The Casimir Effect Explained by the Theory of Entropicity (ToE): How the Universe Presses Back Without Virtual Particles but Through the Entropic Field

The Casimir Effect Explained by Obidi's Theory of Entropicity (ToE): How the Universe Presses Back Without Virtual Particles but Through the Entropic Field 

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1. The Mystery of Empty Space

Physicists have known since the mid-20th century that even an “empty” vacuum can push.
Place two uncharged, perfectly flat metal plates a hair’s breadth apart and—though no light, no air, and no matter lies between them—they will drift together under a measurable force.
This subtle attraction, known as the Casimir effect, is one of the strangest and most elegant predictions of quantum physics, confirmed countless times in the laboratory.

In mainstream quantum field theory, the explanation seems almost mystical:
the vacuum is never truly empty. It seethes with virtual particles that flicker in and out of existence. The plates block some of those quantum vibrations, leaving fewer allowed modes of the electromagnetic field inside than outside. The imbalance of vacuum “pressure” pushes the plates together.

That story works—and the mathematics fits experiments—but it leaves a deeper question hanging:
Why should nothingness have pressure at all?

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2. A New Voice: The Theory of Entropicity

The Theory of Entropicity (ToE), as first formulated and further developed by John Onimisi Obidi, approaches this same effect from an entirely different foundation.
It begins by treating entropy not as a bookkeeping or accounting measure of disorder, but as a real, continuous field that fills the universe. Everything—matter, energy, and even spacetime—is an expression of this field’s structure and curvature.

Instead of particles or forces, ToE starts with entropy itself, flowing and bending like an invisible fabric of information.
From that perspective, what we call “vacuum” is not nothing; it is the most uniform, balanced configuration of the entropic field—a perfectly smooth background state of informational symmetry.

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3. When Boundaries Disturb the Invisible

Now imagine inserting two smooth plates into this tranquil sea of entropy.
Their presence imposes boundaries, limiting how the field can fluctuate between them.
The space inside the narrow gap can no longer support the same range of entropy variations as the open region outside.
Inside the plates, the field’s freedom to express its natural diversity of configurations is reduced.

In ToE language, this means that the entropy density between the plates is slightly lower than the entropy density outside.
The field reacts the only way nature ever reacts when entropy is blocked: it seeks balance.
A gentle flow of the entropic field presses on the plates from both sides, but because the outside region holds more available configurations—more entropy—the outward pressure there is greater.
The result is a net push that drives the plates together.

No virtual particles are required, no flickering quantum foam—only the field’s intrinsic drive toward equilibrium.
The Casimir force becomes an entropic pressure, the universe’s way of smoothing out a tiny wrinkle in its own invisible fabric.

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4. Entropy in Motion

Seen this way, the Casimir effect is a kind of thermodynamic whisper from the cosmos.
Where traditional quantum theory interprets the pressure as the arithmetic difference between zero-point energies, ToE interprets it as a dynamic redistribution of entropy.

The moment you confine space, you confine information; and whenever information is confined, entropy tries to expand again.

The plates, therefore, are not passive.
They act as barriers to the natural breathing motion of the entropic field, and that restriction generates a restoring tension—a gentle pull inward.
What experimentalists measure as a quantum vacuum force is, in ToE terms, the physical expression of entropy’s universal law: systems move toward maximum freedom of configuration.

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5. Why This Matters

This reinterpretation achieves two things that standard quantum field theory struggles to explain intuitively:

1. It removes the need for “virtual” entities.
   Instead of invoking particles that exist fleetingly without observation, ToE grounds the phenomenon in a continuous and ever-present entropic field whose behavior is classical in form but fundamental in scope.

2. It unites the microscopic and macroscopic worlds.
   The same principle that drives heat to flow or gases to expand—the natural increase of entropy—also governs the subtle pressure observed between Casimir plates.
   Thermodynamics and quantum phenomena become two faces of the same entropic dynamics.

