Hamiltonian and physical Hilbert space in polymer quantum mechanics

Abstract

In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics. The kinematical cornerstone of our framework is the so called polymer representation of the Heisenberg-Weyl (H-W) algebra, which is the starting point of the construction. The dynamics is constructed as a continuum limit of effective theories characterized by a scale, and requires a renormalization of the inner product. The result is a physical Hilbert space in which the continuum Hamiltonian can be represented and that is unitarily equivalent to the Schroedinger representation of quantum mechanics. As a concrete implementation of our formalism, the simple harmonic oscillator is fully developed.

19 pages, 2 figures. Comments and references added. Version to be published in CQG

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• Polymer dynamics of isotropic universe in Ashtekar and in volume variables
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• Bianchi I model as a prototype for a cyclical Universe
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