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The Invariant Set Hypothesis: A New Geometric Framework for the Foundations
of Quantum Theory and the Role Played by Gravity
Authors: T.N.Palmer
(Submitted on 5 Dec 2008 (v1 <> ), last
revised 17 Feb 2009 (this version, v3))
Abstract: 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.

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