Fields, formal order, and gauge algebra

The smooth off-shell inputs are

No decay, equation-of-motion quotient, gravitational constraint, or gauge-fixed quotient is imposed. The scalar is massless.

The formal metric jet is

There is no factor $1/2$ in the definition of $k$. Statements modulo ${\kappa }^3$ mean coefficient identities, not convergence of a numerical perturbation series.

Set $x_{\ast }=(4,0,0,0)$ and $r(x)=|x-x_{\ast }{|}_E$, where the Euclidean norm is used only for supports and convolution. Define

Freeze

Then $g=1$ on $r\le 2$, and $\mathrm{supp}g\subset \{r\le 3\}⋐(0,8)\times {\mathbb{R}}^3$.

For a vector field $\xi$, write

The algebra fixed before the calculation is

Indices are raised and lowered with $\eta$. The switch is fixed background data.

For $\xi ,\zeta \in {\mathfrak{g}}_C$,

gives

The scalar transforms by the same formal pullback.