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. PACS: 04.60Kz.
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 scalefree triangle formed by 3 particles in 2-d. The shape spaces (configuration spaces for scalefree systems) both for this and for 4 particles in 1-d are spheres. Three quantities u whose squares sum to one play the role of Cartesian coordinates for an auxiliary surrounding flat 3-space and are then good variables from the perspective of kinematical quantization. These quantities turn out to be obvious ratios of relative separations for 4 particles in 1-d and rather less trivial Dragt-type coordinates in the case of the triangle. In each case, the u furthermore admit useful interpretation as shape quantities; the triangle’s three Dragt-type coordinates are an ellipicity of the system’s moments of inertia, a measure of anisoscelesness and the triangle’s area up to proportionality. 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 mass-weighted shape sphere by its physical interpretation as the triangle, which parallels partitioning the night sky zodiacally and austro-boreally; juxtaposing this with classical trajectories/quantum mechanical wavefunctions permits straightforward reading off of the sequence of triangles in the motion/how probable each triangle is. I explain how features desirable for quantum cosmology cause one’s models to depart in detail from somewhat similar models already studied in molecular physics. I include applications to multiverse modelling and to timeless approaches to the problem of time.
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.