⚛️ quantum physics

Impossible Counterfactuals, Discrete Hilbert Space and Bell's Theorem

This paper proposes a locally realistic model called Rational Mechanics (RaQM), which utilizes a discrete, $p$ -adic Hilbert space to violate Bell's inequality by restricting measurement independence to exact, physically unrealizable settings without denying free will, thereby suggesting that the search for a unified Theory of Everything via high-energy particle accelerators may be futile.

Original authors: Tim Palmer

Published 2026-01-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Tim Palmer

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

The Big Picture: Solving a 50-Year-Old Mystery

For decades, physicists have been puzzled by Bell's Theorem. Experiments show that particles can be "entangled," meaning they seem to communicate instantly across vast distances, defying our normal understanding of space and time. To explain this, most theories suggest one of three "weird" things is happening:

Non-locality: Particles are talking faster than light.
No Realism: Particles don't have definite properties until we look at them.
Conspiracy: The universe is rigged so that experimenters can't actually choose what to measure (a "superdeterministic" conspiracy).

Tim Palmer's paper proposes a fourth way. He suggests that the universe isn't "weird" or "rigged." Instead, the universe is discrete (made of tiny, indivisible chunks) rather than smooth and continuous. Because of this, there are certain measurements that are mathematically impossible to perform, even if we think we are free to choose them.

The Core Idea: The "Pixelated" Universe

Imagine the universe is like a high-resolution digital photo. To our eyes, it looks smooth and continuous. But if you zoom in enough, you see it's actually made of tiny square pixels. You can't have a line that is "half a pixel" wide; it has to be one pixel or two.

Palmer calls his theory Rational Quantum Mechanics (RaQM).

The Rule: In this theory, the "state" of a particle is only mathematically defined if its properties (like angles and probabilities) are rational numbers (fractions like 1/2, 3/4, 7/10).
The Problem: If you try to set up an experiment where the numbers are irrational (like $\pi$ or $\sqrt{2}$ ), the particle's state simply does not exist in that specific configuration. It's not that the particle is hidden; it's that the "world" where that specific measurement happens is mathematically undefined.

The "Impossible Triangle" Analogy

To understand why this breaks Bell's Theorem, imagine a game with three friends: Alice, Bob, and Charlie. They are standing on a giant sphere (the sky).

The Setup: Alice and Bob are far apart. They each have a compass. They want to measure the angle between their compasses.
The "Real" World: Alice picks a direction. Bob picks a direction. They measure the angle. In Palmer's theory, for the universe to "work," the cosine of that angle must be a rational number (a nice fraction).
The "What If" (Counterfactual): Bell's Theorem asks: "What if Bob had chosen a different direction?"
- In standard physics, we assume Bob could have chosen any direction, and the universe would still make sense.
- In Palmer's theory, if Bob changes his direction slightly, he might accidentally pick a direction where the angle with Alice becomes an irrational number.
- The Result: In that specific "what if" scenario, the universe cannot exist. The state of the particles is undefined. It's like trying to build a house on a foundation that doesn't exist.

The "Free Will" Misunderstanding

This is where the paper gets tricky but important. Critics often say, "If you say some measurements are impossible, you are taking away the experimenter's free will!"

Palmer says: No, I am not.

Nominal Accuracy (Free Will): You are completely free to choose your measurement settings. You can say, "I want my compass to point North." You can set it as accurately as your hand allows.
Exact Precision (The Limit): You are not free to control the exact atomic-level position of your compass. Why? Because the universe is constantly being jiggled by things you can't control, like gravitational waves from distant black holes.
The Analogy: Imagine you are trying to balance a pencil on its tip. You can try to balance it perfectly (your free will). But you cannot control the tiny vibrations of the floor or the air currents. Therefore, you can never achieve perfect balance.
The Conclusion: You are free to choose your "nominal" setting (e.g., "North"), but you are never in control of the "exact" setting (North + 0.0000000001 degrees). If that tiny, uncontrollable extra bit makes the math irrational, that specific "perfect" world never happens.

