We introduce a symmetric operad whose algebras are the operator product expansion (OPE) Algebras of quantum fields. There is a natural classical limit for the algebras over this operad and they are commutative associative algebras with derivations. The latter are the algebras of classical fields. In this paper we completely develop our approach to models of quantum fields, which come from vertex algebras in higher dimensions. However, our approach to OPE algebras can be extended to general quantum fields even over curved space–time. We introduce a notion of OPE operations based on the new notion of semi-differential operators. The latter are linear operators \(\varGamma :\mathcal {M}\rightarrow \mathcal {N}\) between two modules of a commutative associative algebra \(\mathcal {A}\), such that for every \(m \in \mathcal {M}\) the assignment \(a\mapsto \varGamma (a \cdot m)\) is a differential operator \(\mathcal {A}\rightarrow \mathcal {N}\) in the usual sense. The residue of a meromorphic function at its pole is an example of a semi-differential operator.

我们介绍了对称算子，其代数是量子场的算子乘积展开（OPE）代数。对于这个算子，代数有一个自然的经典极限，它们是带导数的可交换关联代数。后者是古典领域的代数。在本文中，我们完全开发了量子场模型的方法，该模型来自更高维的顶点代数。但是，即使在弯曲的时空上，我们的OPE代数方法也可以扩展到一般的量子场。我们基于新的半微分运算符概念引入OPE操作的概念。后者是可交换关联代数\（\ mathcal {A} \）的两个模块之间的线性运算符\（\ varGamma：\ mathcal {M} \ rightarrow \ mathcal {N } \），这样，对于每个\（m \ in \ mathcal {M} \），赋值\（a \ mapsto \ varGamma（a \ cdot m）\）就是一个差分运算符\（\ mathcal {A} \ rightarrow \ mathcal { N} \）。亚纯函数极点处的残差是半微分算子的一个示例。