Question, result, and scope

A classical relational construction does not, by itself, specify a quantum product domain. This distinction is especially important when the reference coordinates are obtained by solving a hyperbolic equation with data prescribed on a sharp initial hypersurface.

We consider the geometric functional

where the four functions ${{\mathsf{X}}_{\mathbf{g}}}^{\mu }$ are harmonic for $\mathbf{g}$, with their values and first time derivatives fixed on an initial surface. All classical ingredients used below are specified explicitly in Section sec:classical; no theorem about their diffeomorphism intertwining is assumed or reproved.

The intended second-order interacting coefficient is formally

Before this expression can be constructed term by term, its geometric insertions must be admitted by the chosen domain. That prerequisite fails for the domain studied here.

The main result is Theorem thm:main. For every

the first geometric coefficient

is not a microcausal functional on the full smooth configuration space. The obstruction is a proved wavefront component, rather than a failure to verify a sufficient criterion for multiplying distributions.

We use a finite polynomial-kernel domain. This avoids relying on closure assertions for arbitrary smooth microcausal functionals: Hawkins, Rejzner, and Visser identify a closure problem for that larger class, introduce equicausal functionals, and establish equicausality for Wick polynomials with admissible fixed kernels HRV. The present obstruction is more elementary: it violates the microcausal cone condition itself, and therefore also excludes an equicausal representative that is equal to the original functional off shell.

The gravitational interpretation is deliberately limited. The gauge-fixed field content belongs to perturbative gravity, rather than to a scalar theory with a prescribed external metric. However, admission to a gauge-fixed algebra is not the same as physicality. The latter requires a separate gauge-cohomological analysis, as in the perturbative framework of Brunetti, Fredenhagen, and Rejzner BFR.