Spinor Components, Helicity, and Chirality in Dirac Theory

In the Dirac the spinor components are defined by fermion/antifermion (here labeled as $$+,−$$) and spin component $$S_z$$ ($$↑,↓$$):

$$\begin{pmatrix} \psi_-^\uparrow \\ \psi_-^\downarrow \\ \psi_+^\uparrow \\ \psi_+^\downarrow \end{pmatrix}$$

We understand the first 2 components as the fermion and the last 2 components as the oppositely charged antifermion. So, we can express an electron as:

$$\begin{pmatrix} \psi_-^\uparrow \\ \psi_-^\downarrow \end{pmatrix}$$

We also understand the first 2 components as the projection of the spin on the chosen reference axis. For example, this could be the bi spinor for the same electron but one for an observer at rest and another for an observer in motion.

$$\begin{pmatrix} 1 \\ 0 \end {pmatrix} ; \begin{pmatrix} 0.8 \\ 0.2 \end{pmatrix}$$

The observer at rest sees a 100% probability that the electron is spin up, while the observer in motion sees an 80% probability of seeing the electron with spin up and a 20% probability of seeing the electron with spin down. So depending the frame of reference there are diferences probability to measure the spin.

The same happens with the helicity. It's the projection of the bispinor onto the particle's momentum and it also depends on the chosen frame of reference.

$$\text{Projection} = \psi^\dagger \sigma_i \psi$$

$$\text{Helicity} = \frac{\text{Projection}}{|\mathbf{p}|}$$

So the reference frame at rest gives a helicity of 0 and the system in motion gives a helicity of positive 1.

But what about chirality ? Because according to what I understand, chirality does not depend on the reference system. Where is the chirality information in bispinor?

$$chirality = ψ^† γ₅ ψ$$

I know that chirality can be found in the following way, but why use the gamma_5 matrix?

I think that the clearest answer to your question has to be found in the mathematical formulation of spinors. $$\frac{1}{2}(1\pm\gamma_5)$$ acts as a projector operator allowing to obtain different irreducible representations of the Lorentz group. You will get a lot more intuition on its nature through the framework of Clifford algebra. For example you can have a look at this: