The Invariant Set Postulate: A New Geometric Framework for the Foundations of Quantum Theory and the Role Played by Gravity

TL;DR

This paper introduces the Invariant Set Postulate, proposing a fractal geometric framework on a cosmological scale that offers new insights into quantum phenomena and gravity, challenging conventional quantum theory approaches.

Contribution

It presents a novel geometric model based on fractal invariant sets that unifies quantum mechanics and gravity, providing new interpretations of quantum mysteries and the role of complex numbers.

Findings

Invariant set geometry relates to quantum contextuality.

Fractal structure encodes quantum superposition and measurement.

Gravity manifests as heterogeneities in the fractal geometry.

Abstract

A new law of physics is proposed, defined on the cosmological scale but with significant implications for the microscale. Motivated by nonlinear dynamical systems theory and black-hole thermodynamics, the Invariant Set Postulate proposes that cosmological states of physical reality belong to a non-computable fractal state-space geometry I, invariant under the action of some subordinate deterministic causal dynamics. An exploratory analysis is made of a possible causal realistic framework for quantum physics, based on key properties of I. For example, sparseness is used to relate generic counterfactual states to points not lying on I, thus providing a geometric basis for the essential contextuality of quantum physics and the role of the abstract Hilbert Space in quantum theory. Also, self-similarity, described in a symbolic setting, provides a possible "realistic" perspective on the essential role of complex numbers and quaternions in quantum theory. A new interpretation is given to the standard "mysteries" of quantum theory: superposition, measurement, nonlocality, emergence of classicality and so on. It is proposed that heterogeneities in the fractal geometry of I are manifestations of the phenomenon of gravity. Since quantum theory is inherently blind to the existence of such state-space geometries, the analysis here suggests that attempts to formulate unified theories of physics within a conventional quantum-theoretic framework are misguided, and that a successful quantum theory of gravity should unify the causal non-Euclidean geometry of space time with the atemporal fractal geometry of state space.