Quantum Cosmological Metroland Model

TL;DR

This paper explores a simplified quantum cosmological model called 4-stop metroland, focusing on shape dynamics of four particles on a line, providing exact solutions and insights relevant to quantum gravity and the problem of time.

Contribution

It introduces the N-stop metroland model, analyzing shape-only mechanics of four particles, and extends ideas to N-particle systems with applications to quantum cosmology.

Findings

Exact classical and quantum solutions for 4-stop metroland

Interpretation of shape operators as relative sizes and homogeneity measures

Relevance to the problem of time in quantum gravity

Abstract

Relational particle mechanics is useful for modelling whole-universe issues such as quantum cosmology or the problem of time in quantum gravity, including some aspects outside the reach of comparably complex minisuperspace models. In this article, we consider the mechanics of pure shape and not scale of 4 particles on a line, so that the only physically significant quantities are ratios of relative separations between the constituents' physical objects. Many of our ideas and workings extend to the N-particle case. As such models' configurations resemble depictions of metro lines in public transport maps, we term them `N-stop metrolands'. This 4-stop model's configuration space is a 2-sphere, from which our metroland mechanics interpretation is via the `cubic' tessellation. This model yields conserved quantities which are mathematically SO(3) objects like angular momenta but are physically relative dilational momenta (i.e. coordinates dotted with momenta). We provide and interpret various exact and approximate classical and quantum solutions for 4-stop metroland; from these results one can construct expectations and spreads of shape operators that admit interpretations as relative sizes and the `homogeneity of the model universe's contents', and also objects of significance for the problem of time in quantum gravity (e.g. in the naive Schrodinger and records theory timeless approaches).