Lorenz, G'{o}del and Penrose: New perspectives on determinism and causality in fundamental physics

摘要

Despite being known for his pioneering work on chaotic unpredictability, thekey discovery at the core of meteorologist Ed Lorenz's work is the link betweenspace-time calculus and state-space fractal geometry. Indeed, properties ofLorenz's fractal invariant set relate space-time calculus to deep areas ofmathematics such as G\"{o}del's Incompleteness Theorem. These properties,combined with some recent developments in theoretical and observationalcosmology, motivate what is referred to as the `cosmological invariant setpostulate': that the universe $U$ can be considered a deterministic dynamicalsystem evolving on a causal measure-zero fractal invariant set $I_U$ in itsstate space. Symbolic representations of $I_U$ are constructed explicitly basedon permutation representations of quaternions. The resulting `invariant settheory' provides some new perspectives on determinism and causality infundamental physics. For example, whilst the cosmological invariant set appearsto have a rich enough structure to allow a description of quantum probability,its measure-zero character ensures it is sparse enough to prevent invariant settheory being constrained by the Bell inequality (consistent with a partialviolation of the so-called measurement independence postulate). The primacy ofgeometry as embodied in the proposed theory extends the principles underpinninggeneral relativity. As a result, the physical basis for contemporary programmeswhich apply standard field quantisation to some putative gravitationallagrangian is questioned. Consistent with Penrose's suggestion of adeterministic 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.