Obidi’s Abolition of Duality Through the Haller–Obidi Action (HOA) of the Theory of Entropicity (ToE): A Rigorous Derivation of the de Broglie Wave–Particle Duality, Demonstrating the Duality as a Consequence of the Same Entropic Field of ToE, Thus Giving a New Entropic Interpretation of the Double‑Slit Experiment of Quantum Mechanics Obidi’s Abolition of Duality Through the Haller–Obidi Action (HOA) Derivation of the de Broglie Wave-Particle.Duality Relation from the Theory of Entropicity (ToE)

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Obidi’s-Abolition-of-Duality-Through-the-Haller–Obidi-Action-(HOA)-Derivation-of-the-de-Broglie-Wave-Particle-Duality-Relation-from-the-Theory-of-Entropicity-(ToE).md

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Obidi’s Abolition of Duality Through the Haller–Obidi Action (HOA) of the Theory of Entropicity (ToE): A Rigorous Derivation of the de Broglie Wave–Particle Duality, Demonstrating the Duality as a Consequence of the Same Entropic Field of ToE, Thus Giving a New Entropic Interpretation of the Double‑Slit Experiment of Quantum Mechanics

Obidi’s Abolition of Duality Through the Haller–Obidi Action (HOA) Derivation of the de Broglie Wave-Particle.Duality Relation from the Theory of Entropicity (ToE)

https://github.com/Entropicity/Theory-of-Entropicity-ToE-Research-Lab-The-Aether-Live-Lab-NoteBook/blob/0e432f967fd0446a0344c8408cdfaee19a847a28/markdown-from-clickup-live-lab-notebook/Obidi%E2%80%99s-Abolition-of-Duality-Through-the-Haller%E2%80%93Obidi-Action-(HOA)-Derivation-of-the-de-Broglie-Wave-Particle-Duality-Relation-from-the-Theory-of-Entropicity-(ToE).md

Obidi’s-Abolition-of-Duality-Through-the-Haller–Obidi-Action-(HOA)-Derivation-of-the-de-Broglie-Wave-Particle-Duality-Relation-from-the-Theory-of-Entropicity-(ToE).md

Abstract

This paper presents a rigorous derivation of the de Broglie wave–particle duality relation as a limiting case of the Haller–Obidi Action (HOA) within John Onimisi Obidi’s Theory of Entropicity (ToE). By reducing the universal entropic field $(S(x^{\mu }))$ to its localized worldline limit, the Haller–Obidi Action (HOA) provides a mathematically precise bridge between entropic field dynamics and single‑particle behavior. The derivation demonstrates that the canonical momentum $(p\mu )$ and the wave vector $(k\mu )$ arise from the same entropic gradient, yielding the identity $(p\mu =\hslash k\mu )$.

This identity leads directly to the de Broglie relation $(\lambda =h/p)$. The result abolishes the traditional wave–particle duality by showing that both “wave” and “particle” aspects are emergent manifestations of a single entropic substrate. The double‑slit experiment is reinterpreted as an entropic interference phenomenon, where the entropic field reorganizes itself around informational constraints rather than classical trajectories. This establishes a unified entropic interpretation of quantum behavior.

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1. Introduction

Wave–particle duality has long been regarded as one of the most mysterious features of quantum mechanics. The classical particle picture and the quantum wave picture appear contradictory, yet both are necessary to describe microscopic phenomena. Obidi’s Theory of Entropicity (ToE) resolves this tension by demonstrating that the duality is not fundamental. Instead, both aspects arise from the same entropic field. The Haller–Obidi Action (HOA) provides the mathematical structure that connects the universal entropic field to localized particle behavior. By deriving the de Broglie relation from HOA, we show that wave–particle duality is a limiting case of entropic dynamics.

