What Are the Two Helicities of Electrons and Their Implications?

[cygnet1]

Apparently there are two helicities or "handednesses" of an electron: right and left. A right-handed electron spins in the same direction as its phase advances, whereas the left-handed electron spins opposite to its phase advance. This leads to many questions in my mind, some of which this foum might be able to answer:

1) Do both types of electrons occur in equal numbers, or is there a preponderance of one over the other?

2) Is the helicity of an electron fixed for all time, or is it easily and frequently reversed? If it does reverse, what could cause this reversal?

3) Is it also true that the intrinsic magnetic moments of two electrons having the same spin but opposite helicities point in opposite directions?

4) Can two electrons having the same spin but opposite helicities exist in the same quantum state, or is helicity part of the electron's state?

5) How do we know that there are two helicities of electrons? What experiment confirms this?

Thank you for any replies that might be forthcoming.

 

[t_evans]

Hi Cygnet,

Helicity is (strictly speaking) the projection of the spin onto the direction of momentum. So,

h = s \cdot \hat{p}

Now in the case of a spin 1/2 particle, s can be \pm 1/2, so the helicity can be either value.
There are two limiting cases of interest, firstly the non relativistic, where the spin and the momentum are assumed not to be related to one another, and the electron can have it's spin orientated either way with respect to the momentum.

The other limiting case is the ultrarelativistic case, in the limit as electrons are moving at the speed of light, and the spin is antiparallel the momentum for electrons (negative helicity) and parallel to the momentum for positrons (positive helicity)

1. This in mind, even at low velocity, there is presumably a small preference for right handedness in electrons (and left handedness in positrons), although for most non relativistic purposes this is probably negligible.

2. As clear from above, Helicity is not invariant (although it does commute with the relativistic free particle Hamiltonian, i.e. the Dirac equation, which is why it is useful), so a change of reference frame will change the helicity. For a non interacting particle though, it should be constant. For an interacting particle, I'm not so sure, I've got a feeling the total helicity is conserved in all electroweak interactions, but then this can be freely distributed around the final components.

3. To have opposite helicities but the same spin requires opposite linear momenta, I think that the magnetic moment of an electron is always opposite the direction of it's spin (related by the Bohr magneton and the electron g-factor , about 2).

A positron on the other hand would have the opposite charge and thus the moment would be in the opposite direction, and is sort of the other helicity state (at very high velocities!)

4. Same quantum state implies all the quantum numbers are the same, so same except the spin implies all except the spin. This implies the momentum of the electrons is parallel, so to have opposite helicity requires opposite spin.

5. The evidence for electrons having spin is pretty good, going back to the Stern-Gerlach experiment being an excellent demonstration of the discreteness of angular momentum and thus spin, and thus helicity from the definition above. Further evidence might include the energy levels of atoms requiring a spin degeneracy. And as helicity is just a combination of spin and momentum

 

[cygnet1]

T Evans, many thanks for your thoughful reply. I was beginning to think I had this forum stumped.

Helicity is (strictly speaking) the projection of the spin onto the direction of momentum.

I assume you mean linear momentum here, correct? The angular momentum vector is orthogonal to the linear momentum, so the distinction is important.

Your distinction between the relativistic and non-relativistic cases is interesting. From a classical perspective, all electrons travel at subluminal speeds. However, from a QED perspective, as I understand it, all electrons travel at the speed of light all the time. It is their interaction with the Higgs field that causes them to zigzag in opposite directions and travel at an average velocity slower than light. So when making QED predictions, we should always use negative helicities for electrons?

 

[t_evans]

Hi,

Yes, sorry, I did mean linear momentum. Spin onto angular momentum would be something entirely different, although interesting quantity in it's own right.

I've not heard the Higgs mechanism described in that way before, but then I don't know a great deal about it. Interesting. Just dealing with QED for a minute, the Dirac equation that describes the motion of the free electron certainly refers to it having a rest mass and the possibility of different helicites, and then only being excluded in the ultra relativistic case. The full lagrangian also includes the mass as an empirical parameter, so can refer to slow moving objects too.

From a practical point of view, if the theory in full operates as described (I frankly don't know enough QFT to agree or disagree), then surely the helicity would always be negative and a naive average would always be negative (i.e. when the momentum flips, the spin flips accordingly, so will always come out the same) But then perhaps a proper treatment of working out the time averaged expectation of the operators involved may restore the expected result (i.e. that some average helicity operator can take either value, even if the operator in question only takes one value).

In practice, normally electrons are moving rather fast, otherwise, why are we bothering to do QED in the first place? But in the cases where they are not (or in fact, other charged leptons, which are described in precisely the same way), calculations are done with either helicity state allowed (with one suppressed depending on the momenta)

A good example that comes to mind is the decay of the charged pion (which actually relates to the structure of the electroweak Lagrangian, but what the hey), which looks like

\pi^\pm \rightarrow l^\pm + \nu_l ^\mp

And much prefers to decay to the Muonic rather than electronic mode. Although one might expect it is easier to decay to a lighter particle, it turns the electron (or positron for \pi^+) will be moving at roughly the speed of light, so has a particular helicity, which following through the calculation in full happens to not be preferred by the initial helicity of the involved pion. On the other hand, the Muon, being heavier, generally is moving rather more slowly, and thus can be in the other helicity state, which is preferred by the initial setup of the pion.

Perhaps a less convoluted example would just be that QED should restore QM in the low speed limit, and it's clear from atomic structure that both helicities are allowed for electrons. Furthermore doing the full, QED treatment of the Hydrogen atom, still allows both spin states (although split by a fancier spin-orbit term) of electrons.