On Quantum Indeterminacy

TL;DR

This paper presents a geometric approach to quantum indeterminacy using convex geometry and symplectic topology, deriving uncertainty principles as fundamental phase space constraints.

Contribution

It introduces a novel geometric framework that explains quantum uncertainty without relying on statistical measures, emphasizing symplectic invariants and convex bodies.

Findings

Standard uncertainty inequalities emerge from geometric relations

Robertson-Schrodinger inequalities are natural consequences of the framework

Quantum indeterminacy is shown as a structural property of phase space

Abstract

We introduce a geometric formulation of quantum indeterminacy from which the standard uncertainty inequalities emerge as necessary consequences. Our approach is based on convex geometry in phase space and on methods from symplectic topology, and does not rely on statistical descriptors such as variances or covariances. Instead, we associate to empirical position and momentum data with convex bodies whose mutual relations encode the fundamental constraints of quantum mechanics. The central tools are h-polar duality and symplectic capacities, which provide intrinsic, coordinate-free bounds on admissible phase-space configurations. Within this framework, the Robertson-Schrodinger inequalities arise naturally as manifestations of deeper geometric and topological principles. This perspective suggests that quantum indeterminacy is not primarily a statistical phenomenon, but rather a structural property of phase space governed by symplectic covariance. The results thus provide a unified and conceptually transparent foundation for the uncertainty principle.