# Dealing with multi-index derivatives

Hi, I've been trying to find a way to work with covariant derivatives on non-abelian gauge theories in cadabra.

The issue is cadabra treats derivatives with multiple indices as being a product of derivatives, as discussed in this question:

Is there a simple way to implement this feature other than what is suggested in the link above? Maybe with some lines of python?

Best regards!

Can you give a simple example of the kind of manipulation you would like to do? In many situations you can avoid putting the group index on the derivative itself.

It's true that in many situations it is possible to use matrix notation and avoid writing group indices.

However, I typically deal with expressions of the type

$S = \int d^4 x \chi^{a} D{\mu}^{a b} D{\mu}^{b c} \chi^{c}.$

I am frequently inverting these kind of operators to calculate propagators $< \chi^{a} \chi^{a}>,$ where I take the trace over the gauge group.

This is an example where it would be useful to teach Cadabra to be able deal with $D_{\mu}^{a b}$ as a derivative operator and then make the substitution

$D{\mu}^{a b} \chi^{b} = \delta^{a b} \partial{\mu} \chi^{b} + g f^{a b c} A_{\mu}^{b} \chi^{c}.$

I hope to have explained it concisely, if not, let me know.

Also, if there is a better way to deal with this kind of calculation in Cadabra I'd love to learn.