Two Pillars and an Incompatibility
Contemporary fundamental physics rests on two theories whose empirical successes are considerable. Quantum mechanics, extended to special relativity in the form of quantum field theory, describes the electromagnetic, weak, and strong interactions with remarkable precision; the Standard Model of particle physics is its culmination. General relativity describes the gravitational interaction as a manifestation of the geometry of spacetime; its predictions have been confirmed from the scale of the Solar System up to the recent observations of gravitational waves and black holes.
These two theories, however, rest on conceptual premises that are difficult to reconcile. Quantum field theory is formulated on a background spacetime given a priori, usually Minkowski spacetime: the causal structure, the notion of time, and the measurement of distances are fixed prior to any dynamics. In general relativity, by contrast, the metric is itself the dynamical field: there is no nondynamical geometric structure on which the other fields would propagate. The invariance of the theory under diffeomorphisms of the manifold expresses precisely the absence of any background: the points of spacetime have no physical identity independently of the fields defined on them. Time is no longer an external parameter, as it is in the Schrödinger equation, but one coordinate among others, with no physical meaning of its own. Yet quantum mechanics teaches us that every dynamical field has quantum properties; it is therefore natural to expect that the gravitational field, that is, geometry itself, has them as well. The construction of such a quantum theory of geometry is the object of quantum gravity.
Why a Quantum Theory of Gravity?
Several independent arguments indicate that classical general relativity is not a complete theory. First, the singularity theorems show that the theory predicts its own breakdown, in the form of geodesic incompleteness, near the Big Bang as well as inside black holes, in regions where, in general, the curvature becomes arbitrarily large. Second, the discovery by Bekenstein and Hawking that black holes possess an entropy proportional to the area of their horizon [4, 5] calls for a statistical interpretation in terms of microstates, which only a quantum theory of gravity can provide. Finally, the perturbative quantization of general relativity around Minkowski spacetime leads to a nonrenormalizable theory: ultraviolet divergences appear already at one loop in the presence of matter [6] and at two loops for pure gravity [7].
Quantum-gravitational effects are expected at the Planck scale, characterized by the length and the energy . These scales are beyond the reach of current experiments, which partly explains the coexistence of several research programs. These programs may be divided schematically into two families. Approaches of the first family start from a fixed background. String theory, historically formulated as a perturbative expansion around a given spacetime, is the most extensively studied example; it postulates that the elementary constituents are one-dimensional objects, strings, and aims at a unification of all interactions. Approaches of the second family take background independence as their guiding principle and seek a nonperturbative quantization of geometry: causal dynamical triangulations, causal sets, asymptotic safety, noncommutative geometry, and loop quantum gravity (LQG), which is the subject of this thesis.
Loop Quantum Gravity
Loop quantum gravity is an attempt at a direct quantization of general relativity, without additional assumptions about the unification of the interactions or the existence of extra dimensions or symmetries. Its starting point is the reformulation of general relativity proposed by Ashtekar in 1986 [8, 9], in which the fundamental variable is no longer the metric but a connection, as in Yang–Mills theories. The real version of these variables, due to Barbero [10] and derived from the Holst action [11], introduces a dimensionless parameter, the Barbero–Immirzi parameter [12, 13], which plays a central role in the quantum theory.
The canonical approach may be summarized as follows. Spacetime is split into space and time according to the decomposition of Arnowitt, Deser, and Misner [14, 15]; general relativity then appears as a totally constrained Hamiltonian system, whose Hamiltonian is a linear combination of constraints. In Ashtekar–Barbero variables, these constraints fall into three families: the Gauss constraint, which generates gauge transformations; the diffeomorphism constraint, which generates spatial diffeomorphisms; and the Hamiltonian constraint, which encodes the dynamics. Rather than quantizing the connection and its conjugate momentum at each point, which leads to ill-defined expressions, the theory quantizes their integrated versions: the holonomies of the connection along curves and the fluxes of the electric field across surfaces. These variables require no background metric and form an algebra with well-defined Poisson brackets. Wilson loops, the traces of holonomies along closed loops, gave the theory its name [16, 17].
Imposing the Gauss constraint leads to the basis of spin networks [18, 19], graphs whose links carry representations of and whose nodes carry invariant tensors. This basis diagonalizes the area operator and, for a suitable choice of intertwiners, the volume operator. The spectra of these operators are discrete [20–22], so that the theory predicts a granular structure of space at the Planck scale, with the nodes of spin networks representing quanta of volume and the links representing quanta of area. The Hamiltonian constraint was defined as an operator by Thiemann [23, 24]; its regularization, however, involves ambiguities, and the dynamics remains the least understood part of the theory. The covariant formulation, known as spin foams [25–27], describes the evolution of spin networks as a sum over histories of quantum geometries and provides an alternative formulation of the dynamics [28, 29].
The results of the theory include the computation of black hole entropy by counting the microstates of the horizon [30, 31] and, within loop quantum cosmology, the replacement of the initial singularity by a bounce [32, 33]. Fundamental questions remain open, foremost among them the semiclassical limit of the theory: it has yet to be established that general relativity is indeed the low-energy limit of the quantum dynamics. Like the other approaches to quantum gravity, the theory currently lacks experimental confirmation, although some of its applications, notably in cosmology, lead to potentially observable predictions.
Outline of the Thesis
Chapter 1 collects the necessary classical tools: the formulation of general relativity in terms of tetrads and the spin connection, the gauge symmetries of the theory coupled to matter, the Holst action and the Barbero–Immirzi parameter, and the elements of the Hamiltonian mechanics of constrained systems due to Dirac. Chapter 2 presents the Hamiltonian formulation of general relativity, first in ADM variables and then in Ashtekar–Barbero variables, and brings out the geometric interpretation of the electric field as an area element. Chapter 3 is devoted to quantization: the holonomy–flux algebra, the loop representation, the kinematical Hilbert space, spin networks, geometric operators, and the Hamiltonian constraint. The next two chapters deal with applications. Chapter 4 presents the computation of black hole entropy based on the local first law of Frodden, Ghosh, and Perez [34]. Chapter 5 introduces the spin foam formalism, from the Ponzano–Regge model to the EPRL model. The conclusion summarizes the results obtained and discusses the main open questions. An appendix collects the elements of the representation theory of used in the text.
This thesis draws mainly on the books by Rovelli [1], by Rovelli and Vidotto [2], and by Thiemann [3], on the lecture notes of Doná and Speziale [35] and of Perez [36], and on the reviews [37, 38]; the original references are cited throughout the text.