Fresh perspective on gauging the conformal group

TL;DR

This paper explores the geometric foundations of conformal transformations and introduces extended Weyl gauge theory (eWGT) as a promising gauge theory of the conformal group, addressing limitations of previous approaches.

Contribution

It clarifies the geometric nature of conformal transformations and proposes eWGT as a complete gauge theory of the conformal group, including invariance under inversions and conservation laws.

Findings

eWGT is invariant under full local conformal transformations.

eWGT includes conservation laws extending global conformal invariance.

Standard Weyl gauge theory lacks some properties of a complete conformal gauge theory.

Abstract

We consider the construction of gauge theories of gravity that are invariant under local conformal transformations. We first clarify the geometric nature of global conformal transformations, in both their infinitesimal and finite forms, and the consequences of global conformal invariance for field theories, before reconsidering existing approaches for gauging the conformal group, namely auxiliary conformal gauge theory and biconformal gauge theory, neither of which is generally accepted as a complete solution. We then demonstrate that, provided any matter fields belong to an irreducible representation of the Lorentz group, the recently proposed extended Weyl gauge theory (eWGT) may be considered as an alternative method for gauging the conformal group, since eWGT is invariant under the full set of local conformal transformations, including inversions, as well as possessing conservation laws that provide a natural local generalisation of those satisfied by field theories with global conformal invariance, and also having an `ungauged' limit that corresponds to global conformal transformations. By contrast, although standard Weyl gauge theory also enjoys the first of these properties, it does not share the other two, and so cannot be considered a valid gauge theory of the conformal group.