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The dimension of fields

  1. Jun 23, 2011 #1
    Hi,

    I was not entirely sure where to post this, but I think this will work.

    With the gravitational field we have that
    [tex]g^{\alpha\beta}g_{\alpha\beta}=4[/tex]
    which is the dimension of the manifold I believe. I have normally heard of [itex]g_{\alpha\beta}[/itex] being interpreted as the gravitational field quantity (or the tetrad). For the other fields in physics (like [itex]A_{\mu}[/itex]), how does one compute the dimension, or does such a quantity not exist for anything other than the gravitational field?

    Thanks in advance,
     
  2. jcsd
  3. Jun 23, 2011 #2

    atyy

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    The dimension is a property of the underlying manifold (ie. how many coordinates do you need to specify a point). All fields inherit their dimensionality from the manifold. In string theory, apparently you can get emergent space - where energy becomes a dimension - but I have no understanding of this!
     
  4. Jun 23, 2011 #3
    The contraction of the metric with its inverse indeed gives the dimension of the spacetime you are looking at. You can see this by taking a look at how the metric tensor is contracted twice with some vectorial quantity [itex]A_{\nu}[/itex] in a d-dimensional space:

    [itex]g_{\mu\rho}g^{\mu\nu}A_{\nu}=A_{\rho}[/itex].

    Therefore we can make the identification with a Kronecker delta of dimension d:

    [itex]g_{\mu\rho}g^{\mu\nu}={\delta_{\rho}}^{\nu}[/itex].

    The trace of the Kronecker delta gives just the dimension of the space, therefore

    [itex]g_{\mu\nu}g^{\mu\nu}=d[/itex].
     
    Last edited: Jun 23, 2011
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