Finite-Precision Quantum Mechanics

• URL: http://arxiv.org/abs/2605.19706v2
• Date: Wed, 27 May 2026 13:14:56 GMT
• Title: Finite-Precision Quantum Mechanics
• Abstract summary: We introduce Interval Quantum Mechanics, in which the state of a quantum system is never a point but a quantum parcel.<n>We show that unitary evolution lifts to a deterministic flow on parcels.<n>All empirical predictions of standard quantum mechanics are recovered exactly in the infinite-precision limit.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Standard quantum mechanics is an idealisation based on infinite-precision objects: point states, exact probabilities, and sharp measurements. Yet every real experiment has finite resolution, and for macroscopic systems we never have access to the microscopic state. Following Heisenberg's call for a theory built only on observable quantities, and von Neumann's insight that a complete description of a macroscopic system is neither possible nor necessary, we elevate the macroscopic state to a fundamental concept. We introduce Interval Quantum Mechanics (IQM), in which the state of a quantum system is never a point but a quantum parcel - a basic weak open set of density matrices defined by finitely many open expectation intervals. Such a parcel is the exact mathematical representation of the set of all microscopic states that are compatible with the measured values of a finite set of macroscopic observables. We show that unitary evolution lifts to a deterministic flow on parcels, and that a finite-precision (fuzzy) measurement process is represented by a volume-contracting update that refines the initial parcel into a more constrained open set, strictly increasing the geometric information defined as the Hilbert-Schmidt volume of the parcel. By introducing a second impossible set, we obtain a double-parcel whose information increases monotonically - resolving the von Neumann entropy paradox. The framework eliminates foundational puzzles without additional interpretational assumptions: wave-particle duality becomes a smooth trade-off; Schroedinger's cat is never in a literal superposition; and the spooky action at a distance of entanglement disappears, replaced by a purely epistemic geometric update. All empirical predictions of standard quantum mechanics are recovered exactly in the infinite-precision limit, which is never physically attained.

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