- #1

murillo137

- 1

- 1

I'm going through Peskin & Schroeder's Chapter 19 (Perturbation Theory Anomalies) and it seems to be that equation 19.74 in page 666 has a minus sign missing on the RHS. Namely, I think the correct equation should read

\begin{align}

(i\not\!\! D)^2 = -D^2 - \frac{e}{2}\sigma^{\mu\nu}F_{\mu\nu}

\end{align}

This depends on the convention for the covariant derivative. For Chapter 19, the convention seems to be ##D_{\mu} = \partial_{\mu} + ieA_{\mu}##, with the plus sign, as established in the first line of p. 652, at least for the 2D case which is discussed there. It's also the convention for Chapter 4, when QED is introduced. I don't seem to find any point in Chapter 19 where they switch to a different convention.

We have then:

\begin{align}

(i\not\!\! D)^2 &= - \gamma^{\mu}\gamma^{\nu}(D_{\mu}D_{\nu}) \\

(i\not\!\! D)^2 &= - \frac{1}{2}\{\gamma^{\mu},\gamma^{\nu}\}(D_{\mu}D_{\nu}) - \frac{1}{2}[\gamma^{\mu},\gamma^{\nu}](D_{\mu}D_{\nu})\\

(i\not\!\! D)^2 &= -\frac{1}{2}(2g^{\mu\nu})(D_{\mu}D_{\nu}) - \frac{1}{4}[\gamma^{\mu},\gamma^{\nu}][D_{\mu},D_{\nu}]\\

(i\not\!\! D)^2 &= -D^2 + \frac{i}{2}\sigma^{\mu\nu}[D_{\mu},D_{\nu}],

\end{align}

where ##\sigma^{\mu\nu} = \frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]##. Then, we have

\begin{align}

[D_{\mu},D_{\nu}] &= [\partial_{\mu} + ieA_{\mu}, \partial_{\nu} + ieA_{\nu}] = [\partial_{\mu}, ieA_{\nu}] - [\partial_{\nu}, ieA_{\mu}]\\

[D_{\mu},D_{\nu}] &= ie(\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}) = ieF_{\mu\nu},

\end{align}

such that ##(i\not\!\! D)^2 = -D^2 - \frac{e}{2}\sigma^{\mu\nu}F_{\mu\nu}##.

I looked it up in the errata for the textbook, and there is no mention of this anywhere. Can someone confirm this? Or is there some issue in my derivation?