Weyl Rescalings: A First Order Approach

Abstract

Weyl transformations, --local rescalings in which the metric transforms as $g_{μν}(x)\longrightarrow Ω^2(x)\, g_{μν}(x)$--, are expected to be a symmetry of nature at extremely high energy, that should be (spontaneously) broken at low energy. In Cartan's first order formulation of gravity this translates into the condition that the local frames (vielbein) scale as $e^a(x) \longrightarrow Ω(x) e^a(x)$. In this construction, the Lorentz connection one-form fields $ω^a{}_b$ are independent from $e^a$, and since Lorentz transformations and dilatations commute, $ω$ and $R(ω)$ must be Weyl invariant, while in the second order metric formalism the Ricci tensor undergoes a non-homogeneous transformation. This generates a tension between the two formulations, which should be equivalent in the vanishing torsion limit. Here we show how the puzzle can be resolved taking into account the role of torsion. This also sheds some light on the conformal coupling of matter fields to gravity.