Modular Hamiltonian of the massive scalar field on the half line: A numerical approach

TL;DR

This paper numerically investigates the modular Hamiltonian of a massive free scalar field on a half line, revealing its non-locality near the boundary and for various boundary conditions and masses.

Contribution

It provides the first numerical analysis of the modular Hamiltonian for a massive scalar field with Robin boundary conditions on a half line.

Findings

Modular Hamiltonian is non-local near the boundary for massive scalar fields.

Massless scalar fields with Dirichlet or Neumann conditions have local modular Hamiltonians.

Non-locality persists for all masses when the interval is separated from the boundary.

Abstract

We study the modular Hamiltonian of an interval for the ground state of a massive free scalar field on the half line with Robin boundary conditions, by employing a numerical method. When the interval is adjacent to the boundary, we find numerical evidence that the modular Hamiltonian is non-local, except for the limiting cases of the massless scalar satisfying either Dirichlet or Neumann boundary conditions. When the interval is separated from the boundary, the numerical analysis indicates that the modular Hamiltonian is non-local for all these boundary conditions and any value of the mass.