School of Mathematical & Statistical Sciences Faculty Publications
Document Type
Article
Publication Date
6-2026
Abstract
The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space X as a model for the physical space at short distances. However, we consider the time as a real variable. In this framework, the Dirac–von Neumann formalism can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose R×(R×X)3 as a model of a space–time. For simplicity, we work out our models with R×(R×X) as the configuration space. Quantum mechanics (QM), in the sense of Dirac–von Neumann, on the Hilbert space L2(R×X) is a non-local theory: the Hamiltonians are non-local operators, and thus, spooky action at a distance is allowed. The paradigm asserting that the universe is non-locally real implies that the proposed version of QM admits realism. This version of QM can be specialized to standard QM by using Hamiltonians acting on wavefunctions supported on the region R × R. We apply the developed formalism to the measurement problem. We propose a new mechanism for wavefunction collapse. The mechanism resembles that proposed by Ghirardi, Rimini, and Weber, but there are significant differences. The most important feature is that the Schrödinger equation describes the dynamics at all times, even at the moment of measurement. We also discuss a model for the two-slit experiment in which bright and dark states of light (recently proposed) naturally occur.
Recommended Citation
Zúñiga-Galindo, W. A. "Quantum mechanics, non-locality, and the space discreteness hypothesis." Annals of Physics (2026): 170459. https://doi.org/10.1016/j.aop.2026.170459
Creative Commons License
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Publication Title
Annals of Physics
DOI
10.1016/j.aop.2026.170459
Comments
Original published version available at https://doi.org/10.1016/j.aop.2026.170459