The spinorial Dirac operator

Question

Let $(M,g)$ be a spin manifold. We define the spinors and the spinorial Clifford algebra:

$$\psi . \psi' + \psi' . \psi = 2 < \bar \psi, \psi'>$$

with $\psi,\psi'$ two spinors and

$$(X.\psi).\psi' =\psi .(X. \psi')$$

with $X$ a vector.

$$\psi^{-1}= \frac {\bar \psi}{<\psi,\psi>}$$

We then can define the action of a spinor $\psi$ over a function $f$:

$$\psi (f)=df^* . \psi$$

and a spinorial connection:

$$\nabla_{\psi} (fs)= \psi (f).s + f \nabla_{\psi} (s)$$

with $s$ a section of a module over the spinorial Clifford algebra.

The spinorial Dirac operator is then:

$${\cal D}_{\psi}= \sum_{i \in I} \psi_i . \nabla_{\psi_i}$$

with $(\psi_i)_{i\in I}$ an hermitian basis of the space of spinors.

Has such a construction a physical meaning?