Quasi-equivalence of Gaussian states and energy estimates for functions of modular Hamiltonians
Adriano Chialastri, Ko Sanders
TL;DR
The paper investigates quantitative links between three viewpoints on Gaussian states in the Weyl CCR framework: quasi-equivalence of representations, energy differences on a Cauchy surface, and differences of functions of modular Hamiltonians. Building on Araki–Yamagami and Longo modular theory, it develops explicit Hilbert–Schmidt and trace-class bounds that relate perturbations of the Gaussian inner product $\mu_0$ to differences in polarisation operators and modular data, thereby bounding changes in modular operators by energy differences. These general results are then applied to a real linear scalar field on ultrastatic spacetimes, yielding concrete inequalities for Minkowski-vacuum perturbations and thermal (KMS) states, and providing criteria under which perturbed states remain quasi-equivalent to the vacuum. The outcomes supply a practical toolkit to translate energy shifts into precise control on modular-operator functions, contributing to the understanding of quantum energy inequalities and the connections between relative entropy, energy, and modular theory in quantum field theory.
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.