Relational Quadrilateralland Interpretation of CP^2 and Quotients

Abstract

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.