Feynman famously asserted that interference is the only real mystery in quantum mechanics (QM). It is concluded that the reason for this mystery, and thereby the related mysteries of complementarity, non-commutativity of observables, the uncertainty principle and violation of Bell's equality, is that the axioms of QM depend vitally on the continuum nature of Hilbert Space, deemed unphysical. We develop a theory of quantum physics - Rational Quantum Mechanics (RaQM) - in which Hilbert Space is gravitationally discretised. The key to solving the mysteries of QM in RaQM is a number-theoretic property of the cosine function, concealed in QM when angles range over the continuum. This number-theoretic property describes mathematically the utter indivisibility of the quantum world and implies that the laws of physics are profoundly holistic. We contrast holism with nonlocality. In theories which embrace the continuum, the violation of Bell's inequality requires the laws of physics to be either nonlocal or not realistic; both incomprehensible concepts. By contrast, holism, as embodied in Mach's Principle or in the fractal geometry of a chaotic attractor, is neither incomprehensible nor unphysical. As part of this, we solve the deepest mystery of all; why nature makes use of complex numbers.
Feynman famously asserted that interference is the only real mystery in quantum mechanics (QM). It is concluded that the reason for this mystery, and thereby the related mysteries of complementarity, non-commutativity of observables, the uncertainty principle and violation of Bell's equality, is that the axioms of QM depend vitally on the continuum nature of Hilbert Space, deemed unphysical. We develop a theory of quantum physics - Rational Quantum Mechanics (RaQM) - in which Hilbert Space is gravitationally discretised. The key to solving the mysteries of QM in RaQM is a number-theoretic property of the cosine function, concealed in QM when angles range over the continuum. This number-theoretic property describes mathematically the utter indivisibility of the quantum world and implies that the laws of physics are profoundly holistic. We contrast holism with nonlocality. In theories which embrace the continuum, the violation of Bell's inequality requires the laws of physics to be either nonlocal or not realistic; both incomprehensible concepts. By contrast, holism, as embodied in Mach's Principle or in the fractal geometry of a chaotic attractor, is neither incomprehensible nor unphysical. As part of this, we solve the deepest mystery of all; why nature makes use of complex numbers.
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.
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.
Motivated in part by John Wheeler's assertion that the continuum nature of Hilbert Space conceals the "it-from-bit" information-theoretic character of the quantum wavefunction, a theory of quantum physics (Rational Quantum Mechanics-RaQM) is proposed based on a specific discretization of complex Hilbert Space. The Schrödinger equation is not modified in RaQM, even during measurement. However, the bases in which the quantum state is defined must satisfy certain rational-number constraints. These constraints lead to the notion of finite qubit information capacity [Formula: see text]: For any [Formula: see text] qubit state, there is insufficient information in the [Formula: see text] qubits (linearly growing in [Formula: see text]) to allocate even one bit to each of all [Formula: see text] continuum degrees of freedom (exponentially growing in [Formula: see text]) associated with quantum mechanics/theory (QM, where [Formula: see text]). It is proposed that the discretization of Hilbert Space in RaQM is due to gravity, hence QM is the (singular) continuum limit of RaQM at [Formula: see text]. On this basis, it is estimated that [Formula: see text] lies between about 200 and 400 for current qubit technologies, and will never exceed 1,000. While QM and RaQM are experimentally indistinguishable for small numbers of qubits, RaQM predicts that the exponential advantage of quantum algorithms which, like Shor's, require bases with maximal [Formula: see text]-qubit superposition/entanglement, will have saturated at 1,000 perfect qubits. Hence, insofar as a classical computer will never factor a 2,048-bit RSA integer, RaQM predicts that a quantum computer will not either. This predicted breakdown of QM could be testable in less than 5 y.
Rational Quantum Mechanics (RaQM) is a theory consistent with Gerard ’t Hooft’s proposals about the locally causal nature of quantum physics. Based on a specific discretization of the Riemann Sphere, the quantum state only is only defined in bases where squared amplitudes and complex phases (divided by [Formula: see text]) are rational numbers. These “rational bases” correspond to ’t Hooft’s ontological bases in which the quantum state has a beable representation. Bell’s inequality is violated without breaking realism, locality or free choice. Instead, for each run in a Bell experiment, a number-theoretic property of the cosine function ensures that the squared amplitude of the singlet state is an irrational number in at least one of the two counterfactual pairs of bases needed to derive Bell’s inequality. Hence, the Measurement Independence (MI) assumption is violated — not for the nominal settings under the experimenters’ free control, but for the exact settings which were never under their control anyway. None of the grotesque conspiracies that plague conventional superdeterministic explanations of Bell’s Theorem applies. Bell’s Theorem implies RaQM is a holistic theory. As such, implications for models of cosmology are discussed. An experimental test of RaQM, achievable in about 5 years, is described.
Negating the Measurement Independence assumption (MI) is often referred to as the `third way' to account for the experimental violation of Bell's inequality. However, this route is generally viewed as ludicrously contrived, implying some implausible conspiracy where experimenters are denied the freedom to choose measurement settings as they like. Here, a locally realistic model of quantum physics is developed (Rational Mechanics - RaQM - based on a gravitational discretisation of Hilbert Space) which violates MI without denying free will. Crucially, RaQM distinguishes experimenters' ability to freely choose measurement settings to some nominal accuracy, from an inability to choose exact settings, which were never under their control anyway. In RaQM, Hilbert states are necessarily undefined in bases where squared amplitudes and/or complex phases are irrational numbers. Such `irrational' bases correspond to conceivable but necessarily impossible counterfactual measurements, and are shown to play a ubiquitous role in the analysis of both single- and entangled-particle quantum physics. It is concluded that violation of Bell inequalities can be understood with none of the strange processes historically associated with it. Instead, using concepts from (non-classical) p-adic number theory, we relate RaQM to Bohm and Hiley's concept of a holistic Machian-like Undivided Universe. If this interpretation of Bell's Theorem is correct, building more and more energetic particle accelerators to probe smaller and smaller scales, in the search for a theory which synthesises quantum and gravitational physics and hence a Theory of Everything, may be a fruitless exercise.
Classical physics is traditionally characterized as deterministic and founded on the real number continuum, while quantum mechanics is distinguished by its indeterminism and fundamental uncertainty relations. This apparent dichotomy, however, obscures a deeper continuity. We argue that both frameworks embody a common ontological principle: the physical prohibition of absolute precision. In classical physics, this prohibition is implicit, manifesting in information-theoretic bounds, the non-physicality of infinite-precision initial conditions (chaos theory), and the breakdown of determinism in singular configurations (Norton's dome). These phenomena reveal that the classical reliance on mathematical real numbers with infinite information content is an unphysical idealization. Quantum mechanics, by contrast, does not merely reflect this prohibition; it provides its precise quantitative expression. The Heisenberg uncertainty principle σxσp≥ℏ/2 and the non-commutative structure of observables are the exact mathematical formulation of the fact that physical information is fundamentally finite. The Planck constant ℏ is the universal scale that quantifies the minimal phase space volume—the “atom” of information—below which no further refinement is possible. Thus, quantum mechanics is not a separate theory but the completion of classical physics: it supplies the missing quantification of the principle that nature admits no infinite information density, and consequently, no absolutely precise physical quantities.