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arXiv:gr-qc/0511068 · Classical and Quantum Gravity

Relational particle models: I. Reconciliation with standard classical and quantum theory

Edward Anderson

Published Mar 17, 2006• 35 citations
arXivHTMLPDF

Abstract

This paper concerns the absolute versus relative motion debate. The Barbour and Bertotti 1982 work may be viewed as an indirectly set up relational formulation of a portion of Newtonian mechanics. I consider further direct formulations of this and argue that the portion in question -- universes with zero total angular momentum, that are conservative and with kinetic terms that are (homogeneous) quadratic in their velocities -- is capable of accommodating a wide range of classical physics phenomena. Furthermore, as I develop in Paper II, this relational particle model is a useful toy model for canonical general relativity. I consider what happens if one quantizes relational rather than absolute mechanics, indeed whether the latter is misleading. By exploiting Jacobi coordinates, I show how to access many examples of quantized relational particle models and then interpret these from a relational perspective. By these means, previous suggestions of bad semiclassicality for such models can be eluded. I show how small (particle number) universe relational particle model examples display eigenspectrum truncation, gaps, energy interlocking and counterbalanced total angular momentum. These features mean that these small universe models make interesting toy models for some aspects of closed universe quantum cosmology. While, these features do not compromise the recovery of reality as regards the practicalities of experimentation in a large universe such as our own.

DOI
10.1088/0264-9381/23/7/016

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This paper concerns the absolute versus relative motion debate. The Barbour and Bertotti 1982 work may be viewed as an indirectly set up relational formulation of a portion of Newtonian mechanics. I consider further direct formulations of this and argue that the portion in question -- universes with zero total angular momentum, that are conservative and with kinetic terms that are (homogeneous) quadratic in their velocities -- is capable of accommodating a wide range of classical physics phenomena. Furthermore, as I develop in Paper II, this relational particle model is a useful toy model for canonical general relativity. I consider what happens if one quantizes relational rather than absolute mechanics, indeed whether the latter is misleading. By exploiting Jacobi coordinates, I show how to access many examples of quantized relational particle models and then interpret these from a relational perspective. By these means, previous suggestions of bad semiclassicality for such models can be eluded. I show how small (particle number) universe relational particle model examples display eigenspectrum truncation, gaps, energy interlocking and counterbalanced total angular momentum. These features mean that these small universe models make interesting toy models for some aspects of closed universe quantum cosmology. While, these features do not compromise the recovery of reality as regards the practicalities of experimentation in a large universe such as our own.
We provide the quantum treatment of the relational quadrilateral. The underlying reduced configuration spaces are ℂℙ 2 and the cone over this. We consider exact free and isotropic HO potential cases and perturbations about these. Moreover, our purely relational kinematical quantization is distinct from the usual one for ℂℙ 2 , which turns out to carry absolutist connotations instead. Thus, this paper is the first to note absolute-versus-relational motion distinctions at the kinematical rather than dynamical level. It is also an example of value to the discussion of kinematical quantization along the lines of Isham, 1984. The relational quadrilateral is the simplest RPM whose mathematics is not standard in atomic physics (the triangle and four particles on a line are both based on 𝕊 2 and ℝ 3 mathematics). It is far more typical of the general quantum relational N-a-gon than the previously studied case of the relational triangle. We consider useful integrals as regards perturbation theory and the peaking interpretation of quantum cosmology. We subsequently consider problem of time (PoT) applications of this: quantum Kuchař beables, the Machian version of the semiclassical approach and the timeless naïve Schrödinger interpretation. These go toward extending the combined Machian semiclassical-Histories-Timeless Approach of [Int. J. Mod. Phys. D23 (2014) 1450014] to the case of the quadrilateral, which will be treated in subsequent papers.
This paper provides the quantum treatment of the relational quadrilateral. The underlying reduced configuration spaces are CP2\mathbb{CP}^2CP2 and the cone over this, C(CP2\mathbb{CP}^2CP2). We consider exact free and isotropic HO potential cases and perturbations about these. Moreover, our purely relational kinematical quantization is distinct from the usual one for CP2\mathbb{CP}^2CP2, which turns out to carry absolutist connotations instead. Thus this paper is the first to note absolute-versus-relational motion distinctions at the kinematical rather than dynamical level, and also an example of value to the discussion of kinematical quantization along the lines of Isham 1984. This treatment of the relational quadrilateral is the first relational QM with very new mathematics for a finite QM model, and is far more typical of the general quantum relational NNN-a-gon than the previously-studied case of the relational triangle. We consider useful integrals as regards perturbation theory and the peaking interpretation of quantum cosmology. We subsequently consider problem of time applications of this: quantum Kuchař beables, the Machian version of the semiclassical approach and the timeless naive Schrodinger interpretation. These go toward extending the combined Machian semiclassical-histories-timeless approach of [1] to the case of the quadrilateral, which will be treated in subsequent papers.
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).
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.
The general framework of Entropic Dynamics (ED) is used to construct non-relativistic models of relational Quantum Mechanics from well-known inference principles-probability, entropy and information geometry. Although only partially relational-the absolute structures of simultaneity and Euclidean geometry are still retained-these models provide a useful testing ground for ideas that will prove useful in the context of more realistic relativistic theories. The fact that in ED the positions of particles have definite values, just as in classical mechanics, has allowed us to adapt to the quantum case some intuitions from Barbour and Bertotti's classical framework. Here, however, we propose a new measure of the mismatch between successive states that is adapted to the information metric and the symplectic structures of the quantum phase space. We make explicit that ED is temporally relational and we construct non-relativistic quantum models that are spatially relational with respect to rigid translations and rotations. The ED approach settles the longstanding question of what form the constraints of a classical theory should take after quantization: the quantum constraints that express relationality are to be imposed on expectation values. To highlight the potential impact of these developments, the non-relativistic quantum model is parametrized into a generally covariant form and we show that the ED approach evades the analogue of what in quantum gravity has been called the problem of time.
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.