A C-algebraic Approach to Interacting Quantum Field Theories

• Published in: Communications in mathematical physics Vol. 377; no. 2; pp. 947 - 969
• Main Authors: Buchholz, Detlev, Fredenhagen, Klaus
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2020; Springer Nature B.V
• Subjects: Algebra; Axioms; Classical and Quantum Gravitation; Complex Systems; Field theory; Mathematical and Computational Physics; Mathematical Physics; Operators (mathematics); Physics; Physics and Astronomy; Quantum field theory; Quantum Physics; Quantum theory; Relativistic effects; Relativistic theory; Relativity; Relativity Theory; Scalars; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-020-03700-9

A novel C*-algebraic framework is presented for relativistic quantum field theories, fixed by a Lagrangean. It combines the postulates of local quantum physics, encoded in the Haag–Kastler axioms, with insights gained in the perturbative approach to quantum field theory. Key ingredients are an appropriate version of Bogolubov’s relative S -operators and a reformulation of the Schwinger–Dyson equations. These are used to define for any classical relativistic Lagrangean of a scalar field a non-trivial local net of C*-algebras, encoding the resulting interactions at the quantum level. The construction works in any number of space-time dimensions. It reduces the longstanding existence problem of interacting quantum field theories in physical spacetime to the question of whether the C*-algebras so constructed admit suitable states, such as stable ground and equilibrium states. The method is illustrated on the example of a non-interacting field and it is shown how to pass from it within the algebra to interacting theories by relying on a rigorous local version of the interaction picture.