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

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