Conclusion and Outlook

Summary

This thesis has presented loop quantum gravity as a direct, non-perturbative and background-independent quantization of general relativity. We recall here its main steps.

The formulation of general relativity in terms of tetrads and a spin connection, supplemented by the Holst term, leads to a theory that is classically equivalent to Einstein's theory but whose canonical structure depends on a dimensionless parameter, the Barbero–Immirzi parameter $\gamma$. We have seen that, by virtue of the Nieh–Yan identity, this parameter does not affect the vacuum equations of motion, and that it fixes the ratio between the boost and rotation parts of the momentum conjugate to the connection, ${\vec{{K}}} = \gamma{\vec{{L}}}$.

The $3+1$ decomposition reveals general relativity as a fully constrained system, whose Hamiltonian is a combination of first-class constraints. In Ashtekar–Barbero variables, the phase space is that of an ${\mathrm{SU}}(2)$ gauge theory subject to the Gauss, diffeomorphism and Hamiltonian constraints, and the electric field $E^a_i$ is interpreted as the area element of spatial surfaces.

Quantization based on holonomies and fluxes leads to a kinematical Hilbert space, which is unique under the assumption of diffeomorphism invariance and in which spin networks form an orthonormal basis of the gauge-invariant states. The area and volume operators have discrete spectra on this space: the area of a surface is a sum of quanta $8\pi\gamma{\ell_{\mathrm{P}}}^2\sqrt{j(j+1)}$, and volume is carried by nodes of valence at least four. These results underpin the interpretation of spin networks as quantum states of geometry, composed of grains of volume separated by quanta of area. The Hamiltonian constraint could be defined as an operator by means of Thiemann's construction; its action locally modifies the combinatorial structure of spin networks.

We then developed two applications. The computation of black hole entropy, based on the local first law of Frodden, Ghosh and Perez and on the counting of horizon punctures, reproduces the Bekenstein–Hawking law $S = A/4{\ell_{\mathrm{P}}}^2$ when $\gamma = \gamma_0\simeq0.274$. The relation $E = aA/(8\pi G)$ that underlies this computation itself follows from the simplicity constraint ${\vec{{K}}} = \gamma{\vec{{L}}}$, which provides a consistent link between the canonical structure of the theory and the thermodynamics of horizons. Finally, the spin foam formulation expresses transition amplitudes between quantum geometries as sums over histories of spin networks. From the Ponzano–Regge model, whose semiclassical limit reproduces the Regge action, to the EPRL model, whose boundary state space is that of the canonical theory, this formulation provides a second route to the dynamics.

Open Questions

Loop quantum gravity is not yet a complete theory. We mention here the main open questions, without any claim to exhaustiveness [2, 62, 63].

The dynamics. Thiemann's Hamiltonian constraint involves regularization ambiguities—the choice of representation for the holonomies, the operator ordering, the choice of loops—and it has not been established that it reproduces the classical constraint algebra, with its structure functions. The master constraint program [64] and algebraic quantum gravity [65] aim to overcome these difficulties.

The relation between the canonical and covariant formulations. Although the correspondence is established in three dimensions, where the Ponzano–Regge amplitudes coincide with the physical inner product of the canonical theory [96], it remains incomplete in four dimensions. The existence of several spin foam models, with differing motivations, raises the question of the uniqueness of the dynamics.

The semiclassical limit. It remains to be shown that general relativity is the low-energy limit of the quantum theory, that is, that suitable semiclassical states reproduce, at large scales, a smooth geometry obeying the Einstein equations. The asymptotic results obtained for a 4-simplex [99, 100] constitute an important step, but the question of the continuum limit of an extended spin foam remains open.

Coupling to matter. Gauge fields, fermions and the Higgs field can be coupled to the theory by expressing their actions in Ashtekar–Barbero variables. In the spin network representation, the resulting description resembles that of lattice gauge theories, with the matter fields carried by the nodes and the gauge fields by the links [3]. Coupling to matter is also possible in the spin foam formalism [1]. At this stage, however, the theory imposes no restriction on the matter content.

The Barbero–Immirzi parameter. Its value, which sets the scale of the geometric spectra, is determined only by the black hole entropy computation, and this determination depends on the counting scheme adopted (Section 4.5).

Contact with observation. Like all approaches to quantum gravity, loop quantum gravity does not yet have any experimental confirmation. Loop quantum cosmology, which applies the techniques of the theory to homogeneous models, predicts that the initial singularity is replaced by a quantum bounce [32, 33, 101], and it offers prospects for contact with cosmological observations. The physics of black holes in the final stages of evaporation offers another.

Concluding Remark

The number of open questions should not obscure the merits of the approach. Loop quantum gravity shows that it is possible to construct, without additional assumptions and without background structure, a consistent quantum theory of geometry in which area and volume have discrete spectra and in which black hole entropy receives a microscopic interpretation. These results do not depend on the questions that remain open regarding the dynamics. Nevertheless, no theory of quantum gravity can be regarded as established until it has led to predictions confirmed by experiment. In the absence of such confirmation, the various approaches, whether loop quantum gravity, string theory or the other programs mentioned in the Introduction, each have their strengths and weaknesses, and their comparative study remains fruitful.