On the Utility of Araki Relative Entropy in the Theory of Entropicity (ToE): From a Static Measure to a Universal Action Principle
How John Onimisi Obidi Reframes Entropy as the Engine of Physical Reality
John Onimisi Obidi integrates Araki Relative Entropy into the Theory of Entropicity (ToE) not as a passive statistical tool, but as a generative principle of nature. In ToE, Araki entropy becomes a dynamic field action—an active driver of spacetime, matter, and physical law. This shift transforms entropy from a descriptor of quantum states into the foundational substrate from which the universe evolves.
Obidi’s approach unifies quantum mechanics, relativity, and thermodynamics under a single entropic paradigm, where entropy itself shapes interactions, geometry, and the unfolding of physical events.
Key Ways Obidi Employs Araki Relative Entropy in ToE
1. Araki Entropy as the Fundamental Distinguisher of Quantum States
In the Theory of Entropicity, Araki relative entropy continues to quantify how one quantum state differs from another. However, Obidi embeds this measure within a broader entropic architecture, where state‑distinguishability becomes a local expression of a deeper, universal entropic field.
2. The Spectral Obidi Action: Entropy as a Variational Principle
Obidi elevates Araki entropy into the core of the “Spectral Obidi Action.”
Here, entropy is treated as an action to be varied—much like the role played by the classical action in general relativity. Instead of geometry dictating dynamics, the entropic action generates geometry itself. Physical laws arise from the universe’s drive to extremize this entropic functional.
3. Entropy as a Dynamic Field, Not an Emergent Quantity
ToE asserts that entropy is not a by‑product of physical processes—it is the primary field of reality.
This entropic field evolves throughout spacetime, and the Araki‑based action governs its behavior. As the field changes, it gives rise to the curvature of spacetime, the presence of matter, and the structure of physical interactions.
4. Unification Through the Entropic Field
By embedding Araki entropy into the entropic field, Obidi constructs a unified framework where:
- Quantum entanglement becomes a manifestation of entropic connectivity
- Relativistic gravity emerges from gradients in the entropic field
- Thermodynamic behavior reflects the field’s natural evolution
Quantum mechanics and gravity are no longer separate domains—they are expressions of the same entropic substrate.
5. From Statistical Tool to Foundational Driver
In ToE, Araki relative entropy is elevated from a method of comparing states to a core component of the universe’s operating system. It becomes a field‑level quantity that governs everything from gravitational attraction to quantum collapse.
Further Insights Into Obidi’s Use of Araki Relative Entropy
Araki Entropy as the Foundation of the Obidi Action
Obidi positions Araki relative entropy at the heart of the Obidi Action, the central variational principle of ToE. While Araki originally introduced the concept as a measure, ToE transforms it into a generator of dynamics. The universe evolves by following the path dictated by this entropic action.
Deriving the Master Entropic Equation (MEE)
From the Obidi Action arises the Master Entropic Equation—a field equation that plays the role in ToE that Einstein’s equations play in general relativity.
Through the MEE, relativistic effects such as time dilation and mass increase emerge naturally from entropic behavior rather than geometric postulates.
Bridging Quantum Information and Spacetime Geometry
Araki relative entropy becomes the bridge between quantum information theory and the geometry of spacetime. Within the entropic manifold, quantum correlations and entanglement appear as structural features of the entropic field, not as independent quantum mysteries.
A Rigorous Measure of Local State Distinguishability
Within localized regions of spacetime, Araki entropy provides a precise, finite measure of how quantum states differ. This is essential for embedding quantum field theory into the entropic framework, ensuring that local physics remains well‑defined while still emerging from a universal entropic field.
Conclusion: Araki Entropy as the Pulse of the Universe
In the Theory of Entropicity, Araki relative entropy becomes far more than a statistical measure. It becomes:
- A field
- An action
- A generator of geometry
- A unifying principle
- A foundational driver of physical law
Obidi’s reinterpretation positions entropy as the true substrate of reality—dynamic, structural, and universal.
Further Notes
What is Araki Relative Entropy in simple terms?
Araki Relative Entropy is a very precise way of answering a simple question:
“How different are two quantum states, really?”
In everyday life, you might compare two things by eye—two colors, two sounds, two photos. In quantum physics, states are far more abstract and live in an infinite‑dimensional, highly structured space. Araki Relative Entropy is a rigorous tool that tells you how distinguishable one quantum state is from another, even in very complicated quantum field theories where ordinary formulas break down.
