Measuring finite quantum geometries via quasi-coherent states

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 Rd 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 quasicoherent states is used to extract the local dimension and tangent space of the semi-classical geometry, and provides a measure for the quality and selfconsistency of the semi-classical approximation. The method is discussed and tested with various examples, and implemented in an open-source Mathematica package.