convert the commutator of covariant derivatives acting on spinor into curvature

Question

Hi everyone, I was doing some lengthy computation involving covariant derivatives and gamma matrices acting on spinors. And I often encounter expressions containing the commutator of covariant derivatives. as following

$ R{a b c d} \gamma{a b} \nabla{c}{\nabla{d}{\epsilon}} $

where because of the anti-symmetry on the indices $c$ and $d$ of the Riemann tensor. The summation on theses give rise to the commutator of covariant derivatives acting on spinors and I want to use the identity (or definition) such as

$ [\nabla{c}, {\nabla{d}] {\epsilon}} = \frac{1}{4} R{c d e f} \gamma{e f} \epsilon$

what should i do? of course, I can use the substitute() to convert the whole term like

substitute(, $ R{a b c d} \gamma{a b} \nabla{c}{\nabla{d}{\epsilon}} -> \frac{1}{4} R{a b c d} R{c d e f} \gamma{e f} \epsilon $);

But this is not clever at all and I need to identify manually term by term. Which is obviously impossible in slightly more complicated examples.

Looking forward to learn and to read your suggestions. Thank you if you can give me some hint.

Comments

Hello, Naive suggestion, what about

substitute(ex,$\nabla_{a}{\nabla_{b}{Q??}}->1/2 (\nabla_{a}{\nabla_{b}(Q??)}+\nabla_{b}{\nabla_{a}(Q??)})+R_{a b c d} \gamma^{c d}(Q??)$)
distribute(ex)
canonicalise(ex)

?

---

Thank you for reply,

I actually came to some similar ideas with you :).

But in some cases it creates new covariant derivative terms which do not cancel.

Answer 1

I think that what you are after (but correct me if I'm wrong) is some way to detect the presence of \nabla_{c}{\nabla_{d}{...} where the remainder of that term enforces that c and d are anti-symmetrised?

There isn't currently a pattern that allows you to do that (though I agree it would be nice to have, not just for this purpose). So you'll have to work around it using some additional information. Do you have many different contractions that could lead to this implicit anti-symmetrisation? Or is it always a handful of tensors (Riemann like above plus a few other possibilities)?

Comments

Thanks for your response dear kasper.

You are right, my point is exactly to ask a way to detect the presence of the expression as \nabla{c}{\nabla{d}{...} while the rest of the term enforces c and d being anti-symmetrised. And then I can translate them into Riemann tensor...

Actually, there are several different possibilities which lead to this implicit anti-symmetrisation, just as you mentioned, they are always contractions between tensor indices, such as Riemann tensor, gamma matrices and differential forms etc.