A precise classical smoothing convention

A smoothing convention can be fixed before a quantum state is available.

Choose a real even nonnegative $\rho \in {C_c}^{\infty }({\mathbb{R}}^4)$ with $\int \rho =1$. Let

in the displayed background chart. Choose a spatial cutoff ${\chi }_R$ equal to one on $|\mathbf{x}|\le R$, supported in $|\mathbf{x}|<2R$, and a fixed time cutoff equal to one on a neighborhood of the time projection of $\mathrm{supp}f$.

Apply these cutoffs to field slots before smoothing. The regulated Hessian is

with the fixed time cutoffs understood. For finite $R,\delta >0$, it is a smooth compact test kernel on the two field slots.

This chart smoothing is an explicitly chosen regulator. It is not claimed to be diffeomorphism covariant. Any gauge breaking it causes must be calculated in the quantum construction rather than presumed absent.