Natural extension of the Generalised Uncertainty Principle (0709.1965v2)

Abstract: We discuss a gedanken experiment for the simultaneous measurement of the position and momentum of a particle in de Sitter spacetime. We propose an extension of the so-called generalized uncertainty principle (GUP) which implies the existence of a minimum observable momentum. The new GUP is directly connected to the nonzero cosmological constant, which becomes a necessary ingredient for a more complete picture of the quantum spacetime.

Natural Extension of the Generalised Uncertainty Principle

The paper "Natural extension of the Generalised Uncertainty Principle" by C. Bambi and F.R. Urban investigates the Generalised Uncertainty Principle (GUP) and proposes a further extension in de Sitter spacetime. Central to their exploration is the concept that the GUP can be extended to incorporate the cosmological constant, thereby suggesting the existence of a minimum observable momentum.

Overview

This paper begins by recalling the traditional Uncertainty Principle in quantum mechanics, formulated as $\Delta x \, \Delta p \gtrsim 1$, where $\Delta x$ and $\Delta p$ are the uncertainties in position and momentum, respectively. When gravitational effects are considered at quantum scales, this principle needs revision. This has led to the formulation of the GUP, expressed as $$\Delta x \, \Delta p \gtrsim 1 + \alpha \, L_{Pl}^2 \, (\Delta p)^2$$, incorporating the Planck length $L_{Pl}$.

However, the standard formulation of GUP raises aesthetic and symmetry concerns; it treats position and momentum asymmetrically. Motivated by this asymmetry, the authors propose a further extension: $$\Delta x \, \Delta p \gtrsim 1 + \alpha \, L_{Pl}^2 \, (\Delta p)^2 + \beta \, \frac{(\Delta x)^2}{L_\Lambda^2}$$, where $L_\Lambda$ relates to the de Sitter horizon induced by the cosmological constant $\Lambda$.

Implications and Discussion

Through a gedanken experiment—imagining simultaneous measurement of a particle's position and momentum in de Sitter spacetime—the authors derive the extended GUP. This experiment shows that the non-zero cosmological constant transforms spatiotemporal measurements, leading to the existence of a minimum observable momentum. Their derivation suggests that both the Planck scale and the cosmological constant fundamentally affect quantum spacetime.

The symmetry regained by their proposal not only has theoretical implications but could lead to observable deviations from classical mechanics and alterations in quantum field theory. Such effects might span thermodynamics and statistical mechanics as well, especially as GUP deviations from classical physics are considered both in ultraviolet (small-scale) and infrared (large-scale) regimes.

Practical and Theoretical Implications

Primarily, this research ties the cosmological constant intricately into the structure of spacetime, alongside the gravitational constant. Astrophysical observations indicate a non-zero $\Lambda$, enhancing the pertinence of this theoretical framework. The proposed extension is insightful for considering quantum gravity effects both in early universe cosmology and potential IR signatures.

While the immediate practical applications of this new GUP are limited due to the smallness of $\alpha$ and $\beta$, the theoretical underpinnings provide valuable insights into quantum gravity signatures. Investigations into the quantum mechanics deviations and potential impacts during the universe's inflationary phase motivate further research.

Conclusion

The extended Generalised Uncertainty Principle proposed in this paper suggests profound alterations in our understanding of the quantum spacetime influenced by a cosmological constant. It raises the possibility of observable quantum gravity effects in both small-scale and large-scale spatiotemporal dynamics, promising stimulating advancements in our quest to bridge quantum mechanics and general relativity. Future work may explore the implications of the extended GUP in early cosmological epochs and address its speculative quantum gravity interactions at multiple scales.