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What is BProbeM?

BProbeM, a fuzzy quantum geometry scanner, is a Wolfram Mathematica package. It is the implementation of an algorithm which rasterizes the semi-classical limit of a fuzzy brane described by a set of matrices.

BProbeM is a fork of BProbe by Lukas Schneiderbauer, adding several features for scanning such as the newly added directional scans, improves and completes missing documentation and improves computational time by changing how some internal procedures are handled.

The underlying theory is not described in this document, and for a comprehensive treatment is referred to the paper introducing the original BProbe.

Installation & Getting Started

How to install and use this package is described in Installation & Getting-Started.

How to use this package

The methods constituting this package can be divided into four categories:

• Initialization
  • ProbeInit (arguments, options, example)
• Directional scanning in preparation for the 'actual' scan
  • ProbeDirectionalScan (options, example)
  • ProbeGetDirectionalData(example)
  • ProbeShowDirectionalPlot(example)
• 'Actual' scanning using quasi-coherent state algorithm
  • ProbeScan (options, example)
  • ProbeReset (options, example)
  • ProbeGetPoints (example)
  • ProbeGetTangentspaces (example)
• Helper functions
  • ProbeGetOperator(example)
  • ProbeGetGroundstateEnergy (arguments, example)
  • ProbeGetEnergies (arguments, example)
  • ProbeGetGroundstate (arguments, example)
  • ProbeGetGroundstates (arguments, example)
  • ProbeGetExpectedLocation (arguments, example)
  • MatrixRepSU2 (arguments, example)
  • MatrixRepSU3 (arguments, example)

ProbeScan is the primary scanning utility. It implements the algorithm to rasterize the semi-classical limit of the brane configuration defined by a set of matrices as submanifold of the target space . After scanning, the coordinates of the collected points can be retrieved with the method ProbeGetPoints. Optionally, information about their tangent spaces is obtainable via ProbeGetTangentspaces.

The methods ProbeGetGroundstateEnergy, ProbeGetEnergies, ProbeGetGroundstate, ProbeGetGroundstates and ProbeGetExpectedLocation are helper functions which have nothing to do with the actual scanning procedure. However, they can be used to get specific information of certain points in the target space .

Before the methods with the Probe prefix can be used, the package has to be initialized by calling the method ProbeInit.

In addition, this package provides matrix representations of SU(2) and SU(3) through the methods MatrixRepSU2 and MatrixRepSU3 for convenience, since they exhibit a class of more or less well known fuzzy branes.

For a detailed description of the methods and examples please visit the associated Wiki-pages. They are accompanied with small examples. For larger examples, see the Examples-section.

Examples

Fuzzy 2-sphere

See the attached Wolfram Mathematica notebook on the Fuzzy 2-sphere.

Gaussian Unitary Matrix Distribution

See the attached Wolfram Mathematica notebook on GUE matrices.