Weak Values as Geometric Lenses: Deformations of Hilbert Space and the Emergence of superoscillations

TL;DR

This paper reveals that weak values in quantum mechanics can be understood as geometric deformations of Hilbert space, naturally leading to superoscillations, and unifies these phenomena through a common geometric framework.

Contribution

It introduces a geometric interpretation of weak values as deformations of Hilbert space, explaining superoscillations as a consequence of this structure, which is a novel perspective.

Findings

Weak values are ratios of geometric deformations in Hilbert space.

Superoscillations emerge naturally from the geometric structure of weak values.

The framework unifies weak values and superoscillations under a single geometric principle.

Abstract

The formalism of weak measurement in quantum mechanics has revealed profound connections between measurement theory, quantum foundations, and signal processing. In this paper, we develop a pointer-free derivation of superoscillations, demonstrating that they are a natural and necessary consequence of the geometric structure underlying weak values. We argue that the weak value is best understood as a ratio of geometric deformation, quantifying how an observable transforms the structure of Hilbert space relative to a reference provided by the standard inner product. This deformation acts as a conceptual lens, warping the local structure of quantum states to produce oscillations far exceeding the global Fourier bandwidth. We formalize this by interpreting the weak value as a comparison between a deformed sesquilinear form and the standard one, and explore its deep connections to generalized Rayleigh quotients and the projective geometry of quantum states. This perspective unifies weak values and superoscillations as two facets of a single underlying geometric principle.