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6. The Universe as an Entropic Continuum

In this picture, empty space is not empty at all.
It is a vast, invisible ocean of informational entropy whose smoothness defines spacetime itself.
Every physical object—an atom, a star, a galaxy—is a local disturbance, a region where that entropy field has folded into a more complex pattern.

The Casimir experiment, then, becomes a small-scale demonstration of a cosmic truth:
the universe continually resists confinement.
Whenever boundaries are introduced, entropy flows to equalize them, and that flow manifests as a measurable force.

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7. Beyond Quantum Vacuum

ToE does not discard the mathematics of quantum field theory; it reinterprets it.
The numerical values remain the same—the observed force between the plates is identical—but the story changes.
Where QFT speaks of suppressed electromagnetic modes, ToE speaks of entropic gradients.
Where QFT imagines a restless vacuum filled with virtual quanta, ToE envisions a serene but responsive field of information whose geometry adjusts to any constraint.

This subtle shift reframes our understanding of “nothingness.”
Space is not a stage but an active participant, woven from entropy itself.
The Casimir effect is its quiet response to interference.

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8. A Broader Vision

If the Theory of Entropicity is right, the Casimir effect is not an odd quantum curiosity—it is the first experimentally confirmed whisper of a deeper entropic order.
The same principle might underlie gravity, electromagnetism, and even the emergence of matter.
Everything we observe could be entropy’s attempt to organize itself in the richest, most balanced way possible.

In that light, the plates in a Casimir experiment are not attracting each other through the emptiness of space;
they are being drawn together by the invisible structure that makes space possible.
The universe presses back, not because particles collide, but because entropy seeks harmony.

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9. The Quiet Power of Nothing

ToE’s interpretation of the Casimir effect invites a simple, astonishing conclusion:
what we call “nothing” is alive with structure.
The vacuum’s subtle push is the pulse of the informational field that underlies all things—a reminder that behind every particle, every force, every heartbeat of spacetime, lies the ceaseless balancing act of entropy itself.

When Information Vibrates: Rethinking String Theory Through the Lens of Entropy in Obidi's Theory of Entropicity (ToE)

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1. A Question of What Really Vibrates

In traditional string theory, everything we call “matter” is built from unimaginably tiny strings.
Each string vibrates in spacetime, and its vibrational mode defines what we perceive as an electron, photon, or quark.
Different patterns of vibration create the zoo of particles in the Standard Model.

It’s a breathtaking picture—but it still assumes that spacetime itself already exists, waiting for those strings to dance within it.

But what if spacetime and the strings are not the starting point at all?

What if the act of vibration itself belongs to something deeper and invisible—information?

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2. The Theory of Entropicity (ToE): Reality From Entropy

The Theory of Entropicity (ToE), first formulated and further developed by John Onimisi Obidi as a new theoretical framework in 2025, begins with a radical premise:
Entropy—not energy, not spacetime—is the fundamental field of the universe.

Entropy here doesn’t just mean disorder; it’s treated as an actual field , capable of curvature, flow, and wave-like oscillation.
From this entropic field, geometry, energy, and matter emerge as secondary phenomena.

Where standard physics says “entropy describes matter,” ToE says “matter is structured entropy.”

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3. The Invisible Becomes the Origin

If entropy is the true substrate of existence, then the visible, tangible universe is built from something invisible.
Information geometry already tells us that probability distributions form curved manifolds; ToE extends that concept to physical reality itself.
The curvature of the information manifold—the way entropy changes from point to point—is what we experience as spacetime curvature.

Matter, under this view, is simply localized, compactified information:
stable knots of entropic curvature that persist long enough to look solid.

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4. Strings vs. Entropic Vibrations

Now the key leap.

In string theory:

\text{Particles} = \text{vibrational modes of strings in spacetime.}

In the Theory of Entropicity:

\text{Particles} = \text{vibrational modes of the entropy field } S(x).

