Dirac notation for conjugacy class

[nigelscott]

Is the RHS of the conjugate relationship

Ad(g)x = gxg^-1

from the Lie algebra equivalent to:

<g|λ|g>

In the Dirac notation of quantum mechanics?

I am looking at this in the context of gluons where g is a 3 x 1 basis matrix consisting of components r,g,b, g^-1 is a 1 x 3 matrix consisting of components r^*,g^*,b^* and λ is anyone of the 8 Gell-Mann matrices.

 

[Orodruin]

Please provide a reference to where your notation is taken from and define x.

 

[nigelscott]

Sorry, x should be λ. Thus, Ad(g)λ = gλg^-1. Ad is described here https://en.wikipedia.org/wiki/Adjoint_representation. Thanks.

 

[Orodruin]

g is not a 3x1 matrix. It is an element of the Lie group. The representation is the adjoint representation, which is a representation of a Lie group on its Lie algebra. In the case of SU(3), g is therefore a unitary 3x3 matrix with determinant one.

Edit: Note that the gluon representation of SU(3) is an 8-dimensional one (the adjoint representation of SU(N) is N^2-1-dimensional). The corresponding colour combinations are the traceless colour-anticolour combinations.

 

[nigelscott]

OK. I think I may be confusing things.

\[ \left(\begin{pmatrix} r* & g* & b* \end{pmatrix}λ_i\begin{pmatrix} r \\ g \\ b \end{pmatrix}\right)\]

appears to produce the QM superposition states for the gluons. if g has to be a 3 x 3 invertible matrix then this has to be the bra-ket interpretation. Correct?

 

[Orodruin]

No, this does not have to do with the braket notation. It has to do with how you can construct an SU(3) singlet from an 8-, a 3-, and a 3*-representation.

 

[nigelscott]

OK. I think I may be confusing things.

\begin{pmatrix} r* & g* & b* \end{pmatrix}λ_i\begin{pmatrix} r \\ g \\ b \end{pmatrix}

appears to produce the QM superposition states for the gluons. if g has to be a 3 x 3 invertible matrix then this has to be the bra-ket interpretation. Correct?

 

[nigelscott]

OK. Sorry about the formatting in the previous response. I think I get it now. The product of λ with a column vector gives another column vector. This column vector gets multiplied by the complex conjugate matrix which can be written as a column or row vector. Either way this operation is the tensor product 3 ⊗ 3bar which decomposes into the 1 ⊕ 8. Correct?