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I still have problem clearing once and for all my doubt on the spinor representation. Sorry, but i just cannot catch it.

1)

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Take a left handed spinor, [itex]\chi_L[/itex].

Now, i know it transforms according to the Lorentz group, but why do i have to take the [itex]\Lambda_L[/itex] matrices belonging to [itex]SL(2,\mathbb C)[/itex],

[tex]\chi'=\Lambda_L\chi[/tex]??

Dimensionally it makes sense, it's like

[tex]\left(\begin{align}\chi'_{L1}\\ \chi'_{L2}\end{align}\right)=\left(\begin{align}A &B\\C&D\end{align} \right) \left(\begin{align} \chi_{L1}\\ \chi_{L2} \end{align}\right)[/tex]

but why exacly [itex]SL(2,\mathbb C)[/itex] matrices and not every other generic 2x2 complex matrix?

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2)

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Is it right to say that [itex]\Lambda_L[/itex] is the representation of the lorentz group whichactson spinors?

I have this doubt becaouse i read everywhere that the spinor is a [itex]\left(0,\frac{1}{2}\right)[/itex] representation of the lorentz group, but i'd say that the spinor os the thing on which the [itex]\Lambda_L[/itex] acts, and it is the [itex]\Lambda_L[/itex] itself to be a represetation of the group!!

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# On spinor representations and SL(2,C)

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