Landmark X-F The second-order quantum metric as the retarded M ller response, and the closure of the κ 2 background identity

We complete the ${\hslash }^0$, ${\kappa }^2$ sector of Landmark X-E for the ramp relational scalar, in signature $(+,-,-,-)$, at fixed regulator width. Two items of the X-E sheet are replaced. The gauge fixing of the quantum graviton is the de Donder condition of the classical background metric $\bar{q}=\eta +\kappa h_{\mathrm{cl}}$, with Feynman weight. The second-order quantum metric is defined, not postulated: it is the complete first-order retarded Møller response of the gauge-fixed fluctuation, so that the single tree of X-E becomes three, the matter tree of X-E, a mixed tree bilinear in $(h_{\mathrm{cl}},\hat{h})$, and a self-interaction tree quadratic in $\hat{h}$. No new field is introduced and no ghost is needed.

The mechanism is a commutator identity. Naturality of the Einstein–Hilbert density and of the background gauge-fixing density gives, for the kinetic operator of the fluctuation $\mathcal{Q}[\bar{q}]=P_{\mathrm{gr}}+\kappa {\mathcal{Q}}^{(1)}[h_{\mathrm{cl}}]+O({\kappa }^2)$,

Since ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{P}_{\mathrm{gr}}u=u$ for compactly supported $u$, the retarded response of the fluctuation to a pure-gauge background is ${\mathcal{L}}_{\xi }\hat{h}$ modulo the equation ideal: the transformation law of the field X-E named and did not construct is a theorem about the Møller map. The mixed tree therefore varies by $+{\kappa }^2{\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]$ and cancels X-E's defect exactly; the matter and self trees are inert. The all-tree observable is invariant under the frozen algebra, acting on the classical backgrounds with the quantum fields held fixed, through ${\kappa }^2$ modulo the equation ideal.

Along the way: the exact second-order covariance identity $2{\mathcal{G}}_2(K\xi ,h)={\mathcal{L}}_{\xi }{G}_{\mathrm{lin}}[h]-G_{\mathrm{lin}}[{\mathcal{L}}_{\xi }h]$, which corrects an on-shell shortcut in X-E; the switching of the gravitational self-interaction by the ramp, without which the self tree is undefined; a three-slot admission lemma by a sum-zero argument; and the flat-gauge residual ${\kappa }^2{\hat{A}}_1[{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}[{\mathcal{L}}_{\xi },{\mathcal{F}}^{(0)}]\hat{h}]$, ${\mathcal{F}}^{(0)}=-\frac{1}{4}\mathcal{P}KC$, displayed and not decided. The one-loop terms, ghosts, the master equation with switched matter, positivity, and the regulator limit are named and not constructed.

Figure fig:split-X-F records one view of the same separation. Figure fig:chain-X-F records one view of the same separation. Figure fig:pair-X-F records one view of the same separation. Figure fig:gate-X-F records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-X-F records that separation. The drawing adds no hypothesis.

{Release: Constructed, at fixed regulator and fixed input, through ${\kappa }^2{\hslash }^0$}{ Frozen: X-E's sheet with the background de Donder gauge $\frac{1}{4}\int \sqrt{-\bar{q}}{\bar{q}}_{\mu \nu }{C^{\mu }}_{\bar{q}}{C^{\nu }}_{\bar{q}}$, the background-field prescription, the ramp switch on the gravitational order-$\kappa$ densities, and the Møller-defined response $\hat{k}={{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}\mathsf{S}$.

Proved. F1: the exact second-order covariance identity (eq:cov2). F2: ${\mathcal{F}}^{(0)}=-\frac{1}{4}\mathcal{P}KC$, $P_{\mathrm{gr}}=\frac{1}{2}{G}_{\mathrm{lin}}+{\mathcal{F}}^{(0)}$, the commutator identity (eq:key), and the dynamical jet law (eq:jetlaw). F3: three trees, Definition def:trees; kernels, supports, admission (Lemmas lem:kernels, lem:sumzero, lem:admission), classical values. F4: Theorem thm:main, the background identity through ${\kappa }^2$ modulo $\mathfrak{J}$, quantum fields fixed. F5: the flat-gauge residual, Proposition prop:flat; its class is not decided. F6: VII-C inapplicable, not contradicted.

