Matchete`
Matchete`

DefineCG

DefineCG[symb,{rep1, rep2,…},tensor]

initializes a new Clebsch-Gordan coefficient (CG), symb, with indices in the named representations (ordered). The numerical values of the CG is specified with the tensor argument.

Details and Options

  • DefineCG does not give a return value, but modifies the internal state of Matchete, initializing a new CG coefficient.
  • The index representations specified with the list {rep1, rep2,…} can use any named representations initialized with DefineRepresentation (or automatically through DefineGaugeGroup, or DefineGlobalGroup).
  • tensor is a numerical (exact) SparseArray, to ensure internal consistency between tensors of different CGs, the user is strongly encouraged to obtain them with Generators, StructureConstants, or InvariantTensors.
  • Having defined a CG gives access to the short-hand notations symb[a1, a2,…] and CG[symb, {a1, a2,…}], which is a an ergonomic way to enter the CG.

Examples

Basic Examples  (1)

To set up a CG one must first define a group and one or more representations. In this case, we define a fundamental representation of an SU(2) group:

With that, we may define the Levi-Civita invariant tensor of the fundamental representation. The numerical values of the tensor is determined with InvariantTensors to ensure that the basis is compatible with any other CG we might define:

When the CG has been defined, the user can input it and Matchete will recognize it:

Tech Notes
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  • CG Coefficients
    • ▪
    Group Magic