Wigner-Eckart theorem of SU(3)

Question

I have just come across the Wigner-Eckart theorem and am not sure on how to apply it. How do I find the matrix elements of $\langle u|T_a|v\rangle$ in terms of tensor components and the Gell-Mann matrices, where $T_a$ are the SU(3) generators and $|u\rangle$ and $|v\rangle$ are tensors in the adjoint (8-dimensional) representation of SU(3)?

Answer 1

There are many methods to compute the matrix elements of a simple Lie algebra generators in a given representation. For the problem at hand, I'll try to describe two methods quite in detail and sketch two other methods.

The individual computations actually involve elementary linear algebra and combinatorics, but they are quite lengthy. For example to obtain all matrix elements in the adjoint representation, one needs $ 8 \times 8 \times 8 $ computations

Method-1

The adjoint representation (by definition) can be realized such that the matrix elements with respect to some orthonormal basis are the structure constants

$ \langle u_b | T_a | u_c\rangle = f^a_{bc}$

Method-2

In $SU(3)$, the adjoint representation occurs in the tensor product of the fundamental representation and its dual:

$3\otimes\bar{3}\rightarrow 8 \oplus 1$

It is customary to use the quark and anti-quark names for the basis vectors.

the fundamental representation: $ u : (1,0), d : (-1, 1), s : (0, -1) $

its dual $ \bar{u} : (-1,0), \bar{d} : (1, -1), \bar{s} : (0, 1) $

The weights of the adjoint representation are given by: (please see From Slansky's page 32 equation (5.4))

$ v_1 : (1,1), v_2 : (-1,2), v_3 : (2,-1), v_4 : (0,0), v_5 : (0,0), v_6 : (1,-2), v_7 : (-2,1), v_8 : (-1,-1) $

While, for the nonzero weights, there is only one option to construct the tensor product is:

$ v_1 = u \otimes \bar{s} $

$ v_2 = d \otimes \bar{s} $

$ v_3 = u \otimes \bar{d} $

$ v_6 = s \otimes \bar{d} $

$ v_7 = d \otimes \bar{u} $

$ v_8 = s \otimes \bar{u} $

The zero weight subspace is spanned by $ u \otimes \bar{u}$, $ d \otimes \bar{d}$, $ s \otimes \bar{s}$, but we know from the tensor product decomposition that the weight vectors in this subspace must be orthogonal to the scalar $ u \otimes \bar{u} + d \otimes \bar{d} + s \otimes \bar{s}$, thus we can choose:

$ v_4 = \frac{u \otimes \bar{u} - d \otimes \bar{d}}{\sqrt 2} $

$ v_5 = \frac{u \otimes \bar{u} + d \otimes \bar{d} -2 s \otimes \bar{s}}{\sqrt 6} $

Now, The generators of the Lie algebra on the tensor product have the form:

$ T^a = T_{3}^a \otimes I + I \otimes T_{\bar{3}}^a$,

where, the action on the fundamental representation is through the Gell-mann matrices

$ T_{3}^a = \lambda^a $

and the action on the dual is by means of the negative transpose (not the Hermitian conjugate):

$ T_{\bar{3}}^a = -{\lambda^a}^t$

Now, we are in a position to compute matrix elements, for example:

$ \langle v_1 | T^3 | v_3\rangle = -\langle u \otimes \bar{s} | \lambda^3 \otimes I + I \otimes{ \lambda^3}^t | u \otimes \bar{d} \rangle = - \langle \bar{s} | { \lambda^3}^t |\bar{d} \rangle$

That is, one can compute the matrix elements in the adjoint representation in terms of the matrix elements in the fundamental representation and its dual.

It should be emphasized that the matrices obtained by this method will very likely be different from the ones obtained in the first method but they will be unitarily equivalent.

Method-3

The basis of the adjoint representation space can be written in terms of 8 Young tableaux in which the first row is of length 2 and the second row of length 1. This method allows to write the matrix elements in terms of the matrix elements of a triple tensor product of the fundamental representation. However I won't elaborate an example here.

Method-4

The matrix elements in the Cartan-Weyl basis can be computed in principle from the weight diagram.