Question about coefficients of massless quantum fields

[hgandh]

From Chapter 5.9 Weinberg's QFT Vol 1, massless fields are defined as:
$$\psi_l(x)=(2\pi)^{-3/2}\int d^{3}p\sum_{\sigma}[k a(p,\sigma)u_l(p,\sigma)e^{ipx}+\lambda a^{c\dagger}(p,\sigma)v_l(p,\sigma)e^{-ipx}]$$
With coefficients defined by the conditions:
$$u_{\bar{l}}(p,\sigma) =\sqrt{|k|/p^0} \sum_{l}D_{\bar{l}l}(L(p))u_l(k,\sigma)$$
$$v_{\bar{l}}(p,\sigma) =\sqrt{|k|/p^0} \sum_{l}D_{\bar{l}l}(L(p))v_l(k,\sigma)$$
$$u_{\bar{l}}(p,\sigma) exp(i\sigma \theta(k,W) =\sqrt{|k|/p^0} \sum_{l}D_{\bar{l}l}(W)u_l(k,\sigma)$$
$$v_{\bar{l}}(p,\sigma) exp(-i\sigma \theta(k,W) =\sqrt{|k|/p^0} \sum_{l}D_{\bar{l}l}(W)v_l(k,\sigma)$$
Where $D_{\bar{l}l}(L(p))$ is a general, irreducible representation of the homogenous Lorentz group restricted to standard boosts, $L(p)$ that take the standard momentum $k = (0,0,k)$ into arbitrary momentum $p$ and $D_{\bar{l}l}(W)$ is the Lorentz representation restricted to the little group for massless particles. Now Weinberg says that the equations for $v$ are just the complex conjugates of the equations for $u$ so that we can adjust the constants $k$ and $\lambda$ so that
$$v_l(p,\sigma)=u_l(p,\sigma)^*$$
However, taking the complex conjugates of the equations of $u$:
$$u_{\bar{l}}(p,\sigma)^* =\sqrt{|k|/p^0} \sum_{l}D_{\bar{l}l}(L(p))^*u_l(k,\sigma)^*$$
$$u_{\bar{l}}(p,\sigma)^* exp(-i\sigma \theta(k,W) =\sqrt{|k|/p^0} \sum_{l}D_{\bar{l}l}(W)^*u_l(k,\sigma)^*$$

This is where I get stuck. The above will be true if $D_{\bar{l}l}(L(p))^*=D_{\bar{l}l}(L(p))$ and $D_{\bar{l}l}(W)^*=D_{\bar{l}l}(W)$. However, this does seem to necessarily be true. Is there another way to prove Weinberg's claim?

 

[PeterDonis]

@hgandh , this is not a "B" level question. It probably needs to be "A", which assumes a graduate level knowledge of the subject area. What is your background in this subject area?

 

[hgandh]

@hgandh , this is not a "B" level question. It probably needs to be "A", which assumes a graduate level knowledge of the subject area. What is your background in this subject area?

I should have marked it as "A". I am studying QFT currently with all of the assumed pre requisites and some group theory.

 

[PeterDonis]

I should have marked it as "A".

I have changed the thread level to "A".