Semi-Differential Operators and the Algebra of Operator Product Expansion of Quantum Fields

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 Gamma from M to N between two modules of a commutative associative algebra A, such that for every m belonging to M the assignment "a maps to Gamma(a.m)" is a differential operator from A to N in the usual sense. The residue of a meromorphic function at its pole is an example of a semi-differential operator.