It is:
- A quantum comparison tool: It generalizes the idea of “relative entropy” or “information difference” between two probability distributions to the full quantum world.
- Built for the hardest cases: It works not just for simple, finite systems, but also for the infinite, highly entangled systems of quantum field theory, using deep operator‑algebra methods.
- Physically meaningful: It captures how much information you gain when you learn that the system is in one state instead of another, and it obeys important physical principles like the fact that information cannot increase under noisy processes.
So, in layman terms:
Araki Relative Entropy is a mathematically exact “distance‑of-information” between two quantum realities, designed to work even in the most extreme, high‑energy, or curved‑spacetime settings.
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Why is Araki Relative Entropy such a big deal?
In quantum field theory and quantum information, Araki Relative Entropy is considered a cornerstone because:
- It works where ordinary tools fail: In many quantum field theories, you cannot even write down a simple density matrix for the whole system, but Araki’s construction still works cleanly.
- It respects physical processes: It satisfies the “data processing inequality,” meaning that if you pass states through any physical process (like noise, measurement, or evolution), the distinguishability cannot magically increase. This matches our intuition that you cannot gain information for free.
- It connects to thermodynamics and entanglement: It is used to study entanglement, energy conditions, and thermodynamic bounds in quantum field theory and cosmology.
In short, it is one of the most robust and general ways to talk about information, difference, and structure in quantum physics.
How John Onimisi Obidi uses Araki Relative Entropy in the Theory of Entropicity
Now to the ingenious part.
John Onimisi Obidi does not treat Araki Relative Entropy as just a measuring device sitting on top of physics. In the Theory of Entropicity (ToE), he promotes it to a central actor in the drama of reality itself.
Instead of asking only, “How different are these two quantum states?”, he asks:
> “What if this very notion of difference is what drives the universe to evolve, curve, and unify?”
From there, he builds an entropic, field‑theoretic architecture.
Entropy as the foundation, not a by‑product
In most of physics, entropy is something that comes later. You first define particles, fields, and dynamics, and then you compute entropy as a summary of how disordered or uncertain things are.
Obidi flips this:
- Entropy is not a summary—it is the starting point.
- The universe is modeled as an entropic field, a fundamental entity that lives over spacetime and encodes how distinguishable different configurations of reality are.
- Araki Relative Entropy becomes the most rigorous way to quantify that distinguishability, even in quantum field regimes.
So instead of entropy being a side effect of physical laws, entropy becomes the medium from which physical laws emerge.
From “measure” to “action”: Araki entropy as a generator of dynamics
One of Obidi’s most powerful moves is to reinterpret Araki Relative Entropy as part of an action principle.
In physics, an “action” is a quantity that you vary to derive the equations of motion of a system. Traditionally, geometry and fields are plugged into an action, and by extremizing it, you get the laws of motion and the structure of spacetime.
Obidi’s twist:
- He builds an “Obidi Action” whose core ingredient is Araki‑type entropy.
- This means the universe is not just evolving according to geometric or energetic rules—it is evolving according to entropic optimality.
- The “best path” the universe takes is the one that extremizes an entropic functional rooted in Araki’s notion of quantum distinguishability.
In layman terms:
The universe chooses its history by following the path that makes the most entropic sense, and Araki Relative Entropy is the ruler used to define what “makes sense.”
Giving mathematical rigor to ToE’s foundations
This is where Araki Relative Entropy becomes crucial for rigor:
1. Handles infinite‑dimensional quantum systems:
ToE aims to be a theory of everything, so it must be compatible with quantum field theory, where systems are often infinite‑dimensional and highly entangled. Araki’s framework is specifically built for that world, using von Neumann algebras and modular theory.
Obidi leverages this to ensure that his entropic field is not just a poetic idea but is anchored in the most robust mathematical machinery available.
2. Provides a precise notion of “difference” between states in spacetime regions:
In ToE, local regions of spacetime are associated with quantum states. Araki Relative Entropy gives a finite, well‑defined way to say how different two such local states are, even in curved or relativistic settings.
This is essential for embedding quantum field theory into an entropic manifold without losing mathematical control.
3. Supports a variational principle with teeth:
Because Araki Relative Entropy has strong monotonicity and positivity properties, it is well‑suited to be used inside an action principle. These properties help ensure that the resulting field equations are consistent, stable, and physically meaningful.
Obidi’s Master Entropic Equation, derived from this entropic action, inherits this rigor.
