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.

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.