We discuss some general aspects of commutators of local operators in Lorentzian CFTs, which can be obtained from a suitable analytic continuation of the Euclidean operator product expansion (OPE). Commutators only make sense as distributions, and care has to be taken to extract the right distribution from the OPE. We provide explicit computations in two and four-dimensional CFTs, focusing mainly on commutators of components of the stress-tensor. We rederive several familiar results, such as the canonical commutation relations of free field theory, the local form of the Poincaré algebra, and the Virasoro algebra of two-dimensional CFT. We then consider commutators of light-ray operators built from the stress-tensor. Using simplifying features of the light sheet limit in four-dimensional CFT we provide a direct computation of the BMS algebra formed by a specific set of light-ray operators in theories with no light scalar conformal primaries. In four-dimensional CFT we define a new infinite set of light-ray operators constructed from the stress-tensor, which all have well-defined matrix elements. These are a direct generalization of the two-dimensional Virasoro light-ray operators that are obtained from a conformal embedding of Minkowski space in the Lorentzian cylinder. They obey Hermiticity conditions similar to their two-dimensional analogues, and also share the property that a semi-infinite subset annihilates the vacuum.

我们讨论了 Lorentzian CFT 中局部算子的交换子的一些一般方面，这些方面可以从欧几里得算子乘积展开 (OPE) 的适当解析延拓中获得。换向器仅作为分布才有意义，并且必须小心从 OPE 中提取正确的分布。我们在二维和四维 CFT 中提供显式计算，主要关注应力张量分量的换向器。我们重新推导了几个熟悉的结果，例如自由场理论的规范对易关系、Poincaré 代数的局部形式和二维 CFT 的 Virasoro 代数。然后我们考虑由应力张量构建的光线算子的换向器。使用四维 CFT 中光片极限的简化特征，我们提供了在没有光标量共形原色的理论中由一组特定的光线算子形成的 BMS 代数的直接计算。在四维 CFT 中，我们定义了由应力张量构建的一组新的无限光线算子，它们都具有明确定义的矩阵元素。这些是二维 Virasoro 光线算子的直接推广，这些算子是从 Lorentzian 圆柱体中 Minkowski 空间的共形嵌入中获得的。它们遵循类似于二维类似物的厄密性条件，并且还具有半无限子集消灭真空的特性。在四维 CFT 中，我们定义了由应力张量构建的一组新的无限光线算子，它们都具有明确定义的矩阵元素。这些是二维 Virasoro 光线算子的直接推广，这些算子是从 Lorentzian 圆柱体中 Minkowski 空间的共形嵌入中获得的。它们遵循类似于二维类似物的厄密性条件，并且还具有半无限子集消灭真空的特性。在四维 CFT 中，我们定义了由应力张量构建的一组新的无限光线算子，它们都具有明确定义的矩阵元素。这些是二维 Virasoro 光线算子的直接推广，这些算子是从 Lorentzian 圆柱体中 Minkowski 空间的共形嵌入中获得的。它们遵循类似于二维类似物的厄密性条件，并且还具有半无限子集消灭真空的特性。