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Information Theory of Action : Reconstructing Quantum Dynamics from Inference over Action Space

This paper proposes that quantum mechanics, including complex amplitudes and unitary evolution, emerges as the minimal consistent mathematical framework for describing dynamics when one performs maximum-entropy inference over an action space subject to a finite resolution scale.

Original authors: Fabricio Souza Luiz, Marcos César de Oliveira

Published 2026-02-11

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CC BY 4.0

Original authors: Fabricio Souza Luiz, Marcos César de Oliveira

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out the rules of a mysterious, high-stakes game. Usually, in physics, we start by assuming the rules: "Particles move like waves," or "Everything follows these specific mathematical equations."

This paper, "Information Theory of Action," flips that on its head. The authors argue that we don't need to assume the rules of Quantum Mechanics. Instead, they suggest that if you assume just a few basic facts about how we learn and measure the world, the "weirdness" of quantum physics emerges automatically, like a pattern appearing in a kaleidoscope.

Here is the breakdown of their logic using everyday analogies.

1. The Core Concept: The "Action" Budget

In physics, "Action" is like a "cost" or a "budget" associated with moving from point A to point B. In a classical world (like a car driving on a road), there is only one "cheapest" path—the one that uses the least amount of fuel.

The authors propose that instead of looking at paths, we should look at a "Density of Action States."

The Analogy: Imagine you are at a buffet. You don't just have one way to eat; there are thousands of different combinations of food you could put on your plate. The "Density of Action" is like a menu that tells you how many different ways you can achieve a certain "total calorie count" (the Action) to get from being hungry to being full.

2. The Problem: The "Blurry Vision" (Finite Resolution)

In our daily lives, we think we can measure things with perfect precision. But the authors argue that there is a fundamental "blurriness" to the universe. They use a concept called the Cramér–Rao bound, which is a fancy way of saying: "If you try to measure something too precisely, your information becomes unreliable."

The Analogy: Imagine you are looking at a digital photo. If you zoom in too far, you don't see more detail; you just see blurry pixels. You cannot distinguish between two colors if they are almost identical.
The Quantum Connection: The authors say that "Action" has a pixel size. If the difference between two paths is smaller than this "pixel" (which they identify as Planck’s Constant, $\hbar$ ), the universe literally cannot tell them apart. They become indistinguishable.

3. The Magic Trick: Why Waves? (Complex Amplitudes)

This is the most brilliant part of the paper. They ask: If two paths are so similar that they are "indistinguishable," how should we combine them?

You might think you just add their probabilities (Path A + Path B). But the authors show that if you want to keep your math consistent and ensure that "total probability" always adds up to 100% (Normalization), you cannot use simple addition.

To keep the math from breaking, you are forced to use Complex Numbers (numbers that have a "phase" or a "direction," like a clock hand).

The Analogy: Imagine two people walking toward you. If they are "distinguishable," you just count them: 1 + 1 = 2. But if they are "indistinguishable" (like two waves in a pool), they don't just add up. If one wave is at a peak and the other is in a trough, they cancel each other out (1 + -1 = 0).
The Result: Because of this "blurriness," the universe doesn't use simple numbers; it uses waves (amplitudes). Interference—the hallmark of quantum mechanics—is not a "rule" we added; it is the only logical way to combine information when you can't see the fine details.

4. The Grand Finale: Reconstructing the Universe

Once they establish that we must use these "wave-like" numbers to handle the blurriness, everything else falls into place like dominoes:

The Propagator: The way things move through space.
The Schrödinger Equation: The famous equation that governs quantum particles.
The Uncertainty Principle: The reason we can't know everything at once.

They didn't "invent" these; they derived them. They showed that if you accept that:

Moving has a "cost" (Action).
The universe has a "pixel size" (Resolution).
We must be consistent with our information (Inference).

...then Quantum Mechanics is the only possible way the universe could work.

Summary in one sentence:

Quantum mechanics isn't a set of strange rules imposed on nature; it is the inevitable mathematical consequence of trying to make sense of a world that has a fundamental limit on how much detail we can see.

Technical Summary: Information Theory of Action (ITA)

Title: Information Theory of Action: Reconstructing Quantum Dynamics from Inference over Action Space
Authors: Fabricio Souza Luiz and Marcos César de Oliveira
Core Objective: To reconstruct the entire mathematical formalism of quantum mechanics (complex amplitudes, Hilbert space, unitarity, and the Schrödinger equation) not as a set of independent postulates, but as the necessary consequence of rational inference applied to a fundamental density of action states under finite resolution.

