Conformal covariance of massless free nets

TL;DR

This paper reviews the conformal covariance of massless free nets in quantum physics, providing simplified proofs of their symmetry properties and a new interpretation of their algebraic structure.

Contribution

It offers simplified proofs of conformal covariance for massless free nets and a novel group-theoretical interpretation of their embedding, advancing understanding of their symmetry and algebraic properties.

Findings

Massless canonical representations extend to conformal group representations.

Massless free nets are conformally covariant for any helicity.

The group-theoretical embedding characterizes the free net structure.

Abstract

In the present paper we review in a fibre bundle context the covariant and massless canonical representations of the Poincare' group as well as certain unitary representations of the conformal group (in 4 dimensions). We give a simplified proof of the well-known fact that massless canonical representations with discrete helicity extend to unitary and irreducible representations of the conformal group mentioned before. Further we give a simple new proof that massless free nets for any helicity value are covariant under the conformal group. Free nets are the result of a direct (i.e. independent of any explicit use of quantum fields) and natural way of constructing nets of abstract C*-algebras indexed by open and bounded regions in Minkowski space that satisfy standard axioms of local quantum physics. We also give a group theoretical interpretation of the embedding ${\got I}$ that completely characterizes the free net: it reduces the (algebraically) reducible covariant representation in terms of the unitary canonical ones. Finally, as a consequence of the conformal covariance we also mention for these models some of the expected algebraic properties that are a direct consequence of the conformal covariance (essential duality, PCT--symmetry etc.).