[Paper Review] Scaled Triangleland Model of Quantum Cosmology

Edward Anderson|arXiv (Cornell University)|May 14, 2010

Noncommutative and Quantum Gravity Theories38 references7 citations

TL;DR

This paper introduces a scaled triangleland model in quantum cosmology using relational particle mechanics, modeling a universe as a system of three particles forming a triangle with relative configurations only. By employing harmonic oscillator and cosmologically inspired potentials in spherical and parabolic coordinates, it derives quantum solutions that interpret shape and size via expectation values and spreads, offering insights into timeless dynamics, semiclassical time, and the problem of time in quantum gravity.

ABSTRACT

In scaled relational particle mechanics, only relative times, relative angles and relative separations are meaningful. It arose in the study of the absolute versus relative motion debate. It has then turned out to be a useful toy model of classical and quantum general relativity, such as for investigating conceptual strategies for the problem of time. This paper studies the 3-particle 2-d scaled relational particle model, for which the configurations are scaled triangles. The configuration space for these is R^3 with a conformally flat metric thereupon (it is the cone over the corresponding shape space S^2. I use multiple harmonic oscillator type potentials and other potentials suggested by analogy with cosmology, and solve for some of these by using a partial analogy with the treatment of the atom in spherical and parabolic coordinates. Spherical coordinates are here the total moment of inertia $I$ for radius and two pure-shape coordinates. These are Θ, a function of the ratio of the two relative separations of subsystems, and Φ, the relative angle between the two subsystems. Parabolic coordinates are Φagain and twice the partial moments of inertia of each subsystem. I interpret these solutions using 1) a `Bohr moment of inertia' for the model universe (playing the role of the scalefactor). 2) Expectations and spreads of sizes and shapes. 3) Superimposing the probability density function on the labelled tessellation of the configuration space that encodes meaningful subregions such as collinear configurations, equilateral triangles and isosceles triangles. Applications include hidden time, emergent semiclassical time, timeless and histories theory problem of time strategies, and comparing reduced and Dirac methods of quantization.

Motivation & Objective

• To develop a toy model of quantum cosmology based on scaled relational particle mechanics, focusing on three particles forming a triangle in 2D space.
• To address the problem of time in quantum gravity by analyzing a system without absolute time or background structure.
• To explore how semiclassical time and emergent dynamics can arise from a fundamentally timeless quantum theory.
• To compare reduced and Dirac quantization methods in a relational framework, testing their consistency and physical interpretation.
• To investigate foundational issues such as configuration space localization, shape uniformity, and robustness of predictions under coarse-graining.

Proposed method

• Formulates a scaled relational particle mechanics model with only relative times, angles, and separations as physical degrees of freedom.
• Uses spherical coordinates based on total moment of inertia (I) and two pure-shape variables: Θ (ratio of relative separations) and Φ (relative angle).
• Applies parabolic coordinates based on partial moments of inertia and Φ, enabling separation of variables in the Schrödinger equation.
• Introduces harmonic oscillator-type and cosmologically motivated potentials to model quantum dynamics on the configuration space.
• Solves the quantum Hamiltonian constraint using analogies with atomic physics in spherical and parabolic coordinates.
• Analyzes wavefunctions via expectation values, spreads, and probability density distributions mapped onto a labeled tessellation of configuration space.

Experimental results

Research questions

• RQ1 How can a timeless quantum theory of a universe modeled as a triangle yield emergent semiclassical time?
• RQ2 What role do shape and size expectation values play in interpreting the quantum state of a relational universe?
• RQ3 How do different coordinate systems (spherical vs. parabolic) affect the solvability and physical interpretation of the quantum model?
• RQ4 Can the probability density function reveal meaningful physical regions such as collinear, equilateral, or isosceles configurations?
• RQ5 How do reduced and Dirac quantization methods compare in this relational quantum cosmology framework?

Key findings

• The model successfully solves the Schrödinger equation using separation of variables in both spherical and parabolic coordinates, enabling analytical treatment of shape and size dynamics.
• Expectation values and spreads of shape and size variables provide a physical interpretation of the quantum state, with the 'Bohr moment of inertia' serving as a proxy for the scale factor.
• The probability density function is superimposed on a labeled tessellation of configuration space, clearly identifying regions corresponding to collinear, equilateral, and isosceles configurations.
• The model supports both Hartle-type and Isham–Linden histories theories, enabling a framework for decoherence and records-based time emergence.
• The spherical coordinate approach allows a direct analogy with Halliwell–Hawking’s analysis of inhomogeneities on S³, extending its applicability to shape-space dynamics.
• The model demonstrates robustness in capturing uniformity and perturbations, with equilateral triangles representing the most uniform quantum state, supporting foundational studies of cosmological initial conditions.