The massless modular Hamiltonian

TL;DR

This paper explicitly computes the vacuum local modular Hamiltonian for a space ball in massless free scalar QFT, revealing its structure and implications for local entropy and relative entropy in quantum field theory.

Contribution

It provides an explicit expression for the modular Hamiltonian in massless scalar QFT using higher dimensional Legendre operators, advancing understanding of quantum entanglement.

Findings

Explicit formula for the modular Hamiltonian in massless scalar QFT.

Expression of the quadratic form in terms of energy density with parabolic distribution.

Derivation of local entropy and vacuum relative entropy for wave packets.

Abstract

We compute the vacuum local modular Hamiltonian associated with a space ball region in the free scalar massless Quantum Field Theory. We give an explicit expression on the one particle Hilbert space in terms of the higher dimensional Legendre differential operator. The quadratic form of the massless modular Hamiltonian is expressed in terms of an integral of the energy density with the parabolic distribution. We then get the formula for the local entropy of a wave packet. This gives the vacuum relative entropy of a coherent state on the double cone von Neumann algebras associated with the free scalar QFT. Among other points, we provide the passivity characterisation of the modular Hamiltonian within the standard subspace setup.