Peskin & Schroeder's QFT book page 161

Question

I am trouble with below derivation in Peskin & Schroeder's QFT book on page 161. At the bottom of this page, $$ \begin{aligned} \operatorname{tr}\left[\not p^{\prime} \gamma^{\mu} \not k \gamma^{\nu} \not p \gamma_{\nu} \not k \gamma_{\mu}\right] &=\operatorname{tr}\left[\left(-2 \not p^{\prime}\right) \not k(-2 \not p) \not k\right] \\ &=\operatorname{tr}\left[4 \not p^{\prime} \not k(2 p \cdot k- \not k \not p)\right] \\ &=8 p \cdot k \operatorname{tr}\left[\not p^{\prime} \not k\right] \\ &=32(p \cdot k)\left(p^{\prime} \cdot k\right) . \end{aligned} $$ I am really troubled for the last step derivation, I thought the last coefficient should be 8, not 32, the reason is that $$ \begin{aligned} \operatorname{tr}[\not p^{\prime} \not k]&=\operatorname{tr}[p^{\prime}_{\mu}\gamma^{\mu}k_{\nu}\gamma^{\nu}]\\ &=\operatorname{tr}[p^{\prime}_{\mu}k_{\nu}(2g^{\mu \nu}-\gamma^{\nu}\gamma^{\mu})]\\ &=\operatorname{tr}[2p^{\prime}\cdot k-\not k \not p^{\prime}] \\ &=2p^{\prime}\cdot k - \operatorname{tr}[\not p^{\prime} \not k] \end{aligned} $$ so we can see that $$\operatorname{tr}[\not p^{\prime} \not k]=p^{\prime}\cdot k,$$ so in the last line, the coefficient should be still 8. Would you have some comment on this?

Answer 1

Your second calculation is wrong. Basically because you have to be a little bit more careful what you are taking the trace of! You use the anti-commutation relations for the Dirac matrices, which is commonly written down like $$ \{\gamma^\mu, \gamma^\nu\} = 2 g^{\mu\nu}, $$ which is actually a bit sloppy. There is a $\mathbb I$ missing. You can easily see this observing that on the left there are matrices (which are entries of a 4-vector) and on the right there is only an entry of a tensor so there must be the identity. Hence you end up with $$ \text{tr}(\gamma^\mu \gamma^\nu) = \text{tr}(\{\gamma^\mu, \gamma^\nu\} - \gamma^\mu \gamma^\nu) = \text{tr}(2g^{\mu\nu}\mathbb I) -\text{tr}(\gamma^\mu\gamma^\nu) $$ $$ \implies\text{tr}(\gamma^\mu\gamma^\nu) = \frac{2}{2} \text{tr}(g^{\mu\nu} \mathbb I)=4g^{\mu\nu} $$ See also chapter 3.2 in "An Introduction to Quantum Field Theory" by M. Peskin and D.Schröder. (In my edition it is page 40)