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2. The Universal Entropic Field and the Haller–Obidi Action

In ToE, the fundamental entity is the universal entropic field $(S(x^{\mu }))$. Spacetime, matter, and quantum behavior emerge from gradients and reconfigurations of this field. A localized concentration of entropic information, perceived macroscopically as a particle, traces a worldline parameterized by proper time $(\tau )$. The localized dynamics are governed by the Haller–Obidi covariant Lagrangian

$$[\mathcal{L}ent=m{c}^2-\frac{\hslash }{2}{u}^{\mu }\partial \mu S,]$$

where

• $(m)$ is emergent entropic inertia,

• $(u^{\mu }=d{x}^{\mu }/d\tau )$ is the four‑velocity, and

• $({\partial }_{\mu }S)$ is the entropic gradient. The corresponding action is

$$[SHO=\int \mathcal{L}ent,d\tau .]$$

This action reduces to Haller’s 2015 entropy–action identity in the worldline limit:

$$[H=\frac{2}{\hslash }\int (m{c}^2-L),dt.]$$

The variational condition:

$(\delta S_{HO}=0)$

ensures structural stability of the localized entropic knot against the background flow of the universal field.

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3. Phase Accumulation and the Entropic Wave Function

The entropic field induces a phase modulation in the localized entity. The wave function is defined as

$$[\psi (x^{\mu })=\sqrt{\rho (x^{\mu })}{e}^{i\mathrm{\Phi }(x^{\mu })},]$$

where the phase is given by:

$$[\mathrm{\Phi }(x^{\mu })=\frac{1}{\hslash }{S}_{HO}.]$$

Taking the gradient of the phase yields the four‑wave vector:

$$[k\mu =\partial \mu \mathrm{\Phi }=\frac{1}{\hslash }\frac{\partial S_{HO}}{\partial x^{\mu }}.]$$

Hamilton–Jacobi theory shows that the endpoint variation of the action satisfies

$$[\frac{\partial SHO}{\partial x^{\mu }}=\frac{\partial \mathcal{L}HO}{\partial u^{\mu }}.]$$

Thus,

$$[k\mu =\frac{1}{\hslash }\frac{\partial \mathcal{L}HO}{\partial u^{\mu }}.]$$

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4. Canonical Momentum and the Entropic Gradient

The canonical momentum is defined as

$$[p\mu =-\frac{\partial \mathcal{L}ent}{\partial u^{\mu }}.]$$

Evaluating the derivative yields

$$[\frac{\partial \mathcal{L}ent}{\partial u^{\mu }}=-\frac{\hslash }{2}\partial \mu S,]$$

so that

$$[p\mu =\frac{\hslash }{2}\partial \mu S.]$$

Meanwhile, the localized physical Lagrangian satisfies

$$[\mathcal{L}HO=-\mathcal{L}ent+m{c}^2=\frac{\hslash }{2}{u}^{\nu }{\partial }_{\nu }S,]$$

and therefore

$$[\frac{\partial \mathcal{L}HO}{\partial u^{\mu }}=\frac{\hslash }{2}\partial \mu S.]$$

Substituting into the expression for $(k_{\mu })$ yields the identity

$$[k\mu =\frac{1}{\hslash }p\mu ,]$$

or equivalently,

$$[p\mu =\hslash k\mu .]$$

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5. Derivation of the de Broglie Relation

In the non‑relativistic spatial limit, the momentum and wave vector reduce to

$$[\mathbf{p}=\hslash \mathbf{k}.]$$

The geometric identity for a wave gives

$$[|\mathbf{k}|=\frac{2\pi }{\lambda }.]$$

Substituting into the momentum relation yields

$$[p=\hslash \frac{2\pi }{\lambda }.]$$

Using the definition $(h=2\pi \hslash )$, we obtain

$$[p=\frac{h}{\lambda },]$$

and therefore

$$[\lambda =\frac{h}{p}.]$$

This is the de Broglie wave–particle duality relation, derived as a limiting case of the entropic dynamics of ToE.

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6. Abolition of Duality and the Entropic Interpretation of the Double‑Slit Experiment

The identity $(p\mu =\hslash k\mu )$ shows that momentum and wave behavior arise from the same entropic gradient. The “particle” is a localized entropic knot, while the “wave” is the periodic reconfiguration of the entropic field around that knot. In the double‑slit experiment, the entropic field reorganizes itself along multiple informational pathways. The interference pattern is not due to a particle “behaving like a wave,” but due to the entropic field exploring multiple configurations simultaneously.

