On page 90 of this set of lecture notes on quantum field theory, http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf a simple derivation is given to show that each component Dirac equation solves the klein Gordon equation. In the derivation this identity is used without explanation: $$ \gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu}=1/2\{\gamma^{\mu},\gamma^{\nu}\}\partial_{\mu}\partial_{\nu} $$ The notes in their brevity seem to imply that its obvious or trivial but I cannot see it. The anti-commutation relations of the gamma matrices do not imply this (and indeed the anti-commutation identities are actually used later in the derivation). I thought you might be able to show this using the initial dirac equation expressed in co-variant form but I haven't made much progress there either.
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1 Answer
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$\mu$ and $\nu$ are dummy indices.
So the value of the expression cannot possibly depend on them. Moreover, if you change $\mu\rightarrow\nu$ and $\nu\rightarrow\mu$, the value won't change.
This means:
$$\gamma^\mu\gamma^\nu\partial_\mu\partial_\nu=\gamma^\nu\gamma^\mu\partial_\mu\partial_\nu$$ (Since the partials commute)
Now sum them both and divide by two to obtain the result you wish to get.
(If you don't like $\mu\rightarrow\nu$ and $\nu\rightarrow\mu$, as some people find this confusing at first, you can first do $\nu\rightarrow a$ and $\mu\rightarrow b$ and continue from there)
