# Tests of Quantum Gravity-Induced Non-Locality: Hamiltonian formulation of a non-local harmonic oscillator

## TL;DR

This paper develops a Hamiltonian framework for a non-local harmonic oscillator inspired by quantum gravity, extending previous models and analyzing its interaction with cavity fields to aid experimental constraints.

## Contribution

It introduces a Hamiltonian formulation for a non-local harmonic oscillator coupled to cavity fields, enabling spectrum analysis and perturbative study of interactions.

## Key findings

- Recovered and extended previous non-local oscillator models
- Derived the perturbed energy spectrum of the system
- Formulated Langevin equations for the interacting system

## Abstract

Motivated by the development of on-going optomechanical experiments aimed at constraining non-local effects inspired by some quantum gravity scenarios, the Hamiltonian formulation of a non-local harmonic oscillator, and its coupling to a cavity field mode(s), is investigated. In particular, we consider the previously studied model of non-local oscillators obtained as the non-relativistic limit of a class of non-local Klein-Gordon operators, $f(\Box)$, with $f$ an analytical function. The results of previous works, in which the interaction was not included, are recovered and extended by way of standard perturbation theory. At the same time, the perturbed energy spectrum becomes available in this formulation, and we obtain the Langevin's equations characterizing the interacting system.