Theorem: Non-extension of the one-contraction check

1. The expression ${\mathcal{C}}_{\mathrm{F}}(V,F_c)$ is defined by the sheet for every pair (A1)–(A2), and Propositions prop:check–prop:causal are the restrictions of (T1)–(T3) to this pair; the resulting retarded expression is (eq:Rone).

2. For the pair $(V,V)$ with $V$ as in (A1) and $\mathsf{L}\ne 0$, the two-contraction term that (T3) would prescribe, $\frac{{\hslash }^2}{2}⟨V^{\prime \prime },{{\Delta }_{\mathrm{F}}}^{\otimes 2}{V}^{\prime \prime }⟩$, contains the pointwise product ${\Delta }_{\mathrm{F}}\cdot {\Delta }_{\mathrm{F}}$ of the kernel with itself on $M\times M$. This product is not defined by the wavefront product criterion: at every point of the diagonal, $\mathrm{WF}({\Delta }_{\mathrm{F}})$ contains both $(x,x;\xi ,-\xi )$ and $(x,x;-\xi ,\xi )$, whose sum is zero.

3. By contrast the corresponding Wick term $\frac{{\hslash }^2}{2}⟨V^{\prime \prime },{\Delta }_{+}^{\otimes 2}{V}^{\prime \prime }⟩$ is defined by the same criterion, consistently with the definition of ${\mathcal{A}}_{\mathrm{off}}$.

4. Consequently the sheet of X-B defines no $T_2(V,V)$, no $R(V;V)$, and no $T_n$ for $n\ge 2$ on general polynomial pairs. The one-contraction check is an identity for the pair (A1)–(A2) and is not a family of time-ordered maps. Passing the check for this pair does not pass the requirements (T1)–(T4) for any other pair.

Proof. (i) is Propositions prop:check and prop:causal.

(ii) The kernel of $V^{\prime \prime }=-\mathsf{L}$ is $-{\mathsf{L}}_x{\delta }_{\Delta }(x,y)$. In $⟨V^{\prime \prime },{{\Delta }_{\mathrm{F}}}^{\otimes 2}{V}^{\prime \prime }⟩=\int V^{\prime \prime }(x_1,x_2){\Delta }_{\mathrm{F}}(x_1,y_1){\Delta }_{\mathrm{F}}(x_2,y_2)V^{\prime \prime }(y_1,y_2)$, integrating the first delta kernel against $x_2$ identifies the first slots of the two ${\Delta }_{\mathrm{F}}$ factors (with $\mathsf{L}$ acting on one of them), and integrating the second delta kernel against $y_2$ identifies their second slots. The result is the pairing of the compactly supported coefficient distribution of $\mathsf{L}\otimes \mathsf{L}$ against the restriction to $\{x_1=x_2,y_1=y_2\}$ of ${\Delta }_{\mathrm{F}}(x_1,y_1){\Delta }_{\mathrm{F}}(x_2,y_2)$, that is, against a derivative of the pointwise product ${\Delta }_{\mathrm{F}}\cdot {\Delta }_{\mathrm{F}}$ on $M\times M$. The sufficient condition for the existence of a product $uv$ of distributions is that there be no $(z,\zeta )\in \mathrm{WF}(u)$ with $(z,-\zeta )\in \mathrm{WF}(v)$ Hormander (Thm. 8.2.10). With $u=v={\Delta }_{\mathrm{F}}$ and $z=(x,x)$, $\zeta =(\xi ,-\xi )$, Lemma lem:DF(iv) gives both $(z,\zeta )$ and $(z,-\zeta )$ in $\mathrm{WF}({\Delta }_{\mathrm{F}})$. The criterion fails at every diagonal point.

(iii) Every element of $\mathrm{WF}({\Delta }_{+})$ has a nonzero first covector in one fixed closed half of the null cone (Premise prem:free). If $(z,\zeta )$ and $(z,-\zeta )$ were both in $\mathrm{WF}({\Delta }_{+})$, the first components $\xi$ and $-\xi$ would both lie in that closed half cone, forcing $\xi =0$, which is excluded. Hence the criterion holds and ${\Delta }_{+}\cdot {\Delta }_{+}$ is defined Hormander (Thm. 8.2.10).

(iv) A map $T_2$ satisfying (T3) on $(V,V)$ requires a distribution extending ${\Delta }_{\mathrm{F}}\cdot {\Delta }_{\mathrm{F}}$ across the diagonal; the sheet contains no such extension and no normalization condition (T4) selecting one. The existence of such extensions and their freedom is the renormalization problem of BF, which is not opened here. Without $T_2(V,V)$ there is no $R(V;V)$; without $T_2$ on general pairs there is no family. The last sentence is the content of (i)–(iii): the pair (A1)–(A2) never meets the undefined product because ${F_c}^{(2)}=0$.