Why This Solves the Mystery

Bell's Theorem relies on a mathematical trick: it assumes you can compare the results of three different scenarios (Alice/Bob, Alice/Charlie, Bob/Charlie) all at once for the same particle.

Palmer argues that because of the "pixelated" nature of the universe:

You can measure Alice and Bob together (the math works).
You can measure Alice and Charlie together (the math works).
You can measure Bob and Charlie together (the math works).
BUT, you can never have a single universe where all three pairs are measured simultaneously with the exact precision required to prove Bell's inequality.

The "Impossible Triangle" of angles cannot exist in a single, consistent reality. Therefore, the inequality that proves "spooky action at a distance" is never actually tested in the way Bell thought. The universe isn't non-local; it's just that the "what if" scenarios Bell used in his math are impossible worlds.

The "Fractal" Connection

The paper mentions that the universe might be a fractal (a shape that repeats itself infinitely).

Think of a coastline. From far away, it looks smooth. Up close, it's jagged. Up closer, it's even more jagged.
Palmer suggests the "state space" of the universe (the map of all possible realities) is a fractal.
Most of the "points" on this map (the irrational numbers) are actually holes or gaps. The universe only exists on the "solid" parts (the rational numbers).
This connects to the idea of a "Holistic Universe" (like the one proposed by David Bohm and Basil Hiley), where everything is connected not by magic signals, but because the universe is a single, complex, deterministic machine that only allows certain patterns to exist.

The Bottom Line

Tim Palmer is saying:

We don't need to believe in faster-than-light communication.
We don't need to believe the universe is conspiring against us.
We don't need to believe reality isn't real.
We just need to accept that the universe is discrete (pixelated) and that exact measurements are impossible to control due to gravity and chaos.

Because of this, the "impossible" worlds required to prove Bell's Theorem simply cannot exist. The mystery of quantum entanglement is solved by realizing that the universe has a "resolution limit," and we can't force it to show us pictures it isn't capable of drawing.

Final Thought from the Author:
If this is true, building bigger particle accelerators to find a "Theory of Everything" might be a waste of time. Instead, we should look for the limits of quantum mechanics itself and how gravity might be the key to understanding why the universe is "pixelated" in the first place.

Technical Summary: Impossible Counterfactuals, Discrete Hilbert Space and Bell's Theorem

Problem Statement
The paper addresses the experimental violation of Bell's inequality, which traditionally forces a choice between abandoning locality, realism, or the Measurement Independence (MI) assumption. Abandoning MI is often dismissed as "superdeterminism," implying an implausible conspiracy where experimenters are denied the freedom to choose measurement settings. The author argues that this dismissal stems from a failure to distinguish between the nominal accuracy of measurement settings (which experimenters control) and the exact settings (which are subject to uncontrollable physical factors, such as gravitational waves). The core problem is to construct a locally realistic model that violates Bell's inequality by violating MI for exact settings, without denying free will or invoking non-locality.

Methodology
The paper proposes a model called Rational Quantum Mechanics (RaQM), based on a gravitational discretization of Hilbert Space. The methodology rests on three pillars:

Discretized Hilbert Space: Unlike standard Quantum Mechanics (QM), where Hilbert space is a continuum, RaQM posits that the state space is discrete. A unitary evolved qubit state $|\psi(\theta, \phi)\rangle$ is mathematically defined only in bases where the squared amplitudes ( $\cos^2 \theta/2$ ) and complex phases ( $\phi$ in degrees) are rational numbers.
Rationality Constraints and $p$ -adic Metrics: The model utilizes number theory, specifically Niven's Theorem, which states that the only angles $\phi$ $ϕ$ where both $\cos \phi$ $cos ϕ$ and $\phi/\pi$ $ϕ / π$ are rational are a finite set of specific values. Consequently, for any nominal measurement setting chosen by an experimenter, the corresponding exact setting required to define a quantum state in a counterfactual basis is almost certainly irrational (undefined) due to the density of irrational numbers.
- The paper argues that while physical space distances are measured by the Euclidean metric, distances between states in state space (including counterfactual worlds) are measured by the $p$ -adic metric. In this metric, irrational numbers are undefined, much like quantum states in irrational bases.
Bit String Representation: Quantum states are represented as finite-length bit strings (length $L \approx 10^{100}$ for typical qubits). The hidden variable $\lambda$ corresponds to a specific permutation $\xi$ of these bits, fixed at the system's creation. Measurement outcomes are determined by the bit at a specific position $M(\xi)$ after the application of $\xi$ .

