# Dirac spinor and Lorentz group

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

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.)