Theory of Entropicity (ToE) by John Onimisi Obidi

Bogoliubov Transformation of Observer Dependent Creation and Annihilation Operators of Quantum Field Theory (QFT) and its Resolution in Obidi's Theory of Entropicity (ToE)

References:

https://github.com/Entropicity/Theory-of-Entropicity-ToE-Research-Lab-The-Aether-Live-Lab-NoteBook/blob/680a4b4b33d71942169fd15a2331a31c608f0a82/markdown-from-clickup-live-lab-notebook/How-Obidi's-Theory-of-Entropicity-(ToE)-Resolves-the-Bogoliubov-Transformations-of-Observer-Dependent-Creation-and-Annihilation-Operators-of-Relativistic-Quantum-Information-(RQI)-Theory.md

https://github.com/Entropicity/Theory-of-Entropicity-ToE-Research-Lab-The-Aether-Live-Lab-NoteBook/blob/343e416ad8b6c845cf74d7e06a0d8d84b5b0890f/markdown-from-clickup-live-lab-notebook/How-Obidi's-Theory-of-Entropicity-(ToE)-Resolves-the-Bogoliubov-Transformation-of-Relativistic-Quantum-Information-(RQI)-Theory.md

In John Onimisi Obidi’s Theory of Entropicity (ToE), the conventional paradoxes of the Bogoliubov transformation—such as observer-dependent particle creation (the Unruh and Hawking effects) and the degradation of quantum entanglement—are resolved by demoting spacetime and quantum states to emergent structures derived from a fundamental, ontological entropic field ($S(x)$). [1, 2, 3, 4, 5]

Instead of treating the Bogoliubov transformation as a subjective mixing of creation and annihilation operators across flat and curved geometries, ToE uses an informational-geometric approach where quantum fields are localized manifestations of a unified entropic flow. [5, 6, 7, 8, 9]

1. The Ontological Inversion: Dethroning the Observer

In standard Relativistic Quantum Information (RQI), a Bogoliubov transformation is required because an inertial observer and an accelerated observer (e.g., in Rindler spacetime) do not share the same vacuum state. The transformation mixes particle creation ( $a^{†}$ ) and annihilation ( $a$ ) operators, causing one observer to see a thermal bath of particles where another sees a vacuum. [1, 7, 8, 10, 11] Obidi resolves this apparent contradiction by applying the principle of Ontodynamics. In ToE: [12]

The observer is dethroned: The observer is not a privileged entity that dictates the state of the vacuum through their frame of reference. [5, 13]
Entropy is primary: The underlying reality is a continuous, objective entropic field. [5, 14]
Vacuum states are local capacities: The "vacuum" is simply a state of local entropic equilibrium governed by the Obidi Action Principle (OAP). What standard RQI views as observer-dependent "particle creation" is reinterpreted as the objective, physical redistribution of entropy due to local constraints. [13, 15, 16, 17]

2. The Obidi Transformation as a Disformal Deformation

Rather than relying on the classical unitary Bogoliubov matrices to map non-equivalent Fock spaces, ToE handles the transition between different frames of acceleration via the Obidi Transformation. [18]

Breaking Čencov’s Invariance: Standard information geometry relies on Čencov’s theorem, which restricts statistical metrics. Obidi introduces a controlled "Čencov breaking" via a rank-one disformal deformation. [18]
Information Geometry to Spacetime Geometry: This transformation maps the quantum state space (the Fubini–Study metric) directly into a physical, Lorentzian metric (the Obidi Metric). [18]
Geometric Translation: The operator mixing ( $u, v$ coefficients in Bogoliubov transformations) is absorbed into the geometric curvature of the entropic field. Acceleration dynamically deforms the information-geometric manifold, converting what RQI quantifies as "statistical distinguishability" into actual physical distance and energy density. [4, 9, 19, 20]

3. Resolution of Entanglement Degradation via the "No-Rush" Theorem

In RQI, Bogoliubov transformations across a horizon cause an incurable degradation of quantum resources (like entanglement and mutual information) due to info-scrambling and mode loss. ToE bypasses this degradation using two key principles: [1, 7]

The "No-Rush" Theorem: This theorem dictates that the entropy field cannot rearrange itself instantaneously; the speed of light ( $c$ ) is defined strictly as the Entropic Speed Limit (ESL)—the maximum rate of entropic redistribution. [5, 9, 21]
Entropic Capacity Constraints: When an RQI system undergoes extreme acceleration, the "loss" of entanglement is not a destruction of information, but a conservation necessity. As a system's velocity or acceleration increases, the localized entropic field allocates its finite capacity to maintaining macroscopic motion. This leaves less capacity for internal sub-system correlations, producing the mathematical illusion of decoherence or thermal noise in standard RQI calculations. [15, 22]

Summary of the Resolution

Conceptual Dimension	Relativistic Quantum Information (RQI)	Theory of Entropicity (ToE)
The Vacuum State	Frame-dependent; relative to the observer's path.	Monistic; a real state of local entropic field equilibrium.
Bogoliubov Mixing	Mathematical transformation of operators ( $a \to α u + β^{†} v$ ).	Physical disformal deformation of the underlying information geometry.
Particle Creation	Perceived thermal bath due to horizons/acceleration.	Objective condensation and rearrangement of the entropic field.
Entanglement Loss	Permanent degradation of quantum resources across frames.	Apparent loss due to strict entropic capacity constraints under the "No-Rush" theorem.

By unifying general relativity, quantum mechanics, and information geometry under the Master Entropic Equation (MEE), Obidi demonstrates that the Bogoliubov transformation is not an unresolvable quantum-foundational paradox, but an inevitable consequence of a dynamic, irreversible entropic reality. [13, 18]

If you want to dig deeper into the mathematical mechanics of ToE, let us know if you would like to:

Explore the specific role of the Obidi Curvature Invariant ( $\ln 2$ ) in defining physical distinguishability limits.
Review how the Vuli-Ndlela Integral calculates macroscopic mass from microscopic entropic fluctuations.
Look at how ToE mathematically derives Mercury's perihelion precession without standard Einsteinian gravity. [20, 23, 24, 25]

[1] https://arxiv.org

[2] https://en.wikipedia.org

[3] https://www.researchgate.net

[4] https://medium.com

[5] https://medium.com

[6] https://entropicity.github.io

[7] https://link.aps.org

[8] https://ui.adsabs.harvard.edu

[9] https://medium.com

[10] https://www.mas.ncl.ac.uk

[11] https://carlip.physics.ucdavis.edu

[12] https://www.cambridge.org