Invariant Set Theory

TL;DR

Invariant Set Theory proposes a deterministic, fractal geometric model of the universe that unifies cosmology and quantum physics, explaining quantum phenomena through the geometry of a measure-zero invariant set in state space.

Contribution

It introduces a novel geometric framework based on fractal invariant sets and p-adic metrics, providing a new approach to quantum physics and gravity without fine-tuning.

Findings

Describes quantum observables as arising from the geometry of invariant sets.

Shows how the Dirac equation and Hilbert space emerge in the theory.

Provides a potential pathway to unify quantum mechanics and gravity.

Abstract

Invariant Set Theory (IST) is a realistic, locally causal theory of fundamental physics which assumes a much stronger synergy between cosmology and quantum physics than exists in contemporary theory. In IST the (quasi-cyclic) universe $U$ is treated as a deterministic dynamical system evolving precisely on a measure-zero fractal invariant subset $I_U$ of its state space. In this approach, the geometry of $I_U$, and not a set of differential evolution equations in space-time $\mathcal M_U$, provides the most primitive description of the laws of physics. As such, IST is non-classical. The geometry of $I_U$ is based on Cantor sets of space-time trajectories in state space, homeomorphic to the algebraic set of $p$-adic integers, for large but finite $p$. In IST, the non-commutativity of position and momentum observables arises from number theory - in particular the non-commensurateness of $\phi$ and $\cos \phi$. The complex Hilbert Space and the relativistic Dirac Equation respectively are shown to describe $I_U$, and evolution on $I_U$, in the singular limit of IST at $p=\infty$; particle properties such as de Broglie relationships arise from the helical geometry of trajectories on $I_U$ in the neighbourhood of $\mathcal M_U$. With the p-adic metric as a fundamental measure of distance on $I_U$, certain key perturbations which seem conspiratorially small relative to the more traditional Euclidean metric, take points away from $I_U$ and are therefore unphysically large. This allows (the $\psi$-epistemic) IST to evade the Bell and Pusey et al theorems without fine tuning or other objections. In IST, the problem of quantum gravity becomes one of combining the pseudo-Riemannian metric of $\mathcal M_U$ with the p-adic metric of $I_U$. A generalisation of the field equations of general relativity which can achieve this is proposed.