Where an expression for the (1+1)-dimensional retarded Dirac propagator in position space can be found, especially including the generalized funcion supported on the light-cone?

In particular, is it true that the square of the absolute value of this distribution is ill-defined (hence cannot be interpreted as charge/probability density)?

An expression for the required propagator, i.e., the retarded fundamental solution of the Dirac equation in $(1+1)$-dimensions
$$
\begin{pmatrix}
-m & i\,\partial/\partial t-i\,\partial/\partial x \\
i\,\partial/\partial t+i\,\partial/\partial x & -m
\end{pmatrix}
\begin{pmatrix}
\psi_1 \\ \psi_2
\end{pmatrix}=0
$$
inside the future light cone $t>|x|$ is (see e.g. Eq.~(13) in this paper)
$$
\frac{m}{2}
\begin{pmatrix}
-\frac{t+x}{\sqrt{t^2-x^2}}\,J_1(m\sqrt{t^2-x^2}) &
i\,J_0(m\sqrt{t^2-x^2}) \\
i\,J_0(m\sqrt{t^2-x^2}) &
\frac{-t+x}{\sqrt{t^2-x^2}}\,J_1(m\sqrt{t^2-x^2})
\end{pmatrix}.
$$
The propagator contains also some generalized function supported on the future light cone $t=|x|$, which particularly interested in. A receipt for computation has been given in this answer, but it seems tricky to identify all the resulting delta-functions and their derivatives (and also the retarded propagator has been confused with the Feynman propagator there). Thus a reference to the final expression would be preferrable.

Notice that searching for *retarded* Dirac propagator (vanishing outside the light cone), not just the 'Dirac propagator' or Feynman propagator (not vanishing outside the light cone).

This post imported from StackExchange Physics at 2019-10-30 18:27 (UTC), posted by SE-user Mikhail Skopenkov