Reflection positivity concerns the semi-positive definite nature of expectation values of the type $\langle F\theta_{P}F \rangle \ge 0$. Here, $\theta_{P}$ denotes reflection with respect to the plane $P$. In the above, $F$ is a general function of fields (or spins, etc.) and $\theta_{P}$ is a reflection operator defined by $\theta_{P} F(x) = F(\theta_{P} x)$. That is, $\theta_{P}$ transforms the argument(s) of F to the reflection or mirror image(s) (in the plane $P$) of the coordinate(s) $x$ on which it is defined. The average $\langle \cdot \rangle$ is computed with a measure that is (up to a normalization factor) given by $e^{-H}$ with $H$ a Hamiltonian (or action) that is reflection invariant: $\theta_{P} H = H$.
My naive question is how general is this inequality of $\langle F \theta_{P} F \rangle \ge 0$? Does $F$ need to have its support only on local spatial regions $x$? In simple lattice proofs, I can see how this can be demonstrated by having the plane $P$ lie along a line of sites, etc., with local $F$ and Hamiltonians $H$ that are sums of local terms. Does this trivially hold or fail for a many body system with long range interactions? For which quantum system does it hold?
The literature seems focused on imaginary time questions and spin systems.