Real Weyl Spinors

Real Weyl Spinors can be thought of as \(j=\frac{1}{2}\) representations of SU\((2)_{L/R}\), which come together to form Spin\((4)\simeq\)SU\((2)_L\times\)SU\((2)_R\). They differ from a representation of plain SU(2) in that, they MUST come together to form Spin(4). So really, they are representations for \((j_L,j_R)=(\frac{1}{2},0)\) and \( (0,\frac{1}{2})\) of Spin(4), respectively.  Left-handed spinors are defined first. They have no bar, with indices contracting up to down \(\psi\chi\equiv\psi^\alpha\chi_\alpha\equiv\epsilon^{\alpha\beta}\psi_\beta\chi_\alpha \). Right-handed spinors came late, so their index-free notations have bars, their indices have dots and contract down to up \(\bar{\psi}\bar{\chi}\equiv\bar{\psi}_{\dot{\alpha}}\bar{\chi}^{\dot{\alpha}}\equiv\epsilon_{\dot{\alpha}\dot{\beta}}\psi^{\dot{\beta}}\chi^{\dot{\alpha}} \).

Unfortunately, the spinor metric
$$\epsilon^{\alpha\beta}=\left(
\begin{array}{cc}
0 & 1 \\
-1 & 0 \\
\end{array}
\right),$$
is anti-symmetric, thus \(\epsilon^{\alpha\beta}\psi_\beta\chi_\alpha=-\epsilon^{\alpha\beta}\psi_\alpha\chi_\beta\). Therefore we have to require the real Weyl spinor components to be Grassmann variables which anti-commute \(\psi_\beta\chi_\alpha\equiv-\psi_\alpha\chi_\beta\). Luckily, the above definitions work out such that the real Weyl Spinors commute in index-free notion.

Complex Weyl Spinors

Stick two real Weyl spinors (belonging to the same group, duh) together with an i, and you have got yourself a complex Weyl spinor. Going along with the horrendous abuse of notation, we will call it \(\chi=\theta+i\rho\), where \(\theta\) and \(\rho\) are left-handed real Weyl spinors. Adding to the notation nightmare, let’s also define \(\chi^\dagger\equiv\theta-i\rho\), which has nothing to do with conjugate transpose. It does serve as a memoto that \(\chi^\dagger\) is rand-handed.