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, $g_p$, 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 $g_p$, 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 $I_U$ in cosmological state space, from which space-time and the laws of physics in space-time are emergent.
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.
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.
AbstractStandard quantum mechanics is formulated using complex Hilbert spaces, real-valued probabilities, continuous one-parameter dynamics, and—where particles and fields have continuous degrees of freedom —function spaces constructed over continua such as R3 . In that precise mathematical sense, the real and complex continua are deeply embedded in the standard theory. It does not follow, however, that experiments have demonstrated the physical existence of an uncountable set of exact real numbers, still less that individual quantum systems physically “store” infinitely many digits. Every preparation, control operation, and measurement has finite precision, and many quantum predictions can be approximated arbitrarily closely using countable or finite descriptions.This distinction matters because the continuum is sometimes blamed for quantum theory's conceptual strangeness. That diagnosis is mostly unsupported. Superposition, interference, uncertainty, noncommutativity, entanglement, contextuality, Bell-inequality violations, and no-cloning already occur in finite-dimensional quantum systems and do not depend in any straightforward way on continuous position space or arbitrary irrational numbers. Restricting a Hilbert space to rational, algebraic, computable, or finitely specified states therefore does not automatically produce classical physics. Indeed, dense restricted subsets can approximate ordinary quantum predictions to arbitrary finite accuracy, although turning such a subset into a complete physical theory requires much more: closed dynamics, a composition rule, a measurement theory, robustness, and new empirical predictions.A particularly developed contemporary alternative is Tim Palmer's Invariant Set Theory and its recent formulation as Rational Quantum Mechanics (RaQM). RaQM is not accurately described as simply “quantum mechanics with irrational numbers banned.” It restricts the bases in which states are physically defined through number-theoretic rationality conditions, represents admissible states using finite bit strings, invokes a fractal invariant state space equipped with a non-Euclidean, p-adic-like metric, and violates Bell measurement independence at the level of exact counterfactual settings. Palmer's 2026 proposal further predicts a finite qubit-information capacity and an eventual saturation of quantum computational advantage. These are genuine, potentially falsifiable proposals, but they remain speculative and have not displaced standard quantum theory.The strongest conclusion is consequently asymmetric. There is a serious foundational question about whether the continuum is an ontological feature of nature or an extraordinarily efficient mathematical completion. Finite precision prevents existing experiments from proving a literal continuum. But there is presently little evidence that continuum mathematics causes quantum phenomena. Any discrete replacement must still account quantitatively for interference, contextuality, Bell correlations, entanglement, spectroscopy, atomic structure, and quantum computation. Discreteness can change quantum physics only when it is accompanied by additional physical postulates—and those postulates, rather than discreteness alone, must ultimately face experiment.
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.
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 U is treated as a deterministic dynamical system evolving precisely on a measure-zero fractal invariant subset IU of its state space. In this approach, the geometry of IU, and not a set of differential evolution equations in space-time MU, provides the most primitive description of the laws of physics. As such, IST is non-classical. The geometry of IU is based on Cantor sets of space-time trajectories in state space, homeomorphic to the algebraic set of p-adic integers, for large but finite p. In IST, the non-commutativity of position and momentum observables arises from number theory - in particular the non-commensurateness of ϕ and cosϕ. The complex Hilbert Space and the relativistic Dirac Equation respectively are shown to describe IU, and evolution on IU, in the singular limit of IST at p=∞; particle properties such as de Broglie relationships arise from the helical geometry of trajectories on IU in the neighbourhood of MU. With the p-adic metric as a fundamental measure of distance on IU, certain key perturbations which seem conspiratorially small relative to the more traditional Euclidean metric, take points away from IU and are therefore unphysically large. This allows (the ψ-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 with the p-adic metric of IU. A generalisation of the field equations of general relativity which can achieve this is proposed.
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 p-adic rather than Euclidean where 1/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=∞. A modification of general relativity, consistent with this discretised model of quantum physics, is proposed.
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.