Entropy as a Field: Can the Spectral Obidi Action Go Beyond Araki Relative Entropy? A Most Radical Conceptualization of a Unified Field in the Theory of Entropicity (ToE)

Introduction: The Measure That Became a Candidate for a Field

Entropy has always been one of the most enigmatic concepts in science. From the early days of thermodynamics, where it was introduced as a measure of disorder, to the information age, where Shannon reframed it as a measure of uncertainty, entropy has been treated as a tool rather than a thing. It is something we calculate, something we use to compare states, something we invoke to explain irreversibility. But it has never been considered a field in the same way that electromagnetism or gravity are fields.

The Theory of Entropicity (ToE) challenges this long-standing assumption. It proposes that entropy is not merely a statistical measure but the fundamental continuum of reality itself. At the heart of this proposal lies the Spectral Obidi Action (SOA), an action principle that looks strikingly similar to Araki relative entropy but is claimed to serve a radically different purpose.

This raises a provocative question: is ToE simply repeating Araki’s work under a new name, or does the SOA genuinely open a new path by treating entropy as a field?

Araki Relative Entropy: The Established Framework

To appreciate the novelty of SOA, we need to understand what Araki relative entropy already does. In operator algebra and quantum field theory, Araki relative entropy is defined as:

Araki Relative Entropy

This expression compares two quantum states, ρ\rho and σ\sigma, through the modular operator:

Araki Modular Operator

Its meaning is precise: it quantifies how distinguishable one state is from another. It is a measure of relative information, deeply tied to the structure of von Neumann algebras and modular theory.

Araki entropy has been deployed extensively. It appears in studies of entanglement entropy, in modular Hamiltonians, and in the algebraic formulation of quantum field theory. It is mathematically rigorous, physically interpretable, and widely accepted. But it is always used as a measure. It does not evolve. It does not generate equations of motion. It does not act as a field.

The Spectral Obidi Action: A Radical Reinterpretation

The Spectral Obidi Action (SOA) proposed in the Theory of Entropicity (ToE) is written as:

Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE)

At first glance, this looks like Araki entropy stripped of its dependence on specific states. But ToE interprets it differently. Instead of being a measure of distinguishability, SOA is framed as an action principle — something to be varied, something that generates dynamics.

This is a profound shift in theoretical physics. In physics, an action principle is not just a mathematical curiosity. It is the foundation of dynamics. The Einstein–Hilbert action generates Einstein’s equations. The Yang–Mills action generates the equations of gauge fields. By proposing SOA as an action, ToE is suggesting that entropy itself can be treated as a field variable, with its own equations of motion.

In this framework, entropy is no longer emergent. It is fundamental. And SOA is not isolated — it is coupled with other terms: geometric actions, generalized entropies (Shannon, von Neumann, Rényi, Tsallis, KL, Araki), spectral operator geometry, and causal constraints. Together, these form the Generalized Obidi Action, from which the Master Entropic Equation (MEE) emerges.

Why No One Else Has Tried This

The absence of prior work in this direction is not accidental. Researchers have avoided treating entropy as a field for several reasons.

First, entropy has always been understood as emergent. It arises from coarse-graining, from statistical descriptions, from the loss of information about microstates. To elevate it to a fundamental field risks stripping it of its meaning.

Second, entropy in its traditional forms does not generate dynamics. Araki relative entropy, for example, is relational. It compares states but does not evolve them. It is not designed to produce equations of motion.

Third, the physics community is cautious. Without clear predictions or experimental consequences, entropy-as-field risks being mathematically elegant but physically empty. Researchers prefer frameworks that yield testable results, and entropy has always been seen as a derived quantity rather than a fundamental one.

Is SOA Valid or Useful?

This brings us to the crux of the matter: is the SOA action principle valid, and is it useful?

Validity here means more than mathematical consistency. It requires that varying SOA yields well-defined field equations. It requires that those equations integrate coherently with the rest of physics. And it requires that the framework does not collapse into redundancy with existing measures like Araki entropy.

Usefulness, meanwhile, demands predictive novelty. If SOA can lead to testable predictions — finite-rate entanglement formation, corrections to general relativity, causal bounds — then it offers something new. If it can integrate entropy measures into a unified field theory that explains time’s arrow, spacetime curvature, and quantum coherence, then it is more than a restatement.

But if it cannot, then it risks being a formal repackaging of known entropy measures, elegant but empty.

The Stakes for ToE

The stakes are high. If entropy can be treated as a field, physics could be recast in entropic terms. Spacetime curvature would be understood as entropic geometry. Quantum coherence would emerge from spectral entropy dynamics. Causality would be enforced by finite-rate entropy redistribution. And time’s arrow would be explained as entropic asymmetry.

This would be a profound unification, bringing together thermodynamics, relativity, and quantum mechanics under a single entropic continuum. But it requires ToE to demonstrate that SOA is not just Araki entropy in disguise, but a genuine action principle with predictive power.

Conclusion: A Bold but Risky Leap

The Spectral Obidi Action is bold. It risks redundancy, but it also opens the door to a new way of thinking. No other researchers have tried to recast Araki relative entropy into a field-theoretic action because, in its traditional form, it does not yield physical meaning or dynamics. ToE is unusual in attempting it.

Whether SOA is valid depends on whether ToE can show that entropy-as-field yields new dynamics and testable predictions. If it can, this could mark a turning point in physics. If not, it will remain an elegant but empty reformulation.

Either way, the question is captivating: Can entropy itself be the field that unifies physics?