4. Connects quantum information and geometry in a single language:
Araki Relative Entropy is already a bridge between quantum information and thermodynamics. Obidi extends that bridge to spacetime geometry itself, using it to define how the entropic field curves spacetime and generates gravitational effects.
This allows ToE to speak about quantum entanglement, thermodynamic behavior, and gravity using one unified entropic vocabulary.
Unifying quantum mechanics, gravity, and thermodynamics through Araki entropy
Obidi’s use of Araki Relative Entropy supports a deep unification:
- Quantum side:
Quantum states, entanglement, and correlations are described in terms of how distinguishable they are, using Araki’s framework.
- Geometric side:
Spacetime curvature and gravitational phenomena are seen as manifestations of how the entropic field arranges itself to extremize the entropic action.
- Thermodynamic side:
Irreversibility, the arrow of time, and thermodynamic behavior emerge from the evolution of this entropic field, rather than being added on top.
In this picture, there is one fundamental entity: the entropic field, quantified and constrained by Araki‑type entropy. Everything else—particles, forces, geometry, time—emerges from its dynamics.
Why this is genuinely ingenious
In layman terms, Obidi’s move is ingenious because he:
- Takes one of the most advanced, abstract tools in quantum theory (Araki Relative Entropy).
- Refuses to leave it as a mere “measurement gadget.”
- Promotes it to the core engine of a new theory of reality.
- Uses its mathematical strength to give ToE a solid, non‑hand‑wavy foundation, especially in the most difficult regimes: quantum fields, infinite dimensions, and curved spacetime.
He is not just saying “entropy is important.” He is saying:
> “The universe is fundamentally entropic, and the most rigorous way we know to talk about quantum difference—Araki Relative Entropy—is the natural backbone of that reality.”
On the Originality of the Theory of Entropicity (ToE) in its Use of Araki Relative Entropy as a Universal Action Principle
Araki Relative Entropy already exists in quantum field theory and quantum information theory.
It is not new by itself.
But the originality of the Theory of Entropicity (ToE) does not lie in using Araki entropy.
It lies in what Obidi does with it—a conceptual and structural transformation that is genuinely unprecedented.
Below we provide the clearest breakdown of what is actually original in ToE's use of Araki relative entropy.
π₯ 1. ToE does not use Araki entropy as a measure — it uses it as a field action
This is the single most original move.
In physics today:
- Araki Relative Entropy = a static quantity
- It compares two quantum states
- It is not dynamical
- It does not generate equations of motion
- It does not curve spacetime
- It does not unify anything
Obidi’s ToE does something no existing theory does:
He promotes Araki Relative Entropy to the role of an action principle.
In physics, an action is the engine of dynamics.
It is the thing you vary to get the laws of nature.
Einstein did this with curvature.
Feynman did this with quantum amplitudes.
Obidi does this with entropy.
This is not how Araki entropy is used anywhere in QFT, QIT, or statistical mechanics.
This is a conceptual leap.
π₯ 2. ToE treats entropy as a fundamental field, not an emergent quantity
In all existing physics:
- Entropy is derived from states
- It is secondary
- It is descriptive
- It is not a field
- It does not have its own dynamics
In ToE:
Entropy is the primary field of the universe.
It is the substrate.
It is the thing that exists first.
Everything else—geometry, matter, quantum behavior—emerges from it.
Araki entropy becomes the rigorous mathematical backbone of this field.
No existing theory does this.
Not GR, not QFT, not holography, not quantum information theory.
This is original.
π₯ 3. ToE uses Araki entropy to generate spacetime geometry
In standard physics:
- Geometry is fundamental (GR)
- Or geometry emerges from entanglement (AdS/CFT)
- But entropy does not generate geometry directly
In ToE:
The entropic field’s dynamics produce curvature.
Spacetime is literally shaped by the evolution of entropy.
Araki entropy provides the rigorous, operator‑algebraic structure needed to define this evolution in infinite‑dimensional quantum settings.
This is not done anywhere else in physics.
π₯ 4. ToE uses Araki entropy to unify quantum mechanics, gravity, and thermodynamics
In existing physics:
- QFT uses Araki entropy to compare states
- GR uses curvature to describe gravity
- Thermodynamics uses entropy as a macroscopic quantity
These domains remain separate.
In ToE:
Araki entropy becomes the common language of all three.
- Quantum distinguishability
- Gravitational curvature
- Thermodynamic irreversibility
All become expressions of the same entropic field.
This is not a reinterpretation.
