标题： 哈特尔-霍金无边界提案和霍拉瓦-利夫希茨引力 显示英文标题

标题： Hartle-Hawking No-boundary Proposal and Hořava-Lifshitz Gravity

摘要： 我们研究在霍拉瓦-利夫希茨引力框架下的哈特尔-霍金无边界提议。前者是一个描述宇宙量子创生的重要假说，而后者是一种潜在的量子引力理论，至少在所谓的可投影版本中，它确保了可重整化和单位性。为了简化，我们专注于一个由一组局部宇宙组成的全局宇宙，每个局部宇宙都是闭合的、均匀的和各向同性的。尽管将无边界提议应用于霍拉瓦-利夫希茨引力并不直接，但我们证明可以利用量子引力的洛伦兹路径积分公式在霍拉瓦-利夫希茨引力中提出该提议。在可投影的霍拉瓦-利夫希茨引力中，全局宇宙的无边界波函数不可避免地包含由“暗物质作为积分常数”引起的不同局部宇宙之间的纠缠。另一方面，在不可投影版本中，全局宇宙的无边界波函数只是每个局部宇宙波函数的直接乘积。然后我们讨论在狄利克雷和罗宾边界条件下无边界波函数是如何提出的。对于狄利克雷边界条件，我们指出由于高维算符其有效作用量发散，但原则上可以通过考虑重整化群流来缓解这个问题。然而，利用皮卡德-列夫谢茨理论来确定相关临界点并执行复数时间间隔积分，我们发现只能得到隧穿波函数，正如广义相对论的情况一样。另一方面，对于初始超曲面上具有特定虚数哈勃膨胀率的罗宾边界条件，可以在霍拉瓦-利夫希茨引力中获得无边界波函数。 显示英文摘要

摘要： We study the Hartle-Hawking no-boundary proposal in the framework of Ho\v{r}ava-Lifshitz gravity. The former is a prominent hypothesis that describes the quantum creation of the universe, while the latter is a potential theory of quantum gravity that ensures renormalizability and unitarity, at least in the so-called projectable version. For simplicity, we focus on a global universe composed of a set of local universes each of which is closed, homogeneous and isotropic. Although applying the no-boundary proposal to Ho\v{r}ava-Lifshitz gravity is not straightforward, we demonstrate that the proposal can be formulated within the Ho\v{r}ava-Lifshitz gravity utilizing the Lorentzian path integral formulation of quantum gravity. In projectable Ho\v{r}ava-Lifshitz gravity, the no-boundary wave function of the global universe inevitably contains entanglement between different local universes induced by ``dark matter as integration constant''. On the other hand, in the non-projectable version, the no-boundary wave function of the global universe is simply the direct product of wave functions of each local universe. We then discuss how the no-boundary wave function is formulated under Dirichlet and Robin boundary conditions. For the Dirichlet boundary condition, we point out that its on-shell action diverges due to higher-dimensional operators, but this problem can in principle be ameliorated by taking into account the renormalization group flow. However, utilizing the Picard-Lefschetz theory to identify the relevant critical points and performing the complex lapse integration, we find that only the tunneling wave function can be obtained, as in the case of general relativity. On the other hand, for the Robin boundary condition with a particular imaginary Hubble expansion rate at the initial hypersurface, the no-boundary wave function can be achieved in the Ho\v{r}ava-Lifshitz gravity.