单叶双曲面的Fourier分析与全纯分解
Fourier analysis and holomorphic decomposition on the one-sheeted hyperboloid
摘要：
本文证明了复化单叶二维双曲面的四个特殊管状区域上全纯函数类的Cauchy型积分表示。从物理学的观点来看，这个双曲面可以用来描述二维的德西特宇宙和反德西特宇宙。对于其中的两个管形，称为“洛伦兹管形”，与德西特量子场论有关，在这些域中全纯函数的实双曲面上的边界值允许连续的傅里叶-赫尔加松型变换。对于另外两个管形，称为“手征管形”，与反德西特量子场论相关，在这些域中全纯函数的实数上的边界值允许离散的傅里叶-赫尔加松型变换。在这两种情况下，这些变换的反演公式是通过使用以前的柯西表示的相应类别的功能。将实双曲面上的函数分解为前四个曲面上的全纯函数的边值之和，给出了单曲双曲面的Gelfand-Gindikin程序的完整而明确的处理。
英文摘要：
We prove a Cauchy-type integral representation for classes of functions holomorphic in four priviledged tuboid domains of the complexified one-sheeted two-dimensional hyperboloid. From a physical viewpoint, this hyperboloid can be used for describing both the two-dimensional de Sitter and anti-de Sitter universes. For two of these tuboids, called ``the Lorentz tuboids'' and relevant for de Sitter Quantum Field Theory, the boundary values onto the real hyperboloid of functions holomorphic in these domains admit continuous Fourier-Helgason-type transforms. For the other two tuboids, called the ``chiral tuboids'' and relevant for anti-de Sitter Quantum Field Theory, the boundary values on the reals of functions holomorphic in these domains admit discrete Fourier-Helgason-type transforms. In both cases, the inversion formulae for these transformations are derived by using the previous Cauchy representation for the respective classes of functions. The decomposition of functions on the real hyperboloid into sums of boundary values of holomorphic functions from the previous four tuboids gives a complete and explicit treatment of the Gelfand-Gindikin program for the one-sheeted hyperboloid.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