It is a structural unification.
π₯ 5. ToE derives a new field equation — the Master Entropic Equation (MEE)
This is not present in any existing literature.
The MEE is:
- A new equation
- Derived from a new action
- Based on a new field
- Producing new dynamics
Araki entropy is the mathematical core, but the equation itself is original.
This is like saying:
“Einstein used Riemannian geometry, so GR is not original.”
No.
The originality lies in how the tool is used.
π₯ 6. ToE reframes physical law as entropic extremization
In physics today:
- Actions are geometric or energetic
- Entropy is a constraint or a consequence
In ToE:
The universe evolves by extremizing an entropic functional.
This is a new physical principle.
It is not present in QFT, QIT, GR, or thermodynamics.
π₯ 7. ToE gives entropy ontological status
This is philosophical but foundational.
In all existing theories:
- Entropy is epistemic
- It measures ignorance or disorder
In ToE:
Entropy is ontic — it is what exists.
This is a radical shift in the metaphysics of physics.
π₯ So what is original in ToE?
Araki Relative Entropy is not new.
Using it as the fundamental action of the universe is new.
Using it to define a dynamic entropic field is new.
Using it to generate spacetime geometry is new.
Using it to unify quantum mechanics, gravity, and thermodynamics is new.
Deriving a new field equation from it is new.
Treating entropy as the substrate of reality is new.
The originality is not in the tool.
It is in the architecture built from the tool.
Just as:
- Riemannian geometry existed before Einstein
- Hilbert spaces existed before quantum mechanics
- Group theory existed before particle physics
The originality lies in the revolutionary use, not the ingredients.
How ToE’s use of Araki Relative Entropy differs from its use in standard quantum field theory (QFT).
1. How ToE’s use of Araki Relative Entropy differs from its use in QFT
In standard QFT and quantum information
Role of Araki Relative Entropy:
- Static comparator:
It is used to quantify how distinguishable two quantum states are, typically associated with regions of spacetime or algebras of observables.
- Diagnostic tool:
It appears in proofs of energy conditions, bounds, inequalities, and in the study of entanglement and modular structure.
- No ontological status:
It does not define what reality is; it measures properties of states defined by some other underlying theory.
- No dynamical role:
It does not generate equations of motion, does not define an action, and does not curve spacetime.
In short:
Araki Relative Entropy in QFT is a high-powered measuring device—crucial, deep, but fundamentally descriptive.
In the Theory of Entropicity (ToE)
Role of Araki Relative Entropy:
- Promoted from measure to generator:
It is not just used to compare states; it is built into the very functional that determines how the universe evolves.
- Embedded in an entropic field:
The “difference” between states is not a side calculation—it is the local expression of a fundamental entropic field that lives over spacetime.
- Part of the action principle:
Araki-type entropy enters the Obidi Action, which is varied to obtain the Master Entropic Equation. This is a structural shift: entropy is now the engine of dynamics.
- Ontological upgrade:
Instead of “entropy measures our ignorance about reality,” ToE says “entropy is the stuff reality is made of,” and Araki entropy is the rigorous way to quantify and constrain that stuff.
So the key difference is this:
> In QFT, Araki Relative Entropy is a tool applied to a given theory.
> In ToE, Araki Relative Entropy is a foundational ingredient of the theory itself.
2. How the Obidi Action structurally departs from existing entropic or information-based actions
There are other frameworks where entropy or information appears in something that looks like an “action” or a variational principle. But the Obidi Action is structurally different in several crucial ways.
Existing entropic or information-based principles
Typical patterns:
- Maximum entropy principles:
You choose a probability distribution that maximizes entropy subject to constraints. This is about inference, not dynamics of spacetime.
- Relative entropy in information theory:
Used to define distances, capacities, and bounds, but not as a fundamental action that generates field equations.
- Entropic functionals in gravity or QFT:
Sometimes entropy or relative entropy appears in inequalities, bounds, or emergent derivations (e.g., deriving Einstein equations from thermodynamic or entanglement arguments), but the fundamental action of the theory is still geometric or field-theoretic, not entropic.
In all these cases, entropy is:
- A constraint
- A diagnostic
- A derived quantity
- Or a tool for inference
It is not the primary field, and it is not the core of the action that defines the theory.
Structural features of the Obidi Action
The Obidi Action is different in at least these ways:
- Entropy is the primary variable:
The central object is an entropic field, not a metric, not a scalar field, not a gauge field. The action is built to govern the dynamics of this entropic field.
