Abstract The original “magic identities” are due to J. M. Drummond, J. Henn, V. A. Smirnov and E. Sokatchev; they assert that all n-loop box integrals for four scalar massless particles are equal to each other [3] . The authors give a proof of the magic identities for the Euclidean metric case only and claim that the result is also true in the Minkowski metric. However, the Minkowski case is much more subtle and requires specification of the relative positions of cycles of integration to make these identities correct. In this article we prove the magic identities in the Minkowski metric case and, in particular, specify the cycles of integration. Our proof of magic identities relies on previous results from [7] , [8] , where we give a mathematical interpretation of the n-loop box integrals in the context of representations of a Lie group U ( 2 , 2 ) and quaternionic analysis. The main result of [7] , [8] is a (weaker) operator version of the “magic identities”. No prior knowledge of physics or Feynman diagrams is assumed from the reader. We provide a summary of all relevant results from quaternionic analysis to make the article self-contained.

中文翻译：

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共形四点积分的神奇恒等式；Minkowski 度量案例

摘要 最初的“魔法身份”归功于 JM Drummond、J. Henn、VA Smirnov 和 E. Sokatchev；他们断言四个标量无质量粒子的所有 n 环盒积分彼此相等 [3]。作者仅在欧几里得度量的情况下给出了神奇恒等式的证明，并声称该结果在 Minkowski 度量中也是正确的。然而，Minkowski 的情况要微妙得多，需要指定积分周期的相对位置才能使这些恒等式正确。在本文中，我们证明了 Minkowski 度量案例中的神奇恒等式，特别是指定了积分周期。我们的魔法恒等式证明依赖于之前的结果 [7] , [8] ，我们在李群 U ( 2 , 2) 和四元数分析。[7]、[8] 的主要结果是“魔法身份”的（较弱）算子版本。假设读者不具备物理学或费曼图的先验知识。我们提供了四元数分析的所有相关结果的摘要，以使文章自包含。