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?

  • $\begingroup$ 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. $\endgroup$ – ACuriousMind Jan 6 '16 at 20:42
  • $\begingroup$ @ACuriousMind, yes you are right, but my motivation for asking this question is actually reflected by Timaeus's answer. $\endgroup$ – Wein Eld Jan 6 '16 at 21:09

One sense that is co served is that a photon is always spin 1 and an electronic always spin 1/2 and a Higgs is always spin 0 and those don't ever ever change.

But basically the spin state of a system is just a linear combination of tensor products of a bunch of 1d vectors for every spin 0 particle in the system, a bunch of 2d vectors for every spin 1/2 particle in the system, a bunch of 3d vectors for every spin 1 particle in the system, a bunch of a bunch of 4d vectors for every spin 3/2 particle in the system, and a bunch of 5d vectors for every spin 2 particle in the system.

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.

  • $\begingroup$ 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. $\endgroup$ – Wein Eld Jan 6 '16 at 21:07
  • $\begingroup$ @WeinEld I'm not sure what you are asking about. If the spin were different it would be a different particle. Even a neutrino that flavor oscillates is still described as one neutrino being destroyed and another being created. A particle is what a particle is. That way you can talk about creating it or destroying it and you know what you got or what you lost. $\endgroup$ – Timaeus Jan 6 '16 at 21:11
  • $\begingroup$ It has the similar taste of the supersymmetry in superspace. $\endgroup$ – Wein Eld Jan 6 '16 at 21:12

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