Gauge covariant derivative in curvilinear coordinates

In summary: Your Name]In summary, the gauge field term in polar coordinates should have a factor of 1/r, due to the transformation properties of the gauge field under a change of coordinates. This extra term arises from the transformation of the radial coordinate r.
  • #1
wandering.the.cosmos
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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.
 
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  • #2

Thank you for your question. You are correct in your observation that the gauge field term in polar coordinates should also have a factor of 1/r. This is due to the transformation properties of the gauge field under a change of coordinates.

In Cartesian coordinates, the gauge vector field transforms as A^a_x \rightarrow A^a_x + \partial_x \lambda^a, where \lambda^a is the gauge parameter. However, in polar coordinates, the gauge vector field transforms as A^a_\theta \rightarrow A^a_\theta + \partial_\theta \lambda^a + \frac{1}{r} \partial_r \lambda^a. This extra term arises from the transformation of the radial coordinate r, which is why the gauge field term in polar coordinates has an extra 1/r factor.

I hope this clarifies your confusion. Please let me know if you have any further questions.
 
  • #3


In curvilinear coordinates, the gauge covariant derivative is given by:

D_\mu \phi = \partial_\mu \phi + igA_\mu^aT_a\phi

where \partial_\mu is the partial derivative with respect to the coordinate \mu, and A_\mu^a is the gauge vector field. In polar coordinates, the gauge covariant derivative in the \theta direction would be:

D_\theta \phi = \partial_\theta \phi + igA_\theta^aT_a\phi

The reason why there is no factor of 1/r in front of the gauge field term is because the gauge field itself transforms under the gauge transformation. In other words, the gauge field is not a scalar and its transformation properties must be taken into account. In the case of polar coordinates, the gauge field transforms as:

A_\theta^a \rightarrow A_\theta^a - \frac{1}{r}\partial_\theta \alpha^a

where \alpha^a are the gauge transformation parameters. This transformation cancels out the 1/r factor in the gauge field term, making it consistent with the gauge transformation in curvilinear coordinates.

I hope this helps clarify your confusion.
 

What is a gauge covariant derivative in curvilinear coordinates?

A gauge covariant derivative in curvilinear coordinates is a mathematical operator that is used to describe how a vector or tensor field changes over a curved surface or space. It takes into account the effects of both the curvature of the space and the choice of coordinate system, and is used in many areas of physics such as general relativity and fluid mechanics.

How is a gauge covariant derivative different from a regular derivative?

A regular derivative only takes into account the changes in a field with respect to one coordinate system, while a gauge covariant derivative takes into account the changes with respect to any coordinate system. This makes it more general and applicable to curved spaces.

Why is a gauge covariant derivative important?

A gauge covariant derivative allows us to describe physical phenomena in curved spaces, which is essential in many areas of physics such as general relativity and fluid mechanics. It also helps us to understand the effects of curvature on a system and to make predictions based on these effects.

How is a gauge covariant derivative calculated?

The calculation of a gauge covariant derivative involves taking into account the effects of the curvature of the space and the choice of coordinate system. This is typically done using mathematical tools such as tensor calculus and differential geometry.

What are some applications of gauge covariant derivative in curvilinear coordinates?

Gauge covariant derivatives are used in many areas of physics, including general relativity, fluid mechanics, and electromagnetism. They are also used in engineering and applied mathematics to model and understand complex systems with curved geometry.

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