The Spacetime Picture in Quantum Gravity

We propose an approach which, by combining insights from loop quantum gravity (LQG), Topos theory, non-commutative geometry à la Connes, and spacetime relationalism, provides fertile ground for the search of an adequate spacetime picture in quantum gravity. With this approach, we obtain a novel way of deducing the quantization of the possible values for the area of a surface. One gets the same area values than those from the area operator in standard LQG, but our approach makes a further prediction: some smaller values and sub-divisions are also allowed. In addition, the area arises as a noncommutative distance between two noncommutative points, and thus they should be interpreted as irreducible string-like objects at the physical level (where the area interpretation for the noncommutative distance holds). (paper)