Function Repository Resource:

# WignerMatrix

Get the irreducible group representation of SU(2) for a given angular momentum

Contributed by: Bernd Günther
 ResourceFunction["WignerMatrix"][j,a] computes the Wigner matrix corresponding to the matrix a representing the irreducible Lie group representation corresponding to angular momentum j.

## Details

The angular momentum j should be a non-negative integer or half-integer, while a should be a 2×2 matrix (the representation, though originally defined only on , extends over ).
ResourceFunction["WignerMatrix"] gives a matrix of dimension (2j+1)×(2j+1), whose entries are homogeneous polynomials of degree 2j in the entries of a.

## Examples

### Basic Examples (2)

For j=1/2, WignerMatrix yields the fundamental representation (i.e. the identity function):

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The matrix entries are homogeneous polynomial functions of the elements of the argument matrix :

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### Applications (2)

Obtain the images of the infinitesimal generators using PauliMatrix:

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Verify an identity:

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### Properties and Relations (2)

For j=1 and argument matrices expressed in terms of Euler angles, the result of WignerMatrix is related to EulerMatrix through a similarity transformation:

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For general angular momentum j, the entries of WignerMatrix are given by WignerD functions:

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### Neat Examples (4)

Check the multiplicative property of WignerMatrix:

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The determinant of WignerMatrix with angular momentum j is the j(2j+1)th power of the determinant of the argument matrix:

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The character of the representation, i.e. the trace of WignerMatrix, is given by Tr[WignerMatrix[j,x]]=Det[x]jChebyshevU[2j,Tr[x]/(2Sqrt[Det[x]])]:

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As a matrix with homogeneous polynomial entries, WignerMatrix must satisfy Euler's homogeneity relation:

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bg4math

## Version History

• 1.1.0 – 28 June 2021