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Could someone please explain this 4-vector/tensor notation?

  1. Mar 1, 2009 #1
    If you can answer any of the questions below, your help will be greatly appreciated.

    There's an equation from E&M (I believe the definition of the antisymmetric field tensor) that reads:

    [tex]
    F_{\mu \nu} = \frac{\partial A_\nu}{\partial x^\mu} - \frac{\partial A_\mu}{\partial x^\nu},
    [/tex]

    where (of course) [itex]A[/itex] is the 4-vector potential. I have no idea how to parse this equation...what's going on with taking the partial of [itex]A_\nu[/itex] with respect to [itex]x^\mu[/itex]? What's this notation describing, or telling me to do? I just don't see it.

    Also (and this pertains to relativistic quantum mechanics), what's meant by something like

    [tex]
    \partial_\mu [\partial_\nu \Psi(x^\mu)],
    [/tex]

    where (of course) [itex]\Psi(x^\mu)[/itex] is the wave function? I.e., what does it mean to operate on [itex]\partial_\nu[/itex] of something with [itex]\partial_\mu[/itex]? Particularly when that something I'm operating on is a function of [itex]x^\mu[/itex]? (By the way: In case the notation is unfamiliar to you, [itex]\partial^\mu = \partial / \partial x_\mu[/itex].) Does that turn into something? Can I compactify that? Or do I just have to leave it as [itex]\partial_\mu [\partial_\nu \Psi(x^\mu)][/itex]? Also, would I get an equivalent expression if I interchanged the order of [itex]\partial_\mu[/itex] and [itex]\partial_\nu[/itex]; i.e., if I tried to evaluate [itex]\partial_\nu [\partial_\mu \Psi(x^\mu)][/itex]? Or would I get something quite different?

    Does all of this boil down to "we need two distinct dummy indices, [itex]\mu[/itex] and [itex]\nu[/itex], to indicate that there shouldn't be any summation going on?"

    (P.S. - I would like to thank David J. Griffiths for getting me into the habit of italicizing words I want to emphasize.)
     
  2. jcsd
  3. Mar 1, 2009 #2
    Given two four vectors, one for the field, and one for the spacetime
    position

    [tex]
    \vec{A} = (A_0, -A_1, -A_2, -A_3)
    [/tex]
    [tex]
    \vec{x} = (x^0, x^1, x^2, x^3) = (ct, x, y, z)
    [/tex]

    This says that there's 16 scalar quantities that can be calculated by
    taking derivatives. Two examples are:

    [tex]
    F_{23} =
    \frac{\partial A_3}{\partial x^2} - \frac{\partial A_2}{\partial x^3} =
    \frac{\partial A_3}{\partial y} - \frac{\partial A_2}{\partial z}
    [/tex]

    [tex]
    F_{33} =
    \frac{\partial A_3}{\partial x^3} - \frac{\partial A_3}{\partial x^3} = 0
    [/tex]

    For your other question, I'd assume summation over [itex]\mu[/itex] is implied.
     
  4. Mar 1, 2009 #3
    Thanks a lot for your help!

    What if I confront something like

    [tex]
    \frac{\partial F^{\mu \nu}}{\partial x^\nu} = \partial_\nu F^{\mu \nu}?
    [/tex]

    Does that imply a sum over [itex]\nu[/itex]?
     
  5. Mar 1, 2009 #4

    nicksauce

    User Avatar
    Science Advisor
    Homework Helper

    Yes, if an index is repeated as an upper index and a lower index, it implies a sum over that index.
     
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