Triangleland. II. Quantum Mechanics of Pure Shape

TL;DR

This paper investigates the quantum mechanics of a pure shape relational particle model called triangleland, providing explicit solutions and exploring its implications for the problem of time in quantum gravity.

Contribution

It offers the first detailed quantum analysis of pure shape triangleland using various methods, connecting relational models to quantum gravity conceptual issues.

Findings

Exact and approximate solutions for quantum triangleland.

Analogy to linear rigid rotor in electric field.

Potential for applying methods to other relational models.

Abstract

Relational particle models are of value in the absolute versus relative motion debate. They are also analogous to the dynamical formulation of general relativity, and as such are useful for investigating conceptual strategies proposed for resolving the problem of time in quantum general relativity. Moreover, to date there are few explicit examples of these at the quantum level. In this paper I exploit recent geometrical and classical dynamics work to provide such a study based on reduced quantization in the case of pure shape (no scale) in 2-d for 3 particles (triangleland) with multiple harmonic oscillator type potentials. I explore solutions for these making use of exact, asymptotic, perturbative and numerical methods. An analogy to the mathematics of the linear rigid rotor in a background electric field is useful throughout. I argue that further relational models are accessible by the methods used in this paper, and for specific uses of the models covered by this paper in the investigation of the problem of time (and other conceptual and technical issues) in quantum general relativity.