Theorem: Sharp-boundary obstruction

Fix the field multiplet and the off-shell domain of Section sec:domain. For every $0\ne f\in {C_c}^{\infty }(\{t>0\};\mathbb{R})$:

1. ${{\mathcal{B}}_f}^{(1)}$ is a continuous bilinear functional of smooth $h,\varphi$, with compact functional support contained in ${\mathcal{D}}_{\mathrm{supp}f}$.

2. It is not microcausal and, in particular,

   $${{\mathcal{B}}_f}^{(1)}\notin {\mathcal{P}}_{\mathrm{mc}}.$$

3. There is no admissible quantization of ${{\mathcal{B}}_f}^{(1)}$ satisfying Definition def:normalization.

4. There is no termwise retarded-product construction of (eq:four-pieces-intro) in ${\mathfrak{A}}_{\mathrm{poly}}$ that admits the original geometric insertion coefficientwise and preserves its off-shell classical normalization.

Proof. Continuity of $L$ on smooth fields, locally in the future half-space, follows from the continuous dependence of the inhomogeneous Cauchy problem on its source. Multiplication by $f$, differentiation of $\varphi$, and integration are continuous. This proves the first assertion, including polynomial smoothness. The support statement follows from finite propagation and (eq:DK).

For the second assertion, differentiate with respect to the metric field:

This derivative is independent of $h$.

Choose $\varphi ={\varphi }_f$ from Lemma lem:witness. Then $b=E_{\mathrm{A}}(f\mathrm{d}{\varphi }_f)$ has a nonzero trace at some point of ${\Sigma }_0$. Lemma lem:wavefront shows that $-L^{\ast }(f\mathrm{d}{\varphi }_f)$ has nonzero timelike covectors in its wavefront set. This contradicts the necessary microcausal condition of Lemma lem:derivative-condition.

If an admissible quantization existed, its coefficient of ${\hslash }^0$ would belong to ${\mathcal{P}}_{\mathrm{mc}}$ and would equal ${{\mathcal{B}}_f}^{(1)}$ on every smooth configuration. Differentiating this equality gives the same forbidden derivative, proving the third assertion.

Finally, a coefficientwise construction that admits the geometric insertion in ${\mathfrak{A}}_{\mathrm{poly}}[[\kappa ]]$ must admit its coefficient ${{\mathcal{B}}_f}^{(1)}$, before applying $R_1(V_1;\cdot )$. The third assertion excludes this prerequisite. This proves the fourth assertion under its stated termwise and off-shell requirements.