# Weinberg QFT Chapter 5: gamma matrices consistency

Currently reading through Weinberg's QFT book (Vol. 1) [readable in parts here].

In his derivation of the causal Dirac field in Ch. 5, he chooses his gamma matrices as (5.4.17) \begin{align} \gamma^0&=-i\left[\begin{array}{cc}0&1\\1&0\end{array}\right]\\ \gamma^i&=-i\left[\begin{array}{cc}0&\sigma^i\\\sigma^i&0\end{array}\right] \end{align}

As far as I can see this is just the Weyl basis as I know it from other sources (Peskin & Schroeder, Schwartz) but with an extra factor $-i$ here because of Weinberg's choice of metric $\eta^{\mu\nu}$=diag(-1,1,1,1) in the anticommutation relations for the gamma matrices

$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}.$$

He then goes on to define the matrix

$$\beta=i\gamma^0=\left[\begin{array}{cc}0&1\\ 1&0 \end{array} \right] \,(i)$$

(both on p218 and pxxv) and uses this form of $\beta$ explicitly to derive the zero momentum spinors (5.5.17),(5.5.18).

Then, however, on the next page in front of Eq. (5.5.26), he states $\beta=-i\gamma^0$ and uses that in (5.5.26) and what follows to construct the causal field and spin sums. The form these equations take does seem to depend on the sign chosen for $\beta$ in (i). If I translate his spin sums back to the (more familiar to me) notation of P&S ($\gamma^\mu \rightarrow -i\gamma^\mu,a^\mu b_\mu \rightarrow -a^\mu b_\mu, \beta \rightarrow \gamma^0$) Eqs. (5.5.37),(5.5.38) have the familiar form they take in P&S.

So I'm wondering where the inconsistency went, because using both $\beta=\pm i \gamma^0$ in the same derivation should lead to one...?

[On a sidenote, in that section in Eqs. (5.5.20)-(5.5.23) the factors $(2\pi)^3$ are also all over the place and not consistent]

Ok, tracked the 2005 paperback edition down and indeed those parts have been corrected.

In front of Eq. (5.5.26) he now states $\beta=i\gamma^0$ and Eq. (5.5.26) has now changed to [2005 Paperback]

$$D(L(p))\beta D^{-1}(L(p)) = i{L_\mu}^0(p)\gamma^0=-ip_\mu \gamma^\mu /m,$$ with the sign change in the last step because of his $\eta^{00}=-1$ convention.

The previous edition [199x hardcover] reads

$$D(L(p))\beta D^{-1}(L(p)) = -i{L_\mu}^0(p)\gamma^0=-ip_\mu \gamma^\mu /m,$$ with an extra mistake in the last step giving the same answer so all the derivations after that remain correct. So this mistake in the last step was the inconsistency I was looking for and hadn't spotted.

The corrected edition of 1996 does not have such inconsistencies with the sign. Take the + sign and be sure to check the errata before continuing.

• Is this the 2005 paperback edition you are referring to? I can't find any references to other updated (?) editions. Don't have access to the 2005 edition currently. I googled and checked the publisher site for errata lists for the first edition before posting my question but couldn't track anything down. If there is one available a link is always appreciated. Nov 3 '14 at 11:05