The quantum world is usually drawn with points, but real instruments only see blurry spots. Heisenberg and von Neumann long suspected: precision is unattainable. A new theory suggests thinking of states as 'quantum droplets' — regions of possible values, like a droplet of water under a microscope with its fuzzy boundary.
An unexpected conclusion: if we abandon the illusion of precision, paradoxes vanish by themselves. Schrödinger's cat was never both alive and dead — it's just that both possibilities coexist in the fuzzy droplet, and the 'spooky connection' at a distance reduces to ordinary knowledge updating, no faster than light. Standard quantum mechanics turns out to be just an approximation — as if we considered a real droplet as a perfect sphere.
as written by the authors arXiv:2605.19706
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.
Information in the new theory is simply the volume of the droplet. The more precise the measurement, the smaller the droplet, and knowledge grows — as if fog condenses into a visible form.