Not opened: ${\hslash }^1$, ghosts, BRST/BV with switched matter, positivity, the classical tadpole, gauge-parameter dependence, $\epsilon \to 0$, X-Enter, all orders in $\kappa$.}

Three endings are available:

1. Constructed. The gauge, the response, and all ${\kappa }^2{\hslash }^0$ trees are named, admitted, and the all-tree observable transforms correctly on the frozen algebra through ${\kappa }^2$.

2. Proved obstructed. The all-tree identity acquires a nonzero defect at ${\kappa }^2$, or a tree kernel leaves the declared algebra.

3. Unresolved. A tree is unnamed or its admission is not proved.

The ending reached is (1). The defect of X-E is shown to be a truncation defect of the one-tree observable in the flat gauge, not a field-content obstruction.

A missing object is recorded as missing, not as a no-go. Nothing in this manuscript depends on a state for the graviton.

As in X-E XE:

Indices are moved with $\eta$ unless $q$ or $\bar{q}$ is written. ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ are the Klein–Gordon Green operators, ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}}$ the massless ones. Fix $f\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3;\mathbb{R})$, $g\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3;[0,1])$, a relatively compact plateau $U$ with $\overline{U}\subset P_g=\mathrm{int}\{g=1\}$, ${\mathfrak{g}}_U={C_c}^{\infty }(U;TM)$, and $\delta >0$ with

Fix $0<\epsilon <\delta$. The ramp ${\sigma }_{\epsilon }(t)=\chi (t/\epsilon )$ is X-D's: smooth, ${\sigma }_{\epsilon }=0$ on $t\le 0$, ${\sigma }_{\epsilon }=1$ on $t\ge \epsilon$. In particular ${\sigma }_{\epsilon }=1$ on a neighbourhood of $\{t\ge 2\delta \}\supset \overline{U}\cup \mathrm{supp}g\cup K_f$. ${\mathcal{G}}_U$ is the group generated formally by $\kappa \xi +{\kappa }^2\zeta$, $\xi ,\zeta \in {\mathfrak{g}}_U$. For a vector field $\xi$, $F_s$ denotes its flow and $K\xi :={\mathcal{L}}_{\xi }\eta$, $(K\xi {)}_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$.

The following are copied from X-A to X-E and not re-proved. Equation numbers in brackets refer to X-E.

• Density coefficients $a,b,A,B$ [(2.2)], ramp coordinates ${\mathsf{X}}_1,{\mathsf{X}}_2$ [(2.3)], Euler coefficients ${\mathsf{L}}_{1,g},{\mathsf{L}}_{2,g}$, completed tests $d_1,d_2$ [(2.5)], ${\hat{A}}_1[h]=\hat{\varphi }(d_1[h])$, ${\hat{A}}_2=\hat{\varphi }(d_2)$, $\hat{Q}[h,h]=:\hat{\varphi }(Q[h,h]):$, and the affine-slot lemma $d_2[h,k]=Q[h,h]+d_1[k]$, $d_1$ linear [Lemma 2.1]. The kernel $D_1$ of $d_1$ [(4.8)] is a compactly supported distribution on $M\times M$ with $\mathrm{supp}{D}_1\subset K_f\times (J^{-}(K_f)\cap \{t\ge 0\})$.

• With $\mathcal{E}[q]:=\sqrt{-q}{G}^{\mu \nu }[q]$,

[(4.6)], ${\mathcal{G}}_2(\cdot ,\cdot )$ its symmetric bilinear polarization; $S_{\mathrm{gr}}=-(2{\kappa }^2{)}^{-1}\int \sqrt{-q}R$, $\delta S_{\mathrm{gr}}/\delta h=\frac{1}{2\kappa }\mathcal{E}$ [Lemma 4.1]; $S_{\mathrm{gr}}=S_{\mathrm{FP}}[h]+\kappa (2S_{\mathrm{FP}}(h,k)+S_3[h])+O({\kappa }^2)$, $\delta S_{\mathrm{FP}}/\delta h=\frac{1}{2}{G}_{\mathrm{lin}}[h]$, $\delta S_3/\delta h=\frac{1}{2}{\mathcal{G}}_2[h,h]$ [Lemma 6.1, Sec. 4.6]; $G_{\mathrm{lin}}[K\xi ]=0$; the vertex identity $V_{1,g}[h,\varphi ]=-\frac{1}{2}\int g{h}_{\mu \nu }{T}^{\mu \nu }[\varphi ]$, $\delta V_{1,g}/\delta h=-\frac{1}{2}gT[\varphi ]$, $\delta V_{1,g}/\delta \varphi =-{\mathsf{L}}_{1,g}[h]\varphi$ [Lemma 4.2].

