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

HUERTA, MARINA; VAN DER VELDE, GUIDO

JOURNAL OF HIGH ENERGY PHYSICS - (Online)

ELSEVIER

Lugar: Trieste; Año: 2023

1029-8479

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