发表机构
Universidade do Estado do Rio de Janeiro; Centro Brasileiro de Pesquisas Físicas; ETH Zürich（里约热内卢州立大学; 巴西物理研究中心; 苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究自由马约拉纳场真空中的Bell-CHSH不等式，通过模种子构造双线性可观测量，精确计算Bell参数，证明Tsirelson界不可达，并阐明其与Summers-Werner机制的联系。
AI 中文摘要
本文研究了$1+1$维自由无质量马约拉纳场真空中的Bell-CHSH不等式，将两方的二分可观测量取为分别定域于右、左Rindler楔中的涂抹场偶双线性量。测试函数由模频率变量中的种子通过右楔的Tomita-Takesaki算子生成，而在左楔一侧则通过其伴随算子乘以$i$生成，这体现了扭曲对偶性。对于支撑在正模频率处的种子，两方之间的单粒子重叠不依赖于模权重，我们得到了Bell参数的闭式表达式。当种子是单个紧致凸包的相位旋转时，双线性量在每个楔中生成泡利代数的一个副本，我们精确计算了真空态限制于这两个副本的相关矩阵。最优设置违反经典界所允许的谱宽度是平面设置所能达到的两倍以上。我们还证明了在任何二分可观测量选择下，楔真空态中Tsirelson界均无法达到，且逼近该界迫使Bell矢量的模谱权重集中于模算子的谱点$\lambda=1$处。这精确地确立了自由马约拉纳场的偶扇区如何实现Summers-Werner机制。
英文摘要
In this work we study the Bell-CHSH inequality in the vacuum of the free massive Majorana field in $1+1$ dimensions, taking as the dichotomic observables of the two parties even bilinears of smeared fields localized in the right and in the left Rindler wedges. The test functions are generated from seeds in the modular-frequency variable through the Tomita-Takesaki operator of the right wedge and, on the side of the left wedge, through its adjoint multiplied by $i$, which accounts for twisted duality. For seeds supported at positive modular frequency the one-particle overlaps between the two parties do not depend on the modular weights, and we obtain the Bell parameter in closed form. When the seeds are phase rotations of a single compact bump, the bilinears generate a copy of the Pauli algebra in each wedge, and we compute exactly the correlation matrix of the vacuum restricted to the two copies. The optimal settings violate the classical bound up to spectral widths more than twice as large as those reached with planar settings. We also show that the Tsirelson bound is never attained in the wedge vacuum for any choice of dichotomic observables, and that approaching it forces the modular spectral weight of the Bell vectors to concentrate at the spectral point $λ=1$ of the modular operator. This establishes in a precise way how the even sector of the free Majorana field realizes the Summers-Werner mechanism.
Comments17 pages, 2 figures
