "entropic cost". The interference pattern shifts because the two paths around the solenoid have different entropic weights due to the enclosed magnetic flux, leading to a measurable phase difference. 

Comparison with Standard Physics

Feature	Standard Quantum Mechanics	Theory of Entropicity (ToE)
Agent of Change	Magnetic Vector Potential  $$	Dynamic Entropic Field  $$
Nature of Effect	Non-local or Topological	Localized Informational Interaction
Causality	Phase "picked up" from potential	Result of "No-Rush" synchronization
Speed Limit  $$	Universal Postulate	Maximum rate of entropic rearrangement

The Theory of Entropicity is currently considered an emerging and speculative framework that aims to provide a "why" for phenomena that standard physics treats as postulates. 

If you'd like, we can provide more details on the mathematical foundations of the Obidi Action or how this theory applies to quantum entanglement.

Key Principles of the Theory of Entropicity (ToE): Mathematical and Conceptual Framework, Implications and Applications

The Theory of Entropicity posits that entropy is the fundamental substrate of reality, driving all physical processes and giving rise to spacetime, forces, and quantum phenomena.

Core Concept

The Theory of Entropicity (ToE), formulated by John Onimisi Obidi, redefines entropy as a dynamic, field-like entity rather than a mere statistical measure of disorder or unavailable energy. In this framework, entropy forms the primary ontological substrate, called the Entropic Field, from which geometry, matter, information, and physical laws emerge. This approach challenges traditional physics by treating spacetime, quantum states, and forces as emergent phenomena rather than fundamental primitives. 

entropicity.github.io+2

Key Principles

• Entropy as a Fundamental Force: Entropy actively drives motion, interactions, and the evolution of systems, rather than being a passive consequence of physical processes. 

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• No-Rush Theorem: All physical processes require a finite, non-zero duration; instantaneous interactions are impossible. This principle ensures that events unfold through entropic dynamics. 

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• Emergent Spacetime: Spacetime is not fundamental but arises from the behavior of the entropic field, with gravity interpreted as a consequence of entropy gradients rather than spacetime curvature. 

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• Quantum Phenomena: Wave function collapse, entanglement, and decoherence are explained as entropy-driven processes occurring over finite time intervals, unifying quantum behavior with thermodynamic principles. 

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• Unified Forces: Traditional forces and fields are seen as manifestations of underlying entropic dynamics, potentially offering a framework for unifying physics without relying on separate force postulates. 

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Mathematical and Conceptual Framework

ToE includes a multi-stage, multi-phase diffusion pipeline for formalizing entropic dynamics, along with the Master Entropic Equation and Obidi Field Equations. It introduces concepts like Self-Referential Entropy (SRE), entropic probability, and entropic CPT symmetry, which provide new conservation laws and constraints on physical interactions. These formulations aim to quantify internal entropic feedback and establish universal limits, such as a speed limit for interactions and a thermodynamic uncertainty relation. 

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Implications and Applications

• Relativistic Effects: ToE can reproduce phenomena like Mercury’s perihelion precession without invoking spacetime curvature. 

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• Quantum Information: The theory offers a framework for understanding decoherence, wavefunction collapse, and entropic thresholds in quantum systems. 

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• Broader Applications: Potential applications extend to AI architecture, consciousness studies, and clinical biomarkers, reflecting the theory’s interdisciplinary reach. 

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Summary

The Theory of Entropicity represents a paradigm shift in physics, proposing that entropy is the ultimate driver of reality. By treating entropy as a fundamental field, ToE provides a unified perspective on spacetime, forces, and quantum phenomena, emphasizing finite-duration interactions and emergent structures. While still speculative and under development, it offers a novel lens for understanding the universe and its underlying dynamics. 

entropicity.github.io+3