Clebsch-Gordan Coefficients in Causal Fields I am trying to understand Section 5.7 in Weinberg's field theory book. The task is to construct causal fields transforming according to the general $(A,B)$ representations of the proper orthochronous Lorentz group:
$$\psi_{ab}(x) = (2 \pi)^{-3/2}\sum_\sigma \int \mathrm{d}^3p \left[\kappa \,  a(\mathbf{p},\sigma)\, \mathrm{e}^{\mathrm{i} p \cdot x} u_{ab}(\mathbf{p},\sigma) + \lambda a^{c \dagger}(\mathbf{p},\sigma)\mathrm{e}^{-\mathrm{i} p \cdot x} v_{ab}(\mathbf{p},\sigma)\right].$$
In particular I want to know how to calculate the $u_{ab}(0,\sigma)$'s and how they relate to polarization. Weinberg says that these are simply the Clebsch-Gordan coefficients:
$$u_{ab}(0,\sigma) = \frac{1}{\sqrt{2m}} C_{AB}(j \sigma ; a b)$$
For instance, in the case of a spin-1 massive boson we know that fields get dressed with polarization vectors:
$$\epsilon^\mu(0,0) \sim (0,0,0,1)$$
$$\epsilon^\mu(0,-1) \sim (0,1,-\mathrm{i},0)$$
$$\epsilon^\mu(0,1) \sim (0,1,\mathrm{i},0)$$
I guess we can write these as bispinors via contraction with pauli matrices:
$$u_{ab}(\sigma = \{0, \, 1, \, -1\}) = \left\{\left(
\begin{array}{cc}
 1 & 0 \\
 0 & -1 \\
\end{array}
\right),\left(
\begin{array}{cc}
 0 & 2 \\
 0 & 0 \\
\end{array}
\right),\left(
\begin{array}{cc}
 0 & 0 \\
 2 & 0 \\
\end{array}
\right)\right\}$$
I have three related questions:

*

*Those bispinors are supposedly proportional to $C_{A B}(j \sigma ; ab)$ for some choice of $A \, B, \, j.$ What are the values of $A, \, B, \, j$?

*How do we start with the notion of a "spin-1 massive boson" and deduce the aforementioned values?

*What about a spin-2 massive boson? What are the $C_{A B}(j \sigma ; ab)$ and how do we extract the 5 polarization tensors from them?

There is a similar question ($(A,B)$-Representation of Lorentz Group: Coefficient functions of fields) but the scope of this one is purposefully far more limited since the root of my difficulty with the subject is that the presentation is very general and lacking in explicit calculation.
 A: A massive spin $1$ boson is best represented by a vector field $V^{\mu}$, which transforms with respect to the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group, with the Lorentz constraint $\partial_{\mu}V^{\mu}=0$. The Clebsch-Gordan coefficients that are relevant here are $C_{1/2,1/2}(1\sigma,ab)$
The matrices you arrive at are
$$u_{ab}(\sigma=\{-1,0,1\})=\left\{\begin{pmatrix}1&0\\0&0\end{pmatrix},\begin{pmatrix}0&1/\sqrt2\\1/\sqrt2&0\end{pmatrix},\begin{pmatrix}0&0\\0&1\end{pmatrix}\right\}$$
There isn't an immediate relation between these and what you wrote down.
A massive spin $2$ boson is represented with a symmetric rank two tensor $g_{\mu\nu}$, which transforms with respect to the $(1,1)\oplus(0,0)$ representation of the Lorentz group, with analogous constraints which eliminate the two spin $0$ components and a spin $1$ component. The relevant Clebsch-Gordan coefficients are $C_{1,1}(2\sigma,ab)$
$$u_{ab}(\sigma=\{-2,-1,0,1,2\})=\left\{\begin{pmatrix}1&0&0\\0&0&0\\0&0&0 \end{pmatrix},\begin{pmatrix}0&1/\sqrt2&0\\1/\sqrt2&0&0\\0&0&0 \end{pmatrix},\begin{pmatrix}0&0&1/\sqrt6\\0&\sqrt{2/3}&0\\1/\sqrt6&0&0 \end{pmatrix},\begin{pmatrix}0&0&0\\0&0&1/\sqrt2\\0&1/\sqrt2&0 \end{pmatrix},\begin{pmatrix}0&0&0\\0&0&0\\0&0&1 \end{pmatrix}\right\}$$
