Domain and field/background split

All constructions in this section are local on $U$. Fields are unrestricted smooth local sections before equations are imposed. Backgrounds $h,k$ are fixed smooth covariant symmetric tensors, not dynamical fields and not the host metric.

Declare the formal local reference split

The use of $\bar{q}$ through second order fixes a local reference expansion. It does not assert a new retarded prescription for X-F.

Let $\mathcal{P}(U)[[\kappa ]]$ be the graded differential algebra of local polynomial jet expressions, allowing the formal inverse metric coefficients determined by (eq:split). At each formal order the expressions have finite jet order and finite field degree.

Define $s$ by fixing $\bar{q},h,k$, commuting with total spacetime derivatives, and setting

The Leibniz rule is left graded: $s(FG)=(sF)G+(-1{)}^{|F|}F(sG)$. In particular,

The coefficient operators are explicit:

The leading field differential is exactly (eq:freeBRST).