Proposition: Smooth bisolutions preserve algebraic data but not positivity

Let $P$ be a real scalar field operator on a smooth spacetime. Suppose a bidistribution $W$ is a positive two-point function satisfying

with a specified wavefront set. Let $u$ be a nonzero real smooth solution of $Pu=0$. For real $a$, put

Then:

1. every $W_a$ has the same field equations, antisymmetric part, and wavefront set as $W$;

2. every $W_a$ with $a\ge 0$ is positive;

3. for some $a<0$, $W_a$ is not positive.

Thus those algebraic and singularity data do not uniquely select a positive covariance, and do not by themselves establish positivity.

Proof. The added term is smooth, symmetric, and a bisolution. Adding a smooth distribution does not change the wavefront set. This proves (i).

For a complex compact test $f$,

This is nonnegative when $a\ge 0$. Since $u\ne 0$, choose $f$ with $⟨u,f⟩\ne 0$. The displayed expression becomes negative when

This proves (iii).