Is there a way to split a Lorentz index into spinor indices?

Question

So, I'm working with Gauge Theory Gravity (the one with Geometric Algebra), and I'm simultanesously translating the equations to conventional tensor calculus notation, but in the process I've stumbled into an interesting problem: there is a displacemente field $\overline{h}$ (basically a vierbein) that acts on multivectors, mapping the "flat-space" vectors into "curved space" vectors (actually the background is always flat, but we can interpret it as curved space formalism), and in the formalism this field of vierbein can act on the Dirac spinor $\psi$. But, in conventional notation, the $\overline{h}$ field only has Lorentz vector indices, $\overline{h}^{\mu} _{\alpha}$, so I need to know if there is a way to convert the Lorentz vector indiced into spinor indices for the $\overline{h}$ field to act on the Dirac spinor of conventional notation as in $\overline{h}^{A B} _{C D} \psi ^{C}$, since this action makes perfect sense in Geometric Algebra.

Answer 1

According to your notation (which is different from the standard Gauge Theory Gravity notation, and assuming $\mu$ is the tangent flat spacetime index), the $\overline{h}$ field is translated into a one-form field: $$ \overline{h} = \overline{h}^{\mu} _{\alpha}\gamma_\mu dx^\alpha $$ where $\gamma_\mu$ has spinor indices for the $\overline{h}$ field to act on the Dirac spinor of conventional notation as in $$ \overline{h}^{\mu} _{\alpha} \gamma^{A}_{\mu C}\psi ^{C}dx^\alpha $$

Answer 2

1. Well, one can convert a flat Lorentz index into a pair of left- and right-handed (undotted and dotted) Weyl spinor indices using the Pauli matrices as intertwiners, cf. e.g. this Phys.SE post.

2. Equivalently, the 4-dimensional vector representation of the Lorentz group is isomorphic to the $(\frac{1}{2},\frac{1}{2})$ representation, which is a tensor product of the left- and right-handed Weyl-spinor representation.

   Returning to OP's question, be aware that it is not isomorphic to the Dirac spinor representation $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$, which instead is a direct sum of the left- and right-handed Weyl-spinor representation, cf. e.g. this Phys.SE post.