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First of all, sorry if the question has been asked. I tryied to find some answers but my ignorance goes too deep for any of the previous topics I could find.

I'm completly lost when it comes to the Lorentz/Pointcarré groups representations.

In the case of QFT, this is done to be able to apply Lorentz transformations to mathematical objects such as, vectors, and mostly fields.

Each representations you create must obey the Lorentz Algebra. So far I've been able to find the Lorentz algebra using its representation acting on 4-vectors. In the way we introduced the exponential map, and generators. And some cool theorems.

Since we also wanted to study how fields transform we created the representation for scalar fields, which don't "really" transform (only the coordinates do) so that was easy.

The part that I don't understand is the spinors representation, (1/2,0), (0,1/2) and (1/2,1/2).

I understand we want to create a representation acting on something that would have a spin. So those representations act on Weyl field (left/right handed spinors). I just don't understand what they are and what this representation is.

-(1/2,0) seem to be the eigenvalues of the spin operator, but I've never seen a 1/2 spin transforming into a 0spin

-(1/2,1/2), here does it mean a combination of two 1/2 spins (like a 2 electrons system)?

I guess my problem comes from the fact that I can't understand what the vector space, on which the Lorentz transformation will act, really is.

I hope my question makes sense,

thank you!