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Shape Operators and Zodiac Tessellation: Improved Interpretation of Quantum Cosmological Triangleland

Edward James Anderson

Published Sep 13, 2009• 0 citations
OpenAlex

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 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.

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Edward James Anderson, Damtp Cambridge U. K

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.

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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.
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.
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 III 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.
Relational particle mechanics is useful for modelling whole-universe issues such as quantum cosmology or the problem of time in quantum gravity, including some aspects outside the reach of comparably complex mini-superspace models. In this paper, we consider the mechanics of pure shape and not scale of four particles on a line, so that the only physically significant quantities are ratios of relative separations between the constituents' physical objects. Many of our ideas and workings extend to the N-particle case. As such models' configurations resemble depictions of metro lines in public transport maps, we term them 'N-stop metrolands'. This 4-stop model's configuration space is a 2-sphere, from which our metroland mechanics interpretation is via the 'cubic' tessellation. This model yields conserved quantities which are mathematically SO(3) objects like angular momenta but are physically relative dilational momenta (i.e. coordinates dotted with momenta). We provide and interpret various exact and approximate classical and quantum solutions for 4-stop metroland; from these results one can construct expectations and spreads of shape operators that admit interpretations as relative sizes and the 'homogeneity of the model universe's contents', and also objects of significance for the problem of time in quantum gravity (e.g. in the naïve Schrödinger and records theory timeless approaches).
I consider the momenta and conserved quantities for CP^2 interpreted as the space of quadrilaterals. This builds on seminar I and II's kinematics via making use of MacFarlane's work considering the SU(3)-like (and thus particle physics-like) conserved quantities that occur for CP^2. I perform the additional step of further interpreting that as the configuration space of all relational quadrilaterals and thus an interesting toy model for whole-universe, relational and geometrodynamical-analogue physics. I also provide the Kuchar observables for the quadrilateral, which is a particular resolution of the Problem of Observables. I study HO-like and highly symmetric potentials. I also provide some exact solutions and qualitative behaviours for dynamics on CP^2. In each case, I reinterpret the results in terms of quadrilaterals. This paves the way for the quantum mechanical study of the relational quadrilateral and for investigations of a number of Problem of Time strategies and of a number of other foundational and qualitative investigations of Quantum Cosmology.
Inspired by the central equation of canonical quantum gravity, the timeless Wheeler-DeWitt equation, we investigate timeless properties for quantum cosmological models. In particular we are interested in the following question: What is the probability of the system entering certain regions of its configuration space without reference to time at all? For the classical case this is a property of the entire system trajectories and can be formulated using a phase space probability distribution 1. In the quantum case we propose and investigate a probability which we obtain by adapting the decoherent histories approach to quantum theory 2 to the timeless case and by using the induced inner product on the physical state space of quantum cosmological models 3. For the example of the Klein-Gordon equation we have investigated the case of space–like surfaces and our results agree with the standard operator methods 4. For a more general model we consider a semiclassical approximation as well as the effect of an environment 1. When the histories are decoherent, the probabilities approximately coincide with the classical case, with the phase space probability distribution replaced by the Wigner function of the quantum state. The influence functional of the environment respects the reparametrization invariance of the quantum cosmological Wheeler-DeWitt equation.
We reconsider the problem of the interpretation of the Quantum Theory (QT) in the perspective of the entire universe and of Bphr idea that the classical language is the language of our experience and QT acquires a meaning only with a reference to it. We distinguish a classical or macroscopic level, and a quantum or microscopic one that is perceived only through the modifications that it induces in the first. The macroscopic state of the universe is assumed to be specified by a set of variables, a classical energy momentum tensor and some conserved currents, which are supposed to have a well defined value across the entire space-time. To the energy-momentum tensor a classical metric is related by the Einstein equation. The quantum state and dynamics are expressed by the usual QT formalism in terms of a density operator and the ordinary quantum operators in Heisenberg picture. For the macroscopic variables a basic distribution of probability is postulated in terms of a density and the corresponding quantum operators, so in some way their evolution is driven by the underlying QT. Such postulate essentially replaces the usual elfadjoint operators correspondence. For the Universe we adopt a variance of the ΛCDM model with Omega=1, one single inflaton with an Higgs type potential, the initial time at t=minus infinite. The expectation values of all fundamental fields are supposed to vanish for time going to minus infinite. In the framework the scalar fluctuation in the Cosmic Microwave Background are correctly explained giving appropriate calue to the parameters in the potential. As in more conventional models the absence of the tensor fluctuations remains not understood, if even a quantum metric is introduced. This seems to suggest that Gravity is a pure classical phenomenon, what could be consistently accommodated in our formalism by an appropriate even if somewhat ad hoc assumption