  In perturbative gravity, it is straight-forward to characterize the two local
degrees of freedom of the gravitational field in terms of a mode expansion of
the linearized perturbation. In the non-perturbative regime, we are in a more
difficult position. It is not at all obvious how to construct Dirac observables
that can separate the gauge orbits. Standard procedures rely on asymptotic
boundary conditions or formal Taylor expansions of partial observables along
the gauge orbits. In this paper, we lay out a new non-perturbative lattice
approach to tackle the problem in terms of Ashtekar's self-dual formulation.
Starting from a simplicial decomposition of space, we introduce a local
kinematical phase space at the lattice sites. At each lattice site, we
introduce a set of constraints that replace the generators of the hypersurface
deformation algebra in the continuum. We show that the discretized constraints
close under the Poisson bracket. The resulting reduced phase space describes
two complex physical degrees of freedom representing the two radiative modes at
the discretized level. The paper concludes with a discussion of the key open
problems ahead and the implications for quantum gravity.
