In this paper, we investigate the relationship between quadratic gravity and a restricted Weyl symmetry where a gauge parameter Ω(x) of Weyl transformation satisfies a constraint □Ω = 0 in a curved spacetime. First, we briefly review a model with a restricted gauge symmetry on the basis of QED, where a U(1) gauge parameter 𝜃(x) obeys a similar constraint □𝜃 = 0 in a flat Minkowski spacetime, 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.

中文翻译：

---

二次引力和受限外尔对称

在本文中，我们研究了二次重力与受限外尔对称性之间的关系，其中规范参数Ω(X)的 Weyl 变换满足一个约束□Ω = 0在弯曲的时空中。首先，我们简要回顾一个基于 QED 的受限规范对称模型，其中ü(1)量规参数𝜃(X)服从类似的约束□𝜃 = 0在平坦的闵可夫斯基时空中，并解释受限规范对称性消除了规范场的一种壳上模式，该模式与费曼规范一起仅留下两个横向极化作为物理状态。接下来，表明受限的 Weyl 对称性还消除了平面 Minkowski 背景周围二次引力中偶极场的一个分量，只留下一个标量状态。最后，我们通过使用二次引力的零能量定理证明了受限的外尔对称不能消除静态背景度量中的任何动态自由度。这一事实也适用于欧几里得背景度量而不强加静态条件。