[Paper Review] $p$-adic Distance, Finite Precision and Emergent Superdeterminism: A Number-Theoretic Consistent-Histories Approach to Local Quantum Realism

T. N. Palmer|arXiv (Cornell University)|Sep 24, 2016

Quantum Mechanics and Applications32 references3 citations

TL;DR

This paper proposes a locally causal, realistic quantum theory—Invariant Set Theory—where superdeterminism emerges naturally from number-theoretic constraints on state-space trajectories using the $p$-adic metric. By reinterpreting Hilbert space vectors via finite-bit descriptions of transcendental functions like cosine, the model explains Bell inequality violations without violating experimenter free will, offering a novel, non-conspiratorial resolution to the measurement problem and quantum gravity.

ABSTRACT

Although the notion of superdeterminism can, in principle, account for the violation of the Bell inequalities, this potential explanation has been roundly rejected by the quantum foundations community. The arguments for rejection, one of the most substantive coming from Bell himself, are critically reviewed. In particular, analysis of Bell's argument reveals an implicit unwarranted assumption: that the Euclidean metric is the appropriate yardstick for measuring distances in state space. Bell's argument is largely negated if this yardstick is instead based on the alternative $p$-adic metric. Such a metric, common in number theory, arises naturally when describing chaotic systems which evolve precisely on self-similar invariant sets in their state space. A locally-causal realistic model of quantum entanglement is developed, based on the premise that the laws of physics ultimately derive from an invariant-set geometry in the state space of a deterministic quasi-cyclic mono-universe. Based on this, the notion of a complex Hilbert vector is reinterpreted in terms of an uncertain selection from a finite sample space of states, leading to a novel form of `consistent histories' based on number-theoretic properties of the transcendental cosine function. This leads to novel realistic interpretations of position/momentum non-commutativity, EPR, the Bell Theorem and the Tsirelson bound. In this inherently holistic theory - neither conspiratorial, retrocausal, fine tuned nor nonlocal - superdeterminism is not invoked by fiat but is emergent from these `consistent histories' number-theoretic constraints. Invariant set theory provides new perspectives on many of the contemporary problems at the interface of quantum and gravitational physics, and, if correct, may signal the end of particle physics beyond the Standard Model.

Motivation & Objective

• To re-evaluate superdeterminism as a viable foundation for quantum mechanics, countering widespread rejection in the quantum foundations community.
• To resolve the tension between local realism and Bell inequality violations by replacing the Euclidean metric in state space with the $p$-adic metric.
• To develop a theory where quantum mechanics arises as a singular limit of a deterministic, fractal state-space geometry, preserving local causality.
• To reconcile quantum nonlocality with experimenter free will by grounding it in finite experimental precision and number-theoretic incompatibilities.
• To propose that quantum gravity and dark energy emerge from the geometry of invariant sets in the state space of a deterministic, quasi-cyclic universe.

Proposed method

• Uses the $p$-adic metric as the fundamental distance measure in state space, replacing the Euclidean metric assumed in Bell’s original argument.
• Models the universe as a deterministic, quasi-cyclic system evolving on a self-similar, invariant set $I_U$ that is locally homeomorphic to $\mathbb{Z}_2 \times \mathbb{R}$, where $\mathbb{Z}_2$ is the 2-adic integers.
• Represents quantum states as uncertain selections from finite sample spaces of $2^N$-bit strings, linking complex Hilbert vectors to number-theoretic properties of the cosine function.
• Applies number-theoretic constraints—specifically, the finite describability of $\cos\phi$ and $\phi/\pi$ in bits—to define mutually incompatible consistent histories.
• Derives the Schrödinger and Dirac equations as singular limits as a fractal parameter $N \to \infty$, with quantum mechanics emerging from a deterministic, finite-precision dynamics.
• Models quantum decoherence via chaotic riddled-basin dynamics, leading to natural clustering of trajectories into measurement eigenstates on the invariant set.

Experimental results

Research questions

• RQ1 Can superdeterminism be a viable, non-conspiratorial explanation for quantum nonlocality if the state-space distance metric is redefined from Euclidean to $p$-adic?
• RQ2 How can local realism be preserved while still violating Bell inequalities, without requiring fine-tuning or retrocausality?
• RQ3 What is the role of finite experimental precision in preserving experimenter free will within a deterministic quantum theory?
• RQ4 How do number-theoretic properties of transcendental functions like cosine lead to the structure of quantum mechanics and consistent histories?
• RQ5 Can quantum gravity and dark energy emerge from the geometric structure of a deterministic, fractal state-space invariant set?

Key findings

• The violation of Bell inequalities is not due to nonlocality or conspiracy, but arises from number-theoretic incompatibilities between finite-bit descriptions of $\cos\phi$ and $\phi/\pi$, which are almost always mutually exclusive.
• Experimenter free will is preserved because finite precision prevents direct control over whether $\cos\phi$ is finitely describable, making the outcome statistically unpredictable despite determinism.
• The complex Hilbert space and associated Schrödinger/Dirac equations emerge as a singular limit when the fractal parameter $N \to \infty$, with finite $N$ corresponding to a deterministic, discrete theory.
• Quantum decoherence is explained by chaotic riddled-basin dynamics, where trajectories cluster on the invariant set, forming the physical basis for measurement eigenstates.
• The cosmological constant is naturally small because vacuum fluctuations do not couple to gravity—this arises from the fractal, measure-zero structure of the invariant set, avoiding the 120-order-of-magnitude problem.
• The theory implies the end of particle physics beyond the Standard Model, as no particles (including gravitons) exist—instead, gravity is a geometric phenomenon of clustering on a fractal state-space manifold.