arXiv subjects

T. N. Palmer

Publications and source records attributed to T. N. Palmer.

26 records · Page 2Linked to original sources

arXiv2013

Quantum Theory and The Symbolic Dynamics of Invariant Sets: Towards a Gravitational Theory of the Quantum

A realistic measurement-free theory for the quantum physics of multiple qubits is proposed. This theory is based on a symbolic representation of a fractal state-space geometry which is invariant under the action of deterministic and locally causal dynamics. This symbolic representation is constructed from self-similar families of quaternionic operators. Using number-theoretic properties of the cosine function, the statistical properties of the symbolic representation of the invariant set are shown to be consistent with the contextual requirements of the Kochen-Specker theorem, are not constrained by Bell inequalities, and mirror the statistics of entangled qubits. These number-theoretic properties in turn reflect the sparseness of the invariant set in state space, and relate to the metaphysical notion of counterfactual incompleteness. Using the concept of probability, the complex Hilbert Space can be considered the completion of this symbolic representation into the state space continuum. As a result, it is proposed that the complex Hilbert Space should merely be considered a computational convenience in the light of the algorithmic intractability of the invariant set geometry, and consequently the superposed state should not be considered a fundamental aspect of physical theory. The physical basis for the proposed theory is relativistic gravity; for example the symbols used to describe the invariant set themselves label gravitationally distinct cosmological space-times. This implies that the very notion of a `quantum theory of gravity' may be profoundly misguided - erroneously putting the quantum cart before the gravitational horse. Here some elements of an alternative `gravitational theory of the quantum' are proposed, based on a deterministic and locally causal theory of gravity which extends general relativity by being geometric in both space-time and state space.

quant-ph↗

arXiv2009

Towards the Probabilistic Earth-System Model

Multi-model ensembles provide a pragmatic approach to the representation of model uncertainty in climate prediction. However, such representations are inherently ad hoc, and, as shown, probability distributions of climate variables based on current-generation multi-model ensembles, are not accurate. Results from seasonal re-forecast studies suggest that climate model ensembles based on stochastic-dynamic parametrisation are beginning to outperform multi-model ensembles, and have the potential to become significantly more skilful than multi-model ensembles. The case is made for stochastic representations of model uncertainty in future-generation climate prediction models. Firstly, a guiding characteristic of the scientific method is an ability to characterise and predict uncertainty; individual climate models are not currently able to do this. Secondly, through the effects of noise-induced rectification, stochastic-dynamic parametrisation may provide a (poor man's) surrogate to high resolution. Thirdly, stochastic-dynamic parametrisations may be able to take advantage of the inherent stochasticity of electron flow through certain types of low-energy computer chips, currently under development. These arguments have particular resonance for next-generation Earth-System models, which purport to be comprehensive numerical representations of climate, and where integrations at high resolution may be unaffordable.

physics.ao-ph↗

arXiv2009

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

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. 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 not lying on I, 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, nonlocality, 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.

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arXiv2005

Causal Incompleteness: A New Perspective on Quantum Non-locality

The mathematical notion of incompleteness (eg of rational numbers, Turing-computable functions, and arithmetic proof) does not play a key role in conventional physics. Here, a reformulation of the kinematics of quantum theory is attempted, based on an inherently granular and discontinuous state space, in which the quantum wavefunction is associated with a finite set of finite bit strings, and the unitary transformations of complex Hilbert space are reformulated as finite permutation and related operators incorporating complex and hyper-complex structure. Such a reformulation, consistent with Wheeler's `It from Bit' programme, provides the basis for a novel interpretation of the Bell theorem: that the experimental violation of the Bell inequalities reveals the inevitable incompleteness of the causal structure of physical theory. The kinematic reformulation of quantum theory so developed, provides a new perspective on the age-old dichotomy of free will versus determinism.

