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Impossible Counterfactuals, Discrete Hilbert Space and Bell's Theorem

Tim Palmer

TL;DR

This work argues that Bell-inequality violations need not imply nonlocality or conspiratorial hidden correlations. By introducing Rational Quantum Mechanics (RaQM), a gravity-induced discretisation of Hilbert space where states are only well-defined in rational bases, the author shows MI can fail for exact measurement settings while preserving nominal free choice. The approach leverages number-theoretic constraints (notably Niven's theorem) to derive complementarity and non-commutativity from arithmetic, and employs $p$-adic fractal geometry to motivate a holistic invariant-set view of the universe. If valid, RaQM challenges the emphasis on probing smaller scales with more energetic accelerators and reframes Bell’s theorem as a statement about holism in state space rather than nonlocality. The work situates these ideas within a broader context of Mach’s principle and Bohm–Hiley’s Undivided Universe, suggesting new avenues for testing the limits of quantum mechanics and gravity.

Abstract

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.

Impossible Counterfactuals, Discrete Hilbert Space and Bell's Theorem

TL;DR

This work argues that Bell-inequality violations need not imply nonlocality or conspiratorial hidden correlations. By introducing Rational Quantum Mechanics (RaQM), a gravity-induced discretisation of Hilbert space where states are only well-defined in rational bases, the author shows MI can fail for exact measurement settings while preserving nominal free choice. The approach leverages number-theoretic constraints (notably Niven's theorem) to derive complementarity and non-commutativity from arithmetic, and employs -adic fractal geometry to motivate a holistic invariant-set view of the universe. If valid, RaQM challenges the emphasis on probing smaller scales with more energetic accelerators and reframes Bell’s theorem as a statement about holism in state space rather than nonlocality. The work situates these ideas within a broader context of Mach’s principle and Bohm–Hiley’s Undivided Universe, suggesting new avenues for testing the limits of quantum mechanics and gravity.

Abstract

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) -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.
Paper Structure (12 sections, 38 equations, 3 figures)

This paper contains 12 sections, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Measurement Independence and Counterfactual Definiteness
  3. Discretised Hilbert Space - Rational Quantum Mechanics
  4. Preamble
  5. Single-particle discretisation
  6. Multiple Qubits and EPR/Bell Locality
  7. Complementarity and Non-commutativity from Number Theory
  8. Complementarity
  9. Non-Commutativity
  10. $p$-adic Numbers and Fractal Geometry
  11. Bell's Theorem
  12. Discussion and Conclusions

Figures (3)

  • Figure 1: In QM, complementarity implies that simultaneous measurement outcomes do not exist for a quantum system in a Mach-Zehnder interferometer with a) the second half silvered mirror in place (a wave-like interferometric measurement), and b) removed (a particle-like which-way measurement). In RaQM, such complementarity is a mathematical consequence of a number-theoretic incommensurateness between angles and their cosines: Niven's Theorem Niven.
  • Figure 2: A spin-$1/2$ quantum system is prepared 'up' by Stern-Gerlach device $SG_A$. The spin-up output of $SG_A$ is fed into device $SG_B$ and the spin-up output of $SG_B$ is fed into $SG_C$. The experimenter is free to choose the nominal orientations of these apparatuses as they like. However, according to the rationality constraints (\ref{['ratSG2']}), Niven's Theorem prevents the simultaneous counterfactual world where $SG_B$ and $SG_C$ are swapped from having a well-defined measurement outcome.
  • Figure 3: An impossible spherical triangle, where the cosine of the angular length of each side of the triangle on the unit sphere is rational, and the internal angles are rational in degrees. The impossible triangle provides the basis for understanding both single particle non-commutativity and violation of Bell's inequality for entangled particles. In practice, for the sequential SG experiment, $\phi_B$ is close but not exactly equal to $180^\circ$.