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Hamiltonian and physical Hilbert space in polymer quantum mechanics

Alejandro Corichi·Tatjana Vukasinac·Jose A. Zapata
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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.

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Alejandro Corichi, Tatjana Vukasinac, Jose A. Zapata. 2007-02-07. Hamiltonian and physical Hilbert space in polymer quantum mechanics. https://doi.org/10.1088/0264-9381%2F24%2F6%2F008

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