quant-ph↗

arXiv2005

Quantum Reality, Complex Numbers and the Meteorological Butterfly Effect

A not-too-technical version of the paper: "A Granular Permutation-based Representation of Complex Numbers and Quaternions: Elements of a Realistic Quantum Theory" - Proc. Roy. Soc.A (2004) 460, 1039-1055. The phrase "meteorological butterfly effect" is introduced to illustrate, not the familiar loss of predictability in low-dimensional chaos, but the much less familiar and much more radical paradigm of the finite-time predictability horizon, associated with upscale transfer of uncertainty in certain multi-scale systems. This motivates a novel reinterpretation of unit complex numbers (and quaternions) in terms of a family of self-similar permutation operators. A realistic deterministic kinematic reformulation of the foundations of quantum theory is given using this reinterpretation of complex numbers. Using a property of the cosine function not normally encountered in physics, that it is irrational for all dyadic rational angles between 0 and pi/2, this reformulation is shown to have the emergent property of counterfactual indefiniteness and is therefore not non-locally causal.

quant-ph↗

arXiv2002

A Realistic Deterministic Quantum Theory Using Borelian-Normal Numbers

The elements of a deterministic quantum theory are developed, which reformulates and extends standard quantum theory. The proposed theory is `realistic' in the sense that in it, a general M-level quantum state is represented by a single real number r. Surprising as it may seem, this real number is shown to contain the same probabilistic information as the standard Hilbert-space state, plus additional information from which measurement outcome is determined. A crucial concept in achieving this is that of Borelian (number-theoretic) normality. The essential role of complex numbers in standard quantum theory is subsumed by the action of a set of self-similar permutation operators on the digits and places of the base-M expansion of a base-M Borelian-normal r; these permutation operators are shown to have complex structure and leave invariant the normality of the underlying real number. The set of real numbers generated by these permutations defines not only the Hilbert space of standard quantum theory, but also, in addition, the sample space from which quantum measurement outcomes can be objectively determined. Dynamical real-number state reduction is precisely described by deterministic number-theoretic operators that reduce the degree of normality of r; from the degree of normality one can infer the standard quantum-theoretic trace rule for measurement probability. All the foundational difficulties of standard quantum theory are described in terms of the proposed theory. It is shown that the real-number states of the proposed theory are precisely its beables.

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arXiv2001

Formulation of Quantum Theory Using Computable and Non-Computable Real Numbers

It is shown that in two-state quantum theory, a generic quantum state can be described by a non-computable real number. In terms of this, the criterion for measurement outcome is simply and deterministically defined. This demonstration is based on a construction of the Riemann sphere whose points represent, not complex numbers, but divergent sequences with bivalent elements. Complex structure arises from self-similar properties of a set of operators which generate these sequences. In general, a rotation of (the coordinates of) the sphere maps a computable real to a non-computable real. This is interpreted physically as a mapping of a physically-measurable state to a counterfactual state. Implications for non-locality, null measurements, many worlds and so on, are discussed. The possible role of the Euler equation as the counterpart of the Schrodinger equation for real-number quantum state evolution is also outlined.

quant-ph↗

arXiv1995

A Local Deterministic Model of Quantum Spin Measurement

The conventional view, that Einstein was wrong to believe that quantum physics is local and deterministic, is challenged. A parametrised model, Q, for the state vector evolution of spin 1/2 particles during measurement is developed. Q draws on recent work on so-called riddled basins in dynamical systems theory, and is local, deterministic, nonlinear and time asymmetric. Moreover the evolution of the state vector to one of two chaotic attractors (taken to represent observed spin states) is effectively uncomputable. Motivation for this model arises from Penrose's speculations about the nature and role of quantum gravity. Although the evolution of Q's state vector is uncomputable, the probability that the system will evolve to one of the two attractors is computable. These probabilities correspond quantitatively to the statistics of spin 1/2 particles. In an ensemble sense the evolution of the state vector towards an attractor can be described by a diffusive random walk. Bell's theorem and a version of the Bell-Kochen_specker quantum entanglement paradox are discussed. It is shown that proving an inconsistency with locality demands the existence of definite truth values to certain counterfactual propositions. In Q these deterministic propositions are physically uncomputable and no non-algorithmic solution is either known or suspected. Adapting the mathematical formalist approach, the non-existence of definite truth values to such counterfactual propositions is posited. No inconsistency with experiment is found. Hence Q is not necessarily constrained by Bell's inequality.

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