Quasi-equivalence of Gaussian states and energy estimates for functions of modular Hamiltonians

TL;DR

This paper investigates quantitative relations between Gaussian states of free scalar quantum fields, focusing on energy differences and functions of modular Hamiltonians, extending known qualitative connections with new estimates.

Contribution

It provides new quantitative estimates linking energy differences and functions of modular Hamiltonians for Gaussian states in quantum field theory.

Findings

Established bounds on differences of functions of modular Hamiltonians

Quantified relations between state quasi-equivalence and energy differences

Extended qualitative results to quantitative estimates in QFT

Abstract

To compare two Gaussian states of the Weyl-CCR algebra of a free scalar QFT we study three closely related perspectives: (i) quasi-equivalence of the GNS-representations, (ii) differences of the total energy (on some Cauchy surface), and (iii) differences between functions of the modular Hamiltonians. (For perspective (ii) we will only consider real linear free scalar quantum fields on ultrastatic spacetimes.) These three perspectives are known to be related qualitatively, due to work of Araki and Yamagami, Verch and Longo. Our aim is to investigate quantitative relations, including in particular estimates of differences between functions of modular Hamiltonians in terms of energy differences.