# Covariant Derivative and Gauge Covariant Derivative

1. Jun 15, 2011

### 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!)?

Last edited: Jun 15, 2011