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The concept of minimum length, widely accepted as a low-energy effect ofquantum gravity, manifests itself in quantum mechanics through generalizeduncertainty principles. Curved momentum space, on the other hand, is at theheart of similar applications such as doubly special relativity. We introduce aduality between theories yielding generalized uncertainty principles andquantum mechanics on nontrivial momentum space. In particular, we findcanonically conjugate variables which map the former into the latter. In thatvein, we explicitly derive the vielbein corresponding to a generic generalizeduncertainty principle in $d$ dimensions. Assuming the predominantly usedquadratic form of the modification, the curvature tensor in momentum space isproportional to the noncommutativity of the coordinates in the modifiedHeisenberg algebra. Yet, the metric is non-Euclidean even in the flat casecorresponding to commutative space, because the resulting momentum basis isnoncanonical. These insides are used to constrain the curvature and thedeviation from the canonical basis.

Generalized uncertainty principle or curved momentum space?