Abstract

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.