Obidi's Conceptual and Mathematical Leap in his Transformation of Araki Relative Entropy into an Action Principle in Modern Theoretical Physics.
Traditionally the Araki relative entropy,
S(\rho || \sigma) = \text{Tr}\big[ \rho (\ln \rho - \ln \sigma) \big],
is a state functional, not an action. It measures the distinguishability between two quantum states and ; it has no dynamical term, no variational principle, and no kinetic component. In other words — it can tell us how different two configurations are, but it cannot by itself tell us how one evolves into the other.
1. Why Araki Relative Entropy Is Not an Action
An action in physics (such as the Einstein–Hilbert or Dirac actions) encodes dynamics: the equations of motion come from minimizing or extremizing the action functional.
Araki relative entropy, by contrast, is static: it is defined between two fixed density matrices and quantifies the information-theoretic distance between them.
Mathematically:
- The Araki functional is non-symmetric and positive-definite.
- It satisfies monotonicity and convexity properties, but
- It lacks any dependence on time derivatives or geometric flow terms like , , or curvature integrals.
Hence, it is a metric measure, not a Lagrangian density.
2. What the Theory of Entropicity (ToE) Does Differently
In Obidi’s Theory of Entropicity (ToE), this insight is precisely where the new physics begins.
ToE accepts that Araki relative entropy can only compare states —
but then extends it by embedding it into an action-like structure that governs how one entropic configuration transforms into another.
In ToE, the Spectral Obidi Action is defined conceptually as:
\mathcal{A}_{\text{ToE}}[S] = \int \! \Big( \mathcal{L}_{\text{geom}}(S, \nabla S) + \lambda \, D(S_1 || S_2) \Big) \, d^4x,
where:
- is the entropic field,
- is a relative curvature functional analogous in form to Araki entropy,
- contains the dynamical (kinetic and curvature) terms,
- and sets the coupling between distinguishability and dynamical evolution.
Thus, ToE promotes the measure of distinguishability to a source term in the action.
This bridges geometry (the dynamics of curvature) and information (the measure of distinguishability).
3. The Conceptual and Mathematical Leap of Obidi
So:
- Araki relative entropy tells us how much two configurations differ.
- The Obidi Action tells us how one configuration evolves toward or away from another through curvature flow.
In ToE, the evolution of the entropic field is driven by minimizing the total distinguishability integrated over spacetime —
that is, the universe tends toward minimal distinguishability curvature configurations (stable ln 2 separations).
This is how ln 2 [the Obidi Curvature Invariant (OCI)] arises as a stationary curvature invariant: the smallest distinguishable separation between two configurations of the entropic field.
4. Why This Matters
By moving from a comparative functional (Araki) to a variational functional (Obidi),
ToE introduces the missing dynamical law that connects entropy, information, and geometry.
It thus parallels the step from:
- metric geometry → General Relativity (via Einstein–Hilbert Action),
- quantum states → quantum dynamics (via Schrödinger or Dirac Action),
- distinguishability → evolution of curvature (via the Obidi Action).
So, the Araki relative entropy cannot itself be an action.
But in the Theory of Entropicity (ToE), it becomes a term within an entropic action —
providing the geometric “potential” that drives the field dynamics of . This is one of Obidi's conceptual and mathematical leaps in his formulation of the Theory of Entropicity (ToE).
Mathematical Form of The Obidi Action and Field Equations With Araki Relative Entropy
1. What makes an “action” an action
In any field theory an action has to integrate a Lagrangian density that depends on the field itself and, crucially, on its spacetime variations. The prototype is
\mathcal A[\phi]=\int \mathcal L(\phi,\partial_\mu\phi)\,d^4x .
Varying with respect to produces equations of motion.
An information–geometric quantity such as the Araki relative entropy,
S(\rho\Vert\sigma)=\operatorname{Tr}[\rho(\ln\rho-\ln\sigma)],
has no such derivatives—it compares two fixed operators. On its own it cannot generate dynamics.
2. Making distinguishability a potential term
If we treat an entropy field as the fundamental variable, the natural way to incorporate “relative entropy” information is to let it weight how far the present configuration is from a local reference configuration . A simple covariant scalar capturing that idea is
D(x)=S(x)\ln\!\frac{S(x)}{S_0(x)}-S(x)+S_0(x),
the continuum analogue of Kullback–Leibler divergence.
is positive, vanishes when , and grows with distinguishability. This plays the role of a potential energy density.
3. The Spectral Obidi Action
The full action must also contain a geometric or “kinetic” term that governs how the field varies through spacetime.
A minimal generally–covariant form is
\boxed{
\mathcal A_{\text{ToE}}[S]
=\int d^4x\,\sqrt{-g}\,
\Big[
\frac{\alpha}{2}R[g]
-\frac{\beta}{2}\,g^{\mu\nu}\nabla_\mu S\nabla_\nu S
-\lambda\,D(S,S_0)
\Big] .
