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Gauge covariant derivative in curvilinear coordinates

  1. Apr 30, 2007 #1
    If we work in cartesian coordinates, we say for instance, that

    [tex]D_x \phi = \left( \frac{\partial}{\partial x} + i g \sum_a T_a A^a_x \right) \phi[/tex]

    where g is the gauge coupling, and [itex]\{T^a\}[/itex] are the generators of the gauge group, and [itex]\{A^a_\mu\}[/itex] is the gauge vector field.

    But what happens when we go to curvilinear coordinates. Specifically, suppose we're in 2 space dimensions, do we have, in polar coordinates

    [tex]D_\theta \phi = \left( \frac{\partial}{\partial \theta} + i g \sum_a T_a A^a_\theta \right) \phi[/tex]?

    The reason why I ask is Sidney Coleman says in his lectures on what he calls lumps, that

    [tex]e_\theta \cdot D \phi = \left(\frac{1}{r} \frac{\partial}{\partial \theta} + i g \sum_a T_a A^a_\theta \right) \phi[/tex]

    I'm not sure why the gauge field term does not also have a factor of 1/r.

    I must be missing something elementary.
    Last edited: Apr 30, 2007
  2. jcsd
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