[Paper Review] Invariant Set Theory and the Symbolism of Quantum Measurement

T. N. Palmer|arXiv (Cornell University)|Feb 24, 2015

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TL;DR

This paper proposes Invariant Set Theory, a deterministic, realist framework for quantum physics where the universe evolves on a fractal invariant set $I_U$ in state space. By encoding quaternionic multiplication symbolically and taking a singular limit as a fractal parameter $N \to \infty$, the theory derives Hilbert space structure and non-commutativity from number-theoretic incommensurateness, yielding quantum phenomena without nonlocality or retrocausality.

ABSTRACT

Elements of a novel theory of quantum physics are developed, synthesising the role of symbolism in describing quantum measurement and in the topological representation of fractal invariant sets in nonlinear dynamical systems theory. In this synthesis, the universe $U$ is treated as an isolated deterministic dynamical system evolving precisely on a measure-zero fractal invariant subset $I_U$ of its state space. A non-classical approach to the physics of $U$ is developed by treating the geometry of $I_U$ as more primitive than dynamical evolution equations on $I_U$. A specific symbolic representation of $I_U$ is constructed which encodes quaternionic multiplication and from which the statistical properties of complex Hilbert Space vectors are emergent. The Hilbert Space itself arises as the singular limit of Invariant Set Theory as a fractal parameter $N ightarrow \infty$. Although the Hilbert Space of quantum theory is counterfactually complete, the measure-zero set $I_U$ is counterfactually incomplete, no matter how large is $N$. Such incompleteness allows reinterpretations of familiar quantum phenomena, consistent with realism and local causality. The non-computable nature of $I_U$ ensures that these reinterpretations are neither conspiratorial nor retrocausal and, through a homeomorphism with the ring of $2^N$-adic integers, are robust to noise and hence not fine tuned. The non-commutativity of Hilbert Space observables emerges from the symbolic representation of $I_U$ through the generic number-theoretic incommensurateness of $ϕ/π$ and $\cos ϕ$. Invariant Set Theory implies a much stronger synergy between cosmology and quantum physics than exists in contemporary theory, suggesting a novel approach to synthesising gravitational and quantum physics and providing new perspectives on the dark universe and information loss in black holes.

Motivation & Objective

• To develop a deterministic, realist alternative to quantum mechanics that avoids nonlocality and retrocausality.
• To unify quantum physics and cosmology by treating the universe's state space as a measure-zero fractal invariant set.
• To explain the emergence of Hilbert space and non-commutative observables from geometric and number-theoretic structures on $I_U$.
• To ensure robustness against noise through a homeomorphism with $2^N$-adic integers, avoiding fine-tuning.

Proposed method

• Model the universe $U$ as a deterministic dynamical system evolving on a measure-zero fractal invariant set $I_U$ in state space.
• Construct a symbolic representation of $I_U$ encoding quaternionic multiplication using number-theoretic properties.
• Derive the statistical structure of quantum states as the singular limit of $I_U$ as a fractal parameter $N \to \infty$, yielding Hilbert space.
• Establish non-commutativity through the incommensurateness of $\phi/\pi$ and $\cos\phi$ in the symbolic representation.
• Use a homeomorphism between $I_U$ and the ring of $2^N$-adic integers to ensure robustness to noise and avoid fine-tuning.
• Treat the geometry of $I_U$ as more fundamental than dynamical equations, making it the primitive structure of physical law.

Experimental results

Research questions

• RQ1 How can quantum mechanics be derived from a deterministic, realist theory based on fractal geometry?
• RQ2 What is the origin of Hilbert space structure in a non-probabilistic, geometric framework?
• RQ3 How does non-commutativity of observables emerge from number-theoretic incommensurateness rather than postulate?
• RQ4 Why is the universe's state space counterfactually incomplete, and how does this enable local causal explanations of quantum phenomena?
• RQ5 Can a theory based on a measure-zero set avoid conspiratorial or retrocausal features while remaining robust to noise?

Key findings

• The Hilbert space of quantum mechanics emerges as a singular limit of Invariant Set Theory as the fractal parameter $N \to \infty$.
• Non-commutativity of observables arises from the generic incommensurateness of $\phi/\pi$ and $\cos\phi$ in the symbolic representation of $I_U$.
• $I_U$ is counterfactually incomplete, even for arbitrarily large $N$, allowing local causal explanations of quantum phenomena.
• The theory is robust to noise due to a homeomorphism with the ring of $2^N$-adic integers, ensuring it is not fine-tuned.
• The theory preserves realism and local causality without requiring nonlocality or retrocausality.
• The symbolic representation of $I_U$ encodes quaternionic multiplication, from which the statistical properties of complex Hilbert space vectors are emergent.