# Weyl Spinors & Helicity: Explained

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• kelly0303
In summary, a Weyl spinor is a 2-dimensional spinor that describes massless particles with definite helicities. It has two spin components, but the helicity can only be either +1/2 for the right-handed or -1/2 for the left-handed Weyl spinor. This is demonstrated by the math behind the Weyl equation. However, for massless neutrinos, which are described by left-handed Weyl spinors, there is currently no experimental evidence for the existence of right-handed neutrinos (or left-handed anti-neutrinos). It is possible to represent a left-handed neutrino as (1,0)^T and a right-handed anti-neutrino as (0,1f

#### kelly0303

Hello! So Weyl spinors are 2 dimensional spinors which describe massless particles and have definite helicities. So if we have a right handed Weyl spinor going along the positive x-axis, it's spin will always point along the positive x-axis too. I am a bit confused how can an object have 2 spin states, yet it's spin can point only in a given direction. I guess I am missing something. Can someone explain to me how does this work?

For a Weyl spinor each spin component can take two values ##\pm 1/2##, but the helicity can only be either +1/2 for the right-handed or -1/2 for the left-handed Weyl spinor. The math is nicely summarized at Wikipedia:

https://en.wikipedia.org/wiki/Weyl_equation

For a Weyl spinor each spin component can take two values ##\pm 1/2##, but the helicity can only be either +1/2 for the right-handed or -1/2 for the left-handed Weyl spinor. The math is nicely summarized at Wikipedia:

https://en.wikipedia.org/wiki/Weyl_equation
Thank you for your reply. But I am still not sure I understand. A right handed Weyl spinor, will have it's spin pointing along its direction of motion (say x) during its entire existence. Now in the case of an electron, if the spin is along x and I measure it along z, I get half of the time +1/2 and half of the time -1/2. But in the case of the Weyl spinor, I am not even sure what I get. Due to the fact that it is a spin half particle (I think they even used it to describe neutrino) I would expect to also get 50-50 up and down along z. But having right polarization all the time it's spin can't be along the z axis, as it has to be all the time along the x axis. I am just so confused.

A right handed Weyl spinor, will have it's spin pointing along its direction of motion (say x) during its entire existence.
The components of a right-handed Weyl spinor are the right-handed particle and the left-handed anti-particle.

In the case of massless neutrinos, they would be left-handed Weyl spinors containing the left-handed neutrino and the right-handed anti-neutrino. We currently have no experimental evidence for the existence of right-handed neutrinos (or left-handed anti-neutrinos).

The components of a right-handed Weyl spinor are the right-handed particle and the left-handed anti-particle.

In the case of massless neutrinos, they would be left-handed Weyl spinors containing the left-handed neutrino and the right-handed anti-neutrino. We currently have no experimental evidence for the existence of right-handed neutrinos (or left-handed anti-neutrinos).
So for example ##(1,0)^T## can represent a left-handed neutrino and ##(0,1)^T## a right-handed anti-neutrino. Is this right? But can we tell anything about its spin along a given axis, other than the one along it moves? As I said before, if you measure its spin on a direction perpendicular to the direction of motion, what would you get?