# Transverse neutrino spin

1. Jul 9, 2015

### VantagePoint72

Neutrinos are always left handed* and so, if you set up a coordinate system with the z-axis pointing in the direction of a neutrino's momentum, any measurement of its spin's z-component will always yield $-\hbar/2$. What if you measured the spin of the neutrino along the x or y axes? Or is there some reason why such a measurement is impossible? If you measured a neutrino along the x axis and got, say, $+\hbar/2$, that would project the neutrino's spin state into $\frac{1}{\sqrt{2}}(|z+\rangle + |z-\rangle)$. But that's not possible, or else a subsequent measurement of its spin along the z axis would have a non-zero (50%, in fact) probability of yielding $+\hbar/2$, in which case you would then have a right handed neutrino.

*Edit: And, just to clarify, assume we are talking about massless Standard Model neutrinos, for which helicity is a good quantum number, rather than physical neutrinos with a small mass.

Last edited: Jul 9, 2015
2. Jul 9, 2015

Staff Emeritus
You are confusing helicity and chirality.

3. Jul 9, 2015

### VantagePoint72

No, I'm well aware of the distinction. Standard Model neutrinos, being massless, are left handed in both the chirality and helicity sense. I'm obviously talking about helicity.

Edit: Judging from the time stamps, maybe you did not see my edit in the original post in which I clarified I was asking about massless Standard Model neutrinos, not the physical ones we observe that have very small mass and thus for which chirality and helicity will not be the same.

Last edited: Jul 9, 2015
4. Jul 10, 2015

Staff Emeritus
Then how exactly do you measure helicity? Remember, any interaction with matter will project out the left-handed chiral component.

5. Jul 10, 2015

### VantagePoint72

So there's just no possible experiment that would allow you to measure the spin of a neutrino orthogonal to its momentum? Like, only the component that determines helicity can couple to matter? Or is it just not right to think about the spin of a neutrino, period, and we should only be talking about its helicity?

6. Jul 10, 2015

### RGevo

As vanadium suggests, since the neutrinos only sit in the left handed doublet, every weak interaction must project out this component.

If you want to make a measurement of a neutrino, it has to be via a weak interaction. So I think you answer your own question now :)

7. Jul 11, 2015

### VantagePoint72

I'm not following, this is just repeating what Vanadium said. I'm still not fully understanding the implications of this statement. As I asked after his post: "So there's just no possible experiment that would allow you to measure the spin of a neutrino orthogonal to its momentum? Like, only the component that determines helicity can couple to matter?" It's not clear to me if you were saying the answer to that question is yes.

Would it be possible to have a particle like the SM neutrino, massless and chiral, but also with an electric charge? If so, what would happen if you did a Stern-Gerlach type experiment with it to measure its x- and y- spin components? Or is a chiral particle necessarily charge neutral since EM is CP-invariant?

Last edited: Jul 11, 2015
8. Jul 11, 2015

### RGevo

Yes, sorry if my statement wasn't very useful.

I wonder if hypothetically one could use lepton W decay, make measurements on the charged lepton and use momentum and spin conservation? If one could produce a beam of left handed polarised Ws for example.

9. Jul 11, 2015

Staff Emeritus
This is getting more and more complicated. So you want two fields - one massless, left-handed, with electric and weak charge, and another massless, right-handed, with electric but no weak charge. Why would people consider these the same particle?

10. Jul 11, 2015

### strangerep

For massless fields, helicity and chirality coincide.

But also, massless fields don't have the same degrees of freedom as a massive field. For a massive field, we can Lorentz-boost to its rest frame, and find that its remaining (spacetime) symmetries correspond to SO(3), i.e., ordinary spatial rotations. So it makes sense to talk about spin components orthogonal to the linear momentum.

For massless fields, the situation is rather different since we cannot Lorentz-boost to a rest frame. One finds a different group E(2), involving 1 rotation generator and 2 translation-like generators (being the remnants of 2 of the rotation generators of SO(3) that remain in this case). Since we don't observe fields with such properties, one postulates that physical massless fields must transform trivially under those generators. I.e., those 2 translation-like degrees of freedom are "amputated" by hand, leaving just 1 degree of (rotational) freedom around the direction of momentum. For more detail, try Weinberg vol 1.

