haushofer said:
I'm also used to that last interpretation in the context of supergravity,
Where in supergravity do you find such bi-spinor interpretation?
e.g. in the susy-transfo of vielbeine.
Under local supersymmetry, the frame field transforms as
\delta_{\epsilon}e_{\mu}{}^{a} = \frac{1}{2}\left( \epsilon^{\beta} \ (\sigma^{a})_{\beta \dot{\alpha}} \ \bar{\psi}_{\mu}^{\dot{\alpha}} + \bar{\epsilon}_{\dot{\beta}} \ (\bar{\sigma}^{a})^{\alpha \dot{\beta}} \ \psi_{\alpha \mu} \right) , where \psi_{\alpha \mu} \in (\frac{1}{2} , 1) and \bar{\psi}_{\mu}^{\dot{\alpha}} \in (1 , \frac{1}{2}), are Weyl
spinor-vectors, i.e., Lorentz vectors taking values in the 2-dimenstional spin space \mathbb{C}^{2}. Introducing the Majorana
bispinor-vector (the superpartner of the gravitational field or the gravitino field) \Psi_{\mu} = \begin{pmatrix} \psi_{\mu \alpha} \\ \bar{\psi}^{\dot{\alpha}}_{\mu} \end{pmatrix} \in \left(\frac{1}{2} , 1 \right) \oplus \left(1 , \frac{1}{2} \right) , the Majorana
bispinor \bar{\Upsilon} = \left( \epsilon^{\beta} , \bar{\epsilon}_{\dot{\beta}}\right), and the 4 \times 4 Dirac matrices \gamma^{a} = \begin{pmatrix} 0 & (\sigma^{a})_{\beta \dot{\alpha}} \\ (\bar{\sigma}^{a})^{\dot{\beta}\alpha} & 0 \end{pmatrix} , we can write \delta_{\epsilon}e_{\mu}{}^{a} = \frac{1}{2} \bar{\Upsilon} \gamma^{a} \Psi_{\mu} . The frame field can be used to convert the world index (\mu) on the gravitinos field to a tangent-space index (a): \psi_{\alpha}^{a} = e_{\mu}{}^{a} \psi_{\alpha}^{\mu}, and the local Lorentz index can be converted into a pair of spinor indices: \psi_{\alpha}{}^{\dot{\beta} \beta} = (\bar{\sigma}^{a})^{\dot{\beta}\beta} \psi_{\alpha}^{a}. Thus, the gravitino can be described by a Majorana
bispinor-vector field \Psi_{\mu}(x) or, equivalently, by a pair of (mixed) rank-
3 spin tensor fields.