Solving the Mysteries of Quantum Mechanics: Why Nature Abhors a Continuum

TL;DR

RaQM replaces the continuum complex Hilbert space with a gravity-induced, discretised Riemann Sphere, defining i constructively as a permutation/negation operator on L-bit strings. By enforcing rationality constraints via Niven’s theorem and the Impossible Triangle Corollary, RaQM attributes quantum mysteries to the limitations of exact rational bases, producing a holistic, nonlocal-free account of interference, complementarity, uncertainty, and non-commutativity. The framework reproduces QM correlations in many regimes while offering testable differences at extreme scales, and it provides a natural justification for using complex-like structure without committing to the full continuum of $\\mathbb{C}$. This approach ties the foundations of quantum theory to gravity and p-adic geometry, suggesting that holism, rather than nonlocality, underpins the quantum world and that complex numbers reflect the indivisibility of quantum states. The discretised, gappy Hilbert space thus offers a route to unifying quantum and gravitational physics, with tangible implications for future experiments on large-scale quantum devices.

Abstract

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.

Figures (5)

• Figure 1: An impossible spherical triangle, where the cosine of the angular length of each side of the triangle on the unit sphere is rational, and the internal angles are rational in degrees. The Impossible Triangle Corollary (to Niven's Theorem) provides the basis for understanding the mysteries of QM.
• Figure 2: In RaQM, the Uncertainty Principle $\Delta S_x \Delta S_y \ge \frac{\hbar}{2} \overline S_z$ for a single qubit arises from the trigonometry of spherical triangles and the correspondence between points on the discretised Riemann sphere with coordinates satisfying the rationality conditions and corresponding bit strings.
• Figure 3: A spin-$1/2$ quantum system is prepared 'up' by Stern-Gerlach device $SG_A$. The spin-up output of $SG_A$ is fed into device $SG_B$ and the spin-up output of $SG_B$ is fed into $SG_C$. The experimenter is free to choose the nominal orientations of these apparatuses as they like. However, according to the rationality constraints (\ref{['ratSG2']}), Niven's Theorem prevents the simultaneous counterfactual world where $SG_B$ and $SG_C$ are swapped from having a well-defined measurement outcome.
• Figure 4: a) The discretised Riemann Sphere at $L=2$, represented as four 2-bit strings on a complete great circle at $\phi=0,\pi$ through the poles. b) A schematic illustration of how to construct the discretised Riemann Sphere for $L=4$, again on the great circle through $\phi=0,\pi$, from two $L=2$ great circles as in a). See the text for details. This also illustrates the construction of an $L=2^M$ discretised Riemann Sphere on the great circle at $\phi=0, \pi$, from two $L/2$-bit discretised great circles.
• Figure 5: The full 3D discretised Riemann Sphere for $L=4$ - where, consistent with quaternionic structure (see text), rotation about the polar axes corresponding to a cyclic permutation of the 4-bit strings.