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

February 16, 2019

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.

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