We consider modified Weyl gravity where a Dirac spinor field is nonminimally coupled to gravity. It is assumed that such modified gravity is some approximation for the description of quantum gravitational effects related to the gravitating spinor field. It is shown that such a theory contains solutions for a class of metrics which are conformally equivalent to the Hopf metric on the Hopf fibration. For this case, we obtain a full discrete spectrum of the solutions and show that they can be related to the Hopf invariant on the Hopf fibration. The expression for the spin operator in the Hopf coordinates is obtained. It is demonstrated that this class of conformally equivalent metrics contains the following: (a) a metric describing a toroidal wormhole without exotic matter; (b) a cosmological solution with a bounce and inflation and (c) a transition with a change in metric signature. A physical discussion of the results is given.

中文翻译：

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F(B2) 修正的外尔引力中的旋量场解

我们考虑修正的外尔重力，其中狄拉克旋量场与重力非最小耦合。假设这种修改后的引力是描述与引力旋量场相关的量子引力效应的某种近似。结果表明，这种理论包含一类度量的解，这些度量与 Hopf 纤维化上的 Hopf 度量等价。对于这种情况，我们获得了解的完整离散谱，并表明它们可以与 Hopf 纤维化的 Hopf 不变量相关。得到 Hopf 坐标中自旋算子的表达式。已经证明，这类共形等效度量包含以下内容： (a) 描述没有外来物质的环形虫洞的度量；(b) 具有反弹和膨胀的宇宙学解决方案和 (c) 具有度量特征变化的过渡。给出了结果的物理讨论。