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

• Published in: Communications in mathematical physics 2021-05, Vol.384 (1), p.201-244
• Main Author: Nikolov, Nikolay M.
• Format: Article
• Language: English
• Subjects: Algebra; Classical and Quantum Gravitation; Complex Systems; Differential equations; Linear operators; Mathematical and Computational Physics; Mathematical Physics; Meromorphic functions; Operators (mathematics); Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-021-04051-9

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 Γ : M → N between two modules of a commutative associative algebra A , such that for every m ∈ M the assignment a ↦ Γ ( a · m ) is a differential operator A → N in the usual sense. The residue of a meromorphic function at its pole is an example of a semi-differential operator.