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 -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 Omega(x) obeys the condition g(mu nu)del(mu)del(nu)Omega = 0. We refer to the latter as restricted Weyl invariance. We show that all the dimensionless geometric terms such as R-2, R mu nu R mu nu and (RR mu nu sigma tau)-R-mu nu sigma 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.