# Gauge covariant derivative in curvilinear coordinates

1. Apr 30, 2007

### wandering.the.cosmos

If we work in cartesian coordinates, we say for instance, that

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

where g is the gauge coupling, and $\{T^a\}$ are the generators of the gauge group, and $\{A^a_\mu\}$ 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

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

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

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

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