Relational Quadrilateralland. I. The Classical Theory

TL;DR

This paper extends relational particle mechanics models from triangle to quadrilateral configurations, exploring classical dynamics and shape space geometry to better understand structure formation and the Problem of Time in quantum gravity.

Contribution

It introduces the classical analysis of quadrilateralland, a more complex relational model with CP^2 shape space, advancing the study of structure and constraints relevant to quantum gravity.

Findings

Classical solution of quadrilateralland dynamics.

Identification of CP^2 as the shape space for quadrilaterals.

Foundation for quantum analysis in Paper II.

Abstract

Relational particle mechanics models bolster the relational side of the absolute versus relational motion debate, and are additionally toy models for the dynamical formulation of General Relativity and its Problem of Time. They cover two aspects that the more commonly studied minisuperspace General Relativity models do not: 1) by having a nontrivial notion of structure and thus of cosmological structure formation and of localized records. 2) They have linear as well as quadratic constraints, which is crucial as regards modelling many Problem of Time facets. I previously solved relational triangleland classically, quantum mechanically and as regards a local resolution of the Problem of Time. This rested on triangleland's shape space being S^2 with isometry group SO(3), allowing for use of widely-known Geometry, Methods and Atomic/Molecular Physics analogies. I now extend this work to the relational quadrilateral, which is far more typical of the general N-a-gon, represents a `diagonal to nondiagonal Bianchi IX minisuperspace' step-up in complexity, and encodes further Problem of Time subtleties. The shape space now being CP^2 with isometry group SU(3)/Z_3, I now need to draw on Geometry, Shape Statistics and Particle Physics to solve this model; this is therefore an interdisciplinary paper. This Paper treats quadrilateralland at the classical level, and then Paper II provides a quantum treatment.