The localized knot emerges only upon measurement, when the entropic field collapses into a single configuration. Thus, wave–particle duality is abolished; both aspects are entropic manifestations of the same underlying field.

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7. Conclusion

The Haller–Obidi Action provides a rigorous mathematical bridge between the universal entropic field and localized particle behavior. By deriving the de Broglie relation from HOA, we demonstrate that wave–particle duality is not fundamental but emergent.

The double‑slit experiment becomes an entropic interference phenomenon, and quantum behavior is reinterpreted as the dynamics of informational gradients.

Obidi’s Theory of Entropicity (ToE) thus provides a unified entropic foundation for quantum mechanics, relativity, and emergent spacetime.

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Scholium

🔥 Obidi’s Abolition of Duality Through the Haller–Obidi Action (HOA) of the Theory of Entropicity (ToE):

A Rigorous Derivation of the de Broglie Wave–Particle Duality, Demonstrating the Duality as a Consequence of the Same Entropic Field of ToE, Thus Giving a New Entropic Interpretation of the Double‑Slit Experiment of Quantum Mechanics

🌌 A New Paradigm in Quantum Foundations

Wave–particle duality has long been treated as a mysterious contradiction at the heart of quantum mechanics.

Obidi’s ToE shows that this duality isn't fundamental at all. Both “wave” and “particle” behaviors emerge from the same entropic field, and the Haller–Obidi Action (HOA) provides the mathematical bridge that makes this explicit.

🧩 The Entropic Field and the Localized Knot

In ToE, the universe is governed by a continuous entropic field S(xᵐᵘ). A particle is simply a localized knot of this field. Its dynamics follow the Haller–Obidi Action:

Sᴴᴼ = ∫ (m c² − (ħ/2) uᵐᵘ ∂ᵐᵘ S) dτ

This reduces to Haller’s 2015 entropy–action identity in the worldline limit. The phase of the quantum state is the accumulated entropic action:

Φ = (1/ħ) Sᴴᴼ

⚡ Momentum and Wave Vector: One Entropic Origin

Taking the gradient of the phase gives the wave vector:

kᵐᵘ = ∂ᵐᵘ Φ

Varying the entropic Lagrangian yields the canonical momentum:

pᵐᵘ = (ħ/2) ∂ᵐᵘ S

The endpoint variation of the action shows these two quantities share the same entropic origin:

pᵐᵘ = ħ kᵐᵘ

This is the entropic identity that abolishes duality.

🌊 Deriving the de Broglie Wave-Particle Duality Relation from Entropy

In the spatial limit, the identity becomes:

p = ħ k

Wave geometry defines the wave‑vector as:

k = 2π / λ

And the Planck constant satisfies:

h = 2π ħ

Substituting these into the entropic momentum relation yields the classical de Broglie wave-particle duality relation of Quantum Mechanics:

λ = h / p

Thus, the de Broglie wavelength isn't a mystical dual property of matter. It is the spatial periodicity of the entropic field as it reorganizes around a moving informational knot.

🎯 The Double‑Slit Experiment of Quantum Mechanics Reinterpreted Through the Lens of the Theory of Entropicity (ToE)

Under ToE, the double‑slit experiment is not a particle “behaving like a wave.” It's the same entropic field exploring multiple entropic informational configurations simultaneously. The interference pattern is the field’s thermodynamic negotiation, not a paradox of nature.

🚀 Closing Insight

Obidi’s HOA shows that wave–particle duality is abolished. Both behaviors are entropic manifestations of the same underlying field. Quantum mechanics becomes a natural consequence of informational geometry, not a collection of mysteries.

Matter, therefore, does not behave like a particle and also like a wave; but it is the same Entropic Field which projects itself as a particle and as a wave, resolving into a singular aspect upon crossing a given entropic threshold linked to the Obidi Curvature Invariant (OCI).