Band structure in the polymer quantization of the harmonic oscillator

TL;DR

This paper analyzes the spectrum of the polymer quantized harmonic oscillator, revealing band structures similar to periodic potentials, which could impact scalar field quantization and statistical mechanics.

Contribution

It provides a detailed comparison of the spectrum in polymer and standard quantizations, highlighting the band structure due to non-separability of the Hilbert space.

Findings

Spectrum exhibits band structure similar to periodic potentials.

Non-separability leads to non-trivial spectral features.

Implications for scalar field polymer quantization and statistical mechanics.

Abstract

We discuss the detailed structure of the spectrum of the Hamiltonian for the polymerized harmonic oscillator and compare it with the spectrum in the standard quantization. As we will see the non-separability of the Hilbert space implies that the point spectrum consists of bands similar to the ones appearing in the treatment of periodic potentials. This feature of the spectrum of the polymeric harmonic oscillator may be relevant for the discussion of the polymer quantization of the scalar field and may have interesting consequences for the statistical mechanics of these models.