We argue that the spectrally cut-off Gaussian free field \(\Phi _\Lambda \) on a compact Riemannian manifold or on \({\textbf {R}}^d\) cannot satisfy the spatial Markov property. Moreover, when the manifold is reflection positive, we show that \(\Phi _\Lambda \) fails to be reflection positive. We explain the difficulties one encounters when trying to deduce the reflection positivity property of the measure exp \((-\Vert \rho \Phi _\Lambda \Vert _{L^4}^4) \mu _{\text {GFF}}(\textrm{d}\Phi )\) from the reflection positivity property of the Gaussian free field measure \(\mu _{\text {GFF}}\) in a naive way. These issues are probably well-known to experts of constructive quantum field theory but to our knowledge, no detailed account can be found in the literature. Our pedagogical note aims to fill this small gap.
