= Solution
A two-component <Weyl spinor> is a complex spinor transforming in one of the two inequivalent <group representations> $(1/2,0)$ and $(0,1/2)$ of the <Lorentz spinor double cover> $\mathrm{SL}(2,\mathbb C)$. An undotted left-handed field $\chi_\alpha$ transforms by a matrix $M\in\mathrm{SL}(2,\mathbb C)$; its <complex conjugate> $\bar\chi_{\dot\alpha}$ transforms by $M^*$. Dotted and undotted indices distinguish the two <chirality> representations. Raising a dotted index uses the invariant epsilon tensor and converts the <complex conjugate> transformation into the equivalent inverse-adjoint form. Rotations act by the same <SU(2)> generators on the two chiralities, while boosts have opposite generators, so the two complex representations cannot be identified by a complex-linear <Lorentz invariant> map.
The antisymmetric <invariant tensors> $\epsilon_{\alpha\beta}$ and $\epsilon_{\dot\alpha\dot\beta}$ raise and lower spinor indices; $M^T\epsilon M=\epsilon$ proves invariance of the contracted bilinear $\chi\psi=\chi^\alpha\psi_\alpha$. The components of a fermionic <Weyl spinor> are odd <Grassmann variables>, so this bilinear is symmetric under exchange of the two spinors: the antisymmetric epsilon sign cancels the Grassmann sign. In particular, choosing $\epsilon_{12}=1$ and $\epsilon^{12}=-1$ gives $\chi\chi=2\chi_1\chi_2$, generally nonzero. This <Weyl spinor bilinear exchange identity> permits a <Majorana mass term> for a single suitable <Weyl spinor>. With mostly-minus <metric signature> and $\bar\sigma^\mu=(I,-\boldsymbol\sigma)$, the free massless kinetic term is $i\chi^\dagger\bar\sigma^\mu\partial_\mu\chi$, and its field <equation of motion> is $\boxed{\bar\sigma^\mu\partial_\mu\chi=0}$.
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