free fermion

The conformally invariant Dirac field $\psi = \bigl( \begin{smallmatrix} \psi_R\\ \psi_L \end{smallmatrix} \bigr)\,$ in $2\,$ -dimensional Minkowski space has action:

$S\,$	$= \int\!d^2x\, \psi^\dagger \gamma^0 \gamma^\mu \partial_\mu \psi\,$ ,
	$= \int\!d^2z\, \psi^\dagger_R \bar\partial \psi_R + \psi_L^\dagger \bar\partial \psi_L\,$ .