differential geometry - Covariant derivative for spinor fields

scalars (spin-0) derivatives is expressed as:

$$\nabla_{i} \phi = \frac{\partial \phi}{ \partial x_{i}}.$$

vector (spin-1) derivatives are expressed as:

$$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ \partial x_{i}} + \Gamma^k_{m i} V^m.$$

My Question: What is the expression for covariant derivatives of spinor (spin-1/2) quantities?

Answer

There is an interesting way to look at Christoffel connections with spinor fields. The usual Dirac operator is written as $\gamma^\mu\partial_\mu$. It is interesting to change this to $\partial_\mu(\gamma^\mu\psi)$. This then becomes

$$\partial_\mu(\gamma^\mu\psi)~=~ \gamma^\mu\partial_\mu~+~(\partial_\mu\gamma^\mu)\psi.$$ The anticommutator $\{\gamma^\mu,~\gamma^\nu\}~=~2g^{\mu\nu}$ and the covariant constancy of the metric gives $\partial_\mu\gamma^\mu~=~\Gamma^\mu_{\mu\sigma}\gamma^\sigma$. So we may then write the Dirac operator in this different form as $$\delta_\nu^\mu\partial_\mu(\gamma^\nu\psi)~=~ \delta^\mu_\nu \gamma^\nu\partial_\mu\psi~+~\delta^\mu_\nu \Gamma^\nu_{\mu\sigma}\gamma^\sigma\psi.$$ Now if you peel off the Kronecker delta you have a covariant derivative of the spinor field.

What this means is that in general the Clifford algebra $CL(3,1)$ representation of the Dirac matrices is local. The connection coefficient can then be seen as due to transition functions between these representations, so the differential produces connection coefficients.