The fall and the rise of Weyl gauge theory

TL;DR

This paper reviews the evolution of Weyl gauge theory from its origins to a modern, anomaly-free formulation with a geometric interpretation and implications for gravity and cosmology.

Contribution

It presents a modern, physically consistent Weyl gauge theory of gravity with a geometric foundation, including a Weyl-Dirac-Born-Infeld extension.

Findings

The theory is the only gauge theory of spacetime symmetry with a physical gauge boson.

It is Weyl anomaly-free and has an exact geometric interpretation.

It generates Einstein-Hilbert action and a positive cosmological constant in broken phase.

Abstract

In 1918 Weyl introduced Weyl conformal geometry and its associated quadratic action which was the first gauge theory, of a spacetime symmetry, the Weyl gauge theory (of dilatations and Poincar\'e symmetry). The initial physical interpretation of his theory was however short-lived and led to the downfall of Weyl geometry as a physical theory. We review how this action was re-born into a physical Weyl gauge theory of gravity. This is the only gauge theory of a spacetime symmetry with a physical gauge boson, is Weyl anomaly-free, has {\it exact} geometric interpretation, with all scales of geometric origin, and generates Einstein-Hilbert action and a positive cosmological constant in its spontaneously broken phase. A more fundamental Weyl-Dirac-Born-Infeld gauge theory action exists in Weyl geometry, that does not need a UV regularisation, of which the (geometrically regularised) Weyl gauge theory is the leading order.