I am a bit confused about how quantum angular momentum is conserved in decay processes, especially when massless particles are involved.

Suppose we have a spin 1/2 particle that decays into a photon and another spin 1/2 particle (I am taking problem 2.4 from Advanced Quantum Mechanics by Sakurai, the reason I started asking myself these questions). Also, I let the z axis be taken in the direction in which the initial particle has a definite AM, so $|\psi\rangle = |+1/2_{z}\rangle$.

Now, the initial state has definite AM and if I am not mistaken so should the final state, by AM conservation. But I just don't see how this can be the case in a general scenario.

Firstly, I consider the case where both particles are ejected along the z axis in opposite directions. The photon carries AM in the direction of propagation. I can (albeit only intuitively) see that in this case a way of conserving AM would be that the photon is R-polarised and travels towards positive z, and the spin 1/2 particle travels towards negative z with its spin pointing downward, so that the total AM is $J=1-1/2=+1/2=J_{0}$. In this case, however, I am not sure about how one would go about addition of AM with CG coefficients. I want to reproduce the initial state $|J=1/2,M=1/2\rangle$ by combining a spin 1 and a spin 1/2 particle. The resulting state, however, includes a state vector where $m=0$ for the photon, which is not a valid state. Therefore I am not sure how to express my previous idea in a more mathematical way. I thought that maybe the $m=0$ state that appears for the photon is actually a superposition of the $m=1$ and $m=-1$ states, but the way I understood the CG coefficients until now was as a way to construct states with definite AM by using other states of definite AM. So my first question is how we get such a state by coupling spins when one of the particles is massless.

I am even more confused when it comes to the particles being ejected with an angle $\theta$ with respect to the initial z axis. The situation I defined before (R photon with spin up, spin 1/2 particle in opposite direction of movement and spin) gives us a definite 1/2 spin but now along an axis that is not the z axis anymore. So it would seem that it is required for the particles to have the mentioned spin along z. But the photon carries AM in its direction of propagation only, so that is obviously not possible. My second question is therefore how it is possible to achieve a state with definite AM in the z direction, just like the first one, when the particles (maybe massless) are moving with an angle with respect to the original spin direction.

Thank you in advance!


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