The Invariant Set Hypothesis: A New Geometric Framework for the Foundations of Quantum Theory and the Role Played by Gravity. Ted Palmer studied general relativity at the University of Oxford, working under the same PhD adviser as Stephen Hawking. He has worked for the last 20 years as a leading mathematical climatologist.

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 D. The Invariant Set Hypothesis is motivated by key results in nonlinear dynamical-systems 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 not on 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.

The 29 page Invariant Set Hypothesis paper is here

New Scientist has coverage of using fractals to make sense of the quantum world.

Principles of invariance and symmetry lie at the heart of the foundations of

physics. We have introduced a new type of invariance; the Invariant Set

Hypothesis subordinates the notion of the differential equation and elevates

as primitive the notion of fractal state space geometry in defining the notion

of physical reality. It is suggested that this has profound implications for our

understanding of quantum theory as discussed at length in the body of this

paper.The Invariant Set Hypothesis is motivated by two quite disparate ideas in

physics. Firstly, certain nonlinear dynamical systems have measure-zero,

nowhere-dense, self-similar non-computational invariant sets. Secondly, the

behaviour of extreme gravitationally bound systems is described by the

irreversible laws of thermodynamics at a fundamental rather than

phenomenological level.General relativity has already elevated geometry as a key concept for

investigating the causal structure of space time. The Invariant Set Hypothesis

similarly elevates geometry as a key concept for understanding the atemporal

structure of quantum physics.In the 1960s, the introduction of global space-time geometric and topological

methods, transformed our understanding of gravitational physics in space

time (Penrose, 1965). It is proposed that the introduction of global geometric

and topological methods in state space, may similarly transform our

understanding of quantum physics and the role of gravity in quantum

physics. Combining these disparate forms of geometry may provide the

missing element needed to advance the search for a unified theory of basic

physics.