Modular flow of excited states

We develop new techniques for studying the modular and the relative modular flows of general excited states. We show that the class of states obtained by acting on the vacuum (or any cyclic and separating state) with invertible operators from the algebra of a region is dense in the Hilbert space. This enables us to express the modular and the relative modular operators, as well as the relative entropies of generic excited states in terms of the vacuum modular operator and the operator that creates the state. In particular, the modular and the relative modular flows of any state can be expanded in terms of the modular flow of operators in vacuum. We illustrate the formalism with simple examples including states close to the vacuum, and coherent and squeezed states in generalized free field theory.

Hypo-EP Matrices of Adjointable Operators on Hilbert C ∗ -Modules

This paper introduces and studies hypo-EP matrices of adjointable operators on Hilbert C ∗ -modules, based on the generalized Schur complement. The necessary and sufficient conditions for some modular operator matrices to be hypo-EP are given, and some special circumstances are also analyzed. Furthermore, an application of the EP operator in operator equations is given.

On the Modular Operator of Mutli-component Regions in Chiral CFT

We introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

ON LOCALIZATION AND POSITION OPERATORS IN MÖBIUS-COVARIANT THEORIES

Some years ago it was shown that, in some cases, a notion of locality can arise from the symmetry group of the theory [1–3], i.e. in an intrinsic way. In particular, when the Möbius covariance is present, it is possible to associate some particular transformations with the Tomita–Takesaki modular operator and conjugation of a specific interval of an abstract circle. In this context we propose a way to define an operator representing the coordinate conjugated to the modular transformations. Remarkably this coordinate turns out to be compatible with the abstract notion of locality. Finally a concrete example concerning a quantum particle on a line is given.