Key Contributions

Reinterpretation of Measurement Independence (MI): The paper distinguishes between nominal settings ( $A_{nom}, B_{nom}$ ), which experimenters can choose freely, and exact settings ( $A, B$ ), which they cannot. RaQM violates MI only for exact settings ( $\rho(\lambda|A, B) \neq \rho(\lambda)$ ), while preserving it for nominal settings ( $\rho(\lambda|A_{nom}, B_{nom}) = \rho(\lambda)$ ). This avoids the "conspiracy" objection because experimenters are free to choose any nominal setting; they simply cannot guarantee the exact setting required to make a counterfactual world mathematically well-defined.
Resolution of Non-Commutativity and Complementarity: Using Niven's Theorem, the paper demonstrates that for a single particle, it is mathematically impossible for a quantum state to be simultaneously well-defined in two non-commuting bases (e.g., wave-like vs. particle-like measurements) if the exact angles involved satisfy the rationality constraints. This explains complementarity and non-commutativity as a consequence of number-theoretic incommensurateness rather than fundamental indeterminism.
Local Realism without Non-Locality: The model is explicitly EPR/Bell local. Alice's measurement outcome is determined by her local setting and the shared hidden variable $\lambda$ (the permutation $\xi$ ). Bob's choice of a different nominal setting does not alter Alice's bit string or outcome.

Results
The paper demonstrates that Bell's inequality is violated because the mathematical proof of the inequality requires the simultaneous existence of well-defined outcomes for three different measurement settings (an "impossible triangle" of settings) for a single hidden variable $\lambda$ .

In RaQM, for a given run of an experiment with fixed $\lambda$ , if Alice and Bob choose nominal settings $A_{nom}$ and $B_{nom}$ such that the exact settings $A$ and $B$ satisfy the rationality condition, the correlation is defined.
However, to satisfy Bell's inequality, one must also consider counterfactual worlds where Bob chooses $C_{nom}$ (with exact setting $C$ ) while Alice keeps $A$ , and where Alice chooses $C_{nom}$ while Bob keeps $B$ .
Due to the constraints of Niven's Theorem and the $p$ -adic metric, it is mathematically impossible for the exact settings $A, B,$ and $C$ to simultaneously satisfy the rationality conditions required to define the quantum state in all three bases.
Therefore, the terms in the Bell inequality summation (specifically the counterfactual terms) are mathematically undefined for a specific $\lambda$ . The inequality cannot be derived because the necessary counterfactual definiteness (CD) does not hold.

Significance and Claims
The paper claims that the violation of Bell's inequality does not necessitate "weird" phenomena such as non-locality, retrocausality, wormholes, or the branching of worlds. Instead, it suggests that the laws of physics are holistic, akin to Mach's Principle, where the structure of state space (a fractal invariant set) constrains which counterfactuals are physically possible.

Holistic Universe: The model aligns with Bohm and Hiley's concept of an "Undivided Universe," where the universe is a deterministic chaotic system evolving on a fractal invariant set in state space.
Implications for Fundamental Physics: If RaQM is correct, the search for a "Theory of Everything" via increasingly energetic particle accelerators to probe smaller scales may be fruitless. The synthesis of quantum and gravitational physics may instead require understanding the limits of QM itself and the role of gravity in defining the state space geometry.
Nature of Reality: The paper posits that the continuum limit of QM is a singular limit ( $G=0$ ). The "gappy" nature of the discretized Hilbert space, driven by the weakness of gravity, prevents the construction of classical systems with these properties, explaining why the macroscopic world appears classical while the microscopic world exhibits quantum behavior.

The author concludes that Bell's Theorem is not a proof of non-locality but a constraint on Counterfactual Definiteness arising from the number-theoretic properties of a discretized, gravitationally induced state space.