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The key open problem in the theory is to prove that loop gravity gives general relativity correctly in the low-energy, or classical, limit. Naively, this is the case, because replacing the quantum operators in the dynamical equations with their classical counterpart yields the equations of classical general relativity. But the background independent structure of the quantum theory is peculiar and unconventional, and a more convincing proof is certainly needed. Furthermore, we need to understand how the low-energy limit emerges from the background-independent theory in order to correct the low-order quantum corrections to classical general relativity.
The kinematics of the theory is well understood both physically (quanta of area and volume, discrete geometry) and from the mathematical point of view. The part of the theory that is not yet fully under control is the dynamics, which is determined by the Hamiltonian constraint. A plausible candidate for the quantum Hamiltonian constraint is available, and provides a well-defined theory, but a number of variants are under investigation, both as far as the operator and the general formalism are concerned. The question is whether the theory defined has the correct classical limit (see n-point functions, below).
Finding a version of the loop quantum gravity Hamiltonian and a spin-foam theory, in four dimensions, that could be proven to be equivalent, would be a major step ahead in understanding the theory. Substantial progress in this direction has been recently obtained with the introduction of the new vertex that corrects the difficulties of the Barrett–Crane one (see Section 6.7) and appears to provide a direct relation between the spin foam and the loop-gravity language. The viability of these new models now needs to be investigated in depth.
In four dimensions, the coupling of matter in the spin foams and in the group field theory formalisms needs to be better understood.
In the Hamiltonian formulation, the Lorentzian version of the Hamiltonian constraint is well defined (see [294]), but it has not been much studied yet. A number of Lorentzian spin-foam models have been investigated [65, 231, 51, 4, 5, 7], including with the presence of a cosmological constant [211]. Some of these have been proven to be finite [93, 86]. But it is not yet clear what is the best approach for defining the physical theory.
There is another possibility. In conventional quantum field theory, the procedure of defining the physical theory as an analytic continuation of an Euclidean theory is extremely effective. The conventional Wick rotation does not make sense in general relativity, therefore this procedure cannot be generalized to gravity naively. But this does not imply that an appropriate generalization of this procedure could not be found, perhaps in terms of an analytic continuation in some physical boundary time variable. For instance, if the calculations of the n-point functions from Euclidean loop quantum gravity converge to the n-point functions of Euclidean perturbative quantum gravity, then it would make sense to analytically continue in the n-point functions temporal argument, and this is precisely a physical boundary time in the nonperturbative theory [69]. In other words, Euclidean loop quantum gravity might still be a tool for defining the physical theory, and not simply a useful example of background independent quantum field theory.
Quantum cosmology is developing rapidly. It faces two major challenges: to prove that the models used truly derive from full loop quantum gravity, and to develop the models to the point at which they could lead to predictions that can be compared with cosmological observations. This is probably the best hope for finding a window to the empirical verification of loop quantum gravity.
The derivation of the Bekenstein–Hawking entropy formula is a major success of loop gravity, but much remains to be understood. A clean derivation from the full quantum theory is not yet available. Such a derivation would require us to understand what, precisely, the event horizon in the quantum theory is. In other words, given a quantum state of the geometry, we should be able to define and “locate” its horizon (or the structure that replaces it in the quantum theory). To do so, we should understand how to effectively deal with the quantum dynamics, how to describe the classical limit (in order to find the quantum states corresponding to classical black-hole solutions), as well as how to describe asymptotically-flat quantum states.
Besides these formal issues, at the root of the black-hole entropy puzzle is a basic physical problem, which, to my understanding, is still open. The problem is to understand how we can use basic thermodynamical and statistical ideas and techniques in a general covariant context.8 To appreciate the difficulty, notice that statistical mechanics makes heavy use of the notion of energy (for example, in the definition of the canonical or microcanonical ensembles); but there is no natural local notion of energy associated to a black hole (or there are too many of such notions). Energy is an extremely slippery notion in gravity. Thus, how do we define the statistical ensemble? In other words, to compute the entropy (for example, in the microcanonical ensemble) of a normal system, we count the states with a given energy. In GR we should count the states with a given what? One may say black-hole states with a given area. But why? We understand why the number of states with given energy governs the thermodynamical behavior of normal systems. But why should the number of states with given area govern the thermodynamical behavior of the system, namely govern its heat exchanges with the exterior? For a discussion of this last point, see [254].
Computing scattering amplitudes from loop gravity is of interest for a number of reasons. First, it allows a connection with the conventional language of particle physics to be established, and therefore also to compare the loop theory with other approaches to quantum gravity. Second, it provides a direct test of the low-energy limit of the theory. Third, it opens the way to a systematical computation of the quantum corrections to general relativity. The calculation of these n-point functions has begun, but it is laborious, and much remains to be done to set up a consistent general formalism. One of the results of these calculations has been to rule out the Barrett–Crane vertex as a way to define the dynamics of loop gravity [2, 3], opening the way to the development of the new vertex.
Sundance Bilson-Thompson, Fotini Markopoulou and Lee Smolin, have recently introduced the intriguing idea of the possibility of describing fermion degrees of freedom in terms of the braiding of the spin networks [70]. The idea is old; it can be traced to Lord Kelvin, who suggested that the stability of atoms could be understood if atoms are different knots of vortex lines in the ether. It is soon to be understood if this idea can work in loop gravity, but the possibility is obviously very interesting.
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