Null twisted geometries

We define and investigate a quantization of null hypersurfaces in the context of loop quantum gravity on a fixed graph. The main tool we use is the parametrization of the theory in terms of twistors, which has already proved useful in discussing the interpretation of spin networks as the quantization of twisted geometries. The classical formalism can be extended in a natural way to null hypersurfaces, with the Euclidean polyhedra replaced by null polyhedra with spacelike faces, and SU(2) by the little group ISO(2). The main difference is that the simplicity constraints present in the formalism are all first class, and the symplectic reduction selects only the helicity subgroup of the little group. As a consequence, information on the shapes of the polyhedra is lost, and the result is a much simpler, Abelian geometric picture. It can be described by a Euclidean singular structure on the two-dimensional spacelike surface defined by a foliation of spacetime by null hypersurfaces. This geometric structure is naturally decomposed into a conformal metric and scale factors, forming locally conjugate pairs. Proper action-angle variables on the gauge-invariant phase space are described by the eigenvectors of the Laplacian of the dual graph. We also identify the variables of the phase space amenable to characterize the extrinsic geometry of the foliation. Finally, we quantize the phase space and its algebra using Dirac’s algorithm, obtaining a notion of spin networks for null hypersurfaces. Such spin networks are labeled by SO(2) quantum numbers and are embedded nontrivially in the unitary, infinite-dimensional irreducible representations of the Lorentz group.

Figure 1

A foliation of spacetime by null hypersurfaces.

Figure 2

From half links $(z,\tilde{z})$ to links $(j,\xi )$ and to loops $(J,\mathrm{\Phi })$.

Figure 3

The deficit angle $(2\pi -\mathrm{\Phi })$ and the scale $J$ of the cone.