A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle

Abstract

In this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator, [X^,P^]=iħ(1+βp2), may have different physical consequences such as having no minimal length at all. These differences depend on how the position and/or momentum operators are modified rather than only on the resulting modified commutator. This provides guidance when constructing GUP models since it distinguishes those GUPs that have a minimal length scale, as suggested by some broad arguments about quantum gravity, versus GUPs without a minimal length scale.

The more things change the more they stay the same: Minimum lengths with unmodified uncertainty principle and dispersion relation

“…However, as shown in [11,12], certain modified operators do not lead to a minimum length despite satisfying the same GUP.…”

Near horizon thermodynamics of hairy black holes from gravitational decoupling

“…Combining Eqs. ( 33),( 34) and (35), and neglecting higher orders of β, the generalized Dirac equation in curved spacetime is found [42,44]…”

“…On the other hand, it is usually believed that a minimal length in the spacetime is related to a generalization of the uncertainty principle in the quantum realm in a plethora of theories and models [28][29][30][31][32][33][34]. The link between them may be heuristically described [35] by noticing that in natural units the Schwarzschild radius, r s , scales as r s ∼ M . In higher energies, where small length scales are scrutinized, the previous relation would be transposed to ∆x ∼ ∆p, so that the typical product ∆x∆p would have a correction proportional to ∆p 2 .…”

Fermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP case

“…Generalizations of (the Heisenberg) uncertainty principle (GUPs) have been quite fruitful in providing predictions for quantum gravity effects, without specifying the mathematical structure of the models, although most, see e.g. [1][2][3][4][5][6], sharing the minimum length scale property, expected to be of the order of the Planck length L P ∼ √ ℏG c 3 , see also [7][8][9]. Nevertheless, the uncertainty relationship between two physical quantities is closely related to their commutation relation, making the deformation of quantum phase space and non-commutative (NC) quantum space-times a natural background for GUP theories.…”