Generalized uncertainty principle or curved momentum space?

TL;DR

This paper explores a duality between generalized uncertainty principles and quantum mechanics on curved momentum space, deriving geometric structures and constraints that connect quantum gravity effects with nontrivial momentum geometries.

Contribution

It introduces a duality framework linking generalized uncertainty principles to quantum mechanics on curved momentum space, deriving the associated vielbein and analyzing curvature implications.

Findings

Curvature tensor in momentum space relates to noncommutativity of coordinates.

The metric remains non-Euclidean even in flat space due to noncanonical momentum basis.

Constraints on curvature and basis deviations are derived from the duality.

Abstract

The concept of minimum length, widely accepted as a low-energy effect of quantum gravity, manifests itself in quantum mechanics through generalized uncertainty principles. Curved momentum space, on the other hand, is at the heart of similar applications such as doubly special relativity. We introduce a duality between theories yielding generalized uncertainty principles and quantum mechanics on nontrivial momentum space. In particular, we find canonically conjugate variables which map the former into the latter. In that vein, we explicitly derive the vielbein corresponding to a generic generalized uncertainty principle in $d$ dimensions. Assuming the predominantly used quadratic form of the modification, the curvature tensor in momentum space is proportional to the noncommutativity of the coordinates in the modified Heisenberg algebra. Yet, the metric is non-Euclidean even in the flat case corresponding to commutative space, because the resulting momentum basis is noncanonical. These insides are used to constrain the curvature and the deviation from the canonical basis.