• $\mathcal{P}u=u-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }u$, ${\mathcal{P}}^2=1$; $C^{\mu }[u]={\partial }_{\nu }(\mathcal{P}u{)}^{\nu \mu }$; ${S^{(0)}}_{\mathrm{gf}}[u]=\frac{1}{4}\int C_{\mu }{C}^{\mu }$;

[Lemma 6.3]. Here $(KC{)}_{\mu \nu }={\partial }_{\mu }{C}_{\nu }+{\partial }_{\nu }{C}_{\mu }$.

• The multiplet Wick algebra $\mathfrak{W}$ of $(\hat{\varphi },\hat{h})$ with Hadamard matrix of zero cross blocks, ${\mathsf{H}}_{\mathrm{gr}}=-4\mathcal{P}\otimes {\mathsf{H}}_0$, and the closed two-sided equation ideal $\mathfrak{J}$ generated by $\hat{\varphi }(P_{m}v)$, $\hat{h}(P_{\mathrm{gr}}w)$ [Sec. 6.4]. The admission criterion of X-B XB,BF: a Wick monomial of degree $n$ in the multiplet belongs to $\mathfrak{W}$ iff its kernel $u\in {\mathcal{D}}^{\prime }(M^n)$ is compactly supported and

${\overline{V}}_{\pm }$ the closed future/past covector cones including zero. The wavefront bound $\mathrm{WF}({{\mathsf{E}}_0}^{\mathrm{R}})\subset \{(x,\xi ;y,-\xi ):\xi \ne 0\}$ [Lemma 4.13, DH], and the facts that elements of $\mathfrak{J}$ vanish on all pairs of smooth free solutions [Lemma 4.17(ii)].

• Theorem 4.18 of X-E: with $h=h_{\mathrm{cl}}+\hat{h}$, under $h_{\mathrm{cl}}^{\prime }=h_{\mathrm{cl}}+K\xi$, $k_{\mathrm{cl}}^{\prime }=k_{\mathrm{cl}}+{\mathcal{L}}_{\xi }{h}_{\mathrm{cl}}+K\zeta +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$, $\hat{\varphi },\hat{h}$ fixed,

Proposition 4.21 of X-E: for $m=0$, \(=

ial_1\), $h=x^1{\delta }^2\otimes {\delta }^2$, ${\varphi }_0=t{x}^1$, $⟨d_1[{\mathcal{L}}_{\xi }h],{\varphi }_0⟩=2\int f{\partial }_1{\partial }_t({{\mathsf{E}}_0}^{\mathrm{R}}{)}^2\chi \ne 0$ for suitable $f,g,\delta$.

• Lemma 5.1 and Proposition 5.2 of X-E: for a diffeomorphism $F$ equal to the identity outside a compact set, a natural Lagrangian density integrates to the same value before and after pull-back, and its Euler–Lagrange tensor density is natural, $E[F^{\ast }\text{fields}]=F^{\ast }E[\text{fields}]$.

