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9 Short Summary and Conclusion

The mathematics of loop quantum gravity is solidly defined, and is understood from several alternative points of view. Long-standing problems, such as the lack of a scalar product, the difficulty 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 a well-defined Hilbert space ℋ that carries a representation of the basic operators. A convenient orthonormal basis in ℋ 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 6.4). These states give a description of quantum spacetime in terms of discrete excitations of the geometry. More precisely, in terms of elementary excitations carrying discretized quanta of area (Section 7).

The dynamics can be coded into the Hamiltonian constraint. A well-defined version of this constraint exists, and thus a complete and consistent theory exists, but it is not easy to extract physics from this theory and proof that the classical limit of this theory is correct 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 at the root of the finiteness of the theory. 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 is given by the spin-foam formalism (Section 6.7). The group field theory formalisms (Section 6.8) provides a dual formulation of the theory that generates the spin-foam sum. In 4D, the precise relation between the different formalisms is under investigation.

A key physical result is given by the explicit computation of the eigenvalues of area and volume (see Equation (33View Equation)). This result is at the basis of the physical picture of a discrete spacetime. To be sure, discreteness is in the quantum mechanical sense: generically physical space is not an ensemble of quanta, but a continuous probabilistic superposition of ensembles of quanta.

The theory has numerous physical applications, including quantum cosmology, black hole physics and others.

The two main (related) open problems are to understand the description of the low energy regime within the theory and to find the correct version of the dynamics, either via the Hamiltonian constraint or via a spin-foam vertex.

The history of quantum gravity is a sequence of moments of great excitement followed by bitter disappointment. I distinctively remember, as a young student, listening to a very famous physicist announcing at a major conference that quantum gravity was solved (I think it was the turn of supergravity). The list of theories that claimed to be final and have then ended up forgotten or superseded is a reason for embarrassment for the theoretical physics community, in my opinion.

In my view, loop quantum gravity is the best we can do so far in trying to understand quantum spacetime, from a nonperturbative, background-independent point of view. Theoretically, we have reasons to suspect that this approach could represent a consistent quantum theory with the correct classical limit. The theory yields a definite physical picture of quantum spacetime and definite quantitative predictions, but a systematic way of extracting physical information is still lacking. Experimentally, there is no support for the theory, neither direct nor indirect. The spectra of area and volume computed within the theory could, or could not, be physically correct. I hope we may find a way to know in the not too distant future.

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