通过模理论研究Araki相对熵的质量依赖性
Mass Dependence of Araki Relative Entropy through Modular Theory
- UERJ – Universidade do Estado do Rio de Janeiro（里约热内卢州立大学）
- CBPF – Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过模理论构造质量参数化的楔形标准子空间向量族，研究Araki相对熵对质量的非平凡依赖，证明其有限正性、大质量抑制及中等质量峰值等性质。
AI中文摘要：
基于相干态Araki相对熵的已建立单粒子公式，我们研究其值如何获得对标量场质量的非平凡依赖性。对于属于单粒子希尔伯特空间的标准子空间$H_m$的局域化向量$h$，已知的二次型表达式为：$S_{H_m}(h)=-\langle h,\log\delta_{H_m}\\,h\rangle$。我们的贡献是显式构造一个楔形标准子空间的以质量为指标的向量族，该表达式在此向量族上进行求值。质量壳映射$h_m=E_mf$组织了关于快度代表的四个结构条件——壳上依赖性、受控的无质量边界值、大实快度下的衰减以及Bisognano--Wichmann带解析性——并且我们展示了一个完整的快度波函数，由双光锥相位、两个sinc因子和高斯对构成，它满足这些条件以及尖锐的局域化准则：在整个Bisognano--Wichmann带内的Hardy型$L^2$控制以及精确的Tomita边界关系。因此，该向量族对于每个$m>0$都属于$H_m(\W_R)$，其Araki相对熵是有限且严格正的，具有使正性明显的精确谱表示。熵在大质量时被强烈抑制，在$1+1$维中于中等质量处达到最大值，并在$m\to0^+$时沿模流收敛到有限值。该构造通过横向质量逐纤维扩展到$1{+}d$维。
英文摘要：
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: $S_{H_m}(h)=-\langle h,\logδ_{H_m}\,h\rangle$. 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 $m\to0^+$. The construction extends fiberwise to $1{+}d$ dimensions through the transverse mass.