${\mathcal{L}}_{\xi }$ acts on covariant symmetric two-tensors by $({\mathcal{L}}_{\xi }u{)}_{\mu \nu }={\xi }^{\lambda }{\partial }_{\lambda }{u}_{\mu \nu }+u_{\lambda \nu }{\partial }_{\mu }{\xi }^{\lambda }+u_{\mu \lambda }{\partial }_{\nu }{\xi }^{\lambda }$ and on contravariant symmetric tensor densities of weight one by ${\mathcal{L}}_{\xi }{T}^{\mu \nu }={\partial }_{\lambda }({\xi }^{\lambda }{T}^{\mu \nu })-T^{\lambda \nu }{\partial }_{\lambda }{\xi }^{\mu }-T^{\mu \lambda }{\partial }_{\lambda }{\xi }^{\nu }$. For the flow $F_s$ of $\xi$, $\frac{d}{ds}{F}_s^{\ast }T=F_s^{\ast }{\mathcal{L}}_{\xi }T$ for every tensor density $T$. Metrics and fluctuations are covariant tensors; Euler derivatives with respect to them, in particular $\mathcal{E}$, $G_{\mathrm{lin}}$, ${\mathcal{G}}_2$, $P_{\mathrm{gr}}u$, and the gauge-fixing derivative $\mathcal{F}$ below, are contravariant tensor densities. When ${\mathcal{L}}_{\xi }$ is written on either side of such an operator, it acts in the type of its argument. The two-jet law of xe:ward is the expansion $F^{\ast }(\eta +\kappa h_{\mathrm{cl}}+{\kappa }^2{k}_{\mathrm{cl}})=\eta +\kappa h_{\mathrm{cl}}^{\prime }+{\kappa }^2{k}_{\mathrm{cl}}^{\prime }+O({\kappa }^3)$ for $F=\exp (\kappa \xi +{\kappa }^2\zeta )$.

For a Lorentzian metric $\bar{q}$ with Levi-Civita connection $\bar{\nabla }$ and a smooth symmetric $u$,

and $\mathcal{F}[u;\bar{q}]:=\delta S_{\mathrm{gf}}[u;\bar{q}]/\delta u_{\mu \nu }$ its Euler derivative with respect to the fluctuation, a contravariant symmetric tensor density, defined through compactly supported variations of $u$. At $\bar{q}=\eta$, ${C^{\mu }}_{\eta }=C^{\mu }$ and $S_{\mathrm{gf}}[u;\eta ]={S^{(0)}}_{\mathrm{gf}}[u]$ of xe:gf. For $\bar{q}=\eta +\kappa h_{\mathrm{cl}}$ write

with ${\mathcal{F}}^{(1)}[w]u$ linear in $w$ and in $u$.

The gauge condition is imposed on the quantum fluctuation only; the classical backgrounds remain unconstrained. The dependence of the gauge fixing on $k_{\mathrm{cl}}$ begins at order ${\kappa }^2$ in $\mathcal{F}$ and enters the observable below at order ${\kappa }^3$; it is dropped.

with $\hat{\varphi }$ the free Klein–Gordon field, $\hat{h}$ the free flat gauge-fixed graviton of xe:algebra, $h_{\mathrm{cl}},k_{\mathrm{cl}}$ smooth classical off-shell backgrounds, and $\hat{k}$ defined as follows.

The kinetic operator of the fluctuation at background $\bar{q}$ is the Euler derivative of the part of $S_{\mathrm{gr}}[\bar{q}+\kappa u]+S_{\mathrm{gf}}[u;\bar{q}]$ quadratic in $u$,

where the expansion and the identification ${\mathcal{Q}}^{(1)}[h_{\mathrm{cl}}]u={\mathcal{G}}_2(h_{\mathrm{cl}},u)+{\mathcal{F}}^{(1)}[h_{\mathrm{cl}}]u$ are Lemma lem:fluct.

The switched source is

and the second-order quantum metric is its retarded response,

Two prescriptions are part of the definition. First, the background-field prescription: the terms of the fluctuation equation that do not depend on $u$, the Einstein tensor density of the off-shell background, are dropped (Lemma lem:fluct, Remark rem:tadpole). Second, the switch: the order-$\kappa$ gravitational and gauge-fixing sources are multiplied by X-D's ramp ${\sigma }_{\epsilon }$, so that $\mathsf{S}$ is past-compact and ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}\mathsf{S}$ is defined by the retarded propagator alone, in the same place where X-A's coordinate equation $\square {{\mathsf{X}}_1}^{\rho }=-{\partial }_{\mu }({\sigma }_{\epsilon }{a}^{\mu \rho })$ multiplies its source. The switch acts on the source, not on a density; the two differ by terms supported in the layer $0<t<\epsilon$, which are not used.

{}1.12

No state for the graviton is an input. The graviton Hadamard form ${\mathsf{H}}_{\mathrm{gr}}$ enters only the definition of Wick ordering of the graviton-quadratic monomials $\hat{Q}[\hat{h},\hat{h}]$ and ${\mathsf{T}}_{\mathrm{self}}$; no computed quantity, no classical value, and no step of Theorem thm:main depends on it.

