Chirality vs Helicity: Explaining the Difference

[gremezd]

What is the difference between chirality and helicity? I know, that helicity shows the handedness of the particle. For example, we have a right-handed electron if its spin is aligned in the direction of its propagation. The concept of chirality is a bit similar to that of helicity, but something more abstract, I guess. Could maybe anyone explain what the chirality is and tell its difference from helicity?

 

[humanino]

Helicity is the projection of the spin on the momentum direction. Chirality refers to the transformation laws, or the representation of the Lorentz group the particle belongs to. If a particle is massless, the situation is simple : helicity can only take the two extreme +S and -S, and all observer agree on that. Therefore, helicity and chirality can be considered identical in the massless case. But for a massive particle, a Loretz boost can switch helicity. On the wiki page, they call helicity the "apparent chirality" in that case, which seems appropriate to me.

 

[gremezd]

OK. Let's think of a massive electron. As long as it cannot propagate with the speed of light, it cannot have definite helicity, as we can boost to the other frame, where our electron's helicity would change. But again, as long as our electron is massive, it will have both parts, the left-handed and right-handed, of the Dirac spinor. So, how can we decide if our electron is of right-handed chirality or that of left-handed?

 

[Creator]

So, how can we decide if our electron is of right-handed chirality or that of left-handed?

I suppose by measuring its deflection in an inhomogenous magnetic field...? ( AKA, Stern Gerlach).


...

 

[Meir Achuz]

OK. Let's think of a massive electron. As long as it cannot propagate with the speed of light, it cannot have definite helicity, as we can boost to the other frame, where our electron's helicity would change. But again, as long as our electron is massive, it will have both parts, the left-handed and right-handed, of the Dirac spinor. So, how can we decide if our electron is of right-handed chirality or that of left-handed?

As you point out, an electron can be given any helicity by boosting.
Chirality refers to the natural handedness a particle takes in the weak interaction.
For an electron this is left handed corresponding the the V-A interaction.
In etymology, the two words mean just about the same thing.
In 1957, the Physical Review made a decision after some discussion which word would mean which physical concept.

 

[gremezd]

Clem, do you mean that the value of chirality (left-handed or right-handed) is "god given" to a particular electron and cannot change in its lifetime? Or, as you say, depending on interaction, all electrons are left-handed or all right-handed? (I just try to create some image of what chirality is in my mind, as I meet this word quite often and cannot grasp it decently..)

 

[Meir Achuz]

Think of it like a naturally right-handed person who bats left handed sometimes.
If baseball doesn't help, consider that "left-handed" has two different meanings for an electron.
Chirality was coined to describe how it naturally entered the weak interaction.
Helicity was used to describe its spin angular momentum ion the direction of its momentum.