For the covariant spinor derivative we need to introduce a connection which can parallel transport a spinor. Such a connection takes values in the Lie-algebra of the group the spinor transforms under. Then we have:

$$ D_i \psi = \partial_i \psi + g A_i^I T_I \psi $$

Here $T_I$ are the generators of the lie-algebra and are matrix valued. We have suppressed spinorial indices. Writing them out explicitly we get:

$$ D_i \psi_a = \partial_i \psi_a + g A_{i\,I} T^I{}_a{}^b \psi_b $$

For eg, for $SU(2)$ the lie-algebra generators are given by the three pauli matrices $\sigma_x,\sigma_y, \sigma_z$ which then act on two component spinors. If you wish to work with four-component spinors $\psi_A$, transforming under the Lorentz group, the relevant generators are those of $SO(3,1)$. You can find these in Peskin and Schroeder, page 41.

There are relations between the spin connection, the christoffel connection and the metric but this is the definition of the spin connection.

This post imported from StackExchange Physics at 2014-04-01 16:35 (UCT), posted by SE-user user346