The switch on the gravitational source is a frozen input. It is not derived from a switched action, and no claim is made about the layer terms by which a switched density would differ from it.

VII-C VIIC is audited in Section sec:F6. Its assumptions are copied, not inferred. X-D's convergence theorem is not used. The regulator is not removed.

Let $F$ be a diffeomorphism of $M$ equal to the identity outside a compact set. For all smooth Lorentzian $q,\bar{q}$ and smooth symmetric $u$,

as contravariant symmetric tensor densities.

Proof. $\sqrt{-q}R[q]$ and ${\mathfrak{L}}_{\mathrm{gf}}[u;\bar{q}]=\frac{1}{4}\sqrt{-\bar{q}}{\bar{q}}_{\mu \nu }{C^{\mu }}_{\bar{q}}[u]{C^{\nu }}_{\bar{q}}[u]$ are natural scalar densities: they are built from their arguments by tensor operations and the Levi-Civita connection of $\bar{q}$, so $\omega [F^{\ast }\text{fields}]=F^{\ast }\omega [\text{fields}]$. The argument of X-E's Proposition Covariance of the Euler operators applies verbatim: differentiate $\int \omega [F^{\ast }(\text{fields}+s\delta )]=\int \omega [\text{fields}+s\delta ]$ at $s=0$ for compactly supported variations $\delta$ of $q$, respectively of $u$, change variables by $F$ on the left, and use that the pairing of a tensor density with a tensor is invariant. The third identity follows from the first two by differentiating the first along $q=\bar{q}+su$ at $s=0$, since $F^{\ast }$ is linear.

For every smooth vector field $\xi$ and smooth symmetric $u$,

where ${\mathcal{L}}_{\xi }$ acts on $G_{\mathrm{lin}}[u]$ as on a contravariant tensor density and on $u$ as on a covariant tensor.

Proof. Both sides of (eq:cov2) are bidifferential expressions in $(\xi ,u)$ with constant coefficients, so the identity at a point $x$ involves only the germs of $\xi$ and $u$ at $x$; it suffices to prove it for compactly supported $\xi$. Let $F_s$ be the flow of such a $\xi$; $F_s$ is the identity outside a compact set, $F_s^{\ast }\eta =\eta +sK\xi +O(s^2)$, and $\frac{d}{ds}{F}_s^{\ast }T{|}_{s=0}={\mathcal{L}}_{\xi }T$ for every tensor density. By Lemma lem:natural with $q=\eta +\lambda u$,

Expand both sides by xe:einstein with $\kappa h=sK\xi +\lambda u+O(s^2)$, and ${\kappa }^{2}k$ collecting the $O(s^2)$ remainder of $F_s^{\ast }\eta$ and the $s\lambda$ term:

The $s{\lambda }^0$ coefficient gives $G_{\mathrm{lin}}[K\xi ]=0$. The $s\lambda$ coefficient gives $G_{\mathrm{lin}}[{\mathcal{L}}_{\xi }u]+2{\mathcal{G}}_2(K\xi ,u)={\mathcal{L}}_{\xi }{G}_{\mathrm{lin}}[u]$.

On $G_{\mathrm{lin}}[u]=0$, (eq:cov2) is the formula $G_{\mathrm{lin}}[{\mathcal{L}}_{\xi }u]=-2{\mathcal{G}}_2(K\xi ,u)$ quoted in X-E's closing remark of E4. Off it the term ${\mathcal{L}}_{\xi }{G}_{\mathrm{lin}}[u]$ is present. For the free gauge-fixed graviton, $P_{\mathrm{gr}}\hat{h}=0$ gives $G_{\mathrm{lin}}[\hat{h}]=\frac{1}{2}\mathcal{P}KC[\hat{h}]$ by (eq:xe-gf), which does not vanish unless $C[\hat{h}]=0$; the shortcut therefore does not apply to $\hat{h}$, and the exact form is used below.

