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

TL;DR

This paper investigates the limitations of the spectral cut-off Gaussian free field (GFF) in satisfying key properties like the Markov property and reflection positivity, highlighting challenges in constructing certain quantum field measures.

Contribution

It provides a detailed analysis of why the spectrally cut-off GFF cannot satisfy the Markov property or reflection positivity, clarifying issues in quantum field theory measure constructions.

Findings

Spectrally cut-off GFF does not satisfy the spatial Markov property.

Spectrally cut-off GFF fails to be reflection positive on reflection positive manifolds.

Difficulties in deriving reflection positivity for certain measures from GFF measure.

Abstract

We argue that the spectrally cut-off Gaussian free field $\Phi_\Lambda$ on a compact Riemannian manifold or on $\mathbb{R}^n$ 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$(-\|\rho\Phi_\Lambda\|_{L^4}^4) \mu_{\text{GFF}}(d\Phi)$ 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.