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arXiv:1709.00329

A Gravitational Theory of the Quantum

T. N. Palmer

Published Aug 30, 2017• 1 citation
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Abstract

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 $I_U$ 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 $I_U$, a non-Euclidean (and hence non-classical) state-space metric $g_p$ is defined, related to the $p$-adic metric of number theory where $p$ is a large but finite Pythagorean prime. Uncertain states on $I_U$ are described using complex Hilbert states, but only if their squared amplitudes are rational and corresponding complex phase angles are rational multiples of $2 π$. Such Hilbert states are necessarily $g_p$-distant from states with either irrational squared amplitudes or irrational phase angles. The gappy fractal nature of $I_U$ 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 $g_p$-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 $I_U$. The Schrödinger and Dirac equations describe evolution on $I_U$ in the singular limit at $p=\infty$. By contrast, an extension of the Einstein field equations on $I_U$ is proposed which reduces smoothly to general relativity as $p \rightarrow \infty$. Novel proposals for the dark universe and the elimination of classical space-time singularities are given and experimental implications outlined.

DOI
10.48550/arxiv.1709.00329

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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 IUI_UIU​ 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 IUI_UIU​, a non-Euclidean (and hence non-classical) state-space metric gpg_pgp​ is defined, related to the ppp-adic metric of number theory where ppp is a large but finite Pythagorean prime. Uncertain states on IUI_UIU​ are described using complex Hilbert states, but only if their squared amplitudes are rational and corresponding complex phase angles are rational multiples of 2π2 π2π. Such Hilbert states are necessarily gpg_pgp​-distant from states with either irrational squared amplitudes or irrational phase angles. The gappy fractal nature of IUI_UIU​ 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 gpg_pgp​-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 IUI_UIU​. The Schrödinger and Dirac equations describe evolution on IUI_UIU​ in the singular limit at p=∞p=\inftyp=∞. By contrast, an extension of the Einstein field equations on IUI_UIU​ is proposed which reduces smoothly to general relativity as p→∞p \rightarrow \inftyp→∞. Novel proposals for the dark universe and the elimination of classical space-time singularities are given and experimental implications outlined.
Invariant Set Theory (IST) is a realistic, locally causal theory of fundamental physics which assumes a much stronger synergy between cosmology and quantum physics than exists in contemporary theory. In IST the (quasi-cyclic) universe UUU is treated as a deterministic dynamical system evolving precisely on a measure-zero fractal invariant subset IUI_UIU​ of its state space. In this approach, the geometry of IUI_UIU​, and not a set of differential evolution equations in space-time MU\mathcal M_UMU​, provides the most primitive description of the laws of physics. As such, IST is non-classical. The geometry of IUI_UIU​ is based on Cantor sets of space-time trajectories in state space, homeomorphic to the algebraic set of ppp-adic integers, for large but finite ppp. In IST, the non-commutativity of position and momentum observables arises from number theory - in particular the non-commensurateness of ϕϕϕ and cos⁡ϕ\cos ϕcosϕ. The complex Hilbert Space and the relativistic Dirac Equation respectively are shown to describe IUI_UIU​, and evolution on IUI_UIU​, in the singular limit of IST at p=∞p=\inftyp=∞; particle properties such as de Broglie relationships arise from the helical geometry of trajectories on IUI_UIU​ in the neighbourhood of MU\mathcal M_UMU​. With the p-adic metric as a fundamental measure of distance on IUI_UIU​, certain key perturbations which seem conspiratorially small relative to the more traditional Euclidean metric, take points away from IUI_UIU​ and are therefore unphysically large. This allows (the ψψψ-epistemic) IST to evade the Bell and Pusey et al theorems without fine tuning or other objections. In IST, the problem of quantum gravity becomes one of combining the pseudo-Riemannian metric of MU\mathcal M_UMU​ with the p-adic metric of IUI_UIU​. A generalisation of the field equations of general relativity which can achieve this is proposed.
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 UUU is treated as an isolated deterministic dynamical system evolving precisely on a measure-zero fractal invariant subset IUI_UIU​ of its state space. A non-classical approach to the physics of UUU is developed by treating the geometry of IUI_UIU​ as more primitive than dynamical evolution equations on IUI_UIU​. A specific symbolic representation of IUI_UIU​ 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→∞N \rightarrow \inftyN→∞. Although the Hilbert Space of quantum theory is counterfactually complete, the measure-zero set IUI_UIU​ is counterfactually incomplete, no matter how large is NNN. Such incompleteness allows reinterpretations of familiar quantum phenomena, consistent with realism and local causality. The non-computable nature of IUI_UIU​ ensures that these reinterpretations are neither conspiratorial nor retrocausal and, through a homeomorphism with the ring of 2N2^N2N-adic integers, are robust to noise and hence not fine tuned. The non-commutativity of Hilbert Space observables emerges from the symbolic representation of IUI_UIU​ through the generic number-theoretic incommensurateness of ϕ/πϕ/πϕ/π and cos⁡ϕ\cos ϕcosϕ. 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.