11. Jul 11, 2015

### VantagePoint72

I'm not following. What do you mean two fields? I was just asking a separate question to try to see if I was understanding your previous answer, sorry if that was not clear. Forgetting about neutrinos, suppose we had a particle that was like a neutrino—massless and left handed—but that also has an electric charge. That would allow us to easily measure its spin along axes orthogonal to its momentum with a Stern-Gerlach device. As I said earlier, that seems weird to me because I don't understand how a measurement of such a spin state makes sense, since it would require the particle to have a non-zero right-handed component. If you think this example is introducing unnecessary complication, that's fine, ignore it, it was just meant to illustrate what I'm asking. Again, I'm just trying to understand whether it makes sense to talk about the spin states of neutrinos along axes orthogonal to their momentum. The example I suggested was just me taking a stab at proposing a way that such a quantity could be easily measurable.

@strangerep, I understand (at least, at a fairly basic level) the mechanics of how spin and helicity arise as irreps of the Poincare group (and I did note right at the outset that I understand chirality and helicity coincide for massless particles). That's not really the issue. Take the photon, for example. It's two helicity states correspond to the two circular polarization states of classical EM. It can also be in superpositions of these states—in particular, in those superpositions that correspond to classical horizontal/vertical and diagonal/antidiagonal polarization states and which, up to a phase, have exactly the same form as the spin states of massive spin half particles along the x- and y-axes, respectively. Maybe this is where the problem lies—I've often heard colleagues in quantum optics talking about measuring the "spin along the x- and y-axes of a photon" just as they would the spin of an electron, but maybe this is just sloppy language that should be avoided? I.e. it's taking the fact that the polarization states of a photon and the spin states of a massive spin-half particle are both described by Bloch sphere too literally?

My confusion, as I described in the above response to Vanadium, is that while it should not (as far I can tell) be possible to have a neutrino in a superposition of helicities, since they are always left handed, if its spin orthogonal to its momentum were well-defined, such a thing seems like it could be measured. So, I'm getting the feeling is that the answer is that it just does not make sense to talk about the spin of a neutrino orthogonal to its to momentum (and so, presumably, nothing would happen if an electrically charged version of a neutrino were fired through a Stern-Gerlach device).

I hope that was somewhat clear—I'm sort of talking this out to myself as I go—so if it sounds like I'm on the right track, let me know. I suppose maybe a there's simpler question I can distill all this down to, as I alluded to above, that might make things clearer for me:

If a photon (or just generally any massless spin-1 particle) is propagating along the z-axis of a coordinate system, does it make sense to talk about its spin along the x- and y-axes? Or, as I suggested, is that just a mistaken notion based on the fact that the photon's polarization states are described quantum mechanically in the same way as the components of a massive spin-1/2 particle's spin? If, indeed, it does not make sense to talk about such spin states for a photon than I think my question about neutrinos is answered too.

Last edited: Jul 11, 2015
12. Jul 11, 2015

### VantagePoint72

Follow up: I've poked around a bit more, and from here: https://en.wikipedia.org/wiki/Spin_angular_momentum_of_light it seems like my confusion did indeed stem from taking the similarity of two level helicity systems with two level spin systems (and, of course, all other two level systems) too literally. So, I think I've answered my own question(s) at this point. Neutrinos simply do not have intrinsic angular momentum orthogonal to their linear momentum. If I'm mistaken on that point, let me know, but otherwise thanks for the answers that pointed me in the right direction.

13. Jul 12, 2015

### strangerep

Sounds like you're heading in the right direction. There's one additional feature (which may or may not be relevant): photons also "do not have intrinsic angular momentum orthogonal to their linear momentum", but can have intrinsic angular momentum either parallel or antiparallel to the linear momentum -- hence can have polarizations along x/y directions by suitable superpositions of these LH and RH states. But neutrinos only have intrinsic angular momentum which is left-handed wrt their linear momentum -- which makes it rather difficult to construct neutrino states analogous to the linearly polarized photon states...

14. Jul 12, 2015

### VantagePoint72

That makes sense, and is in line with what I was thinking. Thanks.