  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.
