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Lorentz transformations and vector fields

  1. Feb 26, 2016 #1
    Hi Everyone.

    There is an equation which I have known for a long time but quite never used really. Now I have doubts I really understand it. Consider the unitary operator implementing a Lorentz transformation. Many books show the following equation for vector fields:

    [tex] U(\Lambda)^{-1}A^\mu U(\Lambda)=\Lambda^\mu_{..\nu} A^\nu[/tex]

    The operator U should be a matrix with the dimensions corresponding to the representation of the object being transformed. Consider the spinor case for example!

    I am getting confused by this. Should not the index on A on the left side be involved in a summation with one of the indices of U?
  2. jcsd
  3. Feb 26, 2016 #2
    I believe that the LHS is just the generic notation that [tex]A^\mu[/tex] is undergoing a symmetry transformation. That is U just represents a certain symmetry group. In order to perform the transformation itself, you must choose a representation for that group, which in the vector representation of the Lorentz group is [tex]\Lambda^\mu_{..\nu}[/tex]. It only makes sense for a representation to have indices because that is an actual matrix.

    My jargon may be off, but that is the way I understand it.
    Last edited by a moderator: Feb 26, 2016
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