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A Completeness relation for SO(N)

  1. Jul 1, 2016 #1
    Hello everyone,
    I was wondering if anyone knows what the completeness relation for the fundamental representation of SO(N) is.
    For example, in the SU(N) we know that, if [itex]T^a_{ij}[/itex] are the generators of the fundamental representation then we have the following relation
    $$
    T^a_{ij}T^a_{km}=\frac{1}{2}\left(\delta_{im}\delta_{jk}-\frac{1}{N}\delta_{ij}\delta_{km}\right)
    $$
    This follows from the fact that the [itex]T^a[/itex], together with the identity form a complete basis for the [itex]N\times N[/itex] complex matrices.

    Does anyone know how to find the analogous for SO(N) (if any)?

    Thanks a lot!
     
  2. jcsd
  3. Jul 2, 2016 #2

    jambaugh

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    Science Advisor
    Gold Member

    There is not an analogue. See http://repositorio.unesp.br/bitstream/handle/11449/23433/WOS000234099500010.pdf?sequence=1 [Broken] by C. C. Nishi
    equation 22 and the paragraph following it.
     
    Last edited by a moderator: May 8, 2017
  4. Nov 24, 2016 #3
    Such a relation you can obtain in fundamental rep by yourself just using definition of generators
    $$
    (\lambda_{ab})_{cd}=-i(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}) \in so(n).
    $$
    By contracting with an other $\lambda_{ab})_{$ you get
    $$
    (\lambda_{ab})_{cd}(\lambda_{ab})_{ef}=-2(\delta_{ce}\delta_{df}-\delta_{cf}\delta_{de}).
    $$

    In spinor rep it's going to be more complicated.
     
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