The synthesis of quantum and gravitational physics is sought through a finite, realistic, locally causal theory where gravity plays a vital role not only during decoherent measurement but also during non-decoherent unitary evolution. Invariant set theory is built on geometric properties of a compact fractal-like subset
IU of cosmological state space on which the universe is assumed to evolve and from which the laws of physics are assumed to derive. Consistent with the primacy of
IU, a non-Euclidean (and hence non-classical) state-space metric
gp is defined, related to the
p-adic metric of number theory where
p is a large but finite Pythagorean prime. Uncertain states on
IU are described using complex Hilbert states, but only if their squared amplitudes are rational and corresponding complex phase angles are rational multiples of
2π. Such Hilbert states are necessarily
gp-distant from states with either irrational squared amplitudes or irrational phase angles. The gappy fractal nature of
IU accounts for quantum complementarity and is characterised numerically by a generic number-theoretic incommensurateness between rational angles and rational cosines of angles. The Bell inequality, whose violation would be inconsistent with local realism, is shown to be
gp-distant from all forms of the inequality that are violated in any finite-precision experiment. The delayed-choice paradox is resolved through the computational irreducibility of
IU. The Schrödinger and Dirac equations describe evolution on
IU in the singular limit at
p=∞. By contrast, an extension of the Einstein field equations on
IU is proposed which reduces smoothly to general relativity as
p→∞. Novel proposals for the dark universe and the elimination of classical space-time singularities are given and experimental implications outlined.
Elements of a novel theory of quantum physics are developed, synthesising the role of symbolism in describing quantum measurement and in the topological representation of fractal invariant sets in nonlinear dynamical systems theory. In this synthesis, the universe
U is treated as an isolated deterministic dynamical system evolving precisely on a measure-zero fractal invariant subset
IU of its state space. A non-classical approach to the physics of
U is developed by treating the geometry of
IU as more primitive than dynamical evolution equations on
IU. A specific symbolic representation of
IU is constructed which encodes quaternionic multiplication and from which the statistical properties of complex Hilbert Space vectors are emergent. The Hilbert Space itself arises as the singular limit of Invariant Set Theory as a fractal parameter
N→∞. Although the Hilbert Space of quantum theory is counterfactually complete, the measure-zero set
IU is counterfactually incomplete, no matter how large is
N. Such incompleteness allows reinterpretations of familiar quantum phenomena, consistent with realism and local causality. The non-computable nature of
IU ensures that these reinterpretations are neither conspiratorial nor retrocausal and, through a homeomorphism with the ring of
2N-adic integers, are robust to noise and hence not fine tuned. The non-commutativity of Hilbert Space observables emerges from the symbolic representation of
IU through the generic number-theoretic incommensurateness of
ϕ/π and
cosϕ. Invariant Set Theory implies a much stronger synergy between cosmology and quantum physics than exists in contemporary theory, suggesting a novel approach to synthesising gravitational and quantum physics and providing new perspectives on the dark universe and information loss in black holes.
A new law of physics is proposed, defined on the cosmological scale but with significant implications for the microscale. Motivated by nonlinear dynamical systems theory and black-hole thermodynamics, the Invariant Set Postulate proposes that cosmological states of physical reality belong to a non-computable fractal state-space geometry I , invariant under the action of some subordinate deterministic causal dynamics D I . An exploratory analysis is made of a possible causal realistic framework for quantum physics based on key properties of I . For example, sparseness is used to relate generic counterfactual states to points p ∉ I of unreality, thus providing a geometric basis for the essential contextuality of quantum physics and the role of the abstract Hilbert Space in quantum theory. Also, self-similarity, described in a symbolic setting, provides a possible realistic perspective on the essential role of complex numbers and quaternions in quantum theory. A new interpretation is given to the standard ‘mysteries’ of quantum theory: superposition, measurement, non-locality, emergence of classicality and so on. It is proposed that heterogeneities in the fractal geometry of I are manifestations of the phenomenon of gravity. Since quantum theory is inherently blind to the existence of such state-space geometries, the analysis here suggests that attempts to formulate unified theories of physics within a conventional quantum-theoretic framework are misguided, and that 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 task is not to make sense of the quantum axioms by heaping more structure, more definitions, more science fiction imagery on top of them, but to throw them away wholesale and start afresh. We should be relentless in asking ourselves: From what deep physical principles might we derive this exquisite structure? These principles should be crisp, they should be compelling. They should stir the soul. Chris Fuchs ( Gilder 2008 , p. 335)
We analyze from the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry, we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many "asymptotically" equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well-defined measure of entanglement) of the state Var [v] attains maximum. We provide an algorithm for finding critical sets of Var [v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using some developments in the momentum map geometry known as the Kirwan–Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.
Hardy's axiomatic approach to the quantum theory of discrete Hilbert Spaces reveals that just one principle distinguishes it from classical probability theory: there should be continuous (and hence infinitesimal) reversible transformations between any pair of pure states - the single word `continuous' giving rise to quantum theory. This raises the question: Can one formulate a finite theory of qubit physics (FTQP) - necessary different from quantum theory - which can replicate the tested predictions of quantum theory of qubits to experimental accuracy? Here we show that an FTQP based on complex Hilbert vectors with rational squared amplitudes and rational phase angles is possible, provided the metric of state space,
gp, is based on
p-adic rather than Euclidean distance. A key number theorem describing an incompatibility between rational angles and rational cosines accounts for quantum complementarity in this FTQP. Dynamical evolution is described by a deterministic mapping on the set of
p-adic integers and the measurement problem is trivially solved in terms of a nonlinear clustering of states in state space. Based on
gp, causal deterministic analyses of quantum interferometry, GHZ, the sequential Stern-Gerlach experiment, Leggett-Garg and the Bell Theorem are described. The close relationship between fractals and
p-adic integers suggest the existence of a primal fractal-like 'invariant set' geometry
IU in cosmological state space, from which space-time and the laws of physics in space-time are emergent.