}
Here
- – curvature scalar of the metric induced by the entropic field,
- – coupling constants,
- – the distinguishability potential introduced above.
The first term gives geometric dynamics (like Einstein–Hilbert),
the second term supplies the “kinetic” energy of entropy variations,
and the third term penalizes deviation from equilibrium curvature .
4. Field equation: the Master Entropic Equation
Varying this action with respect to gives
\beta\,\nabla_\mu\nabla^\mu S
=\lambda\,\frac{\partial D}{\partial S}
=\lambda\,\ln\!\frac{S}{S_0}.
This is a nonlinear wave equation whose equilibrium points satisfy .
Small oscillations around equilibrium obey
\nabla_\mu\nabla^\mu\,\delta S
=\frac{\lambda}{\beta S_0}\,\delta S ,
showing that departures from equilibrium propagate as curvature waves with an effective mass term .
If we now couple back to the metric through the first term, the resulting Einstein–like equations read
G_{\mu\nu}
=\frac{1}{\alpha}
\Big[
\beta\,(\nabla_\mu S\nabla_\nu S-\tfrac{1}{2}g_{\mu\nu}(\nabla S)^2)
+\lambda\,g_{\mu\nu}D(S,S_0)
\Big],
which reduce to Einstein’s equations in the smooth limit where and . The ln 2 unit of distinguishability enters through the normalization of : when , .
5. Physical meaning
- The Araki–type term is no longer a static comparison; it is a potential driving the field toward equilibrium.
- The Spectral Obidi Action unifies geometry (), dynamics (), and information ().
- Extremizing this action yields both the Master Entropic Equation for and Einstein–like equations for .
In this construction the constant appears as the first non-zero minimum of the potential , identifying it with the smallest distinguishable curvature fold—the Obidi Curvature Invariant.
6. Summary
The essential move is that ToE does not turn Araki relative entropy itself into an action; rather, it uses its functional form as the potential term within an action that already contains dynamical curvature and derivative terms.
That is what makes the resulting Spectral Obidi Action a genuine and really powerful variational principle capable of producing equations of motion and, in the appropriate limit, the familiar dynamics of general relativity and thermodynamics.
From the Obidi Action and Field Equations of the Theory of Entropicity (ToE), the Einstein Field Equations of General Relativity and the Schrödinger Equation of Quantum Mechanics emerge after proper approximations.
How Quantizing Small Oscillations of S(x) in the Obidi Action Leads to a Wave Equation Analogous to the Schrödinger and Dirac Equations and Thereby Links the Theory of Entropicity (ToE) with Quantum Mechanics
Let us now see what it would actually mean to quantize small oscillations of the entropy field that appear in the Obidi action we wrote earlier.
1. Linearising the entropic field
Start with the background that satisfies the equilibrium (or “classical”) Master Entropic Equation,
\beta\,\nabla_\mu\nabla^\mu S_0=\lambda\,\ln\!\frac{S_0}{S_\text{ref}} .
Now write the full field as a small fluctuation about that background,
S(x)=S_0(x)+\delta S(x),\qquad |\delta S|\ll S_0 .
Expanding the distinguishability potential to second order gives
D(S,S_0)\approx
\frac{1}{2S_0}(\delta S)^2 ,
so in the small-fluctuation limit the Lagrangian density in the Spectral Obidi Action becomes, up to an overall ,
\mathcal L_{\text{quad}}
=
-\frac{\beta}{2}g^{\mu\nu}\nabla_\mu\delta S\,\nabla_\nu\delta S
-\frac{\lambda}{2S_0}(\delta S)^2 .
2. The field equation for small oscillations
Varying this quadratic Lagrangian gives the wave equation
\beta\,\nabla_\mu\nabla^\mu \delta S
+\frac{\lambda}{S_0}\,\delta S=0 .
In flat spacetime, with metric signature , this reduces to
\partial_t^2\delta S
-c_S^2\nabla^2\delta S
+\omega_S^2\,\delta S=0 ,
c_S^2=1,
\qquad
\omega_S^2=\frac{\lambda}{\beta S_0}.
This is a Klein–Gordon-type equation describing harmonic oscillations of the entropic field with an effective “mass”
m_S=\hbar\,\sqrt{\frac{\lambda}{\beta S_0}}/c^2 .
3. Canonical quantisation
To quantise, we promote and its conjugate momentum
\Pi(x)=\frac{\partial\mathcal L}{\partial(\partial_t\delta S)}
=\beta\,\partial_t\delta S
[\delta S(\mathbf x),\Pi(\mathbf y)]=i\hbar\,\delta^{(3)}(\mathbf x-\mathbf y).
