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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.

Broad arguments indicate that quantum gravity should have a minimal length scale. In this paper, we construct a minimum length model by generalizing the time-position and energy–momentum operators while keeping much of the structure of quantum mechanics and relativity intact: the standard position-momentum commutator, the special relativistic time-position, and energy–momentum relationships all remain the same. Since the time-position and energy–momentum relationships for the modified operators remain the same, we retain a form of Lorentz symmetry. This avoids the constraints on these theories coming from lack of photon dispersion while holding the potential to address the Greisen–Zatsepin–Kuzmin (GZK) puzzle of ultra high-energy cosmic rays.

The existence of a small, non-zero cosmological constant is one of the major puzzles in fundamental physics. Naively, quantum field theory arguments would imply a cosmological constant which is up to 10,120 times larger than the observed one. It is believed a comprehensive theory of quantum gravity would resolve this enormous mismatch between theory and observation. In this work, we study the ability of generalized uncertainty principle (GUP) models, which are phenomenologically motivated models of quantum gravity, to address the cosmological constant problem. In particular, we focus on how these GUP models may change the phase space of QFT, and how this affects the momentum space integration of the zero-point energies of normal modes of fields. We point out several issues that make it unlikely that GUP models, in their current form, would be able to adequately address the cosmological constant problem.

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