Answering them, on the other hand, has proved to be surprisingly difficult. The main reason, I believe, is the inadequacy of standard techniques. More precisely, to examine the microscopic structure of geometry, we must treat Einstein gravity quantum mechanically, i.e., construct at least the basics of a quantum theory of the gravitational field. With this general philosophy, in this article I will summarize the picture of quantum geometry that has emerged from a specific approach to quantum gravity. This approach is non-perturbative.
Discreteness is then hard to unravel.
In the non-perturbative approach, by contrast, there is no background metric at all. All we have is a bare manifold to start with. Consequently, the description can not refer to a background metric. As we will see, this fact leads one to Hilbert spaces of quantum states which are quite different from the familiar Fock spaces of particle physics. Microscopically, geometry is rather like a polymer.