Theorem: Same-class obstruction to an exact positive physical restriction

Let $W_{\mathcal{B}}\in {\mathfrak{W}}_{\mathcal{B}}$, and choose $U,f$ as in Lemma lem:TT. Put

and define

Then:

1. ${W_{\mathcal{B}}}^{\mathrm{bad}}$ has the same field equations, canonical antisymmetric part, adjoint condition, and Hadamard singularities;

2. ${W_{\mathcal{B}}}^{\mathrm{bad}}(f,f)=-1$;

3. no positive physical state can have this exact restriction on the invariant observable $f$.

The class of $f$ in (eq:quotient) is nonzero.

Proof. The added kernel is smooth, symmetric, real, and a bisolution of $\mathcal{B}$. Thus it changes none of the singular or antisymmetric data. The definition of $\lambda$ gives $q+\lambda c^2=-1$.

If $f=\mathcal{H}a$ for compact $a$, formal self-adjointness and $\mathcal{H}U=0$ would give $(U,f{)}_0=0$, a contradiction. Hence $f$ is a nonzero invariant observable. Proposition prop:necessary excludes the exact positive restriction.