2009/09/30 by Edward Anderson
Physics and Astronomy · #Black Holes and Theoretical Physics #Differential geometry #Interpretation (philosophy) #Measure (data warehouse) #Noncommutative and Quantum Gravity Theories #Quantum #Quantum and Classical Electrodynamics #Quantum cosmology #Quantum geometry #Quantum gravity #Quantum process #Realization (probability) #gr-qc
paper · pdf · doi:10.1007/s10714-010-1140-z
published as Gen.Rel.Grav.43:1529-1548,2011 · A shorter version, as per the first stage in the refereeing process, and containing some new references
arxiv created 2010/10/18 · openalex publication_date 2011/01/28 · arxiv updated 2011/04/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.