I have two points not clear in Peskin & Schroeder's QFT on page 166.
- On Figure 5.6, "Since helicity is conserved, a unit of spin angular momentum is converted to orbital angular momentum", I am really puzzled on this sentence. In my understanding, in this Center-of-Mass frame, the initial total spin is zero, also the finial total spin is zero. For the orbital angular momentum, I see that the "back-scattering" probability is large, so their need to have some orbital angular momentum in principle. But I am really puzzled on how this one unit of spin angular momentum converted to orbital angular momentum. By the wave, the book mentioned the final states is a "p-wave", also what is this "p-wave"?
- I also troubled with the sign in eq.(5.101), I thought their need to have a "minus" sign, the reason as follows:
for the equation above (5.101), I thought it should be
$$
\bar{\sigma}\cdot (p-k^{\prime})\simeq -\bar{\sigma}^{1}(p-k^{\prime})^{1}=\sigma^{1}\cdot (-\omega\chi)
$$
so it has a "minus" sign difference with book, and we have used the approximation
$$
p-k^{\prime}\simeq (0,-\omega \chi,0,0)
$$
If you have any comments on above questions, I am really appreciate it.
Best Answer
The total spin about the z-axis in the initial state is +1/2 (photons have spin 1), and in the final state it is -1/2 (if $\chi=0$, to be exact), so there must be some additional +1 orbital angular momentum about the z-axis in the final state compared to whatever was there in the initial state.
The amplitude is calculated in terms of well defined initial 3-momentum pointing along the z-axis, and final momentum pointing in a direction specified by the angle $\chi$. But to really talk about orbital angular momentum we ought to expand in terms of spherical harmonics. Consider a table of spherical harmonics. Notice that all the spherical harmonics vanish at $\theta=0$ unless $m=0$, so even though there is not a well-defined initial total orbital angular momentum $l$, there is definitely initially zero z-component.
So we know the $z$ component in the final state should be $m=1$, and if you again take a look at that table you'll notice all of the $m=1$ spherical harmonics are proportional to $-\sin\theta\approx \chi$, which is consistent with Peskin and Schroeder's calculation. By the way ``p-wave" just means the $l=1$ orbital angular momentum state, and it is a slight abuse of terminology in my opinion since what is really well-defined is that $m=1$, not the $l$ value.