Consistent discretizations: the basic idea

There has long been the hope that lattice methods could be used as a non-perturbative approach to Quantum Gravity. This is in part based on the fact that lattice methods have been quite successful in the treatment of quantum chromodynamics. However, one needs to recall that one of the appeals of lattice methods in QCD is that they are gauge invariant regularization methods. In the gravitational context this is not the case. As soon as one discretizes space-time one breaks the invariance under diffeomorphisms, the symmetry of most gravitational theories of interest. As such, lattice methods in the gravitational context face unique challenges. For instance, in the path integral context, since the lattices break some of the symmetries of the theory, this may complicate the use of the Fadeev–Popov technique. In the canonical approach if one discretizes the constraints and equations of motion, the resulting discrete equations are inconsistent: they cannot be solved simultaneously. A related problem is that the discretized constraints fail to close a constraint algebra.

To address these problems we have proposed a different methodology for discretizing gravitational theories (or to use a different terminology “to put gravity on the lattice”). The methodology is related to a discretization technique that has existed for a while in the context of unconstrained theories called “variational integrators”. In a nutshell, the technique consists in discretizing the action of the theory and working from it the discrete equations of motion.