What is the relationship between polarization vectors and spin in QFT?

[RicardoMP]

Homework Statement

I'm told to consider the polarization vector $$s_L^\mu=(\gamma \beta, \gamma \vec\beta/\beta)$$, which is longitudinal ($\vec s_L||\vec\beta$, where $\beta$ is the relative velocity in a Lorentz boost), and that I want to show that $s^\mu_L$ satisfies $s^2=-1$.

Relevant Equations

$$s_L^\mu=(\gamma \beta, \gamma \vec\beta/\beta)$$
$s^\mu=(0,\vec s)$ and $|\vec s|=1$

I'm looking forward to have a better understanding of the polarization vector in quantum field theory in order to solve a particular problem.
In class and in several textbooks I see that $s^\mu=(0,\vec s)$ and $|\vec s|=1$. Are polarizations vectors defined to have no temporal component in Minkowski space and for its modulus to be 1? If I square the longitudinal part I get 0 for which I assume that the only contribution to $s^2$ comes from the transverse part($s^\mu=s^\mu_L+s^\mu_T$).
How is this polarization vector related to spin and what does it represent in a particle's state?

Thank you and stay safe.

 

[George Jones]

I want to show that $s^\mu_L$ satisfies $s^2=-1$.

Do you know how to calculate $s^\mu_L \left(s_L \right)_\mu$?