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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, adding several features for scanning such as the newly added directional scans, improves and completes missing documentation and improves typical scenario computation times by changing how some internal procedures are handled.

The underlying theory is not described in this documentation, 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 commands ProbeDirectionalScan, ProbeGetDirectionalData, ProbeShowDirectionalPlot are used for preliminary directional scanning which is very useful for helping with the appropriate selection of starting points and parameters in preparation for the actual scan. Getting familiar with this procedure will be crucial for the sensible use of the package on non-trivial examples. On trivial examples, ProbeScan can manage on its own, but with more complicated structures manual focusing can increase scan quality quite drastically.

The commands ProbeGetOperator,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. Two larger examples that go through a typical application process including the thought process of BProbeM are provided in the Examples-section.

Large 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.