Proposition: A well-defined self-product kernel

For $f,f^{\prime }\in {C_c}^{\infty }(M_{+})$, the following compact distributional polynomial is well defined:

It is the Wick contraction formula for the proposed $B_f{\star }_H{B}_{f^{\prime }}$. The potential ${\hslash }^2$ coefficient vanishes.

Proof. For fixed smooth $h$, the factor multiplying the differentiated Hadamard distribution in (eq:self) is a smooth compact test function. Hence the pairing exists.

The map from $h$ to $w[h]$, restricted to any fixed compact observation region, is continuous in the smooth topology and depends only on a fixed compact causal region. Polarization of (eq:self) therefore gives a continuous bilinear form on the corresponding smooth graviton configurations. The Schwartz kernel theorem gives a compact distributional quadratic kernel.

There are no $h$–$\varphi$ contractions. Every term involving a $w$–$w$ contraction vanishes by Proposition prop:tensor. This leaves exactly (eq:self).