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arXiv 2112.04522gr-qchep-th
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宇宙的波函数作为最终膨胀宇宙的总和

Wave function of the Universe as a sum over eventually inflating universes

Karthik Rajeev

备注：
16 pages, 3 figures, published version
发表信息：
Phys. Rev. D 106 (2022), 023511
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摘要：

我们考虑这样一个建议：把宇宙的波函数定义为最终膨胀的时空的总和。在minisuperspace模型中，我们明确地表明，一个简单的家庭的初始条件，参数化的正实数$a_0$，可以施加实现这一规定。由此产生的波函数被发现是成比例的Hartle-Hawking波函数和它的依赖于$a_0$是通过一个整体的相位因子。出于这一观察，我们问是否有可能分析扩展$a_0$到一个扩展区域$\bar{\mathcal{D}}$在复杂的$a_0-$平面，同时保留哈特尔-霍金形式的波函数。我们使用的路径积分的收敛条件和最近的一个定理，由于Kontsevich和Segal，进一步扩展威滕，明确找到$\bar{\mathcal{D}}$。有趣的是，$\bar{\mathcal{D}}$边界上的一个特殊点恢复了精确的无边界波函数。接下来，我们证明了我们的处方导致一个家庭的量子态的微扰，从而产生标度不变的功率谱。如果我们要求在$\bar{\mathcal{D}}$中的“无边界点”处有一个匹配条件，作为我们处方的一个额外成分，那么我们会收敛到微扰的唯一量子态。

英文摘要：

We consider a proposal to define the wave function of the Universe as a sum over spacetimes that eventually inflate. In the minisuperspace model, we explicitly show that a simple family of initial conditions, parametrized by a positive real number $a_0$, can be imposed to realise this prescription. The resulting wave function is found to be proportional to the Hartle-Hawking wave function and its dependence on $a_0$ is only through an overall phase factor. Motivated by this observation, we ask whether it is possible to analytically extend $a_0$ to an extended region $\bar{\mathcal{D}}$ in complex $a_0-$plane, while retaining the Hartle-Hawking form of the wave function. We use the condition for convergence of path integral and a recent theorem due to Kontsevich and Segal, further extended by Witten, to explicitly find $\bar{\mathcal{D}}$. Interestingly, a special point on the boundary of $\bar{\mathcal{D}}$ recovers the exact no-boundary wave function. Following that, we show that our prescription leads to a family of quantum states for the perturbations, which give rise to scale-invariant power spectra. If we demand, as an extra ingredient to our prescription, a matching condition at the "no-boundary point" in $\bar{\mathcal{D}}$, we converge on a unique quantum state for the perturbations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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