Members: K. Mecke, T. Thiemann
Among the constraints that govern the canonical formulation of General Relativity the Hamiltonian constraint is the most difficult one to solve and even to define. It is this constraint that implements the dynamics of General Relativity. Within the context of Loop Quantum Gravity, the Hamiltonian constraint has been quantised [25], [26], [27], however, its complexity prohibits the possibility of analytic solutions.To make progress, numerical methods can be used, utilising well established algorithms known from Statistical Physics, Euclidian Field Theory and Lattice Gauge Theory [28] which certainly will require supercomputing.
Complementary to this Hamiltonian description, in recent years path integral formulations of Loop Quantum Gravity have been considered, called spin foam models [29], that generalise state sum models [30] for Topological Field Theories [31]. Also here the state sums are too complex to be carried out analytically thus again requiring supercomputing. In addition, an open issue in Quantum Geometry is the meaning of Wilson renormalisation and effective actions which in usual Quantum Field Theory describe the relevant physics at different length scales. The challenge is that Quantum Geometry is a background independent theory which has no background length scale that usually governs block spin transformations. The methods from Statistical Field Theory may therefore be used to some extent but certainly have to be generalised.