Instanton solutions on the polymer harmonic oscillator

TL;DR

This paper uses instanton methods to compute the energy band structure of the polymer harmonic oscillator, revealing a pure point spectrum and the influence of lattice parameters, with implications for quantum systems with discrete spatial structures.

Contribution

It introduces instanton techniques to analyze the polymer harmonic oscillator and explores the effects of lattice parameters on its spectral properties.

Findings

First allowed energy band computed using instanton methods

Spectrum is consistent with quantum pendulum band structure but is purely discrete

Emergence of infinite degeneracy as the lattice spacing approaches zero

Abstract

It is computed, using instanton methods, the first allowed energy band for the polymer harmonic oscillator. The result is consistent with the band structure of the standard quantum pendulum but with pure point spectrum. An effective infinite degeneracy emerges in the formal limit $\mu/l_0 \to 0$ where $l_0$ is the characteristic length of the vacuum eigenfunction of a quantum harmonic oscillator. As an additional result, it is shown along the article the role played by the lattice reference point $\lambda$ in the full quantization of the polymer harmonic oscillator.