Proof. $\delta {S^{(0)}}_{\mathrm{gf}}=\frac{1}{2}\int C_{\mu }{\partial }_{\nu }(\mathcal{P}\delta u{)}^{\nu \mu }=-\frac{1}{2}\int {\partial }_{(\nu }{C}_{\mu )}(\mathcal{P}\delta u{)}^{\nu \mu }=-\frac{1}{4}\int (\mathcal{P}KC{)}^{\nu \mu }\delta u_{\nu \mu }$, using the symmetry of $\mathcal{P}$. With (eq:xe-gf), $\frac{1}{2}{G}_{\mathrm{lin}}[u]=\mathcal{P}(-\frac{1}{4}\square u+\frac{1}{4}KC[u])$, and the $KC$ terms cancel. Finally $(\mathcal{P}K\xi {)}^{\nu \mu }={\partial }^{\nu }{\xi }^{\mu }+{\partial }^{\mu }{\xi }^{\nu }-{\eta }^{\nu \mu }\partial \cdot \xi$, whose divergence is $\square {\xi }^{\mu }$.

In particular $P_{\mathrm{gr}}K\xi =\frac{1}{2}{G}_{\mathrm{lin}}[K\xi ]+{\mathcal{F}}^{(0)}K\xi =-\frac{1}{4}\mathcal{P}K\square \xi$, which is X-E's Proposition No compactly supported residual gauge; that proposition stands unchanged on this sheet.

With $q=\bar{q}+\kappa u+{\kappa }^2{k}_{\mathrm{cl}}$, $\bar{q}=\eta +\kappa h_{\mathrm{cl}}$, and $S_{\mathrm{q}}:=S_{\mathrm{gr}}[q]+S_{\mathrm{gf}}[u;\bar{q}]+{S^{\mathrm{sc}}}_g[q,\varphi ]$,

and (eq:Q-def) holds. For each smooth $w$, ${\mathcal{Q}}^{(1)}[w]$ is a linear differential operator of order $\le 2$ in $u$, with coefficients linear in $(w,\partial w,{\partial }^{2}w)$; in particular $\mathrm{supp}({\mathcal{Q}}^{(1)}[w]u)\subset \mathrm{supp}w$. Explicitly, ${\mathcal{F}}^{(1)}[h_{\mathrm{cl}}]u=\delta \int {{\mathfrak{L}}_{\mathrm{gf}}}^{(1)}/\delta u$ with

$C^{\mu }=C^{\mu }[u]$, $\bar{u}=\mathcal{P}u$, $H_{\mathrm{cl}}={\mathrm{tr}}_{\eta }{h}_{\mathrm{cl}}$, ${\Gamma }_1$ X-E's linearized Christoffel symbol.

Proof. $\delta S_{\mathrm{gr}}/\delta u=\frac{1}{2\kappa }\mathcal{E}[q]$ by the chain rule from xe:einstein. With $\kappa h=\kappa (h_{\mathrm{cl}}+u)$, ${\kappa }^{2}k={\kappa }^2{k}_{\mathrm{cl}}$, $\mathcal{E}=\kappa G_{\mathrm{lin}}[h_{\mathrm{cl}}+u]+{\kappa }^2(G_{\mathrm{lin}}[k_{\mathrm{cl}}]+{\mathcal{G}}_2[h_{\mathrm{cl}}+u,h_{\mathrm{cl}}+u])+O({\kappa }^3)$; divide by $2\kappa$ and polarize ${\mathcal{G}}_2$. The matter term is X-E's $\delta {S^{\mathrm{sc}}}_g/\delta q=-\frac{1}{2}g\sqrt{-q}T[q,\varphi ]$ times $\kappa$. The gauge-fixing term is (eq:F-expansion). Grouping by powers of $\kappa$ and using (eq:F0) gives (eq:fluct) and (eq:Q1); the $u$-linear part of the right side of (eq:fluct) is by construction $\mathcal{Q}[\bar{q}]u$, which proves (eq:Q-def). For (eq:Lgf1): $\sqrt{-\bar{q}}=1+\kappa b_1[h_{\mathrm{cl}}]+O({\kappa }^2)$, ${\bar{q}}^{\mu \nu }={\eta }^{\mu \nu }-\kappa {h_{\mathrm{cl}}}^{\mu \nu }+O({\kappa }^2)$, so ${\overline{u^{\rho \nu }}}_{\bar{q}}={\bar{u}}^{\rho \nu }+\kappa {\beta }^{\rho \nu }+O({\kappa }^2)$; ${\bar{\nabla }}_{\rho }{\bar{u}}^{\rho \nu }={\partial }_{\rho }{\bar{u}}^{\rho \nu }+{\bar{\Gamma }}_{\rho \lambda }^{\rho }{\bar{u}}^{\lambda \nu }+{\bar{\Gamma }}_{\rho \lambda }^{\nu }{\bar{u}}^{\rho \lambda }$ with $\bar{\Gamma }=\kappa {\Gamma }_1[h_{\mathrm{cl}}]+O({\kappa }^2)$ and ${\Gamma }_1{}_{\rho \lambda }^{\rho }=\frac{1}{2}{\partial }_{\lambda }{H}_{\mathrm{cl}}$. Insert into $\frac{1}{4}\sqrt{-\bar{q}}{\bar{q}}_{\mu \nu }{C^{\mu }}_{\bar{q}}{C^{\nu }}_{\bar{q}}$. ${\mathcal{G}}_2(h_{\mathrm{cl}},u)$ is the polarization of a bidifferential expression of total order two with constant coefficients; ${{\mathfrak{L}}_{\mathrm{gf}}}^{(1)}$ is quadratic in $(u,\partial u)$ with coefficients linear in $(h_{\mathrm{cl}},\partial h_{\mathrm{cl}})$, so its Euler derivative is of order $\le 2$ in $u$ with coefficients linear in $(h_{\mathrm{cl}},\partial h_{\mathrm{cl}},{\partial }^2{h}_{\mathrm{cl}})$.

