Polymer quantum mechanics and its continuum limit
Abstract
A rather nonstandard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation, has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schrödinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schrödinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schrödinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle, and a simple cosmological model.
 Publication:

Physical Review D
 Pub Date:
 August 2007
 DOI:
 10.1103/PhysRevD.76.044016
 arXiv:
 arXiv:0704.0007
 Bibcode:
 2007PhRvD..76d4016C
 Keywords:

 04.60.Pp;
 04.60.Ds;
 04.60.Nc;
 Loop quantum gravity quantum geometry spin foams;
 Canonical quantization;
 Lattice and discrete methods;
 General Relativity and Quantum Cosmology
 EPrint:
 16 pages, no figures. Typos corrected to match published version