Dealing with multi-index derivatives

Question

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:

https://cadabra.science/qa/84/derivatives-with-multiple-indices?show=84#q84

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!

Comments

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.

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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.