The minimal length: a cut-off in disguise?

TL;DR

This paper demonstrates that the minimal-length concept in quantum gravity can be understood as a cut-off in wave-number space, revealing that some models allow arbitrary position measurement precision despite the minimal length hypothesis.

Contribution

It establishes the equivalence between minimal-length modifications and wave-number space cut-offs, generalizes the concept to multiple dimensions and geometries, and clarifies the kinematical nature of minimal length.

Findings

Minimal length corresponds to a wave-number space cut-off.

Some models permit arbitrary position measurement precision.

The minimal length is a purely kinematical concept.

Abstract

The minimal-length paradigm, a possible implication of quantum gravity at low energies, is commonly understood as a phenomenological modification of Heisenberg's uncertainty relation. We show that this modification is equivalent to a cut-off in the space conjugate to the position representation, i.e. the space of wave numbers, which does not necessarily correspond to momentum space. This result is generalized to several dimensions and noncommutative geometries once a suitable definition of the wave number is provided. Furthermore, we find a direct relation between the ensuing bound in wave-number space and the minimal-length scale. For scenarios in which the existence of the minimal length cannot be explicitly verified, the proposed framework can be used to clarify the situation. Indeed, applying it to common models, we find that one of them does, against all expectations, allow for arbitrary precision in position measurements. In closing, we comment on general implications of our findings for the field. In particular, we point out that the minimal length is purely kinematical such that, effectively, there is only one model of minimal-length quantum mechanics.