Analyticity and crossing properties of four-point function are investigated in conformal field theories in the frameworks of Wightman axioms. A Hermitian scalar conformal field, satisfying the Wightman axioms, is considered. The crucial role of microcausality in deriving analyticity domains is discussed and domains of analyticity are presented. A pair of permuted Wightman functions are envisaged. The crossing property is derived by appealing to the technique of analytic completion for the pair of permuted Wightman functions. The operator product expansion of a pair of scalar fields is studied and analyticity property of the matrix elements of composite fields, appearing in the operator product expansion, is investigated. An integral representation is presented for the commutator of composite fields where microcausality is a key ingredient. Three fundamental theorems of axiomatic local field theories; namely, PCT theorem, the theorem proving equivalence between PCT theorem and weak local commutativity and the edge-of-the-wedge theorem are invoked to derive a conformal bootstrap equation rigorously.

中文翻译：

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共形场论中的因果性、交叉性和解析性

在 Wightman 公理的框架下，在共形场论中研究了四点函数的解析性和交叉性质。考虑满足 Wightman 公理的 Hermitian 标量共形场。讨论了微因果关系在推导分析域中的关键作用，并提出了分析域。设想了一对置换的 Wightman 函数。交叉属性是通过对一对置换 Wightman 函数的解析完成技术得出的。研究了一对标量域的算子积展开，研究了在算子积展开中出现的复合域的矩阵元素的解析性质。为复合域的交换子提供了一个积分表示，其中微因果性是一个关键因素。公理化局部场论的三个基本定理；即PCT定理、证明PCT定理与弱局部交换律等价的定理和楔边定理严格推导出保形自举方程。