Geometry, backgrounds, regulator, and algebra

Let

The quantum field is defined on $M$. The slab $M_T=[0,T]\times {\mathbb{R}}^3$ and its initial surface belong to the classical coordinate prescription.

The classical backgrounds are arbitrary smooth real symmetric tensors $h,k$, and

Indices are moved with $\eta$. Background topology means convergence of all derivatives on compact subsets of $M$.

Fix $0<\epsilon <T$ and the same ramp

Thus ${\sigma }_{\epsilon }$ is smooth and flat at zero. Fix the same $g\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3;[0,1])$, with $P_g=\mathrm{int}\{g=1\}$, and

The formal group ${\mathcal{G}}_U$ consists of pullbacks generated by $\kappa \xi +{\kappa }^2\zeta$, $\xi ,\zeta \in {\mathfrak{g}}_U$.

For the corresponding flow $F$, the complete metric law is

The second line is essential; $k=0$ does not describe a second-order pure-gauge jet when $h=K\xi$.