Fock representation of gravitational boundary modes and the discreteness of the area spectrum

In this article, we study the quantum theory of gravitational boundary modes on a null surface. These boundary modes are given by a spinor and a spinor-valued two-form, which enter the gravitational boundary term for self-dual gravity. Using a Fock representation, we quantise the boundary fields, and show that the area of a two-dimensional cross section turns into the difference of two number operators. The spectrum is discrete, and it agrees with the one known from loop quantum gravity with the correct dependence on the Barbero--Immirzi parameter. No discrete structures (such as spin network functions, or triangulations of space) are ever required---the entire derivation happens at the level of the continuum theory. In addition, the area spectrum is manifestly Lorentz invariant.