9 Short Summary and Conclusion 

In this section, I very briefly summarize the state of loop quantum gravity and its main results. The mathematics of the theory is solidly defined, and is understood from several alternative points of view. Long standing problems such as the lack of a scalar product, the difficult of controlling the overcompleteness of the loop basis and the problem of implementing the reality condition in the quantum theory have been successfully solved or sidestepped. The kinematics is given by the Hilbert space , defined in Section 6.2, which carries a representation of the basic operators: the loop operators (22 - 23). A convenient orthonormal basis in is provided by the spin network states, defined in Section 6.3 . The diffeomorphism invariant states are given by the s-knot states, and the structure and properties of the (diff-invariant) quantum states of the geometry are quite well understood (Section 6.8). These states give a description of quantum spacetime in terms of polymer-like excitations of the geometry. More precisely, in terms of elementary excitations carrying discretized quanta of area.

The dynamics is coded into the hamiltonian constraint. A well defined version of this constraint exists (see equation (35)), and thus a consistent theory exists, but a proof that the classical limit of this theory is classical general relativity is still lacking. Alternative versions of the hamiltonian constraint have been proposed and are under investigation. In all these cases, the hamiltonian has the crucial properties of acting on nodes only. This implies that its action is naturally discrete and combinatorial. This fact is possibly at the roots of the finiteness of the theory. A large class of physical states which are exact solutions of the dynamics are given by s-knots without nodes; other exact states are related to knot theory invariants (Section 7.1).

The theory can be extended to include matter, and there are strong indications that ultraviolet divergences do not appear. A spacetime covariant version of the theory, in the form of a topological sum over surfaces, is under development (Section 6.10).

The main physical results derived so far from the theory are given by the explicit computation of the eigenvalues of area and volume, some of which are given in equations (41 - 45), and a derivation of the black hole entropy formula (Section 41). The two main (related) open problems are to understand the description of the low energy regime within the theory and to choose the correct version of the hamiltonian constraint.

• 9.1 Conclusion

Loop Quantum Gravity Carlo Rovelli http://www.livingreviews.org/lrr-1998-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de