Generalized Uncertainty Principle: from the harmonic oscillator to a QFT toy model

TL;DR

This paper explores the effects of a polynomial Generalized Uncertainty Principle on the harmonic oscillator and constructs a related quantum field theoretic toy model, revealing non-trivial spectral modifications.

Contribution

It introduces a new formalism for a polynomial GUP, derives the exact algebra and spectrum for the harmonic oscillator, and develops a QFT toy model based on GUP.

Findings

Energy spectrum is significantly altered by GUP.

New ladder operators are constructed that exactly factorize the Hamiltonian.

A quantum field theoretic toy model incorporating GUP is developed.

Abstract

Several models of quantum gravity predict the emergence of a minimal length at Planck scale. This is commonly taken into consideration by modifying the Heisenberg Uncertainty Principle into the Generalized Uncertainty Principle. In this work, we study the implications of a polynomial Generalized Uncertainty Principle on the harmonic oscillator. We revisit both the analytic and algebraic methods, deriving the exact form of the generalized Heisenberg algebra in terms of the new position and momentum operators. We show that the energy spectrum and eigenfunctions are affected in a non-trivial way. Furthermore, a new set of ladder operators is derived which factorize the Hamiltonian exactly. The above formalism is finally exploited to construct a quantum field theoretic toy model based on the Generalized Uncertainty Principle.