Wedge reflection positivity

TL;DR

The paper establishes wedge reflection positivity (WRP) as a real-time analogue of Euclidean reflection positivity by formulating positivity inequalities for Wightman functions with wedge-reflected arguments. The core method proves a positive-definite quadratic form for scalar fields using analyticity of Wightman functions and a special complex Lorentz transformation that implements the wedge reflection, with spin being handled in a subsequent subsection through representation-theoretic adjustments. It then connects WRP to the Tomita-Takesaki modular framework and the Bisognano-Wichmann theorem, identifying the modular reflection with a geometric wedge reflection and deriving positivity from the positivity of $e^{\pi K_1}$, thereby linking WRp to wedge TCP and potential odd-dimensional generalizations. The work suggests applications to reconstructing QFT data from wedge-based correlations and to understanding the relationship between vacuum entanglement (as captured by Renyi entropies) and local operator content, outlining future directions toward a reconstruction theorem grounded in WRP.

Abstract

We show there is a positivity property for Wightman functions which is analogous to the reflection positivity for the euclidean ones. The role of euclidean time reflections is played here by the wedge reflections, which change the sign of the time and one of the spatial coordinates.