Quantum Structure of Geometry: Loopy and fuzzy?

In any attempt to build a quantum theory of gravity, a central issue is to unravel the structure of space-time at the smallest scale. Of particular relevance is the possible definition of coordinate functions within the theory and the study of their algebraic properties, such as non-commutativity. Here we approach this issue from the perspective of loop quantum gravity and the picture of quantum geometry that the formalism offers. In particular, as we argue here, this emerging picture has two main elements: i) The nature of the quantum geometry at Planck scale is one-dimensional, polymeric with quantized geometrical quantities and; ii) Appropriately defined operators corresponding to coordinates by means of intrinsic, relational, constructions become non-commuting. This particular feature of the operators, that operationally localize points on space, gives rise to an emerging geometry that is also, in a precise sense, fuzzy.