Finite-Precision Quantum Mechanics

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 convex weak open set of density matrices defined by finitely many open expectation intervals, representing exactly the set of microscopic states compatible with a finite set of macroscopic measurements. We show that unitary evolution lifts to a deterministic flow on parcels, and that a fuzzy measurement is represented by a volume-contracting update, strictly increasing the geometric information defined as the inverse of the Hilbert-Schmidt volume. By introducing a second impossible set we obtain a double-parcel whose geometric information increases monotonically -- dissolving the entropy stagnation problem, since von Neumann entropy is defined on point states that no finite-precision experiment can certify. The framework reformulates foundational puzzles without additional interpretational assumptions: wave-particle duality becomes a smooth trade-off; Schroedinger's cat is described by a parcel containing many microscopically distinct states; and spooky action at a distance disappears, replaced by a purely epistemic geometric update. All empirical predictions of standard quantum mechanics are recovered in the infinite-precision limit, which is never physically attained.