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In this paper, a version of polymer quantum mechanics, which is inspired byloop quantum gravity, is considered and shown to be equivalent, in a precisesense, to the standard, experimentally tested, Schroedinger quantum mechanics.The kinematical cornerstone of our framework is the so called polymerrepresentation of the Heisenberg-Weyl (H-W) algebra, which is the startingpoint of the construction. The dynamics is constructed as a continuum limit ofeffective theories characterized by a scale, and requires a renormalization ofthe inner product. The result is a physical Hilbert space in which thecontinuum Hamiltonian can be represented and that is unitarily equivalent tothe Schroedinger representation of quantum mechanics. As a concreteimplementation of our formalism, the simple harmonic oscillator is fullydeveloped.Comment: 19 pages, 2 figures. Comments and references added. Version to be  published in CQ

Hamiltonian and physical Hilbert space in polymer quantum mechanics