The kinematics of the theory is well understood, both physically (quanta of area and volume, polymer-like geometry) and from the mathematical point of view (, s-knot states, area and volume operators). The part of the theory which 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 the operator introduced by Thiemann [206, 201, 202]. The commutators of the Thiemann operator with itself and with the diffeomorphism constraints close, and therefore the operator defines a complete and consistent quantum theory. However, doubts have been raised on the physical correctness of this theory, and some variants of the operator have been considered.
The doubts originate from various considerations. First, Lewandowski, Marolf and others have stressed the fact the quantum constraint algebra closes, but it is not isomorphic to the classical constraint algebra of GR . Recently, a detailed analysis of this problem has been completed by Marolf, Lewandowski, Gambini and Pullin . The failure to reproduce the classical constraint algebra has been disputed, and is not necessarily a problem, since the only strict requirement on the quantum theory, besides consistency, is that its gauge invariant physical predictions match the ones of classical general relativity in the appropriate limit. Still, the difference in the algebras may be seen as circumstantial evidence (not a proof) for the failure of the classical limit. The issue is technically delicate and still controversial. I hope I will be able to say something more definitive in the next update of this review.
Second, Brügmann  and Smolin  have pointed out a sort of excess ``locality'' in the form of the operator, which, intuitively, seems in contradiction with the propagation properties of the Einstein equations. Finally, by translating the Thiemann operator into a spacetime covariant four-dimensional formalism, Reisenberger and Rovelli have noticed a suspicious lack of manifest 4-d covariance in the action of the operator , a fact pointing again to the possibility of anomalies in the quantum constraint algebra.
Motivated by these doubts, several variants of Thiemann's operator have been suggested. The original Thiemann's operators is constructed using the volume operator. There are two versions of the volume operator in the literature: , introduced in  and , introduced in [14, 16, 15]. See  for a detailed comparison. Originally, Thiemann thought that using in the hamiltonian constraint would yield difficulties, but it later became clear that this is not the case . Both versions of the volume can be used in the definition, yielding two alternative versions of the hamiltonian . Next, in its simplest version the operator is non-symmetric. Since the classical hamiltonian constraint is real (on SU (2) gauge invariant states), one might expect a corresponding self-adjoint quantum operator. Accordingly, several ways of symmetrizing the operator have been considered (see a list in ). Next, Smolin has considered some ad hoc modifications of the constraint in . Finally, the spacetime covariant formalism in  naturally suggest a ``covariantisation'' of the operator, described in  under the name of ``crossing symmetry''. This covariantisation amounts to adding to the vertex described in Figure 3 the vertices, described in Figure 5, which are simply obtained by re-orienting Figure 3 in spacetime.
A full comparative analysis of this various proposals would be of great interest.
Ultimately, the final tests of any proposal for the hamiltonian constraint operator must be consistency and a correct classical limit. Thus, the solution of the hamiltonian constraint puzzle is likely to be subordinate to the solution of the problem of extracting the classical limit (of the dynamics) from the theory.
The basics of the description of matter in the loop formalism have been established in [150, 151, 133, 26, 207, 200]. Work needs to be done in order to develop a full description of the basic matter couplings. In particular, there are strong recurring indications that the Planck scale discreteness naturally cuts the traditional quantum field infinities off. In particular, in , Thiemann argues that the Hamiltonian constraint governing the coordinate time evolution of the Yang-Mills field is a well defined operator (I recall that, due to the ultraviolet divergences, no rigorously well defined hamiltonian operator for conventional Yang-Mills theory is known in 4 dimensions.) If these indications are confirmed, the result would be very remarkable. What is still missing are calculational techniques that could allow us to connect the well-defined constraint with finite observables quantities such as scattering amplitudes.
In my view, the development of continuous spacetime formalisms, [181, 160, 33, 144], is one of the most promising areas of development of the theory, because it might be the key for addressing most of the open problems. First, a spacetime formalism frees us from the obscurities of the frozen time formalism, and allows an intuitive, Feynman-style, description of the dynamics of quantum spacetime. I think that the classical limit, the quantum description of black holes, or graviton-graviton scattering, just to mention a few examples, could be addressed much more easily in the covariant picture. Second, it allows the general ideas of Hartle  and Isham [119, 120, 123, 122] on the interpretation of generally covariant quantum theories to be applied in loop quantum gravity. This could drastically simplify the complications of the canonical way of dealing with general covariant observables [169, 167]. Third, the spacetime formalism should suggest solutions to the problem of selecting the correct hamiltonian constraint: it is usually easier to deal with invariances in the Lagrangian rather than in the hamiltonian formalisms. The spacetime formalism is just born, and much has to be done. See the original papers for suggestions and open problems.
The derivation of the Bekenstein-Hawking entropy formula is a major success of loop quantum 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 whatever structure 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 roots of the black hole entropy puzzle there 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. To appreciate the difficulty, notice that statistical mechanics makes heavy use of the notion of energy (say 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? Put in other words: To compute the entropy (say in the microcanonical) 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 so? We do 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? A tentative physical discussion of this last point can be found in .
Assume we pick a specific hamiltonian constraint. Then we have, in principle, a well defined quantum theory. How do we extract physical information from it? Some physical consequences of the theory, such as the area and volume eigenvalues, or the entropy formula, have been extracted from the theory by various ad hoc methods. But is there a general technique, say corresponding to the traditional QFT perturbation expansion of the S matrix, for describing the full dynamics of the gravitational field? Presumably, such general techniques should involve some kind of expansion, since we could not hope to solve the theory exactly. Attempts to define physical expansions have been initiated in  and, in different form, in . Ideally, one would want a general scheme for computing transition amplitudes in some expansion parameter around some state. Computing scattering amplitudes would be of particular interest, in order to make connection with particle physics language and to compare the theory with string predictions.
Finally, to prove that loop quantum gravity is a valuable candidate for describing quantum spacetime, we need to prove that its classical limit is GR (or at least overlaps GR in the regime where GR is well tested). The traditional connection between loop quantum gravity and classical GR is via the notion of weave, a quantum state that ``looks semiclassical'' at distances large compared to the Planck scale. However, the weaves studied so far [23, 99] are 3d weaves, in the sense that they are eigenstates of the three dimensional metric. Such a state corresponds to an eigenstate of the position for a particle. Classical behavior is recovered not by these states but rather by wave packets which have small spread in position as well as in momentum. Similarly, the quantum Minkowski spacetime should have small spread in the three metric as well as in its momentum - as the quantum electromagnetic vacuum has small quantum spread in the electric and magnetic field. To recover classical GR from loop quantum gravity, we must understand such states. Preliminary investigation in this direction can be found in [126, 125], but these papers are now several years old, and they were written before the more recent solidification of the basics of the theory. Another direction consists in the direct study of coherent states in the state space of the theory.
As these brief notes indicate, the various open problems in loop quantum gravity are interconnected. In a sense, loop quantum gravity grew aiming at the nonperturbative regime, and the physical results obtained so far are in this regime. The main issue is then to recover the long distance behavior of the theory. That is, to study its classical limit and the dynamics of the low energy excitations over a semiclassical background. Understanding this aspect of the theory would assure us that the theory we are dealing with is indeed a quantum theory of the gravitational field, would allow us to understand quantum black holes, would clarify the origin of infinities in the matter hamiltonians and so on. Still, in other words, what mostly needs to be understood is the structure of the (Minkowski) vacuum in loop quantum gravity.
|Loop Quantum Gravity
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