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 $$ S_F(x,y)=\left(\frac{\gamma^\mu(x-y)^\mu}{|x-y|^5}+\frac{m}{|x-y|^3}\right)J_1(m|x-y|). $$$$ S_F(x-y)=\left(\frac{\gamma^\mu(x-y)_\mu}{|x-y|^5}+\frac{m}{|x-y|^3}\right)J_1(m|x-y|). $$ 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$$\gamma^\mu(x-y)_\mu$ as matrix $\gamma^\mu$, every element of which is multiplied by $(x-y)^\mu$$(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?