Gauging the spacetime metric -- looking back and forth a century later

TL;DR

This paper reviews the historical development and renewed interest in Weyl geometry, a generalized geometric framework linking scale invariance to fundamental physics, from its origins to modern applications in cosmology and particle physics.

Contribution

It provides a comprehensive overview of Weyl geometry's evolution and its recent resurgence in theoretical physics and cosmology over the past century.

Findings

Weyl geometry influenced early unified field theories.

Recent studies connect Weyl geometry to modern cosmological models.

Weyl's ideas continue to inspire research in fundamental physics.

Abstract

H. Weyl's proposal of 1918 for generalizing Riemannian geometry by local scale gauge (later called {\em Weyl geometry}) was motivated by mathematical, philosophical and physical considerations. It was the starting point of his unified field theory of electromagnetism and gravity. After getting disillusioned with this research program and after the rise of a convincing alternative for the gauge idea by translating it to the phase of wave functions and spinor fields in quantum mechanics, Weyl no longer considered the original scale gauge as physically relevant. About the middle of the last century the question of conformal and/or local scale gauge transformation were reconsidered by different authors in high energy physics (Bopp, Wess, et al.) and, independently, in gravitation theory (Jordan, Fierz, Brans, Dicke). In this context Weyl geometry attracted new interest among different groups of physicists (Omote/Utiyama/Kugo, Dirac/Canuto/Maeder, Ehlers/Pirani/Schild and others), often by hypothesizing a new scalar field linked to gravity and/or high energy physics. Although not crowned by immediate success, this ``retake'' of Weyl geometrical methods lives on and has been extended a century after Weyl's first proposal of his basic geometrical structure. It finds new interest in present day studies of elementary particle physics, cosmology, and philosophy of physics.