Null twisted geometries

Abstract

We define and investigate a quantisation of null hypersurfaces in the context of loop quantum gravity on a fixed graph. The main tool we use is the parametrisation 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 space-like faces, and SU(2) by the little group ISO(2). The main difference is that the simplicity constraints present in the formalims 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 an Euclidean singular structure on the 2-dimensional space-like surface defined by a foliation of space-time 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 quantise the phase space and its algebra using Dirac's algorithm, obtaining a notion of spin networks for null hypersurfaces. Such spin networks are labelled by SO(2) quantum numbers, and are embedded non-trivially in the unitary, infinite-dimensional irreducible representations of the Lorentz group.Comment: 22 pages, 3 figures. v2: minor corrections, improved presentation in section 4, references update

Sachs’ free data in real connection variables

“…The identification of the connection constraint-free data as null rotations means that the degrees of freedom form a group, albeit non-compact, hence one could try to use loop quantum gravity quantization techniques without introducing the Immirzi parameter. Some of the corner data, which we did not investigate here, have already be shown to lead to a quantization of the area [22,50,59]. A quantization of the connection description of the radiative degrees of freedom can lead to new insights both for loop quantum gravity and for asymptotic quantisations based on a Fock space.…”

“…Quantising with analogue connection methods the constraint-free data on null foliations would allow us to study the quantum structure of the physical degrees of freedom directly. 2 As a preliminary result in this direction, it was shown in [22] that at the kinematical level, discretisations of the 2d space-like metric have quantum area operators with a discrete spectrum given by the helicity quantum numbers. A stronger more recent result appeared in [23], based on covariant phase space methods and a spinorial boundary term, confirming the discrete area spectrum without a discretisation.…”

“…22 Next, let us look at the tertiary constraints in its form (3.23), and project it in the same way on S:…”

Octahedron of complex null rays and conformal symmetry breaking

“…See also[24] for related reductions to the little groups ISO(2) and SU (1, 1) stabilizing resp. a null and a space-like direction.…”

Discrete gravity as a topological field theory with light-like curvature defects

“…In a Hamiltonian approach, the three-manifold Σ is often required to be a Cauchy hypersurface as in e.g [36][37][38],. a generalisation to null surfaces was proposed as well cf [39]…”