Exact inverses, not positivity by terminology

The pairings used for $\mathcal{L}$ and $Q$ are indefinite. An algebraic Hadamard contraction can nevertheless be constructed. Positivity on a physical quotient is a different problem.

For a real self-adjoint action operator $P=\mathcal{B},Q$, choose an exact Feynman inverse $F_P$ such that

on compact tests. Choose the Feynman orientation for which $i(F_P-G_P^{-})$ has positive-frequency Hadamard wavefront. This is the convention appropriate to the action operator whose scalar principal differential part is ${\square }_{\bar{g}}$, with signature $(-,+,+,+)$. Under the opposite operator-sign convention, the familiar formula has the opposite prefactor.