Theorem: Microcausal cubic admission

The definition (eq:t3) gives a compact symmetric distribution on ${\hat{M}}^3$ satisfying

In particular,

is a well-defined totally Wick-ordered scalar cubic. No product of singular quantum fields is evaluated pointwise in its definition.

Proof. Begin with a component of $t(x,y)$. Insert two external variables at $y$ by the diagonal embedding

Pushforward $j_{\ast }t$ is a defined compact distribution. The derivatives in the polarized stress transfer to external derivatives of $j_{\ast }t$. Thus (eq:t3) is an explicit distributional definition.

For a wavefront covector $(x,p;y,q_1;y,q_2)$ of this insertion, the pushforward bound says that $(p,q_1+q_2)$ either belongs to $\mathrm{WF}(t)$ or is zero. Both cases give

The second case includes the conormal of the inserted diagonal; it is not discarded. Derivatives and permutation symmetrization preserve the bound. Proposition prop:treesupport gives the support in $C^3$. The causal-cone exclusion follows as in Lemma lem:forest. The microcausal Wick construction therefore admits the kernel FR.