arXiv 2412.197822024-12hep-thastro-ph.COgr-qc

宇宙学微扰与Wheeler DeWitt方程的交汇

Cosmological perturbations meet Wheeler DeWitt

Federico Piazza, Siméon Vareilles

备注：: 32 pages, two figures. Comments welcome. v2: several minor improvements, references added

阅读 PDF

arXiv 页面 ↗HTML 原文 ↗

AI总结：

该研究通过Wheeler-DeWitt方程的近似解，与宇宙学微扰理论对比，在迷你超空间中引入小耦合常数，建立半经典波函数与标准微扰波函数的关系，并用两个例子验证，澄清条件概率与规范固定问题。

摘要：

我们研究Wheeler DeWitt (WdW)方程的近似解，并将其与宇宙学微扰理论的标准结果进行比较。在迷你超空间中，我们引入一个无量纲引力耦合常数$α$，它通常非常小，在WKB展开中起到类似$\hbar$的作用。我们寻求形如$Ψ= e^{iS/α} ψ$的解，它们是给定经典背景时空最接近的量子类比。函数$S$满足Hamilton-Jacobi方程，而$ψ$服从类Schrödinger方程，并可赋予概率解释。半经典极限提示了$ψ$与标准微扰理论波函数$ψ_P$之间的特定关系。我们在两个主要例子中验证了这一关系：一个具有纯指数势的标量场（已知其简单标度解）和一个在场空间原点附近展开的慢滚场景。每个例子都在两种不同的规范中计算，即迷你超空间中对应的幺正规范和空间平坦规范。我们讨论了相对于经典背景轨迹的可能偏离，以及WdW方程中存在但微扰方法中不存在的更高阶“时间”导数项。我们阐明了波函数的条件概率内容，以及这与微扰理论中标准规范固定过程的关系。

英文摘要：

We study approximate solutions of the Wheeler DeWitt (WdW) equation and compare them with the standard results of cosmological perturbation theory. In mini-superspace, we introduce a dimensionless gravitational coupling $α$ that is typically very small and functions like $\hbar$ in a WKB expansion. We seek solutions of the form $Ψ= e^{iS/α} ψ$ that are the closest quantum analog of a given classical background spacetime. The function $S$ satisfies the Hamilton-Jacobi equation, while $ψ$ obeys a Schrödinger-like equation and can be given a probabilistic interpretation. The semiclassical limit suggests a specific relation between $ψ$ and the standard perturbation-theory wavefunction $ψ_P$. We verify this relation in two main examples: a scalar field with a purely exponential potential, of which simple scaling solutions are known and a slow-roll scenario expanded in the vicinity of the origin in field space. Each example is worked out in two different gauges, that are the minisuperspace equivalent of unitary gauge and spatially flat gauge. We discuss possible deviations from the classical background trajectory as well as the higher ``time" derivative terms that are present in the WdW equation but not in the perturbative approach. We clarify the \emph{conditional probability} content of the wavefunctions and how this is related with the standard gauge fixing procedure in perturbation theory.

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。

AI总结：

摘要：

英文摘要：

相关论文