Abstract

It is arguably one of the main achievements of loop quantum gravity to have demonstrated that space itself may have an atomic structure. One of the key open problems of the theory is to reconcile the discrete spectra of geometric observables (such as area and volume) with general relativity in the continuum. In this talk, I present recent progress regarding this issue: I will show that the loop gravity discreteness of space can be understood from a conventional Fock quantisation of gravitational boundary modes on a null surface. These boundary modes are found by considering a quasi-local Hamiltonian analysis, where general relativity is treated as a Hamiltonian system in domains with inner null boundaries. The presence of such null boundaries requires then an additional boundary term in the action. Using Ashtekar’s original SL(2,C) self-dual variables, I will explain that the natural such boundary term is nothing but a kinetic term for a spinor (defining the null flag of the boundary) and a spinor-valued two-form, which are both intrinsic to the boundary. Finally, we will turn to the quantum theory and I will explain how the cross-sectional area two-form on the null surface turns into the difference of two number operators. The resulting area spectrum is discrete. Spin networks or triangulations of space do not enter the construction.