Restricted Weyl invariance in four-dimensional curved spacetime

TL;DR

This paper explores a special form of Weyl invariance in four-dimensional curved spacetime, called restricted Weyl invariance, which applies to certain dimensionless actions and geometric terms, revealing unique mathematical properties.

Contribution

It introduces and characterizes restricted Weyl invariance, a novel symmetry condition for dimensionless actions in four-dimensional curved spacetime, and analyzes its mathematical properties and distinctions from other invariances.

Findings

Restricted Weyl invariance applies when the conformal factor satisfies a specific differential equation.

All quadratic curvature terms are restricted Weyl invariant.

Restricted Weyl transformations have well-defined composition and inverse operations.

Abstract

We discuss the physics of {\it 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 $g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\Omega=0$. We refer to the latter as {\it restricted Weyl invariance}. We show that all the dimensionless geometric terms such as $R^2$, $R_{\mu\nu}R^{\mu\nu}$ and $R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}$ 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.