Smallest Relational Mechanics Model of Quantum Cosmology

TL;DR

This paper develops a relational particle mechanics model with three particles in one dimension, providing insights into quantum cosmology by combining shape and scale variables, and offering a new conceptual framework for inhomogeneity and structure formation.

Contribution

It introduces a relational model with shape-scale split variables that parallels Friedmann equations and incorporates semiclassical approximation for quantum cosmology.

Findings

Scale acts as an approximate timestandard.

Model captures inhomogeneity and structure formation.

Provides conceptual insights into nontrivial linear constraints.

Abstract

Relational particle mechanics are models in which there is, overall, no time, position, orientation (nor, sometimes, scale). They are useful for whole-universe modelling - the setting for quantum cosmology. This note concerns 3 particles in 1d in shape-scale split variables. The scale part parallels certain Friedmann equations, while in this note the shape part involves functions on the circle. The scale part is taken to be `heavy' and `slow' so the semiclassical approach applies and scale provides an approximate timestandard with repect to which the light physics runs. Relational particle mechanics moreover provide conceptual models of inhomogeneity, structure formation and nontrivial linear constraints (minisuperspace models do not and midisuperspace models only do at the cost of substantial complications).