In this paper, we elucidate the relation between the restricted Weyl symmetry and spontaneous symmetry breakdown of conformal symmetry. Using a scalar–tensor gravity, we show that the restricted Weyl symmetry leads to spontaneous symmetry breakdown of a global scale symmetry when the vacuum expectation value of a scalar field takes a nonzero value. It is then shown that this spontaneous symmetry breakdown induces spontaneous symmetry breakdown of special conformal symmetry in a flat Minkowski spacetime, but the resultant Nambu–Goldstone boson is not an independent physical mode but expressed in terms of the derivative of the dilaton which is the Nambu–Goldstone boson of the global scale symmetry. In other words, the theories which are invariant under the general coordinate transformation and the restricted Weyl transformation exhibit a Nambu–Goldstone phase where both special conformal transformation and dilatation are spontaneously broken while preserving the Poincaré symmetry.

中文翻译：

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受限的外尔对称性和共形对称性的自发对称性破坏

在本文中，我们阐明了受限的外尔对称性与共形对称性的自发对称性破坏之间的关系。使用标量张量引力，我们表明当标量场的真空期望值取非零值时，受限的 Weyl 对称性会导致全局尺度对称性的自发对称性破坏。然后表明，这种自发的对称性破坏导致了平坦 Minkowski 时空中特殊共形对称性的自发对称性破坏，但由此产生的 Nambu-Goldstone 玻色子不是独立的物理模式，而是以膨胀子的导数表示，即 Nambu ——全球尺度对称的戈德斯通玻色子。换一种说法，