Covariant Derivative and 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!)?

 

[eljose79]

I can provide some insight into the difference between the usual covariant derivative and the gauge covariant derivative.

The usual covariant derivative, defined by \nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial x^{\mu}}+\Gamma_{\mu\nu}^{\beta}u^{\nu}\right) e_{\beta}, is used in General Relativity and takes into account the curvature of spacetime. It allows us to define a covariant derivative for a vector field in a curved space, where the basis vectors may vary from point to point. This is necessary because in a curved space, the concept of parallel transport (maintaining the direction of a vector while moving along a curve) is not well-defined. The Christoffel symbols, \Gamma_{\mu\nu}^{\beta}, represent the connection coefficients that account for the curvature of spacetime.

On the other hand, the gauge covariant derivative, defined by D_{\mu}=\partial_{\mu}-iW_{\mu}^a(x)\,T^a, is used in gauge theories such as the Standard Model of particle physics. It is a generalization of the usual derivative that takes into account the local symmetry of the system. The gauge fields, W_{\mu}^a(x), represent the gauge bosons that mediate the interactions between particles, and the T^a are the generators of the gauge group. The gauge covariant derivative ensures that the equations describing the behavior of particles under these interactions are invariant under local gauge transformations.

While the usual covariant derivative has a clear geometric interpretation, the gauge covariant derivative is more abstract and does not have a direct geometric interpretation. It is primarily used to maintain the gauge invariance of equations in gauge theories. However, some researchers have proposed geometric interpretations for gauge theories, such as the idea of a gauge field as a connection on a fiber bundle. But these interpretations are not universally accepted and are still an area of active research.

In summary, the main difference between the two derivatives is their purpose and the systems in which they are used. The usual covariant derivative accounts for the curvature of spacetime, while the gauge covariant derivative maintains the local symmetry of gauge theories. Both are important tools in their respective fields and have different interpretations.