# Question about coefficients of massless quantum fields

• A
• hgandh
In summary, in Chapter 5.9 of Weinberg's QFT Vol 1, massless fields are defined as a sum of two terms, one involving a creation operator and the other involving an annihilation operator. The coefficients for these terms are defined by certain conditions, and the equations for the annihilation operator are the complex conjugates of the equations for the creation operator. Weinberg claims that this can be proven by adjusting the constants in the equations, but it is not necessarily true and may require further proof.

#### 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?

@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?

PeterDonis said:
@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.

hgandh said:
I should have marked it as "A".

I have changed the thread level to "A".