  Phenomenological approaches to quantum gravity implement a minimum resolvable
length-scale but do not link it to an underlying formalism describing geometric
superpositions. Here, we introduce an intuitive approach in which points in the
classical spatial background are delocalised, or `smeared', giving rise to an
entangled superposition of geometries. The model uses additional degrees of
freedom to parameterise the superposed classical backgrounds. Our formalism
contains both minimum length and minimum momentum resolutions and we naturally
identify the former with the Planck length. In addition, we argue that the
minimum momentum is determined by the de Sitter scale, and may be identified
with the effects of dark energy in the form of a cosmological constant. Within
the new formalism, we obtain both the generalised uncertainty principle (GUP)
and extended uncertainty principle (EUP), which may be combined to give an
uncertainty relation that is symmetric in position and momentum. Crucially, our
approach does not imply a significant modification of the position-momentum
commutator, which remains proportional to the identity matrix. It therefore
yields generalised uncertainty relations without violating the equivalence
principle, in contradistinction to existing models based on nonlinear
dispersion relations. Implications for cosmology and the black hole uncertainty
principle correspondence are briefly discussed, and prospects for future work
on the smeared-space model are outlined.
