World-Line Actions in Weyl Geometry
arXiv:2607.26868
The paper derives a dimensionless, Weyl‑invariant world‑line action for a particle in Weyl geometry, showing that proper time cannot be defined in the symmetric phase but reappears after spontaneous symmetry breaking, and presents an equivalent quadratic action using an einbein.
Abstract
In this note we construct, from a gauge theory perspective, the world-line action for a particle moving on a time-like curve in Weyl geometry. The action we find is dimensionless, Weyl invariant, additive and, in general, non-local due to an open Wilson line which we have to add in order to account for a general Weyl field. In special cases, this Wilson line can be local, but the geometry becomes integrable. The action can not be used to measure the proper time as it is dimensionless and no mass parameter is allowed in the symmetric phase of the theory. We show that the usual conditions for defining proper time: affine parametrization, dimension of time and additivity supplemented by the requirement of Weyl invariance can not be fulfilled simultaneously and therefore no satisfactory notion of proper time exists in the symmetric phase. Under spontaneous symmetry breaking the particle acquires a mass, the action becomes Riemannian and the proper time can be again defined. We also construct a classically equivalent quadratic action by using an einbein on the world-line and show that the non-locality can be seen to arise from integrating out a constrained field.
20 pages