Restricted Weyl invariance in four-dimensional curved spacetime

We discuss the physics of restricted Weyl invariance, a symmetry of dimensionless actions in four-dimensional curved space time. When we study a scalar field nonminimally coupled to gravity with Weyl(conformal) weight of $\ensuremath{-}1$ (i.e. scalar field with the usual two-derivative kinetic term), we find that dimensionless terms are either fully Weyl invariant or are Weyl invariant if the conformal factor $\mathrm{\ensuremath{\Omega}}(x)$ obeys the condition ${g}^{\ensuremath{\mu}\ensuremath{\nu}}{\ensuremath{\nabla}}_{\ensuremath{\mu}}{\ensuremath{\nabla}}_{\ensuremath{\nu}}\mathrm{\ensuremath{\Omega}}=0$. We refer to the latter as restricted Weyl invariance. We show that all the dimensionless geometric terms such as ${R}^{2}$, ${R}_{\ensuremath{\mu}\ensuremath{\nu}}{R}^{\ensuremath{\mu}\ensuremath{\nu}}$ and ${R}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\sigma}\ensuremath{\tau}}{R}^{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\sigma}\ensuremath{\tau}}$ are restricted Weyl invariant. Restricted Weyl transformations possesses nice mathematical properties such as the existence of a composition and an inverse in four-dimensional space-time. We exemplify the distinction among rigid Weyl invariance, restricted Weyl invariance and the full Weyl invariance in dimensionless actions constructed out of scalar fields and vector fields with Weyl weight zero.

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