Epistemic Horizons From Deterministic Laws: Lessons From a Nomic Toy Theory

Imagine you are trying to take a perfect photograph of a speeding race car. In the world of everyday physics, we usually assume that if you have a fast enough camera and a steady enough hand, you can capture the car's exact location and its exact speed at the same time. If you know where it is and how fast it's going, you can predict exactly where it will be a second from now. This idea, that the universe is a giant clockwork machine where everything is knowable if you just look hard enough, is the traditional view of classical mechanics.

But there's a famous twist in the story of physics called the "epistemic horizon." Think of this as a foggy window that no matter how hard you wipe, you can never clear completely. In the strange world of quantum mechanics, this fog is real: you simply cannot know certain pairs of facts about a particle (like its position and speed) at the same time. The more you know about one, the less you know about the other. This isn't just because our tools are bad; it's a fundamental rule of nature. Scientists have long wondered: Is this fog just a quirk of the quantum world, or could it happen even in a perfectly predictable, "classical" universe? If we built a world with strict, unbreakable laws, could an observer still be forced to remain ignorant of some details?

This paper, titled "Epistemic Horizons From Deterministic Laws," dives into that very question. The authors, Johannes Fankhauser, Tomáš Gonda, and Gemma De les Coves, construct a simple, made-up universe they call "nomic toy theory." In this universe, everything follows strict, predictable rules—there is no randomness, no magic, and no quantum weirdness. However, they introduce a special rule for how "observers" (who are just physical objects like the things they are watching) gather information. They prove that even in this perfectly logical, clockwork world, an observer hits a hard wall. They can never learn everything about a system at once. If they try to measure one property, they inevitably lose the ability to know another.

The authors show that this limitation isn't because the universe is chaotic or because the observer is clumsy. It's because of the way the observer and the object interact. In their model, an observer is like a detective who can only see the world through a specific pair of glasses. These glasses only let certain types of information pass through. If the detective tries to look at a "position" clue, the glasses blur the "speed" clue, and vice versa. The paper proves mathematically that this happens even though the underlying laws of the toy universe are completely deterministic. The result is a fascinating discovery: the "fog" of uncertainty doesn't require a quantum universe to exist; it can emerge naturally from the simple fact that an observer is a physical part of the system they are trying to study.

The Detective and the Foggy Glasses

To understand how this works, let's imagine a game played in a giant, flat playground. In this game, there are two types of characters: Objects (the things being studied) and Subjects (the detectives trying to learn about them).

In a normal, boring physics class, a detective could walk up to an Object, measure its "Position" (where it is), and then measure its "Momentum" (how fast and in what direction it's moving) without changing anything. They could write down both numbers and know the Object's entire secret life.

But in the authors' "nomic toy theory," the rules are different. Here, the detective is not a ghost floating outside the game; they are a physical object made of the same stuff as the Object. To learn something, the detective must bump into the Object. This bump is a physical interaction, like two billiard balls colliding.

The authors set up a specific rule for the detectives: A detective only "knows" what they can see on their own dashboard. Let's call this dashboard the "Manifest Variable." For our detective, this dashboard only shows their own "Position." They have no dashboard for their own "Momentum." They are blind to their own speed.

Now, imagine the detective wants to learn the Position of the Object. They line up their dashboard with the Object and perform a specific "measurement interaction" (a carefully choreographed collision). Because of the strict laws of this toy universe, this collision does two things:

1. It updates the detective's dashboard to show the Object's Position.
2. It accidentally kicks the Object's Momentum, scrambling it based on how fast the detective was moving.

Here is the kicker: Because the detective doesn't know their own speed (their Momentum), they cannot calculate how hard they kicked the Object. They see the new Position on their dashboard, but the Object's Momentum is now a mystery.

The paper proves a powerful theorem: You can only measure things that are "compatible." In this toy world, "compatible" means the two things you want to know don't fight each other. If you try to measure the Position and the Momentum at the same time, the math of the universe says it's impossible. The interaction that reveals the Position must scramble the Momentum, and there is no way to fix it because the detective is blind to their own speed.

The "Glasses" Analogy

Think of the detective's ability to learn as wearing a pair of special sunglasses.

• If you wear Vertical Glasses, you can see the "Vertical" properties of the world clearly, but everything "Horizontal" looks like a blur.
• If you wear Horizontal Glasses, you see the Horizontal properties, but the Vertical ones are a blur.

In this theory, the detective can choose which glasses to wear (which measurement to perform). But they can never wear both at once. The paper shows that in this specific, deterministic world, the laws of physics act like those glasses. The "fog" isn't because the world is random; it's because the act of looking changes what you see, and the looker is too tied up in the system to correct for the change.

What This Means for the Big Picture

The authors' main finding is that uncertainty can arise from pure logic and interaction, not just from quantum magic. They built a world where everything is determined by strict laws (like a perfect computer simulation), yet the "players" in that world are fundamentally limited in what they can know.

They explicitly rule out the idea that this limitation is just a lack of technology. It's not that the detective's tools are too slow; it's that the tools are the detective. If the detective tries to fix the problem by measuring their own speed first, the paper shows that this just leads to a new set of problems. You can't measure your own speed without messing up the thing you're trying to measure, and you can't measure your own speed without knowing your own position, which you can't know without messing up your speed. It's a perfect loop of ignorance.

This connects to a famous idea called Spekkens' Toy Theory. The authors show that their deterministic "nomic toy theory" is the hidden engine behind Spekkens' theory. Spekkens' theory was a puzzle that said, "Let's pretend we can't know everything, and see what happens." The authors of this paper say, "We don't need to pretend. If we build a world where observers are physical systems, the 'pretend' rule happens automatically."

So, the next time you hear that the universe is fuzzy and uncertain, remember this: maybe the fuzziness isn't because the universe is broken or random. Maybe it's just because we are inside the machine, wearing glasses that only let us see one side of the picture at a time. The authors have proven that even in a perfectly clear, logical world, the act of looking creates a horizon we cannot cross.