Modular Hamiltonian of the scalar field in the semi infinite line: dimensional reduction for spherically symmetric regions

TL;DR

This paper derives a local expression for the modular Hamiltonian of dimensionally reduced scalar fields in semi-infinite lines, linking it to conformal theories and analyzing entanglement entropy contributions across dimensions.

Contribution

It demonstrates that the modular Hamiltonian for reduced scalar theories is a local energy density operator derived from the conformal parent theory, extending understanding of entanglement in non-conformal settings.

Findings

Modular Hamiltonian is local and related to the conformal parent theory.

Analytic expression for entanglement entropy spectrum.

Recovery of conformal anomaly and universal constants in entropy.

Abstract

We focus our attention on the one dimensional scalar theories that result from dimensionally reducing the free scalar field theory in arbitrary d dimensions. As is well known, after integrating out the angular coordinates, the free scalar theory can be expressed as an infinite sum of theories living in the semi-infinite line, labeled by the angular modes $\{l, \vec{m}\}$. We show that their modular Hamiltonian in an interval attached to the origin is, in turn, the one obtained from the dimensional reduction of the modular Hamiltonian of the conformal parent theory in a sphere. Remarkably, this is a local expression in the energy density, as happens in the conformal case, although the resulting one dimensional theories are clearly not conformal. We support this result by analyzing the symmetries of these theories, which turn out to be a portion of the original conformal group, and proving that the reduced modular Hamiltonian is in fact the operator generating the modular flow in the interval. By studying the spectrum of these modular Hamiltonians, we also provide an analytic expression for the associated entanglement entropy. Finally, extending the radial regularization scheme originally introduced by Srednicki, we sum over the angular modes to successfully recover the conformal anomaly in the entropy logarithmic coefficient in even dimensions, as well as the universal constant $F$ term in $d = 3$.