Spectrally cut-off GFF, regularized Φ⁴ measure, and reflection positivity

2023/12/24 by Ismaël Bailleul, Bailleul, Ismael, Nguyen Viet Dang +7 · 1 citation
Mathematics · #81T08 81T20 28C20 46T12 #Advanced Mathematical Physics Problems #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Probability (math.PR)

paper · doi:10.48550/arxiv.2312.15511

openalex publication_date 2023/12/24 · openalex created_date 2024/01/18 · openalex updated_date 2026/07/28

Abstract

We argue that the spectrally cut-off Gaussian free field ΦΛ on a compact Riemannian manifold or on ℝn cannot satisfy the spatial Markov property. Moreover, when the manifold is reflection positive, we show that ΦΛ fails to be reflection positive. We explain the difficulties one encounters when trying to deduce the reflection positivity property of the measure exp(-‖ρΦΛ‖L44) μGFF(dΦ) from the reflection positivity property of the Gaussian free field measure μGFF 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.

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