That single shift changes the ontology of physics.
Instead of one-dimensional objects oscillating inside a pre-existing space, information itself is vibrating, generating space and time as by-products of its motion.
The geometry isn’t the stage for vibration—it’s the echo of vibration.

This interpretation somehow keeps string theory’s mathematics (but of course with entropic corrections and reformulations) but gives it new philosophical meaning.
Where string theory asks how geometry shapes vibration, ToE asks how vibration of information shapes geometry. In ToE, information not only has geometry: The Theory of Entropicity ToE now demands that information itself (via entropy) also vibrates.

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5. Matter as Compactified Entropy

If this perspective is right, every particle we observe—electrons, quarks, even Higgs bosons—is a localized standing wave of the entropy field.
Their apparent “mass” and “charge” arise from how entropy folds and loops back on itself in compact regions of information space.
What we call massive matter is the densest, most compactified form of entropy;
what we call radiation is entropy in motion, propagating freely.

This aligns with a simple but powerful statement:

The universe’s solidity is an illusion created by the stability of invisible information patterns.

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6. Implications for Physics

1. A Deeper Foundation for the Standard Model
   The model’s particle fields could be emergent solutions of a single entropic equation, rather than separate fundamental entities.

2. A Bridge to Quantum Gravity
   If spacetime curvature is entropic curvature, then general relativity and quantum theory may already share a common informational base.

3. A New View on String Theory
   Strings and branes would represent geometrical projections of information vibrations.
   The extra “compact dimensions” of string theory might correspond to informational compactifications—dimensions of entropy rather than space.

4. The Arrow of Time Built In
   Because entropy flows irreversibly, ToE naturally incorporates time’s one-way direction into fundamental physics.

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7. Philosophical Consequences

This view turns the usual hierarchy upside down.
Instead of matter → energy → information → entropy, the order reverses:

\text{Entropy (information)} \;\Rightarrow\; 
\text{Energy and geometry} \;\Rightarrow\; 
\text{Matter and forces.}

Reality becomes a process of information crystallizing into the shapes we observe.
Physical existence is a visible language spoken by invisible data.

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8. Where Thought Meets Physics

None of this displaces the mathematical successes of string theory or the Standard Model.
Rather, it reframes them as phenomenological layers within a deeper informational universe.
If ToE’s premise holds, then the ultimate “theory of everything” would not describe how matter moves through space, but how information becomes matter by forming space.

That possibility is as humbling as it is thrilling.
It means that behind the universe’s tangible architecture lies something that can’t be touched or seen—only inferred:
a silent, self-organizing ocean of entropy whose vibrations create the music of reality.

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Author’s Note

This essay presents a conceptual interpretation inspired by recent work on information geometry and entropy-based field theories, particularly the Theory of Entropicity (ToE).

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The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory

Dedication

This work is respectfully and wholeheartedly dedicated to Professor Tadashi Takayanagi, Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Japan. 

His groundbreaking contributions to holography, quantum entanglement, and the profound interplay between geometry and information have transformed the landscape of modern theoretical physics. 

From the AdS/BCFT correspondence to the pioneering formulation of pseudo-entropy and the illumination of entanglement structures in de Sitter holography, his ideas have continually opened new conceptual horizons for understanding spacetime, duality, and the quantum foundations of geometry. 

The Theory of Entropicity (ToE) is offered here in admiration and profound appreciation of his influence and in recognition of the intellectual foundations he has laid for future generations of researchers in quantum gravity, holography, and the physics of information.

Prologue

This paper presents a systematic comparison between the recently developed pseudo entropy framework of Takayanagi, Kusuki, and Tamaoka and the Theory of Entropicity (ToE). While pseudo–entropy has revealed a remarkable boundary route to the linearized Einstein equation in dS3, the Theory of Entropicity proposes a far more fundamental idea: that entropy is not a boundary diagnostic of geometry, but the underlying field from which geometry, matter, motion, and time themselves emerge. The discussion that follows demonstrates how the pseudo–entropy program fits naturally within the broader structure of ToE, and how the ToE framework generalizes, extends, and ultimately surpasses it.

