[Paper Review] Generalised Uncertainty Relations and the Problem of Dark Energy

Matthew J. Lake|arXiv (Cornell University)|Dec 27, 2021

Quantum Mechanics and Applications4 citations

TL;DR

This paper proposes a novel quantum gravity model where generalised uncertainty principles (GUP, EUP, EGUP) emerge from a quantum superposition of spacetime geometries, without modifying the canonical Heisenberg algebra. By treating spacetime as a quantum object with intrinsic geometric fluctuations, the model derives an effective dark energy density matching the observed value (~10⁻³⁰ g·cm⁻³), with spatial oscillations at ~0.1 mm wavelength, offering a testable signature distinct from standard cosmological models.

ABSTRACT

We outline a new model in which generalised uncertainty relations, that govern the behaviour of microscopic world, and dark energy, that determines the large-scale evolution of the Universe, are intrinsically linked via the quantum properties of space-time. In this approach the background is treated as a genuinely quantum object, with an associated state vector, and additional fluctuations of the geometry naturally give rise to the extended generalised uncertainty principle (EGUP). An effective dark energy density then emerges from the field that minimises the modified uncertainty relations. These results are obtained via modifications of the canonical quantum operators, but without modifications of the canonical Heisenberg algebra, allowing many well known problems associated with existing GUP models to be circumvented.

Motivation & Objective

• To resolve the long-standing problem of dark energy's origin by linking it to quantum gravity effects.
• To derive the generalized uncertainty principle (GUP, EUP, EGUP) from a rigorous quantum formalism without altering the canonical Heisenberg commutator.
• To explain the observed dark energy density (~10⁻³⁰ g·cm⁻³) as an emergent consequence of geometric fluctuations in a quantum spacetime.
• To avoid the pathologies of conventional GUP models—such as Lorentz violation and the soccer ball problem—by preserving the canonical algebra.
• To propose that spacetime is quantized at a scale β ≈ ħ × 10⁻⁶¹, distinct from matter’s Planck-scale quantization.

Proposed method

• The model treats spacetime as a quantum superposition of geometries, introducing nonlocal quantum geometry via a smeared spatial point representation.
• It defines position and momentum uncertainties as standard deviations from a generalized probability distribution incorporating both wavefunction diffusion and geometric fluctuations.
• The canonical Heisenberg commutator is preserved, but the effective Planck constant is rescaled as ħ → ħ(1 + δ), with δ ≈ √(ħ³GΛ/c³) ≈ 10⁻⁶¹.
• The extended generalized uncertainty principle (EGUP) is derived from the modified uncertainty product, incorporating both gravitational (GUP) and dark energy (EUP) effects.
• The field minimizing the product of generalized uncertainties yields an effective dark energy density matching the observed value ρΛ ≈ 10⁻³⁰ g·cm⁻³.
• The model introduces separate quantization scales: ħ for matter and β ≈ ħ × 10⁻⁶¹ for geometry, resolving the vacuum energy problem.

Experimental results

Research questions

• RQ1 Can the observed dark energy density emerge naturally from quantum spacetime fluctuations without modifying the canonical commutator?
• RQ2 How can the generalized uncertainty principle (GUP, EUP, EGUP) be derived from a consistent quantum formalism of spacetime geometry?
• RQ3 What is the origin of the cosmological constant in a quantum gravity framework that avoids the standard vacuum energy divergence?
• RQ4 Can a model preserve the canonical Heisenberg algebra while still generating measurable GUP-like effects and dark energy?
• RQ5 What is the physical significance of a fundamental quantum of action β ≪ ħ for geometric degrees of freedom?

Key findings

• The effective dark energy density derived from the EGUP matches the observed value ρΛ ≈ 10⁻³⁰ g·cm⁻³ on large scales.
• The model predicts spatial oscillations in the dark energy density with a wavelength of approximately 0.1 mm, offering a testable signature.
• The effective Planck constant is rescaled as ħ → ħ(1 + δ), with δ ≈ 10⁻⁶¹, preserving the canonical Heisenberg algebra and avoiding known pathologies of GUP models.
• The vacuum energy problem is resolved by introducing a distinct quantization scale β ≈ ħ × 10⁻⁶¹ for spacetime geometry, distinct from matter’s Planck-scale quantization.
• The model implies a minimum energy density in nature, identified as dark energy, which is necessary for the momentum space representation to remain well-defined.
• The formalism naturally leads to a nonlocal quantum geometry, where each spatial point is smeared over a volume comparable to the Planck volume.