I'm going over some algebra in chapter 5 of Weinberg's QFT.

On page 192 we have Lorentz invariance of the free field

$$\tag{5.1.4} \psi_l^+(x)\sum_{\sigma n}\int d^3p\,\,u_l(x;p,\sigma,n)\,a(p,\sigma,n)$$

which is $$\tag{5.1.6} U_0(\Lambda,a)\psi_l^+(x) U_0^{-1}(\Lambda,a)=\sum_{\overline{l}}D_{l\overline{l}}(\Lambda^{-1})\psi_{\overline{l}}^+(\Lambda x+a)$$

This implies (unnumbered equation on page 194)

$$\sum_{\overline{l}}D_{l\overline{l}}(\Lambda^{-1})u_{\overline{l}}(\Lambda x+b;\Lambda p,\sigma,n)=\sqrt{p^0/(\Lambda p)^0}\times \sum_{\overline{\sigma}}D_{\sigma\overline{\sigma}}^{(j_n)}(W^{-1}(\Lambda,p))\exp(i(\Lambda p)\cdot b)u_l(x;p,\overline{\sigma},n)$$

which Weinberg claims to be equivalent to

$$\tag{5.1.13} \sum_{\overline{\sigma}}u_{\overline{l}}(\Lambda x+b;\Lambda p,\overline{\sigma},n)D_{\overline{\sigma}\sigma}(W(\Lambda,p))=\sqrt{p^0/(\Lambda p)^0}\times \sum_{\overline{l}}D_{\overline{l}l}^{(j_n)}(\Lambda)\exp(i(\Lambda p)\cdot b)u_l(x;p,\sigma,n)$$

But I checked my algebra several times, shouldn't $D_{\overline{\sigma}\sigma}^{(j_n)}$ on the right hand side of 5.1.13 be actually $D_{\sigma\overline{\sigma}}^{(j_n)}$?

(I think this post agrees with me https://www.physicsoverflow.org/25639/errata-of-weinbergs-qft-textbooks-2005-edition)

But the problem is that if we assume there is a typo, later in section 5.5 with Dirac fields, this messes up the algebra on page 220 with the unnumbered equation

$$\sum_{\overline{\sigma}}u_{\overline{m}\pm}(0,\overline{\sigma})J_{\overline{\sigma}\sigma}^{(j)}=\sum_{m}\frac{1}{2}\sigma_{\overline{m}m}u_{m\pm}(0,\sigma)$$ the coefficients $u(0,\sigma)$ Weinberg obtained would be wrong. Something similar happens with the construction of spin 1 vector fields. My question is: is there a typo in 5.1.13?


I checked the derivation again. Now I'm positive the typo occurs in the unnumbered equation on page 194, instead of

$$\sum_{\overline{l}}D_{l\overline{l}}(\Lambda^{-1})u_{\overline{l}}(\Lambda x+b;\Lambda p,\sigma,n)=\sqrt{p^0/(\Lambda p)^0}\times \sum_{\overline{\sigma}}D_{\sigma\overline{\sigma}}^{(j_n)}(W^{-1}(\Lambda,p))\exp(i(\Lambda p)\cdot b)u_l(x;p,\overline{\sigma},n)$$

it should be

$$\sum_{\overline{l}}D_{l\overline{l}}(\Lambda^{-1})u_{\overline{l}}(\Lambda x+b;\Lambda p,\sigma,n)=\sqrt{p^0/(\Lambda p)^0}\times \sum_{\overline{\sigma}}D_{\overline{\sigma}\sigma}^{(j_n)}(W^{-1}(\Lambda,p))\exp(i(\Lambda p)\cdot b)u_l(x;p,\overline{\sigma},n)$$

Note on the right hand side the subscript of $D$ is wrong in the book. Anyone can confirm if I'm correct?


1 Answer 1


The answer is in the compilation of typos given by https://sites.google.com/site/jiayiyanghbar/notes-i-wrote, there is a link in https://www.physicsoverflow.org/25639/errata-of-weinbergs-qft-textbooks-2005-edition . The equations 5.1.11 and 5.1.12 are correct. When the adjoint of the equation 4.1.12 is taken, one needs to take an hermitian conjugate of the creation operator, and complex conjugate of the elements of $D_{\overline{\sigma}\sigma}$ function (indices do not interchange). And there is a typo in unnumbered equations on page 194 after the phrase "it is necessary and sufficient that" - the indices of $D_{\sigma\overline{\sigma}}$ functions must be interchanged in both equations. Equations 5.1.13 and 5.1.14 are correct.


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