关系四边形与CP ^2和商的解释
Relational Quadrilateralland Interpretation of CP^2 and Quotients
摘要：
我调查定性重要区域的配置空间的经典和量子力学的关系四边形在2-d。这是相对的，因为只有相对的间隔比率、相对的角度和相对的时间才是重要的。这种关联粒子力学模型与广义相对论的几何动力学表述有许多相似之处。因此，它们适合作为玩具模型来研究1）量子引力策略中的时间问题，特别是无时间、半经典和历史理论方法以及这些方法的组合。2)各种其他量子宇宙学问题，如结构形成/不均匀性和均匀态的意义。关系四边形是更有用的，在这些方面比以前调查的简单RPM的，因为它具有线性约束，非平凡的子系统和它的配置空间是一个非平凡的复杂的投影空间。本文研究了共线构形的子流形、单粒子碰撞的子流形、正方形构形以及近似共线、近似正方形和近似三角形等构形空间的区域。我认为这两个镜像识别和未识别的形状，以及可区分和不可区分的粒子标签。我发现下面的CP^2坐标系对这些目的很有用。1)柯伊伯坐标，对于四边形，它是雅可比向量的大小和它们之间形成的三角形的不等度。2)Gibbons-Pope型坐标，对于四边形，它是两个子系统之间相对角度的和与差、两个子系统大小的比值以及子系统所占宇宙模型的比例。
英文摘要：
I investigate qualitatively significant regions of the configuration space for the classical and quantum mechanics of the relational quadrilateral in 2-d. This is relational in the sense that only relative ratios of separations, relative angles and relative times are significant. Such relational particle mechanics models have many analogies with the geometrodynamical formulation of general relativity. Thus, they are suitable as toy models for studying 1) problem of time in quantum gravity strategies, in particular timeless, semiclassical and histories theory approaches and combinations of these. 2) Various other quantum-cosmological issues, such as structure formation/inhomogeneity and the significance of uniform states. The relational quadrilateral is more useful in these respects than previously investigated simpler RPM's as it possesses linear constraints, nontrivial subsystems and its configuration space is a nontrivial complex-projective space. In this paper, I investigate the submanifold of collinear configurations, the submanifolds with a single-particle collision, the square configurations and regions of the configuration space for which such as approximate collinearity, approximate squareness and approximate triangularity hold. I consider both mirror image identified and unidentified shapes, as well as both distinguishable and indistinguishable particle labels. I find the following CP^2 coordinate systems to be useful to these ends. 1) Kuiper's coordinates, which, for quadrilateralland, are magnitudes of the Jacobi vectors and anisoscelesnesses of the triangles formed between them. 2) Gibbons-Pope type coordinates, which, for quadrilateralland, are the sum and difference of relative angles between subsystems, the ratio of size of 2 subsystems and the proportion of universe model occupied by the subsystems.
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