Measuring finite Quantum Geometries via Quasi-Coherent States

TL;DR

This paper introduces a systematic numerical method using quasi-coherent states to determine and measure the geometry of finite quantum or fuzzy spaces, including their semi-classical limits.

Contribution

It develops a general approach to extract geometric information from finite-dimensional matrices representing quantum geometries, applicable beyond standard fuzzy spaces.

Findings

Successfully recovers semi-classical geometry from matrix models

Provides a measure of the quality of semi-classical approximation

Implemented as an open-source Mathematica package

Abstract

We develop a systematic approach to determine and measure numerically the geometry of generic quantum or "fuzzy" geometries realized by a set of finite-dimensional hermitian matrices. The method is designed to recover the semi-classical limit of quantized symplectic spaces embedded in $\mathbb{R}^d$ including the well-known examples of fuzzy spaces, but it applies much more generally. The central tool is provided by quasi-coherent states, which are defined as ground states of Laplace- or Dirac operators corresponding to localized point branes in target space. The displacement energy of these quasi-coherent states is used to extract the local dimension and tangent space of the semi-classical geometry, and provides a measure for the quality and self-consistency of the semi-classical approximation. The method is discussed and tested with various examples, and implemented in an open-source Mathematica package.