Quasi-Coherent States on Deformed Quantum Geometries

TL;DR

This paper refines and implements a numerical approach to associate classical geometries with matrix configurations from quantum models, revealing their structure as smooth manifolds and exploring phenomena like oxidation.

Contribution

It develops a computational method to analyze deformations of matrix configurations, improving understanding of their classical geometric interpretation and stability.

Findings

Classical spaces can be embedded as smooth manifolds in complex projective space.

Foliations are necessary to handle the oxidation phenomenon.

Numerical methods enable visualization and quantitative analysis of semiclassical limits.

Abstract

Matrix configurations coming from matrix models comprise many important aspects of modern physics. They represent special quantum spaces and are thus strongly related to noncommutative geometry. In order to establish a semiclassical limit that allows to extract their intuitive geometrical content, this thesis analyzes and refines an approach that associates a classical geometry to a given matrix configuration, based on quasi-coherent states. While, so far, the approach is only well understood for very specific cases, in this work it is reviewed and implemented on a computer, allowing the numerical investigation of deformations of these cases. It is proven that the classical space can be made into a smooth manifold immersed into complex projective space. Further, the necessity for the consideration of foliations thereof is shown in order to deal with the observed and subsequently described phenomenon called oxidation. The developed numerical methods allow the visualization of the semiclassical limit as well as quantitative calculations. Explicit examples suggest the stability under perturbations of the refined approach and highlight the physical interpretation of the construction. All this supports a better understanding of the geometrical content of arbitrary matrix configurations as well as their classical interpretation and establishes the determination of important quantities.