Polymer quantization versus the Snyder noncommutative space

TL;DR

This paper explores a noncanonical polymer quantum mechanics framework, revealing modifications to the Heisenberg algebra and potential links to string theory's generalized uncertainty principles, while maintaining similar physical results to standard quantum mechanics.

Contribution

It introduces a noncanonical Hilbert space representation of polymer quantum mechanics and connects it with Snyder-deformed algebras and generalized uncertainty principles.

Findings

Modified Heisenberg algebra leads to a generalized uncertainty principle.

Noncanonical Poisson brackets relate to canonical ones via Darboux transformation.

Translation group remains undeformed in the Snyder-deformed algebra.

Abstract

We study a noncanonical Hilbert space representation of the polymer quantum mechanics. It is shown that Heisenberg algebra get some modifications in the constructed setup from which a generalized uncertainty principle will naturally come out. Although the extracted physical results are the same as those obtained from the standard canonical representation, the noncanonical representation may be notable in view of its possible connection with the generalized uncertainty theories suggested by string theory. In this regard, by considering an Snyder-deformed Heisenberg algebra we show that since the translation group is not deformed it can be identified with a polymer-modified Heisenberg algebra. In classical level, it is shown the noncanonical Poisson brackets are related to their canonical counterparts by means of a Darboux transformation on the corresponding phase space.