The terms $\frac{1}{2}{G}_{\mathrm{lin}}[h_{\mathrm{cl}}]+\frac{\kappa }{2}(G_{\mathrm{lin}}[k_{\mathrm{cl}}]+{\mathcal{G}}_2[h_{\mathrm{cl}},h_{\mathrm{cl}}])$ of (eq:fluct) are the Einstein tensor density of the off-shell background. The background-field prescription of Definition def:content drops them. The remaining equation, with $u=\hat{h}+\kappa \hat{k}$ and $P_{\mathrm{gr}}\hat{h}=0$, reads at order $\kappa$

whose right side, with $\varphi \to \hat{\varphi }$, Wick ordered, and switched by ${\sigma }_{\epsilon }$ on the gravitational terms, is $\mathsf{S}$ of (eq:source). The definition $\hat{k}={{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}\mathsf{S}$ is therefore the first-order retarded Møller correction of the metric slot, as X-E's $k_1$ was its matter part; $k_1={{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}(\frac{1}{2}g\Theta )$ is the first term of $\hat{k}$.

For every smooth vector field $\xi$ and smooth symmetric $u$,

In the flat gauge of X-E, ${\mathcal{F}}^{(1)}\equiv 0$ and ${{\mathcal{Q}}^{(1)}}_{\mathrm{flat}}[K\xi ]={\mathcal{G}}_2(K\xi ,\cdot )=\frac{1}{2}[{\mathcal{L}}_{\xi },G_{\mathrm{lin}}]=[{\mathcal{L}}_{\xi },P_{\mathrm{gr}}]-[{\mathcal{L}}_{\xi },{\mathcal{F}}^{(0)}]$.

Proof. As in Lemma lem:cov2, reduce to compactly supported $\xi$ and let $F_s$ be its flow. Lemma lem:natural gives $\mathcal{F}[F_s^{\ast }u;F_s^{\ast }\eta ]=F_s^{\ast }\mathcal{F}[u;\eta ]$. Differentiate at $s=0$ using (eq:F-expansion) with $\kappa h_{\mathrm{cl}}\to sK\xi$: ${\mathcal{F}}^{(1)}[K\xi ]u+{\mathcal{F}}^{(0)}{\mathcal{L}}_{\xi }u={\mathcal{L}}_{\xi }{\mathcal{F}}^{(0)}u$. The second identity follows from the first, (eq:cov2), (eq:Q1), and (eq:F0); or directly from the third identity of Lemma lem:natural. The flat-gauge statement is (eq:cov2) with (eq:F0).

If $\xi$ is a Killing field of $\eta$, both sides of (eq:key) vanish: $K\xi =0$, and $P_{\mathrm{gr}}$ has constant coefficients and is Poincaré covariant. For $\xi \in {\mathfrak{g}}_U$ neither side vanishes in general; this is the statement, already in X-E, that $\hat{h}\mapsto \hat{h}+\kappa {\mathcal{L}}_{\xi }\hat{h}$ is not an automorphism of the free gauge-fixed algebra. The present sheet does not transform $\hat{h}$.

