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A Dirac spinor transforms as $(1/2,0)\oplus (0,1/2)$, so it combines the two Weyl chiralities into a parity-invariant representation.

A coordinate transformation $x\mapsto Px$ conjugates an infinitesimal generator, hence $M^{\mu \nu }\mapsto P{M}^{\mu \nu }P$. It fixes the rotation generators $J_i$ and negates the boosts $K_i$. Therefore it exchanges $A_i$ and $B_i$, and the parity action on a Lorentz representation isAn irreducible representation is parity invariant precisely when $j_L=j_R$. If $j_L\ne j_R$, parity invariance requires the reducible sum $(j_L,j_R)\oplus (j_R,j_L)$. Thus the two Weyl-spinor representations $(1/2,0)$ and $(0,1/2)$ are exchanged, while their direct sum is the parity-invariant Dirac spinor.

Solved by gpt-5.6-sol high.

In the chiral coordinate $y^{\mu }=x^{\mu }+i\bar{\theta }{\bar{\sigma }}^{\mu }\theta$, a chiral superfield isHere $\varphi$ is a complex scalar, ${\psi }_{\alpha }$ is a two-component Weyl spinor, and $F$ is a complex auxiliary field. Off shell, $\varphi$ and $F$ contribute two real bosonic components each, while $\psi$ has two complex Grassmann components. Thus there are four real bosonic and four real fermionic degrees of freedom.

Solved by gpt-5.6-sol high.