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

References in corpus (1)

• Quantum Nature of the Big Bang: Improved dynamics

Cited by in corpus (6)

• Polymer Quantum Mechanics and its Continuum Limit
• Bare Action and Regularized Functional Integral of Asymptotically Safe Quantum Gravity
• Quantization of the Jackiw-Teitelboim model
• Polymer Quantum Dynamics of the Taub Universe
• Holography from loop quantum gravity
• Loop quantum gravity and Planck-size black hole entropy