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Boost generator transforms as vector under rotations

  1. Sep 5, 2010 #1
    Hi,

    I've read quite a few times now in group theory and QFT books that [tex] [X_i,Y_j]=i\epsilon_{ijk}Y_k[/tex] can be regarded as saying that [tex] \vec{Y} [/tex], the vector of boost generators transforms as a vector under rotations (where X are SO(3) generators).

    I don't really understand why this implies this fact, perhaps some could enlighten me.

    Thanks
     
  2. jcsd
  3. Sep 5, 2010 #2

    dextercioby

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    What's the definition in quantum mechanics for a vector operator ?
     
  4. Sep 7, 2010 #3
    Well I'm guessing this is it, but I don't really understand why.
     
  5. Sep 7, 2010 #4
    A 3-vector is a set of three quantities that transform correctly under rotations. In a Hilbert space the unitary rotation operators are

    [tex]U(R)=1+i\mathbf{a}\mathbf{J}[/tex]

    where J is the total angular momentum and R is a clockwise rotation of angle |a| around the a/|a| direction. A vector operator Y transforms like

    [tex]U(R)\mathbf{Y}U^\dagger(R)=R\mathbf{Y}[/tex]

    If you expand this in first order taylor series (i.e. if you consider infinitesimal rotations), you'll find the commutation relations you mentioned.
     
  6. Sep 9, 2010 #5
    Thanks alot, that has cleared it up for me.
     
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