Triangleland. II. Quantum Mechanics of Pure Shape

Abstract

Relational particle models are of value in the absolute versus relative motion debate. They are also analogous to the dynamical formulation of general relativity, and as such are useful for investigating conceptual strategies proposed for resolving the problem of time in quantum general relativity. Moreover, to date there are few explicit examples of these at the quantum level. In this paper I exploit recent geometrical and classical dynamics work to provide such a study based on reduced quantization in the case of pure shape (no scale) in 2-d for 3 particles (triangleland) with multiple harmonic oscillator type potentials. I explore solutions for these making use of exact, asymptotic, perturbative and numerical methods. An analogy to the mathematics of the linear rigid rotor in a background electric field is useful throughout. I argue that further relational models are accessible by the methods used in this paper, and for specific uses of the models covered by this paper in the investigation of the problem of time (and other conceptual and technical issues) in quantum general relativity.

Journal Reference added, minor updates to References and Figures

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Cited by in corpus (16)

• New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split
• How does geometry affect quantum gases?
• Triangleland. I. Classical dynamics with exchange of relative angular momentum
• Records Theory
• Approaching the Problem of Time with a Combined Semiclassical-Records-Histories Scheme
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• Seminar on Records Theory
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• Quantum Cosmological Metroland Model
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• Quantum gases on a torus
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