# Is intrinsic spin conserved?

In general, we have total angular momentum, which is the sum of spin angular momentum and orbital angular momentum, and we know that in isolated systems the total angular momentum is conserved. Now, what can we say about spin angular momentum? Is that conserved?

• I'm not sure what you mean by "intrinsic spin". For example, what is the intrinsic spin of a system with two electrons? – AccidentalFourierTransform Feb 11 '17 at 17:00
• I meant that if we have total angular momentum which is the sum of spin angular momentum and orbital angular momentum and if we know that in isolated system, total angular momentum is conserved, now what can we say about spin angular momentum? is that conserved? – S.CH Feb 15 '17 at 9:49
• Total angular momentum $J$ is always conserved in a rotationally-invariant system, but $L$ and $S$ individually need not be. An example is a system of a photon and an atom, in which the photon is absorbed by the atom and the photon's one unit of spin angular momentum becomes one unit of orbital angular momentum of an electron in the atom. – Rococo Feb 18 '17 at 19:22
• Also, even the question of whether a generic EM field can always be decomposed into orbital and spin angular momenta is a somewhat thorny issue, see for example iopscience.iop.org/article/10.1088/1367-2630/16/9/093037/meta . – Rococo Feb 18 '17 at 20:03

## 2 Answers

The (intrinsic) spin has a meaning but only in the rest frame of the particle. In particular most 1-particle states can be classified as representations of the Poincaré-group by its mass and its (intrinsic) spin. In that case (rest frame of the particle), I guess, the spin is conserved. In case the particle is moving, the (intrinsic) spin looses its meaning as the spin operator no longer commutes with the Hamilton operator. However, the projection of the spin on the axis of motion $\bf{s}\cdot\bf{n}$ still is conserved. Therefore this quantity got its own name "helicity".

Is intrinsic spin conserved?

Generally speaking, yes. It's conserved like angular momentum is conserved. See the Einstein-de Haas effect which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". Also see Hans Ohanian’s 1984 paper what is spin? Note this bit: “the means for filling the gap have been at hand since 1939, when Belinfante established that the spin could be regarded as due to a circulating flow of energy”.

That's referring to the Poynting vector, which Feynman mentioned in his lectures: “Suppose we take the example of a point charge sitting near the center of a bar magnet, as shown in Fig. 27–6. Everything is at rest, so the energy is not changing with time. Also, E and B are quite static. But the Poynting vector says that there is a flow of energy, because there is an E × B that is not zero. If you look at the energy flow, you find that it just circulates around and around. There isn’t any change in the energy anywhere - everything which flows into one volume flows out again. It is like incompressible water flowing around. So there is a circulation of energy in this so-called static condition. How absurd it gets!" Only it isn't absurd.

In general, we have total angular momentum, which is the sum of spin angular momentum and orbital angular momentum, and we know that in isolated systems the total angular momentum is conserved. Now, what can we say about spin angular momentum? Is that conserved?

Generally speaking, yes. The positron has the opposite chirality to the electron. When they annihilate their spins cancel. So spin is conserved like charge is conserved. However don't forget that a flywheel can shatter, with pieces flying outwards in all directions. There appears to be no angular momentum any more. There may be some way to do something similar in particle physics, and black holes are a little counter-intuitive. So I hesitate to say spin angular momentum is always conserved in every conceivable situation.

• I believe you have this somewhat backward. The Einstein-de Haas effect shows that change of spin along some direction can be converted into change of angular momentum along that direction. This implies that they are not separately conserved, contrary to your first statement. – Rococo Feb 18 '17 at 19:25