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arXiv 0909.2436gr-qc
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量子宇宙都市模型

Quantum Cosmological Metroland Model

Edward Anderson, Anne Franzen

摘要:

关系粒子力学对于量子宇宙学或量子引力中的时间问题等全宇宙问题的建模非常有用,包括一些复杂的迷你超空间模型无法触及的方面。在这篇文章中,我们考虑了一条线上4个粒子的纯形状而不是尺度的力学,因此唯一有物理意义的量是组成物质之间的相对距离的比率。我们的许多想法和工作延伸到N粒子的情况。由于这种模型的配置类似于公共交通地图中的地铁线路,我们称之为“N站地铁”。这个4站模型的配置空间是一个2球,从我们的都市力学解释是通过'立方'镶嵌。这个模型产生的守恒量在数学上是SO(3)对象,比如角动量,但在物理上是相对膨胀动量(即用动量点缀的坐标)。我们提供和解释各种精确的和近似的经典和量子的解决方案4停止metroland;从这些结果可以构建的期望和传播的形状算子,承认解释为相对大小和“均匀的模型宇宙的内容”,也对象的重要性问题的时间在量子引力(例如,在天真的薛定谔和记录理论的无时间的方法)。

英文摘要:

Relational particle mechanics is useful for modelling whole-universe issues such as quantum cosmology or the problem of time in quantum gravity, including some aspects outside the reach of comparably complex minisuperspace models. In this article, we consider the mechanics of pure shape and not scale of 4 particles on a line, so that the only physically significant quantities are ratios of relative separations between the constituents' physical objects. Many of our ideas and workings extend to the N-particle case. As such models' configurations resemble depictions of metro lines in public transport maps, we term them `N-stop metrolands'. This 4-stop model's configuration space is a 2-sphere, from which our metroland mechanics interpretation is via the `cubic' tessellation. This model yields conserved quantities which are mathematically SO(3) objects like angular momenta but are physically relative dilational momenta (i.e. coordinates dotted with momenta). We provide and interpret various exact and approximate classical and quantum solutions for 4-stop metroland; from these results one can construct expectations and spreads of shape operators that admit interpretations as relative sizes and the `homogeneity of the model universe's contents', and also objects of significance for the problem of time in quantum gravity (e.g. in the naive Schrodinger and records theory timeless approaches).

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