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

TL;DR

This paper develops a continuum, Lorentz-invariant quantisation of gravitational boundary modes on a null surface by introducing boundary spinors and their conjugate momentum, then successfully recasts the cross-sectional area as the difference of two number operators via a Landau-type representation. The resulting area spectrum is discrete and proportional to the Barbero–Immirzi parameter, matching the loop quantum gravity area spectrum up to ordering, all without discretising space. By imposing quantum reality conditions, the boundary states reduce to representations with ρ = β j and align with Ashtekar–Lewandowski-type boundary states, establishing a covariant bridge between boundary spinor quantisation and standard LQG. The work demonstrates that area quantisation can emerge from continuum boundary dynamics on null surfaces while preserving Lorentz invariance and avoiding spin networks or triangulations in the bulk. Overall, it provides a coherent, boundary-first route to a discrete, Lorentz-invariant geometric spectrum compatible with non-perturbative quantum gravity frameworks.

Abstract

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.

Figures (2)

• Figure 1: In loop quantum gravity the quantum states of the gravitational field are built from gravitational Wilson lines (lying in a three-dimensional spatial hypersurface). These Wilson lines can hit a two-dimensional boundary $\mathcal{C}$, where they create a surface charge, namely a spinor-valued surface density $\pi_A$.
• Figure 2: We are considering the gravitational field in a four-dimensional causal region $\mathcal{M}$, whose boundary has four components, namely the three-dimensional null surfaces $\mathcal{N}_+$ and $\mathcal{N}_-$, which have the topology of a cylinder $[0,1]\times S^2$, and the spacelike disks $\varSigma_-$ and $\varSigma_+$ at the top and bottom. The boundary has three corners, which appear as the boundary of the boundary, namely $\partial\mathcal{N}_+=\mathcal{C}_+\cup\mathcal{C}_o^{-1}$ and $\partial\mathcal{N}_-=\mathcal{C}_o\cup\mathcal{C}_-^{-1}$. All these manifolds carry an orientation, which is induced from the bulk: $\partial\mathcal{M}={\varSigma}_{-}^{-1}\cup\mathcal{N}_-\cup\mathcal{N}_+\cup\varSigma_+$.