Spectrally cut-off GFF, regularized $Φ^4$ measure, and reflection positivity

Abstract: We argue that the spectrally cut-off Gaussian free field $\Phi_\Lambda$ on a compact Riemannian manifold or on $\mathbb{R}<sup>n$ 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$$(-|\rho\Phi_\Lambda|<em>{L^4}⁴⁾ \mu</em>{\text{GFF}}(d\Phi)$$ 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 litterature. Our pedagogical note aims to fill this small gap.