1. The Problem: Conceptual Fragmentation in Quantum Mechanics

The authors identify a gap in the foundations of quantum mechanics: the standard Hilbert-space formalism treats several key elements—linear state structure, complex numbers, the Born rule, and the measurement postulate—as independent axioms. This obscures the logical interrelations between them. Furthermore, while the action ( $S$ ) is central to classical mechanics and Feynman’s path integrals, the path integral itself relies on a formal measure that lacks a rigorous operational definition. The authors seek to bridge this gap by asking if quantum mechanics can be derived from more primitive, information-theoretic principles.

2. Methodology: The Information-Theoretic Reconstruction

The ITA framework moves from dynamical structure to probabilistic representation. The derivation follows a rigorous logical chain:

Axiomatic Foundation: The theory starts with three minimal axioms:
1. Configuration Space ( $Q$ ): The kinematical arena.
2. Action Space ( $A$ ): A scalar, additive space representing accumulated action.
3. Density of Action States ( $g(A; b, T|a)$ ): A structural object (analogous to the density of states in statistical mechanics) that encodes the multiplicity of ways a system can evolve from $a$ to $b$ with total action $A$ . It is defined by its composition law (convolution), which reflects the additivity of action.
Maximum Entropy (MaxEnt) Inference: Given the multiplicity $g$ , the authors use Jaynes’ principle to assign probabilities $P(A, b|a)$ to dynamical alternatives. This results in an exponential family distribution: $P \propto g e^{-\eta A}$ .
The Cramér–Rao Bound & Indistinguishability: Using Fisher information, the authors derive a fundamental limit on action resolution: $\Delta A_{min} \sim 1/\eta$ . This establishes that action differences below a certain scale are operationally indistinguishable.
Consistency Requirements: The authors apply a Cox-type argument to determine how these indistinguishable alternatives must combine. They prove that to maintain probability normalization and respect action additivity, the only consistent representation is through complex amplitudes ( $\chi(A) = e^{i\eta A}$ ).

3. Key Contributions and Results

Emergence of Complex Amplitudes and Unitarity: Unlike previous reconstructions that postulate interference, ITA derives complex numbers as the unique continuous representation compatible with the requirement that probabilities must sum to unity (normalization) while allowing for the coherent combination of indistinguishable alternatives. Unitarity ( $|e^{i\eta A}| = 1$ ) is a derived theorem, not an axiom.
Derivation of the Propagator: The quantum propagator $K(b|a)$ is shown to be the Fourier transform of the action density $g$ . The authors prove that this propagator satisfies the semigroup property, which is the mathematical prerequisite for time evolution.
Identification of $\hbar$ : The parameter $\eta$ (the Lagrange multiplier from MaxEnt) is shown to govern both the resolution scale and the phase accumulation. By matching the predicted interference patterns (e.g., in a double-slit setup) with empirical data, the authors identify $\eta = 1/\hbar$ . Thus, $\hbar$ is interpreted as the fundamental information-theoretic resolution scale of action.
Derivation of the Schrödinger Equation:
- The Lagrangian emerges from the short-time limit of the mean action.
- The Gaussian form of the short-time propagator is derived from the Lévy–Khintchine theorem (as a consequence of the composition law and finite variance).
- The Schrödinger equation and canonical commutation relations ( $[\hat{x}, \hat{p}] = i\hbar$ ) emerge as the continuum limit of this propagator.
The Classical Limit: The classical limit ( $g \to \delta(A - S_{cl})$ ) is derived as a consequence of the stationary phase principle in the regime of large action, rather than being assumed.

4. Significance and Implications

Foundational Unity: The paper provides a "top-down" logical derivation where the complex-valued nature of quantum mechanics is revealed to be a requirement of consistent inference rather than a mysterious property of nature.
Minimalism: It reduces the "bag of postulates" in quantum mechanics to a much smaller set of information-theoretic and symmetry-based principles.
Conceptual Shift: It shifts the view of quantum mechanics from a theory of "wave-like particles" to a theory of rational inference under finite resolution. It suggests that quantum interference is the unavoidable way to represent dynamical alternatives when the action cannot be resolved with infinite precision.
Mathematical Rigor: By using the density of action states $g$ as a primitive, the authors avoid the mathematical ambiguities associated with the formal measure in the standard Feynman path integral.