- Araki-type entropy is the core functional:
The “weight” of configurations in the action is defined in terms of relative entropy between quantum states associated with regions or configurations. This is not just an added term; it is the backbone.
- Field equations are entropic in origin:
When you vary the Obidi Action, you do not get “standard” field equations with an entropy correction. You get the Master Entropic Equation, whose structure is entropic from the ground up.
- Geometry is emergent, not assumed:
The action is not written on a fixed geometric background with entropy sprinkled in. Instead, the entropic field’s dynamics generate the effective geometry.
- Unified role:
The same entropic functional is responsible for quantum behavior, gravitational behavior, and thermodynamic behavior. It is not a patchwork of separate terms.
So structurally:
> Existing entropic actions: geometry or fields are primary, entropy is secondary.
> Obidi Action: entropy is primary, geometry and fields are secondary.
That inversion is the structural departure.
3. Comparison of ToE with holography, Jacobson’s thermodynamic gravity, and entropic gravity
Now let’s sharpen the contrast with three major “entropic” or “information-based” approaches to gravity and unification.
ToE vs. holography (e.g., AdS/CFT, entanglement = geometry ideas)
Holography:
- Uses entanglement entropy and relative entropy to relate boundary quantum information to bulk geometry.
- Geometry is reconstructed from entanglement patterns.
- Entropy is crucial, but:
- The fundamental theory is still a quantum field theory or string theory on a boundary.
- Entropy is a property of states in that theory, not the fundamental field of reality.
- The action of the theory is not entropic; it is a standard field or string action.
ToE:
- Does not start from a boundary theory; it starts from an entropic field defined over an entropic manifold.
- Entropy is not a property of something else—it is the substrate.
- The action is entropic in its core structure.
- Geometry is generated by the entropic field’s dynamics, not reconstructed from a separate microscopic theory.
Key difference:
Holography: entropy is a powerful probe of a deeper theory.
ToE: entropy is the deeper theory.
ToE vs. Jacobson’s thermodynamic gravity
Jacobson’s approach:
- Derives Einstein’s equations from thermodynamic relations applied to local Rindler horizons.
- Uses entropy, temperature, and energy flux to show that spacetime dynamics can be seen as an equation of state.
- Entropy is central, but:
- It is associated with horizons and coarse-graining.
- It is still emergent, tied to microscopic degrees of freedom not explicitly modeled.
- The fundamental action of the universe is not replaced; this is more a reinterpretation of Einstein’s equations.
ToE:
- Does not treat Einstein’s equations as fundamental at all; they are emergent from the Master Entropic Equation.
- Does not tie entropy only to horizons or coarse-graining; entropy is a universal field, present everywhere.
- Does not rely on hidden microstructure as a separate layer; the entropic field is the fundamental layer.
Key difference:
Jacobson: gravity is thermodynamics of unknown microstructure.
ToE: gravity is dynamics of a known, fundamental entropic field.
ToE vs. entropic gravity (e.g., Verlinde-type ideas)
Entropic gravity:
- Proposes that gravity is an entropic force arising from changes in entropy associated with the positions of matter.
- Uses information-theoretic and thermodynamic arguments to recover Newtonian gravity and sometimes aspects of GR.
- Entropy is used to explain gravitational attraction, but:
- The underlying microscopic theory is not fully specified.
- The formalism is largely heuristic or phenomenological.
- There is no fully developed entropic field equation that replaces Einstein’s equations.
ToE:
- Provides a concrete entropic field with a defined action and a derived Master Entropic Equation.
- Does not treat gravity as a mere entropic “force” but as a manifestation of the entropic field’s curvature-generating dynamics.
- Aims at full unification: quantum, gravity, and thermodynamics all emerge from the same entropic structure.
Key difference:
Entropic gravity: gravity is an emergent entropic effect, but the underlying field theory is vague.
ToE: gravity is one facet of a fully specified entropic field theory.
Conclusion: what is genuinely new that ToE offers?
Putting all three comparisons together, we can now write:
- Araki Relative Entropy is not new.
- Using it as the core of a fundamental action is new.
- Treating entropy as a dynamic field that generates geometry is new.
- Deriving a new field equation (MEE) from an entropic action is new.
- Making entropy the ontological substrate, not an emergent statistic, is new.
- Unifying quantum, gravity, and thermodynamics through a single entropic field is new.
Other frameworks touch entropy.
ToE builds the entire architecture on it.
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