A named classical symbol space

Let ${\mathcal{P}}_{\mathrm{mc}}$ be the space of finite polynomial functionals on smooth configurations whose compact distributional derivative kernels satisfy

Here ${\tilde{V}}_{\pm }$ are the closed causal covector cones with zero covectors included; the total zero covector is omitted from a wavefront set. Field indices are finite dimensional and do not change the criterion.

This is a classical symbol declaration. No Hadamard product, locally convex quantum completion, or inclusion into ${\mathcal{A}}_0$ is contained in the definition.

Let ${G_0}^{\mathrm{R}}(x-z)$ be the retarded wave kernel and define

Integration by parts, with the flat ramp removing the initial boundary term, gives

All integrals are restricted to the slab where necessary.

The bare insertion is

Ordering the variables as scalar slot $x$ and metric slot $z$, its kernel is