# Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion

January 20, 2000

The perturbative treatment of quantum field theory is formulated within the
framework of algebraic quantum field theory. We show that the algebra of
interacting fields is additive, i.e. fully determined by its subalgebras
associated to arbitrary small subregions of Minkowski space. We also give an
algebraic formulation of the loop expansion by introducing a projective system
${\mathcal A}^{(n)}$ of observables ``up to $n$ loops'' where ${\mathcal A}^{(0)}$ is
the Poisson algebra of the classical field theory. Finally we give a local
algebraic formulation for two cases of the quantum action principle and compare
it with the usual formulation in terms of Green's functions.

Keywords:

*none*