On the reality of space-time geometry and the wavefunction

G. Papini

Jincheng Zhang

In conventional quantum mechanics the trajectory of a matter wave is described implicitly by the phase of its wavefunction, yet no explicit geometric quantity has been associated with the propagation path itself. This paper introduces the concept of matter wave propagation curvature, denoted K_psi, as an intrinsic geometric measure of the bending of a quantum trajectory in configuration space. Unlike the curvature of space‑time employed in general relativity, K_psi depends solely on the wavefunction's phase structure and is independent of the underlying metric. We derive K_psi from the gradient of the phase of the wavefunction, show that it satisfies a conservation law along stationary paths, and discuss its relationship to the effective potential and external fields. The curvature framework predicts observable deviations in large‑mass interference experiments, suggesting a new route to probe the geometric nature of quantum states and to explore the possibility of a wavefunction‑based geometry. The potential to uncover a deeper layer of quantum reality is highlighted, pointing toward a Nobel‑potential discovery of wavefunction geometry.

Jincheng Zhang

In conventional quantum mechanics the trajectory of a matter wave is described implicitly by the phase of its wavefunction, yet no explicit geometric quantity has been associated with the propagation path itself. This paper introduces the concept of matter wave propagation curvature, denoted K_psi, as an intrinsic geometric measure of the bending of a quantum trajectory in configuration space. Unlike the curvature of space‑time employed in general relativity, K_psi depends solely on the wavefunction's phase structure and is independent of the underlying metric. We derive K_psi from the gradient of the phase of the wavefunction, show that it satisfies a conservation law along stationary paths, and discuss its relationship to the effective potential and external fields. The curvature framework predicts observable deviations in large‑mass interference experiments, suggesting a new route to probe the geometric nature of quantum states and to explore the possibility of a wavefunction‑based geometry. The potential to uncover a deeper layer of quantum reality is highlighted, pointing toward a Nobel‑potential discovery of wavefunction geometry.

Roger Boudet

T. N. Palmer

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 $p=\infty$. A modification of general relativity, consistent with this discretised model of quantum physics, is proposed.

James Thomas Wheeler