WKB近似下Bianchi IX模型角点的量子动力学
Quantum dynamics of the corner of the Bianchi IX model in WKB approximation
摘要：
在本文中，我们分析了Bianchi IX宇宙动力学的角落区域相关的潜在的长期的空间曲率诱导在Minissuperspace。研究是在真空中进行的，并且存在无质量标量场$ϕ$和宇宙学常数项$Λ$。我们调查的动力学WKB的情况下，各向同性的Misner变量（体积）和两个各向异性的（和$ϕ$当存在）的一个被处理在一个半经典的水平，而其余的各向异性的自由度，一个被困在角落里，被描述在一个纯量子水平。对于频率依赖于时间的谐振势，量子动力学总是归结为依赖于时间的薛定谔方程。真空的情况下处理的限制collpasing和膨胀的宇宙，而$ϕ$和$Λ$的存在下的动力学研究$t \rightarrow \infty $。在这两种分析中，各向异性变量的量子动力学与其概率密度的衰减标准差相关联，对应于相关量子各向异性的抑制。在真空的情况下，角配置成为一个吸引子的动力学和演化类似于一个非奇异初始宇宙的限制之一的陶布宇宙学。这表明，如果比安奇动力学进入足够的潜在的角落，然后初始的奇异性被删除，并出现了陶伯图片。当$ϕ$存在时，很好地模拟了暴胀宇宙的De-Sitter阶段。在这里，我们表明，无论是经典和量子各向异性指数抑制，使由此产生的动力学对应于一个各向同性封闭罗伯逊-沃克几何。
英文摘要：
In this paper we analyse the Bianchi IX Universe dynamics within the corner region associated to the potential term which the spatial curvature induces in the Minisuperspace. The study is done in the vacuum and in the presence of a massless scalar field $ϕ$ and a cosmological constant term $Λ$. We investigate the dynamics in terms of WKB scenario for which the isotropic Misner variables (the volume) and one of the two anisotropic ones (and $ϕ$ when present) are treated on a semi-classical level, while the remaining anisotropy degree of freedom, the one trapped in the corner, is described on a pure quantum level. The quantum dynamics always reduces to the one of a time-dependent Schröedinger equation for a harmonic potential with a time dependent frequency. The vacuum case is treated in the limits of a collpasing and an expanding Universe, while the dynamics in presence of $ϕ$ and $Λ$ is studied for $t \rightarrow \infty $. In both analysis the quantum dynamics of the anisotropy variable is associated to a decaying standard deviation of its probability density, corresponding to a suppression of the quantum anisotropy associated. In the vacuum case, the corner configuration becomes an attractor for the dynamics and the evolution resembles that one of a Taub cosmology in the limit of a non-singular initial Universe. This suggests that if the Bianchi dynamics enters enough the potential corner then the initial singularity is removed and a Taub picture emerges. The case when $ϕ$ is present well mimics the De-Sitter phase of an inflationary Universe. Here we show that both the classical and quantum anisotropies are exponentially suppressed, so that the resulting dynamics corresponds to an isotropic closed Robertson-Walker geometry.
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