Representation theory MOC

Representation operator

Given a unitary representation and some representation (usually an irrep) , a representation operator¹ transforming in is a linear map satisfying² rep

for all and . If is an irrep we denote a corresponding representation operator as .

Warning

Every operator transforms in a representation

Fixed basis

The properties and motivation for a representation operator become clearer when a basis is fixed for . It is common to think of a representation operator as a set of irreducible operators³ corresponding to each basis vector. The condition above thence becomes

which is essentially the statement that the transform like the basis vectors under . This is a direct generalisation of the Tensor operator (including scalar and vector operators), which transform in the standard representation of .

Properties

• If is the trivial representation, then is just an operator that commutes with .
• Irreducible operators applied to an irreducible orthonormal basis transform in the product representation
• Wigner-Eckart theorem

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Footnotes

1. Keppeler refers to this as a set of irreducible operators ↩

2. 2015, Introduction to tensors and group theory for physicists, §6.2, p. 276ff ↩

3. 2023, Groups and representations, §4.2, p. 54 ↩