Lorenz, Gödel and Penrose: New perspectives on determinism and causality in fundamental physics

Abstract

Despite being known for his pioneering work on chaotic unpredictability, the key discovery at the core of meteorologist Ed Lorenz's work is the link between space-time calculus and state-space fractal geometry. Indeed, properties of Lorenz's fractal invariant set relate space-time calculus to deep areas of mathematics such as Gödel's Incompleteness Theorem. These properties, combined with some recent developments in theoretical and observational cosmology, motivate what is referred to as the `cosmological invariant set postulate': that the universe $U$ can be considered a deterministic dynamical system evolving on a causal measure-zero fractal invariant set $I_U$ in its state space. Symbolic representations of $I_U$ are constructed explicitly based on permutation representations of quaternions. The resulting `invariant set theory' provides some new perspectives on determinism and causality in fundamental physics. For example, whilst the cosmological invariant set appears to have a rich enough structure to allow a description of quantum probability, its measure-zero character ensures it is sparse enough to prevent invariant set theory being constrained by the Bell inequality (consistent with a partial violation of the so-called measurement independence postulate). The primacy of geometry as embodied in the proposed theory extends the principles underpinning general relativity. As a result, the physical basis for contemporary programmes which apply standard field quantisation to some putative gravitational lagrangian is questioned. Consistent with Penrose's suggestion of a deterministic but non-computable theory of fundamental physics, a `gravitational theory of the quantum' is proposed based on the geometry of $I_U$, with potential observational consequences for the dark universe.

This manuscript has been accepted for publication in Contemporary Physics and is based on the author's 9th Dennis Sciama Lecture, given in Oxford and Trieste

References in corpus (2)

• Local deterministic model of singlet state correlations based on relaxing measurement independence
• Relaxed Bell inequalities and Kochen-Specker theorems

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• Experimental Non-Violation of the Bell Inequality
• Invariant Set Theory and the Symbolism of Quantum Measurement
• $p$-adic Distance, Finite Precision and Emergent Superdeterminism: A Number-Theoretic Consistent-Histories Approach to Local Quantum Realism
• A Gravitational Theory of the Quantum
• Bell's Conspiracy, Schrödinger's Black Cat and Global Invariant Sets
• Invariant Set Theory
• Invariant Set Theory: Violating Measurement Independence without Fine Tuning, Conspiracy, Constraints on Free Will or Retrocausality