Hamiltonian and physical Hilbert space in polymer quantum mechanics

TL;DR

This paper demonstrates that polymer quantum mechanics, inspired by loop quantum gravity, can be rigorously shown to be equivalent to standard Schrödinger quantum mechanics through a continuum limit and renormalization, exemplified by the harmonic oscillator.

Contribution

It establishes a precise equivalence between polymer quantum mechanics and Schrödinger quantum mechanics, including the construction of a physical Hilbert space and continuum Hamiltonian.

Findings

Polymer representation of the Heisenberg-Weyl algebra is equivalent to the Schrödinger representation.

Continuum limit and renormalization produce a physical Hilbert space with standard quantum dynamics.

The harmonic oscillator is fully realized within this formalism.

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.