# (A,B)-Representation of Lorentz Group: Coefficient functions of fields

+ 1 like - 0 dislike
110 views

I have a question regarding the construction of general causal fields in Weinberg's book on quantum field theory.

In his conventions a field that transforms according to the irreducible (A,B) representation of the Lorentz group is given by (eq.5.7.1) $$\psi_{ab}=(2\pi)^{-3/2}\sum_{\sigma}\int d^3p\left[\kappa a(\boldsymbol{p},\sigma)e^{ipx}u_{ab}(\boldsymbol{p},\sigma)+\lambda a^{c\dagger}(\boldsymbol{p},\sigma)e^{-ipx}v_{ab}(\boldsymbol{p},\sigma)\right]\, .$$ Here, $a$ and $a^{\dagger}$ are the usual creation and annihilation operators, $u_{ab}$ and $v_{ab}$ are coefficients carrying an irreducible representation of the Lorentz group, and $\kappa$ and $\lambda$ are coefficients.

The zero-momentum coefficients $u_{ab}(0,\sigma)$ have to fulfill the conditions $$\sum_{\bar{\sigma}}u_{\bar{a}\bar{b}}(0,\bar{\sigma})\boldsymbol{J}^{(j)}_{\bar{\sigma}\sigma}=\sum_{ab}\mathcal{J}_{\bar{a}\bar{b},ab}u_{ab}(0,\sigma)$$ $$-\sum_{\bar{\sigma}}v_{\bar{a}\bar{b}}(0,\bar{\sigma})\boldsymbol{J}^{(j)*}_{\bar{\sigma}\sigma}=\sum_{ab}\mathcal{J}_{\bar{a}\bar{b},ab}v_{ab}(0,\sigma),$$ where $\boldsymbol{J}^{(j)}_{\bar{\sigma}\sigma}$ are the angular momentum matrices in the$j$- representations of the rotation group, and $\mathcal{J}_{\bar{a}\bar{b},ab}v_{ab}(0,\sigma)$ are the angular momentum matrices in the $(A,B)$ representation of the Lorentz-group.

Weinberg shows that $u_{ab}(0,\sigma)$ is given by $$u_{ab}(0,\sigma)=(2m)^{-1/2}C_{AB}(j\sigma;ab)\, ,$$ where $C_{AB}(j\sigma;ab)$ is the Clebsch-Gordan coefficient and the normalization was chosen for convenience. However, when I try to calculate the coefficient $u_{ab}$ in the $(1/2,1/2)$ representation and want to relate them to the $u^{\mu}$ obtained when working directly in the vector representation of the Lorentz group I cannot reproduce them. , where $$u^{\mu}(0,\sigma=0)=(2m)^{-1/2}\begin{pmatrix}0\\0\\0\\1\end{pmatrix}\qquad u^{\mu}(0,\sigma=1)=-\frac{1}{\sqrt{2}}(2m)^{-1/2}\begin{pmatrix}0\\1\\+i\\0\end{pmatrix}$$ $$u^{\mu}(0,\sigma=-1)=\frac{1}{\sqrt{2}}(2m)^{-1/2}\begin{pmatrix}0\\1\\-i\\0\end{pmatrix}\, .$$ What is the procedure of translating from (A,B) to a mixture of Lorentz indices and spinor indices in more general cases, such as in the Rarita-Schwinger field?

This post imported from StackExchange Physics at 2014-08-07 15:37 (UCT), posted by SE-user Lurianus

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.