Proposition: Scope of the constructed objects

Each of the objects

of X-B has the following properties.

1. It is a finite sum of elements $\hat{\varphi }(d)$ (or $F_d$ before passage to the quotient) with $d\in \mathcal{D}$; its field degree is one.

2. The test $d$ depends polynomially on $(h,k)$, of degree at most two in $h$ and one in $k$, and continuously in the smooth background topology at fixed $\epsilon$.

3. No $\hslash$ occurs in $d$. The constant $\hslash$ and the kernel ${\Delta }_{+}$ enter X-B only through the commutator $[\hat{\varphi }(c),\hat{\varphi }(d)]=i\hslash ⟨c,{\Delta }_{S}d⟩$ and through the one-contraction check of Section sec:boundary, in which ${\Delta }_{+}$ cancels.

4. Every two-point kernel that occurs is a scalar Green kernel (${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$, ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$, ${{\mathsf{E}}_0}^{\mathrm{R}}$) applied to a compactly supported smooth source, or ${\Delta }_{+}$ as in (iii). Every background slot is evaluated on a classical configuration; no background slot is contracted against another background slot through a kernel.

5. No functional of $h$ or $k$ alone (in particular no gravitational action) is formed.

Proof. (i)–(ii) for ${\mathcal{O}}_{\kappa }$: the $c_j$ are the frozen tests of X-B, Proposition prop:linear, each in ${\mathcal{D}}_K(M)$, with $c_1$ linear in $h$ and $c_2$ quadratic in $h$ and linear in $k$ through ${\mathsf{X}}_1[h]$ and ${\mathsf{X}}_2[h,k]$; continuity is the handoff statement. For $R_1$: $d_c[h]=-{\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$ is linear in $h$ through the coefficients of (eq:Lone), and ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$ is smooth, so $d_c[h]\in \mathcal{D}$ with support in $\mathrm{supp}g$; continuity is X-B, Theorem thm:retarded. For ${\hat{A}}_1$ and $\mathcal{M}$: sums and compositions of the previous two with $c$ replaced by $f$ or $c_1[h]$; the degree in $h$ of $\mathcal{M}$ is two. For ${\delta }_{\xi }{\hat{A}}_1$: X-B, (eq:defect-class-ref) below is the difference of two such elements. Field degree one is the statement that each object is a class of a functional $F_d$, which has first derivative $d$ and vanishing higher derivatives.

(iii) The tests $d$ are defined by (eq:Rone) and the formulas for $c_j$, none of which contains $\hslash$ or ${\Delta }_{+}$. X-B uses ${\Delta }_{+}$ in the Wick product and in the one-contraction check; Proposition prop:check below shows that the check is independent of ${\Delta }_{+}$.

(iv) By inspection of Definition def:sheet: the kernels ${G_0}^{\mathrm{R}}$ inside ${\mathsf{X}}_1$, ${\mathsf{X}}_2$, and ${\Lambda }_{\epsilon }$ are applied to the classical sources ${\partial }_{\mu }({\sigma }_{\epsilon }{a_1}^{\mu \rho }[h])$ and their second-order analogues; ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ is applied to $c$. The background slots of $c_2$ are evaluated on $(h,h)$ and $k$: two slots of the same kernel are both evaluated classically; neither is integrated against a two-point distribution in the metric variables.

(v) Every object has field degree one in $\phi$; a functional of $h,k$ alone would have degree zero.