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

TL;DR

This paper formulates perturbative quantum field theory within algebraic quantum field theory, showing the algebra of interacting fields is determined by small regions and introducing an algebraic loop expansion framework.

Contribution

It provides an algebraic formulation of the loop expansion and demonstrates the additive property of the algebra of interacting fields in Minkowski space.

Findings

Algebra of interacting fields is determined by subalgebras of small regions.

Introduces a projective system for observables up to n loops.

Provides a local algebraic formulation of the quantum action principle.

Abstract

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 ${\cal A}^{(n)}$ of observables ``up to $n$ loops'' where ${\cal A}^{(0)}$ 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.