Scaling Algebras and Renormalization Group in Algebraic Quantum Field Theory. II. Instructive Examples

TL;DR

This paper applies the scaling algebra framework to free field theories in various dimensions, revealing how short-distance limits relate massive and massless theories and uncovering new structures like non-trivial centers in one dimension.

Contribution

It demonstrates the application of scaling algebra methods to explicit free field models, showing how short-distance limits produce known and novel algebraic structures.

Findings

For s=2,3, the scaling limit yields the algebra of massless free fields.

In s=1, the algebra acquires a non-trivial center, indicating charged states.

The results provide insights into the short-distance behavior of quantum field theories.

Abstract

The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of massive free field theory in s = 1,2 and 3 spatial dimensions. Not quite unexpectedly, one obtains for s = 2,3 in the scaling (short distance) limit the algebra of local observables in massless free field theory. The case s =1 offers, however, some surprises. There the algebra of observables acquires in the scaling limit a non-trivial center and describes charged physical states satisfying Gauss' law. The latter result is of relevance for the interpretation of the Schwinger model at short distances and illustrates the conceptual and computational virtues of the method.