# Questions about fundamental solutions and propagators for the Dirac operator

I thought that propagator is a synonym for fundamental solution. But that seems not to be the case since in this answer it is said that an expression with delta function on a surface has to be differentiated to obtain a propagator for Dirac operator. Looks like this propagator doesn't satisfy the equation with Dirac delta function in the rhs.

The primary question: what is the fundamental solution for the Dirac equation and how it is related to the propagator? I'm interested in an explicit formula for the fundamental solution.

Also I've found a propagator of the form $$S_F(x-y)=\left(\frac{\gamma^\mu(x-y)_\mu}{|x-y|^5}+\frac{m}{|x-y|^3}\right)J_1(m|x-y|).$$ But then I put $$y=0$$ and apply the Dirac operator $$i \gamma^\mu \partial_\mu -m$$ to it, the result is not zero. Here $$\gamma^\mu$$ are standard gamma matrices.

Is $$S_F$$ indeed a propagator?

In calculations, in the formula for $$S_F$$ I treated the expression $$\gamma^\mu(x-y)_\mu$$ as matrix $$\gamma^\mu$$, every element of which is multiplied by $$(x-y)_\mu$$ and the term $$m/|x-y|^3$$ was added to all elements of $$4\times4$$ matrix. Is this right or I'm missing something here?