Theorem: Background identity through κ 2 0

Let $\xi ,\zeta \in {\mathfrak{g}}_p$ act on the classical backgrounds by (eq:backgroundlaw), and hold $\mathbf{\varphi },\mathbf{r},H,f,g,\sigma$ fixed. Then

This is the direct identity for the specified tree-order normal polynomial. It is not an interacting quantum Ward or BRST theorem.

Proof. The completed scalar part with independent free graviton has variation (eq:one-defect). The matter and self trees have no classical-background dependence and hence are inert.

The mixed tree depends linearly on $h$. By Theorem thm:jet,

where Proposition prop:remainderideal handles the last term.

Its first term cancels the negative term in (eq:one-defect). The scalar-slot representatives in that identity are already in $\mathcal{J}$, by Lemma lem:forest. No second-order parameter acts on a quantum field at this order.