Meteorology is a wonderfully interdisciplinary subject. But can nonlinear thinking about predictability of weather and climate contribute usefully to issues in fundamental physics? Although this might seem extremely unlikely at first sight, an attempt is made to answer the question positively. The long-standing conceptual problems of quantum theory are outlined, focusing on indeterminacy and nonlocal causality, problems that led Einstein to reject quantum mechanics as a fundamental theory of physics (a glossary of some of the key terms used in this paper is given in the sidebar). These conceptual problems are considered in the light of both low-order chaos and the more radical (and less well known) paradigm of the finite-time predictability horizon associated with the self-similar upscale cascade of uncertainty in a turbulent fluid. The analysis of these dynamical systems calls into doubt one of the key pieces of logic used in quantum nonlocality theorems: that of counterfactual reasoning. By considering an idealization of the upscale cascade (which provides a novel representation of complex numbers and quaternions), a case is made for reinterpreting the quantum wave function as a set of intricately encoded binary sequences. In this reinterpretation, it is argued that the quantum world has no need for dice-playing deities, undead cats, multiple universes, or “spooky action at a distance.”
It is suggested that the long–standing conceptual difficulties of quantum theory (the measurement problem and associated concept of probability on the one hand and the notions of entanglement and non–locality on the other) arise fundamentally from the role played by the continuum field C of complex numbers in the axioms of standard quantum theory. An alternative representation of the 2Nth roots of unity is developed, based on a self–similar family P of permutation operators, acting on the digits and places of the binary expansion of a real number r0, generically Borel normal. Based on r0 and P, an analytically irregular real–valued function 0≤ r < 1 is constructed on S2, which plays the role of the Riemann sphere. As a result of its irregularity (or, equivalently, granularity), r is only uniquely definable on a countable set of (‘rational’) meridians of S2. P is further generalized to describe permutation–operator representations of the quaternions, from which transformations of r under rotations of the sphere can be defined.Using P, a proposal for a constructive realistic deterministic single–world quantum theory is given. The ‘realistic’ roots–of–unity permutation operator encodes the apparent stochasticity of the complex phase function riEt/ħ in the standard quantum theoretic wave function; similarly, the quaternionic permutations encode ‘realistically’ the spatial entanglement relationships in the wave function. Based on this formalism, finite samples of values of r on the rational meridians are used to define a frequentist–based probability measure consistent with quantum–theoretic probability and correlation. The granularity of r is crucial in accounting for the theory's evasion of the Bell inequalities. Standard quantum phenomena are discussed from the perspective of the proposed theory. Some emphasis is given to the issues of contextuality and non–locality in the proposed theory.
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.
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.
Why does quantum theory need the complex numbers? With a view toward answering this question, I argue that the usual Hilbert-space formalism is a special case of the general method of Markovian embeddings. I then describe the `indivisible interpretation' of quantum theory, according to which a quantum system can be regarded as an `indivisible' stochastic process unfolding in an old-fashioned configuration space, with wave functions and other exotic Hilbert-space ingredients demoted from having an ontological status. The complex numbers end up being necessary to ensure that the Hilbert-space formalism is indeed a Markovian embedding.
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.