Shape Quantities for Relational Quadrilateralland

Abstract

I investigate useful shape quantities for the classical and quantum mechanics of the relational quadrilateral in 2-d. This is relational in the sense that only relative times, relative ratios of separations and relative angles are significant. Relational particle mechanics models such as this paper's have many analogies with the geometrodynamical formulation of general relativity. This renders them suitable as toy models for 1) studying Problem of Time in Quantum Gravity strategies, in particular timeless, semiclassical and histories theory approaches and combinations of these. 2) For consideration of various other quantum-cosmological issues, such as structure formation/inhomogeneity and notions of uniform states and their significance. The relational quadrilateral is more useful in these respects than previously investigated simpler RPM's due to simultaneously possessing linear constraints, nontrivial subsystems and nontrivial complex-projective mathematics. Such shape have been found to be useful in simpler relational models such as the relational triangle and in 1-d.

Seminar I on relational quadrilaterals (2-d 4-body problem as a whole-universe model in the absense of all absolute connotations). The set of shape quantities given here was found to be incomplete, as pointed out and completed in arXiv:1202.4186

References in corpus (12)

• The Problem of Time in Quantum Gravity
• The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena
• Constraints and gauge transformations: Dirac's theorem is not always valid
• Foundations of Relational Particle Dynamics
• New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split
• Triangleland. I. Classical dynamics with exchange of relative angular momentum
• Does relationalism alone control geometrodynamics with sources?
• On the Semiclassical Approach to Quantum Cosmology
• Seminar on Records Theory
• Relational mechanics of shape and scale
• Scaled Triangleland Model of Quantum Cosmology
• Relational Quadrilateralland Interpretation of CP^2 and Quotients

Cited by in corpus (5)

• Approaching the Problem of Time with a Combined Semiclassical-Records-Histories Scheme
• On the Semiclassical Approach to Quantum Cosmology
• Scaled Triangleland Model of Quantum Cosmology
• Relational Quadrilateralland. Analogues of Isospin and Hypercharge
• Relational Quadrilateralland Interpretation of CP^2 and Quotients