Generalized uncertainty principle or curved momentum space?

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.