# Shape Space Methods for Quantum Cosmological Triangleland

## TL;DR

This paper explores the classical and quantum dynamics of a scale-free triangle model in quantum cosmology, utilizing shape space techniques and interpreting shape quantities as operators to understand quantum states and trajectories.

## Contribution

It introduces shape space methods from 4-particle models to 3-particle triangle models, providing new shape quantities and their quantum operator interpretations.

## Key findings

- Shape quantities as quantum operators reveal state properties
- Tessellation of shape space aids in interpreting trajectories and wavefunctions
- Applications to timeless quantum cosmology approaches

## Abstract

With toy modelling of conceptual aspects of quantum cosmology and the problem of time in quantum gravity in mind, I study the classical and quantum dynamics of the pure-shape (i.e. scale-free) triangle formed by 3 particles in 2-d. I do so by importing techniques to the triangle model from the corresponding 4 particles in 1-d model, using the fact that both have 2-spheres for shape spaces, though the latter has a trivial realization whilst the former has a more involved Hopf (or Dragt) type realization. I furthermore interpret the ensuing Dragt-type coordinates as shape quantities: a measure of anisoscelesness, the ellipticity of the base and apex's moments of inertia, and a quantity proportional to the area of the triangle. I promote these quantities at the quantum level to operators whose expectation and spread are then useful in understanding the quantum states of the system. Additionally, I tessellate the 2-sphere by its physical interpretation as the shape space of triangles, and then use this as a back-cloth from which to read off the interpretation of dynamical trajectories, potentials and wavefunctions. I include applications to timeless approaches to the problem of time and to the role of uniform states in quantum cosmological modelling.