One of the main open problems in loop quantum gravity is to reconcile the fundamental quantum discreteness of space with general relativity in the continuum. In this talk, I present recent progress regarding this issue: I will explain, in particular, how the discrete spectra of geometric observables that we find in loop gravity can be understood from a conventional Fock quantisation of gravitational edge modes on a null surface boundary. On a technical level, 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 additional boundary terms 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. The simplest observable on the boundary phase space is the cross sectional area two-form, which generates dilatations of the boundary spinors. In quantum theory, the corresponding area operator turns into the difference of two number operators. The area spectrum is discrete without ever introducing spin networks or triangulations of space. I will also comment on a similar construction in three euclidean spacetime dimensions, where the discreteness of length follows from the quantisation of gravitational edge modes on a one-dimensional cross section of the boundary.
The talk is based on my recent papers: arXiv:1804.08643 and arXiv:1706.00479.
- Relativity Seminar