Expanding in plane-wave modes gives
\delta S(x)=\!\!\int\!\!\frac{d^3k}{(2\pi)^3\sqrt{2\omega_k}}
\Big(
a_{\mathbf k}\,e^{-i\omega_k t+i\mathbf k\cdot\mathbf x}
+a_{\mathbf k}^\dagger\,e^{i\omega_k t-i\mathbf k\cdot\mathbf x}
\Big),
\omega_k^2=c_S^2k^2+\omega_S^2.
The operators and create and annihilate quanta of entropic curvature—small ripples in the field. These excitations are sometimes called entropions in the ToE language: the elementary carriers of entropic disturbance.
4. Relation to quantum wave equations
When spatial variations are slow and curvature coupling is weak (w small), each mode satisfies an approximate Schrödinger-type equation for its amplitude ,
i\hbar\,\partial_t\psi
=-\frac{\hbar^2}{2m_S}\nabla^2\psi
+V_{\text{eff}}(\mathbf x)\psi ,
where arises from the background .
Thus, quantum mechanics appears as the low-energy, single-quantum limit of the entropic field dynamics: wave-like behaviour of small curvature disturbances in an otherwise classical entropic background.
5. Conceptual picture
- The classical ToE equations describe how the mean entropic curvature shapes geometry and gravitation.
- The quantum limit describes small fluctuations of that curvature—the entropic analogue of matter waves.
- Planck’s constant sets the scale of quantisation, while the invariant fixes the smallest distinguishable amplitude of a curvature fold.
In this view, the Schrödinger and Dirac equations are not independent postulates but linearised, quantised limits of the broader entropic field theory embodied by the Spectral Obidi Action (SOA).
That completes the logical chain of Obidi's recovery of Quantum Mechanics from his Theory of Entropicity (ToE):
starting from a geometric action for the entropy field, linearising to find its wave equation, and quantising to recover the familiar quantum mechanical dynamics.
1. The Schrödinger equation and its meaning in ordinary physics
In standard quantum theory,
i\hbar\,\frac{\partial\psi}{\partial t}
=-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi ,
It is linear and norm-preserving because of the underlying symmetries: conservation of probability, Galilean invariance, and the role of Planck’s constant as the quantum of action.
What it does not explain is why nature uses complex amplitudes, why has the value it does, or how the quantum wave relates to thermodynamic or gravitational structure.
2. What the Obidi Action introduces
In the Theory of Entropicity, the basic variable is not but the entropic field , a continuous measure of information curvature.
The Spectral Obidi Action includes three parts:
\mathcal A_{\text{ToE}}
=\int\! d^4x\,\sqrt{-g}\,
\Big[
\frac{\alpha}{2}R[g]
-\frac{\beta}{2}(\nabla S)^2
-\lambda\,D(S,S_0)
\Big],
where the last term measures distinguishability from a local reference configuration.
When is expanded about equilibrium and quantised, its small fluctuations obey a linear wave equation.
That wave equation takes the same mathematical form as the Schrödinger or Klein–Gordon equation, but here its coefficients are not inserted by hand; they are derived from thermodynamic parameters of the entropic field:
i\hbar_{\text{eff}}\frac{\partial\psi}{\partial t}
=-\frac{\hbar_{\text{eff}}^2}{2m_S}\nabla^2\psi+V_{\text{ent}}\psi,
\hbar_{\text{eff}}^2 = 2\beta k_B T_S \ln2 .
Thus Planck’s constant appears as an effective constant of entropic curvature, linking information geometry and quantum action.
3. Physical interpretation supplied by ToE
-
Origin of h-bar :
In ToE, arises from the product of entropy, temperature, and the invariant ; it represents the minimal exchange of “entropic action,” the smallest reversible fold the field can make. Conventional quantum theory takes as empirical; ToE attempts to interpret it geometrically. -
Why the wave function is complex:
The field supports oscillatory solutions because entropy and energy are conjugate variables; combining them naturally yields pairs of harmonic modes that combine as complex exponentials. Complex amplitudes thus represent oscillations in entropic curvature, not abstract probability postulates. -
Conservation of probability:
The conservation law follows from the symmetry of the Obidi Action under global phase rotation of the oscillatory modes—Noether’s theorem provides the continuity equation automatically. -
Statistical interpretation:
Because encodes information density, represents the local informational weight, connecting thermodynamic and probabilistic views.
4. What this accomplishes conceptually
The Obidi Action therefore does more than reproduce Schrödinger’s equation:
- It roots the equation in an informational-geometric field rather than postulating it.
- It derives and the linear complex structure from the properties of entropy flow.
- It links quantum behaviour with gravitational and thermodynamic curvature through the same invariant ln 2.
- It predicts possible corrections if the entropic field deviates strongly from equilibrium—something testable in principle.
In summary, any action can be tuned to yield Schrödinger’s equation, but the Theory of Entropicity (ToE) is able to explain its ingredients:
why action is quantised, why complex amplitudes describe probability, and how thermodynamics, gravity, and quantum mechanics can all be faces of the same informational curvature field.
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