Modular structure of the Weyl algebra

TL;DR

This paper analyzes the modular Hamiltonian of Gaussian states on the Weyl algebra, providing criteria for state equivalence, automorphism properties, and explicit formulas for local entropy in 2D free quantum field theory.

Contribution

It introduces new criteria for Gaussian state equivalence and automorphism innerness, and describes the vacuum modular Hamiltonian in 2D massless free QFT, extending existing results.

Findings

Criteria for local equivalence of Gaussian states

Conditions for Bogoliubov automorphisms to be weakly inner

Explicit formula for local entropy of massless wave packets

Abstract

We study the modular Hamiltonian associated with a Gaussian state on the Weyl algebra. We obtain necessary/sufficient criteria for the local equivalence of Gaussian states, independently of the classical results by Araki and Yamagami, Van Daele, Holevo. We then present a criterion for a Bogoliubov automorphism to be weakly inner in the GNS representation. We also describe the vacuum modular Hamiltonian associated with a time-zero interval in the scalar, massless, free QFT in two spacetime dimensions, thus complementing the recent results in higher space dimensions. In particular, we have the formula for the local entropy of a one-dimensional massless wave packet and Araki's vacuum relative entropy of a coherent state on a double cone von Neumann algebra.