A C*-algebraic Approach to Interacting Quantum Field Theories
Abstract
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.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- March 2020
- DOI:
- 10.1007/s00220-020-03700-9
- arXiv:
- arXiv:1902.06062
- Bibcode:
- 2020CMaPh.377..947B
- Keywords:
-
- Mathematical Physics;
- High Energy Physics - Theory;
- Mathematics - Operator Algebras;
- Quantum Physics
- E-Print:
- 28 pages, 4 figures, v3: sign error corrected