My question is: Does angular momentum conservation considerations alone forbid a spin-1/2 particle from decaying into two spin-1/2 particles?
According to this Phys.SE post, the answer seems to argue that you cannot add two spin-1/2 particles and get a spin-1/2 particle. Yes, I understand that the tensor product of two spin-1/2 particles can be re-organised as a direct sum of a spin-1 space and spin-0 space. However, I disagree that this forbids the decay, because orbital angular momentum could change such that spin+orbital angular momentum ends up being conserved. Hence I believe that under angular momentum conservation considerations alone, it is still possible to for the decay to happen. Is this reasoning correct?
Best Answer
No. The total orbital angular momentum is always an integer. Adding the orbital and spin angular momenta for a spin-$\frac{1}{2}$ particle will produce another half-integer value, $j=\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$. In contrast, the total angular momentum of two fermions (orbital, plus the first spin-$\frac{1}{2}$, plus the second spin-$\frac{1}{2}$) is necessarily an integer. So the total angular momentum cannot be the same between the two arrangements. It is impossible to transition from a state with an odd number of spin-$\frac{1}{2}$ particles to a state with an even number.