2016/01/31 by Lukas Schneiderbauer, Harold C. Steinacker, Harold C Steinacker · 28 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Dimension (graph theory) #Dirac (video compression format) #Displacement (psychology) #Hermitian matrix #Measure (data warehouse) #Noncommutative and Quantum Gravity Theories #Point (geometry) #Quantum #Quantum Mechanics and Non-Hermitian Physics #Symplectic geometry #hep-th
paper · pdf · doi:10.1088/1751-8113/49/28/285301
published in Journal of Physics A Mathematical and Theoretical 49(28), 285301 (Institute of Physics) · 41 pages, 14 figures. V2: discussion of Dirac operator improved, published version
arxiv created 2016/05/23 · openalex publication_date 2016/05/31 · arxiv updated 2016/06/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 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.