  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.
