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Dirac notation for conjugacy class

  1. Sep 18, 2016 #1
    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 any one of the 8 Gell-Mann matrices.
     
  2. jcsd
  3. Sep 19, 2016 #2

    Orodruin

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    Please provide a reference to where your notation is taken from and define x.
     
  4. Sep 19, 2016 #3
  5. Sep 19, 2016 #4

    Orodruin

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    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.
     
  6. Sep 19, 2016 #5
    OK. I think I may be confusing things.

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

    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?
     
    Last edited: Sep 19, 2016
  7. Sep 19, 2016 #6

    Orodruin

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    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.
     
  8. Sep 19, 2016 #7
     
  9. Sep 19, 2016 #8
    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?
     
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