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

The Invariant Set Hypothesis proposes that states of physical reality belong to, and are governed by, a non-computable fractal subset I of state space, invariant under the action of some subordinate deterministic causal dynamics I D . The Invariant Set Hypothesis is motivated by key results in nonlinear dynamicalsystems theory, and black-hole thermodynamics. The elements of a reformulation of quantum theory are developed using two key properties of I : sparseness and self-similarity. Sparseness is used to relate counterfactual states to points p I ∉ thus providing a basis for understanding the essential contextuality of quantum physics. Self similarity is used to relate the quantum state to oscillating coarse-grain probability mixtures based on fractal partitions of I , thus providing the basis for understanding the notion of quantum coherence. Combining these, an entirely analysis is given of the standard “mysteries” of quantum theory: superposition, nonlocality, measurement, emergence of classicality, the ontology of uncertainty and so on It is proposed that gravity plays a key role in generating the fractal geometry of I . Since quantum theory does not itself recognise the existence of such a state-space geometry, the results here suggest that attempts to formulate unified theories of physics within a quantum theoretic framework are misguided; rather, a successful quantum theory of gravity should unify the causal non-euclidean geometry of space time with the atemporal fractal geometry of state space.