Lemma: Products on the punctured space

For all multi-indices $\alpha ,\beta$, the product ${q^{\circ }}_{\alpha \beta }={\partial }^{\alpha }S\cdot {\partial }^{\beta }S$ exists in ${\mathcal{D}}^{\prime }({\mathbb{R}}^4\setminus \{0\})$, and ${q^{\circ }}_{\alpha \beta }={q^{\circ }}_{\beta \alpha }$.

Proof. As in XVI-K, Lemma lem:prod and Remark rem:transfer: Fredenhagen–Rejzner, arXiv:1208.1428v2, Section 7 (text after (77)), for $x\ne y$ the wavefront set of ${{\Delta }^{\mathrm{FR}}}_F$ has covectors cotangent to a null geodesic with matching time orientation. Pulling back by $\Phi (x,y)=x-y$ and restricting to the difference variable, every covector of $\mathrm{WF}({\Phi }^{\ast }S)$ has the form $(k,-k)$ with $k\ne 0$ (Hörmander, Theorem 8.2.4); the naive claim $k+k^{\prime }\ne 0$ for an unordered pair of Feynman covectors is therefore not the right condition in the difference variable. The transferred criterion (XVI-K, Remark rem:transfer) is that two admissible wavefront covectors of the factors ${\partial }^{\alpha }S$ and ${\partial }^{\beta }S$ at the same difference point $z\ne 0$ cannot sum to zero (same time orientation when both are nonzero). Hence Hörmander's Theorem 8.2.10 applies on ${\mathbb{R}}^4\setminus \{0\}$. By XVI-K, eq:S, $S=-{{\Delta }^{\mathrm{FR}}}_F$; derivatives do not enlarge wavefront sets. Commutativity is a property of that product.