For past-compact smooth $u$, ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{P}_{\mathrm{gr}}u=u$. For compactly supported $w$ and past-compact $S$, $\int w\cdot {{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}S=\int {{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w\cdot S$. For a covariant $v$ and a density $T$, one of them compactly supported, $\int v\cdot {\mathcal{L}}_{\xi }T=-\int {\mathcal{L}}_{\xi }v\cdot T$.

Proof. ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{P}_{\mathrm{gr}}u=\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}\square \mathcal{P}u=\mathcal{PP}u=u$, since $\mathcal{P}$ is constant and ${{\mathsf{E}}_0}^{\mathrm{R}}\square v=v$ for past-compact $v$. The second statement is the transposition of ${{\mathsf{E}}_0}^{\mathrm{R}}$, both integrals being over the compact set $\mathrm{supp}{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w\cap \mathrm{supp}S$. The third: $v\cdot T$ is a scalar density, ${\mathcal{L}}_{\xi }(v\cdot T)={\partial }_{\lambda }({\xi }^{\lambda }v\cdot T)$ integrates to zero, and ${\mathcal{L}}_{\xi }$ is a derivation.

Let $\xi \in {\mathfrak{g}}_U$. With $\hat{\varphi }$ and $\hat{h}$ held fixed,

and for every compactly supported smooth symmetric $w$,

Consequently the total second-order metric $k=k_{\mathrm{cl}}+\hat{k}$ transforms under the frozen action of ${\mathcal{G}}_U$ by the complete two-jet law of X-A,

modulo $\mathfrak{J}$ after smearing, although $\hat{h}$ itself is not transformed.

Proof. $\mathsf{S}$ depends on $h_{\mathrm{cl}}$ only through the linear term $-{\sigma }_{\epsilon }{\mathcal{Q}}^{(1)}[h_{\mathrm{cl}}]\hat{h}$, so $\mathsf{S}[h_{\mathrm{cl}}+K\xi ]-\mathsf{S}[h_{\mathrm{cl}}]=-{\sigma }_{\epsilon }{\mathcal{Q}}^{(1)}[K\xi ]\hat{h}=-{\mathcal{Q}}^{(1)}[K\xi ]\hat{h}$, because $\mathrm{supp}{\mathcal{Q}}^{(1)}[K\xi ]\hat{h}\subset \mathrm{supp}\xi \subset U\subset \{{\sigma }_{\epsilon }=1\}$ by Lemma lem:fluct and (eq:separation). By (eq:key), $-{\mathcal{Q}}^{(1)}[K\xi ]\hat{h}=P_{\mathrm{gr}}({\mathcal{L}}_{\xi }\hat{h})-{\mathcal{L}}_{\xi }(P_{\mathrm{gr}}\hat{h})$. Both terms are compactly supported in $\mathrm{supp}\xi$. Apply ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$ and Lemma lem:DRP: ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{P}_{\mathrm{gr}}{\mathcal{L}}_{\xi }\hat{h}={\mathcal{L}}_{\xi }\hat{h}$. This is (eq:jetlaw). For (eq:jetlaw-ideal), $\int w\cdot {{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h}=\int {{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w\cdot {\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h}=-\int {\mathcal{L}}_{\xi }{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w\cdot P_{\mathrm{gr}}\hat{h}=-\int P_{\mathrm{gr}}({\mathcal{L}}_{\xi }{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w)\cdot \hat{h}$, by the last two statements of Lemma lem:DRP and the formal self-adjointness of $P_{\mathrm{gr}}$; ${\mathcal{L}}_{\xi }{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w$ is smooth and supported in $\mathrm{supp}\xi$, so $\hat{h}(P_{\mathrm{gr}}{\mathcal{L}}_{\xi }{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w)$ is a generator of $\mathfrak{J}$. The last statement combines (eq:jetlaw) with the frozen law $k_{\mathrm{cl}}\mapsto k_{\mathrm{cl}}+{\mathcal{L}}_{\xi }{h}_{\mathrm{cl}}+K\zeta +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$.