Undergrad Questions about Helicity & Chirality: Answers Explained

[Silviu]

Hello! I have some questions about helicity and chirality: So I understand how is helicity defined and that it has eigenvalues of 1 or -1. But can a particle (massless) have mixed helicity? Like the spin not to be along the direction of motion? (I assume it can but I want to make sure, because in all the articles I read they talk only about 1 and -1 values for it). I understand (I hope) the meaning of chirality but I am not really sure how does it affects the weak interaction. From what I understand, an electron is a superposition of left and right chirality electrons and when you measure the chirality, you pick one of them. So when an electron interacts with a W boson (let's say) does it pick a chirality and something happens only if the left chirality is chosen? Or if an electron gets out of a weak interaction, you are sure it is left handed? And in general, if you have an electron in lab, how can you find out it's chirality? Thank you!

 

[vanhees71]

Sure, if you have a proper superposition of the two helicity eigenvectors the helicity becomes undetermined,
$$|\psi \rangle=c_1 |h=1 \rangle + c_2 |h=-1 \rangle.$$
If we normalize it such that $|c_1|^2+|c_2|^2=1$ the probability to find a particle with helicity $1$ when measuring helicity, is $|c_1|^2$ and to find it with helicity $-1$ is $|c_2|^2$. If not either $c_1$ or $c_2$ are 0, the particle's helicity is indetermined.