# A C*-algebraic approach to interacting quantum field theories

## TL;DR

This paper introduces a new C*-algebraic framework for relativistic quantum field theories that combines local quantum physics principles with perturbative insights, enabling the construction of interacting models across dimensions.

## Contribution

It develops a novel C*-algebraic approach to define local nets of algebras for interacting quantum fields from classical Lagrangians, addressing the existence problem in quantum field theory.

## Key findings

- Constructs local C*-algebras for scalar fields in any dimension.
- Shows how to incorporate interactions using a local interaction picture.
- Reduces the existence problem to finding suitable states on the constructed algebras.

## 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 spacetimeto 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.