It is said that in 1+1 dimension, if we take $\gamma^0=i\sigma^2$ and $\gamma^1=\sigma^1$, then the two components of dirac spinor $\psi_L$(upper component) and $\psi_R$(lower component) decouple in the Dirac equation $(i\gamma^u\partial_u-m)\psi=0$ But, the Pauli matrices $\sigma^1$ and $\sigma^2$ are off-diagonal matrices. So the components of the spinor $\psi$ clearly mixes together in the equation. So what does it mean by the components decouple?
Also I think that it is wrong to choose $\gamma^0$ and $\gamma^1$ like above. In order to satisfy the Clifford algebra, I think it must be that $\gamma^0=\sigma^2$ and $\gamma^1=i\sigma^1$. Is this correct?