Quadratic Gravity and Restricted Weyl Symmetry

TL;DR

This paper explores how a restricted Weyl symmetry constrains degrees of freedom in quadratic gravity, showing it reduces some modes but cannot eliminate all dynamical degrees in static or Euclidean backgrounds.

Contribution

It demonstrates that restricted Weyl symmetry removes one component of the dipole field in quadratic gravity but cannot eliminate all dynamical modes in certain backgrounds.

Findings

Restricted Weyl symmetry reduces one scalar mode in quadratic gravity.

It cannot remove all dynamical degrees of freedom in static backgrounds.

The symmetry's effect is limited in Euclidean and non-static backgrounds.

Abstract

We investigate the relationship between quadratic gravity and a restricted Weyl symmetry where a gauge parameter $\Omega(x)$ of Weyl transformation satisfies a constraint $\Box \Omega = 0$ in a curved space-time. First, we briefly review a model with a restricted gauge symmetry on the basis of QED where a $U(1)$ gauge parameter $\theta(x)$ obeys a similar constraint $\Box \theta = 0$ in a flat Minkowski space-time, and explain that the restricted gauge symmetry removes one on-shell mode of gauge field, which together with the Feynman gauge leaves only two transverse polarizations as physical states. Next, it is shown that the restricted Weyl symmetry also eliminates one component of a dipole field in quadratic gravity around a flat Minkowski background, leaving only a single scalar state. Finally, we show that the restricted Weyl symmetry cannot remove any dynamical degrees of freedom in static background metrics by using the zero-energy theorem of quadratic gravity. This fact also holds for the Euclidean background metrics without imposing the static condition.