With toy modelling of conceptual aspects of quantum cosmology and the problem of time in quantum gravity in mind, I study the classical and quantum dynamics of the pure-shape (i.e. scale-free) triangle formed by 3 particles in 2-d. I do so by importing techniques to the triangle model from the corresponding 4 particles in 1-d model, using the fact that both have 2-spheres for shape spaces, though the latter has a trivial realization whilst the former has a more involved Hopf (or Dragt) type realization. I furthermore interpret the ensuing Dragt-type coordinates as shape quantities: a measure of anisoscelesness, the ellipticity of the base and apex's moments of inertia, and a quantity proportional to the area of the triangle. I promote these quantities at the quantum level to operators whose expectation and spread are then useful in understanding the quantum states of the system. Additionally, I tessellate the 2-sphere by its physical interpretation as the shape space of triangles, and then use this as a back-cloth from which to read off the interpretation of dynamical trajectories, potentials and wavefunctions. I include applications to timeless approaches to the problem of time and to the role of uniform states in quantum cosmological modelling.
Relational Quadrilateralland. I. The Classical Theory
相对粒子力学模型支持了绝对运动与相对运动之争的关系方面，并且是广义相对论及其时间问题的动力学公式的玩具模型。它们涵盖了两个更普遍研究的小超空间广义相对论模型所没有的方面：1）具有非平凡的结构概念，从而具有宇宙结构形成和定域记录的概念。2)它们具有线性和二次约束，这对于建模许多时间问题方面至关重要。我以前解决了关系三角和经典，量子力学和关于时间问题的局部解决方案。这依赖于三角形的形状空间是S^2，等距群SO（3），允许使用广为人知的几何，方法和原子/分子物理学类比。我现在将这项工作扩展到关系四边形，这是更典型的一般N-a-gon，代表了一个“对角到非对角Bianchi IX minissuperspace”的复杂性，并编码进一步的时间问题的微妙之处。形状空间现在是CP ^2，等距群SU（3）/Z_3，我现在需要利用几何学，形状统计学和粒子物理学来解决这个模型;因此，这是一个跨学科的论文。本文在经典水平上处理四边形和，然后论文II提供了量子处理。
Harish-Chandra Research Institute (HRI)（哈里什-昌德拉研究所）；Homi Bhabha National Institute (HBNI)（霍米·巴巴国立研究所）；School of Mathematics and Natural Sciences, Chanakya University（查纳基亚大学数学与自然科学学院）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·派尔斯理论物理中心）
