Quantum points/patterns, Part 2. From quantum points to quantum patterns via multiresolution

TL;DR

This paper introduces a multiresolution sheaf-based framework for quantum states, enabling detailed analysis of complex quantum behaviors like entanglement, chaos, and decoherence through localized eigenmodes.

Contribution

It develops a novel sheaf-theoretic approach to quantum states, extending the framework to describe complex quantum patterns and behaviors beyond traditional wave functions.

Findings

Methods for modeling localized and chaotic quantum patterns

Description of entanglement and decoherence via nonlinear eigenmodes

Framework for analyzing quantum hierarchies with sheaf theory

Abstract

It is obvious that we still have not any unified framework covering a zoo of interpretations of Quantum Mechanics, as well as satisfactory understanding of main ingredients of a phenomena like entanglement. The starting point is an idea to describe properly the key ingredient of the area, namely point/particle-like objects (physical quantum points/particles or, at least, structureless but quantum objects) and to change point (wave) functions by sheaves to the sheaf wave functions (Quantum Sheaves). In such an approach Quantum States are sections of the coherent sheaves or contravariant functors from the kinematical category describing space-time to other one, Quantum Dynamical Category, properly describing the complex dynamics of Quantum Patterns. The objects of this category are some filtrations on the functional realization of Hilbert space of Quantum States. In this Part 2, the sequel of Part 1, we present a family of methods which can describe important details of complex behaviour in quantum ensembles: the creation of nontrivial patterns, localized, chaotic, entangled or decoherent, from the fundamental basic localized (nonlinear) eigenmodes (in contrast with orthodox gaussian-like) in various collective models arising from the quantum hierarchies described by Wigner-like equations.