Impossible Counterfactuals, Discrete Hilbert Space and Bell's Theorem

Negating the Measurement Independence assumption (MI) is often referred to as the third way' to account for the experimental violation of Bell's inequality. However, this route is generally viewed as ludicrously contrived, implying some implausible conspiracy where experimenters are denied the freedom to choose measurement settings as they like. Here, a locally realistic model of quantum physics is developed (Rational Mechanics - RaQM - based on a gravitational discretisation of Hilbert Space) which violates MI without denying free will. Crucially, RaQM distinguishes experimenters' ability to freely choose measurement settings to some nominal accuracy, from an inability to choose exact settings, which were never under their control anyway. In RaQM, Hilbert states are necessarily undefined in bases where squared amplitudes and/or complex phases are irrational numbers. Such irrational’ bases correspond to conceivable but necessarily impossible counterfactual measurements, and are shown to play a ubiquitous role in the analysis of both single- and entangled-particle quantum physics. It is concluded that violation of Bell inequalities can be understood with none of the strange processes historically associated with it. Instead, using concepts from (non-classical) $p$-adic number theory, we relate RaQM to Bohm and Hiley’s concept of a holistic Machian-like Undivided Universe. If this interpretation of Bell’s Theorem is correct, building more and more energetic particle accelerators to probe smaller and smaller scales, in the search for a theory which synthesises quantum and gravitational physics and hence a Theory of Everything, may be a fruitless exercise.

💡 Research Summary

Tim Palmer’s paper proposes a novel locally realistic framework called Rational Quantum Mechanics (RaQM) that seeks to explain the experimental violation of Bell’s inequality without invoking non‑locality, super‑deterministic conspiracies, or other “weird” quantum phenomena. The central idea is that Hilbert space is not a continuous mathematical ideal but is discretised by gravity at an extremely fine scale. In this discretised space, quantum states are only mathematically defined in bases where the squared amplitudes are rational numbers and the complex phases (expressed in degrees) are rational multiples of 2π. Bases that would require irrational amplitudes or phases correspond to “impossible” counterfactual measurements – they are conceivable but physically unattainable.

The paper distinguishes between nominal measurement settings (the values experimenters can freely choose, e.g., “set the interferometer arms equal”) and exact settings (the precise values that would be required for a perfect mathematical description). Because of unavoidable gravitational disturbances (e.g., passing gravitational waves), experimenters cannot control the exact settings; they can only achieve them to a finite precision. Consequently, the usual Measurement Independence (MI) condition ρ(λ|A,B)=ρ(λ) holds for nominal settings but fails for exact settings. This nuanced violation of MI preserves free will while allowing a locally realistic model to reproduce Bell‑type correlations.

Mathematically, RaQM represents complex numbers and quaternions as permutation/negation operators acting on finite‑length bit strings of length L, where L is a large integer set by the gravitational discretisation (estimated around 10^100 for a typical qubit). A qubit state |ψ(θ,φ)⟩ is encoded as a bit string whose proportion of +1’s equals cos²(θ/2) and whose cyclic permutation count encodes the phase φ. The hidden permutation ξ plays the role of a hidden variable that determines which bit of the string is read out as the measurement outcome. For multi‑qubit systems, the same ξ acts on each string, providing the necessary correlations for entangled states. When L is sufficiently large, the model reproduces the full Hilbert space dimension of N qubits (2^N).

A crucial technical ingredient is the use of a p‑adic metric on state space rather than the usual Euclidean metric. In the p‑adic sense, two rational numbers that differ by a very small p‑adic distance are considered “close,” which naturally accommodates the distinction between nominal and exact settings. This metric also underlies the claim that RaQM is not fine‑tuned: the set of rational bases is dense in the p‑adic topology, so the theory does not require an implausible conspiracy between hidden variables and measurement choices.

Applying this framework to Bell experiments, Palmer shows that the counterfactual outcomes required for the standard Bell derivation are undefined because they would involve irrational amplitudes or phases. Hence the Bell inequality does not apply, and its violation does not imply any non‑local influence. The model reproduces the quantum predictions for entangled pairs while maintaining locality and a genuine notion of free choice (at the nominal level).

The paper further connects RaQM to Bohm and Hiley’s “Undivided Universe” by interpreting the p‑adic fractal structure of state space as a holistic invariant set. In this view, the laws of physics are globally constrained rather than locally independent, echoing Mach’s principle.

Finally, Palmer argues that RaQM makes experimentally testable predictions: the finite information capacity L should manifest as deviations from perfect Born‑rule statistics in high‑precision quantum‑computing experiments, and gravitational‑wave‑induced phase jitter could be used to probe the nominal‑exact distinction. If confirmed, the pursuit of ever higher‑energy particle accelerators to probe smaller scales may be misguided, as the fundamental discreteness set by gravity would limit the resolution of physical law.