Conformal Transformations as Observables

TL;DR

This paper constructs a unique inner representation of the conformal group in quantum field theory that implements automorphisms on von Neumann algebras, revealing that conformal transformations are weak limits of local observables.

Contribution

It introduces the Borchers-Sugawara construction, a novel method for representing conformal transformations as observables in quantum field theory.

Findings

Conformal transformations with positive energy are weak limit points of local observables.

The construction satisfies the spectrum condition and acts trivially on invariant vectors.

Implications for chiral subnets are discussed.

Abstract

C denotes either the conformal group in 3+1 dimensions, or in one chiral dimension. Let U be a unitary, strongly continuous representation of C satisfying the spectrum condition and inducing, by its adjoint action, automorphisms of a v.Neumann algebra A. We construct the unique inner representation U^A of the universal covering group of C implementing these automorphisms. U^A satisfies the spectrum condition and acts trivially on any U-invariant vector. This means in particular: Conformal transformations of a field theory having positive energy are weak limit points of local observables. Some immediate implications for chiral subnets are given. We propose the name ``Borchers-Sugawara construction''.