Mass Dependence of Araki Relative Entropy through Modular Theory
Abstract
Building on the established one-particle formula for the Araki relative entropy of coherent states, we study how its value acquires a nontrivial dependence on the mass of the scalar field. For a localized vector h belonging to the standard subspace Hm of the one-particle Hilbert space, the known quadratic-form expression is: SHm(h)=−⟨ h,logδHm h⟩. Our contribution is to construct explicitly a mass-indexed family of vectors of the wedge standard subspace on which this expression is evaluated. The mass-shell map hm=Emf organizes four structural conditions on rapidity representatives---on-shell dependence, a controlled massless boundary value, decay for large real rapidity, and Bisognano--Wichmann strip analyticity---and we exhibit an entire rapidity wave function, built from a doubled light-cone phase, two sinc factors, and a Gaussian pair, that satisfies them together with the sharp localization criterion: Hardy-type L2 control throughout the Bisognano--Wichmann strip and the exact Tomita boundary relation. The family therefore belongs to Hm(WR) for every m>0, and its Araki relative entropy is finite and strictly positive, with an exact spectral representation that makes positivity manifest. The entropy is strongly suppressed at large mass, attains a maximum at intermediate mass in 1+1 dimension, and converges to a finite value along the modular flow as m→0+. The construction extends fiberwise to 1+d dimensions through the transverse mass.
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BibTeX
@misc{guimaraes2026mass,
title = {{Mass Dependence of Araki Relative Entropy through Modular Theory}},
author = {M. S. Guimaraes and I. Roditi and S. P. Sorella},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12676},
eprint = {2609.12676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12676},
}
RIS
TY - UNPB
TI - Mass Dependence of Araki Relative Entropy through Modular Theory
AU - M. S. Guimaraes
AU - I. Roditi
AU - S. P. Sorella
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12676
UR - https://arxiv.org/abs/2609.12676
AB - Building on the established one-particle formula for the Araki relative entropy of coherent states, we study how its value acquires a nontrivial dependence on the mass of the scalar field. For a localized vector h belonging to the standard subspace H_m of the one-particle Hilbert space, the known quadratic-form expression is: S_H_m(h)=−⟨ h,logδ_H_m h⟩. Our contribution is to construct explicitly a mass-indexed family of vectors of the wedge standard subspace on which this expression is evaluated. The mass-shell map h_m=E_mf organizes four structural conditions on rapidity representatives---on-shell dependence, a controlled massless boundary value, decay for large real rapidity, and Bisognano--Wichmann strip analyticity---and we exhibit an entire rapidity wave function, built from a doubled light-cone phase, two sinc factors, and a Gaussian pair, that satisfies them together with the sharp localization criterion: Hardy-type L^2 control throughout the Bisognano--Wichmann strip and the exact Tomita boundary relation. The family therefore belongs to H_m(W_R) for every m>0, and its Araki relative entropy is finite and strictly positive, with an exact spectral representation that makes positivity manifest. The entropy is strongly suppressed at large mass, attains a maximum at intermediate mass in 1+1 dimension, and converges to a finite value along the modular flow as m→0^+. The construction extends fiberwise to 1+d dimensions through the transverse mass.
ER -