# Is spin angular momentum conserved?

According to the Noether theorem, we only have the conserved quantity $$J+S,$$ where $J$ is the orbital angular momentum and $S$ is the spin angular momentum. But I am always impressed that the spin is a conserved quantity such as in the case of EPR paradox. One may think that in the EPR paradox, we only deal with non-relativistic system and spin $S$ decouples from angular momentum $J$. But even in QFT, spin is always used to characterize the particle state and so is also treated as conserved quantity.

Could you please point out what did I miss in the analysis?

• Spin is not always treated as conserved! Consider, for instance, the case of an electron in an atom absorbing a photon. The photon has spin-1, but the spin of the electron is not going to change. – ACuriousMind Jan 6 '16 at 20:42
• @ACuriousMind, yes you are right, but my motivation for asking this question is actually reflected by Timaeus's answer. – Wein Eld Jan 6 '16 at 21:09

But if a particle is created or destroyed, then the spin is gone. As for whether $J=L+S$ is conserved, it's not even an observable. As for observables just as $J^2$ or $J_z$ whether they are conserved just depends on whether they commute with the Hamiltonian.
• Your answer is very enlightening. Regarding the first point ( the sense of conservation of the intrinsic property of quantum field), is there anything hide there? I mean, is there something kind of operator representing conserved quantity analogous the angular momentum in the representation of $SO(3)$ in the QM case. The quantity should not simply be the spin, because it should correspond the field rather the one-particle state. – Wein Eld Jan 6 '16 at 21:07