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Differentiating a 'vector'

  1. Feb 15, 2010 #1
    Again, I'm not sure whether this is the best place to post this question but its to do with gauge transformations, etc.

    The question itself is rather stupid...

    If we have a matrix U(g) (a Lie Group) and a vector φ in C (which is a scalar in spacetime) - does it make sense to use the chain rule thus:

    [tex]{\partial}_\mu (U(g) \phi) = U(g){\partial}_\mu \phi + ({\partial}_\mu U(g)) \phi[/tex]

    We are separately differentiating a matrix and vector - this seems very odd to me.
  2. jcsd
  3. Feb 16, 2010 #2
    Look at it from a component point of view. The i'th component of the vector [itex]\phi' = U(g)\phi[/itex] is

    [tex]\phi'_{i} = \sum_j U(g)_{ij}\phi_j[/itex]

    This is simply a sum of differentiable stuff. So differentiating gives

    [tex]{\partial}_\mu \phi'_{i} = {\partial}_\mu\left(\sum_j U(g)_{ij}\phi_j\right) = \sum_j \left({\partial}_\mu U(g)_{ij}\right)\phi_j + \sum_j U(g)_{ij}\left({\partial}_\mu\phi_j\right)[/itex]

    Now you can identify the first term with [itex]({\partial}_\mu U(g)) \phi[/itex] and the second with [itex]U(g){\partial}_\mu \phi[/itex]
  4. Feb 16, 2010 #3
    That makes sense, and doesn't seem stupid to me.
  5. Feb 16, 2010 #4
    thanks xempa - convincing explanation:)
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