Entanglement in $(1+1)$-dimensional Free Scalar Field Theory: Tiptoeing between Continuum and Discrete Formulations

TL;DR

This paper reviews methods for calculating entanglement entropy in (1+1)-dimensional free scalar field theory, bridging discrete and continuum approaches, and explicitly derives the modular Hamiltonian in a specific coordinate system.

Contribution

It demonstrates how discretized theory methods can be applied to continuum theories and explicitly derives the modular Hamiltonian without modular flow.

Findings

Entanglement entropy for finite interval calculated.

Identifications between discretized and continuum methods established.

Modular Hamiltonian expressed as a free field Hamiltonian on the Rindler wedge.

Abstract

We review some classic works on ground state entanglement entropy in $(1+1)$-dimensional free scalar field theory. We point out identifications between the methods for the calculation of entanglement entropy and we show how the formalism developed for the discretized theory can be utilized in order to obtain results in the continuous theory. We specify the entanglement spectrum and we calculate the entanglement entropy for the theory defined on an interval of finite length $L$. Finally, we derive the modular Hamiltonian directly, without using the modular flow, via the continuous limit of the expressions obtained in the discretized theory. In a specific coordinate system, the modular Hamiltonian assumes the form of a free field Hamiltonian on the Rindler wedge.