Remark: The source is retained

The triple $({\gamma }_{0}h,{\gamma }_{1}h,j)$ is a presentation of the same off-shell field, not a new collection of propagating fields. In particular, $j$ is not set to zero.

For linear source smearings, the bisolution identities imply

Thus such source variables are central for the free Wick product. This does not make them zero classical functionals, and it does not by itself define products involving arbitrary singular source-sector kernels.