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, gpg_pgp​, is based on ppp-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 ppp-adic integers and the measurement problem is trivially solved in terms of a nonlinear clustering of states in state space. Based on gpg_pgp​, 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 ppp-adic integers suggest the existence of a primal fractal-like 'invariant set' geometry IUI_UIU​ in cosmological state space, from which space-time and the laws of physics in space-time are emergent.
Invariant Set (IS) theory is a locally causal ontic theory of physics based on the Cosmological Invariant Set postulate that the universe UUU can be considered a deterministic dynamical system evolving precisely on a (suitably constructed) fractal dynamically invariant set in UUU's state space. IS theory violates the Bell inequalities by violating Measurement Independence. Despite this, IS theory is not fine tuned, is not conspiratorial, does not constrain experimenter free will and does not invoke retrocausality. The reasons behind these claims are discussed in this paper. These arise from properties not found in conventional ontic models: the invariant set has zero measure in its Euclidean embedding space, has Cantor Set structure homeomorphic to the p-adic integers (p⋙0p \ggg 0p⋙0) and is non-computable. In particular, it is shown that the p-adic metric encapulates the physics of the Cosmological Invariant Set postulate, and provides the technical means to demonstrate no fine tuning or conspiracy. Quantum theory can be viewed as the singular limit of IS theory when when ppp is set equal to infinity. Since it is based around a top-down constraint from cosmology, IS theory suggests that gravitational and quantum physics will be unified by a gravitational theory of the quantum, rather than a quantum theory of gravity. Some implications arising from such a perspective are discussed.
This paper draws on a number of Roger Penrose's ideas—including the non-Hamiltonian phase-space flow of the Hawking Box, conformal cyclic cosmology, non-computability, and gravitationally induced quantum state reduction—in order to propose a radically unconventional approach to quantum gravity: Invariant Set Theory (IST). In IST, the fundamental laws of physics describe the geometry of the phase portrait of the universe as a whole: “quantum” process is associated with fine-scale fractal geometry, “gravitational” process with larger-scale heterogeneous geometry. With this, it becomes possible to explain the experimental violation of Bell inequalities without having to abandon key ingredients of general relativity: determinism and local causality. Ensembles in IST can be described by complex Hilbert states over a finite set ℂp of complex numbers, where p is a large finite integer. The quantum mechanics of finite-dimensional Hilbert spaces is emergent as a singular limit when p→∞. A small modification to the field equations of general relativity is proposed to make it consistent with IST.
The properties which give quantum mechanics its unique character - unitarity, complementarity, non-commutativity, uncertainty, nonlocality - derive from the algebraic structure of Hermitian operators acting on the wavefunction in complex Hilbert space. Because of this, the wavefunction cannot be shown to describe an ensemble of deterministic states where uncertainty simply reflects a lack of knowledge about which ensemble member describes reality. This has led to endless debates about the ontology of quantum mechanics. Here we derive these same quantum properties from number theoretic attributes of trigonometric functions applied to an explicitly ensemble-based representation of discretised complex Hilbert states. To avoid fine-tuning, the metric on state space must be ppp-adic rather than Euclidean where 1/p1/p1/p determines the fineness of the discretisation. This hints at both the existence of an underpinning fractal state-space geometry onto which states of the world are constrained. In this model, violation of Bell inequalities is a manifestation of this geometric constraint and does not imply a breakdown of local space-time causality. Because the discretised wavefunction describes an ensemble of states, there is no collapse of the wavefunction. Instead measurement describes a nonlinear clustering of state-space trajectories on the state-space geometry. In this model, systems with mass greater than the Planck mass will not exhibit quantum properties and instead behave classically. The geometric constraint suggests that the exponential increase in the size of state space with qubit number may break down with qubit numbers as small as a few hundred. Quantum mechanics is itself a singular limit of this number-theoretic model at p=∞p=\inftyp=∞. A modification of general relativity, consistent with this discretised model of quantum physics, is proposed.