[Paper Review] Generalized Uncertainty Principle: from the harmonic oscillator to a QFT toy model

Pasquale Bosso, Giuseppe Gaetano Luciano|arXiv (Cornell University)|Sep 30, 2021

Noncommutative and Quantum Gravity Theories77 references43 citations

TL;DR

This paper develops a polynomial Generalized Uncertainty Principle (GUP) framework for the harmonic oscillator, deriving exact energy spectra and eigenfunctions via analytic and algebraic methods. It constructs exact ladder operators that factorize the Hamiltonian and extends the formalism to a 1+1-dimensional quantum field theory toy model based on the GUP, showing consistent modifications to the Klein-Gordon field quantization under minimal length effects.

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.

Motivation & Objective

• To develop a generalized uncertainty principle (GUP) framework based on a polynomial deformation of the Heisenberg algebra.
• To analytically solve the GUP-modified harmonic oscillator, deriving exact energy eigenvalues and eigenfunctions.
• To construct exact ladder operators that factorize the GUP Hamiltonian, enabling algebraic treatment of the system.
• To extend the GUP formalism to a quantum field theory (QFT) toy model for a real scalar field in 1+1 dimensions.
• To demonstrate that GUP corrections induce non-trivial, non-perturbative modifications to the quantum harmonic oscillator and field-theoretic spectra.

Proposed method

• Formalism based on a deformed commutator [q̂, p̂] = iℏf(p̂), where f(p̂) is a polynomial function modeling minimal length effects.
• Derivation of generalized position and momentum operators in momentum representation using operator ordering prescriptions (A = 0, 1, 1/2).
• Solution of the stationary Schrödinger equation for the GUP harmonic oscillator using analytic methods, yielding modified energy levels and wavefunctions.
• Construction of exact ladder operators (C and C†) that satisfy [C, C†] = g(H) - H and factorize the Hamiltonian H.
• Adaptation of the algebraic method to define coherent states and probability distributions in the presence of minimal length.
• Extension to QFT by quantizing the Klein-Gordon field with GUP-deformed commutators, leading to modified field Hamiltonian density and spectrum.

Experimental results

Research questions

• RQ1 How do polynomial GUP deformations modify the energy spectrum and eigenfunctions of the harmonic oscillator?
• RQ2 Can exact ladder operators be derived for the GUP-modified harmonic oscillator that exactly factorize the Hamiltonian?
• RQ3 What is the structure of the generalized Heisenberg algebra under polynomial GUP deformation, and how does it differ from standard quantum mechanics?
• RQ4 How can the GUP formalism be consistently extended to a quantum field theory toy model in 1+1 dimensions?
• RQ5 What are the implications of minimal length effects on the field-theoretic spectrum and dynamics in a GUP-deformed scalar field?

Key findings

• The energy spectrum of the GUP harmonic oscillator is modified in a non-perturbative, non-trivial way, with corrections dependent on the polynomial form of f(p̂).
• Exact eigenfunctions are derived that depend on the expectation values of position and momentum, enabling precise trajectory analysis in coherent states.
• A new set of ladder operators is constructed that exactly factorize the GUP Hamiltonian, providing a complete algebraic solution to the system.
• The generalized Heisenberg algebra is derived in terms of the new position and momentum operators, with non-canonical commutation relations encoded in f(p̂).
• The QFT toy model for a real scalar field in 1+1 dimensions exhibits modified field commutators and Hamiltonian density, with spectrum shifted by GUP corrections.
• The field-theoretic extension preserves consistency with the GUP framework, showing that GUP effects propagate from single-particle to many-body systems via the deformed algebra.