有限外尔多项式与相对论标量量子场论中齐雷尔森界的趋近方法
Finite Weyl polynomials and the approach to Tsirelson's bound in relativistic scalar quantum field theory
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中文总结 AI 辅助
该研究在1+1维自由有质量标量场中构造有限外尔多项式形式的Bell-CHSH不等式算符，证明其族上确界为2√2，且有限成员可达2.80027，突破了高斯表示下结果低于2的限制。
中文摘要 AI 辅助
我们针对1+1维自由有质量标量场的Bell-CHSH不等式，构造了四个有界厄米算符，每个均为幺正外尔算子的有限多项式，且这些算符定域于互补楔形区域中。奇数阶外尔谐波在每个楔形区域提供两个完全反对易的轴可观测量，而在洛伦兹 boost 生成元谱上具有紧支集的归一化波包则给出非零带宽下的精确模内积。所得Bell-CHSH关联函数为有限双重求和：每个轴含6个外尔项的算符已给出值2.14885。利用符号函数sgn(cos(x))的归一化Fejér逼近，我们证明该有限多项式族的上确界等于2√2，尽管无有限成员达到此值，其中一个511阶的例子给出值2.80027。我们还指出，在中心准自由态中，四个最终设置为单个正交分量有界函数的Bell-CHSH测试具有共同高斯表示，因此结果保持在2以下；而有限外尔构造可避免此限制，因为Bob的最终设置会混合两个非对易轴可观测量。
英文摘要
We construct four bounded Hermitian operators, each one a finite polynomial in the unitary Weyl operators, for the Bell-CHSH inequality in a free massive scalar field in $1+1$ dimensions. The operators are localized in complementary wedges. Odd Weyl harmonics provide two exactly anticommuting axis observables in each wedge, while normalizable packets with compact support in the spectrum of the boost generator give exact modular inner products at nonvanishing bandwidth. The resulting Bell-CHSH correlator is a finite double sum. An operator with six Weyl terms per axis already gives $2.14885$. Using normalized Fejér approximants of $\operatorname{sgn}(\cos(x))$, we show that the supremum over the finite-polynomial family equals $2\sqrt{2}$, although no finite member attains it; a degree-$511$ example gives $2.80027$. We also point out that, in a centered quasifree state, a Bell-CHSH test whose four final settings are bounded functions of individual quadratures admits one common Gaussian representation and therefore stays below $2$. The finite-Weyl construction avoids this restriction because Bob's final settings mix two noncommuting axis observables.
