Propagators in Polymer Quantum Mechanics

TL;DR

This paper computes polymer quantum propagators for a free particle and a particle in a box, demonstrating their properties and reduction to standard quantum mechanics in the appropriate limit, thus extending polymer quantization techniques.

Contribution

It provides explicit calculations of polymer propagators for simple systems and analyzes their properties and limits, advancing the application of polymer quantum mechanics.

Findings

Polymer propagators satisfy standard quantum properties.

They reduce to Schrödinger propagators as the polymer scale parameter approaches zero.

The method of images applies to polymer propagators for a particle in a box.

Abstract

Polymer Quantum Mechanics is based on some of the techniques used in the loop quantization of gravity that are adapted to describe systems possessing a finite number of degrees of freedom. It has been used in two ways: on one hand it has been used to represent some aspects of the loop quantization in a simpler context, and, on the other, it has been applied to each of the infinite mechanical modes of other systems. Indeed, this polymer approach was recently implemented for the free scalar field propagator. In this work we compute the polymer propagators of the free particle and a particle in a box; amusingly, just as in the non polymeric case, the one of the particle in a box may be computed also from that of the free particle using the method of images. We verify the propagators hereby obtained satisfy standard properties such as: consistency with initial conditions, composition and Green's function character. Furthermore they are also shown to reduce to the usual Schr\"odinger propagators in the limit of small parameter $\mu_0$, the length scale introduced in the polymer dynamics and which plays a role analog of that of Planck length in Quantum Gravity.