Given a number field $K$ with a Hecke character $\chi$, for each place $\nu$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $\chi$. These theories appear in the study of $p$‑adic string theory and $p$‑adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of $\chi$ We find that the Green’s function is given by the local functional equation for Zeta integrals. Furthermore, considering all places $\nu$, the field theory two-point functions corresponding to the Green’s functions satisfy an adelic product formula, which is equivalent to the global functional equation for Zeta integrals. In particular, this points out a role of Tate’s thesis in adelic physics.

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弗拉基米洛夫导数的格林函数和泰特的论文

给定一个带有 Hecke 字符 $\chi$ 的数域 $K$，对于每个位置 $\nu$，我们研究自由标量场理论，其动力学项由与 $\chi$ 的局部分量相关的正则化 Vladimirov 导数给出. 这些理论出现在 $p$-adic 弦理论和 $p$-adic AdS/CFT 对应的研究中。我们根据 $\chi$ 的局部分量的傅立叶共轭证明了正则化 Vladimirov 导数的公式。我们发现格林函数由 Zeta 积分的局部函数方程给出。此外，考虑到所有位置$\nu$，格林函数对应的场论两点函数满足一个adelic积公式，它等价于Zeta积分的全局函数方程。特别是，这指出了泰特的论文在 adelic 物理学中的作用。