Quantum Gravity-Inspired Dispersion Relations Remain Hyperbolic across Major Models, Ensuring Stable Propagation

Modified dispersion relations, which alter the fundamental relationship between energy and momentum, frequently appear in theories attempting to reconcile quantum mechanics with gravity, often presenting challenges for standard analytical techniques. Gines R. Perez Teruel, alongside colleagues, investigates these relations using a geometric approach, analysing the curvature of the surfaces representing energy and momentum. This work demonstrates that dispersion relations derived from Loop Quantum Gravity consistently exhibit stable, hyperbolic propagation without instabilities or new scales, even in regimes where standard methods fail. The team extends this geometric framework to analyse a broad range of modified dispersion relations, establishing universal criteria for assessing stability and revealing the robustness of those arising from quantum gravity theories.

Nometric modified dispersion relations appear in several quantum gravity theories, including causal set theory and loop quantum gravity, while loop quantum gravity yields corrections to energy and momentum relationships. Scientists have developed a geometric framework to analyse these modified dispersion relations, focusing on the intrinsic curvature of the energy-momentum surfaces that represent them. Negative curvature ensures stable propagation of signals, while changes in curvature or the appearance of critical points indicate potential instabilities or the existence of new energy scales. Applying this method comprehensively to modified dispersion relations derived from loop quantum gravity, researchers demonstrate that these relations remain hyperbolic, guaranteeing stable propagation, across the entire range of energies relevant to observational experiments.

Geometric Criteria for Modified Dispersion Relation Viability

This research introduces a geometric framework for analysing modified dispersion relations, key predictions of many quantum gravity theories. Instead of relying on approximations, the authors focus on the intrinsic geometry of the energy-momentum surface defined by these relations. This allows them to establish general criteria for viability, independent of specific theoretical assumptions. The goal is to provide a universal method for constraining quantum gravity models based on their geometric properties. The framework centres on modified dispersion relations, which suggest that the speed of light may vary at extremely high energies.

Scientists analyse the energy-momentum surface defined by these relations, examining its intrinsic geometry, including curvature and topology. Viability criteria include hyperbolicity, ensuring well-behaved signal propagation, and the absence of critical points, which could indicate new, unphysical energy scales. The research specifically applies this framework to modified dispersion relations arising from loop quantum gravity, including those based on polymer quantization, holonomy corrections, and inverse-triad structures. The primary achievement is the development of a coordinate-independent, geometric approach to analysing modified dispersion relations, offering a more robust and universal method than traditional techniques.

The authors establish clear geometric criteria for determining whether a modified dispersion relation is physically viable. They demonstrate that loop quantum gravity-motivated models remain intrinsically stable and consistent with observations within this framework. Furthermore, the research derives model-independent bounds on parameters in various modified dispersion relations based solely on geometric consistency. This provides constraints even when the underlying theory is not fully understood, and allows for a classification of modified dispersion relations based on their geometric properties, providing a deeper understanding of their behaviour.

These findings offer stronger constraints on quantum gravity theories, suggesting the robustness of loop quantum gravity-motivated models. The framework provides a direct link between theoretical predictions and observational constraints, extending beyond the limitations of effective field theory. This work focuses on the intrinsic curvature of energy-momentum surfaces associated with these modified dispersion relations, establishing that negative curvature ensures stable propagation while curvature changes or critical points indicate instabilities or new energy scales. The team measured constraints on logarithmic, exponential, and trigonometric modified dispersion relations, establishing a unified and coordinate-independent assessment of stability and invariant scales. For logarithmic modified dispersion relations, the analysis reveals a lower bound of Λ 10 13 , 10 14 GeV for photons with energies of 100, 300 TeV.

Exponential modified dispersion relations are constrained by thresholds determined by photon, neutrino, and ultra-high-energy cosmic ray energies, resulting in a minimum value of M 10 8 , 10 9 GeV for μ approximately equal to 1 and photon energies around 100, 300 TeV. Trigonometric modified dispersion relations are constrained to λ -6 GeV -1 for photons in the TeV, PeV range. These results, presented as exclusion lines for logarithmic and exponential modified dispersion relations, represent the first model-independent bounds on non-analytic modified dispersion relations. The team established that loop quantum gravity predicts modified dispersion relations that are intrinsically stable and phenomenologically robust, remaining hyperbolic throughout the observational window. This geometric analysis provides a unifying criterion for constraining quantum gravity scenarios, extending beyond standard effective field theory methods and offering a bridge between quantum gravity phenomenology and high-energy observations. Researchers developed a method representing these relations as surfaces embedded in energy-momentum space, allowing them to assess stability and identify new momentum scales using geometric invariants like curvature. This approach successfully unifies the analysis of polynomial, non-polynomial, and factorizable dispersion relations, extending beyond the limitations of traditional effective field theory techniques. The team exhaustively applied this geometric method to modified dispersion relations derived from loop quantum gravity, including those with polymeric, holonomy-corrected, inverse-triad, and semiclassical forms.

Results demonstrate that all these dispersion relations remain hyperbolic and stable across relevant energy regimes, without exhibiting critical points that would indicate instability. This confirms the robustness of these models and provides a coordinate-independent assessment of their properties. This work establishes a powerful new tool for probing the fundamental nature of spacetime and testing the predictions of quantum gravity theories.