Polymer-Fourier quantization of the scalar field revisited

TL;DR

This paper revisits the polymer quantization of scalar field Fourier modes, replacing the standard positive linear functional with a singular one, leading to a different symmetry group and a new irreducible representation of the CCR.

Contribution

It introduces a novel polymer Fourier quantization approach using a singular positive linear functional, resulting in a different symmetry group and representation.

Findings

The symmetry group is the volume-preserving diffeomorphisms subgroup SDiff(R^4).

The resulting representation of CCR is non-unitarily equivalent to the standard Fock representation.

Comparison between Poincaré invariant Fock vacuum and polymer Fourier vacuum was performed.

Abstract

The Polymer Quantization of the Fourier modes of the real scalar field is studied within algebraic scheme. We replace the positive linear functional of the standard Poincar\'e invariant quantization by a singular one. This singular positive linear functional is constructed as mimicking the singular limit of the complex structure of the Poincar\'e invariant Fock quantization. The resulting symmetry group of such Polymer Quantization is the subgroup $\mbox{SDiff}(\mathbb{R}^4)$ which is a subgroup of $\mbox{Diff}(\mathbb{R}^4)$ formed by spatial volume preserving diffeomorphisms. In consequence, this yields an entirely different irreducible representation of the Canonical Commutation Relations, non-unitary equivalent to the standard Fock representation. We also compared the Poincar\'e invariant Fock vacuum with the Polymer Fourier vacuum.