Polymer Quantum Mechanics as a Deformation Quantization

TL;DR

This paper explores the polymer representation of quantum mechanics using deformation quantization, constructing Wigner functions and star-products, and demonstrating consistency with loop quantum cosmology and minimal length theories.

Contribution

It introduces a deformation quantization framework for polymer quantum mechanics, including Wigner functions and star-products, linking it to loop quantum cosmology and minimal length principles.

Findings

Wigner function constructed as a distributional limit of Schrödinger representation

Polymer star-product satisfies Bohr's correspondence principle

Derived a generalized uncertainty principle consistent with minimal length theories

Abstract

We analyze the polymer representation of quantum mechanics within the deformation quantization formalism. In particular, we construct the Wigner function and the star-product for the polymer representation as a distributional limit of the Schr\"odinger representation for the Weyl algebra in a Gaussian weighted measure, and we observe that the quasi-probability distribution limit of this Schr\"odinger representation agrees with the Wigner function for Loop Quantum Cosmology. Further, the introduced polymer star-product fulfills Bohr's correspondence principle even though not all the operators are well defined in the polymer representation. Finally, within our framework, we also derive a generalized uncertainty principle which is consistent to the ones usually obtained in theories assuming a fundamental minimal length in their formulation.