Critical reassessment of the restricted Weyl symmetry

TL;DR

This paper critically examines the concept of restricted Weyl symmetry in scalar-tensor theories, revealing that it is not a true symmetry of the equations of motion due to nonlocal effects and boundary considerations.

Contribution

The paper clarifies that restricted Weyl transformations are not genuine symmetries of the full scalar-tensor system, correcting previous assumptions.

Findings

Restricted Weyl transformations are nonlocal due to conformal factor restrictions.

Equations of motion are not invariant under restricted Weyl transformations.

Boundary terms cannot be ignored in the presence of nonlocal transformations.

Abstract

A class of globally scale-invariant scalar-tensor theories have been proposed to be invariant under a larger class of transformations that take the form of local Weyl transformations supplemented by a restriction that the conformal factor satisfies a covariant Klein-Gordon equation. The action of these theories indeed seems to be invariant under such transformations up to boundary terms, this property being referred to as ``restricted Weyl symmetry''. However, we find that corresponding equations of motion are not invariant under these transformations. This is a paradox, that is explained by realizing that the restriction condition on the conformal factor forces the restricted Weyl transformation to be a nonlocal transformation. For nonlocal transformations would-be boundary terms cannot in general be discarded from the action. Moreover, variations of trajectories cannot be assumed to vanish at boundaries of the action when deriving equations of motion. We illustrate both of these less known properties by considering a series of simple examples. Finally, we apply these observations to the case of globally scale-invariant scalar-tensor theories to demonstrate that restricted Weyl transformations are, in fact, not symmetries of the full system.