Covariant derivative vs Gauge Covariant derivative

[lennyleonard]

As you may guess from the title this question is about the covariant derivatives, more precisely about the difference between the usual covariant derivative, the one used in General Relativity defined by:$$\nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial x^{\mu}}+\Gamma_{\mu\nu}^{\beta}u^{\nu}\right) e_{\beta}$$and the Gauge covariant derivative, defined by $$D_{\mu}=\partial_{\mu}-iW_{\mu}^a(x)\,T^a$$where the $W_{\mu}^a(x)$s are the gauge fields and the $T^a$s are the generators of the Lie algebra.

They seem quite different to me: the former deals whit the fact that the basis vectors may vary from point to point (like the polar basis vectors): it has therefore a very simple geometrical interpretation.

The latter instead have been introduced (as far as I know) to make gauge invariant (according to the gauge group concerned) the equations to which it's applied, but I don't see any geometrical wiew to this, although I guess it has to have one!

Can you tell me what is it (if there actually is one!)?

 

[lavinia]

As you may guess from the title this question is about the covariant derivatives, more precisely about the difference between the usual covariant derivative, the one used in General Relativity defined by:$$\nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial x^{\mu}}+\Gamma_{\mu\nu}^{\beta}u^{\nu}\right) e_{\beta}$$and the Gauge covariant derivative, defined by $$D_{\mu}=\partial_{\mu}-iW_{\mu}^a(x)\,T^a$$where the $W_{\mu}^a(x)$s are the gauge fields and the $T^a$s are the generators of the Lie algebra.

They seem quite different to me: the former deals whit the fact that the basis vectors may vary from point to point (like the polar basis vectors): it has therefore a very simple geometrical interpretation.

The latter instead have been introduced (as far as I know) to make gauge invariant (according to the gauge group concerned) the equations to which it's applied, but I don't see any geometrical wiew to this, although I guess it has to have one!

Can you tell me what is it (if there actually is one!)?

I am not familiar with this formulation of covariant derivative so I may not be answering your question. With this caveat in mind, my understanding is that classical connections of vector bundles are Levi-Cevita connections. This means that the connection is symmetric and is compatible with a Riemannian metric or I guess in the case of the Theory of Relativity with a Minkowski type metric.

In general though a connection only requires an idea of parallel translation but the idea of parallel may not refer to length or angle measurements on the vectors.
This more general idea, called a gauge field - I think, is usually described as an operator on sections of the bundle that maps them into the tensor product of the bundle of sections with the cotangent bundle. In this form, one must show that there is a second operator that gives the curvature 2 forms of the connection.

The only examples I know of are the formulation of Maxwell's equations as a connection on a complex line bundle and flat bundles on Riemann surfaces that are not compatible with any metric.

Compatibility with a metric is a strong condition. If you review your General Relativity you will see that the connection is determined by the metric and symmetry and one finds the Chistoffel symbols by solving equations involving the metric. In a gauge field there is no such procedure.

 

[henry_m]

They are both examples of what a differential geometer would call 'connections on a principal bundle'.

The rough idea of what's going on is that you want a derivative that gives you the components of an object which behave properly under some set of transformations. In the first case, this is the changes of basis, and the second it is the gauge transformations. Now, because the basis change/gauge transformations can depend on the point in spacetime, partial derivatives are no good. You have to add in an extra term to fix it, and define the transformation properties of that term so you get the right answer. In the first case, this is the Christoffel symbol, and in the second the gauge field.

This all looks much nicer and more natural in an abstract, coordinate free formalism but that requires quite a lot of work to develop.

 

[lennyleonard]

Thanks a lot for your help fellas!

 

[blue_raver22]

The covariant derivative and gauge covariant derivative may seem quite different, but they both serve important purposes in different areas of physics. The covariant derivative, as you mentioned, takes into account the fact that basis vectors may vary from point to point, and it allows us to define a derivative that is independent of the choice of coordinates. This is crucial in General Relativity, where the metric tensor varies from point to point and we need a way to take derivatives that are consistent with this variation.

On the other hand, the gauge covariant derivative is used in gauge theories, such as the Standard Model of particle physics. In these theories, we have local gauge symmetries, which means that the equations of motion should remain unchanged under certain transformations. The gauge covariant derivative is a way to make sure that the equations of motion are gauge invariant, by introducing gauge fields and generators of the Lie algebra. So, while the covariant derivative has a clear geometric interpretation, the gauge covariant derivative is more abstract and is used to satisfy the requirements of gauge invariance in these theories.

However, there are some connections between the two concepts. In some cases, the gauge fields can be interpreted as connections on a fiber bundle, and the gauge covariant derivative can be seen as a generalization of the covariant derivative in this context. Additionally, both the covariant derivative and gauge covariant derivative involve the concept of parallel transport, where we move vectors or fields along a path while maintaining their intrinsic properties.

In summary, while the covariant derivative and gauge covariant derivative may have different origins and purposes, they both play important roles in understanding and describing the fundamental laws of nature.