The pseudo–entropy construction shows that a non–Hermitian generalization of entanglement entropy in a two–dimensional CFT satisfies a first law whose bulk dual reproduces the perturbative Einstein equation in dS3. Moreover, infinitesimal variations of pseudo–entropy obey a Klein–Gordon equation on a kinematic dS2 space, suggesting the emergence of time from Euclidean CFT data. In this paper, we reinterpret these results within the Theory of Entropicity by showing that the same Klein–Gordon structure appears as the boundary–projected, linearized limit of the Master Entropic Equation derived from the Local Obidi Action. Thus, what pseudo–entropy identifies kinematically from the boundary, ToE generates dynamically in the bulk through the entropic field S(x).

The conceptual innovations embedded in the mathematical structure of the Theory of Entropicity (ToE) introduce a framework that resonates with, and extends beyond, the established formalisms of modern physics. These formulations yield results with potential implications for the Standard Model and, in particular, for the interpretation of the Higgs field. They demonstrate how comparable mathematical architectures—such as those that govern mass generation and field curvature—may naturally emerge within an alternative informational or thermodynamic context.

Within this perspective, the appearance of an effective mass term derived from the curvature of an entropy potential mirrors the function of the Higgs potential, which generates particle masses through spontaneous symmetry breaking. This parallel does not supplant the Higgs mechanism but suggests that similar structures can arise from purely entropic or information-geometric principles.

In this broader interpretation, the Higgs field and its associated boson exemplify a particular mani festation of a deeper and more general phenomenon: the emergence of inertial mass from curvature in an underlying scalar potential—be it physical, geometric, or entropic in origin. Consequently, the ToE framework expands the conceptual boundaries of field theory, proposing that mass, symmetry, and curvature can all be understood as distinct expressions of entropy-driven geometry.

More broadly still, this view intimates that the fundamental dynamical laws of the Standard Model may share profound structural affinities with thermodynamic and informational principles. It implies that the long-standing division between energy-based and entropy-based descriptions of nature may be less absolute than once believed, hinting at a deeper unifying language underlying both.

The current work further embeds pseudo–entropy into a broader landscape of entropic approaches— Jacobson’s thermodynamic derivation of Einstein equations, Padmanabhan’s emergent spacetime, Verlinde’s entropic gravity, Caticha’s entropic inference, and Bianconi’s metric relative entropy. Where these earlier programs emphasize information, thermodynamics, or emergence, ToE provides a uni fying ontological principle: entropy itself is the fundamental field of the universe. By promoting the modular–like operator ∆ to a dynamical object through the Spectral Obidi Action, ToE offers a natural explanation of dark matter, dark energy, and vacuum entropic pressure—domains entirely absent from the pseudo–entropy framework. This paper shows explicitly how Bianconi’s relative–entropy action and the Takayanagi–Kusuki–Tamaoka pseudo–entropy construction both appear as limiting cases of the Obidi Actions.

Finally, we demonstrate that ToE provides a unified entropic–spectral variational principle in which bosons and fermions arise from the same foundational structure. The spectral interpretation of bosonic actions, the Dirac–based fermionic bilinears, and geometric actions such as Einstein–Hilbert and Yang–Mills all emerge as projections of the Local and Spectral Obidi Actions. This paper therefore positions pseudo–entropy not as an alternative to ToE, but as a special holographic shadow of a deeper entropic field theory.

In this sense, the present work does not merely compare two independent approaches. Rather, it establishes a hierarchical synthesis: pseudo–entropy reconstructs gravity from boundary information, while the Theory of Entropicity constructs gravity, geometry, quantum structure, and temporal dynamics from an underlying entropic field. This paper argues that pseudo–entropy is best understood not as a standalone gravitational principle, but as a boundary manifestation of the universal entropic dynamics formulated by the Theory of Entropicity (ToE).

