Quantum points/patterns, Part 1. From geometrical points to quantum points in a sheaf framework

TL;DR

This paper proposes a sheaf-theoretic framework for quantum states, aiming to deepen understanding of quantum phenomena and address interpretational issues in quantum mechanics.

Contribution

It introduces a novel sheaf-based approach to quantum states using multiscale filtrations in infinite-dimensional Hilbert spaces, unifying various symmetries behind quantum phenomena.

Findings

Sheaf-based quantum states can model nonlocal phenomena.

Multiscale filtrations reveal hidden symmetries in quantum systems.

The framework offers new insights into faster-than-light propagation.

Abstract

We consider some generalization of the theory of quantum states, which is based on the analysis of long standing problems and unsatisfactory situation with the possible interpretations of quantum mechanics. We demonstrate that the consideration of quantum states as sheaves can provide, in principle, more deep understanding of some phenomena. The key ingredients of the proposed construction are the families of sections of sheaves with values in the category of the functional realizations of infinite-dimensional Hilbert spaces with special (multiscale) filtration. Three different symmetries, kinematical (on space-time), hidden/dynamical (on sections of sheaves), unified (on filtration of the full scale of spaces) are generic objects generating the full zoo of quantum phenomena (e.g., faster than light propagation).