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How does the reducibility of lorentz group ensures that there exist two types of particle i.e left-handed and right-handed (weyl spinors)?

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The Lie algebra of the Lorentz group can be understood as two copies of the $SU(2)$ Lie algebra. Representations of $SU(2)$ can be labelled by half integers $(0,1/2,1,\ldots)$. (This should be familiar from discussions of angular momentum in quantum mechanics.)

This implies that we can label representations of the Lorentz group by using two labels $(i,j)$, where $i$ corresponds to the first $SU(2)$ copy and $j$ to the second one.

The lowest dimensional representations are $$ (0,0),\quad (\frac{1}{2},0),\quad(0,\frac{1}{2}),\quad(\frac{1}{2},\frac{1}{2}) .$$ The $(\frac{1}{2},0)$ representation acts on left-chiral spinors, while the $0,\frac{1}{2})$ representation acts on right-chiral spinors. ($(0,0)$ is the scalar representationa and $(\frac{1}{2},\frac{1}{2})$ is the four-vector representation.)

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  • $\begingroup$ My question in other words was how does irreducibility of lorentz group ensures elementary particle? How can we reconcile this two fact? Can we use the view points of conservation law? $\endgroup$ – Sakh10 Nov 20 '19 at 5:54

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