Peskin equation 6.38 In Peskin and Schroeder's QFT book, page 189, equation 6.38, how do they get from the first line to the second line?
In particular, I am stuck on the transition from what I perceive to be:
$$ k'_\alpha \gamma^\alpha m \gamma^\mu + m k_\alpha \gamma^\alpha \gamma^\mu $$
into:
$$ -2m(k+k')^\mu $$
what am I missing?
I thought it might be using the Dirac equation because it works on $u(p)$, but that can't be it since $k\neq p$. Also couldn't figure out how to use the anticommutation relations of the gamma matrices.
 A: Verifying this in its entirety is tedious but good practice, so here's the skeleton of what you need to do without giving it all away:


*

*Recall the fundamental structure relation
\begin{align}
  \{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}
\end{align}
where, as usual, there is an identity matrix implicit on the right hand side.

*The expressions you really want to compare are the expression on the first line which reads
\begin{align}
  -ig_{\nu\rho}(-ie\gamma^\nu)i(k_\alpha\gamma^\alpha + m)\gamma^u i (k_\beta \gamma^\beta+m)(-ie\gamma^\rho)
\end{align}
and the expression on the second line which reads
\begin{align}
  2ie^2(k_\alpha\gamma^\alpha\gamma^\mu\gamma^\beta k'_\beta -2m(k+k')^\mu+ m^2\gamma^\mu)
\end{align}

*It's useful to match the stuff in each line that doesn't depend on $k$ and $k'$ first, and then match the stuff that does depend on $k$ and $k'$.  For example, the term on the first line that doesn't have $k$ and $k'$ in it is
\begin{align}
  -ie^2g_{\nu\rho}\gamma^\nu\gamma^\mu\gamma^\rho m^2
\end{align}
while the stuff on the second line that doesn't have $k$ and $k'$ in it is
\begin{align}
  2ie^2m^2\gamma^\mu
\end{align}
These things are the same since
\begin{align}
  g_{\nu\rho}\gamma^\nu\gamma^\mu\gamma^\rho
&= g_{\nu\rho}\gamma^\nu(\{\gamma^\mu, \gamma^\rho\} - \gamma^\rho\gamma^\mu) \\
&= g_{\nu\rho}\gamma^\nu(2g^{\mu\rho}-\gamma^\rho\gamma^\mu) \\
&= 2\gamma^\mu-g_{\nu\rho}\gamma^\nu\gamma^\rho\gamma^\mu \\
&= 2\gamma^\mu-\frac{1}{2}(g_{\nu\rho}\gamma^\nu\gamma^\rho + g_{\rho\nu}\gamma^\rho\gamma^\nu)\gamma^\mu \\
&= 2\gamma^\mu - \frac{1}{2}g_{\nu\rho}\{\gamma^\nu,\gamma^\rho\}\gamma^\mu \\
&= 2\gamma^\mu - \frac{1}{2}g_{\nu\rho}(2g^{\nu\rho})\gamma^\mu \\
&= 2\gamma^\mu-4\gamma^\mu \\
&= -2\gamma^\mu
\end{align}

*Do a similar (but more tedious) thing for the stuff that depends on $k$ and $k'$.