Abstract

The recent work of Takayanagi, Kusuki, and Tamaoka has introduced the concept of holographic pseudo-entropy in non-unitary CFT2 and demonstrated a striking equivalence: the first law of pseudo entropy is precisely dual to the linearized Einstein equation in three-dimensional de Sitter space (dS3) once one allows complexified extremal surfaces in the bulk. Moreover, variations of pseudo-entropy obey a Klein–Gordon equation on the kinematical space dS2, offering an emergent time structure arising from an Euclidean boundary theory.

In this paper we show that while the holographic pseudo-entropy program represents an important boundary diagnostic of gravitational dynamics, it remains a restricted kinematical construction tied to holography, non-unitary conformal field theories, and perturbative de Sitter gravity. By contrast, the Theory of Entropicity (ToE) treats entropy S(x) as the fundamental physical field of nature, endowed with a local variational principle (the Local Obidi Action) and a spectral variational principle (the Spectral Obidi Action). From these actions one derives the Master Entropic Equation, entropic geodesics, irreversible dynamics, and a unified description of gravity, time, quantum processes, and information geometry.

The goal of this work is threefold. First, we present a precise and self-contained exposition of the Takayanagi–Kusuki–Tamaoka framework. Second, we develop the Theory of Entropicity as a universal entropic field theory whose dynamics extend far beyond the holographic pseudo-entropy correspondence. Third, we provide a systematic comparison showing how ToE absorbs pseudo-entropy as a special boundary manifestation of a deeper entropic field, thereby revealing why pseudo-entropy reproduces only the linearized sector of gravitational physics while ToE yields a fully nonlinear, time-asymmetric, and information-geometric unification of physical law.

App Deployment on the Theory of Entropicity (ToE):

App on the Theory of Entropicity (ToE): Click or Open on web browser (a GitHub Deployment - WIP): Theory of Entropicity (ToE)

https://phjob7.github.io/JOO_1PUBLIC/index.html

 

Sources — help

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References

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2. Obidi, John Onimisi. (12th November, 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Cambridge University. https//doi.org/10.33774/coe-2025-g7ztq
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4. Obidi, John Onimisi. 2025. On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Cambridge University. Published October 17, 2025. https://doi.org/10.33774/coe-2025-1dsrv
5. Obidi, John Onimisi (17th October 2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Figshare. https://doi.org/10.6084/m9.figshare.30337396.v2
6. Obidi, John Onimisi. 2025. A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE): Crucial Elements of ToE as a Field Theory. Cambridge University. Published October 20, 2025. https://doi.org/10.33774/coe-2025-bpvf3
7. Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1
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10. Obidi, John Onimisi. (28 October, 2025). The Theory of Entropicity (ToE) Derives and Explains Mass Increase, Time Dilation and Length Contraction in Einstein’s Theory of Relativity (ToR): ToE Applies Logical Entropic Concepts and Principles to Verify Einstein’s Relativity. Cambridge University. https://doi.org/10.33774/coe-2025-6wrkm
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Further Resources on the Theory of Entropicity (ToE):

1. Website: Theory of Entropicity ToE — https://theoryofentropicity.blogspot.com
2. LinkedIn: Theory of Entropicity ToE — https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true
3. Notion-1: Theory of Entropicity (ToE)
4. Notion-2: Theory of Entropicity (ToE)
5. Notion-3: Theory of Entropicity (ToE)
6. Notion-4: Theory of Entropicity (ToE)
7. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack
8. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium
9. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
10. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
11. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).
12. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
13. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
14. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
15. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia
16. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate
17. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare
18. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea
19. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN
20. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).
21. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar
22. IJCSRR: International Journal of Current Science Research and Review - Theory of Entropicity (ToE) - John Onimisi Obidi | IJCSRR
23. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)