Kiefer C. Quantum gravity

198 QUANTUM GRAVITY WITH CONNECTIONS AND LOOPS
states; cf. Thiemann (2001, 2003). In order to avoid some of these problems, the
‘master constraint programme’ has been suggested (Thiemann 2006). There one
combines the smeared Hamiltonian constraints for all smearing functions into
one single constraint. It is, however, premature to make any judgement about
the success of this programme.
Alternative approaches to the quantum Hamiltonian constraint are the ‘spinfoam
models’, which use a path-integral type of approach employing the evolution
of spin networks in ‘time’. We shall not discuss this here, but refer the reader to
the literature; cf. Rovelli (2004), Perez (2006), and Nicolai et al. (2006).
To summarize, the main results in the loop or connection representation have
been found on the kinematical level. This holds in particular for the discrete spectra
of geometric operators. The main open problem is the correct implementation
(and solution) of the Hamiltonian constraint.